File size: 2,573 Bytes
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namespace OAI
namespace PartitionPilot.Forcing.SetFormula
def checkStateFormula : SetFormula 3 :=
ex (ex (.conj (orderedPairFormula 2 1 0) (checkStepFormula 4 3 1 0)))
def checkCertificateFormula : SetFormula 2 := postfixedFormula checkStateFormula
def uniformCheckFormula : SetFormula 3 :=
ex (.conj (binaryApplication checkCertificateFormula Fin.elim0 0 3) (pairMembershipFormula 1 2 0))
end PartitionPilot.Forcing.SetFormula
namespace PartitionPilot.Forcing.TransitiveGround
open SetRealization SetFormula
universe u
variable (M : TransitiveGround.{u})
theorem realize_checkState
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(top z R : ZFSet.{u}) (htop : top ∈ M.sets) (hz : z ∈ M.sets) (hr : R ∈ M.sets) :
Realize M.sets checkStateFormula (extend z (extend R (extend top Fin.elim0))) ↔
∃ x ∈ M.sets, ∃ y ∈ M.sets, z = ZFSet.pair x y ∧ M.RawCheckStep top R x y := by
rw [checkStateFormula, realize_ex]
apply exists_congr
intro x
apply and_congr_right
intro hx
rw [realize_ex]
apply exists_congr
intro y
apply and_congr_right
intro hy
let e := extend y (extend x (extend z (extend R (extend top Fin.elim0))))
change (Realize M.sets _ e ∧ Realize M.sets _ e) ↔ _
rw [M.realize_orderedPairFormula pairing 2 1 0 e hz hx hy,
M.realize_checkStepFormula pairing 4 3 1 0 e htop hr hx hy]
rfl
theorem checkCertificate_sound {P : Type u} (s : Presentation P) (one : P)
(pairing : Realize M.sets SetFormula.pairing Fin.elim0) (hone : s.encode one ∈ M.sets)
{R : ZFSet.{u}} (hr : R ∈ M.sets)
(certificate : Realize M.sets checkCertificateFormula (extend R (extend (s.encode one) Fin.elim0))) :
∀ x ∈ M.sets, ∀ y ∈ M.sets, ZFSet.pair x y ∈ R → y = (Name.check s one x).val := by
have post : ∀ x ∈ M.sets, ∀ y ∈ M.sets, ZFSet.pair x y ∈ R →
M.RawCheckStep (s.encode one) R x y := by
intro x _ y _ rel
have step := (realize_imp M.sets _ _ _).1 (certificate _ (M.transitive hr rel)) rel
obtain ⟨v, _, w, _, eq, raw⟩ := (M.realize_checkState pairing (s.encode one) _ R
hone (M.transitive hr rel) hr).1 step
obtain ⟨rfl, rfl⟩ := ZFSet.pair_inj.1 eq
exact raw
intro x
induction x using ZFSet.inductionOn with
| h x ih =>
intro hx y hy rel
exact M.rawCheckStep_eq s one (post x hx y hy rel)
(fun v hv w hw relation => ih v hv (M.transitive hx hv) w hw relation)
end PartitionPilot.Forcing.TransitiveGround
end OAI
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