| import OAI.Combinatorics.MatchingPSD.SpectralMultiplicity |
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| namespace OAI |
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| universe u_1 u_2 u_3 |
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| namespace MatchingSmoothing |
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| namespace Pairing |
| variable {V : Type u_1} [Fintype V] [DecidableEq V] |
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| noncomputable instance : Fintype (Pairing V) := Fintype.ofFinite _ |
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|
| instance relabelMulAction : MulAction (Equiv.Perm V) (Pairing V) where |
| smul := relabel |
| one_smul := relabel_one |
| mul_smul := relabel_mul |
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|
| omit [Fintype V] [DecidableEq V] in |
| lemma smul_eq_relabel (p : Equiv.Perm V) (M : Pairing V) : p • M = relabel p M := rfl |
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| noncomputable def table (M : Pairing V) (f : V → Fin 3) (t : Fin 6) : ℕ := |
| histogram (fun v => threeEdgeType (f v, f (M.1 v))) t / 2 |
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|
| omit [DecidableEq V] in |
| lemma table_relabel (p : Equiv.Perm V) (M : Pairing V) (f : V → Fin 3) : |
| (relabel p M).table f = M.table (f ∘ p) := by |
| funext t |
| unfold table |
| congr 1 |
| have he : (fun v => threeEdgeType (f v, f ((relabel p M).1 v))) ∘ p = |
| (fun v => threeEdgeType ((f ∘ p) v, (f ∘ p) (M.1 v))) := by |
| funext v |
| simp only [Function.comp_apply, relabel_apply] |
| rw [Equiv.Perm.inv_def, p.symm_apply_apply] |
| rw [← he, histogram_comp_perm] |
|
|
| end Pairing |
|
|
| section StandardTable |
| variable {H : Type u_2} [Fintype H] [DecidableEq H] |
|
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| def standardPairing : Pairing (H × Bool) := |
| ⟨Equiv.prodCongr (Equiv.refl H) Equiv.boolNot, |
| by rintro ⟨i,b⟩; cases b <;> rfl, |
| by rintro ⟨i,b⟩ h; have he := congrArg Prod.snd h; cases b <;> contradiction⟩ |
|
|
| omit [DecidableEq H] in |
| lemma standardPairing_table (f : H × Bool → Fin 3) : |
| (standardPairing (H := H)).table f = threeTable f := by |
| funext t |
| have he : histogram (fun v => threeEdgeType (f v, f ((standardPairing (H := H)).1 v))) t = |
| 2 * threeTable f t := by |
| have hr : (histogram (fun v => threeEdgeType (f v, |
| f ((standardPairing (H := H)).1 v))) t : ℝ) = 2 * threeTable f t := by |
| simp only [histogram_cast_sum, Fintype.sum_prod_type, Fintype.sum_bool, |
| standardPairing, Equiv.prodCongr_apply] |
| change (∑ i, ((if threeEdgeType (f (i,true),f (i,false)) = t then (1 : ℝ) else 0) + |
| (if threeEdgeType (f (i,false),f (i,true)) = t then 1 else 0))) = _ |
| simp_rw [threeEdgeType_symm (f (_,true)) (f (_,false))] |
| rw [Finset.sum_add_distrib] |
| change (∑ i, if threeEdgeType (f (i,false),f (i,true)) = t then (1 : ℝ) else 0) + |
| (∑ i, if threeEdgeType (f (i,false),f (i,true)) = t then (1 : ℝ) else 0) = _ |
| rw [show (threeTable f t : ℝ) = |
| ∑ i, if threeEdgeType (f (i,false),f (i,true)) = t then 1 else 0 from |
| histogram_cast_sum (threeEdgeType ∘ pairColorEquiv f) t] |
| ring |
| exact_mod_cast hr |
| change _ / 2 = _ |
| rw [he, Nat.mul_div_right _ (by decide : 0 < 2)] |
|
|
| end StandardTable |
|
|
| section MatchingIncidence |
| variable {H : Type u_3} [Fintype H] [DecidableEq H] |
|
|
| noncomputable def tableRelation (f₀ : H × Bool → Fin 3) : |
| Pairing (H × Bool) → Colorings (H × Bool) (Fin 3) (histogram f₀) → Prop := |
| fun M z => M.table z.1 = threeTable f₀ |
|
|
| noncomputable def coloringPerm (s : Fin 3 → ℕ) (p : Equiv.Perm (H × Bool)) : |
| Equiv.Perm (Colorings (H × Bool) (Fin 3) s) where |
| toFun z := ⟨z.1 ∘ p, (histogram_comp_perm z.1 p).trans z.2⟩ |
| invFun z := ⟨z.1 ∘ p.symm, (histogram_comp_perm z.1 p.symm).trans z.2⟩ |
| left_inv z := by apply Subtype.ext; funext v; simp |
| right_inv z := by apply Subtype.ext; funext v; simp |
|
|
| omit [DecidableEq H] in |
| lemma commonIncidence_table_relabel (f₀ : H × Bool → Fin 3) |
| (p : Equiv.Perm (H × Bool)) (M N : Pairing (H × Bool)) : |
| commonIncidence (tableRelation f₀) (Pairing.relabel p M) (Pairing.relabel p N) = |
| commonIncidence (tableRelation f₀) M N := by |
| classical |
| unfold commonIncidence |
| congr 1 |
| apply Fintype.card_congr |
| apply (coloringPerm (histogram f₀) p).subtypeEquiv |
| intro z |
| change ((Pairing.relabel p M).table z.1 = _ ∧ (Pairing.relabel p N).table z.1 = _) ↔ |
| (M.table (z.1 ∘ p) = _ ∧ N.table (z.1 ∘ p) = _) |
| simp only [Pairing.table_relabel] |
|
|
| lemma commonIncidence_standard (f₀ : H × Bool → Fin 3) (hH : 0 < Fintype.card H) |
| (p : Equiv.Perm (H × Bool)) : |
| commonIncidence (tableRelation f₀) standardPairing (Pairing.relabel p standardPairing) = |
| (Fintype.card {f : H × Bool → Fin 3 |
| threeTable f = threeTable f₀ ∧ threeTable (f ∘ p) = threeTable f₀} : ℝ) := by |
| classical |
| unfold commonIncidence |
| congr 1 |
| apply Fintype.card_congr |
| refine { toFun := fun z => ⟨z.1.1, ?_⟩ |
| invFun := fun f => ⟨⟨f.1, threeTable_histogram_eq hH f.2.1⟩, ?_⟩ |
| left_inv := fun _ => rfl |
| right_inv := fun _ => rfl } |
| · simpa only [tableRelation, Pairing.table_relabel, standardPairing_table] using z.2 |
| · simpa only [tableRelation, Pairing.table_relabel, standardPairing_table] using f.2 |
|
|
| lemma matching_table_trace_square (f₀ : H × Bool → Fin 3) (hH : 0 < Fintype.card H) |
| (hf₀ : ∀ i, 0 < histogram f₀ i) : |
| let D : ℝ := Fintype.card {f : H × Bool → Fin 3 |
| Matrix.trace (incidenceOperator (tableRelation f₀) D * |
| incidenceOperator (tableRelation f₀) D) ≤ (Fintype.card H + 1 : ℝ) ^ 33 := by |
| classical |
| dsimp only |
| let D : ℝ := Fintype.card {f : H × Bool → Fin 3 |
| change Matrix.trace (incidenceOperator (tableRelation f₀) D * |
| incidenceOperator (tableRelation f₀) D) ≤ _ |
| have he := trace_normalized_invariant_kernel_sq (G := Equiv.Perm (H × Bool)) |
| (standardPairing (H := H)) (fun M => Pairing.exists_relabel _ M) |
| (incidenceKernel (tableRelation f₀) D) |
| (incidenceKernel_symm (tableRelation f₀) D) |
| (fun p M N => by |
| change (Fintype.card _ : ℝ) / D ^ 2 * commonIncidence _ (Pairing.relabel p M) |
| (Pairing.relabel p N) = (Fintype.card _ : ℝ) / D ^ 2 * commonIncidence _ M N |
| rw [commonIncidence_table_relabel]) |
| change Matrix.trace (incidenceOperator (tableRelation f₀) D * |
| incidenceOperator (tableRelation f₀) D) = _ at he |
| rw [he] |
| have hp : (0 : ℝ) < Fintype.card (Equiv.Perm (H × Bool)) := Nat.cast_pos.mpr Fintype.card_pos |
| apply (div_le_iff₀ hp).mpr |
| simp only [incidenceKernel, Pairing.smul_eq_relabel, |
| commonIncidence_standard f₀ hH] |
| exact (three_table_kernel_moment f₀ hH hf₀).trans_eq (mul_comm _ _) |
|
|
| end MatchingIncidence |
| end MatchingSmoothing |
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| end OAI |
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