File size: 3,569 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 66 67 68 69 70 71 72 73 | import OAI.Computability.DegreeRigidity.Syntax.SatisfactionSystemCertificate
namespace OAI
namespace TuringRigidity.RelativeConstructible
open ElementaryModel BoundedSetTheory TransitiveNameModel SentenceCoding
open BoundedDefinability SetModelFunctions
universe u
def DefSystem (Q B A G T H Z D : ZFSet.{u}) : Prop :=
SatisfactionSystem Q B A G T H Z ∧
(∀ S ∈ D, DefinitionWitness Q B G A T Z S) ∧
∀ c ∈ ZFSet.omega, ∀ t ∈ T, ParameterAdequate Q B G c t →
∃ S ∈ D, DefinesSubset A G T Z c t S
theorem defSystem_definable {M : ZFSet.{u}} (C : Context M)
{Q B : ZFSet.{u}} (hQ : Q ∈ M) (hB : B ∈ M) (a G T H Z D : ℕ) :
Definable M (fun e => DefSystem Q B (e a) (e G) (e T) (e H) (e Z) (e D)) := by
have hn := defAllMem (definitionWitness_variable C hQ hB (a+1) (G+1) (T+1) (Z+1) 0) D
have hc := defAllParam (defAllMem (defImp (parameterAdequate_variable C hQ hB (G+2) 1 0)
((definesSubset_variable C (a+3) (G+3) (T+3) (Z+3) 2 1 0).existsMem (D+2))) (T+1)) C.omega_mem
exact (satisfactionSystem_definable C hQ hB a G T H Z).and (hn.and hc)
theorem DefSystem.exact {Q B A G T H Z D : ZFSet.{u}}
(hQ : ∀ p : SentenceForm, family p ∈ Q)
(hB : ∀ p : SentenceForm, supportGraph p ∈ B)
(h : DefSystem Q B A G T H Z D) : D = definablePower A := by
obtain ⟨ht,hz,hg,_⟩ := h.1.exact hQ hB
subst T; subst Z
apply ZFSet.ext; intro S
rw [mem_definablePower]
constructor
· intro hS
exact (definitionWitness_spec Q B G A S hQ hB hg).mp (h.2.1 S hS)
· rintro ⟨n,p,v,hp,hv,rfl⟩
obtain ⟨S,hS,hs⟩ := h.2.2 _ ((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⟩)
exact (definesSubset_spec A G S hg p v hv hp).mp hs ▸ hS
theorem defSystem_canonical (Q B A G H : 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)
(hH : ∀ p : SentenceForm, finiteSatisfaction A (subformulas p) ∈ H) :
DefSystem Q B A G (tupleSpace A) H (satisfactionSet A) (definablePower A) := by
refine ⟨satisfactionSystem_canonical Q B A G H hQ hB hG hH,?_,?_⟩
· intro S hS
exact (definitionWitness_spec Q B G A S hQ hB hG).mpr ((mem_definablePower A S).mp hS)
· intro c _ t ht ha
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 ⟨definedSubset A p v,(mem_definablePower A _).mpr ⟨n,p,v,hp,hv,rfl⟩,
(definesSubset_spec A G _ hG p v hv hp).mpr rfl⟩
theorem internal_defSystem (M : ZFSet.{u}) (hM : Transitive M) (hT : SourceT M)
(Q B A : ZFSet.{u}) (hA : A ∈ M)
(hQ : ∀ p : SentenceForm, family p ∈ Q)
(hB : ∀ p : SentenceForm, supportGraph p ∈ B) :
∃ G ∈ M, ∃ T ∈ M, ∃ H ∈ M, ∃ Z ∈ M,
DefSystem Q B A G T H Z (definablePower A) := by
let C : Context M := ⟨hM,hT.pairing,hT.union,hT.powerSet,hT.separation.finitePrefix.bounded,hT.infinity⟩
obtain ⟨G,hG,hg⟩ := internal_tuple_graph_bound C hA
obtain ⟨H,hH,_,hh⟩ := internal_certificate_bound M A hM hT hA
exact ⟨G,hG,tupleSpace A,tupleSpace_mem M A hM hT hA,H,hH,
satisfactionSet A,satisfactionSet_mem M A hM hT hA,
defSystem_canonical Q B A G H hQ hB hg (fun p => hh (subformulas p))⟩
end TuringRigidity.RelativeConstructible
end OAI
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