mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Ideal.Maps

Goal state before:
pp: R : Type u
S : Type v
F : Type u_1
inst✝² : Ring R
inst✝¹ : Semiring S
inst✝ : FunLike F R S
rc : RingHomClass F R S
f : F
⊢ Function.Injective ⇑f ↔ ker f = ⊥

Tactic:
by
  rw [SetLike.ext'_iff, ker_eq, Set.ext_iff]
  exact injective_iff_map_eq_zero' f

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

Goal state before:
pp: case refine_1
R : Type u_4
M : Type u_5
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
H : annihilator R M ≤ ⊥
r s : R
H' : ∀ (a : M), r • a = s • a
⊢ r = s

Tactic:
rw [← sub_eq_zero]

Goal state after:
case refine_1
R : Type u_4
M : Type u_5
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
H : annihilator R M ≤ ⊥
r s : R
H' : ∀ (a : M), r • a = s • a
⊢ r - s = 0
Module: Mathlib.RingTheory.Ideal.Maps

Goal state before:
pp: case mpr
R : Type u_1
M : Type u_2
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u_4
inst✝ : Nonempty ι
r : R
h : ∀ (m : M), r • m = 0
⊢ ∀ (m : ι →₀ M), r • m = 0

Tactic:
· intro m; ext i; exact h _

Goal state after:
Module: Mathlib.Algebra.EuclideanDomain.Basic

Goal state before:
pp: case pos
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x y : R
h : gcd x y = 0
⊢ gcd x y * (x * y / gcd x y) = x * y

Tactic:
rw [h, zero_mul]

Goal state after:
case pos
R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
x y : R
h : gcd x y = 0
⊢ 0 = x * y
Module: Mathlib.RingTheory.Ideal.Operations

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
I J : Ideal R
⊢ IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1

Tactic:
by
  rw [← add_eq_one_iff, isCoprime_iff_add]

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

Goal state before:
pp: α : Type u_2
C : Type u_4
inst✝² : SetLike C α
inst✝¹ : GroupWithZero α
inst✝ : SubmonoidClass C α
S : C
a : ↥S
ha : ↑a ≠ 0
⊢ ↑a ≠ 0 ∧ (↑a)⁻¹ ∈ S ↔ (↑a)⁻¹ ∈ S

Tactic:
and_iff_right ha

Goal state after:
α : Type u_2
C : Type u_4
inst✝² : SetLike C α
inst✝¹ : GroupWithZero α
inst✝ : SubmonoidClass C α
S : C
a : ↥S
ha : ↑a ≠ 0
⊢ (↑a)⁻¹ ∈ S ↔ (↑a)⁻¹ ∈ S
Module: Mathlib.Algebra.GCDMonoid.Basic

Goal state before:
pp: α : Type u_2
inst✝¹ : CommMonoidWithZero α
inst✝ : GCDMonoid α
h : gcd 0 0 = 0
⊢ 0 = gcd 0 0 * 1

Tactic:
rw [h, zero_mul]

Goal state after:
Module: Mathlib.RingTheory.PrincipalIdealDomain

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : IsBezout R
inst✝ : IsDomain R
a b c : R
hab : IsCoprime a b
k : ℕ
h : a * b = c ^ k
⊢ ∃ d, Associated (d ^ k) a

Tactic:
by
  classical
  let := IsBezout.toGCDDomain R
  exact exists_associated_pow_of_mul_eq_pow ((gcd_isUnit_iff _ _).mpr hab) h

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

Goal state before:
pp: l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
m' : o → Type u_5
n' : o → Type u_6
R : Type u_7
S : Type u_8
α : Type v
β : Type w
γ : Type u_9
inst✝ : IsEmpty n
M N : Matrix m n α
⊢ M = N

Tactic:
ext i j
    exact isEmptyElim j

Goal state after:
Module: Mathlib.Algebra.GCDMonoid.Basic

Goal state before:
pp: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : GCDMonoid α
⊢ ∀ {a b c : Associates α}, a ∣ c → a ∣ b → a ∣ Quotient.map₂ gcd ⋯ c b

Tactic:
by
    rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ hac hbc
    exact mk_le_mk_of_dvd (dvd_gcd (dvd_of_mk_le_mk hac) (dvd_of_mk_le_mk hbc))

Goal state after:
Module: Mathlib.Algebra.GCDMonoid.Basic

Goal state before:
pp: case mk.mk
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : GCDMonoid α
a✝ : Associates α
a : α
b✝ : Associates α
b : α
⊢ Quotient.map₂ gcd ⋯ (Quot.mk (⇑(Associated.setoid α)) a) (Quot.mk (⇑(Associated.setoid α)) b) *
      Quotient.map₂ lcm ⋯ (Quot.mk (⇑(Associated.setoid α)) a) (Quot.mk (⇑(Associated.setoid α)) b) =
    Quot.mk (⇑(Associated.setoid α)) a * Quot.mk (⇑(Associated.setoid α)) b

Tactic:
exact Quotient.sound <| gcd_mul_lcm _ _

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

Goal state before:
pp: R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type w
inst✝ : RankCondition R
J : Set M
v : Basis ι R M
hJ : span R J = ⊤
this : Nontrivial R
val✝ : Infinite ↑J
S : ↑J → Set ι := fun j ↦ ↑(v.repr ↑j).support
S' : ↑J → Set M := fun j ↦ ⇑v '' S j
b : M
hb : b ∈ range ⇑v
i : ι
hi : v i = b
⊢ b ∈ ⋃ j, S' j

Tactic:
have : span R J ≤ comap v.repr.toLinearMap (Finsupp.supported R R (⋃ j, S j)) :=
        span_le.2 fun j hj x hx => ⟨_, ⟨⟨j, hj⟩, rfl⟩, hx⟩

Goal state after:
R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type w
inst✝ : RankCondition R
J : Set M
v : Basis ι R M
hJ : span R J = ⊤
this✝ : Nontrivial R
val✝ : Infinite ↑J
S : ↑J → Set ι := fun j ↦ ↑(v.repr ↑j).support
S' : ↑J → Set M := fun j ↦ ⇑v '' S j
b : M
hb : b ∈ range ⇑v
i : ι
hi : v i = b
this : span R J ≤ comap (↑v.repr) (Finsupp.supported R R (⋃ j, S j))
⊢ b ∈ ⋃ j, S' j
Module: Mathlib.Data.Matrix.Diagonal

Goal state before:
pp: l : Type u_1
m : Type u_2
α : Type v
inst✝² : Zero α
inst✝¹ : DecidableEq m
inst✝ : DecidableEq l
d : m → α
e : l → m
he : Function.Injective e
i j : l
⊢ (diagonal d).submatrix e e i j = diagonal (d ∘ e) i j

Tactic:
rw [submatrix_apply]
    by_cases h : i = j
    · rw [h, diagonal_apply_eq, diagonal_apply_eq, Function.comp_apply]
    · rw [diagonal_apply_ne _ h, diagonal_apply_ne _ (he.ne h)]

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

Goal state before:
pp: R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : StrongRankCondition R
ι : Type v
v : Basis ι R M
this : Nontrivial R
⊢ LinearIndepOn R id (range ⇑v)

Tactic:
rw [LinearIndepOn]
          convert! v.reindexRange.linearIndependent
          simp

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

Goal state before:
pp: m : Type u_2
n : Type u_3
α : Type v
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : NonUnitalNonAssocSemiring α
u : m → α
x : n → α
e : n ≃ m
⊢ u ⬝ᵥ x ∘ ⇑e.symm = u ∘ ⇑e ⬝ᵥ x

Tactic:
by
  simpa only [Equiv.symm_symm] using (comp_equiv_symm_dotProduct u x e.symm).symm

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

Goal state before:
pp: n : Type u_3
α : Type v
inst✝² : Fintype n
inst✝¹ : NonAssocSemiring α
inst✝ : DecidableEq n
i : n
v : n → α
⊢ Pi.single i 1 ⬝ᵥ v = v i

Tactic:
by
  rw [single_dotProduct, one_mul]

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

Goal state before:
pp: m : Type u_2
n : Type u_3
R : Type u_7
α : Type v
inst✝³ : NonUnitalNonAssocSemiring α
inst✝² : Fintype n
inst✝¹ : DistribSMul R α
inst✝ : IsScalarTower R α α
b : R
M : Matrix m n α
v : n → α
⊢ (b • M) *ᵥ v = b • M *ᵥ v

Tactic:
ext
  exact smul_dotProduct _ _ _

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

Goal state before:
pp: m : Type u_2
n : Type u_3
R : Type u_7
α : Type v
inst✝³ : NonUnitalNonAssocSemiring α
inst✝² : Fintype m
inst✝¹ : DistribSMul R α
inst✝ : IsScalarTower R α α
b : R
v : m → α
M : Matrix m n α
⊢ (b • v) ᵥ* M = b • v ᵥ* M

Tactic:
ext
  exact smul_dotProduct _ _ _

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

Goal state before:
pp: R : Type u_3
A : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
S : Subalgebra R A
⊢ S.subtype.rangeS = S.toSubsemiring

Tactic:
Subsemiring.rangeS_subtype

Goal state after:
R : Type u_3
A : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
S : Subalgebra R A
⊢ S.toSubsemiring = S.toSubsemiring
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
s : Set A
x : A
f : A →ₗ[R] B
hf : ∀ (a₁ a₂ : A), f (a₁ * a₂) = f a₁ * f a₂
r : R
this : f 1 ∈ adjoin R (⇑f '' (s ∪ {1}))
⊢ f ((algebraMap R A) r) = r • f 1

Tactic:
rw [algebraMap_eq_smul_one]

Goal state after:
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
r : R
this : f 1 ∈ adjoin R (⇑f '' (s ∪ {1}))
⊢ f (r • 1) = r • f 1
Module: Mathlib.LinearAlgebra.TensorProduct.Associator

Goal state before:
pp: R : Type u_1
inst✝⁸ : CommSemiring R
M : Type u_5
N : Type u_6
P : Type u_7
Q : Type u_8
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid N
inst✝⁵ : AddCommMonoid P
inst✝⁴ : AddCommMonoid Q
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : Module R Q
inst✝ : Module R P
⊢ tensorTensorTensorComm R M N P Q ≪≫ₗ tensorTensorTensorComm R M P N Q = LinearEquiv.refl R (M ⊗[R] N ⊗[R] (P ⊗[R] Q))

Tactic:
rw [← tensorTensorTensorComm_symm]
  exact LinearEquiv.symm_trans_self _

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

Goal state before:
pp: case h
R : Type u_1
inst✝² : Semiring R
P : Type u_2
inst✝¹ : AddCommMonoid P
inst✝ : Module R P
ι : Type u_8
b : Basis ι R P
m : P
⊢ ((b.repr m).sum fun a b_1 ↦ b_1 • b a) = m

Tactic:
exact b.linearCombination_repr m

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁹ : CommSemiring R
inst✝⁸ : Semiring S
inst✝⁷ : Algebra R S
M : Type u_3
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module R M
inst✝⁴ : Module S M
inst✝³ : IsScalarTower R S M
N : Type u_4
inst✝² : AddCommMonoid N
inst✝¹ : Module R N
ι : Type u_5
inst✝ : DecidableEq ι
i : ι
m : M
n : N
⊢ (finsuppRight R S M N ι).symm (Finsupp.single i (m ⊗ₜ[R] n)) = m ⊗ₜ[R] Finsupp.single i n

Tactic:
simp [LinearEquiv.symm_apply_eq]

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁷ : CommSemiring R
inst✝⁶ : Semiring S
inst✝⁵ : Algebra R S
M : Type u_3
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : Module S M
inst✝¹ : IsScalarTower R S M
ι : Type u_5
inst✝ : DecidableEq ι
m : M
p : ι →₀ R
i : ι
⊢ ((finsuppScalarRight R S M ι) (m ⊗ₜ[R] p)) i = (p.sum fun i n ↦ Finsupp.single i (n • m)) i

Tactic:
rw [finsuppScalarRight_apply_tmul_apply, Finsupp.sum_apply,
    Finsupp.sum_eq_single i (fun _ _ ↦ Finsupp.single_eq_of_ne') (by simp), Finsupp.single_eq_same]

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

Goal state before:
pp: case refine_2
R : Type u
inst✝² : Semiring R
ι : Type v
M : ι → Type w
inst✝¹ : (i : ι) → AddCommMonoid (M i)
inst✝ : (i : ι) → Module R (M i)
H : ∀ (i : ι), Projective R (M i)
⊢ Projective R (⨁ (i : ι), M i)

Tactic:
let e : (⨁ i, M i) ≃ₗ[R] Π₀ i, M i := .refl ..

Goal state after:
case refine_2
R : Type u
inst✝² : Semiring R
ι : Type v
M : ι → Type w
inst✝¹ : (i : ι) → AddCommMonoid (M i)
inst✝ : (i : ι) → Module R (M i)
H : ∀ (i : ι), Projective R (M i)
e : (⨁ (i : ι), M i) ≃ₗ[R] Π₀ (i : ι), M i := LinearEquiv.refl R (⨁ (i : ι), M i)
⊢ Projective R (⨁ (i : ι), M i)
Module: Mathlib.Algebra.Star.Basic

Goal state before:
pp: R : Type u
inst✝¹ : Mul R
inst✝ : StarMul R
x y : R
⊢ star (star x * y) = star y * x

Tactic:
by rw [star_mul, star_star]

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

Goal state before:
pp: R : Type u
inst✝¹ : Mul R
inst✝ : StarMul R
x y : R
⊢ star (x * star y) = y * star x

Tactic:
by rw [star_mul, star_star]

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

Goal state before:
pp: R : Type u_1
inst✝³ : Monoid R
inst✝² : StarMul R
r : R
inst✝¹ : Invertible r
inst✝ : Invertible (star r)
⊢ star ⅟r = ⅟(star r)

Tactic:
← mul_one (star (⅟r)),

Goal state after:
R : Type u_1
inst✝³ : Monoid R
inst✝² : StarMul R
r : R
inst✝¹ : Invertible r
inst✝ : Invertible (star r)
⊢ star ⅟r * 1 = ⅟(star r)
Module: Mathlib.LinearAlgebra.Matrix.ConjTranspose

Goal state before:
pp: l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
m' : o → Type u_5
n' : o → Type u_6
R : Type u_7
S : Type u_8
α : Type v
β : Type w
γ : Type u_9
inst✝⁷ : Fintype n
inst✝⁶ : CommSemiring R
inst✝⁵ : StarRing R
inst✝⁴ : TrivialStar R
inst✝³ : Semiring α
inst✝² : StarRing α
inst✝¹ : Algebra R α
inst✝ : StarModule R α
r : R
M : Matrix n n α
⊢ (conjTransposeRingEquiv n α).toFun (r • M) = r • (conjTransposeRingEquiv n α).toFun M

Tactic:
change conjTransposeRingEquiv n α (r • M) = r • conjTransposeRingEquiv n α M
    simp

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
R : Type u_3
R₂ : Type u_4
M : Type u_5
M' : Type u_6
v : ι → M
inst✝⁴ : Ring R
inst✝³ : CommRing R₂
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Module R₂ M
x✝ y✝ : M
b✝ : Basis ι R M
n : ℕ
N : Submodule R M
y : M
b : Basis (Fin n) R ↥N
hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0
hsp : ∀ (z : M), ∃ c, z + c • y ∈ N
span_b : N = span R (Set.range (⇑N.subtype ∘ ⇑b))
c : R
x : M
hx : x ∈ N
hc : c • y + x = 0
⊢ c = 0

Tactic:
exact hli c x hx hc

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
R : Type u_3
R₂ : Type u_4
M : Type u_5
M' : Type u_6
v : ι → M
inst✝⁴ : Ring R
inst✝³ : CommRing R₂
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Module R₂ M
x✝ y✝ : M
b✝ : Basis ι R M
n : ℕ
N : Submodule R M
b : Basis (Fin n) R ↥N
y : M
hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0
hsp : ∀ (z : M), ∃ c, z + c • y ∈ N
span_b : N = span R (Set.range (⇑N.subtype ∘ ⇑b))
c : R
x : M
hx : x ∈ N
hc : c • y + x = 0
⊢ c = 0

Tactic:
exact hli c x hx hc

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

Goal state before:
pp: case inr.inr
n : Type u_3
α : Type v
inst✝¹ : DecidableEq n
inst✝ : Zero α
v : n → α
i : n
x : α
j k : n
hjk : j ≠ k
hki : k ≠ i
⊢ (diagonal v).updateCol i (Pi.single i x) j k = 0

Tactic:
· rw [updateCol_ne hki, diagonal_apply_ne _ hjk]

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Tower

Goal state before:
pp: R : Type u_1
M : Type u_2
A : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p q : Submodule R M
h : p ≤ q
⊢ (AlgebraTensorModule.map LinearMap.id q.subtype ∘ₗ AlgebraTensorModule.map LinearMap.id (inclusion h)).range ≤
    baseChange A q

Tactic:
apply LinearMap.range_comp_le_range

Goal state after:
Module: Mathlib.Algebra.AffineMonoid.Irreducible

Goal state before:
pp: M : Type u_1
inst✝² : CancelCommMonoid M
inst✝¹ : Subsingleton Mˣ
inst✝ : Monoid.FG M
S : Finset M
hSgen : closure ↑S = ⊤
hSmax : ∀ ⦃y : Finset M⦄, (fun S ↦ closure ↑S = ⊤) y → y ⊆ S → S ⊆ y
r : M
hrS : r ∈ ↑S
hr₀ : r ≠ 1
hrirred : ¬IsUnit r → ∃ x x_1, ∃ (_ : r = x * x_1), ¬IsUnit x ∧ ¬IsUnit x_1
a b : M
hr✝ : r = a * b
ha : ¬IsUnit a
hb : ¬IsUnit b
m n : M → ℕ
hm : a = r ^ m r * ∏ s ∈ S \ {r}, s ^ m s
hn : b = r ^ n r * ∏ s ∈ S \ {r}, s ^ n s
hr : m r + n r = 0
hr' : r = (∏ s ∈ S \ {r}, s ^ m s) * ∏ s ∈ S \ {r}, s ^ n s
⊢ (∏ s ∈ S \ {r}, s ^ m s) * ∏ s ∈ S \ {r}, s ^ n s ∈
    ↑(closure (↑S \ {(∏ s ∈ S \ {r}, s ^ m s) * ∏ s ∈ S \ {r}, s ^ n s}))

Tactic:
refine mul_mem ?_ ?_ <;> refine prod_mem _ fun s hs ↦ pow_mem (subset_closure ?_) _

Goal state after:
case refine_1
M : Type u_1
inst✝² : CancelCommMonoid M
inst✝¹ : Subsingleton Mˣ
inst✝ : Monoid.FG M
S : Finset M
hSgen : closure ↑S = ⊤
hSmax : ∀ ⦃y : Finset M⦄, (fun S ↦ closure ↑S = ⊤) y → y ⊆ S → S ⊆ y
r : M
hrS : r ∈ ↑S
hr₀ : r ≠ 1
hrirred : ¬IsUnit r → ∃ x x_1, ∃ (_ : r = x * x_1), ¬IsUnit x ∧ ¬IsUnit x_1
a b : M
hr✝ : r = a * b
ha : ¬IsUnit a
hb : ¬IsUnit b
m n : M → ℕ
hm : a = r ^ m r * ∏ s ∈ S \ {r}, s ^ m s
hn : b = r ^ n r * ∏ s ∈ S \ {r}, s ^ n s
hr : m r + n r = 0
hr' : r = (∏ s ∈ S \ {r}, s ^ m s) * ∏ s ∈ S \ {r}, s ^ n s
s : M
hs : s ∈ S \ {r}
⊢ s ∈ ↑S \ {(∏ s ∈ S \ {r}, s ^ m s) * ∏ s ∈ S \ {r}, s ^ n s}

case refine_2
M : Type u_1
inst✝² : CancelCommMonoid M
inst✝¹ : Subsingleton Mˣ
inst✝ : Monoid.FG M
S : Finset M
hSgen : closure ↑S = ⊤
hSmax : ∀ ⦃y : Finset M⦄, (fun S ↦ closure ↑S = ⊤) y → y ⊆ S → S ⊆ y
r : M
hrS : r ∈ ↑S
hr₀ : r ≠ 1
hrirred : ¬IsUnit r → ∃ x x_1, ∃ (_ : r = x * x_1), ¬IsUnit x ∧ ¬IsUnit x_1
a b : M
hr✝ : r = a * b
ha : ¬IsUnit a
hb : ¬IsUnit b
m n : M → ℕ
hm : a = r ^ m r * ∏ s ∈ S \ {r}, s ^ m s
hn : b = r ^ n r * ∏ s ∈ S \ {r}, s ^ n s
hr : m r + n r = 0
hr' : r = (∏ s ∈ S \ {r}, s ^ m s) * ∏ s ∈ S \ {r}, s ^ n s
s : M
hs : s ∈ S \ {r}
⊢ s ∈ ↑S \ {(∏ s ∈ S \ {r}, s ^ m s) * ∏ s ∈ S \ {r}, s ^ n s}
Module: Mathlib.LinearAlgebra.Matrix.ToLin

Goal state before:
pp: R : Type u_1
inst✝² : Semiring R
l : Type u_2
m : Type u_3
n : Type u_4
inst✝¹ : Fintype m
inst✝ : DecidableEq m
f g : (m → R) →ₗ[R] n → R
⊢ (fun i j ↦ (f + g) ((single R (fun x ↦ R) i) 1) j) =
    (fun i j ↦ f ((single R (fun x ↦ R) i) 1) j) + fun i j ↦ g ((single R (fun x ↦ R) i) 1) j

Tactic:
by
    ext i j
    simp only [Pi.add_apply, LinearMap.add_apply, Matrix.add_apply]

Goal state after:
Module: Mathlib.Algebra.Field.Subfield.Basic

Goal state before:
pp: case neg
K✝ : Type u
L : Type v
M : Type w
inst✝³ : DivisionRing K✝
inst✝² : DivisionRing L
inst✝¹ : DivisionRing M
s✝¹ : Set K✝
K : Type u
inst✝ : Field K
s✝ : Subfield K
s : Set K
nx : K
hnx : nx ∈ Subring.closure s
dx : K
hdx : dx ∈ Subring.closure s
x_mem : nx / dx ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}
ny : K
hny : ny ∈ Subring.closure s
dy : K
hdy : dy ∈ Subring.closure s
y_mem : ny / dy ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}
hx0 : ¬dx = 0
hy0 : ¬dy = 0
⊢ nx / dx + ny / dy ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}

Tactic:
exact
      ⟨nx * dy + dx * ny, Subring.add_mem _ (Subring.mul_mem _ hnx hdy) (Subring.mul_mem _ hdx hny),
        dx * dy, Subring.mul_mem _ hdx hdy, (div_add_div nx ny hx0 hy0).symm⟩

Goal state after:
Module: Mathlib.Algebra.Field.Subfield.Basic

Goal state before:
pp: K✝ : Type u
L : Type v
M : Type w
inst✝³ : DivisionRing K✝
inst✝² : DivisionRing L
inst✝¹ : DivisionRing M
s✝¹ : Set K✝
K : Type u
inst✝ : Field K
s✝ : Subfield K
s : Set K
x y : K
hy : y ∈ Subring.closure s
z : K
hz : z ∈ Subring.closure s
x_eq : y / z = x
⊢ -x ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}

Tactic:
exact ⟨-y, Subring.neg_mem _ hy, z, hz, x_eq ▸ neg_div _ _⟩

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
⊢ (toMatrixAlgEquiv v₁) id = 1

Tactic:
LinearMap.toMatrixAlgEquiv,

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
⊢ (AlgEquiv.ofLinearEquiv (toMatrix v₁ v₁) ⋯ ⋯) id = 1
Module: Mathlib.LinearAlgebra.FreeModule.PID

Goal state before:
pp: case neg
R : Type u_2
inst✝⁵ : CommRing R
M : Type u_3
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : IsPrincipalIdealRing R
inst✝¹ : IsDomain R
ι : Type u_4
inst✝ : Finite ι
b : Basis ι R M
this : DecidableEq M
val✝ : Fintype ι
N : Submodule R M
ih : ∀ N' ≤ N, ∀ x ∈ N, (∀ (c : R), ∀ y ∈ N', c • x + y = 0 → c = 0) → ∃ n, Nonempty (Basis (Fin n) R ↥N')
b' : Basis (Fin (Fintype.card ι)) R ↥⊤ := (b.reindex (Fintype.equivFin ι)).map (LinearEquiv.ofTop ⊤ ⋯).symm
N_bot : ¬N = ⊥
⊢ ∃ n, Nonempty (Basis (Fin n) R ↥N)

Tactic:
obtain ⟨y, -, a, hay, M', -, N', N'_le_N, -, -, ay_ortho, h'⟩ :=
    Submodule.basis_of_pid_aux ⊤ N b' N_bot le_top

Goal state after:
case neg
R : Type u_2
inst✝⁵ : CommRing R
M : Type u_3
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : IsPrincipalIdealRing R
inst✝¹ : IsDomain R
ι : Type u_4
inst✝ : Finite ι
b : Basis ι R M
this : DecidableEq M
val✝ : Fintype ι
N : Submodule R M
ih : ∀ N' ≤ N, ∀ x ∈ N, (∀ (c : R), ∀ y ∈ N', c • x + y = 0 → c = 0) → ∃ n, Nonempty (Basis (Fin n) R ↥N')
b' : Basis (Fin (Fintype.card ι)) R ↥⊤ := (b.reindex (Fintype.equivFin ι)).map (LinearEquiv.ofTop ⊤ ⋯).symm
N_bot : ¬N = ⊥
y : M
a : R
hay : a • y ∈ N
M' N' : Submodule R M
N'_le_N : N' ≤ N
ay_ortho : ∀ (c : R), ∀ z ∈ N', c • a • y + z = 0 → c = 0
h' :
  ∀ (n' : ℕ) (bN' : Basis (Fin n') R ↥N'),
    ∃ bN,
      ∀ (m' : ℕ) (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R ↥M'),
        ∃ (hnm : n' + 1 ≤ m' + 1),
          ∃ bM,
            ∀ (as : Fin n' → R),
              (∀ (i : Fin n'), ↑(bN' i) = as i • ↑(bM' (Fin.castLE hn'm' i))) →
                ∃ as', ∀ (i : Fin (n' + 1)), ↑(bN i) = as' i • ↑(bM (Fin.castLE hnm i))
⊢ ∃ n, Nonempty (Basis (Fin n) R ↥N)
Module: Mathlib.RingTheory.Adjoin.Basic

Goal state before:
pp: R : Type uR
S : Type uS
A : Type uA
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring S
inst✝⁴ : Semiring A
inst✝³ : Algebra R S
inst✝² : Algebra R A
inst✝¹ : Algebra S A
inst✝ : IsScalarTower R S A
s : Set A
⊢ ↑(adjoin R s) ⊆ ↑(adjoin S s)

Tactic:
· rw [← Subalgebra.coe_restrictScalars R (S := S), SetLike.coe_subset_coe]
    exact adjoin_le subset_adjoin

Goal state after:
Module: Mathlib.RingTheory.Adjoin.Basic

Goal state before:
pp: R : Type uR
S : Type uS
A : Type uA
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring S
inst✝⁴ : Semiring A
inst✝³ : Algebra R S
inst✝² : Algebra R A
inst✝¹ : Algebra S A
inst✝ : IsScalarTower R S A
s : Set A
x : S
⊢ (IsScalarTower.toAlgHom R S A).toRingHom x ∈ Subalgebra.restrictScalars R (adjoin S s)

Tactic:
exact algebraMap_mem (adjoin S s) x

Goal state after:
Module: Mathlib.RingTheory.Localization.FractionRing

Goal state before:
pp: R : Type u_1
inst✝¹⁰ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
P : Type u_3
inst✝⁷ : CommRing P
A : Type u_4
inst✝⁶ : CommRing A
K : Type u_5
inst✝⁵ : CommRing K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : IsDomain A
⊢ IsFractionRing.inv A 0 = 0

Tactic:
IsFractionRing.inv

Goal state after:
R : Type u_1
inst✝¹⁰ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
P : Type u_3
inst✝⁷ : CommRing P
A : Type u_4
inst✝⁶ : CommRing A
K : Type u_5
inst✝⁵ : CommRing K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : IsDomain A
⊢ (if h : 0 = 0 then 0 else mk' K ↑(sec A⁰ 0).2 ⟨(sec A⁰ 0).1, ⋯⟩) = 0
Module: Mathlib.RingTheory.Adjoin.Basic

Goal state before:
pp: R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
s : NonUnitalSubalgebra R A
⊢ R ∙ 1 ⊔ span R ↑(Subsemigroup.closure ↑s) = R ∙ 1 ⊔ s.toSubmodule

Tactic:
← NonUnitalAlgebra.adjoin_eq_span,

Goal state after:
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
s : NonUnitalSubalgebra R A
⊢ R ∙ 1 ⊔ (NonUnitalAlgebra.adjoin R ↑s).toSubmodule = R ∙ 1 ⊔ s.toSubmodule
Module: Mathlib.RingTheory.Localization.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
M : Submonoid R
hM : M ≤ nonZeroDivisors R
a c : R
b d : ↥M
x✝ : ∃ a_1 ∈ M, a_1 * (↑d * a) = a_1 * (↑b * c)
u : R
hu : u ∈ M
h : u * (↑d * a) = u * (↑b * c)
hu' : ∀ (x : R), u * x = 0 → x = 0
⊢ a * ↑d = ↑b * c

Tactic:
rw [← sub_eq_zero]

Goal state after:
R : Type u_1
inst✝ : CommRing R
M : Submonoid R
hM : M ≤ nonZeroDivisors R
a c : R
b d : ↥M
x✝ : ∃ a_1 ∈ M, a_1 * (↑d * a) = a_1 * (↑b * c)
u : R
hu : u ∈ M
h : u * (↑d * a) = u * (↑b * c)
hu' : ∀ (x : R), u * x = 0 → x = 0
⊢ a * ↑d - ↑b * c = 0
Module: Mathlib.RingTheory.Localization.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Semiring A
inst✝⁹ : Semiring B
S : Type u_4
inst✝⁸ : CommSemiring S
inst✝⁷ : Algebra R S
M : Submonoid R
inst✝⁶ : IsLocalization M S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f✝ : A →ₐ[R] B
f : S →ₐ[R] B := f✝.comp (IsScalarTower.toAlgHom R S A)
g : S →ₐ[R] B := IsScalarTower.toAlgHom R S B
this : f.comp (algebraMap R S) = g.comp (algebraMap R S)
⊢ f = g

Tactic:
apply IsLocalization.algHom_ext M

Goal state after:
R : Type u_1
A : Type u_2
B : Type u_3
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Semiring A
inst✝⁹ : Semiring B
S : Type u_4
inst✝⁸ : CommSemiring S
inst✝⁷ : Algebra R S
M : Submonoid R
inst✝⁶ : IsLocalization M S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f✝ : A →ₐ[R] B
f : S →ₐ[R] B := ⋯
g : S →ₐ[R] B := ⋯
this : f.comp (algebraMap R S) = g.comp (algebraMap R S)
⊢ f.comp (Algebra.algHom R R S) = g.comp (Algebra.algHom R R S)