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

Goal state before:
pp: α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : IsCancelMulZero α
inst✝ : DecidableEq α
lcm : α → α → α
dvd_lcm_left : ∀ (a b : α), a ∣ lcm a b
dvd_lcm_right : ∀ (a b : α), b ∣ lcm a b
lcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a
exists_gcd : ∀ (a b : α), lcm a b ∣ a * b := fun a b ↦ lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)
a b c : α
ac : a ∣ c
ab : a ∣ b
h✝ : ¬c = 0
h_1 : ¬b = 0
con : lcm c b = 0
h : lcm c b ∣ c * b
⊢ False

Tactic:
rw [con, zero_dvd_iff, mul_eq_zero] at h

Goal state after:
α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : IsCancelMulZero α
inst✝ : DecidableEq α
lcm : α → α → α
dvd_lcm_left : ∀ (a b : α), a ∣ lcm a b
dvd_lcm_right : ∀ (a b : α), b ∣ lcm a b
lcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a
exists_gcd : ∀ (a b : α), lcm a b ∣ a * b := fun a b ↦ lcm_dvd (Dvd.intro b rfl) (Dvd.intro_left a rfl)
a b c : α
ac : a ∣ c
ab : a ∣ b
h✝ : ¬c = 0
h_1 : ¬b = 0
con : lcm c b = 0
h : c = 0 ∨ b = 0
⊢ False
Module: Mathlib.Data.Matrix.Diagonal

Goal state before:
pp: n : Type u_3
α : Type v
inst✝¹ : DecidableEq n
inst✝ : Zero α
d : n → α
i x✝ : n
⊢ (diagonal d).col i x✝ = Pi.single i (d i) x✝

Tactic:
simp +contextual [diagonal, Pi.single_apply]

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.StrongRankCondition

Goal state before:
pp: R : Type u
M : Type v
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : StrongRankCondition R
ι : Type w
b : Basis ι R M
inst✝ : Infinite ι
κ : Type w
v : κ → M
i : LinearIndependent R v
h : #(Finset ι) < #κ
Φ : κ → Finset ι := ⋯
s : Finset ι
w : Infinite ↑(Φ ⁻¹' {s})
v' : ↑(Φ ⁻¹' {s}) → M := ⋯
i' : LinearIndependent (ι := ↑(Φ ⁻¹' {s})) R v'
⊢ range v' ⊆ ↑(span R ↑(Finset.image (⇑b) s))

Tactic:
rintro m ⟨⟨p, ⟨rfl⟩⟩, rfl⟩

Goal state after:
R : Type u
M : Type v
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : StrongRankCondition R
ι : Type w
b : Basis ι R M
inst✝ : Infinite ι
κ : Type w
v : κ → M
i : LinearIndependent R v
h : #(Finset ι) < #κ
Φ : κ → Finset ι := fun k ↦ (b.repr (v k)).support
p : κ
w : Infinite ↑(Φ ⁻¹' {Φ p})
v' : ↑(Φ ⁻¹' {Φ p}) → M := fun k ↦ v ↑k
i' : LinearIndependent (ι := ↑(Φ ⁻¹' {Φ p})) R v'
⊢ v' ⟨p, ⋯⟩ ∈ ↑(span R ↑(Finset.image (⇑b) (Φ p)))
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: ι : Type u_1
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
a b : ι
I : Ideal R
t : Finset ι
hat : a ∉ t
hp : ∀ i ∈ insert a t, i ≠ a → i ≠ b → (f i).IsPrime
h : ↑I ⊆ ⋃ i ∈ ↑(insert a t), ↑(f i)
has : a ∈ insert a t
hbt : b ∉ t
j : ι
⊢ j ∈ t → (f j).IsPrime

Tactic:
hj

Goal state after:
ι : Type u_1
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
a b : ι
I : Ideal R
t : Finset ι
hat : a ∉ t
hp : ∀ i ∈ insert a t, i ≠ a → i ≠ b → (f i).IsPrime
h : ↑I ⊆ ⋃ i ∈ ↑(insert a t), ↑(f i)
has : a ∈ insert a t
hbt : b ∉ t
j : ι
hj : j ∈ t
⊢ (f j).IsPrime
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: ι : Type u_1
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
a b : ι
I : Ideal R
t : Finset ι
hbt : b ∉ t
hp : ∀ i ∈ insert b t, i ≠ a → i ≠ b → (f i).IsPrime
h : ↑I ⊆ ⋃ i ∈ ↑(insert b t), ↑(f i)
has : a ∉ insert b t
hbs : b ∈ insert b t
j : ι
⊢ j ∈ t → (f j).IsPrime

Tactic:
hj

Goal state after:
ι : Type u_1
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
a b : ι
I : Ideal R
t : Finset ι
hbt : b ∉ t
hp : ∀ i ∈ insert b t, i ≠ a → i ≠ b → (f i).IsPrime
h : ↑I ⊆ ⋃ i ∈ ↑(insert b t), ↑(f i)
has : a ∉ insert b t
hbs : b ∈ insert b t
j : ι
hj : j ∈ t
⊢ (f j).IsPrime
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: ι : Type u_1
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
a b : ι
I : Ideal R
i : ι
t : Finset ι
left✝ : i ∉ t
hp : ∀ i_1 ∈ insert i t, i_1 ≠ a → i_1 ≠ b → (f i_1).IsPrime
h : ↑I ⊆ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1)
has : a ∉ insert i t
hbs : b ∉ insert i t
his : i ∈ insert i t
j : ι
⊢ j ∈ t → (f j).IsPrime

Tactic:
hj

Goal state after:
ι : Type u_1
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
a b : ι
I : Ideal R
i : ι
t : Finset ι
left✝ : i ∉ t
hp : ∀ i_1 ∈ insert i t, i_1 ≠ a → i_1 ≠ b → (f i_1).IsPrime
h : ↑I ⊆ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1)
has : a ∉ insert i t
hbs : b ∉ insert i t
his : i ∈ insert i t
j : ι
hj : j ∈ t
⊢ (f j).IsPrime
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: ι : Type u_1
M : Type u_2
inst✝² : AddCommGroup M
R : Type u_3
inst✝¹ : CommRing R
inst✝ : Module R M
I : Ideal R
v : ι → M
a : ι →₀ R
ha : ∀ (i : ι), a i ∈ I
⊢ ∀ (a : ι), ↑0 • v a = 0

Tactic:
· exact fun _ => zero_smul _ _

Goal state after:
Module: Mathlib.Data.Matrix.Mul

Goal state before:
pp: n : Type u_3
R : Type u_7
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : NonUnitalNonAssocSemiring R
v : n → R
j : n
x : R
i : n
⊢ Pi.single j x i * v i = Pi.single j (x * v j) i

Tactic:
exact Pi.apply_single (fun i x => x * v i) (fun i => zero_mul _) j x i

Goal state after:
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
s : Set R
I J : Ideal R
m : ↑s → ℕ
a b : ↑s → R
ha : ∀ (z : ↑s), a z ∈ I
hb : ∀ (z : ↑s), b z ∈ J
heq : ∀ (z : ↑s), a z + b z = ↑z ^ m z
z : R
hz : z ∈ s
⊢ a ⟨z, hz⟩ + b ⟨z, hz⟩ ∈ span (Set.range a) ⊔ J

Tactic:
exact Ideal.add_mem _ (mem_sup_left (subset_span ⟨⟨z, hz⟩, rfl⟩)) (mem_sup_right <| hb _)

Goal state after:
Module: Mathlib.Algebra.Order.Group.Finset

Goal state before:
pp: case refine_2
α : Type u_1
inst✝ : DecidableEq α
s : Multiset α
a : α
H : s.toFinset = {a}
x : α
⊢ count x s = count x (s.card • {a})

Tactic:
rw [count_nsmul, count_singleton]
    by_cases hx : x = a
    · simp_rw [hx, ite_true, mul_one, count_eq_card]
      intro y hy
      rw [← mem_toFinset, H, Finset.mem_singleton] at hy
      exact hy.symm
    have hx' : x ∉ s := fun h' ↦ hx <| by rwa [← mem_toFinset, H, Finset.mem_singleton] at h'
    simp_rw [count_eq_zero_of_notMem hx', hx, ite_false, Nat.mul_zero]

Goal state after:
Module: Mathlib.Data.Finsupp.Multiset

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
a a✝ : α
⊢ (toFinsupp {a}) a✝ = (Finsupp.single a 1) a✝

Tactic:
toFinsupp_apply,

Goal state after:
α : Type u_1
inst✝ : DecidableEq α
a a✝ : α
⊢ count a✝ {a} = (Finsupp.single a 1) a✝
Module: Mathlib.RingTheory.TwoSidedIdeal.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : NonUnitalNonAssocRing R
I✝ : TwoSidedIdeal R
inst✝ : Nontrivial R
I J : RingCon R
h : I ≠ J
⊢ { ringCon := I } ≠ { ringCon := J }

Tactic:
contrapose h

Goal state after:
R : Type u_1
inst✝¹ : NonUnitalNonAssocRing R
I✝ : TwoSidedIdeal R
inst✝ : Nontrivial R
I J : RingCon R
h : { ringCon := I } = { ringCon := J }
⊢ I = J
Module: Mathlib.LinearAlgebra.Dimension.Free

Goal state before:
pp: R : Type u
M : Type v
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Free R M
inst✝ : StrongRankCondition R
h : 0 < finrank R M
⊢ Module.Finite R M

Tactic:
contrapose h

Goal state after:
R : Type u
M : Type v
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Free R M
inst✝ : StrongRankCondition R
h : ¬Module.Finite R M
⊢ ¬0 < finrank R M
Module: Mathlib.LinearAlgebra.Dimension.Free

Goal state before:
pp: R : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : StrongRankCondition R
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Free R M
d1 : finrank R M = 1
u : M →ₗ[R] M
⊢ ∃! c, u = c • LinearMap.id

Tactic:
let e := (nonempty_linearEquiv_of_finrank_eq_one d1).some

Goal state after:
R : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : StrongRankCondition R
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Free R M
d1 : finrank R M = 1
u : M →ₗ[R] M
e : R ≃ₗ[R] M := ⋯.some
⊢ ∃! c, u = c • LinearMap.id
Module: Mathlib.Algebra.DirectSum.Basic

Goal state before:
pp: ι : Type u_3
α : ι → Type u_4
β : ι → Type u_5
inst✝¹ : (i : ι) → AddCommMonoid (α i)
inst✝ : (i : ι) → AddCommMonoid (β i)
f : (i : ι) → α i →+ β i
x y : ⨁ (i : ι), α i
⊢ (map f) x = (map f) y ↔ ∀ (i : ι), (f i) (x i) = (f i) (y i)

Tactic:
by
  simp_rw [DirectSum.ext_iff, map_apply]

Goal state after:
Module: Mathlib.Algebra.Algebra.NonUnitalSubalgebra

Goal state before:
pp: R : Type u
A : Type v
inst✝⁴ : CommSemiring R
inst✝³ : NonUnitalNonAssocSemiring A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
S T : NonUnitalSubalgebra R A
⊢ ∀ {x : A}, x ∈ T → x ∈ S ⊔ T

Tactic:
by
  rw [← SetLike.le_def]
  exact le_sup_right

Goal state after:
Module: Mathlib.Algebra.Algebra.Subalgebra.Lattice

Goal state before:
pp: case mem
R : Type uR
A : Type uA
B : Type uB
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
s : Set A
x : A
f : A →ₗ[R] B
hf : ∀ (a₁ a₂ : A), f (a₁ * a₂) = f a₁ * f a₂
a : A
ha : a ∈ s
⊢ f a ∈ adjoin R (⇑f '' (s ∪ {1}))

Tactic:
exact subset_adjoin ⟨a, ⟨Set.subset_union_left ha, rfl⟩⟩

Goal state after:
Module: Mathlib.Algebra.Algebra.Subalgebra.Lattice

Goal state before:
pp: R : Type uR
A : Type uA
B : Type uB
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
e : A →ₐ[R] B
x : A
⊢ Subalgebra.map e R[x] = R[e x]

Tactic:
by
  rw [map_adjoin, Set.image_singleton]

Goal state after:
Module: Mathlib.LinearAlgebra.DirectSum.Finsupp

Goal state before:
pp: R : Type u_1
ι : Type u_5
κ : Type u_6
inst✝ : CommSemiring R
i : ι × κ
r : R
⊢ (finsuppTensorFinsupp' R ι κ).symm (Finsupp.single i (1 * r)) = Finsupp.single i.1 1 ⊗ₜ[R] Finsupp.single i.2 r

Tactic:
exact finsuppTensorFinsupp'_symm_single_mul R ι κ i _ _

Goal state after:
Module: Mathlib.LinearAlgebra.DirectSum.Finsupp

Goal state before:
pp: R : Type u_1
ι : Type u_5
κ : Type u_6
inst✝ : CommSemiring R
i : ι × κ
r : R
⊢ (finsuppTensorFinsupp' R ι κ).symm (Finsupp.single i (r * 1)) = Finsupp.single i.1 r ⊗ₜ[R] Finsupp.single i.2 1

Tactic:
exact finsuppTensorFinsupp'_symm_single_mul R ι κ i _ _

Goal state after:
Module: Mathlib.Algebra.Star.SelfAdjoint

Goal state before:
pp: R : Type u_1
inst✝¹ : AddMonoid R
inst✝ : StarAddMonoid R
x y : R
hx : IsSelfAdjoint x
hy : IsSelfAdjoint y
⊢ IsSelfAdjoint (x + y)

Tactic:
simp only [isSelfAdjoint_iff, star_add, hx.star_eq, hy.star_eq]

Goal state after:
Module: Mathlib.LinearAlgebra.Basis.SMul

Goal state before:
pp: ι : Type u_1
R : Type u_2
M : Type u_4
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
G : Type u_5
inst✝³ : Group G
inst✝² : SMul G R
inst✝¹ : MulAction G M
inst✝ : IsScalarTower G R M
v : ι → M
hv : span R (range v) = ⊤
w : ι → G
j : M
⊢ j ∈ ⊤ → j ∈ span R (range (w • v))

Tactic:
hj

Goal state after:
ι : Type u_1
R : Type u_2
M : Type u_4
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
G : Type u_5
inst✝³ : Group G
inst✝² : SMul G R
inst✝¹ : MulAction G M
inst✝ : IsScalarTower G R M
v : ι → M
hv : span R (range v) = ⊤
w : ι → G
j : M
hj : j ∈ ⊤
⊢ j ∈ span R (range (w • v))
Module: Mathlib.LinearAlgebra.Matrix.RowCol

Goal state before:
pp: m : Type u_2
n : Type u_3
α : Type v
inst✝¹ : DecidableEq n
inst✝ : Mul α
u : m → α
v : n → α
j : n
a : α
⊢ vecMulVec u (Function.update v j a) = (vecMulVec u v).updateCol j (MulOpposite.op a • u)

Tactic:
by
  ext i j'
  obtain rfl | hi := eq_or_ne j' j
  · simp [vecMulVec_apply]
  · simp [vecMulVec_apply, hi]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Notation

Goal state before:
pp: α : Type u
inst✝ : Zero α
d : Fin 1 → α
⊢ diagonal d = !![d 0]

Tactic:
by
  simp [← Matrix.ext_iff]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Notation

Goal state before:
pp: α : Type u
inst✝ : Zero α
d : Fin 2 → α
⊢ diagonal d = !![d 0, 0; 0, d 1]

Tactic:
by
  simp [← Matrix.ext_iff]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.ToLin

Goal state before:
pp: R : Type u_1
inst✝⁷ : CommSemiring R
m : Type u_3
n : Type u_4
inst✝⁶ : Fintype n
inst✝⁵ : Finite m
inst✝⁴ : DecidableEq n
M₁ : Type u_5
M₂ : Type u_6
inst✝³ : AddCommMonoid M₁
inst✝² : AddCommMonoid M₂
inst✝¹ : Module R M₁
inst✝ : Module R M₂
v₁ : Basis n R M₁
v₂ : Basis m R M₂
f : M₁ →ₗ[R] M₂
x : M₁
i : m
⊢ (toLin' (((v₁.equivFun.arrowCongr v₂.equivFun).trans toMatrix') f)) (⇑(v₁.repr x)) i = (v₂.repr (f x)) i

Tactic:
LinearEquiv.trans_apply,

Goal state after:
R : Type u_1
inst✝⁷ : CommSemiring R
m : Type u_3
n : Type u_4
inst✝⁶ : Fintype n
inst✝⁵ : Finite m
inst✝⁴ : DecidableEq n
M₁ : Type u_5
M₂ : Type u_6
inst✝³ : AddCommMonoid M₁
inst✝² : AddCommMonoid M₂
inst✝¹ : Module R M₁
inst✝ : Module R M₂
v₁ : Basis n R M₁
v₂ : Basis m R M₂
f : M₁ →ₗ[R] M₂
x : M₁
i : m
⊢ (toLin' (toMatrix' ((v₁.equivFun.arrowCongr v₂.equivFun) f))) (⇑(v₁.repr x)) i = (v₂.repr (f x)) i
Module: Mathlib.GroupTheory.MonoidLocalization.MonoidWithZero

