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
inst_1 : Nontrivial S
inst_2 : IsDedekindDomain S
inst_3 : Free ℤ S
inst_4 : Module.Finite ℤ S
inst_5 : CharZero S
n : ℕ
hn : n > 0
f : (I : Ideal S) → Ideal (HasQuotient.Quotient S (B := Ideal S) (span {(↑n : S)})) :=
  fun (I : Ideal S) ↦
    map (F := S →+* HasQuotient.Quotient S (B := Ideal S) (span {(↑n : S)})) (Quotient.mk (span {(↑n : S)})) I
inr : (f '' {I : Ideal S | (absNorm : (a : Ideal S) → ℕ) I = n}).Finite
⊢ Set.Finite.{u_1} (α := Ideal S) {I : Ideal S | (absNorm : (a : Ideal S) → ℕ) I = n}
have inr₁ : Set.InjOn f {I : Ideal S | (absNorm : (a : Ideal S) → ℕ) I = n}
Tactic: refine Set.Finite.of_finite_image (f := f) ?_ ?_
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
inst_1 : Nontrivial S
inst_2 : IsDedekindDomain S
inst_3 : Free ℤ S
inst_4 : Module.Finite ℤ S
inst_5 : CharZero S
n : ℕ
hn : n > 0
f : (I : Ideal S) → Ideal (HasQuotient.Quotient S (B := Ideal S) (span {(↑n : S)})) :=
  fun (I : Ideal S) ↦
    map (F := S →+* HasQuotient.Quotient S (B := Ideal S) (span {(↑n : S)})) (Quotient.mk (span {(↑n : S)})) I
this : Finite.{u_1 + 1} (HasQuotient.Quotient S (B := Ideal S) (span {(↑n : S)}))
g : Ideal (HasQuotient.Quotient S (B := Ideal S) (span {(↑n : S)})) →
  Set (HasQuotient.Quotient S (B := Ideal S) (span {(↑n : S)})) :=
  SetLike.coe
⊢ (f '' {I : Ideal S | (absNorm : (a : Ideal S) → ℕ) I = n}).Finite
have refine : (g '' f '' {I : Ideal S | (absNorm : (a : Ideal S) → ℕ) I = n}).Finite
Tactic: refine Set.Finite.of_finite_image (f := g) ?_ SetLike.coe_injective.injOn
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
inst_1 : Nontrivial S
inst_2 : IsDedekindDomain S
inst_3 : Free ℤ S
inst_4 : Module.Finite ℤ S
inst_5 : CharZero S
n : ℕ
hn : n > 0
f : (I : Ideal S) → Ideal (HasQuotient.Quotient S (B := Ideal S) (span {(↑n : S)})) :=
  fun (I : Ideal S) ↦
    map (F := S →+* HasQuotient.Quotient S (B := Ideal S) (span {(↑n : S)})) (Quotient.mk (span {(↑n : S)})) I
⊢ (f '' {I : Ideal S | (absNorm : (a : Ideal S) → ℕ) I = n}).Finite
have inr : Finite.{u_1 + 1} (HasQuotient.Quotient S (B := Ideal S) (span {(↑n : S)}))
Tactic: suffices Finite (S ⧸ @Ideal.span S _ {↑n}) by
        let g := ((↑) : Ideal (S ⧸ @Ideal.span S _ {↑n}) → Set (S ⧸ @Ideal.span S _ {↑n}))
        refine Set.Finite.of_finite_image (f := g) ?_ SetLike.coe_injective.injOn
        exact Set.Finite.subset Set.finite_univ (Set.subset_univ _)
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
inst_1 : Nontrivial S
inst_2 : IsDedekindDomain S
inst_3 : Free ℤ S
inst_4 : Module.Finite ℤ S
inst_5 : CharZero S
n : ℕ
hn : n > 0
f : (I : Ideal S) → Ideal (HasQuotient.Quotient S (B := Ideal S) (span {(↑n : S)})) :=
  fun (I : Ideal S) ↦
    map (F := S →+* HasQuotient.Quotient S (B := Ideal S) (span {(↑n : S)})) (Quotient.mk (span {(↑n : S)})) I
⊢ Finite.{u_1 + 1} (HasQuotient.Quotient S (B := Ideal S) (span {(↑n : S)}))
have inr : Int.natAbs ((Algebra.norm ℤ : (a : S) → ℤ) (↑n : S)) ≠ 0
Tactic: rw [← absNorm_ne_zero_iff, absNorm_span_singleton]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
inst_1 : Nontrivial S
inst_2 : IsDedekindDomain S
inst_3 : Free ℤ S
inst_4 : Module.Finite ℤ S
inst_5 : CharZero S
n : ℕ
hn : n > 0
f : (I : Ideal S) → Ideal (HasQuotient.Quotient S (B := Ideal S) (span {(↑n : S)})) :=
  fun (I : Ideal S) ↦
    map (F := S →+* HasQuotient.Quotient S (B := Ideal S) (span {(↑n : S)})) (Quotient.mk (span {(↑n : S)})) I
I : Ideal S
