Documentation

Mathlib.Order.SymmDiff

Symmetric difference and bi-implication #

This file defines the symmetric difference and bi-implication operators in (co-)Heyting algebras.

Examples #

Some examples are

Main declarations #

In generalized Boolean algebras, the symmetric difference operator is:

Notations #

References #

The proof of associativity follows the note "Associativity of the Symmetric Difference of Sets: A Proof from the Book" by John McCuan:

Tags #

boolean ring, generalized boolean algebra, boolean algebra, symmetric difference, bi-implication, Heyting

def symmDiff {α : Type u_2} [Sup α] [SDiff α] (a : α) (b : α) :
α

The symmetric difference operator on a type with and \ is (A \ B) ⊔ (B \ A).

Equations
Instances For
    def bihimp {α : Type u_2} [Inf α] [HImp α] (a : α) (b : α) :
    α

    The Heyting bi-implication is (b ⇨ a) ⊓ (a ⇨ b). This generalizes equivalence of propositions.

    Equations
    Instances For

      Notation for symmDiff

      Equations
      Instances For
        theorem symmDiff_def {α : Type u_2} [Sup α] [SDiff α] (a : α) (b : α) :
        symmDiff a b = a \ b b \ a
        theorem bihimp_def {α : Type u_2} [Inf α] [HImp α] (a : α) (b : α) :
        bihimp a b = (b a) (a b)
        theorem symmDiff_eq_Xor' (p : Prop) (q : Prop) :
        symmDiff p q = Xor' p q
        @[simp]
        theorem bihimp_iff_iff {p : Prop} {q : Prop} :
        bihimp p q (p q)
        @[simp]
        theorem Bool.symmDiff_eq_xor (p : Bool) (q : Bool) :
        symmDiff p q = (p ^^ q)
        @[simp]
        theorem toDual_symmDiff {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a : α) (b : α) :
        OrderDual.toDual (symmDiff a b) = bihimp (OrderDual.toDual a) (OrderDual.toDual b)
        @[simp]
        theorem ofDual_bihimp {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a : αᵒᵈ) (b : αᵒᵈ) :
        OrderDual.ofDual (bihimp a b) = symmDiff (OrderDual.ofDual a) (OrderDual.ofDual b)
        theorem symmDiff_comm {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a : α) (b : α) :
        instance symmDiff_isCommutative {α : Type u_2} [GeneralizedCoheytingAlgebra α] :
        Std.Commutative fun (x1 x2 : α) => symmDiff x1 x2
        Equations
        • =
        @[simp]
        theorem symmDiff_self {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a : α) :
        @[simp]
        theorem symmDiff_bot {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a : α) :
        @[simp]
        theorem bot_symmDiff {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a : α) :
        @[simp]
        theorem symmDiff_eq_bot {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a : α} {b : α} :
        symmDiff a b = a = b
        theorem symmDiff_of_le {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a : α} {b : α} (h : a b) :
        symmDiff a b = b \ a
        theorem symmDiff_of_ge {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a : α} {b : α} (h : b a) :
        symmDiff a b = a \ b
        theorem symmDiff_le {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a : α} {b : α} {c : α} (ha : a b c) (hb : b a c) :
        theorem symmDiff_le_iff {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a : α} {b : α} {c : α} :
        symmDiff a b c a b c b a c
        @[simp]
        theorem symmDiff_le_sup {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a : α} {b : α} :
        symmDiff a b a b
        theorem symmDiff_eq_sup_sdiff_inf {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a : α) (b : α) :
        symmDiff a b = (a b) \ (a b)
        theorem Disjoint.symmDiff_eq_sup {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a : α} {b : α} (h : Disjoint a b) :
        symmDiff a b = a b
