internlm/Lean-Github huggingface dataset:
Theorem: obscure'
Proof state before:
case intro.intro.intro.intro.refine'_2
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
⊢ q = 3 * b * (a - b) * (a + b)
Tactic:
rw [hq']
Proof state after:
case intro.intro.intro.intro.refine'_2
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
⊢ 3 * a ^ 2 * b - 3 * b ^ 3 = 3 * b * (a - b) * (a + b)
Theorem: obscure'
Proof state before:
case intro.intro.intro.intro.refine'_2
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
⊢ 3 * a ^ 2 * b - 3 * b ^ 3 = 3 * b * (a - b) * (a + b)
Tactic:
ring
Proof state after:
no goals
Theorem: obscure'
Proof state before:
case intro.intro.intro.intro.refine'_3
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
⊢ IsCoprime a b
Tactic:
apply isCoprime_of_dvd
Proof state after:
case intro.intro.intro.intro.refine'_3.nonzero
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
⊢ ¬(a = 0 ∧ b = 0)

case intro.intro.intro.intro.refine'_3.H
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
⊢ ∀ z ∈ nonunits ℤ, z ≠ 0 → z ∣ a → ¬z ∣ b
Theorem: obscure'
Proof state before:
case intro.intro.intro.intro.refine'_3.H
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
⊢ ∀ z ∈ nonunits ℤ, z ≠ 0 → z ∣ a → ¬z ∣ b
Tactic:
rintro k hknu - hkdvdleft hkdvdright
Proof state after:
case intro.intro.intro.intro.refine'_3.H
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
k : ℤ
hknu : k ∈ nonunits ℤ
hkdvdleft : k ∣ a
hkdvdright : k ∣ b
⊢ False
Theorem: obscure'
Proof state before:
case intro.intro.intro.intro.refine'_3.H
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
k : ℤ
hknu : k ∈ nonunits ℤ
hkdvdleft : k ∣ a
hkdvdright : k ∣ b
⊢ False
Tactic:
apply hknu
Proof state after:
case intro.intro.intro.intro.refine'_3.H
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
k : ℤ
hknu : k ∈ nonunits ℤ
hkdvdleft : k ∣ a
hkdvdright : k ∣ b
⊢ IsUnit k
Theorem: obscure'
Proof state before:
case intro.intro.intro.intro.refine'_3.H
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
k : ℤ
hknu : k ∈ nonunits ℤ
hkdvdleft : k ∣ a
hkdvdright : k ∣ b
⊢ IsUnit k
Tactic:
apply hcoprime.isUnit_of_dvd'
Proof state after:
case intro.intro.intro.intro.refine'_3.H.ha
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
k : ℤ
hknu : k ∈ nonunits ℤ
hkdvdleft : k ∣ a
hkdvdright : k ∣ b
⊢ k ∣ p

case intro.intro.intro.intro.refine'_3.H.hb
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
k : ℤ
hknu : k ∈ nonunits ℤ
hkdvdleft : k ∣ a
hkdvdright : k ∣ b
⊢ k ∣ q
Theorem: obscure'
Proof state before:
case intro.intro.intro.intro.refine'_3.nonzero
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
⊢ ¬(a = 0 ∧ b = 0)
Tactic:
rintro ⟨rfl, rfl⟩
Proof state after:
case intro.intro.intro.intro.refine'_3.nonzero.intro
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
hp' : p = 0 ^ 3 - 9 * 0 * 0 ^ 2
hq' : q = 3 * 0 ^ 2 * 0 - 3 * 0 ^ 3
⊢ False
Theorem: obscure'
Proof state before:
case intro.intro.intro.intro.refine'_3.nonzero.intro
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
hp' : p = 0 ^ 3 - 9 * 0 * 0 ^ 2
hq' : q = 3 * 0 ^ 2 * 0 - 3 * 0 ^ 3
⊢ False
Tactic:
norm_num at hp'
Proof state after:
case intro.intro.intro.intro.refine'_3.nonzero.intro
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
hq' : q = 3 * 0 ^ 2 * 0 - 3 * 0 ^ 3
hp' : p = 0
⊢ False
Theorem: obscure'
Proof state before:
case intro.intro.intro.intro.refine'_3.nonzero.intro
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
hq' : q = 3 * 0 ^ 2 * 0 - 3 * 0 ^ 3
hp' : p = 0
⊢ False
Tactic:
contradiction
Proof state after:
no goals
Theorem: obscure'
Proof state before:
case intro.intro.intro.intro.refine'_3.H.ha
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
k : ℤ
hknu : k ∈ nonunits ℤ
hkdvdleft : k ∣ a
hkdvdright : k ∣ b
⊢ k ∣ p
Tactic:
rw [hp']
Proof state after:
case intro.intro.intro.intro.refine'_3.H.ha
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
k : ℤ
hknu : k ∈ nonunits ℤ
hkdvdleft : k ∣ a
hkdvdright : k ∣ b
⊢ k ∣ a ^ 3 - 9 * a * b ^ 2
Theorem: obscure'
Proof state before:
case intro.intro.intro.intro.refine'_3.H.ha
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
k : ℤ
hknu : k ∈ nonunits ℤ
hkdvdleft : k ∣ a
hkdvdright : k ∣ b
⊢ k ∣ a ^ 3 - 9 * a * b ^ 2
Tactic:
apply dvd_sub
Proof state after:
case intro.intro.intro.intro.refine'_3.H.ha.h₁
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
k : ℤ
hknu : k ∈ nonunits ℤ
hkdvdleft : k ∣ a
hkdvdright : k ∣ b
⊢ k ∣ a ^ 3

case intro.intro.intro.intro.refine'_3.H.ha.h₂
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
k : ℤ
hknu : k ∈ nonunits ℤ
hkdvdleft : k ∣ a
hkdvdright : k ∣ b
⊢ k ∣ 9 * a * b ^ 2
Theorem: obscure'
Proof state before:
case intro.intro.intro.intro.refine'_3.H.ha.h₁
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
k : ℤ
hknu : k ∈ nonunits ℤ
hkdvdleft : k ∣ a
hkdvdright : k ∣ b
⊢ k ∣ a ^ 3
Tactic:
exact dvd_pow hkdvdleft (by norm_num)
Proof state after:
no goals
Theorem: obscure'
Proof state before:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
k : ℤ
hknu : k ∈ nonunits ℤ
hkdvdleft : k ∣ a
hkdvdright : k ∣ b
⊢ 3 ≠ 0
Tactic:
norm_num
Proof state after:
no goals
Theorem: obscure'
Proof state before:
case intro.intro.intro.intro.refine'_3.H.ha.h₂
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
k : ℤ
hknu : k ∈ nonunits ℤ
hkdvdleft : k ∣ a
hkdvdright : k ∣ b
⊢ k ∣ 9 * a * b ^ 2
Tactic:
rw [mul_comm (9 : ℤ), mul_assoc]
Proof state after:
case intro.intro.intro.intro.refine'_3.H.ha.h₂
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
k : ℤ
hknu : k ∈ nonunits ℤ
hkdvdleft : k ∣ a
hkdvdright : k ∣ b
⊢ k ∣ a * (9 * b ^ 2)
Theorem: obscure'
Proof state before:
case intro.intro.intro.intro.refine'_3.H.ha.h₂
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
k : ℤ
hknu : k ∈ nonunits ℤ
hkdvdleft : k ∣ a
hkdvdright : k ∣ b
⊢ k ∣ a * (9 * b ^ 2)
Tactic:
exact hkdvdleft.mul_right _
Proof state after:
no goals
Theorem: obscure'
Proof state before:
case intro.intro.intro.intro.refine'_3.H.hb
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
k : ℤ
hknu : k ∈ nonunits ℤ
hkdvdleft : k ∣ a
hkdvdright : k ∣ b
⊢ k ∣ q
Tactic:
rw [hq']
Proof state after:
case intro.intro.intro.intro.refine'_3.H.hb
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
k : ℤ
hknu : k ∈ nonunits ℤ
hkdvdleft : k ∣ a
hkdvdright : k ∣ b
⊢ k ∣ 3 * a ^ 2 * b - 3 * b ^ 3
Theorem: obscure'
Proof state before:
case intro.intro.intro.intro.refine'_3.H.hb
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
k : ℤ
hknu : k ∈ nonunits ℤ
hkdvdleft : k ∣ a
hkdvdright : k ∣ b
⊢ k ∣ 3 * a ^ 2 * b - 3 * b ^ 3
Tactic:
apply dvd_sub
Proof state after:
case intro.intro.intro.intro.refine'_3.H.hb.h₁
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
k : ℤ
hknu : k ∈ nonunits ℤ
hkdvdleft : k ∣ a
hkdvdright : k ∣ b
⊢ k ∣ 3 * a ^ 2 * b

case intro.intro.intro.intro.refine'_3.H.hb.h₂
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
k : ℤ
hknu : k ∈ nonunits ℤ
hkdvdleft : k ∣ a
hkdvdright : k ∣ b
⊢ k ∣ 3 * b ^ 3
Theorem: obscure'
Proof state before:
case intro.intro.intro.intro.refine'_3.H.hb.h₁
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
k : ℤ
hknu : k ∈ nonunits ℤ
hkdvdleft : k ∣ a
hkdvdright : k ∣ b
⊢ k ∣ 3 * a ^ 2 * b
Tactic:
exact hkdvdright.mul_left _
Proof state after:
no goals
Theorem: obscure'
Proof state before:
case intro.intro.intro.intro.refine'_3.H.hb.h₂
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
k : ℤ
hknu : k ∈ nonunits ℤ
hkdvdleft : k ∣ a
hkdvdright : k ∣ b
⊢ k ∣ 3 * b ^ 3
Tactic:
exact (hkdvdright.pow (by norm_num)).mul_left _
Proof state after:
no goals
Theorem: obscure'
Proof state before:
case neg
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
haparity : ¬Even a
hbparity : ¬Even b
⊢ Even a ↔ ¬Even b
Tactic:
exfalso
Proof state after:
case neg
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
haparity : ¬Even a
hbparity : ¬Even b
⊢ False
Theorem: obscure'
Proof state before:
case neg
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
haparity : ¬Even a
hbparity : ¬Even b
⊢ False
Tactic:
have : Even p := by
  rw [hp']
  have : ¬ Even (9 : ℤ) := by decide
  simp [this, haparity, hbparity, three_ne_zero, parity_simps]
Proof state after:
case neg
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
haparity : ¬Even a
hbparity : ¬Even b
this : Even p
⊢ False
Theorem: obscure'
Proof state before:
case neg
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
haparity : ¬Even a
hbparity : ¬Even b
this : Even p
⊢ False
Tactic:
have : Even q := by
  rw [hq']
  simp [haparity, hbparity, three_ne_zero, parity_simps]
Proof state after:
case neg
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
haparity : ¬Even a
hbparity : ¬Even b
this✝ : Even p
this : Even q
⊢ False
Theorem: obscure'
Proof state before:
case neg
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
haparity : ¬Even a
hbparity : ¬Even b
this✝ : Even p
this : Even q
⊢ False
Tactic:
tauto
Proof state after:
no goals
Theorem: obscure'
Proof state before:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
haparity : ¬Even a
hbparity : ¬Even b
⊢ Even p
Tactic:
rw [hp']
Proof state after:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
haparity : ¬Even a
hbparity : ¬Even b
⊢ Even (a ^ 3 - 9 * a * b ^ 2)
Theorem: obscure'
Proof state before:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
haparity : ¬Even a
hbparity : ¬Even b
⊢ Even (a ^ 3 - 9 * a * b ^ 2)
Tactic:
have : ¬ Even (9 : ℤ) := by decide
Proof state after:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
haparity : ¬Even a
hbparity : ¬Even b
this : ¬Even 9
⊢ Even (a ^ 3 - 9 * a * b ^ 2)
Theorem: obscure'
Proof state before:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
haparity : ¬Even a
hbparity : ¬Even b
this : ¬Even 9
⊢ Even (a ^ 3 - 9 * a * b ^ 2)
Tactic:
simp [this, haparity, hbparity, three_ne_zero, parity_simps]
Proof state after:
no goals
Theorem: obscure'
Proof state before:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
haparity : ¬Even a
hbparity : ¬Even b
⊢ ¬Even 9
Tactic:
decide
Proof state after:
no goals
Theorem: obscure'
Proof state before:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
haparity : ¬Even a
hbparity : ¬Even b
this : Even p
⊢ Even q
Tactic:
rw [hq']
Proof state after:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
haparity : ¬Even a
hbparity : ¬Even b
this : Even p
⊢ Even (3 * a ^ 2 * b - 3 * b ^ 3)
Theorem: obscure'
Proof state before:
p q : ℤ
hp : p ≠ 0
hcoprime : IsCoprime p q
hparity : Even p ↔ ¬Even q
u : ℤ
hu : p ^ 2 + 3 * q ^ 2 = u ^ 3
a b : ℤ
hp' : p = a ^ 3 - 9 * a * b ^ 2
hq' : q = 3 * a ^ 2 * b - 3 * b ^ 3
haparity : ¬Even a
hbparity : ¬Even b
this : Even p
⊢ Even (3 * a ^ 2 * b - 3 * b ^ 3)
Tactic:
simp [haparity, hbparity, three_ne_zero, parity_simps]
Proof state after:
no goals
Theorem: Int.eq_pow_of_mul_eq_pow_odd
Proof state before:
a b c : ℤ
hab : IsCoprime a b
k : ℕ
hk : Odd k
h : a * b = c ^ k
⊢ (∃ d, a = d ^ k) ∧ ∃ e, b = e ^ k
Tactic:
obtain ⟨k, rfl⟩ := hk.exists_bit1
Proof state after:
case intro
a b c : ℤ
hab : IsCoprime a b
k : ℕ
hk : Odd (bit1 k)
h : a * b = c ^ bit1 k
⊢ (∃ d, a = d ^ bit1 k) ∧ ∃ e, b = e ^ bit1 k
Theorem: Int.eq_pow_of_mul_eq_pow_odd
Proof state before:
case intro
a b c : ℤ
hab : IsCoprime a b
k : ℕ
hk : Odd (bit1 k)
h : a * b = c ^ bit1 k
⊢ (∃ d, a = d ^ bit1 k) ∧ ∃ e, b = e ^ bit1 k
Tactic:
exact Int.eq_pow_of_mul_eq_pow_bit1 hab h
Proof state after:
no goals
Theorem: Int.cube_of_coprime
Proof state before:
a b c s : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hcoprimeab : IsCoprime a b
hcoprimeac : IsCoprime a c
hcoprimebc : IsCoprime b c
hs : a * b * c = s ^ 3
⊢ ∃ A B C, A ≠ 0 ∧ B ≠ 0 ∧ C ≠ 0 ∧ a = A ^ 3 ∧ b = B ^ 3 ∧ c = C ^ 3
Tactic:
have : Odd 3 := by decide
Proof state after:
a b c s : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hcoprimeab : IsCoprime a b
hcoprimeac : IsCoprime a c
hcoprimebc : IsCoprime b c
hs : a * b * c = s ^ 3
this : Odd 3
⊢ ∃ A B C, A ≠ 0 ∧ B ≠ 0 ∧ C ≠ 0 ∧ a = A ^ 3 ∧ b = B ^ 3 ∧ c = C ^ 3
Theorem: Int.cube_of_coprime
Proof state before:
a b c s : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hcoprimeab : IsCoprime a b
hcoprimeac : IsCoprime a c
hcoprimebc : IsCoprime b c
hs : a * b * c = s ^ 3
this : Odd 3
⊢ ∃ A B C, A ≠ 0 ∧ B ≠ 0 ∧ C ≠ 0 ∧ a = A ^ 3 ∧ b = B ^ 3 ∧ c = C ^ 3
Tactic:
obtain ⟨⟨AB, HAB⟩, ⟨C, HC⟩⟩ :=
  Int.eq_pow_of_mul_eq_pow_odd (IsCoprime.mul_left hcoprimeac hcoprimebc) this hs
Proof state after:
case intro.intro.intro
a b c s : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hcoprimeab : IsCoprime a b
hcoprimeac : IsCoprime a c
hcoprimebc : IsCoprime b c
hs : a * b * c = s ^ 3
this : Odd 3
AB : ℤ
HAB : a * b = AB ^ 3
C : ℤ
HC : c = C ^ 3
⊢ ∃ A B C, A ≠ 0 ∧ B ≠ 0 ∧ C ≠ 0 ∧ a = A ^ 3 ∧ b = B ^ 3 ∧ c = C ^ 3
Theorem: Int.cube_of_coprime
Proof state before:
case intro.intro.intro
a b c s : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hcoprimeab : IsCoprime a b
hcoprimeac : IsCoprime a c
hcoprimebc : IsCoprime b c
hs : a * b * c = s ^ 3
this : Odd 3
AB : ℤ
HAB : a * b = AB ^ 3
C : ℤ
HC : c = C ^ 3
⊢ ∃ A B C, A ≠ 0 ∧ B ≠ 0 ∧ C ≠ 0 ∧ a = A ^ 3 ∧ b = B ^ 3 ∧ c = C ^ 3
Tactic:
obtain ⟨⟨A, HA⟩, ⟨B, HB⟩⟩ := Int.eq_pow_of_mul_eq_pow_odd hcoprimeab this HAB
Proof state after:
case intro.intro.intro.intro.intro.intro
a b c s : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hcoprimeab : IsCoprime a b
hcoprimeac : IsCoprime a c
hcoprimebc : IsCoprime b c
hs : a * b * c = s ^ 3
this : Odd 3
AB : ℤ
HAB : a * b = AB ^ 3
C : ℤ
HC : c = C ^ 3
A : ℤ
HA : a = A ^ 3
B : ℤ
HB : b = B ^ 3
⊢ ∃ A B C, A ≠ 0 ∧ B ≠ 0 ∧ C ≠ 0 ∧ a = A ^ 3 ∧ b = B ^ 3 ∧ c = C ^ 3
Theorem: Int.cube_of_coprime
Proof state before:
case intro.intro.intro.intro.intro.intro
a b c s : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hcoprimeab : IsCoprime a b
hcoprimeac : IsCoprime a c
hcoprimebc : IsCoprime b c
hs : a * b * c = s ^ 3
this : Odd 3
AB : ℤ
HAB : a * b = AB ^ 3
C : ℤ
HC : c = C ^ 3
A : ℤ
HA : a = A ^ 3
B : ℤ
HB : b = B ^ 3
⊢ ∃ A B C, A ≠ 0 ∧ B ≠ 0 ∧ C ≠ 0 ∧ a = A ^ 3 ∧ b = B ^ 3 ∧ c = C ^ 3
Tactic:
refine' ⟨A, B, C, _, _, _, HA, HB, HC⟩ <;> apply ne_zero_pow three_ne_zero
Proof state after:
case intro.intro.intro.intro.intro.intro.refine'_1
a b c s : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hcoprimeab : IsCoprime a b
hcoprimeac : IsCoprime a c
hcoprimebc : IsCoprime b c
hs : a * b * c = s ^ 3
this : Odd 3
AB : ℤ
HAB : a * b = AB ^ 3
C : ℤ
HC : c = C ^ 3
A : ℤ
HA : a = A ^ 3
B : ℤ
HB : b = B ^ 3
⊢ A ^ 3 ≠ 0

case intro.intro.intro.intro.intro.intro.refine'_2
a b c s : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hcoprimeab : IsCoprime a b
hcoprimeac : IsCoprime a c
hcoprimebc : IsCoprime b c
hs : a * b * c = s ^ 3
this : Odd 3
AB : ℤ
HAB : a * b = AB ^ 3
C : ℤ
HC : c = C ^ 3
A : ℤ
HA : a = A ^ 3
B : ℤ
HB : b = B ^ 3
⊢ B ^ 3 ≠ 0

case intro.intro.intro.intro.intro.intro.refine'_3
a b c s : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hcoprimeab : IsCoprime a b
hcoprimeac : IsCoprime a c
hcoprimebc : IsCoprime b c
hs : a * b * c = s ^ 3
this : Odd 3
AB : ℤ
HAB : a * b = AB ^ 3
C : ℤ
HC : c = C ^ 3
A : ℤ
HA : a = A ^ 3
B : ℤ
HB : b = B ^ 3
⊢ C ^ 3 ≠ 0
Theorem: Int.cube_of_coprime
Proof state before:
a b c s : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hcoprimeab : IsCoprime a b
hcoprimeac : IsCoprime a c
hcoprimebc : IsCoprime b c
hs : a * b * c = s ^ 3
⊢ Odd 3
Tactic:
decide
Proof state after:
no goals
Theorem: Int.cube_of_coprime
Proof state before:
case intro.intro.intro.intro.intro.intro.refine'_1
a b c s : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hcoprimeab : IsCoprime a b
hcoprimeac : IsCoprime a c
hcoprimebc : IsCoprime b c
hs : a * b * c = s ^ 3
this : Odd 3
AB : ℤ
HAB : a * b = AB ^ 3
C : ℤ
HC : c = C ^ 3
A : ℤ
HA : a = A ^ 3
B : ℤ
HB : b = B ^ 3
⊢ A ^ 3 ≠ 0
Tactic:
rwa [← HA]
Proof state after:
no goals
Theorem: Int.cube_of_coprime
Proof state before:
case intro.intro.intro.intro.intro.intro.refine'_2
a b c s : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hcoprimeab : IsCoprime a b
hcoprimeac : IsCoprime a c
hcoprimebc : IsCoprime b c
hs : a * b * c = s ^ 3
this : Odd 3
AB : ℤ
HAB : a * b = AB ^ 3
C : ℤ
HC : c = C ^ 3
A : ℤ
HA : a = A ^ 3
B : ℤ
HB : b = B ^ 3
⊢ B ^ 3 ≠ 0
Tactic:
rwa [← HB]
Proof state after:
no goals
Theorem: Int.cube_of_coprime
Proof state before:
case intro.intro.intro.intro.intro.intro.refine'_3
a b c s : ℤ
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hcoprimeab : IsCoprime a b
hcoprimeac : IsCoprime a c
hcoprimebc : IsCoprime b c
hs : a * b * c = s ^ 3
this : Odd 3
AB : ℤ
HAB : a * b = AB ^ 3
C : ℤ
HC : c = C ^ 3
A : ℤ
HA : a = A ^ 3
B : ℤ
HB : b = B ^ 3
⊢ C ^ 3 ≠ 0
Tactic:
rwa [← HC]
Proof state after:
no goals
Theorem: Int.gcd1_coprime12
Proof state before:
u v : ℤ
huvcoprime : IsCoprime u v
notdvd_2_2 : ¬2 ∣ u - 3 * v
not_3_dvd_2 : ¬3 ∣ u - 3 * v
⊢ IsCoprime (2 * u) (u - 3 * v)
Tactic:
apply isCoprime_of_prime_dvd
Proof state after:
case nonzero
u v : ℤ
huvcoprime : IsCoprime u v
notdvd_2_2 : ¬2 ∣ u - 3 * v
not_3_dvd_2 : ¬3 ∣ u - 3 * v
⊢ ¬(2 * u = 0 ∧ u - 3 * v = 0)

case H
u v : ℤ
huvcoprime : IsCoprime u v
notdvd_2_2 : ¬2 ∣ u - 3 * v
not_3_dvd_2 : ¬3 ∣ u - 3 * v
⊢ ∀ (z : ℤ), Prime z → z ∣ 2 * u → ¬z ∣ u - 3 * v
Theorem: Int.gcd1_coprime12
Proof state before:
case H
u v : ℤ
huvcoprime : IsCoprime u v
notdvd_2_2 : ¬2 ∣ u - 3 * v
not_3_dvd_2 : ¬3 ∣ u - 3 * v
⊢ ∀ (z : ℤ), Prime z → z ∣ 2 * u → ¬z ∣ u - 3 * v
Tactic:
intro k hkprime hkdvdleft hkdvdright
Proof state after:
case H
u v : ℤ
huvcoprime : IsCoprime u v
notdvd_2_2 : ¬2 ∣ u - 3 * v
not_3_dvd_2 : ¬3 ∣ u - 3 * v
k : ℤ
hkprime : Prime k
hkdvdleft : k ∣ 2 * u
hkdvdright : k ∣ u - 3 * v
⊢ False
Theorem: Int.gcd1_coprime12
Proof state before:
case H
u v : ℤ
huvcoprime : IsCoprime u v
notdvd_2_2 : ¬2 ∣ u - 3 * v
not_3_dvd_2 : ¬3 ∣ u - 3 * v
k : ℤ
hkprime : Prime k
hkdvdleft : k ∣ 2 * u
hkdvdright : k ∣ u - 3 * v
⊢ False
Tactic:
apply hkprime.not_unit
Proof state after:
case H
u v : ℤ
huvcoprime : IsCoprime u v
notdvd_2_2 : ¬2 ∣ u - 3 * v
not_3_dvd_2 : ¬3 ∣ u - 3 * v
k : ℤ
hkprime : Prime k
hkdvdleft : k ∣ 2 * u
hkdvdright : k ∣ u - 3 * v
⊢ IsUnit k