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f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 | import OAI.Probability.FreeIsing.Model
namespace OAI
namespace Problem367
noncomputable section
open Relation
variable {d : ℕ}
theorem ConnectedFinset.image {D : Finset (TreeVertex d)}
(hD : ConnectedFinset D) (g : TreeAut d) :
ConnectedFinset (D.image g.toEquiv) := by
classical
refine ⟨hD.1.image _, ?_⟩
intro u hu v hv
obtain ⟨u, huD, rfl⟩ := Finset.mem_image.mp hu
obtain ⟨v, hvD, rfl⟩ := Finset.mem_image.mp hv
exact ReflTransGen.lift g.toEquiv
(fun x y hxy => ⟨Finset.mem_image.mpr ⟨x, hxy.1, rfl⟩,
Finset.mem_image.mpr ⟨y, hxy.2.1, rfl⟩,
(g.map_adj x y).mpr hxy.2.2⟩) _ _ (hD.2 u huD v hvD)
namespace CylinderAut
private theorem adj_symm {u v : TreeVertex d} (h : treeAdj u v) : treeAdj v u :=
h.symm
private theorem length_ne_of_adj {u v : TreeVertex d} (h : treeAdj u v) :
u.1.length ≠ v.1.length := by
rcases h with ⟨c, hc⟩ | ⟨c, hc⟩
all_goals
have hlen := congrArg List.length hc
simp only [List.length_append, List.length_singleton] at hlen
omega
def orient (e : TreeVertex d × TreeVertex d) : TreeVertex d × TreeVertex d :=
if e.1.1.length < e.2.1.length then e else e.swap
private theorem orient_swap (e : TreeVertex d × TreeVertex d)
(hne : e.1.1.length ≠ e.2.1.length) :
orient e.swap = orient e := by
unfold orient
simp only [Prod.fst_swap, Prod.snd_swap, Prod.swap_swap]
split_ifs <;> first | rfl | (exfalso; omega)
private theorem mem_inducedEdges {D : Finset (TreeVertex d)}
{e : TreeVertex d × TreeVertex d} :
e ∈ inducedEdges D ↔ e.1 ∈ D ∧ e.2 ∈ D ∧ treeAdj e.1 e.2 ∧
e.1.1.length < e.2.1.length := by
classical
simp [inducedEdges, and_assoc]
private theorem orient_mem {D : Finset (TreeVertex d)}
{e : TreeVertex d × TreeVertex d} (h₁ : e.1 ∈ D) (h₂ : e.2 ∈ D)
(ha : treeAdj e.1 e.2) : orient e ∈ inducedEdges D := by
classical
unfold orient
split_ifs with h
· exact mem_inducedEdges.mpr ⟨h₁, h₂, ha, h⟩
· exact mem_inducedEdges.mpr ⟨h₂, h₁, adj_symm ha,
lt_of_le_of_ne (le_of_not_gt h) (length_ne_of_adj ha).symm⟩
private theorem orient_eq_of_mem {D : Finset (TreeVertex d)}
{e : TreeVertex d × TreeVertex d} (he : e ∈ inducedEdges D) :
orient e = e := by
simp [orient, (mem_inducedEdges.mp he).2.2.2]
def push (g : TreeAut d) (e : TreeVertex d × TreeVertex d) :
TreeVertex d × TreeVertex d :=
orient (g.toEquiv e.1, g.toEquiv e.2)
def pull (g : TreeAut d) (e : TreeVertex d × TreeVertex d) :
TreeVertex d × TreeVertex d :=
orient (g.toEquiv.symm e.1, g.toEquiv.symm e.2)
private theorem push_mem (g : TreeAut d) {D : Finset (TreeVertex d)}
{e : TreeVertex d × TreeVertex d} (he : e ∈ inducedEdges D) :
push g e ∈ inducedEdges (D.image g.toEquiv) := by
classical
obtain ⟨h₁, h₂, ha, _⟩ := mem_inducedEdges.mp he
exact orient_mem (Finset.mem_image.mpr ⟨e.1, h₁, rfl⟩)
(Finset.mem_image.mpr ⟨e.2, h₂, rfl⟩) ((g.map_adj _ _).mpr ha)
private theorem pull_push (g : TreeAut d) {D : Finset (TreeVertex d)}
{e : TreeVertex d × TreeVertex d} (he : e ∈ inducedEdges D) :
pull g (push g e) = e := by
have hedir := (mem_inducedEdges.mp he).2.2.2
unfold push pull orient
by_cases h : (g.toEquiv e.1).1.length < (g.toEquiv e.2).1.length
all_goals simp [h, hedir, not_lt_of_ge (le_of_lt hedir)]
private theorem pull_mem (g : TreeAut d) {D : Finset (TreeVertex d)}
{e : TreeVertex d × TreeVertex d} (he : e ∈ inducedEdges (D.image g.toEquiv)) :
pull g e ∈ inducedEdges D := by
classical
obtain ⟨h₁, h₂, ha, _⟩ := mem_inducedEdges.mp he
have hp₁ : g.toEquiv.symm e.1 ∈ D := by
obtain ⟨x, hx, hxe⟩ := Finset.mem_image.mp h₁
simpa [← hxe] using hx
have hp₂ : g.toEquiv.symm e.2 ∈ D := by
obtain ⟨x, hx, hxe⟩ := Finset.mem_image.mp h₂
simpa [← hxe] using hx
have hpa : treeAdj (g.toEquiv.symm e.1) (g.toEquiv.symm e.2) := by
apply (g.map_adj _ _).mp
simpa using ha
exact orient_mem hp₁ hp₂ hpa
private theorem push_pull (g : TreeAut d) {D : Finset (TreeVertex d)}
{e : TreeVertex d × TreeVertex d} (he : e ∈ inducedEdges (D.image g.toEquiv)) :
push g (pull g e) = e := by
have hedir := (mem_inducedEdges.mp he).2.2.2
unfold push pull orient
by_cases h : (g.toEquiv.symm e.1).1.length < (g.toEquiv.symm e.2).1.length
all_goals simp [h, hedir, not_lt_of_ge (le_of_lt hedir)]
theorem prod_inducedEdges_image {M : Type*} [CommMonoid M]
(g : TreeAut d) (D : Finset (TreeVertex d))
(f : TreeVertex d → TreeVertex d → M)
(hf : ∀ u v, f u v = f v u) :
(∏ e ∈ inducedEdges (D.image g.toEquiv),
f (g.toEquiv.symm e.1) (g.toEquiv.symm e.2)) =
∏ e ∈ inducedEdges D, f e.1 e.2 := by
classical
symm
apply Finset.prod_bij (fun e _ => push g e)
· intro e he
exact push_mem g he
· intro e₁ he₁ e₂ he₂ heq
have h := congrArg (pull g) heq
simpa only [pull_push g he₁, pull_push g he₂] using h
· intro e he
exact ⟨pull g e, pull_mem g he, push_pull g he⟩
· intro e he
unfold push orient
split_ifs
· simp
· simpa using hf e.1 e.2
end CylinderAut
theorem freeCylinderWeight_image_relabel (g : TreeAut d) (theta : ℝ)
(D : Finset (TreeVertex d)) (eta : TreeVertex d → IsingSpin) :
freeCylinderWeight theta (D.image g.toEquiv) (relabel g eta) =
freeCylinderWeight theta D eta := by
classical
unfold freeCylinderWeight
apply congrArg ENNReal.ofReal
apply congrArg ((1 / 2 : ℝ) * ·)
exact CylinderAut.prod_inducedEdges_image g D
(fun u v => (1 + (if eta u = eta v then theta else -theta)) / 2)
(by intro u v; simp only [eq_comm])
private def cylinderInverseAut (g : TreeAut d) : TreeAut d where
toEquiv := g.toEquiv.symm
map_adj := by
intro u v
simpa using (g.map_adj (g.toEquiv.symm u) (g.toEquiv.symm v)).symm
private theorem measurable_spin_cylinder (D : Finset (TreeVertex d))
(eta : TreeVertex d → IsingSpin) :
MeasurableSet {x : TreeVertex d → IsingSpin | ∀ v ∈ D, x v = eta v} := by
have hsingle (s : IsingSpin) : MeasurableSet ({s} : Set IsingSpin) := by
trivial
convert D.measurableSet_biInter
(fun v _ => (measurable_pi_apply v) (hsingle (eta v))) using 1
ext x
simp
theorem HasFreeIsingLaw.map_relabel {theta : ℝ}
{mu : MeasureTheory.Measure (TreeVertex d → IsingSpin)}
(hmu : HasFreeIsingLaw theta mu) (g : TreeAut d) :
HasFreeIsingLaw theta (MeasureTheory.Measure.map (relabel g) mu) := by
classical
intro D hD eta
have hm : Measurable (relabel g : (TreeVertex d → IsingSpin) → _) :=
measurable_pi_iff.mpr (fun v => measurable_pi_apply (g.toEquiv.symm v))
rw [MeasureTheory.Measure.map_apply hm (measurable_spin_cylinder D eta)]
let gi := cylinderInverseAut g
have hpre : relabel g ⁻¹' {x : TreeVertex d → IsingSpin | ∀ v ∈ D, x v = eta v} =
{x : TreeVertex d → IsingSpin | ∀ v ∈ D.image gi.toEquiv,
x v = relabel gi eta v} := by
ext x
constructor
· intro hx v hv
obtain ⟨u, hu, rfl⟩ := Finset.mem_image.mp hv
simpa [relabel, gi, cylinderInverseAut] using hx u hu
· intro hx v hv
have hh := hx (gi.toEquiv v) (Finset.mem_image.mpr ⟨v, hv, rfl⟩)
simpa [relabel, gi, cylinderInverseAut] using hh
rw [hpre, hmu _ (hD.image gi), freeCylinderWeight_image_relabel]
end
end Problem367
end OAI
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