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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 | import OAI.Computability.DegreeRigidity.SetModels.ChoiceTraceFormula
namespace OAI
namespace TuringRigidity.BoundedSetTheory
open TransitiveNameModel RelationCollapse
universe u
namespace InternalWellOrder
theorem adjoin_prior (f i x k y : ZFSet.{u}) (hk : k ∈ i) :
ZFSet.pair k y ∈ f ∪ ({ZFSet.pair i x} : ZFSet.{u}) ↔ ZFSet.pair k y ∈ f := by
rw [ZFSet.mem_union,ZFSet.mem_singleton]
constructor
· rintro (h|h)
· exact h
· obtain ⟨rfl,rfl⟩ := ZFSet.pair_inj.mp h
exact False.elim ((lt_irrefl _) (ZFSet.rank_lt_of_mem hk))
· exact Or.inl
theorem Trace.adjoin {α a q s f i x r : ZFSet.{u}} (hf : Trace α a q s f)
(hi : i ∈ α) (hit : Transitive i) (hx : x ∈ a) (hr : r ∈ q)
(hrem : Remaining a f i r) (hrx : ZFSet.pair r x ∈ s)
(hprev : ∀ j ∈ i, ∃ y ∈ a, ZFSet.pair j y ∈ f)
(hdom : ∀ j y, ZFSet.pair j y ∈ f → j ∈ i) :
Trace α a q s (f ∪ ({ZFSet.pair i x} : ZFSet.{u})) := by
refine ⟨?_,?_,?_⟩
· intro z hz
rcases ZFSet.mem_union.mp hz with hz|hz
· exact hf.shape z hz
· obtain rfl := ZFSet.mem_singleton.mp hz
exact ⟨i,hi,x,hx,rfl⟩
· intro j hj y hy hjy k hkj
rcases ZFSet.mem_union.mp hjy with hjy|hjy
· obtain ⟨z,hz,hkz⟩ := hf.initial j hj y hy hjy k hkj
exact ⟨z,hz,ZFSet.mem_union.mpr (Or.inl hkz)⟩
· have he := ZFSet.mem_singleton.mp hjy
obtain ⟨rfl,rfl⟩ := ZFSet.pair_inj.mp he
obtain ⟨z,hz,hkz⟩ := hprev k hkj
exact ⟨z,hz,ZFSet.mem_union.mpr (Or.inl hkz)⟩
· intro j hj y hy hjy
rcases ZFSet.mem_union.mp hjy with hjy|hjy
· obtain ⟨b,hb,hbrem,hby⟩ := hf.choice j hj y hy hjy
refine ⟨b,hb,⟨hbrem.1,?_⟩,hby⟩
intro z hz
rw [hbrem.2 z hz]
apply not_congr
apply exists_congr; intro k
apply and_congr_right; intro hkj
exact (adjoin_prior f i x k z (hit j (hdom j y hjy) k hkj)).symm
· have he := ZFSet.mem_singleton.mp hjy
obtain ⟨rfl,rfl⟩ := ZFSet.pair_inj.mp he
refine ⟨r,hr,⟨hrem.1,?_⟩,hrx⟩
intro z hz
rw [hrem.2 z hz]
apply not_congr
apply exists_congr; intro k
apply and_congr_right; intro hki
exact (adjoin_prior f j y k z hki).symm
theorem Trace.domain_before {α a q s f i : ZFSet.{u}} (hα : α.IsOrdinal)
(hf : Trace α a q s f) (hi : i ∈ α) (hmissing : ¬ ∃ x ∈ a, ZFSet.pair i x ∈ f)
(j y : ZFSet.{u}) (hjy : ZFSet.pair j y ∈ f) : j ∈ i := by
have hj := (hf.pair_mem hjy).1
have hy := (hf.pair_mem hjy).2
rcases (hα.mem hj).mem_trichotomous (hα.mem hi) with hji|he|hij
· exact hji
· subst j; exact False.elim (hmissing ⟨y,hy,hjy⟩)
· exact False.elim (hmissing (hf.initial j hj y hy hjy i hij))
theorem FunctionGraph.exists_presentation {d r f : ZFSet.{u}} (hf : TransitiveNameModel.FunctionGraph d r f) :
∃ j : ZFSet.{u} → ZFSet.{u}, Presents d f j ∧ ∀ x ∈ d, j x ∈ r := by
classical
let j := fun x => if hx : x ∈ d then (hf.2 x hx).choose else ∅
have hj (x : ZFSet.{u}) (hx : x ∈ d) : j x ∈ r ∧ ZFSet.pair x (j x) ∈ f := by
simp only [j,dite_eq_left hx]
exact ⟨(hf.2 x hx).choose_spec.1,(hf.2 x hx).choose_spec.2.1⟩
refine ⟨j,?_,fun x hx => (hj x hx).1⟩
intro z
constructor
· intro hz
obtain ⟨x,hx,y,_,rfl⟩ := hf.1 z hz
exact ⟨x,hx,congrArg (ZFSet.pair x) (hf.functional hx hz (hj x hx).2)⟩
· rintro ⟨x,hx,rfl⟩
exact (hj x hx).2
end InternalWellOrder
end TuringRigidity.BoundedSetTheory
end OAI
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