From mathcomp Require Import all_ssreflect finmap.
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "_ + _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ - _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ <= _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ < _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ >= _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ > _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ <= _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ < _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ <= _ < _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ < _ < _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ * _" was already used in scope nat_scope.
[notation-overridden,parsing]
From hanoi Require Import extra star gdist lhanoi3 ghanoi ghanoi4 shanoi.
From mathcomp Require Import zify.

Open Scope nat_scope.

Set Implicit Arguments.
Unset Strict Implicit.

(******************************************************************************)
(*                                                                            *)
(*  This file proves the formula that gives the distance between two perfect  *)
(*  configurations for the star puzzle. It follows the proof given by Thierry *)
(*  Bousch in  La tour de Stockmeyer                                          *)
(*                                                                            *)
(*                                                                            *)
(******************************************************************************)

Ltac Zify.zify_post_hook ::=
  zify_ssreflect.SsreflectZifyInstances.divZ_modZ_to_equations.

Section sHanoi4.

Local Notation "c1 `-->_s c2" := (smove c1 c2)
    (format "c1  `-->_s  c2", at level 60).
Local Notation "c1 `-->*_s c2" := (connect smove c1 c2) 
    (format "c1  `-->*_s  c2", at level 60).

(* the peg at the center is 0 *)
Let p0 : peg 4 := ord0.

(* specialize version *)
Lemma pE4 (p1 p2 p3 : peg 4) p : 
   p1 != p2 -> p1 != p3 -> p2 != p3 -> 
   p1 != p0 -> p2 != p0 -> p3 != p0 ->
  [\/ p = p0, p = p1, p = p2 | p = p3].
p0:=ord0:peg 4
p1, p2, p3, p:peg 4
p1 != p2 ->
p1 != p3 ->
p2 != p3 ->
p1 != p0 ->
p2 != p0 ->
p3 != p0 -> [\/ p = p0, p = p1, p = p2 | p = p3]
Proof.
2 
rewrite /p0.
4
p1 != p2 ->
p1 != p3 ->
p2 != p3 ->
p1 != ord0 ->
p2 != ord0 ->
p3 != ord0 -> [\/ p = ord0, p = p1, p = p2 | p = p3]
by case: (peg4E p1) => ->; rewrite ?eqxx //; 
   case: (peg4E p2) => ->; rewrite ?eqxx //; 
   case: (peg4E p3) => ->; rewrite ?eqxx //; 
   case: (peg4E p) => ->; rewrite ?eqxx // => *; 
   try ((by apply: Or41) || (by apply: Or42) || 
        (by apply: Or43) || (by apply: Or44)).
Qed.

(* Lifting lhanoi3 to shanoi4 *)

Section lift.

Variable p1 p2 p3 : peg 4.

Hypothesis p1Dp0 : p1 != p0.
Hypothesis p2Dp0 : p2 != p0.
Hypothesis p3Dp0 : p3 != p0.
Hypothesis p1Dp2 : p1 != p2.
Hypothesis p1Dp3 : p1 != p3.
Hypothesis p2Dp3 : p2 != p3.

Variable n m : nat.

Definition clmap (c : configuration 3 n) : configuration 4 n :=
 [ffun i =>
    if c i == 0 :> nat then p1
    else if c i == 1 :> nat then p0 else p2].

Definition clmerge (c : configuration 3 n) : configuration 4 (n + m) :=
  cmerge (clmap c) `c[p3].

Lemma on_top_clmerge x c : on_top x c -> on_top (tlshift m x) (clmerge c).
5
p1, p2, p3:peg 4
p1Dp0:p1 != p0
p2Dp0:p2 != p0
p3Dp0:p3 != p0
p1Dp2:p1 != p2
p1Dp3:p1 != p3
p2Dp3:p2 != p3
n, m:nat
x:disk n
c:configuration 3 n
on_top (q:=3) (n:=n) x c ->
on_top (q:=4) (n:=n + m) (tlshift m x) (clmerge c)
Proof.
e
move=> /on_topP oH; apply/on_topP => d.
51112131415161718191a
oH:forall d1 : ordinal_finType n,
c x = c d1 -> x <= d1
d:ordinal_finType (n + m)
clmerge c (tlshift m x) = clmerge c d ->
tlshift m x <= d
rewrite !ffunE tsplit_tlshift /=.
20
clmap c x =
match tsplit d with
| inl j => clmap c j
| inr j => `c[p3] j
end -> x + m <= d
case: tsplitP => y; rewrite !ffunE => ->.
51112131415161718191a2122
y:'I_m
(if c x == 0 then p1 else if c x == 1 then p0 else p2) =
p3 -> x + m <= y
51112131415161718191a2122
y:'I_n
(if c x == 0 then p1 else if c x == 1 then p0 else p2) =
(if c y == 0 then p1 else if c y == 1 then p0 else p2) ->
x + m <= y + m
  case: (_ == _) => [/eqP|]; first by rewrite (negPf p1Dp3).
2b
(if c x == 1 then p0 else p2) = p3 -> x + m <= y
2e
  case: (_ == _) => /eqP; first by rewrite eq_sym (negPf p3Dp0).
2b
p2 == p3 -> x + m <= y
2e
  by rewrite (negPf p2Dp3).
30
32
case: eqP => [cxE0|cxD0].
51112131415161718191a212231
cxE0:c x = 0
p1 =
(if c y == 0 then p1 else if c y == 1 then p0 else p2) ->
x + m <= y + m
51112131415161718191a212231
cxD0:c x <> 0
(if c x == 1 then p0 else p2) =
(if c y == 0 then p1 else if c y == 1 then p0 else p2) ->
x + m <= y + m
  case: eqP => [cyE0 _| _].
51112131415161718191a21223142
cyE0:c y = 0
x + m <= y + m
41
p1 = (if c y == 1 then p0 else p2) -> x + m <= y + m
45
    by rewrite leq_add2r oH //; apply: val_inj; rewrite /= cyE0.
41
51
44
  by case: (_ == _) => /eqP; rewrite (negPf p1Dp0, negPf p1Dp2).
46
48
case: eqP => [cxE1|cxD1].
51112131415161718191a21223147
cxE1:c x = 1
p0 =
(if c y == 0 then p1 else if c y == 1 then p0 else p2) ->
x + m <= y + m
51112131415161718191a21223147
cxD1:c x <> 1
p2 =
(if c y == 0 then p1 else if c y == 1 then p0 else p2) ->
x + m <= y + m
  case: (_ == _) => [/eqP|]; first by rewrite eq_sym (negPf p1Dp0).
5b
p0 = (if c y == 1 then p0 else p2) -> x + m <= y + m
5e
  case: eqP => [cyE1 _|_ /eqP].
51112131415161718191a212231475c
cyE1:c y = 1
4e
5b
p0 == p2 -> x + m <= y + m
5f
    by rewrite leq_add2r oH //; apply: val_inj; rewrite /= cyE1.
5b
6e
5e
  by rewrite eq_sym (negPf p2Dp0).
60
62
have cxE2 : c x = 2 :> nat by case: (c x) cxD0 cxD1 => [] [|[|[]]].
51112131415161718191a2122314761
cxE2:c x = 2
62
case: eqP => [_ /eqP|cyD0].
78
p2 == p1 -> x + m <= y + m
51112131415161718191a212231476179
cyD0:c y <> 0
p2 = (if c y == 1 then p0 else p2) -> x + m <= y + m
  by rewrite eq_sym (negPf p1Dp2).
80
82
case: eqP => [_ /eqP|cyD1 _].
80
p2 == p0 -> x + m <= y + m
51112131415161718191a21223147617981
cyD1:c y <> 1
4e
  by rewrite (negPf p2Dp0).
8c
4e
have cyE2 : c y = 2 :> nat by case: (c y) cyD0 cyD1 => [] [|[|[]]].
51112131415161718191a212231476179818d
cyE2:c y = 2
4e
by rewrite leq_add2r oH //; apply: val_inj; rewrite /= cyE2.
Qed.

Lemma move_clmerge c1 c2 : lmove c1 c2 ->  clmerge c1 `-->_s clmerge c2.
51112131415161718
c1, c2:configuration 3 n
lmove c1 c2 -> clmerge c1 `-->_s clmerge c2
Proof.
97
move=> /moveP [d1 [H1d1 H2d1 H3d1 H4d1]].
511121314151617189a
d1:ordinal_finType n
H1d1:lrel (n:=3) (c1 d1) (c2 d1)
H2d1:forall d2 : ordinal_finType n,
d1 != d2 -> c1 d2 = c2 d2
H3d1:on_top (q:=3) (n:=n) d1 c1
H4d1:on_top (q:=3) (n:=n) d1 c2
clmerge c1 `-->_s clmerge c2
apply/moveP; exists (tlshift m d1); split => //.
a0
srel (n:=4) (clmerge c1 (tlshift m d1))
  (clmerge c2 (tlshift m d1))
a0
forall d2 : ordinal_finType (n + m),
tlshift m d1 != d2 -> clmerge c1 d2 = clmerge c2 d2
a0
on_top (q:=4) (n:=n + m) (tlshift m d1) (clmerge c1)
a0
on_top (q:=4) (n:=n + m) (tlshift m d1) (clmerge c2)
-
a8 move: H1d1; rewrite !ffunE tsplit_tlshift !ffunE.
511121314151617189aa1a3a4a5
lrel (n:=3) (c1 d1) (c2 d1) ->
srel (n:=4)
  (if c1 d1 == 0
   then p1
   else if c1 d1 == 1 then p0 else p2)
  (if c2 d1 == 0
   then p1
   else if c2 d1 == 1 then p0 else p2)
ab
  rewrite /lrel /= /srel /ghanoi.srel /=.
b6
((c1 d1).+1 == c2 d1) || ((c2 d1).+1 == c1 d1) ->
((if c1 d1 == 0
  then p1
  else if c1 d1 == 1 then p0 else p2)
 != (if c2 d1 == 0
     then p1
     else if c2 d1 == 1 then p0 else p2)) &&
((if c1 d1 == 0
  then p1
  else if c1 d1 == 1 then p0 else p2) *
 (if c2 d1 == 0
  then p1
  else if c2 d1 == 1 then p0 else p2) == 0)
ab
  case: (c1 d1) => [] [|[|[|]]] //=.
b6
0 < 3 ->
(1 == c2 d1) || ((c2 d1).+1 == 0) ->
(p1
 != (if c2 d1 == 0
     then p1
     else if c2 d1 == 1 then p0 else p2)) &&
(p1 *
 (if c2 d1 == 0
  then p1
  else if c2 d1 == 1 then p0 else p2) == 0)
b6
1 < 3 ->
(2 == c2 d1) || ((c2 d1).+1 == 1) ->
(p0
 != (if c2 d1 == 0
     then p1
     else if c2 d1 == 1 then p0 else p2)) &&
(0 *
 (if c2 d1 == 0
  then p1
  else if c2 d1 == 1 then p0 else p2) == 0)
b6
2 < 3 ->
(3 == c2 d1) || ((c2 d1).+1 == 2) ->
(p2
 != (if c2 d1 == 0
     then p1
     else if c2 d1 == 1 then p0 else p2)) &&
(p2 *
 (if c2 d1 == 0
  then p1
  else if c2 d1 == 1 then p0 else p2) == 0)
acaeb0
  -
bd case: (c2 d1) => [] [|[|[|]]] //=.
b6
1 < 3 -> 0 < 3 -> true -> (p1 != p0) && (p1 * 0 == 0)
c0
    by rewrite p1Dp0 muln0.
b6
c2
c3acaeb0
  -
cb case: (c2 d1) => [] [|[|[|]]] //= _ _ _.
b6
(p0 != p1) && (0 * p1 == 0)
b6
(p0 != p2) && (0 * p2 == 0)
c3acaeb0
      by rewrite eq_sym p1Dp0.
b6
d5
cd
    by rewrite eq_sym p2Dp0.
b6
c4
ab
  case: (c2 d1) => [] [|[|[|]]] //= _ _ _.
b6
(p2 != p0) && (p2 * 0 == 0)
ab
  by rewrite p2Dp0 muln0.
a0
ad
aeb0
-
e1 move=> d2; rewrite !ffunE; case: tsplitP => j; rewrite !ffunE //.
511121314151617189aa1a2a3a4a5
d2:ordinal_finType (n + m)
j:'I_n
d2 = j + m ->
tlshift m d1 != d2 ->
(if c1 j == 0
 then p1
 else if c1 j == 1 then p0 else p2) =
(if c2 j == 0
 then p1
 else if c2 j == 1 then p0 else p2)
e3
  move=> d2E /eqP/val_eqP.
511121314151617189aa1a2a3a4a5e9ea
d2E:d2 = j + m
val (tlshift m d1) != val d2 ->
(if c1 j == 0
 then p1
 else if c1 j == 1 then p0 else p2) =
(if c2 j == 0
 then p1
 else if c2 j == 1 then p0 else p2)
e3
  rewrite /= d2E eqn_add2r => H.
511121314151617189aa1a2a3a4a5e9eaf0
H:d1 != j
(if c1 j == 0
 then p1
 else if c1 j == 1 then p0 else p2) =
(if c2 j == 0
 then p1
 else if c2 j == 1 then p0 else p2)
e3
  by rewrite (@H2d1 j).
a0
af
b0
-
f9 by apply: on_top_clmerge.
a0
b1
by apply: on_top_clmerge.
Qed.

Lemma path_clmerge c cs :
  path (move (@lrel 3)) c cs ->
  path smove (clmerge c) [seq clmerge c | c <- cs].
511121314151617181a
cs:seq (configuration 3 n)
path (move (q:=3) (lrel (n:=3))) c cs ->
path smove (clmerge c) [seq clmerge c | c <- cs]
Proof.
101
elim: cs c => //= c1 cs IH c2 /andP[mH pH].
51112131415161718
c1:configuration 3 n
104
IH:forall c : configuration 3 n,
path (move (q:=3) (lrel (n:=3))) c cs ->
path smove (clmerge c)
  [seq clmerge c0 | c0 <- cs]
c2:configuration 3 n
mH:move (q:=3) (lrel (n:=3)) c2 c1
pH:path (move (q:=3) (lrel (n:=3))) c1 cs
(clmerge c2 `-->_s clmerge c1) &&
path smove (clmerge c1) [seq clmerge c | c <- cs]
by rewrite move_clmerge // IH.
Qed.

Lemma gdist_clmerge c1 c2 :
  `d[clmerge c1, clmerge c2]_smove <= `d[c1, c2]_(move (@lrel 3)).
99
`d[clmerge c1, clmerge c2]_smove <=
`d[c1, c2]_(move (q:=3) (lrel (n:=3)))
Proof.
112
have /gpath_connect[pa1 pa1H] := move_lconnect3 c1 c2.
511121314151617189a
pa1:seq
  (finfun_finType (ordinal_finType n)
     (ordinal_finType 3))
pa1H:gpath lmove c1 c2 pa1
114
rewrite (gpath_dist pa1H) -(size_map clmerge).
119
`d[clmerge c1, clmerge c2]_smove <=
size [seq clmerge i | i <- pa1]
apply: gdist_path_le; first by rewrite path_clmerge // (gpath_path pa1H).
119
last (clmerge c1) [seq clmerge i | i <- pa1] =
clmerge c2
by rewrite [LHS]last_map (gpath_last pa1H).
Qed.

End lift.


(* We first prove the lower bound *)

Lemma leq_shanoi4 p1 p2 n :
   p1 != p0 -> p2 != p0 -> p1 != p2 -> 
   `d[`c[p1, n],`c[p2, n]]_smove <= (S_[1] n).*2.
5
p1, p2:ordinal_eqType 4
n:nat
p1 != p0 ->
p2 != p0 ->
p1 != p2 ->
`d[`c[p1 , n], `c[p2 , n]]_smove <= (S_[1] n).*2
Proof.
125
elim/ltn_ind : n p1 p2 => [] [|n] IH p1 p2 p1Dp0 p2Dp0 p1Dp2.
5
IH:forall m : nat,
m < 0 ->
forall p1 p2 : ordinal_eqType 4,
p1 != p0 ->
p2 != p0 ->
p1 != p2 ->
`d[`c[p1], `c[p2]]_smove <= (S_[1] m).*2
128121315
`d[`c[p1], `c[p2]]_smove <= (S_[1] 0).*2
5129
IH:forall m : nat,
m < n.+1 ->
forall p1 p2 : ordinal_eqType 4,
p1 != p0 ->
p2 != p0 ->
p1 != p2 ->
`d[`c[p1], `c[p2]]_smove <= (S_[1] m).*2
128121315
`d[`c[p1], `c[p2]]_smove <= (S_[1] n.+1).*2
  rewrite alphaS_small double0.
12f
`d[`c[p1], `c[p2]]_smove <= 0
132
  rewrite (_ : `c[p1] = `c[p2]) ?gdist0 //.
12f
`c[p1] = `c[p2]
132
  by apply/ffunP=> [] [].
134
136
rewrite S1E S1_bigmin -muln2 -bigminMr.
134
`d[`c[p1], `c[p2]]_smove <=
\min_(i <= n) ((S1 i).*2 + ((3 ^ (n.+1 - i)).-1)./2) *
              2
apply/bigmin_leqP => i iLn.
5129135128121315
i:nat
iLn:i <= n
`d[`c[p1], `c[p2]]_smove <=
((S1 i).*2 + ((3 ^ (n.+1 - i)).-1)./2) * 2
have iLn1 : i <= n.+1 by apply: leq_trans iLn _.
512913512812131514a14b
iLn1:i <= n.+1
14c
rewrite -{-5}(subnK iLn1).
14e
rewrite eq_sym in p2Dp0.
5129135128121514a14b151
p2Dp0:p0 != p2
14c
have [p3 [[p3Dp2 p3Dp0 p3Dp1] _]] := peg4comp3 p1Dp0 p1Dp2 p2Dp0.
5129135128121514a14b151157
p3:peg 4
p3Dp2:p3 != p2
14
p3Dp1:p3 != p1
14c
rewrite eq_sym in p2Dp0.
5129135128121514a14b15115c15d1415e13
14c
pose p1n := `c[p1, n.+1 - i].
5129135128121514a14b15115c15d1415e13
p1n:=`c[p1 , n.+1 - i]:configuration 4 (n.+1 - i)
14c
pose p2n := `c[p2, n.+1 - i].
5129135128121514a14b15115c15d1415e13167
p2n:=`c[p2 , n.+1 - i]:configuration 4 (n.+1 - i)
14c
pose p3n := `c[p3, n.+1 - i].
5129135128121514a14b15115c15d1415e1316716c
p3n:=`c[p3 , n.+1 - i]:configuration 4 (n.+1 - i)
14c
pose p1i := `c[p1, i].
5129135128121514a14b15115c15d1415e1316716c171
p1i:=`c[p1 , i]:configuration 4 i
14c
pose p2i := `c[p2, i].
5129135128121514a14b15115c15d1415e1316716c171176
p2i:=`c[p2 , i]:configuration 4 i
14c
pose p3i := `c[p3, i].
5129135128121514a14b15115c15d1415e1316716c17117617b
p3i:=`c[p3 , i]:configuration 4 i
14c
pose c1 := cmerge p1n p1i.
5129135128121514a14b15115c15d1415e1316716c17117617b180
c1:=cmerge (q:=4) (m:=n.+1 - i) (n:=i) p1n p1i:{ffun forall x : ordinal_finType (n.+1 - i + i),
      (fun=> peg 4) x}
14c
pose c2 := cmerge p1n p3i.
5129135128121514a14b15115c15d1415e1316716c17117617b180185
c2:=cmerge (q:=4) (m:=n.+1 - i) (n:=i) p1n p3i:{ffun forall x : ordinal_finType (n.+1 - i + i),
      (fun=> peg 4) x}
14c
pose c3 := cmerge p2n p3i.
5129135128121514a14b15115c15d1415e1316716c17117617b18018518a
c3:=cmerge (q:=4) (m:=n.+1 - i) (n:=i) p2n p3i:{ffun forall x : ordinal_finType (n.+1 - i + i),
      (fun=> peg 4) x}
14c
pose c4 := cmerge p2n p2i.
5129135128121514a14b15115c15d1415e1316716c17117617b18018518a18f
c4:=cmerge (q:=4) (m:=n.+1 - i) (n:=i) p2n p2i:{ffun forall x : ordinal_finType (n.+1 - i + i),
      (fun=> peg 4) x}
14c
have <- : c1 = `c[p1].
193
c1 = `c[p1]
193
`d[c1, `c[p2]]_smove <=
((S1 i).*2 + ((3 ^ (n.+1 - i)).-1)./2) * 2
  apply/ffunP => j; rewrite /c1 !ffunE.
5129135128121514a14b15115c15d1415e1316716c17117617b18018518a18f194
j:ordinal_finType (n.+1 - i + i)
match tsplit j with
| inl j => p1n j
| inr j => p1i j
end = p1
199
  by case: tsplitP => k; rewrite !ffunE.
193
19b
have <- : c4 = `c[p2].
193
c4 = `c[p2]
193
`d[c1, c4]_smove <=
((S1 i).*2 + ((3 ^ (n.+1 - i)).-1)./2) * 2
  apply/ffunP => j; rewrite /c1 !ffunE.
19f
match tsplit j with
| inl j => p2n j
| inr j => p2i j
end = p2
1a9
  by case: tsplitP => k; rewrite !ffunE.
193
1ab
apply: leq_trans 
   (_ : _ <= `d[c1, c2]_smove + `d[c2, c3]_smove + `d[c3, c4]_smove) _.
193
`d[c1, c4]_smove <=
`d[c1, c2]_smove + `d[c2, c3]_smove + `d[c3, c4]_smove
193
`d[c1, c2]_smove + `d[c2, c3]_smove + `d[c3, c4]_smove <=
((S1 i).*2 + ((3 ^ (n.+1 - i)).-1)./2) * 2
  apply: leq_trans (leq_add (gdist_triangular  _ _ _ _) (leqnn _)).
193
`d[c1, c4]_smove <=
`d[c1, c3]_smove + `d[c3, c4]_smove
1b7
  apply: gdist_triangular.
193
1b9
rewrite -addnn !mulnDl [X in _ <= X]addnAC !muln2.
193
`d[c1, c2]_smove + `d[c2, c3]_smove + `d[c3, c4]_smove <=
(S1 i).*2 + (((3 ^ (n.+1 - i)).-1)./2).*2 + (S1 i).*2
repeat apply: leq_add; first 2 last.
193
`d[c3, c4]_smove <= (S1 i).*2
193
`d[c1, c2]_smove <= (S1 i).*2
193
`d[c2, c3]_smove <= (((3 ^ (n.+1 - i)).-1)./2).*2
-
1c6 apply: leq_trans (gdist_merger _ _ _) _; first by apply: sirr.
193
connect (move (q:=4) (srel (n:=4))) p3i p2i
193
`d[p3i, p2i]_(move (q:=4) (srel (n:=4))) <= (S1 i).*2
1ca1cc
    by apply: shanoi_connect.
193
1d5
1c9
  by rewrite -S1E; apply: IH.
193
1cb
1cc
-
1da apply: leq_trans (gdist_merger _ _ _) _; first by apply: sirr.
193
connect (move (q:=4) (srel (n:=4))) p1i p3i
193
`d[p1i, p3i]_(move (q:=4) (srel (n:=4))) <= (S1 i).*2
1cc
    by apply: shanoi_connect.
193
1e4
1dc
  by rewrite -S1E; apply: IH => //; rewrite eq_sym.
193
1cd
rewrite div2K; last first.
193
~~ odd (3 ^ (n.+1 - i)).-1
193
`d[c2, c3]_smove <= (3 ^ (n.+1 - i)).-1
  rewrite -subn1 oddB ?expn_gt0 // addbT.
193
~~ ~~ odd (3 ^ (n.+1 - i))
1ef
  by rewrite oddX orbT.
193
1f1
have -> : c2 = clmerge p1 p2 p3 _ `c[ord0].
193
c2 = clmerge p1 p2 p3 (n:=n.+1 - i) i `c[ord0]
193
`d[clmerge p1 p2 p3 (n:=n.+1 - i) i `c[ord0], c3]_smove <=
(3 ^ (n.+1 - i)).-1
  apply/ffunP=> x; rewrite !ffunE.
5129135128121514a14b15115c15d1415e1316716c17117617b18018518a18f194
x:ordinal_finType (n.+1 - i + i)
match tsplit x with
| inl j => p1n j
| inr j => p3i j
end =
match tsplit x with
| inl j => clmap p1 p2 (n:=n.+1 - i) `c[ord0] j
| inr j => `c[p3] j
end
1fd
  by case: tsplitP => // j; rewrite !ffunE /=.
193
1ff
have -> : c3 = clmerge p1 p2 p3 _ `c[inord 2].
193
c3 = clmerge p1 p2 p3 (n:=n.+1 - i) i `c[inord 2]
193
`d[clmerge p1 p2 p3 (n:=n.+1 - i) i `c[ord0], 
clmerge p1 p2 p3 (n:=n.+1 - i) i `c[
  inord 2]]_smove <= (3 ^ (n.+1 - i)).-1
  apply/ffunP=> x; rewrite !ffunE.
203
match tsplit x with
| inl j => p2n j
| inr j => p3i j
end =
match tsplit x with
| inl j => clmap p1 p2 (n:=n.+1 - i) `c[inord 2] j
| inr j => `c[p3] j
end
20d
  by case: tsplitP => // j; rewrite !ffunE /= inordK.
193
`d[clmerge p1 p2 p3 (n:=n.+1 - i) i `c[ord0], clmerge
                                                p1 p2
                                                p3
                                                (n:=
                                                n.+1 -
                                                i) i
                                                `c[
                                                inord
                                                 2]]_smove <=
(3 ^ (n.+1 - i)).-1
apply: leq_trans (gdist_clmerge _ _ _ _ _ _ _ _ _) _ => //; 
  try by rewrite eq_sym.
193
`d[`c[ord0], `c[inord 2]]_(move (q:=3) (lrel (n:=3))) <=
(3 ^ (n.+1 - i)).-1
rewrite gdist_lhanoi3p /lrel /= inordK //=.
193
(3 ^ (n.+1 - i)).-1 * (ord0 != inord 2) <=
(3 ^ (n.+1 - i)).-1
case: eqP=> [/val_eqP/=|]; first by rewrite inordK.
193
ord0 <> inord 2 ->
(3 ^ (n.+1 - i)).-1 * ~~ false <= (3 ^ (n.+1 - i)).-1
by rewrite muln1.
Qed.

(* *)
(* Distance for a list *)
Fixpoint distanceL n (l : seq (configuration 4 n)) : nat := 
  if l is a :: l1 then
    if l1 is b :: _ then `d[a, b]_smove + distanceL l1
    else 0
  else 0.

Notation " D[ l ] " := (distanceL l)
  (format " D[ l ]").

(* Alternating peg : depending on the parity of n it is p1 or p2 *)
Definition apeg p1 p2 n : peg 4 := if odd n then p2 else p1.
Notation " a[ p1 , p2 ] " := (apeg p1 p2)
  (format " a[ p1 ,  p2 ]").
 
Lemma apeg32E a p1 p2 : a[p1, p2] (3 * a).-2 = a[p1, p2] a.
5
a:nat
p1, p2:peg 4
 a[p1, p2] (3 * a).-2 =  a[p1, p2] a
Proof.
225
case: a => // a.
5229228
 a[p1, p2] (3 * a.+1).-2 =  a[p1, p2] a.+1
rewrite /apeg -subn2 oddB; last by rewrite mulnS.
22f
(if odd (3 * a.+1) (+) odd 2 then p2 else p1) =
(if odd a.+1 then p2 else p1)
by rewrite oddM /=; case: odd.
Qed.

Lemma apeg0 p1 p2 : a[p1, p2] 0 = p1.
5229
 a[p1, p2] 0 = p1
Proof.
236 by []. Qed.

Lemma apegS p1 p2 n : a[p1, p2] n.+1 = a[p2, p1] n.
5229129
 a[p1, p2] n.+1 =  a[p2, p1] n
Proof.
23c by rewrite /apeg /=; case: odd. Qed.

Lemma apegO p1 p2 n : a[p1, p2] n = a[p2, p1] (~~ (odd n)).
23e
 a[p1, p2] n =  a[p2, p1] (~~ odd n)
Proof.
242 by rewrite /apeg /=; case: odd. Qed.

Lemma apeg_double p1 p2 n : a[p1, p2] (n.*2) = p1.
23e
 a[p1, p2] n.*2 = p1
Proof.
247 by rewrite apegO odd_double. Qed.

Lemma apeg_neq p1 p2 p3 n : p1 != p3 -> p2 != p3 -> a[p1, p2] n != p3.
5
p1, p2, p3:ordinal_eqType 4
129
p1 != p3 -> p2 != p3 ->  a[p1, p2] n != p3
Proof.
24c by rewrite /apeg /=; case: odd; rewrite // eq_sym. Qed.

Lemma apeg_eqC p1 p2 n : (a[p1, p2] n == a[p2, p1] n) = (p1 == p2).
23e
( a[p1, p2] n ==  a[p2, p1] n) = (p1 == p2)
Proof.
253 by rewrite /apeg /=; case: odd; rewrite // eq_sym. Qed.

Lemma apegD p1 p2 m n : a[p1, p2] (m + n) = a[ a[p1, p2] n, a[p2, p1] n] m.
5229
m, n:nat
 a[p1, p2] (m + n) =  a[ a[p1, p2] n,  a[p2, p1] n] m
Proof.
258 by rewrite /apeg oddD; do 2 case: odd. Qed.

Lemma apegMr p1 p2 m n : odd m -> a[p1, p2] (m * n) = a[p1, p2] n.
25a
odd m ->  a[p1, p2] (m * n) =  a[p1, p2] n
Proof.
25f by rewrite /apeg; rewrite oddM => ->. Qed.

Lemma codom_apeg (A : finType) (f : {ffun A -> _}) p1 p2 n : 
  (codom f \subset [::  a[p1, p2] n;  a[p2, p1] n]) =
  (codom f \subset [:: p1; p2]).
5
A:finType
f:{ffun A -> ordinal_finType 4}
229129
(codom f \subset [::  a[p1, p2] n;  a[p2, p1] n]) =
(codom f \subset [:: p1; p2]) 
Proof.
264 by rewrite /apeg; case: odd; rewrite // codom_subC. Qed.

(* This is lemma 2.1 *)
Lemma sgdist_pair n (p2 : peg 4) (u v : configuration 4 n.+1) 
    (p1 := u ldisk) (p3 := v ldisk): 
   {st : (configuration 4 n * configuration 4 n) |
    p1 != p2 -> p1 != p3 -> p2 != p3 -> 
    p1 != p0 -> p2 != p0 -> p3 != p0 ->
     [/\ codom st.1 \subset [:: p2; p3], codom st.2 \subset [:: p1; p2] & 
         `d[u, v]_smove = D[[:: ↓[u]; st.1; st.2; ↓[v]]].+2]}.
5129
p2:peg 4
u, v:configuration 4 n.+1
p1:=u ldisk:peg 4
p3:=v ldisk:peg 4
{st : configuration 4 n * configuration 4 n
| p1 != p2 ->
  p1 != p3 ->
  p2 != p3 ->
  p1 != p0 ->
  p2 != p0 ->
  p3 != p0 ->
  [/\ codom st.1 \subset [:: p2; p3],
      codom st.2 \subset [:: p1; p2]
    & `d[u, v]_smove =
       D[[:: ↓[u]; st.1; st.2; ↓[v]]].+2]}
Proof.
26c
case: eqP => [_|/eqP p1Dp2]; first by exists (↓[u], ↓[v]).
512926f27027127215
{st : configuration 4 n * configuration 4 n
| ~~ false ->
  p1 != p3 ->
  p2 != p3 ->
  p1 != p0 ->
  p2 != p0 ->
  p3 != p0 ->
  [/\ codom st.1 \subset [:: p2; p3],
      codom st.2 \subset [:: p1; p2]
    & `d[u, v]_smove =
       D[[:: ↓[u]; st.1; st.2; ↓[v]]].+2]}
case: eqP => [_|/eqP p1Dp3]; first by exists (↓[u], ↓[v]).
512926f2702712721516
{st : configuration 4 n * configuration 4 n
| ~~ false ->
  ~~ false ->
  p2 != p3 ->
  p1 != p0 ->
  p2 != p0 ->
  p3 != p0 ->
  [/\ codom st.1 \subset [:: p2; p3],
      codom st.2 \subset [:: p1; p2]
    & `d[u, v]_smove =
       D[[:: ↓[u]; st.1; st.2; ↓[v]]].+2]}
case: eqP => [_|/eqP p2Dp3]; first by exists (↓[u], ↓[v]).
512926f270271272151617
{st : configuration 4 n * configuration 4 n
| ~~ false ->
  ~~ false ->
  ~~ false ->
  p1 != p0 ->
  p2 != p0 ->
  p3 != p0 ->
  [/\ codom st.1 \subset [:: p2; p3],
      codom st.2 \subset [:: p1; p2]
    & `d[u, v]_smove =
       D[[:: ↓[u]; st.1; st.2; ↓[v]]].+2]}
case: eqP => [_|/eqP p1Dp0]; first by exists (↓[u], ↓[v]).
512926f27027127215161712
{st : configuration 4 n * configuration 4 n
| ~~ false ->
  ~~ false ->
  ~~ false ->
  ~~ false ->
  p2 != p0 ->
  p3 != p0 ->
  [/\ codom st.1 \subset [:: p2; p3],
      codom st.2 \subset [:: p1; p2]
    & `d[u, v]_smove =
       D[[:: ↓[u]; st.1; st.2; ↓[v]]].+2]}
case: eqP => [_|/eqP p2Dp0]; first by exists (↓[u], ↓[v]).
512926f2702712721516171213
{st : configuration 4 n * configuration 4 n
| ~~ false ->
  ~~ false ->
  ~~ false ->
  ~~ false ->
  ~~ false ->
  p3 != p0 ->
  [/\ codom st.1 \subset [:: p2; p3],
      codom st.2 \subset [:: p1; p2]
    & `d[u, v]_smove =
       D[[:: ↓[u]; st.1; st.2; ↓[v]]].+2]}
case: eqP => [_|/eqP p3Dp0]; first by exists (↓[u], ↓[v]).
512926f270271272151617121314
{st : configuration 4 n * configuration 4 n
| ~~ false ->
  ~~ false ->
  ~~ false ->
  ~~ false ->
  ~~ false ->
  ~~ false ->
  [/\ codom st.1 \subset [:: p2; p3],
      codom st.2 \subset [:: p1; p2]
    & `d[u, v]_smove =
       D[[:: ↓[u]; st.1; st.2; ↓[v]]].+2]}
have pE p : [\/ p = p0, p = p1, p = p2 | p = p3] by apply: pE4.
512926f270271272151617121314
pE:forall p : peg 4,
[\/ p = p0, p = p1, p = p2 | p = p3]
292
have [cs /= csH] := gpath_connect (shanoi_connect u v).
512926f270271272151617121314297
cs:seq
  (finfun_finType (ordinal_finType n.+1)
     (ordinal_finType 4))
csH:gpath smove u v cs
{st : configuration 4 n * configuration 4 n
| true ->
  true ->
  true ->
  true ->
  true ->
  true ->
  [/\ codom st.1 \subset [:: p2; p3],
      codom st.2 \subset [:: p1; p2]
    & `d[u, v]_smove =
      (`d[↓[u], st.1]_smove +
       (`d[st.1, st.2]_smove +
        (`d[st.2, ↓[v]]_smove + 0))).+2]}
have vIcs : v \in cs.
29b
v \in cs
512926f27027127215161712131429729c29d
vIcs:v \in cs
29e
  have := mem_last u cs; rewrite (gpath_last csH).
29b
v \in u :: cs -> v \in cs
2a3
  rewrite inE; case: eqP => //  vEu.
512926f27027127215161712131429729c29d
vEu:v = u
true || (v \in cs) -> v \in cs
2a3
  by case/eqP : p1Dp3; rewrite /p3 vEu.
2a5
29e
case: (@split_first _ cs (fun c => c ldisk != p1)) => [|[[sp cs1] cs2]].
2a5
~~
all
  [predC fun
           c : finfun_finType (ordinal_finType n.+1)
                 (ordinal_finType 4) => c ldisk != p1]
  cs
512926f27027127215161712131429729c29d2a6
sp:finfun_finType (ordinal_finType n.+1)
  (ordinal_finType 4)
cs1, cs2:seq
  (finfun_finType (ordinal_finType n.+1)
     (ordinal_finType 4))
[/\ all
      [predC fun
               c : finfun_finType
                     (ordinal_finType n.+1)
                     (ordinal_finType 4) =>
             c ldisk != p1] 
      (sp, cs1, cs2).1.2,
    (sp, cs1, cs2).1.1 ldisk != p1
  & cs =
    (sp, cs1, cs2).1.2 ++
    (sp, cs1, cs2).1.1 :: (sp, cs1, cs2).2] ->
{st : configuration 4 n * configuration 4 n
| true ->
  true ->
  true ->
  true ->
  true ->
  true ->
  [/\ codom st.1 \subset [:: p2; p3],
      codom st.2 \subset [:: p1; p2]
    & `d[u, v]_smove =
      (`d[↓[u], st.1]_smove +
       (`d[st.1, st.2]_smove +
        (`d[st.2, ↓[v]]_smove + 0))).+2]}
  apply/negP=> /allP /(_ _ vIcs).
2a5
[predC fun
         c : finfun_finType (ordinal_finType n.+1)
               (ordinal_finType 4) => c ldisk != p1] v ->
False
2b8
  rewrite /= -topredE /= negbK.
2a5
v ldisk == p1 -> False
2b8
  by rewrite eq_sym (negPf p1Dp3).
2ba
[/\ all
      [predC fun
               c : finfun_finType
                     (ordinal_finType n.+1)
                     (ordinal_finType 4) =>
             c ldisk != p1] (sp, cs1, cs2).1.2,
    (sp, cs1, cs2).1.1 ldisk != p1
  & cs =
    (sp, cs1, cs2).1.2 ++
    (sp, cs1, cs2).1.1 :: (sp, cs1, cs2).2] ->
{st : configuration 4 n * configuration 4 n
| true ->
  true ->
  true ->
  true ->
  true ->
  true ->
  [/\ codom st.1 \subset [:: p2; p3],
      codom st.2 \subset [:: p1; p2]
    & `d[u, v]_smove =
      (`d[↓[u], st.1]_smove +
       (`d[st.1, st.2]_smove +
        (`d[st.2, ↓[v]]_smove + 0))).+2]}
case=> /allP spH spLE csE.
512926f27027127215161712131429729c29d2a62bb2bc
spH:{in (sp, cs1, cs2).1.2,
  forall
    x : finfun_finType (ordinal_finType n.+1)
          (ordinal_finType 4),
  [predC fun
           c : finfun_finType
                 (ordinal_finType n.+1)
                 (ordinal_finType 4) =>
         c ldisk != p1] x}
spLE:(sp, cs1, cs2).1.1 ldisk != p1
csE:cs =
(sp, cs1, cs2).1.2 ++
(sp, cs1, cs2).1.1 :: (sp, cs1, cs2).2
29e
pose s := last u cs1.
512926f27027127215161712131429729c29d2a62bb2bc2ce2cf2d0
s:=last u cs1:configuration 4 n.+1
29e
have sMsp : s `-->_s sp.
2d4
s `-->_s sp
512926f27027127215161712131429729c29d2a62bb2bc2ce2cf2d02d5
sMsp:s `-->_s sp
29e
  by have := gpath_path csH; rewrite csE cat_path => /and3P[].
2dc
29e
have sLp1 : s ldisk = p1.
2dc
s ldisk = p1
512926f27027127215161712131429729c29d2a62bb2bc2ce2cf2d02d52dd
sLp1:s ldisk = p1
29e
  move: (spH s); rewrite /s; case: (cs1) => //= c cs3 /(_ (mem_last _ _)).
512926f27027127215161712131429729c29d2a62bb2bc2ce2cf2d02d52dd
c:{ffun 'I_n.+1 -> 'I_4}
cs3:seq {ffun 'I_n.+1 -> 'I_4}
last c cs3
  \notin (fun c : {ffun 'I_n.+1 -> 'I_4} =>
          c ldisk != p1) -> last c cs3 ldisk = p1
2e5
  by rewrite -topredE negbK => /eqP.
2e7
29e
have spLp0 : sp ldisk = p0.
2e7
sp ldisk = p0
512926f27027127215161712131429729c29d2a62bb2bc2ce2cf2d02d52dd2e8
spLp0:sp ldisk = p0
29e
  apply/eqP.
2e7
sp ldisk == p0
2f7
  have /(_ ldisk) := move_diskr sMsp.
2e7
(s ldisk != sp ldisk ->
 srel (n:=4) (s ldisk) (sp ldisk)) -> sp ldisk == p0
2f7
  rewrite eq_sym sLp1 => /(_ spLE) /andP[].
2e7
p1 != sp ldisk -> p1 * sp ldisk == 0 -> sp ldisk == p0
2f7
  case: (_ =P p0) => // /val_eqP /=.
2e7
sp ldisk != 0 ->
p1 != sp ldisk -> p1 * sp ldisk == 0 -> false
2f7
  have /eqP/val_eqP/= := p1Dp0.
2e7
p1 != 0 ->
sp ldisk != 0 ->
p1 != sp ldisk -> p1 * sp ldisk == 0 -> false
2f7
  by case: (p1 : nat) => // k; case: (sp _ : nat).
2f9
29e
have sCd : codom (↓[s]) \subset [:: p2; p3].
2f9
codom ↓[s] \subset [:: p2; p3]
512926f27027127215161712131429729c29d2a62bb2bc2ce2cf2d02d52dd2e82fa
sCd:codom ↓[s] \subset [:: p2; p3]
29e
  apply/subsetP=> i; rewrite !inE => /codomP[j] ->.
512926f27027127215161712131429729c29d2a62bb2bc2ce2cf2d02d52dd2e82fa
i:ordinal_finType 4
j:ordinal_finType n
(↓[s] j == p2) || (↓[s] j == p3)
316
  case: (pE (↓[s] j)) => [||->|->]; rewrite ?eqxx ?orbT //.
31d
↓[s] j = p0 -> (↓[s] j == p2) || (↓[s] j == p3)
31d
↓[s] j = p1 -> (↓[s] j == p2) || (↓[s] j == p3)
317
    rewrite ffunE trshift_lift /=.
31d
s (lift ord_max j) = p0 ->
(s (lift ord_max j) == p2)
|| (s (lift ord_max j) == p3)
325
    move/eqP; rewrite -spLp0 -[_ == _]negbK; case/negP.
31d
s (lift ord_max j) != sp ldisk
325
    apply: move_on_toplDr sMsp _ _; first by rewrite sLp1 spLp0.
512926f27027127215161712131429729c29d2a62bb2bc2ce2cf2d02d52e82fa31e31f
lift ord_max j <= ldisk
325
    by rewrite /= /bump [n <= j]leqNgt ltn_ord add0n ltnW.
31d
327
316
  rewrite ffunE trshift_lift /=.
31d
s (lift ord_max j) = p1 ->
(s (lift ord_max j) == p2)
|| (s (lift ord_max j) == p3)
316
  move/eqP; rewrite -sLp1 -[_ == _]negbK; case/negP.
31d
s (lift ord_max j) != s ldisk
316
  apply: move_on_toplDl sMsp _ _; first by rewrite sLp1 spLp0.
333
lift ord_max j < ldisk
316
  by rewrite /= /bump [n <= j]leqNgt ltn_ord add0n.
318
29e
have vIcs2 : v \in cs2.
318
v \in cs2
512926f27027127215161712131429729c29d2a62bb2bc2ce2cf2d02d52dd2e82fa319
vIcs2:v \in cs2
29e
  move: vIcs; rewrite csE !(inE, mem_cat) => /or3P[|/eqP vEs|] //.
512926f27027127215161712131429729c29d2bb2bc2ce2cf2d02d52dd2e82fa319
v \in (sp, cs1, cs2).1.2 -> v \in cs2
512926f27027127215161712131429729c29d2bb2bc2ce2cf2d02d52dd2e82fa319
vEs:v = (sp, cs1, cs2).1.1
34a
34c
    by move=> /spH; rewrite /= -topredE negbK eq_sym (negPf p1Dp3).
356
34a
34b
  by case/eqP : p3Dp0; rewrite /p3 vEs.
34d
29e
case: (@split_last _ (sp :: cs2) (fun c => c ldisk != p3)) => 
     [|/= [[t cs3] cs4]/=].
34d
~~
all
  [predC fun
           c : finfun_finType (ordinal_finType n.+1)
                 (ordinal_finType 4) => c ldisk != p3]
  (sp :: cs2)
512926f27027127215161712131429729c29d2a62bb2bc2ce2cf2d02d52dd2e82fa31934e
t:{ffun 'I_n.+1 -> 'I_4}
cs3, cs4:seq {ffun 'I_n.+1 -> 'I_4}
[/\ t ldisk != p3,
    all
      [predC fun c : {ffun 'I_n.+1 -> 'I_4} =>
             c ldisk != p3] cs4
  & sp :: cs2 = cs3 ++ t :: cs4] ->
{st : configuration 4 n * configuration 4 n
| true ->
  true ->
  true ->
  true ->
  true ->
  true ->
  [/\ codom st.1 \subset [:: p2; p3],
      codom st.2 \subset [:: p1; p2]
    & `d[u, v]_smove =
      (`d[↓[u], st.1]_smove +
       (`d[st.1, st.2]_smove +
        (`d[st.2, ↓[v]]_smove + 0))).+2]}
  apply/negP=> /allP /(_ _ (mem_head _ _)).
34d
[predC fun
         c : finfun_finType (ordinal_finType n.+1)
               (ordinal_finType 4) => c ldisk != p3]
  sp -> False
362
  by rewrite /= -topredE /= negbK spLp0 eq_sym (negPf p3Dp0).
364
367
case=> // tLE tH scs2E.
512926f27027127215161712131429729c29d2a62bb2bc2ce2cf2d02d52dd2e82fa31934e365366
tLE:t ldisk != p3
tH:all
  [predC fun c : {ffun 'I_n.+1 -> 'I_4} =>
         c ldisk != p3] cs4
scs2E:sp :: cs2 = cs3 ++ t :: cs4
29e
have vIcs4 : v \in cs4.
372
v \in cs4
512926f27027127215161712131429729c29d2a62bb2bc2ce2cf2d02d52dd2e82fa31934e365366373374375
vIcs4:v \in cs4
29e
  have := gpath_last csH; rewrite csE scs2E !last_cat /=.
372
last t cs4 = v -> v \in cs4
37a
  case: (cs4) => /= [tEv|c1 cs5 <-]; last by apply: mem_last.
512926f27027127215161712131429729c29d2a62bb2bc2ce2cf2d02d52dd2e82fa31934e365366373374375
tEv:t = v
v \in [::]
37a
  by case/eqP: tLE; rewrite tEv.
37c
29e
  case: cs4 tH scs2E vIcs4 => // tp cs4 /allP tH scs2E vItpcs4.
512926f27027127215161712131429729c29d2a62bb2bc2ce2cf2d02d52dd2e82fa31934e3652ee373
tp:{ffun 'I_n.+1 -> 'I_4}
cs4:seq {ffun 'I_n.+1 -> 'I_4}
tH:{in tp :: cs4,
  forall
    x : finfun_eqType (ordinal_finType n.+1)
          (ordinal_eqType 4),
  [predC fun c : {ffun 'I_n.+1 -> 'I_4} =>
         c ldisk != p3] x}
scs2E:sp :: cs2 = cs3 ++ [:: t, tp & cs4]
vItpcs4:v \in tp :: cs4
29e
have tMtp : t `-->_s tp.
38e
t `-->_s tp
512926f27027127215161712131429729c29d2a62bb2bc2ce2cf2d02d52dd2e82fa31934e3652ee37338f390391392393
tMtp:t `-->_s tp
29e
  by have := gpath_path csH; rewrite csE scs2E !cat_path /= => /and5P[].
39a
29e
have tpLp3 : tp ldisk = p3.
39a
tp ldisk = p3
512926f27027127215161712131429729c29d2a62bb2bc2ce2cf2d02d52dd2e82fa31934e3652ee37338f39039139239339b
tpLp3:tp ldisk = p3
29e
  by move: (tH _ (mem_head _ _)); rewrite /= -topredE negbK => /eqP.
3a5
29e
have tLp0 : t ldisk = p0.
3a5
t ldisk = p0
512926f27027127215161712131429729c29d2a62bb2bc2ce2cf2d02d52dd2e82fa31934e3652ee37338f39039139239339b3a6
tLp0:t ldisk = p0
29e
  apply/eqP.
3a5
t ldisk == p0
3ae
  have /(_ ldisk) := move_diskr tMtp.
3a5
(t ldisk != tp ldisk ->
 srel (n:=4) (t ldisk) (tp ldisk)) -> t ldisk == p0
3ae
  rewrite tpLp3 => /(_ tLE) /andP[].
3a5
t ldisk != p3 -> t ldisk * p3 == 0 -> t ldisk == p0
3ae
  case: (_ =P p0) => // /val_eqP /=.
3a5
t ldisk != 0 ->
t ldisk != p3 -> t ldisk * p3 == 0 -> false
3ae
  have /eqP/val_eqP/= := p3Dp0.
3a5
p3 != 0 ->
t ldisk != 0 ->
t ldisk != p3 -> t ldisk * p3 == 0 -> false
3ae
  by case: (p3 : nat) => // k; case: (t _ : nat).
3b0
29e
have tCd : codom (↓[t]) \subset [:: p1; p2].
3b0
codom ↓[t] \subset [:: p1; p2]
512926f27027127215161712131429729c29d2a62bb2bc2ce2cf2d02d52dd2e82fa31934e3652ee37338f39039139239339b3a63b1
tCd:codom ↓[t] \subset [:: p1; p2]
29e
  apply/subsetP=> i; rewrite !inE => /codomP[j] ->.
512926f27027127215161712131429729c29d2a62bb2bc2ce2cf2d02d52dd2e82fa31934e3652ee37338f39039139239339b3a63b131e31f
(↓[t] j == p1) || (↓[t] j == p2)
3cd
  case: (pE (↓[t] j)) => [|->|->|]; rewrite ?eqxx ?orbT //.
3d4
↓[t] j = p0 -> (↓[t] j == p1) || (↓[t] j == p2)
3d4
↓[t] j = p3 -> (↓[t] j == p1) || (↓[t] j == p2)
3ce
    rewrite ffunE trshift_lift /=.
3d4
t (lift ord_max j) = p0 ->
(t (lift ord_max j) == p1)
|| (t (lift ord_max j) == p2)
3da
    move/eqP; rewrite -tLp0 -[_ == _]negbK; case/negP.
3d4
t (lift ord_max j) != t ldisk
3da
    apply: move_on_toplDl tMtp _ _; first by rewrite tpLp3 tLp0 eq_sym.
512926f27027127215161712131429729c29d2a62bb2bc2ce2cf2d02d52dd2e82fa31934e3652ee37338f3903913923933a63b131e31f
343
3da
    by rewrite /= /bump [n <= j]leqNgt ltn_ord add0n.
3d4
3dc
3cd
  rewrite ffunE trshift_lift /=.
3d4
t (lift ord_max j) = p3 ->
(t (lift ord_max j) == p1)
|| (t (lift ord_max j) == p2)
3cd
  move/eqP; rewrite -tpLp3 -[_ == _]negbK; case/negP.
3d4
t (lift ord_max j) != tp ldisk
3cd
  apply: move_on_toplDr tMtp _ _; first by rewrite tpLp3 tLp0 eq_sym.
3e8
334
3cd
  by rewrite /= /bump [n <= j]leqNgt ltn_ord add0n ltnW.
3cf
29e
exists (↓[s], ↓[t]); split => //=.
512926f27027127215161712131429729c29d2a62bb2bc2ce2cf2d02d52dd2e82fa31934e3652ee37338f39039139239339b3a63b13d0
H, H0, H1, H2, H3, H4:true
`d[u, v]_smove =
(`d[↓[u], ↓[s]]_smove +
 (`d[↓[s], ↓[t]]_smove + (`d[↓[t], ↓[v]]_smove + 0))).+2
rewrite addn0.
3fd
`d[u, v]_smove =
(`d[↓[u], ↓[s]]_smove +
 (`d[↓[s], ↓[t]]_smove + `d[↓[t], ↓[v]]_smove)).+2
move: csH; rewrite csE => csH.
512926f27027127215161712131429729c2a62bb2bc2ce2cf2d02d52dd2e82fa31934e3652ee37338f39039139239339b3a63b13d03fe
csH:gpath smove u v
  ((sp, cs1, cs2).1.2 ++
   (sp, cs1, cs2).1.1 :: (sp, cs1, cs2).2)
403
rewrite (gdist_cat csH) -/s.
407
`d[u, s]_smove + `d[s, v]_smove =
(`d[↓[u], ↓[s]]_smove +
 (`d[↓[s], ↓[t]]_smove + `d[↓[t], ↓[v]]_smove)).+2
move: csH => /gpath_catr; rewrite -/s => csH.
512926f27027127215161712131429729c2a62bb2bc2ce2cf2d02d52dd2e82fa31934e3652ee37338f39039139239339b3a63b13d03fe
csH:gpath smove s v
  ((sp, cs1, cs2).1.1 :: (sp, cs1, cs2).2)
40c
rewrite gdist_cunlift_eq //; try apply: sirr; last by apply: shanoi_connect.
410
`d[↓[u], ↓[s]]_(move (q:=4) (srel (n:=4))) +
`d[s, v]_smove =
(`d[↓[u], ↓[s]]_smove +
 (`d[↓[s], ↓[t]]_smove + `d[↓[t], ↓[v]]_smove)).+2
rewrite -!addnS.
410
`d[↓[u], ↓[s]]_(move (q:=4) (srel (n:=4))) +
`d[s, v]_smove =
`d[↓[u], ↓[s]]_smove +
(`d[↓[s], ↓[t]]_smove + (`d[↓[t], ↓[v]]_smove).+2)
congr (_ + _).
410
`d[s, v]_smove =
`d[↓[s], ↓[t]]_smove + (`d[↓[t], ↓[v]]_smove).+2
have ->: ↓[s] = ↓[sp].
410
↓[s] = ↓[sp]
410
`d[s, v]_smove =
`d[↓[sp], ↓[t]]_smove + (`d[↓[t], ↓[v]]_smove).+2
  have sLDspL : s ldisk != sp ldisk by rewrite sLp1 spLp0.
512926f27027127215161712131429729c2a62bb2bc2ce2cf2d02d52dd2e82fa31934e3652ee37338f39039139239339b3a63b13d03fe411
sLDspL:s ldisk != sp ldisk
421
422
  apply/ffunP => i; rewrite !ffunE.
512926f27027127215161712131429729c2a62bb2bc2ce2cf2d02d52dd2e82fa31934e3652ee37338f39039139239339b3a63b13d03fe411429
i:ordinal_finType n
s (trshift 1 i) = sp (trshift 1 i)
422
  apply: move_disk1 sMsp sLDspL _.
512926f27027127215161712131429729c2a62bb2bc2ce2cf2d02d52e82fa31934e3652ee37338f39039139239339b3a63b13d03fe41142e
ldisk != trshift 1 i
422
  by apply/negP => /eqP/val_eqP /=; rewrite eqn_leq leqNgt ltn_ord.
410
424
rewrite (gdist_cons csH) addnS; congr (_).+1.
410
`d[(sp, cs1, cs2).1.1, v]_smove =
`d[↓[sp], ↓[t]]_smove + (`d[↓[t], ↓[v]]_smove).+1
move: csH => /gpath_consr csH.
512926f27027127215161712131429729c2a62bb2bc2ce2cf2d02d52dd2e82fa31934e3652ee37338f39039139239339b3a63b13d03fe
csH:gpath smove (sp, cs1, cs2).1.1 v
  (sp, cs1, cs2).2
43b
have ctEctp : ↓[t] = ↓[tp].
43f
↓[t] = ↓[tp]
512926f27027127215161712131429729c2a62bb2bc2ce2cf2d02d52dd2e82fa31934e3652ee37338f39039139239339b3a63b13d03fe440
ctEctp:↓[t] = ↓[tp]
43b
  have tLDtpL : t ldisk != tp ldisk by rewrite tLp0 tpLp3 eq_sym.
512926f27027127215161712131429729c2a62bb2bc2ce2cf2d02d52dd2e82fa31934e3652ee37338f39039139239339b3a63b13d03fe440
tLDtpL:t ldisk != tp ldisk
444
445
  apply/ffunP => i; rewrite !ffunE.
512926f27027127215161712131429729c2a62bb2bc2ce2cf2d02d52dd2e82fa31934e3652ee37338f39039139239339b3a63b13d03fe44044d42e
t (trshift 1 i) = tp (trshift 1 i)
445
  apply: move_disk1 tMtp tLDtpL _.
512926f27027127215161712131429729c2a62bb2bc2ce2cf2d02d52dd2e82fa31934e3652ee37338f3903913923933a63b13d03fe44042e
434
445
  by apply/negP => /eqP/val_eqP /=; rewrite eqn_leq leqNgt ltn_ord.
447
43b
case: cs3 scs2E => [[spEt cs2E]|c3 cs3 /= [spE cs2E]].
512926f27027127215161712131429729c2a62bb2bc2ce2cf2d02d52dd2e82fa31934e36537338f39039139339b3a63b13d03fe440448
spEt:sp = t
cs2E:cs2 = tp :: cs4
43b
512926f27027127215161712131429729c2a62bb2bc2ce2cf2d02d52dd2e82fa31934e36537338f39039139339b3a63b13d03fe440448
c3:{ffun 'I_n.+1 -> 'I_4}
2ee
spE:sp = c3
cs2E:cs2 = cs3 ++ [:: t, tp & cs4]
`d[sp, v]_smove =
`d[↓[sp], ↓[t]]_smove + (`d[↓[t], ↓[v]]_smove).+1
  move: csH; rewrite spEt cs2E => csH.
512926f27027127215161712131429729c2a62bb2bc2ce2cf2d02d52dd2e82fa31934e36537338f39039139339b3a63b13d03fe44845e45f
csH:gpath smove (t, cs1, tp :: cs4).1.1 v
  (t, cs1, tp :: cs4).2
`d[(t, cs1, tp :: cs4).1.1, v]_smove =
`d[↓[t], ↓[t]]_smove + (`d[↓[t], ↓[v]]_smove).+1
460
  rewrite gdist0 add0n (gdist_cons csH) ctEctp.
46a
(`d[tp, v]_smove).+1 = (`d[↓[tp], ↓[v]]_smove).+1
460
  move: csH => /gpath_consr csH.
512926f27027127215161712131429729c2a62bb2bc2ce2cf2d02d52dd2e82fa31934e36537338f39039139339b3a63b13d03fe44845e45f
csH:gpath smove tp v cs4
470
460
  rewrite gdist_cunlift_eq //; try by apply: sirr.
474
connect (move (q:=4) (srel (n:=4))) tp v
460
  by apply: shanoi_connect.
462
466
move: csH; rewrite cs2E -cat_rcons => csH.
512926f27027127215161712131429729c2a62bb2bc2ce2cf2d02d52dd2e82fa31934e36537338f39039139339b3a63b13d03fe4484632ee464465
csH:gpath smove
  (sp, cs1, rcons cs3 t ++ tp :: cs4).1.1 v
  (sp, cs1, rcons cs3 t ++ tp :: cs4).2
466
rewrite (gdist_cat csH) last_rcons.
480
`d[(sp, cs1, rcons cs3 t ++ tp :: cs4).1.1, t]_smove +
`d[t, v]_smove =
`d[↓[sp], ↓[t]]_smove + (`d[↓[t], ↓[v]]_smove).+1
rewrite gdist_cunlift_eq //; try apply: sirr; last 2 first.
480
connect (move (q:=4) (srel (n:=4)))
  (sp, cs1, rcons cs3 t ++ tp :: cs4).1.1 t
480
(sp, cs1, rcons cs3 t ++ tp :: cs4).1.1 ldisk =
t ldisk
480
`d[↓[(sp, cs1, rcons cs3 t ++ tp :: cs4).1.1], ↓[t]]_
(move (q:=4) (srel (n:=4))) + `d[t, v]_smove =
`d[↓[sp], ↓[t]]_smove + (`d[↓[t], ↓[v]]_smove).+1
-
487 by apply: shanoi_connect.
480
48c
48d
-
491 by rewrite tLp0.
480
48e
congr (_ + _).
480
`d[t, v]_smove = (`d[↓[t], ↓[v]]_smove).+1
move: csH => /gpath_catr; rewrite last_rcons => csH.
512926f27027127215161712131429729c2a62bb2bc2ce2cf2d02d52dd2e82fa31934e36537338f39039139339b3a63b13d03fe4484632ee464465
csH:gpath smove t v (tp :: cs4)
49b
rewrite (gdist_cons csH); congr (_).+1.
49f
`d[tp, v]_smove = `d[↓[t], ↓[v]]_smove
move: csH => /gpath_consr csH.
512926f27027127215161712131429729c2a62bb2bc2ce2cf2d02d52dd2e82fa31934e36537338f39039139339b3a63b13d03fe4484632ee464465475
4a4
rewrite ctEctp gdist_cunlift_eq //; try apply: sirr.
4a8
479
by apply: shanoi_connect.
Qed.

Definition sgdist1 n (p2 : peg 4) (u v : configuration 4 n.+1) :=
 let: (exist (x, _) _) := sgdist_pair p2 u v in x.

Definition sgdist2 n (p2 : peg 4) (u v : configuration 4 n.+1) :=
 let: (exist (_, x) _) := sgdist_pair p2 u v in x.

Lemma sgdistE n (p2 : peg 4) (u v : configuration 4 n.+1) 
    (p1 := u ldisk) (p3 := v ldisk): 
    p1 != p2 -> p1 != p3 -> p2 != p3 -> 
    p1 != p0 -> p2 != p0 -> p3 != p0 -> 
   [/\ codom (sgdist1 p2 u v) \subset [:: p2; p3], 
       codom (sgdist2 p2 u v) \subset [:: p1; p2] & 
       `d[u, v]_smove = D[[:: ↓[u]; sgdist1 p2 u v; sgdist2 p2 u v; ↓[v]]].+2].
26e
p1 != p2 ->
p1 != p3 ->
p2 != p3 ->
p1 != p0 ->
p2 != p0 ->
p3 != p0 ->
[/\ codom (sgdist1 (n:=n) p2 u v) \subset [:: p2; p3],
    codom (sgdist2 (n:=n) p2 u v) \subset [:: p1; p2]
  & `d[u, v]_smove =
     D[[:: ↓[u]; sgdist1 (n:=n) p2 u v;
           sgdist2 (n:=n) p2 u v; ↓[v]]].+2]
Proof.
4ad
move=> p1Dp2 p1Dp3 p2Dp3 p1Dp0 p2Dp0 p3Dp0.
291
[/\ codom (sgdist1 (n:=n) p2 u v) \subset [:: p2; p3],
    codom (sgdist2 (n:=n) p2 u v) \subset [:: p1; p2]
  & `d[u, v]_smove =
     D[[:: ↓[u]; sgdist1 (n:=n) p2 u v;
           sgdist2 (n:=n) p2 u v; ↓[v]]].+2]
by rewrite /sgdist1 /sgdist2; case: sgdist_pair  => [] [x y] [].
Qed.

(* The beta function *)
Definition beta n l k := 
  if ((1 < l) && (k == n.-1)) then α_[l] k else (α_[1] k).*2.

Local Notation "β_[ n , l ]" := (beta n l) (format "β_[ n ,  l ]" ).

Lemma leq_beta n l k : α_[l] k <= β_[n, l] k.
5
n, l, k:nat
α_[l] k <= β_[n, l] k
Proof.
4b6
rewrite /beta; case: (_ < _) => /=; last first.
4b8
α_[l] k <= (α_[1] k).*2
4b8
α_[l] k <=
(if k == n.-1 then α_[l] k else (α_[1] k).*2)
  by apply: bound_alphaL.
4b8
4c2
by case: (_ == _) => //=; apply: bound_alphaL.
Qed.

Lemma geq_beta n l k : β_[n, l] k <= (α_[1] k).*2.
5
n, l:nat
k:nat_eqType
β_[n, l] k <= (α_[1] k).*2
Proof.
4c7
rewrite /beta; case: (_ < _) => //=.
4c9
(if k == n.-1 then α_[l] k else (α_[1] k).*2) <=
(α_[1] k).*2
by case: (_ == _) => //=; apply: bound_alphaL.
Qed.

Lemma beta1E n k : β_[n, 1] k = (α_[1] k).*2.
51294cb
β_[n, 1] k = (α_[1] k).*2
Proof.
4d3 by rewrite /beta ltnn. Qed.

Definition sp n (a : configuration 4 n) l p := 
  \sum_(k < n) (a k != p) * β_[n, l] k.
 
Lemma sum_beta_S n l (a : configuration 4 n.+1) p : 1 < l ->
  sp a l p =
  \sum_(k < n) ((↓[a] k  != p) * (α_[1] k).*2) + (a ord_max != p) * α_[l] n.
54ca
a:configuration 4 n.+1
p:ordinal_eqType 4
1 < l ->
sp (n:=n.+1) a l p =
\sum_(k < n) (↓[a] k != p) * (α_[1] k).*2 +
(a ord_max != p) * α_[l] n
Proof.
4d9
move=> l_gt1; rewrite /sp.
54ca4dc4dd
l_gt1:1 < l
\sum_(k < n.+1) (a k != p) * β_[n.+1, l] k =
\sum_(k < n) (↓[a] k != p) * (α_[1] k).*2 +
(a ord_max != p) * α_[l] n
rewrite big_ord_recr /= /beta l_gt1 eqxx /=; congr (_ + _).
4e3
\sum_(i < n)
   (a (widen_ord (m:=n.+1) (leqnSn n) i) != p) *
   (if i == n then α_[l] i else (α_[1] i).*2) =
\sum_(k < n) (↓[a] k != p) * (α_[1] k).*2
apply: eq_bigr => i _; rewrite !ffunE; congr ((a _ != _) * _).
54ca4dc4dd4e4
i:'I_n
widen_ord (m:=n.+1) (leqnSn n) i = trshift 1 i
4ed
(if i == n then α_[l] i else (α_[1] i).*2) =
(α_[1] i).*2
  by apply: val_inj.
4ed
4f2
by rewrite eqn_leq [_ <= i]leqNgt ltn_ord andbF.
Qed.

Lemma leq_sum_beta n l a : 
  \sum_(k < n) a k * β_[n, l] k <= \sum_(k < n) a k * (α_[1] k).*2.
54ca
a:'I_n -> nat
\sum_(k < n) a k * β_[n, l] k <=
\sum_(k < n) a k * (α_[1] k).*2
Proof.
4f7
by apply: leq_sum => i _; rewrite leq_mul2l geq_beta orbT.
Qed.

Lemma sum_alpha_diffE n (f : configuration 4 n.+1) (p1 p2 : peg 4) v1 v2 :
  1 < v1 -> 1 < v2 -> p1 != p2 -> codom f \subset [:: p1; p2] ->
  sp f v1 p1 + sp f v2 p2 <= (S_[1] n).*2 + α_[maxn v1 v2] n.
5129
f:configuration 4 n.+1
229
v1, v2:nat
1 < v1 ->
1 < v2 ->
p1 != p2 ->
codom f \subset [:: p1; p2] ->
sp (n:=n.+1) f v1 p1 + sp (n:=n.+1) f v2 p2 <=
(S_[1] n).*2 + α_[maxn v1 v2] n
Proof.
4fe
move=> v1_gt1 v2_gt1 p1Dp2 cH.
5129501229502
v1_gt1:1 < v1
v2_gt1:1 < v2
15
cH:codom f \subset [:: p1; p2]
sp (n:=n.+1) f v1 p1 + sp (n:=n.+1) f v2 p2 <=
(S_[1] n).*2 + α_[maxn v1 v2] n
rewrite !sum_beta_S //.
508
\sum_(k < n) (↓[f] k != p1) * (α_[1] k).*2 +
(f ord_max != p1) * α_[v1] n +
(\sum_(k < n) (↓[f] k != p2) * (α_[1] k).*2 +
 (f ord_max != p2) * α_[v2] n) <=
(S_[1] n).*2 + α_[maxn v1 v2] n
rewrite -addnA -addnCA addnC !addnA -addnA leq_add //.
508
\sum_(k < n) (↓[f] k != p1) * (α_[1] k).*2 +
\sum_(k < n) (↓[f] k != p2) * (α_[1] k).*2 <=
(S_[1] n).*2
508
(f ord_max != p2) * α_[v2] n +
(f ord_max != p1) * α_[v1] n <= 
α_[maxn v1 v2] n
  rewrite leq_eqVlt; apply/orP; left; apply/eqP.
508
\sum_(k < n) (↓[f] k != p1) * (α_[1] k).*2 +
\sum_(k < n) (↓[f] k != p2) * (α_[1] k).*2 =
(S_[1] n).*2
515
  rewrite -addnn !dsum_alphaLE -!big_split /=; apply: eq_bigr => i _.
512950122950250950a1550b4ee
(↓[f] i != p1) * (α_[1] i).*2 +
(↓[f] i != p2) * (α_[1] i).*2 = α_[1] i + α_[1] i
515
  rewrite !ffunE; set v := trshift _ _.
512950122950250950a1550b4ee
v:=trshift 1 i:'I_(1 + n)
(f v != p1) * (α_[1] i).*2 +
(f v != p2) * (α_[1] i).*2 = α_[1] i + α_[1] i
515
  have /subsetP := cH => /(_ (f v) (codom_f _ _)).
524
f v \in [:: p1; p2] ->
(f v != p1) * (α_[1] i).*2 +
(f v != p2) * (α_[1] i).*2 = α_[1] i + α_[1] i
515
  rewrite !inE addnn.
524
(f v == p1) || (f v == p2) ->
(f v != p1) * (α_[1] i).*2 +
(f v != p2) * (α_[1] i).*2 = (α_[1] i).*2
515
  case: eqP => /= x1; case: eqP => /= x2; rewrite ?(mul1n, add0n, addn0) //.
512950122950250950a1550b4ee525
x1:f v = p1
x2:f v = p2
true -> 0 * (α_[1] i).*2 = (α_[1] i).*2
515
  by move/eqP: x2; rewrite x1 (negPf p1Dp2).
508
(f ord_max != p2) * α_[v2] n +
(f ord_max != p1) * α_[v1] n <= α_[maxn v1 v2] n
  have /subsetP/(_ (f ord_max) (codom_f _ _)) := cH.
508
f ord_max \in [:: p1; p2] ->
(f ord_max != p2) * α_[v2] n +
(f ord_max != p1) * α_[v1] n <= α_[maxn v1 v2] n
rewrite !inE.
508
(f ord_max == p1) || (f ord_max == p2) ->
(f ord_max != p2) * α_[v2] n +
(f ord_max != p1) * α_[v1] n <= α_[maxn v1 v2] n
case: eqP => /= x1; case: eqP => /= x2 // _; rewrite ?(mul1n, add0n, addn0) //;
     apply: (increasingE (increasing_alphaL_l _)).
512950122950250950a1550b
x1:f ord_max = p1
x2:f ord_max <> p2
v2 <= maxn v1 v2
512950122950250950a1550b
x1:f ord_max <> p1
x2:f ord_max = p2
v1 <= maxn v1 v2
  by apply: leq_maxr.
54b
54e
by apply: leq_maxl.
Qed.

(* This corresponds to 4.1 *)
Section Case1.

Definition sd n l (u : {ffun 'I_l.+1 -> configuration 4 n}) :=
    \sum_(i < l) `d[u (inord i), u (inord i.+1)]_smove.

Variable n : nat.
Hypothesis IH: 
     forall (l : nat) (p1 p2 p3 : ordinal_eqType 4)
       (u : {ffun 'I_l.+1 -> configuration 4 n.+1}),
     p1 != p2 ->
     p1 != p3 ->
     p2 != p3 ->
     p1 != p0 ->
     p2 != p0 ->
     p3 != p0 ->
     (forall k : 'I_l.+1,
      0 < k < l -> codom (u k) \subset [:: p2; a[p1, p3] k]) ->
     (S_[l] n.+1).*2 <= sd u + 
                     sp (u ord0) l (a[p1, p3] 0) + 
                     sp (u ord_max) l (a[p1, p3] l).
 
Variable l : nat.
Variables p1 p2 p3: peg 4.
Variable u : {ffun 'I_l.+2 -> configuration 4 n.+2}.
Hypothesis p1Dp2 : p1 != p2.
Hypothesis p1Dp3 : p1 != p3.
Hypothesis p2Dp3 : p2 != p3.
Hypothesis p1Dp0 : p1 != p0.
Hypothesis p2Dp0 : p2 != p0.
Hypothesis p3Dp0 : p3 != p0.
Hypothesis cH : forall k : 'I_l.+2,
     0 < k < l.+1 -> codom (u k) \subset [:: p2; a[p1, p3] k].

Let u':= ([ffun i => ↓[u i]] : {ffun 'I_l.+2 -> configuration 4 n.+1})
 : {ffun 'I_l.+2 -> configuration 4 n.+1}.

Let H: forall k : 'I_l.+2,
    0 < k < l.+1 -> codom (u' k) \subset [:: p2; a[p1, p3] k].
5129
IH:forall (l : nat) (p1 p2 p3 : ordinal_eqType 4)
  (u : {ffun 'I_l.+1 -> configuration 4 n.+1}),
p1 != p2 ->
p1 != p3 ->
p2 != p3 ->
p1 != p0 ->
p2 != p0 ->
p3 != p0 ->
(forall k : 'I_l.+1,
 0 < k < l ->
 codom (u k) \subset [:: p2;  a[p1, p3] k]) ->
(S_[l] n.+1).*2 <=
sd (n:=n.+1) u +
sp (n:=n.+1) (u ord0) l ( a[p1, p3] 0) +
sp (n:=n.+1) (u ord_max) l ( a[p1, p3] l)
l:nat
11
u:{ffun 'I_l.+2 -> configuration 4 n.+2}
151617121314
cH:forall k : 'I_l.+2,
0 < k < l.+1 ->
codom (u k) \subset [:: p2;  a[p1, p3] k]
u':=([ffun i => ↓[u i]]
   : {ffun 'I_l.+2 -> configuration 4 n.+1})
  : {ffun 'I_l.+2 -> configuration 4 n.+1}:{ffun 'I_l.+2 -> configuration 4 n.+1}
forall k : 'I_l.+2,
0 < k < l.+1 ->
codom (u' k) \subset [:: p2;  a[p1, p3] k]
Proof.
553 by move=> k kH; rewrite ffunE; apply/codom_liftr/cH. Qed.

Let apeg13D2 a : a[p1, p3] a != p2.
51295565571155815161712131455955a
H:forall k : 'I_l.+2,
0 < k < l.+1 ->
codom (u' k) \subset [:: p2;  a[p1, p3] k]
228
 a[p1, p3] a != p2
Proof.
55e by rewrite /apeg; case: odd; rewrite // eq_sym. Qed.

Let apeg13D0 a : a[p1, p3] a != p0.
51295565571155815161712131455955a561
apeg13D2:forall a : nat,  a[p1, p3] a != p2
228
 a[p1, p3] a != p0
Proof.
565 by rewrite /apeg; case: odd; rewrite // eq_sym. Qed.

Hypothesis KH1 : u ord0 ord_max = p1.
Hypothesis KH2 : u ord_max ord_max != a[p3, p1] l.
Hypothesis l_gt0 : l != 0.

