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

Goal state before:
pp: case mp
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
a✝ b✝ : Type u
c✝ : CommRing a✝
d✝ : CommRing b✝
e✝ : a✝ →+* b✝
⊢ (∀ (s : Set b✝), Ideal.span s = ⊤ → (∀ (r : ↑s), P ((algebraMap b✝ (Localization.Away ↑r)).comp e✝)) → P e✝) →
    ∀ (s : Finset b✝), Ideal.span ↑s = ⊤ → (∀ (r : ↥s), P ((algebraMap b✝ (Localization.Away ↑r)).comp e✝)) → P e✝

Tactic:
intro h

Goal state after:
case mp
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
a✝ b✝ : Type u
c✝ : CommRing a✝
d✝ : CommRing b✝
e✝ : a✝ →+* b✝
h : ∀ (s : Set b✝), Ideal.span s = ⊤ → (∀ (r : ↑s), P ((algebraMap b✝ (Localization.Away ↑r)).comp e✝)) → P e✝
⊢ ∀ (s : Finset b✝), Ideal.span ↑s = ⊤ → (∀ (r : ↥s), P ((algebraMap b✝ (Localization.Away ↑r)).comp e✝)) → P e✝
Module: Mathlib.RingTheory.LocalProperties.Basic

Goal state before:
pp: case mpr
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
a✝ b✝ : Type u
c✝ : CommRing a✝
d✝ : CommRing b✝
e✝ : a✝ →+* b✝
⊢ (∀ (s : Finset b✝), Ideal.span ↑s = ⊤ → (∀ (r : ↥s), P ((algebraMap b✝ (Localization.Away ↑r)).comp e✝)) → P e✝) →
    ∀ (s : Set b✝), Ideal.span s = ⊤ → (∀ (r : ↑s), P ((algebraMap b✝ (Localization.Away ↑r)).comp e✝)) → P e✝

Tactic:
intro h

Goal state after:
case mpr
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
a✝ b✝ : Type u
c✝ : CommRing a✝
d✝ : CommRing b✝
e✝ : a✝ →+* b✝
h : ∀ (s : Finset b✝), Ideal.span ↑s = ⊤ → (∀ (r : ↥s), P ((algebraMap b✝ (Localization.Away ↑r)).comp e✝)) → P e✝
⊢ ∀ (s : Set b✝), Ideal.span s = ⊤ → (∀ (r : ↑s), P ((algebraMap b✝ (Localization.Away ↑r)).comp e✝)) → P e✝
Module: Mathlib.RingTheory.Localization.AtPrime.Basic

Goal state before:
pp: case refine_2
S : Type u_6
R : Type u_7
inst✝⁸ : CommRing R
p : Ideal R
inst✝⁷ : p.IsMaximal
Sₚ : Type u_9
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing Sₚ
inst✝³ : Algebra S Sₚ
inst✝² : Algebra R Sₚ
inst✝¹ : IsLocalization (algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝ : IsScalarTower R S Sₚ
x : S
hx : x ∈ comap (algebraMap S Sₚ) (Ideal.map (algebraMap S Sₚ) pS)
α : R
hα : α ∉ p
hαx : α • x ∈ pS
β : R
hβ : (Ideal.Quotient.mk p) β = ((Ideal.Quotient.mk p) α)⁻¹
⊢ β * α = 1 + (β * α - 1)

Tactic:
· rw [add_sub_cancel]

Goal state after:
Module: Mathlib.RingTheory.Localization.AtPrime.Basic

Goal state before:
pp: 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
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
← map_eq_maximalIdeal p,

Goal state after:
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
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Module: Mathlib.RingTheory.Localization.AtPrime.Basic

Goal state before:
pp: case refine_2
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 : R
hβ : (Ideal.Quotient.mk p) β = ((Ideal.Quotient.mk p) α)⁻¹
⊢ α * β = 1 + (α * β - 1)

Tactic:
· rw [add_sub_cancel]

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

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
⊢ toLaurent X = T 1

Tactic:
by
  have : (Polynomial.X : R[X]) = monomial 1 1 := by simp [← C_mul_X_pow_eq_monomial]
  simp [this, Polynomial.toLaurent_C_mul_T]

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

Goal state before:
pp: m : Type u
n : Type v
α : Type w
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
inst✝ : CommRing α
A : Matrix m m α
e : n ≃ m
b : n → α
⊢ (A.submatrix ⇑e ⇑e).cramer b = A.cramer (b ∘ ⇑e.symm) ∘ ⇑e

Tactic:
by
  ext i
  simp_rw [Function.comp_apply, cramer_apply, updateCol_submatrix_equiv,
    det_submatrix_equiv_self e, Function.comp_def]

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

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
a : R
a0 : a ≠ 0
⊢ (C a * T 0).degree = 0

Tactic:
exact degree_C_mul_T 0 a a0

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
this : b = ∑ i, b i • Pi.single i 1
k : n
⊢ A.cramer (∑ i, b i • Pi.single i 1) k = ((of fun i ↦ Aᵀᵀ.cramer (Pi.single i 1))ᵀ *ᵥ b) k

Tactic:
simp [mulVec, dotProduct, mul_comm]

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

Goal state before:
pp: n : Type v
inst✝³ : DecidableEq n
inst✝² : Fintype n
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
M : Matrix n n R
⊢ f.mapMatrix M.adjugate = (f.mapMatrix M).adjugate

Tactic:
ext i k

Goal state after:
n : Type v
inst✝³ : DecidableEq n
inst✝² : Fintype n
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
M : Matrix n n R
i k : n
⊢ f.mapMatrix M.adjugate i k = (f.mapMatrix M).adjugate i k
Module: Mathlib.LinearAlgebra.Matrix.NonsingularInverse

Goal state before:
pp: n : Type u'
α : Type v
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix n n α
h : IsUnit A.det
⊢ IsUnit Aᵀ.det

Tactic:
rw [det_transpose]

Goal state after:
n : Type u'
α : Type v
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix n n α
h : IsUnit A.det
⊢ IsUnit A.det
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 B : Matrix n n α
g : Matrix n n α → Matrix n n α[X] := fun M ↦ M.map ⇑C + X • 1
f' : Matrix n n α[X] →+* Matrix n n α := (evalRingHom 0).mapMatrix
f'_inv : ∀ (M : Matrix n n α), f' (g M) = M
f'_adj : ∀ (M : Matrix n n α), f' (g M).adjugate = M.adjugate
f'_g_mul : ∀ (M N : Matrix n n α), f' (g M * g N) = M * N
hu : ∀ (M : Matrix n n α), IsRegular (g M).det
⊢ (A * B).adjugate = B.adjugate * A.adjugate

Tactic:
rw [← f'_adj, ← f'_adj, ← f'_adj, ← f'.map_mul, ←
    adjugate_mul_distrib_aux _ _ (hu A).left (hu B).left, RingHom.map_adjugate,
    RingHom.map_adjugate, f'_inv, f'_g_mul]

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

Goal state before:
pp: m : Type u
inst✝² : DecidableEq m
K : Type u_3
inst✝¹ : Field K
inst✝ : Fintype m
A : Matrix m m K
⊢ LinearIndependent K A.col ↔ IsUnit A

Tactic:
by
  rw [← row_transpose, linearIndependent_rows_iff_isUnit, isUnit_transpose]

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

Goal state before:
pp: case e_f.e_a.e_f
m : Type u_3
R : Type v
inst✝³ : CommRing R
inst✝² : DecidableEq m
inst✝¹ : Fintype m
M : Matrix m m R
p : m → Prop
inst✝ : DecidablePred p
σ : Equiv.Perm ({ a // p a } ⊕ { a // ¬p a })
x y : { a // p a } ⊕ { a // ¬p a }
hy : σ x = y
⊢ (reindex (Equiv.sumCompl p).symm (Equiv.sumCompl p).symm) M y x =
    fromBlocks (M.toBlock p p) (M.toBlock p fun j ↦ ¬p j) (M.toBlock (fun j ↦ ¬p j) p)
      (M.toBlock (fun j ↦ ¬p j) fun j ↦ ¬p j) y x

Tactic:
cases x

Goal state after:
case e_f.e_a.e_f.inl
m : Type u_3
R : Type v
inst✝³ : CommRing R
inst✝² : DecidableEq m
inst✝¹ : Fintype m
M : Matrix m m R
p : m → Prop
inst✝ : DecidablePred p
σ : Equiv.Perm ({ a // p a } ⊕ { a // ¬p a })
y : { a // p a } ⊕ { a // ¬p a }
val✝ : { a // p a }
hy : σ (Sum.inl val✝) = y
⊢ (reindex (Equiv.sumCompl p).symm (Equiv.sumCompl p).symm) M y (Sum.inl val✝) =
    fromBlocks (M.toBlock p p) (M.toBlock p fun j ↦ ¬p j) (M.toBlock (fun j ↦ ¬p j) p)
      (M.toBlock (fun j ↦ ¬p j) fun j ↦ ¬p j) y (Sum.inl val✝)

case e_f.e_a.e_f.inr
m : Type u_3
R : Type v
inst✝³ : CommRing R
inst✝² : DecidableEq m
inst✝¹ : Fintype m
M : Matrix m m R
p : m → Prop
inst✝ : DecidablePred p
σ : Equiv.Perm ({ a // p a } ⊕ { a // ¬p a })
y : { a // p a } ⊕ { a // ¬p a }
val✝ : { a // ¬p a }
hy : σ (Sum.inr val✝) = y
⊢ (reindex (Equiv.sumCompl p).symm (Equiv.sumCompl p).symm) M y (Sum.inr val✝) =
    fromBlocks (M.toBlock p p) (M.toBlock p fun j ↦ ¬p j) (M.toBlock (fun j ↦ ¬p j) p)
      (M.toBlock (fun j ↦ ¬p j) fun j ↦ ¬p j) y (Sum.inr val✝)
Module: Mathlib.LinearAlgebra.Matrix.Block

Goal state before:
pp: α : Type u_1
m : Type u_3
R : Type v
M : Matrix m m R
b : m → α
inst✝⁴ : CommRing R
inst✝³ : DecidableEq m
inst✝² : Fintype m
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i ↦ b i < k
⊢ (M⁻¹.toBlock (fun i ↦ k ≤ b i) fun i ↦ b i < k) = 0

Tactic:
let q i := ¬b i < k

Goal state after:
α : Type u_1
m : Type u_3
R : Type v
M : Matrix m m R
b : m → α
inst✝⁴ : CommRing R
inst✝³ : DecidableEq m
inst✝² : Fintype m
inst✝¹ : LinearOrder α
inst✝ : Invertible M
hM : M.BlockTriangular b
k : α
p : m → Prop := fun i ↦ b i < k
q : m → Prop := fun i ↦ ¬b i < k
⊢ (M⁻¹.toBlock (fun i ↦ k ≤ b i) fun i ↦ b i < k) = 0
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
⊢ (↑h.unit * N * ↑h.unit⁻¹).trace = N.trace

Tactic:
trace_units_conj

Goal state after:
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
⊢ N.trace = N.trace
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
n : Type u_4
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
f : R →+* S
⊢ (M.map ⇑f).charmatrix.det = Polynomial.map f M.charpoly

Tactic:
charmatrix_map,

Goal state after:
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
n : Type u_4
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
f : R →+* S
⊢ (M.charmatrix.map (Polynomial.map f)).det = Polynomial.map f M.charpoly
Module: Mathlib.RingTheory.MatrixPolynomialAlgebra

Goal state before:
pp: n : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
R : Type u_3
inst✝ : CommRing R
A : Matrix n n R[X]
M : Matrix n n R
⊢ (eval 0 (matPolyEquiv A) +
        eval 0 (matPolyEquiv ((algebraMap R[X] (Matrix n n R[X])) X) * matPolyEquiv (M.map ⇑C))).det =
    (eval 0 (matPolyEquiv A)).det

Tactic:
simp only [Algebra.algebraMap_eq_smul_one, matPolyEquiv_smul_one, map_X, X_mul, eval_mul_X,
    mul_zero, add_zero]

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

Goal state before:
pp: l : Type u_1
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁷ : Fintype l
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : DecidableEq l
inst✝³ : DecidableEq m
inst✝² : DecidableEq n
inst✝¹ : CommRing α
A : Matrix m m α
B : Matrix m n α
D : Matrix n n α
inst✝ : Invertible (fromBlocks A B 0 D)
⊢ (⅟(fromBlocks A B 0 D)).toBlocks₁₁ * A = 1

Tactic:
have := invOf_mul_self (fromBlocks A B 0 D)
      rw [← fromBlocks_toBlocks (⅟(fromBlocks A B 0 D)), fromBlocks_multiply] at this
      simpa only [Matrix.toBlocks_fromBlocks₁₁, Matrix.mul_zero, add_zero, ← fromBlocks_one] using
        congr_arg Matrix.toBlocks₁₁ this

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

Goal state before:
pp: R : Type u_1
inst✝¹ : Ring R
inst✝ : Nontrivial R
⊢ ¬IsNilpotent (-1)

Tactic:
intro h

Goal state after:
R : Type u_1
inst✝¹ : Ring R
inst✝ : Nontrivial R
h : IsNilpotent (-1)
⊢ False
Module: Mathlib.RingTheory.Nilpotent.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Ring R
inst✝ : Nontrivial R
n : ℕ
⊢ (-1) ^ n ≠ 0

Tactic:
intro h

Goal state after:
R : Type u_1
inst✝¹ : Ring R
inst✝ : Nontrivial R
n : ℕ
h : (-1) ^ n = 0
⊢ False
Module: Mathlib.RingTheory.Nilpotent.Basic

Goal state before:
pp: R : Type u_1
x y : R
inst✝ : Semiring R
h_comm : Commute x y
m n k : ℕ
hx : x ^ m = 0
hy : y ^ n = 0
h : m + n ≤ k + 1
⊢ ∀ x_1 ∈ Finset.antidiagonal k, k.choose x_1.1 • (x ^ x_1.1 * y ^ x_1.2) = 0

Tactic:
rintro ⟨i, j⟩ hij

Goal state after:
R : Type u_1
x y : R
inst✝ : Semiring R
h_comm : Commute x y
m n k : ℕ
hx : x ^ m = 0
hy : y ^ n = 0
h : m + n ≤ k + 1
i j : ℕ
hij : (i, j) ∈ Finset.antidiagonal k
⊢ k.choose (i, j).1 • (x ^ (i, j).1 * y ^ (i, j).2) = 0
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap

Goal state before:
pp: case mp
ι : Type u_1
inst✝⁴ : Fintype ι
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
b : ι → M
inst✝ : DecidableEq ι
A : Matrix ι ι R
f : Module.End R M
⊢ Represents b A f → ∀ (j : ι), ∑ i, A i j • b i = f (b j)

Tactic:
intro h

Goal state after:
case mp
ι : Type u_1
inst✝⁴ : Fintype ι
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
b : ι → M
inst✝ : DecidableEq ι
A : Matrix ι ι R
f : Module.End R M
h : Represents b A f
⊢ ∀ (j : ι), ∑ i, A i j • b i = f (b j)
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap

Goal state before:
pp: case mpr
ι : Type u_1
inst✝⁴ : Fintype ι
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
b : ι → M
inst✝ : DecidableEq ι
A : Matrix ι ι R
f : Module.End R M
⊢ (∀ (j : ι), ∑ i, A i j • b i = f (b j)) → Represents b A f

Tactic:
intro h

Goal state after:
case mpr
ι : Type u_1
inst✝⁴ : Fintype ι
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
b : ι → M
inst✝ : DecidableEq ι
A : Matrix ι ι R
f : Module.End R M
h : ∀ (j : ι), ∑ i, A i j • b i = f (b j)
⊢ Represents b A f
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap

Goal state before:
pp: M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
inst✝ : Module.Finite R M
f : Module.End R M
I : Ideal R
hI : range f ≤ I • ⊤
h✝ : Subsingleton R
⊢ Monic 0 ∧ natDegree 0 = ⊤.spanFinrank ∧ (∀ (k : ℕ), coeff 0 k ∈ I ^ (natDegree 0 - k)) ∧ (aeval f) 0 = 0

Tactic:
by simp [nontriviality]

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

Goal state before:
pp: case a
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
h2 : ⨅ s, ⨅ (_ : span R s = p), s.encard ≠ ⊤
⊢ ⨅ s, (↑↑s).encard ≤ ⨅ s, ⨅ (_ : span R s = p), s.encard

Tactic:
refine le_iInf fun s ↦ le_iInf fun h ↦ ?_
      by_cases hs : s.Finite
      · exact iInf_le_of_le ⟨hs.toFinset, by simpa⟩ (by simp)
      · rw [Set.Infinite.encard_eq hs]
        exact OrderTop.le_top _

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

Goal state before:
pp: case mp
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
a : Cardinal.{max u v}
⊢ lift.{v, u} p.spanRank ≤ a → ∃ s, lift.{v, u} #↑s ≤ a ∧ span R s = p

Tactic:
intro h

Goal state after:
case mp
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
a : Cardinal.{max u v}
h : lift.{v, u} p.spanRank ≤ a
⊢ ∃ s, lift.{v, u} #↑s ≤ a ∧ span R s = p
Module: Mathlib.Algebra.Module.SpanRank

Goal state before:
pp: case mp
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I : Submodule R M
⊢ I.spanRank = 0 → I = ⊥

Tactic:
intro h

Goal state after:
case mp
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I : Submodule R M
h : I.spanRank = 0
⊢ I = ⊥
Module: Mathlib.Algebra.Polynomial.Eval.Subring

Goal state before:
pp: case mpr
R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : Semiring S
f : R →+* S
p : S[X]
⊢ (∀ (n : ℕ), p.coeff n ∈ f.rangeS) → p ∈ (mapRingHom f).rangeS

Tactic:
intro h

Goal state after:
case mpr
R : Type u
S : Type v
inst✝¹ : Semiring R
inst✝ : Semiring S
f : R →+* S
p : S[X]
h : ∀ (n : ℕ), p.coeff n ∈ f.rangeS
⊢ p ∈ (mapRingHom f).rangeS
Module: Mathlib.Algebra.Polynomial.Lifts

Goal state before:
pp: case mp
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
p : S[X]
⊢ (∀ (n : ℕ), p.coeff n ∈ Set.range ⇑f) → ↑p.coeffs ⊆ Set.range ⇑f

Tactic:
intro h

Goal state after:
case mp
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
p : S[X]
h : ∀ (n : ℕ), p.coeff n ∈ Set.range ⇑f
⊢ ↑p.coeffs ⊆ Set.range ⇑f
Module: Mathlib.Algebra.Polynomial.Lifts

Goal state before:
pp: case mpr
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
p : S[X]
⊢ ↑p.coeffs ⊆ Set.range ⇑f → ∀ (n : ℕ), p.coeff n ∈ Set.range ⇑f

Tactic:
intro h

Goal state after:
case mpr
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
p : S[X]
h : ↑p.coeffs ⊆ Set.range ⇑f
⊢ ∀ (n : ℕ), p.coeff n ∈ Set.range ⇑f
Module: Mathlib.Algebra.Polynomial.Lifts

Goal state before:
pp: case inr
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
n : ℕ
a : R
hl : (monomial n) (f a) ∈ lifts f
h : f a ≠ 0
⊢ ∃ q, map f q = (monomial n) (f a) ∧ q.degree = ((monomial n) (f a)).degree

Tactic:
refine ⟨monomial n a, map_monomial f, ?_⟩

Goal state after:
case inr
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
n : ℕ
a : R
hl : (monomial n) (f a) ∈ lifts f
h : f a ≠ 0
⊢ ((monomial n) a).degree = ((monomial n) (f a)).degree
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: case h₁
R : Type u_1
inst✝ : Semiring R
k : ℕ
f : R[X]
n : ℕ
⊢ n + k ∉ f.support → ((monomial (n + k - k)) (↑((n + k).choose k) * f.coeff (n + k))).coeff n = 0

Tactic:
intro h

Goal state after:
case h₁
R : Type u_1
inst✝ : Semiring R
k : ℕ
f : R[X]
n : ℕ
h : n + k ∉ f.support
⊢ ((monomial (n + k - k)) (↑((n + k).choose k) * f.coeff (n + k))).coeff n = 0
Module: Mathlib.RingTheory.Polynomial.ScaleRoots

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
p : R[X]
hp : p ≠ 0
s : R
⊢ p.scaleRoots s ≠ 0

Tactic:
intro h

Goal state after:
R : Type u_1
inst✝ : Semiring R
p : R[X]
hp : p ≠ 0
s : R
h : p.scaleRoots s = 0
⊢ False
Module: Mathlib.RingTheory.Polynomial.ScaleRoots

Goal state before:
pp: case neg
R : Type u_1
inst✝ : Semiring R
p : R[X]
s : R
this : (a : Prop) → Decidable a
hp : ¬p = 0
⊢ p.leadingCoeff ≠ 0

Tactic:
intro h

Goal state after:
case neg
R : Type u_1
inst✝ : Semiring R
p : R[X]
s : R
this : (a : Prop) → Decidable a
hp : ¬p = 0
h : p.leadingCoeff = 0
⊢ False
Module: Mathlib.RingTheory.Polynomial.ScaleRoots

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹ : Semiring R
inst✝ : Semiring S
p : R[X]
x : R
f : R →+* S
h : f p.leadingCoeff ≠ 0
⊢ map f (p.scaleRoots x) = (map f p).scaleRoots (f x)

Tactic:
ext
  simp [Polynomial.natDegree_map_of_leadingCoeff_ne_zero _ h]

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

Goal state before:
pp: R : Type u_1
inst✝ : Ring R
T : Subring R
p : (↥T)[X]
n : ℕ
⊢ p.coeff n = 0 → 0 = ↑(p.coeff n)

Tactic:
intro h

Goal state after:
R : Type u_1
inst✝ : Ring R
T : Subring R
p : (↥T)[X]
n : ℕ
h : p.coeff n = 0
⊢ 0 = ↑(p.coeff n)
Module: Mathlib.RingTheory.Polynomial.Subring

Goal state before:
pp: R : Type u_1
inst✝ : Ring R
T : Subring R
p : (↥T)[X]
⊢ ↑(ofSubring T p).coeffs ⊆ ↑T

Tactic:
intro i hi
  simp only [coeffs, Set.mem_image, mem_support_iff, Ne, Finset.mem_coe,
    (Finset.coe_image)] at hi
  rcases hi with ⟨n, _, h'n⟩
  rw [← h'n, coeff_ofSubring]
  exact Subtype.mem (coeff p n : T)

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

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : IsDomain A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : IsDomain B
inst✝³ : Algebra R B
inst✝² : Algebra A B
inst✝¹ : FaithfulSMul A B
inst✝ : IsScalarTower R A B
f : R[X]
hf : (map (algebraMap R A) f).Splits
⊢ ⇑(algebraMap A B) '' f.rootSet A = f.rootSet B

Tactic:
rw [rootSet, ← Finset.coe_image, ← Multiset.toFinset_map, hf.map_aroots_algebraMap, ← rootSet]

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

Goal state before:
pp: case refine_1
R : Type u_1
inst✝¹ : CommRing R
S : Type u_2
inst✝ : CommRing S
f : R →+* S
I : Ideal S
r : S
r_non_zero_divisor : ∀ {x : S}, x * r = 0 → x = 0
hr : r ∈ I
p : R[X]
⊢ 0 ≠ 0 → eval₂ f r 0 = 0 → ∃ i, coeff 0 i ≠ 0 ∧ coeff 0 i ∈ comap f I

Tactic:
intro h

Goal state after:
case refine_1
R : Type u_1
inst✝¹ : CommRing R
S : Type u_2
inst✝ : CommRing S
f : R →+* S
I : Ideal S
r : S
r_non_zero_divisor : ∀ {x : S}, x * r = 0 → x = 0
hr : r ∈ I
p : R[X]
h : 0 ≠ 0
⊢ eval₂ f r 0 = 0 → ∃ i, coeff 0 i ≠ 0 ∧ coeff 0 i ∈ comap f I
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
⊢ Nontrivial N → Nontrivial (N ⊗[R] M)

Tactic:
intro h

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
h : Nontrivial N
⊢ Nontrivial (N ⊗[R] M)
Module: Mathlib.RingTheory.Algebraic.Integral

Goal state before:
pp: R : Type u_1
A : Type u_3
inst✝³ : CommRing R
inst✝² : Ring A
inst✝¹ : Algebra R A
z : A
inst✝ : IsReduced R
x✝ : ∃ y, y ≠ 0 ∧ IsIntegral R (y • z)
w✝ : R
hy : w✝ ≠ 0
int : IsIntegral R (w✝ • z)
⊢ ¬IsNilpotent w✝

Tactic:
by rwa [isNilpotent_iff_eq_zero]

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

Goal state before:
pp: case mp
R : Type u_3
S : Type u_4
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
I : Ideal R
hf₁ : RingHom.ker f ≤ I
hf₂ : f.IsIntegral
⊢ ¬I = ⊤ → ¬map f I = ⊤

Tactic:
intro h

Goal state after:
case mp
R : Type u_3
S : Type u_4
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
I : Ideal R
hf₁ : RingHom.ker f ≤ I
hf₂ : f.IsIntegral
h : ¬I = ⊤
⊢ ¬map f I = ⊤