openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Transfinite /GroundDefinableRefinementHistories.lean
| import OAI.SetTheory.PartitionPrinciple.Definability.RefinementDataHistoryEntryFormulas | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing | |
| open SetFormula | |
| universe u | |
| namespace SetFormula | |
| noncomputable def refinementDataHistoryIndicesSetFormula {n : Nat} | |
| (output stages history : Fin n) : SetFormula n := | |
| exactSetFormula output | |
| (BoundedFormula.refinementDataHistoryIndicesEntryFormula stages.succ history.succ 0).toFormula | |
| noncomputable def refinementDataHistoryLevelsSetFormula {n : Nat} | |
| (output stages history T : Fin n) : SetFormula n := | |
| exactSetFormula output | |
| (BoundedFormula.refinementDataHistoryLevelsEntryFormula stages.succ history.succ T.succ 0).toFormula | |
| end SetFormula | |
| namespace TransitiveGround | |
| variable (M : TransitiveGround.{u}) | |
| variable (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) | |
| include pairing unions powers separation | |
| theorem realize_refinementDataHistoryIndicesSetFormula {n : Nat} | |
| (output stages history : Fin n) (env : Fin n → ZFSet.{u}) (members : ∀ j, env j ∈ M.sets) : | |
| Realize M.sets (refinementDataHistoryIndicesSetFormula output stages history) env ↔ | |
| env output = refinementDataHistoryIndices (env stages) (env history) := by | |
| have ground := M.refinementDataHistoryIndices_mem pairing unions powers separation | |
| (members stages) (members history) | |
| apply M.realize_exactSetFormula output _ env (members output) | |
| (fun _ member => M.transitive ground member) | |
| intro z hz | |
| exact (BoundedFormula.realize_toFormula M _ _ (Fin.cases hz members)).trans | |
| (BoundedFormula.truth_refinementDataHistoryIndicesEntryFormula stages.succ history.succ 0 _) | |
| theorem realize_refinementDataHistoryLevelsSetFormula {n : Nat} | |
| (output stages history T : Fin n) (env : Fin n → ZFSet.{u}) (members : ∀ j, env j ∈ M.sets) : | |
| Realize M.sets (refinementDataHistoryLevelsSetFormula output stages history T) env ↔ | |
| env output = refinementDataHistoryLevels (env stages) (env history) (env T) := by | |
| have ground := M.refinementDataHistoryLevels_mem pairing unions powers separation | |
| (members stages) (members history) (members T) | |
| apply M.realize_exactSetFormula output _ env (members output) | |
| (fun _ member => M.transitive ground member) | |
| intro z hz | |
| exact (BoundedFormula.realize_toFormula M _ _ (Fin.cases hz members)).trans | |
| (BoundedFormula.truth_refinementDataHistoryLevelsEntryFormula stages.succ history.succ T.succ 0 _) | |
| noncomputable def definable_refinementDataHistoryIndices : GroundDefinableFunction M | |
| (fun env : Fin 2 → ZFSet.{u} => refinementDataHistoryIndices (env 0) (env 1)) where | |
| formula := refinementDataHistoryIndicesSetFormula 0 1 2 | |
| closed := fun _ members => M.refinementDataHistoryIndices_mem pairing unions powers separation | |
| (members 0) (members 1) | |
| correct := fun env members z hz => M.realize_refinementDataHistoryIndicesSetFormula | |
| pairing unions powers separation 0 1 2 (extend z env) (Fin.cases hz members) | |
| noncomputable def definable_refinementDataHistoryLevels : GroundDefinableFunction M | |
| (fun env : Fin 3 → ZFSet.{u} => refinementDataHistoryLevels (env 0) (env 1) (env 2)) where | |
| formula := refinementDataHistoryLevelsSetFormula 0 1 2 3 | |
| closed := fun _ members => M.refinementDataHistoryLevels_mem pairing unions powers separation | |
| (members 0) (members 1) (members 2) | |
| correct := fun env members z hz => M.realize_refinementDataHistoryLevelsSetFormula | |
| pairing unions powers separation 0 1 2 3 (extend z env) (Fin.cases hz members) | |
| end TransitiveGround | |
| end PartitionPilot.Forcing | |
| end | |
| end OAI | |