File size: 3,292 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 | import OAI.Computability.DegreeRigidity.OrdinalCodes.CanonicalAdditionSyntax
namespace OAI
namespace TuringRigidity.RelativeConstructible
open ElementaryModel BoundedSetTheory TransitiveNameModel SetModelArithmetic
universe u
def squareFormula (w d : ℕ) : Formula :=
.conj (.allMem d (.existsMem (w+1) (.existsMem (w+2) (.orderedPair 2 1 0))))
(.allMem w (.allMem (w+1) (.existsMem (d+2) (.orderedPair 0 2 1))))
theorem squareFormula_spec (w d : ℕ) (e : ℕ → ZFSet.{u}) :
(squareFormula w d).Eval e ↔ e d = ZFSet.prod (e w) (e w) := by
simp only [squareFormula,Formula.Eval,Formula.eval_allMem,Formula.eval_orderedPair,cons_zero,cons_succ]
constructor
· rintro ⟨h1,h2⟩
apply ZFSet.ext; intro x
constructor
· intro hx
exact ZFSet.mem_prod.mpr (h1 x hx)
· intro hx
obtain ⟨a,ha,b,hb,rfl⟩ := ZFSet.mem_prod.mp hx
obtain ⟨z,hz,rfl⟩ := h2 a ha b hb
exact hz
· intro h
rw [h]
exact ⟨fun _ hx => ZFSet.mem_prod.mp hx,
fun a ha b hb => ⟨ZFSet.pair a b,ZFSet.pair_mem_prod.mpr ⟨ha,hb⟩,rfl⟩⟩
noncomputable def bindArithmetic (p : SentenceForm) : SentenceForm :=
.ex (.conj (canonicalOmega 0)
(.ex (.conj (fromBounded (squareFormula 1 0))
(.ex (.conj (fromBounded (additionSystemFormula 2 1 0)) p)))))
theorem bindArithmetic_spec (M : ZFSet.{u}) (hM : Transitive M)
(hw : ZFSet.omega.{u} ∈ M) (hd : pairNumbers.{u} ∈ M) (ha : additionSet.{u} ∈ M)
(p : SentenceForm) (e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ M) :
(bindArithmetic p).Sat (M : Set ZFSet) e ↔
p.Sat (M : Set ZFSet) (cons additionSet (cons pairNumbers (cons ZFSet.omega e))) := by
have hc (x : ZFSet.{u}) (hx : x ∈ M) : ∀ i, cons x e i ∈ M := by
intro i; cases i; exact hx; exact he _
have hwf (x : ZFSet.{u}) (hx : x ∈ M) := canonicalOmega_spec M hM hw 0 (cons x e) (hc x hx)
have hdf (d : ZFSet.{u}) (hdM : d ∈ M) :
(fromBounded (squareFormula 1 0)).Sat (M : Set ZFSet) (cons d (cons ZFSet.omega e)) ↔ d = pairNumbers := by
rw [bounded_sat,Formula.absolute _ M hM _
(by intro i; rcases i with _|_|i; exact hdM; exact hw; exact he i),squareFormula_spec]
rfl
have haf (a : ZFSet.{u}) (haM : a ∈ M) :
(fromBounded (additionSystemFormula 2 1 0)).Sat (M : Set ZFSet)
(cons a (cons pairNumbers (cons ZFSet.omega e))) ↔ a = additionSet := by
rw [bounded_sat,Formula.absolute _ M hM _
(by intro i; rcases i with _|_|_|i; exact haM; exact hd; exact hw; exact he i)]
exact additionSystemFormula_unique 2 1 0 _ rfl rfl
change (∃ w ∈ M, (canonicalOmega 0).Sat (M : Set ZFSet) (cons w e) ∧
∃ d ∈ M, (fromBounded (squareFormula 1 0)).Sat (M : Set ZFSet) (cons d (cons w e)) ∧
∃ a ∈ M, (fromBounded (additionSystemFormula 2 1 0)).Sat (M : Set ZFSet) (cons a (cons d (cons w e))) ∧
p.Sat (M : Set ZFSet) (cons a (cons d (cons w e)))) ↔ _
constructor
· rintro ⟨w,hwM,hwS,d,hdM,hdS,a,haM,haS,hp⟩
obtain rfl := (hwf w hwM).mp hwS
obtain rfl := (hdf d hdM).mp hdS
obtain rfl := (haf a haM).mp haS
exact hp
· intro hp
exact ⟨_,hw,(hwf _ hw).mpr rfl,_,hd,(hdf _ hd).mpr rfl,_,ha,(haf _ ha).mpr rfl,hp⟩
end TuringRigidity.RelativeConstructible
end OAI
|