File size: 2,416 Bytes
9dbc64e | 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 | import OAI.Computability.DegreeRigidity.Effective.CodingAgreement
namespace OAI
namespace TuringRigidity.CodingOnePoint
open CodingForcing CodingAgreement
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
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
end OAI
|