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8df23f4 | 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 | import OAI.Computability.DegreeRigidity.Syntax.CheckedMembershipSeparation
import OAI.Computability.DegreeRigidity.Syntax.SigmaDisjunction
namespace OAI
namespace TuringRigidity.CheckedForcingRelation
open TransitiveNameModel BoundedSetTheory CountableForcing RecursiveNames AtomicForcing
variable {c : ZFSet.{0}} [Preorder (Conditions c)] [Top (Conditions c)]
noncomputable def relation (a : ZFSet.{0}) (τ : Name (Conditions c)) : ZFSet.{0} :=
(ZFSet.prod a c).sep (fun z => ∃ x ∈ a, ∃ p : Conditions c,
z = ZFSet.pair x (label c p) ∧ MemForces (Name.check x) τ p)
theorem pair_relation (a x : ZFSet.{0}) (τ : Name (Conditions c)) (p : Conditions c) :
ZFSet.pair x (label c p) ∈ relation a τ ↔ x ∈ a ∧ MemForces (Name.check x) τ p := by
rw [relation,ZFSet.mem_sep]
constructor
· rintro ⟨_,y,hy,q,he,hf⟩
have hpq := label_injective c (ZFSet.pair_inj.mp he).2
have hxy := (ZFSet.pair_inj.mp he).1
subst y; subst q; exact ⟨hy,hf⟩
· rintro ⟨hx,hf⟩
exact ⟨ZFSet.mem_prod.mpr ⟨x,hx,_,label_mem c p,rfl⟩,x,hx,p,rfl,hf⟩
theorem relation_mem (M a : ZFSet.{0}) (hM : Transitive M) (hT : SourceT M)
(hc : c ∈ M) (ha : a ∈ M) {o : ZFSet.{0}} (hoM : o ∈ M)
(ho : ∀ p q : Conditions c, ZFSet.pair (label c p) (label c q) ∈ o ↔ p ≤ q)
(τ : Name (Conditions c)) (hτ : τ.encode (label c) ∈ M) : relation a τ ∈ M := by
let e := cons a (cons c (cons o (cons (τ.encode (label c)) (fun _ => label c ⊤))))
have he : ∀ i, e i ∈ M := by
intro i; rcases i with _|_|_|_|i
exact ha; exact hc; exact hoM; exact hτ; exact hM c hc _ (label_mem c ⊤)
let ren : ℕ → ℕ := fun | 0 => 1 | 1 => 4 | 2 => 5 | 3 => 0 | 4 => 6 | _ => 7
let φ : SigmaFormula := .existsMem 1 (.existsMem 3 (.andBounded (.orderedPair 2 1 0)
(FullSetForcing.checkedMembershipSigma.rename ren)))
have hpM := product_mem M hM hT.pairing hT.union hT.powerSet hT.separation.finitePrefix.bounded ha hc
obtain ⟨R,hRM,hR⟩ := hT.separation.finitePrefix φ e he (ZFSet.prod a c) hpM
have spec (x y z : ZFSet.{0}) (hx : x ∈ M) (hy : y ∈ M)
(p : Conditions c) (hpy : label c p = y) :
(FullSetForcing.checkedMembershipSigma.rename ren).Realize M (cons y (cons x (cons z e))) ↔
MemForces (Name.check x) τ p := by
rw [SigmaFormula.realize_rename,FullSetForcing.checkedMembershipSigma_spec]
have hv : ∀ i, (cons y (cons x (cons z e)) ∘ ren) i ∈ M := by
intro i; rcases i with _|_|_|_|_|i
exact hx; exact hc; exact hoM; exact hy; exact hτ; exact he 4
exact FullSetForcing.checkedMembership_spec M hM hT ho τ p x _ hv
rfl rfl rfl hpy.symm rfl rfl
have pairSpec (x y z : ZFSet.{0}) (hx : x ∈ M) (hy : y ∈ M) (hz : z ∈ M) :
(BoundedSetTheory.Formula.orderedPair 2 1 0).Realize M (cons y (cons x (cons z e))) ↔
z = ZFSet.pair x y := by
have hv : ∀ i, cons y (cons x (cons z e)) i ∈ M := by
intro i; rcases i with _|_|_|i; exact hy; exact hx; exact hz; exact he i
exact (BoundedSetTheory.Formula.absolute _ M hM _ hv).trans
(by simp only [BoundedSetTheory.Formula.eval_orderedPair,cons_zero,cons_succ])
have heq : R = relation a τ := by
apply ZFSet.ext; intro z
constructor
· intro hzR
have hz := hM R hRM z hzR
obtain ⟨hzprod,hf⟩ := (hR z hz).mp hzR
simp only [φ,SigmaFormula.realize_existsMem,SigmaFormula.realize_andBounded,
cons_zero,cons_succ,e] at hf
obtain ⟨x,hx,hxa,y,hy,hyc,hzxy,hf⟩ := hf
obtain ⟨p,hpy⟩ := label_surjective c hyc
exact ZFSet.mem_sep.mpr ⟨hzprod,x,hxa,p,((pairSpec x y z hx hy hz).mp hzxy).trans (congrArg (ZFSet.pair x) hpy.symm),
(spec x y z hx hy p hpy).mp hf⟩
· intro hzR
obtain ⟨hzprod,x,hxa,p,hzxy,hf⟩ := ZFSet.mem_sep.mp hzR
have hz := hM _ hpM z hzprod
have hx := hM a ha x hxa
have hy := hM c hc _ (label_mem c p)
apply (hR z hz).mpr
refine ⟨hzprod,?_⟩
simp only [φ,SigmaFormula.realize_existsMem,SigmaFormula.realize_andBounded,
cons_zero,cons_succ,e]
exact ⟨x,hx,hxa,label c p,hy,label_mem c p,(pairSpec x (label c p) z hx hy hz).mpr hzxy,
(spec x (label c p) z hx hy p rfl).mpr hf⟩
exact heq ▸ hRM
end TuringRigidity.CheckedForcingRelation
end OAI
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