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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 | 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
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