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