mathlib documentation

algebra.group.pi

Pi instances for groups and monoids #

This file defines instances for group, monoid, semigroup and related structures on Pi types.

@[instance]
def pi.add_semigroup {I : Type u} {f : I → Type v} [Π (i : I), add_semigroup (f i)] :
add_semigroup (Π (i : I), f i)
@[instance]
def pi.semigroup {I : Type u} {f : I → Type v} [Π (i : I), semigroup (f i)] :
semigroup (Π (i : I), f i)
Equations
@[instance]
def pi.semigroup_with_zero {I : Type u} {f : I → Type v} [Π (i : I), semigroup_with_zero (f i)] :
semigroup_with_zero (Π (i : I), f i)
Equations
@[instance]
def pi.comm_semigroup {I : Type u} {f : I → Type v} [Π (i : I), comm_semigroup (f i)] :
comm_semigroup (Π (i : I), f i)
Equations
@[instance]
def pi.add_comm_semigroup {I : Type u} {f : I → Type v} [Π (i : I), add_comm_semigroup (f i)] :
add_comm_semigroup (Π (i : I), f i)
@[instance]
def pi.mul_one_class {I : Type u} {f : I → Type v} [Π (i : I), mul_one_class (f i)] :
mul_one_class (Π (i : I), f i)
Equations
@[instance]
def pi.add_zero_class {I : Type u} {f : I → Type v} [Π (i : I), add_zero_class (f i)] :
add_zero_class (Π (i : I), f i)
@[instance]
def pi.add_monoid {I : Type u} {f : I → Type v} [Π (i : I), add_monoid (f i)] :
add_monoid (Π (i : I), f i)
@[instance]
def pi.monoid {I : Type u} {f : I → Type v} [Π (i : I), monoid (f i)] :
monoid (Π (i : I), f i)
Equations
@[simp]
theorem pi.pow_apply {I : Type u} {f : I → Type v} (x : Π (i : I), f i) (i : I) [Π (i : I), monoid (f i)] (n : ) :
(x ^ n) i = x i ^ n
@[instance]
def pi.comm_monoid {I : Type u} {f : I → Type v} [Π (i : I), comm_monoid (f i)] :
comm_monoid (Π (i : I), f i)
Equations
@[instance]
def pi.add_comm_monoid {I : Type u} {f : I → Type v} [Π (i : I), add_comm_monoid (f i)] :
add_comm_monoid (Π (i : I), f i)
@[instance]
def pi.sub_neg_add_monoid {I : Type u} {f : I → Type v} [Π (i : I), sub_neg_monoid (f i)] :
sub_neg_monoid (Π (i : I), f i)
@[instance]
def pi.div_inv_monoid {I : Type u} {f : I → Type v} [Π (i : I), div_inv_monoid (f i)] :
div_inv_monoid (Π (i : I), f i)
Equations
@[instance]
def pi.add_group {I : Type u} {f : I → Type v} [Π (i : I), add_group (f i)] :
add_group (Π (i : I), f i)
@[instance]
def pi.group {I : Type u} {f : I → Type v} [Π (i : I), group (f i)] :
group (Π (i : I), f i)
Equations
@[instance]
def pi.add_comm_group {I : Type u} {f : I → Type v} [Π (i : I), add_comm_group (f i)] :
add_comm_group (Π (i : I), f i)
@[instance]
def pi.comm_group {I : Type u} {f : I → Type v} [Π (i : I), comm_group (f i)] :
comm_group (Π (i : I), f i)
Equations
@[instance]
def pi.add_left_cancel_semigroup {I : Type u} {f : I → Type v} [Π (i : I), add_left_cancel_semigroup (f i)] :
add_left_cancel_semigroup (Π (i : I), f i)
@[instance]
def pi.left_cancel_semigroup {I : Type u} {f : I → Type v} [Π (i : I), left_cancel_semigroup (f i)] :
left_cancel_semigroup (Π (i : I), f i)
Equations
@[instance]
def pi.right_cancel_semigroup {I : Type u} {f : I → Type v} [Π (i : I), right_cancel_semigroup (f i)] :