        theorem symmDiff_sdiff {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a : α) (b : α) (c : α) :
        symmDiff a b \ c = a \ (b c) b \ (a c)
        @[simp]
        theorem symmDiff_sdiff_inf {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a : α) (b : α) :
        symmDiff a b \ (a b) = symmDiff a b
        @[simp]
        theorem symmDiff_sdiff_eq_sup {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a : α) (b : α) :
        symmDiff a (b \ a) = a b
        @[simp]
        theorem sdiff_symmDiff_eq_sup {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a : α) (b : α) :
        symmDiff (a \ b) b = a b
        @[simp]
        theorem symmDiff_sup_inf {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a : α) (b : α) :
        symmDiff a b a b = a b
        @[simp]
        theorem inf_sup_symmDiff {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a : α) (b : α) :
        a b symmDiff a b = a b
        @[simp]
        theorem symmDiff_symmDiff_inf {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a : α) (b : α) :
        symmDiff (symmDiff a b) (a b) = a b
        @[simp]
        theorem inf_symmDiff_symmDiff {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a : α) (b : α) :
        symmDiff (a b) (symmDiff a b) = a b
        theorem symmDiff_triangle {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a : α) (b : α) (c : α) :
        theorem le_symmDiff_sup_right {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a : α) (b : α) :
        a symmDiff a b b
        theorem le_symmDiff_sup_left {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a : α) (b : α) :
        b symmDiff a b a
        @[simp]
        theorem toDual_bihimp {α : Type u_2} [GeneralizedHeytingAlgebra α] (a : α) (b : α) :
        OrderDual.toDual (bihimp a b) = symmDiff (OrderDual.toDual a) (OrderDual.toDual b)
        @[simp]
        theorem ofDual_symmDiff {α : Type u_2} [GeneralizedHeytingAlgebra α] (a : αᵒᵈ) (b : αᵒᵈ) :
        OrderDual.ofDual (symmDiff a b) = bihimp (OrderDual.ofDual a) (OrderDual.ofDual b)
        theorem bihimp_comm {α : Type u_2} [GeneralizedHeytingAlgebra α] (a : α) (b : α) :
        bihimp a b = bihimp b a
        instance bihimp_isCommutative {α : Type u_2} [GeneralizedHeytingAlgebra α] :
        Std.Commutative fun (x1 x2 : α) => bihimp x1 x2
        Equations
        • =
        @[simp]
        theorem bihimp_self {α : Type u_2} [GeneralizedHeytingAlgebra α] (a : α) :
        @[simp]
        theorem bihimp_top {α : Type u_2} [GeneralizedHeytingAlgebra α] (a : α) :
        @[simp]
        theorem top_bihimp {α : Type u_2} [GeneralizedHeytingAlgebra α] (a : α) :
        @[simp]
        theorem bihimp_eq_top {α : Type u_2} [GeneralizedHeytingAlgebra α] {a : α} {b : α} :
        bihimp a b = a = b
        theorem bihimp_of_le {α : Type u_2} [GeneralizedHeytingAlgebra α] {a : α} {b : α} (h : a b) :
        bihimp a b = b a
        theorem bihimp_of_ge {α : Type u_2} [GeneralizedHeytingAlgebra α] {a : α} {b : α} (h : b a) :
        bihimp a b = a b
        theorem le_bihimp {α : Type u_2} [GeneralizedHeytingAlgebra α] {a : α} {b : α} {c : α} (hb : a b c) (hc : a c b) :
        a bihimp b c
        theorem le_bihimp_iff {α : Type u_2} [GeneralizedHeytingAlgebra α] {a : α} {b : α} {c : α} :
        a bihimp b c a b c a c b
        @[simp]
        theorem inf_le_bihimp {α : Type u_2} [GeneralizedHeytingAlgebra α] {a : α} {b : α} :
        a b bihimp a b
        theorem bihimp_eq_inf_himp_inf {α : Type u_2} [GeneralizedHeytingAlgebra α] (a : α) (b : α) :
        bihimp a b = a b a b
        theorem Codisjoint.bihimp_eq_inf {α : Type u_2} [GeneralizedHeytingAlgebra α] {a : α} {b : α} (h : Codisjoint a b) :
        bihimp a b = a b
        theorem himp_bihimp {α : Type u_2} [GeneralizedHeytingAlgebra α] (a : α) (b : α) (c : α) :
        a bihimp b c = (a c b) (a b c)
        @[simp]
        theorem sup_himp_bihimp {α : Type u_2} [GeneralizedHeytingAlgebra α] (a : α) (b : α) :
        a b bihimp a b = bihimp a b
        @[simp]
        theorem bihimp_himp_eq_inf {α : Type u_2} [GeneralizedHeytingAlgebra α] (a : α) (b : α) :
        bihimp a (a b) = a b
        @[simp]
        theorem himp_bihimp_eq_inf {α : Type u_2} [GeneralizedHeytingAlgebra α] (a : α) (b : α) :
        bihimp (b a) b = a b
        @[simp]
        theorem bihimp_inf_sup {α : Type u_2} [GeneralizedHeytingAlgebra α] (a : α) (b : α) :
        bihimp a b (a b) = a b
        @[simp]
        theorem sup_inf_bihimp {α : Type u_2} [GeneralizedHeytingAlgebra α] (a : α) (b : α) :
        (a b) bihimp a b = a b
        @[simp]
        theorem bihimp_bihimp_sup {α : Type u_2} [GeneralizedHeytingAlgebra α] (a : α) (b : α) :
        bihimp (bihimp a b) (a b) = a b
        @[simp]
        theorem sup_bihimp_bihimp {α : Type u_2} [GeneralizedHeytingAlgebra α] (a : α) (b : α) :
        bihimp (a b) (bihimp a b) = a b
        theorem bihimp_triangle {α : Type u_2} [GeneralizedHeytingAlgebra α] (a : α) (b : α) (c : α) :
        bihimp a b bihimp b c bihimp a c
        @[simp]
        theorem symmDiff_top' {α : Type u_2} [CoheytingAlgebra α] (a : α) :
        @[simp]
        theorem top_symmDiff' {α : Type u_2} [CoheytingAlgebra α] (a : α) :
        @[simp]
        theorem hnot_symmDiff_self {α : Type u_2} [CoheytingAlgebra α] (a : α) :
        @[simp]
        theorem symmDiff_hnot_self {α : Type u_2} [CoheytingAlgebra α] (a : α) :
        theorem IsCompl.symmDiff_eq_top {α : Type u_2} [CoheytingAlgebra α] {a : α} {b : α} (h : IsCompl a b) :
        @[simp]
        theorem bihimp_bot {α : Type u_2} [HeytingAlgebra α] (a : α) :
        @[simp]
        theorem bot_bihimp {α : Type u_2} [HeytingAlgebra α] (a : α) :
        @[simp]
        theorem compl_bihimp_self {α : Type u_2} [HeytingAlgebra α] (a : α) :
        @[simp]
        theorem bihimp_hnot_self {α : Type u_2} [HeytingAlgebra α] (a : α) :
        theorem IsCompl.bihimp_eq_bot {α : Type u_2} [HeytingAlgebra α] {a : α} {b : α} (h : IsCompl a b) :
        @[simp]
        theorem sup_sdiff_symmDiff {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) (b : α) :
        (a b) \ symmDiff a b = a b
        theorem disjoint_symmDiff_inf {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) (b : α) :
        Disjoint (symmDiff a b) (a b)
        theorem inf_symmDiff_distrib_left {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) (b : α) (c : α) :
        a symmDiff b c = symmDiff (a b) (a c)
        theorem inf_symmDiff_distrib_right {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) (b : α) (c : α) :
        symmDiff a b c = symmDiff (a c) (b c)
        theorem sdiff_symmDiff {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) (b : α) (c : α) :
        c \ symmDiff a b = c a b c \ a c \ b
        theorem sdiff_symmDiff' {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) (b : α) (c : α) :
        c \ symmDiff a b = c a b c \ (a b)
        @[simp]
        theorem symmDiff_sdiff_left {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) (b : α) :
        symmDiff a b \ a = b \ a
        @[simp]
        theorem symmDiff_sdiff_right {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) (b : α) :
        symmDiff a b \ b = a \ b
        @[simp]
        theorem sdiff_symmDiff_left {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) (b : α) :
        a \ symmDiff a b = a b
        @[simp]
        theorem sdiff_symmDiff_right {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) (b : α) :
        b \ symmDiff a b = a b
        theorem symmDiff_eq_sup {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) (b : α) :
