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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 | import OAI.Computability.DegreeRigidity.Syntax.SigmaHull
import OAI.Computability.DegreeRigidity.SetModels.CollapseCoding
import Mathlib.Data.List.Basic
namespace OAI
namespace TuringRigidity.BoundedSetTheory
open TransitiveNameModel RelationCollapse
universe u
namespace FiniteTuple
noncomputable def indexSpace (t : ZFSet.{u}) : ZFSet.{u} :=
ZFSet.prod {natSet 0} t ∪ ZFSet.prod {natSet 1} ZFSet.omega
noncomputable def finiteIndex (t : ZFSet.{u}) (xs : List ZFSet.{u}) (x : ZFSet.{u}) : ZFSet.{u} := by
classical
exact if x ∈ t then ZFSet.pair (natSet 0) x else ZFSet.pair (natSet 1) (natSet (xs.idxOf x))
theorem indexSpace_mem (M : ZFSet.{u}) (hM : Transitive M)
(hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M) (hS : Separation M)
(hω : ZFSet.omega.{u} ∈ M) {t : ZFSet.{u}} (ht : t ∈ M) : indexSpace t ∈ M := by
have hn (n : ℕ) : natSet.{u} n ∈ M := hM _ hω _ ((mem_omega _).mpr ⟨n,rfl⟩)
exact binary_union_mem M hM hP hU
(product_mem M hM hP hU hPow hS (singleton_mem M hM hP (hn 0)) ht)
(product_mem M hM hP hU hPow hS (singleton_mem M hM hP (hn 1)) hω)
theorem finiteIndex_mem (t : ZFSet.{u}) (xs : List ZFSet.{u}) (x : ZFSet.{u}) :
finiteIndex t xs x ∈ indexSpace t := by
classical
unfold finiteIndex
by_cases hx : x ∈ t
· rw [ite_eq_left hx]
exact ZFSet.mem_union.mpr (Or.inl (ZFSet.mem_prod.mpr
⟨_,ZFSet.mem_singleton.mpr rfl,x,hx,rfl⟩))
· rw [ite_eq_right hx]
exact ZFSet.mem_union.mpr (Or.inr (ZFSet.mem_prod.mpr
⟨_,ZFSet.mem_singleton.mpr rfl,_,(mem_omega _).mpr ⟨xs.idxOf x,rfl⟩,rfl⟩))
theorem finiteIndex_injective (t : ZFSet.{u}) (xs : List ZFSet.{u})
(x : ZFSet.{u}) (hx : x ∈ t ∪ elements xs) (y : ZFSet.{u}) (_hy : y ∈ t ∪ elements xs)
(he : finiteIndex t xs x = finiteIndex t xs y) : x = y := by
classical
unfold finiteIndex at he
by_cases hxt : x ∈ t <;> by_cases hyt : y ∈ t
· rw [ite_eq_left hxt,ite_eq_left hyt] at he
exact (ZFSet.pair_inj.mp he).2
· rw [ite_eq_left hxt,ite_eq_right hyt] at he
have hf := natSet_injective (ZFSet.pair_inj.mp he).1
omega
· rw [ite_eq_right hxt,ite_eq_left hyt] at he
have hf := natSet_injective (ZFSet.pair_inj.mp he).1
omega
· rw [ite_eq_right hxt,ite_eq_right hyt] at he
have hidx := natSet_injective (ZFSet.pair_inj.mp he).2
have hxl : x ∈ xs := (mem_elements x xs).mp ((ZFSet.mem_union.mp hx).resolve_left hxt)
exact (List.idxOf_inj hxl).mp hidx
theorem internal_finite_extension_injection (M : ZFSet.{u}) (hM : Transitive M)
(hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M)
(hS : Separation M) (hR : SigmaReplacement M) (hω : ZFSet.omega.{u} ∈ M)
{t : ZFSet.{u}} (ht : t ∈ M) (xs : List ZFSet.{u}) (hxs : ∀ x ∈ xs, x ∈ M) :
∃ f ∈ M, Presents (t ∪ elements xs) f (finiteIndex t xs) := by
classical
have hn (n : ℕ) : natSet.{u} n ∈ M := hM _ hω _ ((mem_omega _).mpr ⟨n,rfl⟩)
have hk := indexSpace_mem M hM hP hU hPow hS hω ht
let j := finiteIndex t xs
have hj (x : ZFSet.{u}) : j x ∈ M := hM _ hk _ (finiteIndex_mem t xs x)
let φ : Formula := .existsMem 2 (.conj (.orderedPair 1 2 0) (.orderedPair 0 4 2))
let e := cons (indexSpace t) (fun _ => natSet 0)
have he : ∀ i, e i ∈ M := by intro i; cases i with
| zero => exact hk
| succ => exact hn 0
obtain ⟨b,hb,hbdef⟩ := replacement_image M hM hR (.bounded φ) e he ht
(fun x => ZFSet.pair x (ZFSet.pair (natSet 0) x))
(fun x hx => orderedPair_mem M hM hP (hM t ht x hx)
(orderedPair_mem M hM hP (hn 0) (hM t ht x hx))) (by
intro x hx y hy
change φ.Realize M (cons y (cons x e)) ↔ _
rw [Formula.absolute _ M hM _ (by
intro i; rcases i with _|i; exact hy
rcases i with _|i; exact hM t ht x hx
exact he i)]
simp only [φ,Formula.Eval,Formula.eval_orderedPair,cons_zero,cons_succ,e]
constructor
· rintro ⟨v,_,hy,rfl⟩; exact hy
· intro hy
have hjx := finiteIndex_mem t xs x
rw [finiteIndex,ite_eq_left hx] at hjx
exact ⟨_,hjx,hy,rfl⟩)
let f := b ∪ elements (xs.map (fun x => ZFSet.pair x (j x)))
have hf : f ∈ M := binary_union_mem M hM hP hU hb
(elements_mem M hM hP hU (hn 0) _ (by
intro z hz
obtain ⟨x,hx,rfl⟩ := List.mem_map.mp hz
exact orderedPair_mem M hM hP (hxs x hx) (hj x)))
refine ⟨f,hf,?_⟩
intro z
change z ∈ b ∪ elements (xs.map (fun x => ZFSet.pair x (j x))) ↔ _
rw [ZFSet.mem_union,hbdef,mem_elements,List.mem_map]
constructor
· rintro (⟨x,hx,rfl⟩|⟨x,hx,rfl⟩)
· refine ⟨x,ZFSet.mem_union.mpr (Or.inl hx),?_⟩
simp only [finiteIndex,ite_eq_left hx]
· exact ⟨x,ZFSet.mem_union.mpr (Or.inr ((mem_elements x xs).mpr hx)),rfl⟩
· rintro ⟨x,hx,rfl⟩
rcases ZFSet.mem_union.mp hx with hx|hx
· apply Or.inl
exact ⟨x,hx,by simp only [finiteIndex,ite_eq_left hx]⟩
· exact Or.inr ⟨x,(mem_elements x xs).mp hx,rfl⟩
end FiniteTuple
end TuringRigidity.BoundedSetTheory
end OAI
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