awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
j i : ι
hij : i ≤ j
⊢ Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 (f '' Set.Iic j)) (gramSchmidt 𝕜 f i)
have rw : Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 (f '' Set.Iic j))
  (f i - ∑ i_1 ∈ Finset.Iio i, (starProjection (𝕜 ∙ gramSchmidt 𝕜 f i_1) : (a : E) → E) (f i))
Tactic: rw [gramSchmidt_def 𝕜 f i]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
j i : ι
hij : i ≤ j
⊢ Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 (f '' Set.Iic j))
    (f i - ∑ i_1 ∈ Finset.Iio i, (starProjection (𝕜 ∙ gramSchmidt 𝕜 f i_1) : (a : E) → E) (f i))
have simp_rw : Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 (f '' Set.Iic j))
  (f i -
    ∑ x ∈ Finset.Iio i,
      (⟪gramSchmidt 𝕜 f x, f i⟫_𝕜 / (↑(HPow.hPow (β := ℕ) ‖gramSchmidt 𝕜 f x‖ 2) : 𝕜)) • gramSchmidt 𝕜 f x)
Tactic: simp_rw [starProjection_singleton]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
j i : ι
hij : i ≤ j
⊢ Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 (f '' Set.Iic j))
    (f i -
      ∑ x ∈ Finset.Iio i,
        (⟪gramSchmidt 𝕜 f x, f i⟫_𝕜 / (↑(HPow.hPow (β := ℕ) ‖gramSchmidt 𝕜 f x‖ 2) : 𝕜)) • gramSchmidt 𝕜 f x)
have refine : ∀ (k : ι) (hk : k ∈ Finset.Iio i),
  Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 (f '' Set.Iic j))
    ((⟪gramSchmidt 𝕜 f k, f i⟫_𝕜 / (↑(HPow.hPow (β := ℕ) ‖gramSchmidt 𝕜 f k‖ 2) : 𝕜)) • gramSchmidt 𝕜 f k)
Tactic: refine Submodule.sub_mem _ (subset_span (mem_image_of_mem _ hij))
    (Submodule.sum_mem _ fun k hk => ?_)
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
hf : Pairwise fun (x1 x2 : ι) ↦ ⟪f x1, f x2⟫_𝕜 = 0
⊢ gramSchmidt 𝕜 f = f
have h : ∀ (i : ι), gramSchmidt 𝕜 f i = f i
Tactic: ext i
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
hf : Pairwise fun (x1 x2 : ι) ↦ ⟪f x1, f x2⟫_𝕜 = 0
i : ι
⊢ gramSchmidt 𝕜 f i = f i
have h : f i - ∑ i_1 ∈ Finset.Iio i, (starProjection (𝕜 ∙ gramSchmidt 𝕜 f i_1) : (a : E) → E) (f i) = f i
Tactic: rw [gramSchmidt_def]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
hf : Pairwise fun (x1 x2 : ι) ↦ ⟪f x1, f x2⟫_𝕜 = 0
i : ι
⊢ f i - ∑ i_1 ∈ Finset.Iio i, (starProjection (𝕜 ∙ gramSchmidt 𝕜 f i_1) : (a : E) → E) (f i) = f i
have trans : f i - ∑ i_1 ∈ Finset.Iio i, (starProjection (𝕜 ∙ gramSchmidt 𝕜 f i_1) : (a : E) → E) (f i) = f i - 0
Tactic: trans f i - 0
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
hf : Pairwise fun (x1 x2 : ι) ↦ ⟪f x1, f x2⟫_𝕜 = 0
i : ι
trans : f i - ∑ i_1 ∈ Finset.Iio i, (starProjection (𝕜 ∙ gramSchmidt 𝕜 f i_1) : (a : E) → E) (f i) = f i - 0
⊢ f i - ∑ i_1 ∈ Finset.Iio i, (starProjection (𝕜 ∙ gramSchmidt 𝕜 f i_1) : (a : E) → E) (f i) = f i
have trans₁ : f i - 0 = f i
Tactic: trans f i - 0
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
hf : Pairwise fun (x1 x2 : ι) ↦ ⟪f x1, f x2⟫_𝕜 = 0
i : ι
⊢ f i - ∑ i_1 ∈ Finset.Iio i, (starProjection (𝕜 ∙ gramSchmidt 𝕜 f i_1) : (a : E) → E) (f i) = f i - 0
