openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Models /GroundMainFinalStructuralRowNonchoice.lean
| import OAI.SetTheory.PartitionPrinciple.Geometry.GroundInternalNoChooserConsequences | |
| import OAI.SetTheory.PartitionPrinciple.Geometry.GroundMainFinalStructuralRowNoChooser | |
| namespace OAI | |
| open scoped ZFSet | |
| namespace PartitionPilot.Forcing.GroundMainPrefix.FinalSupportFulfillment | |
| open SetFormula SetRealization | |
| universe u v | |
| variable {M : TransitiveGround.{u}} {K Labels C D rD : ZFSet.{u}} | |
| variable {pairing : Realize M.sets SetFormula.pairing Fin.elim0} | |
| variable {unions : Realize M.sets SetFormula.union Fin.elim0} | |
| variable {powers : Realize M.sets powerSet Fin.elim0} {separation : SeparationSchema M.sets} | |
| variable {I : Type v} [Small.{u} I] [Preorder I] | |
| variable {run : GroundMainPrefix M K Labels C D rD pairing unions powers separation I} | |
| variable {level : GroundMainCandidateLevel M K} {i₀ : I} | |
| variable (fulfilled : run.FinalSupportFulfillment level i₀) | |
| theorem structural_lifted_rows_not_wellorderable_not_choice | |
| (collection : CollectionSchema M.sets) | |
| (infinite : Realize M.sets infinity Fin.elim0) | |
| (choice : Realize M.sets SetFormula.choice Fin.elim0) | |
| (small : M.sets.Countable) (smallD : M.groundEmbeddable D ZFSet.omega) | |
| (irreflexive : ∀ q ∈ D, ZFSet.pair q q ∉ rD) | |
| (columns : ZFSet.{u}) (hColumns : columns ∈ M.sets) | |
| (uncountable : ¬ M.groundEmbeddable columns ZFSet.omega) | |
| {mu : ZFSet.{u}} (hmu : mu ∈ M.sets) (muOrdinal : ordinalSetProperty mu) : | |
| let geometry := run.finalGeometry i₀ | |
| letI := geometry.group_code.toGroup | |
| letI := geometry.meet.order.toPartialOrder | |
| letI := geometry.decodedBottom | |
| let s := SetMembers.presentation geometry.carrier | |
| let b := SetMembers.presentation geometry.actors | |
| let a := geometry.decodedAction | |
| let directed := geometry.decoded_directed fulfilled.directed | |
| let hL := geometry.carrier_ground | |
| let hG := geometry.actors_ground | |
| let hC := hColumns | |
| let order : s.conditionOrder ∈ M.sets := geometry.decoded_order_eq.symm ▸ geometry.order_ground | |
| let graph : a.actionGraph s b ∈ M.sets := geometry.decoded_action_eq.symm ▸ geometry.action_ground | |
| let nodeMaps := M.conditionMapFamily_mem_of_actionGraph a s b pairing unions powers separation | |
| collection hL hG graph | |
| let supports := geometry.decoded_supportBasis_ground pairing unions powers separation | |
| letI := M.smallFunctionOrder (ZFSet.prod s.nonbottom.carrier columns) (natSetCode 2) ZFSet.omega | |
| letI := M.smallFunctionTop (ZFSet.prod s.nonbottom.carrier columns) (natSetCode 2) ZFSet.omega | |
| ⟨s.carrier, hL⟩ | |
| let act := M.internalRowAction a s b columns pairing unions powers separation infinite hL hC hG graph | |
| let c := SetMembers.presentation (M.smallPartialFunctions | |
| (ZFSet.prod s.nonbottom.carrier columns) (natSetCode 2) ZFSet.omega) | |
| let rows := M.binaryRowNameGraph s.nonbottom.carrier columns ⊤ | |
| let S := M.rowPositionSequences s.carrier s.conditionOrder s.nonbottom.carrier mu | |
| let T := rowTupleTerms (c.encode ⊤) mu s.nonbottom.carrier (⋃₀ ⋃₀ rows) rows S | |
| ∀ valid : ∀ z ∈ T, IsName c z, | |
| let lifted := Name.renamedFamily c.pairEmpty id T valid | |
| let squared := orderedPairNameFamily (c.pairEmpty.encode ⊤) lifted lifted | |
| ∀ (hLifted : lifted ∈ M.sets) (hSquared : squared ∈ M.sets) | |
| (valid1 : ∀ z ∈ lifted, IsName c.pairEmpty z) | |
| (valid2 : ∀ z ∈ squared, IsName c.pairEmpty z) | |
| (closed1 : ∀ g (tau : Name c.pairEmpty), tau.val ∈ lifted → (act.act c.pairEmpty g tau).val ∈ lifted) | |
| (closed2 : ∀ g (tau : Name c.pairEmpty), tau.val ∈ squared → (act.act c.pairEmpty g tau).val ∈ squared) | |
| (children1 : ∀ tau : Name c.pairEmpty, tau.val ∈ lifted → | |
| HereditarilySymmetric act c.pairEmpty (a.principalSupportFamily ⊥ directed) tau) | |
| (children2 : ∀ tau : Name c.pairEmpty, tau.val ∈ squared → | |
| HereditarilySymmetric act c.pairEmpty (a.principalSupportFamily ⊥ directed) tau) | |
| (stages history : ZFSet.{u}) (hs : stages ∈ M.sets) (hh : history ∈ M.sets) | |
| (stagesOrdinal : ordinalSetProperty stages) | |
| (trace : OrdinalRecursionTrace (fun level prior stage => stage = | |
| M.structuralForcingStage c.carrier c.conditionOrder (c.encode ⊤) (act.actionGraph c b) | |
| b.carrier s.carrier s.conditionOrder (a.actionGraph s b) | |
| (supportBasisCode s b a.principalFix) lifted squared level prior) stages history), | |
| let coordinates := M.prod_mem pairing unions powers separation (M.nonbottom_carrier_mem s separation hL) hC | |
| let two := M.natSetCode_mem separation infinite 2 | |
| let carrier := M.countablePartialFunctions_mem pairing unions powers separation infinite coordinates two | |
| let forcingOrder := M.countableFunctionOrder_mem pairing unions powers separation infinite coordinates two | |
| let forcingAction := M.internalRowActionGraph_mem a s b columns pairing unions powers separation infinite hL hC hG graph | |
| let omega := M.omega_mem separation infinite | |
| let topMem := c.encode_mem ⊤ | |
| let topRows : ∀ r ∈ c.carrier, ZFSet.pair r (c.encode ⊤) ∈ c.conditionOrder := | |
| fun r hr => by | |
| obtain ⟨r, rfl⟩ := (c.mem_carrier r).1 hr | |
| exact (c.pair_mem_conditionOrder r ⊤).2 le_top | |
| let packed := M.structuralRefinementHistory b.carrier s.carrier s.conditionOrder | |
| (supportBasisCode s b a.principalFix) lifted squared stages history | |
| let upperQ := M.refinementHistoryForcing c.carrier lifted stages packed | |
| letI := M.refinementHistoryPartialOrder c.conditionOrder_code lifted stages packed | |
| let upperAct := M.semanticStructuralAction c act b s a ⊥ directed lifted squared (c.encode ⊤) | |
| pairing unions powers separation collection carrier forcingOrder forcingAction hG hL order graph | |
| nodeMaps supports hLifted hSquared omega valid1 valid2 closed1 closed2 children1 children2 | |
| topMem topRows hs hh stagesOrdinal trace | |
| let upper := SetMembers.presentation upperQ | |
| let recursive := M.structuralForcingTrace_recursiveOrder c.carrier c.conditionOrder (c.encode ⊤) | |
| (act.actionGraph c b) b.carrier s.carrier s.conditionOrder (a.actionGraph s b) | |
| (supportBasisCode s b a.principalFix) lifted squared stagesOrdinal trace | |
| let inclusion := M.refinementHistoryRowEmbedding c lifted stages packed recursive stagesOrdinal carrier | |
| ∀ (H : ConditionFilter (SetMembers upperQ)), | |
| H.GenericFor (formulaFamily (M.nameDomain upper)) → | |
| let N := M.symmetricExtension upper upperAct (a.principalSupportFamily ⊥ directed) H | |
| let rowSet := Name.eval {p | H.mem p} | |
| (Name.rename upper inclusion (M.internalRowSetName s columns hL)) | |
| ZFRealization N.sets → | |
| rowSet ∈ N.sets ∧ | |
| (¬ ∃ index ∈ N.sets, ordinalSetProperty index ∧ | |
| ∃ enumeration ∈ N.sets, ZFSet.IsFunc index rowSet enumeration ∧ | |
| ∀ x ∈ rowSet, ∃ i ∈ index, ZFSet.pair i x ∈ enumeration) ∧ | |
| ¬ Realize N.sets SetFormula.choice Fin.elim0 := by | |
| let geometry := run.finalGeometry i₀ | |
| let := geometry.group_code.toGroup | |
| let := geometry.meet.order.toPartialOrder | |
| let := geometry.decodedBottom | |
| let s := SetMembers.presentation geometry.carrier | |
| let b := SetMembers.presentation geometry.actors | |
| let a := geometry.decodedAction | |
| let directed := geometry.decoded_directed fulfilled.directed | |
| let hL := geometry.carrier_ground | |
| let hG := geometry.actors_ground | |
| let hC := hColumns | |
| let order : s.conditionOrder ∈ M.sets := geometry.decoded_order_eq.symm ▸ geometry.order_ground | |
| let graph : a.actionGraph s b ∈ M.sets := geometry.decoded_action_eq.symm ▸ geometry.action_ground | |
| let nodeMaps := M.conditionMapFamily_mem_of_actionGraph a s b pairing unions powers separation | |
| collection hL hG graph | |
| let supports := geometry.decoded_supportBasis_ground pairing unions powers separation | |
| let := M.smallFunctionOrder (ZFSet.prod s.nonbottom.carrier columns) (natSetCode 2) ZFSet.omega | |
| let := M.smallFunctionTop (ZFSet.prod s.nonbottom.carrier columns) (natSetCode 2) ZFSet.omega | |
| ⟨s.carrier, hL⟩ | |
| let act := M.internalRowAction a s b columns pairing unions powers separation infinite hL hC hG graph | |
| let c := SetMembers.presentation (M.smallPartialFunctions | |
| (ZFSet.prod s.nonbottom.carrier columns) (natSetCode 2) ZFSet.omega) | |
| let rows := M.binaryRowNameGraph s.nonbottom.carrier columns ⊤ | |
| let S := M.rowPositionSequences s.carrier s.conditionOrder s.nonbottom.carrier mu | |
| let T := rowTupleTerms (c.encode ⊤) mu s.nonbottom.carrier (⋃₀ ⋃₀ rows) rows S | |
| dsimp only | |
| intro valid | |
| let lifted := Name.renamedFamily c.pairEmpty id T valid | |
| let squared := orderedPairNameFamily (c.pairEmpty.encode ⊤) lifted lifted | |
| intro hLifted hSquared valid1 valid2 closed1 closed2 children1 children2 | |
| stages history hs hh stagesOrdinal trace | |
| let coordinates := M.prod_mem pairing unions powers separation (M.nonbottom_carrier_mem s separation hL) hC | |
| let two := M.natSetCode_mem separation infinite 2 | |
| let carrier := M.countablePartialFunctions_mem pairing unions powers separation infinite coordinates two | |
| let forcingOrder := M.countableFunctionOrder_mem pairing unions powers separation infinite coordinates two | |
| let forcingAction := M.internalRowActionGraph_mem a s b columns pairing unions powers separation infinite hL hC hG graph | |
| let omega := M.omega_mem separation infinite | |
| let topMem := c.encode_mem ⊤ | |
| let topRows : ∀ r ∈ c.carrier, ZFSet.pair r (c.encode ⊤) ∈ c.conditionOrder := | |
| fun r hr => by | |
| obtain ⟨r, rfl⟩ := (c.mem_carrier r).1 hr | |
| exact (c.pair_mem_conditionOrder r ⊤).2 le_top | |
| let packed := M.structuralRefinementHistory b.carrier s.carrier s.conditionOrder | |
| (supportBasisCode s b a.principalFix) lifted squared stages history | |
| let upperQ := M.refinementHistoryForcing c.carrier lifted stages packed | |
| let := M.refinementHistoryPartialOrder c.conditionOrder_code lifted stages packed | |
| let upperAct := M.semanticStructuralAction c act b s a ⊥ directed lifted squared (c.encode ⊤) | |
| pairing unions powers separation collection carrier forcingOrder forcingAction hG hL order graph | |
| nodeMaps supports hLifted hSquared omega valid1 valid2 closed1 closed2 children1 children2 | |
| topMem topRows hs hh stagesOrdinal trace | |
| let upper := SetMembers.presentation upperQ | |
| let recursive := M.structuralForcingTrace_recursiveOrder c.carrier c.conditionOrder (c.encode ⊤) | |
| (act.actionGraph c b) b.carrier s.carrier s.conditionOrder (a.actionGraph s b) | |
| (supportBasisCode s b a.principalFix) lifted squared stagesOrdinal trace | |
| let inclusion := M.refinementHistoryRowEmbedding c lifted stages packed recursive stagesOrdinal carrier | |
| intro H ordinary zf | |
| have noChooser := fulfilled.structural_lifted_rows_no_chooser collection infinite choice small | |
| smallD irreflexive columns hColumns uncountable hmu muOrdinal valid hLifted hSquared | |
| valid1 valid2 closed1 closed2 children1 children2 stages history hs hh stagesOrdinal trace H ordinary | |
| exact ⟨noChooser.1, | |
| TransitiveGround.noChooser_no_ordinal_surjection_not_choice | |
| (M.symmetricExtension upper upperAct (a.principalSupportFamily ⊥ directed) H) | |
| zf noChooser.1 noChooser.2⟩ | |
| end PartitionPilot.Forcing.GroundMainPrefix.FinalSupportFulfillment | |
| end OAI | |