Goal state before:
pp: M : Type u_1
inst✝² : CommMonoidWithZero M
S : Submonoid M
N : Type u_2
inst✝¹ : CommMonoidWithZero N
f : S.LocalizationMap N
inst✝ : NoZeroDivisors M
⊢ NoZeroDivisors N

Tactic:
refine noZeroDivisors_iff_forall_mem_nonZeroDivisors.mpr fun n hn ↦ ?_

Goal state after:
M : Type u_1
inst✝² : CommMonoidWithZero M
S : Submonoid M
N : Type u_2
inst✝¹ : CommMonoidWithZero N
f : S.LocalizationMap N
inst✝ : NoZeroDivisors M
n : N
hn : n ≠ 0
⊢ n ∈ nonZeroDivisors N
Module: Mathlib.GroupTheory.MonoidLocalization.Divisibility

Goal state before:
pp: M : Type u_1
N : Type u_2
inst✝¹ : CommMonoid M
S : Submonoid M
inst✝ : CommMonoid N
f : S.LocalizationMap N
m : M
⊢ IsUnit (f m) ↔ ∃ s ∈ S, m ∣ s

Tactic:
refine ⟨fun h ↦ ?_, fun ⟨m, hm, dvd⟩ ↦ isUnit_of_dvd_unit (map_dvd _ dvd) (f.map_units ⟨m, hm⟩)⟩
  have ⟨s, hxs⟩ := isUnit_iff_dvd_one.mp h
  have ⟨⟨r, s⟩, hrs⟩ := f.surj s
  replace hrs := congr(f m * $hrs)
  rw [← mul_assoc, ← hxs, one_mul, ← map_mul] at hrs
  have ⟨s', eq⟩ := f.eq_iff_exists.mp hrs
  exact ⟨s' * s, mul_mem s'.2 s.2, _, mul_left_comm _ m _ ▸ eq⟩

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.ToLin

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommSemiring R
n : Type u_4
inst✝³ : Fintype n
inst✝² : DecidableEq n
M₁ : Type u_5
inst✝¹ : AddCommMonoid M₁
inst✝ : Module R M₁
v₁ : Basis n R M₁
f : M₁ →ₗ[R] M₁
⊢ (toLinAlgEquiv v₁) ((toMatrixAlgEquiv v₁) f) = f

Tactic:
← Matrix.toLinAlgEquiv_symm,

Goal state after:
R : Type u_1
inst✝⁴ : CommSemiring R
n : Type u_4
inst✝³ : Fintype n
inst✝² : DecidableEq n
M₁ : Type u_5
inst✝¹ : AddCommMonoid M₁
inst✝ : Module R M₁
v₁ : Basis n R M₁
f : M₁ →ₗ[R] M₁
⊢ (toLinAlgEquiv v₁) ((toLinAlgEquiv v₁).symm f) = f
Module: Mathlib.LinearAlgebra.Matrix.ToLin

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommSemiring R
n : Type u_4
inst✝³ : Fintype n
inst✝² : DecidableEq n
M₁ : Type u_5
inst✝¹ : AddCommMonoid M₁
inst✝ : Module R M₁
v₁ : Basis n R M₁
M : Matrix n n R
⊢ (toMatrixAlgEquiv v₁) ((toLinAlgEquiv v₁) M) = M

Tactic:
← Matrix.toLinAlgEquiv_symm,

Goal state after:
R : Type u_1
inst✝⁴ : CommSemiring R
n : Type u_4
inst✝³ : Fintype n
inst✝² : DecidableEq n
M₁ : Type u_5
inst✝¹ : AddCommMonoid M₁
inst✝ : Module R M₁
v₁ : Basis n R M₁
M : Matrix n n R
⊢ (toLinAlgEquiv v₁).symm ((toLinAlgEquiv v₁) M) = M
Module: Mathlib.RingTheory.OreLocalization.Ring

Goal state before:
pp: R : Type u_1
inst✝⁵ : Semiring R
S : Submonoid R
inst✝⁴ : OreSet S
X : Type u_2
inst✝³ : AddCommMonoid X
inst✝² : Module R X
R₀ : Type ?u.16
inst✝¹ : CommSemiring R₀
inst✝ : Algebra R₀ R
r : R₀
x : OreLocalization S R
⊢ (numeratorRingHom.comp (algebraMap R₀ R)) r * x = x * (numeratorRingHom.comp (algebraMap R₀ R)) r

Tactic:
by
    induction x using OreLocalization.ind with | _ r₁ s₁
    dsimp
    rw [mul_div_one, oreDiv_mul_char _ _ _ _ (algebraMap R₀ R r) s₁ (Algebra.commutes _ _).symm,
      Algebra.commutes, mul_one]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.ToLin

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : Semiring S
inst✝⁶ : Algebra R S
m : Type u_3
inst✝⁵ : Fintype m
inst✝⁴ : DecidableEq m
b : Basis m R S
G : Type u_4
inst✝³ : Group G
inst✝² : DistribMulAction G S
inst✝¹ : SMulCommClass G R S
inst✝ : SMulCommClass G S S
g : G
x : S
i✝ j✝ : m
⊢ (((DistribMulAction.toLinearEquiv R S g).symm ≪≫ₗ b.repr) (x * g • b j✝)) i✝ = (b.repr (x * b j✝)) i✝

Tactic:
LinearEquiv.trans_apply,

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁸ : CommSemiring R
inst✝⁷ : Semiring S
inst✝⁶ : Algebra R S
m : Type u_3
inst✝⁵ : Fintype m
inst✝⁴ : DecidableEq m
b : Basis m R S
G : Type u_4
inst✝³ : Group G
inst✝² : DistribMulAction G S
inst✝¹ : SMulCommClass G R S
inst✝ : SMulCommClass G S S
g : G
x : S
i✝ j✝ : m
⊢ (b.repr ((DistribMulAction.toLinearEquiv R S g).symm (x * g • b j✝))) i✝ = (b.repr (x * b j✝)) i✝
Module: Mathlib.RingTheory.Localization.Basic

Goal state before:
pp: case refine_1
ι : Type u_1
R : ι → Type u_2
inst✝ : (i : ι) → CommSemiring (R i)
i : ι
S : Submonoid (R i)
T : Submonoid ((i : ι) → R i) := Submonoid.comap (Pi.evalRingHom R i) S
x₁ x₂ : Localization (Submonoid.comap (Pi.evalRingHom R i) S)
eq : (mapPiEvalRingHom S) x₁ = (mapPiEvalRingHom S) x₂
⊢ x₁ = x₂

pp: case refine_2
ι : Type u_1
R : ι → Type u_2
inst✝ : (i : ι) → CommSemiring (R i)
i : ι
S : Submonoid (R i)
T : Submonoid ((i : ι) → R i) := Submonoid.comap (Pi.evalRingHom R i) S
x : Localization S
⊢ ∃ a, (mapPiEvalRingHom S) a = x

Tactic:
· obtain ⟨r₁, s₁, rfl⟩ := exists_mk'_eq T x₁
    obtain ⟨r₂, s₂, rfl⟩ := exists_mk'_eq T x₂
    simp_rw [map_mk'] at eq
    rw [IsLocalization.eq] at eq ⊢
    obtain ⟨s, hs⟩ := eq
    refine ⟨⟨update 0 i s, by apply update_self i s.1 0 ▸ s.2⟩, funext fun j ↦ ?_⟩
    obtain rfl | ne := eq_or_ne j i
    · simpa using hs
    · simp [update_of_ne ne]

Goal state after:
case refine_2
ι : Type u_1
R : ι → Type u_2
inst✝ : (i : ι) → CommSemiring (R i)
i : ι
S : Submonoid (R i)
T : Submonoid ((i : ι) → R i) := Submonoid.comap (Pi.evalRingHom R i) S
x : Localization S
⊢ ∃ a, (mapPiEvalRingHom S) a = x
Module: Mathlib.RingTheory.TensorProduct.Finite

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Finite R M
inst✝ : Nontrivial M
n : ℕ
f : Fin n → M
hf : Submodule.span R (Set.range f) = ⊤
s : Set ℕ := {m | Submodule.span R (f '' Fin.val ⁻¹' Set.Iio m) ≠ ⊤}
⊢ ∀ x ∈ s, x < n

Tactic:
refine fun x hx ↦ lt_iff_not_ge.mpr fun e ↦ ?_

Goal state after:
R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Finite R M
inst✝ : Nontrivial M
n : ℕ
f : Fin n → M
hf : Submodule.span R (Set.range f) = ⊤
s : Set ℕ := {m | Submodule.span R (f '' Fin.val ⁻¹' Set.Iio m) ≠ ⊤}
x : ℕ
hx : x ∈ s
e : n ≤ x
⊢ False
Module: Mathlib.Algebra.Ring.TransferInstance

Goal state before:
pp: α : Type u_1
β : Type u_2
e : α ≃ β
inst✝ : AddMonoidWithOne β
n : ℕ
⊢ e (e.invFun ↑(n + 1)) = e (e.invFun ↑n + 1)

Tactic:
by simp [add_def, one_def]

Goal state after:
Module: Mathlib.Algebra.Algebra.Spectrum.Quasispectrum

Goal state before:
pp: F : Type u_3
R : Type u_4
S : Type u_5
A : Type u_6
B : Type u_7
inst✝¹² : CommSemiring R
inst✝¹¹ : Semiring S
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : NonUnitalRing B
inst✝⁸ : Module R S
inst✝⁷ : Module S A
inst✝⁶ : Module R A
inst✝⁵ : Module S B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R S A
inst✝² : IsScalarTower R S B
inst✝¹ : FunLike F A B
inst✝ : NonUnitalAlgHomClass F S A B
φ : F
a : A
x : R
hx : IsUnit x
this : IsQuasiregular (-(⋯.unit⁻¹ • a))
⊢ IsQuasiregular (-(⋯.unit⁻¹ • φ a))

Tactic:
rw [Units.smul_def, ← smul_one_smul S] at this ⊢

Goal state after:
F : Type u_3
R : Type u_4
S : Type u_5
A : Type u_6
B : Type u_7
inst✝¹² : CommSemiring R
inst✝¹¹ : Semiring S
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : NonUnitalRing B
inst✝⁸ : Module R S
inst✝⁷ : Module S A
inst✝⁶ : Module R A
inst✝⁵ : Module S B
inst✝⁴ : Module R B
inst✝³ : IsScalarTower R S A
inst✝² : IsScalarTower R S B
inst✝¹ : FunLike F A B
inst✝ : NonUnitalAlgHomClass F S A B
φ : F
a : A
x : R
hx : IsUnit x
this : IsQuasiregular (-((↑⋯.unit⁻¹ • 1) • a))
⊢ IsQuasiregular (-((↑⋯.unit⁻¹ • 1) • φ a))
Module: Mathlib.Algebra.Algebra.Unitization

Goal state before:
pp: case inl_add_inr
S : Type u_1
R : Type u_2
A : Type u_3
inst✝¹² : CommSemiring S
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : NonUnitalSemiring A
inst✝⁹ : Module R A
inst✝⁸ : SMulCommClass R A A
inst✝⁷ : IsScalarTower R A A
B : Type u_4
inst✝⁶ : Semiring B
inst✝⁵ : Algebra S B
inst✝⁴ : Algebra S R
inst✝³ : DistribMulAction S A
inst✝² : IsScalarTower S R A
C : Type u_5
inst✝¹ : Semiring C
inst✝ : Algebra R C
φ : A →ₙₐ[R] C
y : Unitization R A
r✝ : R
a✝ : A
⊢ (algebraMap R C) (inl r✝ + ↑a✝ + y).toProd.1 + φ (inl r✝ + ↑a✝ + y).toProd.2 =
    (algebraMap R C) (inl r✝ + ↑a✝).toProd.1 + φ (inl r✝ + ↑a✝).toProd.2 + ((algebraMap R C) y.toProd.1 + φ y.toProd.2)

Tactic:
induction y with
      | inl_add_inr =>
        simp only [fst_add, fst_inl, fst_inr, add_zero, map_add, snd_add, snd_inl, snd_inr,
          zero_add, φ.map_add]
        rw [add_add_add_comm]

Goal state after:
Module: Mathlib.Algebra.Order.Star.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝³ : NonUnitalSemiring R
inst✝² : PartialOrder R
inst✝¹ : StarRing R
inst✝ : StarOrderedRing R
x✝ y✝ x y : R
hle : x ≤ y
z : R
⊢ x + z ≤ y + z

Tactic:
rw [StarOrderedRing.le_iff] at hle ⊢

Goal state after:
R : Type u_1
A : Type u_2
inst✝³ : NonUnitalSemiring R
inst✝² : PartialOrder R
inst✝¹ : StarRing R
inst✝ : StarOrderedRing R
x✝ y✝ x y : R
hle : ∃ p ∈ AddSubmonoid.closure (range fun s ↦ star s * s), y = x + p
z : R
⊢ ∃ p ∈ AddSubmonoid.closure (range fun s ↦ star s * s), y + z = x + z + p
Module: Mathlib.LinearAlgebra.Dimension.Constructions

Goal state before:
pp: R : Type u
M : Type v
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
M' : Submodule R M
ι₁ : Type u_2
ι₂ : Type u_3
f : ι₁ → ↥M'
hf : LinearIndependent R f
g : ι₂ → M
hg : LinearIndependent R (Submodule.Quotient.mk ∘ g)
c : ι₂ →₀ R
h₁ : (c.sum fun i a ↦ a • g i) ∈ span R (range fun x ↦ ↑(f x))
h₂ : (c.sum fun i a ↦ a • g i) ∈ span R (range g)
this : (c.sum fun i a ↦ a • g i) ∈ M'
⊢ (c.sum fun i a ↦ a • g i) = 0

Tactic:
simp_rw [← Quotient.mk_eq_zero, ← mkQ_apply, map_finsuppSum, map_smul, mkQ_apply] at this

Goal state after:
R : Type u
M : Type v
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
M' : Submodule R M
ι₁ : Type u_2
ι₂ : Type u_3
f : ι₁ → ↥M'
hf : LinearIndependent R f
g : ι₂ → M
hg : LinearIndependent R (Submodule.Quotient.mk ∘ g)
c : ι₂ →₀ R
h₁ : (c.sum fun i a ↦ a • g i) ∈ span R (range fun x ↦ ↑(f x))
h₂ : (c.sum fun i a ↦ a • g i) ∈ span R (range g)
this : (c.sum fun a b ↦ b • Submodule.Quotient.mk (g a)) = 0
⊢ (c.sum fun i a ↦ a • g i) = 0
Module: Mathlib.LinearAlgebra.Dimension.Constructions

Goal state before:
pp: R : Type u_2
V : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup V
inst✝ : Module R V
W : Submodule R V
m : Type u_4
n : Type u_5
bW : Basis m R ↥W
bQ : Basis n R (V ⧸ W)
v : V
hv : v ∈ W
j : n
⊢ ((bW.sumQuot bQ).repr v) (Sum.inr j) = 0

Tactic:
by
  suffices W.mkQ v = 0 by simp [sumQuot_repr_inr, this]
  aesop

Goal state after:
Module: Mathlib.Algebra.Star.NonUnitalSubalgebra

Goal state before:
pp: R : Type u
A : Type v
inst✝⁷ : CommSemiring R
inst✝⁶ : StarRing R
inst✝⁵ : NonUnitalSemiring A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : StarModule R A
S T : NonUnitalStarSubalgebra R A
⊢ ∀ {x : A}, x ∈ T → x ∈ S ⊔ T

Tactic:
by
  rw [← SetLike.le_def]
  exact le_sup_right

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.Finite

Goal state before:
pp: R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Nontrivial R
h : 0 < finrank R M
⊢ Nontrivial M

Tactic:
contrapose! h; exact finrank_zero_of_subsingleton.le

Goal state after:
Module: Mathlib.Algebra.Star.NonUnitalSubalgebra

Goal state before:
pp: R : Type u
A : Type v
inst✝⁵ : CommSemiring R
inst✝⁴ : NonUnitalSemiring A
inst✝³ : StarRing A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
s : Set A
⊢ ↑(centralizer R ↑(centralizer R s)) = (s ∪ star s).centralizer.centralizer

Tactic:
rw [coe_centralizer, StarMemClass.star_coe_eq, Set.union_self, coe_centralizer]

Goal state after: