awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
a c : S
a_mem : a ∈ P ^ i
c_mem : c ∈ P ^ i
this : P ^ (i + 1) = span {a} ⊔ P ^ (i + 1)
rw : a ∈ span {a} ⊔ P ^ (i + 1)
⊢ a ∈ P ^ (i + 1)

Tactic:
rw [this]
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
a c : S
a_mem : a ∈ P ^ i
c_mem : c ∈ P ^ i
this : P ^ (i + 1) = span {a} ⊔ P ^ (i + 1)
⊢ 0 ∈ P ^ (i + 1)

Tactic:
simp
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
a c : S
a_mem : a ∈ P ^ i
c_mem : c ∈ P ^ i
this : P ^ (i + 1) = span {a} ⊔ P ^ (i + 1)
⊢ a + 0 = a

Tactic:
simp
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
a c : S
a_mem : a ∈ P ^ i
c_mem : c ∈ P ^ i
this : P ^ (i + 1) = span {a} ⊔ P ^ (i + 1)
⊢ a ∈ span {a} ⊔ P ^ (i + 1)

Tactic:
exact mem_sup.mpr ⟨a, mem_span_singleton_self a, 0, by simp, by simp⟩

Premises used: Iff.mpr, And.intro, Exists.intro, Submodule.mem_sup, OfNat.ofNat, Ideal.mem_span_singleton_self
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
inst_1 : IsDedekindDomain S
P : Ideal S
P_prime : P.IsPrime
hP : P ≠ ⊥
i : ℕ
a b : S
a_notMem : a ∉ P ^ (i + 1)
ab_mem : a * b ∈ P ^ (i + 1)
simp :
  ∀ (a_notMem : ¬P ^ i * P ∣ span {a}) (ab_mem : P ^ i * P ∣ HMul.hMul (α := Ideal S) (span {a}) (span {b})),
    P ∣ span {b}
⊢ b ∈ P

Tactic:
simp only [← Ideal.span_singleton_le_iff_mem, ← Ideal.dvd_iff_le, pow_succ, ←
    Ideal.span_singleton_mul_span_singleton] at a_notMem ab_mem ⊢

Premises used: Ideal.dvd_iff_le, pow_succ, Ideal.span_singleton_le_iff_mem, Ideal.span_singleton_mul_span_singleton
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
inst_1 : IsDedekindDomain S
P : Ideal S
P_prime : P.IsPrime
hP : P ≠ ⊥
i : ℕ
a b : S
a_notMem : ¬P ^ i * P ∣ span {a}
ab_mem : P ^ i * P ∣ HMul.hMul (α := Ideal S) (span {a}) (span {b})
⊢ P ∣ span {b}

Tactic:
(prime_pow_succ_dvd_mul (Ideal.prime_of_isPrime hP P_prime) ab_mem).resolve_left a_notMem

Premises used: Set.Mem, prime_pow_succ_dvd_mul, Or.resolve_left, Ideal.prime_of_isPrime
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
inst_1 : IsDedekindDomain S
P : Ideal S
P_prime : P.IsPrime
hP : P ≠ ⊥
i : ℕ
a b : S
a_notMem : ¬P ^ i * P ∣ span {a}
ab_mem : P ^ i * P ∣ HMul.hMul (α := Ideal S) (span {a}) (span {b})
⊢ P ∣ span {b}

Tactic:
exact (prime_pow_succ_dvd_mul (Ideal.prime_of_isPrime hP P_prime) ab_mem).resolve_left a_notMem

Premises used: prime_pow_succ_dvd_mul, Or.resolve_left, Ideal.prime_of_isPrime
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
a d d' e e' : S
a_notMem : a ∉ P ^ (i + 1)
e_mem : e ∈ P ^ (i + 1)
e'_mem : e' ∈ P ^ (i + 1)
h : a * d + e - (a * d' + e') ∈ P ^ (i + 1)
⊢ a * d + e - (a * d' + e') + (e' - e) ∈ P ^ (i + 1)

Tactic:
Ideal.add_mem _ h (Ideal.sub_mem _ e'_mem e_mem)

Premises used: Ideal.add_mem, Ideal.sub_mem
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
a d d' e e' : S
a_notMem : a ∉ P ^ (i + 1)
e_mem : e ∈ P ^ (i + 1)
e'_mem : e' ∈ P ^ (i + 1)
h : a * d + e - (a * d' + e') ∈ P ^ (i + 1)
h₁ : a * (d - d') = a * d + e - (a * d' + e') + (e' - e)
⊢ a * (d - d') ∈ P ^ (i + 1)

Tactic:
convert! Ideal.add_mem _ h (Ideal.sub_mem _ e'_mem e_mem) using 1

Premises used: Ideal.add_mem, Ideal.sub_mem
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
a d d' e e' : S
a_notMem : a ∉ P ^ (i + 1)
e_mem : e ∈ P ^ (i + 1)
e'_mem : e' ∈ P ^ (i + 1)
h : a * d + e - (a * d' + e') ∈ P ^ (i + 1)
⊢ a * (d - d') = a * d + e - (a * d' + e') + (e' - e)

Tactic:
ring
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
a d d' e e' : S
a_notMem : a ∉ P ^ (i + 1)
e_mem : e ∈ P ^ (i + 1)
e'_mem : e' ∈ P ^ (i + 1)
h : a * d + e - (a * d' + e') ∈ P ^ (i + 1)
⊢ a * (d - d') ∈ P ^ (i + 1)

Tactic:
have h' : a * (d - d') ∈ P ^ (i + 1) := by
    convert! Ideal.add_mem _ h (Ideal.sub_mem _ e'_mem e_mem) using 1
    ring
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
a d d' e e' : S
a_notMem : a ∉ P ^ (i + 1)
e_mem : e ∈ P ^ (i + 1)
e'_mem : e' ∈ P ^ (i + 1)
h : a * d + e - (a * d' + e') ∈ P ^ (i + 1)
h' : a * (d - d') ∈ P ^ (i + 1)
⊢ d - d' ∈ P

Tactic:
Ideal.mem_prime_of_mul_mem_pow hP a_notMem h'

Premises used: Ideal.mem_prime_of_mul_mem_pow
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
a d d' e e' : S
a_notMem : a ∉ P ^ (i + 1)
e_mem : e ∈ P ^ (i + 1)
e'_mem : e' ∈ P ^ (i + 1)
h : a * d + e - (a * d' + e') ∈ P ^ (i + 1)
h' : a * (d - d') ∈ P ^ (i + 1)
⊢ d - d' ∈ P

Tactic:
exact Ideal.mem_prime_of_mul_mem_pow hP a_notMem h'

Premises used: Ideal.mem_prime_of_mul_mem_pow
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
⊢ cardQuot.{u_1, u_1} (R := S) (M := S) (HPow.hPow (β := ℕ) P 0) =
    HPow.hPow (β := ℕ) (cardQuot.{u_1, u_1} (R := S) (M := S) P) 0

Tactic:
simp
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
ih : cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ i) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ i
⊢ cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ (i + 1)) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ (i + 1)

Tactic:
?_
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
succ :
  ∀ (i : ℕ) (ih : cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ i) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ i),
    cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ (i + 1)) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ (i + 1)
⊢ cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ i) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ i

Tactic:
induction i with
  | zero => simp
  | succ i ih => ?_

Premises used: Submodule.zero
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
ih : cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ i) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ i
⊢ P ^ (i + 1) < P ^ i

