# Theory Coxeter

theory Coxeter
imports Chamber
```section ‹Coxeter systems and complexes›

text ‹
A Coxeter system is a group that affords a presentation, where each generator is of order two,
and each relator is an alternating word of even length in two generators.
›

theory Coxeter
imports Chamber

begin

subsection ‹Coxeter-like systems›

text ‹First we work in a group generated by elements of order two.›

subsubsection ‹Locale definition and basic facts›

locale PreCoxeterSystem =
assumes genset_order2: "s∈S ⟹ add_order s = 2"
begin

abbreviation "W ≡ ⟨S⟩"
abbreviation "S_length  ≡ word_length S"
abbreviation "S_reduced_for ≡ reduced_word_for S"
abbreviation "S_reduced ≡ reduced_word S"
abbreviation "relfun ≡ λs t. add_order (s+t)"

lemma no_zero_genset: "0∉S"
proof
assume "0∈S"
ultimately show False using genset_order2 by simp
qed

lemma genset_order2_add: "s∈S ⟹ s + s = 0"

lemma relfun_S: "s∈S ⟹ relfun s s = 1"

lemma relfun_eq1: "⟦ s∈S; relfun s t = 1 ⟧ ⟹ t=s"

lemma S_relator_list: "s∈S ⟹ pair_relator_list s s = [s,s]"
using relfun_S alternating_list2 by simp

lemma S_sym: "T ⊆ S ⟹ uminus ` T ⊆ T"
using genset_uminus by auto

lemmas special_subgroup_eq_sum_list =
genby_sym_eq_sum_lists[OF S_sym]
lemmas genby_S_reduced_word_for_arg_min =
reduced_word_for_genby_sym_arg_min[OF S_sym]
lemmas in_genby_S_reduced_letter_set =
in_genby_sym_imp_in_reduced_letter_set[OF S_sym]

end (* context PreCoxeterSystem *)

subsubsection ‹Special cosets›

text ‹
From a Coxeter system we will eventually construct an associated chamber complex. To do so, we
will consider the collection of special cosets: left cosets of subgroups generated by subsets of
the generating set @{term S}. This collection forms a poset under the supset relation that, under
a certain extra assumption, can be used to form a simplicial complex whose poset of simplices
is isomorphic to this poset of special cosets.  In the literature, groups generated by subsets of
@{term S} are often referred to as parabolic subgroups of @{term W}, and their cosets as parabolic
cosets, but following Garrett \cite{Garrett:Buildings} we have opted for the names special
subgroups and special cosets.
›

context PreCoxeterSystem
begin

definition special_cosets :: "'w set set"
where "special_cosets ≡ (⋃T∈Pow S. (⋃w∈W. { w +o ⟨T⟩ }))"
abbreviation "𝒫 ≡ special_cosets"

lemma special_cosetsI: "T∈Pow S ⟹ w∈W ⟹ w +o ⟨T⟩ ∈ 𝒫"
using special_cosets_def by auto

lemma special_coset_singleton: "w∈W ⟹ {w}∈𝒫"
using special_cosetsI genby_lcoset_empty[of w] by fastforce

lemma special_coset_nempty: "X∈𝒫 ⟹ X ≠ {}"
using special_cosets_def genby_lcoset_refl by fastforce

lemma special_subgroup_special_coset: "T∈Pow S ⟹ ⟨T⟩ ∈ 𝒫"
using genby_0_closed special_cosetsI[of T] by fastforce

lemma special_cosets_lcoset_closed: "w∈W ⟹ X∈𝒫 ⟹ w +o X ∈ 𝒫"

lemma special_cosets_lcoset_shift: "w∈W ⟹ ((+o) w) ` 𝒫 = 𝒫"
using special_cosets_lcoset_closed genby_uminus_closed

lemma special_cosets_has_bottom: "supset_has_bottom 𝒫"
proof (rule ordering.has_bottomI, rule supset_poset)
show "W∈𝒫" using special_subgroup_special_coset by fast
next
fix X assume X: "X∈𝒫"
from this obtain w T where wT: "w∈W" "T∈Pow S" "X = w +o ⟨T⟩"
using special_cosets_def by auto
thus "X ⊆ W" using genby_mono[of T] genby_lcoset_closed[of w] by auto
qed

lemma special_cosets_bottom: "supset_bottom 𝒫 = W"
proof (rule supset_bottomI[THEN sym])
fix X assume "X∈𝒫"
from this obtain w T where "w∈W" "T∈Pow S" "X = w +o ⟨T⟩"
using special_cosets_def by auto
thus "X⊆W"
using genby_mono[of T S] set_plus_mono[of "⟨T⟩" W] genby_lcoset_el_reduce
by    force

end (* context PreCoxeterSystem *)

subsubsection ‹Transfer from the free group over generators›

text ‹
We form a set of relators and show that it and @{term S} form a
@{const GroupWithGeneratorsRelators}. The associated quotient group @{term G} maps surjectively
onto @{term W}. In the ‹CoxeterSystem› locale below, this correspondence will be assumed
to be injective as well.
›

context PreCoxeterSystem
begin

abbreviation R :: "'w list set" where "R ≡ (⋃s∈S. ⋃t∈S. {pair_relator_list s t})"
abbreviation "P ≡ map (charpair S) ` R"
abbreviation "P' ≡ GroupWithGeneratorsRelators.P' S R"
abbreviation "Q ≡ GroupWithGeneratorsRelators.Q S R"
abbreviation "G ≡ GroupWithGeneratorsRelators.G S R"
abbreviation "relator_freeword ≡
GroupWithGeneratorsRelators.relator_freeword S"
abbreviation pair_relator_freeword :: "'w ⇒ 'w ⇒ 'w freeword"
where "pair_relator_freeword s t ≡ Abs_freelist (pair_relator_list s t)"

abbreviation "freeliftid ≡ freeword_funlift id"

abbreviation induced_id :: "'w freeword set permutation ⇒ 'w"
where "induced_id ≡ GroupWithGeneratorsRelators.induced_id S R"

lemma S_relator_freeword: "s∈S ⟹ pair_relator_freeword s s = s[+]s"

lemma map_charpair_map_pairtrue_R:
"s∈S ⟹ t∈S ⟹
map (charpair S) (pair_relator_list s t) = map pairtrue (pair_relator_list s t)"
using set_alternating_list map_charpair_uniform by fastforce

lemma relator_freeword:
"s∈S ⟹ t∈S ⟹
pair_relator_freeword s t = relator_freeword (pair_relator_list s t)"
using set_alternating_list
arg_cong[OF map_charpair_map_pairtrue_R, of s t Abs_freeword]
by    fastforce

lemma relator_freewords: "Abs_freelist ` R = P'"
using relator_freeword by force

lemma GroupWithGeneratorsRelators_S_R: "GroupWithGeneratorsRelators S R"
proof
fix rs assume rs: "rs∈R"
hence rs': "rs ∈ lists S" using set_alternating_list by fast
from rs' show "rs ∈ lists (S ∪ uminus ` S)" by fast
from rs show "sum_list rs = 0" using sum_list_pair_relator_list by fast
from rs' show "proper_signed_list (map (charpair S) rs)"
using proper_signed_list_map_uniform_snd
arg_cong[of "map (charpair S) rs" "map pairtrue rs" proper_signed_list]
by    fastforce
qed

lemmas GroupByPresentation_S_P =
GroupWithGeneratorsRelators.GroupByPresentation_S_P[
OF GroupWithGeneratorsRelators_S_R
]

lemmas Q_FreeS = GroupByPresentation.Q_FreeS[OF GroupByPresentation_S_P]

lemma relator_freeword_Q: "s∈S ⟹ t∈S ⟹ pair_relator_freeword s t ∈ Q"
using relator_freeword
GroupByPresentation.relators[OF GroupByPresentation_S_P]
by    fastforce

lemmas P'_FreeS =
GroupWithGeneratorsRelators.P'_FreeS[
OF GroupWithGeneratorsRelators_S_R
]

lemmas GroupByPresentationInducedFun_S_P_id =
GroupWithGeneratorsRelators.GroupByPresentationInducedFun_S_P_id[
OF GroupWithGeneratorsRelators_S_R
]

lemma rconj_relator_freeword:
"⟦ s∈S; t∈S; proper_signed_list xs; fst ` set xs ⊆ S ⟧ ⟹
rconjby (Abs_freeword xs) (pair_relator_freeword s t) ∈ Q"
using GroupWithGeneratorsRelators.rconj_relator_freeword[
OF GroupWithGeneratorsRelators_S_R
]
relator_freeword
by    force

lemma lconjby_Abs_freelist_relator_freeword:
"⟦ s∈S; t∈S; xs∈lists S ⟧ ⟹
lconjby (Abs_freelist xs) (pair_relator_freeword s t) ∈ Q"
using GroupWithGeneratorsRelators.lconjby_Abs_freelist_relator_freeword[
OF GroupWithGeneratorsRelators_S_R
]
relator_freeword
by    force

lemma Abs_freelist_rev_append_alternating_list_in_Q:
assumes "s∈S" "t∈S"
shows "Abs_freelist (rev (alternating_list n s t) @ alternating_list n s t) ∈ Q"
proof (induct n)
case (Suc m)
define u where "u = (if even m then s else t)"
define x where "x = Abs_freelist (rev (alternating_list m s t) @ alternating_list m s t)"
from u_def x_def assms have
"Abs_freelist (rev (alternating_list (Suc m) s t) @
alternating_list (Suc m) s t) =
(pair_relator_freeword u u) + rconjby (Abs_freeletter u) x"
using Abs_freelist_append[of
"u # rev (alternating_list m s t) @ alternating_list m s t"
"[u]"
]
Abs_freelist_Cons[of
u
"rev (alternating_list m s t) @ alternating_list m s t"
]
moreover from Suc assms u_def x_def have "rconjby (Abs_freeletter u) x ∈ Q"
using Abs_freeletter_in_FreeGroup_iff[of _ S]
FreeGroup_genby_set_lconjby_set_rconjby_closed
by    fastforce
ultimately show ?case
using u_def assms relator_freeword_Q genby_add_closed by fastforce
qed (simp add: zero_freeword.abs_eq[THEN sym] genby_0_closed)

"proper_signed_list xs ⟹ fst ` set xs ⊆ S ⟹
- Abs_freelistfst xs + Abs_freeword xs ∈ Q"
proof (induct xs)
case (Cons x xs)
from Cons(2) have 1:
"- Abs_freelistfst (x#xs) + Abs_freeword (x#xs) =
-Abs_freelistfst xs + -Abs_freeletter (fst x)
+ Abs_freeword [x] + Abs_freeword xs"
using Abs_freelist_Cons[of "fst x" "map fst xs"]
show ?case
proof (cases "snd x")
case True
with Cons show ?thesis
using 1
Abs_freeletter_prod_conv_Abs_freeword
binrelchain_Cons_reduce
)
next
case False
define s where "s = fst x"
with Cons(3) have s_S: "s∈S" by simp
define q where "q = rconjby (Abs_freelistfst xs) (pair_relator_freeword s s)"
from s_def False Cons(3) have
"- Abs_freelistfst (x#xs) + Abs_freeword (x#xs) =
-Abs_freelistfst xs + -pair_relator_freeword s s + Abs_freeword xs"
using 1 surjective_pairing[of x] S_relator_freeword[of s]
uminus_Abs_freeword_singleton[of s False, THEN sym]
with q_def have 2:
"- Abs_freelistfst (x#xs) + Abs_freeword (x#xs) =
-q + (-Abs_freelistfst xs + Abs_freeword xs)"
moreover from q_def s_def Cons(3) have "-q∈Q"
using proper_signed_list_map_uniform_snd[of True "map fst xs"]
rconj_relator_freeword genby_uminus_closed
by    fastforce
moreover from Cons have "-Abs_freelistfst xs + Abs_freeword xs ∈ Q"
ultimately show ?thesis using genby_add_closed by simp
qed
qed (simp add: zero_freeword.abs_eq[THEN sym] genby_0_closed)

lemma Q_freelist_freeword':
"⟦ proper_signed_list xs; fst ` set xs ⊆ S; Abs_freelistfst xs ∈ Q ⟧ ⟹
Abs_freeword xs ∈ Q"
by    fastforce

lemma Q_freelist_freeword:
"c ∈ FreeGroup S ⟹ Abs_freelist (map fst (freeword c)) ∈ Q ⟹ c ∈ Q"
using freeword FreeGroupD Q_freelist_freeword' freeword_inverse[of c]
by     fastforce

text ‹
Here we show that the lift of the identity map to the free group on @{term S} is really just
summation.
›

lemma freeliftid_Abs_freeword_conv_sum_list:
"proper_signed_list xs ⟹ fst ` set xs ⊆ S ⟹
freeliftid (Abs_freeword xs) = sum_list (map fst xs)"
using freeword_funlift_Abs_freeword[of xs id] genset_uminus
sum_list_map_cong[of xs "apply_sign id" fst]
by    fastforce

end (* context PreCoxeterSystem *)

subsubsection ‹Words in generators containing alternating subwords›

text ‹
Besides cancelling subwords equal to relators, the primary algebraic manipulation in seeking to
reduce a word in generators in a Coxeter system is to reverse the order of alternating subwords
of half the length of the associated relator, in order to create adjacent repeated letters that
can be cancelled. Here we detail the mechanics of such manipulations.
