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

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def symmDiff {α : Type u_1} [inst : Sup α] [inst : SDiff α] (a : α) (b : α) :
α

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

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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.

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def «term_∆_» :
Lean.TrailingParserDescr

Notation for symmDiff

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def «term_⇔_» :
Lean.TrailingParserDescr

Notation for bihimp

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

Prod #

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

Pi #

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theorem Pi.symmDiff_def {ι : Type u_2} {π : ι → Type u_1} [inst : (i : ι) → GeneralizedCoheytingAlgebra (π i)] (a : (i : ι) → π i) (b : (i : ι) → π i) :
a ∆ b = fun i => a i ∆ b i
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theorem Pi.bihimp_def {ι : Type u_2} {π : ι → Type u_1} [inst : (i : ι) → GeneralizedHeytingAlgebra (π i)] (a : (i : ι) → π i) (b : (i : ι) → π i) :
a ⇔ b = fun i => a i ⇔ b i
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@[simp]
theorem Pi.symmDiff_apply {ι : Type u_2} {π : ι → Type u_1} [inst : (i : ι) → GeneralizedCoheytingAlgebra (π i)] (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
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@[simp]
theorem Pi.bihimp_apply {ι : Type u_2} {π : ι → Type u_1} [inst : (i : ι) → GeneralizedHeytingAlgebra (π i)] (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