mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Monad.Limits

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
T : Monad C
J : Type u
inst✝² : Category.{v, u} J
D : J ⥤ T.Algebra
c : Cocone (D ⋙ T.forget)
t : IsColimit c
inst✝¹ : PreservesColimit (D ⋙ T.forget) T.toFunctor
inst✝ : PreservesColimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor
⊢ T.μ.app c.pt ≫ lambda c t = T.map (lambda c t) ≫ lambda c t

Tactic:
refine (isColimitOfPreserves _ (isColimitOfPreserves _ t)).hom_ext fun j => ?_

Goal state after:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
T : Monad C
J : Type u
inst✝² : Category.{v, u} J
D : J ⥤ T.Algebra
c : Cocone (D ⋙ T.forget)
t : IsColimit c
inst✝¹ : PreservesColimit (D ⋙ T.forget) T.toFunctor
inst✝ : PreservesColimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor
j : J
⊢ (T.mapCocone (T.mapCocone c)).ι.app j ≫ T.μ.app c.pt ≫ lambda c t =
    (T.mapCocone (T.mapCocone c)).ι.app j ≫ T.map (lambda c t) ≫ lambda c t
Module: Mathlib.CategoryTheory.Limits.Shapes.Reflexive

Goal state before:
pp: case one.zero.leftCompReflexion
C : Type u
inst✝ : Category.{v, u} C
p : Set (StructuredArrow WalkingReflexivePair.one inclusionWalkingReflexivePair)
h₁ : StructuredArrow.mk (𝟙 WalkingReflexivePair.one) ∈ p
h₂ : ∀ {j₁ j₂ : StructuredArrow WalkingReflexivePair.one inclusionWalkingReflexivePair} (x : j₁ ⟶ j₂), j₁ ∈ p ↔ j₂ ∈ p
hₗ : StructuredArrow.mk left ∈ p
hᵣ : StructuredArrow.mk right ∈ p
⊢ { left := { as := PUnit.unit }, right := zero, hom := leftCompReflexion } ∈ p

Tactic:
exact (h₂ (StructuredArrow.homMk .left)).2 hₗ

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Submodule

Goal state before:
pp: R : Type u_1
M : Type u_3
N : Type u_4
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : AddCommMonoid N
inst✝² : Module R M
inst✝¹ : Module R N
p : Submonoid R
f : M →ₗ[R] N
inst✝ : IsLocalizedModule p f
⊢ localized₀ p f ⊤ = ⊤

Tactic:
rw [← top_le_iff]
  rintro x _
  obtain ⟨⟨x, s⟩, rfl⟩ := IsLocalizedModule.mk'_surjective p f x
  exact ⟨x, trivial, s, rfl⟩

Goal state after:
Module: Mathlib.RingTheory.Ideal.MinimalPrime.Basic

Goal state before:
pp: case mp
R : Type u_1
inst✝ : CommSemiring R
I : Ideal R
e : I.minimalPrimes = ∅
h : ¬I = ⊤
M : Ideal R
hM : M.IsMaximal
hM' : I ≤ M
p : Ideal R
hp : p ∈ I.minimalPrimes ∧ p ≤ M
⊢ False

Tactic:
rw [e] at hp

Goal state after:
case mp
R : Type u_1
inst✝ : CommSemiring R
I : Ideal R
e : I.minimalPrimes = ∅
h : ¬I = ⊤
M : Ideal R
hM : M.IsMaximal
hM' : I ≤ M
p : Ideal R
hp : p ∈ ∅ ∧ p ≤ M
⊢ False
Module: Mathlib.RingTheory.Ideal.Quotient.Nilpotent

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
I : Ideal R
⊢ (RingHom.ker (Quotient.mk I)).IsRadical ↔ IsReduced (R ⧸ I)

Tactic:
exact RingHom.ker_isRadical_iff_reduced_of_surjective Quotient.mk_surjective

Goal state after:
Module: Mathlib.Algebra.Module.LocalizedModule.Submodule

Goal state before:
pp: R✝ : Type u_1
S✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
inst✝²¹ : CommSemiring R✝
inst✝²⁰ : CommSemiring S✝
inst✝¹⁹ : AddCommMonoid M✝
inst✝¹⁸ : AddCommMonoid N✝
inst✝¹⁷ : Module R✝ M✝
inst✝¹⁶ : Module R✝ N✝
inst✝¹⁵ : Algebra R✝ S✝
inst✝¹⁴ : Module S✝ N✝
inst✝¹³ : IsScalarTower R✝ S✝ N✝
p✝ : Submonoid R✝
inst✝¹² : IsLocalization p✝ S✝
f✝ : M✝ →ₗ[R✝] N✝
inst✝¹¹ : IsLocalizedModule p✝ f✝
M'✝¹ M'' : Submodule R✝ M✝
R : Type u_5
S : Type u_6
M : Type u_7
N : Type u_8
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup N
inst✝⁶ : Module R M
inst✝⁵ : Module R N
inst✝⁴ : Algebra R S
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
p : Submonoid R
inst✝¹ : IsLocalization p S
f : M →ₗ[R] N
inst✝ : IsLocalizedModule p f
M'✝ M' : Submodule R M
m n : M
e :
  (Submodule.toLocalizedQuotient' S p f M') (Submodule.Quotient.mk m) =
    (Submodule.toLocalizedQuotient' S p f M') (Submodule.Quotient.mk n)
⊢ f (m - n) ∈ ?m.478

Tactic:
by simpa [Submodule.Quotient.eq] using! e

Goal state after:
Module: Mathlib.RingTheory.PolynomialAlgebra

Goal state before:
pp: R : Type u_1
A : Type u_3
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
x : A[X]
⊢ (toFunAlgHom R A) (invFun R A x) = x

Tactic:
refine Polynomial.induction_on' x ?_ ?_
  · intro p q hp hq
    simp only [invFun_add, map_add, hp, hq]
  · intro n a
    rw [invFun_monomial, Algebra.TensorProduct.tmul_pow,
        one_pow, Algebra.TensorProduct.tmul_mul_tmul, mul_one, one_mul, toFunAlgHom_apply_tmul,
        X_pow_eq_monomial, sum_monomial_index] <;>
      simp

Goal state after:
Module: Mathlib.RingTheory.PolynomialAlgebra

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring S
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : Algebra R S
inst✝¹ : Algebra S A
inst✝ : IsScalarTower R S A
x✝¹ : R[X]
x✝ : A[X]
⊢ (mapRingHom (algebraMap R A)) x✝¹ * x✝ = x✝ * (mapRingHom (algebraMap R A)) x✝¹

Tactic:
ext; rw [coeff_mul, ← Finset.Nat.sum_antidiagonal_swap, coeff_mul]; simp [Algebra.commutes]

Goal state after:
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
⊢ (Ideal.Quotient.mk (maximalIdeal Rₚ)) ((algebraMap R Rₚ) (x * s')) =
    (Ideal.Quotient.mk (maximalIdeal Rₚ)) (mk' Rₚ x s)

Tactic:
rw [← sub_eq_zero, ← map_sub, Ideal.Quotient.eq_zero_iff_mem]

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
⊢ (algebraMap R Rₚ) (x * s') - mk' Rₚ x s ∈ maximalIdeal Rₚ
Module: Mathlib.RingTheory.EssentialFiniteness

Goal state before:
pp: case h
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : CommRing T
inst✝² : Algebra R S
inst✝¹ : Algebra R T
inst✝ : EssFiniteType R S
f g : S →ₐ[R] T
H : ∀ s ∈ finset R S, f s = g s
⊢ adjoin R {x | ↑x ∈ finset R S} = ⊤

pp: case hs
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : CommRing T
inst✝² : Algebra R S
inst✝¹ : Algebra R T
inst✝ : EssFiniteType R S
f g : S →ₐ[R] T
H : ∀ s ∈ finset R S, f s = g s
⊢ Set.EqOn ⇑(f.comp (IsScalarTower.toAlgHom R (↥(subalgebra R S)) S))
    ⇑(g.comp (IsScalarTower.toAlgHom R (↥(subalgebra R S)) S)) {x | ↑x ∈ finset R S}

Tactic:
· exact adjoin_mem_finset R S

Goal state after:
case hs
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : CommRing T
inst✝² : Algebra R S
inst✝¹ : Algebra R T
inst✝ : EssFiniteType R S
f g : S →ₐ[R] T
H : ∀ s ∈ finset R S, f s = g s
⊢ Set.EqOn ⇑(f.comp (IsScalarTower.toAlgHom R (↥(subalgebra R S)) S))
    ⇑(g.comp (IsScalarTower.toAlgHom R (↥(subalgebra R S)) S)) {x | ↑x ∈ finset R S}
Module: Mathlib.LinearAlgebra.Matrix.Polynomial

Goal state before:
pp: case inr.inr
n : Type u_1
α : Type u_2
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A : Matrix n n α
h✝ : Nontrivial α
h : (X • Matrix.map 1 ⇑C + A.map ⇑C).det.natDegree < Fintype.card n
⊢ (X • 1 + A.map ⇑C).det.coeff (X • 1 + A.map ⇑C).det.natDegree = (X • 1 + A.map ⇑C).det.coeff (Fintype.card n)

Tactic:
· -- contradiction. we have a hypothesis that the degree is less than |n|
    -- but we know that coeff _ n = 1
    have H := coeff_eq_zero_of_natDegree_lt h
    rw [coeff_det_X_add_C_card] at H
    simp at H

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

Goal state before:
pp: case mul_T
R : Type u_1
inst✝ : Semiring R
Q : R[T;T⁻¹] → Prop
Qf : ∀ (f : R[X]), Q (toLaurent f)
QT : ∀ (f : R[T;T⁻¹]), Q (f * T 1) → Q f
f : R[X]
n : ℕ
⊢ Q (toLaurent f * T (-↑n))

Tactic:
induction n with
  | zero => simpa only [Nat.cast_zero, neg_zero, T_zero, mul_one] using Qf _
  | succ n hn => convert QT _ _; simpa

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 j : n
σ : Perm n
a✝ : σ ∈ univ
h : ¬i = σ j
⊢ Pi.single i 1 (σ j) = 0

Tactic:
Pi.single_eq_of_ne' h

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
⊢ 0 = 0
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
e : n ≃ m
A : Matrix m m α
i j : n
this : (fun j ↦ Pi.single i 1 (e.symm j)) = Pi.single (e i) 1
⊢ ((A.submatrix ⇑e ⇑e).updateRow j (Pi.single i 1)).det = ((A.updateRow (e j) (Pi.single (e i) 1)).submatrix ⇑e ⇑e).det

Tactic:
updateRow_submatrix_equiv,

Goal state after:
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
e : n ≃ m
A : Matrix m m α
i j : n
this : (fun j ↦ Pi.single i 1 (e.symm j)) = Pi.single (e i) 1
⊢ ((A.updateRow (e j) fun j ↦ Pi.single i 1 (e.symm j)).submatrix ⇑e ⇑e).det =
    ((A.updateRow (e j) (Pi.single (e i) 1)).submatrix ⇑e ⇑e).det
Module: Mathlib.Algebra.Polynomial.Laurent

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
n : ℤ
a : R
⊢ (C a * T n).degree ≤ ↑n

Tactic:
by
  by_cases a0 : a = 0
  · simp only [a0, map_zero, zero_mul, degree_zero, bot_le]
  · exact (degree_C_mul_T n a a0).le

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

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

Tactic:
suffices
      t.toMatrix *
            ((L.map toMatrix).prod * (L.reverse.map (toMatrix ∘ TransvectionStruct.inv)).prod) *
          t.inv.toMatrix = 1
      by simpa [Matrix.mul_assoc]

Goal state after:
case cons
n : Type u_1
R : Type u₂
inst✝² : DecidableEq n
inst✝¹ : CommRing R
inst✝ : Fintype n
t : TransvectionStruct n R
L : List (TransvectionStruct n R)
IH : (List.map toMatrix L).prod * (List.map (toMatrix ∘ TransvectionStruct.inv) L.reverse).prod = 1
⊢ t.toMatrix * ((List.map toMatrix L).prod * (List.map (toMatrix ∘ TransvectionStruct.inv) L.reverse).prod) *
      t.inv.toMatrix =
    1
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_card : Fintype.card n - Nat.succ 0 + Nat.succ 0 = Fintype.card n
A' : Matrix n n (MvPolynomial (n × n) ℤ) := mvPolynomialX n n ℤ
⊢ (A' * A'.adjugate).det = A'.det ^ Fintype.card n

Tactic:
rw [mul_adjugate A', det_smul, det_one, mul_one]

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
h : Fintype.card n ≠ 1
⊢ A.adjugate.adjugate = A.det ^ (Fintype.card n - 2) • A

Tactic:
rcases h_card : Fintype.card n with _ | n'

