-- Module: Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional
Goal:
F : Type u_3
inst : NormedAddCommGroup F
inst_1 : InnerProductSpace ℝ F
inst_2 : FiniteDimensional ℝ F
n : ℕ
φ : F ≃ₗᵢ[ℝ] F
hn : finrank.{0, u_3} ℝ ↥(↑(ContinuousLinearMap.id ℝ F - (↑(↑φ : F ≃L[ℝ] F) : F →L[ℝ] F)) : F →ₗ[ℝ] F).kerᗮ ≤ n
zero :
  ∀ (φ : F ≃ₗᵢ[ℝ] F)
    (hn : finrank.{0, u_3} ℝ ↥(↑(ContinuousLinearMap.id ℝ F - (↑(↑φ : F ≃L[ℝ] F) : F →L[ℝ] F)) : F →ₗ[ℝ] F).kerᗮ ≤ 0),
    ∃ (l : List F), l.length ≤ 0 ∧ φ = List.prod (List.map (fun (v : F) ↦ reflection (ℝ ∙ v)ᗮ) l)
succ :
  ∀ (n : ℕ)
    (IH :
      ∀ (φ : F ≃ₗᵢ[ℝ] F)
        (hn :
          finrank.{0, u_3} ℝ ↥(↑(ContinuousLinearMap.id ℝ F - (↑(↑φ : F ≃L[ℝ] F) : F →L[ℝ] F)) : F →ₗ[ℝ] F).kerᗮ ≤ n),
        ∃ (l : List F), l.length ≤ n ∧ φ = List.prod (List.map (fun (v : F) ↦ reflection (ℝ ∙ v)ᗮ) l))
    (φ : F ≃ₗᵢ[ℝ] F)
    (hn :
      finrank.{0, u_3} ℝ ↥(↑(ContinuousLinearMap.id ℝ F - (↑(↑φ : F ≃L[ℝ] F) : F →L[ℝ] F)) : F →ₗ[ℝ] F).kerᗮ ≤ n + 1),
    ∃ (l : List F), l.length ≤ n + 1 ∧ φ = List.prod (List.map (fun (v : F) ↦ reflection (ℝ ∙ v)ᗮ) l)
⊢ ∃ (l : List F), l.length ≤ n ∧ φ = List.prod (List.map (fun (v : F) ↦ reflection (ℝ ∙ v)ᗮ) l)

Tactic:
induction n generalizing φ with
  | zero => -- Base case: `n = 0`, the fixed subspace is the whole space, so `φ = id`
    refine ⟨[], rfl.le, show φ = 1 from ?_⟩
    have : (ContinuousLinearMap.id ℝ F - φ).ker = ⊤ := by
      rwa [le_zero_iff, finrank_eq_zero, 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
  | succ n IH =>
    -- Inductive step.  Let `W` be the fixed subspace of `φ`.  We suppose its complement to have
    -- dimension at most n + 1.
    let W := (ContinuousLinearMap.id ℝ F - φ).ker
    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 lia : 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]
      exact v.prop w hw
    have hv' : (v : F) ∉ W := by
      intro h
      exact hv ((mem_left_iff_eq_zero_of_disjoint W.orthogonal_disjoint).mp h)
    -- Let `ρ` be the reflection in `v - φ v`; this is designed to swap `v` and `φ v`
    let x : F := v - φ v
    let ρ := (ℝ ∙ x)ᗮ.reflection
    -- Notation: Let `V` be the fixed subspace of `φ.trans ρ`
    let V := (ContinuousLinearMap.id ℝ F - φ.trans ρ).ker
    have hV : ∀ w, ρ (φ w) = w → w ∈ V := by
      intro w hw
      change w - ρ (φ w) = 0
      rw [sub_eq_zero, hw]
    -- Everything fixed by `φ` is fixed by `φ.trans ρ`
    have H₂V : W ≤ V := by
      intro w hw
      apply hV
      rw [hW w hw]
      refine reflection_mem_subspace_eq_self ?_
      rw [mem_orthogonal_singleton_iff_inner_left]
      exact Submodule.sub_mem _ v.prop hφv _ hw
    -- `v` is also fixed by `φ.trans ρ`
    have H₁V : (v : F) ∈ V := by
      apply hV
      have : ρ v = φ v := reflection_sub (φ.norm_map v).symm
      rw [← this]
      exact reflection_reflection _ _
    -- By dimension-counting, the complement of the fixed subspace of `φ.trans ρ` has dimension at
    -- most `n`
    have : finrank ℝ Vᗮ ≤ n := by
      change finrank ℝ Wᗮ ≤ n + 1 at hn
      have : finrank ℝ W + 1 ≤ finrank ℝ V :=
        finrank_lt_finrank_of_lt ((SetLike.lt_iff_le_and_exists (B := F)).2 ⟨H₂V, v, H₁V, hv'⟩)
      have : finrank ℝ V + finrank ℝ Vᗮ = finrank ℝ F := V.finrank_add_finrank_orthogonal
      have : finrank ℝ W + finrank ℝ Wᗮ = finrank ℝ F := W.finrank_add_finrank_orthogonal
      lia
    -- So apply the inductive hypothesis to `φ.trans ρ`
    obtain ⟨l, hl, hφl⟩ := IH (ρ * φ) this
    -- Prepend `ρ` to the factorization into reflections obtained for `φ.trans ρ`; this gives a
    -- factorization into reflections for `φ`.
    refine ⟨x::l, Nat.succ_le_succ hl, ?_⟩
    rw [List.map_cons, List.prod_cons]
    have := congr_arg (ρ * ·) hφl
    dsimp only at this
    rwa [← mul_assoc, reflection_mul_reflection, one_mul] at this

Premises used: LinearMap.zero_apply, List.map_cons, sub_eq_zero, mul_assoc, Submodule.reflection_mul_reflection, ContinuousLinearMap.coe_sub, LinearMap.sub_apply, Submodule.zero, List.prod_cons, one_mul