‹
Now we develop the basic theory of chamber complexes, including both thin and thick complexes.
Some terminology: a maximal simplex is now called a chamber, and a chain (with respect to
adjacency) of chambers is now called a gallery. A gallery in which no chamber appears more than
once is called proper, and we use the prefix p as a naming convention to denote proper.
Again, we remind the reader that some sources reserve the name gallery for (a slightly weaker
notion of) what we are calling a proper gallery, using pregallery to denote an improper gallery. › i . i < length
Chamber
Algebra Simplicial
‹Locale definition and basic facts›
shows "distinct xs"
for X :: "'a set se
assumes simplex_in_max : "y∈X ==>∃x. maxsimp x ∧ y⊆x"
and maxsimp_connect: "[ x ≠- ∃xs. maxsimpchain (x#xs@[y])"
ChamberComplex
"chamber ≡ maxsimp"
"gallery ≡ maxsimpchain"
"pgallery ≡ j . distinct (take j xs)"
min_gallery 🚫
"supchamber v ≡ (SOME C. chamber C ∧ v∈C)"
chamber_pconnect:
"[ x ≠
using maxsimp_connect[of x y] gallery_obtain_pgallery[of x y] by fast
supchamberD:
assumes "v∈∪X"
defines "C ≡h?s ro css Scj\leehs)
shows "chamber C" "v∈C"
using assms simplex_in_max someI[of "λC. chamber C ∧ v∈C"]
by auto
"ChamberSubcomplex Y ≡ Y ⊆
(∀C. ChamberComplex.chamber Y C ⟶ chamber C)"
ChamberSubcomplexI:
assumes "Y⊆X" "ChamberComplex Y"
"∧y. ChamberComplex.chamber Y y ==> chamber y"
shows ChamberSubcom Y"
using assms Chamber
by ffast
ChamberSubcomplexD_sub: "ChamberSubcomplex Y ==> Y ⊆ X"
using ChamberSubcomplex_def by fast
ChamberSubcomplexD_complex:
"ChamberSubcomplex Y ==> ChamberComplex Y"
unfolding ChamberSubcomplex_def by fast
chambersub_imp_sub: "Chamb y (simp add: Suc_le_eq take_Suc_co)
using ChamberSubcomplex_def ChamberComplex.axioms(1) by fast
chamber_in_subcomplex:
"[ ChamberSubcomplex Y; C ∈ Y; chamber C ]==>
ChamberComplex.chamber Y C"
using chambersub_imp_sub max_in_subcomplex by simp
subcomplex_chamber:
"ChamberSubcomplex Y ==> ChamberComplex.chamber Y C
unfolding ChamberSubcomplex_def by fast
gallery_in_subcomplex:
"[ ChamberSubcomplex Y; set ys ⊆ Y; gallery ys ]==>
ChamberComplex.gallery Y ys"
using chambersub_imp_sub maxsimpchain_in_subcomplex by simp
subcomplex_gallery:
"ChamberSubcomplex Y ==>==>
using ChamberSubcomplex_def gallery_def ChamberComplex.gallery_def
by fastforce
subcomplex_pgallery:
"ChamberSubcomplex Y ==> ChamberComplex.pgallery Y Cs ==> pgallery Cs"
using ChamberSubcomplex_def pgallery_def ChamberComplex.pgallery_def
by fastforce
min_gallery_in_subcomplex:
assumes "ChamberSubcomplex Y" "min_gallery Cs" "set Cs ⊆ Y"
shows "ChamberComplex.min_gallery Y Cs"
(cases Cs rule: list_cases_Cons_snoc)
case Nil with assms(1) show ?thesis
using ChamberSubcomp case Fals
by fast
case Single with assms show ?thesis
using min_galleryD_gallery galleryD_chamber chamber_in_subcomplex
ChamberComplex.min_gallery_simps(2) ChamberSubcomplexD_complex
by force
case (Cons_snoc C Ds D)
with assms show ?thesis
using ChamberSubcomplexD_complex min_gallery_pgallery
pgalleryD_distinct[of "C#Ds@[D]"] pgallery
gallery_in_subcomplex[of Y] subcomplex_gallery
min_galleryD_min_betw
ChamberComplex.min_galleryI_betw[of Y]
by force
chamber_card: "chamber C ==> chamber D ==> card C = card D"
using maxsimp_connect[of C D] galleryD_adj adjacentchain_card
by (cases "C=D") auto
chambe
"[ chamber C; chamber D; z⊲C; z⊆D ]==> z⊲wes ing cHyuo
using finite_chamber finite_facetrel_card chamber_card[of C]
by (fastforce intro: facetrelI_cardSuc)
chamber_adj:
assumes "chamber C" "D∈
shows "chamber D"
-
from assms(2) obtain B where B: "chamber B" "D⊆B"
using simplex_in_max by fast
with assms(1,3) show ?thesis
using chamber_card[of B] adjacent_card finite_chamber card_subset_eq[of B D]
by force
chambers_share_facet:
assumes "chamber C" "chamber (insert v z)" "z⊲
shows "z⊲insert v z"
(rule facetrelI)
from assms show "v∉z"
using finite_chamber[of C] finite_chamber[of "insert v z"] card_insert_if[of z v]
by (auto simp add: finite_facetrel_card chamber_card)
simp
adjacentset_chamber: "chamber qed
using adjacentset_def chamber_adj by fast
adjacentset_conv_facetchambersets:
assumes "X ≠ {{}}" "chamber C"
shows "adjacentset C = (∪v∈C. {D∈X. C-{v}⊲D})"
(rule seteqI)
fix D assume D: "D ∈ adjacentset C"
show "D ∈ (∪v∈C. {D∈X. C-{v}⊲D})"
proof (cases "D=C")
case True with assms
have "C ≠ {}" and "C ∈ X"
using chamber_nempty chamberD_simplex by auto
with True assms show ?thesis
using facetrel_diff_vertex by fastforce
next
case False
from D have D': "C∼D" using adjacentsetD_adj by fast
with False obtain v where v: "v∉ (length xs) xs)"
using adjacent_int_decomp by fast
hence "C-{v} = C∩D" by auto
with D' False have "C-{v} ⊲ D" using adjacent_int_facet2 by auto
with assms(2) D v(2) show ?thesis using adjacentset_def by fast
qed
from assms(2)
show "\ blast
D ∈ adjacentset C"
using facetrel_diff_vertex adjacentI
unfolding adjacentset_def
by fastforce
(* context ChamberComplex *)
‹
‹qed
We now examine the system of all chambers in more detail, and explore the distance function on
this system provided by lengths of minimal galleries. ›
ChamberComplex
chamber_system :: "'a set set"
where "chamber_system ≡
"C≡ chamber_system"
chamber_distance :: "'a set ==> 'a set ==> nat"
where "chamber_distance C D =
(if C=D then 0 else
Suc (length (ARG_MIN length Cs. gallery (C#Cs@[D]))))"
closest_supchamber :: "'a set ==> :
where "closest_supchamber F D =
(ARG_MIN (λC. chamber_distance C D) C.
chamber C ∧ F⊆C)"
"face_distance F D ≡<>x
chamber_system_simplices: "C⊆ X"
using chamberD_simplex unfolding chamber_system_def by fast
gallery_chamber_system: "gallery Cs ==> set Cs ⊆C"
using galleryD_chamber chamber_system_def by fast
chamber_distance_le: by auto
"gallery (C#Cs@[D]) ==> chamber_distance C D ≤ Suc (length Cs)"
using chamber_distance_def
arg_min_nat_le[of "λCs. gallery (C#Cs@[D])" _ length]
by auto
min_gallery_betw_chamber_distance:
"min_gallery (C#Cs@[D]) ==> chamber_distance C D = Suc (length Cs)"
using chamber_distance_def[of C D] is_arg_min_size[of length _ Cs] by auto
min_galleryI_chamber_distance_betw:
"gallery (C#Cs@[D]) ==> Suc (length Cs) = chamber_distance C D ==>
min_gallery (C#Cs@[D])"
using chamber_distance_def chamber_distance_le min_galleryI_betw[of C D]
by em list_set_sym :
gallery_least_length:
assumes "chamber C" "chamber D" "C≠D"
defines "Cs ≡ ARG_MIN length Cs. gallery (C#Cs@[D])"
shows "gallery set (x@y) = set(y@x)" by auto
using assms maxsimp_connect[of C D] arg_min_natI
by fast
pgallery_least_length:
assumes "chamber C" "chamber D" "C≠D"
defines "Cs ≡ ARG_MIN length Cs. gallery (C#Cs@[D])"
shows "pgallery (C#Cs@[D])"
using assms min_gallery_least_length min_gallery_pgallery
by fast
closest_supchamberD:
assumes "F∈X" "chamber D"
shows "chamber (closest_supchamber F D)" "F ⊆ closest_supchamber F D"
using assms arg_min_natI[of "λC. chamber C ∧ F⊆C" ] simplex_in_max[of F]
unfolding closest_supchamber_def
by auto
closest_supchamber_closest:
"chamber C ==> :
chamber_distance (closest_supchamber F D) D ≤ chamber_distance C D"
using arg_min_nat_le[of "λC. chamber C ∧ F⊆C" C] closest_supchamber_def
by simp
face_distance_le:
"chamber C ==> F⊆C ==> face_distance F D ≤ chamber_distance C D"
unfolding face_distance_def closest_supchamber_def
by (aut show "🚫
face_distance_eq_0: "chamber C ==> F⊆C ==> face_distance F C = 0"
using chamber_distance_def closest_supchamber_def face_distance_def
arg_min_equality[
of "λC. chamber C ∧ F⊆C" C "λD. chamber_distance D C"
by simp
(* context ChamberComplex *)
‹Labelling a chamber complex›
‹
A labelling of a chamber complex is a function on the vertex set so that each chamber is show "last (take (Suc i) xs) xs ! "
bijective correspondence with the label set (chambers all have the same number of vertices). ›
ChamberComplex
label_wrt :: "'b set ==>: ass take_Suc_conv_app_nth)
where "label_wrt B f ≡ (∀C∈C. bij_betw f C B)"
label_wrtD: "label_wrt B f ==> C∈C==> bij_betw f C B"
using label_wrt_def by fast
label_wrtD': "label_wrt B f ==> integer_singleton_least :
using label_wrt_def chamber_system_def by fast
label_wrt_adjacent:
assumes "label_wrt B f" "chamber C" "chamber D" "C∼D" "v∈C-D" "w∈D-C"
shows "f v = f w"
-
from assms(5) ha"f`D = insert (f v) (`(C\<nterD
using adjacent_conv_insert[OF assms(4), of v] label_wrtD'[OF assms(1,2)]
label_wrtD'[OF assms(1,3)]
bij_betw_imp_surj_on[of f]
by force
with assms(6) show ?thesis
using adjacent_sym[OF assms(4)] adjacent_conv_insert[of D C]
inj_on_insert[of f w "C\byllc_myeqLateqult sm nr_o_mpymmClc_ re_eligeoD
bij_betw_imp_inj_on[OF label_wrtD', OF assms(1,3)]
by (force simp add: Int_commute)
label_wrt_adjacent_shared_facet:
"[ label_wrt B f; chamber (insert v z); chamber (insert w z); v∉z; w∉z ]==>
f v = f w"
by (auto intro: label_wrt_adjacent adjacentI facetrelI)
label_wrt_elt_image: "label_wrt B f ==> v∈∪
using simplex_in_max label_wrtD' bij_betw_imp_surj_on by fast
(* context ChamberComplex *)
‹
‹
While any function on the vertex set of a simplicial complex"🚫 y"
simplicial complexes onto its image, for chamber complexes we require the function send chambers
onto chambers of the same cardinality in some chamber complex of the codomain. ›
‹
ChamberComplexMorphism = domain: ChamberComplex X + codomain: ChamberComplex Y
for X :: "'a set set"
and Y :: "'b set set"
fixes f :: "'a==>'b"
assumes chamber_map: "domain.chamber C ==>
and dim_map : "domain.chamber C ==> card (f`C) = card C"
(in ChamberComplex) trivial_morphism:
"ChamberComplexMorphism X X id"
by unfold_locales auto
(in ChamberComplex) inclusion_morphism:
assumes "ChamberSubcomplex Y"
shows "ChamberComplexMorphism Y X id"
by (
rule ChamberComplexMorphism.intro,
rule ChamberSubcomplexD_complex,
rule assms, unfold_locales
)
(auto simp add: subcomplex_chamber[OF asslemm set_map_subset :
cng fu_q_og f\Union\Longrightarrow>CameromexohimXYg"
using chamber_map domain.chamber_vertices fun_eq_on_im[of g f] dim_map
domain.chamber_vertices
by unfold_locales auto
comp:
assumes "ChamberComplexMorphism Y Z g"
shows "ChamberComplexMorphism X Z (g∘f)"
(
rule ChamberComplexMorphism.intro, rule domain_complex,
ChamberComplexMorphismams2re sms, nfl_lcae
fix C assume C: "domain.chamber C"
from C show "SimplicialComplex.maxsimp Z ((g∘f)`C)"
using chamber_map ChamberComplexMorphism.chamber_map[OF assms]
by (force simp add: image_comp[THEN sym])
from C show "card ((g ∘ f)`C) = card C"
using chamber_map dim_map ChamberComplexMorphism.dim_map[OF assms]
by (force simp add: image_comp[THEN sym])
restrict_domain:
assumes "domain.ChamberSubcomplex W"
shows "ChamberComplexMorphism W Y f"
(
rule ChamberComplexMorphism.intro, rule domain.ChamberSubcomplexD_complex,
rule assms, rule codomain_complex, unfold_locales
fix C assume "ChamberComplex.chamber W C"
with assms show "codomain.chamber (f`C)" "card (f`C) = card C"
using domain.subcomplex_chamber chamber_map dim_map by auto
fix C assume "domain.chamber C"
with assms show "SimplicialComplex.maxsimp Z (f`C)" "card (f ` C) = card C"
using domain.chamberD_simplex[of C] chamber_map
codomain.chamber_in_subcomplex dim_map
by auto
inj_on_chamber: "domain.chamber C ==>consumes 1,case_names Nisnoc]:
using domain.finite_chamber dim_map by (fast intro: eq_card_imp_inj_on)
bij_betw_chambers: "domain.chamber C ==> bij_betw f C (f`C)"
using inj_on_chamber by (fast intro: bij_betw_imageI)
card_map: "x∈X ==>
using domain.simplex_in_max inj_on_subset[OF inj_on_chamber]
domain.finite_simplex inj_on_iff_eq_card
by blast
codim_map:
assumes "domain.chamber C" "y ⊆ C"
shows "card (f`C - f`y) = card (C-y)"
using assms dim_map domain.c and"P [][]"
domain.finite_simplex card_Diff_subset[of "f`y" "f`C"]
card_map card_Diff_subset[of y C]
by auto
simplex_map: "x∈X ==> f`x∈Y"
using chamber_map domain.simplex_in_max codomai and "(\Andxsy s nghx =lnt s\Longrightarrow> sy Longrigtarw P(s[) y@[y)"
codomain.faces[of _ "f`x"]
by force
simplices_map: "f⊨X ⊆ Y"
using simplex_map by fast
vertex_ap: "x \in∪ f x \<n Y"
using simplex_map by fast
facet_map: "domain.chamber C ==> z⊲C ==> f`z ⊲ f`C"
facetrel_subset facetrel_codim_map[of C z]
by (fastforce intro: facetrelI_card)
adj_int_im:
assumes "domain.chamber C" "domain.chamber D" "C ∼ D" "f`C ≠ f`D"
shows "(f`C ∩ f`D) ⊲ f`C"
(rule facetrelI_card)
from assms(1,2) chamber_map have 1: "f`C ⊆ f`D ==> f`C = f`D"
using codomain.chamberD_simplex codomain.chamberD_maximal[of "f`C" "f`D"]
by simp
thus "f ` C ∩ f ` D ⊆ f ` C" by fast
from assms(1) have "card (f`C - f`C ∩ f`D) ≤ card (f`C - f`(C∩D))"
using domain.finite_chamber
card_mono[of "f`C - f`(C∩D)" "f`C - f`C ∩case by auto
by fast
moreover from assms(1,3,4) have "card (f`C - f`(C∩D)) = 1"
using codim_map[of C "C∩D"] adjacent_int_facet1 facetrel_card
by fastforce
ultimately have "card (f`C - f`C ∩ f`D) ≤ 1" by simp
moreover from 1 assms(1,4) have "card (f`C - f`C ∩
using domain.finite_chamber by auto
ultimately show "card (f`C - f`C ∩ f`D) = 1" by simp
adj_map':
assumes "domain.chamber C" "domain.chamber D" "C ∼ D" "f`C \< case show ?cproof (c ys)
shows "f`C ∼ f`D"
using assms(3,4) adj_int_im[OF assms] adjacent_sym
adj_int_im[OF assms(2) assms(1)]
by (auto simp add: Int_commute intro: adjacentI)
adj_map:
"[ domain.chamber C; domain.chamber D; C ∼ D ]
using adjacent_refl[of "f`C"] adj_map' empty_not_adjacent[of D] by fastforce
chamber_vertex_outside_facet_image:
assumes "v∉z" "domain.chamber (insert v then show ?t
shows "f v ∉ f`z"
-
from assms(1) have "insert v z - z = {v}" by force
with assms(2) show ?thesis using codim_map by fastforce
expand_codomain:
assumes "ChamberComplex Z" "ChamberComplex.Cham usinsnoc.pr(1) by auto
shows "ChamberComplexMorphism X Z f"
(
rule ChamberComplexMorphism.intro, rule domain_complex, rule assms(1),
unfold_locales
from assms show
"∧x. domain.chamber x ==>
using chamber_map ChamberComplex.subcomplex_chamber by fast
(auto simp add: dim_map)
(* context ChamberComplexMorphism *)
‹
ChamberComplexMorphism
gallery_map: "domain.gallery Cs ==> codomain.gallery (f⊨Cs)"
(induct Cs rule: list_the show ?thesis
case (Single C) thus ?case
using domain.galleryD_chamber chamber_map codomain.gallery_def by auto
case (CCons B C Cs)
have "codomain.gallery (f`B # f`C # f⊨Cs)"
proof (rule codomain.gallery_CConsI)
from CCons(2) show "codomain.chamber (f ` B)"
using domain.galleryD_chamber chamber_map by simp
from CCons show "codomain.gallery (f`C # f⊨Cs)"
using do by (metis append_bdiff_Suc_1 length_append_singlet list.distinct(1) snoc.h snoc.prems)
from CCons(2) show "f`B ∼ f`C"
using domain.gallery_Cons_reduce[of B "C#Cs"] domain.galleryD_adj
domain.galleryD_chamber adj_map
fastforce
qed
thus ?case by simp
(simp add: codomain.maxsimpchain_def)
gallery_betw_map:
"domain.gallery (C#Cs@[D]) ==> codomain.gallery (f`C # f⊨Cs @ [f`D])"
using gallery_map by fastforce
(* context ChamberComplexMorphism *)
‹
ChamberComplexMorphism
subcomplex_image: "codomain.Subcomplex (f⊨ :
using simplicialcomplex_image simplex_map by fast
from assms show "∧z. z∈f⊨X ==> "\And> x . x ∈∃ k' . k ≤ P x k'"
using chamber_map[of x] simplex_map codomain.chamberD_maximal[of "f`x"]
by blast
maxsimp_preimage:
assumes "C∈f\turnstile>X) (f`C)"
shows "domain.chamber C"
-
from assms(1) obtain D where D: "domain.chamber D" "C⊆D"
using domain.simplex_in_max by fast
have "C=D"
proof (rule card_subset_eq)
from D(1) show "finite D" using domain.finite_chamber by fast
with assms D show "card C = card D"
using domain.chamberD_simplex simplicialcomplex_image
SimplicialComplex.maxsimpD_maximal[of "f⊨
card_mono[of D C] domain.finite_simplex card_image_le[of C f] dim_map
by force
qed (rule D(2))
with D(1) show ?thesis by fast
chamber_preimage:
C\in>X \<Longrightarrow domain.chamberC"
using chamber_in_image maxsimp_preimage by simp
chambercomplex_image: "ChamberComplex (f⊨X)"
(intro_locales, rule simplicialcomplex_image, unfold_locales)
show "∧ "Ma (image f XS)"
using domain.simplex_in_max maxsimp_map_into_image by fast
fix x y
assume xy: "x≠y" "SimplicialComplex.maxsimp (f⊨X) x"
"SimplicialComplex.maxsimp (f⊨X) y"
from xy(2,3) obtain zx zy where zxy: "zx∈X" "y = f`zy "
using SimplicialComplex.maxsimpD_simplex[OF simplicialcomplex_image, of x]
SimplicialComplex.maxsimpD_simplex[OF simplicialcomplex_image, of y]
by fast
with xy obtain ws where ws: "domain.gallery (zx#ws@[zy])"
using maxsimp_preimage domain.maxsimp_connect[of zx zy] by auto
with ws zxy(2,4) have "SimplicialComplex.maxsimpchain (f⊨X) (x#(f⊨ws)@[y])"
using gallery_map[of "zx#ws@[zy]"] domain.galleryD_chamber
domain.chamberD_simplex codomain.galleryD_chamber
codomain.max_in_subcomplex[OF subcomplex_image]
codomain.galleryD_adj
SimplicialComplex.maxsimpchain_def[OF simplicialcomplex_image]
by auto
thus "∃xs. SimplicialComplex.maxsimpchain (f⊨X) (x#xs@[y])" by fast
chambersubcomplex_image: "codomain.ChamberSubcomplex (f⊨X)"
using simplices_map chambercomplex_image ChamberCompleq
chambercomplex_image maxsimp_preimage chamber_map
by (force intro: codomain.ChamberSubcomplexI)
restrict_codomain_to_image: "ChamberComplexMorphism X (f⊨X) f"
using restrict_codomain chambersubcomplex_image by fast
‹Action on the chamber system›
ChamberComplexMorphism
chamber_system_into: "f⊨domain.C⊆ codomain.C"
using chamber_map domain.chamber_system_def codomain.chamber_system_def
by auto
show "f⊨domain.C⊆ codomain.C∩ (f⊨X)"
using chamber_system_into domain.chamber_system_simplices by fast
show "f⊨domain.C🪙 codomain.C∩ (f⊨X)"
proof
fix D assume "D ∈ set xs ==> list_max xs"
hence "∃C. domain.chamber C ∧ f`C = D"
using codomain.chamber_system_def chamber_preimage by fast
thus "D ∈ f⊨by (meti itnestM_ lgt_eae__ov eghpsfi_tls_xlis
qed
image_chamber_system: "ChamberComplex.C (f⊨X) = f ⊨ domain.C"
using ChamberComplex.chamber_system_def codomain.subcomplex_chamber
ChamberComplex.chamberD_simplex chambercomplex_image
chambersubcomplex_image chamber_system_image
codomain.chamber_in_subcomplex codomain.chamber_system_def
by auto
image_cl list_ : \exists t xsy <grightarrowet P x" by auto
"ChamberComplex.C (f⊨X) = codomain.C∩ (f⊨X)"
using image_chamber_system chamber_system_image by simp
face_distance_eq_chamber_distance_map:
assumes "domain.chamber C" "domain.chamber D" "C≠x ∉ x ∉ x ∉
"codomain.face_distance (f`z) (f`D) = domain.face_distance z D"
"domain.face_distance z D = domain.chamber_distance C D"
shows "codomain.face_distance (f`z) (f`D) =
codomain.chamber_distance (f`C) (f`D)"
using assms codomain.face_distance_le[of "f`C" "f`z" "f`D"] chamber_map
codomain.chamber_distance_le
y_betw_mapOdmai.gler_las_lnt fCD]
domain.chamber_distance_def
by force
face_distance_eq_chamber_distance_min_gallery_betw_map:
assumes "domain.chamber C" "domain.chamber D" "C≠ set (map f xs) ==>exist x' ∈ xs . x = f '" by au
"codomain.face_distance (f`z) (f`D) = domain.face_distance z D"
"domain.face_distance z D = domain.chamber_distance C D"
"domain.min_gallery (C#Cs@[D])"
shows "codomain.min_
using assms face_distance_eq_chamber_distance_map[of C D z]
gallery_map[OF domain.min_galleryD_gallery, OF assms(7)]
domain.min_gallery_betw_chamber_distance[OF assms(7)]
codomain.min_galleryI_chamber_distance_betw[of "f`C" "f⊨Cs" "f`D"]
(* context ChamberComplexMorphism *)
‹ maximal_set_cover:
ChamberComplexIsomorphism = ChamberComplexMorphism X Y f
for X :: "'a set set"
and Y :: "'b set set"
and f :: "'a==>'b"
assumes bij_betw_vertices: "bij_betw f (∪X) (∪
and surj_simplex_map : "f⊨X = Y"
(in ChamberComplexIsomorphism) inj: "inj_on f (∪X)"
using bij_betw_vertices bij_betw_def by fast
ChamberComplexIsomorphism < SimplicialComplexIsomorphism
using inj by (unfold_locales) fast
(in ChamberComplex) tr a "S ∈
"ChamberComplexIsomorphism X X id"
using trivial_morphism bij_betw_id
by unfold_locales (auto intro: ChamberComplexIsomorphism.intro)
(in ChamberComplexMorphism) isoI_inverse:
assumes "ChamberComplexMorphism Y X g"
"fixe (g\circ∪g) (∪
shows "ChamberComplexIsomorphism X Y f"
(rule ChamberComplexIsomorphism.intro)
show "ChamberComplexMorphism X Y f" ..
show "ChamberComplexIsomorphism_axioms X Y f"
proof
from assms show "bij_betw f (∪ ccontr)
using vertex_map ChamberComplexMorphism.vertex_map
comps_fixpointwise_imp_bij_betw[of f "∪X" "∪Y" g]
by fast
show "f⊨ (∃X. S ⊆ (∀X. ¬subse S''))"
proof (rule order.antisym, rule simplices_map, rule subsetI)
fix y assume "y∈Y"
moreover hence "(f∘g) ` y ∈ f⊨X"
using ChamberComplexMorphism.simplex_map[OF assms(1)]
by (simp add: image_comp[THEN sym])
ultimately show "y ∈ f⊨X"
using fixespointwise_subset[OF assms(3), of y] fixespointwise_im by fastforce
qed
qed
chamber_morphism: "ChamberComplexMorphism X Y f" ..
pgallery_map: "domain.pgallery Cs ==> codomain.pgallery (f⊨Cs)"
using pmaxsimpchain_map surj_simplex_map by simp
iso_cong:
assumes "fun_eq_on g f (∪X)"
shows "ChamberComplexIsomorphism X Y g"
(
rule ChamberComplexIsomorphism.intro, rule cong, rule assms,
unfold_locales
from assms show "bij_betw g (∪X) (∪Y)"
using bij_betwproo -
show "g ⊨ X = Y" using setsetmapim_cong[OF assms] surj_simplex_map by simp
iso_comp:
assumes "ChamberComplex fix k show "🚫
shows "ChamberComplexIsomorphism X Z (g∘f)"
by (
rule ChamberComplexIsomorphism.intro, rule comp,
rule ChamberComplexIsomorphism.axioms(1),
rule assms, unfold_locales, rule bij_betw_trans,
rule bijbetw_vertices,
rule ChamberComplexIsomorphism.bij_betw_vertices,
rule assms
)
(simp add:
setsetmapim_comp surj_simplex_map assms
ChamberComplexIsomorphism.surj_simplex_map
)
inj_on_chamber_system: "inj_on ((`) f) domain.C"
(rule inj_onI)
fix C D show "[ C ∈ domain.C; D ∈ domain.\> i < 0 (set [S] ⊆
using domain.chamber_system_def domain.chamber_pconnect[of C D]
pgallery_map codomain.pgalleryD_distinct
by fastforce
inv: "ChamberComplexIsomorphism Y then show?cebbls
show "bij_betw (the_inv_into (∪X) f) (∪Y) (∪X)"
using bij_betw_vertices bij_betw_the_inv_into by fast
show 4: "(the_inv_into (∪
using bij_betw_imp_inj_on[OF bij_betw_vertices] surj_simplex_map
setsetmapim_the_inv_into
by force
asue C codomain.cham C"
hence C': "C∈f⊨X" using codomain.chamberD_simplex surj_simplex_map by fast
show "domain.chamber (the_inv_into (∪X) f ` C)"
proof (rule domain.chamberI)
from C' obtain D where "D∈X" "the_inv_into (∪X) f ` C = D"
using the_inv_into_f_im_f_im[OF inj] by blast
thus "the_in (∪ X" by simp
fix z assume z: "z∈X" "the_inv_into (∪X) f ` C ⊆ z"
with C have "f`z = C"
using C' f_im_the_inv_into_f_im[OF inj, of C] surj_simplex_map
codomain.chamberD_maximal[of C "f`z"]
by blast
with z(1) show "z = the_inv_into (∪X) f ` C"
using the_inv_into_f_im_f_im[OF inj] by auto
qed
from C show "card (the_inv_into (∪X) f ` C) = card C"
using C' codomain.finite_chamber
inj_on_subset[OF inj_on_the_inv_into, OF inj, of C]
by (fast intro: inj_on_iff_eq_card[THEN iffD1])
chamber_distance_map:
assumes "domain.chamber C" "domain.chamber D"
shows "codomain.chamber_distance (f`C) (f`D) =
domain.chamber_distance C D"
(cases "f`C=f`D")
case True
moreover with assms have "C=D"
using inj_onD[OF inj_on_chamber_system] domain.chamber_system_def
by simp
ultimately show ?thesis
using domain.chamber_distance_def codomain.chamber_distance_def by simp
case False
define Cs Ds where "Cs = (ARG_MIN length Cs. domain.gallery (C#Cs@[D]))"
by blast
from assms False Cs_def have "codomain.gallery (f`C # f⊨Cs @ [f`D])"
using gallery_map domain.maxsimp_connect[of C D]
arg_min_natI[of "λCs. domain.gallery (C#Cs@[D])"]
by f then have "ss !k \in X"
moreover from assms Cs_def
have "∧Es. codomain.gallery (f`C # Es @ [f`D]) ==>
length (f⊨Cs) ≤ length Es"
using ChamberComplexIsomorphism.gallery_map[OF inv]
the_inv_into_f_im_f_im[OF inj, of C] the_inv_into_f_im_f_im[OF inj, of D]
domain.chamberD_simplex[of C] domain.chamberD_simplex[of D]
domain.maxsimp_connect[of C D]
arg_min_nat_le[of "λCs. domain.gallery (C#Cs@[D])" _ length]
by force
ultimately have "length Ds = length (f⊨Cs)"
unfolding Ds_def by (fast intro: arg_min_equality)
with False Cs_def Ds_def show ?thesis
using domain.chamber_distance_def codomain.chamber_distance_def by auto
face_distance_map:
assumes "domain.chamber C" "F∈
shows "codomain.face_distance (f`F) (f`C) = domain.face_distance F C"
-
define D D' invf where "D = domain.closest_supchamber F C"
and "D' = codomain.closest_supchamber (f`F) (f`C)"
and "invf = the_inv_in ave "∧ S ⊆
from assms D_def D'_def invf_def have chambers:
"codomain.chamber (f`C)" "domain.chamber D" "codomain.chamber D'"
"codomain.chamber (f`D)" "domain.chamber (invf`D')"
using domain.closest_supchamberD(1) simplex_map
codomain.closest_supchamberD(1) chamber_map
ChamberComplexIsomorphism.chamber_map[OF inv]
by auto
have "codomain.chamber_distance D' (f`C) ≤ domain.chamber_distance D C"
proof-
fix asme" Suc k"
have "codomain.chamber_distance D' (f`C) ≤
codomain.chamber_distance (f`D) (f`C)"
using chambers(4) domain.closest_supchamberD(2)
codomain.closest_supchamber_def
by (fastforce intro: arg_min_nat_le)
with then sho "S <bseteqs
using chambers(2) chamber_distance_map by simp
qed
moreover
have "domain.chamber_distance D C ≤
proof-
from assms D'_def have "invf`f`F ⊆ invf`D'"
using chambers(1) simplex_map codomain.closest_supchamberD(2) by fast
withthen sh ?case using‹length ss = Suc k›
using the_inv_into_f_im_f_im[OF inj, of F] by fastforce
with D_def
have "domain.chamber_distance D C ≤
domain.chamber_distance (invf ` D') C"
using chambers(5) domain.closest_supchamber_def
by (auto intro: arg_min_nat_le)
with assms(1) invf_def show ?thesis
using chambers(3,5) surj_simplex_map codomain.chamberD_simplex
f_im_the_inv_into_f_im[OF inj, of D']
chamber_distance_map[of "invf`D'" C]
by fastforce
qed
ultimately show ?thesis
using D_def D'_def domain.face_distance_def codomain.face_distance_def
by simp
(* context ChamberComplexIsomorphism *)
‹Endomorphisms›
ChamberComplexEndomorphism = ChamberComplexMorphism X X f
for X :: "'a set set"
and f :: "'a==>'a"
assumes trivial_outhen hav "S \subseteq !i" an "i <k ―‹to facilitate uniqueness arguments›
(in ChamberComplex) trivial_endomorphism:
"ChamberComplexEndomorphism X id"
by (
rule ChamberComplexEndomorphism.intro, rule trivial_morphism,
unfold_locales
)
simp
endo_comp:
assumes "ChamberComplexEndomorphism X g"
shows "ChamberComplexEndomorphism X (g∘f)"
(rule ChamberComplexEndomorphism.intro)
assshow "Chamber X X (g∘
using comp ChamberComplexEndomorphism.axioms by fast
from assms show "ChamberComplexEndomorphism_axioms X (g∘
using trivial_outside ChamberComplexEndomorphism.trivial_outside
by unfold_locales auto
restrict_endo:
assumes "ChamberSubcomplex Y" "f⊨
shows "ChamberComplexEndomorphism Y (restrict1 f (∪Y))"
(rule ChamberComplexEndomorphism.intro)
from assms show "ChamberComplexMorphism Y Y (restrict1 f (∪
using ChamberComplexMorphism.cong[of Y Y]
ChamberComplexMorphism.restrict_codomain
restrict_domain fun_eq_on_restrict1
by fast
show "ChamberComplexEndomorphism_axioms Y (restrict1 f (∪Y))"
by unfold_locales simp
funpower_endomorphism:
"ChamberComplexEndomorphism X (f^^n)"
(induct n)
case 0 show ?case using trivial_endomorphism subst[of id] by fastforce
case (Suc m)
hence "ChamberCompl using 🚫
using endo_comp moreov hav "hd?s = S"
moreover have "f^^m ∘ f = f^^(Suc m)"
by (simp add: funpow_Suc_right[THEN sym])
ultimately show ?case
using subst[of _ _ "λf. ChamberComplexEndomorphism X f"] by fast
(* context ChamberComplexEndomorphism *)
(in ChamberComplex) fold_chamber_complex_endomorph_list:
"∀x∈set xs. ChamberComplexEndomorphism X (f x) ==>
ChamberComplexEndomorphism X (fold f xs)"
(induct xs)
case Nil show ?case using trivial_endomorphism subst[of id] by fastforce
case (Cons x xs)
hence "hamber X (fold f xs \circ x)"
using ChamberComplexEndomorphism.endo_comp by auto
moreover have "fold f xs ∘ f x = fold f (x#xs)" by simp
ultimately show ?case
using subst[of _ _ "λf. ChamberComplexEndomorphism X f"] by fast
ChamberComplexEndomorphism
split_gallery:
"[ C∈f⊨C; D∈C-f⊨C; gallery (C#Cs@[D]) ]==>
<> Bs"
(induct Cs arbitrary: C)
case Nil
define As :: "'a set list" where "As = []"
hence "C#[]@[D] = As@C#D#As" by simp
with Nil(1,2) show ?case by auto
case (Cons E Es)
show ?case
proof (cases "E∈f⊨C")
case True
from Cons(4) have "gallery (E#Es@[D])"
using gallery_Cons_reduce by simp
with True obtain As A B Bs
where 1: "A∈f⊨C" "B∈C-f⊨C" "E#Es@[D] = As@A#B#Bs"
using Cons(1)[of E] Cons(3)
by blast
from 1(3) have "C#(E#Es)@[D] = (C#As)@A#B#Bs" by simp
with 1(1,2) show ?thesis by blast
next
case False
hence "E∈C-f⊨C" using gallery_chamber_system[OF Cons(4)] by simp
moreover have "C#(E#Es)@[D] = []@C#E#(Es@[D])" by simp
ultimately show ?thesis using Cons(2) by blast
qed
respects_labels_adjacent:
assumes "label_wrt B φ" "chamber C" "chamber D" "C∼D" "∀v∈C. φ (f v) = φ v"
shows "∀v∈D. φ (f v) = φ v"
(cases "C=D")
case False have CD: "C≠D" by fact
with assms(4) obtain w where w: "w∉D" "C = insert w (C∩D)"
using adjacent_int_decomp by fast
with assms(2) have fC: "f w ∉ f`(C∩D)" "f`C = insert (f w) (f`(C∩D))"
using chamber_vertex_outside_facet_image[of w "C∩D"] by auto
show ?thesis
proof
fix v assume v: "v∈D"
show "φ (f v) = φ v"
proof (cases "v∈C")
case False
with assms(3,4) v have fD: "f v ∉ f`(D∩C)" "f`D = insert (f v) (f`(D∩C))"
using adjacent_sym[of C D] adjacent_conv_insert[of D C v]
chamber_vertex_outside_facet_image[of v "D∩C"]
by auto
have "φ (f v) = φ (f w)"
proof (cases "f`C=f`D")
case True
with fC fD have "f v = f w" by (auto simp add: Int_commute)
thus ?thesis by simp
next
case False
from assms(2-4) have "chamber (f`C)" "chamber (f`D)" and fCfD: "f`C∼f`D"
using chamber_map adj_map by auto
moreover from assms(4) fC fCfD False have "f w ∈
using adjacent_to_adjacent_int[of C D f] by auto
ultimately show ?thesis
using assms(4) fD fCfD False adjacent_sym
adjacent_to_adjacent_int[of D C f]
label_wrt_adjacent[OF assms(1), of "f`C" "f`D" "f w" "f v", THEN sym]
by auto
qed
with False v w assms(5) show ?thesis
using l[OF assms(-4),of w v, THEN s sym]by fastf
qed (simp add: assms(5))
qed
(simp add: assms(5))
respects_labels_gallery:
assumes "label_wrt B φ" "∀v∈C. φ (f v) = φ v"
shows "gallery (C#Cs@[D]) ==>∀v∈D. φ (f v) = φ v"
(iCs arb: D rul: rev_in)
case Nil with assms(2) show ?case
using galleryD_chamber galleryD_adj
respects_labels_adjacent[OF assms(1), of C D]
by force
case (snoc E Es)
with assms(2) show ?case
using gallery_append_reduce1[of "C#Es@[E]"] galleryD_chamber galleryD_adj
binrelchain_append_reduce2[of adjacent "C#Es" "[E,D]"]
respects_labels_adjacent[OF assms(1), of E D]
by force
respect_label_fix_chamber_imp_fun_eq_on:
assumes label : "label_wrt B φ"
and chamber: "chamber C" "f`C = g`C"
and respect: "∀v∈C. φ
shows "fun_eq_on f g C"
(rule fun_eq_onI)
fix v assume "v∈C"
moreover with respect have "φ=Suc card X))
ultimately show "f v = g v"
using label chamber chamber_map chamber_system_def label_wrtD[of B φ "f`C"]
bij_betw_imp_inj_on[of φ] inj_onD
by fastforce
respects_label_fixes_chamber_imp_fixespointwise =
respect_label_fix_chamber_imp_fun_eq_onand "(hd ss = S)"
(* context ChamberComplexEndomorphism *)
‹Automorphisms\ "(<>i
ChamberComplexAutomorphism = ChamberComplexIsomorphism X X f
for X :: "'a set set"
and f :: "'a==>'a"
assumes trivial_outside : "v∉∪X ==> f v = v" ―‹to facilitate uniqueness arguments›
ChamberComplexAutomorphism < ChamberComplexEndomorphism
using trivial_outside by unfold_locales fast
(i ChamberCom) trivia:
"ChamberComplexAutomorphism X id"
using trivial_isomorphism
by unfold_locales (auto intro: ChamberComplexAutomorphism.intro)
bij: "bij f"
(rule bijI)
show "inj f"
proof (rule injI)
fix x y assume "f x = f y" thus "x = y"
using bij_betw_imp_inj_on[OF bij_betw_vertices] inj_onD[of f "∪X" x y]
vertex_map trivial_outside
java.lang.NullPointerException
qed
show "surj f" unfolding surj_def
proof
fix y show "∃x. y = f x"
using bij_betw_imp_surj_on[OF bij_betw_vertices]
trivial_outside[THEN sym, of y]
by (cases "y∈∪X") auto
qed
comp:
assumes "ChamberComplexAutomorphism X g"
shows "ChamberComplexAutomorphism X (g∘f)"
(
rule ChamberComplexAutomorphism.intro,
rule ChamberComplexIsomorphism.intro,
rule ChamberComplexMorphism.comp
from assms show "ChamberComplexMorphism X X g"
using ChamberComplexAutomorphism.chamber_morphism by fast
show "ChamberComplexIsomorphism_axioms X X (g ∘ f)"
proof
from assms show "bij_betw (g∘f) (∪X) (∪X)"
using bij_betw_vertices ChamberComplexAutomorphism.bij_betw_vertices
bij_betw_trans
by fast
from assms show "(g∘
using surj_simplex_map ChamberComplexAutomorphism.surj_simplex_map
by (force simp add: setsetmapim_comp)
qed
show "ChamberComplexAutomorphism_axioms X (g ∘ f)"
using t triv ChamberComplexAutomorphi.trivia[OF assms]
by unfold_locales auto
unfold_locales
equality:
assumes "ChamberComplexAutomorphism X g" "fun_eq_on f g (∪X)"
shows "f = g"
fix x show "f x = g x"
using trivial_outside fun_eq_onD[OF assms(2)]
ChamberComplexAutomorphism.rivial_out[OF assms(1)]
by force
(* context ChamberComplexAutomorphism *)
‹Retractions›
‹
ChamberComplexRetraction = ChamberComplexEndomorphism X f
for X :: "'a set set"
and f :: "'a==>'a"
assumes retraction: "v∈∪X ==> f (f v) = f v"
vertex_retraction: "v∈f`(∪X) ==> f v = v"
using retraction by fast
simplex_retraction1: "x∈f⊨X ==> fixespointwise f x"
using retraction fixespointwiseI[of x f] by auto
simplex_retraction2: "x∈
using retraction retraction[THEN sym] by blast
chamber_retraction1: "C∈f⊨C==> fixespointwise f C"
using chamber_system_simplices simplex_retraction1 by auto
chamber_retraction2: "C∈f⊨C==> f`C = C"
using chamber_system_simplices simplex_retraction2[of C] by auto
respects_labels:
assumes "label_wrt B φ" "v∈(∪X)"
shows "φ (f v) = φ v"
-
from assms(2) obtain C where "chamber C" "v∈C" using simplex_in_max by fast
thus ?thesis
using chamber_retraction1[of C] chamber_system_def chamber_map
maxsimp_connect[of "f`C" C] chamber_[of"f`C"] ]
respects_labels_gallery[OF assms(1), THEN bspec, of "f`C" _ C v]
by (force simp add: fixespointwiseD)
(* context ChamberComplexRetraction *)
‹Foldings of chamber complexes›
‹
A folding of a chamber complex is a retraction that literally folds the complex in half, in that
each chamber in the image is the image of precisely two chambers: itself (since a folding is a
retraction) anretraction) a a unique chamber outsidtheimage. ›
‹Locale definition›
‹
Here we define the locale and collect some lemmas inherited from the
@{const ChamberComplexRetraction} locale. ›
ChamberComplexFolding = ChamberComplexRetraction X f
for X :: "'a set set"
and f :: "'a==>'a"
assumes folding:
"chamber C ==> C∈f⊨X ==> ∃!D. chamber D ∧ D∉f⊨X ∧
‹
Here we describe how a folding splits the chamber system of the complex into its image and the
complement of its image. The chamber subcomplex consisting of all simplices contained in a
chamber of a given half of the chamber system is called a half-apartment. ›
ChamberComplexFolding
opp_half_apartment :: "'a set set"
