mathlib documentation

algebra.group.inj_surj

Lifting algebraic data classes along injective/surjective maps #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. https://github.com/leanprover-community/mathlib4/pull/707 Any changes to this file require a corresponding PR to mathlib4.

This file provides definitions that are meant to deal with situations such as the following:

Suppose that G is a group, and H is a type endowed with has_one H, has_mul H, and has_inv H. Suppose furthermore, that f : G → H is a surjective map that respects the multiplication, and the unit elements. Then H satisfies the group axioms.

The relevant definition in this case is function.surjective.group. Dually, there is also function.injective.group. And there are versions for (additive) (commutative) semigroups/monoids.

Injective #

@[protected, reducible]
def function.injective.semigroup {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [semigroup M₂] (f : M₁ M₂) (hf : function.injective f) (mul : (x y : M₁), f (x * y) = f x * f y) :

A type endowed with * is a semigroup, if it admits an injective map that preserves * to a semigroup. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.injective.add_semigroup {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [add_semigroup M₂] (f : M₁ M₂) (hf : function.injective f) (mul : (x y : M₁), f (x + y) = f x + f y) :

A type endowed with + is an additive semigroup, if it admits an injective map that preserves + to an additive semigroup.

Equations
@[protected, reducible]
def function.injective.add_comm_semigroup {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [add_comm_semigroup M₂] (f : M₁ M₂) (hf : function.injective f) (mul : (x y : M₁), f (x + y) = f x + f y) :

A type endowed with + is an additive commutative semigroup, if it admits an injective map that preserves + to an additive commutative semigroup.

Equations
@[protected, reducible]
def function.injective.comm_semigroup {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [comm_semigroup M₂] (f : M₁ M₂) (hf : function.injective f) (mul : (x y : M₁), f (x * y) = f x * f y) :

A type endowed with * is a commutative semigroup, if it admits an injective map that preserves * to a commutative semigroup. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.injective.left_cancel_semigroup {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [left_cancel_semigroup M₂] (f : M₁ M₂) (hf : function.injective f) (mul : (x y : M₁), f (x * y) = f x * f y) :

A type endowed with * is a left cancel semigroup, if it admits an injective map that preserves * to a left cancel semigroup. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.injective.add_left_cancel_semigroup {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [add_left_cancel_semigroup M₂] (f : M₁ M₂) (hf : function.injective f) (mul : (x y : M₁), f (x + y) = f x + f y) :

A type endowed with + is an additive left cancel semigroup, if it admits an injective map that preserves + to an additive left cancel semigroup.

Equations
@[protected, reducible]
def function.injective.right_cancel_semigroup {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [right_cancel_semigroup M₂] (f : M₁ M₂) (hf : function.injective f) (mul : (x y : M₁), f (x * y) = f x * f y) :

A type endowed with * is a right cancel semigroup, if it admits an injective map that preserves * to a right cancel semigroup. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.injective.add_right_cancel_semigroup {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [add_right_cancel_semigroup M₂] (f : M₁ M₂) (hf : function.injective f) (mul : (x y : M₁), f (x + y) = f x + f y) :

A type endowed with + is an additive right cancel semigroup, if it admits an injective map that preserves + to an additive right cancel semigroup.

Equations
@[protected, reducible]
def function.injective.add_zero_class {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [has_zero M₁] [add_zero_class M₂] (f : M₁ M₂) (hf : function.injective f) (one : f 0 = 0) (mul : (x y : M₁), f (x + y) = f x + f y) :

A type endowed with 0 and + is an add_zero_class, if it admits an injective map that preserves 0 and + to an add_zero_class.

