Documentation

Foundation.Modal.PLoN.Completeness

@[reducible, inline]
abbrev LO.Modal.PLoN.CanonicalFrame {α : Type u} (H : Hilbert α) [Nonempty (MCT H)] :
Equations
@[reducible, inline]
abbrev LO.Modal.PLoN.CanonicalModel {α : Type u} (H : Hilbert α) [Nonempty (MCT H)] :
Equations
theorem LO.Modal.PLoN.truthlemma {α : Type u} [DecidableEq α] {H : Hilbert α} [Nonempty (MCT H)] [H.HasNecessitation] {φ : Formula α} {Ω : (CanonicalModel H).World} :
Ω φ φ Ω.theory
theorem LO.Modal.PLoN.complete_of_mem_canonicalFrame {α : Type u} [DecidableEq α] {H : Hilbert α} [Nonempty (MCT H)] [H.HasNecessitation] {φ : Formula α} {𝔽 : FrameClass α} (hFC : CanonicalFrame H 𝔽) :
𝔽 φH ⊢! φ
theorem LO.Modal.PLoN.instComplete_of_mem_canonicalFrame {α : Type u} [DecidableEq α] {H : Hilbert α} [Nonempty (MCT H)] [H.HasNecessitation] {𝔽 : FrameClass α} (hFC : CanonicalFrame H 𝔽) :
Complete H 𝔽