| import Mathlib | |
| import OAI.Geometry.Borsuk.Simplicial.NormMap | |
| namespace OAI | |
| namespace BorsukNine.SimplicialChains | |
| open Finset | |
| variable {V W : Type*} [DecidableEq V] [DecidableEq W] | |
| omit [DecidableEq V] in | |
| lemma mem_support_simplex (K : PreAbstractSimplicialComplex V) (s : Finset V) | |
| (t : {s : Finset V // s ∈ K}) : | |
| t ∈ (simplex K s).support ↔ t.val = s := by | |
| classical | |
| by_cases hs : s ∈ K | |
| · rw [simplex_of_mem K hs, Finsupp.support_single _ one_ne_zero] | |
| simp only [Finset.mem_singleton, Subtype.ext_iff] | |
| · simp only [simplex, dite_eq_right hs, Finsupp.support_zero, Finset.notMem_empty, false_iff] | |
| intro he | |
| exact hs (he ▸ t.property) | |
| omit [DecidableEq V] [DecidableEq W] in | |
| lemma support_linearMap (K : PreAbstractSimplicialComplex V) (L : PreAbstractSimplicialComplex W) | |
| (T : Chain K →ₗ[F₂] Chain L) (c : Chain K) | |
| {t : {s : Finset W // s ∈ L}} (ht : t ∈ (T c).support) : | |
| ∃ s ∈ c.support, t ∈ (T (simplex K s.val)).support := by | |
| classical | |
| by_contra hn | |
| push Not at hn | |
| have hh : (T c) t = 0 := by | |
| conv_lhs => rw [chain_eq_sum K c] | |
| rw [map_sum, Finsupp.finsetSum_apply] | |
| apply Finset.sum_eq_zero | |
| intro s hs | |
| rw [map_smul, Finsupp.smul_apply, Finsupp.notMem_support_iff.mp (hn s hs), smul_zero] | |
| exact (Finsupp.mem_support_iff.mp ht) hh | |
| lemma support_cone (K : PreAbstractSimplicialComplex V) (a : V) (c : Chain K) | |
| {t : {s : Finset V // s ∈ K}} (ht : t ∈ (cone K a c).support) : | |
| ∃ s ∈ c.support, a ∉ s.val ∧ t.val = insert a s.val := by | |
| obtain ⟨s, hs, ht⟩ := support_linearMap K K (cone K a) c ht | |
| rw [cone_simplex K a s.property] at ht | |
| split_ifs at ht with ha | |
| · simp at ht | |
| · exact ⟨s, hs, ha, (mem_support_simplex K _ t).mp ht⟩ | |
| lemma cone_degree (K : PreAbstractSimplicialComplex V) (a : V) (d : ℕ) : | |
| degreeChains K d ≤ (degreeChains K (d+1)).comap (cone K a) := by | |
| intro c hc t ht | |
| obtain ⟨s, hs, ha, he⟩ := support_cone K a c ht | |
| change t.val.card = d+1 | |
| rw [he, Finset.card_insert_of_notMem ha, hc hs] | |
| end BorsukNine.SimplicialChains | |
| end OAI | |