awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.LinearAlgebra.RootSystem.WeylGroup (goal)
Goal:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : AddCommGroup N
inst_4 : Module R N
P : RootPairing ι R M N
w : P.Aut
hw : w ∈ P.weylGroup
⊢ Membership.mem (α := M ≃ₗ[R] M) (γ := Subgroup (M ≃ₗ[R] M)) (Subgroup.closure (range P.reflection))
    ((((Equiv.weightHom P).restrict P.weylGroup : ↥P.weylGroup →* M ≃ₗ[R] M) : (a : ↥P.weylGroup) → M ≃ₗ[R] M)
        (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) w hw) :
      M ≃ₗ[R] M)
have refine_2 : Membership.mem (α := M ≃ₗ[R] M) (γ := Subgroup (M ≃ₗ[R] M)) (Subgroup.closure (range P.reflection))
  ((((Equiv.weightHom P).restrict P.weylGroup : ↥P.weylGroup →* M ≃ₗ[R] M) : (a : ↥P.weylGroup) → M ≃ₗ[R] M)
      (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) 1 sorry) :
    M ≃ₗ[R] M)
Tactic: induction hw using Subgroup.closure_induction'' with
    | one =>
      change ((Equiv.weightHom P).restrict P.weylGroup) 1 ∈ _
      simp only [map_one, one_mem]
    | mem w' hw' =>
      obtain ⟨i, rfl⟩ := hw'
      simp only [MonoidHom.restrict_apply, Equiv.weightHom_apply, Equiv.reflection_weightEquiv]
      simpa only [reflection_mem_weylGroup] using Subgroup.subset_closure (mem_range_self i)
    | inv_mem w' hw' =>
      obtain ⟨i, rfl⟩ := hw'
      simp only [Equiv.reflection_inv, MonoidHom.restrict_apply, Equiv.weightHom_apply,
        Equiv.reflection_weightEquiv]
      simpa only [reflection_mem_weylGroup] using Subgroup.subset_closure (mem_range_self i)
    | mul w₁ w₂ hw₁ hw₂ h₁ h₂ =>
      simpa only [← Submonoid.mk_mul_mk _ w₁ w₂ hw₁ hw₂, map_mul] using Subgroup.mul_mem _ h₁ h₂
Cleared hypotheses: hw
-- Lean 4 proof step from Mathlib.LinearAlgebra.RootSystem.WeylGroup (goal)
Goal:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : AddCommGroup N
inst_4 : Module R N
P : RootPairing ι R M N
w : P.Aut
hw : w ∈ P.weylGroup
refine_2 :
  Membership.mem (α := M ≃ₗ[R] M) (γ := Subgroup (M ≃ₗ[R] M)) (Subgroup.closure (range P.reflection))
    ((((Equiv.weightHom P).restrict P.weylGroup : ↥P.weylGroup →* M ≃ₗ[R] M) : (a : ↥P.weylGroup) → M ≃ₗ[R] M)
        (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) 1 ⋯) :
      M ≃ₗ[R] M)
⊢ Membership.mem (α := M ≃ₗ[R] M) (γ := Subgroup (M ≃ₗ[R] M)) (Subgroup.closure (range P.reflection))
    ((((Equiv.weightHom P).restrict P.weylGroup : ↥P.weylGroup →* M ≃ₗ[R] M) : (a : ↥P.weylGroup) → M ≃ₗ[R] M)
        (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) w hw) :
      M ≃ₗ[R] M)
have refine_2₁ : ∀ (w' : P.Aut) (hw' : w' ∈ range (Equiv.reflection P)),
  Membership.mem (α := M ≃ₗ[R] M) (γ := Subgroup (M ≃ₗ[R] M)) (Subgroup.closure (range P.reflection))
    ((((Equiv.weightHom P).restrict P.weylGroup : ↥P.weylGroup →* M ≃ₗ[R] M) : (a : ↥P.weylGroup) → M ≃ₗ[R] M)
