mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α β : Type u
m : Type u → Type u
inst✝⁴ : Applicative m
F : α → m β
F✝ G✝ : Type u → Type u
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
α✝ β✝ γ✝ : Type u
f : β✝ → F✝ γ✝
g : α✝ → G✝ β✝
x✝ x y : FreeMagma α✝
ih1 : traverse (Functor.Comp.mk ∘ Functor.map f ∘ g) x = Functor.Comp.mk (traverse f <$> traverse g x)
ih2 : traverse (Functor.Comp.mk ∘ Functor.map f ∘ g) y = Functor.Comp.mk (traverse f <$> traverse g y)
⊢ traverse (Functor.Comp.mk ∘ Functor.map f ∘ g) (x * y) = Functor.Comp.mk (traverse f <$> traverse g (x * y))

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
L : List (α × Bool)
inst✝ : DecidableEq α
⊢ Red L (reduce L)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
p : Prop
L2 L3 : List (α × Bool)
x✝¹ : α
x✝ : Bool
h : reduce [] = L2 ++ (x✝¹, x✝) :: (x✝¹, !x✝) :: L3
⊢ p

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α β : Type u
m : Type u → Type u
inst✝ : Applicative m
F : α → m β
α✝ β✝ : Type u
f : α✝ → β✝
x✝ x y : FreeMagma α✝
ih1 : traverse (pure ∘ f) x = pure (f <$> x)
ih2 : traverse (pure ∘ f) y = pure (f <$> y)
⊢ traverse (pure ∘ f) (x * y) = pure (f <$> (x * y))

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α✝ β✝ : Type u
m : Type u → Type u
inst✝⁴ : Applicative m
F : α✝ → m β✝
F✝ G✝ : Type u → Type u
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
η : ApplicativeTransformation F✝ G✝
α β : Type u
f : α → F✝ β
x✝ : FreeMagma α
x : α
⊢ (fun {α} ↦ η.app α) (traverse f (pure x)) = traverse ((fun {α} ↦ η.app α) ∘ f) (pure x)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α✝ β✝ : Type u
m : Type u → Type u
inst✝⁴ : Applicative m
F : α✝ → m β✝
F✝ G✝ : Type u → Type u
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
η : ApplicativeTransformation F✝ G✝
α β : Type u
f : α → F✝ β
x✝ x y : FreeMagma α
ih1 : (fun {α} ↦ η.app α) (traverse f x) = traverse ((fun {α} ↦ η.app α) ∘ f) x
ih2 : (fun {α} ↦ η.app α) (traverse f y) = traverse ((fun {α} ↦ η.app α) ∘ f) y
⊢ (fun {α} ↦ η.app α) (traverse f (x * y)) = traverse ((fun {α} ↦ η.app α) ∘ f) (x * y)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.FiveLemma

Goal state before:
pp: M₁ : Type u_1
M₂ : Type u_2
M₃ : Type u_3
M₄ : Type u_4
N₁ : Type u_6
N₂ : Type u_7
N₃ : Type u_8
N₄ : Type u_9
inst✝⁷ : Group M₁
inst✝⁶ : Group M₂
inst✝⁵ : Group M₃
inst✝⁴ : Group M₄
inst✝³ : Group N₁
inst✝² : Group N₂
inst✝¹ : Group N₃
inst✝ : Group N₄
f₁ : M₁ →* M₂
f₂ : M₂ →* M₃
f₃ : M₃ →* M₄
g₁ : N₁ →* N₂
g₂ : N₂ →* N₃
g₃ : N₃ →* N₄
i₁ : M₁ →* N₁
i₂ : M₂ →* N₂
i₃ : M₃ →* N₃
i₄ : M₄ →* N₄
hc₁ : g₁.comp i₁ = i₂.comp f₁
hc₂ : g₂.comp i₂ = i₃.comp f₂
hc₃ : g₃.comp i₃ = i₄.comp f₃
hf₂ : Function.MulExact ⇑f₂ ⇑f₃
hg₁ : Function.MulExact ⇑g₁ ⇑g₂
hg₂ : Function.MulExact ⇑g₂ ⇑g₃
hi₁ : Function.Surjective ⇑i₁
hi₃ : Function.Surjective ⇑i₃
hi₄ : Function.Injective ⇑i₄
x : N₂
y : M₃
hy : i₃ y = g₂ x
⊢ f₃ y = 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.FiveLemma

