mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Lie.Killing

Goal state before:
pp: K : Type u_2
L : Type u_3
inst✝⁴ : Field K
inst✝³ : LieRing L
inst✝² : LieAlgebra K L
inst✝¹ : IsKilling K L
inst✝ : Module.Finite K L
I : LieIdeal K L
this : Disjoint I (killingCompl K L I)
⊢ IsCompl I (killingCompl K L I)

Tactic:
by
    rwa [← LieSubmodule.isCompl_toSubmodule, I.toSubmodule_killingCompl,
      LinearMap.BilinForm.isCompl_orthogonal_iff_disjoint (LieModule.traceForm_isSymm K L L).isRefl,
      ← I.toSubmodule_killingCompl, LieSubmodule.disjoint_toSubmodule]

Goal state after:
Module: Mathlib.Algebra.Lie.TraceForm

Goal state before:
pp: case mem
R : Type u_1
L : Type u_3
M : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
h : traceForm R L M = 0
y : End R M
hy : ∀ z ∈ φ.range, ⁅y, z⁆ ∈ φ.range
x a b c : L
hbc : φ c = ⁅y, φ b⁆
⊢ (trace R M) (φ ⁅a, b⁆ * y) = 0

Tactic:
replace hbc : ⁅φ b, y⁆ = -φ c := by rw [hbc, Module.End.instLieRingModule_eq, lie_skew]

Goal state after:
case mem
R : Type u_1
L : Type u_3
M : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
h : traceForm R L M = 0
y : End R M
hy : ∀ z ∈ φ.range, ⁅y, z⁆ ∈ φ.range
x a b c : L
hbc : ⁅φ b, y⁆ = -φ c
⊢ (trace R M) (φ ⁅a, b⁆ * y) = 0
Module: Mathlib.Algebra.Lie.TraceForm

Goal state before:
pp: case succ
R : Type u_1
L : Type u_3
M : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : LieRing L
inst✝⁵ : LieAlgebra R L
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
inst✝¹ : LieModule R L M
inst✝ : LieRing.IsNilpotent L
B : LinearMap.BilinForm R M
m₀ m₁ : M
hm₀ : m₀ ∈ genWeightSpace M 0
hm₁ : m₁ ∈ posFittingComp R L M
x : L
k : ℕ
ih : ∀ (m n : M), (B m) ((φ x ^ k) n) = (-1) ^ k • (B ((φ x ^ k) m)) n
m n : M
hB : ∀ (m : M), (B m) ((φ x) n) = -1 • (B ((φ x) m)) n
⊢ (B m) ((φ x ^ (k + 1)) n) = (-1) ^ (k + 1) • (B ((φ x ^ (k + 1)) m)) n

Tactic:
have : (-1 : R) ^ k • (-1 : R) = (-1 : R) ^ (k + 1) := by rw [pow_succ (-1 : R), smul_eq_mul]

Goal state after:
case succ
R : Type u_1
L : Type u_3
M : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : LieRing L
inst✝⁵ : LieAlgebra R L
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
inst✝¹ : LieModule R L M
inst✝ : LieRing.IsNilpotent L
B : LinearMap.BilinForm R M
m₀ m₁ : M
hm₀ : m₀ ∈ genWeightSpace M 0
hm₁ : m₁ ∈ posFittingComp R L M
x : L
k : ℕ
ih : ∀ (m n : M), (B m) ((φ x ^ k) n) = (-1) ^ k • (B ((φ x ^ k) m)) n
m n : M
hB : ∀ (m : M), (B m) ((φ x) n) = -1 • (B ((φ x) m)) n
this : (-1) ^ k • -1 = (-1) ^ (k + 1)
⊢ (B m) ((φ x ^ (k + 1)) n) = (-1) ^ (k + 1) • (B ((φ x ^ (k + 1)) m)) n
Module: Mathlib.Algebra.Lie.TraceForm

Goal state before:
pp: R : Type u_1
L : Type u_3
M : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
k : ℕ
x : L
hk : 1 ≤ k
hx : x ∈ lowerCentralSeries R L L k
⊢ (trace R M) (φ x) = 0

Tactic:
replace hx : x ∈ lowerCentralSeries R L L 1 := antitone_lowerCentralSeries _ _ _ hk hx

Goal state after:
R : Type u_1
L : Type u_3
M : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
k : ℕ
x : L
hk : 1 ≤ k
hx : x ∈ lowerCentralSeries R L L 1
⊢ (trace R M) (φ x) = 0
Module: Mathlib.Algebra.Lie.TraceForm

Goal state before:
pp: R : Type u_1
L : Type u_3
M : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule R L M
inst✝³ : LieRing.IsNilpotent L
inst✝² : IsDomain R
inst✝¹ : Free R M
inst✝ : Module.Finite R M
z x : L
A : Type u_1 := AlgebraicClosure (FractionRing R)
hz : 1 ⊗ₜ[R] z ∈ lowerCentralSeries A (A ⊗[R] L) (A ⊗[R] L) 1
hzc : ∀ (x : L), ⁅x, z⁆ = 0
y u v : A ⊗[R] L
hu : ⁅u, 1 ⊗ₜ[R] z⁆ = 0
hv : ⁅v, 1 ⊗ₜ[R] z⁆ = 0
⊢ ⁅u + v, 1 ⊗ₜ[R] z⁆ = 0

Tactic:
by simp [A, hu, hv]

Goal state after:
Module: Mathlib.Algebra.Lie.TraceForm

Goal state before:
pp: R : Type u_1
L : Type u_3
M : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule R L M
inst✝³ : Free R M
inst✝² : Module.Finite R M
inst✝¹ : IsDomain R
inst✝ : IsPrincipalIdealRing R
N : LieSubmodule R L M
I : LieIdeal R L
h : I ≤ N.idealizer
x y : L
hy : y ∈ I
hy' : ∀ m ∈ N, (φ x ∘ₗ φ y) m ∈ N
this : ∀ (m : M), ⁅x, ⁅y, m⁆⁆ ∈ N
⊢ (trace R M) (φ x ∘ₗ φ y) = (trace R ↥N) ((φ x ∘ₗ φ y).restrict hy')

Tactic:
have : (trace R { x // x ∈ N }) ((φ x ∘ₗ φ y).restrict _) = (trace R M) (φ x ∘ₗ φ y) :=
      (φ x ∘ₗ φ y).trace_restrict_eq_of_forall_mem _ this
    simp [this]

Goal state after:
Module: Mathlib.Algebra.Lie.TraceForm

Goal state before:
pp: K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁰ : LieRing L
inst✝⁹ : AddCommGroup M
inst✝⁸ : LieRingModule L M
inst✝⁷ : Field K
inst✝⁶ : LieAlgebra K L
inst✝⁵ : Module K M
inst✝⁴ : LieModule K L M
inst✝³ : FiniteDimensional K M
inst✝² : LieRing.IsNilpotent L
inst✝¹ : LinearWeights K L M
inst✝ : IsTriangularizable K L M
⊢ traceForm K L M =
    ∑ χ with χ.IsNonZero,
      finrank K ↥(genWeightSpace M ⇑χ) • (Weight.toLinear K L M χ).smulRight (Weight.toLinear K L M χ)

Tactic:
classical
  suffices ∑ χ ∈ {χ : Weight K L M | χ.IsZero}, finrank K (genWeightSpace M χ) •
      (χ : L →ₗ[K] K).smulRight (χ : L →ₗ[K] K) = 0 by
    rw [traceForm_eq_sum_finrank_nsmul,
      ← Finset.sum_filter_add_sum_filter_not (p := fun χ : Weight K L M ↦ χ.IsNonZero)]
    simp [this]
  refine Finset.sum_eq_zero fun χ hχ ↦ ?_
  replace hχ : (χ : L →ₗ[K] K) = 0 := by simpa [← Weight.coe_toLinear_eq_zero_iff] using hχ
  simp [hχ]

Goal state after:
Module: Mathlib.Algebra.Lie.Weights.Killing

Goal state before:
pp: K : Type u_2
L : Type u_3
inst✝⁷ : LieRing L
inst✝⁶ : Field K
inst✝⁵ : LieAlgebra K L
inst✝⁴ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝³ : H.IsCartanSubalgebra
inst✝² : IsKilling K L
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : CharZero K
α : Weight K (↥H) L
hα : ¬α.IsZero
⊢ IsCompl Weight.ker (K ∙ coroot α)

Tactic:
rw [← coe_corootSpace_eq_span_singleton]

