internlm/Lean-Github huggingface dataset:
Theorem: eq_or_eq_conj_iff_associated'_of_nonneg
Proof state before:
case neg.inr.h.calc_1
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
v : ℤ
hv1 : ↑u = ↑v
hxre : ¬x.re = 0
habsv : v = -1
hu : -x = A
⊢ A.re = -x.re
Tactic:
simp only [← hu, Zsqrtd.neg_re]
Proof state after:
no goals
Theorem: eq_or_eq_conj_iff_associated'_of_nonneg
Proof state before:
case neg.inr.h.calc_2
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
v : ℤ
hv1 : ↑u = ↑v
hxre : ¬x.re = 0
habsv : v = -1
hu : -x = A
⊢ -x.re < 0
Tactic:
simp only [neg_neg_iff_pos]
Proof state after:
case neg.inr.h.calc_2
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
v : ℤ
hv1 : ↑u = ↑v
hxre : ¬x.re = 0
habsv : v = -1
hu : -x = A
⊢ 0 < x.re
Theorem: eq_or_eq_conj_iff_associated'_of_nonneg
Proof state before:
case neg.inr.h.calc_2
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
v : ℤ
hv1 : ↑u = ↑v
hxre : ¬x.re = 0
habsv : v = -1
hu : -x = A
⊢ 0 < x.re
Tactic:
exact lt_of_le_of_ne hx (Ne.symm hxre)
Proof state after:
no goals
Theorem: eq_or_eq_conj_iff_associated'_of_nonneg
Proof state before:
case mp.inr.intro.intro.intro
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
hu : x * ↑u = star A
v : ℤ
hv1 : ↑u = ↑v
hv2 : abs v = 1
⊢ x = A ∨ x = star A
Tactic:
by_cases hxre : x.re = 0
Proof state after:
case pos
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
hu : x * ↑u = star A
v : ℤ
hv1 : ↑u = ↑v
hv2 : abs v = 1
hxre : x.re = 0
⊢ x = A ∨ x = star A

case neg
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
hu : x * ↑u = star A
v : ℤ
hv1 : ↑u = ↑v
hv2 : abs v = 1
hxre : ¬x.re = 0
⊢ x = A ∨ x = star A
Theorem: eq_or_eq_conj_iff_associated'_of_nonneg
Proof state before:
case pos
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
hu : x * ↑u = star A
v : ℤ
hv1 : ↑u = ↑v
hv2 : abs v = 1
hxre : x.re = 0
⊢ x = A ∨ x = star A
Tactic:
convert(eq_or_eq_conj_of_associated_of_re_zero hxre ⟨u, hu⟩).symm
Proof state after:
case h.e'_1.h.e'_3
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
hu : x * ↑u = star A
v : ℤ
hv1 : ↑u = ↑v
hv2 : abs v = 1
hxre : x.re = 0
⊢ A = star (star A)
Theorem: eq_or_eq_conj_iff_associated'_of_nonneg
Proof state before:
case h.e'_1.h.e'_3
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
hu : x * ↑u = star A
v : ℤ
hv1 : ↑u = ↑v
hv2 : abs v = 1
hxre : x.re = 0
⊢ A = star (star A)
Tactic:
rw [star_star]
Proof state after:
no goals
Theorem: eq_or_eq_conj_iff_associated'_of_nonneg
Proof state before:
case neg
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
hu : x * ↑u = star A
v : ℤ
hv1 : ↑u = ↑v
hv2 : abs v = 1
hxre : ¬x.re = 0
⊢ x = A ∨ x = star A
Tactic:
rw [hv1] at hu
Proof state after:
case neg
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
v : ℤ
hu : x * ↑v = star A
hv1 : ↑u = ↑v
hv2 : abs v = 1
hxre : ¬x.re = 0
⊢ x = A ∨ x = star A
Theorem: eq_or_eq_conj_iff_associated'_of_nonneg
Proof state before:
case neg
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
v : ℤ
hu : x * ↑v = star A
hv1 : ↑u = ↑v
hv2 : abs v = 1
hxre : ¬x.re = 0
⊢ x = A ∨ x = star A
Tactic:
rw [abs_eq <| zero_le_one' ℤ] at hv2
Proof state after:
case neg
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
v : ℤ
hu : x * ↑v = star A
hv1 : ↑u = ↑v
hv2 : v = 1 ∨ v = -1
hxre : ¬x.re = 0
⊢ x = A ∨ x = star A
Theorem: eq_or_eq_conj_iff_associated'_of_nonneg
Proof state before:
case neg.inr
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
v : ℤ
hu : x * ↑v = star A
hv1 : ↑u = ↑v
hxre : ¬x.re = 0
habsv : v = -1
⊢ x = A ∨ x = star A
Tactic:
exfalso
Proof state after:
case neg.inr.h
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
v : ℤ
hu : x * ↑v = star A
hv1 : ↑u = ↑v
hxre : ¬x.re = 0
habsv : v = -1
⊢ False
Theorem: eq_or_eq_conj_iff_associated'_of_nonneg
Proof state before:
case neg.inr.h
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
v : ℤ
hu : x * ↑v = star A
hv1 : ↑u = ↑v
hxre : ¬x.re = 0
habsv : v = -1
⊢ False
Tactic:
simp only [habsv, mul_one, Int.cast_one, mul_neg, Int.cast_neg] at hu
Proof state after:
case neg.inr.h
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
v : ℤ
hv1 : ↑u = ↑v
hxre : ¬x.re = 0
habsv : v = -1
hu : -x = star A
⊢ False
Theorem: eq_or_eq_conj_iff_associated'_of_nonneg
Proof state before:
case neg.inr.h
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
v : ℤ
hv1 : ↑u = ↑v
hxre : ¬x.re = 0
habsv : v = -1
hu : -x = star A
⊢ False
Tactic:
apply lt_irrefl (0 : ℤ)
Proof state after:
case neg.inr.h
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
v : ℤ
hv1 : ↑u = ↑v
hxre : ¬x.re = 0
habsv : v = -1
hu : -x = star A
⊢ 0 < 0
Theorem: eq_or_eq_conj_iff_associated'_of_nonneg
Proof state before:
case neg.inr.h
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
v : ℤ
hv1 : ↑u = ↑v
hxre : ¬x.re = 0
habsv : v = -1
hu : -x = star A
⊢ 0 < 0
Tactic:
calc
  0 ≤ A.re := hA
  _ = -x.re := ?_
  _ < 0 := ?_
Proof state after:
case neg.inr.h.calc_1
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
v : ℤ
hv1 : ↑u = ↑v
hxre : ¬x.re = 0
habsv : v = -1
hu : -x = star A
⊢ A.re = -x.re

case neg.inr.h.calc_2
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
v : ℤ
hv1 : ↑u = ↑v
hxre : ¬x.re = 0
habsv : v = -1
hu : -x = star A
⊢ -x.re < 0
Theorem: eq_or_eq_conj_iff_associated'_of_nonneg
Proof state before:
case neg.inr.h.calc_1
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
v : ℤ
hv1 : ↑u = ↑v
hxre : ¬x.re = 0
habsv : v = -1
hu : -x = star A
⊢ A.re = -x.re
Tactic:
rw [← star_star A, ← hu]
Proof state after:
case neg.inr.h.calc_1
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
v : ℤ
hv1 : ↑u = ↑v
hxre : ¬x.re = 0
habsv : v = -1
hu : -x = star A
⊢ (star (-x)).re = -x.re
Theorem: eq_or_eq_conj_iff_associated'_of_nonneg
Proof state before:
case neg.inr.h.calc_1
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
v : ℤ
hv1 : ↑u = ↑v
hxre : ¬x.re = 0
habsv : v = -1
hu : -x = star A
⊢ (star (-x)).re = -x.re
Tactic:
simp only [Zsqrtd.neg_re, Zsqrtd.star_re]
Proof state after:
no goals
Theorem: eq_or_eq_conj_iff_associated'_of_nonneg
Proof state before:
case neg.inr.h.calc_2
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
v : ℤ
hv1 : ↑u = ↑v
hxre : ¬x.re = 0
habsv : v = -1
hu : -x = star A
⊢ -x.re < 0
Tactic:
simp only [neg_neg_iff_pos]
Proof state after:
case neg.inr.h.calc_2
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
v : ℤ
hv1 : ↑u = ↑v
hxre : ¬x.re = 0
habsv : v = -1
hu : -x = star A
⊢ 0 < x.re
Theorem: eq_or_eq_conj_iff_associated'_of_nonneg
Proof state before:
case neg.inr.h.calc_2
x A : ℤ√(-3)
hx : 0 ≤ x.re
hA : 0 ≤ A.re
u : (ℤ√(-3))ˣ
v : ℤ
hv1 : ↑u = ↑v
hxre : ¬x.re = 0
habsv : v = -1
hu : -x = star A
⊢ 0 < x.re
Tactic:
apply lt_of_le_of_ne hx (Ne.symm hxre)
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
⊢ ∃ g, factorization' hcoprime = 3 • g
Tactic:
obtain ⟨h1, h2⟩ := factorization'_prop hcoprime
Proof state after:
case intro
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
h1 : a = Multiset.prod (factorization' hcoprime) ∨ a = -Multiset.prod (factorization' hcoprime)
h2 : ∀ (pq : ℤ√(-3)), pq ∈ factorization' hcoprime → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
⊢ ∃ g, factorization' hcoprime = 3 • g
Theorem: step5'
Proof state before:
case intro
a : ℤ√(-3)
r : ℤ
hcoprime : IsCoprime a.re a.im
hcube : r ^ 3 = Zsqrtd.norm a
h1 : a = Multiset.prod (factorization' hcoprime) ∨ a = -Multiset.prod (factorization' hcoprime)
h2 : ∀ (pq : ℤ√(-3)), pq ∈ factorization' hcoprime → 0 ≤ pq.re ∧ pq.im ≠ 0 ∧ OddPrimeOrFour (Zsqrtd.norm pq)
⊢ ∃ g, factorization' hcoprime = 3 • g
Tactic:
set f := factorization' hcoprime with hf
Proof state after:
case 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
⊢ ∃ g, f = 3 • g
Theorem: step5'
Proof state before:
case 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
⊢ ∃ g, f = 3 • g
Tactic:
apply Multiset.exists_smul_of_dvd_count
Proof state after:
case intro.h
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
⊢ ∀ (a : ℤ√(-3)), a ∈ f → 3 ∣ Multiset.count a f
Theorem: step5'
Proof state before:
case intro.h
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
⊢ ∀ (a : ℤ√(-3)), a ∈ f → 3 ∣ Multiset.count a f
Tactic:
intro x hx
Proof state after:
case intro.h
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
⊢ 3 ∣ Multiset.count x f
Theorem: step5'
Proof state before:
case intro.h
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
⊢ 3 ∣ Multiset.count x f
Tactic:
convert dvd_mul_right 3 (Multiset.count (Zsqrtd.norm x) (factorsOddPrimeOrFour r))
Proof state after:
case h.e'_4
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
⊢ Multiset.count x f = 3 * Multiset.count (Zsqrtd.norm x) (factorsOddPrimeOrFour r)
Theorem: step5'
Proof state before:
case h.e'_4
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
⊢ Multiset.count x f = 3 * Multiset.count (Zsqrtd.norm x) (factorsOddPrimeOrFour r)
Tactic:
have : Even (evenFactorExp r) :=
  by
  suffices Even (3 * evenFactorExp r)
    by
    rw [Nat.even_mul] at this
    apply this.resolve_left
    norm_num
  rw [← evenFactorExp.pow r 3, hcube]
  exact factors_2_even' hcoprime
Proof state after:
case h.e'_4
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 = 3 * Multiset.count (Zsqrtd.norm x) (factorsOddPrimeOrFour r)
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
⊢ x = A ↔ Zsqrtd.norm x = Zsqrtd.norm A

case h.e'_4.calc_2
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) (Multiset.map Zsqrtd.norm f) =
    Multiset.countp (fun x_1 => Zsqrtd.norm x = x_1) (factorsOddPrimeOrFour (Zsqrtd.norm a))
Tactic:
swap
Proof state after:
case h.e'_4.calc_2
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) (Multiset.map Zsqrtd.norm f) =
    Multiset.countp (fun x_1 => Zsqrtd.norm x = x_1) (factorsOddPrimeOrFour (Zsqrtd.norm a))

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
⊢ x = A ↔ Zsqrtd.norm x = Zsqrtd.norm A
Theorem: step5'
Proof state before:
case h.e'_4.calc_2
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) (Multiset.map Zsqrtd.norm f) =
    Multiset.countp (fun x_1 => Zsqrtd.norm x = x_1) (factorsOddPrimeOrFour (Zsqrtd.norm a))

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
⊢ x = A ↔ Zsqrtd.norm x = Zsqrtd.norm A
Tactic:
show
  Multiset.countp (Eq x.norm) (Multiset.map Zsqrtd.norm f) =
    Multiset.countp (Eq x.norm) (factorsOddPrimeOrFour a.norm)
Proof state after:
case h.e'_4.calc_2
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 (Eq (Zsqrtd.norm x)) (Multiset.map Zsqrtd.norm f) =
    Multiset.countp (Eq (Zsqrtd.norm x)) (factorsOddPrimeOrFour (Zsqrtd.norm a))

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
⊢ x = A ↔ Zsqrtd.norm x = Zsqrtd.norm A
Theorem: step5'
Proof state before:
case h.e'_4.calc_2
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 (Eq (Zsqrtd.norm x)) (Multiset.map Zsqrtd.norm f) =
    Multiset.countp (Eq (Zsqrtd.norm x)) (factorsOddPrimeOrFour (Zsqrtd.norm a))

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
⊢ x = A ↔ Zsqrtd.norm x = Zsqrtd.norm A
Tactic:
congr
Proof state after:
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)

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
⊢ x = A ↔ Zsqrtd.norm x = Zsqrtd.norm A
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
⊢ x = A ↔ Zsqrtd.norm x = Zsqrtd.norm A
Tactic:
have h2x := h2 x hx
Proof state after:
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)
⊢ x = A ↔ Zsqrtd.norm x = Zsqrtd.norm A
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)
⊢ x = A ↔ Zsqrtd.norm x = Zsqrtd.norm A
Tactic:
refine' ⟨congr_arg _, fun h => _⟩
Proof state after:
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
⊢ x = A
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
⊢ x = A
Tactic:
have hassoc := factorization'.associated'_of_norm_eq hcoprime hx HA h
Proof state after:
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
⊢ x = A
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
⊢ x = A
Tactic:
have eq_or_eq_conj := (eq_or_eq_conj_iff_associated'_of_nonneg h2x.1 (h2 A HA).1).mp hassoc
Proof state after:
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
⊢ x = A
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
⊢ x = A
Tactic:
refine' eq_or_eq_conj.resolve_right fun H => _
Proof state after:
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