have e_a : Eq.{u_2 + 1} (α := E) (∑ i_1 ∈ Finset.Iio i, (starProjection (𝕜 ∙ gramSchmidt 𝕜 f i_1) : (a : E) → E) (f i)) 0
Tactic: congr
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
hf : Pairwise fun (x1 x2 : ι) ↦ ⟪f x1, f x2⟫_𝕜 = 0
i : ι
⊢ Eq.{u_2 + 1} (α := E) (∑ i_1 ∈ Finset.Iio i, (starProjection (𝕜 ∙ gramSchmidt 𝕜 f i_1) : (a : E) → E) (f i)) 0
have e_a : ∀ x ∈ Finset.Iio i, Eq (α := E) ((starProjection (𝕜 ∙ gramSchmidt 𝕜 f x) : (a : E) → E) (f i) : E) (0 : E)
Tactic: apply Finset.sum_eq_zero
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
hf : Pairwise fun (x1 x2 : ι) ↦ ⟪f x1, f x2⟫_𝕜 = 0
i j : ι
hj : j ∈ Finset.Iio i
⊢ Eq (α := E) ((starProjection (𝕜 ∙ gramSchmidt 𝕜 f j) : (a : E) → E) (f i) : E) (0 : E)
have e_a : Eq (α := ↥(𝕜 ∙ gramSchmidt 𝕜 f j))
  ((orthogonalProjection (𝕜 ∙ gramSchmidt 𝕜 f j) : (a : E) → ↥(𝕜 ∙ gramSchmidt 𝕜 f j)) (f i) : ↥(𝕜 ∙ gramSchmidt 𝕜 f j))
  (0 : ↥(𝕜 ∙ gramSchmidt 𝕜 f j))
Tactic: rw [Submodule.starProjection_apply, Submodule.coe_eq_zero]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
hf : Pairwise fun (x1 x2 : ι) ↦ ⟪f x1, f x2⟫_𝕜 = 0
i j : ι
hj : j ∈ Finset.Iio i
this : span 𝕜 (f '' Set.Iic j) ⟂ 𝕜 ∙ f i
⊢ Eq (α := ↥(𝕜 ∙ gramSchmidt 𝕜 f j))
    ((orthogonalProjection (𝕜 ∙ gramSchmidt 𝕜 f j) : (a : E) → ↥(𝕜 ∙ gramSchmidt 𝕜 f j)) (f i) :
      ↥(𝕜 ∙ gramSchmidt 𝕜 f j))
    (0 : ↥(𝕜 ∙ gramSchmidt 𝕜 f j))
have hv : Membership.mem (γ := Submodule 𝕜 E) (𝕜 ∙ gramSchmidt 𝕜 f j)ᗮ (f i)
Tactic: apply orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
hf : Pairwise fun (x1 x2 : ι) ↦ ⟪f x1, f x2⟫_𝕜 = 0
i j : ι
hj : j ∈ Finset.Iio i
this : span 𝕜 (f '' Set.Iic j) ⟂ 𝕜 ∙ f i
⊢ Membership.mem (γ := Submodule 𝕜 E) (𝕜 ∙ gramSchmidt 𝕜 f j)ᗮ (f i)
have hv : Membership.mem (γ := Submodule 𝕜 E) (𝕜 ∙ f i)ᗮ (gramSchmidt 𝕜 f j)
Tactic: rw [mem_orthogonal_singleton_iff_inner_left, ← mem_orthogonal_singleton_iff_inner_right]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
hf : Pairwise fun (x1 x2 : ι) ↦ ⟪f x1, f x2⟫_𝕜 = 0
i j : ι
hj : j ∈ Finset.Iio i
⊢ Eq (α := ↥(𝕜 ∙ gramSchmidt 𝕜 f j))
    ((orthogonalProjection (𝕜 ∙ gramSchmidt 𝕜 f j) : (a : E) → ↥(𝕜 ∙ gramSchmidt 𝕜 f j)) (f i) :
      ↥(𝕜 ∙ gramSchmidt 𝕜 f j))
    (0 : ↥(𝕜 ∙ gramSchmidt 𝕜 f j))
have e_a : span 𝕜 (f '' Set.Iic j) ⟂ 𝕜 ∙ f i
Tactic: suffices span 𝕜 (f '' Set.Iic j) ⟂ 𝕜 ∙ f i by
      apply orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero
      rw [mem_orthogonal_singleton_iff_inner_left, ← mem_orthogonal_singleton_iff_inner_right]
      exact this (gramSchmidt_mem_span 𝕜 f (le_refl j))
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
hf : Pairwise fun (x1 x2 : ι) ↦ ⟪f x1, f x2⟫_𝕜 = 0
i j : ι
hj : j ∈ Finset.Iio i
⊢ span 𝕜 (f '' Set.Iic j) ⟂ 𝕜 ∙ f i
have e_a : ∀ ⦃u : E⦄, u ∈ f '' Set.Iic j → ∀ ⦃v : E⦄, Membership.mem (γ := Set E) {f i} v → ⟪u, v⟫_𝕜 = 0
Tactic: rw [isOrtho_span]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
hf : Pairwise fun (x1 x2 : ι) ↦ ⟪f x1, f x2⟫_𝕜 = 0
i j : ι
hj : j ∈ Finset.Iio i
k : ι
hk : k ∈ Set.Iic j
⊢ ⟪f k, f i⟫_𝕜 = 0
have e_a : k ≠ i
Tactic: apply hf
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
n : ι
h₀ : LinearIndependent (ι := { x : ι // x ∈ Set.Iic n }) 𝕜 (f ∘ Subtype.val (p := fun (x : ι) ↦ x ∈ Set.Iic n))
h : gramSchmidt 𝕜 f n = 0
⊢ Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 (f '' Set.Iio n)) (f n)
have rw : Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 (gramSchmidt 𝕜 f '' Set.Iio n))
  (∑ i ∈ Finset.Iio n, (starProjection (𝕜 ∙ gramSchmidt 𝕜 f i) : (a : E) → E) (f n))
Tactic: rw [← span_gramSchmidt_Iio 𝕜 f n, gramSchmidt_def' 𝕜 f, h, zero_add]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
n : ι
h₀ : LinearIndependent (ι := { x : ι // x ∈ Set.Iic n }) 𝕜 (f ∘ Subtype.val (p := fun (x : ι) ↦ x ∈ Set.Iic n))
h : gramSchmidt 𝕜 f n = 0
⊢ Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 (gramSchmidt 𝕜 f '' Set.Iio n))
    (∑ i ∈ Finset.Iio n, (starProjection (𝕜 ∙ gramSchmidt 𝕜 f i) : (a : E) → E) (f n))
have apply : ∀ c ∈ Finset.Iio n,
  Membership.mem (α := E) (γ := Submodule 𝕜 E) (span 𝕜 (gramSchmidt 𝕜 f '' Set.Iio n))
    ((starProjection (𝕜 ∙ gramSchmidt 𝕜 f c) : (a : E) → E) (f n) : E)
Tactic: apply Submodule.sum_mem _ _
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
n : ι
h₀ : LinearIndependent (ι := { x : ι // x ∈ Set.Iic n }) 𝕜 (f ∘ Subtype.val (p := fun (x : ι) ↦ x ∈ Set.Iic n))
h : gramSchmidt 𝕜 f n = 0
a : ι
ha : a ∈ Finset.Iio n
⊢ Membership.mem (α := E) (γ := Submodule 𝕜 E) (span 𝕜 (gramSchmidt 𝕜 f '' Set.Iio n))
    ((starProjection (𝕜 ∙ gramSchmidt 𝕜 f a) : (a : E) → E) (f n) : E)
have simp : Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 (gramSchmidt 𝕜 f '' Set.Iio n))
  ((⟪gramSchmidt 𝕜 f a, f n⟫_𝕜 / (↑(HPow.hPow (β := ℕ) ‖gramSchmidt 𝕜 f a‖ 2) : 𝕜)) • gramSchmidt 𝕜 f a)
Tactic: simp only [starProjection_singleton]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
n : ι
h₀ : LinearIndependent (ι := { x : ι // x ∈ Set.Iic n }) 𝕜 (f ∘ Subtype.val (p := fun (x : ι) ↦ x ∈ Set.Iic n))
h : gramSchmidt 𝕜 f n = 0
a : ι
ha : a ∈ Finset.Iio n
⊢ Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 (gramSchmidt 𝕜 f '' Set.Iio n))
    ((⟪gramSchmidt 𝕜 f a, f n⟫_𝕜 / (↑(HPow.hPow (β := ℕ) ‖gramSchmidt 𝕜 f a‖ 2) : 𝕜)) • gramSchmidt 𝕜 f a)
have apply : Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 (gramSchmidt 𝕜 f '' Set.Iio n)) (gramSchmidt 𝕜 f a)
Tactic: apply Submodule.smul_mem _ _ _
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (hypothesis)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
n : ι
h₀ : LinearIndependent (ι := { x : ι // x ∈ Set.Iic n }) 𝕜 (f ∘ Subtype.val (p := fun (x : ι) ↦ x ∈ Set.Iic n))
h : gramSchmidt 𝕜 f n = 0
a : ι
ha : a ∈ Finset.Iio n
⊢ Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 (gramSchmidt 𝕜 f '' Set.Iio n)) (gramSchmidt 𝕜 f a)
have ha₁ : a < n
Tactic: rw [Finset.mem_Iio] at ha
Cleared hypotheses: ha
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (hypothesis)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
n : ι
h₀ : LinearIndependent (ι := { x : ι // x ∈ Set.Iic n }) 𝕜 (f ∘ Subtype.val (p := fun (x : ι) ↦ x ∈ Set.Iic n))
h : gramSchmidt 𝕜 f n = 0
⊢ False
have h₁ : Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 (f '' Set.Iio n)) (f n)
Tactic: have h₁ : f n ∈ span 𝕜 (f '' Set.Iio n) := by
    rw [← span_gramSchmidt_Iio 𝕜 f n, gramSchmidt_def' 𝕜 f, h, zero_add]
    apply Submodule.sum_mem _ _
    intro a ha
    simp only [starProjection_singleton]
    apply Submodule.smul_mem _ _ _
    rw [Finset.mem_Iio] at ha
    exact subset_span ⟨a, ha, by rfl⟩
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
n : ι
h₀ : LinearIndependent (ι := { x : ι // x ∈ Set.Iic n }) 𝕜 (f ∘ Subtype.val (p := fun (x : ι) ↦ x ∈ Set.Iic n))
h : gramSchmidt 𝕜 f n = 0
h₁ : Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 (f '' Set.Iio n)) (f n)
⊢ Membership.mem (γ := Submodule 𝕜 E)
    (span 𝕜
      (Set.image (α := { x : ι // x ∈ Set.Iic n })
        (f ∘ Subtype.val (p := fun (x : ι) ↦ x ∈ Set.Iic n) : { x : ι // x ∈ Set.Iic n } → E)
        (Set.Iio (Subtype.mk (p := fun (x : ι) ↦ x ∈ Set.Iic n) n ⋯) : Set { x : ι // x ∈ Set.Iic n })))
    (Function.comp (α := { x : ι // x ∈ Set.Iic n }) f
      (Subtype.val (p := fun (x : ι) ↦ x ∈ Set.Iic n) : (self : { x : ι // x ∈ Set.Iic n }) → ι)
      (Subtype.mk (p := fun (x : ι) ↦ x ∈ Set.Iic n) n ⋯ : { x : ι // x ∈ Set.Iic n }))
have rw : Membership.mem (γ := Submodule 𝕜 E)
  (span 𝕜
    (f ''
      Set.image (α := { x : ι // x ∈ Set.Iic n })
        (Subtype.val (p := fun (x : ι) ↦ x ∈ Set.Iic n) : (self : { x : ι // x ∈ Set.Iic n }) → ι)
        (Set.Iio (Subtype.mk (p := fun (x : ι) ↦ x ∈ Set.Iic n) n sorry) : Set { x : ι // x ∈ Set.Iic n })))
  (Function.comp (α := { x : ι // x ∈ Set.Iic n }) f
    (Subtype.val (p := fun (x : ι) ↦ x ∈ Set.Iic n) : (self : { x : ι // x ∈ Set.Iic n }) → ι)
    (Subtype.mk (p := fun (x : ι) ↦ x ∈ Set.Iic n) n sorry : { x : ι // x ∈ Set.Iic n }))
Tactic: rw [image_comp]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (hypothesis)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
n : ι
h₀ : LinearIndependent (ι := { x : ι // x ∈ Set.Iic n }) 𝕜 (f ∘ Subtype.val (p := fun (x : ι) ↦ x ∈ Set.Iic n))
h : gramSchmidt 𝕜 f n = 0
h₁ : Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 (f '' Set.Iio n)) (f n)
⊢ False
have h₂ : Membership.mem (γ := Submodule 𝕜 E)
  (span 𝕜
    (Set.image (α := { x : ι // x ∈ Set.Iic n })
      (f ∘ Subtype.val (p := fun (x : ι) ↦ x ∈ Set.Iic n) : { x : ι // x ∈ Set.Iic n } → E)
      (Set.Iio (Subtype.mk (p := fun (x : ι) ↦ x ∈ Set.Iic n) n sorry) : Set { x : ι // x ∈ Set.Iic n })))
  (Function.comp (α := { x : ι // x ∈ Set.Iic n }) f
    (Subtype.val (p := fun (x : ι) ↦ x ∈ Set.Iic n) : (self : { x : ι // x ∈ Set.Iic n }) → ι)
    (Subtype.mk (p := fun (x : ι) ↦ x ∈ Set.Iic n) n sorry : { x : ι // x ∈ Set.Iic n }))
Tactic: have h₂ : (f ∘ ((↑) : Set.Iic n → ι)) ⟨n, le_refl n⟩ ∈
      span 𝕜 (f ∘ ((↑) : Set.Iic n → ι) '' Set.Iio ⟨n, le_refl n⟩) := by
    rw [image_comp]
    simpa using h₁
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
n : ι
h₀ : LinearIndependent (ι := { x : ι // x ∈ Set.Iic n }) 𝕜 (f ∘ Subtype.val (p := fun (x : ι) ↦ x ∈ Set.Iic n))
h : gramSchmidt 𝕜 f n = 0
h₁ : Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 (f '' Set.Iio n)) (f n)
h₂ :
  Membership.mem (γ := Submodule 𝕜 E)
    (span 𝕜
      (Set.image (α := { x : ι // x ∈ Set.Iic n })
        (f ∘ Subtype.val (p := fun (x : ι) ↦ x ∈ Set.Iic n) : { x : ι // x ∈ Set.Iic n } → E)
        (Set.Iio (Subtype.mk (p := fun (x : ι) ↦ x ∈ Set.Iic n) n ⋯) : Set { x : ι // x ∈ Set.Iic n })))
    (Function.comp (α := { x : ι // x ∈ Set.Iic n }) f
      (Subtype.val (p := fun (x : ι) ↦ x ∈ Set.Iic n) : (self : { x : ι // x ∈ Set.Iic n }) → ι)
      (Subtype.mk (p := fun (x : ι) ↦ x ∈ Set.Iic n) n ⋯ : { x : ι // x ∈ Set.Iic n }))
⊢ False
have apply : (Subtype.mk (p := fun (x : ι) ↦ x ∈ Set.Iic n) n sorry : { x : ι // x ∈ Set.Iic n }) ∉
  (Set.Iio (Subtype.mk (p := fun (x : ι) ↦ x ∈ Set.Iic n) n sorry) : Set { x : ι // x ∈ Set.Iic n })
Tactic: apply LinearIndependent.notMem_span_image h₀ _ h₂
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (hypothesis)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
i j : ι
hij : i < j
b : Basis ι 𝕜 E
⊢ Eq.{u_1 + 1} (α := 𝕜) (((b.repr : (a : E) → ι →₀ 𝕜) (gramSchmidt 𝕜 (⇑b : (a : ι) → E) i) : (a : ι) → 𝕜) j : 𝕜) (0 : 𝕜)
have this : Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 (gramSchmidt 𝕜 (⇑b : (a : ι) → E) '' Set.Iio j))
  (gramSchmidt 𝕜 (⇑b : (a : ι) → E) i)
Tactic: have : gramSchmidt 𝕜 b i ∈ span 𝕜 (gramSchmidt 𝕜 b '' Set.Iio j) :=
    subset_span ((Set.mem_image _ _ _).2 ⟨i, hij, rfl⟩)
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (hypothesis)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
i j : ι
hij : i < j
b : Basis ι 𝕜 E
this :
  Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 (gramSchmidt 𝕜 (⇑b : (a : ι) → E) '' Set.Iio j))
    (gramSchmidt 𝕜 (⇑b : (a : ι) → E) i)
⊢ Eq.{u_1 + 1} (α := 𝕜) (((b.repr : (a : E) → ι →₀ 𝕜) (gramSchmidt 𝕜 (⇑b : (a : ι) → E) i) : (a : ι) → 𝕜) j : 𝕜) (0 : 𝕜)
have this₁ : Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 ((⇑b : (a : ι) → E) '' Set.Iio j)) (gramSchmidt 𝕜 (⇑b : (a : ι) → E) i)
Tactic: have : gramSchmidt 𝕜 b i ∈ span 𝕜 (b '' Set.Iio j) := by rwa [← span_gramSchmidt_Iio 𝕜 b j]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (hypothesis)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
i j : ι
hij : i < j
b : Basis ι 𝕜 E
this_1 :
  Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 (gramSchmidt 𝕜 (⇑b : (a : ι) → E) '' Set.Iio j))
    (gramSchmidt 𝕜 (⇑b : (a : ι) → E) i)
this :
  Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 ((⇑b : (a : ι) → E) '' Set.Iio j)) (gramSchmidt 𝕜 (⇑b : (a : ι) → E) i)
⊢ Eq.{u_1 + 1} (α := 𝕜) (((b.repr : (a : E) → ι →₀ 𝕜) (gramSchmidt 𝕜 (⇑b : (a : ι) → E) i) : (a : ι) → 𝕜) j : 𝕜) (0 : 𝕜)
have this₁ : (↑(Finsupp.support.{u_3, u_1} (M := 𝕜) ((b.repr : (a : E) → ι →₀ 𝕜) (gramSchmidt 𝕜 (⇑b : (a : ι) → E) i))) : Set ι) ⊆
  Set.Iio j
Tactic: have : ↑(b.repr (gramSchmidt 𝕜 b i)).support ⊆ Set.Iio j :=
    Basis.repr_support_subset_of_mem_span b (Set.Iio j) this
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
n : ι
hn : gramSchmidtNormed 𝕜 f n ≠ 0
⊢ ‖gramSchmidtNormed 𝕜 f n‖ = 1
have rw : ∀ (hn : HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) (↑‖gramSchmidt 𝕜 f n‖ : 𝕜)⁻¹ (gramSchmidt 𝕜 f n) ≠ 0),
  ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) (↑‖gramSchmidt 𝕜 f n‖ : 𝕜)⁻¹ (gramSchmidt 𝕜 f n)‖ = 1
Tactic: rw [gramSchmidtNormed] at *
Cleared hypotheses: hn
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
n : ι
hn : HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) (↑‖gramSchmidt 𝕜 f n‖ : 𝕜)⁻¹ (gramSchmidt 𝕜 f n) ≠ 0
⊢ ‖HSMul.hSMul.{u_1, u_2, u_2} (α := 𝕜) (↑‖gramSchmidt 𝕜 f n‖ : 𝕜)⁻¹ (gramSchmidt 𝕜 f n)‖ = 1
have rw : gramSchmidt 𝕜 f n ≠ 0
Tactic: rw [norm_smul_inv_norm]
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
h₀ : LinearIndependent 𝕜 f
⊢ (∀ (i : ι), ‖gramSchmidtNormed 𝕜 f i‖ = 1) ∧
    Pairwise fun (i j : ι) ↦ ⟪gramSchmidtNormed 𝕜 f i, gramSchmidtNormed 𝕜 f j⟫_𝕜 = 0
have left : ∀ (i : ι), ‖gramSchmidtNormed 𝕜 f i‖ = 1
Tactic: constructor
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
h₀ : LinearIndependent 𝕜 f
left : ∀ (i : ι), ‖gramSchmidtNormed 𝕜 f i‖ = 1
⊢ (∀ (i : ι), ‖gramSchmidtNormed 𝕜 f i‖ = 1) ∧
    Pairwise fun (i j : ι) ↦ ⟪gramSchmidtNormed 𝕜 f i, gramSchmidtNormed 𝕜 f j⟫_𝕜 = 0
have right : Pairwise fun (i j : ι) ↦ ⟪gramSchmidtNormed 𝕜 f i, gramSchmidtNormed 𝕜 f j⟫_𝕜 = 0
Tactic: constructor
-- Lean 4 proof step from Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho (goal)
Goal:
𝕜 : Type u_1
E : Type u_2
inst : RCLike 𝕜
inst_1 : NormedAddCommGroup E
inst_2 : InnerProductSpace 𝕜 E
ι : Type u_3
inst_3 : LinearOrder ι
inst_4 : LocallyFiniteOrderBot ι
inst_5 : WellFoundedLT ι
f : ι → E
h₀ : LinearIndependent 𝕜 f
i j : ι
hij : i ≠ j
⊢ ⟪gramSchmidtNormed 𝕜 f i, gramSchmidtNormed 𝕜 f j⟫_𝕜 = 0
have right : gramSchmidt 𝕜 f j = 0 ∨ gramSchmidt 𝕜 f i = 0 ∨ ⟪gramSchmidt 𝕜 f i, gramSchmidt 𝕜 f j⟫_𝕜 = 0
Tactic: simp only [gramSchmidtNormed, inner_smul_left, inner_smul_right, RCLike.conj_inv,
      RCLike.conj_ofReal, mul_eq_zero, inv_eq_zero, RCLike.ofReal_eq_zero, norm_eq_zero]