l3lab/ntp-mathlib huggingface dataset:
/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/
theorem sum_Ico_by_parts (hmn : m < n) :
    ∑ i in Ico m n, f i • g i =
      f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by
  have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by
    rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn), ← sum_Ico_add']
    simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,
      tsub_eq_zero_iff_le, add_tsub_cancel_right]
  have h₂ :
    (∑ i in Ico (m + 1) n, f i • G (i + 1)) =
      (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by
    rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),
      Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]
  rw [sum_eq_sum_Ico_succ_bot hmn]
  -- porting note: the following used to be done with `conv`
  have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =
             (Finset.sum (Ico (m + 1) n) fun i =>
                f i • ((Finset.sum (Finset.range (i + 1)) g) -
                        (Finset.sum (Finset.range i) g))) := by
    congr; funext; rw [← sum_range_succ_sub_sum g]
  rw [h₃]
  simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]
  -- porting note: the following used to be done with `conv`
  have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +
      f (n - 1) • Finset.sum (range n) fun i => g i) -
      f m • Finset.sum (range (m + 1)) fun i => g i) -
      Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =
      f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +
      Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -
      f (i + 1) • (range (i + 1)).sum g) := by
    rw [← add_sub, add_comm, ← add_sub, ← sum_sub_distrib]
  rw [h₄]
  have : ∀ i, f i • G (i + 1) - f (i + 1) • G (i + 1) = -((f (i + 1) - f i) • G (i + 1)) := by
    

Proof state:
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : ℕ → R
g : ℕ → M
m n : ℕ
hmn : m < n
h₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i
h₂ :
  ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =
    ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -
      f m • ∑ i in range (m + 1), g i
h₃ :
  ∑ i in Ico (m + 1) n, f i • g i = ∑ i in Ico (m + 1) n, f i • (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)
h₄ :
  ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -
        f m • ∑ i in range (m + 1), g i -
      ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i =
    f (n - 1) • Finset.sum (range n) g - f m • Finset.sum (range (m + 1)) g +
      ∑ i in Ico m (n - 1), (f i • Finset.sum (range (i + 1)) g - f (i + 1) • Finset.sum (range (i + 1)) g)
⊢ ∀ (i : ℕ),
    f i • ∑ i in range (i + 1), g i - f (i + 1) • ∑ i in range (i + 1), g i =
      -((f (i + 1) - f i) • ∑ i in range (i + 1), g i)

Next tactic:
intro i
/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/
theorem sum_Ico_by_parts (hmn : m < n) :
    ∑ i in Ico m n, f i • g i =
      f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by
  have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by
    rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn), ← sum_Ico_add']
    simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,
      tsub_eq_zero_iff_le, add_tsub_cancel_right]
  have h₂ :
    (∑ i in Ico (m + 1) n, f i • G (i + 1)) =
      (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by
    rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),
      Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]
  rw [sum_eq_sum_Ico_succ_bot hmn]
  -- porting note: the following used to be done with `conv`
  have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =
             (Finset.sum (Ico (m + 1) n) fun i =>
                f i • ((Finset.sum (Finset.range (i + 1)) g) -
                        (Finset.sum (Finset.range i) g))) := by
    congr; funext; rw [← sum_range_succ_sub_sum g]
  rw [h₃]
  simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]
  -- porting note: the following used to be done with `conv`
  have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +
      f (n - 1) • Finset.sum (range n) fun i => g i) -
      f m • Finset.sum (range (m + 1)) fun i => g i) -
      Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =
      f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +
      Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -
      f (i + 1) • (range (i + 1)).sum g) := by
    rw [← add_sub, add_comm, ← add_sub, ← sum_sub_distrib]
  rw [h₄]
  have : ∀ i, f i • G (i + 1) - f (i + 1) • G (i + 1) = -((f (i + 1) - f i) • G (i + 1)) := by
    intro i
    

Proof state:
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : ℕ → R
g : ℕ → M
m n : ℕ
hmn : m < n
h₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i
h₂ :
  ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =
    ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -
      f m • ∑ i in range (m + 1), g i
h₃ :
  ∑ i in Ico (m + 1) n, f i • g i = ∑ i in Ico (m + 1) n, f i • (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)
h₄ :
  ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -
        f m • ∑ i in range (m + 1), g i -
      ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i =
    f (n - 1) • Finset.sum (range n) g - f m • Finset.sum (range (m + 1)) g +
      ∑ i in Ico m (n - 1), (f i • Finset.sum (range (i + 1)) g - f (i + 1) • Finset.sum (range (i + 1)) g)
i : ℕ
⊢ f i • ∑ i in range (i + 1), g i - f (i + 1) • ∑ i in range (i + 1), g i =
    -((f (i + 1) - f i) • ∑ i in range (i + 1), g i)

Next tactic:
rw [sub_smul]
/-- **Summation by parts**, also known as **Abel's lemma** or an **Abel transformation** -/
theorem sum_Ico_by_parts (hmn : m < n) :
    ∑ i in Ico m n, f i • g i =
      f (n - 1) • G n - f m • G m - ∑ i in Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) := by
  have h₁ : (∑ i in Ico (m + 1) n, f i • G i) = ∑ i in Ico m (n - 1), f (i + 1) • G (i + 1) := by
    rw [← Nat.sub_add_cancel (Nat.one_le_of_lt hmn), ← sum_Ico_add']
    simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero,
      tsub_eq_zero_iff_le, add_tsub_cancel_right]
  have h₂ :
    (∑ i in Ico (m + 1) n, f i • G (i + 1)) =
      (∑ i in Ico m (n - 1), f i • G (i + 1)) + f (n - 1) • G n - f m • G (m + 1) := by
    rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),
      Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]
  rw [sum_eq_sum_Ico_succ_bot hmn]
  -- porting note: the following used to be done with `conv`
  have h₃: (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =
             (Finset.sum (Ico (m + 1) n) fun i =>
                f i • ((Finset.sum (Finset.range (i + 1)) g) -
                        (Finset.sum (Finset.range i) g))) := by
    congr; funext; rw [← sum_range_succ_sub_sum g]
  rw [h₃]
  simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]
  -- porting note: the following used to be done with `conv`
  have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +
      f (n - 1) • Finset.sum (range n) fun i => g i) -
      f m • Finset.sum (range (m + 1)) fun i => g i) -
      Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =
      f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +
      Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -
      f (i + 1) • (range (i + 1)).sum g) := by
    rw [← add_sub, add_comm, ← add_sub, ← sum_sub_distrib]
  rw [h₄]
  have : ∀ i, f i • G (i + 1) - f (i + 1) • G (i + 1) = -((f (i + 1) - f i) • G (i + 1)) := by
    intro i
    rw [sub_smul]
    

Proof state:
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : ℕ → R
g : ℕ → M
m n : ℕ
hmn : m < n
h₁ : ∑ i in Ico (m + 1) n, f i • ∑ i in range i, g i = ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i
h₂ :
  ∑ i in Ico (m + 1) n, f i • ∑ i in range (i + 1), g i =
    ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -
      f m • ∑ i in range (m + 1), g i
h₃ :
  ∑ i in Ico (m + 1) n, f i • g i = ∑ i in Ico (m + 1) n, f i • (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)
h₄ :
  ∑ i in Ico m (n - 1), f i • ∑ i in range (i + 1), g i + f (n - 1) • ∑ i in range n, g i -
        f m • ∑ i in range (m + 1), g i -
      ∑ i in Ico m (n - 1), f (i + 1) • ∑ i in range (i + 1), g i =
    f (n - 1) • Finset.sum (range n) g - f m • Finset.sum (range (m + 1)) g +
      ∑ i in Ico m (n - 1), (f i • Finset.sum (range (i + 1)) g - f (i + 1) • Finset.sum (range (i + 1)) g)
i : ℕ
⊢ f i • ∑ i in range (i + 1), g i - f (i + 1) • ∑ i in range (i + 1), g i =
    -(f (i + 1) • ∑ i in range (i + 1), g i - f i • ∑ i in range (i + 1), g i)

Next tactic:
abel
/-- **Summation by parts** for ranges -/
theorem sum_range_by_parts :
    ∑ i in range n, f i • g i =
      f (n - 1) • G n - ∑ i in range (n - 1), (f (i + 1) - f i) • G (i + 1) := by
  

Proof state:
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : ℕ → R
g : ℕ → M
m n : ℕ
⊢ ∑ i in range n, f i • g i =
    f (n - 1) • ∑ i in range n, g i - ∑ i in range (n - 1), (f (i + 1) - f i) • ∑ i in range (i + 1), g i

Next tactic:
by_cases hn : n = 0
/-- **Summation by parts** for ranges -/
theorem sum_range_by_parts :
    ∑ i in range n, f i • g i =
      f (n - 1) • G n - ∑ i in range (n - 1), (f (i + 1) - f i) • G (i + 1) := by
  by_cases hn : n = 0
  · 

Proof state:
case pos
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : ℕ → R
g : ℕ → M
m n : ℕ
hn : n = 0
⊢ ∑ i in range n, f i • g i =
    f (n - 1) • ∑ i in range n, g i - ∑ i in range (n - 1), (f (i + 1) - f i) • ∑ i in range (i + 1), g i

Next tactic:
simp [hn]
/-- **Summation by parts** for ranges -/
theorem sum_range_by_parts :
    ∑ i in range n, f i • g i =
      f (n - 1) • G n - ∑ i in range (n - 1), (f (i + 1) - f i) • G (i + 1) := by
  by_cases hn : n = 0
  · simp [hn]
  · 

Proof state:
case neg
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : ℕ → R
g : ℕ → M
m n : ℕ
hn : ¬n = 0
⊢ ∑ i in range n, f i • g i =
    f (n - 1) • ∑ i in range n, g i - ∑ i in range (n - 1), (f (i + 1) - f i) • ∑ i in range (i + 1), g i

Next tactic:
rw [range_eq_Ico, sum_Ico_by_parts f g (Nat.pos_of_ne_zero hn), sum_range_zero, smul_zero,
      sub_zero, range_eq_Ico]
/-- Let `S / R` be a ring extension and `x : S`, then the conductor of `R<x>` is the
    biggest ideal of `S` contained in `R<x>`. -/
def conductor (x : S) : Ideal S where
  carrier := {a | ∀ b : S, a * b ∈ R<x>}
  zero_mem' b := by simpa only [zero_mul] using Subalgebra.zero_mem _
  add_mem' ha hb c := by 

Proof state:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x a✝ b✝ : S
ha : a✝ ∈ {a | ∀ (b : S), a * b ∈ adjoin R {x}}
hb : b✝ ∈ {a | ∀ (b : S), a * b ∈ adjoin R {x}}
c : S
⊢ (a✝ + b✝) * c ∈ adjoin R {x}

Next tactic:
simpa only [add_mul] using Subalgebra.add_mem _ (ha c) (hb c)
/-- Let `S / R` be a ring extension and `x : S`, then the conductor of `R<x>` is the
    biggest ideal of `S` contained in `R<x>`. -/
def conductor (x : S) : Ideal S where
  carrier := {a | ∀ b : S, a * b ∈ R<x>}
  zero_mem' b := by 

Proof state:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x b : S
⊢ 0 * b ∈ adjoin R {x}

Next tactic:
simpa only [zero_mul] using Subalgebra.zero_mem _
/-- Let `S / R` be a ring extension and `x : S`, then the conductor of `R<x>` is the
    biggest ideal of `S` contained in `R<x>`. -/
def conductor (x : S) : Ideal S where
  carrier := {a | ∀ b : S, a * b ∈ R<x>}
  zero_mem' b := by simpa only [zero_mul] using Subalgebra.zero_mem _
  add_mem' ha hb c := by simpa only [add_mul] using Subalgebra.add_mem _ (ha c) (hb c)
  smul_mem' c a ha b := by 

Proof state:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x c a : S
ha :
  a ∈
    {
          toAddSubsemigroup :=
            { carrier := {a | ∀ (b : S), a * b ∈ adjoin R {x}},
              add_mem' :=
                (_ :
                  ∀ {a b : S},
                    a ∈ {a | ∀ (b : S), a * b ∈ adjoin R {x}} →
                      b ∈ {a | ∀ (b : S), a * b ∈ adjoin R {x}} → ∀ (c : S), (a + b) * c ∈ adjoin R {x}) },
          zero_mem' := (_ : ∀ (b : S), 0 * b ∈ adjoin R {x}) }.toAddSubsemigroup.carrier
b : S
⊢ c • a * b ∈ adjoin R {x}

Next tactic:
simpa only [smul_eq_mul, mul_left_comm, mul_assoc] using ha (c * b)
theorem conductor_subset_adjoin : (conductor R x : Set S) ⊆ R<x> := fun y hy => by
  

Proof state:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x y : S
hy : y ∈ ↑(conductor R x)
⊢ y ∈ ↑(adjoin R {x})

Next tactic:
simpa only [mul_one] using hy 1
theorem conductor_eq_top_of_adjoin_eq_top (h : R<x> = ⊤) : conductor R x = ⊤ := by
  

Proof state:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
h : adjoin R {x} = ⊤
⊢ conductor R x = ⊤

Next tactic:
simp only [Ideal.eq_top_iff_one, mem_conductor_iff, h, mem_top, forall_const]
/-- This technical lemma tell us that if `C` is the conductor of `R<x>` and `I` is an ideal of `R`
  then `p * (I * S) ⊆ I * R<x>` for any `p` in `C ∩ R` -/
theorem prod_mem_ideal_map_of_mem_conductor {p : R} {z : S}
    (hp : p ∈ Ideal.comap (algebraMap R S) (conductor R x)) (hz' : z ∈ I.map (algebraMap R S)) :
    algebraMap R S p * z ∈ algebraMap R<x> S '' ↑(I.map (algebraMap R R<x>)) := by
  

Proof state:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
hz' : z ∈ Ideal.map (algebraMap R S) I
⊢ (algebraMap R S) p * z ∈ ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)

Next tactic:
rw [Ideal.map, Ideal.span, Finsupp.mem_span_image_iff_total] at hz'
/-- This technical lemma tell us that if `C` is the conductor of `R<x>` and `I` is an ideal of `R`
  then `p * (I * S) ⊆ I * R<x>` for any `p` in `C ∩ R` -/
theorem prod_mem_ideal_map_of_mem_conductor {p : R} {z : S}
    (hp : p ∈ Ideal.comap (algebraMap R S) (conductor R x)) (hz' : z ∈ I.map (algebraMap R S)) :
    algebraMap R S p * z ∈ algebraMap R<x> S '' ↑(I.map (algebraMap R R<x>)) := by
  rw [Ideal.map, Ideal.span, Finsupp.mem_span_image_iff_total] at hz'
  

Proof state:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
hz' : ∃ l ∈ Finsupp.supported S S ↑I, (Finsupp.total R S S ⇑(algebraMap R S)) l = z
⊢ (algebraMap R S) p * z ∈ ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)

Next tactic:
obtain ⟨l, H, H'⟩ := hz'
/-- This technical lemma tell us that if `C` is the conductor of `R<x>` and `I` is an ideal of `R`
  then `p * (I * S) ⊆ I * R<x>` for any `p` in `C ∩ R` -/
theorem prod_mem_ideal_map_of_mem_conductor {p : R} {z : S}
    (hp : p ∈ Ideal.comap (algebraMap R S) (conductor R x)) (hz' : z ∈ I.map (algebraMap R S)) :
    algebraMap R S p * z ∈ algebraMap R<x> S '' ↑(I.map (algebraMap R R<x>)) := by
  rw [Ideal.map, Ideal.span, Finsupp.mem_span_image_iff_total] at hz'
  obtain ⟨l, H, H'⟩ := hz'
  

Proof state:
case intro.intro
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (Finsupp.total R S S ⇑(algebraMap R S)) l = z
⊢ (algebraMap R S) p * z ∈ ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)

Next tactic:
rw [Finsupp.total_apply] at H'
/-- This technical lemma tell us that if `C` is the conductor of `R<x>` and `I` is an ideal of `R`
  then `p * (I * S) ⊆ I * R<x>` for any `p` in `C ∩ R` -/
theorem prod_mem_ideal_map_of_mem_conductor {p : R} {z : S}
    (hp : p ∈ Ideal.comap (algebraMap R S) (conductor R x)) (hz' : z ∈ I.map (algebraMap R S)) :
    algebraMap R S p * z ∈ algebraMap R<x> S '' ↑(I.map (algebraMap R R<x>)) := by
  rw [Ideal.map, Ideal.span, Finsupp.mem_span_image_iff_total] at hz'
  obtain ⟨l, H, H'⟩ := hz'
  rw [Finsupp.total_apply] at H'
  

Proof state:
case intro.intro
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (Finsupp.sum l fun i a => a • (algebraMap R S) i) = z
⊢ (algebraMap R S) p * z ∈ ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)

Next tactic:
rw [← H', mul_comm, Finsupp.sum_mul]
/-- This technical lemma tell us that if `C` is the conductor of `R<x>` and `I` is an ideal of `R`
  then `p * (I * S) ⊆ I * R<x>` for any `p` in `C ∩ R` -/
theorem prod_mem_ideal_map_of_mem_conductor {p : R} {z : S}
    (hp : p ∈ Ideal.comap (algebraMap R S) (conductor R x)) (hz' : z ∈ I.map (algebraMap R S)) :
    algebraMap R S p * z ∈ algebraMap R<x> S '' ↑(I.map (algebraMap R R<x>)) := by
  rw [Ideal.map, Ideal.span, Finsupp.mem_span_image_iff_total] at hz'
  obtain ⟨l, H, H'⟩ := hz'
  rw [Finsupp.total_apply] at H'
  rw [← H', mul_comm, Finsupp.sum_mul]
  

Proof state:
case intro.intro
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (Finsupp.sum l fun i a => a • (algebraMap R S) i) = z
⊢ (Finsupp.sum l fun a c => c • (algebraMap R S) a * (algebraMap R S) p) ∈
    ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)

Next tactic:
have lem : ∀ {a : R}, a ∈ I → l a • algebraMap R S a * algebraMap R S p ∈
      algebraMap R<x> S '' I.map (algebraMap R R<x>) := by
    intro a ha
    rw [Algebra.id.smul_eq_mul, mul_assoc, mul_comm, mul_assoc, Set.mem_image]
    refine Exists.intro
        (algebraMap R R<x> a * ⟨l a * algebraMap R S p,
          show l a * algebraMap R S p ∈ R<x> from ?h⟩) ?_
    case h =>
      rw [mul_comm]
      exact mem_conductor_iff.mp (Ideal.mem_comap.mp hp) _
    · refine ⟨?_, ?_⟩
      · rw [mul_comm]
        apply Ideal.mul_mem_left (I.map (algebraMap R R<x>)) _ (Ideal.mem_map_of_mem _ ha)
      · simp only [RingHom.map_mul, mul_comm (algebraMap R S p) (l a)]
        rfl
/-- This technical lemma tell us that if `C` is the conductor of `R<x>` and `I` is an ideal of `R`
  then `p * (I * S) ⊆ I * R<x>` for any `p` in `C ∩ R` -/
theorem prod_mem_ideal_map_of_mem_conductor {p : R} {z : S}
    (hp : p ∈ Ideal.comap (algebraMap R S) (conductor R x)) (hz' : z ∈ I.map (algebraMap R S)) :
    algebraMap R S p * z ∈ algebraMap R<x> S '' ↑(I.map (algebraMap R R<x>)) := by
  rw [Ideal.map, Ideal.span, Finsupp.mem_span_image_iff_total] at hz'
  obtain ⟨l, H, H'⟩ := hz'
  rw [Finsupp.total_apply] at H'
  rw [← H', mul_comm, Finsupp.sum_mul]
  have lem : ∀ {a : R}, a ∈ I → l a • algebraMap R S a * algebraMap R S p ∈
      algebraMap R<x> S '' I.map (algebraMap R R<x>) := by
    

Proof state:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (Finsupp.sum l fun i a => a • (algebraMap R S) i) = z
⊢ ∀ {a : R},
    a ∈ I →
      l a • (algebraMap R S) a * (algebraMap R S) p ∈
        ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)

Next tactic:
intro a ha
/-- This technical lemma tell us that if `C` is the conductor of `R<x>` and `I` is an ideal of `R`
  then `p * (I * S) ⊆ I * R<x>` for any `p` in `C ∩ R` -/
theorem prod_mem_ideal_map_of_mem_conductor {p : R} {z : S}
    (hp : p ∈ Ideal.comap (algebraMap R S) (conductor R x)) (hz' : z ∈ I.map (algebraMap R S)) :
    algebraMap R S p * z ∈ algebraMap R<x> S '' ↑(I.map (algebraMap R R<x>)) := by
  rw [Ideal.map, Ideal.span, Finsupp.mem_span_image_iff_total] at hz'
  obtain ⟨l, H, H'⟩ := hz'
  rw [Finsupp.total_apply] at H'
  rw [← H', mul_comm, Finsupp.sum_mul]
  have lem : ∀ {a : R}, a ∈ I → l a • algebraMap R S a * algebraMap R S p ∈
      algebraMap R<x> S '' I.map (algebraMap R R<x>) := by
    intro a ha
    

Proof state:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (Finsupp.sum l fun i a => a • (algebraMap R S) i) = z
a : R
ha : a ∈ I
⊢ l a • (algebraMap R S) a * (algebraMap R S) p ∈
    ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)

Next tactic:
rw [Algebra.id.smul_eq_mul, mul_assoc, mul_comm, mul_assoc, Set.mem_image]
/-- This technical lemma tell us that if `C` is the conductor of `R<x>` and `I` is an ideal of `R`
  then `p * (I * S) ⊆ I * R<x>` for any `p` in `C ∩ R` -/
theorem prod_mem_ideal_map_of_mem_conductor {p : R} {z : S}
    (hp : p ∈ Ideal.comap (algebraMap R S) (conductor R x)) (hz' : z ∈ I.map (algebraMap R S)) :
    algebraMap R S p * z ∈ algebraMap R<x> S '' ↑(I.map (algebraMap R R<x>)) := by
  rw [Ideal.map, Ideal.span, Finsupp.mem_span_image_iff_total] at hz'
  obtain ⟨l, H, H'⟩ := hz'
  rw [Finsupp.total_apply] at H'
  rw [← H', mul_comm, Finsupp.sum_mul]
  have lem : ∀ {a : R}, a ∈ I → l a • algebraMap R S a * algebraMap R S p ∈
      algebraMap R<x> S '' I.map (algebraMap R R<x>) := by
    intro a ha
    rw [Algebra.id.smul_eq_mul, mul_assoc, mul_comm, mul_assoc, Set.mem_image]
    

Proof state:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (Finsupp.sum l fun i a => a • (algebraMap R S) i) = z
a : R
ha : a ∈ I
⊢ ∃ x_1 ∈ ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I),
    (algebraMap (↥(adjoin R {x})) S) x_1 = (algebraMap R S) a * ((algebraMap R S) p * l a)

Next tactic:
refine Exists.intro
        (algebraMap R R<x> a * ⟨l a * algebraMap R S p,
          show l a * algebraMap R S p ∈ R<x> from ?h⟩) ?_