| import Mathlib |
| import OAI.Computability.MinUncut.Search.InnerParameters |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped BigOperators |
| namespace MinUncut.Outer |
| open MinUncut.Inner |
|
|
| lemma exists_list_scale (M u θ : ℝ) (q : ℕ) {ε : ℝ} (hε : 0<ε) : |
| ∃ k : ℕ, listTerm M u θ q k < ε := by |
| let C : ℝ := (⌈M^2/u^2⌉₊ : ℝ)*(⌈M^2/θ^2⌉₊ : ℝ) |
| have hC : 0≤C := by dsimp [C]; positivity |
| obtain ⟨k,hk⟩ := shared_hint_rate_vanishes q (show 0<ε/(C+1) by positivity) |
| refine ⟨k,?_⟩ |
| have hr := pow_nonneg (hintRate_nonnegative q) k |
| have hx := (lt_div_iff₀ (show 0<C+1 by positivity)).mp hk |
| change C*hintRate q^k<ε |
| nlinarith |
|
|
| lemma exists_remainder_scales {M θ p : ℝ} (hθ : 0<θ) (hp : 0<p) : |
| ∃ u : ℚ, 0<(u:ℝ) ∧ (u:ℝ)<1 ∧ ∀ q k : ℕ, |
| ∃ t : ℕ, 1≤t ∧ k≤t ∧ remainderTerm M u θ q k t < p/2 := by |
| let C : ℝ := 256/θ^4 |
| have hC : 0<C := by dsimp [C]; positivity |
| obtain ⟨u,hu,hub⟩ := exists_rat_btwn (show (0:ℝ)< min 1 (p/(8*C*(M^2+1))) by positivity) |
| obtain ⟨hu1,huB⟩ := lt_min_iff.mp hub |
| have hsmall : C*(u:ℝ)^2*M^2<p/8 := by |
| have hb := (lt_div_iff₀ (show (0:ℝ)<8*C*(M^2+1) by positivity)).mp huB |
| have hs : (u:ℝ)^2≤(u:ℝ) := by nlinarith |
| have h1 := mul_le_mul_of_nonneg_right hs (show 0≤C*M^2 by positivity) |
| nlinarith |
| refine ⟨u,hu,hu1,?_⟩ |
| intro q k |
| let R : ℝ := (M+M^2/(u:ℝ))^4 |
| have hR : 0≤R := by dsimp [R]; positivity |
| let e : ℝ := p/(8*C*(R+1)) |
| have he : 0<e := by dsimp [e]; positivity |
| let B : ℝ := ((2:ℝ)^(4*(q+3)*k)-1)*(k:ℝ)^2 |
| obtain ⟨t,ht⟩ := exists_nat_gt (max (k+1:ℝ) (B/e^2)) |
| have htk : (k:ℝ)+1<t := lt_of_le_of_lt (le_max_left _ _) ht |
| have ht0 : (0:ℝ)<t := by have := Nat.cast_nonneg (α := ℝ) k; linarith |
| have hb : B/(t:ℝ)<e^2 := by |
| apply (div_lt_iff₀ ht0).mpr |
| have hx := (div_lt_iff₀ (show 0<e^2 by positivity)).mp (lt_of_le_of_lt (le_max_right _ _) ht) |
| nlinarith |
| have hs := (Real.sqrt_lt' he).mpr hb |
| have hs0 := Real.sqrt_nonneg (B/(t:ℝ)) |
| have hr : C*Real.sqrt (B/(t:ℝ))*R<p/8 := by |
| have hx := (lt_div_iff₀ (show 0<8*C*(R+1) by positivity)).mp hs |
| nlinarith |
| refine ⟨t,by exact_mod_cast (show (1:ℝ)≤t by have := Nat.cast_nonneg (α := ℝ) k; linarith), |
| by exact_mod_cast (show (k:ℝ)≤t by linarith),?_⟩ |
| change C*((u:ℝ)^2*M^2+Real.sqrt (B/(t:ℝ))*R)<p/2 |
| nlinarith |
|
|
| structure OuterParameters (J : ℝ) (P : InnerParameters J) where |
| u : ℚ |
| b4 : ℚ |
| k : ℕ |
| t : ℕ |
| hu : 0<(u:ℝ) |
| hb4 : 0<(b4:ℝ) |
| ht : 1≤t |
| hk : k≤t |
| hgap : listTerm (Real.sqrt (P.n^P.m:ℕ)/sourceSigma J) u P.θ |
| (Fintype.card (Row P.m P.n)-1) k + |
| (64/(P.θ^2*(sourceSigma J)^2))*J*b4 + |
| remainderTerm (Real.sqrt (P.n^P.m:ℕ)/sourceSigma J) u P.θ |
| (Fintype.card (Row P.m P.n)-1) k t < P.p |
|
|
| lemma exists_outerParameters {J : ℝ} (hJ : 1≤J) (P : InnerParameters J) : |
| Nonempty (OuterParameters J P) := by |
| have hJ0 : 0<J := by linarith |
| have hσ := sourceSigma_pos hJ |
| have hp := P.hp |
| have hθ := P.hθ |
| let M : ℝ := Real.sqrt (P.n^P.m:ℕ)/sourceSigma J |
| let q := Fintype.card (Row P.m P.n)-1 |
| obtain ⟨u,hu,_,hrem⟩ := exists_remainder_scales (M := M) P.hθ P.hp |
| obtain ⟨b,hb,hbB⟩ := exists_rat_btwn (show (0:ℝ)<(P.p:ℝ)*P.θ^2*(sourceSigma J)^2/(256*J) by positivity) |
| have hbudget : (64/(P.θ^2*(sourceSigma J)^2))*J*(b:ℝ)<(P.p:ℝ)/4 := by |
| have hx := (lt_div_iff₀ (show 0<256*J by positivity)).mp hbB |
| have he : (64/(P.θ^2*(sourceSigma J)^2))*J*(b:ℝ) = (64*J*(b:ℝ))/(P.θ^2*(sourceSigma J)^2) := by ring |
| rw [he] |
| apply (div_lt_iff₀ (show 0<P.θ^2*(sourceSigma J)^2 by positivity)).mpr |
| nlinarith |
| obtain ⟨k,hk⟩ := exists_list_scale M u P.θ q (show 0<(P.p:ℝ)/4 by positivity) |
| obtain ⟨t,ht,hkt,hR⟩ := hrem q k |
| exact ⟨⟨u,b,k,t,hu,hb,ht,hkt,by dsimp [M,q] at hk hR; linarith⟩⟩ |
|
|
| end MinUncut.Outer |
|
|
| end |
| end |
|
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| end OAI |
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