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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.Effective.FiniteExtensionCode
namespace OAI
namespace TuringRigidity.BoundedSetTheory
open TransitiveNameModel RelationCollapse
universe u
namespace FiniteTuple
inductive FiniteCode (t : ZFSet.{u}) : ZFSet.{u} → Prop
| atom (x : ZFSet.{u}) (hx : x ∈ t) : FiniteCode t x
| empty : FiniteCode t ∅
| pair {x y : ZFSet.{u}} : FiniteCode t x → FiniteCode t y → FiniteCode t {x,y}
theorem FiniteCode.orderedPair {t x y : ZFSet.{u}} (hx : FiniteCode t x) (hy : FiniteCode t y) :
FiniteCode t (ZFSet.pair x y) := by
have hs : FiniteCode t ({x} : ZFSet.{u}) := by simpa using hx.pair hx
exact hs.pair (hx.pair hy)
theorem FiniteCode.mem_model (M : ZFSet.{u}) (hM : Transitive M) (hP : Pairing M)
(h0 : (∅ : ZFSet.{u}) ∈ M) {t x : ZFSet.{u}} (ht : t ∈ M) (hx : FiniteCode t x) : x ∈ M := by
induction hx with
| atom x hx => exact hM t ht x hx
| empty => exact h0
| pair _ _ ihx ihy => exact pair_mem M hM hP ihx ihy
theorem transitive_union {a b : ZFSet.{u}} (ha : Transitive a) (hb : Transitive b) :
Transitive (a ∪ b) := by
intro x hx y hy
rcases ZFSet.mem_union.mp hx with hx|hx
· exact ZFSet.mem_union.mpr (Or.inl (ha x hx y hy))
· exact ZFSet.mem_union.mpr (Or.inr (hb x hx y hy))
theorem transitive_adjoin {a z : ZFSet.{u}} (ha : Transitive a) (hz : z ⊆ a) :
Transitive (a ∪ ({z} : ZFSet.{u})) := by
intro x hx y hy
rcases ZFSet.mem_union.mp hx with hx|hx
· exact ZFSet.mem_union.mpr (Or.inl (ha x hx y hy))
· obtain rfl := ZFSet.mem_singleton.mp hx
exact ZFSet.mem_union.mpr (Or.inl (hz hy))
theorem FiniteCode.finite_transitive_extension (M : ZFSet.{u}) (hM : Transitive M)
(hP : Pairing M) (h0 : (∅ : ZFSet.{u}) ∈ M)
{t x : ZFSet.{u}} (ht : t ∈ M) (htt : Transitive t) (hx : FiniteCode t x) :
∃ xs : List ZFSet.{u}, (∀ y ∈ xs, y ∈ M) ∧ x ∈ t ∪ elements xs ∧
Transitive (t ∪ elements xs) := by
induction hx with
| atom x hx =>
refine ⟨[],by simp,?_,?_⟩
· exact ZFSet.mem_union.mpr (Or.inl hx)
· have heq : t ∪ elements [] = t := by
apply ZFSet.ext; intro z; simp [ZFSet.mem_union,elements]
exact heq.symm ▸ htt
| empty =>
refine ⟨[∅],?_,?_,?_⟩
· intro y hy; obtain rfl := List.mem_singleton.mp hy; exact h0
· exact ZFSet.mem_union.mpr (Or.inr ((mem_elements _ _).mpr List.mem_cons_self))
· have heq : t ∪ elements [∅] = t ∪ ({∅} : ZFSet.{u}) := by
apply ZFSet.ext; intro z; simp [ZFSet.mem_union,elements]
rw [heq]
exact transitive_adjoin htt (by simp)
| @pair x y hx hy ihx ihy =>
obtain ⟨xs,hxs,hxxs,htxs⟩ := ihx
obtain ⟨ys,hys,hyys,htys⟩ := ihy
let zs := xs ++ ys
have hb : t ∪ elements zs = (t ∪ elements xs) ∪ (t ∪ elements ys) := by
apply ZFSet.ext; intro z
simp only [ZFSet.mem_union,mem_elements,List.mem_append,zs]
tauto
have hbt : Transitive (t ∪ elements zs) := hb.symm ▸ transitive_union htxs htys
have hxb : x ∈ t ∪ elements zs := hb.symm ▸ ZFSet.mem_union.mpr (Or.inl hxxs)
have hyb : y ∈ t ∪ elements zs := hb.symm ▸ ZFSet.mem_union.mpr (Or.inr hyys)
have hp : ({x,y} : ZFSet.{u}) ⊆ t ∪ elements zs := by
intro z hz; rcases ZFSet.mem_pair.mp hz with rfl|rfl
· exact hxb
· exact hyb
refine ⟨({x,y} : ZFSet.{u}) :: zs,?_,?_,?_⟩
· intro z hz; rcases List.mem_cons.mp hz with rfl|hz
· exact pair_mem M hM hP (hx.mem_model M hM hP h0 ht) (hy.mem_model M hM hP h0 ht)
· rcases List.mem_append.mp hz with hz|hz
· exact hxs z hz
· exact hys z hz
· exact ZFSet.mem_union.mpr (Or.inr ((mem_elements _ _).mpr List.mem_cons_self))
· have heq : t ∪ elements (({x,y} : ZFSet.{u})::zs) =
(t ∪ elements zs) ∪ ({({x,y} : ZFSet.{u})} : ZFSet.{u}) := by
apply ZFSet.ext; intro z
simp only [ZFSet.mem_union,mem_elements,List.mem_cons,ZFSet.mem_singleton]
tauto
rw [heq]
exact transitive_adjoin hbt hp
theorem internal_finite_code_bound (M t : ZFSet.{u}) (hM : Transitive M)
(hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M)
(hS : SigmaSeparation M) (hR : SigmaReplacement M) (hI : Infinity M)
(ht : t ∈ M) (htt : Transitive t) :
∃ K ∈ M, ∀ x, FiniteCode t x → x ∈ K := by
have hω := omega_mem M hM hS.bounded hI
have h0 : (∅ : ZFSet.{u}) ∈ M := hM _ hω _ ZFSet.omega_zero
have hk := indexSpace_mem M hM hP hU hPow hS.bounded hω ht
obtain ⟨K,hK,hbound⟩ := internal_small_transitive_bound M (indexSpace t) hM hP hU hPow hS hR hk
refine ⟨K,hK,?_⟩
intro x hx
obtain ⟨xs,hxs,hxD,hDT⟩ := hx.finite_transitive_extension M hM hP h0 ht htt
have hD := binary_union_mem M hM hP hU ht (elements_mem M hM hP hU h0 xs hxs)
obtain ⟨f,hf,hfg⟩ := internal_finite_extension_injection M hM hP hU hPow hS.bounded hR hω ht xs hxs
exact hbound _ hD hDT f hf (finiteIndex t xs) hfg
(fun y _ => finiteIndex_mem t xs y) (finiteIndex_injective t xs) hxD
end FiniteTuple
end TuringRigidity.BoundedSetTheory
end OAI
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