hI : Membership.mem (γ := Set (Ideal S)) {I : Ideal S | (absNorm : (a : Ideal S) → ℕ) I = n} I
J : Ideal S
hJ : Membership.mem (γ := Set (Ideal S)) {I : Ideal S | (absNorm : (a : Ideal S) → ℕ) I = n} J
h : f I = f J
⊢ I = J
have inr : Eq (α := Ideal S)
  (comap (F := S →+* HasQuotient.Quotient S (B := Ideal S) (span {(↑n : S)})) (Quotient.mk (span {(↑n : S)}))
    (map (F := S →+* HasQuotient.Quotient S (B := Ideal S) (span {(↑n : S)})) (Quotient.mk (span {(↑n : S)})) I))
  (comap (F := S →+* HasQuotient.Quotient S (B := Ideal S) (span {(↑n : S)})) (Quotient.mk (span {(↑n : S)}))
    (map (F := S →+* HasQuotient.Quotient S (B := Ideal S) (span {(↑n : S)})) (Quotient.mk (span {(↑n : S)})) J))
Tactic: rw [← comap_map_mk (span_singleton_absNorm_le I), ← hI.symm, ←
        comap_map_mk (span_singleton_absNorm_le J), ← hJ.symm]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
inst_1 : Nontrivial S
inst_2 : IsDedekindDomain S
inst_3 : Free ℤ S
inst_4 : Module.Finite ℤ S
inst_5 : CharZero S
n : ℕ
⊢ Eq.{u_1 + 1} (α := Set (Ideal S)) {I : Ideal S | (absNorm : (a : Ideal S) → ℕ) I ≤ n}
    (⋃ i ∈ Set.Icc 0 n, {I : Ideal S | (absNorm : (a : Ideal S) → ℕ) I = i})
have h : ∀ (x : Ideal S),
  Membership.mem (γ := Set (Ideal S)) {I : Ideal S | (absNorm : (a : Ideal S) → ℕ) I ≤ n} x ↔
    Membership.mem (γ := Set (Ideal S)) (⋃ i ∈ Set.Icc 0 n, {I : Ideal S | (absNorm : (a : Ideal S) → ℕ) I = i}) x
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
inst_1 : Nontrivial S
inst_2 : IsDedekindDomain S
inst_3 : Free ℤ S
inst_4 : Module.Finite ℤ S
inst_5 : CharZero S
n : ℕ
⊢ Set.Finite.{u_1} (α := Ideal S) {I : Ideal S | (absNorm : (a : Ideal S) → ℕ) I ≤ n}
have rw : Set.Finite.{u_1} (α := Ideal S) (⋃ i ∈ Set.Icc 0 n, {I : Ideal S | (absNorm : (a : Ideal S) → ℕ) I = i})
Tactic: rw [show {I : Ideal S | Ideal.absNorm I ≤ n} =
    (⋃ i ∈ Set.Icc 0 n, {I : Ideal S | Ideal.absNorm I = i}) by ext; simp]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (hypothesis)
Goal:
S : Type u_1
inst : CommRing S
inst_1 : Nontrivial S
inst_2 : IsDedekindDomain S
inst_3 : Free ℤ S
inst_4 : Module.Finite ℤ S
inst_5 : CharZero S
n : ℕ
⊢ Set.Finite.{u_1} (α := ↥(Ideal S)⁰) {I : ↥(Ideal S)⁰ | (absNorm : (a : Ideal S) → ℕ) (↑I : Ideal S) ≤ n}
have this : Finite.{max 1 (u_1 + 1)}
  { I : Ideal S // Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
Tactic: have : Finite {I : Ideal S // I ∈ (Ideal S)⁰ ∧ absNorm I ≤ n} :=
    (finite_setOf_absNorm_le n).subset fun _ ⟨_, h⟩ ↦ h
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (hypothesis)
Goal:
S : Type u_1
inst : CommRing S
inst_1 : Nontrivial S
inst_2 : IsDedekindDomain S
inst_3 : Free ℤ S
inst_4 : Module.Finite ℤ S
n : ℕ
inst_5 : CharZero S
⊢ Nat.card.{u_1} { I : Ideal S // (absNorm : (a : Ideal S) → ℕ) I ≤ n } =
    Nat.card.{u_1} { I : ↥(Ideal S)⁰ // (absNorm : (a : Ideal S) → ℕ) (↑I : Ideal S) ≤ n } + 1
have this : Finite.{max 1 (u_1 + 1)}
  { I : Ideal S // Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
Tactic: have : Finite {I : Ideal S // I ∈ (Ideal S)⁰ ∧ absNorm I ≤ n} :=
    (finite_setOf_absNorm_le n).subset fun _ ⟨_, h⟩ ↦ h
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (hypothesis)
Goal:
S : Type u_1
inst : CommRing S
inst_1 : Nontrivial S
inst_2 : IsDedekindDomain S
inst_3 : Free ℤ S
inst_4 : Module.Finite ℤ S
n : ℕ
inst_5 : CharZero S
this :
  Finite.{max 1 (u_1 + 1)}
    { I : Ideal S // Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
⊢ Nat.card.{u_1} { I : Ideal S // (absNorm : (a : Ideal S) → ℕ) I ≤ n } =
