Documentation

Foundation.Modal.System.K4

source
def LO.System.imply_BoxBoxdot_Box {S : Type u_1} {F : Type u_2} [BasicModalLogicalConnective F] [DecidableEq F] [System F S] {𝓒 : S} [System.K4 𝓒] {Ο† : F} :
𝓒 ⊒ β–‘βŠ‘Ο† ➝ β–‘Ο†
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    @[simp]
    theorem LO.System.imply_boxboxdot_box {S : Type u_1} {F : Type u_2} [BasicModalLogicalConnective F] [DecidableEq F] [System F S] {𝓒 : S} [System.K4 𝓒] {Ο† : F} :
    𝓒 ⊒! β–‘βŠ‘Ο† ➝ β–‘Ο†
    source
    def LO.System.imply_Box_BoxBoxdot {S : Type u_1} {F : Type u_2} [BasicModalLogicalConnective F] [DecidableEq F] [System F S] {𝓒 : S} [System.K4 𝓒] {Ο† : F} :
    𝓒 ⊒ β–‘Ο† ➝ β–‘βŠ‘Ο†
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      @[simp]
      theorem LO.System.imply_box_boxboxdot! {S : Type u_1} {F : Type u_2} [BasicModalLogicalConnective F] [DecidableEq F] [System F S] {𝓒 : S} [System.K4 𝓒] {Ο† : F} :
      𝓒 ⊒! β–‘Ο† ➝ β–‘βŠ‘Ο†
      source
      def LO.System.imply_Box_BoxBoxdot' {S : Type u_1} {F : Type u_2} [BasicModalLogicalConnective F] [DecidableEq F] [System F S] {𝓒 : S} [System.K4 𝓒] {Ο† : F} (h : 𝓒 ⊒ β–‘Ο†) :
      𝓒 ⊒ β–‘βŠ‘Ο†
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        def LO.System.imply_Box_BoxBoxdot'! {S : Type u_1} {F : Type u_2} [BasicModalLogicalConnective F] [DecidableEq F] [System F S] {𝓒 : S} [System.K4 𝓒] {Ο† : F} (h : 𝓒 ⊒! β–‘Ο†) :
        𝓒 ⊒! β–‘βŠ‘Ο†
        Equations
        • β‹― = β‹―
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          def LO.System.iff_Box_BoxBoxdot {S : Type u_1} {F : Type u_2} [BasicModalLogicalConnective F] [DecidableEq F] [System F S] {𝓒 : S} [System.K4 𝓒] {Ο† : F} :
          𝓒 ⊒ β–‘Ο† β­€ β–‘βŠ‘Ο†
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            @[simp]
            theorem LO.System.iff_box_boxboxdot! {S : Type u_1} {F : Type u_2} [BasicModalLogicalConnective F] [DecidableEq F] [System F S] {𝓒 : S} [System.K4 𝓒] {Ο† : F} :
            𝓒 ⊒! β–‘Ο† β­€ β–‘βŠ‘Ο†
            source
            def LO.System.iff_Box_BoxdotBox {S : Type u_1} {F : Type u_2} [BasicModalLogicalConnective F] [DecidableEq F] [System F S] {𝓒 : S} [System.K4 𝓒] {Ο† : F} :
            𝓒 ⊒ β–‘Ο† β­€ βŠ‘β–‘Ο†
            Equations
            • One or more equations did not get rendered due to their size.
            Instances For
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              @[simp]
              theorem LO.System.iff_box_boxdotbox! {S : Type u_1} {F : Type u_2} [BasicModalLogicalConnective F] [DecidableEq F] [System F S] {𝓒 : S} [System.K4 𝓒] {Ο† : F} :
              𝓒 ⊒! β–‘Ο† β­€ βŠ‘β–‘Ο†
              source
              def LO.System.iff_Boxdot_BoxdotBoxdot {S : Type u_1} {F : Type u_2} [BasicModalLogicalConnective F] [DecidableEq F] [System F S] {𝓒 : S} [System.K4 𝓒] {Ο† : F} :
              𝓒 ⊒ βŠ‘Ο† β­€ βŠ‘βŠ‘Ο†
              Equations
              • One or more equations did not get rendered due to their size.
              Instances For
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                @[simp]
                theorem LO.System.iff_boxdot_boxdotboxdot {S : Type u_1} {F : Type u_2} [BasicModalLogicalConnective F] [DecidableEq F] [System F S] {𝓒 : S} [System.K4 𝓒] {Ο† : F} :
                𝓒 ⊒! βŠ‘Ο† β­€ βŠ‘βŠ‘Ο†
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                def LO.System.boxdotAxiomFour {S : Type u_1} {F : Type u_2} [BasicModalLogicalConnective F] [DecidableEq F] [System F S] {𝓒 : S} [System.K4 𝓒] {Ο† : F} :
                𝓒 ⊒ βŠ‘Ο† ➝ βŠ‘βŠ‘Ο†
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                  @[simp]
                  theorem LO.System.boxdot_axiomFour! {S : Type u_1} {F : Type u_2} [BasicModalLogicalConnective F] [DecidableEq F] [System F S] {𝓒 : S} [System.K4 𝓒] {Ο† : F} :
                  𝓒 ⊒! βŠ‘Ο† ➝ βŠ‘βŠ‘Ο†