Goal state after:
K : Type u_2
L : Type u_3
inst✝⁷ : LieRing L
inst✝⁶ : Field K
inst✝⁵ : LieAlgebra K L
inst✝⁴ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝³ : H.IsCartanSubalgebra
inst✝² : IsKilling K L
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : CharZero K
α : Weight K (↥H) L
hα : ¬α.IsZero
⊢ IsCompl Weight.ker ↑(corootSpace ⇑α)
Module: Mathlib.Algebra.Lie.Weights.Killing

Goal state before:
pp: K : Type u_2
L : Type u_3
inst✝⁷ : LieRing L
inst✝⁶ : Field K
inst✝⁵ : LieAlgebra K L
inst✝⁴ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝³ : H.IsCartanSubalgebra
inst✝² : IsKilling K L
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : CharZero K
α : Weight K (↥H) L
hα : α.IsNonZero
e f : L
heα : e ∈ rootSpace H ⇑α
hfα : f ∈ rootSpace H (-⇑α)
hef : ⁅e, f⁆ = ((killingForm K L) e) f • ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))
h : ↥H
ht : IsSl2Triple (↑h) e f
key : α h = 2
this : ∃ s, s • h = coroot α
⊢ h = coroot α

Tactic:
by
    obtain ⟨s, hs⟩ := this
    replace this : s = 1 := by simpa [root_apply_coroot hα, key] using congr_arg α hs
    rwa [this, one_smul] at hs

Goal state after:
Module: Mathlib.Algebra.Lie.Basis.Basic

Goal state before:
pp: case lie
ι : Type u_1
R : Type u_2
L : Type u_3
inst✝⁶ : Finite ι
inst✝⁵ : CommRing R
inst✝⁴ : LieRing L
inst✝³ : LieAlgebra R L
H : LieSubalgebra R L
b : Basis ι H
inst✝² : Fintype ι
inst✝¹ : IsDomain R
inst✝ : CharZero R
this : IsLieAbelian ↥H := isLieAbelian_cartan b
x u v : L
hx✝ : u ∈ lieSpan R L (range b.e)
hy✝ : v ∈ lieSpan R L (range b.e)
hv : v ∈ ⨆ n, ⨆ (_ : n ≠ 0), rootSpace H (∑ i, n i • ⇑(b.baseSupp i))
s : Set (↥H → R) := {χ | ∃ n, n ≠ 0 ∧ χ = ⇑(∑ i, n i • b.baseSupp i)}
hs : ∀ χ₁ ∈ s, ∀ χ₂ ∈ s, χ₁ + χ₂ ∈ s
e : ↑{n | n ≠ 0} ≃ ↑s := Equiv.ofBijective (fun n ↦ ⟨∑ i, ↑n i • ⇑(b.baseSupp i), ⋯⟩) ⋯
hu : u ∈ ⨆ χ ∈ s, rootSpace H χ
⊢ ⁅u, v⁆ ∈ ⨆ n, ⨆ (_ : n ≠ 0), rootSpace H (∑ i, n i • ⇑(b.baseSupp i))

Tactic:
replace hv : v ∈ ⨆ χ, ⨆ (_ : χ ∈ s), rootSpace H χ := by
      convert! hv; rw [iSup_subtype', iSup_subtype', ← e.iSup_comp]; rfl

Goal state after:
case lie
ι : Type u_1
R : Type u_2
L : Type u_3
inst✝⁶ : Finite ι
inst✝⁵ : CommRing R
inst✝⁴ : LieRing L
inst✝³ : LieAlgebra R L
H : LieSubalgebra R L
b : Basis ι H
inst✝² : Fintype ι
inst✝¹ : IsDomain R
inst✝ : CharZero R
this : IsLieAbelian ↥H := isLieAbelian_cartan b
x u v : L
hx✝ : u ∈ lieSpan R L (range b.e)
hy✝ : v ∈ lieSpan R L (range b.e)
s : Set (↥H → R) := {χ | ∃ n, n ≠ 0 ∧ χ = ⇑(∑ i, n i • b.baseSupp i)}
hs : ∀ χ₁ ∈ s, ∀ χ₂ ∈ s, χ₁ + χ₂ ∈ s
e : ↑{n | n ≠ 0} ≃ ↑s := Equiv.ofBijective (fun n ↦ ⟨∑ i, ↑n i • ⇑(b.baseSupp i), ⋯⟩) ⋯
hu : u ∈ ⨆ χ ∈ s, rootSpace H χ
hv : v ∈ ⨆ χ ∈ s, rootSpace H χ
⊢ ⁅u, v⁆ ∈ ⨆ n, ⨆ (_ : n ≠ 0), rootSpace H (∑ i, n i • ⇑(b.baseSupp i))
Module: Mathlib.RingTheory.Polynomial.Chebyshev