›

context PreCoxeterSystem
begin

lemma sum_list_pair_relator_halflist_flip:
"s∈S ⟹ t∈S ⟹
sum_list (pair_relator_halflist s t) = sum_list (pair_relator_halflist t s)"
alternating_order2_even_cancel_right[of s t "2*(relfun s t)"]

definition flip_altsublist_adjacent :: "'w list ⇒ 'w list ⇒ bool"
≡ ∃s t as bs. ss = as @ (pair_relator_halflist s t) @ bs ∧
ts = as @ (pair_relator_halflist t s) @ bs"

"ss = as @ (pair_relator_halflist s t) @ bs ⟹
ts = as @ (pair_relator_halflist t s) @ bs ⟹

proof-
from assms obtain s t as bs
where ssts: "ss = as @ (pair_relator_halflist s t) @ bs"
"ts = as @ (pair_relator_halflist t s) @ bs"
by    auto
from ssts have
"a#ss = (a#as) @ (pair_relator_halflist s t) @ bs"
"a#ts = (a#as) @ (pair_relator_halflist t s) @ bs"
by auto
thus ?thesis by (fast intro: flip_altsublist_adjacentI)
qed

lemma flip_altsublist_chain_map_Cons_grow:
"flip_altsublist_chain tss ⟹ flip_altsublist_chain (map ((#) t) tss)"
by  (induct tss rule: list_induct_CCons)
)

"ss ≠ [] ⟹ ss∈lists S ⟹ flip_altsublist_adjacent ss ss"
proof (induct ss rule: list_nonempty_induct)
case (single s)
hence "[s] = [] @ pair_relator_halflist s s @ []"
using relfun_S by simp
thus ?case by (fast intro: flip_altsublist_adjacentI)
next
case cons thus ?case using flip_altsublist_adjacent_Cons_grow by simp
qed

lemma rev_flip_altsublist_chain:
"flip_altsublist_chain xss ⟹ flip_altsublist_chain (rev xss)"
by    (induct xss rule: list_induct_CCons) auto

assumes "ss∈lists S" "flip_altsublist_adjacent ss ts"
shows   "set ts = set ss"
proof-
from assms obtain s t as bs where ssts:
"ss = as @ (pair_relator_halflist s t) @ bs"
"ts = as @ (pair_relator_halflist t s) @ bs"
by    auto
with assms(1) show ?thesis
using set_alternating_list2[of "relfun s t" s t]
set_alternating_list2[of "relfun t s" t s]
by    (cases "relfun s t" rule: nat_cases_2Suc) auto
qed

"∀ss∈lists S. ∀ts. flip_altsublist_adjacent ss ts ⟶ set ts = set ss"

"ss ∈ lists S ⟹ flip_altsublist_adjacent ss ts ⟹ ts ∈ lists S"

"∀ss∈lists S. ∀ts. flip_altsublist_adjacent ss ts ⟶ ts ∈ lists S"

lemma flip_altsublist_chain_lists:
"ss ∈ lists S ⟹ flip_altsublist_chain (ss#xss@[ts]) ⟹ ts ∈ lists S"
binrelchain_propcong_Cons_snoc[of
"λss. ss∈lists S" flip_altsublist_adjacent ss xss ts
]
by    fast

lemmas flip_altsublist_chain_funcong_Cons_snoc =

lemmas flip_altsublist_chain_set =
flip_altsublist_chain_funcong_Cons_snoc[
]

"flip_altsublist_adjacent ss ts ⟹ length ts = length ss"

lemmas flip_altsublist_chain_length =
binrelchain_funcong_Cons_snoc[
]

assumes "ss ∈ lists S" "flip_altsublist_adjacent ss ts"
shows "sum_list ts = sum_list ss"
proof-
from assms(2) obtain s t as bs where stasbs:
"ss = as @ (pair_relator_halflist s t) @ bs"
"ts = as @ (pair_relator_halflist t s) @ bs"
by    auto
show ?thesis
proof (cases "relfun s t")
next
case Suc
with assms stasbs have "s∈S" "t∈S"
using set_alternating_list1[of "add_order (s+t)" s t]
by    auto
with stasbs show ?thesis
using sum_list_pair_relator_halflist_flip by simp
qed
qed

"∀ss∈lists S. ∀ts. flip_altsublist_adjacent ss ts ⟶ sum_list ts = sum_list ss"

"S_reduced_for w ss ⟹ flip_altsublist_adjacent ss ts ⟹ S_reduced_for w ts"
using reduced_word_for_lists[of S] reduced_word_for_sum_list
by    (fastforce intro: reduced_word_forI_compare)

fixes   as bs s t
defines xs: "xs ≡ as @ pair_relator_halflist s t @ bs"
and     ys: "ys ≡ as @ pair_relator_halflist t s @ bs"
assumes Axs: "Abs_freelist xs ∈ Q"
shows "Abs_freelist ys ∈ Q"
proof-
define X Y A B half_st half2_st half_ts
where "X = Abs_freelist xs"
and "Y = Abs_freelist ys"
and "A = Abs_freelist as"
and "B = Abs_freelist bs"
and "half_st = Abs_freelist (pair_relator_halflist s t)"
and "half2_st = Abs_freelist (pair_relator_halflist2 s t)"
and "half_ts = Abs_freelist (pair_relator_halflist t s)"
define z where "z = -half2_st + B"
define w1 w2 where "w1 = rconjby z (pair_relator_freeword s t)"
and "w2 = Abs_freelist (rev (pair_relator_halflist t s) @ pair_relator_halflist t s)"
define w3 where "w3 = rconjby B w2"

from w1_def z_def
have  w1': "w1 = rconjby B (lconjby half2_st (pair_relator_freeword s t))"
hence "-w1 = rconjby B (lconjby half2_st (-pair_relator_freeword s t))"
using lconjby_uminus[of "half2_st"] by (simp add: rconjby_uminus[THEN sym])
moreover from X_def xs A_def half_st_def B_def have "X = A + B + rconjby B half_st"
)
ultimately have
"X + -w1 = A + B +
( rconjby B (half_st + (half2_st + -pair_relator_freeword s t - half2_st)) )"
moreover from w2_def half2_st_def half_ts_def have "w2 = half2_st + half_ts"
Abs_freelist_append[THEN sym]
pair_relator_halflist2_conv_rev_pair_relator_halflist
)
ultimately have
"X + -w1 + w3 = A + B + (rconjby B (-half2_st + (half2_st + half_ts)))"
of half_st half2_st "-pair_relator_freeword s t" "-half2_st"
]
Abs_freelist_append[THEN sym] pair_relator_halflist_append
)
hence Y': "Y = X - w1 + w3"
using A_def half_ts_def B_def ys Y_def
Abs_freelist_append_append[THEN sym]
)

from Axs have xs_S: "xs ∈ lists S" using Q_FreeS FreeGroupD_transfer' by fast
have "w1∈Q ∧ w3∈Q"
proof (cases "relfun s t")
case 0 with w1_def w2_def w3_def show ?thesis using genby_0_closed
zero_freeword.abs_eq[THEN sym]
)
next
case (Suc m) have m: "add_order (s+t) = Suc m" by fact
have st: "{s,t} ⊆ S"
proof (cases m)
case 0 with m xs xs_S show ?thesis
using set_alternating_list1 relfun_eq1 by force
next
case Suc with m xs xs_S show ?thesis
using set_alternating_list2[of "add_order (s+t)" s t] by fastforce
qed
from xs xs_S B_def have B_S: "B ∈ FreeGroup S"
using Abs_freelist_in_FreeGroup[of bs S] by simp
moreover from w2_def have "w2∈Q"
using st Abs_freelist_rev_append_alternating_list_in_Q[of t s "add_order (t+s)"]
by    fast
ultimately have "w3 ∈ Q"
using w3_def FreeGroup_genby_set_lconjby_set_rconjby_closed by fast
moreover from half2_st_def have "w1 ∈ Q"
using w1' st B_S alternating_list_in_lists[of s S] alternating_list_in_lists[of t S]
lconjby_Abs_freelist_relator_freeword[of s t]
by    (force intro: FreeGroup_genby_set_lconjby_set_rconjby_closed)
ultimately show ?thesis by fast
qed
with X_def Y_def Axs show ?thesis
using Y' genby_diff_closed[of X] genby_add_closed[of "X-w1" _ w3] by simp

qed

"Abs_freelist ss ∈ Q ⟹ flip_altsublist_adjacent ss ts ⟹ Abs_freelist ts ∈ Q"

lemma flip_altsublist_chain_G_in_Q:
"⟦ Abs_freelist ss ∈ Q; flip_altsublist_chain (ss#xss@[ts]) ⟧ ⟹ Abs_freelist ts ∈ Q"
binrelchain_propcong_Cons_snoc[of
"λss. Abs_freelist ss ∈ Q"
]
by    fast

lemma alternating_S_no_flip:
assumes "s∈S" "t∈S" "n > 0" "n < relfun s t ∨ relfun s t = 0"
shows   "sum_list (alternating_list n s t) ≠ sum_list (alternating_list n t s)"
proof
assume "sum_list (alternating_list n s t) = sum_list (alternating_list n t s)"
hence "sum_list (alternating_list n s t) + - sum_list (alternating_list n t s) = 0"
by simp
with assms(1,2) have "sum_list (alternating_list (2*n) s t) = 0"
by  (cases "even n")
sum_list.append[THEN sym]
alternating_list_append mult_2
)
qed

lemma exchange_alternating_not_in_alternating:
assumes "n ≥ 2" "n < relfun s t ∨ relfun s t = 0"
"S_reduced_for w (alternating_list n s t @ cs)"
"alternating_list n s t @ cs = xs@[x]@ys" "S_reduced_for w (t#xs@ys)"
shows   "length xs ≥ n"
proof-
from assms(1) obtain m k where n: "n = Suc m" and m: "m = Suc k"
using gr0_implies_Suc by fastforce
define altnst altnts altmts altkst
where "altnst = alternating_list n s t"
and "altnts = alternating_list n t s"
and "altmts = alternating_list m t s"
and "altkst = alternating_list k s t"
from altnst_def altmts_def n have altnmst: "altnst = s # altmts"
using alternating_list_Suc_Cons[of m] by fastforce
with assms(3) altnst_def have s_S: "s∈S" using reduced_word_for_lists by fastforce
from assms(5) have t_S: "t∈S" using reduced_word_for_lists by fastforce
from m altnmst altmts_def altkst_def have altnkst: "altnst = s # t # altkst"
using alternating_list_Suc_Cons by fastforce
have "¬ length xs < n"
proof (cases "Suc (length xs) = n")
case True
with assms(4,5) n altnts_def have flip: "S_reduced_for w (altnts @ cs)"
using length_alternating_list[of n s t]
same_length_eq_append[of altnts "xs@[x]" cs ys]
alternating_list_Suc_Cons[of m t s]
by    auto
from altnst_def have "sum_list altnst = sum_list altnts"
using reduced_word_for_sum_list[OF assms(3)]
reduced_word_for_sum_list[OF flip]
by    auto
with n assms(2) altnst_def altnts_def show ?thesis
using alternating_S_no_flip[OF s_S t_S] by fast
next
case False show ?thesis
proof (cases xs ys rule: two_lists_cases_snoc_Cons)
case Nil1
from Nil1(1) assms(4) altnkst altnst_def have "ys = t # altkst @ cs" by auto
with Nil1(1) assms(5) show ?thesis
contains_order2_nreduced[of t S "[]" "altkst@cs"]
reduced_word_for_imp_reduced_word
by    force
next
case Nil2 with assms(4) altnst_def False show ?thesis
using length_append[of altnst cs]
next
case (snoc_Cons us u z zs)
with assms(4,5) altnst_def
have  1: "altnst @ cs = us@[u,x,z]@zs" "S_reduced_for w (t#us@[u,z]@zs)"
by    auto
from 1(1) snoc_Cons(1) False altnst_def show ?thesis
using take_append[of n altnst cs] take_append[of n "us@[u,x,z]" zs]
set_alternating_list[of n s t]
alternating_list_alternates[of n s t us u]
reduced_word_for_imp_reduced_word[OF 1(2)]
contains_order2_nreduced[of u S "t#us"]
qed
qed
thus ?thesis by fastforce
qed

end (* context PreCoxeterSystem *)

subsubsection ‹Preliminary facts on the word problem›

text ‹
The word problem seeks criteria for determining whether two words over the generator set represent
the same element in @{term W}. Here we establish one direction of the word problem, as well as a
preliminary step toward the other direction.
›

context PreCoxeterSystem
begin

lemmas flip_altsublist_chain_sum_list =
― ‹This lemma represents one direction in the word problem: if a word in generators can be
transformed into another by a sequence of manipulations, each of which consists of replacing a
half-relator subword by its reversal, then the two words sum to the same element of @{term W}.›

lemma reduced_word_problem_eq_hd_step:
assumes step: "⋀y ss ts. ⟦
S_length y < S_length w; y≠0; S_reduced_for y ss; S_reduced_for y ts
⟧ ⟹ ∃xss. flip_altsublist_chain (ss # xss @ [ts])"
and     set_up: "S_reduced_for w (a#ss)" "S_reduced_for w (a#ts)"
shows   "∃xss. flip_altsublist_chain ((a#ss) # xss @ [a#ts])"
proof (cases "ss=ts")
case True
with set_up(1) have "flip_altsublist_chain ((a#ss) # [] @ [a#ts])"
thus ?thesis by fast
next
case False
define y where "y = sum_list ss"
with set_up(1) have ss: "S_reduced_for y ss"
using reduced_word_for_imp_reduced_word reduced_word_Cons_reduce by fast
moreover from y_def ss have ts: "S_reduced_for y ts"
using reduced_word_for_sum_list[OF set_up(1)]
reduced_word_for_sum_list[OF set_up(2)]
reduced_word_for_eq_length[OF set_up(1) set_up(2)]
reduced_word_for_lists[OF set_up(2)]
by    (auto intro: reduced_word_forI_compare)
moreover from ss set_up(1) have "S_length y < S_length w"
using reduced_word_for_length reduced_word_for_length by fastforce
moreover from False have "y ≠ 0"
using ss ts reduced_word_for_0_imp_nil reduced_word_for_0_imp_nil by fastforce
ultimately show ?thesis
using step flip_altsublist_chain_map_Cons_grow by fastforce
qed

end (* context PreCoxeterSystem *)

subsubsection ‹Preliminary facts related to the deletion condition›

text ‹
The deletion condition states that in a Coxeter system, every non-reduced word in the generating
set can be shortened to an equivalent word by deleting some particular pair of letters. This
condition is both necessary and sufficient for a group generated by elements of order two to be a
Coxeter system. Here we establish some facts related to the deletion condition that are true in
any group generated by elements of order two.
›

context PreCoxeterSystem
begin

abbreviation "ℋ ≡ (⋃w∈W. lconjby w ` S)" ― ‹the set of reflections›

abbreviation "lift_signed_lconjperm ≡ freeword_funlift signed_lconjpermutation"

lemma lconjseq_reflections: "ss∈lists S ⟹ set (lconjseq ss) ⊆ ℋ"
using special_subgroup_eq_sum_list[of S]
by    (induct ss rule: rev_induct) (auto simp add: lconjseq_snoc)

lemma deletion':
"ss ∈ lists S ⟹ ¬ distinct (lconjseq ss) ⟹
∃a b as bs cs. ss = as @ [a] @ bs @ [b] @ cs ∧
sum_list ss = sum_list (as@bs@cs)"
proof (induct ss)
case (Cons s ss)
show ?case
proof (cases "distinct (lconjseq ss)")
case True with Cons(2,3) show ?thesis
using subset_inj_on[OF lconjby_inj, of "set (lconjseq ss)" s]
distinct_map[of "lconjby s"]
next
case False
with Cons(1,2) obtain a b as bs cs where
"s#ss = (s#as) @ [a] @ bs @ [b] @ cs"
"sum_list (s#ss) = sum_list ((s#as) @ bs @ cs)"
by    auto
thus ?thesis by fast
qed
qed simp

lemma S_reduced_imp_distinct_lconjseq':
assumes "ss ∈ lists S" "¬ distinct (lconjseq ss)"
shows "¬ S_reduced ss"
proof
assume ss: "S_reduced ss"
from assms obtain as a bs b cs
where decomp: "ss = as @ [a] @ bs @ [b] @ cs"
"sum_list ss = sum_list (as@bs@cs)"
using deletion'[of ss]
by    fast
from ss decomp assms(1) show False
using reduced_word_for_minimal[of S _ ss "as@bs@cs"] by auto
qed

lemma S_reduced_imp_distinct_lconjseq: "S_reduced ss ⟹ distinct (lconjseq ss)"
using reduced_word_for_lists S_reduced_imp_distinct_lconjseq' by fast

lemma permutation_lift_signed_lconjperm_eq_signed_list_lconjaction':
"proper_signed_list xs ⟹ fst ` set xs ⊆ S ⟹
permutation (lift_signed_lconjperm (Abs_freeword xs)) =
signed_list_lconjaction (map fst xs)"
proof (induct xs)
case Nil
have "Abs_freeword ([]::'w signed list) = (0::'w freeword)"
using zero_freeword.abs_eq by simp
thus ?case by (simp add: zero_permutation.rep_eq freeword_funlift_0)
next
case (Cons x xs)
obtain s b where x: "x=(s,b)" by fastforce
with Cons show ?case
using Abs_freeword_Cons[of x xs]
binrelchain_Cons_reduce[of nflipped_signed x xs]
by    (cases b)
freeword_funlift_Abs_freeletter
Abs_permutation_inverse uminus_permutation.rep_eq
the_inv_signed_lconjaction_by_order2
freeword_funlift_uminus_Abs_freeletter
)
qed

lemma permutation_lift_signed_lconjperm_eq_signed_list_lconjaction:
"x ∈ FreeGroup S ⟹
permutation (lift_signed_lconjperm x) =
signed_list_lconjaction (map fst (freeword x))"
using freeword FreeGroup_def[of S] freeword_inverse[of x]
permutation_lift_signed_lconjperm_eq_signed_list_lconjaction'
by    force

lemma even_count_lconjseq_rev_relator:
"s∈S ⟹ t∈S ⟹ even (count_list (lconjseq (rev (pair_relator_list s t))) x)"
using even_count_lconjseq_alternating_order2[of t]

lemma GroupByPresentationInducedFun_S_R_signed_lconjaction:
"GroupByPresentationInducedFun S P signed_lconjpermutation"
proof (intro_locales, rule GroupByPresentation_S_P, unfold_locales)
fix ps assume ps: "ps∈P"
define r where "r = Abs_freeword ps"
with ps have r: "r∈P'" by fast
then obtain s t where st: "s∈S" "t∈S" "r = pair_relator_freeword s t"
using relator_freewords by fast
from r st(3)
have 1: "permutation (lift_signed_lconjperm r) =
signed_list_lconjaction (pair_relator_list s t)"
using P'_FreeS
permutation_lift_signed_lconjperm_eq_signed_list_lconjaction
Abs_freelist_inverse[of "pair_relator_list s t"]
map_fst_map_const_snd[of True "pair_relator_list s t"]
by    force
have "permutation (lift_signed_lconjperm r) = id"
proof
fix x
show "lift_signed_lconjperm r → x = id x"
proof
show "snd (freeword_funlift signed_lconjpermutation r → x) = snd (id x)"
set_alternating_list[of "2*relfun s t" s t]
signed_list_lconjaction_snd[of "pair_relator_list s t" x]
by    fastforce
qed (simp add: 1 signed_list_lconjaction_fst sum_list_pair_relator_list)
qed
moreover
have  "permutation (0::'w signed permutation) = (id::'w signed ⇒ 'w signed)"
using zero_permutation.rep_eq
by    fast
ultimately show "lift_signed_lconjperm r = 0"
using permutation_inject by fastforce
qed

end (* context PreCoxeterSystem *)

subsection ‹Coxeter-like systems with deletion›

text ‹
Here we add the so-called deletion condition as an assumption, and explore its consequences.
›

subsubsection ‹Locale definition›

locale PreCoxeterSystemWithDeletion = PreCoxeterSystem S
+ assumes deletion:
"ss ∈ lists S ⟹ ¬ reduced_word S ss ⟹
∃a b as bs cs. ss = as @ [a] @ bs @ [b] @ cs ∧
sum_list ss = sum_list (as@bs@cs)"

subsubsection ‹Consequences of the deletion condition›

context PreCoxeterSystemWithDeletion
begin

lemma deletion_reduce:
"ss ∈ lists S ⟹ ∃ts. ts ∈ ssubseqs ss ∩ reduced_words_for S (sum_list ss)"
proof (cases "S_reduced ss")
case True
thus  "ss ∈ lists S ⟹
∃ts. ts ∈ ssubseqs ss ∩ reduced_words_for S (sum_list ss)"
next
case False
have "ss ∈ lists S ⟹ ¬ S_reduced ss ⟹
∃ts. ts ∈ ssubseqs ss ∩ reduced_words_for S (sum_list ss)"
proof (induct ss rule: length_induct)
fix xs::"'w list"
assume xs:
"∀ys. length ys < length xs ⟶ ys ∈ lists S ⟶ ¬ S_reduced ys
⟶ (∃ts. ts ∈ ssubseqs ys ∩ reduced_words_for S (sum_list ys))"
"xs ∈ lists S" "¬ S_reduced xs"
from xs(2,3) obtain as a bs b cs
where asbscs: "xs = as@[a]@bs@[b]@cs" "sum_list xs = sum_list (as@bs@cs)"
using deletion[of xs]
by    fast
show "∃ts. ts ∈ ssubseqs xs ∩ reduced_words_for S (sum_list xs)"
proof (cases "S_reduced (as@bs@cs)")
case True with asbscs xs(2) show ?thesis
using delete2_ssubseqs by fastforce
next
case False
moreover from asbscs(1) xs(2)
have  "length (as@bs@cs) < length xs" "as@bs@cs ∈ lists S"
by    auto
ultimately obtain ts
where ts: "ts ∈ ssubseqs (as@bs@cs) ∩
reduced_words_for S (sum_list (as@bs@cs))"
using xs(1,2) asbscs(1)
by    fast
with asbscs show ?thesis
using delete2_ssubseqs[of as bs cs a b] ssubseqs_subset by auto
qed
qed
with False
show  "ss ∈ lists S ⟹
∃ts. ts ∈ ssubseqs ss ∩ reduced_words_for S (sum_list ss)"
by    fast
qed

lemma deletion_reduce':
"ss ∈ lists S ⟹ ∃ts∈reduced_words_for S (sum_list ss). set ts ⊆ set ss"
using deletion_reduce[of ss] subseqs_powset[of ss] by auto

end (* context PreCoxeterSystemWithDeletion *)

subsubsection ‹The exchange condition›

text ‹
The exchange condition states that, given a reduced word in the generators, if prepending a
letter to the word does not remain reduced, then the new word can be shortened to a word
equivalent to the original one by deleting some letter other than the prepended one. Thus, one
able to exchange some letter for the addition of a desired letter at the beginning of a word,
without changing the elemented represented.
›

context PreCoxeterSystemWithDeletion
begin

lemma exchange:
assumes "s∈S" "S_reduced_for w ss" "¬ S_reduced (s#ss)"
shows   "∃t as bs. ss = as@t#bs ∧ reduced_word_for S w (s#as@bs)"
proof-
from assms(2) have ss_lists: "ss ∈ lists S" using reduced_word_for_lists by fast
with assms(1) have "s#ss ∈ lists S" by simp
with assms(3) obtain a b as bs cs
where del:  "s#ss = as @ [a] @ bs @ [b] @ cs"
"sum_list (s#ss) = sum_list (as@bs@cs)"
using deletion[of "s#ss"]
by    fastforce
show ?thesis
proof (cases as)
case Nil with assms(1,2) del show ?thesis
by    (fastforce intro: reduced_word_forI_compare)
next
case (Cons d ds) with del assms(2) show ?thesis
using ss_lists reduced_word_for_imp_reduced_word
reduced_word_for_minimal[of S "sum_list ss" ss "ds@bs@cs"]
by    fastforce
qed
qed

assumes "reduced_word_for S w (s#as)" "reduced_word_for S w cs"
shows   "∃a ds es. cs = ds@[a]@es ∧ reduced_word_for S w (s#ds@es)"
proof-
from assms(1) have s_S: "s∈S" using reduced_word_for_lists by fastforce
moreover have "¬ S_reduced (s#cs)"
proof (rule not_reduced_word_for)
show "as ∈ lists S" using reduced_word_for_lists[OF assms(1)] by simp
from assms(1,2) show "sum_list as = sum_list (s#cs)"
by    fastforce
from assms(1,2) show "length as < length (s#cs)"
using reduced_word_for_length[of S w] by fastforce
qed
ultimately obtain a ds es
where "cs = ds@[a]@es" "reduced_word_for S w (s#ds@es)"
using assms(2) exchange[of s w cs]
by    auto
thus ?thesis by fast
qed

end (* context PreCoxeterSystemWithDeletion *)

subsubsection ‹More on words in generators containing alternating subwords›

text ‹
Here we explore more of the mechanics of manipulating words over @{term S} that contain
alternating subwords, in preparation of the word problem.
›

context PreCoxeterSystemWithDeletion
begin

assumes "s≠t" "reduced_word_for S w (t#bs)" "n < relfun s t ∨ relfun s t = 0"
"reduced_word_for S w (alternating_list n s t @ cs)"
shows   "∃ds. reduced_word_for S w (alternating_list (Suc n) t s @ ds)"
proof-
define altnst where "altnst = alternating_list n s t"
with assms(2,4) obtain x xs ys
where xxsys: "altnst @ cs = xs@[x]@ys" "reduced_word_for S w (t#xs@ys)"
by    fast
show ?thesis
proof (cases n rule: nat_cases_2Suc)
case 0 with xxsys(2) show ?thesis by auto
next
case 1 with assms(1,4) xxsys altnst_def show ?thesis
using reduced_word_for_sum_list[of S w "s#cs"]
reduced_word_for_sum_list[of S w "t#cs"]
by    (cases xs) auto
next
case (SucSuc k)
with assms(3,4) xxsys altnst_def have "length xs ≥ n"
using exchange_alternating_not_in_alternating by simp
moreover define ds where "ds = take (length xs - n) cs"
ultimately have "t#xs@ys = alternating_list (Suc n) t s @ ds @ ys"
using xxsys(1) altnst_def take_append[of "length xs" altnst cs]
alternating_list_Suc_Cons[of n t]
with xxsys(2) show ?thesis by auto
qed
qed

assumes "s≠t" "reduced_word_for S w (s#as)" "reduced_word_for S w (t#bs)"
shows "n ≤ relfun s t ∨ relfun s t = 0 ⟹ (∃cs.
reduced_word_for S w (alternating_list n s t @ cs) ∨
reduced_word_for S w (alternating_list n t s @ cs)
)"
proof (induct n)
case 0 from assms(2) show ?case by auto
next
case (Suc m) thus ?case
OF assms(1)[THEN not_sym] assms(2), of m
]
by    fastforce
qed

assumes "s≠t" "reduced_word_for S w (s#as)" "reduced_word_for S w (t#bs)"
shows "∃cs. reduced_word_for S w (pair_relator_halflist s t @ cs)"
proof-
define altst altts
where "altst = pair_relator_halflist s t"
and "altts = pair_relator_halflist t s"
then obtain cs
where cs: "reduced_word_for S w (altst @ cs) ∨
reduced_word_for S w (altts @ cs)"
by    auto
moreover from altst_def altts_def
have  "reduced_word_for S w (altts @ cs) ⟹ reduced_word_for S w (altst @ cs)"
using reduced_word_for_lists[OF assms(2)] reduced_word_for_lists[OF assms(3)]
ultimately show "∃cs. reduced_word_for S w (altst @ cs)" by fast
qed

assumes "s≠t" "reduced_word_for S w (s#as)" "reduced_word_for S w (t#bs)"
shows   "relfun s t ≠ 0"
proof
assume 1: "relfun s t = 0"
define altst altts
where "altst = alternating_list (Suc (S_length w)) s t"
and "altts = alternating_list (Suc (S_length w)) t s"
with 1 obtain cs
where "reduced_word_for S w (altst @ cs) ∨
reduced_word_for S w (altts @ cs)"
by    fast
moreover from altst_def altts_def
have  "length (altst @ cs) > S_length w"
"length (altts @ cs) > S_length w"
using length_alternating_list[of _ s] length_alternating_list[of _ t]
by    auto
ultimately show False using reduced_word_for_length by fastforce
qed

end (* context PreCoxeterSystemWithDeletion *)

subsubsection ‹The word problem›

text ‹Here we establish the other direction of the word problem for reduced words.›

context PreCoxeterSystemWithDeletion
begin

lemma reduced_word_problem_ConsCons_step:
assumes "⋀y ss ts. ⟦ S_length y < S_length w; y≠0; reduced_word_for S y ss;
reduced_word_for S y ts ⟧ ⟹ ∃xss. flip_altsublist_chain (ss # xss @ [ts])"
"reduced_word_for S w (a#as)" "reduced_word_for S w (b#bs)" "a≠b"
shows   "∃xss. flip_altsublist_chain ((a#as)#xss@[b#bs])"
proof-
from assms(2-4) obtain cs
where cs: "reduced_word_for S w (pair_relator_halflist a b @ cs)"
by    fast
define rs us where "rs = pair_relator_halflist a b @ cs"
and "us = pair_relator_halflist b a @ cs"
from assms(2,3) have a_S: "a∈S" and b_S: "b∈S"
using reduced_word_for_lists[of S _ "a#as"] reduced_word_for_lists[of S _ "b#bs"]
by    auto
from assms(2-4) have "relfun a b ≠ 0"
from this obtain k where k: "relfun a b = Suc k"
using not0_implies_Suc by auto
define qs vs
where "qs = alternating_list k b a @ cs"
and "vs = alternating_list k a b @ cs"
with k rs_def us_def have rs': "rs = a # qs" and us': "us = b # vs"
from assms(1,2) cs rs_def rs'
have  startlink: "as ≠ qs ⟹ ∃xss. flip_altsublist_chain ((a#as) # xss @ [rs])"
using reduced_word_problem_eq_hd_step
by    fastforce
from assms(1,3) rs_def cs us'
have  endlink: "bs ≠ vs ⟹ ∃xss. flip_altsublist_chain (us # xss @ [b#bs])"
reduced_word_problem_eq_hd_step[of w]
by    auto
show ?thesis
proof (cases "as = qs" "bs = vs" rule: two_cases)
case both
with rs' us' have "flip_altsublist_chain ((a#as) # [] @ [b#bs])"
thus ?thesis by fast
next
case one
with rs' obtain xss
where "flip_altsublist_chain ((a#as) # (us # xss) @ [b#bs])"
by    auto
thus ?thesis by fast
next
case other
from other(1) obtain xss where "flip_altsublist_chain ((a#as) # xss @ [rs])"
have  "flip_altsublist_chain ((a#as) # (xss@[rs]) @ [b#bs])"
by    simp
thus ?thesis by fast
next
case neither
from neither(1) obtain xss
where "flip_altsublist_chain ((a#as) # xss @ [rs])"
by    fast
with neither(2) obtain yss
where "flip_altsublist_chain ((a#as) # (xss @ [rs,us] @ yss) @ [b#bs])"
by    auto
thus ?thesis by fast
qed
qed

lemma reduced_word_problem:
"⟦ w≠0; reduced_word_for S w ss; reduced_word_for S w ts ⟧ ⟹
∃xss. flip_altsublist_chain (ss#xss@[ts])"
proof (induct w arbitrary: ss ts rule: measure_induct_rule[of "S_length"])
case (less w)
show ?case
proof (cases ss ts rule: two_lists_cases_Cons_Cons)
case Nil1 from Nil1(1) less(2,3) show ?thesis
using reduced_word_for_sum_list by fastforce
next
case Nil2 from Nil2(2) less(2,4) show ?thesis
using reduced_word_for_sum_list by fastforce
next
case (ConsCons a as b bs)
show ?thesis
proof (cases "a=b")
case True with less ConsCons show ?thesis
using reduced_word_problem_eq_hd_step[of w] by auto
next
case False with less ConsCons show ?thesis
using reduced_word_problem_ConsCons_step[of w] by simp
qed
qed
qed

lemma reduced_word_letter_set:
assumes "S_reduced_for w ss"
shows "reduced_letter_set S w = set ss"
proof (cases "w=0")
case True with assms show ?thesis
using reduced_word_for_0_imp_nil[of S ss] reduced_letter_set_0 by simp
next
case False
show ?thesis
proof
from assms show "set ss ⊆ reduced_letter_set S w" by fast
show "reduced_letter_set S w ⊆ set ss"
proof
fix x assume "x ∈ reduced_letter_set S w"
from this obtain ts where "reduced_word_for S w ts" "x ∈ set ts" by fast
with False assms show "x ∈ set ss"
using reduced_word_for_lists[of S _ ss] reduced_word_problem[of w ss]
flip_altsublist_chain_set
by    force
qed
qed
qed

end (* context PreCoxeterSystemWithDeletion *)

subsubsection ‹Special subgroups and cosets›

text ‹
Recall that special subgroups are those generated by subsets of the generating set @{term S}.
Here we show that the presence of the deletion condition guarantees that the collection of
special subgroups and their left cosets forms a poset under reverse inclusion that satisfies the
necessary properties to ensure that the poset of simplices in the associated simplicial complex
is isomorphic to this poset of special cosets.
›

context PreCoxeterSystemWithDeletion
begin

lemma special_subgroup_int_S:
assumes "T ∈ Pow S"
shows   "⟨T⟩ ∩ S = T"
proof
show "⟨T⟩ ∩ S ⊆ T"
proof
fix t assume t: "t ∈ ⟨T⟩ ∩ S"
with assms obtain ts where ts: "ts ∈ lists T" "t = sum_list ts"
using special_subgroup_eq_sum_list[of T] by fast
with assms obtain us
where us: "reduced_word_for S (sum_list ts) us" "set us ⊆ set ts"
using deletion_reduce'[of ts]
by    auto
with no_zero_genset ts(2) t have "length us = 1"
using reduced_word_for_lists[of S _ us] reduced_word_for_sum_list[of S _ us]
reduced_word_for_imp_reduced_word[of S _ us] el_reduced[of S]
by    auto
with us ts show "t∈T"
using reduced_word_for_sum_list[of S _ us] by (cases us) auto
qed
from assms show "T ⊆ ⟨T⟩ ∩ S" using genby_genset_subset by fast
qed

lemma special_subgroup_inj: "inj_on genby (Pow S)"
using special_subgroup_int_S inj_on_inverseI[of _ "λW. W∩S"] by fastforce

lemma special_subgroup_genby_subset_ordering_iso:
"subset_ordering_iso (Pow S) genby"
proof (unfold_locales, rule genby_mono, simp, rule special_subgroup_inj)
fix X Y assume XY: "X ∈ genby ` Pow S" "Y ∈ genby ` Pow S" "X⊆Y"
from XY(1,2) obtain TX TY
where "TX∈Pow S" "X = ⟨TX⟩" "TY∈Pow S" "Y = ⟨TY⟩"
by    auto
hence "the_inv_into (Pow S) genby X = X∩S"
"the_inv_into (Pow S) genby Y = Y∩S"
using the_inv_into_f_f[OF special_subgroup_inj] special_subgroup_int_S
by    auto
with XY(3)
show  "the_inv_into (Pow S) genby X ⊆ the_inv_into (Pow S) genby Y"
by    auto
qed

lemmas special_subgroup_genby_rev_mono
= OrderingSetIso.rev_ordsetmap[OF special_subgroup_genby_subset_ordering_iso]

lemma special_subgroup_word_length:
assumes "T∈Pow S" "w∈⟨T⟩"
shows   "word_length T w = S_length w"
proof-
from assms obtain ts where ts: "ts ∈ lists T" "w = sum_list ts"
using special_subgroup_eq_sum_list by auto
with assms(1) obtain us where "us ∈ ssubseqs ts" "S_reduced_for w us"
using deletion_reduce[of ts] by fast
with assms(1) ts(1) show ?thesis
using     ssubseqs_lists[of ts] reduced_word_for_sum_list
is_arg_min_size_subprop[of length "word_for S w" us "word_for T w"]
unfolding reduced_word_for_def word_length_def
by        fast
qed

lemma S_subset_reduced_imp_S_reduced:
"T∈Pow S ⟹ reduced_word T ts ⟹ S_reduced ts"
using reduced_word_for_lists reduced_word_for_lists[of T _ ts]
reduced_word_for_length[of T "sum_list ts" ts] special_subgroup_eq_sum_list[of T]
special_subgroup_word_length[of T "sum_list ts"]
by    (fastforce intro: reduced_word_forI_length)

lemma smallest_genby: "T∈Pow S ⟹ w∈⟨T⟩ ⟹ reduced_letter_set S w ⊆ T"
using genby_S_reduced_word_for_arg_min[of T]
reduced_word_for_imp_reduced_word[of T w]
S_subset_reduced_imp_S_reduced[of T "arg_min length (word_for T w)"]
reduced_word_for_sum_list[of T] reduced_word_for_lists reduced_word_letter_set
by    fastforce

lemma special_cosets_below_in:
assumes "w∈W" "T ∈ Pow S"
shows   "𝒫.⊇(w +o ⟨T⟩) = (⋃R∈(Pow S).⊇T. {w +o ⟨R⟩})"
proof (rule seteqI)
fix A assume "A ∈ 𝒫.⊇(w +o ⟨T⟩)"
hence A: "A∈𝒫" "A ⊇ (w +o ⟨T⟩)" by auto
from A(1) obtain R w' where "R∈Pow S" "A = w' +o ⟨R⟩"
using special_cosets_def by auto
with A(2) assms(2) show "A ∈ (⋃R∈(Pow S).⊇T. {w +o ⟨R⟩})"
using genby_lcoset_subgroup_imp_eq_reps[of w T w' R]
lcoset_eq_reps_subset[of w "⟨T⟩"]
special_subgroup_genby_rev_mono[of T R]
by    auto
next
fix B assume "B ∈ (⋃R∈(Pow S).⊇T. {w +o ⟨R⟩})"
from this obtain R where R: " R ∈ (Pow S).⊇T" "B = w +o ⟨R⟩" by auto
moreover hence "B ⊇ w +o ⟨T⟩"
using genby_mono elt_set_plus_def[of w] by auto
ultimately show "B ∈ special_cosets .⊇ (w +o ⟨T⟩)"
using assms(1) special_cosetsI by auto
qed

lemmas special_coset_inj
= comp_inj_on[OF special_subgroup_inj, OF inj_inj_on, OF lcoset_inj_on]

lemma special_coset_eq_imp_eq_gensets:
"⟦ T1∈Pow S; T2∈Pow S; w1 +o ⟨T1⟩ = w2 +o ⟨T2⟩ ⟧ ⟹ T1=T2"
using set_plus_rearrange2[of "-w1" w1 "⟨T1⟩"]
set_plus_rearrange2[of "-w1" w2 "⟨T2⟩"]
genby_lcoset_subgroup_imp_eq_reps[of 0 T1 "-w1+w2" T2]
inj_onD[OF special_subgroup_inj]
by    force

lemma special_subgroup_special_coset_subset_ordering_iso:
"subset_ordering_iso (genby ` Pow S) ((+o) w)"
proof
show "⋀a b. a ⊆ b ⟹ w +o a ⊆ w +o b" using elt_set_plus_def by auto
show 2: "inj_on ((+o) w) (genby ` Pow S)"
using lcoset_inj_on inj_inj_on by fast
show  "⋀a b. a ∈ (+o) w ` genby ` Pow S ⟹
b ∈ (+o) w ` genby ` Pow S ⟹
a ⊆ b ⟹
the_inv_into (genby ` Pow S) ((+o) w) a ⊆
the_inv_into (genby ` Pow S) ((+o) w) b"
proof-
fix a b
assume ab : "a ∈ (+o) w ` genby ` Pow S" "b ∈ (+o) w ` genby ` Pow S"
and  a_b: "a⊆b"
from ab obtain Ta Tb
where "Ta∈Pow S" "a = w +o ⟨Ta⟩" "Tb∈Pow S" "b = w +o ⟨Tb⟩"
by    auto
with a_b
show  "the_inv_into (genby ` Pow S) ((+o) w) a ⊆
the_inv_into (genby ` Pow S) ((+o) w) b"
using the_inv_into_f_eq[OF 2] lcoset_eq_reps_subset[of w "⟨Ta⟩" "⟨Tb⟩"]
by    simp
qed
qed

lemma special_coset_subset_ordering_iso:
"subset_ordering_iso (Pow S) ((+o) w ∘ genby)"
using special_subgroup_genby_subset_ordering_iso
special_subgroup_special_coset_subset_ordering_iso
by    (fast intro: OrderingSetIso.iso_comp)

lemmas special_coset_subset_rev_mono =
OrderingSetIso.rev_ordsetmap[OF special_coset_subset_ordering_iso]

lemma special_coset_below_in_subset_ordering_iso:
"subset_ordering_iso ((Pow S).⊇T) ((+o) w ∘ genby)"
using special_coset_subset_ordering_iso by (auto intro: OrderingSetIso.iso_subset)

lemma special_coset_below_in_supset_ordering_iso:
"OrderingSetIso (⊇) (⊃) (⊇) (⊃) ((Pow S).⊇T) ((+o) w ∘ genby)"
using special_coset_below_in_subset_ordering_iso OrderingSetIso.iso_dual by fast

lemma special_coset_pseudominimals:
assumes "supset_pseudominimal_in 𝒫 X"
shows   "∃w s. w∈W ∧ s∈S ∧ X = w +o ⟨S-{s}⟩"
proof-
from assms have "X∈𝒫" using supset_pseudominimal_inD1 by fast
from this obtain w T where wT: "w∈W" "T∈Pow S" "X = w +o ⟨T⟩"
using special_cosets_def by auto
show ?thesis
proof (cases "T=S")
case True with wT(1,3) assms show ?thesis
using genby_lcoset_el_reduce supset_pseudominimal_ne_bottom
special_cosets_bottom
by    fast
next
case False
with wT(2) obtain s where s: "s∈S" "T ⊆ S-{s}" by fast
from s(2) wT(1,3) assms have "X ⊆ w +o ⟨S-{s}⟩"
using genby_mono by auto
moreover from assms wT(1) s(1) have "¬ X ⊂ w +o ⟨S-{s}⟩"
using special_cosetsI[of _ w]
supset_pseudominimal_inD2[of 𝒫 X "w +o ⟨S-{s}⟩"]
lcoset_eq_reps[of w _ "⟨S⟩"]
inj_onD[OF special_subgroup_inj, of "S-{s}" S]
by    (auto simp add: special_cosets_bottom genby_lcoset_el_reduce)
ultimately show ?thesis using wT(1) s(1) by fast
qed
qed

lemma special_coset_pseudominimal_in_below_in:
assumes "w∈W" "T∈Pow S" "supset_pseudominimal_in (𝒫.⊇(w +o ⟨T⟩)) X"
shows   "∃s∈S-T. X = w +o ⟨S-{s}⟩"
proof-
from assms obtain v s where vs: "v∈W" "s∈S" "X = v +o ⟨S-{s}⟩"
using special_cosets_has_bottom special_cosetsI[of T w]
supset_has_bottom_pseudominimal_in_below_in
special_coset_pseudominimals
by    force
from assms(3) have X: "X ⊇ w +o ⟨T⟩"
using supset_pseudominimal_inD1 by fast
with vs(3) have 1: "X = w +o ⟨S-{s}⟩"
using genby_lcoset_subgroup_imp_eq_reps[of w T v "S-{s}"] by fast
with X assms have "T ⊆ S-{s}"
using special_cosetsI special_coset_subset_rev_mono[of T "S-{s}"]
by    fastforce
with vs(2) show ?thesis using 1 by fast
qed

lemma exclude_one_is_pseudominimal:
assumes "w∈W" "t∈S"
shows   "supset_pseudominimal_in 𝒫 (w +o ⟨S-{t}⟩)"
proof (rule supset_pseudominimal_inI, rule special_cosetsI)
show "w ∈ W" by fact
from assms have "w +o ⟨S - {t}⟩ ≠ W"
using genby_lcoset_el_reduce[of w] lcoset_eq_reps[of w _ W]
inj_onD[OF special_subgroup_inj, of "S-{t}" S]
by    auto
thus "w +o ⟨S - {t}⟩ ≠ supset_bottom 𝒫"
using special_cosets_bottom by fast
next
fix X assume X: "X∈𝒫" "w +o ⟨S - {t}⟩ ⊂ X"
with assms(1) have "X ∈ (⋃R∈(Pow S).⊇(S-{t}). {w +o ⟨R⟩})"
using subst[OF special_cosets_below_in, of w "S-{t}" "λA. X∈A"] by fast
from this obtain R where R: "R ∈ (Pow S).⊇(S-{t})" "X = w +o ⟨R⟩" by auto
from R(2) X(2) have "R ≠ S-{t}" by fast
with R(1) have "R=S" by auto
with assms(1) R(2) show "X = supset_bottom 𝒫"
using genby_lcoset_el_reduce special_cosets_bottom by fast
qed fast

lemma exclude_one_is_pseudominimal_in_below_in:
"⟦ w∈W; T∈Pow S; s∈S-T ⟧ ⟹
supset_pseudominimal_in (𝒫.⊇(w +o ⟨T⟩)) (w +o ⟨S-{s}⟩)"
using special_cosets_has_bottom special_cosetsI
exclude_one_is_pseudominimal[of w s]
genby_mono[of T "S-{s}"]
supset_has_bottom_pseudominimal_in_below_inI[
of 𝒫 "w +o ⟨T⟩" "w +o ⟨S-{s}⟩"
]
by    auto

lemma glb_special_subset_coset:
assumes   wTT': "w∈ W" "T ∈ Pow S" "T' ∈ Pow S"
defines   U: "U ≡ T ∪ T' ∪ reduced_letter_set S w"
shows     "supset_glbound_in_of 𝒫 ⟨T⟩ (w +o ⟨T'⟩) ⟨U⟩"
proof (rule supset_glbound_in_ofI)

from wTT'(2,3) U show "⟨U⟩ ∈ 𝒫"
using reduced_letter_set_subset[of S] special_subgroup_special_coset by simp

show "supset_lbound_of ⟨T⟩ (w +o ⟨T'⟩) ⟨U⟩"
proof (rule supset_lbound_ofI)
from U show "⟨T⟩ ⊆ ⟨U⟩" using genby_mono[of T U] by fast
show "w +o ⟨T'⟩ ⊆ ⟨U⟩"
proof
fix x assume "x ∈ w +o ⟨T'⟩"
with wTT'(3) obtain y where y: "y ∈ ⟨T'⟩" "x = w + y"
using elt_set_plus_def[of w] by auto
with wTT'(1) U show "x ∈ ⟨U⟩"
using in_genby_S_reduced_letter_set genby_mono[of _ U]
genby_mono[of T' U] genby_add_closed[of w U y]
by    auto
qed
qed

next

fix X assume X: "X∈𝒫" "supset_lbound_of ⟨T⟩ (w +o ⟨T'⟩) X"
from X(1) obtain v R where vR: "R∈Pow S" "X = v +o ⟨R⟩"
using special_cosets_def by auto
from X(2) have X': "X ⊇ ⟨T⟩" "X ⊇ w +o ⟨T'⟩"
using supset_lbound_of_def[of _ _ X] by auto
from X'(1) vR(2) have R: "X = ⟨R⟩"
using genby_0_closed genby_lcoset_el_reduce0 by fast
with X'(2) have w: "w∈⟨R⟩" using genby_0_closed lcoset_refl by fast
have "T' ⊆ R"
proof (
rule special_subgroup_genby_rev_mono, rule wTT'(3), rule vR(1), rule subsetI
)
fix x assume "x ∈ ⟨T'⟩"
with X'(2) R show "x ∈ ⟨R⟩"
using elt_set_plus_def[of w "⟨T'⟩"] w genby_uminus_add_closed[of "w" R "w+x"]
by    auto
qed
with X'(1) wTT'(2) vR(1) show "⟨U⟩⊆X"
using special_subgroup_genby_rev_mono[of T R] w smallest_genby U R
genby_mono[of _ R]
by    simp

qed

lemma glb_special_subset_coset_ex:
assumes   "w∈ W" "T ∈ Pow S" "T' ∈ Pow S"
shows     "∃B. supset_glbound_in_of 𝒫 ⟨T⟩ (w +o ⟨T'⟩) B"
using     glb_special_subset_coset[OF assms]
by        fast

lemma special_cosets_have_glbs:
assumes "X∈𝒫" "Y∈𝒫"
shows   "∃B. supset_glbound_in_of 𝒫 X Y B"
proof-
from assms obtain wx Tx wy Ty
where X: "wx ∈ W" "Tx ∈ Pow S" "X = wx +o ⟨Tx⟩"
and   Y: "wy ∈ W" "Ty ∈ Pow S" "Y = wy +o ⟨Ty⟩"
using special_cosets_def
by    auto
from X(1,2) Y(1,2) obtain A
where A: "supset_glbound_in_of 𝒫 ⟨Tx⟩ ((-wx+wy) +o ⟨Ty⟩) A"
using genby_uminus_add_closed[of wx] glb_special_subset_coset_ex by fastforce
from X(1,3) Y(3) have "supset_glbound_in_of 𝒫 X Y (wx +o A)"
using supset_glbound_in_of_lcoset_shift[OF A, of wx]
by    (auto simp add: set_plus_rearrange2 special_cosets_lcoset_shift)
thus ?thesis by fast
qed

end (* context PreCoxeterSystemWithDeletion *)

subsection ‹Coxeter systems›

subsubsection ‹Locale definition and transfer from the associated free group›

text ‹
Now we consider groups generated by elements of order two with an additional assumption to ensure
that the natural correspondence between the group @{term W} and the group presentation on the
generating set @{term S} and its relations is bijective. Below, such groups will be shown to
satisfy the deletion condition.
›

locale CoxeterSystem = PreCoxeterSystem S
+ assumes induced_id_inj: "inj_on induced_id G"

lemma (in PreCoxeterSystem) CoxeterSystemI:
assumes "⋀g. g∈G ⟹ induced_id g = 0 ⟹ g=0"
shows   "CoxeterSystem S"
proof
from assms have "GroupIso G induced_id"
using GroupWithGeneratorsRelators_S_R
GroupWithGeneratorsRelators.induced_id_hom_surj(1)
by    (fast intro: GroupHom.isoI)
thus "inj_on induced_id G" using GroupIso.inj_on by fast
qed

context CoxeterSystem
begin

abbreviation "inv_induced_id ≡ GroupPresentation.inv_induced_id S R"

lemma GroupPresentation_S_R: "GroupPresentation S R"
by  (
intro_locales, rule GroupWithGeneratorsRelators_S_R,
unfold_locales, rule induced_id_inj
)

lemmas inv_induced_id_sum_list =
GroupPresentation.inv_induced_id_sum_list_S[OF GroupPresentation_S_R]

end (* context CoxeterSystem *)

subsubsection ‹The deletion condition is necessary›

text ‹
Call an element of @{term W} a reflection if it is a conjugate of a generating element (and so
is also of order two). Here we use the action of words over @{term S} on such reflections to show
that Coxeter systems satisfy the deletion condition.
›

context CoxeterSystem
begin

abbreviation "induced_signed_lconjperm ≡
GroupByPresentationInducedFun.induced_hom S P signed_lconjpermutation"

definition flipped_reflections :: "'w ⇒ 'w set"
where "flipped_reflections w ≡
{t∈ℋ. induced_signed_lconjperm (inv_induced_id (-w)) →
(t,True) = (rconjby w t, False)}"

lemma induced_signed_lconjperm_inv_induced_id_sum_list:
"ss ∈ lists S ⟹ induced_signed_lconjperm (inv_induced_id (sum_list ss)) =
sum_list (map signed_lconjpermutation ss)"
inv_induced_id_sum_list Abs_freelist_in_FreeGroup
GroupByPresentationInducedFun.induced_hom_Abs_freelist_conv_sum_list[
OF GroupByPresentationInducedFun_S_R_signed_lconjaction
]
)

lemma induced_signed_eq_lconjpermutation:
"ss ∈ lists S ⟹
permutation (induced_signed_lconjperm (inv_induced_id (sum_list ss))) =
signed_list_lconjaction ss"
proof (induct ss)
case Nil
have "permutation (induced_signed_lconjperm (inv_induced_id (sum_list []))) = id"
using induced_signed_lconjperm_inv_induced_id_sum_list[of "[]"]
zero_permutation.rep_eq
by    simp
thus ?case by fastforce
next
case (Cons s ss)
from Cons(2)
have "induced_signed_lconjperm (inv_induced_id (sum_list (s#ss))) =
signed_lconjpermutation s + sum_list (map signed_lconjpermutation ss)"
using induced_signed_lconjperm_inv_induced_id_sum_list[of "s#ss"]
by    simp
with Cons(2) have
"permutation (induced_signed_lconjperm (inv_induced_id (sum_list (s#ss)))) =
permutation (signed_lconjpermutation s) ∘
permutation (induced_signed_lconjperm (inv_induced_id (sum_list ss)))"
using plus_permutation.rep_eq induced_signed_lconjperm_inv_induced_id_sum_list
by    simp
with Cons show ?case
using bij_signed_lconjaction[of s] Abs_permutation_inverse by fastforce
qed

lemma flipped_reflections_odd_lconjseq:
assumes "ss∈lists S"
shows   "flipped_reflections (sum_list ss) = {t∈ℋ. odd (count_list (lconjseq ss) t)}"
proof (rule seteqI)
fix t assume "t ∈ flipped_reflections (sum_list ss)"
moreover with assms
have  "snd (signed_list_lconjaction (rev ss) (t,True)) = False"
induced_signed_eq_lconjpermutation[of "rev ss"]
by    force
ultimately show "t ∈ {t∈ℋ. odd (count_list (lconjseq ss) t)}"
signed_list_lconjaction_snd[of "rev ss"]
by    auto
next
fix t assume t: "t ∈ {t∈ℋ. odd (count_list (lconjseq ss) t)}"
with assms
have  "signed_list_lconjaction (rev ss) (t,True) =
(rconjby (sum_list ss) t, False)"
signed_list_lconjaction_fst[of "rev ss"]
uminus_sum_list_order2[of ss, THEN sym]
by    (auto intro: prod_eqI)
with t assms show "t ∈ flipped_reflections (sum_list ss)"
uminus_sum_list_order2 flipped_reflections_def
by    fastforce
qed

lemma flipped_reflections_in_lconjseq:
"ss∈lists S ⟹ flipped_reflections (sum_list ss) ⊆ set (lconjseq ss)"
using flipped_reflections_odd_lconjseq odd_n0 count_notin[of _ "lconjseq ss"]
by    fastforce

lemma flipped_reflections_distinct_lconjseq_eq_lconjseq:
assumes "ss∈lists S" "distinct (lconjseq ss)"
shows   "flipped_reflections (sum_list ss) = set (lconjseq ss)"
proof
from assms(1) show "flipped_reflections (sum_list ss) ⊆ set (lconjseq ss)"
using flipped_reflections_in_lconjseq by fast
show "flipped_reflections (sum_list ss) ⊇ set (lconjseq ss)"
proof
fix t assume "t ∈ set (lconjseq ss)"
moreover with assms(2) have "count_list (lconjseq ss) t = 1"
ultimately show "t ∈ flipped_reflections (sum_list ss)"
using assms(1) flipped_reflections_odd_lconjseq lconjseq_reflections
by    fastforce
qed
qed

lemma flipped_reflections_reduced_eq_lconjseq:
"S_reduced ss ⟹ flipped_reflections (sum_list ss) = set (lconjseq ss)"
using reduced_word_for_lists[of S] S_reduced_imp_distinct_lconjseq
flipped_reflections_distinct_lconjseq_eq_lconjseq
by    fast

lemma card_flipped_reflections:
assumes "w∈W"
shows "card (flipped_reflections w) = S_length w"
proof-
define ss where "ss = arg_min length (word_for S w)"
with assms have "S_reduced_for w ss"
using genby_S_reduced_word_for_arg_min by simp
thus ?thesis
using reduced_word_for_sum_list flipped_reflections_reduced_eq_lconjseq
S_reduced_imp_distinct_lconjseq distinct_card length_lconjseq[of ss]
reduced_word_for_length
by    fastforce
qed

end (* context CoxeterSystem *)

sublocale CoxeterSystem < PreCoxeterSystemWithDeletion
proof
fix ss assume ss: "ss ∈ lists S" "¬ S_reduced ss"
define w where "w = sum_list ss"
with ss(1)
have  "distinct (lconjseq ss) ⟹ card (flipped_reflections w) = length ss"
flipped_reflections_distinct_lconjseq_eq_lconjseq distinct_card
length_lconjseq)
moreover from w_def ss have "length ss > S_length w" using word_length_lt by fast
moreover from w_def ss(1) have "card (flipped_reflections w) = S_length w"
using special_subgroup_eq_sum_list card_flipped_reflections by fast
ultimately have "¬ distinct (lconjseq ss)" by auto
with w_def ss
show  "∃a b as bs cs. ss = as @ [a] @ bs @ [b] @ cs ∧
sum_list ss = sum_list (as @ bs @ cs)"
using deletion'
by    fast
qed

subsubsection ‹The deletion condition is sufficient›

text ‹
Now we come full circle and show that a pair consisting of a group and a generating set of
order-two elements that satisfies the deletion condition affords a presentation that makes it a
Coxeter system.
›

context PreCoxeterSystemWithDeletion
begin

lemma reducible_by_flipping:
"ss ∈ lists S ⟹ ¬ S_reduced ss ⟹
∃xss as t bs. flip_altsublist_chain (ss # xss @ [as@[t,t]@bs])"
proof (induct ss)
case (Cons s ss)
show ?case
proof (cases "S_reduced ss")
case True
define w where "w = sum_list ss"
with True have ss_red_w: "reduced_word_for S w ss" by fast
moreover from Cons(2) have "s∈S" by simp
ultimately obtain as bs where asbs: "reduced_word_for S w (s#as@bs)"
using Cons(3) exchange by fast
show ?thesis
proof (cases "w=0")
case True with asbs show ?thesis
using reduced_word_for_0_imp_nil by fast
next
case False
from this obtain xss where "flip_altsublist_chain (ss # xss @ [s#as@bs])"
using ss_red_w asbs reduced_word_problem by fast
hence "flip_altsublist_chain (
(s#ss) # map ((#) s) xss @ [[]@[s,s]@(as@bs)]
)"
using flip_altsublist_chain_map_Cons_grow by fastforce
thus ?thesis by fast
qed
next
case False
with Cons(1,2) obtain xss as t bs
where "flip_altsublist_chain (
(s#ss) # map ((#) s) xss @ [(s#as)@[t,t]@bs]
)"
using flip_altsublist_chain_map_Cons_grow
by    fastforce
thus ?thesis by fast
qed

lemma freeliftid_kernel':
"ss ∈ lists S ⟹ sum_list ss = 0 ⟹ Abs_freelist ss ∈ Q"
proof (induct ss rule: length_induct)
fix ss
assume step: "∀ts. length ts < length ss ⟶ ts ∈ lists S ⟶
sum_list ts = 0 ⟶ Abs_freelist ts ∈ Q"
and set_up: "ss ∈ lists S" "sum_list ss = 0"
show "Abs_freelist ss ∈ Q"
proof (cases "ss=[]")
case True thus ?thesis
using genby_0_closed[of "⋃w∈FreeGroup S. lconjby w ` P'"]
next
case False
with set_up obtain xss as t bs
where xss: "flip_altsublist_chain (ss # xss @ [as@[t,t]@bs])"
using sum_list_zero_nreduced reducible_by_flipping[of ss]
by    fast
with set_up
have  astbs:  "length (as@[t,t]@bs) = length ss"
"as@[t,t]@bs ∈ lists S"
"sum_list (as@[t,t]@bs) = 0"
using flip_altsublist_chain_length[of ss xss "as@[t,t]@bs"]
flip_altsublist_chain_sum_list[of ss xss "as@[t,t]@bs"]
flip_altsublist_chain_lists[of ss xss "as@[t,t]@bs"]
by    auto
have listsS: "as ∈ lists S" "t∈S" "bs∈lists S" using astbs(2) by auto
have "sum_list as + (t + t + sum_list bs) = 0"
hence "sum_list (as@bs) = 0"
moreover have "length (as@bs) < length ss" using astbs(1) by simp
moreover have "as@bs ∈ lists S" using listsS(1,3) by simp
ultimately have "Abs_freelist (as@bs) ∈ Q" using step by fast
hence "Abs_freelist as + pair_relator_freeword t t +
(- Abs_freelist as + (Abs_freelist as + Abs_freelist bs)) ∈ Q"
using listsS(1,2) lconjby_Abs_freelist_relator_freeword[of t t as]
hence "Abs_freelist as + Abs_freelist [t,t] + Abs_freelist bs ∈ Q"
thus ?thesis
using Abs_freelist_append_append[of as "[t,t]" bs]
rev_flip_altsublist_chain[OF xss]
flip_altsublist_chain_G_in_Q[of "as@[t,t]@bs" "rev xss" ss]
by    simp
qed
qed

lemma freeliftid_kernel:
assumes "c ∈ FreeGroup S" "freeliftid c = 0"
shows   "c∈Q"
proof-
from assms(2) have "freeliftid (Abs_freeword (freeword c)) = 0"
with assms(1) have "sum_list (map fst (freeword c)) = 0"
using FreeGroup_def freeword freeliftid_Abs_freeword_conv_sum_list by fastforce
with assms(1) show ?thesis
using FreeGroup_def freeliftid_kernel'[of "map fst (freeword c)"]
Q_freelist_freeword
by    fastforce
qed

lemma induced_id_kernel:
"c ∈ FreeGroup S ⟹ induced_id (⌈FreeGroup S|c|Q⌉) = 0 ⟹ c∈Q"
freeliftid_kernel
GroupByPresentationInducedFun.induced_hom_equality[
OF GroupByPresentationInducedFun_S_P_id
]
)

theorem CoxeterSystem: "CoxeterSystem S"
proof (rule CoxeterSystemI)
fix x assume x: "x∈G" "induced_id x = 0"
from x(1) obtain c where "c ∈ FreeGroup S" "x = (⌈FreeGroup S|c|Q⌉)"
using Group.quotient_group_UN FreeGroup_Group by fast
with x(2) show "x=0"
using induced_id_kernel
Group.quotient_identity_rule[OF FreeGroup_Group]
GroupByPresentation.Q_subgroup_FreeS[OF GroupByPresentation_S_P]
GroupByPresentation.normal_Q[OF GroupByPresentation_S_P]
by    auto
qed

end (* context PreCoxeterSystemWithDeletion *)

subsubsection ‹The Coxeter system associated to a thin chamber complex with many foldings›

text ‹
We now show that the fundamental automorphisms in a thin chamber complex with many foldings
satisfy the deletion condition, and hence form a Coxeter system.
›

context ThinChamberComplexManyFoldings
begin

lemma not_reduced_word_not_min_gallery:
assumes "ss ∈ lists S" "¬ reduced_word S ss"
shows   "¬ min_gallery (map (λw. w`→C0) (sums ss))"
proof (cases ss rule: list_cases_Cons_snoc)
case Nil with assms(2) show ?thesis using nil_reduced_word_for_0 by auto
next
case (Single s) with assms show ?thesis
using zero_notin_S reduced_word_singleton[of s S] by fastforce
next
case (Cons_snoc s ts t) have ss: "ss = s#ts@[t]" by fact
define Ms where "Ms = map (λw. w`→C0) (map ((+) s) (sums ts))"
with ss
have  C0_ms_ss_C0: "map (λw. w`→C0) (sums ss) =
C0 # Ms @ [sum_list ss `→ C0]"
define rs where "rs = arg_min length (word_for S (sum_list ss))"
with assms(1) have rs: "rs ∈ lists S" "sum_list rs = sum_list ss"
using arg_min_natI[of "λrs. word_for S (sum_list ss) rs" ss length] by auto
show ?thesis
proof (cases rs rule: list_cases_Cons_snoc)
case Nil
hence "sum_list ss `→ C0 = C0"
using rs(2) by (fastforce simp add: zero_permutation.rep_eq)
with C0_ms_ss_C0 show ?thesis by simp
next
case (Single r)
from Single have "min_gallery [C0,r`→C0]"
fundchamber_S_image_neq_fundchamber
with Single C0_ms_ss_C0 Ms_def show ?thesis
using rs(2) min_galleryD_min_betw[of C0 Ms "sum_list ss `→ C0" "[]"]
min_galleryD_gallery
next
case (Cons_snoc p qs q)
define Ns where "Ns = map (λw. w`→C0) (map ((+) p) (sums qs))"
from assms rs_def have "length rs < length ss"
using word_length_lt[of ss S]
reduced_word_for_length reduced_word_for_arg_min[of ss S]
by    force
with Cons_snoc ss Ms_def Ns_def have "length Ns < length Ms"
moreover from Ns_def Cons_snoc
have  "gallery (C0 # Ns @ [sum_list ss `→ C0])"
using rs S_list_image_gallery[of rs]
by    (auto simp add: sums_snoc zero_permutation.rep_eq)
ultimately show ?thesis using C0_ms_ss_C0 not_min_galleryI_betw by auto
qed
qed

lemma S_list_not_min_gallery_double_split:
assumes "ss ∈ lists S" "ss≠[]" "¬ min_gallery (map (λw. w`→C0) (sums ss))"
shows
"∃f g as s bs t cs.
(f,g)∈foldpairs ∧
sum_list as `→ C0 ∈ f⊢𝒞 ∧
sum_list (as@[s]) `→ C0 ∈ g⊢𝒞 ∧
sum_list (as@[s]@bs) `→ C0 ∈ g⊢𝒞 ∧
sum_list (as@[s]@bs@[t]) `→ C0 ∈ f⊢𝒞 ∧
ss = as@[s]@bs@[t]@cs"
proof-
define Cs where "Cs = map (λw. w`→C0) (sums ss)"
moreover from assms(1) Cs_def have "gallery Cs"
using S_list_image_gallery by fastforce
moreover from assms(1) Cs_def have "{} ∉ set (wall_crossings Cs)"
using S_list_image_crosses_walls by fastforce
ultimately obtain f g As A B Bs E F Fs
where   fg          : "(f,g)∈foldpairs"
and     sep         : "A∈f⊢𝒞" "B∈g⊢𝒞" "E∈g⊢𝒞" "F∈f⊢𝒞"
and     decomp_cases:
"Cs = As@[A,B,F]@Fs ∨ Cs = As@[A,B]@Bs@[E,F]@Fs"
using   assms(3) not_min_gallery_double_split[of Cs]
by      blast
show ?thesis
proof (cases "Cs = As@[A,B,F]@Fs")
case True
define bs :: "'a permutation list" where "bs = []"
from True Cs_def obtain as s t cs where
"ss = as@[s,t]@cs" "A = sum_list as `→ C0" "B = sum_list (as@[s]) `→ C0"
"F = sum_list (as@[s,t]) `→ C0"
using pullback_sums_map_middle3[of "λw. w`→C0" ss As A B F Fs]
by    auto
with sep(1,2,4) bs_def have
"sum_list as `→ C0 ∈ f⊢𝒞" "sum_list (as@[s]) `→ C0 ∈ g⊢𝒞"
"sum_list (as@[s]@bs) `→ C0 ∈ g⊢𝒞" "sum_list (as@[s]@bs@[t]) `→ C0 ∈ f⊢𝒞"
"ss = as@[s]@bs@[t]@cs"
by auto
with fg show ?thesis by blast
next
case False
with Cs_def decomp_cases obtain as s bs t cs where
"ss = as@[s]@bs@[t]@cs" "A = sum_list as `→ C0" "B = sum_list (as@[s]) `→ C0"
"E = sum_list (as@[s]@bs) `→ C0" "F = sum_list (as@[s]@bs@[t]) `→ C0"
using pullback_sums_map_double_middle2[
of "λw. w`→C0" ss As A B Bs E F Fs
]
by    auto
with sep have
"sum_list as `→ C0 ∈ f⊢𝒞" "sum_list (as@[s]) `→ C0 ∈ g⊢𝒞"
"sum_list (as@[s]@bs) `→ C0 ∈ g⊢𝒞" "sum_list (as@[s]@bs@[t]) `→ C0 ∈ f⊢𝒞"
"ss = as@[s]@bs@[t]@cs"
by auto
with fg show ?thesis by blast
qed
qed

lemma fold_end_sum_chain_fg:
fixes   f g :: "'a⇒'a"
defines 𝗌  : "𝗌 ≡ induced_automorph f g"
assumes fg :  "(f,g) ∈ foldpairs"
and     as :  "as ∈ lists S"
and     s  :  "s∈S"
and     sep:  "sum_list as `→ C0 ∈ f⊢𝒞" "sum_list (as@[s]) `→ C0 ∈ g⊢𝒞"
shows   "bs ∈lists S ⟹
𝗌 ` sum_list (as@[s]@bs) `→ C0 = sum_list (as@bs) `→ C0"
proof-
from fg obtain C where C: "OpposedThinChamberComplexFoldings X f g C"
using foldpairs_def by fast
show "bs ∈lists S ⟹ 𝗌 ` sum_list (as@[s]@bs) `→ C0 = sum_list (as@bs) `→ C0"
proof (induct bs rule: rev_induct)
case Nil
from 𝗌 as s sep C show ?case
using sum_list_S_in_W[of as] sum_list_append[of as "[s]"]
)
next
case (snoc b bs)
define bC0 B where "bC0 = b`→C0" and "B = sum_list (as@bs) `→ C0"
define y where "y = C0∩bC0"
define z z'
where "z = 𝗌 ` sum_list (as@[s]@bs) `→ y"
and "z' = sum_list (as@bs) `→ y"

from snoc B_def have B': "𝗌 ` sum_list (as@[s]@bs) `→ C0 = B" by simp

obtain φ where φ: "label_wrt C0 φ" using ex_label_map by fast
from bC0_def y_def snoc(2) obtain u where u: "bC0 = insert u y"
fundchamber_S_image_neq_fundchamber
define v v'
where "v = 𝗌 (sum_list (as@[s]@bs) → u)"
and "v' = sum_list (as@bs) → u"

from bC0_def u v_def z_def v'_def z'_def
have  ins_vz : "𝗌 ` sum_list (as@[s]@bs@[b]) `→ C0 = insert v z"
and   ins_vz': "sum_list (as@bs@[b]) `→ C0 = insert v' z'"
using image_insert[of "permutation (sum_list (as@[s]@bs))" u y, THEN sym]
image_insert[
of 𝗌 "sum_list (as@[s]@bs)→u" "sum_list (as@[s]@bs)`→y",
THEN sym]
image_insert[of "permutation (sum_list (as@bs))" u y, THEN sym]
by    (auto simp add: plus_permutation.rep_eq image_comp)

from as s snoc(2) have sums:
"sum_list (as@[s]@bs) ∈ W" "sum_list (as@bs) ∈ W"
"sum_list (as@[s]@bs@[b]) ∈ W" "sum_list (as@bs@[b]) ∈ W"
using sum_list_S_in_W[of "as@[s]@bs"] sum_list_S_in_W[of "as@bs"]
sum_list_S_in_W[of "as@[s]@bs@[b]"] sum_list_S_in_W[of "as@bs@[b]"]
by    auto
from u bC0_def snoc(2) have u: "u∈⋃X"
using fundchamber_S_chamber[of b] chamberD_simplex[of bC0] by auto
moreover from as s snoc(2) u have "sum_list (as@[s]@bs) → u ∈ ⋃X"
using sums(1)
ChamberComplexEndomorphism.vertex_map[OF W_endomorphism]
by    fastforce
ultimately have "φ v = φ v'"
using 𝗌 v_def v'_def sums(1,2) W_respects_labels[OF φ, of "sum_list (as@[s]@bs)" u]
W_respects_labels[OF φ, of "sum_list (as@bs)" u]
OpposedThinChamberComplexFoldings.indaut_resplabels[
OF C φ
]
by    simp

moreover from 𝗌 have "chamber (insert v z)" "chamber (insert v' z')"
using sums(3,4)
fundchamber_W_image_chamber[of "sum_list (as@[s]@bs@[b])"]
OpposedThinChamberComplexFoldings.indaut_chmap[
OF C
]
fundchamber_W_image_chamber[of "sum_list (as@bs@[b])"]
by    (auto simp add: ins_vz[THEN sym] ins_vz'[THEN sym])

moreover from y_def z_def z'_def bC0_def B_def snoc(2) 𝗌 have "z⊲B" "z'⊲B"
fundchamber_S_image_neq_fundchamber[of b]
W_endomorphism[of "sum_list (as@bs)"]
W_endomorphism[of "sum_list (as@[s]@bs)"]
fundchamber fundchamber_W_image_chamber[of "sum_list (as@[s]@bs)"]
ChamberComplexEndomorphism.facet_map[of X]
OpposedThinChamberComplexFoldings.indaut_morph[
OF C
]
ChamberComplexEndomorphism.facet_map[
of X 𝗌 "sum_list (as@[s]@bs) `→ C0"
]
by    auto

moreover from snoc(2) B_def 𝗌 have "insert v z ≠ B" "insert v' z' ≠ B"
using sum_list_append[of "as@[s]@bs" "[b]"] sum_list_append[of "as@bs" "[b]"]
fundchamber_next_WS_image_neq[of b "sum_list (as@[s]@bs)"]
fundchamber_next_WS_image_neq[of b "sum_list (as@bs)"]
OpposedThinChamberComplexFoldings.indaut_aut[
OF C
]
ChamberComplexAutomorphism.bij bij_is_inj B'
inj_eq_image[
of 𝗌 "sum_list (as@[s]@bs@[b]) `→ C0" "sum_list (as@[s]@bs) `→ C0"
]
by    (auto simp add: ins_vz[THEN sym] ins_vz'[THEN sym])

ultimately show ?case
using B_def sums(2) fundchamber_W_image_chamber[of "sum_list (as@bs)"]
label_wrt_eq_on_adjacent_vertex[OF φ, of v v' B z z']
by    (auto simp add: ins_vz[THEN sym] ins_vz'[THEN sym])
qed
qed

lemma fold_end_sum_chain_gf:
fixes   f g :: "'a⇒'a"
defines "𝗌 ≡ induced_automorph f g"
assumes fg :  "(f,g) ∈ foldpairs"
and     "as ∈ lists S" "s∈S" "bs ∈lists S"
"sum_list as `→ C0 ∈ g⊢𝒞"
"sum_list (as@[s]) `→ C0 ∈ f⊢𝒞"
shows   "𝗌 ` sum_list (as@[s]@bs) `→ C0 = sum_list (as@bs) `→ C0"
proof-
from fg obtain C where C: "OpposedThinChamberComplexFoldings X f g C"
using foldpairs_def by fast
from assms show ?thesis
using foldpairs_sym fold_end_sum_chain_fg[of g f as s bs]
OpposedThinChamberComplexFoldings.induced_automorphism_sym[OF C]
by    simp
qed

lemma fold_middle_sum_chain:
assumes fg :  "(f,g) ∈ foldpairs"
and     S  :  "as ∈ lists S" "s∈S" "bs ∈ lists S" "t∈S" "cs ∈lists S"
and     sep:  "sum_list as `→ C0 ∈ f⊢𝒞"
"sum_list (as@[s]) `→ C0 ∈ g⊢𝒞"
"sum_list (as@[s]@bs) `→ C0 ∈ g⊢𝒞" "sum_list (as@[s]@bs@[t]) `→ C0 ∈ f⊢𝒞"
shows   "sum_list (as@[s]@bs@[t]@cs) `→ C0 = sum_list (as@bs@cs) `→ C0"
proof-
define 𝗌 where "𝗌 = induced_automorph f g"
from fg obtain C
where "OpposedThinChamberComplexFoldings X f g C"
using foldpairs_def
by    fast
then have "id ` sum_list (as@[s]@bs@[t]@cs) `→ C0 = sum_list (as@bs@cs) `→ C0"
using 𝗌_def fg S sep fold_end_sum_chain_gf[of f g "as@[s]@bs" t cs]
fold_end_sum_chain_fg[of f g as s "bs@cs"]
image_comp[THEN sym]
OpposedThinChamberComplexFoldings.indaut_order2[
THEN sym]
)
thus ?thesis by simp
qed

lemma S_list_not_min_gallery_deletion:
fixes ss :: "'a permutation list"
defines w : "w ≡ sum_list ss"
assumes ss: "ss∈lists S" "ss≠[]" "¬ min_gallery (map (λw. w`→C0) (sums ss))"
shows "∃a b as bs cs. ss = as@[a]@bs@[b]@cs ∧ w = sum_list (as@bs@cs)"
proof-
from w ss(1) have w_W: "w∈W" using sum_list_S_in_W by fast

define Cs where "Cs = map (λw. w`→C0) (sums ss)"
from ss obtain f g as s bs t cs
where fg    : "(f,g)∈foldpairs"
and   sep   : "sum_list as `→ C0 ∈ f⊢𝒞"
"sum_list (as@[s]) `→ C0 ∈ g⊢𝒞"
"sum_list (as@[s]@bs) `→ C0 ∈ g⊢𝒞"
"sum_list (as@[s]@bs@[t]) `→ C0 ∈ f⊢𝒞"
and   decomp: "ss = as@[s]@bs@[t]@cs"
using S_list_not_min_gallery_double_split[of ss]
by    blast
from fg sep decomp w ss(1)
have  "w`→C0 = sum_list (as@bs@cs) `→ C0"
using fold_middle_sum_chain
by    auto
with ss(1) decomp have "w = sum_list (as@bs@cs)"
using w_W sum_list_S_in_W[of "as@bs@cs"]
by    (auto intro: inj_onD fundchamber_W_image_inj_on)
with decomp show ?thesis by fast
qed

lemma deletion:
"ss ∈ lists S ⟹ ¬ reduced_word S ss ⟹
∃a b as bs cs. ss = as@[a]@bs@[b]@cs ∧ sum_list ss = sum_list (as@bs@cs)"
using nil_reduced_word_for_0[of S] not_reduced_word_not_min_gallery
S_list_not_min_gallery_deletion
by    fastforce

lemma PreCoxeterSystemWithDeletion: "PreCoxeterSystemWithDeletion S"
using S_add_order2 deletion by unfold_locales simp

lemma CoxeterSystem: "CoxeterSystem S"
using PreCoxeterSystemWithDeletion
PreCoxeterSystemWithDeletion.CoxeterSystem
by    fast

end (* context ThinChamberComplexManyFoldings *)

subsection ‹Coxeter complexes›

subsubsection ‹Locale and complex definitions›

text ‹
Now we add in the assumption that the generating set is finite, and construct the associated
Coxeter complex from the poset of special cosets.
›

locale CoxeterComplex = CoxeterSystem S
+ assumes finite_genset: "finite S"
begin

definition TheComplex :: "'w set set set"
where "TheComplex ≡ ordering.PosetComplex (⊇) (⊃) 𝒫"
abbreviation "Σ ≡ TheComplex"

end (* context CoxeterComplex *)

subsubsection ‹As a simplicial complex›

text ‹
Here we record the fact that the Coxeter complex associated to a Coxeter system is a simplicial
complex, and note that the poset of special cosets is complex-like. This last fact allows us to
reason about the complex by reasoning about the poset, via the poset isomorphism
@{const ComplexLikePoset.smap}.
›

context CoxeterComplex
begin

lemma simplex_like_special_cosets:
assumes "X∈𝒫"
shows "supset_simplex_like (𝒫.⊇X)"
proof-
have image_eq_UN: "⋀f A. f ` A = (⋃x∈A. {f x})" by blast

from assms obtain w T where "w∈W" "T ∈ Pow S" "X = w +o ⟨T⟩"
using special_cosets_def by auto
thus ?thesis
using image_eq_UN[where f= "(+o) w ∘ genby"]
finite_genset simplex_like_pow_above_in
OrderingSetIso.simplex_like_map[
OF special_coset_below_in_supset_ordering_iso, of T w
]
special_cosets_below_in
by    force
qed

lemma SimplicialComplex_Σ: "SimplicialComplex Σ"
unfolding TheComplex_def
proof (rule ordering.poset_is_SimplicialComplex)
show "ordering (⊇) (⊃)" ..
show "∀X∈𝒫. supset_simplex_like (𝒫.⊇X)"
using simplex_like_special_cosets by fast
qed

lemma ComplexLikePoset_special_cosets: "ComplexLikePoset (⊇) (⊃) 𝒫"
using simplex_like_special_cosets special_cosets_has_bottom special_cosets_have_glbs
by    unfold_locales

abbreviation "smap ≡ ordering.poset_simplex_map (⊇) (⊃) 𝒫"

lemmas smap_def = ordering.poset_simplex_map_def[OF supset_poset, of 𝒫]

lemma ordsetmap_smap: "⟦ X∈𝒫; Y∈𝒫; X⊇Y ⟧ ⟹ smap X ⊆ smap Y"
using ComplexLikePoset.ordsetmap_smap[OF ComplexLikePoset_special_cosets]
smap_def
by    simp

lemma rev_ordsetmap_smap: "⟦ X∈𝒫; Y∈𝒫; smap X ⊆ smap Y ⟧ ⟹ X⊇Y"
using ComplexLikePoset.rev_ordsetmap_smap[
OF ComplexLikePoset_special_cosets
]
smap_def
by    simp

lemma smap_onto_PosetComplex: "smap ` 𝒫 = Σ"
using ComplexLikePoset.smap_onto_PosetComplex[
OF ComplexLikePoset_special_cosets
]
smap_def TheComplex_def
by    simp

lemmas simplices_conv_special_cosets = smap_onto_PosetComplex[THEN sym]

lemma smap_into_PosetComplex: "X∈𝒫 ⟹ smap X ∈ Σ"
using smap_onto_PosetComplex by fast

lemma smap_pseudominimal:
"w∈W ⟹ s∈S ⟹ smap (w +o ⟨S-{s}⟩) = {w +o ⟨S-{s}⟩}"
using smap_def[of "w +o ⟨S-{s}⟩"]
special_coset_pseudominimal_in_below_in[of w "S-{s}"]
exclude_one_is_pseudominimal_in_below_in[of w "S-{s}"]
by    auto

lemma exclude_one_notin_smap_singleton:
"s∈S ⟹ w +o ⟨S-{s}⟩ ∉ smap (w +o ⟨{s}⟩)"
using smap_def[of "w +o ⟨{s}⟩"]
supset_pseudominimal_inD1[of "𝒫.⊇(w +o ⟨{s}⟩)" "w +o ⟨S-{s}⟩"]
special_coset_subset_rev_mono[of "{s}" "S-{s}"]
by    auto

lemma maxsimp_vertices: "w∈W ⟹ s∈S ⟹ w +o ⟨S-{s}⟩ ∈ smap {w}"
using special_cosetsI[of "S-{s}"] special_coset_singleton
ordsetmap_smap[of "w +o ⟨S-{s}⟩"] smap_pseudominimal

lemma maxsimp_singleton:
assumes "w∈W"
shows   "SimplicialComplex.maxsimp Σ (smap {w})"
proof (rule SimplicialComplex.maxsimpI, rule SimplicialComplex_Σ)
from assms show "smap {w} ∈ Σ"
using special_coset_singleton smap_into_PosetComplex by fast
next
fix z assume z: "z∈Σ" "smap {w} ⊆ z"
from z(1) obtain X where X: "X∈𝒫" "z = smap X"
using simplices_conv_special_cosets by auto
with assms z(2) have "X = {w}"
using special_coset_singleton rev_ordsetmap_smap special_coset_nempty by fast
with X(2) show "z = smap {w}" by fast
qed

lemma maxsimp_is_singleton:
assumes "SimplicialComplex.maxsimp Σ x"
shows   "∃w∈W. smap {w} = x"
proof-
from assms obtain X where X: "X∈𝒫" "smap X = x"
using SimplicialComplex.maxsimpD_simplex[OF SimplicialComplex_Σ]
simplices_conv_special_cosets
by    auto
from X(1) obtain w T where wT: "w∈W" "T∈Pow S" "X = w +o ⟨T⟩"
using special_cosets_def by auto
from wT(1) have "{w}∈𝒫" using special_coset_singleton by fast
moreover with X wT(3) have "x ⊆ smap {w}"
using genby_lcoset_refl ordsetmap_smap by fast
ultimately show ?thesis
using assms wT(1) smap_into_PosetComplex
SimplicialComplex.maxsimpD_maximal[OF SimplicialComplex_Σ]
by    fast
qed

lemma maxsimp_vertex_conv_special_coset:
"w∈W ⟹ X ∈ smap {w} ⟹ ∃s∈S. X = w +o ⟨S-{s}⟩"
using smap_def special_coset_pseudominimal_in_below_in[of w "{}"]

lemma vertices: "w∈W ⟹ s∈S ⟹ w +o ⟨S-{s}⟩ ∈ ⋃Σ"
using maxsimp_singleton SimplicialComplex.maxsimpD_simplex[OF SimplicialComplex_Σ]
maxsimp_vertices
by    fast

lemma smap0_conv_special_subgroups:
"smap 0 = (λs. ⟨S - {s}⟩) ` S"
using genby_0_closed maxsimp_vertices maxsimp_vertex_conv_special_coset
by    force

lemma S_bij_betw_chamber0: "bij_betw (λs. ⟨S-{s}⟩) S (smap 0)"
unfolding bij_betw_def
proof
show "inj_on (λs. ⟨S-{s}⟩) S"
proof (rule inj_onI)
fix s t show "⟦ s∈S; t∈S; ⟨S-{s}⟩ = ⟨S-{t}⟩ ⟧ ⟹ s = t"
using inj_onD[OF special_subgroup_inj, of "S-{s}" "S-{t}"] by fast
qed
qed (rule smap0_conv_special_subgroups[THEN sym])

lemma smap_singleton_conv_W_image:
"w∈W ⟹ smap {w} = ((+o) w) ` (smap 0)"
using genby_0_closed[of S] maxsimp_vertices[of 0] maxsimp_vertices[of w]
maxsimp_vertex_conv_special_coset
by    force

lemma W_lcoset_bij_betw_singletons:
assumes "w∈W"
shows   "bij_betw ((+o) w) (smap 0) (smap {w})"
unfolding bij_betw_def
proof (rule conjI, rule inj_onI)
fix X Y assume XY: "X ∈ smap 0" "Y ∈ smap 0" "w +o X = w +o Y"
from XY(1,2) obtain sx sy where "X = ⟨S-{sx}⟩" "Y = ⟨S-{sy}⟩"
using maxsimp_vertex_conv_special_coset[of 0 X]
maxsimp_vertex_conv_special_coset[of 0 Y] genby_0_closed[of S]
by    auto
with XY(3) show "X=Y"
using inj_onD[OF special_coset_inj, of w "S-{sx}" "S-{sy}"] by force
qed (rule smap_singleton_conv_W_image[THEN sym], rule assms)

lemma facets:
assumes "w∈W" "s∈S"
shows "smap (w +o ⟨{s}⟩) ⊲ smap {w}"
proof (
rule facetrelI, rule exclude_one_notin_smap_singleton, rule assms(2),
rule order_antisym
)

show "smap {w} ⊆ insert (w +o ⟨S - {s}⟩) (smap (w +o ⟨{s}⟩))"
proof
fix X assume "X ∈ smap {w}"
with assms(1) obtain t where "t∈S" "X = w +o ⟨S-{t}⟩"
using maxsimp_vertex_conv_special_coset by fast
with assms show "X∈ insert (w +o ⟨S - {s}⟩) (smap (w +o ⟨{s}⟩))"
using exclude_one_is_pseudominimal_in_below_in smap_def
by    (cases "t=s") auto
qed

from assms show "smap {w} ⊇ insert (w +o ⟨S - {s}⟩) (smap (w +o ⟨{s}⟩))"
using genby_lcoset_refl special_cosetsI[of "{s}"] special_coset_singleton
ordsetmap_smap maxsimp_vertices
by    fast

qed

lemma facets': "w∈W ⟹ s∈S ⟹ smap {w,w+s} ⊲ smap {w}"

lemma adjacent: "w∈W ⟹ s∈S ⟹ smap {w+s} ∼ smap {w}"
using facets'[of w s] genby_genset_closed genby_add_closed[of w S]
facets'[of "w+s" s]
by    (
)

lemma singleton_adjacent_0: "s∈S ⟹ smap {s} ∼ smap 0"
using genby_genset_closed genby_0_closed facets'[of 0] facets'[of s]

end (* context CoxeterComplex *)

subsubsection ‹As a chamber complex›

text ‹Now we verify that a Coxeter complex is a chamber complex.›

context CoxeterComplex
begin

abbreviation "chamber ≡ SimplicialComplex.maxsimp Σ"
abbreviation "gallery ≡ SimplicialComplex.maxsimpchain Σ"

lemmas chamber_singleton                 = maxsimp_singleton
lemmas chamber_vertex_conv_special_coset = maxsimp_vertex_conv_special_coset
lemmas chamber_vertices                  = maxsimp_vertices
lemmas chamber_is_singleton              = maxsimp_is_singleton

lemmas faces       = SimplicialComplex.faces            [OF SimplicialComplex_Σ]
lemmas gallery_def = SimplicialComplex.maxsimpchain_def [OF SimplicialComplex_Σ]
lemmas gallery_rev = SimplicialComplex.maxsimpchain_rev [OF SimplicialComplex_Σ]

lemmas chamberD_simplex =
SimplicialComplex.maxsimpD_simplex[OF SimplicialComplex_Σ]
lemmas gallery_CConsI =
SimplicialComplex.maxsimpchain_CConsI[OF SimplicialComplex_Σ]

lemmas gallery_overlap_join =
SimplicialComplex.maxsimpchain_overlap_join[OF SimplicialComplex_Σ]

lemma word_gallery_to_0:
"ss ≠ [] ⟹ ss∈ lists S ⟹ ∃xs. gallery (smap {sum_list ss} # xs @ [smap 0])"
proof (induct ss rule: rev_nonempty_induct)
case (single s)
hence "gallery (smap {sum_list [s]} # [] @ [smap 0])"
using genby_genset_closed genby_0_closed chamber_singleton
by    auto
thus ?case by fast
next
case (snoc s ss)
from snoc(2,3) obtain xs where "gallery (smap {sum_list ss} # xs @ [smap 0])"
by auto
moreover from snoc(3) have "chamber (smap {sum_list (ss@[s])})"
using special_subgroup_eq_sum_list chamber_singleton by fast
ultimately
have  "gallery (smap {sum_list (ss@[s])} #
(smap {sum_list ss} # xs) @ [smap 0])"
using snoc(3) special_subgroup_eq_sum_list adjacent[of "sum_list ss" s]
by    (auto intro: gallery_CConsI)
thus ?case by fast
qed

lemma gallery_to_0:
assumes "w∈W" "w≠0"
shows   "∃xs. gallery (smap {w} # xs @ [smap 0])"
proof-
from assms(1) obtain ss where ss: "ss∈lists S" "w = sum_list ss"
using special_subgroup_eq_sum_list by auto
with assms(2) show ?thesis using word_gallery_to_0[of ss] by fastforce
qed

lemma ChamberComplex_Σ: "ChamberComplex Σ"
proof (intro_locales, rule SimplicialComplex_Σ, unfold_locales)
fix y assume "y∈Σ"
from this obtain X where X: "X∈𝒫" "y = smap X"
using simplices_conv_special_cosets by auto
from X(1) obtain w T where "w∈W" "X = w +o ⟨T⟩"
using special_cosets_def by auto
with X show "∃x. chamber x ∧ y ⊆ x"
using genby_lcoset_refl special_coset_singleton ordsetmap_smap
chamber_singleton
by    fastforce
next
fix x y
assume xy: "x≠y" "chamber x" "chamber y"
from xy(2,3) obtain w w'
where ww': "w∈W" "x = smap {w}" "w'∈W" "y = smap {w'}"
using chamber_is_singleton
by    blast
show "∃zs. gallery (x # zs @ [y])"
proof (cases "w=0" "w'=0" rule: two_cases)
case both with xy(1) ww'(2,4) show ?thesis by fast
next
case one with ww'(2-4) show ?thesis
using gallery_to_0 gallery_rev by fastforce
next
case other with ww'(1,2,4) show ?thesis using gallery_to_0 by auto
next
case neither
from this ww' obtain xs ys
where "gallery (x # xs @ [smap 0])" "gallery (smap 0 # ys @ [y])"
using gallery_to_0 gallery_rev
by    force
hence "gallery (x # (xs @ smap 0 # ys) @ [y])"
using gallery_overlap_join[of "x#xs"] by simp
thus ?thesis by fast
qed
qed

lemma card_chamber: "chamber x ⟹ card x = card S"
using bij_betw_same_card[OF S_bij_betw_chamber0] chamber_singleton
genby_0_closed[of S]
ChamberComplex.chamber_card[OF ChamberComplex_Σ, of "smap 0"]
by    simp

lemma vertex_conv_special_coset:
"X∈⋃Σ ⟹ ∃w s. w∈W ∧ s∈S ∧ X = w +o ⟨S-{s}⟩"
using ChamberComplex.simplex_in_max[OF ChamberComplex_Σ] chamber_is_singleton
chamber_vertex_conv_special_coset
by    fast

end (* context CoxeterComplex *)

subsubsection ‹The Coxeter complex associated to a thin chamber complex with many foldings›

text ‹
Having previously verified that the fundamental automorphisms in a thin chamber complex with many
foldings form a Coxeter system, we now record the existence of a chamber complex isomorphism onto
the associated Coxeter complex.
›

context ThinChamberComplexManyFoldings
begin

lemma CoxeterComplex: "CoxeterComplex S"
by  (
rule CoxeterComplex.intro, rule CoxeterSystem, unfold_locales,
rule finite_S
)

abbreviation "Σ ≡ CoxeterComplex.TheComplex S"

lemma S_list_not_min_gallery_not_reduced:
assumes "ss≠[]" "¬ min_gallery (map (λw. w`→C0) (sums ss))"
shows "¬ reduced_word S ss"
proof (cases "ss∈lists S")
case True
obtain a b as bs cs
where "ss = as@[a]@bs@[b]@cs" "sum_list ss = sum_list (as@bs@cs)"
using S_list_not_min_gallery_deletion [OF True assms]
by blast
with True show ?thesis using not_reduced_word_for[of "as@bs@cs"] by auto
next
case False thus ?thesis using reduced_word_for_lists by fast
qed

lemma reduced_S_list_min_gallery:
"ss≠[] ⟹ reduced_word S ss ⟹ min_gallery (map (λw. w`→C0) (sums ss))"
using S_list_not_min_gallery_not_reduced by fast

lemma fundchamber_vertex_stabilizer1:
fixes   t
defines v:  "v ≡ fundantivertex t"
assumes tw: "t∈S" "w∈W" "w→v = v"
shows   "w ∈ ⟨S-{t}⟩"
proof-
from v tw(1) have v_C0: "v∈C0" using fundantivertex by simp
define ss where "ss = arg_min length (word_for S w)"
moreover
have "reduced_word S ss ⟹ sum_list ss → v = v ⟹ sum_list ss ∈ ⟨S-{t}⟩"
proof (induct ss)
case (Cons s ss)
from Cons(2) have s_S: "s∈S" using reduced_word_for_lists by fastforce
from this obtain f g
where fg: "(f,g)∈fundfoldpairs" "s = Abs_induced_automorph f g"
by    auto
from fg(1) have opp_fg: "OpposedThinChamberComplexFoldings X f g C0"
using fundfoldpairs_def by auto
define Cs where "Cs = map (λw. w`→C0) (sums (s#ss))"
with Cons(2) have minCs: "min_gallery Cs"
using reduced_S_list_min_gallery by fast
have sv: "s→v = v"
proof (cases ss rule: rev_cases)
case Nil with Cons(3) show ?thesis by simp
next
case (snoc ts t)
define Ms Cn
where "Ms = map (λw. w`→C0) (map ((+) s) (sums ts))"
and "Cn = sum_list (s#ss) `→ C0"
with snoc Cs_def have "Cs = C0 # Ms @ [Cn]"
with minCs Cs_def fg have "C0∈f⊢𝒞" "Cn∈g⊢𝒞"
using sums_Cons_conv_append_tl[THEN sym, of s ss]
wall_crossings_subset_walls_betw[of C0 Ms Cn] fundfoldpairs_def
OpposedThinChamberComplexFoldings.basech_halfchsys(1)[
OF opp_fg
]
OpposedThinChamberComplexFoldings.separated_by_this_wall_fg[
OF opp_fg, of C0 Cn
]
moreover from Cons(3) Cn_def have "v∈Cn" using v_C0 by force
ultimately show "s→v = v"
using v_C0 fg
OpposedThinChamberComplexFoldings.indaut_wallvertex[
OF opp_fg
]
qed
moreover from Cons(3) have "0 → sum_list ss → v = s→v"
using s_S
ultimately have "sum_list ss → v = v" by (simp add: zero_permutation.rep_eq)
with Cons(1,2) have "sum_list ss ∈ ⟨S-{t}⟩"
using reduced_word_Cons_reduce by auto
moreover from tw(1) v have "s∈⟨S-{t}⟩"
using sv s_S genby_genset_closed[of s "S-{t}"] fundantivertex_unstable
by    fastforce
ultimately show ?case using genby_add_closed by simp
ultimately show ?thesis
using tw(2,3) reduced_word_for_genby_sym_arg_min[OF S_sym]
reduced_word_for_sum_list
by    fastforce
qed

lemma fundchamber_vertex_stabilizer2:
assumes s: "s∈S"
defines v: "v ≡ fundantivertex s"
shows   "w ∈ ⟨S-{s}⟩ ⟹ w→v = v"
proof (erule genby.induct)
show "0→v = v" by (simp add: zero_permutation.rep_eq)
next
fix t assume "t∈S-{s}"
moreover with s v have "v∈C0∩t`→C0"
fundchamber_S_image_neq_fundchamber[THEN not_sym]
not_the1[OF adj_antivertex, of C0 "t`→C0" v] fundantivertex
unfolding fundantivertex_def
by        auto
ultimately show "t→v = v"
using S_fixespointwise_fundchamber_image_int fixespointwiseD by fastforce
next
fix w w' assume ww': "w→v = v" "w'→v = v"
from ww'(2) have "(-w')→v = id v"
using plus_permutation.rep_eq[of "-w'" w']
by    (auto simp add: zero_permutation.rep_eq[THEN sym])
with ww'(1) show "(w-w')→v = v"
using plus_permutation.rep_eq[of w "-w'"] by simp
qed

lemma label_wrt_special_coset1:
assumes "label_wrt C0 φ" "fixespointwise φ C0" "w0∈W" "s∈S"
defines "v ≡ fundantivertex s"
shows   "{w∈W. w → φ (w0→v) = w0→v} = w0 +o ⟨S-{s}⟩"
proof-
from assms(4,5) have v_C0: "v∈C0" using fundantivertex[of s] by simp
show ?thesis
proof (rule seteqI)
fix w assume "w∈{w∈W. w→(φ (w0→v)) = w0→v}"
hence w: "w∈W" "w→(φ (w0→v)) = w0→v" by auto
from assms(2,3) have "(-w0 + w) → v = 0→v"
using w(2) v_C0 fundchamber chamberD_simplex
W_respects_labels[OF assms(1)] plus_permutation.rep_eq[of "-w0" w0]
by    (fastforce simp add: plus_permutation.rep_eq fixespointwiseD)
with assms(3-5) show "w ∈ w0 +o ⟨S-{s}⟩"
using w(1) genby_uminus_add_closed[of w0 S w]
fundchamber_vertex_stabilizer1
by    (force simp add: zero_permutation.rep_eq elt_set_plus_def)
next
fix w assume w: "w ∈ w0 +o ⟨S-{s}⟩"
from this obtain w1 where w1: "w1 ∈ ⟨S-{s}⟩" "w = w0 + w1"
using elt_set_plus_def by blast
moreover with w assms(3) have w_W: "w∈W"
using genby_mono[of "S-{s}" S] genby_add_closed by fastforce
ultimately show "w∈{w∈W. w→(φ (w0→v)) = w0→v}"
using assms(2-5) v_C0 fundchamber chamberD_simplex
W_respects_labels[OF assms(1), of w0 v]
fundchamber_vertex_stabilizer2[of s w1]
by    (fastforce simp add: fixespointwiseD plus_permutation.rep_eq)
qed
qed

lemma label_wrt_special_coset1':
assumes "label_wrt C0 φ" "fixespointwise φ C0" "w0∈W" "v∈C0"
defines "s ≡ fundantipermutation v"
shows   "{w∈W. w → φ (w0→v) = w0→v} = w0 +o ⟨S-{s}⟩"
using   assms fundantipermutation1 fundantivertex_bij_betw
bij_betw_f_the_inv_into_f label_wrt_special_coset1[of φ w0 s]
by      fastforce

lemma label_wrt_special_coset2':
assumes "label_wrt C0 φ" "fixespointwise φ C0" "w0∈W" "v ∈ w0`→C0"
defines "s ≡ fundantipermutation (φ v)"
shows   "{w∈W. w → φ v = v} = w0 +o ⟨S-{s}⟩"
using   assms fundchamber chamberD_simplex W_respects_labels
label_wrt_special_coset1'[OF assms(1-3)]
by      (fastforce simp add: fixespointwiseD) (* slow *)

lemma label_stab_map_W_fundchamber_image:
assumes "label_wrt C0 φ" "fixespointwise φ C0" "w0∈W"
defines "ψ ≡ λv. {w∈W. w→(φ v) = v}"
shows   "ψ`(w0`→C0) = CoxeterComplex.smap S {w0}"
proof (rule seteqI)
from assms
show  "⋀x. x ∈ CoxeterComplex.smap S {w0} ⟹ x ∈ ψ`(w0`→C0)"
using CoxeterComplex.chamber_vertex_conv_special_coset[
OF CoxeterComplex, of w0
]
label_wrt_special_coset1 fundantivertex
by    fastforce
next
fix x assume "x∈ ψ`(w0`→C0)"
from this obtain v where v: "v∈w0`→C0" "x = ψ v" by fast
with assms have "x = w0 +o ⟨S-{fundantipermutation (φ v)}⟩"
using label_wrt_special_coset2' by fast
moreover from v(1) assms(3) have "v∈⋃X"
using fundchamber chamberD_simplex W_endomorphism
ChamberComplexEndomorphism.vertex_map
by    fastforce
ultimately show "x ∈ CoxeterComplex.smap S {w0}"
using assms(1,3) label_wrt_elt_image fundantipermutation1
CoxeterComplex.chamber_vertices[OF CoxeterComplex]
by    fastforce
qed

lemma label_stab_map_chamber_map:
assumes φ: "label_wrt C0 φ" "fixespointwise φ C0"
and     C: "chamber C"
defines ψ: "ψ ≡ λv. {w∈W. w→(φ v) = v}"
shows   "CoxeterComplex.chamber S (ψ`C)"
proof-
from C obtain w where w: "w∈W" "C = w`→C0"
using chamber_eq_W_image by fast
with φ ψ have "ψ`C = CoxeterComplex.smap S {w}"
using label_stab_map_W_fundchamber_image by simp
with w(1) show ?thesis
using CoxeterComplex.chamber_singleton[OF CoxeterComplex] by simp
qed

lemma label_stab_map_inj_on_vertices:
assumes φ: "label_wrt C0 φ" "fixespointwise φ C0"
defines ψ: "ψ ≡ λv. {w∈W. w→(φ v) = v}"
shows "inj_on ψ (⋃X)"
proof (rule inj_onI)
fix v1 v2 assume v: "v1∈⋃X" "v2∈⋃X" "ψ v1 = ψ v2"
from v(1,2) have φv: "φ v1 ∈ C0" "φ v2 ∈ C0"
using label_wrt_elt_image[OF φ(1)] by auto
define s1 s2 where "s1 = fundantipermutation (φ v1)" and "s2 = fundantipermutation (φ v2)"
from v(1,2) obtain w1 w2 where "w1∈W" "v1∈w1`→C0" "w2∈W" "v2∈w2`→C0"
using simplex_in_max chamber_eq_W_image by blast
with assms s1_def s2_def have ψv: "ψ v1 = w1 +o ⟨S-{s1}⟩" "ψ v2 = w2 +o ⟨S-{s2}⟩"
using label_wrt_special_coset2' by auto
with v(3) have "w1 +o ⟨S-{s1}⟩ = w2 +o ⟨S-{s2}⟩"
using label_wrt_special_coset2' by auto
with s1_def s2_def have "φ v1 = φ v2"
using PreCoxeterSystemWithDeletion.special_coset_eq_imp_eq_gensets[
OF PreCoxeterSystemWithDeletion, of "S-{s1}" "S-{s2}" w1 w2
]
φv fundantipermutation1[of "φ v1"] fundantipermutation1[of "φ v2"]
bij_betw_f_the_inv_into_f[OF fundantivertex_bij_betw, of "φ v1"]
bij_betw_f_the_inv_into_f[OF fundantivertex_bij_betw, of "φ v2"]
by    fastforce
with v(3) ψ show "v1=v2"
using ψv(1) genby_0_closed[of "S-{s1}"] lcoset_refl[of "⟨S-{s1}⟩" w1]
by    fastforce
qed

lemma label_stab_map_surj_on_vertices:
assumes "label_wrt C0 φ" "fixespointwise φ C0"
defines "ψ ≡ λv. {w∈W. w→(φ v) = v}"
shows   "ψ`(⋃X) = ⋃Σ"
proof (rule seteqI)
fix u assume "u ∈ ψ`(⋃X)"
from this obtain v where v: "v∈⋃X" "u = ψ v" by fast
from v(1) obtain w where "w∈W" "v∈w`→C0"
using simplex_in_max chamber_eq_W_image by blast
with assms v show "u∈⋃Σ"
using label_wrt_special_coset2' label_wrt_elt_image[OF assms(1)]
fundantipermutation1 CoxeterComplex.vertices[OF CoxeterComplex]
by    auto
next
fix u assume "u∈⋃Σ"
from this obtain w s where "w∈W" "s∈S" "u = w +o ⟨S-{s}⟩"
using CoxeterComplex.vertex_conv_special_coset[OF CoxeterComplex] by blast
with assms show "u ∈ ψ`(⋃X)"
using label_wrt_special_coset1 fundantivertex fundchamber chamberD_simplex
W_endomorphism ChamberComplexEndomorphism.vertex_map
by    fast
qed

lemma label_stab_map_bij_betw_vertices:
assumes   "label_wrt C0 φ" "fixespointwise φ C0"
defines   "ψ ≡ λv. {w∈W. w→(φ v) = v}"
shows     "bij_betw ψ (⋃X) (⋃Σ)"
unfolding bij_betw_def
using     assms label_stab_map_inj_on_vertices label_stab_map_surj_on_vertices
by        auto

lemma label_stab_map_bij_betw_W_chambers:
assumes   "label_wrt C0 φ" "fixespointwise φ C0" "w0∈W"
defines   "ψ ≡ λv. {w∈W. w→(φ v) = v}"
shows     "bij_betw ψ (w0`→C0) (CoxeterComplex.smap S {w0})"
unfolding bij_betw_def
proof (rule conjI, rule inj_on_inverseI)
define f1 f2
where "f1 = the_inv_into (CoxeterComplex.smap S 0) ((+o) w0)"
and "f2 = the_inv_into S (λs. ⟨S-{s}⟩)"
define g where "g = ((→) w0) ∘ fundantivertex ∘ f2 ∘ f1"

from assms(3) have inj_opw0: "inj_on ((+o) w0) (CoxeterComplex.smap S 0)"
using bij_betw_imp_inj_on[OF CoxeterComplex.W_lcoset_bij_betw_singletons]
CoxeterComplex
by    fast
have inj_genby_minus_s: "inj_on (λs. ⟨S-{s}⟩) S"
using bij_betw_imp_inj_on[OF CoxeterComplex.S_bij_betw_chamber0]
CoxeterComplex
by    fast

fix v assume v: "v∈w0`→C0"
from this obtain v0 where v0: "v0∈C0" "v = w0→v0" by fast
from v0(1) have fap_v0: "fundantipermutation v0 ∈ S"
using fundantipermutation1 by auto
with assms(3)
have  v0': "⟨S-{fundantipermutation v0}⟩ ∈ CoxeterComplex.smap S 0"
using genby_0_closed[of S]
CoxeterComplex.chamber_vertices[OF CoxeterComplex, of 0]
by    simp

from v0 assms have "ψ v = w0 +o ⟨S-{fundantipermutation v0}⟩"
using label_wrt_special_coset1' by simp
with f1_def assms(3) f2_def v0 g_def show "g (ψ v) = v"
using v0' fap_v0 the_inv_into_f_f[OF inj_opw0]
the_inv_into_f_f[OF inj_genby_minus_s]
bij_betw_f_the_inv_into_f[OF fundantivertex_bij_betw]
by    simp
next
from assms show "ψ`(w0`→C0) = CoxeterComplex.smap S {w0}"
using label_stab_map_W_fundchamber_image by simp
qed

lemma label_stab_map_surj_on_simplices:
assumes φ: "label_wrt C0 φ" "fixespointwise φ C0"
defines ψ: "ψ ≡ λv. {w∈W. w→(φ v) = v}"
shows   "ψ ⊢ X = Σ"
proof (rule seteqI)
fix y assume "y ∈ ψ ⊢ X"
from this obtain x where x: "x∈X" "y = ψ ` x" by fast
from x(1) obtain C where "chamber C" "x⊆C" using simplex_in_max by fast
with assms x(2) show "y ∈ Σ"
using label_stab_map_chamber_map
CoxeterComplex.chamberD_simplex[OF CoxeterComplex]
CoxeterComplex.faces[OF CoxeterComplex, of "ψ`C" y]
by    auto
next
fix y assume "y ∈ Σ"
from this obtain z where z: "CoxeterComplex.chamber S z" "y⊆z"
using ChamberComplex.simplex_in_max[
OF CoxeterComplex.ChamberComplex_Σ,
OF CoxeterComplex
]
by    fast
from z(1) obtain w where w: "w∈W" "z = CoxeterComplex.smap S {w}"
using CoxeterComplex.chamber_is_singleton[OF CoxeterComplex] by fast
with assms have "bij_betw ψ (w`→C0) z"
using label_stab_map_bij_betw_W_chambers by fast
hence 1: "bij_betw ((`) ψ) (Pow (w`→C0)) (Pow z)"
using bij_betw_imp_bij_betw_Pow by fast
define x where x: "x ≡ the_inv_into (Pow (w`→C0)) ((`) ψ) y"
with z(2) have "x ⊆ w`→C0" using bij_betw_the_inv_into_onto[OF 1] by auto
with w(1) have "x∈X"
using faces fundchamber_W_image_chamber chamberD_simplex
by    fastforce
moreover from x z(2) have "y = ψ ` x"
using bij_betw_f_the_inv_into_f[OF 1] by simp
ultimately show "y ∈ ψ ⊢ X" by fast
qed

lemma label_stab_map_iso_to_coxeter_complex:
assumes "label_wrt C0 φ" "fixespointwise φ C0"
defines "ψ ≡ λv. {w∈W. w→(φ v) = v}"
shows   "ChamberComplexIsomorphism X Σ ψ"
proof (
rule ChamberComplexIsomorphism.intro,
rule ChamberComplexMorphism.intro
)
show "ChamberComplex X" ..
show "ChamberComplex Σ"
using CoxeterComplex CoxeterComplex.ChamberComplex_Σ by fast
from assms show "ChamberComplexMorphism_axioms X Σ ψ"
using label_stab_map_chamber_map
CoxeterComplex.card_chamber[OF CoxeterComplex]
card_S_chamber
by    unfold_locales auto
from assms show "ChamberComplexIsomorphism_axioms X Σ ψ"
using label_stab_map_bij_betw_vertices label_stab_map_surj_on_simplices
by    unfold_locales auto
qed

lemma ex_iso_to_coxeter_complex':
"∃ψ. ChamberComplexIsomorphism X (CoxeterComplex.TheComplex S) ψ"
using CoxeterComplex ex_label_retraction label_stab_map_iso_to_coxeter_complex
by    force

lemma ex_iso_to_coxeter_complex:
"∃S::'a permutation set. CoxeterComplex S ∧
(∃ψ. ChamberComplexIsomorphism X (CoxeterComplex.TheComplex S) ψ)"
using CoxeterComplex ex_iso_to_coxeter_complex' by fast

end (* context ThinChamberComplexManyFoldings *)

end (* theory *)
```