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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 | import OAI.SetTheory.PartitionPrinciple.Geometry.GroundBinaryRowRecordedPreparationBounds
namespace OAI
section
namespace PartitionPilot.Forcing.TransitiveGround
open SetRealization SetFormula
universe u
variable (M : TransitiveGround.{u}) {L : Type u} [PartialOrder L] [OrderBot L]
variable (s : Presentation L) (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 (choice : Realize M.sets SetFormula.choice Fin.elim0)
variable (hL : s.carrier ∈ M.sets) (order : s.conditionOrder ∈ M.sets) (hC : C ∈ M.sets)
variable (directed : ∀ x y : L, ∃ z, x ≤ z ∧ y ≤ z)
variable (countable_bound : ∀ W ∈ M.sets, W ⊆ s.carrier → M.groundEmbeddable W ZFSet.omega →
∃ D : L, ∀ x : L, s.encode x ∈ W → x ≤ D)
include pairing unions powers separation collection infinite choice hL order hC directed countable_bound
theorem binaryRowStructuralCondition_nameSupportBound {mu : ZFSet.{u}} (hmu : mu ∈ M.sets)
(one : SetMembers (M.smallPartialFunctions
(ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega)) :
let c := SetMembers.presentation (M.smallPartialFunctions
(ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega)
let rows := M.binaryRowNameGraph s.nonbottom.carrier C one
let S := M.rowPositionSequences s.carrier s.conditionOrder s.nonbottom.carrier mu
let T := rowTupleTerms (c.encode one) mu s.nonbottom.carrier (ZFSet.sUnion (ZFSet.sUnion rows)) rows S
∀ valid : ∀ z ∈ T, IsName c z,
let lifted := Name.renamedFamily c.pairEmpty id T valid
let tuples := liftedRowTupleFamily c (c.encode one) mu s.nonbottom.carrier
(ZFSet.sUnion (ZFSet.sUnion rows)) rows S valid
∀ (_hLifted : lifted ∈ M.sets) (G basis T2 stages history : ZFSet.{u}),
G ∈ M.sets → basis ∈ M.sets → T2 ∈ M.sets → stages ∈ M.sets → history ∈ M.sets →
let I := refinementDataHistoryIndices stages
(M.structuralRefinementHistory G s.carrier s.conditionOrder basis lifted T2 stages history)
let anchors := refinementIndexFieldGraph I s.carrier 2
∀ (p : SetMembers (M.smallPartialFunctions
(ZFSet.prod s.nonbottom.carrier C) (natSetCode 2) ZFSet.omega)) (A : ZFSet.{u}),
ZFSet.pair (c.encode p) A ∈
M.structuralForcingCarrier c.carrier G s.carrier s.conditionOrder basis lifted T2 stages history →
∀ {G P : Type u} [Group G] [PartialOrder P] [OrderTop P]
(lower : Presentation P) (action : PosetAction G P) (nodeAction : PosetAction G L)
(base : L) (names : Fin 2 → Name lower),
(∀ i, HereditarilySymmetric action lower (nodeAction.principalSupportFamily base directed) (names i)) →
∀ E : L, ∃ D : L,
(∀ x : L, s.encode x ∈ rowUsedRows s.nonbottom.carrier C (natSetCode 2) (SetMembers.raw p) → x ≤ D) ∧
(∀ z ∈ A, ZFSet.pair z (s.encode D) ∈
stageOneDataBounds A I lifted S s.carrier mu s.nonbottom.carrier s.conditionOrder anchors tuples) ∧
E ≤ D ∧ ∀ i, nodeAction.principalFix D ≤ action.stabilizer lower (names i) := by
intro c rows S T valid lifted tuples hLifted Gcode basis T2 stages history
hG hb hT2 hs hh I anchors row A member G P instG instP instTop lower action nodeAction base names symmetric E
obtain ⟨first, firstSupport⟩ := (symmetric 0).symmetric
obtain ⟨second, secondSupport⟩ := (symmetric 1).symmetric
obtain ⟨common, firstBelow, secondBelow⟩ := directed first second
let extra : ZFSet.{u} := {s.encode E, s.encode common}
have hE := M.transitive hL (s.encode_mem E)
have hCommon := M.transitive hL (s.encode_mem common)
have extraGround : extra ∈ M.sets := M.unorderedPair_mem pairing hE hCommon
have extraSubset : extra ⊆ s.carrier := by
intro x hx
rcases ZFSet.mem_pair.1 hx with rfl | rfl
· exact s.encode_mem E
· exact s.encode_mem common
have extraSmall : M.groundEmbeddable extra ZFSet.omega :=
M.unorderedPair_countable pairing unions powers separation infinite hE hCommon
obtain ⟨D, rowBound, assignmentBound, extraBound⟩ :=
M.binaryRowStructuralConditionDataBound s C pairing unions powers separation collection
infinite choice hL order hC directed countable_bound hmu one valid hLifted
Gcode basis T2 stages history hG hb hT2 hs hh row A member extra extraGround extraSubset extraSmall
have commonBelow : common ≤ D := extraBound common (ZFSet.mem_pair.2 (Or.inr rfl))
have atCommon : ∀ i : Fin 2, nodeAction.principalFix common ≤ action.stabilizer lower (names i) :=
Fin.cases ((nodeAction.principalFix_antitone firstBelow).trans firstSupport)
(Fin.cases ((nodeAction.principalFix_antitone secondBelow).trans secondSupport) (fun j => Fin.elim0 j))
exact ⟨D, rowBound, assignmentBound, extraBound E (ZFSet.mem_pair.2 (Or.inl rfl)),
fun i => (nodeAction.principalFix_antitone commonBelow).trans (atCommon i)⟩
end PartitionPilot.Forcing.TransitiveGround
end
end OAI
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