@[reducible, inline]
Equations
- LO.Modal.Hilbert.Kripke.canonicalFrame H = LO.Modal.Kripke.Frame.mk (LO.Modal.MCT H) fun (Ω₁ Ω₂ : LO.Modal.MCT H) => □''⁻¹Ω₁.theory ⊆ Ω₂.theory
@[reducible, inline]
Equations
- One or more equations did not get rendered due to their size.
@[reducible]
instance
LO.Modal.Hilbert.Kripke.instSemanticsFormulaNatWorldCanonicalModel
{H : Hilbert ℕ}
[H.IsNormal]
[H.Consistent]
:
Semantics (Formula ℕ) (canonicalModel H).World
theorem
LO.Modal.Hilbert.Kripke.truthlemma
{H : Hilbert ℕ}
[H.IsNormal]
[H.Consistent]
{φ : Formula ℕ}
{Ω : (canonicalModel H).World}
:
theorem
LO.Modal.Hilbert.Kripke.iff_valid_on_canonicalModel_deducible
{H : Hilbert ℕ}
[H.IsNormal]
[H.Consistent]
{φ : Formula ℕ}
:
canonicalModel H ⊧ φ ↔ H ⊢! φ
theorem
LO.Modal.Hilbert.Kripke.realize_axiomset_of_self_canonicalModel
{H : Hilbert ℕ}
[H.IsNormal]
[H.Consistent]
:
canonicalModel H ⊧* H.axioms
theorem
LO.Modal.Hilbert.Kripke.realize_theory_of_self_canonicalModel
{H : Hilbert ℕ}
[H.IsNormal]
[H.Consistent]
:
theorem
LO.Modal.Hilbert.Kripke.complete_of_canonical
{H : Hilbert ℕ}
[H.IsNormal]
[H.Consistent]
{φ : Formula ℕ}
{C : Kripke.FrameClass}
(hFC : canonicalFrame H ∈ C)
:
theorem
LO.Modal.Hilbert.Kripke.instCompleteOfCanonical
{H : Hilbert ℕ}
[H.IsNormal]
[H.Consistent]
{C : Kripke.FrameClass}
(hC : canonicalFrame H ∈ C)
:
Complete H C