l3lab/ntp-mathlib huggingface dataset:
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · rintro ⟨a, ha, rfl⟩
    exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _
  refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _
  · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff]
    rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩
    refine' ⟨Finsupp.single i y, fun j => _, _⟩
    · letI := Classical.decEq ι
      rw [Finsupp.single_apply]
      split_ifs
      · assumption
      · 

Proof state:
case neg
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
y : R
hy : y ∈ I
i j : ι
this : DecidableEq ι := Classical.decEq ι
h✝ : ¬i = j
⊢ 0 ∈ I

Next tactic:
exact I.zero_mem
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · rintro ⟨a, ha, rfl⟩
    exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _
  refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _
  · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff]
    rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩
    refine' ⟨Finsupp.single i y, fun j => _, _⟩
    · letI := Classical.decEq ι
      rw [Finsupp.single_apply]
      split_ifs
      · assumption
      · exact I.zero_mem
    

Proof state:
case mp.refine'_1.intro.intro.intro.intro.intro.refine'_2
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
y : R
hy : y ∈ I
i : ι
⊢ (Finsupp.sum (fun₀ | i => y) fun i c => c • f i) = y • f i

Next tactic:
refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · rintro ⟨a, ha, rfl⟩
    exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _
  refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _
  · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff]
    rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩
    refine' ⟨Finsupp.single i y, fun j => _, _⟩
    · letI := Classical.decEq ι
      rw [Finsupp.single_apply]
      split_ifs
      · assumption
      · exact I.zero_mem
    refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _
    

Proof state:
case mp.refine'_1.intro.intro.intro.intro.intro.refine'_2
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
y : R
hy : y ∈ I
i : ι
⊢ (fun i y => y • f i) i 0 = 0

Next tactic:
simp
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · rintro ⟨a, ha, rfl⟩
    exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _
  refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _
  · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff]
    rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩
    refine' ⟨Finsupp.single i y, fun j => _, _⟩
    · letI := Classical.decEq ι
      rw [Finsupp.single_apply]
      split_ifs
      · assumption
      · exact I.zero_mem
    refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _
    simp
  · 

Proof state:
case mp.refine'_2
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
⊢ ∃ a, ∃ (_ : ∀ (i : ι), a i ∈ I), (Finsupp.sum a fun i c => c • f i) = 0

Next tactic:
exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · rintro ⟨a, ha, rfl⟩
    exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _
  refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _
  · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff]
    rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩
    refine' ⟨Finsupp.single i y, fun j => _, _⟩
    · letI := Classical.decEq ι
      rw [Finsupp.single_apply]
      split_ifs
      · assumption
      · exact I.zero_mem
    refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _
    simp
  · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩
  · 

Proof state:
case mp.refine'_3
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
⊢ ∀ (x y : M),
    (∃ a, ∃ (_ : ∀ (i : ι), a i ∈ I), (Finsupp.sum a fun i c => c • f i) = x) →
      (∃ a, ∃ (_ : ∀ (i : ι), a i ∈ I), (Finsupp.sum a fun i c => c • f i) = y) →
        ∃ a, ∃ (_ : ∀ (i : ι), a i ∈ I), (Finsupp.sum a fun i c => c • f i) = x + y

Next tactic:
rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · rintro ⟨a, ha, rfl⟩
    exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _
  refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _
  · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff]
    rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩
    refine' ⟨Finsupp.single i y, fun j => _, _⟩
    · letI := Classical.decEq ι
      rw [Finsupp.single_apply]
      split_ifs
      · assumption
      · exact I.zero_mem
    refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _
    simp
  · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩
  · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩
    

Proof state:
case mp.refine'_3.intro.intro.intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
ax : ι →₀ R
hax : ∀ (i : ι), ax i ∈ I
ay : ι →₀ R
hay : ∀ (i : ι), ay i ∈ I
⊢ ∃ a,
    ∃ (_ : ∀ (i : ι), a i ∈ I),
      (Finsupp.sum a fun i c => c • f i) = (Finsupp.sum ax fun i c => c • f i) + Finsupp.sum ay fun i c => c • f i

Next tactic:
refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · rintro ⟨a, ha, rfl⟩
    exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _
  refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _
  · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff]
    rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩
    refine' ⟨Finsupp.single i y, fun j => _, _⟩
    · letI := Classical.decEq ι
      rw [Finsupp.single_apply]
      split_ifs
      · assumption
      · exact I.zero_mem
    refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _
    simp
  · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩
  · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩
    refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> 

Proof state:
case mp.refine'_3.intro.intro.intro.intro.refine'_1
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
ax : ι →₀ R
hax : ∀ (i : ι), ax i ∈ I
ay : ι →₀ R
hay : ∀ (i : ι), ay i ∈ I
⊢ ∀ (a : ι), 0 • f a = 0

Next tactic:
intros
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · rintro ⟨a, ha, rfl⟩
    exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _
  refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _
  · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff]
    rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩
    refine' ⟨Finsupp.single i y, fun j => _, _⟩
    · letI := Classical.decEq ι
      rw [Finsupp.single_apply]
      split_ifs
      · assumption
      · exact I.zero_mem
    refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _
    simp
  · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩
  · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩
    refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> 

Proof state:
case mp.refine'_3.intro.intro.intro.intro.refine'_2
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
ax : ι →₀ R
hax : ∀ (i : ι), ax i ∈ I
ay : ι →₀ R
hay : ∀ (i : ι), ay i ∈ I
⊢ ∀ (a : ι) (b₁ b₂ : R), (b₁ + b₂) • f a = b₁ • f a + b₂ • f a

Next tactic:
intros
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · rintro ⟨a, ha, rfl⟩
    exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _
  refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _
  · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff]
    rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩
    refine' ⟨Finsupp.single i y, fun j => _, _⟩
    · letI := Classical.decEq ι
      rw [Finsupp.single_apply]
      split_ifs
      · assumption
      · exact I.zero_mem
    refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _
    simp
  · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩
  · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩
    refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;>
      

Proof state:
case mp.refine'_3.intro.intro.intro.intro.refine'_1
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
ax : ι →₀ R
hax : ∀ (i : ι), ax i ∈ I
ay : ι →₀ R
hay : ∀ (i : ι), ay i ∈ I
a✝ : ι
⊢ 0 • f a✝ = 0

Next tactic:
simp only [zero_smul, add_smul]
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · rintro ⟨a, ha, rfl⟩
    exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _
  refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _
  · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff]
    rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩
    refine' ⟨Finsupp.single i y, fun j => _, _⟩
    · letI := Classical.decEq ι
      rw [Finsupp.single_apply]
      split_ifs
      · assumption
      · exact I.zero_mem
    refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _
    simp
  · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩
  · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩
    refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;>
      

Proof state:
case mp.refine'_3.intro.intro.intro.intro.refine'_2
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
ax : ι →₀ R
hax : ∀ (i : ι), ax i ∈ I
ay : ι →₀ R
hay : ∀ (i : ι), ay i ∈ I
a✝ : ι
b₁✝ b₂✝ : R
⊢ (b₁✝ + b₂✝) • f a✝ = b₁✝ • f a✝ + b₂✝ • f a✝

Next tactic:
simp only [zero_smul, add_smul]
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · rintro ⟨a, ha, rfl⟩
    exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _
  refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _
  · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff]
    rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩
    refine' ⟨Finsupp.single i y, fun j => _, _⟩
    · letI := Classical.decEq ι
      rw [Finsupp.single_apply]
      split_ifs
      · assumption
      · exact I.zero_mem
    refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _
    simp
  · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩
  · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩
    refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;>
      simp only [zero_smul, add_smul]
  · 

Proof state:
case mp.refine'_4
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
⊢ ∀ (a : R) (x : M),
    (∃ a, ∃ (_ : ∀ (i : ι), a i ∈ I), (Finsupp.sum a fun i c => c • f i) = x) →
      ∃ a_2, ∃ (_ : ∀ (i : ι), a_2 i ∈ I), (Finsupp.sum a_2 fun i c => c • f i) = a • x

Next tactic:
rintro c x ⟨a, ha, rfl⟩
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · rintro ⟨a, ha, rfl⟩
    exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _
  refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _
  · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff]
    rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩
    refine' ⟨Finsupp.single i y, fun j => _, _⟩
    · letI := Classical.decEq ι
      rw [Finsupp.single_apply]
      split_ifs
      · assumption
      · exact I.zero_mem
    refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _
    simp
  · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩
  · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩
    refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;>
      simp only [zero_smul, add_smul]
  · rintro c x ⟨a, ha, rfl⟩
    

Proof state:
case mp.refine'_4.intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
c : R
a : ι →₀ R
ha : ∀ (i : ι), a i ∈ I
⊢ ∃ a_1, ∃ (_ : ∀ (i : ι), a_1 i ∈ I), (Finsupp.sum a_1 fun i c => c • f i) = c • Finsupp.sum a fun i c => c • f i

Next tactic:
refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · rintro ⟨a, ha, rfl⟩
    exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _
  refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _
  · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff]
    rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩
    refine' ⟨Finsupp.single i y, fun j => _, _⟩
    · letI := Classical.decEq ι
      rw [Finsupp.single_apply]
      split_ifs
      · assumption
      · exact I.zero_mem
    refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _
    simp
  · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩
  · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩
    refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;>
      simp only [zero_smul, add_smul]
  · rintro c x ⟨a, ha, rfl⟩
    refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩
    

Proof state:
case mp.refine'_4.intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
c : R
a : ι →₀ R
ha : ∀ (i : ι), a i ∈ I
⊢ (Finsupp.sum (c • a) fun i c => c • f i) = c • Finsupp.sum a fun i c => c • f i

Next tactic:
rw [Finsupp.sum_smul_index, Finsupp.smul_sum]
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · rintro ⟨a, ha, rfl⟩
    exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _
  refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _
  · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff]
    rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩
    refine' ⟨Finsupp.single i y, fun j => _, _⟩
    · letI := Classical.decEq ι
      rw [Finsupp.single_apply]
      split_ifs
      · assumption
      · exact I.zero_mem
    refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _
    simp
  · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩
  · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩
    refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;>
      simp only [zero_smul, add_smul]
  · rintro c x ⟨a, ha, rfl⟩
    refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩
    rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> 

Proof state:
case mp.refine'_4.intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
c : R
a : ι →₀ R
ha : ∀ (i : ι), a i ∈ I
⊢ (Finsupp.sum a fun i a => (c * a) • f i) = Finsupp.sum a fun a b => c • b • f a

Next tactic:
intros
/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,
then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/
theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) :
    x ∈ I • span R (Set.range f) ↔
      ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by
  constructor; swap
  · rintro ⟨a, ha, rfl⟩
    exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _
  refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _
  · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff]
    rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩
    refine' ⟨Finsupp.single i y, fun j => _, _⟩
    · letI := Classical.decEq ι
      rw [Finsupp.single_apply]
      split_ifs
      · assumption
      · exact I.zero_mem
    refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _
    simp
  · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩
  · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩
    refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;>
      simp only [zero_smul, add_smul]
  · rintro c x ⟨a, ha, rfl⟩
    refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩
    rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> 

Proof state:
case mp.refine'_4.intro.intro
R : Type u
M : Type v
F : Type u_1
G : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
I J : Ideal R
N P : Submodule R M
S : Set R
T : Set M
M' : Type w
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
ι : Type u_3
f : ι → M
x : M
hx : x ∈ I • span R (Set.range f)
c : R
a : ι →₀ R
ha : ∀ (i : ι), a i ∈ I
⊢ ∀ (i : ι), 0 • f i = 0

Next tactic:
intros