Documentation

Foundation.Modal.VanBentham.StandardTranslation

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inductive LO.FirstOrder.Language.Frame.Rel :
ℕ → Type
  • pred : ℕ → Rel 1

    propositional variable

  • lt : Rel 2

    binary relation

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    @[reducible]
    def LO.FirstOrder.Language.frame :
    Language

    Language of Kripke frame for basic modal logic.

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      def LO.FirstOrder.Language.term𝓛𝓕 :
      Lean.ParserDescr
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        instance LO.FirstOrder.Language.instLTFrame :
        𝓛𝓕.LT
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        def LO.FirstOrder.Frame.forces (a : ℕ) :
        Semiformula.Operator 𝓛𝓕 1
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          def LO.FirstOrder.Frame.«first_order_formula_⊩_» :
          Lean.ParserDescr
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            def LO.FirstOrder.Frame.transitive :
            Sentence 𝓛𝓕
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              def LO.FirstOrder.Frame.monotone (a : ℕ) :
              Sentence 𝓛𝓕
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                def LO.Modal.standardTranslation :
                NNFormula ℕ → FirstOrder.Semisentence 𝓛𝓕 1
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                  def LO.Modal.«term_¹» :
                  Lean.TrailingParserDescr
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                    instance LO.Modal.Kripke.FirstOrder.instStructureFrameWorld {M : Model} :
                    FirstOrder.Structure 𝓛𝓕 M.World
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                    @[simp]
                    theorem LO.Modal.Kripke.FirstOrder.forces_iff_val {M : Model} {x : M.World} {a : ℕ} :
                    (FirstOrder.Frame.forces a).val ![x] ↔ M.Val x a
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                    @[simp]
                    theorem LO.Modal.Kripke.FirstOrder.lt_iff_rel {M : Model} {x y : M.World} :
                    op(<).val ![x, y] ↔ x ≺ y
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                    theorem LO.Modal.Kripke.FirstOrder.correspondence_satisfies {M : Model} {x : M.World} {φ : NNFormula ℕ} :
                    x ⊧ φ ↔ M.World ⊧/![x] φ¹

                    BdRV Prop 2.47 (i)

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                    theorem LO.Modal.Kripke.FirstOrder.correspondence_validOnModel {M : Model} {φ : NNFormula ℕ} :
                    M ⊧ φ ↔ M.World ⊧ₘ₀ ∀' φ¹

                    BdRV Prop 2.47 (ii)