| import OAI.Combinatorics.MatchingPSD.GlobalGeometry |
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| namespace OAI |
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| universe u_1 u_2 u_3 u_4 |
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| namespace MatchingSmoothing |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
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| section ColorTransport |
| variable {A : Type u_1} {B : Type u_2} {C : Type u_3} [Fintype A] [Fintype B] [Fintype C] [DecidableEq C] |
| noncomputable def colorEquiv (f : A → C) (g : B → C) (h : histogram f = histogram g) : A ≃ B := |
| (Equiv.sigmaFiberEquiv f).symm.trans |
| ((Equiv.sigmaCongrRight (fun c => Fintype.equivOfCardEq (congrFun h c))).trans |
| (Equiv.sigmaFiberEquiv g)) |
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|
| omit [Fintype C] in |
| lemma colorEquiv_spec (f : A → C) (g : B → C) (h : histogram f = histogram g) (a : A) : |
| g (colorEquiv f g h a) = f a := by |
| exact (Fintype.equivOfCardEq (congrFun h (f a)) ⟨a,rfl⟩).property |
| end ColorTransport |
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|
| def typePair : Fin 6 → Fin 3 × Fin 3 := ![(0,0),(0,1),(0,2),(1,1),(1,2),(2,2)] |
| abbrev TypedEdges (E : Fin 6 → ℕ) := (t : Fin 6) × Fin (E t) |
| def edgeColor (E : Fin 6 → ℕ) : TypedEdges E × Bool → Fin 3 := |
| fun v => if v.2 then (typePair v.1.1).2 else (typePair v.1.1).1 |
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| def endpointSizes (E : Fin 6 → ℕ) : Fin 3 → ℕ := |
| ![2*E 0+E 1+E 2, E 1+2*E 3+E 4, E 2+E 4+2*E 5] |
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| lemma edgeColor_histogram (E : Fin 6 → ℕ) : histogram (edgeColor E) = endpointSizes E := by |
| funext c |
| apply Nat.cast_injective (R := ℝ) |
| rw [histogram_cast_sum] |
| simp only [Fintype.sum_prod_type, Fintype.sum_sigma, Fintype.sum_bool] |
| fin_cases c <;> simp [edgeColor, endpointSizes, typePair, Fin.sum_univ_succ] <;> ring |
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| lemma edgeColor_cross (E : Fin 6 → ℕ) (i : Fin 3) : |
| crossCoordinate (edgeColor E) i (standardPairing (H := TypedEdges E)) = |
| (![E 1,E 2,E 4] i : ℝ) := by |
| simp only [crossCoordinate, tableCoordinate, Fintype.sum_prod_type, |
| Fintype.sum_sigma, Fintype.sum_bool] |
| fin_cases i <;> simp [crossPair, standardPairing, edgeColor, typePair, Fin.sum_univ_succ] |
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| lemma exists_pairing_of_endpoints {V : Type u_4} [Fintype V] |
| (z : V → Fin 3) (E : Fin 6 → ℕ) (hz : histogram z = endpointSizes E) : |
| ∃ M : Pairing V, ∀ i : Fin 3, crossCoordinate z i M = (![E 1,E 2,E 4] i : ℝ) := by |
| let e := colorEquiv (edgeColor E) z ((edgeColor_histogram E).trans hz.symm) |
| have he : z ∘ e = edgeColor E := funext (colorEquiv_spec _ _ _) |
| refine ⟨Pairing.transport e standardPairing, ?_⟩ |
| intro i |
| rw [crossCoordinate, tableCoordinate_transport, he] |
| exact edgeColor_cross E i |
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|
| abbrev TripleVertices (s : ℕ) := (Fin (2*s+1) ⊕ Fin (2*s+1)) ⊕ (Fin s × Bool) |
| def tripleColor {s : ℕ} : TripleVertices s → Fin 3 := |
| Sum.elim (Sum.elim (fun _ => 0) (fun _ => 1)) (fun _ => 2) |
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| lemma tripleColor_histogram (s : ℕ) : |
| histogram (@tripleColor s) = ![2*s+1,2*s+1,2*s] := by |
| funext c |
| apply Nat.cast_injective (R := ℝ) |
| rw [histogram_cast_sum] |
| fin_cases c <;> simp [tripleColor, Fintype.sum_sum_type, Nat.cast_add, Nat.cast_mul]; ring |
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| lemma triple_card (s : ℕ) : Fintype.card (TripleVertices s) = 2*(3*s+1) := by |
| simp only [TripleVertices, Fintype.card_sum, Fintype.card_fin, Fintype.card_prod, |
| Fintype.card_bool] |
| omega |
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| noncomputable def tripleEquiv (s : ℕ) : TripleVertices s ≃ Vertices (3*s+1) := |
| Fintype.equivOfCardEq (by |
| change Fintype.card (TripleVertices s) = Fintype.card (Fin (3*s+1) × Bool) |
| rw [triple_card, Fintype.card_prod, Fintype.card_fin, Fintype.card_bool] |
| omega) |
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| lemma triple_histogram_pos {s : ℕ} (hs : 0 < s) (i : Fin 3) : |
| 0 < histogram (@tripleColor s) i := by |
| rw [tripleColor_histogram] |
| fin_cases i <;> simp; omega |
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| end MatchingSmoothing |
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| end OAI |
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