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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 | import OAI.Computability.DegreeRigidity.Constructibility.UniformInternalSatisfaction
namespace OAI
namespace TuringRigidity.RelativeConstructible
open ElementaryModel BoundedSetTheory TransitiveNameModel SentenceCoding
open BoundedDefinability SetModelFunctions
universe u
variable {M : ZFSet.{u}}
def ParameterAdequate (Q B G c t : ZFSet.{u}) : Prop :=
∃ n ∈ ZFSet.omega, ∃ g ∈ G, ∃ b ∈ ZFSet.omega,
t = ZFSet.pair n g ∧ BoundWitness Q B c b ∧ ∀ x ∈ b, x ∈ n ∨ x = n
theorem parameterAdequate_definable (C : Context M) {Q B G : ZFSet.{u}}
(hQ : Q ∈ M) (hB : B ∈ M) (hG : G ∈ M) (c t : ℕ) :
Definable M (fun e => ParameterAdequate Q B G (e c) (e t)) :=
((((defOrderedPair C (t+3) 2 1).and ((boundWitness_definable C hQ hB (c+3) 0).and
(defAllMem (defOr (member_definable C 0 3) (equal_definable C 0 3)) 0))).existsParam
C.omega_mem).existsParam hG).existsParam C.omega_mem
theorem parameterAdequate_spec (Q B G c : ZFSet.{u})
(hQ : ∀ p : SentenceForm, family p ∈ Q)
(hB : ∀ p : SentenceForm, supportGraph p ∈ B)
{n : ℕ} (v : Fin n → ZFSet.{u}) (hG : tupleGraph v ∈ G) :
ParameterAdequate Q B G c (tupleCode v) ↔
∃ p : SentenceForm, c = natSet (Encodable.encode p) ∧ p.bound ≤ n+1 := by
constructor
· rintro ⟨m,_,g,_,b,_,he,hb,hle⟩
have hm : m = natSet n := (ZFSet.pair_inj.mp he).1.symm
obtain ⟨p,hp,rfl⟩ := (boundWitness_spec Q B c b hQ hB).mp hb
refine ⟨p,hp,(natSet_subset_iff p.bound (n+1)).mp ?_⟩
intro x hx
have h := hle x hx
rw [hm] at h
exact ZFSet.mem_insert_iff.mpr h.symm
· rintro ⟨p,rfl,hp⟩
refine ⟨natSet n,(mem_omega _).mpr ⟨n,rfl⟩,tupleGraph v,hG,
natSet p.bound,(mem_omega _).mpr ⟨p.bound,rfl⟩,rfl,
(boundWitness_spec Q B _ _ hQ hB).mpr ⟨p,rfl,rfl⟩,?_⟩
intro x hx
have hs := (natSet_subset_iff p.bound (n+1)).mpr hp hx
exact (ZFSet.mem_insert_iff.mp hs).symm
def DefinesSubset (A G T Z c t S : ZFSet.{u}) : Prop :=
S ⊆ A ∧ ∀ x ∈ A, x ∈ S ↔
∃ s ∈ T, TuplePrefix A G t x s ∧ ZFSet.pair c s ∈ Z
theorem definesSubset_definable (C : Context M) {G : ZFSet.{u}}
(hG : G ∈ M) (a T Z c t S : ℕ) :
Definable M (fun e => DefinesSubset (e a) G (e T) (e Z) (e c) (e t) (e S)) :=
(defSubset C S a).and (defAllMem (defIff (member_definable C 0 (S+1))
(((tuplePrefix_definable C hG (a+2) (t+2) 1 0).and
(defPairMem C (c+2) 0 (Z+2))).existsMem (T+1))) a)
theorem definesSubset_spec (A G S : ZFSet.{u})
(hG : ∀ (n : ℕ) (v : Fin n → ZFSet.{u}), (∀ i, v i ∈ A) → tupleGraph v ∈ G)
(p : SentenceForm) {n : ℕ} (v : Fin n → ZFSet.{u}) (hv : ∀ i, v i ∈ A)
(hb : p.bound ≤ n+1) :
DefinesSubset A G (tupleSpace A) (satisfactionSet A)
(natSet (Encodable.encode p)) (tupleCode v) S ↔ S = definedSubset A p v := by
have hstep (x : ZFSet.{u}) (hx : x ∈ A) :
(∃ s ∈ tupleSpace A, TuplePrefix A G (tupleCode v) x s ∧
ZFSet.pair (natSet (Encodable.encode p)) s ∈ satisfactionSet A) ↔
p.Sat (A : Set ZFSet) (cons x (tupleEnv v)) := by
let w : Fin (n+1) → ZFSet.{u} := Fin.cases (motive := fun _ => ZFSet.{u}) x v
have hw : ∀ i : Fin (n+1), w i ∈ A := fun i => Fin.cases hx hv i
have ht : tupleCode w ∈ tupleSpace A := (mem_tupleSpace A _).mpr ⟨n+1,w,hw,rfl⟩
have he : tupleEnv w = cons x (tupleEnv v) := tupleEnv_cons x v
constructor
· rintro ⟨s,_,hs,hp⟩
have hs' := (tuplePrefix_spec A G hG v hv x s hx).mp hs
rw [hs'] at hp
exact he ▸ (satisfactionSet_spec A p w hw hb).mp hp
· intro hp
refine ⟨tupleCode w,ht,(tuplePrefix_spec A G hG v hv x _ hx).mpr rfl,?_⟩
exact (satisfactionSet_spec A p w hw hb).mpr (he ▸ hp)
constructor
· rintro ⟨hsub,hdef⟩
apply ZFSet.ext; intro x
rw [mem_definedSubset]
exact ⟨fun hx => ⟨hsub hx,(hstep x (hsub hx)).mp ((hdef x (hsub hx)).mp hx)⟩,
fun ⟨hx,hp⟩ => (hdef x hx).mpr ((hstep x hx).mpr hp)⟩
· rintro rfl
refine ⟨fun x hx => (mem_definedSubset A x p v).mp hx |>.1,?_⟩
intro x hx
rw [mem_definedSubset]
exact (and_iff_right hx).trans (hstep x hx).symm
def DefinitionWitness (Q B G A T Z S : ZFSet.{u}) : Prop :=
∃ c ∈ ZFSet.omega, ∃ t ∈ T,
ParameterAdequate Q B G c t ∧ DefinesSubset A G T Z c t S
theorem definitionWitness_definable (C : Context M) {Q B G : ZFSet.{u}}
(hQ : Q ∈ M) (hB : B ∈ M) (hG : G ∈ M) (a T Z S : ℕ) :
Definable M (fun e => DefinitionWitness Q B G (e a) (e T) (e Z) (e S)) :=
(((parameterAdequate_definable C hQ hB hG 1 0).and
(definesSubset_definable C hG (a+2) (T+2) (Z+2) 1 0 (S+2))).existsMem (T+1)).existsParam C.omega_mem
theorem definitionWitness_spec (Q B G A S : ZFSet.{u})
(hQ : ∀ p : SentenceForm, family p ∈ Q)
(hB : ∀ p : SentenceForm, supportGraph p ∈ B)
(hG : ∀ (n : ℕ) (v : Fin n → ZFSet.{u}), (∀ i, v i ∈ A) → tupleGraph v ∈ G) :
DefinitionWitness Q B G A (tupleSpace A) (satisfactionSet A) S ↔ DefinableOver A S := by
constructor
· rintro ⟨c,_,t,ht,ha,hd⟩
obtain ⟨n,v,hv,rfl⟩ := (mem_tupleSpace A t).mp ht
obtain ⟨p,rfl,hp⟩ := (parameterAdequate_spec Q B G c hQ hB v (hG n v hv)).mp ha
exact ⟨n,p,v,hp,hv,(definesSubset_spec A G S hG p v hv hp).mp hd⟩
· rintro ⟨n,p,v,hp,hv,rfl⟩
exact ⟨_,(mem_omega _).mpr ⟨Encodable.encode p,rfl⟩,
_,(mem_tupleSpace A _).mpr ⟨n,v,hv,rfl⟩,
(parameterAdequate_spec Q B G _ hQ hB v (hG n v hv)).mpr ⟨p,rfl,hp⟩,
(definesSubset_spec A G _ hG p v hv hp).mpr rfl⟩
end TuringRigidity.RelativeConstructible
end OAI
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