Documentation

Foundation.Modal.LogicSymbol

class LO.Box (F : Type u_1) :
Type u_1
Instances
    @[simp]
    theorem LO.Box.box_injective {F : Type u_1} [self : LO.Box F] :
    @[reducible, match_pattern, inline]
    abbrev LO.Box.boxdot {F : Type u_1} [LO.Box F] [LO.Wedge F] (p : F) :
    F
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      @[reducible, inline]
      abbrev LO.Box.multibox {F : Type u_1} [LO.Box F] (n : ) :
      FF
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        • One or more equations did not get rendered due to their size.
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          class LO.Box.Subclosed {F : Type u_1} [LO.Box F] (C : FProp) :
          • box_closed : ∀ {p : F}, C (p)C p
          Instances
            theorem LO.Box.Subclosed.box_closed {F : Type u_1} [LO.Box F] {C : FProp} [self : LO.Box.Subclosed C] {p : F} :
            C (p)C p
            @[simp]
            theorem LO.Box.box_injective' {F : Type u_1} [LO.Box F] {p : F} {q : F} :
            p = q p = q
            @[simp]
            theorem LO.Box.multibox_succ {F : Type u_1} [LO.Box F] {p : F} {n : } :
            @[simp]
            theorem LO.Box.multibox_injective {F : Type u_1} [LO.Box F] {n : } :
            Function.Injective fun (x : F) => □^[n]x
            @[simp]
            theorem LO.Box.multimop_injective' {F : Type u_1} [LO.Box F] {p : F} {q : F} {n : } :
            □^[n]p = □^[n]q p = q
            class LO.Dia (F : Type u_1) :
            Type u_1
            Instances
              @[simp]
              theorem LO.Dia.dia_injective {F : Type u_1} [self : LO.Dia F] :
              @[reducible, match_pattern, inline]
              abbrev LO.Dia.diadot {F : Type u_1} [LO.Dia F] [LO.Vee F] (p : F) :
              F
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                @[reducible, inline]
                abbrev LO.Dia.multidia {F : Type u_1} [LO.Dia F] (n : ) :
                FF
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                  • One or more equations did not get rendered due to their size.
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                    class LO.Dia.Subclosed {F : Type u_1} [LO.Dia F] [LO.LogicalConnective F] (C : FProp) :
                    • dia_closed : ∀ {p : F}, C (p)C p
                    Instances
                      theorem LO.Dia.Subclosed.dia_closed {F : Type u_1} [LO.Dia F] [LO.LogicalConnective F] {C : FProp} [self : LO.Dia.Subclosed C] {p : F} :
                      C (p)C p
                      @[simp]
                      theorem LO.Dia.dia_injective' {F : Type u_1} [LO.Dia F] {p : F} {q : F} :
                      p = q p = q
                      @[simp]
                      theorem LO.Dia.multidia_succ {F : Type u_1} [LO.Dia F] {p : F} {n : } :
                      @[simp]
                      theorem LO.Dia.multidia_injective {F : Type u_1} [LO.Dia F] {n : } :
                      Function.Injective fun (x : F) => ◇^[n]x
                      @[simp]
                      theorem LO.Dia.multidia_injective' {F : Type u_1} [LO.Dia F] {p : F} {q : F} {n : } :
                      ◇^[n]p = ◇^[n]q p = q
                      class LO.DiaAbbrev (F : Type u_1) [LO.Box F] [LO.Dia F] [LO.Tilde F] :
                      Instances
                        theorem LO.DiaAbbrev.dia_abbrev {F : Type u_1} [LO.Box F] [LO.Dia F] [LO.Tilde F] [self : LO.DiaAbbrev F] {p : F} :
                        Instances
                          @[simp]
                          @[simp]
                          @[reducible, inline]
                          abbrev Set.multibox {F : Type u_1} [LO.Box F] (n : ) :
                          Set FSet F
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                            • One or more equations did not get rendered due to their size.
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                              @[reducible, inline]
                              abbrev Set.multidia {F : Type u_1} [LO.Dia F] (n : ) :
                              Set FSet F
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                                • One or more equations did not get rendered due to their size.
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                                  @[reducible, inline]
                                  abbrev Set.box {F : Type u_1} [LO.Box F] :
                                  Set FSet F
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                                    @[reducible, inline]
                                    abbrev Set.dia {F : Type u_1} [LO.Dia F] :
                                    Set FSet F
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                                      @[reducible, inline]
                                      abbrev Set.premultibox {F : Type u_1} [LO.Box F] (n : ) :
                                      Set FSet F
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                                        • One or more equations did not get rendered due to their size.
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                                          @[reducible, inline]
                                          abbrev Set.premultidia {F : Type u_1} [LO.Dia F] (n : ) :
                                          Set FSet F
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                                            • One or more equations did not get rendered due to their size.
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                                              @[reducible, inline]
                                              abbrev Set.prebox {F : Type u_1} [LO.Box F] :
                                              Set FSet F
                                              Equations
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                                                @[reducible, inline]
                                                abbrev Set.predia {F : Type u_1} [LO.Dia F] :
                                                Set FSet F
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                                                  @[simp]
                                                  theorem Set.eq_box_multibox_one {F : Type u_1} [LO.Box F] {s : Set F} :
                                                  @[simp]
                                                  theorem Set.eq_dia_multidia_one {F : Type u_1} [LO.Dia F] {s : Set F} :
                                                  @[simp]
                                                  @[simp]
                                                  @[simp]
                                                  theorem Set.multibox_subset_mono {F : Type u_1} [LO.Box F] {s : Set F} {t : Set F} {n : } (h : s t) :
                                                  @[simp]
                                                  theorem Set.box_subset_mono {F : Type u_1} [LO.Box F] {s : Set F} {t : Set F} (h : s t) :
                                                  @[simp]
                                                  theorem Set.multidia_subset_mono {F : Type u_1} [LO.Dia F] {s : Set F} {t : Set F} {n : } (h : s t) :
                                                  @[simp]
                                                  theorem Set.dia_subset_mono {F : Type u_1} [LO.Dia F] {s : Set F} {t : Set F} (h : s t) :
                                                  @[simp]
                                                  theorem Set.premultibox_subset_mono {F : Type u_1} [LO.Box F] {s : Set F} {t : Set F} {n : } (h : s t) :
                                                  @[simp]
                                                  theorem Set.prebox_subset_mono {F : Type u_1} [LO.Box F] {s : Set F} {t : Set F} (h : s t) :
                                                  @[simp]
                                                  theorem Set.premultidia_subset_mono {F : Type u_1} [LO.Dia F] {s : Set F} {t : Set F} {n : } (h : s t) :
                                                  @[simp]
                                                  theorem Set.predia_subset_mono {F : Type u_1} [LO.Dia F] {s : Set F} {t : Set F} (h : s t) :
                                                  @[simp]
                                                  theorem Set.iff_mem_premultibox {F : Type u_1} [LO.Box F] {s : Set F} {p : F} {n : } :
                                                  @[simp]
                                                  theorem Set.iff_mem_multibox {F : Type u_1} [LO.Box F] {s : Set F} {n : } {p : F} :
                                                  @[simp]
                                                  theorem Set.iff_mem_premultidia {F : Type u_1} [LO.Dia F] {s : Set F} {p : F} {n : } :
                                                  @[simp]
                                                  theorem Set.iff_mem_multidia {F : Type u_1} [LO.Dia F] {s : Set F} {n : } {p : F} :
                                                  theorem Set.subset_premulitibox_iff_multibox_subset {F : Type u_1} [LO.Box F] {s : Set F} {t : Set F} {n : } (h : s □''⁻¹^[n]t) :
                                                  theorem Set.subset_prebox_iff_box_subset {F : Type u_1} [LO.Box F] {s : Set F} {t : Set F} (h : s □''⁻¹t) :
                                                  theorem Set.subset_premultidia_iff_multidia_subset {F : Type u_1} [LO.Dia F] {s : Set F} {t : Set F} {n : } (h : s ◇''⁻¹^[n]t) :
                                                  theorem Set.subset_predia_iff_dia_subset {F : Type u_1} [LO.Dia F] {s : Set F} {t : Set F} (h : s ◇''⁻¹t) :
                                                  theorem Set.subset_multibox_iff_premulitibox_subset {F : Type u_1} [LO.Box F] {s : Set F} {t : Set F} {n : } (h : s □''^[n]t) :
                                                  theorem Set.subset_box_iff_prebox_subset {F : Type u_1} [LO.Box F] {s : Set F} {t : Set F} (h : s □''t) :
                                                  theorem Set.subset_multidia_iff_premultidia_subset {F : Type u_1} [LO.Dia F] {s : Set F} {t : Set F} {n : } (h : s ◇''^[n]t) :
                                                  theorem Set.subset_dia_iff_predia_subset {F : Type u_1} [LO.Dia F] {s : Set F} {t : Set F} (h : s ◇''t) :
                                                  theorem Set.forall_multibox_of_subset_multibox {F : Type u_1} [LO.Box F] {s : Set F} {t : Set F} {n : } (h : s □''^[n]t) (p : F) :
                                                  p sqt, p = □^[n]q
                                                  theorem Set.forall_box_of_subset_box {F : Type u_1} [LO.Box F] {s : Set F} {t : Set F} (h : s □''t) (p : F) :
                                                  p sqt, p = q
                                                  theorem Set.forall_multidia_of_subset_multidia {F : Type u_1} [LO.Dia F] {s : Set F} {t : Set F} {n : } (h : s ◇''^[n]t) (p : F) :
                                                  p sqt, p = ◇^[n]q
                                                  theorem Set.forall_dia_of_subset_dia {F : Type u_1} [LO.Dia F] {s : Set F} {t : Set F} (h : s ◇''t) (p : F) :
                                                  p sqt, p = q
                                                  theorem Set.eq_prebox_box_of_subset_prebox {F : Type u_1} [LO.Box F] {s : Set F} {t : Set F} (h : s □''t) :
                                                  theorem Set.eq_predia_dia_of_subset_predia {F : Type u_1} [LO.Dia F] {s : Set F} {t : Set F} (h : s ◇''t) :
                                                  @[reducible, inline]
                                                  abbrev Finset.multibox {F : Type u_1} [DecidableEq F] [LO.Box F] (n : ) :
                                                  Finset FFinset F
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                                                    @[reducible, inline]
                                                    abbrev Finset.multidia {F : Type u_1} [DecidableEq F] [LO.Dia F] (n : ) :
                                                    Finset FFinset F
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                                                      @[reducible, inline]
                                                      abbrev Finset.box {F : Type u_1} [DecidableEq F] [LO.Box F] :
                                                      Finset FFinset F
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                                                        @[reducible, inline]
                                                        abbrev Finset.dia {F : Type u_1} [DecidableEq F] [LO.Dia F] :
                                                        Finset FFinset F
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                                                          @[reducible, inline]
                                                          noncomputable abbrev Finset.premultibox {F : Type u_1} [LO.Box F] (n : ) :
                                                          Finset FFinset F
                                                          Equations
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                                                            @[reducible, inline]
                                                            noncomputable abbrev Finset.premultidia {F : Type u_1} [LO.Dia F] (n : ) :
                                                            Finset FFinset F
                                                            Equations
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                                                              @[reducible, inline]
                                                              noncomputable abbrev Finset.prebox {F : Type u_1} [LO.Box F] :
                                                              Finset FFinset F
                                                              Equations
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                                                                @[reducible, inline]
                                                                noncomputable abbrev Finset.predia {F : Type u_1} [LO.Dia F] :
                                                                Finset FFinset F
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                                                                  @[simp]
                                                                  theorem Finset.eq_box_multibox_one {F : Type u_1} [DecidableEq F] [LO.Box F] {s : Finset F} :
                                                                  s.box = Finset.multibox 1 s
                                                                  @[simp]
                                                                  theorem Finset.eq_dia_multidia_one {F : Type u_1} [DecidableEq F] [LO.Dia F] {s : Finset F} :
                                                                  s.dia = Finset.multidia 1 s
                                                                  @[simp]
                                                                  theorem Finset.eq_prebox_premultibox_one {F : Type u_1} [LO.Box F] {s : Finset F} :
                                                                  s.prebox = Finset.premultibox 1 s
                                                                  @[simp]
                                                                  theorem Finset.eq_predia_premultidia_one {F : Type u_1} [LO.Dia F] {s : Finset F} :
                                                                  s.predia = Finset.premultidia 1 s
                                                                  theorem Finset.multibox_coe {F : Type u_1} [DecidableEq F] [LO.Box F] {s : Finset F} {n : } :
                                                                  (Finset.multibox n s) = □''^[n]s
                                                                  theorem Finset.box_coe {F : Type u_1} [DecidableEq F] [LO.Box F] {s : Finset F} :
                                                                  s.box = □''s
                                                                  theorem Finset.multidia_coe {F : Type u_1} [DecidableEq F] [LO.Dia F] {s : Finset F} {n : } :
                                                                  (Finset.multidia n s) = ◇''^[n]s
                                                                  theorem Finset.dia_coe {F : Type u_1} [DecidableEq F] [LO.Dia F] {s : Finset F} :
                                                                  s.dia = ◇''s
                                                                  theorem Finset.multibox_mem_coe {F : Type u_1} [DecidableEq F] [LO.Box F] {s : Finset F} {n : } {p : F} :
                                                                  theorem Finset.box_mem_coe {F : Type u_1} [DecidableEq F] [LO.Box F] {s : Finset F} {p : F} :
                                                                  p s.box p □''s
                                                                  theorem Finset.multidia_mem_coe {F : Type u_1} [DecidableEq F] [LO.Dia F] {s : Finset F} {n : } {p : F} :
                                                                  theorem Finset.dia_mem_coe {F : Type u_1} [DecidableEq F] [LO.Dia F] {s : Finset F} {p : F} :
                                                                  p s.dia p ◇''s
                                                                  theorem Finset.premultibox_coe {F : Type u_1} [LO.Box F] {s : Finset F} {n : } :
                                                                  theorem Finset.prebox_coe {F : Type u_1} [LO.Box F] {s : Finset F} :
                                                                  s.prebox = □''⁻¹s
                                                                  theorem Finset.premultidia_coe {F : Type u_1} [LO.Dia F] {s : Finset F} {n : } :
                                                                  theorem Finset.predia_coe {F : Type u_1} [LO.Dia F] {s : Finset F} :
                                                                  s.predia = ◇''⁻¹s
                                                                  theorem Finset.prebox_box_eq_of_subset_box {F : Type u_1} [DecidableEq F] [LO.Box F] {s : Finset F} {t : Set F} (hs : s □''t) :
                                                                  s.prebox.box = s
                                                                  theorem Finset.predia_dia_eq_of_subset_dia {F : Type u_1} [DecidableEq F] [LO.Dia F] {s : Finset F} {t : Set F} (hs : s ◇''t) :
                                                                  s.predia.dia = s
                                                                  @[reducible, inline]
                                                                  noncomputable abbrev List.multibox {F : Type u_1} [DecidableEq F] [LO.Box F] (n : ) :
                                                                  List FList F
                                                                  Equations
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                                                                    • One or more equations did not get rendered due to their size.
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                                                                      @[reducible, inline]
                                                                      noncomputable abbrev List.multidia {F : Type u_1} [DecidableEq F] [LO.Dia F] (n : ) :
                                                                      List FList F
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                                                                        • One or more equations did not get rendered due to their size.
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                                                                          @[reducible, inline]
                                                                          noncomputable abbrev List.box {F : Type u_1} [DecidableEq F] [LO.Box F] :
                                                                          List FList F
                                                                          Equations
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                                                                            @[reducible, inline]
                                                                            noncomputable abbrev List.dia {F : Type u_1} [DecidableEq F] [LO.Dia F] :
                                                                            List FList F
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                                                                              @[reducible, inline]
                                                                              noncomputable abbrev List.premultibox {F : Type u_1} [DecidableEq F] [LO.Box F] (n : ) :
                                                                              List FList F
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                                                                              Instances For
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                                                                                • One or more equations did not get rendered due to their size.
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                                                                                  @[reducible, inline]
                                                                                  noncomputable abbrev List.premultidia {F : Type u_1} [DecidableEq F] [LO.Dia F] (n : ) :
                                                                                  List FList F
                                                                                  Equations
                                                                                  Instances For
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                                                                                    • One or more equations did not get rendered due to their size.
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                                                                                      @[reducible, inline]
                                                                                      noncomputable abbrev List.prebox {F : Type u_1} [DecidableEq F] [LO.Box F] :
                                                                                      List FList F
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                                                                                        @[reducible, inline]
                                                                                        noncomputable abbrev List.predia {F : Type u_1} [DecidableEq F] [LO.Dia F] :
                                                                                        List FList F
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                                                                                          @[simp]
                                                                                          theorem List.eq_box_multibox_one {F : Type u_1} [DecidableEq F] [LO.Box F] {l : List F} :
                                                                                          @[simp]
                                                                                          theorem List.eq_dia_multidia_one {F : Type u_1} [DecidableEq F] [LO.Dia F] {l : List F} :
                                                                                          @[simp]
                                                                                          theorem List.multibox_nil {F : Type u_1} [DecidableEq F] [LO.Box F] {n : } :
                                                                                          □'^[n][] = []
                                                                                          @[simp]
                                                                                          theorem List.box_nil {F : Type u_1} [DecidableEq F] [LO.Box F] :
                                                                                          □'[] = []
                                                                                          @[simp]
                                                                                          theorem List.multidia_nil {F : Type u_1} [DecidableEq F] [LO.Dia F] {n : } :
                                                                                          ◇'^[n][] = []
                                                                                          @[simp]
                                                                                          theorem List.dia_nil {F : Type u_1} [DecidableEq F] [LO.Dia F] :
                                                                                          ◇'[] = []
                                                                                          @[simp]
                                                                                          theorem List.premultibox_nil {F : Type u_1} [DecidableEq F] [LO.Box F] {n : } :
                                                                                          @[simp]
                                                                                          theorem List.prebox_nil {F : Type u_1} [DecidableEq F] [LO.Box F] :
                                                                                          @[simp]
                                                                                          theorem List.premultidia_nil {F : Type u_1} [DecidableEq F] [LO.Dia F] {n : } :
                                                                                          @[simp]
                                                                                          theorem List.predia_nil {F : Type u_1} [DecidableEq F] [LO.Dia F] :
                                                                                          @[simp]
                                                                                          theorem List.multibox_single {F : Type u_1} [DecidableEq F] [LO.Box F] {p : F} {n : } :
                                                                                          □'^[n][p] = [□^[n]p]
                                                                                          @[simp]
                                                                                          theorem List.box_single {F : Type u_1} [DecidableEq F] [LO.Box F] {p : F} :
                                                                                          □'[p] = [p]
                                                                                          @[simp]
                                                                                          theorem List.multidia_single {F : Type u_1} [DecidableEq F] [LO.Dia F] {p : F} {n : } :
                                                                                          ◇'^[n][p] = [◇^[n]p]
                                                                                          @[simp]
                                                                                          theorem List.dia_single {F : Type u_1} [DecidableEq F] [LO.Dia F] {p : F} :
                                                                                          ◇'[p] = [p]
                                                                                          theorem List.multibox_cons {F : Type u_1} [DecidableEq F] [LO.Box F] {l : List F} {p : F} {n : } (hl : pl) :
                                                                                          (□'^[n](p :: l)).Perm (□^[n]p :: □'^[n]l)
                                                                                          theorem List.box_cons {F : Type u_1} [DecidableEq F] [LO.Box F] {l : List F} {p : F} (hl : pl) :
                                                                                          (□'(p :: l)).Perm (p :: □'l)
                                                                                          theorem List.multidia_cons {F : Type u_1} [DecidableEq F] [LO.Dia F] {l : List F} {p : F} {n : } (hl : pl) :
                                                                                          (◇'^[n](p :: l)).Perm (◇^[n]p :: ◇'^[n]l)
                                                                                          theorem List.dia_cons {F : Type u_1} [DecidableEq F] [LO.Dia F] {l : List F} {p : F} (hl : pl) :
                                                                                          (◇'(p :: l)).Perm (p :: ◇'l)
                                                                                          theorem List.forall_multibox_of_subset_multibox {F : Type u_1} [LO.Box F] {l : List F} {s : Set F} {n : } (h : pl, p □''^[n]s) (p : F) :
                                                                                          p lqs, p = □^[n]q
                                                                                          theorem List.forall_box_of_subset_box {F : Type u_1} [LO.Box F] {l : List F} {s : Set F} (h : pl, p □''s) (p : F) :
                                                                                          p lqs, p = q
                                                                                          theorem List.forall_multidia_of_subset_multidia {F : Type u_1} [LO.Dia F] {l : List F} {s : Set F} {n : } (h : pl, p ◇''^[n]s) (p : F) :
                                                                                          p lqs, p = ◇^[n]q
                                                                                          theorem List.forall_dia_of_subset_dia {F : Type u_1} [LO.Dia F] {l : List F} {s : Set F} (h : pl, p ◇''s) (p : F) :
                                                                                          p lqs, p = q