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namespace OAI
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_successor_assignment_label_injection
(collection : CollectionSchema M.sets)
(infinite : Realize M.sets infinity Fin.elim0)
(choice : Realize M.sets SetFormula.choice Fin.elim0)
(labels : Labels = M.groundOmegaOne pairing unions powers separation collection infinite)
(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)
{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 rows.sUnion.sUnion 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
∀ (alpha : ZFSet.{u}) (member : alpha ∈ stages),
let hAlpha := M.transitive hs member
let cutOrdinal := ordinalSetProperty_member stagesOrdinal member
let subset := (And.left stagesOrdinal) alpha member
let lowerPacked := graphRestriction alpha packed
let Q := M.refinementHistoryForcing c.carrier lifted alpha lowerPacked
letI := M.refinementHistoryPartialOrder c.conditionOrder_code lifted alpha lowerPacked
let historyRun := 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
letI := M.refinementHistoryOrderTop c.conditionOrder_code lifted alpha lowerPacked
(historyRun.carriers.restrict cutOrdinal subset) cutOrdinal carrier (c.encode ⊤) topMem topRows
let lower := SetMembers.presentation Q
let localAct := M.semanticStructuralPrefixAction 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 hh trace hAlpha cutOrdinal subset
let localValid := M.refinementHistoryRowNameFamily_valid c
(historyRun.carriers.restrict cutOrdinal subset) cutOrdinal carrier lifted valid1
let upperSet := M.refinementHistoryForcing c.carrier lifted stages packed
letI := M.refinementHistoryPartialOrder c.conditionOrder_code lifted stages packed
letI := M.refinementHistoryOrderTop c.conditionOrder_code lifted stages packed
historyRun.carriers stagesOrdinal carrier (c.encode ⊤) topMem topRows
let upper := SetMembers.presentation upperSet
let hPacked := M.structuralRefinementHistory_mem pairing unions powers separation collection
![b.carrier, s.carrier, s.conditionOrder, supportBasisCode s b a.principalFix,
lifted, squared, stages, history]
(Fin.cases hG (Fin.cases hL (Fin.cases order (Fin.cases supports
(Fin.cases hLifted (Fin.cases hSquared (Fin.cases hs (Fin.cases hh (fun j => Fin.elim0 j)))))))))
let witnesses := M.structuralForcingTrace_witnesses 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 mono := M.structuralHistory_rawTests_mono b s a ⊥ directed lifted squared stages history
pairing unions powers separation collection hG hL supports hh hLifted witnesses
let inc := RecursiveRefinementOrderHistory.inclusionFunction historyRun stagesOrdinal cutOrdinal subset
pairing unions powers separation infinite choice hs hPacked hLifted hAlpha carrier (c.encode ⊤)
topMem topRows mono
∀ (_successor : insert alpha alpha = stages)
(V E : SetMembers geometry.carrier) (U Rf Rc : (M.nameDomain lower).names),
let idx := refinementIndexCode alpha (s.encode V) (s.encode E) U.val.val Rf.val.val Rc.val.val
idx ∈ M.niceRefinementIndices localAct lower s a alpha lifted squared localValid →
∀ (H : ConditionFilter (SetMembers upperSet)), H.GenericFor (formulaFamily (M.nameDomain upper)) →
let label := Name.refinementAssignmentLabel upper inc ⊤ lifted localValid idx Rf.val Rc.val
let J := Name.eval {p | H.mem p} label
let coarse := Name.eval {q | H.mem (inc q)} (Name.refinementQuotient lifted localValid U.val Rc.val)
let fine := Name.eval {q | H.mem (inc q)} (Name.refinementQuotient lifted localValid U.val Rf.val)
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
InjectionCode coarse fine J ∧
J ∈ (M.symmetricExtension upper upperAct (a.principalSupportFamily ⊥ directed) H).sets := 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 rows.sUnion.sUnion 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
intro alpha member
let hAlpha := M.transitive hs member
let cutOrdinal := ordinalSetProperty_member stagesOrdinal member
let subset := (And.left stagesOrdinal) alpha member
let lowerPacked := graphRestriction alpha packed
let Q := M.refinementHistoryForcing c.carrier lifted alpha lowerPacked
let := M.refinementHistoryPartialOrder c.conditionOrder_code lifted alpha lowerPacked
let historyRun := 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 := M.refinementHistoryOrderTop c.conditionOrder_code lifted alpha lowerPacked
(historyRun.carriers.restrict cutOrdinal subset) cutOrdinal carrier (c.encode ⊤) topMem topRows
let lower := SetMembers.presentation Q
let localAct := M.semanticStructuralPrefixAction 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 hh trace hAlpha cutOrdinal subset
let localValid := M.refinementHistoryRowNameFamily_valid c
(historyRun.carriers.restrict cutOrdinal subset) cutOrdinal carrier lifted valid1
let upperSet := M.refinementHistoryForcing c.carrier lifted stages packed
let := M.refinementHistoryPartialOrder c.conditionOrder_code lifted stages packed
let := M.refinementHistoryOrderTop c.conditionOrder_code lifted stages packed
historyRun.carriers stagesOrdinal carrier (c.encode ⊤) topMem topRows
let upper := SetMembers.presentation upperSet
let hPacked := M.structuralRefinementHistory_mem pairing unions powers separation collection
![b.carrier, s.carrier, s.conditionOrder, supportBasisCode s b a.principalFix,
lifted, squared, stages, history]
(Fin.cases hG (Fin.cases hL (Fin.cases order (Fin.cases supports
(Fin.cases hLifted (Fin.cases hSquared (Fin.cases hs (Fin.cases hh (fun j => Fin.elim0 j)))))))))
let witnesses := M.structuralForcingTrace_witnesses 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 mono := M.structuralHistory_rawTests_mono b s a ⊥ directed lifted squared stages history
pairing unions powers separation collection hG hL supports hh hLifted witnesses
let inc := RecursiveRefinementOrderHistory.inclusionFunction historyRun stagesOrdinal cutOrdinal subset
pairing unions powers separation infinite choice hs hPacked hLifted hAlpha carrier (c.encode ⊤)
topMem topRows mono
intro successor V E U Rf Rc index H ordinary
have total := fulfilled.structural_successor_assignment_label_total collection infinite choice labels
small smallD irreflexive columns hColumns hmu muOrdinal valid hLifted hSquared
valid1 valid2 closed1 closed2 children1 children2
stages history hs hh stagesOrdinal trace alpha member successor V E U Rf Rc index H ordinary
exact M.semanticStructuralAssignmentLabel_injection_package 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 member infinite choice V E U Rf Rc index H ordinary total
end PartitionPilot.Forcing.GroundMainPrefix.FinalSupportFulfillment
end OAI
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