        (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) w' sorry) :
      M ≃ₗ[R] M)
Tactic: induction hw using Subgroup.closure_induction'' with
    | one =>
      change ((Equiv.weightHom P).restrict P.weylGroup) 1 ∈ _
      simp only [map_one, one_mem]
    | mem w' hw' =>
      obtain ⟨i, rfl⟩ := hw'
      simp only [MonoidHom.restrict_apply, Equiv.weightHom_apply, Equiv.reflection_weightEquiv]
      simpa only [reflection_mem_weylGroup] using Subgroup.subset_closure (mem_range_self i)
    | inv_mem w' hw' =>
      obtain ⟨i, rfl⟩ := hw'
      simp only [Equiv.reflection_inv, MonoidHom.restrict_apply, Equiv.weightHom_apply,
        Equiv.reflection_weightEquiv]
      simpa only [reflection_mem_weylGroup] using Subgroup.subset_closure (mem_range_self i)
    | mul w₁ w₂ hw₁ hw₂ h₁ h₂ =>
      simpa only [← Submonoid.mk_mul_mk _ w₁ w₂ hw₁ hw₂, map_mul] using Subgroup.mul_mem _ h₁ h₂
Cleared hypotheses: hw
-- Lean 4 proof step from Mathlib.LinearAlgebra.RootSystem.WeylGroup (goal)
Goal:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : AddCommGroup N
inst_4 : Module R N
P : RootPairing ι R M N
w : P.Aut
hw : w ∈ P.weylGroup
refine_2 :
  Membership.mem (α := M ≃ₗ[R] M) (γ := Subgroup (M ≃ₗ[R] M)) (Subgroup.closure (range P.reflection))
    ((((Equiv.weightHom P).restrict P.weylGroup : ↥P.weylGroup →* M ≃ₗ[R] M) : (a : ↥P.weylGroup) → M ≃ₗ[R] M)
        (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) 1 ⋯) :
      M ≃ₗ[R] M)
refine_2₁ :
  ∀ (w' : P.Aut) (hw' : w' ∈ range (Equiv.reflection P)),
    Membership.mem (α := M ≃ₗ[R] M) (γ := Subgroup (M ≃ₗ[R] M)) (Subgroup.closure (range P.reflection))
      ((((Equiv.weightHom P).restrict P.weylGroup : ↥P.weylGroup →* M ≃ₗ[R] M) : (a : ↥P.weylGroup) → M ≃ₗ[R] M)
          (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) w' ⋯) :
        M ≃ₗ[R] M)
⊢ Membership.mem (α := M ≃ₗ[R] M) (γ := Subgroup (M ≃ₗ[R] M)) (Subgroup.closure (range P.reflection))
    ((((Equiv.weightHom P).restrict P.weylGroup : ↥P.weylGroup →* M ≃ₗ[R] M) : (a : ↥P.weylGroup) → M ≃ₗ[R] M)
        (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) w hw) :
      M ≃ₗ[R] M)
have refine_2₂ : ∀ (w' : P.Aut) (hw' : w' ∈ range (Equiv.reflection P)),
  Membership.mem (α := M ≃ₗ[R] M) (γ := Subgroup (M ≃ₗ[R] M)) (Subgroup.closure (range P.reflection))
    ((((Equiv.weightHom P).restrict P.weylGroup : ↥P.weylGroup →* M ≃ₗ[R] M) : (a : ↥P.weylGroup) → M ≃ₗ[R] M)
        (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) w'⁻¹ sorry) :
      M ≃ₗ[R] M)
Tactic: induction hw using Subgroup.closure_induction'' with
    | one =>
      change ((Equiv.weightHom P).restrict P.weylGroup) 1 ∈ _
      simp only [map_one, one_mem]
    | mem w' hw' =>
      obtain ⟨i, rfl⟩ := hw'
