openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /SetModels /ChoiceTraceExtension.lean
| 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 | |