mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Localization.AtPrime.Basic

Goal state before:
pp: case h
R✝ : Type u_1
inst✝²⁷ : CommSemiring R✝
S✝ : Type u_2
inst✝²⁶ : CommSemiring S✝
inst✝²⁵ : Algebra R✝ S✝
P✝ : Type u_3
inst✝²⁴ : CommSemiring P✝
p✝ : Ideal R✝
inst✝²³ : p✝.IsPrime
Rₚ✝ : Type u_4
inst✝²² : CommSemiring Rₚ✝
inst✝²¹ : Algebra R✝ Rₚ✝
inst✝²⁰ : IsLocalization.AtPrime Rₚ✝ p✝
inst✝¹⁹ : IsLocalRing Rₚ✝
Sₚ✝ : Type u_5
inst✝¹⁸ : CommSemiring Sₚ✝
inst✝¹⁷ : Algebra S✝ Sₚ✝
inst✝¹⁶ : IsLocalization (algebraMapSubmonoid S✝ p✝.primeCompl) Sₚ✝
inst✝¹⁵ : Algebra Rₚ✝ Sₚ✝
P : Ideal S✝
S : Type u_6
R : Type u_7
inst✝¹⁴ : CommRing R
p : Ideal R
inst✝¹³ : p.IsMaximal
Rₚ : Type u_8
inst✝¹² : CommRing Rₚ
inst✝¹¹ : Algebra R Rₚ
inst✝¹⁰ : IsLocalization.AtPrime Rₚ p
inst✝⁹ : IsLocalRing Rₚ
Sₚ : Type u_9
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
h : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
x : S
s : ↥(algebraMapSubmonoid S p.primeCompl)
α : R
hα : α ∉ p
e : (algebraMap R S) α = ↑s
β γ : R
hγ : γ ∈ p
hβ : α * β = 1 + γ
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
← sub_eq_zero,

Goal state after:
case h
R✝ : Type u_1
inst✝²⁷ : CommSemiring R✝
S✝ : Type u_2
inst✝²⁶ : CommSemiring S✝
inst✝²⁵ : Algebra R✝ S✝
P✝ : Type u_3
inst✝²⁴ : CommSemiring P✝
p✝ : Ideal R✝
inst✝²³ : p✝.IsPrime
Rₚ✝ : Type u_4
inst✝²² : CommSemiring Rₚ✝
inst✝²¹ : Algebra R✝ Rₚ✝
inst✝²⁰ : IsLocalization.AtPrime Rₚ✝ p✝
inst✝¹⁹ : IsLocalRing Rₚ✝
Sₚ✝ : Type u_5
inst✝¹⁸ : CommSemiring Sₚ✝
inst✝¹⁷ : Algebra S✝ Sₚ✝
inst✝¹⁶ : IsLocalization (algebraMapSubmonoid S✝ p✝.primeCompl) Sₚ✝
inst✝¹⁵ : Algebra Rₚ✝ Sₚ✝
P : Ideal S✝
S : Type u_6
R : Type u_7
inst✝¹⁴ : CommRing R
p : Ideal R
inst✝¹³ : p.IsMaximal
Rₚ : Type u_8
inst✝¹² : CommRing Rₚ
inst✝¹¹ : Algebra R Rₚ
inst✝¹⁰ : IsLocalization.AtPrime Rₚ p
inst✝⁹ : IsLocalRing Rₚ
Sₚ : Type u_9
inst✝⁸ : CommRing S
inst✝⁷ : Algebra R S
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
h : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
x : S
s : ↥(algebraMapSubmonoid S p.primeCompl)
α : R
hα : α ∉ p
e : (algebraMap R S) α = ↑s
β γ : R
hγ : γ ∈ p
hβ : α * β = 1 + γ
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Module: Mathlib.LinearAlgebra.Dimension.Localization

Goal state before:
pp: case a
R : Type uR
S : Type uS
N : Type uN
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : Algebra R S
inst✝² : Module S N
inst✝¹ : IsScalarTower R S N
p : Submonoid R
inst✝ : IsLocalization p S
hp : p ≤ R⁰
h✝ : Nontrivial R
inj : Function.Injective ⇑(algebraMap R S)
s : Set N
hs : LinearIndepOn R id s
⊢ #↑↑⟨s, hs⟩ ≤ Module.rank S N

Tactic:
have := inj.nontrivial

Goal state after:
case a
R : Type uR
S : Type uS
N : Type uN
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : Algebra R S
inst✝² : Module S N
inst✝¹ : IsScalarTower R S N
p : Submonoid R
inst✝ : IsLocalization p S
hp : p ≤ R⁰
h✝ : Nontrivial R
inj : Function.Injective ⇑(algebraMap R S)
s : Set N
hs : LinearIndepOn R id s
this : Nontrivial S
⊢ #↑↑⟨s, hs⟩ ≤ Module.rank S N
Module: Mathlib.Algebra.Polynomial.Laurent

Goal state before:
pp: case refine_1
R : Type u_1
inst✝ : Semiring R
⊢ ∀ (p q : R[T;T⁻¹]),
    (∃ n f', toLaurent f' = p * T ↑n) → (∃ n f', toLaurent f' = q * T ↑n) → ∃ n f', toLaurent f' = (p + q) * T ↑n

Tactic:
rintro f g ⟨m, fn, hf⟩ ⟨n, gn, hg⟩

Goal state after:
case refine_1
R : Type u_1
inst✝ : Semiring R
f g : R[T;T⁻¹]
m : ℕ
fn : R[X]
hf : toLaurent fn = f * T ↑m
n : ℕ
gn : R[X]
hg : toLaurent gn = g * T ↑n
⊢ ∃ n f', toLaurent f' = (f + g) * T ↑n
Module: Mathlib.LinearAlgebra.Matrix.Adjugate

Goal state before:
pp: n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A : Matrix n n α
b : n → α
i : n
h : ∀ (j : n), b j = A j i
⊢ A.cramer b = Pi.single i A.det

Tactic:
← transpose_transpose A,

Goal state after:
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A : Matrix n n α
b : n → α
i : n
h : ∀ (j : n), b j = A j i
⊢ Aᵀᵀ.cramer b = Pi.single i Aᵀᵀ.det
Module: Mathlib.Algebra.Polynomial.Laurent

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
a : R
⊢ (C a).degree = (C a * T 0).degree

Tactic:
T_zero,

Goal state after:
R : Type u_1
inst✝ : Semiring R
a : R
⊢ (C a).degree = (C a * 1).degree
Module: Mathlib.Algebra.Polynomial.Laurent

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
p : R[X]
n : ℕ
⊢ IsLocalization.mk' R[T;T⁻¹] p ⟨X ^ n, ⋯⟩ * T ↑n = toLaurent p * T 0

Tactic:
T_zero,

Goal state after:
R : Type u_1
inst✝ : CommSemiring R
p : R[X]
n : ℕ
⊢ IsLocalization.mk' R[T;T⁻¹] p ⟨X ^ n, ⋯⟩ * T ↑n = toLaurent p * 1
Module: Mathlib.Algebra.Polynomial.Laurent

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
n : ℕ
⊢ IsLocalization.mk' R[T;T⁻¹] 1 ⟨X ^ n, ⋯⟩ = T (-↑n)

Tactic:
by
  rw [mk'_eq 1 n, toLaurent_one, one_mul]

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

Goal state before:
pp: n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A : Matrix n n α
b : n → α
⊢ (A.det • 1) *ᵥ b = A.det • b

Tactic:
smul_mulVec,

Goal state after:
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A : Matrix n n α
b : n → α
⊢ A.det • 1 *ᵥ b = A.det • b
Module: Mathlib.LinearAlgebra.Matrix.Transvection

Goal state before:
pp: 𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
i : Fin r
H : ∀ k ≤ r, ((List.drop k (listTransvecCol M)).prod * M) (inl i) (inr ()) = if k ≤ ↑i then 0 else M (inl i) (inr ())
⊢ ((listTransvecCol M).prod * M) (inl i) (inr ()) = 0

Tactic:
by
    simpa [List.drop] using H 0

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

Goal state before:
pp: n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A : Matrix n n α
i : n
this✝ : Nonempty n
n' : ℕ
hn' : Fintype.card n = n'.succ
x✝ : Trunc (n ≃ Fin n'.succ)
e : n ≃ Fin n'.succ
A' : Matrix (Fin n'.succ) (Fin n'.succ) α := (reindex e e) A
this : A.det = ∑ x, (reindex e e) A (e i) x * (reindex e e) A.adjugate x (e i)
⊢ A.det = ∑ j, A i j * A.adjugate j i

Tactic:
reindex_apply,

Goal state after:
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A : Matrix n n α
i : n
this✝ : Nonempty n
n' : ℕ
hn' : Fintype.card n = n'.succ
x✝ : Trunc (n ≃ Fin n'.succ)
e : n ≃ Fin n'.succ
A' : Matrix (Fin n'.succ) (Fin n'.succ) α := (reindex e e) A
this : A.det = ∑ x, A.submatrix (⇑e.symm) (⇑e.symm) (e i) x * A.adjugate.submatrix (⇑e.symm) (⇑e.symm) x (e i)
⊢ A.det = ∑ j, A i j * A.adjugate j i
Module: Mathlib.LinearAlgebra.Matrix.Invertible

Goal state before:
pp: m : Type u_1
n : Type u_2
α : Type u_3
inst✝⁷ : Fintype n
inst✝⁶ : DecidableEq n
inst✝⁵ : Fintype m
inst✝⁴ : DecidableEq m
inst✝³ : Ring α
A : Matrix n n α
U : Matrix n m α
C : Matrix m m α
V : Matrix m n α
inst✝² : Invertible A
inst✝¹ : Invertible C
inst✝ : Invertible (⅟C + V * ⅟A * U)
⊢ 1 + ⅟A * U * C * V - ⅟A * U * ⅟(⅟C + V * ⅟A * U) * (V + V * ⅟A * U * C * V) =
    1 + ⅟A * U * C * V - ⅟A * U * ⅟(⅟C + V * ⅟A * U) * (⅟C + V * ⅟A * U) * C * V

Tactic:
simp only [Matrix.mul_add, Matrix.add_mul, ← Matrix.mul_assoc,
        Matrix.invOf_mul_cancel_right]

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

Goal state before:
pp: l : Type u_1
m : Type u
n : Type u'
α : Type v
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : CommRing α
A B : Matrix n n α
inst✝ : Invertible A.det
⊢ A * ⅟A.det • A.adjugate = 1

Tactic:
rw [mul_smul_comm, mul_adjugate, smul_smul, invOf_mul_self, one_smul]

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

Goal state before:
pp: case pos.intro
n : Type u'
α : Type v
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix n n α
h_det : IsUnit A.det
val✝ : Invertible A
⊢ ⅟A = A⁻¹ʳ

Tactic:
Ring.inverse_invertible

Goal state after:
case pos.intro
n : Type u'
α : Type v
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix n n α
h_det : IsUnit A.det
val✝ : Invertible A
⊢ ⅟A = ⅟A
Module: Mathlib.LinearAlgebra.Matrix.Adjugate

