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8e4d533 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | import Mathlib
import OAI.Geometry.Borsuk.Support.SixFaces
namespace OAI
open Set MeasureTheory
namespace BorsukNine
open scoped RealInnerProductSpace
attribute [local instance] Classical.propDecidable
section Within
variable {α : Type*} [DecidableEq α]
noncomputable def withinLabels (C U : Finset α) : Finset C := U.subtype (· ∈ C)
@[simp] lemma mem_withinLabels {C U : Finset α} {i : C} :
i ∈ withinLabels C U ↔ i.val ∈ U := Finset.mem_subtype
lemma withinLabels_card {C U : Finset α} (hUC : U ⊆ C) : (withinLabels C U).card = U.card := by
rw [withinLabels,Finset.card_subtype,Finset.filter_true_of_mem]
exact fun _ hi => hUC hi
lemma withinLabels_compl (C U : Finset α) :
withinLabels C (C \ U) = (withinLabels C U)ᶜ := by
ext i
simp [i.property]
lemma withinLabels_insert (C U : Finset α) (i : C) :
withinLabels C (insert i.val U) = insert i (withinLabels C U) := by
ext j
simp only [mem_withinLabels,Finset.mem_insert,Subtype.ext_iff]
end Within
section FacetSix
variable {f : Vector4 → Fin 10 → ℝ}
lemma Admissible.facet_complement_exclusion (hf : Admissible f) {S U : Finset (Fin 10)}
(hS : SupportFace f S) (hU : SupportFace f U) : ¬ SupportFace f (Sᶜ \ U) := by
classical
rintro ⟨v,hv,hCv⟩
obtain ⟨w,hw,hSw⟩ := hS
obtain ⟨u,hu,hUu⟩ := hU
obtain ⟨i,hi⟩ := hf.exists_label_absent_dim (n := 3) (by decide) ![w,u,v] (by
intro j
fin_cases j <;> simp [hw,hu,hv])
by_cases hiS : i ∈ S
· exact hi 0 (hSw hiS)
by_cases hiU : i ∈ U
· exact hi 1 (hUu hiU)
exact hi 2 (hCv (Finset.mem_sdiff.mpr ⟨Finset.mem_compl.mpr hiS,hiU⟩))
lemma restricted_face_image {C U : Finset (Fin 10)} (hUC : U ⊆ C)
(e : EuclideanSpace ℝ (Fin 3) →ₗᵢ[ℝ] Vector4)
(hU : SupportFace (fun x (i : C) => f (e x) i.val) (withinLabels C U)) :
SupportFace f U := by
obtain ⟨x,hx,hUx⟩ := hU
refine ⟨e x,by simpa using hx,?_⟩
intro i hi
exact (mem_positiveSupport _ _ _).mpr ((mem_positiveSupport _ _ (⟨i,hUC hi⟩ : C)).mp
(hUx (mem_withinLabels.mpr hi)))
lemma Admissible.facet_six_properties (hf : Admissible f) {S : Finset (Fin 10)}
(hSc : S.card = 4) (hSf : SupportFace f S) :
(∀ U ⊆ Sᶜ, U.card = 3 → (SupportFace f U ↔ ¬ SupportFace f (Sᶜ \ U))) ∧
(∀ P ⊆ Sᶜ, P.card = 2 →
((Sᶜ \ P).filter (fun w => SupportFace f (insert w P))).card = 2) := by
classical
obtain ⟨w,hw,hSw⟩ := hSf
obtain ⟨e,he⟩ := orthogonal_isometry 3 hw
let C : Finset (Fin 10) := Sᶜ
let g : EuclideanSpace ℝ (Fin 3) → C → ℝ := fun x i => f (e x) i.val
have hCc : C.card = 6 := by dsimp [C]; rw [Finset.card_compl,Fintype.card_fin,hSc]
have hcard : Fintype.card C = 6 := by simpa using hCc
have hzero : ∀ x, ‖x‖ = 1 → ∀ i ∉ C, f (e x) i = 0 := by
intro x hx i hi
have hiS : i ∈ S := by simpa [C] using hi
have hwpos : 0 < f w i := (mem_positiveSupport _ _ _).mp (hSw hiS)
exact le_antisymm (le_of_not_gt (fun hp => hf.orthogonal w (e x) hw
(by simpa using hx) (he x) i ⟨hwpos,hp⟩)) (hf.nonneg _ (by simpa using hx) i)
have hg : Admissible g := (hf.comp_isometry e).restrict_labels C hzero
have hmap {U : Finset (Fin 10)} (hUC : U ⊆ C) :
SupportFace g (withinLabels C U) → SupportFace f U := restricted_face_image hUC e
have hfaces {U : Finset (Fin 10)} (hUC : U ⊆ C) (hUc : U.card = 3) :
SupportFace g (withinLabels C U) ↔ SupportFace f U := by
refine ⟨hmap hUC,?_⟩
intro hU
apply (hg.six_complementary hcard (by rw [withinLabels_card hUC,hUc])).mpr
rw [← withinLabels_compl]
intro hcomp
exact hf.facet_complement_exclusion ⟨w,hw,hSw⟩ hU (hmap Finset.sdiff_subset hcomp)
constructor
· intro U hUC hUc
have hcUc : (C \ U).card = 3 := by rw [Finset.card_sdiff_of_subset hUC,hCc,hUc]
rw [← hfaces hUC hUc,← hfaces Finset.sdiff_subset hcUc,withinLabels_compl]
exact hg.six_complementary hcard (by rw [withinLabels_card hUC,hUc])
· intro P hPC hPc
let P' := withinLabels C P
have hP'c : P'.card = 2 := by rw [withinLabels_card hPC,hPc]
have hinsert {i : C} (hi : i ∉ P') :
SupportFace g (insert i P') ↔ SupportFace f (insert i.val P) := by
have hiP : i.val ∉ P := by simpa only [P',mem_withinLabels] using hi
have hh := hfaces (Finset.insert_subset i.property hPC)
(by rw [Finset.card_insert_of_notMem hiP,hPc])
simpa only [withinLabels_insert] using hh
have hc := hg.six_pair_count hcard hP'c
have hcount : ((P'ᶜ).filter (fun i => SupportFace g (insert i P'))).card =
((C \ P).filter (fun i => SupportFace f (insert i P))).card := by
apply Finset.card_bij (fun i _ => i.val)
· intro i hi
obtain ⟨hiP,hif⟩ := Finset.mem_filter.mp hi
have hiP' := Finset.mem_compl.mp hiP
exact Finset.mem_filter.mpr ⟨Finset.mem_sdiff.mpr ⟨i.property,by
simpa only [P',mem_withinLabels] using hiP'⟩,(hinsert hiP').mp hif⟩
· intro i _ j _ hij
exact Subtype.ext hij
· intro i hi
obtain ⟨hiC,hif⟩ := Finset.mem_filter.mp hi
obtain ⟨hiC,hiP⟩ := Finset.mem_sdiff.mp hiC
have hiP' : (⟨i,hiC⟩ : C) ∉ P' := by simpa only [P',mem_withinLabels] using hiP
exact ⟨⟨i,hiC⟩,Finset.mem_filter.mpr ⟨Finset.mem_compl.mpr hiP',(hinsert hiP').mpr hif⟩,rfl⟩
exact hcount.symm.trans hc
end FacetSix
end BorsukNine
end OAI
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