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namespace OAI
namespace TuringRigidity.BoundedDecoding
open AntichainParameters SetCoding RelationCoding
def AntichainBelow (p : AntichainCode) (y : Degree) : Prop :=
p.bound ≤ y ∧ p.left ≤ y ∧ p.right ≤ y
def LocalNontrivial (y b g₀ g₁ x : Degree) : Prop :=
x ≤ b ∧ ∃ z : Degree, z ≤ y ∧ z ≤ g₀ ⊔ x ∧ z ≤ g₁ ⊔ x ∧ ¬ z ≤ x
def LocalAntichain (y : Degree) (p : AntichainCode) (x : Degree) : Prop :=
LocalNontrivial y p.bound p.left p.right x ∧
∀ w : Degree, w ≤ y → LocalNontrivial y p.bound p.left p.right w →
w ≤ x → x ≤ w
theorem local_nontrivial_iff (y b g₀ g₁ x : Degree) (hb : b ≤ y) (hg₀ : g₀ ≤ y) :
LocalNontrivial y b g₀ g₁ x ↔ NontrivialCommonLower b g₀ g₁ x := by
constructor
· rintro ⟨hx, z, _, hz₀, hz₁, hn⟩
exact ⟨hx, z, hz₀, hz₁, hn⟩
· rintro ⟨hx, z, hz₀, hz₁, hn⟩
exact ⟨hx, z, hz₀.trans (sup_le hg₀ (hx.trans hb)), hz₀, hz₁, hn⟩
theorem local_antichain_iff (y : Degree) (p : AntichainCode) (hp : AntichainBelow p y)
(x : Degree) : LocalAntichain y p x ↔ p.Holds x := by
unfold LocalAntichain AntichainCode.Holds CodedAntichain
simp only [local_nontrivial_iff y p.bound p.left p.right _ hp.1 hp.2.1]
constructor
· rintro ⟨hx, hmin⟩
exact ⟨hx, fun w hw hle => hmin w (hw.1.trans hp.1) hw hle⟩
· rintro ⟨hx, hmin⟩
exact ⟨hx, fun w _ hw hle => hmin w hw hle⟩
def SetBelow (p : SetCode) (y : Degree) : Prop :=
p.bound ≤ y ∧ AntichainBelow p.tags y ∧ AntichainBelow p.decorated y
def LocalGraph (y : Degree) (p : SetCode) (x c : Degree) : Prop :=
x ≤ p.bound ∧ LocalAntichain y p.tags c ∧ LocalAntichain y p.decorated (x ⊔ c)
def LocalSet (y : Degree) (p : SetCode) (x : Degree) : Prop :=
∃ c : Degree, c ≤ y ∧ LocalGraph y p x c
theorem local_graph_iff (y : Degree) (p : SetCode) (hp : SetBelow p y) (x c : Degree) :
LocalGraph y p x c ↔ p.Graph x c := by
simp only [LocalGraph, SetCode.Graph, local_antichain_iff y p.tags hp.2.1,
local_antichain_iff y p.decorated hp.2.2]
theorem graph_bounded {y : Degree} {p : SetCode} (hp : SetBelow p y)
{x c : Degree} (h : p.Graph x c) : x ≤ y ∧ c ≤ y :=
⟨h.1.trans hp.1, h.2.1.1.1.trans hp.2.1.1⟩
theorem local_set_iff (y : Degree) (p : SetCode) (hp : SetBelow p y) (x : Degree) :
LocalSet y p x ↔ p.Holds x := by
simp only [LocalSet, local_graph_iff y p hp, SetCode.Holds]
exact ⟨fun ⟨c, _, hc⟩ => ⟨c, hc⟩, fun ⟨c, hc⟩ => ⟨c, (graph_bounded hp hc).2, hc⟩⟩
def RelationBelow {n : ℕ} (p : RelationCode n) (y : Degree) : Prop :=
(∀ i, SetBelow (p.coordinates i) y) ∧ AntichainBelow p.tuples y
def LocalRelation {n : ℕ} (y : Degree) (p : RelationCode n) (v : Fin n → Degree) : Prop :=
∃ c : Fin n → Degree, (∀ i, c i ≤ y ∧ LocalGraph y (p.coordinates i) (v i) (c i)) ∧
LocalAntichain y p.tuples (Finset.univ.sup c)
theorem local_relation_iff {n : ℕ} (y : Degree) (p : RelationCode n)
(hp : RelationBelow p y) (v : Fin n → Degree) :
LocalRelation y p v ↔ p.Holds v := by
unfold LocalRelation RelationCode.Holds
simp only [local_antichain_iff y p.tuples hp.2, local_graph_iff y _ (hp.1 _)]
constructor
· rintro ⟨c, hc, ht⟩
exact ⟨c, fun i => (hc i).2, ht⟩
· rintro ⟨c, hc, ht⟩
exact ⟨c, fun i => ⟨(graph_bounded (hp.1 i) (hc i)).2, hc i⟩, ht⟩
end TuringRigidity.BoundedDecoding
end OAI
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