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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 | import OAI.SetTheory.PartitionPrinciple.Definability.GroundSemanticFairPreparationLimits
namespace OAI
section
namespace PartitionPilot.Forcing.TransitiveGround
open SetRealization SetFormula
universe u
variable (M : TransitiveGround.{u}) {G L : Type u} [Group G] [PartialOrder L]
variable (t : Presentation G) (b : Presentation L) (d : PosetAction G L)
variable (base : L) (directed : ∀ x y : L, ∃ z, x ≤ z ∧ y ≤ z)
theorem countableRow_semanticStructuralTrace_exact_assignment_union
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0)
(powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets)
(collection : CollectionSchema M.sets)
(infinite : Realize M.sets infinity Fin.elim0) (choice : Realize M.sets SetFormula.choice Fin.elim0)
(X : ZFSet.{u}) (hX : X ∈ M.sets) :
letI := M.smallFunctionOrder X (natSetCode 2) ZFSet.omega
let c := SetMembers.presentation (M.smallPartialFunctions X (natSetCode 2) ZFSet.omega)
∀ (a : PosetAction G (SetMembers (M.smallPartialFunctions X (natSetCode 2) ZFSet.omega)))
(T T2 rowTop : ZFSet.{u}),
a.actionGraph c t ∈ M.sets →
t.carrier ∈ M.sets → b.carrier ∈ M.sets → b.conditionOrder ∈ M.sets →
d.actionGraph b t ∈ M.sets → d.conditionMapFamily b t ∈ M.sets →
supportBasisCode b t d.principalFix ∈ M.sets → T ∈ M.sets → T2 ∈ M.sets →
(∀ z ∈ T, IsName c.pairEmpty z) → (∀ z ∈ T2, IsName c.pairEmpty z) →
(∀ g (tau : Name c.pairEmpty), tau.val ∈ T → (a.act c.pairEmpty g tau).val ∈ T) →
(∀ g (tau : Name c.pairEmpty), tau.val ∈ T2 → (a.act c.pairEmpty g tau).val ∈ T2) →
(∀ tau : Name c.pairEmpty, tau.val ∈ T →
HereditarilySymmetric a c.pairEmpty (d.principalSupportFamily base directed) tau) →
(∀ tau : Name c.pairEmpty, tau.val ∈ T2 →
HereditarilySymmetric a c.pairEmpty (d.principalSupportFamily base directed) tau) →
rowTop ∈ c.carrier → (∀ row ∈ c.carrier, ZFSet.pair row rowTop ∈ c.conditionOrder) →
∀ stages history : ZFSet.{u}, stages ∈ M.sets → history ∈ M.sets →
ordinalSetProperty stages →
OrdinalRecursionTrace (fun level prior stage => stage =
M.structuralForcingStage c.carrier c.conditionOrder rowTop (a.actionGraph c t)
t.carrier b.carrier b.conditionOrder (d.actionGraph b t)
(supportBasisCode b t d.principalFix) T T2 level prior) stages history →
let packed := M.structuralRefinementHistory t.carrier b.carrier b.conditionOrder
(supportBasisCode b t d.principalFix) T T2 stages history
letI := M.refinementHistoryPartialOrder c.conditionOrder_code T stages packed
let Q := M.refinementHistoryForcing c.carrier T stages packed
let B := M.groundSmallSubsets (stageOneAssignments (refinementDataHistoryIndices stages packed) T) ZFSet.omega
let shape := M.refinementHistoryForcing_shape c.carrier T stages packed
let row := flatCarrierRow c Q B shape
let data := flatCarrierData c Q B shape
∀ seq : Nat → SetMembers Q, groundSequenceGraph seq ∈ M.sets → Antitone seq →
∃ q : SetMembers (M.smallPartialFunctions X (natSetCode 2) ZFSet.omega),
(∀ n, q ≤ row (seq n)) ∧ ∃ p : SetMembers Q,
SetMembers.raw p = ZFSet.pair (c.encode q)
(stageOneAssignmentUnion (fun n => SetMembers.raw (data (seq n)))) ∧
(∀ n, p ≤ seq n) ∧
SetMembers.raw (data p) = stageOneAssignmentUnion (fun n => SetMembers.raw (data (seq n))) ∧
stageOneAssignmentUnion (fun n => SetMembers.raw (data (seq n))) ∈ M.sets ∧
∀ z, z ∈ SetMembers.raw (data p) ↔ ∃ n, z ∈ SetMembers.raw (data (seq n)) := by
let := M.smallFunctionOrder X (natSetCode 2) ZFSet.omega
let c := SetMembers.presentation (M.smallPartialFunctions X (natSetCode 2) ZFSet.omega)
dsimp only
intro a T T2 rowTop action group nodes nodeOrder nodeAction nodeMaps supports hT hT2
valid1 valid2 closed1 closed2 children1 children2 topMem topRows stages history hs hh ordinal trace
have two := M.natSetCode_mem separation infinite 2
have carrier := M.countablePartialFunctions_mem pairing unions powers separation infinite hX two
have order := M.countableFunctionOrder_mem pairing unions powers separation infinite hX two
have rowClosed := M.countable_function_internally_closed pairing unions powers separation infinite choice hX two
exact M.semanticStructuralTrace_exact_assignment_union c a t b d base directed T T2 rowTop
pairing unions powers separation collection carrier order action group nodes nodeOrder nodeAction
nodeMaps supports hT hT2 (M.omega_mem separation infinite) valid1 valid2 closed1 closed2
children1 children2 topMem topRows hs hh ordinal trace infinite choice rowClosed
end PartitionPilot.Forcing.TransitiveGround
end
end OAI
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