l3lab/ntp-mathlib huggingface dataset:
theorem gramSchmidt_mem_span (f : ι → E) :
    ∀ {j i}, i ≤ j → gramSchmidt 𝕜 f i ∈ span 𝕜 (f '' Set.Iic j) := by
  intro j i hij
  rw [gramSchmidt_def 𝕜 f i]
  simp_rw [orthogonalProjection_singleton]
  refine' Submodule.sub_mem _ (subset_span (mem_image_of_mem _ hij))
    (Submodule.sum_mem _ fun k hk => _)
  let hkj : k < j := (Finset.mem_Iio.1 hk).trans_le hij
  exact smul_mem _ _
    (span_mono (image_subset f <| Iic_subset_Iic.2 hkj.le) <| gramSchmidt_mem_span _ le_rfl)
termination_by _ => j
decreasing_by 

Proof state:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
_x✝ : (j : ι) ×' (i : ι) ×' i ≤ j
a✝² :
  ∀ (y : (j : ι) ×' (i : ι) ×' i ≤ j),
    (invImage (fun a => PSigma.casesOn a fun j snd => PSigma.casesOn snd fun i snd => j) IsWellOrder.toHasWellFounded).1
        y _x✝ →
      gramSchmidt 𝕜 f y.2.1 ∈ span 𝕜 (f '' Set.Iic y.1)
j : ι
i✝ : (i : ι) ×' i ≤ j
a✝¹ :
  ∀ (y : (j : ι) ×' (i : ι) ×' i ≤ j),
    (invImage (fun a => PSigma.casesOn a fun j snd => PSigma.casesOn snd fun i snd => j) IsWellOrder.toHasWellFounded).1
        y { fst := j, snd := i✝ } →
      gramSchmidt 𝕜 f y.2.1 ∈ span 𝕜 (f '' Set.Iic y.1)
i : ι
hij : i ≤ j
a✝ :
  ∀ (y : (j : ι) ×' (i : ι) ×' i ≤ j),
    (invImage (fun a => PSigma.casesOn a fun j snd => PSigma.casesOn snd fun i snd => j) IsWellOrder.toHasWellFounded).1
        y { fst := j, snd := { fst := i, snd := hij } } →
      gramSchmidt 𝕜 f y.2.1 ∈ span 𝕜 (f '' Set.Iic y.1)
k : ι
hk : k ∈ Finset.Iio i
hkj : k < j := LT.lt.trans_le (Finset.mem_Iio.mp hk) hij
⊢ (invImage (fun a => PSigma.casesOn a fun j snd => PSigma.casesOn snd fun i snd => j) IsWellOrder.toHasWellFounded).1
    { fst := k, snd := { fst := k, snd := (_ : k ≤ k) } } { fst := j, snd := { fst := i, snd := hij } }

Next tactic:
exact hkj
theorem gramSchmidt_of_orthogonal {f : ι → E} (hf : Pairwise fun i j => ⟪f i, f j⟫ = 0) :
    gramSchmidt 𝕜 f = f := by
  

Proof state:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
hf : Pairwise fun i j => inner (f i) (f j) = 0
⊢ gramSchmidt 𝕜 f = f

Next tactic:
ext i
theorem gramSchmidt_of_orthogonal {f : ι → E} (hf : Pairwise fun i j => ⟪f i, f j⟫ = 0) :
    gramSchmidt 𝕜 f = f := by
  ext i
  

Proof state:
case h
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
hf : Pairwise fun i j => inner (f i) (f j) = 0
i : ι
⊢ gramSchmidt 𝕜 f i = f i

Next tactic:
rw [gramSchmidt_def]
theorem gramSchmidt_of_orthogonal {f : ι → E} (hf : Pairwise fun i j => ⟪f i, f j⟫ = 0) :
    gramSchmidt 𝕜 f = f := by
  ext i
  rw [gramSchmidt_def]
  

Proof state:
case h
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
hf : Pairwise fun i j => inner (f i) (f j) = 0
i : ι
⊢ f i - ∑ i_1 in Finset.Iio i, ↑((orthogonalProjection (span 𝕜 {gramSchmidt 𝕜 f i_1})) (f i)) = f i

Next tactic:
trans f i - 0
theorem gramSchmidt_of_orthogonal {f : ι → E} (hf : Pairwise fun i j => ⟪f i, f j⟫ = 0) :
    gramSchmidt 𝕜 f = f := by
  ext i
  rw [gramSchmidt_def]
  trans f i - 0
  · 

Proof state:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
hf : Pairwise fun i j => inner (f i) (f j) = 0
i : ι
⊢ f i - ∑ i_1 in Finset.Iio i, ↑((orthogonalProjection (span 𝕜 {gramSchmidt 𝕜 f i_1})) (f i)) = f i - 0

Next tactic:
congr
theorem gramSchmidt_of_orthogonal {f : ι → E} (hf : Pairwise fun i j => ⟪f i, f j⟫ = 0) :
    gramSchmidt 𝕜 f = f := by
  ext i
  rw [gramSchmidt_def]
  trans f i - 0
  · congr
    

Proof state:
case e_a
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
hf : Pairwise fun i j => inner (f i) (f j) = 0
i : ι
⊢ ∑ i_1 in Finset.Iio i, ↑((orthogonalProjection (span 𝕜 {gramSchmidt 𝕜 f i_1})) (f i)) = 0

Next tactic:
apply Finset.sum_eq_zero
theorem gramSchmidt_of_orthogonal {f : ι → E} (hf : Pairwise fun i j => ⟪f i, f j⟫ = 0) :
    gramSchmidt 𝕜 f = f := by
  ext i
  rw [gramSchmidt_def]
  trans f i - 0
  · congr
    apply Finset.sum_eq_zero
    

Proof state:
case e_a.h
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
hf : Pairwise fun i j => inner (f i) (f j) = 0
i : ι
⊢ ∀ x ∈ Finset.Iio i, ↑((orthogonalProjection (span 𝕜 {gramSchmidt 𝕜 f x})) (f i)) = 0

Next tactic:
intro j hj
theorem gramSchmidt_of_orthogonal {f : ι → E} (hf : Pairwise fun i j => ⟪f i, f j⟫ = 0) :
    gramSchmidt 𝕜 f = f := by
  ext i
  rw [gramSchmidt_def]
  trans f i - 0
  · congr
    apply Finset.sum_eq_zero
    intro j hj
    

Proof state:
case e_a.h
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
hf : Pairwise fun i j => inner (f i) (f j) = 0
i j : ι
hj : j ∈ Finset.Iio i
⊢ ↑((orthogonalProjection (span 𝕜 {gramSchmidt 𝕜 f j})) (f i)) = 0

Next tactic:
rw [coe_eq_zero]
theorem gramSchmidt_of_orthogonal {f : ι → E} (hf : Pairwise fun i j => ⟪f i, f j⟫ = 0) :
    gramSchmidt 𝕜 f = f := by
  ext i
  rw [gramSchmidt_def]
  trans f i - 0
  · congr
    apply Finset.sum_eq_zero
    intro j hj
    rw [coe_eq_zero]
    

Proof state:
case e_a.h
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
hf : Pairwise fun i j => inner (f i) (f j) = 0
i j : ι
hj : j ∈ Finset.Iio i
⊢ (orthogonalProjection (span 𝕜 {gramSchmidt 𝕜 f j})) (f i) = 0

Next 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]
      rw [← mem_orthogonal_singleton_iff_inner_right]
      exact this (gramSchmidt_mem_span 𝕜 f (le_refl j))
theorem gramSchmidt_of_orthogonal {f : ι → E} (hf : Pairwise fun i j => ⟪f i, f j⟫ = 0) :
    gramSchmidt 𝕜 f = f := by
  ext i
  rw [gramSchmidt_def]
  trans f i - 0
  · congr
    apply Finset.sum_eq_zero
    intro j hj
    rw [coe_eq_zero]
    suffices span 𝕜 (f '' Set.Iic j) ⟂ 𝕜 ∙ f i by
      

Proof state:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
hf : Pairwise fun i j => inner (f i) (f j) = 0
i j : ι
hj : j ∈ Finset.Iio i
this : span 𝕜 (f '' Set.Iic j) ⟂ span 𝕜 {f i}
⊢ (orthogonalProjection (span 𝕜 {gramSchmidt 𝕜 f j})) (f i) = 0

Next tactic:
apply orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero
theorem gramSchmidt_of_orthogonal {f : ι → E} (hf : Pairwise fun i j => ⟪f i, f j⟫ = 0) :
    gramSchmidt 𝕜 f = f := by
  ext i
  rw [gramSchmidt_def]
  trans f i - 0
  · congr
    apply Finset.sum_eq_zero
    intro j hj
    rw [coe_eq_zero]
    suffices span 𝕜 (f '' Set.Iic j) ⟂ 𝕜 ∙ f i by
      apply orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero
      

Proof state:
case hv
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
hf : Pairwise fun i j => inner (f i) (f j) = 0
i j : ι
hj : j ∈ Finset.Iio i
this : span 𝕜 (f '' Set.Iic j) ⟂ span 𝕜 {f i}
⊢ f i ∈ (span 𝕜 {gramSchmidt 𝕜 f j})ᗮ

Next tactic:
rw [mem_orthogonal_singleton_iff_inner_left]
theorem gramSchmidt_of_orthogonal {f : ι → E} (hf : Pairwise fun i j => ⟪f i, f j⟫ = 0) :
    gramSchmidt 𝕜 f = f := by
  ext i
  rw [gramSchmidt_def]
  trans f i - 0
  · congr
    apply Finset.sum_eq_zero
    intro j hj
    rw [coe_eq_zero]
    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]
      

Proof state:
case hv
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
hf : Pairwise fun i j => inner (f i) (f j) = 0
i j : ι
hj : j ∈ Finset.Iio i
this : span 𝕜 (f '' Set.Iic j) ⟂ span 𝕜 {f i}
⊢ inner (f i) (gramSchmidt 𝕜 f j) = 0

Next tactic:
rw [← mem_orthogonal_singleton_iff_inner_right]
theorem gramSchmidt_of_orthogonal {f : ι → E} (hf : Pairwise fun i j => ⟪f i, f j⟫ = 0) :
    gramSchmidt 𝕜 f = f := by
  ext i
  rw [gramSchmidt_def]
  trans f i - 0
  · congr
    apply Finset.sum_eq_zero
    intro j hj
    rw [coe_eq_zero]
    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]
      rw [← mem_orthogonal_singleton_iff_inner_right]
      

Proof state:
case hv
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
hf : Pairwise fun i j => inner (f i) (f j) = 0
i j : ι
hj : j ∈ Finset.Iio i
this : span 𝕜 (f '' Set.Iic j) ⟂ span 𝕜 {f i}
⊢ gramSchmidt 𝕜 f j ∈ (span 𝕜 {f i})ᗮ

Next tactic:
exact this (gramSchmidt_mem_span 𝕜 f (le_refl j))
theorem gramSchmidt_of_orthogonal {f : ι → E} (hf : Pairwise fun i j => ⟪f i, f j⟫ = 0) :
    gramSchmidt 𝕜 f = f := by
  ext i
  rw [gramSchmidt_def]
  trans f i - 0
  · congr
    apply Finset.sum_eq_zero
    intro j hj
    rw [coe_eq_zero]
    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]
      rw [← mem_orthogonal_singleton_iff_inner_right]
      exact this (gramSchmidt_mem_span 𝕜 f (le_refl j))
    

Proof state:
case e_a.h
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
hf : Pairwise fun i j => inner (f i) (f j) = 0
i j : ι
hj : j ∈ Finset.Iio i
⊢ span 𝕜 (f '' Set.Iic j) ⟂ span 𝕜 {f i}

Next tactic:
rw [isOrtho_span]
theorem gramSchmidt_of_orthogonal {f : ι → E} (hf : Pairwise fun i j => ⟪f i, f j⟫ = 0) :
    gramSchmidt 𝕜 f = f := by
  ext i
  rw [gramSchmidt_def]
  trans f i - 0
  · congr
    apply Finset.sum_eq_zero
    intro j hj
    rw [coe_eq_zero]
    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]
      rw [← mem_orthogonal_singleton_iff_inner_right]
      exact this (gramSchmidt_mem_span 𝕜 f (le_refl j))
    rw [isOrtho_span]
    

Proof state:
case e_a.h
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
hf : Pairwise fun i j => inner (f i) (f j) = 0
i j : ι
hj : j ∈ Finset.Iio i
⊢ ∀ ⦃u : E⦄, u ∈ f '' Set.Iic j → ∀ ⦃v : E⦄, v ∈ {f i} → inner u v = 0

Next tactic:
rintro u ⟨k, hk, rfl⟩ v (rfl : v = f i)
theorem gramSchmidt_of_orthogonal {f : ι → E} (hf : Pairwise fun i j => ⟪f i, f j⟫ = 0) :
    gramSchmidt 𝕜 f = f := by
  ext i
  rw [gramSchmidt_def]
  trans f i - 0
  · congr
    apply Finset.sum_eq_zero
    intro j hj
    rw [coe_eq_zero]
    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]
      rw [← mem_orthogonal_singleton_iff_inner_right]
      exact this (gramSchmidt_mem_span 𝕜 f (le_refl j))
    rw [isOrtho_span]
    rintro u ⟨k, hk, rfl⟩ v (rfl : v = f i)
    

Proof state:
case e_a.h.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
hf : Pairwise fun i j => inner (f i) (f j) = 0
i j : ι
hj : j ∈ Finset.Iio i
k : ι
hk : k ∈ Set.Iic j
⊢ inner (f k) (f i) = 0

Next tactic:
apply hf
theorem gramSchmidt_of_orthogonal {f : ι → E} (hf : Pairwise fun i j => ⟪f i, f j⟫ = 0) :
    gramSchmidt 𝕜 f = f := by
  ext i
  rw [gramSchmidt_def]
  trans f i - 0
  · congr
    apply Finset.sum_eq_zero
    intro j hj
    rw [coe_eq_zero]
    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]
      rw [← mem_orthogonal_singleton_iff_inner_right]
      exact this (gramSchmidt_mem_span 𝕜 f (le_refl j))
    rw [isOrtho_span]
    rintro u ⟨k, hk, rfl⟩ v (rfl : v = f i)
    apply hf
    

Proof state:
case e_a.h.intro.intro.a
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
hf : Pairwise fun i j => inner (f i) (f j) = 0
i j : ι
hj : j ∈ Finset.Iio i
k : ι
hk : k ∈ Set.Iic j
⊢ k ≠ i

Next tactic:
exact (lt_of_le_of_lt hk (Finset.mem_Iio.mp hj)).ne
theorem gramSchmidt_of_orthogonal {f : ι → E} (hf : Pairwise fun i j => ⟪f i, f j⟫ = 0) :
    gramSchmidt 𝕜 f = f := by
  ext i
  rw [gramSchmidt_def]
  trans f i - 0
  · congr
    apply Finset.sum_eq_zero
    intro j hj
    rw [coe_eq_zero]
    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]
      rw [← mem_orthogonal_singleton_iff_inner_right]
      exact this (gramSchmidt_mem_span 𝕜 f (le_refl j))
    rw [isOrtho_span]
    rintro u ⟨k, hk, rfl⟩ v (rfl : v = f i)
    apply hf
    exact (lt_of_le_of_lt hk (Finset.mem_Iio.mp hj)).ne
  · 

Proof state:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
hf : Pairwise fun i j => inner (f i) (f j) = 0
i : ι
⊢ f i - 0 = f i

Next tactic:
simp
theorem gramSchmidt_ne_zero_coe {f : ι → E} (n : ι)
    (h₀ : LinearIndependent 𝕜 (f ∘ ((↑) : Set.Iic n → ι))) : gramSchmidt 𝕜 f n ≠ 0 := by
  

Proof state:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
n : ι
h₀ : LinearIndependent (ι := { x // x ∈ Set.Iic n }) 𝕜 (f ∘ Subtype.val)
⊢ gramSchmidt 𝕜 f n ≠ 0

Next tactic:
by_contra h
theorem gramSchmidt_ne_zero_coe {f : ι → E} (n : ι)
    (h₀ : LinearIndependent 𝕜 (f ∘ ((↑) : Set.Iic n → ι))) : gramSchmidt 𝕜 f n ≠ 0 := by
  by_contra h
  

Proof state:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
n : ι
h₀ : LinearIndependent (ι := { x // x ∈ Set.Iic n }) 𝕜 (f ∘ Subtype.val)
h : gramSchmidt 𝕜 f n = 0
⊢ False

Next 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 [Set.mem_image, Set.mem_Iio, orthogonalProjection_singleton]
    apply Submodule.smul_mem _ _ _
    rw [Finset.mem_Iio] at ha
    refine' subset_span ⟨a, ha, by rfl⟩
theorem gramSchmidt_ne_zero_coe {f : ι → E} (n : ι)
    (h₀ : LinearIndependent 𝕜 (f ∘ ((↑) : Set.Iic n → ι))) : gramSchmidt 𝕜 f n ≠ 0 := by
  by_contra h
  have h₁ : f n ∈ span 𝕜 (f '' Set.Iio n) := by
    

Proof state:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
n : ι
h₀ : LinearIndependent (ι := { x // x ∈ Set.Iic n }) 𝕜 (f ∘ Subtype.val)
h : gramSchmidt 𝕜 f n = 0
⊢ f n ∈ span 𝕜 (f '' Set.Iio n)

Next tactic:
rw [← span_gramSchmidt_Iio 𝕜 f n, gramSchmidt_def' 𝕜 f, h, zero_add]
theorem gramSchmidt_ne_zero_coe {f : ι → E} (n : ι)
    (h₀ : LinearIndependent 𝕜 (f ∘ ((↑) : Set.Iic n → ι))) : gramSchmidt 𝕜 f n ≠ 0 := by
  by_contra h
  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]
    

Proof state:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
n : ι
h₀ : LinearIndependent (ι := { x // x ∈ Set.Iic n }) 𝕜 (f ∘ Subtype.val)
h : gramSchmidt 𝕜 f n = 0
⊢ ∑ i in Finset.Iio n, ↑((orthogonalProjection (span 𝕜 {gramSchmidt 𝕜 f i})) (f n)) ∈
    span 𝕜 (gramSchmidt 𝕜 f '' Set.Iio n)

Next tactic:
apply Submodule.sum_mem _ _
theorem gramSchmidt_ne_zero_coe {f : ι → E} (n : ι)
    (h₀ : LinearIndependent 𝕜 (f ∘ ((↑) : Set.Iic n → ι))) : gramSchmidt 𝕜 f n ≠ 0 := by
  by_contra h
  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 _ _
    

Proof state:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
n : ι
h₀ : LinearIndependent (ι := { x // x ∈ Set.Iic n }) 𝕜 (f ∘ Subtype.val)
h : gramSchmidt 𝕜 f n = 0
⊢ ∀ c ∈ Finset.Iio n,
    ↑((orthogonalProjection (span 𝕜 {gramSchmidt 𝕜 f c})) (f n)) ∈ span 𝕜 (gramSchmidt 𝕜 f '' Set.Iio n)

Next tactic:
intro a ha
theorem gramSchmidt_ne_zero_coe {f : ι → E} (n : ι)
    (h₀ : LinearIndependent 𝕜 (f ∘ ((↑) : Set.Iic n → ι))) : gramSchmidt 𝕜 f n ≠ 0 := by
  by_contra h
  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
    

Proof state:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
n : ι
h₀ : LinearIndependent (ι := { x // x ∈ Set.Iic n }) 𝕜 (f ∘ Subtype.val)
h : gramSchmidt 𝕜 f n = 0
a : ι
ha : a ∈ Finset.Iio n
⊢ ↑((orthogonalProjection (span 𝕜 {gramSchmidt 𝕜 f a})) (f n)) ∈ span 𝕜 (gramSchmidt 𝕜 f '' Set.Iio n)

Next tactic:
simp only [Set.mem_image, Set.mem_Iio, orthogonalProjection_singleton]
theorem gramSchmidt_ne_zero_coe {f : ι → E} (n : ι)
    (h₀ : LinearIndependent 𝕜 (f ∘ ((↑) : Set.Iic n → ι))) : gramSchmidt 𝕜 f n ≠ 0 := by
  by_contra h
  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 [Set.mem_image, Set.mem_Iio, orthogonalProjection_singleton]
    

Proof state:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
n : ι
h₀ : LinearIndependent (ι := { x // x ∈ Set.Iic n }) 𝕜 (f ∘ Subtype.val)
h : gramSchmidt 𝕜 f n = 0
a : ι
ha : a ∈ Finset.Iio n
⊢ (inner (gramSchmidt 𝕜 f a) (f n) / ↑(‖gramSchmidt 𝕜 f a‖ ^ 2)) • gramSchmidt 𝕜 f a ∈
    span 𝕜 (gramSchmidt 𝕜 f '' Set.Iio n)

Next tactic:
apply Submodule.smul_mem _ _ _
theorem gramSchmidt_ne_zero_coe {f : ι → E} (n : ι)
    (h₀ : LinearIndependent 𝕜 (f ∘ ((↑) : Set.Iic n → ι))) : gramSchmidt 𝕜 f n ≠ 0 := by
  by_contra h
  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 [Set.mem_image, Set.mem_Iio, orthogonalProjection_singleton]
    apply Submodule.smul_mem _ _ _
    

Proof state:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
n : ι
h₀ : LinearIndependent (ι := { x // x ∈ Set.Iic n }) 𝕜 (f ∘ Subtype.val)
h : gramSchmidt 𝕜 f n = 0
a : ι
ha : a ∈ Finset.Iio n
⊢ gramSchmidt 𝕜 f a ∈ span 𝕜 (gramSchmidt 𝕜 f '' Set.Iio n)

Next tactic:
rw [Finset.mem_Iio] at ha
theorem gramSchmidt_ne_zero_coe {f : ι → E} (n : ι)
    (h₀ : LinearIndependent 𝕜 (f ∘ ((↑) : Set.Iic n → ι))) : gramSchmidt 𝕜 f n ≠ 0 := by
  by_contra h
  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 [Set.mem_image, Set.mem_Iio, orthogonalProjection_singleton]
    apply Submodule.smul_mem _ _ _
    rw [Finset.mem_Iio] at ha
    

Proof state:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
n : ι
h₀ : LinearIndependent (ι := { x // x ∈ Set.Iic n }) 𝕜 (f ∘ Subtype.val)
h : gramSchmidt 𝕜 f n = 0
a : ι
ha : a < n
⊢ gramSchmidt 𝕜 f a ∈ span 𝕜 (gramSchmidt 𝕜 f '' Set.Iio n)

Next tactic:
refine' subset_span ⟨a, ha, by rfl⟩
theorem gramSchmidt_ne_zero_coe {f : ι → E} (n : ι)
    (h₀ : LinearIndependent 𝕜 (f ∘ ((↑) : Set.Iic n → ι))) : gramSchmidt 𝕜 f n ≠ 0 := by
  by_contra h
  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 [Set.mem_image, Set.mem_Iio, orthogonalProjection_singleton]
    apply Submodule.smul_mem _ _ _
    rw [Finset.mem_Iio] at ha
    refine' subset_span ⟨a, ha, by 

Proof state:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : IsWellOrder ι fun x x_1 => x < x_1
f : ι → E
n : ι
h₀ : LinearIndependent (ι := { x // x ∈ Set.Iic n }) 𝕜 (f ∘ Subtype.val)
h : gramSchmidt 𝕜 f n = 0
a : ι
ha : a < n
⊢ gramSchmidt 𝕜 f a = gramSchmidt 𝕜 f a

Next tactic:
rfl