# Iterates of monoid homomorphisms #

Iterate of a monoid homomorphism is a monoid homomorphism but it has a wrong type, so Lean can't apply lemmas like MonoidHom.map_one to f^[n] 1. Though it is possible to define a monoid structure on the endomorphisms, quite often we do not want to convert from M →* M to Monoid.End M and from f^[n] to f^n just to apply a simple lemma.

So, we restate standard map_* lemmas under names iterate_map_*.

We also prove formulas for iterates of add/mul left/right.

## Tags #

homomorphism, iterate

theorem hom_coe_pow {M : Type u_1} {F : Type u_5} [] (c : FMM) (h1 : c 1 = id) (hmul : ∀ (f g : F), c (f * g) = c f c g) (f : F) (n : ) :
c (f ^ n) = (c f)^[n]

An auxiliary lemma that can be used to prove ⇑(f ^ n) = ⇑f^[n].

@[simp]
theorem iterate_map_add {M : Type u_5} {F : Type u_6} [Add M] [FunLike F M M] [AddHomClass F M M] (f : F) (n : ) (x : M) (y : M) :
(⇑f)^[n] (x + y) = (⇑f)^[n] x + (⇑f)^[n] y
@[simp]
theorem iterate_map_mul {M : Type u_5} {F : Type u_6} [Mul M] [FunLike F M M] [MulHomClass F M M] (f : F) (n : ) (x : M) (y : M) :
(⇑f)^[n] (x * y) = (⇑f)^[n] x * (⇑f)^[n] y
@[simp]
theorem iterate_map_zero {M : Type u_5} {F : Type u_6} [Zero M] [FunLike F M M] [ZeroHomClass F M M] (f : F) (n : ) :
(⇑f)^[n] 0 = 0
@[simp]
theorem iterate_map_one {M : Type u_5} {F : Type u_6} [One M] [FunLike F M M] [OneHomClass F M M] (f : F) (n : ) :
(⇑f)^[n] 1 = 1
@[simp]
theorem iterate_map_neg {M : Type u_5} {F : Type u_6} [] [FunLike F M M] [] (f : F) (n : ) (x : M) :
(⇑f)^[n] (-x) = -(⇑f)^[n] x
@[simp]
theorem iterate_map_inv {M : Type u_5} {F : Type u_6} [] [FunLike F M M] [] (f : F) (n : ) (x : M) :
(⇑f)^[n] x⁻¹ = ((⇑f)^[n] x)⁻¹
@[simp]
theorem iterate_map_sub {M : Type u_5} {F : Type u_6} [] [FunLike F M M] [] (f : F) (n : ) (x : M) (y : M) :
(⇑f)^[n] (x - y) = (⇑f)^[n] x - (⇑f)^[n] y
@[simp]
theorem iterate_map_div {M : Type u_5} {F : Type u_6} [] [FunLike F M M] [] (f : F) (n : ) (x : M) (y : M) :
(⇑f)^[n] (x / y) = (⇑f)^[n] x / (⇑f)^[n] y
@[simp]
theorem iterate_map_nsmul {M : Type u_5} {F : Type u_6} [] [FunLike F M M] [] (f : F) (n : ) (x : M) (k : ) :
(⇑f)^[n] (k x) = k (⇑f)^[n] x
@[simp]
theorem iterate_map_pow {M : Type u_5} {F : Type u_6} [] [FunLike F M M] [] (f : F) (n : ) (x : M) (k : ) :
(⇑f)^[n] (x ^ k) = (⇑f)^[n] x ^ k
@[simp]
theorem iterate_map_zsmul {M : Type u_5} {F : Type u_6} [] [FunLike F M M] [] (f : F) (n : ) (x : M) (k : ) :
(⇑f)^[n] (k x) = k (⇑f)^[n] x
@[simp]
theorem iterate_map_zpow {M : Type u_5} {F : Type u_6} [] [FunLike F M M] [] (f : F) (n : ) (x : M) (k : ) :
(⇑f)^[n] (x ^ k) = (⇑f)^[n] x ^ k
@[simp]
theorem vadd_iterate {G : Type u_3} {H : Type u_4} [] (a : G) (n : ) [] :
(fun (x : H) => a +ᵥ x)^[n] = fun (x : H) => n a +ᵥ x
@[simp]
theorem smul_iterate {G : Type u_3} {H : Type u_4} [] (a : G) (n : ) [] :
(fun (x : H) => a x)^[n] = fun (x : H) => a ^ n x
theorem vadd_iterate_apply {G : Type u_3} {H : Type u_4} [] (a : G) (n : ) [] {b : H} :
(fun (x : H) => a +ᵥ x)^[n] b = n a +ᵥ b
theorem smul_iterate_apply {G : Type u_3} {H : Type u_4} [] (a : G) (n : ) [] {b : H} :
(fun (x : H) => a x)^[n] b = a ^ n b
@[simp]
theorem add_left_iterate {G : Type u_3} [] (a : G) (n : ) :
(fun (x : G) => a + x)^[n] = fun (x : G) => n a + x
@[simp]
theorem mul_left_iterate {G : Type u_3} [] (a : G) (n : ) :
(fun (x : G) => a * x)^[n] = fun (x : G) => a ^ n * x
@[simp]
theorem add_right_iterate {G : Type u_3} [] (a : G) (n : ) :
(fun (x : G) => x + a)^[n] = fun (x : G) => x + n a
@[simp]
theorem mul_right_iterate {G : Type u_3} [] (a : G) (n : ) :
(fun (x : G) => x * a)^[n] = fun (x : G) => x * a ^ n
theorem add_right_iterate_apply_zero {G : Type u_3} [] (a : G) (n : ) :
(fun (x : G) => x + a)^[n] 0 = n a
theorem mul_right_iterate_apply_one {G : Type u_3} [] (a : G) (n : ) :
(fun (x : G) => x * a)^[n] 1 = a ^ n
@[simp]
theorem nsmul_iterate {G : Type u_3} [] (n : ) (j : ) :
(fun (x : G) => n x)^[j] = fun (x : G) => n ^ j x
@[simp]
theorem pow_iterate {G : Type u_3} [] (n : ) (j : ) :
(fun (x : G) => x ^ n)^[j] = fun (x : G) => x ^ n ^ j
@[simp]
theorem zsmul_iterate {G : Type u_3} [] (n : ) (j : ) :
(fun (x : G) => n x)^[j] = fun (x : G) => n ^ j x
@[simp]
theorem zpow_iterate {G : Type u_3} [] (n : ) (j : ) :
(fun (x : G) => x ^ n)^[j] = fun (x : G) => x ^ n ^ j
theorem AddSemiconjBy.function_semiconj_add_left {G : Type u_3} [] {a : G} {b : G} {c : G} (h : ) :
Function.Semiconj (fun (x : G) => a + x) (fun (x : G) => b + x) fun (x : G) => c + x
theorem SemiconjBy.function_semiconj_mul_left {G : Type u_3} [] {a : G} {b : G} {c : G} (h : SemiconjBy a b c) :
Function.Semiconj (fun (x : G) => a * x) (fun (x : G) => b * x) fun (x : G) => c * x
theorem AddCommute.function_commute_add_left {G : Type u_3} [] {a : G} {b : G} (h : ) :
Function.Commute (fun (x : G) => a + x) fun (x : G) => b + x
theorem Commute.function_commute_mul_left {G : Type u_3} [] {a : G} {b : G} (h : Commute a b) :
Function.Commute (fun (x : G) => a * x) fun (x : G) => b * x
theorem AddSemiconjBy.function_semiconj_add_right_swap {G : Type u_3} [] {a : G} {b : G} {c : G} (h : ) :
Function.Semiconj (fun (x : G) => x + a) (fun (x : G) => x + c) fun (x : G) => x + b
theorem SemiconjBy.function_semiconj_mul_right_swap {G : Type u_3} [] {a : G} {b : G} {c : G} (h : SemiconjBy a b c) :
Function.Semiconj (fun (x : G) => x * a) (fun (x : G) => x * c) fun (x : G) => x * b
theorem AddCommute.function_commute_add_right {G : Type u_3} [] {a : G} {b : G} (h : ) :
Function.Commute (fun (x : G) => x + a) fun (x : G) => x + b
theorem Commute.function_commute_mul_right {G : Type u_3} [] {a : G} {b : G} (h : Commute a b) :
Function.Commute (fun (x : G) => x * a) fun (x : G) => x * b