openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Forcing /GroundCompressionSerializationTransport.lean
| import OAI.SetTheory.PartitionPrinciple.Transfinite.GroundSerializedCompressionChronology | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing.GroundCompressionOrigin | |
| open SetFormula | |
| universe u | |
| variable {M : TransitiveGround.{u}} {initial : RecursiveStateData.{u}} | |
| variable {P0 r0 μ0 A0 τ I Sources Reps G m inv one T Filters C Q qlt original Targets Separators base old T0 : ZFSet.{u}} | |
| variable {history : GroundChronologicalHistory M T0 C Q qlt} | |
| variable (origin : GroundCompressionOrigin M initial P0 r0 μ0 A0 τ I Sources Reps G m inv one T Filters) | |
| variable (compression : GroundChronologicalCompression M initial I Sources Reps G m one T τ | |
| P0 C Q qlt original Targets Separators Filters base old T0 history) | |
| variable (prior : GroundSerializedChronology M T0 C Q qlt) | |
| include origin in | |
| theorem exists_serializedCompression | |
| (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) (infinite : Realize M.sets infinity Fin.elim0) | |
| (choice : Realize M.sets SetFormula.choice Fin.elim0) | |
| (hI : I ∈ M.sets) (hSources : Sources ∈ M.sets) (hReps : Reps ∈ M.sets) | |
| (hG : G ∈ M.sets) (hT : T ∈ M.sets) (hBase : base ∈ T0) | |
| (constant : RecursiveTemplateConstancy P0 r0 μ0 initial) | |
| (initialDomain : initial.D = P0) (initialOrder : initial.rD = r0) (initialMeet : initial.μD = μ0) | |
| (priorSpace : history.space base = P0) (priorOrder : history.order base = r0) | |
| (priorMeet : history.meetGraph base = μ0) (priorBirth : history.birth base = old) | |
| (agreement : ∀ j s, s ∈ T0 → prior.fields.value j s = history.fieldValue j s) : | |
| ∃ serial : GroundSerializedChronology M | |
| (insert compression.born.births.limit compression.born.births.limit) C Q qlt, | |
| serial.code ∈ M.sets ∧ ∀ j s, s ∈ insert compression.born.births.limit compression.born.births.limit → | |
| serial.fields.value j s = compression.history.fieldValue j s := by | |
| obtain ⟨seed, state, equal⟩ := origin | |
| cases equal | |
| let data := compression.born.profiled.heighted.completion.algebra.union | |
| let births := compression.born.births | |
| let heights := compression.born.profiled.heighted.heights | |
| obtain ⟨serial, ground, values⟩ := state.exists_serializedPrefix data births heights prior | |
| pairing unions powers separation collection infinite choice hI hSources hReps hG hT hBase | |
| constant initialDomain initialOrder initialMeet | |
| ((agreement 0 base hBase).trans priorSpace) ((agreement 1 base hBase).trans priorOrder) | |
| ((agreement 2 base hBase).trans priorMeet) ((agreement 3 base hBase).trans priorBirth) | |
| refine ⟨serial, ground, ?_⟩ | |
| intro j s hs | |
| rw [values j s hs] | |
| have oldValue (a : ZFSet.{u}) (before : a ∈ base ∨ a = base) : | |
| (state.prefixFields data births heights prior.toHistory).fieldValue j a = | |
| compression.history.fieldValue j a := by | |
| have ha : a ∈ T0 := before.elim (fun h => prior.ordinal.1 base hBase a h) (fun h => h.symm ▸ hBase) | |
| have left := state.prefixFields_old data births heights prior.toHistory before | |
| have right := compression.old_values a before | |
| have eLeft : (state.prefixFields data births heights prior.toHistory).fieldValue j a = | |
| prior.fields.value j a := | |
| Fin.cases left.1 (Fin.cases left.2.1 (Fin.cases left.2.2.1 | |
| (Fin.cases left.2.2.2.1 (Fin.cases left.2.2.2.2 (fun i => Fin.elim0 i))))) j | |
| have eRight : compression.history.fieldValue j a = history.fieldValue j a := | |
| Fin.cases right.1 (Fin.cases right.2.1 (Fin.cases right.2.2.1 | |
| (Fin.cases right.2.2.2.1 (Fin.cases right.2.2.2.2 (fun i => Fin.elim0 i))))) j | |
| exact eLeft.trans ((agreement j a ha).trans eRight.symm) | |
| rcases ZFSet.mem_insert_iff.1 hs with rfl | hs | |
| · have left := state.prefixFields_limit data births heights prior.toHistory | |
| have right := compression.limit_values | |
| exact Fin.cases (left.1.trans right.1.symm) | |
| (Fin.cases (left.2.1.trans right.2.1.symm) (Fin.cases (left.2.2.1.trans right.2.2.1.symm) | |
| (Fin.cases (left.2.2.2.1.trans right.2.2.2.1.symm) | |
| (Fin.cases (left.2.2.2.2.trans right.2.2.2.2.symm) (fun i => Fin.elim0 i))))) j | |
| · rcases (births.times.mem_limit_iff s).1 hs with before | equal | ⟨n, rfl⟩ | |
| · exact oldValue s (Or.inl before) | |
| · exact oldValue s (Or.inr equal) | |
| · have left := state.prefixFields_stamp data births heights prior.toHistory n | |
| have right := compression.stamp_values n | |
| exact Fin.cases (left.1.trans right.1.symm) | |
| (Fin.cases (left.2.1.trans right.2.1.symm) (Fin.cases (left.2.2.1.trans right.2.2.1.symm) | |
| (Fin.cases (left.2.2.2.1.trans right.2.2.2.1.symm) | |
| (Fin.cases (left.2.2.2.2.trans right.2.2.2.2.symm) (fun i => Fin.elim0 i))))) j | |
| end PartitionPilot.Forcing.GroundCompressionOrigin | |
| end | |
| end OAI | |