Goal state after:
case zero
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
h_card : Fintype.card n = 0
⊢ A.adjugate.adjugate = A.det ^ (0 - 2) • A

case succ
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
⊢ A.adjugate.adjugate = A.det ^ (n' + 1 - 2) • A
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'
⊢ (A'.det ^ (Fintype.card n - 2) • A').map ⇑(MvPolynomial.aeval fun p ↦ A p.1 p.2) =
    (MvPolynomial.aeval fun p ↦ A p.1 p.2) ((mvPolynomialX n n ℤ).det ^ (Fintype.card n - 2)) •
      (mvPolynomialX n n ℤ).map ⇑(MvPolynomial.aeval fun p ↦ A p.1 p.2)

Tactic:
Matrix.map_smul' _ _ _ (map_mul _)

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) (A'.det ^ (Fintype.card n - 2)) •
      A'.map ⇑(MvPolynomial.aeval fun p ↦ A p.1 p.2) =
    (MvPolynomial.aeval fun p ↦ A p.1 p.2) ((mvPolynomialX n n ℤ).det ^ (Fintype.card n - 2)) •
      (mvPolynomialX n n ℤ).map ⇑(MvPolynomial.aeval fun p ↦ A p.1 p.2)
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.row ↔ IsUnit A

Tactic:
← col_transpose,

Goal state after:
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 Aᵀ.col ↔ IsUnit A
Module: Mathlib.LinearAlgebra.Matrix.Transvection

Goal state before:
pp: case nil.cons
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
hdiag : ∀ (D : n → 𝕜), (diagonal D).det = M.det → P (diagonal D)
htransvec : ∀ (t : TransvectionStruct n 𝕜), P t.toMatrix
hmul : ∀ (A B : Matrix n n 𝕜), P A → P B → P (A * B)
L L' : List (TransvectionStruct n 𝕜)
D : n → 𝕜
h : M = (List.map toMatrix L).prod * diagonal D * (List.map toMatrix L').prod
PD : P (diagonal D)
t : TransvectionStruct n 𝕜
L₂ : List (TransvectionStruct n 𝕜)
IH : ∀ (E : Matrix n n 𝕜), P E → P (E * (List.map toMatrix L₂).prod)
E : Matrix n n 𝕜
PE : P E
⊢ P (E * (List.map toMatrix (t :: L₂)).prod)

Tactic:
simp only [← Matrix.mul_assoc, List.prod_cons, List.map]

Goal state after:
case nil.cons
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
hdiag : ∀ (D : n → 𝕜), (diagonal D).det = M.det → P (diagonal D)
htransvec : ∀ (t : TransvectionStruct n 𝕜), P t.toMatrix
hmul : ∀ (A B : Matrix n n 𝕜), P A → P B → P (A * B)
L L' : List (TransvectionStruct n 𝕜)
D : n → 𝕜
h : M = (List.map toMatrix L).prod * diagonal D * (List.map toMatrix L').prod
PD : P (diagonal D)
t : TransvectionStruct n 𝕜
L₂ : List (TransvectionStruct n 𝕜)
IH : ∀ (E : Matrix n n 𝕜), P E → P (E * (List.map toMatrix L₂).prod)
E : Matrix n n 𝕜
PE : P E
⊢ P (E * t.toMatrix * (List.map toMatrix L₂).prod)
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✝⁸ : Fintype l
inst✝⁷ : Fintype m
inst✝⁶ : Fintype n
inst✝⁵ : DecidableEq l
inst✝⁴ : DecidableEq m
inst✝³ : DecidableEq n
inst✝² : CommRing α
A : Matrix m m α
B : Matrix m n α
C : Matrix n m α
D : Matrix n n α
inst✝¹ : Invertible A
inst✝ : Invertible (fromBlocks A B C D)
iABCD' : Invertible ((fromBlocks A B C D).submatrix ⇑(Equiv.sumComm n m) ⇑(Equiv.sumComm n m)) :=
  (fromBlocks A B C D).submatrixEquivInvertible (Equiv.sumComm n m) (Equiv.sumComm n m)
iDCBA : Invertible (fromBlocks D C B A) := iABCD'.copy (fromBlocks D C B A) ⋯
⊢ Invertible (D - C * ⅟A * B)

Tactic:
exact invertibleOfFromBlocks₂₂Invertible D C B A

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

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsReduced R
p : R[X]
hpl : 0 < p.natDegree
⊢ ¬IsUnit p

Tactic:
by
  simp only [ne_eq, isNilpotent_iff_eq_zero, not_and, not_forall, exists_prop,
    Polynomial.isUnit_iff_coeff_isUnit_isNilpotent]
  intro _
  refine ⟨p.natDegree, hpl.ne', ?_⟩
  contrapose! hpl
  simp only [coeff_natDegree, leadingCoeff_eq_zero] at hpl
  simp [hpl]

