statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
one_mul : ∀ a : M, 1 * a = a | mul_one_class.one_mul | lemma | one_mul | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_one : ∀ a : M, a * 1 = a | mul_one_class.mul_one | lemma | mul_one | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_one_class.to_is_left_id : is_left_id M (*) 1 | ⟨ mul_one_class.one_mul ⟩ | instance | mul_one_class.to_is_left_id | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_one_class.to_is_right_id : is_right_id M (*) 1 | ⟨ mul_one_class.mul_one ⟩ | instance | mul_one_class.to_is_right_id | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
npow_rec [has_one M] [has_mul M] : ℕ → M → M | | 0 a := 1
| (n+1) a := a * npow_rec n a | def | npow_rec | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | The fundamental power operation in a monoid. `npow_rec n a = a*a*...*a` n times.
Use instead `a ^ n`, which has better definitional behavior. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nsmul_rec [has_zero M] [has_add M] : ℕ → M → M | | 0 a := 0
| (n+1) a := a + nsmul_rec n a | def | nsmul_rec | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | The fundamental scalar multiplication in an additive monoid. `nsmul_rec n a = a+a+...+a` n
times. Use instead `n • a`, which has better definitional behavior. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
try_refl_tac : tactic unit | `[intros; refl] | def | try_refl_tac | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | `try_refl_tac` solves goals of the form `∀ a b, f a b = g a b`,
if they hold by definition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid (M : Type u) extends add_semigroup M, add_zero_class M | (nsmul : ℕ → M → M := nsmul_rec)
(nsmul_zero' : ∀ x, nsmul 0 x = 0 . try_refl_tac)
(nsmul_succ' : ∀ (n : ℕ) x, nsmul n.succ x = x + nsmul n x . try_refl_tac) | class | add_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"add_semigroup",
"add_zero_class",
"nsmul_rec",
"try_refl_tac"
] | An `add_monoid` is an `add_semigroup` with an element `0` such that `0 + a = a + 0 = a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid (M : Type u) extends semigroup M, mul_one_class M | (npow : ℕ → M → M := npow_rec)
(npow_zero' : ∀ x, npow 0 x = 1 . try_refl_tac)
(npow_succ' : ∀ (n : ℕ) x, npow n.succ x = x * npow n x . try_refl_tac) | class | monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"mul_one_class",
"npow_rec",
"semigroup",
"try_refl_tac"
] | A `monoid` is a `semigroup` with an element `1` such that `1 * a = a * 1 = a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid.has_pow {M : Type*} [monoid M] : has_pow M ℕ | ⟨λ x n, monoid.npow n x⟩ | instance | monoid.has_pow | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid.has_smul_nat {M : Type*} [add_monoid M] : has_smul ℕ M | ⟨add_monoid.nsmul⟩ | instance | add_monoid.has_smul_nat | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"add_monoid",
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
npow_eq_pow (n : ℕ) (x : M) : monoid.npow n x = x^n | rfl | lemma | npow_eq_pow | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_zero (a : M) : a^0 = 1 | monoid.npow_zero' _ | theorem | pow_zero | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_succ (a : M) (n : ℕ) : a^(n+1) = a * a^n | monoid.npow_succ' n a | theorem | pow_succ | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_inv_eq_right_inv {a b c : M} (hba : b * a = 1) (hac : a * c = 1) : b = c | by rw [←one_mul c, ←hba, mul_assoc, hac, mul_one b] | lemma | left_inv_eq_right_inv | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"mul_assoc",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_comm_monoid (M : Type u) extends add_monoid M, add_comm_semigroup M | class | add_comm_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"add_comm_semigroup",
"add_monoid"
] | An additive commutative monoid is an additive monoid with commutative `(+)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_monoid (M : Type u) extends monoid M, comm_semigroup M | class | comm_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"comm_semigroup",
"monoid"
] | A commutative monoid is a monoid with commutative `(*)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_left_cancel_monoid (M : Type u) extends add_left_cancel_semigroup M, add_monoid M | class | add_left_cancel_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"add_left_cancel_semigroup",
"add_monoid"
] | An additive monoid in which addition is left-cancellative.
Main examples are `ℕ` and groups. This is the right typeclass for many sum lemmas, as having a zero
is useful to define the sum over the empty set, so `add_left_cancel_semigroup` is not enough. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_cancel_monoid (M : Type u) extends left_cancel_semigroup M, monoid M | class | left_cancel_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"left_cancel_semigroup",
"monoid"
] | A monoid in which multiplication is left-cancellative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_right_cancel_monoid (M : Type u) extends add_right_cancel_semigroup M, add_monoid M | class | add_right_cancel_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"add_monoid",
"add_right_cancel_semigroup"
] | An additive monoid in which addition is right-cancellative.
Main examples are `ℕ` and groups. This is the right typeclass for many sum lemmas, as having a zero
is useful to define the sum over the empty set, so `add_right_cancel_semigroup` is not enough. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_cancel_monoid (M : Type u) extends right_cancel_semigroup M, monoid M | class | right_cancel_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"monoid",
"right_cancel_semigroup"
] | A monoid in which multiplication is right-cancellative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_cancel_monoid (M : Type u)
extends add_left_cancel_monoid M, add_right_cancel_monoid M | class | add_cancel_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"add_left_cancel_monoid",
"add_right_cancel_monoid"
] | An additive monoid in which addition is cancellative on both sides.
Main examples are `ℕ` and groups. This is the right typeclass for many sum lemmas, as having a zero
is useful to define the sum over the empty set, so `add_right_cancel_semigroup` is not enough. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_monoid (M : Type u) extends left_cancel_monoid M, right_cancel_monoid M | class | cancel_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"left_cancel_monoid",
"right_cancel_monoid"
] | A monoid in which multiplication is cancellative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_cancel_comm_monoid (M : Type u) extends add_left_cancel_monoid M, add_comm_monoid M | class | add_cancel_comm_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"add_comm_monoid",
"add_left_cancel_monoid"
] | Commutative version of `add_cancel_monoid`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_comm_monoid (M : Type u) extends left_cancel_monoid M, comm_monoid M | class | cancel_comm_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"comm_monoid",
"left_cancel_monoid"
] | Commutative version of `cancel_monoid`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_comm_monoid.to_cancel_monoid (M : Type u) [cancel_comm_monoid M] :
cancel_monoid M | { .. ‹cancel_comm_monoid M›, .. comm_semigroup.is_left_cancel_mul.to_is_right_cancel_mul M } | instance | cancel_comm_monoid.to_cancel_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"cancel_comm_monoid",
"cancel_monoid",
"comm_semigroup.is_left_cancel_mul.to_is_right_cancel_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_monoid.to_is_cancel_mul (M : Type u) [cancel_monoid M] : is_cancel_mul M | { mul_left_cancel := cancel_monoid.mul_left_cancel,
mul_right_cancel := cancel_monoid.mul_right_cancel } | instance | cancel_monoid.to_is_cancel_mul | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"cancel_monoid",
"is_cancel_mul",
"mul_left_cancel",
"mul_right_cancel"
] | Any `cancel_monoid M` satisfies `is_cancel_mul M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zpow_rec {M : Type*} [has_one M] [has_mul M] [has_inv M] : ℤ → M → M | | (int.of_nat n) a := npow_rec n a
| -[1+ n] a := (npow_rec n.succ a) ⁻¹ | def | zpow_rec | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"npow_rec"
] | The fundamental power operation in a group. `zpow_rec n a = a*a*...*a` n times, for integer `n`.
Use instead `a ^ n`, which has better definitional behavior. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zsmul_rec {M : Type*} [has_zero M] [has_add M] [has_neg M]: ℤ → M → M | | (int.of_nat n) a := nsmul_rec n a
| -[1+ n] a := - (nsmul_rec n.succ a) | def | zsmul_rec | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"nsmul_rec"
] | The fundamental scalar multiplication in an additive group. `zsmul_rec n a = a+a+...+a` n
times, for integer `n`. Use instead `n • a`, which has better definitional behavior. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_involutive_neg (A : Type*) extends has_neg A | (neg_neg : ∀ x : A, - -x = x) | class | has_involutive_neg | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | Auxiliary typeclass for types with an involutive `has_neg`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_involutive_inv (G : Type*) extends has_inv G | (inv_inv : ∀ x : G, x⁻¹⁻¹ = x) | class | has_involutive_inv | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"inv_inv"
] | Auxiliary typeclass for types with an involutive `has_inv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_inv (a : G) : a⁻¹⁻¹ = a | has_involutive_inv.inv_inv _ | lemma | inv_inv | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_inv_monoid (G : Type u) extends monoid G, has_inv G, has_div G | (div := λ a b, a * b⁻¹)
(div_eq_mul_inv : ∀ a b : G, a / b = a * b⁻¹ . try_refl_tac)
(zpow : ℤ → G → G := zpow_rec)
(zpow_zero' : ∀ (a : G), zpow 0 a = 1 . try_refl_tac)
(zpow_succ' :
∀ (n : ℕ) (a : G), zpow (int.of_nat n.succ) a = a * zpow (int.of_nat n) a . try_refl_tac)
(zpow_neg' :
∀ (n : ℕ) (a : G), zpow (-[1+... | class | div_inv_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"div_eq_mul_inv",
"monoid",
"try_refl_tac",
"zpow_rec"
] | A `div_inv_monoid` is a `monoid` with operations `/` and `⁻¹` satisfying
`div_eq_mul_inv : ∀ a b, a / b = a * b⁻¹`.
This deduplicates the name `div_eq_mul_inv`.
The default for `div` is such that `a / b = a * b⁻¹` holds by definition.
Adding `div` as a field rather than defining `a / b := a * b⁻¹` allows us to
avoid ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_neg_monoid (G : Type u) extends add_monoid G, has_neg G, has_sub G | (sub := λ a b, a + -b)
(sub_eq_add_neg : ∀ a b : G, a - b = a + -b . try_refl_tac)
(zsmul : ℤ → G → G := zsmul_rec)
(zsmul_zero' : ∀ (a : G), zsmul 0 a = 0 . try_refl_tac)
(zsmul_succ' :
∀ (n : ℕ) (a : G), zsmul (int.of_nat n.succ) a = a + zsmul (int.of_nat n) a . try_refl_tac)
(zsmul_neg' :
∀ (n : ℕ) (a : G), zsmu... | class | sub_neg_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"add_monoid",
"try_refl_tac",
"zsmul_rec"
] | A `sub_neg_monoid` is an `add_monoid` with unary `-` and binary `-` operations
satisfying `sub_eq_add_neg : ∀ a b, a - b = a + -b`.
The default for `sub` is such that `a - b = a + -b` holds by definition.
Adding `sub` as a field rather than defining `a - b := a + -b` allows us to
avoid certain classes of unification ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
div_inv_monoid.has_pow {M} [div_inv_monoid M] : has_pow M ℤ | ⟨λ x n, div_inv_monoid.zpow n x⟩ | instance | div_inv_monoid.has_pow | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"div_inv_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_neg_monoid.has_smul_int {M} [sub_neg_monoid M] : has_smul ℤ M | ⟨sub_neg_monoid.zsmul⟩ | instance | sub_neg_monoid.has_smul_int | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"has_smul",
"sub_neg_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_eq_pow (n : ℤ) (x : G) : div_inv_monoid.zpow n x = x^n | rfl | lemma | zpow_eq_pow | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_zero (a : G) : a ^ (0:ℤ) = 1 | div_inv_monoid.zpow_zero' a | theorem | zpow_zero | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_coe_nat (a : G) : ∀ n : ℕ, a ^ (n:ℤ) = a ^ n | | 0 := (zpow_zero _).trans (pow_zero _).symm
| (n + 1) :=
calc a ^ (↑(n + 1) : ℤ) = a * a ^ (n : ℤ) : div_inv_monoid.zpow_succ' _ _
... = a * a ^ n : congr_arg ((*) a) (zpow_coe_nat n)
... = a ^ (n + 1) : (pow_succ _ _).symm | theorem | zpow_coe_nat | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"pow_succ",
"pow_zero",
"zpow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_of_nat (a : G) (n : ℕ) : a ^ (int.of_nat n) = a ^ n | zpow_coe_nat a n | theorem | zpow_of_nat | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"zpow_coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_neg_succ_of_nat (a : G) (n : ℕ) : a ^ -[1+n] = (a ^ (n + 1))⁻¹ | by { rw ← zpow_coe_nat, exact div_inv_monoid.zpow_neg' n a } | theorem | zpow_neg_succ_of_nat | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"zpow_coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_mul_inv (a b : G) : a / b = a * b⁻¹ | div_inv_monoid.div_eq_mul_inv _ _ | lemma | div_eq_mul_inv | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | Dividing by an element is the same as multiplying by its inverse.
This is a duplicate of `div_inv_monoid.div_eq_mul_inv` ensuring that the types unfold better. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg_zero_class (G : Type*) extends has_zero G, has_neg G | (neg_zero : -(0 : G) = 0) | class | neg_zero_class | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | Typeclass for expressing that `-0 = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_neg_zero_monoid (G : Type*) extends sub_neg_monoid G, neg_zero_class G | class | sub_neg_zero_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"neg_zero_class",
"sub_neg_monoid"
] | A `sub_neg_monoid` where `-0 = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_one_class (G : Type*) extends has_one G, has_inv G | (inv_one : (1 : G)⁻¹ = 1) | class | inv_one_class | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"inv_one"
] | Typeclass for expressing that `1⁻¹ = 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
div_inv_one_monoid (G : Type*) extends div_inv_monoid G, inv_one_class G | class | div_inv_one_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"div_inv_monoid",
"inv_one_class"
] | A `div_inv_monoid` where `1⁻¹ = 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_one : (1 : G)⁻¹ = 1 | inv_one_class.inv_one | lemma | inv_one | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtraction_monoid (G : Type u) extends sub_neg_monoid G, has_involutive_neg G | (neg_add_rev (a b : G) : -(a + b) = -b + -a)
/- Despite the asymmetry of `neg_eq_of_add`, the symmetric version is true thanks to the
involutivity of negation. -/
(neg_eq_of_add (a b : G) : a + b = 0 → -a = b) | class | subtraction_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"has_involutive_neg",
"sub_neg_monoid"
] | A `subtraction_monoid` is a `sub_neg_monoid` with involutive negation and such that
`-(a + b) = -b + -a` and `a + b = 0 → -a = b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
division_monoid (G : Type u) extends div_inv_monoid G, has_involutive_inv G | (mul_inv_rev (a b : G) : (a * b)⁻¹ = b⁻¹ * a⁻¹)
/- Despite the asymmetry of `inv_eq_of_mul`, the symmetric version is true thanks to the
involutivity of inversion. -/
(inv_eq_of_mul (a b : G) : a * b = 1 → a⁻¹ = b) | class | division_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"div_inv_monoid",
"has_involutive_inv",
"inv_eq_of_mul",
"mul_inv_rev"
] | A `division_monoid` is a `div_inv_monoid` with involutive inversion and such that
`(a * b)⁻¹ = b⁻¹ * a⁻¹` and `a * b = 1 → a⁻¹ = b`.
This is the immediate common ancestor of `group` and `group_with_zero`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_inv_rev (a b : G) : (a * b)⁻¹ = b⁻¹ * a⁻¹ | division_monoid.mul_inv_rev _ _ | lemma | mul_inv_rev | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_of_mul_eq_one_right : a * b = 1 → a⁻¹ = b | division_monoid.inv_eq_of_mul _ _ | lemma | inv_eq_of_mul_eq_one_right | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtraction_comm_monoid (G : Type u) extends subtraction_monoid G, add_comm_monoid G | class | subtraction_comm_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"add_comm_monoid",
"subtraction_monoid"
] | Commutative `subtraction_monoid`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
division_comm_monoid (G : Type u) extends division_monoid G, comm_monoid G | class | division_comm_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"comm_monoid",
"division_monoid"
] | Commutative `division_monoid`.
This is the immediate common ancestor of `comm_group` and `comm_group_with_zero`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group (G : Type u) extends div_inv_monoid G | (mul_left_inv : ∀ a : G, a⁻¹ * a = 1) | class | group | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"div_inv_monoid",
"mul_left_inv"
] | A `group` is a `monoid` with an operation `⁻¹` satisfying `a⁻¹ * a = 1`.
There is also a division operation `/` such that `a / b = a * b⁻¹`,
with a default so that `a / b = a * b⁻¹` holds by definition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_group (A : Type u) extends sub_neg_monoid A | (add_left_neg : ∀ a : A, -a + a = 0) | class | add_group | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"sub_neg_monoid"
] | An `add_group` is an `add_monoid` with a unary `-` satisfying `-a + a = 0`.
There is also a binary operation `-` such that `a - b = a + -b`,
with a default so that `a - b = a + -b` holds by definition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
group.to_monoid (G : Type u) [group G] : monoid G | @div_inv_monoid.to_monoid _ (@group.to_div_inv_monoid _ _) | def | group.to_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"group",
"monoid"
] | Abbreviation for `@div_inv_monoid.to_monoid _ (@group.to_div_inv_monoid _ _)`.
Useful because it corresponds to the fact that `Grp` is a subcategory of `Mon`.
