awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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
e_a : ∀ x ∈ Finset.Iio i, Eq (α := E) ((starProjection (𝕜 ∙ gramSchmidt 𝕜 f x) : (a : E) → E) (f i) : E) (0 : E)
⊢ 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:
apply Finset.sum_eq_zero

Premises used: Finset.sum_eq_zero
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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
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))
⊢ Eq (α := E) ((starProjection (𝕜 ∙ gramSchmidt 𝕜 f j) : (a : E) → E) (f i) : E) (0 : E)

Tactic:
rw [Submodule.starProjection_apply, Submodule.coe_eq_zero]

Premises used: Submodule.starProjection_apply, Submodule.coe_eq_zero
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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
hv : Membership.mem (γ := Submodule 𝕜 E) (𝕜 ∙ gramSchmidt 𝕜 f j)ᗮ (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))

Tactic:
apply orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero

Premises used: Submodule.orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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
hv : Membership.mem (γ := Submodule 𝕜 E) (𝕜 ∙ f i)ᗮ (gramSchmidt 𝕜 f j)
⊢ Membership.mem (γ := Submodule 𝕜 E) (𝕜 ∙ gramSchmidt 𝕜 f j)ᗮ (f i)

Tactic:
rw [mem_orthogonal_singleton_iff_inner_left, ← mem_orthogonal_singleton_iff_inner_right]

Premises used: Submodule.mem_orthogonal_singleton_iff_inner_right, Submodule.mem_orthogonal_singleton_iff_inner_left
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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 ∙ f i)ᗮ (gramSchmidt 𝕜 f j)

Tactic:
this (gramSchmidt_mem_span 𝕜 f (le_refl j))

Premises used: le_refl, InnerProductSpace.gramSchmidt_mem_span
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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 ∙ f i)ᗮ (gramSchmidt 𝕜 f j)

Tactic:
exact this (gramSchmidt_mem_span 𝕜 f (le_refl j))

Premises used: le_refl, InnerProductSpace.gramSchmidt_mem_span
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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
e_a : 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))

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))
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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
e_a : ∀ ⦃u : E⦄, u ∈ f '' Set.Iic j → ∀ ⦃v : E⦄, Membership.mem (γ := Set E) {f i} v → ⟪u, v⟫_𝕜 = 0
⊢ span 𝕜 (f '' Set.Iic j) ⟂ 𝕜 ∙ f i

Tactic:
rw [isOrtho_span]

Premises used: Submodule.isOrtho_span
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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
e_a : k ≠ i
⊢ ⟪f k, f i⟫_𝕜 = 0

Tactic:
apply hf
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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
⊢ k ≠ i

Tactic:
(lt_of_le_of_lt hk (Finset.mem_Iio.mp hj)).ne

Premises used: lt_of_le_of_lt, Iff.mp, Finset.mem_Iio, LT.lt.ne
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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
⊢ k ≠ i

Tactic:
exact (lt_of_le_of_lt hk (Finset.mem_Iio.mp hj)).ne

Premises used: lt_of_le_of_lt, Iff.mp, Finset.mem_Iio, LT.lt.ne
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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 - 0 = f i

Tactic:
simp
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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))
by_contra : ∀ (h : gramSchmidt 𝕜 f n = 0), False
⊢ gramSchmidt 𝕜 f n ≠ 0

Tactic:
by_contra h
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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
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))
⊢ Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 (f '' Set.Iio n)) (f n)

Tactic:
rw [← span_gramSchmidt_Iio 𝕜 f n, gramSchmidt_def' 𝕜 f, h, zero_add]

Premises used: zero_add, InnerProductSpace.gramSchmidt_def', InnerProductSpace.span_gramSchmidt_Iio
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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
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)
⊢ 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:
apply Submodule.sum_mem _ _

Premises used: Submodule.sum_mem
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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
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)
⊢ 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)

Tactic:
simp only [starProjection_singleton]

Premises used: Submodule.starProjection_singleton
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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
apply : Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 (gramSchmidt 𝕜 f '' Set.Iio n)) (gramSchmidt 𝕜 f a)
⊢ 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:
apply Submodule.smul_mem _ _ _

Premises used: Submodule.smul_mem
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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
⊢ a < n

Tactic:
rw [Finset.mem_Iio] at ha

Premises used: Finset.mem_Iio
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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 < n
⊢ gramSchmidt 𝕜 f a = gramSchmidt 𝕜 f a

Tactic:
rfl
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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 < n
⊢ Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 (gramSchmidt 𝕜 f '' Set.Iio n)) (gramSchmidt 𝕜 f a)

Tactic:
exact subset_span ⟨a, ha, by rfl⟩
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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)

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⟩
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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)
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 ⋯) : 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 }))
⊢ 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 }))

Tactic:
rw [image_comp]

Premises used: Set.image_comp
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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
      (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 ⋯) : 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 }))

Tactic:
simpa using h₁
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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 }))

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
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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 }))
apply :
  (Subtype.mk (p := fun (x : ι) ↦ x ∈ Set.Iic n) n ⋯ : { x : ι // x ∈ Set.Iic n }) ∉
    (Set.Iio (Subtype.mk (p := fun (x : ι) ↦ x ∈ Set.Iic n) n ⋯) : Set { x : ι // x ∈ Set.Iic n })
⊢ False

Tactic:
apply LinearIndependent.notMem_span_image h₀ _ h₂

Premises used: LinearIndependent.notMem_span_image
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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 }))
⊢ (Subtype.mk (p := fun (x : ι) ↦ x ∈ Set.Iic n) n ⋯ : { x : ι // x ∈ Set.Iic n }) ∉
    (Set.Iio (Subtype.mk (p := fun (x : ι) ↦ x ∈ Set.Iic n) n ⋯) : Set { x : ι // x ∈ Set.Iic n })

Tactic:
simp only [Set.mem_Iio, lt_self_iff_false, not_false_iff]

Premises used: Set.mem_Iio, lt_self_iff_false, not_false_iff
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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 𝕜 f
⊢ gramSchmidt 𝕜 f n ≠ 0

Tactic:
gramSchmidt_ne_zero_coe _ (LinearIndependent.comp h₀ _ Subtype.coe_injective)

Premises used: Subtype.coe_injective, InnerProductSpace.gramSchmidt_ne_zero_coe, LinearIndependent.comp
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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
⊢ 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⟩)
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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 rwa rwa₁ rwa₂ rwa₃ :
  Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 (gramSchmidt 𝕜 (⇑b : (a : ι) → E) '' Set.Iio j))
    (gramSchmidt 𝕜 (⇑b : (a : ι) → E) i)
⊢ Membership.mem (γ := Submodule 𝕜 E) (span 𝕜 ((⇑b : (a : ι) → E) '' Set.Iio j)) (gramSchmidt 𝕜 (⇑b : (a : ι) → E) i)

Tactic:
rwa [← span_gramSchmidt_Iio 𝕜 b j]

Premises used: InnerProductSpace.span_gramSchmidt_Iio, DFunLike.coe
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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)
⊢ 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]
-- Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
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)
⊢ (↑(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