awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.IntegralClosure.Algebra.Ideal
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
I : Ideal R
x : S
hx :
  IsIntegral.{u_1, u_2}
    (↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))
    (HMul.hMul (α := S[X]) (β := S[X]) ((C : (a : S) → S[X]) x : S[X]) X)
h : Nontrivial R
inr :
  ∀
    (p :
      (↥(Algebra.adjoin R
            {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))[X])
    (hp : p.Monic)
    (e :
      eval₂
          (algebraMap
            (↥(Algebra.adjoin R
                {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))
            S[X])
          (HMul.hMul (α := S[X]) (β := S[X]) ((C : (a : S) → S[X]) x : S[X]) X) p =
        0),
    ∃ (p : R[X]),
      p.Monic ∧
        Eq.{u_2 + 1} (α := S) ((aeval x : (a : R[X]) → S) p : S) (0 : S) ∧ ∀ (i : ℕ), p.coeff i ∈ I ^ (p.natDegree - i)
⊢ ∃ (p : R[X]),
    p.Monic ∧
      Eq.{u_2 + 1} (α := S) ((aeval x : (a : R[X]) → S) p : S) (0 : S) ∧ ∀ (i : ℕ), p.coeff i ∈ I ^ (p.natDegree - i)

Tactic:
obtain ⟨p, hp, e⟩ := hx
-- Module: Mathlib.RingTheory.IntegralClosure.Algebra.Ideal
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
I : Ideal R
x : S
h : Nontrivial R
p :
  (↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))[X]
hp : p.Monic
e :
  eval₂
      (algebraMap
        (↥(Algebra.adjoin R
            {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))
        S[X])
      (HMul.hMul (α := S[X]) (β := S[X]) ((C : (a : S) → S[X]) x : S[X]) X) p =
    0
q : R[X] :=
  ∑ i ∈ Finset.range (p.natDegree + 1),
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) ((↑(p.coeff i) : R[X]).coeff (p.natDegree - i)) : R[X])
      (HPow.hPow (α := R[X]) X i)
⊢ q.natDegree = p.natDegree

Tactic:
natDegree_eq_of_le_of_coeff_ne_zero (natDegree_sum_le_of_forall_le _ _ ?_) ?_

Premises used: Polynomial.natDegree_sum_le_of_forall_le, Polynomial.natDegree_eq_of_le_of_coeff_ne_zero
-- Module: Mathlib.RingTheory.IntegralClosure.Algebra.Ideal
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
I : Ideal R
x : S
h : Nontrivial R
p :
  (↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))[X]
hp : p.Monic
e :
  eval₂
      (algebraMap
        (↥(Algebra.adjoin R
            {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))
        S[X])
      (HMul.hMul (α := S[X]) (β := S[X]) ((C : (a : S) → S[X]) x : S[X]) X) p =
    0
q : R[X] :=
  ∑ i ∈ Finset.range (p.natDegree + 1),
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) ((↑(p.coeff i) : R[X]).coeff (p.natDegree - i)) : R[X])
      (HPow.hPow (α := R[X]) X i)
refine_1 :
  ∀ i ∈ Finset.range (p.natDegree + 1),
    (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) ((↑(p.coeff i) : R[X]).coeff (p.natDegree - i)) : R[X])
          (HPow.hPow (α := R[X]) X i)).natDegree ≤
      p.natDegree
refine_2 : q.coeff p.natDegree ≠ 0
⊢ q.natDegree = p.natDegree

Tactic:
refine natDegree_eq_of_le_of_coeff_ne_zero (natDegree_sum_le_of_forall_le _ _ ?_) ?_

Premises used: Polynomial.natDegree_sum_le_of_forall_le, Polynomial.natDegree_eq_of_le_of_coeff_ne_zero
-- Module: Mathlib.RingTheory.IntegralClosure.Algebra.Ideal
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
I : Ideal R
x : S
h : Nontrivial R
p :
  (↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))[X]
hp : p.Monic
e :
  eval₂
      (algebraMap
        (↥(Algebra.adjoin R
            {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))
        S[X])
      (HMul.hMul (α := S[X]) (β := S[X]) ((C : (a : S) → S[X]) x : S[X]) X) p =
    0
q : R[X] :=
  ∑ i ∈ Finset.range (p.natDegree + 1),
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) ((↑(p.coeff i) : R[X]).coeff (p.natDegree - i)) : R[X])
      (HPow.hPow (α := R[X]) X i)
i : ℕ
hi : i ∈ Finset.range (p.natDegree + 1)
⊢ i ≤ p.natDegree

Tactic:
simpa [Nat.lt_succ_iff] using hi

Premises used: Nat.lt_succ_iff
-- Module: Mathlib.RingTheory.IntegralClosure.Algebra.Ideal
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
I : Ideal R
x : S
h : Nontrivial R
p :
  (↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))[X]
hp : p.Monic
e :
  eval₂
      (algebraMap
        (↥(Algebra.adjoin R
            {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))
        S[X])
      (HMul.hMul (α := S[X]) (β := S[X]) ((C : (a : S) → S[X]) x : S[X]) X) p =
    0
q : R[X] :=
  ∑ i ∈ Finset.range (p.natDegree + 1),
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) ((↑(p.coeff i) : R[X]).coeff (p.natDegree - i)) : R[X])
      (HPow.hPow (α := R[X]) X i)
⊢ ∀ i ∈ Finset.range (p.natDegree + 1),
    (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) ((↑(p.coeff i) : R[X]).coeff (p.natDegree - i)) : R[X])
          (HPow.hPow (α := R[X]) X i)).natDegree ≤
      p.natDegree

Tactic:
exact fun i hi ↦ (natDegree_C_mul_X_pow_le _ _).trans (by simpa [Nat.lt_succ_iff] using hi)

Premises used: LE.le.trans, Polynomial.natDegree_C_mul_X_pow_le
-- Module: Mathlib.RingTheory.IntegralClosure.Algebra.Ideal
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
I : Ideal R
x : S
h : Nontrivial R
p :
  (↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))[X]
hp : p.Monic
e :
  eval₂
      (algebraMap
        (↥(Algebra.adjoin R
            {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))
        S[X])
      (HMul.hMul (α := S[X]) (β := S[X]) ((C : (a : S) → S[X]) x : S[X]) X) p =
    0
q : R[X] :=
  ∑ i ∈ Finset.range (p.natDegree + 1),
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) ((↑(p.coeff i) : R[X]).coeff (p.natDegree - i)) : R[X])
      (HPow.hPow (α := R[X]) X i)
⊢ q.coeff p.natDegree ≠ 0

Tactic:
simp [q, hp]
-- Module: Mathlib.RingTheory.IntegralClosure.Algebra.Ideal
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
I : Ideal R
x : S
h : Nontrivial R
p :
  (↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))[X]
hp : p.Monic
e :
  eval₂
      (algebraMap
        (↥(Algebra.adjoin R
            {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))
        S[X])
      (HMul.hMul (α := S[X]) (β := S[X]) ((C : (a : S) → S[X]) x : S[X]) X) p =
    0
q : R[X] :=
  ∑ i ∈ Finset.range (p.natDegree + 1),
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) ((↑(p.coeff i) : R[X]).coeff (p.natDegree - i)) : R[X])
      (HPow.hPow (α := R[X]) X i)
⊢ q.natDegree = p.natDegree

Tactic:
have hq : q.natDegree = p.natDegree := by
    refine natDegree_eq_of_le_of_coeff_ne_zero (natDegree_sum_le_of_forall_le _ _ ?_) ?_
    · exact fun i hi ↦ (natDegree_C_mul_X_pow_le _ _).trans (by simpa [Nat.lt_succ_iff] using hi)
    · simp [q, hp]
-- Module: Mathlib.RingTheory.IntegralClosure.Algebra.Ideal
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
I : Ideal R
x : S
h : Nontrivial R
p :
  (↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))[X]
hp : p.Monic
e :
  eval₂
      (algebraMap
        (↥(Algebra.adjoin R
            {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))
        S[X])
      (HMul.hMul (α := S[X]) (β := S[X]) ((C : (a : S) → S[X]) x : S[X]) X) p =
    0
q : R[X] :=
  ∑ i ∈ Finset.range (p.natDegree + 1),
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) ((↑(p.coeff i) : R[X]).coeff (p.natDegree - i)) : R[X])
      (HPow.hPow (α := R[X]) X i)
hq : q.natDegree = p.natDegree
⊢ ∃ (p : R[X]),
    p.Monic ∧
      Eq.{u_2 + 1} (α := S) ((aeval x : (a : R[X]) → S) p : S) (0 : S) ∧ ∀ (i : ℕ), p.coeff i ∈ I ^ (p.natDegree - i)

Tactic:
⟨q, ?_, ?_, ?_⟩

Premises used: And.intro, Exists.intro
-- Module: Mathlib.RingTheory.IntegralClosure.Algebra.Ideal
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
I : Ideal R
x : S
h : Nontrivial R
p :
  (↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))[X]
hp : p.Monic
e :
  eval₂
      (algebraMap
        (↥(Algebra.adjoin R
            {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))
        S[X])
      (HMul.hMul (α := S[X]) (β := S[X]) ((C : (a : S) → S[X]) x : S[X]) X) p =
    0
q : R[X] :=
  ∑ i ∈ Finset.range (p.natDegree + 1),
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) ((↑(p.coeff i) : R[X]).coeff (p.natDegree - i)) : R[X])
      (HPow.hPow (α := R[X]) X i)
hq : q.natDegree = p.natDegree
inr : q.Monic
inr₁ : Eq.{u_2 + 1} (α := S) ((aeval x : (a : R[X]) → S) q : S) (0 : S)
inr₂ : ∀ (i : ℕ), q.coeff i ∈ I ^ (q.natDegree - i)
⊢ ∃ (p : R[X]),
    p.Monic ∧
      Eq.{u_2 + 1} (α := S) ((aeval x : (a : R[X]) → S) p : S) (0 : S) ∧ ∀ (i : ℕ), p.coeff i ∈ I ^ (p.natDegree - i)

Tactic:
refine ⟨q, ?_, ?_, ?_⟩
-- Module: Mathlib.RingTheory.IntegralClosure.Algebra.Ideal
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
I : Ideal R
x : S
h : Nontrivial R
p :
  (↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))[X]
hp : p.Monic
e :
  eval₂
      (algebraMap
        (↥(Algebra.adjoin R
            {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))
        S[X])
      (HMul.hMul (α := S[X]) (β := S[X]) ((C : (a : S) → S[X]) x : S[X]) X) p =
    0
q : R[X] :=
  ∑ i ∈ Finset.range (p.natDegree + 1),
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) ((↑(p.coeff i) : R[X]).coeff (p.natDegree - i)) : R[X])
      (HPow.hPow (α := R[X]) X i)
hq : q.natDegree = p.natDegree
⊢ q.coeff p.natDegree = 1

Tactic:
simp [q, hp]
-- Module: Mathlib.RingTheory.IntegralClosure.Algebra.Ideal
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
I : Ideal R
x : S
h : Nontrivial R
p :
  (↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))[X]
hp : p.Monic
e :
  eval₂
      (algebraMap
        (↥(Algebra.adjoin R
            {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))
        S[X])
      (HMul.hMul (α := S[X]) (β := S[X]) ((C : (a : S) → S[X]) x : S[X]) X) p =
    0
q : R[X] :=
  ∑ i ∈ Finset.range (p.natDegree + 1),
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) ((↑(p.coeff i) : R[X]).coeff (p.natDegree - i)) : R[X])
      (HPow.hPow (α := R[X]) X i)
hq : q.natDegree = p.natDegree
⊢ q.Monic

Tactic:
simpa [← hq] using show q.coeff p.natDegree = 1 by simp [q, hp]
-- Module: Mathlib.RingTheory.IntegralClosure.Algebra.Ideal
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
I : Ideal R
x : S
h : Nontrivial R
p :
  (↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))[X]
hp : p.Monic
e :
  eval₂
      (algebraMap
        (↥(Algebra.adjoin R
            {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))
        S[X])
      (HMul.hMul (α := S[X]) (β := S[X]) ((C : (a : S) → S[X]) x : S[X]) X) p =
    0
q : R[X] :=
  ∑ i ∈ Finset.range (p.natDegree + 1),
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) ((↑(p.coeff i) : R[X]).coeff (p.natDegree - i)) : R[X])
      (HPow.hPow (α := R[X]) X i)
hq : q.natDegree = p.natDegree
⊢ (eval₂
          (algebraMap
            (↥(Algebra.adjoin R
                {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))
            S[X])
          (HMul.hMul (α := S[X]) (β := S[X]) ((C : (a : S) → S[X]) x : S[X]) X) p).coeff
      p.natDegree =
    coeff 0 p.natDegree

Tactic:
replace e := congr(($e).coeff p.natDegree)
-- Module: Mathlib.RingTheory.IntegralClosure.Algebra.Ideal
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
I : Ideal R
x : S
h : Nontrivial R
p :
  (↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))[X]
hp : p.Monic
q : R[X] :=
  ∑ i ∈ Finset.range (p.natDegree + 1),
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) ((↑(p.coeff i) : R[X]).coeff (p.natDegree - i)) : R[X])
      (HPow.hPow (α := R[X]) X i)
hq : q.natDegree = p.natDegree
e :
  (eval₂
          (algebraMap
            (↥(Algebra.adjoin R
                {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))
            S[X])
          (HMul.hMul (α := S[X]) (β := S[X]) ((C : (a : S) → S[X]) x : S[X]) X) p).coeff
      p.natDegree =
    coeff 0 p.natDegree
⊢ Eq.{u_2 + 1} (α := S)
    (∑ b ∈ Finset.range (p.natDegree + 1),
      (HMul.hMul (α := S[X])
            ((algebraMap
                    (↥(Algebra.adjoin R
                        {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))
                    S[X] :
                  (a : ↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, (C : (a : R) → R[X]) r * X = x})) → S[X])
                (p.coeff b) :
              S[X])
            (HMul.hMul (α := S[X]) (β := S[X]) ((C : (a : S) → S[X]) x : S[X]) X ^ b)).coeff
        p.natDegree)
    0

Tactic:
simp only [eval₂_eq_sum_range, finsetSum_coeff, coeff_zero] at e

Premises used: Polynomial.finsetSum_coeff, Polynomial.coeff_zero, Polynomial.eval₂_eq_sum_range
-- Module: Mathlib.RingTheory.IntegralClosure.Algebra.Ideal
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
I : Ideal R
x : S
h : Nontrivial R
p :
  (↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))[X]
hp : p.Monic
q : R[X] :=
  ∑ i ∈ Finset.range (p.natDegree + 1),
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) ((↑(p.coeff i) : R[X]).coeff (p.natDegree - i)) : R[X])
      (HPow.hPow (α := R[X]) X i)
hq : q.natDegree = p.natDegree
e :
  Eq.{u_2 + 1} (α := S)
    (∑ b ∈ Finset.range (p.natDegree + 1),
      (HMul.hMul (α := S[X])
            ((algebraMap
                    (↥(Algebra.adjoin R
                        {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))
                    S[X] :
                  (a : ↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, (C : (a : R) → R[X]) r * X = x})) → S[X])
                (p.coeff b) :
              S[X])
            (HMul.hMul (α := S[X]) (β := S[X]) ((C : (a : S) → S[X]) x : S[X]) X ^ b)).coeff
        p.natDegree)
    0
inr :
  Eq.{u_2 + 1} (α := S)
    (∑ x_1 ∈ Finset.range (p.natDegree + 1),
      HMul.hMul (α := S) ((algebraMap R S : (a : R) → S) ((↑(p.coeff x_1) : R[X]).coeff (p.natDegree - x_1)) : S)
        (x ^ x_1))
    0
⊢ Eq.{u_2 + 1} (α := S) ((aeval x : (a : R[X]) → S) q : S) (0 : S)

Tactic:
simp only [q, map_sum, map_mul, aeval_C, map_pow, aeval_X]

Premises used: map_pow, map_sum, map_mul, Polynomial.aeval_C, Polynomial.aeval_X
-- Module: Mathlib.RingTheory.IntegralClosure.Algebra.Ideal
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
I : Ideal R
x : S
h : Nontrivial R
p :
  (↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))[X]
hp : p.Monic
q : R[X] :=
  ∑ i ∈ Finset.range (p.natDegree + 1),
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) ((↑(p.coeff i) : R[X]).coeff (p.natDegree - i)) : R[X])
      (HPow.hPow (α := R[X]) X i)
hq : q.natDegree = p.natDegree
e :
  Eq.{u_2 + 1} (α := S)
    (∑ b ∈ Finset.range (p.natDegree + 1),
      (HMul.hMul (α := S[X])
            ((algebraMap
                    (↥(Algebra.adjoin R
                        {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))
                    S[X] :
                  (a : ↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, (C : (a : R) → R[X]) r * X = x})) → S[X])
                (p.coeff b) :
              S[X])
            (HMul.hMul (α := S[X]) (β := S[X]) ((C : (a : S) → S[X]) x : S[X]) X ^ b)).coeff
        p.natDegree)
    0
⊢ Eq.{u_2 + 1} (α := S)
    (∑ x_1 ∈ Finset.range (p.natDegree + 1),
      HMul.hMul (α := S) ((algebraMap R S : (a : R) → S) ((↑(p.coeff x_1) : R[X]).coeff (p.natDegree - x_1)) : S)
        (x ^ x_1))
    0

Tactic:
(Finset.sum_congr rfl fun i hi ↦ ?_).trans e

Premises used: Finset.sum_congr, rfl, Eq.trans
-- Module: Mathlib.RingTheory.IntegralClosure.Algebra.Ideal
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
I : Ideal R
x : S
h : Nontrivial R
p :
  (↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))[X]
hp : p.Monic
q : R[X] :=
  ∑ i ∈ Finset.range (p.natDegree + 1),
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) ((↑(p.coeff i) : R[X]).coeff (p.natDegree - i)) : R[X])
      (HPow.hPow (α := R[X]) X i)
hq : q.natDegree = p.natDegree
e :
  Eq.{u_2 + 1} (α := S)
    (∑ b ∈ Finset.range (p.natDegree + 1),
      (HMul.hMul (α := S[X])
            ((algebraMap
                    (↥(Algebra.adjoin R
                        {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))
                    S[X] :
                  (a : ↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, (C : (a : R) → R[X]) r * X = x})) → S[X])
                (p.coeff b) :
              S[X])
            (HMul.hMul (α := S[X]) (β := S[X]) ((C : (a : S) → S[X]) x : S[X]) X ^ b)).coeff
        p.natDegree)
    0
inr :
  ∀ (i : ℕ) (hi : i ∈ Finset.range (p.natDegree + 1)),
    HMul.hMul (α := S) ((algebraMap R S : (a : R) → S) ((↑(p.coeff i) : R[X]).coeff (p.natDegree - i)) : S) (x ^ i) =
      (HMul.hMul (α := S[X])
            ((algebraMap
                    (↥(Algebra.adjoin R
                        {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))
                    S[X] :
                  (a : ↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, (C : (a : R) → R[X]) r * X = x})) → S[X])
                (p.coeff i) :
              S[X])
            (HMul.hMul (α := S[X]) (β := S[X]) ((C : (a : S) → S[X]) x : S[X]) X ^ i)).coeff
        p.natDegree
⊢ Eq.{u_2 + 1} (α := S)
    (∑ x_1 ∈ Finset.range (p.natDegree + 1),
      HMul.hMul (α := S) ((algebraMap R S : (a : R) → S) ((↑(p.coeff x_1) : R[X]).coeff (p.natDegree - x_1)) : S)
        (x ^ x_1))
    0

Tactic:
refine (Finset.sum_congr rfl fun i hi ↦ ?_).trans e

Premises used: Finset.sum_congr, rfl, Eq.trans
-- Module: Mathlib.RingTheory.IntegralClosure.Algebra.Ideal
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
I : Ideal R
x : S
h : Nontrivial R
p :
  (↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))[X]
hp : p.Monic
q : R[X] :=
  ∑ i ∈ Finset.range (p.natDegree + 1),
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) ((↑(p.coeff i) : R[X]).coeff (p.natDegree - i)) : R[X])
      (HPow.hPow (α := R[X]) X i)
hq : q.natDegree = p.natDegree
e :
  Eq.{u_2 + 1} (α := S)
    (∑ b ∈ Finset.range (p.natDegree + 1),
      (HMul.hMul (α := S[X])
            ((algebraMap
                    (↥(Algebra.adjoin R
                        {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))
                    S[X] :
                  (a : ↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, (C : (a : R) → R[X]) r * X = x})) → S[X])
                (p.coeff b) :
              S[X])
            (HMul.hMul (α := S[X]) (β := S[X]) ((C : (a : S) → S[X]) x : S[X]) X ^ b)).coeff
        p.natDegree)
    0
i : ℕ
hi : i ∈ Finset.range (p.natDegree + 1)
⊢ i ≤ p.natDegree

Tactic:
simp only [Finset.mem_range, Nat.lt_succ_iff] at hi

Premises used: Finset.mem_range, Nat.lt_succ_iff
-- Module: Mathlib.RingTheory.IntegralClosure.Algebra.Ideal
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
I : Ideal R
x : S
h : Nontrivial R
p :
  (↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))[X]
hp : p.Monic
q : R[X] :=
  ∑ i ∈ Finset.range (p.natDegree + 1),
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) ((↑(p.coeff i) : R[X]).coeff (p.natDegree - i)) : R[X])
      (HPow.hPow (α := R[X]) X i)
hq : q.natDegree = p.natDegree
e :
  Eq.{u_2 + 1} (α := S)
    (∑ b ∈ Finset.range (p.natDegree + 1),
      (HMul.hMul (α := S[X])
            ((algebraMap
                    (↥(Algebra.adjoin R
                        {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))
                    S[X] :
                  (a : ↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, (C : (a : R) → R[X]) r * X = x})) → S[X])
                (p.coeff b) :
              S[X])
            (HMul.hMul (α := S[X]) (β := S[X]) ((C : (a : S) → S[X]) x : S[X]) X ^ b)).coeff
        p.natDegree)
    0
i : ℕ
hi : i ≤ p.natDegree
inr :
  HMul.hMul (β := S) (x ^ i) ((algebraMap R S : (a : R) → S) ((↑(p.coeff i) : R[X]).coeff (p.natDegree - i)) : S) =
    x ^ i *
      coeff
        ((algebraMap
              (↥(Algebra.adjoin R
                  {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))
              S[X] :
            (a : ↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, (C : (a : R) → R[X]) r * X = x})) → S[X])
          (p.coeff i))
        (p.natDegree - i)
⊢ HMul.hMul (α := S) ((algebraMap R S : (a : R) → S) ((↑(p.coeff i) : R[X]).coeff (p.natDegree - i)) : S) (x ^ i) =
    (HMul.hMul (α := S[X])
          ((algebraMap
                  (↥(Algebra.adjoin R
                      {x : R[X] | ∃ r ∈ I, HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) r : R[X]) X = x}))
                  S[X] :
                (a : ↥(Algebra.adjoin R {x : R[X] | ∃ r ∈ I, (C : (a : R) → R[X]) r * X = x})) → S[X])
              (p.coeff i) :
            S[X])
          (HMul.hMul (α := S[X]) (β := S[X]) ((C : (a : S) → S[X]) x : S[X]) X ^ i)).coeff
      p.natDegree

Tactic:
rw [mul_pow, mul_left_comm, ← map_pow, coeff_C_mul, coeff_mul_X_pow', if_pos hi, mul_comm]

Premises used: Polynomial.coeff_C_mul, map_pow, mul_left_comm, if_pos, mul_comm, Polynomial.coeff_mul_X_pow', mul_pow