| import Mathlib |
| import OAI.Probability.SKGap.Localization.SelectionPartition |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| namespace SKGapCutoff.Static |
| universe u |
| variable {Ω : Type u} {n : Ω→ℕ} |
| lemma UniformWeak.finset_sum {τ : Type*} {P : ∀a,Observables (n a)} |
| {F : τ→∀a,Observables (n a)} (s : Finset τ) (h : ∀t∈s,UniformWeak P (F t)) : |
| UniformWeak P (fun a x=>∑t∈s,F t a x) := by |
| classical |
| induction s using Finset.induction_on with |
| | empty=>simpa using UniformWeak.zero P |
| | @insert t s ht ih=> |
| simpa only [Finset.sum_insert ht] using (h t (Finset.mem_insert_self ..)).add |
| (ih (fun r hr=>h r (Finset.mem_insert_of_mem hr))) |
| lemma UniformSquare.finset_sum {τ : Type*} {P : ∀a,Observables (n a)} |
| {F : τ→∀a,Observables (n a)} (s : Finset τ) (h : ∀t∈s,UniformSquare P (F t)) : |
| UniformSquare P (fun a x=>∑t∈s,F t a x) := by |
| classical |
| induction s using Finset.induction_on with |
| | empty=>simpa using UniformSquare.zero P |
| | @insert t s ht ih=> |
| simpa only [Finset.sum_insert ht] using (h t (Finset.mem_insert_self ..)).add |
| (ih (fun r hr=>h r (Finset.mem_insert_of_mem hr))) |
| end SKGapCutoff.Static |
| namespace SKGapCutoff.Recipe |
| open Primary Static |
| variable {n M : ℕ} |
|
|
| def constantRecipe (j : ℝ) (J : Interaction n) (h e : Fin n→ℝ) (M : ℕ) : |
| OrdinaryData n (Fin M) (Fin M) Unit where |
| J:=J |
| j:=j |
| H:=fun q=>fld j J h q.val |
| predecessor:=fun q=>mag j J h q.val |
| θ:=fun q x=>j*(onsager j J h (q.val+1) x-onsager j J h q.val x) |
| seed:=fun _=>e |
| seedFunction:=fun _ _ _ _=>1 |
| seedDerivative:=fun _ _ _ _=>0 |
| auxFunction:=fun _ _ _ _=>0 |
| auxDerivative:=fun _ _ _ _=>0 |
|
|
| lemma constantRecipe_source (j : ℝ) (J : Interaction n) (h e : Fin n→ℝ) (M : ℕ) : |
| (constantRecipe j J h e M).source 0=fun _=>e := by |
| rw [OrdinaryData.source_eq] |
| ext x i |
| simp [OrdinaryData.sourceOf,constantRecipe,OrdinaryData.seedCoefficient,OrdinaryData.auxCoefficient,coefficient] |
|
|
| lemma constantRecipe_coefficients (j : ℝ) (J : Interaction n) (h e : Fin n→ℝ) (M N : ℕ) (x : Spin n) : |
| (constantRecipe j J h e M).CoefficientClass N x 1 := by |
| refine ⟨le_rfl,?_,?_,?_,?_⟩ |
| · intro a ha s; exact (segmentRegular_const _ _ x 1).mono zero_le_one |
| · intro a ha b; exact (segmentRegular_const _ _ x 0).mono zero_le_one |
| · intros; norm_num [SegmentValue,constantRecipe] |
| · intros; norm_num [SegmentValue,constantRecipe] |
|
|
| lemma constantRecipe_admissible (j : ℝ) (J : Interaction n) (h e : Fin n→ℝ) (M N p : ℕ) : |
| (constantRecipe j J h e M).Admissible N p := by |
| intro a ha q hq |
| constructor <;> intro s <;> ext x i <;> |
| simp [OrdinaryData.seedPartial,OrdinaryData.auxPartial,localPartial,constantRecipe,partialAt] |
|
|
| universe u |
| variable {Ω : Type u} {N : Ω→ℕ} {j R B W C : ℝ} |
| variable {J : ∀a,Interaction (N a)} {h e : ∀a,Fin (N a)→ℝ} |
|
|
| lemma constantRecipe_family (he : ∀a,vectorNorm (e a)≤1) (M p : ℕ) : |
| FamilyRecipe j J h (fun a=>constantRecipe j (J a) (h a) (e a) M) 0 p 2 := by |
| refine ⟨fun _=>rfl,fun _=>rfl,fun _ _=>rfl,fun _ _=>rfl,fun _ _=>rfl, |
| ⟨1,zero_le_one,?_⟩,fun a x=>(constantRecipe_coefficients j (J a) (h a) (e a) M 0 x).mono (by norm_num), |
| fun a=>constantRecipe_admissible j (J a) (h a) (e a) M 0 p⟩ |
| intro a; simpa [constantRecipe] using he a |
|
|
| lemma primary_residual_telescope (j : ℝ) (J : Interaction n) (h : Fin n→ℝ) (d : ℕ) (x : Spin n) (i : Fin n) : |
| (∑k∈Finset.range d,residual j J h k x i)=spin x i-mag j J h d x i := by |
| induction d with |
| | zero=>simp [mag_zero] |
| | succ d ih=>rw [Finset.sum_range_succ,ih];dsimp [Primary.residual];ring |
|
|
| theorem gibbs_constant_residuals (d : ℕ) (hR : 0≤R) (hB : 0≤B) (hW : 0≤W) (hC : 0≤C) |
| (hn : ∀a,0<N a) (hJ : ∀a,(J a).IsSymm) (hdiag : ∀a i,J a i i=0) |
| (hevent : ∀a,RecipeMatrixEvent j R (residualCoefficientBudget j 2 d 0) B W C (d+1) (2*d) (J a)) |
| (he : ∀a,vectorNorm (e a)≤1) (k : ℕ) (hk : k≤d) : |
| UniformWeak (fun a=>fieldGibbs (J a) (h a)) |
| (fun a x=>∑i,(spin x i-mag j (J a) (h a) k x i)*e a i) ∧ |
| UniformSquare (fun a=>fieldGibbs (J a) (h a)) |
| (fun a x=>∑i,(spin x i-mag j (J a) (h a) k x i)*e a i) := by |
| have H (l : ℕ) (hl : l<k):= |
| gibbs_ordinary_residuals l 0 (by norm_num : (2:ℝ)≤2) (by omega : l<d+1) |
| hR hB hW hC hn hJ hdiag |
| (fun a=>(hevent a).mono (residualCoefficientBudget_depth_mono j (by omega)) (by omega)) |
| (fun a=>constantRecipe j (J a) (h a) (e a) (d+1)) (constantRecipe_family he (d+1) l) |
| have H' (l : ℕ) (hl : l<k) : |
| UniformWeak (fun a=>fieldGibbs (J a) (h a)) (fun a x=>∑i,residual j (J a) (h a) l x i*e a i) ∧ |
| UniformSquare (fun a=>fieldGibbs (J a) (h a)) (fun a x=>∑i,residual j (J a) (h a) l x i*e a i) := by |
| simpa only [constantRecipe_source] using H l hl |
| have hh (a : Ω) (x : Spin (N a)) : |
| (∑l∈Finset.range k,∑i,residual j (J a) (h a) l x i*e a i)= |
| ∑i,(spin x i-mag j (J a) (h a) k x i)*e a i := by |
| rw [Finset.sum_comm] |
| simp only [←Finset.sum_mul,primary_residual_telescope] |
| exact ⟨((UniformWeak.finset_sum (Finset.range k) (fun l hl=>(H' l (Finset.mem_range.mp hl)).1))).congr hh, |
| ((UniformSquare.finset_sum (Finset.range k) (fun l hl=>(H' l (Finset.mem_range.mp hl)).2))).congr hh⟩ |
| |
| end SKGapCutoff.Recipe |
| |
| end |
| end |
| |
| section |
| |
| noncomputable section |
| open scoped BigOperators |
| namespace SKGapCutoff.Static |
| universe u |
| variable {Ω : Type u} {n : Ω→ℕ} |
| lemma UniformMultiplier.neg {F : ∀a,Observables (n a)} (hF : UniformMultiplier F) : |
| UniformMultiplier (fun a x=>-F a x) := by |
| obtain ⟨B,L,hB,hL,hF,hd⟩:=hF |
| refine ⟨B,L,hB,hL,fun a x=>by simpa using hF a x,fun a x=>?_⟩ |
| have he (i : Fin (n a)) : halfDiff i (fun y=>-F a y) x= -halfDiff i (F a) x := by unfold halfDiff;ring |
| simpa only [he,neg_sq] using hd a x |
| lemma UniformMultiplier.mul {F G : ∀a,Observables (n a)} (hF : UniformMultiplier F) |
| (hG : UniformMultiplier G) : UniformMultiplier (fun a x=>F a x*G a x) := by |
| obtain ⟨B,L,hB,hL,hF,hdF⟩:=hF |
| obtain ⟨D,K,hD,hK,hG,hdG⟩:=hG |
| exact ⟨B*D,Real.sqrt (2*B^2*K^2+2*D^2*L^2),mul_nonneg hB hD,Real.sqrt_nonneg _, |
| fun a=>(scalar_product_bounds (F a) (G a) hB hL hD hK (hF a) (hdF a) (hG a) (hdG a)).1, |
| fun a=>(scalar_product_bounds (F a) (G a) hB hL hD hK (hF a) (hdF a) (hG a) (hdG a)).2⟩ |
| lemma UniformMultiplier.finset_prod {τ : Type*} {F : τ→∀a,Observables (n a)} (s : Finset τ) |
| (h : ∀t∈s,UniformMultiplier (F t)) : UniformMultiplier (fun a x=>∏t∈s,F t a x) := by |
| classical |
| induction s using Finset.induction_on with |
| | empty=>simpa using UniformMultiplier.const (n:=n) 1 |
| | @insert t s ht ih=> |
| simpa only [Finset.prod_insert ht] using (h t (Finset.mem_insert_self ..)).mul |
| (ih (fun r hr=>h r (Finset.mem_insert_of_mem hr))) |
| end SKGapCutoff.Static |
| namespace SKGapCutoff.Recipe |
| open Primary Static |
| variable {n : ℕ} |
| |
| def residualDerivativeBudget (j R B : ℝ) (k : ℕ) : ℝ := |
| differentiationBound j R B k+differentiationBound j R B (k+1) |
| |
| lemma residualDerivativeBudget_nonneg {j R B : ℝ} (hR : 0≤R) (hB : 0≤B) (k : ℕ) : |
| 0≤residualDerivativeBudget j R B k := |
| add_nonneg (differentiationBound_nonneg hR hB k) (differentiationBound_nonneg hR hB (k+1)) |
| |
| lemma residual_derivative_bound (hn : 0<n) {j R B : ℝ} (hR : 0≤R) (hB : 0≤B) |
| (J : Interaction n) (h : Fin n→ℝ) (hJ : SKGap.opNorm J≤R) (k : ℕ) |
| (hformal : ∀x l,l≤k+1→ShapeBound (formalField j J h x l) B) (x : Spin n) : |
| SKGap.opNorm (derivativeMatrix (residual j J h k) x)≤residualDerivativeBudget j R B k := by |
| have hm (l : ℕ) (hl : l≤k+1) : |
| SKGap.opNorm (derivativeMatrix (mag j J h l) x)≤differentiationBound j R B l := |
| (primary_differentiation J h x hn hR hB hJ l (fun t ht=>hformal x t (ht.trans hl))).2.2.1 |
| rw [show derivativeMatrix (residual j J h k) x= |
| derivativeMatrix (mag j J h k) x-derivativeMatrix (mag j J h (k+1)) x from derivativeMatrix_sub _ _ x] |
| simp only [opNorm_eq,map_sub] |
| exact (norm_sub_le _ _).trans (add_le_add (hm _ (by omega)) (hm _ le_rfl)) |
| |
| universe u |
| variable {Ω : Type u} {N : Ω→ℕ} {j R B ρ : ℝ} {M : ℕ} |
| variable {J : ∀a,Interaction (N a)} {h : ∀a,Fin (N a)→ℝ} |
| |
| lemma residualCutoff_uniform (hρ : 0<ρ) (hR : 0≤R) (hB : 0≤B) |
| (hn : ∀a,0<N a) (hop : ∀a,SKGap.opNorm (J a)≤R) |
| (hformal : ∀a x l,l<M→ShapeBound (formalField j (J a) (h a) x l) B) |
| (k : ℕ) (hk : k+2<M) : |
| UniformMultiplier (fun a=>residualCutoff ρ j (J a) (h a) k) := by |
| obtain ⟨K,hK⟩:=residualScalarCutoff_lipschitz |
| let A:=residualDerivativeBudget j R B k |
| let D:=residualDerivativeBudget j R B (k+1) |
| have hA : 0≤A:=residualDerivativeBudget_nonneg hR hB k |
| have hD : 0≤D:=residualDerivativeBudget_nonneg hR hB (k+1) |
| let L:=2*(K:ℝ)/ρ^2*((4*A+2*A^2)+(4*D+2*D^2)) |
| have hL : 0≤L:=by dsimp [L];positivity |
| refine ⟨1,L,zero_le_one,hL,fun a x=>?_,fun a x=>?_⟩ |
| · rw [abs_of_nonneg (residualCutoff_bounds ρ j (J a) (h a) k x).1] |
| exact (residualCutoff_bounds ρ j (J a) (h a) k x).2 |
| · have H:=residualCutoff_gradient (hn a) hρ hK (J a) (h a) k x hA hD |
| (residual_derivative_bound (hn a) hR hB (J a) (h a) (hop a) k (fun x l hl=>hformal a x l (by omega)) x) |
| (residual_derivative_bound (hn a) hR hB (J a) (h a) (hop a) (k+1) (fun x l hl=>hformal a x l (by omega)) x) |
| have hone : 1≤Real.sqrt (N a:ℝ) := Real.one_le_sqrt.mpr (by exact_mod_cast hn a) |
| have hd : ‖derivativeVector (residualCutoff ρ j (J a) (h a) k) x‖≤L := |
| H.trans (div_le_self hL hone) |
| simpa only [norm_derivativeVector_sq] using pow_le_pow_left₀ (norm_nonneg _) hd 2 |
| |
| lemma cutoffWeight_uniform {C : ℕ→∀a,Observables (N a)} (k : ℕ) |
| (hC : ∀l≤k,UniformMultiplier (C l)) : |
| UniformMultiplier (fun a x=>cutoffWeight (fun l=>C l a x) k) := by |
| exact (hC k le_rfl).mul (UniformMultiplier.finset_prod (Finset.range k) |
| (fun l hl=>(UniformMultiplier.const 1).add (hC l (Finset.mem_range.mp hl).le).neg)) |
| |
| lemma residualWeight_uniform (hρ : 0<ρ) (hR : 0≤R) (hB : 0≤B) |
| (hn : ∀a,0<N a) (hop : ∀a,SKGap.opNorm (J a)≤R) |
| (hformal : ∀a x l,l<M→ShapeBound (formalField j (J a) (h a) x l) B) |
| (k : ℕ) (hk : k+2<M) : |
| UniformMultiplier (fun a x=>cutoffWeight (fun l=>residualCutoff ρ j (J a) (h a) l x) k) := by |
| apply cutoffWeight_uniform |
| intro l hl |
| exact residualCutoff_uniform hρ hR hB hn hop hformal l (by omega) |
| |
| end SKGapCutoff.Recipe |
| |
| end |
| end |
| |
| end OAI |
| |
