internlm/Lean-Github huggingface dataset:
Theorem: step1a
Proof state before:
case intro
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
this✝ : Even a.re ↔ Even a.im
hre : 2 ∣ a.re
him : 2 ∣ a.im
this : IsUnit 2
⊢ False
Tactic:
rw [isUnit_iff_dvd_one] at this
Proof state after:
case intro
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
this✝ : Even a.re ↔ Even a.im
hre : 2 ∣ a.re
him : 2 ∣ a.im
this : 2 ∣ 1
⊢ False
Theorem: step1a
Proof state before:
case intro
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
this✝ : Even a.re ↔ Even a.im
hre : 2 ∣ a.re
him : 2 ∣ a.im
this : 2 ∣ 1
⊢ False
Tactic:
norm_num at this
Proof state after:
no goals
Theorem: step1a
Proof state before:
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
⊢ Even a.re ↔ Even a.im
Tactic:
simpa [Zsqrtd.norm_def, parity_simps] using heven
Proof state after:
no goals
Theorem: step1
Proof state before:
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
⊢ ∃ u, a = { re := 1, im := 1 } * u ∨ a = { re := 1, im := -1 } * u
Tactic:
obtain ⟨ha, hb⟩ := step1a hcoprime heven
Proof state after:
case intro
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
⊢ ∃ u, a = { re := 1, im := 1 } * u ∨ a = { re := 1, im := -1 } * u
Theorem: step1
Proof state before:
case intro
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
⊢ ∃ u, a = { re := 1, im := 1 } * u ∨ a = { re := 1, im := -1 } * u
Tactic:
obtain hdvd | hdvd := Int.four_dvd_add_or_sub_of_odd ha hb
Proof state after:
case intro.inl
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
hdvd : 4 ∣ a.re + a.im
⊢ ∃ u, a = { re := 1, im := 1 } * u ∨ a = { re := 1, im := -1 } * u

case intro.inr
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
hdvd : 4 ∣ a.re - a.im
⊢ ∃ u, a = { re := 1, im := 1 } * u ∨ a = { re := 1, im := -1 } * u
Theorem: step1
Proof state before:
case intro.inl
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
hdvd : 4 ∣ a.re + a.im
⊢ ∃ u, a = { re := 1, im := 1 } * u ∨ a = { re := 1, im := -1 } * u
Tactic:
obtain ⟨v, hv⟩ := hdvd
Proof state after:
case intro.inl.intro
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
v : ℤ
hv : a.re + a.im = 4 * v
⊢ ∃ u, a = { re := 1, im := 1 } * u ∨ a = { re := 1, im := -1 } * u
Theorem: step1
Proof state before:
case intro.inl.intro
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
v : ℤ
hv : a.re + a.im = 4 * v
⊢ ∃ u, a = { re := 1, im := 1 } * u ∨ a = { re := 1, im := -1 } * u
Tactic:
rw [← eq_sub_iff_add_eq] at hv
Proof state after:
case intro.inl.intro
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
v : ℤ
hv : a.re = 4 * v - a.im
⊢ ∃ u, a = { re := 1, im := 1 } * u ∨ a = { re := 1, im := -1 } * u
Theorem: step1
Proof state before:
case intro.inl.intro
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
v : ℤ
hv : a.re = 4 * v - a.im
⊢ ∃ u, a = { re := 1, im := 1 } * u ∨ a = { re := 1, im := -1 } * u
Tactic:
use ⟨v - a.im, v⟩
Proof state after:
case intro.inl.intro
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
v : ℤ
hv : a.re = 4 * v - a.im
⊢ a = { re := 1, im := 1 } * { re := v - a.im, im := v } ∨ a = { re := 1, im := -1 } * { re := v - a.im, im := v }
Theorem: step1
Proof state before:
case intro.inl.intro
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
v : ℤ
hv : a.re = 4 * v - a.im
⊢ a = { re := 1, im := 1 } * { re := v - a.im, im := v } ∨ a = { re := 1, im := -1 } * { re := v - a.im, im := v }
Tactic:
right
Proof state after:
case intro.inl.intro.h
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
v : ℤ
hv : a.re = 4 * v - a.im
⊢ a = { re := 1, im := -1 } * { re := v - a.im, im := v }
Theorem: step1
Proof state before:
case intro.inl.intro.h
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
v : ℤ
hv : a.re = 4 * v - a.im
⊢ a = { re := 1, im := -1 } * { re := v - a.im, im := v }
Tactic:
rw [Zsqrtd.ext, hv, Zsqrtd.mul_re, Zsqrtd.mul_im]
Proof state after:
case intro.inl.intro.h
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
v : ℤ
hv : a.re = 4 * v - a.im
⊢ 4 * v - a.im =
      { re := 1, im := -1 }.re * { re := v - a.im, im := v }.re +
        -3 * { re := 1, im := -1 }.im * { re := v - a.im, im := v }.im ∧
    a.im =
      { re := 1, im := -1 }.re * { re := v - a.im, im := v }.im +
        { re := 1, im := -1 }.im * { re := v - a.im, im := v }.re
Theorem: step1
Proof state before:
case intro.inl.intro.h
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
v : ℤ
hv : a.re = 4 * v - a.im
⊢ 4 * v - a.im =
      { re := 1, im := -1 }.re * { re := v - a.im, im := v }.re +
        -3 * { re := 1, im := -1 }.im * { re := v - a.im, im := v }.im ∧
    a.im =
      { re := 1, im := -1 }.re * { re := v - a.im, im := v }.im +
        { re := 1, im := -1 }.im * { re := v - a.im, im := v }.re
Tactic:
dsimp only
Proof state after:
case intro.inl.intro.h
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
v : ℤ
hv : a.re = 4 * v - a.im
⊢ 4 * v - a.im = 1 * (v - a.im) + -3 * -1 * v ∧ a.im = 1 * v + -1 * (v - a.im)
Theorem: step1
Proof state before:
case intro.inl.intro.h
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
v : ℤ
hv : a.re = 4 * v - a.im
⊢ 4 * v - a.im = 1 * (v - a.im) + -3 * -1 * v ∧ a.im = 1 * v + -1 * (v - a.im)
Tactic:
constructor <;> ring
Proof state after:
no goals
Theorem: step1
Proof state before:
case intro.inr
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
hdvd : 4 ∣ a.re - a.im
⊢ ∃ u, a = { re := 1, im := 1 } * u ∨ a = { re := 1, im := -1 } * u
Tactic:
obtain ⟨v, hv⟩ := hdvd
Proof state after:
case intro.inr.intro
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
v : ℤ
hv : a.re - a.im = 4 * v
⊢ ∃ u, a = { re := 1, im := 1 } * u ∨ a = { re := 1, im := -1 } * u
Theorem: step1
Proof state before:
case intro.inr.intro
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
v : ℤ
hv : a.re - a.im = 4 * v
⊢ ∃ u, a = { re := 1, im := 1 } * u ∨ a = { re := 1, im := -1 } * u
Tactic:
rw [sub_eq_iff_eq_add] at hv
Proof state after:
case intro.inr.intro
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
v : ℤ
hv : a.re = 4 * v + a.im
⊢ ∃ u, a = { re := 1, im := 1 } * u ∨ a = { re := 1, im := -1 } * u
Theorem: step1
Proof state before:
case intro.inr.intro
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
v : ℤ
hv : a.re = 4 * v + a.im
⊢ ∃ u, a = { re := 1, im := 1 } * u ∨ a = { re := 1, im := -1 } * u
Tactic:
use ⟨a.im + v, -v⟩
Proof state after:
case intro.inr.intro
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
v : ℤ
hv : a.re = 4 * v + a.im
⊢ a = { re := 1, im := 1 } * { re := a.im + v, im := -v } ∨ a = { re := 1, im := -1 } * { re := a.im + v, im := -v }
Theorem: step1
Proof state before:
case intro.inr.intro
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
v : ℤ
hv : a.re = 4 * v + a.im
⊢ a = { re := 1, im := 1 } * { re := a.im + v, im := -v } ∨ a = { re := 1, im := -1 } * { re := a.im + v, im := -v }
Tactic:
left
Proof state after:
case intro.inr.intro.h
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
v : ℤ
hv : a.re = 4 * v + a.im
⊢ a = { re := 1, im := 1 } * { re := a.im + v, im := -v }
Theorem: step1
Proof state before:
case intro.inr.intro.h
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
v : ℤ
hv : a.re = 4 * v + a.im
⊢ a = { re := 1, im := 1 } * { re := a.im + v, im := -v }
Tactic:
rw [Zsqrtd.ext, hv, Zsqrtd.mul_re, Zsqrtd.mul_im]
Proof state after:
case intro.inr.intro.h
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
v : ℤ
hv : a.re = 4 * v + a.im
⊢ 4 * v + a.im =
      { re := 1, im := 1 }.re * { re := a.im + v, im := -v }.re +
        -3 * { re := 1, im := 1 }.im * { re := a.im + v, im := -v }.im ∧
    a.im =
      { re := 1, im := 1 }.re * { re := a.im + v, im := -v }.im +
        { re := 1, im := 1 }.im * { re := a.im + v, im := -v }.re
Theorem: step1
Proof state before:
case intro.inr.intro.h
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
v : ℤ
hv : a.re = 4 * v + a.im
⊢ 4 * v + a.im =
      { re := 1, im := 1 }.re * { re := a.im + v, im := -v }.re +
        -3 * { re := 1, im := 1 }.im * { re := a.im + v, im := -v }.im ∧
    a.im =
      { re := 1, im := 1 }.re * { re := a.im + v, im := -v }.im +
        { re := 1, im := 1 }.im * { re := a.im + v, im := -v }.re
Tactic:
dsimp only
Proof state after:
case intro.inr.intro.h
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
v : ℤ
hv : a.re = 4 * v + a.im
⊢ 4 * v + a.im = 1 * (a.im + v) + -3 * 1 * -v ∧ a.im = 1 * -v + 1 * (a.im + v)
Theorem: step1
Proof state before:
case intro.inr.intro.h
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
ha : Odd a.re
hb : Odd a.im
v : ℤ
hv : a.re = 4 * v + a.im
⊢ 4 * v + a.im = 1 * (a.im + v) + -3 * 1 * -v ∧ a.im = 1 * -v + 1 * (a.im + v)
Tactic:
constructor <;> ring
Proof state after:
no goals
Theorem: step1'
Proof state before:
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
⊢ ∃ u, IsCoprime u.re u.im ∧ Zsqrtd.norm a = 4 * Zsqrtd.norm u
Tactic:
obtain ⟨u', hu'⟩ := step1 hcoprime heven
Proof state after:
case intro
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
u' : ℤ√(-3)
hu' : a = { re := 1, im := 1 } * u' ∨ a = { re := 1, im := -1 } * u'
⊢ ∃ u, IsCoprime u.re u.im ∧ Zsqrtd.norm a = 4 * Zsqrtd.norm u
Theorem: step1'
Proof state before:
case intro
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
u' : ℤ√(-3)
hu' : a = { re := 1, im := 1 } * u' ∨ a = { re := 1, im := -1 } * u'
⊢ ∃ u, IsCoprime u.re u.im ∧ Zsqrtd.norm a = 4 * Zsqrtd.norm u
Tactic:
refine' ⟨u', _, _⟩
Proof state after:
case intro.refine'_1
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
u' : ℤ√(-3)
hu' : a = { re := 1, im := 1 } * u' ∨ a = { re := 1, im := -1 } * u'
⊢ IsCoprime u'.re u'.im

case intro.refine'_2
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
u' : ℤ√(-3)
hu' : a = { re := 1, im := 1 } * u' ∨ a = { re := 1, im := -1 } * u'
⊢ Zsqrtd.norm a = 4 * Zsqrtd.norm u'
Theorem: step1'
Proof state before:
case intro.refine'_1
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
u' : ℤ√(-3)
hu' : a = { re := 1, im := 1 } * u' ∨ a = { re := 1, im := -1 } * u'
⊢ IsCoprime u'.re u'.im
Tactic:
apply Zsqrtd.coprime_of_dvd_coprime hcoprime
Proof state after:
case intro.refine'_1
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
u' : ℤ√(-3)
hu' : a = { re := 1, im := 1 } * u' ∨ a = { re := 1, im := -1 } * u'
⊢ u' ∣ a
Theorem: step1'
Proof state before:
case intro.refine'_1
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
u' : ℤ√(-3)
hu' : a = { re := 1, im := 1 } * u' ∨ a = { re := 1, im := -1 } * u'
⊢ u' ∣ a
Tactic:
obtain rfl | rfl := hu' <;> apply dvd_mul_left
Proof state after:
no goals
Theorem: step1'
Proof state before:
case intro.refine'_2.inr
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
u' : ℤ√(-3)
hu' : a = { re := 1, im := -1 } * u'
⊢ Zsqrtd.norm a = 4 * Zsqrtd.norm u'
Tactic:
rw [hu', Zsqrtd.norm_mul]
Proof state after:
case intro.refine'_2.inr
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
u' : ℤ√(-3)
hu' : a = { re := 1, im := -1 } * u'
⊢ Zsqrtd.norm { re := 1, im := -1 } * Zsqrtd.norm u' = 4 * Zsqrtd.norm u'
Theorem: step1'
Proof state before:
case intro.refine'_2.inr
a : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
heven : Even (Zsqrtd.norm a)
u' : ℤ√(-3)
hu' : a = { re := 1, im := -1 } * u'
⊢ Zsqrtd.norm { re := 1, im := -1 } * Zsqrtd.norm u' = 4 * Zsqrtd.norm u'
Tactic:
congr
Proof state after:
no goals
Theorem: step1''
Proof state before:
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
⊢ ∃ u, IsCoprime u.re u.im ∧ (a = p * u ∨ a = star p * u) ∧ Zsqrtd.norm a = 4 * Zsqrtd.norm u
Tactic:
obtain ⟨u, h2⟩ := step1 hcoprime heven
Proof state after:
case intro
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
h2 : a = { re := 1, im := 1 } * u ∨ a = { re := 1, im := -1 } * u
⊢ ∃ u, IsCoprime u.re u.im ∧ (a = p * u ∨ a = star p * u) ∧ Zsqrtd.norm a = 4 * Zsqrtd.norm u
Theorem: step1''
Proof state before:
case intro
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
h2 : a = { re := 1, im := 1 } * u ∨ a = { re := 1, im := -1 } * u
⊢ ∃ u, IsCoprime u.re u.im ∧ (a = p * u ∨ a = star p * u) ∧ Zsqrtd.norm a = 4 * Zsqrtd.norm u
Tactic:
set q : ℤ√(-3) := ⟨1, 1⟩
Proof state after:
case intro
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = { re := 1, im := -1 } * u
⊢ ∃ u, IsCoprime u.re u.im ∧ (a = p * u ∨ a = star p * u) ∧ Zsqrtd.norm a = 4 * Zsqrtd.norm u
Theorem: step1''
Proof state before:
case intro
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = { re := 1, im := -1 } * u
⊢ ∃ u, IsCoprime u.re u.im ∧ (a = p * u ∨ a = star p * u) ∧ Zsqrtd.norm a = 4 * Zsqrtd.norm u
Tactic:
replace h2 : a = q * u ∨ a = star q * u := h2
Proof state after:
case intro
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
⊢ ∃ u, IsCoprime u.re u.im ∧ (a = p * u ∨ a = star p * u) ∧ Zsqrtd.norm a = 4 * Zsqrtd.norm u
Theorem: step1''
Proof state before:
case intro
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
⊢ ∃ u, IsCoprime u.re u.im ∧ (a = p * u ∨ a = star p * u) ∧ Zsqrtd.norm a = 4 * Zsqrtd.norm u
Tactic:
obtain ⟨hp', hq'⟩ := Spts.four hp hq
Proof state after:
case intro.intro
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hp' : abs p.re = 1
hq' : abs p.im = 1
⊢ ∃ u, IsCoprime u.re u.im ∧ (a = p * u ∨ a = star p * u) ∧ Zsqrtd.norm a = 4 * Zsqrtd.norm u
Theorem: step1''
Proof state before:
case intro.intro
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hp' : abs p.re = 1
hq' : abs p.im = 1
⊢ ∃ u, IsCoprime u.re u.im ∧ (a = p * u ∨ a = star p * u) ∧ Zsqrtd.norm a = 4 * Zsqrtd.norm u
Tactic:
refine' ⟨p.re * u, _, _, _⟩
Proof state after:
case intro.intro.refine'_1
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hp' : abs p.re = 1
hq' : abs p.im = 1
⊢ IsCoprime (↑p.re * u).re (↑p.re * u).im

case intro.intro.refine'_2
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hp' : abs p.re = 1
hq' : abs p.im = 1
⊢ a = p * (↑p.re * u) ∨ a = star p * (↑p.re * u)

case intro.intro.refine'_3
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hp' : abs p.re = 1
hq' : abs p.im = 1
⊢ Zsqrtd.norm a = 4 * Zsqrtd.norm (↑p.re * u)
Theorem: step1''
Proof state before:
case intro.intro.refine'_1
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hp' : abs p.re = 1
hq' : abs p.im = 1
⊢ IsCoprime (↑p.re * u).re (↑p.re * u).im
Tactic:
rw [← Int.isUnit_iff_abs_eq] at hp'
Proof state after:
case intro.intro.refine'_1
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hp' : IsUnit p.re
hq' : abs p.im = 1
⊢ IsCoprime (↑p.re * u).re (↑p.re * u).im
Theorem: step1''
Proof state before:
case intro.intro.refine'_1
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hp' : IsUnit p.re
hq' : abs p.im = 1
⊢ IsCoprime (↑p.re * u).re (↑p.re * u).im
Tactic:
rw [Zsqrtd.smul_re, Zsqrtd.smul_im, isCoprime_mul_unit_left hp']
Proof state after:
case intro.intro.refine'_1
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hp' : IsUnit p.re
hq' : abs p.im = 1
⊢ IsCoprime u.re u.im
Theorem: step1''
Proof state before:
case intro.intro.refine'_1
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hp' : IsUnit p.re
hq' : abs p.im = 1
⊢ IsCoprime u.re u.im
Tactic:
apply Zsqrtd.coprime_of_dvd_coprime hcoprime
Proof state after:
case intro.intro.refine'_1
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hp' : IsUnit p.re
hq' : abs p.im = 1
⊢ u ∣ a
Theorem: step1''
Proof state before:
case intro.intro.refine'_1
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hp' : IsUnit p.re
hq' : abs p.im = 1
⊢ u ∣ a
Tactic:
obtain rfl | rfl := h2 <;> apply dvd_mul_left
Proof state after:
no goals
Theorem: step1''
Proof state before:
case intro.intro.refine'_2
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hp' : abs p.re = 1
hq' : abs p.im = 1
⊢ a = p * (↑p.re * u) ∨ a = star p * (↑p.re * u)
Tactic:
rw [abs_eq <| zero_le_one' ℤ] at hp' hq'
Proof state after:
case intro.intro.refine'_2
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hp' : p.re = 1 ∨ p.re = -1
hq' : p.im = 1 ∨ p.im = -1
⊢ a = p * (↑p.re * u) ∨ a = star p * (↑p.re * u)
Theorem: step1''
Proof state before:
case intro.intro.refine'_2.inl
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hq' : p.im = 1 ∨ p.im = -1
hp' : p.re = 1
h4 : p = q ∨ p = star q
⊢ a = p * (↑p.re * u) ∨ a = star p * (↑p.re * u)
Tactic:
simp only [hp', one_mul, Int.cast_one]
Proof state after:
case intro.intro.refine'_2.inl
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hq' : p.im = 1 ∨ p.im = -1
hp' : p.re = 1
h4 : p = q ∨ p = star q
⊢ a = p * u ∨ a = star p * u
Theorem: step1''
Proof state before:
case intro.intro.refine'_2.inl
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hq' : p.im = 1 ∨ p.im = -1
hp' : p.re = 1
h4 : p = q ∨ p = star q
⊢ a = p * u ∨ a = star p * u
Tactic:
cases h4 <;> cases h2 <;> simp [*, or_comm]
Proof state after:
no goals
Theorem: step1''
Proof state before:
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hq' : p.im = 1 ∨ p.im = -1
hp' : p.re = 1
⊢ p.im = -1 → p = star q
Tactic:
intro h5
Proof state after:
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hq' : p.im = 1 ∨ p.im = -1
hp' : p.re = 1
h5 : p.im = -1
⊢ p = star q
Theorem: step1''
Proof state before:
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hq' : p.im = 1 ∨ p.im = -1
hp' : p.re = 1
h5 : p.im = -1
⊢ p = star q
Tactic:
simp [Zsqrtd.ext, hp', h5]
Proof state after:
no goals
Theorem: step1''
Proof state before:
case intro.intro.refine'_2.inr
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hq' : p.im = 1 ∨ p.im = -1
hp' : p.re = -1
h4 : p = -q ∨ p = -star q
⊢ a = p * (↑p.re * u) ∨ a = star p * (↑p.re * u)
Tactic:
simp only [hp', one_mul, Zsqrtd.norm_neg, Int.cast_one, Int.cast_neg, neg_one_mul]
Proof state after:
case intro.intro.refine'_2.inr
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hq' : p.im = 1 ∨ p.im = -1
hp' : p.re = -1
h4 : p = -q ∨ p = -star q
⊢ a = p * -u ∨ a = star p * -u
Theorem: step1''
Proof state before:
case intro.intro.refine'_2.inr
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hq' : p.im = 1 ∨ p.im = -1
hp' : p.re = -1
h4 : p = -q ∨ p = -star q
⊢ a = p * -u ∨ a = star p * -u
Tactic:
cases h4 <;> cases h2 <;> simp [*]
Proof state after:
no goals
Theorem: step1''
Proof state before:
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hq' : p.im = 1 ∨ p.im = -1
hp' : p.re = -1
⊢ p = -q ∨ p = -star q
Tactic:
apply Or.imp _ _ hq'.symm
Proof state after:
a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hq' : p.im = 1 ∨ p.im = -1
hp' : p.re = -1
⊢ p.im = -1 → p = -q

a p : ℤ√(-3)
hcoprime : IsCoprime a.re a.im
hp : Zsqrtd.norm p = 4
hq : p.im ≠ 0
heven : Even (Zsqrtd.norm a)
u : ℤ√(-3)
q : ℤ√(-3) := { re := 1, im := 1 }
h2 : a = q * u ∨ a = star q * u
hq' : p.im = 1 ∨ p.im = -1
hp' : p.re = -1
⊢ p.im = 1 → p = -star q