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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 | import OAI.Computability.DegreeRigidity.SetModels.InternalBooleanDense
import OAI.Computability.DegreeRigidity.SetModels.InternalCountableMap
namespace OAI
namespace TuringRigidity.CountedRegularBasis
open TransitiveNameModel BoundedSetTheory InternalRegularOperations InternalRegularAlgebra
open InternalBooleanDense
def basicFormula (B p U : ℕ) : Formula :=
.conj (.member U B) (.conj (.member p U)
(.allMem B (.imp (.member (p+1) 0) (.subset (U+1) 0))))
theorem basicFormula_spec (M c B : ZFSet.{0}) (hM : Transitive M) (hT : SourceT M)
(hc : c ∈ M) (hB : ∀ U, U ∈ B ↔ U ∈ M ∧ IsCode c U)
(b p U : ℕ) (e : ℕ → ZFSet.{0}) (hb : e b = B) (hp : e p ∈ c) :
(basicFormula b p U).Eval e ↔ e U = basicCode c (e p) := by
have hbasic : basicCode c (e p) ∈ B :=
(hB _).mpr ⟨basicCode_mem M c _ hM hT hc hp,regular_isCode c _⟩
simp only [basicFormula,Formula.Eval,Formula.eval_allMem,Formula.eval_imp,
Formula.eval_subset,cons_zero,cons_succ,hb]
constructor
· rintro ⟨hU,hpU,hmin⟩
have h₁ := hmin _ hbasic (mem_basicCode c _ hp)
have h₂ := (basicCode_le_iff c _ _ hp ((hB _).mp hU).2).mpr hpU
exact ZFSet.ext (fun z => ⟨fun hz => h₁ hz,fun hz => h₂ hz⟩)
· rintro h; rw [h]
exact ⟨hbasic,mem_basicCode c _ hp,fun V hV hpV =>
(basicCode_le_iff c _ V hp ((hB V).mp hV).2).mpr hpV⟩
theorem internal_basis (M c : ZFSet.{0}) (hM : Transitive M) (hT : SourceT M)
(hc : c ∈ M) :
∃ B ∈ M, (∀ U, U ∈ B ↔ U ∈ M ∧ IsCode c U) ∧
∃ S ∈ M, (∀ U, U ∈ S ↔ ∃ p ∈ c, U = basicCode c p) ∧
∃ g ∈ M, FunctionGraph c S g ∧
(∀ p ∈ c, ∀ U, ZFSet.pair p U ∈ g ↔ U = basicCode c p) ∧
(∀ U ∈ S, ∃ p ∈ c, ZFSet.pair p U ∈ g) := by
obtain ⟨B,hBM,hB⟩ := internal_algebra M c hM hT hc
let e := cons c (fun _ => B)
have he : ∀ i, e i ∈ M := by intro i; cases i; exact hc; exact hBM
let S := B.sep (fun U => ∃ p ∈ c, U = basicCode c p)
have hs (U : ZFSet.{0}) : U ∈ S ↔ ∃ p ∈ c, U = basicCode c p := by
change (U ∈ B.sep _) ↔ _
rw [ZFSet.mem_sep]
exact ⟨And.right,fun h@⟨p,hp,hU⟩ => ⟨hU ▸ (hB _).mpr
⟨basicCode_mem M c p hM hT hc hp,regular_isCode c _⟩,h⟩⟩
have hφ (U : ZFSet.{0}) : (Formula.existsMem 1 (basicFormula 3 0 1)).Eval (cons U e) ↔
∃ p ∈ c, U = basicCode c p := by
change (∃ p ∈ c, (basicFormula 3 0 1).Eval (cons p (cons U e))) ↔ _
exact exists_congr (fun p => and_congr_right (fun hp =>
basicFormula_spec M c B hM hT hc hB 3 0 1 _ rfl hp))
have hSM : S ∈ M := by
simpa only [hφ,S] using sep_mem M hM hT.separation.finitePrefix.bounded
(.existsMem 1 (basicFormula 3 0 1)) e he hBM
let v := cons c (cons S (fun _ => B))
have hv : ∀ i, v i ∈ M := by
intro i; rcases i with _|_|i; exact hc; exact hSM; exact hBM
let ψ : Formula := .existsMem 1 (.existsMem 3 (.conj (.orderedPair 2 1 0) (basicFormula 5 1 0)))
let g := (ZFSet.prod c S).sep (fun z => ψ.Eval (cons z v))
have hgM : g ∈ M := sep_mem M hM hT.separation.finitePrefix.bounded ψ v hv
(product_mem M hM hT.pairing hT.union hT.powerSet hT.separation.finitePrefix.bounded hc hSM)
have hψ (z : ZFSet.{0}) : ψ.Eval (cons z v) ↔
∃ p ∈ c, ∃ U ∈ S, z = ZFSet.pair p U ∧ U = basicCode c p := by
simp only [ψ,Formula.Eval,Formula.eval_orderedPair,cons_zero,cons_succ,v]
apply exists_congr; intro p
apply and_congr_right; intro hp
apply exists_congr; intro U
apply and_congr_right; intro _
exact and_congr_right (fun _ => basicFormula_spec M c B hM hT hc hB 5 1 0 _ rfl hp)
have hgs (p U : ZFSet.{0}) (hp : p ∈ c) :
ZFSet.pair p U ∈ g ↔ U = basicCode c p := by
simp only [g,ZFSet.mem_sep,hψ]
constructor
· rintro ⟨_,p',_,U',_,he,hU⟩
obtain ⟨rfl,rfl⟩ := ZFSet.pair_inj.mp he
exact hU
· intro hU
have hUS := (hs U).mpr ⟨p,hp,hU⟩
exact ⟨ZFSet.pair_mem_prod.mpr ⟨hp,hUS⟩,p,hp,U,hUS,rfl,hU⟩
refine ⟨B,hBM,hB,S,hSM,hs,g,hgM,⟨?_,?_⟩,(fun p hp U => hgs p U hp),?_⟩
· intro z hz; exact ZFSet.mem_prod.mp (ZFSet.mem_sep.mp hz).1
· intro p hp
exact ⟨basicCode c p,(hs _).mpr ⟨p,hp,rfl⟩,(hgs p _ hp).mpr rfl,
fun U _ hU => (hgs p U hp).mp hU⟩
· intro U hU
obtain ⟨p,hp,he⟩ := (hs U).mp hU
exact ⟨p,hp,(hgs p U hp).mpr he⟩
theorem counted_basis (M c : ZFSet.{0}) (hM : Transitive M) (hT : SourceT M)
(hc : c ∈ M) (hct : InternallyCountable M c) :
∃ S ∈ M, InternallyCountable M S ∧
(∀ U, U ∈ S ↔ ∃ p ∈ c, U = basicCode c p) ∧
(∀ U ∈ S, IsCode c U ∧ U ≠ ∅) ∧
(∀ U, IsCode c U → U ≠ ∅ → ∃ V ∈ S, V ⊆ U) := by
classical
obtain ⟨B,_,_,S,hS,hSs,g,hg,hgf,_,hgo⟩ := internal_basis M c hM hT hc
refine ⟨S,hS,InternalCountableMap.countable_of_surjection M c S g hM hT hc hS hg hgf hgo hct,
hSs,?_,?_⟩
· intro U hU
obtain ⟨p,hp,rfl⟩ := (hSs U).mp hU
exact ⟨regular_isCode c _,fun h => ZFSet.notMem_empty p (h ▸ mem_basicCode c p hp)⟩
· intro U hU hne
have hex : ∃ p, p ∈ U := by
by_contra h
exact hne (ZFSet.ext (fun p => ⟨fun hp => False.elim (h ⟨p,hp⟩),
fun hp => False.elim (ZFSet.notMem_empty p hp)⟩))
obtain ⟨p,hp⟩ := hex
exact ⟨basicCode c p,(hSs _).mpr ⟨p,hU.1 hp,rfl⟩,
(basicCode_le_iff c p U (hU.1 hp) hU).mpr hp⟩
end TuringRigidity.CountedRegularBasis
end OAI
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