| import OAI.Computability.DegreeRigidity.SetModels.RelationCoding |
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| namespace OAI |
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| namespace TuringRigidity.BoundedDecoding |
| open AntichainParameters SetCoding RelationCoding |
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| def AntichainBelow (p : AntichainCode) (y : Degree) : Prop := |
| p.bound ≤ y ∧ p.left ≤ y ∧ p.right ≤ y |
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| def LocalNontrivial (y b g₀ g₁ x : Degree) : Prop := |
| x ≤ b ∧ ∃ z : Degree, z ≤ y ∧ z ≤ g₀ ⊔ x ∧ z ≤ g₁ ⊔ x ∧ ¬ z ≤ x |
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|
| 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 |
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| 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⟩ |
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| 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⟩ |
|
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| def SetBelow (p : SetCode) (y : Degree) : Prop := |
| p.bound ≤ y ∧ AntichainBelow p.tags y ∧ AntichainBelow p.decorated y |
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| def LocalGraph (y : Degree) (p : SetCode) (x c : Degree) : Prop := |
| x ≤ p.bound ∧ LocalAntichain y p.tags c ∧ LocalAntichain y p.decorated (x ⊔ c) |
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| def LocalSet (y : Degree) (p : SetCode) (x : Degree) : Prop := |
| ∃ c : Degree, c ≤ y ∧ LocalGraph y p x c |
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| 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] |
|
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| 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⟩ |
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| 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⟩⟩ |
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| def RelationBelow {n : ℕ} (p : RelationCode n) (y : Degree) : Prop := |
| (∀ i, SetBelow (p.coordinates i) y) ∧ AntichainBelow p.tuples y |
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| 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) |
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| 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 |
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| end OAI |
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