openAI-math-dataset / lean /OAI /Combinatorics /IndependentSets /Reduction /FiniteParameterSelection.lean
| import Mathlib | |
| import OAI.Combinatorics.IndependentSets.PCP.LabelCoverData | |
| import OAI.Combinatorics.IndependentSets.Reduction.FullAbsorptionSquare | |
| namespace OAI | |
| namespace LargeIndependentSets | |
| open Coefficient | |
| open scoped Classical BigOperators | |
| theorem finite_parameter_selection {ε β L : ℝ} (hε : 0 < ε) (hβ : 0 < β) | |
| (hL : 0 ≤ L) (K : ℕ) : | |
| ∃ m : ℕ, 0 < m ∧ ∃ ρ : ℚ, 0 < ρ ∧ ρ < 1 ∧ | |
| L/m < β/2 ∧ (ρ:ℝ)*K < ε ∧ | |
| ∃ s : ℕ, 2 ≤ s ∧ (1-levelMass m ρ ⟨m,by omega⟩)^s < ε ∧ | |
| ∃ γ : ℝ, 0 < γ ∧ (s:ℝ)*(2*γ)^2/β^2 < ε ∧ | |
| ∃ k : ℕ, 1 ≤ k ∧ | |
| 2*L/(m*2^k) < γ ∧ 2*L*s/(2^k) < ε := by | |
| obtain ⟨m,hm⟩ := exists_nat_gt (max (1:ℝ) (2*L/β)) | |
| have hm1 : (1:ℝ) < m := (le_max_left _ _).trans_lt hm | |
| have hmp : 0 < m := by exact_mod_cast (lt_trans (by norm_num : (0:ℝ) < 1) hm1) | |
| have hmβ : 2*L < (m:ℝ)*β := (div_lt_iff₀ hβ).mp ((le_max_right _ _).trans_lt hm) | |
| have hstep : L/m < β/2 := (div_lt_iff₀ (by positivity : (0:ℝ) < m)).mpr (by nlinarith) | |
| obtain ⟨ρ,hρ,hρb⟩ := exists_pos_rat_lt (show (0:ℝ) < min 1 (ε/(K+1)) by positivity) | |
| have hρR : (0:ℝ) < ρ := by exact_mod_cast hρ | |
| have hρ1 : ρ < 1 := by exact_mod_cast (hρb.trans_le (min_le_left _ _)) | |
| have hρK : (ρ:ℝ)*K < ε := by | |
| have h := (lt_div_iff₀ (by positivity : (0:ℝ) < K+1)).mp (hρb.trans_le (min_le_right _ _)) | |
| nlinarith | |
| have hpm := levelMass_pos m hρR ⟨m,by omega⟩ | |
| have hpm1 : levelMass m ρ ⟨m,by omega⟩ ≤ 1 := by | |
| rw [← levelMass_total m hρR.le] | |
| exact Finset.single_le_sum (fun k _ => levelMass_nonneg m hρR.le k) (Finset.mem_univ _) | |
| have hpow := (tendsto_pow_atTop_nhds_zero_of_lt_one (sub_nonneg.mpr hpm1) | |
| (show 1-levelMass m ρ ⟨m,by omega⟩ < 1 by linarith)).eventually_lt_const hε | |
| obtain ⟨s,hs,hsb⟩ := (hpow.and (Filter.eventually_ge_atTop 2)).exists | |
| let γ : ℝ := Real.sqrt (ε*β^2/(8*(s+1))) | |
| have hγ : 0 < γ := Real.sqrt_pos.mpr (by positivity) | |
| have hγsq : γ^2 = ε*β^2/(8*(s+1)) := Real.sq_sqrt (by positivity) | |
| have hzero : (s:ℝ)*(2*γ)^2/β^2 < ε := by | |
| apply (div_lt_iff₀ (sq_pos_of_pos hβ)).mpr | |
| have he := (div_mul_cancel₀ (ε*β^2) (by positivity : (8*(s+1):ℝ) ≠ 0)) | |
| rw [← hγsq] at he | |
| nlinarith [sq_nonneg γ, sq_pos_of_pos hβ, show (0:ℝ) ≤ s by positivity] | |
| have hdy := tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0:ℝ) ≤ 1/2) | |
| (by norm_num : (1/2:ℝ) < 1) | |
| have hz : 0 < min (γ/(2*L+1)) (ε/(2*L*s+1)) := by positivity | |
| obtain ⟨k,hk,hkp⟩ := ((hdy.eventually_lt_const hz).and (Filter.eventually_ge_atTop 1)).exists | |
| have hpoweq : (1/2:ℝ)^k = 1/(2^k:ℝ) := by rw [div_pow,one_pow] | |
| rw [hpoweq] at hk | |
| have hkg : 2*L/(m*2^k) < γ := by | |
| have he := (lt_div_iff₀ (by positivity : (0:ℝ) < 2*L+1)).mp (hk.trans_le (min_le_left _ _)) | |
| have hpos : (0:ℝ) < 2^k := by positivity | |
| have hh := (div_lt_iff₀ hpos).mp (show (2*L+1)/2^k < γ by simpa [div_eq_mul_inv, mul_comm] using he) | |
| apply (div_lt_iff₀ (by positivity : (0:ℝ) < m*2^k)).mpr | |
| nlinarith [mul_pos hγ hpos] | |
| have hke : 2*L*s/2^k < ε := by | |
| have he := (lt_div_iff₀ (by positivity : (0:ℝ) < 2*L*s+1)).mp (hk.trans_le (min_le_right _ _)) | |
| have hpos : (0:ℝ) < 2^k := by positivity | |
| have hh := (div_lt_iff₀ hpos).mp (show (2*L*s+1)/2^k < ε by simpa [div_eq_mul_inv, mul_comm] using he) | |
| exact (div_lt_iff₀ hpos).mpr (by linarith) | |
| exact ⟨m,hmp,ρ,hρ,hρ1,hstep,hρK,s,hsb,hs,γ,hγ,hzero,k,hkp,hkg,hke⟩ | |
| lemma final_soundness_choice {ε : ℝ} (hε : 0 < ε) (r d : ℕ) : | |
| ∃ σ : ℚ, 0 < σ ∧ σ < 1 ∧ (4:ℝ)^r*d^2*σ < ε := by | |
| obtain ⟨σ,hσ,hσb⟩ := exists_pos_rat_lt | |
| (show (0:ℝ) < min 1 (ε/((4:ℝ)^r*d^2+1)) by positivity) | |
| have hσR : (0:ℝ) < σ := by exact_mod_cast hσ | |
| refine ⟨σ,hσ,?_,?_⟩ | |
| · exact_mod_cast hσb.trans_le (min_le_left _ _) | |
| · have h := (lt_div_iff₀ (by positivity : (0:ℝ) < (4:ℝ)^r*d^2+1)).mp | |
| (hσb.trans_le (min_le_right _ _)) | |
| nlinarith | |
| end LargeIndependentSets | |
| namespace LargeIndependentSets.PhaseTest | |
| open scoped BigOperators Classical | |
| open Coefficient BooleanJunta | |
| variable {ι : Type} [Fintype ι] [DecidableEq ι] | |
| variable {M : ι → Type} [∀ j, Fintype (M j)] | |
| variable {κ : Type} [Fintype κ] | |
| def gridLocation (p : ∀ j, κ → M j) {m P : ℕ} | |
| (t : ι → Fin (m+1)) (θ : Sigma M → Fin P) : κ → ZMod (m*P) := | |
| fun k => ∑ j, ((t j).val * (θ ⟨j,p j k⟩).val : ℕ) | |
| omit [DecidableEq ι] [∀ index, Fintype (M index)] [Fintype κ] in | |
| lemma gridLocation_phase (p : ∀ j, κ → M j) {m P : ℕ} | |
| (hm : 0 < m) (hP : 0 < P) (t : ι → Fin (m+1)) (θ : Sigma M → Fin P) : | |
| letI : NeZero (m*P) := ⟨Nat.ne_of_gt (Nat.mul_pos hm hP)⟩ | |
| torusGrid (gridLocation p t θ) = | |
| phase p (fun j => value (t j)) (fun c => (θ c).val/(P:ℝ)) := by | |
| let : NeZero (m*P) := ⟨Nat.ne_of_gt (Nat.mul_pos hm hP)⟩ | |
| funext k | |
| simp only [torusGrid, gridLocation, map_sum, phase] | |
| apply Finset.sum_congr rfl | |
| intro j _ | |
| rw [ZMod.toAddCircle_natCast] | |
| congr 1 | |
| simp only [value, Nat.cast_mul] | |
| ring | |
| omit [DecidableEq ι] [∀ index, Fintype (M index)] [Fintype κ] in | |
| lemma gridLocation_cube (p : ∀ j, κ → M j) {m : ℕ} | |
| (hm : 0 < m) (k : ℕ) (t : ι → Fin (m+1)) (v : Sigma M → Cube k) : | |
| letI : NeZero (m*2^k) := ⟨Nat.ne_of_gt (Nat.mul_pos hm (by positivity))⟩ | |
| torusGrid (gridLocation p t (fun c => gridEquiv k (v c))) = | |
| phase p (fun j => value (t j)) (fun c => scalarCell (v c) 0) := by | |
| rw [gridLocation_phase p hm (by positivity)] | |
| congr 1 | |
| funext c | |
| exact by simpa using (scalarCell_zero_grid (v c)).symm | |
| end LargeIndependentSets.PhaseTest | |
| end OAI | |