Documentation

Foundation.Modal.PLoN.Semantics

structure LO.Modal.PLoN.Frame (α : Type u_2) :
Type (max (u_1 + 1) u_2)
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    @[reducible, inline]
    noncomputable abbrev LO.Modal.PLoN.Frame.default {α : Type u_1} {F : LO.Modal.PLoN.Frame α} :
    F.World
    Equations
    • LO.Modal.PLoN.Frame.default = .some
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      Equations
      • LO.Modal.PLoN.instCoeFunFrameForallFormulaForallWorldForallProp = { coe := LO.Modal.PLoN.Frame.Rel }
      @[reducible, inline]
      abbrev LO.Modal.PLoN.Frame.Rel' {α : Type u_1} {F : LO.Modal.PLoN.Frame α} (p : LO.Modal.Formula α) (x : F.World) (y : F.World) :
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        • One or more equations did not get rendered due to their size.
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          structure LO.Modal.PLoN.FiniteFrame (α : Type u_1) extends LO.Modal.PLoN.Frame :
          Type (max u_1 (u_2 + 1))
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            @[reducible, inline]
            abbrev LO.Modal.PLoN.FrameClass (α : Type u_1) :
            Type (max u_1 (u_2 + 1))
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              @[reducible, inline]
              abbrev LO.Modal.PLoN.Valuation {α : Type u_2} (F : LO.Modal.PLoN.Frame α✝) (α : Type u_1) :
              Type (max u_1 u_3)
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                structure LO.Modal.PLoN.Model (α : Type u_1) :
                Type (max u_1 (u_2 + 1))
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                  @[reducible, inline]
                  abbrev LO.Modal.PLoN.Model.World {α : Type u_1} (M : LO.Modal.PLoN.Model α) :
                  Type u_2
                  Equations
                  • M.World = M.Frame.World
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                    Equations
                    • LO.Modal.PLoN.instCoeSortModelType = { coe := LO.Modal.PLoN.Model.World }
                    theorem LO.Modal.Formula.PLoN.Satisfies.box_def {α : Type u_1} {M : LO.Modal.PLoN.Model α} {x : M.World} {p : LO.Modal.Formula α} :
                    x p ∀ (y : M.Frame.World), LO.Modal.PLoN.Frame.Rel' p x yy p
                    theorem LO.Modal.Formula.PLoN.Satisfies.not_def {α : Type u_1} {M : LO.Modal.PLoN.Model α} {x : M.World} {p : LO.Modal.Formula α} :
                    x p ¬x p
                    Equations
                    • LO.Modal.Formula.PLoN.Satisfies.instNotWorld = { realize_not := }
                    theorem LO.Modal.Formula.PLoN.Satisfies.imp_def {α : Type u_1} {M : LO.Modal.PLoN.Model α} {x : M.World} {p : LO.Modal.Formula α} {q : LO.Modal.Formula α} :
                    x p q x px q
                    Equations
                    • LO.Modal.Formula.PLoN.Satisfies.instImpWorld = { realize_imp := }
                    theorem LO.Modal.Formula.PLoN.Satisfies.or_def {α : Type u_1} {M : LO.Modal.PLoN.Model α} {x : M.World} {p : LO.Modal.Formula α} {q : LO.Modal.Formula α} :
                    x p q x p x q
                    Equations
                    • LO.Modal.Formula.PLoN.Satisfies.instOrWorld = { realize_or := }
                    theorem LO.Modal.Formula.PLoN.Satisfies.and_def {α : Type u_1} {M : LO.Modal.PLoN.Model α} {x : M.World} {p : LO.Modal.Formula α} {q : LO.Modal.Formula α} :
                    x p q x p x q
                    Equations
                    • LO.Modal.Formula.PLoN.Satisfies.instAndWorld = { realize_and := }
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                    • LO.Modal.Formula.PLoN.Satisfies.instTarskiWorld = LO.Semantics.Tarski.mk
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                      • LO.Modal.Formula.PLoN.ValidOnModel.instBotModel = { realize_bot := }
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                        • LO.Modal.Formula.PLoN.ValidOnFrame.instBotFrame = { realize_bot := }
                        theorem LO.Modal.Formula.PLoN.ValidOnFrame.mdp {α : Type u_1} {F : LO.Modal.PLoN.Frame α} {p : LO.Modal.Formula α} {q : LO.Modal.Formula α} (hpq : F p q) (hp : F p) :
                        F q
                        Equations
                        theorem LO.Modal.Formula.PLoN.ValidOnFrameClass.mdp {α : Type u_1} {𝔽 : LO.Modal.PLoN.FrameClass α} {p : LO.Modal.Formula α} {q : LO.Modal.Formula α} (hpq : 𝔽 p q) (hp : 𝔽 p) :
                        𝔽 q
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                          @[reducible, inline]
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                            theorem LO.Modal.PLoN.N_defines {α : Type u_1} :
                            𝐍.DefinesPLoNFrameClass (LO.Modal.PLoN.AllFrameClass α)