Equations
@[protected, reducible]
def function.injective.mul_one_class {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [has_one M₁] [mul_one_class M₂] (f : M₁ M₂) (hf : function.injective f) (one : f 1 = 1) (mul : (x y : M₁), f (x * y) = f x * f y) :

A type endowed with 1 and * is a mul_one_class, if it admits an injective map that preserves 1 and * to a mul_one_class. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.injective.add_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [has_zero M₁] [has_smul M₁] [add_monoid M₂] (f : M₁ M₂) (hf : function.injective f) (one : f 0 = 0) (mul : (x y : M₁), f (x + y) = f x + f y) (npow : (x : M₁) (n : ), f (n x) = n f x) :

A type endowed with 0 and + is an additive monoid, if it admits an injective map that preserves 0 and + to an additive monoid. This version takes a custom nsmul as a [has_smul ℕ M₁] argument.

Equations
@[protected, reducible]
def function.injective.monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [has_one M₁] [has_pow M₁ ] [monoid M₂] (f : M₁ M₂) (hf : function.injective f) (one : f 1 = 1) (mul : (x y : M₁), f (x * y) = f x * f y) (npow : (x : M₁) (n : ), f (x ^ n) = f x ^ n) :
monoid M₁

A type endowed with 1 and * is a monoid, if it admits an injective map that preserves 1 and * to a monoid. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.injective.add_monoid_with_one {M₂ : Type u_2} {M₁ : Type u_1} [has_zero M₁] [has_one M₁] [has_add M₁] [has_smul M₁] [has_nat_cast M₁] [add_monoid_with_one M₂] (f : M₁ M₂) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1) (add : (x y : M₁), f (x + y) = f x + f y) (nsmul : (x : M₁) (n : ), f (n x) = n f x) (nat_cast : (n : ), f n = n) :

A type endowed with 0, 1 and + is an additive monoid with one, if it admits an injective map that preserves 0, 1 and + to an additive monoid with one. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.injective.left_cancel_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [has_one M₁] [has_pow M₁ ] [left_cancel_monoid M₂] (f : M₁ M₂) (hf : function.injective f) (one : f 1 = 1) (mul : (x y : M₁), f (x * y) = f x * f y) (npow : (x : M₁) (n : ), f (x ^ n) = f x ^ n) :

A type endowed with 1 and * is a left cancel monoid, if it admits an injective map that preserves 1 and * to a left cancel monoid. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.injective.add_left_cancel_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [has_zero M₁] [has_smul M₁] [add_left_cancel_monoid M₂] (f : M₁ M₂) (hf : function.injective f) (one : f 0 = 0) (mul : (x y : M₁), f (x + y) = f x + f y) (npow : (x : M₁) (n : ), f (n x) = n f x) :

A type endowed with 0 and + is an additive left cancel monoid, if it admits an injective map that preserves 0 and + to an additive left cancel monoid.

Equations
@[protected, reducible]
def function.injective.right_cancel_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [has_one M₁] [has_pow M₁ ] [right_cancel_monoid M₂] (f : M₁ M₂) (hf : function.injective f) (one : f 1 = 1) (mul : (x y : M₁), f (x * y) = f x * f y) (npow : (x : M₁) (n : ), f (x ^ n) = f x ^ n) :

A type endowed with 1 and * is a right cancel monoid, if it admits an injective map that preserves 1 and * to a right cancel monoid. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.injective.add_right_cancel_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [has_zero M₁] [has_smul M₁] [add_right_cancel_monoid M₂] (f : M₁ M₂) (hf : function.injective f) (one : f 0 = 0) (mul : (x y : M₁), f (x + y) = f x + f y) (npow : (x : M₁) (n : ), f (n x) = n f x) :

A type endowed with 0 and + is an additive left cancel monoid, if it admits an injective map that preserves 0 and + to an additive left cancel monoid.

Equations
@[protected, reducible]
def function.injective.add_cancel_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [has_zero M₁] [has_smul M₁] [add_cancel_monoid M₂] (f : M₁ M₂) (hf : function.injective f) (one : f 0 = 0) (mul : (x y : M₁), f (x + y) = f x + f y) (npow : (x : M₁) (n : ), f (n x) = n f x) :

A type endowed with 0 and + is an additive left cancel monoid, if it admits an injective map that preserves 0 and + to an additive left cancel monoid.

Equations
@[protected, reducible]
def function.injective.cancel_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [has_one M₁] [has_pow M₁ ] [cancel_monoid M₂] (f : M₁ M₂) (hf : function.injective f) (one : f 1 = 1) (mul : (x y : M₁), f (x * y) = f x * f y) (npow : (x : M₁) (n : ), f (x ^ n) = f x ^ n) :

A type endowed with 1 and * is a cancel monoid, if it admits an injective map that preserves 1 and * to a cancel monoid. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.injective.add_comm_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [has_zero M₁] [has_smul M₁] [add_comm_monoid M₂] (f : M₁ M₂) (hf : function.injective f) (one : f 0 = 0) (mul : (x y : M₁), f (x + y) = f x + f y) (npow : (x : M₁) (n : ), f (n x) = n f x) :

A type endowed with 0 and + is an additive commutative monoid, if it admits an injective map that preserves 0 and + to an additive commutative monoid.

Equations
@[protected, reducible]
def function.injective.comm_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [has_one M₁] [has_pow M₁ ] [comm_monoid M₂] (f : M₁ M₂) (hf : function.injective f) (one : f 1 = 1) (mul : (x y : M₁), f (x * y) = f x * f y) (npow : (x : M₁) (n : ), f (x ^ n) = f x ^ n) :

A type endowed with 1 and * is a commutative monoid, if it admits an injective map that preserves 1 and * to a commutative monoid. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.injective.add_cancel_comm_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [has_zero M₁] [has_smul M₁] [add_cancel_comm_monoid M₂] (f : M₁ M₂) (hf : function.injective f) (one : f 0 = 0) (mul : (x y : M₁), f (x + y) = f x + f y) (npow : (x : M₁) (n : ), f (n x) = n f x) :

A type endowed with 0 and + is an additive cancel commutative monoid, if it admits an injective map that preserves 0 and + to an additive cancel commutative monoid.

Equations
@[protected, reducible]
def function.injective.cancel_comm_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [has_one M₁] [has_pow M₁ ] [cancel_comm_monoid M₂] (f : M₁ M₂) (hf : function.injective f) (one : f 1 = 1) (mul : (x y : M₁), f (x * y) = f x * f y) (npow : (x : M₁) (n : ), f (x ^ n) = f x ^ n) :

A type endowed with 1 and * is a cancel commutative monoid, if it admits an injective map that preserves 1 and * to a cancel commutative monoid. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.injective.has_involutive_inv {M₂ : Type u_2} {M₁ : Type u_1} [has_inv M₁] [has_involutive_inv M₂] (f : M₁ M₂) (hf : function.injective f) (inv : (x : M₁), f x⁻¹ = (f x)⁻¹) :

A type has an involutive inversion if it admits a surjective map that preserves ⁻¹ to a type which has an involutive inversion.

Equations
@[protected, reducible]
def function.injective.has_involutive_neg {M₂ : Type u_2} {M₁ : Type u_1} [has_neg M₁] [has_involutive_neg M₂] (f : M₁ M₂) (hf : function.injective f) (inv : (x : M₁), f (-x) = -f x) :

A type has an involutive negation if it admits a surjective map that preserves ⁻¹ to a type which has an involutive inversion.

Equations
@[protected, reducible]
def function.injective.sub_neg_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [has_zero M₁] [has_smul M₁] [has_neg M₁] [has_sub M₁] [has_smul M₁] [sub_neg_monoid M₂] (f : M₁ M₂) (hf : function.injective f) (one : f 0 = 0) (mul : (x y : M₁), f (x + y) = f x + f y) (inv : (x : M₁), f (-x) = -f x) (div : (x y : M₁), f (x - y) = f x - f y) (npow : (x : M₁) (n : ), f (n x) = n f x) (zpow : (x : M₁) (n : ), f (n x) = n f x) :

A type endowed with 0, +, unary -, and binary - is a sub_neg_monoid if it admits an injective map that preserves 0, +, unary -, and binary - to a sub_neg_monoid. This version takes custom nsmul and zsmul as [has_smul ℕ M₁] and [has_smul ℤ M₁] arguments.

Equations
@[protected, reducible]
def function.injective.div_inv_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [has_one M₁] [has_pow M₁ ] [has_inv M₁] [has_div M₁] [has_pow M₁ ] [div_inv_monoid M₂] (f : M₁ M₂) (hf : function.injective f) (one : f 1 = 1) (mul : (x y : M₁), f (x * y) = f x * f y) (inv : (x : M₁), f x⁻¹ = (f x)⁻¹) (div : (x y : M₁), f (x / y) = f x / f y) (npow : (x : M₁) (n : ), f (x ^ n) = f x ^ n) (zpow : (x : M₁) (n : ), f (x ^ n) = f x ^ n) :

A type endowed with 1, *, ⁻¹, and / is a div_inv_monoid if it admits an injective map that preserves 1, *, ⁻¹, and / to a div_inv_monoid. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.injective.subtraction_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [has_zero M₁] [has_smul M₁] [has_neg M₁] [has_sub M₁] [has_smul M₁] [subtraction_monoid M₂] (f : M₁ M₂) (hf : function.injective f) (one : f 0 = 0) (mul : (x y : M₁), f (x + y) = f x + f y) (inv : (x : M₁), f (-x) = -f x) (div : (x y : M₁), f (x - y) = f x - f y) (npow : (x : M₁) (n : ), f (n x) = n f x) (zpow : (x : M₁) (n : ), f (n x) = n f x) :

A type endowed with 0, +, unary -, and binary - is a subtraction_monoid if it admits an injective map that preserves 0, +, unary -, and binary - to a subtraction_monoid. This version takes custom nsmul and zsmul as [has_smul ℕ M₁] and [has_smul ℤ M₁] arguments.

Equations
@[protected, reducible]
def function.injective.division_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [has_one M₁] [has_pow M₁ ] [has_inv M₁] [has_div M₁] [has_pow M₁ ] [division_monoid M₂] (f : M₁ M₂) (hf : function.injective f) (one : f 1 = 1) (mul : (x y : M₁), f (x * y) = f x * f y) (inv : (x : M₁), f x⁻¹ = (f x)⁻¹) (div : (x y : M₁), f (x / y) = f x / f y) (npow : (x : M₁) (n : ), f (x ^ n) = f x ^ n) (zpow : (x : M₁) (n : ), f (x ^ n) = f x ^ n) :

A type endowed with 1, *, ⁻¹, and / is a division_monoid if it admits an injective map that preserves 1, *, ⁻¹, and / to a division_monoid.

Equations
@[protected, reducible]
def function.injective.division_comm_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [has_one M₁] [has_pow M₁ ] [has_inv M₁] [has_div M₁] [has_pow M₁ ] [division_comm_monoid M₂] (f : M₁ M₂) (hf : function.injective f) (one : f 1 = 1) (mul : (x y : M₁), f (x * y) = f x * f y) (inv : (x : M₁), f x⁻¹ = (f x)⁻¹) (div : (x y : M₁), f (x / y) = f x / f y) (npow : (x : M₁) (n : ), f (x ^ n) = f x ^ n) (zpow : (x : M₁) (n : ), f (x ^ n) = f x ^ n) :

A type endowed with 1, *, ⁻¹, and / is a division_comm_monoid if it admits an injective map that preserves 1, *, ⁻¹, and / to a division_comm_monoid. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.injective.subtraction_comm_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [has_zero M₁] [has_smul M₁] [has_neg M₁] [has_sub M₁] [has_smul M₁] [subtraction_comm_monoid M₂] (f : M₁ M₂) (hf : function.injective f) (one : f 0 = 0) (mul : (x y : M₁), f (x + y) = f x + f y) (inv : (x : M₁), f (-x) = -f x) (div : (x y : M₁), f (x - y) = f x - f y) (npow : (x : M₁) (n : ), f (n x) = n f x) (zpow : (x : M₁) (n : ), f (n x) = n f x) :

A type endowed with 0, +, unary -, and binary - is a subtraction_comm_monoid if it admits an injective map that preserves 0, +, unary -, and binary - to a subtraction_comm_monoid. This version takes custom nsmul and zsmul as [has_smul ℕ M₁] and [has_smul ℤ M₁] arguments.

Equations
@[protected, reducible]
def function.injective.group {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [has_one M₁] [has_pow M₁ ] [has_inv M₁] [has_div M₁] [has_pow M₁ ] [group M₂] (f : M₁ M₂) (hf : function.injective f) (one : f 1 = 1) (mul : (x y : M₁), f (x * y) = f x * f y) (inv : (x : M₁), f x⁻¹ = (f x)⁻¹) (div : (x y : M₁), f (x / y) = f x / f y) (npow : (x : M₁) (n : ), f (x ^ n) = f x ^ n) (zpow : (x : M₁) (n : ), f (x ^ n) = f x ^ n) :
group M₁

A type endowed with 1, * and ⁻¹ is a group, if it admits an injective map that preserves 1, * and ⁻¹ to a group. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.injective.add_group {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [has_zero M₁] [has_smul M₁] [has_neg M₁] [has_sub M₁] [has_smul M₁] [add_group M₂] (f : M₁ M₂) (hf : function.injective f) (one : f 0 = 0) (mul : (x y : M₁), f (x + y) = f x + f y) (inv : (x : M₁), f (-x) = -f x) (div : (x y : M₁), f (x - y) = f x - f y) (npow : (x : M₁) (n : ), f (n x) = n f x) (zpow : (x : M₁) (n : ), f (n x) = n f x) :

A type endowed with 0 and + is an additive group, if it admits an injective map that preserves 0 and + to an additive group.

Equations
@[protected, reducible]
def function.injective.add_group_with_one {M₂ : Type u_2} {M₁ : Type u_1} [has_zero M₁] [has_one M₁] [has_add M₁] [has_smul M₁] [has_neg M₁] [has_sub M₁] [has_smul M₁] [has_nat_cast M₁] [has_int_cast M₁] [add_group_with_one M₂] (f : M₁ M₂) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1) (add : (x y : M₁), f (x + y) = f x + f y) (neg : (x : M₁), f (-x) = -f x) (sub : (x y : M₁), f (x - y) = f x - f y) (nsmul : (x : M₁) (n : ), f (n x) = n f x) (zsmul : (x : M₁) (n : ), f (n x) = n f x) (nat_cast : (n : ), f n = n) (int_cast : (n : ), f n = n) :

A type endowed with 0, 1 and + is an additive group with one, if it admits an injective map that preserves 0, 1 and + to an additive group with one. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.injective.comm_group {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₁] [has_one M₁] [has_pow M₁ ] [has_inv M₁] [has_div M₁] [has_pow M₁ ] [comm_group M₂] (f : M₁ M₂) (hf : function.injective f) (one : f 1 = 1) (mul : (x y : M₁), f (x * y) = f x * f y) (inv : (x : M₁), f x⁻¹ = (f x)⁻¹) (div : (x y : M₁), f (x / y) = f x / f y) (npow : (x : M₁) (n : ), f (x ^ n) = f x ^ n) (zpow : (x : M₁) (n : ), f (x ^ n) = f x ^ n) :

A type endowed with 1, * and ⁻¹ is a commutative group, if it admits an injective map that preserves 1, * and ⁻¹ to a commutative group. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.injective.add_comm_group {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₁] [has_zero M₁] [has_smul M₁] [has_neg M₁] [has_sub M₁] [has_smul M₁] [add_comm_group M₂] (f : M₁ M₂) (hf : function.injective f) (one : f 0 = 0) (mul : (x y : M₁), f (x + y) = f x + f y) (inv : (x : M₁), f (-x) = -f x) (div : (x y : M₁), f (x - y) = f x - f y) (npow : (x : M₁) (n : ), f (n x) = n f x) (zpow : (x : M₁) (n : ), f (n x) = n f x) :

A type endowed with 0 and + is an additive commutative group, if it admits an injective map that preserves 0 and + to an additive commutative group.

Equations

Surjective #

@[protected, reducible]
def function.surjective.add_semigroup {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₂] [add_semigroup M₁] (f : M₁ M₂) (hf : function.surjective f) (mul : (x y : M₁), f (x + y) = f x + f y) :

A type endowed with + is an additive semigroup, if it admits a surjective map that preserves + from an additive semigroup.

Equations
@[protected, reducible]
def function.surjective.semigroup {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₂] [semigroup M₁] (f : M₁ M₂) (hf : function.surjective f) (mul : (x y : M₁), f (x * y) = f x * f y) :

A type endowed with * is a semigroup, if it admits a surjective map that preserves * from a semigroup. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.surjective.comm_semigroup {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₂] [comm_semigroup M₁] (f : M₁ M₂) (hf : function.surjective f) (mul : (x y : M₁), f (x * y) = f x * f y) :

A type endowed with * is a commutative semigroup, if it admits a surjective map that preserves * from a commutative semigroup. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.surjective.add_comm_semigroup {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₂] [add_comm_semigroup M₁] (f : M₁ M₂) (hf : function.surjective f) (mul : (x y : M₁), f (x + y) = f x + f y) :

A type endowed with + is an additive commutative semigroup, if it admits a surjective map that preserves + from an additive commutative semigroup.

Equations
@[protected, reducible]
def function.surjective.mul_one_class {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₂] [has_one M₂] [mul_one_class M₁] (f : M₁ M₂) (hf : function.surjective f) (one : f 1 = 1) (mul : (x y : M₁), f (x * y) = f x * f y) :

A type endowed with 1 and * is a mul_one_class, if it admits a surjective map that preserves 1 and * from a mul_one_class. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.surjective.add_zero_class {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₂] [has_zero M₂] [add_zero_class M₁] (f : M₁ M₂) (hf : function.surjective f) (one : f 0 = 0) (mul : (x y : M₁), f (x + y) = f x + f y) :

A type endowed with 0 and + is an add_zero_class, if it admits a surjective map that preserves 0 and + to an add_zero_class.

Equations
@[protected, reducible]
def function.surjective.monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₂] [has_one M₂] [has_pow M₂ ] [monoid M₁] (f : M₁ M₂) (hf : function.surjective f) (one : f 1 = 1) (mul : (x y : M₁), f (x * y) = f x * f y) (npow : (x : M₁) (n : ), f (x ^ n) = f x ^ n) :
monoid M₂

A type endowed with 1 and * is a monoid, if it admits a surjective map that preserves 1 and * to a monoid. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.surjective.add_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₂] [has_zero M₂] [has_smul M₂] [add_monoid M₁] (f : M₁ M₂) (hf : function.surjective f) (one : f 0 = 0) (mul : (x y : M₁), f (x + y) = f x + f y) (npow : (x : M₁) (n : ), f (n x) = n f x) :

A type endowed with 0 and + is an additive monoid, if it admits a surjective map that preserves 0 and + to an additive monoid. This version takes a custom nsmul as a [has_smul ℕ M₂] argument.

Equations
@[protected, reducible]
def function.surjective.add_monoid_with_one {M₁ : Type u_1} {M₂ : Type u_2} [has_zero M₂] [has_one M₂] [has_add M₂] [has_smul M₂] [has_nat_cast M₂] [add_monoid_with_one M₁] (f : M₁ M₂) (hf : function.surjective f) (zero : f 0 = 0) (one : f 1 = 1) (add : (x y : M₁), f (x + y) = f x + f y) (nsmul : (x : M₁) (n : ), f (n x) = n f x) (nat_cast : (n : ), f n = n) :

A type endowed with 0, 1 and + is an additive monoid with one, if it admits a surjective map that preserves 0, 1 and * from an additive monoid with one. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.surjective.add_comm_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₂] [has_zero M₂] [has_smul M₂] [add_comm_monoid M₁] (f : M₁ M₂) (hf : function.surjective f) (one : f 0 = 0) (mul : (x y : M₁), f (x + y) = f x + f y) (npow : (x : M₁) (n : ), f (n x) = n f x) :

A type endowed with 0 and + is an additive commutative monoid, if it admits a surjective map that preserves 0 and + to an additive commutative monoid.

Equations
@[protected, reducible]
def function.surjective.comm_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₂] [has_one M₂] [has_pow M₂ ] [comm_monoid M₁] (f : M₁ M₂) (hf : function.surjective f) (one : f 1 = 1) (mul : (x y : M₁), f (x * y) = f x * f y) (npow : (x : M₁) (n : ), f (x ^ n) = f x ^ n) :

A type endowed with 1 and * is a commutative monoid, if it admits a surjective map that preserves 1 and * from a commutative monoid. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.surjective.has_involutive_inv {M₁ : Type u_1} {M₂ : Type u_2} [has_inv M₂] [has_involutive_inv M₁] (f : M₁ M₂) (hf : function.surjective f) (inv : (x : M₁), f x⁻¹ = (f x)⁻¹) :

A type has an involutive inversion if it admits a surjective map that preserves ⁻¹ to a type which has an involutive inversion.

Equations
@[protected, reducible]
def function.surjective.has_involutive_neg {M₁ : Type u_1} {M₂ : Type u_2} [has_neg M₂] [has_involutive_neg M₁] (f : M₁ M₂) (hf : function.surjective f) (inv : (x : M₁), f (-x) = -f x) :

A type has an involutive negation if it admits a surjective map that preserves ⁻¹ to a type which has an involutive inversion.

Equations
@[protected, reducible]
def function.surjective.div_inv_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₂] [has_one M₂] [has_pow M₂ ] [has_inv M₂] [has_div M₂] [has_pow M₂ ] [div_inv_monoid M₁] (f : M₁ M₂) (hf : function.surjective f) (one : f 1 = 1) (mul : (x y : M₁), f (x * y) = f x * f y) (inv : (x : M₁), f x⁻¹ = (f x)⁻¹) (div : (x y : M₁), f (x / y) = f x / f y) (npow : (x : M₁) (n : ), f (x ^ n) = f x ^ n) (zpow : (x : M₁) (n : ), f (x ^ n) = f x ^ n) :

A type endowed with 1, *, ⁻¹, and / is a div_inv_monoid if it admits a surjective map that preserves 1, *, ⁻¹, and / to a div_inv_monoid. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.surjective.sub_neg_monoid {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₂] [has_zero M₂] [has_smul M₂] [has_neg M₂] [has_sub M₂] [has_smul M₂] [sub_neg_monoid M₁] (f : M₁ M₂) (hf : function.surjective f) (one : f 0 = 0) (mul : (x y : M₁), f (x + y) = f x + f y) (inv : (x : M₁), f (-x) = -f x) (div : (x y : M₁), f (x - y) = f x - f y) (npow : (x : M₁) (n : ), f (n x) = n f x) (zpow : (x : M₁) (n : ), f (n x) = n f x) :

A type endowed with 0, +, unary -, and binary - is a sub_neg_monoid if it admits a surjective map that preserves 0, +, unary -, and binary - to a sub_neg_monoid.

Equations
@[protected, reducible]
def function.surjective.group {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₂] [has_one M₂] [has_pow M₂ ] [has_inv M₂] [has_div M₂] [has_pow M₂ ] [group M₁] (f : M₁ M₂) (hf : function.surjective f) (one : f 1 = 1) (mul : (x y : M₁), f (x * y) = f x * f y) (inv : (x : M₁), f x⁻¹ = (f x)⁻¹) (div : (x y : M₁), f (x / y) = f x / f y) (npow : (x : M₁) (n : ), f (x ^ n) = f x ^ n) (zpow : (x : M₁) (n : ), f (x ^ n) = f x ^ n) :
group M₂

A type endowed with 1, * and ⁻¹ is a group, if it admits a surjective map that preserves 1, * and ⁻¹ to a group. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.surjective.add_group {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₂] [has_zero M₂] [has_smul M₂] [has_neg M₂] [has_sub M₂] [has_smul M₂] [add_group M₁] (f : M₁ M₂) (hf : function.surjective f) (one : f 0 = 0) (mul : (x y : M₁), f (x + y) = f x + f y) (inv : (x : M₁), f (-x) = -f x) (div : (x y : M₁), f (x - y) = f x - f y) (npow : (x : M₁) (n : ), f (n x) = n f x) (zpow : (x : M₁) (n : ), f (n x) = n f x) :

A type endowed with 0 and + is an additive group, if it admits a surjective map that preserves 0 and + to an additive group.

Equations
@[protected]
def function.surjective.add_group_with_one {M₁ : Type u_1} {M₂ : Type u_2} [has_zero M₂] [has_one M₂] [has_add M₂] [has_neg M₂] [has_sub M₂] [has_smul M₂] [has_smul M₂] [has_nat_cast M₂] [has_int_cast M₂] [add_group_with_one M₁] (f : M₁ M₂) (hf : function.surjective f) (zero : f 0 = 0) (one : f 1 = 1) (add : (x y : M₁), f (x + y) = f x + f y) (neg : (x : M₁), f (-x) = -f x) (sub : (x y : M₁), f (x - y) = f x - f y) (nsmul : (x : M₁) (n : ), f (n x) = n f x) (zsmul : (x : M₁) (n : ), f (n x) = n f x) (nat_cast : (n : ), f n = n) (int_cast : (n : ), f n = n) :

A type endowed with 0, 1, + is an additive group with one, if it admits a surjective map that preserves 0, 1, and + to an additive group with one. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.surjective.comm_group {M₁ : Type u_1} {M₂ : Type u_2} [has_mul M₂] [has_one M₂] [has_pow M₂ ] [has_inv M₂] [has_div M₂] [has_pow M₂ ] [comm_group M₁] (f : M₁ M₂) (hf : function.surjective f) (one : f 1 = 1) (mul : (x y : M₁), f (x * y) = f x * f y) (inv : (x : M₁), f x⁻¹ = (f x)⁻¹) (div : (x y : M₁), f (x / y) = f x / f y) (npow : (x : M₁) (n : ), f (x ^ n) = f x ^ n) (zpow : (x : M₁) (n : ), f (x ^ n) = f x ^ n) :

A type endowed with 1, *, ⁻¹, and / is a commutative group, if it admits a surjective map that preserves 1, *, ⁻¹, and / from a commutative group. See note [reducible non-instances].

Equations
@[protected, reducible]
def function.surjective.add_comm_group {M₁ : Type u_1} {M₂ : Type u_2} [has_add M₂] [has_zero M₂] [has_smul M₂] [has_neg M₂] [has_sub M₂] [has_smul M₂] [add_comm_group M₁] (f : M₁ M₂) (hf : function.surjective f) (one : f 0 = 0) (mul : (x y : M₁), f (x + y) = f x + f y) (inv : (x : M₁), f (-x) = -f x) (div : (x y : M₁), f (x - y) = f x - f y) (npow : (x : M₁) (n : ), f (n x) = n f x) (zpow : (x : M₁) (n : ), f (n x) = n f x) :

A type endowed with 0 and + is an additive commutative group, if it admits a surjective map that preserves 0 and + to an additive commutative group.

Equations