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