openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Magnetic /MagneticConstrainedPressureLower.lean
| import OAI.Probability.InvariantIsing.Magnetic.MagneticPhysicalIncrementLower | |
| import OAI.Probability.InvariantIsing.Cavity.ConsecutivePhysicalIncrement | |
| namespace OAI | |
| /-! Physical constrained pressures inherit the magnetic variational lower | |
| bound from the proved block increment, with actual minimizing perturbations. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron Filter | |
| open scoped Topology BigOperators | |
| namespace InvariantIsing | |
| theorem magnetic_physical_constrained_pressure_lower | |
| (hhaar : HaarConcentrationInput) (hgauss : GaussianLipschitzVarianceInput) | |
| (hpub : PanchenkoTalagrandRestrictedFieldPairInput) | |
| {m : ℕ} (hm : 2 ≤ m) (ρ lam : Fin m → ℝ) (hρ : ∀ a, 0 < ρ a) (hsum : ∑ a, ρ a=1) | |
| {K : ℝ} (hK : 0 ≤ K) (hlam : ∀ a, |lam a| ≤ K) | |
| (amax : Fin m) (hmax : ∀ a, lam a ≤ lam amax) | |
| (μ : (M : ℕ) → Measure (Orthogonal M)) [∀ M, IsProbabilityMeasure (μ M)] | |
| [∀ M, (μ M).IsMulRightInvariant] | |
| (N : ℕ → ℕ) (hN : ∀ r, 0 < N r) (hNlim : Tendsto N atTop atTop) | |
| (spec : ℕ → Fin m → ℕ) (hsp : ∀ r a, 0 < spec r a) (hspec : ∀ r, ∑ a, spec r a=N r) | |
| (hρspec : ∀ r a, (spec r a : ℝ)=(N r : ℝ)*ρ a) | |
| {A : Type*} [Fintype A] [DecidableEq A] | |
| (group : ∀ r, Fin (N r) → A) (k : ℕ → A → ℕ) | |
| (hk : ∀ r a, k r a ≤ spinGroupSize (group r) a) | |
| (γ mag : A → ℝ) (hγ : ∀ a, 0 ≤ γ a) (hγsum : ∑ a, γ a=1) | |
| (hcount : ∀ r a, (spinGroupSize (group r) a : ℝ)=N r*γ a) | |
| {s : ℝ} (hs : s < 1) (hmag : ∀ a, |mag a| ≤ s) | |
| (hc : ∀ r a, (k r a : ℝ)=spinGroupSize (group r) a*((1+mag a)/2)) | |
| (L : ℝ) (hL : L < (magneticVariationalFunctional (finiteR ρ lam hρ hsum) γ mag).toReal) : | |
| ∃ r, ∀ ε > 0, ∀ᶠ j in atTop, | |
| let q := m*N r-N r+N r+3 | |
| let M := (q+j)*N r | |
| let g := cavityRationalLabel (by omega : 0 < m) (spec r) (hspec r) M | |
| L-ε ≤ ∫ V, restrictedRotatedPressure | |
| (consecutiveBlockConstraint (N r) (q+j) (spinGroupSlice (group r) (k r))) | |
| (fun i => lam (g i)) (matrixRotation V⁻¹) (fun _ => 0) ∂μ M := by | |
| let ν := cavityOrientedFamily μ | |
| have hν M : (ν M).IsMulLeftInvariant := cavityOrientedFamily_leftInvariant μ M | |
| let g := fun r M => cavityRationalLabel (by omega : 0 < m) (spec r) (hspec r) M | |
| let eig := fun r M i => lam (g r M i) | |
| let I := fun r M => cavitySpectralGroup (g r M) | |
| let C := fun r => spinGroupSlice (group r) (k r) | |
| have hC r : (C r).Nonempty := spinGroupSlice_nonempty (group r) (k r) (hk r) | |
| let q := fun r => m*N r-N r+N r+3 | |
| choose U V hU hV hmin using fun r => repeated_prior_minimizing_sequence hhaar hgauss | |
| ν hν (eig r) (fun _ _ => 0) (I r) (C r) (hC r) | |
| let u : (r j : ℕ) → Fin (cavityRationalSize (N r) (m*N r-N r) j) → ℝ := | |
| fun r j => U r (j+q r) | |
| let v := fun r j => V r (j+q r) | |
| have hum r j i : u r j i ∈ Set.Icc (1 : ℝ) 2 := hU r (j+q r) i | |
| have hvm r j a : v r j a ∈ Set.Icc (1 : ℝ) 2 := hV r (j+q r) a | |
| have hmm r j (u' : Fin (cavityRationalSize (N r) (m*N r-N r) j) → ℝ) (v' : Fin m → ℝ) | |
| (hu' : ∀ i, u' i ∈ Set.Icc (1 : ℝ) 2) (hv' : ∀ a, v' a ∈ Set.Icc (1 : ℝ) 2) : | |
| rationalConstrainedObjective hm (hN r) (spec r) (hspec r) (C r) (hC r) μ lam j (u r j) (v r j) ≤ | |
| rationalConstrainedObjective hm (hN r) (spec r) (hspec r) (C r) (hC r) μ lam j u' v' := by | |
| have hdim : 3 ≤ (j+q r)*N r := cavityRationalSize_ge_three (N r) (m*N r-N r) j (hN r) | |
| have hh := hmin r (j+q r) hdim u' v' hu' hv' | |
| have hp : 0 < (j+q r)*N r := by omega | |
| simpa only [repeatedPriorObjective, rationalConstrainedObjective, ν, | |
| cavityOrientedFamily_pos μ hp, eig, I, g, u, v, q, cavityRationalSize] using hh | |
| obtain ⟨r,hr⟩ := magnetic_rational_increment_eventually_lower hhaar hgauss hpub hm | |
| ρ lam hρ hsum hK hlam amax hmax μ N hN hNlim spec hsp hspec hρspec | |
| group k hk γ mag hγ hγsum hcount hs hmag hc u v hum hvm hmm L hL | |
| have hdim j : 3 ≤ (q r+j)*N r := by | |
| simpa only [q, cavityRationalSize, Nat.add_comm] using cavityRationalSize_ge_three (N r) (m*N r-N r) j (hN r) | |
| have hh := consecutive_envelope_pressure_lower hhaar hgauss (hN r) (by simpa using hdim 0) | |
| ν (fun M _ => hν M) (eig r) (fun _ _ => 0) (I r) (C r) (hC r) | |
| (fun j => U r (q r+j)) (fun j => V r (q r+j)) | |
| (fun j => hU r (q r+j)) (fun j => hV r (q r+j)) | |
| (fun j => hmin r (q r+j) (hdim j)) L (by | |
| intro ε hε | |
| filter_upwards [hr] with j hj | |
| have he := consecutive_prior_increment_physical hhaar hgauss μ (eig r) (I r) (C r) (hC r) | |
| (fun j => U r (q r+j)) (fun j => V r (q r+j)) (fun j => hU r (q r+j)) j (hdim j) | |
| rw [he] | |
| have hi : rationalRestrictedIncrement hm (hN r) (spec r) (hspec r) (C r) (hC r) μ lam (u r) (v r) j = | |
| repeatedRotationIncrement μ (eig r) (I r) (C r) (hC r) (q r+j) (U r (q r+j)) (V r (q r+j)) := by | |
| change repeatedRotationIncrement μ (eig r) (I r) (C r) (hC r) (j+q r) | |
| (U r (j+q r)) (V r (j+q r)) = _ | |
| rw [Nat.add_comm j (q r)] | |
| rw [hi] at hj | |
| have hj' := (sub_le_self L hε.le).trans hj.le | |
| simpa only [div_eq_mul_inv, mul_comm] using hj') | |
| refine ⟨r,?_⟩ | |
| intro ε hε | |
| filter_upwards [hh ε hε] with j hj | |
| dsimp only [ν] at hj | |
| rw [cavityOrientedFamily_pos μ (by have := hdim j; omega), cavity_oriented_restricted_pressure] at hj | |
| exact hj | |
| end InvariantIsing | |
| end | |
| end OAI | |