| import OAI.Probability.InvariantIsing.Fields.FieldSpatialParameterOrder |
| import OAI.Probability.InvariantIsing.Fields.FieldTangentOrder |
| import OAI.Probability.InvariantIsing.Fields.FieldAffineShape |
|
|
| namespace OAI |
|
|
| /-! The actual internal-height comparison in `fld:mixed`. The fixed |
| tail is the finite log-cosh recursion. Increasing one height adds its |
| parameter to one variance and subtracts it from the next variance. -/ |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory IsingPerceptron Set |
| open scoped NNReal |
|
|
| namespace InvariantIsing |
|
|
| def fieldAdjacentTail (a b : ℝ) (L : List FieldAffineStep) |
| (hb : ∀ av ∈ L, 0 < av.base) (hs : ∀ av ∈ L, av.slope = 0) : |
| FieldSmoothFamily (Ioo (-a) b) := |
| fieldAffineFamily (Ioo (-a) b) isOpen_Ioo L (fun av hav _ _ => by |
| rw [hs av hav, zero_mul, add_zero] |
| exact hb av hav) |
|
|
| def fieldAdjacentInner (a b η : ℝ) (L : List FieldAffineStep) |
| (hb : ∀ av ∈ L, 0 < av.base) (hs : ∀ av ∈ L, av.slope = 0) : |
| FieldSmoothFamily (Ioo (-a) b) := |
| (fieldAdjacentTail a b L hb hs).transform isOpen_Ioo b (-1) η (fun t ht => by |
| have h := ht.2 |
| linarith) |
|
|
| def fieldAdjacentFamily (a b ζ η : ℝ) (L : List FieldAffineStep) |
| (hb : ∀ av ∈ L, 0 < av.base) (hs : ∀ av ∈ L, av.slope = 0) : |
| FieldSmoothFamily (Ioo (-a) b) := |
| (fieldAdjacentInner a b η L hb hs).transform isOpen_Ioo a 1 ζ (fun t ht => by |
| have h := ht.1 |
| linarith) |
|
|
| lemma fieldAdjacentTail_stationary (a b : ℝ) (L : List FieldAffineStep) |
| (hb : ∀ av ∈ L, 0 < av.base) (hs : ∀ av ∈ L, av.slope = 0) (p : ℝ × ℝ) : |
| (fieldAdjacentTail a b L hb hs).T p = 0 := |
| fieldAffineFamily_stationary (Ioo (-a) b) isOpen_Ioo L _ hs p |
|
|
| lemma fieldAdjacentInner_shape (a b η : ℝ) (L : List FieldAffineStep) |
| (hb : ∀ av ∈ L, 0 < av.base) (hs : ∀ av ∈ L, av.slope = 0) |
| (hη : 0 < η) (hζ : ∀ av ∈ L, 0 < av.exponent) (t : ℝ) : |
| let G := fieldAdjacentInner a b η L hb hs |
| Function.Even (fun z => G.U (t, z)) ∧ |
| Function.Odd (fun z => G.X (t, z)) ∧ |
| MonotoneOn (fun z => G.X (t, z)) (Ici 0) ∧ |
| (∀ z ∈ Ici (0 : ℝ), 0 ≤ G.X (t, z)) ∧ (∀ z, |G.X (t, z)| ≤ 1) := by |
| let av : FieldAffineStep := ⟨b, -1, η⟩ |
| have hv : ∀ v ∈ av :: L, ∀ q ∈ Ioo (-a) b, 0 < v.base + v.slope * q := by |
| intro v hv q hq |
| rcases List.mem_cons.mp hv with rfl | hv |
| · dsimp only [av] |
| linarith [hq.2] |
| · rw [hs v hv, zero_mul, add_zero] |
| exact hb v hv |
| have he : ∀ v ∈ av :: L, 0 < v.exponent := by |
| intro v hv |
| rcases List.mem_cons.mp hv with rfl | hv |
| · exact hη |
| · exact hζ v hv |
| change Function.Even (fun z => (fieldAffineFamily (Ioo (-a) b) isOpen_Ioo (av :: L) hv).U (t, z)) ∧ _ |
| exact fieldAffineFamily_shape (Ioo (-a) b) isOpen_Ioo (av :: L) hv he t |
|
|
| lemma fieldAdjacentFamily_shape (a b ζ η : ℝ) (L : List FieldAffineStep) |
| (hb : ∀ av ∈ L, 0 < av.base) (hs : ∀ av ∈ L, av.slope = 0) |
| (hζ : 0 < ζ) (hη : 0 < η) (htail : ∀ av ∈ L, 0 < av.exponent) (t : ℝ) : |
| let G := fieldAdjacentFamily a b ζ η L hb hs |
| Function.Even (fun z => G.U (t, z)) ∧ |
| Function.Odd (fun z => G.X (t, z)) ∧ |
| MonotoneOn (fun z => G.X (t, z)) (Ici 0) ∧ |
| (∀ z ∈ Ici (0 : ℝ), 0 ≤ G.X (t, z)) ∧ (∀ z, |G.X (t, z)| ≤ 1) := by |
| let av : FieldAffineStep := ⟨a, 1, ζ⟩ |
| let bv : FieldAffineStep := ⟨b, -1, η⟩ |
| have hv : ∀ v ∈ av :: bv :: L, ∀ q ∈ Ioo (-a) b, 0 < v.base + v.slope * q := by |
| intro v hv q hq |
| rcases List.mem_cons.mp hv with rfl | hv |
| · dsimp only [av] |
| linarith [hq.1] |
| · rcases List.mem_cons.mp hv with rfl | hv |
| · dsimp only [bv] |
| linarith [hq.2] |
| · rw [hs v hv, zero_mul, add_zero] |
| exact hb v hv |
| have he : ∀ v ∈ av :: bv :: L, 0 < v.exponent := by |
| intro v hv |
| rcases List.mem_cons.mp hv with rfl | hv |
| · exact hζ |
| · rcases List.mem_cons.mp hv with rfl | hv |
| · exact hη |
| · exact htail v hv |
| change Function.Even (fun z => (fieldAffineFamily (Ioo (-a) b) isOpen_Ioo (av :: bv :: L) hv).U (t, z)) ∧ _ |
| exact fieldAffineFamily_shape (Ioo (-a) b) isOpen_Ioo (av :: bv :: L) hv he t |
|
|
| theorem fieldAdjacentFamily_tangent_antitone (a b ζ η : ℝ) (L : List FieldAffineStep) |
| (hb : ∀ av ∈ L, 0 < av.base) (hs : ∀ av ∈ L, av.slope = 0) |
| (hη : 0 < η) (hζη : ζ ≤ η) (htail : ∀ av ∈ L, 0 < av.exponent) |
| {t : ℝ} (ht : t ∈ Ioo (-a) b) : |
| AntitoneOn (fun z => (fieldAdjacentFamily a b ζ η L hb hs).T (t, z)) (Ici 0) := by |
| let F := fieldAdjacentTail a b L hb hs |
| let G := fieldAdjacentInner a b η L hb hs |
| have hg := fieldAdjacentInner_shape a b η L hb hs hη htail t |
| have hU : Measurable (fun z => G.U (t, z)) := G.mU.comp (by fun_prop) |
| have hX : Measurable (fun z => G.X (t, z)) := G.mX.comp (by fun_prop) |
| have hshape := field_negative_square_average_shape (a + t) ζ ((η - ζ) / 2) |
| (by linarith) hU hg.1 (G.growth t) hX hg.2.1 hg.2.2.1 hg.2.2.2.1 hg.2.2.2.2 |
| have heq : (fun z => (fieldAdjacentFamily a b ζ η L hb hs).T (t, z)) = |
| (fun z => -((η - ζ) / 2) * gaussianTiltAverage (a + t) ζ |
| (fun y => G.U (t, y)) (fun y => (G.X (t, y)) ^ 2) z) := by |
| funext z |
| have h := F.adjacent_tangent isOpen_Ioo a b ζ η |
| (fun q hq => by linarith [hq.1]) (fun q hq => by linarith [hq.2]) |
| (p := (t, z)) ht (fun y => fieldAdjacentTail_stationary a b L hb hs (t, y)) |
| simpa only [F, G, fieldAdjacentFamily, fieldAdjacentInner, neg_div] using h |
| rw [heq] |
| exact hshape.2 |
|
|
| theorem fieldAdjacent_earlier_squares_antitone (a b ζ η : ℝ) (L : List FieldAffineStep) |
| (hb : ∀ av ∈ L, 0 < av.base) (hs : ∀ av ∈ L, av.slope = 0) |
| (hζ : 0 < ζ) (hζη : ζ ≤ η) (htail : ∀ av ∈ L, 0 < av.exponent) |
| (P : List (ℝ × ℝ≥0)) (hP : ∀ av ∈ P, 0 < av.1) |
| {t s : ℝ} (ht : t ∈ Ioo (-a) b) (hss : s ∈ Ioo (-a) b) (hts : t ≤ s) : |
| let G := fieldAdjacentFamily a b ζ η L hb hs |
| ∀ i z, fieldScalarSquares P (fun y => G.U (s, y)) (fun y => G.X (s, y)) i z ≤ |
| fieldScalarSquares P (fun y => G.U (t, y)) (fun y => G.X (t, y)) i z := by |
| have hη : 0 < η := lt_of_lt_of_le hζ hζη |
| let G := fieldAdjacentFamily a b ζ η L hb hs |
| have hshape (q : ℝ) := fieldAdjacentFamily_shape a b ζ η L hb hs hζ hη htail q |
| exact G.earlier_squares_antitone_parameter (convex_Ioo (-a) b) |
| (fun q hq => fieldAdjacentFamily_tangent_antitone a b ζ η L hb hs hη hζη htail hq) |
| (fun q _ => (hshape q).1) (fun q _ => (hshape q).2.1) |
| (fun q _ => (hshape q).2.2.1) (fun q _ => (hshape q).2.2.2.1) |
| (fun p => (hshape p.1).2.2.2.2 p.2) P hP ht hss hts |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
|
|