Goal state before:
pp: n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A : Matrix n n α
h : Fintype.card n ≠ 1
n✝ : ℕ
h_card : Fintype.card n = n✝ + 1 + 1
A' : Matrix n n (MvPolynomial (n × n) ℤ) := mvPolynomialX n n ℤ
this : A'.adjugate.adjugate = A'.det ^ (Fintype.card n - 2) • A'
⊢ (MvPolynomial.aeval fun p ↦ A p.1 p.2).mapMatrix (A'.det ^ (Fintype.card n - 2) • A') =
    (MvPolynomial.aeval fun p ↦ A p.1 p.2) (mvPolynomialX n n ℤ).det ^ (Fintype.card n - 2) •
      (MvPolynomial.aeval fun p ↦ A p.1 p.2).mapMatrix (mvPolynomialX n n ℤ)

Tactic:
← map_pow (MvPolynomial.aeval fun p : n × n ↦ A p.1 p.2),

Goal state after:
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A : Matrix n n α
h : Fintype.card n ≠ 1
n✝ : ℕ
h_card : Fintype.card n = n✝ + 1 + 1
A' : Matrix n n (MvPolynomial (n × n) ℤ) := mvPolynomialX n n ℤ
this : A'.adjugate.adjugate = A'.det ^ (Fintype.card n - 2) • A'
⊢ (MvPolynomial.aeval fun p ↦ A p.1 p.2).mapMatrix (A'.det ^ (Fintype.card n - 2) • A') =
    (MvPolynomial.aeval fun p ↦ A p.1 p.2) ((mvPolynomialX n n ℤ).det ^ (Fintype.card n - 2)) •
      (MvPolynomial.aeval fun p ↦ A p.1 p.2).mapMatrix (mvPolynomialX n n ℤ)
Module: Mathlib.LinearAlgebra.Matrix.Transvection

Goal state before:
pp: case neg.inr.hnc
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
i : Fin r
n : ℕ
IH : n ≤ r → (M * (List.take n (listTransvecRow M)).prod) (inr ()) (inl i) = if n ≤ ↑i then M (inr ()) (inl i) else 0
hk : n + 1 ≤ r
hnr : n < r
n' : Fin r := ⟨n, hnr⟩
A : (listTransvecRow M)[n]? = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))
h : ¬n' = i
hni : n ≠ ↑i
hi : ↑i < n + 1
⊢ ¬n ≤ ↑i

Tactic:
simpa only [hni.symm, not_le, or_false] using! Nat.lt_succ_iff_lt_or_eq.1 hi

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

Goal state before:
pp: m : Type u
α : Type v
inst✝² : CommRing α
inst✝¹ : Fintype m
inst✝ : DecidableEq m
M : Matrix m m α
h : IsUnit M
N : Matrix m m α
⊢ (M * N * M⁻¹).trace = N.trace

Tactic:
rw [← h.unit_spec, ← coe_units_inv, trace_units_conj]

Goal state after:
Module: Mathlib.RingTheory.MatrixPolynomialAlgebra

Goal state before:
pp: case refine_3
R : Type u_1
inst✝² : CommSemiring R
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
m : Matrix n n R[X]
k : ℕ
i j i' j' : n
x : R[X]
⊢ (matPolyEquiv (single i' j' x)).coeff k i j = (single i' j' x i j).coeff k

Tactic:
rw [matPolyEquiv_coeff_apply_aux_2]

Goal state after:
case refine_3
R : Type u_1
inst✝² : CommSemiring R
n : Type w
inst✝¹ : DecidableEq n
inst✝ : Fintype n
m : Matrix n n R[X]
k : ℕ
i j i' j' : n
x : R[X]
⊢ single i' j' (x.coeff k) i j = (single i' j' x i j).coeff k
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
m : Type u_3
n : Type u_4
inst✝³ : DecidableEq m
inst✝² : DecidableEq n
inst✝¹ : Fintype m
inst✝ : Fintype n
A : Matrix m n R
B : Matrix n m R
M : Matrix (m ⊕ n) (m ⊕ n) R[X] := fromBlocks ((scalar m) X) (A.map ⇑C) (B.map ⇑C) 1
N : Matrix (m ⊕ n) (m ⊕ n) R[X] := fromBlocks (-1) 0 (B.map ⇑C) (-(scalar n) X)
hMN : M * N = fromBlocks (-(scalar m) X + (A * B).map ⇑C) (-X • A.map ⇑C) 0 (-(scalar n) X)
hNM : N * M = fromBlocks (-(scalar m) X) (-A.map ⇑C) 0 ((B * A).map ⇑C - (scalar n) X)
⊢ (M * N).det = (-1) ^ (Fintype.card m + Fintype.card n) * (X ^ Fintype.card n * ((scalar m) X - (A * B).map ⇑C).det)

