| import OAI.Probability.CriticalZ3.SeedQuality |
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| namespace OAI |
|
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| universe uOmega |
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| open MeasureTheory ProbabilityTheory Classical Filter |
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| open scoped ENNReal BigOperators Matrix |
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| namespace MeasureTheory |
| open scoped _root_.MeasureTheory |
| variable {Ω : Type uOmega} [MeasurableSpace Ω] {μ : _root_.MeasureTheory.Measure Ω} [_root_.MeasureTheory.IsFiniteMeasure μ] |
| theorem finite_measureReal_mono_ae {A B : Set Ω} (h : A ≤ᵐ[μ] B) : |
| μ.real A ≤ μ.real B := |
| ENNReal.toReal_le_toReal (_root_.MeasureTheory.measure_ne_top μ A) (_root_.MeasureTheory.measure_ne_top μ B) |>.mpr (_root_.MeasureTheory.measure_mono_ae h) |
| end MeasureTheory |
|
|
| namespace CriticalZ3.LatticeGraph |
| open MeasureTheory ProbabilityTheory Classical InfiniteConditioning |
| variable (G : LatticeGraph) (m : Model) |
|
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| theorem entrant_exterior {C : Box} {ω : m.Config} {o : G.Node} {y : Vertex} |
| (hy : y ∈ G.entrants m C ω o) : C.Exterior y := |
| (Finset.mem_filter.mp hy).2.choose_spec.1 |
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| theorem entrant_node {C : Box} {ω : m.Config} {o : G.Node} {y : Vertex} |
| (hy : y ∈ G.entrants m C ω o) : y ∈ G.vertices := (Finset.mem_filter.mp hy).1 |
|
|
| /-- Actual separated seed trials under an arbitrary fixed outside pattern. -/ |
| theorem fixed_large_entrants_failure (p : m.Bit → Set.Icc (0 : ℝ) 1) |
| (q : Set.Icc (0 : ℝ) 1) (hq : 0 < (q : ℝ)) |
| (C : Box) (r K : ℕ) (hK : 0 < K) |
| (hw : ∀ i, C.lower i+r ≤ C.upper i-r) (hC : G.ContainsBox (C.expand 1)) |
| (o : G.Node) (T : Set Vertex) (ξ : m.Config) |
| (hp : ∀ e ∈ (C.expand 1).bits m, p e = q) {γ τ : ℝ} (hτ : 0 < τ) |
| (hqual : ∀ y, C.Exterior y → (ProductCylinder.law p).real |
| (allValue ((C.seedBox r y).bits m) true ∩ |
| {ω | conditionalProbability p (C.bits m) |
| {ω | G.Meets m ω (G.relayNode C r hw (hC.mono G (fun _ => C.contains_expand 1)) y) T} ω ≤ 1-τ}) |
| ≤ (q : ℝ)^Box.seedCard m r*γ/τ) |
| (hlarge : K*(8*r+5)^3 ≤ (G.entrants m C ξ o).card) : |
| (ProductCylinder.law (fixedParameter p (G.outsideBits m C) ξ)).real |
| {ω | ¬G.Meets m ω o (G.goodRelays m p (C.bits m) T τ ω)} ≤ |
| 4/(K*(q : ℝ)^Box.successCard m r)+2*γ/τ := by |
| let pf := fixedParameter p (G.outsideBits m C) ξ |
| let μ := ProductCylinder.law pf |
| let c := (q : ℝ)^Box.successCard m r |
| obtain ⟨t,ht,hcard,hsep⟩ := Box.separated_subset (G.entrants m C ξ o) r K hlarge |
| let v (y : t) := G.relayNode C r hw (hC.mono G (fun _ => C.contains_expand 1)) y.1 |
| let F (y : t) := {ω | conditionalProbability p (C.bits m) {ω | G.Meets m ω (v y) T} ω ≤ 1-τ} |
| let A (y : t) := allValue (C.successBits r m y.1) true |
| let B (y : t) := A y ∩ F y |
| have hy (y : t) : C.Exterior y.1 := G.entrant_exterior m (ht y.2) |
| have hF (y : t) : ProductCylinder.DeterminedBy (F := Prop) (C.bits m) (fun ω => ω ∈ F y) := by |
| intro ω ω' h |
| simp only [F,Set.mem_ofPred_eq,conditionalProbability_determined p (C.bits m) _ ω ω' h] |
| have hfresh (y : t) : ∀ e ∈ C.successBits r m y.1, pf e = q := by |
| intro e he |
| change fixedParameter p (G.outsideBits m C) ξ e = q |
| rw [fixedParameter,ite_eq_right (Finset.disjoint_left.mp (G.successBits_disjoint_outside m C r hw (hy y)) he)] |
| exact hp e (C.successBits_contained r m hw (hy y) he) |
| have hmass (y : t) : μ.real (A y) = c := by |
| rw [allValue_probability_constant pf _ q (hfresh y)] |
| simp only [bitProbability,ite_true,C.successBits_card r m hw (hy y)] |
| rfl |
| have hpair (y z : t) (hyz : y ≠ z) : μ.real (A y ∩ A z) ≤ c*c := by |
| have hd := C.successBits_disjoint r m hw (hy y) (hy z) |
| (hsep y y.2 z z.2 (fun h => hyz (Subtype.ext h))) |
| have hind := indepSet_determined pf _ _ hd (A y) (A z) |
| (allValue_determined _ _) (allValue_determined _ _) |
| have hh : μ.real (A y ∩ A z) = μ.real (A y)*μ.real (A z) := by |
| simp only [Measure.real_def,μ,hind.measure_inter_eq_mul,ENNReal.toReal_mul] |
| rw [hh,hmass y,hmass z] |
| have hbad (y : t) : μ.real (B y) ≤ c*γ/τ := |
| G.fixed_bad_success_mass m p q C r hw (hy y) ξ hp (F y) (hF y) (hqual y (hy y)) |
| have hbound := TrialCounts.no_good_probability μ A B |
| (fun _ => measurableSet_allValue _ _) |
| (fun y => (measurableSet_allValue _ _).inter (measurableSet_conditional_test _ _ _ _)) |
| (c := c) (γ := γ) (τ := τ) |
| (by rw [Fintype.card_coe,hcard]; exact mul_pos (Nat.cast_pos.mpr hK) (pow_pos hq _)) |
| hτ hmass hpair hbad |
| rw [Fintype.card_coe,hcard] at hbound |
| apply (finite_measureReal_mono_ae (μ := μ) ?_).trans hbound |
| filter_upwards [ae_fixed_determined p (G.outsideBits m C) ξ |
| (fun ω => G.entrants m C ω o) (G.entrants_determined m C o)] with ω hω |
| intro hfail y hAy |
| refine ⟨hAy,?_⟩ |
| by_contra hnot |
| have hg : 1-τ < conditionalProbability p (C.bits m) {ω | G.Meets m ω (v y) T} ω := |
| lt_of_not_ge hnot |
| have hyω : y.1 ∈ G.entrants m C ω o := hω.symm ▸ ht y.2 |
| let yn : G.Node := ⟨y.1,G.entrant_node m hyω⟩ |
| obtain ⟨u,hu,hconn⟩ := G.success_reaches_center m C r hw hC (y := yn) hyω hAy |
| have huv : u = v y := Subtype.ext hu |
| apply hfail |
| refine ⟨u,?_,hconn⟩ |
| exact ⟨v y,congrArg Subtype.val huv.symm,hg⟩ |
|
|
| end CriticalZ3.LatticeGraph |
|
|
| namespace CriticalZ3.LatticeGraph |
| open MeasureTheory ProbabilityTheory Classical InfiniteConditioning |
| variable (G : LatticeGraph) (m : Model) |
|
|
| theorem measurableSet_large_entrants (C : Box) (o : G.Node) (L : ℕ) : |
| MeasurableSet {ω | L ≤ (G.entrants m C ω o).card} := by |
| apply ProductCylinder.measurableSet_determined (G.outsideBits m C) |
| intro ω ω' h |
| simp only [Set.mem_ofPred_eq,G.entrants_determined m C o ω ω' h] |
|
|
| /-- Integrating the exact separated-trial estimate, including histories with |
| fewer than L entries. -/ |
| theorem large_entrants_failure (p : m.Bit → Set.Icc (0 : ℝ) 1) |
| (q : Set.Icc (0 : ℝ) 1) (hq : 0 < (q : ℝ)) |
| (C : Box) (r K : ℕ) (hK : 0 < K) |
| (hw : ∀ i, C.lower i+r ≤ C.upper i-r) (hC : G.ContainsBox (C.expand 1)) |
| (o : G.Node) (T : Set Vertex) |
| (hp : ∀ e ∈ (C.expand 1).bits m, p e = q) {γ τ : ℝ} (hγ : 0 ≤ γ) (hτ : 0 < τ) |
| (hqual : ∀ y, C.Exterior y → (ProductCylinder.law p).real |
| (allValue ((C.seedBox r y).bits m) true ∩ |
| {ω | conditionalProbability p (C.bits m) |
| {ω | G.Meets m ω (G.relayNode C r hw (hC.mono G (fun _ => C.contains_expand 1)) y) T} ω ≤ 1-τ}) |
| ≤ (q : ℝ)^Box.seedCard m r*γ/τ) : |
| (ProductCylinder.law p).real |
| {ω | K*(8*r+5)^3 ≤ (G.entrants m C ω o).card ∧ |
| ¬G.Meets m ω o (G.goodRelays m p (C.bits m) T τ ω)} ≤ |
| 4/(K*(q : ℝ)^Box.successCard m r)+2*γ/τ := by |
| let E : Set m.Config := {ω | K*(8*r+5)^3 ≤ (G.entrants m C ω o).card ∧ |
| ¬G.Meets m ω o (G.goodRelays m p (C.bits m) T τ ω)} |
| have hm : MeasurableSet E := (G.measurableSet_large_entrants m C o _).inter |
| (G.measurableSet_meets_goodRelays m p (C.bits m) o T τ).compl |
| have hr := select_probability_le p (G.outsideBits m C) (fun _ => E) (fun _ => hm) |
| (b := 4/(K*(q : ℝ)^Box.successCard m r)+2*γ/τ) ?_ |
| · exact hr |
| · intro a |
| let ξ := ProductCylinder.extend (G.outsideBits m C) a |
| by_cases hlarge : K*(8*r+5)^3 ≤ (G.entrants m C ξ o).card |
| · exact (measureReal_mono (fun _ h => h.2)).trans |
| (G.fixed_large_entrants_failure m p q hq C r K hK hw hC o T ξ hp hτ hqual hlarge) |
| · have he : (ProductCylinder.law (fixedParameter p (G.outsideBits m C) ξ)).real E = 0 := by |
| have hz : (ProductCylinder.law (fixedParameter p (G.outsideBits m C) ξ)) E = 0 := by |
| apply measure_eq_zero_iff_ae_notMem.mpr |
| filter_upwards [ae_fixed_determined p (G.outsideBits m C) ξ |
| (fun ω => G.entrants m C ω o) (G.entrants_determined m C o)] with ω hω |
| intro h |
| exact hlarge (hω ▸ h.1) |
| simp only [Measure.real_def,hz,ENNReal.toReal_zero] |
| rw [he] |
| positivity |
|
|
| end CriticalZ3.LatticeGraph |
|
|
| namespace CriticalZ3 |
| open MeasureTheory ProbabilityTheory Classical InfiniteConditioning |
| namespace Box |
|
|
| theorem entryBit_internal_expand {C : Box} {y : Vertex} (hy : C.Exterior y) (m : Model) : |
| C.entryBit m y ∈ (C.expand 1).bits m := by |
| cases m |
| · exact LatticeGraph.incidence_internal (exterior_expand hy) |
| (C.contains_expand 1 (entry_spec hy).1) (neighborBond_incidence (entry_spec hy).2) |
| · exact (mem_vertices _ _).mpr (C.contains_expand 1 (entry_spec hy).1) |
| end Box |
|
|
| namespace LatticeGraph |
| variable (G : LatticeGraph) (m : Model) |
|
|
| /-- The shell + separated-seed + interior-quality extension, with all probability |
| terms explicit. The input seed events are the exact cube-restricted events. -/ |
| theorem extension_bound (p : m.Bit → Set.Icc (0 : ℝ) 1) |
| (q : Set.Icc (0 : ℝ) 1) (hq : 0 < (q : ℝ)) {p₀ : ℝ} (hqp : (q : ℝ) ≤ p₀) (hp₀ : p₀ < 1) |
| (B : Box) (r K R : ℕ) (hK : 0 < K) (hrR : r < R) |
| (hG : G.ContainsBox (B.expand R)) (o : G.Node) (T : Set Vertex) |
| (ho : ¬(B.expand R).Contains o.1) |
| (hp : ∀ e ∈ (B.expand R).bits m, p e = q) |
| {γ τ : ℝ} (hγ : 0 ≤ γ) (hτ : 0 < τ) |
| (hreach : ∀ c, (B.expand R).Contains c → ∃ ℓ : ℕ, |
| r ≤ ℓ ∧ G.ContainsBox (Box.cube c ℓ) ∧ |
| 1-γ ≤ (ProductCylinder.law p).real (m.SeedReaches c r ℓ T)) : |
| (ProductCylinder.law p).real |
| {ω | G.Meets m ω o {x | B.Contains x} ∧ ¬G.Meets m ω o T} ≤ |
| 1/((1-p₀)^(K*(8*r+5)^3)*(R-r : ℕ)) + |
| 4/(K*(q : ℝ)^Box.successCard m r)+2*γ/τ+τ := by |
| have hsub (j : ℕ) (hj : j < R) : ∀ x, ((B.expand j).expand 1).Contains x → |
| (B.expand R).Contains x := by |
| intro x hx |
| rw [Box.expand_expand] at hx |
| exact B.expand_mono (by omega) hx |
| obtain ⟨j,hj,hlayer⟩ := G.exists_shell m p B o q hqp hp₀ (K*(8*r+5)^3) r R hrR ho (by |
| intro j hj y hy |
| exact hp _ (Box.bits_mono m (hsub j (Finset.mem_Ico.mp hj).2) (Box.entryBit_internal_expand hy m))) |
| let C := B.expand j |
| have hjR : j < R := (Finset.mem_Ico.mp hj).2 |
| have hw : ∀ i, C.lower i+r ≤ C.upper i-r := B.expanded_seed_width (Finset.mem_Ico.mp hj).1 |
| have hC : G.ContainsBox (C.expand 1) := hG.mono G (hsub j hjR) |
| have hpC : ∀ e ∈ (C.expand 1).bits m, p e = q := fun e he => |
| hp e (Box.bits_mono m (hsub j hjR) he) |
| have hqual (y : Vertex) (_hy : C.Exterior y) : (ProductCylinder.law p).real |
| (allValue ((C.seedBox r y).bits m) true ∩ |
| {ω | conditionalProbability p (C.bits m) |
| {ω | G.Meets m ω (G.relayNode C r hw (hC.mono G (fun _ => C.contains_expand 1)) y) T} ω ≤ 1-τ}) |
| ≤ (q : ℝ)^Box.seedCard m r*γ/τ := by |
| let c := C.relayCenter r y |
| have hc : C.Contains c := C.seedBox_contained r hw y _ (Box.center_mem_cube _ _) |
| obtain ⟨ℓ,hrℓ,hcube,hprob⟩ := hreach c (B.expand_mono hjR.le hc) |
| have hmass : (ProductCylinder.law p).real (allValue ((C.seedBox r y).bits m) true) = |
| (q : ℝ)^Box.seedCard m r := by |
| rw [allValue_probability_constant p _ q (by |
| intro e he |
| exact hpC e (Box.bits_mono m (fun x hx => C.contains_expand 1 |
| (C.seedBox_contained r hw y x hx)) he))] |
| simp only [bitProbability,ite_true,Box.seedBox,Box.cube_bits_card,Box.seedCard] |
| have hh := G.seed_conditional_quality m p C T hrℓ (C.seedBox_contained r hw y) |
| hcube (G.relayNode C r hw (hC.mono G (fun _ => C.contains_expand 1)) y) rfl hτ hprob |
| change (ProductCylinder.law p).real |
| (allValue ((C.seedBox r y).bits m) true ∩ _) ≤ |
| (ProductCylinder.law p).real (allValue ((C.seedBox r y).bits m) true)*γ/τ at hh |
| rwa [hmass] at hh |
| have hlarge := G.large_entrants_failure m p q hq C r K hK hw hC o T hpC hγ hτ hqual |
| have hrelay := G.random_relay_failure m p (C.bits m) o T hτ.le |
| let E₁ : Set m.Config := {ω | G.Meets m ω o {x | B.Contains x} ∧ |
| (G.entrants m C ω o).card < K*(8*r+5)^3} |
| let E₂ : Set m.Config := {ω | K*(8*r+5)^3 ≤ (G.entrants m C ω o).card ∧ |
| ¬G.Meets m ω o (G.goodRelays m p (C.bits m) T τ ω)} |
| let E₃ : Set m.Config := {ω | G.Meets m ω o (G.goodRelays m p (C.bits m) T τ ω) ∧ |
| ¬G.Meets m ω o T} |
| have hcontain : {ω | G.Meets m ω o {x | B.Contains x} ∧ ¬G.Meets m ω o T} ⊆ (E₁ ∪ E₂) ∪ E₃ := by |
| intro ω hω |
| by_cases hn : (G.entrants m C ω o).card < K*(8*r+5)^3 |
| · exact Or.inl (Or.inl ⟨hω.1,hn⟩) |
| · by_cases hg : G.Meets m ω o (G.goodRelays m p (C.bits m) T τ ω) |
| · exact Or.inr ⟨hg,hω.2⟩ |
| · exact Or.inl (Or.inr ⟨le_of_not_gt hn,hg⟩) |
| have hh := (measureReal_mono (μ := ProductCylinder.law p) hcontain).trans |
| ((measureReal_union_le (E₁ ∪ E₂) E₃).trans |
| (add_le_add (measureReal_union_le E₁ E₂) le_rfl)) |
| have := hh.trans (add_le_add (add_le_add hlayer hlarge) hrelay) |
| simpa only [add_assoc] using this |
|
|
| end LatticeGraph |
| end CriticalZ3 |
|
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| end OAI |
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|