Goal state before:
pp: M₁ : Type u_1
M₂ : Type u_2
M₃ : Type u_3
M₄ : Type u_4
N₁ : Type u_6
N₂ : Type u_7
N₃ : Type u_8
N₄ : Type u_9
inst✝⁷ : Group M₁
inst✝⁶ : Group M₂
inst✝⁵ : Group M₃
inst✝⁴ : Group M₄
inst✝³ : Group N₁
inst✝² : Group N₂
inst✝¹ : Group N₃
inst✝ : Group N₄
f₁ : M₁ →* M₂
f₂ : M₂ →* M₃
f₃ : M₃ →* M₄
g₁ : N₁ →* N₂
g₂ : N₂ →* N₃
g₃ : N₃ →* N₄
i₁ : M₁ →* N₁
i₂ : M₂ →* N₂
i₃ : M₃ →* N₃
i₄ : M₄ →* N₄
hc₁ : g₁.comp i₁ = i₂.comp f₁
hc₂ : g₂.comp i₂ = i₃.comp f₂
hc₃ : g₃.comp i₃ = i₄.comp f₃
hf₂ : Function.MulExact ⇑f₂ ⇑f₃
hg₁ : Function.MulExact ⇑g₁ ⇑g₂
hg₂ : Function.MulExact ⇑g₂ ⇑g₃
hi₁ : Function.Surjective ⇑i₁
hi₃ : Function.Surjective ⇑i₃
hi₄ : Function.Injective ⇑i₄
x : N₂
a : M₂
hy : i₃ (f₂ a) = g₂ x
⊢ g₂ (x / i₂ a) = 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α : Type u
inst✝ : Mul α
x y : AssocQuotient α
⊢ AssocQuotient α

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α : Type u
inst✝¹ : Mul α
β : Type v
inst✝ : Semigroup β
f✝ f : α →ₙ* β
x : AssocQuotient α
⊢ ∀ (a b : α), AssocRel α a b → f a = f b

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α : Type u
x y : FreeSemigroup α
⊢ (x * y).length = x.length + y.length

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α : Type u
β : Type v
inst✝ : Mul β
f g : FreeSemigroup α →ₙ* β
h : ⇑f ∘ of = ⇑g ∘ of
x✝ : FreeSemigroup α
x : α
y : FreeSemigroup α
hx : f (of x) = g (of x)
hy : f y = g y
⊢ f (of x * y) = g (of x * y)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α : Type u
β : Type v
inst✝ : Semigroup β
f✝ f : α → β
⊢ ∀ (x y : FreeSemigroup α),
    List.foldl (fun a b ↦ a * f b) (f (x * y).head) (x * y).tail =
      List.foldl (fun a b ↦ a * f b) (f x.head) x.tail * List.foldl (fun a b ↦ a * f b) (f y.head) y.tail

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.FiveLemma

Goal state before:
pp: M₁ : Type u_1
M₂ : Type u_2
M₃ : Type u_3
M₄ : Type u_4
N₁ : Type u_6
N₂ : Type u_7
N₃ : Type u_8
N₄ : Type u_9
inst✝⁷ : Group M₁
inst✝⁶ : Group M₂
inst✝⁵ : Group M₃
inst✝⁴ : Group M₄
inst✝³ : Group N₁
inst✝² : Group N₂
inst✝¹ : Group N₃
inst✝ : Group N₄
f₁ : M₁ →* M₂
f₂ : M₂ →* M₃
f₃ : M₃ →* M₄
g₁ : N₁ →* N₂
g₂ : N₂ →* N₃
g₃ : N₃ →* N₄
i₁ : M₁ →* N₁
i₂ : M₂ →* N₂
i₃ : M₃ →* N₃
i₄ : M₄ →* N₄
hc₁ : g₁.comp i₁ = i₂.comp f₁
hc₂ : g₂.comp i₂ = i₃.comp f₂
hc₃ : g₃.comp i₃ = i₄.comp f₃
hf₂ : Function.MulExact ⇑f₂ ⇑f₃
hg₁ : Function.MulExact ⇑g₁ ⇑g₂
hg₂ : Function.MulExact ⇑g₂ ⇑g₃
hi₁ : Function.Surjective ⇑i₁
hi₃ : Function.Surjective ⇑i₃
hi₄ : Function.Injective ⇑i₄
⊢ Function.Surjective ⇑i₂

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α : Type u
β : Type v
inst✝ : Semigroup β
f : α → β
x : α
y : FreeSemigroup α
⊢ (lift f) (of x * y) = f x * (lift f) y

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α : Type u
β : Type v
f : α → β
x✝ : FreeSemigroup α
x : α
y : FreeSemigroup α
hx : ((map f) (of x)).length = (of x).length
hy : ((map f) y).length = y.length
⊢ ((map f) (of x * y)).length = (of x * y).length

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.FiveLemma

Goal state before:
pp: M₁ : Type u_1
M₂ : Type u_2
M₃ : Type u_3
N₁ : Type u_6
N₂ : Type u_7
N₃ : Type u_8
inst✝⁵ : Group M₁
inst✝⁴ : Group M₂
inst✝³ : Group M₃
inst✝² : Group N₁
inst✝¹ : Group N₂
inst✝ : Group N₃
f₁ : M₁ →* M₂
f₂ : M₂ →* M₃
g₁ : N₁ →* N₂
g₂ : N₂ →* N₃
i₁ : M₁ →* N₁
i₂ : M₂ →* N₂
i₃ : M₃ →* N₃
hc₁ : g₁.comp i₁ = i₂.comp f₁
hc₂ : g₂.comp i₂ = i₃.comp f₂
hf₁ : Function.MulExact ⇑f₁ ⇑f₂
hg₁ : Function.MulExact ⇑g₁ ⇑g₂
hi₂ : Function.Surjective ⇑i₂
hi₃ : Function.Injective ⇑i₃
hg₀ : Function.Injective ⇑g₁
⊢ comp 1 1 = i₁.comp 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α β α✝ : Type u
x✝ : FreeSemigroup α✝
x : α✝
y : FreeSemigroup α✝
ih1 : id <$> pure x = pure x
ih2 : id <$> y = y
⊢ id <$> (pure x * y) = pure x * y

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.FiveLemma

Goal state before:
pp: M₁ : Type u_1
M₂ : Type u_2
M₃ : Type u_3
N₁ : Type u_6
N₂ : Type u_7
N₃ : Type u_8
inst✝⁵ : Group M₁
inst✝⁴ : Group M₂
inst✝³ : Group M₃
inst✝² : Group N₁
inst✝¹ : Group N₂
inst✝ : Group N₃
f₁ : M₁ →* M₂
f₂ : M₂ →* M₃
g₁ : N₁ →* N₂
g₂ : N₂ →* N₃
i₁ : M₁ →* N₁
i₂ : M₂ →* N₂
i₃ : M₃ →* N₃
hc₁ : g₁.comp i₁ = i₂.comp f₁
hc₂ : g₂.comp i₂ = i₃.comp f₂
hf₁ : Function.MulExact ⇑f₁ ⇑f₂
hg₁ : Function.MulExact ⇑g₁ ⇑g₂
hi₂ : Function.Surjective ⇑i₂
hi₃ : Function.Injective ⇑i₃
hg₀ : Function.Injective ⇑g₁
⊢ Function.Surjective ⇑i₁

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α β α✝ β✝ γ✝ : Type u
x✝ : FreeSemigroup α✝
g : α✝ → FreeSemigroup β✝
f : β✝ → FreeSemigroup γ✝
x : α✝
y : FreeSemigroup α✝
ih1 : pure x >>= g >>= f = pure x >>= fun x ↦ g x >>= f
ih2 : y >>= g >>= f = y >>= fun x ↦ g x >>= f
⊢ pure x * y >>= g >>= f = pure x * y >>= fun x ↦ g x >>= f

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
p : Prop
x : α
b : Bool
L1 L2 L3 : List (α × Bool)
x' : α
b' : Bool
⊢ reduce ((x, b) :: L1) = L2 ++ (x', b') :: (x', !b') :: L3 → p

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
L₁ L₂ : List (α × Bool)
inst✝ : DecidableEq α
H : Red (reduce L₁) L₂
⊢ reduce L₁ = L₂

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
⊢ ∀ {x : FreeGroup α}, mk x.toWord = x

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
⊢ Function.Injective toWord

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
⊢ ∀ (x : FreeGroup α), reduce x.toWord = x.toWord

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
x y : FreeGroup α
⊢ (x * y).toWord = reduce (x.toWord ++ y.toWord)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α β : Type u
m : Type u → Type u
inst✝ : Applicative m
F : α → m β
α✝ : Type u
x✝ : FreeSemigroup α✝
x : α✝
y : FreeSemigroup α✝
ih1 : traverse pure (of x) = pure (of x)
ih2 : traverse pure y = pure y
⊢ traverse pure (of x * y) = pure (of x * y)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α β : Type u
m : Type u → Type u
inst✝⁴ : Applicative m
F : α → m β
F✝ G✝ : Type u → Type u
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
α✝ β✝ γ✝ : Type u
f : β✝ → F✝ γ✝
g : α✝ → G✝ β✝
x✝ : FreeSemigroup α✝
x : α✝
⊢ traverse (Functor.Comp.mk ∘ Functor.map f ∘ g) (pure x) = Functor.Comp.mk (traverse f <$> traverse g (pure x))

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
x : FreeGroup α
n : ℕ
⊢ (x ^ n).toWord = reduce (List.replicate n x.toWord).flatten

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.FiveLemma

Goal state before:
pp: M₁ : Type u_1
M₂ : Type u_2
M₃ : Type u_3
M₄ : Type u_4
N₁ : Type u_6
N₂ : Type u_7
N₃ : Type u_8
N₄ : Type u_9
inst✝⁷ : Group M₁
inst✝⁶ : Group M₂
inst✝⁵ : Group M₃
inst✝⁴ : Group M₄
inst✝³ : Group N₁
inst✝² : Group N₂
inst✝¹ : Group N₃
inst✝ : Group N₄
f₁ : M₁ →* M₂
f₂ : M₂ →* M₃
f₃ : M₃ →* M₄
g₁ : N₁ →* N₂
g₂ : N₂ →* N₃
g₃ : N₃ →* N₄
i₁ : M₁ →* N₁
i₂ : M₂ →* N₂
i₃ : M₃ →* N₃
i₄ : M₄ →* N₄
hc₁ : g₁.comp i₁ = i₂.comp f₁
hc₂ : g₂.comp i₂ = i₃.comp f₂
hc₃ : g₃.comp i₃ = i₄.comp f₃
hf₁ : Function.MulExact ⇑f₁ ⇑f₂
hf₂ : Function.MulExact ⇑f₂ ⇑f₃
hg₁ : Function.MulExact ⇑g₁ ⇑g₂
hi₁ : Function.Surjective ⇑i₁
hi₂ : Function.Injective ⇑i₂
hi₄ : Function.Injective ⇑i₄
m : M₃
hm : i₃ m = 1
h : i₄ (f₃ m) = 1
⊢ f₃ m = 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
a : α
n : ℕ
⊢ (of a ^ n).toWord = List.replicate n (a, true)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
w : List (α × Bool)
⊢ reduce (invRev w) = invRev (reduce w)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
x : FreeGroup α
⊢ x⁻¹.toWord = invRev x.toWord

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.FiveLemma

Goal state before:
pp: M₁ : Type u_1
M₂ : Type u_2
M₃ : Type u_3
M₄ : Type u_4
N₁ : Type u_6
N₂ : Type u_7
N₃ : Type u_8
N₄ : Type u_9
inst✝⁷ : Group M₁
inst✝⁶ : Group M₂
inst✝⁵ : Group M₃
inst✝⁴ : Group M₄
inst✝³ : Group N₁
inst✝² : Group N₂
inst✝¹ : Group N₃
inst✝ : Group N₄
f₁ : M₁ →* M₂
f₂ : M₂ →* M₃
f₃ : M₃ →* M₄
g₁ : N₁ →* N₂
g₂ : N₂ →* N₃
g₃ : N₃ →* N₄
i₁ : M₁ →* N₁
i₂ : M₂ →* N₂
i₃ : M₃ →* N₃
i₄ : M₄ →* N₄
hc₁ : g₁.comp i₁ = i₂.comp f₁
hc₂ : g₂.comp i₂ = i₃.comp f₂
hc₃ : g₃.comp i₃ = i₄.comp f₃
hf₁ : Function.MulExact ⇑f₁ ⇑f₂
hf₂ : Function.MulExact ⇑f₂ ⇑f₃
hg₁ : Function.MulExact ⇑g₁ ⇑g₂
hi₁ : Function.Surjective ⇑i₁
hi₂ : Function.Injective ⇑i₂
hi₄ : Function.Injective ⇑i₄
m : M₃
hm : i₃ m = 1
⊢ f₃ m = 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
L₁ L₂ : List (α × Bool)
inst✝ : DecidableEq α
⊢ reduce (reduce L₁ ++ reduce L₂) = reduce (L₁ ++ L₂)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
L : List (α × Bool)
inst✝ : DecidableEq α
a : α × Bool
⊢ reduce (a :: reduce L) = reduce (a :: L)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
L : List (α × Bool)
inst✝ : DecidableEq α
⊢ reduce (invRev L ++ L) = []

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α β : Type u
m : Type u → Type u
inst✝⁴ : Applicative m
F : α → m β
F✝ G✝ : Type u → Type u
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
α✝ β✝ γ✝ : Type u
f : β✝ → F✝ γ✝
g : α✝ → G✝ β✝
x✝ : FreeSemigroup α✝
x : α✝
y : FreeSemigroup α✝
ih1 : traverse (Functor.Comp.mk ∘ Functor.map f ∘ g) (pure x) = Functor.Comp.mk (traverse f <$> traverse g (pure x))
ih2 : traverse (Functor.Comp.mk ∘ Functor.map f ∘ g) y = Functor.Comp.mk (traverse f <$> traverse g y)
⊢ traverse (Functor.Comp.mk ∘ Functor.map f ∘ g) (pure x * y) = Functor.Comp.mk (traverse f <$> traverse g (pure x * y))

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
x y : FreeGroup α
⊢ x * y = mk (x.toWord ++ y.toWord)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
x y : FreeGroup α
⊢ (x * y).toWord <+ x.toWord ++ y.toWord

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
L : List (α × Bool)
inst✝ : DecidableEq α
h : IsReduced L
⊢ reduce L = L

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.FiveLemma

Goal state before:
pp: M₁ : Type u_1
M₂ : Type u_2
M₃ : Type u_3
M₄ : Type u_4
N₁ : Type u_6
N₂ : Type u_7
N₃ : Type u_8
N₄ : Type u_9
inst✝⁷ : Group M₁
inst✝⁶ : Group M₂
inst✝⁵ : Group M₃
inst✝⁴ : Group M₄
inst✝³ : Group N₁
inst✝² : Group N₂
inst✝¹ : Group N₃
inst✝ : Group N₄
f₁ : M₁ →* M₂
f₂ : M₂ →* M₃
f₃ : M₃ →* M₄
g₁ : N₁ →* N₂
g₂ : N₂ →* N₃
g₃ : N₃ →* N₄
i₁ : M₁ →* N₁
i₂ : M₂ →* N₂
i₃ : M₃ →* N₃
i₄ : M₄ →* N₄
hc₁ : g₁.comp i₁ = i₂.comp f₁
hc₂ : g₂.comp i₂ = i₃.comp f₂
hc₃ : g₃.comp i₃ = i₄.comp f₃
hf₁ : Function.MulExact ⇑f₁ ⇑f₂
hf₂ : Function.MulExact ⇑f₂ ⇑f₃
hg₁ : Function.MulExact ⇑g₁ ⇑g₂
hi₁ : Function.Surjective ⇑i₁
hi₂ : Function.Injective ⇑i₂
hi₄ : Function.Injective ⇑i₄
x : M₂
hm : i₃ (f₂ x) = 1
⊢ g₂ ?m.183 = 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α β : Type u
m : Type u → Type u
inst✝ : Applicative m
F : α → m β
α✝ β✝ : Type u
f : α✝ → β✝
x✝ : FreeSemigroup α✝
x : α✝
y : FreeSemigroup α✝
ih1 : traverse (pure ∘ f) (of x) = pure (f <$> of x)
ih2 : traverse (pure ∘ f) y = pure (f <$> y)
⊢ traverse (pure ∘ f) (of x * y) = pure (f <$> (of x * y))

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α✝ β✝ : Type u
m : Type u → Type u
inst✝⁴ : Applicative m
F : α✝ → m β✝
F✝ G✝ : Type u → Type u
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
η : ApplicativeTransformation F✝ G✝
α β : Type u
f : α → F✝ β
x✝ : FreeSemigroup α
x : α
⊢ (fun {α} ↦ η.app α) (traverse f (pure x)) = traverse ((fun {α} ↦ η.app α) ∘ f) (pure x)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
L : List (α × Bool)
inst✝ : DecidableEq α
h : reduce L = L
⊢ IsReduced L

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
x : FreeGroup α
⊢ IsReduced x.toWord

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α✝ β✝ : Type u
m : Type u → Type u
inst✝⁴ : Applicative m
F : α✝ → m β✝
F✝ G✝ : Type u → Type u
inst✝³ : Applicative F✝
inst✝² : Applicative G✝
inst✝¹ : LawfulApplicative F✝
inst✝ : LawfulApplicative G✝
η : ApplicativeTransformation F✝ G✝
α β : Type u
f : α → F✝ β
x✝ : FreeSemigroup α
x : α
y : FreeSemigroup α
ih1 : (fun {α} ↦ η.app α) (traverse f (pure x)) = traverse ((fun {α} ↦ η.app α) ∘ f) (pure x)
ih2 : (fun {α} ↦ η.app α) (traverse f y) = traverse ((fun {α} ↦ η.app α) ∘ f) y
⊢ (fun {α} ↦ η.app α) (traverse f (pure x * y)) = traverse ((fun {α} ↦ η.app α) ∘ f) (pure x * y)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
a : α
this : DecidableEq α := Classical.decEq α
⊢ toWord 1 ≠ (of a).toWord

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
x : FreeGroup α
⊢ x⁻¹.norm = x.norm

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α : Type u
β : Type v
f : α → β
⊢ toFreeSemigroup.comp (map f) = (FreeSemigroup.map f).comp toFreeSemigroup

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
x : FreeGroup α
⊢ x.norm = 0 ↔ x = 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
x y : FreeGroup α
⊢ (x * y).norm = (mk (x.toWord ++ y.toWord)).norm

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
a : α
n : ℕ
⊢ (of a ^ n).norm = n

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Nonempty α
⊢ Function.Surjective norm

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α : Type u
x✝ x y : FreeMagma α
hx : (toFreeSemigroup x).length = x.length
hy : (toFreeSemigroup y).length = y.length
⊢ (toFreeSemigroup (x * y)).length = (x * y).length

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α : Type u
⊢ (FreeSemigroup.lift (⇑Magma.AssocQuotient.of ∘ FreeMagma.of)).comp
      (Magma.AssocQuotient.lift FreeMagma.toFreeSemigroup) =
    MulHom.id (Magma.AssocQuotient (FreeMagma α))

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Free

Goal state before:
pp: α : Type u
⊢ (Magma.AssocQuotient.lift FreeMagma.toFreeSemigroup).comp
      (FreeSemigroup.lift (⇑Magma.AssocQuotient.of ∘ FreeMagma.of)) =
    MulHom.id (FreeSemigroup α)

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Free

Goal state before:
pp: α : Type u
inst✝ : Nonempty α
⊢ Infinite (FreeGroup α)

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Free

Goal state before:
pp: case a.inl
α : Type u
inst✝ : Nonempty α
hα : #α < ℵ₀
⊢ max (#α * 2) ℵ₀ = max #α ℵ₀

Tactic:
simp only [hα.le, max_eq_right, max_eq_right_iff]

Goal state after:
case a.inl
α : Type u
inst✝ : Nonempty α
hα : #α < ℵ₀
⊢ #α * 2 ≤ ℵ₀
Module: Mathlib.SetTheory.Cardinal.Free

Goal state before:
pp: α : Type u
inst✝ : Nonempty α
hα : ℵ₀ ≤ #α
⊢ 2 ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Free

Goal state before:
pp: α : Type u
inst✝ : Nonempty α
⊢ #(FreeGroup α) = max #α ℵ₀

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.FreeAlgebra.Cardinality

Goal state before:
pp: R : Type u
inst✝² : CommSemiring R
X : Type v
inst✝¹ : Nonempty X
inst✝ : Nontrivial R
⊢ #(FreeAlgebra R X) = max (max (Cardinal.lift.{v, u} #R) (Cardinal.lift.{u, v} #X)) ℵ₀

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.FreeAlgebra.Cardinality

Goal state before:
pp: R : Type u
inst✝¹ : CommSemiring R
X : Type v
inst✝ : IsEmpty X
⊢ #(FreeAlgebra R X) = Cardinal.lift.{v, u} #R

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.FreeAlgebra.Cardinality

Goal state before:
pp: R : Type u
inst✝¹ : CommSemiring R
X : Type v
inst✝ : Subsingleton R
⊢ #(FreeAlgebra R X) = 1

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Free

Goal state before:
pp: α : Type u
inst✝ : Nonempty α
⊢ #(FreeAbelianGroup α) = max #α ℵ₀

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.FreeAlgebra.Cardinality

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
X : Type v
⊢ #(FreeAlgebra R X) ≤ max (max (Cardinal.lift.{v, u} #R) (Cardinal.lift.{u, v} #X)) ℵ₀

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.FiveLemma

Goal state before:
pp: M₁ : Type u_1
M₂ : Type u_2
M₃ : Type u_3
M₄ : Type u_4
N₁ : Type u_6
N₂ : Type u_7
N₃ : Type u_8
N₄ : Type u_9
inst✝⁷ : Group M₁
inst✝⁶ : Group M₂
inst✝⁵ : Group M₃
inst✝⁴ : Group M₄
inst✝³ : Group N₁
inst✝² : Group N₂
inst✝¹ : Group N₃
inst✝ : Group N₄
f₁ : M₁ →* M₂
f₂ : M₂ →* M₃
f₃ : M₃ →* M₄
g₁ : N₁ →* N₂
g₂ : N₂ →* N₃
g₃ : N₃ →* N₄
i₁ : M₁ →* N₁
i₂ : M₂ →* N₂
i₃ : M₃ →* N₃
i₄ : M₄ →* N₄
hc₁ : g₁.comp i₁ = i₂.comp f₁
hc₂ : g₂.comp i₂ = i₃.comp f₂
hc₃ : g₃.comp i₃ = i₄.comp f₃
hf₁ : Function.MulExact ⇑f₁ ⇑f₂
hf₂ : Function.MulExact ⇑f₂ ⇑f₃
hg₁ : Function.MulExact ⇑g₁ ⇑g₂
hi₁ : Function.Surjective ⇑i₁
hi₂ : Function.Injective ⇑i₂
hi₄ : Function.Injective ⇑i₄
⊢ Function.Injective ⇑i₃

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.FreeAlgebra.Cardinality

Goal state before:
pp: R : Type u
inst✝² : CommSemiring R
X : Type u
inst✝¹ : Nonempty X
inst✝ : Nontrivial R
⊢ #(FreeAlgebra R X) = max (max #R #X) ℵ₀

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Free

Goal state before:
pp: α : Type u
⊢ #(FreeRing α) = max #α ℵ₀

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.FiveLemma

Goal state before:
pp: M₁ : Type u_1
M₂ : Type u_2
M₃ : Type u_3
N₁ : Type u_6
N₂ : Type u_7
N₃ : Type u_8
inst✝⁵ : Group M₁
inst✝⁴ : Group M₂
inst✝³ : Group M₃
inst✝² : Group N₁
inst✝¹ : Group N₂
inst✝ : Group N₃
f₁ : M₁ →* M₂
f₂ : M₂ →* M₃
g₁ : N₁ →* N₂
g₂ : N₂ →* N₃
i₁ : M₁ →* N₁
i₂ : M₂ →* N₂
i₃ : M₃ →* N₃
hc₁ : g₁.comp i₁ = i₂.comp f₁
hc₂ : g₂.comp i₂ = i₃.comp f₂
hf₁ : Function.MulExact ⇑f₁ ⇑f₂
hg₁ : Function.MulExact ⇑g₁ ⇑g₂
hi₁ : Function.Surjective ⇑i₁
hi₂ : Function.Injective ⇑i₂
hf₂ : Function.Surjective ⇑f₂
⊢ comp 1 i₃ = comp 1 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.FreeAlgebra.Cardinality

Goal state before:
pp: R : Type u
inst✝¹ : CommSemiring R
X : Type u
inst✝ : IsEmpty X
⊢ #(FreeAlgebra R X) = #R

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.FreeAlgebra.Cardinality

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
X : Type u
⊢ #(FreeAlgebra R X) ≤ max (max #R #X) ℵ₀

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.FiveLemma

Goal state before:
pp: M₁ : Type u_1
M₂ : Type u_2
M₃ : Type u_3
N₁ : Type u_6
N₂ : Type u_7
N₃ : Type u_8
inst✝⁵ : Group M₁
inst✝⁴ : Group M₂
inst✝³ : Group M₃
inst✝² : Group N₁
inst✝¹ : Group N₂
inst✝ : Group N₃
f₁ : M₁ →* M₂
f₂ : M₂ →* M₃
g₁ : N₁ →* N₂
g₂ : N₂ →* N₃
i₁ : M₁ →* N₁
i₂ : M₂ →* N₂
i₃ : M₃ →* N₃
hc₁ : g₁.comp i₁ = i₂.comp f₁
hc₂ : g₂.comp i₂ = i₃.comp f₂
hf₁ : Function.MulExact ⇑f₁ ⇑f₂
hg₁ : Function.MulExact ⇑g₁ ⇑g₂
hi₁ : Function.Surjective ⇑i₁
hi₂ : Function.Injective ⇑i₂
hf₂ : Function.Surjective ⇑f₂
y : M₃
⊢ 1 y = 1 ↔ y ∈ Set.range ⇑f₂

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Free

Goal state before:
pp: α : Type u
⊢ #(FreeCommRing α) = max #α ℵ₀

Tactic:
by

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Free

Goal state before:
pp: α : Type u
inst✝ : Nonempty α
hR : Finite α
x : Fintype α
⊢ Fintype.card α ≠ 0

Tactic:
by

Goal state after: