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

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
c₁ c₂ c₃ : R
q : Basis A c₁ c₂ c₃
⊢ q.i * (q.i * q.j) = c₁ • q.j + c₂ • (q.i * q.j)

Tactic:
← mul_assoc,

Goal state after:
R : Type u_1
A : Type u_2
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
c₁ c₂ c₃ : R
q : Basis A c₁ c₂ c₃
⊢ q.i * q.i * q.j = c₁ • q.j + c₂ • (q.i * q.j)
Module: Mathlib.Algebra.QuaternionBasis

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
c₁ c₂ c₃ : R
q : Basis A c₁ c₂ c₃
⊢ q.j * (q.i * q.j) = (c₂ * c₃) • 1 - c₃ • q.i

Tactic:
← mul_assoc,

Goal state after:
R : Type u_1
A : Type u_2
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
c₁ c₂ c₃ : R
q : Basis A c₁ c₂ c₃
⊢ q.j * q.i * q.j = (c₂ * c₃) • 1 - c₃ • q.i
Module: Mathlib.Algebra.QuaternionBasis

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
c₁ c₂ c₃ : R
q : Basis A c₁ c₂ c₃
⊢ q.i * ((c₂ • q.j - q.i * q.j) * q.j) = -((c₁ * c₃) • 1)

Tactic:
← mul_assoc,

Goal state after:
R : Type u_1
A : Type u_2
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
c₁ c₂ c₃ : R
q : Basis A c₁ c₂ c₃
⊢ q.i * (c₂ • q.j - q.i * q.j) * q.j = -((c₁ * c₃) • 1)
Module: Mathlib.Algebra.Polynomial.PartialFractions

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
K : Type u_2
inst✝² : CommRing K
inst✝¹ : Algebra R[X] K
inst✝ : FaithfulSMul R[X] K
ι : Type u_3
s : Finset ι
g : ι → R[X]
hg : ∀ i ∈ s, (g i).Monic
hgg : (↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)
n : ι → ℕ
gi : ι → K
hgi : ∀ i ∈ s, gi i * (algebraMap R[X] K) (g i) = 1
q₁ q₂ : R[X]
r₁ r₂ : (i : ι) → Fin (n i) → R[X]
hr₁ : ∀ i ∈ s, ∀ (j : Fin (n i)), (r₁ i j).degree < (g i).degree
hr₂ : ∀ i ∈ s, ∀ (j : Fin (n i)), (r₂ i j).degree < (g i).degree
hf :
  (algebraMap R[X] K) q₁ + ∑ i ∈ s, ∑ j, (algebraMap R[X] K) (r₁ i j) * gi i ^ (↑j + 1) =
    (algebraMap R[X] K) q₂ + ∑ i ∈ s, ∑ j, (algebraMap R[X] K) (r₂ i j) * gi i ^ (↑j + 1)
q : R[X]
r : (i : ι) → Fin (n i) → R[X]
⊢ (algebraMap R[X] K) q * ∏ i ∈ s, (algebraMap R[X] K) (g i) ^ n i +
      ∑ x ∈ s, ∑ i, (algebraMap R[X] K) (r x i) * gi x ^ (↑i + 1) * ∏ i ∈ s, (algebraMap R[X] K) (g i) ^ n i =
    (algebraMap R[X] K) (q * ∏ i ∈ s, g i ^ n i + ∑ i ∈ s, ∑ j, r i j.rev * g i ^ ↑j * ∏ i ∈ s.erase i, g i ^ n i)

Tactic:
map_add,

Goal state after:
R : Type u_1
inst✝³ : CommRing R
K : Type u_2
inst✝² : CommRing K
inst✝¹ : Algebra R[X] K
inst✝ : FaithfulSMul R[X] K
ι : Type u_3
s : Finset ι
g : ι → R[X]
hg : ∀ i ∈ s, (g i).Monic
hgg : (↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)
n : ι → ℕ
gi : ι → K
hgi : ∀ i ∈ s, gi i * (algebraMap R[X] K) (g i) = 1
q₁ q₂ : R[X]
r₁ r₂ : (i : ι) → Fin (n i) → R[X]
hr₁ : ∀ i ∈ s, ∀ (j : Fin (n i)), (r₁ i j).degree < (g i).degree
hr₂ : ∀ i ∈ s, ∀ (j : Fin (n i)), (r₂ i j).degree < (g i).degree
hf :
  (algebraMap R[X] K) q₁ + ∑ i ∈ s, ∑ j, (algebraMap R[X] K) (r₁ i j) * gi i ^ (↑j + 1) =
    (algebraMap R[X] K) q₂ + ∑ i ∈ s, ∑ j, (algebraMap R[X] K) (r₂ i j) * gi i ^ (↑j + 1)
q : R[X]
r : (i : ι) → Fin (n i) → R[X]
⊢ (algebraMap R[X] K) q * ∏ i ∈ s, (algebraMap R[X] K) (g i) ^ n i +
      ∑ x ∈ s, ∑ i, (algebraMap R[X] K) (r x i) * gi x ^ (↑i + 1) * ∏ i ∈ s, (algebraMap R[X] K) (g i) ^ n i =
    (algebraMap R[X] K) (q * ∏ i ∈ s, g i ^ n i) +
      (algebraMap R[X] K) (∑ i ∈ s, ∑ j, r i j.rev * g i ^ ↑j * ∏ i ∈ s.erase i, g i ^ n i)
Module: Mathlib.Algebra.Polynomial.PartialFractions

