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e5a97c6 | 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 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 | import OAI.Combinatorics.MatchingPSD.MatchingFibers
namespace OAI
universe u_1 u_2 u_3 u_4 u_5 u_6 u_7
namespace MatchingSmoothing
open scoped BigOperators
attribute [local instance] Classical.propDecidable
section ThreeCellGeometry
variable {A : Type u_1} {B : Type u_2} {C : Type u_3} [Fintype A] [Fintype B] [Fintype C]
def abColor : A ⊕ B → Fin 3 := Sum.elim (fun _ => 0) (fun _ => 1)
def threeColor : (A ⊕ B) ⊕ C → Fin 3 := Sum.elim abColor (fun _ => 2)
noncomputable def crossVector {V : Type u_4} [Fintype V] (z : V → Fin 3)
(M : Pairing V) : Fin 3 → ℝ := fun i => crossCoordinate z i M
lemma crossVector_sum (M : Pairing (A ⊕ B)) (N : Pairing C) :
crossVector threeColor (Pairing.sum M N) = ![tableCoordinate abColor (0,1) M,0,0] := by
have h02 : tableCoordinate (abColor : A ⊕ B → Fin 3) (0,2) M = 0 := by
apply (tableCoordinate_eq_zero_iff _ _ _).mpr
intro v hv
have hh := hv.2
cases he : M.1 v <;> simp [he, abColor] at hh
have h12 : tableCoordinate (abColor : A ⊕ B → Fin 3) (1,2) M = 0 := by
apply (tableCoordinate_eq_zero_iff _ _ _).mpr
intro v hv
have hh := hv.2
cases he : M.1 v <;> simp [he, abColor] at hh
funext i
fin_cases i <;>
simp [crossVector, crossCoordinate, crossPair, threeColor,
tableCoordinate_sum, tableCoordinate_constant, h02, h12]
lemma threeColor_closed (M : Pairing ((A ⊕ B) ⊕ C))
(h1 : crossCoordinate threeColor 1 M = 0) (h2 : crossCoordinate threeColor 2 M = 0) :
∀ a : A ⊕ B, ∃ a' : A ⊕ B, M.1 (.inl a) = .inl a' := by
have h1' := (tableCoordinate_eq_zero_iff _ _ _).mp h1
have h2' := (tableCoordinate_eq_zero_iff _ _ _).mp h2
intro a
cases hm : M.1 (.inl a) with
| inl a' => exact ⟨a',rfl⟩
| inr c =>
cases a with
| inl a => exact False.elim (h1' (.inl (.inl a)) ⟨rfl, by rw [hm]; rfl⟩)
| inr b => exact False.elim (h2' (.inl (.inr b)) ⟨rfl, by rw [hm]; rfl⟩)
noncomputable def starFiberEquiv (x : ℝ) :
{M : Pairing (A ⊕ B) // tableCoordinate abColor (0,1) M = x} × Pairing C ≃
{M : Pairing ((A ⊕ B) ⊕ C) // crossVector threeColor M = ![x,0,0]} :=
Pairing.sumFiberEquiv _ _
(fun M N => by rw [crossVector_sum]; simp)
(fun M hM => threeColor_closed M
(by have := congrFun hM 1; exact this)
(by have := congrFun hM 2; exact this))
end ThreeCellGeometry
section GlobalExtension
variable {D : Type u_5} {C : Type u_6} {I : Type u_7} [DecidableEq I]
def insertCellEquiv (i : I) : C ⊕ ({j : I // j ≠ i} × C) ≃ I × C where
toFun v := v.elim (fun c => (i,c)) (fun jc => (jc.1.1,jc.2))
invFun v := if h : v.1 = i then .inl v.2 else .inr (⟨v.1,h⟩,v.2)
left_inv v := by
cases v with
| inl c => simp
| inr jc => simp [jc.1.2]
right_inv v := by
rcases v with ⟨j,c⟩
by_cases h : j = i <;> simp_all
def localSplitEquiv (i : I) : (D ⊕ C) ⊕ ({j : I // j ≠ i} × C) ≃ D ⊕ (I × C) :=
(Equiv.sumAssoc D C _).trans (Equiv.sumCongr (Equiv.refl D) (insertCellEquiv i))
def extendLocal (i : I) (M : Pairing (D ⊕ C)) (N : {j : I // j ≠ i} → Pairing C) :
Pairing (D ⊕ (I × C)) :=
Pairing.transport (localSplitEquiv i) (Pairing.sum M (Pairing.cellwise N))
def liftLocal (i : I) (p : Equiv.Perm (D ⊕ C)) : Equiv.Perm (D ⊕ (I × C)) :=
(localSplitEquiv i).symm.trans
((Equiv.sumCongr p (Equiv.refl _)).trans (localSplitEquiv i))
lemma extendLocal_relabel (i : I) (p : Equiv.Perm (D ⊕ C))
(M : Pairing (D ⊕ C)) (N : {j : I // j ≠ i} → Pairing C) :
extendLocal i (Pairing.relabel p M) N = Pairing.relabel (liftLocal i p) (extendLocal i M N) := by
rw [extendLocal, Pairing.sum_relabel_left, Pairing.transport_relabel]
rfl
lemma localSplit_inl (i : I) (d : D) :
localSplitEquiv (C := C) i (.inl (.inl d)) = .inl d := rfl
lemma localSplit_cell (i : I) (c : C) :
localSplitEquiv (D := D) i (.inl (.inr c)) = .inr (i,c) := rfl
lemma localSplit_outside (i : I) (v : {j : I // j ≠ i} × C) :
localSplitEquiv (D := D) i (.inr v) = .inr (v.1.1,v.2) := rfl
lemma extendLocal_sum (i : I) (M : Pairing D) (N : I → Pairing C) :
extendLocal i (Pairing.sum M (N i)) (fun j => N j.1) =
Pairing.sum M (Pairing.cellwise N) := by
apply Pairing.ext
intro v
obtain ⟨v,rfl⟩ := (localSplitEquiv (D := D) (C := C) i).surjective v
simp only [extendLocal, Pairing.transport_apply, Equiv.symm_apply_apply]
cases v with
| inl v =>
cases v with
| inl d => rfl
| inr c => rfl
| inr jc => rfl
end GlobalExtension
end MatchingSmoothing
end OAI
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