import Mathlib import OAI.Computability.DirectedFeedback.RankGraph.PrefixBit namespace OAI section noncomputable section open scoped BigOperators noncomputable section open scoped Classical BigOperators noncomputable section open scoped Classical noncomputable section open scoped Classical noncomputable section open scoped Classical BigOperators noncomputable section open scoped BigOperators noncomputable section open scoped BigOperators open DirectedFeedback.SourceProbability noncomputable section open scoped Classical BigOperators noncomputable section open scoped Classical BigOperators noncomputable section open scoped Classical BigOperators noncomputable section open scoped Classical BigOperators noncomputable section open scoped Classical BigOperators noncomputable section open scoped Classical BigOperators noncomputable section open scoped Classical BigOperators noncomputable section open scoped Classical BigOperators noncomputable section open scoped Classical BigOperators noncomputable section open scoped Classical BigOperators noncomputable section open scoped Classical BigOperators noncomputable section open scoped Classical noncomputable section open scoped Classical BigOperators noncomputable section open scoped Classical BigOperators noncomputable section open scoped Classical BigOperators noncomputable section open scoped Classical BigOperators noncomputable section open scoped Classical BigOperators noncomputable section noncomputable section open scoped Classical BigOperators noncomputable section open scoped Classical BigOperators noncomputable section open scoped Classical BigOperators noncomputable section open scoped Classical BigOperators noncomputable section open scoped Classical BigOperators noncomputable section open scoped Classical BigOperators noncomputable section open scoped Classical BigOperators noncomputable section open scoped BigOperators noncomputable section open scoped BigOperators noncomputable section open scoped BigOperators open scoped Classical noncomputable section open scoped Classical BigOperators noncomputable section open scoped Classical BigOperators namespace DirectedFeedback.Construction open DirectedFeedback.SourceProbability FiniteDistribution open RankGraph Prefix PrefixExperiment Games Ranks variable {U V E X Y : Type} [Fintype U] [Fintype V] [Fintype E] [Fintype X] [Fintype Y] [Nonempty X] [Nonempty Y] variable {M T N : ℕ} [NeZero M] [NeZero T] variable (G : Game U V E X Y) (ψLaw : FiniteDistribution (Emb (rankLength X T) N)) variable (hYX : Fintype.card Y ≤ Fintype.card X) (fiber : E → X ≃ Y × Bool) variable (deleted : OutputVertex (M := M) G ψLaw → Bool) variable (ord : OutputVertex (M := M) G ψLaw → ℕ) variable (hord : ∀ {a b}, deleted a = false → deleted b = false → OutputArc G ψLaw hYX fiber a b → ord a < ord b) variable (hrank : ∀ r, deleted (.inl r) = false) variable {η ρ : ℝ} (hη : 0 ≤ η) variable (haccuracy : 3*η*(rankLength X T : ℝ)^3*2^((Fintype.card X)^T) ≤ ρ) variable (hs : Spreads ψLaw η) include hord hrank hη haccuracy hs in theorem big_badness : ((common (M := M) G ψLaw).product (bigMarginal G)).probability (fun a => decide (¬bigGood G ψLaw ord a.1 a.2)) ≤ ρ := by let prefLaw := (uniform (Fin T)).sigma (fun t => (atomLaw (bigMarginal G) (subsetLaw (X := X) (M := M))).iid t.val) have hh := sampled_goodness hYX rankLength_ge hη haccuracy ψLaw hs (prefLaw.product (bigMarginal G)) (fun a => Sum.inl ⟨⟨a.1.1.val+1, by have := a.1.1.isLt; omega⟩, Fin.snoc (vertices a.1.2) a.2⟩) (rankOrder G ψLaw ord) (rankOrder_arc G ψLaw hYX fiber deleted ord hord hrank) simp only [probability_eq_expect, expectation_product, common, commonLaw, bigGood, prefLaw] at hh ⊢ apply le_trans (le_of_eq ?_) hh apply expectation_congr intro psi apply expectation_congr intro pre apply expectation_congr intro u rfl include hord hrank hη haccuracy hs in theorem small_badness : ((common (M := M) G ψLaw).product (smallMarginal G)).probability (fun a => decide (¬smallGood G ψLaw ord a.1 a.2)) ≤ ρ := by let prefLaw := (uniform (Fin T)).sigma (fun t => (atomLaw (bigMarginal G) (subsetLaw (X := X) (M := M))).iid t.val) have hh := sampled_goodness hYX rankLength_ge hη haccuracy ψLaw hs (prefLaw.product (smallMarginal G)) (fun a => Sum.inr ⟨a.1.1,vertices a.1.2,a.2⟩) (rankOrder G ψLaw ord) (rankOrder_arc G ψLaw hYX fiber deleted ord hord hrank) simp only [probability_eq_expect, expectation_product, common, commonLaw, smallGood, prefLaw] at hh ⊢ apply le_trans (le_of_eq ?_) hh apply expectation_congr intro psi apply expectation_congr intro pre apply expectation_congr intro v rfl include hord hrank hη haccuracy hs in theorem edge_badness : ((common (M := M) G ψLaw).product G.edgeLaw).probability (fun a => decide (¬bigGood G ψLaw ord a.1 (G.left a.2))) ≤ ρ ∧ ((common (M := M) G ψLaw).product G.edgeLaw).probability (fun a => decide (¬smallGood G ψLaw ord a.1 (G.right a.2))) ≤ ρ := by constructor · have hh := big_badness G ψLaw hYX fiber deleted ord hord hrank hη haccuracy hs simpa only [probability_eq_expect, expectation_product, bigMarginal, expectation_pushforward] using hh · have hh := small_badness G ψLaw hYX fiber deleted ord hord hrank hη haccuracy hs simpa only [probability_eq_expect, expectation_product, smallMarginal, expectation_pushforward] using hh end DirectedFeedback.Construction noncomputable section open scoped Classical BigOperators namespace DirectedFeedback.Soundness open DirectedFeedback.SourceProbability open FiniteDistribution variable {Ω : Type*} [Fintype Ω] theorem acceptance_forces_tail (μ : FiniteDistribution Ω) (p : ℝ) (hp : 0 < p) (a b B : Ω → ℝ) (ha0 : ∀ ω, 0 ≤ a ω) (ha1 : ∀ ω, a ω ≤ 1) (hb1 : ∀ ω, b ω ≤ 1) (hbB : ∀ ω, b ω ≤ 2 * p * B ω) (hlower : p / 4 ≤ μ.expectation a) (hcomp : μ.expectation (fun ω => a ω * (1 - b ω)) ≤ p / 16) : p / 8 ≤ μ.probability (fun ω => decide (1 / (4 * p) ≤ B ω)) := by have hp4 : 0 < 4 * p := by positivity have hpoint (ω) : a ω ≤ 2 * (a ω * (1 - b ω)) + if 1 / (4 * p) ≤ B ω then 1 else 0 := by by_cases hh : 1 / (4 * p) ≤ B ω · simp only [hh, ↓reduceIte] have hz : 0 ≤ a ω * (1 - b ω) := mul_nonneg (ha0 ω) (sub_nonneg.mpr (hb1 ω)) linarith [ha1 ω] · simp only [hh, ↓reduceIte, add_zero] have hsmall : 4 * p * B ω < 1 := by have hh' := (lt_div_iff₀ hp4).mp (lt_of_not_ge hh) nlinarith have hbhalf : b ω < 1 / 2 := by nlinarith [hbB ω] nlinarith [mul_nonneg (ha0 ω) (show 0 ≤ 1 / 2 - b ω by linarith)] have he := μ.expectation_mono hpoint have heq : μ.expectation (fun ω => 2 * (a ω * (1 - b ω)) + if 1 / (4 * p) ≤ B ω then 1 else 0) = 2 * μ.expectation (fun ω => a ω * (1 - b ω)) + μ.probability (fun ω => decide (1 / (4 * p) ≤ B ω)) := by unfold expectation probability simp only [mul_add, Finset.sum_add_distrib, decide_eq_true_eq, mul_ite, mul_one, mul_zero] congr 1 rw [Finset.mul_sum] apply Finset.sum_congr rfl intro ω _ ring rw [heq] at he linarith theorem geometric_half_sum (n : ℕ) : ∑ i ∈ Finset.range (n + 1), (2 : ℝ)^i / 2 = 2^n - 1 / 2 := by induction n with | zero => norm_num | succ n ih => rw [Finset.sum_range_succ, ih, pow_succ]; ring theorem dyadic_layer_cake (M : ℕ) (x : ℝ) (hx : 0 ≤ x) : (∑ i ∈ Finset.range M, (2 : ℝ)^i / 2 * if 2^i ≤ x then 1 else 0) ≤ x := by induction M with | zero => simpa using hx | succ n ih => by_cases hh : (2 : ℝ)^n ≤ x · calc _ ≤ ∑ i ∈ Finset.range (n + 1), (2 : ℝ)^i / 2 := by apply Finset.sum_le_sum intro i _ split_ifs · simp · simp only [mul_zero] positivity _ = 2^n - 1 / 2 := geometric_half_sum n _ ≤ x := by linarith · simpa only [Finset.sum_range_succ, hh, ↓reduceIte, mul_zero, add_zero] using ih theorem dyadic_tails_expectation (μ : FiniteDistribution Ω) (M : ℕ) (B : Ω → ℝ) (hB : ∀ ω, 0 ≤ B ω) (htail : ∀ i < M, (2 : ℝ)⁻¹ ^ (i + 2) / 8 ≤ μ.probability (fun ω => decide (2^i ≤ B ω))) : (M : ℝ) / 64 ≤ μ.expectation B := by have he := μ.expectation_mono (fun ω => dyadic_layer_cake M (B ω) (hB ω)) have heq : μ.expectation (fun ω => ∑ i ∈ Finset.range M, (2 : ℝ)^i / 2 * if 2^i ≤ B ω then 1 else 0) = ∑ i ∈ Finset.range M, (2 : ℝ)^i / 2 * μ.probability (fun ω => decide (2^i ≤ B ω)) := by unfold expectation probability simp only [Finset.mul_sum, decide_eq_true_eq, mul_ite, mul_one, mul_zero] rw [Finset.sum_comm] apply Finset.sum_congr rfl intro i _ apply Finset.sum_congr rfl intro ω _ split_ifs <;> ring rw [heq] at he calc _ = ∑ i ∈ Finset.range M, (2 : ℝ)^i / 2 * ((2 : ℝ)⁻¹ ^ (i + 2) / 8) := by have hterm (i : ℕ) : (2 : ℝ)^i / 2 * ((2 : ℝ)⁻¹ ^ (i + 2) / 8) = 1 / 64 := by rw [inv_pow, pow_add] have hz : (2 : ℝ)^i ≠ 0 := by positivity field_simp ring simp_rw [hterm] simp ring _ ≤ _ := (Finset.sum_le_sum (fun i hi => mul_le_mul_of_nonneg_left (htail i (Finset.mem_range.mp hi)) (by positivity))).trans he theorem dyadic_contradiction (μ : FiniteDistribution Ω) (D : ℝ) (hD : 0 ≤ D) (M : ℕ) (hM : (M : ℝ) = 128 * (D + 1)) (B : Ω → ℝ) (hB0 : ∀ ω, 0 ≤ B ω) (hB : μ.expectation B ≤ D + 1) (a b : Fin M → Ω → ℝ) (ha0 : ∀ i ω, 0 ≤ a i ω) (ha1 : ∀ i ω, a i ω ≤ 1) (hb1 : ∀ i ω, b i ω ≤ 1) (hbB : ∀ i ω, b i ω ≤ 2 * ((2 : ℝ)⁻¹ ^ (i.val + 2)) * B ω) (hlower : ∀ i, (2 : ℝ)⁻¹ ^ (i.val + 2) / 4 ≤ μ.expectation (a i)) (hcomp : ∀ i, μ.expectation (fun ω => a i ω * (1 - b i ω)) ≤ (2 : ℝ)⁻¹ ^ (i.val + 2) / 16) : False := by have ht (i : ℕ) (hi : i < M) := acceptance_forces_tail μ ((2 : ℝ)⁻¹ ^ (i + 2)) (by positivity) (a ⟨i, hi⟩) (b ⟨i, hi⟩) B (ha0 ⟨i, hi⟩) (ha1 ⟨i, hi⟩) (hb1 ⟨i, hi⟩) (hbB ⟨i, hi⟩) (hlower ⟨i, hi⟩) (hcomp ⟨i, hi⟩) have hthreshold (i : ℕ) : 1 / (4 * ((2 : ℝ)⁻¹ ^ (i + 2))) = 2^i := by rw [pow_add, inv_pow] have hz : (2 : ℝ)^i ≠ 0 := by positivity field_simp ring simp_rw [hthreshold] at ht have hh := dyadic_tails_expectation μ M B hB0 ht rw [hM] at hh linarith end DirectedFeedback.Soundness noncomputable section open scoped Classical BigOperators namespace DirectedFeedback.Construction open DirectedFeedback.SourceProbability FiniteDistribution open PrefixExperiment Pivotal Games variable {M : ℕ} [NeZero M] def lastBias (M : ℕ) : ℝ := (2:ℝ)⁻¹^(M+1) omit [NeZero M] in theorem lastBias_le (i : Fin M) : lastBias M ≤ bias i := by apply pow_le_pow_of_le_one (by norm_num) (by norm_num) have := i.isLt omega end DirectedFeedback.Construction noncomputable section open scoped Classical BigOperators namespace DirectedFeedback.Construction open DirectedFeedback.SourceProbability FiniteDistribution open RankGraph Prefix PrefixExperiment Games Pivotal Ranks variable {U V E X Y : Type} [Fintype U] [Fintype V] [Fintype E] [Fintype X] [Fintype Y] [Nonempty X] [Nonempty Y] variable {M T N : ℕ} [NeZero M] [NeZero T] theorem weighted_soundness (G : Game U V E X Y) (ψLaw : FiniteDistribution (Emb (rankLength X T) N)) (hYX : Fintype.card Y ≤ Fintype.card X) (fiber : E → X ≃ Y × Bool) (hproj : ∀ e x, (fiber e x).1 = G.project e x) (D H K η ρ γ θ : ℝ) (hD : 0 ≤ D) (hH : 0 < H) (hK : 0 < K) (hη : 0 ≤ η) (hγ : 0 < γ) (hM : (M:ℝ) = 128*(D+1)) (haccuracy : 3*η*(rankLength X T : ℝ)^3*2^((Fintype.card X)^T) ≤ ρ) (hspreads : Spreads ψLaw η) (hρ : Fintype.card X * ρ ≤ 1) (hρ0 : 0 ≤ ρ) (hfull : ρ+D/H ≤ 1/2) (J : ℕ) (hJ : 1 ≤ J) (hJunta : DyadicJuntaBound M ((D+1)/γ) γ J) (hcomp : 2*ρ + M*(2:ℝ)^T*D/K + 6*γ + θ*(J:ℝ)^2 ≤ lastBias M/16) (hsound : G.Sound θ) (F : Finset (OutputVertex (M := M) G ψLaw)) (hF : FeedbackR (OutputArc G ψLaw hYX fiber) F) : D < totalDeletionCost G ψLaw D H K (fun v => decide (v ∈ F)) := by by_contra hn have hcost := not_lt.mp hn let deleted : OutputVertex (M := M) G ψLaw → Bool := fun v => decide (v ∈ F) obtain ⟨ord, hord', _⟩ := exists_topological_order (OutputArc G ψLaw hYX fiber) F hF have hord : ∀ {a b}, deleted a = false → deleted b = false → OutputArc G ψLaw hYX fiber a b → ord a < ord b := by intro a b ha hb hab exact hord' a b (by simpa [deleted] using ha) (by simpa [deleted] using hb) hab have hrank := rank_retained G ψLaw D H K hD hH hK deleted hcost obtain ⟨hf,hb,hv,hc⟩ := family_costs G ψLaw deleted D H K hD hH.le hK.le hcost have hbu := big_badness G ψLaw hYX fiber deleted ord hord hrank hη haccuracy hspreads have hbv := small_badness G ψLaw hYX fiber deleted ord hord hrank hη haccuracy hspreads have hρY : (Fintype.card Y:ℝ)*ρ ≤ 1 := (mul_le_mul_of_nonneg_right (by exact_mod_cast hYX) hρ0).trans hρ let fx : X := Classical.choice inferInstance let fy : Y := Classical.choice inferInstance have hBu' := big_budget G ψLaw hYX fiber deleted ord hord hrank fx D ρ hρ hbu hb have hBv' := small_budget G ψLaw hYX fiber deleted ord hord hrank fy D ρ hρY hbv hv have hBu : ((common (M := M) G ψLaw).product G.edgeLaw).expectation (bigBudget G ψLaw hYX fiber deleted ord hord hrank fx) ≤ D+1 := by simpa only [bigBudget, expectation_product, bigMarginal, expectation_pushforward] using hBu' have hBv : ((common (M := M) G ψLaw).product G.edgeLaw).expectation (smallBudget G ψLaw hYX fiber deleted ord hord hrank fy) ≤ D+1 := by simpa only [smallBudget, expectation_product, smallMarginal, expectation_pushforward] using hBv' have hbad := full_badness G ψLaw hYX fiber deleted ord hord hrank hη haccuracy hspreads have hgoodprob := favorable_probability G ψLaw deleted ord D H ρ hH hf hbad hfull obtain ⟨hebu,hebv⟩ := edge_badness G ψLaw hYX fiber deleted ord hord hrank hη haccuracy hspreads apply Soundness.dyadic_contradiction ((common (M := M) G ψLaw).product G.edgeLaw) D hD M hM (smallBudget G ψLaw hYX fiber deleted ord hord hrank fy) (fun a => budget_nonneg _) hBv (bigAcceptance G ψLaw hYX fiber deleted ord hord hrank fx) (smallAcceptance G ψLaw hYX fiber deleted ord hord hrank fy) · intro i a exact probability_nonnegative _ _ · intro i a exact probability_le_one _ _ · intro i a exact probability_le_one _ _ · intro i a have hp := smallFunction_properties G ψLaw hYX fiber deleted ord hord hrank fy a.1 (G.right a.2) exact acceptance_le_mul_budget _ hp.1 hp.2.1 (2*bias i) (mul_nonneg (by norm_num) (bias_pos i).le) (by linarith [bias_le_quarter i]) · intro i have ha := full_acceptance G ψLaw hYX fiber deleted ord hord hrank fx hgoodprob i simpa only [bigAcceptance, expectation_product, bigMarginal, expectation_pushforward, bias] using ha · intro i have hh := independent_comparison G ψLaw hYX fiber deleted ord hord hrank fx fy i D K ρ γ θ hD hK hγ hsound hproj J hJ hJunta hBu hBv hc hebu hebv exact hh.trans (hcomp.trans (div_le_div_of_nonneg_right (lastBias_le i) (by norm_num))) end DirectedFeedback.Construction noncomputable section open scoped Classical BigOperators namespace DirectedFeedback.SourceProbability.FiniteDistribution variable {I : Type*} [Fintype I] theorem expectation_iid_prod (μ : FiniteDistribution I) (T : ℕ) (f : I → ℝ) : (μ.iid T).expectation (fun a => ∏ t, f (a t)) = (μ.expectation f)^T := by simp only [expectation, iid] simpa only [Finset.prod_mul_distrib] using (Fintype.sum_pow (fun i => μ.weight i * f i) T).symm theorem probability_sublevel_exp (μ : FiniteDistribution I) (count : I → ℝ) (b : ℝ) : μ.probability (fun a => decide (count a < b)) ≤ Real.exp b * μ.expectation (fun a => Real.exp (-count a)) := by rw [probability_eq_expect, ← expectation_mul_left] apply expectation_mono intro a by_cases ha : count a < b · simp only [ha, decide_true, ↓reduceIte] rw [← Real.exp_add] exact Real.one_le_exp_iff.mpr (by linarith) · simp only [ha, decide_false, Bool.false_eq_true, ↓reduceIte] positivity end DirectedFeedback.SourceProbability.FiniteDistribution namespace DirectedFeedback.Pivotal open DirectedFeedback.SourceProbability FiniteDistribution def countTrue {T : ℕ} (a : Fin T → Bool) : ℝ := ∑ t, bitValue (a t) theorem exp_neg_count {T : ℕ} (a : Fin T → Bool) : Real.exp (-countTrue a) = ∏ t, Real.exp (-bitValue (a t)) := by simp only [countTrue, ← Finset.sum_neg_distrib, Real.exp_sum] theorem expMoment_count (a : ℝ) (ha0 : 0 ≤ a) (ha1 : a ≤ 1) (T : ℕ) : ((coin a ha0 ha1).iid T).expectation (fun x => Real.exp (-countTrue x)) = (1-a+a*Real.exp (-1))^T := by simp_rw [exp_neg_count] rw [expectation_iid_prod (coin a ha0 ha1) T (fun b => Real.exp (-bitValue b))] congr 1 simp [expectation, coin, bitValue] ring theorem expMoment_count_le (a : ℝ) (ha0 : 0 ≤ a) (ha1 : a ≤ 1) (T : ℕ) : ((coin a ha0 ha1).iid T).expectation (fun x => Real.exp (-countTrue x)) ≤ Real.exp (-(a*T)*(1-Real.exp (-1))) := by rw [expMoment_count] have hbase : 0 ≤ 1-a+a*Real.exp (-1) := by positivity have hbound : 1-a+a*Real.exp (-1) ≤ Real.exp (a*(Real.exp (-1)-1)) := by nlinarith [Real.add_one_le_exp (a*(Real.exp (-1)-1))] have hp := pow_le_pow_left₀ hbase hbound T calc _ ≤ (Real.exp (a*(Real.exp (-1)-1)))^T := hp _ = Real.exp (-(a*T)*(1-Real.exp (-1))) := by rw [← Real.exp_nat_mul] congr 1 ring theorem exp_neg_one_le : Real.exp (-1) ≤ (3:ℝ)/8 := by have h := Real.sum_le_exp_of_nonneg (show (0:ℝ) ≤ 1 by norm_num) 4 norm_num [Finset.sum_range_succ] at h rw [Real.exp_neg] rw [inv_eq_one_div] exact (div_le_iff₀ (Real.exp_pos 1)).mpr (by linarith) theorem bernoulli_lower_tail (a : ℝ) (ha0 : 0 ≤ a) (ha1 : a ≤ 1) (T : ℕ) : ((coin a ha0 ha1).iid T).probability (fun x => decide (countTrue x < (a*T)/2)) ≤ Real.exp (-(a*T)/8) := by calc _ ≤ Real.exp ((a*T)/2) * ((coin a ha0 ha1).iid T).expectation (fun x => Real.exp (-countTrue x)) := probability_sublevel_exp _ _ _ _ ≤ Real.exp ((a*T)/2) * Real.exp (-(a*T)*(1-Real.exp (-1))) := mul_le_mul_of_nonneg_left (expMoment_count_le a ha0 ha1 T) (Real.exp_nonneg _) _ ≤ Real.exp (-(a*T)/8) := by rw [← Real.exp_add] apply Real.exp_le_exp.mpr have hμ : 0 ≤ a*T := mul_nonneg ha0 (Nat.cast_nonneg _) nlinarith [mul_le_mul_of_nonneg_left exp_neg_one_le hμ] end DirectedFeedback.Pivotal noncomputable section open scoped Classical BigOperators namespace DirectedFeedback.SourceProbability.FiniteDistribution variable {Ω : Type*} [Fintype Ω] theorem pushforward_bool (μ : FiniteDistribution Ω) (event : Ω → Bool) (p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) (he : μ.probability event = p) : μ.pushforward event = DirectedFeedback.Pivotal.coin p hp0 hp1 := by apply eq_of_weight_eq intro b have ht : (μ.pushforward event).weight true = p := by calc _ = μ.probability event := by unfold pushforward probability apply Finset.sum_congr rfl intro x _ cases event x <;> simp _ = p := he have hn := (μ.pushforward event).normalized simp only [Fintype.sum_bool] at hn cases b · change (μ.pushforward event).weight false = 1-p linarith · exact ht end DirectedFeedback.SourceProbability.FiniteDistribution namespace DirectedFeedback.Pivotal open DirectedFeedback.SourceProbability FiniteDistribution variable {Ω : Type*} [Fintype Ω] theorem iid_event_lower_tail (μ : FiniteDistribution Ω) (event : Ω → Bool) (p : ℝ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) (he : μ.probability event = p) (T : ℕ) : (μ.iid T).probability (fun a => decide (countTrue (fun t => event (a t)) < p*T/2)) ≤ Real.exp (-(p*T)/8) := by rw [← probability_pushforward (μ.iid T) (fun a t => event (a t)) (fun b => decide (countTrue b < p*T/2)), ← iid_pushforward, pushforward_bool μ event p hp0 hp1 he] exact bernoulli_lower_tail p hp0 hp1 T end DirectedFeedback.Pivotal noncomputable section open scoped Classical BigOperators namespace DirectedFeedback.Coupling open DirectedFeedback.SourceProbability FiniteDistribution variable {X Y : Type} [Fintype X] [Fintype Y] omit [Fintype X] in theorem support_inclusion (e : X ≃ Y × Bool) (p : ℝ) (hp0 : 0 ≤ p) (hp : p ≤ 1/2) (z : Y → Block) (hz : (law p hp0 hp).weight z ≠ 0) (x : X) : bigBits e z x = true → smallBits z (e x).1 = true := by have hprod : (∏ y, blockWeight p (z y)) ≠ 0 := hz have hblock : blockWeight p (z (e x).1) ≠ 0 := by exact (Finset.prod_ne_zero_iff.mp hprod) _ (Finset.mem_univ _) intro hx apply block_inclusion p _ hblock change (if (e x).2 = true then (z (e x).1).1.2 else (z (e x).1).1.1) = true at hx split at hx · exact Or.inr hx · exact Or.inl hx end DirectedFeedback.Coupling namespace DirectedFeedback.Construction open DirectedFeedback.SourceProbability FiniteDistribution open RankGraph Prefix PrefixExperiment Games variable {U V E X Y : Type} [Fintype U] [Fintype V] [Fintype E] [Fintype X] [Fintype Y] [Nonempty X] [Nonempty Y] variable {M T N : ℕ} [NeZero M] [NeZero T] variable (G : Game U V E X Y) (ψLaw : FiniteDistribution (Emb (rankLength X T) N)) variable (fiber : E → X ≃ Y × Bool) (hproject : ∀ e x, (fiber e x).1 = G.project e x) omit [Fintype V] [Nonempty X] [Nonempty Y] in theorem comparison_support_coupling (d : ComparisonSupport (M := M) G ψLaw) : (Coupling.law (Y := Y) (bias d.val.2.2.1) (bias_pos _).le (by linarith [bias_le_quarter d.val.2.2.1])).weight d.val.2.2.2.2 ≠ 0 := by intro hz have he : (comparisonLaw G ψLaw).weight d.val = 0 := by simp only [comparisonLaw, sigma, product, hz, mul_zero] linarith [d.property] include hproject in omit [Fintype V] [Nonempty X] [Nonempty Y] in theorem comparison_label_increases (ellU : U → X) (ellV : V → Y) (d : ComparisonSupport (M := M) G ψLaw) (hsat : G.project d.val.2.1 (ellU (G.left d.val.2.1)) = ellV (G.right d.val.2.1)) : eval ellU ellV (comparison G fiber d.val).source < eval ellU ellV (comparison G fiber d.val).target := by let pref : Fin d.val.1.2.1.val → X := fun j => ellU (vertices d.val.1.2.2 j) let xu := ellU (G.left d.val.2.1) let yv := ellV (G.right d.val.2.1) have himp : Coupling.bigBits (fiber d.val.2.1) d.val.2.2.2.2 xu = true → Coupling.smallBits d.val.2.2.2.2 yv = true := by have he : (fiber d.val.2.1 xu).1 = yv := (hproject _ _).trans hsat rw [← he] exact Coupling.support_inclusion _ _ _ _ _ (comparison_support_coupling G ψLaw d) _ have hlabels : labels ellU ellV (.inl ⟨⟨d.val.1.2.1.val+1, by exact Nat.succ_lt_succ d.val.1.2.1.isLt⟩, Fin.snoc (vertices d.val.1.2.2) (G.left d.val.2.1)⟩) = Fin.snoc pref xu := by funext j refine Fin.lastCases ?_ (fun i => ?_) j · simp [labels, pref, xu] · simp [labels, pref, xu] simp only [comparison, twoLevel, eval_vertex, hlabels, Set.mem_preimage, splitLast, Equiv.coe_fn_mk, Fin.init_snoc, Fin.snoc_last] change (if (pref,yv) ∈ conditionedSet (cell d.val.1.2.2) d.val.2.2.2.1 {y | Coupling.smallBits d.val.2.2.2.2 y = true} then d.val.1.1.val ⟨1,_⟩ else d.val.1.1.val ⟨16,_⟩) < (if (pref,xu) ∈ conditionedSet (cell d.val.1.2.2) d.val.2.2.2.1 {x | Coupling.bigBits (fiber d.val.2.1) d.val.2.2.2.2 x = true} then d.val.1.1.val ⟨2,_⟩ else d.val.1.1.val ⟨17,_⟩) by_cases hbefore : (cell d.val.1.2.2 pref).val < d.val.2.2.2.1.val · simp only [conditionedSet, Set.mem_ofPred_eq, hbefore, true_or, ↓reduceIte] apply d.val.1.1.property; change (1 : ℕ) < 2; omega · by_cases heq : cell d.val.1.2.2 pref = d.val.2.2.2.1 · simp only [conditionedSet, Set.mem_ofPred_eq, heq, lt_self_iff_false, false_or, true_and] by_cases hu : Coupling.bigBits (fiber d.val.2.1) d.val.2.2.2.2 xu = true · simp only [hu, himp hu, ↓reduceIte] apply d.val.1.1.property; change (1 : ℕ) < 2; omega · simp only [hu] split · apply d.val.1.1.property; change (1 : ℕ) < 17; omega · apply d.val.1.1.property; change (16 : ℕ) < 17; omega · simp only [conditionedSet, Set.mem_ofPred_eq, hbefore, false_or, heq, false_and, ↓reduceIte] apply d.val.1.1.property; change (16 : ℕ) < 17; omega end DirectedFeedback.Construction noncomputable section open scoped Classical BigOperators namespace DirectedFeedback.Construction open DirectedFeedback.SourceProbability FiniteDistribution open PrefixExperiment Pivotal Games variable {U V E X Y : Type} [Fintype U] [Fintype V] [Fintype E] [Fintype X] [Fintype Y] [Nonempty X] [Nonempty Y] variable {M T N : ℕ} [NeZero M] [NeZero T] omit [NeZero M] [Nonempty X] in theorem subset_eval_probability (i : Fin M) (x : X) : (subsetLaw i).probability (fun b => b x) = bias i := by rw [probability_eq_expect] unfold subsetLaw Pivotal.law rw [expectation_table_eval (fun _ : X => coin (bias i) (bias_pos i).le (by linarith [bias_le_quarter i])) x (fun b : Bool => if b then 1 else 0)] simp [expectation, coin] omit [Fintype V] [Fintype Y] [Nonempty X] [Nonempty Y] in theorem slot_label_probability (G : Game U V E X Y) (ell : U → X) (i : Fin M) : (atomLaw (bigMarginal G) subsetLaw).probability (fun a => decide (a.2.1 = i) && a.2.2 (ell a.1)) = bias i / M := by rw [probability_eq_expect] have hind (a : Slot U X M) : (if decide (a.2.1 = i) && a.2.2 (ell a.1) then (1:ℝ) else 0) = if a.2.1 = i then (if a.2.2 (ell a.1) then 1 else 0) else 0 := by by_cases hi : a.2.1 = i <;> cases a.2.2 (ell a.1) <;> simp [hi] simp_rw [hind] have he := fresh_index_expectation (bigMarginal G) subsetLaw i (fun u b => if b (ell u) then 1 else 0) calc _ = (1 / (Fintype.card (Fin M) : ℝ)) * (bigMarginal G).expectation (fun u => (subsetLaw i).expectation (fun b => if b (ell u) then 1 else 0)) := by convert he using 1 apply expectation_congr intro a by_cases hi : a.2.1 = i <;> simp [hi] _ = bias i / M := by simp_rw [← probability_eq_expect, subset_eval_probability, expectation_const] simp [div_eq_mul_inv, mul_comm] omit [NeZero T] [Fintype V] [Fintype Y] [Nonempty X] [Nonempty Y] in theorem fullCount_lower_tail (G : Game U V E X Y) (ell : U → X) (i : Fin M) : ((atomLaw (bigMarginal G) subsetLaw).iid T).probability (fun a => decide ((fullCount a i (fun t => ell (vertices a t)) : ℝ) < bias i*T/(2*M))) ≤ Real.exp (-(bias i*T/(8*M))) := by have hM : (0:ℝ) < M := by exact_mod_cast Nat.pos_of_ne_zero (NeZero.ne M) have hM' : (1:ℝ) ≤ M := by exact_mod_cast Nat.pos_of_ne_zero (NeZero.ne M) have hp0 : 0 ≤ bias i / M := (div_pos (bias_pos i) hM).le have hp1 : bias i / M ≤ 1 := (div_le_one hM).mpr (by linarith [bias_le_quarter i]) have hh := iid_event_lower_tail (atomLaw (bigMarginal G) subsetLaw) (fun a => decide (a.2.1 = i) && a.2.2 (ell a.1)) (bias i / M) hp0 hp1 (slot_label_probability G ell i) T have hc (a : PrefixData U X M T) : countTrue (fun t => decide ((a t).2.1 = i) && (a t).2.2 (ell (a t).1)) = (fullCount a i (fun t => ell (vertices a t)) : ℝ) := by simp only [countTrue, fullCount, Finset.card_filter, Nat.cast_sum, Nat.cast_ite, Nat.cast_one, Nat.cast_zero] apply Finset.sum_congr rfl intro t _ by_cases hi : (a t).2.1 = i <;> cases hb : (a t).2.2 (ell (a t).1) <;> simp [hi,hb,bitValue,vertices] simp_rw [hc] at hh have he1 : bias i / (M:ℝ) * T / 2 = bias i*T/(2*M) := by ring have he2 : -(bias i / (M:ℝ) * T) / 8 = -(bias i*T/(8*M)) := by ring simpa only [he1, he2] using hh omit [Fintype V] [Fintype Y] [Nonempty X] [Nonempty Y] [NeZero T] in theorem fullStars_probability (G : Game U V E X Y) (ψLaw : FiniteDistribution (Emb (rankLength X T) N)) (ell : U → X) : (fullLaw (M := M) G ψLaw).probability (fun d => decide ((fun t => ell (vertices d.2 t)) ∈ fullStars d.2)) ≤ M * Real.exp (-(lastBias M * T/(8*M))) := by rw [fullLaw, probability_eq_expect, expectation_product] dsimp only have he : (ψLaw.expectation fun _ => ((atomLaw (bigMarginal G) (subsetLaw (M := M))).iid T).probability (fun a => decide ((fun t => ell (vertices a t)) ∈ fullStars a))) = ((atomLaw (bigMarginal G) (subsetLaw (M := M))).iid T).probability (fun a => decide ((fun t => ell (vertices a t)) ∈ fullStars a)) := expectation_const _ _ apply le_trans (le_of_eq (Eq.trans ?_ he)) swap · simp only [probability_eq_expect] have hu := probability_exists_le_sum ((atomLaw (bigMarginal G) (subsetLaw (M := M))).iid T) (fun i a => decide ((fullCount a i (fun t => ell (vertices a t)) : ℝ) < bias i*T/(2*M))) simp only [decide_eq_true_eq] at hu apply le_trans (b := ∑ i : Fin M, ((atomLaw (bigMarginal G) (subsetLaw (M := M))).iid T).probability (fun a => decide ((fullCount a i (fun t => ell (vertices a t)) : ℝ) < bias i*T/(2*M)))) · have hev : (fun a : PrefixData U X M T => decide ((fun t => ell (vertices a t)) ∈ fullStars a)) = (fun a => decide (∃ i : Fin M, (fullCount a i (fun t => ell (vertices a t)) : ℝ) < bias i*T/(2*M))) := by funext a apply Bool.eq_iff_iff.mpr rw [decide_eq_true_eq, decide_eq_true_eq] rfl rw [hev] exact hu calc _ ≤ ∑ i : Fin M, Real.exp (-(bias i*T/(8*M))) := Finset.sum_le_sum (fun i _ => fullCount_lower_tail G ell i) _ ≤ ∑ _i : Fin M, Real.exp (-(lastBias M*T/(8*M))) := by apply Finset.sum_le_sum intro i _ apply Real.exp_le_exp.mpr apply neg_le_neg exact div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_right (lastBias_le i) (Nat.cast_nonneg _)) (by positivity) _ = _ := by simp end DirectedFeedback.Construction noncomputable section open scoped Classical BigOperators namespace DirectedFeedback.Pivotal open DirectedFeedback.SourceProbability FiniteDistribution variable {I : Type} [Fintype I] [DecidableEq I] [Nonempty I] theorem labelOrder_label_probability (x : I) : labelOrderLaw.probability (fun o : LabelOrder I => decide (x = o.1)) = 1 / Fintype.card I := by rw [probability_eq_expect, labelOrderLaw, expectation_sigma] simp only [expectation_const, decide_eq_true_eq] rw [expectation_uniform] simp end DirectedFeedback.Pivotal namespace DirectedFeedback.Construction open DirectedFeedback.SourceProbability FiniteDistribution open RankGraph Prefix PrefixExperiment Games Pivotal variable {U V E X Y : Type} [Fintype U] [Fintype V] [Fintype E] [Fintype X] [Fintype Y] [Nonempty X] [Nonempty Y] variable {M T N : ℕ} [NeZero M] [NeZero T] variable (G : Game U V E X Y) (ψLaw : FiniteDistribution (Emb (rankLength X T) N)) variable (hYX : Fintype.card Y ≤ Fintype.card X) (fiber : E → X ≃ Y × Bool) variable (ellU : U → X) (ellV : V → Y) def labelDeleted : OutputVertex (M := M) G ψLaw → Bool := fun v => decide (TestGraph.LabelDeleted (wildSupport G ψLaw hYX) (compSupport G ψLaw fiber) ellU ellV v) def labelFeedback : Finset (OutputVertex (M := M) G ψLaw) := Finset.univ.filter (TestGraph.LabelDeleted (wildSupport G ψLaw hYX) (compSupport G ψLaw fiber) ellU ellV) theorem labelFeedback_spec : FeedbackR (OutputArc G ψLaw hYX fiber) (labelFeedback (M := M) G ψLaw hYX fiber ellU ellV) := by exact TestGraph.labelDeleted_feedback _ _ ellU ellV theorem labelFeedback_indicator : (fun v => decide (v ∈ labelFeedback (M := M) G ψLaw hYX fiber ellU ellV)) = labelDeleted (M := M) G ψLaw hYX fiber ellU ellV := by funext v simp only [labelFeedback, Finset.mem_filter, Finset.mem_univ, true_and, labelDeleted] theorem full_label_cost : (fullLaw (M := M) G ψLaw).probability (fullDeleted G ψLaw (labelDeleted (M := M) G ψLaw hYX fiber ellU ellV)) ≤ M * Real.exp (-(lastBias M*T/(8*M))) := by rw [fullDeleted, probability_extend] have he : (fullLaw (M := M) G ψLaw).supported.probability (fun d => labelDeleted (M := M) G ψLaw hYX fiber ellU ellV (.inr (.inl (.inl d)))) = (fullLaw (M := M) G ψLaw).probability (fun d => decide ((fun t => ellU (vertices d.2 t)) ∈ fullStars d.2)) := by exact probability_supported (fullLaw (M := M) G ψLaw) (fun d => decide ((fun t => ellU (vertices d.2 t)) ∈ fullStars d.2)) rw [he] exact fullStars_probability G ψLaw ellU theorem big_label_cost : Fintype.card X * (bigLaw (M := M) G ψLaw).probability (bigDeleted G ψLaw (labelDeleted (M := M) G ψLaw hYX fiber ellU ellV)) = 1 := by rw [bigDeleted, probability_extend] have he : (bigLaw (M := M) G ψLaw).supported.probability (fun d => labelDeleted (M := M) G ψLaw hYX fiber ellU ellV (.inr (.inl (.inr (.inl d))))) = (bigLaw (M := M) G ψLaw).probability (fun d => decide (ellU d.2.1 = d.2.2.1)) := by convert probability_supported (bigLaw (M := M) G ψLaw) (fun d => decide (ellU d.2.1 = d.2.2.1)) using 1 apply congrArg (fun f => (bigLaw (M := M) G ψLaw).supported.probability f) funext d apply Bool.eq_iff_iff.mpr simp only [labelDeleted, decide_eq_true_eq] change (ellU (Fin.snoc (α := fun _ => U) (vertices d.val.1.2.2) d.val.2.1 (Fin.last d.val.1.2.1.val)) = d.val.2.2.1) ↔ _ simp only [Fin.snoc_last] rw [he, bigLaw, probability_eq_expect] simp only [expectation_product] simp_rw [← probability_eq_expect, labelOrder_label_probability, expectation_const] have hc : (Fintype.card X:ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero field_simp theorem small_label_cost : Fintype.card Y * (smallLaw (M := M) G ψLaw).probability (smallDeleted G ψLaw (labelDeleted (M := M) G ψLaw hYX fiber ellU ellV)) = 1 := by rw [smallDeleted, probability_extend] have he : (smallLaw (M := M) G ψLaw).supported.probability (fun d => labelDeleted (M := M) G ψLaw hYX fiber ellU ellV (.inr (.inl (.inr (.inr d))))) = (smallLaw (M := M) G ψLaw).probability (fun d => decide (ellV d.2.1 = d.2.2.1)) := by exact probability_supported (smallLaw (M := M) G ψLaw) (fun d => decide (ellV d.2.1 = d.2.2.1)) rw [he, smallLaw, probability_eq_expect] simp only [expectation_product] simp_rw [← probability_eq_expect, labelOrder_label_probability, expectation_const] have hc : (Fintype.card Y:ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero field_simp include fiber in theorem comparison_label_cost (hproject : ∀ e x, (fiber e x).1 = G.project e x) : (comparisonLaw (M := M) G ψLaw).probability (comparisonDeleted G ψLaw (labelDeleted (M := M) G ψLaw hYX fiber ellU ellV)) ≤ 1-G.score ellU ellV := by rw [comparisonDeleted, probability_extend] have hm := probability_mono (μ := (comparisonLaw (M := M) G ψLaw).supported) (event := fun d => labelDeleted (M := M) G ψLaw hYX fiber ellU ellV (.inr (.inr d))) (event' := fun d => decide (G.project d.val.2.1 (ellU (G.left d.val.2.1)) ≠ ellV (G.right d.val.2.1))) (by intro d hd apply decide_eq_true intro hs have hi := comparison_label_increases G ψLaw fiber hproject ellU ellV d hs have hd' : TestGraph.LabelDeleted (wildSupport G ψLaw hYX) (compSupport G ψLaw fiber) ellU ellV (.inr (.inr d)) := of_decide_eq_true hd exact not_le.mpr hi hd') apply hm.trans_eq have he := probability_supported (comparisonLaw (M := M) G ψLaw) (fun d => decide (G.project d.2.1 (ellU (G.left d.2.1)) ≠ ellV (G.right d.2.1))) rw [he, comparisonLaw, probability_eq_expect, expectation_sigma] trans (common (M := M) G ψLaw).expectation (fun _ => G.edgeLaw.probability (fun e => decide (G.project e (ellU (G.left e)) ≠ ellV (G.right e)))) · apply expectation_congr intro d rw [expectation_product, probability_eq_expect] apply expectation_congr intro e exact expectation_const ((uniform (Fin M)).sigma (fun i => (uniform (Fin (cellCount d.2.1.val))).product (Coupling.law (Y := Y) (bias i) (bias_pos i).le (by linarith [bias_le_quarter i])))) (if decide (G.project e (ellU (G.left e)) ≠ ellV (G.right e)) then 1 else 0) rw [expectation_const] have hh := probability_not G.edgeLaw (fun e => decide (G.project e (ellU (G.left e)) = ellV (G.right e))) simpa only [decide_not, Game.score] using hh theorem weighted_completeness (hproject : ∀ e x, (fiber e x).1 = G.project e x) (D H K : ℝ) (hH : 0 ≤ H) (hK : 0 ≤ K) : totalDeletionCost G ψLaw D H K (labelDeleted (M := M) G ψLaw hYX fiber ellU ellV) ≤ H * (M * Real.exp (-(lastBias M*T/(8*M)))) + 2 + K*(1-G.score ellU ellV) := by rw [deletionCost_decomposition] have hr : (∑ r : Rank U V X Y T N, if labelDeleted (M := M) G ψLaw hYX fiber ellU ellV (.inl r) then D+1 else 0) = 0 := by apply Finset.sum_eq_zero intro r _ have h : labelDeleted (M := M) G ψLaw hYX fiber ellU ellV (.inl r) = false := by cases e : labelDeleted (M := M) G ψLaw hYX fiber ellU ellV (.inl r) with | false => rfl | true => exact False.elim (of_decide_eq_true e : TestGraph.LabelDeleted (wildSupport G ψLaw hYX) (compSupport G ψLaw fiber) ellU ellV (.inl r)) rw [h] rfl rw [hr, zero_add, big_label_cost, small_label_cost] have hf := mul_le_mul_of_nonneg_left (full_label_cost (M := M) G ψLaw hYX fiber ellU ellV) hH have hc := mul_le_mul_of_nonneg_left (comparison_label_cost (M := M) G ψLaw hYX fiber ellU ellV hproject) hK linarith end DirectedFeedback.Construction noncomputable section open scoped Classical BigOperators namespace DirectedFeedback.Construction open Pivotal structure Parameters (D : ℕ) where M : ℕ T : ℕ K : ℕ J : ℕ M_eq : M = 128*(D+1) T_pos : 0 < T K_pos : 0 < K J_pos : 1 ≤ J theta : ℚ theta_pos : 0 < theta theta_half : (theta : ℝ) < 1/2 fullChernoff : (8*((D:ℝ)+1)) * (M * Real.exp (-(lastBias M*T/(8*M)))) ≤ 1 comparison : (M:ℝ)*2^T*D/K ≤ lastBias M/1000 junta : DyadicJuntaBound M (((D:ℝ)+1)/(lastBias M/1000)) (lastBias M/1000) J Ktheta : (K:ℝ)*theta ≤ 1 Jtheta : (theta:ℝ)*(J:ℝ)^2 ≤ lastBias M/1000 theorem exists_parameters (D : ℕ) : Nonempty (Parameters D) := by let M := 128*(D+1) let H : ℝ := 8*((D:ℝ)+1) let p := lastBias M let γ := p/1000 have hM : (0:ℝ) < M := by dsimp [M]; positivity have hH : 0 < H := by dsimp [H]; positivity have hp : 0 < p := by dsimp [p, lastBias]; positivity have hg : 0 < γ := by dsimp [γ]; positivity obtain ⟨T, hT⟩ := exists_nat_gt (8*M*(H*M)/p) have hT0 : 0 < T := by have ht : (0:ℝ) < T := lt_of_le_of_lt (by positivity) hT exact_mod_cast ht have hprefix : H * (M * Real.exp (-(p*T/(8*M)))) ≤ 1 := by have harg : H*M ≤ p*T/(8*M) := by apply (le_div_iff₀ (by positivity)).mpr have hh := (div_lt_iff₀ hp).mp hT nlinarith calc _ = (H*M)/Real.exp (p*T/(8*M)) := by rw [Real.exp_neg]; ring _ ≤ 1 := (div_le_one (Real.exp_pos _)).mpr (harg.trans (by linarith [Real.add_one_le_exp (p*T/(8*M))])) obtain ⟨K, hK⟩ := exists_nat_gt (max 0 ((M:ℝ)*2^T*D/γ)) have hK0 : (0:ℝ) < K := lt_of_le_of_lt (le_max_left _ _) hK have hKnat : 0 < K := by exact_mod_cast hK0 have hcomparison : (M:ℝ)*2^T*D/K ≤ γ := by apply (div_le_iff₀ hK0).mpr have hh := (div_lt_iff₀ hg).mp (lt_of_le_of_lt (le_max_right _ _) hK) linarith obtain ⟨J,hJ,hjunta⟩ := uniform_dyadic_junta M (((D:ℝ)+1)/γ) γ (by positivity) hg have hJ0 : (0:ℝ) < J := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hJ) let t : ℝ := min (1/4) (min (1/K) (γ/(J:ℝ)^2)) have ht : 0 < t := by dsimp [t]; positivity obtain ⟨θ,hθ0,hθ⟩ := exists_rat_btwn ht have hθnat : (0:ℚ) < θ := by exact_mod_cast hθ0 have ht1 : (θ:ℝ) < 1/4 := hθ.trans_le (min_le_left _ _) have htK : (θ:ℝ) < 1/(K:ℝ) := hθ.trans_le ((min_le_right _ _).trans (min_le_left _ _)) have htJ : (θ:ℝ) < γ/(J:ℝ)^2 := hθ.trans_le ((min_le_right _ _).trans (min_le_right _ _)) refine ⟨⟨M,T,K,J,rfl,hT0,hKnat,hJ,θ,hθnat,by linarith,hprefix,hcomparison,hjunta,?_,?_⟩⟩ · have hh := (lt_div_iff₀ hK0).mp htK nlinarith · exact ((lt_div_iff₀ (sq_pos_of_pos hJ0)).mp htJ).le namespace Parameters variable {D : ℕ} (P : Parameters D) instance : NeZero P.M := ⟨by rw [P.M_eq]; omega⟩ instance : NeZero P.T := ⟨ne_of_gt P.T_pos⟩ theorem gamma_pos : 0 < lastBias P.M/1000 := by unfold lastBias; positivity theorem gamma_le : lastBias P.M/1000 ≤ 1/1000 := by have hp : lastBias P.M ≤ 1 := pow_le_one₀ (by norm_num) (by norm_num) linarith theorem full_error (ρ : ℝ) (hρ : ρ ≤ lastBias P.M/1000) : ρ + (D:ℝ)/(8*((D:ℝ)+1)) ≤ 1/2 := by have hsmall : (D:ℝ)/(8*((D:ℝ)+1)) ≤ 1/8 := by apply (div_le_iff₀ (by positivity)).mpr nlinarith linarith [P.gamma_le] theorem product_error (ρ : ℝ) (hρ : ρ ≤ lastBias P.M/1000) : 2*ρ + P.M*(2:ℝ)^P.T*D/P.K + 6*(lastBias P.M/1000) + (P.theta:ℝ)*(P.J:ℝ)^2 ≤ lastBias P.M/16 := by have hp : 0 < lastBias P.M := by unfold lastBias; positivity linarith [P.comparison,P.Jtheta] end Parameters end DirectedFeedback.Construction noncomputable section open scoped Classical BigOperators namespace DirectedFeedback.Construction open DirectedFeedback.SourceProbability FiniteDistribution open RankGraph Prefix PrefixExperiment Games Pivotal Ranks structure RankParameters {D : ℕ} (P : Parameters D) (X : Type) [Fintype X] where N : ℕ law : FiniteDistribution (Emb (rankLength X P.T) N) eta : ℝ rho : ℝ eta_pos : 0 < eta rho_nonneg : 0 ≤ rho rho_small : rho ≤ lastBias P.M/1000 rho_alphabet : Fintype.card X * rho ≤ 1 accuracy : 3*eta*(rankLength X P.T : ℝ)^3*2^((Fintype.card X)^P.T) ≤ rho spreads : Spreads law eta rational : ∀ ψ, ∃ q : ℚ, law.weight ψ = q theorem exists_rankParameters {D : ℕ} (P : Parameters D) (X : Type) [Fintype X] [Nonempty X] : Nonempty (RankParameters P X) := by let m : ℝ := Fintype.card X let ρ := min (lastBias P.M/1000) (1/m) let L := rankLength X P.T let η := ρ / (3*(L:ℝ)^3*2^((Fintype.card X)^P.T)) have hm : 0 < m := by dsimp [m]; exact_mod_cast Fintype.card_pos have hρ : 0 < ρ := lt_min P.gamma_pos (div_pos (by norm_num) hm) have hL : 3 ≤ L := by dsimp [L, rankLength]; omega have hLc : (0:ℝ) < L := by exact_mod_cast (by omega : 0 < L) have hη : 0 < η := by dsimp [η]; positivity obtain ⟨N,_,μ,hs,hr⟩ := exists_spreading_law L hL η hη refine ⟨⟨N,μ,η,ρ,hη,hρ.le,min_le_left _ _,?_,?_,hs,hr⟩⟩ · have hh := min_le_right (lastBias P.M/1000) (1/m) have hb := (le_div_iff₀ hm).mp hh dsimp [m] at hb ⊢ nlinarith · dsimp only [η] have hden : (3:ℝ)*(L:ℝ)^3*2^((Fintype.card X)^P.T) ≠ 0 := by positivity field_simp ring_nf rfl variable {U V E X Y : Type} [Fintype U] [Fintype V] [Fintype E] [Fintype X] [Fintype Y] [Nonempty X] [Nonempty Y] end DirectedFeedback.Construction end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end OAI