Documentation

Foundation.Modal.Kripke.Completeness

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    @[simp]
    theorem LO.Modal.Kripke.CanonicalFrame.rel_def_box {α : Type u} {Λ : LO.Modal.Hilbert α} [Nonempty (LO.Modal.MCT Λ)] {Ω₁ : (LO.Modal.Kripke.CanonicalFrame Λ).World} {Ω₂ : (LO.Modal.Kripke.CanonicalFrame Λ).World} :
    Ω₁ Ω₂ ∀ {p : LO.Modal.Formula α}, p Ω₁.theoryp Ω₂.theory
    theorem LO.Modal.Kripke.CanonicalFrame.multirel_def_multibox {α : Type u} [DecidableEq α] {Λ : LO.Modal.Hilbert α} [Λ.IsNormal] [Nonempty (LO.Modal.MCT Λ)] {Ω₁ : (LO.Modal.Kripke.CanonicalFrame Λ).World} {Ω₂ : (LO.Modal.Kripke.CanonicalFrame Λ).World} {n : } :
    Ω₁ ≺^[n] Ω₂ ∀ {p : LO.Modal.Formula α}, □^[n]p Ω₁.theoryp Ω₂.theory
    theorem LO.Modal.Kripke.CanonicalFrame.multirel_def_multibox' {α : Type u} [DecidableEq α] {Λ : LO.Modal.Hilbert α} [Λ.IsNormal] [Nonempty (LO.Modal.MCT Λ)] {Ω₁ : (LO.Modal.Kripke.CanonicalFrame Λ).World} {Ω₂ : (LO.Modal.Kripke.CanonicalFrame Λ).World} {n : } :
    Ω₁ ≺^[n] Ω₂ ∀ {p : LO.Modal.Formula α}, p □''⁻¹^[n]Ω₁.theoryp Ω₂.theory
    theorem LO.Modal.Kripke.CanonicalFrame.multirel_def_multidia {α : Type u} [DecidableEq α] {Λ : LO.Modal.Hilbert α} [Λ.IsNormal] [Nonempty (LO.Modal.MCT Λ)] {Ω₁ : (LO.Modal.Kripke.CanonicalFrame Λ).World} {Ω₂ : (LO.Modal.Kripke.CanonicalFrame Λ).World} {n : } :
    Ω₁ ≺^[n] Ω₂ ∀ {p : LO.Modal.Formula α}, p Ω₂.theory◇^[n]p Ω₁.theory
    theorem LO.Modal.Kripke.CanonicalFrame.rel_def_dia {α : Type u} [DecidableEq α] {Λ : LO.Modal.Hilbert α} [Λ.IsNormal] [Nonempty (LO.Modal.MCT Λ)] {Ω₁ : (LO.Modal.Kripke.CanonicalFrame Λ).World} {Ω₂ : (LO.Modal.Kripke.CanonicalFrame Λ).World} :
    Ω₁ Ω₂ ∀ {p : LO.Modal.Formula α}, p Ω₂.theoryp Ω₁.theory
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      @[reducible]
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      • LO.Modal.Kripke.CanonicalModel.instSemanticsFormulaWorld = LO.Modal.Formula.Kripke.Satisfies.semantics
      theorem LO.Modal.Kripke.truthlemma {α : Type u} [DecidableEq α] {Λ : LO.Modal.Hilbert α} [Λ.IsNormal] [Nonempty (LO.Modal.MCT Λ)] {p : LO.Modal.Formula α} {Ω : (LO.Modal.Kripke.CanonicalModel Λ).World} :
      Ω p p Ω.theory
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