Iterations of a function #

In this file we prove simple properties of Nat.iterate f n a.k.a. f^[n]:

iterate_zero, iterate_succ, iterate_succ', iterate_add, iterate_mul: formulas for f^[0], f^[n+1] (two versions), f^[n+m], and f^[n*m];
iterate_id : id^[n]=id;
Injective.iterate, Surjective.iterate, Bijective.iterate : iterates of an injective/surjective/bijective function belong to the same class;
LeftInverse.iterate, RightInverse.iterate, Commute.iterate_left, Commute.iterate_right, Commute.iterate_iterate: some properties of pairs of functions survive under iterations
iterate_fixed, Function.Semiconj.iterate_*, Function.Semiconj₂.iterate: if f fixes a point (resp., semiconjugates unary/binary operations), then so does f^[n].

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def Nat.iterate {α : Sort u} (op : α → α) :

ℕ → α → α

Iterate a function.

Equations

op ^[0] x = x
op ^[Nat.succ k] x = op ^[k] (op x)

Instances For

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def «term_^[_]» :

Lean.TrailingParserDescr

Iterate a function.

Equations

One or more equations did not get rendered due to their size.

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@[simp]

theorem Function.iterate_zero {α : Type u} (f : α → α) :

f^[0] = id

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theorem Function.iterate_zero_apply {α : Type u} (f : α → α) (x : α) :

f ^[0] x = x

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@[simp]

theorem Function.iterate_succ {α : Type u} (f : α → α) (n : ℕ) :

f^[Nat.succ n] = f^[n] ∘ f

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theorem Function.iterate_succ_apply {α : Type u} (f : α → α) (n : ℕ) (x : α) :

f ^[Nat.succ n] x = f ^[n] (f x)

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@[simp]

theorem Function.iterate_id {α : Type u} (n : ℕ) :

id^[n] = id

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theorem Function.iterate_add {α : Type u} (f : α → α) (m : ℕ) (n : ℕ) :

f^[m + n] = f^[m] ∘ f^[n]

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theorem Function.iterate_add_apply {α : Type u} (f : α → α) (m : ℕ) (n : ℕ) (x : α) :

f ^[m + n] x = f ^[m] (f ^[n] x)

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@[simp]

theorem Function.iterate_one {α : Type u} (f : α → α) :

f^[1] = f

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theorem Function.iterate_mul {α : Type u} (f : α → α) (m : ℕ) (n : ℕ) :

f^[m * n] = f^[m]^[n]

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theorem Function.iterate_fixed {α : Type u} {f : α → α} {x : α} (h : f x = x) (n : ℕ) :

f ^[n] x = x

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theorem Function.Injective.iterate {α : Type u} {f : α → α} (Hinj : Function.Injective f) (n : ℕ) :

Function.Injective f^[n]

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theorem Function.Surjective.iterate {α : Type u} {f : α → α} (Hsurj : Function.Surjective f) (n : ℕ) :

Function.Surjective f^[n]

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theorem Function.Bijective.iterate {α : Type u} {f : α → α} (Hbij : Function.Bijective f) (n : ℕ) :

Function.Bijective f^[n]

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theorem Function.Semiconj.iterate_right {α : Type u} {β : Type v} {f : α → β} {ga : α → α} {gb : β → β} (h : Function.Semiconj f ga gb) (n : ℕ) :

Function.Semiconj f ga^[n] gb^[n]

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theorem Function.Semiconj.iterate_left {α : Type u} {f : α → α} {g : ℕ → α → α} (H : ∀ (n : ℕ), Function.Semiconj f (g n) (g (n + 1))) (n : ℕ) (k : ℕ) :

Function.Semiconj f^[n] (g k) (g (n + k))

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theorem Function.Commute.iterate_right {α : Type u} {f : α → α} {g : α → α} (h : Function.Commute f g) (n : ℕ) :

Function.Commute f g^[n]

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theorem Function.Commute.iterate_left {α : Type u} {f : α → α} {g : α → α} (h : Function.Commute f g) (n : ℕ) :

Function.Commute f^[n] g

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theorem Function.Commute.iterate_iterate {α : Type u} {f : α → α} {g : α → α} (h : Function.Commute f g) (m : ℕ) (n : ℕ) :

Function.Commute f^[m] g^[n]

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theorem Function.Commute.iterate_eq_of_map_eq {α : Type u} {f : α → α} {g : α → α} (h : Function.Commute f g) (n : ℕ) {x : α} (hx : f x = g x) :

f ^[n] x = g ^[n] x

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theorem Function.Commute.comp_iterate {α : Type u} {f : α → α} {g : α → α} (h : Function.Commute f g) (n : ℕ) :

(f ∘ g)^[n] = f^[n] ∘ g^[n]

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theorem Function.Commute.iterate_self {α : Type u} (f : α → α) (n : ℕ) :

Function.Commute f^[n] f

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theorem Function.Commute.self_iterate {α : Type u} (f : α → α) (n : ℕ) :

Function.Commute f f^[n]

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theorem Function.Commute.iterate_iterate_self {α : Type u} (f : α → α) (m : ℕ) (n : ℕ) :

Function.Commute f^[m] f^[n]

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theorem Function.Semiconj₂.iterate {α : Type u} {f : α → α} {op : α → α → α} (hf : Function.Semiconj₂ f op op) (n : ℕ) :

Function.Semiconj₂ f^[n] op op

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theorem Function.iterate_succ' {α : Type u} (f : α → α) (n : ℕ) :

f^[Nat.succ n] = f ∘ f^[n]

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theorem Function.iterate_succ_apply' {α : Type u} (f : α → α) (n : ℕ) (x : α) :

f ^[Nat.succ n] x = f (f ^[n] x)

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theorem Function.iterate_pred_comp_of_pos {α : Type u} (f : α → α) {n : ℕ} (hn : 0 < n) :

f^[Nat.pred n] ∘ f = f^[n]

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theorem Function.comp_iterate_pred_of_pos {α : Type u} (f : α → α) {n : ℕ} (hn : 0 < n) :

f ∘ f^[Nat.pred n] = f^[n]

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def Function.Iterate.rec {α : Type u} (p : α → Sort u_1) {f : α → α} (h : (a : α) → p a → p (f a)) {a : α} (ha : p a) (n : ℕ) :

p (f ^[n] a)

A recursor for the iterate of a function.

Equations

Function.Iterate.rec p h ha 0 = ha
Function.Iterate.rec p h ha (Nat.succ m) = Function.Iterate.rec p h (h a ha) m

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theorem Function.Iterate.rec_zero {α : Type u} (p : α → Sort u_1) {f : α → α} (h : (a : α) → p a → p (f a)) {a : α} (ha : p a) :

Function.Iterate.rec p h ha 0 = ha

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theorem Function.LeftInverse.iterate {α : Type u} {f : α → α} {g : α → α} (hg : Function.LeftInverse g f) (n : ℕ) :

Function.LeftInverse g^[n] f^[n]

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theorem Function.RightInverse.iterate {α : Type u} {f : α → α} {g : α → α} (hg : Function.RightInverse g f) (n : ℕ) :

Function.RightInverse g^[n] f^[n]

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theorem Function.iterate_comm {α : Type u} (f : α → α) (m : ℕ) (n : ℕ) :

f^[n]^[m] = f^[m]^[n]

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theorem Function.iterate_commute {α : Type u} (m : ℕ) (n : ℕ) :

Function.Commute (fun (f : α → α) => f^[m]) fun (f : α → α) => f^[n]

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theorem Function.iterate_add_eq_iterate {α : Type u} {f : α → α} {m : ℕ} {n : ℕ} {a : α} (hf : Function.Injective f) :

f ^[m + n] a = f ^[n] a ↔ f ^[m] a = a

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theorem Function.iterate_cancel_of_add {α : Type u} {f : α → α} {m : ℕ} {n : ℕ} {a : α} (hf : Function.Injective f) :

f ^[m + n] a = f ^[n] a → f ^[m] a = a

Alias of the forward direction of Function.iterate_add_eq_iterate.

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theorem Function.iterate_cancel {α : Type u} {f : α → α} {m : ℕ} {n : ℕ} {a : α} (hf : Function.Injective f) (ha : f ^[m] a = f ^[n] a) :

f ^[m - n] a = a

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theorem Function.involutive_iff_iter_2_eq_id {α : Sort u_1} {f : α → α} :

Function.Involutive f ↔ f^[2] = id

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theorem List.foldl_const {α : Type u} {β : Type v} (f : α → α) (a : α) (l : List β) :

List.foldl (fun (b : α) (x : β) => f b) a l = f ^[List.length l] a

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theorem List.foldr_const {α : Type u} {β : Type v} (f : β → β) (b : β) (l : List α) :

List.foldr (fun (x : α) => f) b l = f ^[List.length l] b