# 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

• The symmetric difference of two sets is the set of elements that are in either but not both.
• The symmetric difference on propositions is Xor'.
• The symmetric difference on Bool is Bool.xor.
• The equivalence of propositions. Two propositions are equivalent if they imply each other.
• The symmetric difference translates to addition when considering a Boolean algebra as a Boolean ring.

## Main declarations #

• symmDiff: The symmetric difference operator, defined as (a \ b) ⊔ (b \ a)⊔ (b \ a)
• bihimp: The bi-implication operator, defined as (b ⇨ a) ⊓ (a ⇨ b)⇨ a) ⊓ (a ⇨ b)⊓ (a ⇨ b)⇨ b)

In generalized Boolean algebras, the symmetric difference operator is:

• symmDiff_comm: commutative, and
• symmDiff_assoc: associative.

## Notations #

• a ∆ b∆ b: symmDiff a b
• a ⇔ b⇔ b: bihimp a b

## 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_1} [inst : Sup α] [inst : ] (a : α) (b : α) :
α

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

Equations
def bihimp {α : Type u_1} [inst : Inf α] [inst : HImp α] (a : α) (b : α) :
α

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

Equations

Notation for symmDiff

Equations

Notation for bihimp

Equations
theorem symmDiff_def {α : Type u_1} [inst : Sup α] [inst : ] (a : α) (b : α) :
a b = a \ b b \ a
theorem bihimp_def {α : Type u_1} [inst : Inf α] [inst : HImp α] (a : α) (b : α) :
a b = (b a) (a b)
theorem symmDiff_eq_Xor' (p : Prop) (q : Prop) :
p q = Xor' p q
@[simp]
theorem bihimp_iff_iff {p : Prop} {q : Prop} :
p q (p q)
@[simp]
theorem Bool.symmDiff_eq_xor (p : Bool) (q : Bool) :
p q = xor p q
@[simp]
theorem toDual_symmDiff {α : Type u_1} [inst : ] (a : α) (b : α) :
OrderDual.toDual (a b) = OrderDual.toDual a OrderDual.toDual b
@[simp]
theorem ofDual_bihimp {α : Type u_1} [inst : ] (a : αᵒᵈ) (b : αᵒᵈ) :
OrderDual.ofDual (a b) = OrderDual.ofDual a OrderDual.ofDual b
theorem symmDiff_comm {α : Type u_1} [inst : ] (a : α) (b : α) :
a b = b a
instance symmDiff_isCommutative {α : Type u_1} [inst : ] :
IsCommutative α fun x x_1 => x x_1
Equations
@[simp]
theorem symmDiff_self {α : Type u_1} [inst : ] (a : α) :
a a =
@[simp]
theorem symmDiff_bot {α : Type u_1} [inst : ] (a : α) :
a = a
@[simp]
theorem bot_symmDiff {α : Type u_1} [inst : ] (a : α) :
a = a
@[simp]
theorem symmDiff_eq_bot {α : Type u_1} [inst : ] {a : α} {b : α} :
a b = a = b
theorem symmDiff_of_le {α : Type u_1} [inst : ] {a : α} {b : α} (h : a b) :
a b = b \ a
theorem symmDiff_of_ge {α : Type u_1} [inst : ] {a : α} {b : α} (h : b a) :
a b = a \ b
theorem symmDiff_le {α : Type u_1} [inst : ] {a : α} {b : α} {c : α} (ha : a b c) (hb : b a c) :
a b c
theorem symmDiff_le_iff {α : Type u_1} [inst : ] {a : α} {b : α} {c : α} :
a b c a b c b a c
@[simp]
theorem symmDiff_le_sup {α : Type u_1} [inst : ] {a : α} {b : α} :
a b a b
theorem symmDiff_eq_sup_sdiff_inf {α : Type u_1} [inst : ] (a : α) (b : α) :
a b = (a b) \ (a b)