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
⊢ ⨅ s, ⨅ (_ : span R s = p), s.encard ≤ toENat (⨅ s, #↑↑s)

Tactic:
have := congrFun toENat_comp_ofENat.{u}.symm (⨅ (s : Set M) (_ : span R s = p), s.encard)
    rw [id_eq] at this; rw [this]
    refine toENat.monotone' (le_ciInf fun s ↦ ?_)
    have : ofENat.{u} (⨅ (s' : Set M), ⨅ (_ : span R s' = p), s'.encard) ≤ ofENat s.1.encard :=
      ofENatHom.monotone' (le_trans (ciInf_le' _ s.1) (ciInf_le' _ s.2))
    apply le_trans this
    rw [Set.encard, ENat.card]
    exact Cardinal.ofENat_toENat_le _

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

Goal state before:
pp: case refine_1
R : Type u_1
M : Type u
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : RankCondition R
ι : Type u
v : ι → M
hv : Function.Injective v
hs : LinearIndependent R v
x : { s // span R s = span R (Set.range v) }
this : #↑↑x = #↑(Subtype.val ⁻¹' ↑x)
⊢ #ι ≤ #↑(Set.range ⇑(Module.Basis.span hs))

pp: case refine_2
R : Type u_1
M : Type u
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : RankCondition R
ι : Type u
v : ι → M
hv : Function.Injective v
hs : LinearIndependent R v
x : { s // span R s = span R (Set.range v) }
this : #↑↑x = #↑(Subtype.val ⁻¹' ↑x)
⊢ span R (Subtype.val ⁻¹' ↑x) = ⊤

Tactic:
· rw [mk_range_eq]
    exact .of_comp (f := Subtype.val) (by convert! hv; ext; simp [Module.Basis.span_apply])

Goal state after:
case refine_2
R : Type u_1
M : Type u
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : RankCondition R
ι : Type u
v : ι → M
hv : Function.Injective v
hs : LinearIndependent R v
x : { s // span R s = span R (Set.range v) }
this : #↑↑x = #↑(Subtype.val ⁻¹' ↑x)
⊢ span R (Subtype.val ⁻¹' ↑x) = ⊤
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
    obtain ⟨s, ⟨hs₁, hs₂⟩⟩ :=
      (FG.spanRank_le_iff_exists_span_set_card_le I (a := 0)).mp (by rw [h])
    simp only [nonpos_iff_eq_zero, mk_eq_zero_iff, Set.isEmpty_coe_sort] at hs₁
    simp_all

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

Goal state before:
pp: R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
h : p.FG
heq : p.generators = ∅
⊢ span R ∅ = ⊥

Tactic:
span_empty

Goal state after:
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
h : p.FG
heq : p.generators = ∅
⊢ ⊥ = ⊥
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: case succ
R : Type u
inst✝ : CommRing R
n : ℕ
IH : ∀ (M : Matrix (Fin n) (Fin n) R), eval 0 (derivative (1 + X • M.map ⇑C).det) = M.trace
M : Matrix (Fin (n + 1)) (Fin (n + 1)) R
⊢ eval 0 (derivative (1 + X • M.map ⇑C).det) = M.trace

Tactic:
det_succ_row_zero,

Goal state after:
case succ
R : Type u
inst✝ : CommRing R
n : ℕ
IH : ∀ (M : Matrix (Fin n) (Fin n) R), eval 0 (derivative (1 + X • M.map ⇑C).det) = M.trace
M : Matrix (Fin (n + 1)) (Fin (n + 1)) R
⊢ eval 0
      (derivative (∑ j, (-1) ^ ↑j * (1 + X • M.map ⇑C) 0 j * ((1 + X • M.map ⇑C).submatrix Fin.succ j.succAbove).det)) =
    M.trace
Module: Mathlib.Algebra.Module.SpanRank

Goal state before:
pp: case hs
R : Type u_1
M : Type u_2
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : RankCondition R
ι : Type u_3
v : Basis ι R M
⊢ LinearIndepOn R id (Set.range ⇑v)

Tactic:
exact v.linearIndependent.linearIndepOn_id

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

Goal state before:
pp: R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
p : S[X]
hlifts : ∀ (n : ℕ), p.coeff n ∈ Set.range ⇑f
g : ℕ → R := fun k ↦ Exists.choose ⋯
hg : ∀ (k : ℕ), f (g k) = p.coeff k
q : R[X] := ∑ k ∈ p.support, (monomial k) (g k)
hq : map f q = p
⊢ ∃ q, map f q = p ∧ q.support = p.support

Tactic:
have hq' : q.support = p.support := by
    simp_rw [Finset.ext_iff, mem_support_iff, q, finsetSum_coeff, coeff_monomial,
      Finset.sum_ite_eq', ite_ne_right_iff, mem_support_iff, and_iff_left_iff_imp, not_imp_not]
    exact fun k h ↦ by rw [← hg, h, map_zero]

Goal state after:
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
p : S[X]
hlifts : ∀ (n : ℕ), p.coeff n ∈ Set.range ⇑f
g : ℕ → R := fun k ↦ Exists.choose ⋯
hg : ∀ (k : ℕ), f (g k) = p.coeff k
q : R[X] := ∑ k ∈ p.support, (monomial k) (g k)
hq : map f q = p
hq' : q.support = p.support
⊢ ∃ q, map f q = p ∧ q.support = p.support
Module: Mathlib.RingTheory.IntegralClosure.Algebra.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : IsDomain A
M : Type u_5
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module A M
inst✝¹ : IsScalarTower R A M
inst✝ : Module.IsTorsionFree A M
N : Submodule R M
hN : N ≠ ⊥
hN' : N.FG
x : A
hx : ∀ n ∈ N, x • n ∈ N
A' : Subalgebra R A :=
  { carrier := {x | ∀ n ∈ N, x • n ∈ N}, mul_mem' := ⋯, one_mem' := ⋯, add_mem' := ⋯, zero_mem' := ⋯,
    algebraMap_mem' := ⋯ }
f : ↥A' →ₐ[R] Module.End R ↥N :=
  AlgHom.ofLinearMap { toFun := fun x ↦ (DistribSMul.toLinearMap R M x).restrict ⋯, map_add' := ⋯, map_smul' := ⋯ } ⋯ ⋯
a : M
ha₁ : a ∈ N
ha₂ : a ≠ 0
this : Function.Injective ⇑f
⊢ IsIntegral R ⟨x, hx⟩

Tactic:
← isIntegral_algHom_iff f this

Goal state after:
R : Type u_1
A : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : IsDomain A
M : Type u_5
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module A M
inst✝¹ : IsScalarTower R A M
inst✝ : Module.IsTorsionFree A M
N : Submodule R M
hN : N ≠ ⊥
hN' : N.FG
x : A
hx : ∀ n ∈ N, x • n ∈ N
A' : Subalgebra R A :=
  { carrier := {x | ∀ n ∈ N, x • n ∈ N}, mul_mem' := ⋯, one_mem' := ⋯, add_mem' := ⋯, zero_mem' := ⋯,
    algebraMap_mem' := ⋯ }
f : ↥A' →ₐ[R] Module.End R ↥N :=
  AlgHom.ofLinearMap { toFun := fun x ↦ (DistribSMul.toLinearMap R M x).restrict ⋯, map_add' := ⋯, map_smul' := ⋯ } ⋯ ⋯
a : M
ha₁ : a ∈ N
ha₂ : a ≠ 0
this : Function.Injective ⇑f
⊢ IsIntegral R (f ⟨x, hx⟩)
Module: Mathlib.Algebra.Polynomial.Taylor

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
r : R
f : R[X]
⊢ ((taylor r) f).coeff 1 = eval r (derivative f)

Tactic:
by
  rw [taylor_coeff, hasseDeriv_one]

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

Goal state before:
pp: R : Type u_1
inst✝ : CommSemiring R
p : R[X]
x y : R
⊢ ∃ c, c * y ^ 2 + eval x (derivative p) * y + eval x p = eval (x + y) p

Tactic:
by
  have this t :
      (taylor x p).eval t =
      ∑ i ∈ Finset.range ((taylor x p).natDegree + 2), (taylor x p).coeff i * t ^ i :=
    (taylor x p).eval_eq_sum_range' (n := (taylor x p).natDegree + 2) (by lia) t
  rw [add_comm, ← p.taylor_eval x y, this, Finset.sum_range_succ', Finset.sum_range_succ']
  use ∑ i ∈ Finset.range p.natDegree, (taylor x p).coeff (i + 2) * y ^ i
  simp [pow_succ, mul_assoc, Finset.sum_mul]

Goal state after: