openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Transfinite /GroundCountableRowExactUnions.lean
| import OAI.SetTheory.PartitionPrinciple.Definability.GroundSemanticFairPreparationLimits | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing.TransitiveGround | |
| open SetRealization SetFormula | |
| universe u | |
| variable (M : TransitiveGround.{u}) {G L : Type u} [Group G] [PartialOrder L] | |
| variable (t : Presentation G) (b : Presentation L) (d : PosetAction G L) | |
| variable (base : L) (directed : ∀ x y : L, ∃ z, x ≤ z ∧ y ≤ z) | |
| theorem countableRow_semanticStructuralTrace_exact_assignment_union | |
| (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) | |
| (X : ZFSet.{u}) (hX : X ∈ M.sets) : | |
| letI := M.smallFunctionOrder X (natSetCode 2) ZFSet.omega | |
| let c := SetMembers.presentation (M.smallPartialFunctions X (natSetCode 2) ZFSet.omega) | |
| ∀ (a : PosetAction G (SetMembers (M.smallPartialFunctions X (natSetCode 2) ZFSet.omega))) | |
| (T T2 rowTop : ZFSet.{u}), | |
| a.actionGraph c t ∈ M.sets → | |
| t.carrier ∈ M.sets → b.carrier ∈ M.sets → b.conditionOrder ∈ M.sets → | |
| d.actionGraph b t ∈ M.sets → d.conditionMapFamily b t ∈ M.sets → | |
| supportBasisCode b t d.principalFix ∈ M.sets → T ∈ M.sets → T2 ∈ M.sets → | |
| (∀ z ∈ T, IsName c.pairEmpty z) → (∀ z ∈ T2, IsName c.pairEmpty z) → | |
| (∀ g (tau : Name c.pairEmpty), tau.val ∈ T → (a.act c.pairEmpty g tau).val ∈ T) → | |
| (∀ g (tau : Name c.pairEmpty), tau.val ∈ T2 → (a.act c.pairEmpty g tau).val ∈ T2) → | |
| (∀ tau : Name c.pairEmpty, tau.val ∈ T → | |
| HereditarilySymmetric a c.pairEmpty (d.principalSupportFamily base directed) tau) → | |
| (∀ tau : Name c.pairEmpty, tau.val ∈ T2 → | |
| HereditarilySymmetric a c.pairEmpty (d.principalSupportFamily base directed) tau) → | |
| rowTop ∈ c.carrier → (∀ row ∈ c.carrier, ZFSet.pair row rowTop ∈ c.conditionOrder) → | |
| ∀ stages history : ZFSet.{u}, stages ∈ M.sets → history ∈ M.sets → | |
| ordinalSetProperty stages → | |
| OrdinalRecursionTrace (fun level prior stage => stage = | |
| M.structuralForcingStage c.carrier c.conditionOrder rowTop (a.actionGraph c t) | |
| t.carrier b.carrier b.conditionOrder (d.actionGraph b t) | |
| (supportBasisCode b t d.principalFix) T T2 level prior) stages history → | |
| let packed := M.structuralRefinementHistory t.carrier b.carrier b.conditionOrder | |
| (supportBasisCode b t d.principalFix) T T2 stages history | |
| letI := M.refinementHistoryPartialOrder c.conditionOrder_code T stages packed | |
| let Q := M.refinementHistoryForcing c.carrier T stages packed | |
| let B := M.groundSmallSubsets (stageOneAssignments (refinementDataHistoryIndices stages packed) T) ZFSet.omega | |
| let shape := M.refinementHistoryForcing_shape c.carrier T stages packed | |
| let row := flatCarrierRow c Q B shape | |
| let data := flatCarrierData c Q B shape | |
| ∀ seq : Nat → SetMembers Q, groundSequenceGraph seq ∈ M.sets → Antitone seq → | |
| ∃ q : SetMembers (M.smallPartialFunctions X (natSetCode 2) ZFSet.omega), | |
| (∀ n, q ≤ row (seq n)) ∧ ∃ p : SetMembers Q, | |
| SetMembers.raw p = ZFSet.pair (c.encode q) | |
| (stageOneAssignmentUnion (fun n => SetMembers.raw (data (seq n)))) ∧ | |
| (∀ n, p ≤ seq n) ∧ | |
| SetMembers.raw (data p) = stageOneAssignmentUnion (fun n => SetMembers.raw (data (seq n))) ∧ | |
| stageOneAssignmentUnion (fun n => SetMembers.raw (data (seq n))) ∈ M.sets ∧ | |
| ∀ z, z ∈ SetMembers.raw (data p) ↔ ∃ n, z ∈ SetMembers.raw (data (seq n)) := by | |
| let := M.smallFunctionOrder X (natSetCode 2) ZFSet.omega | |
| let c := SetMembers.presentation (M.smallPartialFunctions X (natSetCode 2) ZFSet.omega) | |
| dsimp only | |
| intro a T T2 rowTop action group nodes nodeOrder nodeAction nodeMaps supports hT hT2 | |
| valid1 valid2 closed1 closed2 children1 children2 topMem topRows stages history hs hh ordinal trace | |
| have two := M.natSetCode_mem separation infinite 2 | |
| have carrier := M.countablePartialFunctions_mem pairing unions powers separation infinite hX two | |
| have order := M.countableFunctionOrder_mem pairing unions powers separation infinite hX two | |
| have rowClosed := M.countable_function_internally_closed pairing unions powers separation infinite choice hX two | |
| exact M.semanticStructuralTrace_exact_assignment_union c a t b d base directed T T2 rowTop | |
| pairing unions powers separation collection carrier order action group nodes nodeOrder nodeAction | |
| nodeMaps supports hT hT2 (M.omega_mem separation infinite) valid1 valid2 closed1 closed2 | |
| children1 children2 topMem topRows hs hh ordinal trace infinite choice rowClosed | |
| end PartitionPilot.Forcing.TransitiveGround | |
| end | |
| end OAI | |