mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.FieldTheory.Finite.Basic

Goal state before:
pp: K : Type u_1
R : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Fintype K
inst✝³ : CommRing R
inst✝² : Algebra K R
p : ℕ
inst✝¹ : ExpChar R p
inst✝ : PerfectRing R p
p' : ℕ
left✝ : CharP K p'
n : ℕ+
hp : Nat.Prime p'
card_eq : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ Function.Bijective ⇑(frobeniusAlgHom K R)

Tactic:
rw [coe_frobeniusAlgHom, card_eq]

Goal state after:
K : Type u_1
R : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Fintype K
inst✝³ : CommRing R
inst✝² : Algebra K R
p : ℕ
inst✝¹ : ExpChar R p
inst✝ : PerfectRing R p
p' : ℕ
left✝ : CharP K p'
n : ℕ+
hp : Nat.Prime p'
card_eq : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ Function.Bijective fun x ↦ x ^ p' ^ ↑n
Module: Mathlib.FieldTheory.Finite.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
f : K[X]
p : ℕ
hp✝ : CharP K p
n : ℕ
npos : 0 < n
hp : Nat.Prime p
hn : q = p ^ ↑⟨n, npos⟩
⊢ (expand K q) f = f ^ q

Tactic:
have : Fact p.Prime := ⟨hp⟩

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : Fintype K
f : K[X]
p : ℕ
hp✝ : CharP K p
n : ℕ
npos : 0 < n
hp : Nat.Prime p
hn : q = p ^ ↑⟨n, npos⟩
this : Fact (Nat.Prime p)
⊢ (expand K q) f = f ^ q
Module: Mathlib.FieldTheory.Finite.Basic

Goal state before:
pp: case succ
n✝ : ℕ
x : (ZMod (n✝ + 1))ˣ
⊢ x ^ φ (n✝ + 1) = 1

Tactic:
· rw [← card_units_eq_totient, pow_card_eq_one]

Goal state after:
Module: Mathlib.FieldTheory.Finite.Basic

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
a : ZMod p
⊢ a ^ (p - 1) = if a ≠ 0 then 1 else 0

Tactic:
split_ifs with ha

Goal state after:
case pos
p : ℕ
inst✝ : Fact (Nat.Prime p)
a : ZMod p
ha : a ≠ 0
⊢ a ^ (p - 1) = 1

case neg
p : ℕ
inst✝ : Fact (Nat.Prime p)
a : ZMod p
ha : ¬a ≠ 0
⊢ a ^ (p - 1) = 0
Module: Mathlib.FieldTheory.Finite.Basic

Goal state before:
pp: p : ℕ
hp : Nat.Prime p
n : ℤ
hpn : IsCoprime n ↑p
⊢ n ^ (p - 1) ≡ 1 [ZMOD ↑p]

Tactic:
have : Fact p.Prime := ⟨hp⟩

Goal state after:
p : ℕ
hp : Nat.Prime p
n : ℤ
hpn : IsCoprime n ↑p
this : Fact (Nat.Prime p)
⊢ n ^ (p - 1) ≡ 1 [ZMOD ↑p]
Module: Mathlib.FieldTheory.Finite.Basic

Goal state before:
pp: p : ℕ
hp : Nat.Prime p
n : ℤ
⊢ n ^ p ≡ n [ZMOD ↑p]

Tactic:
have : Fact p.Prime := ⟨hp⟩

Goal state after:
p : ℕ
hp : Nat.Prime p
n : ℤ
this : Fact (Nat.Prime p)
⊢ n ^ p ≡ n [ZMOD ↑p]
Module: Mathlib.LinearAlgebra.FreeModule.Finite.Quotient

Goal state before:
pp: ι : Type u_1
R : Type u_2
M : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
inst✝ : Finite ι
N : Submodule R M
b : Basis ι R M
h : finrank R ↥N = finrank R M
this : Fintype ι
a : ι → R := smithNormalFormCoeffs b h
b' : Basis ι R M := smithNormalFormTopBasis b h
ab : Basis ι R ↥N := smithNormalFormBotBasis b h
ab_eq : ∀ (i : ι), ↑((smithNormalFormBotBasis b h) i) = smithNormalFormCoeffs b h i • (smithNormalFormTopBasis b h) i
mem_I_iff : ∀ (x : M), x ∈ N ↔ ∀ (i : ι), a i ∣ (b'.repr x) i
N' : Submodule R (ι → R) := pi Set.univ fun i ↦ R ∙ a i
x : ι → R
⊢ x ∈ map (↑b'.equivFun) N ↔ x ∈ N'

