fact
stringlengths 6
3.84k
| type
stringclasses 11
values | library
stringclasses 32
values | imports
listlengths 1
14
| filename
stringlengths 20
95
| symbolic_name
stringlengths 1
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| docstring
stringlengths 7
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⌀ |
|---|---|---|---|---|---|---|
@[to_additive]
FreeMagma (α : Type u) : Type u
| of : α → FreeMagma α
| mul : FreeMagma α → FreeMagma α → FreeMagma α
deriving DecidableEq
compile_inductive% FreeMagma
|
inductive
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
FreeMagma
|
If `α` is a type, then `FreeMagma α` is the free magma generated by `α`.
This is a magma equipped with a function `FreeMagma.of : α → FreeMagma α` which has
the following universal property: if `M` is any magma, and `f : α → M` is any function,
then this function is the composite of `FreeMagma.of` and a unique multiplicative homomorphism
`FreeMagma.lift f : FreeMagma α →ₙ* M`.
A typical element of `FreeMagma α` is a formal non-associative product of
elements of `α`. For example if `x` and `y` are terms of type `α` then `x * ((y * y) * x)` is a
"typical" element of `FreeMagma α`.
One can think of `FreeMagma α` as the type of binary trees with leaves labelled by `α`.
In general, no pair of distinct elements in `FreeMagma α` will commute.
|
@[to_additive (attr := simp)]
mul_eq (x y : FreeMagma α) : mul x y = x * y := rfl
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
mul_eq
| null |
@[to_additive (attr := elab_as_elim, induction_eliminator)
/-- Recursor for `FreeAddMagma` using `x + y` instead of `FreeAddMagma.add x y`. -/]
recOnMul {C : FreeMagma α → Sort l} (x) (ih1 : ∀ x, C (of x))
(ih2 : ∀ x y, C x → C y → C (x * y)) : C x :=
FreeMagma.recOn x ih1 ih2
@[to_additive (attr := ext 1100)]
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
recOnMul
|
Recursor for `FreeMagma` using `x * y` instead of `FreeMagma.mul x y`.
|
hom_ext {β : Type v} [Mul β] {f g : FreeMagma α →ₙ* β} (h : f ∘ of = g ∘ of) : f = g :=
(DFunLike.ext _ _) fun x ↦ recOnMul x (congr_fun h) <| by intros; simp only [map_mul, *]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
hom_ext
| null |
FreeMagma.liftAux {α : Type u} {β : Type v} [Mul β] (f : α → β) : FreeMagma α → β
| FreeMagma.of x => f x
| x * y => liftAux f x * liftAux f y
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
FreeMagma.liftAux
|
Lifts a function `α → β` to a magma homomorphism `FreeMagma α → β` given a magma `β`.
|
FreeAddMagma.liftAux {α : Type u} {β : Type v} [Add β] (f : α → β) : FreeAddMagma α → β
| FreeAddMagma.of x => f x
| x + y => liftAux f x + liftAux f y
attribute [to_additive existing] FreeMagma.liftAux
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
FreeAddMagma.liftAux
|
Lifts a function `α → β` to an additive magma homomorphism `FreeAddMagma α → β` given
an additive magma `β`.
|
@[to_additive (attr := simps symm_apply)
/-- The universal property of the free additive magma expressing its adjointness. -/]
lift : (α → β) ≃ (FreeMagma α →ₙ* β) where
toFun f :=
{ toFun := liftAux f
map_mul' := fun _ _ ↦ rfl }
invFun F := F ∘ of
@[to_additive (attr := simp)]
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
lift
|
The universal property of the free magma expressing its adjointness.
|
lift_of (x) : lift f (of x) = f x := rfl
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
lift_of
| null |
lift_comp_of : lift f ∘ of = f := rfl
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
lift_comp_of
| null |
lift_comp_of' (f : FreeMagma α →ₙ* β) : lift (f ∘ of) = f := lift.apply_symm_apply f
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
lift_comp_of'
| null |
@[to_additive /-- The unique additive magma homomorphism `FreeAddMagma α → FreeAddMagma β` that
sends each `of x` to `of (f x)`. -/]
map (f : α → β) : FreeMagma α →ₙ* FreeMagma β := lift (of ∘ f)
@[to_additive (attr := simp)]
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
map
|
The unique magma homomorphism `FreeMagma α →ₙ* FreeMagma β` that sends
each `of x` to `of (f x)`.
|
map_of (x) : map f (of x) = of (f x) := rfl
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
map_of
| null |
@[to_additive (attr := elab_as_elim)
/-- Recursor on `FreeAddMagma` using `pure` instead of `of`. -/]
protected recOnPure {C : FreeMagma α → Sort l} (x) (ih1 : ∀ x, C (pure x))
(ih2 : ∀ x y, C x → C y → C (x * y)) : C x :=
FreeMagma.recOnMul x ih1 ih2
@[to_additive (attr := simp)]
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
recOnPure
|
Recursor on `FreeMagma` using `pure` instead of `of`.
|
protected map_pure (f : α → β) (x) : (f <$> pure x : FreeMagma β) = pure (f x) := rfl
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
map_pure
| null |
map_mul' (f : α → β) (x y : FreeMagma α) : f <$> (x * y) = f <$> x * f <$> y := rfl
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
map_mul'
| null |
pure_bind (f : α → FreeMagma β) (x) : pure x >>= f = f x := rfl
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
pure_bind
| null |
mul_bind (f : α → FreeMagma β) (x y : FreeMagma α) : x * y >>= f = (x >>= f) * (y >>= f) :=
rfl
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
mul_bind
| null |
pure_seq {α β : Type u} {f : α → β} {x : FreeMagma α} : pure f <*> x = f <$> x := rfl
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
pure_seq
| null |
mul_seq {α β : Type u} {f g : FreeMagma (α → β)} {x : FreeMagma α} :
f * g <*> x = (f <*> x) * (g <*> x) := rfl
@[to_additive]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
mul_seq
| null |
instLawfulMonad : LawfulMonad FreeMagma.{u} := LawfulMonad.mk'
(pure_bind := fun _ _ ↦ rfl)
(bind_assoc := fun x f g ↦ FreeMagma.recOnPure x (fun _ ↦ rfl) fun x y ih1 ih2 ↦ by
rw [mul_bind, mul_bind, mul_bind, ih1, ih2])
(id_map := fun x ↦ FreeMagma.recOnPure x (fun _ ↦ rfl) fun x y ih1 ih2 ↦ by
rw [map_mul', ih1, ih2])
|
instance
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
instLawfulMonad
| null |
protected FreeMagma.traverse {m : Type u → Type u} [Applicative m] {α β : Type u}
(F : α → m β) : FreeMagma α → m (FreeMagma β)
| FreeMagma.of x => FreeMagma.of <$> F x
| x * y => (· * ·) <$> x.traverse F <*> y.traverse F
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
FreeMagma.traverse
|
`FreeMagma` is traversable.
|
protected FreeAddMagma.traverse {m : Type u → Type u} [Applicative m] {α β : Type u}
(F : α → m β) : FreeAddMagma α → m (FreeAddMagma β)
| FreeAddMagma.of x => FreeAddMagma.of <$> F x
| x + y => (· + ·) <$> x.traverse F <*> y.traverse F
attribute [to_additive existing] FreeMagma.traverse
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
FreeAddMagma.traverse
|
`FreeAddMagma` is traversable.
|
@[to_additive (attr := simp)]
traverse_pure (x) : traverse F (pure x : FreeMagma α) = pure <$> F x := rfl
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
traverse_pure
| null |
traverse_pure' : traverse F ∘ pure = fun x ↦ (pure <$> F x : m (FreeMagma β)) := rfl
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
traverse_pure'
| null |
traverse_mul (x y : FreeMagma α) :
traverse F (x * y) = (· * ·) <$> traverse F x <*> traverse F y := rfl
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
traverse_mul
| null |
traverse_mul' :
Function.comp (traverse F) ∘ (HMul.hMul : FreeMagma α → FreeMagma α → FreeMagma α) = fun x y ↦
(· * ·) <$> traverse F x <*> traverse F y := rfl
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
traverse_mul'
| null |
traverse_eq (x) : FreeMagma.traverse F x = traverse F x := rfl
@[to_additive (attr := deprecated "Use map_pure and seq_pure" (since := "2025-05-21"))]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
traverse_eq
| null |
mul_map_seq (x y : FreeMagma α) :
((· * ·) <$> x <*> y : Id (FreeMagma α)) = (x * y : FreeMagma α) := rfl
@[to_additive]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
mul_map_seq
| null |
protected FreeMagma.repr {α : Type u} [Repr α] : FreeMagma α → Lean.Format
| FreeMagma.of x => repr x
| x * y => "( " ++ x.repr ++ " * " ++ y.repr ++ " )"
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
FreeMagma.repr
|
Representation of an element of a free magma.
|
protected FreeAddMagma.repr {α : Type u} [Repr α] : FreeAddMagma α → Lean.Format
| FreeAddMagma.of x => repr x
| x + y => "( " ++ x.repr ++ " + " ++ y.repr ++ " )"
attribute [to_additive existing] FreeMagma.repr
@[to_additive]
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
FreeAddMagma.repr
|
Representation of an element of a free additive magma.
|
FreeMagma.length {α : Type u} : FreeMagma α → ℕ
| FreeMagma.of _x => 1
| x * y => x.length + y.length
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
FreeMagma.length
|
Length of an element of a free magma.
|
FreeAddMagma.length {α : Type u} : FreeAddMagma α → ℕ
| FreeAddMagma.of _x => 1
| x + y => x.length + y.length
attribute [to_additive existing (attr := simp)] FreeMagma.length
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
FreeAddMagma.length
|
Length of an element of a free additive magma.
|
@[to_additive /-- The length of an element of a free additive magma is positive. -/]
FreeMagma.length_pos {α : Type u} (x : FreeMagma α) : 0 < x.length :=
match x with
| FreeMagma.of _ => Nat.succ_pos 0
| mul y z => Nat.add_pos_left (length_pos y) z.length
|
lemma
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
FreeMagma.length_pos
|
The length of an element of a free magma is positive.
|
AddMagma.AssocRel (α : Type u) [Add α] : α → α → Prop
| intro : ∀ x y z, AddMagma.AssocRel α (x + y + z) (x + (y + z))
| left : ∀ w x y z, AddMagma.AssocRel α (w + (x + y + z)) (w + (x + (y + z)))
|
inductive
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
AddMagma.AssocRel
|
Associativity relations for an additive magma.
|
@[to_additive AddMagma.AssocRel /-- Associativity relations for an additive magma. -/]
Magma.AssocRel (α : Type u) [Mul α] : α → α → Prop
| intro : ∀ x y z, Magma.AssocRel α (x * y * z) (x * (y * z))
| left : ∀ w x y z, Magma.AssocRel α (w * (x * y * z)) (w * (x * (y * z)))
|
inductive
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
Magma.AssocRel
|
Associativity relations for a magma.
|
@[to_additive AddMagma.FreeAddSemigroup /-- Additive semigroup quotient of an additive magma. -/]
AssocQuotient (α : Type u) [Mul α] : Type u :=
Quot <| AssocRel α
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
AssocQuotient
|
Semigroup quotient of a magma.
|
@[to_additive]
quot_mk_assoc (x y z : α) : Quot.mk (AssocRel α) (x * y * z) = Quot.mk _ (x * (y * z)) :=
Quot.sound (AssocRel.intro _ _ _)
@[to_additive]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
quot_mk_assoc
| null |
quot_mk_assoc_left (x y z w : α) :
Quot.mk (AssocRel α) (x * (y * z * w)) = Quot.mk _ (x * (y * (z * w))) :=
Quot.sound (AssocRel.left _ _ _ _)
@[to_additive]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
quot_mk_assoc_left
| null |
@[to_additive /-- Embedding from additive magma to its free additive semigroup. -/]
of : α →ₙ* AssocQuotient α where toFun := Quot.mk _; map_mul' _x _y := rfl
@[to_additive]
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
of
|
Embedding from magma to its free semigroup.
|
@[to_additive (attr := elab_as_elim, induction_eliminator)]
protected induction_on {C : AssocQuotient α → Prop} (x : AssocQuotient α)
(ih : ∀ x, C (of x)) : C x := Quot.induction_on x ih
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
induction_on
| null |
@[to_additive (attr := ext 1100)]
hom_ext {f g : AssocQuotient α →ₙ* β} (h : f.comp of = g.comp of) : f = g :=
(DFunLike.ext _ _) fun x => AssocQuotient.induction_on x <| DFunLike.congr_fun h
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
hom_ext
| null |
@[to_additive (attr := simps symm_apply) /-- Lifts an additive magma homomorphism `α → β` to an
additive semigroup homomorphism `AddMagma.AssocQuotient α → β` given an additive semigroup `β`. -/]
lift : (α →ₙ* β) ≃ (AssocQuotient α →ₙ* β) where
toFun f :=
{ toFun := fun x ↦
Quot.liftOn x f <| by rintro a b (⟨c, d, e⟩ | ⟨c, d, e, f⟩) <;> simp only [map_mul, mul_assoc]
map_mul' := fun x y ↦ Quot.induction_on₂ x y (map_mul f) }
invFun f := f.comp of
@[to_additive (attr := simp)]
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
lift
|
Lifts a magma homomorphism `α → β` to a semigroup homomorphism `Magma.AssocQuotient α → β`
given a semigroup `β`.
|
lift_of (x : α) : lift f (of x) = f x := rfl
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
lift_of
| null |
lift_comp_of : (lift f).comp of = f := lift.symm_apply_apply f
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
lift_comp_of
| null |
lift_comp_of' (f : AssocQuotient α →ₙ* β) : lift (f.comp of) = f := lift.apply_symm_apply f
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
lift_comp_of'
| null |
@[to_additive /-- From an additive magma homomorphism `α → β` to an additive semigroup homomorphism
`AddMagma.AssocQuotient α → AddMagma.AssocQuotient β`. -/]
map : AssocQuotient α →ₙ* AssocQuotient β := lift (of.comp f)
@[to_additive (attr := simp)]
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
map
|
From a magma homomorphism `α →ₙ* β` to a semigroup homomorphism
`Magma.AssocQuotient α →ₙ* Magma.AssocQuotient β`.
|
map_of (x) : map f (of x) = of (f x) := rfl
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
map_of
| null |
FreeAddSemigroup (α : Type u) where
/-- The head of the element -/
head : α
/-- The tail of the element -/
tail : List α
compile_inductive% FreeAddSemigroup
|
structure
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
FreeAddSemigroup
|
If `α` is a type, then `FreeAddSemigroup α` is the free additive semigroup generated by `α`.
This is an additive semigroup equipped with a function
`FreeAddSemigroup.of : α → FreeAddSemigroup α` which has the following universal property:
if `M` is any additive semigroup, and `f : α → M` is any function,
then this function is the composite of `FreeAddSemigroup.of` and a unique semigroup homomorphism
`FreeAddSemigroup.lift f : FreeAddSemigroup α →ₙ+ M`.
A typical element of `FreeAddSemigroup α` is a nonempty formal sum of elements of `α`.
For example if `x` and `y` are terms of type `α` then `x + y + y + x` is a
"typical" element of `FreeAddSemigroup α`. In particular if `α` is empty
then `FreeAddSemigroup α` is also empty, and if `α` has one term
then `FreeAddSemigroup α` is isomorphic to `ℕ+`.
If `α` has two or more terms then `FreeAddSemigroup α` is not commutative.
One can think of `FreeAddSemigroup α` as the type of nonempty lists of `α`, with addition
given by concatenation.
|
@[to_additive (attr := ext)]
FreeSemigroup (α : Type u) where
/-- The head of the element -/
head : α
/-- The tail of the element -/
tail : List α
compile_inductive% FreeSemigroup
|
structure
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
FreeSemigroup
|
If `α` is a type, then `FreeSemigroup α` is the free semigroup generated by `α`.
This is a semigroup equipped with a function `FreeSemigroup.of : α → FreeSemigroup α` which has
the following universal property: if `M` is any semigroup, and `f : α → M` is any function,
then this function is the composite of `FreeSemigroup.of` and a unique semigroup homomorphism
`FreeSemigroup.lift f : FreeSemigroup α →ₙ* M`.
A typical element of `FreeSemigroup α` is a nonempty formal product of elements of `α`.
For example if `x` and `y` are terms of type `α` then `x * y * y * x` is a
"typical" element of `FreeSemigroup α`. In particular if `α` is empty
then `FreeSemigroup α` is also empty, and if `α` has one term
then `FreeSemigroup α` is isomorphic to `Multiplicative ℕ+`.
If `α` has two or more terms then `FreeSemigroup α` is not commutative.
One can think of `FreeSemigroup α` as the type of nonempty lists of `α`, with multiplication
given by concatenation.
|
@[to_additive (attr := simp)]
head_mul (x y : FreeSemigroup α) : (x * y).1 = x.1 := rfl
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
head_mul
| null |
tail_mul (x y : FreeSemigroup α) : (x * y).2 = x.2 ++ y.1 :: y.2 := rfl
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
tail_mul
| null |
mk_mul_mk (x y : α) (L1 L2 : List α) : mk x L1 * mk y L2 = mk x (L1 ++ y :: L2) := rfl
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
mk_mul_mk
| null |
@[to_additive (attr := simps) /-- The embedding `α → FreeAddSemigroup α`. -/]
of (x : α) : FreeSemigroup α := ⟨x, []⟩
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
of
|
The embedding `α → FreeSemigroup α`.
|
@[to_additive /-- Length of an element of free additive semigroup -/]
length (x : FreeSemigroup α) : ℕ := x.tail.length + 1
@[to_additive (attr := simp)]
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
length
|
Length of an element of free semigroup.
|
length_mul (x y : FreeSemigroup α) : (x * y).length = x.length + y.length := by
simp [length, Nat.add_right_comm, List.length, List.length_append]
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
length_mul
| null |
length_of (x : α) : (of x).length = 1 := rfl
@[to_additive]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
length_of
| null |
@[to_additive (attr := elab_as_elim, induction_eliminator)
/-- Recursor for free additive semigroup using `of` and `+`. -/]
protected recOnMul {C : FreeSemigroup α → Sort l} (x) (ih1 : ∀ x, C (of x))
(ih2 : ∀ x y, C (of x) → C y → C (of x * y)) : C x :=
FreeSemigroup.recOn x fun f s ↦
List.recOn s ih1 (fun hd tl ih f ↦ ih2 f ⟨hd, tl⟩ (ih1 f) (ih hd)) f
@[to_additive (attr := ext 1100)]
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
recOnMul
|
Recursor for free semigroup using `of` and `*`.
|
hom_ext {β : Type v} [Mul β] {f g : FreeSemigroup α →ₙ* β} (h : f ∘ of = g ∘ of) : f = g :=
(DFunLike.ext _ _) fun x ↦
FreeSemigroup.recOnMul x (congr_fun h) fun x y hx hy ↦ by simp only [map_mul, *]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
hom_ext
| null |
@[to_additive (attr := simps symm_apply) /-- Lifts a function `α → β` to an additive semigroup
homomorphism `FreeAddSemigroup α → β` given an additive semigroup `β`. -/]
lift : (α → β) ≃ (FreeSemigroup α →ₙ* β) where
toFun f :=
{ toFun := fun x ↦ x.2.foldl (fun a b ↦ a * f b) (f x.1)
map_mul' := fun x y ↦ by
simp [head_mul, tail_mul, ← List.foldl_map, List.foldl_append, List.foldl_cons,
List.foldl_assoc] }
invFun f := f ∘ of
@[to_additive (attr := simp)]
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
lift
|
Lifts a function `α → β` to a semigroup homomorphism `FreeSemigroup α → β` given
a semigroup `β`.
|
lift_of (x : α) : lift f (of x) = f x := rfl
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
lift_of
| null |
lift_comp_of : lift f ∘ of = f := rfl
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
lift_comp_of
| null |
lift_comp_of' (f : FreeSemigroup α →ₙ* β) : lift (f ∘ of) = f := hom_ext rfl
@[to_additive]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
lift_comp_of'
| null |
lift_of_mul (x y) : lift f (of x * y) = f x * lift f y := by rw [map_mul, lift_of]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
lift_of_mul
| null |
@[to_additive /-- The unique additive semigroup homomorphism that sends `of x` to `of (f x)`. -/]
map : FreeSemigroup α →ₙ* FreeSemigroup β :=
lift <| of ∘ f
@[to_additive (attr := simp)]
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
map
|
The unique semigroup homomorphism that sends `of x` to `of (f x)`.
|
map_of (x) : map f (of x) = of (f x) := rfl
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
map_of
| null |
length_map (x) : (map f x).length = x.length :=
FreeSemigroup.recOnMul x (fun _ ↦ rfl) (fun x y hx hy ↦ by simp only [map_mul, length_mul, *])
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
length_map
| null |
@[to_additive (attr := elab_as_elim) /-- Recursor that uses `pure` instead of `of`. -/]
recOnPure {C : FreeSemigroup α → Sort l} (x) (ih1 : ∀ x, C (pure x))
(ih2 : ∀ x y, C (pure x) → C y → C (pure x * y)) : C x :=
FreeSemigroup.recOnMul x ih1 ih2
@[to_additive (attr := simp)]
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
recOnPure
|
Recursor that uses `pure` instead of `of`.
|
protected map_pure (f : α → β) (x) : (f <$> pure x : FreeSemigroup β) = pure (f x) := rfl
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
map_pure
| null |
map_mul' (f : α → β) (x y : FreeSemigroup α) : f <$> (x * y) = f <$> x * f <$> y :=
map_mul (map f) _ _
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
map_mul'
| null |
pure_bind (f : α → FreeSemigroup β) (x) : pure x >>= f = f x := rfl
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
pure_bind
| null |
mul_bind (f : α → FreeSemigroup β) (x y : FreeSemigroup α) :
x * y >>= f = (x >>= f) * (y >>= f) := map_mul (lift f) _ _
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
mul_bind
| null |
pure_seq {f : α → β} {x : FreeSemigroup α} : pure f <*> x = f <$> x := rfl
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
pure_seq
| null |
mul_seq {f g : FreeSemigroup (α → β)} {x : FreeSemigroup α} :
f * g <*> x = (f <*> x) * (g <*> x) := mul_bind _ _ _
@[to_additive]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
mul_seq
| null |
instLawfulMonad : LawfulMonad FreeSemigroup.{u} := LawfulMonad.mk'
(pure_bind := fun _ _ ↦ rfl)
(bind_assoc := fun x g f ↦
recOnPure x (fun _ ↦ rfl) fun x y ih1 ih2 ↦ by rw [mul_bind, mul_bind, mul_bind, ih1, ih2])
(id_map := fun x ↦ recOnPure x (fun _ ↦ rfl) fun x y ih1 ih2 ↦ by rw [map_mul', ih1, ih2])
|
instance
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
instLawfulMonad
| null |
@[to_additive /-- `FreeAddSemigroup` is traversable. -/]
protected traverse {m : Type u → Type u} [Applicative m] {α β : Type u}
(F : α → m β) (x : FreeSemigroup α) : m (FreeSemigroup β) :=
recOnPure x (fun x ↦ pure <$> F x) fun _x _y ihx ihy ↦ (· * ·) <$> ihx <*> ihy
@[to_additive]
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
traverse
|
`FreeSemigroup` is traversable.
|
@[to_additive (attr := simp)]
traverse_pure (x) : traverse F (pure x : FreeSemigroup α) = pure <$> F x := rfl
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
traverse_pure
| null |
traverse_pure' : traverse F ∘ pure = fun x ↦ (pure <$> F x : m (FreeSemigroup β)) := rfl
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
traverse_pure'
| null |
@[to_additive (attr := simp)]
traverse_mul (x y : FreeSemigroup α) :
traverse F (x * y) = (· * ·) <$> traverse F x <*> traverse F y :=
let ⟨x, L1⟩ := x
let ⟨y, L2⟩ := y
List.recOn L1 (fun _ ↦ rfl)
(fun hd tl ih x ↦ show
(· * ·) <$> pure <$> F x <*> traverse F (mk hd tl * mk y L2) =
(· * ·) <$> ((· * ·) <$> pure <$> F x <*> traverse F (mk hd tl)) <*> traverse F (mk y L2)
by rw [ih]; simp only [Function.comp_def, (mul_assoc _ _ _).symm, functor_norm])
x
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
traverse_mul
| null |
traverse_mul' :
Function.comp (traverse F) ∘ (HMul.hMul : FreeSemigroup α → FreeSemigroup α → FreeSemigroup α) =
fun x y ↦ (· * ·) <$> traverse F x <*> traverse F y :=
funext fun x ↦ funext fun y ↦ traverse_mul F x y
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
traverse_mul'
| null |
@[to_additive (attr := simp)]
traverse_eq (x) : FreeSemigroup.traverse F x = traverse F x := rfl
@[to_additive (attr := deprecated "Use map_pure and seq_pure" (since := "2025-05-21"))]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
traverse_eq
| null |
mul_map_seq (x y : FreeSemigroup α) :
((· * ·) <$> x <*> y : Id (FreeSemigroup α)) = (x * y : FreeSemigroup α) := rfl
@[to_additive]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
mul_map_seq
| null |
@[to_additive /-- The canonical additive morphism from `FreeAddMagma α` to `FreeAddSemigroup α`. -/]
toFreeSemigroup : FreeMagma α →ₙ* FreeSemigroup α := FreeMagma.lift FreeSemigroup.of
@[to_additive (attr := simp)]
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
toFreeSemigroup
|
The canonical multiplicative morphism from `FreeMagma α` to `FreeSemigroup α`.
|
toFreeSemigroup_of (x : α) : toFreeSemigroup (of x) = FreeSemigroup.of x := rfl
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
toFreeSemigroup_of
| null |
toFreeSemigroup_comp_of : @toFreeSemigroup α ∘ of = FreeSemigroup.of := rfl
@[to_additive]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
toFreeSemigroup_comp_of
| null |
toFreeSemigroup_comp_map (f : α → β) :
toFreeSemigroup.comp (map f) = (FreeSemigroup.map f).comp toFreeSemigroup := by ext1; rfl
@[to_additive]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
toFreeSemigroup_comp_map
| null |
toFreeSemigroup_map (f : α → β) (x : FreeMagma α) :
toFreeSemigroup (map f x) = FreeSemigroup.map f (toFreeSemigroup x) :=
DFunLike.congr_fun (toFreeSemigroup_comp_map f) x
@[to_additive (attr := simp)]
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
toFreeSemigroup_map
| null |
length_toFreeSemigroup (x : FreeMagma α) : (toFreeSemigroup x).length = x.length :=
FreeMagma.recOnMul x (fun _ ↦ rfl) fun x y hx hy ↦ by
rw [map_mul, FreeSemigroup.length_mul, hx, hy]; rfl
|
theorem
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
length_toFreeSemigroup
| null |
@[to_additive /-- Isomorphism between `AddMagma.AssocQuotient (FreeAddMagma α)` and
`FreeAddSemigroup α`. -/]
FreeMagmaAssocQuotientEquiv (α : Type u) :
Magma.AssocQuotient (FreeMagma α) ≃* FreeSemigroup α :=
(Magma.AssocQuotient.lift FreeMagma.toFreeSemigroup).toMulEquiv
(FreeSemigroup.lift (Magma.AssocQuotient.of ∘ FreeMagma.of))
(by ext; rfl)
(by ext1; rfl)
|
def
|
Algebra
|
[
"Mathlib.Algebra.Group.Basic",
"Mathlib.Algebra.Group.Equiv.Defs",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Tactic.AdaptationNote"
] |
Mathlib/Algebra/Free.lean
|
FreeMagmaAssocQuotientEquiv
|
Isomorphism between `Magma.AssocQuotient (FreeMagma α)` and `FreeSemigroup α`.
|
Pre
| of : X → Pre
| ofScalar : R → Pre
| add : Pre → Pre → Pre
| mul : Pre → Pre → Pre
|
inductive
|
Algebra
|
[
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.FreeMonoid.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors"
] |
Mathlib/Algebra/FreeAlgebra.lean
|
Pre
|
This inductive type is used to express representatives of the free algebra.
|
hasCoeGenerator : Coe X (Pre R X) := ⟨of⟩
|
def
|
Algebra
|
[
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.FreeMonoid.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors"
] |
Mathlib/Algebra/FreeAlgebra.lean
|
hasCoeGenerator
|
Coercion from `X` to `Pre R X`. Note: Used for notation only.
|
hasCoeSemiring : Coe R (Pre R X) := ⟨ofScalar⟩
|
def
|
Algebra
|
[
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.FreeMonoid.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors"
] |
Mathlib/Algebra/FreeAlgebra.lean
|
hasCoeSemiring
|
Coercion from `R` to `Pre R X`. Note: Used for notation only.
|
hasMul : Mul (Pre R X) := ⟨mul⟩
|
def
|
Algebra
|
[
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.FreeMonoid.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors"
] |
Mathlib/Algebra/FreeAlgebra.lean
|
hasMul
|
Multiplication in `Pre R X` defined as `Pre.mul`. Note: Used for notation only.
|
hasAdd : Add (Pre R X) := ⟨add⟩
|
def
|
Algebra
|
[
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.FreeMonoid.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors"
] |
Mathlib/Algebra/FreeAlgebra.lean
|
hasAdd
|
Addition in `Pre R X` defined as `Pre.add`. Note: Used for notation only.
|
hasZero : Zero (Pre R X) := ⟨ofScalar 0⟩
|
def
|
Algebra
|
[
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.FreeMonoid.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors"
] |
Mathlib/Algebra/FreeAlgebra.lean
|
hasZero
|
Zero in `Pre R X` defined as the image of `0` from `R`. Note: Used for notation only.
|
hasOne : One (Pre R X) := ⟨ofScalar 1⟩
|
def
|
Algebra
|
[
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.FreeMonoid.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors"
] |
Mathlib/Algebra/FreeAlgebra.lean
|
hasOne
|
One in `Pre R X` defined as the image of `1` from `R`. Note: Used for notation only.
|
hasSMul : SMul R (Pre R X) := ⟨fun r m ↦ mul (ofScalar r) m⟩
|
def
|
Algebra
|
[
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.FreeMonoid.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors"
] |
Mathlib/Algebra/FreeAlgebra.lean
|
hasSMul
|
Scalar multiplication defined as multiplication by the image of elements from `R`.
Note: Used for notation only.
|
liftFun {A : Type*} [Semiring A] [Algebra R A] (f : X → A) :
Pre R X → A
| .of t => f t
| .add a b => liftFun f a + liftFun f b
| .mul a b => liftFun f a * liftFun f b
| .ofScalar c => algebraMap _ _ c
|
def
|
Algebra
|
[
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.FreeMonoid.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors"
] |
Mathlib/Algebra/FreeAlgebra.lean
|
liftFun
|
Given a function from `X` to an `R`-algebra `A`, `lift_fun` provides a lift of `f` to a function
from `Pre R X` to `A`. This is mainly used in the construction of `FreeAlgebra.lift`.
|
Rel : Pre R X → Pre R X → Prop
| add_scalar {r s : R} : Rel (↑(r + s)) (↑r + ↑s)
| mul_scalar {r s : R} : Rel (↑(r * s)) (↑r * ↑s)
| central_scalar {r : R} {a : Pre R X} : Rel (r * a) (a * r)
| add_assoc {a b c : Pre R X} : Rel (a + b + c) (a + (b + c))
| add_comm {a b : Pre R X} : Rel (a + b) (b + a)
| zero_add {a : Pre R X} : Rel (0 + a) a
| mul_assoc {a b c : Pre R X} : Rel (a * b * c) (a * (b * c))
| one_mul {a : Pre R X} : Rel (1 * a) a
| mul_one {a : Pre R X} : Rel (a * 1) a
| left_distrib {a b c : Pre R X} : Rel (a * (b + c)) (a * b + a * c)
| right_distrib {a b c : Pre R X} :
Rel ((a + b) * c) (a * c + b * c)
| zero_mul {a : Pre R X} : Rel (0 * a) 0
| mul_zero {a : Pre R X} : Rel (a * 0) 0
| add_compat_left {a b c : Pre R X} : Rel a b → Rel (a + c) (b + c)
| add_compat_right {a b c : Pre R X} : Rel a b → Rel (c + a) (c + b)
| mul_compat_left {a b c : Pre R X} : Rel a b → Rel (a * c) (b * c)
| mul_compat_right {a b c : Pre R X} : Rel a b → Rel (c * a) (c * b)
|
inductive
|
Algebra
|
[
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.FreeMonoid.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors"
] |
Mathlib/Algebra/FreeAlgebra.lean
|
Rel
|
An inductively defined relation on `Pre R X` used to force the initial algebra structure on
the associated quotient.
|
FreeAlgebra :=
Quot (FreeAlgebra.Rel R X)
|
def
|
Algebra
|
[
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.FreeMonoid.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors"
] |
Mathlib/Algebra/FreeAlgebra.lean
|
FreeAlgebra
|
If `α` is a type, and `R` is a commutative semiring, then `FreeAlgebra R α` is the
free (unital, associative) `R`-algebra generated by `α`.
This is an `R`-algebra equipped with a function `FreeAlgebra.ι R : α → FreeAlgebra R α` which has
the following universal property: if `A` is any `R`-algebra, and `f : α → A` is any function,
then this function is the composite of `FreeAlgebra.ι R` and a unique `R`-algebra homomorphism
`FreeAlgebra.lift R f : FreeAlgebra R α →ₐ[R] A`.
A typical element of `FreeAlgebra R α` is an `R`-linear
combination of formal products of elements of `α`.
For example if `x` and `y` are terms of type `α` and `a`, `b` are terms of type `R` then
`(3 * a * a) • (x * y * x) + (2 * b + 1) • (y * x) + (a * b * b + 3)` is a
"typical" element of `FreeAlgebra R α`. In particular if `α` is empty
then `FreeAlgebra R α` is isomorphic to `R`, and if `α` has one term `t`
then `FreeAlgebra R α` is isomorphic to the polynomial ring `R[t]`.
If `α` has two or more terms then `FreeAlgebra R α` is not commutative.
One can think of `FreeAlgebra R α` as the free non-commutative polynomial ring
with coefficients in `R` and variables indexed by `α`.
|
instSMul {A} [CommSemiring A] [Algebra R A] : SMul R (FreeAlgebra A X) where
smul r := Quot.map (HMul.hMul (algebraMap R A r : Pre A X)) fun _ _ ↦ Rel.mul_compat_right
|
instance
|
Algebra
|
[
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Algebra.Algebra.Subalgebra.Lattice",
"Mathlib.Algebra.FreeMonoid.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors"
] |
Mathlib/Algebra/FreeAlgebra.lean
|
instSMul
| null |
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