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e3e0a0d | 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 | import OAI.SetTheory.PartitionPrinciple.Geometry.GroundContainmentOrders
namespace OAI
section
namespace PartitionPilot.Forcing.BoundedFormula
open SetFormula
universe u
def upwardDirectedFormula {n} (l r : Fin n) : BoundedFormula n :=
.allMem l (.allMem l.succ (exMem l.succ.succ
(.conj (pairMembershipFormula 2 0 r.succ.succ.succ)
(pairMembershipFormula 1 0 r.succ.succ.succ))))
theorem truth_upwardDirectedFormula {n} (l r : Fin n) (env : Fin n → ZFSet.{u}) :
Truth (upwardDirectedFormula l r) env ↔
∀ x ∈ env l, ∀ y ∈ env l, ∃ z ∈ env l,
ZFSet.pair x z ∈ env r ∧ ZFSet.pair y z ∈ env r := by
simp only [upwardDirectedFormula, Truth, truth_exMem, truth_pairMembershipFormula]
rfl
end PartitionPilot.Forcing.BoundedFormula
namespace PartitionPilot.Forcing.TransitiveGround
open SetRealization SetFormula
universe u
variable (M : TransitiveGround.{u})
theorem realize_upwardDirectedCode {L R : ZFSet.{u}} (hL : L ∈ M.sets) (hR : R ∈ M.sets) :
Realize M.sets (BoundedFormula.upwardDirectedFormula 0 1).toFormula
(extend L (extend R Fin.elim0)) ↔
∀ x ∈ L, ∀ y ∈ L, ∃ z ∈ L, ZFSet.pair x z ∈ R ∧ ZFSet.pair y z ∈ R :=
(BoundedFormula.realize_toFormula M _ _
(Fin.cases hL (Fin.cases hR (fun j => Fin.elim0 j)))).trans
(BoundedFormula.truth_upwardDirectedFormula 0 1 _)
theorem decoded_upward_directed {L R : ZFSet.{u}} (order : PartialOrderCode L R)
(hL : L ∈ M.sets) (hR : R ∈ M.sets)
(laws : Realize M.sets (BoundedFormula.upwardDirectedFormula 0 1).toFormula
(extend L (extend R Fin.elim0))) :
letI := order.toPartialOrder
∀ C D : SetMembers L, ∃ E, C ≤ E ∧ D ≤ E := by
let := order.toPartialOrder
intro C D
obtain ⟨z, hz, cz, dz⟩ := (M.realize_upwardDirectedCode hL hR).1 laws
_ (SetMembers.raw_mem C) _ (SetMembers.raw_mem D)
refine ⟨SetMembers.ofMem z hz, ?_, ?_⟩
· change ZFSet.pair (SetMembers.raw C) (SetMembers.raw (SetMembers.ofMem z hz)) ∈ R
rw [SetMembers.raw_ofMem]
exact cz
· change ZFSet.pair (SetMembers.raw D) (SetMembers.raw (SetMembers.ofMem z hz)) ∈ R
rw [SetMembers.raw_ofMem]
exact dz
end PartitionPilot.Forcing.TransitiveGround
end
end OAI
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