@[reducible, inline]
abbrev
LO.Modal.Kripke.CanonicalFrame
{α : Type u}
(Λ : LO.Modal.Hilbert α)
[Nonempty (LO.Modal.MCT Λ)]
:
Equations
- LO.Modal.Kripke.CanonicalFrame Λ = LO.Kripke.Frame.mk (LO.Modal.MCT Λ) fun (Ω₁ Ω₂ : LO.Modal.MCT Λ) => □''⁻¹Ω₁.theory ⊆ Ω₂.theory
Instances For
@[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}
:
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 : ℕ}
:
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 : ℕ}
:
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 : ℕ}
:
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}
:
@[reducible, inline]
abbrev
LO.Modal.Kripke.CanonicalModel
{α : Type u}
(Λ : LO.Modal.Hilbert α)
[Nonempty (LO.Modal.MCT Λ)]
:
Equations
- LO.Modal.Kripke.CanonicalModel Λ = { Frame := LO.Modal.Kripke.CanonicalFrame Λ, Valuation := fun (Ω : (LO.Modal.Kripke.CanonicalFrame Λ).World) (a : α) => LO.Modal.Formula.atom a ∈ Ω.theory }
Instances For
@[reducible]
instance
LO.Modal.Kripke.CanonicalModel.instSemanticsFormulaWorld
{α : Type u}
{Λ : LO.Modal.Hilbert α}
[Nonempty (LO.Modal.MCT Λ)]
:
LO.Semantics (LO.Modal.Formula α) (LO.Modal.Kripke.CanonicalModel Λ).World
Equations
- 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}
:
theorem
LO.Modal.Kripke.iff_valid_on_canonicalModel_deducible
{α : Type u}
[DecidableEq α]
{Λ : LO.Modal.Hilbert α}
[Λ.IsNormal]
[Nonempty (LO.Modal.MCT Λ)]
{p : LO.Modal.Formula α}
:
LO.Modal.Kripke.CanonicalModel Λ ⊧ p ↔ Λ ⊢! p
theorem
LO.Modal.Kripke.realize_axiomset_of_self_canonicalModel
{α : Type u}
[DecidableEq α]
{Λ : LO.Modal.Hilbert α}
[Λ.IsNormal]
[Nonempty (LO.Modal.MCT Λ)]
:
theorem
LO.Modal.Kripke.realize_theory_of_self_canonicalModel
{α : Type u}
[DecidableEq α]
{Λ : LO.Modal.Hilbert α}
[Λ.IsNormal]
[Nonempty (LO.Modal.MCT Λ)]
:
theorem
LO.Modal.Kripke.complete_of_mem_canonicalFrame
{α : Type u}
[DecidableEq α]
{Λ : LO.Modal.Hilbert α}
[Λ.IsNormal]
{p : LO.Modal.Formula α}
[Nonempty (LO.Modal.MCT Λ)]
{𝔽 : LO.Kripke.FrameClass}
(hFC : LO.Modal.Kripke.CanonicalFrame Λ ∈ 𝔽)
:
instance
LO.Modal.Kripke.instComplete_of_mem_canonicalFrame
{α : Type u}
[DecidableEq α]
{Λ : LO.Modal.Hilbert α}
[Λ.IsNormal]
[Nonempty (LO.Modal.MCT Λ)]
(𝔽 : LO.Kripke.FrameClass)
(hFC : LO.Modal.Kripke.CanonicalFrame Λ ∈ 𝔽)
:
LO.Complete Λ 𝔽#α
Equations
- ⋯ = ⋯
Equations
- ⋯ = ⋯