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

Goal state before:
pp: case refine_2
R : Type u_1
inst✝⁴ : CommRing R
A : Type u_2
inst✝³ : CommRing A
inst✝² : Algebra R A
L : Type u_3
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x✝³ x✝² : LieDerivation R L L
z x✝¹ x✝ : A ⊗[R] L
hx :
  { toFun := ⇑(LinearMap.lTensor A ↑⁅x✝³, x✝²⁆), map_add' := ⋯, map_smul' := ⋯, leibniz' := ⋯ } x✝¹ =
    ⁅{ toFun := ⇑(LinearMap.lTensor A ↑x✝³), map_add' := ⋯, map_smul' := ⋯, leibniz' := ⋯ },
        { toFun := ⇑(LinearMap.lTensor A ↑x✝²), map_add' := ⋯, map_smul' := ⋯, leibniz' := ⋯ }⁆
      x✝¹
hy :
  { toFun := ⇑(LinearMap.lTensor A ↑⁅x✝³, x✝²⁆), map_add' := ⋯, map_smul' := ⋯, leibniz' := ⋯ } x✝ =
    ⁅{ toFun := ⇑(LinearMap.lTensor A ↑x✝³), map_add' := ⋯, map_smul' := ⋯, leibniz' := ⋯ },
        { toFun := ⇑(LinearMap.lTensor A ↑x✝²), map_add' := ⋯, map_smul' := ⋯, leibniz' := ⋯ }⁆
      x✝
⊢ { toFun := ⇑(LinearMap.lTensor A ↑⁅x✝³, x✝²⁆), map_add' := ⋯, map_smul' := ⋯, leibniz' := ⋯ } (x✝¹ + x✝) =
    ⁅{ toFun := ⇑(LinearMap.lTensor A ↑x✝³), map_add' := ⋯, map_smul' := ⋯, leibniz' := ⋯ },
        { toFun := ⇑(LinearMap.lTensor A ↑x✝²), map_add' := ⋯, map_smul' := ⋯, leibniz' := ⋯ }⁆
      (x✝¹ + x✝)

Tactic:
· simp_all [sub_add_sub_comm]

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

Goal state before:
pp: case mp
R : Type u₁
L : Type u₂
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
A : Type u₃
inst✝¹ : Ring A
inst✝ : Algebra R A
f : L →ₗ⁅R⁆ A
g : UniversalEnvelopingAlgebra R L →ₐ[R] A
h : ⇑g ∘ ⇑(ι R) = ⇑f
x✝ : L
⊢ ((lift R).symm g) x✝ = f x✝

Tactic:
simp [← h]

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

Goal state before:
pp: case mpr
R : Type u₁
L : Type u₂
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
A : Type u₃
inst✝¹ : Ring A
inst✝ : Algebra R A
f : L →ₗ⁅R⁆ A
g : UniversalEnvelopingAlgebra R L →ₐ[R] A
h : (lift R).symm g = f
x✝ : L
⊢ (⇑g ∘ ⇑(ι R)) x✝ = f x✝

Tactic:
simp [← h]

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

Goal state before:
pp: R : Type u
X : Type v
inst✝² : CommRing R
L : Type w
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
f : X → L
⊢ ∀ (x y : FreeLieAlgebra R X),
    Quot.liftOn (x + y) ⇑(liftAux R f) ⋯ = Quot.liftOn x ⇑(liftAux R f) ⋯ + Quot.liftOn y ⇑(liftAux R f) ⋯

Tactic:
by rintro ⟨a⟩ ⟨b⟩; rw [← liftAux_map_add]; rfl

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

Goal state before:
pp: case hadd
ι : Type u_1
σ : Type u_2
R : Type u_3
L : Type u_4
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
ℒ : ι → Submodule R L
inst✝ : GradedLieAlgebra ℒ
φ : ι →+ R
y : ⨁ (i : ι), ↥(ℒ i)
hM :
  ∀ (k : ι) (b : ⨁ (i : ι), ↥(ℒ i)),
    (decompose ℒ).symm b ∈ ℒ k →
      (toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ (φ i • LinearMap.id)) b = φ k • b
j i : ι
a : ↥(ℒ i)
f : ⨁ (i : ι), ↥(ℒ i)
a✝¹ : f i = 0
a✝ : a ≠ 0
ih :
  ↑(((toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ Module.End.smulLeft (φ i) ⋯) ⁅f, y⁆) j) =
    ↑((⁅f, (toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ Module.End.smulLeft (φ i) ⋯) y⁆ -
          ⁅y, (toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ Module.End.smulLeft (φ i) ⋯) f⁆)
        j)
⊢ ↑(((toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ Module.End.smulLeft (φ i) ⋯)
          ⁅(of (fun i ↦ ↥(ℒ i)) i) a + f, y⁆)
        j) =
    ↑((⁅(of (fun i ↦ ↥(ℒ i)) i) a + f,
            (toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ Module.End.smulLeft (φ i) ⋯) y⁆ -
          ⁅y,
            (toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ Module.End.smulLeft (φ i) ⋯)
              ((of (fun i ↦ ↥(ℒ i)) i) a + f)⁆)
        j)

Tactic:
simp only [Module.End.smulLeft_eq, DirectSum.sub_apply, AddSubgroupClass.coe_sub] at ih

Goal state after:
case hadd
ι : Type u_1
σ : Type u_2
R : Type u_3
L : Type u_4
inst✝⁵ : DecidableEq ι
inst✝⁴ : AddCommMonoid ι
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
ℒ : ι → Submodule R L
inst✝ : GradedLieAlgebra ℒ
φ : ι →+ R
y : ⨁ (i : ι), ↥(ℒ i)
hM :
  ∀ (k : ι) (b : ⨁ (i : ι), ↥(ℒ i)),
    (decompose ℒ).symm b ∈ ℒ k →
      (toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ (φ i • LinearMap.id)) b = φ k • b
j i : ι
a : ↥(ℒ i)
f : ⨁ (i : ι), ↥(ℒ i)
a✝¹ : f i = 0
a✝ : a ≠ 0
ih :
  ↑(((toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ (φ i • LinearMap.id)) ⁅f, y⁆) j) =
    ↑(⁅f, (toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ (φ i • LinearMap.id)) y⁆ j) -
      ↑(⁅y, (toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ (φ i • LinearMap.id)) f⁆ j)
⊢ ↑(((toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ Module.End.smulLeft (φ i) ⋯)
          ⁅(of (fun i ↦ ↥(ℒ i)) i) a + f, y⁆)
        j) =
    ↑((⁅(of (fun i ↦ ↥(ℒ i)) i) a + f,
            (toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ Module.End.smulLeft (φ i) ⋯) y⁆ -
          ⁅y,
            (toModule R ι (⨁ (i : ι), ↥(ℒ i)) fun i ↦ lof R ι (fun x ↦ ↥(ℒ x)) i ∘ₗ Module.End.smulLeft (φ i) ⋯)
              ((of (fun i ↦ ↥(ℒ i)) i) a + f)⁆)
        j)
Module: Mathlib.Algebra.Lie.SemiDirect

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : LieRing K
inst✝² : LieAlgebra R K
L : Type u_3
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
ψ : L →ₗ⁅R⁆ LieDerivation R K K
x✝² x✝¹ x✝ : K ⋊⁅ψ⁆ L
⊢ ⁅x✝², ⁅x✝¹, x✝⁆⁆ = ⁅⁅x✝², x✝¹⁆, x✝⁆ + ⁅x✝¹, ⁅x✝², x✝⁆⁆

Tactic:
by simp; grind [lie_skew]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Determinant.Bird.Correctness

Goal state before:
pp: n : ℕ
i : Fin n
⊢ S 0 i = {![]}

Tactic:
ext; simp [mem_S_iff, Fin.strictMono_iff_lt_succ, eq_iff_true_of_subsingleton]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Determinant.Bird.Correctness

Goal state before:
pp: case refine_1
n p : ℕ
i : Fin n
α : Fin (p + 1) → Fin n
hα : StrictMono (Fin.cons i α)
⊢ StrictMono (Fin.cons (α 0) (Fin.tail α))

Tactic:
simp only [Fin.cons_self_tail]
    exact hα.comp Fin.strictMono_succ

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Determinant.Bird.Correctness

Goal state before:
pp: case h.refine_1.refine_1
n p : ℕ
i : Fin n
α : Fin p → Fin n
k : Fin n
hik : i < k
hk : k ∉ Set.range α
t : Fin (p + 1) := ⨅ j ∈ {j | k < α j}, j.castSucc
t_eq : t = ⨅ j ∈ {j | k < α j}, j.castSucc
hα : (∀ (j : Fin p), i < α j) ∧ StrictMono α
j : Fin p
x : Fin (p + 1)
hjx : j.castSucc < x
h : ∀ (i : Fin p), k < α i → x ≤ i.castSucc
⊢ α j < k

Tactic:
contrapose! h

Goal state after:
case h.refine_1.refine_1
n p : ℕ
i : Fin n
α : Fin p → Fin n
k : Fin n
hik : i < k
hk : k ∉ Set.range α
t : Fin (p + 1) := ⨅ j ∈ {j | k < α j}, j.castSucc
t_eq : t = ⨅ j ∈ {j | k < α j}, j.castSucc
hα : (∀ (j : Fin p), i < α j) ∧ StrictMono α
j : Fin p
x : Fin (p + 1)
hjx : j.castSucc < x
h : k ≤ α j
⊢ ∃ i, k < α i ∧ i.castSucc < x
Module: Mathlib.RingTheory.FractionalIdeal.Basic

Goal state before:
pp: case refine_1
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
b : P
hb : b ∈ I * J
m : P
hm : m ∈ I
n : P
hn : n ∈ J
n' : R
hn' : (algebraMap R P) n' = aJ • n
⊢ n' • m ∈ I

Tactic:
exact Submodule.smul_mem _ _ hm

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

Goal state before:
pp: case mp.refine_1
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
J : FractionalIdeal S P
hJ : J ≤ 1
a b : R
ha : a ∈ {x | (algebraMap R P) x ∈ J}
hb : b ∈ {x | (algebraMap R P) x ∈ J}
⊢ (algebraMap R P) a + (algebraMap R P) b ∈ J

Tactic:
exact J.val.add_mem ha hb

Goal state after:
Module: Mathlib.RingTheory.ChainOfDivisors

Goal state before:
pp: case neg.refine_2
M : Type u_1
inst✝⁴ : CommMonoidWithZero M
inst✝³ : IsCancelMulZero M
N : Type u_2
inst✝² : CommMonoidWithZero N
inst✝¹ : UniqueFactorizationMonoid N
inst✝ : UniqueFactorizationMonoid M
m : Associates M
n : Associates N
hn : n ≠ 0
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ i, r = c₁ i
c₂ : Fin (s + 1) → Associates N := fun t ↦ ↑(d ⟨c₁ t, ⋯⟩)
c₂_def : ∀ (t : Fin (s + 1)), c₂ t = ↑(d ⟨c₁ t, ⋯⟩)
r : Associates N
hr : r ≤ ↑(d ⟨c₁ 1 ^ s, hs'⟩)
this : r ≤ n
⊢ d.symm ⟨r, this⟩ ≤ d.symm (d ⟨c₁ 1 ^ s, hs'⟩)

Tactic:
rw [d.symm.le_iff_le]

Goal state after:
case neg.refine_2
M : Type u_1
inst✝⁴ : CommMonoidWithZero M
inst✝³ : IsCancelMulZero M
N : Type u_2
inst✝² : CommMonoidWithZero N
inst✝¹ : UniqueFactorizationMonoid N
inst✝ : UniqueFactorizationMonoid M
m : Associates M
n : Associates N
hn : n ≠ 0
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
s : ℕ
hs : ¬s = 0
c₁ : Fin (s + 1) → Associates M
hp : c₁ 1 ∈ normalizedFactors m
hs' : c₁ 1 ^ s ≤ m
hc₁' : StrictMono c₁
hc₁'' : ∀ {r : Associates M}, r ≤ c₁ 1 ^ s ↔ ∃ i, r = c₁ i
c₂ : Fin (s + 1) → Associates N := fun t ↦ ↑(d ⟨c₁ t, ⋯⟩)
c₂_def : ∀ (t : Fin (s + 1)), c₂ t = ↑(d ⟨c₁ t, ⋯⟩)
r : Associates N
hr : r ≤ ↑(d ⟨c₁ 1 ^ s, hs'⟩)
this : r ≤ n
⊢ ⟨r, this⟩ ≤ d ⟨c₁ 1 ^ s, hs'⟩
Module: Mathlib.RingTheory.ChainOfDivisors

Goal state before:
pp: case refine_1
M : Type u_1
inst✝⁴ : CommMonoidWithZero M
inst✝³ : IsCancelMulZero M
N : Type u_2
inst✝² : CommMonoidWithZero N
inst✝¹ : UniqueFactorizationMonoid N
inst✝ : UniqueFactorizationMonoid M
m : Associates M
n : Associates N
hn : n ≠ 0
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
hp : 1 ∈ normalizedFactors m
⊢ False

Tactic:
exact (prime_of_normalized_factor 1 hp).not_isUnit isUnit_one

Goal state after:
Module: Mathlib.RingTheory.ChainOfDivisors

Goal state before:
pp: M : Type u_1
inst✝⁶ : CommMonoidWithZero M
inst✝⁵ : IsCancelMulZero M
N : Type u_2
inst✝⁴ : CommMonoidWithZero N
inst✝³ : Subsingleton Mˣ
inst✝² : Subsingleton Nˣ
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : UniqueFactorizationMonoid N
m p : M
n : N
hm : m ≠ 0
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : { l // l ∣ m } ≃ { l // l ∣ n }
hd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'
⊢ Associates.mk p ∈ Set.Iic (Associates.mk m)

Tactic:
exact mk_dvd_mk.mpr (dvd_of_mem_normalizedFactors hp)

Goal state after:
Module: Mathlib.RingTheory.ChainOfDivisors

Goal state before:
pp: M : Type u_1
inst✝⁶ : CommMonoidWithZero M
inst✝⁵ : IsCancelMulZero M
N : Type u_2
inst✝⁴ : CommMonoidWithZero N
inst✝³ : Subsingleton Mˣ
inst✝² : Subsingleton Nˣ
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : UniqueFactorizationMonoid N
m p : M
n : N
hm : m ≠ 0
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : { l // l ∣ m } ≃ { l // l ∣ n }
hd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'
this✝ :
  Associates.mk ↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), ⋯⟩) =
    ↑((mkFactorOrderIsoOfFactorDvdEquiv hd) ⟨associatesEquivOfUniqueUnits.symm p, ⋯⟩)
this : DecidableEq (Associates M) := Classical.decEq (Associates M)
⊢ associatesEquivOfUniqueUnits.symm p ∈ normalizedFactors (Associates.mk m)

Tactic:
obtain ⟨q, hq, hq'⟩ :=
    exists_mem_normalizedFactors_of_dvd (mk_ne_zero.mpr hm)
      (prime_mk.mpr (prime_of_normalized_factor p (by convert! hp))).irreducible
      (mk_le_mk_of_dvd (dvd_of_mem_normalizedFactors hp))

Goal state after:
M : Type u_1
inst✝⁶ : CommMonoidWithZero M
inst✝⁵ : IsCancelMulZero M
N : Type u_2
inst✝⁴ : CommMonoidWithZero N
inst✝³ : Subsingleton Mˣ
inst✝² : Subsingleton Nˣ
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : UniqueFactorizationMonoid N
m p : M
n : N
hm : m ≠ 0
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : { l // l ∣ m } ≃ { l // l ∣ n }
hd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'
this✝ :
  Associates.mk ↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), ⋯⟩) =
    ↑((mkFactorOrderIsoOfFactorDvdEquiv hd) ⟨associatesEquivOfUniqueUnits.symm p, ⋯⟩)
this : DecidableEq (Associates M) := Classical.decEq (Associates M)
q : Associates M
hq : q ∈ normalizedFactors (Associates.mk m)
hq' : Associated (Associates.mk p) q
⊢ associatesEquivOfUniqueUnits.symm p ∈ normalizedFactors (Associates.mk m)
Module: Mathlib.Algebra.Lie.Weights.IsSimple

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
inst✝² : IsKilling K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
q : Submodule K (Dual K ↥H)
χ : Weight K (↥H) L
x_χ m_α : L
hx_χ : x_χ ∈ genWeightSpace L ⇑χ
α : Weight K (↥H) L
hαq : Weight.toLinear K (↥H) L α ∈ q
w_plus : Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α ≠ 0
w_minus : Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α ≠ 0
w_chi : Weight.toLinear K (↥H) L χ ≠ 0
m_pos m_neg : L
y : ↥H
h_bracket_sum : ⁅x_χ, m_α⁆ = ⁅x_χ, m_pos⁆ + ⁅x_χ, m_neg⁆ + ⁅x_χ, ↑y⁆
h_pos_containment : ⁅x_χ, m_pos⁆ ∈ genWeightSpace L (⇑χ + ⇑α)
h_neg_containment : ⁅x_χ, m_neg⁆ ∈ genWeightSpace L (⇑χ - ⇑α)
h_chi_in_q : Weight.toLinear K (↥H) L χ ∈ q
h_h_containment : ⁅x_χ, ↑y⁆ ∈ genWeightSpace L ⇑χ
⊢ ⁅x_χ, m_α⁆ ∈
    genWeightSpace L ⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) ⊔
        genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ) - ⇑(Weight.toLinear K (↥H) L α)) ⊔
      genWeightSpace L ⇑χ

Tactic:
h_bracket_sum

Goal state after:
K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
inst✝² : IsKilling K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
q : Submodule K (Dual K ↥H)
χ : Weight K (↥H) L
x_χ m_α : L
hx_χ : x_χ ∈ genWeightSpace L ⇑χ
α : Weight K (↥H) L
hαq : Weight.toLinear K (↥H) L α ∈ q
w_plus : Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α ≠ 0
w_minus : Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α ≠ 0
w_chi : Weight.toLinear K (↥H) L χ ≠ 0
m_pos m_neg : L
y : ↥H
h_bracket_sum : ⁅x_χ, m_α⁆ = ⁅x_χ, m_pos⁆ + ⁅x_χ, m_neg⁆ + ⁅x_χ, ↑y⁆
h_pos_containment : ⁅x_χ, m_pos⁆ ∈ genWeightSpace L (⇑χ + ⇑α)
h_neg_containment : ⁅x_χ, m_neg⁆ ∈ genWeightSpace L (⇑χ - ⇑α)
h_chi_in_q : Weight.toLinear K (↥H) L χ ∈ q
h_h_containment : ⁅x_χ, ↑y⁆ ∈ genWeightSpace L ⇑χ
⊢ ⁅x_χ, m_pos⁆ + ⁅x_χ, m_neg⁆ + ⁅x_χ, ↑y⁆ ∈
    genWeightSpace L ⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) ⊔
        genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ) - ⇑(Weight.toLinear K (↥H) L α)) ⊔
      genWeightSpace L ⇑χ
Module: Mathlib.Algebra.Lie.Weights.IsSimple

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
inst✝² : IsKilling K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
q : Submodule K (Dual K ↥H)
hq : ∀ (i : ↥LieSubalgebra.root), q ∈ End.invtSubmodule ↑((rootSystem H).reflection i)
χ : Weight K (↥H) L
x_χ m_α : L
hx_χ : x_χ ∈ genWeightSpace L ⇑χ
α : Weight K (↥H) L
hαq : Weight.toLinear K (↥H) L α ∈ q
hα₀ : α.IsNonZero
w_plus : Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α ≠ 0
w_minus : Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α ≠ 0
w_chi : Weight.toLinear K (↥H) L χ ≠ 0
m_pos m_neg : L
y : ↥H
hy : y ∈ corootSpace ⇑α
h_bracket_sum : ⁅x_χ, m_α⁆ = ⁅x_χ, m_pos⁆ + ⁅x_χ, m_neg⁆ + ⁅x_χ, ↑y⁆
h_pos_containment : ⁅x_χ, m_pos⁆ ∈ genWeightSpace L (⇑χ + ⇑α)
h_neg_containment : ⁅x_χ, m_neg⁆ ∈ genWeightSpace L (⇑χ - ⇑α)
h_chi_not_in_q : Weight.toLinear K (↥H) L χ ∉ q
S : RootPairing (↥LieSubalgebra.root) K (Dual K ↥H) ↥H := rootSystem H
exists_root_index : ∀ (γ : Weight K (↥H) L), γ.IsNonZero → ∃ i, S.root i = Weight.toLinear K (↥H) L γ
h_plus_ne_bot : ¬genWeightSpace L ⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) = ⊥
γ : Weight K (↥H) L :=
  { toFun := ⇑(Weight.toLinear K (↥H) L χ) + ⇑(Weight.toLinear K (↥H) L α), genWeightSpace_ne_bot' := h_plus_ne_bot }
hγ_nonzero : γ.IsNonZero
i : ↥LieSubalgebra.root
hi : S.root i = Weight.toLinear K (↥H) L χ
j : ↥LieSubalgebra.root
hj : S.root j = Weight.toLinear K (↥H) L α
⊢ Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α ∈ Set.range ⇑S.root

Tactic:
exact ⟨⟨γ, by simpa [LieSubalgebra.root]⟩, rfl⟩

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: case pos
A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
p I : Ideal A
hI : I ≠ ⊥
hp : p = 0
⊢ p ∈ normalizedFactors I ↔ p.IsPrime ∧ p ∣ I

Tactic:
rw [← zero_eq_bot] at hI
    simp only [hp, zero_notMem_normalizedFactors, zero_dvd_iff, hI, false_iff, not_and,
      not_false_eq_true, implies_true]

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

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
inst✝² : IsKilling K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
q : Submodule K (Dual K ↥H)
hq : ∀ (i : ↥LieSubalgebra.root), q ∈ End.invtSubmodule ↑((rootSystem H).reflection i)
χ : Weight K (↥H) L
x_χ m_α : L
hx_χ : x_χ ∈ genWeightSpace L ⇑χ
α : Weight K (↥H) L
hαq : Weight.toLinear K (↥H) L α ∈ q
hα₀ : α.IsNonZero
w_plus : Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α ≠ 0
w_minus : Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α ≠ 0
w_chi : Weight.toLinear K (↥H) L χ ≠ 0
m_pos m_neg : L
y : ↥H
hy : y ∈ corootSpace ⇑α
h_bracket_sum : ⁅x_χ, m_α⁆ = ⁅x_χ, m_pos⁆ + ⁅x_χ, m_neg⁆ + ⁅x_χ, ↑y⁆
h_pos_containment : ⁅x_χ, m_pos⁆ ∈ genWeightSpace L (⇑χ + ⇑α)
h_neg_containment : ⁅x_χ, m_neg⁆ ∈ genWeightSpace L (⇑χ - ⇑α)
h_chi_not_in_q : Weight.toLinear K (↥H) L χ ∉ q
S : RootPairing (↥LieSubalgebra.root) K (Dual K ↥H) ↥H := rootSystem H
exists_root_index : ∀ (γ : Weight K (↥H) L), γ.IsNonZero → ∃ i, S.root i = Weight.toLinear K (↥H) L γ
h_plus_bot : genWeightSpace L ⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) = ⊥
h_minus_ne_bot : ¬genWeightSpace L ⇑(Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α) = ⊥
γ : Weight K (↥H) L :=
  { toFun := ⇑(Weight.toLinear K (↥H) L χ) - ⇑(Weight.toLinear K (↥H) L α), genWeightSpace_ne_bot' := h_minus_ne_bot }
hγ_nonzero : γ.IsNonZero
i : ↥LieSubalgebra.root
hi : S.root i = Weight.toLinear K (↥H) L χ
j : ↥LieSubalgebra.root
hj : S.root j = Weight.toLinear K (↥H) L (-α)
⊢ Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α ∈ Set.range ⇑S.root

Tactic:
exact ⟨⟨γ, by simpa [LieSubalgebra.root]⟩, rfl⟩

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
⊢ normalize ?m.44 = I ⊔ J

pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := ⋯
⊢ Ideal A

pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
⊢ gcd I J ∣ ?m.44

pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
⊢ ?m.44 ∣ gcd I J

Tactic:
normalize_eq

Goal state after:
A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
⊢ ?m.51 = I ⊔ J

A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := ⋯
⊢ Ideal A

A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
⊢ gcd I J ∣ ?m.51

A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
⊢ ?m.51 ∣ gcd I J
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
hgcd : gcd I J = I ⊔ J
⊢ lcm I J = I ⊓ J

Tactic:
lcm_eq_normalize _ _,

Goal state after:
A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
hgcd : gcd I J = I ⊔ J
⊢ normalize ?m.122 = I ⊓ J

A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := ⋯
hgcd : gcd I J = I ⊔ J
⊢ Ideal A

A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
hgcd : gcd I J = I ⊔ J
⊢ lcm I J ∣ ?m.122

A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
hgcd : gcd I J = I ⊔ J
⊢ ?m.122 ∣ lcm I J
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
hgcd : gcd I J = I ⊔ J
⊢ normalize ?m.122 = I ⊓ J

pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := ⋯
hgcd : gcd I J = I ⊔ J
⊢ Ideal A

pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
hgcd : gcd I J = I ⊔ J
⊢ lcm I J ∣ ?m.122

pp: A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
hgcd : gcd I J = I ⊔ J
⊢ ?m.122 ∣ lcm I J

Tactic:
normalize_eq

Goal state after:
A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
hgcd : gcd I J = I ⊔ J
⊢ ?m.129 = I ⊓ J

A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := ⋯
hgcd : gcd I J = I ⊔ J
⊢ Ideal A

A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
hgcd : gcd I J = I ⊔ J
⊢ lcm I J ∣ ?m.129

A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
I J : Ideal A
this : NormalizedGCDMonoid (Ideal A) := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
hgcd : gcd I J = I ⊔ J
⊢ ?m.129 ∣ lcm I J
Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
inst✝ : NormalizationMonoid R
r d : R
hr : r ≠ 0
hd : d ∈ normalizedFactors r
⊢ emultiplicity d r = emultiplicity (↑((normalizedFactorsEquivSpanNormalizedFactors hr) ⟨d, hd⟩)) (span {r})

Tactic:
simp only [normalizedFactorsEquivSpanNormalizedFactors, emultiplicity_eq_emultiplicity_span,
    Subtype.coe_mk, Equiv.ofBijective_apply]

Goal state after: