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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 | import OAI.Computability.DegreeRigidity.SetModels.CollapseCollection
namespace OAI
namespace TuringRigidity.RelationCollapse
open TransitiveNameModel BoundedSetTheory
universe u
def Presents (a f : ZFSet.{u}) (j : ZFSet.{u} → ZFSet.{u}) : Prop :=
∀ z, z ∈ f ↔ ∃ x ∈ a, z = ZFSet.pair x (j x)
theorem Presents.pair_iff {a f : ZFSet.{u}} {j : ZFSet.{u} → ZFSet.{u}}
(hf : Presents a f j) (x y : ZFSet.{u}) :
ZFSet.pair x y ∈ f ↔ x ∈ a ∧ y = j x := by
rw [hf]
constructor
· rintro ⟨z,hz,he⟩
obtain ⟨rfl,rfl⟩ := ZFSet.pair_inj.mp he
exact ⟨hz,rfl⟩
· rintro ⟨hx,rfl⟩
exact ⟨x,hx,rfl⟩
noncomputable def codedRelation (d a f : ZFSet.{u}) : ZFSet.{u} :=
ZFSet.sep (fun z => ∃ u ∈ d, ∃ v ∈ d, z = ZFSet.pair u v ∧
∃ x ∈ a, ∃ y ∈ a, x ∈ y ∧ ZFSet.pair x u ∈ f ∧ ZFSet.pair y v ∈ f) (ZFSet.prod d d)
def codedFormula : Formula :=
.existsMem 1 (.existsMem 2 (.conj (.orderedPair 2 1 0)
(.existsMem 4 (.existsMem 5 (.conj (.member 1 0)
(.conj (.pairMem 1 3 7) (.pairMem 0 2 7)))))))
theorem codedRelation_mem (M : ZFSet.{u}) (hM : Transitive M)
(hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M) (hS : Separation M)
{d a f : ZFSet.{u}} (hd : d ∈ M) (ha : a ∈ M) (hf : f ∈ M) :
codedRelation d a f ∈ M := by
let e := cons d (cons a (cons f (fun _ => d)))
have he : ∀ i, e i ∈ M := by
intro i; rcases i with _|i; exact hd
rcases i with _|i; exact ha
rcases i with _|i; exact hf
exact hd
simpa only [codedFormula,Formula.Eval,Formula.eval_orderedPair,Formula.eval_pairMem,
e,cons_zero,cons_succ,codedRelation] using
sep_mem M hM hS codedFormula e he (product_mem M hM hP hU hPow hS hd hd)
theorem codedRelation_on (d a f : ZFSet.{u}) : On d (codedRelation d a f) :=
fun _ hz => (ZFSet.mem_sep.mp hz).1
theorem codedRelation_pair {d a f : ZFSet.{u}} {j : ZFSet.{u} → ZFSet.{u}}
(hf : Presents a f j) (hj : ∀ x ∈ a, j x ∈ d) (u v : ZFSet.{u}) :
ZFSet.pair u v ∈ codedRelation d a f ↔
∃ x ∈ a, ∃ y ∈ a, x ∈ y ∧ u = j x ∧ v = j y := by
rw [codedRelation,ZFSet.mem_sep]
constructor
· rintro ⟨_,u',_,v',_,hp,x,hx,y,hy,hxy,hxu,hyv⟩
obtain ⟨rfl,rfl⟩ := ZFSet.pair_inj.mp hp
exact ⟨x,hx,y,hy,hxy,((hf.pair_iff x u).mp hxu).2,((hf.pair_iff y v).mp hyv).2⟩
· rintro ⟨x,hx,y,hy,hxy,rfl,rfl⟩
exact ⟨ZFSet.mem_prod.mpr ⟨_,hj x hx,_,hj y hy,rfl⟩,
_,hj x hx,_,hj y hy,rfl,x,hx,y,hy,hxy,
(hf.pair_iff x _).mpr ⟨hx,rfl⟩,(hf.pair_iff y _).mpr ⟨hy,rfl⟩⟩
theorem codedRelation_wellFounded {d a f : ZFSet.{u}} {j : ZFSet.{u} → ZFSet.{u}}
(hf : Presents a f j) (hj : ∀ x ∈ a, j x ∈ d)
(hi : ∀ x ∈ a, ∀ y ∈ a, j x = j y → x = y) :
WellFounded (Rel d (codedRelation d a f)) := by
classical
let F : ZFSet.{u} → ZFSet.{u} := fun u =>
if h : ∃ x ∈ a, j x = u then h.choose else ∅
have hF (x : ZFSet.{u}) (hx : x ∈ a) : F (j x) = x := by
have h : ∃ y ∈ a, j y = j x := ⟨x,hx,rfl⟩
simp only [F,dite_eq_left h]
exact hi _ h.choose_spec.1 x hx h.choose_spec.2
apply (show WellFounded (fun y x => F y ∈ F x) from InvImage.wf F ZFSet.mem_wf).mono
intro u v huv
obtain ⟨x,hx,y,hy,hxy,rfl,rfl⟩ := (codedRelation_pair hf hj u v).mp huv.2
simpa only [hF x hx,hF y hy] using hxy
theorem codedRelation_value {d a f : ZFSet.{u}} {j : ZFSet.{u} → ZFSet.{u}}
(ha : Transitive a) (hf : Presents a f j) (hj : ∀ x ∈ a, j x ∈ d)
(hi : ∀ x ∈ a, ∀ y ∈ a, j x = j y → x = y)
(wf : WellFounded (Rel d (codedRelation d a f)))
(x : ZFSet.{u}) (hx : x ∈ a) : value d (codedRelation d a f) wf (j x) = x := by
induction x using ZFSet.inductionOn with
| h x ih =>
apply ZFSet.ext
intro z
rw [mem_value]
constructor
· rintro ⟨u,_,hux,hz⟩
obtain ⟨y,hy,w,hw,hyw,rfl,hwx⟩ := (codedRelation_pair hf hj u (j x)).mp hux
have hwx' : x = w := hi x hx w hw hwx
subst w
rw [ih y hyw hy] at hz
exact hz ▸ hyw
· intro hz
have hza := ha x hx z hz
refine ⟨j z,hj z hza,(codedRelation_pair hf hj _ _).mpr
⟨z,hza,x,hx,hz,rfl,rfl⟩,?_⟩
exact (ih z hz hza).symm
theorem internal_small_transitive_bound (M d : ZFSet.{u}) (hM : Transitive M)
(hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M)
(hS : SigmaSeparation M) (hR : SigmaReplacement M) (hd : d ∈ M) :
∃ W ∈ M, ∀ a ∈ M, Transitive a → ∀ f ∈ M, ∀ j : ZFSet.{u} → ZFSet.{u},
Presents a f j → (∀ x ∈ a, j x ∈ d) →
(∀ x ∈ a, ∀ y ∈ a, j x = j y → x = y) → a ⊆ W := by
obtain ⟨W,hW,hbound⟩ := internal_value_bound M d hM hP hU hPow hS hR hd
refine ⟨W,hW,?_⟩
intro a ha hat f hf j hfj hj hi x hx
have hr := codedRelation_mem M hM hP hU hPow hS.bounded hd ha hf
have wf := codedRelation_wellFounded hfj hj hi
have hv := hbound _ hr (codedRelation_on d a f) wf (j x) (hj x hx)
rw [codedRelation_value hat hfj hj hi wf x hx] at hv
exact hv
end TuringRigidity.RelationCollapse
end OAI
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