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
K : Type u_2
inst✝² : CommRing K
inst✝¹ : Algebra R[X] K
inst✝ : FaithfulSMul R[X] K
ι : Type u_3
s : Finset ι
g : ι → R[X]
hg : ∀ i ∈ s, (g i).Monic
hgg : (↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)
n : ι → ℕ
gi : ι → K
hgi : ∀ i ∈ s, gi i * (algebraMap R[X] K) (g i) = 1
q₁ q₂ : R[X]
r₁ r₂ : (i : ι) → Fin (n i) → R[X]
hr₁ : ∀ i ∈ s, ∀ (j : Fin (n i)), (r₁ i j).degree < (g i).degree
hr₂ : ∀ i ∈ s, ∀ (j : Fin (n i)), (r₂ i j).degree < (g i).degree
hf :
  (algebraMap R[X] K) q₁ + ∑ i ∈ s, ∑ j, (algebraMap R[X] K) (r₁ i j) * gi i ^ (↑j + 1) =
    (algebraMap R[X] K) q₂ + ∑ i ∈ s, ∑ j, (algebraMap R[X] K) (r₂ i j) * gi i ^ (↑j + 1)
q : R[X]
r : (i : ι) → Fin (n i) → R[X]
i : ι
hi : i ∈ s
j : Fin (n i)
⊢ (algebraMap R[X] K) (r i j.rev) * gi i ^ (↑j.rev + 1) *
      ((algebraMap R[X] K) (g i) ^ n i * ∏ x ∈ s.erase i, (algebraMap R[X] K) (g x) ^ n x) =
    (algebraMap R[X] K) (r i j.rev) * (algebraMap R[X] K) (g i) ^ ↑j * ∏ x ∈ s.erase i, (algebraMap R[X] K) (g x) ^ n x

Tactic:
← mul_assoc,

Goal state after:
R : Type u_1
inst✝³ : CommRing R
K : Type u_2
inst✝² : CommRing K
inst✝¹ : Algebra R[X] K
inst✝ : FaithfulSMul R[X] K
ι : Type u_3
s : Finset ι
g : ι → R[X]
hg : ∀ i ∈ s, (g i).Monic
hgg : (↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)
n : ι → ℕ
gi : ι → K
hgi : ∀ i ∈ s, gi i * (algebraMap R[X] K) (g i) = 1
q₁ q₂ : R[X]
r₁ r₂ : (i : ι) → Fin (n i) → R[X]
hr₁ : ∀ i ∈ s, ∀ (j : Fin (n i)), (r₁ i j).degree < (g i).degree
hr₂ : ∀ i ∈ s, ∀ (j : Fin (n i)), (r₂ i j).degree < (g i).degree
hf :
  (algebraMap R[X] K) q₁ + ∑ i ∈ s, ∑ j, (algebraMap R[X] K) (r₁ i j) * gi i ^ (↑j + 1) =
    (algebraMap R[X] K) q₂ + ∑ i ∈ s, ∑ j, (algebraMap R[X] K) (r₂ i j) * gi i ^ (↑j + 1)
q : R[X]
r : (i : ι) → Fin (n i) → R[X]
i : ι
hi : i ∈ s
j : Fin (n i)
⊢ (algebraMap R[X] K) (r i j.rev) * gi i ^ (↑j.rev + 1) * (algebraMap R[X] K) (g i) ^ n i *
      ∏ x ∈ s.erase i, (algebraMap R[X] K) (g x) ^ n x =
    (algebraMap R[X] K) (r i j.rev) * (algebraMap R[X] K) (g i) ^ ↑j * ∏ x ∈ s.erase i, (algebraMap R[X] K) (g x) ^ n x
Module: Mathlib.Algebra.Polynomial.PartialFractions

