Equations
- LO.Modal.NNFormula.Kripke.Satisfies M x (LO.Modal.NNFormula.atom a) = M.Val x a
- LO.Modal.NNFormula.Kripke.Satisfies M x (LO.Modal.NNFormula.natom a) = ¬M.Val x a
- LO.Modal.NNFormula.Kripke.Satisfies M x LO.Modal.NNFormula.verum = True
- LO.Modal.NNFormula.Kripke.Satisfies M x LO.Modal.NNFormula.falsum = False
- LO.Modal.NNFormula.Kripke.Satisfies M x (φ.or ψ) = (LO.Modal.NNFormula.Kripke.Satisfies M x φ ∨ LO.Modal.NNFormula.Kripke.Satisfies M x ψ)
- LO.Modal.NNFormula.Kripke.Satisfies M x (φ.and ψ) = (LO.Modal.NNFormula.Kripke.Satisfies M x φ ∧ LO.Modal.NNFormula.Kripke.Satisfies M x ψ)
- LO.Modal.NNFormula.Kripke.Satisfies M x φ.box = ∀ (y : M.World), x ≺ y → LO.Modal.NNFormula.Kripke.Satisfies M y φ
- LO.Modal.NNFormula.Kripke.Satisfies M x φ.dia = ∃ (y : M.World), x ≺ y ∧ LO.Modal.NNFormula.Kripke.Satisfies M y φ
Equations
- LO.Modal.NNFormula.Kripke.Satisfies.semantics = { Realize := fun (x : M.World) => LO.Modal.NNFormula.Kripke.Satisfies M x }
theorem
LO.Modal.NNFormula.Kripke.Satisfies.iff_models
{φ : NNFormula ℕ}
{M : Kripke.Model}
{x : M.World}
:
@[simp]
@[simp]
theorem
LO.Modal.NNFormula.Kripke.Satisfies.box_def
{φ : NNFormula ℕ}
{M : Kripke.Model}
{x : M.World}
:
theorem
LO.Modal.NNFormula.Kripke.Satisfies.neg_def
{φ : NNFormula ℕ}
{M : Kripke.Model}
{x : M.World}
:
theorem
LO.Modal.NNFormula.Kripke.Satisfies.imp_def
{φ ψ : NNFormula ℕ}
{M : Kripke.Model}
{x : M.World}
:
instance
LO.Modal.NNFormula.Kripke.Satisfies.instTarskiWorldNat
{M : Kripke.Model}
:
Semantics.Tarski M.World
Equations
- LO.Modal.NNFormula.Kripke.ValidOnModel M φ = ∀ (x : M.World), LO.Modal.NNFormula.Kripke.Satisfies M x φ
Equations
- LO.Modal.NNFormula.Kripke.ValidOnModel.semantics = { Realize := fun (M : LO.Modal.Kripke.Model) => LO.Modal.NNFormula.Kripke.ValidOnModel M }
@[simp]
theorem
LO.Modal.NNFormula.Kripke.ValidOnModel.iff_models
{φ : NNFormula ℕ}
{M : Kripke.Model}
:
M ⊧ φ ↔ ValidOnModel M φ
Equations
- LO.Modal.NNFormula.Kripke.ValidOnFrame F φ = ∀ (V : LO.Modal.Kripke.Valuation F), { toFrame := F, Val := V } ⊧ φ
Equations
- LO.Modal.NNFormula.Kripke.ValidOnFrame.semantics = { Realize := fun (F : LO.Modal.Kripke.Frame) => LO.Modal.NNFormula.Kripke.ValidOnFrame F }
@[simp]
theorem
LO.Modal.NNFormula.Kripke.ValidOnFrame.iff_models
{φ : NNFormula ℕ}
{F : Kripke.Frame}
:
F ⊧ φ ↔ ValidOnFrame F φ
Equations
- LO.Modal.NNFormula.Kripke.ValidOnFrameClass C φ = ∀ {F : LO.Modal.Kripke.Frame}, F ∈ C → LO.Modal.NNFormula.Kripke.ValidOnFrame F φ
Equations
- LO.Modal.NNFormula.Kripke.ValidOnFrameClass.semantics = { Realize := fun (C : LO.Modal.Kripke.FrameClass) => LO.Modal.NNFormula.Kripke.ValidOnFrameClass C }
@[simp]
theorem
LO.Modal.NNFormula.Kripke.ValidOnFrameClass.iff_models
{φ : NNFormula ℕ}
{C : Kripke.FrameClass}
:
C ⊧ φ ↔ ValidOnFrameClass C φ
theorem
LO.Modal.NNFormula.Kripke.Satisfies.toFormula
{φ : NNFormula ℕ}
{M : Kripke.Model}
{x : M.World}
:
Satisfies M x φ ↔ Formula.Kripke.Satisfies M x φ.toFormula
theorem
LO.Modal.NNFormula.Kripke.ValidOnModel.toFormula
{φ : NNFormula ℕ}
{M : Kripke.Model}
:
ValidOnModel M φ ↔ Formula.Kripke.ValidOnModel M φ.toFormula
theorem
LO.Modal.NNFormula.Kripke.ValidOnFrame.toFormula
{φ : NNFormula ℕ}
{F : Kripke.Frame}
:
ValidOnFrame F φ ↔ Formula.Kripke.ValidOnFrame F φ.toFormula
theorem
LO.Modal.NNFormula.Kripke.ValidOnFrameClass.toFormula
{φ : NNFormula ℕ}
{C : Kripke.FrameClass}
:
ValidOnFrameClass C φ ↔ Formula.Kripke.ValidOnFrameClass C φ.toFormula
theorem
LO.Modal.Formula.Kripke.Satisfies.toNNFormula
{φ : Formula ℕ}
{M : Kripke.Model}
{x : M.World}
:
Satisfies M x φ ↔ NNFormula.Kripke.Satisfies M x φ.toNNFormula
theorem
LO.Modal.Formula.Kripke.ValidOnModel.toNNFormula
{φ : Formula ℕ}
{M : Kripke.Model}
:
ValidOnModel M φ ↔ NNFormula.Kripke.ValidOnModel M φ.toNNFormula
theorem
LO.Modal.Formula.Kripke.ValidOnFrame.toNNFormula
{φ : Formula ℕ}
{F : Kripke.Frame}
:
ValidOnFrame F φ ↔ NNFormula.Kripke.ValidOnFrame F φ.toNNFormula
theorem
LO.Modal.Formula.Kripke.ValidOnFrameClass.toNNFormula
{φ : Formula ℕ}
{C : Kripke.FrameClass}
:
ValidOnFrameClass C φ ↔ NNFormula.Kripke.ValidOnFrameClass C φ.toNNFormula