Semiconjugate elements of a semigroup #

Main definitions #

We say that x is semiconjugate to y by a (SemiconjBy a x y), if a * x = y * a. In this file we provide operations on SemiconjBy _ _ _.

In the names of these operations, we treat a as the “left” argument, and both x and y as “right” arguments. This way most names in this file agree with the names of the corresponding lemmas for Commute a b = SemiconjBy a b b. As a side effect, some lemmas have only _right version.

Lean does not immediately recognise these terms as equations, so for rewriting we need syntax like rw [(h.pow_right 5).eq] rather than just rw [h.pow_right 5].

This file provides only basic operations (mul_left, mul_right, inv_right etc). Other operations (pow_right, field inverse etc) are in the files that define corresponding notions.

def AddSemiconjBy {M : Type u_2} [Add M] (a : M) (x : M) (y : M) :

x is additive semiconjugate to y by a if a + x = y + a

Equations
Instances For
def SemiconjBy {M : Type u_2} [Mul M] (a : M) (x : M) (y : M) :

x is semiconjugate to y by a, if a * x = y * a.

Equations
Instances For
theorem AddSemiconjBy.eq {S : Type u_1} [Add S] {a : S} {x : S} {y : S} (h : ) :
a + x = y + a

Equality behind AddSemiconjBy a x y; useful for rewriting.

theorem SemiconjBy.eq {S : Type u_1} [Mul S] {a : S} {x : S} {y : S} (h : SemiconjBy a x y) :
a * x = y * a

Equality behind SemiconjBy a x y; useful for rewriting.

@[simp]
theorem AddSemiconjBy.add_right {S : Type u_1} [] {a : S} {x : S} {y : S} {x' : S} {y' : S} (h : ) (h' : AddSemiconjBy a x' y') :
AddSemiconjBy a (x + x') (y + y')

If a semiconjugates x to y and x' to y', then it semiconjugates x + x' to y + y'.

@[simp]
theorem SemiconjBy.mul_right {S : Type u_1} [] {a : S} {x : S} {y : S} {x' : S} {y' : S} (h : SemiconjBy a x y) (h' : SemiconjBy a x' y') :
SemiconjBy a (x * x') (y * y')

If a semiconjugates x to y and x' to y', then it semiconjugates x * x' to y * y'.

theorem AddSemiconjBy.add_left {S : Type u_1} [] {a : S} {b : S} {x : S} {y : S} {z : S} (ha : ) (hb : ) :
AddSemiconjBy (a + b) x z

If b semiconjugates x to y and a semiconjugates y to z, then a + b semiconjugates x to z.

theorem SemiconjBy.mul_left {S : Type u_1} [] {a : S} {b : S} {x : S} {y : S} {z : S} (ha : SemiconjBy a y z) (hb : SemiconjBy b x y) :
SemiconjBy (a * b) x z

If b semiconjugates x to y and a semiconjugates y to z, then a * b semiconjugates x to z.

abbrev AddSemiconjBy.transitive.match_1 {S : Type u_1} [] (motive : (x x_1 x_2 : S) → (∃ (c : S), AddSemiconjBy c x x_1)(∃ (c : S), AddSemiconjBy c x_1 x_2)Prop) :
∀ (x x_1 x_2 : S) (x_3 : ∃ (c : S), AddSemiconjBy c x x_1) (x_4 : ∃ (c : S), AddSemiconjBy c x_1 x_2), (∀ (x x_5 x_6 x_7 : S) (hx : AddSemiconjBy x_7 x x_5) (y : S) (hy : AddSemiconjBy y x_5 x_6), motive x x_5 x_6 )motive x x_1 x_2 x_3 x_4
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• =
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theorem AddSemiconjBy.transitive {S : Type u_1} [] :
Transitive fun (a b : S) => ∃ (c : S),

The relation “there exists an element that semiconjugates a to b” on an additive semigroup is transitive.

theorem SemiconjBy.transitive {S : Type u_1} [] :
Transitive fun (a b : S) => ∃ (c : S), SemiconjBy c a b

The relation “there exists an element that semiconjugates a to b” on a semigroup is transitive.

@[simp]
theorem AddSemiconjBy.zero_right {M : Type u_2} [] (a : M) :

Any element semiconjugates 0 to 0.

@[simp]
theorem SemiconjBy.one_right {M : Type u_2} [] (a : M) :

Any element semiconjugates 1 to 1.

@[simp]
theorem AddSemiconjBy.zero_left {M : Type u_2} [] (x : M) :

Zero semiconjugates any element to itself.

@[simp]
theorem SemiconjBy.one_left {M : Type u_2} [] (x : M) :

One semiconjugates any element to itself.

theorem AddSemiconjBy.reflexive {M : Type u_2} [] :
Reflexive fun (a b : M) => ∃ (c : M),

The relation “there exists an element that semiconjugates a to b” on an additive monoid (or, more generally, on an AddZeroClass type) is reflexive.

theorem SemiconjBy.reflexive {M : Type u_2} [] :
Reflexive fun (a b : M) => ∃ (c : M), SemiconjBy c a b

The relation “there exists an element that semiconjugates a to b” on a monoid (or, more generally, on MulOneClass type) is reflexive.

@[simp]
theorem AddSemiconjBy.nsmul_right {M : Type u_2} [] {a : M} {x : M} {y : M} (h : ) (n : ) :
AddSemiconjBy a (n x) (n y)
@[simp]
theorem SemiconjBy.pow_right {M : Type u_2} [] {a : M} {x : M} {y : M} (h : SemiconjBy a x y) (n : ) :
SemiconjBy a (x ^ n) (y ^ n)
theorem AddSemiconjBy.conj_mk {G : Type u_3} [] (a : G) (x : G) :
AddSemiconjBy a x (a + x + -a)

a semiconjugates x to a + x + -a.

theorem SemiconjBy.conj_mk {G : Type u_3} [] (a : G) (x : G) :
SemiconjBy a x (a * x * a⁻¹)

a semiconjugates x to a * x * a⁻¹.

@[simp]
theorem AddSemiconjBy.conj_iff {G : Type u_3} [] {a : G} {x : G} {y : G} {b : G} :
AddSemiconjBy (b + a + -b) (b + x + -b) (b + y + -b)
@[simp]
theorem SemiconjBy.conj_iff {G : Type u_3} [] {a : G} {x : G} {y : G} {b : G} :
SemiconjBy (b * a * b⁻¹) (b * x * b⁻¹) (b * y * b⁻¹) SemiconjBy a x y
@[simp]
theorem addSemiconjBy_iff_eq {M : Type u_2} {a : M} {x : M} {y : M} :
x = y
@[simp]
theorem semiconjBy_iff_eq {M : Type u_2} [] {a : M} {x : M} {y : M} :
SemiconjBy a x y x = y