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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 | import OAI.SetTheory.PartitionPrinciple.Transfinite.GroundOrdinalEnumerationFamilies
namespace OAI
namespace PartitionPilot.Forcing.TransitiveGround
open SetRealization SetFormula
universe u
variable (M : TransitiveGround.{u}) {G P L : Type u}
variable [Group G] [PartialOrder P] [PartialOrder L] [OrderBot L]
variable (a : PosetAction G P) (s : Presentation P) (t : Presentation G)
variable (b : Presentation L) (d : PosetAction G L)
variable (directed : ∀ x y : L, ∃ z, x ≤ z ∧ y ≤ z)
theorem exists_groundSupportEnumerationChoices
(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) (choice : Realize M.sets SetFormula.choice Fin.elim0)
(carrier : s.carrier ∈ M.sets) (group : t.carrier ∈ M.sets)
(nodes : b.carrier ∈ M.sets) (nodeOrder : b.conditionOrder ∈ M.sets)
(supports : supportBasisCode b t d.principalFix ∈ M.sets)
(maps : a.conditionMapFamily s t ∈ M.sets)
(W : ZFSet.{u}) (hW : W ∈ M.sets) (valid : ∀ z ∈ W, IsName s z)
(symmetric : ∀ tau : Name s, tau.val ∈ W →
HereditarilySymmetric a s (d.principalSupportFamily ⊥ directed) tau) :
∃ supportChoice ∈ M.sets, ∃ B ∈ M.sets, ∃ enumerationChoice ∈ M.sets,
ZFSet.IsFunc W b.carrier supportChoice ∧ ZFSet.IsFunc b.carrier B enumerationChoice ∧
let selected := relationCompose W b.carrier B supportChoice enumerationChoice
selected ∈ M.sets ∧ ZFSet.IsFunc W B selected ∧
∀ (tau : Name s) z, ZFSet.pair tau.val z ∈ selected →
∃ D : L, ZFSet.pair tau.val (b.encode D) ∈ supportChoice ∧
ZFSet.pair (b.encode D) z ∈ enumerationChoice ∧
d.principalFix D ≤ a.stabilizer s tau ∧
∃ delta ∈ M.sets, ∃ f ∈ M.sets, z = ZFSet.pair delta f ∧
ordinalSetProperty delta ∧
InjectionCode delta (relationFiber b.nonbottom.carrier
(relationConverse b.carrier b.carrier b.conditionOrder) (b.encode D)) f ∧
∀ v : L, v ≠ ⊥ → v ≤ D → ∃ i ∈ delta, ZFSet.pair i (b.encode v) ∈ f := by
have hNonbottom := M.nonbottom_carrier_mem b separation nodes
let reverse := relationConverse b.carrier b.carrier b.conditionOrder
have hReverse := M.relationConverse_mem pairing unions powers separation nodes nodes nodeOrder
let fibers := M.relationFiberFamily b.carrier b.nonbottom.carrier reverse
have hFibers : fibers ∈ M.sets := M.relationFiberFamily_mem pairing unions powers separation
nodes hNonbottom hReverse
have fibersFunction := M.relationFiberFamily_function separation nodes hNonbottom hReverse
have hPower := M.groundPower_mem powers hNonbottom
obtain ⟨B, hB, enumerations, hEnumerations, enumFunction, enumValues⟩ :=
M.exists_groundOrdinalEnumerationFamily pairing unions powers separation collection choice
(M.groundPower b.nonbottom.carrier) hPower
let enumerationChoice := relationCompose b.carrier (M.groundPower b.nonbottom.carrier) B fibers enumerations
have enumerationGround : enumerationChoice ∈ M.sets := M.relationCompose_mem pairing unions powers
separation nodes hPower hB hFibers hEnumerations
have enumerationFunction : ZFSet.IsFunc b.carrier B enumerationChoice :=
relationCompose_function_of_subset fibersFunction enumFunction (fun _ h => h)
obtain ⟨supportChoice, hSupportChoice, supportFunction, supported⟩ :=
M.exists_groundNameSupportChoice a s t b d ⊥ directed pairing unions powers separation collection
choice carrier group nodes supports maps W hW valid symmetric
refine ⟨supportChoice, hSupportChoice, B, hB, enumerationChoice, enumerationGround,
supportFunction, enumerationFunction, ?_⟩
dsimp only
refine ⟨M.relationCompose_mem pairing unions powers separation hW nodes hB hSupportChoice enumerationGround,
relationCompose_function_of_subset supportFunction enumerationFunction (fun _ h => h), ?_⟩
intro tau z entry
obtain ⟨_, _, v, hv, supportEdge, enumEdge⟩ := (pair_mem_relationCompose _ _ _ _ _ _ _).1 entry
obtain ⟨D, rfl⟩ := (b.mem_carrier v).1 hv
have fixes := (supported tau D supportEdge).2
obtain ⟨_, _, ideal, _, idealEdge, chosen⟩ := (pair_mem_relationCompose _ _ _ _ _ _ _).1 enumEdge
have idealEq := ((M.pair_mem_relationFiberFamily b.carrier b.nonbottom.carrier reverse
(b.encode D) ideal).1 idealEdge).2.2
obtain ⟨delta, hDelta, f, hf, shape, ordinal, injection, onto⟩ := enumValues ideal z chosen
rw [idealEq] at injection onto
refine ⟨D, supportEdge, enumEdge, fixes, delta, hDelta, f, hf, shape, ordinal, injection, ?_⟩
intro v nonzero below
have member : b.encode v ∈ relationFiber b.nonbottom.carrier reverse (b.encode D) := by
apply (mem_relationFiber _ _ _ _).2
refine ⟨?_, (pair_mem_relationConverse _ _ _ _ _).2
⟨b.encode_mem v, b.encode_mem D, (b.pair_mem_conditionOrder v D).2 below⟩⟩
exact (b.nonbottom.mem_carrier _).2 ⟨⟨v, nonzero⟩, rfl⟩
exact onto (b.encode v) member
end PartitionPilot.Forcing.TransitiveGround
end OAI
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