File size: 3,703 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 72 73 74 75 76 77 78 79 80 81 | import OAI.Computability.DegreeRigidity.SetModels.ChoiceTrace
namespace OAI
namespace TuringRigidity.BoundedSetTheory
open TransitiveNameModel
universe u
namespace InternalWellOrder
open Formula
def remainingFormula (a f i r : ℕ) : Formula :=
.conj (.subset r a) (.allMem a (.iff (.member 0 (r+1))
(.neg (.existsMem (i+1) (.pairMem 0 1 (f+2))))))
theorem eval_remainingFormula (a f i r : ℕ) (e : ℕ → ZFSet.{u}) :
(remainingFormula a f i r).Eval e ↔ Remaining (e a) (e f) (e i) (e r) := by
simp only [remainingFormula,Remaining,Formula.Eval,eval_subset,eval_allMem,eval_iff,
eval_pairMem,cons_zero,cons_succ]
def traceFormula (α a q s f : ℕ) : Formula := .conj
(.allMem f (.existsMem (α+1) (.existsMem (a+2) (.orderedPair 2 1 0))))
(.conj
(.allMem α (.allMem (a+1) (.imp (.pairMem 1 0 (f+2))
(.allMem 1 (.existsMem (a+3) (.pairMem 1 0 (f+4)))))))
(.allMem α (.allMem (a+1) (.imp (.pairMem 1 0 (f+2))
(.existsMem (q+2) (.conj (remainingFormula (a+3) (f+3) 2 0) (.pairMem 0 1 (s+3))))))))
theorem eval_traceFormula (α a q s f : ℕ) (e : ℕ → ZFSet.{u}) :
(traceFormula α a q s f).Eval e ↔ Trace (e α) (e a) (e q) (e s) (e f) := by
simp only [traceFormula,Formula.Eval,eval_allMem,eval_imp,eval_orderedPair,
eval_pairMem,eval_remainingFormula,cons_zero,cons_succ]
exact ⟨fun ⟨hshape,hinitial,hchoice⟩ => ⟨hshape,hinitial,hchoice⟩,
fun h => ⟨h.shape,h.initial,h.choice⟩⟩
theorem Trace.subset {α a q s f : ZFSet.{u}} (hf : Trace α a q s f) : f ⊆ ZFSet.prod α a :=
fun z hz => ZFSet.mem_prod.mpr (hf.shape z hz)
theorem internal_maximal_trace (M : ZFSet.{u}) (hM : Transitive M)
(hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M) (hS : Separation M)
{α a q s : ZFSet.{u}} (hαM : α ∈ M) (ha : a ∈ M) (hq : q ∈ M) (hsM : s ∈ M)
(hα : Transitive α) (hs : ChoiceGraph q s) :
∃ F ∈ M, Trace α a q s F ∧ ∀ f ∈ M, Trace α a q s f → f ⊆ F := by
have hp := product_mem M hM hP hU hPow hS hαM ha
obtain ⟨P,hP0,hPdef⟩ := internal_power M hM hPow hp
let e := cons α (cons a (cons q (fun _ => s)))
have he : ∀ i, e i ∈ M := by
intro i; rcases i with _|i; exact hαM
rcases i with _|i; exact ha
rcases i with _|i; exact hq
exact hsM
let C := ZFSet.sep (fun f => (traceFormula 1 2 3 4 0).Eval (cons f e)) P
have hC : C ∈ M := sep_mem M hM hS (traceFormula 1 2 3 4 0) e he hP0
have hCdef (f : ZFSet.{u}) : f ∈ C ↔ f ∈ M ∧ Trace α a q s f := by
rw [ZFSet.mem_sep,hPdef,eval_traceFormula]
constructor
· rintro ⟨⟨hf,_⟩,ht⟩; exact ⟨hf,ht⟩
· rintro ⟨hf,ht⟩; exact ⟨⟨hf,ht.subset⟩,ht⟩
refine ⟨ZFSet.sUnion C,union_mem M hM hU hC,Trace.sUnion hα hs (fun f hf => ((hCdef f).mp hf).2),?_⟩
intro f hf htrace z hz
exact ZFSet.mem_sUnion.mpr ⟨f,(hCdef f).mpr ⟨hf,htrace⟩,hz⟩
noncomputable def residual (a f i : ZFSet.{u}) : ZFSet.{u} :=
ZFSet.sep (fun x => ¬ ∃ j ∈ i, ZFSet.pair j x ∈ f) a
theorem residual_remaining (a f i : ZFSet.{u}) : Remaining a f i (residual a f i) := by
refine ⟨fun _ hx => (ZFSet.mem_sep.mp hx).1,?_⟩
intro x hx
exact ZFSet.mem_sep.trans (and_iff_right hx)
theorem residual_mem (M : ZFSet.{u}) (hM : Transitive M) (hS : Separation M)
{a f i : ZFSet.{u}} (ha : a ∈ M) (hf : f ∈ M) (hi : i ∈ M) : residual a f i ∈ M := by
simpa only [residual,Formula.Eval,eval_pairMem,cons_zero,cons_succ] using
sep_mem M hM hS (.neg (.existsMem 2 (.pairMem 0 1 2))) (cons f (fun _ => i))
(by intro j; cases j <;> assumption) ha
end InternalWellOrder
end TuringRigidity.BoundedSetTheory
end OAI
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