      simp only [MonoidHom.restrict_apply, Equiv.weightHom_apply, Equiv.reflection_weightEquiv]
      simpa only [reflection_mem_weylGroup] using Subgroup.subset_closure (mem_range_self i)
    | inv_mem w' hw' =>
      obtain ⟨i, rfl⟩ := hw'
      simp only [Equiv.reflection_inv, MonoidHom.restrict_apply, Equiv.weightHom_apply,
        Equiv.reflection_weightEquiv]
      simpa only [reflection_mem_weylGroup] using Subgroup.subset_closure (mem_range_self i)
    | mul w₁ w₂ hw₁ hw₂ h₁ h₂ =>
      simpa only [← Submonoid.mk_mul_mk _ w₁ w₂ hw₁ hw₂, map_mul] using Subgroup.mul_mem _ h₁ h₂
Cleared hypotheses: hw
-- Lean 4 proof step from Mathlib.LinearAlgebra.RootSystem.WeylGroup (goal)
Goal:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : AddCommGroup N
inst_4 : Module R N
P : RootPairing ι R M N
⊢ MonoidHom.range ((Equiv.coweightHom P).restrict P.weylGroup) =
    Subgroup.closure (range (MulOpposite.op ∘ P.coreflection))
have refine_1 : range (MulOpposite.op ∘ P.coreflection) ⊆
  (↑(MonoidHom.range ((Equiv.coweightHom P).restrict P.weylGroup)) : Set (N ≃ₗ[R] N)ᵐᵒᵖ)
Tactic: refine (Subgroup.closure_eq_of_le _ ?_ ?_).symm
-- Lean 4 proof step from Mathlib.LinearAlgebra.RootSystem.WeylGroup (goal)
Goal:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : AddCommGroup N
inst_4 : Module R N
P : RootPairing ι R M N
refine_1 :
  range (MulOpposite.op ∘ P.coreflection) ⊆
    (↑(MonoidHom.range ((Equiv.coweightHom P).restrict P.weylGroup)) : Set (N ≃ₗ[R] N)ᵐᵒᵖ)
⊢ MonoidHom.range ((Equiv.coweightHom P).restrict P.weylGroup) =
    Subgroup.closure (range (MulOpposite.op ∘ P.coreflection))
have refine_2 : MonoidHom.range ((Equiv.coweightHom P).restrict P.weylGroup) ≤
  Subgroup.closure (range (MulOpposite.op ∘ P.coreflection))
Tactic: refine (Subgroup.closure_eq_of_le _ ?_ ?_).symm
-- Lean 4 proof step from Mathlib.LinearAlgebra.RootSystem.WeylGroup (goal)
Goal:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : AddCommGroup N
inst_4 : Module R N
P : RootPairing ι R M N
i : ι
⊢ (MulOpposite.op ∘ P.coreflection) i ∈
    (↑(MonoidHom.range ((Equiv.coweightHom P).restrict P.weylGroup)) : Set (N ≃ₗ[R] N)ᵐᵒᵖ)
have refine_1 : ∃ x ∈ P.weylGroup, (Equiv.coweightHom P : (a : P.Aut) → (N ≃ₗ[R] N)ᵐᵒᵖ) x = (MulOpposite.op ∘ P.coreflection) i
Tactic: simp only [MonoidHom.restrict_range, Subgroup.coe_map, mem_image,
      SetLike.mem_coe]
-- Lean 4 proof step from Mathlib.LinearAlgebra.RootSystem.WeylGroup (goal)
Goal:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : AddCommGroup N
inst_4 : Module R N
P : RootPairing ι R M N
i : ι
⊢ ∃ x ∈ P.weylGroup, (Equiv.coweightHom P : (a : P.Aut) → (N ≃ₗ[R] N)ᵐᵒᵖ) x = (MulOpposite.op ∘ P.coreflection) i
have h : Equiv.reflection P i ∈ P.weylGroup ∧
  (Equiv.coweightHom P : (a : P.Aut) → (N ≃ₗ[R] N)ᵐᵒᵖ) (Equiv.reflection P i) = (MulOpposite.op ∘ P.coreflection) i
Tactic: use Equiv.reflection P i
-- Lean 4 proof step from Mathlib.LinearAlgebra.RootSystem.WeylGroup (goal)
Goal:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : AddCommGroup N
inst_4 : Module R N
P : RootPairing ι R M N
w w' : P.Aut
hw' : w' ∈ range (Equiv.reflection P)
⊢ Membership.mem (α := (N ≃ₗ[R] N)ᵐᵒᵖ) (γ := Subgroup (N ≃ₗ[R] N)ᵐᵒᵖ)
    (Subgroup.closure (range (MulOpposite.op ∘ P.coreflection)))
    ((((Equiv.coweightHom P).restrict P.weylGroup : ↥P.weylGroup →* (N ≃ₗ[R] N)ᵐᵒᵖ) :
          (a : ↥P.weylGroup) → (N ≃ₗ[R] N)ᵐᵒᵖ)
        (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) w' ⋯) :
      (N ≃ₗ[R] N)ᵐᵒᵖ)
have refine_2 : ∀ (i : ι),
  Membership.mem (α := (N ≃ₗ[R] N)ᵐᵒᵖ) (γ := Subgroup (N ≃ₗ[R] N)ᵐᵒᵖ)
    (Subgroup.closure (range (MulOpposite.op ∘ P.coreflection)))
    ((((Equiv.coweightHom P).restrict P.weylGroup : ↥P.weylGroup →* (N ≃ₗ[R] N)ᵐᵒᵖ) :
          (a : ↥P.weylGroup) → (N ≃ₗ[R] N)ᵐᵒᵖ)
        (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) (Equiv.reflection P i) sorry) :
      (N ≃ₗ[R] N)ᵐᵒᵖ)
Tactic: obtain ⟨i, rfl⟩ := hw'
Cleared hypotheses: w', hw'
-- Lean 4 proof step from Mathlib.LinearAlgebra.RootSystem.WeylGroup (goal)
Goal:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : AddCommGroup N
inst_4 : Module R N
P : RootPairing ι R M N
w : P.Aut
i : ι
⊢ Membership.mem (α := (N ≃ₗ[R] N)ᵐᵒᵖ) (γ := Subgroup (N ≃ₗ[R] N)ᵐᵒᵖ)
    (Subgroup.closure (range (MulOpposite.op ∘ P.coreflection)))
    ((((Equiv.coweightHom P).restrict P.weylGroup : ↥P.weylGroup →* (N ≃ₗ[R] N)ᵐᵒᵖ) :
          (a : ↥P.weylGroup) → (N ≃ₗ[R] N)ᵐᵒᵖ)
        (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) (Equiv.reflection P i) ⋯) :
      (N ≃ₗ[R] N)ᵐᵒᵖ)
have refine_2 : Membership.mem (γ := Subgroup (N ≃ₗ[R] N)ᵐᵒᵖ) (Subgroup.closure (range (MulOpposite.op ∘ P.coreflection)))
  (MulOpposite.op (P.coreflection i))
Tactic: simp only [MonoidHom.restrict_apply, Equiv.coweightHom_apply, Equiv.reflection_coweightEquiv]
-- Lean 4 proof step from Mathlib.LinearAlgebra.RootSystem.WeylGroup (goal)
Goal:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : AddCommGroup N
inst_4 : Module R N
P : RootPairing ι R M N
w w' : P.Aut
hw' : w' ∈ range (Equiv.reflection P)
⊢ Membership.mem (α := (N ≃ₗ[R] N)ᵐᵒᵖ) (γ := Subgroup (N ≃ₗ[R] N)ᵐᵒᵖ)
    (Subgroup.closure (range (MulOpposite.op ∘ P.coreflection)))
    ((((Equiv.coweightHom P).restrict P.weylGroup : ↥P.weylGroup →* (N ≃ₗ[R] N)ᵐᵒᵖ) :
          (a : ↥P.weylGroup) → (N ≃ₗ[R] N)ᵐᵒᵖ)
        (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) w'⁻¹ ⋯) :
      (N ≃ₗ[R] N)ᵐᵒᵖ)
have refine_2 : ∀ (i : ι),
  Membership.mem (α := (N ≃ₗ[R] N)ᵐᵒᵖ) (γ := Subgroup (N ≃ₗ[R] N)ᵐᵒᵖ)
    (Subgroup.closure (range (MulOpposite.op ∘ P.coreflection)))
    ((((Equiv.coweightHom P).restrict P.weylGroup : ↥P.weylGroup →* (N ≃ₗ[R] N)ᵐᵒᵖ) :
          (a : ↥P.weylGroup) → (N ≃ₗ[R] N)ᵐᵒᵖ)
        (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) (Equiv.reflection P i)⁻¹ sorry) :
      (N ≃ₗ[R] N)ᵐᵒᵖ)
Tactic: obtain ⟨i, rfl⟩ := hw'
Cleared hypotheses: w', hw'
-- Lean 4 proof step from Mathlib.LinearAlgebra.RootSystem.WeylGroup (goal)
Goal:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : AddCommGroup N
inst_4 : Module R N
P : RootPairing ι R M N
w : P.Aut
i : ι
⊢ Membership.mem (α := (N ≃ₗ[R] N)ᵐᵒᵖ) (γ := Subgroup (N ≃ₗ[R] N)ᵐᵒᵖ)
    (Subgroup.closure (range (MulOpposite.op ∘ P.coreflection)))
    ((((Equiv.coweightHom P).restrict P.weylGroup : ↥P.weylGroup →* (N ≃ₗ[R] N)ᵐᵒᵖ) :
          (a : ↥P.weylGroup) → (N ≃ₗ[R] N)ᵐᵒᵖ)
        (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) (Equiv.reflection P i)⁻¹ ⋯) :
      (N ≃ₗ[R] N)ᵐᵒᵖ)
have refine_2 : Membership.mem (γ := Subgroup (N ≃ₗ[R] N)ᵐᵒᵖ) (Subgroup.closure (range (MulOpposite.op ∘ P.coreflection)))
  (MulOpposite.op (P.coreflection i))
Tactic: simp only [Equiv.reflection_inv, MonoidHom.restrict_apply, Equiv.coweightHom_apply,
        Equiv.reflection_coweightEquiv]
-- Lean 4 proof step from Mathlib.LinearAlgebra.RootSystem.WeylGroup (goal)
Goal:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : AddCommGroup N
inst_4 : Module R N
P : RootPairing ι R M N
w : P.Aut
hw : w ∈ P.weylGroup
⊢ Membership.mem (α := (N ≃ₗ[R] N)ᵐᵒᵖ) (γ := Subgroup (N ≃ₗ[R] N)ᵐᵒᵖ)
    (Subgroup.closure (range (MulOpposite.op ∘ P.coreflection)))
    ((((Equiv.coweightHom P).restrict P.weylGroup : ↥P.weylGroup →* (N ≃ₗ[R] N)ᵐᵒᵖ) :
          (a : ↥P.weylGroup) → (N ≃ₗ[R] N)ᵐᵒᵖ)
        (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) w hw) :
      (N ≃ₗ[R] N)ᵐᵒᵖ)
have refine_2 : Membership.mem (α := (N ≃ₗ[R] N)ᵐᵒᵖ) (γ := Subgroup (N ≃ₗ[R] N)ᵐᵒᵖ)
  (Subgroup.closure (range (MulOpposite.op ∘ P.coreflection)))
  ((((Equiv.coweightHom P).restrict P.weylGroup : ↥P.weylGroup →* (N ≃ₗ[R] N)ᵐᵒᵖ) : (a : ↥P.weylGroup) → (N ≃ₗ[R] N)ᵐᵒᵖ)
      (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) 1 sorry) :
    (N ≃ₗ[R] N)ᵐᵒᵖ)
Tactic: induction hw using Subgroup.closure_induction'' with
    | one =>
      change ((Equiv.coweightHom P).restrict P.weylGroup) 1 ∈ _
      simp only [map_one, one_mem]
    | mem w' hw' =>
      obtain ⟨i, rfl⟩ := hw'
      simp only [MonoidHom.restrict_apply, Equiv.coweightHom_apply, Equiv.reflection_coweightEquiv]
      simpa only [reflection_mem_weylGroup] using Subgroup.subset_closure (mem_range_self i)
    | inv_mem w' hw' =>
      obtain ⟨i, rfl⟩ := hw'
      simp only [Equiv.reflection_inv, MonoidHom.restrict_apply, Equiv.coweightHom_apply,
        Equiv.reflection_coweightEquiv]
      simpa only [reflection_mem_weylGroup] using Subgroup.subset_closure (mem_range_self i)
    | mul w₁ w₂ hw₁ hw₂ h₁ h₂ =>
      simpa only [← Submonoid.mk_mul_mk _ w₁ w₂ hw₁ hw₂, map_mul] using Subgroup.mul_mem _ h₁ h₂
Cleared hypotheses: hw
-- Lean 4 proof step from Mathlib.LinearAlgebra.RootSystem.WeylGroup (goal)
Goal:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : AddCommGroup N
inst_4 : Module R N
P : RootPairing ι R M N
w : P.Aut
hw : w ∈ P.weylGroup
refine_2 :
  Membership.mem (α := (N ≃ₗ[R] N)ᵐᵒᵖ) (γ := Subgroup (N ≃ₗ[R] N)ᵐᵒᵖ)
    (Subgroup.closure (range (MulOpposite.op ∘ P.coreflection)))
    ((((Equiv.coweightHom P).restrict P.weylGroup : ↥P.weylGroup →* (N ≃ₗ[R] N)ᵐᵒᵖ) :
          (a : ↥P.weylGroup) → (N ≃ₗ[R] N)ᵐᵒᵖ)
        (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) 1 ⋯) :
      (N ≃ₗ[R] N)ᵐᵒᵖ)
⊢ Membership.mem (α := (N ≃ₗ[R] N)ᵐᵒᵖ) (γ := Subgroup (N ≃ₗ[R] N)ᵐᵒᵖ)
    (Subgroup.closure (range (MulOpposite.op ∘ P.coreflection)))
    ((((Equiv.coweightHom P).restrict P.weylGroup : ↥P.weylGroup →* (N ≃ₗ[R] N)ᵐᵒᵖ) :
          (a : ↥P.weylGroup) → (N ≃ₗ[R] N)ᵐᵒᵖ)
        (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) w hw) :
      (N ≃ₗ[R] N)ᵐᵒᵖ)
have refine_2₁ : ∀ (w' : P.Aut) (hw' : w' ∈ range (Equiv.reflection P)),
  Membership.mem (α := (N ≃ₗ[R] N)ᵐᵒᵖ) (γ := Subgroup (N ≃ₗ[R] N)ᵐᵒᵖ)
    (Subgroup.closure (range (MulOpposite.op ∘ P.coreflection)))
    ((((Equiv.coweightHom P).restrict P.weylGroup : ↥P.weylGroup →* (N ≃ₗ[R] N)ᵐᵒᵖ) :
          (a : ↥P.weylGroup) → (N ≃ₗ[R] N)ᵐᵒᵖ)
        (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) w' sorry) :
      (N ≃ₗ[R] N)ᵐᵒᵖ)
Tactic: induction hw using Subgroup.closure_induction'' with
    | one =>
      change ((Equiv.coweightHom P).restrict P.weylGroup) 1 ∈ _
      simp only [map_one, one_mem]
    | mem w' hw' =>
      obtain ⟨i, rfl⟩ := hw'
      simp only [MonoidHom.restrict_apply, Equiv.coweightHom_apply, Equiv.reflection_coweightEquiv]
      simpa only [reflection_mem_weylGroup] using Subgroup.subset_closure (mem_range_self i)
    | inv_mem w' hw' =>
      obtain ⟨i, rfl⟩ := hw'
      simp only [Equiv.reflection_inv, MonoidHom.restrict_apply, Equiv.coweightHom_apply,
        Equiv.reflection_coweightEquiv]
      simpa only [reflection_mem_weylGroup] using Subgroup.subset_closure (mem_range_self i)
    | mul w₁ w₂ hw₁ hw₂ h₁ h₂ =>
      simpa only [← Submonoid.mk_mul_mk _ w₁ w₂ hw₁ hw₂, map_mul] using Subgroup.mul_mem _ h₁ h₂
Cleared hypotheses: hw
-- Lean 4 proof step from Mathlib.LinearAlgebra.RootSystem.WeylGroup (goal)
Goal:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : AddCommGroup N
inst_4 : Module R N
P : RootPairing ι R M N
w : P.Aut
hw : w ∈ P.weylGroup
refine_2 :
  Membership.mem (α := (N ≃ₗ[R] N)ᵐᵒᵖ) (γ := Subgroup (N ≃ₗ[R] N)ᵐᵒᵖ)
    (Subgroup.closure (range (MulOpposite.op ∘ P.coreflection)))
    ((((Equiv.coweightHom P).restrict P.weylGroup : ↥P.weylGroup →* (N ≃ₗ[R] N)ᵐᵒᵖ) :
          (a : ↥P.weylGroup) → (N ≃ₗ[R] N)ᵐᵒᵖ)
        (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) 1 ⋯) :
      (N ≃ₗ[R] N)ᵐᵒᵖ)
refine_2₁ :
  ∀ (w' : P.Aut) (hw' : w' ∈ range (Equiv.reflection P)),
    Membership.mem (α := (N ≃ₗ[R] N)ᵐᵒᵖ) (γ := Subgroup (N ≃ₗ[R] N)ᵐᵒᵖ)
      (Subgroup.closure (range (MulOpposite.op ∘ P.coreflection)))
      ((((Equiv.coweightHom P).restrict P.weylGroup : ↥P.weylGroup →* (N ≃ₗ[R] N)ᵐᵒᵖ) :
            (a : ↥P.weylGroup) → (N ≃ₗ[R] N)ᵐᵒᵖ)
          (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) w' ⋯) :
        (N ≃ₗ[R] N)ᵐᵒᵖ)
⊢ Membership.mem (α := (N ≃ₗ[R] N)ᵐᵒᵖ) (γ := Subgroup (N ≃ₗ[R] N)ᵐᵒᵖ)
    (Subgroup.closure (range (MulOpposite.op ∘ P.coreflection)))
    ((((Equiv.coweightHom P).restrict P.weylGroup : ↥P.weylGroup →* (N ≃ₗ[R] N)ᵐᵒᵖ) :
          (a : ↥P.weylGroup) → (N ≃ₗ[R] N)ᵐᵒᵖ)
        (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) w hw) :
      (N ≃ₗ[R] N)ᵐᵒᵖ)
have refine_2₂ : ∀ (w' : P.Aut) (hw' : w' ∈ range (Equiv.reflection P)),
  Membership.mem (α := (N ≃ₗ[R] N)ᵐᵒᵖ) (γ := Subgroup (N ≃ₗ[R] N)ᵐᵒᵖ)
    (Subgroup.closure (range (MulOpposite.op ∘ P.coreflection)))
    ((((Equiv.coweightHom P).restrict P.weylGroup : ↥P.weylGroup →* (N ≃ₗ[R] N)ᵐᵒᵖ) :
          (a : ↥P.weylGroup) → (N ≃ₗ[R] N)ᵐᵒᵖ)
        (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) w'⁻¹ sorry) :
      (N ≃ₗ[R] N)ᵐᵒᵖ)
Tactic: induction hw using Subgroup.closure_induction'' with
    | one =>
      change ((Equiv.coweightHom P).restrict P.weylGroup) 1 ∈ _
      simp only [map_one, one_mem]
    | mem w' hw' =>
      obtain ⟨i, rfl⟩ := hw'
      simp only [MonoidHom.restrict_apply, Equiv.coweightHom_apply, Equiv.reflection_coweightEquiv]
      simpa only [reflection_mem_weylGroup] using Subgroup.subset_closure (mem_range_self i)
    | inv_mem w' hw' =>
      obtain ⟨i, rfl⟩ := hw'
      simp only [Equiv.reflection_inv, MonoidHom.restrict_apply, Equiv.coweightHom_apply,
        Equiv.reflection_coweightEquiv]
      simpa only [reflection_mem_weylGroup] using Subgroup.subset_closure (mem_range_self i)
    | mul w₁ w₂ hw₁ hw₂ h₁ h₂ =>
      simpa only [← Submonoid.mk_mul_mk _ w₁ w₂ hw₁ hw₂, map_mul] using Subgroup.mul_mem _ h₁ h₂
Cleared hypotheses: hw
-- Lean 4 proof step from Mathlib.LinearAlgebra.RootSystem.WeylGroup (goal)
Goal:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : AddCommGroup N
inst_4 : Module R N
P : RootPairing ι R M N
⊢ P.weylGroupToPerm.range = Subgroup.closure (range P.reflectionPerm)
have refine_1 : range P.reflectionPerm ⊆ (↑P.weylGroupToPerm.range : Set (ι ≃ ι))
Tactic: refine (Subgroup.closure_eq_of_le _ ?_ ?_).symm
-- Lean 4 proof step from Mathlib.LinearAlgebra.RootSystem.WeylGroup (goal)
Goal:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : AddCommGroup N
inst_4 : Module R N
P : RootPairing ι R M N
refine_1 : range P.reflectionPerm ⊆ (↑P.weylGroupToPerm.range : Set (ι ≃ ι))
⊢ P.weylGroupToPerm.range = Subgroup.closure (range P.reflectionPerm)
have refine_2 : P.weylGroupToPerm.range ≤ Subgroup.closure (range P.reflectionPerm)
Tactic: refine (Subgroup.closure_eq_of_le _ ?_ ?_).symm
-- Lean 4 proof step from Mathlib.LinearAlgebra.RootSystem.WeylGroup (goal)
Goal:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : AddCommGroup N
inst_4 : Module R N
P : RootPairing ι R M N
i : ι
⊢ P.reflectionPerm i ∈ (↑P.weylGroupToPerm.range : Set (ι ≃ ι))
have refine_1 : ∃ x ∈ P.weylGroup, (Equiv.indexHom P : (a : P.Aut) → ι ≃ ι) x = P.reflectionPerm i
Tactic: simp only [MonoidHom.restrict_range, Subgroup.coe_map, mem_image,
      SetLike.mem_coe]
-- Lean 4 proof step from Mathlib.LinearAlgebra.RootSystem.WeylGroup (goal)
Goal:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : AddCommGroup N
inst_4 : Module R N
P : RootPairing ι R M N
i : ι
⊢ ∃ x ∈ P.weylGroup, (Equiv.indexHom P : (a : P.Aut) → ι ≃ ι) x = P.reflectionPerm i
have h : Equiv.reflection P i ∈ P.weylGroup ∧
  (Equiv.indexHom P : (a : P.Aut) → ι ≃ ι) (Equiv.reflection P i) = P.reflectionPerm i
Tactic: use Equiv.reflection P i
-- Lean 4 proof step from Mathlib.LinearAlgebra.RootSystem.WeylGroup (goal)
Goal:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : AddCommGroup N
inst_4 : Module R N
P : RootPairing ι R M N
w w' : P.Aut
hw' : w' ∈ range (Equiv.reflection P)
⊢ Membership.mem (α := ι ≃ ι) (γ := Subgroup (ι ≃ ι)) (Subgroup.closure (range P.reflectionPerm))
    ((P.weylGroupToPerm : (a : ↥P.weylGroup) → ι ≃ ι) (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) w' ⋯) :
      ι ≃ ι)
have refine_2 : ∀ (i : ι),
  Membership.mem (α := ι ≃ ι) (γ := Subgroup (ι ≃ ι)) (Subgroup.closure (range P.reflectionPerm))
    ((P.weylGroupToPerm : (a : ↥P.weylGroup) → ι ≃ ι)
        (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) (Equiv.reflection P i) sorry) :
      ι ≃ ι)
Tactic: obtain ⟨i, rfl⟩ := hw'
Cleared hypotheses: w', hw'
-- Lean 4 proof step from Mathlib.LinearAlgebra.RootSystem.WeylGroup (goal)
Goal:
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
inst_3 : AddCommGroup N
inst_4 : Module R N
P : RootPairing ι R M N
w : P.Aut
i : ι
⊢ Membership.mem (α := ι ≃ ι) (γ := Subgroup (ι ≃ ι)) (Subgroup.closure (range P.reflectionPerm))
    ((P.weylGroupToPerm : (a : ↥P.weylGroup) → ι ≃ ι)
        (Subtype.mk (p := fun (x : P.Aut) ↦ x ∈ P.weylGroup) (Equiv.reflection P i) ⋯) :
      ι ≃ ι)
have refine_2 : Membership.mem (γ := Subgroup (ι ≃ ι)) (Subgroup.closure (range P.reflectionPerm)) (P.reflectionPerm i)
Tactic: simp only [MonoidHom.restrict_apply, Equiv.indexHom_apply, Equiv.reflection_indexEquiv]