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

Goal state before:
pp: R : Type u_2
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 : Module.Finite R L
inst✝² : Free R L
inst✝¹ : Module.Finite R M
inst✝ : Free R M
x y : L
r : R
b : Basis (ChooseBasisIndex R L) R L := chooseBasis R L
aux : (fun i ↦ (b.repr y) i * r + (b.repr x) i) = ⇑(b.repr (r • y + x))
⊢ Polynomial.map ((evalRingHom r).comp ↑(MvPolynomial.aeval fun i ↦ C ((b.repr y) i) * X + C ((b.repr x) i)))
      ((↑φ).polyCharpoly b) =
    charpoly (φ (r • y + x))

Tactic:
← coe_aeval_eq_evalRingHom,

Goal state after:
R : Type u_2
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 : Module.Finite R L
inst✝² : Free R L
inst✝¹ : Module.Finite R M
inst✝ : Free R M
x y : L
r : R
b : Basis (ChooseBasisIndex R L) R L := chooseBasis R L
aux : (fun i ↦ (b.repr y) i * r + (b.repr x) i) = ⇑(b.repr (r • y + x))
⊢ Polynomial.map ((↑(aeval r)).comp ↑(MvPolynomial.aeval fun i ↦ C ((b.repr y) i) * X + C ((b.repr x) i)))
      ((↑φ).polyCharpoly b) =
    charpoly (φ (r • y + x))
Module: Mathlib.LinearAlgebra.SymplecticGroup

Goal state before:
pp: l : Type u_1
inst✝² : DecidableEq l
inst✝¹ : Fintype l
R : Type u_3
inst✝ : Field R
A C : Matrix l l R
hker : ∀ (x : l → R), A • x = 0 → C • x = 0 → x = 0
hsymm : Aᵀ * C = Cᵀ * A
⊢ ∃ X, X.IsSymm ∧ IsUnit (A + X * C)

Tactic:
rcases exists_rank_normal_form C with ⟨V, U, s, hV, hU, heq⟩

Goal state after:
l : Type u_1
inst✝² : DecidableEq l
inst✝¹ : Fintype l
R : Type u_3
inst✝ : Field R
A C : Matrix l l R
hker : ∀ (x : l → R), A • x = 0 → C • x = 0 → x = 0
hsymm : Aᵀ * C = Cᵀ * A
V U : Matrix l l R
s : l ≃ Fin C.rank ⊕ Fin (Fintype.card l - C.rank)
hV : IsUnit V
hU : IsUnit U
heq : V * C * U = (fromBlocks 1 0 0 0).submatrix ⇑s ⇑s
⊢ ∃ X, X.IsSymm ∧ IsUnit (A + X * C)
Module: Mathlib.LinearAlgebra.Matrix.Rank

Goal state before:
pp: m : Type um
m₀ : Type um₀
n : Type un
n₀ : Type un₀
R : Type uR
inst✝² : Fintype n
inst✝¹ : Fintype n₀
inst✝ : CommSemiring R
em : m ≃ m₀
en : n ≃ n₀
A : Matrix m n R
⊢ finrank R ↥(Submodule.map (↑(LinearEquiv.funCongrLeft R R em.symm)) (Submodule.map A.mulVecLin ⊤)) =
    finrank R ↥A.mulVecLin.range

Tactic:
Submodule.map_top,

Goal state after:
m : Type um
m₀ : Type um₀
n : Type un
n₀ : Type un₀
R : Type uR
inst✝² : Fintype n
inst✝¹ : Fintype n₀
inst✝ : CommSemiring R
em : m ≃ m₀
en : n ≃ n₀
A : Matrix m n R
⊢ finrank R ↥(Submodule.map (↑(LinearEquiv.funCongrLeft R R em.symm)) A.mulVecLin.range) = finrank R ↥A.mulVecLin.range
Module: Mathlib.LinearAlgebra.Matrix.Rank

Goal state before:
pp: m : Type um
R : Type uR
inst✝² : Field R
inst✝¹ : Fintype m
inst✝ : DecidableEq m
M : Matrix m m R
L L' : List (TransvectionStruct m R)
D : m → R
hM0 : M = (List.map toMatrix L).prod * diagonal D * (List.map toMatrix L').prod
E : m → R := fun i ↦ if D i = 0 then 1 else (D i)⁻¹
E_def : E = fun i ↦ if D i = 0 then 1 else (D i)⁻¹
s : Finset m := {i | D i ≠ 0}
s_def : s = {i | D i ≠ 0}
V : Matrix m m R := diagonal E * (List.map (toMatrix ∘ TransvectionStruct.inv) L.reverse).prod
V_def : V = diagonal E * (List.map (toMatrix ∘ TransvectionStruct.inv) L.reverse).prod
U : Matrix m m R := (List.map (toMatrix ∘ TransvectionStruct.inv) L'.reverse).prod
U_def : U = (List.map (toMatrix ∘ TransvectionStruct.inv) L'.reverse).prod
hUdet : IsUnit U.det
hVdet : IsUnit V.det
hM : V * M * U = diagonal fun i ↦ if i ∈ s then 1 else 0
hs : s.card = M.rank
e : m ≃ Fin M.rank ⊕ Fin (Fintype.card m - M.rank) :=
  (Equiv.sumCompl fun x ↦ x ∈ s).symm.trans ((Finset.equivFinOfCardEq hs).sumCongr (Fintype.equivFinOfCardEq ⋯))
he : e = (Equiv.sumCompl fun x ↦ x ∈ s).symm.trans ((Finset.equivFinOfCardEq hs).sumCongr (Fintype.equivFinOfCardEq ⋯))
⊢ (diagonal fun i ↦ if i ∈ s then 1 else 0) = (diagonal (Sum.elim (fun x ↦ 1) fun x ↦ 0)).submatrix ⇑e ⇑e

Tactic:
submatrix_diagonal_equiv

Goal state after:
m : Type um
R : Type uR
inst✝² : Field R
inst✝¹ : Fintype m
inst✝ : DecidableEq m
M : Matrix m m R
L L' : List (TransvectionStruct m R)
D : m → R
hM0 : M = (List.map toMatrix L).prod * diagonal D * (List.map toMatrix L').prod
E : m → R := fun i ↦ if D i = 0 then 1 else (D i)⁻¹
E_def : E = fun i ↦ if D i = 0 then 1 else (D i)⁻¹
s : Finset m := {i | D i ≠ 0}
s_def : s = {i | D i ≠ 0}
V : Matrix m m R := diagonal E * (List.map (toMatrix ∘ TransvectionStruct.inv) L.reverse).prod
V_def : V = diagonal E * (List.map (toMatrix ∘ TransvectionStruct.inv) L.reverse).prod
U : Matrix m m R := (List.map (toMatrix ∘ TransvectionStruct.inv) L'.reverse).prod
U_def : U = (List.map (toMatrix ∘ TransvectionStruct.inv) L'.reverse).prod
hUdet : IsUnit U.det
hVdet : IsUnit V.det
hM : V * M * U = diagonal fun i ↦ if i ∈ s then 1 else 0
hs : s.card = M.rank
e : m ≃ Fin M.rank ⊕ Fin (Fintype.card m - M.rank) :=
  (Equiv.sumCompl fun x ↦ x ∈ s).symm.trans ((Finset.equivFinOfCardEq hs).sumCongr (Fintype.equivFinOfCardEq ⋯))
he : e = (Equiv.sumCompl fun x ↦ x ∈ s).symm.trans ((Finset.equivFinOfCardEq hs).sumCongr (Fintype.equivFinOfCardEq ⋯))
⊢ (diagonal fun i ↦ if i ∈ s then 1 else 0) = diagonal ((Sum.elim (fun x ↦ 1) fun x ↦ 0) ∘ ⇑e)
Module: Mathlib.Algebra.Lie.CartanExists

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : LieRing L
inst✝¹ : LieAlgebra K L
inst✝ : Module.Finite K L
hLK : ↑(finrank K L) ≤ #K
U : LieSubalgebra K L
x : L
hxU : x ∈ U
y : L
hyU : y ∈ U
Ex : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩
Ey : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y, ⋯⟩
hUle : U ≤ ↑Ex
hmin : ∀ E ≤ Ex, Ex ≤ E
E : LieSubmodule K (↥U) L :=
  let __src := engel K x;
  { toSubmodule := __src.toSubmodule, lie_mem := ⋯ }
hx₀ : x ≠ 0
Q : Type u_2 := L ⧸ E
r : ℕ := finrank K ↥E
hr : r < finrank K L
x' : ↥U := ⟨x, hxU⟩
y' : ↥U := ⟨y, hyU⟩
u : ↥U := y' - x'
χ : K[X][X] := lieCharpoly K (↥E) x' u
ψ : K[X][X] := lieCharpoly K Q x' u
i : ℕ
hi : i < r
hi0 : i ≠ 0
hψ : constantCoeff ψ ≠ 0
s : Finset K
hs : r ≤ s.card
hsψ : ∀ α ∈ s, eval α (constantCoeff ψ) ≠ 0
α : K
hα : α ∈ s
v : ↥U := α • u + x'
z : L
hz : z ∈ engel K ↑v
⊢ z ∈ E

Tactic:
← LieSubmodule.Quotient.mk_eq_zero

Goal state after:
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : LieRing L
inst✝¹ : LieAlgebra K L
inst✝ : Module.Finite K L
hLK : ↑(finrank K L) ≤ #K
U : LieSubalgebra K L
x : L
hxU : x ∈ U
y : L
hyU : y ∈ U
Ex : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩
Ey : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y, ⋯⟩
hUle : U ≤ ↑Ex
hmin : ∀ E ≤ Ex, Ex ≤ E
E : LieSubmodule K (↥U) L :=
  let __src := engel K x;
  { toSubmodule := __src.toSubmodule, lie_mem := ⋯ }
hx₀ : x ≠ 0
Q : Type u_2 := L ⧸ E
r : ℕ := finrank K ↥E
hr : r < finrank K L
x' : ↥U := ⟨x, hxU⟩
y' : ↥U := ⟨y, hyU⟩
u : ↥U := y' - x'
χ : K[X][X] := lieCharpoly K (↥E) x' u
ψ : K[X][X] := lieCharpoly K Q x' u
i : ℕ
hi : i < r
hi0 : i ≠ 0
hψ : constantCoeff ψ ≠ 0
s : Finset K
hs : r ≤ s.card
hsψ : ∀ α ∈ s, eval α (constantCoeff ψ) ≠ 0
α : K
hα : α ∈ s
v : ↥U := α • u + x'
z : L
hz : z ∈ engel K ↑v
⊢ (LieSubmodule.Quotient.mk' E) z = 0
Module: Mathlib.LinearAlgebra.SymplecticGroup

Goal state before:
pp: l : Type u_1
R : Type u_2
inst✝³ : DecidableEq l
inst✝² : Fintype l
inst✝¹ : CommRing R
inst✝ : IsLocalRing R
M : Matrix (l ⊕ l) (l ⊕ l) R
hM : M ∈ symplecticGroup l R
A : Matrix l l R := M.toBlocks₁₁
B : Matrix l l R := M.toBlocks₁₂
C : Matrix l l R := M.toBlocks₂₁
D : Matrix l l R := M.toBlocks₂₂
⊢ M.det = 1

Tactic:
obtain ⟨X, hX_symm, hA_isUnit⟩ := exists_symmetric_X_isUnit_det_add_mul_of_symplectic <|
    M.fromBlocks_toBlocks ▸ hM

Goal state after:
l : Type u_1
R : Type u_2
inst✝³ : DecidableEq l
inst✝² : Fintype l
inst✝¹ : CommRing R
inst✝ : IsLocalRing R
M : Matrix (l ⊕ l) (l ⊕ l) R
hM : M ∈ symplecticGroup l R
A : Matrix l l R := M.toBlocks₁₁
B : Matrix l l R := M.toBlocks₁₂
C : Matrix l l R := M.toBlocks₂₁
D : Matrix l l R := M.toBlocks₂₂
X : Matrix l l R
hX_symm : X.IsSymm
hA_isUnit : IsUnit (M.toBlocks₁₁ + X * M.toBlocks₂₁).det
⊢ M.det = 1
Module: Mathlib.Algebra.Lie.CartanExists

Goal state before:
pp: case h
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : LieRing L
inst✝¹ : LieAlgebra K L
inst✝ : Module.Finite K L
h : ↑(finrank K L) ≤ #K
x : L
hx : IsRegular K x
y✝ : L
hy✝ : y✝ ∈ engel K x
Ex : ↑{x_1 | ∃ z ∈ engel K x, engel K z = x_1} := ⟨engel K x, ⋯⟩
y : L
hy : y ∈ engel K x
hyx : ⟨engel K y, ⋯⟩ ≤ Ex
⊢ finrank K ↥(engel K x) ≤ finrank K ↥(engel K y)

Tactic:
(isRegular_iff_finrank_engel_eq_rank K x).mp hx

Goal state after:
case h
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : LieRing L
inst✝¹ : LieAlgebra K L
inst✝ : Module.Finite K L
h : ↑(finrank K L) ≤ #K
x : L
hx : IsRegular K x
y✝ : L
hy✝ : y✝ ∈ engel K x
Ex : ↑{x_1 | ∃ z ∈ engel K x, engel K z = x_1} := ⟨engel K x, ⋯⟩
y : L
hy : y ∈ engel K x
hyx : ⟨engel K y, ⋯⟩ ≤ Ex
⊢ rank K L ≤ finrank K ↥(engel K y)
Module: Mathlib.Algebra.Lie.Weights.IsSimple

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : LieRing L
inst✝⁵ : LieAlgebra K L
inst✝⁴ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝³ : H.IsCartanSubalgebra
inst✝² : CharZero K
inst✝¹ : IsKilling K L
inst✝ : IsTriangularizable K (↥H) L
I : LieIdeal K L
⊢ I.rootSpan ∈ (rootSystem H).invtRootSubmodule

Tactic:
rw [RootPairing.mem_invtRootSubmodule_iff]
  intro β
  rw [Module.End.mem_invtSubmodule, rootSpan, Submodule.span_le]
  rintro - ⟨α, hα, rfl⟩
  rw [SetLike.mem_coe, Submodule.mem_comap, LinearEquiv.coe_coe, ← RootPairing.root_reflectionPerm]
  exact Submodule.subset_span ⟨_, (I.reflectionPerm_mem_rootSet_iff α β).mpr hα, rfl⟩

Goal state after:
Module: Mathlib.RingTheory.ChainOfDivisors

Goal state before:
pp: case succ.refine_2
M : Type u_1
inst✝¹ : CommMonoidWithZero M
inst✝ : IsCancelMulZero M
q : Associates M
hq : q ≠ 0
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
i : ℕ
hi : i < n + 1 + 1
hb : c ⟨i, hi⟩ < c 1
⊢ c ⟨i, hi⟩ = ⊥

Tactic:
cases i

Goal state after:
case succ.refine_2.zero
M : Type u_1
inst✝¹ : CommMonoidWithZero M
inst✝ : IsCancelMulZero M
q : Associates M
hq : q ≠ 0
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
hi : 0 < n + 1 + 1
hb : c ⟨0, hi⟩ < c 1
⊢ c ⟨0, hi⟩ = ⊥

case succ.refine_2.succ
M : Type u_1
inst✝¹ : CommMonoidWithZero M
inst✝ : IsCancelMulZero M
q : Associates M
hq : q ≠ 0
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
n✝ : ℕ
hi : n✝ + 1 < n + 1 + 1
hb : c ⟨n✝ + 1, hi⟩ < c 1
⊢ c ⟨n✝ + 1, hi⟩ = ⊥
Module: Mathlib.Algebra.Lie.Weights.IsSimple

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : LieRing L
inst✝⁵ : LieAlgebra K L
inst✝⁴ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝³ : H.IsCartanSubalgebra
inst✝² : CharZero K
inst✝¹ : IsKilling K L
inst✝ : IsTriangularizable K (↥H) L
I : LieIdeal K L
α : ↥LieSubalgebra.root
hα : α ∈ I.rootSet
β : ↥LieSubalgebra.root
hβ : β ∉ I.rootSet
h_ker : coroot ↑α ∈ Weight.ker
⊢ coroot ↑β ∈ (traceForm K (↥H) L).orthogonal (K ∙ coroot ↑α)

Tactic:
LinearMap.BilinForm.orthogonal_span_singleton_eq_toLin_ker,

Goal state after:
K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : LieRing L
inst✝⁵ : LieAlgebra K L
inst✝⁴ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝³ : H.IsCartanSubalgebra
inst✝² : CharZero K
inst✝¹ : IsKilling K L
inst✝ : IsTriangularizable K (↥H) L
I : LieIdeal K L
α : ↥LieSubalgebra.root
hα : α ∈ I.rootSet
β : ↥LieSubalgebra.root
hβ : β ∉ I.rootSet
h_ker : coroot ↑α ∈ Weight.ker
⊢ coroot ↑β ∈ ((traceForm K (↥H) L).toLinHomAux₁ (coroot ↑α)).ker
Module: Mathlib.RingTheory.ChainOfDivisors

Goal state before:
pp: M : Type u_1
inst✝ : CommMonoidWithZero M
q : Associates M
n : ℕ
c : Fin (n + 1) → Associates M
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
m : Finset (Associates M)
hm : ∀ r ∈ m, r ≤ q
⊢ ∀ r ≤ q, r ∈ Finset.image c Finset.univ

Tactic:
intro r hr

Goal state after:
M : Type u_1
inst✝ : CommMonoidWithZero M
q : Associates M
n : ℕ
c : Fin (n + 1) → Associates M
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
m : Finset (Associates M)
hm : ∀ r ∈ m, r ≤ q
r : Associates M
hr : r ≤ q
⊢ r ∈ Finset.image c Finset.univ
Module: Mathlib.RingTheory.ChainOfDivisors

Goal state before:
pp: M : Type u_1
inst✝¹ : CommMonoidWithZero M
inst✝ : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ ↑i
⊢ Finset.image c Finset.univ ⊆ Finset.image (fun m ↦ c 1 ^ ↑m) Finset.univ

Tactic:
intro r hr

Goal state after:
M : Type u_1
inst✝¹ : CommMonoidWithZero M
inst✝ : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ ↑i
r : Associates M
hr : r ∈ Finset.image c Finset.univ
⊢ r ∈ Finset.image (fun m ↦ c 1 ^ ↑m) Finset.univ
Module: Mathlib.RingTheory.FractionalIdeal.Inverse

Goal state before:
pp: K : Type u_3
inst✝⁴ : Field K
R₁ : Type u_4
inst✝³ : CommRing R₁
inst✝² : IsDomain R₁
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
I : FractionalIdeal R₁⁰ K
hI : I ≠ ⊥
i : K
hi : i ∈ I
⊢ Submodule.map (Algebra.linearMap R₁ K) ⊤ ≤ 1

Tactic:
Submodule.map_top,

Goal state after:
K : Type u_3
inst✝⁴ : Field K
R₁ : Type u_4
inst✝³ : CommRing R₁
inst✝² : IsDomain R₁
inst✝¹ : Algebra R₁ K
inst✝ : IsFractionRing R₁ K
I : FractionalIdeal R₁⁰ K
hI : I ≠ ⊥
i : K
hi : i ∈ I
⊢ (Algebra.linearMap R₁ K).range ≤ 1
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
S : Submonoid R
P : Type u_2
inst✝⁴ : CommRing P
inst✝³ : Algebra R P
inst✝² : IsLocalization S P
inst✝¹ : IsDomain R
inst✝ : Module.IsTorsionFree R P
x y : P
⊢ ⟨R ∙ x, ⋯⟩ = ⟨R ∙ y, ⋯⟩ ↔ R ∙ x = R ∙ y

Tactic:
exact Subtype.mk_eq_mk

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)
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
h_bracket_decomp :
  ⁅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
I : LieSubmodule K (↥H) L := ⨆ β, sl2SubmoduleOfRoot ⋯
genWeightSpace_le_I : ∀ β_lin ∈ q, β_lin ≠ 0 → genWeightSpace L ⇑β_lin ≤ I
h_plus_contain : genWeightSpace L ⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) ≤ I
h_minus_contain : genWeightSpace L ⇑(Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α) ≤ I
h_chi_contain : genWeightSpace L ⇑(Weight.toLinear K (↥H) L χ) ≤ I
⊢ ⁅x_χ, m_α⁆ ∈ ⨆ α, sl2SubmoduleOfRoot ⋯

Tactic:
exact sup_le (sup_le h_plus_contain h_minus_contain) h_chi_contain h_bracket_decomp

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

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
I : FractionalIdeal S P
z✝ : P
h : z✝ ∈ I
⊢ (algebraMap R P) ↑I.den * z✝ ∈ I

Tactic:
rw [← Algebra.smul_def]

Goal state after:
R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
I : FractionalIdeal S P
z✝ : P
h : z✝ ∈ I
⊢ ↑I.den • z✝ ∈ I
Module: Mathlib.RingTheory.FractionalIdeal.Operations

Goal state before:
pp: R : Type u_1
inst✝³ : CommRing R
S : Submonoid R
P : Type u_2
inst✝² : CommRing P
inst✝¹ : Algebra R P
inst✝ : IsLocalization S P
I : FractionalIdeal S P
z✝ : P
h : z✝ ∈ I
⊢ ↑I.den • z✝ ∈ I

Tactic:
exact Submodule.smul_mem _ _ h

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

Goal state before:
pp: case refine_2.refine_4
A : Type u_2
K : Type u_3
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
J : Ideal A
hJ : J ≠ ⊤
ι : Type u_4
s : Finset ι
f : ι → K
j : ι
hjs : j ∈ s
hjf : f j ≠ 0
I : FractionalIdeal A⁰ K := spanFinset A s f
hI0 : I ≠ 0
this : ↑J / I < I⁻¹
left✝ : ↑J / I ≤ I⁻¹
a : K
hI : ∀ y ∈ I, a * y ∈ 1
hpI : ∀ i ∈ s, a * f i ∈ ↑J
y : K
hy : y ∈ I
⊢ ∀ (a_1 : A), ∀ x ∈ Submodule.span A (f '' ↑s), a * x ∈ ↑J → a * a_1 • x ∈ ↑J

Tactic:
intro b x _ hx

Goal state after:
case refine_2.refine_4
A : Type u_2
K : Type u_3
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
J : Ideal A
hJ : J ≠ ⊤
ι : Type u_4
s : Finset ι
f : ι → K
j : ι
hjs : j ∈ s
hjf : f j ≠ 0
I : FractionalIdeal A⁰ K := spanFinset A s f
hI0 : I ≠ 0
this : ↑J / I < I⁻¹
left✝ : ↑J / I ≤ I⁻¹
a : K
hI : ∀ y ∈ I, a * y ∈ 1
hpI : ∀ i ∈ s, a * f i ∈ ↑J
y : K
hy : y ∈ I
b : A
x : K
hx✝ : x ∈ Submodule.span A (f '' ↑s)
hx : a * x ∈ ↑J
⊢ a * b • x ∈ ↑J
Module: Mathlib.Algebra.Module.GradedModule

Goal state before:
pp: ιA : Type u_1
ιM : Type u_2
R : Type u_3
A : Type u_4
M : Type u_5
σ : Type u_6
σ' : Type u_7
inst✝¹⁵ : AddMonoid ιA
inst✝¹⁴ : AddAction ιA ιM
inst✝¹³ : CommSemiring R
inst✝¹² : Semiring A
inst✝¹¹ : Algebra R A
𝓐 : ιA → σ'
inst✝¹⁰ : SetLike σ' A
𝓜 : ιM → σ
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module A M
inst✝⁷ : SetLike σ M
inst✝⁶ : AddSubmonoidClass σ' A
inst✝⁵ : AddSubmonoidClass σ M
inst✝⁴ : SetLike.GradedSMul 𝓐 𝓜
inst✝³ : DecidableEq ιA
inst✝² : DecidableEq ιM
inst✝¹ : GradedRing 𝓐
inst✝ : Decomposition 𝓜
h : Module A (⨁ (i : ιM), ↥(𝓜 i)) := isModule 𝓐 𝓜
x : A
y : M
i : ιA
_hi : i ∈ DFinsupp.support ((decompose 𝓐) x)
⊢ (decomposeAddEquiv 𝓜) (↑(((decompose 𝓐) x) i) • y) = (RingHom.id A) ↑(((decompose 𝓐) x) i) • (decomposeAddEquiv 𝓜) y

Tactic:
rw [RingHom.id_apply, ← DirectSum.sum_support_decompose 𝓜 y, map_sum, Finset.smul_sum, map_sum,
    Finset.smul_sum]

Goal state after:
ιA : Type u_1
ιM : Type u_2
R : Type u_3
A : Type u_4
M : Type u_5
σ : Type u_6
σ' : Type u_7
inst✝¹⁵ : AddMonoid ιA
inst✝¹⁴ : AddAction ιA ιM
inst✝¹³ : CommSemiring R
inst✝¹² : Semiring A
inst✝¹¹ : Algebra R A
𝓐 : ιA → σ'
inst✝¹⁰ : SetLike σ' A
𝓜 : ιM → σ
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module A M
inst✝⁷ : SetLike σ M
inst✝⁶ : AddSubmonoidClass σ' A
inst✝⁵ : AddSubmonoidClass σ M
inst✝⁴ : SetLike.GradedSMul 𝓐 𝓜
inst✝³ : DecidableEq ιA
inst✝² : DecidableEq ιM
inst✝¹ : GradedRing 𝓐
inst✝ : Decomposition 𝓜
h : Module A (⨁ (i : ιM), ↥(𝓜 i)) := isModule 𝓐 𝓜
x : A
y : M
i : ιA
_hi : i ∈ DFinsupp.support ((decompose 𝓐) x)
⊢ ∑ x_1 ∈ DFinsupp.support ((decompose 𝓜) y),
      (decomposeAddEquiv 𝓜) (↑(((decompose 𝓐) x) i) • ↑(((decompose 𝓜) y) x_1)) =
    ∑ x_1 ∈ DFinsupp.support ((decompose 𝓜) y), ↑(((decompose 𝓐) x) i) • (decomposeAddEquiv 𝓜) ↑(((decompose 𝓜) y) x_1)
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_bot : genWeightSpace L ⇑(Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α) = ⊥
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
h_eq : i = j
⊢ Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α = 0

Tactic:
← hi,

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)
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_bot : genWeightSpace L ⇑(Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α) = ⊥
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
h_eq : i = j
⊢ S.root i - Weight.toLinear K (↥H) L α = 0