openAI-math-dataset / lean /OAI /Probability /SKGap /Localization /BufferedSelectionDiagnostics.lean
| import Mathlib | |
| import OAI.Probability.SKGap.Localization.LocalWeakBase | |
| import OAI.Probability.SKGap.Localization.BufferedFlip | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open scoped BigOperators Matrix.Norms.Frobenius | |
| namespace SKGapCutoff.Recipe | |
| open SKGap.Stein Primary Static | |
| variable {n : ℕ} | |
| lemma vector_flip_additive (F : VectorFields n) (x : Spin n) (k : Fin n) {B : ℝ} | |
| (hD : SKGap.opNorm (derivativeMatrix F x)≤B) : | |
| vectorNorm (F (flip x k))≤vectorNorm (F x)+2*B := by | |
| have H : vectorNorm (F (flip x k))≤vectorNorm (F x)+vectorNorm (F (flip x k)-F x) := by | |
| calc | |
| _ = vectorNorm (F x+(F (flip x k)-F x)) := by congr 1;ext i;simp | |
| _ ≤ _ := SKGap.vectorNorm_add_le _ _ | |
| have H' := (vector_flip_opNorm F x k).trans | |
| (mul_le_mul_of_nonneg_left hD (by norm_num : (0:ℝ)≤2)) | |
| linarith | |
| lemma vector_neighborhood_twice_buffer (F : VectorFields n) (E : Set (Spin n)) | |
| {B ρ : ℝ} (hB : 0≤B) (hD : ∀x,SKGap.opNorm (derivativeMatrix F x)≤B) | |
| (hsmall : 4*B≤ρ*Real.sqrt (n:ℝ)) | |
| (hF : ∀x∈E,vectorNorm (F x)≤ρ*Real.sqrt (n:ℝ)) (x : Spin n) | |
| (hx : x∈flipNeighborhood E) : | |
| vectorNorm (F x)≤(2*ρ)*Real.sqrt (n:ℝ) ∧ | |
| ∀k,vectorNorm (F (flip x k))≤(2*ρ)*Real.sqrt (n:ℝ) := by | |
| have hbase : vectorNorm (F x)≤ρ*Real.sqrt (n:ℝ)+2*B := by | |
| rcases hx with hx|⟨k,hk⟩ | |
| · exact (hF x hx).trans (le_add_of_nonneg_right (by positivity)) | |
| · have H:=vector_flip_additive F (flip x k) k (hD (flip x k)) | |
| simp only [flip_flip] at H | |
| have H' := hF _ hk | |
| linarith | |
| refine ⟨by linarith,?_⟩ | |
| intro k | |
| have H:=vector_flip_additive F x k (hD x) | |
| linarith | |
| theorem buffered_selected_initial_diagnostics {j K B A ε c r₀ R ρ : ℝ} | |
| (hj : 0≤j) (hK : 0≤K) (hB : 0≤B) (hA : 1≤A) (hc : 0<c) | |
| (hr₀ : 0<r₀) (hρ : 0<ρ) (hε : 0≤ε) (hAR : Real.exp (R/2)≤A) | |
| (hbuffer : (K+4*j)*(2*ρ)<r₀) | |
| (hR : (1+2*j)*(1+(1+(K+3*j)/c)*(K+4*j))*(2*ρ)≤R) | |
| (hsmall : (3*Real.exp (R/2)+2)*((1+2*j)*(1+(1+(K+3*j)/c)*(K+4*j))*(2*ρ))≤ε) | |
| (himplicit : (1+2*j)*(1+(1+(K+3*j)/c)*(K+4*j))*(2*ρ)≤ | |
| c/(2*(|j| *Real.exp (R/2)*(3*Real.exp (R/2)+16)+1))) | |
| (k : ℕ) (f : KernelExpr) : | |
| ∃V≥0,∃S≥0,∃F≥0,∀n : ℕ,0<n→∀(J : Interaction n) (h : Fin n→ℝ), | |
| J.IsSymm→SKGap.opNorm J≤K→¬SKGap.rootBad j A ε c r₀ J h→ | |
| (∀x l,l<k+3→ShapeBound (formalField j J h x l) B)→ | |
| 4*(residualDerivativeBudget j K B k+residualDerivativeBudget j K B (k+1))≤ρ*Real.sqrt (n:ℝ)→ | |
| ∃r:Fin n→ℝ,SKGap.tapField j J h r=0 ∧ (∀v,SKGap.tapField j J h v=0→v=r) ∧ | |
| ∀e:Fin n→ℝ,vectorNorm e≤1→∃w y : VectorFields n,∃c₀:Spin n→ℝ, | |
| ∀x:Spin n,x∈flipNeighborhood {v | residualCutoff ρ j J h k v≠0}→ | |
| let a:=fun v=>j*(1-Primary.onsager j J h (k+1) v-SKGap.overlap r) | |
| LiteralEquations J j f (fld j J h (k+1)) (mag j J h (k+1)) w y a c₀ r e x ∧ | |
| LiteralInitialDiagnostics | |
| (literalInitial J j (Real.sqrt (n:ℝ)) f (fld j J h (k+1)) (mag j J h (k+1)) a c₀ r e) | |
| w y (primaryTree j J h x (k+1)) x V S F (Real.exp (R/2)) := by | |
| let η:=(1+2*j)*(1+(1+(K+3*j)/c)*(K+4*j))*(2*ρ) | |
| let C:=localPrimaryBudget j K B (k+3) | |
| let T:=(1+|j|)*C | |
| have hη : 0≤η:=by dsimp [η];positivity | |
| have hC : 0≤C:=localPrimaryBudget_nonneg hK hB _ | |
| have hT : 0≤T:=by dsimp [T];positivity | |
| obtain ⟨V,hV,S,hS,F,hF,hdiag⟩:=literal_initial_diagnostics j R η c T K (2*C) C f hj hc hη hT | |
| (mul_nonneg (by norm_num) hC) himplicit | |
| refine ⟨V,hV,S,hS,F,hF,?_⟩ | |
| intro n hn J h hJ hop hbad hformal hdim | |
| have hact : ∀v,vectorNorm (J.mulVec v)≤K*vectorNorm v := fun v=> | |
| (vectorNorm_matrix_mul J v).trans (mul_le_mul_of_nonneg_right hop (vectorNorm_nonneg v)) | |
| obtain ⟨r,hr,hu,hstable⟩:=stable_primary_buffer hn hj hK hA hc hr₀ | |
| (mul_nonneg (by norm_num) hρ.le) hε hAR hbuffer hR hsmall J hJ h hact hbad | |
| refine ⟨r,hr,hu,?_⟩ | |
| intro e he | |
| let a:Spin n→ℝ:=fun v=>j*(1-Primary.onsager j J h (k+1) v-SKGap.overlap r) | |
| obtain ⟨w,y,c₀,heq,hd⟩:=hdiag n hn J (fld j J h (k+1)) (mag j J h (k+1)) a r e hJ hop he | |
| refine ⟨w,y,c₀,?_⟩ | |
| intro x hx | |
| let BB:=residualDerivativeBudget j K B k+residualDerivativeBudget j K B (k+1) | |
| have hBB : 0≤BB:=add_nonneg (residualDerivativeBudget_nonneg hK hB _) | |
| (residualDerivativeBudget_nonneg hK hB _) | |
| have hD₀ (v : Spin n) : SKGap.opNorm (derivativeMatrix (Primary.residual j J h k) v)≤BB := | |
| (residual_derivative_bound hn hK hB J h hop k | |
| (fun v l hl=>hformal v l (by omega)) v).trans | |
| (le_add_of_nonneg_right (residualDerivativeBudget_nonneg hK hB _)) | |
| have hD₁ (v : Spin n) : SKGap.opNorm (derivativeMatrix (Primary.residual j J h (k+1)) v)≤BB := | |
| (residual_derivative_bound hn hK hB J h hop (k+1) | |
| (fun v l hl=>hformal v l (by omega)) v).trans | |
| (le_add_of_nonneg_left (residualDerivativeBudget_nonneg hK hB _)) | |
| have H0:=vector_neighborhood_twice_buffer (Primary.residual j J h k) | |
| {v | residualCutoff ρ j J h k v≠0} hBB hD₀ hdim | |
| (fun v hv=>(residualCutoff_support hn hρ J h k v hv).1) x hx | |
| have H1:=vector_neighborhood_twice_buffer (Primary.residual j J h (k+1)) | |
| {v | residualCutoff ρ j J h k v≠0} hBB hD₁ hdim | |
| (fun v hv=>(residualCutoff_support hn hρ J h k v hv).2) x hx | |
| have H:=hstable k x H0.1 H1.1 | |
| have Hf (v:Fin n):=hstable k (flip x v) (H0.2 v) (H1.2 v) | |
| have hb:=primary_literal_derivative_diagnostics hn hK hB J h hop k hformal r x | |
| change LiteralEquations J j f (fld j J h (k+1)) (mag j J h (k+1)) w y a c₀ r e x ∧ _ | |
| exact ⟨heq x H,hd x (primaryTree j J h x (k+1)) H Hf hb.1 hb.2.1 | |
| (primaryState_mag_bounded j J h (k+1) x) hb.2.2.1 hb.2.2.2.1 hb.2.2.2.2.1 hb.2.2.2.2.2⟩ | |
| end SKGapCutoff.Recipe | |
| end | |
| end | |
| section | |
| noncomputable section | |
| open scoped BigOperators | |
| namespace SKGapCutoff.Recipe | |
| open SKGap.Stein Primary | |
| variable {n : ℕ} | |
| def literalCoefficientBudget (f : KernelExpr) (j R B : ℝ) : ℝ := | |
| 1+|initialRegularBudget f j R B|+f.mass*Real.exp (R/2)+ | |
| (1+|j| *|B|)*phiExpr.mass*Real.exp (R/2) | |
| lemma literalInitial_coefficientClass (J : Interaction n) (j d : ℝ) (f : KernelExpr) | |
| (z m : VectorFields n) (a c : Spin n→ℝ) (r e : Fin n→ℝ) (x : Spin n) | |
| {R B : ℝ} (hB : 0≤B) (ha : |a x|≤R) (haf : ∀k,|a (flip x k)|≤R) | |
| (hc : |d*c x|≤B) (hcf : ∀k,|d*c (flip x k)|≤B) (N : ℕ) : | |
| (literalInitial J j d f z m a c r e).CoefficientClass N x | |
| (literalCoefficientBudget f j R B) := by | |
| let D:=literalInitial J j d f z m a c r e | |
| let K:=literalCoefficientBudget f j R B | |
| have hreg:=literalInitial_regular J j d f z m a c r e x hB ha haf hc hcf | |
| have h1 : 1≤K := by | |
| have h2 : 0≤f.mass*Real.exp (R/2):=mul_nonneg f.mass_nonneg (Real.exp_pos _).le | |
| have h3 : 0≤(1+|j| *|B|)*phiExpr.mass*Real.exp (R/2):=by positivity [phiExpr.mass_nonneg] | |
| dsimp [K,literalCoefficientBudget];linarith [abs_nonneg (initialRegularBudget f j R B)] | |
| have hR : initialRegularBudget f j R B≤K := by | |
| have h0:=le_abs_self (initialRegularBudget f j R B) | |
| have h2 : 0≤f.mass*Real.exp (R/2):=mul_nonneg f.mass_nonneg (Real.exp_pos _).le | |
| have h3 : 0≤(1+|j| *|B|)*phiExpr.mass*Real.exp (R/2):=by positivity [phiExpr.mass_nonneg] | |
| dsimp [K,literalCoefficientBudget];linarith | |
| have hF : f.mass*Real.exp (R/2)≤K := by | |
| have h3 : 0≤(1+|j| *|B|)*phiExpr.mass*Real.exp (R/2):=by positivity [phiExpr.mass_nonneg] | |
| dsimp [K,literalCoefficientBudget];linarith [abs_nonneg (initialRegularBudget f j R B)] | |
| have hP : phiExpr.mass*Real.exp (R/2)≤K := by | |
| have h2 : 0≤f.mass*Real.exp (R/2):=mul_nonneg f.mass_nonneg (Real.exp_pos _).le | |
| have h3 : 0≤|j| *|B| *(phiExpr.mass*Real.exp (R/2)):=by positivity [phiExpr.mass_nonneg] | |
| dsimp [K,literalCoefficientBudget];nlinarith [abs_nonneg (initialRegularBudget f j R B)] | |
| have hJP : |j| *B*(phiExpr.mass*Real.exp (R/2))≤K := by | |
| have h2 : 0≤f.mass*Real.exp (R/2):=mul_nonneg f.mass_nonneg (Real.exp_pos _).le | |
| have h3 : 0≤phiExpr.mass*Real.exp (R/2):=by positivity [phiExpr.mass_nonneg] | |
| dsimp [K,literalCoefficientBudget];rw [abs_of_nonneg hB] | |
| nlinarith [abs_nonneg (initialRegularBudget f j R B)] | |
| refine ⟨h1,?_,?_,?_,?_⟩ | |
| · intro q hq b | |
| simpa only [D,literalInitial] using (hreg.1 b).mono hR | |
| · intro q hq b | |
| simpa only [D,literalInitial] using hreg.2.mono hR | |
| · intro q hq b i k t ht | |
| cases b | |
| · exact (initialKernel_segment_value f D.H D.θ x r R ha haf i k t ht).trans hF | |
| · change |(-j)*(_)*initialKernel phiExpr (r i) _|≤K | |
| rw [abs_mul,abs_mul,abs_neg] | |
| exact (mul_le_mul (mul_le_mul_of_nonneg_left | |
| (parameter_segment_value D.H D.θ x true hc hcf i k t ht) (abs_nonneg _)) | |
| (initialKernel_segment_value phiExpr D.H D.θ x r R ha haf i k t ht) | |
| (abs_nonneg _) (mul_nonneg (abs_nonneg _) hB)).trans hJP | |
| · intro q hq b i k t ht | |
| exact (initialKernel_segment_value phiExpr D.H D.θ x r R ha haf i k t ht).trans hP | |
| lemma literalInitial_seed_bound (J : Interaction n) (j : ℝ) (f : KernelExpr) | |
| (z m : VectorFields n) (a c : Spin n→ℝ) (r e : Fin n→ℝ) | |
| (hn : 0<n) (he : vectorNorm e≤1) : | |
| (∑b,vectorNorm ((literalInitial J j (Real.sqrt (n:ℝ)) f z m a c r e).seed b))≤2 := by | |
| have hd : 0<Real.sqrt (n:ℝ):=Real.sqrt_pos.mpr (Nat.cast_pos.mpr hn) | |
| have hb : vectorNorm (fun i=>Real.tanh (r i)/Real.sqrt (n:ℝ))≤1 := by | |
| have hv:=vectorNorm_tanh r | |
| have hscale : (fun i=>Real.tanh (r i)/Real.sqrt (n:ℝ))= | |
| (Real.sqrt (n:ℝ))⁻¹ • (fun i=>Real.tanh (r i)):=by ext i;simp [div_eq_mul_inv,mul_comm] | |
| rw [hscale,vectorNorm_smul,abs_of_nonneg (inv_nonneg.mpr hd.le)] | |
| calc | |
| _ ≤ (Real.sqrt (n:ℝ))⁻¹*Real.sqrt (n:ℝ) := mul_le_mul_of_nonneg_left hv (inv_nonneg.mpr hd.le) | |
| _ = 1 := inv_mul_cancel₀ hd.ne' | |
| simp only [Fintype.sum_bool,literalInitial,Bool.false_eq_true,↓reduceIte] | |
| change vectorNorm (fun i=>Real.tanh (r i)/Real.sqrt (n:ℝ))+vectorNorm e≤2 | |
| linarith | |
| end SKGapCutoff.Recipe | |
| end | |
| end | |
| end OAI | |