Documentation

Foundation.Modal.Kripke.GL.MDP

source
@[reducible, inline]
abbrev LO.Modal.Kripke.GL_MDPCounterexampleFrame (F₁ : LO.Kripke.FiniteTransitiveTree) (F₂ : LO.Kripke.FiniteTransitiveTree) :
LO.Kripke.FiniteTransitiveTree
Equations
  • One or more equations did not get rendered due to their size.
Instances For
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    instance LO.Modal.Kripke.GL_MDPCounterexampleFrame.instCoeWorld {F₁ : LO.Kripke.FiniteTransitiveTree} {F₂ : LO.Kripke.FiniteTransitiveTree} :
    Coe F₁.World (LO.Modal.Kripke.GL_MDPCounterexampleFrame F₁ F₂).World
    Equations
    • LO.Modal.Kripke.GL_MDPCounterexampleFrame.instCoeWorld = { coe := Sum.inr ∘ Sum.inl }
    source
    instance LO.Modal.Kripke.GL_MDPCounterexampleFrame.instCoeWorld_1 {F₁ : LO.Kripke.FiniteTransitiveTree} {F₂ : LO.Kripke.FiniteTransitiveTree} :
    Coe F₂.World (LO.Modal.Kripke.GL_MDPCounterexampleFrame F₁ F₂).World
    Equations
    • LO.Modal.Kripke.GL_MDPCounterexampleFrame.instCoeWorld_1 = { coe := Sum.inr ∘ Sum.inr }
    source
    def LO.Modal.Kripke.GL_MDPCounterexampleFrame.p_morphism₁ {F₁ : LO.Kripke.FiniteTransitiveTree} {F₂ : LO.Kripke.FiniteTransitiveTree} :
    F₁.toFrame →ₚ (LO.Modal.Kripke.GL_MDPCounterexampleFrame F₁ F₂).toFrame
    Equations
    • LO.Modal.Kripke.GL_MDPCounterexampleFrame.p_morphism₁ = { toFun := fun (x : F₁.World) => Sum.inr (Sum.inl x), forth := ⋯, back := ⋯ }
    Instances For
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      def LO.Modal.Kripke.GL_MDPCounterexampleFrame.p_morphism₂ {F₁ : LO.Kripke.FiniteTransitiveTree} {F₂ : LO.Kripke.FiniteTransitiveTree} :
      F₂.toFrame →ₚ (LO.Modal.Kripke.GL_MDPCounterexampleFrame F₁ F₂).toFrame
      Equations
      • LO.Modal.Kripke.GL_MDPCounterexampleFrame.p_morphism₂ = { toFun := fun (x : F₂.World) => Sum.inr (Sum.inr x), forth := ⋯, back := ⋯ }
      Instances For
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        theorem LO.Modal.Kripke.GL_MDPCounterexampleFrame.through_original_root {F₁ : LO.Kripke.FiniteTransitiveTree} {F₂ : LO.Kripke.FiniteTransitiveTree} {x : (LO.Modal.Kripke.GL_MDPCounterexampleFrame F₁ F₂).World} (h : (LO.Modal.Kripke.GL_MDPCounterexampleFrame F₁ F₂).root ≺ x) :
        (x = (Sum.inr ∘ Sum.inl) F₁.root ∨ Sum.inr (Sum.inl F₁.root) ≺ x) ∨ x = (Sum.inr ∘ Sum.inr) F₂.root ∨ Sum.inr (Sum.inr F₂.root) ≺ x
        source
        @[reducible, inline]
        abbrev LO.Modal.Kripke.GL_MDPCounterexampleModel {α : Type u_1} (M₁ : LO.Kripke.FiniteTransitiveTreeModel α) (M₂ : LO.Kripke.FiniteTransitiveTreeModel α) :
        LO.Kripke.FiniteTransitiveTreeModel α
        Equations
        • One or more equations did not get rendered due to their size.
        Instances For
          source
          instance LO.Modal.Kripke.GL_MDPCounterexampleModel.instCoeWorld {α : Type u_1} {M₁ : LO.Kripke.FiniteTransitiveTreeModel α} {M₂ : LO.Kripke.FiniteTransitiveTreeModel α} :
          Coe M₁.World (LO.Modal.Kripke.GL_MDPCounterexampleModel M₁ M₂).World
          Equations
          • LO.Modal.Kripke.GL_MDPCounterexampleModel.instCoeWorld = { coe := Sum.inr ∘ Sum.inl }
          source
          instance LO.Modal.Kripke.GL_MDPCounterexampleModel.instCoeWorld_1 {α : Type u_1} {M₁ : LO.Kripke.FiniteTransitiveTreeModel α} {M₂ : LO.Kripke.FiniteTransitiveTreeModel α} :
          Coe M₂.World (LO.Modal.Kripke.GL_MDPCounterexampleModel M₁ M₂).World
          Equations
          • LO.Modal.Kripke.GL_MDPCounterexampleModel.instCoeWorld_1 = { coe := Sum.inr ∘ Sum.inr }
          source
          def LO.Modal.Kripke.GL_MDPCounterexampleModel.p_morphism₁ {α : Type u_1} {M₁ : LO.Kripke.FiniteTransitiveTreeModel α} {M₂ : LO.Kripke.FiniteTransitiveTreeModel α} :
          M₁.toModel →ₚ (LO.Modal.Kripke.GL_MDPCounterexampleModel M₁ M₂).toModel
          Equations
          Instances For
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            theorem LO.Modal.Kripke.GL_MDPCounterexampleModel.modal_equivalence_original_world₁ {α : Type u_1} {M₁ : LO.Kripke.FiniteTransitiveTreeModel α} {M₂ : LO.Kripke.FiniteTransitiveTreeModel α} {x : M₁.toModel.World} :
            x ↭ (Sum.inr ∘ Sum.inl) x
            source
            def LO.Modal.Kripke.GL_MDPCounterexampleModel.p_morphism₂ {α : Type u_1} {M₁ : LO.Kripke.FiniteTransitiveTreeModel α} {M₂ : LO.Kripke.FiniteTransitiveTreeModel α} :
            M₂.toModel →ₚ (LO.Modal.Kripke.GL_MDPCounterexampleModel M₁ M₂).toModel
            Equations
            Instances For
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              theorem LO.Modal.Kripke.GL_MDPCounterexampleModel.modal_equivalence_original_world₂ {α : Type u_1} {M₁ : LO.Kripke.FiniteTransitiveTreeModel α} {M₂ : LO.Kripke.FiniteTransitiveTreeModel α} {x : M₂.toModel.World} :
              x ↭ (Sum.inr ∘ Sum.inr) x
              source
              theorem LO.Modal.GL_MDP_Aux {α : Type u_1} [Inhabited α] [DecidableEq α] {X : LO.Modal.Theory α} {p₁ : LO.Modal.Formula α} {p₂ : LO.Modal.Formula α} (h : (□''X) *⊢[𝐆𝐋]! □p₁ ⋎ □p₂) :
              (□''X) *⊢[𝐆𝐋]! □p₁ ∨ (□''X) *⊢[𝐆𝐋]! □p₂
              source
              theorem LO.Modal.GL_MDP {α : Type u_1} [Inhabited α] [DecidableEq α] {p₁ : LO.Modal.Formula α} {p₂ : LO.Modal.Formula α} (h : 𝐆𝐋 ⊢! □p₁ ⋎ □p₂) :
              𝐆𝐋 ⊢! p₁ ∨ 𝐆𝐋 ⊢! p₂
              source
              instance LO.Modal.instModalDisjunctiveFormulaHilbertGL {α : Type u_1} [Inhabited α] [DecidableEq α] :
              LO.System.ModalDisjunctive 𝐆𝐋
              Equations
              • ⋯ = ⋯