openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Reduction /CompletionOutputLemmas.lean
| import Mathlib.Data.Fin.Embedding | |
| import Mathlib.Tactic.Linarith | |
| import OAI.Computability.PerfectCompleteness.Machines.EncodingLemmas | |
| import OAI.Computability.PerfectCompleteness.Reduction.CanonicalGameLemmas | |
| namespace OAI | |
| namespace PerfectCompleteness.CompletionOutput | |
| open scoped BigOperators Classical | |
| open UniqueGamesTheorem.Foundations.Games | |
| open WeightRounding CompletionSoundness CompletionRounding | |
| open CompletionSoundness.LegalProjectionGame | |
| noncomputable section | |
| variable {m l r q : Nat} {L : Fin l → Type*} {R : Fin r → Type*} | |
| [∀ x, Fintype (L x)] | |
| (G : LegalProjectionGame (Fin m) (Fin l) (Fin r) L R) | |
| (legalL : ∀ x, L x ↪ Fin (2 * q)) (legalR : ∀ y, R y ↪ Fin q) | |
| (hsmall : ∀ e b, Fintype.card {a : L (G.left e) // G.projection e a = b} ≤ 2) | |
| def family : G.CompletionFamily legalL legalR := | |
| Classical.choose (G.exists_completionFamily_bounded legalL legalR hsmall (by simp)) | |
| theorem family_size (e : Fin m) : | |
| (family G legalL legalR hsmall).size e ≤ | |
| max 1 (2 * q - Fintype.card (L (G.left e))) := by | |
| simpa only [family, Fintype.card_fin] using | |
| Classical.choose_spec | |
| (G.exists_completionFamily_bounded legalL legalR hsmall (by simp)) e | |
| theorem family_total_size (hq : 0 < q) : | |
| (∑ e, (family G legalL legalR hsmall).size e) ≤ m * (2 * q) := by | |
| calc | |
| _ ≤ ∑ _e : Fin m, 2 * q := by | |
| apply Finset.sum_le_sum | |
| intro e _ | |
| exact (family_size G legalL legalR hsmall e).trans | |
| (max_le (by omega) (Nat.sub_le _ _)) | |
| _ = m * (2 * q) := by simp | |
| def construct (w : PositiveWeights m) (δ : ℚ) (hδ : 0 < δ) : Instance q := | |
| CompletionRounding.output w (family G legalL legalR hsmall) δ hδ | |
| @[simp] theorem construct_leftVertices (w : PositiveWeights m) (δ : ℚ) (hδ : 0 < δ) : | |
| (construct G legalL legalR hsmall w δ hδ).leftVertices = l := rfl | |
| @[simp] theorem construct_rightVertices (w : PositiveWeights m) (δ : ℚ) (hδ : 0 < δ) : | |
| (construct G legalL legalR hsmall w δ hδ).rightVertices = r := rfl | |
| theorem construct_edge_count (w : PositiveWeights m) (δ : ℚ) (hδ : 0 < δ) | |
| (hq : 0 < q) : | |
| (construct G legalL legalR hsmall w δ hδ).edges.length ≤ | |
| ⌈(3 : ℚ) / δ⌉₊ * m * (2 * q) := by | |
| exact (output_edge_count_le w (family G legalL legalR hsmall) δ hδ).trans | |
| (by simpa only [Nat.mul_assoc] using | |
| Nat.mul_le_mul_left ⌈(3 : ℚ) / δ⌉₊ (family_total_size G legalL legalR hsmall hq)) | |
| theorem construct_word_count (w : PositiveWeights m) (δ : ℚ) (hδ : 0 < δ) | |
| (hq : 0 < q) : | |
| (Encoding.gameWords (construct G legalL legalR hsmall w δ hδ)).length ≤ | |
| 4 + (⌈(3 : ℚ) / δ⌉₊ * m * (2 * q)) * (2 * q + 2) := by | |
| rw [Encoding.gameWords_length] | |
| exact Nat.add_le_add_left | |
| (Nat.mul_le_mul_right (2 * q + 2) (construct_edge_count G legalL legalR hsmall w δ hδ hq)) 4 | |
| variable [∀ y, Fintype (R y)] [∀ x, Nonempty (L x)] [∀ y, Nonempty (R y)] | |
| theorem construct_value_le (w : PositiveWeights m) | |
| (hweights : ∀ e, G.occurrences.weight e = (w.weight e : ℝ)) | |
| (δ : ℚ) (hδ : 0 < δ) (hq : 0 < q) : | |
| (construct G legalL legalR hsmall w δ hδ).value ≤ | |
| G.value + G.occurrences.expectation | |
| (fun e => 2 / (2 * q - Fintype.card (L (G.left e)) : Nat)) + (δ : ℝ) / 3 := | |
| output_value_le w (family G legalL legalR hsmall) hweights δ hδ hq | |
| theorem construct_value_le_of_residual_bound (w : PositiveWeights m) | |
| (hweights : ∀ e, G.occurrences.weight e = (w.weight e : ℝ)) | |
| (δ : ℚ) (hδ : 0 < δ) (hq : 0 < q) (η : ℝ) | |
| (hη : ∀ e, 2 / (2 * q - Fintype.card (L (G.left e)) : Nat) ≤ η) : | |
| (construct G legalL legalR hsmall w δ hδ).value ≤ G.value + η + (δ : ℝ) / 3 := | |
| output_value_le_of_residual_bound w (family G legalL legalR hsmall) hweights δ hδ hq η hη | |
| theorem construct_perfectlyComplete (w : PositiveWeights m) | |
| (hweights : ∀ e, G.occurrences.weight e = (w.weight e : ℝ)) | |
| (δ : ℚ) (hδ : 0 < δ) (hq : 0 < q) (hcomplete : G.value = 1) : | |
| PerfectlyComplete (construct G legalL legalR hsmall w δ hδ) := by | |
| apply ((construct G legalL legalR hsmall w δ hδ).perfectlyComplete_iff_value_eq_one hq).2 | |
| exact output_value_eq_one w (family G legalL legalR hsmall) hweights δ hδ hq hcomplete | |
| theorem construct_soundAt (w : PositiveWeights m) | |
| (hweights : ∀ e, G.occurrences.weight e = (w.weight e : ℝ)) | |
| (δ : ℚ) (hδ : 0 < δ) (hq : 0 < q) (η : ℝ) | |
| (hη : ∀ e, 2 / (2 * q - Fintype.card (L (G.left e)) : Nat) ≤ η) | |
| (d : RationalThreshold) | |
| (hbudget : G.value + η + (δ : ℝ) / 3 ≤ (d.numerator : ℝ) / d.denominator) : | |
| SoundAt d (construct G legalL legalR hsmall w δ hδ) := by | |
| apply ((construct G legalL legalR hsmall w δ hδ).soundAt_iff_value_le hq d).2 | |
| exact (construct_value_le_of_residual_bound G legalL legalR hsmall w hweights | |
| δ hδ hq η hη).trans hbudget | |
| end | |
| end PerfectCompleteness.CompletionOutput | |
| namespace PerfectCompleteness.CompletionParameters | |
| open scoped BigOperators Classical | |
| open UniqueGamesTheorem.Foundations.Games | |
| open WeightRounding CompletionSoundness CompletionOutput | |
| noncomputable section | |
| theorem residual_le_third {q n : Nat} {δ : ℝ} (hq : 0 < q) | |
| (hn : n ≤ q) (hδ : 0 < δ) (hbudget : 6 ≤ δ * q) : | |
| 2 / (2 * q - n : Nat) ≤ δ / 3 := by | |
| have hden : q ≤ 2 * q - n := by omega | |
| have hden_real : (q : ℝ) ≤ (2 * q - n : Nat) := by exact_mod_cast hden | |
| have hpos : (0 : ℝ) < (2 * q - n : Nat) := by | |
| exact_mod_cast lt_of_lt_of_le hq hden | |
| rw [div_le_iff₀ hpos] | |
| have hmul := mul_le_mul_of_nonneg_left hden_real (le_of_lt hδ) | |
| nlinarith | |
| def embedFin (T : Type*) [Fintype T] (n : Nat) (h : Fintype.card T ≤ n) : T ↪ Fin n := | |
| (Fintype.equivFin T).toEmbedding.trans (Fin.castLEEmb h) | |
| section Game | |
| variable {m l r q : Nat} {L : Fin l → Type*} {R : Fin r → Type*} | |
| [∀ x, Fintype (L x)] [∀ y, Fintype (R y)] | |
| [∀ x, Nonempty (L x)] [∀ y, Nonempty (R y)] | |
| (G : LegalProjectionGame (Fin m) (Fin l) (Fin r) L R) | |
| (legalL : ∀ x, L x ↪ Fin (2 * q)) (legalR : ∀ y, R y ↪ Fin q) | |
| (hsmall : ∀ e b, Fintype.card {a : L (G.left e) // G.projection e a = b} ≤ 2) | |
| theorem construct_value_le_delta (w : PositiveWeights m) | |
| (hweights : ∀ e, G.occurrences.weight e = (w.weight e : ℝ)) | |
| (δ : ℚ) (hδ : 0 < δ) (hq : 0 < q) | |
| (hleft : ∀ x, Fintype.card (L x) ≤ q) | |
| (hbudget : (6 : ℝ) ≤ (δ : ℝ) * q) | |
| (hsound : G.value ≤ (δ : ℝ) / 3) : | |
| (construct G legalL legalR hsmall w δ hδ).value ≤ (δ : ℝ) := by | |
| have hδreal : (0 : ℝ) < (δ : ℝ) := by exact_mod_cast hδ | |
| have hres : ∀ e, 2 / (2 * q - Fintype.card (L (G.left e)) : Nat) ≤ (δ : ℝ) / 3 := | |
| fun e => residual_le_third hq (hleft (G.left e)) hδreal hbudget | |
| have h := construct_value_le_of_residual_bound G legalL legalR hsmall | |
| w hweights δ hδ hq ((δ : ℝ) / 3) hres | |
| linarith | |
| theorem construct_soundAt_of_budget (w : PositiveWeights m) | |
| (hweights : ∀ e, G.occurrences.weight e = (w.weight e : ℝ)) | |
| (δ : ℚ) (hδ : 0 < δ) (hq : 0 < q) | |
| (hleft : ∀ x, Fintype.card (L x) ≤ q) | |
| (hbudget : (6 : ℝ) ≤ (δ : ℝ) * q) | |
| (hsound : G.value ≤ (δ : ℝ) / 3) (d : RationalThreshold) | |
| (hthreshold : (δ : ℝ) ≤ (d.numerator : ℝ) / d.denominator) : | |
| SoundAt d (construct G legalL legalR hsmall w δ hδ) := by | |
| apply ((construct G legalL legalR hsmall w δ hδ).soundAt_iff_value_le hq d).2 | |
| exact (construct_value_le_delta G legalL legalR hsmall w hweights | |
| δ hδ hq hleft hbudget hsound).trans hthreshold | |
| end Game | |
| theorem exists_target {m l r q : Nat} {L : Fin l → Type*} {R : Fin r → Type*} | |
| [∀ x, Fintype (L x)] [∀ y, Fintype (R y)] | |
| [∀ x, Nonempty (L x)] [∀ y, Nonempty (R y)] | |
| (G : LegalProjectionGame (Fin m) (Fin l) (Fin r) L R) | |
| (w : PositiveWeights m) | |
| (hweights : ∀ e, G.occurrences.weight e = (w.weight e : ℝ)) | |
| (hsmall : ∀ e b, Fintype.card {a : L (G.left e) // G.projection e a = b} ≤ 2) | |
| (hleft : ∀ x, Fintype.card (L x) ≤ q) | |
| (hright : ∀ y, Fintype.card (R y) ≤ q) | |
| (δ : ℚ) (hδ : 0 < δ) (hq : 0 < q) | |
| (hbudget : (6 : ℝ) ≤ (δ : ℝ) * q) : | |
| ∃ target : Instance q, | |
| target.leftVertices = l ∧ target.rightVertices = r ∧ | |
| target.edges.length ≤ ⌈(3 : ℚ) / δ⌉₊ * m * (2 * q) ∧ | |
| (G.value = 1 → PerfectlyComplete target) ∧ | |
| (G.value ≤ (δ : ℝ) / 3 → target.value ≤ (δ : ℝ)) := by | |
| let legalL : ∀ x, L x ↪ Fin (2 * q) := | |
| fun x => embedFin (L x) (2 * q) ((hleft x).trans (by omega)) | |
| let legalR : ∀ y, R y ↪ Fin q := fun y => embedFin (R y) q (hright y) | |
| refine ⟨construct G legalL legalR hsmall w δ hδ, rfl, rfl, | |
| construct_edge_count G legalL legalR hsmall w δ hδ hq, ?_, ?_⟩ | |
| · exact construct_perfectlyComplete G legalL legalR hsmall w hweights δ hδ hq | |
| · exact construct_value_le_delta G legalL legalR hsmall w hweights δ hδ hq hleft hbudget | |
| end | |
| end PerfectCompleteness.CompletionParameters | |
| end OAI | |