internlm/Lean-Github huggingface dataset:
Theorem: step5'
Proof state before:
case h.e'_4.calc_1
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
A : ℤ√(-3)
HA : A ∈ f
h2x : 0 ≤ x.re ∧ x.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm x)
h : Zsqrtd.norm x = Zsqrtd.norm A
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
⊢ False
Tactic:
apply no_conj f
Proof state after:
case h.e'_4.calc_1.hp
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
A : ℤ√(-3)
HA : A ∈ f
h2x : 0 ≤ x.re ∧ x.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm x)
h : Zsqrtd.norm x = Zsqrtd.norm A
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
⊢ ¬IsUnit ?m.291656

case h.e'_4.calc_1.hcoprime
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
A : ℤ√(-3)
HA : A ∈ f
h2x : 0 ≤ x.re ∧ x.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm x)
h : Zsqrtd.norm x = Zsqrtd.norm A
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
⊢ IsCoprime (Multiset.prod f).re (Multiset.prod f).im

case h.e'_4.calc_1.a
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
A : ℤ√(-3)
HA : A ∈ f
h2x : 0 ≤ x.re ∧ x.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm x)
h : Zsqrtd.norm x = Zsqrtd.norm A
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
⊢ ?m.291656 ∈ f ∧ star ?m.291656 ∈ f

a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
A : ℤ√(-3)
HA : A ∈ f
h2x : 0 ≤ x.re ∧ x.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm x)
h : Zsqrtd.norm x = Zsqrtd.norm A
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
⊢ ℤ√(-3)
Theorem: step5'
Proof state before:
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
⊢ Even (evenFactorExp r)
Tactic:
suffices Even (3 * evenFactorExp r)
  by
  rw [Nat.even_mul] at this
  apply this.resolve_left
  norm_num
Proof state after:
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
⊢ Even (3 * evenFactorExp r)
Theorem: step5'
Proof state before:
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
⊢ Even (3 * evenFactorExp r)
Tactic:
rw [← evenFactorExp.pow r 3, hcube]
Proof state after:
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
⊢ Even (evenFactorExp (Zsqrtd.norm a))
Theorem: step5'
Proof state before:
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
⊢ Even (evenFactorExp (Zsqrtd.norm a))
Tactic:
exact factors_2_even' hcoprime
Proof state after:
no goals
Theorem: step5'
Proof state before:
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (3 * evenFactorExp r)
⊢ Even (evenFactorExp r)
Tactic:
rw [Nat.even_mul] at this
Proof state after:
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even 3 ∨ Even (evenFactorExp r)
⊢ Even (evenFactorExp r)
Theorem: step5'
Proof state before:
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even 3 ∨ Even (evenFactorExp r)
⊢ Even (evenFactorExp r)
Tactic:
apply this.resolve_left
Proof state after:
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even 3 ∨ Even (evenFactorExp r)
⊢ ¬Even 3
Theorem: step5'
Proof state before:
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even 3 ∨ Even (evenFactorExp r)
⊢ ¬Even 3
Tactic:
norm_num
Proof state after:
no goals
Theorem: step5'
Proof state before:
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
⊢ Multiset.count x f = ↑Multiset.card (Multiset.filter (fun x_1 => x = x_1) f)
Tactic:
rw [Multiset.count, Multiset.countp_eq_card_filter]
Proof state after:
no goals
Theorem: step5'
Proof state before:
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
⊢ Multiset.countp (fun x_1 => Zsqrtd.norm x = x_1) (factorsOddPrimeOrFour (Zsqrtd.norm a)) =
    Multiset.count (Zsqrtd.norm x) (factorsOddPrimeOrFour (Zsqrtd.norm a))
Tactic:
rw [Multiset.count]
Proof state after:
no goals
Theorem: step5'
Proof state before:
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
⊢ Multiset.count (Zsqrtd.norm x) (factorsOddPrimeOrFour (Zsqrtd.norm a)) =
    Multiset.count (Zsqrtd.norm x) (factorsOddPrimeOrFour (r ^ 3))
Tactic:
rw [hcube]
Proof state after:
no goals
Theorem: step5'
Proof state before:
case h.e'_4.calc_2.e_s
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
⊢ Multiset.map Zsqrtd.norm f = factorsOddPrimeOrFour (Zsqrtd.norm a)
Tactic:
apply factorsOddPrimeOrFour.unique hcoprime
Proof state after:
case h.e'_4.calc_2.e_s.hf
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
⊢ ∀ (x : ℤ), x ∈ Multiset.map Zsqrtd.norm f → OddPrimeOrFour x

case h.e'_4.calc_2.e_s.hf'
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
⊢ ∀ (x : ℤ), x ∈ Multiset.map Zsqrtd.norm f → 0 ≤ x

case h.e'_4.calc_2.e_s.hassoc
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
⊢ Associated (Multiset.prod (Multiset.map Zsqrtd.norm f)) (Zsqrtd.norm a)
Theorem: step5'
Proof state before:
case h.e'_4.calc_2.e_s.hf
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
⊢ ∀ (x : ℤ), x ∈ Multiset.map Zsqrtd.norm f → OddPrimeOrFour x
Tactic:
intro x hx
Proof state after:
case h.e'_4.calc_2.e_s.hf
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x✝ : ℤ√(-3)
hx✝ : x✝ ∈ f
this : Even (evenFactorExp r)
x : ℤ
hx : x ∈ Multiset.map Zsqrtd.norm f
⊢ OddPrimeOrFour x
Theorem: step5'
Proof state before:
case h.e'_4.calc_2.e_s.hf
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x✝ : ℤ√(-3)
hx✝ : x✝ ∈ f
this : Even (evenFactorExp r)
x : ℤ
hx : x ∈ Multiset.map Zsqrtd.norm f
⊢ OddPrimeOrFour x
Tactic:
obtain ⟨y, hy, rfl⟩ := Multiset.mem_map.mp hx
Proof state after:
case h.e'_4.calc_2.e_s.hf.intro.intro
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx✝ : x ∈ f
this : Even (evenFactorExp r)
y : ℤ√(-3)
hy : y ∈ f
hx : Zsqrtd.norm y ∈ Multiset.map Zsqrtd.norm f
⊢ OddPrimeOrFour (Zsqrtd.norm y)
Theorem: step5'
Proof state before:
case h.e'_4.calc_2.e_s.hf.intro.intro
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx✝ : x ∈ f
this : Even (evenFactorExp r)
y : ℤ√(-3)
hy : y ∈ f
hx : Zsqrtd.norm y ∈ Multiset.map Zsqrtd.norm f
⊢ OddPrimeOrFour (Zsqrtd.norm y)
Tactic:
exact (h2 y hy).2.2
Proof state after:
no goals
Theorem: step5'
Proof state before:
case h.e'_4.calc_2.e_s.hf'
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
⊢ ∀ (x : ℤ), x ∈ Multiset.map Zsqrtd.norm f → 0 ≤ x
Tactic:
intro x hx
Proof state after:
case h.e'_4.calc_2.e_s.hf'
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x✝ : ℤ√(-3)
hx✝ : x✝ ∈ f
this : Even (evenFactorExp r)
x : ℤ
hx : x ∈ Multiset.map Zsqrtd.norm f
⊢ 0 ≤ x
Theorem: step5'
Proof state before:
case h.e'_4.calc_2.e_s.hf'
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x✝ : ℤ√(-3)
hx✝ : x✝ ∈ f
this : Even (evenFactorExp r)
x : ℤ
hx : x ∈ Multiset.map Zsqrtd.norm f
⊢ 0 ≤ x
Tactic:
obtain ⟨y, -, rfl⟩ := Multiset.mem_map.mp hx
Proof state after:
case h.e'_4.calc_2.e_s.hf'.intro.intro
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx✝ : x ∈ f
this : Even (evenFactorExp r)
y : ℤ√(-3)
hx : Zsqrtd.norm y ∈ Multiset.map Zsqrtd.norm f
⊢ 0 ≤ Zsqrtd.norm y
Theorem: step5'
Proof state before:
case h.e'_4.calc_2.e_s.hf'.intro.intro
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx✝ : x ∈ f
this : Even (evenFactorExp r)
y : ℤ√(-3)
hx : Zsqrtd.norm y ∈ Multiset.map Zsqrtd.norm f
⊢ 0 ≤ Zsqrtd.norm y
Tactic:
apply Zsqrtd.norm_nonneg
Proof state after:
case h.e'_4.calc_2.e_s.hf'.intro.intro.hd
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx✝ : x ∈ f
this : Even (evenFactorExp r)
y : ℤ√(-3)
hx : Zsqrtd.norm y ∈ Multiset.map Zsqrtd.norm f
⊢ -3 ≤ 0
Theorem: step5'
Proof state before:
case h.e'_4.calc_2.e_s.hf'.intro.intro.hd
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx✝ : x ∈ f
this : Even (evenFactorExp r)
y : ℤ√(-3)
hx : Zsqrtd.norm y ∈ Multiset.map Zsqrtd.norm f
⊢ -3 ≤ 0
Tactic:
norm_num
Proof state after:
no goals
Theorem: step5'
Proof state before:
case h.e'_4.calc_2.e_s.hassoc
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
⊢ Associated (Multiset.prod (Multiset.map Zsqrtd.norm f)) (Zsqrtd.norm a)
Tactic:
rw [prod_map_norm]
Proof state after:
case h.e'_4.calc_2.e_s.hassoc
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
⊢ Associated (Zsqrtd.norm (Multiset.prod f)) (Zsqrtd.norm a)
Theorem: step5'
Proof state before:
case h.e'_4.calc_2.e_s.hassoc
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
⊢ Associated (Zsqrtd.norm (Multiset.prod f)) (Zsqrtd.norm a)
Tactic:
cases' h1 with h1 h1 <;> rw [h1]
Proof state after:
case h.e'_4.calc_2.e_s.hassoc.inr
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
h1 : a = -Multiset.prod f
⊢ Associated (Zsqrtd.norm (Multiset.prod f)) (Zsqrtd.norm (-Multiset.prod f))
Theorem: step5'
Proof state before:
case h.e'_4.calc_2.e_s.hassoc.inr
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
h1 : a = -Multiset.prod f
⊢ Associated (Zsqrtd.norm (Multiset.prod f)) (Zsqrtd.norm (-Multiset.prod f))
Tactic:
rw [Zsqrtd.norm_neg]
Proof state after:
no goals
Theorem: step5'
Proof state before:
case h.e'_4.calc_1.hp
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
A : ℤ√(-3)
HA : A ∈ f
h2x : 0 ≤ x.re ∧ x.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm x)
h : Zsqrtd.norm x = Zsqrtd.norm A
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
⊢ ¬IsUnit ?m.291656
Tactic:
intro HH
Proof state after:
case h.e'_4.calc_1.hp
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
A : ℤ√(-3)
HA : A ∈ f
h2x : 0 ≤ x.re ∧ x.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm x)
h : Zsqrtd.norm x = Zsqrtd.norm A
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
HH : IsUnit ?m.291656
⊢ False
Theorem: step5'
Proof state before:
case h.e'_4.calc_1.hp
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
A : ℤ√(-3)
HA : A ∈ f
h2x : 0 ≤ x.re ∧ x.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm x)
h : Zsqrtd.norm x = Zsqrtd.norm A
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
HH : IsUnit ?m.291656
⊢ False
Tactic:
apply h2x.2.1
Proof state after:
case h.e'_4.calc_1.hp
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
A : ℤ√(-3)
HA : A ∈ f
h2x : 0 ≤ x.re ∧ x.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm x)
h : Zsqrtd.norm x = Zsqrtd.norm A
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
HH : IsUnit ?m.291656
⊢ x.im = 0
Theorem: step5'
Proof state before:
case h.e'_4.calc_1.hp
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
A : ℤ√(-3)
HA : A ∈ f
h2x : 0 ≤ x.re ∧ x.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm x)
h : Zsqrtd.norm x = Zsqrtd.norm A
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
HH : IsUnit ?m.291656
⊢ x.im = 0
Tactic:
rw [Zsqrt3.isUnit_iff] at HH
Proof state after:
case h.e'_4.calc_1.hp
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
A : ℤ√(-3)
HA : A ∈ f
h2x : 0 ≤ x.re ∧ x.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm x)
h : Zsqrtd.norm x = Zsqrtd.norm A
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
HH : abs ?m.291680.re = 1 ∧ ?m.291680.im = 0
⊢ x.im = 0

a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = Multiset.prod f ∨ a = -Multiset.prod f
h2 : ∀ (pq : ℤ√(-3)), pq ∈ f → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
A : ℤ√(-3)
HA : A ∈ f
h2x : 0 ≤ x.re ∧ x.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm x)
h : Zsqrtd.norm x = Zsqrtd.norm A
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
⊢ ℤ√(-3)