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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 | import OAI.Computability.DegreeRigidity.Constructibility.DefSystemCertificate
namespace OAI
namespace TuringRigidity.RelativeConstructible
open ElementaryModel BoundedSetTheory TransitiveNameModel SentenceCoding
open BoundedDefinability SetModelFunctions
universe u
theorem tupleSpace_mem_of_context (M A : ZFSet.{u}) (C : Context M) (hA : A ∈ M) : tupleSpace A ∈ M := by
have hS := C.separation
have hω := C.omega_mem
obtain ⟨Q,hQM,hQdef⟩ := internal_power M C.transitive C.power
(product_mem M C.transitive C.pairing C.union C.power hS hω hA)
have hQ (n : ℕ) (v : Fin n → ZFSet.{u}) (hv : ∀ i, v i ∈ A) : tupleGraph v ∈ Q := by
apply (hQdef _).mpr
refine ⟨tupleGraph_mem M C.transitive C.pairing C.union hω v (fun i => C.transitive A hA _ (hv i)),?_⟩
intro z hz
obtain ⟨i,rfl⟩ := ZFSet.mem_range.mp hz
exact ZFSet.mem_prod.mpr ⟨_,(mem_omega _).mpr ⟨i.val,rfl⟩,_,hv i,rfl⟩
let e := cons ZFSet.omega (cons A (fun _ => Q))
have he : ∀ i, e i ∈ M := by intro i; rcases i with _|_|i; exact hω; exact hA; exact hQM
have hs := sep_mem M C.transitive hS (tupleFormula 1 2 3 0) e he
(product_mem M C.transitive C.pairing C.union C.power hS hω hQM)
have heq : (ZFSet.prod ZFSet.omega Q).sep
(fun t => (tupleFormula 1 2 3 0).Eval (cons t e)) = tupleSpace A := by
apply ZFSet.ext; intro t
rw [ZFSet.mem_sep,tupleFormula_spec 1 2 3 0 (cons t e) rfl hQ]
change (t ∈ ZFSet.prod ZFSet.omega Q ∧ t ∈ tupleSpace A) ↔ t ∈ tupleSpace A
refine ⟨And.right,fun ht => ⟨?_,ht⟩⟩
obtain ⟨n,v,hv,rfl⟩ := (mem_tupleSpace A t).mp ht
exact ZFSet.mem_prod.mpr ⟨natSet n,(mem_omega _).mpr ⟨n,rfl⟩,tupleGraph v,hQ n v hv,rfl⟩
exact heq ▸ hs
theorem truthSlice_mem_of_context (M A : ZFSet.{u}) (C : Context M) (hA : A ∈ M) (p : SentenceForm) : truthSlice A p ∈ M := by
have hS := C.separation
have hω := C.omega_mem
have hn (n : ℕ) : natSet.{u} n ∈ M := C.transitive _ hω _ ((mem_omega _).mpr ⟨n,rfl⟩)
obtain ⟨Q,hQM,hQdef⟩ := internal_power M C.transitive C.power
(product_mem M C.transitive C.pairing C.union C.power hS hω hA)
have hQ {n : ℕ} (v : Fin n → ZFSet.{u}) (hv : ∀ i, v i ∈ A) : tupleGraph v ∈ Q := by
apply (hQdef _).mpr
refine ⟨tupleGraph_mem M C.transitive C.pairing C.union hω v (fun i => C.transitive A hA _ (hv i)),?_⟩
intro z hz
obtain ⟨i,rfl⟩ := ZFSet.mem_range.mp hz
exact ZFSet.mem_prod.mpr ⟨_,(mem_omega _).mpr ⟨i.val,rfl⟩,_,hv i,rfl⟩
let e := cons ZFSet.omega (cons A (cons Q (cons (natSet p.bound) natSet)))
have he : ∀ i, e i ∈ M := by
intro i; rcases i with _|_|_|_|i
exact hω; exact hA; exact hQM; exact hn _; exact hn _
let φ := tupleTruth p 1 2 3 4 0 (fun i => i+5)
have hs := sep_mem M C.transitive hS φ e he (tupleSpace_mem_of_context M A C hA)
have heq : (tupleSpace A).sep (fun t => φ.Eval (cons t e)) = truthSlice A p := by
apply ZFSet.ext; intro t
rw [truthSlice,ZFSet.mem_sep,ZFSet.mem_sep]
apply and_congr_right; intro ht
obtain ⟨n,v,hv,rfl⟩ := (mem_tupleSpace A t).mp ht
exact (tupleTruth_spec p 1 2 3 4 0 (fun i => i+5) (cons (tupleCode v) e)
rfl rfl (fun _ _ => rfl) v hv rfl (hQ v hv)).trans (satisfactionSet_record A p v hv).symm
exact heq ▸ hs
theorem finiteSatisfaction_mem_of_context (M A : ZFSet.{u}) (C : Context M) (hA : A ∈ M) (ps : List SentenceForm) : finiteSatisfaction A ps ∈ M := by
have hω := C.omega_mem
have hn (n : ℕ) : natSet.{u} n ∈ M := C.transitive _ hω _ ((mem_omega _).mpr ⟨n,rfl⟩)
apply union_mem M C.transitive C.union
apply finite_range_mem M C.transitive C.pairing C.union (hn 0)
intro i
exact product_mem M C.transitive C.pairing C.union C.power C.separation
(singleton_mem M C.transitive C.pairing (hn _)) (truthSlice_mem_of_context M A C hA ps[i])
theorem internal_certificate_bound_of_context (M A : ZFSet.{u}) (C : Context M) (hA : A ∈ M) :
∃ Q ∈ M,
(∀ S, S ∈ Q ↔ S ∈ M ∧ S ⊆ ZFSet.prod ZFSet.omega (tupleSpace A)) ∧
∀ ps : List SentenceForm, finiteSatisfaction A ps ∈ Q := by
obtain ⟨Q,hQ,hdef⟩ := internal_power M C.transitive C.power
(product_mem M C.transitive C.pairing C.union C.power C.separation
(C.omega_mem) (tupleSpace_mem_of_context M A C hA))
refine ⟨Q,hQ,hdef,fun ps => (hdef _).mpr ⟨finiteSatisfaction_mem_of_context M A C hA ps,?_⟩⟩
exact fun z hz => satisfactionSet_subset_product A (finiteSatisfaction_subset A ps hz)
theorem satisfactionSet_mem_of_context (M A : ZFSet.{u}) (C : Context M) (hA : A ∈ M) : satisfactionSet A ∈ M := by
obtain ⟨Q,hQ,_,hf⟩ := internal_family_bound M C.transitive C.pairing C.union C.power C.omega_mem
obtain ⟨B,hB,hb⟩ := internal_support_bound C
obtain ⟨G,hG,hg⟩ := internal_tuple_graph_bound C hA
obtain ⟨H,hH,_,hh⟩ := internal_certificate_bound_of_context M A C hA
have ht := tupleSpace_mem_of_context M A C hA
let e := cons A (cons (tupleSpace A) (fun _ => ZFSet.omega))
have he : ∀ i, e i ∈ M := by
intro i; rcases i with _|_|i
· exact hA
· exact ht
· exact C.omega_mem
have hd := satisfactionWitness_definable C hQ hB hG hH 1 2 0
have hs := hd.sep_mem C e he (C.prod_mem C.omega_mem ht)
have heq : (ZFSet.prod ZFSet.omega (tupleSpace A)).sep
(fun z => SatisfactionWitness Q B G H A (tupleSpace A) z) = satisfactionSet A := by
apply ZFSet.ext; intro z
rw [ZFSet.mem_sep,satisfactionWitness_spec Q B G H A z hf hb hg (fun p => hh (subformulas p))]
exact ⟨And.right,fun hz => ⟨satisfactionSet_subset_product A hz,hz⟩⟩
exact heq ▸ hs
theorem definablePower_mem_of_context (M A : ZFSet.{u}) (C : Context M) (hA : A ∈ M) : definablePower A ∈ M := by
obtain ⟨Q,hQ,_,hf⟩ := internal_family_bound M C.transitive C.pairing C.union C.power C.omega_mem
obtain ⟨B,hB,hb⟩ := internal_support_bound C
obtain ⟨G,hG,hg⟩ := internal_tuple_graph_bound C hA
have ht := tupleSpace_mem_of_context M A C hA
have hz := satisfactionSet_mem_of_context M A C hA
obtain ⟨P,hP,_,hsub⟩ := definablePower_internal_bound M A C.transitive C.separation C.power hA
let e := cons A (cons (tupleSpace A) (cons (satisfactionSet A) (fun _ => ZFSet.omega)))
have he : ∀ i, e i ∈ M := by
intro i; rcases i with _|_|_|i
· exact hA
· exact ht
· exact hz
· exact C.omega_mem
have hd := definitionWitness_definable C hQ hB hG 1 2 3 0
have hs := hd.sep_mem C e he hP
have heq : P.sep (fun S => DefinitionWitness Q B G A (tupleSpace A) (satisfactionSet A) S) =
definablePower A := by
apply ZFSet.ext; intro S
rw [ZFSet.mem_sep,definitionWitness_spec Q B G A S hf hb hg,← mem_definablePower]
exact ⟨And.right,fun h => ⟨hsub h,h⟩⟩
exact heq ▸ hs
theorem internal_defSystem_of_context (M : ZFSet.{u}) (C : Context 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
obtain ⟨G,hG,hg⟩ := internal_tuple_graph_bound C hA
obtain ⟨H,hH,_,hh⟩ := internal_certificate_bound_of_context M A C hA
exact ⟨G,hG,tupleSpace A,tupleSpace_mem_of_context M A C hA,H,hH,
satisfactionSet A,satisfactionSet_mem_of_context M A C 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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