openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Transfinite /GroundChronologicalCompressionCoinitiality.lean
| import OAI.SetTheory.PartitionPrinciple.Geometry.CompressionExternalTileOverlaps | |
| import OAI.SetTheory.PartitionPrinciple.Transfinite.GroundSerializedChronologyTimeEquality | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing | |
| universe u | |
| structure GroundChronologicalCompression (M : TransitiveGround.{u}) (initial : RecursiveStateData.{u}) | |
| (I Sources Reps G m one T τ P0 C Q qlt original Targets Separators Filters base old T0 : ZFSet.{u}) | |
| (prior : GroundChronologicalHistory M T0 C Q qlt) where | |
| born : GroundBornCompression M initial I Sources Reps G m one T τ P0 C Q qlt original | |
| Targets Separators Filters base old | |
| history : GroundChronologicalHistory M (insert born.births.limit born.births.limit) C Q qlt | |
| original_height : prior.height base = original | |
| old_values : ∀ s, s ∈ base ∨ s = base → | |
| history.space s = prior.space s ∧ history.order s = prior.order s ∧ | |
| history.meetGraph s = prior.meetGraph s ∧ history.birth s = prior.birth s ∧ | |
| history.height s = prior.height s | |
| stamp_values : ∀ n, | |
| history.space (compressionStamp base n) = (initial.iterate M I Sources Reps G T (n + 1)).D ∧ | |
| history.order (compressionStamp base n) = (initial.iterate M I Sources Reps G T (n + 1)).rD ∧ | |
| history.meetGraph (compressionStamp base n) = (initial.iterate M I Sources Reps G T (n + 1)).μD ∧ | |
| history.birth (compressionStamp base n) = | |
| graphRestriction (initial.iterate M I Sources Reps G T (n + 1)).D born.births.birth ∧ | |
| history.height (compressionStamp base n) = born.profiled.heighted.heights.function.natValue (n + 1) | |
| limit_values : | |
| history.space born.births.limit = born.profiled.heighted.completion.algebra.union.fields 0 ∧ | |
| history.order born.births.limit = born.profiled.heighted.completion.algebra.union.fields 1 ∧ | |
| history.meetGraph born.births.limit = born.profiled.heighted.completion.algebra.union.fields 2 ∧ | |
| history.birth born.births.limit = born.births.birth ∧ | |
| history.height born.births.limit = born.profiled.heighted.heights.height | |
| end PartitionPilot.Forcing | |
| namespace PartitionPilot.Forcing.GroundChronologicalCompression | |
| open SetFormula | |
| universe u | |
| variable {M N : TransitiveGround.{u}} {initial : RecursiveStateData.{u}} | |
| variable {I Sources Reps G m one T τ P0 C Q qlt original Targets Separators Filters base old T0 : ZFSet.{u}} | |
| variable {prior : GroundChronologicalHistory M T0 C Q qlt} | |
| variable (compression : GroundChronologicalCompression M initial I Sources Reps G m one T τ | |
| P0 C Q qlt original Targets Separators Filters base old T0 prior) | |
| theorem exists_extension_basis | |
| (pairing : Realize M.sets SetFormula.pairing Fin.elim0) | |
| (unions : Realize M.sets SetFormula.union Fin.elim0) | |
| (includes : M.sets ⊆ N.sets) | |
| (pairingN : Realize N.sets SetFormula.pairing Fin.elim0) | |
| (unionsN : Realize N.sets SetFormula.union Fin.elim0) | |
| (powersN : Realize N.sets powerSet Fin.elim0) (separationN : SeparationSchema N.sets) | |
| (infiniteN : Realize N.sets SetFormula.infinity Fin.elim0) | |
| (choiceN : Realize N.sets SetFormula.choice Fin.elim0) | |
| (hC : C ∈ M.sets) (hQ : Q ∈ M.sets) | |
| (smallQ : M.groundEmbeddable Q ZFSet.omega) | |
| (irreflexive : ∀ q ∈ Q, ZFSet.pair q q ∉ qlt) | |
| {F : ZFSet.{u}} (hF : F ∈ N.sets) | |
| (filter : MeetFilterCode (compression.born.profiled.heighted.completion.algebra.union.fields 0) | |
| (compression.born.profiled.heighted.completion.algebra.union.fields 1) | |
| (compression.born.profiled.heighted.completion.algebra.union.fields 2) F) : | |
| ∃ B ∈ N.sets, ∃ f ∈ N.sets, InjectionCode B ZFSet.omega f ∧ B ⊆ F ∧ | |
| ∀ x ∈ F, ∃ b ∈ B, | |
| ZFSet.pair b x ∈ compression.born.profiled.heighted.completion.algebra.union.fields 1 := by | |
| have filterAtLimit : MeetFilterCode (compression.history.space compression.born.births.limit) | |
| (compression.history.order compression.born.births.limit) | |
| (compression.history.meetGraph compression.born.births.limit) F := by | |
| simpa only [compression.limit_values.1, compression.limit_values.2.1, compression.limit_values.2.2.1] using filter | |
| have basis := compression.history.exists_extension_basis includes pairingN unionsN powersN | |
| separationN infiniteN choiceN | |
| (M.insert_mem pairing unions compression.born.births.limit_ground compression.born.births.limit_ground) | |
| hC hQ (ordinalSetProperty_succ compression.born.births.times.ordinal) smallQ irreflexive | |
| (ZFSet.mem_insert compression.born.births.limit compression.born.births.limit) hF filterAtLimit | |
| simpa only [compression.limit_values.2.1] using basis | |
| end PartitionPilot.Forcing.GroundChronologicalCompression | |
| end | |
| end OAI | |