        symmDiff a b = a b Disjoint a b
        @[simp]
        theorem le_symmDiff_iff_left {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) (b : α) :
        @[simp]
        theorem le_symmDiff_iff_right {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) (b : α) :
        theorem symmDiff_symmDiff_left {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) (b : α) (c : α) :
        symmDiff (symmDiff a b) c = a \ (b c) b \ (a c) c \ (a b) a b c
        theorem symmDiff_symmDiff_right {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) (b : α) (c : α) :
        symmDiff a (symmDiff b c) = a \ (b c) b \ (a c) c \ (a b) a b c
        theorem symmDiff_assoc {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) (b : α) (c : α) :
        instance symmDiff_isAssociative {α : Type u_2} [GeneralizedBooleanAlgebra α] :
        Std.Associative fun (x1 x2 : α) => symmDiff x1 x2
        Equations
        • =
        theorem symmDiff_left_comm {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) (b : α) (c : α) :
        theorem symmDiff_right_comm {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) (b : α) (c : α) :
        theorem symmDiff_symmDiff_symmDiff_comm {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) (b : α) (c : α) (d : α) :
        @[simp]
        theorem symmDiff_symmDiff_cancel_left {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) (b : α) :
        symmDiff a (symmDiff a b) = b
        @[simp]
        theorem symmDiff_symmDiff_cancel_right {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) (b : α) :
        symmDiff (symmDiff b a) a = b
        @[simp]
        theorem symmDiff_symmDiff_self' {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) (b : α) :
        symmDiff (symmDiff a b) a = b
        theorem symmDiff_left_involutive {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) :
        Function.Involutive fun (x : α) => symmDiff x a
        theorem symmDiff_right_involutive {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) :
        Function.Involutive fun (x : α) => symmDiff a x
        theorem symmDiff_left_injective {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) :
        Function.Injective fun (x : α) => symmDiff x a
        theorem symmDiff_right_injective {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) :
        Function.Injective fun (x : α) => symmDiff a x
        theorem symmDiff_left_surjective {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) :
        Function.Surjective fun (x : α) => symmDiff x a
        theorem symmDiff_right_surjective {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α) :
        Function.Surjective fun (x : α) => symmDiff a x
        @[simp]
        theorem symmDiff_left_inj {α : Type u_2} [GeneralizedBooleanAlgebra α] {a : α} {b : α} {c : α} :
        symmDiff a b = symmDiff c b a = c
        @[simp]
        theorem symmDiff_right_inj {α : Type u_2} [GeneralizedBooleanAlgebra α] {a : α} {b : α} {c : α} :
        symmDiff a b = symmDiff a c b = c
        @[simp]
        theorem symmDiff_eq_left {α : Type u_2} [GeneralizedBooleanAlgebra α] {a : α} {b : α} :
        symmDiff a b = a b =
        @[simp]
        theorem symmDiff_eq_right {α : Type u_2} [GeneralizedBooleanAlgebra α] {a : α} {b : α} :
        symmDiff a b = b a =
        theorem Disjoint.symmDiff_left {α : Type u_2} [GeneralizedBooleanAlgebra α] {a : α} {b : α} {c : α} (ha : Disjoint a c) (hb : Disjoint b c) :
        theorem Disjoint.symmDiff_right {α : Type u_2} [GeneralizedBooleanAlgebra α] {a : α} {b : α} {c : α} (ha : Disjoint a b) (hb : Disjoint a c) :
        theorem symmDiff_eq_iff_sdiff_eq {α : Type u_2} [GeneralizedBooleanAlgebra α] {a : α} {b : α} {c : α} (ha : a c) :
        symmDiff a b = c c \ a = b

        CogeneralizedBooleanAlgebra isn't actually a typeclass, but the lemmas in here are dual to the GeneralizedBooleanAlgebra ones

        @[simp]
        theorem inf_himp_bihimp {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) :
        bihimp a b a b = a b
        theorem codisjoint_bihimp_sup {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) :
        Codisjoint (bihimp a b) (a b)
        @[simp]
        theorem himp_bihimp_left {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) :
        a bihimp a b = a b
        @[simp]
        theorem himp_bihimp_right {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) :
        b bihimp a b = b a
        @[simp]
        theorem bihimp_himp_left {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) :
        bihimp a b a = a b
        @[simp]
        theorem bihimp_himp_right {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) :
        bihimp a b b = a b
        @[simp]
        theorem bihimp_eq_inf {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) :
        bihimp a b = a b Codisjoint a b
        @[simp]
        theorem bihimp_le_iff_left {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) :
        @[simp]
        theorem bihimp_le_iff_right {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) :
        theorem bihimp_assoc {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) (c : α) :
        bihimp (bihimp a b) c = bihimp a (bihimp b c)
        instance bihimp_isAssociative {α : Type u_2} [BooleanAlgebra α] :
        Std.Associative fun (x1 x2 : α) => bihimp x1 x2
        Equations
        • =
        theorem bihimp_left_comm {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) (c : α) :
        bihimp a (bihimp b c) = bihimp b (bihimp a c)
        theorem bihimp_right_comm {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) (c : α) :
        bihimp (bihimp a b) c = bihimp (bihimp a c) b
        theorem bihimp_bihimp_bihimp_comm {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) (c : α) (d : α) :
        bihimp (bihimp a b) (bihimp c d) = bihimp (bihimp a c) (bihimp b d)
        @[simp]
        theorem bihimp_bihimp_cancel_left {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) :
        bihimp a (bihimp a b) = b
        @[simp]
        theorem bihimp_bihimp_cancel_right {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) :
        bihimp (bihimp b a) a = b
        @[simp]
        theorem bihimp_bihimp_self {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) :
        bihimp (bihimp a b) a = b
        theorem bihimp_left_involutive {α : Type u_2} [BooleanAlgebra α] (a : α) :
        Function.Involutive fun (x : α) => bihimp x a
        theorem bihimp_right_involutive {α : Type u_2} [BooleanAlgebra α] (a : α) :
        Function.Involutive fun (x : α) => bihimp a x
        theorem bihimp_left_injective {α : Type u_2} [BooleanAlgebra α] (a : α) :
        Function.Injective fun (x : α) => bihimp x a
        theorem bihimp_right_injective {α : Type u_2} [BooleanAlgebra α] (a : α) :
        Function.Injective fun (x : α) => bihimp a x
        theorem bihimp_left_surjective {α : Type u_2} [BooleanAlgebra α] (a : α) :
        Function.Surjective fun (x : α) => bihimp x a
        theorem bihimp_right_surjective {α : Type u_2} [BooleanAlgebra α] (a : α) :
        Function.Surjective fun (x : α) => bihimp a x
        @[simp]
        theorem bihimp_left_inj {α : Type u_2} [BooleanAlgebra α] {a : α} {b : α} {c : α} :
        bihimp a b = bihimp c b a = c
        @[simp]
        theorem bihimp_right_inj {α : Type u_2} [BooleanAlgebra α] {a : α} {b : α} {c : α} :
        bihimp a b = bihimp a c b = c
        @[simp]
        theorem bihimp_eq_left {α : Type u_2} [BooleanAlgebra α] {a : α} {b : α} :
        bihimp a b = a b =
        @[simp]
        theorem bihimp_eq_right {α : Type u_2} [BooleanAlgebra α] {a : α} {b : α} :
        bihimp a b = b a =
        theorem Codisjoint.bihimp_left {α : Type u_2} [BooleanAlgebra α] {a : α} {b : α} {c : α} (ha : Codisjoint a c) (hb : Codisjoint b c) :
        theorem Codisjoint.bihimp_right {α : Type u_2} [BooleanAlgebra α] {a : α} {b : α} {c : α} (ha : Codisjoint a b) (hb : Codisjoint a c) :
        theorem symmDiff_eq {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) :
        symmDiff a b = a b b a
        theorem bihimp_eq {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) :
        bihimp a b = (a b) (b a)
        theorem symmDiff_eq' {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) :
        symmDiff a b = (a b) (a b)
        theorem bihimp_eq' {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) :
        bihimp a b = a b a b
        theorem symmDiff_top {α : Type u_2} [BooleanAlgebra α] (a : α) :
        theorem top_symmDiff {α : Type u_2} [BooleanAlgebra α] (a : α) :
        @[simp]
        theorem compl_symmDiff {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) :
        (symmDiff a b) = bihimp a b
        @[simp]
        theorem compl_bihimp {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) :
        (bihimp a b) = symmDiff a b
        @[simp]
        theorem compl_symmDiff_compl {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) :
        @[simp]
        theorem compl_bihimp_compl {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) :
        @[simp]
        theorem symmDiff_eq_top {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) :
        @[simp]
        theorem bihimp_eq_bot {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) :
        @[simp]
        theorem compl_symmDiff_self {α : Type u_2} [BooleanAlgebra α] (a : α) :
        @[simp]
        theorem symmDiff_compl_self {α : Type u_2} [BooleanAlgebra α] (a : α) :
        theorem symmDiff_symmDiff_right' {α : Type u_2} [BooleanAlgebra α] (a : α) (b : α) (c : α) :
        theorem Disjoint.le_symmDiff_sup_symmDiff_left {α : Type u_2} [BooleanAlgebra α] {a : α} {b : α} {c : α} (h : Disjoint a b) :
        theorem Disjoint.le_symmDiff_sup_symmDiff_right {α : Type u_2} [BooleanAlgebra α] {a : α} {b : α} {c : α} (h : Disjoint b c) :
        theorem Codisjoint.bihimp_inf_bihimp_le_left {α : Type u_2} [BooleanAlgebra α] {a : α} {b : α} {c : α} (h : Codisjoint a b) :
        bihimp a c bihimp b c c
        theorem Codisjoint.bihimp_inf_bihimp_le_right {α : Type u_2} [BooleanAlgebra α] {a : α} {b : α} {c : α} (h : Codisjoint b c) :
        bihimp a b bihimp a c a

