feat(library/algebra/group_power): add times (additive variant of power)

library/algebra/group.lean

@@ -351,6 +351,11 @@ structure comm_group [class] (A : Type) extends group A, comm_monoid A
 structure add_group [class] (A : Type) extends add_monoid A, has_neg A :=
 (add_left_inv : ∀a, add (neg a) a = zero)
 
+definition add_group.to_group {A : Type} [s : add_group A] : group A :=
+⦃ group, add_monoid.to_monoid,
+  mul_left_inv := add_group.add_left_inv ⦄
+
+
 section add_group
 
   variables [s : add_group A]
@@ -579,7 +584,6 @@ section add_comm_group
 
   theorem sub_eq_sub_add_sub (a b c : A) : a - b = c - b + (a - c) :=
     by rewrite [add_sub, sub_add_cancel] ⬝ !add.comm
-
 end add_comm_group
 
 definition group_of_add_group (A : Type) [G : add_group A] : group A :=

library/algebra/group_power.lean

@@ -167,4 +167,71 @@ theorem pow_two_add (n : ℕ) : pow 2 n + pow 2 n = pow 2 (succ n) :=
 
 end ordered_ring
 
+/- additive monoid -/
+
+section add_monoid
+variable [s : add_monoid A]
+include s
+local attribute add_monoid.to_monoid [trans-instance]
+open nat
+
+definition times : ℕ → A → A := λ n a, pow a n
+
+infix [priority algebra.prio] `*` := times
+
+theorem zero_times (a : A) : (0:ℕ) * a = 0 := pow_zero a
+theorem succ_times (n : ℕ) (a : A) : succ n * a = times n a + a := pow_succ a n
+
+theorem succ_times' (n : ℕ) (a : A) : times (succ n) a = a + (times n a) := pow_succ' a n
+
+theorem times_zero (n : ℕ) : n * 0 = (0:A) := one_pow n
+
+theorem one_times (a : A) : 1 * a = a := pow_one a
+
+theorem add_times (m n : ℕ) (a : A) : (m + n) * a = (m * a) + (n * a) := pow_add a m n
+
+theorem mul_times (m n : ℕ) (a : A) : (m * n) * a = m * (n * a) := eq.subst (nat.mul.comm n m) (pow_mul a n m)
+
+theorem times_comm (m n : ℕ) (a : A) : (m * a) + (n * a) = (n * a) + (m * a) := pow_comm a m n
+
+end add_monoid
+
+/- additive commutative monoid -/
+
+section add_comm_monoid
+open nat
+variable [s : add_comm_monoid A]
+include s
+local attribute add_comm_monoid.to_comm_monoid [trans-instance]
+
+theorem times_add (n : ℕ) (a b : A) : n * (a + b) = (n * a) + (n * b) := mul_pow a b n
+
+end add_comm_monoid
+
+section add_group
+variable [s : add_group A]
+include s
+local attribute add_group.to_group [trans-instance]
+
+section nat
+open nat
+theorem times_neg (n : ℕ) (a : A) : n * (-a) = -(n * a) := inv_pow a n
+
+theorem sub_times {m n : ℕ} (a : A) (H : m ≥ n) : (m - n) * a = (m * a) + -(n * a) := pow_sub a H
+
+theorem times_neg_comm (m n : ℕ) (a : A) : (m * (-a)) + (n * a) = (n * a) + (m * (-a)) := pow_inv_comm a m n
+
+end nat
+
+open int
+
+definition itimes : ℤ → A → A := λ i a, ipow a i
+
+theorem add_itimes (i j : ℤ) (a : A) : itimes (i + j) a = itimes i a + itimes j a :=
+  ipow_add a i j
+
+theorem itimes_comm (i j : ℤ) (a : A) : itimes i a + itimes j a = itimes j a + itimes i a := ipow_comm a i j
+
+end add_group
+
 end algebra