Lemma case1 :
  (S_[l.+1] n.+2).*2 <= 
  sd u + sp (u ord0) l.+1 p1 + 
         sp (u ord_max) l.+1 (a[p3, p1] l).
51295565571155815161712131455955a561568
apeg13D0:forall a : nat,  a[p1, p3] a != p0
KH1:u ord0 ord_max = p1
KH2:u ord_max ord_max !=  a[p3, p1] l
l_gt0:l != 0
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u + sp (n:=n.+2) (u ord0) l.+1 p1 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
Proof.
56c
have il1E : inord l.+1 = ord_max :> 'I_l.+2
  by apply/val_eqP; rewrite /= inordK.
51295565571155815161712131455955a56156856f570571572
il1E:inord l.+1 = ord_max
573
have iE : exists i, u (inord i) ord_max != a[p1, p3] i.
578
exists i : nat, u (inord i) ord_max !=  a[p1, p3] i
51295565571155815161712131455955a56156856f570571572579
iE:exists i : nat,
  u (inord i) ord_max !=  a[p1, p3] i
573
  by exists l.+1; rewrite apegS il1E.
580
573
case: (@ex_minnP _ iE) => a aH aMin.
51295565571155815161712131455955a56156856f570571572579581228
aH:u (inord a) ord_max !=  a[p1, p3] a
aMin:forall n0 : nat,
u (inord n0) ord_max !=  a[p1, p3] n0 ->
a <= n0
573
have aMin1 i : i < a -> u (inord i) ord_max = a[p1, p3] i.
51295565571155815161712131455955a56156856f57057157257958122858958a14a
i < a -> u (inord i) ord_max =  a[p1, p3] i
51295565571155815161712131455955a56156856f57057157257958122858958a
aMin1:forall i : nat,
i < a -> u (inord i) ord_max =  a[p1, p3] i
573
  by case: (u (inord i) ord_max =P a[p1, p3] i) => // /eqP /aMin; case: leqP.
592
573
have aLld1 : a <= l.+1 by apply: aMin; rewrite apegS il1E.
51295565571155815161712131455955a56156856f57057157257958122858958a593
aLld1:a <= l.+1
573
have a_gt0 : 0 < a by case: (a) aH; rewrite // apeg0 -KH1 inord_eq0 //= eqxx.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b
a_gt0:0 < a
573
pose ai : 'I_l.+2 := inord a.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a0
ai:=inord a : 'I_l.+2:'I_l.+2
573
have aiE : ai = a :> nat by rewrite inordK.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a5
aiE:ai = a
573
pose b := l.+1 - a.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa
b:=l.+1 - a:nat
573
pose bi : 'I_l.+2 := inord b.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af
bi:=inord b : 'I_l.+2:'I_l.+2
573
have biE : bi = b :> nat by rewrite inordK // /b ltn_subLR // leq_addl.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b4
biE:bi = b
573
have l_ggt0 : 0 < l by case: l l_gt0.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b9
l_ggt0:0 < l
573
have [/andP[a_gt1 aLlm1]|] := boolP (2 <= a <= l.-1).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be
a_gt1:1 < a
aLlm1:a <= l.-1
573
5bd
~~ (1 < a <= l.-1) ->
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u + sp (n:=n.+2) (u ord0) l.+1 p1 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
  have aLl : a <= l by rewrite (leq_trans aLlm1) // -subn1 leq_subr.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c4
aLl:a <= l
573
5c5
  have b_gt1 : 1 < b by rewrite leq_subRL // addn2 ltnS -[l]prednK.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc
b_gt1:1 < b
573
5c5
  have uaiLEp2 : u ai ldisk = p2.
5d0
u ai ldisk = p2
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d1
uaiLEp2:u ai ldisk = p2
573
5c6
    have /cH/subsetP/(_ (u ai ldisk) (codom_f _ _)) : 0 < ai < l.+1.
5d0
0 < ai < l.+1
5d0
u ai ldisk \in [:: p2;  a[p1, p3] ai] ->
u ai ldisk = p2
5d75c6
      by rewrite aiE (leq_trans _ a_gt1) //= ltnS (leq_trans aLl1)
                 // ssrnat.leq_pred.
5d0
5e0
5d6
    by rewrite !inE aiE (negPf aH) orbF => /eqP.
5d8
573
5c5
  pose si i : configuration 4 n.+1 := 
     sgdist1 p2 (u (inord i)) (u (inord i.+1)).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d9
si:=(fun i : nat =>
 sgdist1 (n:=n.+1) p2 (u (inord i))
   (u (inord i.+1)))
  : nat -> configuration 4 n.+1:nat -> configuration 4 n.+1
573
5c5
  pose ti i : configuration 4 n.+1 := 
     sgdist2 p2 (u (inord i)) (u (inord i.+1)).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb
ti:=(fun i : nat =>
 sgdist2 (n:=n.+1) p2 (u (inord i))
   (u (inord i.+1)))
  : nat -> configuration 4 n.+1:nat -> configuration 4 n.+1
573
5c5
  have sitiH i : i < a.-1 ->
     [/\ codom (si i) \subset [:: p2; u (inord i.+1) ldisk ], 
         codom (ti i) \subset [:: u (inord i) ldisk; p2] & 
         `d[u (inord i), u (inord i.+1)]_smove = 
           D[[:: ↓[u (inord i)]; si i; ti i; ↓[u (inord i.+1)]]].+2].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f014a
i < a.-1 ->
[/\ codom (si i) \subset [:: p2; u (inord i.+1) ldisk],
    codom (ti i) \subset [:: u (inord i) ldisk; p2]
  & `d[u (inord i), u (inord i.+1)]_smove =
     D[[:: ↓[u (inord i)]; si i; ti i;
           ↓[u (inord i.+1)]]].+2]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f0
sitiH:forall i : nat,
i < a.-1 ->
[/\ codom (si i) \subset
    [:: p2; u (inord i.+1) ldisk],
    codom (ti i) \subset
    [:: u (inord i) ldisk; p2]
  & `d[u (inord i), u (inord i.+1)]_smove =
     D[[:: ↓[u (inord i)]; 
          si i; ti i; ↓[
          u (inord i.+1)]]].+2]
573
5c6
    move=> iLa.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f014a
iLa:i < a.-1
[/\ codom (si i) \subset [:: p2; u (inord i.+1) ldisk],
    codom (ti i) \subset [:: u (inord i) ldisk; p2]
  & `d[u (inord i), u (inord i.+1)]_smove =
     D[[:: ↓[u (inord i)]; si i; ti i;
           ↓[u (inord i.+1)]]].+2]
5f6
    have iLa1 : i < a by rewrite (leq_trans iLa) // ssrnat.leq_pred.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f014a5fe
iLa1:i < a
5ff
5f6
    apply: sgdistE => //.
603
u (inord i) ldisk != p2
603
u (inord i) ldisk != u (inord i.+1) ldisk
603
p2 != u (inord i.+1) ldisk
603
u (inord i) ldisk != p0
603
u (inord i.+1) ldisk != p0
5f75c6
    -
606 by rewrite aMin1.
603
60b
60c60e6105f75c6
    -
614 rewrite !aMin1 //; last by rewrite -[a]prednK.
603
 a[p1, p3] i !=  a[p1, p3] i.+1
616
      by rewrite apegS apeg_eqC.
603
60d
60e6105f75c6
    -
61d by rewrite aMin1 1?eq_sym // -[a]prednK.
603
60f
6105f75c6
    -
622 by rewrite aMin1.
603
611
5f6
    by rewrite aMin1 // -[a]prednK.
5f8
573
5c5
  pose sam1 : configuration 4 n.+1 := 
     sgdist1 (a[p1, p3] a) (u (inord a.-1)) (u (inord a)).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f9
sam1:=sgdist1 (n:=n.+1) ( a[p1, p3] a)
  (u (inord a.-1)) (u (inord a))
  : configuration 4 n.+1:configuration 4 n.+1
573
5c5
  pose tam1 : configuration 4 n.+1 := 
     sgdist2 (a[p1, p3] a) (u (inord a.-1)) (u (inord a)).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f9630
tam1:=sgdist2 (n:=n.+1) ( a[p1, p3] a)
  (u (inord a.-1)) (u (inord a))
  : configuration 4 n.+1:configuration 4 n.+1
573
5c5
  have [sam1C tam1C duam1ua1E] : 
     [/\ codom sam1 \subset [:: a[p1, p3] a; u (inord a) ldisk ], 
         codom tam1 \subset [:: u (inord a.-1) ldisk; a[p1, p3] a] & 
         `d[u (inord a.-1), u (inord a)]_smove = 
           D[[:: ↓[u (inord a.-1)]; sam1; tam1; ↓[u (inord a)]]].+2].
634
[/\ codom sam1 \subset
    [::  a[p1, p3] a; u (inord a) ldisk],
    codom tam1 \subset
    [:: u (inord a.-1) ldisk;  a[p1, p3] a]
  & `d[u (inord a.-1), u (inord a)]_smove =
     D[[:: ↓[u (inord a.-1)]; sam1; tam1;
           ↓[u (inord a)]]].+2]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f9630635
sam1C:codom sam1 \subset
[::  a[p1, p3] a; u (inord a) ldisk]
tam1C:codom tam1 \subset
[:: u (inord a.-1) ldisk;  a[p1, p3] a]
duam1ua1E:`d[u (inord a.-1), u (inord a)]_smove =
 D[[:: ↓[u (inord a.-1)]; sam1; tam1;
       ↓[u (inord a)]]].+2
573
5c6
    apply: sgdistE => //.
634
u (inord a.-1) ldisk !=  a[p1, p3] a
634
u (inord a.-1) ldisk != u (inord a) ldisk
634
 a[p1, p3] a != u (inord a) ldisk
634
u (inord a.-1) ldisk != p0
634
u (inord a) ldisk != p0
63b5c6
    -
641 by rewrite aMin1 ?prednK // -{2}[a]prednK //= apegS apeg_eqC.
634
646
64764964b63b5c6
    -
64f by rewrite uaiLEp2 aMin1 ?prednK.
634
648
64964b63b5c6
    -
654 by rewrite uaiLEp2.
634
64a
64b63b5c6
    -
659 by rewrite aMin1 ?prednK.
634
64c
63a
    by rewrite uaiLEp2.
63c
573
5c5 
  have {}tam1C : codom tam1 \subset [:: p1; p3].
63c
codom tam1 \subset [:: p1; p3]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f
tam1C:codom tam1 \subset [:: p1; p3]
573
5c6
    move: tam1C; rewrite aMin1; last by rewrite -{2}[a]prednK.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f
codom tam1 \subset [::  a[p1, p3] a.-1;  a[p1, p3] a] ->
codom tam1 \subset [:: p1; p3]
667
    rewrite -subn1 apegO oddB //= [a[_, _]a]apegO; case: odd => //=.
66e
codom tam1 \subset [::  a[p3, p1] 0;  a[p3, p1] 1] ->
codom tam1 \subset [:: p1; p3]
667
    by rewrite codom_subC.
669
573
5c5
  pose u1 := 
    [ffun i =>
    if ((i : 'I_(3 * a).-2.+1) == 3 * a.-1 :>nat) then ↓[u (inord a.-1)] 
    else if (i == (3 * a.-1).+1 :>nat) then sam1
    else if (i %% 3) == 0 then ↓[u (inord (i %/ 3))]
    else if (i %% 3) == 1 then si (i %/ 3)
    else ti (i %/ 3)].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a
u1:=[ffun i => if
            (i : 'I_(3 * a).-2.+1) == 
            3 * a.-1 :> nat
           then ↓[u (inord a.-1)]
           else
            if i == (3 * a.-1).+1 :> nat
            then sam1
            else
             if i %% 3 == 0
             then ↓[u (inord (i %/ 3))]
             else
              if i %% 3 == 1
              then si (i %/ 3)
              else ti (i %/ 3)]:{ffun forall x : ordinal_finType (3 * a).-2.+1,
      (fun=> configuration 4 n.+1) x}
573
5c5
  have u10E : u1 ord0 = ↓[u ord0].
67a
u1 ord0 = ↓[u ord0]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b
u10E:u1 ord0 = ↓[u ord0]
573
5c6
    rewrite ffunE /= ifN ?inord_eq0 //.
67a
0 != 3 * a.-1
680
    by rewrite neq_ltn muln_gt0 /= -ltnS prednK //= a_gt1.
682
573
5c5
  have uiME : u1 ord_max = sam1.
682
u1 ord_max = sam1
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b683
uiME:u1 ord_max = sam1
573
5c6
    rewrite ffunE /= eqn_leq leqNgt -{2}[a]prednK // mulnS.
682
(if
  ~~ (3 * a.-1 < (3 + 3 * a.-1).-2) &&
  (3 * a.-1 <= (3 * a).-2)
 then ↓[u (inord a.-1)]
 else
  if (3 * a).-2 == (3 * a.-1).+1
  then sam1
  else
   if (3 * a).-2 %% 3 == 0
   then ↓[u (inord ((3 * a).-2 %/ 3))]
   else
    if (3 * a).-2 %% 3 == 1
    then si ((3 * a).-2 %/ 3)
    else ti ((3 * a).-2 %/ 3)) = sam1
68f
    rewrite addSn add2n ltnS leqnn /=.
682
(if (3 * a).-2 == (3 * a.-1).+1
 then sam1
 else
  if (3 * a).-2 %% 3 == 0
  then ↓[u (inord ((3 * a).-2 %/ 3))]
  else
   if (3 * a).-2 %% 3 == 1
   then si ((3 * a).-2 %/ 3)
   else ti ((3 * a).-2 %/ 3)) = sam1
68f
    by rewrite -{1}[a]prednK // mulnS eqxx.
691
573
5c5
  pose u2 :=  [ffun i  : 'I_3 =>  
    if i == 0 :> nat then sam1 else
    if i == 1 :> nat then tam1 else ↓[u ai]].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b683692
u2:=[ffun i => if i == 0 :> nat
           then sam1
           else
            if i == 1 :> nat
            then tam1
            else ↓[u ai]]:{ffun forall x : ordinal_finType 3,
      (fun=> configuration 4 n.+1) x}
573
5c5
  pose u3 := [ffun i => ↓[u (inord ((i : 'I_b.+1) + a))]].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a2
u3:=[ffun i => ↓[u (inord ((i : 'I_b.+1) + a))]]:{ffun forall x : ordinal_finType b.+1,
      (fun=> configuration 4 n.+1) x}
573
5c5
  have P1 : a.*2 + sd u1 + sd u2 + sd u3 <= sd u.
6a6
a.*2 + sd (n:=n.+1) u1 + sd (n:=n.+1) u2 +
sd (n:=n.+1) u3 <= sd (n:=n.+2) u
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a7
P1:a.*2 + sd (n:=n.+1) u1 + sd (n:=n.+1) u2 +
sd (n:=n.+1) u3 <= sd (n:=n.+2) u
573
5c6 
    have G b1 : b1 = a ->  
      \sum_(i < (3 * b1).-2) `d[u1 (inord i), u1 (inord i.+1)]_smove =
      \sum_(i < (3 * a.-1)) `d[u1 (inord i), u1 (inord i.+1)]_smove +
      `d[↓[u (inord a.-1)], sam1]_smove.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a7
b1:nat
b1 = a ->
\sum_(i < (3 * b1).-2)
   `d[u1 (inord i), u1 (inord i.+1)]_smove =
\sum_(i < 3 * a.-1)
   `d[u1 (inord i), u1 (inord i.+1)]_smove +
`d[↓[u (inord a.-1)], sam1]_smove
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a7
G:forall b1 : nat,
b1 = a ->
\sum_(i < (3 * b1).-2)
   `d[u1 (inord i), u1 (inord i.+1)]_smove =
\sum_(i < 3 * a.-1)
   `d[u1 (inord i), u1 (inord i.+1)]_smove +
`d[↓[u (inord a.-1)], sam1]_smove
6ab
6ad5c6
      move=> b1Ea.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a76b4
b1Ea:b1 = a
\sum_(i < (3 * b1).-2)
   `d[u1 (inord i), u1 (inord i.+1)]_smove =
\sum_(i < 3 * a.-1)
   `d[u1 (inord i), u1 (inord i.+1)]_smove +
`d[↓[u (inord a.-1)], sam1]_smove
6b6
      have ta2E : (3 * b1).-2 = (3 * a.-1).+1.
6bd
(3 * b1).-2 = (3 * a.-1).+1
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a76b46be
ta2E:(3 * b1).-2 = (3 * a.-1).+1
6bf
6b76ad5c6
        by rewrite b1Ea -{1}[a]prednK // mulnS.
6c6
6bf
6b6
      rewrite ta2E big_ord_recr /=; congr (_ + `d[_,_]_smove).
6c6
u1 (inord (3 * a.-1)) = ↓[u (inord a.-1)]
6c6
u1 (inord (3 * a.-1).+1) = sam1
6b76ad5c6
        by rewrite ffunE ifT// inordK // -{2}b1Ea ?ta2E.
6c6
6d1
6b6
      rewrite ffunE inordK; last by rewrite -{2}b1Ea ?ta2E.
6c6
(if (3 * a.-1).+1 == 3 * a.-1
 then ↓[u (inord a.-1)]
 else
  if (3 * a.-1).+1 == (3 * a.-1).+1
  then sam1
  else
   if (3 * a.-1).+1 %% 3 == 0
   then ↓[u (inord ((3 * a.-1).+1 %/ 3))]
   else
    if (3 * a.-1).+1 %% 3 == 1
    then si ((3 * a.-1).+1 %/ 3)
    else ti ((3 * a.-1).+1 %/ 3)) = sam1
6b6
      by rewrite eqn_leq ltnn /= eqxx.
6b8
6ab
6ac
    rewrite {}[sd u1]G //.
6a6
a.*2 +
(\sum_(i < 3 * a.-1)
    `d[u1 (inord i), u1 (inord i.+1)]_smove +
 `d[↓[u (inord a.-1)], sam1]_smove) + sd (n:=n.+1) u2 +
sd (n:=n.+1) u3 <= sd (n:=n.+2) u
6ac
    have -> := @sum3E _ (fun i => `d[u1 (inord i), u1 (inord i.+1)]_smove).
6a6
a.*2 +
(\sum_(i < a.-1)
    (`d[u1 (inord (3 * i)), u1 (inord (3 * i).+1)]_smove +
     `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
     `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove) +
 `d[↓[u (inord a.-1)], sam1]_smove) + sd (n:=n.+1) u2 +
sd (n:=n.+1) u3 <= sd (n:=n.+2) u
6ac
    have -> : a.*2 = 2 + \sum_(i < a.-1) 2.
6a6
a.*2 = 2 + \sum_(i < a.-1) 2
6a6
2 + \sum_(i < a.-1) 2 +
(\sum_(i < a.-1)
    (`d[u1 (inord (3 * i)), 
     u1 (inord (3 * i).+1)]_smove +
     `d[u1 (inord (3 * i).+1), 
     u1 (inord (3 * i).+2)]_smove +
     `d[u1 (inord (3 * i).+2), 
     u1 (inord (3 * i).+3)]_smove) +
 `d[↓[u (inord a.-1)], sam1]_smove) + 
sd (n:=n.+1) u2 + sd (n:=n.+1) u3 <= 
sd (n:=n.+2) u
6ad5c6
      rewrite sum_nat_const /= cardT size_enum_ord.
6a6
a.*2 = 2 + a.-1 * 2
6e8
      by rewrite muln2 -(doubleD 1) add1n prednK.
6a6
2 + \sum_(i < a.-1) 2 +
(\sum_(i < a.-1)
    (`d[u1 (inord (3 * i)), u1 (inord (3 * i).+1)]_smove +
     `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
     `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove) +
 `d[↓[u (inord a.-1)], sam1]_smove) + sd (n:=n.+1) u2 +
sd (n:=n.+1) u3 <= sd (n:=n.+2) u
6ac
    rewrite !addnA -[2 + _ + _]addnA -big_split /=.
6a6
2 +
\sum_(i < a.-1)
   (2 +
    (`d[u1 (inord (3 * i)), u1 (inord (3 * i).+1)]_smove +
     `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
     `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove)) +
`d[↓[u (inord a.-1)], sam1]_smove + sd (n:=n.+1) u2 +
sd (n:=n.+1) u3 <= sd (n:=n.+2) u
6ac
    rewrite [2 + _]addnC -!addnA 2![X in _ + X <= _]addnA addnA.
6a6
\sum_(i < a.-1)
   (2 +
    (`d[u1 (inord (3 * i)), u1 (inord (3 * i).+1)]_smove +
     `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
     `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove)) +
(2 + `d[↓[u (inord a.-1)], sam1]_smove +
 sd (n:=n.+1) u2) + sd (n:=n.+1) u3 <= sd (n:=n.+2) u
6ac
    have -> : sd u =
      \sum_(i < a.-1) (`d[u (inord i), u (inord i.+1)]_smove) + 
      `d[u (inord a.-1), u ai]_smove +
      \sum_(i < b) `d[u (inord (a +i)), u (inord (a + i.+1))]_smove.
6a6
sd (n:=n.+2) u =
\sum_(i < a.-1) `d[u (inord i), u (inord i.+1)]_smove +
`d[u (inord a.-1), u ai]_smove +
\sum_(i < b)
   `d[u (inord (a + i)), u (inord (a + i.+1))]_smove
6a6
\sum_(i < a.-1)
   (2 +
    (`d[u1 (inord (3 * i)), 
     u1 (inord (3 * i).+1)]_smove +
     `d[u1 (inord (3 * i).+1), 
     u1 (inord (3 * i).+2)]_smove +
     `d[u1 (inord (3 * i).+2), 
     u1 (inord (3 * i).+3)]_smove)) +
(2 + `d[↓[u (inord a.-1)], sam1]_smove +
 sd (n:=n.+1) u2) + sd (n:=n.+1) u3 <=
\sum_(i < a.-1) `d[u (inord i), u (inord i.+1)]_smove +
`d[u (inord a.-1), u ai]_smove +
\sum_(i < b)
   `d[u (inord (a + i)), 
u (inord (a + i.+1))]_smove
6ad5c6
      have F := big_mkord xpredT 
                    (fun i => `d[u (inord i), u (inord i.+1)]_smove).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a7
F:forall (n0 : nat) (n1 : nat -> nat -> nat)
  (n2 : nat),
\big[n1/n0]_(0 <= i < n2)
   `d[u (inord i), u (inord i.+1)]_smove =
\big[n1/n0]_(i < n2) `d[
u (inord i), u (inord i.+1)]_smove
6fe
6ff
      rewrite -{}[sd u]F.
6a6
\sum_(0 <= i < l.+1)
   `d[u (inord i), u (inord i.+1)]_smove =
\sum_(i < a.-1) `d[u (inord i), u (inord i.+1)]_smove +
`d[u (inord a.-1), u ai]_smove +
\sum_(i < b)
   `d[u (inord (a + i)), u (inord (a + i.+1))]_smove
6ff
      rewrite (big_cat_nat _ _ _ (_ : _ <= (a.-1).+1)) //=; last first.
6a6
a.-1 < l.+1
6a6
\sum_(0 <= i < a.-1.+1)
   `d[u (inord i), u (inord i.+1)]_smove +
\sum_(a.-1.+1 <= i < l.+1)
   `d[u (inord i), u (inord i.+1)]_smove =
\sum_(i < a.-1) `d[u (inord i), u (inord i.+1)]_smove +
`d[u (inord a.-1), u ai]_smove +
\sum_(i < b)
   `d[u (inord (a + i)), 
u (inord (a + i.+1))]_smove
7006ad5c6
         by rewrite prednK.
6a6
\sum_(0 <= i < a.-1.+1)
   `d[u (inord i), u (inord i.+1)]_smove +
\sum_(a.-1.+1 <= i < l.+1)
   `d[u (inord i), u (inord i.+1)]_smove =
\sum_(i < a.-1) `d[u (inord i), u (inord i.+1)]_smove +
`d[u (inord a.-1), u ai]_smove +
\sum_(i < b)
   `d[u (inord (a + i)), u (inord (a + i.+1))]_smove
6ff
      rewrite big_mkord big_ord_recr /= prednK //.
6a6
\sum_(i < a.-1) `d[u (inord i), u (inord i.+1)]_smove +
`d[u (inord a.-1), u (inord a)]_smove +
\sum_(a <= i < l.+1)
   `d[u (inord i), u (inord i.+1)]_smove =
\sum_(i < a.-1) `d[u (inord i), u (inord i.+1)]_smove +
`d[u (inord a.-1), u ai]_smove +
\sum_(i < b)
   `d[u (inord (a + i)), u (inord (a + i.+1))]_smove
6ff
      rewrite -{9}[a]add0n big_addn -/b big_mkord.
6a6
\sum_(i < a.-1) `d[u (inord i), u (inord i.+1)]_smove +
`d[u (inord a.-1), u (inord a)]_smove +
\sum_(i < b)
   `d[u (inord (i + a)), u (inord (i + a).+1)]_smove =
\sum_(i < a.-1) `d[u (inord i), u (inord i.+1)]_smove +
`d[u (inord a.-1), u ai]_smove +
\sum_(i < b)
   `d[u (inord (a + i)), u (inord (a + i.+1))]_smove
6ff
      congr (_ + _ + _).
6a6
\sum_(i < b)
   `d[u (inord (i + a)), u (inord (i + a).+1)]_smove =
\sum_(i < b)
   `d[u (inord (a + i)), u (inord (a + i.+1))]_smove
6ff
      by apply: eq_bigr => i; rewrite addnC addnS.
6a6
\sum_(i < a.-1)
   (2 +
    (`d[u1 (inord (3 * i)), u1 (inord (3 * i).+1)]_smove +
     `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
     `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove)) +
(2 + `d[↓[u (inord a.-1)], sam1]_smove +
 sd (n:=n.+1) u2) + sd (n:=n.+1) u3 <=
\sum_(i < a.-1) `d[u (inord i), u (inord i.+1)]_smove +
`d[u (inord a.-1), u ai]_smove +
\sum_(i < b)
   `d[u (inord (a + i)), u (inord (a + i.+1))]_smove
6ac
    apply: leq_add; first apply: leq_add.
6a6
\sum_(i < a.-1)
   (2 +
    (`d[u1 (inord (3 * i)), u1 (inord (3 * i).+1)]_smove +
     `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
     `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove)) <=
\sum_(i < a.-1) `d[u (inord i), u (inord i.+1)]_smove
6a6
2 + `d[↓[u (inord a.-1)], sam1]_smove +
sd (n:=n.+1) u2 <= `d[u (inord a.-1), 
u ai]_smove
6a6
sd (n:=n.+1) u3 <=
\sum_(i < b)
   `d[u (inord (a + i)), 
u (inord (a + i.+1))]_smove
6ad5c6
    -
727 rewrite leq_eqVlt; apply/orP; left; apply/eqP.
6a6
\sum_(i < a.-1)
   (2 +
    (`d[u1 (inord (3 * i)), u1 (inord (3 * i).+1)]_smove +
     `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
     `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove)) =
\sum_(i < a.-1) `d[u (inord i), u (inord i.+1)]_smove
72a
      apply: eq_bigr => i _.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a7
i:'I_a.-1
2 +
(`d[u1 (inord (3 * i)), u1 (inord (3 * i).+1)]_smove +
 `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
 `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
72a
      have -> : u1 (inord (3 * i)) = ↓[u (inord i)].
737
u1 (inord (3 * i)) = ↓[u (inord i)]
737
2 +
(`d[↓[u (inord i)], u1 (inord (3 * i).+1)]_smove +
 `d[u1 (inord (3 * i).+1), 
 u1 (inord (3 * i).+2)]_smove +
 `d[u1 (inord (3 * i).+2), 
 u1 (inord (3 * i).+3)]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
72b72d6ad5c6
        rewrite ffunE inordK; last first.
737
3 * i < (3 * a).-2.+1
737
(if 3 * i == 3 * a.-1
 then ↓[u (inord a.-1)]
 else
  if 3 * i == (3 * a.-1).+1
  then sam1
  else
   if (3 * i) %% 3 == 0
   then ↓[u (inord ((3 * i) %/ 3))]
   else
    if (3 * i) %% 3 == 1
    then si ((3 * i) %/ 3)
    else ti ((3 * i) %/ 3)) = ↓[
u (inord i)]
73f72b72d6ad5c6
          rewrite ltnS (leq_trans (_ : _ <= 3 * (a.-1))) //.
737
3 * i <= 3 * a.-1
737
3 * a.-1 <= (3 * a).-2
74673f72b72d6ad5c6
            by rewrite leq_mul2l /= ltnW.
737
74e
745
          by rewrite -{2}[a]prednK // mulnS addSn add2n /= ltnW.
737
(if 3 * i == 3 * a.-1
 then ↓[u (inord a.-1)]
 else
  if 3 * i == (3 * a.-1).+1
  then sam1
  else
   if (3 * i) %% 3 == 0
   then ↓[u (inord ((3 * i) %/ 3))]
   else
    if (3 * i) %% 3 == 1
    then si ((3 * i) %/ 3)
    else ti ((3 * i) %/ 3)) = ↓[u (inord i)]
73e
        rewrite eqn_mul2l /= eqn_leq [_ <= i]leqNgt ltn_ord andbF.
737
(if 3 * i == (3 * a.-1).+1
 then sam1
 else
  if (3 * i) %% 3 == 0
  then ↓[u (inord ((3 * i) %/ 3))]
  else
   if (3 * i) %% 3 == 1
   then si ((3 * i) %/ 3)
   else ti ((3 * i) %/ 3)) = ↓[u (inord i)]
73e
        rewrite eqn_leq ltn_mul2l /= ltnNge [i <= _]ltnW // andbF.
737
(if (3 * i) %% 3 == 0
 then ↓[u (inord ((3 * i) %/ 3))]
 else
  if (3 * i) %% 3 == 1
  then si ((3 * i) %/ 3)
  else ti ((3 * i) %/ 3)) = ↓[u (inord i)]
73e
        by rewrite mod3E eqxx mulKn.
737
2 +
(`d[↓[u (inord i)], u1 (inord (3 * i).+1)]_smove +
 `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
 `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
72a
      have -> : u1 (inord (3 * i).+1) = si i.
737
u1 (inord (3 * i).+1) = si i
737
2 +
(`d[↓[u (inord i)], si i]_smove +
 `d[si i, u1 (inord (3 * i).+2)]_smove +
 `d[u1 (inord (3 * i).+2), 
 u1 (inord (3 * i).+3)]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
72b72d6ad5c6
        rewrite ffunE inordK; last first.
737
(3 * i).+1 < (3 * a).-2.+1
737
(if (3 * i).+1 == 3 * a.-1
 then ↓[u (inord a.-1)]
 else
  if (3 * i).+1 == (3 * a.-1).+1
  then sam1
  else
   if (3 * i).+1 %% 3 == 0
   then ↓[u (inord ((3 * i).+1 %/ 3))]
   else
    if (3 * i).+1 %% 3 == 1
    then si ((3 * i).+1 %/ 3)
    else ti ((3 * i).+1 %/ 3)) = 
si i
76772b72d6ad5c6
          rewrite ltnS (leq_trans (_ : _ <= 3 * (a.-1))) //.
737
3 * i < 3 * a.-1
74d76e76772b72d6ad5c6
            by rewrite ltn_mul2l /=.
75176d
          by rewrite -{2}[a]prednK // mulnS addSn add2n /= ltnW.
737
(if (3 * i).+1 == 3 * a.-1
 then ↓[u (inord a.-1)]
 else
  if (3 * i).+1 == (3 * a.-1).+1
  then sam1
  else
   if (3 * i).+1 %% 3 == 0
   then ↓[u (inord ((3 * i).+1 %/ 3))]
   else
    if (3 * i).+1 %% 3 == 1
    then si ((3 * i).+1 %/ 3)
    else ti ((3 * i).+1 %/ 3)) = si i
766
        rewrite eqn_leq [_ <= _.+1]leqNgt.
737
(if (3 * i < 3 * a.-1) && ~~ ((3 * i).+1 < 3 * a.-1)
 then ↓[u (inord a.-1)]
 else
  if (3 * i).+1 == (3 * a.-1).+1
  then sam1
  else
   if (3 * i).+1 %% 3 == 0
   then ↓[u (inord ((3 * i).+1 %/ 3))]
   else
    if (3 * i).+1 %% 3 == 1
    then si ((3 * i).+1 %/ 3)
    else ti ((3 * i).+1 %/ 3)) = si i
766
        rewrite (leq_trans (_ : (3 * i).+2 <= (3 * i.+1))) ?andbF //;
               last 2 first.
737
(3 * i).+1 < 3 * i.+1
737
3 * i.+1 <= 3 * a.-1
737
(if (3 * i).+1 == (3 * a.-1).+1
 then sam1
 else
  if (3 * i).+1 %% 3 == 0
  then ↓[u (inord ((3 * i).+1 %/ 3))]
  else
   if (3 * i).+1 %% 3 == 1
   then si ((3 * i).+1 %/ 3)
   else ti ((3 * i).+1 %/ 3)) = 
si i
76772b72d6ad5c6
        -
780 by rewrite mulnS addSn add2n.
737
785
78676772b72d6ad5c6
        -
78a by rewrite leq_mul2l /=.
737
(if (3 * i).+1 == (3 * a.-1).+1
 then sam1
 else
  if (3 * i).+1 %% 3 == 0
  then ↓[u (inord ((3 * i).+1 %/ 3))]
  else
   if (3 * i).+1 %% 3 == 1
   then si ((3 * i).+1 %/ 3)
   else ti ((3 * i).+1 %/ 3)) = si i
766
        rewrite eqn_leq !ltnS [_ <= 3 * i]leq_mul2l.
737
(if (3 * i <= 3 * a.-1) && ((3 == 0) || (a.-1 <= i))
 then sam1
 else
  if (3 * i).+1 %% 3 == 0
  then ↓[u (inord ((3 * i).+1 %/ 3))]
  else
   if (3 * i).+1 %% 3 == 1
   then si ((3 * i).+1 %/ 3)
   else ti ((3 * i).+1 %/ 3)) = si i
766
        by rewrite [_ <= i]leqNgt ltn_ord andbF mod3E /= div3E.
737
2 +
(`d[↓[u (inord i)], si i]_smove +
 `d[si i, u1 (inord (3 * i).+2)]_smove +
 `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
72a
      have -> : u1 (inord (3 * i).+2) = ti i.
737
u1 (inord (3 * i).+2) = ti i
737
2 +
(`d[↓[u (inord i)], si i]_smove + `d[si i, ti i]_smove +
 `d[ti i, u1 (inord (3 * i).+3)]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
72b72d6ad5c6
        rewrite ffunE inordK; last first.
737
(3 * i).+2 < (3 * a).-2.+1
737
(if (3 * i).+2 == 3 * a.-1
 then ↓[u (inord a.-1)]
 else
  if (3 * i).+2 == (3 * a.-1).+1
  then sam1
  else
   if (3 * i).+2 %% 3 == 0
   then ↓[u (inord ((3 * i).+2 %/ 3))]
   else
    if (3 * i).+2 %% 3 == 1
    then si ((3 * i).+2 %/ 3)
    else ti ((3 * i).+2 %/ 3)) = 
ti i
79f72b72d6ad5c6
          rewrite ltnS (leq_trans (_ : _ <= 3 * (a.-1))) //.
737
(3 * i).+1 < 3 * a.-1
74d7a679f72b72d6ad5c6
            rewrite (leq_trans (_ : _ <= 3 * (i.+1))) //.
781
78474d7a679f72b72d6ad5c6
              by rewrite mulnS addSn add2n.
78b7ac
            by rewrite leq_mul2l /=.
7517a5
          by rewrite -{2}[a]prednK // mulnS addSn add2n /= ltnW.
737
(if (3 * i).+2 == 3 * a.-1
 then ↓[u (inord a.-1)]
 else
  if (3 * i).+2 == (3 * a.-1).+1
  then sam1
  else
   if (3 * i).+2 %% 3 == 0
   then ↓[u (inord ((3 * i).+2 %/ 3))]
   else
    if (3 * i).+2 %% 3 == 1
    then si ((3 * i).+2 %/ 3)
    else ti ((3 * i).+2 %/ 3)) = ti i
79e
        rewrite eqn_leq [_ <= _.+1]leqNgt.
737
(if
  ((3 * i).+1 < 3 * a.-1) &&
  ~~ ((3 * i).+2 < 3 * a.-1)
 then ↓[u (inord a.-1)]
 else
  if (3 * i).+2 == (3 * a.-1).+1
  then sam1
  else
   if (3 * i).+2 %% 3 == 0
   then ↓[u (inord ((3 * i).+2 %/ 3))]
   else
    if (3 * i).+2 %% 3 == 1
    then si ((3 * i).+2 %/ 3)
    else ti ((3 * i).+2 %/ 3)) = ti i
79e
        rewrite -[(3 * i).+3](mulnDr 3 1 i) leq_mul2l ltn_ord andbF.
737
(if (3 * i).+2 == (3 * a.-1).+1
 then sam1
 else
  if (3 * i).+2 %% 3 == 0
  then ↓[u (inord ((3 * i).+2 %/ 3))]
  else
   if (3 * i).+2 %% 3 == 1
   then si ((3 * i).+2 %/ 3)
   else ti ((3 * i).+2 %/ 3)) = ti i
79e
        rewrite eqn_leq !ltnS [_ <= _.+1]leqNgt.
737
(if (3 * i < 3 * a.-1) && ~~ ((3 * i).+1 < 3 * a.-1)
 then sam1
 else
  if (3 * i).+2 %% 3 == 0
  then ↓[u (inord ((3 * i).+2 %/ 3))]
  else
   if (3 * i).+2 %% 3 == 1
   then si ((3 * i).+2 %/ 3)
   else ti ((3 * i).+2 %/ 3)) = ti i
79e
        rewrite (leq_trans (_ : (3 * i).+1 < 3 * (i.+1))) ?andbF //;
             last 2 first.
781
784
737
(if (3 * i).+2 %% 3 == 0
 then ↓[u (inord ((3 * i).+2 %/ 3))]
 else
  if (3 * i).+2 %% 3 == 1
  then si ((3 * i).+2 %/ 3)
  else ti ((3 * i).+2 %/ 3)) = 
ti i
79f72b72d6ad5c6
        -
7c5 by rewrite mulnS addSn add2n.
78b
7c779f72b72d6ad5c6
        -
7cb by rewrite leq_mul2l /=.
737
(if (3 * i).+2 %% 3 == 0
 then ↓[u (inord ((3 * i).+2 %/ 3))]
 else
  if (3 * i).+2 %% 3 == 1
  then si ((3 * i).+2 %/ 3)
  else ti ((3 * i).+2 %/ 3)) = ti i
79e
        by rewrite mod3E /= div3E.
737
7a0
72a
      have -> : u1 (inord (3 * i).+3) = ↓[u (inord i.+1)].
737
u1 (inord (3 * i).+3) = ↓[u (inord i.+1)]
737
2 +
(`d[↓[u (inord i)], si i]_smove + `d[si i, ti i]_smove +
 `d[ti i, ↓[u (inord i.+1)]]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
72b72d6ad5c6
        have -> : (3 * i).+3 = 3 * (i.+1) by rewrite mulnS addSn add2n.
737
u1 (inord (3 * i.+1)) = ↓[u (inord i.+1)]
7d9
        rewrite ffunE inordK; last first.
737
3 * i.+1 < (3 * a).-2.+1
737
(if 3 * i.+1 == 3 * a.-1
 then ↓[u (inord a.-1)]
 else
  if 3 * i.+1 == (3 * a.-1).+1
  then sam1
  else
   if (3 * i.+1) %% 3 == 0
   then ↓[u (inord ((3 * i.+1) %/ 3))]
   else
    if (3 * i.+1) %% 3 == 1
    then si ((3 * i.+1) %/ 3)
    else ti ((3 * i.+1) %/ 3)) = ↓[
u (inord i.+1)]
7da72b72d6ad5c6
          rewrite ltnS (leq_trans (_ : _ <= 3 * (a.-1))) //.
78b
74d7e57da72b72d6ad5c6
            by rewrite leq_mul2l /=.
7517e4
          by rewrite -{2}[a]prednK // mulnS addSn add2n /= ltnW.
737
(if 3 * i.+1 == 3 * a.-1
 then ↓[u (inord a.-1)]
 else
  if 3 * i.+1 == (3 * a.-1).+1
  then sam1
  else
   if (3 * i.+1) %% 3 == 0
   then ↓[u (inord ((3 * i.+1) %/ 3))]
   else
    if (3 * i.+1) %% 3 == 1
    then si ((3 * i.+1) %/ 3)
    else ti ((3 * i.+1) %/ 3)) = ↓[u (inord i.+1)]
7d9
        rewrite eqn_mul2l /=; case: eqP => [->//|_].
737
(if 3 * i.+1 == (3 * a.-1).+1
 then sam1
 else
  if (3 * i.+1) %% 3 == 0
  then ↓[u (inord ((3 * i.+1) %/ 3))]
  else
   if (3 * i.+1) %% 3 == 1
   then si ((3 * i.+1) %/ 3)
   else ti ((3 * i.+1) %/ 3)) = ↓[u (inord i.+1)]
7d9
        rewrite eqn_leq [_ <= 3 * i.+1]leqNgt ltnS leq_mul2l /= ltn_ord andbF.
737
(if (3 * i.+1) %% 3 == 0
 then ↓[u (inord ((3 * i.+1) %/ 3))]
 else
  if (3 * i.+1) %% 3 == 1
  then si ((3 * i.+1) %/ 3)
  else ti ((3 * i.+1) %/ 3)) = ↓[u (inord i.+1)]
7d9
        by rewrite mod3E /= div3E.
737
7db
72a
      case: (sitiH i) => // _ _ ->.
737
2 +
(`d[↓[u (inord i)], si i]_smove + `d[si i, ti i]_smove +
 `d[ti i, ↓[u (inord i.+1)]]_smove) =
 D[[:: ↓[u (inord i)]; si i; ti i; ↓[u (inord i.+1)]]].+2
72a
      by rewrite add2n /= addnA addn0.
6a6
2 + `d[↓[u (inord a.-1)], sam1]_smove +
sd (n:=n.+1) u2 <= `d[u (inord a.-1), u ai]_smove
72d6ad5c6
    -
800 rewrite leq_eqVlt; apply/orP; left; apply/eqP.
6a6
2 + `d[↓[u (inord a.-1)], sam1]_smove +
sd (n:=n.+1) u2 = `d[u (inord a.-1), u ai]_smove
803
      rewrite /sd !big_ord_recr big_ord0 //= add0n !addnA.
6a6
2 + `d[↓[u (inord a.-1)], sam1]_smove +
`d[u2 (inord 0), u2 (inord 1)]_smove +
`d[u2 (inord 1), u2 (inord 2)]_smove =
`d[u (inord a.-1), u ai]_smove
803
      have -> : u2 (inord 0) = sam1 by rewrite ffunE /= inordK.
6a6
2 + `d[↓[u (inord a.-1)], sam1]_smove +
`d[sam1, u2 (inord 1)]_smove +
`d[u2 (inord 1), u2 (inord 2)]_smove =
`d[u (inord a.-1), u ai]_smove
803
      have -> : u2 (inord 1) = tam1 by rewrite ffunE /= inordK.
6a6
2 + `d[↓[u (inord a.-1)], sam1]_smove +
`d[sam1, tam1]_smove + `d[tam1, u2 (inord 2)]_smove =
`d[u (inord a.-1), u ai]_smove
803
      have -> : u2 (inord 2) = ↓[u ai] by rewrite ffunE /= inordK.
6a6
2 + `d[↓[u (inord a.-1)], sam1]_smove +
`d[sam1, tam1]_smove + `d[tam1, ↓[u ai]]_smove =
`d[u (inord a.-1), u ai]_smove
803
      by move: duam1ua1E; rewrite /= addn0 add2n !addnA !addSn.
6a6
sd (n:=n.+1) u3 <=
\sum_(i < b)
   `d[u (inord (a + i)), u (inord (a + i.+1))]_smove
6ac
    apply: leq_sum => i _.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a7
i:ordinal_finType b
`d[u3 (inord i), u3 (inord i.+1)]_smove <=
`d[u (inord (a + i)), u (inord (a + i.+1))]_smove
6ac
    rewrite ffunE inordK; last by rewrite ltnS ltnW.
820
`d[↓[u (inord (i + a))], u3 (inord i.+1)]_smove <=
`d[u (inord (a + i)), u (inord (a + i.+1))]_smove
6ac
    rewrite ffunE inordK; last by rewrite ltnS.
820
`d[↓[u (inord (i + a))], ↓[u (inord (i.+1 + a))]]_smove <=
`d[u (inord (a + i)), u (inord (a + i.+1))]_smove
6ac
    rewrite addSn [i + _]addnC addnS.
820
`d[↓[u (inord (a + i))], ↓[u (inord (a + i).+1)]]_smove <=
`d[u (inord (a + i)), u (inord (a + i).+1)]_smove
6ac
    by apply/gdist_cunlift/shanoi_connect.
6ae
573
5c5
  set x1P1 := sd _ in P1; set x2P1 := sd _ in P1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a7
x1P1:=sd (n:=n.+1) u1:nat
x2P1:=sd (n:=n.+1) u2:nat
P1:a.*2 + x1P1 + x2P1 + sd (n:=n.+1) u3 <=
sd (n:=n.+2) u
573
5c5
  set x3P1 := sd _ in P1; set xP1 := sd _ in P1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a7836837
x3P1:=sd (n:=n.+1) u3:nat
xP1:=sd (n:=n.+2) u:nat
P1:a.*2 + x1P1 + x2P1 + x3P1 <= xP1
573
5c5
  rewrite -/xP1.
83c
(S_[l.+1] n.+2).*2 <=
xP1 + sp (n:=n.+2) (u ord0) l.+1 p1 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
5c5
  have cH2 (k : 'I_(3 * a).-2.+1) : 
      0 < k < (3 * a).-2 -> codom (u1 k) \subset [:: p2; a[p1, p3] k].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f
k:'I_(3 * a).-2.+1
0 < k < (3 * a).-2 ->
codom (u1 k) \subset [:: p2;  a[p1, p3] k]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f
cH2:forall k : 'I_(3 * a).-2.+1,
0 < k < (3 * a).-2 ->
codom (u1 k) \subset [:: p2;  a[p1, p3] k]
843
5c6
    move=> /andP[k_gt0 k_lt].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f848
k_gt0:0 < k
k_lt:k < (3 * a).-2
codom (u1 k) \subset [:: p2;  a[p1, p3] k]
84a
    have kL3a1 : k <= (3 * a.-1).
851
k <= 3 * a.-1
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f848852853
kL3a1:k <= 3 * a.-1
854
84b5c6
      by rewrite -ltnS (leq_trans k_lt) // -{1}[a]prednK // mulnS.
85b
854
84a
    have kLa1 : k %/ 3 <= a.-1.
85b
k %/ 3 <= a.-1
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f84885285385c
kLa1:k %/ 3 <= a.-1
854
84b5c6
      by rewrite -(mulKn a.-1 (isT : 0 < 3)) leq_div2r.
866
854
84a
    rewrite ffunE; case: eqP => [kH|/eqP kH].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f84885285385c867
kH:k = 3 * a.-1
codom ↓[u (inord a.-1)] \subset [:: p2;  a[p1, p3] k]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f84885285385c867
kH:k != 3 * a.-1
codom
  (if k == (3 * a.-1).+1
   then sam1
   else
    if k %% 3 == 0
    then ↓[u (inord (k %/ 3))]
    else
     if k %% 3 == 1 then si (k %/ 3) else ti (k %/ 3)) \subset
[:: p2;  a[p1, p3] k]
84b5c6
      rewrite kH apegMr //.
86e
codom ↓[u (inord a.-1)] \subset
[:: p2;  a[p1, p3] a.-1]
871
      have := @H (inord a.-1); rewrite inordK //.
86e
(0 < a.-1 < l.+1 ->
 codom (u' (inord a.-1)) \subset
 [:: p2;  a[p1, p3] a.-1]) ->
codom ↓[u (inord a.-1)] \subset
[:: p2;  a[p1, p3] a.-1]
86e
a.-1 < l.+2
87284b5c6
        rewrite ffunE; apply.
86e
0 < a.-1 < l.+1
87e
        by rewrite -ltnS prednK // a_gt1.
86e
880
871
      by rewrite prednK // (leq_trans aLld1).
873
875
84a
    have k3La2 : k %/ 3 <= a.-2.
873
k %/ 3 <= a.-2
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f84885285385c867874
k3La2:k %/ 3 <= a.-2
875
84b5c6
      rewrite -ltnS [a.-2.+1]prednK //; last first.
873
0 < a.-1
873
k %/ 3 < a.-1
89084b5c6
        by rewrite -subn1 ltn_subRL.
873
899
88f
      rewrite ltn_divLR // [_ * 3]mulnC.
873
k < 3 * a.-1
88f
      by move: kL3a1 kH; case: ltngtP.
891
875
84a
    rewrite eqn_leq [_ <= k]leqNgt (leq_trans k_lt) ?andbF //; last first.
891
(3 * a).-2 <= (3 * a.-1).+1
891
codom
  (if k %% 3 == 0
   then ↓[u (inord (k %/ 3))]
   else
    if k %% 3 == 1 then si (k %/ 3) else ti (k %/ 3)) \subset
[:: p2;  a[p1, p3] k]
84b5c6
      by rewrite -{1}[a]prednK // mulnS addSn add2n.
891
8aa
84a
    have k3La : k %/ 3 <= a.
891
k %/ 3 <= a
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f84885285385c867874892
k3La:k %/ 3 <= a
8aa
84b5c6
      by rewrite (leq_trans kLa1) // -subn1 leq_subr.
8b4
8aa
84a
    case: eqP => k3mH.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f84885285385c8678748928b5
k3mH:k %% 3 = 0
codom ↓[u (inord (k %/ 3))] \subset
[:: p2;  a[p1, p3] k]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f84885285385c8678748928b5
k3mH:k %% 3 <> 0
codom
  (if k %% 3 == 1 then si (k %/ 3) else ti (k %/ 3)) \subset
[:: p2;  a[p1, p3] k]
84b5c6
      apply: codom_liftr.
8bc
codom (u (inord (k %/ 3))) \subset
[:: p2;  a[p1, p3] k]
8bf
      rewrite {2}(divn_eq k 3) k3mH addn0 [in a[_,_] _]mulnC apegMr //.
8bc
codom (u (inord (k %/ 3))) \subset
[:: p2;  a[p1, p3] (k %/ 3)]
8bf
      have := @cH (inord (k %/3)); rewrite inordK; last first.
8bc
k %/ 3 < l.+2
8bc
(0 < k %/ 3 < l.+1 ->
 codom (u (inord (k %/ 3))) \subset
 [:: p2;  a[p1, p3] (k %/ 3)]) ->
codom (u (inord (k %/ 3))) \subset
[:: p2;  a[p1, p3] (k %/ 3)]
8c084b5c6
        by rewrite (leq_ltn_trans k3La).
8bc
8d2
8bf
      apply.
8bc
0 < k %/ 3 < l.+1
8bf
      rewrite (leq_ltn_trans k3La) ?ltnS //.
8bc
(0 < k %/ 3) && true
8bf
      move: k_gt0; rewrite andbT {1}(divn_eq k 3) k3mH addn0 muln_gt0.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f84885385c8678748928b58bd
(0 < k %/ 3) && (0 < 3) -> 0 < k %/ 3
8bf
      by case/andP.
8c1
8c3
84a
    case: eqP => k3mH1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f84885285385c8678748928b58c2
k3mH1:k %% 3 = 1
codom (si (k %/ 3)) \subset [:: p2;  a[p1, p3] k]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f84885285385c8678748928b58c2
k3mH1:k %% 3 <> 1
codom (ti (k %/ 3)) \subset [:: p2;  a[p1, p3] k]
84b5c6
      case: (sitiH ((k %/ 3))) => /=.
8e9
899
8e9
codom (si (k %/ 3)) \subset
[:: p2; u (inord (k %/ 3).+1) ldisk] ->
codom (ti (k %/ 3)) \subset
[:: u (inord (k %/ 3)) ldisk; p2] ->
`d[u (inord (k %/ 3)), u (inord (k %/ 3).+1)]_smove =
(`d[↓[u (inord (k %/ 3))], 
 si (k %/ 3)]_smove +
 (`d[si (k %/ 3), ti (k %/ 3)]_smove +
  (`d[ti (k %/ 3), ↓[u (inord (k %/ 3).+1)]]_smove + 0))).+2 ->
codom (si (k %/ 3)) \subset [:: p2;  a[p1, p3] k]
8ed84b5c6
        by rewrite -[a.-1]prednK // -ltnS prednK.
8e9
codom (si (k %/ 3)) \subset
[:: p2; u (inord (k %/ 3).+1) ldisk] ->
codom (ti (k %/ 3)) \subset
[:: u (inord (k %/ 3)) ldisk; p2] ->
`d[u (inord (k %/ 3)), u (inord (k %/ 3).+1)]_smove =
(`d[↓[u (inord (k %/ 3))], si (k %/ 3)]_smove +
 (`d[si (k %/ 3), ti (k %/ 3)]_smove +
  (`d[ti (k %/ 3), ↓[u (inord (k %/ 3).+1)]]_smove + 0))).+2 ->
codom (si (k %/ 3)) \subset [:: p2;  a[p1, p3] k]
8ec
      set pp := u _ _; (suff <- : pp = a[p1, p3] k by []); rewrite {}/pp.
8e9
u (inord (k %/ 3).+1) ldisk =  a[p1, p3] k
8ec
      set i := inord _; have <- : a[p1, p3] i = a[p1, p3] k.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f84885285385c8678748928b58c28ea
i:=inord (k %/ 3).+1:'I_l.+2
 a[p1, p3] i =  a[p1, p3] k
902
u i ldisk =  a[p1, p3] i
8ed84b5c6
        rewrite (divn_eq k 3) k3mH1 addn1 apegS mulnC apegMr //.
902
 a[p1, p3] i =  a[p3, p1] (k %/ 3)
905
        by rewrite /i !inordK ?apegS //= !ltnS (leq_trans k3La).
902
907
8ec
      have H2 : (k %/ 3).+1 < l.+2 by rewrite !ltnS (leq_trans k3La).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f84885285385c8678748928b58c28ea903
H2:(k %/ 3).+1 < l.+2
907
8ec
      rewrite -(@aMin1 i); last first.
912
i < a
912
u i ldisk = u (inord i) ord_max
8ed84b5c6
        rewrite !inordK // -{2}[a]prednK // ltnS // -[a.-1]prednK // .
912
896
918
        by rewrite -subn1 subn_gt0.
912
91a
8ec
      congr (u _ _).
912
i = inord i
8ec
      by apply/val_eqP; rewrite /= !inordK.
8ee
8f0
84a
    rewrite codom_subC.
8ee
codom (ti (k %/ 3)) \subset [::  a[p1, p3] k; p2]
84a
    case: (sitiH ((k %/ 3))) => /= [|_].
8ee
899
8ee
codom (ti (k %/ 3)) \subset
[:: u (inord (k %/ 3)) ldisk; p2] ->
`d[u (inord (k %/ 3)), u (inord (k %/ 3).+1)]_smove =
(`d[↓[u (inord (k %/ 3))], 
 si (k %/ 3)]_smove +
 (`d[si (k %/ 3), ti (k %/ 3)]_smove +
  (`d[ti (k %/ 3), ↓[u (inord (k %/ 3).+1)]]_smove + 0))).+2 ->
codom (ti (k %/ 3)) \subset [::  a[p1, p3] k; p2]
84b5c6
      by rewrite -[a.-1]prednK // -ltnS prednK.
8ee
codom (ti (k %/ 3)) \subset
[:: u (inord (k %/ 3)) ldisk; p2] ->
`d[u (inord (k %/ 3)), u (inord (k %/ 3).+1)]_smove =
(`d[↓[u (inord (k %/ 3))], si (k %/ 3)]_smove +
 (`d[si (k %/ 3), ti (k %/ 3)]_smove +
  (`d[ti (k %/ 3), ↓[u (inord (k %/ 3).+1)]]_smove + 0))).+2 ->
codom (ti (k %/ 3)) \subset [::  a[p1, p3] k; p2]
84a
    set pp := u _ _; (suff <- : pp = a[p1, p3] k by []); rewrite {}/pp.
8ee
u (inord (k %/ 3)) ldisk =  a[p1, p3] k
84a
    set i := inord _; have <- : a[p1, p3] i = a[p1, p3] k.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f84885285385c8678748928b58c28ef
i:=inord (k %/ 3):'I_l.+2
904
93d
907
84b5c6
      have k3mH2 : k %% 3 = 2.
93d
k %% 3 = 2
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f84885285385c8678748928b58c28ef93e
k3mH2:k %% 3 = 2
904
94084b5c6
        have := ltn_mod k 3; move: k3mH k3mH1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f84885285385c8678748928b593e
k %% 3 <> 0 ->
k %% 3 <> 1 -> (k %% 3 < 3) = (0 < 3) -> k %% 3 = 2
945
        by case: modn => // [] [|[]].
947
904
93f
      rewrite (divn_eq k 3) k3mH2 addn2 !apegS mulnC apegMr //.
947
 a[p1, p3] i =  a[p1, p3] (k %/ 3)
93f
        by rewrite /i !inordK ?apegS //= !ltnS (leq_trans k3La).
93d
907
84a
      have H2 : (k %/ 3) < l.+2 by rewrite !ltnS (leq_trans k3La).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f84885285385c8678748928b58c28ef93e
H2:k %/ 3 < l.+2
907
84a
      rewrite -(@aMin1 i); last by rewrite !inordK // -{2}[a]prednK .
95b
91a
84a
      congr (u _ _).
95b
924
84a
      by apply/val_eqP; rewrite /= !inordK.
84c
843
5c5
  have {cH2}P2 := IH p1Dp2 p1Dp3 p2Dp3 p1Dp0 p2Dp0 p3Dp0 cH2.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f
P2:(S_[(3 * a).-2] n.+1).*2 <=
sd (n:=n.+1) u1 +
sp (n:=n.+1) (u1 ord0) (3 * a).-2 ( a[p1, p3] 0) +
sp (n:=n.+1) (u1 ord_max) 
  (3 * a).-2 ( a[p1, p3] (3 * a).-2)
843
5c5
  have a3B2_gt1 : 1 < (3 * a).-2.
969
1 < (3 * a).-2
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f96a
a3B2_gt1:1 < (3 * a).-2
843
5c6
    rewrite -subn2 leq_subRL.
969
2 + 2 <= 3 * a
969
1 < 3 * a
9705c6
      by rewrite (leq_trans (_ : 4 <= 3 * 2)) // leq_mul2l.
969
979
96f
    by rewrite (leq_trans (_ : 2 <= 3 * 2)) // leq_mul2l.
971
843
5c5
  rewrite u10E uiME apeg0 in P2.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f972
P2:(S_[(3 * a).-2] n.+1).*2 <=
sd (n:=n.+1) u1 +
sp (n:=n.+1) ↓[u ord0] (3 * a).-2 p1 +
sp (n:=n.+1) sam1 (3 * a).-2
  ( a[p1, p3] (3 * a).-2)
843
5c5
  have {}P2 := leq_trans P2 (leq_add (leq_add (leqnn _) (leq_sum_beta _ _))
                          (leqnn _)).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f972
P2:(S_[(3 * a).-2] n.+1).*2 <=
sd (n:=n.+1) u1 +
\sum_(k < n.+1)
   (↓[u ord0] k != p1) * (α_[1] k).*2 +
sp (n:=n.+1) sam1 (3 * a).-2
  ( a[p1, p3] (3 * a).-2)
843
5c5
  pose pa := a[p1, p3] a; pose paS := a[p1, p3] a.+1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f972989
pa:= a[p1, p3] a:peg 4
paS:= a[p1, p3] a.+1:peg 4
843
5c5
  have cH3 (k : 'I_3) : 
    0 < k < 2 -> codom (u2 k) \subset [:: paS; a[p2, pa] k].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298998e98f
k:'I_3
0 < k < 2 ->
codom (u2 k) \subset [:: paS;  a[p2, pa] k]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298998e98f
cH3:forall k : 'I_3,
0 < k < 2 ->
codom (u2 k) \subset [:: paS;  a[p2, pa] k]
843
5c6
    case: k => [] [|[|]] //= iH _.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298998e98f
iH:1 < 3
codom (u2 (Ordinal (n:=3) (m:=1) iH)) \subset
[:: paS;  a[p2, pa] 1]
996
    by rewrite /pa /paS !apegS apeg0 ffunE /= codom_apeg codom_subC.
998
843
5c5
  rewrite -/x1P1 in P2.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98f999
P2:(S_[(3 * a).-2] n.+1).*2 <=
x1P1 +
\sum_(k < n.+1)
   (↓[u ord0] k != p1) * (α_[1] k).*2 +
sp (n:=n.+1) sam1 (3 * a).-2
  ( a[p1, p3] (3 * a).-2)
843
5c5
  have {cH3}P3 : 
    (S_[2] n.+1).*2 <= sd u2 + sp (u2 ord0) 2 (a[p2, pa] 0) +
                       sp (u2 ord_max) 2 (a[p2, pa] 2).
9a6
(S_[2] n.+1).*2 <=
sd (n:=n.+1) u2 +
sp (n:=n.+1) (u2 ord0) 2 ( a[p2, pa] 0) +
sp (n:=n.+1) (u2 ord_max) 2 ( a[p2, pa] 2)
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98f9a7
P3:(S_[2] n.+1).*2 <=
sd (n:=n.+1) u2 +
sp (n:=n.+1) (u2 ord0) 2 ( a[p2, pa] 0) +
sp (n:=n.+1) (u2 ord_max) 2 ( a[p2, pa] 2)
843
5c6
    apply: IH cH3 => //; try by rewrite /pa /paS // eq_sym.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98f9a7
paS != pa
9ac
    by rewrite /pa /paS apegS apeg_eqC eq_sym.
9ae
843
5c5
  rewrite !ffunE /= apeg0 -/x2P1 in P3.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98f9a7
P3:(S_[2] n.+1).*2 <=
x2P1 + sp (n:=n.+1) sam1 2 p2 +
sp (n:=n.+1) ↓[u ai] 2 ( a[p2, pa] 2)
843
5c5
  have cH4 (k : 'I_b.+1) :
      0 < k < b -> codom (u3 k) \subset [:: p2; a[pa, paS] k].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98f9a79bc
k:'I_b.+1
0 < k < b ->
codom (u3 k) \subset [:: p2;  a[pa, paS] k]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98f9a79bc
cH4:forall k : 'I_b.+1,
0 < k < b ->
codom (u3 k) \subset [:: p2;  a[pa, paS] k]
843
5c6
      rewrite /b .
9c0
0 < k < l.+1 - a ->
codom (u3 k) \subset [:: p2;  a[pa, paS] k]
9c3
    move=> /andP[k_gt0 kLb].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98f9a79bc9c1852
kLb:k < l.+1 - a
codom (u3 k) \subset [:: p2;  a[pa, paS] k]
9c3
    have kBound : 0 < k + a < l.+1.
9ce
0 < k + a < l.+1
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98f9a79bc9c18529cf
kBound:0 < k + a < l.+1
9d0
9c45c6
      rewrite (leq_trans a_gt0) ?leq_addl //=.
9ce
k + a < l.+1
9d5
      by move: kLb; rewrite /b ltn_subRL addnC.
9d7
9d0
9c3
    rewrite ffunE codom_liftr //.
9d7
codom (u (inord (k + a))) \subset
[:: p2;  a[pa, paS] k]
9c3
    have := @cH (inord (k + a)).
9d7
(0 < inord (k + a) < l.+1 ->
 codom (u (inord (k + a))) \subset
 [:: p2;  a[p1, p3] (inord (k + a))]) ->
codom (u (inord (k + a))) \subset
[:: p2;  a[pa, paS] k]
9c3
    rewrite /pa /paS apegS inordK; first by rewrite -apegD; apply.
9d7
k + a < l.+2
9c3
    by rewrite ltnW // ltnS; case/andP: kBound.
9c5
843
5c5
  have {cH4}P4 : 
   (S_[b] n.+1).*2 <= sd u3 + sp (u3 ord0) b (a[pa, paS] 0) +
                      sp (u3 ord_max) b (a[pa, paS] b).
9c5
(S_[b] n.+1).*2 <=
sd (n:=n.+1) u3 +
sp (n:=n.+1) (u3 ord0) b ( a[pa, paS] 0) +
sp (n:=n.+1) (u3 ord_max) b ( a[pa, paS] b)
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98f9a79bc
P4:(S_[b] n.+1).*2 <=
sd (n:=n.+1) u3 +
sp (n:=n.+1) (u3 ord0) b ( a[pa, paS] 0) +
sp (n:=n.+1) (u3 ord_max) b ( a[pa, paS] b)
843
5c6
    apply: IH cH4 => //; try by rewrite /pa /paS // eq_sym.
9bb
pa != paS
9f3
    by rewrite /pa /paS apegS apeg_eqC.
9f5
843
5c5
  have {}P4 := leq_trans P4 (leq_add (leqnn _) (leq_sum_beta _ _)).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98f9a79bc
P4:(S_[b] n.+1).*2 <=
sd (n:=n.+1) u3 +
sp (n:=n.+1) (u3 ord0) b ( a[pa, paS] 0) +
\sum_(k < n.+1)
   (u3 ord_max k !=  a[pa, paS] b) * (α_[1] k).*2
843
5c5
  rewrite apeg0 /= (_ : a[_, _] b = a[p3, p1] l) in P4; last first.
9bb
 a[pa, paS] b =  a[p3, p1] l
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98f9a79bc
P4:(S_[b] n.+1).*2 <=
sd (n:=n.+1) u3 + sp (n:=n.+1) (u3 ord0) b pa +
\sum_(k < n.+1)
   (u3 ord_max k !=  a[p3, p1] l) * (α_[1] k).*2
843
5c6
    by rewrite /pa /paS apegS -apegD subnK ?apegS.
a09
843
5c5
  rewrite !ffunE /= add0n -/ai in P4.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98f9a79bc
P4:(S_[b] n.+1).*2 <=
sd (n:=n.+1) u3 + sp (n:=n.+1) ↓[u ai] b pa +
\sum_(k < n.+1)
   (↓[u (inord (b + a))] k !=  a[p3, p1] l) *
   (α_[1] k).*2
843
5c5
  rewrite -/x3P1 subnK // (_ : inord l.+1 = ord_max) in P4; last first.
9bb
inord l.+1 = ord_max
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98f9a79bc
P4:(S_[b] n.+1).*2 <=
x3P1 + sp (n:=n.+1) ↓[u ai] b pa +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) *
   (α_[1] k).*2
843
5c6
    by apply/val_eqP; rewrite /= inordK.
a19
843
5c5
  rewrite [X in _ <= _ + X + _]sum_beta_S //= KH1 eqxx addn0.
a19
(S_[l.+1] n.+2).*2 <=
xP1 +
\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
5c5
  set xS := \sum_(_ < _) _ in P2; rewrite -/xS.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98f9bca1a
xS:=\sum_(H < n.+1) (↓[u ord0] H != p1) * (α_[1] H).*2:nat
P2:(S_[(3 * a).-2] n.+1).*2 <=
x1P1 + xS +
sp (n:=n.+1) sam1 (3 * a).-2
  ( a[p1, p3] (3 * a).-2)
(S_[l.+1] n.+2).*2 <=
xP1 + xS +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
5c5
  rewrite [X in _ <= _ + _ + X]sum_beta_S //= KH2 mul1n.
a25
(S_[l.+1] n.+2).*2 <=
xP1 + xS +
(\sum_(k < n.+1)
    (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2 +
 α_[l.+1] n.+1)
5c5
  set yS := \sum_(_ < _) _ in P4; rewrite -/yS.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98f9bca26a27
yS:=\sum_(H < n.+1)
   (↓[u ord_max] H !=  a[p3, p1] l) * (α_[1] H).*2:nat
P4:(S_[b] n.+1).*2 <=
x3P1 + sp (n:=n.+1) ↓[u ai] b pa + yS
(S_[l.+1] n.+2).*2 <= xP1 + xS + (yS + α_[l.+1] n.+1)
5c5
  set x1S := sp _ _ _ in P2; set y1S := sp _ _ _ in P3.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a32
x1S:=sp (n:=n.+1) sam1 (3 * a).-2
  ( a[p1, p3] (3 * a).-2):nat
P2:(S_[(3 * a).-2] n.+1).*2 <= x1P1 + xS + x1S
y1S:=sp (n:=n.+1) sam1 2 p2:nat
P3:(S_[2] n.+1).*2 <=
x2P1 + y1S +
sp (n:=n.+1) ↓[u ai] 2 ( a[p2, pa] 2)
a33
5c5
  have x1Sy1SE : x1S + y1S <= (S_[1] n).*2 + α_[maxn (3 * a).-2 2] n.
a37
x1S + y1S <= (S_[1] n).*2 + α_[maxn (3 * a).-2 2] n
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a32a38a39a3aa3b
x1Sy1SE:x1S + y1S <=
(S_[1] n).*2 + α_[maxn (3 * a).-2 2] n
a33
5c6
    apply: sum_alpha_diffE => //.
a37
codom sam1 \subset [::  a[p1, p3] (3 * a).-2; p2]
a40
    rewrite apeg32E //.
a37
codom sam1 \subset [::  a[p1, p3] a; p2]
a40
    by move: sam1C; rewrite -uiME uaiLEp2.
a42
a33
5c5
  rewrite (maxn_idPl _) // in x1Sy1SE.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a32a38a39a3aa3b
x1Sy1SE:x1S + y1S <= (S_[1] n).*2 + α_[(3 * a).-2] n
a33
5c5
  set x2S := sp _ _ _ in P3; set y2S := sp _ _ _ in P4.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a38a39a3aa53
x2S:=sp (n:=n.+1) ↓[u ai] 2 ( a[p2, pa] 2):nat
P3:(S_[2] n.+1).*2 <= x2P1 + y1S + x2S
y2S:=sp (n:=n.+1) ↓[u ai] b pa:nat
P4:(S_[b] n.+1).*2 <= x3P1 + y2S + yS
a33
5c5
  have x2y2SE : x2S + y2S <= (S_[1] n).*2 + α_[maxn 2 b] n.
a57
x2S + y2S <= (S_[1] n).*2 + α_[maxn 2 b] n
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a38a39a3aa53a58a59a5aa5b
x2y2SE:x2S + y2S <= (S_[1] n).*2 + α_[maxn 2 b] n
a33
5c6
    apply: sum_alpha_diffE => //; first by rewrite /pa eq_sym.
a57
codom ↓[u ai] \subset [::  a[p2, pa] 2; pa]
a60
    have /H : 0 < (inord a : 'I_l.+2) < l.+1 by rewrite inordK // a_gt0.
a57
codom (u' (inord a)) \subset
[:: p2;  a[p1, p3] (inord a)] ->
codom ↓[u ai] \subset [::  a[p2, pa] 2; pa]
a60
    by rewrite ffunE inordK.
a62
a33
5c5
  rewrite -addnn {1}dsum_alphaL_S in P2.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a38a3aa53a58a59a5aa5ba63
P2:S_[(3 * a).-2] n + α_[(3 * a).-2] n +
S_[(3 * a).-2] n.+1 <= x1P1 + xS + x1S
a33
5c5
  rewrite -addnn {1}dsum_alphaL_S in P3.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a38a3aa53a58a5aa5ba63a73
P3:S_[2] n + α_[2] n + S_[2] n.+1 <=
x2P1 + y1S + x2S
a33
5c5
  rewrite -addnn {1}dsum_alphaL_S.
a77
S_[l.+1] n.+1 + α_[l.+1] n.+1 + S_[l.+1] n.+2 <=
xP1 + xS + (yS + α_[l.+1] n.+1)
5c5
  have F4 k : S_[a.-1] k.+1 <= S_[(3 *a).-2] k + a.-1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a38a3aa53a58a5aa5ba63a73a78
k:nat
S_[a.-1] k.+1 <= S_[(3 * a).-2] k + a.-1
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a38a3aa53a58a5aa5ba63a73a78
F4:forall k : nat,
S_[a.-1] k.+1 <= S_[(3 * a).-2] k + a.-1
a7c
5c6
    apply: leq_trans (dsum_alpha3l _ _) _.
a80
S_[3 * a.-1] k + a.-1 <= S_[(3 * a).-2] k + a.-1
a83
    rewrite leq_add2r.
a80
S_[3 * a.-1] k <= S_[(3 * a).-2] k
a83
    apply: (increasingE (increasing_dsum_alphaL_l _)).
a80
74e
a83
    by rewrite -{2}[a]prednK // mulnS addSn add2n /=.
a85
a7c
5c5
  have F4n := F4 n.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a38a3aa53a58a5aa5ba63a73a78a86
F4n:S_[a.-1] n.+1 <= S_[(3 * a).-2] n + a.-1
a7c
5c5
  have {F4}F4n1 := F4 n.+1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a38a3aa53a58a5aa5ba63a73a78a99
F4n1:S_[a.-1] n.+2 <= S_[(3 * a).-2] n.+1 + a.-1
a7c
5c5
  have F5 k :
    S_[l.+1] k.+1 + S_[1] k.+1 + S_[b.+1] k.+1 <=
    S_[a.-1] k.+1 + S_[b.+2] k.+1 + S_[b] k + (α_[1] k).*2.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a38a3aa53a58a5aa5ba63a73a78a99a9ea81
S_[l.+1] k.+1 + S_[1] k.+1 + S_[b.+1] k.+1 <=
S_[a.-1] k.+1 + S_[b.+2] k.+1 + S_[b] k + (α_[1] k).*2
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a38a3aa53a58a5aa5ba63a73a78a99a9e
F5:forall k : nat,
S_[l.+1] k.+1 + S_[1] k.+1 + S_[b.+1] k.+1 <=
S_[a.-1] k.+1 + S_[b.+2] k.+1 + S_[b] k +
(α_[1] k).*2
a7c
5c6
    have <- : a + b = l.+1 by rewrite addnC subnK.
aa2
S_[a + b] k.+1 + S_[1] k.+1 + S_[b.+1] k.+1 <=
S_[a.-1] k.+1 + S_[b.+2] k.+1 + S_[b] k + (α_[1] k).*2
aa4
    rewrite -[X in _ <= X]addnA (leq_add _ (dsum_alphaL_alpha k b_gt1)) //.
aa2
S_[a + b] k.+1 + S_[1] k.+1 <=
S_[a.-1] k.+1 + S_[b.+2] k.+1
aa4
    have := concaveEk1 1 a.-2 b.+1 (concave_dsum_alphaL_l k.+1).
aa2
S_[1 + a.-2 + b.+1] k.+1 + S_[1] k.+1 <=
S_[1 + b.+1] k.+1 + S_[1 + a.-2] k.+1 ->
S_[a + b] k.+1 + S_[1] k.+1 <=
S_[a.-1] k.+1 + S_[b.+2] k.+1
aa4
    rewrite add1n prednK; last by rewrite -subn1 ltn_subRL addn0.
aa2
S_[a.-1 + b.+1] k.+1 + S_[1] k.+1 <=
S_[1 + b.+1] k.+1 + S_[a.-1] k.+1 ->
S_[a + b] k.+1 + S_[1] k.+1 <=
S_[a.-1] k.+1 + S_[b.+2] k.+1
aa4
    by rewrite -addSnnS prednK // add1n [X in _ <= X]addnC.
aa6
a7c
5c5
  have F5n := F5 n.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a38a3aa53a58a5aa5ba63a73a78a99a9eaa7
F5n:S_[l.+1] n.+1 + S_[1] n.+1 + S_[b.+1] n.+1 <=
S_[a.-1] n.+1 + S_[b.+2] n.+1 + S_[b] n +
(α_[1] n).*2
a7c
5c5
  have {F5}F5n1 := F5 n.+1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a38a3aa53a58a5aa5ba63a73a78a99a9eabf
F5n1:S_[l.+1] n.+2 + S_[1] n.+2 + S_[b.+1] n.+2 <=
S_[a.-1] n.+2 + S_[b.+2] n.+2 + S_[b] n.+1 +
(α_[1] n.+1).*2
a7c
5c5
  have F6 k : S_[b.+2] k + S_[1] k <= S_[b.+1] k + S_[2] k.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a38a3aa53a58a5aa5ba63a73a78a99a9eabfac4a81
S_[b.+2] k + S_[1] k <= S_[b.+1] k + S_[2] k
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a38a3aa53a58a5aa5ba63a73a78a99a9eabfac4
F6:forall k : nat,
S_[b.+2] k + S_[1] k <= S_[b.+1] k + S_[2] k
a7c
5c6
    by have := concaveEk1 1 1 b (concave_dsum_alphaL_l k).
acc
a7c
5c5
  have F6n := F6 n.+1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a38a3aa53a58a5aa5ba63a73a78a99a9eabfac4acd
F6n:S_[b.+2] n.+1 + S_[1] n.+1 <=
S_[b.+1] n.+1 + S_[2] n.+1
a7c
5c5
  have {F6}F6n1 := F6 n.+2.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a38a3aa53a58a5aa5ba63a73a78a99a9eabfac4ad5
F6n1:S_[b.+2] n.+2 + S_[1] n.+2 <=
S_[b.+1] n.+2 + S_[2] n.+2
a7c
5c5
  have F72 := dsum_alphaL_S 2 n.+1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a38a3aa53a58a5aa5ba63a73a78a99a9eabfac4ad5ada
F72:S_[2] n.+2 = S_[2] n.+1 + α_[2] n.+1
a7c
5c5
  have F721 := dsum_alphaL_S 2 n.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a38a3aa53a58a5aa5ba63a73a78a99a9eabfac4ad5adaadf
F721:S_[2] n.+1 = S_[2] n + α_[2] n
a7c
5c5
  have F711 := dsum_alphaL_S 1 n.+1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a38a3aa53a58a5aa5ba63a73a78a99a9eabfac4ad5adaadfae4
F711:S_[1] n.+2 = S_[1] n.+1 + α_[1] n.+1
a7c
5c5
  have F71 := dsum_alphaL_S 1 n.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a38a3aa53a58a5aa5ba63a73a78a99a9eabfac4ad5adaadfae4ae9
F71:S_[1] n.+1 = S_[1] n + α_[1] n
a7c
5c5
  have F73 := dsum_alphaL_S b n.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a38a3aa53a58a5aa5ba63a73a78a99a9eabfac4ad5adaadfae4ae9aee
F73:S_[b] n.+1 = S_[b] n + α_[b] n
a7c
5c5  
  have F8 : α_[2] n.+1 <= α_[1] n.+2.
af2
α_[2] n.+1 <= α_[1] n.+2
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a38a3aa53a58a5aa5ba63a73a78a99a9eabfac4ad5adaadfae4ae9aeeaf3
F8:α_[2] n.+1 <= α_[1] n.+2
a7c
5c6
    have ->: α_[1] n.+2 = α_[3] n.+1 by rewrite alphaL_3E.
af2
α_[2] n.+1 <= α_[3] n.+1
af8
    by apply: increasing_alphaL_l.
afa
a7c
5c5
  have F9 : α_[1] (n.+2) <= (α_[1] n).*2.+2.
afa
α_[1] n.+2 <= (α_[1] n).*2.+2
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a38a3aa53a58a5aa5ba63a73a78a99a9eabfac4ad5adaadfae4ae9aeeaf3afb
F9:α_[1] n.+2 <= (α_[1] n).*2.+2
a7c
5c6
    apply: leq_trans (leqnSn _).
afa
α_[1] n.+2 <= (α_[1] n).*2.+1
b07
    by apply: alphaL_4_5.
b09
a7c
5c5
  rewrite (maxn_idPr _) // in x1Sy1SE x2y2SE.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be5c35c45cc5d15d95eb5f05f963063563d63f66a67b6836926a26a783683783d83e83f97298e98fa26a31a38a3aa58a5aa5ba73a78a99a9eabfac4ad5adaadfae4ae9aeeaf3afbb0aa53
x2y2SE:x2S + y2S <= (S_[1] n).*2 + α_[b] n
a7c
5c5
  by lia.
5bd
5c7
rewrite negb_and -leqNgt -ltnNge => oH.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95be
oH:(a <= 1) || (l.-1 < a)
573
have [/andP[a_gt1 /eqP bE1]|] := boolP ((1 < a) && (b == 1)).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3
bE1:b = 1
573
b1d
~~ ((1 < a) && (b == 1)) ->
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u + sp (n:=n.+2) (u ord0) l.+1 p1 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
(* 2 <= a and b = 1 *)
  have am1_gt0 : 0 < a.-1 by rewrite -subn1 subn_gt0.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23
am1_gt0:0 < a.-1
573
b24
  have lEa : l = a.
b2a
l = a
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2b
lEa:l = a
573
b25
    have : a + b = l.+1 by rewrite addnC subnK //.
b2a
a + b = l.+1 -> l = a
b30
    by rewrite bE1 addn1 => [] [].
b32
573
b24
  have uaiLEp2 : u ai ldisk = p2.
b32
5d5
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d9
573
b25
    have /cH/subsetP/(_ (u ai ldisk) (codom_f _ _)) : 0 < ai < l.+1.
b32
5dd
b32
5e0
b3fb25
      by rewrite aiE a_gt0 lEa /=.
b32
5e0
b3e
    by rewrite !inE aiE (negPf aH) orbF => /eqP.
b40
573
b24
  pose si i : configuration 4 n.+1 := 
     sgdist1 p2 (u (inord i)) (u (inord i.+1)).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb
573
b24
  pose ti i : configuration 4 n.+1 := 
     sgdist2 p2 (u (inord i)) (u (inord i.+1)).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f0
573
b24
  have sitiH i : i < a.-1 ->
     [/\ codom (si i) \subset [:: p2; u (inord i.+1) ldisk ], 
         codom (ti i) \subset [:: u (inord i) ldisk; p2] & 
         `d[u (inord i), u (inord i.+1)]_smove = 
           D[[:: ↓[u (inord i)]; si i; ti i; ↓[u (inord i.+1)]]].+2].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f014a
5f5
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f9
573
b25
    move=> iLa.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f014a5fe
5ff
b58
    have iLa1 : i < a by rewrite (leq_trans iLa) // ssrnat.leq_pred.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f014a5fe604
5ff
b58
    apply: sgdistE => //.
b62
608
b62
60b
b62
60d
b62
60f
b62
611
b59b25
    -
b64 by rewrite aMin1.
b62
60b
b68b69b6ab59b25
    -
b6d rewrite !aMin1 //; last by rewrite -[a]prednK.
b62
61b
b6f
      by rewrite apegS apeg_eqC.
b62
60d
b69b6ab59b25
    -
b75 by rewrite aMin1 1?eq_sym // -[a]prednK.
b62
60f
b6ab59b25
    -
b7a by rewrite aMin1.
b62
611
b58
    by rewrite aMin1 // -[a]prednK.
b5a
573
b24
  pose pa := a[p1, p3] a.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998e
573
b24
  pose sam1 : configuration 4 n.+1 := sgdist1 pa (u (inord a.-1)) (u (inord a)).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998e
sam1:=sgdist1 (n:=n.+1) pa (u (inord a.-1))
  (u (inord a))
  : configuration 4 n.+1:configuration 4 n.+1
573
b24
  pose tam1 : configuration 4 n.+1 := sgdist2 pa (u (inord a.-1)) (u (inord a)).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8c
tam1:=sgdist2 (n:=n.+1) pa (u (inord a.-1))
  (u (inord a))
  : configuration 4 n.+1:configuration 4 n.+1
573
b24
  have [sam1C tam1C duam1ua1E] : 
     [/\ codom sam1 \subset [:: pa; u (inord a) ldisk ], 
         codom tam1 \subset [:: u (inord a.-1) ldisk; pa] & 
         `d[u (inord a.-1), u (inord a)]_smove = 
           D[[:: ↓[u (inord a.-1)]; sam1; tam1; ↓[u (inord a)]]].+2].
b90
[/\ codom sam1 \subset [:: pa; u (inord a) ldisk],
    codom tam1 \subset [:: u (inord a.-1) ldisk; pa]
  & `d[u (inord a.-1), u (inord a)]_smove =
     D[[:: ↓[u (inord a.-1)]; sam1; tam1;
           ↓[u (inord a)]]].+2]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91
sam1C:codom sam1 \subset [:: pa; u (inord a) ldisk]
tam1C:codom tam1 \subset
[:: u (inord a.-1) ldisk; pa]
63f
573
b25
    apply: sgdistE; rewrite /pa //.
b90
643
b90
646
b90
648
b90
64a
b90
64c
b97b25
    -
b9c by rewrite aMin1 ?prednK // -{2}[a]prednK //= apegS apeg_eqC.
b90
646
ba0ba1ba2b97b25
    -
ba5 by rewrite uaiLEp2 aMin1 // prednK.
b90
648
ba1ba2b97b25
    -
baa by rewrite uaiLEp2.
b90
64a
ba2b97b25
    -
baf by rewrite aMin1 // prednK.
b90
64c
b96
    by rewrite uaiLEp2.
b98
573
b24
  have {}tam1C : codom tam1 \subset [:: p1; p3].
b98
666
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a
573
b25
    move: tam1C; rewrite aMin1; last by rewrite -{2}[a]prednK.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f
codom tam1 \subset [::  a[p1, p3] a.-1; pa] ->
codom tam1 \subset [:: p1; p3]
bbc
    by rewrite /pa -{2}[a]prednK // apegS codom_apeg.
bbe
573
b24
  pose u1 := 
    [ffun i =>
    if ((i : 'I_(3 * a).-2.+1) == 3 * a.-1 :>nat) then ↓[u (inord a.-1)] 
    else if (i == (3 * a.-1).+1 :>nat) then sam1
    else if (i %% 3) == 0 then ↓[u (inord (i %/ 3))]
    else if (i %% 3) == 1 then si (i %/ 3)
    else ti (i %/ 3)].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b
573
b24
  have u10E : u1 ord0 = ↓[u ord0].
bca
67f
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683
573
b25
    rewrite ffunE /= ifN ?inord_eq0 //.
bca
687
bce
    by rewrite neq_ltn muln_gt0 /= -ltnS prednK //= a_gt1.
bd0
573
b24
  have uiME : u1 ord_max = sam1.
bd0
68e
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692
573
b25
    rewrite ffunE /= eqn_leq leqNgt -{2}[a]prednK // mulnS.
bd0
696
bda
    rewrite addSn add2n ltnS leqnn /=.
bd0
69a
bda
    by rewrite -{1}[a]prednK // mulnS eqxx.
bdc
573
b24
  pose u2 := 
    [ffun i : 'I_4 =>  
    if i == 0 :> nat then sam1 else
    if i == 1 :> nat then tam1 else
    if i == 2 :> nat then ↓[u ai] else ↓[u (inord (l.+1))]].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692
u2:=[ffun i => if i == 0 :> nat
           then sam1
           else
            if i == 1 :> nat
            then tam1
            else
             if i == 2 :> nat
             then ↓[u ai]
             else ↓[u (inord l.+1)]]:{ffun forall x : ordinal_finType 4,
      (fun=> configuration 4 n.+1) x}
573
b24
  have P1 :  a.*2 + sd u1 + sd u2 <= sd u.
be9
a.*2 + sd (n:=n.+1) u1 + sd (n:=n.+1) u2 <=
sd (n:=n.+2) u
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea
P1:a.*2 + sd (n:=n.+1) u1 + sd (n:=n.+1) u2 <=
sd (n:=n.+2) u
573
b25
    have G b1 : b1 = a ->  
      \sum_(i < (3 * b1).-2) `d[u1 (inord i), u1 (inord i.+1)]_smove =
      \sum_(i < (3 * a.-1)) `d[u1 (inord i), u1 (inord i.+1)]_smove +
      `d[↓[u (inord a.-1)], sam1]_smove.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea6b4
6b5
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea6b9
bee
bf0b25
      move=> b1Ea.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea6b46be
6bf
bf7
      have ta2E : (3 * b1).-2 = (3 * a.-1).+1.
bfd
6c3
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea6b46be6c7
6bf
bf8bf0b25
        by rewrite b1Ea -{1}[a]prednK // mulnS.
c03
6bf
bf7
      rewrite ta2E big_ord_recr /=; congr (_ + `d[_,_]_smove).
c03
6ce
c03
6d1
bf8bf0b25
        by rewrite ffunE ifT// inordK // -{2}b1Ea ?ta2E.
c03
6d1
bf7
      rewrite ffunE inordK; last by rewrite -{2}b1Ea ?ta2E.
c03
6d8
bf7
      by rewrite eqn_leq ltnn /= eqxx.
bf9
bee
bef
    rewrite {}[sd u1]G //.
be9
a.*2 +
(\sum_(i < 3 * a.-1)
    `d[u1 (inord i), u1 (inord i.+1)]_smove +
 `d[↓[u (inord a.-1)], sam1]_smove) + sd (n:=n.+1) u2 <=
sd (n:=n.+2) u
bef
    have -> := @sum3E _ (fun i => `d[u1 (inord i), u1 (inord i.+1)]_smove).
be9
a.*2 +
(\sum_(i < a.-1)
    (`d[u1 (inord (3 * i)), u1 (inord (3 * i).+1)]_smove +
     `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
     `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove) +
 `d[↓[u (inord a.-1)], sam1]_smove) + sd (n:=n.+1) u2 <=
sd (n:=n.+2) u
bef
    have -> : a.*2 = 2 + \sum_(i < a.-1) 2.
be9
6e7
be9
2 + \sum_(i < a.-1) 2 +
(\sum_(i < a.-1)
    (`d[u1 (inord (3 * i)), 
     u1 (inord (3 * i).+1)]_smove +
     `d[u1 (inord (3 * i).+1), 
     u1 (inord (3 * i).+2)]_smove +
     `d[u1 (inord (3 * i).+2), 
     u1 (inord (3 * i).+3)]_smove) +
 `d[↓[u (inord a.-1)], sam1]_smove) + 
sd (n:=n.+1) u2 <= sd (n:=n.+2) u
bf0b25
      rewrite sum_nat_const /= cardT size_enum_ord.
be9
6ee
c20
      by rewrite muln2 -(doubleD 1) add1n prednK.
be9
2 + \sum_(i < a.-1) 2 +
(\sum_(i < a.-1)
    (`d[u1 (inord (3 * i)), u1 (inord (3 * i).+1)]_smove +
     `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
     `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove) +
 `d[↓[u (inord a.-1)], sam1]_smove) + sd (n:=n.+1) u2 <=
sd (n:=n.+2) u
bef
    rewrite !addnA -[2 + _ + _]addnA -big_split /=.
be9
2 +
\sum_(i < a.-1)
   (2 +
    (`d[u1 (inord (3 * i)), u1 (inord (3 * i).+1)]_smove +
     `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
     `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove)) +
`d[↓[u (inord a.-1)], sam1]_smove + sd (n:=n.+1) u2 <=
sd (n:=n.+2) u
bef
    rewrite [sd u2]big_ord_recr /=.
be9
2 +
\sum_(i < a.-1)
   (2 +
    (`d[u1 (inord (3 * i)), u1 (inord (3 * i).+1)]_smove +
     `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
     `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove)) +
`d[↓[u (inord a.-1)], sam1]_smove +
(\sum_(i < 2) `d[u2 (inord i), u2 (inord i.+1)]_smove +
 `d[u2 (inord 2), u2 (inord 3)]_smove) <=
sd (n:=n.+2) u
bef
    rewrite [u2 (inord 2)]ffunE inordK //=.
be9
2 +
\sum_(i < a.-1)
   (2 +
    (`d[u1 (inord (3 * i)), u1 (inord (3 * i).+1)]_smove +
     `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
     `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove)) +
`d[↓[u (inord a.-1)], sam1]_smove +
(\sum_(i < 2) `d[u2 (inord i), u2 (inord i.+1)]_smove +
 `d[↓[u ai], u2 (inord 3)]_smove) <= sd (n:=n.+2) u
bef
    rewrite [u2 (inord 3)]ffunE inordK //=.
be9
2 +
\sum_(i < a.-1)
   (2 +
    (`d[u1 (inord (3 * i)), u1 (inord (3 * i).+1)]_smove +
     `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
     `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove)) +
`d[↓[u (inord a.-1)], sam1]_smove +
(\sum_(i < 2) `d[u2 (inord i), u2 (inord i.+1)]_smove +
 `d[↓[u ai], ↓[u (inord l.+1)]]_smove) <=
sd (n:=n.+2) u
bef
    rewrite [2 + _]addnC -!addnA 2![X in _ + X <= _]addnA addnA.
be9
\sum_(i < a.-1)
   (2 +
    (`d[u1 (inord (3 * i)), u1 (inord (3 * i).+1)]_smove +
     `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
     `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove)) +
(2 + `d[↓[u (inord a.-1)], sam1]_smove +
 \sum_(i < 2) `d[u2 (inord i), u2 (inord i.+1)]_smove) +
`d[↓[u ai], ↓[u (inord l.+1)]]_smove <= sd (n:=n.+2) u
bef
    have -> : sd u =
      \sum_(i < a.-1) (`d[u (inord i), u (inord i.+1)]_smove) + 
      `d[u (inord a.-1), u ai]_smove + `d[u ai, u (inord ai.+1)]_smove.
be9
sd (n:=n.+2) u =
\sum_(i < a.-1) `d[u (inord i), u (inord i.+1)]_smove +
`d[u (inord a.-1), u ai]_smove +
`d[u ai, u (inord ai.+1)]_smove
be9
\sum_(i < a.-1)
   (2 +
    (`d[u1 (inord (3 * i)), 
     u1 (inord (3 * i).+1)]_smove +
     `d[u1 (inord (3 * i).+1), 
     u1 (inord (3 * i).+2)]_smove +
     `d[u1 (inord (3 * i).+2), 
     u1 (inord (3 * i).+3)]_smove)) +
(2 + `d[↓[u (inord a.-1)], sam1]_smove +
 \sum_(i < 2) `d[u2 (inord i), u2 (inord i.+1)]_smove) +
`d[↓[u ai], ↓[u (inord l.+1)]]_smove <=
\sum_(i < a.-1) `d[u (inord i), u (inord i.+1)]_smove +
`d[u (inord a.-1), u ai]_smove +
`d[u ai, u (inord ai.+1)]_smove
bf0b25
      rewrite -lEa (_ : ai = inord l); last first.
be9
ai = inord l
be9
sd (n:=n.+2) u =
\sum_(i < l.-1) `d[u (inord i), u (inord i.+1)]_smove +
`d[u (inord l.-1), u (inord l)]_smove +
`d[u (inord l), u (inord (inord l).+1)]_smove
c43bf0b25
        by apply/val_eqP; rewrite /= aiE inordK // lEa.
be9
c4b
c42
      case: l l_gt0 u => //= l1 _ f.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea
l1:nat
f:{ffun 'I_l1.+3 -> configuration 4 n.+2}
sd (n:=n.+2) f =
\sum_(i < l1) `d[f (inord i), f (inord i.+1)]_smove +
`d[f (inord l1), f (inord l1.+1)]_smove +
`d[f (inord l1.+1), f (inord (inord l1.+1).+1)]_smove
c42
      rewrite /sd 2!big_ord_recr //=; congr (_ + _ + `d[f _, f _]_smove).
c52
inord l1.+2 = inord (inord l1.+1).+1
c42
      by apply/val_eqP; rewrite /= !inordK.
be9
\sum_(i < a.-1)
   (2 +
    (`d[u1 (inord (3 * i)), u1 (inord (3 * i).+1)]_smove +
     `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
     `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove)) +
(2 + `d[↓[u (inord a.-1)], sam1]_smove +
 \sum_(i < 2) `d[u2 (inord i), u2 (inord i.+1)]_smove) +
`d[↓[u ai], ↓[u (inord l.+1)]]_smove <=
\sum_(i < a.-1) `d[u (inord i), u (inord i.+1)]_smove +
`d[u (inord a.-1), u ai]_smove +
`d[u ai, u (inord ai.+1)]_smove
bef
    apply: leq_add; first apply: leq_add.
be9
729
be9
2 + `d[↓[u (inord a.-1)], sam1]_smove +
\sum_(i < 2) `d[u2 (inord i), u2 (inord i.+1)]_smove <=
`d[u (inord a.-1), u ai]_smove
be9
`d[↓[u ai], ↓[u (inord l.+1)]]_smove <=
`d[u ai, u (inord ai.+1)]_smove
bf0b25
    -
c5f rewrite leq_eqVlt; apply/orP; left; apply/eqP.
be9
733
c61
      apply: eq_bigr => i _.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea738
739
c61
      have -> : u1 (inord (3 * i)) = ↓[u (inord i)].
c6d
73d
c6d
740
c62c64bf0b25
        rewrite ffunE inordK; last first.
c6d
744
c6d
747
c72c62c64bf0b25
          rewrite ltnS (leq_trans (_ : _ <= 3 * (a.-1))) //.
c6d
74b
c6d
74e
c77c72c62c64bf0b25
            by rewrite leq_mul2l /= ltnW.
c6d
74e
c76
          by rewrite -{2}[a]prednK // mulnS addSn add2n /= ltnW.
c6d
755
c71
        rewrite eqn_mul2l /= eqn_leq [_ <= i]leqNgt ltn_ord andbF.
c6d
759
c71
        rewrite eqn_leq ltn_mul2l /= ltnNge [i <= _]ltnW // andbF.
c6d
75d
c71
        by rewrite mod3E /= div3E.
c6d
761
c61
      have -> : u1 (inord (3 * i).+1) = si i.
c6d
765
c6d
768
c62c64bf0b25
        rewrite ffunE inordK; last first.
c6d
76c
c6d
76f
c90c62c64bf0b25
          rewrite ltnS (leq_trans (_ : _ <= 3 * (a.-1))) //.
c6d
773
c7cc95c90c62c64bf0b25
            by rewrite ltn_mul2l /=.
c7fc94
          by rewrite -{2}[a]prednK // mulnS addSn add2n /= ltnW.
c6d
77a
c8f
        rewrite eqn_leq [_ <= _.+1]leqNgt.
c6d
77e
c8f
        rewrite (leq_trans (_ : (3 * i).+2 <= (3 * i.+1))) ?andbF //;
               last 2 first.
c6d
782
c6d
785
c6d
787
c90c62c64bf0b25
        -
ca3 by rewrite mulnS addSn add2n.
c6d
785
ca7c90c62c64bf0b25
        -
caa by rewrite leq_mul2l /=.
c6d
791
c8f
        rewrite eqn_leq !ltnS [_ <= 3 * i]leq_mul2l.
c6d
795
c8f
        by rewrite [_ <= i]leqNgt ltn_ord andbF mod3E /= div3E.
c6d
799
c61
      have -> : u1 (inord (3 * i).+2) = ti i.
c6d
79d
c6d
7a0
c62c64bf0b25
        rewrite ffunE inordK; last first.
c6d
7a4
c6d
7a7
cbbc62c64bf0b25
          rewrite ltnS (leq_trans (_ : _ <= 3 * (a.-1))) //.
c6d
7ab
c7ccc0cbbc62c64bf0b25
            rewrite (leq_trans (_ : _ <= 3 * (i.+1))) //.
ca4
ca6c7ccc0cbbc62c64bf0b25
              by rewrite mulnS addSn add2n.
cabcc4
            by rewrite leq_mul2l /=.
c7fcbf
          by rewrite -{2}[a]prednK // mulnS addSn add2n /= ltnW.
c6d
7b7
cba
        rewrite eqn_leq [_ <= _.+1]leqNgt.
c6d
7bb
cba
        rewrite -[(3 * i).+3](mulnDr 3 1 i) leq_mul2l ltn_ord andbF.
c6d
7bf
cba
        rewrite eqn_leq !ltnS [_ <= _.+1]leqNgt.
c6d
7c3
cba
        rewrite (leq_trans (_ : (3 * i).+1 < 3 * (i.+1))) ?andbF //;
             last 2 first.
ca4
ca6
c6d
7c8
cbbc62c64bf0b25
        -
cd9 by rewrite mulnS addSn add2n.
cab
cdbcbbc62c64bf0b25
        -
cde by rewrite leq_mul2l /=.
c6d
7d1
cba
        by rewrite mod3E /= div3E.
c6d
7a0
c61
      have -> : u1 (inord (3 * i).+3) = ↓[u (inord i.+1)].
c6d
7d8
c6d
7db
c62c64bf0b25
        have -> : (3 * i).+3 = 3 * (i.+1) by rewrite mulnS addSn add2n.
c6d
7df
cea
        rewrite ffunE inordK; last first.
c6d
7e3
c6d
7e6
cebc62c64bf0b25
          rewrite ltnS (leq_trans (_ : _ <= 3 * (a.-1))) //.
cab
c7ccf3cebc62c64bf0b25
            by rewrite leq_mul2l /=.
c7fcf2
          by rewrite -{2}[a]prednK // mulnS addSn add2n /= ltnW.
c6d
7ef
cea
        rewrite eqn_mul2l /=; case: eqP => [->//|_].
c6d
7f3
cea
        rewrite eqn_leq [_ <= 3 * i.+1]leqNgt ltnS leq_mul2l /= ltn_ord andbF.
c6d
7f7
cea
        by rewrite mod3E /= div3E.
c6d
7db
c61
      case: (sitiH i) => //= _ _.
c6d
`d[u (inord i), u (inord i.+1)]_smove =
(`d[↓[u (inord i)], si i]_smove +
 (`d[si i, ti i]_smove +
  (`d[ti i, ↓[u (inord i.+1)]]_smove + 0))).+2 ->
2 +
(`d[↓[u (inord i)], si i]_smove + `d[si i, ti i]_smove +
 `d[ti i, ↓[u (inord i.+1)]]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
c61
      by rewrite !addnA addn0 add2n.
be9
c63
c64bf0b25
    -
d0a rewrite leq_eqVlt; apply/orP; left; apply/eqP.
be9
2 + `d[↓[u (inord a.-1)], sam1]_smove +
\sum_(i < 2) `d[u2 (inord i), u2 (inord i.+1)]_smove =
`d[u (inord a.-1), u ai]_smove
d0c
      rewrite !big_ord_recr big_ord0 //= add0n !addnA.
be9
80c
d0c
      have -> : u2 (inord 0) = sam1 by rewrite ffunE /= inordK.
be9
810
d0c
      have -> : u2 (inord 1) = tam1 by rewrite ffunE /= inordK.
be9
814
d0c
      have -> : u2 (inord 2) = ↓[u ai] by rewrite ffunE /= inordK.
be9
818
d0c
      by move: duam1ua1E; rewrite /= addn0 add2n !addnA !addSn.
be9
c65
bef
    rewrite (_ : inord l.+1 = inord ai.+1); last first.
be9
inord l.+1 = inord ai.+1
be9
`d[↓[u ai], ↓[u (inord ai.+1)]]_smove <=
`d[u ai, u (inord ai.+1)]_smove
bf0b25
      by apply/val_eqP; rewrite /= !inordK // lEa.
be9
d27
bef
    by apply/gdist_cunlift/shanoi_connect.
bf1
573
b24
  set x1P1 := sd _ in P1; set x2P1 := sd _ in P1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea836837
P1:a.*2 + x1P1 + x2P1 <= sd (n:=n.+2) u
573
b24
  set xP1 := sd _ in P1; rewrite -/xP1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783e
P1:a.*2 + x1P1 + x2P1 <= xP1
843
b24
  have cH2 (k : 'I_(3 * a).-2.+1) : 
      0 < k < (3 * a).-2 -> codom (u1 k) \subset [:: p2; a[p1, p3] k].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed37848
849
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3784d
843
b25
    move=> /andP[k_gt0 k_lt].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed37848852853
854
d3c
    have kL3a1 : k <= (3 * a.-1).
d42
858
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3784885285385c
854
d3db25
      by rewrite -ltnS (leq_trans k_lt) // -{1}[a]prednK // mulnS.
d48
854
d3c
    have kLa1 : k %/ 3 <= a.-1.
d48
863
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3784885285385c867
854
d3db25
      by rewrite -(mulKn a.-1 (isT : 0 < 3)) leq_div2r.
d51
854
d3c
    rewrite ffunE; case: eqP => [kH|/eqP kH].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3784885285385c86786f
870
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3784885285385c867874
875
d3db25
      rewrite kH apegMr //.
d58
879
d59
      have := @H (inord a.-1); rewrite inordK //.
d58
87d
d58
880
d5ad3db25
        rewrite ffunE; apply.
d58
884
d62
        by rewrite -ltnS prednK // a_gt1.
d58
880
d59
      by rewrite prednK // (leq_trans aLld1).
d5b
875
d3c
    have k3La2 : k %/ 3 <= a.-2.
d5b
88e
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3784885285385c867874892
875
d3db25
      rewrite -ltnS [a.-2.+1]prednK //.
d5b
899
d70
      rewrite ltn_divLR // [_ * 3]mulnC.
d5b
8a0
d70
      by move: kL3a1 kH; case: ltngtP.
d72
875
d3c
    rewrite eqn_leq [_ <= k]leqNgt (leq_trans k_lt) ?andbF //; last first.
d72
8a7
d72
8aa
d3db25
      by rewrite -{1}[a]prednK // mulnS addSn add2n.
d72
8aa
d3c
    have k3La : k %/ 3 <= a.
d72
8b1
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3784885285385c8678748928b5
8aa
d3db25
      by rewrite (leq_trans kLa1) // -subn1 leq_subr.
d89
8aa
d3c
    case: eqP => k3mH.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3784885285385c8678748928b58bd
8be
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3784885285385c8678748928b58c2
8c3
d3db25
      apply: codom_liftr.
d90
8c7
d91
      rewrite {2}(divn_eq k 3) k3mH addn0 [in a[_,_] _]mulnC apegMr //.
d90
8cb
d91
      have := @cH (inord (k %/3)); rewrite inordK; last first.
d90
8cf
d90
8d2
d92d3db25
        by rewrite (leq_ltn_trans k3La).
d90
8d2
d91
      apply.
d90
8d9
d91
      rewrite (leq_ltn_trans k3La) ?ltnS ?lEa //.
d90
8dd
d91
      move: k_gt0; rewrite andbT {1}(divn_eq k 3) k3mH addn0 muln_gt0.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3784885385c8678748928b58bd
8e2
d91
      by case/andP.
d93
8c3
d3c
    case: eqP => k3mH1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3784885285385c8678748928b58c28ea
8eb
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3784885285385c8678748928b58c28ef
8f0
d3db25
      case: (sitiH (k %/ 3)); first by rewrite -[a.-1]prednK.
db2
codom (si (k %/ 3)) \subset
[:: p2; u (inord (k %/ 3).+1) ldisk] ->
codom (ti (k %/ 3)) \subset
[:: u (inord (k %/ 3)) ldisk; p2] ->
`d[u (inord (k %/ 3)), u (inord (k %/ 3).+1)]_smove =
 D[[:: ↓[u (inord (k %/ 3))]; si (k %/ 3);
       ti (k %/ 3); ↓[u (inord (k %/ 3).+1)]]].+2 ->
codom (si (k %/ 3)) \subset [:: p2;  a[p1, p3] k]
db3
      set pp := u _ _; (suff <- : pp = a[p1, p3] k by []); rewrite {}/pp.
db2
8fe
db3
      set i := inord _; have <- : a[p1, p3] i = a[p1, p3] k.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3784885285385c8678748928b58c28ea903
904
dc0
907
db4d3db25
        rewrite (divn_eq k 3) k3mH1 addn1 apegS mulnC apegMr //.
dc0
90b
dc1
        by rewrite /i !inordK ?apegS //= !ltnS lEa.
dc0
907
db3  
      have H2 : (k %/ 3).+1 < l.+2 by rewrite !ltnS lEa (leq_trans k3La).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3784885285385c8678748928b58c28ea903913
907
db3
      rewrite -(@aMin1 i); last first.
dcc
917
dcc
91a
db4d3db25
        by rewrite !inordK // -{2}[a]prednK // ltnS // -[a.-1]prednK.
dcc
91a
db3
      congr (u _ _).
dcc
924
db3
      by apply/val_eqP; rewrite /= !inordK.
db5
8f0
d3c
    rewrite codom_subC.
db5
92b
d3c
    case: (sitiH (k %/ 3)) => [|_]; first by rewrite -[a.-1]prednK.
db5
codom (ti (k %/ 3)) \subset
[:: u (inord (k %/ 3)) ldisk; p2] ->
`d[u (inord (k %/ 3)), u (inord (k %/ 3).+1)]_smove =
 D[[:: ↓[u (inord (k %/ 3))]; si (k %/ 3);
       ti (k %/ 3); ↓[u (inord (k %/ 3).+1)]]].+2 ->
codom (ti (k %/ 3)) \subset [::  a[p1, p3] k; p2]
d3c
    set pp := u _ _; (suff <- : pp = a[p1, p3] k by []); rewrite {}/pp.
db5
939
d3c
    set i := inord _; have <- : a[p1, p3] i = a[p1, p3] k.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3784885285385c8678748928b58c28ef93e
904
de8
907
d3db25
      have k3mH2 : k %% 3 = 2.
de8
944
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3784885285385c8678748928b58c28ef93e948
904
dead3db25
        have := ltn_mod k 3; move: k3mH k3mH1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3784885285385c8678748928b593e
94d
dee
        by case: modn => // [] [|[]].
df0
904
de9
      rewrite (divn_eq k 3) k3mH2 addn2 !apegS mulnC apegMr //.
df0
954
de9
        by rewrite /i !inordK ?apegS //= !ltnS (leq_trans k3La).
de8
907
d3c
      have H2 : (k %/ 3) < l.+2 by rewrite !ltnS (leq_trans k3La).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3784885285385c8678748928b58c28ef93e95c
907
d3c
      rewrite -(@aMin1 i); last by rewrite !inordK // -{2}[a]prednK .
e01
91a
d3c
      congr (u _ _).
e01
924
d3c
      by apply/val_eqP; rewrite /= !inordK.
d3e
843
b24
  have {cH2}P2 := IH p1Dp2 p1Dp3 p2Dp3 p1Dp0 p2Dp0 p3Dp0 cH2.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3796a
843
b24
  have a3B2_gt1 : 1 < (3 * a).-2.
e0e
96e
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3796a972
843
b25
    rewrite -subn2 leq_subRL.
e0e
976
e0e
979
e13b25
      by rewrite (leq_trans (_ : 4 <= 3 * 2)) // leq_mul2l.
e0e
979
e12
    by rewrite (leq_trans (_ : 2 <= 3 * 2)) // leq_mul2l.
e14
843
b24
  rewrite u10E uiME apeg0 in P2.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed37972984
843
b24
  have {}P2 := leq_trans P2 (leq_add (leq_add (leqnn _) (leq_sum_beta _ _))
                          (leqnn _)).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed37972989
843
b24
  rewrite -/x1P1 in P2.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed379729a7
843
b24
  pose paS := a[p1, p3] a.+1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed379729a798f
843
b24
  have cH3 (k : 'I_4) :
    0 < k < 3 -> codom (u2 k) \subset [:: pa; a[p2, paS] k].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed379729a798f
k:'I_4
0 < k < 3 ->
codom (u2 k) \subset [:: pa;  a[p2, paS] k]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed379729a798f
cH3:forall k : 'I_4,
0 < k < 3 ->
codom (u2 k) \subset [:: pa;  a[p2, paS] k]
843
b25
    case: k => [] [|[|[|]]] //= iH _; rewrite ffunE /=.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed379729a798f
iH:1 < 4
codom tam1 \subset [:: pa;  a[p2, paS] 1]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed379729a798f
iH:2 < 4
codom ↓[u ai] \subset [:: pa;  a[p2, paS] 2]
e37b25
      by have := tam1C; rewrite /pa /paS !(apegS, apeg0) codom_apeg.
e42
e44
e36
    apply: codom_liftr.
e42
codom (u ai) \subset [:: pa;  a[p2, paS] 2]
e36
    have /cH : 0 < ai < l.+1 by rewrite aiE a_gt0 // lEa /=.
e42
codom (u ai) \subset [:: p2;  a[p1, p3] ai] ->
codom (u ai) \subset [:: pa;  a[p2, paS] 2]
e36
    by rewrite /pa /paS !(apegS, apeg0) aiE codom_subC.
e38
843
b24
  have {cH3}P3 : 
    (S_[3] n.+1).*2 <= sd u2 + sp (u2 ord0) 3 (a[p2, paS] 0) +
                       sp (u2 ord_max) 3 (a[p2, paS] 3).
e38
(S_[3] n.+1).*2 <=
sd (n:=n.+1) u2 +
sp (n:=n.+1) (u2 ord0) 3 ( a[p2, paS] 0) +
sp (n:=n.+1) (u2 ord_max) 3 ( a[p2, paS] 3)
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed379729a798f
P3:(S_[3] n.+1).*2 <=
sd (n:=n.+1) u2 +
sp (n:=n.+1) (u2 ord0) 3 ( a[p2, paS] 0) +
sp (n:=n.+1) (u2 ord_max) 3 ( a[p2, paS] 3)
843
b25
    apply: IH cH3 => //; try by rewrite /pa /paS // eq_sym.
e2f
9fa
e57
    by rewrite /paS apegS apeg_eqC.
e59
843
b24
  have {}P3 := leq_trans P3 (leq_add (leqnn _) (leq_sum_beta _ _)).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed379729a798f
P3:(S_[3] n.+1).*2 <=
sd (n:=n.+1) u2 +
sp (n:=n.+1) (u2 ord0) 3 ( a[p2, paS] 0) +
\sum_(k < n.+1)
   (u2 ord_max k !=  a[p2, paS] 3) * (α_[1] k).*2
843
b24
  rewrite /pa /paS -lEa !(apegS, apeg0) in P3.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed379729a798f
P3:(S_[3] n.+1).*2 <=
sd (n:=n.+1) u2 + sp (n:=n.+1) (u2 ord0) 3 p2 +
\sum_(k < n.+1)
   (u2 ord_max k !=  a[p3, p1] l) * (α_[1] k).*2
843
b24
  rewrite !ffunE /= -/x2P1 (_ : inord l.+1 = ord_max) in P3; last first.
e2f
a16
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed379729a798f
P3:(S_[3] n.+1).*2 <=
x2P1 + sp (n:=n.+1) sam1 3 p2 +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) *
   (α_[1] k).*2
843
b25
    by apply/val_eqP; rewrite /= inordK.
e70
843
b24
  rewrite [X in _ <= _ + X + _]sum_beta_S //= KH1 eqxx addn0.
e70
a21
b24
  set xS := \sum_(_ < _) _ in P2; rewrite -/xS.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3797298fe71a26a27
a28
b24
  rewrite [X in _ <= _ + _ + X]sum_beta_S //= KH2 mul1n.
e7b
a2c
b24
  set yS := \sum_(i < _) _ in P3; rewrite -/yS.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3797298fa26a27
yS:=\sum_(i < n.+1)
   (↓[u ord_max] i !=  a[p3, p1] l) * (α_[1] i).*2:nat
P3:(S_[3] n.+1).*2 <=
x2P1 + sp (n:=n.+1) sam1 3 p2 + yS
a33
b24
  set x1S := sp _ _  _ in P2; set y1S := sp _ _ _ in P3.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3797298fa26e83a38a39
y1S:=sp (n:=n.+1) sam1 3 p2:nat
P3:(S_[3] n.+1).*2 <= x2P1 + y1S + yS
a33
b24
  have x1Sy1SE : x1S + y1S <= (S_[1] n).*2 + α_[maxn (3 * a).-2 3] n.
e88
x1S + y1S <= (S_[1] n).*2 + α_[maxn (3 * a).-2 3] n
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3797298fa26e83a38a39e89e8a
x1Sy1SE:x1S + y1S <=
(S_[1] n).*2 + α_[maxn (3 * a).-2 3] n
a33
b25
    apply: sum_alpha_diffE => //.
e88
a47
e8f
    by move: sam1C; rewrite -uiME uaiLEp2 /pa apeg32E.
e91
a33
b24
  rewrite (maxn_idPl _) // in x1Sy1SE; last first.
e88
2 < (3 * a).-2
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3797298fa26e83a38a39e89e8aa53
a33
b25
    by rewrite -subn2 ltn_subRL ltnW // (leq_mul2l 3 2).
e9f
a33
b24
  rewrite -addnn {1}dsum_alphaL_S in P2.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3797298fa26e83a38e89e8aa53a73
a33
b24
  rewrite -addnn {1}dsum_alphaL_S.
ea6
a7c
b24
  have F4 k : S_[a.-1] k.+1 <= S_[(3 *a).-2] k + a.-1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3797298fa26e83a38e89e8aa53a73a81
a82
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3797298fa26e83a38e89e8aa53a73a86
a7c
b25
    apply: leq_trans (dsum_alpha3l _ _) _.
ead
a8a
eae
    rewrite leq_add2r.
ead
a8e
eae
    apply: (increasingE (increasing_dsum_alphaL_l _)).
ead
74e
eae
    by rewrite -{2}[a]prednK // mulnS addSn add2n /=.
eb0
a7c
b24
  have F4n := F4 n.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3797298fa26e83a38e89e8aa53a73a86a99
a7c
b24
  have {F4}F4n1 := F4 n.+1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3797298fa26e83a38e89e8aa53a73a99a9e
a7c
b24
  have F6 k : S_[a.+1] k + S_[1] k <= S_[a.-1] k + S_[3] k.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3797298fa26e83a38e89e8aa53a73a99a9ea81
S_[a.+1] k + S_[1] k <= S_[a.-1] k + S_[3] k
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3797298fa26e83a38e89e8aa53a73a99a9e
F6:forall k : nat,
S_[a.+1] k + S_[1] k <= S_[a.-1] k + S_[3] k
a7c
b25
    have := concaveEk1 1 a.-2 2 (concave_dsum_alphaL_l k).
ec8
S_[1 + a.-2 + 2] k + S_[1] k <=
S_[1 + 2] k + S_[1 + a.-2] k ->
S_[a.+1] k + S_[1] k <= S_[a.-1] k + S_[3] k
eca
    by rewrite add1n addn2 !prednK // [_ + S_[3] _]addnC.
ecc
a7c
b24
  have F6n := F6 n.+1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3797298fa26e83a38e89e8aa53a73a99a9eecd
F6n:S_[a.+1] n.+1 + S_[1] n.+1 <=
S_[a.-1] n.+1 + S_[3] n.+1
a7c
b24
  have {F6}F6n1 := F6 n.+2.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3797298fa26e83a38e89e8aa53a73a99a9eed9
F6n1:S_[a.+1] n.+2 + S_[1] n.+2 <=
S_[a.-1] n.+2 + S_[3] n.+2
a7c
b24
  have F72 := dsum_alphaL_S 3 n.+1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3797298fa26e83a38e89e8aa53a73a99a9eed9ede
F72:S_[3] n.+2 = S_[3] n.+1 + α_[3] n.+1
a7c
b24
  have F721 := dsum_alphaL_S 3 n.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3797298fa26e83a38e89e8aa53a73a99a9eed9edeee3
F721:S_[3] n.+1 = S_[3] n + α_[3] n
a7c
b24
  have F711 := dsum_alphaL_S 1 n.+1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3797298fa26e83a38e89e8aa53a73a99a9eed9edeee3ee8ae9
a7c
b24
  have F71 := dsum_alphaL_S 1 n.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3797298fa26e83a38e89e8aa53a73a99a9eed9edeee3ee8ae9aee
a7c
b24 
  have F8 : α_[1] n.+2 = α_[3] n.+1 by rewrite alphaL_3E.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3797298fa26e83a38e89e8aa53a73a99a9eed9edeee3ee8ae9aee
F8:α_[1] n.+2 = α_[3] n.+1
a7c
b24
  have F9 : α_[1] (n.+2) <= (α_[1] n).*2.+2.
ef4
b06
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e5c3b23b2bb335d95eb5f05f998eb8cb91b9963f66a67b683692bea83683783ed3797298fa26e83a38e89e8aa53a73a99a9eed9edeee3ee8ae9aeeef5b0a
a7c
b25
    apply: leq_trans (leqnSn _).
ef4
b0e
ef9
    by apply: alphaL_4_5.
efb
a7c
b24
  by subst l; lia.
b1d
b26
rewrite negb_and -leqNgt => o1H.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1e
o1H:(a <= 1) || (b != 1)
573
have [/eqP aE1| aD1] := boolP (a == 1).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09
aE1:a = 1
573
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09
aD1:a != 1
573
  have b_gt0 : 0 < b by rewrite /b aE1 subn1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0e
b_gt0:0 < b
573
f0f
  have bE : b = l by rewrite /b aE1 subn1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef17
bE:b = l
573
f0f
  move: b_gt0; rewrite leq_eqVlt => /orP[/eqP bE1 | b_gt1]; last first.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1
573
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c
bE1:1 = b
573
f10
    have u1C : u (inord 1) ldisk = p2.
f20
u (inord 1) ldisk = p2
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1
u1C:u (inord 1) ldisk = p2
573
f22f10
      move: aH; rewrite aE1 apegS apeg0.
51295565571155815161712131455955a56156856f57057157257958122858a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1
u (inord 1) ord_max != p3 -> u (inord 1) ldisk = p2
f29 
      have /cH : 0 < (inord 1 : 'I_l.+2) < l.+1 by rewrite inordK.
f30
codom (u (inord 1)) \subset
[:: p2;  a[p1, p3] (inord 1)] ->
u (inord 1) ord_max != p3 -> u (inord 1) ldisk = p2
f29
      move=> /subsetP/(_ (u (inord 1) ldisk) (codom_f _ _)).
f30
u (inord 1) ldisk \in [:: p2;  a[p1, p3] (inord 1)] ->
u (inord 1) ord_max != p3 -> u (inord 1) ldisk = p2
f29
      by rewrite !inE => /orP[] /eqP-> //; rewrite !inordK // apegS apeg0 eqxx.
f2b
573
f21
    pose s0 : configuration 4 n.+1 :=  sgdist1 p3 (u ord0) (u (inord 1)).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2c
s0:=sgdist1 (n:=n.+1) p3 (u ord0) (u (inord 1))
  : configuration 4 n.+1:configuration 4 n.+1
573
f21
    pose t0 : configuration 4 n.+1 :=  sgdist2 p3 (u ord0) (u (inord 1)).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41
t0:=sgdist2 (n:=n.+1) p3 (u ord0) (u (inord 1))
  : configuration 4 n.+1:configuration 4 n.+1
573
f21
    have [s0C t0C ds0t0E] :
        [/\ codom s0 \subset [:: p3; u (inord 1) ldisk ], 
         codom t0 \subset [:: u ord0 ldisk; p3] & 
         `d[u ord0, u (inord 1)]_smove = 
           D[[:: ↓[u ord0]; s0; t0; ↓[u (inord 1)]]].+2].
f45
[/\ codom s0 \subset [:: p3; u (inord 1) ldisk],
    codom t0 \subset [:: u ord0 ldisk; p3]
  & `d[u ord0, u (inord 1)]_smove =
     D[[:: ↓[u ord0]; s0; t0; ↓[u (inord 1)]]].+2]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46
s0C:codom s0 \subset [:: p3; u (inord 1) ldisk]
t0C:codom t0 \subset [:: u ord0 ldisk; p3]
ds0t0E:`d[u ord0, u (inord 1)]_smove =
 D[[:: ↓[u ord0]; s0; t0; ↓[u (inord 1)]]].+2
573
f22f10
      apply: sgdistE => //.
f45
u ord0 ldisk != p3
f45
u ord0 ldisk != u (inord 1) ldisk
f45
p3 != u (inord 1) ldisk
f45
u ord0 ldisk != p0
f45
u (inord 1) ldisk != p0
f4cf22f10
      -
f52 by rewrite  /= KH1.
f45
f57
f58f5af5cf4cf22f10
      -
f60 by rewrite KH1 u1C.
f45
f59
f5af5cf4cf22f10
      -
f65 by rewrite u1C eq_sym.
f45
f5b
f5cf4cf22f10
      -
f6a by rewrite KH1.
f45
f5d
f4b
      by rewrite u1C.
f4d
573
f21
    pose u1 := 
      [ffun i : 'I_4 =>  
      if i == 0 :> nat then ↓[u ord0] else
      if i == 1 :> nat then s0 else
      if i == 2 :> nat then t0 else ↓[u (inord 1)]].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50
u1:=[ffun i => if i == 0 :> nat
           then ↓[u ord0]
           else
            if i == 1 :> nat
            then s0
            else
             if i == 2 :> nat
             then t0
             else ↓[u (inord 1)]]:{ffun forall x : ordinal_finType 4,
      (fun=> configuration 4 n.+1) x}
573
f21
    pose u2 := [ffun i => ↓[u (inord ((i : 'I_b.+1) + 1))]].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78
u2:=[ffun i => ↓[u (inord ((i : 'I_b.+1) + 1))]]:{ffun forall x : ordinal_finType b.+1,
      (fun=> configuration 4 n.+1) x}
573
f21
    have P1 : 2 + sd u1 + sd u2 <= sd u.
f7c
2 + sd (n:=n.+1) u1 + sd (n:=n.+1) u2 <=
sd (n:=n.+2) u
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d
P1:2 + sd (n:=n.+1) u1 + sd (n:=n.+1) u2 <=
sd (n:=n.+2) u
573
f22f10
      have -> : sd u = 
          `d[u ord0, u (inord 1)]_smove +
          \sum_(i < b) `d[u (inord (a +i)), u (inord (a + i.+1))]_smove.
f7c
sd (n:=n.+2) u =
`d[u ord0, u (inord 1)]_smove +
\sum_(i < b)
   `d[u (inord (a + i)), u (inord (a + i.+1))]_smove
f7c
2 + sd (n:=n.+1) u1 + sd (n:=n.+1) u2 <=
`d[u ord0, u (inord 1)]_smove +
\sum_(i < b)
   `d[u (inord (a + i)), 
u (inord (a + i.+1))]_smove
f83f22f10
        rewrite /sd big_ord_recl /= bE.
f7c
`d[u (inord 0), u (inord 1)]_smove +
\sum_(i < l)
   `d[u (inord (bump 0 i)), u (inord (bump 0 i).+1)]_smove =
`d[u ord0, u (inord 1)]_smove +
\sum_(i < l)
   `d[u (inord (a + i)), u (inord (a + i.+1))]_smove
f8a
        congr (`d[u _,u _]_smove + _).
f7c
inord 0 = ord0
f7c
\sum_(i < l)
   `d[u (inord (bump 0 i)), 
u (inord (bump 0 i).+1)]_smove =
\sum_(i < l)
   `d[u (inord (a + i)), 
u (inord (a + i.+1))]_smove
f8bf83f22f10
          by apply/val_eqP; rewrite /= inordK.
f7c
\sum_(i < l)
   `d[u (inord (bump 0 i)), u (inord (bump 0 i).+1)]_smove =
\sum_(i < l)
   `d[u (inord (a + i)), u (inord (a + i.+1))]_smove
f8a
        apply: eq_bigr => i _.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d
i:'I_l
`d[u (inord (bump 0 i)), u (inord (bump 0 i).+1)]_smove =
`d[u (inord (a + i)), u (inord (a + i.+1))]_smove
f8a
        congr (`d[u _,u _]_smove); apply/val_eqP; rewrite aE1 /= inordK //=.
f9f
bump 0 i < l.+2
f9f
(bump 0 i).+1 < l.+2
f8bf83f22f10
          by rewrite /bump add1n !ltnS ltnW.
f9f
fa8
f8a
        by rewrite /bump add1n !ltnS.
f7c
2 + sd (n:=n.+1) u1 + sd (n:=n.+1) u2 <=
`d[u ord0, u (inord 1)]_smove +
\sum_(i < b)
   `d[u (inord (a + i)), u (inord (a + i.+1))]_smove
f82
      apply: leq_add.
f7c
2 + sd (n:=n.+1) u1 <= `d[u ord0, u (inord 1)]_smove
f7c
sd (n:=n.+1) u2 <=
\sum_(i < b)
   `d[u (inord (a + i)), 
u (inord (a + i.+1))]_smove
f83f22f10
        rewrite leq_eqVlt; apply/orP; left; apply/eqP.
f7c
2 + sd (n:=n.+1) u1 = `d[u ord0, u (inord 1)]_smove
fb4
        rewrite /sd !big_ord_recr /= big_ord0 add0n.
f7c
2 +
(`d[u1 (inord 0), u1 (inord 1)]_smove +
 `d[u1 (inord 1), u1 (inord 2)]_smove +
 `d[u1 (inord 2), u1 (inord 3)]_smove) =
`d[u ord0, u (inord 1)]_smove
fb4
        rewrite !ffunE !inordK //= add2n.
f7c
(`d[↓[u ord0], s0]_smove + `d[s0, t0]_smove +
 `d[t0, ↓[u (inord 1)]]_smove).+2 =
`d[u ord0, u (inord 1)]_smove
fb4
        by rewrite ds0t0E /= addn0 !addnA.
f7c
sd (n:=n.+1) u2 <=
\sum_(i < b)
   `d[u (inord (a + i)), u (inord (a + i.+1))]_smove
f82
      apply: leq_sum => i _.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d821
`d[u2 (inord i), u2 (inord i.+1)]_smove <=
`d[u (inord (a + i)), u (inord (a + i.+1))]_smove
f82
      rewrite ffunE inordK; last by rewrite ltnS ltnW.
fca
`d[↓[u (inord (i + 1))], u2 (inord i.+1)]_smove <=
`d[u (inord (a + i)), u (inord (a + i.+1))]_smove
f82
      rewrite ffunE inordK; last by rewrite ltnS.
fca
`d[↓[u (inord (i + 1))], ↓[u (inord (i.+1 + 1))]]_smove <=
`d[u (inord (a + i)), u (inord (a + i.+1))]_smove
f82
      rewrite aE1 addSn [i + _]addnC addnS.
fca
`d[↓[u (inord (1 + i))], ↓[u (inord (1 + i).+1)]]_smove <=
`d[u (inord (1 + i)), u (inord (1 + i).+1)]_smove
f82
      by apply/gdist_cunlift/shanoi_connect.
f84
573
f21
    set x1P1 := sd _ in P1; set x2P1 := sd _ in P1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d836837
P1:2 + x1P1 + x2P1 <= sd (n:=n.+2) u
573
f21
    set xP1 := sd _ in P1; rewrite -/xP1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d83683783e
P1:2 + x1P1 + x2P1 <= xP1
843
f21
    have cH2 (k : 'I_4) :
      0 < k < 3 -> codom (u1 k) \subset [:: p3; a[p1, p2] k].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d83683783efe4e34
0 < k < 3 ->
codom (u1 k) \subset [:: p3;  a[p1, p2] k]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d83683783efe4
cH2:forall k : 'I_4,
0 < k < 3 ->
codom (u1 k) \subset [:: p3;  a[p1, p2] k]
843
f22f10 
      case: k => [] [|[|[|]]] //= iH _; rewrite ffunE /= !(apegS, apeg0).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d83683783efe4e3e
codom s0 \subset [:: p3; p2]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d83683783efe4e43
codom t0 \subset [:: p3; p1]
febf22f10
        by have := s0C; rewrite u1C.
ff5
ff6
fea
      by rewrite codom_subC -KH1.
fec
843
f21
    have {cH2}P2 : 
      (S_[3] n.+1).*2 <= sd u1 + sp ↓[u (inord 0)] 3 (a[p1, p2] 0) +
                         sp ↓[u (inord 1)] 3 (a[p1, p2] 3).
fec
(S_[3] n.+1).*2 <=
sd (n:=n.+1) u1 +
sp (n:=n.+1) ↓[u (inord 0)] 3 ( a[p1, p2] 0) +
sp (n:=n.+1) ↓[u (inord 1)] 3 ( a[p1, p2] 3)
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d83683783efe4
P2:(S_[3] n.+1).*2 <=
sd (n:=n.+1) u1 +
sp (n:=n.+1) ↓[u (inord 0)] 3 ( a[p1, p2] 0) +
sp (n:=n.+1) ↓[u (inord 1)] 3 ( a[p1, p2] 3)
843
f22f10
      apply: leq_trans (IH _ _ _ _ _ _ cH2) _ => //.
fec
p3 != p2
fec
sd (n:=n.+1) u1 +
sp (n:=n.+1) (u1 ord0) 3 ( a[p1, p2] 0) +
sp (n:=n.+1) (u1 ord_max) 3 ( a[p1, p2] 3) <=
sd (n:=n.+1) u1 +
sp (n:=n.+1) ↓[u (inord 0)] 3 ( a[p1, p2] 0) +
sp (n:=n.+1) ↓[u (inord 1)] 3 ( a[p1, p2] 3)
1002f22f10
        by rewrite eq_sym.
fec
100b
1001
      repeat apply: leq_add => //; rewrite leq_eqVlt;
             apply/orP; left; apply/eqP.
fec
sp (n:=n.+1) (u1 ord0) 3 ( a[p1, p2] 0) =
sp (n:=n.+1) ↓[u (inord 0)] 3 ( a[p1, p2] 0)
fec
sp (n:=n.+1) (u1 ord_max) 3 ( a[p1, p2] 3) =
sp (n:=n.+1) ↓[u (inord 1)] 3 ( a[p1, p2] 3)
1002f22f10
        by apply: eq_bigr => i _; rewrite !ffunE inord_eq0.
fec
1015
1001
      by apply: eq_bigr => i _; rewrite !ffunE.
1003
843
f21
    have {}P2 := leq_trans P2 
               (leq_add (leq_add (leqnn _) (leq_sum_beta _ _)) (leqnn _)).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d83683783efe4
P2:(S_[3] n.+1).*2 <=
sd (n:=n.+1) u1 +
\sum_(k < n.+1)
   (↓[u (inord 0)] k !=  a[p1, p2] 0) *
   (α_[1] k).*2 +
sp (n:=n.+1) ↓[u (inord 1)] 3 ( a[p1, p2] 3)
843
f21
    rewrite -/x2P1 (inord_eq0) // -[in inord 1]aE1 !(apegS, apeg0) in P2.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d83683783efe4
P2:(S_[3] n.+1).*2 <=
sd (n:=n.+1) u1 +
\sum_(k < n.+1)
   (↓[u ord0] k != p1) * (α_[1] k).*2 +
sp (n:=n.+1) ↓[u (inord a)] 3 p2
843
f21
    have cH3 (k : 'I_b.+1) :
      0 < k < b -> codom (u2 k) \subset [:: p2; a[p3, p1] k].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d83683783efe410259c1
0 < k < b ->
codom (u2 k) \subset [:: p2;  a[p3, p1] k]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d83683783efe41025
cH3:forall k : 'I_b.+1,
0 < k < b ->
codom (u2 k) \subset [:: p2;  a[p3, p1] k]
843
f22f10
      rewrite [X in k < X]bE => /andP[k_gt0 kLb].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d83683783efe410259c1852
kLb:k < l
codom (u2 k) \subset [:: p2;  a[p3, p1] k]
102b
      rewrite ffunE addn1; apply: codom_liftr.
1032
codom (u (inord k.+1)) \subset [:: p2;  a[p3, p1] k]
102b
      have := @cH (inord k.+1).
1032
(0 < inord k.+1 < l.+1 ->
 codom (u (inord k.+1)) \subset
 [:: p2;  a[p1, p3] (inord k.+1)]) ->
codom (u (inord k.+1)) \subset [:: p2;  a[p3, p1] k]
102b
      rewrite inordK ?apegS //=; first by apply.
1032
k.+1 < l.+2
102b
      by rewrite ltnS (leq_trans kLb).
102d
843
f21
    have {cH3}P3 : 
     (S_[b] n.+1).*2 <= sd u2 + sp ↓[u (inord 1)] b p3 + 
                        sp ↓[u ord_max] b (a[p3, p1] b).
102d
(S_[b] n.+1).*2 <=
sd (n:=n.+1) u2 + sp (n:=n.+1) ↓[u (inord 1)] b p3 +
sp (n:=n.+1) ↓[u ord_max] b ( a[p3, p1] b)
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d83683783efe41025
P3:(S_[b] n.+1).*2 <=
sd (n:=n.+1) u2 +
sp (n:=n.+1) ↓[u (inord 1)] b p3 +
sp (n:=n.+1) ↓[u ord_max] b ( a[p3, p1] b)
843
f22f10
      apply: leq_trans (IH _ _ _ _ _ _ cH3) _ => //; try by rewrite eq_sym.
102d
sd (n:=n.+1) u2 +
sp (n:=n.+1) (u2 ord0) b ( a[p3, p1] 0) +
sp (n:=n.+1) (u2 ord_max) b ( a[p3, p1] b) <=
sd (n:=n.+1) u2 + sp (n:=n.+1) ↓[u (inord 1)] b p3 +
sp (n:=n.+1) ↓[u ord_max] b ( a[p3, p1] b)
1048
      repeat apply: leq_add => //; rewrite leq_eqVlt; 
         apply/orP; left; apply/eqP; apply: eq_bigr => i _.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d83683783efe41025102e
i:ordinal_finType n.+1
(u2 ord0 i !=  a[p3, p1] 0) * β_[n.+1, b] i =
(↓[u (inord 1)] i != p3) * β_[n.+1, b] i
1053
(u2 ord_max i !=  a[p3, p1] b) * β_[n.+1, b] i =
(↓[u ord_max] i !=  a[p3, p1] b) * β_[n.+1, b] i
1049f22f10
        by rewrite !ffunE.
1053
1058
1048
      rewrite !ffunE /= addn1 bE; congr ((u _ _ != _) * _).
1053
a16
1048
      by apply/val_eqP; rewrite /= inordK.
104a
843
f21
    have {}P3 := leq_trans P3 (leq_add (leqnn _) (leq_sum_beta _ _)).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d83683783efe41025
P3:(S_[b] n.+1).*2 <=
sd (n:=n.+1) u2 +
sp (n:=n.+1) ↓[u (inord 1)] b p3 +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] b) *
   (α_[1] k).*2
843
f21
    rewrite -/x2P1 (_ : a[p3, p1] b = a[p3, p1] l) in P3; last first.
1024
 a[p3, p1] b =  a[p3, p1] l
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d83683783efe41025
P3:(S_[b] n.+1).*2 <=
x2P1 + sp (n:=n.+1) ↓[u (inord 1)] b p3 +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) *
   (α_[1] k).*2
843
f22f10
      by rewrite bE.
106d
843
f21
    rewrite -[in inord 1]aE1 in P3.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d83683783efe41025
P3:(S_[b] n.+1).*2 <=
x2P1 + sp (n:=n.+1) ↓[u (inord a)] b p3 +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) *
   (α_[1] k).*2
843
f21
    rewrite !sum_beta_S // KH1 eqxx mul0n addn0 KH2 mul1n.
1075
(S_[l.+1] n.+2).*2 <=
xP1 +
\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
(\sum_(k < n.+1)
    (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2 +
 α_[l.+1] n.+1)
f21
    set xS := \sum_(i < _) _ in P2; rewrite -/xS.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d83683783efe41076
xS:=\sum_(i < n.+1) (↓[u ord0] i != p1) * (α_[1] i).*2:nat
P2:(S_[3] n.+1).*2 <=
sd (n:=n.+1) u1 + xS +
sp (n:=n.+1) ↓[u (inord a)] 3 p2
a2c
f21
    set yS := \sum_(i < _) _ in P3; rewrite -/yS.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d83683783efe4107f1080e83
P3:(S_[b] n.+1).*2 <=
x2P1 + sp (n:=n.+1) ↓[u (inord a)] b p3 + yS
a33
f21
    set x1S := sp _ _ _ in P2; set y1S := sp _ _ _ in P3.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d83683783efe4107fe83
x1S:=sp (n:=n.+1) ↓[u (inord a)] 3 p2:nat
P2:(S_[3] n.+1).*2 <= sd (n:=n.+1) u1 + xS + x1S
y1S:=sp (n:=n.+1) ↓[u (inord a)] b p3:nat
P3:(S_[b] n.+1).*2 <= x2P1 + y1S + yS
a33
f21
    have x1Sy1SE : x1S + y1S <= (S_[1] n).*2 + α_[maxn 3 b] n.
1089
x1S + y1S <= (S_[1] n).*2 + α_[maxn 3 b] n
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d83683783efe4107fe83108a108b108c108d
x1Sy1SE:x1S + y1S <= (S_[1] n).*2 + α_[maxn 3 b] n
a33
f22f10
      apply: sum_alpha_diffE => //.
1089
codom ↓[u (inord a)] \subset [:: p2; p3]
1092
      apply: codom_liftr.
1089
codom (u (inord a)) \subset [:: p2; p3]
1092
      have /cH : 0 < (inord a : 'I_l.+2) < l.+1 by rewrite aE1 inordK.
1089
codom (u (inord a)) \subset
[:: p2;  a[p1, p3] (inord a)] ->
codom (u (inord a)) \subset [:: p2; p3]
1092
      by rewrite inordK aE1.
1094
a33
f21
    rewrite -/x1P1 in P2.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1c5d1f2cf41f46f4ef4ff50f78f7d83683783efe4107fe83108a108c108d1095
P2:(S_[3] n.+1).*2 <= x1P1 + xS + x1S
a33
f21
    rewrite -addnn {1}dsum_alphaL_S -bE.
10a8
S_[b.+1] n.+1 + α_[b.+1] n.+1 + S_[b.+1] n.+2 <=
xP1 + xS + (yS + α_[b.+1] n.+1)
f21
    suff :
      S_[b.+1] n.+1 + S_[b.+1] n.+2 + 2 * S_[1] n + α_[maxn 3 b] n <=
      2 * S_[3] n.+1 + 2 * S_[b] n.+1 + 2.
10a8
S_[b.+1] n.+1 + S_[b.+1] n.+2 + 2 * S_[1] n +
α_[maxn 3 b] n <= 2 * S_[3] n.+1 + 2 * S_[b] n.+1 + 2 ->
S_[b.+1] n.+1 + α_[b.+1] n.+1 + S_[b.+1] n.+2 <=
xP1 + xS + (yS + α_[b.+1] n.+1)
10a8
S_[b.+1] n.+1 + S_[b.+1] n.+2 + 2 * S_[1] n +
α_[maxn 3 b] n <= 2 * S_[3] n.+1 + 2 * S_[b] n.+1 + 2
f22f10
      by lia.
10a8
10b4
f21
    move: b_gt1; rewrite leq_eqVlt => /orP[/eqP bE2|b_gt2].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf2cf41f46f4ef4ff50f78f7d83683783efe4107fe83108a108c108d109510a9
bE2:2 = b
10b4
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf2cf41f46f4ef4ff50f78f7d83683783efe4107fe83108a108c108d109510a9
b_gt2:2 < b
10b4
f22f10
      rewrite -bE2 (maxn_idPl _) //.
10bb
S_[3] n.+1 + S_[3] n.+2 + 2 * S_[1] n + α_[3] n <=
2 * S_[3] n.+1 + 2 * S_[2] n.+1 + 2
10bd
      rewrite [S_[3] n.+2]dsum_alphaL_S.
10bb
S_[3] n.+1 + (S_[3] n.+1 + α_[3] n.+1) + 2 * S_[1] n +
α_[3] n <= 2 * S_[3] n.+1 + 2 * S_[2] n.+1 + 2
10bd
      have G1 := alphaL_3E n.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf2cf41f46f4ef4ff50f78f7d83683783efe4107fe83108a108c108d109510a910bc
G1:α_[3] n = α_[1] n.+1
10c8
10bd
      have G2 := alphaL_3E n.+1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf2cf41f46f4ef4ff50f78f7d83683783efe4107fe83108a108c108d109510a910bc10cd
G2:α_[3] n.+1 = α_[1] n.+2
10c8
10bd
      have F := leq_dsum_alpha_2l_1 n.+1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf2cf41f46f4ef4ff50f78f7d83683783efe4107fe83108a108c108d109510a910bc10cd10d2
F:S_[1] n.+1 + S_[1] n.+2 <= (S_[2] n.+1).*2.+1
10c8
10bd
      rewrite 3!dsum_alphaL_S in F *.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf2cf41f46f4ef4ff50f78f7d83683783efe4107fe83108a108c108d109510a910bc10cd10d2
F:S_[1] n + α_[1] n +
(S_[1] n + α_[1] n + α_[1] n.+1) <=
(S_[2] n.+1).*2.+1
10c8
10bd
      by have F1 := alphaL_4_5 n; lia.
10bf
10b4
f21
    rewrite (maxn_idPr _) //.
10bf
S_[b.+1] n.+1 + S_[b.+1] n.+2 + 2 * S_[1] n + α_[b] n <=
2 * S_[3] n.+1 + 2 * S_[b] n.+1 + 2
f21
    have A0 := dsum_alpha3_S n.+1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf2cf41f46f4ef4ff50f78f7d83683783efe4107fe83108a108c108d109510a910c0
A0:S_[1] n.+2 = (S_[3] n.+1).+1
10e3
f21
    have A1 := dsum_alphaL_S 1 n.+1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf2cf41f46f4ef4ff50f78f7d83683783efe4107fe83108a108c108d109510a910c010e8
A1:S_[1] n.+2 = S_[1] n.+1 + α_[1] n.+1
10e3
f21
    have A2 := dsum_alphaL_S 1 n.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf2cf41f46f4ef4ff50f78f7d83683783efe4107fe83108a108c108d109510a910c010e810ed
A2:S_[1] n.+1 = S_[1] n + α_[1] n
10e3
f21
    have A3 := dsum_alphaL_S b n.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf2cf41f46f4ef4ff50f78f7d83683783efe4107fe83108a108c108d109510a910c010e810ed10f2
A3:S_[b] n.+1 = S_[b] n + α_[b] n
10e3
f21
    have A4 := @dsum_alphaL_alpha b n.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf2cf41f46f4ef4ff50f78f7d83683783efe4107fe83108a108c108d109510a910c010e810ed10f210f7
A4:1 < b -> S_[b.+1] n.+1 <= S_[b] n + (α_[1] n).*2
10e3
f21
    have A5 := @dsum_alphaL_alpha b n.+1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf2cf41f46f4ef4ff50f78f7d83683783efe4107fe83108a108c108d109510a910c010e810ed10f210f710fc
A5:1 < b ->
S_[b.+1] n.+2 <= S_[b] n.+1 + (α_[1] n.+1).*2
10e3
f21
    by lia.
f23
573
f0f
  have /cH : 0 < (inord 1 : 'I_l.+2) < l.+1 by rewrite inordK.
f23
codom (u (inord 1)) \subset
[:: p2;  a[p1, p3] (inord 1)] ->
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u + sp (n:=n.+2) (u ord0) l.+1 p1 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
f0f
  move=> /subsetP /(_ (u (inord 1) ldisk) (codom_f _ _)).
f23
u (inord 1) ldisk \in [:: p2;  a[p1, p3] (inord 1)] ->
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u + sp (n:=n.+2) (u ord0) l.+1 p1 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
f0f
  rewrite !inE inordK // apegS apeg0.
f23
(u (inord 1) ldisk == p2) || (u (inord 1) ldisk == p3) ->
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u + sp (n:=n.+2) (u ord0) l.+1 p1 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
f0f
  have := aH; rewrite (_ : inord a = inord 1); last first.
f23
inord a = inord 1
f23
u (inord 1) ord_max !=  a[p1, p3] a ->
(u (inord 1) ldisk == p2) || (u (inord 1) ldisk == p3) ->
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u + sp (n:=n.+2) (u ord0) l.+1 p1 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
f10
    by apply/val_eqP; rewrite /= !inordK // aE1.
f23
1117
f0f
  rewrite aE1 apegS => /negPf->; rewrite orbF => /eqP u1LE.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf24
u1LE:u (inord 1) ldisk = p2
573
f0f 
  case: (@sgdistE _ p3 (u ord0) (u (inord 1))) => //.
111e
f54
111e
f57
111e
f59
111e
f5b
111e
f5d
111e
codom (sgdist1 (n:=n.+1) p3 (u ord0) (u (inord 1))) \subset
[:: p3; u (inord 1) ldisk] ->
codom (sgdist2 (n:=n.+1) p3 (u ord0) (u (inord 1))) \subset
[:: u ord0 ldisk; p3] ->
`d[u ord0, u (inord 1)]_smove =
 D[[:: ↓[u ord0];
       sgdist1 (n:=n.+1) p3 (u ord0) (u (inord 1));
       sgdist2 (n:=n.+1) p3 (u ord0) (u (inord 1));
       ↓[u (inord 1)]]].+2 ->
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u + sp (n:=n.+2) (u ord0) l.+1 p1 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
f10
  -
1121 by rewrite KH1.
111e
f57
1125112611271128f10
  -
112c by rewrite KH1 u1LE.
111e
f59
112611271128f10
  -
1131 by rewrite u1LE eq_sym.
111e
f5b
11271128f10
  -
1136 by rewrite KH1.
111e
f5d
1128f10
  by rewrite u1LE.
111e
1129
f0f
  set s0 := sgdist1 _ _ _; set t0 := sgdist2 _ _ _.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf24111f
s0:=sgdist1 (n:=n.+1) p3 (u ord0) (u (inord 1)):configuration 4 n.+1
t0:=sgdist2 (n:=n.+1) p3 (u ord0) (u (inord 1)):configuration 4 n.+1
codom s0 \subset [:: p3; u (inord 1) ldisk] ->
codom t0 \subset [:: u ord0 ldisk; p3] ->
`d[u ord0, u (inord 1)]_smove =
 D[[:: ↓[u ord0]; s0; t0; ↓[u (inord 1)]]].+2 ->
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u + sp (n:=n.+2) (u ord0) l.+1 p1 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
f0f
  move=> s0C t0C du0u1; rewrite /= !addnA addn0 in du0u1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf24111f11451146f4ef4f
du0u1:`d[u ord0, u (inord 1)]_smove =
(`d[↓[u ord0], s0]_smove + `d[s0, t0]_smove +
 `d[t0, ↓[u (inord 1)]]_smove).+2
573
f0f
  pose u1 := 
    [ffun i =>  
      if (i : 'I_5) == 0 :> nat then ↓[u ord0] else
      if i == 1 :> nat then s0 else
      if i == 2 :> nat then t0 else 
      if i == 3 :> nat then ↓[u (inord 1)] else ↓[u (inord 2)]].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf24111f11451146f4ef4f114c
u1:=[ffun i => if (i : 'I_5) == 0 :> nat
           then ↓[u ord0]
           else
            if i == 1 :> nat
            then s0
            else
             if i == 2 :> nat
             then t0
             else
              if i == 3 :> nat
              then ↓[u (inord 1)]
              else ↓[u (inord 2)]]:{ffun forall x : ordinal_finType 5,
      (fun=> configuration 4 n.+1) x}
573
f0f
  have P1 : 2 + sd u1 <= sd u.
1150
2 + sd (n:=n.+1) u1 <= sd (n:=n.+2) u
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf24111f11451146f4ef4f114c1151
P1:2 + sd (n:=n.+1) u1 <= sd (n:=n.+2) u
573
f10
    rewrite /sd 2![in X in X <= _]big_ord_recr 
                  [X in _ <= X]big_ord_recr /= !addnA.
1150
2 +
\sum_(i < 2) `d[u1 (inord i), u1 (inord i.+1)]_smove +
`d[u1 (inord 2), u1 (inord 3)]_smove +
`d[u1 (inord 3), u1 (inord 4)]_smove <=
\sum_(i < l) `d[u (inord i), u (inord i.+1)]_smove +
`d[u (inord l), u (inord l.+1)]_smove
1156
    apply: leq_add; last first.
1150
`d[u1 (inord 3), u1 (inord 4)]_smove <=
`d[u (inord l), u (inord l.+1)]_smove
1150
2 +
\sum_(i < 2) `d[u1 (inord i), u1 (inord i.+1)]_smove +
`d[u1 (inord 2), u1 (inord 3)]_smove <=
\sum_(i < l) `d[u (inord i), u (inord i.+1)]_smove
1157f10
      apply: leq_trans (gdist_cunlift _); last by apply: shanoi_connect.
1150
`d[u1 (inord 3), u1 (inord 4)]_smove <=
`d[↓[u (inord l)], ↓[u (inord l.+1)]]_(move (q:=4)
                                         (srel (n:=4)))
1162
      rewrite leq_eqVlt; apply/orP; left; apply/eqP.
1150
`d[u1 (inord 3), u1 (inord 4)]_smove =
`d[↓[u (inord l)], ↓[u (inord l.+1)]]_(move (q:=4)
                                         (srel (n:=4)))
1162
      rewrite !ffunE ![in LHS]inordK //=.
1150
`d[↓[u (inord 1)], ↓[u (inord 2)]]_smove =
`d[↓[u (inord l)], ↓[u (inord l.+1)]]_(move (q:=4)
                                         (srel (n:=4)))
1162
      by congr(`d[↓[u _],↓[u _]]_smove); 
         apply/val_eqP; rewrite /= !inordK //= -bE bE1.
1150
1164
1156
    rewrite leq_eqVlt; apply/orP; left; apply/eqP.
1150
2 +
\sum_(i < 2) `d[u1 (inord i), u1 (inord i.+1)]_smove +
`d[u1 (inord 2), u1 (inord 3)]_smove =
\sum_(i < l) `d[u (inord i), u (inord i.+1)]_smove
1156
    rewrite !big_ord_recr big_ord0 /= !ffunE !inordK //= add0n.
1150
2 + (`d[↓[u ord0], s0]_smove + `d[s0, t0]_smove) +
`d[t0, ↓[u (inord 1)]]_smove =
\sum_(i < l) `d[u (inord i), u (inord i.+1)]_smove
1156
    rewrite -addnA add2n -du0u1.
1150
`d[u ord0, u (inord 1)]_smove =
\sum_(i < l) `d[u (inord i), u (inord i.+1)]_smove
1156
    move: bE; rewrite -bE1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef24111f11451146f4ef4f114c1151
1 = l ->
`d[u ord0, u (inord 1)]_smove =
\sum_(i < l) `d[u (inord i), u (inord i.+1)]_smove
1156
    case: (l) u => // [] [] // uu _; rewrite big_ord_recl big_ord0 addn0 /=.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef24111f11451146f4ef4f114c1151
uu:{ffun 'I_3 -> configuration 4 n.+2}
`d[uu ord0, uu (inord 1)]_smove =
`d[uu (inord 0), uu (inord 1)]_smove
1156
    by congr(`d[uu _, uu _]_smove); apply/val_eqP; rewrite /= !inordK.
1158
573
f0f
  have cH1 (k : 'I_5) : 
    0 < k < 4 -> codom (u1 k) \subset [:: p3; a[p1, p2] k].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf24111f11451146f4ef4f114c11511159
k:'I_5
0 < k < 4 ->
codom (u1 k) \subset [:: p3;  a[p1, p2] k]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf24111f11451146f4ef4f114c11511159
cH1:forall k : 'I_5,
0 < k < 4 ->
codom (u1 k) \subset [:: p3;  a[p1, p2] k]
573
f10
    case: k => [] [|[|[|[|]]]] //= iH _; rewrite !ffunE !(apegS, apeg0).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf24111f11451146f4ef4f114c11511159
iH:1 < 5
codom
  (if Ordinal (n:=5) (m:=1) iH == 0
   then ↓[u ord0]
   else
    if Ordinal (n:=5) (m:=1) iH == 1
    then s0
    else
     if Ordinal (n:=5) (m:=1) iH == 2
     then t0
     else
      if Ordinal (n:=5) (m:=1) iH == 3
      then ↓[u (inord 1)]
      else ↓[u (inord 2)]) \subset [:: p3; p2]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf24111f11451146f4ef4f114c11511159
iH:2 < 5
codom
  (if Ordinal (n:=5) (m:=2) iH == 0
   then ↓[u ord0]
   else
    if Ordinal (n:=5) (m:=2) iH == 1
    then s0
    else
     if Ordinal (n:=5) (m:=2) iH == 2
     then t0
     else
      if Ordinal (n:=5) (m:=2) iH == 3
      then ↓[u (inord 1)]
      else ↓[u (inord 2)]) \subset [::
 p3; p1]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf24111f11451146f4ef4f114c11511159
iH:3 < 5
codom
  (if Ordinal (n:=5) (m:=3) iH == 0
   then ↓[u ord0]
   else
    if Ordinal (n:=5) (m:=3) iH == 1
    then s0
    else
     if Ordinal (n:=5) (m:=3) iH == 2
     then t0
     else
      if Ordinal (n:=5) (m:=3) iH == 3
      then ↓[u (inord 1)]
      else ↓[u (inord 2)]) \subset [::
 p3; p2]
1195f10
    -
1199 by have := s0C; rewrite u1LE.
11a0
codom
  (if Ordinal (n:=5) (m:=2) iH == 0
   then ↓[u ord0]
   else
    if Ordinal (n:=5) (m:=2) iH == 1
    then s0
    else
     if Ordinal (n:=5) (m:=2) iH == 2
     then t0
     else
      if Ordinal (n:=5) (m:=2) iH == 3
      then ↓[u (inord 1)]
      else ↓[u (inord 2)]) \subset [:: p3; p1]
11a31195f10
    -
11a9 by rewrite codom_subC -KH1.
11a4
codom
  (if Ordinal (n:=5) (m:=3) iH == 0
   then ↓[u ord0]
   else
    if Ordinal (n:=5) (m:=3) iH == 1
    then s0
    else
     if Ordinal (n:=5) (m:=3) iH == 2
     then t0
     else
      if Ordinal (n:=5) (m:=3) iH == 3
      then ↓[u (inord 1)]
      else ↓[u (inord 2)]) \subset [:: p3; p2]
1194
    rewrite codom_subC.
11a4
codom ↓[u (inord 1)] \subset [:: p2; p3]
1194
    apply/subsetP=> i /codomP[x ->]; rewrite !ffunE.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf24111f11451146f4ef4f114c1151115911a531e
x:ordinal_finType n.+1
u (inord 1) (trshift 1 x) \in [:: p2; p3]
1194
    have /cH: 0 < (inord 1 : 'I_l.+2) < l.+1 by rewrite inordK //.
11b9
codom (u (inord 1)) \subset
[:: p2;  a[p1, p3] (inord 1)] ->
u (inord 1) (trshift 1 x) \in [:: p2; p3]
1194
    by rewrite inordK // => 
        /subsetP /(_ (u (inord 1) (trshift 1 x)) (codom_f _ _)).
1196
573
f0f
  have {cH1}P2 :
     (S_[4] n.+1).*2 <= sd u1 + sp (u1 ord0) 4 p1 + sp (u1 ord_max) 4 p1.
1196
(S_[4] n.+1).*2 <=
sd (n:=n.+1) u1 + sp (n:=n.+1) (u1 ord0) 4 p1 +
sp (n:=n.+1) (u1 ord_max) 4 p1
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf24111f11451146f4ef4f114c11511159
P2:(S_[4] n.+1).*2 <=
sd (n:=n.+1) u1 + sp (n:=n.+1) (u1 ord0) 4 p1 +
sp (n:=n.+1) (u1 ord_max) 4 p1
573
f10
    by apply: IH cH1 => //; rewrite eq_sym.
11c9
573
f0f
  set xP := sd _; rewrite -/xP in P1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf24111f11451146f4ef4f114c115111ca
xP:=sd (n:=n.+2) u:nat
P1:2 + sd (n:=n.+1) u1 <= xP
(S_[l.+1] n.+2).*2 <=
xP + sp (n:=n.+2) (u ord0) l.+1 p1 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
f0f
  have {}P2 := 
    leq_trans P2 (leq_add (leq_add (leqnn _ ) (leq_sum_beta _ _))
                          (leq_sum_beta _ _)).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf24111f11451146f4ef4f114c115111d211d3
P2:(S_[4] n.+1).*2 <=
sd (n:=n.+1) u1 +
\sum_(k < n.+1) (u1 ord0 k != p1) * (α_[1] k).*2 +
\sum_(k < n.+1)
   (u1 ord_max k != p1) * (α_[1] k).*2
11d4
f0f
  set x1P := sd _ in P1; rewrite -/x1P in P2.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf24111f11451146f4ef4f114c115111d2
x1P:=sd (n:=n.+1) u1:nat
P1:2 + x1P <= xP
P2:(S_[4] n.+1).*2 <=
x1P +
\sum_(k < n.+1) (u1 ord0 k != p1) * (α_[1] k).*2 +
\sum_(k < n.+1)
   (u1 ord_max k != p1) * (α_[1] k).*2
11d4
f0f
  rewrite !ffunE /= (_ : inord 2 = ord_max) in P2; last first.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf24111f11451146f4ef4f114c115111d211de11df
inord 2 = ord_max
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf24111f11451146f4ef4f114c115111d211de11df
P2:(S_[4] n.+1).*2 <=
x1P +
\sum_(k < n.+1)
   (↓[u ord0] k != p1) * (α_[1] k).*2 +
\sum_(k < n.+1)
   (↓[u ord_max] k != p1) * (α_[1] k).*2
11d4
f10
    by apply: val_inj; rewrite /= inordK // -bE bE1.
11e8
11d4
f0f
  rewrite {2}(_ : p1 = a[p3, p1] l) in P2; last first.
11e4
p1 =  a[p3, p1] l
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf24111f11451146f4ef4f114c115111d211de11df
P2:(S_[4] n.+1).*2 <=
x1P +
\sum_(k < n.+1)
   (↓[u ord0] k != p1) * (α_[1] k).*2 +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) *
   (α_[1] k).*2
11d4
f10
    by rewrite -bE -bE1.
11f3
11d4
f0f
  set x1S := \sum_(_ < _) _ in P2; set x2S := \sum_(_ < _) _ in P2.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f0ef1cf24111f11451146f4ef4f114c115111d211de11df
x1S:=\sum_(H < n.+1)
   (↓[u ord0] H != p1) * (α_[1] H).*2:nat
x2S:=\sum_(H < n.+1)
   (↓[u ord_max] H !=  a[p3, p1] l) *
   (α_[1] H).*2:nat
P2:(S_[4] n.+1).*2 <= x1P + x1S + x2S
11d4
f0f
  rewrite !sum_beta_S //= KH1 KH2 !eqxx !addn0 mul1n.
11fb
(S_[l.+1] n.+2).*2 <=
xP +
\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
(\sum_(k < n.+1)
    (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2 +
 α_[l.+1] n.+1)
f0f
  rewrite -/x1S -/x2S -!addnA.
11fb
(S_[l.+1] n.+2).*2 <=
xP + (x1S + (x2S + α_[l.+1] n.+1))
f0f
  apply: leq_trans (leq_add P1 (leqnn _)).
11fb
(S_[l.+1] n.+2).*2 <=
2 + x1P + (x1S + (x2S + α_[l.+1] n.+1))
f0f
  rewrite -!addnA addnC !addnA -addnA.
11fb
(S_[l.+1] n.+2).*2 <=
x1P + x1S + x2S + (α_[l.+1] n.+1 + 2)
f0f
  apply: leq_trans (leq_add P2 (leqnn _)).
11fb
(S_[l.+1] n.+2).*2 <=
(S_[4] n.+1).*2 + (α_[l.+1] n.+1 + 2)
f0f
  rewrite -bE -bE1 -addnn {2}dsum_alphaL_S addnA.
11fb
S_[2] n.+2 + S_[2] n.+1 + α_[2] n.+1 <=
(S_[4] n.+1).*2 + (α_[2] n.+1 + 2)
f0f
  rewrite [_ + 2]addnC !addnA leq_add2r addnC.
11fb
S_[2] n.+1 + S_[2] n.+2 <= (S_[4] n.+1).*2 + 2
f0f
  by apply: leq_dsum_alpha_2l_l 2 n.+1.
f11
573
have aEl : a = l.+1.
f11
a = l.+1
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f12
aEl:a = l.+1
573
  apply/eqP; rewrite eqn_leq aLld1 /=.
f11
l < a
1222
  case/orP: oH; first by case: (a) aD1 a_gt0 => [|[|]].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95bef09f12
l.-1 < a -> l < a
1222
  case/orP: o1H; first by case: (a) aD1 a_gt0 => [|[|]].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95bef12
b != 1 -> l.-1 < a -> l < a
1222
  rewrite /b prednK // => H1 lLa; move: lLa H1.
1232
l <= a -> l.+1 - a != 1 -> l < a
1222
  by rewrite leq_eqVlt => /orP[] // /eqP ->; rewrite subSn // subnn eqxx.
1224
573
have bE0 : b = 0 by rewrite /b aEl subnn.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225
bE0:b = 0
573
pose si i : configuration 4 n.+1 := 
  sgdist1 p2 (u (inord i)) (u (inord i.+1)).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb
573
pose ti i : configuration 4 n.+1 := 
   sgdist2 p2 (u (inord i)) (u (inord i.+1)).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f0
573
have sitiH i : i < l ->
   [/\ codom (si i) \subset [:: p2; u (inord i.+1) ldisk ], 
       codom (ti i) \subset [:: u (inord i) ldisk; p2] & 
       `d[u (inord i), u (inord i.+1)]_smove = 
         D[[:: ↓[u (inord i)]; si i; ti i; ↓[u (inord i.+1)]]].+2].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f014a
i < l ->
[/\ codom (si i) \subset [:: p2; u (inord i.+1) ldisk],
    codom (ti i) \subset [:: u (inord i) ldisk; p2]
  & `d[u (inord i), u (inord i.+1)]_smove =
     D[[:: ↓[u (inord i)]; si i; ti i;
           ↓[u (inord i.+1)]]].+2]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f0
sitiH:forall i : nat,
i < l ->
[/\ codom (si i) \subset
    [:: p2; u (inord i.+1) ldisk],
    codom (ti i) \subset
    [:: u (inord i) ldisk; p2]
  & `d[u (inord i), u (inord i.+1)]_smove =
     D[[:: ↓[u (inord i)]; 
          si i; ti i; ↓[
          u (inord i.+1)]]].+2]
573
  move=> iLl.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f014a
iLl:i < l
5ff
124d
  have iLl1 : i < l.+1 by rewrite (leq_trans iLl).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f014a1255
iLl1:i < l.+1
5ff
124d
  apply: sgdistE => //.
1259
608
1259
60b
1259
60d
1259
60f
1259
611
124e
  -
125c by rewrite aMin1 // aEl.
1259
60b
126012611262124e
  -
1265 rewrite !aMin1 ?aEl //.
1259
61b
1267
    by rewrite apegS apeg_eqC.
1259
60d
12611262124e
  -
126d by rewrite aMin1 ?aEl // eq_sym.
1259
60f
1262124e
  -
1272 by rewrite aMin1 // aEl.
1259
611
124d
  by rewrite aMin1 // aEl.
124f
573
pose u1 := 
  [ffun i =>
    if ((i : 'I_(3 * l).+2) == (3 * l).+1 :>nat) then ↓[u ai] 
    else if (i %% 3) == 0 then ↓[u (inord (i %/ 3))]
    else if (i %% 3) == 1 then si (i %/ 3)
    else ti (i %/ 3)].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f01250
u1:=[ffun i => if
            (i : 'I_(3 * l).+2) == 
            (3 * l).+1 :> nat
           then ↓[u ai]
           else
            if i %% 3 == 0
            then ↓[u (inord (i %/ 3))]
            else
             if i %% 3 == 1
             then si (i %/ 3)
             else ti (i %/ 3)]:{ffun forall x : ordinal_finType (3 * l).+2,
      (fun=> configuration 4 n.+1) x}
573
have u10E : u1 ord0 = ↓[u ord0].
127f
67f
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f012501280683
573
  rewrite ffunE /=.
127f
↓[u (inord (0 %/ 3))] = ↓[u ord0]
1284
  by congr (↓[u _]); apply/val_eqP; rewrite /= inordK.
1286
573
have u1mE : u1 ord_max = ↓[u ord_max].
1286
u1 ord_max = ↓[u ord_max]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f012501280683
u1mE:u1 ord_max = ↓[u ord_max]
573
  rewrite ffunE eqxx.
1286
↓[u ai] = ↓[u ord_max]
1292
  by congr (↓[u _]); apply/val_eqP/eqP; rewrite /= inordK.
1294
573
have P1 : l.*2 + sd u1 <= sd u.
1294
l.*2 + sd (n:=n.+1) u1 <= sd (n:=n.+2) u
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f0125012806831295
P1:l.*2 + sd (n:=n.+1) u1 <= sd (n:=n.+2) u
573
  rewrite /sd !big_ord_recr /=.
1294
l.*2 +
(\sum_(i < 3 * l)
    `d[u1 (inord i), u1 (inord i.+1)]_smove +
 `d[u1 (inord (3 * l)), u1 (inord (3 * l).+1)]_smove) <=
\sum_(i < l) `d[u (inord i), u (inord i.+1)]_smove +
`d[u (inord l), u (inord l.+1)]_smove
12a1
  rewrite addnA; apply: leq_add; last first.
1294
`d[u1 (inord (3 * l)), u1 (inord (3 * l).+1)]_smove <=
`d[u (inord l), u (inord l.+1)]_smove
1294
l.*2 +
\sum_(i < 3 * l)
   `d[u1 (inord i), u1 (inord i.+1)]_smove <=
\sum_(i < l) `d[u (inord i), u (inord i.+1)]_smove
12a2 
    apply: leq_trans (gdist_cunlift _); last by apply: shanoi_connect.
1294
`d[u1 (inord (3 * l)), u1 (inord (3 * l).+1)]_smove <=
`d[↓[u (inord l)], ↓[u (inord l.+1)]]_(move (q:=4)
                                         (srel (n:=4)))
12ad
    rewrite leq_eqVlt; apply/orP; left; apply/eqP.
1294
`d[u1 (inord (3 * l)), u1 (inord (3 * l).+1)]_smove =
`d[↓[u (inord l)], ↓[u (inord l.+1)]]_(move (q:=4)
                                         (srel (n:=4)))
12ad
    rewrite !ffunE !inordK ?eqxx //.
1294
`d[if 3 * l == (3 * l).+1
   then ↓[u ai]
   else
    if (3 * l) %% 3 == 0
    then ↓[u (inord ((3 * l) %/ 3))]
    else
     if (3 * l) %% 3 == 1
     then si ((3 * l) %/ 3)
     else ti ((3 * l) %/ 3), ↓[u ai]]_smove =
`d[↓[u (inord l)], ↓[u (inord l.+1)]]_(move (q:=4)
                                         (srel (n:=4)))
12ad
    rewrite eqn_leq [_ <= _ * _]leqNgt leqnn andbF mod3E /= div3E.
1294
`d[↓[u (inord l)], ↓[u ai]]_smove =
`d[↓[u (inord l)], ↓[u (inord l.+1)]]_(move (q:=4)
                                         (srel (n:=4)))
12ad
    congr (`d[↓[u _], ↓[u _]]_smove); apply/val_eqP; rewrite /= ?mulKn //.
1294
ai == inord l.+1
12ad
    by rewrite inordK // aiE aEl.
1294
12af
12a1
  rewrite leq_eqVlt; apply/orP; left; apply/eqP.
1294
l.*2 +
\sum_(i < 3 * l)
   `d[u1 (inord i), u1 (inord i.+1)]_smove =
\sum_(i < l) `d[u (inord i), u (inord i.+1)]_smove
12a1
  have -> := @sum3E _ (fun i => `d[u1 (inord i), u1 (inord i.+1)]_smove).
1294
l.*2 +
\sum_(i < l)
   (`d[u1 (inord (3 * i)), u1 (inord (3 * i).+1)]_smove +
    `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
    `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove) =
\sum_(i < l) `d[u (inord i), u (inord i.+1)]_smove
12a1
  have -> : l.*2 = \sum_(i < l) 2.
1294
l.*2 = \sum_(i < l) 2
1294
\sum_(i < l) 2 +
\sum_(i < l)
   (`d[u1 (inord (3 * i)), 
    u1 (inord (3 * i).+1)]_smove +
    `d[u1 (inord (3 * i).+1), 
    u1 (inord (3 * i).+2)]_smove +
    `d[u1 (inord (3 * i).+2), 
    u1 (inord (3 * i).+3)]_smove) =
\sum_(i < l) `d[u (inord i), u (inord i.+1)]_smove
12a2
    by rewrite sum_nat_const /= cardT size_enum_ord muln2.
1294
\sum_(i < l) 2 +
\sum_(i < l)
   (`d[u1 (inord (3 * i)), u1 (inord (3 * i).+1)]_smove +
    `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
    `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove) =
\sum_(i < l) `d[u (inord i), u (inord i.+1)]_smove
12a1
  rewrite -big_split /=; apply: eq_bigr => i _.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f0125012806831295fa0
739
12a1
  rewrite ffunE !inordK //; last first.
12dd
3 * i < (3 * l).+2
12dd
2 +
(`d[if 3 * i == (3 * l).+1
    then ↓[u ai]
    else
     if (3 * i) %% 3 == 0
     then ↓[u (inord ((3 * i) %/ 3))]
     else
      if (3 * i) %% 3 == 1
      then si ((3 * i) %/ 3)
      else ti ((3 * i) %/ 3), 
 u1 (inord (3 * i).+1)]_smove +
 `d[u1 (inord (3 * i).+1), 
 u1 (inord (3 * i).+2)]_smove +
 `d[u1 (inord (3 * i).+2), 
 u1 (inord (3 * i).+3)]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
12a2
    by rewrite ltnS ltnW // ltnS leq_mul2l /= ltnW.
12dd
2 +
(`d[if 3 * i == (3 * l).+1
    then ↓[u ai]
    else
     if (3 * i) %% 3 == 0
     then ↓[u (inord ((3 * i) %/ 3))]
     else
      if (3 * i) %% 3 == 1
      then si ((3 * i) %/ 3)
      else ti ((3 * i) %/ 3), u1 (inord (3 * i).+1)]_smove +
 `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
 `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
12a1
  rewrite eqn_leq [_ <= _ * _]leqNgt ltnS leq_mul2l //=.
12dd
2 +
(`d[if (3 * i <= (3 * l).+1) && ~~ (i <= l)
    then ↓[u ai]
    else
     if (3 * i) %% 3 == 0
     then ↓[u (inord ((3 * i) %/ 3))]
     else
      if (3 * i) %% 3 == 1
      then si ((3 * i) %/ 3)
      else ti ((3 * i) %/ 3), u1 (inord (3 * i).+1)]_smove +
 `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
 `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
12a1
  rewrite (ltnW (ltn_ord _)) andbF mod3E /= div3E.
12dd
761
12a1
  rewrite ffunE !inordK //; last first.
12dd
(3 * i).+1 < (3 * l).+2
12dd
2 +
(`d[↓[u (inord i)], if (3 * i).+1 == (3 * l).+1
                    then ↓[u ai]
                    else
                     if (3 * i).+1 %% 3 == 0
                     then
                      ↓[
                     u (inord ((3 * i).+1 %/ 3))]
                     else
                      if (3 * i).+1 %% 3 == 1
                      then si ((3 * i).+1 %/ 3)
                      else ti ((3 * i).+1 %/ 3)]_smove +
 `d[if (3 * i).+1 == (3 * l).+1
    then ↓[u ai]
    else
     if (3 * i).+1 %% 3 == 0
     then ↓[u (inord ((3 * i).+1 %/ 3))]
     else
      if (3 * i).+1 %% 3 == 1
      then si ((3 * i).+1 %/ 3)
      else ti ((3 * i).+1 %/ 3), 
 u1 (inord (3 * i).+2)]_smove +
 `d[u1 (inord (3 * i).+2), 
 u1 (inord (3 * i).+3)]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
12a2
    by rewrite !ltnS leq_mul2l /= ltnW.
12dd
2 +
(`d[↓[u (inord i)], if (3 * i).+1 == (3 * l).+1
                    then ↓[u ai]
                    else
                     if (3 * i).+1 %% 3 == 0
                     then
                      ↓[u (inord ((3 * i).+1 %/ 3))]
                     else
                      if (3 * i).+1 %% 3 == 1
                      then si ((3 * i).+1 %/ 3)
                      else ti ((3 * i).+1 %/ 3)]_smove +
 `d[if (3 * i).+1 == (3 * l).+1
    then ↓[u ai]
    else
     if (3 * i).+1 %% 3 == 0
     then ↓[u (inord ((3 * i).+1 %/ 3))]
     else
      if (3 * i).+1 %% 3 == 1
      then si ((3 * i).+1 %/ 3)
      else ti ((3 * i).+1 %/ 3), u1 (inord (3 * i).+2)]_smove +
 `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
12a1
  rewrite eqn_leq [_.+1 <= _]leqNgt !ltnS leq_mul2l /= [_ <= i]leqNgt.
12dd
2 +
(`d[↓[u (inord i)], if
                     ~~ (3 * l < 3 * i) && ~~ (i < l)
                    then ↓[u ai]
                    else
                     if (3 * i).+1 %% 3 == 0
                     then
                      ↓[u (inord ((3 * i).+1 %/ 3))]
                     else
                      if (3 * i).+1 %% 3 == 1
                      then si ((3 * i).+1 %/ 3)
                      else ti ((3 * i).+1 %/ 3)]_smove +
 `d[if ~~ (3 * l < 3 * i) && ~~ (i < l)
    then ↓[u ai]
    else
     if (3 * i).+1 %% 3 == 0
     then ↓[u (inord ((3 * i).+1 %/ 3))]
     else
      if (3 * i).+1 %% 3 == 1
      then si ((3 * i).+1 %/ 3)
      else ti ((3 * i).+1 %/ 3), u1 (inord (3 * i).+2)]_smove +
 `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
12a1
  rewrite ltn_ord andbF mod3E /= div3E -[(_ * _).+3](mulnDr 3 1 i) add1n.
12dd
2 +
(`d[↓[u (inord i)], si i]_smove +
 `d[si i, u1 (inord (3 * i).+2)]_smove +
 `d[u1 (inord (3 * i).+2), u1 (inord (3 * i.+1))]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
12a1
  rewrite ffunE !inordK //; last first.
12dd
(3 * i).+2 < (3 * l).+2
12dd
2 +
(`d[↓[u (inord i)], si i]_smove +
 `d[si i, if (3 * i).+2 == (3 * l).+1
          then ↓[u ai]
          else
           if (3 * i).+2 %% 3 == 0
           then ↓[u (inord ((3 * i).+2 %/ 3))]
           else
            if (3 * i).+2 %% 3 == 1
            then si ((3 * i).+2 %/ 3)
            else ti ((3 * i).+2 %/ 3)]_smove +
 `d[if (3 * i).+2 == (3 * l).+1
    then ↓[u ai]
    else
     if (3 * i).+2 %% 3 == 0
     then ↓[u (inord ((3 * i).+2 %/ 3))]
     else
      if (3 * i).+2 %% 3 == 1
      then si ((3 * i).+2 %/ 3)
      else ti ((3 * i).+2 %/ 3), 
 u1 (inord (3 * i.+1))]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
12a2
    by rewrite!ltnS ltn_mul2l /=.
12dd
2 +
(`d[↓[u (inord i)], si i]_smove +
 `d[si i, if (3 * i).+2 == (3 * l).+1
          then ↓[u ai]
          else
           if (3 * i).+2 %% 3 == 0
           then ↓[u (inord ((3 * i).+2 %/ 3))]
           else
            if (3 * i).+2 %% 3 == 1
            then si ((3 * i).+2 %/ 3)
            else ti ((3 * i).+2 %/ 3)]_smove +
 `d[if (3 * i).+2 == (3 * l).+1
    then ↓[u ai]
    else
     if (3 * i).+2 %% 3 == 0
     then ↓[u (inord ((3 * i).+2 %/ 3))]
     else
      if (3 * i).+2 %% 3 == 1
      then si ((3 * i).+2 %/ 3)
      else ti ((3 * i).+2 %/ 3), u1 (inord (3 * i.+1))]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
12a1
  rewrite !mod3E /= !div3E ifN; last first.
12dd
(3 * i).+2 != (3 * l).+1
12dd
2 +
(`d[↓[u (inord i)], si i]_smove + `d[si i, ti i]_smove +
 `d[ti i, u1 (inord (3 * i.+1))]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
12a2
    case: eqP => // eH.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f0125012806831295fa0
eH:(3 * i).+2 = (3 * l).+1
~~ true
1312
    have : (3 * i).+2 %% 3 = (3 * l).+1 %% 3 by rewrite eH.
1318
(3 * i).+2 = (3 * l).+1 %[mod 3] -> ~~ true
1312
    by rewrite !mod3E.
12dd
1314
12a1
  rewrite ffunE inordK; last first.
12dd
3 * i.+1 < (3 * l).+2
12dd
2 +
(`d[↓[u (inord i)], si i]_smove + `d[si i, ti i]_smove +
 `d[ti i, if 3 * i.+1 == (3 * l).+1
          then ↓[u ai]
          else
           if (3 * i.+1) %% 3 == 0
           then ↓[u (inord ((3 * i.+1) %/ 3))]
           else
            if (3 * i.+1) %% 3 == 1
            then si ((3 * i.+1) %/ 3)
            else ti ((3 * i.+1) %/ 3)]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
12a2
    by rewrite ltnS ltnW // ltnS leq_mul2l /=.
12dd
1328
12a1
  rewrite mod3E /= div3E ifN.
12dd
7db
12dd
3 * i.+1 != (3 * l).+1
12a2
    case: (sitiH i) => //= _ _ ->.
12dd
2 +
(`d[↓[u (inord i)], si i]_smove + `d[si i, ti i]_smove +
 `d[ti i, ↓[u (inord i.+1)]]_smove) =
(`d[↓[u (inord i)], si i]_smove +
 (`d[si i, ti i]_smove +
  (`d[ti i, ↓[u (inord i.+1)]]_smove + 0))).+2
132f
    by rewrite /= !addnA addn0.
12dd
1331
12a1
  case: eqP => // eH.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f0125012806831295fa0
eH:3 * i.+1 = (3 * l).+1
131a
12a1
  have : (3 * i.+1) %% 3 = (3 * l).+1 %% 3 by rewrite eH.
133c
3 * i.+1 = (3 * l).+1 %[mod 3] -> ~~ true
12a1
  by rewrite !mod3E.
12a3
573
have cH1 (k : 'I_(3 * l).+2) : 
    0 < k < (3 * l).+1 -> codom (u1 k) \subset [:: p2; a[p1, p3] k].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f012501280683129512a4
k:'I_(3 * l).+2
0 < k < (3 * l).+1 ->
codom (u1 k) \subset [:: p2;  a[p1, p3] k]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f012501280683129512a4
cH1:forall k : 'I_(3 * l).+2,
0 < k < (3 * l).+1 ->
codom (u1 k) \subset [:: p2;  a[p1, p3] k]
573
  rewrite !ffunE.
1348
0 < k < (3 * l).+1 ->
codom
  (if k == (3 * l).+1
   then ↓[u ai]
   else
    if k %% 3 == 0
    then ↓[u (inord (k %/ 3))]
    else
     if k %% 3 == 1 then si (k %/ 3) else ti (k %/ 3)) \subset
[:: p2;  a[p1, p3] k]
134b
  case: eqP => [->|/eqP kD3l1]; first by rewrite ltnn andbF.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f012501280683129512a41349
kD3l1:k != (3 * l).+1
0 < k < (3 * l).+1 ->
codom
  (if k %% 3 == 0
   then ↓[u (inord (k %/ 3))]
   else
    if k %% 3 == 1 then si (k %/ 3) else ti (k %/ 3)) \subset
[:: p2;  a[p1, p3] k]
134b
  case: eqP => [km3E0|/eqP km3D0].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f012501280683129512a413491357
km3E0:k %% 3 = 0
0 < k < (3 * l).+1 ->
codom ↓[u (inord (k %/ 3))] \subset
[:: p2;  a[p1, p3] k]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f012501280683129512a413491357
km3D0:k %% 3 != 0
0 < k < (3 * l).+1 ->
codom
  (if k %% 3 == 1 then si (k %/ 3) else ti (k %/ 3)) \subset
[:: p2;  a[p1, p3] k]
134c
    rewrite {1 2 4}(divn_eq k 3) km3E0 addn0.
135c
0 < k %/ 3 * 3 < (3 * l).+1 ->
codom ↓[u (inord (k %/ 3))] \subset
[:: p2;  a[p1, p3] (k %/ 3 * 3)]
135f
    rewrite ltnS mulnC muln_gt0 leq_mul2l /= => /andP[k3_gt0 k3Ll].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f012501280683129512a413491357135d
k3_gt0:0 < k %/ 3
k3Ll:k %/ 3 <= l
codom ↓[u (inord (k %/ 3))] \subset
[:: p2;  a[p1, p3] (3 * (k %/ 3))]
135f
    rewrite apegMr //.
136b
codom ↓[u (inord (k %/ 3))] \subset
[:: p2;  a[p1, p3] (k %/ 3)]
135f
    have:= @H (inord (k %/ 3)); rewrite ffunE inordK //; last first.
136b
8cf
136b
(0 < k %/ 3 < l.+1 ->
 codom ↓[u (inord (k %/ 3))] \subset
 [:: p2;  a[p1, p3] (k %/ 3)]) ->
codom ↓[u (inord (k %/ 3))] \subset
[:: p2;  a[p1, p3] (k %/ 3)]
1360134c
      by apply: leq_trans k3Ll _.
136b
1378
135f
    by apply; rewrite k3_gt0.
1361
1363
134b
  case: eqP => [km3E1|/eqP km3D1].
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f012501280683129512a4134913571362
km3E1:k %% 3 = 1
0 < k < (3 * l).+1 ->
codom (si (k %/ 3)) \subset [:: p2;  a[p1, p3] k]
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f012501280683129512a4134913571362
km3D1:k %% 3 != 1
0 < k < (3 * l).+1 ->
codom (ti (k %/ 3)) \subset [:: p2;  a[p1, p3] k]
134c
    rewrite {1 2 4}(divn_eq k 3) km3E1 addn1 !ltnS /= mulnC ltn_mul2l /= => H1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f012501280683129512a41349135713621383
H1:k %/ 3 < l
codom (si (k %/ 3)) \subset
[:: p2;  a[p1, p3] (3 * (k %/ 3)).+1]
1385
    case: (sitiH (k %/ 3)) => //.
138d
codom (si (k %/ 3)) \subset
[:: p2; u (inord (k %/ 3).+1) ldisk] ->
codom (ti (k %/ 3)) \subset
[:: u (inord (k %/ 3)) ldisk; p2] ->
`d[u (inord (k %/ 3)), u (inord (k %/ 3).+1)]_smove =
 D[[:: ↓[u (inord (k %/ 3))]; si (k %/ 3);
       ti (k %/ 3); ↓[u (inord (k %/ 3).+1)]]].+2 ->
codom (si (k %/ 3)) \subset
[:: p2;  a[p1, p3] (3 * (k %/ 3)).+1]
1385
    by rewrite apegS apegMr // aMin1 ?apegS // aEl ltnS.
1387
1389
134b
  have km3E2 : k %% 3 = 2.
1387
944
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f012501280683129512a41349135713621388
km3E2:k %% 3 = 2
1389
134c
    by case: (_ %% _) (ltn_mod k 3) km3D0 km3D1 => // [] [|[|]].
139c
1389
134b
  rewrite {1 2 4}(divn_eq k 3) km3E2 addn2 /= ltnS => H1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f012501280683129512a41349135713621388139d
H1:(k %/ 3 * 3).+1 < 3 * l
codom (ti (k %/ 3)) \subset
[:: p2;  a[p1, p3] (k %/ 3 * 3).+2]
134b
  have {}H1 : k %/ 3 < l.
13a4
k %/ 3 < l
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f012501280683129512a41349135713621388139d138e
13a6
134c
    by rewrite -[_ < _](ltn_mul2r 3) [l * 3]mulnC (leq_ltn_trans _ H1).
13ad
13a6
134b
  rewrite codom_subC.
13ad
codom (ti (k %/ 3)) \subset
[::  a[p1, p3] (k %/ 3 * 3).+2; p2]
134b
  case: (sitiH (k %/ 3)) => // _.
13ad
codom (ti (k %/ 3)) \subset
[:: u (inord (k %/ 3)) ldisk; p2] ->
`d[u (inord (k %/ 3)), u (inord (k %/ 3).+1)]_smove =
 D[[:: ↓[u (inord (k %/ 3))]; si (k %/ 3);
       ti (k %/ 3); ↓[u (inord (k %/ 3).+1)]]].+2 ->
codom (ti (k %/ 3)) \subset
[::  a[p1, p3] (k %/ 3 * 3).+2; p2]
134b
  rewrite !apegS [_ * 3]mulnC apegMr //.
13ad
codom (ti (k %/ 3)) \subset
[:: u (inord (k %/ 3)) ldisk; p2] ->
`d[u (inord (k %/ 3)), u (inord (k %/ 3).+1)]_smove =
 D[[:: ↓[u (inord (k %/ 3))]; si (k %/ 3);
       ti (k %/ 3); ↓[u (inord (k %/ 3).+1)]]].+2 ->
codom (ti (k %/ 3)) \subset
[::  a[p1, p3] (k %/ 3); p2]
134b
  by rewrite aMin1 // (leq_trans H1) // aEl.
134d
573
have {cH1}P2 : 
   (S_[(3 * l).+1] n.+1).*2 <= sd u1 + sp (u1 ord0) (3 * l).+1 p1 +
                               sp (u1 ord_max) (3 * l).+1 
                                               (a[p1, p3] (3 * l).+1).
134d
(S_[(3 * l).+1] n.+1).*2 <=
sd (n:=n.+1) u1 + sp (n:=n.+1) (u1 ord0) (3 * l).+1 p1 +
sp (n:=n.+1) (u1 ord_max) (3 * l).+1
  ( a[p1, p3] (3 * l).+1)
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f012501280683129512a4
P2:(S_[(3 * l).+1] n.+1).*2 <=
sd (n:=n.+1) u1 +
sp (n:=n.+1) (u1 ord0) (3 * l).+1 p1 +
sp (n:=n.+1) (u1 ord_max) 
  (3 * l).+1 ( a[p1, p3] (3 * l).+1)
573
  by apply: IH cH1.
13c6
573
 have {}P2 := leq_trans P2 (leq_add (leq_add (leqnn _) (leq_sum_beta _ _))
                          (leq_sum_beta _ _)).
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f012501280683129512a4
P2:(S_[(3 * l).+1] n.+1).*2 <=
sd (n:=n.+1) u1 +
\sum_(k < n.+1) (u1 ord0 k != p1) * (α_[1] k).*2 +
\sum_(k < n.+1)
   (u1 ord_max k !=  a[p1, p3] (3 * l).+1) *
   (α_[1] k).*2
573
set xP := sd _; rewrite -/xP in P1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f012501280683129513cf11d2
P1:l.*2 + sd (n:=n.+1) u1 <= xP
11d4
set x1P := sd _ in P1.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f012501280683129513cf11d211de
P1:l.*2 + x1P <= xP
11d4
rewrite -/x1P (_ : a[p1, p3] (3 * l).+1 = a[p3, p1] l) in P2; last first.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f012501280683129511d211de13d9
 a[p1, p3] (3 * l).+1 =  a[p3, p1] l
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f012501280683129511d211de13d9
P2:(S_[(3 * l).+1] n.+1).*2 <=
x1P +
\sum_(k < n.+1) (u1 ord0 k != p1) * (α_[1] k).*2 +
\sum_(k < n.+1)
   (u1 ord_max k !=  a[p3, p1] l) * (α_[1] k).*2
11d4
  by rewrite apegS apegMr.
13e1
11d4
rewrite !sum_beta_S // KH1 KH2 !eqxx addn0 mul1n.
13e1
1202
rewrite !ffunE /= eqxx inord_eq0 // (_ : ai = ord_max) in P2; last first.
13dd
ai = ord_max
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f012501280683129511d211de13d9
P2:(S_[(3 * l).+1] n.+1).*2 <=
x1P +
\sum_(k < n.+1)
   (↓[u ord0] k != p1) * (α_[1] k).*2 +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) *
   (α_[1] k).*2
1202
  by apply: val_inj; rewrite /= inordK.
13ef
1202
set x1S := \sum_(_ < _) _ in P2; set x2S := \sum_(_ < _) _ in P2.
51295565571155815161712131455955a56156856f57057157257958122858958a59359b5a05a55aa5af5b45b95beb1ef09f121225123f5eb5f012501280683129511d211de13d911fc11fd
P2:(S_[(3 * l).+1] n.+1).*2 <= x1P + x1S + x2S
1202
rewrite -/x1S -/x2S .
13f7
(S_[l.+1] n.+2).*2 <= xP + x1S + (x2S + α_[l.+1] n.+1)
rewrite -addnn {1}dsum_alphaL_S addnAC !addnA leq_add2r.
13f7
S_[l.+1] n.+1 + S_[l.+1] n.+2 <= xP + x1S + x2S
rewrite -[X in _ <= X]addnA.
13f7
S_[l.+1] n.+1 + S_[l.+1] n.+2 <= xP + (x1S + x2S)
apply: leq_trans (leq_add P1 (leqnn _)).
13f7
S_[l.+1] n.+1 + S_[l.+1] n.+2 <=
l.*2 + x1P + (x1S + x2S)
rewrite -[X in _ <= X]addnA [x1P + _]addnA.
13f7
S_[l.+1] n.+1 + S_[l.+1] n.+2 <=
l.*2 + (x1P + x1S + x2S)
apply: leq_trans (leq_add (leqnn _) P2).
13f7
S_[l.+1] n.+1 + S_[l.+1] n.+2 <=
l.*2 + (S_[(3 * l).+1] n.+1).*2
apply: leq_trans (leq_dsum_alpha_2l_l _ _) _.
13f7
(S_[(l.+1).*2] n.+1).*2 + l.+1 <=
l.*2 + (S_[(3 * l).+1] n.+1).*2
rewrite addnC; apply: leq_add.
13f7
l < l.*2
13f7
(S_[(l.+1).*2] n.+1).*2 <= (S_[(3 * l).+1] n.+1).*2
  by rewrite -addnn -addn1 leq_add.
13f7
141b
rewrite leq_double.
13f7
S_[(l.+1).*2] n.+1 <= S_[(3 * l).+1] n.+1
apply: (increasingE (increasing_dsum_alphaL_l _)).
13f7
(l.+1).*2 <= (3 * l).+1
by rewrite doubleS ltnS -mul2n ltn_mul2r l_ggt0.
Qed.

End Case1.

(* This corresponds to the subcase of 4.2 when E is empty *)
Section Case2.

Variable n : nat.

Hypothesis IH:
     forall (l : nat) (p1 p2 p3 : ordinal_eqType 4)
       (u : {ffun 'I_l.+1 -> configuration 4 n.+1}),
     p1 != p2 ->
     p1 != p3 ->
     p2 != p3 ->
     p1 != p0 ->
     p2 != p0 ->
     p3 != p0 ->
     (forall k : 'I_l.+1,
      0 < k < l -> codom (u k) \subset [:: p2; a[p1, p3] k]) ->
     (S_[l] n.+1).*2 <= sd u + sp (u ord0) l (a[p1, p3] 0) +
                            sp (u ord_max) l (a[p1, p3] l).

Lemma case2 l (u :  {ffun 'I_l.+2 -> configuration 4 n.+2}) 
              (p1 p2 p3: peg 4) :
   p1 != p2 -> p1 != p3 -> p2 != p3 ->
   p1 != p0 -> p2 != p0 -> p3 != p0 ->
  (forall k : 'I_l.+2,
     0 < k < l.+1 -> codom (u k) \subset [:: p2; a[p1, p3] k]) ->
  (u ord0 ord_max = p1) ->
  (u ord_max ord_max = a[p3, p1] l) ->
  (forall k, u k ldisk = a[p1, p3] k) ->
  (S_[l.+1] n.+2).*2 <= sd u + sp (u ord0) l.+1 p1
                             + sp (u ord_max) l.+1 (a[p3, p1] l).
512955655755811
p1 != p2 ->
p1 != p3 ->
p2 != p3 ->
p1 != p0 ->
p2 != p0 ->
p3 != p0 ->
(forall k : 'I_l.+2,
 0 < k < l.+1 ->
 codom (u k) \subset [:: p2;  a[p1, p3] k]) ->
u ord0 ord_max = p1 ->
u ord_max ord_max =  a[p3, p1] l ->
(forall k : ordinal_finType l.+2,
 u k ldisk =  a[p1, p3] k) ->
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u + sp (n:=n.+2) (u ord0) l.+1 p1 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
Proof.
1428
move=> p1Dp2 p1Dp3 p2Dp3 p1Dp0 p2Dp0 p3Dp0.
512955655755811151617121314
(forall k : 'I_l.+2,
 0 < k < l.+1 ->
 codom (u k) \subset [:: p2;  a[p1, p3] k]) ->
u ord0 ord_max = p1 ->
u ord_max ord_max =  a[p3, p1] l ->
(forall k : ordinal_finType l.+2,
 u k ldisk =  a[p1, p3] k) ->
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u + sp (n:=n.+2) (u ord0) l.+1 p1 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
move=> /= cH KH1 KH2 ukLE.
512955655755811151617121314559570
KH2:u ord_max ord_max =  a[p3, p1] l
ukLE:forall k : 'I_l.+2, u k ldisk =  a[p1, p3] k
573
pose u':= ([ffun i => ↓[u i]] : {ffun 'I_l.+2 -> configuration 4 n.+1})
 : {ffun 'I_l.+2 -> configuration 4 n.+1}.
5129556557558111516171213145595701436143755a
573
have H: forall k : 'I_l.+2,
    0 < k < l.+1 -> codom (u' k) \subset [:: p2; a[p1, p3] k].
143b
55b
5129556557558111516171213145595701436143755a561
573
  by move=> k kH; rewrite ffunE; apply/codom_liftr/cH.
1441
573
have apegpaD2 a : a[p1, p3] a != p2.
5129556557558111516171213145595701436143755a561228
562
5129556557558111516171213145595701436143755a561
apegpaD2:forall a : nat,  a[p1, p3] a != p2
573
  by rewrite /apeg; case: odd; rewrite // eq_sym.
144b
573
have apegpaD0 a : a[p1, p3] a != p0.
5129556557558111516171213145595701436143755a561144c228
569
5129556557558111516171213145595701436143755a561144c
apegpaD0:forall a : nat,  a[p1, p3] a != p0
573
  by rewrite /apeg; case: odd; rewrite // eq_sym.
1456
573
pose si i : configuration 4 n.+1 := 
    sgdist1 p2 (u (inord i)) (u (inord i.+1)).
5129556557558111516171213145595701436143755a561144c14575eb
573
pose ti i : configuration 4 n.+1 := 
    sgdist2 p2 (u (inord i)) (u (inord i.+1)).
5129556557558111516171213145595701436143755a561144c14575eb5f0
573
have sitiH i : i < l.+1 ->
    [/\ codom (si i) \subset [:: p2; u (inord i.+1) ldisk ], 
        codom (ti i) \subset [:: u (inord i) ldisk; p2] & 
        `d[u (inord i), u (inord i.+1)]_smove = 
          D[[:: ↓[u (inord i)]; si i; ti i; ↓[u (inord i.+1)]]].+2].
5129556557558111516171213145595701436143755a561144c14575eb5f014a
i < l.+1 ->
[/\ codom (si i) \subset [:: p2; u (inord i.+1) ldisk],
    codom (ti i) \subset [:: u (inord i) ldisk; p2]
  & `d[u (inord i), u (inord i.+1)]_smove =
     D[[:: ↓[u (inord i)]; si i; ti i;
           ↓[u (inord i.+1)]]].+2]
5129556557558111516171213145595701436143755a561144c14575eb5f0
sitiH:forall i : nat,
i < l.+1 ->
[/\ codom (si i) \subset
    [:: p2; u (inord i.+1) ldisk],
    codom (ti i) \subset
    [:: u (inord i) ldisk; p2]
  & `d[u (inord i), u (inord i.+1)]_smove =
     D[[:: ↓[u (inord i)]; 
          si i; ti i; ↓[
          u (inord i.+1)]]].+2]
573
  move=> iLl.
5129556557558111516171213145595701436143755a561144c14575eb5f014a
iLl:i < l.+1
5ff
1468
  have i1Ll1 : i <= l by [].
5129556557558111516171213145595701436143755a561144c14575eb5f014a1470
i1Ll1:i <= l
5ff
1468
  apply: sgdistE; rewrite ?ukLE //.
1474
 a[p1, p3] (inord i) !=  a[p1, p3] (inord i.+1)
1474
p2 !=  a[p1, p3] (inord i.+1)
1469
    rewrite !inordK //=; last by rewrite ltnS ltnW // ltnS.
1474
61b
147a
    by rewrite apegS apeg_eqC.
1474
147c
1468
  by rewrite eq_sym.
146a
573
pose u1 := 
  [ffun i =>
  if ((i : 'I_(3 * l.+1).+1) %% 3) == 0 then ↓[u (inord (i %/ 3))]
  else if (i %% 3) == 1 then si (i %/ 3)
  else ti (i %/ 3)].
5129556557558111516171213145595701436143755a561144c14575eb5f0146b
u1:=[ffun i => if (i : 'I_(3 * l.+1).+1) %% 3 == 0
           then ↓[u (inord (i %/ 3))]
           else
            if i %% 3 == 1
            then si (i %/ 3)
            else ti (i %/ 3)]:{ffun forall x : ordinal_finType (3 * l.+1).+1,
      (fun=> configuration 4 n.+1) x}
573
have u10E : u1 ord0 = ↓[u ord0].
1489
67f
5129556557558111516171213145595701436143755a561144c14575eb5f0146b148a683
573
  rewrite ffunE /=.
1489
128a
148e
  by congr (↓[u _]); apply/val_eqP; rewrite /= inordK.
1490
573
have u1ME : u1 ord_max = ↓[u ord_max].
1490
1291
5129556557558111516171213145595701436143755a561144c14575eb5f0146b148a683
u1ME:u1 ord_max = ↓[u ord_max]
573
  rewrite ffunE /= mod3E /= div3E.
1490
↓[u (inord l.+1)] = ↓[u ord_max]
149a
  by congr (↓[u _]); apply/val_eqP; rewrite /= inordK.
149c
573
have P1 : l.+1.*2 + sd u1 = sd u.
149c
(l.+1).*2 + sd (n:=n.+1) u1 = sd (n:=n.+2) u
5129556557558111516171213145595701436143755a561144c14575eb5f0146b148a683149d
P1:(l.+1).*2 + sd (n:=n.+1) u1 = sd (n:=n.+2) u
573
  rewrite /sd.
149c
(l.+1).*2 +
\sum_(i < 3 * l.+1)
   `d[u1 (inord i), u1 (inord i.+1)]_smove =
\sum_(i < l.+1) `d[u (inord i), u (inord i.+1)]_smove
14a9
  have -> := @sum3E _ (fun i => `d[u1 (inord i), u1 (inord i.+1)]_smove).
149c
(l.+1).*2 +
\sum_(i < l.+1)
   (`d[u1 (inord (3 * i)), u1 (inord (3 * i).+1)]_smove +
    `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
    `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove) =
\sum_(i < l.+1) `d[u (inord i), u (inord i.+1)]_smove
14a9
  have -> : l.+1.*2 = \sum_(i < l.+1) 2.
149c
(l.+1).*2 = \sum_(i < l.+1) 2
149c
\sum_(i < l.+1) 2 +
\sum_(i < l.+1)
   (`d[u1 (inord (3 * i)), 
    u1 (inord (3 * i).+1)]_smove +
    `d[u1 (inord (3 * i).+1), 
    u1 (inord (3 * i).+2)]_smove +
    `d[u1 (inord (3 * i).+2), 
    u1 (inord (3 * i).+3)]_smove) =
\sum_(i < l.+1) `d[u (inord i), u (inord i.+1)]_smove
14aa
    by rewrite sum_nat_const /= cardT size_enum_ord muln2.
149c
\sum_(i < l.+1) 2 +
\sum_(i < l.+1)
   (`d[u1 (inord (3 * i)), u1 (inord (3 * i).+1)]_smove +
    `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
    `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove) =
\sum_(i < l.+1) `d[u (inord i), u (inord i.+1)]_smove
14a9
  rewrite -big_split /=; apply: eq_bigr => i _.
5129556557558111516171213145595701436143755a561144c14575eb5f0146b148a683149d
i:'I_l.+1
739
14a9
  rewrite ffunE !inordK //; last first.
14c3
3 * i < (3 * l.+1).+1
14c3
2 +
(`d[if (3 * i) %% 3 == 0
    then ↓[u (inord ((3 * i) %/ 3))]
    else
     if (3 * i) %% 3 == 1
     then si ((3 * i) %/ 3)
     else ti ((3 * i) %/ 3), 
 u1 (inord (3 * i).+1)]_smove +
 `d[u1 (inord (3 * i).+1), 
 u1 (inord (3 * i).+2)]_smove +
 `d[u1 (inord (3 * i).+2), 
 u1 (inord (3 * i).+3)]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
14aa
    by rewrite ltnS leq_mul2l ltnW.
14c3
2 +
(`d[if (3 * i) %% 3 == 0
    then ↓[u (inord ((3 * i) %/ 3))]
    else
     if (3 * i) %% 3 == 1
     then si ((3 * i) %/ 3)
     else ti ((3 * i) %/ 3), u1 (inord (3 * i).+1)]_smove +
 `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
 `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
14a9
  rewrite mod3E /= div3E.
14c3
761
14a9
  rewrite ffunE inordK; last first.
14c3
(3 * i).+1 < (3 * l.+1).+1
14c3
2 +
(`d[↓[u (inord i)], if (3 * i).+1 %% 3 == 0
                    then
                     ↓[u (inord ((3 * i).+1 %/ 3))]
                    else
                     if (3 * i).+1 %% 3 == 1
                     then si ((3 * i).+1 %/ 3)
                     else ti ((3 * i).+1 %/ 3)]_smove +
 `d[if (3 * i).+1 %% 3 == 0
    then ↓[u (inord ((3 * i).+1 %/ 3))]
    else
     if (3 * i).+1 %% 3 == 1
     then si ((3 * i).+1 %/ 3)
     else ti ((3 * i).+1 %/ 3), 
 u1 (inord (3 * i).+2)]_smove +
 `d[u1 (inord (3 * i).+2), 
 u1 (inord (3 * i).+3)]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
14aa
    by rewrite ltnS ltn_mul2l /=.
14c3
2 +
(`d[↓[u (inord i)], if (3 * i).+1 %% 3 == 0
                    then
                     ↓[u (inord ((3 * i).+1 %/ 3))]
                    else
                     if (3 * i).+1 %% 3 == 1
                     then si ((3 * i).+1 %/ 3)
                     else ti ((3 * i).+1 %/ 3)]_smove +
 `d[if (3 * i).+1 %% 3 == 0
    then ↓[u (inord ((3 * i).+1 %/ 3))]
    else
     if (3 * i).+1 %% 3 == 1
     then si ((3 * i).+1 %/ 3)
     else ti ((3 * i).+1 %/ 3), u1 (inord (3 * i).+2)]_smove +
 `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
14a9
  rewrite mod3E /= div3E.
14c3
799
14a9
  rewrite ffunE inordK; last first.
14c3
(3 * i).+2 < (3 * l.+1).+1
14c3
2 +
(`d[↓[u (inord i)], si i]_smove +
 `d[si i, if (3 * i).+2 %% 3 == 0
          then ↓[u (inord ((3 * i).+2 %/ 3))]
          else
           if (3 * i).+2 %% 3 == 1
           then si ((3 * i).+2 %/ 3)
           else ti ((3 * i).+2 %/ 3)]_smove +
 `d[if (3 * i).+2 %% 3 == 0
    then ↓[u (inord ((3 * i).+2 %/ 3))]
    else
     if (3 * i).+2 %% 3 == 1
     then si ((3 * i).+2 %/ 3)
     else ti ((3 * i).+2 %/ 3), 
 u1 (inord (3 * i).+3)]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
14aa
    by rewrite -[_.+3](mulnDr 3 1) ltnW // ltnS leq_mul2l /= add1n.
14c3
2 +
(`d[↓[u (inord i)], si i]_smove +
 `d[si i, if (3 * i).+2 %% 3 == 0
          then ↓[u (inord ((3 * i).+2 %/ 3))]
          else
           if (3 * i).+2 %% 3 == 1
           then si ((3 * i).+2 %/ 3)
           else ti ((3 * i).+2 %/ 3)]_smove +
 `d[if (3 * i).+2 %% 3 == 0
    then ↓[u (inord ((3 * i).+2 %/ 3))]
    else
     if (3 * i).+2 %% 3 == 1
     then si ((3 * i).+2 %/ 3)
     else ti ((3 * i).+2 %/ 3), u1 (inord (3 * i).+3)]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
14a9
  rewrite mod3E /= div3E.
14c3
7a0
14a9
  rewrite ffunE inordK; last first.
14c3
(3 * i).+3 < (3 * l.+1).+1
14c3
2 +
(`d[↓[u (inord i)], si i]_smove + `d[si i, ti i]_smove +
 `d[ti i, if (3 * i).+3 %% 3 == 0
          then ↓[u (inord ((3 * i).+3 %/ 3))]
          else
           if (3 * i).+3 %% 3 == 1
           then si ((3 * i).+3 %/ 3)
           else ti ((3 * i).+3 %/ 3)]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
14aa
    by rewrite -[(_ * _).+3](mulnDr 3 1) ltnS leq_mul2l /=.
14c3
14f5
14a9
  rewrite -[(_ * _).+3](mulnDr 3 1) mod3E /= div3E.
14c3
2 +
(`d[↓[u (inord i)], si i]_smove + `d[si i, ti i]_smove +
 `d[ti i, ↓[u (inord (1 + i))]]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
14a9
  case: (sitiH i) => // _ _.
14c3
`d[u (inord i), u (inord i.+1)]_smove =
 D[[:: ↓[u (inord i)]; si i; ti i; ↓[u (inord i.+1)]]].+2 ->
2 +
(`d[↓[u (inord i)], si i]_smove + `d[si i, ti i]_smove +
 `d[ti i, ↓[u (inord (1 + i))]]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
14a9
  by rewrite /= add2n addn0 !addnA add1n.
14ab
573
have cH1 (k : 'I_(3 * l.+1).+1) : 
  0 < k < 3 * l.+1 -> codom (u1 k) \subset [:: p2; a[p1, p3] k].
5129556557558111516171213145595701436143755a561144c14575eb5f0146b148a683149d14ac
k:'I_(3 * l.+1).+1
0 < k < 3 * l.+1 ->
codom (u1 k) \subset [:: p2;  a[p1, p3] k]
5129556557558111516171213145595701436143755a561144c14575eb5f0146b148a683149d14ac
cH1:forall k : 'I_(3 * l.+1).+1,
0 < k < 3 * l.+1 ->
codom (u1 k) \subset [:: p2;  a[p1, p3] k]
573
  rewrite !ffunE.
1507
0 < k < 3 * l.+1 ->
codom
  (if k %% 3 == 0
   then ↓[u (inord (k %/ 3))]
   else
    if k %% 3 == 1 then si (k %/ 3) else ti (k %/ 3)) \subset
[:: p2;  a[p1, p3] k]
150a
  case: eqP => [km3E0|/eqP km3D0].
5129556557558111516171213145595701436143755a561144c14575eb5f0146b148a683149d14ac1508135d
0 < k < 3 * l.+1 ->
codom ↓[u (inord (k %/ 3))] \subset
[:: p2;  a[p1, p3] k]
5129556557558111516171213145595701436143755a561144c14575eb5f0146b148a683149d14ac15081362
0 < k < 3 * l.+1 ->
codom
  (if k %% 3 == 1 then si (k %/ 3) else ti (k %/ 3)) \subset
[:: p2;  a[p1, p3] k]
150b
    rewrite {1 2 4}(divn_eq k 3) km3E0 addn0.
1515
0 < k %/ 3 * 3 < 3 * l.+1 ->
codom ↓[u (inord (k %/ 3))] \subset
[:: p2;  a[p1, p3] (k %/ 3 * 3)]
1517
    rewrite muln_gt0 andbT [_ * 3]mulnC ltn_mul2l /= => /andP[k3_gt0 k3Ll].
5129556557558111516171213145595701436143755a561144c14575eb5f0146b148a683149d14ac1508135d136c
k3Ll:k %/ 3 < l.+1
136e
1517
    rewrite apegMr //.
1522
1372
1517
    have:= @H (inord (k %/ 3)); rewrite ffunE inordK //; last first.
1522
8cf
1522
1378
1518150b
      by apply: leq_trans k3Ll _.
1522
1378
1517
    by apply; rewrite k3_gt0.
1519
151a
150a
  case: eqP => [km3E1|/eqP km3D1].
5129556557558111516171213145595701436143755a561144c14575eb5f0146b148a683149d14ac150813621383
0 < k < 3 * l.+1 ->
codom (si (k %/ 3)) \subset [:: p2;  a[p1, p3] k]
5129556557558111516171213145595701436143755a561144c14575eb5f0146b148a683149d14ac150813621388
0 < k < 3 * l.+1 ->
codom (ti (k %/ 3)) \subset [:: p2;  a[p1, p3] k]
150b
    rewrite {1 2 4}(divn_eq k 3) km3E1 addn1 /= => H1.
5129556557558111516171213145595701436143755a561144c14575eb5f0146b148a683149d14ac150813621383
H1:(k %/ 3 * 3).+1 < 3 * l.+1
codom (si (k %/ 3)) \subset
[:: p2;  a[p1, p3] (k %/ 3 * 3).+1]
1537
    have : 3 * (k %/ 3)  < 3 * l.+1.
153e
3 * (k %/ 3) < 3 * l.+1
153e
3 * (k %/ 3) < 3 * l.+1 ->
codom (si (k %/ 3)) \subset
[:: p2;  a[p1, p3] (k %/ 3 * 3).+1]
1538150b
      apply: leq_trans H1; first by rewrite mulnC ltnW.
153e
1547
1537
      rewrite ltn_mul2l /= => k3Ll1.
5129556557558111516171213145595701436143755a561144c14575eb5f0146b148a683149d14ac150813621383153f
k3Ll1:k %/ 3 < l.+1
1540
1537
    case: (sitiH (k %/ 3)) => //.
154e
codom (si (k %/ 3)) \subset
[:: p2; u (inord (k %/ 3).+1) ldisk] ->
codom (ti (k %/ 3)) \subset
[:: u (inord (k %/ 3)) ldisk; p2] ->
`d[u (inord (k %/ 3)), u (inord (k %/ 3).+1)]_smove =
 D[[:: ↓[u (inord (k %/ 3))]; si (k %/ 3);
       ti (k %/ 3); ↓[u (inord (k %/ 3).+1)]]].+2 ->
codom (si (k %/ 3)) \subset
[:: p2;  a[p1, p3] (k %/ 3 * 3).+1]
1537
    by rewrite apegS [_ * 3]mulnC apegMr // ukLE inordK // apegS.
1539
153a
150a
  have km3E2 : k %% 3 = 2.
1539
944
5129556557558111516171213145595701436143755a561144c14575eb5f0146b148a683149d14ac150813621388139d
153a
150b
    by case: (_ %% _) (ltn_mod k 3) km3D0 km3D1 => // [] [|[|]].
155c
153a
150a
  rewrite {1 2 4}(divn_eq k 3) km3E2 addn2 /= -[_.+3](mulnDl 1 _ 3).
155c
(1 + k %/ 3) * 3 <= 3 * l.+1 ->
codom (ti (k %/ 3)) \subset
[:: p2;  a[p1, p3] (k %/ 3 * 3).+2]
150a
  rewrite mulnC leq_mul2l /= add1n => H1.
5129556557558111516171213145595701436143755a561144c14575eb5f0146b148a683149d14ac150813621388139d
H1:k %/ 3 < l.+1
13a6
150a
  rewrite codom_subC.
1567
13b4
150a
  case: (sitiH (k %/ 3)) => // _ sH _; move: sH.
1567
codom (ti (k %/ 3)) \subset
[:: u (inord (k %/ 3)) ldisk; p2] ->
codom (ti (k %/ 3)) \subset
[::  a[p1, p3] (k %/ 3 * 3).+2; p2]
150a
  by rewrite ukLE // inordK ?(leq_trans H1) // !apegS [_ * 3]mulnC apegMr.
150c
573
have {cH1}P2 : 
    (S_[3 * l.+1] n.+1).*2 <= sd u1 + sp (u1 ord0) (3 * l.+1) p1 
                                    + sp (u1 ord_max) (3 * l.+1)
                                         (a[p1, p3] (3 * l.+1)).
150c
(S_[3 * l.+1] n.+1).*2 <=
sd (n:=n.+1) u1 + sp (n:=n.+1) (u1 ord0) (3 * l.+1) p1 +
sp (n:=n.+1) (u1 ord_max) (3 * l.+1)
  ( a[p1, p3] (3 * l.+1))
5129556557558111516171213145595701436143755a561144c14575eb5f0146b148a683149d14ac
P2:(S_[3 * l.+1] n.+1).*2 <=
sd (n:=n.+1) u1 +
sp (n:=n.+1) (u1 ord0) (3 * l.+1) p1 +
sp (n:=n.+1) (u1 ord_max) 
  (3 * l.+1) ( a[p1, p3] (3 * l.+1))
573
  by apply: IH cH1.
1579
573
have {}P2 := leq_trans P2 (leq_add (leq_add (leqnn _) (leq_sum_beta _ _))
                        (leq_sum_beta _ _)).
5129556557558111516171213145595701436143755a561144c14575eb5f0146b148a683149d14ac
P2:(S_[3 * l.+1] n.+1).*2 <=
sd (n:=n.+1) u1 +
\sum_(k < n.+1) (u1 ord0 k != p1) * (α_[1] k).*2 +
\sum_(k < n.+1)
   (u1 ord_max k !=  a[p1, p3] (3 * l.+1)) *
   (α_[1] k).*2
573
rewrite [X in _ <= _ + X + _]big_ord_recr /= ukLE eqxx addn0.
1581
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u +
\sum_(i < n.+1)
   (u ord0 (widen_ord (m:=n.+2) (leqnSn n.+1) i) != p1) *
   β_[n.+2, l.+1] i +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
rewrite [X in _ <= _ + X]big_ord_recr /= ukLE apegS eqxx addn0.
1581
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u +
\sum_(i < n.+1)
   (u ord0 (widen_ord (m:=n.+2) (leqnSn n.+1) i) != p1) *
   β_[n.+2, l.+1] i +
\sum_(i < n.+1)
   (u ord_max (widen_ord (m:=n.+2) (leqnSn n.+1) i)
    !=  a[p3, p1] l) * β_[n.+2, l.+1] i
set x1S := \sum_(_ < n.+1) _ in P2; set x2S := \sum_(_ < n.+1) _ in P2.
5129556557558111516171213145595701436143755a561144c14575eb5f0146b148a683149d14ac
x1S:=\sum_(H < n.+1) (u1 ord0 H != p1) * (α_[1] H).*2:nat
x2S:=\sum_(H < n.+1)
   (u1 ord_max H !=  a[p1, p3] (3 * l.+1)) *
   (α_[1] H).*2:nat
P2:(S_[3 * l.+1] n.+1).*2 <=
sd (n:=n.+1) u1 + x1S + x2S
158a
rewrite [X in _ <= _ + X + _](_ : _ = x1S); last first.
158e
\sum_(i < n.+1)
   (u ord0 (widen_ord (m:=n.+2) (leqnSn n.+1) i) != p1) *
   β_[n.+2, l.+1] i = x1S
158e
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u + x1S +
\sum_(i < n.+1)
   (u ord_max (widen_ord (m:=n.+2) (leqnSn n.+1) i)
    !=  a[p3, p1] l) * β_[n.+2, l.+1] i
  apply: eq_bigr => i _.
5129556557558111516171213145595701436143755a561144c14575eb5f0146b148a683149d14ac158f15901591
i:'I_n.+1
(u ord0 (widen_ord (m:=n.+2) (leqnSn n.+1) i) != p1) *
β_[n.+2, l.+1] i = (u1 ord0 i != p1) * (α_[1] i).*2
1596
  rewrite u10E ffunE /beta ifF; last first.
159c
(1 < l.+1) && (i == n.+2.-1) = false
159c
(u ord0 (widen_ord (m:=n.+2) (leqnSn n.+1) i) != p1) *
(α_[1] i).*2 =
(u ord0 (trshift 1 i) != p1) * (α_[1] i).*2
1597
    by rewrite eqn_leq [_ <= i]leqNgt ltn_ord !andbF.
159c
15a5
1596
  by congr ((u _ _ != _) * _); apply/val_eqP; rewrite /=.
158e
1598
rewrite [X in _ <= _ + X](_ : _ = x2S); last first.
158e
\sum_(i < n.+1)
   (u ord_max (widen_ord (m:=n.+2) (leqnSn n.+1) i)
    !=  a[p3, p1] l) * β_[n.+2, l.+1] i = x2S
158e
(S_[l.+1] n.+2).*2 <= sd (n:=n.+2) u + x1S + x2S
  apply: eq_bigr => i _.
159c
(u ord_max (widen_ord (m:=n.+2) (leqnSn n.+1) i)
 !=  a[p3, p1] l) * β_[n.+2, l.+1] i =
(u1 ord_max i !=  a[p1, p3] (3 * l.+1)) * (α_[1] i).*2
15b0
  rewrite u1ME ffunE /beta ifF; last first.
15a1
159c
(u ord_max (widen_ord (m:=n.+2) (leqnSn n.+1) i)
 !=  a[p3, p1] l) * (α_[1] i).*2 =
(u ord_max (trshift 1 i) !=  a[p1, p3] (3 * l.+1)) *
(α_[1] i).*2
15b1
    by rewrite eqn_leq [_ <= i]leqNgt ltn_ord !andbF.
159c
15bb
15b0
  congr ((u _ _ != _) * _); first by apply/val_eqP.
159c
 a[p3, p1] l =  a[p1, p3] (3 * l.+1)
15b0
  by rewrite apegMr // apegS.
158e
15b2
rewrite -P1 -!addnA addnC !addnA.
158e
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+1) u1 + x1S + x2S + (l.+1).*2
apply: leq_trans (leq_add P2 (leqnn _)).
158e
(S_[l.+1] n.+2).*2 <=
(S_[3 * l.+1] n.+1).*2 + (l.+1).*2
by rewrite -doubleD leq_double dsum_alpha3l.
Qed.

End Case2.

(* This corresponds to 4.2 *)
Section Case3.

Variable n : nat.

Hypothesis IH:
  forall (l : nat) (p1 p2 p3 : ordinal_eqType 4)
    (u : {ffun 'I_l.+1 -> configuration 4 n.+1}),
  p1 != p2 ->
  p1 != p3 ->
  p2 != p3 ->
  p1 != p0 ->
  p2 != p0 ->
  p3 != p0 ->
  (forall k : 'I_l.+1,
  0 < k < l -> codom (u k) \subset [:: p2; a[p1, p3] k]) ->
  (S_[l] n.+1).*2 <= sd u + sp (u ord0) l p1 + sp (u ord_max) l (a[p1, p3] l).

Lemma case3 l (u :  {ffun 'I_l.+2 -> configuration 4 n.+2}) 
              (p1 p2 p3: peg 4) :
   p1 != p2 -> p1 != p3 -> p2 != p3 ->
   p1 != p0 -> p2 != p0 -> p3 != p0 ->
  (forall k : 'I_l.+2,
     0 < k < l.+1 -> codom (u k) \subset [:: p2; a[p1, p3] k]) ->
  (u ord0 ord_max = p1) ->
  (u ord_max ord_max = a[p3, p1] l) ->
  (S_[l.+1] n.+2).*2 <= sd u + sp (u ord0) l.+1 p1 
                             + sp (u ord_max) l.+1 (a[p3, p1] l).
5129
IH:forall (l : nat) (p1 p2 p3 : ordinal_eqType 4)
  (u : {ffun 'I_l.+1 -> configuration 4 n.+1}),
p1 != p2 ->
p1 != p3 ->
p2 != p3 ->
p1 != p0 ->
p2 != p0 ->
p3 != p0 ->
(forall k : 'I_l.+1,
 0 < k < l ->
 codom (u k) \subset [:: p2;  a[p1, p3] k]) ->
(S_[l] n.+1).*2 <=
sd (n:=n.+1) u + sp (n:=n.+1) (u ord0) l p1 +
sp (n:=n.+1) (u ord_max) l ( a[p1, p3] l)
55755811
p1 != p2 ->
p1 != p3 ->
p2 != p3 ->
p1 != p0 ->
p2 != p0 ->
p3 != p0 ->
(forall k : 'I_l.+2,
 0 < k < l.+1 ->
 codom (u k) \subset [:: p2;  a[p1, p3] k]) ->
u ord0 ord_max = p1 ->
u ord_max ord_max =  a[p3, p1] l ->
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u + sp (n:=n.+2) (u ord0) l.+1 p1 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
Proof.
15cf
move=> p1Dp2 p1Dp3 p2Dp3 p1Dp0 p2Dp0 p3Dp0.
512915d255755811151617121314
(forall k : 'I_l.+2,
 0 < k < l.+1 ->
 codom (u k) \subset [:: p2;  a[p1, p3] k]) ->
u ord0 ord_max = p1 ->
u ord_max ord_max =  a[p3, p1] l ->
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u + sp (n:=n.+2) (u ord0) l.+1 p1 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
move=> cH KH1 KH2.
512915d2557558111516171213145595701436
573
pose u':= ([ffun i => ↓[u i]] : {ffun 'I_l.+2 -> configuration 4 n.+1})
 : {ffun 'I_l.+2 -> configuration 4 n.+1}.
512915d255755811151617121314559570143655a
573
have H: forall k : 'I_l.+2,
    0 < k < l.+1 -> codom (u' k) \subset [:: p2; a[p1, p3] k].
15e1
55b
512915d255755811151617121314559570143655a561
573
  by move=> k kH; rewrite ffunE; apply/codom_liftr/cH.
15e7
573
have apegpaD2 a : a[p1, p3] a != p2.
512915d255755811151617121314559570143655a561228
562
512915d255755811151617121314559570143655a561144c
573
  by rewrite /apeg; case: odd; rewrite // eq_sym.
15f1
573
have apegpaD0 a : a[p1, p3] a != p0.
512915d255755811151617121314559570143655a561144c228
569
512915d255755811151617121314559570143655a561144c1457
573
  by rewrite /apeg; case: odd; rewrite // eq_sym.
15fb
573
case: (pickP (fun i => u i ldisk != a[p1, p3] i)) => [x xH|pH]; last first.
512915d255755811151617121314559570143655a561144c1457
pH:(fun i : ordinal_finType l.+2 =>
 u i ldisk !=  a[p1, p3] i) =1 xpred0
573
512915d255755811151617121314559570143655a561144c1457
x:ordinal_finType l.+2
xH:u x ldisk !=  a[p1, p3] x
573
  have {pH}ukLE k : u k ldisk = a[p1, p3] k by have := pH k; case: eqP.
512915d255755811151617121314559570143655a561144c1457
ukLE:forall k : ordinal_finType l.+2,
u k ldisk =  a[p1, p3] k
573
1604
  have := case2 _ _ _ _ _ _ _ cH KH1 KH2 ukLE.
160c
((forall (l : nat) (p1 p2 p3 : ordinal_eqType 4)
    (u : {ffun 'I_l.+1 -> configuration 4 n.+1}),
  p1 != p2 ->
  p1 != p3 ->
  p2 != p3 ->
  p1 != p0 ->
  p2 != p0 ->
  p3 != p0 ->
  (forall k : 'I_l.+1,
   0 < k < l ->
   codom (u k) \subset [:: p2;  a[p1, p3] k]) ->
  (S_[l] n.+1).*2 <=
  sd (n:=n.+1) u +
  sp (n:=n.+1) (u ord0) l ( a[p1, p3] 0) +
  sp (n:=n.+1) (u ord_max) l ( a[p1, p3] l)) ->
 p1 != p2 ->
 p1 != p3 ->
 p2 != p3 ->
 p1 != p0 ->
 p2 != p0 ->
 p3 != p0 ->
 (S_[l.+1] n.+2).*2 <=
 sd (n:=n.+2) u + sp (n:=n.+2) (u ord0) l.+1 p1 +
 sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)) ->
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u + sp (n:=n.+2) (u ord0) l.+1 p1 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
1604
  by apply.
1606
573
have exH : exists x, u (inord x) ldisk != a[p1, p3] x.
1606
exists x : nat, u (inord x) ldisk !=  a[p1, p3] x
512915d255755811151617121314559570143655a561144c145716071608
exH:exists x : nat,
  u (inord x) ldisk !=  a[p1, p3] x
573
  exists x; rewrite (_ : inord x = x) //.
1606
inord x = x
1619
  by apply/val_eqP; rewrite /= inordK.
161b
573
case: (@ex_minnP _ exH) => a aH {exH}aMin.
512915d255755811151617121314559570143655a561144c145716071608228
aH:u (inord a) ldisk !=  a[p1, p3] a
aMin:forall n0 : nat,
u (inord n0) ldisk !=  a[p1, p3] n0 -> a <= n0
573
have exL : exists x, (x <= l) && (u (inord x) ord_max != a[p1, p3] x).
1627
exists x : nat,
  (x <= l) && (u (inord x) ord_max !=  a[p1, p3] x)
512915d255755811151617121314559570143655a561144c14571607160822816281629
exL:exists x : nat,
  (x <= l) &&
  (u (inord x) ord_max !=  a[p1, p3] x)
573
  exists x; apply/andP; split => //.
1627
x <= l
1627
u (inord x) ord_max !=  a[p1, p3] x
162f
    have := ltn_ord x.
1627
x < l.+2 -> x <= l
1636
     rewrite ltnS leq_eqVlt => /orP[] // /eqP xEl1.
512915d255755811151617121314559570143655a561144c14571607160822816281629
xEl1:x = l.+1
1635
1636
    case/eqP: xH.
512915d255755811151617121314559570143655a561144c14571607228162816291641
u x ldisk =  a[p1, p3] x
1636
    rewrite (_ : x = ord_max); last by apply/val_eqP/eqP.
1645
u ord_max ldisk =  a[p1, p3] ord_max
1636
    by apply/eqP/val_eqP; rewrite /= apegS.
1627
1638
162e
  by rewrite (_ : inord x = x) //; apply/val_eqP; rewrite /= inordK.
1630
573
have exL1 y : (y <= l) && (u (inord y) ord_max != a[p1, p3] y) -> y <= l.
512915d255755811151617121314559570143655a561144c145716071608228162816291631
y:nat
(y <= l) && (u (inord y) ord_max !=  a[p1, p3] y) ->
y <= l
512915d255755811151617121314559570143655a561144c145716071608228162816291631
exL1:forall y : nat,
(y <= l) &&
(u (inord y) ord_max !=  a[p1, p3] y) -> 
y <= l
573
  by case/andP.
1659
573
pose a1 := ex_maxn exL exL1.
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a
a1:=ex_maxn
  (P:=fun x : nat =>
      (x <= l) &&
      (u (inord x) ord_max !=  a[p1, p3] x))
  (m:=l) exL exL1:nat
573
have a1H: u (inord a1) ldisk != a[p1, p3] a1.
1661
u (inord a1) ldisk !=  a[p1, p3] a1
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662
a1H:u (inord a1) ldisk !=  a[p1, p3] a1
573
  by rewrite /a1; case: ex_maxnP => i /andP[].
1669
573
have a1Ll : a1 <= l by rewrite /a1; case: ex_maxnP => i /andP[].
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a
a1Ll:a1 <= l
573
have a1Max i : a1 < i < l.+2 -> u (inord i) ord_max = a[p1, p3] i.
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a167214a
a1 < i < l.+2 -> u (inord i) ord_max =  a[p1, p3] i
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672
a1Max:forall i : nat,
a1 < i < l.+2 ->
u (inord i) ord_max =  a[p1, p3] i
573
  case/andP=> a1Li; rewrite ltnS leq_eqVlt => /orP[/eqP iEl1|iLl].
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a167214a
a1Li:a1 < i
iEl1:i = l.+1
u (inord i) ord_max =  a[p1, p3] i
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a167214a16801470
1682
1679
    rewrite iEl1 apegS (_ : inord l.+1 = ldisk) //.
167f
inord l.+1 = ldisk
1683
    by apply/val_eqP; rewrite /= inordK.
1685
1682
1678
  move: a1Li; rewrite /a1; case: ex_maxnP => j _ /(_ i).
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a167214a1470
j:nat
((i <= l) && (u (inord i) ord_max !=  a[p1, p3] i) ->
 i <= j) ->
j < i -> u (inord i) ord_max =  a[p1, p3] i
1678
  by case: eqP; rewrite // andbT => _ /(_ iLl); rewrite ltnNge => ->.
167a
573
have aLld1 : a <= l.+1.
167a
a <= l.+1
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b
573
  rewrite -ltnS; apply: leq_trans (ltn_ord x).
167a
a < x.+1
169a
  rewrite ltnS; apply: aMin;rewrite (_ : inord x = x) //.
512915d255755811151617121314559570143655a561144c14571607160822816281631165a1662166a1672167b
1620
169a
  by apply/val_eqP; rewrite /= inordK.
169c
573
(* Showing the symmetry of the problem *)
wlog aLl1Ba1 : u p1 p3 p1Dp3 p2Dp3 p3Dp0 apegpaD2 apegpaD0 p1Dp2 p1Dp0 
   {u' H xH exL exL1}cH KH1 KH2 a a1 aH aLld1 aMin a1H a1Ll a1Max /
  a <= l.+1 - a1.
169c
(forall (u : {ffun 'I_l.+2 -> configuration 4 n.+2})
   (p1 p3 : peg 4),
 p1 != p3 ->
 p2 != p3 ->
 p3 != p0 ->
 (forall a : nat,  a[p1, p3] a != p2) ->
 (forall a : nat,  a[p1, p3] a != p0) ->
 p1 != p2 ->
 p1 != p0 ->
 (forall k : 'I_l.+2,
  0 < k < l.+1 ->
  codom (u k) \subset [:: p2;  a[p1, p3] k]) ->
 u ord0 ord_max = p1 ->
 u ord_max ord_max =  a[p3, p1] l ->
 forall a a1 : nat,
 u (inord a) ldisk !=  a[p1, p3] a ->
 a <= l.+1 ->
 (forall n0 : nat,
  u (inord n0) ldisk !=  a[p1, p3] n0 -> a <= n0) ->
 u (inord a1) ldisk !=  a[p1, p3] a1 ->
 a1 <= l ->
 (forall i : nat,
  a1 < i < l.+2 -> u (inord i) ord_max =  a[p1, p3] i) ->
 a <= l.+1 - a1 ->
 (S_[l.+1] n.+2).*2 <=
 sd (n:=n.+2) u + sp (n:=n.+2) (u ord0) l.+1 p1 +
 sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)) ->
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u + sp (n:=n.+2) (u ord0) l.+1 p1 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
512915d255726f131607558
p1, p3:peg 4
161714144c145715125595701436
a, a1:nat
162859b1629166a1672167b
aLl1Ba1:a <= l.+1 - a1
573
  move=> wH.
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b
wH:forall
  (u : {ffun 'I_l.+2 -> configuration 4 n.+2})
  (p1 p3 : peg 4),
p1 != p3 ->
p2 != p3 ->
p3 != p0 ->
(forall a : nat,  a[p1, p3] a != p2) ->
(forall a : nat,  a[p1, p3] a != p0) ->
p1 != p2 ->
p1 != p0 ->
(forall k : 'I_l.+2,
 0 < k < l.+1 ->
 codom (u k) \subset [:: p2;  a[p1, p3] k]) ->
u ord0 ord_max = p1 ->
u ord_max ord_max =  a[p3, p1] l ->
forall a a1 : nat,
u (inord a) ldisk !=  a[p1, p3] a ->
a <= l.+1 ->
(forall n0 : nat,
 u (inord n0) ldisk !=  a[p1, p3] n0 -> a <= n0) ->
u (inord a1) ldisk !=  a[p1, p3] a1 ->
a1 <= l ->
(forall i : nat,
 a1 < i < l.+2 ->
 u (inord i) ord_max =  a[p1, p3] i) ->
a <= l.+1 - a1 ->
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u + sp (n:=n.+2) (u ord0) l.+1 p1 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
573
16ac
  case: (leqP a (l.+1 -a1)) => [aLl1Ba1|l1Ba1La].
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16b616b1
573
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16b6
l1Ba1La:l.+1 - a1 < a
573
16ad
    by apply: wH aLl1Ba1.
16bd
573
16ac
  have a_gt0 : 0 < a by apply: leq_ltn_trans l1Ba1La.
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16b616be5a0
573
16ac
  pose u1 : {ffun 'I_l.+2 -> configuration 4 n.+2} := 
     [ffun i : 'I_l.+2 => u (inord (l.+1 - i))].
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16b616be5a0
u1:=[ ffun i : 'I_l.+2 => u (inord (l.+1 - i))]
  : {ffun 'I_l.+2 -> configuration 4 n.+2}:{ffun 'I_l.+2 -> configuration 4 n.+2}
573
16ac
  have -> : sd u = sd u1.
16c9
sd (n:=n.+2) u = sd (n:=n.+2) u1
16c9
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u1 + sp (n:=n.+2) (u ord0) l.+1 p1 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
16ad
    have F : injective (fun i : 'I_l.+1 => (inord (l - i) : 'I_l.+1)).
16c9
injective (fun i : 'I_l.+1 => inord (l - i))
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16b616be5a016ca
F:injective (fun i : 'I_l.+1 => inord (l - i))
16ce
16d016ad
      move=> i j /val_eqP.
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16b616be5a016ca
i, j:'I_l.+1
val (inord (l - i)) == val (inord (l - j)) -> i = j
16d6
        have lBiLl (i1 : 'I_l.+1) : l - i1 < l.+1.
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16b616be5a016ca
i, j, i1:'I_l.+1
l - i1 < l.+1
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16b616be5a016ca16de
lBiLl:forall i1 : 'I_l.+1, l - i1 < l.+1
16df
16d716d016ad
          by rewrite ltn_subLR ?leq_addl // -ltnS.
16e8
16df
16d6
      rewrite /= !inordK // => /eqP liE; apply/val_eqP => /=.
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16b616be5a016ca16de16e9
liE:l - i = l - j
i == j
16d6
      rewrite -(subKn (_ : i <= l)); last by rewrite -ltnS.
16f0
l - (l - i) == j
16d6
      by rewrite liE subKn // -ltnS.
16d8
16ce
16cf
    rewrite [sd u](reindex_inj F) /=.
16d8
\sum_(j < l.+1)
   `d[u (inord (inord (l - j))), u
                                   (inord
                                      (inord (l - j)).+1)]_smove =
sd (n:=n.+2) u1
16cf
    apply: eq_bigr => i _; rewrite !ffunE.
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16b616be5a016ca16d914c4
`d[u (inord (inord (l - i))), u
                                (inord
                                   (inord (l - i)).+1)]_smove =
`d[u (inord (l.+1 - inord i)), u
                                 (inord
                                    (l.+1 - inord i.+1))]_smove
16cf
    have iLl2 : i < l.+2 by apply: leq_trans (ltn_ord _) _.
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16b616be5a016ca16d914c4
iLl2:i < l.+2
1702
16cf
    have lBiLl : l - i < l.+1 by rewrite ltn_subLR ?leq_addl // -ltnS.
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16b616be5a016ca16d914c41707
lBiLl:l - i < l.+1
1702
16cf
    have iLl : i <= l by rewrite -ltnS.
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16b616be5a016ca16d914c41707170c
iLl:i <= l
1702
16cf
    rewrite gdistC; last by apply/move_sym/ssym.
1710
`d[u (inord (inord (l - i)).+1), u
                                   (inord
                                      (inord (l - i)))]_smove =
`d[u (inord (l.+1 - inord i)), u
                                 (inord
                                    (l.+1 - inord i.+1))]_smove
16cf
    congr (`d[u _,u _]_smove).
1710
inord (inord (l - i)).+1 = inord (l.+1 - inord i)
1710
inord (inord (l - i)) = inord (l.+1 - inord i.+1)
16d016ad
      by apply/val_eqP; rewrite /= !inordK ?subSn.
1710
171c
16cf
    by apply/val_eqP; rewrite /= !inordK // ?subSS ltnS // ltnW.
16c9
16d1
16ac
  pose p1' := a[p1, p3] l; pose p3' := a[p3, p1] l.
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16b616be5a016ca
p1':= a[p1, p3] l:peg 4
p3':= a[p3, p1] l:peg 4
16d1
16ac
  have -> : sp (u ord0) l.+1 p1 = sp (u1 ord_max) l.+1 (a[p1', p3'] l).
1726
sp (n:=n.+2) (u ord0) l.+1 p1 =
sp (n:=n.+2) (u1 ord_max) l.+1 ( a[p1', p3'] l)
1726
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u1 +
sp (n:=n.+2) (u1 ord_max) l.+1 ( a[p1', p3'] l) +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
16ad 
    apply: eq_bigr => i _; congr (_ * _).
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16b616be5a016ca17271728
i:ordinal_finType n.+2
(u ord0 i != p1) = (u1 ord_max i !=  a[p1', p3'] l)
172d
    rewrite -apegD addnn apegO odd_double /= apegS apeg0.
1733
(u ord0 i != p1) = (u1 ord_max i != p1)
172d
    by rewrite ffunE subnn inord_eq0.
1726
172f
16ac
  have -> : sp (u ord_max) l.+1 (a[p3, p1] l) = sp (u1 ord0) l.+1 p3'.
1726
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l) =
sp (n:=n.+2) (u1 ord0) l.+1 p3'
1726
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u1 +
sp (n:=n.+2) (u1 ord_max) l.+1 ( a[p1', p3'] l) +
sp (n:=n.+2) (u1 ord0) l.+1 p3'
16ad
    apply: eq_bigr => i _; congr (_ * _).
1733
(u ord_max i !=  a[p3, p1] l) = (u1 ord0 i != p3')
1741
    rewrite ffunE subn0.
1733
(u ord_max i !=  a[p3, p1] l) =
(u (inord l.+1) i != p3')
1741
    suff -> : inord l.+1 = ord_max :> 'I_l.+2 by [].
1733
a16
1741 
    by apply/val_eqP; rewrite /= inordK.
1726
1743
16ac
  rewrite addnAC.
1726
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u1 + sp (n:=n.+2) (u1 ord0) l.+1 p3' +
sp (n:=n.+2) (u1 ord_max) l.+1 ( a[p1', p3'] l)
16ac
  have FF : (l.+1 - a1) <= l.+1 - (l.+1 - a).
1726
l.+1 - a1 <= l.+1 - (l.+1 - a)
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16b616be5a016ca17271728
FF:l.+1 - a1 <= l.+1 - (l.+1 - a)
1755
16ad
    by rewrite leq_sub2l // leq_subLR addnC -leq_subLR ltnW.
175c
1755
16ac
  have a1Ll1 : a1 <= l.+1 by rewrite (leq_trans a1Ll).
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16b616be5a016ca17271728175d
a1Ll1:a1 <= l.+1
1755
16ac
  apply: wH FF => //.
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16be5a016ca172717281765
p3' != p1'
1769
p2 != p1'
1769
p1' != p0
1769
forall a : nat,  a[p3', p1'] a != p2
1769
forall a : nat,  a[p3', p1'] a != p0
1769
p3' != p2
1769
p3' != p0
1769
forall k : 'I_l.+2,
0 < k < l.+1 ->
codom (u1 k) \subset [:: p2;  a[p3', p1'] k]
1769
u1 ord0 ord_max = p3'
1769
u1 ord_max ord_max =  a[p1', p3'] l
1769
u1 (inord (l.+1 - a1)) ldisk
!=  a[p3', p1'] (l.+1 - a1)
1769
l.+1 - a1 <= l.+1
1769
forall n0 : nat,
u1 (inord n0) ldisk !=  a[p3', p1'] n0 ->
l.+1 - a1 <= n0
1769
u1 (inord (l.+1 - a)) ldisk !=  a[p3', p1'] (l.+1 - a)
1769
l.+1 - a <= l
1769
forall i : nat,
l.+1 - a < i < l.+2 ->
u1 (inord i) ord_max =  a[p3', p1'] i
16ad
  -
1767 by rewrite apeg_eqC eq_sym.
1769
176d
176e17701772177417761778177a177c177e1780178217841786178816ad
  -
178c by rewrite eq_sym apeg_neq // eq_sym.
1769
176f
17701772177417761778177a177c177e1780178217841786178816ad
  -
1791 by rewrite apeg_neq // eq_sym.
1769
1771
1772177417761778177a177c177e1780178217841786178816ad
  -
1796 by move=> b; rewrite !apeg_neq // eq_sym.
1769
1773
177417761778177a177c177e1780178217841786178816ad
  -
179b by move=> b; rewrite !apeg_neq // eq_sym.
1769
1775
17761778177a177c177e1780178217841786178816ad
  -
17a0 by rewrite apeg_neq // eq_sym.
1769
1777
1778177a177c177e1780178217841786178816ad
  -
17a5 by rewrite apeg_neq // eq_sym.
1769
1779
177a177c177e1780178217841786178816ad
  -
17aa move=> k /andP[kH1 kH2]; rewrite ffunE.
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16be5a016ca172717281765
k:'I_l.+2
kH1:0 < k
kH2:k < l.+1
codom (u (inord (l.+1 - k))) \subset
[:: p2;  a[p3', p1'] k]
17ac
    have F1 : k <= l.+1 by apply: ltnW.
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16be5a016ca17271728176517b217b317b4
F1:k <= l.+1
17b5
17ac
    have F2 : l.+1 - k < l.+2.
17b9
l.+1 - k < l.+2
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16be5a016ca17271728176517b217b317b417ba
F2:l.+1 - k < l.+2
17b5
177a177c177e1780178217841786178816ad
      by rewrite ltn_subLR // addnS ltnS leq_addl.
17c1
17b5
17ac
    have -> :  a[p3', p1'] k = a[p1, p3] (inord (l.+1 - k) : 'I_l.+2) .
17c1
 a[p3', p1'] k =  a[p1, p3] (inord (l.+1 - k))
17c1
codom (u (inord (l.+1 - k))) \subset
[:: p2;  a[p1, p3] (inord (l.+1 - k))]
177a177c177e1780178217841786178816ad
      rewrite inordK // -apegD -apegS -addnS.
17c1
 a[p1, p3] (k + l.+1) =  a[p1, p3] (l.+1 - k)
17ca
      by rewrite [LHS]apegO [RHS]apegO oddD oddB //= addbC.
17c1
17cc
17ac
    apply: cH; rewrite inordK // subn_gt0 // ltn_subLR // kH2 //=.
512915d255755811151617121314570143655a561144c145716071608228162816291631165a1662166a1672167b59b16be5a016ca17271728176517b217b317b417ba17c2
l.+1 < k + l.+1
17ac
    by rewrite addnS ltnS -{1}[l.+1]add1n leq_add2r.
1769
177b
177c177e1780178217841786178816ad
  -
17da rewrite !ffunE /p1' subn0 -[RHS]KH2; congr (u _ _); apply/val_eqP.
1769
val (inord l.+1) == val ord_max
17dc
    by rewrite /= inordK.
1769
177d
177e1780178217841786178816ad
  -
17e3 by rewrite !ffunE /p1' subnn inord_eq0 // KH1 /p3' -apegD addnn apeg_double.
1769
177f
1780178217841786178816ad
  -
17e8 rewrite ffunE inordK //=; last by rewrite ltnS leq_subr.
1769
u (inord (l.+1 - (l.+1 - a1))) ldisk
!=  a[p3', p1'] (l.+1 - a1)
17ea 
    rewrite subKn // -apegD apegO oddD oddB //= addbAC !addNb addbb /=.
1769
u (inord a1) ldisk !=  a[p1, p3] (~~ ~~ odd a1)
17ea
    by rewrite -oddS -apegO apegS.
1769
1781
178217841786178816ad
  -
17f5 by rewrite leq_subr.
1769
1783
17841786178816ad
  -
17fa move=> k; rewrite ffunE.
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16be5a016ca172717281765a81
u (inord (l.+1 - inord k)) ldisk !=  a[p3', p1'] k ->
l.+1 - a1 <= k
17fc
    case: leqP => //; rewrite ltn_subRL addnC -ltn_subRL => kH.
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16be5a016ca172717281765a81
kH:a1 < l.+1 - k
u (inord (l.+1 - inord k)) ldisk !=  a[p3', p1'] k ->
false
17fc
    have kLl1 : k < l.+1 by rewrite -subn_gt0 (leq_ltn_trans _ kH).
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16be5a016ca172717281765a811807
kLl1:k < l.+1
1808
17fc
      rewrite inordK; last by rewrite ltnS ltnW.
180c
u (inord (l.+1 - k)) ldisk !=  a[p3', p1'] k -> false
17fc
      have /a1Max-> : a1 < l.+1 - k < l.+2 by rewrite kH /= ltnS leq_subr.
180c
 a[p1, p3] (l.+1 - k) !=  a[p3', p1'] k -> false
17fc
    case/eqP; rewrite -apegD apegO oddB 1?ltnW // -oddD -apegO.
180c
 a[p1, p3] (l.+1 + k) =  a[p3, p1] (k + l)
17fc
    by rewrite addSn addnC apegS.
1769
1785
1786178816ad
  -
181b rewrite ffunE inordK ?ltnS ?leq_subr //.
1769
u (inord (l.+1 - (l.+1 - a))) ldisk
!=  a[p3', p1'] (l.+1 - a)
181d
    rewrite subKn //.
1769
u (inord a) ldisk !=  a[p3', p1'] (l.+1 - a)
181d
    rewrite -apegD apegO oddD oddB //= !addNb addbAC addbb /=.
1769
u (inord a) ldisk !=  a[p1, p3] (~~ ~~ odd a)
181d
    by rewrite -oddS -apegO apegS.
1769
1787
178816ad
  -
182c by rewrite -[a]prednK // subSS leq_subr.
1769
1789
16ac
  move=> k /andP[l1BaLk kLl2]; rewrite ffunE.
512915d255755811151617121314559570143655a561144c145716071608228162816291631165a1662166a1672167b59b16be5a016ca172717281765a81
l1BaLk:l.+1 - a < k
kLl2:k < l.+2
u (inord (l.+1 - inord k)) ord_max =  a[p3', p1'] k
16ac
  rewrite inordK //; apply/eqP; case: eqP => // /eqP.
1836
u (inord (l.+1 - k)) ord_max !=  a[p3', p1'] k ->
false
16ac
  suff -> : a[p3', p1'] k = a[p1, p3] (l.+1 - k).
1836
u (inord (l.+1 - k)) ord_max !=  a[p1, p3] (l.+1 - k) ->
false
1836
 a[p3', p1'] k =  a[p1, p3] (l.+1 - k)
16ad
    by move=> /aMin; rewrite leqNgt ltn_subLR // addnC -ltn_subLR // l1BaLk.
1836
1844
16ac
  by rewrite [RHS]apegO oddB // -oddD -apegO addSn apegS addnC apegD.
16ae
573
pose u':= ([ffun i => ↓[u i]] : {ffun 'I_l.+2 -> configuration 4 n.+1})
 : {ffun 'I_l.+2 -> configuration 4 n.+1}.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a
573
have H: forall k : 'I_l.+2,
    0 < k < l.+1 -> codom (u' k) \subset [:: p2; a[p1, p3] k].
184e
55b
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a561
573
  by move=> k kH; rewrite ffunE; apply/codom_liftr/cH.
1854
573
have aMin1 i : i < a -> u (inord i) ord_max = a[p1, p3] i.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a56114a
58f
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a561593
573
  by case: (u (inord i) ord_max =P a[p1, p3] i) => // /eqP /aMin; case: leqP.
185e
573
have a_gt0 : 0 < a.
185e
0 < a
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a0
573
  have : 0 <= a by [].
185e
0 <= a -> 0 < a
1866
  rewrite leq_eqVlt => /orP[/eqP aE0|] //.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a561593
aE0:0 = a
1865
1866
  by case/eqP: aH; rewrite inord_eq0 // -aE0.
1868
573
pose ai : 'I_l.+2 := inord a.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a5
573
have aiE : ai = a :> nat by rewrite inordK.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa
573
have aLa1 : a <= a1.
187c
a <= a1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa
aLa1:a <= a1
573
  case: leqP => // => a1La.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa
a1La:a1 < a
false
1881
  by case/eqP: aH; apply: a1Max; rewrite a1La.
1883
573
have uaLEp2 : u (inord a) ldisk = p2.
1883
u (inord a) ldisk = p2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884
uaLEp2:u (inord a) ldisk = p2
573
  have /cH : 0 < (inord a : 'I_l.+2) < l.+1.
1883
0 < inord a < l.+1
1883
codom (u (inord a)) \subset
[:: p2;  a[p1, p3] (inord a)] ->
u (inord a) ldisk = p2
1893
    by rewrite inordK // a_gt0 /= (leq_ltn_trans _ (_ : a1 < l.+1)).
1883
189c
1892
  move=> /subsetP/(_ (u (inord a) ldisk) (codom_f _ _)).
1883
u (inord a) ldisk \in [:: p2;  a[p1, p3] (inord a)] ->
u (inord a) ldisk = p2
1892
  by rewrite !inE inordK // (negPf aH) orbF => /eqP.
1894
573
have ua1LEp2 : u (inord a1) ldisk = p2.
1894
u (inord a1) ldisk = p2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa18841895
ua1LEp2:u (inord a1) ldisk = p2
573
  have /cH : 0 < (inord a1 : 'I_l.+2) < l.+1.
1894
0 < inord a1 < l.+1
1894
codom (u (inord a1)) \subset
[:: p2;  a[p1, p3] (inord a1)] ->
u (inord a1) ldisk = p2
18ac
    by rewrite inordK // !ltnS (leq_trans a1Ll) // (leq_trans a_gt0).
1894
18b5
18ab
  move=> /subsetP/(_ (u (inord a1) ldisk) (codom_f _ _)).
1894
u (inord a1) ldisk \in [:: p2;  a[p1, p3] (inord a1)] ->
u (inord a1) ldisk = p2
18ab
  by rewrite !inE inordK ?ltnS ?(leq_trans a1Ll) // (negPf a1H) orbF => /eqP.
18ad
573
have a1_gt0 : 0 < a1 by apply: leq_trans a_gt0 _.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae
a1_gt0:0 < a1
573
pose b := a1 - a.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c4
b:=a1 - a:nat
573
pose c := l.+1 - a1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c9
c:=l.+1 - a1:nat
573
have c_gt0 : 0 < c by rewrite subn_gt0.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce
c_gt0:0 < c
573
have l1E : l.+1 = a + b + c.
18d2
l.+1 = a + b + c
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d3
l1E:l.+1 = a + b + c
573
  by rewrite /b /c [a + _]addnC subnK // addnC subnK // ltnW.
18da
573
pose si i : configuration 4 n.+1 := 
   sgdist1 p2 (u (inord i)) (u (inord i.+1)).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb
573
pose ti i : configuration 4 n.+1 := 
   sgdist2 p2 (u (inord i)) (u (inord i.+1)).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f0
573
have sitiH i : i < a.-1 ->
   [/\ codom (si i) \subset [:: p2; u (inord i.+1) ldisk ], 
       codom (ti i) \subset [:: u (inord i) ldisk; p2] & 
       `d[u (inord i), u (inord i.+1)]_smove = 
         D[[:: ↓[u (inord i)]; si i; ti i; ↓[u (inord i.+1)]]].+2].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f014a
5f5
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f9
573
  move=> iLa.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f014a5fe
5ff
18eb
  have iLa1 : i < a by rewrite (leq_trans iLa) // ssrnat.leq_pred.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f014a5fe604
5ff
18eb
  apply: sgdistE; rewrite ?aMin1 // -1?[a]prednK //.
18f5
61b
18f5
p2 !=  a[p1, p3] i.+1
18ec
    by rewrite apegS apeg_eqC.
18f5
18fb
18eb
  by rewrite eq_sym.
18ed
573
pose pa := a[p1, p3] a.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998e
573
pose sam1 : configuration 4 n.+1 := sgdist1 pa (u (inord a.-1)) (u (inord a)).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8c
573
pose tam1 : configuration 4 n.+1 := sgdist2 pa (u (inord a.-1)) (u (inord a)).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb91
573
have [sam1C tam1C duam1ua1E] : 
   [/\ codom sam1 \subset [:: pa; u (inord a) ldisk ], 
       codom tam1 \subset [:: u (inord a.-1) ldisk; pa] & 
       `d[u (inord a.-1), u (inord a)]_smove = 
         D[[:: ↓[u (inord a.-1)]; sam1; tam1; ↓[u (inord a)]]].+2].
190d
b95
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb91b99b9a63f
573
  apply: sgdistE; rewrite /pa ?uaLEp2 ?aMin1 ?prednK //.
190d
 a[p1, p3] a.-1 !=  a[p1, p3] a
1911
  by rewrite -{2}(prednK a_gt0) apegS apeg_eqC.
1913
573
rewrite uaLEp2 in sam1C.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb91b9a63f
sam1C:codom sam1 \subset [:: pa; p2]
573
have {}tam1C : codom tam1 \subset [:: p1; p3].
191e
666
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a
573
  move: tam1C; rewrite aMin1; last by rewrite -{2}[a]prednK.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f
bc3
1923
  by rewrite /pa  -{2}[a]prednK // apegS codom_apeg.
1925
573
pose u1 := 
  [ffun i : 'I_(3 * a).+1 =>
  if (i %% 3) == 0 then ↓[u (inord (i %/ 3))]
  else if (i %% 3) == 1 then 
     (if i == (3 * a.-1).+1 :> nat then sam1 else si (i %/ 3))
  else 
     (if i == (3 * a.-1).+2 :> nat then tam1 else ti (i %/ 3))].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a
u1:=[ffun i => if i %% 3 == 0
           then ↓[u (inord (i %/ 3))]
           else
            if i %% 3 == 1
            then
             if i == (3 * a.-1).+1 :> nat
             then sam1
             else si (i %/ 3)
            else
             if i == (3 * a.-1).+2 :> nat
             then tam1
             else ti (i %/ 3)]:{ffun forall x : ordinal_finType (3 * a).+1,
      (fun=> configuration 4 n.+1) x}
573
have u10E : u1 ord0 = ↓[u ord0] by rewrite ffunE /= inord_eq0.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683
573
have uiME : u1 ord_max = ↓[u (inord a)] by rewrite ffunE /= mod3E /= div3E.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683
uiME:u1 ord_max = ↓[u (inord a)]
573
have P1 :  a.*2 + sd u1 = \sum_(i < a) `d[u (inord i), u (inord i.+1)]_smove.
1939
a.*2 + sd (n:=n.+1) u1 =
\sum_(i < a) `d[u (inord i), u (inord i.+1)]_smove
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a
P1:a.*2 + sd (n:=n.+1) u1 =
\sum_(i < a)
   `d[u (inord i), u (inord i.+1)]_smove
573
  rewrite /sd.
1939
a.*2 +
\sum_(i < 3 * a)
   `d[u1 (inord i), u1 (inord i.+1)]_smove =
\sum_(i < a) `d[u (inord i), u (inord i.+1)]_smove
193f
  have -> := @sum3E _ (fun i => `d[u1 (inord i), u1 (inord i.+1)]_smove).
1939
a.*2 +
\sum_(i < a)
   (`d[u1 (inord (3 * i)), u1 (inord (3 * i).+1)]_smove +
    `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
    `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove) =
\sum_(i < a) `d[u (inord i), u (inord i.+1)]_smove
193f
  have -> : a.*2 = \sum_(i < a) 2.
1939
a.*2 = \sum_(i < a) 2
1939
\sum_(i < a) 2 +
\sum_(i < a)
   (`d[u1 (inord (3 * i)), 
    u1 (inord (3 * i).+1)]_smove +
    `d[u1 (inord (3 * i).+1), 
    u1 (inord (3 * i).+2)]_smove +
    `d[u1 (inord (3 * i).+2), 
    u1 (inord (3 * i).+3)]_smove) =
\sum_(i < a) `d[u (inord i), u (inord i.+1)]_smove
1940
    by rewrite sum_nat_const /= cardT size_enum_ord muln2.
1939
\sum_(i < a) 2 +
\sum_(i < a)
   (`d[u1 (inord (3 * i)), u1 (inord (3 * i).+1)]_smove +
    `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
    `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove) =
\sum_(i < a) `d[u (inord i), u (inord i.+1)]_smove
193f
  rewrite -big_split /=.
1939
\sum_(i < a)
   (2 +
    (`d[u1 (inord (3 * i)), u1 (inord (3 * i).+1)]_smove +
     `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
     `d[u1 (inord (3 * i).+2), u1 (inord (3 * i).+3)]_smove)) =
\sum_(i < a) `d[u (inord i), u (inord i.+1)]_smove
193f
  apply: eq_bigr => i _.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a
i:'I_a
739
193f
  have -> : u1 (inord (3 * i)) = ↓[u (inord i)].
195d
73d
195d
740
1940
    rewrite ffunE inordK; last by rewrite ltnS leq_mul2l /= ltnW.
195d
(if (3 * i) %% 3 == 0
 then ↓[u (inord ((3 * i) %/ 3))]
 else
  if (3 * i) %% 3 == 1
  then
   if 3 * i == (3 * a.-1).+1
   then sam1
   else si ((3 * i) %/ 3)
  else
   if 3 * i == (3 * a.-1).+2
   then tam1
   else ti ((3 * i) %/ 3)) = ↓[u (inord i)]
1962
    by rewrite mod3E /= div3E.
195d
761
193f
  have -> : u1 (inord (3 * i).+3) = ↓[u (inord i.+1)].
195d
7d8
195d
2 +
(`d[↓[u (inord i)], u1 (inord (3 * i).+1)]_smove +
 `d[u1 (inord (3 * i).+1), 
 u1 (inord (3 * i).+2)]_smove +
 `d[u1 (inord (3 * i).+2), ↓[
 u (inord i.+1)]]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
1940
    rewrite -[(_ * _).+3](mulnDr 3 1).
195d
u1 (inord (3 * (1 + i))) = ↓[u (inord i.+1)]
196e
    rewrite ffunE inordK; last by rewrite ltnS leq_mul2l /=.
195d
(if (3 * (1 + i)) %% 3 == 0
 then ↓[u (inord ((3 * (1 + i)) %/ 3))]
 else
  if (3 * (1 + i)) %% 3 == 1
  then
   if 3 * (1 + i) == (3 * a.-1).+1
   then sam1
   else si ((3 * (1 + i)) %/ 3)
  else
   if 3 * (1 + i) == (3 * a.-1).+2
   then tam1
   else ti ((3 * (1 + i)) %/ 3)) = ↓[u (inord i.+1)]
196e
    by rewrite mod3E /= div3E.
195d
2 +
(`d[↓[u (inord i)], u1 (inord (3 * i).+1)]_smove +
 `d[u1 (inord (3 * i).+1), u1 (inord (3 * i).+2)]_smove +
 `d[u1 (inord (3 * i).+2), ↓[u (inord i.+1)]]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
193f
  have [/eqP iEa| iDa] := boolP (i == a.-1 :> nat).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a195e
iEa:i = a.-1
197c
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a195e
iDa:i != a.-1
1970
1940
    have -> : u1 (inord (3 * i).+1) = sam1.
1980
u1 (inord (3 * i).+1) = sam1
1980
2 +
(`d[↓[u (inord i)], sam1]_smove +
 `d[sam1, u1 (inord (3 * i).+2)]_smove +
 `d[u1 (inord (3 * i).+2), ↓[
 u (inord i.+1)]]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
19831940
      rewrite ffunE inordK; last by rewrite ltnS ltn_mul2l /=.
1980
(if (3 * i).+1 %% 3 == 0
 then ↓[u (inord ((3 * i).+1 %/ 3))]
 else
  if (3 * i).+1 %% 3 == 1
  then
   if (3 * i).+1 == (3 * a.-1).+1
   then sam1
   else si ((3 * i).+1 %/ 3)
  else
   if (3 * i).+1 == (3 * a.-1).+2
   then tam1
   else ti ((3 * i).+1 %/ 3)) = sam1
198a
      rewrite mod3E /= div3E.
1980
(if (3 * i).+1 == (3 * a.-1).+1 then sam1 else si i) =
sam1
198a
      by rewrite iEa eqxx.
1980
2 +
(`d[↓[u (inord i)], sam1]_smove +
 `d[sam1, u1 (inord (3 * i).+2)]_smove +
 `d[u1 (inord (3 * i).+2), ↓[u (inord i.+1)]]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
1982
    have -> : u1 (inord (3 * i).+2) = tam1.
1980
u1 (inord (3 * i).+2) = tam1
1980
2 +
(`d[↓[u (inord i)], sam1]_smove + `d[sam1, tam1]_smove +
 `d[tam1, ↓[u (inord i.+1)]]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
19831940
      rewrite ffunE inordK; last first.
1980
(3 * i).+2 < (3 * a).+1
1980
(if (3 * i).+2 %% 3 == 0
 then ↓[u (inord ((3 * i).+2 %/ 3))]
 else
  if (3 * i).+2 %% 3 == 1
  then
   if (3 * i).+2 == (3 * a.-1).+1
   then sam1
   else si ((3 * i).+2 %/ 3)
  else
   if (3 * i).+2 == (3 * a.-1).+2
   then tam1
   else ti ((3 * i).+2 %/ 3)) = tam1
199e19831940
        by rewrite -[_.+3](mulnDr 3 1) ltnW // add1n ltnS leq_mul2l /=.
1980
19a6
199d
      rewrite mod3E /= div3E.
1980
(if (3 * i).+2 == (3 * a.-1).+2 then tam1 else ti i) =
tam1
199d
      by rewrite iEa eqxx.
1980
199f
1982
    by rewrite iEa prednK // duam1ua1E /= -!addnA add2n !addnA addn0.
1984
197c
193f
  have -> : u1 (inord (3 * i).+1) = si i.
1984
765
1984
2 +
(`d[↓[u (inord i)], si i]_smove +
 `d[si i, u1 (inord (3 * i).+2)]_smove +
 `d[u1 (inord (3 * i).+2), ↓[
 u (inord i.+1)]]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
1940
    rewrite ffunE inordK; last by rewrite ltnS ltn_mul2l /=.
1984
(if (3 * i).+1 %% 3 == 0
 then ↓[u (inord ((3 * i).+1 %/ 3))]
 else
  if (3 * i).+1 %% 3 == 1
  then
   if (3 * i).+1 == (3 * a.-1).+1
   then sam1
   else si ((3 * i).+1 %/ 3)
  else
   if (3 * i).+1 == (3 * a.-1).+2
   then tam1
   else ti ((3 * i).+1 %/ 3)) = si i
19b7
    rewrite mod3E /= div3E.
1984
(if (3 * i).+1 == (3 * a.-1).+1 then sam1 else si i) =
si i
19b7
    by rewrite [_.+1 == _.+1]eqn_mul2l /= (negPf iDa).
1984
2 +
(`d[↓[u (inord i)], si i]_smove +
 `d[si i, u1 (inord (3 * i).+2)]_smove +
 `d[u1 (inord (3 * i).+2), ↓[u (inord i.+1)]]_smove) =
`d[u (inord i), u (inord i.+1)]_smove
193f
  have -> : u1 (inord (3 * i).+2) = ti i.
1984
79d
1984
7db
1940
    rewrite ffunE inordK; last first.
1984
19a3
1984
(if (3 * i).+2 %% 3 == 0
 then ↓[u (inord ((3 * i).+2 %/ 3))]
 else
  if (3 * i).+2 %% 3 == 1
  then
   if (3 * i).+2 == (3 * a.-1).+1
   then sam1
   else si ((3 * i).+2 %/ 3)
  else
   if (3 * i).+2 == (3 * a.-1).+2
   then tam1
   else ti ((3 * i).+2 %/ 3)) = 
ti i
19ca1940
      by rewrite -[_.+3](mulnDr 3 1) ltnW // ltnS add1n leq_mul2l /=.
1984
(if (3 * i).+2 %% 3 == 0
 then ↓[u (inord ((3 * i).+2 %/ 3))]
 else
  if (3 * i).+2 %% 3 == 1
  then
   if (3 * i).+2 == (3 * a.-1).+1
   then sam1
   else si ((3 * i).+2 %/ 3)
  else
   if (3 * i).+2 == (3 * a.-1).+2
   then tam1
   else ti ((3 * i).+2 %/ 3)) = ti i
19c9
    rewrite mod3E /= div3E.
1984
(if (3 * i).+2 == (3 * a.-1).+2 then tam1 else ti i) =
ti i
19c9
    by rewrite [_.+1 == _.+1]eqn_mul2l /= (negPf iDa).
1984
7db
193f
  case: (sitiH i) => // [|_ _ ->].
1984
i < a.-1
1984
7fe
1940
    rewrite -ltnS prednK //.
1984
i.+1 < a
19e0
    have := ltn_ord i; rewrite leq_eqVlt => /orP[] //.
1984
i.+1 == a -> i.+1 < a
19e0
    by rewrite -{2}[a]prednK // [_ == _](negPf iDa).
1984
7fe
193f
  by rewrite /= -!addnA add2n addn0.
1941
573
have sitiHb i : a1 < i < l.+1 ->
   [/\ codom (si i) \subset [:: p2; u (inord i.+1) ldisk ], 
       codom (ti i) \subset [:: u (inord i) ldisk; p2] & 
       `d[u (inord i), u (inord i.+1)]_smove = 
         D[[:: ↓[u (inord i)]; si i; ti i; ↓[u (inord i.+1)]]].+2].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194214a
a1 < i < l.+1 ->
[/\ codom (si i) \subset [:: p2; u (inord i.+1) ldisk],
    codom (ti i) \subset [:: u (inord i) ldisk; p2]
  & `d[u (inord i), u (inord i.+1)]_smove =
     D[[:: ↓[u (inord i)]; si i; ti i;
           ↓[u (inord i.+1)]]].+2]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a1942
sitiHb:forall i : nat,
a1 < i < l.+1 ->
[/\ codom (si i) \subset
    [:: p2; u (inord i.+1) ldisk],
    codom (ti i) \subset
    [:: u (inord i) ldisk; p2]
  & `d[u (inord i), u (inord i.+1)]_smove =
     D[[:: ↓[u (inord i)]; 
          si i; ti i; ↓[
          u (inord i.+1)]]].+2]
573
  move=> /andP[a1Li iLl].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194214a16801470
5ff
19f5
  apply: sgdistE => //; 
      rewrite !a1Max ?a1Li ?(leq_trans a1Li) // ?(leq_trans iLl) //.
19fc
61b
19fc
18fb
19f6
    by rewrite apegS apeg_eqC.
19fc
18fb
19f5
  by rewrite eq_sym.
19f7
573
pose sa1 : configuration 4 n.+1 := 
  sgdist1 (a[p1, p3] a1) (u (inord a1)) (u (inord a1.+1)).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f8
sa1:=sgdist1 (n:=n.+1) ( a[p1, p3] a1) 
  (u (inord a1)) (u (inord a1.+1))
  : configuration 4 n.+1:configuration 4 n.+1
573
pose ta1 : configuration 4 n.+1 := 
  sgdist2 (a[p1, p3] a1) (u (inord a1)) (u (inord a1.+1)).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c
ta1:=sgdist2 (n:=n.+1) ( a[p1, p3] a1) 
  (u (inord a1)) (u (inord a1.+1))
  : configuration 4 n.+1:configuration 4 n.+1
573
have [sa1C ta1C dua1uad1E] : 
   [/\ codom sa1 \subset [:: a[p1, p3] a1; u (inord a1.+1) ldisk ], 
       codom ta1 \subset [:: u (inord a1) ldisk; a[p1, p3] a1] & 
       `d[u (inord a1), u (inord a1.+1)]_smove = 
         D[[:: ↓[u (inord a1)]; sa1 ; ta1; ↓[u (inord a1.+1)]]].+2].
1a10
[/\ codom sa1 \subset
    [::  a[p1, p3] a1; u (inord a1.+1) ldisk],
    codom ta1 \subset
    [:: u (inord a1) ldisk;  a[p1, p3] a1]
  & `d[u (inord a1), u (inord a1.+1)]_smove =
     D[[:: ↓[u (inord a1)]; sa1; ta1;
           ↓[u (inord a1.+1)]]].+2]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a11
sa1C:codom sa1 \subset
[::  a[p1, p3] a1; u (inord a1.+1) ldisk]
ta1C:codom ta1 \subset
[:: u (inord a1) ldisk;  a[p1, p3] a1]
dua1uad1E:`d[u (inord a1), u (inord a1.+1)]_smove =
 D[[:: ↓[u (inord a1)]; sa1; ta1;
       ↓[u (inord a1.+1)]]].+2
573
  apply: sgdistE; rewrite ?ua1LEp2 ?a1Max ?leqnn // 1?eq_sym //.
1a10
 a[p1, p3] a1.+1 !=  a[p1, p3] a1
1a16
  by rewrite apegS apeg_eqC eq_sym.
1a18
573
rewrite ua1LEp2 in ta1C.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a191a1b
ta1C:codom ta1 \subset [:: p2;  a[p1, p3] a1]
573
have {}sa1C : codom sa1 \subset [:: p1; p3].
1a26
codom sa1 \subset [:: p1; p3]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a27
sa1C:codom sa1 \subset [:: p1; p3]
573
  move: sa1C; rewrite a1Max //; last by rewrite leqnn.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a27
codom sa1 \subset [::  a[p1, p3] a1;  a[p1, p3] a1.+1] ->
codom sa1 \subset [:: p1; p3]
1a2c
  by rewrite apegS codom_apeg.
1a2e
573
pose u2 := 
  [ffun i : 'I_(3 * c).+1 =>
  if (i %% 3) == 0 then ↓[u (inord (a1 +(i %/ 3)))]
  else if (i %% 3) == 1 then 
     (if i == 1 :> nat then sa1 else si (a1 + (i %/ 3)))
  else 
     (if i == 2 :> nat then ta1 else ti (a1 + (i %/ 3)))].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f
u2:=[ffun i => if i %% 3 == 0
           then ↓[u (inord (a1 + i %/ 3))]
           else
            if i %% 3 == 1
            then
             if i == 1 :> nat
             then sa1
             else si (a1 + i %/ 3)
            else
             if i == 2 :> nat
             then ta1
             else ti (a1 + i %/ 3)]:{ffun forall x : ordinal_finType (3 * c).+1,
      (fun=> configuration 4 n.+1) x}
573
have u20E : u2 ord0 = ↓[u (inord a1)] by rewrite ffunE /= addn0.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c
u20E:u2 ord0 = ↓[u (inord a1)]
573
have u2ME : u2 ord_max = ↓[u ord_max].
1a40
u2 ord_max = ↓[u ord_max]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a41
u2ME:u2 ord_max = ↓[u ord_max]
573
  rewrite ffunE /= mod3E /= div3E addnC subnK //.
1a40
14a1
1a40
a1 <= l.+1
1a47
    by rewrite (_ : inord _ = ord_max) //; apply/val_eqP; rewrite /= inordK.
1a40
1a4f
1a46
  by rewrite ltnW // ltnS.
1a48
573
have P2 : c.*2 + sd u2 = 
  \sum_(a1 <= i < l.+1)  `d[u (inord i), u (inord i.+1)]_smove.
1a48
c.*2 + sd (n:=n.+1) u2 =
\sum_(a1 <= i < l.+1)
   `d[u (inord i), u (inord i.+1)]_smove
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a49
P2:c.*2 + sd (n:=n.+1) u2 =
\sum_(a1 <= i < l.+1)
   `d[u (inord i), u (inord i.+1)]_smove
573
  have -> : \sum_(a1 <= i < l.+1) `d[u (inord i), u (inord i.+1)]_smove =
            \sum_(i < c) `d[u (inord (a1 + i)), u (inord (a1 + i).+1)]_smove.
1a48
\sum_(a1 <= i < l.+1)
   `d[u (inord i), u (inord i.+1)]_smove =
\sum_(i < c)
   `d[u (inord (a1 + i)), u (inord (a1 + i).+1)]_smove
1a48
c.*2 + sd (n:=n.+1) u2 =
\sum_(i < c)
   `d[u (inord (a1 + i)), 
u (inord (a1 + i).+1)]_smove
1a5b
    rewrite -{1}[a1]add0n big_addn big_mkord.
1a48
\sum_(i < l.+1 - a1)
   `d[u (inord (i + a1)), u (inord (i + a1).+1)]_smove =
\sum_(i < c)
   `d[u (inord (a1 + i)), u (inord (a1 + i).+1)]_smove
1a62
    by apply: eq_bigr => i _; rewrite addnC.
1a48
c.*2 + sd (n:=n.+1) u2 =
\sum_(i < c)
   `d[u (inord (a1 + i)), u (inord (a1 + i).+1)]_smove
1a5a
  rewrite /sd.
1a48
c.*2 +
\sum_(i < 3 * c)
   `d[u2 (inord i), u2 (inord i.+1)]_smove =
\sum_(i < c)
   `d[u (inord (a1 + i)), u (inord (a1 + i).+1)]_smove
1a5a
  have -> := @sum3E _ (fun i => `d[u2 (inord i), u2 (inord i.+1)]_smove).
1a48
c.*2 +
\sum_(i < c)
   (`d[u2 (inord (3 * i)), u2 (inord (3 * i).+1)]_smove +
    `d[u2 (inord (3 * i).+1), u2 (inord (3 * i).+2)]_smove +
    `d[u2 (inord (3 * i).+2), u2 (inord (3 * i).+3)]_smove) =
\sum_(i < c)
   `d[u (inord (a1 + i)), u (inord (a1 + i).+1)]_smove
1a5a
  have -> : c.*2 = \sum_(i < c) 2.
1a48
c.*2 = \sum_(i < c) 2
1a48
\sum_(i < c) 2 +
\sum_(i < c)
   (`d[u2 (inord (3 * i)), 
    u2 (inord (3 * i).+1)]_smove +
    `d[u2 (inord (3 * i).+1), 
    u2 (inord (3 * i).+2)]_smove +
    `d[u2 (inord (3 * i).+2), 
    u2 (inord (3 * i).+3)]_smove) =
\sum_(i < c)
   `d[u (inord (a1 + i)), 
u (inord (a1 + i).+1)]_smove
1a5b
    by rewrite sum_nat_const /= cardT size_enum_ord muln2.
1a48
\sum_(i < c) 2 +
\sum_(i < c)
   (`d[u2 (inord (3 * i)), u2 (inord (3 * i).+1)]_smove +
    `d[u2 (inord (3 * i).+1), u2 (inord (3 * i).+2)]_smove +
    `d[u2 (inord (3 * i).+2), u2 (inord (3 * i).+3)]_smove) =
\sum_(i < c)
   `d[u (inord (a1 + i)), u (inord (a1 + i).+1)]_smove
1a5a
  rewrite -big_split /=.
1a48
\sum_(i < c)
   (2 +
    (`d[u2 (inord (3 * i)), u2 (inord (3 * i).+1)]_smove +
     `d[u2 (inord (3 * i).+1), u2 (inord (3 * i).+2)]_smove +
     `d[u2 (inord (3 * i).+2), u2 (inord (3 * i).+3)]_smove)) =
\sum_(i < c)
   `d[u (inord (a1 + i)), u (inord (a1 + i).+1)]_smove
1a5a
  apply: eq_bigr => i _.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a49
i:'I_c
2 +
(`d[u2 (inord (3 * i)), u2 (inord (3 * i).+1)]_smove +
 `d[u2 (inord (3 * i).+1), u2 (inord (3 * i).+2)]_smove +
 `d[u2 (inord (3 * i).+2), u2 (inord (3 * i).+3)]_smove) =
`d[u (inord (a1 + i)), u (inord (a1 + i).+1)]_smove
1a5a
  have -> : u2 (inord (3 * i)) = ↓[u (inord (a1 + i))].
1a87
u2 (inord (3 * i)) = ↓[u (inord (a1 + i))]
1a87
2 +
(`d[↓[u (inord (a1 + i))], 
 u2 (inord (3 * i).+1)]_smove +
 `d[u2 (inord (3 * i).+1), 
 u2 (inord (3 * i).+2)]_smove +
 `d[u2 (inord (3 * i).+2), 
 u2 (inord (3 * i).+3)]_smove) =
`d[u (inord (a1 + i)), u (inord (a1 + i).+1)]_smove
1a5b
    rewrite ffunE inordK; last by rewrite ltnS leq_mul2l /= ltnW.
1a87
(if (3 * i) %% 3 == 0
 then ↓[u (inord (a1 + (3 * i) %/ 3))]
 else
  if (3 * i) %% 3 == 1
  then
   if 3 * i == 1 then sa1 else si (a1 + (3 * i) %/ 3)
  else
   if 3 * i == 2 then ta1 else ti (a1 + (3 * i) %/ 3)) =
↓[u (inord (a1 + i))]
1a8e
    by rewrite mod3E /= div3E.
1a87
2 +
(`d[↓[u (inord (a1 + i))], u2 (inord (3 * i).+1)]_smove +
 `d[u2 (inord (3 * i).+1), u2 (inord (3 * i).+2)]_smove +
 `d[u2 (inord (3 * i).+2), u2 (inord (3 * i).+3)]_smove) =
`d[u (inord (a1 + i)), u (inord (a1 + i).+1)]_smove
1a5a
  have -> : u2 (inord (3 * i).+3) = ↓[u (inord (a1 + i).+1)].
1a87
u2 (inord (3 * i).+3) = ↓[u (inord (a1 + i).+1)]
1a87
2 +
(`d[↓[u (inord (a1 + i))], 
 u2 (inord (3 * i).+1)]_smove +
 `d[u2 (inord (3 * i).+1), 
 u2 (inord (3 * i).+2)]_smove +
 `d[u2 (inord (3 * i).+2), ↓[
 u (inord (a1 + i).+1)]]_smove) =
`d[u (inord (a1 + i)), u (inord (a1 + i).+1)]_smove
1a5b
    rewrite -[(_ * _).+3](mulnDr 3 1).
1a87
u2 (inord (3 * (1 + i))) = ↓[u (inord (a1 + i).+1)]
1a9d
    rewrite ffunE inordK; last by rewrite ltnS leq_mul2l /=.
1a87
(if (3 * (1 + i)) %% 3 == 0
 then ↓[u (inord (a1 + (3 * (1 + i)) %/ 3))]
 else
  if (3 * (1 + i)) %% 3 == 1
  then
   if 3 * (1 + i) == 1
   then sa1
   else si (a1 + (3 * (1 + i)) %/ 3)
  else
   if 3 * (1 + i) == 2
   then ta1
   else ti (a1 + (3 * (1 + i)) %/ 3)) =
↓[u (inord (a1 + i).+1)]
1a9d
    by rewrite mod3E /= div3E // add1n addnS.
1a87
2 +
(`d[↓[u (inord (a1 + i))], u2 (inord (3 * i).+1)]_smove +
 `d[u2 (inord (3 * i).+1), u2 (inord (3 * i).+2)]_smove +
 `d[u2 (inord (3 * i).+2), ↓[u (inord (a1 + i).+1)]]_smove) =
`d[u (inord (a1 + i)), u (inord (a1 + i).+1)]_smove
1a5a
  have [/eqP iEa| iDa] := boolP (i == 0 :> nat).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a88
iEa:i = 0
1aab
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a88
iDa:i != 0
1a9f
1a5b
    have -> : u2 (inord (3 * i).+1) = sa1.
1aaf
u2 (inord (3 * i).+1) = sa1
1aaf
2 +
(`d[↓[u (inord (a1 + i))], sa1]_smove +
 `d[sa1, u2 (inord (3 * i).+2)]_smove +
 `d[u2 (inord (3 * i).+2), ↓[
 u (inord (a1 + i).+1)]]_smove) =
`d[u (inord (a1 + i)), u (inord (a1 + i).+1)]_smove
1ab21a5b
      rewrite ffunE inordK; last by rewrite ltnS ltn_mul2l /=.
1aaf
(if (3 * i).+1 %% 3 == 0
 then ↓[u (inord (a1 + (3 * i).+1 %/ 3))]
 else
  if (3 * i).+1 %% 3 == 1
  then
   if (3 * i).+1 == 1
   then sa1
   else si (a1 + (3 * i).+1 %/ 3)
  else
   if (3 * i).+1 == 2
   then ta1
   else ti (a1 + (3 * i).+1 %/ 3)) = sa1
1ab9
      rewrite mod3E /=.
1aaf
(if (3 * i).+1 == 1
 then sa1
 else si (a1 + (3 * i).+1 %/ 3)) = sa1
1ab9
      by rewrite iEa eqxx.
1aaf
2 +
(`d[↓[u (inord (a1 + i))], sa1]_smove +
 `d[sa1, u2 (inord (3 * i).+2)]_smove +
 `d[u2 (inord (3 * i).+2), ↓[u (inord (a1 + i).+1)]]_smove) =
`d[u (inord (a1 + i)), u (inord (a1 + i).+1)]_smove
1ab1
    have -> : u2 (inord (3 * i).+2) = ta1.
1aaf
u2 (inord (3 * i).+2) = ta1
1aaf
2 +
(`d[↓[u (inord (a1 + i))], sa1]_smove +
 `d[sa1, ta1]_smove +
 `d[ta1, ↓[u (inord (a1 + i).+1)]]_smove) =
`d[u (inord (a1 + i)), u (inord (a1 + i).+1)]_smove
1ab21a5b
      rewrite ffunE inordK; last first.
1aaf
(3 * i).+2 < (3 * c).+1
1aaf
(if (3 * i).+2 %% 3 == 0
 then ↓[u (inord (a1 + (3 * i).+2 %/ 3))]
 else
  if (3 * i).+2 %% 3 == 1
  then
   if (3 * i).+2 == 1
   then sa1
   else si (a1 + (3 * i).+2 %/ 3)
  else
   if (3 * i).+2 == 2
   then ta1
   else ti (a1 + (3 * i).+2 %/ 3)) = ta1
1acd1ab21a5b
        by rewrite -[_.+3](mulnDr 3 1) ltnW // add1n ltnS leq_mul2l /=.
1aaf
1ad5
1acc
      rewrite mod3E /=.
1aaf
(if (3 * i).+2 == 2
 then ta1
 else ti (a1 + (3 * i).+2 %/ 3)) = ta1
1acc
      by rewrite iEa eqxx.
1aaf
1ace
1ab1
    by rewrite iEa !addn0 // dua1uad1E /= -!addnA add2n !addnA addn0.
1ab3
1aab
1a5a
  have -> : u2 (inord (3 * i).+1) = si (a1 + i).
1ab3
u2 (inord (3 * i).+1) = si (a1 + i)
1ab3
2 +
(`d[↓[u (inord (a1 + i))], 
 si (a1 + i)]_smove +
 `d[si (a1 + i), u2 (inord (3 * i).+2)]_smove +
 `d[u2 (inord (3 * i).+2), ↓[
 u (inord (a1 + i).+1)]]_smove) =
`d[u (inord (a1 + i)), u (inord (a1 + i).+1)]_smove
1a5b
    rewrite ffunE inordK; last by rewrite ltnS ltn_mul2l /=.
1ab3
(if (3 * i).+1 %% 3 == 0
 then ↓[u (inord (a1 + (3 * i).+1 %/ 3))]
 else
  if (3 * i).+1 %% 3 == 1
  then
   if (3 * i).+1 == 1
   then sa1
   else si (a1 + (3 * i).+1 %/ 3)
  else
   if (3 * i).+1 == 2
   then ta1
   else ti (a1 + (3 * i).+1 %/ 3)) = si (a1 + i)
1ae7
    rewrite mod3E /= div3E.
1ab3
(if (3 * i).+1 == 1 then sa1 else si (a1 + i)) =
si (a1 + i)
1ae7
    by rewrite [_.+1 == _.+1](eqn_mul2l 3 _ 0) /= (negPf iDa).
1ab3
2 +
(`d[↓[u (inord (a1 + i))], si (a1 + i)]_smove +
 `d[si (a1 + i), u2 (inord (3 * i).+2)]_smove +
 `d[u2 (inord (3 * i).+2), ↓[u (inord (a1 + i).+1)]]_smove) =
`d[u (inord (a1 + i)), u (inord (a1 + i).+1)]_smove
1a5a
  have -> : u2 (inord (3 * i).+2) = ti (a1 + i).
1ab3
u2 (inord (3 * i).+2) = ti (a1 + i)
1ab3
2 +
(`d[↓[u (inord (a1 + i))], 
 si (a1 + i)]_smove +
 `d[si (a1 + i), ti (a1 + i)]_smove +
 `d[ti (a1 + i), ↓[u (inord (a1 + i).+1)]]_smove) =
`d[u (inord (a1 + i)), u (inord (a1 + i).+1)]_smove
1a5b
    rewrite ffunE inordK; last first.
1ab3
1ad2
1ab3
(if (3 * i).+2 %% 3 == 0
 then ↓[u (inord (a1 + (3 * i).+2 %/ 3))]
 else
  if (3 * i).+2 %% 3 == 1
  then
   if (3 * i).+2 == 1
   then sa1
   else si (a1 + (3 * i).+2 %/ 3)
  else
   if (3 * i).+2 == 2
   then ta1
   else ti (a1 + (3 * i).+2 %/ 3)) = 
ti (a1 + i)
1afb1a5b
      by rewrite -[_.+3](mulnDr 3 1) ltnW // ltnS add1n leq_mul2l /=.
1ab3
(if (3 * i).+2 %% 3 == 0
 then ↓[u (inord (a1 + (3 * i).+2 %/ 3))]
 else
  if (3 * i).+2 %% 3 == 1
  then
   if (3 * i).+2 == 1
   then sa1
   else si (a1 + (3 * i).+2 %/ 3)
  else
   if (3 * i).+2 == 2
   then ta1
   else ti (a1 + (3 * i).+2 %/ 3)) = ti (a1 + i)
1afa
    rewrite mod3E /= div3E.
1ab3
(if (3 * i).+2 == 2 then ta1 else ti (a1 + i)) =
ti (a1 + i)
1afa
    by rewrite [_.+1 == _.+1](eqn_mul2l 3 _ 0) /= (negPf iDa).
1ab3
2 +
(`d[↓[u (inord (a1 + i))], si (a1 + i)]_smove +
 `d[si (a1 + i), ti (a1 + i)]_smove +
 `d[ti (a1 + i), ↓[u (inord (a1 + i).+1)]]_smove) =
`d[u (inord (a1 + i)), u (inord (a1 + i).+1)]_smove
1a5a
  case: (sitiHb (a1 + i)) => // [|_ _ ->].
1ab3
a1 < a1 + i < l.+1
1ab3
2 +
(`d[↓[u (inord (a1 + i))], 
 si (a1 + i)]_smove +
 `d[si (a1 + i), ti (a1 + i)]_smove +
 `d[ti (a1 + i), ↓[u (inord (a1 + i).+1)]]_smove) =
 D[[:: ↓[u (inord (a1 + i))]; 
      si (a1 + i); ti (a1 + i);
       ↓[u (inord (a1 + i).+1)]]].+2
1a5b
    rewrite -addn1 leq_add2l lt0n iDa /= -[l.+1](subnK (_ : a1 <= l.+1)).
1ab3
a1 + i < l.+1 - a1 + a1
1ab3
1a4f
1b141a5b
      by rewrite addnC ltn_add2r.
1ab3
1a4f
1b13
    by rewrite ltnW.
1ab3
2 +
(`d[↓[u (inord (a1 + i))], si (a1 + i)]_smove +
 `d[si (a1 + i), ti (a1 + i)]_smove +
 `d[ti (a1 + i), ↓[u (inord (a1 + i).+1)]]_smove) =
 D[[:: ↓[u (inord (a1 + i))]; si (a1 + i);
       ti (a1 + i); ↓[u (inord (a1 + i).+1)]]].+2
1a5a
  by rewrite /= -!addnA add2n addn0.
1a5c
573
have P3 : 
     \sum_(a <= i < a1) `d[↓[u (inord i)], ↓[u (inord i.+1)]]_smove <=
     \sum_(a <= i < a1) `d[u (inord i), u (inord i.+1)]_smove.
1a5c
\sum_(a <= i < a1)
   `d[↓[u (inord i)], ↓[u (inord i.+1)]]_smove <=
\sum_(a <= i < a1)
   `d[u (inord i), u (inord i.+1)]_smove
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d
P3:\sum_(a <= i < a1)
   `d[↓[u (inord i)], ↓[
u (inord i.+1)]]_smove <=
\sum_(a <= i < a1)
   `d[u (inord i), u (inord i.+1)]_smove
573
  apply: leq_sum => i _.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d14a
`d[↓[u (inord i)], ↓[u (inord i.+1)]]_smove <=
`d[u (inord i), u (inord i.+1)]_smove
1b2a
  by apply: gdist_cunlift; apply: shanoi_connect.
1b2c
573
have SH : 2 <= l -> 
  (S_[l.+1] n.+2).*2  <= S_[l.+1] n.+2 + S_[l] n.+1 + (α_[1] n.+1).*2.
1b2c
1 < l ->
(S_[l.+1] n.+2).*2 <=
S_[l.+1] n.+2 + S_[l] n.+1 + (α_[1] n.+1).*2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d
SH:1 < l ->
(S_[l.+1] n.+2).*2 <=
S_[l.+1] n.+2 + S_[l] n.+1 + (α_[1] n.+1).*2
573
  move=> l_gt2.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d
l_gt2:1 < l
(S_[l.+1] n.+2).*2 <=
S_[l.+1] n.+2 + S_[l] n.+1 + (α_[1] n.+1).*2
1b3a
  rewrite -addnn -!addnA leq_add2l.
1b41
S_[l.+1] n.+2 <= S_[l] n.+1 + (α_[1] n.+1).*2
1b3a
  by apply: dsum_alphaL_alpha.
1b3c
573
have [/andP[a_gt1 c_gt1]|] := boolP ((1 < a) && (1 < c)).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c3
c_gt1:1 < c
573
1b3c
~~ ((1 < a) && (1 < c)) ->
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u + sp (n:=n.+2) (u ord0) l.+1 p1 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
(* This is 4.2.1 *)
  have l_gt0 : 0 < l by rewrite -ltnS l1E (leq_trans c_gt1) // leq_addl.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f
l_gt0:0 < l
573
1b50
  rewrite !sum_beta_S //= KH1 KH2 !eqxx !addn0.
1b56
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u +
\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2
1b50 
  pose u1l := [ffun i : 'I_(3 * a).-2.+1 => u1 (inord i)].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b57
u1l:=[ ffun i : 'I_(3 * a).-2.+1 => u1 (inord i)]:{ffun forall x : ordinal_finType (3 * a).-2.+1,
      (fun i : 'I_(3 * a).-2.+1 =>
       (fun=> configuration 4 n.+1) (inord i)) x}
1b5b
1b50
  have a3_gt1 : 1 < (3 * a).-2 by rewrite -(subnK a_gt1) addn2 !mulnS.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b60
a3_gt1:1 < (3 * a).-2
1b5b
1b50
  have c3_gt1 : 1 < (3 * c).-2 by rewrite -(subnK c_gt1) addn2 !mulnS.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b65
c3_gt1:1 < (3 * c).-2
1b5b
1b50
  have cH1 (k : 'I_(3 * a).-2.+1) : 
    0 < k < (3 * a).-2 -> codom (u1l k) \subset [:: p2; a[p1, p3] k].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a848
0 < k < (3 * a).-2 ->
codom (u1l k) \subset [:: p2;  a[p1, p3] k]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a
cH1:forall k : 'I_(3 * a).-2.+1,
0 < k < (3 * a).-2 ->
codom (u1l k) \subset [:: p2;  a[p1, p3] k]
1b5b
1b51
    move=> /andP[k_gt0 k3L3a].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a848852
k3L3a:k < (3 * a).-2
codom (u1l k) \subset [:: p2;  a[p1, p3] k]
1b70
    have k3La1 : k %/ 3 <= a.-1.
1b77
863
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a8488521b78
k3La1:k %/ 3 <= a.-1
1b79
1b711b51
      have kL3a : k <= 3 * (a.-1).
1b77
858
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a8488521b78
kL3a:k <= 3 * a.-1
863
1b7e1b711b51
        rewrite -[a.-1]subn1 mulnBr -ltnS -[3 *1]/(2.+1) subnSK.
1b77
k < 3 * a - 2
1b77
2 < 3 * a
1b851b7e1b711b51
          by rewrite subn2.
1b77
1b8e
1b84
        by rewrite (leq_mul2l 3 1).
1b86
863
1b7d
      by rewrite -[a.-1](mulKn _ (_ : 0 < 3)) // leq_div2r.
1b7f
1b79
1b70
    have k3Ll : k %/ 3 < l.
1b7f
13aa
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a8488521b781b80
k3Ll:k %/ 3 < l
1b79
1b711b51
      rewrite -ltnS l1E (leq_trans (_ : _ < a.-1.+3)) //; last first.
1b7f
a.-1.+2 < a + b + c
1b7f
(k %/ 3).+1 < a.-1.+3
1b9c1b711b51
        rewrite prednK // addnAC -addn2 (leq_trans (_ : _ <= a + c)) //.
1b7f
a + 2 <= a + c
1b7f
a + c <= a + c + b
1ba41b9c1b711b51
          by rewrite leq_add2l.
1b7f
1bac
1ba3
        by rewrite leq_addr.
1b7f
1ba5
1b9b
      by rewrite !ltnS ltnW // ltnS.
1b9d
1b79
1b70
    rewrite !ffunE inordK //; last first.
1b9d
k < (3 * a).+1
1b9d
codom
  (if k %% 3 == 0
   then ↓[u (inord (k %/ 3))]
   else
    if k %% 3 == 1
    then
     if k == (3 * a.-1).+1 then sam1 else si (k %/ 3)
    else
     if k == (3 * a.-1).+2 then tam1 else ti (k %/ 3)) \subset
[:: p2;  a[p1, p3] k]
1b711b51
      rewrite (leq_trans (ltn_ord _)) // -subn2 ltn_subLR.
1b9d
3 * a < 2 + (3 * a).+1
1b9d
979
1bbb1b711b51
        by rewrite add2n ltnS // ltnW.
1b9d
979
1bba
      by rewrite (leq_trans (_ : _ < 3 * 1)) // leq_mul2l.
1b9d
1bbc
1b70
    rewrite {8}(divn_eq k 3) apegD [_ * 3]mulnC apegMr //=.
1b9d
codom
  (if k %% 3 == 0
   then ↓[u (inord (k %/ 3))]
   else
    if k %% 3 == 1
    then
     if k == (3 * a.-1).+1 then sam1 else si (k %/ 3)
    else
     if k == (3 * a.-1).+2 then tam1 else ti (k %/ 3)) \subset
[:: p2;
     a[ a[p1, p3] (k %% 3),  a[p3, p1] (k %% 3)]
      (k %/ 3)]
1b70
    case: eqP => [km3E0 /= |/eqP km3D0].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a8488521b781b801b9e135d
codom ↓[u (inord (k %/ 3))] \subset
[:: p2;
     a[ a[p1, p3] (k %% 3),  a[p3, p1] (k %% 3)]
      (k %/ 3)]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a8488521b781b801b9e1362
codom
  (if k %% 3 == 1
   then
    if k == (3 * a.-1).+1 then sam1 else si (k %/ 3)
   else
    if k == (3 * a.-1).+2 then tam1 else ti (k %/ 3)) \subset
[:: p2;
     a[ a[p1, p3] (k %% 3), 
       a[p3, p1] (k %% 3)] 
      (k %/ 3)]
1b711b51
      have k3_gt0 : 0 < k %/ 3.
1bd0
0 < k %/ 3
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a8488521b781b801b9e135d136c
codom ↓[u (inord (k %/ 3))] \subset
[:: p2;
     a[ a[p1, p3] (k %% 3), 
       a[p3, p1] (k %% 3)] 
      (k %/ 3)]
1bd31b711b51
        case: (k: nat) k_gt0 km3E0  => // [] [|[|k1 _ _]] //.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a8481b781b801b9e
k1:nat
0 < k1.+3 %/ 3
1bda
        by rewrite (divnMDl 1 k1 (_ : 0 < 3)).
1bdc
1bd1
1bd2
      rewrite km3E0 !apeg0.
1bdc
1372
1bd2
      have /H : 0 < (inord (k %/ 3) : 'I_l.+2) < l.+1.
1bdc
0 < inord (k %/ 3) < l.+1
1bdc
codom (u' (inord (k %/ 3))) \subset
[:: p2;  a[p1, p3] (inord (k %/ 3))] ->
codom ↓[u (inord (k %/ 3))] \subset
[:: p2;  a[p1, p3] (k %/ 3)]
1bd31b711b51
        rewrite inordK //.
1bdc
8d9
1bdc
8cf
1bef1bd31b711b51
          by rewrite k3_gt0 /= (leq_trans k3Ll).
1bdc
8cf
1bee
        by rewrite (leq_trans k3Ll) // ltnW.
1bdc
1bf0
1bd2
      by rewrite ffunE inordK // (leq_trans k3Ll) // ltnW .
1bd4
codom
  (if k %% 3 == 1
   then
    if k == (3 * a.-1).+1 then sam1 else si (k %/ 3)
   else
    if k == (3 * a.-1).+2 then tam1 else ti (k %/ 3)) \subset
[:: p2;
     a[ a[p1, p3] (k %% 3),  a[p3, p1] (k %% 3)]
      (k %/ 3)]
1b70
    case: eqP => [km3E1|/eqP km3D1].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a8488521b781b801b9e13621383
codom
  (if k == (3 * a.-1).+1 then sam1 else si (k %/ 3)) \subset
[:: p2;
     a[ a[p1, p3] (k %% 3),  a[p3, p1] (k %% 3)]
      (k %/ 3)]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a8488521b781b801b9e13621388
codom
  (if k == (3 * a.-1).+2 then tam1 else ti (k %/ 3)) \subset
[:: p2;
     a[ a[p1, p3] (k %% 3), 
       a[p3, p1] (k %% 3)] 
      (k %/ 3)]
1b711b51
      rewrite km3E1 !apegS !apeg0; case: eqP => [->|kE3a1].
1c03
codom sam1 \subset
[:: p2;  a[p3, p1] ((3 * a.-1).+1 %/ 3)]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a8488521b781b801b9e13621383
kE3a1:k <> (3 * a.-1).+1
codom (si (k %/ 3)) \subset
[:: p2;  a[p3, p1] (k %/ 3)]
1c061b711b51
        rewrite (_ : (3 * a.-1).+1 = a.-1 * 3 + 1); last by rewrite addn1 mulnC.
1c03
codom sam1 \subset
[:: p2;  a[p3, p1] ((a.-1 * 3 + 1) %/ 3)]
1c0d
        rewrite divnMDl // divn_small // addn0.
1c03
codom sam1 \subset [:: p2;  a[p3, p1] a.-1]
1c0d
        by have := sam1C; rewrite codom_subC /pa -{1}[a]prednK //= apegS.
1c0f
1c11
1c05
      move: k3La1; rewrite leq_eqVlt => /orP[/eqP k3Ea1| k3La].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a8488521b781b9e136213831c10
k3Ea1:k %/ 3 = a.-1
1c11
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a8488521b781b9e136213831c10
k3La:k %/ 3 < a.-1
1c11
1c061b711b51
        case: kE3a1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a8488521b781b9e136213831c21
k = (3 * a.-1).+1
1c22
        by rewrite (divn_eq k 3) km3E1 k3Ea1 mulnC addn1.
1c24
1c11
1c05
      case: (sitiH (k %/ 3)) => // siH _ _; move: siH.
1c24
codom (si (k %/ 3)) \subset
[:: p2; u (inord (k %/ 3).+1) ldisk] ->
codom (si (k %/ 3)) \subset
[:: p2;  a[p3, p1] (k %/ 3)]
1c05
      by rewrite aMin1 ?apegS // -{2}[a]prednK .
1c07
codom
  (if k == (3 * a.-1).+2 then tam1 else ti (k %/ 3)) \subset
[:: p2;
     a[ a[p1, p3] (k %% 3),  a[p3, p1] (k %% 3)]
      (k %/ 3)]
1b70
    have km3E2 : k %% 3 = 2.
1c07
944
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a8488521b781b801b9e13621388139d
1c08
1b711b51
      by case: (_ %% _) (ltn_mod k 3) km3D0 km3D1 => // [] [|[|]].
1c3b
1c35
1b70
    rewrite km3E2 !apegS !apeg0; case: eqP => [kE3a1|kD3a1].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a8488521b781b801b9e13621388139d
kE3a1:k = (3 * a.-1).+2
codom tam1 \subset [:: p2;  a[p1, p3] (k %/ 3)]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a8488521b781b801b9e13621388139d
kD3a1:k <> (3 * a.-1).+2
codom (ti (k %/ 3)) \subset
[:: p2;  a[p1, p3] (k %/ 3)]
1b711b51
      rewrite ltnNge in k3L3a; case/negP: k3L3a.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a8488521b801b9e13621388139d1c43
(3 * a).-2 <= k
1c45
      by rewrite kE3a1 -{1}[a]prednK // mulnS.
1c47
1c49
1b70
    move: k3La1; rewrite leq_eqVlt => /orP[/eqP k3Ea1| k3La].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a8488521b781b9e13621388139d1c481c21
1c49
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a8488521b781b9e13621388139d1c481c25
1c49
1b711b51
      case: kD3a1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a8488521b781b9e13621388139d1c21
k = (3 * a.-1).+2
1c56
      by rewrite (divn_eq k 3) km3E2 k3Ea1 mulnC addn2.
1c58
1c49
1b70
    rewrite codom_subC.
1c58
codom (ti (k %/ 3)) \subset
[::  a[p1, p3] (k %/ 3); p2]
1b70
    case: (sitiH (k %/ 3)) => // _ tiH _; move: tiH.
1c58
codom (ti (k %/ 3)) \subset
[:: u (inord (k %/ 3)) ldisk; p2] ->
codom (ti (k %/ 3)) \subset
[::  a[p1, p3] (k %/ 3); p2]
1b70
    by rewrite aMin1 // -{2}[a]prednK // ltnS ltnW.
1b72
1b5b
1b50
    have {cH1}P4 : 
      (S_[(3 * a).-2] n.+1).*2 <=
      \sum_(i < (3 * a).-2) `d[u1 (inord i), u1 (inord i.+1)]_smove +
      \sum_(k < n.+1) (↓[u ord0] k != p1) * β_[n.+1, (3 * a).-2] k +
      \sum_(k < n.+1) (sam1 k != pa) * β_[n.+1, (3 * a).-2] k.
1b72
(S_[(3 * a).-2] n.+1).*2 <=
\sum_(i < (3 * a).-2)
   `d[u1 (inord i), u1 (inord i.+1)]_smove +
\sum_(k < n.+1)
   (↓[u ord0] k != p1) * β_[n.+1, (3 * a).-2] k +
\sum_(k < n.+1)
   (sam1 k != pa) * β_[n.+1, (3 * a).-2] k
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a
P4:(S_[(3 * a).-2] n.+1).*2 <=
\sum_(i < (3 * a).-2)
   `d[u1 (inord i), u1 (inord i.+1)]_smove +
\sum_(k < n.+1)
   (↓[u ord0] k != p1) * β_[n.+1, (3 * a).-2] k +
\sum_(k < n.+1)
   (sam1 k != pa) * β_[n.+1, (3 * a).-2] k
1b5b
1b51
    apply: leq_trans (IH _ _ _ _ _ _ cH1) _ => //.
1b72
sd (n:=n.+1) u1l +
sp (n:=n.+1) (u1l ord0) (3 * a).-2 p1 +
sp (n:=n.+1) (u1l ord_max) (3 * a).-2
  ( a[p1, p3] (3 * a).-2) <=
\sum_(i < (3 * a).-2)
   `d[u1 (inord i), u1 (inord i.+1)]_smove +
\sum_(k < n.+1)
   (↓[u ord0] k != p1) * β_[n.+1, (3 * a).-2] k +
\sum_(k < n.+1)
   (sam1 k != pa) * β_[n.+1, (3 * a).-2] k
1c70
    rewrite leq_eqVlt; apply/orP; left; apply/eqP.
1b72
sd (n:=n.+1) u1l +
sp (n:=n.+1) (u1l ord0) (3 * a).-2 p1 +
sp (n:=n.+1) (u1l ord_max) (3 * a).-2
  ( a[p1, p3] (3 * a).-2) =
\sum_(i < (3 * a).-2)
   `d[u1 (inord i), u1 (inord i.+1)]_smove +
\sum_(k < n.+1)
   (↓[u ord0] k != p1) * β_[n.+1, (3 * a).-2] k +
\sum_(k < n.+1)
   (sam1 k != pa) * β_[n.+1, (3 * a).-2] k
1c70
    congr (_ + _ + _); apply: eq_bigr => i _.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a1b73
i:ordinal_finType (3 * a).-2
`d[u1l (inord i), u1l (inord i.+1)]_smove =
`d[u1 (inord i), u1 (inord i.+1)]_smove
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a1b731054
(u1l ord0 i != p1) * β_[n.+1, (3 * a).-2] i =
(↓[u ord0] i != p1) * β_[n.+1, (3 * a).-2] i
1c84
(u1l ord_max i !=  a[p1, p3] (3 * a).-2) *
β_[n.+1, (3 * a).-2] i =
(sam1 i != pa) * β_[n.+1, (3 * a).-2] i
1c711b51
    -
1c7d by rewrite ![u1l _]ffunE !inordK // ltnS // ltnW.
1c84
1c85
1c861c711b51
    -
1c8a by rewrite ffunE inord_eq0 // ffunE /= inord_eq0.
1c84
1c87
1c70
    rewrite !ffunE /= inordK //; last first.
1c84
(3 * a).-2 < (3 * a).+1
1c84
((if (3 * a).-2 %% 3 == 0
  then ↓[u (inord ((3 * a).-2 %/ 3))]
  else
   if (3 * a).-2 %% 3 == 1
   then
    if (3 * a).-2 == (3 * a.-1).+1
    then sam1
    else si ((3 * a).-2 %/ 3)
   else
    if (3 * a).-2 == (3 * a.-1).+2
    then tam1
    else ti ((3 * a).-2 %/ 3)) i
 !=  a[p1, p3] (3 * a).-2) * 
β_[n.+1, (3 * a).-2] i =
(sam1 i != pa) * β_[n.+1, (3 * a).-2] i
1c711b51
      by rewrite ltnS -subn2 leq_subr.
1c84
((if (3 * a).-2 %% 3 == 0
  then ↓[u (inord ((3 * a).-2 %/ 3))]
  else
   if (3 * a).-2 %% 3 == 1
   then
    if (3 * a).-2 == (3 * a.-1).+1
    then sam1
    else si ((3 * a).-2 %/ 3)
   else
    if (3 * a).-2 == (3 * a.-1).+2
    then tam1
    else ti ((3 * a).-2 %/ 3)) i
 !=  a[p1, p3] (3 * a).-2) * β_[n.+1, (3 * a).-2] i =
(sam1 i != pa) * β_[n.+1, (3 * a).-2] i
1c70
    rewrite (_ : _.-2 = (3 * a.-1).+1); last first.
1c84
(3 * a).-2 = (3 * a.-1).+1
1c84
((if (3 * a.-1).+1 %% 3 == 0
  then ↓[u (inord ((3 * a.-1).+1 %/ 3))]
  else
   if (3 * a.-1).+1 %% 3 == 1
   then
    if (3 * a.-1).+1 == (3 * a.-1).+1
    then sam1
    else si ((3 * a.-1).+1 %/ 3)
   else
    if (3 * a.-1).+1 == (3 * a.-1).+2
    then tam1
    else ti ((3 * a.-1).+1 %/ 3)) i
 !=  a[p1, p3] (3 * a.-1).+1) *
β_[n.+1, (3 * a.-1).+1] i =
(sam1 i != pa) * β_[n.+1, (3 * a.-1).+1] i
1c711b51
      by rewrite -{1}[a]prednK // mulnS.
1c84
1ca2
1c70
    rewrite mod3E /= eqxx.
1c84
(sam1 i !=  a[p1, p3] (3 * a.-1).+1) *
β_[n.+1, (3 * a.-1).+1] i =
(sam1 i != pa) * β_[n.+1, (3 * a.-1).+1] i
1c70
    by rewrite /pa apegS apegMr //= -apegS prednK.
1c72
1b5b
1b50
  have {}P4 := leq_trans P4 (leq_add (leq_add (leqnn _) 
                     (leq_sum_beta _ _)) (leqnn _)).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a
P4:(S_[(3 * a).-2] n.+1).*2 <=
\sum_(i < (3 * a).-2)
   `d[u1 (inord i), u1 (inord i.+1)]_smove +
\sum_(k < n.+1)
   (↓[u ord0] k != p1) * (α_[1] k).*2 +
\sum_(k < n.+1)
   (sam1 k != pa) * β_[n.+1, (3 * a).-2] k
1b5b
1b50
  set x1S := \sum_(_ < _.+1) _ in P4; set x2S := \sum_(_ < _.+1) _ in P4.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc
x2S:=\sum_(H < n.+1)
   (sam1 H != pa) * β_[n.+1, (3 * a).-2] H:nat
P4:(S_[(3 * a).-2] n.+1).*2 <=
\sum_(i < (3 * a).-2)
   `d[u1 (inord i), u1 (inord i.+1)]_smove + x1S +
x2S
1b5b
1b50
  rewrite -/x1S.
1cb5
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u + x1S +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2
1b50
  pose u2r := [ffun i : 'I_(3 * c).-2.+1 => u2 (inord i.+2)].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb7
u2r:=[ ffun i : 'I_(3 * c).-2.+1 => u2 (inord i.+2)]:{ffun forall x : ordinal_finType (3 * c).-2.+1,
      (fun i : 'I_(3 * c).-2.+1 =>
       (fun=> configuration 4 n.+1) (inord i.+2))
        x}
1cbb
1b50
  pose pa1 := a[p1, p3] a1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc0
pa1:= a[p1, p3] a1:peg 4
1cbb
1b50
  pose pa1S := a[p1, p3] a1.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc5
pa1S:= a[p1, p3] a1.+1:peg 4
1cbb
1b50
  have cH1 (k : 'I_(3 * c).-2.+1) : 
    0 < k < (3 * c).-2 -> codom (u2r k) \subset [:: p2; a[pa1, pa1S] k].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca
k:'I_(3 * c).-2.+1
0 < k < (3 * c).-2 ->
codom (u2r k) \subset [:: p2;  a[pa1, pa1S] k]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca
cH1:forall k : 'I_(3 * c).-2.+1,
0 < k < (3 * c).-2 ->
codom (u2r k) \subset [:: p2;  a[pa1, pa1S] k]
1cbb
1b51
    move=> /andP[k_gt0 k3L3a].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1ccf852
k3L3a:k < (3 * c).-2
codom (u2r k) \subset [:: p2;  a[pa1, pa1S] k]
1cd1
    have k2L3c : k.+2 < (3 * c).
1cd8
k.+2 < 3 * c
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1ccf8521cd9
k2L3c:k.+2 < 3 * c
1cda
1cd21b51
      by have := k3L3a; rewrite -{2}subn2 ltn_subRL add2n.
1ce1
1cda
1cd1    
    rewrite !ffunE inordK ?ltnS 1?ltnW //.
1ce1
codom
  (if k.+2 %% 3 == 0
   then ↓[u (inord (a1 + k.+2 %/ 3))]
   else
    if k.+2 %% 3 == 1
    then
     if k.+2 == 1 then sa1 else si (a1 + k.+2 %/ 3)
    else
     if k.+2 == 2 then ta1 else ti (a1 + k.+2 %/ 3)) \subset
[:: p2;  a[pa1, pa1S] k]
1cd1
    case: eqP => [km3E0 /= |/eqP km3D0].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1ccf8521cd91ce2
km3E0:k.+2 %% 3 = 0
codom ↓[u (inord (a1 + k.+2 %/ 3))] \subset
[:: p2;  a[pa1, pa1S] k]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1ccf8521cd91ce2
km3D0:k.+2 %% 3 != 0
codom
  (if k.+2 %% 3 == 1
   then if k.+2 == 1 then sa1 else si (a1 + k.+2 %/ 3)
   else if k.+2 == 2 then ta1 else ti (a1 + k.+2 %/ 3)) \subset
[:: p2;  a[pa1, pa1S] k]
1cd21b51
      have k3_gt0 : 0 < k.+2 %/ 3.
1ced
0 < k.+2 %/ 3
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1ccf8521cd91ce21cee
k3_gt0:0 < k.+2 %/ 3
1cef
1cf11cd21b51
        case: (k: nat) k_gt0 km3E0  => // [] //= k1 _ _.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1ccf1cd91ce21be2
1be3
1cf9
        by rewrite (divnMDl 1 k1 (_ : 0 < 3)).
1cfb
1cef
1cf0
      have a1k3Ll1 : a1 + k.+2 %/ 3 < l.+1.
1cfb
a1 + k.+2 %/ 3 < l.+1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1ccf8521cd91ce21cee1cfc
a1k3Ll1:a1 + k.+2 %/ 3 < l.+1
1cef
1cf11cd21b51
        move: k2L3c.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1ccf8521cd91cee1cfc
k.+2 < 3 * c -> a1 + k.+2 %/ 3 < l.+1
1d08
        rewrite {1}(divn_eq k.+2 3) km3E0 addn0 mulnC ltn_mul2l /= => k2L3c.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1ccf8521cd91cee1cfc
k2L3c:k.+2 %/ 3 < c
1d07
1d08
        rewrite -(subnK (_ : a1 <= l.+1)) ?(leq_trans a1Ll) //.
1d14
a1 + k.+2 %/ 3 < l.+1 - a1 + a1
1d08
        by rewrite addnC ltn_add2r -/c.
1d0a
1cef
1cf0
      rewrite (_ : a[_, _ ] _ = a[p1, p3] (a1 + k.+2 %/ 3)) ; last first.
1d0a
 a[pa1, pa1S] k =  a[p1, p3] (a1 + k.+2 %/ 3)
1d0a
codom ↓[u (inord (a1 + k.+2 %/ 3))] \subset
[:: p2;  a[p1, p3] (a1 + k.+2 %/ 3)]
1cf11cd21b51
        rewrite /pa1 /pa1S apegS -apegD -2!apegS -!addSn addnC.
1d0a
 a[p1, p3] (a1 + k.+2) =  a[p1, p3] (a1 + k.+2 %/ 3)
1d21
        rewrite {1}(divn_eq k.+2 3) km3E0 addn0 apegO.
1d0a
 a[p3, p1] (~~ odd (a1 + k.+2 %/ 3 * 3)) =
 a[p1, p3] (a1 + k.+2 %/ 3)
1d21
        by rewrite oddD oddM andbT -oddD -apegO.
1d0a
1d23
1cf0
      have /H : 0 < (inord (a1 + k.+2 %/ 3) : 'I_l.+2) < l.+1.
1d0a
0 < inord (a1 + k.+2 %/ 3) < l.+1
1d0a
codom (u' (inord (a1 + k.+2 %/ 3))) \subset
[:: p2;  a[p1, p3] (inord (a1 + k.+2 %/ 3))] ->
codom ↓[u (inord (a1 + k.+2 %/ 3))] \subset
[:: p2;  a[p1, p3] (a1 + k.+2 %/ 3)]
1cf11cd21b51
        rewrite inordK; first by rewrite addn_gt0 a1_gt0.
1d0a
a1 + k.+2 %/ 3 < l.+2
1d33
        by rewrite ltnS ltnW.
1d0a
1d35
1cf0
      by rewrite ffunE inordK // ltnS ltnW.
1cf2
1cf4
1cd1
    case: eqP => [km3E1|/eqP km3D1].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1ccf8521cd91ce21cf3
km3E1:k.+2 %% 3 = 1
codom (if k.+2 == 1 then sa1 else si (a1 + k.+2 %/ 3)) \subset
[:: p2;  a[pa1, pa1S] k]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1ccf8521cd91ce21cf3
km3D1:k.+2 %% 3 != 1
codom (if k.+2 == 2 then ta1 else ti (a1 + k.+2 %/ 3)) \subset
[:: p2;  a[pa1, pa1S] k]
1cd21b51
      rewrite eqn_leq ltnNge /=.
1d43
codom (si (a1 + k.+2 %/ 3)) \subset
[:: p2;  a[pa1, pa1S] k]
1d46
      have a1La1k : a1 < a1 + k.+2 %/3.
1d43
a1 < a1 + k.+2 %/ 3
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1ccf8521cd91ce21cf31d44
a1La1k:a1 < a1 + k.+2 %/ 3
1d4e
1d471cd21b51
        by rewrite -{1}(addn0 a1) ltn_add2l divn_gt0.
1d55
1d4e
1d46
      have a1k3Ll : a1 + k.+2 %/ 3 < l.+1.
1d55
1d07
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1ccf8521cd91ce21cf31d441d56
a1k3Ll:a1 + k.+2 %/ 3 < l.+1
1d4e
1d471cd21b51
        move: (ltnW k2L3c).
1d55
k.+1 < 3 * c -> a1 + k.+2 %/ 3 < l.+1
1d5d
        rewrite {1}(divn_eq k.+2 3) km3E1 addn1 mulnC ltn_mul2l /=.
1d55
k.+2 %/ 3 < c -> a1 + k.+2 %/ 3 < l.+1
1d5d
        rewrite -(subnK (_ : a1 <= l.+1)) ?(leq_trans a1Ll) //.
1d55
k.+2 %/ 3 < c -> a1 + k.+2 %/ 3 < l.+1 - a1 + a1
1d5d
        by rewrite addnC ltn_add2r -/c.
1d5f
1d4e
1d46
      rewrite (_ : a[_, _ ] _ = a[p1, p3] (a1 + k.+2)) ; last first.
1d5f
 a[pa1, pa1S] k =  a[p1, p3] (a1 + k.+2)
1d5f
codom (si (a1 + k.+2 %/ 3)) \subset
[:: p2;  a[p1, p3] (a1 + k.+2)]
1d471cd21b51
        by rewrite /pa1S apegS -apegD !addnS addnC !apegS.
1d5f
1d76
1d46
      rewrite (_ : a[_, _ ] _ = a[p3, p1] (a1 + (k.+2) %/ 3)) ; last first.
1d5f
 a[p1, p3] (a1 + k.+2) =  a[p3, p1] (a1 + k.+2 %/ 3)
1d5f
codom (si (a1 + k.+2 %/ 3)) \subset
[:: p2;  a[p3, p1] (a1 + k.+2 %/ 3)]
1d471cd21b51
        rewrite apegO oddD {1}(divn_eq k.+2 3) oddD km3E1 addbT oddM andbT.
1d5f
 a[p3, p1] (~~ (odd a1 (+) ~~ odd (k.+2 %/ 3))) =
 a[p3, p1] (a1 + k.+2 %/ 3)
1d7e
        by rewrite addbN -oddD -oddS -apegO apegS.
1d5f
1d80
1d46
      case: (sitiHb (a1 + k.+2 %/ 3)) => // [| siH _ _].
1d5f
a1 < a1 + k.+2 %/ 3 < l.+1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1ccf8521cd91ce21cf31d441d561d60
siH:codom (si (a1 + k.+2 %/ 3)) \subset
[:: p2; u (inord (a1 + k.+2 %/ 3).+1) ldisk]
1d80
1d471cd21b51
        by rewrite a1La1k.
1d8e
1d80
1d46
      by move: siH; rewrite a1Max ?apegS // ltnS ltnW.
1d48
1d4a
1cd1
    have km3E2 : k.+2 %% 3 = 2.
1d48
k.+2 %% 3 = 2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1ccf8521cd91ce21cf31d49
km3E2:k.+2 %% 3 = 2
1d4a
1cd21b51
      by case: (_ %% _) (ltn_mod k.+2 3) km3D0 km3D1 => // [] [|[|]].
1d9c
1d4a
1cd1
    rewrite ifN; last by case: (k : nat) k_gt0.
1d9c
codom (ti (a1 + k.+2 %/ 3)) \subset
[:: p2;  a[pa1, pa1S] k]
1cd1
    have a1La1k : a1 < a1 + k.+2 %/3.
1d9c
1d52
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1ccf8521cd91ce21cf31d491d9d1d56
1da4
1cd21b51
      by rewrite -{1}(addn0 a1) ltn_add2l divn_gt0.
1daa
1da4
1cd1
    have a1k3Ll : a1 + k.+2 %/ 3 < l.+1.
1daa
1d07
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1ccf8521cd91ce21cf31d491d9d1d561d60
1da4
1cd21b51
      have: k.+2 %/ 3 * 3 < 3 * c.
1daa
k.+2 %/ 3 * 3 < 3 * c
1daa
k.+2 %/ 3 * 3 < 3 * c -> a1 + k.+2 %/ 3 < l.+1
1db21cd21b51
        rewrite ltnW //.
1daa
(k.+2 %/ 3 * 3).+1 < 3 * c
1db8
        move: (ltnW k2L3c).
1daa
k.+1 < 3 * c -> (k.+2 %/ 3 * 3).+1 < 3 * c
1db8
        by rewrite {1}(divn_eq k.+2 3) km3E2 addn2.
1daa
1dba
1db1
      rewrite mulnC ltn_mul2l /=.
1daa
1d68
1db1
      rewrite -(subnK (_ : a1 <= l.+1)) ?(leq_trans a1Ll) //.
1daa
1d6c
1db1
      by rewrite addnC ltn_add2r -/c.
1db3
1da4
1cd1
    rewrite (_ : a[_, _ ] _ = a[p1, p3] (a1 + k.+2)) ; last first.
1db3
1d73
1db3
codom (ti (a1 + k.+2 %/ 3)) \subset
[:: p2;  a[p1, p3] (a1 + k.+2)]
1cd21b51
      by rewrite /pa1 /pa1S apegS -apegD addnC !addnS !apegS.
1db3
1dd4
1cd1
    rewrite (_ : a[_, _ ] _ = a[p1, p3] (a1 + (k.+2) %/ 3)) ; last first.
1db3
1d27
1db3
codom (ti (a1 + k.+2 %/ 3)) \subset
[:: p2;  a[p1, p3] (a1 + k.+2 %/ 3)]
1cd21b51
      rewrite apegO /= oddD {1}(divn_eq k.+2 3) !oddD oddM andbT.
1db3
 a[p3, p1]
  (~~
   (odd a1 (+) (odd (k.+2 %/ 3) (+) odd (k.+2 %% 3)))) =
 a[p1, p3] (a1 + k.+2 %/ 3)
1ddb
      by rewrite km3E2 addbF -oddD -apegO.
1db3
1ddd
1cd1
    rewrite codom_subC.
1db3
codom (ti (a1 + k.+2 %/ 3)) \subset
[::  a[p1, p3] (a1 + k.+2 %/ 3); p2]
1cd1
    case: (sitiHb (a1 + k.+2 %/ 3)) => // [| _ tiH _]; first by rewrite a1La1k.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1ccf8521cd91ce21cf31d491d9d1d561d60
tiH:codom (ti (a1 + k.+2 %/ 3)) \subset
[:: u (inord (a1 + k.+2 %/ 3)) ldisk; p2]
1de8
1cd1
    by move: tiH; rewrite a1Max // a1La1k ltnS ltnW.
1cd3
1cbb
1b50
  have P5 :
    (S_[(3 * c).-2] n.+1).*2 <=       
    \sum_(i < (3 * c).-2) `d[u2 (inord i.+2), u2 (inord i.+3)]_smove +
    sp ta1 (3 * c).-2  pa1 + sp ↓[u ord_max] (3 * c).-2 (a[p3, p1] l).
1cd3
(S_[(3 * c).-2] n.+1).*2 <=
\sum_(i < (3 * c).-2)
   `d[u2 (inord i.+2), u2 (inord i.+3)]_smove +
sp (n:=n.+1) ta1 (3 * c).-2 pa1 +
sp (n:=n.+1) ↓[u ord_max] (3 * c).-2 ( a[p3, p1] l)
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1cd4
P5:(S_[(3 * c).-2] n.+1).*2 <=
\sum_(i < (3 * c).-2)
   `d[u2 (inord i.+2), u2 (inord i.+3)]_smove +
sp (n:=n.+1) ta1 (3 * c).-2 pa1 +
sp (n:=n.+1) ↓[u ord_max] 
  (3 * c).-2 ( a[p3, p1] l)
1cbb
1b51
    apply: leq_trans (IH _ _ _ _ _ _ cH1) _; rewrite /pa1 /pa1S {cH1}//.
1cc9
 a[p1, p3] a1 !=  a[p1, p3] a1.+1
1cc9
p2 !=  a[p1, p3] a1.+1
1cc9
sd (n:=n.+1) u2r +
sp (n:=n.+1) (u2r ord0) (3 * c).-2 ( a[p1, p3] a1) +
sp (n:=n.+1) (u2r ord_max) 
  (3 * c).-2
  ( a[ a[p1, p3] a1,  a[p1, p3] a1.+1] (3 * c).-2) <=
\sum_(i < (3 * c).-2)
   `d[u2 (inord i.+2), u2 (inord i.+3)]_smove +
sp (n:=n.+1) ta1 (3 * c).-2 ( a[p1, p3] a1) +
sp (n:=n.+1) ↓[u ord_max] (3 * c).-2 ( a[p3, p1] l)
1df61b51
    -
1dfa by rewrite apegS apeg_eqC.
1cc9
1dff
1e001df61b51
    -
1e04 by rewrite eq_sym.
1cc9
sd (n:=n.+1) u2r +
sp (n:=n.+1) (u2r ord0) (3 * c).-2 ( a[p1, p3] a1) +
sp (n:=n.+1) (u2r ord_max) (3 * c).-2
  ( a[ a[p1, p3] a1,  a[p1, p3] a1.+1] (3 * c).-2) <=
\sum_(i < (3 * c).-2)
   `d[u2 (inord i.+2), u2 (inord i.+3)]_smove +
sp (n:=n.+1) ta1 (3 * c).-2 ( a[p1, p3] a1) +
sp (n:=n.+1) ↓[u ord_max] (3 * c).-2 ( a[p3, p1] l)
1df5
    repeat apply: leq_add; rewrite leq_eqVlt; apply/orP; left; apply/eqP.
1cc9
sd (n:=n.+1) u2r =
\sum_(i < (3 * c).-2)
   `d[u2 (inord i.+2), u2 (inord i.+3)]_smove
1cc9
sp (n:=n.+1) (u2r ord0) (3 * c).-2 ( a[p1, p3] a1) =
sp (n:=n.+1) ta1 (3 * c).-2 ( a[p1, p3] a1)
1cc9
sp (n:=n.+1) (u2r ord_max) 
  (3 * c).-2
  ( a[ a[p1, p3] a1,  a[p1, p3] a1.+1] (3 * c).-2) =
sp (n:=n.+1) ↓[u ord_max] (3 * c).-2 ( a[p3, p1] l)
1df61b51
    -
1e0d apply: eq_bigr => i _.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca
i:ordinal_finType (3 * c).-2
`d[u2r (inord i), u2r (inord i.+1)]_smove =
`d[u2 (inord i.+2), u2 (inord i.+3)]_smove
1e10
      by rewrite ![u2r _]ffunE !inordK // ltnS // ltnW.
1cc9
1e12
1e131df61b51
    -
1e1d apply: eq_bigr => i _; congr ((_ != _) * _).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1054
u2r ord0 i = ta1 i
1e1f
      rewrite !ffunE /= !inordK //=.
1e24
2 < (3 * c).+1
1e1f
      by rewrite (leq_trans (_ : 2 < (3 * 1).+1)) // ltnS leq_mul2l.
1cc9
sp (n:=n.+1) (u2r ord_max) (3 * c).-2
  ( a[ a[p1, p3] a1,  a[p1, p3] a1.+1] (3 * c).-2) =
sp (n:=n.+1) ↓[u ord_max] (3 * c).-2 ( a[p3, p1] l)
1df5
    apply: eq_bigr => i _; congr ((_ != _) * _).
1e24
u2r ord_max i = ↓[u ord_max] i
1e24
 a[ a[p1, p3] a1,  a[p1, p3] a1.+1] (3 * c).-2 =
 a[p3, p1] l
1df61b51
      rewrite !ffunE /= !prednK ?muln_gt0 //=; last first.
1e24
0 < (3 * c).-1
1e24
(if inord (3 * c) %% 3 == 0
 then ↓[u (inord (a1 + inord (3 * c) %/ 3))]
 else
  if inord (3 * c) %% 3 == 1
  then
   if inord (3 * c) == 1
   then sa1
   else si (a1 + inord (3 * c) %/ 3)
  else
   if inord (3 * c) == 2
   then ta1
   else ti (a1 + inord (3 * c) %/ 3)) i =
u ord_max (trshift 1 i)
1e331df61b51
        by rewrite -subn1 ltn_subRL addn0 (leq_trans (_ : _ < 3 * 1)) // 
                   leq_mul2l.
1e24
1e3b
1e32
      rewrite inordK // mod3E /= div3E addnC subnK; last first.
1e24
1a4f
1e24
↓[u (inord l.+1)] i = u ord_max (trshift 1 i)
1e331df61b51
        by apply: (leq_trans a1Ll).
1e24
1e44
1e32
      by rewrite ffunE; congr (u _ _ ); apply/val_eqP; rewrite /= inordK.
1e24
1e34
1df5
    rewrite -[RHS]apegS -(subnK (_ : a1 <= l.+1)) // -/c.
1e24
 a[ a[p1, p3] a1,  a[p1, p3] a1.+1] (3 * c).-2 =
 a[p1, p3] (c + a1)
1e24
1a4f
1df61b51
      rewrite apegS -apegD apegO oddD -subn2 oddB.
1e24
 a[p3, p1] (~~ (odd (3 * c) (+) odd 2 (+) odd a1)) =
 a[p1, p3] (c + a1)
1e24
1 < 3 * c
1e501df61b51
        by rewrite addbF oddM /= -oddD -apegO.
1e24
1e57
1e4f
      by rewrite (leq_trans (_ : _ < 3 * 1)) // leq_mul2l //.
1e411df5
    by rewrite ltnW.
1df7
1cbb
1b50
  rewrite {cH1}//.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1df8
1cbb
1b50
  have {}P5 := 
    leq_trans P5 (leq_add (leqnn _) (leq_sum_beta _ _)).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca
P5:(S_[(3 * c).-2] n.+1).*2 <=
\sum_(i < (3 * c).-2)
   `d[u2 (inord i.+2), u2 (inord i.+3)]_smove +
sp (n:=n.+1) ta1 (3 * c).-2 pa1 +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) *
   (α_[1] k).*2
1cbb
1b50
  set y1S := \sum_(_ < _) _ in P5; set y2S := sp _ _ _ in P5.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca
y1S:=\sum_(H < (3 * c).-2)
   `d[u2 (inord H.+2), u2 (inord H.+3)]_smove:nat
y2S:=sp (n:=n.+1) ta1 (3 * c).-2 pa1:nat
P5:(S_[(3 * c).-2] n.+1).*2 <=
y1S + y2S +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) *
   (α_[1] k).*2
1cbb
1b50
  have [b_gt1|b_le2] := leqP 2 b.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d1
1cbb
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f
b_le2:b < 2
1cbb
1b51
(*   subcase b >= 2 *)
    pose u3 := [ffun i : 'I_3 =>
                 if i == 0 :> nat then sam1 
                 else if i == 1 :> nat then tam1 else ↓[u (inord a)]].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d1
u3:=[ffun i => if i == 0 :> nat
           then sam1
           else
            if i == 1 :> nat
            then tam1
            else ↓[u (inord a)]]:{ffun forall x : ordinal_finType 3,
      (fun=> configuration 4 n.+1) x}
1cbb
1e74
    have cH6 (k : 'I_3) : 
      0 < k < 2 -> codom (u3 k) \subset [:: p1; a[p2, p3] k].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c994
0 < k < 2 ->
codom (u3 k) \subset [:: p1;  a[p2, p3] k]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c
cH6:forall k : 'I_3,
0 < k < 2 ->
codom (u3 k) \subset [:: p1;  a[p2, p3] k]
1cbb
1e751b51
      move=> /andP[k_gt0 k_lt2].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c994852
k_lt2:k < 2
codom (u3 k) \subset [:: p1;  a[p2, p3] k]
1e82
      rewrite !ffunE.
1e89
codom
  (if k == 0
   then sam1
   else if k == 1 then tam1 else ↓[u (inord a)]) \subset
[:: p1;  a[p2, p3] k]
1e82
      by have ->/= : k = 1 :> nat by case: (k : nat) k_gt0 k_lt2 => // [] [|].
1e84
1cbb
1e74
    have P6 : 
      (S_[2] n.+1).*2 <=
      (`d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove) + sp sam1 2 p2 
                                                + sp ↓[u (inord a)] 2 p2.
1e84
(S_[2] n.+1).*2 <=
`d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove +
sp (n:=n.+1) sam1 2 p2 +
sp (n:=n.+1) ↓[u (inord a)] 2 p2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1e85
P6:(S_[2] n.+1).*2 <=
`d[sam1, tam1]_smove +
`d[tam1, ↓[u (inord a)]]_smove +
sp (n:=n.+1) sam1 2 p2 +
sp (n:=n.+1) ↓[u (inord a)] 2 p2
1cbb
1e751b51
      apply: (leq_trans (IH _ _ _ _ _ _ cH6)) => //.
1e84
p2 != p1
1e84
sd (n:=n.+1) u3 + sp (n:=n.+1) (u3 ord0) 2 p2 +
sp (n:=n.+1) (u3 ord_max) 2 ( a[p2, p3] 2) <=
`d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove +
sp (n:=n.+1) sam1 2 p2 +
sp (n:=n.+1) ↓[u (inord a)] 2 p2
1e981e751b51
        by rewrite eq_sym.
1e84
1ea1
1e97
      repeat apply: leq_add; rewrite leq_eqVlt; apply/orP; left; apply/eqP.
1e84
sd (n:=n.+1) u3 =
`d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove
1e84
sp (n:=n.+1) (u3 ord0) 2 p2 = sp (n:=n.+1) sam1 2 p2
1e84
sp (n:=n.+1) (u3 ord_max) 2 ( a[p2, p3] 2) =
sp (n:=n.+1) ↓[u (inord a)] 2 p2
1e981e751b51
      -
1ea6 by rewrite /sd !big_ord_recr big_ord0 !ffunE !inordK /=.
1e84
1eab
1eac1e981e751b51
      -
1eb0 by apply: eq_bigr => i _; rewrite !ffunE.
1e84
1ead
1e97
      by apply: eq_bigr => i _; rewrite !ffunE.
1e99
1cbb
1e74
    rewrite {cH6}// in P6.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1e9a
1cbb
1e74
    set x3S := sp _ _ _ in P6; set y3S := sp _ _ _ in P6.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c
x3S:=sp (n:=n.+1) sam1 2 p2:nat
y3S:=sp (n:=n.+1) ↓[u (inord a)] 2 p2:nat
P6:(S_[2] n.+1).*2 <=
`d[sam1, tam1]_smove +
`d[tam1, ↓[u (inord a)]]_smove + x3S + y3S
1cbb
1e74
    have x2Sx3SE : x2S + x3S <= (S_[1] n).*2 + α_[maxn (3 * a).-2 2] n.
1ec1
x2S + x3S <= (S_[1] n).*2 + α_[maxn (3 * a).-2 2] n
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec4
x2Sx3SE:x2S + x3S <=
(S_[1] n).*2 + α_[maxn (3 * a).-2 2] n
1cbb
1e751b51
      by apply: sum_alpha_diffE; by rewrite /pa.
1ecb
1cbb
1e74
    rewrite (maxn_idPl _) // in x2Sx3SE.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec4
x2Sx3SE:x2S + x3S <= (S_[1] n).*2 + α_[(3 * a).-2] n
1cbb
1e74
    pose u4 := [ffun i : 'I_b.+1 => ↓[u (inord (a + i))]].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed4
u4:=[ffun i => ↓[u (inord (a + i))]]:{ffun forall x : ordinal_finType b.+1,
      (fun=> configuration 4 n.+1) x}
1cbb
1e74
    pose paS := a[p1, p3] a.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f
1cbb
1e74
    have cH7 (k : 'I_b.+1) : 
      0 < k < b -> codom (u4 k) \subset [:: p2; a[pa, paS] k].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f9c1
0 < k < b ->
codom (u4 k) \subset [:: p2;  a[pa, paS] k]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f
cH7:forall k : 'I_b.+1,
0 < k < b ->
codom (u4 k) \subset [:: p2;  a[pa, paS] k]
1cbb
1e751b51
      move=> /andP[k_gt0 k_ltb].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f9c1852
k_ltb:k < b
codom (u4 k) \subset [:: p2;  a[pa, paS] k]
1ee3
      have akLl : a + k < l.+1.
1eea
a + k < l.+1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f9c18521eeb
akLl:a + k < l.+1
1eec
1ee41e751b51
        rewrite  ltnS (leq_trans _ a1Ll) //.
1eea
a + k <= a1
1ef1
        by rewrite -(subnK aLa1) addnC leq_add2r ltnW.
1ef3
1eec
1ee3
      rewrite !ffunE.
1ef3
codom ↓[u (inord (a + k))] \subset
[:: p2;  a[pa, paS] k]
1ee3
      have /H : 0 < (inord (a + k) : 'I_l.+2) < l.+1.
1ef3
0 < inord (a + k) < l.+1
1ef3
codom (u' (inord (a + k))) \subset
[:: p2;  a[p1, p3] (inord (a + k))] ->
codom ↓[u (inord (a + k))] \subset
[:: p2;  a[pa, paS] k]
1ee41e751b51
        rewrite inordK; first by rewrite addn_gt0 a_gt0.
1ef3
a + k < l.+2
1f04
        by rewrite ltnS ltnW.
1ef3
1f06
1ee3
      rewrite ffunE inordK //; last by rewrite ltnS ltnW.
1ef3
codom ↓[u (inord (a + k))] \subset
[:: p2;  a[p1, p3] (a + k)] ->
codom ↓[u (inord (a + k))] \subset
[:: p2;  a[pa, paS] k]
1ee3
      by rewrite /pa /paS apegS -apegD [k + _]addnC.
1ee5
1cbb
1e74
    have P7 : 
      (S_[b] n.+1).*2 <= 
      \sum_(i < b) `d[↓[u (inord (a + i))], ↓[u (inord (a + i).+1)]]_smove +
      sp ↓[u (inord a)] b pa + sp ↓[u (inord a1)] b (a[p1, p3] a1).
1ee5
(S_[b] n.+1).*2 <=
\sum_(i < b)
   `d[↓[u (inord (a + i))], ↓[u (inord (a + i).+1)]]_smove +
sp (n:=n.+1) ↓[u (inord a)] b pa +
sp (n:=n.+1) ↓[u (inord a1)] b ( a[p1, p3] a1)
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f1ee6
P7:(S_[b] n.+1).*2 <=
\sum_(i < b)
   `d[↓[u (inord (a + i))], ↓[
u (inord (a + i).+1)]]_smove +
sp (n:=n.+1) ↓[u (inord a)] b pa +
sp (n:=n.+1) ↓[u (inord a1)] b ( a[p1, p3] a1)
1cbb
1e751b51
      apply: leq_trans (IH _ _ _ _ _ _ cH7) (leq_add (leq_add _ _) _);
        rewrite /pa /paS // 1?eq_sym //.
1ee5
 a[p1, p3] a.+1 !=  a[p1, p3] a
1ee5
sd (n:=n.+1) u4 <=
\sum_(i < b)
   `d[↓[u (inord (a + i))], ↓[
u (inord (a + i).+1)]]_smove
1ee5
sp (n:=n.+1) (u4 ord0) b ( a[p1, p3] a) <=
sp (n:=n.+1) ↓[u (inord a)] b ( a[p1, p3] a)
1ee5
sp (n:=n.+1) (u4 ord_max) b
  ( a[ a[p1, p3] a,  a[p1, p3] a.+1] b) <=
sp (n:=n.+1) ↓[u (inord a1)] b ( a[p1, p3] a1)
1f1a1e751b51
      -
1f1e by rewrite apegS apeg_eqC eq_sym.
1ee5
sd (n:=n.+1) u4 <=
\sum_(i < b)
   `d[↓[u (inord (a + i))], ↓[u (inord (a + i).+1)]]_smove
1f241f261f1a1e751b51
      -
1f2a apply: leq_sum => i; rewrite !ffunE !inordK ?addnS //.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f1ee6821
i.+1 < b.+1
1f32
i < b.+1
1f241f261f1a1e751b51
          by rewrite ltnS.
1f32
1f36
1f2d
        by rewrite ltnS ltnW.
1ee5
1f25
1f261f1a1e751b51
      -
1f3b by apply: leq_sum => i _; rewrite !ffunE addn0.
1ee5
1f27
1f19
      apply: leq_sum => i _; rewrite !ffunE /= [a + b]addnC subnK //.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f1ee61054
(u (inord a1) (trshift 1 i)
 !=  a[ a[p1, p3] a,  a[p1, p3] a.+1] b) *
β_[n.+1, b] i <=
(u (inord a1) (trshift 1 i) !=  a[p1, p3] a1) *
β_[n.+1, b] i
1f19
      by rewrite apegS -apegD subnK.
1f1b
1cbb
1e74
    set x4S := sp _ _ _ in P7; set y4S := sp _ _ _ in P7.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f1ee6
x4S:=sp (n:=n.+1) ↓[u (inord a)] b pa:nat
y4S:=sp (n:=n.+1) ↓[u (inord a1)] b ( a[p1, p3] a1):nat
P7:(S_[b] n.+1).*2 <=
\sum_(i < b)
   `d[↓[u (inord (a + i))], ↓[
u (inord (a + i).+1)]]_smove + x4S + y4S
1cbb
1e74
    have y3Sx4SE : y3S + x4S <= (S_[1] n).*2 + α_[maxn 2 b] n.
1f4d
y3S + x4S <= (S_[1] n).*2 + α_[maxn 2 b] n
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f1ee61f4e1f4f1f50
y3Sx4SE:y3S + x4S <= (S_[1] n).*2 + α_[maxn 2 b] n
1cbb
1e751b51
      apply: sum_alpha_diffE => //; first by rewrite /pa eq_sym.
1f4d
codom ↓[u (inord a)] \subset [:: p2; pa]
1f55
      apply: codom_liftr.
1f4d
codom (u (inord a)) \subset [:: p2; pa]
1f55
      have := @cH (inord a); rewrite inordK //; apply.
1f4d
0 < a < l.+1
1f55
      by rewrite a_gt0 /= ltnS (leq_trans _ a1Ll).
1f57
1cbb
1e74
    rewrite (maxn_idPr _) // in y3Sx4SE.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f1ee61f4e1f4f1f50
y3Sx4SE:y3S + x4S <= (S_[1] n).*2 + α_[b] n
1cbb
1e74
    pose u5 := [ffun i : 'I_3 =>
                     if i == 0 :> nat then ↓[u (inord a1)]
                 else if i == 1 :> nat then sa1 else ta1].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f1ee61f4e1f4f1f501f6c
u5:=[ffun i => if i == 0 :> nat
           then ↓[u (inord a1)]
           else if i == 1 :> nat then sa1 else ta1]:{ffun forall x : ordinal_finType 3,
      (fun=> configuration 4 n.+1) x}
1cbb
1e74
    have cH8 (k : 'I_3) : 
      0 < k < 2 -> codom (u5 k) \subset [:: p1; a[p2, p3] k].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f1ee61f4e1f4f1f501f6c1f71994
0 < k < 2 ->
codom (u5 k) \subset [:: p1;  a[p2, p3] k]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f1ee61f4e1f4f1f501f6c1f71
cH8:forall k : 'I_3,
0 < k < 2 ->
codom (u5 k) \subset [:: p1;  a[p2, p3] k]
1cbb
1e751b51
      move=> /andP[k_gt0 k_lt2].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f1ee61f4e1f4f1f501f6c1f719948521e8a
codom (u5 k) \subset [:: p1;  a[p2, p3] k]
1f77
      rewrite !ffunE.
1f7e
codom
  (if k == 0
   then ↓[u (inord a1)]
   else if k == 1 then sa1 else ta1) \subset
[:: p1;  a[p2, p3] k]
1f77
      by have ->/= : k = 1 :> nat by case: (k : nat) k_gt0 k_lt2 => // [] [|].
1f79
1cbb
1e74
    have P8 : 
      (S_[2] n.+1).*2 <=
      (`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove) + 
        sp ↓[u (inord a1)] 2 p2 + sp ta1 2 p2.
1f79
(S_[2] n.+1).*2 <=
`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
sp (n:=n.+1) ↓[u (inord a1)] 2 p2 +
sp (n:=n.+1) ta1 2 p2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f1ee61f4e1f4f1f501f6c1f711f7a
P8:(S_[2] n.+1).*2 <=
`d[↓[u (inord a1)], sa1]_smove +
`d[sa1, ta1]_smove +
sp (n:=n.+1) ↓[u (inord a1)] 2 p2 +
sp (n:=n.+1) ta1 2 p2
1cbb
1e751b51
      apply: (leq_trans (IH _ _ _ _ _ _ cH8)) => //; first by rewrite eq_sym.
1f79
sd (n:=n.+1) u5 + sp (n:=n.+1) (u5 ord0) 2 p2 +
sp (n:=n.+1) (u5 ord_max) 2 ( a[p2, p3] 2) <=
`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
sp (n:=n.+1) ↓[u (inord a1)] 2 p2 +
sp (n:=n.+1) ta1 2 p2
1f8b
      repeat apply: leq_add; rewrite leq_eqVlt; apply/orP; left; apply/eqP.
1f79
sd (n:=n.+1) u5 =
`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove
1f79
sp (n:=n.+1) (u5 ord0) 2 p2 =
sp (n:=n.+1) ↓[u (inord a1)] 2 p2
1f79
sp (n:=n.+1) (u5 ord_max) 2 ( a[p2, p3] 2) =
sp (n:=n.+1) ta1 2 p2
1f8c1e751b51
      -
1f94 by rewrite /sd !big_ord_recr big_ord0 !ffunE !inordK /=.
1f79
1f99
1f9a1f8c1e751b51
      -
1f9e by apply: eq_bigr => i _; rewrite !ffunE.
1f79
1f9b
1f8b
      by apply: eq_bigr => i _; rewrite !ffunE.
1f8d
1cbb
1e74
    set x5S := sp _ _ _ in P8; set y5S := sp _ _ _ in P8.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f1ee61f4e1f4f1f501f6c1f711f7a
x5S:=sp (n:=n.+1) ↓[u (inord a1)] 2 p2:nat
y5S:=sp (n:=n.+1) ta1 2 p2:nat
P8:(S_[2] n.+1).*2 <=
`d[↓[u (inord a1)], sa1]_smove +
`d[sa1, ta1]_smove + x5S + y5S
1cbb
1e74
    have y4Sx5SE : y4S + x5S <= (S_[1] n).*2 + α_[maxn b 2] n.
1fab
y4S + x5S <= (S_[1] n).*2 + α_[maxn b 2] n
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f1ee61f4e1f4f1f501f6c1f711f7a1fac1fad1fae
y4Sx5SE:y4S + x5S <= (S_[1] n).*2 + α_[maxn b 2] n
1cbb
1e751b51
      apply: sum_alpha_diffE => //.
1fab
codom ↓[u (inord a1)] \subset [::  a[p1, p3] a1; p2]
1fb3
      apply: codom_liftr; rewrite codom_subC.
1fab
codom (u (inord a1)) \subset [:: p2;  a[p1, p3] a1]
1fb3
      have := @cH (inord a1); rewrite inordK //.
1fab
(0 < a1 < l.+1 ->
 codom (u (inord a1)) \subset [:: p2;  a[p1, p3] a1]) ->
codom (u (inord a1)) \subset [:: p2;  a[p1, p3] a1]
1fab
a1 < l.+2
1fb41e751b51
        by apply; rewrite a1_gt0 /= ltnS.
1fab
1fc5
1fb3
      by rewrite ltnS ltnW.
1fb5
1cbb
1e74
    rewrite (maxn_idPl _) // in y4Sx5SE.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f1ee61f4e1f4f1f501f6c1f711f7a1fac1fad1fae
y4Sx5SE:y4S + x5S <= (S_[1] n).*2 + α_[b] n
1cbb
1e74
    have y2Sy5SE : y2S + y5S <= (S_[1] n).*2 + α_[maxn (3 * c).-2 2] n.
1fcf
y2S + y5S <= (S_[1] n).*2 + α_[maxn (3 * c).-2 2] n
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f1ee61f4e1f4f1f501f6c1f711f7a1fac1fad1fae1fd0
y2Sy5SE:y2S + y5S <=
(S_[1] n).*2 + α_[maxn (3 * c).-2 2] n
1cbb
1e751b51
      apply: sum_alpha_diffE => //; first by rewrite /pa1.
1fcf
codom ta1 \subset [:: pa1; p2]
1fd5
      by rewrite codom_subC.
1fd7
1cbb
1e74
    rewrite (maxn_idPl _) // in y2Sy5SE.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f1ee61f4e1f4f1f501f6c1f711f7a1fac1fad1fae1fd0
y2Sy5SE:y2S + y5S <= (S_[1] n).*2 + α_[(3 * c).-2] n
1cbb
1e74
    rewrite (_ : sd u = 
       \sum_(i < a) `d[u (inord i), u (inord i.+1)]_smove +
       \sum_(a <= i < a1) `d[u (inord i), u (inord i.+1)]_smove +
       \sum_(a1 <= i < l.+1) `d[u (inord i), u (inord i.+1)]_smove); last first.
1fe3
sd (n:=n.+2) u =
\sum_(i < a) `d[u (inord i), u (inord i.+1)]_smove +
\sum_(a <= i < a1)
   `d[u (inord i), u (inord i.+1)]_smove +
\sum_(a1 <= i < l.+1)
   `d[u (inord i), u (inord i.+1)]_smove
1fe3
(S_[l.+1] n.+2).*2 <=
\sum_(i < a) `d[u (inord i), u (inord i.+1)]_smove +
\sum_(a <= i < a1)
   `d[u (inord i), u (inord i.+1)]_smove +
\sum_(a1 <= i < l.+1)
   `d[u (inord i), u (inord i.+1)]_smove + x1S +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2
1e751b51
      pose f i := `d[u (inord i), u (inord i.+1)]_smove.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f1ee61f4e1f4f1f501f6c1f711f7a1fac1fad1fae1fd01fe4
f:=fun i : nat =>
`d[u (inord i), u (inord i.+1)]_smove:nat -> nat
1fe8
1fe9
      rewrite /sd -!(big_mkord xpredT f).
1fef
\sum_(0 <= i < l.+1) f i =
\sum_(0 <= i < a) f i +
\sum_(a <= i < a1)
   `d[u (inord i), u (inord i.+1)]_smove +
\sum_(a1 <= i < l.+1)
   `d[u (inord i), u (inord i.+1)]_smove
1fe9
      by rewrite -!big_cat_nat //= ltnW.
1fe3
1feb
1e74
    rewrite -{}P1 -{}P2.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f1ee61f4e1f4f1f501f6c1f711f7a1fac1fad1fae1fd01fe4
(S_[l.+1] n.+2).*2 <=
a.*2 + sd (n:=n.+1) u1 +
\sum_(a <= i < a1)
   `d[u (inord i), u (inord i.+1)]_smove +
(c.*2 + sd (n:=n.+1) u2) + x1S +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2
1e74
    rewrite [X in _ <= _ + X + _ + _ + _ + _]
       (_ : _  =
              \sum_(i < (3 * a).-2) `d[u1 (inord i), u1 (inord i.+1)]_smove
              + `d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove); 
              last first.
1ffb
sd (n:=n.+1) u1 =
\sum_(i < (3 * a).-2)
   `d[u1 (inord i), u1 (inord i.+1)]_smove +
`d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove
1ffb
(S_[l.+1] n.+2).*2 <=
a.*2 +
(\sum_(i < (3 * a).-2)
    `d[u1 (inord i), u1 (inord i.+1)]_smove +
 `d[sam1, tam1]_smove + `d[tam1, ↓[
 u (inord a)]]_smove) +
\sum_(a <= i < a1)
   `d[u (inord i), u (inord i.+1)]_smove +
(c.*2 + sd (n:=n.+1) u2) + x1S +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2
1e751b51
      pose f i := `d[u1 (inord i), u1 (inord i.+1)]_smove.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f1ee61f4e1f4f1f501f6c1f711f7a1fac1fad1fae1fd01fe4
f:=fun i : nat =>
`d[u1 (inord i), u1 (inord i.+1)]_smove:nat -> nat
2000
2001
      rewrite /sd -!(big_mkord xpredT f) .
2007
\sum_(0 <= i < 3 * a) f i =
\sum_(0 <= i < (3 * a).-2) f i + `d[sam1, tam1]_smove +
`d[tam1, ↓[u (inord a)]]_smove
2001
      rewrite -{1}(subnK (_ : 1 < 3 * a)) //; last first.
2007
979
2007
\sum_(0 <= i < 3 * a - 2 + 2) f i =
\sum_(0 <= i < (3 * a).-2) f i + `d[sam1, tam1]_smove +
`d[tam1, ↓[u (inord a)]]_smove
20021e751b51
        by rewrite -[a]prednK // mulnS.
2007
2012
2001
      rewrite subn2 !addn2 !big_nat_recr /f //=; congr (_ + _ + _).
2007
`d[u1 (inord (3 * a).-2), u1 (inord (3 * a).-2.+1)]_smove =
`d[sam1, tam1]_smove
2007
`d[u1 (inord (3 * a).-2.+1), 
u1 (inord (3 * a).-2.+2)]_smove =
`d[tam1, ↓[u (inord a)]]_smove
20021e751b51
        rewrite ffunE (_ : (3 * a).-2 = (3 * (a.-1)).+1); last first.
2007
1c9f
2007
`d[if inord (3 * a.-1).+1 %% 3 == 0
   then ↓[u (inord (inord (3 * a.-1).+1 %/ 3))]
   else
    if inord (3 * a.-1).+1 %% 3 == 1
    then
     if inord (3 * a.-1).+1 == (3 * a.-1).+1
     then sam1
     else si (inord (3 * a.-1).+1 %/ 3)
    else
     if inord (3 * a.-1).+1 == (3 * a.-1).+2
     then tam1
     else ti (inord (3 * a.-1).+1 %/ 3), 
u1 (inord (3 * a.-1).+2)]_smove = `d[sam1, tam1]_smove
201b20021e751b51
          by rewrite -{1}[a]prednK // mulnS.
2007
`d[if inord (3 * a.-1).+1 %% 3 == 0
   then ↓[u (inord (inord (3 * a.-1).+1 %/ 3))]
   else
    if inord (3 * a.-1).+1 %% 3 == 1
    then
     if inord (3 * a.-1).+1 == (3 * a.-1).+1
     then sam1
     else si (inord (3 * a.-1).+1 %/ 3)
    else
     if inord (3 * a.-1).+1 == (3 * a.-1).+2
     then tam1
     else ti (inord (3 * a.-1).+1 %/ 3), u1
                                           (inord
                                              (3 *
                                               a.-1).+2)]_smove =
`d[sam1, tam1]_smove
201a
        rewrite inordK; last by rewrite ltnS ltn_mul2l /= prednK.
2007
`d[if (3 * a.-1).+1 %% 3 == 0
   then ↓[u (inord ((3 * a.-1).+1 %/ 3))]
   else
    if (3 * a.-1).+1 %% 3 == 1
    then
     if (3 * a.-1).+1 == (3 * a.-1).+1
     then sam1
     else si ((3 * a.-1).+1 %/ 3)
    else
     if (3 * a.-1).+1 == (3 * a.-1).+2
     then tam1
     else ti ((3 * a.-1).+1 %/ 3), u1
                                     (inord
                                        (3 * a.-1).+2)]_smove =
`d[sam1, tam1]_smove
201a
        rewrite mod3E /= eqxx.
2007
`d[sam1, u1 (inord (3 * a.-1).+2)]_smove =
`d[sam1, tam1]_smove
201a
        rewrite ffunE inordK; last first.
2007
(3 * a.-1).+2 < (3 * a).+1
2007
`d[sam1, if (3 * a.-1).+2 %% 3 == 0
         then ↓[u (inord ((3 * a.-1).+2 %/ 3))]
         else
          if (3 * a.-1).+2 %% 3 == 1
          then
           if (3 * a.-1).+2 == (3 * a.-1).+1
           then sam1
           else si ((3 * a.-1).+2 %/ 3)
          else
           if (3 * a.-1).+2 == (3 * a.-1).+2
           then tam1
           else ti ((3 * a.-1).+2 %/ 3)]_smove =
`d[sam1, tam1]_smove
201b20021e751b51
          by rewrite -[_.+3](mulnDr 3 1) add1n prednK // ltnW.
2007
2035
201a
        by rewrite mod3E /= eqxx.
2007
`d[u1 (inord (3 * a).-2.+1), u1 (inord (3 * a).-2.+2)]_smove =
`d[tam1, ↓[u (inord a)]]_smove
2001
      rewrite ffunE (_ : (3 * a).-2 = (3 * (a.-1)).+1); last first.
201f
2007
`d[if inord (3 * a.-1).+2 %% 3 == 0
   then ↓[u (inord (inord (3 * a.-1).+2 %/ 3))]
   else
    if inord (3 * a.-1).+2 %% 3 == 1
    then
     if inord (3 * a.-1).+2 == (3 * a.-1).+1
     then sam1
     else si (inord (3 * a.-1).+2 %/ 3)
    else
     if inord (3 * a.-1).+2 == (3 * a.-1).+2
     then tam1
     else ti (inord (3 * a.-1).+2 %/ 3), 
u1 (inord (3 * a.-1).+3)]_smove =
`d[tam1, ↓[u (inord a)]]_smove
20021e751b51
        by rewrite -{1}[a]prednK // mulnS.
2007
`d[if inord (3 * a.-1).+2 %% 3 == 0
   then ↓[u (inord (inord (3 * a.-1).+2 %/ 3))]
   else
    if inord (3 * a.-1).+2 %% 3 == 1
    then
     if inord (3 * a.-1).+2 == (3 * a.-1).+1
     then sam1
     else si (inord (3 * a.-1).+2 %/ 3)
    else
     if inord (3 * a.-1).+2 == (3 * a.-1).+2
     then tam1
     else ti (inord (3 * a.-1).+2 %/ 3), u1
                                           (inord
                                              (3 *
                                               a.-1).+3)]_smove =
`d[tam1, ↓[u (inord a)]]_smove
2001
      rewrite inordK; last first.
2031
2007
`d[if (3 * a.-1).+2 %% 3 == 0
   then ↓[u (inord ((3 * a.-1).+2 %/ 3))]
   else
    if (3 * a.-1).+2 %% 3 == 1
    then
     if (3 * a.-1).+2 == (3 * a.-1).+1
     then sam1
     else si ((3 * a.-1).+2 %/ 3)
    else
     if (3 * a.-1).+2 == (3 * a.-1).+2
     then tam1
     else ti ((3 * a.-1).+2 %/ 3), 
u1 (inord (3 * a.-1).+3)]_smove =
`d[tam1, ↓[u (inord a)]]_smove
20021e751b51
        by rewrite -{2}[a]prednK // mulnS.
2007
`d[if (3 * a.-1).+2 %% 3 == 0
   then ↓[u (inord ((3 * a.-1).+2 %/ 3))]
   else
    if (3 * a.-1).+2 %% 3 == 1
    then
     if (3 * a.-1).+2 == (3 * a.-1).+1
     then sam1
     else si ((3 * a.-1).+2 %/ 3)
    else
     if (3 * a.-1).+2 == (3 * a.-1).+2
     then tam1
     else ti ((3 * a.-1).+2 %/ 3), u1
                                     (inord
                                        (3 * a.-1).+3)]_smove =
`d[tam1, ↓[u (inord a)]]_smove
2001
      rewrite mod3E /= eqxx.
2007
`d[tam1, u1 (inord (3 * a.-1).+3)]_smove =
`d[tam1, ↓[u (inord a)]]_smove
2001
      rewrite ffunE inordK; last first.
2007
(3 * a.-1).+3 < (3 * a).+1
2007
`d[tam1, if (3 * a.-1).+3 %% 3 == 0
         then ↓[u (inord ((3 * a.-1).+3 %/ 3))]
         else
          if (3 * a.-1).+3 %% 3 == 1
          then
           if (3 * a.-1).+3 == (3 * a.-1).+1
           then sam1
           else si ((3 * a.-1).+3 %/ 3)
          else
           if (3 * a.-1).+3 == (3 * a.-1).+2
           then tam1
           else ti ((3 * a.-1).+3 %/ 3)]_smove =
`d[tam1, ↓[u (inord a)]]_smove
20021e751b51
        by rewrite -{2}[a]prednK // mulnS.
2007
2059
2001
      by rewrite -[(_ * _).+3](mulnDr 3 1) mod3E /= div3E add1n prednK.
1ffb
(S_[l.+1] n.+2).*2 <=
a.*2 +
(\sum_(i < (3 * a).-2)
    `d[u1 (inord i), u1 (inord i.+1)]_smove +
 `d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove) +
\sum_(a <= i < a1)
   `d[u (inord i), u (inord i.+1)]_smove +
(c.*2 + sd (n:=n.+1) u2) + x1S +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2
1e74
    rewrite [X in _ <= _ + (_ + X) + _ + _]
       (_ : _  = `d[↓[u (inord a1)], sa1]_smove +  `d[sa1, ta1]_smove +
         \sum_(i < (3 * c).-2) `d[u2 (inord i.+2), u2 (inord i.+3)]_smove);
        last first.
1ffb
sd (n:=n.+1) u2 =
`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
\sum_(i < (3 * c).-2)
   `d[u2 (inord i.+2), u2 (inord i.+3)]_smove
1ffb
(S_[l.+1] n.+2).*2 <=
a.*2 +
(\sum_(i < (3 * a).-2)
    `d[u1 (inord i), u1 (inord i.+1)]_smove +
 `d[sam1, tam1]_smove + `d[tam1, ↓[
 u (inord a)]]_smove) +
\sum_(a <= i < a1)
   `d[u (inord i), u (inord i.+1)]_smove +
(c.*2 +
 (`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
  \sum_(i < (3 * c).-2)
     `d[u2 (inord i.+2), 
  u2 (inord i.+3)]_smove)) + x1S +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2
1e751b51
      pose f i := `d[u2 (inord i), u2 (inord i.+1)]_smove.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f1ee61f4e1f4f1f501f6c1f711f7a1fac1fad1fae1fd01fe4
f:=fun i : nat =>
`d[u2 (inord i), u2 (inord i.+1)]_smove:nat -> nat
2064
2065
      rewrite /sd -!(big_mkord xpredT f) .
206b
\sum_(0 <= i < 3 * c) f i =
`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
\sum_(i < (3 * c).-2)
   `d[u2 (inord i.+2), u2 (inord i.+3)]_smove
2065
      rewrite -{1}(subnK (_ : 1 < 3 * c)) //; last first.
206b
1e57
206b
\sum_(0 <= i < 3 * c - 2 + 2) f i =
`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
\sum_(i < (3 * c).-2)
   `d[u2 (inord i.+2), u2 (inord i.+3)]_smove
20661e751b51
        by rewrite -[c]prednK // mulnS.
206b
2076
2065
      rewrite subn2 !addn2 !big_nat_recl // /f /= !addnA; congr (_ + _ + _).
206b
`d[u2 (inord 0), u2 (inord 1)]_smove =
`d[↓[u (inord a1)], sa1]_smove
206b
`d[u2 (inord 1), u2 (inord 2)]_smove =
`d[sa1, ta1]_smove
206b
\sum_(0 <= i < (3 * c).-2)
   `d[u2 (inord i.+2), u2 (inord i.+3)]_smove =
\sum_(i < (3 * c).-2)
   `d[u2 (inord i.+2), u2 (inord i.+3)]_smove
20661e751b51
      -
207b rewrite !ffunE !inordK ?addn0 //=.
206b
1 < (3 * c).+1
207e
        by rewrite ltnS muln_gt0.
206b
2080
208120661e751b51
      -
2089 rewrite !ffunE !inordK //=.
206b
1e29
206b
2087
208120661e751b51
          by rewrite ltnW // ltnS (leq_mul2l 3 1 c).
2086208b
        by rewrite ltnS muln_gt0.
206b
2082
2065
      by rewrite big_mkord.
1ffb
(S_[l.+1] n.+2).*2 <=
a.*2 +
(\sum_(i < (3 * a).-2)
    `d[u1 (inord i), u1 (inord i.+1)]_smove +
 `d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove) +
\sum_(a <= i < a1)
   `d[u (inord i), u (inord i.+1)]_smove +
(c.*2 +
 (`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
  \sum_(i < (3 * c).-2)
     `d[u2 (inord i.+2), u2 (inord i.+3)]_smove)) +
x1S +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2
1e74
    rewrite -/y1S.
1ffb
(S_[l.+1] n.+2).*2 <=
a.*2 +
(\sum_(i < (3 * a).-2)
    `d[u1 (inord i), u1 (inord i.+1)]_smove +
 `d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove) +
\sum_(a <= i < a1)
   `d[u (inord i), u (inord i.+1)]_smove +
(c.*2 +
 (`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
  y1S)) + x1S +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2
1e74
    set z1S := \sum_(_ < _) _; set z2S := \sum_(_ < _) _.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f1ee61f4e1f4f1f501f6c1f711f7a1fac1fad1fae1fd01fe4
z1S:=\sum_(H < (3 * a).-2)
   `d[u1 (inord H), u1 (inord H.+1)]_smove:nat
z2S:=\sum_(H < n.+1)
   (↓[u ord_max] H !=  a[p3, p1] l) *
   (α_[1] H).*2:nat
(S_[l.+1] n.+2).*2 <=
a.*2 +
(z1S + `d[sam1, tam1]_smove +
 `d[tam1, ↓[u (inord a)]]_smove) +
\sum_(a <= i < a1)
   `d[u (inord i), u (inord i.+1)]_smove +
(c.*2 +
 (`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
  y1S)) + x1S + z2S
1e74
    set z3S := \sum_(_ <= _ < _) _.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f5d11e7c1ec21ec31ec41ed41ed998f1ee61f4e1f4f1f501f6c1f711f7a1fac1fad1fae1fd01fe420a320a4
z3S:=\sum_(a <= H < a1)
   `d[u (inord H), u (inord H.+1)]_smove:nat
(S_[l.+1] n.+2).*2 <=
a.*2 +
(z1S + `d[sam1, tam1]_smove +
 `d[tam1, ↓[u (inord a)]]_smove) + z3S +
(c.*2 +
 (`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
  y1S)) + x1S + z2S
1e74 
    rewrite -/z1S in P4; rewrite -/z2S in P5; rewrite -/z3S in P3.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ec41ed41ed998f1ee61f4e1f4f1f501f6c1f711f7a1fac1fad1fae1fd01fe420a320a420aa
P4:(S_[(3 * a).-2] n.+1).*2 <= z1S + x1S + x2S
P5:(S_[(3 * c).-2] n.+1).*2 <= y1S + y2S + z2S
P3:\sum_(a <= i < a1)
   `d[↓[u (inord i)], ↓[
u (inord i.+1)]]_smove <= z3S
20ab
1e74
    rewrite -[z1S + _ + _]addnA; set d1 := `d[_,_]_ _ + `d[_,_]_ _;
       rewrite -/d1 in P6.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f501f6c1f711f7a1fac1fad1fae1fd01fe420a320a420aa20b020b120b2
d1:=`d[sam1, tam1]_smove +
`d[tam1, ↓[u (inord a)]]_smove:nat
P6:(S_[2] n.+1).*2 <= d1 + x3S + y3S
(S_[l.+1] n.+2).*2 <=
a.*2 + (z1S + d1) + z3S +
(c.*2 +
 (`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
  y1S)) + x1S + z2S
1e74
    set d2 := `d[_,_]_ _ + `d[_,_]_ _; rewrite -/d2 in P8.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f501f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b220b720b8
d2:=`d[↓[u (inord a1)], sa1]_smove +
`d[sa1, ta1]_smove:nat
P8:(S_[2] n.+1).*2 <= d2 + x5S + y5S
(S_[l.+1] n.+2).*2 <=
a.*2 + (z1S + d1) + z3S + (c.*2 + (d2 + y1S)) + x1S +
z2S
1e74
    set xS := \sum_(_ <= _ < _) _ in P3.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f501f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf
xS:=\sum_(a <= H < a1)
   `d[↓[u (inord H)], ↓[
u (inord H.+1)]]_smove:nat
P3:xS <= z3S
20c0
1e74
    rewrite [X in _ <= X + _ + _](_ : _ = xS) in P7; last first.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c6
\sum_(i < b)
   `d[↓[u (inord (a + i))], ↓[u (inord (a + i).+1)]]_smove =
xS
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c6
P7:(S_[b] n.+1).*2 <= xS + x4S + y4S
20c0
1e751b51
      rewrite /xS -[in RHS](add0n a) big_addn -/b big_mkord.
20ca
\sum_(i < b)
   `d[↓[u (inord (a + i))], ↓[u (inord (a + i).+1)]]_smove =
\sum_(i < b)
   `d[↓[u (inord (i + a))], ↓[u (inord (i + a).+1)]]_smove
20cc
      by apply: eq_bigr => i _; rewrite [a + _]addnC.
20ce
20c0
1e74
    have F1 := dsum_alphaL_S (3 * a).-2 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf
F1:S_[(3 * a).-2] n.+1 =
S_[(3 * a).-2] n + α_[(3 * a).-2] n
20c0
1e74
    have F2 := dsum_alphaL_S (3 * c).-2 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf20db
F2:S_[(3 * c).-2] n.+1 =
S_[(3 * c).-2] n + α_[(3 * c).-2] n
20c0
1e74
    have F3 := dsum_alphaL_S b n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf20db20e0
F3:S_[b] n.+1 = S_[b] n + α_[b] n
20c0
1e74
    have F4_a : 2 * a = 2 + a.-1 + a.-1.
20e4
2 * a = 2 + a.-1 + a.-1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf20db20e020e5
F4_a:2 * a = 2 + a.-1 + a.-1
20c0
1e751b51
      by rewrite -{1}[a]prednK // mul2n -addnn addnS.
20ec
20c0
1e74
    have F4_c : 2 * c = 2 + c.-1 + c.-1.
20ec
2 * c = 2 + c.-1 + c.-1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf20db20e020e520ed
F4_c:2 * c = 2 + c.-1 + c.-1
20c0
1e751b51
      by rewrite -{1}[c]prednK // mul2n -addnn addnS.
20f7
20c0
1e74
    have F4 i k : 0 < i -> S_[i.-1] k.+1 <= S_[(3 *i).-2] k + i.-1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf20db20e020e520ed20f8
i, k:nat
0 < i -> S_[i.-1] k.+1 <= S_[(3 * i).-2] k + i.-1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf20db20e020e520ed20f8
F4:forall i k : nat,
0 < i -> S_[i.-1] k.+1 <= S_[(3 * i).-2] k + i.-1
20c0
1e751b51
      move=> i_gt0; apply: leq_trans (dsum_alpha3l _ _) _.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf20db20e020e520ed20f82100
i_gt0:0 < i
S_[3 * i.-1] k + i.-1 <= S_[(3 * i).-2] k + i.-1
2102
      rewrite leq_add2r.
2109
S_[3 * i.-1] k <= S_[(3 * i).-2] k
2102
      apply: (increasingE (increasing_dsum_alphaL_l _)).
2109
3 * i.-1 <= (3 * i).-2
2102
      by rewrite -{2}[i]prednK // mulnS addSn add2n /=.
2104
20c0
1e74
    have F4an := F4 a n a_gt0.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf20db20e020e520ed20f82105
F4an:S_[a.-1] n.+1 <= S_[(3 * a).-2] n + a.-1
20c0
1e74
    have F4an1 := F4 a n.+1 a_gt0.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf20db20e020e520ed20f82105211b
F4an1:S_[a.-1] n.+2 <= S_[(3 * a).-2] n.+1 + a.-1
20c0
1e74
    have F4cn := F4 c n (c_gt0).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf20db20e020e520ed20f82105211b2120
F4cn:S_[c.-1] n.+1 <= S_[(3 * c).-2] n + c.-1
20c0
1e74
    have F4cn1 := F4 c n.+1 (c_gt0).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf20db20e020e520ed20f82105211b21202125
F4cn1:S_[c.-1] n.+2 <= S_[(3 * c).-2] n.+1 + c.-1
20c0
1e74
    have l_gt1 : 1 < l.
2129
1 < l
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf20db20e020e520ed20f82105211b21202125212a4e4
20c0
1e751b51
      by rewrite -ltnS l1E -[3]/(2 + 0 + 1); rewrite !leq_add.
2131
20c0
1e74
    have F5 := SH l_gt1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf20db20e020e520ed20f82105211b21202125212a4e4
F5:(S_[l.+1] n.+2).*2 <=
S_[l.+1] n.+2 + S_[l] n.+1 + (α_[1] n.+1).*2
20c0
1e74
    have F6 : S_[l.+1] n.+2 + S_[1] n.+2 <= S_[a.-1] n.+2 + S_[(b + c).+2] n.+2.
2138
S_[l.+1] n.+2 + S_[1] n.+2 <=
S_[a.-1] n.+2 + S_[(b + c).+2] n.+2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf20db20e020e520ed20f82105211b21202125212a4e42139
F6:S_[l.+1] n.+2 + S_[1] n.+2 <=
S_[a.-1] n.+2 + S_[(b + c).+2] n.+2
20c0
1e751b51
      rewrite [X in _ <= X]addnC.
2138
S_[l.+1] n.+2 + S_[1] n.+2 <=
S_[(b + c).+2] n.+2 + S_[a.-1] n.+2
213e
      rewrite (_ : l.+1 =  1 + a.-2 + (b + c).+1); last first.
2138
l.+1 = 1 + a.-2 + (b + c).+1
2138
S_[1 + a.-2 + (b + c).+1] n.+2 + S_[1] n.+2 <=
S_[(b + c).+2] n.+2 + S_[a.-1] n.+2
213f1e751b51
        rewrite /b /c add1n -addSnnS !prednK //; last by rewrite -ltnS prednK.
2138
l.+1 = a + (a1 - a + (l.+1 - a1))
214a
        by rewrite addnCA addnA subnK // addnC subnK // ltnW.
2138
214c
213e
      rewrite (_ : a.-1 = 1 + a.-2); last first.
2138
a.-1 = 1 + a.-2
2138
S_[1 + (1 + a.-2).-1 + (b + c).+1] n.+2 + S_[1] n.+2 <=
S_[(b + c).+2] n.+2 + S_[1 + a.-2] n.+2
213f1e751b51
         by rewrite add1n prednK // -ltnS prednK.
2138
215a
213e
      by apply: concaveEk1 (concave_dsum_alphaL_l n.+2).
2140
20c0
1e74
    have F7 : S_[l] n.+1 + S_[1] n.+1 <= S_[a.-1] n.+1 + S_[(b + c).+1] n.+1.
2140
S_[l] n.+1 + S_[1] n.+1 <=
S_[a.-1] n.+1 + S_[(b + c).+1] n.+1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf20db20e020e520ed20f82105211b21202125212a4e421392141
F7:S_[l] n.+1 + S_[1] n.+1 <=
S_[a.-1] n.+1 + S_[(b + c).+1] n.+1
20c0
1e751b51
      rewrite [X in _ <= X]addnC.
2140
S_[l] n.+1 + S_[1] n.+1 <=
S_[(b + c).+1] n.+1 + S_[a.-1] n.+1
2165
      rewrite (_ : l =  1 + a.-2 + (b + c)); last first.
2140
l = 1 + a.-2 + (b + c)
2140
S_[1 + a.-2 + (b + c)] n.+1 + S_[1] n.+1 <=
S_[(b + c).+1] n.+1 + S_[a.-1] n.+1
21661e751b51
        rewrite /b /c subSn // addnS -addSnnS add1n.
2140
l = a.-2.+2 + (a1 - a + (l - a1))
2171
        rewrite !prednK //; last by rewrite -ltnS prednK.
2140
l = a + (a1 - a + (l - a1))
2171
        by rewrite addnCA addnA subnK // addnC subnK // ltnW.
2140
2173
2165
      rewrite (_ : a.-1 = 1 + a.-2); last first.
2140
2157
2140
S_[1 + (1 + a.-2).-1 + (b + c)] n.+1 + S_[1] n.+1 <=
S_[(b + c).+1] n.+1 + S_[1 + a.-2] n.+1
21661e751b51
         by rewrite add1n prednK // -ltnS prednK.
2140
2184
2165
      by apply: concaveEk1 (concave_dsum_alphaL_l n.+1).
2167
20c0
1e74
    have F8 : S_[(b + c).+2] n.+2 + S_[1] n.+2 <= S_[c.-1] n.+2 + S_[b.+4] n.+2.
2167
S_[(b + c).+2] n.+2 + S_[1] n.+2 <=
S_[c.-1] n.+2 + S_[b.+4] n.+2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf20db20e020e520ed20f82105211b21202125212a4e4213921412168
F8:S_[(b + c).+2] n.+2 + S_[1] n.+2 <=
S_[c.-1] n.+2 + S_[b.+4] n.+2
20c0
1e751b51
      rewrite (_ : (b + c).+2 =  1 + b.+3 + (c.-2)); last first.
2167
(b + c).+2 = 1 + b.+3 + c.-2
2167
S_[1 + b.+3 + c.-2] n.+2 + S_[1] n.+2 <=
S_[c.-1] n.+2 + S_[b.+4] n.+2
21901e751b51
        by rewrite add1n 2!addSnnS !prednK // -subn1 ltn_subRL addn0.
2167
2199
218f
      rewrite (_ : c.-1 = 1 + c.-2); last first.
2167
c.-1 = 1 + c.-2
2167
S_[1 + b.+3 + (1 + c.-2).-1] n.+2 + S_[1] n.+2 <=
S_[1 + c.-2] n.+2 + S_[b.+4] n.+2
21901e751b51
        by rewrite add1n prednK // -ltnS prednK.
2167
21a3
218f
      by apply: concaveEk1 (concave_dsum_alphaL_l n.+2).
2191
20c0
1e74
    have F9 : S_[(b + c).+1] n.+1 + S_[1] n.+1 <= S_[c.-1] n.+1 + S_[b.+3] n.+1.
2191
S_[(b + c).+1] n.+1 + S_[1] n.+1 <=
S_[c.-1] n.+1 + S_[b.+3] n.+1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf20db20e020e520ed20f82105211b21202125212a4e42139214121682192
F9:S_[(b + c).+1] n.+1 + S_[1] n.+1 <=
S_[c.-1] n.+1 + S_[b.+3] n.+1
20c0
1e751b51
      rewrite (_ : (b + c).+1 =  1 + b.+2 + (c.-2)); last first.
2191
(b + c).+1 = 1 + b.+2 + c.-2
2191
S_[1 + b.+2 + c.-2] n.+1 + S_[1] n.+1 <=
S_[c.-1] n.+1 + S_[b.+3] n.+1
21af1e751b51
        by rewrite add1n 2!addSnnS !prednK // -subn1 ltn_subRL addn0.
2191
21b8
21ae
      rewrite (_ : c.-1 = 1 + c.-2); last first.
2191
21a0
2191
S_[1 + b.+2 + (1 + c.-2).-1] n.+1 + S_[1] n.+1 <=
S_[1 + c.-2] n.+1 + S_[b.+3] n.+1
21af1e751b51
        by rewrite add1n prednK // -ltnS prednK.
2191
21c1
21ae
      by apply: concaveEk1 (concave_dsum_alphaL_l n.+1).
21b0
20c0
1e74
    have F10 : S_[b.+4] n.+2 <= S_[b.+3] n.+1 + (α_[1] n.+1).*2.
21b0
S_[b.+4] n.+2 <= S_[b.+3] n.+1 + (α_[1] n.+1).*2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf20db20e020e520ed20f82105211b21202125212a4e4213921412168219221b1
F10:S_[b.+4] n.+2 <= S_[b.+3] n.+1 + (α_[1] n.+1).*2
20c0
1e751b51
      by apply: dsum_alphaL_alpha.
21ce
20c0
1e74
    have F11 : S_[b.+3] n.+1 <= S_[b.+2] n + (α_[1] n).*2.
21ce
S_[b.+3] n.+1 <= S_[b.+2] n + (α_[1] n).*2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf20db20e020e520ed20f82105211b21202125212a4e4213921412168219221b121cf
F11:S_[b.+3] n.+1 <= S_[b.+2] n + (α_[1] n).*2
20c0
1e751b51
      by apply: dsum_alphaL_alpha.
21d9
20c0
1e74
    have F12 : S_[b.+2] n + S_[1] n <= S_[b] n + S_[3] n.
21d9
S_[b.+2] n + S_[1] n <= S_[b] n + S_[3] n
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf20db20e020e520ed20f82105211b21202125212a4e4213921412168219221b121cf21da
F12:S_[b.+2] n + S_[1] n <= S_[b] n + S_[3] n
20c0
1e751b51
      rewrite (_ : b.+2 = 1 + 2 + b.-1); last first.
21d9
b.+2 = 1 + 2 + b.-1
21d9
S_[1 + 2 + b.-1] n + S_[1] n <= S_[b] n + S_[3] n
21e31e751b51
        by rewrite add1n addSnnS prednK // ltnW.
21d9
21ec
21e2
      rewrite -{2}[b]prednK; last by apply: ltnW.
21d9
S_[1 + 2 + b.-1] n + S_[1] n <=
S_[b.-1.+1] n + S_[3] n
21e2
      by apply: concaveEk1 (concave_dsum_alphaL_l n).
21e4
20c0
1e74
    have F13 : S_[1] n.+1 = (S_[3] n).+1 by rewrite dsum_alpha3_S.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf20db20e020e520ed20f82105211b21202125212a4e4213921412168219221b121cf21da21e5
F13:S_[1] n.+1 = (S_[3] n).+1
20c0
1e74
    have F14 := leq_dsum_alpha_2l_1 n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf20db20e020e520ed20f82105211b21202125212a4e4213921412168219221b121cf21da21e521fb
F14:S_[1] n.+1 + S_[1] n.+2 <= (S_[2] n.+1).*2.+1
20c0
1e74
    have F15n := dsum_alphaL_S 1 n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf20db20e020e520ed20f82105211b21202125212a4e4213921412168219221b121cf21da21e521fb2200
F15n:S_[1] n.+2 = S_[1] n.+1 + α_[1] n.+1
20c0
1e74
    have F15n1 := dsum_alphaL_S 1 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e5d11e7c1ec21ec31ed41ed998f1ee61f4e1f4f1f6c1f711f7a1fac1fad1fd01fe420a320a420aa20b020b120b720b820be20bf20c520c620cf20db20e020e520ed20f82105211b21202125212a4e4213921412168219221b121cf21da21e521fb22002205
F15n1:S_[1] n.+1 = S_[1] n + α_[1] n
20c0
1e74
    by lia.
1e76
1cbb
1b50
  move: b_le2; rewrite leq_eqVlt => /orP[/eqP[] bE1|].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6fb23
1cbb
1e6c
b.+1 < 2 ->
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u + x1S +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2
1b51
(*  subcase b = 1 *)  
    have a1E : a1 = a.+1 by rewrite -(subnK aLa1) -/b bE1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6fb23
a1E:a1 = a.+1
1cbb
2212
    pose u3 := [ffun i : 'I_4 =>
                 if i == 0 :> nat then sam1 
                 else if i == 1 :> nat then tam1 
                 else if i == 2 :> nat then ↓[u (inord a)]
                 else ↓[u (inord a.+1)]].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6fb232219
u3:=[ffun i => if i == 0 :> nat
           then sam1
           else
            if i == 1 :> nat
            then tam1
            else
             if i == 2 :> nat
             then ↓[u (inord a)]
             else ↓[u (inord a.+1)]]:{ffun forall x : ordinal_finType 4,
      (fun=> configuration 4 n.+1) x}
1cbb
2212
    have cH6 (k : 'I_4) : 
      0 < k < 3 -> codom (u3 k) \subset [:: pa1S; a[p2, pa1] k].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6fb232219221ee34
0 < k < 3 ->
codom (u3 k) \subset [:: pa1S;  a[p2, pa1] k]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6fb232219221e
cH6:forall k : 'I_4,
0 < k < 3 ->
codom (u3 k) \subset [:: pa1S;  a[p2, pa1] k]
1cbb
22131b51
      move=> /andP[k_gt0 k_lt3].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6fb232219221ee34852
k_lt3:k < 3
codom (u3 k) \subset [:: pa1S;  a[p2, pa1] k]
2224
      rewrite !ffunE eqn_leq leqNgt k_gt0 /=.
222b
codom
  (if k == 1
   then tam1
   else
    if k == 2
    then ↓[u (inord a)]
    else ↓[u (inord a.+1)]) \subset
[:: pa1S;  a[p2, pa1] k]
2224
      case: eqP => [->/=|/eqP kD1].
222b
codom tam1 \subset [:: pa1S;  a[p2, pa1] 1]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6fb232219221ee34852222c
kD1:k != 1
codom
  (if k == 2
   then ↓[u (inord a)]
   else ↓[u (inord a.+1)]) \subset
[:: pa1S;  a[p2, pa1] k]
222522131b51
        by rewrite /pa1S !apegS apeg0 codom_apeg codom_subC.
2238
223a
2224
      have ->/= : k = 2 :> nat.
2238
k = 2
2238
codom ↓[u (inord a)] \subset [:: pa1S;  a[p2, pa1] 2]
222522131b51
        by apply/eqP; rewrite eqn_leq -ltnS k_lt3 ltn_neqAle eq_sym k_gt0 kD1.
2238
2244
2224
      apply: codom_liftr; rewrite codom_subC.
2238
codom (u (inord a)) \subset [::  a[p2, pa1] 2; pa1S]
2224
      have /cH : 0 < (inord a : 'I_l.+2) < l.+1.
2238
1899
2238
codom (u (inord a)) \subset
[:: p2;  a[p1, p3] (inord a)] ->
codom (u (inord a)) \subset [::  a[p2, pa1] 2; pa1S]
222522131b51
        rewrite inordK //.
2238
1f64
224f
        by rewrite a_gt0 /= l1E bE1 addn1 leq_addr.
2238
2251
2224
      by rewrite inordK // /pa1S a1E !apegS apeg0.
2226
1cbb
2212
    have P6 : 
      (S_[3] n.+1).*2 <=
      (`d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove + 
          `d[↓[u (inord a)], ↓[u (inord a.+1)]]_smove) +
      sp sam1 3 p2 + sp ↓[u (inord a.+1)] 3 (a[p1, p3] a1).
2226
(S_[3] n.+1).*2 <=
`d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove +
`d[↓[u (inord a)], ↓[u (inord a.+1)]]_smove +
sp (n:=n.+1) sam1 3 p2 +
sp (n:=n.+1) ↓[u (inord a.+1)] 3 ( a[p1, p3] a1)
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6fb232219221e2227
P6:(S_[3] n.+1).*2 <=
`d[sam1, tam1]_smove +
`d[tam1, ↓[u (inord a)]]_smove +
`d[↓[u (inord a)], ↓[u (inord a.+1)]]_smove +
sp (n:=n.+1) sam1 3 p2 +
sp (n:=n.+1) ↓[u (inord a.+1)] 3 ( a[p1, p3] a1)
1cbb
22131b51
      apply: (leq_trans (IH _ _ _ _ _ _ cH6)); rewrite /pa1S /pa1 //.
2226
1dff
2226
p2 !=  a[p1, p3] a1
2226
1a1f
2226
sd (n:=n.+1) u3 + sp (n:=n.+1) (u3 ord0) 3 p2 +
sp (n:=n.+1) (u3 ord_max) 3 ( a[p2,  a[p1, p3] a1] 3) <=
`d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove +
`d[↓[u (inord a)], ↓[u (inord a.+1)]]_smove +
sp (n:=n.+1) sam1 3 p2 +
sp (n:=n.+1) ↓[u (inord a.+1)] 3 ( a[p1, p3] a1)
226022131b51
      -
2264 by rewrite eq_sym.
2226
2268
2269226a226022131b51
      -
226e by rewrite eq_sym.
2226
1a1f
226a226022131b51
      -
2273 by rewrite apegS apeg_eqC eq_sym.
2226
226b
225f
      repeat apply: leq_add; rewrite leq_eqVlt; apply/orP; left; apply/eqP.
2226
sd (n:=n.+1) u3 =
`d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove +
`d[↓[u (inord a)], ↓[u (inord a.+1)]]_smove
2226
sp (n:=n.+1) (u3 ord0) 3 p2 = sp (n:=n.+1) sam1 3 p2
2226
sp (n:=n.+1) (u3 ord_max) 3 ( a[p2,  a[p1, p3] a1] 3) =
sp (n:=n.+1) ↓[u (inord a.+1)] 3 ( a[p1, p3] a1)
226022131b51
      -
227b by rewrite /sd !big_ord_recr big_ord0 !ffunE !inordK /=.
2226
2280
2281226022131b51
      -
2285 by apply: eq_bigr => i _; rewrite !ffunE /=.
2226
2282
225f
      by apply: eq_bigr => i _; rewrite !ffunE.
2261
1cbb
2212
    set x3S := sp _ _ _ in P6; set y3S := sp _ _ _ in P6.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6fb232219221e2227
x3S:=sp (n:=n.+1) sam1 3 p2:nat
y3S:=sp (n:=n.+1) ↓[u (inord a.+1)] 3 ( a[p1, p3] a1):nat
P6:(S_[3] n.+1).*2 <=
`d[sam1, tam1]_smove +
`d[tam1, ↓[u (inord a)]]_smove +
`d[↓[u (inord a)], ↓[u (inord a.+1)]]_smove + x3S +
y3S
1cbb
2212
    have x2Sx3SE : x2S + x3S <= (S_[1] n).*2 + α_[maxn (3 * a).-2 3] n.
2292
x2S + x3S <= (S_[1] n).*2 + α_[maxn (3 * a).-2 3] n
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6fb232219221e2227229322942295
x2Sx3SE:x2S + x3S <=
(S_[1] n).*2 + α_[maxn (3 * a).-2 3] n
1cbb
22131b51
      by apply: sum_alpha_diffE; rewrite /pa.
229c
1cbb
2212
    rewrite (maxn_idPl _) in x2Sx3SE; last first.
2292
e9c
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6fb232219221e22272293229422951ed4
1cbb
22131b51
      by rewrite -subn2  ltn_subRL ltnW // (leq_mul2l 3 2).
22a6
1cbb
2212
    pose u5 := [ffun i : 'I_3 =>
                     if i == 0 :> nat then ↓[u (inord a1)]
                 else if i == 1 :> nat then sa1 else ta1].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6fb232219221e22272293229422951ed41f71
1cbb
2212
    have cH8 (k : 'I_3) : 
      0 < k < 2 -> codom (u5 k) \subset [:: p1; a[p2, p3] k].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6fb232219221e22272293229422951ed41f71994
1f76
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6fb232219221e22272293229422951ed41f711f7a
1cbb
22131b51
      move=> /andP[k_gt0 k_lt2].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6fb232219221e22272293229422951ed41f719948521e8a
1f7f
22b2
      rewrite !ffunE.
22b8
1f83
22b2
      by have ->/= : k = 1 :> nat by case: (k : nat) k_gt0 k_lt2 => // [] [|].
22b4
1cbb
2212
    have P8 : 
      (S_[2] n.+1).*2 <=
      (`d[↓[u (inord a.+1)], sa1]_smove + `d[sa1, ta1]_smove) +
      sp ↓[u (inord a.+1)] 2 p2 + sp ta1 2 p2.
22b4
(S_[2] n.+1).*2 <=
`d[↓[u (inord a.+1)], sa1]_smove + `d[sa1, ta1]_smove +
sp (n:=n.+1) ↓[u (inord a.+1)] 2 p2 +
sp (n:=n.+1) ta1 2 p2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6fb232219221e22272293229422951ed41f711f7a
P8:(S_[2] n.+1).*2 <=
`d[↓[u (inord a.+1)], sa1]_smove +
`d[sa1, ta1]_smove +
sp (n:=n.+1) ↓[u (inord a.+1)] 2 p2 +
sp (n:=n.+1) ta1 2 p2
1cbb
22131b51
      apply: (leq_trans (IH _ _ _ _ _ _ cH8)) => //.
22b4
1e9e
22b4
sd (n:=n.+1) u5 + sp (n:=n.+1) (u5 ord0) 2 p2 +
sp (n:=n.+1) (u5 ord_max) 2 ( a[p2, p3] 2) <=
`d[↓[u (inord a.+1)], sa1]_smove + `d[sa1, ta1]_smove +
sp (n:=n.+1) ↓[u (inord a.+1)] 2 p2 +
sp (n:=n.+1) ta1 2 p2
22c422131b51
        by rewrite eq_sym.
22b4
22cc
22c3
      repeat apply: leq_add; rewrite leq_eqVlt; apply/orP; left; apply/eqP.
22b4
sd (n:=n.+1) u5 =
`d[↓[u (inord a.+1)], sa1]_smove + `d[sa1, ta1]_smove
22b4
sp (n:=n.+1) (u5 ord0) 2 p2 =
sp (n:=n.+1) ↓[u (inord a.+1)] 2 p2
22b4
1f9b
22c422131b51
      -
22d1 by rewrite /sd !big_ord_recr big_ord0 !ffunE !inordK //= a1E.
22b4
22d6
22d722c422131b51
      -
22da by apply: eq_bigr => i _; rewrite !ffunE a1E.
22b4
1f9b
22c3
      by apply: eq_bigr => i _; rewrite !ffunE.
22c5
1cbb
2212
    set x5S := sp _ _ _ in P8; set y5S := sp _ _ _ in P8.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6fb232219221e22272293229422951ed41f711f7a
x5S:=sp (n:=n.+1) ↓[u (inord a.+1)] 2 p2:nat
1fad
P8:(S_[2] n.+1).*2 <=
`d[↓[u (inord a.+1)], sa1]_smove +
`d[sa1, ta1]_smove + x5S + y5S
1cbb
2212
    have y3Sx5SE : y3S + x5S <= (S_[1] n).*2 + α_[3] n.
22e7
y3S + x5S <= (S_[1] n).*2 + α_[3] n
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6fb232219221e22272293229422951ed41f711f7a22e81fad22e9
y3Sx5SE:y3S + x5S <= (S_[1] n).*2 + α_[3] n
1cbb
22131b51
      apply: sum_alpha_diffE => //.
22e7
codom ↓[u (inord a.+1)] \subset [::  a[p1, p3] a1; p2]
22ee
      apply: codom_liftr; rewrite codom_subC.
22e7
codom (u (inord a.+1)) \subset [:: p2;  a[p1, p3] a1]
22ee
      have := @cH (inord a.+1); rewrite inordK ?a1E //.
22e7
(0 < a.+1 < l.+1 ->
 codom (u (inord a.+1)) \subset
 [:: p2;  a[p1, p3] a.+1]) ->
codom (u (inord a.+1)) \subset
[:: p2;  a[p1, p3] a.+1]
22e7
a.+1 < l.+2
22ef22131b51
        by apply; rewrite andTb //= ltnS -a1E.
22e7
2300
22ee
      by rewrite -a1E ltnS ltnW.
22f0
1cbb
2212
    have y5Sy2SE : y5S + y2S <= (S_[1] n).*2 + α_[maxn 2 (3 * c).-2] n.
22f0
y5S + y2S <= (S_[1] n).*2 + α_[maxn 2 (3 * c).-2] n
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6fb232219221e22272293229422951ed41f711f7a22e81fad22e922f1
y5Sy2SE:y5S + y2S <=
(S_[1] n).*2 + α_[maxn 2 (3 * c).-2] n
1cbb
22131b51
      by apply: sum_alpha_diffE; rewrite // /pa1 eq_sym.
230d
1cbb
2212
    rewrite (maxn_idPr _) in y5Sy2SE; last first.
22f0
1 < (3 * c).-2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6fb232219221e22272293229422951ed41f711f7a22e81fad22e922f1
y5Sy2SE:y5S + y2S <= (S_[1] n).*2 + α_[(3 * c).-2] n
1cbb
22131b51
      by rewrite -subn2 ltn_subRL (ltn_mul2l 3 1).
2318
1cbb
2212
    rewrite (_ : sd u = 
       \sum_(i < a) `d[u (inord i), u (inord i.+1)]_smove +
       \sum_(a <= i < a1) `d[u (inord i), u (inord i.+1)]_smove +
       \sum_(a1 <= i < l.+1) `d[u (inord i), u (inord i.+1)]_smove);
        last first.
2318
1fe8
2318
1feb
22131b51
      pose f i := `d[u (inord i), u (inord i.+1)]_smove.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6fb232219221e22272293229422951ed41f711f7a22e81fad22e922f123191ff0
1fe8
2320
      rewrite /sd -!(big_mkord xpredT f).
2325
1ff4
2320
      by rewrite -!big_cat_nat //= ltnW.
2318
1feb
2212
    rewrite -{}P1 -{}P2; applyr P3.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6fb232219221e22272293229422951ed41f711f7a22e81fad22e922f12319
2 * S_[1 + l] (2 + n) <=
2 * a + sd (n:=1 + n) u1 + 2 * c + sd (n:=1 + n) u2 +
x1S +
\sum_(k < 1 + n)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2 +
\sum_(a <= i < a1)
   `d[↓[u (inord i)], ↓[u (inord i.+1)]]_smove
2212
    rewrite (_ : sd u1  =
              \sum_(i < (3 * a).-2) `d[u1 (inord i), u1 (inord i.+1)]_smove
              + `d[sam1, tam1]_smove + 
              `d[tam1, ↓[u (inord a)]]_smove); last first.
232f
2000
232f
2 * S_[1 + l] (2 + n) <=
2 * a +
(\sum_(i < (3 * a).-2)
    `d[u1 (inord i), u1 (inord i.+1)]_smove +
 `d[sam1, tam1]_smove + `d[tam1, ↓[
 u (inord a)]]_smove) + 
2 * c + sd (n:=1 + n) u2 + x1S +
\sum_(k < 1 + n)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2 +
\sum_(a <= i < a1)
   `d[↓[u (inord i)], ↓[
u (inord i.+1)]]_smove
22131b51
      pose f i := `d[u1 (inord i), u1 (inord i.+1)]_smove.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6fb232219221e22272293229422951ed41f711f7a22e81fad22e922f123192008
2000
2334
      rewrite /sd -!(big_mkord xpredT f) .
233a
200c
2334
      rewrite -{1}(subnK (_ : 1 < 3 * a)) //; last first.
233a
979
233a
2012
233522131b51
        by rewrite -[a]prednK // mulnS.
233a
2012
2334
      rewrite subn2 !addn2 !big_nat_recr /f //=; congr (_ + _ + _).
233a
2019
233a
201c
233522131b51
        rewrite ffunE (_ : (3 * a).-2 = (3 * (a.-1)).+1); last first.
233a
1c9f
233a
2022
234a233522131b51
          by rewrite -{1}[a]prednK // mulnS.
233a
2026
2349
        rewrite inordK; last by rewrite ltnS ltn_mul2l /= prednK.
233a
202a
2349
        rewrite mod3E /= eqxx.
233a
202e
2349
        rewrite ffunE inordK; last first.
233a
2032
233a
2035
234a233522131b51
          by rewrite -[_.+3](mulnDr 3 1) add1n prednK // ltnW.
233a
2035
2349
        by rewrite mod3E /= eqxx.
233a
203c
2334
      rewrite ffunE (_ : (3 * a).-2 = (3 * (a.-1)).+1); last first.
234d
233a
2041
233522131b51
        by rewrite -{1}[a]prednK // mulnS.
233a
2045
2334
      rewrite inordK; last first.
235b
233a
204a
233522131b51
        by rewrite -{2}[a]prednK // mulnS.
233a
204e
2334
      rewrite mod3E /= eqxx.
233a
2052
2334
      rewrite ffunE inordK; last first.
233a
2056
233a
2059
233522131b51
        by rewrite -{2}[a]prednK // mulnS.
233a
2059
2334
      by rewrite -[(_ * _).+3](mulnDr 3 1) mod3E /= div3E add1n prednK.
232f
2 * S_[1 + l] (2 + n) <=
2 * a +
(\sum_(i < (3 * a).-2)
    `d[u1 (inord i), u1 (inord i.+1)]_smove +
 `d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove) +
2 * c + sd (n:=1 + n) u2 + x1S +
\sum_(k < 1 + n)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2 +
\sum_(a <= i < a1)
   `d[↓[u (inord i)], ↓[u (inord i.+1)]]_smove
2212
    rewrite (_ : sd u2  = 
       `d[↓[u (inord a1)], sa1]_smove +  `d[sa1, ta1]_smove +
               \sum_(i < (3 * c).-2) 
                  `d[u2 (inord i.+2), u2 (inord i.+3)]_smove); last first.
232f
2064
232f
2 * S_[1 + l] (2 + n) <=
2 * a +
(\sum_(i < (3 * a).-2)
    `d[u1 (inord i), u1 (inord i.+1)]_smove +
 `d[sam1, tam1]_smove + `d[tam1, ↓[
 u (inord a)]]_smove) + 
2 * c +
(`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
 \sum_(i < (3 * c).-2)
    `d[u2 (inord i.+2), 
 u2 (inord i.+3)]_smove) + x1S +
\sum_(k < 1 + n)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2 +
\sum_(a <= i < a1)
   `d[↓[u (inord i)], ↓[
u (inord i.+1)]]_smove
22131b51
      pose f i := `d[u2 (inord i), u2 (inord i.+1)]_smove.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6fb232219221e22272293229422951ed41f711f7a22e81fad22e922f12319206c
2064
2384
      rewrite /sd -!(big_mkord xpredT f) .
238a
2070
2384
      rewrite -{1}(subnK (_ : 1 < 3 * c)) //; last first.
238a
1e57
238a
2076
238522131b51
        by rewrite -[c]prednK // mulnS.
238a
2076
2384
      rewrite subn2 !addn2 !big_nat_recl // /f /= !addnA.
238a
`d[u2 (inord 0), u2 (inord 1)]_smove +
`d[u2 (inord 1), u2 (inord 2)]_smove +
\sum_(0 <= i < (3 * c).-2)
   `d[u2 (inord i.+2), u2 (inord i.+3)]_smove =
`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
\sum_(i < (3 * c).-2)
   `d[u2 (inord i.+2), u2 (inord i.+3)]_smove
2384
       congr (_ + _ + _).
238a
207d
238a
2080
238a
2082
238522131b51
      -
239b rewrite !ffunE !inordK ?addn0 //=.
238a
2087
239d
        by rewrite ltnS muln_gt0.
238a
2080
239f238522131b51
      -
23a5 rewrite !ffunE !inordK //=.
238a
1e29
238a
2087
239f238522131b51
          by rewrite ltnW // ltnS (leq_mul2l 3 1 c).
23a323a7
        by rewrite ltnS muln_gt0.
238a
2082
2384
      by rewrite big_mkord.
232f
2 * S_[1 + l] (2 + n) <=
2 * a +
(\sum_(i < (3 * a).-2)
    `d[u1 (inord i), u1 (inord i.+1)]_smove +
 `d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove) +
2 * c +
(`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
 \sum_(i < (3 * c).-2)
    `d[u2 (inord i.+2), u2 (inord i.+3)]_smove) + x1S +
\sum_(k < 1 + n)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2 +
\sum_(a <= i < a1)
   `d[↓[u (inord i)], ↓[u (inord i.+1)]]_smove
2212
    rewrite (_ : \sum_(_ <= _ < _) _ =
      `d[↓[u (inord a)], ↓[u (inord a.+1)]]_smove); last first.
232f
\sum_(a <= H < a1)
   `d[↓[u (inord H)], ↓[u (inord H.+1)]]_smove =
`d[↓[u (inord a)], ↓[u (inord a.+1)]]_smove
232f
2 * S_[1 + l] (2 + n) <=
2 * a +
(\sum_(i < (3 * a).-2)
    `d[u1 (inord i), u1 (inord i.+1)]_smove +
 `d[sam1, tam1]_smove + `d[tam1, ↓[
 u (inord a)]]_smove) + 
2 * c +
(`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
 \sum_(i < (3 * c).-2)
    `d[u2 (inord i.+2), 
 u2 (inord i.+3)]_smove) + x1S +
\sum_(k < 1 + n)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2 +
`d[↓[u (inord a)], ↓[u (inord a.+1)]]_smove
22131b51
      by rewrite a1E big_nat1.
232f
2 * S_[1 + l] (2 + n) <=
2 * a +
(\sum_(i < (3 * a).-2)
    `d[u1 (inord i), u1 (inord i.+1)]_smove +
 `d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove) +
2 * c +
(`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
 \sum_(i < (3 * c).-2)
    `d[u2 (inord i.+2), u2 (inord i.+3)]_smove) + x1S +
\sum_(k < 1 + n)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2 +
`d[↓[u (inord a)], ↓[u (inord a.+1)]]_smove
2212
    set z1S := \sum_(_ < _) _ in P4; set z2S := \sum_(_ < _) _ in  P5.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e22272293229422951ed41f711f7a22e81fad22e922f1231920a320b020a420b1
23c1
2212
    rewrite -/z1S -/z2S.
23c5
2 * S_[1 + l] (2 + n) <=
2 * a +
(z1S + `d[sam1, tam1]_smove +
 `d[tam1, ↓[u (inord a)]]_smove) + 2 * c +
(`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
 \sum_(i < (3 * c).-2)
    `d[u2 (inord i.+2), u2 (inord i.+3)]_smove) + x1S +
z2S + `d[↓[u (inord a)], ↓[u (inord a.+1)]]_smove
2212
    rewrite -a1E in P8; set d2 := `d[_,_]_ _ + `d[_,_]_ _ in P8; rewrite -/d2.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e22272293229422951ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf
2 * S_[1 + l] (2 + n) <=
2 * a +
(z1S + `d[sam1, tam1]_smove +
 `d[tam1, ↓[u (inord a)]]_smove) + 2 * c +
(d2 +
 \sum_(i < (3 * c).-2)
    `d[u2 (inord i.+2), u2 (inord i.+3)]_smove) + x1S +
z2S + `d[↓[u (inord a)], ↓[u (inord a.+1)]]_smove
2212
    set d1 := `d[_,_]_ _ + `d[_,_]_ _ + `d[_,_]_ _ in P6.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf
d1:=`d[sam1, tam1]_smove +
`d[tam1, ↓[u (inord a)]]_smove +
`d[↓[u (inord a)], ↓[u (inord a.+1)]]_smove:nat
P6:(S_[3] n.+1).*2 <= d1 + x3S + y3S
23ce
2212
    rewrite -/y1S.
23d2
2 * S_[1 + l] (2 + n) <=
2 * a +
(z1S + `d[sam1, tam1]_smove +
 `d[tam1, ↓[u (inord a)]]_smove) + 2 * c + (d2 + y1S) +
x1S + z2S +
`d[↓[u (inord a)], ↓[u (inord a.+1)]]_smove
2212
    changer (2 * a + 2 * c + x1S + y1S + z1S + z2S + d2 + 
              (`d[sam1, tam1]_smove +  `d[tam1, ↓[u (inord a)]]_smove +
               `d[↓[u (inord a)], ↓[u (inord a.+1)]]_smove)).
23d2
2 * S_[1 + l] (2 + n) <=
2 * a + 2 * c + x1S + y1S + z1S + z2S + d2 +
(`d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove +
 `d[↓[u (inord a)], ↓[u (inord a.+1)]]_smove)
2212
    rewrite -/d1.
23d2
2 * S_[1 + l] (2 + n) <=
2 * a + 2 * c + x1S + y1S + z1S + z2S + d2 + d1
2212
    have F1 := dsum_alphaL_S (3 * a).-2 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db
23e0
2212
    have F2 := dsum_alphaL_S (3 * c).-2 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e0
23e0
2212
    have F3 := dsum_alphaL_S 3 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e0
F3:S_[3] n.+1 = S_[3] n + α_[3] n
23e0
2212
    have F4_a : 2 * a = 2 + a.-1 + a.-1.
23ec
20e9
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed
23e0
22131b51
      by rewrite -{1}[a]prednK // mul2n -addnn addnS.
23f3
23e0
2212
    have F4_c : 2 * c = 2 + c.-1 + c.-1.
23f3
20f4
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed20f8
23e0
22131b51
      by rewrite -{1}[c]prednK // mul2n -addnn addnS.
23fc
23e0
2212
    have F4 i k : 0 < i -> S_[i.-1] k.+1 <= S_[(3 *i).-2] k + i.-1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed20f82100
2101
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed20f82105
23e0
22131b51
      move=> i_gt0; apply: leq_trans (dsum_alpha3l _ _) _.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed20f82100210a
210b
2404
      rewrite leq_add2r.
240a
210f
2404
      apply: (increasingE (increasing_dsum_alphaL_l _)).
240a
2113
2404
      by rewrite -{2}[i]prednK // mulnS addSn add2n /=.
2406
23e0
2212
    have F4an := F4 a n a_gt0.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed20f82105211b
23e0
2212
    have F4an1 := F4 a n.+1 a_gt0.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed20f82105211b2120
23e0
2212
    have F4cn := F4 c n (c_gt0).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed20f82105211b21202125
23e0
2212
    have F4cn1 := F4 c n.+1 (c_gt0).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed20f82105211b21202125212a
23e0
2212
    have l_gt1 : 1 < l.
2423
212e
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed20f82105211b21202125212a4e4
23e0
22131b51
      by rewrite -ltnS l1E -[3]/(2 + 0 + 1); rewrite !leq_add.
2429
23e0
2212
    have F5 := SH l_gt1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed20f82105211b21202125212a4e42139
23e0
2212
    have F6n : S_[1] n.+1 = (S_[3] n).+1 by rewrite dsum_alpha3_S.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed20f82105211b21202125212a4e42139
F6n:S_[1] n.+1 = (S_[3] n).+1
23e0
2212
    have F6n1 : S_[1] n.+2 = (S_[3] n.+1).+1 by rewrite dsum_alpha3_S.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed20f82105211b21202125212a4e421392435
F6n1:S_[1] n.+2 = (S_[3] n.+1).+1
23e0
2212
    have am1_gt0 : 0 < a.-1 by rewrite -subn1 ltn_subRL addn0.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed20f82105211b21202125212a4e421392435243ab2b
23e0
2212
    have F7n1 : S_[1] n.+2 + S_[l.+1] n.+2 <= S_[a.-1] n.+2 + S_[c.+3] n.+2.
243e
S_[1] n.+2 + S_[l.+1] n.+2 <=
S_[a.-1] n.+2 + S_[c.+3] n.+2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed20f82105211b21202125212a4e421392435243ab2b
F7n1:S_[1] n.+2 + S_[l.+1] n.+2 <=
S_[a.-1] n.+2 + S_[c.+3] n.+2
23e0
22131b51
      rewrite addnC (_ : l.+1 = 1 + c.+2 + a.-2); last first.
243e
l.+1 = 1 + c.+2 + a.-2
243e
S_[1 + c.+2 + a.-2] n.+2 + S_[1] n.+2 <=
S_[a.-1] n.+2 + S_[c.+3] n.+2
244422131b51
        by rewrite /c a1E add1n !addSnnS !prednK ?subnK // -a1E ltnW.
243e
244d
2443
      rewrite {2}(_ : a.-1 = 1 + a.-2); last by rewrite add1n prednK.
243e
S_[1 + c.+2 + a.-2] n.+2 + S_[1] n.+2 <=
S_[1 + a.-2] n.+2 + S_[c.+3] n.+2
2443
      by apply: concaveEk1 (concave_dsum_alphaL_l n.+2).
2445
23e0
2212
    have F7n : S_[1] n.+1 + S_[l] n.+1 <= S_[a.-1] n.+1 + S_[c.+2] n.+1.
2445
S_[1] n.+1 + S_[l] n.+1 <=
S_[a.-1] n.+1 + S_[c.+2] n.+1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed20f82105211b21202125212a4e421392435243ab2b2446
F7n:S_[1] n.+1 + S_[l] n.+1 <=
S_[a.-1] n.+1 + S_[c.+2] n.+1
23e0
22131b51
      rewrite addnC (_ : l = 1 + c.+1 + a.-2); last first.
2445
l = 1 + c.+1 + a.-2
2445
S_[1 + c.+1 + a.-2] n.+1 + S_[1] n.+1 <=
S_[a.-1] n.+1 + S_[c.+2] n.+1
245d22131b51
        by rewrite /c a1E add1n subSS !addSnnS !prednK ?subnK // -ltnS -a1E ltnW.
2445
2466
245c
      rewrite {2}(_ : a.-1 = 1 + a.-2); last by rewrite add1n prednK.
2445
S_[1 + c.+1 + a.-2] n.+1 + S_[1] n.+1 <=
S_[1 + a.-2] n.+1 + S_[c.+2] n.+1
245c
      by apply: concaveEk1 (concave_dsum_alphaL_l n.+1).
245e
23e0
2212
    have cm1_gt0 : 0 < c.-1 by rewrite -subn1 ltn_subRL addn0.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed20f82105211b21202125212a4e421392435243ab2b2446245f
cm1_gt0:0 < c.-1
23e0
2212
    have F8n1 : S_[1] n.+2 + S_[c.+3] n.+2 <= S_[c.-1] n.+2 + S_[5] n.+2.
2474
S_[1] n.+2 + S_[c.+3] n.+2 <=
S_[c.-1] n.+2 + S_[5] n.+2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed20f82105211b21202125212a4e421392435243ab2b2446245f2475
F8n1:S_[1] n.+2 + S_[c.+3] n.+2 <=
S_[c.-1] n.+2 + S_[5] n.+2
23e0
22131b51
      rewrite addnC (_ : c.+3 = 1 + 4 + c.-2); last first.
2474
c.+3 = 1 + 4 + c.-2
2474
S_[1 + 4 + c.-2] n.+2 + S_[1] n.+2 <=
S_[c.-1] n.+2 + S_[5] n.+2
247b22131b51
        by rewrite add1n 2!addSnnS !prednK.
2474
2484
247a
      rewrite {2}(_ : c.-1 = 1 + c.-2); last by rewrite add1n prednK.
2474
S_[1 + 4 + c.-2] n.+2 + S_[1] n.+2 <=
S_[1 + c.-2] n.+2 + S_[5] n.+2
247a
      by apply: concaveEk1 (concave_dsum_alphaL_l n.+2).
247c
23e0
2212
    have F8n : S_[1] n.+1 + S_[c.+2] n.+1 <= S_[c.-1] n.+1 + S_[4] n.+1.
247c
S_[1] n.+1 + S_[c.+2] n.+1 <=
S_[c.-1] n.+1 + S_[4] n.+1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed20f82105211b21202125212a4e421392435243ab2b2446245f2475247d
F8n:S_[1] n.+1 + S_[c.+2] n.+1 <=
S_[c.-1] n.+1 + S_[4] n.+1
23e0
22131b51
      rewrite addnC (_ : c.+2 = 1 + 3 + c.-2); last first.
247c
c.+2 = 1 + 3 + c.-2
247c
S_[1 + 3 + c.-2] n.+1 + S_[1] n.+1 <=
S_[c.-1] n.+1 + S_[4] n.+1
249422131b51
        by rewrite add1n 2!addSnnS !prednK.
247c
249d
2493
      rewrite {2}(_ : c.-1 = 1 + c.-2); last by rewrite add1n prednK.
247c
S_[1 + 3 + c.-2] n.+1 + S_[1] n.+1 <=
S_[1 + c.-2] n.+1 + S_[4] n.+1
2493
      by apply: concaveEk1 (concave_dsum_alphaL_l n.+1).
2495
23e0
2212
    have F9 : S_[5] n.+2 <= S_[4] n.+1 + (α_[1] n.+1).*2.
2495
S_[5] n.+2 <= S_[4] n.+1 + (α_[1] n.+1).*2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed20f82105211b21202125212a4e421392435243ab2b2446245f2475247d2496
F9:S_[5] n.+2 <= S_[4] n.+1 + (α_[1] n.+1).*2
23e0
22131b51
      by apply: dsum_alphaL_alpha.
24ae
23e0
2212
    have F10 : S_[4] n.+1 <= S_[3] n + (α_[1] n).*2.
24ae
S_[4] n.+1 <= S_[3] n + (α_[1] n).*2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed20f82105211b21202125212a4e421392435243ab2b2446245f2475247d249624af
F10:S_[4] n.+1 <= S_[3] n + (α_[1] n).*2
23e0
22131b51
      by apply: dsum_alphaL_alpha.
24b9
23e0
2212
    have F11 : S_[1] n.+1 = (S_[3] n).+1 by rewrite dsum_alpha3_S.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed20f82105211b21202125212a4e421392435243ab2b2446245f2475247d249624af24ba
F11:S_[1] n.+1 = (S_[3] n).+1
23e0
2212
    have F12 := leq_dsum_alpha_2l_1 n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed20f82105211b21202125212a4e421392435243ab2b2446245f2475247d249624af24ba24c2
F12:S_[1] n.+1 + S_[1] n.+2 <= (S_[2] n.+1).*2.+1
23e0
2212
    have F13n1 := dsum_alphaL_S 1 n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed20f82105211b21202125212a4e421392435243ab2b2446245f2475247d249624af24ba24c224c7
F13n1:S_[1] n.+2 = S_[1] n.+1 + α_[1] n.+1
23e0
2212
    have F13n := dsum_alphaL_S 1 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6eb232219221e2227229322941ed41f711f7a22e81fad22f1231920a320b020a420b120be20bf23d323d420db20e023ed20ed20f82105211b21202125212a4e421392435243ab2b2446245f2475247d249624af24ba24c224c724cc
F13n:S_[1] n.+1 = S_[1] n + α_[1] n
23e0
2212
    rewrite !add1n !add2n.
24d0
2 * S_[l.+1] n.+2 <=
2 * a + 2 * c + x1S + y1S + z1S + z2S + d2 + d1
2212
    by lia.
1e6c
2214
1b50
(* subcase b = 0 *)
  rewrite !ltnS leqn0 => /eqP bE0.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f123f
1cbb
1b50
  have a1E : a1 = a by rewrite -(subnK aLa1) -/b bE0.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f123f
a1E:a1 = a
1cbb
1b50
  pose u3 := [ffun i : 'I_5 =>
                 if i == 0 :> nat then sam1 
                 else if i == 1 :> nat then tam1 
                 else if i == 2 :> nat then ↓[u (inord a)]
                 else if i == 3 :> nat then sa1 
                 else ta1].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f123f24e1
u3:=[ffun i => if i == 0 :> nat
           then sam1
           else
            if i == 1 :> nat
            then tam1
            else
             if i == 2 :> nat
             then ↓[u (inord a)]
             else
              if i == 3 :> nat then sa1 else ta1]:{ffun forall x : ordinal_finType 5,
      (fun=> configuration 4 n.+1) x}
1cbb
1b50
  have cH6 (k : 'I_5) : 
    0 < k < 4 -> codom (u3 k) \subset [:: pa1; a[p2, pa1S] k].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f123f24e124e61192
0 < k < 4 ->
codom (u3 k) \subset [:: pa1;  a[p2, pa1S] k]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f123f24e124e6
cH6:forall k : 'I_5,
0 < k < 4 ->
codom (u3 k) \subset [:: pa1;  a[p2, pa1S] k]
1cbb
1b51
    move=> /andP[k_gt0 k_lt3].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f123f24e124e61192852
k_lt3:k < 4
codom (u3 k) \subset [:: pa1;  a[p2, pa1S] k]
24ec
    rewrite !ffunE eqn_leq leqNgt k_gt0 /=.
24f3
codom
  (if k == 1
   then tam1
   else
    if k == 2
    then ↓[u (inord a)]
    else if k == 3 then sa1 else ta1) \subset
[:: pa1;  a[p2, pa1S] k]
24ec
    case: eqP => [->/=|/eqP kD1].
24f3
codom tam1 \subset [:: pa1;  a[p2, pa1S] 1]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f123f24e124e6119285224f42239
codom
  (if k == 2
   then ↓[u (inord a)]
   else if k == 3 then sa1 else ta1) \subset
[:: pa1;  a[p2, pa1S] k]
24ed1b51
      by rewrite /pa1S !apegS apeg0 codom_apeg.
2500
2501
24ec
    case: eqP =>[->/=|/eqP kD2].
2500
codom ↓[u (inord a)] \subset [:: pa1;  a[p2, pa1S] 2]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f123f24e124e6119285224f42239
kD2:k != 2
codom (if k == 3 then sa1 else ta1) \subset
[:: pa1;  a[p2, pa1S] k]
24ed1b51
      apply: codom_liftr; rewrite codom_subC.
2500
codom (u (inord a)) \subset [::  a[p2, pa1S] 2; pa1]
2509
      have /cH : 0 < (inord a : 'I_l.+2) < l.+1.
2500
1899
2500
codom (u (inord a)) \subset
[:: p2;  a[p1, p3] (inord a)] ->
codom (u (inord a)) \subset [::  a[p2, pa1S] 2; pa1]
250a24ed1b51
        by rewrite inordK // a_gt0 /= -a1E.
2500
2517
2509
      by rewrite inordK // /pa1 a1E.
250b
250d
24ec
    have ->/= : k = 3 :> nat.
250b
k = 3
250b
codom sa1 \subset [:: pa1;  a[p2, pa1S] 3]
24ed1b51
      by case: (k : nat) k_lt3 k_gt0 kD1 kD2 => // [] [|[|[|]]].
250b
2524
24ec
    by rewrite /pa1S !(apegS, apeg0) codom_apeg.
24ee
1cbb
1b50
  have P6 : 
    (S_[4] n.+1).*2 <=
    (`d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove + 
        `d[↓[u (inord a)], sa1]_smove +
        `d[sa1, ta1]_smove) + sp sam1 4 p2 + sp ta1 4 p2.
24ee
(S_[4] n.+1).*2 <=
`d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove +
`d[↓[u (inord a)], sa1]_smove + `d[sa1, ta1]_smove +
sp (n:=n.+1) sam1 4 p2 + sp (n:=n.+1) ta1 4 p2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f123f24e124e624ef
P6:(S_[4] n.+1).*2 <=
`d[sam1, tam1]_smove +
`d[tam1, ↓[u (inord a)]]_smove +
`d[↓[u (inord a)], sa1]_smove +
`d[sa1, ta1]_smove + sp (n:=n.+1) sam1 4 p2 +
sp (n:=n.+1) ta1 4 p2
1cbb
1b51
    apply: (leq_trans (IH _ _ _ _ _ _ cH6)); rewrite /pa1S /pa1 //.
24ee
2268
24ee
1dff
24ee
1dfc
24ee
sd (n:=n.+1) u3 + sp (n:=n.+1) (u3 ord0) 4 p2 +
sp (n:=n.+1) (u3 ord_max) 4
  ( a[p2,  a[p1, p3] a1.+1] 4) <=
`d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove +
`d[↓[u (inord a)], sa1]_smove + `d[sa1, ta1]_smove +
sp (n:=n.+1) sam1 4 p2 + 
sp (n:=n.+1) ta1 4 p2
25301b51
    -
2534 by rewrite eq_sym.
24ee
1dff
2538253925301b51
    -
253d by rewrite eq_sym.
24ee
1dfc
253925301b51
    -
2542 by rewrite apegS apeg_eqC.
24ee
sd (n:=n.+1) u3 + sp (n:=n.+1) (u3 ord0) 4 p2 +
sp (n:=n.+1) (u3 ord_max) 4
  ( a[p2,  a[p1, p3] a1.+1] 4) <=
`d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove +
`d[↓[u (inord a)], sa1]_smove + `d[sa1, ta1]_smove +
sp (n:=n.+1) sam1 4 p2 + sp (n:=n.+1) ta1 4 p2
252f
    repeat apply: leq_add; rewrite leq_eqVlt; apply/orP; left; apply/eqP.
24ee
sd (n:=n.+1) u3 =
`d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove +
`d[↓[u (inord a)], sa1]_smove + `d[sa1, ta1]_smove
24ee
sp (n:=n.+1) (u3 ord0) 4 p2 = sp (n:=n.+1) sam1 4 p2
24ee
sp (n:=n.+1) (u3 ord_max) 4
  ( a[p2,  a[p1, p3] a1.+1] 4) = 
sp (n:=n.+1) ta1 4 p2
25301b51
    -
254b by rewrite /sd !big_ord_recr big_ord0 !ffunE !inordK /= ?a1E.
24ee
2550
255125301b51
    -
2555 by apply: eq_bigr => i _; rewrite !ffunE /=.
24ee
sp (n:=n.+1) (u3 ord_max) 4
  ( a[p2,  a[p1, p3] a1.+1] 4) = sp (n:=n.+1) ta1 4 p2
252f
    by apply: eq_bigr => i _; rewrite !ffunE /= a1E.
2531
1cbb
1b50
  set x3S := sp _ _ _ in P6; set y3S := sp _ _ _ in P6.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f123f24e124e624ef
x3S:=sp (n:=n.+1) sam1 4 p2:nat
y3S:=sp (n:=n.+1) ta1 4 p2:nat
P6:(S_[4] n.+1).*2 <=
`d[sam1, tam1]_smove +
`d[tam1, ↓[u (inord a)]]_smove +
`d[↓[u (inord a)], sa1]_smove +
`d[sa1, ta1]_smove + x3S + y3S
1cbb
1b50
  have x2Sx3SE : x2S + x3S <= (S_[1] n).*2 + α_[maxn (3 * a).-2 4] n.
2563
x2S + x3S <= (S_[1] n).*2 + α_[maxn (3 * a).-2 4] n
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f123f24e124e624ef256425652566
x2Sx3SE:x2S + x3S <=
(S_[1] n).*2 + α_[maxn (3 * a).-2 4] n
1cbb
1b51
    by apply: sum_alpha_diffE; rewrite /pa.
256d
1cbb
1b50
  rewrite (maxn_idPl _) in x2Sx3SE; last first.
2563
3 < (3 * a).-2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f123f24e124e624ef2564256525661ed4
1cbb
1b51
    by rewrite -subn2 ltn_subRL // (leq_mul2l 3 2).
2578
1cbb
1b50
  have y2Sy3SE : y2S + y3S <= (S_[1] n).*2 + α_[maxn (3 * c).-2 4] n.
2578
y2S + y3S <= (S_[1] n).*2 + α_[maxn (3 * c).-2 4] n
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f123f24e124e624ef2564256525661ed4
y2Sy3SE:y2S + y3S <=
(S_[1] n).*2 + α_[maxn (3 * c).-2 4] n
1cbb
1b51
    apply: sum_alpha_diffE => //; first by rewrite /pa1.
2578
1fdc
2580
    by rewrite codom_subC.
2582
1cbb
1b50
  rewrite (maxn_idPl _) in y2Sy3SE; last first.
2578
3 < (3 * c).-2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f123f24e124e624ef2564256525661ed4
y2Sy3SE:y2S + y3S <= (S_[1] n).*2 + α_[(3 * c).-2] n
1cbb
1b51
    by rewrite -subn2 ltn_subRL (leq_mul2l 3 2).
2590
1cbb
1b50
  rewrite (_ : sd u = 
      \sum_(i < a) `d[u (inord i), u (inord i.+1)]_smove +
      \sum_(a1 <= i < l.+1) `d[u (inord i), u (inord i.+1)]_smove); last first.
2590
sd (n:=n.+2) u =
\sum_(i < a) `d[u (inord i), u (inord i.+1)]_smove +
\sum_(a1 <= i < l.+1)
   `d[u (inord i), u (inord i.+1)]_smove
2590
(S_[l.+1] n.+2).*2 <=
\sum_(i < a) `d[u (inord i), u (inord i.+1)]_smove +
\sum_(a1 <= i < l.+1)
   `d[u (inord i), u (inord i.+1)]_smove + x1S +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2
1b51
    pose f i := `d[u (inord i), u (inord i.+1)]_smove.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f123f24e124e624ef2564256525661ed425911ff0
2598
2599
    by rewrite /sd -!(big_mkord xpredT f) a1E -!big_cat_nat.
2590
259b
1b50
  rewrite -{}P1 -{P3}P2.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f123f24e124e624ef2564256525661ed42591
(S_[l.+1] n.+2).*2 <=
a.*2 + sd (n:=n.+1) u1 + (c.*2 + sd (n:=n.+1) u2) +
x1S +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2
1b50
  rewrite (_ : sd u1  =
            \sum_(i < (3 * a).-2) `d[u1 (inord i), u1 (inord i.+1)]_smove
            + `d[sam1, tam1]_smove 
            + `d[tam1, ↓[u (inord a)]]_smove); last first.
25a6
2000
25a6
(S_[l.+1] n.+2).*2 <=
a.*2 +
(\sum_(i < (3 * a).-2)
    `d[u1 (inord i), u1 (inord i.+1)]_smove +
 `d[sam1, tam1]_smove + `d[tam1, ↓[
 u (inord a)]]_smove) + 
(c.*2 + sd (n:=n.+1) u2) + x1S +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2
1b51
    pose f i := `d[u1 (inord i), u1 (inord i.+1)]_smove.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f123f24e124e624ef2564256525661ed425912008
2000
25ab
    rewrite /sd -!(big_mkord xpredT f) .
25b1
200c
25ab
    rewrite -{1}(subnK (_ : 1 < 3 * a)) //; last first.
25b1
979
25b1
2012
25ac1b51
      by rewrite -[a]prednK // mulnS.
25b1
2012
25ab
    rewrite subn2 !addn2 !big_nat_recr /f //=; congr (_ + _ + _).
25b1
2019
25b1
201c
25ac1b51
      rewrite ffunE (_ : (3 * a).-2 = (3 * (a.-1)).+1); last first.
25b1
1c9f
25b1
2022
25c125ac1b51
        by rewrite -{1}[a]prednK // mulnS.
25b1
2026
25c0
      rewrite inordK; last by rewrite ltnS ltn_mul2l /= prednK.
25b1
202a
25c0
      rewrite mod3E /= eqxx.
25b1
202e
25c0
      rewrite ffunE inordK; last first.
25b1
2032
25b1
2035
25c125ac1b51
        by rewrite -[_.+3](mulnDr 3 1) add1n prednK // ltnW.
25b1
2035
25c0
      by rewrite mod3E /= eqxx.
25b1
203c
25ab
    rewrite ffunE (_ : (3 * a).-2 = (3 * (a.-1)).+1); last first.
25c4
25b1
2041
25ac1b51
      by rewrite -{1}[a]prednK // mulnS.
25b1
2045
25ab
    rewrite inordK; last first.
25d2
25b1
204a
25ac1b51
      by rewrite -{2}[a]prednK // mulnS.
25b1
204e
25ab
    rewrite mod3E /= eqxx.
25b1
2052
25ab
    rewrite ffunE inordK; last first.
25b1
2056
25b1
2059
25ac1b51
      by rewrite -{2}[a]prednK // mulnS.
25b1
2059
25ab
    by rewrite -[(_ * _).+3](mulnDr 3 1) mod3E /= div3E add1n prednK.
25a6
(S_[l.+1] n.+2).*2 <=
a.*2 +
(\sum_(i < (3 * a).-2)
    `d[u1 (inord i), u1 (inord i.+1)]_smove +
 `d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove) +
(c.*2 + sd (n:=n.+1) u2) + x1S +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2
1b50
  rewrite [X in _ <= _ + (_ + X) + _ + _](_ : _  = 
      `d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
      \sum_(i < (3 * c).-2) `d[u2 (inord i.+2), u2 (inord i.+3)]_smove);
       last first.
25a6
2064
25a6
(S_[l.+1] n.+2).*2 <=
a.*2 +
(\sum_(i < (3 * a).-2)
    `d[u1 (inord i), u1 (inord i.+1)]_smove +
 `d[sam1, tam1]_smove + `d[tam1, ↓[
 u (inord a)]]_smove) +
(c.*2 +
 (`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
  \sum_(i < (3 * c).-2)
     `d[u2 (inord i.+2), 
  u2 (inord i.+3)]_smove)) + x1S +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2
1b51
    pose f i := `d[u2 (inord i), u2 (inord i.+1)]_smove.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cb71cc01cc51cca1e6d1e6e1e6f123f24e124e624ef2564256525661ed42591206c
2064
25fb
    rewrite /sd -!(big_mkord xpredT f) .
2601
2070
25fb
    rewrite -{1}(subnK (_ : 1 < 3 * c)) //; last first.
2601
1e57
2601
2076
25fc1b51
      by rewrite -[c]prednK // mulnS.
2601
2076
25fb
    rewrite subn2 !addn2 !big_nat_recl // /f /= !addnA; congr (_ + _ + _).
2601
207d
2601
2080
2601
2082
25fc1b51
    -
260e rewrite !ffunE !inordK ?addn0 //=.
2601
2087
2610
      by rewrite ltnS muln_gt0.
2601
2080
261225fc1b51
    -
2618 rewrite !ffunE !inordK //=.
2601
1e29
2601
2087
261225fc1b51
        by rewrite ltnW // ltnS (leq_mul2l 3 1 c).
2616261a
      by rewrite ltnS muln_gt0.
2601
2082
25fb
    by rewrite big_mkord.
25a6
(S_[l.+1] n.+2).*2 <=
a.*2 +
(\sum_(i < (3 * a).-2)
    `d[u1 (inord i), u1 (inord i.+1)]_smove +
 `d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove) +
(c.*2 +
 (`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
  \sum_(i < (3 * c).-2)
     `d[u2 (inord i.+2), u2 (inord i.+3)]_smove)) +
x1S +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2
1b50
  set z1S := \sum_(_ < _) _ in P4; set z2S := \sum_(_ < _) _ in P5.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef2564256525661ed4259120a320b020a420b1
2629
1b50
  rewrite -/z1S -/z2S.
262d
(S_[l.+1] n.+2).*2 <=
a.*2 +
(z1S + `d[sam1, tam1]_smove +
 `d[tam1, ↓[u (inord a)]]_smove) +
(c.*2 +
 (`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
  \sum_(i < (3 * c).-2)
     `d[u2 (inord i.+2), u2 (inord i.+3)]_smove)) +
x1S + z2S
1b50
  rewrite a1E -/y1S.
262d
(S_[l.+1] n.+2).*2 <=
a.*2 +
(z1S + `d[sam1, tam1]_smove +
 `d[tam1, ↓[u (inord a)]]_smove) +
(c.*2 +
 (`d[↓[u (inord a)], sa1]_smove + `d[sa1, ta1]_smove +
  y1S)) + x1S + z2S
1b50
  changer (a.*2 + c.*2 + (`d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove
    + `d[↓[u (inord a)], sa1]_smove + `d[sa1, ta1]_smove) 
    + x1S + y1S + z1S + z2S).
262d
(S_[l.+1] n.+2).*2 <=
a.*2 + c.*2 +
(`d[sam1, tam1]_smove + `d[tam1, ↓[u (inord a)]]_smove +
 `d[↓[u (inord a)], sa1]_smove + `d[sa1, ta1]_smove) +
x1S + y1S + z1S + z2S
1b50
  set d1 := `d[_,_]_ _ + `d[_,_]_ _  + `d[_,_]_ _ + `d[_,_]_ _; 
    rewrite -/d1 in P6.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1
d1:=`d[sam1, tam1]_smove +
`d[tam1, ↓[u (inord a)]]_smove +
`d[↓[u (inord a)], sa1]_smove + `d[sa1, ta1]_smove:nat
P6:(S_[4] n.+1).*2 <= d1 + x3S + y3S
(S_[l.+1] n.+2).*2 <=
a.*2 + c.*2 + d1 + x1S + y1S + z1S + z2S
1b50
  have F1 := dsum_alphaL_S (3 * a).-2 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db
2640
1b50
  have F2 := dsum_alphaL_S (3 * c).-2 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0
2640
1b50
  have F3 := dsum_alphaL_S 4 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0
F3:S_[4] n.+1 = S_[4] n + α_[4] n
2640
1b50
  have F4_a : 2 * a = 2 + a.-1 + a.-1.
264c
20e9
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0264d20ed
2640
1b51
    by rewrite mul2n -{1}[a]prednK // -addnn addnS.
2653
2640
1b50
  have F4_c : 2 * c = 2 + c.-1 + c.-1.
2653
20f4
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0264d20ed20f8
2640
1b51
    by rewrite mul2n -{1}[c]prednK // -addnn addnS.
265c
2640
1b50
  have F4 i k : 0 < i -> S_[i.-1] k.+1 <= S_[(3 *i).-2] k + i.-1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0264d20ed20f82100
2101
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0264d20ed20f82105
2640
1b51
    move=> i_gt0; apply: leq_trans (dsum_alpha3l _ _) _.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0264d20ed20f82100210a
210b
2664
    rewrite leq_add2r.
266a
210f
2664
    apply: (increasingE (increasing_dsum_alphaL_l _)).
266a
2113
2664
    by rewrite -{2}[i]prednK // mulnS addSn add2n /=.
2666
2640
1b50
  have F4an := F4 a n a_gt0.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0264d20ed20f82105211b
2640
1b50
  have F4an1 := F4 a n.+1 a_gt0.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0264d20ed20f82105211b2120
2640
1b50
  have F4cn := F4 c n (c_gt0).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0264d20ed20f82105211b21202125
2640
1b50
  have F4cn1 := F4 c n.+1 (c_gt0).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0264d20ed20f82105211b21202125212a
2640
1b50
  have l_gt1 : 1 < l.
2683
212e
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0264d20ed20f82105211b21202125212a4e4
2640
1b51
    by rewrite -ltnS l1E -[3]/(2 + 0 + 1); rewrite !leq_add.
2689
2640
1b50
  have F5 := SH l_gt1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0264d20ed20f82105211b21202125212a4e42139
2640
1b50
  have am1_gt0 : 0 < a.-1 by rewrite -subn1 ltn_subRL addn0.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0264d20ed20f82105211b21202125212a4e42139b2b
2640
1b50
  have F6n1 : S_[1] n.+2 + S_[l.+1] n.+2 <= S_[a.-1] n.+2 + S_[c.+2] n.+2.
2694
S_[1] n.+2 + S_[l.+1] n.+2 <=
S_[a.-1] n.+2 + S_[c.+2] n.+2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0264d20ed20f82105211b21202125212a4e42139b2b
F6n1:S_[1] n.+2 + S_[l.+1] n.+2 <=
S_[a.-1] n.+2 + S_[c.+2] n.+2
2640
1b51
    rewrite addnC (_ : l.+1 = 1 + c.+1 + a.-2); last first.
2694
l.+1 = 1 + c.+1 + a.-2
2694
S_[1 + c.+1 + a.-2] n.+2 + S_[1] n.+2 <=
S_[a.-1] n.+2 + S_[c.+2] n.+2
269a1b51
      by rewrite  /c add1n 2!addSnnS !prednK ?a1E ?subnK.
2694
26a3
2699
    rewrite {2}(_ : a.-1 = 1 + a.-2); last by rewrite add1n prednK.
2694
S_[1 + c.+1 + a.-2] n.+2 + S_[1] n.+2 <=
S_[1 + a.-2] n.+2 + S_[c.+2] n.+2
2699
    by apply: concaveEk1 (concave_dsum_alphaL_l n.+2).
269b
2640
1b50
  have F6n : S_[1] n.+1 + S_[l] n.+1 <= S_[a.-1] n.+1 + S_[c.+1] n.+1.
269b
S_[1] n.+1 + S_[l] n.+1 <=
S_[a.-1] n.+1 + S_[c.+1] n.+1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0264d20ed20f82105211b21202125212a4e42139b2b269c
F6n:S_[1] n.+1 + S_[l] n.+1 <=
S_[a.-1] n.+1 + S_[c.+1] n.+1
2640
1b51
    rewrite addnC (_ : l = 1 + c + a.-2); last first.
269b
l = 1 + c + a.-2
269b
S_[1 + c + a.-2] n.+1 + S_[1] n.+1 <=
S_[a.-1] n.+1 + S_[c.+1] n.+1
26b31b51
      rewrite  /c add1n a1E -{1}[a]prednK // subSS.
269b
l = (l - a.-1).+1 + a.-2
26ba
      by rewrite !addSnnS !prednK // subnK // -ltnS prednK.
269b
26bc
26b2
    rewrite {2}(_ : a.-1 = 1 + a.-2); last by rewrite add1n prednK.
269b
S_[1 + c + a.-2] n.+1 + S_[1] n.+1 <=
S_[1 + a.-2] n.+1 + S_[c.+1] n.+1
26b2
    by apply: concaveEk1 (concave_dsum_alphaL_l n.+1).
26b4
2640
1b50
  have cm1_gt0 : 0 < c.-1 by rewrite -subn1 ltn_subRL addn0.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0264d20ed20f82105211b21202125212a4e42139b2b269c26b52475
2640
1b50
  have F7n1 : S_[1] n.+2 + S_[c.+2] n.+2 <= S_[c.-1] n.+2 + S_[4] n.+2.
26ce
S_[1] n.+2 + S_[c.+2] n.+2 <=
S_[c.-1] n.+2 + S_[4] n.+2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0264d20ed20f82105211b21202125212a4e42139b2b269c26b52475
F7n1:S_[1] n.+2 + S_[c.+2] n.+2 <=
S_[c.-1] n.+2 + S_[4] n.+2
2640
1b51
    rewrite addnC (_ : c.+2 = 1 + 3 + c.-2); last first.
26ce
249a
26ce
S_[1 + 3 + c.-2] n.+2 + S_[1] n.+2 <=
S_[c.-1] n.+2 + S_[4] n.+2
26d41b51
      by rewrite add1n 2!addSnnS !prednK.
26ce
26dc
26d3
    rewrite {2}(_ : c.-1 = 1 + c.-2); last by rewrite add1n prednK.
26ce
S_[1 + 3 + c.-2] n.+2 + S_[1] n.+2 <=
S_[1 + c.-2] n.+2 + S_[4] n.+2
26d3
    by apply: concaveEk1 (concave_dsum_alphaL_l n.+2).
26d5
2640
1b50
  have F7n : S_[1] n.+1 + S_[c.+1] n.+1 <= S_[c.-1] n.+1 + S_[3] n.+1.
26d5
S_[1] n.+1 + S_[c.+1] n.+1 <=
S_[c.-1] n.+1 + S_[3] n.+1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0264d20ed20f82105211b21202125212a4e42139b2b269c26b5247526d6
F7n:S_[1] n.+1 + S_[c.+1] n.+1 <=
S_[c.-1] n.+1 + S_[3] n.+1
2640
1b51
    rewrite addnC (_ : c.+1 = 1 + 2 + c.-2); last first.
26d5
c.+1 = 1 + 2 + c.-2
26d5
S_[1 + 2 + c.-2] n.+1 + S_[1] n.+1 <=
S_[c.-1] n.+1 + S_[3] n.+1
26ec1b51
      by rewrite add1n 2!addSnnS !prednK.
26d5
26f5
26eb
    rewrite {2}(_ : c.-1 = 1 + c.-2); last by rewrite add1n prednK.
26d5
S_[1 + 2 + c.-2] n.+1 + S_[1] n.+1 <=
S_[1 + c.-2] n.+1 + S_[3] n.+1
26eb
    by apply: concaveEk1 (concave_dsum_alphaL_l n.+1).
26ed
2640
1b50
  have F8 : S_[4] n.+2 <= S_[3] n.+1 + (α_[1] n.+1).*2.
26ed
S_[4] n.+2 <= S_[3] n.+1 + (α_[1] n.+1).*2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0264d20ed20f82105211b21202125212a4e42139b2b269c26b5247526d626ee
F8:S_[4] n.+2 <= S_[3] n.+1 + (α_[1] n.+1).*2
2640
1b51
    by apply: dsum_alphaL_alpha n.+1 (isT : 1 < 3).
2706
2640
1b50
  have F9 : S_[3] n.+1 <= S_[4] n.+1 by case: (concave_dsum_alphaL_l n.+1).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0264d20ed20f82105211b21202125212a4e42139b2b269c26b5247526d626ee2707
F9:S_[3] n.+1 <= S_[4] n.+1
2640
1b50
  have F10n1 := dsum_alphaL_S 1 n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0264d20ed20f82105211b21202125212a4e42139b2b269c26b5247526d626ee2707270f
F10n1:S_[1] n.+2 = S_[1] n.+1 + α_[1] n.+1
2640
1b50
  have F10n := dsum_alphaL_S 1 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0264d20ed20f82105211b21202125212a4e42139b2b269c26b5247526d626ee2707270f2714
F10n:S_[1] n.+1 = S_[1] n + α_[1] n
2640
1b50 
  have F11 := bound_alphaL 2 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0264d20ed20f82105211b21202125212a4e42139b2b269c26b5247526d626ee2707270f27142719
F11:α_[2] n <= (α_[1] n).*2
2640
1b50
  have F12 := alphaL_1_2 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d5c31b4f1b571b601b651b6a11fc1cb61cc01cc51cca1e6d1e6e123f24e124e624ef256425651ed4259120a320b020a420b1263e263f20db20e0264d20ed20f82105211b21202125212a4e42139b2b269c26b5247526d626ee2707270f27142719271e
F12:α_[1] n.+1 <= (α_[1] n).*2
2640
1b50
  by lia.
1b3c
1b52
rewrite negb_and -!leqNgt => acH.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d
acH:(a <= 1) || (c <= 1)
573
(* subcase a = 1 *)
have aLc : a <= c by [].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b
aLc:a <= c
573
have aE1 : a = 1.
272f
a = 1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e
573
  case/orP: acH; case: (a) aLc a_gt0 => // [] [|n1] //.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d
n1:nat
n1.+1 < c -> 0 < n1.+2 -> c <= 1 -> n1.+2 = 1
2735
  by case: (c) => // [] [].
2737
573
have leq_S4 k : 5 * S_[1] k.+1 + S_[1] k.+2 <= 6 * S_[4] k + 9.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0ea81
5 * S_[1] k.+1 + S_[1] k.+2 <= 6 * S_[4] k + 9
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e
leq_S4:forall k : nat,
5 * S_[1] k.+1 + S_[1] k.+2 <=
6 * S_[4] k + 9
573
  have F3 :  4 * S_[3] k + 2 * S_[6] k <= 6 * S_[4] k.
2744
4 * S_[3] k + 2 * S_[6] k <= 6 * S_[4] k
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0ea81
F3:4 * S_[3] k + 2 * S_[6] k <= 6 * S_[4] k
2745
2747
    rewrite -{1}[4]/(2 * 2) -mulnA -mulnDr -{2}[6]/(2 * 3) -mulnA.
2744
2 * (2 * S_[3] k + S_[6] k) <= 2 * (3 * S_[4] k)
274e
    rewrite leq_mul2l /= -(leq_add2r (S_[5] k)).
2744
2 * S_[3] k + S_[6] k + S_[5] k <=
3 * S_[4] k + S_[5] k
274e
    changel (S_[6] k + S_[3] k +  (S_[5] k + S_[3] k)).
2744
S_[6] k + S_[3] k + (S_[5] k + S_[3] k) <=
3 * S_[4] k + S_[5] k
274e
    changer (S_[5] k + S_[4] k +  (S_[4] k + S_[4] k)).
2744
S_[6] k + S_[3] k + (S_[5] k + S_[3] k) <=
S_[5] k + S_[4] k + (S_[4] k + S_[4] k)
274e
    by apply: leq_add; apply: concaveEk1 3 _ _ (concave_dsum_alphaL_l k).
2750
2745
2746
  applyr F3.
2744
5 * S_[1] (1 + k) + S_[1] (2 + k) <=
9 + (4 * S_[3] k + 2 * S_[6] k)
2746
  have F3k : S_[1] k.+1 <= S_[3] k + 1 by apply: dsum_alpha3l.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0ea81
F3k:S_[1] k.+1 <= S_[3] k + 1
2768
2746
  have F3k1 : S_[2] k.+1 <= S_[6] k + 2 by apply: dsum_alpha3l.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0ea81276d
F3k1:S_[2] k.+1 <= S_[6] k + 2
2768
2746
  have F4 := leq_dsum_alpha_2l_1 k.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18d318db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0ea81276d2772
F4:S_[1] k.+1 + S_[1] k.+2 <= (S_[2] k.+1).*2.+1
2768
2746
  by rewrite !add1n !add2n; lia.
2748
573
move: c_gt0; rewrite leq_eqVlt eq_sym => /orP[/eqP cE1|c_gt1].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e2749
cE1:c = 1
573
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f
573
(* This is 4.2.2 a = 1, b = 1 *)
  move : P1 P2 P3 => P1 P2 P3.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d
573
2780
  have l_gt0 : 0 < l by rewrite -ltnS l1E aE1 cE1 addn1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b57
573
2780
  have a1El : a1 = l.
278a
a1 = l
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b57
a1El:a1 = l
573
2781
    by rewrite -(subKn (_ : a1 <= l.+1)) 1?ltnW // -/c cE1 subSS subn0.
2791
573
2780
  have [b_gt1|b_lt2]:= leqP 2 b.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b5727925d1
573
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792
b_lt2:b < 2
573
2781
(*   subcase b >= 2*)
    rewrite (_ : sd u = 
        \sum_(i < a) `d[u (inord i), u (inord i.+1)]_smove +
        \sum_(a <= i < a1) `d[u (inord i), u (inord i.+1)]_smove +
        \sum_(a1 <= i < l.+1) `d[u (inord i), u (inord i.+1)]_smove);
        last first.
2799
1fe8
2799
(S_[l.+1] n.+2).*2 <=
\sum_(i < a) `d[u (inord i), u (inord i.+1)]_smove +
\sum_(a <= i < a1)
   `d[u (inord i), u (inord i.+1)]_smove +
\sum_(a1 <= i < l.+1)
   `d[u (inord i), u (inord i.+1)]_smove +
sp (n:=n.+2) (u ord0) l.+1 p1 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
279b2781
      pose f i := `d[u (inord i), u (inord i.+1)]_smove.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b5727925d11ff0
1fe8
27a1
      rewrite /sd -!(big_mkord xpredT f).
27a7
1ff4
27a1
      by rewrite -!big_cat_nat //= ltnW.
2799
27a3
279a
    rewrite !sum_beta_S // KH1 eqxx addn0 KH2 eqxx addn0.
2799
(S_[l.+1] n.+2).*2 <=
\sum_(i < a) `d[u (inord i), u (inord i.+1)]_smove +
\sum_(a <= i < a1)
   `d[u (inord i), u (inord i.+1)]_smove +
\sum_(a1 <= i < l.+1)
   `d[u (inord i), u (inord i.+1)]_smove +
\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2
279a
    apply: leq_trans (leq_add (leq_add (leq_add (leq_add (leqnn _) P3) 
                       (leqnn _)) (leqnn _)) (leqnn _)); rewrite -{P3}P1 -{}P2.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d1
(S_[l.+1] n.+2).*2 <=
a.*2 + sd (n:=n.+1) u1 +
\sum_(a <= i < a1)
   `d[↓[u (inord i)], ↓[u (inord i.+1)]]_smove +
(c.*2 + sd (n:=n.+1) u2) +
\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2
279a
    set x2S := sd u2; set x0S := sd u1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d1
x2S:=sd (n:=n.+1) u2:nat
x0S:=sd (n:=n.+1) u1:nat
(S_[l.+1] n.+2).*2 <=
a.*2 + x0S +
\sum_(a <= i < a1)
   `d[↓[u (inord i)], ↓[u (inord i.+1)]]_smove +
(c.*2 + x2S) +
\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2
279a
    pose u3 := [ffun i : 'I_b.+1 => ↓[u (inord (a + i))]].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc
u3:=[ffun i => ↓[u (inord (a + i))]]:{ffun forall x : ordinal_finType b.+1,
      (fun=> configuration 4 n.+1) x}
27bd
279a
    rewrite  (_ : \sum_(_ <= _ < _) _ =
        \sum_(i < b) `d[u3 (inord i), u3 (inord i.+1)]_smove); last first.
27c1
\sum_(a <= H < a1)
   `d[↓[u (inord H)], ↓[u (inord H.+1)]]_smove =
\sum_(i < b) `d[u3 (inord i), u3 (inord i.+1)]_smove
27c1
(S_[l.+1] n.+2).*2 <=
a.*2 + x0S +
\sum_(i < b) `d[u3 (inord i), u3 (inord i.+1)]_smove +
(c.*2 + x2S) +
\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2
279b2781
      rewrite -{1}[a]add0n big_addn  -/b big_mkord.
27c1
\sum_(i < b)
   `d[↓[u (inord (i + a))], ↓[u (inord (i + a).+1)]]_smove =
\sum_(i < b) `d[u3 (inord i), u3 (inord i.+1)]_smove
27c7
      apply: eq_bigr => i _.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c2821
`d[↓[u (inord (i + a))], ↓[u (inord (i + a).+1)]]_smove =
`d[u3 (inord i), u3 (inord i.+1)]_smove
27c7
      by rewrite !ffunE addnC !inordK ?(addnS, ltnS) // ltnW.
27c1
27c9
279a
    pose paS := a[p1, p3] a.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f
27c9
279a
    have cH3 (k : 'I_b.+1) : 
      0 < k < b -> codom (u3 k) \subset [:: p2; a[pa, paS] k].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f9c1
9c2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f
cH3:forall k : 'I_b.+1,
0 < k < b ->
codom (u3 k) \subset [:: p2;  a[pa, paS] k]
27c9
279b2781
      move=> /andP[k_gt0 k_ltb].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f9c18521eeb
9d0
27de
      have akLl : a + k < l.+1.
27e5
1ef0
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f9c18521eeb1ef4
9d0
27df279b2781
        rewrite  ltnS (leq_trans _ a1Ll) //.
27e5
1ef8
27e9
        by rewrite -(subnK aLa1) addnC leq_add2r ltnW.
27eb
9d0
27de
      rewrite !ffunE.
27eb
1eff
27de
      have /H : 0 < (inord (a + k) : 'I_l.+2) < l.+1.
27eb
1f03
27eb
1f06
27df279b2781
        rewrite inordK; first by rewrite addn_gt0 a_gt0.
27eb
1f0a
27f8
        by rewrite ltnS ltnW.
27eb
1f06
27de
      rewrite ffunE inordK //; last by rewrite ltnS ltnW.
27eb
1f11
27de
      by rewrite /paS apegS -apegD [k + _]addnC.
27e0
27c9
279a
    pose pa1 := a[p1, p3] a1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5
27c9
279a
    have P3 : 
      (S_[b] n.+1).*2 <=
      \sum_(i < b) `d[↓[u (inord (a + i))], ↓[u (inord (a + i).+1)]]_smove +
      sp ↓[u (inord a)] b pa + sp ↓[u (inord a1)] b pa1.
2809
(S_[b] n.+1).*2 <=
\sum_(i < b)
   `d[↓[u (inord (a + i))], ↓[u (inord (a + i).+1)]]_smove +
sp (n:=n.+1) ↓[u (inord a)] b pa +
sp (n:=n.+1) ↓[u (inord a1)] b pa1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5
P3:(S_[b] n.+1).*2 <=
\sum_(i < b)
   `d[↓[u (inord (a + i))], ↓[
u (inord (a + i).+1)]]_smove +
sp (n:=n.+1) ↓[u (inord a)] b pa +
sp (n:=n.+1) ↓[u (inord a1)] b pa1
27c9
279b2781
      apply: leq_trans (IH _ _ _ _ _ _ cH3) (leq_add (leq_add _ _) _);
        rewrite /pa /paS // 1?eq_sym //.
2809
1f20
2809
sd (n:=n.+1) u3 <=
\sum_(i < b)
   `d[↓[u (inord (a + i))], ↓[
u (inord (a + i).+1)]]_smove
2809
sp (n:=n.+1) (u3 ord0) b ( a[p1, p3] a) <=
sp (n:=n.+1) ↓[u (inord a)] b ( a[p1, p3] a)
2809
sp (n:=n.+1) (u3 ord_max) b
  ( a[ a[p1, p3] a,  a[p1, p3] a.+1] b) <=
sp (n:=n.+1) ↓[u (inord a1)] b pa1
280f279b2781
      -
2813 by rewrite apegS apeg_eqC eq_sym.
2809
sd (n:=n.+1) u3 <=
\sum_(i < b)
   `d[↓[u (inord (a + i))], ↓[u (inord (a + i).+1)]]_smove
2818281a280f279b2781
      -
281e apply: leq_sum => i; rewrite !ffunE !inordK ?addnS //.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5821
1f33
2826
1f36
2818281a280f279b2781
          by rewrite ltnS.
2826
1f36
2821
        by rewrite ltnS ltnW.
2809
2819
281a280f279b2781
      -
282d by apply: leq_sum => i _; rewrite !ffunE addn0.
2809
281b
280e
      apply: leq_sum => i _; rewrite !ffunE /= [a + b]addnC subnK //.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc51054
(u (inord a1) (trshift 1 i)
 !=  a[ a[p1, p3] a,  a[p1, p3] a.+1] b) *
β_[n.+1, b] i <=
(u (inord a1) (trshift 1 i) != pa1) * β_[n.+1, b] i
280e
      by rewrite apegS -apegD subnK.
2810
27c9
279a
    set x1S := \sum_(_ < _) _ in P3; set y2S := sp _ _ _ in P3.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5
x1S:=\sum_(H < b)
   `d[↓[u (inord (a + H))], ↓[
u (inord (a + H).+1)]]_smove:nat
y2S:=sp (n:=n.+1) ↓[u (inord a)] b pa:nat
P3:(S_[b] n.+1).*2 <=
x1S + y2S + sp (n:=n.+1) ↓[u (inord a1)] b pa1
27c9
279a
    set y3S :=  sp _ _ _ in P3.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc528402841
y3S:=sp (n:=n.+1) ↓[u (inord a1)] b pa1:nat
P3:(S_[b] n.+1).*2 <= x1S + y2S + y3S
27c9
279a
    rewrite [\sum_(_ < _) _](_ : _ = x1S); last first.
2846
\sum_(H < b) `d[u3 (inord H), u3 (inord H.+1)]_smove =
x1S
2846
(S_[l.+1] n.+2).*2 <=
a.*2 + x0S + x1S + (c.*2 + x2S) +
\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2
279b2781
      by apply: eq_bigr => i; rewrite !ffunE !inordK ?addnS // ltnS // ltnW.
2846
284f
279a
    set y0S := \sum_(_ < _) _ ; set y5S := \sum_(_ < _) _.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc52840284128472848
y0S:=\sum_(H < n.+1)
   (↓[u ord0] H != p1) * (α_[1] H).*2:nat
y5S:=\sum_(H < n.+1)
   (↓[u ord_max] H !=  a[p3, p1] l) *
   (α_[1] H).*2:nat
(S_[l.+1] n.+2).*2 <=
a.*2 + x0S + x1S + (c.*2 + x2S) + y0S + y5S
279a
    have cH1 (k : 'I_(3 * a).+1) : 
      0 < k < 3 * a -> codom (u1 k) \subset [:: p3; a[p1, p2] k].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858
k:'I_(3 * a).+1
0 < k < 3 * a ->
codom (u1 k) \subset [:: p3;  a[p1, p2] k]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858
cH1:forall k : 'I_(3 * a).+1,
0 < k < 3 * a ->
codom (u1 k) \subset [:: p3;  a[p1, p2] k]
2859
279b2781
      move=> /andP[k_gt0 k_ltb].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858285e852
k_ltb:k < 3 * a
codom (u1 k) \subset [:: p3;  a[p1, p2] k]
2860
      rewrite {2}aE1 muln1 in k_ltb.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858285e852
k_ltb:k < 3
2869
2860
      rewrite ffunE !modn_small // !divn_small // inord_eq0 //.
286d
codom
  (if k == 0
   then ↓[u ord0]
   else
    if k == 1
    then if k == (3 * a.-1).+1 then sam1 else si 0
    else if k == (3 * a.-1).+2 then tam1 else ti 0) \subset
[:: p3;  a[p1, p2] k]
2860
      rewrite [in 3 * a.-1]aE1 /=.
286d
codom
  (if k == 0
   then ↓[u ord0]
   else
    if k == 1
    then if k == (3 * 0).+1 then sam1 else si 0
    else if k == (3 * 0).+2 then tam1 else ti 0) \subset
[:: p3;  a[p1, p2] k]
2860
      case: (k : nat) k_gt0 k_ltb => //= [] [|[|]] //= _ _.
285d
codom sam1 \subset [:: p3;  a[p1, p2] 1]
285d
codom tam1 \subset [:: p3;  a[p1, p2] 2]
2861279b2781
        rewrite !(apegS, apeg0).
285d
codom sam1 \subset [:: p3; p2]
287b
        by have := sam1C; rewrite /pa aE1 apegS apeg0.
285d
287d
2860
      by rewrite !(apegS, apeg0) codom_subC.
2862
2859
279a
    have P1 : 
      (S_[3 * a] n.+1).*2 <= x0S + sp ↓[u ord0] 3 p1 + sp ↓[u (inord a)] 3 p2.
2862
(S_[3 * a] n.+1).*2 <=
x0S + sp (n:=n.+1) ↓[u ord0] 3 p1 +
sp (n:=n.+1) ↓[u (inord a)] 3 p2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc52840284128472848285728582863
P1:(S_[3 * a] n.+1).*2 <=
x0S + sp (n:=n.+1) ↓[u ord0] 3 p1 +
sp (n:=n.+1) ↓[u (inord a)] 3 p2
2859
279b2781
      apply: leq_trans (IH _ _ _ _ _ _ cH1) (leq_add (leq_add _ _) _);
          rewrite // 1?eq_sym //.
2862
sp (n:=n.+1) (u1 ord0) (3 * a) p1 <=
sp (n:=n.+1) ↓[u ord0] 3 p1
2862
sp (n:=n.+1) (u1 ord_max) (3 * a) ( a[p1, p2] (3 * a)) <=
sp (n:=n.+1) ↓[u (inord a)] 3 p2
288d279b2781
        by apply: leq_sum => i _; rewrite ffunE /= inord_eq0 // aE1 muln1.
2862
2896
288c
      by apply: leq_sum => i _; rewrite ffunE /= mod3E /= div3E // aE1 muln1.
288e
2859
279a
    have {}P1 := leq_trans P1 (leq_add (leq_add (leqnn _) (leq_sum_beta _ _))
                          (leqnn _)).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc52840284128472848285728582863
P1:(S_[3 * a] n.+1).*2 <=
x0S +
\sum_(k < n.+1)
   (↓[u ord0] k != p1) * (α_[1] k).*2 +
sp (n:=n.+1) ↓[u (inord a)] 3 p2
2859
279a
    rewrite -/y0S in P1; set y1S := sp _ _ _ in P1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc52840284128472848285728582863
y1S:=sp (n:=n.+1) ↓[u (inord a)] 3 p2:nat
P1:(S_[3 * a] n.+1).*2 <= x0S + y0S + y1S
2859
279a
    rewrite aE1 muln1 in P1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858286328a6
P1:(S_[3] n.+1).*2 <= x0S + y0S + y1S
2859
279a
    have y1Sy2SE : y1S + y2S <= (S_[1] n).*2 + α_[maxn 3 b] n.
28ab
y1S + y2S <= (S_[1] n).*2 + α_[maxn 3 b] n
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858286328a628ac
y1Sy2SE:y1S + y2S <= (S_[1] n).*2 + α_[maxn 3 b] n
2859
279b2781
      apply: sum_alpha_diffE => //; first by rewrite /pa // eq_sym.
28ab
1f5c
28b1
      apply: codom_liftr.
28ab
1f60
28b1
      have /cH : 0 < (inord 1 : 'I_l.+2) < l.+1 by rewrite inordK.
28ab
codom (u (inord 1)) \subset
[:: p2;  a[p1, p3] (inord 1)] ->
codom (u (inord a)) \subset [:: p2; pa]
28b1
      by rewrite /pa aE1 inordK.
28b3
2859
279a
    have cH2 (k : 'I_(3 * c).+1) : 
      0 < k < 3 * c -> 
      codom (u2 k) \subset [:: a[p1, p3] a1; a[p2, a[p1, p3] a1.+1] k].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858286328a628ac28b4
k:'I_(3 * c).+1
0 < k < 3 * c ->
codom (u2 k) \subset
[::  a[p1, p3] a1;  a[p2,  a[p1, p3] a1.+1] k]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858286328a628ac28b4
cH2:forall k : 'I_(3 * c).+1,
0 < k < 3 * c ->
codom (u2 k) \subset
[::  a[p1, p3] a1;  a[p2,  a[p1, p3] a1.+1] k]
2859
279b2781
      move=> /andP[k_gt0 k_ltb].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858286328a628ac28b428c6852
k_ltb:k < 3 * c
codom (u2 k) \subset
[::  a[p1, p3] a1;  a[p2,  a[p1, p3] a1.+1] k]
28c8
      rewrite {2}cE1 muln1 in k_ltb.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858286328a628ac28b428c6852286e
28d1
28c8
      rewrite ffunE !modn_small // !divn_small // addn0.
28d5
codom
  (if k == 0
   then ↓[u (inord a1)]
   else
    if k == 1
    then if k == 1 then sa1 else si a1
    else if k == 2 then ta1 else ti a1) \subset
[::  a[p1, p3] a1;  a[p2,  a[p1, p3] a1.+1] k]
28c8
      case: (k : nat) k_gt0 k_ltb => //= [] [|[|]] //= _ _.
28c5
codom sa1 \subset
[::  a[p1, p3] a1;  a[p2,  a[p1, p3] a1.+1] 1]
28c5
codom ta1 \subset
[::  a[p1, p3] a1;  a[p2,  a[p1, p3] a1.+1] 2]
28c9279b2781
        by rewrite !apegS apeg0 codom_apeg.
28c5
28e0
28c8
      by rewrite !(apegS, apeg0) codom_subC.
28ca
2859
279a
    have P2 : 
      (S_[3 * c] n.+1).*2 <= x2S + sp ↓[u (inord a1)] 3 p2 +
                             sp ↓[u ord_max] 3 (a[p3, p1] l).
28ca
(S_[3 * c] n.+1).*2 <=
x2S + sp (n:=n.+1) ↓[u (inord a1)] 3 p2 +
sp (n:=n.+1) ↓[u ord_max] 3 ( a[p3, p1] l)
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858286328a628ac28b428cb
P2:(S_[3 * c] n.+1).*2 <=
x2S + sp (n:=n.+1) ↓[u (inord a1)] 3 p2 +
sp (n:=n.+1) ↓[u ord_max] 3 ( a[p3, p1] l)
2859
279b2781
      apply: leq_trans (IH _ _ _ _ _ _ cH2) (leq_add (leq_add _ _) _);
          rewrite // 1?eq_sym //.
28ca
1a1f
28ca
sp (n:=n.+1) (u2 ord0) (3 * c) p2 <=
sp (n:=n.+1) ↓[u (inord a1)] 3 p2
28ca
sp (n:=n.+1) (u2 ord_max) 
  (3 * c) ( a[p2,  a[p1, p3] a1.+1] (3 * c)) <=
sp (n:=n.+1) ↓[u ord_max] 3 ( a[p3, p1] l)
28ec279b2781
      -
28f0 by rewrite apegS apeg_eqC eq_sym.
28ca
28f4
28f528ec279b2781
      -
28f9 by apply: leq_sum => i _; rewrite ffunE /= addn0 cE1 muln1.
28ca
sp (n:=n.+1) (u2 ord_max) (3 * c)
  ( a[p2,  a[p1, p3] a1.+1] (3 * c)) <=
sp (n:=n.+1) ↓[u ord_max] 3 ( a[p3, p1] l)
28eb
      apply: leq_sum => i _; rewrite ffunE /= mod3E /= div3E // cE1 muln1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858286328a628ac28b428cb1054
(↓[u (inord (a1 + 1))] i !=  a[p2,  a[p1, p3] a1.+1] 3) *
β_[n.+1, 3] i <=
(↓[u ord_max] i !=  a[p3, p1] l) * β_[n.+1, 3] i
28eb
      rewrite a1El addn1 (_ : inord _ = ord_max) ?apegS //.
2904
a16
28eb
      by apply/val_eqP; rewrite /= inordK.
28ed
2859
279a
    have {}P2 := leq_trans P2 (leq_add (leqnn _) (leq_sum_beta _ _)).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858286328a628ac28b428cb
P2:(S_[3 * c] n.+1).*2 <=
x2S + sp (n:=n.+1) ↓[u (inord a1)] 3 p2 +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) *
   (α_[1] k).*2
2859
279a
    rewrite -/y5S in P2; set y4S := sp _ _ _ in P2.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858286328a628ac28b428cb
y4S:=sp (n:=n.+1) ↓[u (inord a1)] 3 p2:nat
P2:(S_[3 * c] n.+1).*2 <= x2S + y4S + y5S
2859
279a
    have y3Sy4SE : y3S + y4S <= (S_[1] n).*2 + α_[maxn b 3] n.
2914
y3S + y4S <= (S_[1] n).*2 + α_[maxn b 3] n
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858286328a628ac28b428cb29152916
y3Sy4SE:y3S + y4S <= (S_[1] n).*2 + α_[maxn b 3] n
2859
279b2781
      apply: sum_alpha_diffE; rewrite /pa /pa1 //.
2914
1fba
291b
      apply: codom_liftr; rewrite codom_subC.
2914
1fbe
291b
      have /cH : 0 < (inord l : 'I_l.+2) < l.+1 by rewrite inordK // l_gt0 /=.
2914
codom (u (inord l)) \subset
[:: p2;  a[p1, p3] (inord l)] ->
codom (u (inord a1)) \subset [:: p2;  a[p1, p3] a1]
291b
      by rewrite a1El inordK.
291d
2859
279a
    have Eb1 : l = b.+1.
291d
l = b.+1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858286328a628ac28b428cb29152916291e
Eb1:l = b.+1
2859
279b2781
      by have := l1E; rewrite aE1 cE1 addn1 => [] [].
2932
2859
279a
    rewrite Eb1.
2932
(S_[b.+2] n.+2).*2 <=
a.*2 + x0S + x1S + (c.*2 + x2S) + y0S + y5S
279a
    have F1b := dsum_alphaL_S b n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858286328a628ac28b428cb29152916291e2933
F1b:S_[b] n.+1 = S_[b] n + α_[b] n
293a
279a
    have F1n31 : (S_[3] n.+1).+1 = S_[1] n.+2.
293e
(S_[3] n.+1).+1 = S_[1] n.+2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858286328a628ac28b428cb29152916291e2933293f
F1n31:(S_[3] n.+1).+1 = S_[1] n.+2
293a
279b2781
      by rewrite dsum_alpha3_S.
2946
293a
279a
    have F1n1 := dsum_alphaL_S 1 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858286328a628ac28b428cb29152916291e2933293f2947
F1n1:S_[1] n.+1 = S_[1] n + α_[1] n
293a
279a
    have F1n2 := dsum_alphaL_S 1 n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858286328a628ac28b428cb29152916291e2933293f2947294f
F1n2:S_[1] n.+2 = S_[1] n.+1 + α_[1] n.+1
293a
279a
    have b_gt0 : 0 < b by apply: leq_trans b_gt1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858286328a628ac28b428cb29152916291e2933293f2947294f2954f17
293a
279a
    have F2n1 : S_[b.+2] n.+2 <= S_[b.+1] n.+1 + (α_[1] n.+1).*2.
2958
S_[b.+2] n.+2 <= S_[b.+1] n.+1 + (α_[1] n.+1).*2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858286328a628ac28b428cb29152916291e2933293f2947294f2954f17
F2n1:S_[b.+2] n.+2 <=
S_[b.+1] n.+1 + (α_[1] n.+1).*2
293a
279b2781
      by rewrite dsum_alphaL_alpha .
295f
293a
279a
    have F2n : S_[b.+1] n.+1 <= S_[b] n + (α_[1] n).*2.
295f
S_[b.+1] n.+1 <= S_[b] n + (α_[1] n).*2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858286328a628ac28b428cb29152916291e2933293f2947294f2954f172960
F2n:S_[b.+1] n.+1 <= S_[b] n + (α_[1] n).*2
293a
279b2781
      by rewrite dsum_alphaL_alpha.
296a
293a
279a
    have F3n : S_[4] n.+2 <= S_[3] n.+1 + (α_[1] n.+1).*2.
296a
2703
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858286328a628ac28b428cb29152916291e2933293f2947294f2954f172960296b
F3n:S_[4] n.+2 <= S_[3] n.+1 + (α_[1] n.+1).*2
293a
279b2781
      by rewrite dsum_alphaL_alpha.
2974
293a
279a
    have F4n := alphaL_3E n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858286328a628ac28b428cb29152916291e2933293f2947294f2954f172960296b2975
F4n:α_[3] n = α_[1] n.+1
293a
279a
    have F5 := leq_dsum_alpha_2l_1 n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858286328a628ac28b428cb29152916291e2933293f2947294f2954f172960296b2975297d
F5:S_[1] n.+1 + S_[1] n.+2 <= (S_[2] n.+1).*2.+1
293a
279a
    rewrite cE1 !muln1 in P1 P2.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858286328a628b428cb2915291e2933293f2947294f2954f172960296b2975297d298228ac
P2:(S_[3] n.+1).*2 <= x2S + y4S + y5S
293a
279a
    rewrite maxnC in y3Sy4SE.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b5727925d127bb27bc27c298f27e11cc5284028412847284828572858286328a628b428cb29152933293f2947294f2954f172960296b2975297d298228ac2987
y3Sy4SE:y3S + y4S <= (S_[1] n).*2 + α_[maxn 3 b] n
293a
279a
    suff : 
      S_[b.+2] n.+2 + (S_[1] n).*2 + α_[maxn 3 b] n <=
     ((S_[3] n.+1).+1).*2 + S_[b] n.+1
       by lia.
298b
S_[b.+2] n.+2 + (S_[1] n).*2 + α_[maxn 3 b] n <=
((S_[3] n.+1).+1).*2 + S_[b] n.+1
279a
    move: b_gt1; rewrite leq_eqVlt => /orP[/eqP<-|b_gt2]; last first.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b57279227bb27bc27c298f27e11cc5284028412847284828572858286328a628b428cb29152933293f2947294f2954f172960296b2975297d298228ac2987298c10c0
2990
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b57279227bb27bc27c298f27e11cc5284028412847284828572858286328a628b428cb29152933293f2947294f2954f172960296b2975297d298228ac2987298c
S_[4] n.+2 + (S_[1] n).*2 + α_[maxn 3 2] n <=
((S_[3] n.+1).+1).*2 + S_[2] n.+1
279b2781
(*    Subcase b >= 3 *)
      rewrite (maxn_idPr _) //.
2994
S_[b.+2] n.+2 + (S_[1] n).*2 + α_[b] n <=
((S_[3] n.+1).+1).*2 + S_[b] n.+1
2995
      by lia.
2997
2998
279a
(*  Subcase b = 2*)
    rewrite (maxn_idPl _) //.
2997
S_[4] n.+2 + (S_[1] n).*2 + α_[3] n <=
((S_[3] n.+1).+1).*2 + S_[2] n.+1
279a
    suff : 2 * S_[1] n + 2 * α_[1] n.+1 <= S_[1] n.+1 + S_[2] n.+1 + 1
      by lia.
2997
2 * S_[1] n + 2 * α_[1] n.+1 <=
S_[1] n.+1 + S_[2] n.+1 + 1
279a
    have [n_gt3 |] := leqP 4 n; last first.
2997
n < 4 ->
2 * S_[1] n + 2 * α_[1] n.+1 <=
S_[1] n.+1 + S_[2] n.+1 + 1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b57279227bb27bc27c298f27e11cc5284028412847284828572858286328a628b428cb29152933293f2947294f2954f172960296b2975297d298228ac2987298c
n_gt3:3 < n
29a7
279b2781
      by case: n => [|[|[|[|n1]]]]; rewrite // !(alphaS_small, alpha_small).
29ae
29a7
279a
    by have F := alpha_4_3 (n_gt3); lia.
279c
573
2780
  move: b_lt2; rewrite leq_eqVlt => /orP[/eqP [] bE1 | b_lt1].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b23
573
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792
b_lt1:b.+1 < 2
573
2781
(*  subcase b = 1 *)
    pose u1l := 
      [ffun i =>  
        if (i : 'I_5) == 0 :> nat then ↓[u ord0] else
        if i == 1 :> nat then sam1 else
        if i == 2 :> nat then tam1 else 
        if i == 3 :> nat then ↓[u (inord 1)] else ↓[u (inord 2)]].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b23
u1l:=[ffun i => if (i : 'I_5) == 0 :> nat
           then ↓[u ord0]
           else
            if i == 1 :> nat
            then sam1
            else
             if i == 2 :> nat
             then tam1
             else
              if i == 3 :> nat
              then ↓[u (inord 1)]
              else ↓[u (inord 2)]]:{ffun forall x : ordinal_finType 5,
      (fun=> configuration 4 n.+1) x}
573
29ba
    have a1E2 : a1 = 2.
29c1
a1 = 2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b2329c2
a1E2:a1 = 2
573
29bb2781
      by move: (cE1); rewrite /c l1E aE1 bE1 cE1; case: (a1) => // [] [|[|]].
29c9
573
29ba
    have P1l1 : a.*2 + sd u1l <=
      \sum_(i < a1) `d[u (inord i), u (inord i.+1)]_smove.
29c9
a.*2 + sd (n:=n.+1) u1l <=
\sum_(i < a1) `d[u (inord i), u (inord i.+1)]_smove
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b2329c229ca
P1l1:a.*2 + sd (n:=n.+1) u1l <=
\sum_(i < a1)
   `d[u (inord i), u (inord i.+1)]_smove
573
29bb2781
      rewrite /sd a1E2 !big_ord_recr /= !big_ord0 !add0n {1}aE1.
29c9
1.*2 +
(`d[u1l (inord 0), u1l (inord 1)]_smove +
 `d[u1l (inord 1), u1l (inord 2)]_smove +
 `d[u1l (inord 2), u1l (inord 3)]_smove +
 `d[u1l (inord 3), u1l (inord 4)]_smove) <=
`d[u (inord 0), u (inord 1)]_smove +
`d[u (inord 1), u (inord 2)]_smove
29d2
      rewrite !ffunE /= !inordK ?aE1 //= add2n -2!addSn leq_add //; last first.
29c9
`d[↓[u (inord 1)], ↓[u (inord 2)]]_smove <=
`d[u (inord 1), u (inord 2)]_smove
29c9
(`d[↓[u ord0], sam1]_smove + `d[sam1, tam1]_smove +
 `d[tam1, ↓[u (inord 1)]]_smove).+1 <
`d[u (inord 0), u (inord 1)]_smove
29d329bb2781
        by apply/gdist_cunlift/shanoi_connect.
29c9
29e0
29d2
      rewrite (inord_eq0 _ (_ : 0 = 0)) //.
29c9
(`d[↓[u ord0], sam1]_smove + `d[sam1, tam1]_smove +
 `d[tam1, ↓[u (inord 1)]]_smove).+1 <
`d[u ord0, u (inord 1)]_smove
29d2
      rewrite (_ : ord0 = inord a.-1); last first.
29c9
ord0 = inord a.-1
29c9
(`d[↓[u (inord a.-1)], sam1]_smove +
 `d[sam1, tam1]_smove + `d[tam1, ↓[
 u (inord 1)]]_smove).+1 <
`d[u (inord a.-1), u (inord 1)]_smove
29d329bb2781
        by apply/val_eqP; rewrite /= inordK aE1.
29c9
(`d[↓[u (inord a.-1)], sam1]_smove +
 `d[sam1, tam1]_smove + `d[tam1, ↓[u (inord 1)]]_smove).+1 <
`d[u (inord a.-1), u (inord 1)]_smove
29d2
      by rewrite -[in inord 1]aE1 duam1ua1E /= addnA addn0.
29d4
573
29ba
    have cH1 (k : 'I_5) : 
      0 < k < 4 -> codom (u1l k) \subset [:: p3; a[p1, p2] k].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b2329c229ca29d51192
0 < k < 4 ->
codom (u1l k) \subset [:: p3;  a[p1, p2] k]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b2329c229ca29d5
cH1:forall k : 'I_5,
0 < k < 4 ->
codom (u1l k) \subset [:: p3;  a[p1, p2] k]
573
29bb2781
      case: k => [] [|[|[|[|]]]] //= iH _; rewrite !ffunE !(apegS, apeg0).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b2329c229ca29d5119c
codom
  (if Ordinal (n:=5) (m:=1) iH == 0
   then ↓[u ord0]
   else
    if Ordinal (n:=5) (m:=1) iH == 1
    then sam1
    else
     if Ordinal (n:=5) (m:=1) iH == 2
     then tam1
     else
      if Ordinal (n:=5) (m:=1) iH == 3
      then ↓[u (inord 1)]
      else ↓[u (inord 2)]) \subset [:: p3; p2]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b2329c229ca29d511a1
codom
  (if Ordinal (n:=5) (m:=2) iH == 0
   then ↓[u ord0]
   else
    if Ordinal (n:=5) (m:=2) iH == 1
    then sam1
    else
     if Ordinal (n:=5) (m:=2) iH == 2
     then tam1
     else
      if Ordinal (n:=5) (m:=2) iH == 3
      then ↓[u (inord 1)]
      else ↓[u (inord 2)]) \subset [::
 p3; p1]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b2329c229ca29d511a5
codom
  (if Ordinal (n:=5) (m:=3) iH == 0
   then ↓[u ord0]
   else
    if Ordinal (n:=5) (m:=3) iH == 1
    then sam1
    else
     if Ordinal (n:=5) (m:=3) iH == 2
     then tam1
     else
      if Ordinal (n:=5) (m:=3) iH == 3
      then ↓[u (inord 1)]
      else ↓[u (inord 2)]) \subset [::
 p3; p2]
29fc29bb2781
      -
2a00 by have := sam1C; rewrite /pa aE1.
2a06
codom
  (if Ordinal (n:=5) (m:=2) iH == 0
   then ↓[u ord0]
   else
    if Ordinal (n:=5) (m:=2) iH == 1
    then sam1
    else
     if Ordinal (n:=5) (m:=2) iH == 2
     then tam1
     else
      if Ordinal (n:=5) (m:=2) iH == 3
      then ↓[u (inord 1)]
      else ↓[u (inord 2)]) \subset [:: p3; p1]
2a0829fc29bb2781
      -
2a0d by rewrite codom_subC.
2a09
codom
  (if Ordinal (n:=5) (m:=3) iH == 0
   then ↓[u ord0]
   else
    if Ordinal (n:=5) (m:=3) iH == 1
    then sam1
    else
     if Ordinal (n:=5) (m:=3) iH == 2
     then tam1
     else
      if Ordinal (n:=5) (m:=3) iH == 3
      then ↓[u (inord 1)]
      else ↓[u (inord 2)]) \subset [:: p3; p2]
29fb
      apply: codom_liftr; rewrite codom_subC.
2a09
codom (u (inord 1)) \subset [:: p2; p3]
29fb
      have /cH: 0 < (inord 1 : 'I_l.+2) < l.+1 by rewrite inordK.
2a09
codom (u (inord 1)) \subset
[:: p2;  a[p1, p3] (inord 1)] ->
codom (u (inord 1)) \subset [:: p2; p3]
29fb
      by rewrite inordK.
29fd
573
29ba
    have lE2 : l = 2 by move: l1E; rewrite aE1 bE1 cE1 => [] [].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b2329c229ca29d529fe
lE2:l = 2
573
29ba
    have {cH1}P4 :
      (S_[4] n.+1).*2 <= sd u1l + sp ↓[u ord0] 4 p1 +
                                  sp ↓[u (inord a1)] 4 p1.
2a24
(S_[4] n.+1).*2 <=
sd (n:=n.+1) u1l + sp (n:=n.+1) ↓[u ord0] 4 p1 +
sp (n:=n.+1) ↓[u (inord a1)] 4 p1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b2329c229ca29d52a25
P4:(S_[4] n.+1).*2 <=
sd (n:=n.+1) u1l + sp (n:=n.+1) ↓[u ord0] 4 p1 +
sp (n:=n.+1) ↓[u (inord a1)] 4 p1
573
29bb2781
      apply: leq_trans (IH _ _ _ _ _ _ cH1) _ => //.
2a24
1008
2a24
sd (n:=n.+1) u1l + sp (n:=n.+1) (u1l ord0) 4 p1 +
sp (n:=n.+1) (u1l ord_max) 4 ( a[p1, p2] 4) <=
sd (n:=n.+1) u1l + sp (n:=n.+1) ↓[u ord0] 4 p1 +
sp (n:=n.+1) ↓[u (inord a1)] 4 p1
2a2b29bb2781
        by rewrite eq_sym.
2a24
2a33
2a2a
      repeat apply: leq_add => //; apply: leq_sum  => i /= _; rewrite !ffunE //.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b2329c229ca29d529fe2a251054
(u (inord 2) (trshift 1 i) !=  a[p1, p2] 4) *
β_[n.+1, 4] i <=
(u (inord a1) (trshift 1 i) != p1) * β_[n.+1, 4] i
2a2a
      by rewrite a1El -[in inord 2]lE2.
2a2c
573
29ba
    have {}P4 := leq_trans P4 (leq_add (leq_add (leqnn _) (leq_sum_beta _ _))
                                    (leqnn _)).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b2329c229ca29d52a25
P4:(S_[4] n.+1).*2 <=
sd (n:=n.+1) u1l +
\sum_(k < n.+1)
   (↓[u ord0] k != p1) * (α_[1] k).*2 +
sp (n:=n.+1) ↓[u (inord a1)] 4 p1
573
29ba
    set x1S := sd _ in P4.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b2329c229ca29d52a25
x1S:=sd (n:=n.+1) u1l:nat
P4:(S_[4] n.+1).*2 <=
x1S +
\sum_(k < n.+1)
   (↓[u ord0] k != p1) * (α_[1] k).*2 +
sp (n:=n.+1) ↓[u (inord a1)] 4 p1
573
29ba
    set y1S := \sum_(_ < _) _ in P4; set y2S := sp _ _ _ in P4.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b2329c229ca29d52a252a48
y1S:=\sum_(H < n.+1)
   (↓[u ord0] H != p1) * (α_[1] H).*2:nat
y2S:=sp (n:=n.+1) ↓[u (inord a1)] 4 p1:nat
P4:(S_[4] n.+1).*2 <= x1S + y1S + y2S
573
29ba
    have cH2 (k : 'I_(3 * c).+1) : 
      0 < k < 3 * c -> 
      codom (u2 k) \subset [:: a[p1, p3] a1; a[p2, a[p1, p3] a1.+1] k].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b2329c229ca29d52a252a482a4e2a4f2a5028c6
28c7
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b2329c229ca29d52a252a482a4e2a4f2a5028cb
573
29bb2781
      move=> /andP[k_gt0 k_ltb].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b2329c229ca29d52a252a482a4e2a4f2a5028c685228d0
28d1
2a55
      rewrite {2}cE1 muln1 in k_ltb.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b2329c229ca29d52a252a482a4e2a4f2a5028c6852286e
28d1
2a55
      rewrite ffunE !modn_small // !divn_small // addn0.
2a5f
28d9
2a55
      case: (k : nat) k_gt0 k_ltb => //= [] [|[|]] //= _ _.
2a54
28dd
2a54
28e0
2a5629bb2781
        by rewrite a1E2.
2a54
28e0
2a55
      by rewrite codom_subC.
2a57
573
29ba
    have P5 : 
      (S_[3 * c] n.+1).*2 <= sd u2 + sp ↓[u (inord a1)] 3 p2 +
                                     sp ↓[u ord_max] 3 (a[p3, p1] l).
2a57
(S_[3 * c] n.+1).*2 <=
sd (n:=n.+1) u2 + sp (n:=n.+1) ↓[u (inord a1)] 3 p2 +
sp (n:=n.+1) ↓[u ord_max] 3 ( a[p3, p1] l)
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b2329c229ca29d52a252a482a4e2a4f2a5028cb
P5:(S_[3 * c] n.+1).*2 <=
sd (n:=n.+1) u2 +
sp (n:=n.+1) ↓[u (inord a1)] 3 p2 +
sp (n:=n.+1) ↓[u ord_max] 3 ( a[p3, p1] l)
573
29bb2781
      apply: leq_trans (IH _ _ _ _ _ _ cH2) (leq_add (leq_add _ _) _);
          rewrite // 1?eq_sym //.
2a57
1a1f
2a57
28f4
2a57
28f6
2a7329bb2781
      -
2a77 by rewrite a1E2 eq_sym.
2a57
28f4
2a7b2a7329bb2781
      -
2a7e by apply: leq_sum => i _; rewrite ffunE /= addn0 cE1 muln1.
2a57
2900
2a72
      apply: leq_sum => i _; rewrite ffunE /= mod3E /= div3E // cE1 muln1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b2329c229ca29d52a252a482a4e2a4f2a5028cb1054
2905
2a72
      rewrite a1El addn1 (_ : inord _ = ord_max) ?apegS //.
2a88
a16
2a72
      by apply/val_eqP; rewrite /= inordK.
2a74
573
29ba
    have {}P5 := leq_trans P5 (leq_add (leqnn _) (leq_sum_beta _ _)).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b2329c229ca29d52a252a482a4e2a4f2a5028cb
P5:(S_[3 * c] n.+1).*2 <=
sd (n:=n.+1) u2 +
sp (n:=n.+1) ↓[u (inord a1)] 3 p2 +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) *
   (α_[1] k).*2
573
29ba
    rewrite // {1}[in 3 *c]cE1 muln1 in P5.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b2329c229ca29d52a252a482a4e2a4f2a5028cb
P5:(S_[3] n.+1).*2 <=
sd (n:=n.+1) u2 +
sp (n:=n.+1) ↓[u (inord a1)] 3 p2 +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) *
   (α_[1] k).*2
573
29ba
    set x2S := sd _ in P5.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b2329c229ca29d52a252a482a4e2a4f2a5028cb27bb
P5:(S_[3] n.+1).*2 <=
x2S + sp (n:=n.+1) ↓[u (inord a1)] 3 p2 +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) *
   (α_[1] k).*2
573
29ba
    set y3S := sp _ _ _ in P5; set y4S := \sum_(_ < _) _ in P5.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b572792b2329c229ca29d52a252a482a4e2a4f2a5028cb27bb
y3S:=sp (n:=n.+1) ↓[u (inord a1)] 3 p2:nat
y4S:=\sum_(H < n.+1)
   (↓[u ord_max] H !=  a[p3, p1] l) *
   (α_[1] H).*2:nat
P5:(S_[3] n.+1).*2 <= x2S + y3S + y4S
573
29ba
    rewrite /sd -(big_mkord xpredT 
                   (fun i => `d[u (inord i), u (inord i.+1)]_smove)).
2aa1
(S_[l.+1] n.+2).*2 <=
\sum_(0 <= i < l.+1)
   `d[u (inord i), u (inord i.+1)]_smove +
sp (n:=n.+2) (u ord0) l.+1 p1 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
29ba
    rewrite (big_cat_nat _ _ _ (_ : 0 <= a1)) //=; last by apply: ltnW.
2aa1
(S_[l.+1] n.+2).*2 <=
\sum_(0 <= i < a1)
   `d[u (inord i), u (inord i.+1)]_smove +
\sum_(a1 <= i < l.+1)
   `d[u (inord i), u (inord i.+1)]_smove +
sp (n:=n.+2) (u ord0) l.+1 p1 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)
29ba
    rewrite big_mkord -!addnA.
2aa1
(S_[l.+1] n.+2).*2 <=
\sum_(i < a1) `d[u (inord i), u (inord i.+1)]_smove +
(\sum_(a1 <= i < l.+1)
    `d[u (inord i), u (inord i.+1)]_smove +
 (sp (n:=n.+2) (u ord0) l.+1 p1 +
  sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)))
29ba
    apply: leq_trans (leq_add P1l1  (leqnn _)); rewrite -/x1S.
2aa1
(S_[l.+1] n.+2).*2 <=
a.*2 + x1S +
(\sum_(a1 <= i < l.+1)
    `d[u (inord i), u (inord i.+1)]_smove +
 (sp (n:=n.+2) (u ord0) l.+1 p1 +
  sp (n:=n.+2) (u ord_max) l.+1 ( a[p3, p1] l)))
29ba
    rewrite -{P1l1}P2 -/x2S {1}lE2 !sum_beta_S //.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421b2d1b572792b2329c229ca2a252a482a4e2a4f2a5028cb27bb2aa22aa32aa4
(S_[3] n.+2).*2 <=
a.*2 + x1S +
(c.*2 + x2S +
 (\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
  (u ord0 ord_max != p1) * α_[l.+1] n.+1 +
  (\sum_(k < n.+1)
      (↓[u ord_max] k !=  a[p3, p1] l) * (α_[1] k).*2 +
   (u ord_max ord_max !=  a[p3, p1] l) * α_[l.+1] n.+1)))
29ba
    rewrite KH1 KH2 !eqxx !addn0 aE1 -/y1S -/y4S.
2ab8
(S_[3] n.+2).*2 <=
1.*2 + x1S + (c.*2 + x2S + (y1S + y4S))
29ba
    have y2Sy3SE : y2S + y3S <= (S_[1] n).*2 + α_[4] n.
2ab8
y2S + y3S <= (S_[1] n).*2 + α_[4] n
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421b2d1b572792b2329c229ca2a252a482a4e2a4f2a5028cb27bb2aa22aa32aa4
y2Sy3SE:y2S + y3S <= (S_[1] n).*2 + α_[4] n
2abd
29bb2781
      apply: sum_alpha_diffE => //.
2ab8
codom ↓[u (inord a1)] \subset [:: p1; p2]
2ac2
      apply: codom_liftr; rewrite codom_subC.
2ab8
codom (u (inord a1)) \subset [:: p2; p1]
2ac2
      have /cH : 0 < (inord a1 : 'I_l.+2) < l.+1 by rewrite inordK // a1E2 lE2.
2ab8
codom (u (inord a1)) \subset
[:: p2;  a[p1, p3] (inord a1)] ->
codom (u (inord a1)) \subset [:: p2; p1]
2ac2
      by rewrite inordK // a1E2.
2ac4
2abd
29ba
    rewrite {1}aE1 -[1.*2]/2 in P1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b2d1b572792b2329c229ca2a252a482a4e2a4f2a5028cb27bb2aa22aa32aa42ac5
P1:2 + sd (n:=n.+1) u1 =
\sum_(i < a)
   `d[u (inord i), u (inord i.+1)]_smove
2abd
29ba
    rewrite {1}cE1 -[1.*2]/2.
2ad8
(S_[3] n.+2).*2 <= 2 + x1S + (2 + x2S + (y1S + y4S))
29ba
    have F1n4 := dsum_alphaL_S 4 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b2d1b572792b2329c229ca2a252a482a4e2a4f2a5028cb27bb2aa22aa32aa42ac52ad9
F1n4:S_[4] n.+1 = S_[4] n + α_[4] n
2add
29ba
    have F1n3 : (S_[3] n.+2).+1 = S_[1] n.+3.
2ae1
(S_[3] n.+2).+1 = S_[1] n.+3
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b2d1b572792b2329c229ca2a252a482a4e2a4f2a5028cb27bb2aa22aa32aa42ac52ad92ae2
F1n3:(S_[3] n.+2).+1 = S_[1] n.+3
2add
29bb2781
      by rewrite -dsum_alpha3_S.
2ae9
2add
29ba
    have F1n13 : (S_[3] n.+1).+1 = S_[1] n.+2.
2ae9
2943
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b2d1b572792b2329c229ca2a252a482a4e2a4f2a5028cb27bb2aa22aa32aa42ac52ad92ae22aea
F1n13:(S_[3] n.+1).+1 = S_[1] n.+2
2add
29bb2781
      by rewrite -dsum_alpha3_S.
2af3
2add
29ba
    suff : (S_[1] n.+3).*2 + (S_[1] n).*2 <=
             4 + S_[4] n + S_[4] n.+1 + (S_[1] n.+2).*2.
2af3
(S_[1] n.+3).*2 + (S_[1] n).*2 <=
4 + S_[4] n + S_[4] n.+1 + (S_[1] n.+2).*2 ->
(S_[3] n.+2).*2 <= 2 + x1S + (2 + x2S + (y1S + y4S))
2af3
(S_[1] n.+3).*2 + (S_[1] n).*2 <=
4 + S_[4] n + S_[4] n.+1 + (S_[1] n.+2).*2
29bb2781
      by lia.
2af3
2afe
29ba
    case : n => [|[|[|[|n1]]]]; try by rewrite !alphaS_small.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b2d1b572792b2329c229ca2a252a482a4e2a4f2a5028cb27bb2aa22aa32aa42ac52ad92ae22aea2af4273c
(S_[1] n1.+3.+4).*2 + (S_[1] n1.+4).*2 <=
4 + S_[4] n1.+4 + S_[4] n1.+1.+4 + (S_[1] n1.+2.+4).*2
29ba
    set m1 := n1.+3.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b2d1b572792b2329c229ca2a252a482a4e2a4f2a5028cb27bb2aa22aa32aa42ac52ad92ae22aea2af4273c
m1:=n1.+3:nat
(S_[1] m1.+4).*2 + (S_[1] m1.+1).*2 <=
4 + S_[4] m1.+1 + S_[4] m1.+2 + (S_[1] m1.+3).*2
29ba
    have F := leq_S4 m1.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b2d1b572792b2329c229ca2a252a482a4e2a4f2a5028cb27bb2aa22aa32aa42ac52ad92ae22aea2af4273c2b0b
F:5 * S_[1] m1.+2 + S_[1] m1.+3 <=
6 * S_[4] m1.+1 + 9
2b0c
29ba
    have F1 := leq_S4 m1.+2.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b2d1b572792b2329c229ca2a252a482a4e2a4f2a5028cb27bb2aa22aa32aa42ac52ad92ae22aea2af4273c2b0b2b11
F1:5 * S_[1] m1.+3 + S_[1] m1.+4 <=
6 * S_[4] m1.+2 + 9
2b0c
29ba
    have F3n3 := dsum_alphaL_S 1 m1.+3.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b2d1b572792b2329c229ca2a252a482a4e2a4f2a5028cb27bb2aa22aa32aa42ac52ad92ae22aea2af4273c2b0b2b112b16
F3n3:S_[1] m1.+4 = S_[1] m1.+3 + α_[1] m1.+3
2b0c
29ba
    have F3n2 := dsum_alphaL_S 1 m1.+2.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b2d1b572792b2329c229ca2a252a482a4e2a4f2a5028cb27bb2aa22aa32aa42ac52ad92ae22aea2af4273c2b0b2b112b162b1b
F3n2:S_[1] m1.+3 = S_[1] m1.+2 + α_[1] m1.+2
2b0c
29ba
    have F3n1 := dsum_alphaL_S 1 m1.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b2d1b572792b2329c229ca2a252a482a4e2a4f2a5028cb27bb2aa22aa32aa42ac52ad92ae22aea2af4273c2b0b2b112b162b1b2b20
F3n1:S_[1] m1.+2 = S_[1] m1.+1 + α_[1] m1.+1
2b0c
29ba
    have F4m1 : 12 * α_[1] m1.+3 <= 16 * α_[1] m1.+2.
2b24
12 * α_[1] m1.+3 <= 16 * α_[1] m1.+2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b2d1b572792b2329c229ca2a252a482a4e2a4f2a5028cb27bb2aa22aa32aa42ac52ad92ae22aea2af4273c2b0b2b112b162b1b2b202b25
F4m1:12 * α_[1] m1.+3 <= 16 * α_[1] m1.+2
2b0c
29bb2781
      rewrite -[12]/(4 * 3) -mulnA.
2b24
4 * (3 * α_[1] m1.+3) <= (4 * 3).+4 * α_[1] m1.+2
2b2a
      rewrite -[16]/(4 * 4) -mulnA leq_mul2l /=.
2b24
3 * α_[1] m1.+3 <= 4 * α_[1] m1.+2
2b2a
      by rewrite alpha_4_3.
2b2c
2b0c
29ba
    have F4m2 : 9 * α_[1] m1.+2 <= 12 * α_[1] m1.+1.
2b2c
9 * α_[1] m1.+2 <= 12 * α_[1] m1.+1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f1b2d1b572792b2329c229ca2a252a482a4e2a4f2a5028cb27bb2aa22aa32aa42ac52ad92ae22aea2af4273c2b0b2b112b162b1b2b202b252b2d
F4m2:9 * α_[1] m1.+2 <= 12 * α_[1] m1.+1
2b0c
29bb2781
      rewrite -[12]/(3 * 4) -mulnA.
2b2c
9 * α_[1] m1.+2 <= 3 * (4 * α_[1] m1.+1)
2b3d
      rewrite -[9]/(3 * 3) -mulnA leq_mul2l /=.
2b2c
3 * α_[1] m1.+2 <= 4 * α_[1] m1.+1
2b3d
      by rewrite alpha_4_3.
2b3f
2b0c
29ba
    by lia.
29bc
573
2780 
(* subcase b = 1 *)
  have bE0 : b = 0 by case: (b) b_lt1 => //.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b57279229bd123f
573
2780
  have lE1 : l = 1 by have := l1E; rewrite aE1 bE0 cE1 => [] [].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b57279229bd123f
lE1:l = 1
573
2780
  have a1E1 : a1 = 1 by rewrite a1El.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b57279229bd123f2b57
a1E1:a1 = 1
573
2780
  rewrite (_ : a[p3, p1] l = a[p2, p1] l); last by rewrite lE1.
2b5b
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u + sp (n:=n.+2) (u ord0) l.+1 p1 +
sp (n:=n.+2) (u ord_max) l.+1 ( a[p2, p1] l)
2780
  apply: (@case2 _ IH)  => //.
2b5b
p1 !=  a[p1, p3] ?a
2b5b
forall k : 'I_l.+2,
0 < k < l.+1 ->
codom (u k) \subset [::  a[p1, p3] ?a;  a[p1, p2] k]
2b5b
u ord_max ord_max =  a[p2, p1] l
2b5b
forall k : ordinal_finType l.+2,
u k ldisk =  a[p1, p2] k
2781
  -
2b62 by rewrite (_ : a[p1, p3] _ = a[p1, p3] 1).
2b5b
forall k : 'I_l.+2,
0 < k < l.+1 ->
codom (u k) \subset [::  a[p1, p3] 1;  a[p1, p2] k]
2b682b6a2781
  -
2b6e move=> k hK; rewrite {3}lE1 in hK.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b57279229bd123f2b572b5c17b2
hK:0 < k < 2
codom (u k) \subset [::  a[p1, p3] 1;  a[p1, p2] k]
2b71
    have -> : k = inord 1.
2b76
k = inord 1
2b76
codom (u (inord 1)) \subset
[::  a[p1, p3] 1;  a[p1, p2] (inord 1)]
2b682b6a2781
      apply/val_eqP; rewrite /= inordK //.
2b76
k == 1
2b7d
      by case: (k : nat) hK => [] // [|].
2b76
2b7f
2b71
    rewrite inordK //= !(apegS, apeg0) //= codom_subC.
2b76
2a19
2b71
    have := cH (inord 1).
2b76
(0 < inord 1 < l.+1 ->
 codom (u (inord 1)) \subset
 [:: p2;  a[p1, p3] (inord 1)]) ->
codom (u (inord 1)) \subset [:: p2; p3]
2b71
    rewrite inordK // !(apegS, apeg0).
2b76
(0 < 1 < l.+1 ->
 codom (u (inord 1)) \subset [:: p2; p3]) ->
codom (u (inord 1)) \subset [:: p2; p3]
2b71
    by apply.
2b5b
2b69
2b6a2781 
  -
2b93 by rewrite KH2 lE1.
2b5b
2b6b
2780
  move=> k.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e2749277f19421a5d1b2d1b57279229bd123f2b572b5c
k:ordinal_finType l.+2
u k ldisk =  a[p1, p2] k
2780
  have [->|[->|->]] // : k = ord0 \/ k = (inord 1) \/ k = (inord 2).
2b9d
k = ord0 \/ k = inord 1 \/ k = inord 2
2b9d
u (inord 1) ldisk =  a[p1, p2] (inord 1)
2b9d
u (inord 2) ldisk =  a[p1, p2] (inord 2)
2781
  -
2ba1 move: k; rewrite lE1.
2b5b
forall k : ordinal_finType 3,
k = ord0 \/ k = inord 1 \/ k = inord 2
2ba4
    case => [] []; first by left; apply: val_inj => /=.
2b5b
forall (n : nat) (i : n.+1 < 3),
Ordinal (n:=3) (m:=n.+1) i = ord0 \/
Ordinal (n:=3) (m:=n.+1) i = inord 1 \/
Ordinal (n:=3) (m:=n.+1) i = inord 2
2ba4
    case.
2b5b
forall i : 1 < 3,
Ordinal (n:=3) (m:=1) i = ord0 \/
Ordinal (n:=3) (m:=1) i = inord 1 \/
Ordinal (n:=3) (m:=1) i = inord 2
2b5b
forall (n : nat) (i : n.+2 < 3),
Ordinal (n:=3) (m:=n.+2) i = ord0 \/
Ordinal (n:=3) (m:=n.+2) i = inord 1 \/
Ordinal (n:=3) (m:=n.+2) i = inord 2
2ba52ba72781
      by right; left; apply: val_inj; rewrite /= inordK.
2b5b
2bb8
2ba4
    case => //.
2b5b
forall i : 2 < 3,
Ordinal (n:=3) (m:=2) i = ord0 \/
Ordinal (n:=3) (m:=2) i = inord 1 \/
Ordinal (n:=3) (m:=2) i = inord 2
2ba4
    by right; right; apply: val_inj; rewrite /= inordK.
2b9d
2ba6
2ba72781
  by rewrite -[in inord 1]aE1 uaLEp2 aE1 inordK.
2b9d
2ba8
2780
  rewrite {1}(_ : inord 2 = ord_max); last first.
2b9d
11e5
2b9d
u ord_max ldisk =  a[p1, p2] (inord 2)
2781
    by apply/val_eqP; rewrite /= lE1 inordK.
2b9d
2bcc
2780
  by rewrite KH2 lE1 /= inordK.
2782
573
(* This is 4.2.3 a = 1, c >= 2 *)
pose u2r := [ffun i : 'I_(3 * c).-2.+1 => u2 (inord i.+2)].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc0
573
pose pa1 := a[p1, p3] a1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc5
573
pose pa1S := a[p1, p3] a1.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca
573
have cH8 (k : 'I_(3 * c).-2.+1) : 
  0 < k < (3 * c).-2 -> codom (u2r k) \subset [:: p2; a[pa1, pa1S] k].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca1ccf
1cd0
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca
cH8:forall k : 'I_(3 * c).-2.+1,
0 < k < (3 * c).-2 ->
codom (u2r k) \subset [:: p2;  a[pa1, pa1S] k]
573
  move=> /andP[k_gt0 k3L3a].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca1ccf8521cd9
1cda
2be3
  have k2L3c : k.+2 < (3 * c).
2bea
1cde
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca1ccf8521cd91ce2
1cda
2be4
    by have := k3L3a; rewrite -{2}subn2 ltn_subRL add2n.
2bf0
1cda
2be3    
  rewrite !ffunE inordK ?ltnS 1?ltnW //.
2bf0
1ce9
2be3
  case: eqP => [km3E0 /= |/eqP km3D0].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca1ccf8521cd91ce21cee
1cef
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca1ccf8521cd91ce21cf3
1cf4
2be4
    have k3_gt0 : 0 < k.+2 %/ 3.
2bfa
1cf8
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca1ccf8521cd91ce21cee1cfc
1cef
2bfc2be4
      case: (k: nat) k_gt0 km3E0  => // [] //= k1 _ _.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca1ccf1cd91ce21be2
1be3
2c01
      by rewrite (divnMDl 1 k1 (_ : 0 < 3)).
2c03
1cef
2bfb
    have a1k3Ll1 : a1 + k.+2 %/ 3 < l.+1.
2c03
1d07
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca1ccf8521cd91ce21cee1cfc1d0b
1cef
2bfc2be4
      move: k2L3c.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca1ccf8521cd91cee1cfc
1d10
2c0e
      rewrite {1}(divn_eq k.+2 3) km3E0 addn0 mulnC ltn_mul2l /= => k2L3c.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca1ccf8521cd91cee1cfc1d15
1d07
2c0e
      rewrite -(subnK (_ : a1 <= l.+1)) ?(leq_trans a1Ll) //.
2c18
1d19
2c0e
      by rewrite addnC ltn_add2r -/c.
2c10
1cef
2bfb
    rewrite (_ : a[_, _ ] _ = a[p1, p3] (a1 + k.+2)) ; last first.
2c10
1d73
2c10
codom ↓[u (inord (a1 + k.+2 %/ 3))] \subset
[:: p2;  a[p1, p3] (a1 + k.+2)]
2bfc2be4
      by rewrite /pa1S !addnS !apegS -apegD addnC.
2c10
2c24
2bfb
    rewrite (_ : a[_, _ ] _ = a[p1, p3] (a1 + (k.+2) %/ 3)) ; last first.
2c10
1d27
2c10
1d23
2bfc2be4
      rewrite apegO oddD {1}(divn_eq k.+2 3) !oddD oddM andbT -oddD.
2c10
 a[p3, p1]
  (~~ (odd a1 (+) odd (k.+2 %/ 3 + k.+2 %% 3))) =
 a[p1, p3] (a1 + k.+2 %/ 3)
2c2b
      by rewrite km3E0 addn0 -oddD -apegO.
2c10
1d23
2bfb
    have /H : 0 < (inord (a1 + k.+2 %/ 3) : 'I_l.+2) < l.+1.
2c10
1d32
2c10
1d35
2bfc2be4
      rewrite inordK; first by rewrite addn_gt0 a1_gt0.
2c10
1d39
2c37
      by rewrite ltnS ltnW.
2c10
1d35
2bfb
    by rewrite ffunE inordK // ltnS ltnW.
2bfd
1cf4
2be3
  case: eqP => [km3E1|/eqP km3D1].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca1ccf8521cd91ce21cf31d44
1d45
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca1ccf8521cd91ce21cf31d49
1d4a
2be4
    rewrite eqn_leq ltnNge /=.
2c45
1d4e
2c46
    have a1La1k : a1 < a1 + k.+2 %/3.
2c45
1d52
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca1ccf8521cd91ce21cf31d441d56
1d4e
2c472be4
      by rewrite -{1}(addn0 a1) ltn_add2l divn_gt0.
2c51
1d4e
2c46
    have a1k3Ll : a1 + k.+2 %/ 3 < l.+1.
2c51
1d07
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca1ccf8521cd91ce21cf31d441d561d60
1d4e
2c472be4
      move: (ltnW k2L3c).
2c51
1d64
2c58
      rewrite {1}(divn_eq k.+2 3) km3E1 addn1 mulnC ltn_mul2l /=.
2c51
1d68
2c58
      rewrite -(subnK (_ : a1 <= l.+1)) ?(leq_trans a1Ll) //.
2c51
1d6c
2c58
      by rewrite addnC ltn_add2r -/c.
2c5a
1d4e
2c46
    rewrite (_ : a[_, _ ] _ = a[p1, p3] (a1 + k.+2)) ; last first.
2c5a
1d73
2c5a
1d76
2c472be4
      by rewrite /pa1S !addnS !apegS -apegD addnC.
2c5a
1d76
2c46
    rewrite (_ : a[_, _ ] _ = a[p1, p3] (a1 + (k.+2) %/ 3).+1) ; last first.
2c5a
 a[p1, p3] (a1 + k.+2) =
 a[p1, p3] (a1 + k.+2 %/ 3).+1
2c5a
codom (si (a1 + k.+2 %/ 3)) \subset
[:: p2;  a[p1, p3] (a1 + k.+2 %/ 3).+1]
2c472be4
      rewrite apegS apegO {1}(divn_eq k.+2 3) !oddD oddM andbT.
2c5a
 a[p3, p1]
  (~~
   (odd a1 (+) (odd (k.+2 %/ 3) (+) odd (k.+2 %% 3)))) =
 a[p3, p1] (a1 + k.+2 %/ 3)
2c73
      by rewrite km3E1 addbT addbN -oddD -oddS -apegO apegS.
2c5a
2c75
2c46
    case: (sitiHb (a1 + k.+2 %/ 3)) => // [| siH _ _]; first by rewrite a1La1k.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca1ccf8521cd91ce21cf31d441d561d601d8f
2c75
2c46
    by move: siH; rewrite a1Max // ltnS ltnW.
2c48
1d4a
2be3
  have km3E2 : k.+2 %% 3 = 2.
2c48
1d99
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca1ccf8521cd91ce21cf31d491d9d
1d4a
2be4
    by case: (_ %% _) (ltn_mod k.+2 3) km3D0 km3D1 => // [] [|[|]].
2c89
1d4a
2be3
  rewrite ifN; last by case: (k : nat) k_gt0.
2c89
1da4
2be3
  have a1La1k : a1 < a1 + k.+2 %/3.
2c89
1d52
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca1ccf8521cd91ce21cf31d491d9d1d56
1da4
2be4
    by rewrite -{1}(addn0 a1) ltn_add2l divn_gt0.
2c95
1da4
2be3
  have a1k3Ll : a1 + k.+2 %/ 3 < l.+1.
2c95
1d07
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca1ccf8521cd91ce21cf31d491d9d1d561d60
1da4
2be4
    have: k.+2 %/ 3 * 3 < 3 * c.
2c95
1db7
2c95
1dba
2c9d2be4
      rewrite ltnW //.
2c95
1dbe
2ca2
      move: (ltnW k2L3c).
2c95
1dc2
2ca2
      by rewrite {1}(divn_eq k.+2 3) km3E2 addn2.
2c95
1dba
2c9c
    rewrite mulnC ltn_mul2l /=.
2c95
1d68
2c9c
    rewrite -(subnK (_ : a1 <= l.+1)) ?(leq_trans a1Ll) //.
2c95
1d6c
2c9c
    by rewrite addnC ltn_add2r -/c.
2c9e
1da4
2be3
  rewrite (_ : a[_, _ ] _ = a[p1, p3] (a1 + k.+2)) ; last first.
2c9e
1d73
2c9e
1dd4
2be4
    by rewrite !addnS /pa1S !apegS -apegD addnC.
2c9e
1dd4
2be3
  rewrite (_ : a[_, _ ] _ = a[p1, p3] (a1 + (k.+2) %/ 3)) ; last first.
2c9e
1d27
2c9e
1ddd
2be4
    rewrite apegO {1}(divn_eq k.+2 3) !oddD oddM andbT.
2c9e
1de1
2cc1
    by rewrite km3E2 addbF -oddD -apegO.
2c9e
1ddd
2be3
  rewrite codom_subC.
2c9e
1de8
2be3
  case: (sitiHb (a1 + k.+2 %/ 3)) => // [| _ tiH _]; first by rewrite a1La1k.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca1ccf8521cd91ce21cf31d491d9d1d561d601ded
1de8
2be3
  by move: tiH; rewrite a1Max // a1La1k ltnS ltnW.
2be5
573
have c_gt0 : 0 < c by rewrite ltnW.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca2be618d3
573
have P8 :
  (S_[(3 * c).-2] n.+1).*2 <=
  \sum_(i < (3 * c).-2) `d[u2 (inord i.+2), u2 (inord i.+3)]_smove +
  sp ta1 (3 * c).-2 pa1 + sp ↓[u ord_max] (3 * c).-2 (a[p3, p1] l).
2cd6
1df4
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca2be618d3
P8:(S_[(3 * c).-2] n.+1).*2 <=
\sum_(i < (3 * c).-2)
   `d[u2 (inord i.+2), u2 (inord i.+3)]_smove +
sp (n:=n.+1) ta1 (3 * c).-2 pa1 +
sp (n:=n.+1) ↓[u ord_max] 
  (3 * c).-2 ( a[p3, p1] l)
573
  apply: leq_trans (IH _ _ _ _ _ _ cH8) _; rewrite /pa1 /pa1S {cH8}//.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca18d3
1dfc
2ce1
1dff
2ce1
1e01
2cdb
  -
2cdf by rewrite apegS apeg_eqC.
2ce1
1dff
2ce42cdb
  -
2ce7 by rewrite eq_sym.
2ce1
1e0b
2cda
  repeat apply: leq_add; rewrite leq_eqVlt; apply/orP; left; apply/eqP.
2ce1
1e0f
2ce1
1e12
2ce1
1e14
2cdb
  -
2cef apply: eq_bigr => i _.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca18d31e1a
1e1b
2cf1
    by rewrite ![u2r _]ffunE !inordK // ltnS // ltnW.
2ce1
1e12
2cf32cdb
  -
2cfa apply: eq_bigr => i _; congr ((_ != _) * _).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca18d31054
1e25
2cfc
    rewrite !ffunE /= !inordK //=.
2d01
1e29
2cfc
    by rewrite (leq_trans (_ : 2 < (3 * 1).+1)) // ltnS leq_mul2l ltnW.
2ce1
1e2d
2cda
  apply: eq_bigr => i _; congr ((_ != _) * _).
2d01
1e31
2d01
1e34
2cdb
    rewrite !ffunE /= !prednK ?muln_gt0 //=; last first.
2d01
1e38
2d01
1e3b
2d0c2cdb
      by rewrite -subn1 ltn_subRL addn0 (leq_trans (_ : _ < 3 * 1)) //
                 leq_mul2l.
2d01
1e3b
2d0b
    rewrite inordK // mod3E /= div3E addnC subnK; last first.
2d01
1a4f
2d01
1e44
2d0c2cdb
      by apply: (leq_trans a1Ll).
2d01
1e44
2d0b
    by rewrite ffunE; congr (u _ _ ); apply/val_eqP; rewrite /= inordK.
2d01
1e34
2cda
  rewrite /pa1 /pa1S -[RHS]apegS -(subnK (_ : a1 <= l.+1)) // -/c.
2d01
1e4e
2d01
1a4f
2cdb
    rewrite apegS -apegD apegO -subn2 oddD oddB //.
2d01
1e54
2d01
1e57
2d242cdb
      by rewrite oddM /= addbF -oddD -apegO.
2d01
1e57
2d23
    by rewrite (leq_trans (_ : _ < 3 * 1)) // leq_mul2l //.
2d172cda
  by rewrite ltnW.
2cdc
573
rewrite {cH8}//.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca18d32cdd
573
have l_gt0 : 0 < l.
2d35
0 < l
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca18d32cdd1b57
573
  by rewrite -ltnS l1E aE1 add1n addSn ltnS addn_gt0 c_gt0 orbT.
2d3c
573
have c3m2_gt1 : 1 < (3 * c).-2.
2d3c
2315
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca18d32cdd1b57
c3m2_gt1:1 < (3 * c).-2
573
  by rewrite -subn2 ltn_subRL -[2 +1]/(3 * 1) ltn_mul2l.
2d45
573
have {}P8 := leq_trans P8 (leq_add (leq_add (leqnn _) (leqnn _))
                      (leq_sum_beta _ _)).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d46
P8:(S_[(3 * c).-2] n.+1).*2 <=
\sum_(i < (3 * c).-2)
   `d[u2 (inord i.+2), u2 (inord i.+3)]_smove +
sp (n:=n.+1) ta1 (3 * c).-2 pa1 +
\sum_(k < n.+1)
   (↓[u ord_max] k !=  a[p3, p1] l) *
   (α_[1] k).*2
573
set y7S := sp _ _ _ in P8; set y8S := \sum_(_ < n.+1) _ in P8.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a194219f81a0c1a111a1b1a271a2f1a3c1a411a491a5d1b2d1b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d46
y7S:=sp (n:=n.+1) ta1 (3 * c).-2 pa1:nat
y8S:=\sum_(H < n.+1)
   (↓[u ord_max] H !=  a[p3, p1] l) *
   (α_[1] H).*2:nat
P8:(S_[(3 * c).-2] n.+1).*2 <=
\sum_(i < (3 * c).-2)
   `d[u2 (inord i.+2), u2 (inord i.+3)]_smove +
y7S + y8S
573
rewrite !sum_beta_S // KH1 KH2 !eqxx !addn0 -/y8S.
2d52
(S_[l.+1] n.+2).*2 <=
sd (n:=n.+2) u +
\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
y8S
rewrite (_ : sd u =
  \sum_(i < a) `d[u (inord i), u (inord i.+1)]_smove +
  \sum_(a <= i < a1) `d[u (inord i), u (inord i.+1)]_smove +
  \sum_(a1 <= i < l.+1) `d[u (inord i), u (inord i.+1)]_smove); last first.
2d52
1fe8
2d52
(S_[l.+1] n.+2).*2 <=
\sum_(i < a) `d[u (inord i), u (inord i.+1)]_smove +
\sum_(a <= i < a1)
   `d[u (inord i), u (inord i.+1)]_smove +
\sum_(a1 <= i < l.+1)
   `d[u (inord i), u (inord i.+1)]_smove +
\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
y8S
  rewrite /sd -[X in _ = X + _ + _](big_mkord xpredT 
                (fun i => `d[u (inord i), u (inord i.+1)]_smove)).
2d52
\sum_(i < l.+1) `d[u (inord i), u (inord i.+1)]_smove =
\sum_(0 <= i < a)
   `d[u (inord i), u (inord i.+1)]_smove +
\sum_(a <= i < a1)
   `d[u (inord i), u (inord i.+1)]_smove +
\sum_(a1 <= i < l.+1)
   `d[u (inord i), u (inord i.+1)]_smove
2d5d
  by rewrite -!big_cat_nat ?big_mkord //= ltnW.
2d52
2d5f
rewrite -{}P1 -{}P2.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b2d1b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d55
(S_[l.+1] n.+2).*2 <=
a.*2 + sd (n:=n.+1) u1 +
\sum_(a <= i < a1)
   `d[u (inord i), u (inord i.+1)]_smove +
(c.*2 + sd (n:=n.+1) u2) +
\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
y8S
move: P3 => P3.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d551b2d
2d6b
rewrite -2!addnA.
2d6f
(S_[l.+1] n.+2).*2 <=
a.*2 + sd (n:=n.+1) u1 +
\sum_(a <= i < a1)
   `d[u (inord i), u (inord i.+1)]_smove +
(c.*2 + sd (n:=n.+1) u2 +
 (\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
  y8S))
apply: leq_trans (leq_add (leq_add (leqnn _) P3) (leqnn _)); rewrite {P3}//.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d55
(S_[l.+1] n.+2).*2 <=
a.*2 + sd (n:=n.+1) u1 +
\sum_(a <= i < a1)
   `d[↓[u (inord i)], ↓[u (inord i.+1)]]_smove +
(c.*2 + sd (n:=n.+1) u2 +
 (\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
  y8S))
have du2E : `d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove =
             \sum_(i < 2) `d[u2 (inord i), u2 (inord i.+1)]_smove.
2d77
`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove =
\sum_(i < 2) `d[u2 (inord i), u2 (inord i.+1)]_smove
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d55
du2E:`d[↓[u (inord a1)], sa1]_smove +
`d[sa1, ta1]_smove =
\sum_(i < 2)
   `d[u2 (inord i), u2 (inord i.+1)]_smove
(S_[l.+1] n.+2).*2 <=
a.*2 + sd (n:=n.+1) u1 +
\sum_(a <= i < a1)
   `d[↓[u (inord i)], ↓[
u (inord i.+1)]]_smove +
(c.*2 + sd (n:=n.+1) u2 +
 (\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
  y8S))
  rewrite !big_ord_recr /= big_ord0 !ffunE !inordK //= ?addn0 //.
2d77
1e29
2d77
2087
2d7e
    by rewrite ltnW // ltnS (leq_mul2l 3 1).
2d77
2087
2d7d
  by rewrite ltnS muln_gt0.
2d7f
2d78
rewrite (_ : sd u2 =
  (`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove) +
  \sum_(i <  (3 * c).-2) `d[u2 (inord i.+2), u2 (inord i.+3)]_smove);
   last first.
2d7f
2064
2d7f
(S_[l.+1] n.+2).*2 <=
a.*2 + sd (n:=n.+1) u1 +
\sum_(a <= i < a1)
   `d[↓[u (inord i)], ↓[
u (inord i.+1)]]_smove +
(c.*2 +
 (`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
  \sum_(i < (3 * c).-2)
     `d[u2 (inord i.+2), 
  u2 (inord i.+3)]_smove) +
 (\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
  y8S))
  rewrite /sd du2E.
2d7f
\sum_(i < 3 * c)
   `d[u2 (inord i), u2 (inord i.+1)]_smove =
\sum_(i < 2) `d[u2 (inord i), u2 (inord i.+1)]_smove +
\sum_(i < (3 * c).-2)
   `d[u2 (inord i.+2), u2 (inord i.+3)]_smove
2d90
  rewrite -!(big_mkord xpredT 
                (fun i => `d[u2 (inord i), u2 (inord i.+1)]_smove)).
2d7f
\sum_(0 <= i < 3 * c)
   `d[u2 (inord i), u2 (inord i.+1)]_smove =
\sum_(0 <= i < 2)
   `d[u2 (inord i), u2 (inord i.+1)]_smove +
\sum_(i < (3 * c).-2)
   `d[u2 (inord i.+2), u2 (inord i.+3)]_smove
2d90
  rewrite -!(big_mkord xpredT 
                (fun i => `d[u2 (inord i.+2), u2 (inord i.+3)]_smove)).
2d7f
\sum_(0 <= i < 3 * c)
   `d[u2 (inord i), u2 (inord i.+1)]_smove =
\sum_(0 <= i < 2)
   `d[u2 (inord i), u2 (inord i.+1)]_smove +
\sum_(0 <= i < (3 * c).-2)
   `d[u2 (inord i.+2), u2 (inord i.+3)]_smove
2d90
  rewrite[X in _ = _ + X] (eq_bigr
                (fun i => `d[u2 (inord (i + 2)), u2 (inord (i + 2).+1)]_smove)); 
    last by move=> i_; rewrite !addn2.
2d7f
\sum_(0 <= i < 3 * c)
   `d[u2 (inord i), u2 (inord i.+1)]_smove =
\sum_(0 <= i < 2)
   `d[u2 (inord i), u2 (inord i.+1)]_smove +
\sum_(0 <= i < (3 * c).-2)
   `d[u2 (inord (i + 2)), u2 (inord (i + 2).+1)]_smove
2d90
  rewrite -subn2.
2d7f
\sum_(0 <= i < 3 * c)
   `d[u2 (inord i), u2 (inord i.+1)]_smove =
\sum_(0 <= i < 2)
   `d[u2 (inord i), u2 (inord i.+1)]_smove +
\sum_(0 <= i < 3 * c - 2)
   `d[u2 (inord (i + 2)), u2 (inord (i + 2).+1)]_smove
2d90
  have <- := big_addn _ _ _ xpredT 
               (fun i => `d[u2 (inord i), u2 (inord i.+1)]_smove).
2d7f
\sum_(0 <= i < 3 * c)
   `d[u2 (inord i), u2 (inord i.+1)]_smove =
\sum_(0 <= i < 2)
   `d[u2 (inord i), u2 (inord i.+1)]_smove +
\sum_(0 + 2 <= i < 3 * c)
   `d[u2 (inord i), u2 (inord i.+1)]_smove
2d90
  rewrite -big_cat_nat //.
2d7f
1e57
2d90
  by rewrite ltnW // (leq_mul2l 3 1).
2d7f
(S_[l.+1] n.+2).*2 <=
a.*2 + sd (n:=n.+1) u1 +
\sum_(a <= i < a1)
   `d[↓[u (inord i)], ↓[u (inord i.+1)]]_smove +
(c.*2 +
 (`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
  \sum_(i < (3 * c).-2)
     `d[u2 (inord i.+2), u2 (inord i.+3)]_smove) +
 (\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
  y8S))
set x3S := \sum_(_ < _) _ in P8; rewrite -/x3S.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d80
x3S:=\sum_(H < (3 * c).-2)
   `d[u2 (inord H.+2), u2 (inord H.+3)]_smove:nat
P8:(S_[(3 * c).-2] n.+1).*2 <= x3S + y7S + y8S
(S_[l.+1] n.+2).*2 <=
a.*2 + sd (n:=n.+1) u1 +
\sum_(a <= i < a1)
   `d[↓[u (inord i)], ↓[u (inord i.+1)]]_smove +
(c.*2 +
 (`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
  x3S) +
 (\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
  y8S))       
have [|b_gt0] := leqP b 0; last first.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17
2db8
2db5
b <= 0 ->
(S_[l.+1] n.+2).*2 <=
a.*2 + sd (n:=n.+1) u1 +
\sum_(a <= i0 < a1)
   `d[↓[u (inord i0)], ↓[
u (inord i0.+1)]]_smove +
(c.*2 +
 (`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
  x3S) +
 (\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
  y8S))
(* subcase 1 <= b *)
  have cH1 (k : 'I_(3 * a).+1) : 
    0 < k < 3 * a -> codom (u1 k) \subset [:: p3; a[p1, p2] k].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17285e
285f
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f172863
(S_[l.+1] n.+2).*2 <=
a.*2 + sd (n:=n.+1) u1 +
\sum_(a <= i < a1)
   `d[↓[u (inord i)], ↓[
u (inord i.+1)]]_smove +
(c.*2 +
 (`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
  x3S) +
 (\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
  y8S))
2dbe
    move=> /andP[k_gt0 k_ltb].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17285e8522868
2869
2dc4
    rewrite {2}aE1 muln1 in k_ltb.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17285e852286e
2869
2dc4
    rewrite ffunE !modn_small // !divn_small // inord_eq0 //.
2dcf
2872
2dc4
    rewrite [in 3 * a.-1]aE1 /=.
2dcf
2876
2dc4
    case: (k : nat) k_gt0 k_ltb => //= [] [|[|]] //= _ _.
2dc3
287a
2dc3
287d
2dc52dbe
      rewrite apegS apeg0.
2dc3
2881
2dd9
      by have := sam1C; rewrite /pa aE1.
2dc3
287d
2dc4
    by rewrite codom_subC.
2dc6
2db8
2dbd
  set x0S := sd u1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc
(S_[l.+1] n.+2).*2 <=
a.*2 + x0S +
\sum_(a <= i < a1)
   `d[↓[u (inord i)], ↓[u (inord i.+1)]]_smove +
(c.*2 +
 (`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
  x3S) +
 (\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
  y8S))
2dbd
  have P4 : 
    (S_[3 * a] n.+1).*2 <= x0S + sp ↓[u ord0] 3 p1 +
                                 sp ↓[u (inord a)] 3 p2.
2de7
288b
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc
P4:(S_[3 * a] n.+1).*2 <=
x0S + sp (n:=n.+1) ↓[u ord0] 3 p1 +
sp (n:=n.+1) ↓[u (inord a)] 3 p2
(S_[l.+1] n.+2).*2 <=
a.*2 + x0S +
\sum_(a <= i < a1)
   `d[↓[u (inord i)], ↓[
u (inord i.+1)]]_smove +
(c.*2 +
 (`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
  x3S) +
 (\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
  y8S))
2dbe
    apply: leq_trans (IH _ _ _ _ _ _ cH1) (leq_add (leq_add _ _) _);
        rewrite // 1?eq_sym //.
2de7
2893
2de7
2896
2ded2dbe
      by apply: leq_sum => i _; rewrite ffunE /= inord_eq0 // aE1 muln1.
2de7
2896
2dec
    by apply: leq_sum => i _; rewrite ffunE /= mod3E /= div3E // aE1 muln1.
2dee
2de8
2dbd
  have {}P4 := leq_trans P4 (leq_add (leq_add (leqnn _) (leq_sum_beta _ _))
                        (leqnn _)).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc
P4:(S_[3 * a] n.+1).*2 <=
x0S +
\sum_(k < n.+1)
   (↓[u ord0] k != p1) * (α_[1] k).*2 +
sp (n:=n.+1) ↓[u (inord a)] 3 p2
2de8
2dbd
  set y0S := \sum_(_ < _) _ in P4; set y1S := sp _ _ _ in P4.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a6
P4:(S_[3 * a] n.+1).*2 <= x0S + y0S + y1S
2de8
2dbd
  have [b_gt1|b_le2] := leqP 2 b.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d1
2de8
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e051e77
2df0
2dbe
(*  subcase b >= 2 *)  
    pose u3 := [ffun i => ↓[u (inord ((i : 'I_b.+1) + a))]].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a7
2de8
2e0a
    pose paS := a[p1, p3] a.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f
2de8
2e0a
    have cH6 (k : 'I_b.+1) :
        0 < k < b -> codom (u3 k) \subset [:: p2; a[pa, paS] k].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f9c1
9c2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f
cH6:forall k : 'I_b.+1,
0 < k < b ->
codom (u3 k) \subset [:: p2;  a[pa, paS] k]
2df0
2e0b2dbe
      move=> /andP[k_gt0 kLb].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f9c1852
kLb:k < b
9d0
2e19
      have kBound : 0 < k + a < l.+1.
2e20
9d4
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f9c18522e219d8
9d0
2e1a2e0b2dbe
        rewrite (leq_trans a_gt0) ?leq_addl //=.
2e20
9dc
2e25
        move: kLb; rewrite /b ltn_subRL addnC => /leq_trans-> //.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f9c1852
1a4f
2e25
        by rewrite ltnW.
2e27
9d0
2e19
      rewrite ffunE codom_liftr //.
2e27
9e3
2e19
      have := @cH (inord (k + a)).
2e27
9e7
2e19
      rewrite /pa /paS apegS -apegD /= !inordK //; last first.
2e27
9eb
2e27
(0 < k + a < l.+1 ->
 codom (u (inord (k + a))) \subset
 [:: p2;  a[p1, p3] (k + a)]) ->
codom (u (inord (k + a))) \subset
[:: p2;  a[p1, p3] (k + a)]
2e1a2e0b2dbe
        by rewrite ltnW // ltnS; case/andP: kBound.
2e27
2e3d
2e19
      by apply.
2e1b
2de8
2e0a
    have {cH6}P6 : 
    (S_[b] n.+1).*2 <= sd u3 + sp (u3 ord0) b pa + 
                               sp (u3 ord_max) b (a[pa, paS] b).
2e1b
(S_[b] n.+1).*2 <=
sd (n:=n.+1) u3 + sp (n:=n.+1) (u3 ord0) b pa +
sp (n:=n.+1) (u3 ord_max) b ( a[pa, paS] b)
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f
P6:(S_[b] n.+1).*2 <=
sd (n:=n.+1) u3 + sp (n:=n.+1) (u3 ord0) b pa +
sp (n:=n.+1) (u3 ord_max) b ( a[pa, paS] b)
2df0
2e0b2dbe
      apply: IH cH6 => //; try by rewrite /pa /paS // eq_sym.
2e14
9fa
2e48
      by rewrite /paS apegS apeg_eqC.
2e4a
2de8
2e0a
    set x1S := \sum_(_ <= _ < _) _.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f2e4b
x1S:=\sum_(a <= H < a1)
   `d[↓[u (inord H)], ↓[
u (inord H.+1)]]_smove:nat
(S_[l.+1] n.+2).*2 <=
a.*2 + x0S + x1S +
(c.*2 +
 (`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
  x3S) +
 (\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
  y8S))
2e0a
    rewrite !ffunE (_ : sd u3  = x1S) in P6; last first.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f2e56
sd (n:=n.+1) u3 = x1S
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f2e56
P6:(S_[b] n.+1).*2 <=
x1S + sp (n:=n.+1) ↓[u (inord (ord0 + a))] b pa +
sp (n:=n.+1) ↓[u (inord (ord_max + a))] b
  ( a[pa, paS] b)
2e57
2e0b2dbe
      rewrite /x1S -[a]add0n big_addn -/b big_mkord.
2e5b
sd (n:=n.+1) u3 =
\sum_(i < b)
   `d[↓[u (inord (i + a))], ↓[u (inord (i + a).+1)]]_smove
2e5d
      by apply: eq_bigr => i _; rewrite !ffunE !inordK ?ltnS // ltnW.
2e5f
2e57
2e0a
    rewrite add0n subnK // in P6.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f2e56
P6:(S_[b] n.+1).*2 <=
x1S + sp (n:=n.+1) ↓[u (inord a)] b pa +
sp (n:=n.+1) ↓[u (inord a1)] b ( a[pa, paS] b)
2e57
2e0a
    set y2S := sp _ _ _ in P6; set y3S := sp _ _ _ in P6.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f2e562841
y3S:=sp (n:=n.+1) ↓[u (inord a1)] b ( a[pa, paS] b):nat
P6:(S_[b] n.+1).*2 <= x1S + y2S + y3S
2e57
2e0a
    have aLl : a <= l by apply: leq_trans aLa1 _.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f2e5628412e712e725cc
2e57
2e0a
    have y1Sy2SE : y1S + y2S <= (S_[1] n).*2 + α_[maxn 3 b] n.
2e76
28b0
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f2e5628412e712e725cc28b4
2e57
2e0b2dbe
      apply: sum_alpha_diffE => //; first by rewrite /pa eq_sym.
2e76
1f5c
2e7a
      apply/codom_liftr.
2e76
1f60
2e7a
      have /cH : 0 < (inord a : 'I_l.+2) < l.+1 by rewrite inordK // a_gt0.
2e76
codom (u (inord a)) \subset
[:: p2;  a[p1, p3] (inord a)] ->
codom (u (inord a)) \subset [:: p2; pa]
2e7a
      by rewrite inordK // aE1  /=.
2e7c
2e57
2e0a
    pose u5 := [ffun i : 'I_3 =>
                     if i == 0 :> nat then ↓[u (inord a1)]
                     else if i == 1 :> nat then sa1 else ta1].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f2e5628412e712e725cc28b41f71
2e57
2e0a
    have cH7 (k : 'I_3) : 
      0 < k < 2 -> codom (u5 k) \subset [:: p1; a[p2, p3] k].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f2e5628412e712e725cc28b41f71994
1f76
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f2e5628412e712e725cc28b41f71
cH7:forall k : 'I_3,
0 < k < 2 ->
codom (u5 k) \subset [:: p1;  a[p2, p3] k]
2e57
2e0b2dbe
      move=> /andP[k_gt0 k_lt2].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f2e5628412e712e725cc28b41f719948521e8a
1f7f
2e92
      rewrite !ffunE.
2e99
1f83
2e92
      by have ->/= : k = 1 :> nat by case: (k : nat) k_gt0 k_lt2 => // [] [|].
2e94
2e57
2e0a
    have P7 : 
      (S_[2] n.+1).*2 <=
      (`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove) +
         sp ↓[u (inord a1)] 2 p2 + sp ta1 2 p2.
2e94
1f8a
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f2e5628412e712e725cc28b41f712e95
P7:(S_[2] n.+1).*2 <=
`d[↓[u (inord a1)], sa1]_smove +
`d[sa1, ta1]_smove +
sp (n:=n.+1) ↓[u (inord a1)] 2 p2 +
sp (n:=n.+1) ta1 2 p2
2e57
2e0b2dbe
      apply: (leq_trans (IH _ _ _ _ _ _ cH7)) => //.
2e94
1e9e
2e94
1f92
2ea42e0b2dbe
        by rewrite eq_sym.
2e94
1f92
2ea3
      repeat apply: leq_add; rewrite leq_eqVlt; apply/orP; left; apply/eqP.
2e94
1f96
2e94
1f99
2e94
1f9b
2ea42e0b2dbe
      -
2eb0 by rewrite /sd !big_ord_recr big_ord0 !ffunE !inordK /=.
2e94
1f99
2eb42ea42e0b2dbe
      -
2eb7 by apply: eq_bigr => i _; rewrite !ffunE.
2e94
1f9b
2ea3
      by apply: eq_bigr => i _; rewrite !ffunE.
2ea5
2e57
2e0a
    set x2S := (`d[_, _]_ _ + `d[_, _]_ _); rewrite -/x2S in P7.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f2e5628412e712e725cc28b41f712e95
x2S:=`d[↓[u (inord a1)], sa1]_smove +
`d[sa1, ta1]_smove:nat
P7:(S_[2] n.+1).*2 <=
x2S + sp (n:=n.+1) ↓[u (inord a1)] 2 p2 +
sp (n:=n.+1) ta1 2 p2
(S_[l.+1] n.+2).*2 <=
a.*2 + x0S + x1S +
(c.*2 + (x2S + x3S) +
 (\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
  y8S))
2e0a
    set y4S := sp _ _ _ in P7; set y5S := sp _ _ _ in P7.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f2e5628412e712e725cc28b41f712e952ec5
y4S:=sp (n:=n.+1) ↓[u (inord a1)] 2 p2:nat
1fad
P7:(S_[2] n.+1).*2 <= x2S + y4S + y5S
2ec7
2e0a
    have y3Sy4SE : y3S + y4S <= (S_[1] n).*2 + α_[maxn b 2] n.
2ecb
y3S + y4S <= (S_[1] n).*2 + α_[maxn b 2] n
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f2e5628412e712e725cc28b41f712e952ec52ecc1fad2ecd
y3Sy4SE:y3S + y4S <= (S_[1] n).*2 + α_[maxn b 2] n
2ec7
2e0b2dbe
      apply: sum_alpha_diffE => //.
2ecb
 a[pa, paS] b != p2
2ecb
codom ↓[u (inord a1)] \subset [::  a[pa, paS] b; p2]
2ed32e0b2dbe
        by rewrite /paS apegS -apegD apeg_neq // eq_sym.
2ecb
2edc
2ed2
      apply: codom_liftr; rewrite codom_subC.
2ecb
codom (u (inord a1)) \subset [:: p2;  a[pa, paS] b]
2ed2
      have /cH : 0 < (inord a1 : 'I_l.+2) < l.+1.
2ecb
18b2
2ecb
codom (u (inord a1)) \subset
[:: p2;  a[p1, p3] (inord a1)] ->
codom (u (inord a1)) \subset [:: p2;  a[pa, paS] b]
2ed32e0b2dbe
        by rewrite inordK ?a1_gt0 // ltnS (leq_trans a1Ll).
2ecb
2ee9
2ed2
      rewrite inordK ?ltnS ?(leq_trans a1Ll) //.
2ecb
codom (u (inord a1)) \subset [:: p2;  a[p1, p3] a1] ->
codom (u (inord a1)) \subset [:: p2;  a[pa, paS] b]
2ed2
      by rewrite /paS apegS -apegD subnK.
2ed4
2ec7
2e0a
    rewrite (maxn_idPl _) // in y3Sy4SE.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f2e5628412e712e725cc28b41f712e952ec52ecc1fad2ecd
y3Sy4SE:y3S + y4S <= (S_[1] n).*2 + α_[b] n
2ec7
2e0a
    have y5Sy7SE : y5S + y7S <= (S_[1] n).*2 + α_[maxn 2 (3 * c).-2] n.
2ef7
y5S + y7S <= (S_[1] n).*2 + α_[maxn 2 (3 * c).-2] n
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f2e5628412e712e725cc28b41f712e952ec52ecc1fad2ecd2ef8
y5Sy7SE:y5S + y7S <=
(S_[1] n).*2 + α_[maxn 2 (3 * c).-2] n
2ec7
2e0b2dbe
      by apply: sum_alpha_diffE => //; first by rewrite /pa1 eq_sym.
2eff
2ec7
2e0a
    rewrite (maxn_idPr _) // in y5Sy7SE.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e055d16a798f2e5628412e712e725cc28b41f712e952ec52ecc1fad2ecd2ef8
y5Sy7SE:y5S + y7S <= (S_[1] n).*2 + α_[(3 * c).-2] n
2ec7
2e0a
    rewrite -/y0S.
2f07
(S_[l.+1] n.+2).*2 <=
a.*2 + x0S + x1S + (c.*2 + (x2S + x3S) + (y0S + y8S))
2e0a
    rewrite aE1 !muln1 in P4 P6 P7 P8.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f08
P4:(S_[3] n.+1).*2 <= x0S + y0S + y1S
2e722ecd2db7
2f0c
2e0a
    have F1 := dsum_alphaL_S (3 * c).-2 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db7
F1:S_[(3 * c).-2] n.+1 =
S_[(3 * c).-2] n + α_[(3 * c).-2] n
2f0c
2e0a
    have F2 i k : 0 < i -> S_[i.-1] k.+1 <= S_[(3 *i).-2] k + i.-1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f162100
2101
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f16
F2:forall i k : nat,
0 < i -> S_[i.-1] k.+1 <= S_[(3 * i).-2] k + i.-1
2f0c
2e0b2dbe
      move=> i_gt0; apply: leq_trans (dsum_alpha3l _ _) _.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f162100210a
210b
2f1b
      rewrite leq_add2r.
2f22
210f
2f1b
      apply: (increasingE (increasing_dsum_alphaL_l _)).
2f22
2113
2f1b
      by rewrite -{2}[i]prednK // mulnS addSn add2n /=.
2f1d
2f0c
2e0a
    have F2n := F2 c n c_gt0.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f162f1e
F2n:S_[c.-1] n.+1 <= S_[(3 * c).-2] n + c.-1
2f0c
2e0a
    have F2n1 := F2 c n.+1 c_gt0.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f162f1e2f30
F2n1:S_[c.-1] n.+2 <= S_[(3 * c).-2] n.+1 + c.-1
2f0c
2e0a
    have F3 : c = c.-1.+1 by rewrite prednK.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f162f1e2f302f35
F3:c = c.-1.+1
2f0c
2e0a
    have l_gt1 : 1 < l.
2f39
212e
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f162f1e2f302f352f3a4e4
2f0c
2e0b2dbe
      by rewrite -ltnS l1E aE1 add1n addSn ltnS (leq_trans b_gt1) // leq_addr.
2f40
2f0c
2e0a
    have F4 := SH l_gt1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f162f1e2f302f352f3a4e4
F4:(S_[l.+1] n.+2).*2 <=
S_[l.+1] n.+2 + S_[l] n.+1 + (α_[1] n.+1).*2
2f0c
2e0a
    have cB1_gt0 : 0 < c.-1 by rewrite -ltnS prednK.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f162f1e2f302f352f3a4e42f48
cB1_gt0:0 < c.-1
2f0c
2e0a
    have F5 : S_[l.+1] n.+2 + S_[1] n.+2 <= S_[c.-1] n.+2 + S_[b.+3] n.+2.
2f4c
S_[l.+1] n.+2 + S_[1] n.+2 <=
S_[c.-1] n.+2 + S_[b.+3] n.+2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f162f1e2f302f352f3a4e42f482f4d
F5:S_[l.+1] n.+2 + S_[1] n.+2 <=
S_[c.-1] n.+2 + S_[b.+3] n.+2
2f0c
2e0b2dbe
      rewrite [X in _ <= X]addnC.
2f4c
S_[l.+1] n.+2 + S_[1] n.+2 <=
S_[b.+3] n.+2 + S_[c.-1] n.+2
2f52
      rewrite (_ : l.+1 =  1 + c.-2 + b.+2); last first.
2f4c
l.+1 = 1 + c.-2 + b.+2
2f4c
S_[1 + c.-2 + b.+2] n.+2 + S_[1] n.+2 <=
S_[b.+3] n.+2 + S_[c.-1] n.+2
2f532e0b2dbe
        by rewrite l1E aE1 !add1n -addSnnS !prednK // addnC.
2f4c
2f60
2f52
      rewrite (_ : c.-1 = 1 + c.-2); last first.
2f4c
21a0
2f4c
S_[1 + (1 + c.-2).-1 + b.+2] n.+2 + S_[1] n.+2 <=
S_[b.+3] n.+2 + S_[1 + c.-2] n.+2
2f532e0b2dbe
         by rewrite add1n prednK // -ltnS prednK.
2f4c
2f69
2f52
      by apply: concaveEk1 (concave_dsum_alphaL_l n.+2).
2f54
2f0c
2e0a
    have F6 : S_[l] n.+1 + S_[1] n.+1 <= S_[c.-1] n.+1 + S_[b.+2] n.+1.
2f54
S_[l] n.+1 + S_[1] n.+1 <=
S_[c.-1] n.+1 + S_[b.+2] n.+1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f162f1e2f302f352f3a4e42f482f4d2f55
F6:S_[l] n.+1 + S_[1] n.+1 <=
S_[c.-1] n.+1 + S_[b.+2] n.+1
2f0c
2e0b2dbe
      rewrite [X in _ <= X]addnC.
2f54
S_[l] n.+1 + S_[1] n.+1 <=
S_[b.+2] n.+1 + S_[c.-1] n.+1
2f74
      rewrite (_ : l =  1 + c.-2 + b.+1); last first.
2f54
l = 1 + c.-2 + b.+1
2f54
S_[1 + c.-2 + b.+1] n.+1 + S_[1] n.+1 <=
S_[b.+2] n.+1 + S_[c.-1] n.+1
2f752e0b2dbe
        rewrite -[l]/(l.+1.-1) l1E aE1.
2f54
(1 + b + c).-1 = 1 + c.-2 + b.+1
2f80
        by rewrite !add1n -addSnnS !prednK // addnC addnS.
2f54
2f82
2f74
      rewrite (_ : c.-1 = 1 + c.-2); last first.
2f54
21a0
2f54
S_[1 + (1 + c.-2).-1 + b.+1] n.+1 + S_[1] n.+1 <=
S_[b.+2] n.+1 + S_[1 + c.-2] n.+1
2f752e0b2dbe
         by rewrite add1n prednK // -ltnS prednK.
2f54
2f8f
2f74
      by apply: concaveEk1 (concave_dsum_alphaL_l n.+1).
2f76
2f0c
2e0a
    suff : 
         6 * S_[1] n + α_[b] n + α_[maxn 3 b] n + 2 * α_[1] n.+1 + 
         S_[b.+2] n.+1 + S_[b.+3] n.+2 <=
         2 * S_[3] n.+1 + 2 * S_[b] n.+1 + 2 * S_[2] n.+1 + 
         S_[1] n.+1 + S_[1] n.+2 + 4
        by lia.
2f76
6 * S_[1] n + α_[b] n + α_[maxn 3 b] n +
2 * α_[1] n.+1 + S_[b.+2] n.+1 + S_[b.+3] n.+2 <=
2 * S_[3] n.+1 + 2 * S_[b] n.+1 + 2 * S_[2] n.+1 +
S_[1] n.+1 + S_[1] n.+2 + 4
2e0a
    rewrite {F1 F2 F3 F4 F5 F6}.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f302f354e42f4d
2f99
2e0a
    have [b_gt2|b_le3] := leqP 3 b.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f302f354e42f4d10c0
6 * S_[1] n + α_[b] n + α_[b] n + 2 * α_[1] n.+1 +
S_[b.+2] n.+1 + S_[b.+3] n.+2 <=
2 * S_[3] n.+1 + 2 * S_[b] n.+1 + 2 * S_[2] n.+1 +
S_[1] n.+1 + S_[1] n.+2 + 4
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f302f354e42f4d
b_le3:b < 3
6 * S_[1] n + α_[b] n + α_[3] n + 2 * α_[1] n.+1 +
S_[b.+2] n.+1 + S_[b.+3] n.+2 <=
2 * S_[3] n.+1 + 2 * S_[b] n.+1 + 2 * S_[2] n.+1 +
S_[1] n.+1 + S_[1] n.+2 + 4
2e0b2dbe
(*    subcase b >= 3 *)
      have F7 := dsum_alphaL_S b n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f302f354e42f4d10c0
F7:S_[b] n.+1 = S_[b] n + α_[b] n
2fa2
2fa3
      have F8 : S_[b.+3] n.+2 <= S_[b.+2] n.+1 + (α_[1] n.+1).*2.
2fab
S_[b.+3] n.+2 <= S_[b.+2] n.+1 + (α_[1] n.+1).*2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f302f354e42f4d10c02fac
F8:S_[b.+3] n.+2 <= S_[b.+2] n.+1 + (α_[1] n.+1).*2
2fa2
2fa42e0b2dbe
        by apply: dsum_alphaL_alpha.
2fb3
2fa2
2fa3
      have F9 : S_[b.+1] n.+1 <= S_[b] n + (α_[1] n).*2.
2fb3
2967
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f302f354e42f4d10c02fac2fb4
F9:S_[b.+1] n.+1 <= S_[b] n + (α_[1] n).*2
2fa2
2fa42e0b2dbe
        by apply: dsum_alphaL_alpha.
2fbd
2fa2
2fa3
      have F10 : S_[1] n.+1 + S_[b.+2] n.+1 <= S_[b.+1] n.+1 + S_[2] n.+1.
2fbd
S_[1] n.+1 + S_[b.+2] n.+1 <=
S_[b.+1] n.+1 + S_[2] n.+1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f302f354e42f4d10c02fac2fb42fbe
F10:S_[1] n.+1 + S_[b.+2] n.+1 <=
S_[b.+1] n.+1 + S_[2] n.+1
2fa2
2fa42e0b2dbe
        rewrite addnC -[b.+2]/(1 + 1 + b).
2fbd
S_[1 + 1 + b] n.+1 + S_[1] n.+1 <=
S_[b.+1] n.+1 + S_[2] n.+1
2fc6
        by apply: concaveEk1 (concave_dsum_alphaL_l n.+1).
2fc8
2fa2
2fa3
      have F11 := dsum_alpha3_S n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f302f354e42f4d10c02fac2fb42fbe2fc9
F11:S_[1] n.+2 = (S_[3] n.+1).+1
2fa2
2fa3
      have F12 := dsum_alphaL_S 1 n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f302f354e42f4d10c02fac2fb42fbe2fc92fd5
F12:S_[1] n.+2 = S_[1] n.+1 + α_[1] n.+1
2fa2
2fa3
      have F13 := dsum_alphaL_S 1 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f302f354e42f4d10c02fac2fb42fbe2fc92fd52fda
F13:S_[1] n.+1 = S_[1] n + α_[1] n
2fa2
2fa3
      have F14 := alphaL_1_2 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f302f354e42f4d10c02fac2fb42fbe2fc92fd52fda2fdf
F14:α_[1] n.+1 <= (α_[1] n).*2
2fa2
2fa3
      by lia.
2fa5
2fa7
2e0a
(*  subcase b = 2 *)
    have bE2 : b = 2.
2fa5
b = 2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f302f354e42f4d2fa6
bE2:b = 2
2fa7
2e0b2dbe
      by case: (b) b_gt1 b_le3 => // [] [|[|]].
2fee
2fa7
2e0a
    rewrite bE2.
2fee
6 * S_[1] n + α_[2] n + α_[3] n + 2 * α_[1] n.+1 +
S_[4] n.+1 + S_[5] n.+2 <=
2 * S_[3] n.+1 + 2 * S_[2] n.+1 + 2 * S_[2] n.+1 +
S_[1] n.+1 + S_[1] n.+2 + 4
2e0a
    have F1 := dsum_alphaL_S 2 n; rewrite {1}F1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f302f354e42f4d2fa62fef
F1:S_[2] n.+1 = S_[2] n + α_[2] n
6 * S_[1] n + α_[2] n + α_[3] n + 2 * α_[1] n.+1 +
S_[4] n.+1 + S_[5] n.+2 <=
2 * S_[3] n.+1 + 2 * (S_[2] n + α_[2] n) +
2 * S_[2] n.+1 + S_[1] n.+1 + S_[1] n.+2 + 4
2e0a
    have F2 := dsum_alphaL_S 3 n; rewrite {1}F2.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f302f354e42f4d2fa62fef2ffb
F2:S_[3] n.+1 = S_[3] n + α_[3] n
6 * S_[1] n + α_[2] n + α_[3] n + 2 * α_[1] n.+1 +
S_[4] n.+1 + S_[5] n.+2 <=
2 * (S_[3] n + α_[3] n) + 2 * (S_[2] n + α_[2] n) +
2 * S_[2] n.+1 + S_[1] n.+1 + S_[1] n.+2 + 4
2e0a
    have F3 : S_[5] n.+2 <= S_[4] n.+1 + (α_[1] n.+1).*2.
3000
24ab
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f302f354e42f4d2fa62fef2ffb3001
F3:S_[5] n.+2 <= S_[4] n.+1 + (α_[1] n.+1).*2
6 * S_[1] n + α_[2] n + α_[3] n + 2 * α_[1] n.+1 +
S_[4] n.+1 + S_[5] n.+2 <=
2 * (S_[3] n + α_[3] n) + 2 * (S_[2] n + α_[2] n) +
2 * S_[2] n.+1 + S_[1] n.+1 + 
S_[1] n.+2 + 4
2e0b2dbe
      by apply: dsum_alphaL_alpha.
3008
3002
2e0a
    have F4 : S_[4] n.+1 <= S_[3] n + (α_[1] n).*2.
3008
24b6
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f302f354e42f4d2fa62fef2ffb30013009
F4:S_[4] n.+1 <= S_[3] n + (α_[1] n).*2
300a
2e0b2dbe
      by apply: dsum_alphaL_alpha.
3013
3002
2e0a
    have F5 := dsum_alpha3_S n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f302f354e42f4d2fa62fef2ffb300130093014
F5:S_[1] n.+2 = (S_[3] n.+1).+1
3002
2e0a
    have F6 := dsum_alpha3_S n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f302f354e42f4d2fa62fef2ffb300130093014301c
F6:S_[1] n.+1 = (S_[3] n).+1
3002
2e0a
    have F7 := leq_dsum_alpha_2l_1 n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f302f354e42f4d2fa62fef2ffb300130093014301c3021
F7:S_[1] n.+1 + S_[1] n.+2 <= (S_[2] n.+1).*2.+1
3002
2e0a
    have F8 := leq_dsum_alpha_2l_1 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f302f354e42f4d2fa62fef2ffb300130093014301c30213026
F8:S_[1] n + S_[1] n.+1 <= (S_[2] n).*2.+1
3002
2e0a
    have F9 := dsum_alphaL_S 1 n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f302f354e42f4d2fa62fef2ffb300130093014301c30213026302b
F9:S_[1] n.+2 = S_[1] n.+1 + α_[1] n.+1
3002
2e0a
    have F10 := dsum_alphaL_S 1 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f302f354e42f4d2fa62fef2ffb300130093014301c30213026302b3030
F10:S_[1] n.+1 = S_[1] n + α_[1] n
3002
2e0a
    have F11 := alphaL_1_2 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a65d16a798f2e5628412e715cc28b41f712e952ec52ecc1fad2ef82f082f112e722ecd2db72f302f354e42f4d2fa62fef2ffb300130093014301c30213026302b30303035
F11:α_[1] n.+1 <= (α_[1] n).*2
3002
2e0a
    by lia.
2e0c
2de8
2dbd
(* subcase b = 1 *)
  have bE1 : b = 1.
2e0c
b = 1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e051e77b23
2df0
2dbe
    by case: (b) b_gt0 b_le2 => // [] [|].
3044
2de8
2dbd
  have a1E2 : a1 = 2.
3044
29c6
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e051e77b2329ca
2df0
2dbe
    by move: bE1; rewrite /b aE1; case: (a1) => // [] [|[]].
304d
2de8
2dbd
  pose u3 := 
    [ffun i : 'I_4 =>  
    if i == 0 :> nat then ↓[u (inord a)] else
    if i == 1 :> nat then ↓[u (inord a1)] else
    if i == 2 :> nat then sa1 else ta1].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e051e77b2329ca
u3:=[ffun i => if i == 0 :> nat
           then ↓[u (inord a)]
           else
            if i == 1 :> nat
            then ↓[u (inord a1)]
            else
             if i == 2 :> nat then sa1 else ta1]:{ffun forall x : ordinal_finType 4,
      (fun=> configuration 4 n.+1) x}
2de8
2dbd
  set x1S := \sum_(_ <= _ < _) _.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e051e77b2329ca30552e56
2e57
2dbd
  set d1 := `d[_, _]_ _ + `d[_, _]_ _.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e051e77b2329ca30552e56
d1:=`d[↓[u (inord a1)], sa1]_smove +
`d[sa1, ta1]_smove:nat
(S_[l.+1] n.+2).*2 <=
a.*2 + x0S + x1S +
(c.*2 + (d1 + x3S) +
 (\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
  y8S))
2dbd
  rewrite -/y0S.
305d
(S_[l.+1] n.+2).*2 <=
a.*2 + x0S + x1S + (c.*2 + (d1 + x3S) + (y0S + y8S))
2dbd
  changer (a.*2 + x0S + c.*2 + x3S + y0S + y8S + (x1S + d1)).
305d
(S_[l.+1] n.+2).*2 <=
a.*2 + x0S + c.*2 + x3S + y0S + y8S + (x1S + d1)
2dbd
  have -> : x1S + d1 = \sum_(i < 3) `d[u3 (inord i), u3 (inord i.+1)]_smove.
305d
x1S + d1 =
\sum_(i < 3) `d[u3 (inord i), u3 (inord i.+1)]_smove
305d
(S_[l.+1] n.+2).*2 <=
a.*2 + x0S + c.*2 + x3S + y0S + y8S +
\sum_(i < 3) `d[u3 (inord i), u3 (inord i.+1)]_smove
2dbe
    rewrite /x1S aE1 a1E2 big_nat1 /d1 !big_ord_recr big_ord0 /=.
305d
`d[↓[u (inord 1)], ↓[u (inord 2)]]_smove +
(`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove) =
0 + `d[u3 (inord 0), u3 (inord 1)]_smove +
`d[u3 (inord 1), u3 (inord 2)]_smove +
`d[u3 (inord 2), u3 (inord 3)]_smove
306c
    by rewrite !ffunE !inordK //= aE1 a1E2 add0n !addnA.
305d
306e
2dbd
    have l_gt2 : 2 < l by rewrite -ltnS l1E aE1 bE1 !add1n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e051e77b2329ca30552e56305e
l_gt2:2 < l
306e
2dbd
  set x6S := \sum_(_ < 3) _.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e051e77b2329ca30552e56305e307a
x6S:=\sum_(H < 3)
   `d[u3 (inord H), u3 (inord H.+1)]_smove:nat
(S_[l.+1] n.+2).*2 <=
a.*2 + x0S + c.*2 + x3S + y0S + y8S + x6S
2dbd
  have cH6 (k : 'I_4) : 
    0 < k < 3 -> codom (u3 k) \subset [:: pa1; a[pa1S, p2] k].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e051e77b2329ca30552e56305e307a307fe34
0 < k < 3 ->
codom (u3 k) \subset [:: pa1;  a[pa1S, p2] k]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e051e77b2329ca30552e56305e307a307f
cH6:forall k : 'I_4,
0 < k < 3 ->
codom (u3 k) \subset [:: pa1;  a[pa1S, p2] k]
3080
2dbe
    move=> /andP[k_gt0 k_lt3].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e051e77b2329ca30552e56305e307a307fe34852222c
codom (u3 k) \subset [:: pa1;  a[pa1S, p2] k]
3086
    rewrite !ffunE eqn_leq leqNgt k_gt0 /=.
308d
codom
  (if k == 1
   then ↓[u (inord a1)]
   else if k == 2 then sa1 else ta1) \subset
[:: pa1;  a[pa1S, p2] k]
3086
    case: eqP => [->/=|/eqP kD1].
308d
codom ↓[u (inord a1)] \subset [:: pa1;  a[pa1S, p2] 1]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e051e77b2329ca30552e56305e307a307fe34852222c2239
codom (if k == 2 then sa1 else ta1) \subset
[:: pa1;  a[pa1S, p2] k]
30872dbe
      apply: codom_liftr; rewrite codom_subC.
308d
codom (u (inord a1)) \subset [::  a[pa1S, p2] 1; pa1]
3097
      have /cH :  0 < (inord a1 : 'I_l.+2 ) < l.+1.
308d
18b2
308d
codom (u (inord a1)) \subset
[:: p2;  a[p1, p3] (inord a1)] ->
codom (u (inord a1)) \subset [::  a[pa1S, p2] 1; pa1]
309830872dbe
        by rewrite inordK a1E2 // (leq_trans l_gt2).
308d
30a4
3097
      by rewrite inordK // a1E2 (leq_trans l_gt2) // ltnW.
3099
309a
3086
    have -> : k = 2 :> nat.
3099
2241
3099
codom (if 2 == 2 then sa1 else ta1) \subset
[:: pa1;  a[pa1S, p2] 2]
30872dbe
      by case: (k : nat) k_gt0 k_lt3 kD1 => // [] [|[]].
3099
30b0
3086
    by have := sa1C; rewrite eqxx /pa1 /pa1S !(apegS, apeg0) codom_apeg.
3088
3080
2dbd
  have P6 : 
    (S_[3] n.+1).*2 <= x6S + sp (u3 ord0) 3 pa1S + sp (u3 ord_max) 3 p2.
3088
(S_[3] n.+1).*2 <=
x6S + sp (n:=n.+1) (u3 ord0) 3 pa1S +
sp (n:=n.+1) (u3 ord_max) 3 p2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e051e77b2329ca30552e56305e307a307f3089
P6:(S_[3] n.+1).*2 <=
x6S + sp (n:=n.+1) (u3 ord0) 3 pa1S +
sp (n:=n.+1) (u3 ord_max) 3 p2
3080
2dbe
    apply: IH cH6; rewrite /pa1S /pa1 //.
307e
1a1f
30bb
    by rewrite a1E2 eq_sym.
30bd
3080
2dbd
  rewrite !ffunE /= in P6.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e051e77b2329ca30552e56305e307a307f3089
P6:(S_[3] n.+1).*2 <=
x6S + sp (n:=n.+1) ↓[u (inord a)] 3 pa1S +
sp (n:=n.+1) ta1 3 p2
3080
2dbd
  set y2S := sp _ _ _ in P6; set y3S := sp _ _ _ in P6.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e051e77b2329ca30552e56305e307a307f3089
y2S:=sp (n:=n.+1) ↓[u (inord a)] 3 pa1S:nat
y3S:=sp (n:=n.+1) ta1 3 p2:nat
P6:(S_[3] n.+1).*2 <= x6S + y2S + y3S
3080
2dbd
  have y1Sy2SE : y1S + y2S <= (S_[1] n).*2 + α_[3] n.
30cd
y1S + y2S <= (S_[1] n).*2 + α_[3] n
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e051e77b2329ca30552e56305e307a307f308930ce30cf30d0
y1Sy2SE:y1S + y2S <= (S_[1] n).*2 + α_[3] n
3080
2dbe
    apply: sum_alpha_diffE => //; first by rewrite eq_sym /pa1S a1E2.
30cd
codom ↓[u (inord a)] \subset [:: p2; pa1S]
30d5
    have /H : 0 < (inord a : 'I_l.+2) < l.+1; first by rewrite inordK // aE1.
30cd
codom (u' (inord a)) \subset
[:: p2;  a[p1, p3] (inord a)] ->
codom ↓[u (inord a)] \subset [:: p2; pa1S]
30d5
    by rewrite ffunE inordK // /pa1S /= aE1 a1E2.
30d7
3080
2dbd
  have lEc : l = c.+1 by rewrite -[l]/l.+1.-1 l1E aE1 bE1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e051e77b2329ca30552e56305e307a307f308930ce30cf30d030d8
lEc:l = c.+1
3080
2dbd
  rewrite lEc.
30e7
(S_[c.+2] n.+2).*2 <=
a.*2 + x0S + c.*2 + x3S + y0S + y8S + x6S
2dbd
  have y3Sy7SE : y3S + y7S <=  (S_[1] n).*2 + α_[maxn 3 (3 * c).-2] n.
30e7
y3S + y7S <= (S_[1] n).*2 + α_[maxn 3 (3 * c).-2] n
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e051e77b2329ca30552e56305e307a307f308930ce30cf30d030d830e8
y3Sy7SE:y3S + y7S <=
(S_[1] n).*2 + α_[maxn 3 (3 * c).-2] n
30ec
2dbe
    by apply: sum_alpha_diffE => //; first by rewrite /pa1 a1E2 eq_sym.
30f3
30ec
2dbd
  rewrite (maxn_idPr _) in y3Sy7SE; last first.
30e7
2 < (3 * c).-2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7f17286327bc285728a62e051e77b2329ca30552e56305e307a307f308930ce30cf30d030d830e8
y3Sy7SE:y3S + y7S <= (S_[1] n).*2 + α_[(3 * c).-2] n
30ec
2dbe
    by rewrite -subn2 ltn_subRL ltnW // (leq_mul2l 3 2).
30fe
30ec
2dbd
  move: P4 P6 P8 => P4 P6 P8.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a61e77b2329ca30552e56305e307a307f308930ce30cf30d830e830ff2e0530d02db7
30ec
2dbd
  rewrite aE1 !muln1 in P4 P6 P8.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a61e77b2329ca30552e56305e307a307f308930ce30cf30d830e830ff2f1130d02db7
30ec
2dbd
  have F1 := dsum_alphaL_S 3 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a61e77b2329ca30552e56305e307a307f308930ce30cf30d830e830ff2f1130d02db7
F1:S_[3] n.+1 = S_[3] n + α_[3] n
30ec
2dbd
  have F2 := dsum_alphaL_S (3 * c).-2 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a61e77b2329ca30552e56305e307a307f308930ce30cf30d830e830ff2f1130d02db7310f20e0
30ec
2dbd
  have F4 k : S_[c.-1] k.+1 <= S_[(3 *c).-2] k + c.-1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a61e77b2329ca30552e56305e307a307f308930ce30cf30d830e830ff2f1130d02db7310f20e0a81
S_[c.-1] k.+1 <= S_[(3 * c).-2] k + c.-1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a61e77b2329ca30552e56305e307a307f308930ce30cf30d830e830ff2f1130d02db7310f20e0
F4:forall k : nat,
S_[c.-1] k.+1 <= S_[(3 * c).-2] k + c.-1
30ec
2dbe
    apply: leq_trans (dsum_alpha3l _ _) _.
3117
S_[3 * c.-1] k + c.-1 <= S_[(3 * c).-2] k + c.-1
3119
    rewrite leq_add2r.
3117
S_[3 * c.-1] k <= S_[(3 * c).-2] k
3119
    apply: (increasingE (increasing_dsum_alphaL_l _)).
3117
3 * c.-1 <= (3 * c).-2
3119
    by rewrite -{2}[c]prednK // mulnS addSn add2n /=.
311b
30ec
2dbd
  have F4n := F4 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a61e77b2329ca30552e56305e307a307f308930ce30cf30d830e830ff2f1130d02db7310f20e0311c
F4n:S_[c.-1] n.+1 <= S_[(3 * c).-2] n + c.-1
30ec
2dbd
  have F4n1 := F4 n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a61e77b2329ca30552e56305e307a307f308930ce30cf30d830e830ff2f1130d02db7310f20e0311c3130
F4n1:S_[c.-1] n.+2 <= S_[(3 * c).-2] n.+1 + c.-1
30ec
2dbd
  have F5 := prednK c_gt0.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a61e77b2329ca30552e56305e307a307f308930ce30cf30d830e830ff2f1130d02db7310f20e0311c31303135
F5:c.-1.+1 = c
30ec
2dbd
  have l_gt1 : 1 < l by rewrite lEc.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a61e77b2329ca30552e56305e307a307f308930ce30cf30d830e830ff2f1130d02db7310f20e0311c31303135313a4e4
30ec
2dbd
  have F6 : (S_[c.+2] n.+2).*2 <= 
             S_[c.+2] n.+2 + S_[c.+1] n.+1 + (α_[1] n.+1).*2.
313e
(S_[c.+2] n.+2).*2 <=
S_[c.+2] n.+2 + S_[c.+1] n.+1 + (α_[1] n.+1).*2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a61e77b2329ca30552e56305e307a307f308930ce30cf30d830e830ff2f1130d02db7310f20e0311c31303135313a4e4
F6:(S_[c.+2] n.+2).*2 <=
S_[c.+2] n.+2 + S_[c.+1] n.+1 + (α_[1] n.+1).*2
30ec
2dbe
    by have := (SH l_gt1); rewrite lEc.
3145
30ec
2dbd
  have F7 : S_[c.+2] n.+2 + S_[1] n.+2 <= S_[c.-1] n.+2 + S_[4] n.+2.
3145
S_[c.+2] n.+2 + S_[1] n.+2 <=
S_[c.-1] n.+2 + S_[4] n.+2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a61e77b2329ca30552e56305e307a307f308930ce30cf30d830e830ff2f1130d02db7310f20e0311c31303135313a4e43146
F7:S_[c.+2] n.+2 + S_[1] n.+2 <=
S_[c.-1] n.+2 + S_[4] n.+2
30ec
2dbe
    rewrite (_ : c.+2 =  1 + 3 + c.-2); last first.
3145
249a
3145
26dc
314f2dbe
      by rewrite !addSn add0n !prednK ?addn2 // -ltnS prednK.
3145
26dc
314e
    rewrite {2}(_ : c.-1 = 1 + c.-2); last first.
3145
21a0
3145
26e3
314f2dbe
      by rewrite add1n prednK // -ltnS prednK.
3145
26e3
314e
    by apply: concaveEk1 (concave_dsum_alphaL_l n.+2).
3150
30ec
2dbd
  have F8 : S_[c.+1] n.+1 + S_[1] n.+1 <= S_[c.-1] n.+1 + S_[3] n.+1.
3150
S_[c.+1] n.+1 + S_[1] n.+1 <=
S_[c.-1] n.+1 + S_[3] n.+1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a61e77b2329ca30552e56305e307a307f308930ce30cf30d830e830ff2f1130d02db7310f20e0311c31303135313a4e431463151
F8:S_[c.+1] n.+1 + S_[1] n.+1 <=
S_[c.-1] n.+1 + S_[3] n.+1
30ec
2dbe
    rewrite (_ : c.+1 =  1 + 2 + c.-2); last first.
3150
26f2
3150
26f5
316a2dbe
      by rewrite !addSn add0n !prednK ?addn2 // -ltnS prednK.
3150
26f5
3169
    rewrite {2}(_ : c.-1 = 1 + c.-2); last first.
3150
21a0
3150
26fc
316a2dbe
      by rewrite add1n prednK // -ltnS prednK.
3150
26fc
3169
    by apply: concaveEk1 (concave_dsum_alphaL_l n.+1).
316b
30ec
2dbd
  have F9 : S_[4] n.+2 <= S_[3] n.+1 + (α_[1] n.+1).*2.
316b
2703
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a61e77b2329ca30552e56305e307a307f308930ce30cf30d830e830ff2f1130d02db7310f20e0311c31303135313a4e431463151316c
F9:S_[4] n.+2 <= S_[3] n.+1 + (α_[1] n.+1).*2
30ec
2dbe
    by apply: dsum_alphaL_alpha.
3185
30ec
2dbd
  have F10 := dsum_alpha3_S n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a61e77b2329ca30552e56305e307a307f308930ce30cf30d830e830ff2f1130d02db7310f20e0311c31303135313a4e431463151316c3186
F10:S_[1] n.+1 = (S_[3] n).+1
30ec
2dbd
  have F11 := dsum_alpha3_S n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a61e77b2329ca30552e56305e307a307f308930ce30cf30d830e830ff2f1130d02db7310f20e0311c31303135313a4e431463151316c3186318e2fd5
30ec
2dbd
  have F12 := dsum_alphaL_S 1 n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a61e77b2329ca30552e56305e307a307f308930ce30cf30d830e830ff2f1130d02db7310f20e0311c31303135313a4e431463151316c3186318e2fd52fda
30ec
2dbd
  have F13 := dsum_alphaL_S 1 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a61e77b2329ca30552e56305e307a307f308930ce30cf30d830e830ff2f1130d02db7310f20e0311c31303135313a4e431463151316c3186318e2fd52fda2fdf
30ec
2dbd
  have F14 := alphaL_1_2 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6f17286327bc285728a61e77b2329ca30552e56305e307a307f308930ce30cf30d830e830ff2f1130d02db7310f20e0311c31303135313a4e431463151316c3186318e2fd52fda2fdf2fe4
30ec
2dbd
  by lia.
2db5
b <= 0 ->
(S_[l.+1] n.+2).*2 <=
a.*2 + sd (n:=n.+1) u1 +
\sum_(a <= i0 < a1)
   `d[↓[u (inord i0)], ↓[u (inord i0.+1)]]_smove +
(c.*2 +
 (`d[↓[u (inord a1)], sa1]_smove + `d[sa1, ta1]_smove +
  x3S) +
 (\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
  y8S))
(* subcase b = 0 *)
rewrite leqn0 => /eqP bE0.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f
2db8
have a1Ea : a1 = a by rewrite -(subnK aLa1) -/b bE0.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f
a1Ea:a1 = a
2db8
rewrite big_geq ?a1Ea // addn0.
31aa
(S_[l.+1] n.+2).*2 <=
a.*2 + sd (n:=n.+1) u1 +
(c.*2 +
 (`d[↓[u (inord a)], sa1]_smove + `d[sa1, ta1]_smove +
  x3S) +
 (\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
  y8S))
have lEc : l = c by rewrite -[l]/(l.+1.-1) l1E aE1 bE0.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab
lEc:l = c
31af
rewrite {1}lEc.
31b3
(S_[c.+1] n.+2).*2 <=
a.*2 + sd (n:=n.+1) u1 +
(c.*2 +
 (`d[↓[u (inord a)], sa1]_smove + `d[sa1, ta1]_smove +
  x3S) +
 (\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
  y8S))
set d1 := `d[_, _]_ _ + `d[_, _]_ _.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b4
d1:=`d[↓[u (inord a)], sa1]_smove + `d[sa1, ta1]_smove:nat
(S_[c.+1] n.+2).*2 <=
a.*2 + sd (n:=n.+1) u1 +
(c.*2 + (d1 + x3S) +
 (\sum_(k < n.+1) (↓[u ord0] k != p1) * (α_[1] k).*2 +
  y8S))
set x0S := sd _; set y0S := \sum_(_ < _) _.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc2857
(S_[c.+1] n.+2).*2 <=
a.*2 + x0S + (c.*2 + (d1 + x3S) + (y0S + y8S))
changer (a.*2 + c.*2 + x3S + y0S + y8S + (x0S + d1)).
31c2
(S_[c.+1] n.+2).*2 <=
a.*2 + c.*2 + x3S + y0S + y8S + (x0S + d1)
pose u3 := [ffun i : 'I_6 =>
  if i == 0 :> nat then ↓[u ord0] else
  if i == 1 :> nat then sam1 else 
  if i == 2 :> nat then tam1 else 
  if i == 3 :> nat then ↓[u (inord a1)] else 
  if i == 4 :> nat then sa1 else 
  ta1].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc2857
u3:=[ffun i => if i == 0 :> nat
           then ↓[u ord0]
           else
            if i == 1 :> nat
            then sam1
            else
             if i == 2 :> nat
             then tam1
             else
              if i == 3 :> nat
              then ↓[u (inord a1)]
              else
               if i == 4 :> nat then sa1 else ta1]:{ffun forall x : ordinal_finType 6,
      (fun=> configuration 4 n.+1) x}
31c7
have -> : x0S + d1 = sd u3.
31cb
x0S + d1 = sd (n:=n.+1) u3
31cb
(S_[c.+1] n.+2).*2 <=
a.*2 + c.*2 + x3S + y0S + y8S + sd (n:=n.+1) u3
  rewrite /x0S /d1 /sd.
31cb
\sum_(i < 3 * a)
   `d[u1 (inord i), u1 (inord i.+1)]_smove +
(`d[↓[u (inord a)], sa1]_smove + `d[sa1, ta1]_smove) =
\sum_(i < 5) `d[u3 (inord i), u3 (inord i.+1)]_smove
31d1 
  rewrite -(big_mkord xpredT (fun i => `d[u1 (inord i), u1 (inord i.+1)]_ _)).
31cb
\sum_(0 <= i < 3 * a)
   `d[u1 (inord i), u1 (inord i.+1)]_smove +
(`d[↓[u (inord a)], sa1]_smove + `d[sa1, ta1]_smove) =
\sum_(i < 5) `d[u3 (inord i), u3 (inord i.+1)]_smove
31d1
  rewrite {1}aE1 !(big_nat1, big_nat_recr) // !big_ord_recr big_ord0 
          /= ?inordK //= !ffunE ?inordK ?a1Ea ?aE1 //=.
31cb
`d[↓[u (inord (0 %/ 3))], sam1]_smove +
`d[sam1, tam1]_smove +
`d[tam1, ↓[u (inord (3 %/ 3))]]_smove +
(`d[↓[u (inord 1)], sa1]_smove + `d[sa1, ta1]_smove) =
0 + `d[↓[u ord0], sam1]_smove + `d[sam1, tam1]_smove +
`d[tam1, ↓[u (inord 1)]]_smove +
`d[↓[u (inord 1)], sa1]_smove + `d[sa1, ta1]_smove
31d1
  by rewrite add0n !addnA inord_eq0.
31cb
31d3
have cH4 (k : 'I_6) : 
    0 < k < 5 -> codom (u3 k) \subset [:: p3; a[p1, p2] k].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc
k:'I_6
0 < k < 5 ->
codom (u3 k) \subset [:: p3;  a[p1, p2] k]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc
cH4:forall k : 'I_6,
0 < k < 5 ->
codom (u3 k) \subset [:: p3;  a[p1, p2] k]
31d3
    case: k => [] [|[|[|[|[]]]]] //= iH _; rewrite !ffunE !(apegS, apeg0).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc
iH:1 < 6
codom
  (if Ordinal (n:=6) (m:=1) iH == 0
   then ↓[u ord0]
   else
    if Ordinal (n:=6) (m:=1) iH == 1
    then sam1
    else
     if Ordinal (n:=6) (m:=1) iH == 2
     then tam1
     else
      if Ordinal (n:=6) (m:=1) iH == 3
      then ↓[u (inord a1)]
      else
       if Ordinal (n:=6) (m:=1) iH == 4
       then sa1
       else ta1) \subset [:: p3; p2]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc
iH:2 < 6
codom
  (if Ordinal (n:=6) (m:=2) iH == 0
   then ↓[u ord0]
   else
    if Ordinal (n:=6) (m:=2) iH == 1
    then sam1
    else
     if Ordinal (n:=6) (m:=2) iH == 2
     then tam1
     else
      if Ordinal (n:=6) (m:=2) iH == 3
      then ↓[u (inord a1)]
      else
       if Ordinal (n:=6) (m:=2) iH == 4
       then sa1
       else ta1) \subset [::
 p3; p1]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc
iH:3 < 6
codom
  (if Ordinal (n:=6) (m:=3) iH == 0
   then ↓[u ord0]
   else
    if Ordinal (n:=6) (m:=3) iH == 1
    then sam1
    else
     if Ordinal (n:=6) (m:=3) iH == 2
     then tam1
     else
      if Ordinal (n:=6) (m:=3) iH == 3
      then ↓[u (inord a1)]
      else
       if Ordinal (n:=6) (m:=3) iH == 4
       then sa1
       else ta1) \subset [::
 p3; p2]
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc
iH:4 < 6
codom
  (if Ordinal (n:=6) (m:=4) iH == 0
   then ↓[u ord0]
   else
    if Ordinal (n:=6) (m:=4) iH == 1
    then sam1
    else
     if Ordinal (n:=6) (m:=4) iH == 2
     then tam1
     else
      if Ordinal (n:=6) (m:=4) iH == 3
      then ↓[u (inord a1)]
      else
       if Ordinal (n:=6) (m:=4) iH == 4
       then sa1
       else ta1) \subset [::
 p3; p1]
31ea
  -
31ee by have := sam1C; rewrite /pa aE1.
31f5
codom
  (if Ordinal (n:=6) (m:=2) iH == 0
   then ↓[u ord0]
   else
    if Ordinal (n:=6) (m:=2) iH == 1
    then sam1
    else
     if Ordinal (n:=6) (m:=2) iH == 2
     then tam1
     else
      if Ordinal (n:=6) (m:=2) iH == 3
      then ↓[u (inord a1)]
      else
       if Ordinal (n:=6) (m:=2) iH == 4
       then sa1
       else ta1) \subset [:: p3; p1]
31f831fc31ea
  -
3202 by rewrite codom_subC.
31f9
codom
  (if Ordinal (n:=6) (m:=3) iH == 0
   then ↓[u ord0]
   else
    if Ordinal (n:=6) (m:=3) iH == 1
    then sam1
    else
     if Ordinal (n:=6) (m:=3) iH == 2
     then tam1
     else
      if Ordinal (n:=6) (m:=3) iH == 3
      then ↓[u (inord a1)]
      else
       if Ordinal (n:=6) (m:=3) iH == 4
       then sa1
       else ta1) \subset [:: p3; p2]
31fc31ea
  -
3208 rewrite a1Ea aE1.
31f9
codom
  (if Ordinal (n:=6) (m:=3) iH == 0
   then ↓[u ord0]
   else
    if Ordinal (n:=6) (m:=3) iH == 1
    then sam1
    else
     if Ordinal (n:=6) (m:=3) iH == 2
     then tam1
     else
      if Ordinal (n:=6) (m:=3) iH == 3
      then ↓[u (inord 1)]
      else
       if Ordinal (n:=6) (m:=3) iH == 4
       then sa1
       else ta1) \subset [:: p3; p2]
320b
  -
320e have /H : 0 < (inord 1 : 'I_l.+2) < l.+1 by rewrite !inordK.
31f9
codom (u' (inord 1)) \subset
[:: p2;  a[p1, p3] (inord 1)] ->
codom
  (if Ordinal (n:=6) (m:=3) iH == 0
   then ↓[u ord0]
   else
    if Ordinal (n:=6) (m:=3) iH == 1
    then sam1
    else
     if Ordinal (n:=6) (m:=3) iH == 2
     then tam1
     else
      if Ordinal (n:=6) (m:=3) iH == 3
      then ↓[u (inord 1)]
      else
       if Ordinal (n:=6) (m:=3) iH == 4
       then sa1
       else ta1) \subset [:: p3; p2]
320b
    by rewrite ffunE inordK // codom_subC.
31fd
codom
  (if Ordinal (n:=6) (m:=4) iH == 0
   then ↓[u ord0]
   else
    if Ordinal (n:=6) (m:=4) iH == 1
    then sam1
    else
     if Ordinal (n:=6) (m:=4) iH == 2
     then tam1
     else
      if Ordinal (n:=6) (m:=4) iH == 3
      then ↓[u (inord a1)]
      else
       if Ordinal (n:=6) (m:=4) iH == 4
       then sa1
       else ta1) \subset [:: p3; p1]
31e9
  by rewrite codom_subC.
31eb
31d3
have P4 :
  (S_[5] n.+1).*2 <= sd u3 + sp ↓[u ord0] 5 p1 + sp ta1 5 p2.
31eb
(S_[5] n.+1).*2 <=
sd (n:=n.+1) u3 + sp (n:=n.+1) ↓[u ord0] 5 p1 +
sp (n:=n.+1) ta1 5 p2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec
P4:(S_[5] n.+1).*2 <=
sd (n:=n.+1) u3 + sp (n:=n.+1) ↓[u ord0] 5 p1 +
sp (n:=n.+1) ta1 5 p2
31d3
  apply: leq_trans (IH _ _ _ _ _ _ cH4) _ => //; first by rewrite eq_sym.
31eb
sd (n:=n.+1) u3 + sp (n:=n.+1) (u3 ord0) 5 p1 +
sp (n:=n.+1) (u3 ord_max) 5 ( a[p1, p2] 5) <=
sd (n:=n.+1) u3 + sp (n:=n.+1) ↓[u ord0] 5 p1 +
sp (n:=n.+1) ta1 5 p2
3221
  by apply: leq_add; rewrite !ffunE.
3223
31d3
have {}P4 := leq_trans P4 
  (leq_add (leq_add (leqnn _) (leq_sum_beta _ _)) (leqnn _)).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec
P4:(S_[5] n.+1).*2 <=
sd (n:=n.+1) u3 +
\sum_(k < n.+1)
   (↓[u ord0] k != p1) * (α_[1] k).*2 +
sp (n:=n.+1) ta1 5 p2
31d3
rewrite -/y0S in P4; set y1S := sp _ _ _ in P4.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec
y1S:=sp (n:=n.+1) ta1 5 p2:nat
P4:(S_[5] n.+1).*2 <= sd (n:=n.+1) u3 + y0S + y1S
31d3
set x1S := sd _ in P4; rewrite -/x1S.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235
x1S:=sd (n:=n.+1) u3:nat
P4:(S_[5] n.+1).*2 <= x1S + y0S + y1S
(S_[c.+1] n.+2).*2 <=
a.*2 + c.*2 + x3S + y0S + y8S + x1S
have y1Sy7SE : y1S + y7S <= (S_[1] n).*2 + α_[maxn 5 (3 * c).-2] n.
323a
y1S + y7S <= (S_[1] n).*2 + α_[maxn 5 (3 * c).-2] n
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c
y1Sy7SE:y1S + y7S <=
(S_[1] n).*2 + α_[maxn 5 (3 * c).-2] n
323d 
  apply: sum_alpha_diffE => //; first by rewrite /pa1 eq_sym.
3244
323d
rewrite aE1.
3244
(S_[c.+1] n.+2).*2 <=
1.*2 + c.*2 + x3S + y0S + y8S + x1S
have leq_S5 k : 2 * S_[1] k.+1 + S_[1] k.+2 <= 3 * S_[5] k + 6.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c3245a81
2 * S_[1] k.+1 + S_[1] k.+2 <= 3 * S_[5] k + 6
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c3245
leq_S5:forall k : nat,
2 * S_[1] k.+1 + S_[1] k.+2 <=
3 * S_[5] k + 6
324c
  have F8 :  S_[3] k + 2 * S_[6] k <= 3 * S_[5] k.
3250
S_[3] k + 2 * S_[6] k <= 3 * S_[5] k
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c3245a81
F8:S_[3] k + 2 * S_[6] k <= 3 * S_[5] k
3251
3253
    have X1 := concaveEk1 3 2 1 (concave_dsum_alphaL_l k).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c3245a81
X1:S_[3 + 2 + 1] k + S_[3] k <=
S_[3 + 1] k + S_[3 + 2] k
3259
325a
    rewrite -[_ + 1]/6 -[_ + 1]/4  -[_ + 2]/5 in X1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c3245a81
X1:S_[6] k + S_[3] k <= S_[4] k + S_[5] k
3259
325a
    have X2 := concaveEk1 4 1 1 (concave_dsum_alphaL_l k).
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c3245a813267
X2:S_[4 + 1 + 1] k + S_[4] k <=
S_[4 + 1] k + S_[4 + 1] k
3259
325a
    rewrite -[_ + 1]/6 -[_ + 1]/5 in X2.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c3245a813267
X2:S_[6] k + S_[4] k <= S_[5] k + S_[5] k
3259
325a
    by lia.
325c
3251
3252
  have F3k : S_[1] k.+1 <= S_[3] k + 1 by apply: dsum_alpha3l.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c3245a81325d276d
3251
3252
  have F3k1 : S_[2] k.+1 <= S_[6] k + 2 by apply: dsum_alpha3l.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c3245a81325d276d2772
3251
3252
  have F3k2 := leq_dsum_alpha_2l_1 k.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c3245a81325d276d2772
F3k2:S_[1] k.+1 + S_[1] k.+2 <= (S_[2] k.+1).*2.+1
3251
3252
  by lia.
3254
324c
have [c_gt2|c_lt3] := leqP 3 c.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c32453255
c_gt2:2 < c
324c
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c32453255
c_lt3:c < 3
324c
(* Subcase c >= 3 *)
  rewrite (maxn_idPr _) in y1Sy7SE; last first.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c32553289
4 < (3 * c).-2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c32553289
y1Sy7SE:y1S + y7S <= (S_[1] n).*2 + α_[(3 * c).-2] n
324c
328b
    by rewrite -subn2 ltn_subRL (ltn_mul2l 3 2).
3295
324c
328a
  have F1 k : S_[c.-1] k.+1 <= S_[(3 *c).-2] k + c.-1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c325532893296a81
3118
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c325532893296
F1:forall k : nat,
S_[c.-1] k.+1 <= S_[(3 * c).-2] k + c.-1
324c
328b
    apply: leq_trans (dsum_alpha3l _ _) _.
329d
3120
329e
    rewrite leq_add2r.
329d
3124
329e
    apply: (increasingE (increasing_dsum_alphaL_l _)).
329d
3128
329e
    by rewrite -{2}[c]prednK // mulnS addSn add2n /=.
32a0
324c
328a
  have F1n := F1 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c32553289329632a1
F1n:S_[c.-1] n.+1 <= S_[(3 * c).-2] n + c.-1
324c
328a
  have F1n1 := F1 n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c32553289329632a132b2
F1n1:S_[c.-1] n.+2 <= S_[(3 * c).-2] n.+1 + c.-1
324c
328a
  have F2 := dsum_alphaL_S (3 * c).-2 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c32553289329632a132b232b720e0
324c
328a
  have F3 : 2 * c  = 2 + c.-1 + c.-1.
32bb
20f4
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c32553289329632a132b232b720e0
F3:2 * c = 2 + c.-1 + c.-1
324c
328b
    by rewrite mul2n add2n prednK // addSnnS prednK // addnn.
32c1
324c
328a
  have F4 : S_[c.+1] n.+2 <= S_[c] n.+1 + (α_[1] n.+1).*2.
32c1
S_[c.+1] n.+2 <= S_[c] n.+1 + (α_[1] n.+1).*2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c32553289329632a132b232b720e032c2
F4:S_[c.+1] n.+2 <= S_[c] n.+1 + (α_[1] n.+1).*2
324c
328b
    by apply: dsum_alphaL_alpha.
32cc
324c
328a
  have F7n1 : S_[2] n.+2 + S_[c.+1] n.+2 <= S_[c.-1] n.+2 + S_[4] n.+2.
32cc
S_[2] n.+2 + S_[c.+1] n.+2 <=
S_[c.-1] n.+2 + S_[4] n.+2
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c32553289329632a132b232b720e032c232cd
F7n1:S_[2] n.+2 + S_[c.+1] n.+2 <=
S_[c.-1] n.+2 + S_[4] n.+2
324c
328b
    rewrite addnC (_ : c.+1 = 2 + 2 + c.-2.-1); last first.
32cc
c.+1 = 2 + 2 + c.-1.-2
32cc
S_[2 + 2 + c.-1.-2] n.+2 + S_[2] n.+2 <=
S_[c.-1] n.+2 + S_[4] n.+2
32d6328b
       by rewrite !addSn add0n !prednK // -ltnS prednK // -subn1 ltn_subRL //.
32cc
32df
32d5
    rewrite {2}(_ : c.-1 = 2 + c.-1.-2); last first.
32cc
c.-1 = 2 + c.-1.-2
32cc
S_[2 + 2 + c.-1.-2] n.+2 + S_[2] n.+2 <=
S_[2 + c.-1.-2] n.+2 + S_[4] n.+2
32d6328b
      by rewrite !addSn add0n !prednK // -?subn2 -?subn1 ltn_subRL.
32cc
32e9
32d5
    by apply: concaveEk1 (concave_dsum_alphaL_l n.+2).
32d7
324c
328a
  have F7n : S_[2] n.+1 + S_[c] n.+1 <= S_[c.-1] n.+1 + S_[3] n.+1.
32d7
S_[2] n.+1 + S_[c] n.+1 <= S_[c.-1] n.+1 + S_[3] n.+1
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c32553289329632a132b232b720e032c232cd32d8
F7n:S_[2] n.+1 + S_[c] n.+1 <=
S_[c.-1] n.+1 + S_[3] n.+1
324c
328b
    rewrite addnC {1}(_ : c = 2 + 1 + c.-1.-2); last first.
32d7
c = 2 + 1 + c.-1.-2
32d7
S_[2 + 1 + c.-1.-2] n.+1 + S_[2] n.+1 <=
S_[c.-1] n.+1 + S_[3] n.+1
32f5328b
      by rewrite !addSn add0n !prednK // -2!ltnS !prednK // -ltnS prednK.
32d7
32fe
32f4
    rewrite {2}(_ : c.-1 = 2 + c.-1.-2); last first.
32d7
32e6
32d7
S_[2 + 1 + c.-1.-2] n.+1 + S_[2] n.+1 <=
S_[2 + c.-1.-2] n.+1 + S_[3] n.+1
32f5328b
      by rewrite !addSn add0n !prednK // -?subn2 -?subn1 ltn_subRL.
32d7
3307
32f4
    by apply: concaveEk1 (concave_dsum_alphaL_l n.+1).
32f6
324c
328a
  have F8 := leq_S5 n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c32553289329632a132b232b720e032c232cd32d832f7
F8:2 * S_[1] n.+2 + S_[1] n.+3 <= 3 * S_[5] n.+1 + 6
324c
328a
  have F9 : S_[4] n.+2 <= S_[3] n.+1 + (α_[1] n.+1).*2.
3311
2703
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c32553289329632a132b232b720e032c232cd32d832f733123186
324c
328b
    by apply: dsum_alphaL_alpha.
3318
324c
328a
  have F10 := dsum_alpha3_S n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c32553289329632a132b232b720e032c232cd32d832f733123186
F10:S_[1] n.+2 = (S_[3] n.+1).+1
324c
328a
  have F11 := leq_dsum_alpha_2l_1 n.+2.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c32553289329632a132b232b720e032c232cd32d832f7331231863320
F11:S_[1] n.+2 + S_[1] n.+3 <= (S_[2] n.+2).*2.+1
324c
328a
  have F12 := leq_dsum_alpha_2l_1 n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c32553289329632a132b232b720e032c232cd32d832f7331231863320332524c7
324c
328a
  have F13 := dsum_alphaL_S 1 n.+2.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c32553289329632a132b232b720e032c232cd32d832f7331231863320332524c7
F13:S_[1] n.+3 = S_[1] n.+2 + α_[1] n.+2
324c
328a
  have F14 := dsum_alphaL_S 1 n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c32553289329632a132b232b720e032c232cd32d832f7331231863320332524c7332e
F14:S_[1] n.+2 = S_[1] n.+1 + α_[1] n.+1
324c
328a
  have F15 := dsum_alphaL_S 1 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c32553289329632a132b232b720e032c232cd32d832f7331231863320332524c7332e3333
F15:S_[1] n.+1 = S_[1] n + α_[1] n
324c
328a
  have F16 := increasing_alphaL 1 n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c32553289329632a132b232b720e032c232cd32d832f7331231863320332524c7332e33333338
F16:α_[1] n.+1 <= α_[1] n.+2
324c
328a
  suff : 8 * α_[1] n.+1 <= 12 * α_[1] n + 6 by lia.
333c
8 * α_[1] n.+1 <= 12 * α_[1] n + 6
328a
  case: (n) => [|[|[|[|n1]]]]; rewrite ?alpha_small //.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c32553289329632a132b232b720e032c232cd32d832f7331231863320332524c7332e33333338333d273c
8 * α_[1] n1.+1.+4 <= 12 * α_[1] n1.+4 + 6
328a
  have F17 : 3 * α_[1] n1.+1.+4 <= 4 * α_[1] n1.+4.
3345
3 * α_[1] n1.+1.+4 <= 4 * α_[1] n1.+4
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c32553289329632a132b232b720e032c232cd32d832f7331231863320332524c7332e33333338333d273c
F17:3 * α_[1] n1.+1.+4 <= 4 * α_[1] n1.+4
3346
328b
    by rewrite alpha_4_3.
334d
3346
328a
  by lia.
328c
324c
have cE2 : c = 2 by case: (c) c_gt1 c_lt3 => [|[|[|]]].
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c32453255328d
cE2:c = 2
324c
rewrite (maxn_idPl _) // in y1Sy7SE; last by lia.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db62db7123f31ab31b431bd27bc285731cc31ec3235323b323c3255328d3359
y1Sy7SE:y1S + y7S <= (S_[1] n).*2 + α_[5] n
324c
rewrite cE2 -[_.-2]/4 in P8 *.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6123f31ab31b431bd27bc285731cc31ec3235323b323c3255328d3359335e
P8:(S_[4] n.+1).*2 <= x3S + y7S + y8S
(S_[3] n.+2).*2 <= 1.*2 + 2.*2 + x3S + y0S + y8S + x1S
have F1 := dsum_alphaL_S 5 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6123f31ab31b431bd27bc285731cc31ec3235323b323c3255328d3359335e3363
F1:S_[5] n.+1 = S_[5] n + α_[5] n
3364
have F2 := dsum_alpha3_S n.+2.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6123f31ab31b431bd27bc285731cc31ec3235323b323c3255328d3359335e33633369
F2:S_[1] n.+3 = (S_[3] n.+2).+1
3364
have F3 := leq_S5 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6123f31ab31b431bd27bc285731cc31ec3235323b323c3255328d3359335e33633369336e
F3:2 * S_[1] n.+1 + S_[1] n.+2 <= 3 * S_[5] n + 6
3364
have F4 := leq_S5 n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6123f31ab31b431bd27bc285731cc31ec3235323b323c3255328d3359335e33633369336e3373
F4:2 * S_[1] n.+2 + S_[1] n.+3 <= 3 * S_[5] n.+1 + 6
3364
have F5 := leq_S4 n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6123f31ab31b431bd27bc285731cc31ec3235323b323c3255328d3359335e33633369336e33733378
F5:5 * S_[1] n.+2 + S_[1] n.+3 <= 6 * S_[4] n.+1 + 9
3364
have F6 := dsum_alphaL_S 1 n.+2.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6123f31ab31b431bd27bc285731cc31ec3235323b323c3255328d3359335e33633369336e33733378337d
F6:S_[1] n.+3 = S_[1] n.+2 + α_[1] n.+2
3364
have F7 := dsum_alphaL_S 1 n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6123f31ab31b431bd27bc285731cc31ec3235323b323c3255328d3359335e33633369336e33733378337d3382
F7:S_[1] n.+2 = S_[1] n.+1 + α_[1] n.+1
3364
have F8 := dsum_alphaL_S 1 n.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6123f31ab31b431bd27bc285731cc31ec3235323b323c3255328d3359335e33633369336e33733378337d33823387
F8:S_[1] n.+1 = S_[1] n + α_[1] n
3364
have F10 := alphaL_1_2 n.+1.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6123f31ab31b431bd27bc285731cc31ec3235323b323c3255328d3359335e33633369336e33733378337d33823387338c
F10:α_[1] n.+2 <= (α_[1] n.+1).*2
3364
suff : 4 * α_[1] n.+1 <= 6 * α_[1] n + 3 by lia.
3390
4 * α_[1] n.+1 <= 6 * α_[1] n + 3
case: (n) => [|[|[|[|n1]]]]; rewrite ?alpha_small //.
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6123f31ab31b431bd27bc285731cc31ec3235323b323c3255328d3359335e33633369336e33733378337d33823387338c3391273c
4 * α_[1] n1.+1.+4 <= 6 * α_[1] n1.+4 + 3
have F9 : 3 * α_[1] n1.+1.+4 <= 4 * α_[1] n1.+4.
3399
334a
512915d255726f13160755816af161714144c14571512559570143616b0162859b1629166a1672167b16b155a5615935a05a55aa1884189518ae18c418c918ce18db5eb5f05f998eb8cb9163f191f66a1931683193a19f81a0c1a111a1b1a271a2f1a3c1a411a491b3d272b2730f0e27491b4f1cc01cc51cca18d31b572d462d532d542d802db6123f31ab31b431bd27bc285731cc31ec3235323b323c3255328d3359335e33633369336e33733378337d33823387338c3391273c
F9:3 * α_[1] n1.+1.+4 <= 4 * α_[1] n1.+4
339a
  by rewrite alpha_4_3.
33a0
339a
by lia.
Qed.

End Case3.

Lemma main p1 p2 p3 n l (u : {ffun 'I_l.+1 -> configuration 4 n}) :
   p1 != p2 -> p1 != p3 -> p2 != p3 -> 
   p1 != p0 -> p2 != p0 -> p3 != p0 ->
  (forall k : 'I_l.+1, 0 < k < l -> 
                       codom (u k) \subset [:: p2; a[p1, p3] k]) ->
  (S_[l] n).*2 <=  \sum_(i < l) `d[u (inord i), u (inord i.+1)]_smove + 
                    \sum_(k < n) (u ord0 k != p1) * β_[n, l] k +  
                    \sum_(k < n) (u ord_max k != a[p1, p3] l) * β_[n, l] k.
524f4ca
u:{ffun 'I_l.+1 -> configuration 4 n}
p1 != p2 ->
p1 != p3 ->
p2 != p3 ->
p1 != p0 ->
p2 != p0 ->
p3 != p0 ->
(forall k : 'I_l.+1,
 0 < k < l ->
 codom (u k) \subset [:: p2;  a[p1, p3] k]) ->
(S_[l] n).*2 <=
\sum_(i < l) `d[u (inord i), u (inord i.+1)]_smove +
\sum_(k < n) (u ord0 k != p1) * β_[n, l] k +
\sum_(k < n)
   (u ord_max k !=  a[p1, p3] l) * β_[n, l] k
Proof.
33a6
elim: n l p1 p2 p3 u => [|[|n] IH] l p1 p2 p3 
                u p1Dp2 p1Dp3 p2Dp3 p1Dp0 p2Dp0 p3Dp0 cH.
555724f
u:{ffun 'I_l.+1 -> configuration 4 0}
151617121314
cH:forall k : 'I_l.+1,
0 < k < l ->
codom (u k) \subset [:: p2;  a[p1, p3] k]
(S_[l] 0).*2 <=
\sum_(i < l) `d[u (inord i), u (inord i.+1)]_smove +
\sum_(k < 0) (u ord0 k != p1) * β_[0, l] k +
\sum_(k < 0)
   (u ord_max k !=  a[p1, p3] l) * β_[0, l] k
5
IH:forall (l : nat) (p1 p2 p3 : ordinal_eqType 4)
  (u : {ffun 'I_l.+1 -> configuration 4 0}),
p1 != p2 ->
p1 != p3 ->
p2 != p3 ->
p1 != p0 ->
p2 != p0 ->
p3 != p0 ->
(forall k : 'I_l.+1,
 0 < k < l ->
 codom (u k) \subset [:: p2;  a[p1, p3] k]) ->
(S_[l] 0).*2 <=
\sum_(i < l)
   `d[u (inord i), u (inord i.+1)]_smove +
\sum_(k < 0) (u ord0 k != p1) * β_[0, l] k +
\sum_(k < 0)
   (u ord_max k !=  a[p1, p3] l) * β_[0, l] k
55724f
u:{ffun 'I_l.+1 -> configuration 4 1}
15161712131433b1
(S_[l] 1).*2 <=
\sum_(i < l) `d[u (inord i), u (inord i.+1)]_smove +
\sum_(k < 1) (u ord0 k != p1) * β_[1, l] k +
\sum_(k < 1)
   (u ord_max k !=  a[p1, p3] l) * β_[1, l] k
5129
IH:forall (l : nat) (p1 p2 p3 : ordinal_eqType 4)
  (u : {ffun 'I_l.+1 -> configuration 4 n.+1}),
p1 != p2 ->
p1 != p3 ->
p2 != p3 ->
p1 != p0 ->
p2 != p0 ->
p3 != p0 ->
(forall k : 'I_l.+1,
 0 < k < l ->
 codom (u k) \subset [:: p2;  a[p1, p3] k]) ->
(S_[l] n.+1).*2 <=
\sum_(i < l)
   `d[u (inord i), u (inord i.+1)]_smove +
\sum_(k < n.+1) (u ord0 k != p1) * β_[n.+1, l] k +
\sum_(k < n.+1)
   (u ord_max k !=  a[p1, p3] l) * β_[n.+1, l] k
55724f
u:{ffun 'I_l.+1 -> configuration 4 n.+2}
15161712131433b1
(S_[l] n.+2).*2 <=
\sum_(i < l) `d[u (inord i), u (inord i.+1)]_smove +
\sum_(k < n.+2) (u ord0 k != p1) * β_[n.+2, l] k +
\sum_(k < n.+2)
   (u ord_max k !=  a[p1, p3] l) * β_[n.+2, l] k
-
33ad by rewrite /dsum_alphaL /conv /= dsum_alpha_0 muln0.
33b5
33b8
33b9
-
33c0 rewrite /dsum_alphaL /conv /= dsum_alpha_1 dsum_alpha_0 
          muln0 addn0 add0n muln1.
33b5
(minn 1.*2 l).*2 <=
\sum_(i < l) `d[u (inord i), u (inord i.+1)]_smove +
\sum_(k < 1) (u ord0 k != p1) * β_[1, l] k +
\sum_(k < 1)
   (u ord_max k !=  a[p1, p3] l) * β_[1, l] k
33c2
  rewrite [X in _ <= _ + X + _]big_ord_recl big_ord0  /= addn0.
33b5
(minn 1.*2 l).*2 <=
\sum_(i < l) `d[u (inord i), u (inord i.+1)]_smove +
(u ord0 ord0 != p1) * β_[1, l] 0 +
\sum_(k < 1)
   (u ord_max k !=  a[p1, p3] l) * β_[1, l] k
33c2
  rewrite [X in _ <= _ + X]big_ord_recl big_ord0  /= addn0.
33b5
(minn 1.*2 l).*2 <=
\sum_(i < l) `d[u (inord i), u (inord i.+1)]_smove +
(u ord0 ord0 != p1) * β_[1, l] 0 +
(u ord_max ord0 !=  a[p1, p3] l) * β_[1, l] 0
33c2
  have d0fE (c1 c2 : configuration 4 1) : 
    c1 != `c[p0] -> c2 != `c[p0] -> `d[c1, c2]_smove = (c1 != c2).*2.
533b655724f33b715161712131433b1
c1, c2:configuration 4 1
c1 != `c[p0] ->
c2 != `c[p0] -> `d[c1, c2]_smove = (c1 != c2).*2
533b655724f33b715161712131433b1
d0fE:forall c1 c2 : configuration 4 1,
c1 != `c[p0] ->
c2 != `c[p0] ->
`d[c1, c2]_smove = (c1 != c2).*2
33cf
33b9
    move=> c1Dp0 c2Dp0.
533b655724f33b715161712131433b133d4
c1Dp0:c1 != `c[p0]
c2Dp0:c2 != `c[p0]
`d[c1, c2]_smove = (c1 != c2).*2
33d6
    case: (gpath_connect (shanoi_connect c1 c2)) 
              => [] [/gpath_dist H |a [|b p]].
533b655724f33b715161712131433b133d433de33df
H:`d[c1, c2]_smove = size [::]
33e0
533b655724f33b715161712131433b133d433de33df
a:finfun_finType (ordinal_finType 1)
  (ordinal_finType 4)
gpath smove c1 c2 [:: a] ->
`d[c1, c2]_smove = (c1 != c2).*2
533b655724f33b715161712131433b133d433de33df
a, b:finfun_finType (ordinal_finType 1)
  (ordinal_finType 4)
p:seq
  (finfun_finType (ordinal_finType 1)
     (ordinal_finType 4))
gpath smove c1 c2 [:: a, b & p] ->
`d[c1, c2]_smove = (c1 != c2).*2
33d733b9
    -
33e2 by rewrite H; move/eqP: H; rewrite gdist_eq0 => ->.
33e8
33ea
33eb33d733b9
    -
33f2 case/gpathP; rewrite /= andbT => /moveP[z].
533b655724f33b715161712131433b133d433de33df33e9
z:ordinal_finType 1
[/\ srel (n:=4) (c1 z) (a z),
    forall d2 : ordinal_finType 1,
    z != d2 -> c1 d2 = a d2, on_top (q:=4) (n:=1) z c1
  & on_top (q:=4) (n:=1) z a] ->
a = c2 ->
`d[c1, c2]_smove = 1 ->
`d[c1, c2]_smove = (c1 != c2).*2
33f4
      rewrite [z]disk1_all => [] [zH _ _ _] aE.
533b655724f33b715161712131433b133d433de33df33e933fa
zH:srel (n:=4) (c1 sdisk) (a sdisk)
aE:a = c2
`d[c1, c2]_smove = 1 ->
`d[c1, c2]_smove = (c1 != c2).*2
33f4
      case/andP: zH => _; rewrite aE.
533b655724f33b715161712131433b133d433de33df33e933fa3401
c1 sdisk * c2 sdisk == 0 ->
`d[c1, c2]_smove = 1 ->
`d[c1, c2]_smove = (c1 != c2).*2
33f4
      move: c1Dp0 c2Dp0; rewrite !configuration1_eq !Ival_eq /= !ffunE /=.
533b655724f33b715161712131433b133d433e933fa3401
c1 sdisk != 0 ->
c2 sdisk != 0 ->
c1 sdisk * c2 sdisk == 0 ->
`d[c1, c2]_smove = 1 ->
`d[c1, c2]_smove = (c1 sdisk != c2 sdisk).*2
33f4
      by do 2 case: val.
33ec
33ef
33d6
    move=> /gpath_dist; case: eqP => [<-|/eqP c1Dc2]; first by rewrite gdist0.
533b655724f33b715161712131433b133d433de33df33ed33ee
c1Dc2:c1 != c2
`d[c1, c2]_smove = size [:: a, b & p] ->
`d[c1, c2]_smove = (~~ false).*2
33d6
    move=> H; apply/eqP; rewrite eqn_leq {2}H /= andbT.
533b655724f33b715161712131433b133d433de33df33ed33ee3414
H:`d[c1, c2]_smove = size [:: a, b & p]
`d[c1, c2]_smove <= 1.*2
33d6
    have pH : path smove c1 [::`c[p0]; c2].
3419
path smove c1 [:: `c[p0]; c2]
533b655724f33b715161712131433b133d433de33df33ed33ee3414341a
pH:path smove c1 [:: `c[p0]; c2]
341b
33d733b9
      rewrite /=; apply/and3P; split => //.
3419
c1 `-->_s `c[p0]
3419
`c[p0] `-->_s c2
342133d733b9
        apply/moveP; exists ord0; split => //=.
3419
srel (n:=4) (c1 ord0) (`c[p0] ord0)
3419
forall d2 : 'I_1, ord0 != d2 -> c1 d2 = `c[p0] d2
3419
on_top (q:=4) (n:=1) ord0 c1
3419
on_top (q:=4) (n:=1) ord0 `c[p0]
3429342133d733b9
        -
342c rewrite !ffunE; apply/andP; split; last by rewrite muln0.
3419
c1 ord0 != p0
342f
          by move: c1Dp0; rewrite configuration1_eq ffunE.
3419
3431
343234343429342133d733b9
        -
343c by move=> i; rewrite [i]disk1_all.
3419
3433
34343429342133d733b9
        -
3441 by apply/on_topP => j; rewrite [j]disk1_all.
3419
3435
3428
        by apply/on_topP => j; rewrite [j]disk1_all.
3419
342a
3420
      apply/moveP; exists ord0; split => //=.
3419
srel (n:=4) (`c[p0] ord0) (c2 ord0)
3419
forall d2 : 'I_1, ord0 != d2 -> `c[p0] d2 = c2 d2
3434
3419
on_top (q:=4) (n:=1) ord0 c2
342133d733b9
      -
344c rewrite !ffunE; apply/andP; split=> //.
3419
p0 != c2 ord0
344f
        by move: c2Dp0; rewrite configuration1_eq ffunE eq_sym.
3419
3451
34343452342133d733b9
      -
345a by move=> i; rewrite [i]disk1_all.
3447
3452342133d733b9
      -
345f by apply/on_topP => j; rewrite [j]disk1_all.
3419
3453
3420
      by apply/on_topP => j; rewrite [j]disk1_all.
3422
341b
33d6
    rewrite -[1.*2]/(size [:: `c[p0]; c2]).
3422
`d[c1, c2]_smove <= size [:: `c[p0]; c2]
33d6
    by apply: gdist_path_le pH _.
33d8
33cf
33c2
  case: l u cH => [|l] u cH; first by rewrite (minn_idPr _).
533b624f15161712131433d9557
u:{ffun 'I_l.+2 -> configuration 4 1}
559
(minn 1.*2 l.+1).*2 <=
\sum_(i < l.+1) `d[u (inord i), u (inord i.+1)]_smove +
(u ord0 ord0 != p1) * β_[1, l.+1] 0 +
(u ord_max ord0 !=  a[p1, p3] l.+1) * β_[1, l.+1] 0
33c2
  case: l u cH => [|l] u cH.
533b624f15161712131433d9
u:{ffun 'I_2 -> configuration 4 1}
cH:forall k : 'I_2,
0 < k < 1 ->
codom (u k) \subset [:: p2;  a[p1, p3] k]
(minn 1.*2 1).*2 <=
\sum_(i < 1) `d[u (inord i), u (inord i.+1)]_smove +
(u ord0 ord0 != p1) * β_[1, 1] 0 +
(u ord_max ord0 !=  a[p1, p3] 1) * β_[1, 1] 0
533b624f15161712131433d9557
u:{ffun 'I_l.+3 -> configuration 4 1}
cH:forall k : 'I_l.+3,
0 < k < l.+2 ->
codom (u k) \subset [:: p2;  a[p1, p3] k]
(minn 1.*2 l.+2).*2 <=
\sum_(i < l.+2) `d[u (inord i), u (inord i.+1)]_smove +
(u ord0 ord0 != p1) * β_[1, l.+2] 0 +
(u ord_max ord0 !=  a[p1, p3] l.+2) * β_[1, l.+2] 0
33b9
    rewrite (minn_idPr _) //.
3478
1.*2 <=
\sum_(i < 1) `d[u (inord i), u (inord i.+1)]_smove +
(u ord0 ord0 != p1) * β_[1, 1] 0 +
(u ord_max ord0 !=  a[p1, p3] 1) * β_[1, 1] 0
347c
    rewrite big_ord_recl big_ord0 addn0.
3478
1.*2 <=
`d[u (inord ord0), u (inord ord0.+1)]_smove +
(u ord0 ord0 != p1) * β_[1, 1] 0 +
(u ord_max ord0 !=  a[p1, p3] 1) * β_[1, 1] 0
347c
    rewrite inord_eq0 // (_ : inord 1 = ord_max); last first.
3478
inord 1 = ord_max
3478
1.*2 <=
`d[u ord0, u ord_max]_smove +
(u ord0 ord0 != p1) * β_[1, 1] 0 +
(u ord_max ord0 !=  a[p1, p3] 1) * β_[1, 1] 0
347d33b9
      by apply/val_eqP; rewrite /= inordK.
3478
3490
347c
    rewrite /beta !(apegS, apeg0) /= alphaL1E -[_ 0]/1; 
        do 2 case: eqP; rewrite /= ?addn0;
        try by move=> *; rewrite leq_addl.
3478
u ord_max ord0 = p3 ->
u ord0 ord0 = p1 ->
1.*2 <= `d[u ord0, u ord_max]_smove
347c
    move=> u1E u0E.
533b624f15161712131433d93479347a
u1E:u ord_max ord0 = p3
u0E:u ord0 ord0 = p1
1.*2 <= `d[u ord0, u ord_max]_smove
347c
    rewrite d0fE //.
349b
1.*2 <= (u ord0 != u ord_max).*2
349b
u ord0 != `c[p0]
349b
u ord_max != `c[p0]
347d33b9
    -
34a0 by rewrite configuration1_eq u1E u0E p1Dp3.
349b
34a5
34a6347d33b9
    -
34aa by rewrite configuration1_eq ffunE u0E.
349b
34a7
347c
    by rewrite configuration1_eq ffunE u1E.
347e
3481
33c2
  rewrite (minn_idPl _) // -[_.*2]/4.
347e
3 <
\sum_(i < l.+2) `d[u (inord i), u (inord i.+1)]_smove +
(u ord0 ord0 != p1) * β_[1, l.+2] 0 +
(u ord_max ord0 !=  a[p1, p3] l.+2) * β_[1, l.+2] 0
33c2
  rewrite big_ord_recl big_ord_recr /=.
347e
3 <
`d[u (inord 0), u (inord 1)]_smove +
(\sum_(i < l)
    `d[u (inord (bump 0 i)), u (inord (bump 0 i).+1)]_smove +
 `d[u (inord (bump 0 l)), u (inord (bump 0 l).+1)]_smove) +
(u ord0 ord0 != p1) * β_[1, l.+2] 0 +
(u ord_max ord0 !=  a[p1, p3] l.+2) * β_[1, l.+2] 0
33c2
  rewrite -!addnA addnC -!addnA (leq_trans _ (leq_addl _ _)) //.
347e
3 <
`d[u (inord (bump 0 l)), u (inord (bump 0 l).+1)]_smove +
((u ord0 ord0 != p1) * β_[1, l.+2] 0 +
 ((u ord_max ord0 !=  a[p1, p3] l.+2) * β_[1, l.+2] 0 +
  `d[u (inord 0), u (inord 1)]_smove))
33c2
  rewrite !apegS /beta /= alphaL_0 (minn_idPl _) /bump /= ?add1n //.
347e
3 <
`d[u (inord l.+1), u (inord l.+2)]_smove +
((u ord0 ord0 != p1) * 2 +
 ((u ord_max ord0 !=  a[p1, p3] l) * 2 +
  `d[u (inord 0), u (inord 1)]_smove))
33c2
  rewrite addnCA addnC !addnA -addnA -[4]/(2 + 2).
347e
2 + 2 <=
`d[u (inord l.+1), u (inord l.+2)]_smove +
(u ord_max ord0 !=  a[p1, p3] l) * 2 +
(`d[u (inord 0), u (inord 1)]_smove +
 (u ord0 ord0 != p1) * 2)
33c2
  apply: leq_add.
347e
1 <
`d[u (inord l.+1), u (inord l.+2)]_smove +
(u ord_max ord0 !=  a[p1, p3] l) * 2
347e
1 <
`d[u (inord 0), u (inord 1)]_smove +
(u ord0 ord0 != p1) * 2
33b9
    case: eqP => /= [umE|umD]; last by rewrite leq_addl.
533b624f15161712131433d9557347f3480
umE:u ord_max ord0 =  a[p1, p3] l
1 < `d[u (inord l.+1), u (inord l.+2)]_smove + 0 * 2
34cc
    rewrite addn0 d0fE.
34d2
1 < (u (inord l.+1) != u (inord l.+2)).*2
34d2
u (inord l.+1) != `c[p0]
34d2
u (inord l.+2) != `c[p0]
34cd33b9
    -
34d6 case: eqP => // H.
533b624f15161712131433d9557347f348034d3
H:u (inord l.+1) = u (inord l.+2)
1 < (~~ true).*2
34d9
      have /cH :  0 < (inord l.+1 : 'I_l.+3) < l.+2 by rewrite inordK /=.
34e2
codom (u (inord l.+1)) \subset
[:: p2;  a[p1, p3] (inord l.+1)] -> 1 < (~~ true).*2
34d9
      move=> /subsetP/(_ _ (codom_f _ sdisk)).
34e2
u (inord l.+1) sdisk
  \in [:: p2;  a[p1, p3] (inord l.+1)] ->
1 < (~~ true).*2
34d9
      rewrite H (_ : inord l.+2 = ord_max).
34e2
u ord_max sdisk \in [:: p2;  a[p1, p3] (inord l.+1)] ->
1 < (~~ true).*2
34e2
inord l.+2 = ord_max
34da34dc34cd33b9
        rewrite umE inordK // apegS //= !inE.
34e2
( a[p1, p3] l == p2) || ( a[p1, p3] l ==  a[p3, p1] l) ->
1 < 0.*2
34f1
        rewrite (apeg_eqC _ _ _) // (negPf (apeg_neq _ _ _)) //=.
34e2
p1 == p3 -> 1 < 0.*2
34e2
1008
34f234da34dc34cd33b9
          by rewrite (negPf p1Dp3).
34e2
1008
34f1
        by rewrite eq_sym.
34e2
34f3
34d9
      by apply/val_eqP; rewrite /= inordK.
34d2
34db
34dc34cd33b9
    -
3505 rewrite configuration1_eq ffunE.
34d2
u (inord l.+1) sdisk != p0
3507
      case: (_ =P _) => // H.
533b624f15161712131433d9557347f348034d3
H:u (inord l.+1) sdisk = p0
131a
3507
      have /cH :  0 < (inord l.+1 : 'I_l.+3) < l.+2 by rewrite inordK /=.
3510
codom (u (inord l.+1)) \subset
[:: p2;  a[p1, p3] (inord l.+1)] -> ~~ true
3507
      move=> /subsetP/(_ _ (codom_f _ sdisk)).
3510
u (inord l.+1) sdisk
  \in [:: p2;  a[p1, p3] (inord l.+1)] -> ~~ true
3507
      by rewrite H !inE eq_sym (negPf p2Dp0) /= eq_sym (negPf (apeg_neq _ _ _)).
34d2
34dd
34cc
    rewrite configuration1_eq ffunE.
34d2
u (inord l.+2) sdisk != p0
34cc
    rewrite (_ : inord l.+2 = ord_max).
34d2
u ord_max sdisk != p0
34d2
34f3
34cd33b9
      by rewrite umE apeg_neq.
34d2
34f3
34cc
    by apply/val_eqP; rewrite /= inordK.
347e
34ce
33c2
  case: eqP => /= [u0E|u0D]; last by rewrite leq_addl.
533b624f15161712131433d9557347f3480349d
1 < `d[u (inord 0), u (inord 1)]_smove + 0 * 2
33c2
  rewrite addn0 d0fE.
3530
1 < (u (inord 0) != u (inord 1)).*2
3530
u (inord 0) != `c[p0]
3530
u (inord 1) != `c[p0]
33b9
  -
3533 case: eqP => // H.
533b624f15161712131433d9557347f3480349d
H:u (inord 0) = u (inord 1)
34e4
3536
    have /cH :  0 < (inord 1 : 'I_l.+3) < l.+2 by rewrite inordK /=.
353f
codom (u (inord 1)) \subset
[:: p2;  a[p1, p3] (inord 1)] -> 1 < (~~ true).*2
3536
    move=> /subsetP/(_ _ (codom_f _ sdisk)).
353f
u (inord 1) sdisk \in [:: p2;  a[p1, p3] (inord 1)] ->
1 < (~~ true).*2
3536
    rewrite -H inord_eq0 // u0E inordK //= !(apegS, apeg0).
353f
p1 \in [:: p2; p3] -> 1 < 0.*2
3536
    by rewrite !inE (negPf p1Dp2) (negPf p1Dp3).
3530
3538
353933b9
  -
354e by rewrite configuration1_eq ffunE inord_eq0 // u0E.
3530
353a
33c2
  rewrite configuration1_eq ffunE.
3530
u (inord 1) sdisk != p0
33c2
  case: eqP => // H.
533b624f15161712131433d9557347f3480349d
H:u (inord 1) sdisk = p0
131a
33c2
  have /cH :  0 < (inord 1 : 'I_l.+3) < l.+2 by rewrite inordK /=.
355c
codom (u (inord 1)) \subset
[:: p2;  a[p1, p3] (inord 1)] -> ~~ true
33c2
  move=> /subsetP/(_ _ (codom_f _ sdisk)).
355c
u (inord 1) sdisk \in [:: p2;  a[p1, p3] (inord 1)] ->
~~ true
33c2
  by rewrite H !inE eq_sym (negPf p2Dp0) /= eq_sym (negPf (apeg_neq _ _ _)).
33ba
33bd
case: l u cH => [|l] u cH; first by rewrite S0E.
512933bb24f151617121314557558559
(S_[l.+1] n.+2).*2 <=
\sum_(i < l.+1) `d[u (inord i), u (inord i.+1)]_smove +
\sum_(k < n.+2) (u ord0 k != p1) * β_[n.+2, l.+1] k +
\sum_(k < n.+2)
   (u ord_max k !=  a[p1, p3] l.+1) * β_[n.+2, l.+1] k
rewrite apegS.
356c
(S_[l.+1] n.+2).*2 <=
\sum_(i < l.+1) `d[u (inord i), u (inord i.+1)]_smove +
\sum_(k < n.+2) (u ord0 k != p1) * β_[n.+2, l.+1] k +
\sum_(k < n.+2)
   (u ord_max k !=  a[p3, p1] l) * β_[n.+2, l.+1] k
pose u' : {ffun 'I_l.+2 -> configuration 4 n.+1} :=
     [ffun i : 'I_l.+2 => ↓[u i]].
512933bb24f151617121314557558559
u':=[ffun i => ↓[u i]]
  : {ffun 'I_l.+2 -> configuration 4 n.+1}:{ffun 'I_l.+2 -> configuration 4 n.+1}
3571
have H (k : 'I_l.+2) : 
  0 < k < l.+1 -> codom (u' k) \subset [:: p2; a[p1, p3] k].
512933bb24f151617121314557558559357617b2
0 < k < l.+1 ->
codom (u' k) \subset [:: p2;  a[p1, p3] k]
512933bb24f1516171213145575585593576561
3571
  by move=> k1; rewrite ffunE; apply/codom_liftr/cH.
357e
3571 
pose K := (u ord0 ord_max != p1) + 
          (u ord_max ord_max != a[p3, p1] l).
512933bb24f1516171213145575585593576561
K:=(u ord0 ord_max != p1) +
(u ord_max ord_max !=  a[p3, p1] l):nat
3571
have [HK|] := boolP ((K == 2) || ((K == 1) && (l == 0))).
512933bb24f15161712131455755855935765613586
HK:(K == 2) || (K == 1) && (l == 0)
3571
3585
~~ ((K == 2) || (K == 1) && (l == 0)) ->
(S_[l.+1] n.+2).*2 <=
\sum_(i0 < l.+1)
   `d[u (inord i0), u (inord i0.+1)]_smove +
\sum_(k < n.+2) (u ord0 k != p1) * β_[n.+2, l.+1] k +
\sum_(k < n.+2)
   (u ord_max k !=  a[p3, p1] l) * β_[n.+2, l.+1] k
  rewrite dsum_alphaL_S doubleD.
358a
(S_[l.+1] n.+1).*2 + (α_[l.+1] n.+1).*2 <=
\sum_(i < l.+1) `d[u (inord i), u (inord i.+1)]_smove +
\sum_(k < n.+2) (u ord0 k != p1) * β_[n.+2, l.+1] k +
\sum_(k < n.+2)
   (u ord_max k !=  a[p3, p1] l) * β_[n.+2, l.+1] k
358c
  have IH' := IH _ p1 p2 p3 u'
              p1Dp2 p1Dp3 p2Dp3 p1Dp0 p2Dp0 p3Dp0 H.
512933bb24f15161712131455755855935765613586358b
IH':(S_[l.+1] n.+1).*2 <=
\sum_(i < l.+1)
   `d[u' (inord i), u' (inord i.+1)]_smove +
\sum_(k < n.+1)
   (u' ord0 k != p1) * β_[n.+1, l.+1] k +
\sum_(k < n.+1)
   (u' ord_max k !=  a[p1, p3] l.+1) *
   β_[n.+1, l.+1] k
3592
358c
  rewrite apegS in IH'.
512933bb24f15161712131455755855935765613586358b
IH':(S_[l.+1] n.+1).*2 <=
\sum_(i < l.+1)
   `d[u' (inord i), u' (inord i.+1)]_smove +
\sum_(k < n.+1)
   (u' ord0 k != p1) * β_[n.+1, l.+1] k +
\sum_(k < n.+1)
   (u' ord_max k !=  a[p3, p1] l) *
   β_[n.+1, l.+1] k
3592
358c
  rewrite ![\sum_(_ < _.+2) _]big_ord_recr /=.
359b
(S_[l.+1] n.+1).*2 + (α_[l.+1] n.+1).*2 <=
\sum_(i < l.+1) `d[u (inord i), u (inord i.+1)]_smove +
(\sum_(i < n.+1)
    (u ord0 (widen_ord (m:=n.+2) (leqnSn n.+1) i)
     != p1) * β_[n.+2, l.+1] i +
 (u ord0 ord_max != p1) * β_[n.+2, l.+1] n.+1) +
(\sum_(i < n.+1)
    (u ord_max (widen_ord (m:=n.+2) (leqnSn n.+1) i)
     !=  a[p3, p1] l) * β_[n.+2, l.+1] i +
 (u ord_max ord_max !=  a[p3, p1] l) *
 β_[n.+2, l.+1] n.+1)
358c
  set u1 := \sum_ (_ < _) _; set u2 := \sum_ (_ < _) _; set u3 := \sum_ (_ < _) _.
512933bb24f15161712131455755855935765613586358b359c
u1:=\sum_(H < l.+1)
   `d[u (inord H), u (inord H.+1)]_smove:nat
u2:=\sum_(H < n.+1)
   (u ord0 (widen_ord (m:=n.+2) (leqnSn n.+1) H)
    != p1) * β_[n.+2, l.+1] H:nat
u3:=\sum_(H < n.+1)
   (u ord_max
      (widen_ord (m:=n.+2) (leqnSn n.+1) H)
    !=  a[p3, p1] l) * β_[n.+2, l.+1] H:nat
(S_[l.+1] n.+1).*2 + (α_[l.+1] n.+1).*2 <=
u1 +
(u2 + (u ord0 ord_max != p1) * β_[n.+2, l.+1] n.+1) +
(u3 +
 (u ord_max ord_max !=  a[p3, p1] l) *
 β_[n.+2, l.+1] n.+1)
358c
  rewrite -!addnA [_ + (u3 + _)]addnCA 2!addnA.
35a4
(S_[l.+1] n.+1).*2 + (α_[l.+1] n.+1).*2 <=
u1 + u2 + u3 +
((u ord0 ord_max != p1) * β_[n.+2, l.+1] n.+1 +
 (u ord_max ord_max !=  a[p3, p1] l) *
 β_[n.+2, l.+1] n.+1)
358c
  apply: leq_add.
35a4
(S_[l.+1] n.+1).*2 <= u1 + u2 + u3
35a4
(α_[l.+1] n.+1).*2 <=
(u ord0 ord_max != p1) * β_[n.+2, l.+1] n.+1 +
(u ord_max ord_max !=  a[p3, p1] l) *
β_[n.+2, l.+1] n.+1
358d
    apply: leq_trans IH' _.
512933bb24f15161712131455755855935765613586358b35a535a635a7
\sum_(i < l.+1)
   `d[u' (inord i), u' (inord i.+1)]_smove +
\sum_(k < n.+1) (u' ord0 k != p1) * β_[n.+1, l.+1] k +
\sum_(k < n.+1)
   (u' ord_max k !=  a[p3, p1] l) * β_[n.+1, l.+1] k <=
u1 + u2 + u3
35b1
    apply: leq_add; last first.
35b7
\sum_(k < n.+1)
   (u' ord_max k !=  a[p3, p1] l) * β_[n.+1, l.+1] k <=
u3
35b7
\sum_(i < l.+1)
   `d[u' (inord i), u' (inord i.+1)]_smove +
\sum_(k < n.+1) (u' ord0 k != p1) * β_[n.+1, l.+1] k <=
u1 + u2
35b2358d
      apply: leq_sum => i _.
512933bb24f15161712131455755855935765613586358b35a535a635a71054
(u' ord_max i !=  a[p3, p1] l) * β_[n.+1, l.+1] i <=
(u ord_max (widen_ord (m:=n.+2) (leqnSn n.+1) i)
 !=  a[p3, p1] l) * β_[n.+2, l.+1] i
35bd
      apply: leq_mul.
35c3
(u' ord_max i !=  a[p3, p1] l) <=
(u ord_max (widen_ord (m:=n.+2) (leqnSn n.+1) i)
 !=  a[p3, p1] l)
35c3
β_[n.+1, l.+1] i <= β_[n.+2, l.+1] i
35be35b2358d
        rewrite leq_eqVlt; apply/orP; left; apply/eqP.
35c3
(u' ord_max i !=  a[p3, p1] l) =
(u ord_max (widen_ord (m:=n.+2) (leqnSn n.+1) i)
 !=  a[p3, p1] l)
35c9
        by rewrite !ffunE; congr (u _ _ != _); apply/val_eqP.
35c3
35cb
35bd
      rewrite {2}/beta eqn_leq [_ <= i]leqNgt ltn_ord !andbF /=.
35c3
β_[n.+1, l.+1] i <= (α_[1] i).*2
35bd
    apply: geq_beta.
35b7
35bf
35b1
    apply: leq_add; last first.
35b7
\sum_(k < n.+1) (u' ord0 k != p1) * β_[n.+1, l.+1] k <=
u2
35b7
\sum_(i < l.+1)
   `d[u' (inord i), u' (inord i.+1)]_smove <= u1
35b2358d
      apply: leq_sum => i _.
35c3
(u' ord0 i != p1) * β_[n.+1, l.+1] i <=
(u ord0 (widen_ord (m:=n.+2) (leqnSn n.+1) i) != p1) *
β_[n.+2, l.+1] i
35de
      apply: leq_mul.
35c3
(u' ord0 i != p1) <=
(u ord0 (widen_ord (m:=n.+2) (leqnSn n.+1) i) != p1)
35ca35df35b2358d
        rewrite leq_eqVlt; apply/orP; left; apply/eqP.
35c3
(u' ord0 i != p1) =
(u ord0 (widen_ord (m:=n.+2) (leqnSn n.+1) i) != p1)
35e9
        by rewrite !ffunE; congr (u _ _ != _); apply/val_eqP.
35d235de
      rewrite {2}/beta eqn_leq [_ <= i]leqNgt ltn_ord !andbF /=.
35d535de
      apply: geq_beta.
35b7
35e0
35b1
    apply: leq_sum => i _.
512933bb24f15161712131455755855935765613586358b35a535a635a7
i:ordinal_finType l.+1
`d[u' (inord i), u' (inord i.+1)]_smove <=
`d[u (inord i), u (inord i.+1)]_smove
35b1
    rewrite !ffunE.
35f8
1b32
35b1
    by apply/gdist_cunlift/shanoi_connect.
35a4
35b3
358c
  rewrite -mulnDl -/K.
35a4
(α_[l.+1] n.+1).*2 <= K * β_[n.+2, l.+1] n.+1
358c
  case/orP: HK => [/eqP->|/andP[/eqP-> /eqP->]].
512933bb24f15161712131455755855935765613586359c35a535a635a7
(α_[l.+1] n.+1).*2 <= 2 * β_[n.+2, l.+1] n.+1
3608
(α_[1] n.+1).*2 <= 1 * β_[n.+2, 1] n.+1
358d
    by rewrite mul2n leq_double leq_beta.
3608
360c
358c
  by rewrite mul1n /beta /= /alphaL /delta !S1E.
3585
358e
rewrite negb_or negb_and.
3585
(K != 2) && ((K != 1) || (l != 0)) ->
(S_[l.+1] n.+2).*2 <=
\sum_(i0 < l.+1)
   `d[u (inord i0), u (inord i0.+1)]_smove +
\sum_(k < n.+2) (u ord0 k != p1) * β_[n.+2, l.+1] k +
\sum_(k < n.+2)
   (u ord_max k !=  a[p3, p1] l) * β_[n.+2, l.+1] k
have: K <= 2 by rewrite /K; do 2 case: (_ != _).
3585
K <= 2 ->
(K != 2) && ((K != 1) || (l != 0)) ->
(S_[l.+1] n.+2).*2 <=
\sum_(i0 < l.+1)
   `d[u (inord i0), u (inord i0.+1)]_smove +
\sum_(k < n.+2) (u ord0 k != p1) * β_[n.+2, l.+1] k +
\sum_(k < n.+2)
   (u ord_max k !=  a[p3, p1] l) * β_[n.+2, l.+1] k
case: ltngtP; rewrite // ltnS.
3585
K <= 1 ->
true ->
~~ false && ((K != 1) || (l != 0)) ->
(S_[l.+1] n.+2).*2 <=
\sum_(i1 < l.+1)
   `d[u (inord i1), u (inord i1.+1)]_smove +
\sum_(k < n.+2) (u ord0 k != p1) * β_[n.+2, l.+1] k +
\sum_(k < n.+2)
   (u ord_max k !=  a[p3, p1] l) * β_[n.+2, l.+1] k
case: ltngtP => //= KE _ _ l_gt0; last first.
512933bb24f15161712131455755855935765613586
KE:K = 1
572
3571
512933bb24f15161712131455755855935765613586
KE:K < 1
l_gt0:true
3571
  move: KE; rewrite /K.
512933bb24f15161712131455755855935765613586572
(u ord0 ord_max != p1) +
(u ord_max ord_max !=  a[p3, p1] l) = 1 ->
(S_[l.+1] n.+2).*2 <=
\sum_(i < l.+1) `d[u (inord i), u (inord i.+1)]_smove +
\sum_(k < n.+2) (u ord0 k != p1) * β_[n.+2, l.+1] k +
\sum_(k < n.+2)
   (u ord_max k !=  a[p3, p1] l) * β_[n.+2, l.+1] k
3624
  (case: eqP => [KH1|/eqP KH1] /=; case: eqP) => // [/eqP KH2|KH2] _.
512933bb24f15161712131455755855935765613586572570571
3571
512933bb24f15161712131455755855935765613586572
KH1:u ord0 ord_max != p1
1436
3571
3625
    by apply: (@case1 _ IH) cH KH1 KH2 l_gt0.
3634
3571
3624
  pose u1 : {ffun 'I_l.+2 -> configuration 4 n.+2} := 
     [ffun i : 'I_l.+2 => u (inord (l.+1 - i))].
512933bb24f151617121314557558559357656135865723635143616ca
3571
3624
  have -> : \sum_(i < l.+1) `d[u (inord i), u (inord i.+1)]_smove =
            \sum_(i < l.+1) `d[u1 (inord i), u1 (inord i.+1)]_smove.
363c
\sum_(i < l.+1) `d[u (inord i), u (inord i.+1)]_smove =
\sum_(i < l.+1)
   `d[u1 (inord i), u1 (inord i.+1)]_smove
363c
(S_[l.+1] n.+2).*2 <=
\sum_(i < l.+1)
   `d[u1 (inord i), u1 (inord i.+1)]_smove +
\sum_(k < n.+2) (u ord0 k != p1) * β_[n.+2, l.+1] k +
\sum_(k < n.+2)
   (u ord_max k !=  a[p3, p1] l) * β_[n.+2, l.+1] k
3625
    have F : injective (fun i : 'I_l.+1 => (inord (l - i) : 'I_l.+1)).
363c
16d5
512933bb24f151617121314557558559357656135865723635143616ca16d9
3640
36423625
      move=> i j /val_eqP.
512933bb24f151617121314557558559357656135865723635143616ca16de
16df
3647
      have lBiLl (i1 : 'I_l.+1) : l - i1 < l.+1.
512933bb24f151617121314557558559357656135865723635143616ca16e4
16e5
512933bb24f151617121314557558559357656135865723635143616ca16de16e9
16df
364836423625
        by rewrite ltn_subLR ?leq_addl // -ltnS.
3654
16df
3647
      rewrite /= !inordK // => /eqP liE; apply/val_eqP => /=.
512933bb24f151617121314557558559357656135865723635143616ca16de16e916f1
16f2
3647
      rewrite -(subKn (_ : i <= l)); last by rewrite -ltnS.
365b
16f6
3647
      by rewrite liE subKn // -ltnS.
3649
3640
3641
    rewrite (reindex_inj F) /=.
3649
\sum_(j < l.+1)
   `d[u (inord (inord (l - j))), u
                                   (inord
                                      (inord (l - j)).+1)]_smove =
\sum_(i < l.+1)
   `d[u1 (inord i), u1 (inord i.+1)]_smove
3641
    apply: eq_bigr => i _; rewrite !ffunE.
512933bb24f151617121314557558559357656135865723635143616ca16d914c4
1702
3641
    have iLl2 : i < l.+2 by apply: leq_trans (ltn_ord _) _.
512933bb24f151617121314557558559357656135865723635143616ca16d914c41707
1702
3641
    have lBiLl : l - i < l.+1 by rewrite ltn_subLR ?leq_addl // -ltnS.
512933bb24f151617121314557558559357656135865723635143616ca16d914c41707170c
1702
3641
    have iLl : i <= l by rewrite -ltnS.
512933bb24f151617121314557558559357656135865723635143616ca16d914c41707170c1711
1702
3641
    rewrite gdistC; last by apply/move_sym/ssym.
3675
1715
3641
    congr (`d[u _,u _]_smove).
3675
1719
3675
171c
36423625
      by apply/val_eqP; rewrite /= !inordK ?subSn.
3675
171c
3641
    by apply/val_eqP; rewrite /= !inordK // ?subSS ltnS // ltnW.
363c
3643
3624
  pose p1' := a[p3, p1] l.
512933bb24f151617121314557558559357656135865723635143616ca
p1':= a[p3, p1] l:peg 4
3643
3624
  pose p3' := a[p1, p3] l.
512933bb24f151617121314557558559357656135865723635143616ca3688
p3':= a[p1, p3] l:peg 4
3643
3624
  have -> : \sum_(k < n.+2) (u ord0 k != p1) * β_[n.+2, l.+1] k =
            \sum_(k < n.+2) (u1 ord_max k != a[p1', p3'] l.+1) * 
                            β_[n.+2, l.+1] k.
368c
\sum_(k < n.+2) (u ord0 k != p1) * β_[n.+2, l.+1] k =
\sum_(k < n.+2)
   (u1 ord_max k !=  a[p1', p3'] l.+1) *
   β_[n.+2, l.+1] k
368c
(S_[l.+1] n.+2).*2 <=
\sum_(i < l.+1)
   `d[u1 (inord i), u1 (inord i.+1)]_smove +
\sum_(k < n.+2)
   (u1 ord_max k !=  a[p1', p3'] l.+1) *
   β_[n.+2, l.+1] k +
\sum_(k < n.+2)
   (u ord_max k !=  a[p3, p1] l) * β_[n.+2, l.+1] k
3625
    apply: eq_bigr => i _; congr (_ * _).
512933bb24f151617121314557558559357656135865723635143616ca3688368d1734
(u ord0 i != p1) = (u1 ord_max i !=  a[p1', p3'] l.+1)
3692
    rewrite ffunE subnn.
3698
(u ord0 i != p1) =
(u (inord 0) i !=  a[p1', p3'] l.+1)
3692
    by rewrite inord_eq0 // -apegD addSn addnn apegS apeg_double.
368c
3694
3624
  have -> : \sum_(k < n.+2) 
              (u ord_max k != a[p3, p1] l) * β_[n.+2, l.+1] k =
            \sum_(k < n.+2) (u1 ord0 k != p1') * β_[n.+2, l.+1] k.
368c
\sum_(k < n.+2)
   (u ord_max k !=  a[p3, p1] l) * β_[n.+2, l.+1] k =
\sum_(k < n.+2) (u1 ord0 k != p1') * β_[n.+2, l.+1] k
368c
(S_[l.+1] n.+2).*2 <=
\sum_(i < l.+1)
   `d[u1 (inord i), u1 (inord i.+1)]_smove +
\sum_(k < n.+2)
   (u1 ord_max k !=  a[p1', p3'] l.+1) *
   β_[n.+2, l.+1] k +
\sum_(k < n.+2) (u1 ord0 k != p1') * β_[n.+2, l.+1] k
3625
    apply: eq_bigr => i _; congr (_ * _).
3698
(u ord_max i !=  a[p3, p1] l) = (u1 ord0 i != p1')
36a5
    rewrite ffunE subn0.
3698
(u ord_max i !=  a[p3, p1] l) =
(u (inord l.+1) i != p1')
36a5
    suff -> : inord l.+1 = ord_max :> 'I_l.+2 by [].
3698
a16
36a5 
    by apply/val_eqP; rewrite /= inordK.
368c
36a7
3624
  have cH1 (k : 'I_l.+2) : 
     0 < k < l.+1 -> codom (u1 k) \subset [:: p2; a[p1', p3'] k].
512933bb24f151617121314557558559357656135865723635143616ca3688368d17b2
0 < k < l.+1 ->
codom (u1 k) \subset [:: p2;  a[p1', p3'] k]
512933bb24f151617121314557558559357656135865723635143616ca3688368d
cH1:forall k : 'I_l.+2,
0 < k < l.+1 ->
codom (u1 k) \subset [:: p2;  a[p1', p3'] k]
36a7
3625
    move=> /andP[kH1 kH2]; rewrite ffunE.
512933bb24f151617121314557558559357656135865723635143616ca3688368d17b217b317b4
codom (u (inord (l.+1 - k))) \subset
[:: p2;  a[p1', p3'] k]
36bb
    have F1 : k <= l.+1 by apply: ltnW.
512933bb24f151617121314557558559357656135865723635143616ca3688368d17b217b317b417ba
36c3
36bb
    have F2 : l.+1 - k < l.+2.
36c7
17be
512933bb24f151617121314557558559357656135865723635143616ca3688368d17b217b317b417ba17c2
36c3
36bc3625
      by rewrite ltn_subLR // addnS ltnS leq_addl.
36cd
36c3
36bb
    have -> :  a[p1', p3'] k = a[p1, p3] (inord (l.+1 - k) : 'I_l.+2).
36cd
 a[p1', p3'] k =  a[p1, p3] (inord (l.+1 - k))
36cd
17cc
36bc3625
      rewrite -apegD inordK // [RHS]apegO oddB //.
36cd
 a[p3, p1] (k + l) =
 a[p3, p1] (~~ (odd l.+1 (+) odd k))
36d5
      by rewrite -oddD -apegO addSn apegS addnC.
36cd
17cc
36bb
    apply: cH; rewrite inordK // subn_gt0 // ltn_subLR // kH2 //=.
512933bb24f151617121314557558357656135865723635143616ca3688368d17b217b317b417ba17c2
17d8
36bb
    by rewrite addnS ltnS -{1}[l.+1]add1n leq_add2r.
36bd
36a7
3624
  rewrite -addnA [X in _ <= _ + X]addnC addnA apegS.
36bd
(S_[l.+1] n.+2).*2 <=
\sum_(i < l.+1)
   `d[u1 (inord i), u1 (inord i.+1)]_smove +
\sum_(k < n.+2) (u1 ord0 k != p1') * β_[n.+2, l.+1] k +
\sum_(k < n.+2)
   (u1 ord_max k !=  a[p3', p1'] l) * β_[n.+2, l.+1] k
3624
  apply: (@case1 _ IH) cH1 _  _ _ => //.
368c
p1' != p2
368c
p1' != p3'
368c
p2 != p3'
368c
176f
368c
1777
368c
u1 ord0 ord_max = p1'
368c
u1 ord_max ord_max !=  a[p3', p1'] l
3625
  -
36ea by rewrite apeg_neq // eq_sym.
368c
36ef
36f036f236f336f436f63625
  -
36fa by rewrite apeg_eqC // eq_sym.
368c
36f1
36f236f336f436f63625
  -
36ff by rewrite eq_sym apeg_neq // eq_sym.
368c
176f
36f336f436f63625
  -
3704 by rewrite apeg_neq // eq_sym.
368c
1777
36f436f63625
  -
3709 by rewrite apeg_neq // eq_sym.
368c
36f5
36f63625
  -
370e rewrite ffunE subn0.
368c
u (inord l.+1) ord_max = p1'
3710
    suff -> : inord l.+1 = ord_max :> 'I_l.+2 by [].
368c
a16
3710
    by apply/val_eqP; rewrite /= inordK.
368c
36f7
3624
  by rewrite ffunE subnn inord_eq0 // -apegD addnn apeg_double.
3626
3571
move: KE; rewrite /K; do 2 case: eqP => //=.
512933bb24f151617121314557558559357656135863628
u ord_max ord_max =  a[p3, p1] l ->
u ord0 ord_max = p1 ->
0 + 0 < 1 ->
(S_[l.+1] n.+2).*2 <=
\sum_(i < l.+1) `d[u (inord i), u (inord i.+1)]_smove +
\sum_(k < n.+2) (u ord0 k != p1) * β_[n.+2, l.+1] k +
\sum_(k < n.+2)
   (u ord_max k !=  a[p3, p1] l) * β_[n.+2, l.+1] k
move => uME u0E _.
512933bb24f151617121314557558559357656135863628
uME:u ord_max ord_max =  a[p3, p1] l
u0E:u ord0 ord_max = p1
3571
by apply: (case3 IH _ _ _ _ _ _ cH).
Qed.

Lemma main_cor p1 p2 n :
  p1 != p0 -> p2 != p0 -> p1 != p2 ->  
  (S_[1] n).*2 <= `d[`c[p1, n],`c[p2, n]]_smove.
127
p1 != p0 ->
p2 != p0 ->
p1 != p2 ->
(S_[1] n).*2 <= `d[`c[p1 , n], `c[p2 , n]]_smove
Proof.
372b
move=> p1Dp0 p2Dp0 p1Dp2.
5128129121315
(S_[1] n).*2 <= `d[`c[p1 , n], `c[p2 , n]]_smove
rewrite eq_sym in p2Dp0.
51281291215157
3733
have [p3 [[p3Dp2 p3Dp0 p3Dp1] _]] := peg4comp3 p1Dp0 p1Dp2 p2Dp0.
5128129121515715c15d1415e
(S_[1] n).*2 <= `d[`c[p1], `c[p2]]_smove
rewrite eq_sym in p2Dp0.
5128129121515c15d1415e13
373c
pose u := [ffun i : 'I_2 => if i == 0 :> nat then
                             `c[p1, n] else `c[p2, n]].
5128129121515c15d1415e13
u:=[ffun i => if i == 0 :> nat
           then `c[p1 , n]
           else `c[p2 , n]]:{ffun forall x : ordinal_finType 2,
      (fun=> configuration 4 n) x}
373c
suff :
  (S_[1] n).*2 <=
  \sum_(i < 1) `d[u (inord i), u (inord i.+1)]_smove +
  \sum_(k < n) (u ord0 k != p1) * β_[n, 1] k +
  \sum_(k < n) (u ord_max k != p2) * β_[n, 1] k.
3744
(S_[1] n).*2 <=
\sum_(i < 1) `d[u (inord i), u (inord i.+1)]_smove +
\sum_(k < n) (u ord0 k != p1) * β_[n, 1] k +
\sum_(k < n) (u ord_max k != p2) * β_[n, 1] k ->
(S_[1] n).*2 <= `d[`c[p1], `c[p2]]_smove
3744
(S_[1] n).*2 <=
\sum_(i < 1) `d[u (inord i), u (inord i.+1)]_smove +
\sum_(k < n) (u ord0 k != p1) * β_[n, 1] k +
\sum_(k < n) (u ord_max k != p2) * β_[n, 1] k
  rewrite big_ord_recr big_ord0 !ffunE /= inord_eq0 //= add0n.
3744
(S_[1] n).*2 <=
`d[`c[p1], if inord 1 == 0 then `c[p1] else `c[p2]]_smove +
\sum_(k < n) (`c[p1] k != p1) * β_[n, 1] k +
\sum_(k < n) (`c[p2] k != p2) * β_[n, 1] k ->
(S_[1] n).*2 <= `d[`c[p1], `c[p2]]_smove
374a
  rewrite inordK //=.
3744
(S_[1] n).*2 <=
`d[`c[p1], `c[p2]]_smove +
\sum_(k < n) (`c[p1] k != p1) * β_[n, 1] k +
\sum_(k < n) (`c[p2] k != p2) * β_[n, 1] k ->
(S_[1] n).*2 <= `d[`c[p1], `c[p2]]_smove
374a
  rewrite (eq_bigr (fun i : 'I_n => 0)) => [|i _]; last first.
5128129121515c15d1415e1337454ee
(`c[p1] i != p1) * β_[n, 1] i = 0
3744
(S_[1] n).*2 <=
`d[`c[p1], `c[p2]]_smove + \sum_(i < n) 0 +
\sum_(k < n) (`c[p2] k != p2) * β_[n, 1] k ->
(S_[1] n).*2 <= `d[`c[p1], `c[p2]]_smove
374b
    by rewrite !ffunE eqxx.
3744
375c
374a
  rewrite -(big_mkord xpredT (fun=> 0)) sum_nat_const_nat muln0.
3744
(S_[1] n).*2 <=
`d[`c[p1], `c[p2]]_smove + 0 +
\sum_(k < n) (`c[p2] k != p2) * β_[n, 1] k ->
(S_[1] n).*2 <= `d[`c[p1], `c[p2]]_smove
374a
  rewrite (eq_bigr (fun i : 'I_n => 0)) => [|i _]; last first.
3758
(`c[p2] i != p2) * β_[n, 1] i = 0
3744
(S_[1] n).*2 <=
`d[`c[p1], `c[p2]]_smove + 0 + \sum_(i < n) 0 ->
(S_[1] n).*2 <= `d[`c[p1], `c[p2]]_smove
374b
    by rewrite !ffunE eqxx.
3744
376a
374a
  rewrite -(big_mkord xpredT (fun=> 0)) sum_nat_const_nat muln0.
3744
(S_[1] n).*2 <= `d[`c[p1], `c[p2]]_smove + 0 + 0 ->
(S_[1] n).*2 <= `d[`c[p1], `c[p2]]_smove
374a
  by rewrite !addn0.
3744
374c
apply: (@main p1 p3 p2) => //.
3744
p1 != p3
3744
forall k : 'I_2,
0 < k < 1 -> codom (u k) \subset [:: p3;  a[p1, p2] k]
  by rewrite eq_sym.
3744
377b
by case=> [] [].
Qed.

Lemma gdist_shanoi4 (n : nat) (p1 p2 : peg 4) : 
  p1 != p2 -> p1 != p0 -> p2 != p0 -> 
  `d[`c[p1, n], `c[p2, n]]_smove = (S_[1] n).*2.
5129229
p1 != p2 ->
p1 != p0 ->
p2 != p0 ->
`d[`c[p1 , n], `c[p2 , n]]_smove = (S_[1] n).*2
Proof.
3780
move=> p1Dp2 p1Dp0 p2Dp0; apply/eqP; rewrite eqn_leq.
5129229151213
`d[`c[p1], `c[p2]]_smove <= (S_[1] n).*2 <=
`d[`c[p1], `c[p2]]_smove
by rewrite leq_shanoi4 // main_cor.
Qed.

End sHanoi4.