awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.NatInt (goal)
Goal:
P : Ideal ℕ
h : P.IsPrime
h0 : ∃ x ∈ P, x ∉ ⊥
hsp : ¬∃ (p : ℕ), Nat.Prime p ∧ P = span {p}
n : ℕ
hn : Membership.mem (γ := Submodule ℕ ℕ) (maximalIdeal ℕ) n
refine_2 refine_2₁ refine_2₂ : n ∈ P
refine_2₃ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0), n ∈ P
⊢ n ∈ P
have refine_2₄ : let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1), n ∈ P
Tactic: classical
  let p := Nat.find h0
  have ⟨(hp : p ∈ P), (hp0 : p ≠ 0)⟩ := Nat.find_spec h0
  have : p ≠ 1 := ne_of_mem_of_not_mem hp P.one_notMem
  have prime : p.Prime := Nat.prime_iff_not_exists_mul_eq.mpr <| .intro (by lia)
    fun ⟨m, n, hm, hn, eq⟩ ↦ have := mul_ne_zero_iff.mp (eq ▸ hp0)
    (h.mem_or_mem (eq ▸ hp)).elim (Nat.find_min h0 hm ⟨·, this.1⟩) (Nat.find_min h0 hn ⟨·, this.2⟩)
  push Not at hsp
  have ⟨q, hq, hqp⟩ := SetLike.exists_of_lt
    ((P.span_singleton_le_iff_mem.mpr hp).lt_of_ne (hsp p prime).symm)
  obtain rfl | hn1 := eq_or_ne n 0
  · exact Ideal.zero_mem _
  have : n ≠ 1 := Nat.mem_maximalIdeal_iff.mp hn
  have ⟨a, b, eq⟩ := Nat.exists_add_mul_eq_of_gcd_dvd_of_mul_pred_le p q _
    (by simp [prime.coprime_iff_not_dvd.mpr (Ideal.mem_span_singleton.not.mp hqp)])
    (Nat.lt_pow_self (show 1 < n by lia)).le
  exact h.mem_of_pow_mem _ (eq ▸ add_mem (P.mul_mem_left _ hp) (P.mul_mem_left _ hq))
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.NatInt (goal)
Goal:
P : Ideal ℕ
h : P.IsPrime
h0 : ∃ x ∈ P, x ∉ ⊥
hsp : ¬∃ (p : ℕ), Nat.Prime p ∧ P = span {p}
n : ℕ
hn : Membership.mem (γ := Submodule ℕ ℕ) (maximalIdeal ℕ) n
refine_2 refine_2₁ refine_2₂ : n ∈ P
refine_2₃ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0), n ∈ P
refine_2₄ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1), n ∈ P
⊢ n ∈ P
have refine_2₅ : let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1) (prime : Nat.Prime p), n ∈ P
Tactic: classical
  let p := Nat.find h0
  have ⟨(hp : p ∈ P), (hp0 : p ≠ 0)⟩ := Nat.find_spec h0
  have : p ≠ 1 := ne_of_mem_of_not_mem hp P.one_notMem
  have prime : p.Prime := Nat.prime_iff_not_exists_mul_eq.mpr <| .intro (by lia)
    fun ⟨m, n, hm, hn, eq⟩ ↦ have := mul_ne_zero_iff.mp (eq ▸ hp0)
    (h.mem_or_mem (eq ▸ hp)).elim (Nat.find_min h0 hm ⟨·, this.1⟩) (Nat.find_min h0 hn ⟨·, this.2⟩)
  push Not at hsp
  have ⟨q, hq, hqp⟩ := SetLike.exists_of_lt
    ((P.span_singleton_le_iff_mem.mpr hp).lt_of_ne (hsp p prime).symm)
  obtain rfl | hn1 := eq_or_ne n 0
  · exact Ideal.zero_mem _
  have : n ≠ 1 := Nat.mem_maximalIdeal_iff.mp hn
  have ⟨a, b, eq⟩ := Nat.exists_add_mul_eq_of_gcd_dvd_of_mul_pred_le p q _
    (by simp [prime.coprime_iff_not_dvd.mpr (Ideal.mem_span_singleton.not.mp hqp)])
    (Nat.lt_pow_self (show 1 < n by lia)).le
  exact h.mem_of_pow_mem _ (eq ▸ add_mem (P.mul_mem_left _ hp) (P.mul_mem_left _ hq))
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.NatInt (goal)
Goal:
P : Ideal ℕ
h : P.IsPrime
h0 : ∃ x ∈ P, x ∉ ⊥
hsp : ¬∃ (p : ℕ), Nat.Prime p ∧ P = span {p}
n : ℕ
hn : Membership.mem (γ := Submodule ℕ ℕ) (maximalIdeal ℕ) n
refine_2 refine_2₁ refine_2₂ : n ∈ P
refine_2₃ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0), n ∈ P
refine_2₄ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1), n ∈ P
refine_2₅ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1) (prime : Nat.Prime p), n ∈ P
⊢ n ∈ P
have refine_2₆ : let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1) (prime : Nat.Prime p) (hsp : ∀ (p : ℕ) (_ : Nat.Prime p), P ≠ span {p}),
  n ∈ P
Tactic: classical
  let p := Nat.find h0
  have ⟨(hp : p ∈ P), (hp0 : p ≠ 0)⟩ := Nat.find_spec h0
  have : p ≠ 1 := ne_of_mem_of_not_mem hp P.one_notMem
  have prime : p.Prime := Nat.prime_iff_not_exists_mul_eq.mpr <| .intro (by lia)
    fun ⟨m, n, hm, hn, eq⟩ ↦ have := mul_ne_zero_iff.mp (eq ▸ hp0)
    (h.mem_or_mem (eq ▸ hp)).elim (Nat.find_min h0 hm ⟨·, this.1⟩) (Nat.find_min h0 hn ⟨·, this.2⟩)
  push Not at hsp
  have ⟨q, hq, hqp⟩ := SetLike.exists_of_lt
    ((P.span_singleton_le_iff_mem.mpr hp).lt_of_ne (hsp p prime).symm)
  obtain rfl | hn1 := eq_or_ne n 0
  · exact Ideal.zero_mem _
  have : n ≠ 1 := Nat.mem_maximalIdeal_iff.mp hn
  have ⟨a, b, eq⟩ := Nat.exists_add_mul_eq_of_gcd_dvd_of_mul_pred_le p q _
    (by simp [prime.coprime_iff_not_dvd.mpr (Ideal.mem_span_singleton.not.mp hqp)])
    (Nat.lt_pow_self (show 1 < n by lia)).le
  exact h.mem_of_pow_mem _ (eq ▸ add_mem (P.mul_mem_left _ hp) (P.mul_mem_left _ hq))
Cleared hypotheses: hsp
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.NatInt (goal)
Goal:
P : Ideal ℕ
h : P.IsPrime
h0 : ∃ x ∈ P, x ∉ ⊥
hsp : ¬∃ (p : ℕ), Nat.Prime p ∧ P = span {p}
n : ℕ
hn : Membership.mem (γ := Submodule ℕ ℕ) (maximalIdeal ℕ) n
refine_2 refine_2₁ refine_2₂ : n ∈ P
refine_2₃ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0), n ∈ P
refine_2₄ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1), n ∈ P
refine_2₅ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1) (prime : Nat.Prime p), n ∈ P
refine_2₆ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1) (prime : Nat.Prime p) (hsp : ∀ (p : ℕ) (_ : Nat.Prime p), P ≠ span {p}),
    n ∈ P
⊢ n ∈ P
have refine_2₇ : let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1) (prime : Nat.Prime p) (hsp : ∀ (p : ℕ) (_ : Nat.Prime p), P ≠ span {p})
  (q : ℕ) (hq : q ∈ P) (hqp : q ∉ span {p}), n ∈ P
Tactic: classical
  let p := Nat.find h0
  have ⟨(hp : p ∈ P), (hp0 : p ≠ 0)⟩ := Nat.find_spec h0
  have : p ≠ 1 := ne_of_mem_of_not_mem hp P.one_notMem
  have prime : p.Prime := Nat.prime_iff_not_exists_mul_eq.mpr <| .intro (by lia)
    fun ⟨m, n, hm, hn, eq⟩ ↦ have := mul_ne_zero_iff.mp (eq ▸ hp0)
    (h.mem_or_mem (eq ▸ hp)).elim (Nat.find_min h0 hm ⟨·, this.1⟩) (Nat.find_min h0 hn ⟨·, this.2⟩)
  push Not at hsp
  have ⟨q, hq, hqp⟩ := SetLike.exists_of_lt
    ((P.span_singleton_le_iff_mem.mpr hp).lt_of_ne (hsp p prime).symm)
  obtain rfl | hn1 := eq_or_ne n 0
  · exact Ideal.zero_mem _
  have : n ≠ 1 := Nat.mem_maximalIdeal_iff.mp hn
  have ⟨a, b, eq⟩ := Nat.exists_add_mul_eq_of_gcd_dvd_of_mul_pred_le p q _
    (by simp [prime.coprime_iff_not_dvd.mpr (Ideal.mem_span_singleton.not.mp hqp)])
    (Nat.lt_pow_self (show 1 < n by lia)).le
  exact h.mem_of_pow_mem _ (eq ▸ add_mem (P.mul_mem_left _ hp) (P.mul_mem_left _ hq))
Cleared hypotheses: hsp
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.NatInt (goal)
Goal:
P : Ideal ℕ
h : P.IsPrime
h0 : ∃ x ∈ P, x ∉ ⊥
hsp : ¬∃ (p : ℕ), Nat.Prime p ∧ P = span {p}
n : ℕ
hn : Membership.mem (γ := Submodule ℕ ℕ) (maximalIdeal ℕ) n
refine_2 refine_2₁ refine_2₂ : n ∈ P
refine_2₃ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0), n ∈ P
refine_2₄ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1), n ∈ P
refine_2₅ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1) (prime : Nat.Prime p), n ∈ P
refine_2₆ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1) (prime : Nat.Prime p) (hsp : ∀ (p : ℕ) (_ : Nat.Prime p), P ≠ span {p}),
    n ∈ P
refine_2₇ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1) (prime : Nat.Prime p) (hsp : ∀ (p : ℕ) (_ : Nat.Prime p), P ≠ span {p})
    (q : ℕ) (hq : q ∈ P) (hqp : q ∉ span {p}), n ∈ P
⊢ n ∈ P
have refine_2₈ : let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1) (prime : Nat.Prime p) (hsp : ∀ (p : ℕ) (_ : Nat.Prime p), P ≠ span {p})
  (q : ℕ) (hq : q ∈ P) (hqp : q ∉ span {p}) (hn : Membership.mem (γ := Submodule ℕ ℕ) (maximalIdeal ℕ) 0), 0 ∈ P
Tactic: classical
  let p := Nat.find h0
  have ⟨(hp : p ∈ P), (hp0 : p ≠ 0)⟩ := Nat.find_spec h0
  have : p ≠ 1 := ne_of_mem_of_not_mem hp P.one_notMem
  have prime : p.Prime := Nat.prime_iff_not_exists_mul_eq.mpr <| .intro (by lia)
    fun ⟨m, n, hm, hn, eq⟩ ↦ have := mul_ne_zero_iff.mp (eq ▸ hp0)
    (h.mem_or_mem (eq ▸ hp)).elim (Nat.find_min h0 hm ⟨·, this.1⟩) (Nat.find_min h0 hn ⟨·, this.2⟩)
  push Not at hsp
  have ⟨q, hq, hqp⟩ := SetLike.exists_of_lt
    ((P.span_singleton_le_iff_mem.mpr hp).lt_of_ne (hsp p prime).symm)
  obtain rfl | hn1 := eq_or_ne n 0
  · exact Ideal.zero_mem _
  have : n ≠ 1 := Nat.mem_maximalIdeal_iff.mp hn
  have ⟨a, b, eq⟩ := Nat.exists_add_mul_eq_of_gcd_dvd_of_mul_pred_le p q _
    (by simp [prime.coprime_iff_not_dvd.mpr (Ideal.mem_span_singleton.not.mp hqp)])
    (Nat.lt_pow_self (show 1 < n by lia)).le
  exact h.mem_of_pow_mem _ (eq ▸ add_mem (P.mul_mem_left _ hp) (P.mul_mem_left _ hq))
Cleared hypotheses: hsp, n, hn
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.NatInt (goal)
Goal:
P : Ideal ℕ
h : P.IsPrime
h0 : ∃ x ∈ P, x ∉ ⊥
hsp : ¬∃ (p : ℕ), Nat.Prime p ∧ P = span {p}
n : ℕ
hn : Membership.mem (γ := Submodule ℕ ℕ) (maximalIdeal ℕ) n
refine_2 refine_2₁ refine_2₂ : n ∈ P
refine_2₃ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0), n ∈ P
refine_2₄ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1), n ∈ P
refine_2₅ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1) (prime : Nat.Prime p), n ∈ P
refine_2₆ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1) (prime : Nat.Prime p) (hsp : ∀ (p : ℕ) (_ : Nat.Prime p), P ≠ span {p}),
    n ∈ P
refine_2₇ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1) (prime : Nat.Prime p) (hsp : ∀ (p : ℕ) (_ : Nat.Prime p), P ≠ span {p})
    (q : ℕ) (hq : q ∈ P) (hqp : q ∉ span {p}), n ∈ P
refine_2₈ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1) (prime : Nat.Prime p) (hsp : ∀ (p : ℕ) (_ : Nat.Prime p), P ≠ span {p})
    (q : ℕ) (hq : q ∈ P) (hqp : q ∉ span {p}) (hn : Membership.mem (γ := Submodule ℕ ℕ) (maximalIdeal ℕ) 0), 0 ∈ P
⊢ n ∈ P
have refine_2₉ : let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1) (prime : Nat.Prime p) (hsp : ∀ (p : ℕ) (_ : Nat.Prime p), P ≠ span {p})
  (q : ℕ) (hq : q ∈ P) (hqp : q ∉ span {p}) (hn1 : n ≠ 0), n ∈ P
Tactic: classical
  let p := Nat.find h0
  have ⟨(hp : p ∈ P), (hp0 : p ≠ 0)⟩ := Nat.find_spec h0
  have : p ≠ 1 := ne_of_mem_of_not_mem hp P.one_notMem
  have prime : p.Prime := Nat.prime_iff_not_exists_mul_eq.mpr <| .intro (by lia)
    fun ⟨m, n, hm, hn, eq⟩ ↦ have := mul_ne_zero_iff.mp (eq ▸ hp0)
    (h.mem_or_mem (eq ▸ hp)).elim (Nat.find_min h0 hm ⟨·, this.1⟩) (Nat.find_min h0 hn ⟨·, this.2⟩)
  push Not at hsp
  have ⟨q, hq, hqp⟩ := SetLike.exists_of_lt
    ((P.span_singleton_le_iff_mem.mpr hp).lt_of_ne (hsp p prime).symm)
  obtain rfl | hn1 := eq_or_ne n 0
  · exact Ideal.zero_mem _
  have : n ≠ 1 := Nat.mem_maximalIdeal_iff.mp hn
  have ⟨a, b, eq⟩ := Nat.exists_add_mul_eq_of_gcd_dvd_of_mul_pred_le p q _
    (by simp [prime.coprime_iff_not_dvd.mpr (Ideal.mem_span_singleton.not.mp hqp)])
    (Nat.lt_pow_self (show 1 < n by lia)).le
  exact h.mem_of_pow_mem _ (eq ▸ add_mem (P.mul_mem_left _ hp) (P.mul_mem_left _ hq))
Cleared hypotheses: hsp
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.NatInt (goal)
Goal:
P : Ideal ℕ
h : P.IsPrime
h0 : ∃ x ∈ P, x ∉ ⊥
hsp : ¬∃ (p : ℕ), Nat.Prime p ∧ P = span {p}
n : ℕ
hn : Membership.mem (γ := Submodule ℕ ℕ) (maximalIdeal ℕ) n
refine_2 refine_2₁ refine_2₂ : n ∈ P
refine_2₃ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0), n ∈ P
refine_2₄ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1), n ∈ P
refine_2₅ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1) (prime : Nat.Prime p), n ∈ P
refine_2₆ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1) (prime : Nat.Prime p) (hsp : ∀ (p : ℕ) (_ : Nat.Prime p), P ≠ span {p}),
    n ∈ P
refine_2₇ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1) (prime : Nat.Prime p) (hsp : ∀ (p : ℕ) (_ : Nat.Prime p), P ≠ span {p})
    (q : ℕ) (hq : q ∈ P) (hqp : q ∉ span {p}), n ∈ P
refine_2₈ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1) (prime : Nat.Prime p) (hsp : ∀ (p : ℕ) (_ : Nat.Prime p), P ≠ span {p})
    (q : ℕ) (hq : q ∈ P) (hqp : q ∉ span {p}) (hn : Membership.mem (γ := Submodule ℕ ℕ) (maximalIdeal ℕ) 0), 0 ∈ P
refine_2₉ refine_2₁₀ :
  let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
  ∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1) (prime : Nat.Prime p) (hsp : ∀ (p : ℕ) (_ : Nat.Prime p), P ≠ span {p})
    (q : ℕ) (hq : q ∈ P) (hqp : q ∉ span {p}) (hn1 : n ≠ 0), n ∈ P
⊢ n ∈ P
have refine_2₁₁ : let p := Nat.find (p := fun (n : ℕ) ↦ n ∈ P ∧ n ∉ ⊥) h0;
∀ (hp : p ∈ P) (hp0 : p ≠ 0) (this : p ≠ 1) (prime : Nat.Prime p) (hsp : ∀ (p : ℕ) (_ : Nat.Prime p), P ≠ span {p})
  (q : ℕ) (hq : q ∈ P) (hqp : q ∉ span {p}) (hn1 : n ≠ 0) (this : n ≠ 1), n ∈ P
Tactic: classical
  let p := Nat.find h0
  have ⟨(hp : p ∈ P), (hp0 : p ≠ 0)⟩ := Nat.find_spec h0
  have : p ≠ 1 := ne_of_mem_of_not_mem hp P.one_notMem
  have prime : p.Prime := Nat.prime_iff_not_exists_mul_eq.mpr <| .intro (by lia)
    fun ⟨m, n, hm, hn, eq⟩ ↦ have := mul_ne_zero_iff.mp (eq ▸ hp0)
    (h.mem_or_mem (eq ▸ hp)).elim (Nat.find_min h0 hm ⟨·, this.1⟩) (Nat.find_min h0 hn ⟨·, this.2⟩)
  push Not at hsp
  have ⟨q, hq, hqp⟩ := SetLike.exists_of_lt
    ((P.span_singleton_le_iff_mem.mpr hp).lt_of_ne (hsp p prime).symm)
  obtain rfl | hn1 := eq_or_ne n 0
  · exact Ideal.zero_mem _
  have : n ≠ 1 := Nat.mem_maximalIdeal_iff.mp hn
  have ⟨a, b, eq⟩ := Nat.exists_add_mul_eq_of_gcd_dvd_of_mul_pred_le p q _
    (by simp [prime.coprime_iff_not_dvd.mpr (Ideal.mem_span_singleton.not.mp hqp)])
    (Nat.lt_pow_self (show 1 < n by lia)).le
  exact h.mem_of_pow_mem _ (eq ▸ add_mem (P.mul_mem_left _ hp) (P.mul_mem_left _ hq))
Cleared hypotheses: hsp
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.NatInt (goal)
Goal:
⊢ ringKrullDim ℕ = 2
have refine_1 : ∀ (s : LTSeries (PrimeSpectrum ℕ))
  (hs :
    LT.lt (α := WithBot ℕ∞) 2
      (↑(s.length (r :=
            {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
              match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
              | (a, b) => a < b})) :
        WithBot ℕ∞)),
  False
Tactic: refine le_antisymm (iSup_le fun s ↦ le_of_not_gt fun hs ↦ ?_) ?_
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.NatInt (goal)
Goal:
refine_1 :
  ∀ (s : LTSeries (PrimeSpectrum ℕ))
    (hs :
      LT.lt (α := WithBot ℕ∞) 2
        (↑(s.length (r :=
              {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
                match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
                | (a, b) => a < b})) :
          WithBot ℕ∞)),
    False
⊢ ringKrullDim ℕ = 2
have refine_2 : 2 ≤ ringKrullDim ℕ
Tactic: refine le_antisymm (iSup_le fun s ↦ le_of_not_gt fun hs ↦ ?_) ?_
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.NatInt (hypothesis)
Goal:
s : LTSeries (PrimeSpectrum ℕ)
hs :
  LT.lt (α := WithBot ℕ∞) 2
    (↑(s.length (r :=
          {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
            match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
            | (a, b) => a < b})) :
      WithBot ℕ∞)
⊢ False
have hs₁ : 2 <
  s.length (r :=
    {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
      match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
      | (a, b) => a < b})
Tactic: replace hs : 2 < s.length := ENat.coe_lt_coe.mp (WithBot.coe_lt_coe.mp hs)
Cleared hypotheses: hs
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.NatInt (hypothesis)
Goal:
s_1 : LTSeries (PrimeSpectrum ℕ)
hs :
  2 <
    s_1.length (r :=
      {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
        match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
        | (a, b) => a < b})
s : RelSeries (α := PrimeSpectrum ℕ)
  {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
    match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
    | (a, b) => a < b} :=
  RelSeries.take (r :=
    {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
      match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
      | (a, b) => a < b})
    s_1
    (⟨3, ⋯⟩ :
      Fin
        (s_1.length (r :=
            {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
              match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
              | (a, b) => a < b}) +
          1))
⊢ False
have this : NeZero
  (s.length (r :=
    {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
      match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
      | (a, b) => a < b}))
Tactic: have : NeZero s.length := ⟨three_ne_zero⟩
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.NatInt (hypothesis)
Goal:
s_1 : LTSeries (PrimeSpectrum ℕ)
hs :
  2 <
    s_1.length (r :=
      {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
        match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
        | (a, b) => a < b})
s : RelSeries (α := PrimeSpectrum ℕ)
  {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
