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3277efa | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 | 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
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