internlm/Lean-Github huggingface dataset:
Theorem: step5'
Proof state before:
case h.e'_4.calc_2.e_s.hassoc.inr
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h2 : ∀ pq ∈ f, 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour pq.norm.natAbs
hf : f = factorization' hcoprime
x : ℤ√(-3)
hx : x ∈ f
this : Even (evenFactorExp r)
h1 : a = -f.prod
⊢ f.prod.norm.natAbs = (-f.prod).norm.natAbs
Tactic:
rw [Zsqrtd.norm_neg]
Proof state after:
no goals
Theorem: step5'
Proof state before:
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = f.prod ∨ a = -f.prod
h2 : ∀ pq ∈ f, 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour pq.norm.natAbs
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 x.norm.natAbs
h : x = A
⊢ x.norm.natAbs = A.norm.natAbs
Tactic:
rw [h]
Proof state after:
no goals
Theorem: step5'
Proof state before:
case h.e'_4.calc_1.hp
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = f.prod ∨ a = -f.prod
h2 : ∀ pq ∈ f, 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour pq.norm.natAbs
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 x.norm.natAbs
h : Associated x.norm A.norm
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
⊢ ¬IsUnit ?m.81883
Tactic:
intro HH
Proof state after:
case h.e'_4.calc_1.hp
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = f.prod ∨ a = -f.prod
h2 : ∀ pq ∈ f, 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour pq.norm.natAbs
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 x.norm.natAbs
h : Associated x.norm A.norm
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
HH : IsUnit ?m.81883
⊢ False
Theorem: step5'
Proof state before:
case h.e'_4.calc_1.hp
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = f.prod ∨ a = -f.prod
h2 : ∀ pq ∈ f, 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour pq.norm.natAbs
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 x.norm.natAbs
h : Associated x.norm A.norm
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
HH : IsUnit ?m.81883
⊢ False
Tactic:
apply h2x.2.1
Proof state after:
case h.e'_4.calc_1.hp
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = f.prod ∨ a = -f.prod
h2 : ∀ pq ∈ f, 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour pq.norm.natAbs
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 x.norm.natAbs
h : Associated x.norm A.norm
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
HH : IsUnit ?m.81883
⊢ x.im = 0
Theorem: step5'
Proof state before:
case h.e'_4.calc_1.hp
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = f.prod ∨ a = -f.prod
h2 : ∀ pq ∈ f, 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour pq.norm.natAbs
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 x.norm.natAbs
h : Associated x.norm A.norm
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
HH : IsUnit ?m.81883
⊢ x.im = 0
Tactic:
rw [Zsqrt3.isUnit_iff] at HH
Proof state after:
case h.e'_4.calc_1.hp
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = f.prod ∨ a = -f.prod
h2 : ∀ pq ∈ f, 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour pq.norm.natAbs
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 x.norm.natAbs
h : Associated x.norm A.norm
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
HH : |?m.81907.re| = 1 ∧ ?m.81907.im = 0
⊢ x.im = 0

a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = f.prod ∨ a = -f.prod
h2 : ∀ pq ∈ f, 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour pq.norm.natAbs
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 x.norm.natAbs
h : Associated x.norm A.norm
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
⊢ ℤ√(-3)
Theorem: step5'
Proof state before:
case h.e'_4.calc_1.hp
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = f.prod ∨ a = -f.prod
h2 : ∀ pq ∈ f, 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour pq.norm.natAbs
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 x.norm.natAbs
h : Associated x.norm A.norm
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
HH : |?m.81907.re| = 1 ∧ ?m.81907.im = 0
⊢ x.im = 0

a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = f.prod ∨ a = -f.prod
h2 : ∀ pq ∈ f, 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour pq.norm.natAbs
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 x.norm.natAbs
h : Associated x.norm A.norm
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
⊢ ℤ√(-3)
Tactic:
exact HH.2
Proof state after:
no goals
Theorem: step5'
Proof state before:
case h.e'_4.calc_1.hcoprime
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = f.prod ∨ a = -f.prod
h2 : ∀ pq ∈ f, 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour pq.norm.natAbs
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 x.norm.natAbs
h : Associated x.norm A.norm
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
⊢ IsCoprime f.prod.re f.prod.im
Tactic:
cases' h1 with h1 h1 <;> rw [h1] at hcoprime
Proof state after:
case h.e'_4.calc_1.hcoprime.inl
a : ℤ√(-3)
r : ℕ
hcoprime✝ : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3)) := factorization' hcoprime✝
hcoprime : IsCoprime f.prod.re f.prod.im
h2 : ∀ pq ∈ f, 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour pq.norm.natAbs
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 x.norm.natAbs
h : Associated x.norm A.norm
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
h1 : a = f.prod
⊢ IsCoprime f.prod.re f.prod.im

case h.e'_4.calc_1.hcoprime.inr
a : ℤ√(-3)
r : ℕ
hcoprime✝ : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3)) := factorization' hcoprime✝
hcoprime : IsCoprime (-f.prod).re (-f.prod).im
h2 : ∀ pq ∈ f, 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour pq.norm.natAbs
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 x.norm.natAbs
h : Associated x.norm A.norm
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
h1 : a = -f.prod
⊢ IsCoprime f.prod.re f.prod.im
Theorem: step5'
Proof state before:
case h.e'_4.calc_1.hcoprime.inl
a : ℤ√(-3)
r : ℕ
hcoprime✝ : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3)) := factorization' hcoprime✝
hcoprime : IsCoprime f.prod.re f.prod.im
h2 : ∀ pq ∈ f, 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour pq.norm.natAbs
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 x.norm.natAbs
h : Associated x.norm A.norm
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
h1 : a = f.prod
⊢ IsCoprime f.prod.re f.prod.im
Tactic:
exact hcoprime
Proof state after:
no goals
Theorem: step5'
Proof state before:
case h.e'_4.calc_1.hcoprime.inr
a : ℤ√(-3)
r : ℕ
hcoprime✝ : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3)) := factorization' hcoprime✝
hcoprime : IsCoprime (-f.prod).re (-f.prod).im
h2 : ∀ pq ∈ f, 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour pq.norm.natAbs
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 x.norm.natAbs
h : Associated x.norm A.norm
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
h1 : a = -f.prod
⊢ IsCoprime f.prod.re f.prod.im
Tactic:
rwa [← IsCoprime.neg_neg_iff, ← Zsqrtd.neg_im, ← Zsqrtd.neg_re]
Proof state after:
no goals
Theorem: step5'
Proof state before:
case h.e'_4.calc_1.a
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = f.prod ∨ a = -f.prod
h2 : ∀ pq ∈ f, 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour pq.norm.natAbs
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 x.norm.natAbs
h : Associated x.norm A.norm
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
⊢ x ∈ f ∧ star x ∈ f
Tactic:
refine' ⟨hx, _⟩
Proof state after:
case h.e'_4.calc_1.a
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = f.prod ∨ a = -f.prod
h2 : ∀ pq ∈ f, 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour pq.norm.natAbs
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 x.norm.natAbs
h : Associated x.norm A.norm
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
⊢ star x ∈ f
Theorem: step5'
Proof state before:
case h.e'_4.calc_1.a
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3)) := factorization' hcoprime
h1 : a = f.prod ∨ a = -f.prod
h2 : ∀ pq ∈ f, 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour pq.norm.natAbs
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 x.norm.natAbs
h : Associated x.norm A.norm
hassoc : Associated' x A
eq_or_eq_conj : x = A ∨ x = star A
H : x = star A
⊢ star x ∈ f
Tactic:
rwa [H, star_star]
Proof state after:
no goals
Theorem: step5
Proof state before:
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
⊢ ∃ p, a = p ^ 3
Tactic:
obtain ⟨f, hf⟩ := step5' a r hcoprime hcube
Proof state after:
case intro
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3))
hf : factorization' hcoprime = 3 • f
⊢ ∃ p, a = p ^ 3
Theorem: step5
Proof state before:
case intro
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3))
hf : factorization' hcoprime = 3 • f
⊢ ∃ p, a = p ^ 3
Tactic:
obtain ⟨h1, -⟩ := factorization'_prop hcoprime
Proof state after:
case intro.intro
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3))
hf : factorization' hcoprime = 3 • f
h1 : a = (factorization' hcoprime).prod ∨ a = -(factorization' hcoprime).prod
⊢ ∃ p, a = p ^ 3
Theorem: step5
Proof state before:
case intro.intro
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3))
hf : factorization' hcoprime = 3 • f
h1 : a = (factorization' hcoprime).prod ∨ a = -(factorization' hcoprime).prod
⊢ ∃ p, a = p ^ 3
Tactic:
cases h1 with
| inl h1 =>
  use f.prod
  rw [h1, hf, Multiset.prod_nsmul]
| inr h1 =>
  use -f.prod
  rw [h1, hf, Multiset.prod_nsmul, Odd.neg_pow]
  norm_num; decide
Proof state after:
no goals
Theorem: step5
Proof state before:
case intro.intro.inl
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3))
hf : factorization' hcoprime = 3 • f
h1 : a = (factorization' hcoprime).prod
⊢ ∃ p, a = p ^ 3
Tactic:
use f.prod
Proof state after:
case h
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3))
hf : factorization' hcoprime = 3 • f
h1 : a = (factorization' hcoprime).prod
⊢ a = f.prod ^ 3
Theorem: step5
Proof state before:
case h
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3))
hf : factorization' hcoprime = 3 • f
h1 : a = (factorization' hcoprime).prod
⊢ a = f.prod ^ 3
Tactic:
rw [h1, hf, Multiset.prod_nsmul]
Proof state after:
no goals
Theorem: step5
Proof state before:
case intro.intro.inr
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3))
hf : factorization' hcoprime = 3 • f
h1 : a = -(factorization' hcoprime).prod
⊢ ∃ p, a = p ^ 3
Tactic:
use -f.prod
Proof state after:
case h
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3))
hf : factorization' hcoprime = 3 • f
h1 : a = -(factorization' hcoprime).prod
⊢ a = (-f.prod) ^ 3
Theorem: step5
Proof state before:
case h
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3))
hf : factorization' hcoprime = 3 • f
h1 : a = -(factorization' hcoprime).prod
⊢ a = (-f.prod) ^ 3
Tactic:
rw [h1, hf, Multiset.prod_nsmul, Odd.neg_pow]
Proof state after:
case h.a
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3))
hf : factorization' hcoprime = 3 • f
h1 : a = -(factorization' hcoprime).prod
⊢ Odd 3
Theorem: step5
Proof state before:
case h.a
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3))
hf : factorization' hcoprime = 3 • f
h1 : a = -(factorization' hcoprime).prod
⊢ Odd 3
Tactic:
norm_num
Proof state after:
case h.a
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3))
hf : factorization' hcoprime = 3 • f
h1 : a = -(factorization' hcoprime).prod
⊢ ¬Even 3
Theorem: step5
Proof state before:
case h.a
a : ℤ√(-3)
r : ℕ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = a.norm.natAbs
f : Multiset (ℤ√(-3))
hf : factorization' hcoprime = 3 • f
h1 : a = -(factorization' hcoprime).prod
⊢ ¬Even 3
Tactic:
decide
Proof state after:
no goals
Theorem: step6
Proof state before:
a b r : ℤ
hcoprime : IsCoprime a b
hcube : r ^ 3 = a ^ 2 + 3 * b ^ 2
⊢ ∃ p q, a = p ^ 3 - 9 * p * q ^ 2 ∧ b = 3 * p ^ 2 * q - 3 * q ^ 3
Tactic:
set A : ℤ√(-3) := ⟨a, b⟩ with hA
Proof state after:
a b r : ℤ
hcoprime : IsCoprime a b
hcube : r ^ 3 = a ^ 2 + 3 * b ^ 2
A : ℤ√(-3) := { re := a, im := b }
hA : A = { re := a, im := b }
⊢ ∃ p q, a = p ^ 3 - 9 * p * q ^ 2 ∧ b = 3 * p ^ 2 * q - 3 * q ^ 3
Theorem: step6
Proof state before:
a b r : ℤ
hcoprime : IsCoprime a b
hcube : r ^ 3 = a ^ 2 + 3 * b ^ 2
A : ℤ√(-3) := { re := a, im := b }
hA : A = { re := a, im := b }
⊢ ∃ p q, a = p ^ 3 - 9 * p * q ^ 2 ∧ b = 3 * p ^ 2 * q - 3 * q ^ 3
Tactic:
have hcube' : r.natAbs ^ 3 = A.norm.natAbs :=
  by
  rw [← Int.natAbs_pow]
  apply congr_arg
  simp only [hcube, Zsqrtd.norm_def, hA]
  ring
Proof state after:
a b r : ℤ
hcoprime : IsCoprime a b
hcube : r ^ 3 = a ^ 2 + 3 * b ^ 2
A : ℤ√(-3) := { re := a, im := b }
hA : A = { re := a, im := b }
hcube' : r.natAbs ^ 3 = A.norm.natAbs
⊢ ∃ p q, a = p ^ 3 - 9 * p * q ^ 2 ∧ b = 3 * p ^ 2 * q - 3 * q ^ 3
Theorem: step6
Proof state before:
a b r : ℤ
hcoprime : IsCoprime a b
hcube : r ^ 3 = a ^ 2 + 3 * b ^ 2
A : ℤ√(-3) := { re := a, im := b }
hA : A = { re := a, im := b }
hcube' : r.natAbs ^ 3 = A.norm.natAbs
⊢ ∃ p q, a = p ^ 3 - 9 * p * q ^ 2 ∧ b = 3 * p ^ 2 * q - 3 * q ^ 3
Tactic:
obtain ⟨p, hp⟩ := step5 A r.natAbs hcoprime hcube'
Proof state after:
case intro
a b r : ℤ
hcoprime : IsCoprime a b
hcube : r ^ 3 = a ^ 2 + 3 * b ^ 2
A : ℤ√(-3) := { re := a, im := b }
hA : A = { re := a, im := b }
hcube' : r.natAbs ^ 3 = A.norm.natAbs
p : ℤ√(-3)
hp : A = p ^ 3
⊢ ∃ p q, a = p ^ 3 - 9 * p * q ^ 2 ∧ b = 3 * p ^ 2 * q - 3 * q ^ 3
Theorem: step6
Proof state before:
case intro
a b r : ℤ
hcoprime : IsCoprime a b
hcube : r ^ 3 = a ^ 2 + 3 * b ^ 2
A : ℤ√(-3) := { re := a, im := b }
hA : A = { re := a, im := b }
hcube' : r.natAbs ^ 3 = A.norm.natAbs
p : ℤ√(-3)
hp : A = p ^ 3
⊢ ∃ p q, a = p ^ 3 - 9 * p * q ^ 2 ∧ b = 3 * p ^ 2 * q - 3 * q ^ 3
Tactic:
use p.re, p.im
Proof state after:
case h
a b r : ℤ
hcoprime : IsCoprime a b
hcube : r ^ 3 = a ^ 2 + 3 * b ^ 2
A : ℤ√(-3) := { re := a, im := b }
hA : A = { re := a, im := b }
hcube' : r.natAbs ^ 3 = A.norm.natAbs
p : ℤ√(-3)
hp : A = p ^ 3
⊢ a = p.re ^ 3 - 9 * p.re * p.im ^ 2 ∧ b = 3 * p.re ^ 2 * p.im - 3 * p.im ^ 3
Theorem: step6
Proof state before:
case h
a b r : ℤ
hcoprime : IsCoprime a b
hcube : r ^ 3 = a ^ 2 + 3 * b ^ 2
A : ℤ√(-3) := { re := a, im := b }
hA : A = { re := a, im := b }
hcube' : r.natAbs ^ 3 = A.norm.natAbs
p : ℤ√(-3)
hp : A = p ^ 3
⊢ a = p.re ^ 3 - 9 * p.re * p.im ^ 2 ∧ b = 3 * p.re ^ 2 * p.im - 3 * p.im ^ 3
Tactic:
simp only [Zsqrtd.ext_iff, pow_succ', pow_two, Zsqrtd.mul_re, Zsqrtd.mul_im] at hp
Proof state after:
case h
a b r : ℤ
hcoprime : IsCoprime a b
hcube : r ^ 3 = a ^ 2 + 3 * b ^ 2
A : ℤ√(-3) := { re := a, im := b }
hA : A = { re := a, im := b }
hcube' : r.natAbs ^ 3 = A.norm.natAbs
p : ℤ√(-3)
hp :
  a =
      p.re *
          (p.re * (p.re * (p ^ 0).re + -3 * p.im * (p ^ 0).im) + -3 * p.im * (p.re * (p ^ 0).im + p.im * (p ^ 0).re)) +
        -3 * p.im *
          (p.re * (p.re * (p ^ 0).im + p.im * (p ^ 0).re) + p.im * (p.re * (p ^ 0).re + -3 * p.im * (p ^ 0).im)) ∧
    b =
      p.re * (p.re * (p.re * (p ^ 0).im + p.im * (p ^ 0).re) + p.im * (p.re * (p ^ 0).re + -3 * p.im * (p ^ 0).im)) +
        p.im *
          (p.re * (p.re * (p ^ 0).re + -3 * p.im * (p ^ 0).im) + -3 * p.im * (p.re * (p ^ 0).im + p.im * (p ^ 0).re))
⊢ a = p.re ^ 3 - 9 * p.re * p.im ^ 2 ∧ b = 3 * p.re ^ 2 * p.im - 3 * p.im ^ 3
Theorem: step6
Proof state before:
case h
a b r : ℤ
hcoprime : IsCoprime a b
hcube : r ^ 3 = a ^ 2 + 3 * b ^ 2
A : ℤ√(-3) := { re := a, im := b }
hA : A = { re := a, im := b }
hcube' : r.natAbs ^ 3 = A.norm.natAbs
p : ℤ√(-3)
hp :
  a =
      p.re *
          (p.re * (p.re * (p ^ 0).re + -3 * p.im * (p ^ 0).im) + -3 * p.im * (p.re * (p ^ 0).im + p.im * (p ^ 0).re)) +
        -3 * p.im *
          (p.re * (p.re * (p ^ 0).im + p.im * (p ^ 0).re) + p.im * (p.re * (p ^ 0).re + -3 * p.im * (p ^ 0).im)) ∧
    b =
      p.re * (p.re * (p.re * (p ^ 0).im + p.im * (p ^ 0).re) + p.im * (p.re * (p ^ 0).re + -3 * p.im * (p ^ 0).im)) +
        p.im *
          (p.re * (p.re * (p ^ 0).re + -3 * p.im * (p ^ 0).im) + -3 * p.im * (p.re * (p ^ 0).im + p.im * (p ^ 0).re))
⊢ a = p.re ^ 3 - 9 * p.re * p.im ^ 2 ∧ b = 3 * p.re ^ 2 * p.im - 3 * p.im ^ 3
Tactic:
convert hp using 2 <;> simp <;> ring
Proof state after:
no goals
Theorem: step6
Proof state before:
a b r : ℤ
hcoprime : IsCoprime a b
hcube : r ^ 3 = a ^ 2 + 3 * b ^ 2
A : ℤ√(-3) := { re := a, im := b }
hA : A = { re := a, im := b }
⊢ r.natAbs ^ 3 = A.norm.natAbs
Tactic:
rw [← Int.natAbs_pow]
Proof state after:
a b r : ℤ
hcoprime : IsCoprime a b
hcube : r ^ 3 = a ^ 2 + 3 * b ^ 2
A : ℤ√(-3) := { re := a, im := b }
hA : A = { re := a, im := b }
⊢ (r ^ 3).natAbs = A.norm.natAbs
Theorem: step6
Proof state before:
a b r : ℤ
hcoprime : IsCoprime a b
hcube : r ^ 3 = a ^ 2 + 3 * b ^ 2
A : ℤ√(-3) := { re := a, im := b }
hA : A = { re := a, im := b }
⊢ (r ^ 3).natAbs = A.norm.natAbs
Tactic:
apply congr_arg
Proof state after:
case h
a b r : ℤ
hcoprime : IsCoprime a b
hcube : r ^ 3 = a ^ 2 + 3 * b ^ 2
A : ℤ√(-3) := { re := a, im := b }
hA : A = { re := a, im := b }
⊢ r ^ 3 = A.norm
Theorem: step6
Proof state before:
case h
a b r : ℤ
hcoprime : IsCoprime a b
hcube : r ^ 3 = a ^ 2 + 3 * b ^ 2
A : ℤ√(-3) := { re := a, im := b }
hA : A = { re := a, im := b }
⊢ r ^ 3 = A.norm
Tactic:
simp only [hcube, Zsqrtd.norm_def, hA]
Proof state after:
case h
a b r : ℤ
hcoprime : IsCoprime a b
hcube : r ^ 3 = a ^ 2 + 3 * b ^ 2
A : ℤ√(-3) := { re := a, im := b }
hA : A = { re := a, im := b }
⊢ a ^ 2 + 3 * b ^ 2 = a * a - -3 * b * b
Theorem: step6
Proof state before:
case h
a b r : ℤ
hcoprime : IsCoprime a b
hcube : r ^ 3 = a ^ 2 + 3 * b ^ 2
A : ℤ√(-3) := { re := a, im := b }
hA : A = { re := a, im := b }
⊢ a ^ 2 + 3 * b ^ 2 = a * a - -3 * b * b
Tactic:
ring
Proof state after:
no goals
Theorem: FltRegular.CaseI.two_lt
Proof state before:
p : ℕ
hpri : p.Prime
hp5 : 5 ≤ p
⊢ 2 < p
Tactic:
linarith
Proof state after:
no goals
Theorem: FltRegular.CaseI.aux_cong0k₁
Proof state before:
p : ℕ
hpri : p.Prime
k : Fin p
hcong : ↑↑k ≡ -1 [ZMOD ↑p]
⊢ k = ⟨p.pred, ⋯⟩
Tactic:
refine' Fin.ext _
Proof state after:
p : ℕ
hpri : p.Prime
k : Fin p
hcong : ↑↑k ≡ -1 [ZMOD ↑p]
⊢ ↑k = ↑⟨p.pred, ⋯⟩
Theorem: FltRegular.CaseI.aux_cong0k₁
Proof state before:
p : ℕ
hpri : p.Prime
k : Fin p
hcong : ↑↑k ≡ -1 [ZMOD ↑p]
⊢ ↑k = ↑⟨p.pred, ⋯⟩
Tactic:
rw [Fin.val_mk, ← ZMod.val_cast_of_lt (Fin.is_lt k)]
Proof state after:
p : ℕ
hpri : p.Prime
k : Fin p
hcong : ↑↑k ≡ -1 [ZMOD ↑p]
⊢ (↑↑k).val = ↑⟨p.pred, ⋯⟩
Theorem: FltRegular.CaseI.aux_cong0k₁
Proof state before:
p : ℕ
hpri : p.Prime
k : Fin p
hcong : ↑↑k ≡ -1 [ZMOD ↑p]
⊢ (↑↑k).val = ↑⟨p.pred, ⋯⟩
Tactic:
suffices ((k : ℤ) : ZMod p).val = p.pred by simpa
Proof state after:
p : ℕ
hpri : p.Prime
k : Fin p
hcong : ↑↑k ≡ -1 [ZMOD ↑p]
⊢ (↑↑↑k).val = p.pred
Theorem: FltRegular.CaseI.aux_cong0k₁
Proof state before:
p : ℕ
hpri : p.Prime
k : Fin p
hcong : ↑↑k ≡ -1 [ZMOD ↑p]
⊢ (↑↑↑k).val = p.pred
Tactic:
rw [← ZMod.intCast_eq_intCast_iff] at hcong
Proof state after:
p : ℕ
hpri : p.Prime
k : Fin p
hcong : ↑↑↑k = ↑(-1)
⊢ (↑↑↑k).val = p.pred
Theorem: FltRegular.CaseI.aux_cong0k₁
Proof state before:
p : ℕ
hpri : p.Prime
k : Fin p
hcong : ↑↑↑k = ↑(-1)
⊢ (↑↑↑k).val = p.pred
Tactic:
rw [hcong, cast_neg, Int.cast_one, pred_eq_sub_one]
Proof state after:
p : ℕ
hpri : p.Prime
k : Fin p
hcong : ↑↑↑k = ↑(-1)
⊢ (-1).val = p - 1
Theorem: FltRegular.CaseI.aux_cong0k₁
Proof state before:
p : ℕ
hpri : p.Prime
k : Fin p
hcong : ↑↑↑k = ↑(-1)
⊢ (-1).val = p - 1
Tactic:
haveI : NeZero p := ⟨hpri.ne_zero⟩
Proof state after:
p : ℕ
hpri : p.Prime
k : Fin p
hcong : ↑↑↑k = ↑(-1)
this : NeZero p
⊢ (-1).val = p - 1
Theorem: FltRegular.CaseI.aux_cong0k₁
Proof state before:
p : ℕ
hpri : p.Prime
k : Fin p
hcong : ↑↑↑k = ↑(-1)
this : NeZero p
⊢ (-1).val = p - 1
Tactic:
haveI : Fact p.Prime := ⟨hpri⟩
Proof state after:
p : ℕ
hpri : p.Prime
k : Fin p
hcong : ↑↑↑k = ↑(-1)
this✝ : NeZero p
this : Fact p.Prime
⊢ (-1).val = p - 1
Theorem: FltRegular.CaseI.aux_cong0k₁
Proof state before:
p : ℕ
hpri : p.Prime
k : Fin p
hcong : ↑↑↑k = ↑(-1)
this✝ : NeZero p
this : Fact p.Prime
⊢ (-1).val = p - 1
Tactic:
haveI : Fact (1 < p) := ⟨hpri.one_lt⟩
Proof state after:
p : ℕ
hpri : p.Prime
k : Fin p
hcong : ↑↑↑k = ↑(-1)
this✝¹ : NeZero p
this✝ : Fact p.Prime
this : Fact (1 < p)
⊢ (-1).val = p - 1
Theorem: FltRegular.CaseI.aux_cong0k₁
Proof state before:
p : ℕ
hpri : p.Prime
k : Fin p
hcong : ↑↑↑k = ↑(-1)
this✝¹ : NeZero p
this✝ : Fact p.Prime
this : Fact (1 < p)
⊢ (-1).val = p - 1
Tactic:
simp [ZMod.neg_val, ZMod.val_one]
Proof state after:
no goals
Theorem: FltRegular.CaseI.aux_cong0k₁
Proof state before:
p : ℕ
hpri : p.Prime
k : Fin p
hcong : ↑↑k ≡ -1 [ZMOD ↑p]
this : (↑↑↑k).val = p.pred
⊢ (↑↑k).val = ↑⟨p.pred, ⋯⟩
Tactic:
simpa
Proof state after:
no goals
Theorem: FltRegular.CaseI.auxf0k₁
Proof state before:
p : ℕ
hpri : p.Prime
hp5 : 5 ≤ p
b : ℤ
⊢ ∃ i, f0k₁ b p ↑i = 0
Tactic:
refine' ⟨⟨2, two_lt hp5⟩, _⟩
Proof state after:
p : ℕ
hpri : p.Prime
hp5 : 5 ≤ p
b : ℤ
⊢ f0k₁ b p ↑⟨2, ⋯⟩ = 0