        Prod #

        @[simp]
        theorem symmDiff_fst {α : Type u_2} {β : Type u_3} [GeneralizedCoheytingAlgebra α] [GeneralizedCoheytingAlgebra β] (a : α × β) (b : α × β) :
        (symmDiff a b).1 = symmDiff a.1 b.1
        @[simp]
        theorem symmDiff_snd {α : Type u_2} {β : Type u_3} [GeneralizedCoheytingAlgebra α] [GeneralizedCoheytingAlgebra β] (a : α × β) (b : α × β) :
        (symmDiff a b).2 = symmDiff a.2 b.2
        @[simp]
        theorem bihimp_fst {α : Type u_2} {β : Type u_3} [GeneralizedHeytingAlgebra α] [GeneralizedHeytingAlgebra β] (a : α × β) (b : α × β) :
        (bihimp a b).1 = bihimp a.1 b.1
        @[simp]
        theorem bihimp_snd {α : Type u_2} {β : Type u_3} [GeneralizedHeytingAlgebra α] [GeneralizedHeytingAlgebra β] (a : α × β) (b : α × β) :
        (bihimp a b).2 = bihimp a.2 b.2

        Pi #

        theorem Pi.symmDiff_def {ι : Type u_1} {π : ιType u_4} [(i : ι) → GeneralizedCoheytingAlgebra (π i)] (a : (i : ι) → π i) (b : (i : ι) → π i) :
        symmDiff a b = fun (i : ι) => symmDiff (a i) (b i)
        theorem Pi.bihimp_def {ι : Type u_1} {π : ιType u_4} [(i : ι) → GeneralizedHeytingAlgebra (π i)] (a : (i : ι) → π i) (b : (i : ι) → π i) :
        bihimp a b = fun (i : ι) => bihimp (a i) (b i)
        @[simp]
        theorem Pi.symmDiff_apply {ι : Type u_1} {π : ιType u_4} [(i : ι) → GeneralizedCoheytingAlgebra (π i)] (a : (i : ι) → π i) (b : (i : ι) → π i) (i : ι) :
        symmDiff a b i = symmDiff (a i) (b i)
        @[simp]
        theorem Pi.bihimp_apply {ι : Type u_1} {π : ιType u_4} [(i : ι) → GeneralizedHeytingAlgebra (π i)] (a : (i : ι) → π i) (b : (i : ι) → π i) (i : ι) :
        bihimp a b i = bihimp (a i) (b i)