right_cancel_semigroup (Π (i : I), f i)
Equations
@[instance]
def pi.add_right_cancel_semigroup {I : Type u} {f : I → Type v} [Π (i : I), add_right_cancel_semigroup (f i)] :
add_right_cancel_semigroup (Π (i : I), f i)
@[instance]
def pi.add_left_cancel_monoid {I : Type u} {f : I → Type v} [Π (i : I), add_left_cancel_monoid (f i)] :
add_left_cancel_monoid (Π (i : I), f i)
@[instance]
def pi.left_cancel_monoid {I : Type u} {f : I → Type v} [Π (i : I), left_cancel_monoid (f i)] :
left_cancel_monoid (Π (i : I), f i)
Equations
@[instance]
def pi.add_right_cancel_monoid {I : Type u} {f : I → Type v} [Π (i : I), add_right_cancel_monoid (f i)] :
add_right_cancel_monoid (Π (i : I), f i)
@[instance]
def pi.right_cancel_monoid {I : Type u} {f : I → Type v} [Π (i : I), right_cancel_monoid (f i)] :
right_cancel_monoid (Π (i : I), f i)
Equations
@[instance]
def pi.add_cancel_monoid {I : Type u} {f : I → Type v} [Π (i : I), add_cancel_monoid (f i)] :
add_cancel_monoid (Π (i : I), f i)
@[instance]
def pi.cancel_monoid {I : Type u} {f : I → Type v} [Π (i : I), cancel_monoid (f i)] :
cancel_monoid (Π (i : I), f i)
Equations
@[instance]
def pi.cancel_comm_monoid {I : Type u} {f : I → Type v} [Π (i : I), cancel_comm_monoid (f i)] :
cancel_comm_monoid (Π (i : I), f i)
Equations
@[instance]
def pi.add_cancel_comm_monoid {I : Type u} {f : I → Type v} [Π (i : I), add_cancel_comm_monoid (f i)] :
add_cancel_comm_monoid (Π (i : I), f i)
@[instance]
def pi.mul_zero_class {I : Type u} {f : I → Type v} [Π (i : I), mul_zero_class (f i)] :
mul_zero_class (Π (i : I), f i)
Equations
@[instance]
def pi.mul_zero_one_class {I : Type u} {f : I → Type v} [Π (i : I), mul_zero_one_class (f i)] :
mul_zero_one_class (Π (i : I), f i)
Equations
@[instance]
def pi.monoid_with_zero {I : Type u} {f : I → Type v} [Π (i : I), monoid_with_zero (f i)] :
monoid_with_zero (Π (i : I), f i)
Equations
@[instance]
def pi.comm_monoid_with_zero {I : Type u} {f : I → Type v} [Π (i : I), comm_monoid_with_zero (f i)] :
comm_monoid_with_zero (Π (i : I), f i)
Equations
@[simp]
theorem pi.const_one {α : Type u_1} {β : Type u_2} [has_one β] :
@[simp]
theorem pi.const_zero {α : Type u_1} {β : Type u_2} [has_zero β] :
@[simp]
theorem pi.comp_zero {α : Type u_1} {β : Type u_2} {γ : Type u_3} [has_zero β] {f : β → γ} :
f 0 = function.const α (f 0)
@[simp]
theorem pi.comp_one {α : Type u_1} {β : Type u_2} {γ : Type u_3} [has_one β] {f : β → γ} :
f 1 = function.const α (f 1)
@[simp]
theorem pi.zero_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [has_zero γ] {f : α → β} :
0 f = 0
@[simp]
theorem pi.one_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [has_one γ] {f : α → β} :
1 f = 1
@[simp]
theorem pi.eval_add_monoid_hom_apply {I : Type u} (f : I → Type v) [Π (i : I), add_zero_class (f i)] (i : I) (g : Π (i : I), f i) :
def pi.eval_add_monoid_hom {I : Type u} (f : I → Type v) [Π (i : I), add_zero_class (f i)] (i : I) :
(Π (i : I), f i) →+ f i

