awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
P : Ideal S
P_prime : P.IsPrime
inst_1 : IsDedekindDomain S
hP : P ≠ ⊥
i : ℕ
⊢ cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ i) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ i
have 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)
Tactic: induction i with
  | zero => simp
  | succ i ih => ?_
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (hypothesis)
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)
have this : P ^ (i + 1) < P ^ i
Tactic: have : P ^ (i + 1) < P ^ i := Ideal.pow_succ_lt_pow hP i
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
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
⊢ cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ (i + 1)) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ (i + 1)
have 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
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]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
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
⊢ cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ (i + 1)) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ (i + 1)
have 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)
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]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
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)
⊢ cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ (i + 1)) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ (i + 1)
have 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)
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]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
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)
⊢ cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ (i + 1)) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ (i + 1)
have 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)
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]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
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)
⊢ cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ (i + 1)) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ (i + 1)
have 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)
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]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
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)
⊢ cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ (i + 1)) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ (i + 1)
have 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)
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]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
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)
⊢ cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ (i + 1)) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ (i + 1)
have rw₆ : 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)
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]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
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
⊢ cardQuot.{u_1, u_1} (R := S) (M := S) (P ^ (i + 1)) = cardQuot.{u_1, u_1} (R := S) (M := S) P ^ (i + 1)
have succ : ↥(map (mkQ (P ^ i.succ)) (P ^ i)) ≃ S ⧸ P
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]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
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
⊢ ↥(map (mkQ (P ^ i.succ)) (P ^ i)) ≃ S ⧸ P
have succ : (a : S) → (a_mem : a ∈ P ^ i) → (a_notMem : a ∉ P ^ (i + 1)) → ↥(map (mkQ (P ^ i.succ)) (P ^ i)) ≃ S ⧸ P
Tactic: choose a a_mem a_notMem using SetLike.exists_of_lt this
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
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)
⊢ ↥(map (mkQ (P ^ i.succ)) (P ^ i)) ≃ S ⧸ P
have 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
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
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
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
⊢ ↥(map (mkQ (P ^ i.succ)) (P ^ i)) ≃ S ⧸ P
have 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
Tactic: choose k hk_mem hk_eq using fun c' (hc' : c' ∈ map (mkQ (P ^ i.succ)) (P ^ i)) =>
    Submodule.mem_map.mp hc'
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
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'
⊢ ↥(map (mkQ (P ^ i.succ)) (P ^ i)) ≃ S ⧸ P
have succ : Function.Injective (β := S ⧸ P) fun (c' : ↥(map (mkQ (P ^ i.succ)) (P ^ i))) ↦
  Quotient.mk'' (f (k (↑c' : S ⧸ P ^ i.succ) sorry) sorry)
Tactic: refine Equiv.ofBijective (fun c' => Quotient.mk'' (f (k c' c'.prop) (hk_mem c' c'.prop))) ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
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) ⋯) ⋯)
⊢ ↥(map (mkQ (P ^ i.succ)) (P ^ i)) ≃ S ⧸ P
have succ₁ : Function.Surjective (β := S ⧸ P) fun (c' : ↥(map (mkQ (P ^ i.succ)) (P ^ i))) ↦
  Quotient.mk'' (f (k (↑c' : S ⧸ P ^ i.succ) sorry) sorry)
Tactic: refine Equiv.ofBijective (fun c' => Quotient.mk'' (f (k c' c'.prop) (hk_mem c' c'.prop))) ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
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'))
⊢ 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'
have succ : a * f (k c₁' hc₁') sorry + g (k c₁' hc₁') sorry - (a * f (k c₂' hc₂') sorry + g (k c₂' hc₂') sorry) ∈ P ^ i.succ
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')]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
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
have succ : f (k c₁' hc₁') sorry - f (k c₂' hc₂') sorry ∈ P
Tactic: refine Ideal.mul_add_mem_pow_succ_inj _ _ _ _ _ _ a_mem (hg _ _) (hg _ _) ?_
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
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
⊢ ∃ (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'
have 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) sorry) sorry) : Quotient (α := S) (quotientRel (R := S) P)))
        a)
      ⟦d⟧
Tactic: induction d' using Quotient.inductionOn with | _ d
Cleared hypotheses: d'
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (hypothesis)
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
⊢ ∃ (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⟧
have hd' : Membership.mem (α := S ⧸ P ^ i.succ) (γ := Submodule S (S ⧸ P ^ i.succ)) (map (mkQ (P ^ i.succ)) (P ^ i))
  ((mkQ (P ^ i.succ) : (a : S) → S ⧸ P ^ i.succ) (a * d) : S ⧸ P ^ i.succ)
Tactic: have hd' := (mem_map (f := mkQ (P ^ i.succ))).mpr ⟨a * d, Ideal.mul_mem_right d _ a_mem, rfl⟩