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

Goal state before:
pp: case h
R✝ : Type u_1
inst✝¹⁸ : CommSemiring R✝
S✝ : Type u_2
inst✝¹⁷ : CommSemiring S✝
inst✝¹⁶ : Algebra R✝ S✝
P✝ : Type u_3
inst✝¹⁵ : CommSemiring P✝
p✝ : Ideal R✝
inst✝¹⁴ : p✝.IsPrime
Rₚ✝ : Type u_4
inst✝¹³ : CommSemiring Rₚ✝
inst✝¹² : Algebra R✝ Rₚ✝
inst✝¹¹ : IsLocalization.AtPrime Rₚ✝ p✝
inst✝¹⁰ : IsLocalRing Rₚ✝
Sₚ : Type u_5
inst✝⁹ : CommSemiring Sₚ
inst✝⁸ : Algebra S✝ Sₚ
inst✝⁷ : IsLocalization (algebraMapSubmonoid S✝ p✝.primeCompl) Sₚ
inst✝⁶ : Algebra Rₚ✝ Sₚ
P : Ideal S✝
S : Type u_6
R : Type u_7
inst✝⁵ : CommRing R
p : Ideal R
inst✝⁴ : p.IsMaximal
Rₚ : Type u_8
inst✝³ : CommRing Rₚ
inst✝² : Algebra R Rₚ
inst✝¹ : IsLocalization.AtPrime Rₚ p
inst✝ : IsLocalRing Rₚ
x : R
s : ↥p.primeCompl
s' : R
hs : (Ideal.Quotient.mk p) s' = ((Ideal.Quotient.mk p) ↑s)⁻¹
this : (algebraMap R Rₚ) ↑s ∉ maximalIdeal Rₚ
⊢ (algebraMap R Rₚ) ↑s * (algebraMap R Rₚ) (x * s') - mk' Rₚ (↑s * x) s ∈ maximalIdeal Rₚ

Tactic:
IsLocalization.mk'_mul_cancel_left,

Goal state after:
case h
R✝ : Type u_1
inst✝¹⁸ : CommSemiring R✝
S✝ : Type u_2
inst✝¹⁷ : CommSemiring S✝
inst✝¹⁶ : Algebra R✝ S✝
P✝ : Type u_3
inst✝¹⁵ : CommSemiring P✝
p✝ : Ideal R✝
inst✝¹⁴ : p✝.IsPrime
Rₚ✝ : Type u_4
inst✝¹³ : CommSemiring Rₚ✝
inst✝¹² : Algebra R✝ Rₚ✝
inst✝¹¹ : IsLocalization.AtPrime Rₚ✝ p✝
inst✝¹⁰ : IsLocalRing Rₚ✝
Sₚ : Type u_5
inst✝⁹ : CommSemiring Sₚ
inst✝⁸ : Algebra S✝ Sₚ
inst✝⁷ : IsLocalization (algebraMapSubmonoid S✝ p✝.primeCompl) Sₚ
inst✝⁶ : Algebra Rₚ✝ Sₚ
P : Ideal S✝
S : Type u_6
R : Type u_7
inst✝⁵ : CommRing R
p : Ideal R
inst✝⁴ : p.IsMaximal
Rₚ : Type u_8
inst✝³ : CommRing Rₚ
inst✝² : Algebra R Rₚ
inst✝¹ : IsLocalization.AtPrime Rₚ p
inst✝ : IsLocalRing Rₚ
x : R
s : ↥p.primeCompl
s' : R
hs : (Ideal.Quotient.mk p) s' = ((Ideal.Quotient.mk p) ↑s)⁻¹
this : (algebraMap R Rₚ) ↑s ∉ maximalIdeal Rₚ
⊢ (algebraMap R Rₚ) ↑s * (algebraMap R Rₚ) (x * s') - (algebraMap R Rₚ) x ∈ maximalIdeal Rₚ
Module: Mathlib.RingTheory.TensorProduct.Quotient

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
A : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : Algebra S A
inst✝ : IsScalarTower R S A
I : Ideal T
g : (A ⊗[R] T ⧸ ((AlgebraTensorModule.lTensor S A) (↑R (Submodule.subtype I))).range) ≃ₗ[S]
  A ⊗[R] T ⧸ Submodule.restrictScalars S (Ideal.map includeRight I) :=
  ((AlgebraTensorModule.lTensor S A) (↑R (Submodule.subtype I))).range.quotEquivOfEq
    (Submodule.restrictScalars S (Ideal.map includeRight I)) ⋯
a₁ a₂ : A
b₁ b₂ : T ⧸ I
⊢ ↑(AlgebraTensorModule.tensorQuotientEquiv S T A I ≪≫ₗ g) ((a₁ * a₂) ⊗ₜ[R] (b₁ * b₂)) =
    ↑(AlgebraTensorModule.tensorQuotientEquiv S T A I ≪≫ₗ g) (a₁ ⊗ₜ[R] b₁) *
      ↑(AlgebraTensorModule.tensorQuotientEquiv S T A I ≪≫ₗ g) (a₂ ⊗ₜ[R] b₂)

Tactic:
obtain ⟨b₁, rfl⟩ := Ideal.Quotient.mk_surjective b₁

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
A : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : Algebra S A
inst✝ : IsScalarTower R S A
I : Ideal T
g : (A ⊗[R] T ⧸ ((AlgebraTensorModule.lTensor S A) (↑R (Submodule.subtype I))).range) ≃ₗ[S]
  A ⊗[R] T ⧸ Submodule.restrictScalars S (Ideal.map includeRight I) :=
  ((AlgebraTensorModule.lTensor S A) (↑R (Submodule.subtype I))).range.quotEquivOfEq
    (Submodule.restrictScalars S (Ideal.map includeRight I)) ⋯
a₁ a₂ : A
b₂ : T ⧸ I
b₁ : T
⊢ ↑(AlgebraTensorModule.tensorQuotientEquiv S T A I ≪≫ₗ g) ((a₁ * a₂) ⊗ₜ[R] ((Ideal.Quotient.mk I) b₁ * b₂)) =
    ↑(AlgebraTensorModule.tensorQuotientEquiv S T A I ≪≫ₗ g) (a₁ ⊗ₜ[R] (Ideal.Quotient.mk I) b₁) *
      ↑(AlgebraTensorModule.tensorQuotientEquiv S T A I ≪≫ₗ g) (a₂ ⊗ₜ[R] b₂)
Module: Mathlib.RingTheory.Localization.AtPrime.Basic

Goal state before:
pp: R : Type u_7
inst✝⁵ : CommRing R
p : Ideal R
inst✝⁴ : p.IsMaximal
Rₚ : Type u_8
inst✝³ : CommRing Rₚ
inst✝² : Algebra R Rₚ
inst✝¹ : IsLocalization.AtPrime Rₚ p
inst✝ : IsLocalRing Rₚ
x : R
s : ↥p.primeCompl
h₁ : (Ideal.Quotient.mk p) ↑s ≠ 0
⊢ (equivQuotMaximalIdeal p Rₚ).symm ((Ideal.Quotient.mk (maximalIdeal Rₚ)) (mk' Rₚ x s)) =
    (Ideal.Quotient.mk p) x * ((Ideal.Quotient.mk p) ↑s)⁻¹

Tactic:
have h₂ : equivQuotMaximalIdeal p Rₚ (Ideal.Quotient.mk p ↑s) ≠ 0 := by
    rwa [RingEquiv.map_ne_zero_iff]

Goal state after:
R : Type u_7
inst✝⁵ : CommRing R
p : Ideal R
inst✝⁴ : p.IsMaximal
Rₚ : Type u_8
inst✝³ : CommRing Rₚ
inst✝² : Algebra R Rₚ
inst✝¹ : IsLocalization.AtPrime Rₚ p
inst✝ : IsLocalRing Rₚ
x : R
s : ↥p.primeCompl
h₁ : (Ideal.Quotient.mk p) ↑s ≠ 0
h₂ : (equivQuotMaximalIdeal p Rₚ) ((Ideal.Quotient.mk p) ↑s) ≠ 0
⊢ (equivQuotMaximalIdeal p Rₚ).symm ((Ideal.Quotient.mk (maximalIdeal Rₚ)) (mk' Rₚ x s)) =
    (Ideal.Quotient.mk p) x * ((Ideal.Quotient.mk p) ↑s)⁻¹
Module: Mathlib.RingTheory.Localization.AtPrime.Basic

Goal state before:
pp: case h.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 γ : R
hγ : γ ∈ p
hβ : α * β = 1 + γ
⊢ (algebraMap S Sₚ) ↑s * (algebraMap S Sₚ) (β • x) - mk' Sₚ (↑s * x) s = (algebraMap S Sₚ) (γ • x)

Tactic:
IsLocalization.mk'_mul_cancel_left,

Goal state after:
case h.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 γ : R
hγ : γ ∈ p
hβ : α * β = 1 + γ
⊢ (algebraMap S Sₚ) ↑s * (algebraMap S Sₚ) (β • x) - (algebraMap S Sₚ) x = (algebraMap S Sₚ) (γ • x)
Module: Mathlib.LinearAlgebra.Dimension.Localization

Goal state before:
pp: R : Type uR
M : Type uM
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDomain R
M' : Submodule R M
⊢ lift.{max uR uM, uM} (Module.rank R (M ⧸ M') + Module.rank R ↥M') = lift.{max uR uM, uM} (Module.rank R M)

Tactic:
lift_add,

Goal state after:
R : Type uR
M : Type uM
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDomain R
M' : Submodule R M
⊢ lift.{max uM uR, uM} (Module.rank R (M ⧸ M')) + lift.{max uM uR, uM} (Module.rank R ↥M') =
    lift.{max uR uM, uM} (Module.rank R M)
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 j : n
σ : Perm n
a✝ : σ ∈ univ
h : ¬i = σ j
⊢ ∏ j', A.updateCol j (Pi.single i 1) (σ j') j' = 0

Tactic:
apply prod_eq_zero (mem_univ j)

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 j : n
σ : Perm n
a✝ : σ ∈ univ
h : ¬i = σ j
⊢ A.updateCol j (Pi.single i 1) (σ j) j = 0
Module: Mathlib.LinearAlgebra.Matrix.Transvection

Goal state before:
pp: n : Type u_1
R : Type u₂
inst✝² : DecidableEq n
inst✝¹ : CommRing R
inst✝ : Fintype n
L : List (TransvectionStruct n R)
⊢ (List.map toMatrix L).prod.det = 1

Tactic:
induction L with

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

Goal state before:
pp: n : Type u_1
R : Type u₂
inst✝² : DecidableEq n
inst✝¹ : CommRing R
inst✝ : Fintype n
L : List (TransvectionStruct n R)
⊢ (List.map (toMatrix ∘ TransvectionStruct.inv) L.reverse).prod * (List.map toMatrix L).prod = 1

Tactic:
induction L with

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

Goal state before:
pp: n : Type u_1
R : Type u₂
inst✝² : DecidableEq n
inst✝¹ : CommRing R
inst✝ : Fintype n
L : List (TransvectionStruct n R)
⊢ (List.map toMatrix L).prod * (List.map (toMatrix ∘ TransvectionStruct.inv) L.reverse).prod = 1

Tactic:
induction L with

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

Goal state before:
pp: n : Type u_1
p : Type u_2
R : Type u₂
inst✝⁴ : DecidableEq n
inst✝³ : DecidableEq p
inst✝² : CommRing R
inst✝¹ : Fintype n
inst✝ : Fintype p
M : Matrix n n R
L : List (TransvectionStruct n R)
N : Matrix p p R
⊢ (List.map (toMatrix ∘ sumInl p) L).prod * fromBlocks M 0 0 N = fromBlocks ((List.map toMatrix L).prod * M) 0 0 N

Tactic:
induction L with

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

Goal state before:
pp: α : Type w
inst✝ : CommRing α
A : Matrix (Fin 2) (Fin 2) α
i j : Fin 2
⊢ (-1) ^ (↑j + ↑i) * (A.submatrix j.succAbove i.succAbove).det = !![A 1 1, -A 0 1; -A 1 0, A 0 0] i j

