l3lab/ntp-mathlib huggingface dataset:
/-- An element `φ` of the orthogonal group of `F` can be factored as a product of reflections, and
specifically at most as many reflections as the dimension of the complement of the fixed subspace
of `φ`. -/
theorem LinearIsometryEquiv.reflections_generate_dim_aux [FiniteDimensional ℝ F] {n : ℕ}
    (φ : F ≃ₗᵢ[ℝ] F) (hn : finrank ℝ (ker (ContinuousLinearMap.id ℝ F - φ))ᗮ ≤ n) :
    ∃ l : List F, l.length ≤ n ∧ φ = (l.map fun v => reflection (ℝ ∙ v)ᗮ).prod := by
  -- We prove this by strong induction on `n`, the dimension of the orthogonal complement of the
  -- fixed subspace of the endomorphism `φ`
  induction' n with n IH generalizing φ
  · -- Base case: `n = 0`, the fixed subspace is the whole space, so `φ = id`
    refine' ⟨[], rfl.le, show φ = 1 from _⟩
    have : ker (ContinuousLinearMap.id ℝ F - φ) = ⊤ := by
      rwa [Nat.zero_eq, le_zero_iff, Submodule.finrank_eq_zero,
        Submodule.orthogonal_eq_bot_iff] at hn
    symm
    ext x
    have := LinearMap.congr_fun (LinearMap.ker_eq_top.mp this) x
    simpa only [sub_eq_zero, ContinuousLinearMap.coe_sub, LinearMap.sub_apply,
      LinearMap.zero_apply] using this
  · -- Inductive step.  Let `W` be the fixed subspace of `φ`.  We suppose its complement to have
    -- dimension at most n + 1.
    let W := ker (ContinuousLinearMap.id ℝ F - φ)
    have hW : ∀ w ∈ W, φ w = w := fun w hw => (sub_eq_zero.mp hw).symm
    by_cases hn' : finrank ℝ Wᗮ ≤ n
    · 

Proof state:
case pos
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
inst✝ : FiniteDimensional ℝ F
n : ℕ
IH :
  ∀ (φ : F ≃ₗᵢ[ℝ] F),
    finrank ℝ ↥(LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv)))ᗮ ≤ n →
      ∃ l, List.length l ≤ n ∧ φ = List.prod (List.map (fun v => reflection (Submodule.span ℝ {v})ᗮ) l)
φ : F ≃ₗᵢ[ℝ] F
hn : finrank ℝ ↥(LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv)))ᗮ ≤ Nat.succ n
W : Submodule ℝ F := LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv))
hW : ∀ w ∈ W, φ w = w
hn' : finrank ℝ ↥Wᗮ ≤ n
⊢ ∃ l, List.length l ≤ Nat.succ n ∧ φ = List.prod (List.map (fun v => reflection (Submodule.span ℝ {v})ᗮ) l)

Next tactic:
obtain ⟨V, hV₁, hV₂⟩ := IH φ hn'
/-- An element `φ` of the orthogonal group of `F` can be factored as a product of reflections, and
specifically at most as many reflections as the dimension of the complement of the fixed subspace
of `φ`. -/
theorem LinearIsometryEquiv.reflections_generate_dim_aux [FiniteDimensional ℝ F] {n : ℕ}
    (φ : F ≃ₗᵢ[ℝ] F) (hn : finrank ℝ (ker (ContinuousLinearMap.id ℝ F - φ))ᗮ ≤ n) :
    ∃ l : List F, l.length ≤ n ∧ φ = (l.map fun v => reflection (ℝ ∙ v)ᗮ).prod := by
  -- We prove this by strong induction on `n`, the dimension of the orthogonal complement of the
  -- fixed subspace of the endomorphism `φ`
  induction' n with n IH generalizing φ
  · -- Base case: `n = 0`, the fixed subspace is the whole space, so `φ = id`
    refine' ⟨[], rfl.le, show φ = 1 from _⟩
    have : ker (ContinuousLinearMap.id ℝ F - φ) = ⊤ := by
      rwa [Nat.zero_eq, le_zero_iff, Submodule.finrank_eq_zero,
        Submodule.orthogonal_eq_bot_iff] at hn
    symm
    ext x
    have := LinearMap.congr_fun (LinearMap.ker_eq_top.mp this) x
    simpa only [sub_eq_zero, ContinuousLinearMap.coe_sub, LinearMap.sub_apply,
      LinearMap.zero_apply] using this
  · -- Inductive step.  Let `W` be the fixed subspace of `φ`.  We suppose its complement to have
    -- dimension at most n + 1.
    let W := ker (ContinuousLinearMap.id ℝ F - φ)
    have hW : ∀ w ∈ W, φ w = w := fun w hw => (sub_eq_zero.mp hw).symm
    by_cases hn' : finrank ℝ Wᗮ ≤ n
    · obtain ⟨V, hV₁, hV₂⟩ := IH φ hn'
      

