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)
• bihimp: The bi-implication operator, defined as (b ⇨ a) ⊓ (a ⇨ b)

In generalized Boolean algebras, the symmetric difference operator is:

• symmDiff_comm: commutative, and
• symmDiff_assoc: associative.

Notations

• a ∆ b: symmDiff a b
• a ⇔ 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:

• https://people.math.gatech.edu/~mccuan/courses/4317/symmetricdifference.pdf

Tags

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

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

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

Equations

• symmDiff a b = a \ b ⊔ b \ a

def bihimp {α : Type u_2} [Min α] [HImp α] (a b : α) : α

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

Equations

• bihimp a b = (b ⇨ a) ⊓ (a ⇨ b)

def symmDiff.«term_∆_» : Lean.TrailingParserDescr

Notation for symmDiff

Equations

• symmDiff.«term_∆_» = Lean.ParserDescr.trailingNode `symmDiff.«term_∆_» 100 100 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ∆ ") (Lean.ParserDescr.cat `term 101))

def symmDiff.«term_⇔_» : Lean.TrailingParserDescr

Notation for bihimp

Equations

• symmDiff.«term_⇔_» = Lean.ParserDescr.trailingNode `symmDiff.«term_⇔_» 100 100 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ⇔ ") (Lean.ParserDescr.cat `term 101))

theorem symmDiff_def {α : Type u_2} [Max α] [SDiff α] (a b : α) : symmDiff a b = a \ b ⊔ b \ a

theorem bihimp_def {α : Type u_2} [Min α] [HImp α] (a b : α) : bihimp a b = (b ⇨ a) ⊓ (a ⇨ b)

theorem symmDiff_eq_Xor' (p q : Prop) : symmDiff p q = Xor' p q

@[simp]

theorem bihimp_iff_iff {p q : Prop} : bihimp p q ↔ (p ↔ q)

@[simp]

theorem Bool.symmDiff_eq_xor (p q : Bool) : symmDiff p q = (p ^^ q)

@[simp]

theorem toDual_symmDiff {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a 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 b : αᵒᵈ) : OrderDual.ofDual (bihimp a b) = symmDiff (OrderDual.ofDual a) (OrderDual.ofDual b)

theorem symmDiff_comm {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α) : symmDiff a b = symmDiff b a

instance symmDiff_isCommutative {α : Type u_2} [GeneralizedCoheytingAlgebra α] : Std.Commutative fun (x1 x2 : α) => symmDiff x1 x2

@[simp]

theorem symmDiff_self {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a : α) : symmDiff a a = ⊥

@[simp]

theorem symmDiff_bot {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a : α) : symmDiff a ⊥ = a

@[simp]

theorem bot_symmDiff {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a : α) : symmDiff ⊥ a = a

@[simp]

theorem symmDiff_eq_bot {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α} : symmDiff a b = ⊥ ↔ a = b

theorem symmDiff_of_le {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α} (h : a ≤ b) : symmDiff a b = b \ a

theorem symmDiff_of_ge {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α} (h : b ≤ a) : symmDiff a b = a \ b

theorem symmDiff_le {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α} (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : symmDiff a b ≤ c

theorem symmDiff_le_iff {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α} : symmDiff a b ≤ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c

@[simp]

theorem symmDiff_le_sup {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α} : symmDiff a b ≤ a ⊔ b

theorem symmDiff_eq_sup_sdiff_inf {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α) : symmDiff a b = (a ⊔ b) \ (a ⊓ b)

theorem Disjoint.symmDiff_eq_sup {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α} (h : Disjoint a b) : symmDiff a b = a ⊔ b

theorem symmDiff_sdiff {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b c : α) : symmDiff a b \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c)

@[simp]

theorem symmDiff_sdiff_inf {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α) : symmDiff a b \ (a ⊓ b) = symmDiff a b

@[simp]

theorem symmDiff_sdiff_eq_sup {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α) : symmDiff a (b \ a) = a ⊔ b

@[simp]

theorem sdiff_symmDiff_eq_sup {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α) : symmDiff (a \ b) b = a ⊔ b

@[simp]

theorem symmDiff_sup_inf {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α) : symmDiff a b ⊔ a ⊓ b = a ⊔ b

@[simp]

theorem inf_sup_symmDiff {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α) : a ⊓ b ⊔ symmDiff a b = a ⊔ b

@[simp]

theorem symmDiff_symmDiff_inf {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α) : symmDiff (symmDiff a b) (a ⊓ b) = a ⊔ b

@[simp]

theorem inf_symmDiff_symmDiff {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α) : symmDiff (a ⊓ b) (symmDiff a b) = a ⊔ b

theorem symmDiff_triangle {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b c : α) : symmDiff a c ≤ symmDiff a b ⊔ symmDiff b c

theorem le_symmDiff_sup_right {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α) : a ≤ symmDiff a b ⊔ b

theorem le_symmDiff_sup_left {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α) : b ≤ symmDiff a b ⊔ a

@[simp]

theorem toDual_bihimp {α : Type u_2} [GeneralizedHeytingAlgebra α] (a 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 b : αᵒᵈ) : OrderDual.ofDual (symmDiff a b) = bihimp (OrderDual.ofDual a) (OrderDual.ofDual b)

theorem bihimp_comm {α : Type u_2} [GeneralizedHeytingAlgebra α] (a 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

@[simp]

theorem bihimp_self {α : Type u_2} [GeneralizedHeytingAlgebra α] (a : α) : bihimp a a = ⊤

@[simp]

theorem bihimp_top {α : Type u_2} [GeneralizedHeytingAlgebra α] (a : α) : bihimp a ⊤ = a

@[simp]

theorem top_bihimp {α : Type u_2} [GeneralizedHeytingAlgebra α] (a : α) : bihimp ⊤ a = a

@[simp]

theorem bihimp_eq_top {α : Type u_2} [GeneralizedHeytingAlgebra α] {a b : α} : bihimp a b = ⊤ ↔ a = b

theorem bihimp_of_le {α : Type u_2} [GeneralizedHeytingAlgebra α] {a b : α} (h : a ≤ b) : bihimp a b = b ⇨ a

theorem bihimp_of_ge {α : Type u_2} [GeneralizedHeytingAlgebra α] {a b : α} (h : b ≤ a) : bihimp a b = a ⇨ b

theorem le_bihimp {α : Type u_2} [GeneralizedHeytingAlgebra α] {a b 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 b c : α} : a ≤ bihimp b c ↔ a ⊓ b ≤ c ∧ a ⊓ c ≤ b

@[simp]

theorem inf_le_bihimp {α : Type u_2} [GeneralizedHeytingAlgebra α] {a b : α} : a ⊓ b ≤ bihimp a b

theorem bihimp_eq_inf_himp_inf {α : Type u_2} [GeneralizedHeytingAlgebra α] (a b : α) : bihimp a b = a ⊔ b ⇨ a ⊓ b

theorem Codisjoint.bihimp_eq_inf {α : Type u_2} [GeneralizedHeytingAlgebra α] {a b : α} (h : Codisjoint a b) : bihimp a b = a ⊓ b

theorem himp_bihimp {α : Type u_2} [GeneralizedHeytingAlgebra α] (a b c : α) : a ⇨ bihimp b c = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c)

@[simp]

theorem sup_himp_bihimp {α : Type u_2} [GeneralizedHeytingAlgebra α] (a b : α) : a ⊔ b ⇨ bihimp a b = bihimp a b

@[simp]

theorem bihimp_himp_eq_inf {α : Type u_2} [GeneralizedHeytingAlgebra α] (a b : α) : bihimp a (a ⇨ b) = a ⊓ b

@[simp]

theorem himp_bihimp_eq_inf {α : Type u_2} [GeneralizedHeytingAlgebra α] (a b : α) : bihimp (b ⇨ a) b = a ⊓ b

@[simp]

theorem bihimp_inf_sup {α : Type u_2} [GeneralizedHeytingAlgebra α] (a b : α) : bihimp a b ⊓ (a ⊔ b) = a ⊓ b

@[simp]

theorem sup_inf_bihimp {α : Type u_2} [GeneralizedHeytingAlgebra α] (a b : α) : (a ⊔ b) ⊓ bihimp a b = a ⊓ b

@[simp]

theorem bihimp_bihimp_sup {α : Type u_2} [GeneralizedHeytingAlgebra α] (a b : α) : bihimp (bihimp a b) (a ⊔ b) = a ⊓ b

@[simp]

theorem sup_bihimp_bihimp {α : Type u_2} [GeneralizedHeytingAlgebra α] (a b : α) : bihimp (a ⊔ b) (bihimp a b) = a ⊓ b

theorem bihimp_triangle {α : Type u_2} [GeneralizedHeytingAlgebra α] (a b c : α) : bihimp a b ⊓ bihimp b c ≤ bihimp a c

@[simp]

theorem symmDiff_top' {α : Type u_2} [CoheytingAlgebra α] (a : α) : symmDiff a ⊤ = ￢a

@[simp]

theorem top_symmDiff' {α : Type u_2} [CoheytingAlgebra α] (a : α) : symmDiff ⊤ a = ￢a

@[simp]

theorem hnot_symmDiff_self {α : Type u_2} [CoheytingAlgebra α] (a : α) : symmDiff (￢a) a = ⊤

@[simp]

theorem symmDiff_hnot_self {α : Type u_2} [CoheytingAlgebra α] (a : α) : symmDiff a (￢a) = ⊤

theorem IsCompl.symmDiff_eq_top {α : Type u_2} [CoheytingAlgebra α] {a b : α} (h : IsCompl a b) : symmDiff a b = ⊤

@[simp]

theorem bihimp_bot {α : Type u_2} [HeytingAlgebra α] (a : α) : bihimp a ⊥ = aᶜ

@[simp]

theorem bot_bihimp {α : Type u_2} [HeytingAlgebra α] (a : α) : bihimp ⊥ a = aᶜ

@[simp]

theorem compl_bihimp_self {α : Type u_2} [HeytingAlgebra α] (a : α) : bihimp aᶜ a = ⊥

@[simp]

theorem bihimp_hnot_self {α : Type u_2} [HeytingAlgebra α] (a : α) : bihimp a aᶜ = ⊥

theorem IsCompl.bihimp_eq_bot {α : Type u_2} [HeytingAlgebra α] {a b : α} (h : IsCompl a b) : bihimp a b = ⊥

@[simp]

theorem sup_sdiff_symmDiff {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b : α) : (a ⊔ b) \ symmDiff a b = a ⊓ b

theorem disjoint_symmDiff_inf {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b : α) : Disjoint (symmDiff a b) (a ⊓ b)

theorem inf_symmDiff_distrib_left {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b c : α) : a ⊓ symmDiff b c = symmDiff (a ⊓ b) (a ⊓ c)

theorem inf_symmDiff_distrib_right {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b c : α) : symmDiff a b ⊓ c = symmDiff (a ⊓ c) (b ⊓ c)

theorem sdiff_symmDiff {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b c : α) : c \ symmDiff a b = c ⊓ a ⊓ b ⊔ c \ a ⊓ c \ b

theorem sdiff_symmDiff' {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b c : α) : c \ symmDiff a b = c ⊓ a ⊓ b ⊔ c \ (a ⊔ b)

@[simp]

theorem symmDiff_sdiff_left {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b : α) : symmDiff a b \ a = b \ a

@[simp]

theorem symmDiff_sdiff_right {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b : α) : symmDiff a b \ b = a \ b

@[simp]

theorem sdiff_symmDiff_left {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b : α) : a \ symmDiff a b = a ⊓ b

@[simp]

theorem sdiff_symmDiff_right {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b : α) : b \ symmDiff a b = a ⊓ b

theorem symmDiff_eq_sup {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b : α) : symmDiff a b = a ⊔ b ↔ Disjoint a b

@[simp]

theorem le_symmDiff_iff_left {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b : α) : a ≤ symmDiff a b ↔ Disjoint a b

@[simp]

theorem le_symmDiff_iff_right {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b : α) : b ≤ symmDiff a b ↔ Disjoint a b

theorem symmDiff_symmDiff_left {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b 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 b 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 b c : α) : symmDiff (symmDiff a b) c = symmDiff a (symmDiff b c)

instance symmDiff_isAssociative {α : Type u_2} [GeneralizedBooleanAlgebra α] : Std.Associative fun (x1 x2 : α) => symmDiff x1 x2

theorem symmDiff_left_comm {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b c : α) : symmDiff a (symmDiff b c) = symmDiff b (symmDiff a c)

theorem symmDiff_right_comm {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b c : α) : symmDiff (symmDiff a b) c = symmDiff (symmDiff a c) b

theorem symmDiff_symmDiff_symmDiff_comm {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b c d : α) : symmDiff (symmDiff a b) (symmDiff c d) = symmDiff (symmDiff a c) (symmDiff b d)

@[simp]

theorem symmDiff_symmDiff_cancel_left {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b : α) : symmDiff a (symmDiff a b) = b

@[simp]

theorem symmDiff_symmDiff_cancel_right {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b : α) : symmDiff (symmDiff b a) a = b

@[simp]

theorem symmDiff_symmDiff_self' {α : Type u_2} [GeneralizedBooleanAlgebra α] (a 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 b c : α} : symmDiff a b = symmDiff c b ↔ a = c

@[simp]

theorem symmDiff_right_inj {α : Type u_2} [GeneralizedBooleanAlgebra α] {a b c : α} : symmDiff a b = symmDiff a c ↔ b = c

@[simp]

theorem symmDiff_eq_left {α : Type u_2} [GeneralizedBooleanAlgebra α] {a b : α} : symmDiff a b = a ↔ b = ⊥

@[simp]

theorem symmDiff_eq_right {α : Type u_2} [GeneralizedBooleanAlgebra α] {a b : α} : symmDiff a b = b ↔ a = ⊥

theorem Disjoint.symmDiff_left {α : Type u_2} [GeneralizedBooleanAlgebra α] {a b c : α} (ha : Disjoint a c) (hb : Disjoint b c) : Disjoint (symmDiff a b) c

theorem Disjoint.symmDiff_right {α : Type u_2} [GeneralizedBooleanAlgebra α] {a b c : α} (ha : Disjoint a b) (hb : Disjoint a c) : Disjoint a (symmDiff b c)

theorem symmDiff_eq_iff_sdiff_eq {α : Type u_2} [GeneralizedBooleanAlgebra α] {a b 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 b : α) : bihimp a b ⇨ a ⊓ b = a ⊔ b

theorem codisjoint_bihimp_sup {α : Type u_2} [BooleanAlgebra α] (a b : α) : Codisjoint (bihimp a b) (a ⊔ b)

@[simp]

theorem himp_bihimp_left {α : Type u_2} [BooleanAlgebra α] (a b : α) : a ⇨ bihimp a b = a ⇨ b

@[simp]

theorem himp_bihimp_right {α : Type u_2} [BooleanAlgebra α] (a b : α) : b ⇨ bihimp a b = b ⇨ a

@[simp]

theorem bihimp_himp_left {α : Type u_2} [BooleanAlgebra α] (a b : α) : bihimp a b ⇨ a = a ⊔ b

@[simp]

theorem bihimp_himp_right {α : Type u_2} [BooleanAlgebra α] (a b : α) : bihimp a b ⇨ b = a ⊔ b

@[simp]

theorem bihimp_eq_inf {α : Type u_2} [BooleanAlgebra α] (a b : α) : bihimp a b = a ⊓ b ↔ Codisjoint a b

@[simp]

theorem bihimp_le_iff_left {α : Type u_2} [BooleanAlgebra α] (a b : α) : bihimp a b ≤ a ↔ Codisjoint a b

@[simp]

theorem bihimp_le_iff_right {α : Type u_2} [BooleanAlgebra α] (a b : α) : bihimp a b ≤ b ↔ Codisjoint a b

theorem bihimp_assoc {α : Type u_2} [BooleanAlgebra α] (a b 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

theorem bihimp_left_comm {α : Type u_2} [BooleanAlgebra α] (a b c : α) : bihimp a (bihimp b c) = bihimp b (bihimp a c)

theorem bihimp_right_comm {α : Type u_2} [BooleanAlgebra α] (a b c : α) : bihimp (bihimp a b) c = bihimp (bihimp a c) b

theorem bihimp_bihimp_bihimp_comm {α : Type u_2} [BooleanAlgebra α] (a b c 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 b : α) : bihimp a (bihimp a b) = b

@[simp]

theorem bihimp_bihimp_cancel_right {α : Type u_2} [BooleanAlgebra α] (a b : α) : bihimp (bihimp b a) a = b

@[simp]

theorem bihimp_bihimp_self {α : Type u_2} [BooleanAlgebra α] (a 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 b c : α} : bihimp a b = bihimp c b ↔ a = c

@[simp]

theorem bihimp_right_inj {α : Type u_2} [BooleanAlgebra α] {a b c : α} : bihimp a b = bihimp a c ↔ b = c

@[simp]

theorem bihimp_eq_left {α : Type u_2} [BooleanAlgebra α] {a b : α} : bihimp a b = a ↔ b = ⊤

@[simp]

theorem bihimp_eq_right {α : Type u_2} [BooleanAlgebra α] {a b : α} : bihimp a b = b ↔ a = ⊤

theorem Codisjoint.bihimp_left {α : Type u_2} [BooleanAlgebra α] {a b c : α} (ha : Codisjoint a c) (hb : Codisjoint b c) : Codisjoint (bihimp a b) c

theorem Codisjoint.bihimp_right {α : Type u_2} [BooleanAlgebra α] {a b c : α} (ha : Codisjoint a b) (hb : Codisjoint a c) : Codisjoint a (bihimp b c)

theorem symmDiff_eq {α : Type u_2} [BooleanAlgebra α] (a b : α) : symmDiff a b = a ⊓ bᶜ ⊔ b ⊓ aᶜ

theorem bihimp_eq {α : Type u_2} [BooleanAlgebra α] (a b : α) : bihimp a b = (a ⊔ bᶜ) ⊓ (b ⊔ aᶜ)

theorem symmDiff_eq' {α : Type u_2} [BooleanAlgebra α] (a b : α) : symmDiff a b = (a ⊔ b) ⊓ (aᶜ ⊔ bᶜ)

theorem bihimp_eq' {α : Type u_2} [BooleanAlgebra α] (a b : α) : bihimp a b = a ⊓ b ⊔ aᶜ ⊓ bᶜ

theorem symmDiff_top {α : Type u_2} [BooleanAlgebra α] (a : α) : symmDiff a ⊤ = aᶜ

theorem top_symmDiff {α : Type u_2} [BooleanAlgebra α] (a : α) : symmDiff ⊤ a = aᶜ

@[simp]

theorem compl_symmDiff {α : Type u_2} [BooleanAlgebra α] (a b : α) : (symmDiff a b)ᶜ = bihimp a b

@[simp]

theorem compl_bihimp {α : Type u_2} [BooleanAlgebra α] (a b : α) : (bihimp a b)ᶜ = symmDiff a b

@[simp]

theorem compl_symmDiff_compl {α : Type u_2} [BooleanAlgebra α] (a b : α) : symmDiff aᶜ bᶜ = symmDiff a b

@[simp]

theorem compl_bihimp_compl {α : Type u_2} [BooleanAlgebra α] (a b : α) : bihimp aᶜ bᶜ = bihimp a b

@[simp]

theorem symmDiff_eq_top {α : Type u_2} [BooleanAlgebra α] (a b : α) : symmDiff a b = ⊤ ↔ IsCompl a b

@[simp]

theorem bihimp_eq_bot {α : Type u_2} [BooleanAlgebra α] (a b : α) : bihimp a b = ⊥ ↔ IsCompl a b

@[simp]

theorem compl_symmDiff_self {α : Type u_2} [BooleanAlgebra α] (a : α) : symmDiff aᶜ a = ⊤

@[simp]

theorem symmDiff_compl_self {α : Type u_2} [BooleanAlgebra α] (a : α) : symmDiff a aᶜ = ⊤

theorem symmDiff_symmDiff_right' {α : Type u_2} [BooleanAlgebra α] (a b c : α) : symmDiff a (symmDiff b c) = a ⊓ b ⊓ c ⊔ a ⊓ bᶜ ⊓ cᶜ ⊔ aᶜ ⊓ b ⊓ cᶜ ⊔ aᶜ ⊓ bᶜ ⊓ c

theorem Disjoint.le_symmDiff_sup_symmDiff_left {α : Type u_2} [BooleanAlgebra α] {a b c : α} (h : Disjoint a b) : c ≤ symmDiff a c ⊔ symmDiff b c

theorem Disjoint.le_symmDiff_sup_symmDiff_right {α : Type u_2} [BooleanAlgebra α] {a b c : α} (h : Disjoint b c) : a ≤ symmDiff a b ⊔ symmDiff a c

theorem Codisjoint.bihimp_inf_bihimp_le_left {α : Type u_2} [BooleanAlgebra α] {a b 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 b 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 b : α × β) : (symmDiff a b).fst = symmDiff a.fst b.fst

@[simp]

theorem symmDiff_snd {α : Type u_2} {β : Type u_3} [GeneralizedCoheytingAlgebra α] [GeneralizedCoheytingAlgebra β] (a b : α × β) : (symmDiff a b).snd = symmDiff a.snd b.snd

@[simp]

theorem bihimp_fst {α : Type u_2} {β : Type u_3} [GeneralizedHeytingAlgebra α] [GeneralizedHeytingAlgebra β] (a b : α × β) : (bihimp a b).fst = bihimp a.fst b.fst

@[simp]

theorem bihimp_snd {α : Type u_2} {β : Type u_3} [GeneralizedHeytingAlgebra α] [GeneralizedHeytingAlgebra β] (a b : α × β) : (bihimp a b).snd = bihimp a.snd b.snd

Pi

theorem Pi.symmDiff_def {ι : Type u_1} {π : ι → Type u_4} [(i : ι) → GeneralizedCoheytingAlgebra (π i)] (a 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 : ι → Type u_4} [(i : ι) → GeneralizedHeytingAlgebra (π i)] (a 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 : ι → Type u_4} [(i : ι) → GeneralizedCoheytingAlgebra (π i)] (a 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 : ι → Type u_4} [(i : ι) → GeneralizedHeytingAlgebra (π i)] (a b : (i : ι) → π i) (i : ι) : bihimp a b i = bihimp (a i) (b i)