openAI-math-dataset / lean /OAI /Computability /VertexCover /Reduction /PrivatePairProjectionRight.lean
| import Mathlib | |
| import OAI.Computability.VertexCover.Reduction.CoordinateProductInvariant | |
| import OAI.Computability.VertexCover.Reduction.RestrictionStarBudget | |
| namespace OAI | |
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| namespace VertexCover.LabelCover | |
| open VertexCover.Restriction | |
| open EnergyForm.Projection | |
| open scoped BigOperators | |
| noncomputable def privateLocalKey (Φ : LabelCover) {d : ℕ} | |
| (J : Finset (Fin d)) (j : Fin d) (e : PositionPair d) (c : Fin Φ.M) : | |
| Option (Fin Φ.M) × Option (Fin Φ.u) × Option (Fin Φ.v) := by | |
| classical | |
| exact (if e ∈ internalPairs J then none else some c, | |
| if e ∈ internalPairs J ∧ e.1.1 = j then some (Φ.left c) else none, | |
| if e ∈ internalPairs J ∧ e.1.2 = j then some (Φ.right c) else none) | |
| noncomputable def privateSeedKey (Φ : LabelCover) {d : ℕ} | |
| (J : Finset (Fin d)) (j : Fin d) (seed : Φ.Seeds d) := | |
| fun e => Φ.privateLocalKey J j e (seed e) | |
| noncomputable def privateSeedProjection (Φ : LabelCover) {d : ℕ} | |
| (J : Finset (Fin d)) (j : Fin d) : (Φ.restrictionForm d).Projection := | |
| condition (B := Φ.restrictionBase d) (Φ.privateSeedKey J j) | |
| noncomputable def privatePairProjection (Φ : LabelCover) {d : ℕ} | |
| (J : Finset (Fin d)) (j : Fin d) (e : PositionPair d) : | |
| (Φ.restrictionForm d).Projection := | |
| coordinateCondition (Φ.restrictionBase d) ⟨0, Φ.M_pos⟩ e (Φ.privateLocalKey J j e) | |
| theorem privatePairProjection_commute (Φ : LabelCover) {d : ℕ} | |
| (J : Finset (Fin d)) (j : Fin d) (e b : PositionPair d) (f : Φ.RestrictionSpace d) : | |
| Φ.privatePairProjection J j e (Φ.privatePairProjection J j b f) = | |
| Φ.privatePairProjection J j b (Φ.privatePairProjection J j e f) := by | |
| by_cases heb : e = b | |
| · subst b; rfl | |
| · change conditionMap (coordinateKey ⟨0,Φ.M_pos⟩ e (Φ.privateLocalKey J j e)) | |
| (conditionMap (coordinateKey ⟨0,Φ.M_pos⟩ b (Φ.privateLocalKey J j b)) f) = | |
| conditionMap (coordinateKey ⟨0,Φ.M_pos⟩ b (Φ.privateLocalKey J j b)) | |
| (conditionMap (coordinateKey ⟨0,Φ.M_pos⟩ e (Φ.privateLocalKey J j e)) f) | |
| apply conditionMap_coordinate_commute | |
| exact heb | |
| theorem privateSeedProjection_product (Φ : LabelCover) {d : ℕ} | |
| (J : Finset (Fin d)) (j : Fin d) (f : Φ.RestrictionSpace d) : | |
| Φ.privateSeedProjection J j | |
| (productMap (Φ.privatePairProjection J j) Finset.univ.toList f) = | |
| productMap (Φ.privatePairProjection J j) Finset.univ.toList f := by | |
| classical | |
| let F := productMap (Φ.privatePairProjection J j) Finset.univ.toList f | |
| change Φ.privateSeedProjection J j F = F | |
| funext seed | |
| apply conditionMap_of_invariant | |
| intro s t hst | |
| apply invariant_of_coordinate_invariant (Φ.privateLocalKey J j) F _ s t | |
| (fun e => congrFun hst e) | |
| intro e a c hc | |
| have he := productMap_fixed_of_mem (Φ.privatePairProjection J j) | |
| (Φ.privatePairProjection_commute J j) Finset.univ.toList e (by simp) f | |
| change Φ.privatePairProjection J j e F = F at he | |
| rw [← he] | |
| apply conditionMap_eq_of_key_eq | |
| simp only [coordinateKey, Function.update_idem, Function.update_self, hc] | |
| theorem query_eq_questions_iff (Φ : LabelCover) {d : ℕ} (s t : Φ.Seeds d) (j : Fin d) : | |
| Φ.query s j = Φ.query t j ↔ | |
| (∀ k (h : j < k), Φ.left (s ⟨(j,k),h⟩) = Φ.left (t ⟨(j,k),h⟩)) ∧ | |
| (∀ k (h : k < j), Φ.right (s ⟨(k,j),h⟩) = Φ.right (t ⟨(k,j),h⟩)) := by | |
| constructor | |
| · intro h | |
| have hp := eq_of_heq (Sigma.mk.inj h).2 | |
| exact ⟨fun k hk => congrFun (congrArg Prod.fst hp) ⟨k,hk⟩, | |
| fun k hk => congrFun (congrArg Prod.snd hp) ⟨k,hk⟩⟩ | |
| · rintro ⟨hL,hR⟩ | |
| exact Φ.query_eq_of_questions s t j hL hR | |
| theorem privateSeedKey_eq_iff (Φ : LabelCover) {d : ℕ} | |
| (J : Finset (Fin d)) (j : Fin d) (s t : Φ.Seeds d) : | |
| Φ.privateSeedKey J j s = Φ.privateSeedKey J j t ↔ | |
| (∀ e, e ∉ internalPairs J → s e = t e) ∧ Φ.query s j = Φ.query t j := by | |
| classical | |
| constructor | |
| · intro h | |
| have hs : ∀ e, e ∉ internalPairs J → s e = t e := by | |
| intro e he | |
| have hh := congrArg Prod.fst (congrFun h e) | |
| simpa only [privateSeedKey, privateLocalKey, ite_eq_right he, Option.some.injEq] using hh | |
| refine ⟨hs, (Φ.query_eq_questions_iff s t j).mpr ⟨?_, ?_⟩⟩ | |
| · intro k hkj | |
| let e : PositionPair d := ⟨(j,k),hkj⟩ | |
| by_cases he : e ∈ internalPairs J | |
| · have hh := congrArg (fun x => x.2.1) (congrFun h e) | |
| simpa [privateSeedKey, privateLocalKey, he, e] using hh | |
| · exact congrArg Φ.left (hs e he) | |
| · intro k hkj | |
| let e : PositionPair d := ⟨(k,j),hkj⟩ | |
| by_cases he : e ∈ internalPairs J | |
| · have hh := congrArg (fun x => x.2.2) (congrFun h e) | |
| simpa [privateSeedKey, privateLocalKey, he, e] using hh | |
| · exact congrArg Φ.right (hs e he) | |
| · rintro ⟨hs,hq⟩ | |
| obtain ⟨hL,hR⟩ := (Φ.query_eq_questions_iff s t j).mp hq | |
| funext e | |
| by_cases he : e ∈ internalPairs J | |
| · by_cases hlj : e.1.1 = j | |
| · have hrj : e.1.2 ≠ j := by have := e.2; rw [hlj] at this; exact ne_of_gt this | |
| have ht := hL e.1.2 (hlj ▸ e.2) | |
| have hp : (⟨(j,e.1.2), hlj ▸ e.2⟩ : PositionPair d) = e := by | |
| apply Subtype.ext; simp only [← hlj, Prod.mk.eta] | |
| rw [hp] at ht | |
| simp [privateSeedKey, privateLocalKey, he, hlj, hrj, ht] | |
| · by_cases hrj : e.1.2 = j | |
| · have ht := hR e.1.1 (hrj ▸ e.2) | |
| have hp : (⟨(e.1.1,j), hrj ▸ e.2⟩ : PositionPair d) = e := by | |
| apply Subtype.ext; simp only [← hrj, Prod.mk.eta] | |
| rw [hp] at ht | |
| simp [privateSeedKey, privateLocalKey, he, hlj, hrj, ht] | |
| · simp [privateSeedKey, privateLocalKey, he, hlj, hrj] | |
| · simp [privateSeedKey, privateLocalKey, he, hs e he] | |
| theorem privatePairProjection_outside (Φ : LabelCover) {d : ℕ} | |
| (J : Finset (Fin d)) (j : Fin d) (e : PositionPair d) (he : e ∉ internalPairs J) : | |
| Φ.privatePairProjection J j e = ident := by | |
| classical | |
| apply EnergyForm.Projection.ext | |
| intro f | |
| funext seed | |
| change conditionMap (coordinateKey ⟨0,Φ.M_pos⟩ e (Φ.privateLocalKey J j e)) f seed = f seed | |
| apply conditionMap_of_invariant | |
| intro s t hst | |
| obtain ⟨ho,hq⟩ := (coordinateKey_eq_iff ⟨0,Φ.M_pos⟩ e (Φ.privateLocalKey J j e) s t).mp hst | |
| have heq : s e = t e := by | |
| have hh := congrArg Prod.fst hq | |
| simpa [privateLocalKey, he] using hh | |
| have hst' : s = t := by | |
| funext b | |
| by_cases hb : b = e | |
| · subst b; exact heq | |
| · exact ho b hb | |
| rw [hst'] | |
| theorem privatePairProjection_nonincident (Φ : LabelCover) {d : ℕ} | |
| (J : Finset (Fin d)) (j : Fin d) (e : PositionPair d) | |
| (he : e ∈ internalPairs J) (hL : e.1.1 ≠ j) (hR : e.1.2 ≠ j) : | |
| Φ.privatePairProjection J j e = Φ.seedProjection e := by | |
| apply EnergyForm.Projection.ext | |
| intro f | |
| change conditionMap (coordinateKey ⟨0,Φ.M_pos⟩ e (Φ.privateLocalKey J j e)) f = | |
| conditionMap (fun seed : Φ.Seeds d => Function.update seed e ⟨0,Φ.M_pos⟩) f | |
| exact congrArg (fun L => L f) (conditionMap_congr | |
| (coordinateKey ⟨0,Φ.M_pos⟩ e (Φ.privateLocalKey J j e)) | |
| (fun seed : Φ.Seeds d => Function.update seed e ⟨0,Φ.M_pos⟩) (fun s t => by | |
| simp [coordinateKey, privateLocalKey, he, hL, hR])) | |
| theorem privatePairProjection_left (Φ : LabelCover) {d : ℕ} | |
| (J : Finset (Fin d)) (j : Fin d) (e : PositionPair d) | |
| (he : e ∈ internalPairs J) (hL : e.1.1 = j) : | |
| Φ.privatePairProjection J j e = Φ.oppositeProjection e.1.2 e := by | |
| classical | |
| have hR : e.1.2 ≠ j := by rw [← hL]; exact ne_of_gt e.2 | |
| apply EnergyForm.Projection.ext | |
| intro f | |
| change conditionMap (coordinateKey ⟨0,Φ.M_pos⟩ e (Φ.privateLocalKey J j e)) f = | |
| conditionMap (Φ.oppositePairKey e.1.2 e) f | |
| exact congrArg (fun L => L f) (conditionMap_congr | |
| (coordinateKey ⟨0,Φ.M_pos⟩ e (Φ.privateLocalKey J j e)) | |
| (Φ.oppositePairKey e.1.2 e) (fun s t => by | |
| simp only [coordinateKey, privateLocalKey, oppositePairKey, he, hL, and_self, ite_true, hR, and_false, ite_false, Prod.mk.injEq, Option.some.injEq, true_and, and_true, ite_eq_right (Ne.symm hR)] | |
| constructor | |
| · rintro ⟨ho,hq⟩ | |
| exact Prod.ext ho (congrArg Sum.inl hq) | |
| · intro h | |
| exact ⟨congrArg Prod.fst h, Sum.inl.inj (congrArg Prod.snd h)⟩)) | |
| theorem privatePairProjection_right (Φ : LabelCover) {d : ℕ} | |
| (J : Finset (Fin d)) (j : Fin d) (e : PositionPair d) | |
| (he : e ∈ internalPairs J) (hR : e.1.2 = j) : | |
| Φ.privatePairProjection J j e = Φ.oppositeProjection e.1.1 e := by | |
| classical | |
| have hL : e.1.1 ≠ j := by rw [← hR]; exact ne_of_lt e.2 | |
| apply EnergyForm.Projection.ext | |
| intro f | |
| change conditionMap (coordinateKey ⟨0,Φ.M_pos⟩ e (Φ.privateLocalKey J j e)) f = | |
| conditionMap (Φ.oppositePairKey e.1.1 e) f | |
| exact congrArg (fun L => L f) (conditionMap_congr | |
| (coordinateKey ⟨0,Φ.M_pos⟩ e (Φ.privateLocalKey J j e)) | |
| (Φ.oppositePairKey e.1.1 e) (fun s t => by | |
| simp only [coordinateKey, privateLocalKey, oppositePairKey, he, hL, hR, and_false, and_self, ite_false, ite_true, Prod.mk.injEq, Option.some.injEq, true_and] | |
| constructor | |
| · rintro ⟨ho,hq⟩ | |
| exact Prod.ext ho (congrArg Sum.inr hq) | |
| · intro h | |
| exact ⟨congrArg Prod.fst h, Sum.inr.inj (congrArg Prod.snd h)⟩)) | |
| end VertexCover.LabelCover | |
| end | |
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| end OAI | |