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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 | import OAI.SetTheory.PartitionPrinciple.Geometry.GrowthLayerSourceCongruence
namespace OAI
section
namespace PartitionPilot.Forcing
open SetFormula
universe u
structure CountableLayerCode (L birth t D N injection : ZFSet.{u}) : Prop where
small : InjectionCode D N injection
cover : ∀ x ∈ L, ZFSet.pair x t ∈ birth → x ∈ D
namespace BoundedFormula
def countableLayerFormula {n} (L birth t D N injection : Fin n) : BoundedFormula n :=
.conj (injectionFormula D N injection)
(.allMem L (imp (pairMembershipFormula 0 t.succ birth.succ) (.mem 0 D.succ)))
theorem truth_countableLayerFormula {n} (L birth t D N injection : Fin n)
(env : Fin n → ZFSet.{u}) :
Truth (countableLayerFormula L birth t D N injection) env ↔
CountableLayerCode (env L) (env birth) (env t) (env D) (env N) (env injection) := by
simp only [countableLayerFormula, Truth, truth_injectionFormula, truth_imp,
truth_pairMembershipFormula]
exact ⟨fun h => ⟨h.1, h.2⟩, fun h => ⟨h.small, h.cover⟩⟩
theorem realize_countableLayerFormula (M : TransitiveGround.{u}) {n}
(L birth t D N injection : Fin n) (env : Fin n → ZFSet.{u})
(members : ∀ i, env i ∈ M.sets) :
Realize M.sets (countableLayerFormula L birth t D N injection).toFormula env ↔
CountableLayerCode (env L) (env birth) (env t) (env D) (env N) (env injection) :=
(realize_toFormula M _ env members).trans (truth_countableLayerFormula L birth t D N injection env)
end BoundedFormula
namespace CountableLayerCode
def toLayer {M : TransitiveGround.{u}} {L r μ birth t D N injection : ZFSet.{u}}
(code : CountableLayerCode L birth t D N injection)
(Space sourceOrder sourceMeet : ZFSet.{u} → ZFSet.{u})
(hD : D ∈ M.sets) (hInjection : injection ∈ M.sets) (omega : N = ZFSet.omega) :
CodedGrowthLayer M Space sourceOrder sourceMeet L r μ birth t :=
.countablePoints D hD ⟨injection, hInjection, omega ▸ code.small⟩ code.cover
theorem exists_witness {M : TransitiveGround.{u}} {L birth t D : ZFSet.{u}}
(small : M.groundEmbeddable D ZFSet.omega)
(cover : ∀ x ∈ L, ZFSet.pair x t ∈ birth → x ∈ D) :
∃ injection ∈ M.sets, CountableLayerCode L birth t D ZFSet.omega injection := by
obtain ⟨injection, hInjection, code⟩ := small
exact ⟨injection, hInjection, code, cover⟩
end CountableLayerCode
end PartitionPilot.Forcing
end
end OAI
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