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Mathlib.Logic.Function.Conjugate

Semiconjugate and commuting maps #

We define the following predicates:

Function.Semiconj: f : α → β semiconjugates ga : α → α to gb : β → β if f ∘ ga = gb ∘ f;
Function.Semiconj₂: f : α → β semiconjugates a binary operation ga : α → α → α to gb : β → β → β if f (ga x y) = gb (f x) (f y);
Function.Commute: f : α → α commutes with g : α → α if f ∘ g = g ∘ f, or equivalently Semiconj f g g.

def Function.Semiconj {α : Type u_1} {β : Type u_2} (f : α → β) (ga : α → α) (gb : β → β) :

We say that f : α → β semiconjugates ga : α → α to gb : β → β if f ∘ ga = gb ∘ f. We use ∀ x, f (ga x) = gb (f x) as the definition, so given h : Function.Semiconj f ga gb and a : α, we have h a : f (ga a) = gb (f a) and h.comp_eq : f ∘ ga = gb ∘ f.

Equations

Function.Semiconj f ga gb = ∀ (x : α), f (ga x) = gb (f x)

theorem Function.semiconj_iff_comp_eq {α : Type u_1} {β : Type u_2} {f : α → β} {ga : α → α} {gb : β → β} :

Function.Semiconj f ga gb ↔ f ∘ ga = gb ∘ f

Definition of Function.Semiconj in terms of functional equality.

theorem Function.Semiconj.comp_eq {α : Type u_1} {β : Type u_2} {f : α → β} {ga : α → α} {gb : β → β} :

Function.Semiconj f ga gb → f ∘ ga = gb ∘ f

Alias of the forward direction of Function.semiconj_iff_comp_eq.

Definition of Function.Semiconj in terms of functional equality.

theorem Function.Semiconj.eq {α : Type u_1} {β : Type u_2} {f : α → β} {ga : α → α} {gb : β → β} (h : Function.Semiconj f ga gb) (x : α) :

f (ga x) = gb (f x)

theorem Function.Semiconj.comp_right {α : Type u_1} {β : Type u_2} {f : α → β} {ga : α → α} {ga' : α → α} {gb : β → β} {gb' : β → β} (h : Function.Semiconj f ga gb) (h' : Function.Semiconj f ga' gb') :

Function.Semiconj f (ga ∘ ga') (gb ∘ gb')

If f semiconjugates ga to gb and ga' to gb', then it semiconjugates ga ∘ ga' to gb ∘ gb'.

theorem Function.Semiconj.trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} {fab : α → β} {fbc : β → γ} {ga : α → α} {gb : β → β} {gc : γ → γ} (hab : Function.Semiconj fab ga gb) (hbc : Function.Semiconj fbc gb gc) :

Function.Semiconj (fbc ∘ fab) ga gc

If fab : α → β semiconjugates ga to gb and fbc : β → γ semiconjugates gb to gc, then fbc ∘ fab semiconjugates ga to gc.

See also Function.Semiconj.comp_left for a version with reversed arguments.

theorem Function.Semiconj.comp_left {α : Type u_1} {β : Type u_2} {γ : Type u_3} {fab : α → β} {fbc : β → γ} {ga : α → α} {gb : β → β} {gc : γ → γ} (hbc : Function.Semiconj fbc gb gc) (hab : Function.Semiconj fab ga gb) :

Function.Semiconj (fbc ∘ fab) ga gc

If fbc : β → γ semiconjugates gb to gc and fab : α → β semiconjugates ga to gb, then fbc ∘ fab semiconjugates ga to gc.

See also Function.Semiconj.trans for a version with reversed arguments.

Backward compatibility note: before 2024/01/13, this lemma used to have the same order of arguments that Function.Semiconj.trans has now.

theorem Function.Semiconj.id_right {α : Type u_1} {β : Type u_2} {f : α → β} :

Function.Semiconj f id id

Any function semiconjugates the identity function to the identity function.

theorem Function.Semiconj.id_left {α : Type u_1} {ga : α → α} :

Function.Semiconj id ga ga

The identity function semiconjugates any function to itself.

theorem Function.Semiconj.inverses_right {α : Type u_1} {β : Type u_2} {f : α → β} {ga : α → α} {ga' : α → α} {gb : β → β} {gb' : β → β} (h : Function.Semiconj f ga gb) (ha : Function.RightInverse ga' ga) (hb : Function.LeftInverse gb' gb) :

Function.Semiconj f ga' gb'

If f : α → β semiconjugates ga : α → α to gb : β → β, ga' is a right inverse of ga, and gb' is a left inverse of gb, then f semiconjugates ga' to gb' as well.

theorem Function.Semiconj.inverse_left {α : Type u_1} {β : Type u_2} {f : α → β} {ga : α → α} {gb : β → β} {f' : β → α} (h : Function.Semiconj f ga gb) (hf₁ : Function.LeftInverse f' f) (hf₂ : Function.RightInverse f' f) :

Function.Semiconj f' gb ga

If f semiconjugates ga to gb and f' is both a left and a right inverse of f, then f' semiconjugates gb to ga.

theorem Function.Semiconj.option_map {α : Type u_1} {β : Type u_2} {f : α → β} {ga : α → α} {gb : β → β} (h : Function.Semiconj f ga gb) :

Function.Semiconj (Option.map f) (Option.map ga) (Option.map gb)

If f : α → β semiconjugates ga : α → α to gb : β → β, then Option.map f semiconjugates Option.map ga to Option.map gb.

def Function.Commute {α : Type u_1} (f : α → α) (g : α → α) :

Two maps f g : α → α commute if f (g x) = g (f x) for all x : α. Given h : Function.commute f g and a : α, we have h a : f (g a) = g (f a) and h.comp_eq : f ∘ g = g ∘ f.

Equations

Function.Commute f g = Function.Semiconj f g g

theorem Function.Semiconj.commute {α : Type u_1} {f : α → α} {g : α → α} (h : Function.Semiconj f g g) :

Function.Commute f g

Reinterpret Function.Semiconj f g g as Function.Commute f g. These two predicates are definitionally equal but have different dot-notation lemmas.

theorem Function.Commute.semiconj {α : Type u_1} {f : α → α} {g : α → α} (h : Function.Commute f g) :

Function.Semiconj f g g

Reinterpret Function.Commute f g as Function.Semiconj f g g. These two predicates are definitionally equal but have different dot-notation lemmas.

theorem Function.Commute.refl {α : Type u_1} (f : α → α) :

Function.Commute f f

theorem Function.Commute.symm {α : Type u_1} {f : α → α} {g : α → α} (h : Function.Commute f g) :

Function.Commute g f

theorem Function.Commute.comp_right {α : Type u_1} {f : α → α} {g : α → α} {g' : α → α} (h : Function.Commute f g) (h' : Function.Commute f g') :

Function.Commute f (g ∘ g')

If f commutes with g and g', then it commutes with g ∘ g'.

theorem Function.Commute.comp_left {α : Type u_1} {f : α → α} {f' : α → α} {g : α → α} (h : Function.Commute f g) (h' : Function.Commute f' g) :

Function.Commute (f ∘ f') g

If f and f' commute with g, then f ∘ f' commutes with g as well.

theorem Function.Commute.id_right {α : Type u_1} {f : α → α} :

Function.Commute f id

Any self-map commutes with the identity map.

theorem Function.Commute.id_left {α : Type u_1} {f : α → α} :

Function.Commute id f

The identity map commutes with any self-map.

theorem Function.Commute.option_map {α : Type u_1} {f : α → α} {g : α → α} (h : Function.Commute f g) :

Function.Commute (Option.map f) (Option.map g)

If f commutes with g, then Option.map f commutes with Option.map g.

def Function.Semiconj₂ {α : Type u_1} {β : Type u_2} (f : α → β) (ga : α → α → α) (gb : β → β → β) :

A map f semiconjugates a binary operation ga to a binary operation gb if for all x, y we have f (ga x y) = gb (f x) (f y). E.g., a MonoidHom semiconjugates (*) to (*).

Equations

Function.Semiconj₂ f ga gb = ∀ (x y : α), f (ga x y) = gb (f x) (f y)

theorem Function.Semiconj₂.eq {α : Type u_1} {β : Type u_2} {f : α → β} {ga : α → α → α} {gb : β → β → β} (h : Function.Semiconj₂ f ga gb) (x : α) (y : α) :

f (ga x y) = gb (f x) (f y)

theorem Function.Semiconj₂.comp_eq {α : Type u_1} {β : Type u_2} {f : α → β} {ga : α → α → α} {gb : β → β → β} (h : Function.Semiconj₂ f ga gb) :

Function.bicompr f ga = Function.bicompl gb f f

theorem Function.Semiconj₂.id_left {α : Type u_1} (op : α → α → α) :

Function.Semiconj₂ id op op

theorem Function.Semiconj₂.comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β} {ga : α → α → α} {gb : β → β → β} {f' : β → γ} {gc : γ → γ → γ} (hf' : Function.Semiconj₂ f' gb gc) (hf : Function.Semiconj₂ f ga gb) :

Function.Semiconj₂ (f' ∘ f) ga gc

theorem Function.Semiconj₂.isAssociative_right {α : Type u_1} {β : Type u_2} {f : α → β} {ga : α → α → α} {gb : β → β → β} [Std.Associative ga] (h : Function.Semiconj₂ f ga gb) (h_surj : Function.Surjective f) :

Std.Associative gb

theorem Function.Semiconj₂.isAssociative_left {α : Type u_1} {β : Type u_2} {f : α → β} {ga : α → α → α} {gb : β → β → β} [Std.Associative gb] (h : Function.Semiconj₂ f ga gb) (h_inj : Function.Injective f) :

Std.Associative ga

theorem Function.Semiconj₂.isIdempotent_right {α : Type u_1} {β : Type u_2} {f : α → β} {ga : α → α → α} {gb : β → β → β} [Std.IdempotentOp ga] (h : Function.Semiconj₂ f ga gb) (h_surj : Function.Surjective f) :

Std.IdempotentOp gb

theorem Function.Semiconj₂.isIdempotent_left {α : Type u_1} {β : Type u_2} {f : α → β} {ga : α → α → α} {gb : β → β → β} [Std.IdempotentOp gb] (h : Function.Semiconj₂ f ga gb) (h_inj : Function.Injective f) :

Std.IdempotentOp ga