Documentation

Foundation.Modal.Kripke.Completeness

@[reducible, inline]
abbrev LO.Modal.Hilbert.Kripke.canonicalFrame (H : Hilbert ) [H.IsNormal] [H.Consistent] :
Equations
@[simp]
theorem LO.Modal.Hilbert.Kripke.canonicalFrame.rel_def_box {H : Hilbert } [H.IsNormal] [H.Consistent] {Ω₁ Ω₂ : (canonicalFrame H).World} :
Ω₁ Ω₂ ∀ {φ : Formula }, φ Ω₁.theoryφ Ω₂.theory
theorem LO.Modal.Hilbert.Kripke.canonicalFrame.multirel_def_multibox {H : Hilbert } [H.IsNormal] [H.Consistent] {Ω₁ Ω₂ : (canonicalFrame H).World} {n : } :
Ω₁ ≺^[n] Ω₂ ∀ {φ : Formula }, □^[n]φ Ω₁.theoryφ Ω₂.theory
theorem LO.Modal.Hilbert.Kripke.canonicalFrame.multirel_def_multibox' {H : Hilbert } [H.IsNormal] [H.Consistent] {Ω₁ Ω₂ : (canonicalFrame H).World} {n : } :
Ω₁ ≺^[n] Ω₂ ∀ {φ : Formula }, φ □''⁻¹^[n]Ω₁.theoryφ Ω₂.theory
theorem LO.Modal.Hilbert.Kripke.canonicalFrame.multirel_def_multidia {H : Hilbert } [H.IsNormal] [H.Consistent] {Ω₁ Ω₂ : (canonicalFrame H).World} {n : } :
Ω₁ ≺^[n] Ω₂ ∀ {φ : Formula }, φ Ω₂.theory◇^[n]φ Ω₁.theory
theorem LO.Modal.Hilbert.Kripke.canonicalFrame.rel_def_dia {H : Hilbert } [H.IsNormal] [H.Consistent] {Ω₁ Ω₂ : (canonicalFrame H).World} :
Ω₁ Ω₂ ∀ {φ : Formula }, φ Ω₂.theoryφ Ω₁.theory
@[reducible, inline]
abbrev LO.Modal.Hilbert.Kripke.canonicalModel (H : Hilbert ) [H.IsNormal] [H.Consistent] :
Equations
  • One or more equations did not get rendered due to their size.
theorem LO.Modal.Hilbert.Kripke.truthlemma {H : Hilbert } [H.IsNormal] [H.Consistent] {φ : Formula } {Ω : (canonicalModel H).World} :
Ω φ φ Ω.theory
theorem LO.Modal.Hilbert.Kripke.complete_of_canonical {H : Hilbert } [H.IsNormal] [H.Consistent] {φ : Formula } {C : Kripke.FrameClass} (hFC : canonicalFrame H C) :
C φH ⊢! φ
theorem LO.Modal.Hilbert.Kripke.instCompleteOfCanonical {H : Hilbert } [H.IsNormal] [H.Consistent] {C : Kripke.FrameClass} (hC : canonicalFrame H C) :