openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Magnetic /MagneticBoundaryContinuity.lean
| import OAI.Probability.InvariantIsing.Fields.FieldHeightBoundaryContinuity | |
| import OAI.Probability.InvariantIsing.Magnetic.MagneticImplicitBias | |
| import OAI.Probability.InvariantIsing.Magnetic.MagneticFieldLevels | |
| namespace OAI | |
| /-! Continuity of the prescribed-magnetization functional along the strict | |
| height regularization, including coincident heights and zero root variance. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron Set Filter | |
| open scoped NNReal Topology | |
| namespace InvariantIsing | |
| lemma fieldRegularized_depth (h : FieldStep) (t : ℝ) : | |
| (fieldRegularized h t).depth = h.depth := rfl | |
| lemma continuous_fieldRegularizedValue_bias (h : FieldStep) : | |
| Continuous (fun p : ℝ × ℝ => fieldValue (fieldRegularized h p.1) p.2) := by | |
| let L := fieldRegularizedIncrements h | |
| let V := h.height (Fin.last h.depth) + 1 | |
| have hV : 0 ≤ V := add_nonneg (h.nonneg _) zero_le_one | |
| have hL := fieldRegularizedIncrements_positive h | |
| have hc := fieldRegularizedIncrements_continuous h | |
| have hv := fieldRegularizedIncrements_bound h | |
| have hval := continuous_fieldVaryingValue L hL hc hV hv | |
| obtain ⟨hcap, hb⟩ := fieldVaryingValue_bound L hL hV hv | |
| have hroot : Continuous (fun p : ℝ × ℝ => (fieldRegularizedRoot h p.1 : ℝ)) := | |
| (NNReal.continuous_coe.comp (continuous_fieldRegularizedRoot h)).comp continuous_fst | |
| have hi := continuous_field_gaussian_integral (E := ℝ × ℝ) | |
| (U := fun p y => fieldVaryingValue L p.1 y) | |
| (hval.comp (continuous_fst.fst.prodMk continuous_snd)) hroot continuous_snd | |
| hcap zero_le_one (fun p y => by simpa only [one_mul] using hb p.1 y) | |
| have hlast : Continuous (fun p : ℝ × ℝ => | |
| (fieldRegularized h p.1).height (Fin.last h.depth) / 2) := by | |
| change Continuous (fun p : ℝ × ℝ => (h.height (Fin.last h.depth) + | |
| fieldHeightEpsilon p.1 * ((Fin.last h.depth).val + 1 : ℕ)) / 2) | |
| have heps := continuous_fieldHeightEpsilon | |
| fun_prop | |
| convert hi.sub hlast using 1 | |
| funext p | |
| rw [fieldValue_root, ← fieldRegularizedIncrements_eval, | |
| ← fieldVaryingValue_eq_scalar] | |
| simp only [gaussianOperator, ite_true, fieldRegularized, fieldWithHeights, | |
| fieldRegularizedHeight, Fin.val_zero, zero_add, Nat.cast_one, mul_one, | |
| fieldRegularizedRoot, NNReal.coe_mk, Pi.sub_apply, L] | |
| rfl | |
| lemma continuous_fieldRegularizedMean_bias (h : FieldStep) : | |
| Continuous (fun p : ℝ × ℝ => fieldBiasMean (fieldRegularized h p.1) p.2) := by | |
| let L := fieldRegularizedIncrements h | |
| let V := h.height (Fin.last h.depth) + 1 | |
| have hV : 0 ≤ V := add_nonneg (h.nonneg _) zero_le_one | |
| have hL := fieldRegularizedIncrements_positive h | |
| have hc := fieldRegularizedIncrements_continuous h | |
| have hv := fieldRegularizedIncrements_bound h | |
| have hm := continuous_fieldVaryingMean L hL hc hV hv | |
| have hroot : Continuous (fun p : ℝ × ℝ => (fieldRegularizedRoot h p.1 : ℝ)) := | |
| (NNReal.continuous_coe.comp (continuous_fieldRegularizedRoot h)).comp continuous_fst | |
| have hi := continuous_field_gaussian_integral (E := ℝ × ℝ) | |
| (U := fun p y => fieldVaryingMean L p.1 y) | |
| (hm.comp (continuous_fst.fst.prodMk continuous_snd)) hroot continuous_snd | |
| zero_le_one (le_refl (0 : ℝ)) (fun p y => by | |
| simpa only [zero_mul, add_zero] using (fieldVaryingMean_regular L hL p.1).2 y) | |
| convert hi using 1 | |
| funext p | |
| unfold fieldBiasMean | |
| rw [← fieldRegularizedIncrements_eval, ← fieldVaryingValue_eq_scalar, | |
| ← fieldVaryingMean_eq_scalar, | |
| fieldSpinTransition_eq_standard 0 _ (fieldVaryingValue_regular L hL p.1).1 | |
| (fieldVaryingMean_regular L hL p.1).1, gaussianTiltAverage_zero] | |
| simp only [fieldRegularizedRoot, fieldRegularized, fieldWithHeights, | |
| fieldRegularizedHeight, Fin.val_zero, zero_add, Nat.cast_one, mul_one, NNReal.coe_mk] | |
| rfl | |
| lemma continuous_fieldRegularizedLevel_bias (h : FieldStep) (i : Fin (h.depth + 1)) : | |
| Continuous (fun p : ℝ × ℝ => magneticLevelAtBias (fieldRegularized h p.1) p.2 i) := by | |
| let L := fieldRegularizedIncrements h | |
| let V := h.height (Fin.last h.depth) + 1 | |
| have hV : 0 ≤ V := add_nonneg (h.nonneg _) zero_le_one | |
| have hL := fieldRegularizedIncrements_positive h | |
| have hc := fieldRegularizedIncrements_continuous h | |
| have hv := fieldRegularizedIncrements_bound h | |
| let j : Fin (L.length + 1) := ⟨i.val, by | |
| simpa only [L, fieldRegularizedIncrements, List.length_ofFn] using i.isLt⟩ | |
| have hm := continuous_fieldVaryingSquares L hL hc hV hv j | |
| have hroot : Continuous (fun p : ℝ × ℝ => (fieldRegularizedRoot h p.1 : ℝ)) := | |
| (NNReal.continuous_coe.comp (continuous_fieldRegularizedRoot h)).comp continuous_fst | |
| have hi := continuous_field_gaussian_integral (E := ℝ × ℝ) | |
| (U := fun p y => fieldVaryingSquares L p.1 j y) | |
| (hm.comp (continuous_fst.fst.prodMk continuous_snd)) hroot continuous_snd | |
| zero_le_one (le_refl (0 : ℝ)) (fun p y => by | |
| have hb := (fieldVaryingSquares_regular L hL p.1 j).2 y | |
| simpa only [zero_mul, add_zero, abs_of_nonneg hb.1] using hb.2) | |
| convert hi using 1 | |
| funext p | |
| change fieldSpinTransition 0 (NNReal.mk ((fieldRegularized h p.1).height 0) | |
| ((fieldRegularized h p.1).nonneg 0)) | |
| (fieldScalarValue (scalarFieldIncrements (fieldRegularized h p.1)) | |
| (fun z => Real.log (Real.cosh z))) | |
| (fieldScalarSquares (scalarFieldIncrements (fieldRegularized h p.1)) | |
| (fun z => Real.log (Real.cosh z)) Real.tanh | |
| ⟨i.val, by rw [scalarFieldIncrements_length, fieldRegularized_depth]; exact i.isLt⟩) p.2 = _ | |
| have he : fieldScalarSquares (scalarFieldIncrements (fieldRegularized h p.1)) | |
| (fun z => Real.log (Real.cosh z)) Real.tanh | |
| ⟨i.val, by rw [scalarFieldIncrements_length, fieldRegularized_depth]; exact i.isLt⟩ = | |
| fieldVaryingSquares L p.1 j := by | |
| rw [fieldVaryingSquares_eq_scalar] | |
| exact fieldScalarSquares_congr_list (fieldRegularizedIncrements_eval h p.1).symm | |
| (fun z => Real.log (Real.cosh z)) Real.tanh _ _ rfl | |
| rw [he, ← fieldRegularizedIncrements_eval, ← fieldVaryingValue_eq_scalar, | |
| fieldSpinTransition_eq_standard 0 _ (fieldVaryingValue_regular L hL p.1).1 | |
| (fieldVaryingSquares_regular L hL p.1 j).1, gaussianTiltAverage_zero] | |
| simp only [fieldRegularizedRoot, fieldRegularized, fieldWithHeights, | |
| fieldRegularizedHeight, Fin.val_zero, zero_add, Nat.cast_one, mul_one, NNReal.coe_mk] | |
| rfl | |
| lemma continuous_fieldRegularizedMagneticBias (h : FieldStep) {s : ℝ} (hs : |s| < 1) : | |
| Continuous (fun t => magneticBias (fieldRegularized h t) s) := by | |
| apply continuous_iff_continuousAt.mpr | |
| intro t | |
| apply parametric_bias_continuousAt | |
| (A := fun p => fieldBiasMean (fieldRegularized h p.1) p.2) | |
| (s := s) | |
| · intro z | |
| exact ((continuous_fieldRegularizedMean_bias h).comp | |
| (continuous_id.prodMk continuous_const)).continuousAt | |
| · exact Eventually.of_forall (fun q => strictMono_fieldBiasMean (fieldRegularized h q)) | |
| · exact Eventually.of_forall (fun q => fieldBiasMean_magneticBias (fieldRegularized h q) hs) | |
| lemma continuous_fieldRegularizedConstrainedValue (h : FieldStep) {s : ℝ} (hs : |s| < 1) : | |
| Continuous (fun t => constrainedFieldValue (fieldRegularized h t) s) := by | |
| have hb := continuous_fieldRegularizedMagneticBias h hs | |
| have hv := (continuous_fieldRegularizedValue_bias h).comp (continuous_id.prodMk hb) | |
| convert hv.sub (hb.mul_const s) using 1 | |
| funext t | |
| exact constrainedFieldValue_magneticBias (fieldRegularized h t) hs | |
| lemma continuous_fieldRegularizedMagneticLevel (h : FieldStep) {s : ℝ} (hs : |s| < 1) | |
| (i : Fin (h.depth + 1)) : | |
| Continuous (fun t => magneticFieldLevel (fieldRegularized h t) s i) := by | |
| have hb : Continuous (fun t => magneticBias (fieldRegularized h t) s) := | |
| continuous_fieldRegularizedMagneticBias h hs | |
| have hp : Continuous (fun t : ℝ => (t, magneticBias (fieldRegularized h t) s)) := | |
| continuous_id.prodMk hb | |
| have hc : Continuous (fun p : ℝ × ℝ => | |
| magneticLevelAtBias (fieldRegularized h p.1) p.2 i) := | |
| continuous_fieldRegularizedLevel_bias h i | |
| change Continuous (fun t : ℝ => magneticLevelAtBias (fieldRegularized h t) | |
| (magneticBias (fieldRegularized h t) s) i) | |
| simpa only [Function.comp_def] using hc.comp hp | |
| end InvariantIsing | |
| end | |
| end OAI | |