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

Goal state before:
pp: 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
⊢ ⨆ x, rootSpace H (∑ i, ↑x i • ⇑(b.baseSupp i)) = ⨆ x, LieModule.genWeightSpace L ↑(e x)

Tactic:
]

Goal state after:
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
⊢ ⨆ x, rootSpace H (∑ i, ↑x i • ⇑(b.baseSupp i)) = ⨆ x, LieModule.genWeightSpace L ↑(e x)
Module: Mathlib.RingTheory.Polynomial.Chebyshev

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

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
n : ℤ
h : T R (n + 2) = -U R n - X * -U R (n + 1)
⊢ T R (n + 2) = X * U R (n + 1) - U R n
Module: Mathlib.RingTheory.Polynomial.Chebyshev

Goal state before:
pp: case one
R : Type u_1
inst✝ : CommRing R
⊢ 2 • T R 1 ∈ Submodule.span ℕ ((fun m ↦ T R ↑m) '' Set.Icc 0 1)

Tactic:
]

Goal state after:
case one
R : Type u_1
inst✝ : CommRing R
⊢ 2 • T R 1 ∈ Submodule.span ℕ ((fun m ↦ T R ↑m) '' Set.Icc 0 1)
Module: Mathlib.RingTheory.Polynomial.Chebyshev

Goal state before:
pp: case more
R : Type u_1
inst✝ : CommRing R
n : ℕ
h₀ : U R ↑n ∈ Submodule.span ℕ ((fun m ↦ T R ↑m) '' Set.Icc 0 n)
a✝ : U R ↑(n + 1) ∈ Submodule.span ℕ ((fun m ↦ T R ↑m) '' Set.Icc 0 (n + 1))
⊢ 2 • T R (↑n + 2) + U R ↑n ∈ Submodule.span ℕ ((fun m ↦ T R ↑m) '' Set.Icc 0 (n + 2))

Tactic:
]

Goal state after:
case more
R : Type u_1
inst✝ : CommRing R
n : ℕ
h₀ : U R ↑n ∈ Submodule.span ℕ ((fun m ↦ T R ↑m) '' Set.Icc 0 n)
a✝ : U R ↑(n + 1) ∈ Submodule.span ℕ ((fun m ↦ T R ↑m) '' Set.Icc 0 (n + 1))
⊢ 2 • T R (↑n + 2) + U R ↑n ∈ Submodule.span ℕ ((fun m ↦ T R ↑m) '' Set.Icc 0 (n + 2))
Module: Mathlib.RingTheory.Polynomial.Chebyshev

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
⊢ X * C R (-↑0) - C R (-↑0 + 1) = X

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
⊢ X * C R (-↑0) - C R (-↑0 + 1) = X
Module: Mathlib.RingTheory.Polynomial.Chebyshev

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

Tactic:
induction n using Polynomial.Chebyshev.induct with

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

Goal state before:
pp: case inl
R : Type u_1
inst✝ : CommRing R
n : ℤ
h : n = ↑n.natAbs
⊢ C R ↑n.natAbs = C R ↑n.natAbs

Tactic:
]

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

Goal state before:
pp: case inr
R : Type u_1
inst✝ : CommRing R
n : ℤ
h : n = -↑n.natAbs
⊢ C R ↑n.natAbs = C R (-↑n.natAbs)

Tactic:
]

Goal state after:
case inr
R : Type u_1
inst✝ : CommRing R
n : ℤ
h : n = -↑n.natAbs
⊢ C R ↑n.natAbs = C R (-↑n.natAbs)
Module: Mathlib.RingTheory.Polynomial.Chebyshev

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℤ
⊢ (C R n).comp (2 * X) = 2 * T R n

Tactic:
induction n using Polynomial.Chebyshev.induct with

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℤ
⊢ eval 2 (C R n) = 2

Tactic:
induction n using Polynomial.Chebyshev.induct with

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℤ
⊢ eval (-2) (C R n) = 2 * ↑↑n.negOnePow