Proof state:
case pos.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
inst✝ : FiniteDimensional ℝ F
n : ℕ
IH :
  ∀ (φ : F ≃ₗᵢ[ℝ] F),
    finrank ℝ ↥(LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv)))ᗮ ≤ n →
      ∃ l, List.length l ≤ n ∧ φ = List.prod (List.map (fun v => reflection (Submodule.span ℝ {v})ᗮ) l)
φ : F ≃ₗᵢ[ℝ] F
hn : finrank ℝ ↥(LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv)))ᗮ ≤ Nat.succ n
W : Submodule ℝ F := LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv))
hW : ∀ w ∈ W, φ w = w
hn' : finrank ℝ ↥Wᗮ ≤ n
V : List F
hV₁ : List.length V ≤ n
hV₂ : φ = List.prod (List.map (fun v => reflection (Submodule.span ℝ {v})ᗮ) V)
⊢ ∃ l, List.length l ≤ Nat.succ n ∧ φ = List.prod (List.map (fun v => reflection (Submodule.span ℝ {v})ᗮ) l)

Next tactic:
exact ⟨V, hV₁.trans n.le_succ, hV₂⟩
/-- An element `φ` of the orthogonal group of `F` can be factored as a product of reflections, and
specifically at most as many reflections as the dimension of the complement of the fixed subspace
of `φ`. -/
theorem LinearIsometryEquiv.reflections_generate_dim_aux [FiniteDimensional ℝ F] {n : ℕ}
    (φ : F ≃ₗᵢ[ℝ] F) (hn : finrank ℝ (ker (ContinuousLinearMap.id ℝ F - φ))ᗮ ≤ n) :
    ∃ l : List F, l.length ≤ n ∧ φ = (l.map fun v => reflection (ℝ ∙ v)ᗮ).prod := by
  -- We prove this by strong induction on `n`, the dimension of the orthogonal complement of the
  -- fixed subspace of the endomorphism `φ`
  induction' n with n IH generalizing φ
  · -- Base case: `n = 0`, the fixed subspace is the whole space, so `φ = id`
    refine' ⟨[], rfl.le, show φ = 1 from _⟩
    have : ker (ContinuousLinearMap.id ℝ F - φ) = ⊤ := by
      rwa [Nat.zero_eq, le_zero_iff, Submodule.finrank_eq_zero,
        Submodule.orthogonal_eq_bot_iff] at hn
    symm
    ext x
    have := LinearMap.congr_fun (LinearMap.ker_eq_top.mp this) x
    simpa only [sub_eq_zero, ContinuousLinearMap.coe_sub, LinearMap.sub_apply,
      LinearMap.zero_apply] using this
  · -- Inductive step.  Let `W` be the fixed subspace of `φ`.  We suppose its complement to have
    -- dimension at most n + 1.
    let W := ker (ContinuousLinearMap.id ℝ F - φ)
    have hW : ∀ w ∈ W, φ w = w := fun w hw => (sub_eq_zero.mp hw).symm
    by_cases hn' : finrank ℝ Wᗮ ≤ n
    · obtain ⟨V, hV₁, hV₂⟩ := IH φ hn'
      exact ⟨V, hV₁.trans n.le_succ, hV₂⟩
    -- Take a nonzero element `v` of the orthogonal complement of `W`.
    

Proof state:
case neg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
inst✝ : FiniteDimensional ℝ F
n : ℕ
IH :
  ∀ (φ : F ≃ₗᵢ[ℝ] F),
    finrank ℝ ↥(LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv)))ᗮ ≤ n →
      ∃ l, List.length l ≤ n ∧ φ = List.prod (List.map (fun v => reflection (Submodule.span ℝ {v})ᗮ) l)
φ : F ≃ₗᵢ[ℝ] F
hn : finrank ℝ ↥(LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv)))ᗮ ≤ Nat.succ n
W : Submodule ℝ F := LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv))
hW : ∀ w ∈ W, φ w = w
hn' : ¬finrank ℝ ↥Wᗮ ≤ n
⊢ ∃ l, List.length l ≤ Nat.succ n ∧ φ = List.prod (List.map (fun v => reflection (Submodule.span ℝ {v})ᗮ) l)

Next tactic:
haveI : Nontrivial Wᗮ := nontrivial_of_finrank_pos (by linarith [zero_le n] : 0 < finrank ℝ Wᗮ)
/-- An element `φ` of the orthogonal group of `F` can be factored as a product of reflections, and
specifically at most as many reflections as the dimension of the complement of the fixed subspace
of `φ`. -/
theorem LinearIsometryEquiv.reflections_generate_dim_aux [FiniteDimensional ℝ F] {n : ℕ}
    (φ : F ≃ₗᵢ[ℝ] F) (hn : finrank ℝ (ker (ContinuousLinearMap.id ℝ F - φ))ᗮ ≤ n) :
    ∃ l : List F, l.length ≤ n ∧ φ = (l.map fun v => reflection (ℝ ∙ v)ᗮ).prod := by
  -- We prove this by strong induction on `n`, the dimension of the orthogonal complement of the
  -- fixed subspace of the endomorphism `φ`
  induction' n with n IH generalizing φ
  · -- Base case: `n = 0`, the fixed subspace is the whole space, so `φ = id`
    refine' ⟨[], rfl.le, show φ = 1 from _⟩
    have : ker (ContinuousLinearMap.id ℝ F - φ) = ⊤ := by
      rwa [Nat.zero_eq, le_zero_iff, Submodule.finrank_eq_zero,
        Submodule.orthogonal_eq_bot_iff] at hn
    symm
    ext x
    have := LinearMap.congr_fun (LinearMap.ker_eq_top.mp this) x
    simpa only [sub_eq_zero, ContinuousLinearMap.coe_sub, LinearMap.sub_apply,
      LinearMap.zero_apply] using this
  · -- Inductive step.  Let `W` be the fixed subspace of `φ`.  We suppose its complement to have
    -- dimension at most n + 1.
    let W := ker (ContinuousLinearMap.id ℝ F - φ)
    have hW : ∀ w ∈ W, φ w = w := fun w hw => (sub_eq_zero.mp hw).symm
    by_cases hn' : finrank ℝ Wᗮ ≤ n
    · obtain ⟨V, hV₁, hV₂⟩ := IH φ hn'
      exact ⟨V, hV₁.trans n.le_succ, hV₂⟩
    -- Take a nonzero element `v` of the orthogonal complement of `W`.
    haveI : Nontrivial Wᗮ := nontrivial_of_finrank_pos (by 

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
inst✝ : FiniteDimensional ℝ F
n : ℕ
IH :
  ∀ (φ : F ≃ₗᵢ[ℝ] F),
    finrank ℝ ↥(LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv)))ᗮ ≤ n →
      ∃ l, List.length l ≤ n ∧ φ = List.prod (List.map (fun v => reflection (Submodule.span ℝ {v})ᗮ) l)
φ : F ≃ₗᵢ[ℝ] F
hn : finrank ℝ ↥(LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv)))ᗮ ≤ Nat.succ n
W : Submodule ℝ F := LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv))
hW : ∀ w ∈ W, φ w = w
hn' : ¬finrank ℝ ↥Wᗮ ≤ n
⊢ 0 < finrank ℝ ↥Wᗮ

Next tactic:
linarith [zero_le n]
/-- An element `φ` of the orthogonal group of `F` can be factored as a product of reflections, and
specifically at most as many reflections as the dimension of the complement of the fixed subspace
of `φ`. -/
theorem LinearIsometryEquiv.reflections_generate_dim_aux [FiniteDimensional ℝ F] {n : ℕ}
    (φ : F ≃ₗᵢ[ℝ] F) (hn : finrank ℝ (ker (ContinuousLinearMap.id ℝ F - φ))ᗮ ≤ n) :
    ∃ l : List F, l.length ≤ n ∧ φ = (l.map fun v => reflection (ℝ ∙ v)ᗮ).prod := by
  -- We prove this by strong induction on `n`, the dimension of the orthogonal complement of the
  -- fixed subspace of the endomorphism `φ`
  induction' n with n IH generalizing φ
  · -- Base case: `n = 0`, the fixed subspace is the whole space, so `φ = id`
    refine' ⟨[], rfl.le, show φ = 1 from _⟩
    have : ker (ContinuousLinearMap.id ℝ F - φ) = ⊤ := by
      rwa [Nat.zero_eq, le_zero_iff, Submodule.finrank_eq_zero,
        Submodule.orthogonal_eq_bot_iff] at hn
    symm
    ext x
    have := LinearMap.congr_fun (LinearMap.ker_eq_top.mp this) x
    simpa only [sub_eq_zero, ContinuousLinearMap.coe_sub, LinearMap.sub_apply,
      LinearMap.zero_apply] using this
  · -- Inductive step.  Let `W` be the fixed subspace of `φ`.  We suppose its complement to have
    -- dimension at most n + 1.
    let W := ker (ContinuousLinearMap.id ℝ F - φ)
    have hW : ∀ w ∈ W, φ w = w := fun w hw => (sub_eq_zero.mp hw).symm
    by_cases hn' : finrank ℝ Wᗮ ≤ n
    · obtain ⟨V, hV₁, hV₂⟩ := IH φ hn'
      exact ⟨V, hV₁.trans n.le_succ, hV₂⟩
    -- Take a nonzero element `v` of the orthogonal complement of `W`.
    haveI : Nontrivial Wᗮ := nontrivial_of_finrank_pos (by linarith [zero_le n] : 0 < finrank ℝ Wᗮ)
    

Proof state:
case neg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
inst✝ : FiniteDimensional ℝ F
n : ℕ
IH :
  ∀ (φ : F ≃ₗᵢ[ℝ] F),
    finrank ℝ ↥(LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv)))ᗮ ≤ n →
      ∃ l, List.length l ≤ n ∧ φ = List.prod (List.map (fun v => reflection (Submodule.span ℝ {v})ᗮ) l)
φ : F ≃ₗᵢ[ℝ] F
hn : finrank ℝ ↥(LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv)))ᗮ ≤ Nat.succ n
W : Submodule ℝ F := LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv))
hW : ∀ w ∈ W, φ w = w
hn' : ¬finrank ℝ ↥Wᗮ ≤ n
this : Nontrivial ↥Wᗮ
⊢ ∃ l, List.length l ≤ Nat.succ n ∧ φ = List.prod (List.map (fun v => reflection (Submodule.span ℝ {v})ᗮ) l)

Next tactic:
obtain ⟨v, hv⟩ := exists_ne (0 : Wᗮ)
/-- An element `φ` of the orthogonal group of `F` can be factored as a product of reflections, and
specifically at most as many reflections as the dimension of the complement of the fixed subspace
of `φ`. -/
theorem LinearIsometryEquiv.reflections_generate_dim_aux [FiniteDimensional ℝ F] {n : ℕ}
    (φ : F ≃ₗᵢ[ℝ] F) (hn : finrank ℝ (ker (ContinuousLinearMap.id ℝ F - φ))ᗮ ≤ n) :
    ∃ l : List F, l.length ≤ n ∧ φ = (l.map fun v => reflection (ℝ ∙ v)ᗮ).prod := by
  -- We prove this by strong induction on `n`, the dimension of the orthogonal complement of the
  -- fixed subspace of the endomorphism `φ`
  induction' n with n IH generalizing φ
  · -- Base case: `n = 0`, the fixed subspace is the whole space, so `φ = id`
    refine' ⟨[], rfl.le, show φ = 1 from _⟩
    have : ker (ContinuousLinearMap.id ℝ F - φ) = ⊤ := by
      rwa [Nat.zero_eq, le_zero_iff, Submodule.finrank_eq_zero,
        Submodule.orthogonal_eq_bot_iff] at hn
    symm
    ext x
    have := LinearMap.congr_fun (LinearMap.ker_eq_top.mp this) x
    simpa only [sub_eq_zero, ContinuousLinearMap.coe_sub, LinearMap.sub_apply,
      LinearMap.zero_apply] using this
  · -- Inductive step.  Let `W` be the fixed subspace of `φ`.  We suppose its complement to have
    -- dimension at most n + 1.
    let W := ker (ContinuousLinearMap.id ℝ F - φ)
    have hW : ∀ w ∈ W, φ w = w := fun w hw => (sub_eq_zero.mp hw).symm
    by_cases hn' : finrank ℝ Wᗮ ≤ n
    · obtain ⟨V, hV₁, hV₂⟩ := IH φ hn'
      exact ⟨V, hV₁.trans n.le_succ, hV₂⟩
    -- Take a nonzero element `v` of the orthogonal complement of `W`.
    haveI : Nontrivial Wᗮ := nontrivial_of_finrank_pos (by linarith [zero_le n] : 0 < finrank ℝ Wᗮ)
    obtain ⟨v, hv⟩ := exists_ne (0 : Wᗮ)
    

Proof state:
case neg.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
inst✝ : FiniteDimensional ℝ F
n : ℕ
IH :
  ∀ (φ : F ≃ₗᵢ[ℝ] F),
    finrank ℝ ↥(LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv)))ᗮ ≤ n →
      ∃ l, List.length l ≤ n ∧ φ = List.prod (List.map (fun v => reflection (Submodule.span ℝ {v})ᗮ) l)
φ : F ≃ₗᵢ[ℝ] F
hn : finrank ℝ ↥(LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv)))ᗮ ≤ Nat.succ n
W : Submodule ℝ F := LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv))
hW : ∀ w ∈ W, φ w = w
hn' : ¬finrank ℝ ↥Wᗮ ≤ n
this : Nontrivial ↥Wᗮ
v : ↥Wᗮ
hv : v ≠ 0
⊢ ∃ l, List.length l ≤ Nat.succ n ∧ φ = List.prod (List.map (fun v => reflection (Submodule.span ℝ {v})ᗮ) l)

Next tactic:
have hφv : φ v ∈ Wᗮ := by
      intro w hw
      rw [← hW w hw, LinearIsometryEquiv.inner_map_map]
      exact v.prop w hw
/-- An element `φ` of the orthogonal group of `F` can be factored as a product of reflections, and
specifically at most as many reflections as the dimension of the complement of the fixed subspace
of `φ`. -/
theorem LinearIsometryEquiv.reflections_generate_dim_aux [FiniteDimensional ℝ F] {n : ℕ}
    (φ : F ≃ₗᵢ[ℝ] F) (hn : finrank ℝ (ker (ContinuousLinearMap.id ℝ F - φ))ᗮ ≤ n) :
    ∃ l : List F, l.length ≤ n ∧ φ = (l.map fun v => reflection (ℝ ∙ v)ᗮ).prod := by
  -- We prove this by strong induction on `n`, the dimension of the orthogonal complement of the
  -- fixed subspace of the endomorphism `φ`
  induction' n with n IH generalizing φ
  · -- Base case: `n = 0`, the fixed subspace is the whole space, so `φ = id`
    refine' ⟨[], rfl.le, show φ = 1 from _⟩
    have : ker (ContinuousLinearMap.id ℝ F - φ) = ⊤ := by
      rwa [Nat.zero_eq, le_zero_iff, Submodule.finrank_eq_zero,
        Submodule.orthogonal_eq_bot_iff] at hn
    symm
    ext x
    have := LinearMap.congr_fun (LinearMap.ker_eq_top.mp this) x
    simpa only [sub_eq_zero, ContinuousLinearMap.coe_sub, LinearMap.sub_apply,
      LinearMap.zero_apply] using this
  · -- Inductive step.  Let `W` be the fixed subspace of `φ`.  We suppose its complement to have
    -- dimension at most n + 1.
    let W := ker (ContinuousLinearMap.id ℝ F - φ)
    have hW : ∀ w ∈ W, φ w = w := fun w hw => (sub_eq_zero.mp hw).symm
    by_cases hn' : finrank ℝ Wᗮ ≤ n
    · obtain ⟨V, hV₁, hV₂⟩ := IH φ hn'
      exact ⟨V, hV₁.trans n.le_succ, hV₂⟩
    -- Take a nonzero element `v` of the orthogonal complement of `W`.
    haveI : Nontrivial Wᗮ := nontrivial_of_finrank_pos (by linarith [zero_le n] : 0 < finrank ℝ Wᗮ)
    obtain ⟨v, hv⟩ := exists_ne (0 : Wᗮ)
    have hφv : φ v ∈ Wᗮ := by
      

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
inst✝ : FiniteDimensional ℝ F
n : ℕ
IH :
  ∀ (φ : F ≃ₗᵢ[ℝ] F),
    finrank ℝ ↥(LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv)))ᗮ ≤ n →
      ∃ l, List.length l ≤ n ∧ φ = List.prod (List.map (fun v => reflection (Submodule.span ℝ {v})ᗮ) l)
φ : F ≃ₗᵢ[ℝ] F
hn : finrank ℝ ↥(LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv)))ᗮ ≤ Nat.succ n
W : Submodule ℝ F := LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv))
hW : ∀ w ∈ W, φ w = w
hn' : ¬finrank ℝ ↥Wᗮ ≤ n
this : Nontrivial ↥Wᗮ
v : ↥Wᗮ
hv : v ≠ 0
⊢ φ ↑v ∈ Wᗮ

Next tactic:
intro w hw
/-- An element `φ` of the orthogonal group of `F` can be factored as a product of reflections, and
specifically at most as many reflections as the dimension of the complement of the fixed subspace
of `φ`. -/
theorem LinearIsometryEquiv.reflections_generate_dim_aux [FiniteDimensional ℝ F] {n : ℕ}
    (φ : F ≃ₗᵢ[ℝ] F) (hn : finrank ℝ (ker (ContinuousLinearMap.id ℝ F - φ))ᗮ ≤ n) :
    ∃ l : List F, l.length ≤ n ∧ φ = (l.map fun v => reflection (ℝ ∙ v)ᗮ).prod := by
  -- We prove this by strong induction on `n`, the dimension of the orthogonal complement of the
  -- fixed subspace of the endomorphism `φ`
  induction' n with n IH generalizing φ
  · -- Base case: `n = 0`, the fixed subspace is the whole space, so `φ = id`
    refine' ⟨[], rfl.le, show φ = 1 from _⟩
    have : ker (ContinuousLinearMap.id ℝ F - φ) = ⊤ := by
      rwa [Nat.zero_eq, le_zero_iff, Submodule.finrank_eq_zero,
        Submodule.orthogonal_eq_bot_iff] at hn
    symm
    ext x
    have := LinearMap.congr_fun (LinearMap.ker_eq_top.mp this) x
    simpa only [sub_eq_zero, ContinuousLinearMap.coe_sub, LinearMap.sub_apply,
      LinearMap.zero_apply] using this
  · -- Inductive step.  Let `W` be the fixed subspace of `φ`.  We suppose its complement to have
    -- dimension at most n + 1.
    let W := ker (ContinuousLinearMap.id ℝ F - φ)
    have hW : ∀ w ∈ W, φ w = w := fun w hw => (sub_eq_zero.mp hw).symm
    by_cases hn' : finrank ℝ Wᗮ ≤ n
    · obtain ⟨V, hV₁, hV₂⟩ := IH φ hn'
      exact ⟨V, hV₁.trans n.le_succ, hV₂⟩
    -- Take a nonzero element `v` of the orthogonal complement of `W`.
    haveI : Nontrivial Wᗮ := nontrivial_of_finrank_pos (by linarith [zero_le n] : 0 < finrank ℝ Wᗮ)
    obtain ⟨v, hv⟩ := exists_ne (0 : Wᗮ)
    have hφv : φ v ∈ Wᗮ := by
      intro w hw
      

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
inst✝ : FiniteDimensional ℝ F
n : ℕ
IH :
  ∀ (φ : F ≃ₗᵢ[ℝ] F),
    finrank ℝ ↥(LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv)))ᗮ ≤ n →
      ∃ l, List.length l ≤ n ∧ φ = List.prod (List.map (fun v => reflection (Submodule.span ℝ {v})ᗮ) l)
φ : F ≃ₗᵢ[ℝ] F
hn : finrank ℝ ↥(LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv)))ᗮ ≤ Nat.succ n
W : Submodule ℝ F := LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv))
hW : ∀ w ∈ W, φ w = w
hn' : ¬finrank ℝ ↥Wᗮ ≤ n
this : Nontrivial ↥Wᗮ
v : ↥Wᗮ
hv : v ≠ 0
w : F
hw : w ∈ W
⊢ inner w (φ ↑v) = 0

Next tactic:
rw [← hW w hw, LinearIsometryEquiv.inner_map_map]
/-- An element `φ` of the orthogonal group of `F` can be factored as a product of reflections, and
specifically at most as many reflections as the dimension of the complement of the fixed subspace
of `φ`. -/
theorem LinearIsometryEquiv.reflections_generate_dim_aux [FiniteDimensional ℝ F] {n : ℕ}
    (φ : F ≃ₗᵢ[ℝ] F) (hn : finrank ℝ (ker (ContinuousLinearMap.id ℝ F - φ))ᗮ ≤ n) :
    ∃ l : List F, l.length ≤ n ∧ φ = (l.map fun v => reflection (ℝ ∙ v)ᗮ).prod := by
  -- We prove this by strong induction on `n`, the dimension of the orthogonal complement of the
  -- fixed subspace of the endomorphism `φ`
  induction' n with n IH generalizing φ
  · -- Base case: `n = 0`, the fixed subspace is the whole space, so `φ = id`
    refine' ⟨[], rfl.le, show φ = 1 from _⟩
    have : ker (ContinuousLinearMap.id ℝ F - φ) = ⊤ := by
      rwa [Nat.zero_eq, le_zero_iff, Submodule.finrank_eq_zero,
        Submodule.orthogonal_eq_bot_iff] at hn
    symm
    ext x
    have := LinearMap.congr_fun (LinearMap.ker_eq_top.mp this) x
    simpa only [sub_eq_zero, ContinuousLinearMap.coe_sub, LinearMap.sub_apply,
      LinearMap.zero_apply] using this
  · -- Inductive step.  Let `W` be the fixed subspace of `φ`.  We suppose its complement to have
    -- dimension at most n + 1.
    let W := ker (ContinuousLinearMap.id ℝ F - φ)
    have hW : ∀ w ∈ W, φ w = w := fun w hw => (sub_eq_zero.mp hw).symm
    by_cases hn' : finrank ℝ Wᗮ ≤ n
    · obtain ⟨V, hV₁, hV₂⟩ := IH φ hn'
      exact ⟨V, hV₁.trans n.le_succ, hV₂⟩
    -- Take a nonzero element `v` of the orthogonal complement of `W`.
    haveI : Nontrivial Wᗮ := nontrivial_of_finrank_pos (by linarith [zero_le n] : 0 < finrank ℝ Wᗮ)
    obtain ⟨v, hv⟩ := exists_ne (0 : Wᗮ)
    have hφv : φ v ∈ Wᗮ := by
      intro w hw
      rw [← hW w hw, LinearIsometryEquiv.inner_map_map]
      

Proof state:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
inst✝ : FiniteDimensional ℝ F
n : ℕ
IH :
  ∀ (φ : F ≃ₗᵢ[ℝ] F),
    finrank ℝ ↥(LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv)))ᗮ ≤ n →
      ∃ l, List.length l ≤ n ∧ φ = List.prod (List.map (fun v => reflection (Submodule.span ℝ {v})ᗮ) l)
φ : F ≃ₗᵢ[ℝ] F
hn : finrank ℝ ↥(LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv)))ᗮ ≤ Nat.succ n
W : Submodule ℝ F := LinearMap.ker (ContinuousLinearMap.id ℝ F - ↑(ContinuousLinearEquiv.mk φ.toLinearEquiv))
hW : ∀ w ∈ W, φ w = w
hn' : ¬finrank ℝ ↥Wᗮ ≤ n
this : Nontrivial ↥Wᗮ
v : ↥Wᗮ
hv : v ≠ 0
w : F
hw : w ∈ W
⊢ inner w ↑v = 0

Next tactic:
exact v.prop w hw