Documentation

Logic.Modal.Standard.Kripke.Preservation

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    theorem LO.Modal.Standard.Kripke.Model.Bisimulation.atomic {α : Type u_1} {M₁ : LO.Modal.Standard.Kripke.Model α} {M₂ : LO.Modal.Standard.Kripke.Model α} (self : M₁ M₂) {x₁ : M₁.World} {x₂ : M₂.World} {a : α} :
    self.toRel x₁ x₂(M₁.Valuation x₁ a M₂.Valuation x₂ a)
    theorem LO.Modal.Standard.Kripke.Model.Bisimulation.forth {α : Type u_1} {M₁ : LO.Modal.Standard.Kripke.Model α} {M₂ : LO.Modal.Standard.Kripke.Model α} (self : M₁ M₂) {x₁ : M₁.World} {y₁ : M₁.World} {x₂ : M₂.World} :
    self.toRel x₁ x₂LO.Modal.Standard.Kripke.Frame.Rel' x₁ y₁∃ (y₂ : M₂.World), self.toRel y₁ y₂ LO.Modal.Standard.Kripke.Frame.Rel' x₂ y₂
    theorem LO.Modal.Standard.Kripke.Model.Bisimulation.back {α : Type u_1} {M₁ : LO.Modal.Standard.Kripke.Model α} {M₂ : LO.Modal.Standard.Kripke.Model α} (self : M₁ M₂) {x₁ : M₁.World} {x₂ : M₂.World} {y₂ : M₂.World} :
    self.toRel x₁ x₂LO.Modal.Standard.Kripke.Frame.Rel' x₂ y₂∃ (y₁ : M₁.World), self.toRel y₁ y₂ LO.Modal.Standard.Kripke.Frame.Rel' x₁ y₁
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      instance LO.Modal.Standard.Kripke.instCoeFunBisimulationForallWorldForallProp :
      {α : Type u_1} → {M₁ : LO.Modal.Standard.Kripke.Model α} → {M₂ : LO.Modal.Standard.Kripke.Model α} → CoeFun (M₁ M₂) fun (x : M₁ M₂) => M₁.WorldM₂.WorldProp
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      • LO.Modal.Standard.Kripke.instCoeFunBisimulationForallWorldForallProp = { coe := fun (bi : M₁ M₂) => bi.toRel }
      def LO.Modal.Standard.Kripke.ModalEquivalent {α : Type u_1} {M₁ : LO.Modal.Standard.Kripke.Model α} {M₂ : LO.Modal.Standard.Kripke.Model α} (w₁ : M₁.World) (w₂ : M₂.World) :
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          theorem LO.Modal.Standard.Kripke.modal_equivalent_of_bisimilar {α : Type u_1} {M₁ : LO.Modal.Standard.Kripke.Model α} {M₂ : LO.Modal.Standard.Kripke.Model α} (Bi : M₁ M₂) {x₁ : M₁.World} {x₂ : M₂.World} (bisx : Bi.toRel x₁ x₂) :
          x₁ x₂

          As known as p-morphism.

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            theorem LO.Modal.Standard.Kripke.Frame.PseudoEpimorphism.back {F₁ : LO.Modal.Standard.Kripke.Frame} {F₂ : LO.Modal.Standard.Kripke.Frame} (self : F₁ →ₚ F₂) {w : F₁.World} {v : F₂.World} :
            LO.Modal.Standard.Kripke.Frame.Rel' (self.toFun w) v∃ (u : F₁.World), self.toFun u = v LO.Modal.Standard.Kripke.Frame.Rel' w u
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              instance LO.Modal.Standard.Kripke.instCoeFunPseudoEpimorphismForallWorld {F₁ : LO.Modal.Standard.Kripke.Frame} {F₂ : LO.Modal.Standard.Kripke.Frame} :
              CoeFun (F₁ →ₚ F₂) fun (x : F₁ →ₚ F₂) => F₁.WorldF₂.World
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              • LO.Modal.Standard.Kripke.instCoeFunPseudoEpimorphismForallWorld = { coe := fun (f : F₁ →ₚ F₂) => f.toFun }
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              • LO.Modal.Standard.Kripke.Frame.PseudoEpimorphism.id = { toFun := id, forth := , back := }
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                • f.TransitiveClosure F₂_trans = { toFun := f.toFun, forth := , back := }
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                  • f.comp g = { toFun := g.toFun f.toFun, forth := , back := }
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                      theorem LO.Modal.Standard.Kripke.Model.PseudoEpimorphism.atomic {α : Type u_1} {M₁ : LO.Modal.Standard.Kripke.Model α} {M₂ : LO.Modal.Standard.Kripke.Model α} (self : M₁ →ₚ M₂) {w : M₁.World} {a : α} :
                      M₁.Valuation w a M₂.Valuation (self.toFun w) a
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                        instance LO.Modal.Standard.Kripke.instCoeFunPseudoEpimorphismForallWorld_1 :
                        {α : Type u_1} → {M₁ : LO.Modal.Standard.Kripke.Model α} → {M₂ : LO.Modal.Standard.Kripke.Model α} → CoeFun (M₁ →ₚ M₂) fun (x : M₁ →ₚ M₂) => M₁.WorldM₂.World
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                        • LO.Modal.Standard.Kripke.instCoeFunPseudoEpimorphismForallWorld_1 = { coe := fun (f : M₁ →ₚ M₂) => f.toFun }
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                        • f.toFramePseudoEpimorphism = { toFun := f.toFun, forth := , back := }
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                          • LO.Modal.Standard.Kripke.Model.PseudoEpimorphism.id = { toFun := id, forth := , back := , atomic := }
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                            def LO.Modal.Standard.Kripke.Model.PseudoEpimorphism.mkAtomic {α : Type u_1} {M₁ : LO.Modal.Standard.Kripke.Model α} {M₂ : LO.Modal.Standard.Kripke.Model α} (f : M₁.Frame →ₚ M₂.Frame) (atomic : ∀ {w : M₁.Frame.World} {a : α}, M₁.Valuation w a M₂.Valuation (f.toFun w) a) :
                            M₁ →ₚ M₂
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                                • f.Bisimulation = { toRel := Function.graph f.toFun, atomic := , forth := , back := }
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                                  theorem LO.Modal.Standard.Kripke.modal_equivalence_of_modal_morphism {α : Type u_1} {M₁ : LO.Modal.Standard.Kripke.Model α} {M₂ : LO.Modal.Standard.Kripke.Model α} (f : M₁ →ₚ M₂) (w : M₁.World) :
                                  w f.toFun w
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                                    • World : Type u_1
                                    • Rel : Rel self.World self.World
                                    • World_nonempty : Nonempty self.World
                                    • root : self.World
                                    • root_rooted : self.isRooted self.root
                                    • default : self.World
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                                        theorem LO.Modal.Standard.Kripke.Frame.PointGenerated.rooted {F : LO.Modal.Standard.Kripke.Frame} {r : F.World} :
                                        (F.PointGenerated r).isRooted r,
                                        instance LO.Modal.Standard.Kripke.Frame.PointGenerated.instFiniteWorld {F : LO.Modal.Standard.Kripke.Frame} {r : F.World} [Finite F.World] :
                                        Finite (F.PointGenerated r).World
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                                        • LO.Modal.Standard.Kripke.Frame.PointGenerated.instDecidableEqWorld = Subtype.instDecidableEq
                                        @[reducible, inline]
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                                            • Mr = { Frame := (M.Framer).toFrame, Valuation := fun (w : (M.Framer).World) (a : α) => M.Valuation (w) a }
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                                                def LO.Modal.Standard.Kripke.Model.PointGenerated.Bisimulation {α : Type u_1} (M : LO.Modal.Standard.Kripke.Model α) (M_trans : Transitive M.Frame.Rel) (r : M.World) :
                                                (Mr) M
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                                                  theorem LO.Modal.Standard.Kripke.Model.PointGenerated.modal_equivalent_to_root {α : Type u_1} (M : LO.Modal.Standard.Kripke.Model α) (M_trans : Transitive M.Frame.Rel) (r : M.World) :
                                                  r, r
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