| import Mathlib |
| import OAI.Geometry.Borsuk.Simplicial.MemSupportSimplex |
|
|
| namespace OAI |
|
|
| namespace BorsukNine.SimplicialChains |
| open Finset |
| variable {V W : Type*} [DecidableEq V] [DecidableEq W] |
|
|
| noncomputable def subdivSimplex (L : PreAbstractSimplicialComplex W) (b : Finset V → W) |
| (s : Finset V) : Chain L := |
| if _hs : s.Nonempty then |
| if s.card = 1 then simplex L {b s} |
| else cone L (b s) (∑ a ∈ s.attach, subdivSimplex L b (s.erase a.val)) |
| else 0 |
| termination_by s.card |
| decreasing_by exact Finset.card_erase_lt_of_mem a.property |
|
|
| @[simp] lemma subdivSimplex_empty (L : PreAbstractSimplicialComplex W) (b : Finset V → W) : |
| subdivSimplex L b ∅ = 0 := by rw [subdivSimplex]; simp |
|
|
| lemma subdivSimplex_one (L : PreAbstractSimplicialComplex W) (b : Finset V → W) |
| {s : Finset V} (hc : s.card = 1) : subdivSimplex L b s = simplex L {b s} := by |
| have hn : s.Nonempty := Finset.card_pos.mp (by omega) |
| rw [subdivSimplex, dite_eq_left hn, ite_eq_left hc] |
|
|
| lemma subdivSimplex_many (L : PreAbstractSimplicialComplex W) (b : Finset V → W) |
| {s : Finset V} (hn : s.Nonempty) (hc : s.card ≠ 1) : |
| subdivSimplex L b s = cone L (b s) (∑ a ∈ s, subdivSimplex L b (s.erase a)) := by |
| rw [subdivSimplex, dite_eq_left hn, ite_eq_right hc] |
| exact congrArg (cone L (b s)) (Finset.sum_attach s (fun a : V => subdivSimplex L b (s.erase a))) |
|
|
| def vertexCarrier (b : Finset V → W) (s : Finset V) : Set W := |
| {w | ∃ t : Finset V, t.Nonempty ∧ t ⊆ s ∧ b t = w} |
|
|
| omit [DecidableEq V] [DecidableEq W] in |
| lemma vertexCarrier_mono (b : Finset V → W) {s t : Finset V} (hst : s ⊆ t) : |
| vertexCarrier b s ⊆ vertexCarrier b t := by |
| rintro w ⟨u, hu, hus, hb⟩ |
| exact ⟨u, hu, hus.trans hst, hb⟩ |
|
|
| omit [DecidableEq V] [DecidableEq W] in |
| lemma center_mem_carrier (b : Finset V → W) {s : Finset V} (hs : s.Nonempty) : |
| b s ∈ vertexCarrier b s := ⟨s, hs, Finset.Subset.refl _, rfl⟩ |
|
|
| lemma support_sum_chains (L : PreAbstractSimplicialComplex W) {α : Type*} |
| (s : Finset α) (c : α → Chain L) {t : {s : Finset W // s ∈ L}} |
| (ht : t ∈ (∑ a ∈ s, c a).support) : ∃ a ∈ s, t ∈ (c a).support := by |
| classical |
| by_contra hn |
| push Not at hn |
| have hh : (∑ a ∈ s, c a) t = 0 := by |
| rw [Finsupp.finsetSum_apply] |
| exact Finset.sum_eq_zero (fun a ha => Finsupp.notMem_support_iff.mp (hn a ha)) |
| exact Finsupp.mem_support_iff.mp ht hh |
|
|
| lemma subdivSimplex_carrier (L : PreAbstractSimplicialComplex W) (b : Finset V → W) |
| (s : Finset V) {t : {s : Finset W // s ∈ L}} |
| (ht : t ∈ (subdivSimplex L b s).support) : ↑t.val ⊆ vertexCarrier b s := by |
| induction s using Finset.strongInductionOn generalizing t with |
| | _ s ih => |
| rcases s.eq_empty_or_nonempty with rfl | hn |
| · simp at ht |
| by_cases hc : s.card = 1 |
| · rw [subdivSimplex_one L b hc, mem_support_simplex] at ht |
| rw [ht] |
| exact Finset.singleton_subset_set_iff.mpr (center_mem_carrier b hn) |
| · rw [subdivSimplex_many L b hn hc] at ht |
| obtain ⟨u, hu, _, htu⟩ := support_cone L (b s) _ ht |
| obtain ⟨a, ha, hua⟩ := support_sum_chains L s _ hu |
| have hu' := ih (s.erase a) (Finset.erase_ssubset ha) hua |
| rw [htu, Finset.coe_insert] |
| exact Set.insert_subset (center_mem_carrier b hn) |
| (hu'.trans (vertexCarrier_mono b (Finset.erase_subset a s))) |
|
|
| end BorsukNine.SimplicialChains |
|
|
| end OAI |
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|
