File size: 3,394 Bytes
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 | 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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