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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 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 | import OAI.Computability.DegreeRigidity.Constructibility.ConstructionInlining
namespace OAI
namespace TuringRigidity.RelativeConstructible
open ElementaryModel BoundedSetTheory TransitiveNameModel
universe u
inductive ConstructionInput : Type (u+1)
| reals
| stage (o : Ordinal.{u})
| realPath (path : List ℕ)
def pathReal (P : Oracle) : List ℕ → Oracle
| [] => P
| i::is => pathReal (realPart P i) is
noncomputable def ConstructionInput.value (R : ZFSet.{u}) (P : Oracle) :
ConstructionInput.{u} → ZFSet.{u}
| .reals => R
| .stage o => level R o
| .realPath path => realCode (pathReal P path)
def ConstructionInput.prefix (i : ℕ) : ConstructionInput.{u} → ConstructionInput.{u}
| .reals => .reals
| .stage o => .stage o
| .realPath path => .realPath (i::path)
theorem ConstructionInput.prefix_value (d : ConstructionInput.{u})
(R : ZFSet.{u}) (P : Oracle) (i : ℕ) :
(d.prefix i).value R P = d.value R (realPart P i) := by
cases d <;> rfl
def ConstructionInput.Indexed (M : ZFSet.{u}) : ConstructionInput.{u} → Prop
| .reals => True
| .stage o => o.toZFSet ∈ M
| .realPath _ => True
theorem ConstructionInput.prefix_indexed (d : ConstructionInput.{u})
(M : ZFSet.{u}) (i : ℕ) : (d.prefix i).Indexed M ↔ d.Indexed M := by
cases d <;> rfl
noncomputable def Construction.inputData : Construction.{u} → ℕ → ConstructionInput.{u}
| .reals, _ => .reals
| .parameter, _ => .realPath []
| .define o _ _ _, 0 => .stage o
| .define _ n _ c, k+1 =>
if h : (Nat.unpair k).1 < n then
((c ⟨(Nat.unpair k).1,h⟩).inputData (Nat.unpair k).2).prefix (Nat.unpair k).1
else .reals
theorem Construction.inputs_eq_data (t : Construction.{u}) (R : ZFSet.{u})
(P : Oracle) (k : ℕ) : t.inputs R P k = (t.inputData k).value R P := by
induction t generalizing P k with
| reals => rfl
| parameter => rfl
| define o n p c ih =>
cases k with
| zero => rfl
| succ k =>
simp only [Construction.inputs,Construction.inputData]
split
next h => rw [ConstructionInput.prefix_value]; exact ih _ _ _
next => rfl
theorem Construction.inputData_indexed (t : Construction.{u}) (M : ZFSet.{u})
(hi : t.Indexed M) : ∀ k, (t.inputData k).Indexed M := by
induction t with
| reals => exact fun _ => trivial
| parameter => exact fun _ => trivial
| define o n p c ih =>
intro k
cases k with
| zero => exact hi.1
| succ k =>
simp only [Construction.inputData]
split
next h =>
rw [ConstructionInput.prefix_indexed]
exact ih _ (hi.2 _) _
next => trivial
noncomputable def Construction.membershipFormula (t : Construction.{u}) : SentenceForm :=
.ex (.conj (t.formula 0 (fun k => k+2)) (.member 1 0))
theorem Construction.membershipFormula_spec (t : Construction.{u}) (M R : ZFSet.{u})
(hM : Transitive M) (hT : SourceT M) (hR : R ∈ M) (P : Oracle)
(ht : t.Certified R P) (hi : t.Indexed M) (z : ZFSet.{u}) (hz : z ∈ M) :
t.membershipFormula.Sat (M : Set ZFSet) (cons z (t.inputs R P)) ↔
z ∈ t.value R P := by
have hspec (x : ZFSet.{u}) (hx : x ∈ M) :
(t.formula 0 (fun k => k+2)).Sat (M : Set ZFSet)
(cons x (cons z (t.inputs R P))) ↔ x = t.value R P := by
apply t.formula_spec M R hM hT hR P ht hi
· intro i
rcases i with _|_|i
· exact hx
· exact hz
· exact t.inputs_mem M R hM hT hR P ht hi i
· exact fun _ => rfl
change (∃ x ∈ M, (t.formula 0 (fun k => k+2)).Sat (M : Set ZFSet)
(cons x (cons z (t.inputs R P))) ∧ z ∈ x) ↔ _
constructor
· rintro ⟨x,hx,hf,hzx⟩
exact (hspec x hx).mp hf ▸ hzx
· intro hzt
have hv := t.value_mem M R hM hT hR P ht hi
exact ⟨_,hv,(hspec _ hv).mpr rfl,hzt⟩
theorem Construction.finite_membership_definition (t : Construction.{u}) :
∃ (n : ℕ) (p : SentenceForm) (d : Fin n → ConstructionInput.{u}),
p.bound ≤ n+1 ∧
∀ (M R : ZFSet.{u}) (_hM : Transitive M) (_hT : SourceT M) (_hR : R ∈ M)
(P : Oracle) (_ht : t.Certified R P) (_hi : t.Indexed M),
(∀ i, (d i).Indexed M ∧ (d i).value R P ∈ M) ∧
∀ z ∈ M, p.Sat (M : Set ZFSet)
(cons z (tupleEnv (fun i => (d i).value R P))) ↔ z ∈ t.value R P := by
let p := t.membershipFormula
refine ⟨p.bound,p,fun i => t.inputData i,by omega,?_⟩
intro M R hM hT hR P ht hi
refine ⟨fun i => ⟨t.inputData_indexed M hi i,?_⟩,?_⟩
· rw [← t.inputs_eq_data]
exact t.inputs_mem M R hM hT hR P ht hi i
· intro z hz
have hf : p.Sat (M : Set ZFSet)
(cons z (tupleEnv (fun i : Fin p.bound => (t.inputData i).value R P))) ↔
p.Sat (M : Set ZFSet) (cons z (t.inputs R P)) := by
apply p.finite_support
intro i hin
cases i with
| zero => rfl
| succ i =>
simp only [cons_succ]
change tupleEnv (fun j : Fin p.bound => (t.inputData j).value R P) i = t.inputs R P i
rw [tupleEnv, dite_eq_left (show i < p.bound by omega)]
exact (t.inputs_eq_data R P i).symm
exact hf.trans (t.membershipFormula_spec M R hM hT hR P ht hi z hz)
end TuringRigidity.RelativeConstructible
end OAI
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