Tactic:
simp only [N', Submodule.mem_map, Submodule.mem_pi, mem_span_singleton, Set.mem_univ,
      mem_I_iff, smul_eq_mul, forall_true_left, LinearEquiv.coe_coe,
      Basis.equivFun_apply, mul_comm _ (a _), eq_comm (b := (x _))]

Goal state after:
ι : Type u_1
R : Type u_2
M : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
inst✝ : Finite ι
N : Submodule R M
b : Basis ι R M
h : finrank R ↥N = finrank R M
this : Fintype ι
a : ι → R := smithNormalFormCoeffs b h
b' : Basis ι R M := smithNormalFormTopBasis b h
ab : Basis ι R ↥N := smithNormalFormBotBasis b h
ab_eq : ∀ (i : ι), ↑((smithNormalFormBotBasis b h) i) = smithNormalFormCoeffs b h i • (smithNormalFormTopBasis b h) i
mem_I_iff : ∀ (x : M), x ∈ N ↔ ∀ (i : ι), a i ∣ (b'.repr x) i
N' : Submodule R (ι → R) := pi Set.univ fun i ↦ R ∙ a i
x : ι → R
⊢ (∃ y, (∀ (i : ι), a i ∣ (b'.repr y) i) ∧ ⇑(b'.repr y) = x) ↔ ∀ (i : ι), ∃ a_1, x i = a i * a_1
Module: Mathlib.LinearAlgebra.FreeModule.Finite.CardQuotient

Goal state before:
pp: M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Free ℤ M
inst✝² : Module.Finite ℤ M
N : Submodule ℤ M
E : Type u_2
inst✝¹ : EquivLike E M ↥N
inst✝ : AddEquivClass E M ↥N
e : E
b : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := Free.chooseBasis ℤ M
h : finrank ℤ ↥N = finrank ℤ M
a : Free.ChooseBasisIndex ℤ M → ℤ := smithNormalFormCoeffs b h
b' : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := smithNormalFormTopBasis b h
ab : Basis (Free.ChooseBasisIndex ℤ M) ℤ ↥N := smithNormalFormBotBasis b h
ab_eq :
  ∀ (i : Free.ChooseBasisIndex ℤ M),
    ↑((smithNormalFormBotBasis b h) i) = smithNormalFormCoeffs b h i • (smithNormalFormTopBasis b h) i
e' : M ≃ₗ[ℤ] ↥N := b'.equiv ab (Equiv.refl (Free.ChooseBasisIndex ℤ M))
f : M →ₗ[ℤ] M := N.subtype ∘ₗ ↑e'
f_apply : ∀ (x : M), f x = ↑((b'.equiv ab (Equiv.refl (Free.ChooseBasisIndex ℤ M))) x) := fun x ↦ rfl
ha : ∀ (i : Free.ChooseBasisIndex ℤ M), f (b' i) = a i • b' i
⊢ (Matrix.diagonal a).det.natAbs = (∏ i, a i).natAbs

Tactic:
by rw [Matrix.det_diagonal]

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

Goal state before:
pp: G : Type u_1
A : Type u_2
B : Type u_3
K : Type u_4
L : Type u_5
inst✝¹⁸ : Group G
inst✝¹⁷ : CommRing A
inst✝¹⁶ : CommRing B
inst✝¹⁵ : Algebra A B
inst✝¹⁴ : Field K
inst✝¹³ : Field L
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A K
inst✝¹⁰ : Algebra B L
inst✝⁹ : Algebra A L
inst✝⁸ : IsFractionRing A K
inst✝⁷ : IsFractionRing B L
inst✝⁶ : IsScalarTower A K L
inst✝⁵ : IsScalarTower A B L
inst✝⁴ : MulSemiringAction G B
inst✝³ : MulSemiringAction G L
inst✝² : SMulDistribClass G B L
hAB : Algebra.IsInvariant A B G
inst✝¹ : SMulCommClass G A B
inst✝ : Algebra.IsIntegral A B
hc : ∀ (a : A), (algebraMap K L) ((algebraMap A K) a) = (algebraMap B L) ((algebraMap A B) a)
this✝ : Nontrivial A
this : Nontrivial B
x y : B
hy : y ∈ nonZeroDivisors B
h : ∀ (g : G), g • ((algebraMap B L) x / (algebraMap B L) y) = (algebraMap B L) x / (algebraMap B L) y
hy' : (algebraMap B L) y ≠ 0
b : B
a : A
hb : x * (algebraMap A B) a = b * y
ha : (algebraMap B L) ((algebraMap A B) a) ≠ 0
⊢ ∃ a, (algebraMap K L) a = (algebraMap B L) x / (algebraMap B L) y

