| import OAI.NumberTheory.Jacobsthal.Estimates.SuccessfulRewardComparison |
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| namespace OAI |
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| namespace Erdos970 |
| open scoped _root_.Erdos970 |
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| section |
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| namespace NumberTheoryLean.CompactTestGeometry |
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| open _root_.Set _root_.MeasureTheory |
| open FinitePathGeometry FinitePathMeasures PrimeHistories PrimeKilledChain |
| open ActualProcessCoupling ActualFlagInvariant RegeneratingInverseBands |
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| variable {w ell S : ℝ} {start : Node} |
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| noncomputable def primeTest (F : Side → ℝ×ℝ → ℝ) : ChainState w ell S start → ℝ |
| | none => 0 |
| | some h => F h.node.side (h.node.gap,h.node.ratio) |
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| noncomputable def continuousTest (v : ℝ) (F : Side → ℝ×ℝ → ℝ) : CemeteryKernel.Space CostState → ℝ |
| | .inr _ => 0 |
| | .inl z => F (stateSide z.1) (gapValue v z,stateRatio z.1) |
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| theorem phase_uniform_modulus (F : Side → ℝ×ℝ → ℝ) (hF : ∀ i,UniformContinuous (F i)) |
| {η : ℝ} (hη : 0 < η) : ∃ δ : ℝ,0 < δ ∧ ∀ i : Side,∀ x y : ℝ×ℝ, |
| |x.1-y.1| ≤ δ → |x.2-y.2| ≤ δ → |F i x-F i y| ≤ η := by |
| obtain ⟨de,hde,he⟩ := Metric.uniformContinuous_iff_le.mp (hF .even) η hη |
| obtain ⟨do_,hdo,ho⟩ := Metric.uniformContinuous_iff_le.mp (hF .odd) η hη |
| refine ⟨min de do_,lt_min hde hdo,?_⟩ |
| intro i x y hx hy |
| have hdist : dist x y ≤ min de do_ := by rw [Prod.dist_eq,Real.dist_eq,Real.dist_eq]; exact max_le hx hy |
| cases i with |
| | even => simpa only [Real.dist_eq] using he (hdist.trans (min_le_left _ _)) |
| | odd => simpa only [Real.dist_eq] using ho (hdist.trans (min_le_right _ _)) |
| |
| theorem compact_gap_close {K r q d : ℝ} (_hK : 0 ≤ K) (hr : 0 < r) (hq : 0 < q) |
| (hd : |Real.log r-Real.log q| ≤ d) (hd1 : d ≤ 1) (hcompact : r ≤ K ∨ q ≤ K) : |
| r ≤ 3*K ∧ q ≤ 3*K ∧ |r-q| ≤ 3*K*(Real.exp d-1) := by |
| have hrel := NearbyParentGeometry.log_gap_relative hr hq hd |
| have he3 : Real.exp d ≤ 3 := (Real.exp_le_exp.mpr hd1).trans Real.exp_one_lt_three.le |
| have hd0 : 0 ≤ d := (abs_nonneg _).trans hd |
| have he0 : 0 ≤ Real.exp d-1 := sub_nonneg.mpr (Real.one_le_exp_iff.mpr hd0) |
| have hrq : r ≤ 3*q := hrel.1.trans (mul_le_mul_of_nonneg_right he3 hq.le) |
| have hqr : q ≤ 3*r := hrel.2.trans (mul_le_mul_of_nonneg_right he3 hr.le) |
| have hboth : r ≤ 3*K ∧ q ≤ 3*K := by rcases hcompact with h | h <;> constructor <;> linarith |
| refine ⟨hboth.1,hboth.2,abs_le.mpr ⟨?_,?_⟩⟩ |
| · have hh := mul_le_mul_of_nonneg_right hboth.1 he0 |
| nlinarith |
| · have hh := mul_le_mul_of_nonneg_right hboth.2 he0 |
| nlinarith |
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| theorem actual_compact_test_comparison (F : Side → ℝ×ℝ → ℝ) |
| (hF : ∀ i,UniformContinuous (F i)) {K η : ℝ} (hK : 0 ≤ K) (hη : 0 < η) |
| (hsupport : ∀ i : Side,∀ r s : ℝ,K < r → F i (r,s) = 0) : |
| ∃ δ : ℝ,0 < δ ∧ ∀ w ell S : ℝ,∀ start : Node, |
| 0 ≤ ell → 0 < start.gap → Valid start.side start.ratio → |
| ∀ v mesh : ℝ,∀ n : ℕ,mesh ≤ δ → 4*(n:ℝ)*mesh ≤ 1 → |
| 3*K*(Real.exp (4*(n:ℝ)*mesh)-1) ≤ δ → |
| ∀ q : JointState w ell S start,GoodAt v mesh n q → |
| |primeTest F q.1-continuousTest v F q.2| ≤ η*PrimeCompactVisits.visit (3*K) q.1 := by |
| obtain ⟨δ,hδ,hmod⟩ := phase_uniform_modulus F hF hη |
| refine ⟨δ,hδ,?_⟩ |
| intro w ell S start hell hr hs v mesh n hm hd hgap q hgood |
| rcases q with ⟨p,y⟩ |
| cases p with |
| | none => |
| cases y with |
| | inr u => simp [primeTest,continuousTest,PrimeCompactVisits.visit] |
| | inl z => exact False.elim hgood |
| | some h => |
| cases y with |
| | inr u => exact False.elim hgood |
| | inl z => |
| have hh : liveGood v mesh n h z := hgood |
| have hr' : 0 < h.node.gap := terminal_gap_positive hell hr hs h.admissible |
| have hq' : 0 < gapValue v z := Real.exp_pos _ |
| by_cases hz : F h.node.side (h.node.gap,h.node.ratio) = 0 ∧ |
| F (stateSide z.1) (gapValue v z,stateRatio z.1) = 0 |
| · simp only [primeTest,continuousTest,hz.1,hz.2,sub_zero,abs_zero] |
| exact mul_nonneg hη.le (PrimeCompactVisits.visit_nonneg _ _) |
| · have hcomp : h.node.gap ≤ K ∨ gapValue v z ≤ K := by |
| by_contra hn |
| push Not at hn |
| exact hz ⟨hsupport _ _ _ hn.1,hsupport _ _ _ hn.2⟩ |
| have hclose := compact_gap_close hK hr' hq' hh.2.2.2 hd hcomp |
| have hvalue := hmod h.node.side (h.node.gap,h.node.ratio) (gapValue v z,stateRatio z.1) |
| (hclose.2.2.trans hgap) (hh.2.2.1.trans hm) |
| simp only [primeTest,continuousTest,PrimeCompactVisits.visit,ite_eq_left hclose.1,mul_one] |
| simpa only [hh.2.1] using hvalue |
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| end NumberTheoryLean.CompactTestGeometry |
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| end |
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| end Erdos970 |
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| end OAI |
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