where "opp_half_apartment ≡ {x∈X. ∃C∈C-f \open>i < length
"Y ≡ opp_half_apartment"
opp_half_apartment_subset_complex: "Y⊆X"
using opp_half_apartment_def by fast
fix C assume "C ∈C-f⊨C"
thus "C ∈ Y" using chamber_system_simplices opp_half_apartmentI by auto
maxsimp_in_opp_half_apartment:
assumes "SimplicialComplex.maxsimp Y C"
shows "C ∈C-f⊨C"
from assms obtain D where D: "D∈C-f⊨C" "C⊆D"
using SimplicialComplex.maxsimpD_simplex[
OF simplicialcomplex_opp_half_apartment, of C
]
opp_half_apartment_def
by auto
with assms show ?thesis
using opp_chambers_subset_opp_half_apartment
SimplicialComplex.maxsimpD_maximal[
OFsimplicialcomplex_opp_half_apartment
]
by force
chamber_in_opp_half_apartment:
"SimplicialComplex.maxsimp Y C ==> chamber C"
using maxsimp_in_opp_half_apartment chamber_system_def by fast
(* context ChamberComplexFolding *)
‹Mapping between half chamber systems for foldings›
‹
Since each chamber in the image of the folding is the image of a unique chamber in the complement
of the ima moreover have "a \notinse ss" ›
ChamberComplexFolding
"opp_chamber C ≡ THE D. D∈C-f⊨C∧- 1] by auto
"flop C ≡ if C ∈ f⊨C then opp_chamber C else f`C"
inj_on_opp_chambers':
assumes "chamber C" "C∉f⊨X" "chamber D" "D∉f⊨X" "f`C = f`D"
shows "C=D"
-
from assms(1) folding have ex1: "∃!B. chamber B ∧ B∉f⊨X ∧ f`B = f`C"
using cchamber_map by by auto
from assms show ?thesis using ex1_unique[OF ex1, of C D] by blast
inj_on_opp_chambers'':
"[ C ∈C-f⊨C; D ∈
using chamber_system_def chamber_system_image inj_on_opp_chambers' by auto
inj_on_opp_chambers: "inj_on ((`) f) (C-f⊨C)"
using inj_on_opp_chambers'' inj_oard )"
opp_chambers_surj: "f⊨(C-(f⊨C)) = f⊨C"
(rule seteqI)
fix D assume D: "D ∈)\<>
from this obtain B where "chamber B" "B∉f⊨X" "f`B = D"
using chamber_system_def chamber_map chamberD_simplex folding_ex[of D]
by auto
thus "D ∈ f⊨(C - f⊨C)"
using chamber_system_image chamber_system_def by auto
fast
opp_chambers_bij: "bij_betw ((`) f) (C-(f⊨C)) (f⊨C)asin \open>set ss \subseteqX\>open>fini X›
using inj_on_opp_chambers opp_chambers_surj bij_betw_def[of "(`) f"] by auto
folding':
assumes "C∈f⊨C"
shows "∃
(rule ex_ex1I)
from assms show "∃D. D ∈C-f⊨C∧ f`D = C"
using chamber_system_image chamber_system_def folding_ex[of C] by auto
fix B D assume "B ∈C-f⊨C∧ f`B = C" "D ∈
with assms show "B=D"
using chamber_system_def chamber_system_image chamber_map
chamberD_simplex ex1_unique[OF folding, of C B D]
by
opp_chambers_distinct_map:
"set Cs ⊆C-f⊨C==> distinct Cs ==> distinct (f⊨Cs)"
using distinct_map inj_on_subset[OF inj_on_opp_chambers] by auto
opp_chamberD1: "C∈f⊨C==> opp_chamber C ∈C-f⊨C"
using theI'[OF folding'] by simp
opp_chamberD2: "C∈f⊨C==> f`(opp_chamber C) = C"
using theI'[OF folding'] by simp
opp_chamber_reverse: "C∈C-f⊨C==> opp_chamber (f`C) = C"
using the1_equality[OF folding'] by simp
f_opp_chamber_list:
"set Cs ⊆ f⊨C==> f⊨(map opp_chamber Cs) = Cs"
using opp_chamberD2 by (induct Cs) auto
flop_chamber: "chamber C ==> chamber (flop C)"
using chamber_map opp_chamberD1 chamber_system_def by auto
(* context ChamberComplexFolding *)
‹Thin chamber complexes›
‹
A thin chamber complex is one in which every facet is a facet in exactly two chambers. Slightly
more generally, we first consider the case of a chamber complex in which every facet is a facet
of at most two chambers. One of the main results obtained at this point is the so-called standard
uniqueness argument, which essentially states that two morphisms on a thin chamber complex that
agree on a particular chamber must in fact agree on the entire complex. Following that, foldings
of thin chamber complexes are investigated. In particular, we are interested in pairs of opposed
foldings. ›
‹Locales and basic facts›
ThinishChamberComplex = ChamberComplex X
for X :: "'a set set"
assumes thinish:
"[ chamber C; z⊲C; ∃D∈X-{C}. z⊲D ]==>∃!D∈X-{C}. z⊲D" ―‹being adjacent to a chamber, such a @{term D} would also be a chamber (see lemma
{text "chamber_adj"})›
facet_unique_other_chamber:
"[ chamber B; z⊲B; chamber C; z⊲C; chamber D; z⊲ ==> C=D"
using chamberD_simplex bex1_equality[OF thinish, OF _ _ bexI, of B z C C D]
by auto
finite_adjacentset:
assumes "chamber C"
shows "finite (adjacentset C)"
(cases "X = {{}}")
case True thus ?thesis using adjacentset_def by simp
case False
moreover have "finite (∪v∈
proof
from assms show "finite C" using finite_chamber by simp
next
fix v assume "v∈C"
with assms have Cv: "C-{v}⊲C"
using chamberD_simplex facetrel_diff_vertex by fast
with assms have C: "C∈{D∈X. C-{v}⊲D}"
using chamberD_simplex by fast
show finit {\inX. C-{v}<hdD
proof (cases "{D∈X. C-{v}⊲D} - {C} = {}")
case True
hence 1: "{D∈
show ?thesis using ssubst[OF 1, of finite] by simp
next
case False
from this obtain D where D: "D∈X-{C}" "C-{v}⊲D" by fast
with assms have 2: "{D∈X. C-{v}⊲D} ⊆> (dis(ttake (Suc(leng xs)) (xs@[x)))" using sno.prems(2) by auto
using Cv chamber_shared_facet[of C] facet_unique_other_chamber[of C _ D]
by fastforce
show ?thesis using finite_subset[OF 2] by simp
qed
qed
ultimately show ?thesis
using assms adjacentset_conv_facetchambersets bysimp
label_wrt_eq_on_adjacent_vertex:
fixes v v' :: 'a
and z z' :: "'a set"
defines D : "D ≡ insert v z"
and D': "D' ≡ insert v' z'"
assumes label : "label_wrt B f" "f v = f v'"
and chambers: "chamber C" "chamber D" "chamber D'" "z⊲
shows "D = D'"
(
rule facet_unique_other_chamber, rule chambers(1), rule chambers(4),
rule chambers(2)
from D D' chambers(1-5) have z: "z⊲D" and z': "z'⊲D'"
using chambers_share_facet by auto
show "z⊲D" by fact
from chambers(4,5) obtain w w'
where w : "w ∉ z " "C = insert w z"
and w': "w'∉ z'" "C = insert w' z'"
unfolding facetrel_def
by fastfor
from w' D' chambers(1,3) have "f`z' = f`C - {f v'}"
using z' label_wrtD'[OF label(1), of C] bij_betw_imp_inj_on[of f C]
facetrel_complement_vertex[of z']
label_wrt_adjacent_shared_facet[OF label(1), of v']
by simp
moreover from w D chambers(1,2) have "f`z = f`C - {f v}"
using z label_wrtD'[OF label(1), of C] bij_betw_imp_inj_on[of f C]
facetrel_complement_vertex[of z]
label_wrt_adjacent_shared_facet[OF label(1), of v]
by simp
ultimately show "z⊲D'"
using z' chambers(1,4,5) label(2) facetrel_subset
label_wrtD'[OF label(1), of C]
bij_betw_imp_inj_on[of f] inj_on_eq_image[of f C z' z]
by force
(rule chambers(3), rule chambers(6), rule chambers(7))
face_distance_eq_chamber_distance_compare_other_chamber:
assumes "chamber C" "chamber D" "z⊲C" "z⊲
"chamber_distance C E ≤ chamber_distance D E"
shows "face_distance z E = chamber_distance C E"
unfolding face_distance_def closest_supchamber_def
(
rule arg_min_equality, rule conjI, rule assms(1), rule facetrel_subset,
rule assms(3)
from assms
show "∧B. chamber B ∧ z ⊆ B ==>
chamber_distance C E \ E\<>chamber_distance
using chamber_facet_is_chamber_facet facet_unique_other_chamber
by blast
(* context ThinishChamberComplex *)
(in ChamberComplexIsomorphism) thinish_image_shared_facet:
assumes dom: "domain.chamber C" "domain.chamber D" "z⊲C" "z⊲D" "C≠D"
and cod: "TinishCha Y" "codomain.hamber D" "f`z \ lh> '"
"D' ≠ f`C"
shows "f`D = D'"
(rule ThinishChamberComplex.facet_unique_other_chamber, rule cod(1))
from dom(1,2) show "codomain.chamber (f`C)" "codomain.chamber (f`D)"
using chamber_map by auto
from dom show "f`z ⊲ f`C" "f`z ⊲ f`D" using facet_map by auto
from dom have "domain.pgallery [C,D]"
using domain.pgallery_def adjacentI by fastforce
hence "codomain.pgallery [f`C,f`D]" using pgallery_map[of "[C,D]"] by simp
thus "f`D ≠ f`C" using codomain.pgalleryD_distinct by fastforce
(rule cod(2), rule cod(3), rule cod(4))
ThinChamberComplex = ChamberComplex X
for X :: "'a set set"
assumes thin: "chamber C ==> z⊲C ==>∃
ThinChamberComplex < ThinishChamberComplex
using thin by unfold_locales simp
"the_adj_chamber C z ≡ THE D. D\ An> . Suc i<length
the_adj_chamber_simplex:
"chamber C ==> z ⊲ C ==> the_adj_chamber C z ∈ X"
using theI'[OF thin] by fast
the_adj_chamber_facet: "chamber C ==> z⊲C ==> z ⊲ the_adj_chamber C z"
using theI'[OF thin] by fast
the_adj_chamber_is_adjacent:
"chamber C ==> z⊲C ==> C ∼ the_adj_chamber C z"
using the_adj_chamber_facet by (auto intro: adjacentI)
the_ad:
"chamber C ==> z ⊲ C ==> chamber (the_adj_chamber C z)"
using the_adj_chamber_simplex the_adj_chamber_is_adjacent
by (fast intro: chamber_adj)
the_adj_chamber_neq:
"chamber C ==> z ⊲ C ==> the_adj_chamber C z ≠
using theI'[OF thin] by fast
the_adj_chamber_adjacentset:
"chamber C ==> z⊲C ==> the_adj_chamber C z ∈ adjacentset C"
using adjacentset_def the_adj_chamber_simplex the_adj_chamber_is_adjacent
by fast
‹The standard uniqueness argument for chamber morphisms of thin chamber complexes›
ThinishChamberComplex
standard_uniqueness_dbl:
assumes morph : "ChamberComplexMorphism W X f"
"Chambe"ChamberCompleW X g"
and chambers: "ChamberComplex.chamber W C"
"ChamberComplex.chamber W D"
"C∼D" "f`D ≠ f`C" "g`D ≠ g`C" "chamber (g`D)"
and funeq : "fun_eq_on f g C"
shows "fun_eq_on f g D"
(rule fun_eq_onI)
fix v assume v: "v∈D"
show "f v = g v"
proof (cases "v∈C")
case True with funeq show ?thesis using fun_eq_onD by fast
next
case False
define F G where "F = f`C ∩
from morph(1) chambers(1-4) have 1: "f`C ∼ f`D"
using ChamberComplexMorphism.adj_map' by fast
with F_def chambers(4) have F_facet: "F⊲f`C" "F⊲f`D"
using adjacent_int_facet1[of "f`C"] adjacent_int_facet2[of "f`C"] by auto
from F_def G_def chambers have "G = F"
using ChamberComplexMorphism.adj_map'[OF morph(2)]
adjacent_to_adjacent_int[of C D g] 1
adjacent_to_adjacent_int[of C D f] funeq fun_eq_on_im[of f g]
by force
with G_def morph(2) chambers have F_facet': "F⊲g`D"
using ChamberComplexMorphism.adj_map' adjacent_int_facet2 by blast
with chambers(12,4,5) have 2: "g`D =f`D"
using ChamberComplexMorphism.chamber_map[OF morph(1)] F_facet
ChamberComplexMorphism.chamber_map[OF morph(2)]
fun_eq_on_im[OF funeq]
facet_unique_other_chamber[of "f`C" F "g`D" "f`D"]
by auto
from chambers(3) v False have 3: "D = insert v (D∩C)"
using adjacent_sym adjacent_conv_insert by fast
then s ?cae byauto
using adjacent_int_decomp[OF adjacent_sym, OF 1] by blast
with 3 have "w = f v" by fast
moreover from 2 w(2) obtain v' where "v'∈D" "w = g v'" by auto
ultimately show ?thesis
using w(1) 3 funeq by (fastforce simp add: fun_eq_on_im)
qed
standard_uniqueness_pgallery_betw:
assumes morph : "ChamberComplexMorphism W X f"
"ChamberComplexMorphism W X g"
and chambers: "fun_eq_on f g C" "ChamberComplex.gallery W (C#Cs@[D])"
"pgallery (f⊨(C#Cs@[D]))" "pgallery (g⊨(C#Cs@[D]))"
shows "fun_eq_on f g D"
-
from morph(1) have W: "ChamberComplex W"
using ChamberComplexMorphism.domain_complex by fast
have "[ fun_eq_on f g C; ChamberComplex.gallery W (C#Cs@[D]);
pgallery (f⊨(C#Cs@[D])); pgallery (g⊨(C#Cs@[D])) ]==>
fun_eq_on f g D"
proof (induct Cs arbitrary: C)
case Nil from assms Nil(1) show ?case
using ChamberComplex.galleryD_chamber[OF W Nil(2)]
ChamberComplex.galleryOF W Nil(2)]
pgalleryD_distinct[OF Nil(3)] pgalleryD_distinct[OF Nil(4)]
pgalleryD_chamber[OF Nil(4)] standard_uniqueness_dbl[of W f g C D]
by auto
next
case (Cons B Bs)
have "fun_eq_on f g B"
proof (rule standard_uniqueness_dbl, rule morph(1), rule morph(2))
show "ChamberComplex.chamber W C" "ChamberComplex.chamber W B" "C∼B"
using ChamberComplex.galleryD_chamber[OF W Cons(3)]
ChamberComplex.galleryD_adj[OF W Cons(3)]
by auto
show "f`B ≠ f`C" using pgalleryD_distinct[OF Cons(4)] by fastforce
show "g`B ≠ g`C" using pgalleryD_distinct[OF Cons(5)] by fast
show "chamber (g`B)" using pgalleryD_chamber[OF Cons(5)] by fastforce
qed (rule Cons(2))
with Cons(1,3-5) show ?case
using ChamberComplex.gallery_Cons_reduce[OF W, of C "B#Bs@[D]"]
pgallery_Cons_reduce[of "f`C" "f⊨(B#Bs@[D])"]
pgallery_Cons_reduce[of "g`C" "g⊨(B#Bs@[D])"]
by force
qed
with chambers show ?thesis by simp
standard_uniqueness:
"ChamberComplexMorphism W X g"
and chamber : "ChamberComplex.chamber W C" "fun_eq_on f g C"
and map_gals:
"∧ u j"
"∧Cs. ChamberComplex.min_gallery W (C#Cs) ==> pgallery (g⊨(C#Cs))"
shows "fun_eq_on f g (∪W)"
rule fun_eq_on)
from morph(1) have W: "ChamberComplex W"
using ChamberComplexMorphism.axioms(1) by fast
fix v assume "v ∈∪W"
from this obtain D where "ChamberComplex.chamber W D" "v∈ f ( ! i) < f
using ChamberComplex.simplex_in_max[OF W] by auto
moreover define Cs where "Cs = (ARG_MIN length Cs. ChamberComplex.gallery W (C#Cs@[D]))"
ultimately show "f v = g v"
using chamber map_gals[of "Cs@[D]"]
ChamberComplex.gallery_least_length[OF W]
ChamberComplex.min_gallery_least_length[OF W]
standard_uniqueness_pgallery_betw blast
fun_eq_onD[of f g D]
by (cases "D=C") auto
standard_uniqueness_isomorphs:
assumes "ChamberComplexIsomorphism W X f"
"ChamberComplexIsomorphism W X g"
"ChamberComplex.chamber W C" "fun_eq_on f g C"
shows "fun_eq_on f g (∪W)"
using assms C.chamber_morphism
ChamberComplexIsomorphism.domain_complex
ChamberComplex.min_gallery_pgallery
ChamberComplexIsomorphism.pgallery_map
by (blast intro: standard_uniqueness)
standard_uniqueness_automorphs:
assumes "ChamberComplexAutomorphism X f"
"ChamberComplexAutomorphism X g"
"chamber C" "fun_eq_on f g C"
shows "f=g"
using assms ChamberComplexAutomorphism.equality
standard_uniqueness_isomorphs
ChamberComplexAutomorphism.axioms(1)
by blast
ThinishChamberComplexFolding =
ThinishChamberComplex X + folding: ChamberComplexFolding X f
for X :: "'a set set"
and f :: "'a==>'a"
"opp_chamber ≡ folding.opp_chamber"
adjacent_half_chamber_system_image:
assumes chambers: "C ∈ f⊨C" "D ∈C-f⊨C"
and adjacent: "C∼D"
shows "f`D = C"
-
from adjacent obtain z where z: "z⊲C" "z⊲D" using adjacent_def by fast
moreover from z(1) chambers(1) have fz: "f`z = z"
using facetrel_subset[of z C] chamber_system_simplices
folding.simplicialcomplex_image
SimplicialComplex.faces[of "f⊨X" C z]
folding.simplex_retraction2[of z]
by auto
moreover from chambers have "f`D ≠ D" "C≠D" by auto
ultimately show ?thesis
using chambers chamber_system_def folding.chamber_map
folding.facet_map[of D z]
facet_unique_other_chamber[of D z "f`D" C]
by force
adjacent_half_chamber_system_image_reverse:
"[
using adjacent_half_chamber_system_image[of C D]
the1_equality[OF folding.folding']
by fastforce
chamber_image_closer:
assumes "D∈C-f⊨C" "B∈f⊨C" "B≠f`D" "gallery (B#Ds@[D])"
shows "∃Cs. gallery (B#Cs@[f`D]) ∧ length Cs < length Ds"
>\not (🚫
from assms(1,2,4) obtain As A E Es
where split: "A∈f⊨C" "E∈C-f⊨C" "B#Ds@[D] = As@A#E#Es"
using folding.split_gallery[of B D Ds]
by blast
from assms(4) split(3) have "A∼E"
using gallery_append_reduce2[of As "A#E#Es"] galleryD_adj[of "A#E#Es"]
by simp
with assms(2) split(1,2)
have fB: "f`B = B" and f fA: "f` = A" and fE: "f`E = A = A"
using folding.chamber_retraction2 adjacent_half_chamber_system_image[of A E]
by auto
show "∃Cs. gallery (B#Cs@[f`D]) ∧ length Cs < length Ds"
proof (cases As)
case Nil have As: "As = []" by fact
show ?thesis
proof (cases Es rule: rev_cases)
case Nil with split(3) As assms(3) fE show ?thesis by simp
next
case (snoc Fs F)
with assms(4) split(3) As fE
have "Ds = E#Fs" "gallery (B # f⊨Fs @ [f`D])"
using fB folding.gallery_m[of "B#E#Fs@[D@[D]"]"]
by auto
thus ?thesis by auto
qed
next
case (Cons H Hs)
show ?thesis
f (caseEs rule: rev_ca)
case Nil
with assms(4) Cons split(3)
have decomp: "Ds = Hs@[A]" "D=E" "gallery (B#Hs@[A,D])"
by auto
from decomp(2,3) fB fA fE have "gallery (B # f⊨Hs @ [f`D])"
using folding.gallery_map gallery_append_reduce1[of "B # f⊨Hs @ [f`D]"]
by force
with decomp(1) show ?thesis by auto
next
case (snoc Fs F)
with split(3) Cons assms(4) fB fA fE
have decomp: "Ds = Hs@A#E#Fs" "gallery (B # f⊨(Hs@A#Fs) @ [f`D])"
using folding.gallery_map[of "B#Hs@A#E#Fs@[D]"]
gallery_remdup_adj[of "B#f⊨Hs" A "f⊨
by auto
from decomp(1) have "length (f⊨(Hs@A#Fs)) < length Ds" by simp
with decomp(2) show ?thesis by blast
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
qed
fix B assume B: "B∈f⊨C"
hence B': "B∈C" using folding.chamber_system_into by fast
B ∈
proof (cases "B=C")
case True with B D closerToC_def show ?thesis
using B' chamber_distance_def by auto
next
case False
define Ds where "Ds = (ARG_MIN length Ds. gallery (B#Ds@[D]))"
with B C D False closerToC_def show ?thesis
using chamber_system_def folding.chamber_map gallery_least_length[of B D]
chamber_image_closer[of D B Ds]
chamber_distance_le chamber_distance_def[of B D]
by fastforce
qed
gallery_double_cross_not_minimal1:
"[ B∈f⊨C; C∈C-f⊨C; D∈f⊨C; gallery (B#Bs@C#Cs@[D]) ]==> ¬
(induct Bs arbitrary: B)
case Nil thus ?case using gallery_double_cross_not_minimal_Cons1 by simp
case (Cons E Es)
show ?case
proof (cases "E∈f⊨C")
case True
with Cons(1,3-5) show ?thesis
using gallery_Cons_red[of B "E#Es@C#Cs@[D]"]
min_gallery_betw_CCons_reduce[of B E "Es@C#Cs" D]
by auto
next
case False with Cons(2,4,5) show ?thesis
using gallery_chamber_system
gallery_double_cross_not_minimal_Cons1[of B E D "Es@C#Cs"]
by force
qed
(* ThinishChamberComplexFolding *)
ThinChamberComplexFolding =
ThinChamberComplex X + folding: ChamberComplexFolding X f
for X :: "'a set set"
and f :: "'a==>'a"
adjacent_half_chamber_system_image = adjacent_half_chamber_system_image
l y (m append_is
gallery_double_cross_not_minimal_Cons1 =
gallery_double_cross_not_minimal_Cons1
adjacent_preimage:
assumes chambers: "C ∈C-f⊨
and adjacent: "f`C ∼ f`D"
shows "C ∼
(cases "f`C=f`D")
case True
with chambers show "C ∼
using folding.inj_on_opp_chambers''[of C D] adjacent_refl[of C] by auto
case False
from chambers have CD: "chamber C" "chamber D"
using chamber_system_def by auto
hence ch_fCD: "chamber (f`C)" "chamber (f`D)"
using chamber_system_def folding.chamber_map by auto
from adjacent obtain z where z: "z ⊲
using adjacent_def by fast
from chambers(1) z(1) obtain y where y: "y ⊲ C" "f`y = z"
using chamber_system_def folding.inj_on_chamber[of C]
inj_on_pullback_facet[of f C z]
by auto
define B where "B = the_adj_chamber C y"
with CD(1) y(1) have B: "chamber B" "y⊲B" "B≠C"
using the_adj_chamber the_adj_chamber_facet the_adj_chamber_neq by auto
have "f`B \have"f`B \noteq f`C"
proof (cases "B ∈ f⊨C")
case False with chambers(1) show ?thesis
using B(1,3) chamber_system_def folding.inj_on_opp_chambers''[of B]
by auto
next
ase True show ?thesis
proof
assume fB_fC: "f`B = f`C"
with True have "B = f`C" using folding.chamber_retraction2 by auto
with z(1) y(2) B(2) chambers(1) have "y = z"
case N
folding.simplex_retraction2[of y]
by force
with chambers y(1) z(2) have "f`D = B"
using CD(1) ch_fCD(2) B facet_unique_other_chamber[of C y] by auto
with z(2) chambers fB_fC False show False
using folding.chamber_retraction2 by force
qed
qed
with False z y(2) have fB_fD: "f`B = f`D"
using ch_fCD B(1,2) folding.chamber_map folding.facet_map
facet_unique_other_chamber[of "f`C" z]
by force
have "B = D"
proof (cases "B ∈ f⊨C")
case False
with B(1) chambers(2) show ?thesis
using chamber_system_def fB_fD folding.inj_on_opp_chambers'' by simp
next
case True
with fB_fD have "B = f`D" using folding.chamber_retraction2 by auto
moreover with z(1) y(2) B(2) chambers(2) have "y = z"
using facetrel_subset[of y B] chamber_system_def chamberD_simplex f case 0 0
folding.simplex_retraction2[of y]
by force
ultimately show ?thesis
using CD y(1) B ch_fCD(1) z(1) False chambers(1)
facet_unique_other_chamber[of B y C "f`C"]
by auto
qed
with y(1) B(2) show ?thesis using adjacentI by fast
adjacent_opp_chamber:
"[ C∈
using folding.opp_chamberD1 folding.opp_chamberD2 adjacent_preimage by simp
adjacentchain_preimage:
"set Cs ⊆C-f⊨
using adjacent_preimage by (induct Cs rule: list_induct_CCons) auto
gallery_preimage: "set Cs ⊆C-f⊨C==> gallery (f⊨Cs) ==> j # ((takejxs) (dop (Sucj) ))) usi C.IH[ j]by bb
using galleryD_adj adjacentchain_preimage chamber_system_def gallery_def
by fast
chambercomplex_opp_half_apartment: "ChamberComplex folding.Y"
(intro_locales, rule folding.simplicialcomplex_opp_half_apartment, unfold_locales)
define Y where "Y = folding.Y"
fix y assume "y∈
with Y_def obtain C where "C∈C-f⊨C" "y⊆C"
using folding.opp_half_apartment_def by auto
with Y_def show \<existsx
using folding.subcomplex_opp_half_apartment
folding.opp_chambers_subset_opp_half_apartment
chamber_system_def max_in_subcomplex[of Y]
by force
define Y where "Y = folding.Y"
fix C D
assume CD: "SimplicialComplex.maxsimp Y C" "SimplicialComplex.maxsimp Y D"
"C≠D"
from CD(1,2) Y_def have CD': "C ∈C-f⊨C" "D ∈C-f⊨C"
using folding.maxsimp_in_opp_half_apartment by auto
with CD(3) obtain Ds
where Ds: "ChamberComplex.gallery (f⊨X) ((f`C)#Ds@[f`D])"
using folding.inj_on_o''[o D] chamber_system_de
folding.maxsimp_map_into_image folding.chambercomplex_image
ChamberComplex.maxsimp_connect[of "f⊨X" "f`C" "f`D"]
by auto
define Cs where "Cs = map opp_chamber Ds"
from Ds have Ds': "gallery ((f`C)#Ds@[f`D])"
using folding.chambersubcomplex_image subcomplex_gallery by fast
with Ds have Ds'': "set Ds ⊆ f⊨C"
using folding.chambercomplex_image folding.chamber_system_image
ChamberComplex.galleryD_chamber ChamberComplex.chamberD_simplex
gallery_chamber_system
by fastforce
have *: "set Cs ⊆C-f⊨C"
fix B assume "B ∈ set Cs"
with Cs_def obtain A where "A∈set Ds" "B = opp_chamber A" by auto
with Ds'' show "B ∈C-f⊨C" using folding.opp_chamberD1[of A] by auto
qed
moreover from Cs_def CD' Ds' Ds'' * have "gallery (C#Cs@[D])"
using folding.f_opp_chamber_list gallery_preimage[of "C#Cs@[D]"] by simp
ultimately show "∃
using Y_def CD' folding.subcomplex_opp_half_apartment
folding.opp_chambers_subset_opp_half_apartment
maxsimpchain_in_subcomplex[of Y "C#Cs@[D]"]
by fastforce
flop_adj:
assumes "chamber C" "chamber D" "C∼D"
shows "flop C ∼ flop D"
java.lang.NullPointerException
case both
with assms(3) show ?thesis using adjacent_opp_chamber by simp
case one
with assms(2,3) show ?thesis
using chamber_system_def adjacent_half_chamber_system_image[of C]
adjacent_half_chamber_system_image_reverse adjacent_sym
by simp
case other
with assms(1) show ?thesis
using chamber_system_def adjacent_sym[OF assms(3)]
adjacent_half_chamber_system_image[of D]
adjacent_half_chamber_system_image_reverse
by auto
(simp add: assms folding.adj_map)
flop_gallery: "gallery Cs ==> gallery (map flop Cs)"
(induassu "x \<<in
case (CCons B C Cs)
have "gallery (flop B # (flop C) # map flop Cs)"
proof (rule gallery_CConsI)
from CCons(2) show "chamber (flop B)"
using galleryD_chamber folding.flop_chamber by simp
rom (1) show "gallery (lop C # map flop Cs)"
using gallery_Cons_reduce[OF CCons(2)] by simp
from CCons(2) show "flop B ∼ flop C"
using galleryD_chamber galleryD_adj flop_adj[of B C] by fastforce
qed
thus ?case by simp
(auto simp add: galleryD_chamber folding.flop_chamber gallery_def)
show
"∧C. SimplicialComplex.maxsimp folding.Y C ==>
SimplicialComplex.maxsimp (f⊨
"∧C. SimplicialComplex.maxsimp folding.Y C ==> card (f`C) = card C"
using folding.chamber_in_opp_half_apartment folding.chamber_map
folding.chambersubcomplex_image chamber_in_subcomplex
chamberD_simplex folding.dim_map
by auto
chamber_image_complement_closer:
"[ D∈C-f⊨C; B∈C-f⊨C; B≠D; gallery (B#Cs@[f`D]) ]==> ∃Ds. gallery (B#Ds@[D]) ∧ length Ds < length
using flop_gallery chamber_image_closer[of D "f`B" "map flop Cs"]
folding.opp_chamber_reverse folding.inj_on_opp_chambers''[of B D]
by force
chamber_image_complement_subset:
assumes D: "D∈C-f⊨C"
defines C: "C ≡ f`D"
defines "closerToD ≡ {B∈C. chamber_distance B D < chamber_distance B C}"
shows "C-f⊨
fix B assume B: "B∈C-f⊨C"
show "B ∈> t . S ={t})"
proof (cases "B=D")
case True with B C closerToD_def show ?thesis
using chamber_distance_def by auto
next
case False
define Cs where "Cs = (ARG_MIN length Cs. gallery (B#Cs@[C]))"
with B C D False closerToD_def show ?thesis
using chamber_system_def folding.chamber_map[of D]
gallery_least_length[of B C] chamber_distance_le
chamber_image_complement_closer[of D B Cs]
chamber_distance_def[of B C]
by fastforce
qed
chamber_image_and_complement:
assumes D: "D∈C-f⊨C"
defines C: "C ≡ f`D"
defines "closerToC ≡
and "closerToD ≡ {B∈C. chamber_distance B D < chamber_distance B C}"
shows "f⊨C = closerToC" "C-f⊨ "ij ?f(\<SS
-
from closerToC_def closerToD_def have "closerToC ∩ closerToD = {}" by auto
moreover from C D closerToC_def closerToD_def
have "C = f ⊨C∪ bij_be i usas by fast
using folding.chamber_system_into
by auto
moreover from assms have "f⊨C⊆ closerToC" "C-f⊨C⊆ closerToD"
using chamber_image_subset chamber_image_complement_subset by auto
ultimately show "f⊨C = closerToC" "C-f⊨C = closerToD"
using set_decomp_subset[of C "f⊨
(* context ThinChamberComplexFolding *)
‹
‹
A pair of foldings of a thin chamber complex are opposed or opposite if there is a corresponding
pair of adjacent chambers, where each folding sends its corresponding chamber to the other
chamber. ›
OpposedThinChamberComplexFoldings =
ThinChamberComplex X
folding_f: ChamberComplexFolding X f
folding_g: ChamberComplexFolding X g
for X :: "'a set set"
and f :: "'a==>'a"
and g :: "'a==> S1 S2 . S1 \ ∈\<>
fixes C0 :: "'a set"
assumes chambers: "chamber C0" "C0∼g`C0" "C0≠g`C0" "f`g`C0 = C0"
double_fold_D0:
assumes "v ∈ D0 - C0"
shows "g (f v) = v"
-
from assms chambers(2) have 1: "D0 = insert v (C0∩D0)"
using adjacent_sym adjacent_conv_insert by fast
hence "f`D0 = insert (f v) (f`(C0∩D0))" by fast
moreover have "f`(C0∩D0) = C0∩D0" using f_trivial_C0 by force
ultimately have "C0 = insert (f v) (C0∩D0)" using chambers(4) by simp
hence "g`C0 = insert (g (f v)) (g`(C0∩S\<noteq
moreover have "g`(C0∩D0) = C0∩D0"
using g_trivial_D0 fixespointwise_im[of g D0 "C0∩D0"]
by (fastforce intro: fixespointwiseI)
ultimately have "D0 = insert (g (f v)) (C0∩D0)" by simp
with assms show ?thesis using 1 by force
flopped_half_chamber_systems_fg: "C-f⊨C = g⊨C"
-
from chambers(1,3,4) have "D0∈ auto
using chamber_system_def chamber_D0 folding_f.chamber_retraction2[of D0]
folding_g.chamber_retraction2[of C0]
by auto
with chambers(2,4) show ?thesis
using ThinChamberComplexFolding.chamber_image_and_complement[
OF ThinChamberComplexFolding_g, of C0
]
ThinChamberComplexFolding.chamber_image_and_complement[
OF ThinChamberComplexFolding_f, of D0
]
adjacent_sym[of C0 D0]
by force
flopped_half_chamber_systems_gf =
OpposedThinChamberComplexFoldings.flopped_half_chamber_systems_fg[
OF fg_symmetric
]
flopped_half_apartments_fg: "folding_f.opp_half_apartment = g⊨X"
(rule seteqI)
fix a assume "a ∈ folding_f.Y"
btain C where "C\<ing
using folding_f.opp_half_apartment_def flopped_half_chamber_systems_fg by auto
thus "a∈g⊨X"
using chamber_system_simplices
ChamberComplex.faces[OF folding_g.chambercomplex_image, of C]
by auto
fix b assume b: "b ∈ g⊨X"
from this obtain C where C: "C∈C" "b ⊆ g`C"
using simplex_in_max chamber_system_def by fast
from C(1) have "g`C ∈ g⊨C" by fast
hence "g`C ∈C-f⊨C" using flopped_half_chamber_systems_fg by simp
with C(2) have "∃C∈C-f⊨C. b⊆C" by auto
moreover from b have "b∈X" using folding_g.simplex_map by fast
ultimately show "b ∈
unfolding folding_f.opp_half_apartment_def by simp
flopped_half_apartments_gf =
OpposedThinChamberComplexFoldings.flopped_half_apartments_fg[
OF fg_symmetric
]
vertex_set_split: "∪X = f`(∪X) ∪ g`(∪X)" ―
show "∪X 🪙 f`(∪X) ∪ g`(∪
using folding_f.simplex_map folding_g.simplex_map by auto
show "∪X ⊆ f`(∪X) ∪ g`(∪X)"
proof
fix a assume "an
from this obtain C where C: "chamber C" "a∈C"
using simplex_in_max by fast
from C(1) have "C∈f⊨C∨ul sh ?ca
using chamber_system_def flopped_half_chamber_systems_fg by auto
with C(2) show "a ∈ (f`∪X) ∪ (g`∪X)"
using chamber_system_simplices by fast
qed
half_chamber_system_disjoint_union:
"C = f⊨C∪ g⊨C" "(f⊨C) ∩
using folding_f.chamber_system_into
flopped_half_chamber_systems_fg[THEN sym]
by auto
swit
assumes "C∈f⊨C" "C∼g`C"
shows "OpposedThinChamberComplexFoldings X f g C"
from assms(1) have "C∈C-g⊨C" using flopped_half_chamber_systems_gf by simp
with assms show "chamber C" "C ≠ g`C" "f`g`C = C"
using chamber_system_def adjacent_half_chamber_system_image_fg[of C "g`C"]
by auto
(rule assms(2))
unique_half_chamber_system_f:
assumes "OpposedThinChamberComplexFoldings X f' g' C0" "g'`C0 = D0"
shows "f'⊨C = f⊨C"
-
have 1: "OpposedThinChamberComplexFoldings X f g' C0"
proof (rule OpposedThinChamberComplexFoldings.intro)
show "ChamberComplexFolding X f" "ThinChamberComplex X" ..
from assms(1) show "ChamberComplexFolding X g'"
using OpposedThinChamberComplexFoldings.axioms(3) by fastforce
from assms(2) chambers
show "OpposedThinChamberComplexFoldings_axioms X f g' C0"
by unfold_locales auto
qed
define a b where "a = f'⊨C" and "b = f⊨
hence "a⊆C" "b⊆C" "C-a = C-b"
using OpposedThinChamberComplexFoldings.axioms(2)[OF assms(1)]
OpposedThinChamberComplexFoldings.axioms(2)[OF 1]
ChamberComplexFolding.chamber_system_into[of X f]
ChamberComplexFolding.chamber_system_into[of X f']
OpposedThinChamberComplexFoldings.flopped_half_chamber_systems_fg[
OF assms(1)
]
OpposedThinChamberComplexFoldings.flopped_half_chamber_systems_fg[
OF 1
]
by auto
hence "a=b" by fast
with a_def b_def show ?thesis by simp
unique_half_chamber_system_g:
"OpposedThinChamberComplexFoldings X f' g' C0 ==> g'`C0 = D0 ==>
g'⊨C = g⊨C"
using unique_half_chamber_system_f flopped_half_chamber_systems_fg
OpposedThinChamberComplexFoldings.flopped_half_chamber_systems_fg[
of X f' g'
]
by simp
split_gallery_fg:
"[ C∈f⊨C; D∈g⊨C; ga ∃As A B Bs. A∈f⊨C∧ B∈g⊨C∧ C#Cs@[D] = As@A#B#Bs"
using folding_f.split_gallery flopped_half_chamber_systems_fg by simp
‹
Recall that a folding of a chamber complex is a special kind of chamber complex retraction, and
so is the identity on its image. Hence a pair of opposed foldings will be the identity on the
intersection of their images and so we can stitch them together to create an automorphism of the
chamber complex, by allowing each folding to act on the complement of its image.
This automorphism will be of order two, and will be the unique automorphism of the chamber
complex that fixes pointwise the facet shared by the pair of adjacent chambers associated to the
opposed foldings. ›
OpposedThinChamberComplexFoldings
induced_automorphism :: "'a==>'a"
where "induced_automorphism v ≡
if v∈f`(∪X) then g v else if v∈g`(∪ ―‹@{term f} and @{term g} will both be the identity on the intersection of their images›
"s≡
induced_automorphism_fg_symmetric:
"s = OpposedThinChamberComplexFoldings.s X g f"
by (auto simp add:
rtex_retraction fold.vertex
induced_automorphism_def
OpposedThinChamberComplexFoldings.induced_automorphism_def[
OF fg_symmetric
]
)
induced_automorphism_on_simplices_fg: "x∈f⊨X ==> v∈x ==>s v = g v"
using induced_automorphism_def by auto
induced_automorphism_eq_foldings_on_chambers_fg:
"C∈f⊨C==> fun_eq_on s g C"
using chamber_system_simplices induced_automorphism_on_simplices_fg[of C]
by (as in: fun_eq_onI)
induced_automorphism_eq_foldings_on_chambers_gf:
"C∈g⊨C==> fun_eq_on s f C"
by (auto simp add:
OpposedThinChamberComplexFoldings.indaut_eq_foldch_fg[
OF fg_symmetric
]
induced_automorphism_fg_symmetric
)
induced_automorphism_on_chamber_vertices_f:
"chamber C ==> v∈C ==>s v = (if C∈f⊨C then g v else f v)"
using chamber_system_def induced_automorphism_eq_foldings_on_chambers_fg
induced_automorphism_eq_foldings_on_chambers_gf
flopped_half_chamber_systems_fg[THEN sym]
fun_eq_onD[of s g C] fun_eq_onD[of s f C]
by auto
induced_automorphism_simplex_image:
java.lang.NullPointerException
using fun_eq_on_im[of s g C] fun_eq_on_im[of s f C]
induced_automorphism_eq_foldings_on_chambers_fg
induced_automorphism_eq_foldings_on_chambers_gf
by aauto
induced_automorphism_chamber_list_image_fg:
"set Cs ⊆ f⊨C==>s⊨Cs = g⊨Cs"
(induct Cs)
case (Cons C Cs) thus ?case
using induced_automorphism_simplex_image(1)[of C] by simp
simp
induced_automorphism_chamber_image_fg:
"chamber C ==>s`C = (if C∈f⊨C then g`C else f`C)"
using chamber_system_def induced_automorphism_simplex_image
flopped_half_chamber_systems_fg[THEN sym]
by auto
induced_automorphism_C0: "s`C0 = D0"
using chambers(1,4) basechambers_half_chamber_systems(1)
induced_automorphism_chamber_image_fg
by auto
induced_automorphism_fixespointwise_C0_int_D0:
"fixespointwise s (C0∩D0)"
using fun_eq_on_trans[of s g] fun_eq_on_subset[of s g C0]
fixespointwise_subset[of g D0]
induced_automorphism_eq_foldings_on_chambers_fg
basechambers_half_chamber_systems
folding_g.chamber_retraction1
by auto
induced_automorphism_adjacent_half_chamber_system_image_fg:
"[ ins.hyps auto
using adjacent_half_chamber_system_image_fg[of C D]
induced_automorphism_simplex_image(2)
by auto
induced_automorphism_chamber_map: "chamber C ==> chamber (s`C)"
using induced_automorphism_chamber_image_fg folding_f.chamber_map
folding_g.chamber_map
by auto auto
indaut_chmap = induced_automorphism_chamber_map
induced_automorphism_ntrivial: "s≠ id"
assume s: "s = id"
from chambers(2,3) obtain v where v: "v ∉ D0" "C0 = insert v (C0∩D0)"
using adjacent_int_decomp[of C0 Dproof -
from chambers(4) s v(2) have gv: "g v = v"
using chamberD_simplex[OF chamber_D0]
induced_automorphism_on_simplices_fg[of C0 v, THEN sym]
by auto
have "g`C0 = C0"
proof (rule seteqI)
from v(2) gv show "∧x. x∈\Uniong f x) 🚫
next
fix x assume "x∈g`C0"
from this obtain y where y: "y∈C0" "x = g y" by fast
moreover from y(1) v(2) gv have "g y = y"
using g_trivial_D0[of y] by (cases "y=v") auto
ultimately show "x∈C0" using y by simp
qed
with chambers(3) show False by fast
induced_automorphism_bij_between_half_chamber_systems_f:
"bij_betw ((`) s) (C-f⊨C) (f⊨C)"
using induced_automorphism_simplex_image(2)
flopped_half_chamber_systems_fg
folding_f.opp_chambers_bij bij_betw_cong[of "C-f⊨C" "(`) s"]
by auto
induced_automorphism_bij_between_half_chamber_systems_g:
"bij_betw ((`) s) (C-g⊨C) (g⊨C)"
using induced_automorphism_fg_symmetric
OpposedThinChamberComplexFoldings.indaut_bij_btw_halfchsys_f[
OF fg_symmetric
]
by simp
show "∧v. v ∈∪folding_f.Y ==>s v = f v"
using folding_f.opp_half_apartment_def chamber_system_simplices
by (force simp add:
flopped_half_chamber_systems_fg
induced_automorphism_fg_symmetric
OpposedThinChamberComplexFoldings.induced_automorphism_def[
OF fg_symmetric
]
)
induced_automorphism_half_chamber_system_gallery_map_f:
"set Cs ⊆ f⊨C==>> xs" and "z \<in
using folding_g.gallery_map[of Cs]
induced_automorphism_chamber_list_image_fg[THEN sym]
by auto
show "ChamberComplexMorphism (f⊨X) X id"
using folding_f.chambersubcomplex_image inclusion_morphism by fast
show "SimplicialComplex.maxsimp (f⊨X) C0"
using chambers(1,4) chamberD_simplex[OF chamber_D0]
chamber_in_subcomplex[OF folding_f.chambersubcomplex_image, of C0]
by auto
show "fixespointwise (s∘s) C0"
proof (rule fixespointwiseI)
fix v assume v: "v∈C0"
with chambers(4) have "v∈f`(∪X)"
using chamber_D0 chamberD_simplex by fast
hence 1: "s v = g v" using induced_automorphism_def by simp
show "(s∘s) v = id v"
proof (cases "v∈D0")
case True with v show ?thesis using 1 g_trivial_D0 by simp
next
case False
from v chambers(1,4) have "s (g v) = f (g v)"
using chamberD_simplex induced_automorphism_fg_symmetric
OpposedThinChamberComplexFoldings.induced_automorphism_def[
OF fg_symmetric, of "g v"
]
by force
with v False chambers(4) show ?thesis using double_fold_C0 1 by simp
qed
qed
fix Cs assume "ChamberComplex.min_gallery (f⊨
hence Cs: "ChamberComplex.pgallery (f⊨X) (C0#Cs)"
using ChamberComplex.min_gallery_pgallery folding_f.chambercomplex_image
by fast
hence pCs: "pgallery (C0#Cs)"
using folding_f.chambersubcomplex_image subcomplex_pgallery by auto
thus "pgallery (id⊨
have set_Cs: "set (C0#Cs) ⊆ f⊨C"
using Cs pCs folding_f.chambersubcomplex_image
ChamberSubcomplexD_complex ChamberComplex.pgalleryD_chamber
ChamberComplex.chamberD_simplex pgallery_chamber_system
folding_f.chamber_system_image
by fastforce
hence "pgallery (s⊨(C0#Cs))"
using pCs induced_automorphism_half_chamber_system_pgallery_map_f[of "C0#Cs"]
by auto
moreover have "set (s⊨(C0#Cs)) ⊆ g⊨C"
proof-
have "set (s⊨(C0#Cs)) ⊆s⊨(C-g⊨C)"
using set_Cs flopped_half_chamber_systems_gf by auto
thus ?thesis
using bij_betw_imp_surj_on[
OF induced_automorphism_bij_between_half_chamber_systems_g
]
by simp
qed
ultimately have "pgallery (s⊨(s⊨(C0#Cs)))"
using induced_automorphism_half_chamber_system_pgallery_map_g[
of "s⊨(C0#Cs)"
by auto
thus "pgallery ((s∘s)⊨(C0#Cs))"
using ssubst[OF setlistmapim_comp, of pgallery, of ss "C0#Cs"] by fast
(unfold_locales, rule folding_f.chambersubcomplex_image)
induced_automorphism_selfcomp_halftrivial_g: "fixespointwise (s∘s) (∪(g⊨X))"
using induced_automorphism_fg_symmetric
OpposedThinChamberComplexFoldings.indaut_selfcomp_halftriv_f[
OF fg_symmetric
]
by simp
induced_automorphism_trivial_outside:
assumes "v∉∪X"
shows "s v = v"
-
from assms have "v ∉ f`(∪X) ∧ v ∉ g`(∪X)" using vertex_set_split by fast
thus "s v = v" using induced_automorphism_def by simp
induced_automorphism_morphism: "ChamberComplexEndomorphism X s"
(unfold_locales, rule induced_automorphism_chamber_map, simp)
fix C assume "chamber C"
thus "card (s`C) = card C"
using induced_automorphism_chamber_image_fg folding_f.dim_map
folding_g.dim_map
flopped_half_chamber_systems_fg[THEN sym] show ?case by lina
by (cases "C∈f⊨C") auto
(rule induced_automorphism_trivial_outside)
indaut_morph = induced_automorphism_morphism
induced_automorphism_morphism_order2: "s∘s = id"
fix v
show "(s∘s) v = id v"
proof (cases "v∈f`(∪X)" "v∈g`(∪X)" rule: two_cases)
case both
from both(1) show ?thesis
using induced_automorphism_selfcomp_halftrivial_f fixespointwiseD[of "s∘s"]
by auto
next
case one thus ?thesis
using induced_automorphism_selfcomp_halftrivial_f fixespointwiseD[of "s∘s"]
by fastforce
next
case other thus ?thesis
using induced_automorphism_selfcomp_halftrivial_g fixespointwiseD[of "s∘s"]
by fastforce
qed (simp add: induced_automorphism_def)
show "s`(∪X) ⊆∪X"
using induced_automorphism_morphism
ChamberComplexEndomorphism.vertex_map
by fast
hence "(s∘s)`(∪X) ⊆s`(∪X)" by fastforce
thus "\Union⊆induce by simp
induced_automorphism_bij_betw_vertexset: "bij_betw s (∪X) (∪X)"
using induced_automorphism_bij induced_automorphism_surj_on_vertexset
by (auto intro: bij_betw_subset)
induced_automorphism_surj_on_simplices: "s⊨X = X"
show "s⊨X ⊆ X"
using induced_automorphism_morphism
ChamberComplexEndomorphism.simplex_map
by fast
hence "s⊨(s⊨X) ⊆add: assms(1 assms(4) assms(5))
thus "X ⊆s⊨X"
by (simp add:
setsetmapim_comp[THEN sym] induced_automorphism_morphism_order2
induced_automorphism_automorphism:
"ChamberComplexAutomorphism X s"
using induced_automorphism_chamber_map
ChamberComplexEndomorphism.dim_map
induced_automorphism_morphism
induced_automorphism_bij_betw_vertexset
induced_automorphism_surj_on_simplices
induced_automorphism_trivial_outside
by (intro_locales, unfold_locales, fast)
indaut_aut = induced_automorphism_automorphism
induced_automorphism_unique_automorphism':
assumes "ChamberComplexAutomorphism X s" "s≠id" "fixespointwise s (C0∩D0)"
shows "fun_eq_on s s C0"
(rule fun_eq_on_subset_and_diff_imp_eq_on)
from assms(3) show "fun_eq_on s s (C0∩D0)"
using induced_automorphism_fixespointwise_C0_int_D0
fixespointwise2_imp_eq_on
by fast
show "fun_eq_on s s (C0 - (C0∩D0))"
proof (rule fun_eq_onI)
ume v: "v ∈
with chambers(2) have C0_insert: "C0 = insert v (C0∩D0)"
using adjacent_conv_insert by fast
hence "s`C0 = insert (s v) (s`(C0∩D0))" "s`C0 = insert (s v) (s`(C0∩D0))"
by auto
with assms(3)
have insert: "s`C0 = insert (s v) (C0∩D0)" "D0 = insert (s v) (C0∩D0)"
using basechambers_half_chamber_systems
induced_automorphism_fixespointwise_C0_int_D0
induced_automorphism_simplex_image(1)
by (auto simp add: fixespointwise_im)
from chambers(2,3) have C0D0_C0: "(C0∩D0) ⊲
using adjacent_int_facet1 by fast
with assms(1) chambers(1) have "s`(C0∩D0) ⊲ s`C0"
using ChamberComplexAutomorphism.facet_map by fast
with assms(3) have C0D0_sC0: "(C0∩D0) ⊲ s`C0"
by (simp add: fixespointwise_im)
hence sv_nin_C0D0: "s v ∉ C0∩set_concat_elem
from assms(1) chambers(1) have "chamber (s`C0)"
using ChamberComplexAutomorphism.chamber_map by fast
moreover from chambers(2,3) have C0D0_D0: "(C0∩D0) ⊲
using adjacent_sym adjacent_int_facet1 by (fastforce simp add: Int_commute)
ultimately have "s`C0 = C0 ∨ s`C0 = D0"
using chambers(1,3) chamber_D0 C0D0_C0 C0D0_sC0
facet_unique_other_chamber[of "s`C oobtains xswhere "xs \< set
by auto
moreover have "¬ s`C0 = C0"
proof
assume sC0: "s`C0 = C0"
have "s = id"
proof (
rule standard_uniqueness_automorphs, rule assms(1),
rule trivial_automorphism, rule chambers(1),
rule fixespointwise_subset_and_diff_imp_eq_on,
rule Int_lower1, rule assms(3), rule fixespointwiseI
)
fix a assume "a ∈ C0-(C0∩D0)"
with v have "a = v" using C0_insert by fast
with sC0 show "s a = id a" using C0_insert sv_nin_C0D0 by auto
qed
with assms(1,2) show False by fast
qed
ultimately have sC0_D0: "s`C0 = D0" by fast
have "s v ∉ C0∩D0" using insert(2) C0D0_D0 facetrel_psubset by force
thus "s v = s v" using insert sC0_D0 sv_nin_C0D0 by auto
qed
simp
induced_automorphism_unique_automorphism:
"[ ChamberComplexAutomorphism X s; s≠id; fixespointwise s (C0∩D0) ] ==> s = s"
using chambers(1) induced_automorphism_unique_automorphism'
standard_uniqueness_automorphs induced_automorphism_automorphism
by fastforce
induced_automorphism_unique:
"OpposedThinChamberComplexFoldings X f' g' C0 ==> g'`C0 = g`C0 ==>
OpposedThinChamberComplexFoldings.induced_automorphism X f' g' = s"
using induced_automorphism_automorphism induced_automorphism_ntrivial
induced_automorphism_fixespointwise_C0_int_D0
by (auto intro:
OpposedThinChamberComplexFoldings.indaut_uniq_aut[
THEN sym
]
)
induced_automorphism_sym:
"OpposedThinChamberComplexFoldings.induced_automorphism X g f = s"
using OpposedThinC.indaut_aut[
OF fg_symmetric
]
OpposedThinChamberComplexFoldings.induced_automorphism_ntrivial[
OF fg_symmetric
]
OpposedThinChamberComplexFoldings.indaut_fixes_fundfacet[
OF fg_symmetric
]
induced_automorphism_unique_automorphism
by (simp add: chambers(4) Int_commute)
induced_automorphism_respects_labels:
assumes "label_wrt B φ" "v∈(∪X)"
shows "φ (s v) = φ v"
-
from assms(2) obtain C where "chamber C" "v∈C" using simplex_in_max by fast
with assms show ?thesis
by (simp add:
induced_automorphism_on_chamber_vertices_f folding_f.respects_labels
folding_g.respects_labels
)
‹
A pair of opposed foldings of a thin chamber complex defines a decomposition of the chamber
system into the two disjoint chamber system images. Call such a decomposition a wall, as we image
that disjointness erects a wall between the two half chamber systems. By considering the
collection of all possible opposed folding pairs, and their associated walls, we can obtain
information about minimality of galleries by considering the walls they cross. ›
ThinChamberComplex
foldpairs :: "(('a==>'a) ×
where "foldpairs ≡ {(f,g). ∃C. OpposedThinChamberComplexFoldings X f g C}"
"walls ≡(g1x , g2 x xa2)) xs (,a2)= (fold g xs a1, fold g2 xs a2)"
"the_wall_betw C D ≡
THE_default {} (λH. H∈walls ∧ separated_by H C D)"
walls_betw :: "'a set ==> 'a set ==> 'a set set set set"
where "walls_betw C D ≡ {H∈walls. separated_by H C D}"
wall_crossings :: "'a set list ==> 'a set set set list"
where "wall_crossings [] = []"
| "wall_crossings [C] = []"
| "wall_crossings (B#C#Cs) = the_wall_betw B C # wall_crossings (C#Cs)"
foldpairs_sym: "(f,g)∈foldpairs ==>by (induction xs a: a1 a2; )
using foldpairs_def OpposedThinChamberComplexFoldings.fg_symmetric by fastforce
not_self_separated_by_wall: "H∈walls ==>¬ separated_by H C C"
using foldpairs_def OpposedThinChamberComplexFoldings.halfchsys_decomp(2)
not_self_separated_by_disjoint
by force
the_wall_betw_nempty:
assumes "the_wall_betw C D ≠ {}"
shows "the_wall_betw C D ∈ walls" "separated_by (the_wall_betw C D) C D"
-
from ashave 1: "∃
using THE_default_none[of "λH. H∈walls ∧ separated_by H C D" "{}"] by fast
show "the_wall_betw C D ∈ walls" "separated_by (the_wall_betw C D) C D"
using THE_defaultI'[OF 1] by auto
the_wall_betw_self_empty: "the_wall_betw C C = {}"
-
{
assume *: "the_wall_betw C C ≠ {}"
then obtain f g
where "(f,g)∈foldpairs" "the_wall_betw C C = {f⊨C,g⊨C}"
using the_wall_betw_nempty(1)[of C C]
by blast
with * have False
using the_wall_betw_nempty(2)[of C C] foldpairs_def
OpposedThinChamberComplexFoldings.halfchsys_decomp(2)[
of X
]
not_self_separated_by_disjoint[of "f⊨C" "g⊨C"]
by auto
}
thus ?thesis by fast
length_wall_crossings: "length (wall_crossings Cs) = length Cs - 1"
by (induct Cs rule: list_induct_CCons) auto
wall_crossings_snoc:
"wall_crossings (Cs@[D,E]) = wall_crossings (Cs@[D]) @ [the_wall_betw D E]"
by (induct Cs rule: list_induct_CCons) auto
wall_crossings_are_walls:
"H∈set (wall_crossings Cs) ==> H≠{} ==>
(induct Cs arbitrary: H rule: list_induct_CCons)
case (CCons B C Cs) thus ?case
using the_wall_betw_nempty(1)
by (cases "H∈set (wall_crossings (C#Cs))") auto
auto
in_set_wall_crossings_decomp:
"H∈set (wall_cro ∃As A B Bs. Cs = As@[A,B]@Bs ∧ H = the_wall_betw A B"
(induct Cs rule: list_induct_CCons)
case (CCons C D Ds)
show ?case
proof (cases "H ∈ set (wall_crossings (D#Ds))")
case True
with CCons(1) obtain As A B Bs
where "C#(D#Ds) = (C#As)@[A,B]@Bs" "H = the_wall_betw A B"
by fastforce
thus ?thesis by fast
next
case False
with CCons(2) have "C#(D#Ds) = []@[C,D]@Ds" "H = the_wall_betw C D"
by auto
thus ?thesis by fast
qed
auto
(* context ThinChamberComplex *)
OpposedThinChamberComplexFoldings
foldpair: "(f,g)∈foldpairs"
unfolding foldpairs_def
-
have "OpposedThinChamberComplexFoldings X f g C0" ..
thus "(f, g) ∈ {(f, g). ∃C. OpposedThinChamberComplexFoldings X f g C}"
by fast
separated_by_this_wall_fg:
"separated_by {f⊨C,g⊨C} C D ==> C∈f⊨C==> D∈g⊨C"
using separated_by_disjoint[
OF _ half_chamber_system_disjoint_union(2), of C D
]
by fast
separated_by_this_wall_gf =
OpposedThinChamberComplexFoldings.separated_by_this_wall_fg[
OF fg_symmetric
]
induced_automorphism_this_wall_vertex:
assumes "C∈
shows "s v = v"
-
from assms have "s v = g v"
using chamber_system_simplices induced_automorphism_on_simplices_fg
by auto
with assms(2,3) show "s v = v"
using chamber_system_simplices folding_g.retraction by auto
unique_wall:
assumes opp' : "OpposedThinChamberComplexFoldings X f' g' C'"
and chambers: "A∈f⊨C" "A∈f'⊨C" "B∈g\<ultimately
shows "{f⊨
-
from chambers have B: "B=g`A" "B=g'`A"
using adjacent_sym[of A B] adjacent_half_chamber_system_image_gf
OpposedThinChamberComplexFoldings.adjhalfchsys_image_gf[
OF opp'
]
by auto
with chambers(1,2,5)
have A : "OpposedThinChamberComplexFoldings X f g A"
and A': "OpposedThinChamberComplexFoldings X f' g' A"
using switch_basechamber[of A]
OpposedThinChamberComplexFoldings.switch_basechamber[
OF opp', of A
]
by auto
with B show ?thesis
using OpposedThinChamberComplexFoldings.unique_half_chamber_system_f[
OF A A'
]
OpposedThinChamberComplexFoldings.unique_half_chamber_system_g[
OF A A'
]
by auto
(* context OpposedThinChamberComplexFoldings *)
ThinChamberComplex
separated_by_wall_ex_foldpair:
assumes "H∈walls" "separated_by H C D"
shows "∃(f,g)∈foldpairs. H = {f⊨C,g⊨C} ∧ C∈f⊨C∧ D∈g⊨C"
-
from assms(1) obtain f g where fg: "(f,g)∈foldpairs" "H = {f⊨C,g⊨C}" by auto
show ?thesis
proof (cases "C∈f⊨C")
case True
moreover with fg assms(2) have "D∈g⊨C"
using foldpairs_def
OpposedThinChamberComplexFoldings.separated_by_this_wall_[
of X f g _ C D
]
by auto
ultimately show ?thesis using fg by auto
next
case False with assms(2) fg show ?thesis
using foldpairs_sym[of f g] separated_by_in_other[of "f⊨C" "g⊨C" C D] by auto
qed
not_separated_by_wall_ex_foldpair:
assumes chambers: "chamber C" "chamber D"
and wall : "H∈walls" "¬ separated_by H C D"
shows "∃(f,g)∈foldpairs. H = {f⊨C,g⊨
-
from wall(1) obtain f g where fg: "(f,g)∈foldpairs" "H = {f⊨C,g⊨C}" by auto
from fg(1) obtain A where A: "OpposedThinChamberComplexFoldings X f g A"
using foldpairs_def by fast
from chambers have chambers': "C∈f⊨C∨
using chamber_system_def
OpposedThinChamberComplexFoldings.halfchsys_decomp(1)[
OF A
]
by auto
show ?thesis
proof (cases "C∈f⊨C")
case True
moreover with chambers'(2) fg(2) wall(2) have "D∈f⊨C"
unfolding separated_by_def by auto
ultimately show ?thesis using fg by auto
next
case False
with chambers'(1) have "C∈g⊨C" by simp
moreover with chambers'(2) fg(2) wall(2) have "D∈( bya
using insert_commute[of "f⊨C" "g⊨C" "{}"] unfolding separated_by_def by auto
ultimately show ?thesis using fg foldpairs_sym[of f g] by auto
qed
adj_wall_imp_ex1_wall:
assumes adj : "C∼D"
and wall: "H0∈walls" "separated_by H0 C D"
shows "∃!H∈walls. separated_by H C D"
(rule ex1 (ruleex1I, ru conjI rulewall(1), rule wall(2))
fix H assume H: "H∈walls ∧ separated_by H C D"
from this obtain f g
where fg: "(f,g)∈foldpairs" "H={f⊨C,g⊨C}" "C∈f⊨C" "D∈g⊨C"
using separated_by_wall_ex_foldpair[of H C D]
by auto
from wall obtain f0 g0
where f0g0: "(f0,g0)∈foldpairs" "H0={f0⊨C,g0⊨C}" "C∈f0⊨C" "D∈g0⊨C"
using separated_by_wall_ex_foldpair[of H0 C D]
by auto
from fg(1) f0g0(1) obtain A A0
where A : "OpposedThinChamberComplexFoldings X f g A"
and A0: "OpposedThinChamberComplexFoldings X f0 g0 A0"
using foldpairs_def
by auto
from fg(2-4) f0g0(2-4) adj show "H = H0"
using O.unique_wall[OFA0 A] y auto
(* context ThinChamberComplex *)
OpposedThinChamberComplexFoldings
this_wall_betwI:
assumes "C∈f⊨
shows "the_wall_betw C D = {f⊨C,g⊨C}"
(rule THE_default1_equality, rule adj_wall_imp_ex1_wall)
have "OpposedThinChamberComplexFold
thus "{f⊨C,g⊨C}∈walls" using foldpairs_def by auto
moreover from assms(1,2) show "separated_by {f⊨>f x = ›
by (auto intro: separated_byI)
ultimately show "{f⊨C,g⊨C}∈walls ∧ separated_by {f⊨C,g⊨C} C D" by simp
(rule assms(3))
this_wall_betw_basechambers:
"the_wall_betw C0 D0 = {f⊨C,g⊨C}"
using basechambers_half_chamber_systems chambers(2) this_wall_betwI by auto
this_wall_in_crossingsI_fg:
defines H: "H ≡(Suc x)"
assumes D: "D∈g⊨C"
shows "C∈f⊨C==> gallery (C#Cs@[D]) ==> H ∈ set (wall_crossings (C#Cs@[D]))"
(induct Cs arbitrary: C)
case Nil
from Nil(1) assms have "H∈walls" "separated_by H C D"
using foldpair by (auto intro: separated_byI)
thus ?case
using galleryD_adj[OF Nil(2)]
THE_default1_equality[OF adj_wall_imp_ex1_wall]
by auto
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 4
case (Cons B Bs)
show ?case
proof (cases "B∈f⊨C")
case True with Cons(1,3) show ?thesis using gallery_Cons_reduce by simp
next
case False
with Cons(2,3) H have "H∈walls" "separated_by H C B"
using galleryD_chamber[OF Cons(3)] chamber_in_other_half_fg[of B] foldpair
by (auto intro: separated_byI)
thus ?thesis
using galleryD_adj[OF Cons(3)]
THE_default1_equality[OF adj_wall_imp_ex1_wall]
by auto
qed
(* context OpposedThinChamberComplexFoldings *)
(in ThinChamberComplex) walls_betw_subset_wall_crossings:
assumes "gallery (C#Cs@[D])"
shows "walls_betw C D ⊆ set (wall_crossings (C#Cs@[D]))"
fix H assume "H ∈ walls_betw C D"
hence H: "H∈walls" "separated_by H C D" using walls_betw_def by auto
from this obtain f g
where fg: "(f,g)∈foldpairs" "H = {f⊨C,g⊨C}" "C∈f⊨C" "D∈g⊨C"
using separated_by_wall_ex_foldpair[of H C D]
by auto
from fg(1) obtain Z where Z: "OpposedThinChamberComplexFoldings X f g Z"
using foldpairs_def by fast
from assms H(2) fg(2-4) show "H ∈ set (wall_crossings (C#Cs@[D]))"
using OpposedThinChamberComplexFoldings.this_wall_in_crossingsI_fg[OF Z]
by auto
OpposedThinChamberComplexFoldings
qed
"gallery (C#Cs@[D]) ==> C∈f⊨C==> D∈f⊨C==>
{f⊨C,g⊨C}∈set (wall_crossings (C#Cs@[D])) ==> ¬ distinct (wall_crossings (C#Cs@[D]))"
(induct Cs arbitrary: C)
case Nil
hence "{f⊨C,g⊨C} = the_wall_betw C D" by simp
moreover hence "the_wall_betw C D ≠ {}" by fast
ultimately show ?case
using Nil(2,3) the_wall_betw_nempty(2) separated_by_this_wall_fg[of C D]
half_chamber_system_disjoint_union(2)
by auto
case (Cons E Es)
show ?case
proof
assume 1: "distinct (wall_crossings (C # (E # Es) @ [D]))"
show False
proof (
cases "E∈
rule: two_cases
)
case both with Cons(1,2,4) 1 show False
using gallery_Cons_reduce by simp
next
case one
from one(2) Cons(5) have "{f⊨C,g⊨C} = the_wall_betw C E" by simp
moreover hence "the_wall_betw C E ≠ {}" by fast
ultimately show False
using Cons(3) one(1) the_wall_betw_nempty(2)
separated_by_this_wall_fg[of C E]
half_chamber_system_disjoint_union(2)
by auto
next
case other with Cons(3) show False
using 1 galleryD_chamber[OF Cons(2)] galleryD_adj[OF Cons(2)]
chamber_in_other_half_fg this_wall_betwI
by force
next
case neither
from Cons(2) neither(1) have "E\<sing
using galleryD_chamber chamber_in_other_half_fg by auto
with Cons(4) have "separated_by {g⊨C,f⊨C} E D"
by (blast intro: separated_byI)
hence "{f⊨C,g⊨C} ∈ walls_betw E D"
using foldpair walls_betw_def by (auto simp add: insert_commute)
with neither(2) show False
using gallery_Cons_reduce[OF Cons(2)] walls_betw_subset_wall_crossings
by auto
qed
qed
wall_crossings_min_gallery_betwI:
assumes "gallery (C#Cs@[D])"
"distinct (wall_crossings (C#Cs@[D]))"
"∀H∈set (wall_crossings (C#Cs@[D])). separated_by H C D"
shows "min_gallery (C#Cs@[D])"
(rule min_galleryI_betw)
obtain B Bs where BBs: "Cs@[D] = B#Bs" using snoc_conv_cons by fast
define H where "H = the_wall_betw C B"
with BBs assms(3) have 1: "separated_by H C D" by simp
show "C≠D"
proof (cases "H={}")
case True thus ?thesis
using 1 unfolding separated_by_def by simp
next
case False
with H_def have "H ∈ walls" using the_wall_betw_nempty(1) by simp
from this obtain f g
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
using 1 separated_by_wall_ex_foldpair[of H C D]
by auto
thus ?thesis
using foldpairs_def
.(2)[
of X f g
]
by auto
qed
fix Ds assume Ds: "gallery (C # Ds @ [D])"
have "Suc (length Cs) = card (walls_betw C D)"
proof-
from assms(1,3) have "set (wall_crossings (C#Cs@[D])) = walls_betw C D"
using separated_by_not_empty wall_crossings_are_walls[of _ "C#Cs@[D]"]
walls_betw_def
walls_betw_subset_wall_crossings[OF assms(1)]
unfolding separated_by_def
by auto
with assms(2) show ?thesis
using distinct_card[THEN sym] length_wall_crossings by fastforce
qed
moreover have "card (walls_betw C D) ≤ {..l xs - 1"
proof-
from Ds have "card (walls_betw C D) ≤ card (set (wall_crossings (C#Ds@[D])))"
using walls_betw_subset_wall_crossings finite_set card_mono by force
also have "…≤ length (wall_crossings (C#Ds@[D]))"
using card_length by auto
finally show ?thesis using length_wall_crossings by simp
qed
ultimately show "length Cs ≤ length Ds" by simp
(rule assms(1))
ex_nonseparating_wall_imp_wall_crossings_not_distinct:
assumes gal : "gallery (C#Cs@[D])"
and wall: "H∈set (wall_crossings (C#Cs@[D]))" "H≠
"¬ separated_by H C D"
shows "¬ distinct (wall_crossings (C#Cs@[D]))"
-
from assms obtain f g
java.lang.NullPointerException
using wall_crossings_are_walls[of H]
not_separated_by_wall_ex_foldpair[of C D H]
galleryD_chamber
by auto
from fg(1) obtain Z where Z: "OpposedThinChamberComplexFoldings X f g Z"
using foldpairs_def by fast
from wall fg(2-4) show ?thesis
using OpposedThinChamberComplexFoldings.sside_wcrossings_ndistinct_f [
OF Z gal
]
by blast
not_min_gallery_double_crosses_wall:
assumes "gallery Cs" "¬ min_gallery Cs" "{} ∉ set (wall_crossings Cs)"
shows "¬ distinct (wall_crossings Cs)"
(cases Cs rule: list_cases_Cons_snoc)
case Nil with assms(2) show ?thesis by simp
case Single with assms(1,2) show ?thesis using galleryD_chamber by simp
case (Cons_snoc B Bs C)
show ?thesis
proof (cases "B=C")
case True show ?thesis
proof (cases Bs)
case Nil with True Cons_snoc assms(3) show ?thesis
using the_wall_betw_self_empty by simp
next
case (Cons E Es)
define H where "H = the_wall_betw B E"
with Cons have *: "H ∈ set (wall_crossings (B#Bs@[C]))" by simp
moreover from assms(3) Cons_snoc * have "H ≠ {}" by fast
ultimately show ?thesis
using assms(1) Cons_snoc Cons True H_def
the_wall_betw_nempty(1)[of B E] not_self_separated_by_wall[of H B]
ex_nonseparating_wall_imp_wall_crossings_not_distinct[of B Bs C H]
by fast
qed
next
case False
with assms Cons_snoc
have 1: "¬ distinct (wall_crossings Cs) ∨ ¬ (∀H∈set (wall_crossings Cs). separated_by H B C)"
using wall_crossings_min_gallery_betwI
by force
moreover {
assume "¬ (∀H∈set (wall_crossings Cs). separated_by H B C)"
from this obtain H
where H: "H∈set (wall_crossings Cs)" "¬ separated_by H B C"
by auto
moreover from H(1) assms(3) have "H≠{}" by fast
ultimately have ?thesis
using assms(1) Cons_snoc
ex_nonseparating_wall_imp_wall_crossings_not_distinct
by simp
}
ultimately show ?thesis by fast
qed
not_distinct_crossings_split_gallery:
"[ gallery Cs; {} ∉ set (wall_crossings Cs); ¬ distinct (wall_crossings Cs) ]==> ∃f g As A B Bs E F Fs.
java.lang.NullPointerException
( Cs = As@[A,B,F]@Fs ∨ Cs = As@[A,B]@Bs@[E,F]@Fs )"
(induct Cs rule: list_induct_CCons)
show ?case
proof (cases "distinct (wall_crossings (J#Js))")
case False
moreover from CCons(2) have "gallery (J#Js)"
using gallery_Cons_reduce by simp
moreover from CCons(3) have "{} ∉ set (wall_crossings (J#Js))" by simp
ultimately obtain f g As A B Bs E F Fs where split:
"(f,g)∈foldpairs" "A∈f⊨C" "B∈g⊨C" "E∈ (metis as_list(1)image_i se
"J#Js = As@[A,B,F]@Fs ∨ J#Js = As@[A,B]@Bs@[E,F]@Fs"
using CCons(1)
by blast
from split(6)
have "C#J#Js = (C#As)@[A,B,F]@Fs ∨
C#JJs = (C#As)@[A,B]@@Bs@@[E,F]@s"
by simp
with split(1-5) show ?thesis by blast
next
case True
define H where "H = the_wall_betw C J"
with True CCons(4) have "H∈set (wall_crossings (J#Js))" by simp
from this obtain Bs E F Fs
where split1: "J#Js = Bs@[E,F]@Fs" "H = the_wall_betw E F"
using in_set_wall_crossings_decomp
by fast
from H_def split1(2) CCons(3)
have Hwall: "H ∈ walls" "separated_by H C J" "separated_by H E F"
using the_wall_betw_nempty[of C J] the_wall_betw_nempty[of E F]
by auto
from Hwall(1,2) obtain f g
where fg: "(f,g)∈foldpairs" "H={f⊨C,g⊨C}" "C∈f⊨C" "J∈g⊨C"
using separated_by_wall_ex_foldpair[of H C J]
by auto
from fg(1) obtain Z
where Z: "OpposedThinChamberComplexFoldings X f g Z"
using foldpairs_def
by fast
show ?thesis
proof (cases Bs)
case Nil
with CCons(2) Hwall(2,3) fg(2-4) split1(1)
have "F∈f⊨Clambda>x.. as_list_help (p x)) X)) = p ` X
using galleryD_chamber
OpposedThinChamberComplexFoldings.sepwall_chain3_fg(2)[
OF Z, of C J F
]
by auto
with fg(1,3,4) show ?thesis by blast
next
case (Cons K Ks) have Bs: "Bs = K#Ks" by fact
proof (cases "E∈f⊨C")
case True
from CCons(2) split1(1) Bs have "gallery (J#Ks@[E])"
using gallery_Cons_reduce[of C "J#Ks@E#F#Fs"]
gallery_append_reduce1[of "J#Ks@[E]" "F#Fs"]
"> = p X"
with fg(4) True obtain Ls L M Ms
where LsLMMs: "L∈g⊨C" "M∈f⊨C" "J#Ks@[E] = Ls@L#M#Ms"
using OpposedThinChamberComplexFoldings.split_gallery_gf[
OF Z, of J E Ks
]
by blast
show ?thesis
proof (cases Ls)
case Nil
with split1(1) Bs LsLMMs(3)
have "C#J#Js = []@[C,J,M]@(Ms@F#F
by simp
with fg(1,3,4) LsLMMs(2) show ?thesis by blast
next
case (Cons N Ns)
with split1(1) Bs LsLMMs(3)
have "C#J#Js = []@[C,J]@Ns@[L,M]@(Ms@F#Fs)"
by simp
with fg(1,3,4) LsLMMs(1,2) show ?thesis by blast
qed
next
case False
with Hwall(2,3) fg(2) split1(1) Cons
have "E∈g⊨C" "F∈f⊨C" "C#J#Js = []@[C,J]@Ks@[E,F]@Fs"
using OpposedThinChamberComplexFoldings.separated_by_this_wall_fg[
OF Z
]
separated_by_in_other[of "f⊨C" "g⊨C"]
by auto
with fg(1,3,4) show ?thesis by blast
qed
qed
qed
auto
not_min_gallery_double_split:
"[ gallery Cs; ¬ min_gallery Cs; {} ∉ set (wall_crossings Cs) ]🚫 ∃f g As A B Bs E F Fs.
(f,g)∈foldpairs ∧ A∈f⊨
( Cs = As@[A,B,F]@Fs ∨ Cs = As@[A,B]@Bs@[E,F]@Fs )"
using not_min_galnot_distinct_crossings_split
by simp
(* context ThinChamberComplex *)
‹Thin chamber complexes with many foldings›
‹
Here we begin to examine thin chamber complexes in which every pair of adjacent chambers affords a
pair of opposed foldings of the complex. This condition will ultimately be shown to be sufficient
to ensure that a thin chamber complex is isomorphic to some Coxeter complex. ›
‹Locale definition and basic facts›
y (metis lis.set_mp)
for X :: "'a set set"
fixes C0 :: "'a set"
assumes fundchamber: "chamber C0"
and ex_walls :
"[ chamber C; chamber D; C∼D; C≠D ]==> ∃f g. OpposedThinChamberComplexFoldings X f g C ∧ D=g`C"
(in ThinChamberComplex) ThinChamberComplexManyFoldingsI:
assumes "chamber C0"
and "∧C D. [ chamber C; chamber D; C∼D; C≠D ]==> ∃f g. OpposedThinChamberComplexFoldings X f g C ∧ D=g`C"
shows "ThinChamberComplexManyFoldings X C0"
using assms
by (intro_locales, unfold_locales, fast)
(in ThinChamberComplexManyFoldings) wall_crossings_subset_walls_betw:
assumes "min_galery (C#C#Cs@[D])"
shows "set (wall_crossings (C#Cs@[D])) ⊆ walls_betw C D"
fix H assume "H ∈ set (wall_crossings (C#Cs@[D]))"
from this obtain As A B Bs
where H: "C#Cs@[D] = As@[A,B]@Bs" "H=the_wall_betw A B"
using in_set_wall_crossings_decomp
by blast
from assms have pgal: "pgallery (C#Cs@[D])"
using min_gallery_pgallery by fast
with H(1) obtain f g
where fg: "OpposedThinChamberComplexFoldings X f g A" "B=g`A"
using pgalleryD_chamber pgalleryD_adj
binrelchain_append_reduce2[of adjacent As "[A,B]@Bs"]
pgalleryD_distinct[of "As@[A,B]@Bs"] ex_walls[of A B]
by auto
from H(2) fg have H': "A∈
using OpposedThinChamberComplexFoldings.basech_halfchsys[
OF fg(1)
then have "?P !i \noteq !j"
OpposedThinChamberComplexFoldings.chambers(2)[OF fg(1)]
OpposedThinChamberComplexFoldings.this_wall_betwI[OF fg(1)]
foldpairs_def
by auto
have CD: "C ∈ f⊨C∪ g⊨C" "D ∈ f⊨C∪ g⊨C"
using pgal pgalleryD_chamber chamber_system_def
OpposedThinChamberComplexFoldings.halfchsys_decomp(1)[
OF fg(1)
]
by auto
show "H ∈ walls_betw C D"
proof (cases Bs As rule: two_lists_cases_snoc_Cons')
case both_Nil with H show ?thesis
using H'(3) the_wall_betw_nempty[of A B] unfolding walls_betw_def by force
next
case (Nil1 E Es)
show ?thesis
<f<
case True
with Nil1 H(1) have "separated_by H C D"
using H'(2,3) by (auto intro: separated_byI)
thus ?thesis using H'(4) unfolding walls_betw_def by simp
next
case False with assms Nil1 H(1) show ?thesis
using OpposedThinChamberComplexFoldings.foldg[
OF fg(1)
]
CD(1) H'(1,2) pgal pgallery
OpposedThinChamberComplexFoldings.flopped_half_chamber_systems_gf[
OF fg(1)
]
ThinChamberComplexFolding.gallery_double_cross_not_minimal1[
of X g E A B Es "[]"
]
by force
qed
next
case (Nil2 Fs F)
show ?thesis
proof (cases "D∈f⊨C (p x
case True
with assms Nil2 H(1) show ?thesis
using OpposedThinChamberComplexFoldings.foldf[
OF fg(1)
]
H'(1,2) pgal pgallery
OpposedThinChamberComplexFoldings.flopped_half_chamber_systems_fg[
OF fg(1)
]
ThinChamberComplexFolding.gallery_double_cross_not_minimal_Cons1[
of X f
]
by force
next
case False with Nil2 H(1) have "separated_by H C D"
using CD(2) H'(1,3) by (auto intro: separated_byI)
thus ?thesis using H'(4) unfolding walls_betw_def by simp
qed
next
case (snoc_Cons Fs F E Es) show ?thesis
proof (cases "C∈ultimatelyhave "p xi \noteq"
case both thus ?thesis
using H'(3,4) walls_betw_def unfolding separated_by_def by auto
next
case one
with snoc_Cons assms H(1) show ?thesis
using OpposedThinChamberComplexFoldings.foldf[
OF fg(1)
]
CD(2) H'(2) pgal pgallery
OpposedThinChamberComplexFoldings.flopped_half_chamber_systems_fg[
OF fg(1)
]
ThinChamberComplexFolding.gallery_double_cross_not_minimal1[
of X f C B D "Es@[A]"
]
by fastforce
next
case other
with snoc_Cons assms H(1) show ?thesis
using OpposedThinChamberComplexFoldings.ThinChamberComplexFolding_g[
OF fg(1)
]
CD(1) H'(1) pgal pgallery
OpposedThinChamberComplexFoldings.flopped_half_chamber_systems_gf[
OF fg(1)
]
ThinChamberComplexFolding.gallery_double_cross_not_minimal1[
of X g E A F Es "B#Fs"
]
by force
next
case neither
hence "separated_by {g⊨C,f⊨C} C D" using CD by (auto intro: separated_byI)
thus ?thesis
using H'(3,4) walls_betw_def by (auto simp add: insert_commute)
qed
qed
‹The group of automorphisms›
‹
Recall that a pair of opposed foldings of a thin chamber complex can be stitched together to form
an automorphism of the complex. Choosing an arbitrary chamber in the complex to act as a sort of
centre of the complex (referred to as the fundamental chamber), we consider the group (under
composition) generated by the automorphisms afforded by the chambers adjacent to the fundamental
chamber via the pairs of opposed foldings that we have assumed to exist. ›
ThinChamberComplexManyFoldings
fundfoldpairs :: "(('a==>'a)×('a==>'a)) set"
where "fundfoldpairs ≡ {(f,g). OpposedThinChamberComplexFoldings X f g C0}"
"fundadjset ≡p1 a as (4
induced_automorph :: "('a==>'a) ==> ('a==>'a) ==> ('a==> imag)
where "induced_automorph f g ≡
OpposedThinChamberComplexFoldings.induced_automorphism X f g"
Abs_induced_automorph :: "('a==>'a) ==> ('a==>'a) ==> 'a permutation"
"S ≡∪(f,g)∈fundfoldpairs. {Abs_induced_automorph f g}"
"W ≡
fundfoldpairs_induced_autormorph_bij:
"(f,g) ∈ fundfoldpairs ==> bij (induced_automorph f g)"
using OpposedThinChamberComplexFoldings.induced_automorphism_bij
unfolding fundfoldpairs_def
by fast
permutation_conv_induced_automorph =
Abs_permutation_inverse[OF CollectI, OF fundfoldpairs_induced_autormorph_bij]
fundfoldpairs_induced_autormorph_order2:
"(f,g) ∈ fundfoldpairs ==> induced_automorph f g ∘ induced_automorph f g = id"
using OpposedThinChamberComplexFoldings.indaut_order2
unfolding fundfoldpairs_def
by fast
fundfoldpairs_induced_autormorph_ntrivial:
"(f,g) ∈
using OpposedThinChamberComplexFoldings.induced_automorphism_ntrivial
unfolding fundfoldpairs_def
by fast
fundfoldpairs_fundchamber_image:
"(f,g)∈fundfoldpairs ==> Abs_induced_automorph f g `→ C0 = g`C0"
using fundfoldpairs_def
by (simp add:
permutation_conv_induced_automorph
OpposedThinChamberComplexFoldings.induced_automorphism_C0
)
fundfoldpair_fundchamber_in_half_chamber_system_f:
"(f,g)∈fundfoldpairs ==> C0∈f⊨C"
using fundfoldpairs_def
OpposedThinChamberComplexFoldings.basech_halfchsys(1)
by fast
fundfoldpair_unique_half_chamber_system_f:
assumes "(f,g)∈fundfoldpairs" "(f',g')∈fundfoldpairs"
"Abs_induced_automorph f' g' = Abs_induced_automorph f g"
shows "f'⊨C = f⊨C"
-
from assms have "g'`C0 = g`C0"
using fundfoldpairs_fundchamber_image[OF assms(1)]
fundfoldpairs_fundchamber_image[
by simp
with assms show "f'⊨C = f⊨ image :
using fundfoldpairs_def
OpposedThinChamberComplexFoldings.unique_half_chamber_system_f[
of X f g C
]
by auto
fundfoldpair_unique_half_chamber_systems_chamber_ng_f:
assumes "(f,g)∈fundfoldpairs" "(f',g')∈fundfoldpairs"
"Abs_induced_automorph f' g' = Abs_induced_automorph f g"
"chamber C" "C∉g⊨C"
shows "C∈f'⊨C"
using assms(1,3-5) fundfoldpairs_def chamber_system_def
OpposedThinChamberComplexFoldings.flopped_half_chamber_systems_gf[
THEN sym
]
fundfoldpair_unique_half_chamber_system_f[OF assms(1,2)]
by fastforce
the_wall_betw_adj_fundchamber:
"(f,g)∈fundfoldpairs ==>
the_wall_betw C0 (Abs_induced_automorph f g `→ X)
using fundfoldpairs_def
OpposedThinChamberComplexFoldings.this_wall_betw_basechambers
OpposedThinChamberComplexFoldings.induced_automorphism_C0
by (fastforce simp add: permutation_conv_induced_automor case empty
zero_notin_S: "0∉S"
assume "0∈S"
from this obtain f g
where "(f,g)∈fundfoldpairs" "0 = Abs_induced_automorph f g"
by fast
thus F
using Abs_permutation_inject[of id "induced_automorph f g"]
bij_id fundfoldpairs_induced_autormorph_bij
fundfoldpairs_induced_autormorph_ntrivial
by (force simp add: zero_permutation.abs_eq)
S_order2_add: "s∈S ==> s + s = 0"
using fundfoldpairs_induced_autormorph_bij zero_permutation.abs_eq
by (fastforce simp add:
plus_permutation_abs_eq fundfoldpairs_induced_autormorph_order2
)
S_add_order2:
assumes "s∈S"
"add_ord s = 22"
(rule add_order_equality)
from assms show "s+^2 = 0" using S_order2_add by (simp add: nataction_2)
fix m assume "0 < m" "s+^m = 0"
with assms show "2 ≤ m" using zero_notin_S by (cases "m=1") auto
simp
S_endomorphism:
"s∈S ==> ChamberComplexEndomorphism X (permutation s)"
using fundfoldpairs_def
OpposedThinChamberComplexFoldings.induced_automorphism_morphism
by (fastforce simp add: permutation_conv_induced_automorph)
S_list_endomorphism:
"ss∈lists S ==> ChamberComplexEndomorphism X (permutation (sum_list ss))"
by (induct ss)
(auto simp add:
zero_permutation.rep_eq trivial_endomorphism plus_permutation.rep_eq
S_endomorphism ChamberComplexEndomorphism.endo_comp
)
W_endomorphism:
"w∈W ==> ChamberComplexEndomorphism X (permutation w)"
using W_conv_sum_lists S_list_endomorphism by auto
S_automorphism:
"s∈S ==> ChamberComplexAutomorphism X (permutation s)"
using fundfoldpairs_def
OpposedThinChamberComplexFoldings.induced_automorphism_automorphism
by (fastforce simp add: permutation_conv_induced_automorph)
S_list_automorphism:
"ss∈lists S ==> ChamberComplexAutomorphism X (permutation (sum_list ss))"
(indu ss
(auto simp add:
zero_permutation.rep_eq trivial_automorphism plus_permutation.rep_eq
S_automorphism ChamberComplexAutomorphism.comp
)
W_automorphism:
"w∈W ==> ChamberComplexAutomorphism X (permutation w)"
using W_conv_sum_lists S_list_automorphism by auto
S_respects_labels: "[ label_wrt B φ; s∈S; v∈(∪X) ]==> φ (s → v) = φ v"
using fundfoldpairs_def
OpposedThinChamberComplexFoldings.indaut_resplabels[
of X _ _ C0 B φ v
]
by (auto simp add: permutation_conv_induced_automorph)
S_list_respects_labels:
"[ label_wrt B φ; ss∈lists S; v∈(∪X) ]==> φ (sum_list ss → v) = φ v"
using S_endomorphism ChamberComplexEndomorphism.vertex_map[of X]
by (induct ss arbitrary: v rule: rev_induct)
(auto simp add:
plus_permutation.rep_eq S_respects_labels zero_permutation.rep_eq
)
W_respects_labels:
"[ label_wrt B φ; w∈W; v∈(∪X) ]==> φ (w→v) = φ v"
usingu le_Suc by pr
(* context ThinChamberComplexManyFoldings *)
‹Action of the group of automorphisms on the chamber system›
‹
Now we examine the action of the group @{term W} on the chamber system. In particular, we show
that t by auto auto ›
ThinChamberComplexManyFoldings
fundchamber_S_chamber: "s∈S ==> chamber (s`→C0)"
using fundfoldpairs_def
by (fastforce simp add:
fundfoldpairs_fundchamber_image
OpposedThinChamberComplexFoldings.chamber_D0
)
fundchamber_W_image_chamber:
"w∈W ==> chamber (w`→C0)"
using fundchamber W_endomorphism
ChamberComplexEndomorphism.chamber_map
by auto
fundchamber_S_adjacent: "s∈S ==> C0 ∼\Longrightarrow> } <>\
using fundfoldpairs_def
by (auto simp add:
fundfoldpairs_fundchamber_image
OpposedThinChamberComplexFoldings.chambers(2)
)
fundchamber_WS_image_adjacent:
"w∈W ==> s∈S ==> (w`→C0) ∼ ((w+s)`→C0)"
using fundchamber fundchamber_S_adjacent fundchamber_S_chamber
W_endomorphism
ChamberComplexEndomorphism.adj_map[of X "permutation w" C0 "s`→C0"]
by (auto simp add: image_comp plus_permutation.rep_eq)
fundchamber_S_image_neq_fundchamber: "s∈S ==> s`→C0 ≠ C0"
using fundfoldpairs_def OpposedThinChamberComplexFoldings.chambers(3)
by (fastforce simp add: fundfoldpairs_fundchamber_image)
assume "(w+s) `→ C0 = w `→ C0"
with assms show False
using fundchamber_S_image_neq_fundchamber[of s]
by (auto simp add: plus_permutation.rep_eq image_comp permutation_eq_image)
fundchamber_S_fundadjset: "s∈S ==> s`→C0 ∈ fundadjset"
using fundchamber_S_adjacent fundchamber_S_image_neq_fundchamber
fundchamber_S_chamber chamberD_simplex adjacentset_def
by simp
fundadjset_eq_S_image: "D∈fundadjset ==>∃s∈S. D = s`→C0"
using fundchamber adjacentsetD_adj adjacentset_chamber ex_walls[of C0 D]
fundfoldpairs_def fundfoldpairs_fundchamber_image
by blast
S_fixespointwise_fundchamber_image_int:
assumes "s∈S"
shows "fixespointwise ((→) s) (C0∩s`→C0)"
-
from assms(1) obtain f g
where fg: "(f,g)∈
by fast
show ?thesis
proof (rule fixespointwise_cong)
from fg show "fun_eq_on ((→
using permutation_conv_induced_automorph fun_eq_onI by fastforce
from fg show "fixespointwise (induced_automorph f g) (C0∩s`→C0)"
using fundfoldpairs_fundchamber_image fundfoldpairs_def
OpposedThinChamberComplexFoldings.indaut_fixes_fundfacet
by auto
qed
S_fixes_fundchamber_image_int:
"s∈S ==> s`→(C0∩s`→C0) = C0∩s`→C0"
using fixespointwise_im[OF S_fixespointwise_fundchamber_image_int] by simp
fundfacets:
assumes "s∈S"
shows "C0∩s`→C0 ⊲ C0" "C0∩s`→C0 ⊲ s`→C0"
using assms fundchamber_S_adjacent[of s]
fundchamber_S_image_neq_fundchamber[of s]
adjacent_int_facet1[of C0] adjacent_int_facet2[of C0]
by auto
fundadjset_ex1_eq_S_image:
assumes "D∈fundadjset"
shows "∃!s∈S. D = s`→C0"
(rule ex_ex1I)
from assms show "∃s. s∈S ∧ D = s `→ C0"
using fundadjset_eq_S_image by fast
fix s t assume "s∈S ∧ D = s`→C0" "t∈S ∧ D = t`→C0"
hence s: "s∈S" "D = s`→C0"
and t: "t∈
by auto
from s(1) t(1) obtain f g f' g'
(f,g)\in" "s Abs_ind f g"
and "(f',g')∈fundfoldpairs" "t = Abs_induced_automorph f' g'"
by auto
with s(2) t(2) using a(2)
using fundfoldpairs_def fundfoldpairs_fundchamber_image
OpposedThinChamberComplexFoldings.induced_automorphism_unique[
of X f' g' C0 f g
]
by auto
fundchamber_S_image_inj_on: "inj_on (λs. s`→C0) S"
(rule inj_onI)
fix s t assume "s∈S" "t∈S" "s`\ (1 by si
using fundchamber_S_fundadjset
bex1_equality[OF fundadjset_ex1_eq_S_image, of "s`→C0" s t]
by simp
S_list_image_gallery:
"ss∈lists S ==> gallery (map (λw. w`→C0) (sums ss))"
(induct ss rule: list_induct_s using fin by auto
case (Single s) thus ?case
using fundchamber fundchamber_S_chamber fundchamber_S_adjacent
gallery_def
by (fastforce simp add: zero_permutation.rep_eq)
case (ssnoc ss s t)
define Cs D E where "Cs = map (λw. w `→ C0) (sums ss)"
and "D = sum_list (ss@[s]) `→ C0"
and "E = sum_list (ss@[s,t]) `→ ` UNI" and "\And'. x' ∈
with ssnoc have "gallery (Cs@[D,E])"
using sum_list_S_in_W[of "ss@[s,t]"] sum_list_S_in_W[of "ss@[s]"]
fundchamber_W_image_chamber
fundchamber_WS_image_adjacent[of "sum_list (ss@[s])" t]
sum_list_append[of "ss@[s]" "[t]"]
by (auto intro: gallery_snocI simp add: sums_snoc)
with Cs_def D_def E_def show ?case using sums_snoc[of "ss@[s]" t] by (simp add: sums_snoc)
(auto simp add: gallery_def fundchamber zero_permutation.rep_eq)
pgallery_last_eq_W_image:
"pgallery (C0#Cs@[C]) ==>∃w∈W. C = w`→C0"
(induct Cs arbitrary: C rule: rev_induct)
case Nil
hence "C∈fundadjset"
pgal chamberD_s adjac by fastfo
from this obtain s where "s∈S" "C = s`→C0"
using fundadjset_eq_S_image[of C] by auto
thus ?case using genby_genset_closed[of s S] by fast
case (snoc D Ds)
have DC: "chamber D" "chamber C" "D∼C" "D≠C"
using pgallery_def snoc(2)
binrelchain_append_reduce2[of adjacent "C0#Ds" "[D,C]"]
by auto
from snoc obtain w where w: "w∈W" "D = w`→ UUNIV›
using pgallery_append_reduce1[of "C0#Ds@[D]" "[C]"] by force
from w(2) have "(-w)`→D = C0"
by (simp add:
image_comp plus_permutation.rep_eq[THEN sym]
zero_permutation.rep_eq
)
with DC w(1) have "C0 ∼ (-w)`→C" "C0 ≠ (-w)`→C" "(-w)`→C ∈ X"
using genby_uminus_closed W_endomorphism[of "-w"]
ChamberComplexEndomorphism.adj_map[of X _ D C]
permutation_eq_image[of "-w" D] chamberD_simplex[of C]
ChamberComplexEndomorphism.simplex_map[of X "permutation (-w)" C]
by auto
hence "(-w)`→C ∈ fundadjset" using adjacentset_def by fast
from this obtain s where s: "s∈S" "(-w)`→C = s`→C0"
using fundadjset_eq_S_image by force
from s(2) have
"per w ∘
by (simp add: image_comp[THEN sym])
hence "C = (w+s)`→C0"
by (simp add: plus_permutation.rep_eq[THEN sym] zero_permutation.rep_eq)
with w(1) s(1) show ?case
using genby_genset_closed[of s S] genby_add_closed by blast
by (mesonassms() ass(2)infinite le0)
chamber_eq_W_image:
assumes "chamber C"
shows "∃w∈W. C = w`→C0"
(cases "C=C0")
case True
hence "0∈
using genby_0_closed by (auto simp add: zero_permutation.rep_eq)
thus ?thesis by fast
case False with assms show ?thesis
using fundchamber chamber_pconnect pgallery_last_eq_W_image by blast
S_list_image_crosses_walls:
"ss ∈ lists S ==> {} ∉ set (wall_crossings (map (λw. w`→C0) (sums ss)))"
(induct ss rule: list_induct_ssnoc)
case (Single s) thus ?case
using fundchamber fundchamber_S_chamber fundchamber_S_adjacent
fundchamber_S_image_neq_fundchamber[of s]qed
OpposedThinChamberComplexFoldings.this_wall_betw_basechambers
by (force simp add: zero_permutation.rep_eq)
case (ssnoc ss s t)
moreover
by
moreover from ssnoc(2) A_def B_def obtain f g
where "OpposedThinChamberComplexFoldings X f g A" "B=g`A"
using sum_list_S_in_W[of "ss@[s]"] sum_list_S_in_W[of "ss@[s,t]"]
fundchamber_W_image_chamber sum_list_append[of "ss@[s]" "[t]"]
fundchamber_next_WS_image_neq[of t "sum_list (ss@[s])"]
fundchamber_WS_image_adjacent[of "sum_list (ss@[s])" t]
ex_walls[of A B]
finite :
ultimately show ?case
using OpposedThinChamberComplexFoldings.this_wall_betw_basechambers
sums_snoc[of "ss@[s]" t]
by (force simp add: sums_snoc wall_crossings_snoc)
(simp add: zero_permutation.rep_eq)
(* context ThinChamberComplexManyFoldings *)
‹A labelling by the vertices of the fundamental chamber›
Here we show that by repeatedly applying the composition of all the elements in the collection
@{term S} of fundamental automorphisms, we can retract the entire chamber complex onto the
fundamental chamber. This retraction provides a means of labelling the chamber complex, using the
vertices of the fundamental chamber as labels. ›
ThinChamberComplexManyFoldings
Spair :: "'a permutation ==> ('a==>'a)×('a==>'a)"
where "Spair s ≡
SOME fg. fg ∈ fundfoldpairs ∧ s = case_prod Abs_induced_automorph fg"
Spair_fundfoldpair: "s∈S ==> Spair s ∈ fundfoldpairs"
using Spair_def
someI_ex[of
"λfg. fg ∈ fundfoldpairs ∧
s = case_prod Abs_induced_automorph fg"
]
by auto
Spair_induced_automorph:
"s∈S ==> s = case_prod Abs_induced_automorph (Spair s)"
using Spair_def
someI_ex[of
"λfg. fg ∈ fundfoldpairs ∧
s = case_prod Abs_induced_automorph fg"
]
by auto
S_list_pgallery_decomp1:
assumes ss: "set ss hen show ?thesis using inser.IH by blast
shows "∃s∈set ss. ∃C∈set Cs. ∀(f,g)∈fundfoldpairs.
s = Abs_induced_automorph f g ⟶ C ∈ g⊨ next
(cases Cs)
case (Cons D Ds)
with gal(2) have "D∈ case False
using pgallery_def chamberD_simplex adjacentset_def by fastforce
from this obtain s where s: "s∈S" "D = s`→C0"
using fundadjset_eq_S_image by blast
from s(2) have
"∀(f,g)∈fundfoldpairs. s = Abs_induced_automorph f g ⟶ D∈g⊨C"
using fu
OpposedThinChamberComplexFoldings.basechambers_half_chamber_systems(2)
by auto
with s(1) ss Cons show ?thesis by auto
(simp add: gal(1))
S_list_pgallery_decomp2:
assumes "set ss = S" "Cs≠[]" "pgallery (C0#Cs)"
shows
"∃rs s ts. ss = rs@s#ts ∧
java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
s = Abs_induced_automorph f g ⟶ C ∈ g⊨C) ∧
(∀r∈set rs. ∀C∈set Cs. ∀(f,g)∈fundfoldpairs.
r = Abs_induced_automorph f g ⟶ C∈f⊨C)"
-
from assms obtain rs s ts where rs_s_ts:
"ss = rs@s#ts"
"∃C∈set Cs. ∀(f,g)∈fundfoldpairs.
s = Abs_induced_automorph f g ⟶ C ∈ g⊨C"
"∀r∈set rs. ∀C∈set Cs. ¬ (∀(f,g)∈fundfoldpairs. r = Abs_induced_automorph f g ⟶ C ∈ g⊨C)"
using split_list_first_prop[OF S_list_pgallery_decomp1, of ss Cs]
by auto
have "∀r∈set rs. ∀C∈set Cs. ∀(f,g)∈fundfoldpairs.
then have "card (g ` insex F) =ca (g `F)"
proof (rule ballI, rule ballI, rule prod_ballI, rule impI)
fix r C f g
assume "r ∈ set rs" "C ∈ set Cs" "(f,g)∈fundfoldpairs"
"r = Abs_induced_automorph f g"
with rs_s_ts(3) assms(3) show "C∈f⊨C"
using pgalleryD_chamber
fundfoldpair_unique_half_chamber_systems_chamber_ng_f[
of _ _ f g C
]
by fastforce
qed
rs_s_ts((12 ?thesis by auto
S_list_pgallery_decomp3:
assumes "set ss = S" "Cs≠[]" "pgallery (C0#Cs)"
shows
"∃rs s ts As B Bs. ss = rs@s#ts ∧
(∀(f,g)∈fundfoldpairs. s = Abs_induced_automorph f g ⟶ B∈g⊨C) ∧
(∀A∈set As. ∀(f,g)∈fundfoldpairs.
s = Abs_induced_automorph f g ⟶ A∈f⊨C) ∧
(∀r∈set rs. ∀C∈set Cs. ∀ ` )))\close lessI less_ir)
r = Abs_induced_automorph f g ⟶ C∈f⊨C)"
-
from assms obtain rs s ts where rs_s_ts:
"ss = rs@s#ts"
"∃B∈set Cs. ∀(f,g)∈fundfoldpairs. s = Abs_induced_automorph f g ⟶
"∀r∈set rs. ∀B∈set Cs. ∀(f,g)∈fundfoldpairs.
r = Abs_induced_automorph f g ⟶ B∈f⊨C"
using S_list_pgallery_decomp2[of ss Cs]
by auto
obtain As B Bs where As_B_Bs:
"Cs = As@B#Bs"
"∀(f,g)∈fundfoldpairs. s = Abs_induced_automorph f g ⟶ B ∈
"∀A∈set As. ∃(f,g)∈fundfoldpairs. s = Abs_induced_automorph f g ∧ A∉g⊨C"
using split_list_first_prop[OF rs_s_ts(2)]
by fastforce
from As_B_Bs(1,3) assms(3)
have "∀A∈set As. ∀(f,g)∈fundfoldpairs.
s = Abs_induced_automorph f g ⟶ A∈f⊨C"
using pgalleryD_chamber
fundfoldpair_unique_half_chamber_systems_chamber_ng_f
by auto
with rs_s_ts(1,3) As_B_Bs(1,2) show ?thesis by fast
fundfold_trivial_fC:
"r∈S ==>∀(f,g)∈fundfoldpairs. r = Abs_induced_automorph f g ⟶ C∈f⊨C==>
fst (Spair r) ` C = C"
using Spair_fundfoldpair[of r] Spair_induced_automorph[of r] fundfoldpairs_def
OpposedThinChamberComplexFoldings.axioms(2)[
of X "fst (Spair r)" "snd (Spair r)" C0
]
ChamberComplexFolding.chamber_retraction2[of X "fst (Spair r)" C]
by fastforce
fundfold_comp_trivial_fC:
"set rs ⊆ S ==> ∀r∈set rs. ∀(f,g)∈fundfoldpairs.
r = Abs_induced_automorph f g ⟶ C∈f⊨C==>
fold fst (map Spair rs) ` C = C"
(induct rs)
case (Cons r rs)
have "fold fst (map Spair (r#rs)) ` C =
fold fst (map Spair rs) ` fst (Spair r) ` C"
by (simp add: image_comp)
also from Cons have "… = C" by (simp add: fundfold_trivial_fC)
finally show ?case by fast
simp
fundfold_trivial_fC_list:
"r∈S ==> ∀C∈set Cs. ∀(f,g)∈fundfoldpairs.
r = Abs_induced_automorph f g ⟶ C∈f⊨C==>
fst (Spair r) ⊨ Cs = Cs"
using fundfold_trivial_fC by (induct Cs) auto
fundfold_comp_trivial_fC_list:
"set rs ⊆ S ==> ∀r∈set rs. ∀C∈set Cs. ∀(f,g)∈fundfoldpairs.
r = Abs_induced_automorph f g ⟶ C∈f⊨C==>
fold froof -
(induct rs Cs rule: list_induct2')
case (4 r rs C Cs)
from 4(3)
have r: "∀D∈set (C#Cs). ∀(f,g)∈fundfoldpairs.
r = Abs_induced_automorph f g ⟶ D∈f⊨C"
by simp
from 4(2)
have "fold fst (map Spair (r#rs)) ⊨ (C#Cs) =
map ((`) (fold fst (map Spair rs))) (fst (Spair r) ⊨ (C#Cs))"
by (auto simp add: image_comp)
also from 4 have "… = C#Cs"
using fundfold_trivial_fC_list[of r "C#Cs"]
by (simp add: fundfold_comp_trivial_fC)
finally show ?case by fast
auto
fundfold_gallery_map:
"s∈S ==> gallery Cs ==> gallery (fst (Spair s) ⊨ Cs)"
using Spair_fundfoldpair fundfoldpairs_def
OpposedThinChamberComplexFoldings.axioms(2)
ChamberComplexFolding.gallery_map[of X "fst (Spair s)"]
by fastforce
fundfold_comp_gallery_map:
assumes pregal: "gallery Cs"
shows "set ss ⊆ S ==> gallery (fold fst (map Spair ss) ⊨ Cs)"
(induct ss rule: rev_induct)
case (snoc s ss)
hence 1: "gallery (fst (Spair s) ⊨ (fold fst (map Spair ss) ⊨ Cs))"
using fundfold_gallery_map by fastforce
have 2: "fst (Spair s) ⊨ (fold fst (map Spair ss) ⊨ Cs) =
fold fst (map Spair (ss@[s])) ⊨ Cs"
by (simp add: image_comp)
show ?case using 1 subst[OF 2, of gallery, OF 1] by fast
(simp add: pregal galleryD_adj)
fundfold_comp_pgallery_ex_funpow:
assumes ss: "set ss = S"
∃n. (fold fst (map Spair ss) ^^ n) ` C = C0"
(induct Cs arbitrary: C rule: length_induct)
fix Cs C
assume step : "∀ys. length ys < length
(∀x. pgallery (C0 # ys @ [x]) ⟶
(∃n. (fold fst (map Spair ss) ^^ n) ` x = C0))"
and set_up: "pgallery (C0#Cs@[C])"
from ss set_up obtain rs s ts As B Bs where decomps:
"ss = rs@s#ts" "Cs@[C] = As@B#Bs"
"∀(f,g)∈
"∀A∈set As. ∀(f,g)∈fundfoldpairs. s = Abs_induced_automorph f g ⟶ A∈f⊨C"
"∀r∈proo -
r = Abs_induced_automorph f g ⟶ D∈f⊨C"
using S_list_pgallery_decomp3[of ss "Cs@[C]"]
by fastforce
obtain Es E where EsE: "C0#As = Es@[E]" using cons_conv_snoc by fast
have EsE_s_fC:
"∀A∈set (Es@[E]). ∀(f,g)∈fundfoldpairs.
s = Abs_induced_automorph f g ⟶ A∈f⊨C"
proof (rule ballI)
fix A assume "A∈set (Es@[E])"
with EsE decomps(4)
show "∀(f, g)∈fundfoldpairs. s = Abs_induced_automorph f g ⟶ A ∈ f ⊨C"
using fundfoldpair_fundchamber_in_half_chamber_system_f
set_ConsD[of A C0 As]
by auto
qed
moreover from decomps(2) EsE
have decomp2: "C0#Cs@[C] = Es@E#B#Bs"
by ed
moreover from ss decomps(1) have "s∈S" by auto
ultimately have sB: "fst (Spair s) ` B = E"
using set_up decomps(3) Spair_fundfoldpair[of s]
Spair_induced_automorph[of s] fundfoldpairs_def
pgalleryD_adj
binrelchain_append_reduce2[of adjacent Es "E#B#Bs"]
OpposedThinChamberComplexFoldings.adjacent_half_chamber_system_image_fg[
of X "fst (Spair s)" "snd (Spair s)" C0 E B
]
by auto
show "∃n. (fold fst (map Spair ss) ^^ n) ` C = C0"
proof (cases "Es=[] ∧ Bs = []")
case True
from decomps(5) have
"∀,g∈C∈
by auto
with decomps(1) ss
have "fold fst (map Spair ss) ` C = fold fst (map Spair ts) ` fst (Spair s) ` C"
using fundfold_comp_trivial_fC[of rs C]
by (fastforce simp add: image_comp[THEN sym])
moreover
have "∀r∈set ts. ∀(f,g)∈fundfoldpairs.
r = Abs_induced_automorph f g ⟶ C0∈f⊨C"
using fundfoldpair_fundchamber_in_half_chamber_system_f
by fast
ultimately have "(fold fst (map Spair ss) ^^ 1) ` C = C0"
using True decomps(1,2) ss EsE sB fundfold_comp_trivial_fC[of ts C0]
fundfold_comp_trivial_fC[of ts C0]
by fastforce
thus ?thesis by fast
next
show ?thesis
proof (cases "fold fst (map Spair ss) ` C = C0")
case True
hence "(fold fst (map Spair ss) ^^ 1) ` C = C0" by simp
thus ?thesis by fast
next
case False
from decomps(5) have C0CsC_rs_fC:
"∀r∈set rs. ∀D∈set (C0#Cs@[C]). ∀(f,g)∈fundfoldpairs.
r = Abs_induced_automorph f g ⟶ D∈| "f (Suk) < +
using fundfoldpair_fundchamber_in_half_chamber_system_f
by auto
from decomps(1)
have "fold fst (map Spair (rs@[s])) ⊨ (C0#Cs@[C]) =
fst (Spair s) ⊨ (fold fst (map Spair rs) ⊨ (C0#Cs@[C]))"
also from ss decomps(1)
have "… = fst (Spair s) ⊨ (C0#Cs@[C])"
using C0CsC_rs_fC fundfold_comp_trivial_fC_list[of rs "C0#Cs@[C]"]
by fastforce
also from decomp2 have "… = fst (Spair s) ⊨ (Es@E#B#Bs)"
by (simp a by blast
finally
have "fold fst (map Spair (rs@[s])) ⊨ (C0#Cs@[C]) =
Es @ E # E # fst (Spair s) ⊨ Bs"
using decomps(1) ss sB EsE_s_fC fundfold_trivial_fC_list[of s "Es@[E]"]
by fastforce
with set_up ss decomps(1)
have gal: " then sh show ?case pr cases
using pgallery fundfold_comp_gallery_map[of _ "rs@[s]"]
gallery_remdup_adj[of Es E "fst (Spair s) ⊨ Bs"]
by fastforce
from EsBs decomp2 EsE
have "∃Zs. length Zs < length
Es @ E # fst (Spair s) ⊨ Bs = C0 # Zs @ [fst (Spair s) ` C]"
using sB
by (cases Bs Es rule: two_lists_cases_snoc_Cons') auto
from this obtain Zs where Zs:
"length Zs < length Cs"
"Es @ E # fst (Spair s) ⊨ Bs = C0 # Zs @ [fst (Spair s) ` C]"
fas
define Ys where "Ys = fold fst (map Spair ts) ⊨ Zs"
with Zs(2) have
"fold fst (map Spair ts) ⊨ (Es @ E # fst (Spair s) ⊨ Bs) =
fold fst (map Spair ts) ` C0 # Ys @ [fold fst (map Spair (s#ts)) ` C]"
by (simp add: image_comp)
moreover
"\<>r
r = Abs_induced_automorph f g ⟶ C0∈f⊨C"
using fundfoldpair_fundchamber_in_half_chamber_system_f
by fast
ultimately have
"fold fst (map Spair ts) ⊨ (Es @ E # fst (Spair s) ⊨ Bs) =
C0 # Ys @ [fold fst (map Spair (s#ts)) ` fold fst (map Spair rs) ` C]"
case 2
fundfold_comp_trivial_fC[of rs C]
by fastforce
with decomps(1) ss obtain Xs where Xs:
"length Xs ≤ length Ys"
"pgallery (C0 # Xs @ [fold fst (map Spair ss) ` C])"
using gal fundfold_comp_gallery_map[of "Es @ E # fst (Spair s) ⊨ Bs" ts]
gallery_obtain_pgallery[OF False[THEN not_sym]]
by (fastforce simp add: image_comp)
from Ys_def Xs(1) Zs(1) have "length Xs < length Cs" by simp
with Xs(2) obtain n where "(fold fst (map Spair ss) ^^ (Suc n)) ` C = C0"
using step by (force simp add: image_comp funpow_Suc_right[THEN sym])
thus ?thesis by fast
qed
qed
fundfold_comp_chamber_ex_funpow:
assumes ss: "set ss = S" and C: "chamber C"
shows "∃n. (fold fst (map Spair ss) ^^ n) ` C = C0"
(cases "C=C0")
case True
hence "(fold fst (map Spair ss) ^^ 0) ` C = C0" by simp
thus ?thesis by fast
case False with fundchamber assms show ?thesis
using chamber_pconnect[of C0 C] fundfold_comp_pgallery_ex_funpow
by fastforce
fundfold_comp_fixespointwise_C0:
assumes "set ss ⊆ S"
shows "fixespointwise (fold fst (map Spair ss)) C0"
(rule fold_fixespointwise, rule ballI)
fix fg assume "fg ∈
from this obtain s where "s∈set ss" "fg = Spair s" by auto
with assms
have fg': "OpposedThinChamberComplexFoldings X (fst fg) (snd fg) C0"
using Spair_fundfoldpair fundfoldpairs_def
by fastforce
show "fixespointwise (fst fg) C0"
using OpposedThinChamberComplexFoldings.axioms(2)[OF fg']
OpposedThinChamberComplexFoldings.chamber_D0[OF fg']
OpposedThinChamberComplexFoldings.chambers(4)[OF fg']
chamber_system_def
ChamberComplexFolding.chamber_retraction1[of X "fst fg" C0]
by auto
fundfold_comp_endomorphism:
assumes "set ss ⊆ S"
shows "ChamberComplexEndomorphism X (fold fst (map Spair ss))"
(rule fold_chamber_complex_endomorph_list, rule ballI)
fix fg assume fg: "fg ∈set (map Spair ss)"
from this obtain s where "s∈set ss" "fg = Spair s" by auto
with assms show "ChamberComplexEndomorphism X (fst fg)"
using Spair_fundfoldpair
OpposedThinChamberComplexFoldings.axioms(2)[of X]
ChamberComplexFolding.axioms(1)[of X]
ChamberComplexRetraction.axioms(1)[of X]
unfolding fundfoldpairs_def
by fastforce
finite_S: "finite S"
using fundchamber_S_fundadjset fundchamber finite_adjacentset
by (blast intro: inj_on_finite fundchamber_S_image_inj_on)
ex_label_retraction: "∃φ. label_wrt C0 φ ∧ fixespointwise φ C0"
-
obtain ss where ss: "set ss = S" using finite_S finite_list by fastforce
define fgs where "fgs = map Spair ss" ―‹for @{term "fg ∈ set fgs"}, have @{term "(fst fg) ` D = C0"} for some @{term "D ∈ fundajdset"}›
define ψ where "ψ
define vdist where "vdist v = (LEAST n. (ψ^^n) v ∈ C0)" for v
define φ where "φ v = (ψ^^(vdist v)) v" for v
have "label_wrt C0 φ"
unfolding label_wrt_def
proof
fix C assume C: "C∈C
show "bij_betw φ C C0"
proof-
from ψ_def fgs_def ss C obtain m where m: "(ψ^^m)`C = C0"
using chamber_system_def fundfold_comp_chamber_ex_funpow by fastforce
have "∧v. v∈C ==> (ψ^^m) v = φ v"
proof-
fix v assume v: "v∈C"
define n where "n = (LEAST n. (ψ^^n) v ∈ C0)"
from v m φ_def vdist_def n_def have "m ≥ n" "φ v ∈ C0"
using Least_le[of "λn. (ψ^^n) v ∈ C0" m]
LeastI_ex[of "λ using assms by simp
by auto
then show "(ψ^^m) v = φ v"
using ss ψ_def fgs_def φ_def vdist_def n_def funpow_add[of "m-n" n ψ]
fundfold_comp_fixespointwise_C0
funpower_fixespointwise fixespointwiseD
by fastforce
qed
with C m ss ψ_def fgs_def show ?thesis
using chamber_system_def fundchamber fundfold_comp_endomorphism
ChamberComplexEndomorphism.funpower_endomorphism[of X]
ChamberComplexEndomorphism.bij_betw_chambers[of X]
bij_betw_cong[of C "ψ^^m" φ C0]
by fastforce
qed
qed
moreover from vdist_def φ_def have "fixespointwise φ C0"
using Least_eq_0 by (fastforce intro: fixespointwiseI)
ultimately show ?thesis by fast
ex_label_map: "∃φ. label_wrt C0 φ"
using ex_label_retraction by fast
(* context ThinChamberComplexManyFoldings *)
‹More on the action of the group of automorphisms on chambers›
‹
Recall that we have already verified that @{term W} acts transitively on the chamber system. We
now use the labelling of the chamber complex examined in the previous section to show that this
action is simply transitive. ›
ThinChamberComplexManyFoldings
fundchamber_W_image_ker:
assumes
shows "w = 0"
-
obtain φ : "fold (|\union xs ffUnion fset_oflist xs"
have "fixespointwise (permutation w) C0"
using W_respects_labels[OF φ assms(1)] chamberD_simplex[OF fundchamber]
ChamberComplexEndomorphism.respects_label_fixes_chamber_imp_fixespointwise[
OF W_endomorphism, OF assms(1) φ fundchamber assms(2)
]
by fast
with assms(1) show ?thesis
using fundchamber W_automorphism trivial_automorphism
standard_uniqueness_automorphs
permutation_inject[of w 0]
by (auto simp add: zero_permutation.rep_eq)
fundchamber_W_image_inj_on:
"inj_on (λw. w`→C0) W"
(rule inj_onI)
fix w w' assume ww': "w∈W" "w'∈W" "w`→C0 = w'`→C0"
from ww'(3) have "(-w')`→ = -w')`→
with ww'(1,2) show "w = w'"
using fundchamber_W_image_ker[of "-w'+w"] add.assoc[of w' "-w'" w]
by (simp add:
image_comp plus_permutation.rep_eq[THEN sym]
zero_permutation.rep_eq genby_uminus_add_closed
)
(* context ThinChamberComplexManyFoldings *)
‹A bijection between the fundamental chamber and the set of generating automorphisms›
‹
Removing a single vertex from the fundamental chamber determines a facet, a facet in the
fundamental chamber determines an adjacent chamber (since our complex is thin), and a chamber
adjacent to the fundamental chamber determines an automorphism (via some pair of opposed foldings)
in our generating set @{term S}. Here we show that this correspondence is bijective. ›
ThinChamberComplexManyFoldings
fundantivertex :: "'a permutation ==>
where "fundantivertex s ≡ (THE v. v ∈ C0-s`→C0)"
"fundantipermutation ≡ the_inv_into S fundantivertex"
fundantivertex: "s∈S ==> fundantivertex s ∈ C0-s`→C0"
using fundcham[of s]
fundchamber_S_image_neq_fundchamber[of s]
fundantivertex_def[of s] theI'[OF adj_antivertex]
by auto
fundantivertex_fundchamber_decomp:
"s∈S ==> C0 = insert (fundantivertex s) (C0∩s`→C0)"
using fundchamber_S_adjacent[of s]
fundchamber_S_image_neq_fundchamber[of s]
fundantivertex[of s] adjacent_conv_insert[of C0]
by auto
fundantivertex_unstable:
"s∈S ==> s → fundantivertex s ≠ fundantivertex s"
foldr_funion_fmember : "B \subseteqfoldr (|🚫
image_insert[of "(→) s" "fundantivertex s" "C0∩s`→C0"]
S_fixes_fundchamber_image_int fundchamber_S_image_neq_fundchamber
by fastforce
fundantivertex_inj_on: "inj_on fundantivertex S"
(rule inj_onI)
fix s t assume st: "s∈S" "t∈S" "fundantivertex s = fundantivertex t"
hence "insert (fundantivertex s) (C0∩s`→C0) =
insert (fundantiver
using fundantivertex_fundchamber_decomp[of s]
fundantivertex_fundchamber_decomp[of t]
by auto
moreover from st
have "fundantivertex s ∉ C0∩s`→C0" "fundantivertex s ∉ C0∩t`→C0"
using fundantivertex[of s] fundantivertex[of t]
by auto
ultimately have "C0∩s`→C0 = C0∩t`→C0"
using insert_subset_equality[of "fundantivertex s"] by simp
with st(1,2) show "s=t"
using fundchamber fundchamber_S_chamber[of s] fundchamber_S_chamber[of t]
fundfacets[of s] fundfacets(2)[of t]
fundchamber_S_image_neq_fundchamber[of s]
fundchamber_S_image_neq_fundchamber[of t]
facet_unique_other_chamber[of C0 "C0∩s`→C0" "s`→C0" "t`→C0"]
genby_genset_closed[of _ S]
inj_onD[OF fundchamber_W_image_inj_on, of s t]
by auto
fundantivertex_surj_on: "fundantivertex ` S = C0"
(rule seteqI)
show "∧v. v ∈ fundantivertex ` S ==> v∈C0" using fundantivertex by fast
fix v assume v: "v∈C0"
define D where "D = the_adj_chamber C0 (C0-{v})"
with v have "D∈fundadjset"
using fundchamber facetrel_diff_vertex the_adj_chamber_adjacentset
the_adj_chamber_neq
by fastforce
from this obtain s where s: "s∈S" "D = s`→C0"
using fundadjset_eq_S_image by blast
with v D_def [abs_def] have "fundantivertex s = v"
using fundchamber fundchamber_S_adjacent
fundchamber_S_image_neq_fundchamber[of s]
facetrel_diff_vertex[of v C0]
the_adj_chamber_facet facetrel_def[of "C0-{v}" D]
unfolding fundantivertex_d
by (force intro: the1_equality[OF adj_antivertex])
with s(1) show "v ∈ fundantivertex ` S" by fast
fundantivertex_bij_betw: "bij_betw fundantivertex S C0"
unfolding bij_betw_def
using fundantivertex_inj_on fundantivertex_surj_on
by fast
card_S_fundchamber: "card S = card C0"
using bij_betw_same_card[OF fundantivertex_bij_betw] by fast
card_S_chamber:
"chamber C ==> card C = card S"
using fundchamber chamber_card[of C0 C] card_S_fundchamber by auto
fundantipermutation1:
"v∈C0 ==> fundantipermutation v ∈ S"
using fundantivertex_surj_on the_inv_into_into[OF fundantivertex_inj_on] by blast
(* context ThinChamberComplexManyFoldings *)
‹Thick chamber complexes›
‹
A thick chamber complex is one in which every facet is a facet of at least three chambers. ›
ThickChamberComplex = ChamberComplex X
for X :: "'a set set"
assumes thick:
"chamber C ==> z⊲C ==> ∃D E. D∈X-{C} ∧ z⊲D ∧
some_third_chamber :: "'a set ==>
where "some_third_chamber C D z ≡ SOME E. E∈X-{C,D} ∧ z⊲E"
facet_ex_third_chamber: "chamber C ==> z⊲C ==>∃E∈X-{C,D}. z⊲E"
using thick[of C z] by auto
some_third_chamberD_facet:
"chamber C ==> z⊲C ==> z ⊲ some_third_chamber C D z"
using facet_ex_third_chamber[of C z D] someI_ex[of "λE. E∈X-{C,D} ∧ z⊲E"]
some_third_chamber_def
by auto
somsomethird_cham:
"chamber C ==> z⊲C ==> some_third_chamber C D z ∈ X"
using facet_ex_third_chamber[of C z D] someI_ex[of "λE. E∈X-{C,D} ∧ z⊲E"]
some_third_chamber_def
by auto
some_third_chamberD_adj:
"chamber C ==> z⊲C ==> C ∼ some_third_chamber C D z"
using some_third_chamberD_facet by (fast intro: adjacentI)
chamber_some_third_chamber:
"chamber C ==> z⊲C ==> chamber (some_third_chamber C D z)"
using chamber_adj some_third_chamberD_simplex some_third_chamberD_adj
by fast
some_third_chamberD_ne:
assumes "chamber C" "z⊲C"
shows "some_third_chamber C D z ≠ C" "some_third_chamber C D z ≠ D"
using assms facet_ex_third_chamber[of C z D]
someI_ex[of "λE. E∈X-{C,D} ∧ z⊲E"] some_third_chamber_def
by auto
(* context ThickChamberComplex *)
(* theory *)
Messung V0.5 in Prozent
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(vorverarbeitet am 2026-06-13)
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