Tactic:
have hxy : algebraMap B L x / algebraMap B L y =
    algebraMap B L b / algebraMap B L (algebraMap A B a) := by
    rw [div_eq_div_iff hy' ha, ← map_mul, hb, map_mul]

Goal state after:
G : Type u_1
A : Type u_2
B : Type u_3
K : Type u_4
L : Type u_5
inst✝¹⁸ : Group G
inst✝¹⁷ : CommRing A
inst✝¹⁶ : CommRing B
inst✝¹⁵ : Algebra A B
inst✝¹⁴ : Field K
inst✝¹³ : Field L
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A K
inst✝¹⁰ : Algebra B L
inst✝⁹ : Algebra A L
inst✝⁸ : IsFractionRing A K
inst✝⁷ : IsFractionRing B L
inst✝⁶ : IsScalarTower A K L
inst✝⁵ : IsScalarTower A B L
inst✝⁴ : MulSemiringAction G B
inst✝³ : MulSemiringAction G L
inst✝² : SMulDistribClass G B L
hAB : Algebra.IsInvariant A B G
inst✝¹ : SMulCommClass G A B
inst✝ : Algebra.IsIntegral A B
hc : ∀ (a : A), (algebraMap K L) ((algebraMap A K) a) = (algebraMap B L) ((algebraMap A B) a)
this✝ : Nontrivial A
this : Nontrivial B
x y : B
hy : y ∈ nonZeroDivisors B
h : ∀ (g : G), g • ((algebraMap B L) x / (algebraMap B L) y) = (algebraMap B L) x / (algebraMap B L) y
hy' : (algebraMap B L) y ≠ 0
b : B
a : A
hb : x * (algebraMap A B) a = b * y
ha : (algebraMap B L) ((algebraMap A B) a) ≠ 0
hxy : (algebraMap B L) x / (algebraMap B L) y = (algebraMap B L) b / (algebraMap B L) ((algebraMap A B) a)
⊢ ∃ a, (algebraMap K L) a = (algebraMap B L) x / (algebraMap B L) y
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative

Goal state before:
pp: α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
⊢ IsRelPrime (p ^ i p) (∏ p' ∈ s, p' ^ i p')

Tactic:
have hp := is_prime _ (Finset.mem_insert_self _ _)

Goal state after:
α : Type u_1
inst✝² : CommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : DecidableEq α
s : Finset α
i : α → ℕ
p : α
hps : p ∉ s
is_prime : ∀ q ∈ insert p s, Prime q
is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'
hp : Prime p
⊢ IsRelPrime (p ^ i p) (∏ p' ∈ s, p' ^ i p')
Module: Mathlib.LinearAlgebra.FreeModule.Finite.CardQuotient

Goal state before:
pp: E : Type u_1
inst✝² : AddCommGroup E
L₁ L₂ : AddSubgroup E
H : L₁ ≤ L₂
ι : Type u_2
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
b₁ : Basis ι ℤ ↥(toIntSubmodule L₁)
b₂ : Basis ι ℤ ↥(toIntSubmodule L₂)
⊢ L₁.relIndex L₂ = (b₂.det fun i ↦ ⟨↑(b₁ i), ⋯⟩).natAbs

Tactic:
rw [relIndex, index_eq_natAbs_det b₂ _ (b₁.map (addSubgroupOfEquivOfLe H).toIntLinearEquiv.symm)]

Goal state after:
E : Type u_1
inst✝² : AddCommGroup E
L₁ L₂ : AddSubgroup E
H : L₁ ≤ L₂
ι : Type u_2
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
b₁ : Basis ι ℤ ↥(toIntSubmodule L₁)
b₂ : Basis ι ℤ ↥(toIntSubmodule L₂)
⊢ (b₂.det fun x ↦ ↑((b₁.map (addSubgroupOfEquivOfLe H).toIntLinearEquiv.symm) x)).natAbs =
    (b₂.det fun i ↦ ⟨↑(b₁ i), ⋯⟩).natAbs
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: S : Type u_1
inst✝¹ : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst✝ : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
a c : S
a_mem : a ∈ P ^ i
a_notMem : a ∉ P ^ (i + 1)
c_mem : c ∈ P ^ i
⊢ P ^ i = span {a} ⊔ P ^ (i + 1)

Tactic:
refine (Ideal.eq_prime_pow_of_succ_lt_of_le hP (lt_of_le_of_ne le_sup_right ?_)
    (sup_le (Ideal.span_le.mpr (Set.singleton_subset_iff.mpr a_mem))
      (Ideal.pow_succ_lt_pow hP i).le)).symm

Goal state after:
S : Type u_1
inst✝¹ : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst✝ : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
a c : S
a_mem : a ∈ P ^ i
a_notMem : a ∉ P ^ (i + 1)
c_mem : c ∈ P ^ i
⊢ P ^ (i + 1) ≠ span {a} ⊔ P ^ (i + 1)
Module: Mathlib.RingTheory.FiniteStability

Goal state before:
pp: case refine_2
R : Type w₁
inst✝⁵ : CommRing R
A : Type w₂
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
B : Type w₃
inst✝² : CommRing B
inst✝¹ : Algebra R B
inst✝ : FinitePresentation R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] A
hsurj : Function.Surjective ⇑f
hfg : (RingHom.ker f.toRingHom).FG
g : B ⊗[R] MvPolynomial (Fin n) R →ₐ[B] B ⊗[R] A := TensorProduct.map (AlgHom.id B B) f
hgsurj : Function.Surjective ⇑g
hker_eq : RingHom.ker g = Ideal.map TensorProduct.includeRight (RingHom.ker f)
hfgg : (RingHom.ker g).FG
g' : MvPolynomial (Fin n) B →ₐ[B] B ⊗[R] A := g.comp ↑(MvPolynomial.algebraTensorAlgEquiv R B).symm
⊢ (RingHom.ker ↑↑(MvPolynomial.algebraTensorAlgEquiv R B).symm).FG

Tactic:
change Ideal.FG (RingHom.ker (AlgEquiv.symm (MvPolynomial.algebraTensorAlgEquiv R B)))
    simp only [RingHom.ker_equiv]
    exact Submodule.fg_bot

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm

Goal state before:
pp: case succ.refine_1
S : Type u_1
inst✝¹ : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst✝ : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
ih : cardQuot (P ^ i) = cardQuot P ^ i
this : P ^ (i + 1) < P ^ i
a : S
a_mem : a ∈ P ^ i
a_notMem : a ∉ P ^ (i + 1)
f g : (c : S) → c ∈ P ^ i → S
hg : ∀ (c : S) (hc : c ∈ P ^ i), g c hc ∈ P ^ (i + 1)
hf : ∀ (c : S) (hc : c ∈ P ^ i), a * f c hc + g c hc = c
k : (c' : S ⧸ P ^ i.succ) → c' ∈ map (mkQ (P ^ i.succ)) (P ^ i) → S
hk_mem : ∀ (c' : S ⧸ P ^ i.succ) (hc' : c' ∈ map (mkQ (P ^ i.succ)) (P ^ i)), k c' hc' ∈ P ^ i
hk_eq : ∀ (c' : S ⧸ P ^ i.succ) (hc' : c' ∈ map (mkQ (P ^ i.succ)) (P ^ i)), (mkQ (P ^ i.succ)) (k c' hc') = c'
c₁' : S ⧸ P ^ i.succ
hc₁' : c₁' ∈ map (mkQ (P ^ i.succ)) (P ^ i)
c₂' : S ⧸ P ^ i.succ
hc₂' : c₂' ∈ map (mkQ (P ^ i.succ)) (P ^ i)
h : (fun c' ↦ Quotient.mk'' (f (k ↑c' ⋯) ⋯)) ⟨c₁', hc₁'⟩ = (fun c' ↦ Quotient.mk'' (f (k ↑c' ⋯) ⋯)) ⟨c₂', hc₂'⟩
⊢ k c₁' hc₁' - k c₂' hc₂' ∈ P ^ i.succ

Tactic:
← hf _ (hk_mem _ hc₁'),

Goal state after:
case succ.refine_1
S : Type u_1
inst✝¹ : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst✝ : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
ih : cardQuot (P ^ i) = cardQuot P ^ i
this : P ^ (i + 1) < P ^ i
a : S
a_mem : a ∈ P ^ i
a_notMem : a ∉ P ^ (i + 1)
f g : (c : S) → c ∈ P ^ i → S
hg : ∀ (c : S) (hc : c ∈ P ^ i), g c hc ∈ P ^ (i + 1)
hf : ∀ (c : S) (hc : c ∈ P ^ i), a * f c hc + g c hc = c
k : (c' : S ⧸ P ^ i.succ) → c' ∈ map (mkQ (P ^ i.succ)) (P ^ i) → S
hk_mem : ∀ (c' : S ⧸ P ^ i.succ) (hc' : c' ∈ map (mkQ (P ^ i.succ)) (P ^ i)), k c' hc' ∈ P ^ i
hk_eq : ∀ (c' : S ⧸ P ^ i.succ) (hc' : c' ∈ map (mkQ (P ^ i.succ)) (P ^ i)), (mkQ (P ^ i.succ)) (k c' hc') = c'
c₁' : S ⧸ P ^ i.succ
hc₁' : c₁' ∈ map (mkQ (P ^ i.succ)) (P ^ i)
c₂' : S ⧸ P ^ i.succ
hc₂' : c₂' ∈ map (mkQ (P ^ i.succ)) (P ^ i)
h : (fun c' ↦ Quotient.mk'' (f (k ↑c' ⋯) ⋯)) ⟨c₁', hc₁'⟩ = (fun c' ↦ Quotient.mk'' (f (k ↑c' ⋯) ⋯)) ⟨c₂', hc₂'⟩
⊢ a * f (k c₁' hc₁') ⋯ + g (k c₁' hc₁') ⋯ - k c₂' hc₂' ∈ P ^ i.succ
Module: Mathlib.RingTheory.LocalRing.Length

Goal state before:
pp: case neg
A : Type u_1
B : Type u_2
M : Type u_3
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : IsLocalRing A
inst✝⁶ : IsLocalRing B
inst✝⁵ : Algebra A B
inst✝⁴ : IsLocalHom (algebraMap A B)
inst✝³ : AddCommGroup M
inst✝² : Module A M
inst✝¹ : Module B M
inst✝ : IsScalarTower A B M
h : ¬IsFiniteLength B M
this : ¬IsFiniteLength A M
⊢ length A M = length B M * length (ResidueField A) (ResidueField B)

Tactic:
← length_ne_top_iff,

Goal state after:
case neg
A : Type u_1
B : Type u_2
M : Type u_3
inst✝⁹ : CommRing A
inst✝⁸ : CommRing B
inst✝⁷ : IsLocalRing A
inst✝⁶ : IsLocalRing B
inst✝⁵ : Algebra A B
inst✝⁴ : IsLocalHom (algebraMap A B)
inst✝³ : AddCommGroup M
inst✝² : Module A M
inst✝¹ : Module B M
inst✝ : IsScalarTower A B M
h : ¬length B M ≠ ⊤
this : ¬length A M ≠ ⊤
⊢ length A M = length B M * length (ResidueField A) (ResidueField B)
Module: Mathlib.RingTheory.LocalRing.Length

Goal state before:
pp: case neg
A : Type u_1
B : Type u_2
M : Type u_3
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : IsLocalRing A
inst✝⁵ : IsLocalRing B
inst✝⁴ : Algebra A B
inst✝³ : IsLocalHom (algebraMap A B)
inst✝² : AddCommGroup M
inst✝¹ : Module A M
inst✝ : Flat A B
h : ¬IsFiniteLength A M
this : ¬IsFiniteLength B (B ⊗[A] M)
⊢ length B (B ⊗[A] M) = length A M * length B (B ⧸ Ideal.map (algebraMap A B) (maximalIdeal A))

Tactic:
← length_ne_top_iff,

Goal state after:
case neg
A : Type u_1
B : Type u_2
M : Type u_3
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : IsLocalRing A
inst✝⁵ : IsLocalRing B
inst✝⁴ : Algebra A B
inst✝³ : IsLocalHom (algebraMap A B)
inst✝² : AddCommGroup M
inst✝¹ : Module A M
inst✝ : Flat A B
h : ¬length A M ≠ ⊤
this : ¬length B (B ⊗[A] M) ≠ ⊤
⊢ length B (B ⊗[A] M) = length A M * length B (B ⧸ Ideal.map (algebraMap A B) (maximalIdeal A))
Module: Mathlib.RingTheory.Ideal.IdempotentFG

Goal state before:
pp: case mpr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
I : Ideal R
h : I.FG
⊢ I = ⊥ ∨ I = ⊤ → IsIdempotentElem I

Tactic:
rintro (rfl | rfl) <;> simp [IsIdempotentElem]

Goal state after:
Module: Mathlib.RingTheory.LocalRing.ResidueField.Instances

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
inst✝¹² : CommRing R
inst✝¹¹ : CommRing A
inst✝¹⁰ : CommRing B
inst✝⁹ : Algebra R A
inst✝⁸ : Algebra A B
inst✝⁷ : Algebra R B
inst✝⁶ : IsScalarTower R A B
p : Ideal A
q : Ideal B
inst✝⁵ : q.LiesOver p
inst✝⁴ : p.IsMaximal
inst✝³ : q.IsMaximal
inst✝² : Algebra (Localization.AtPrime p) (Localization.AtPrime q)
inst✝¹ : Localization.AtPrime.IsLiesOverAlgebra p q
inst✝ : Algebra.IsSeparable p.ResidueField q.ResidueField
x : A
⊢ (RingEquiv.ofBijective (algebraMap (B ⧸ q) q.ResidueField) ⋯)
      (((algebraMap (A ⧸ p) (B ⧸ q)).comp ↑(RingEquiv.ofBijective (algebraMap (A ⧸ p) p.ResidueField) ⋯).symm)
        ((RingEquiv.ofBijective (algebraMap (A ⧸ p) p.ResidueField) ⋯) ((Ideal.Quotient.mk p) x))) =
    (RingEquiv.ofBijective (algebraMap (B ⧸ q) q.ResidueField) ⋯)
      (((↑(RingEquiv.ofBijective (algebraMap (B ⧸ q) q.ResidueField) ⋯).symm).comp
          (algebraMap p.ResidueField q.ResidueField))
        ((RingEquiv.ofBijective (algebraMap (A ⧸ p) p.ResidueField) ⋯) ((Ideal.Quotient.mk p) x)))

Tactic:
simp only [RingHom.coe_comp, RingHom.coe_coe, Function.comp_apply, RingEquiv.symm_apply_apply,
    RingEquiv.apply_symm_apply]

Goal state after:
R : Type u_1
A : Type u_2
B : Type u_3
inst✝¹² : CommRing R
inst✝¹¹ : CommRing A
inst✝¹⁰ : CommRing B
inst✝⁹ : Algebra R A
inst✝⁸ : Algebra A B
inst✝⁷ : Algebra R B
inst✝⁶ : IsScalarTower R A B
p : Ideal A
q : Ideal B
inst✝⁵ : q.LiesOver p
inst✝⁴ : p.IsMaximal
inst✝³ : q.IsMaximal
inst✝² : Algebra (Localization.AtPrime p) (Localization.AtPrime q)
inst✝¹ : Localization.AtPrime.IsLiesOverAlgebra p q
inst✝ : Algebra.IsSeparable p.ResidueField q.ResidueField
x : A
⊢ (RingEquiv.ofBijective (algebraMap (B ⧸ q) q.ResidueField) ⋯)
      ((algebraMap (A ⧸ p) (B ⧸ q)) ((Ideal.Quotient.mk p) x)) =
    (algebraMap p.ResidueField q.ResidueField)
      ((RingEquiv.ofBijective (algebraMap (A ⧸ p) p.ResidueField) ⋯) ((Ideal.Quotient.mk p) x))
Module: Mathlib.NumberTheory.RamificationInertia.Ramification

Goal state before:
pp: case pos
R : Type u
inst✝³ : CommRing R
S : Type v
inst✝² : CommRing S
inst✝¹ : Algebra R S
p : Ideal R
P : Ideal S
inst✝ : IsDedekindDomain S
hp : map f p ≠ ⊥
hP : P.IsPrime
hP₂ : P = ⊥
⊢ p.ramificationIdx' P = multiplicity P (map f p)

Tactic:
rw [hP₂, ← Ideal.zero_eq_bot, multiplicity_zero_eq_zero_of_ne_zero _ hp]

Goal state after:
case pos
R : Type u
inst✝³ : CommRing R
S : Type v
inst✝² : CommRing S
inst✝¹ : Algebra R S
p : Ideal R
P : Ideal S
inst✝ : IsDedekindDomain S
hp : map f p ≠ ⊥
hP : P.IsPrime
hP₂ : P = ⊥
⊢ p.ramificationIdx' 0 = 0
Module: Mathlib.RingTheory.Unramified.Basic

Goal state before:
pp: R : Type u_1
inst✝⁷ : CommRing R
A : Type u_2
B : Type u_3
inst✝⁶ : CommRing A
inst✝⁵ : Algebra R A
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : FormallyUnramified R A
e : A ≃ₐ[R] B
C : Type u_3
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[R] C
e' : (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂
⊢ f₁.comp (AlgHom.id R B) = f₂

Tactic:
← f₂.comp_id,

Goal state after:
R : Type u_1
inst✝⁷ : CommRing R
A : Type u_2
B : Type u_3
inst✝⁶ : CommRing A
inst✝⁵ : Algebra R A
inst✝⁴ : CommRing B
inst✝³ : Algebra R B
inst✝² : FormallyUnramified R A
e : A ≃ₐ[R] B
C : Type u_3
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[R] C
e' : (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂
⊢ f₁.comp (AlgHom.id R B) = f₂.comp (AlgHom.id R B)
Module: Mathlib.RingTheory.QuasiFinite.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝¹² : CommRing R
inst✝¹¹ : CommRing S
inst✝¹⁰ : CommRing T
inst✝⁹ : Algebra R S
inst✝⁸ : Algebra R T
inst✝⁷ : Algebra S T
inst✝⁶ : IsScalarTower R S T
p : Ideal R
inst✝⁵ : p.IsPrime
P : Ideal S
inst✝⁴ : P.IsPrime
inst✝³ : P.LiesOver p
inst✝² : QuasiFiniteAt R P
inst✝¹ : Algebra (Localization.AtPrime p) (Localization.AtPrime P)
inst✝ : Localization.AtPrime.IsLiesOverAlgebra p P
⊢ Module.Finite p.ResidueField P.ResidueField

Tactic:
let m := IsLocalRing.maximalIdeal (Localization.AtPrime P)
  let : m.LiesOver p := .trans _ P _
  let := Localization.AtPrime.algebraOfLiesOver p m
  let := Localization.AtPrime.algebraOfLiesOver P m
  let e := AlgEquiv.ofBijective (IsScalarTower.toAlgHom p.ResidueField P.ResidueField
    m.ResidueField) ((RingHom.surjectiveOnStalks_of_isLocalization
        P.primeCompl _).residueFieldMap_bijective P m (m.over_def P))
  exact .of_surjective e.symm.toLinearMap e.symm.surjective

Goal state after:
Module: Mathlib.RingTheory.Kaehler.JacobiZariski

Goal state before:
pp: R : Type u₁
S : Type u₂
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u₃
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
ι : Type w₁
σ : Type w₂
Q : Generators S T ι
P : Generators R S σ
⊢ LinearMap.fst T Q.toExtension.CotangentSpace (T ⊗[S] P.toExtension.CotangentSpace) ∘ₗ ↑(compEquiv Q P) =
    Extension.CotangentSpace.map (Q.ofComp P).toExtensionHom

Tactic:
apply (Q.comp P).cotangentSpaceBasis.ext

Goal state after:
R : Type u₁
S : Type u₂
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u₃
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
ι : Type w₁
σ : Type w₂
Q : Generators S T ι
P : Generators R S σ
⊢ ∀ (i : ι ⊕ σ),
    (LinearMap.fst T Q.toExtension.CotangentSpace (T ⊗[S] P.toExtension.CotangentSpace) ∘ₗ ↑(compEquiv Q P))
        ((Q.comp P).cotangentSpaceBasis i) =
      (Extension.CotangentSpace.map (Q.ofComp P).toExtensionHom) ((Q.comp P).cotangentSpaceBasis i)
Module: Mathlib.RingTheory.Smooth.Basic

Goal state before:
pp: R : Type u_4
inst✝² : CommRing R
A : Type u_6
inst✝¹ : CommRing A
inst✝ : Algebra R A
h : Function.Surjective ⇑(algebraMap R A)
g : A →ₐ[R] R ⧸ RingHom.ker (ofId R A).toRingHom ^ 2
hg : (ofId R A).kerSquareLift.comp g = AlgHom.id R A
e : A ≃ₐ[R] R ⧸ RingHom.ker (algebraMap R A) ^ 2 := AlgEquiv.ofAlgHom g (ofId R A).kerSquareLift ⋯ hg
⊢ Function.Injective ⇑↑↑e

Tactic:
by exact e.injective

Goal state after:
Module: Mathlib.RingTheory.Kaehler.JacobiZariski

Goal state before:
pp: R : Type u₁
S : Type u₂
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u₃
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
ι : Type w₁
σ : Type w₂
Q : Generators S T ι
P : Generators R S σ
⊢ Function.Surjective ⇑(Extension.CotangentSpace.map (Q.ofComp P).toExtensionHom)

Tactic:
← fst_compEquiv

Goal state after:
R : Type u₁
S : Type u₂
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u₃
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
ι : Type w₁
σ : Type w₂
Q : Generators S T ι
P : Generators R S σ
⊢ Function.Surjective
    ⇑(LinearMap.fst T Q.toExtension.CotangentSpace (T ⊗[S] P.toExtension.CotangentSpace) ∘ₗ ↑(compEquiv Q P))
Module: Mathlib.RingTheory.Kaehler.JacobiZariski

Goal state before:
pp: R : Type u₁
S : Type u₂
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u₃
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
ι : Type w₁
σ : Type w₂
Q : Generators S T ι
P : Generators R S σ
⊢ Function.Exact
    ⇑(↑(compEquiv Q P).symm ∘ₗ LinearMap.inr T Q.toExtension.CotangentSpace (T ⊗[S] P.toExtension.CotangentSpace))
    ⇑(LinearMap.fst T Q.toExtension.CotangentSpace (T ⊗[S] P.toExtension.CotangentSpace) ∘ₗ ↑(compEquiv Q P).symm.symm)

Tactic:
rw [LinearEquiv.conj_exact_iff_exact]

Goal state after:
R : Type u₁
S : Type u₂
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u₃
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
ι : Type w₁
σ : Type w₂
Q : Generators S T ι
P : Generators R S σ
⊢ Function.Exact ⇑(LinearMap.inr T Q.toExtension.CotangentSpace (T ⊗[S] P.toExtension.CotangentSpace))
    ⇑(LinearMap.fst T Q.toExtension.CotangentSpace (T ⊗[S] P.toExtension.CotangentSpace))
Module: Mathlib.RingTheory.Smooth.Kaehler

Goal state before:
pp: R : Type u_1
P : Type u_2
S : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
l : S ⊗[P] Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (algebraMap P S))
hl : l ∘ₗ kerToTensor R P S = LinearMap.id
σ : S → P
hσ : ∀ (x : S), (algebraMap P S) (σ x) = x
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
hf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥
⊢ σ 0 - ↑(l (1 ⊗ₜ[P] (D R P) (σ 0))) = 0

Tactic:
by simp [sectionOfRetractionKerToTensorAux_prop l hl (σ 0) 0 (by simp [hσ])]

Goal state after:
Module: Mathlib.RingTheory.Etale.Kaehler

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R S
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
inst✝ : Algebra.FormallyEtale S T
s : S
⊢ (D R T) ((algebraMap S T) s) = (mapBaseChange R S T) (1 ⊗ₜ[S] (D R S) s)

Tactic:
mapBaseChange_tmul,

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R S
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
inst✝ : Algebra.FormallyEtale S T
s : S
⊢ (D R T) ((algebraMap S T) s) = 1 • (map R R S T) ((D R S) s)