    Nat.card.{u_1} { I : ↥(Ideal S)⁰ // (absNorm : (a : Ideal S) → ℕ) (↑I : Ideal S) ≤ n } + 1
have this₁ : Finite.{max 1 (u_1 + 1)} { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
Tactic: have : Finite {I : Ideal S // I ∉ (Ideal S)⁰ ∧ absNorm I ≤ n} :=
    (finite_setOf_absNorm_le n).subset fun _ ⟨_, h⟩ ↦ h
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
inst_1 : Nontrivial S
inst_2 : IsDedekindDomain S
inst_3 : Free ℤ S
inst_4 : Module.Finite ℤ S
n : ℕ
inst_5 : CharZero S
this_1 :
  Finite.{max 1 (u_1 + 1)}
    { I : Ideal S // Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
this : Finite.{max 1 (u_1 + 1)} { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
⊢ Nat.card.{u_1} { I : Ideal S // (absNorm : (a : Ideal S) → ℕ) I ≤ n } =
    Nat.card.{u_1} { I : ↥(Ideal S)⁰ // (absNorm : (a : Ideal S) → ℕ) (↑I : Ideal S) ≤ n } + 1
have rw : Nat.card.{u_1} { I : Ideal S // (absNorm : (a : Ideal S) → ℕ) I ≤ n } =
  Nat.card.{u_1}
      { x : Ideal S // Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ x ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n } +
    1
Tactic: rw [Nat.card_congr (Equiv.subtypeSubtypeEquivSubtypeInter (fun I ↦ I ∈ (Ideal S)⁰)
    (fun I ↦ absNorm I ≤ n))]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
inst_1 : Nontrivial S
inst_2 : IsDedekindDomain S
inst_3 : Free ℤ S
inst_4 : Module.Finite ℤ S
n : ℕ
inst_5 : CharZero S
this_1 :
  Finite.{max 1 (u_1 + 1)}
    { I : Ideal S // Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
this : Finite.{max 1 (u_1 + 1)} { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
⊢ { I : Ideal S // (absNorm : (a : Ideal S) → ℕ) I ≤ n } ≃
    { I : Ideal S // Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n } ⊕
      { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
have refine_1 : ∀ (x : Ideal S),
  (absNorm : (a : Ideal S) → ℕ) x ≤ n ↔
    Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ x ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n ∨
      x ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n
Tactic: refine (Equiv.subtypeEquivRight ?_).trans (subtypeOrEquiv _ _ ?_)
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
inst_1 : Nontrivial S
inst_2 : IsDedekindDomain S
inst_3 : Free ℤ S
inst_4 : Module.Finite ℤ S
n : ℕ
inst_5 : CharZero S
this_1 :
  Finite.{max 1 (u_1 + 1)}
    { I : Ideal S // Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
this : Finite.{max 1 (u_1 + 1)} { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
refine_1 :
  ∀ (x : Ideal S),
    (absNorm : (a : Ideal S) → ℕ) x ≤ n ↔
      Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ x ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n ∨
        x ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n
⊢ { I : Ideal S // (absNorm : (a : Ideal S) → ℕ) I ≤ n } ≃
    { I : Ideal S // Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n } ⊕
      { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
have refine_2 : Disjoint.{u_1} (α := Ideal S → Prop)
  (fun (I : Ideal S) ↦ Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n)
  fun (I : Ideal S) ↦ I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n
Tactic: refine (Equiv.subtypeEquivRight ?_).trans (subtypeOrEquiv _ _ ?_)
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
inst_1 : Nontrivial S
inst_2 : IsDedekindDomain S
inst_3 : Free ℤ S
inst_4 : Module.Finite ℤ S
n : ℕ
inst_5 : CharZero S
this_1 :
  Finite.{max 1 (u_1 + 1)}
    { I : Ideal S // Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
this : Finite.{max 1 (u_1 + 1)} { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
x : Ideal S
⊢ (absNorm : (a : Ideal S) → ℕ) x ≤ n ↔
    Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ x ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n ∨
      x ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n
have refine_1 : (absNorm : (a : Ideal S) → ℕ) x ≤ n ↔
  (Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ x ∨ x ∉ (Ideal S)⁰) ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n
Tactic: simp_rw [← or_and_right, em, true_and]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
inst_1 : Nontrivial S
inst_2 : IsDedekindDomain S
inst_3 : Free ℤ S
inst_4 : Module.Finite ℤ S
n : ℕ
inst_5 : CharZero S
this_1 :
  Finite.{max 1 (u_1 + 1)}
    { I : Ideal S // Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
this : Finite.{max 1 (u_1 + 1)} { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
x : Ideal S
refine_1 :
  (absNorm : (a : Ideal S) → ℕ) x ≤ n ↔
    (Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ x ∨ x ∉ (Ideal S)⁰) ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n
⊢ (absNorm : (a : Ideal S) → ℕ) x ≤ n ↔
    Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ x ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n ∨
      x ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n
have refine_1₁ : (absNorm : (a : Ideal S) → ℕ) x ≤ n ↔ True ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n
Tactic: simp_rw [← or_and_right, em, true_and]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
inst_1 : Nontrivial S
inst_2 : IsDedekindDomain S
inst_3 : Free ℤ S
inst_4 : Module.Finite ℤ S
n : ℕ
inst_5 : CharZero S
this_1 :
  Finite.{max 1 (u_1 + 1)}
    { I : Ideal S // Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
this : Finite.{max 1 (u_1 + 1)} { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
I : Ideal S
⊢ ¬((Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n) ∧
      I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n)
have tauto : Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I →
  (absNorm : (a : Ideal S) → ℕ) I ≤ n → I ∉ (Ideal S)⁰ → (absNorm : (a : Ideal S) → ℕ) I ≤ n → False
Tactic: tauto
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
inst_1 : Nontrivial S
inst_2 : IsDedekindDomain S
inst_3 : Free ℤ S
inst_4 : Module.Finite ℤ S
n : ℕ
inst_5 : CharZero S
this_1 :
  Finite.{max 1 (u_1 + 1)}
    { I : Ideal S // Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
this : Finite.{max 1 (u_1 + 1)} { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
e : { I : Ideal S // (absNorm : (a : Ideal S) → ℕ) I ≤ n } ≃
  { I : Ideal S // Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n } ⊕
    { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n } :=
  Equiv.trans
    (Equiv.subtypeEquivRight (p := fun (I : Ideal S) ↦ (absNorm : (a : Ideal S) → ℕ) I ≤ n) (q := fun (x : Ideal S) ↦
      Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ x ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n ∨
        x ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n)
      ⋯)
    (subtypeOrEquiv
      (fun (I : Ideal S) ↦ Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n)
      (fun (I : Ideal S) ↦ I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n) ⋯)
⊢ Nat.card.{u_1} { I : Ideal S // (absNorm : (a : Ideal S) → ℕ) I ≤ n } =
    Nat.card.{u_1}
        { x : Ideal S //
          Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ x ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n } +
      1
have simp_rw : Nat.card.{u_1} { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n } = 1
Tactic: simp_rw [Nat.card_congr e, Nat.card_sum, add_right_inj]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
inst_1 : Nontrivial S
inst_2 : IsDedekindDomain S
inst_3 : Free ℤ S
inst_4 : Module.Finite ℤ S
n : ℕ
inst_5 : CharZero S
this_1 :
  Finite.{max 1 (u_1 + 1)}
    { I : Ideal S // Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
this : Finite.{max 1 (u_1 + 1)} { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
e : { I : Ideal S // (absNorm : (a : Ideal S) → ℕ) I ≤ n } ≃
  { I : Ideal S // Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n } ⊕
    { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n } :=
  Equiv.trans
    (Equiv.subtypeEquivRight (p := fun (I : Ideal S) ↦ (absNorm : (a : Ideal S) → ℕ) I ≤ n) (q := fun (x : Ideal S) ↦
      Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ x ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n ∨
        x ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n)
      ⋯)
    (subtypeOrEquiv
      (fun (I : Ideal S) ↦ Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n)
      (fun (I : Ideal S) ↦ I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n) ⋯)
| Nat.card.{u_1} { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
have h : ∀ (I : Ideal S), I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n
Tactic: enter [1, 1, I]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
inst_1 : Nontrivial S
inst_2 : IsDedekindDomain S
inst_3 : Free ℤ S
inst_4 : Module.Finite ℤ S
n : ℕ
inst_5 : CharZero S
this_1 :
  Finite.{max 1 (u_1 + 1)}
    { I : Ideal S // Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
this : Finite.{max 1 (u_1 + 1)} { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
e : { I : Ideal S // (absNorm : (a : Ideal S) → ℕ) I ≤ n } ≃
  { I : Ideal S // Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n } ⊕
    { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n } :=
  Equiv.trans
    (Equiv.subtypeEquivRight (p := fun (I : Ideal S) ↦ (absNorm : (a : Ideal S) → ℕ) I ≤ n) (q := fun (x : Ideal S) ↦
      Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ x ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n ∨
        x ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n)
      ⋯)
    (subtypeOrEquiv
      (fun (I : Ideal S) ↦ Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n)
      (fun (I : Ideal S) ↦ I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n) ⋯)
I : Ideal S
h : Eq (α := ℕ) ((absNorm : (a : Ideal S) → ℕ) I : ℕ) (0 : ℕ)
⊢ (absNorm : (a : Ideal S) → ℕ) I ≤ n
have rw : 0 ≤ n
Tactic: rw [h]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
inst_1 : Nontrivial S
inst_2 : IsDedekindDomain S
inst_3 : Free ℤ S
inst_4 : Module.Finite ℤ S
n : ℕ
inst_5 : CharZero S
this_1 :
  Finite.{max 1 (u_1 + 1)}
    { I : Ideal S // Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
this : Finite.{max 1 (u_1 + 1)} { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
e : { I : Ideal S // (absNorm : (a : Ideal S) → ℕ) I ≤ n } ≃
  { I : Ideal S // Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n } ⊕
    { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n } :=
  Equiv.trans
    (Equiv.subtypeEquivRight (p := fun (I : Ideal S) ↦ (absNorm : (a : Ideal S) → ℕ) I ≤ n) (q := fun (x : Ideal S) ↦
      Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ x ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n ∨
        x ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n)
      ⋯)
    (subtypeOrEquiv
      (fun (I : Ideal S) ↦ Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n)
      (fun (I : Ideal S) ↦ I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n) ⋯)
I : Ideal S
| I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n
have h : I = ⊥
Tactic: rw [← absNorm_ne_zero_iff_mem_nonZeroDivisors, ne_eq, not_not, and_iff_left_iff_imp.mpr
      (fun h ↦ by rw [h]; exact Nat.zero_le n), absNorm_eq_zero_iff]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
inst_1 : Nontrivial S
inst_2 : IsDedekindDomain S
inst_3 : Free ℤ S
inst_4 : Module.Finite ℤ S
n : ℕ
inst_5 : CharZero S
this_1 :
  Finite.{max 1 (u_1 + 1)}
    { I : Ideal S // Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
this : Finite.{max 1 (u_1 + 1)} { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
e : { I : Ideal S // (absNorm : (a : Ideal S) → ℕ) I ≤ n } ≃
  { I : Ideal S // Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n } ⊕
    { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n } :=
  Equiv.trans
    (Equiv.subtypeEquivRight (p := fun (I : Ideal S) ↦ (absNorm : (a : Ideal S) → ℕ) I ≤ n) (q := fun (x : Ideal S) ↦
      Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ x ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n ∨
        x ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n)
      ⋯)
    (subtypeOrEquiv
      (fun (I : Ideal S) ↦ Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n)
      (fun (I : Ideal S) ↦ I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n) ⋯)
| Nat.card.{u_1} { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
have h : ∀ (I : Ideal S), I = ⊥
Tactic: enter [1, 1, I]
    rw [← absNorm_ne_zero_iff_mem_nonZeroDivisors, ne_eq, not_not, and_iff_left_iff_imp.mpr
      (fun h ↦ by rw [h]; exact Nat.zero_le n), absNorm_eq_zero_iff]
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.Norm.AbsNorm (goal)
Goal:
S : Type u_1
inst : CommRing S
inst_1 : Nontrivial S
inst_2 : IsDedekindDomain S
inst_3 : Free ℤ S
inst_4 : Module.Finite ℤ S
n : ℕ
inst_5 : CharZero S
this_1 :
  Finite.{max 1 (u_1 + 1)}
    { I : Ideal S // Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
this : Finite.{max 1 (u_1 + 1)} { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n }
e : { I : Ideal S // (absNorm : (a : Ideal S) → ℕ) I ≤ n } ≃
  { I : Ideal S // Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n } ⊕
    { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n } :=
  Equiv.trans
    (Equiv.subtypeEquivRight (p := fun (I : Ideal S) ↦ (absNorm : (a : Ideal S) → ℕ) I ≤ n) (q := fun (x : Ideal S) ↦
      Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ x ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n ∨
        x ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) x ≤ n)
      ⋯)
    (subtypeOrEquiv
      (fun (I : Ideal S) ↦ Membership.mem (γ := Submonoid (Ideal S)) (Ideal S)⁰ I ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n)
      (fun (I : Ideal S) ↦ I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n) ⋯)
⊢ Nat.card.{u_1} { I : Ideal S // I ∉ (Ideal S)⁰ ∧ (absNorm : (a : Ideal S) → ℕ) I ≤ n } = 1
have conv_lhs : Nat.card.{u_1} { I : Ideal S // I = ⊥ } = 1
Tactic: conv_lhs =>
    enter [1, 1, I]
    rw [← absNorm_ne_zero_iff_mem_nonZeroDivisors, ne_eq, not_not, and_iff_left_iff_imp.mpr
      (fun h ↦ by rw [h]; exact Nat.zero_le n), absNorm_eq_zero_iff]