Tactic:
rw [hMN, det_fromBlocks_zero₂₁, neg_add_eq_sub, ← neg_sub, det_neg]

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
m : Type u_3
n : Type u_4
inst✝³ : DecidableEq m
inst✝² : DecidableEq n
inst✝¹ : Fintype m
inst✝ : Fintype n
A : Matrix m n R
B : Matrix n m R
M : Matrix (m ⊕ n) (m ⊕ n) R[X] := fromBlocks ((scalar m) X) (A.map ⇑C) (B.map ⇑C) 1
N : Matrix (m ⊕ n) (m ⊕ n) R[X] := fromBlocks (-1) 0 (B.map ⇑C) (-(scalar n) X)
hMN : M * N = fromBlocks (-(scalar m) X + (A * B).map ⇑C) (-X • A.map ⇑C) 0 (-(scalar n) X)
hNM : N * M = fromBlocks (-(scalar m) X) (-A.map ⇑C) 0 ((B * A).map ⇑C - (scalar n) X)
⊢ (-1) ^ Fintype.card m * ((scalar m) X - (A * B).map ⇑C).det * (-(scalar n) X).det =
    (-1) ^ (Fintype.card m + Fintype.card n) * (X ^ Fintype.card n * ((scalar m) X - (A * B).map ⇑C).det)
Module: Mathlib.LinearAlgebra.Matrix.Transvection

Goal state before:
pp: n : Type u_1
𝕜 : Type u_3
inst✝² : Field 𝕜
inst✝¹ : DecidableEq n
inst✝ : Fintype n
P : Matrix n n 𝕜 → Prop
M : Matrix n n 𝕜
hMdet : M.det ≠ 0
hdiag : ∀ (D : n → 𝕜), (diagonal D).det ≠ 0 → P (diagonal D)
htransvec : ∀ (t : TransvectionStruct n 𝕜), P t.toMatrix
hmul : ∀ (A B : Matrix n n 𝕜), A.det ≠ 0 → B.det ≠ 0 → P A → P B → P (A * B)
Q : Matrix n n 𝕜 → Prop := fun N ↦ N.det ≠ 0 ∧ P N
A B : Matrix n n 𝕜
QA : Q A
QB : Q B
⊢ (A * B).det ≠ 0

Tactic:
by simp [QA.1, QB.1]

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

Goal state before:
pp: case hmul
n : Type u_1
𝕜 : Type u_3
inst✝² : Field 𝕜
inst✝¹ : DecidableEq n
inst✝ : Fintype n
P : Matrix n n 𝕜 → Prop
M : Matrix n n 𝕜
hMdet : M.det ≠ 0
hdiag : ∀ (D : n → 𝕜), (diagonal D).det ≠ 0 → P (diagonal D)
htransvec : ∀ (t : TransvectionStruct n 𝕜), P t.toMatrix
hmul : ∀ (A B : Matrix n n 𝕜), A.det ≠ 0 → B.det ≠ 0 → P A → P B → P (A * B)
Q : Matrix n n 𝕜 → Prop := ⋯
⊢ ∀ (A B : Matrix n n 𝕜), Q A → Q B → Q (A * B)

Tactic:
· intro A B QA QB
      exact ⟨by simp [QA.1, QB.1], hmul A B QA.1 QB.1 QA.2 QB.2⟩

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Nilpotent

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
P : R[X]
| P

Tactic:
rw [← modByMonic_add_div P X]