theorem Disjoint.symmDiff_eq_sup {α : Type u_1} [inst : ] {a : α} {b : α} (h : Disjoint a b) :
a b = a b
theorem symmDiff_sdiff {α : Type u_1} [inst : ] (a : α) (b : α) (c : α) :
a b \ c = a \ (b c) b \ (a c)
@[simp]
theorem symmDiff_sdiff_inf {α : Type u_1} [inst : ] (a : α) (b : α) :
a b \ (a b) = a b
@[simp]
theorem symmDiff_sdiff_eq_sup {α : Type u_1} [inst : ] (a : α) (b : α) :
a (b \ a) = a b
@[simp]
theorem sdiff_symmDiff_eq_sup {α : Type u_1} [inst : ] (a : α) (b : α) :
(a \ b) b = a b
@[simp]
theorem symmDiff_sup_inf {α : Type u_1} [inst : ] (a : α) (b : α) :
a b a b = a b
@[simp]
theorem inf_sup_symmDiff {α : Type u_1} [inst : ] (a : α) (b : α) :
a b a b = a b
@[simp]
theorem symmDiff_symmDiff_inf {α : Type u_1} [inst : ] (a : α) (b : α) :
a b (a b) = a b
@[simp]
theorem inf_symmDiff_symmDiff {α : Type u_1} [inst : ] (a : α) (b : α) :
(a b) (a b) = a b
theorem symmDiff_triangle {α : Type u_1} [inst : ] (a : α) (b : α) (c : α) :
a c a b b c
@[simp]
theorem toDual_bihimp {α : Type u_1} [inst : ] (a : α) (b : α) :
OrderDual.toDual (a b) = OrderDual.toDual a OrderDual.toDual b
@[simp]
theorem ofDual_symmDiff {α : Type u_1} [inst : ] (a : αᵒᵈ) (b : αᵒᵈ) :
OrderDual.ofDual (a b) = OrderDual.ofDual a OrderDual.ofDual b
theorem bihimp_comm {α : Type u_1} [inst : ] (a : α) (b : α) :
a b = b a
instance bihimp_isCommutative {α : Type u_1} [inst : ] :
IsCommutative α fun x x_1 => x x_1
Equations
@[simp]
theorem bihimp_self {α : Type u_1} [inst : ] (a : α) :
a a =
@[simp]
theorem bihimp_top {α : Type u_1} [inst : ] (a : α) :
a = a
@[simp]
theorem top_bihimp {α : Type u_1} [inst : ] (a : α) :
a = a
@[simp]
theorem bihimp_eq_top {α : Type u_1} [inst : ] {a : α} {b : α} :
a b = a = b
theorem bihimp_of_le {α : Type u_1} [inst : ] {a : α} {b : α} (h : a b) :
a b = b a
theorem bihimp_of_ge {α : Type u_1} [inst : ] {a : α} {b : α} (h : b a) :
a b = a b
theorem le_bihimp {α : Type u_1} [inst : ] {a : α} {b : α} {c : α} (hb : a b c) (hc : a c b) :
a b c
theorem le_bihimp_iff {α : Type u_1} [inst : ] {a : α} {b : α} {c : α} :
a b c a b c a c b
@[simp]
theorem inf_le_bihimp {α : Type u_1} [inst : ] {a : α} {b : α} :
a b a b
theorem bihimp_eq_inf_himp_inf {α : Type u_1} [inst : ] (a : α) (b : α) :
a b = a b a b
theorem Codisjoint.bihimp_eq_inf {α : Type u_1} [inst : ] {a : α} {b : α} (h : ) :
a b = a b
theorem himp_bihimp {α : Type u_1} [inst : ] (a : α) (b : α) (c : α) :
a b c = (a c b) (a b c)
@[simp]
theorem sup_himp_bihimp {α : Type u_1} [inst : ] (a : α) (b : α) :
a b a b = a b
@[simp]
theorem bihimp_himp_eq_inf {α : Type u_1} [inst : ] (a : α) (b : α) :
a (a b) = a b
@[simp]
theorem himp_bihimp_eq_inf {α : Type u_1} [inst : ] (a : α) (b : α) :
(b a) b = a b
@[simp]
theorem bihimp_inf_sup {α : Type u_1} [inst : ] (a : α) (b : α) :
a b (a b) = a b
@[simp]
theorem sup_inf_bihimp {α : Type u_1} [inst : ] (a : α) (b : α) :
(a b) a b = a b
@[simp]
theorem bihimp_bihimp_sup {α : Type u_1} [inst : ] (a : α) (b : α) :
a b (a b) = a b
@[simp]
theorem sup_bihimp_bihimp {α : Type u_1} [inst : ] (a : α) (b : α) :