Not an instance since it duplicates `@div_inv_monoid.to_monoid _ (@group.to_div_inv_monoid _ _)`.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_left_inv : ∀ a : G, a⁻¹ * a = 1 | group.mul_left_inv | lemma | mul_left_inv | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_self (a : G) : a⁻¹ * a = 1 | mul_left_inv a | lemma | inv_mul_self | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"mul_left_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_of_mul (h : a * b = 1) : a⁻¹ = b | left_inv_eq_right_inv (inv_mul_self a) h | lemma | inv_eq_of_mul | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"inv_mul_self",
"left_inv_eq_right_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_inv (a : G) : a * a⁻¹ = 1 | by rw [←mul_left_inv a⁻¹, inv_eq_of_mul (mul_left_inv a)] | lemma | mul_right_inv | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"inv_eq_of_mul",
"mul_left_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_self (a : G) : a * a⁻¹ = 1 | mul_right_inv a | lemma | mul_inv_self | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"mul_right_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_cancel_left (a b : G) : a⁻¹ * (a * b) = b | by rw [←mul_assoc, mul_left_inv, one_mul] | lemma | inv_mul_cancel_left | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"mul_left_inv",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_cancel_left (a b : G) : a * (a⁻¹ * b) = b | by rw [←mul_assoc, mul_right_inv, one_mul] | lemma | mul_inv_cancel_left | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"mul_right_inv",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_cancel_right (a b : G) : a * b * b⁻¹ = a | by rw [mul_assoc, mul_right_inv, mul_one] | lemma | mul_inv_cancel_right | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"mul_assoc",
"mul_one",
"mul_right_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_cancel_right (a b : G) : a * b⁻¹ * b = a | by rw [mul_assoc, mul_left_inv, mul_one] | lemma | inv_mul_cancel_right | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"mul_assoc",
"mul_left_inv",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group.to_division_monoid : division_monoid G | { inv_inv := λ a, inv_eq_of_mul (mul_left_inv a),
mul_inv_rev := λ a b, inv_eq_of_mul $ by rw [mul_assoc, mul_inv_cancel_left, mul_right_inv],
inv_eq_of_mul := λ _ _, inv_eq_of_mul,
..‹group G› } | instance | group.to_division_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"division_monoid",
"inv_eq_of_mul",
"inv_inv",
"mul_assoc",
"mul_inv_cancel_left",
"mul_inv_rev",
"mul_left_inv",
"mul_right_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group.to_cancel_monoid : cancel_monoid G | { mul_right_cancel := λ a b c h, by rw [← mul_inv_cancel_right a b, h, mul_inv_cancel_right],
mul_left_cancel := λ a b c h, by rw [← inv_mul_cancel_left a b, h, inv_mul_cancel_left],
..‹group G› } | instance | group.to_cancel_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"cancel_monoid",
"inv_mul_cancel_left",
"mul_inv_cancel_right",
"mul_left_cancel",
"mul_right_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group.to_div_inv_monoid_injective {G : Type*} :
function.injective (@group.to_div_inv_monoid G) | by { rintros ⟨⟩ ⟨⟩ ⟨⟩, refl } | lemma | group.to_div_inv_monoid_injective | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_group (G : Type u) extends group G, comm_monoid G | class | comm_group | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"comm_monoid",
"group"
] | A commutative group is a group with commutative `(*)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_comm_group (G : Type u) extends add_group G, add_comm_monoid G | class | add_comm_group | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"add_comm_monoid",
"add_group"
] | An additive commutative group is an additive group with commutative `(+)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_group.to_group_injective {G : Type u} :
function.injective (@comm_group.to_group G) | by { rintros ⟨⟩ ⟨⟩ ⟨⟩, refl } | lemma | comm_group.to_group_injective | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_group.to_cancel_comm_monoid : cancel_comm_monoid G | { ..‹comm_group G›,
..group.to_cancel_monoid } | instance | comm_group.to_cancel_comm_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"cancel_comm_monoid",
"group.to_cancel_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_group.to_division_comm_monoid : division_comm_monoid G | { ..‹comm_group G›,
..group.to_division_monoid } | instance | comm_group.to_division_comm_monoid | algebra.group | src/algebra/group/defs.lean | [
"tactic.basic",
"logic.function.basic"
] | [
"division_comm_monoid",
"group.to_division_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid.ext {M : Type u} ⦃m₁ m₂ : monoid M⦄ (h_mul : m₁.mul = m₂.mul) : m₁ = m₂ | begin
have h₁ : (@monoid.to_mul_one_class _ m₁).one = (@monoid.to_mul_one_class _ m₂).one,
from congr_arg (@mul_one_class.one M) (mul_one_class.ext h_mul),
set f : @monoid_hom M M (@monoid.to_mul_one_class _ m₁) (@monoid.to_mul_one_class _ m₂) :=
{ to_fun := id, map_one' := h₁, map_mul' := λ x y, congr_fun ... | lemma | monoid.ext | algebra.group | src/algebra/group/ext.lean | [
"algebra.hom.group"
] | [
"monoid",
"monoid_hom",
"monoid_hom.map_pow",
"mul_one_class.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_monoid.to_monoid_injective {M : Type u} :
function.injective (@comm_monoid.to_monoid M) | begin
rintros ⟨⟩ ⟨⟩ h,
congr'; injection h,
end | lemma | comm_monoid.to_monoid_injective | algebra.group | src/algebra/group/ext.lean | [
"algebra.hom.group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_monoid.ext {M : Type*} ⦃m₁ m₂ : comm_monoid M⦄ (h_mul : m₁.mul = m₂.mul) : m₁ = m₂ | comm_monoid.to_monoid_injective $ monoid.ext h_mul | lemma | comm_monoid.ext | algebra.group | src/algebra/group/ext.lean | [
"algebra.hom.group"
] | [
"comm_monoid",
"comm_monoid.to_monoid_injective",
"monoid.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_cancel_monoid.to_monoid_injective {M : Type u} :
function.injective (@left_cancel_monoid.to_monoid M) | begin
rintros ⟨⟩ ⟨⟩ h,
congr'; injection h,
end | lemma | left_cancel_monoid.to_monoid_injective | algebra.group | src/algebra/group/ext.lean | [
"algebra.hom.group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_cancel_monoid.ext {M : Type u} ⦃m₁ m₂ : left_cancel_monoid M⦄
(h_mul : m₁.mul = m₂.mul) : m₁ = m₂ | left_cancel_monoid.to_monoid_injective $ monoid.ext h_mul | lemma | left_cancel_monoid.ext | algebra.group | src/algebra/group/ext.lean | [
"algebra.hom.group"
] | [
"left_cancel_monoid",
"left_cancel_monoid.to_monoid_injective",
"monoid.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_cancel_monoid.to_monoid_injective {M : Type u} :
function.injective (@right_cancel_monoid.to_monoid M) | begin
rintros ⟨⟩ ⟨⟩ h,
congr'; injection h,
end | lemma | right_cancel_monoid.to_monoid_injective | algebra.group | src/algebra/group/ext.lean | [
"algebra.hom.group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_cancel_monoid.ext {M : Type u} ⦃m₁ m₂ : right_cancel_monoid M⦄
(h_mul : m₁.mul = m₂.mul) : m₁ = m₂ | right_cancel_monoid.to_monoid_injective $ monoid.ext h_mul | lemma | right_cancel_monoid.ext | algebra.group | src/algebra/group/ext.lean | [
"algebra.hom.group"
] | [
"monoid.ext",
"right_cancel_monoid",
"right_cancel_monoid.to_monoid_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_monoid.to_left_cancel_monoid_injective {M : Type u} :
function.injective (@cancel_monoid.to_left_cancel_monoid M) | begin
rintros ⟨⟩ ⟨⟩ h,
congr'; injection h,
end | lemma | cancel_monoid.to_left_cancel_monoid_injective | algebra.group | src/algebra/group/ext.lean | [
"algebra.hom.group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_monoid.ext {M : Type*} ⦃m₁ m₂ : cancel_monoid M⦄
(h_mul : m₁.mul = m₂.mul) : m₁ = m₂ | cancel_monoid.to_left_cancel_monoid_injective $ left_cancel_monoid.ext h_mul | lemma | cancel_monoid.ext | algebra.group | src/algebra/group/ext.lean | [
"algebra.hom.group"
] | [
"cancel_monoid",
"cancel_monoid.to_left_cancel_monoid_injective",
"left_cancel_monoid.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_comm_monoid.to_comm_monoid_injective {M : Type u} :
function.injective (@cancel_comm_monoid.to_comm_monoid M) | begin
rintros ⟨⟩ ⟨⟩ h,
congr'; injection h,
end | lemma | cancel_comm_monoid.to_comm_monoid_injective | algebra.group | src/algebra/group/ext.lean | [
"algebra.hom.group"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_comm_monoid.ext {M : Type*} ⦃m₁ m₂ : cancel_comm_monoid M⦄
(h_mul : m₁.mul = m₂.mul) : m₁ = m₂ | cancel_comm_monoid.to_comm_monoid_injective $ comm_monoid.ext h_mul | lemma | cancel_comm_monoid.ext | algebra.group | src/algebra/group/ext.lean | [
"algebra.hom.group"
] | [
"cancel_comm_monoid",
"cancel_comm_monoid.to_comm_monoid_injective",
"comm_monoid.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_inv_monoid.ext {M : Type*} ⦃m₁ m₂ : div_inv_monoid M⦄ (h_mul : m₁.mul = m₂.mul)
(h_inv : m₁.inv = m₂.inv) : m₁ = m₂ | begin
have h₁ : (@div_inv_monoid.to_monoid _ m₁).one = (@div_inv_monoid.to_monoid _ m₂).one,
from congr_arg (@monoid.one M) (monoid.ext h_mul),
set f : @monoid_hom M M (by letI := m₁; apply_instance) (by letI := m₂; apply_instance) :=
{ to_fun := id, map_one' := h₁, map_mul' := λ x y, congr_fun (congr_fun h... | lemma | div_inv_monoid.ext | algebra.group | src/algebra/group/ext.lean | [
"algebra.hom.group"
] | [
"div_inv_monoid",
"map_div'",
"monoid.ext",
"monoid_hom",
"monoid_hom.map_zpow'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group.ext {G : Type*} ⦃g₁ g₂ : group G⦄ (h_mul : g₁.mul = g₂.mul) : g₁ = g₂ | begin
set f := @monoid_hom.mk' G G (by letI := g₁; apply_instance) g₂ id
(λ a b, congr_fun (congr_fun h_mul a) b),
exact group.to_div_inv_monoid_injective (div_inv_monoid.ext h_mul
(funext $ @monoid_hom.map_inv G G g₁ (@group.to_division_monoid _ g₂) f))
end | lemma | group.ext | algebra.group | src/algebra/group/ext.lean | [
"algebra.hom.group"
] | [
"div_inv_monoid.ext",
"group",
"group.to_div_inv_monoid_injective",
"group.to_division_monoid",
"monoid_hom.map_inv",
"monoid_hom.mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_group.ext {G : Type*} ⦃g₁ g₂ : comm_group G⦄
(h_mul : g₁.mul = g₂.mul) : g₁ = g₂ | comm_group.to_group_injective $ group.ext h_mul | lemma | comm_group.ext | algebra.group | src/algebra/group/ext.lean | [
"algebra.hom.group"
] | [
"comm_group",
"comm_group.to_group_injective",
"group.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semigroup [semigroup M₂] (f : M₁ → M₂) (hf : injective f)
(mul : ∀ x y, f (x * y) = f x * f y) :
semigroup M₁ | { mul_assoc := λ x y z, hf $ by erw [mul, mul, mul, mul, mul_assoc],
..‹has_mul M₁› } | def | function.injective.semigroup | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"mul_assoc",
"semigroup"
] | A type endowed with `*` is a semigroup,
if it admits an injective map that preserves `*` to a semigroup.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comm_semigroup [comm_semigroup M₂] (f : M₁ → M₂) (hf : injective f)
(mul : ∀ x y, f (x * y) = f x * f y) :
comm_semigroup M₁ | { mul_comm := λ x y, hf $ by erw [mul, mul, mul_comm],
.. hf.semigroup f mul } | def | function.injective.comm_semigroup | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"comm_semigroup",
"mul_comm"
] | 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]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_cancel_semigroup [left_cancel_semigroup M₂] (f : M₁ → M₂) (hf : injective f)
(mul : ∀ x y, f (x * y) = f x * f y) :
left_cancel_semigroup M₁ | { mul := (*),
mul_left_cancel := λ x y z H, hf $ (mul_right_inj (f x)).1 $ by erw [← mul, ← mul, H]; refl,
.. hf.semigroup f mul } | def | function.injective.left_cancel_semigroup | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"left_cancel_semigroup",
"mul_left_cancel",
"mul_right_inj"
] | 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]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_cancel_semigroup [right_cancel_semigroup M₂] (f : M₁ → M₂) (hf : injective f)
(mul : ∀ x y, f (x * y) = f x * f y) :
right_cancel_semigroup M₁ | { mul := (*),
mul_right_cancel := λ x y z H, hf $ (mul_left_inj (f y)).1 $ by erw [← mul, ← mul, H]; refl,
.. hf.semigroup f mul } | def | function.injective.right_cancel_semigroup | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"mul_left_inj",
"mul_right_cancel",
"right_cancel_semigroup"
] | 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]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_one_class [mul_one_class M₂] (f : M₁ → M₂) (hf : injective f)
(one : f 1 = 1) (mul : ∀ x y, f (x * y) = f x * f y) :
mul_one_class M₁ | { one_mul := λ x, hf $ by erw [mul, one, one_mul],
mul_one := λ x, hf $ by erw [mul, one, mul_one],
..‹has_one M₁›, ..‹has_mul M₁› } | def | function.injective.mul_one_class | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"mul_one",
"mul_one_class",
"one_mul"
] | 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]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid [monoid M₂] (f : M₁ → M₂) (hf : injective f)
(one : f 1 = 1) (mul : ∀ x y, f (x * y) = f x * f y) (npow : ∀ x (n : ℕ), f (x ^ n) = f x ^ n) :
monoid M₁ | { npow := λ n x, x ^ n,
npow_zero' := λ x, hf $ by erw [npow, one, pow_zero],
npow_succ' := λ n x, hf $ by erw [npow, pow_succ, mul, npow],
.. hf.semigroup f mul, .. hf.mul_one_class f one mul } | def | function.injective.monoid | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"monoid",
"pow_succ",
"pow_zero"
] | 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]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid_with_one {M₁}
[has_zero M₁] [has_one M₁] [has_add M₁] [has_smul ℕ M₁] [has_nat_cast M₁]
[add_monoid_with_one M₂] (f : M₁ → M₂) (hf : injective f)
(zero : f 0 = 0) (one : f 1 = 1) (add : ∀ x y, f (x + y) = f x + f y)
(nsmul : ∀ x (n : ℕ), f (n • x) = n • f x)
(nat_cast : ∀ n : ℕ, f n = n) :
add_mo... | { nat_cast := coe,
nat_cast_zero := hf (by erw [nat_cast, nat.cast_zero, zero]),
nat_cast_succ := λ n, hf (by erw [nat_cast, nat.cast_succ, add, one, nat_cast]),
one := 1, .. hf.add_monoid f zero add nsmul } | def | function.injective.add_monoid_with_one | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"add_monoid_with_one",
"has_nat_cast",
"has_smul",
"nat.cast_succ",
"nat.cast_zero"
] | 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]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_cancel_monoid [left_cancel_monoid M₂] (f : M₁ → M₂) (hf : injective f)
(one : f 1 = 1) (mul : ∀ x y, f (x * y) = f x * f y) (npow : ∀ x (n : ℕ), f (x ^ n) = f x ^ n) :
left_cancel_monoid M₁ | { .. hf.left_cancel_semigroup f mul, .. hf.monoid f one mul npow } | def | function.injective.left_cancel_monoid | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"left_cancel_monoid"
] | 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]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
right_cancel_monoid [right_cancel_monoid M₂] (f : M₁ → M₂) (hf : injective f)
(one : f 1 = 1) (mul : ∀ x y, f (x * y) = f x * f y) (npow : ∀ x (n : ℕ), f (x ^ n) = f x ^ n) :
right_cancel_monoid M₁ | { .. hf.right_cancel_semigroup f mul, .. hf.monoid f one mul npow } | def | function.injective.right_cancel_monoid | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"right_cancel_monoid"
] | 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]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cancel_monoid [cancel_monoid M₂] (f : M₁ → M₂) (hf : injective f)
(one : f 1 = 1) (mul : ∀ x y, f (x * y) = f x * f y) (npow : ∀ x (n : ℕ), f (x ^ n) = f x ^ n) :
cancel_monoid M₁ | { .. hf.left_cancel_monoid f one mul npow, .. hf.right_cancel_monoid f one mul npow } | def | function.injective.cancel_monoid | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"cancel_monoid"
] | 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]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comm_monoid [comm_monoid M₂] (f : M₁ → M₂) (hf : injective f)
(one : f 1 = 1) (mul : ∀ x y, f (x * y) = f x * f y) (npow : ∀ x (n : ℕ), f (x ^ n) = f x ^ n) :
comm_monoid M₁ | { .. hf.comm_semigroup f mul, .. hf.monoid f one mul npow } | def | function.injective.comm_monoid | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"comm_monoid"
] | 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]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_comm_monoid_with_one {M₁} [has_zero M₁] [has_one M₁] [has_add M₁] [has_smul ℕ M₁]
[has_nat_cast M₁] [add_comm_monoid_with_one M₂] (f : M₁ → M₂) (hf : injective f) (zero : f 0 = 0)
(one : f 1 = 1) (add : ∀ x y, f (x + y) = f x + f y) (nsmul : ∀ x (n : ℕ), f (n • x) = n • f x)
(nat_cast : ∀ n : ℕ, f n = n) :
... | { ..hf.add_monoid_with_one f zero one add nsmul nat_cast, ..hf.add_comm_monoid f zero add nsmul } | def | function.injective.add_comm_monoid_with_one | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"add_comm_monoid_with_one",
"has_nat_cast",
"has_smul"
] | A type endowed with `0`, `1` and `+` is an additive commutative monoid with one, if it admits an
injective map that preserves `0`, `1` and `+` to an additive commutative monoid with one.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cancel_comm_monoid [cancel_comm_monoid M₂] (f : M₁ → M₂) (hf : injective f)
(one : f 1 = 1) (mul : ∀ x y, f (x * y) = f x * f y) (npow : ∀ x (n : ℕ), f (x ^ n) = f x ^ n) :
cancel_comm_monoid M₁ | { .. hf.left_cancel_semigroup f mul, .. hf.comm_monoid f one mul npow } | def | function.injective.cancel_comm_monoid | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"cancel_comm_monoid"
] | 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]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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