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caec230 | 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 | import Mathlib
import OAI.MathematicalPhysics.PEPSFilters.LocalOperators
namespace OAI
/-! Clipped rectangle contours and physical boundary counts. -/
noncomputable section
open scoped BigOperators ComplexOrder
open scoped BigOperators ComplexOrder Matrix.Norms.L2Operator
open scoped BigOperators
open scoped Topology
open Filter
open scoped MatrixOrder
open scoped BigOperators Matrix.Norms.L2Operator
open scoped ComplexOrder BigOperators Matrix.Norms.L2Operator
open Matrix
open Filter Topology
open PolynomialPEPS.PinnedEntropy
namespace PolynomialPEPS.Subvolume.RectangleContours
open scoped BigOperators
variable {L : ℕ}
def rectangle (lo₁ hi₁ lo₂ hi₂ : ℤ) (n : ℕ) : Finset (Vertex L) := by
classical
exact Finset.univ.filter (fun v =>
lo₁-(n:ℤ)≤(v.1.val:ℤ) ∧ (v.1.val:ℤ)≤hi₁+n ∧
lo₂-(n:ℤ)≤(v.2.val:ℤ) ∧ (v.2.val:ℤ)≤hi₂+n)
@[simp] lemma mem_rectangle (lo₁ hi₁ lo₂ hi₂ : ℤ) (n : ℕ) (v : Vertex L) :
v∈rectangle lo₁ hi₁ lo₂ hi₂ n ↔
lo₁-(n:ℤ)≤(v.1.val:ℤ) ∧ (v.1.val:ℤ)≤hi₁+n ∧
lo₂-(n:ℤ)≤(v.2.val:ℤ) ∧ (v.2.val:ℤ)≤hi₂+n := by
classical
simp [rectangle]
lemma monotone_rectangle (lo₁ hi₁ lo₂ hi₂ : ℤ) :
Monotone (rectangle (L := L) lo₁ hi₁ lo₂ hi₂) := by
intro n m hnm v hv
rw [mem_rectangle] at hv ⊢
omega
lemma neighbor_in_succ (lo₁ hi₁ lo₂ hi₂ : ℤ) (n : ℕ) (v w : Vertex L)
(h : ForwardAdjacent v w ∨ ForwardAdjacent w v)
(hv : v∈rectangle lo₁ hi₁ lo₂ hi₂ n) :
w∈rectangle lo₁ hi₁ lo₂ hi₂ (n+1) := by
rw [mem_rectangle] at hv ⊢
rcases h with (⟨hx,hy⟩|⟨hx,hy⟩)|(⟨hx,hy⟩|⟨hx,hy⟩)
all_goals
simp only [Fin.ext_iff] at hx hy
omega
lemma crossing_enters_next (lo₁ hi₁ lo₂ hi₂ : ℤ) (n : ℕ) (e : Edge L)
(h : (e.val.1∈rectangle lo₁ hi₁ lo₂ hi₂ n ∧ e.val.2∉rectangle lo₁ hi₁ lo₂ hi₂ n) ∨
(e.val.1∉rectangle lo₁ hi₁ lo₂ hi₂ n ∧ e.val.2∈rectangle lo₁ hi₁ lo₂ hi₂ n)) :
e.val.1∈rectangle lo₁ hi₁ lo₂ hi₂ (n+1) ∧
e.val.2∈rectangle lo₁ hi₁ lo₂ hi₂ (n+1) := by
rcases h with h|h
· exact ⟨monotone_rectangle lo₁ hi₁ lo₂ hi₂ (Nat.le_succ n) h.1,
neighbor_in_succ lo₁ hi₁ lo₂ hi₂ n e.val.1 e.val.2 (Or.inl e.property) h.1⟩
· exact ⟨neighbor_in_succ lo₁ hi₁ lo₂ hi₂ n e.val.2 e.val.1 (Or.inr e.property) h.2,
monotone_rectangle lo₁ hi₁ lo₂ hi₂ (Nat.le_succ n) h.2⟩
theorem unique_crossing (lo₁ hi₁ lo₂ hi₂ : ℤ) (e : Edge L) (j k : ℕ)
(hj : (e.val.1∈rectangle lo₁ hi₁ lo₂ hi₂ j ∧ e.val.2∉rectangle lo₁ hi₁ lo₂ hi₂ j) ∨
(e.val.1∉rectangle lo₁ hi₁ lo₂ hi₂ j ∧ e.val.2∈rectangle lo₁ hi₁ lo₂ hi₂ j))
(hk : (e.val.1∈rectangle lo₁ hi₁ lo₂ hi₂ k ∧ e.val.2∉rectangle lo₁ hi₁ lo₂ hi₂ k) ∨
(e.val.1∉rectangle lo₁ hi₁ lo₂ hi₂ k ∧ e.val.2∈rectangle lo₁ hi₁ lo₂ hi₂ k)) : j=k := by
wlog hle : j≤k generalizing j k
· exact (this k j hk hj (by omega)).symm
by_contra hne
have hlt : j+1≤k := by omega
have hboth := crossing_enters_next lo₁ hi₁ lo₂ hi₂ j e hj
have h₁ := monotone_rectangle lo₁ hi₁ lo₂ hi₂ hlt hboth.1
have h₂ := monotone_rectangle lo₁ hi₁ lo₂ hi₂ hlt hboth.2
rcases hk with hk|hk
· exact hk.2 h₂
· exact hk.1 h₁
private lemma orthogonal_coordinate_bound (lo₁ hi₁ lo₂ hi₂ : ℤ) (n r : ℕ)
(h₁ : hi₁-lo₁+1≤r) (h₂ : hi₂-lo₂+1≤r) (e : Edge L)
(he : (e.val.1∈rectangle lo₁ hi₁ lo₂ hi₂ n ∧ e.val.2∉rectangle lo₁ hi₁ lo₂ hi₂ n) ∨
(e.val.1∉rectangle lo₁ hi₁ lo₂ hi₂ n ∧ e.val.2∈rectangle lo₁ hi₁ lo₂ hi₂ n)) :
let z : ℤ := if e.val.1.1.val+1=e.val.2.1.val then
(e.val.1.2.val:ℤ)-(lo₂-n) else (e.val.1.1.val:ℤ)-(lo₁-n)
0≤z ∧ z<r+2*n := by
simp only [mem_rectangle] at he
rcases e.property with ⟨hx,hy⟩|⟨hx,hy⟩
· simp only [hx,ite_true]
simp only [Fin.ext_iff] at hy
omega
· simp only [Fin.ext_iff] at hx
have hx' : ¬e.val.1.1.val+1=e.val.2.1.val := by omega
simp only [hx',ite_false]
omega
private def boundaryEncoding (lo₁ hi₁ lo₂ hi₂ : ℤ) (n r : ℕ)
(h₁ : hi₁-lo₁+1≤r) (h₂ : hi₂-lo₂+1≤r)
(e : {e : Edge L //
(e.val.1∈rectangle lo₁ hi₁ lo₂ hi₂ n ∧ e.val.2∉rectangle lo₁ hi₁ lo₂ hi₂ n) ∨
(e.val.1∉rectangle lo₁ hi₁ lo₂ hi₂ n ∧ e.val.2∈rectangle lo₁ hi₁ lo₂ hi₂ n)}) :
Bool × Bool × Fin (r+2*n) := by
classical
let z : ℤ := if e.val.val.1.1.val+1=e.val.val.2.1.val then
(e.val.val.1.2.val:ℤ)-(lo₂-n) else (e.val.val.1.1.val:ℤ)-(lo₁-n)
have hz := orthogonal_coordinate_bound lo₁ hi₁ lo₂ hi₂ n r h₁ h₂ e.val e.property
refine ⟨decide (e.val.val.1.1.val+1=e.val.val.2.1.val),
decide (e.val.val.1∈rectangle lo₁ hi₁ lo₂ hi₂ n), z.toNat, ?_⟩
change 0≤z ∧ z<(r:ℤ)+2*(n:ℤ) at hz
omega
private lemma boundaryEncoding_injective (lo₁ hi₁ lo₂ hi₂ : ℤ) (n r : ℕ)
(h₁ : hi₁-lo₁+1≤r) (h₂ : hi₂-lo₂+1≤r) :
Function.Injective (boundaryEncoding (L:=L) lo₁ hi₁ lo₂ hi₂ n r h₁ h₂) := by
classical
intro e f hef
have he := e.property
have hf := f.property
have hae := e.val.property
have haf := f.val.property
have hze := orthogonal_coordinate_bound lo₁ hi₁ lo₂ hi₂ n r h₁ h₂ e.val e.property
have hzf := orthogonal_coordinate_bound lo₁ hi₁ lo₂ hi₂ n r h₁ h₂ f.val f.property
have ht := congrArg Prod.fst hef
have hi := congrArg (fun x => x.2.1) hef
have hz := congrArg (fun x => x.2.2.val) hef
simp only [boundaryEncoding,decide_eq_decide] at ht hi hz
apply Subtype.ext
apply Subtype.ext
apply Prod.ext <;> apply Prod.ext <;> apply Fin.ext
all_goals
simp only [mem_rectangle] at he hf hi
simp only [ForwardAdjacent,Fin.ext_iff] at hae haf
split_ifs at hze hzf hz <;> omega
theorem boundaryCard_le (lo₁ hi₁ lo₂ hi₂ : ℤ) (n r : ℕ)
(h₁ : hi₁-lo₁+1≤r) (h₂ : hi₂-lo₂+1≤r) :
boundaryCard (rectangle (L:=L) lo₁ hi₁ lo₂ hi₂ n)≤4*(r+2*n) := by
have h := Fintype.card_le_of_injective _
(boundaryEncoding_injective (L:=L) lo₁ hi₁ lo₂ hi₂ n r h₁ h₂)
simpa only [boundaryCard,Fintype.card_prod,Fintype.card_bool,Fintype.card_fin,
← mul_assoc,Nat.reduceMul] using h
end PolynomialPEPS.Subvolume.RectangleContours
end
end OAI
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