awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.IsSimple (hypothesis)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : CharZero K
inst_2 : LieRing L
inst_3 : LieAlgebra K L
inst_4 : FiniteDimensional K L
inst_5 : IsKilling K L
H : LieSubalgebra K L
inst_6 : H.IsCartanSubalgebra
inst_7 : IsTriangularizable K (↥H) L
q : Submodule K (Dual.{u_1, u_2} K ↥H)
hq :
  ∀ (i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))),
    q ∈ End.invtSubmodule (↑(RootPairing.reflection (rootSystem H) i) : Dual K ↥H →ₗ[K] Dual K ↥H)
χ : Weight K (↥H) L
x_χ m_α : L
hx_χ : Membership.mem (γ := LieSubmodule K (↥H) L) (genWeightSpace L (⇑χ : (a : ↥H) → K)) x_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 : Membership.mem (γ := LieIdeal K ↥H) (corootSpace (⇑α : (a : ↥H) → K)) y
h_bracket_sum : ⁅x_χ, m_α⁆ = ⁅x_χ, m_pos⁆ + ⁅x_χ, m_neg⁆ + ⁅x_χ, (↑y : L)⁆
h_pos_containment :
  Membership.mem (γ := LieSubmodule K (↥H) L)
    (genWeightSpace L (HAdd.hAdd (α := (a : ↥H) → K) (⇑χ : (a : ↥H) → K) (⇑α : (a : ↥H) → K))) ⁅x_χ, m_pos⁆
h_neg_containment :
  Membership.mem (γ := LieSubmodule K (↥H) L)
    (genWeightSpace L (HSub.hSub (α := (a : ↥H) → K) (⇑χ : (a : ↥H) → K) (⇑α : (a : ↥H) → K))) ⁅x_χ, m_neg⁆
h_chi_not_in_q : Weight.toLinear K (↥H) L χ ∉ q
S : RootPairing (↥(LieSubalgebra.root : Finset (Weight K (↥H) L))) K (Dual.{u_1, u_2} K ↥H) ↥H := rootSystem H
exists_root_index :
  ∀ (γ : Weight K (↥H) L) (hγ : γ.IsNonZero),
    ∃ (i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))),
      (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) i = Weight.toLinear K (↥H) L γ
h_plus_bot :
  Eq (α := LieSubmodule K (↥H) L)
    (genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) : (a : ↥H) → K)) ⊥
h_minus_ne_bot :
  ¬Eq (α := LieSubmodule K (↥H) L)
      (genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α) : (a : ↥H) → K)) ⊥
γ : Weight K (↥H) L :=
  {
    toFun :=
      HSub.hSub (α := (a : ↥H) → K) (⇑(Weight.toLinear K (↥H) L χ) : (a : ↥H) → K)
        (⇑(Weight.toLinear K (↥H) L α) : (a : ↥H) → K),
    genWeightSpace_ne_bot' := h_minus_ne_bot }
⊢ False
have hγ_nonzero : γ.IsNonZero
Tactic: have hγ_nonzero : γ.IsNonZero := Weight.coe_toLinear_ne_zero_iff.mp w_minus
-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.IsSimple (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : CharZero K
inst_2 : LieRing L
inst_3 : LieAlgebra K L
inst_4 : FiniteDimensional K L
inst_5 : IsKilling K L
H : LieSubalgebra K L
inst_6 : H.IsCartanSubalgebra
inst_7 : IsTriangularizable K (↥H) L
q : Submodule K (Dual.{u_1, u_2} K ↥H)
hq :
  ∀ (i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))),
    q ∈ End.invtSubmodule (↑(RootPairing.reflection (rootSystem H) i) : Dual K ↥H →ₗ[K] Dual K ↥H)
χ : Weight K (↥H) L
x_χ m_α : L
hx_χ : Membership.mem (γ := LieSubmodule K (↥H) L) (genWeightSpace L (⇑χ : (a : ↥H) → K)) x_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 : Membership.mem (γ := LieIdeal K ↥H) (corootSpace (⇑α : (a : ↥H) → K)) y
h_bracket_sum : ⁅x_χ, m_α⁆ = ⁅x_χ, m_pos⁆ + ⁅x_χ, m_neg⁆ + ⁅x_χ, (↑y : L)⁆
h_pos_containment :
  Membership.mem (γ := LieSubmodule K (↥H) L)
    (genWeightSpace L (HAdd.hAdd (α := (a : ↥H) → K) (⇑χ : (a : ↥H) → K) (⇑α : (a : ↥H) → K))) ⁅x_χ, m_pos⁆
h_neg_containment :
  Membership.mem (γ := LieSubmodule K (↥H) L)
    (genWeightSpace L (HSub.hSub (α := (a : ↥H) → K) (⇑χ : (a : ↥H) → K) (⇑α : (a : ↥H) → K))) ⁅x_χ, m_neg⁆
h_chi_not_in_q : Weight.toLinear K (↥H) L χ ∉ q
S : RootPairing (↥(LieSubalgebra.root : Finset (Weight K (↥H) L))) K (Dual.{u_1, u_2} K ↥H) ↥H := rootSystem H
exists_root_index :
  ∀ (γ : Weight K (↥H) L) (hγ : γ.IsNonZero),
    ∃ (i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))),
      (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) i = Weight.toLinear K (↥H) L γ
h_plus_bot :
  Eq (α := LieSubmodule K (↥H) L)
    (genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) : (a : ↥H) → K)) ⊥
h_minus_ne_bot :
  ¬Eq (α := LieSubmodule K (↥H) L)
      (genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α) : (a : ↥H) → K)) ⊥
γ : Weight K (↥H) L :=
  {
    toFun :=
      HSub.hSub (α := (a : ↥H) → K) (⇑(Weight.toLinear K (↥H) L χ) : (a : ↥H) → K)
        (⇑(Weight.toLinear K (↥H) L α) : (a : ↥H) → K),
    genWeightSpace_ne_bot' := h_minus_ne_bot }
hγ_nonzero : γ.IsNonZero
⊢ False
have obtain : ∀ (i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L)))
  (hi : (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) i = Weight.toLinear K (↥H) L χ), False
Tactic: obtain ⟨i, hi⟩ := exists_root_index χ (Weight.coe_toLinear_ne_zero_iff.mp w_chi)
-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.IsSimple (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : CharZero K
inst_2 : LieRing L
inst_3 : LieAlgebra K L
inst_4 : FiniteDimensional K L
inst_5 : IsKilling K L
H : LieSubalgebra K L
inst_6 : H.IsCartanSubalgebra
inst_7 : IsTriangularizable K (↥H) L
q : Submodule K (Dual.{u_1, u_2} K ↥H)
hq :
  ∀ (i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))),
    q ∈ End.invtSubmodule (↑(RootPairing.reflection (rootSystem H) i) : Dual K ↥H →ₗ[K] Dual K ↥H)
χ : Weight K (↥H) L
x_χ m_α : L
hx_χ : Membership.mem (γ := LieSubmodule K (↥H) L) (genWeightSpace L (⇑χ : (a : ↥H) → K)) x_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 : Membership.mem (γ := LieIdeal K ↥H) (corootSpace (⇑α : (a : ↥H) → K)) y
h_bracket_sum : ⁅x_χ, m_α⁆ = ⁅x_χ, m_pos⁆ + ⁅x_χ, m_neg⁆ + ⁅x_χ, (↑y : L)⁆
h_pos_containment :
  Membership.mem (γ := LieSubmodule K (↥H) L)
    (genWeightSpace L (HAdd.hAdd (α := (a : ↥H) → K) (⇑χ : (a : ↥H) → K) (⇑α : (a : ↥H) → K))) ⁅x_χ, m_pos⁆
h_neg_containment :
  Membership.mem (γ := LieSubmodule K (↥H) L)
    (genWeightSpace L (HSub.hSub (α := (a : ↥H) → K) (⇑χ : (a : ↥H) → K) (⇑α : (a : ↥H) → K))) ⁅x_χ, m_neg⁆
h_chi_not_in_q : Weight.toLinear K (↥H) L χ ∉ q
S : RootPairing (↥(LieSubalgebra.root : Finset (Weight K (↥H) L))) K (Dual.{u_1, u_2} K ↥H) ↥H := rootSystem H
exists_root_index :
  ∀ (γ : Weight K (↥H) L) (hγ : γ.IsNonZero),
    ∃ (i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))),
      (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) i = Weight.toLinear K (↥H) L γ
h_plus_bot :
  Eq (α := LieSubmodule K (↥H) L)
    (genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) : (a : ↥H) → K)) ⊥
h_minus_ne_bot :
  ¬Eq (α := LieSubmodule K (↥H) L)
      (genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α) : (a : ↥H) → K)) ⊥
γ : Weight K (↥H) L :=
  {
    toFun :=
      HSub.hSub (α := (a : ↥H) → K) (⇑(Weight.toLinear K (↥H) L χ) : (a : ↥H) → K)
        (⇑(Weight.toLinear K (↥H) L α) : (a : ↥H) → K),
    genWeightSpace_ne_bot' := h_minus_ne_bot }
hγ_nonzero : γ.IsNonZero
i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))
hi : (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) i = Weight.toLinear K (↥H) L χ
⊢ False
have obtain : ∀ (j : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L)))
  (hj : (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) j = Weight.toLinear K (↥H) L (-α)), False
Tactic: obtain ⟨j, hj⟩ := exists_root_index (-α) (Weight.IsNonZero.neg hα₀)
-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.IsSimple (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : CharZero K
inst_2 : LieRing L
inst_3 : LieAlgebra K L
inst_4 : FiniteDimensional K L
inst_5 : IsKilling K L
H : LieSubalgebra K L
inst_6 : H.IsCartanSubalgebra
inst_7 : IsTriangularizable K (↥H) L
q : Submodule K (Dual.{u_1, u_2} K ↥H)
hq :
  ∀ (i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))),
    q ∈ End.invtSubmodule (↑(RootPairing.reflection (rootSystem H) i) : Dual K ↥H →ₗ[K] Dual K ↥H)
χ : Weight K (↥H) L
x_χ m_α : L
hx_χ : Membership.mem (γ := LieSubmodule K (↥H) L) (genWeightSpace L (⇑χ : (a : ↥H) → K)) x_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 : Membership.mem (γ := LieIdeal K ↥H) (corootSpace (⇑α : (a : ↥H) → K)) y
h_bracket_sum : ⁅x_χ, m_α⁆ = ⁅x_χ, m_pos⁆ + ⁅x_χ, m_neg⁆ + ⁅x_χ, (↑y : L)⁆
h_pos_containment :
  Membership.mem (γ := LieSubmodule K (↥H) L)
    (genWeightSpace L (HAdd.hAdd (α := (a : ↥H) → K) (⇑χ : (a : ↥H) → K) (⇑α : (a : ↥H) → K))) ⁅x_χ, m_pos⁆
h_neg_containment :
  Membership.mem (γ := LieSubmodule K (↥H) L)
    (genWeightSpace L (HSub.hSub (α := (a : ↥H) → K) (⇑χ : (a : ↥H) → K) (⇑α : (a : ↥H) → K))) ⁅x_χ, m_neg⁆
h_chi_not_in_q : Weight.toLinear K (↥H) L χ ∉ q
S : RootPairing (↥(LieSubalgebra.root : Finset (Weight K (↥H) L))) K (Dual.{u_1, u_2} K ↥H) ↥H := rootSystem H
exists_root_index :
  ∀ (γ : Weight K (↥H) L) (hγ : γ.IsNonZero),
    ∃ (i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))),
      (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) i = Weight.toLinear K (↥H) L γ
h_plus_bot :
  Eq (α := LieSubmodule K (↥H) L)
    (genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) : (a : ↥H) → K)) ⊥
h_minus_bot :
  Eq (α := LieSubmodule K (↥H) L)
    (genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α) : (a : ↥H) → K)) ⊥
⊢ Membership.mem (γ := LieSubmodule K (↥H) L)
    (⨆ (α : { α : Weight K (↥H) L // Weight.toLinear K (↥H) L α ∈ q ∧ α.IsNonZero }), sl2SubmoduleOfRoot ⋯) ⁅x_χ, m_α⁆
have obtain : ∀ (i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L)))
  (hi : (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) i = Weight.toLinear K (↥H) L χ),
  Membership.mem (γ := LieSubmodule K (↥H) L)
    (⨆ (α : { α : Weight K (↥H) L // Weight.toLinear K (↥H) L α ∈ q ∧ α.IsNonZero }), sl2SubmoduleOfRoot sorry) ⁅x_χ, m_α⁆
Tactic: obtain ⟨i, hi⟩ := exists_root_index χ (Weight.coe_toLinear_ne_zero_iff.mp w_chi)
-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.IsSimple (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : CharZero K
inst_2 : LieRing L
inst_3 : LieAlgebra K L
inst_4 : FiniteDimensional K L
inst_5 : IsKilling K L
H : LieSubalgebra K L
inst_6 : H.IsCartanSubalgebra
inst_7 : IsTriangularizable K (↥H) L
q : Submodule K (Dual.{u_1, u_2} K ↥H)
hq :
  ∀ (i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))),
    q ∈ End.invtSubmodule (↑(RootPairing.reflection (rootSystem H) i) : Dual K ↥H →ₗ[K] Dual K ↥H)
χ : Weight K (↥H) L
x_χ m_α : L
hx_χ : Membership.mem (γ := LieSubmodule K (↥H) L) (genWeightSpace L (⇑χ : (a : ↥H) → K)) x_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 : Membership.mem (γ := LieIdeal K ↥H) (corootSpace (⇑α : (a : ↥H) → K)) y
h_bracket_sum : ⁅x_χ, m_α⁆ = ⁅x_χ, m_pos⁆ + ⁅x_χ, m_neg⁆ + ⁅x_χ, (↑y : L)⁆
h_pos_containment :
  Membership.mem (γ := LieSubmodule K (↥H) L)
    (genWeightSpace L (HAdd.hAdd (α := (a : ↥H) → K) (⇑χ : (a : ↥H) → K) (⇑α : (a : ↥H) → K))) ⁅x_χ, m_pos⁆
h_neg_containment :
  Membership.mem (γ := LieSubmodule K (↥H) L)
    (genWeightSpace L (HSub.hSub (α := (a : ↥H) → K) (⇑χ : (a : ↥H) → K) (⇑α : (a : ↥H) → K))) ⁅x_χ, m_neg⁆
h_chi_not_in_q : Weight.toLinear K (↥H) L χ ∉ q
S : RootPairing (↥(LieSubalgebra.root : Finset (Weight K (↥H) L))) K (Dual.{u_1, u_2} K ↥H) ↥H := rootSystem H
exists_root_index :
  ∀ (γ : Weight K (↥H) L) (hγ : γ.IsNonZero),
    ∃ (i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))),
      (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) i = Weight.toLinear K (↥H) L γ
h_plus_bot :
  Eq (α := LieSubmodule K (↥H) L)
    (genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) : (a : ↥H) → K)) ⊥
h_minus_bot :
  Eq (α := LieSubmodule K (↥H) L)
    (genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α) : (a : ↥H) → K)) ⊥
i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))
hi : (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) i = Weight.toLinear K (↥H) L χ
⊢ Membership.mem (γ := LieSubmodule K (↥H) L)
    (⨆ (α : { α : Weight K (↥H) L // Weight.toLinear K (↥H) L α ∈ q ∧ α.IsNonZero }), sl2SubmoduleOfRoot ⋯) ⁅x_χ, m_α⁆
have obtain : ∀ (j : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L)))
  (hj : (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) j = Weight.toLinear K (↥H) L α),
  Membership.mem (γ := LieSubmodule K (↥H) L)
    (⨆ (α : { α : Weight K (↥H) L // Weight.toLinear K (↥H) L α ∈ q ∧ α.IsNonZero }), sl2SubmoduleOfRoot sorry) ⁅x_χ, m_α⁆
Tactic: obtain ⟨j, hj⟩ := exists_root_index α hα₀
-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.IsSimple (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : CharZero K
inst_2 : LieRing L
inst_3 : LieAlgebra K L
inst_4 : FiniteDimensional K L
inst_5 : IsKilling K L
H : LieSubalgebra K L
inst_6 : H.IsCartanSubalgebra
inst_7 : IsTriangularizable K (↥H) L
q : Submodule K (Dual.{u_1, u_2} K ↥H)
hq :
  ∀ (i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))),
    q ∈ End.invtSubmodule (↑(RootPairing.reflection (rootSystem H) i) : Dual K ↥H →ₗ[K] Dual K ↥H)
χ : Weight K (↥H) L
x_χ m_α : L
hx_χ : Membership.mem (γ := LieSubmodule K (↥H) L) (genWeightSpace L (⇑χ : (a : ↥H) → K)) x_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 : Membership.mem (γ := LieIdeal K ↥H) (corootSpace (⇑α : (a : ↥H) → K)) y
h_bracket_sum : ⁅x_χ, m_α⁆ = ⁅x_χ, m_pos⁆ + ⁅x_χ, m_neg⁆ + ⁅x_χ, (↑y : L)⁆
h_pos_containment :
  Membership.mem (γ := LieSubmodule K (↥H) L)
    (genWeightSpace L (HAdd.hAdd (α := (a : ↥H) → K) (⇑χ : (a : ↥H) → K) (⇑α : (a : ↥H) → K))) ⁅x_χ, m_pos⁆
h_neg_containment :
  Membership.mem (γ := LieSubmodule K (↥H) L)
    (genWeightSpace L (HSub.hSub (α := (a : ↥H) → K) (⇑χ : (a : ↥H) → K) (⇑α : (a : ↥H) → K))) ⁅x_χ, m_neg⁆
h_chi_not_in_q : Weight.toLinear K (↥H) L χ ∉ q
S : RootPairing (↥(LieSubalgebra.root : Finset (Weight K (↥H) L))) K (Dual.{u_1, u_2} K ↥H) ↥H := rootSystem H
exists_root_index :
  ∀ (γ : Weight K (↥H) L) (hγ : γ.IsNonZero),
    ∃ (i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))),
      (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) i = Weight.toLinear K (↥H) L γ
h_plus_bot :
  Eq (α := LieSubmodule K (↥H) L)
    (genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) : (a : ↥H) → K)) ⊥
h_minus_bot :
  Eq (α := LieSubmodule K (↥H) L)
    (genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α) : (a : ↥H) → K)) ⊥
i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))
hi : (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) i = Weight.toLinear K (↥H) L χ
j : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))
hj : (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) j = Weight.toLinear K (↥H) L α
⊢ S.pairing i j = 0
have hp : i ≠ j
Tactic: apply RootPairing.pairing_eq_zero_of_add_notMem_of_sub_notMem S
-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.IsSimple (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : CharZero K
inst_2 : LieRing L
inst_3 : LieAlgebra K L
inst_4 : FiniteDimensional K L
inst_5 : IsKilling K L
H : LieSubalgebra K L
inst_6 : H.IsCartanSubalgebra
inst_7 : IsTriangularizable K (↥H) L
q : Submodule K (Dual.{u_1, u_2} K ↥H)
hq :
  ∀ (i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))),
    q ∈ End.invtSubmodule (↑(RootPairing.reflection (rootSystem H) i) : Dual K ↥H →ₗ[K] Dual K ↥H)
χ : Weight K (↥H) L
x_χ m_α : L
hx_χ : Membership.mem (γ := LieSubmodule K (↥H) L) (genWeightSpace L (⇑χ : (a : ↥H) → K)) x_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 : Membership.mem (γ := LieIdeal K ↥H) (corootSpace (⇑α : (a : ↥H) → K)) y
h_bracket_sum : ⁅x_χ, m_α⁆ = ⁅x_χ, m_pos⁆ + ⁅x_χ, m_neg⁆ + ⁅x_χ, (↑y : L)⁆
h_pos_containment :
  Membership.mem (γ := LieSubmodule K (↥H) L)
    (genWeightSpace L (HAdd.hAdd (α := (a : ↥H) → K) (⇑χ : (a : ↥H) → K) (⇑α : (a : ↥H) → K))) ⁅x_χ, m_pos⁆
h_neg_containment :
  Membership.mem (γ := LieSubmodule K (↥H) L)
    (genWeightSpace L (HSub.hSub (α := (a : ↥H) → K) (⇑χ : (a : ↥H) → K) (⇑α : (a : ↥H) → K))) ⁅x_χ, m_neg⁆
h_chi_not_in_q : Weight.toLinear K (↥H) L χ ∉ q
S : RootPairing (↥(LieSubalgebra.root : Finset (Weight K (↥H) L))) K (Dual.{u_1, u_2} K ↥H) ↥H := rootSystem H
exists_root_index :
  ∀ (γ : Weight K (↥H) L) (hγ : γ.IsNonZero),
    ∃ (i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))),
      (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) i = Weight.toLinear K (↥H) L γ
h_plus_bot :
  Eq (α := LieSubmodule K (↥H) L)
    (genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) : (a : ↥H) → K)) ⊥
h_minus_bot :
  Eq (α := LieSubmodule K (↥H) L)
    (genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α) : (a : ↥H) → K)) ⊥
i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))
hi : (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) i = Weight.toLinear K (↥H) L χ
j : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))
hj : (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) j = Weight.toLinear K (↥H) L α
hp : i ≠ j
⊢ S.pairing i j = 0
have hn : Ne (α := Dual.{u_1, u_2} K ↥H) ((S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) i : Dual.{u_1, u_2} K ↥H)
  (-(S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) j : Dual.{u_1, u_2} K ↥H)
Tactic: apply RootPairing.pairing_eq_zero_of_add_notMem_of_sub_notMem S
-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.IsSimple (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : CharZero K
inst_2 : LieRing L
inst_3 : LieAlgebra K L
inst_4 : FiniteDimensional K L
inst_5 : IsKilling K L
H : LieSubalgebra K L
inst_6 : H.IsCartanSubalgebra
inst_7 : IsTriangularizable K (↥H) L
q : Submodule K (Dual.{u_1, u_2} K ↥H)
hq :
  ∀ (i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))),
    q ∈ End.invtSubmodule (↑(RootPairing.reflection (rootSystem H) i) : Dual K ↥H →ₗ[K] Dual K ↥H)
χ : Weight K (↥H) L
x_χ m_α : L
hx_χ : Membership.mem (γ := LieSubmodule K (↥H) L) (genWeightSpace L (⇑χ : (a : ↥H) → K)) x_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 : Membership.mem (γ := LieIdeal K ↥H) (corootSpace (⇑α : (a : ↥H) → K)) y
h_bracket_sum : ⁅x_χ, m_α⁆ = ⁅x_χ, m_pos⁆ + ⁅x_χ, m_neg⁆ + ⁅x_χ, (↑y : L)⁆
h_pos_containment :
  Membership.mem (γ := LieSubmodule K (↥H) L)
    (genWeightSpace L (HAdd.hAdd (α := (a : ↥H) → K) (⇑χ : (a : ↥H) → K) (⇑α : (a : ↥H) → K))) ⁅x_χ, m_pos⁆
h_neg_containment :
  Membership.mem (γ := LieSubmodule K (↥H) L)
    (genWeightSpace L (HSub.hSub (α := (a : ↥H) → K) (⇑χ : (a : ↥H) → K) (⇑α : (a : ↥H) → K))) ⁅x_χ, m_neg⁆
h_chi_not_in_q : Weight.toLinear K (↥H) L χ ∉ q
S : RootPairing (↥(LieSubalgebra.root : Finset (Weight K (↥H) L))) K (Dual.{u_1, u_2} K ↥H) ↥H := rootSystem H
exists_root_index :
  ∀ (γ : Weight K (↥H) L) (hγ : γ.IsNonZero),
    ∃ (i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))),
      (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) i = Weight.toLinear K (↥H) L γ
h_plus_bot :
  Eq (α := LieSubmodule K (↥H) L)
    (genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) : (a : ↥H) → K)) ⊥
h_minus_bot :
  Eq (α := LieSubmodule K (↥H) L)
    (genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α) : (a : ↥H) → K)) ⊥
i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))
hi : (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) i = Weight.toLinear K (↥H) L χ
j : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))
hj : (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) j = Weight.toLinear K (↥H) L α
hp : i ≠ j
hn :
  Ne (α := Dual.{u_1, u_2} K ↥H) ((S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) i : Dual.{u_1, u_2} K ↥H)
    (-(S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) j : Dual.{u_1, u_2} K ↥H)
⊢ S.pairing i j = 0
have h_add : HAdd.hAdd (α := Dual.{u_1, u_2} K ↥H) (β := Dual.{u_1, u_2} K ↥H)
    ((S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) i : Dual.{u_1, u_2} K ↥H)
    ((S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) j : Dual.{u_1, u_2} K ↥H) ∉
  Set.range (ι := ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))) (⇑S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H)
Tactic: apply RootPairing.pairing_eq_zero_of_add_notMem_of_sub_notMem S
-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.IsSimple (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : CharZero K
inst_2 : LieRing L
inst_3 : LieAlgebra K L
inst_4 : FiniteDimensional K L
inst_5 : IsKilling K L
H : LieSubalgebra K L
inst_6 : H.IsCartanSubalgebra
inst_7 : IsTriangularizable K (↥H) L
q : Submodule K (Dual.{u_1, u_2} K ↥H)
hq :
  ∀ (i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))),
    q ∈ End.invtSubmodule (↑(RootPairing.reflection (rootSystem H) i) : Dual K ↥H →ₗ[K] Dual K ↥H)
χ : Weight K (↥H) L
x_χ m_α : L
hx_χ : Membership.mem (γ := LieSubmodule K (↥H) L) (genWeightSpace L (⇑χ : (a : ↥H) → K)) x_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 : Membership.mem (γ := LieIdeal K ↥H) (corootSpace (⇑α : (a : ↥H) → K)) y
h_bracket_sum : ⁅x_χ, m_α⁆ = ⁅x_χ, m_pos⁆ + ⁅x_χ, m_neg⁆ + ⁅x_χ, (↑y : L)⁆
h_pos_containment :
  Membership.mem (γ := LieSubmodule K (↥H) L)
    (genWeightSpace L (HAdd.hAdd (α := (a : ↥H) → K) (⇑χ : (a : ↥H) → K) (⇑α : (a : ↥H) → K))) ⁅x_χ, m_pos⁆
h_neg_containment :
  Membership.mem (γ := LieSubmodule K (↥H) L)
    (genWeightSpace L (HSub.hSub (α := (a : ↥H) → K) (⇑χ : (a : ↥H) → K) (⇑α : (a : ↥H) → K))) ⁅x_χ, m_neg⁆
h_chi_not_in_q : Weight.toLinear K (↥H) L χ ∉ q
S : RootPairing (↥(LieSubalgebra.root : Finset (Weight K (↥H) L))) K (Dual.{u_1, u_2} K ↥H) ↥H := rootSystem H
exists_root_index :
  ∀ (γ : Weight K (↥H) L) (hγ : γ.IsNonZero),
    ∃ (i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))),
      (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) i = Weight.toLinear K (↥H) L γ
h_plus_bot :
  Eq (α := LieSubmodule K (↥H) L)
    (genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) : (a : ↥H) → K)) ⊥
h_minus_bot :
  Eq (α := LieSubmodule K (↥H) L)
    (genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α) : (a : ↥H) → K)) ⊥
i : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))
hi : (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) i = Weight.toLinear K (↥H) L χ
j : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L))
hj : (S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) j = Weight.toLinear K (↥H) L α
h_eq : i = j
⊢ Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α = 0
have rw : HSub.hSub (α := LinearMap.{u_1, u_1, u_2, u_1} (RingHom.id K) (↥H) K)
    ((S.root : (a : ↥LieSubalgebra.root) → Dual K ↥H) i : Dual.{u_1, u_2} K ↥H) (Weight.toLinear K (↥H) L α) =
  0
Tactic: rw [← hi, ← hj, h_eq, sub_self]