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
K : Type u_2
inst✝² : CommRing K
inst✝¹ : Algebra R[X] K
inst✝ : FaithfulSMul R[X] K
ι : Type u_3
s : Finset ι
g : ι → R[X]
hg : ∀ i ∈ s, (g i).Monic
hgg : (↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)
n : ι → ℕ
gi : ι → K
hgi : ∀ i ∈ s, gi i * (algebraMap R[X] K) (g i) = 1
q₁ q₂ : R[X]
r₁ r₂ : (i : ι) → Fin (n i) → R[X]
hr₁ : ∀ i ∈ s, ∀ (j : Fin (n i)), (r₁ i j).degree < (g i).degree
hr₂ : ∀ i ∈ s, ∀ (j : Fin (n i)), (r₂ i j).degree < (g i).degree
hf :
  (algebraMap R[X] K) q₁ + ∑ i ∈ s, ∑ j, (algebraMap R[X] K) (r₁ i j) * gi i ^ (↑j + 1) =
    (algebraMap R[X] K) q₂ + ∑ i ∈ s, ∑ j, (algebraMap R[X] K) (r₂ i j) * gi i ^ (↑j + 1)
q : R[X]
r : (i : ι) → Fin (n i) → R[X]
i : ι
hi : i ∈ s
j : Fin (n i)
⊢ gi i ^ (↑j.rev + 1) * (gi i ^ ↑j * (algebraMap R[X] K) (g i) ^ ↑j) * (algebraMap R[X] K) (g i) ^ n i =
    (algebraMap R[X] K) (g i) ^ ↑j

Tactic:
← mul_assoc,

Goal state after:
R : Type u_1
inst✝³ : CommRing R
K : Type u_2
inst✝² : CommRing K
inst✝¹ : Algebra R[X] K
inst✝ : FaithfulSMul R[X] K
ι : Type u_3
s : Finset ι
g : ι → R[X]
hg : ∀ i ∈ s, (g i).Monic
hgg : (↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)
n : ι → ℕ
gi : ι → K
hgi : ∀ i ∈ s, gi i * (algebraMap R[X] K) (g i) = 1
q₁ q₂ : R[X]
r₁ r₂ : (i : ι) → Fin (n i) → R[X]
hr₁ : ∀ i ∈ s, ∀ (j : Fin (n i)), (r₁ i j).degree < (g i).degree
hr₂ : ∀ i ∈ s, ∀ (j : Fin (n i)), (r₂ i j).degree < (g i).degree
hf :
  (algebraMap R[X] K) q₁ + ∑ i ∈ s, ∑ j, (algebraMap R[X] K) (r₁ i j) * gi i ^ (↑j + 1) =
    (algebraMap R[X] K) q₂ + ∑ i ∈ s, ∑ j, (algebraMap R[X] K) (r₂ i j) * gi i ^ (↑j + 1)
q : R[X]
r : (i : ι) → Fin (n i) → R[X]
i : ι
hi : i ∈ s
j : Fin (n i)
⊢ gi i ^ (↑j.rev + 1) * gi i ^ ↑j * (algebraMap R[X] K) (g i) ^ ↑j * (algebraMap R[X] K) (g i) ^ n i =
    (algebraMap R[X] K) (g i) ^ ↑j
Module: Mathlib.Algebra.QuaternionBasis

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : Ring A
inst✝² : Ring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
c₁ c₂ c₃ : R
q : Basis A c₁ c₂ c₃
F : A →ₐ[R] B
⊢ F (c₁ • 1 + c₂ • q.i) = c₁ • 1 + c₂ • F q.i

Tactic:
map_add,

Goal state after:
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : Ring A
inst✝² : Ring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
c₁ c₂ c₃ : R
q : Basis A c₁ c₂ c₃
F : A →ₐ[R] B
⊢ F (c₁ • 1) + F (c₂ • q.i) = c₁ • 1 + c₂ • F q.i
Module: Mathlib.Algebra.SkewMonoidAlgebra.Support

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝³ : Monoid G
inst✝² : Semiring k
inst✝¹ : MulSemiringAction G k
f : SkewMonoidAlgebra k G
inst✝ : DecidableEq G
r : k
a : G
⊢ image₂ (fun x1 x2 ↦ x1 * x2) {a} f.support ⊆ image (fun x ↦ a * x) f.support

Tactic:
rw [Finset.image₂_singleton_left]

Goal state after:
Module: Mathlib.Algebra.RingQuot

Goal state before:
pp: R : Type uR
inst✝ : Semiring R
r : R → R → Prop
⊢ Relation.EqvGen (Rel r) = RingConGen.Rel r

Tactic:
ext x₁ x₂

Goal state after:
R : Type uR
inst✝ : Semiring R
r : R → R → Prop
x₁ x₂ : R
⊢ Relation.EqvGen (Rel r) x₁ x₂ ↔ RingConGen.Rel r x₁ x₂
Module: Mathlib.Algebra.RingQuot

Goal state before:
pp: R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r✝ r : R → R → Prop
⊢ ∀ (x : RingQuot r), 0 • x = 0

Tactic:
rintro ⟨⟨⟩⟩
    simp only [smul_quot, zero_smul, zero_quot]

Goal state after:
Module: Mathlib.Algebra.RingQuot

Goal state before:
pp: case add_left
R : Type uR
inst✝⁴ : Semiring R
S : Type uS
inst✝³ : CommSemiring S
T : Type uT
A : Type uA
inst✝² : Semiring A
inst✝¹ : Algebra S A
r✝ : R → R → Prop
inst✝ : Semiring T
r : R → R → Prop
f : R →+* T
h : ∀ ⦃x y : R⦄, r x y → f x = f y
x : RingQuot r
a✝² b✝¹ a✝¹ b✝ c✝ : R
a✝ : Rel r a✝¹ b✝
r' : f a✝¹ = f b✝
⊢ f (a✝¹ + c✝) = f (b✝ + c✝)

Tactic:
map_add,

Goal state after:
case add_left
R : Type uR
inst✝⁴ : Semiring R
S : Type uS
inst✝³ : CommSemiring S
T : Type uT
A : Type uA
inst✝² : Semiring A
inst✝¹ : Algebra S A
r✝ : R → R → Prop
inst✝ : Semiring T
r : R → R → Prop
f : R →+* T
h : ∀ ⦃x y : R⦄, r x y → f x = f y
x : RingQuot r
a✝² b✝¹ a✝¹ b✝ c✝ : R
a✝ : Rel r a✝¹ b✝
r' : f a✝¹ = f b✝
⊢ f a✝¹ + f c✝ = f (b✝ + c✝)
Module: Mathlib.Algebra.RingQuot

Goal state before:
pp: case add_left
R : Type uR
inst✝⁴ : Semiring R
S : Type uS
inst✝³ : CommSemiring S
T : Type uT
A : Type uA
inst✝² : Semiring A
inst✝¹ : Algebra S A
r✝ : R → R → Prop
inst✝ : Semiring T
r : R → R → Prop
f : R →+* T
h : ∀ ⦃x y : R⦄, r x y → f x = f y
x : RingQuot r
a✝² b✝¹ a✝¹ b✝ c✝ : R
a✝ : Rel r a✝¹ b✝
r' : f a✝¹ = f b✝
⊢ f a✝¹ + f c✝ = f (b✝ + c✝)

Tactic:
map_add,

Goal state after:
case add_left
R : Type uR
inst✝⁴ : Semiring R
S : Type uS
inst✝³ : CommSemiring S
T : Type uT
A : Type uA
inst✝² : Semiring A
inst✝¹ : Algebra S A
r✝ : R → R → Prop
inst✝ : Semiring T
r : R → R → Prop
f : R →+* T
h : ∀ ⦃x y : R⦄, r x y → f x = f y
x : RingQuot r
a✝² b✝¹ a✝¹ b✝ c✝ : R
a✝ : Rel r a✝¹ b✝
r' : f a✝¹ = f b✝
⊢ f a✝¹ + f c✝ = f b✝ + f c✝
Module: Mathlib.Algebra.RingQuot

Goal state before:
pp: case refine_1
R : Type uR
inst✝⁵ : Semiring R
S : Type uS
inst✝⁴ : CommSemiring S
T : Type uT
A : Type uA
inst✝³ : Semiring A
inst✝² : Algebra S A
r✝ : R → R → Prop
inst✝¹ : Semiring T
B : Type uR
inst✝ : CommRing B
r : B → B → Prop
x : B
h✝ : x ∈ Ideal.ofRel r
y a b : B
h : r a b
su : a = y + b
⊢ (mkRingHom r) y = 0

Tactic:
rw [← sub_eq_iff_eq_add] at su

Goal state after:
case refine_1
R : Type uR
inst✝⁵ : Semiring R
S : Type uS
inst✝⁴ : CommSemiring S
T : Type uT
A : Type uA
inst✝³ : Semiring A
inst✝² : Algebra S A
r✝ : R → R → Prop
inst✝¹ : Semiring T
B : Type uR
inst✝ : CommRing B
r : B → B → Prop
x : B
h✝ : x ∈ Ideal.ofRel r
y a b : B
h : r a b
su : a - b = y
⊢ (mkRingHom r) y = 0
Module: Mathlib.Algebra.SkewPolynomial.Basic

Goal state before:
pp: case succ
R : Type u_1
inst✝¹ : Semiring R
inst✝ : MulSemiringAction (Multiplicative ℕ) R
n : ℕ
ih : ∀ {p : SkewPolynomial R}, X ^ n * p = (p.sum fun a b ↦ (monomial a) ((⇑φ)^[n] b)) * X ^ n
p : SkewPolynomial R
⊢ X ^ n * ((p.sum fun a b ↦ (monomial a) (φ b)) * X) = (p.sum fun a b ↦ (monomial a) ((⇑φ)^[n + 1] b)) * X ^ (n + 1)

Tactic:
← mul_assoc,

Goal state after:
case succ
R : Type u_1
inst✝¹ : Semiring R
inst✝ : MulSemiringAction (Multiplicative ℕ) R
n : ℕ
ih : ∀ {p : SkewPolynomial R}, X ^ n * p = (p.sum fun a b ↦ (monomial a) ((⇑φ)^[n] b)) * X ^ n
p : SkewPolynomial R
⊢ (X ^ n * p.sum fun a b ↦ (monomial a) (φ b)) * X = (p.sum fun a b ↦ (monomial a) ((⇑φ)^[n + 1] b)) * X ^ (n + 1)
Module: Mathlib.Algebra.SkewPolynomial.Basic

Goal state before:
pp: R : Type u_1
n : ℕ
inst✝ : Semiring R
a : R
⊢ (C a).coeff n = if n = 0 then a else 0

Tactic:
convert! coeff_monomial using 2

Goal state after:
case e'_3.h₁
R : Type u_1
n : ℕ
inst✝ : Semiring R
a : R
⊢ n = 0 ↔ 0 = n
Module: Mathlib.Algebra.SkewPolynomial.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
a : R
inst✝ : MulSemiringAction (Multiplicative ℕ) R
n : ℕ
⊢ C a * (X ^ n * X) = (monomial (n + 1)) a

Tactic:
← mul_assoc,

Goal state after:
R : Type u_1
inst✝¹ : Semiring R
a : R
inst✝ : MulSemiringAction (Multiplicative ℕ) R
n : ℕ
⊢ C a * X ^ n * X = (monomial (n + 1)) a
Module: Mathlib.Algebra.Star.UnitaryStarAlgAut

Goal state before:
pp: R : Type u_2
inst✝⁴ : Semiring R
inst✝³ : StarMul R
S : Type u_3
inst✝² : CommSemiring S
inst✝¹ : Algebra S R
inst✝ : Algebra.IsCentral S R
u v : ↥(unitary R)
⊢ (∀ (a : R), a * ↑(toUnits v)⁻¹ = ↑(toUnits v)⁻¹ * (↑(toUnits u) * (a * ↑(toUnits u)⁻¹))) ↔
    ∃ α, ↑(toUnits u) = α • ↑(toUnits v)

Tactic:
← mul_assoc,

Goal state after:
R : Type u_2
inst✝⁴ : Semiring R
inst✝³ : StarMul R
S : Type u_3
inst✝² : CommSemiring S
inst✝¹ : Algebra S R
inst✝ : Algebra.IsCentral S R
u v : ↥(unitary R)
⊢ (∀ (a : R), a * ↑(toUnits v)⁻¹ = ↑(toUnits v)⁻¹ * ↑(toUnits u) * a * ↑(toUnits u)⁻¹) ↔
    ∃ α, ↑(toUnits u) = α • ↑(toUnits v)
Module: Mathlib.Algebra.Star.UnitaryStarAlgAut

Goal state before:
pp: R : Type u_3
S : Type u_4
inst✝⁸ : Ring R
inst✝⁷ : StarMul R
inst✝⁶ : CommRing S
inst✝⁵ : StarMul S
inst✝⁴ : Algebra S R
inst✝³ : StarModule S R
inst✝² : Algebra.IsCentral S R
inst✝¹ : IsCancelMulZero S
inst✝ : Module.IsTorsionFree S R
u v : ↥(unitary R)
⊢ (∀ (a : R), a * ↑(toUnits v)⁻¹ = ↑(toUnits v)⁻¹ * (↑(toUnits u) * (a * ↑(toUnits u)⁻¹))) ↔ ∃ α, u = α • v

Tactic:
← mul_assoc,

Goal state after:
R : Type u_3
S : Type u_4
inst✝⁸ : Ring R
inst✝⁷ : StarMul R
inst✝⁶ : CommRing S
inst✝⁵ : StarMul S
inst✝⁴ : Algebra S R
inst✝³ : StarModule S R
inst✝² : Algebra.IsCentral S R
inst✝¹ : IsCancelMulZero S
inst✝ : Module.IsTorsionFree S R
u v : ↥(unitary R)
⊢ (∀ (a : R), a * ↑(toUnits v)⁻¹ = ↑(toUnits v)⁻¹ * ↑(toUnits u) * a * ↑(toUnits u)⁻¹) ↔ ∃ α, u = α • v
Module: Mathlib.Algebra.Symmetrized

Goal state before:
pp: case calc_1
α : Type u_1
inst✝¹ : Ring α
inst✝ : Invertible 2
a b : αˢʸᵐ
commute_half_left : ∀ (a : α), ⅟2 * a = a * ⅟2
⊢ sym
      (⅟2 *
        (⅟2 * (unsym a * unsym b + unsym b * unsym a) * unsym (a * a) +
          unsym (a * a) * (⅟2 * (unsym a * unsym b + unsym b * unsym a)))) =
    sym
      (⅟2 * ⅟2 *
        (unsym a * unsym b * unsym (a * a) + unsym b * unsym a * unsym (a * a) + unsym (a * a) * unsym a * unsym b +
          unsym (a * a) * unsym b * unsym a))

Tactic:
← mul_assoc,

Goal state after:
case calc_1
α : Type u_1
inst✝¹ : Ring α
inst✝ : Invertible 2
a b : αˢʸᵐ
commute_half_left : ∀ (a : α), ⅟2 * a = a * ⅟2
⊢ sym
      (⅟2 *
        (⅟2 * (unsym a * unsym b + unsym b * unsym a) * unsym (a * a) +
          unsym (a * a) * ⅟2 * (unsym a * unsym b + unsym b * unsym a))) =
    sym
      (⅟2 * ⅟2 *
        (unsym a * unsym b * unsym (a * a) + unsym b * unsym a * unsym (a * a) + unsym (a * a) * unsym a * unsym b +
          unsym (a * a) * unsym b * unsym a))
Module: Mathlib.Algebra.Symmetrized

Goal state before:
pp: case calc_1
α : Type u_1
inst✝¹ : Ring α
inst✝ : Invertible 2
a b : αˢʸᵐ
commute_half_left : ∀ (a : α), ⅟2 * a = a * ⅟2
⊢ sym
      (⅟2 *
        (⅟2 *
          ((unsym a * unsym b + unsym b * unsym a) * unsym (a * a) +
            unsym (a * a) * (unsym a * unsym b + unsym b * unsym a)))) =
    sym
      (⅟2 * ⅟2 *
        (unsym a * unsym b * unsym (a * a) + unsym b * unsym a * unsym (a * a) + unsym (a * a) * unsym a * unsym b +
          unsym (a * a) * unsym b * unsym a))

Tactic:
← mul_assoc,

Goal state after:
case calc_1
α : Type u_1
inst✝¹ : Ring α
inst✝ : Invertible 2
a b : αˢʸᵐ
commute_half_left : ∀ (a : α), ⅟2 * a = a * ⅟2
⊢ sym
      (⅟2 * ⅟2 *
        ((unsym a * unsym b + unsym b * unsym a) * unsym (a * a) +
          unsym (a * a) * (unsym a * unsym b + unsym b * unsym a))) =
    sym
      (⅟2 * ⅟2 *
        (unsym a * unsym b * unsym (a * a) + unsym b * unsym a * unsym (a * a) + unsym (a * a) * unsym a * unsym b +
          unsym (a * a) * unsym b * unsym a))
Module: Mathlib.Algebra.Symmetrized

Goal state before:
pp: case calc_1
α : Type u_1
inst✝¹ : Ring α
inst✝ : Invertible 2
a b : αˢʸᵐ
commute_half_left : ∀ (a : α), ⅟2 * a = a * ⅟2
⊢ sym
      (⅟2 * ⅟2 *
        (unsym a * unsym b * unsym (a * a) + unsym b * unsym a * unsym (a * a) + unsym (a * a) * (unsym a * unsym b) +
          unsym (a * a) * (unsym b * unsym a))) =
    sym
      (⅟2 * ⅟2 *
        (unsym a * unsym b * unsym (a * a) + unsym b * unsym a * unsym (a * a) + unsym (a * a) * unsym a * unsym b +
          unsym (a * a) * unsym b * unsym a))

Tactic:
← mul_assoc,

Goal state after:
case calc_1
α : Type u_1
inst✝¹ : Ring α
inst✝ : Invertible 2
a b : αˢʸᵐ
commute_half_left : ∀ (a : α), ⅟2 * a = a * ⅟2
⊢ sym
      (⅟2 * ⅟2 *
        (unsym a * unsym b * unsym (a * a) + unsym b * unsym a * unsym (a * a) + unsym (a * a) * unsym a * unsym b +
          unsym (a * a) * (unsym b * unsym a))) =
    sym
      (⅟2 * ⅟2 *
        (unsym a * unsym b * unsym (a * a) + unsym b * unsym a * unsym (a * a) + unsym (a * a) * unsym a * unsym b +
          unsym (a * a) * unsym b * unsym a))
Module: Mathlib.Algebra.Symmetrized

Goal state before:
pp: case calc_2
α : Type u_1
inst✝¹ : Ring α
inst✝ : Invertible 2
a b : αˢʸᵐ
commute_half_left : ∀ (a : α), ⅟2 * a = a * ⅟2
⊢ ⅟2 * ⅟2 *
      (unsym a * unsym b * unsym (a * a) + unsym b * unsym a * unsym (a * a) + unsym (a * a) * unsym a * unsym b +
        unsym (a * a) * unsym b * unsym a) =
    ⅟2 *
      (unsym a * (⅟2 * (unsym b * unsym (a * a) + unsym (a * a) * unsym b)) +
        ⅟2 * (unsym b * unsym (a * a) + unsym (a * a) * unsym b) * unsym a)

Tactic:
← mul_assoc,

Goal state after:
case calc_2
α : Type u_1
inst✝¹ : Ring α
inst✝ : Invertible 2
a b : αˢʸᵐ
commute_half_left : ∀ (a : α), ⅟2 * a = a * ⅟2
⊢ ⅟2 * ⅟2 *
      (unsym a * unsym b * unsym (a * a) + unsym b * unsym a * unsym (a * a) + unsym (a * a) * unsym a * unsym b +
        unsym (a * a) * unsym b * unsym a) =
    ⅟2 *
      (unsym a * ⅟2 * (unsym b * unsym (a * a) + unsym (a * a) * unsym b) +
        ⅟2 * (unsym b * unsym (a * a) + unsym (a * a) * unsym b) * unsym a)
Module: Mathlib.Algebra.Symmetrized

Goal state before:
pp: case calc_2
α : Type u_1
inst✝¹ : Ring α
inst✝ : Invertible 2
a b : αˢʸᵐ
commute_half_left : ∀ (a : α), ⅟2 * a = a * ⅟2
⊢ ⅟2 * ⅟2 *
      (unsym a * unsym b * unsym (a * a) + unsym b * unsym a * unsym (a * a) + unsym (a * a) * unsym a * unsym b +
        unsym (a * a) * unsym b * unsym a) =
    ⅟2 * ⅟2 *
      (unsym a * (unsym b * unsym (a * a)) + unsym a * (unsym (a * a) * unsym b) + unsym b * unsym (a * a) * unsym a +
        unsym (a * a) * unsym b * unsym a)

Tactic:
← mul_assoc,

Goal state after:
case calc_2
α : Type u_1
inst✝¹ : Ring α
inst✝ : Invertible 2
a b : αˢʸᵐ
commute_half_left : ∀ (a : α), ⅟2 * a = a * ⅟2
⊢ ⅟2 * ⅟2 *
      (unsym a * unsym b * unsym (a * a) + unsym b * unsym a * unsym (a * a) + unsym (a * a) * unsym a * unsym b +
        unsym (a * a) * unsym b * unsym a) =
    ⅟2 * ⅟2 *
      (unsym a * unsym b * unsym (a * a) + unsym a * (unsym (a * a) * unsym b) + unsym b * unsym (a * a) * unsym a +
        unsym (a * a) * unsym b * unsym a)
Module: Mathlib.Algebra.Symmetrized

Goal state before:
pp: case calc_2
α : Type u_1
inst✝¹ : Ring α
inst✝ : Invertible 2
a b : αˢʸᵐ
commute_half_left : ∀ (a : α), ⅟2 * a = a * ⅟2
⊢ ⅟2 * ⅟2 *
      (unsym a * unsym b * (unsym a * unsym a) + unsym b * unsym a * (unsym a * unsym a) +
          unsym a * unsym a * unsym a * unsym b +
        unsym a * unsym a * unsym b * unsym a) =
    ⅟2 * ⅟2 *
      (unsym a * unsym b * (unsym a * unsym a) + unsym a * (unsym a * unsym a) * unsym b +
          unsym b * (unsym a * unsym a) * unsym a +
        unsym a * unsym a * unsym b * unsym a)

Tactic:
← mul_assoc,

Goal state after:
case calc_2
α : Type u_1
inst✝¹ : Ring α
inst✝ : Invertible 2
a b : αˢʸᵐ
commute_half_left : ∀ (a : α), ⅟2 * a = a * ⅟2
⊢ ⅟2 * ⅟2 *
      (unsym a * unsym b * unsym a * unsym a + unsym b * unsym a * (unsym a * unsym a) +
          unsym a * unsym a * unsym a * unsym b +
        unsym a * unsym a * unsym b * unsym a) =
    ⅟2 * ⅟2 *
      (unsym a * unsym b * unsym a * unsym a + unsym a * (unsym a * unsym a) * unsym b +
          unsym b * (unsym a * unsym a) * unsym a +
        unsym a * unsym a * unsym b * unsym a)
Module: Mathlib.Algebra.Symmetrized

Goal state before:
pp: case calc_2
α : Type u_1
inst✝¹ : Ring α
inst✝ : Invertible 2
a b : αˢʸᵐ
commute_half_left : ∀ (a : α), ⅟2 * a = a * ⅟2
⊢ ⅟2 * ⅟2 *
      (unsym a * unsym b * unsym a * unsym a + unsym b * unsym a * (unsym a * unsym a) +
          unsym a * unsym a * unsym a * unsym b +
        unsym a * unsym a * unsym b * unsym a) =
    ⅟2 * ⅟2 *
      (unsym a * unsym b * unsym a * unsym a + unsym a * (unsym a * unsym a) * unsym b +
          unsym b * (unsym a * unsym a) * unsym a +
        unsym a * unsym a * unsym b * unsym a)

Tactic:
← mul_assoc,

Goal state after:
case calc_2
α : Type u_1
inst✝¹ : Ring α
inst✝ : Invertible 2
a b : αˢʸᵐ
commute_half_left : ∀ (a : α), ⅟2 * a = a * ⅟2
⊢ ⅟2 * ⅟2 *
      (unsym a * unsym b * unsym a * unsym a + unsym b * unsym a * unsym a * unsym a +
          unsym a * unsym a * unsym a * unsym b +
        unsym a * unsym a * unsym b * unsym a) =
    ⅟2 * ⅟2 *
      (unsym a * unsym b * unsym a * unsym a + unsym a * (unsym a * unsym a) * unsym b +
          unsym b * (unsym a * unsym a) * unsym a +
        unsym a * unsym a * unsym b * unsym a)
Module: Mathlib.Algebra.Symmetrized

Goal state before:
pp: case calc_2
α : Type u_1
inst✝¹ : Ring α
inst✝ : Invertible 2
a b : αˢʸᵐ
commute_half_left : ∀ (a : α), ⅟2 * a = a * ⅟2
⊢ ⅟2 * ⅟2 *
      (unsym a * unsym b * unsym a * unsym a + unsym b * unsym a * unsym a * unsym a +
          unsym a * unsym a * unsym a * unsym b +
        unsym a * unsym a * unsym b * unsym a) =
    ⅟2 * ⅟2 *
      (unsym a * unsym b * unsym a * unsym a + unsym a * (unsym a * unsym a) * unsym b +
          unsym b * (unsym a * unsym a) * unsym a +
        unsym a * unsym a * unsym b * unsym a)

Tactic:
← mul_assoc,

Goal state after:
case calc_2
α : Type u_1
inst✝¹ : Ring α
inst✝ : Invertible 2
a b : αˢʸᵐ
commute_half_left : ∀ (a : α), ⅟2 * a = a * ⅟2
⊢ ⅟2 * ⅟2 *
      (unsym a * unsym b * unsym a * unsym a + unsym b * unsym a * unsym a * unsym a +
          unsym a * unsym a * unsym a * unsym b +
        unsym a * unsym a * unsym b * unsym a) =
    ⅟2 * ⅟2 *
      (unsym a * unsym b * unsym a * unsym a + unsym a * unsym a * unsym a * unsym b +
          unsym b * (unsym a * unsym a) * unsym a +
        unsym a * unsym a * unsym b * unsym a)
Module: Mathlib.Algebra.Symmetrized

Goal state before:
pp: case calc_2
α : Type u_1
inst✝¹ : Ring α
inst✝ : Invertible 2
a b : αˢʸᵐ
commute_half_left : ∀ (a : α), ⅟2 * a = a * ⅟2
⊢ ⅟2 * ⅟2 *
      (unsym a * unsym b * unsym a * unsym a + unsym b * unsym a * unsym a * unsym a +
          unsym a * unsym a * unsym a * unsym b +
        unsym a * unsym a * unsym b * unsym a) =
    ⅟2 * ⅟2 *
      (unsym a * unsym b * unsym a * unsym a + unsym a * unsym a * unsym a * unsym b +
          unsym b * (unsym a * unsym a) * unsym a +
        unsym a * unsym a * unsym b * unsym a)

Tactic:
← mul_assoc,

Goal state after:
case calc_2
α : Type u_1
inst✝¹ : Ring α
inst✝ : Invertible 2
a b : αˢʸᵐ
commute_half_left : ∀ (a : α), ⅟2 * a = a * ⅟2
⊢ ⅟2 * ⅟2 *
      (unsym a * unsym b * unsym a * unsym a + unsym b * unsym a * unsym a * unsym a +
          unsym a * unsym a * unsym a * unsym b +
        unsym a * unsym a * unsym b * unsym a) =
    ⅟2 * ⅟2 *
      (unsym a * unsym b * unsym a * unsym a + unsym a * unsym a * unsym a * unsym b +
          unsym b * unsym a * unsym a * unsym a +
        unsym a * unsym a * unsym b * unsym a)