Goal state after:
R : Type u_1
inst✝ : CommRing R
P : R[X]
| P %ₘ X + X * (P /ₘ X)
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: case e'_3
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
e : Matrix n n R ≃ₗ[R] Matrix (Fin (Fintype.card n)) (Fin (Fintype.card n)) R :=
  reindexLinearEquiv R R (Fintype.equivFin n) (Fintype.equivFin n)
⊢ ∑ i, M.diag ((Fintype.equivFin n).symm i) = ∑ x, ((reindex (Fintype.equivFin n) (Fintype.equivFin n)) M).diag x

Tactic:
reindex_apply,

Goal state after:
case e'_3
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
e : Matrix n n R ≃ₗ[R] Matrix (Fin (Fintype.card n)) (Fin (Fintype.card n)) R :=
  reindexLinearEquiv R R (Fintype.equivFin n) (Fintype.equivFin n)
⊢ ∑ i, M.diag ((Fintype.equivFin n).symm i) =
    ∑ x, (M.submatrix ⇑(Fintype.equivFin n).symm ⇑(Fintype.equivFin n).symm).diag x
Module: Mathlib.RingTheory.Polynomial.ScaleRoots

Goal state before:
pp: case pos
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h₁ : p.natDegree ≤ n
h₂ : n = p.natDegree
⊢ p.coeff n = p.leadingCoeff

Tactic:
subst h₂; rfl

Goal state after:
Module: Mathlib.RingTheory.Polynomial.ScaleRoots

Goal state before:
pp: case inr.inr
R : Type u_1
inst✝ : CommRing R
p : R[X]
r a : R
hr : IsLeftRegular r
h✝ : Nontrivial R
hp : p ≠ 0
q : R[X]
e : p = (X - C a) ^ rootMultiplicity a p * q
hq : ¬X - C a ∣ q
⊢ rootMultiplicity (r * a) (p.scaleRoots r) = rootMultiplicity a p

Tactic:
have hq0 : q ≠ 0 := by contrapose hp; simp_all

Goal state after:
case inr.inr
R : Type u_1
inst✝ : CommRing R
p : R[X]
r a : R
hr : IsLeftRegular r
h✝ : Nontrivial R
hp : p ≠ 0
q : R[X]
e : p = (X - C a) ^ rootMultiplicity a p * q
hq : ¬X - C a ∣ q
hq0 : q ≠ 0
⊢ rootMultiplicity (r * a) (p.scaleRoots r) = rootMultiplicity a p
Module: Mathlib.Algebra.Polynomial.Splits

Goal state before:
pp: case inl
R : Type u_1
inst✝ : CommSemiring R
f g : R[X]
hf : f.Splits
hg✝ : g.natDegree ≤ 1
h : Invertible g.leadingCoeff
hg : g.natDegree < 1
⊢ (f.comp g).Splits

Tactic:
rw [eq_C_of_natDegree_eq_zero (Nat.lt_one_iff.mp hg)]

Goal state after:
case inl
R : Type u_1
inst✝ : CommSemiring R
f g : R[X]
hf : f.Splits
hg✝ : g.natDegree ≤ 1
h : Invertible g.leadingCoeff
hg : g.natDegree < 1
⊢ (f.comp (C (g.coeff 0))).Splits
Module: Mathlib.Algebra.Polynomial.Splits

Goal state before:
pp: case inr
R : Type u_1
inst✝ : Ring R
f : R[X]
hf : f.Splits
a : R
⊢ ((X + C a).comp (-X)).Splits

Tactic:
rw [add_comp, X_comp, C_comp, neg_add_eq_sub, ← neg_sub]

Goal state after:
case inr
R : Type u_1
inst✝ : Ring R
f : R[X]
hf : f.Splits
a : R
⊢ (-(X - C a)).Splits
Module: Mathlib.Algebra.Polynomial.Splits

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
f : R[X]
inst✝ : IsDomain R
hf : f.Splits
⊢ f.nextCoeff = -f.leadingCoeff * f.roots.sum

Tactic:
conv_lhs => rw [hf.eq_prod_roots]
  simp [Multiset.sum_map_neg', monic_X_sub_C, Monic.nextCoeff_multiset_prod]

Goal state after:
Module: Mathlib.Algebra.Polynomial.Splits

Goal state before:
pp: case pos
R : Type u_1
inst✝¹ : CommRing R
f : R[X]
inst✝ : IsDomain R
a : R
hf : ((X - C a) * f).Splits
hf₀ : f = 0
⊢ f.Splits

pp: case neg
R : Type u_1
inst✝¹ : CommRing R
f : R[X]
inst✝ : IsDomain R
a : R
hf : ((X - C a) * f).Splits
hf₀ : ¬f = 0
⊢ f.Splits

Tactic:
· aesop

Goal state after:
case neg
R : Type u_1
inst✝¹ : CommRing R
f : R[X]
inst✝ : IsDomain R
a : R
hf : ((X - C a) * f).Splits
hf₀ : ¬f = 0
⊢ f.Splits
Module: Mathlib.RingTheory.Localization.Integral

Goal state before:
pp: R : Type u_5
S : Type u_6
inst✝⁷ : CommSemiring R
inst✝⁶ : CommSemiring S
f : R →+* S
x : S
p : R[X]
hf : eval₂ f x p = 0
M : Submonoid R
hM : p.leadingCoeff ∈ M
Rₘ : Type u_7
Sₘ : Type u_8
inst✝⁵ : CommRing Rₘ
inst✝⁴ : CommRing Sₘ
inst✝³ : Algebra R Rₘ
inst✝² : IsLocalization M Rₘ
inst✝¹ : Algebra S Sₘ
inst✝ : IsLocalization (Submonoid.map f M) Sₘ
⊢ (IsLocalization.map Sₘ f ⋯).IsIntegralElem ((algebraMap S Sₘ) x)

Tactic:
by_cases triv : (1 : Rₘ) = 0

Goal state after:
case pos
R : Type u_5
S : Type u_6
inst✝⁷ : CommSemiring R
inst✝⁶ : CommSemiring S
f : R →+* S
x : S
p : R[X]
hf : eval₂ f x p = 0
M : Submonoid R
hM : p.leadingCoeff ∈ M
Rₘ : Type u_7
Sₘ : Type u_8
inst✝⁵ : CommRing Rₘ
inst✝⁴ : CommRing Sₘ
inst✝³ : Algebra R Rₘ
inst✝² : IsLocalization M Rₘ
inst✝¹ : Algebra S Sₘ
inst✝ : IsLocalization (Submonoid.map f M) Sₘ
triv : 1 = 0
⊢ (IsLocalization.map Sₘ f ⋯).IsIntegralElem ((algebraMap S Sₘ) x)

case neg
R : Type u_5
S : Type u_6
inst✝⁷ : CommSemiring R
inst✝⁶ : CommSemiring S
f : R →+* S
x : S
p : R[X]
hf : eval₂ f x p = 0
M : Submonoid R
hM : p.leadingCoeff ∈ M
Rₘ : Type u_7
Sₘ : Type u_8
inst✝⁵ : CommRing Rₘ
inst✝⁴ : CommRing Sₘ
inst✝³ : Algebra R Rₘ
inst✝² : IsLocalization M Rₘ
inst✝¹ : Algebra S Sₘ
inst✝ : IsLocalization (Submonoid.map f M) Sₘ
triv : ¬1 = 0
⊢ (IsLocalization.map Sₘ f ⋯).IsIntegralElem ((algebraMap S Sₘ) x)
Module: Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic

Goal state before:
pp: R : Type u_1
S : Type u_4
T : Type u_5
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : CommRing T
f : R →+* S
g : S →+* T
x : T
h : (g.comp f).IsIntegralElem x
p : R[X]
hp : p.Monic
hp' : eval₂ (g.comp f) x p = 0
⊢ eval₂ g x (Polynomial.map f p) = 0

Tactic:
rwa [← eval₂_map] at hp'

Goal state after:
Module: Mathlib.Algebra.Polynomial.Splits

Goal state before:
pp: case inl
R : Type u_1
inst✝ : Field R
⊢ Splits 0

pp: case inr
R : Type u_1
inst✝ : Field R
f : R[X]
hf : ∀ {g : R[X]}, Irreducible g → g ∣ f → g.degree = 1
⊢ f.Splits

Tactic:
· aesop

Goal state after:
case inr
R : Type u_1
inst✝ : Field R
f : R[X]
hf : ∀ {g : R[X]}, Irreducible g → g ∣ f → g.degree = 1
⊢ f.Splits
Module: Mathlib.RingTheory.Flat.FaithfullyFlat.Basic

Goal state before:
pp: R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : FaithfullyFlat R M
N : Type u_1
inst✝¹ : AddCommGroup N
inst✝ : Module R N
⊢ Subsingleton (N ⊗[R] M) → Subsingleton N

Tactic:
contrapose!

Goal state after:
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : FaithfullyFlat R M
N : Type u_1
inst✝¹ : AddCommGroup N
inst✝ : Module R N
⊢ Nontrivial N → Nontrivial (N ⊗[R] M)
Module: Mathlib.RingTheory.Flat.FaithfullyFlat.Basic

Goal state before:
pp: R : Type u
M : Type v
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : FaithfullyFlat R M
N : Type u_1
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : Subsingleton (M ⊗[R] N)
this : Subsingleton (N ⊗[R] M)
⊢ Subsingleton N

Tactic:
apply rTensor_reflects_triviality R M

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

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
S : Type u_2
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Algebra.IsIntegral R S
I : Ideal S
hI : I.IsMaximal
this : (comap (algebraMap R S) I).IsPrime
⊢ IsField (R ⧸ comap (algebraMap R S) I)

Tactic:
exact isField_of_isIntegral_of_isField
    algebraMap_quotient_injective (by rwa [← Quotient.maximal_ideal_iff_isField_quotient])

Goal state after:
Module: Mathlib.RingTheory.Flat.FaithfullyFlat.Basic

Goal state before:
pp: R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_1
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : FaithfullyFlat R M
⊢ Nontrivial (M ⊗[R] N) ↔ Nontrivial N

Tactic:
contrapose!

Goal state after:
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_1
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : FaithfullyFlat R M
⊢ Subsingleton (M ⊗[R] N) ↔ Subsingleton N
Module: Mathlib.RingTheory.Flat.FaithfullyFlat.Basic

Goal state before:
pp: R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_1
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : FaithfullyFlat R N
⊢ Nontrivial (M ⊗[R] N) ↔ Nontrivial M

Tactic:
contrapose!

Goal state after:
R : Type u
M : Type v
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_1
inst✝² : AddCommGroup N
inst✝¹ : Module R N
inst✝ : FaithfullyFlat R N
⊢ Subsingleton (M ⊗[R] N) ↔ Subsingleton M
Module: Mathlib.RingTheory.Spectrum.Prime.Basic

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
⊢ IsEmpty (PrimeSpectrum R) ↔ Subsingleton R

Tactic:
contrapose!

Goal state after:
R : Type u
inst✝ : CommSemiring R
⊢ Nonempty (PrimeSpectrum R) ↔ Nontrivial R
Module: Mathlib.RingTheory.Spectrum.Prime.Basic

Goal state before:
pp: case mp
R : Type u
inst✝ : CommSemiring R
I : Ideal R
⊢ zeroLocus ↑I = ∅ → I = ⊤

Tactic:
contrapose!

Goal state after:
case mp
R : Type u
inst✝ : CommSemiring R
I : Ideal R
⊢ I ≠ ⊤ → (zeroLocus ↑I).Nonempty