Tactic:
have : P ^ (i + 1) < P ^ i := Ideal.pow_succ_lt_pow hP i
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
ih : cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ i) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ i
this : P ^ (i + 1) < P ^ i
hquot : ↥(map (mkQ (P ^ i.succ)) (P ^ i)) ≃ S ⧸ P
rw :
  cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ (i + 1)) =
    cardQuot.{u_1, u_1} (R := S) (M := S) P * cardQuot.{u_1, u_1} (R := S) (M := S) P ^ i
rw₁ :
  cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ (i + 1)) =
    cardQuot.{u_1, u_1} (R := S) (M := S) P * cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ i)
rw₂ :
  Nat.card.{u_1} (S ⧸ P ^ i.succ) =
    cardQuot.{u_1, u_1} (R := S) (M := S) P * cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ i)
rw₃ :
  Nat.card.{u_1} ↥(map (mkQ (P ^ i.succ)) (P ^ i)) * Nat.card.{u_1} (S ⧸ P ^ i) =
    cardQuot.{u_1, u_1} (R := S) (M := S) P * cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ i)
rw₄ :
  Nat.card.{u_1} ↥(map (mkQ (P ^ i.succ)) (P ^ i)) * Nat.card.{u_1} (S ⧸ P ^ i) =
    cardQuot.{u_1, u_1} (R := S) (M := S) P * Nat.card.{u_1} (S ⧸ P ^ i)
rw₅ :
  Nat.card.{u_1} ↥(map (mkQ (P ^ i.succ)) (P ^ i)) * Nat.card.{u_1} (S ⧸ P ^ i) =
    Nat.card.{u_1} (S ⧸ P) * Nat.card.{u_1} (S ⧸ P ^ i)
rw₆ rw₇ : Nat.card.{u_1} (S ⧸ P) * Nat.card.{u_1} (S ⧸ P ^ i) = Nat.card.{u_1} (S ⧸ P) * Nat.card.{u_1} (S ⧸ P ^ i)
⊢ cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ (i + 1)) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ (i + 1)

Tactic:
rw [pow_succ' (cardQuot P), ← ih, cardQuot_apply (P ^ i.succ), ←
      card_quotient_mul_card_quotient (P ^ i) (P ^ i.succ) this.le, cardQuot_apply (P ^ i),
      cardQuot_apply P, Nat.card_congr hquot]

Premises used: LT.lt.le, Submodule.card_quotient_mul_card_quotient, Submodule.cardQuot, Nat.card_congr, HPow.hPow, Nat.succ, pow_succ', Submodule.cardQuot_apply
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
ih : cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ i) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ i
this : P ^ (i + 1) < P ^ i
succ : ↥(map (mkQ (P ^ i.succ)) (P ^ i)) ≃ S ⧸ P
⊢ cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ (i + 1)) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ (i + 1)

Tactic:
suffices hquot : map (P ^ i.succ).mkQ (P ^ i) ≃ S ⧸ P by
    rw [pow_succ' (cardQuot P), ← ih, cardQuot_apply (P ^ i.succ), ←
      card_quotient_mul_card_quotient (P ^ i) (P ^ i.succ) this.le, cardQuot_apply (P ^ i),
      cardQuot_apply P, Nat.card_congr hquot]
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
ih : cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ i) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ i
this : P ^ (i + 1) < P ^ i
succ : (a : S) → (a_mem : a ∈ P ^ i) → (a_notMem : a ∉ P ^ (i + 1)) → ↥(map (mkQ (P ^ i.succ)) (P ^ i)) ≃ S ⧸ P
⊢ ↥(map (mkQ (P ^ i.succ)) (P ^ i)) ≃ S ⧸ P

Tactic:
choose a a_mem a_notMem using SetLike.exists_of_lt this

Premises used: SetLike.exists_of_lt
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
ih : cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ i) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ i
this : P ^ (i + 1) < P ^ i
a : S
a_mem : a ∈ P ^ i
a_notMem : a ∉ P ^ (i + 1)
succ :
  (f g : (c : S) → (hc : c ∈ P ^ i) → S) →
    (hg : ∀ (c : S) (hc : c ∈ P ^ i), g c hc ∈ P ^ (i + 1)) →
      (hf : ∀ (c : S) (hc : c ∈ P ^ i), a * f c hc + g c hc = c) → ↥(map (mkQ (P ^ i.succ)) (P ^ i)) ≃ S ⧸ P
⊢ ↥(map (mkQ (P ^ i.succ)) (P ^ i)) ≃ S ⧸ P

Tactic:
choose f g hg hf using fun c (hc : c ∈ P ^ i) =>
    Ideal.exists_mul_add_mem_pow_succ hP a c a_mem a_notMem hc

Premises used: Ideal.exists_mul_add_mem_pow_succ
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
ih : cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ i) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ i
this : P ^ (i + 1) < P ^ i
a : S
a_mem : a ∈ P ^ i
a_notMem : a ∉ P ^ (i + 1)
f g : (c : S) → (hc : c ∈ P ^ i) → S
hg : ∀ (c : S) (hc : c ∈ P ^ i), g c hc ∈ P ^ (i + 1)
hf : ∀ (c : S) (hc : c ∈ P ^ i), a * f c hc + g c hc = c
succ :
  (k :
      (c' : S ⧸ P ^ i.succ) →
        (hc' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c') → S) →
    (hk_mem :
        ∀ (c' : S ⧸ P ^ i.succ)
          (hc' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c'),
          k c' hc' ∈ P ^ i) →
      (hk_eq :
          ∀ (c' : S ⧸ P ^ i.succ)
            (hc' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c'),
            (mkQ (P ^ i.succ) : (a : S) → S ⧸ P ^ i.succ) (k c' hc') = c') →
        ↥(map (mkQ (P ^ i.succ)) (P ^ i)) ≃ S ⧸ P
⊢ ↥(map (mkQ (P ^ i.succ)) (P ^ i)) ≃ S ⧸ P

Tactic:
choose k hk_mem hk_eq using fun c' (hc' : c' ∈ map (mkQ (P ^ i.succ)) (P ^ i)) =>
    Submodule.mem_map.mp hc'

Premises used: Submodule.mem_map, Iff.mp
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
ih : cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ i) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ i
this : P ^ (i + 1) < P ^ i
a : S
a_mem : a ∈ P ^ i
a_notMem : a ∉ P ^ (i + 1)
f g : (c : S) → (hc : c ∈ P ^ i) → S
hg : ∀ (c : S) (hc : c ∈ P ^ i), g c hc ∈ P ^ (i + 1)
hf : ∀ (c : S) (hc : c ∈ P ^ i), a * f c hc + g c hc = c
k :
  (c' : S ⧸ P ^ i.succ) →
    (hc' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c') → S
hk_mem :
  ∀ (c' : S ⧸ P ^ i.succ)
    (hc' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c'), k c' hc' ∈ P ^ i
hk_eq :
  ∀ (c' : S ⧸ P ^ i.succ)
    (hc' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c'),
    (mkQ (P ^ i.succ) : (a : S) → S ⧸ P ^ i.succ) (k c' hc') = c'
succ :
  Function.Injective (β := S ⧸ P) fun (c' : ↥(map (mkQ (P ^ i.succ)) (P ^ i))) ↦
    Quotient.mk'' (f (k (↑c' : S ⧸ P ^ i.succ) ⋯) ⋯)
succ₁ :
  Function.Surjective (β := S ⧸ P) fun (c' : ↥(map (mkQ (P ^ i.succ)) (P ^ i))) ↦
    Quotient.mk'' (f (k (↑c' : S ⧸ P ^ i.succ) ⋯) ⋯)
⊢ ↥(map (mkQ (P ^ i.succ)) (P ^ i)) ≃ S ⧸ P

Tactic:
refine Equiv.ofBijective (fun c' => Quotient.mk'' (f (k c' c'.prop) (hk_mem c' c'.prop))) ⟨?_, ?_⟩
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
ih : cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ i) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ i
this : P ^ (i + 1) < P ^ i
a : S
a_mem : a ∈ P ^ i
a_notMem : a ∉ P ^ (i + 1)
f g : (c : S) → (hc : c ∈ P ^ i) → S
hg : ∀ (c : S) (hc : c ∈ P ^ i), g c hc ∈ P ^ (i + 1)
hf : ∀ (c : S) (hc : c ∈ P ^ i), a * f c hc + g c hc = c
k :
  (c' : S ⧸ P ^ i.succ) →
    (hc' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c') → S
hk_mem :
  ∀ (c' : S ⧸ P ^ i.succ)
    (hc' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c'), k c' hc' ∈ P ^ i
hk_eq :
  ∀ (c' : S ⧸ P ^ i.succ)
    (hc' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c'),
    (mkQ (P ^ i.succ) : (a : S) → S ⧸ P ^ i.succ) (k c' hc') = c'
c₁' : S ⧸ P ^ i.succ
hc₁' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c₁'
c₂' : S ⧸ P ^ i.succ
hc₂' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c₂'
h :
  Eq.{u_1 + 1} (α := S ⧸ P)
    ((fun (c' : ↥(map (mkQ (P ^ i.succ)) (P ^ i))) ↦
        (Quotient.mk'' (f (k (↑c' : S ⧸ P ^ i.succ) ⋯) ⋯) : Quotient (α := S) (quotientRel (R := S) P)))
      (Subtype.mk (p := fun (x : S ⧸ P ^ i.succ) ↦
        Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) x) c₁' hc₁'))
    ((fun (c' : ↥(map (mkQ (P ^ i.succ)) (P ^ i))) ↦
        (Quotient.mk'' (f (k (↑c' : S ⧸ P ^ i.succ) ⋯) ⋯) : Quotient (α := S) (quotientRel (R := S) P)))
      (Subtype.mk (p := fun (x : S ⧸ P ^ i.succ) ↦
        Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) x) c₂' hc₂'))
succ : a * f (k c₁' hc₁') ⋯ + g (k c₁' hc₁') ⋯ - (a * f (k c₂' hc₂') ⋯ + g (k c₂' hc₂') ⋯) ∈ P ^ i.succ
⊢ Subtype.mk (p := fun (x : S ⧸ P ^ i.succ) ↦
      Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) x) c₁' hc₁' =
    Subtype.mk (p := fun (x : S ⧸ P ^ i.succ) ↦
      Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) x) c₂' hc₂'

Tactic:
rw [Subtype.mk_eq_mk, ← hk_eq _ hc₁', ← hk_eq _ hc₂', mkQ_apply, mkQ_apply,
      Submodule.Quotient.eq, ← hf _ (hk_mem _ hc₁'), ← hf _ (hk_mem _ hc₂')]

Premises used: Submodule.Quotient.eq, Submodule.mkQ_apply, Subtype.mk_eq_mk
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
ih : cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ i) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ i
this : P ^ (i + 1) < P ^ i
a : S
a_mem : a ∈ P ^ i
a_notMem : a ∉ P ^ (i + 1)
f g : (c : S) → (hc : c ∈ P ^ i) → S
hg : ∀ (c : S) (hc : c ∈ P ^ i), g c hc ∈ P ^ (i + 1)
hf : ∀ (c : S) (hc : c ∈ P ^ i), a * f c hc + g c hc = c
k :
  (c' : S ⧸ P ^ i.succ) →
    (hc' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c') → S
hk_mem :
  ∀ (c' : S ⧸ P ^ i.succ)
    (hc' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c'), k c' hc' ∈ P ^ i
hk_eq :
  ∀ (c' : S ⧸ P ^ i.succ)
    (hc' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c'),
    (mkQ (P ^ i.succ) : (a : S) → S ⧸ P ^ i.succ) (k c' hc') = c'
c₁' : S ⧸ P ^ i.succ
hc₁' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c₁'
c₂' : S ⧸ P ^ i.succ
hc₂' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c₂'
h :
  Eq.{u_1 + 1} (α := S ⧸ P)
    ((fun (c' : ↥(map (mkQ (P ^ i.succ)) (P ^ i))) ↦
        (Quotient.mk'' (f (k (↑c' : S ⧸ P ^ i.succ) ⋯) ⋯) : Quotient (α := S) (quotientRel (R := S) P)))
      (Subtype.mk (p := fun (x : S ⧸ P ^ i.succ) ↦
        Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) x) c₁' hc₁'))
    ((fun (c' : ↥(map (mkQ (P ^ i.succ)) (P ^ i))) ↦
        (Quotient.mk'' (f (k (↑c' : S ⧸ P ^ i.succ) ⋯) ⋯) : Quotient (α := S) (quotientRel (R := S) P)))
      (Subtype.mk (p := fun (x : S ⧸ P ^ i.succ) ↦
        Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) x) c₂' hc₂'))
⊢ a * f (k c₁' hc₁') ⋯ + g (k c₁' hc₁') ⋯ - (a * f (k c₂' hc₂') ⋯ + g (k c₂' hc₂') ⋯) ∈ P ^ i.succ

Tactic:
Ideal.mul_add_mem_pow_succ_inj _ _ _ _ _ _ a_mem (hg _ _) (hg _ _) ?_

Premises used: Set.Mem, Ideal.mul_add_mem_pow_succ_inj
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
ih : cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ i) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ i
this : P ^ (i + 1) < P ^ i
a : S
a_mem : a ∈ P ^ i
a_notMem : a ∉ P ^ (i + 1)
f g : (c : S) → (hc : c ∈ P ^ i) → S
hg : ∀ (c : S) (hc : c ∈ P ^ i), g c hc ∈ P ^ (i + 1)
hf : ∀ (c : S) (hc : c ∈ P ^ i), a * f c hc + g c hc = c
k :
  (c' : S ⧸ P ^ i.succ) →
    (hc' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c') → S
hk_mem :
  ∀ (c' : S ⧸ P ^ i.succ)
    (hc' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c'), k c' hc' ∈ P ^ i
hk_eq :
  ∀ (c' : S ⧸ P ^ i.succ)
    (hc' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c'),
    (mkQ (P ^ i.succ) : (a : S) → S ⧸ P ^ i.succ) (k c' hc') = c'
c₁' : S ⧸ P ^ i.succ
hc₁' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c₁'
c₂' : S ⧸ P ^ i.succ
hc₂' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c₂'
h :
  Eq.{u_1 + 1} (α := S ⧸ P)
    ((fun (c' : ↥(map (mkQ (P ^ i.succ)) (P ^ i))) ↦
        (Quotient.mk'' (f (k (↑c' : S ⧸ P ^ i.succ) ⋯) ⋯) : Quotient (α := S) (quotientRel (R := S) P)))
      (Subtype.mk (p := fun (x : S ⧸ P ^ i.succ) ↦
        Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) x) c₁' hc₁'))
    ((fun (c' : ↥(map (mkQ (P ^ i.succ)) (P ^ i))) ↦
        (Quotient.mk'' (f (k (↑c' : S ⧸ P ^ i.succ) ⋯) ⋯) : Quotient (α := S) (quotientRel (R := S) P)))
      (Subtype.mk (p := fun (x : S ⧸ P ^ i.succ) ↦
        Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) x) c₂' hc₂'))
succ : f (k c₁' hc₁') ⋯ - f (k c₂' hc₂') ⋯ ∈ P
⊢ a * f (k c₁' hc₁') ⋯ + g (k c₁' hc₁') ⋯ - (a * f (k c₂' hc₂') ⋯ + g (k c₂' hc₂') ⋯) ∈ P ^ i.succ

Tactic:
refine Ideal.mul_add_mem_pow_succ_inj _ _ _ _ _ _ a_mem (hg _ _) (hg _ _) ?_

Premises used: Ideal.mul_add_mem_pow_succ_inj
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
ih : cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ i) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ i
this : P ^ (i + 1) < P ^ i
a : S
a_mem : a ∈ P ^ i
a_notMem : a ∉ P ^ (i + 1)
f g : (c : S) → (hc : c ∈ P ^ i) → S
hg : ∀ (c : S) (hc : c ∈ P ^ i), g c hc ∈ P ^ (i + 1)
hf : ∀ (c : S) (hc : c ∈ P ^ i), a * f c hc + g c hc = c
k :
  (c' : S ⧸ P ^ i.succ) →
    (hc' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c') → S
hk_mem :
  ∀ (c' : S ⧸ P ^ i.succ)
    (hc' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c'), k c' hc' ∈ P ^ i
hk_eq :
  ∀ (c' : S ⧸ P ^ i.succ)
    (hc' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c'),
    (mkQ (P ^ i.succ) : (a : S) → S ⧸ P ^ i.succ) (k c' hc') = c'
c₁' : S ⧸ P ^ i.succ
hc₁' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c₁'
c₂' : S ⧸ P ^ i.succ
hc₂' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c₂'
h :
  Eq.{u_1 + 1} (α := S ⧸ P)
    ((fun (c' : ↥(map (mkQ (P ^ i.succ)) (P ^ i))) ↦
        (Quotient.mk'' (f (k (↑c' : S ⧸ P ^ i.succ) ⋯) ⋯) : Quotient (α := S) (quotientRel (R := S) P)))
      (Subtype.mk (p := fun (x : S ⧸ P ^ i.succ) ↦
        Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) x) c₁' hc₁'))
    ((fun (c' : ↥(map (mkQ (P ^ i.succ)) (P ^ i))) ↦
        (Quotient.mk'' (f (k (↑c' : S ⧸ P ^ i.succ) ⋯) ⋯) : Quotient (α := S) (quotientRel (R := S) P)))
      (Subtype.mk (p := fun (x : S ⧸ P ^ i.succ) ↦
        Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) x) c₂' hc₂'))
⊢ f (k c₁' hc₁') ⋯ - f (k c₂' hc₂') ⋯ ∈ P

Tactic:
simpa only [Submodule.Quotient.mk''_eq_mk, Submodule.Quotient.mk''_eq_mk,
      Submodule.Quotient.eq] using h

Premises used: Submodule.Quotient.eq, Submodule.Quotient.mk''_eq_mk
-- Module: Mathlib.RingTheory.Ideal.Norm.AbsNorm
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
ih : cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ i) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ i
this : P ^ (i + 1) < P ^ i
a : S
a_mem : a ∈ P ^ i
a_notMem : a ∉ P ^ (i + 1)
f g : (c : S) → (hc : c ∈ P ^ i) → S
hg : ∀ (c : S) (hc : c ∈ P ^ i), g c hc ∈ P ^ (i + 1)
hf : ∀ (c : S) (hc : c ∈ P ^ i), a * f c hc + g c hc = c
k :
  (c' : S ⧸ P ^ i.succ) →
    (hc' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c') → S
hk_mem :
  ∀ (c' : S ⧸ P ^ i.succ)
    (hc' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c'), k c' hc' ∈ P ^ i
hk_eq :
  ∀ (c' : S ⧸ P ^ i.succ)
    (hc' : Membership.mem (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i)) c'),
    (mkQ (P ^ i.succ) : (a : S) → S ⧸ P ^ i.succ) (k c' hc') = c'
d' : S ⧸ P
succ :
  ∀ (d : S),
    ∃ (a : ↥(map (mkQ (P ^ i.succ)) (P ^ i))),
      Eq (α := S ⧸ P)
        ((fun (c' : ↥(map (mkQ (P ^ i.succ)) (P ^ i))) ↦
            (Quotient.mk'' (f (k (↑c' : S ⧸ P ^ i.succ) ⋯) ⋯) : Quotient (α := S) (quotientRel (R := S) P)))
          a)
        ⟦d⟧
⊢ ∃ (a : ↥(map (mkQ (P ^ i.succ)) (P ^ i))),
    (fun (c' : ↥(map (mkQ (P ^ i.succ)) (P ^ i))) ↦
          (Quotient.mk'' (f (k (↑c' : S ⧸ P ^ i.succ) ⋯) ⋯) : Quotient (α := S) (quotientRel (R := S) P)))
        a =
      d'

Tactic:
induction d' using Quotient.inductionOn with | _ d