    match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
    | (a, b) => a < b} :=
  RelSeries.take (r :=
    {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
      match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
      | (a, b) => a < b})
    s_1
    (⟨3, ⋯⟩ :
      Fin
        (s_1.length (r :=
            {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
              match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
              | (a, b) => a < b}) +
          1))
this :
  NeZero
    (s.length (r :=
      {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
        match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
        | (a, b) => a < b}))
⊢ False
have h1 : ⊥ <
  (s.toFun (r :=
      {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
        match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
        | (a, b) => a < b})
      1).asIdeal
Tactic: have h1 : ⊥ < (s 1).asIdeal := bot_le.trans_lt (s.step 0)
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.NatInt (goal)
Goal:
s_1 : LTSeries (PrimeSpectrum ℕ)
hs :
  2 <
    s_1.length (r :=
      {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
        match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
        | (a, b) => a < b})
s : RelSeries (α := PrimeSpectrum ℕ)
  {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
    match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
    | (a, b) => a < b} :=
  RelSeries.take (r :=
    {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
      match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
      | (a, b) => a < b})
    s_1
    (⟨3, ⋯⟩ :
      Fin
        (s_1.length (r :=
            {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
              match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
              | (a, b) => a < b}) +
          1))
this :
  NeZero
    (s.length (r :=
      {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
        match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
        | (a, b) => a < b}))
h1 :
  ⊥ <
    (s.toFun (r :=
        {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
          match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
          | (a, b) => a < b})
        1).asIdeal
⊢ False
have refine_1 : ∀
  (hmax :
    (s.toFun (r :=
          {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
            match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
            | (a, b) => a < b})
          1).asIdeal =
      maximalIdeal ℕ),
  False
Tactic: obtain hmax | ⟨p, hp, hsp⟩ := (Ideal.isPrime_nat_iff.mp (s 1).2).resolve_left h1.ne'
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.NatInt (goal)
Goal:
s_1 : LTSeries (PrimeSpectrum ℕ)
hs :
  2 <
    s_1.length (r :=
      {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
        match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
        | (a, b) => a < b})
s : RelSeries (α := PrimeSpectrum ℕ)
  {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
    match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
    | (a, b) => a < b} :=
  RelSeries.take (r :=
    {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
      match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
      | (a, b) => a < b})
    s_1
    (⟨3, ⋯⟩ :
      Fin
        (s_1.length (r :=
            {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
              match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
              | (a, b) => a < b}) +
          1))
this :
  NeZero
    (s.length (r :=
      {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
        match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
        | (a, b) => a < b}))
h1 :
  ⊥ <
    (s.toFun (r :=
        {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
          match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
          | (a, b) => a < b})
        1).asIdeal
refine_1 :
  ∀
    (hmax :
      (s.toFun (r :=
            {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
              match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
              | (a, b) => a < b})
            1).asIdeal =
        maximalIdeal ℕ),
    False
⊢ False
have refine_1₁ : ∀ (p : ℕ) (hp : Nat.Prime p)
  (hsp :
    (s.toFun (r :=
          {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
            match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
            | (a, b) => a < b})
          1).asIdeal =
      span {p}),
  False
Tactic: obtain hmax | ⟨p, hp, hsp⟩ := (Ideal.isPrime_nat_iff.mp (s 1).2).resolve_left h1.ne'
-- Lean 4 proof step from Mathlib.RingTheory.Ideal.NatInt (goal)
Goal:
s_1 : LTSeries (PrimeSpectrum ℕ)
hs :
  2 <
    s_1.length (r :=
      {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
        match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
        | (a, b) => a < b})
s : RelSeries (α := PrimeSpectrum ℕ)
  {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
    match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
    | (a, b) => a < b} :=
  RelSeries.take (r :=
    {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
      match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
      | (a, b) => a < b})
    s_1
    (⟨3, ⋯⟩ :
      Fin
        (s_1.length (r :=
            {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
              match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
              | (a, b) => a < b}) +
          1))
this :
  NeZero
    (s.length (r :=
      {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
        match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
        | (a, b) => a < b}))
h1 :
  ⊥ <
    (s.toFun (r :=
        {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
          match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
          | (a, b) => a < b})
        1).asIdeal
p : ℕ
hp : Nat.Prime p
hsp :
  (s.toFun (r :=
        {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
          match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
          | (a, b) => a < b})
        1).asIdeal =
    span {p}
⊢ False
have refine_1 : ∀
  (hmax :
    (s.toFun (r :=
          {x : PrimeSpectrum ℕ × PrimeSpectrum ℕ |
            match (motive := PrimeSpectrum ℕ × PrimeSpectrum ℕ → Prop) x with
            | (a, b) => a < b})
          2).asIdeal =
      maximalIdeal ℕ),
  False
Tactic: obtain hmax | ⟨q, hq, hsq⟩ :=
      (Ideal.isPrime_nat_iff.mp (s 2).2).resolve_left (h1.trans (s.step 1)).ne'