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
⊢ 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 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
Tactic: have exists_root_index (γ : Weight K H L) (hγ : γ.IsNonZero) : ∃ i, S.root i = ↑γ :=
    ⟨⟨γ, by simpa [LieSubalgebra.root]⟩, rfl⟩
-- 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
⊢ Eq (α := LieSubmodule K (↥H) L)
    (genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α) : (a : ↥H) → K)) ⊥
have by_contra : ∀
  (h_plus_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)) ⊥),
  False
Tactic: by_contra h_plus_ne_bot
-- 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_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 :=
      HAdd.hAdd (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_plus_ne_bot }
⊢ False
have hγ_nonzero : γ.IsNonZero
Tactic: have hγ_nonzero : γ.IsNonZero := Weight.coe_toLinear_ne_zero_iff.mp w_plus
-- 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_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 :=
      HAdd.hAdd (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_plus_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_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 :=
      HAdd.hAdd (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_plus_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 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_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 :=
      HAdd.hAdd (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_plus_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
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 := Set (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))
    (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))
have rw : Membership.mem (γ := Set (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))
  (Weight.toLinear K (↥H) L χ + Weight.toLinear K (↥H) L α)
Tactic: rw [hi, hj]
-- 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_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 :=
      HAdd.hAdd (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_plus_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
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
have h_sum_in_range : Membership.mem (γ := Set (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))
  (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))
Tactic: have h_sum_in_range : S.root i + S.root j ∈ Set.range S.root := by
      rw [hi, hj]
      exact ⟨⟨γ, by simpa [LieSubalgebra.root]⟩, rfl⟩
-- 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
⊢ 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 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)) ⊥
Tactic: have h_plus_bot : genWeightSpace L (χ.toLinear + α.toLinear) = ⊥ := by
    by_contra h_plus_ne_bot
    let γ : Weight K H L := ⟨χ.toLinear + α.toLinear, h_plus_ne_bot⟩
    have hγ_nonzero : γ.IsNonZero := Weight.coe_toLinear_ne_zero_iff.mp w_plus
    obtain ⟨i, hi⟩ := exists_root_index χ (Weight.coe_toLinear_ne_zero_iff.mp w_chi)
    obtain ⟨j, hj⟩ := exists_root_index α hα₀
    have h_sum_in_range : S.root i + S.root j ∈ Set.range S.root := by
      rw [hi, hj]
      exact ⟨⟨γ, by simpa [LieSubalgebra.root]⟩, rfl⟩
    have h_equiv := RootPairing.root_mem_submodule_iff_of_add_mem_invtSubmodule
      ⟨q, by rw [RootPairing.mem_invtRootSubmodule_iff]; exact hq⟩ h_sum_in_range
    rw [hi] at h_equiv
    exact h_chi_not_in_q (h_equiv.mpr (by rw [hj]; exact hαq))
-- 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)) ⊥
⊢ Eq (α := LieSubmodule K (↥H) L)
    (genWeightSpace L (⇑(Weight.toLinear K (↥H) L χ - Weight.toLinear K (↥H) L α) : (a : ↥H) → K)) ⊥
have by_contra : ∀
  (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)) ⊥),
  False
Tactic: by_contra h_minus_ne_bot