Evaluation of functions into an indexed collection of additive monoids at a point is an additive monoid homomorphism. This is function.eval i as an add_monoid_hom.

def pi.eval_monoid_hom {I : Type u} (f : I → Type v) [Π (i : I), mul_one_class (f i)] (i : I) :
(Π (i : I), f i) →* f i

Evaluation of functions into an indexed collection of monoids at a point is a monoid homomorphism. This is function.eval i as a monoid_hom.

Equations
@[simp]
theorem pi.eval_monoid_hom_apply {I : Type u} (f : I → Type v) [Π (i : I), mul_one_class (f i)] (i : I) (g : Π (i : I), f i) :
def pi.const_add_monoid_hom (α : Type u_1) (β : Type u_2) [add_zero_class β] :
β →+ α → β

function.const as an add_monoid_hom.

def pi.const_monoid_hom (α : Type u_1) (β : Type u_2) [mul_one_class β] :
β →* α → β

function.const as a monoid_hom.

Equations
@[simp]
theorem pi.const_monoid_hom_apply (α : Type u_1) (β : Type u_2) [mul_one_class β] (a : β) (ᾰ : α) :
(pi.const_monoid_hom α β) a = function.const α a
@[simp]
theorem pi.const_add_monoid_hom_apply (α : Type u_1) (β : Type u_2) [add_zero_class β] (a : β) (ᾰ : α) :
def monoid_hom.coe_fn (α : Type u_1) (β : Type u_2) [mul_one_class α] [comm_monoid β] :
→* β) →* α → β

Coercion of a monoid_hom into a function is itself a monoid_hom.

See also monoid_hom.eval.

Equations
@[simp]
theorem monoid_hom.coe_fn_apply (α : Type u_1) (β : Type u_2) [mul_one_class α] [comm_monoid β] (g : α →* β) (ᾰ : α) :
(monoid_hom.coe_fn α β) g = g ᾰ
def add_monoid_hom.coe_fn (α : Type u_1) (β : Type u_2) [add_zero_class α] [add_comm_monoid β] :
→+ β) →+ α → β

Coercion of an add_monoid_hom into a function is itself a add_monoid_hom.

See also add_monoid_hom.eval.

@[simp]
theorem add_monoid_hom.coe_fn_apply (α : Type u_1) (β : Type u_2) [add_zero_class α] [add_comm_monoid β] (g : α →+ β) (ᾰ : α) :
(add_monoid_hom.coe_fn α β) g = g ᾰ
def add_monoid_hom.comp_left {α : Type u_1} {β : Type u_2} [add_zero_class α] [add_zero_class β] (f : α →+ β) (I : Type u_3) :
(I → α) →+ I → β

Additive monoid homomorphism between the function spaces I → α and I → β, induced by an additive monoid homomorphism f between α and β

@[simp]
theorem monoid_hom.comp_left_apply {α : Type u_1} {β : Type u_2} [mul_one_class α] [mul_one_class β] (f : α →* β) (I : Type u_3) (h : I → α) (ᾰ : I) :
(f.comp_left I) h = (f h)
@[simp]
theorem add_monoid_hom.comp_left_apply {α : Type u_1} {β : Type u_2} [add_zero_class α] [add_zero_class β] (f : α →+ β) (I : Type u_3) (h : I → α) (ᾰ : I) :
(f.comp_left I) h = (f h)
def monoid_hom.comp_left {α : Type u_1} {β : Type u_2} [mul_one_class α] [mul_one_class β] (f : α →* β) (I : Type u_3) :
(I → α) →* I → β

Monoid homomorphism between the function spaces I → α and I → β, induced by a monoid homomorphism f between α and β.

Equations
@[simp]
theorem zero_hom.single_apply {I : Type u} (f : I → Type v) [decidable_eq I] [Π (i : I), has_zero (f i)] (i : I) (x : f i) (i_1 : I) :
(zero_hom.single f i) x i_1 = pi.single i x i_1
def zero_hom.single {I : Type u} (f : I → Type v) [decidable_eq I] [Π (i : I), has_zero (f i)] (i : I) :
zero_hom (f i) (Π (i : I), f i)

The zero-preserving homomorphism including a single value into a dependent family of values, as functions supported at a point.

This is the zero_hom version of pi.single.

Equations
def add_monoid_hom.single {I : Type u} (f : I → Type v) [decidable_eq I] [Π (i : I), add_zero_class (f i)] (i : I) :
f i →+ Π (i : I), f i

The additive monoid homomorphism including a single additive monoid into a dependent family of additive monoids, as functions supported at a point.

This is the add_monoid_hom version of pi.single.

Equations
@[simp]
theorem add_monoid_hom.single_apply {I : Type u} (f : I → Type v) [decidable_eq I] [Π (i : I), add_zero_class (f i)] (i : I) (x : f i) (i_1 : I) :
@[simp]
theorem mul_hom.single_apply {I : Type u} (f : I → Type v) [decidable_eq I] [Π (i : I), mul_zero_class (f i)] (i : I) (x : f i) (i_1 : I) :
(mul_hom.single f i) x i_1 = pi.single i x i_1
def mul_hom.single {I : Type u} (f : I → Type v) [decidable_eq I] [Π (i : I), mul_zero_class (f i)] (i : I) :
mul_hom (f i) (Π (i : I), f i)

The multiplicative homomorphism including a single mul_zero_class into a dependent family of mul_zero_classes, as functions supported at a point.

This is the mul_hom version of pi.single.

Equations
theorem pi.single_add {I : Type u} {f : I → Type v} [decidable_eq I] [Π (i : I), add_zero_class (f i)] (i : I) (x y : f i) :
pi.single i (x + y) = pi.single i x + pi.single i y
theorem pi.single_neg {I : Type u} {f : I → Type v} [decidable_eq I] [Π (i : I), add_group (f i)] (i : I) (x : f i) :
theorem pi.single_sub {I : Type u} {f : I → Type v} [decidable_eq I] [Π (i : I), add_group (f i)] (i : I) (x y : f i) :
pi.single i (x - y) = pi.single i x - pi.single i y
theorem pi.single_mul {I : Type u} {f : I → Type v} [decidable_eq I] [Π (i : I), mul_zero_class (f i)] (i : I) (x y : f i) :
pi.single i (x * y) = (pi.single i x) * pi.single i y
theorem pi.update_eq_sub_add_single {I : Type u} {f : I → Type v} (i : I) [decidable_eq I] [Π (i : I), add_group (f i)] (g : Π (i : I), f i) (x : f i) :
function.update g i x = g - pi.single i (g i) + pi.single i x
theorem set.piecewise_add {I : Type u} {f : I → Type v} [Π (i : I), has_add (f i)] (s : set I) [Π (i : I), decidable (i s)] (f₁ f₂ g₁ g₂ : Π (i : I), f i) :
s.piecewise (f₁ + f₂) (g₁ + g₂) = s.piecewise f₁ g₁ + s.piecewise f₂ g₂
theorem set.piecewise_mul {I : Type u} {f : I → Type v} [Π (i : I), has_mul (f i)] (s : set I) [Π (i : I), decidable (i s)] (f₁ f₂ g₁ g₂ : Π (i : I), f i) :
s.piecewise (f₁ * f₂) (g₁ * g₂) = (s.piecewise f₁ g₁) * s.piecewise f₂ g₂
theorem pi.piecewise_inv {I : Type u} {f : I → Type v} [Π (i : I), has_inv (f i)] (s : set I) [Π (i : I), decidable (i s)] (f₁ g₁ : Π (i : I), f i) :
s.piecewise f₁⁻¹ g₁⁻¹ = (s.piecewise f₁ g₁)⁻¹
theorem pi.piecewise_neg {I : Type u} {f : I → Type v} [Π (i : I), has_neg (f i)] (s : set I) [Π (i : I), decidable (i s)] (f₁ g₁ : Π (i : I), f i) :
s.piecewise (-f₁) (-g₁) = -s.piecewise f₁ g₁
theorem pi.piecewise_sub {I : Type u} {f : I → Type v} [Π (i : I), has_sub (f i)] (s : set I) [Π (i : I), decidable (i s)] (f₁ f₂ g₁ g₂ : Π (i : I), f i) :
s.piecewise (f₁ - f₂) (g₁ - g₂) = s.piecewise f₁ g₁ - s.piecewise f₂ g₂
theorem pi.piecewise_div {I : Type u} {f : I → Type v} [Π (i : I), has_div (f i)] (s : set I) [Π (i : I), decidable (i s)] (f₁ f₂ g₁ g₂ : Π (i : I), f i) :
s.piecewise (f₁ / f₂) (g₁ / g₂) = s.piecewise f₁ g₁ / s.piecewise f₂ g₂
@[simp]
theorem function.extend_by_one.hom_apply {ι : Type u} {η : Type v} (R : Type w) (s : ι → η) [mul_one_class R] (f : ι → R) (ᾰ : η) :
def function.extend_by_zero.hom {ι : Type u} {η : Type v} (R : Type w) (s : ι → η) [add_zero_class R] :
(ι → R) →+ η → R

function.extend s f 0 as a bundled hom.

@[simp]
theorem function.extend_by_zero.hom_apply {ι : Type u} {η : Type v} (R : Type w) (s : ι → η) [add_zero_class R] (f : ι → R) (ᾰ : η) :
def function.extend_by_one.hom {ι : Type u} {η : Type v} (R : Type w) (s : ι → η) [mul_one_class R] :
(ι → R) →* η → R

function.extend s f 1 as a bundled hom.

Equations