mathlib-initiative/mathlib-tactics huggingface dataset:
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 ((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)) = 0
this :
  (cartanEquivDual H).symm (Weight.toLinear K (↥H) L α) ∈
    Weight.ker ⊓ (LieIdeal.toLieSubalgebra K (↥H) (corootSpace ⇑α)).toSubmodule
⊢ α.IsZero

Tactic:
by

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
⊢ α ((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)) ≠ 0

Tactic:
by

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
⊢ α (coroot α) = 2

Tactic:
by

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
⊢ coroot α = 0 ↔ α.IsZero

Tactic:
by

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

Goal state before:
pp: K : Type u_2
L : Type u_3
inst✝⁸ : LieRing L
inst✝⁷ : Field K
inst✝⁶ : LieAlgebra K L
inst✝⁵ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝⁴ : H.IsCartanSubalgebra
inst✝³ : IsKilling K L
inst✝² : IsTriangularizable K (↥H) L
inst✝¹ : CharZero K
inst✝ : Nontrivial L
⊢ coroot 0 = 0

Tactic:
by

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
α' : ↥H := (cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)
this : K ∙ coroot α = K ∙ α'
⊢ ↑(corootSpace ⇑α) = K ∙ coroot α

Tactic:
by

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
α' : ↥H := (cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)
⊢ IsUnit (2 * (α α')⁻¹)

Tactic:
by

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
⊢ ↑(corootSpace ⇑α) = K ∙ coroot α

Tactic:
by

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
x : ↥H
h_mem : x ∈ corootSpace ⇑α
h_two : α x = 2
h₀ : ¬α.IsZero
⊢ x ∈ K ∙ coroot α

Tactic:
by

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
h₀ : ¬α.IsZero
t : K
h_two : α (t • coroot α) = 2
h_mem : t • coroot α ∈ K ∙ coroot α
this : t = 1
⊢ t • coroot α = coroot α

Tactic:
by

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
x : ↥H
h_mem : x ∈ corootSpace ⇑α
h_two : α x = 2
⊢ x = coroot α

Tactic:
by

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
⊢ corootSpace ⇑α = ⊥ ↔ α.IsZero

Tactic:
by

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

Goal state before:
pp: motive : ℤ → Prop
zero : motive 0
one : motive 1
add_two : ∀ (n : ℕ), motive (↑n + 1) → motive ↑n → motive (↑n + 2)
neg_add_one : ∀ (n : ℕ), motive (-↑n) → motive (-↑n + 1) → motive (-↑n - 1)
n : ℕ
hn : motive (-↑n)
hnm : motive (-↑n + 1)
⊢ motive (Int.negSucc n)

Tactic:
by

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

Goal state before:
pp: motive : ℤ → Prop
zero : motive 0
one : motive 1
add_two : ∀ (n : ℕ), motive (↑n + 1) → motive ↑n → motive (↑n + 2)
neg : ∀ (n : ℤ), motive n → motive (-n)
n : ℤ
h : motive (-n)
⊢ motive n

Tactic:
by

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
m : ℕ
S_eval_t_sub_two :
  ∀ (k : ℤ), Polynomial.eval t (S R (k - 2)) = t * Polynomial.eval t (S R (k - 1)) - Polynomial.eval t (S R k)
S_eval_t_sq_add_S_eval_t_sq :
  ∀ (k : ℤ),
    Polynomial.eval t (S R k) ^ 2 + Polynomial.eval t (S R (k + 1)) ^ 2 -
        t * Polynomial.eval t (S R k) * Polynomial.eval t (S R (k + 1)) =
      1
k : ℤ := ↑m / 2
ih :
  ((reflection hf * reflection hg) ^ m) z =
    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 k) + Polynomial.eval t (S R ((↑m - 2) / 2)))) •
        ((f y * g z - f z) • x - g z • y)
e : ℤ := ↑m % 2
⊢ 2 ≠ 0

Tactic:
by

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
⊢ Weight.toLinear K (↥H) L α ≠ 0

Tactic:
by

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
⊢ corootSpace ⇑α ≠ ⊥

Tactic:
by

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

Goal state before:
pp: motive : ℤ → Prop
zero : motive 0
one : motive 1
add_two : ∀ (n : ℕ), motive (↑n + 1) → motive ↑n → motive (↑n + 2)
neg : ∀ (n : ℤ), motive n → motive (-n)
⊢ ∀ (a : ℤ), motive a

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
k : ℕ
⊢ T R (-↑(k + 1) + 2) = 2 * X * T R (-↑(k + 1) + 1) - T R (-↑(k + 1))

Tactic:
by

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
⊢ IsCompl Weight.ker (K ∙ coroot α)

Tactic:
by

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

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

Tactic:
by

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
m : ℕ
S_eval_t_sub_two :
  ∀ (k : ℤ), Polynomial.eval t (S R (k - 2)) = t * Polynomial.eval t (S R (k - 1)) - Polynomial.eval t (S R k)
S_eval_t_sq_add_S_eval_t_sq :
  ∀ (k : ℤ),
    Polynomial.eval t (S R k) ^ 2 + Polynomial.eval t (S R (k + 1)) ^ 2 -
        t * Polynomial.eval t (S R k) * Polynomial.eval t (S R (k + 1)) =
      1
k : ℤ := ↑m / 2
ih :
  ((reflection hf * reflection hg) ^ m) z =
    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 k) + Polynomial.eval t (S R ((↑m - 2) / 2)))) •
        ((f y * g z - f z) • x - g z • y)
e : ℤ := ↑m % 2
⊢ e = 0 ∨ e = 1

Tactic:
by

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

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

Tactic:
by

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

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

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
⊢ T R 0 = 1

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
⊢ T R 1 = X

Tactic:
by

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
x y : ↥H
hx : x ∈ Weight.ker
z : ↥H
u : L
hu : u ∈ rootSpace H ⇑α
v : L
huv : ⁅u, v⁆ = ↑z
⊢ α x = 0

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
this : 2 * X - X = X
⊢ 2 * X * T R (-↑0) - T R (-↑0 + 1) = X

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
⊢ T R (-1) = X

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
⊢ T R 2 = 2 * X ^ 2 - 1

Tactic:
by

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
x y : ↥H
hx : x ∈ Weight.ker
hy : y ∈ span K {x | ∃ y ∈ rootSpace H ⇑α, ∃ z ∈ rootSpace H (-⇑α), ⁅y, z⁆ = ↑x}
⊢ ((traceForm K (↥H) L) x) y = 0

Tactic:
induction hy using Submodule.span_induction with
  | mem z hz =>
    obtain ⟨u, hu, v, -, huv⟩ := hz
    change killingForm K L (x : L) (z : L) = 0
    replace hx : α x = 0 := by simpa using hx
    rw [← huv, ← traceForm_apply_lie_apply, ← LieSubalgebra.coe_bracket_of_module,
      lie_eq_smul_of_mem_rootSpace hu, hx, zero_smul, map_zero, LinearMap.zero_apply]
  | zero => simp
  | add _ _ _ _ hx hy => simp [hx, hy]
  | smul _ _ _ hz => simp [hz]

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
x y : ↥H
hx : x ∈ Weight.ker
hy : y ∈ K ∙ coroot α
⊢ ((traceForm K (↥H) L) x) y = 0

Tactic:
by

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
⊢ coroot α = 0

Tactic:
by

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
hα' : coroot α = 0
⊢ Weight.ker = ⊤

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℤ
⊢ T R ↑n.natAbs = T R n

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
R : Type u_2
L : Type u_3
inst✝³ : Finite ι
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
H : LieSubalgebra R L
b : Basis ι H
⊢ iSup ![H.toLieSubmodule, b.borelLower, b.borelUpper] = ⊤

Tactic:
by

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

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

Tactic:
by

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
⊢ (traceForm K (↥H) L).orthogonal (K ∙ coroot α) = Weight.ker

Tactic:
by

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 β : Weight K (↥H) L
h : α = β
⊢ coroot α = coroot β

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
R : Type u_2
L : Type u_3
inst✝⁴ : Finite ι
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
H : LieSubalgebra R L
b : Basis ι H
inst✝ : Fintype ι
i j : ι
e : ↥H.toSubmodule ≃ₗ[R] ↥(Submodule.span R (range b.h)) :=
  LinearEquiv.ofEq H.toSubmodule (Submodule.span R (range b.h)) ⋯
f : ι → R := fun k ↦ b.A i k • ((Basis.span ⋯).repr (e (b.h' j))) k
⊢ f = fun k ↦ if j = k then ↑(b.A i k) else 0

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℕ
ih1 : eval 0 (T R (↑n + 1)) = ↑(if Even (↑n + 1) then ↑((↑n + 1) / 2).negOnePow else 0)
ih2 : eval 0 (T R ↑n) = ↑(if Even ↑n then ↑(↑n / 2).negOnePow else 0)
⊢ (↑n + 2) / 2 = ↑n / 2 + 1

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
R : Type u_2
L : Type u_3
inst✝⁴ : Finite ι
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
H : LieSubalgebra R L
b : Basis ι H
inst✝ : Fintype ι
i j : ι
⊢ (b.baseSupp i) (b.h' j) = ↑(b.A i j)

Tactic:
by

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 β : Weight K (↥H) L
hyp : coroot α = coroot β
hα : α.IsZero
⊢ β.IsZero

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℕ
ih1 : eval 0 (T R (-↑n)) = ↑(if Even (-↑n) then ↑(-↑n / 2).negOnePow else 0)
ih2 : eval 0 (T R (-↑n + 1)) = ↑(if Even (-↑n + 1) then ↑((-↑n + 1) / 2).negOnePow else 0)
⊢ (-↑n + 1) / 2 = (-↑n - 1) / 2 + 1

Tactic:
by

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 β : Weight K (↥H) L
hyp : coroot α = coroot β
hα : ¬α.IsZero
⊢ β.IsNonZero

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
R : Type u_2
L : Type u_3
inst✝⁴ : Finite ι
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
H : LieSubalgebra R L
b : Basis ι H
inst✝ : Fintype ι
b₁ : Module.Basis ι R ↥H := (Basis.span ⋯).map (LinearEquiv.ofEq H.toSubmodule (Submodule.span R (range b.h)) ⋯).symm
b₂ : Module.Basis ι R ↥H :=
  (Basis.span ⋯).map (LinearEquiv.ofEq H.toSubmodule (Submodule.span R (range b.symm.h)) ⋯).symm
this : b₁.coord = -b₂.coord
⊢ b.symm.baseSupp = -b.baseSupp

Tactic:
by

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 β : Weight K (↥H) L
hyp : coroot α = coroot β
hα : ¬α.IsZero
hβ : β.IsNonZero
⊢ Weight.ker = Weight.ker

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
R : Type u_2
L : Type u_3
inst✝⁴ : Finite ι
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
H : LieSubalgebra R L
b : Basis ι H
inst✝ : Fintype ι
⊢ b.symm.baseSupp = -b.baseSupp

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℤ
⊢ eval 0 (T R n) = ↑(if Even n then ↑(n / 2).negOnePow else 0)

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℤ
hn : Even n
⊢ eval 0 (T R n) = ↑↑(n / 2).negOnePow

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℤ
⊢ eval 0 (T R (2 * n)) = ↑↑n.negOnePow

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℤ
hn : Odd n
⊢ eval 0 (T R n) = 0

Tactic:
by

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 β : Weight K (↥H) L
hyp : coroot α = coroot β
hα : ¬α.IsZero
hβ : β.IsNonZero
this✝ : Weight.ker = Weight.ker
this : Weight.toLinear K (↥H) L α = Weight.toLinear K (↥H) L β
⊢ α = β

Tactic:
by

Goal state after:
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
⊢ ((Int.castRingHom R).mapMatrix b.A).Nondegenerate

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NeZero 2
n : ℕ
ih1 : (T R (↑n + 1)).degree = ↑(↑n + 1).natAbs
ih2 : (T R ↑n).degree = ↑(↑n).natAbs
⊢ (2 * X * T R (↑n + 1)).degree = ↑(n + 2)

Tactic:
by

Goal state after:
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
⊢ LinearIndependent R b.baseSupp

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
R : Type u_2
L : Type u_3
inst✝⁴ : Finite ι
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
H : LieSubalgebra R L
b : Basis ι H
inst✝ : Fintype ι
i : ι
x : L
hx : x ∈ H
⊢ x ∈ Submodule.span R (range b.h)

Tactic:
by

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 β : Weight K (↥H) L
⊢ coroot α = coroot β ↔ α = β

Tactic:
by

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
e : L
heα : e ∈ rootSpace H ⇑α
he₀ : e ≠ 0
⊢ ∃ f ∈ rootSpace H (-⇑α), ((killingForm K L) e) f ≠ 0

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NeZero 2
n : ℤ
⊢ (T R n).degree = ↑n.natAbs

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NeZero 2
n : ℕ
ih1 : (T R (↑n + 1)).leadingCoeff = 2 ^ ((↑n + 1).natAbs - 1)
ih2 : (T R ↑n).leadingCoeff = 2 ^ ((↑n).natAbs - 1)
⊢ leadingCoeff 2 = 2

Tactic:
by

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
e : L
heα : e ∈ rootSpace H ⇑α
he₀ : e ≠ 0
f' : L
hfα : f' ∈ rootSpace H (-⇑α)
hf : ((killingForm K L) e) f' ≠ 0
hef : ⁅e, f'⁆ = ((killingForm K L) e) f' • ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))
h : ↥H := ⟨⁅e, f'⁆, ⋯⟩
⊢ h = ((killingForm K L) e) f' • (cartanEquivDual H).symm (Weight.toLinear K (↥H) L α)

Tactic:
by

Goal state after: