openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Models /GroundChoiceEnumerationTraces.lean
| import OAI.SetTheory.PartitionPrinciple.Transfinite.GroundOrdinalEnumerationFamilies | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing.TransitiveGround | |
| open SetFormula | |
| universe u | |
| variable (M : TransitiveGround.{u}) | |
| theorem exists_choiceEnumerationTrace | |
| (pairing : Realize M.sets SetFormula.pairing Fin.elim0) | |
| (unions : Realize M.sets SetFormula.union Fin.elim0) | |
| (powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets) | |
| (collection : CollectionSchema M.sets) | |
| {A c T : ZFSet.{u}} (hA : A ∈ M.sets) (hc : c ∈ M.sets) | |
| (hT : T ∈ M.sets) (ordinal : ordinalSetProperty T) | |
| (code : ChoiceFunctionCode (M.nonemptyGroundSubsets A) c) : | |
| ∃ f ∈ M.sets, | |
| OrdinalRecursionTrace (ChoiceEnumerationStep A (M.nonemptyGroundSubsets A) c) T f := by | |
| let F := M.nonemptyGroundSubsets A | |
| have hF : F ∈ M.sets := M.nonemptyGroundSubsets_mem powers separation hA | |
| let params := extend A (extend F (extend c Fin.elim0)) | |
| have members : ∀ i, params i ∈ M.sets := | |
| Fin.cases hA (Fin.cases hF (Fin.cases hc (fun i => Fin.elim0 i))) | |
| let formula : BoundedFormula 6 := BoundedFormula.choiceEnumerationStepFormula 3 4 5 0 1 2 | |
| have recognize (a p y : ZFSet.{u}) (ha : a ∈ M.sets) (hp : p ∈ M.sets) | |
| (hy : y ∈ M.sets) : | |
| Realize M.sets formula.toFormula (extend a (extend p (extend y params))) ↔ | |
| ChoiceEnumerationStep A F c a p y := | |
| (BoundedFormula.realize_toFormula M formula _ | |
| (Fin.cases ha (Fin.cases hp (Fin.cases hy members)))).trans | |
| (BoundedFormula.truth_choiceEnumerationStepFormula 3 4 5 0 1 2 _) | |
| have closure : GroundOrdinalStepClosure M T formula.toFormula params := by | |
| intro a ha p hp _ | |
| have haM := M.transitive hT ha | |
| obtain ⟨y, hy, step, unique⟩ := | |
| M.choiceEnumerationStep_exists_unique separation hA haM hp code | |
| exact ⟨y, hy, (recognize a p y haM hp hy).2 step, | |
| fun z hz relation => unique z ((recognize a p z haM hp hz).1 relation)⟩ | |
| obtain ⟨f, hf, trace⟩ := M.exists_ordinalRecursionTrace pairing unions separation collection | |
| hT ordinal formula.toFormula params members closure | |
| refine ⟨f, hf, trace.1, ?_⟩ | |
| intro a ha y entry | |
| have step := trace.2 a ha y entry | |
| exact (recognize a _ y (M.transitive hT ha) step.1 step.2.1).1 step.2.2 | |
| end PartitionPilot.Forcing.TransitiveGround | |
| end | |
| end OAI | |