Tactic:
fin_cases i

Goal state after:
case «0»
α : Type w
inst✝ : CommRing α
A : Matrix (Fin 2) (Fin 2) α
j : Fin 2
⊢ (-1) ^ (↑j + ↑((fun i ↦ i) ⟨0, ⋯⟩)) * (A.submatrix j.succAbove ((fun i ↦ i) ⟨0, ⋯⟩).succAbove).det =
    !![A 1 1, -A 0 1; -A 1 0, A 0 0] ((fun i ↦ i) ⟨0, ⋯⟩) j

case «1»
α : Type w
inst✝ : CommRing α
A : Matrix (Fin 2) (Fin 2) α
j : Fin 2
⊢ (-1) ^ (↑j + ↑((fun i ↦ i) ⟨1, ⋯⟩)) * (A.submatrix j.succAbove ((fun i ↦ i) ⟨1, ⋯⟩).succAbove).det =
    !![A 1 1, -A 0 1; -A 1 0, A 0 0] ((fun i ↦ i) ⟨1, ⋯⟩) j
Module: Mathlib.LinearAlgebra.Matrix.Transvection

Goal state before:
pp: n : Type u_1
p : Type u_2
R : Type u₂
inst✝⁴ : DecidableEq n
inst✝³ : DecidableEq p
inst✝² : CommRing R
inst✝¹ : Fintype n
inst✝ : Fintype p
e : n ≃ p
L : List (TransvectionStruct n R)
⊢ (List.map (toMatrix ∘ reindexEquiv e) L).prod = (reindexAlgEquiv R R e) (List.map toMatrix L).prod

Tactic:
induction L with

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

Goal state before:
pp: α : Type w
inst✝ : CommRing α
A : Matrix (Fin 3) (Fin 3) α
i j : Fin 3
⊢ (-1) ^ (↑j + ↑i) *
      (A.submatrix j.succAbove i.succAbove 0 0 * A.submatrix j.succAbove i.succAbove 1 1 -
        A.submatrix j.succAbove i.succAbove 0 1 * A.submatrix j.succAbove i.succAbove 1 0) =
    !![A 1 1 * A 2 2 - A 1 2 * A 2 1, -(A 0 1 * A 2 2) + A 0 2 * A 2 1, A 0 1 * A 1 2 - A 0 2 * A 1 1;
        -(A 1 0 * A 2 2) + A 1 2 * A 2 0, A 0 0 * A 2 2 - A 0 2 * A 2 0, -(A 0 0 * A 1 2) + A 0 2 * A 1 0;
        A 1 0 * A 2 1 - A 1 1 * A 2 0, -(A 0 0 * A 2 1) + A 0 1 * A 2 0, A 0 0 * A 1 1 - A 0 1 * A 1 0]
      i j

Tactic:
fin_cases i

Goal state after:
case «0»
α : Type w
inst✝ : CommRing α
A : Matrix (Fin 3) (Fin 3) α
j : Fin 3
⊢ (-1) ^ (↑j + ↑((fun i ↦ i) ⟨0, ⋯⟩)) *
      (A.submatrix j.succAbove ((fun i ↦ i) ⟨0, ⋯⟩).succAbove 0 0 *
          A.submatrix j.succAbove ((fun i ↦ i) ⟨0, ⋯⟩).succAbove 1 1 -
        A.submatrix j.succAbove ((fun i ↦ i) ⟨0, ⋯⟩).succAbove 0 1 *
          A.submatrix j.succAbove ((fun i ↦ i) ⟨0, ⋯⟩).succAbove 1 0) =
    !![A 1 1 * A 2 2 - A 1 2 * A 2 1, -(A 0 1 * A 2 2) + A 0 2 * A 2 1, A 0 1 * A 1 2 - A 0 2 * A 1 1;
        -(A 1 0 * A 2 2) + A 1 2 * A 2 0, A 0 0 * A 2 2 - A 0 2 * A 2 0, -(A 0 0 * A 1 2) + A 0 2 * A 1 0;
        A 1 0 * A 2 1 - A 1 1 * A 2 0, -(A 0 0 * A 2 1) + A 0 1 * A 2 0, A 0 0 * A 1 1 - A 0 1 * A 1 0]
      ((fun i ↦ i) ⟨0, ⋯⟩) j

case «1»
α : Type w
inst✝ : CommRing α
A : Matrix (Fin 3) (Fin 3) α
j : Fin 3
⊢ (-1) ^ (↑j + ↑((fun i ↦ i) ⟨1, ⋯⟩)) *
      (A.submatrix j.succAbove ((fun i ↦ i) ⟨1, ⋯⟩).succAbove 0 0 *
          A.submatrix j.succAbove ((fun i ↦ i) ⟨1, ⋯⟩).succAbove 1 1 -
        A.submatrix j.succAbove ((fun i ↦ i) ⟨1, ⋯⟩).succAbove 0 1 *
          A.submatrix j.succAbove ((fun i ↦ i) ⟨1, ⋯⟩).succAbove 1 0) =
    !![A 1 1 * A 2 2 - A 1 2 * A 2 1, -(A 0 1 * A 2 2) + A 0 2 * A 2 1, A 0 1 * A 1 2 - A 0 2 * A 1 1;
        -(A 1 0 * A 2 2) + A 1 2 * A 2 0, A 0 0 * A 2 2 - A 0 2 * A 2 0, -(A 0 0 * A 1 2) + A 0 2 * A 1 0;
        A 1 0 * A 2 1 - A 1 1 * A 2 0, -(A 0 0 * A 2 1) + A 0 1 * A 2 0, A 0 0 * A 1 1 - A 0 1 * A 1 0]
      ((fun i ↦ i) ⟨1, ⋯⟩) j

case «2»
α : Type w
inst✝ : CommRing α
A : Matrix (Fin 3) (Fin 3) α
j : Fin 3
⊢ (-1) ^ (↑j + ↑((fun i ↦ i) ⟨2, ⋯⟩)) *
      (A.submatrix j.succAbove ((fun i ↦ i) ⟨2, ⋯⟩).succAbove 0 0 *
          A.submatrix j.succAbove ((fun i ↦ i) ⟨2, ⋯⟩).succAbove 1 1 -
        A.submatrix j.succAbove ((fun i ↦ i) ⟨2, ⋯⟩).succAbove 0 1 *
          A.submatrix j.succAbove ((fun i ↦ i) ⟨2, ⋯⟩).succAbove 1 0) =
    !![A 1 1 * A 2 2 - A 1 2 * A 2 1, -(A 0 1 * A 2 2) + A 0 2 * A 2 1, A 0 1 * A 1 2 - A 0 2 * A 1 1;
        -(A 1 0 * A 2 2) + A 1 2 * A 2 0, A 0 0 * A 2 2 - A 0 2 * A 2 0, -(A 0 0 * A 1 2) + A 0 2 * A 1 0;
        A 1 0 * A 2 1 - A 1 1 * A 2 0, -(A 0 0 * A 2 1) + A 0 1 * A 2 0, A 0 0 * A 1 1 - A 0 1 * A 1 0]
      ((fun i ↦ i) ⟨2, ⋯⟩) j
Module: Mathlib.LinearAlgebra.Matrix.Invertible

Goal state before:
pp: m : Type u_1
n : Type u_2
α : Type u_3
inst✝⁸ : Fintype n
inst✝⁷ : DecidableEq n
inst✝⁶ : Fintype m
inst✝⁵ : DecidableEq m
inst✝⁴ : Ring α
A : Matrix n n α
U : Matrix n m α
C : Matrix m m α
V : Matrix m n α
inst✝³ : Invertible A
inst✝² : Invertible C
inst✝¹ : Invertible (⅟C + V * ⅟A * U)
inst✝ : Invertible (A + U * C * V)
this : Invertible (A + U * C * V) := A.invertibleAddMulMul U C V
⊢ ⅟(A + U * C * V) = ⅟A - ⅟A * U * ⅟(⅟C + V * ⅟A * U) * V * ⅟A

Tactic:
convert! (rfl : ⅟(A + U * C * V) = _)

Goal state after:
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 + C * V * ⅟A * U * C)
inst✝ : Invertible (A + U * C * V)
this : Invertible (A + U * C * V) := A.invertibleAddMulMul' U C V
⊢ ⅟(A + U * C * V) = ⅟A - ⅟A * U * C * ⅟(C + C * V * ⅟A * U * C) * C * V * ⅟A

Tactic:
convert! (rfl : ⅟(A + U * C * V) = _)

Goal state after:
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
⊢ A * A⁻¹ * A⁻¹⁻¹ = A

Tactic:
rw [Matrix.mul_assoc, A⁻¹.mul_nonsing_inv (A.isUnit_nonsing_inv_det h), Matrix.mul_one]

Goal state after:
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
⊢ IsUnit A⁻¹ ↔ IsUnit A

Tactic:
simp_rw [isUnit_iff_isUnit_det, isUnit_nonsing_inv_det_iff]

Goal state after:
Module: Mathlib.RingTheory.MatrixPolynomialAlgebra

Goal state before:
pp: case refine_2
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 : n
p q : Matrix n n R[X]
hp : (matPolyEquiv p).coeff k i j = (p i j).coeff k
hq : (matPolyEquiv q).coeff k i j = (q i j).coeff k
⊢ (matPolyEquiv (p + q)).coeff k i j = ((p + q) i j).coeff k

Tactic:
simp [hp, hq]

Goal state after:
Module: Mathlib.RingTheory.MatrixAlgebra

Goal state before:
pp: n : Type u_3
R : Type u_5
A : Type u_7
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : DecidableEq n
inst✝ : Fintype n
a : A
M : Matrix n n A
⊢ invFun n R A (a • M) = a ⊗ₜ[R] 1 * invFun n R A M

Tactic:
simp [invFun, Finset.mul_sum]

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

Goal state before:
pp: m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m n α
B : Matrix n m α
⊢ (1 + -(A * B)).det = (1 - B * A).det

Tactic:
← Matrix.neg_mul,

Goal state after:
m : Type u_2
n : Type u_3
α : Type u_4
inst✝⁴ : Fintype m
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix m n α
B : Matrix n m α
⊢ (1 + -A * B).det = (1 - B * A).det
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
  exact not_isNilpotent_neg_one ⟨n, h⟩

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
i j : n
⊢ (M.charmatrix i j).natDegree ≤ if i = j then 1 else 0

Tactic:
by
  split_ifs with h <;> simp [h, natDegree_X_le]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
k : ℕ
h : Fintype.card n - 1 ≤ k
⊢ (M.charpoly - ∏ i, (X - C (M i i))).degree < ↑k

Tactic:
apply lt_of_lt_of_le (charpoly_sub_diagonal_degree_lt M) ?_

Goal state after:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
k : ℕ
h : Fintype.card n - 1 ≤ k
⊢ ↑(Fintype.card n - 1) ≤ ↑k
Module: Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_4
inst✝² : CommRing R
inst✝¹ : Ring S
inst✝ : Ring T
f : R →+* S
g : S →+* T
hg : Function.Injective ⇑g
x : S
p : R[X]
hp : p.Monic
hx : eval₂ (g.comp f) (g x) p = 0
⊢ g (eval₂ f x p) = g 0

Tactic:
by simp [Polynomial.hom_eval₂, hx]

Goal state after:
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]
h : ∀ (n : ℕ), p.coeff n ∈ f.rangeS
⊢ ∑ i ∈ Finset.range (p.natDegree + 1), C (p.coeff i) * X ^ i ∈ (mapRingHom f).rangeS

Tactic:
refine (mapRingHom f).rangeS.sum_mem ?_

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
⊢ ∀ c ∈ Finset.range (p.natDegree + 1), C (p.coeff c) * X ^ c ∈ (mapRingHom f).rangeS
Module: Mathlib.Algebra.Polynomial.Lifts

Goal state before:
pp: case inr.ha
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
⊢ a ≠ 0

Tactic:
exact mt (fun ha ↦ ha ▸ map_zero f) h

Goal state after:
Module: Mathlib.Data.Nat.Choose.Vandermonde

Goal state before:
pp: m n k : ℕ
⊢ ∀ x ∈ antidiagonal k, ((X + 1) ^ m).coeff x.1 * ((X + 1) ^ n).coeff x.2 = m.choose x.1 * n.choose x.2

Tactic:
simp only [coeff_X_add_one_pow, cast_id, imp_true_iff]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
a✝ : Nontrivial R
t : R[T;T⁻¹] := T 1
t_inv : R[T;T⁻¹] := T (-1)
p : R[T;T⁻¹] := ((scalar n) t - M.map ⇑LaurentPolynomial.C).det
q : R[T;T⁻¹] := (1 - (scalar n) t * M.map ⇑LaurentPolynomial.C).det
ht : t_inv * t = 1
hp : toLaurentAlg M.charpoly = p
hq : toLaurentAlg M.charpolyRev = q
this : t_inv ^ Fintype.card n * p = invert q
⊢ M.charpoly.reverse = M.charpolyRev

Tactic:
by
    apply toLaurent_injective
    rwa [toLaurent_reverse, ← coe_toLaurentAlg, hp, hq, ← involutive_invert.injective.eq_iff,
      map_mul, involutive_invert p, charpoly_natDegree_eq_dim,
      ← mul_one (Fintype.card n : ℤ), ← T_pow, map_pow, invert_T, mul_comm]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: case inr
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
hM : IsNilpotent M
h✝ : Nonempty n
⊢ IsNilpotent M.trace

Tactic:
suffices IsNilpotent (coeff (charpolyRev M) 1) by simpa using this

Goal state after:
case inr
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
hM : IsNilpotent M
h✝ : Nonempty n
⊢ IsNilpotent (M.charpolyRev.coeff 1)
Module: Mathlib.Algebra.Polynomial.Taylor

Goal state before:
pp: case pos
R : Type u_1
inst✝ : Semiring R
r : R
f : R[X]
hf : f = 0
⊢ coeff 0 (natDegree 0) = leadingCoeff 0

Tactic:
coeff_natDegree

Goal state after:
case pos
R : Type u_1
inst✝ : Semiring R
r : R
f : R[X]
hf : f = 0
⊢ leadingCoeff 0 = leadingCoeff 0
Module: Mathlib.Algebra.Polynomial.Splits

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommSemiring R
f : R[X]
A : Type u_2
B : Type u_3
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
hf : (map (algebraMap R A) f).Splits
e : A →ₐ[R] B
⊢ (map (algebraMap R B) f).Splits

Tactic:
rw [← e.comp_algebraMap, ← map_map]
  apply hf.map

Goal state after:
Module: Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
S T : Subalgebra R A
⊢ Algebra.IsIntegral R ↥(S ⊔ T) ↔ Algebra.IsIntegral R ↥S ∧ Algebra.IsIntegral R ↥T

Tactic:
by
  simp_rw [← le_integralClosure_iff_isIntegral, sup_le_iff]

Goal state after:
Module: Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic

Goal state before:
pp: case pos
R : Type u_1
S : Type u_4
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
p : R[X]
x : S
h : eval₂ f x p = 0
h' : ¬1 ≤ p.natDegree
hp : f (p.coeff p.natDegree) = 0
⊢ f.IsIntegralElem (f p.leadingCoeff * x)

Tactic:
coeff_natDegree

Goal state after:
case pos
R : Type u_1
S : Type u_4
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
p : R[X]
x : S
h : eval₂ f x p = 0
h' : ¬1 ≤ p.natDegree
hp : f p.leadingCoeff = 0
⊢ f.IsIntegralElem (f p.leadingCoeff * x)
Module: Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing A
inst✝⁹ : CommRing B
inst✝⁸ : Algebra R B
inst✝⁷ : Algebra A B
inst✝⁶ : IsIntegralClosure A R B
S : Type u_4
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : Algebra S B
inst✝² : IsScalarTower R S B
inst✝¹ : Algebra R A
inst✝ : IsScalarTower R A B
isIntegral : Algebra.IsIntegral R S
x : R
⊢ mk' A ((algebraMap S B) ((algebraMap R S) x)) ⋯ = (algebraMap R A) x

Tactic:
by simp_rw [← IsScalarTower.algebraMap_apply, mk'_algebraMap]

Goal state after: