import Mathlib import OAI.Probability.Contingency.LawComposition import OAI.Probability.Contingency.RootTransport namespace OAI /-! Child Log Concavity. -/ noncomputable section open scoped BigOperators open Filter open scoped BigOperators open QuadraticMap Filter namespace Contingency theorem bool_signature_determinant (M : Bool → Bool → ℝ) (hs : sigPos (matrixForm M) ≤ 1) (hm : M true false = M false true) (ha : 0 < M true true) : M true true * M false false ≤ M true false ^ 2 := by let e : Bool → ℝ := fun b => if b then 1 else 0 let v : Bool → ℝ := fun b => if b then 0 else 1 have he : matrixForm M e = M true true := by simp [matrixForm_apply, e] have hv : matrixForm M v = M false false := by simp [matrixForm_apply, v] have he' : 0 < matrixForm M e := he ▸ ha have hve : (matrixForm M).associated v e = M true false := by rw [QuadraticMap.associated_apply] norm_num simp [matrixForm_apply,e,v,hm] ring have h := positive_projection (matrixForm M) hs e v he' rw [he,hv,hve] at h have := (le_div_iff₀ ha).mp h nlinarith def childProfile (C j : ℤ) : Bool → ℤ := fun b => if b then j else C-j theorem signature_child_logconcave (C : ℕ) (f : (Bool → ℤ) → ℝ) (hp : PositiveSlice f (fun _ => (C:ℤ)) C) (hs : HasSignature f (fun _ => (C:ℤ)) C) (j : ℤ) (hj : 1 ≤ j) (hjC : j < C) : f (childProfile C (j-1)) * f (childProfile C (j+1)) ≤ f (childProfile C j)^2 := by let p : Bool → ℤ := fun b => if b then j+1 else (C:ℤ)-j+1 have hb : InBox (fun _ => (C:ℤ)) p := by intro b; cases b <;> dsimp [p] <;> omega have ht : (∑ b, p b) = (C:ℤ)+2 := by simp [p]; omega have hrem (i k : Bool) : removePair p i k = childProfile C (j+1 - (if i then 1 else 0) - (if k then 1 else 0)) := by funext b cases i <;> cases k <;> cases b <;> simp [removePair,p,childProfile] <;> omega have ha : 0 < f (childProfile C (j-1)) := by apply hp.1 · intro b; cases b <;> dsimp [childProfile] <;> omega · simp [childProfile] have h := bool_signature_determinant (twoRemoval f p) (hs p hb ht) (by simp only [twoRemoval]; rw [removePair_comm]) (by simpa [twoRemoval,hrem] using ha) simpa [twoRemoval,hrem] using h theorem positive_logconcave_min (f : ℤ → ℝ) (a b j : ℤ) (haj : a ≤ j) (hjb : j ≤ b) (hp : ∀ i ∈ Finset.Icc a b, 0 < f i) (hl : ∀ i, a < i → i < b → f (i-1)*f (i+1) ≤ f i ^ 2) : min (f a) (f b) ≤ f j := by classical by_contra hh have hh' : f j < min (f a) (f b) := lt_of_not_ge hh obtain ⟨k,hk,hmin⟩ := Finset.exists_min_image (Finset.Icc a b) f ⟨j,Finset.mem_Icc.mpr ⟨haj,hjb⟩⟩ let A := (Finset.Icc a b).filter (fun i => f i = f k) have hA : A.Nonempty := ⟨k,by simp [A,hk]⟩ let i := A.min' hA have hi : i ∈ A := Finset.min'_mem A hA have hi' := Finset.mem_filter.mp hi have hib := Finset.mem_Icc.mp hi'.1 have hik : f i = f k := hi'.2 have hfi : f i < min (f a) (f b) := by rw [hik]; exact lt_of_le_of_lt (hmin j (Finset.mem_Icc.mpr ⟨haj,hjb⟩)) hh' have hia : a < i := by have hne : i ≠ a := by intro h; rw [h] at hfi; exact (not_lt_of_ge (min_le_left _ _)) hfi omega have hib' : i < b := by have hne : i ≠ b := by intro h; rw [h] at hfi; exact (not_lt_of_ge (min_le_right _ _)) hfi omega have himem : i-1 ∈ Finset.Icc a b := Finset.mem_Icc.mpr (by omega) have hinmem : i+1 ∈ Finset.Icc a b := Finset.mem_Icc.mpr (by omega) have hprev : f i < f (i-1) := by have hle : f i ≤ f (i-1) := hik ▸ hmin (i-1) himem apply lt_of_le_of_ne hle intro he have hm : i-1 ∈ A := Finset.mem_filter.mpr ⟨himem,he.symm.trans hik⟩ have := Finset.min'_le A (i-1) hm change i ≤ i-1 at this omega have hnext : f i ≤ f (i+1) := hik ▸ hmin (i+1) hinmem have hpos := hp i hi'.1 have hloc := hl i hia hib' have hx := mul_lt_mul_of_pos_right hprev hpos have hy := mul_le_mul_of_nonneg_left hnext (le_of_lt (lt_trans hpos hprev)) nlinarith theorem signature_child_min (C : ℕ) (f : (Bool → ℤ) → ℝ) (hp : PositiveSlice f (fun _ => (C:ℤ)) C) (hs : HasSignature f (fun _ => (C:ℤ)) C) (a b j : ℤ) (ha : 0 ≤ a) (haj : a ≤ j) (hjb : j ≤ b) (hb : b ≤ C) : min (f (childProfile C a)) (f (childProfile C b)) ≤ f (childProfile C j) := by apply positive_logconcave_min (fun i => f (childProfile C i)) a b j haj hjb · intro i hi have hi' := Finset.mem_Icc.mp hi apply hp.1 · intro k; cases k <;> dsimp [childProfile] <;> omega · simp [childProfile] · intro i hi hi' exact signature_child_logconcave C f hp hs i (by omega) (by omega) theorem zero_child_min (C : ℕ) (f : ℝ → (Bool → ℤ) → ℝ) (hc : ∀ x, Continuous (fun η => f η x)) (hp : ∀ η, 0 < η → η < 1 → PositiveSlice (f η) (fun _ => (C:ℤ)) C) (hs : ∀ η, 0 < η → η < 1 → HasSignature (f η) (fun _ => (C:ℤ)) C) (a b j : ℤ) (ha : 0 ≤ a) (haj : a ≤ j) (hjb : j ≤ b) (hb : b ≤ C) : min (f 0 (childProfile C a)) (f 0 (childProfile C b)) ≤ f 0 (childProfile C j) := by apply le_of_tendsto_of_tendsto (((hc _).continuousAt.min (hc _).continuousAt).comp softSequence_tendsto) ((hc _).continuousAt.tendsto.comp softSequence_tendsto) exact Filter.Eventually.of_forall fun k => signature_child_min C (f (softSequence k)) (hp _ (softSequence_range k).1 (softSequence_range k).2) (hs _ (softSequence_range k).1 (softSequence_range k).2) a b j ha haj hjb hb end Contingency open scoped BigOperators namespace Contingency section FiniteVariance variable {α : Type*} [Fintype α] noncomputable def totalWeight (w : α → ℝ) : ℝ := ∑ x, w x noncomputable def weightedMean (w H : α → ℝ) : ℝ := (∑ x, w x*H x) / totalWeight w noncomputable def weightedVariance (w H : α → ℝ) : ℝ := ∑ x, w x * (H x-weightedMean w H)^2 theorem centered_sum (w H : α → ℝ) (hw : totalWeight w ≠ 0) : (∑ x, w x*(H x-weightedMean w H))=0 := by simp only [mul_sub,Finset.sum_sub_distrib,← Finset.sum_mul] change (∑ x, w x*H x) - totalWeight w*((∑ x, w x*H x)/totalWeight w) = 0 field_simp ring theorem variance_translate (w H : α → ℝ) (hw : totalWeight w ≠ 0) (a : ℝ) : (∑ x, w x*(H x-a)^2) = weightedVariance w H + totalWeight w*(weightedMean w H-a)^2 := by have h (x : α) : w x*(H x-a)^2 = w x*(H x-weightedMean w H)^2 + 2*(weightedMean w H-a)*(w x*(H x-weightedMean w H)) + w x*(weightedMean w H-a)^2 := by ring simp only [h,Finset.sum_add_distrib,← Finset.mul_sum,← Finset.sum_mul] rw [centered_sum w H hw] simp [weightedVariance,totalWeight] theorem variance_secondMoment (w H : α → ℝ) (hw : totalWeight w ≠ 0) : weightedVariance w H = (∑ x, w x*H x^2) - (∑ x, w x*H x)^2 / totalWeight w := by have h := variance_translate w H hw 0 simp only [sub_zero] at h rw [weightedMean] at h have he : totalWeight w*((∑ x, w x*H x)/totalWeight w)^2 = (∑ x, w x*H x)^2 / totalWeight w := by field_simp rw [he] at h linarith theorem variance_pair_formula (w H : α → ℝ) (hw : totalWeight w ≠ 0) : weightedVariance w H = (∑ a, ∑ b, w a*w b*(H a-H b)^2) / (2*totalWeight w) := by have hexp (a b : α) : w a*w b*(H a-H b)^2 = (w a*H a^2)*w b + w a*(w b*H b^2) -2*(w a*H a)*(w b*H b) := by ring simp only [hexp,Finset.sum_sub_distrib,Finset.sum_add_distrib, ← Finset.mul_sum,← Finset.sum_mul] rw [variance_secondMoment w H hw] unfold totalWeight at * field_simp [hw] ring theorem variance_nonneg (w H : α → ℝ) (hw : ∀ x, 0 ≤ w x) : 0 ≤ weightedVariance w H := Finset.sum_nonneg fun x _ => mul_nonneg (hw x) (sq_nonneg _) theorem variance_le_card_sq (w H : α → ℝ) (hw : 0 < totalWeight w) (B : ℝ) (hB : 0 ≤ B) (hp : ∀ a b, w a*w b/totalWeight w*(H a-H b)^2 ≤ B) : weightedVariance w H ≤ (Fintype.card α:ℝ)^2*B := by rw [variance_pair_formula w H hw.ne'] have hb : (∑ a, ∑ b, w a*w b*(H a-H b)^2) ≤ (Fintype.card α:ℝ)^2*(totalWeight w*B) := by calc _ ≤ ∑ a : α, ∑ b : α, totalWeight w*B := by apply Finset.sum_le_sum; intro a ha apply Finset.sum_le_sum; intro b hb have h := (div_le_iff₀ hw).mp (show w a*w b*(H a-H b)^2/totalWeight w ≤ B by simpa only [div_mul_eq_mul_div] using hp a b) nlinarith _ = _ := by simp; ring apply (div_le_iff₀ (mul_pos (by norm_num) hw)).mpr have hc : 0 ≤ (Fintype.card α:ℝ)^2 := sq_nonneg _ nlinarith [mul_nonneg hc hB] end FiniteVariance theorem interval_square_bound (H : ℕ → ℝ) (a b : ℕ) (hab : a ≤ b) : (H a-H b)^2 ≤ (b-a:ℕ)*(∑ j ∈ Finset.Ico a b, (H j-H (j+1))^2) := by have htel : (∑ j ∈ Finset.Ico a b, (H j-H (j+1))) = H a-H b := by calc _ = -(∑ j ∈ Finset.Ico a b, (H (j+1)-H j)) := by rw [← Finset.sum_neg_distrib] apply Finset.sum_congr rfl; intro j hj; ring _ = H a-H b := by rw [Finset.sum_Ico_sub H hab]; ring rw [← htel] calc _ ≤ (∑ j ∈ Finset.Ico a b, (1:ℝ)) * ∑ j ∈ Finset.Ico a b, (H j-H (j+1))^2 := by apply Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul · intros; norm_num · intros; positivity · intros; simp _ = _ := by simp theorem child_pair_bound (U : ℕ) (z H : ℕ → ℝ) (Z A E : ℝ) (hZ : 0 < Z) (hz : ∀ j ≤ U, 0 ≤ z j) (hzZ : ∀ j ≤ U, z j ≤ Z) (hA : 0 ≤ A) (hE : 0 ≤ E) (hmin : ∀ a b j, a ≤ j → j ≤ b → b ≤ U → min (z a) (z b) ≤ z j) (hadj : ∀ j < U, min (z j) (z (j+1))*(H j-H (j+1))^2 ≤ A*E) (a b : ℕ) (hab : a ≤ b) (hbU : b ≤ U) : z a*z b/Z*(H a-H b)^2 ≤ (U:ℝ)^2*A*E := by have haU := hab.trans hbU have hzmin : 0 ≤ min (z a) (z b) := le_min (hz a haU) (hz b hbU) have hpmin : z a*z b/Z ≤ min (z a) (z b) := by apply le_min · apply (div_le_iff₀ hZ).mpr exact mul_le_mul_of_nonneg_left (hzZ b hbU) (hz a haU) · apply (div_le_iff₀ hZ).mpr nlinarith [mul_le_mul_of_nonneg_right (hzZ a haU) (hz b hbU)] have hedge (j : ℕ) (hj : j ∈ Finset.Ico a b) : min (z a) (z b)*(H j-H (j+1))^2 ≤ A*E := by have hj' := Finset.mem_Ico.mp hj have hm : min (z a) (z b) ≤ min (z j) (z (j+1)) := by apply le_min · exact hmin a b j hj'.1 (by omega) hbU · exact hmin a b (j+1) (by omega) (by omega) hbU exact (mul_le_mul_of_nonneg_right hm (sq_nonneg _)).trans (hadj j (by omega)) have hs : min (z a) (z b)*(∑ j ∈ Finset.Ico a b, (H j-H (j+1))^2) ≤ ((b-a:ℕ):ℝ)*(A*E) := by rw [Finset.mul_sum] calc _ ≤ ∑ j ∈ Finset.Ico a b, A*E := Finset.sum_le_sum hedge _ = _ := by simp have hlen : ((b-a:ℕ):ℝ) ≤ U := by exact_mod_cast (show b-a ≤ U by omega) have hlen0 : (0:ℝ) ≤ ((b-a:ℕ):ℝ) := Nat.cast_nonneg _ have hlenSq : (((b-a:ℕ):ℝ))^2 ≤ (U:ℝ)^2 := by exact sq_le_sq₀ hlen0 (Nat.cast_nonneg U) |>.mpr hlen calc _ ≤ min (z a) (z b)*(H a-H b)^2 := mul_le_mul_of_nonneg_right hpmin (sq_nonneg _) _ ≤ min (z a) (z b)*(((b-a:ℕ):ℝ)*∑ j ∈ Finset.Ico a b, (H j-H (j+1))^2) := mul_le_mul_of_nonneg_left (interval_square_bound H a b hab) hzmin _ = ((b-a:ℕ):ℝ)*(min (z a) (z b)*∑ j ∈ Finset.Ico a b, (H j-H (j+1))^2) := by ring _ ≤ ((b-a:ℕ):ℝ)*(((b-a:ℕ):ℝ)*(A*E)) := mul_le_mul_of_nonneg_left hs hlen0 _ ≤ (U:ℝ)^2*A*E := by nlinarith [mul_le_mul_of_nonneg_right hlenSq (mul_nonneg hA hE)] end Contingency open scoped BigOperators namespace Contingency.FiniteChain structure Chain (V : Type*) [Fintype V] where π : V → ℝ P : V → V → ℝ positive : ∀ x, 0 < π x normalized : ∑ x, π x = 1 nonneg : ∀ x y, 0 ≤ P x y stochastic : ∀ x, ∑ y, P x y = 1 reversible : ∀ x y, π x*P x y = π y*P y x lazy : ∀ x, 1/2 ≤ P x x namespace Chain noncomputable section variable {V : Type*} [Fintype V] (C : Chain V) def mean (f : V → ℝ) : ℝ := ∑ x, C.π x*f x def inner (f g : V → ℝ) : ℝ := ∑ x, C.π x*f x*g x def normSq (f : V → ℝ) : ℝ := ∑ x, C.π x*f x^2 def step (f : V → ℝ) (x : V) : ℝ := ∑ y, C.P x y*f y def energy (f : V → ℝ) : ℝ := (∑ x, ∑ y, C.π x*C.P x y*(f x-f y)^2)/2 theorem stationary (y : V) : (∑ x, C.π x*C.P x y)=C.π y := by simp_rw [C.reversible] rw [← Finset.mul_sum,C.stochastic,mul_one] theorem normSq_nonneg (f : V → ℝ) : 0≤C.normSq f := Finset.sum_nonneg fun x _ => mul_nonneg (C.positive x).le (sq_nonneg _) theorem inner_self (f : V → ℝ) : C.inner f f=C.normSq f := by simp only [inner,normSq,pow_two,mul_assoc] theorem step_mean (f : V → ℝ) : C.mean (C.step f)=C.mean f := by simp only [mean,step,Finset.mul_sum] rw [Finset.sum_comm] simp only [← mul_assoc,← Finset.sum_mul,C.stationary] theorem step_selfadjoint (f g : V → ℝ) : C.inner f (C.step g)=C.inner (C.step f) g := by simp only [inner,step,Finset.mul_sum,Finset.sum_mul] rw [Finset.sum_comm] apply Finset.sum_congr rfl; intro x _ apply Finset.sum_congr rfl; intro y _ calc C.π y*f y*(C.P y x*g x) = (C.π y*C.P y x)*f y*g x := by ring _ = _ := by rw [C.reversible]; ring theorem inner_comm (f g : V → ℝ) : C.inner f g=C.inner g f := by unfold inner apply Finset.sum_congr rfl; intro x _; ring theorem edge_sums_left (f : V → ℝ) : (∑ x, ∑ y, C.π x*C.P x y*f x^2)=C.normSq f := by simp only [mul_right_comm (C.π _) (C.P _ _),← Finset.mul_sum,C.stochastic,mul_one,normSq] theorem edge_sums_right (f : V → ℝ) : (∑ x, ∑ y, C.π x*C.P x y*f y^2)=C.normSq f := by rw [Finset.sum_comm] simp only [← Finset.sum_mul,C.stationary,normSq] theorem edge_cross (f g : V → ℝ) : (∑ x, ∑ y, C.π x*C.P x y*f x*g y)=C.inner f (C.step g) := by simp only [inner,step,Finset.mul_sum] apply Finset.sum_congr rfl; intro x _ apply Finset.sum_congr rfl; intro y _; ring theorem energy_identity (f : V → ℝ) : C.energy f=C.normSq f-C.inner f (C.step f) := by have h (x y) : C.π x*C.P x y*(f x-f y)^2 = C.π x*C.P x y*f x^2+C.π x*C.P x y*f y^2 - 2*(C.π x*C.P x y*f x*f y) := by ring simp only [energy,h,Finset.sum_sub_distrib,Finset.sum_add_distrib,← Finset.mul_sum] rw [C.edge_sums_left,C.edge_sums_right,C.edge_cross] ring theorem step_positive (f : V → ℝ) : 0≤C.inner f (C.step f) := by classical have hn : 0≤∑ x, ∑ y, C.π x*(C.P x y-if x=y then 1/2 else 0)*(f x+f y)^2 := by apply Finset.sum_nonneg; intro x _ apply Finset.sum_nonneg; intro y _ apply mul_nonneg · apply mul_nonneg (C.positive x).le by_cases h : x=y · subst y; simp only [ite_true]; linarith [C.lazy x] · simp only [ite_eq_right h,sub_zero]; exact C.nonneg x y · exact sq_nonneg _ have he : (∑ x, ∑ y, C.π x*(C.P x y-if x=y then 1/2 else 0)*(f x+f y)^2) = 2*C.inner f (C.step f) := by have h (x y) : C.π x*(C.P x y-if x=y then 1/2 else 0)*(f x+f y)^2 = (C.π x*C.P x y*f x^2 + C.π x*C.P x y*f y^2 + 2*(C.π x*C.P x y*f x*f y)) - (if x=y then 2*(C.π x*f x^2) else 0) := by by_cases hh : x=y · subst y; simp only [ite_true]; ring · simp only [ite_eq_right hh]; ring simp only [h,Finset.sum_sub_distrib,Finset.sum_add_distrib,← Finset.mul_sum] simp only [Finset.sum_ite_eq,Finset.mem_univ,ite_true] rw [C.edge_sums_left,C.edge_sums_right,C.edge_cross] simp only [normSq,← Finset.mul_sum]; ring rw [he] at hn linarith theorem step_cauchy (f g : V → ℝ) : (C.inner f (C.step g))^2 ≤ C.inner f (C.step f)*C.inner g (C.step g) := by have hh (t : ℝ) : 0 ≤ C.inner f (C.step f)*t^2 + 2*C.inner f (C.step g)*t+C.inner g (C.step g) := by have h := C.step_positive (fun x => t*f x+g x) have he : C.inner (fun x => t*f x+g x) (C.step (fun x => t*f x+g x)) = C.inner f (C.step f)*t^2 + (C.inner f (C.step g)+C.inner g (C.step f))*t+C.inner g (C.step g) := by simp only [inner,step,Finset.mul_sum,Finset.sum_mul,← Finset.sum_add_distrib] apply Finset.sum_congr rfl; intro x _ apply Finset.sum_congr rfl; intro y _ ring rw [he,C.inner_comm g (C.step f),← C.step_selfadjoint f g] at h nlinarith only [h] have hd := discrim_le_zero (by intro t simpa only [pow_two] using hh t) unfold discrim at hd nlinarith only [hd] end end Chain end Contingency.FiniteChain open scoped BigOperators namespace Contingency.FiniteChain.Chain noncomputable section variable {V : Type*} [Fintype V] (C : Chain V) def variance (f : V → ℝ) : ℝ := ∑ x, C.π x*(f x-C.mean f)^2 theorem variance_centered (f : V → ℝ) (h : C.mean f=0) : C.variance f=C.normSq f := by simp only [variance,h,sub_zero,normSq] theorem gap_inner (K : ℝ) (hK : 1≤K) (hvar : ∀ f : V → ℝ, C.variance f ≤ K*C.energy f) (f : V → ℝ) (hf : C.mean f=0) : C.inner f (C.step f) ≤ (1-1/K)*C.normSq f := by have h := hvar f rw [C.variance_centered f hf,C.energy_identity] at h have hk0 : 0 f | t+1, f => C.step (iterate t f) theorem iterate_mean (t : ℕ) (f : V → ℝ) : C.mean (C.iterate t f)=C.mean f := by induction t with | zero => rfl | succ t ih => simpa only [iterate,C.step_mean] using ih theorem iterate_normSq (K : ℝ) (hK : 1≤K) (hvar : ∀ f : V → ℝ, C.variance f ≤ K*C.energy f) (t : ℕ) (f : V → ℝ) (hf : C.mean f=0) : C.normSq (C.iterate t f) ≤ (1-1/K)^(2*t)*C.normSq f := by induction t with | zero => simp [iterate] | succ t ih => have hg := C.gap_normSq K hK hvar (C.iterate t f) (by rw [C.iterate_mean,hf]) have hi := mul_le_mul_of_nonneg_left ih (sq_nonneg (1-1/K)) simp only [iterate] calc _ ≤ (1-1/K)^2*C.normSq (C.iterate t f) := hg _ ≤ (1-1/K)^2*((1-1/K)^(2*t)*C.normSq f) := hi _ = _ := by rw [show 2*(t+1)=2+2*t by omega,pow_add]; ring def law [DecidableEq V] : ℕ → V → V → ℝ | 0, x, y => if x=y then 1 else 0 | t+1, x, y => ∑ z, law t x z*C.P z y def initialDensity [DecidableEq V] (x y : V) : ℝ := (if x=y then 1 else 0)/C.π x-1 theorem density_mean [DecidableEq V] (x : V) : C.mean (C.initialDensity x)=0 := by simp only [mean,initialDensity,mul_sub,mul_one,Finset.sum_sub_distrib] rw [C.normalized] simp only [ite_div,mul_ite,mul_zero,zero_div,Finset.sum_ite_eq, Finset.mem_univ,ite_true] field_simp [(C.positive x).ne'] ring theorem density_normSq [DecidableEq V] (x : V) : C.normSq (C.initialDensity x)=1/C.π x-1 := by have h (y) : C.π y*(C.initialDensity x y)^2 = (if x=y then 1/C.π x-2 else 0)+C.π y := by by_cases hh : x=y · subst y simp only [initialDensity,ite_true] field_simp [(C.positive x).ne'] ring · simp [initialDensity,hh] simp only [normSq,h,Finset.sum_add_distrib,Finset.sum_ite_eq, Finset.mem_univ,ite_true,C.normalized] ring theorem density_law [DecidableEq V] (t : ℕ) (x y : V) : C.π y*(1+C.iterate t (C.initialDensity x) y)=C.law t x y := by induction t generalizing y with | zero => by_cases h : x=y · subst y simp [iterate,initialDensity,law,(C.positive x).ne'] · simp [iterate,initialDensity,law,h] | succ t ih => simp only [iterate,step,law] calc _ = ∑ z, C.π y*C.P y z*(1+C.iterate t (C.initialDensity x) z) := by simp only [mul_add,mul_one,Finset.sum_add_distrib,← Finset.mul_sum,C.stochastic] simp only [mul_assoc,← Finset.mul_sum] _ = ∑ z, (C.π z*(1+C.iterate t (C.initialDensity x) z))*C.P z y := by apply Finset.sum_congr rfl; intro z _ rw [C.reversible]; ring _ = _ := by simp_rw [ih] def tv [DecidableEq V] (t : ℕ) (x : V) : ℝ := (∑ y, |C.law t x y-C.π y|)/2 theorem tv_density [DecidableEq V] (t : ℕ) (x : V) : C.tv t x = (∑ y, C.π y*|C.iterate t (C.initialDensity x) y|)/2 := by unfold tv congr 1 apply Finset.sum_congr rfl; intro y _ rw [← C.density_law] have he : C.π y*(1+C.iterate t (C.initialDensity x) y)-C.π y= C.π y*C.iterate t (C.initialDensity x) y := by ring rw [he,abs_mul,abs_of_pos (C.positive y)] theorem weighted_abs_sq (f : V → ℝ) : (∑ x, C.π x*|f x|)^2 ≤ C.normSq f := by calc _ ≤ (∑ x, C.π x)*(∑ x, C.π x*f x^2) := by apply Finset.sum_sq_le_sum_mul_sum_of_sq_le_mul · intro x _; exact (C.positive x).le · intro x _; exact mul_nonneg (C.positive x).le (sq_nonneg _) · intro x _; rw [mul_pow,sq_abs]; nlinarith only [] _ = _ := by rw [C.normalized,one_mul]; rfl theorem tv_sq [DecidableEq V] (t : ℕ) (x : V) : (C.tv t x)^2 ≤ C.normSq (C.iterate t (C.initialDensity x))/4 := by rw [C.tv_density] have h := C.weighted_abs_sq (C.iterate t (C.initialDensity x)) nlinarith only [h] theorem mixing_bound [DecidableEq V] (K : ℝ) (hK : 1≤K) (hvar : ∀ f : V → ℝ, C.variance f ≤ K*C.energy f) (t : ℕ) (x : V) : C.tv t x ≤ (C.π x)⁻¹*Real.exp (-(t:ℝ)/K) := by have hk0 : 0 (C.positive y).le) (Finset.mem_univ x) _ = 1 := C.normalized have hi : 1≤(C.π x)⁻¹ := (one_le_inv₀ (C.positive x)).mpr hπ have hic : (1/C.π x-1)/4≤((C.π x)⁻¹)^2 := by simp only [one_div] nlinarith only [hi,sq_nonneg ((C.π x)⁻¹-1)] have he0 : 0≤(Real.exp (-(t:ℝ)/K))^2 := sq_nonneg _ have hfinal := mul_le_mul_of_nonneg_right hic he0 have htv := C.tv_sq t x have hi0 : 0≤C.tv t x := by unfold tv; positivity have hout0 : 0≤(C.π x)⁻¹*Real.exp (-(t:ℝ)/K) := by positivity apply (sq_le_sq₀ hi0 hout0).mp nlinarith only [htv,hn,hp,hfinal] theorem mixing_bound_min [DecidableEq V] (K : ℝ) (hK : 1≤K) (hvar : ∀ f : V → ℝ, C.variance f ≤ K*C.energy f) (pmin : ℝ) (hpmin : 0