| import OAI.SetTheory.PartitionPrinciple.Definability.RawCheckFormulas |
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| namespace OAI |
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| namespace PartitionPilot.Forcing.SetFormula |
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| def checkStateFormula : SetFormula 3 := |
| ex (ex (.conj (orderedPairFormula 2 1 0) (checkStepFormula 4 3 1 0))) |
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| def checkCertificateFormula : SetFormula 2 := postfixedFormula checkStateFormula |
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| def uniformCheckFormula : SetFormula 3 := |
| ex (.conj (binaryApplication checkCertificateFormula Fin.elim0 0 3) (pairMembershipFormula 1 2 0)) |
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| end PartitionPilot.Forcing.SetFormula |
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|
| namespace PartitionPilot.Forcing.TransitiveGround |
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|
| 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 |
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| 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) |
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| end PartitionPilot.Forcing.TransitiveGround |
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| end OAI |
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|
