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import Mathlib.Data.List.Sort
namespace OAI
/-! Finite common refinements, built only by duplicating constant-height cells. -/
noncomputable section
open MeasureTheory IsingPerceptron Set
open scoped NNReal
namespace InvariantIsing
def fieldCutSet (h : FieldStep) : Finset ℝ := by
classical
exact Finset.univ.image h.cut
@[simp] lemma mem_fieldCutSet (h : FieldStep) (s : ℝ) :
s ∈ fieldCutSet h ↔ ∃ i, h.cut i = s := by
classical
simp [fieldCutSet]
lemma fieldCutSet_bounds (h : FieldStep) {s : ℝ} (hs : s ∈ fieldCutSet h) :
s ∈ Icc (0 : ℝ) 1 := by
obtain ⟨i, rfl⟩ := (mem_fieldCutSet h s).mp hs
constructor
· rw [← h.first]; exact h.ordered_cut.monotone (Fin.zero_le i)
· rw [← h.last]; exact h.ordered_cut.monotone (Fin.le_last i)
lemma fieldCutSet_insertCut (h : FieldStep) (i : Fin (h.depth + 1)) (c : ℝ)
(hc₀ : h.cut i.castSucc < c) (hc₁ : c < h.cut i.succ) :
fieldCutSet (fieldInsertCut h i c hc₀ hc₁) = insert c (fieldCutSet h) := by
classical
have he := Fin.range_insertNth i.castSucc.succ c h.cut
ext s
simp only [mem_fieldCutSet, Finset.mem_insert]
change (∃ j, Fin.insertNth (α := fun _ => ℝ) i.castSucc.succ c h.cut j = s) ↔
s = c ∨ ∃ j, h.cut j = s
simpa only [Set.insert, Set.mem_ofPred_eq, Set.mem_range] using Set.ext_iff.mp he s
inductive FieldRefines : FieldStep → FieldStep → Prop
| refl (h : FieldStep) : FieldRefines h h
| insert {h k : FieldStep} (hk : FieldRefines h k) (i : Fin (k.depth + 1))
(c : ℝ) (hc₀ : k.cut i.castSucc < c) (hc₁ : c < k.cut i.succ) :
FieldRefines h (fieldInsertCut k i c hc₀ hc₁)
lemma FieldRefines.trans {h k l : FieldStep} (hhk : FieldRefines h k)
(hkl : FieldRefines k l) : FieldRefines h l := by
induction hkl with
| refl => exact hhk
| insert hkl i c hc₀ hc₁ ih => exact FieldRefines.insert ih i c hc₀ hc₁
lemma FieldRefines.value {h k : FieldStep} (hk : FieldRefines h k) (z : ℝ) :
fieldValue k z = fieldValue h z := by
induction hk with
| refl => rfl
| insert hk i c hc₀ hc₁ ih => exact (fieldValue_insertCut _ i c hc₀ hc₁ z).trans ih
lemma FieldRefines.field_ae {h k : FieldStep} (hk : FieldRefines h k) :
fieldFunction k =ᵐ[pathMeasure] fieldFunction h := by
induction hk with
| refl => exact Filter.EventuallyEq.rfl
| insert hk i c hc₀ hc₁ ih => exact (fieldFunction_insertCut_ae _ i c hc₀ hc₁).trans ih
lemma FieldRefines.magnetization_ae {h k : FieldStep} (hk : FieldRefines h k) :
(fieldMagnetizationPath k).val =ᵐ[pathMeasure] (fieldMagnetizationPath h).val := by
induction hk with
| refl => exact Filter.EventuallyEq.rfl
| insert hk i c hc₀ hc₁ ih =>
exact (fieldMagnetizationPath_insertCut_ae _ i c hc₀ hc₁).trans ih
lemma FieldRefines.pairing {h k : FieldStep} (hk : FieldRefines h k) (p : OverlapPath) :
fieldPairing p k = fieldPairing p h := by
apply integral_congr_ae
filter_upwards [hk.field_ae] with s hs
rw [hs]
lemma field_exists_refinement (h : FieldStep) (S : Finset ℝ)
(hS : ∀ s ∈ S, s ∈ Icc (0 : ℝ) 1) :
∃ k, FieldRefines h k ∧ fieldCutSet k = fieldCutSet h ∪ S := by
classical
induction S using Finset.induction_on with
| empty => exact ⟨h, FieldRefines.refl h, by simp⟩
| @insert c S hc ih =>
obtain ⟨k, hk, hcuts⟩ := ih (fun s hs => hS s (Finset.mem_insert_of_mem hs))
by_cases hck : c ∈ fieldCutSet k
· refine ⟨k, hk, ?_⟩
rw [hcuts] at hck ⊢
ext s
simp only [Finset.mem_union, Finset.mem_insert]
constructor
· intro hs; rcases hs with hs | hs
· exact Or.inl hs
· exact Or.inr (Or.inr hs)
· intro hs; rcases hs with hs | rfl | hs
· exact Or.inl hs
· simpa only [Finset.mem_union] using hck
· exact Or.inr hs
· have hcb := hS c (Finset.mem_insert_self c S)
have hc0 : 0 < c := by
have hne : c ≠ 0 := by
intro he; apply hck
apply (mem_fieldCutSet k c).mpr
exact ⟨0, k.first.trans he.symm⟩
exact lt_of_le_of_ne hcb.1 (Ne.symm hne)
have hc1 : c < 1 := by
have hne : c ≠ 1 := by
intro he; apply hck
apply (mem_fieldCutSet k c).mpr
exact ⟨Fin.last (k.depth + 1), k.last.trans he.symm⟩
exact lt_of_le_of_ne hcb.2 hne
obtain ⟨i, hi₀, hi₁⟩ := finite_partition_cell k.cut
(by simpa only [k.first] using hc0.le) (by simpa only [k.last] using hc1)
have hi : k.cut i.castSucc < c := lt_of_le_of_ne hi₀ (by
intro he; exact hck ((mem_fieldCutSet k c).mpr ⟨i.castSucc, he⟩))
refine ⟨fieldInsertCut k i c hi hi₁, FieldRefines.insert hk i c hi hi₁, ?_⟩
rw [fieldCutSet_insertCut, hcuts]
ext s
simp only [Finset.mem_insert, Finset.mem_union]
tauto
lemma field_cutList_eq_of_set_eq {h k : FieldStep} (he : fieldCutSet h = fieldCutSet k) :
List.ofFn h.cut = List.ofFn k.cut := by
have hh : (List.ofFn h.cut).SortedLT :=
List.sortedLT_iff_pairwise.mpr (List.pairwise_ofFn.mpr h.ordered_cut)
have hk : (List.ofFn k.cut).SortedLT :=
List.sortedLT_iff_pairwise.mpr (List.pairwise_ofFn.mpr k.ordered_cut)
apply hh.eq_of_mem_iff hk
intro s
have hs := Finset.ext_iff.mp he s
simpa only [mem_fieldCutSet, List.mem_ofFn] using hs
lemma field_common_refinement (h k : FieldStep) :
∃ h' k', FieldRefines h h' ∧ FieldRefines k k' ∧
List.ofFn h'.cut = List.ofFn k'.cut := by
obtain ⟨h', hh', hhcut⟩ := field_exists_refinement h (fieldCutSet k)
(fun _ hs => fieldCutSet_bounds k hs)
obtain ⟨k', hk', hkcut⟩ := field_exists_refinement k (fieldCutSet h)
(fun _ hs => fieldCutSet_bounds h hs)
refine ⟨h', k', hh', hk', field_cutList_eq_of_set_eq ?_⟩
rw [hhcut, hkcut, Finset.union_comm]
end InvariantIsing
end
end OAI
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