Goal state before:
pp: case add_two
R : Type u_1
inst✝ : CommRing R
n : ℕ
ih1 : eval 2 (C R (↑n + 1)) = 2
ih2 : eval 2 (C R ↑n) = 2
⊢ eval 2 (C R (↑n + 2)) = 2

Tactic:
simp [C_add_two, ih1, ih2]

Goal state after:
case add_two
R : Type u_1
inst✝ : CommRing R
n : ℕ
ih1 : eval 2 (C R (↑n + 1)) = 2
ih2 : eval 2 (C R ↑n) = 2
⊢ 2 * 2 - 2 = 2
Module: Mathlib.RingTheory.Polynomial.Chebyshev

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℤ
⊢ S R (-n - 1) = -S R (n - 1)

Tactic:
induction n using Polynomial.Chebyshev.induct with
  | zero => simp
  | one => simp
  | add_two n ih1 ih2 =>
    have h₁ := S_add_one R n
    have h₂ := S_sub_two R (-n - 1)
    linear_combination (norm := ring_nf) (X : R[X]) * ih1 - ih2 + h₁ + h₂
  | neg_add_one n ih1 ih2 =>
    have h₁ := S_eq R n
    have h₂ := S_sub_two R (-n)
    linear_combination (norm := ring_nf) (X : R[X]) * ih1 - ih2 + h₁ + h₂

Goal state after:
Module: Mathlib.LinearAlgebra.RootSystem.Defs

Goal state before:
pp: ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : RootPairing ι R M N
i j k : ι
⊢ P.pairing j ((P.reflectionPerm i) k) = P.pairing ((P.reflectionPerm i) j) k

Tactic:
by
  simp only [pairing, root', coroot_reflectionPerm, root_reflectionPerm]
  simp [coreflection_apply_coroot, reflection_apply_root, mul_comm]

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℤ
⊢ (1 - X ^ 2) * derivative (T R (n + 1)) = (↑n + 1) * (T R n - X * T R (n + 1))

Tactic:
have H₁ := one_sub_X_sq_mul_U_eq_pol_in_T R n

Goal state after:
R : Type u_1
inst✝ : CommRing R
n : ℤ
H₁ : (1 - X ^ 2) * U R n = X * T R (n + 1) - T R (n + 2)
⊢ (1 - X ^ 2) * derivative (T R (n + 1)) = (↑n + 1) * (T R n - X * T R (n + 1))
Module: Mathlib.LinearAlgebra.Matrix.BaseChange

Goal state before:
pp: case intro
m : Type u_1
n : Type u_2
L : Type u_3
inst✝³ : Finite m
inst✝² : Fintype n
inst✝¹ : DecidableEq m
inst✝ : Field L
e : m ≃ n
K : Subfield L
A : Matrix m n L
B : Matrix n m L
hAB : A * B = 1
h_mem : ∀ (i : m) (j : n), A i j ∈ K
i : n
j : m
val✝ : Fintype m
A' : Matrix m m ↥K := of fun i j ↦ ⟨A.submatrix id (⇑e) i j, ⋯⟩
hA' : A'.map ⇑K.subtype = A.submatrix id ⇑e
hA : IsUnit A'
B' : Matrix m m ↥K
hB : A' * B' = 1
⊢ B i j ∈ K

Tactic:
suffices (B'.submatrix e.symm id).map K.subtype = B by simp [← this]

Goal state after:
case intro
m : Type u_1
n : Type u_2
L : Type u_3
inst✝³ : Finite m
inst✝² : Fintype n
inst✝¹ : DecidableEq m
inst✝ : Field L
e : m ≃ n
K : Subfield L
A : Matrix m n L
B : Matrix n m L
hAB : A * B = 1
h_mem : ∀ (i : m) (j : n), A i j ∈ K
i : n
j : m
val✝ : Fintype m
A' : Matrix m m ↥K := of fun i j ↦ ⟨A.submatrix id (⇑e) i j, ⋯⟩
hA' : A'.map ⇑K.subtype = A.submatrix id ⇑e
hA : IsUnit A'
B' : Matrix m m ↥K
hB : A' * B' = 1
⊢ (B'.submatrix (⇑e.symm) id).map ⇑K.subtype = B
Module: Mathlib.RingTheory.Polynomial.Chebyshev

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℤ
k : ℕ
x : R
⊢ (1 - x ^ 2) * eval x ((⇑derivative)^[k + 2] (T R n)) =
    (2 * ↑k + 1) * x * eval x ((⇑derivative)^[k + 1] (T R n)) - (↑n ^ 2 - ↑k ^ 2) * eval x ((⇑derivative)^[k] (T R n))

Tactic:
by
  have h := congr_arg (fun (p : R[X]) => p.eval x) <|
    one_sub_X_sq_mul_iterate_derivative_T_eq_poly_in_T n k
  simp only [eval_mul, eval_sub, eval_one, eval_pow,
    eval_X, eval_add, eval_ofNat, eval_natCast, eval_intCast] at h
  linear_combination (norm := (push_cast; ring_nf)) h

Goal state after:
Module: Mathlib.RingTheory.Length

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Free R M
⊢ length R M = Cardinal.toENat (Module.rank R M) * length R R

Tactic:
by
  let b := Module.Free.chooseBasis R M
  nontriviality R
  nontriviality M
  by_cases H : Module.length R R = ⊤
  · simp [b.repr.length_eq, H, rank_pos_of_free.ne']
  rw [← ne_eq, Module.length_ne_top_iff, isFiniteLength_iff_isNoetherian_isArtinian] at H
  cases H
  let b := Module.Free.chooseBasis R M
  rw [b.repr.length_eq, Module.length_finsupp, Free.rank_eq_card_chooseBasisIndex, ENat.card]

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

Goal state before:
pp: case refine_1.h
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_5
inst✝¹ : AddCommGroup N
inst✝ : Module R N
x✝¹ : AdicCompletion I M
h : ∀ (n : ℕ), x✝¹ ≡ 0 [SMOD I ^ n • ⊤]
n : ℕ
r : R
hr : r ∈ I ^ n
x : AdicCompletion I M
x✝ : x ∈ ⊤
a : M
⊢ r • Quotient.mk'' a = 0 n

Tactic:
| _ a
       => 

Goal state after:
case refine_1.h
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_5
inst✝¹ : AddCommGroup N
inst✝ : Module R N
x✝¹ : AdicCompletion I M
h : ∀ (n : ℕ), x✝¹ ≡ 0 [SMOD I ^ n • ⊤]
n : ℕ
r : R
hr : r ∈ I ^ n
x : AdicCompletion I M
x✝ : x ∈ ⊤
a : M
⊢ r • Quotient.mk'' a = 0 n
Module: Mathlib.RingTheory.DiscreteValuationRing.Basic

Goal state before:
pp: R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : IsLocalRing R
inst✝ : IsDomain R
ϖ : R
hϖ : ϖ ≠ 0
h : maximalIdeal R = span {ϖ}
h2 : ¬IsUnit ϖ
a b : R
⊢ ϖ = a * b → IsUnit a ∨ IsUnit b

Tactic:
hab

Goal state after:
R : Type u_1
inst✝² : CommSemiring R
inst✝¹ : IsLocalRing R
inst✝ : IsDomain R
ϖ : R
hϖ : ϖ ≠ 0
h : maximalIdeal R = span {ϖ}
h2 : ¬IsUnit ϖ
a b : R
hab : ϖ = a * b
⊢ IsUnit a ∨ IsUnit b
Module: Mathlib.RingTheory.Valuation.Basic

Goal state before:

Tactic:
v x = v (x + y - y) := by simp
    _ ≤ max (v <| x + y) (v y) := map_sub _ _ _
    _ < v x := max_lt h' vyx

Goal state after:
Module: Mathlib.RingTheory.KrullDimension.Zero

Goal state before:
pp: R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsReduced R
inst✝ : Nontrivial R
⊢ Subsingleton (PrimeSpectrum R) ↔ IsField R

Tactic:
by
  refine ⟨fun H ↦ ?_, fun H ↦ letI := H.toField; inferInstance⟩
  have : Subsingleton (MaximalSpectrum R) := MaximalSpectrum.toPrimeSpectrum_injective.subsingleton
  have : IsLocalRing R := .of_singleton_maximalSpectrum
  exact Ring.KrullDimLE.isField_of_isReduced

Goal state after:
Module: Mathlib.RingTheory.Jacobson.Ring

Goal state before:
pp: case refine_1
R : Type u_1
inst✝⁶ : CommRing R
Rₘ : Type u_3
Sₘ : Type u_4
inst✝⁵ : CommRing Rₘ
inst✝⁴ : CommRing Sₘ
P : Ideal R[X]
pX : R[X]
hpX : pX ∈ P
inst✝³ : Algebra (R ⧸ comap C P) Rₘ
inst✝² : IsLocalization.Away (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff Rₘ
inst✝¹ : Algebra (R[X] ⧸ P) Sₘ
inst✝ :
  IsLocalization
    (Submonoid.map (quotientMap P C ⋯) (Submonoid.powers (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff)) Sₘ
P' : Ideal R := comap C P
M : Submonoid (R ⧸ P') := Submonoid.powers (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff
M' : Submonoid (R[X] ⧸ P) :=
  Submonoid.map (quotientMap P C ⋯) (Submonoid.powers (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff)
φ : R ⧸ P' →+* R[X] ⧸ P := quotientMap P C ⋯
φ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯
hφ' : φ.comp (Ideal.Quotient.mk P') = (Ideal.Quotient.mk P).comp C
p : Sₘ
p' q : R[X] ⧸ P
hq : q ∈ M'
hp : p * (algebraMap (R[X] ⧸ P) Sₘ) q = (algebraMap (R[X] ⧸ P) Sₘ) p'
⊢ ∀ x ∈ ⇑((algebraMap (R[X] ⧸ P) Sₘ).comp (Ideal.Quotient.mk P)) '' insert X {p | p.degree ≤ 0}, φ'.IsIntegralElem x

Tactic:
rintro x ⟨p, hp, rfl⟩

Goal state after:
case refine_1
R : Type u_1
inst✝⁶ : CommRing R
Rₘ : Type u_3
Sₘ : Type u_4
inst✝⁵ : CommRing Rₘ
inst✝⁴ : CommRing Sₘ
P : Ideal R[X]
pX : R[X]
hpX : pX ∈ P
inst✝³ : Algebra (R ⧸ comap C P) Rₘ
inst✝² : IsLocalization.Away (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff Rₘ
inst✝¹ : Algebra (R[X] ⧸ P) Sₘ
inst✝ :
  IsLocalization
    (Submonoid.map (quotientMap P C ⋯) (Submonoid.powers (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff)) Sₘ
P' : Ideal R := comap C P
M : Submonoid (R ⧸ P') := Submonoid.powers (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff
M' : Submonoid (R[X] ⧸ P) :=
  Submonoid.map (quotientMap P C ⋯) (Submonoid.powers (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff)
φ : R ⧸ P' →+* R[X] ⧸ P := quotientMap P C ⋯
φ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯
hφ' : φ.comp (Ideal.Quotient.mk P') = (Ideal.Quotient.mk P).comp C
p✝ : Sₘ
p' q : R[X] ⧸ P
hq : q ∈ M'
hp✝ : p✝ * (algebraMap (R[X] ⧸ P) Sₘ) q = (algebraMap (R[X] ⧸ P) Sₘ) p'
p : R[X]
hp : p ∈ insert X {p | p.degree ≤ 0}
⊢ φ'.IsIntegralElem (((algebraMap (R[X] ⧸ P) Sₘ).comp (Ideal.Quotient.mk P)) p)
Module: Mathlib.RingTheory.Valuation.Basic

Goal state before:
pp: R : Type u_3
Γ₀ : Type u_4
inst✝¹ : Ring R
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
x : R
⊢ v.restrict x = 1 ↔ v x = 1

Tactic:
by
  simp [restrict_def, restrict₀_eq_one_iff]

Goal state after:
Module: Mathlib.RingTheory.Jacobson.Ring

Goal state before:
pp: case refine_2
R : Type u_1
inst✝⁶ : CommRing R
Rₘ : Type u_3
Sₘ : Type u_4
inst✝⁵ : CommRing Rₘ
inst✝⁴ : CommRing Sₘ
P : Ideal R[X]
pX : R[X]
hpX : pX ∈ P
inst✝³ : Algebra (R ⧸ comap C P) Rₘ
inst✝² : IsLocalization.Away (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff Rₘ
inst✝¹ : Algebra (R[X] ⧸ P) Sₘ
inst✝ :
  IsLocalization
    (Submonoid.map (quotientMap P C ⋯) (Submonoid.powers (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff)) Sₘ
P' : Ideal R := comap C P
M : Submonoid (R ⧸ P') := Submonoid.powers (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff
M' : Submonoid (R[X] ⧸ P) :=
  Submonoid.map (quotientMap P C ⋯) (Submonoid.powers (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff)
φ : R ⧸ P' →+* R[X] ⧸ P := quotientMap P C ⋯
φ' : Rₘ →+* Sₘ := IsLocalization.map Sₘ φ ⋯
hφ' : φ.comp (Ideal.Quotient.mk P') = (Ideal.Quotient.mk P).comp C
p : Sₘ
p' q : R[X] ⧸ P
hq : q ∈ M'
hp : p * (algebraMap (R[X] ⧸ P) Sₘ) q = (algebraMap (R[X] ⧸ P) Sₘ) p'
⊢ (algebraMap (R[X] ⧸ P) Sₘ) p' ∈
    Subring.closure (⇑((algebraMap (R[X] ⧸ P) Sₘ).comp (Ideal.Quotient.mk P)) '' insert X {p | p.degree ≤ 0})

Tactic:
obtain ⟨p, rfl⟩ := Ideal.Quotient.mk_surjective p'
    rw [← RingHom.comp_apply]
    apply Subring.mem_closure_image_of
    apply Polynomial.mem_closure_X_union_C

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

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
⊢ (fun I ↦ IsPrincipal I ∧ I ∈ (Ideal R)⁰) = fun I ↦ I ∈ (Ideal R)⁰ ∧ IsPrincipal I

Tactic:
simp_rw [and_comm]

Goal state after:
Module: Mathlib.RingTheory.Jacobson.Ring

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
P : Ideal R[X]
hP : P.IsMaximal
inst✝¹ : IsJacobsonRing R
inst✝ : Nontrivial R
hP' : ∀ (x : R), C x ∈ P → x = 0
P' : Ideal R := ⋯
hP'_prime : P'.IsPrime
m : R[X]
hmem_P : m ∈ P
hm : ⟨m, hmem_P⟩ ≠ 0
hm' : m ≠ 0
φ : R ⧸ P' →+* R[X] ⧸ P := ⋯
a : R ⧸ P' := ⋯
M : Submonoid (R ⧸ P') := ⋯
hp0 : a ≠ 0
hM : 0 ∉ M
M' : Submonoid (R[X] ⧸ P) := ⋯
hM' : 0 ∉ M'
bot_maximal : ⊥.IsMaximal := ⋯
⊢ IsField (R[X] ⧸ P)

Tactic:
rwa [← Quotient.maximal_ideal_iff_isField_quotient, ← bot_quotient_isMaximal_iff]

Goal state after:
Module: Mathlib.Algebra.Ring.SumsOfSquares

Goal state before:
pp: T : Type u_2
inst✝ : CommSemiring T
⊢ closure {x | IsSquare x} = sumSq T

Tactic:
apply_fun toNonUnitalSubsemiring using toNonUnitalSubsemiring_injective

Goal state after:
T : Type u_2
inst✝ : CommSemiring T
⊢ (closure {x | IsSquare x}).toNonUnitalSubsemiring = (sumSq T).toNonUnitalSubsemiring
Module: Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear

Goal state before:
pp: ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹³ : CommRing R
inst✝¹² : AddCommGroup M
inst✝¹¹ : Module R M
inst✝¹⁰ : AddCommGroup N
inst✝⁹ : Module R N
P : RootPairing ι R M N
S : Type u_5
inst✝⁸ : CommRing S
inst✝⁷ : Algebra S R
inst✝⁶ : FaithfulSMul S R
inst✝⁵ : Module S M
inst✝⁴ : IsScalarTower S R M
inst✝³ : Module S N
inst✝² : IsScalarTower S R N
inst✝¹ : P.IsValuedIn S
inst✝ : Fintype ι
x : ↥(P.rootSpan S)
⊢ (P.PolarizationIn S) x = P.Polarization ↑x

Tactic:
simp only [PolarizationIn, LinearMap.coe_sum, LinearMap.coe_comp, Finset.sum_apply, comp_apply,
    LinearMap.toSpanSingleton_apply, Polarization_apply]

Goal state after:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹³ : CommRing R
inst✝¹² : AddCommGroup M
inst✝¹¹ : Module R M
inst✝¹⁰ : AddCommGroup N
inst✝⁹ : Module R N
P : RootPairing ι R M N
S : Type u_5
inst✝⁸ : CommRing S
inst✝⁷ : Algebra S R
inst✝⁶ : FaithfulSMul S R
inst✝⁵ : Module S M
inst✝⁴ : IsScalarTower S R M
inst✝³ : Module S N
inst✝² : IsScalarTower S R N
inst✝¹ : P.IsValuedIn S
inst✝ : Fintype ι
x : ↥(P.rootSpan S)
⊢ ∑ x_1, (P.coroot'In S x_1) x • P.coroot x_1 = ∑ i, (P.coroot' i) ↑x • P.coroot i
Module: Mathlib.RingTheory.Valuation.Basic

Goal state before:
pp: case mpr.mp.inl
K : Type u_1
inst✝² : DivisionRing K
Γ₀ : Type u_4
Γ'₀ : Type u_5
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, v x = 1 ↔ v' x = 1
x : K
hx : v x ≤ 1
hx' : v x < 1
this : v' (1 + x) = 1
⊢ v' (-1 + (1 + x)) ≤ 1

Tactic:
refine le_trans (v'.map_add _ _) ?_

Goal state after:
case mpr.mp.inl
K : Type u_1
inst✝² : DivisionRing K
Γ₀ : Type u_4
Γ'₀ : Type u_5
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, v x = 1 ↔ v' x = 1
x : K
hx : v x ≤ 1
hx' : v x < 1
this : v' (1 + x) = 1
⊢ max (v' (-1)) (v' (1 + x)) ≤ 1
Module: Mathlib.Algebra.Module.Submodule.Union

Goal state before:
pp: ι : Type u_1
K : Type u_2
M : Type u_3
inst✝² : Field K
inst✝¹ : AddCommGroup M
inst✝ : Module K M
p : ι → Submodule K M
h₁ : ∀ (i : ι), p i ≠ ⊤
j : ι
s : Finset ι
hj : j ∉ s
hs : s.Nonempty
h₂ : ↑s.card + 1 < ENat.card K
hj' : ↑(p j) ∪ ⋃ i ∈ s, ↑(p i) = univ
x : M
hx : x ∈ p j
hx₀ : x ≠ 0
y : M
hy : y ∉ p j
hy₀ : y ≠ 0
sxy : Set M := {x_1 | ∃ t, ∃ (_ : t ≠ 0), x + t • y = x_1}
hsxy : sxy ⊆ ⋃ i ∈ s, ↑(p i)
this : ∃ k ∈ s, ∃ z₁ ∈ sxy, ∃ z₂ ∈ sxy, z₁ ≠ z₂ ∧ z₁ ∈ p k ∧ z₂ ∈ p k
⊢ ∃ k ∈ s, ∃ t₁ t₂, t₁ ≠ t₂ ∧ x + t₁ • y ∈ p k ∧ x + t₂ • y ∈ p k

Tactic:
obtain ⟨k, hk, -, ⟨t₁, -, rfl⟩, -, ⟨t₂, -, rfl⟩, htne, ht₁, ht₂⟩ := this
        exact ⟨k, hk, t₁, t₂, by aesop, ht₁, ht₂⟩

Goal state after:
Module: Mathlib.LinearAlgebra.RootSystem.Reduced

Goal state before:
pp: ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
P : RootPairing ι R M N
i j : ι
inst✝³ : Finite ι
inst✝² : CharZero R
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R M
⊢ P.coxeterWeight i j = 4 ↔ ¬LinearIndependent R ![P.root i, P.root j]

Tactic:
rw [P.linearIndependent_iff_coxeterWeight_ne_four, not_not]

Goal state after: