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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 | import OAI.Computability.DegreeRigidity.Syntax.AtomicCertificate
namespace OAI
namespace TuringRigidity.AtomicForcing
open Set RecursiveNames TransitiveNameModel BoundedSetTheory
universe u
variable {c : ZFSet.{u}} [Preorder (Conditions c)]
def Atomic (isMem : Bool) (a b : Name (Conditions c)) (p : Conditions c) : Prop :=
if isMem then MemForces a b p else EqForces a b p
private theorem eval_matrix (isMem : Bool) (d k o f : ZFSet.{u})
(ho : ∀ r s : Conditions c, ZFSet.pair (label c r) (label c s) ∈ o ↔ r ≤ s)
(a b : Name (Conditions c)) (p : Conditions c) (tail : ℕ → ZFSet.{u}) :
(AtomicFormula.matrix isMem).Eval (cons f (cons k (cons d (cons (label c p)
(cons (a.encode (label c)) (cons (b.encode (label c)) (cons c (cons o tail)))))))) ↔
Transitive d ∧ a.encode (label c) ∈ d ∧ b.encode (label c) ∈ d ∧
k = ZFSet.prod d d ∧ Graph d c o f ∧ Atomic isMem a b p := by
let e := cons f (cons k (cons d (cons (label c p)
(cons (a.encode (label c)) (cons (b.encode (label c)) (cons c (cons o tail)))))))
have query (hd : Transitive d) (ha : a.encode (label c) ∈ d)
(hb : b.encode (label c) ∈ d) (hk : k = ZFSet.prod d d) (hf : Graph d c o f) :
(AtomicFormula.query isMem 2 6 1 7 0 4 5 3).Eval e ↔ Atomic isMem a b p := by
cases isMem
· simp only [AtomicFormula.query,Atomic,Bool.false_eq_true,ite_false,AtomicFormula.eval_triple]
change Triple k f (a.encode (label c)) (b.encode (label c)) (label c p) ↔ _
rw [hk]
exact (triple_pair ha hb).trans (hf.correct hd ho a b ha hb p)
· exact eval_membership hd ho hf a b ha hb p e 2 6 1 7 0 4 5 3
⟨rfl,rfl,hk,rfl,rfl,rfl,rfl,rfl⟩
change (AtomicFormula.matrix isMem).Eval e ↔ _
simp only [AtomicFormula.matrix,Formula.Eval,Formula.eval_transitive,
AtomicFormula.eval_product,cons_zero,cons_succ,e]
constructor
· rintro ⟨hd,ha,hb,hk,hgraph,hquery⟩
have hf := (AtomicFormula.eval_graph 2 6 1 7 0 e hk).mp hgraph
exact ⟨hd,ha,hb,hk,hf,(query hd ha hb hk hf).mp hquery⟩
· rintro ⟨hd,ha,hb,hk,hf,hq⟩
exact ⟨hd,ha,hb,hk,(AtomicFormula.eval_graph 2 6 1 7 0 e hk).mpr hf,
(query hd ha hb hk hf).mpr hq⟩
theorem realize_atomicCertificate (M : ZFSet.{u}) (hM : Transitive M)
(hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M) (hS : Separation M)
(hR : SigmaReplacement M) (hI : Infinity M) (hc : c ∈ M)
{o : ZFSet.{u}} (hoM : o ∈ M)
(ho : ∀ r s : Conditions c, ZFSet.pair (label c r) (label c s) ∈ o ↔ r ≤ s)
(isMem : Bool) (a b : Name (Conditions c))
(ha : a.encode (label c) ∈ M) (hb : b.encode (label c) ∈ M) (p : Conditions c) :
(AtomicFormula.certificate isMem).Realize M (cons (label c p)
(cons (a.encode (label c)) (cons (b.encode (label c)) (cons c (cons o (fun _ => c)))))) ↔
Atomic isMem a b p := by
have hp : label c p ∈ M := hM c hc _ (label_mem c p)
let e := cons (label c p) (cons (a.encode (label c))
(cons (b.encode (label c)) (cons c (cons o (fun _ => c)))))
have he : ∀ i, e i ∈ M := by
intro i
rcases i with _|i; exact hp
rcases i with _|i; exact ha
rcases i with _|i; exact hb
rcases i with _|i; exact hc
rcases i with _|i; exact hoM
exact hc
have matrix (d k f : ZFSet.{u}) (hd : d ∈ M) (hk : k ∈ M) (hf : f ∈ M) :
(AtomicFormula.matrix isMem).Realize M (cons f (cons k (cons d e))) ↔
Transitive d ∧ a.encode (label c) ∈ d ∧ b.encode (label c) ∈ d ∧
k = ZFSet.prod d d ∧ Graph d c o f ∧ Atomic isMem a b p := by
have he' : ∀ i, cons f (cons k (cons d e)) i ∈ M := by
intro i
rcases i with _|i; exact hf
rcases i with _|i; exact hk
rcases i with _|i; exact hd
exact he i
exact (Formula.absolute _ M hM _ he').trans (eval_matrix isMem d k o f ho a b p _)
change (∃ d ∈ M, ∃ k ∈ M, ∃ f ∈ M,
(AtomicFormula.matrix isMem).Realize M (cons f (cons k (cons d e)))) ↔ _
constructor
· rintro ⟨d,hd,k,hk,f,hf,hm⟩
exact ((matrix d k f hd hk hf).mp hm).2.2.2.2.2
· intro h
obtain ⟨d,hd,hdT,hdab⟩ := internal_transitive_container M hM hP hU hS hR hI
(pair_mem M hM hP ha hb)
have had : a.encode (label c) ∈ d := hdT _ hdab _ (ZFSet.mem_pair.mpr (Or.inl rfl))
have hbd : b.encode (label c) ∈ d := hdT _ hdab _ (ZFSet.mem_pair.mpr (Or.inr rfl))
have hk := product_mem M hM hP hU hPow hS hd hd
obtain ⟨f,hf,hfG⟩ := internal_atomic_graph M hM hP hU hPow hS hd hc hoM
exact ⟨d,hd,_,hk,f,hf,(matrix d _ f hd hk hf).mpr ⟨hdT,had,hbd,rfl,hfG,h⟩⟩
end TuringRigidity.AtomicForcing
end OAI
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