(a b) (a b) = a b
theorem bihimp_triangle {α : Type u_1} [inst : ] (a : α) (b : α) (c : α) :
a b b c a c
@[simp]
theorem symmDiff_top' {α : Type u_1} [inst : ] (a : α) :
@[simp]
theorem top_symmDiff' {α : Type u_1} [inst : ] (a : α) :
@[simp]
theorem hnot_symmDiff_self {α : Type u_1} [inst : ] (a : α) :
(a) a =
@[simp]
theorem symmDiff_hnot_self {α : Type u_1} [inst : ] (a : α) :
a (a) =
theorem IsCompl.symmDiff_eq_top {α : Type u_1} [inst : ] {a : α} {b : α} (h : IsCompl a b) :
a b =
@[simp]
theorem bihimp_bot {α : Type u_1} [inst : ] (a : α) :
@[simp]
theorem bot_bihimp {α : Type u_1} [inst : ] (a : α) :
@[simp]
theorem compl_bihimp_self {α : Type u_1} [inst : ] (a : α) :
@[simp]
theorem bihimp_hnot_self {α : Type u_1} [inst : ] (a : α) :
theorem IsCompl.bihimp_eq_bot {α : Type u_1} [inst : ] {a : α} {b : α} (h : IsCompl a b) :
a b =
@[simp]
theorem sup_sdiff_symmDiff {α : Type u_1} [inst : ] (a : α) (b : α) :
(a b) \ a b = a b
theorem disjoint_symmDiff_inf {α : Type u_1} [inst : ] (a : α) (b : α) :
Disjoint (a b) (a b)
theorem inf_symmDiff_distrib_left {α : Type u_1} [inst : ] (a : α) (b : α) (c : α) :
a b c = (a b) (a c)
theorem inf_symmDiff_distrib_right {α : Type u_1} [inst : ] (a : α) (b : α) (c : α) :
a b c = (a c) (b c)
theorem sdiff_symmDiff {α : Type u_1} [inst : ] (a : α) (b : α) (c : α) :
c \ a b = c a b c \ a c \ b
theorem sdiff_symmDiff' {α : Type u_1} [inst : ] (a : α) (b : α) (c : α) :
c \ a b = c a b c \ (a b)
@[simp]
theorem symmDiff_sdiff_left {α : Type u_1} [inst : ] (a : α) (b : α) :
a b \ a = b \ a
@[simp]
theorem symmDiff_sdiff_right {α : Type u_1} [inst : ] (a : α) (b : α) :
a b \ b = a \ b
@[simp]
theorem sdiff_symmDiff_left {α : Type u_1} [inst : ] (a : α) (b : α) :
a \ a b = a b
@[simp]
theorem sdiff_symmDiff_right {α : Type u_1} [inst : ] (a : α) (b : α) :
b \ a b = a b
theorem symmDiff_eq_sup {α : Type u_1} [inst : ] (a : α) (b : α) :
a b = a b Disjoint a b
@[simp]
theorem le_symmDiff_iff_left {α : Type u_1} [inst : ] (a : α) (b : α) :
a a b Disjoint a b
@[simp]
theorem le_symmDiff_iff_right {α : Type u_1} [inst : ] (a : α) (b : α) :
b a b Disjoint a b
theorem symmDiff_symmDiff_left {α : Type u_1} [inst : ] (a : α) (b : α) (c : α) :
a b c = a \ (b c) b \ (a c) c \ (a b) a b c
theorem symmDiff_symmDiff_right {α : Type u_1} [inst : ] (a : α) (b : α) (c : α) :
a (b c) = a \ (b c) b \ (a c) c \ (a b) a b c
theorem symmDiff_assoc {α : Type u_1} [inst : ] (a : α) (b : α) (c : α) :
a b c = a (b c)
instance symmDiff_isAssociative {α : Type u_1} [inst : ] :
IsAssociative α fun x x_1 => x x_1
Equations
theorem symmDiff_left_comm {α : Type u_1} [inst : ] (a : α) (b : α) (c : α) :
a (b c) = b (a c)
theorem symmDiff_right_comm {α : Type u_1} [inst : ] (a : α) (b : α) (c : α) :
a b c = a c b
theorem symmDiff_symmDiff_symmDiff_comm {α : Type u_1} [inst : ] (a : α) (b : α) (c : α) (d : α) :
a b (c d) = a c (b d)
@[simp]
theorem symmDiff_symmDiff_cancel_left {α : Type u_1} [inst : ] (a : α) (b : α) :
a (a b) = b
@[simp]
theorem symmDiff_symmDiff_cancel_right {α : Type u_1} [inst : ] (a : α) (b : α) :
b a a = b
@[simp]
theorem symmDiff_symmDiff_self' {α : Type u_1} [inst : ] (a : α) (b : α) :
a b a = b
theorem symmDiff_left_involutive {α : Type u_1} [inst : ] (a : α) :
theorem symmDiff_right_involutive {α : Type u_1} [inst : ] (a : α) :
Function.Involutive ((fun x x_1 => x x_1) a)
theorem symmDiff_left_injective {α : Type u_1} [inst : ] (a : α) :
Function.Injective fun x => x a
theorem symmDiff_right_injective {α : Type u_1} [inst : ] (a : α) :
Function.Injective ((fun x x_1 => x x_1) a)
theorem symmDiff_left_surjective {α : Type u_1} [inst : ] (a : α) :
theorem symmDiff_right_surjective {α : Type u_1} [inst : ] (a : α) :
Function.Surjective ((fun x x_1 => x x_1) a)
@[simp]
theorem symmDiff_left_inj {α : Type u_1} [inst : ] {a : α} {b : α} {c : α} :
a b = c b a = c
@[simp]
theorem symmDiff_right_inj {α : Type u_1} [inst : ] {a : α} {b : α} {c : α} :
a b = a c b = c
@[simp]
theorem symmDiff_eq_left {α : Type u_1} [inst : ] {a : α} {b : α} :
a b = a b =
@[simp]
theorem symmDiff_eq_right {α : Type u_1} [inst : ] {a : α} {b : α} :
a b = b a =
theorem Disjoint.symmDiff_left {α : Type u_1} [inst : ] {a : α} {b : α} {c : α} (ha : Disjoint a c) (hb : Disjoint b c) :
Disjoint (a b) c
theorem Disjoint.symmDiff_right {α : Type u_1} [inst : ] {a : α} {b : α} {c : α} (ha : Disjoint a b) (hb : Disjoint a c) :
Disjoint a (b c)
theorem symmDiff_eq_iff_sdiff_eq {α : Type u_1} [inst : ] {a : α} {b : α} {c : α} (ha : a c) :
a b = c c \ a = b
@[simp]
theorem inf_himp_bihimp {α : Type u_1} [inst : ] (a : α) (b : α) :
a b a b = a b
theorem codisjoint_bihimp_sup {α : Type u_1} [inst : ] (a : α) (b : α) :
Codisjoint (a b) (a b)
@[simp]
theorem himp_bihimp_left {α : Type u_1} [inst : ] (a : α) (b : α) :
a a b = a b
@[simp]
theorem himp_bihimp_right {α : Type u_1} [inst : ] (a : α) (b : α) :
b a b = b a
@[simp]
theorem bihimp_himp_left {α : Type u_1} [inst : ] (a : α) (b : α) :
a b a = a b
@[simp]
theorem bihimp_himp_right {α : Type u_1} [inst : ] (a : α) (b : α) :
a b b = a b
@[simp]
theorem bihimp_eq_inf {α : Type u_1} [inst : ] (a : α) (b : α) :
a b = a b
@[simp]
theorem bihimp_le_iff_left {α : Type u_1} [inst : ] (a : α) (b : α) :
a b a
@[simp]
theorem bihimp_le_iff_right {α : Type u_1} [inst : ] (a : α) (b : α) :
a b b
theorem bihimp_assoc {α : Type u_1} [inst : ] (a : α) (b : α) (c : α) :
a b c = a (b c)
instance bihimp_isAssociative {α : Type u_1} [inst : ] :
IsAssociative α fun x x_1 => x x_1
Equations
theorem bihimp_left_comm {α : Type u_1} [inst : ] (a : α) (b : α) (c : α) :
a (b c) = b (a c)
theorem bihimp_right_comm {α : Type u_1} [inst : ] (a : α) (b : α) (c : α) :
a b c = a c b
theorem bihimp_bihimp_bihimp_comm {α : Type u_1} [inst : ] (a : α) (b : α) (c : α) (d : α) :
a b (c d) = a c (b d)
@[simp]
theorem bihimp_bihimp_cancel_left {α : Type u_1} [inst : ] (a : α) (b : α) :
a (a b) = b
@[simp]
theorem bihimp_bihimp_cancel_right {α : Type u_1} [inst : ] (a : α) (b : α) :
b a a = b
@[simp]
theorem bihimp_bihimp_self {α : Type u_1} [inst : ] (a : α) (b : α) :
a b a = b
theorem bihimp_left_involutive {α : Type u_1} [inst : ] (a : α) :
theorem bihimp_right_involutive {α : Type u_1} [inst : ] (a : α) :
Function.Involutive ((fun x x_1 => x x_1) a)
theorem bihimp_left_injective {α : Type u_1} [inst : ] (a : α) :
Function.Injective fun x => x a
theorem bihimp_right_injective {α : Type u_1} [inst : ] (a : α) :
Function.Injective ((fun x x_1 => x x_1) a)
theorem bihimp_left_surjective {α : Type u_1} [inst : ] (a : α) :
theorem bihimp_right_surjective {α : Type u_1} [inst : ] (a : α) :
Function.Surjective ((fun x x_1 => x x_1) a)
@[simp]
theorem bihimp_left_inj {α : Type u_1} [inst : ] {a : α} {b : α} {c : α} :
a b = c b a = c
@[simp]
theorem bihimp_right_inj {α : Type u_1} [inst : ] {a : α} {b : α} {c : α} :
a b = a c b = c
@[simp]
theorem bihimp_eq_left {α : Type u_1} [inst : ] {a : α} {b : α} :
a b = a b =
@[simp]
theorem bihimp_eq_right {α : Type u_1} [inst : ] {a : α} {b : α} :
a b = b a =
theorem Codisjoint.bihimp_left {α : Type u_1} [inst : ] {a : α} {b : α} {c : α} (ha : ) (hb : ) :
Codisjoint (a b) c
theorem Codisjoint.bihimp_right {α : Type u_1} [inst : ] {a : α} {b : α} {c : α} (ha : ) (hb : ) :
Codisjoint a (b c)
theorem symmDiff_eq {α : Type u_1} [inst : ] (a : α) (b : α) :
a b = a b b a
theorem bihimp_eq {α : Type u_1} [inst : ] (a : α) (b : α) :
a b = (a b) (b a)
theorem symmDiff_eq' {α : Type u_1} [inst : ] (a : α) (b : α) :
a b = (a b) (a b)
theorem bihimp_eq' {α : Type u_1} [inst : ] (a : α) (b : α) :
a b = a b a b
theorem symmDiff_top {α : Type u_1} [inst : ] (a : α) :
theorem top_symmDiff {α : Type u_1} [inst : ] (a : α) :
@[simp]
theorem compl_symmDiff {α : Type u_1} [inst : ] (a : α) (b : α) :
(a b) = a b
@[simp]
theorem compl_bihimp {α : Type u_1} [inst : ] (a : α) (b : α) :
(a b) = a b
@[simp]
theorem compl_symmDiff_compl {α : Type u_1} [inst : ] (a : α) (b : α) :
a b = a b
@[simp]
theorem compl_bihimp_compl {α : Type u_1} [inst : ] (a : α) (b : α) :
a b = a b
@[simp]
theorem symmDiff_eq_top {α : Type u_1} [inst : ] (a : α) (b : α) :
a b = IsCompl a b
@[simp]
theorem bihimp_eq_bot {α : Type u_1} [inst : ] (a : α) (b : α) :
a b = IsCompl a b
@[simp]
theorem compl_symmDiff_self {α : Type u_1} [inst : ] (a : α) :
@[simp]
theorem symmDiff_compl_self {α : Type u_1} [inst : ] (a : α) :
theorem symmDiff_symmDiff_right' {α : Type u_1} [inst : ] (a : α) (b : α) (c : α) :
a (b c) = a b c a b c a b c a b c
theorem Disjoint.le_symmDiff_sup_symmDiff_left {α : Type u_1} [inst : ] {a : α} {b : α} {c : α} (h : Disjoint a b) :
c a c b c
theorem Disjoint.le_symmDiff_sup_symmDiff_right {α : Type u_1} [inst : ] {a : α} {b : α} {c : α} (h : Disjoint b c) :
a a b a c
theorem Codisjoint.bihimp_inf_bihimp_le_left {α : Type u_1} [inst : ] {a : α} {b : α} {c : α} (h : ) :
a c b c c
theorem Codisjoint.bihimp_inf_bihimp_le_right {α : Type u_1} [inst : ] {a : α} {b : α} {c : α} (h : ) :
a b a c a

### Prod #

@[simp]
theorem symmDiff_fst {α : Type u_1} {β : Type u_2} [inst : ] [inst : ] (a : α × β) (b : α × β) :
(a b).fst = a.fst b.fst
@[simp]
theorem symmDiff_snd {α : Type u_1} {β : Type u_2} [inst : ] [inst : ] (a : α × β) (b : α × β) :
(a b).snd = a.snd b.snd
@[simp]
theorem bihimp_fst {α : Type u_1} {β : Type u_2} [inst : ] [inst : ] (a : α × β) (b : α × β) :
(a b).fst = a.fst b.fst
@[simp]
theorem bihimp_snd {α : Type u_1} {β : Type u_2} [inst : ] [inst : ] (a : α × β) (b : α × β) :
(a b).snd = a.snd b.snd

### Pi #

theorem Pi.symmDiff_def {ι : Type u_2} {π : ιType u_1} [inst : (i : ι) → ] (a : (i : ι) → π i) (b : (i : ι) → π i) :
a b = fun i => a i b i
theorem Pi.bihimp_def {ι : Type u_2} {π : ιType u_1} [inst : (i : ι) → ] (a : (i : ι) → π i) (b : (i : ι) → π i) :
a b = fun i => a i b i
@[simp]
theorem Pi.symmDiff_apply {ι : Type u_2} {π : ιType u_1} [inst : (i : ι) → ] (a : (i : ι) → π i) (b : (i : ι) → π i) (i : ι) :
(((i : ι) → π i) Pi.instSupForAll) Pi.sdiff a b i = a i b i
@[simp]
theorem Pi.bihimp_apply {ι : Type u_2} {π : ιType u_1} [inst : (i : ι) → ] (a : (i : ι) → π i) (b : (i : ι) → π i) (i : ι) :
(((i : ι) → π i) Pi.instInfForAll) Pi.instHImpForAll a b i = a i b i