Tactic:
induction n using Polynomial.Chebyshev.induct with

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

Goal state before:
pp: R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : LieRing L
inst✝⁷ : LieAlgebra R 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
h : ↥H
h' e f : L
ht : IsSl2Triple h' e f
heα : e ∈ rootSpace H ⇑α
hfα : f ∈ rootSpace H (-⇑α)
c₁ c₂ c₃ : K
hx : c₁ • e + c₂ • f + c₃ • ⁅e, f⁆ ∈ sl2SubalgebraOfRoot hα
⊢ ∃ c₁_1 c₂_1 c₃_1, c₁ • ⁅↑h, e⁆ + c₂ • ⁅↑h, f⁆ + c₃ • ⁅↑h, ⁅e, f⁆⁆ = c₁_1 • e + c₂_1 • f + c₃_1 • ⁅e, f⁆

Tactic:
]

Goal state after:
R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : LieRing L
inst✝⁷ : LieAlgebra R 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
h : ↥H
h' e f : L
ht : IsSl2Triple h' e f
heα : e ∈ rootSpace H ⇑α
hfα : f ∈ rootSpace H (-⇑α)
c₁ c₂ c₃ : K
hx : c₁ • e + c₂ • f + c₃ • ⁅e, f⁆ ∈ sl2SubalgebraOfRoot hα
⊢ ∃ c₁_1 c₂_1 c₃_1, c₁ • ⁅↑h, e⁆ + c₂ • ⁅↑h, f⁆ + c₃ • ⁅↑h, ⁅e, f⁆⁆ = c₁_1 • e + c₂_1 • f + c₃_1 • ⁅e, f⁆
Module: Mathlib.RingTheory.Polynomial.Chebyshev

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : Invertible 2
n : ℤ
⊢ T R n = T R n

Tactic:
]

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

Goal state before:
pp: R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : LieRing L
inst✝⁷ : LieAlgebra R 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
h : ↥H
h' e f : L
ht : IsSl2Triple h' e f
heα : e ∈ rootSpace H ⇑α
hfα : f ∈ rootSpace H (-⇑α)
c₁ c₂ c₃ : K
hx : c₁ • e + c₂ • f + c₃ • ⁅e, f⁆ ∈ sl2SubalgebraOfRoot hα
he_wt : ⁅↑h, e⁆ = α h • e
hf_wt : ⁅↑h, f⁆ = (-α) h • f
⊢ ↑(coroot α) ∈ H.toLieSubmodule

Tactic:
]

Goal state after:
R : Type u_1
K : Type u_2
L : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : LieRing L
inst✝⁷ : LieAlgebra R 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
h : ↥H
h' e f : L
ht : IsSl2Triple h' e f
heα : e ∈ rootSpace H ⇑α
hfα : f ∈ rootSpace H (-⇑α)
c₁ c₂ c₃ : K
hx : c₁ • e + c₂ • f + c₃ • ⁅e, f⁆ ∈ sl2SubalgebraOfRoot hα
he_wt : ⁅↑h, e⁆ = α h • e
hf_wt : ⁅↑h, f⁆ = (-α) h • f
⊢ ↑(coroot α) ∈ H.toLieSubmodule
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

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℤ
⊢ eval 2 (S R n) = ↑n + 1

Tactic:
induction n using Polynomial.Chebyshev.induct with

Goal state after:
Module: Mathlib.LinearAlgebra.Reflection

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x y : M
f g : Dual R M
hf : f x = 2
hg : g y = 2
z : M
t : R
ht : t = f y * g x - 2
a✝ :
  ∀ (n : ℕ),
    ((reflection hf * reflection hg) ^ ↑n) z =
      z +
          (Polynomial.eval t (S R ((↑n - 2) / 2)) *
              (Polynomial.eval t (S R ((↑n - 1) / 2)) + Polynomial.eval t (S R ((↑n - 3) / 2)))) •
            ((g x * f z - g z) • y - f z • x) +
        (Polynomial.eval t (S R ((↑n - 1) / 2)) *
            (Polynomial.eval t (S R (↑n / 2)) + Polynomial.eval t (S R ((↑n - 2) / 2)))) •
          ((f y * g z - f z) • x - g z • y)
m : ℕ
⊢ t = f y * g x - 2

Tactic:
]

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x y : M
f g : Dual R M
hf : f x = 2
hg : g y = 2
z : M
t : R
ht : t = f y * g x - 2
a✝ :
  ∀ (n : ℕ),
    ((reflection hf * reflection hg) ^ ↑n) z =
      z +
          (Polynomial.eval t (S R ((↑n - 2) / 2)) *
              (Polynomial.eval t (S R ((↑n - 1) / 2)) + Polynomial.eval t (S R ((↑n - 3) / 2)))) •
            ((g x * f z - g z) • y - f z • x) +
        (Polynomial.eval t (S R ((↑n - 1) / 2)) *
            (Polynomial.eval t (S R (↑n / 2)) + Polynomial.eval t (S R ((↑n - 2) / 2)))) •
          ((f y * g z - f z) • x - g z • y)
m : ℕ
⊢ t = f y * g x - 2
Module: Mathlib.RingTheory.Polynomial.Chebyshev

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℤ
⊢ eval (-2) (S R n) = ↑↑n.negOnePow * (↑n + 1)

Tactic:
induction n using Polynomial.Chebyshev.induct with

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℤ
⊢ (S R n).comp (2 * X) = U R n

Tactic:
induction n using Polynomial.Chebyshev.induct with

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

Goal state before:
pp: case refine_1.refine_3
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
h' e f : L
ht : IsSl2Triple h' e f
heα : e ∈ rootSpace H ⇑α
hfα : f ∈ rootSpace H (-⇑α)
c₁ c₂ c₃ : K
hx : c₁ • e + c₂ • f + c₃ • ⁅e, f⁆ ∈ sl2SubalgebraOfRoot hα
⊢ c₃ • ↑(coroot α) = c₃ • ↑(coroot α)

Tactic:
]

Goal state after:
Module: Mathlib.LinearAlgebra.Reflection

Goal state before:
pp: case neg
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x y : M
f g : Dual R M
hf : f x = 2
hg : g y = 2
z : M
t : R
ht : t = f y * g x - 2
a✝ :
  ∀ (n : ℕ),
    ((reflection hf * reflection hg) ^ ↑n) z =
      z +
          (Polynomial.eval t (S R ((↑n - 2) / 2)) *
              (Polynomial.eval t (S R ((↑n - 1) / 2)) + Polynomial.eval t (S R ((↑n - 3) / 2)))) •
            ((g x * f z - g z) • y - f z • x) +
        (Polynomial.eval t (S R ((↑n - 1) / 2)) *
            (Polynomial.eval t (S R (↑n / 2)) + Polynomial.eval t (S R ((↑n - 2) / 2)))) •
          ((f y * g z - f z) • x - g z • y)
m : ℕ
ht' : t = g x * f y - 2
⊢ z +
        (Polynomial.eval t (S R ((↑m - 1) / 2)) *
            (Polynomial.eval t (S R (↑m / 2)) + Polynomial.eval t (S R ((↑m - 2) / 2)))) •
          ((g x * f z - g z) • y - f z • x) +
      (Polynomial.eval t (S R ((↑m - 2) / 2)) *
          (Polynomial.eval t (S R ((↑m - 1) / 2)) + Polynomial.eval t (S R ((↑m - 3) / 2)))) •
        ((f y * g z - f z) • x - g z • y) =
    z +
        (Polynomial.eval t (S R ((-↑m - 2) / 2)) *
            (Polynomial.eval t (S R ((-↑m - 1) / 2)) + Polynomial.eval t (S R ((-↑m - 3) / 2)))) •
          ((g x * f z - g z) • y - f z • x) +
      (Polynomial.eval t (S R ((-↑m - 1) / 2)) *
          (Polynomial.eval t (S R (-↑m / 2)) + Polynomial.eval t (S R ((-↑m - 2) / 2)))) •
        ((f y * g z - f z) • x - g z • y)

Tactic:
]

Goal state after:
case neg
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x y : M
f g : Dual R M
hf : f x = 2
hg : g y = 2
z : M
t : R
ht : t = f y * g x - 2
a✝ :
  ∀ (n : ℕ),
    ((reflection hf * reflection hg) ^ ↑n) z =
      z +
          (Polynomial.eval t (S R ((↑n - 2) / 2)) *
              (Polynomial.eval t (S R ((↑n - 1) / 2)) + Polynomial.eval t (S R ((↑n - 3) / 2)))) •
            ((g x * f z - g z) • y - f z • x) +
        (Polynomial.eval t (S R ((↑n - 1) / 2)) *
            (Polynomial.eval t (S R (↑n / 2)) + Polynomial.eval t (S R ((↑n - 2) / 2)))) •
          ((f y * g z - f z) • x - g z • y)
m : ℕ
ht' : t = g x * f y - 2
⊢ z +
        (Polynomial.eval t (S R ((↑m - 1) / 2)) *
            (Polynomial.eval t (S R (↑m / 2)) + Polynomial.eval t (S R ((↑m - 2) / 2)))) •
          ((g x * f z - g z) • y - f z • x) +
      (Polynomial.eval t (S R ((↑m - 2) / 2)) *
          (Polynomial.eval t (S R ((↑m - 1) / 2)) + Polynomial.eval t (S R ((↑m - 3) / 2)))) •
        ((f y * g z - f z) • x - g z • y) =
    z +
        (Polynomial.eval t (S R ((-↑m - 2) / 2)) *
            (Polynomial.eval t (S R ((-↑m - 1) / 2)) + Polynomial.eval t (S R ((-↑m - 3) / 2)))) •
          ((g x * f z - g z) • y - f z • x) +
      (Polynomial.eval t (S R ((-↑m - 1) / 2)) *
          (Polynomial.eval t (S R (-↑m / 2)) + Polynomial.eval t (S R ((-↑m - 2) / 2)))) •
        ((f y * g z - f z) • x - g z • y)
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
x h' e f : L
ht : IsSl2Triple h' e f
heα : e ∈ rootSpace H ⇑α
hfα : f ∈ rootSpace H (-⇑α)
hx : x ∈ genWeightSpace L ⇑α ⊔ genWeightSpace L (-⇑α) ⊔ corootSubmodule α
β : Weight K (↥H) L
hβ : β.IsNonZero
y g : L
hy : y ∈ genWeightSpace L ⇑β
hg : g ∈ rootSpace H ⇑β
hg_ne_zero : g ≠ 0
⊢ ¬g = 0

Tactic:
]

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.IsNonZero
x h' e f : L
ht : IsSl2Triple h' e f
heα : e ∈ rootSpace H ⇑α
hfα : f ∈ rootSpace H (-⇑α)
hx : x ∈ genWeightSpace L ⇑α ⊔ genWeightSpace L (-⇑α) ⊔ corootSubmodule α
β : Weight K (↥H) L
hβ : β.IsNonZero
y g : L
hy : y ∈ genWeightSpace L ⇑β
hg : g ∈ rootSpace H ⇑β
hg_ne_zero : g ≠ 0
⊢ ¬g = 0
Module: Mathlib.Algebra.Lie.Basis.Basic

Goal state before:
pp: ι : Type u_1
R : Type u_2
L : Type u_3
inst✝⁷ : Finite ι
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
H : LieSubalgebra R L
b : Basis ι H
inst✝³ : Fintype ι
inst✝² : IsDomain R
inst✝¹ : CharZero R
inst✝ : IsTorsionFree R L
this : IsLieAbelian ↥H := isLieAbelian_cartan b
U : LieSubmodule R (↥H) L := ⨆ n, ⨆ (_ : n ≠ 0), rootSpace H (∑ i, n i • ⇑((-b.baseSupp) i))
hU : U = ⨆ n, ⨆ (_ : n ≠ 0), rootSpace H (∑ i, n i • ⇑((-b.baseSupp) i))
V : LieSubmodule R (↥H) L := ⨆ n, ⨆ (_ : n ≠ 0), rootSpace H (∑ i, n i • ⇑(b.baseSupp i))
hV : V = ⨆ n, ⨆ (_ : n ≠ 0), rootSpace H (∑ i, n i • ⇑(b.baseSupp i))
s0 : Set (↥H → R) := {0}
hs0 : s0 = {0}
sU : Set (↥H → R) := {f | ∃ n, n ≠ 0 ∧ f = ⇑(∑ i, n i • (-b.baseSupp) i)}
hsU : sU = {f | ∃ n, n ≠ 0 ∧ f = ⇑(∑ i, n i • (-b.baseSupp) i)}
sV : Set (↥H → R) := {f | ∃ n, n ≠ 0 ∧ f = ⇑(∑ i, n i • b.baseSupp i)}
hsV : sV = {f | ∃ n, n ≠ 0 ∧ f = ⇑(∑ i, n i • b.baseSupp i)}
hs0' : rootSpace H 0 = ⨆ i ∈ s0, LieModule.genWeightSpace L i
hsU' : U = ⨆ i ∈ sU, LieModule.genWeightSpace L i
hsV' : V = ⨆ i ∈ sV, LieModule.genWeightSpace L i
n : ι → ℕ
hn : n ≠ 0
hg : -⇑(∑ i, n i • b.baseSupp i) ∈ sU
contra : ∑ i, n i • b.baseSupp i = 0
⊢ False

Tactic:
]

Goal state after:
ι : Type u_1
R : Type u_2
L : Type u_3
inst✝⁷ : Finite ι
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
H : LieSubalgebra R L
b : Basis ι H
inst✝³ : Fintype ι
inst✝² : IsDomain R
inst✝¹ : CharZero R
inst✝ : IsTorsionFree R L
this : IsLieAbelian ↥H := isLieAbelian_cartan b
U : LieSubmodule R (↥H) L := ⨆ n, ⨆ (_ : n ≠ 0), rootSpace H (∑ i, n i • ⇑((-b.baseSupp) i))
hU : U = ⨆ n, ⨆ (_ : n ≠ 0), rootSpace H (∑ i, n i • ⇑((-b.baseSupp) i))
V : LieSubmodule R (↥H) L := ⨆ n, ⨆ (_ : n ≠ 0), rootSpace H (∑ i, n i • ⇑(b.baseSupp i))
hV : V = ⨆ n, ⨆ (_ : n ≠ 0), rootSpace H (∑ i, n i • ⇑(b.baseSupp i))
s0 : Set (↥H → R) := {0}
hs0 : s0 = {0}
sU : Set (↥H → R) := {f | ∃ n, n ≠ 0 ∧ f = ⇑(∑ i, n i • (-b.baseSupp) i)}
hsU : sU = {f | ∃ n, n ≠ 0 ∧ f = ⇑(∑ i, n i • (-b.baseSupp) i)}
sV : Set (↥H → R) := {f | ∃ n, n ≠ 0 ∧ f = ⇑(∑ i, n i • b.baseSupp i)}
hsV : sV = {f | ∃ n, n ≠ 0 ∧ f = ⇑(∑ i, n i • b.baseSupp i)}
hs0' : rootSpace H 0 = ⨆ i ∈ s0, LieModule.genWeightSpace L i
hsU' : U = ⨆ i ∈ sU, LieModule.genWeightSpace L i
hsV' : V = ⨆ i ∈ sV, LieModule.genWeightSpace L i
n : ι → ℕ
hn : n ≠ 0
hg : -⇑(∑ i, n i • b.baseSupp i) ∈ sU
contra : ∑ i, n i • b.baseSupp i = 0
⊢ False
Module: Mathlib.RingTheory.Polynomial.Chebyshev

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℤ
⊢ 2 * S R (n + 1) = X * S R n + C R (n + 1)

Tactic:
induction n using Polynomial.Chebyshev.induct with

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

Goal state before:
pp: case refine_2
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
h' e f : L
ht : IsSl2Triple h' e f
heα : e ∈ rootSpace H ⇑α
hfα : f ∈ rootSpace H (-⇑α)
aux :
  ∀ {β : Weight K (↥H) L},
    β.IsNonZero → ∀ {y g : L}, y ∈ genWeightSpace L ⇑β → g ∈ rootSpace H ⇑β → g ≠ 0 → ∃ c, y = c • g
c₁ : K
hx_pos : c₁ • e ∈ ↑(genWeightSpace L ⇑α)
c₂ : K
hx_neg : c₂ • f ∈ ↑(genWeightSpace L (-⇑α))
hx_αneg : c₁ • e + c₂ • f ∈ ↑(genWeightSpace L ⇑α ⊔ genWeightSpace L (-⇑α))
c₃ : K
hy_coroot : c₃ • coroot α ∈ ↑↑(corootSpace ⇑α)
hx :
  c₁ • e + c₂ • f + ↑H.toLieSubmodule.incl (c₃ • coroot α) ∈
    genWeightSpace L ⇑α ⊔ genWeightSpace L (-⇑α) ⊔ corootSubmodule α
⊢ ∃ c₁_1 c₂_1 c₃_1, c₁ • e + c₂ • f + ↑H.toLieSubmodule.incl (c₃ • coroot α) = c₁_1 • e + c₂_1 • f + c₃_1 • ⁅e, f⁆

Tactic:
]

Goal state after:
case refine_2
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
h' e f : L
ht : IsSl2Triple h' e f
heα : e ∈ rootSpace H ⇑α
hfα : f ∈ rootSpace H (-⇑α)
aux :
  ∀ {β : Weight K (↥H) L},
    β.IsNonZero → ∀ {y g : L}, y ∈ genWeightSpace L ⇑β → g ∈ rootSpace H ⇑β → g ≠ 0 → ∃ c, y = c • g
c₁ : K
hx_pos : c₁ • e ∈ ↑(genWeightSpace L ⇑α)
c₂ : K
hx_neg : c₂ • f ∈ ↑(genWeightSpace L (-⇑α))
hx_αneg : c₁ • e + c₂ • f ∈ ↑(genWeightSpace L ⇑α ⊔ genWeightSpace L (-⇑α))
c₃ : K
hy_coroot : c₃ • coroot α ∈ ↑↑(corootSpace ⇑α)
hx :
  c₁ • e + c₂ • f + ↑H.toLieSubmodule.incl (c₃ • coroot α) ∈
    genWeightSpace L ⇑α ⊔ genWeightSpace L (-⇑α) ⊔ corootSubmodule α
⊢ ∃ c₁_1 c₂_1 c₃_1, c₁ • e + c₂ • f + ↑H.toLieSubmodule.incl (c₃ • coroot α) = c₁_1 • e + c₂_1 • f + c₃_1 • ⁅e, f⁆
Module: Mathlib.RingTheory.Polynomial.Chebyshev

Goal state before:
pp: R : Type u_1
R' : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing R'
f : R →+* R'
n : ℤ
⊢ map f (T R n) = T R' n

Tactic:
induction n using Polynomial.Chebyshev.induct with

Goal state after:
Module: Mathlib.LinearAlgebra.Reflection

Goal state before:
pp: case succ
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x y : M
f g : Dual R M
hf : f x = 2
hg : g y = 2
t : R
ht : t = f y * g x - 2
S_eval_t_sub_two :
  ∀ (k : ℤ),
    Polynomial.eval t (S R (k - 2)) = (f y * g x - 2) * Polynomial.eval t (S R (k - 1)) - Polynomial.eval t (S R k)
m : ℕ
ih :
  ((reflection hf * reflection hg) ^ ↑m) x =
    (Polynomial.eval t (S R ↑m) + Polynomial.eval t (S R (↑m - 1))) • x + (Polynomial.eval t (S R (↑m - 1)) * -g x) • y
⊢ (reflection hf)
      ((reflection hg)
        ((Polynomial.eval t (S R ↑m) + Polynomial.eval t (S R (↑m - 1))) • x +
          (Polynomial.eval t (S R (↑m - 1)) * -g x) • y)) =
    (Polynomial.eval t (S R (1 + ↑m)) + Polynomial.eval t (S R (1 + ↑m - 1))) • x +
      (Polynomial.eval t (S R (1 + ↑m - 1)) * -g x) • y

Tactic:
]

Goal state after:
case succ
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x y : M
f g : Dual R M
hf : f x = 2
hg : g y = 2
t : R
ht : t = f y * g x - 2
S_eval_t_sub_two :
  ∀ (k : ℤ),
    Polynomial.eval t (S R (k - 2)) = (f y * g x - 2) * Polynomial.eval t (S R (k - 1)) - Polynomial.eval t (S R k)
m : ℕ
ih :
  ((reflection hf * reflection hg) ^ ↑m) x =
    (Polynomial.eval t (S R ↑m) + Polynomial.eval t (S R (↑m - 1))) • x + (Polynomial.eval t (S R (↑m - 1)) * -g x) • y
⊢ (reflection hf)
      ((reflection hg)
        ((Polynomial.eval t (S R ↑m) + Polynomial.eval t (S R (↑m - 1))) • x +
          (Polynomial.eval t (S R (↑m - 1)) * -g x) • y)) =
    (Polynomial.eval t (S R (1 + ↑m)) + Polynomial.eval t (S R (1 + ↑m - 1))) • x +
      (Polynomial.eval t (S R (1 + ↑m - 1)) * -g x) • y
Module: Mathlib.RingTheory.Polynomial.Chebyshev

Goal state before:
pp: R : Type u_1
R' : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing R'
f : R →+* R'
n : ℤ
⊢ map f (U R n) = U R' n

Tactic:
induction n using Polynomial.Chebyshev.induct with

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.IsNonZero
⊢ genWeightSpace L ⇑α ⊔ genWeightSpace L (-⇑α) ⊔ corootSubmodule α ≠ ⊥

Tactic:
]

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.IsNonZero
⊢ genWeightSpace L ⇑α ⊔ genWeightSpace L (-⇑α) ⊔ corootSubmodule α ≠ ⊥
Module: Mathlib.RingTheory.Polynomial.Chebyshev

Goal state before:
pp: R : Type u_1
R' : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing R'
f : R →+* R'
n : ℤ
⊢ map f (C R n) = C R' n

Tactic:
induction n using Polynomial.Chebyshev.induct with

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
⊢ ∑ χ with χ.IsNonZero,
      finrank K ↥(genWeightSpace L ⇑χ) • (Weight.toLinear K (↥H) L χ).smulRight (Weight.toLinear K (↥H) L χ) =
    ∑ α ∈ LieSubalgebra.root, (Weight.toLinear K (↥H) L α).smulRight (Weight.toLinear K (↥H) L α)

Tactic:
]

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
⊢ ∑ χ with χ.IsNonZero,
      finrank K ↥(genWeightSpace L ⇑χ) • (Weight.toLinear K (↥H) L χ).smulRight (Weight.toLinear K (↥H) L χ) =
    ∑ α ∈ LieSubalgebra.root, (Weight.toLinear K (↥H) L α).smulRight (Weight.toLinear K (↥H) L α)
Module: Mathlib.RingTheory.Polynomial.Chebyshev

Goal state before:
pp: R : Type u_1
R' : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing R'
f : R →+* R'
n : ℤ
⊢ map f (S R n) = S R' n

Tactic:
induction n using Polynomial.Chebyshev.induct with

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

Goal state before:
pp: R : Type u_1
R' : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing R'
inst✝ : Algebra R R'
x : R'
n : ℤ
⊢ eval x (T R' n) = eval x (T R' n)

Tactic:
]

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

Goal state before:
pp: R : Type u_1
R' : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing R'
inst✝ : Algebra R R'
x : R'
n : ℤ
⊢ eval x (U R' n) = eval x (U R' n)

Tactic:
]

Goal state after: