| import OAI.Computability.DegreeRigidity.Effective.CodingAgreement |
|
|
| namespace OAI |
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| namespace TuringRigidity.CodingOnePoint |
| open CodingForcing CodingAgreement |
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|
| theorem prefix_set_of_le {s t : List Bool} (h : s <+: t) (m : ℕ) |
| (hm : s.length ≤ m) (b : Bool) : s <+: t.set m b := by |
| obtain ⟨u, rfl⟩ := h |
| rw [List.set_append_right _ _ hm] |
| exact List.prefix_append _ _ |
|
|
| theorem set_prefix_set {s t : List Bool} (h : s <+: t) (m : ℕ) |
| (hm : m < s.length) (b : Bool) : s.set m b <+: t.set m b := by |
| obtain ⟨u, rfl⟩ := h |
| rw [List.set_append_left _ _ hm] |
| exact List.prefix_append _ _ |
|
|
| def changeLeft (q : Condition) (m : ℕ) (b : Bool) : Condition where |
| left := q.left.set m b |
| right := q.right |
| sameLength := by simpa using q.sameLength |
| active := q.active |
|
|
| theorem changeLeft_extends {A : ℕ → Oracle} {p q : Condition} (hpq : Extends A p q) |
| (m : ℕ) (hm : p.left.length ≤ m) (hnc : ¬ CodingLocation A p m) (b : Bool) : |
| Extends A p (changeLeft q m b) := by |
| refine ⟨prefix_set_of_le hpq.1 m hm b, hpq.2.1, hpq.2.2.1, ?_⟩ |
| intro j hj hjlen |
| have hne : m ≠ j := by |
| intro h |
| subst j |
| exact hnc hj |
| have heq := hpq.2.2.2 j hj (by simpa [changeLeft] using hjlen) |
| simpa [changeLeft, List.getD_eq_getElem?_getD, List.getElem?_set_ne hne] using heq |
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|
| theorem split_coding_or_divergence {A : ℕ → Oracle} {Y : Oracle} {e₀ e₁ : OracleCode} |
| {p r : Condition} (hn : NoDisagreement A Y e₀ e₁ p) (hpr : Extends A p r) |
| (m : ℕ) (hm : p.left.length ≤ m) (hmr : m < r.left.length) (b : Bool) |
| (n v w : ℕ) (hvw : v ≠ w) |
| (hv : v ∈ CommonIdeal.run Y e₀ r.left n) |
| (hw : w ∈ CommonIdeal.run Y e₀ (r.left.set m b) n) : |
| CodingLocation A p m ∨ |
| ∀ q, Extends A r q → ∀ c, c ∉ CommonIdeal.run Y e₁ q.right n := by |
| by_cases hc : CodingLocation A p m |
| · exact Or.inl hc |
| · right |
| intro q hrq c hvc |
| have hpq := extends_trans hpr hrq |
| have hpc := changeLeft_extends hpq m hm hc b |
| have hvq := CommonIdeal.run_mono hrq.1 n v hv |
| have hwq := CommonIdeal.run_mono (set_prefix_set hrq.1 m hmr b) n w hw |
| have hveq : v = c := by |
| by_contra hne |
| exact hn q hpq ⟨n, v, c, hne, hvq, hvc⟩ |
| have hweq : w = c := by |
| by_contra hne |
| exact hn (changeLeft q m b) hpc ⟨n, w, c, hne, hwq, hvc⟩ |
| exact hvw (hveq.trans hweq.symm) |
|
|
| end TuringRigidity.CodingOnePoint |
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| end OAI |
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