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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 | import OAI.SetTheory.PartitionPrinciple.Definability.GroundSemanticAssignmentLabelDomainTotality
namespace OAI
open scoped ZFSet
namespace PartitionPilot.Forcing.TransitiveGround
open SetRealization SetFormula
universe u
variable (M : TransitiveGround.{u}) {L G : Type u}
variable [PartialOrder L] [OrderBot L] [Group G]
variable (a : PosetAction G L) (s : Presentation L) (b : Presentation G) (C : 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 (collection : CollectionSchema M.sets)
variable (infinite : Realize M.sets infinity Fin.elim0)
variable (hL : s.carrier ∈ M.sets) (order : s.conditionOrder ∈ M.sets)
variable (hC : C ∈ M.sets) (hG : b.carrier ∈ M.sets)
variable (graph : a.actionGraph s b ∈ M.sets)
variable (nodeMaps : a.conditionMapFamily s b ∈ M.sets)
variable (supports : supportBasisCode s b a.principalFix ∈ M.sets)
variable (directed : ∀ x y : L, ∃ z, x ≤ z ∧ y ≤ z)
include collection order nodeMaps supports
theorem binaryRowStructuralAssignmentImage_dataBound {mu : ZFSet.{u}} (hmu : mu ∈ M.sets)
(muOrdinal : ordinalSetProperty mu) :
letI := M.smallFunctionOrder (ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega
letI := M.smallFunctionTop (ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega
⟨s.carrier, hL⟩
let act := M.internalRowAction a s b C pairing unions powers separation infinite hL hC hG graph
let c := SetMembers.presentation (M.smallPartialFunctions
(ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega)
let rows := M.binaryRowNameGraph s.nonbottom.carrier C ⊤
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
let tuples := liftedRowTupleFamily c (c.encode ⊤) mu s.nonbottom.carrier (⋃₀ ⋃₀ rows) rows S valid
∀ (_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 C 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 I := refinementDataHistoryIndices stages packed
let anchors := refinementIndexFieldGraph I s.carrier 2
∀ (g : G) (D : L) (A : ZFSet.{u}), A ∈ M.sets →
(∀ z ∈ A, ZFSet.pair z (s.encode D) ∈ stageOneDataBounds A I lifted S s.carrier mu
s.nonbottom.carrier s.conditionOrder anchors tuples) →
let movedA := M.structuralAssignmentImage b.carrier s.carrier s.conditionOrder (a.actionGraph s b)
(supportBasisCode s b a.principalFix) lifted squared stages history (b.encode g) A
movedA ∈ M.sets ∧
stageOneDataBounds movedA I lifted S s.carrier mu s.nonbottom.carrier s.conditionOrder anchors tuples ∈ M.sets ∧
∀ z ∈ movedA, ZFSet.pair z (s.encode (a.iso g D)) ∈
stageOneDataBounds movedA I lifted S s.carrier mu s.nonbottom.carrier s.conditionOrder anchors tuples := by
let := M.smallFunctionOrder (ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega
let := M.smallFunctionTop (ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega
⟨s.carrier, hL⟩
intro act c rows S T valid lifted squared tuples hLifted hSquared valid1 valid2 closed1 closed2
children1 children2 stages history hs hh stagesOrdinal trace coordinates two carrier forcingOrder
forcingAction omega topMem topRows packed I anchors
have hNonbottom := M.nonbottom_carrier_mem s separation hL
have hS := M.rowPositionSequences_mem pairing unions powers separation hL order hNonbottom hmu
have hTuples := M.binaryLiftedRowTupleFamily_mem pairing unions powers separation collection
infinite hL order hNonbottom hC hmu ⊤ valid
exact M.semanticStructuralAssignmentImage_dataBound 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 S mu tuples hS hmu hTuples
(fun f z entry => ((pair_mem_relationCompose S _ lifted _ _ f z).1 entry).2.1)
(fun g f hf tau entry => ⟨M.rowPositionSequenceAction_mem a s b pairing unions powers separation
hL hG graph hmu muOrdinal g hf,
M.binaryLiftedRowTupleFamily_action a s b C pairing unions powers separation infinite
hL hC hG graph hmu muOrdinal valid g f tau entry⟩)
end PartitionPilot.Forcing.TransitiveGround
end OAI
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