internlm/Lean-Github huggingface dataset:
Theorem: ℤθ.unit_pow_three_pow_1
Proof state before:
case zero
a b c : ℤθ
⊢ ∃ a b c, (↑(unit ^ 3)).f = 1 + 3 + 3 ^ 2 * a ∧ (↑(unit ^ 3)).g = 3 ^ 2 * b ∧ (↑(unit ^ 3)).h = 3 + 3 ^ 2 * c
Tactic:
change
  ∃ a b c : ℤ,
    ((unit : ℤθ) ^ 3).f = 1 + 3 + 3 ^ 2 * a ∧
      ((unit : ℤθ) ^ 3).g = 3 ^ 2 * b ∧ ((unit : ℤθ) ^ 3).h = 3 + 3 ^ 2 * c
Proof state after:
case zero
a b c : ℤθ
⊢ ∃ a b c, (↑unit ^ 3).f = 1 + 3 + 3 ^ 2 * a ∧ (↑unit ^ 3).g = 3 ^ 2 * b ∧ (↑unit ^ 3).h = 3 + 3 ^ 2 * c
Theorem: ℤθ.unit_pow_three_pow_1
Proof state before:
case zero
a b c : ℤθ
⊢ ∃ a b c, (↑unit ^ 3).f = 1 + 3 + 3 ^ 2 * a ∧ (↑unit ^ 3).g = 3 ^ 2 * b ∧ (↑unit ^ 3).h = 3 + 3 ^ 2 * c
Tactic:
rw [unit_cubed]
Proof state after:
case zero
a b c : ℤθ
⊢ ∃ a b c,
    { f := -23, g := -63, h := -15 }.f = 1 + 3 + 3 ^ 2 * a ∧
      { f := -23, g := -63, h := -15 }.g = 3 ^ 2 * b ∧ { f := -23, g := -63, h := -15 }.h = 3 + 3 ^ 2 * c
Theorem: ℤθ.unit_pow_three_pow_1
Proof state before:
case zero
a b c : ℤθ
⊢ ∃ a b c,
    { f := -23, g := -63, h := -15 }.f = 1 + 3 + 3 ^ 2 * a ∧
      { f := -23, g := -63, h := -15 }.g = 3 ^ 2 * b ∧ { f := -23, g := -63, h := -15 }.h = 3 + 3 ^ 2 * c
Tactic:
dsimp
Proof state after:
case zero
a b c : ℤθ
⊢ ∃ a b c, -23 = 1 + 3 + 3 ^ 2 * a ∧ -63 = 3 ^ 2 * b ∧ -15 = 3 + 3 ^ 2 * c
Theorem: ℤθ.unit_pow_three_pow_1
Proof state before:
case zero
a b c : ℤθ
⊢ ∃ a b c, -23 = 1 + 3 + 3 ^ 2 * a ∧ -63 = 3 ^ 2 * b ∧ -15 = 3 + 3 ^ 2 * c
Tactic:
use-3
Proof state after:
case h
a b c : ℤθ
⊢ ∃ b c, -23 = 1 + 3 + 3 ^ 2 * -3 ∧ -63 = 3 ^ 2 * b ∧ -15 = 3 + 3 ^ 2 * c
Theorem: ℤθ.unit_pow_three_pow_1
Proof state before:
case h
a b c : ℤθ
⊢ ∃ b c, -23 = 1 + 3 + 3 ^ 2 * -3 ∧ -63 = 3 ^ 2 * b ∧ -15 = 3 + 3 ^ 2 * c
Tactic:
use-7
Proof state after:
case h
a b c : ℤθ
⊢ ∃ c, -23 = 1 + 3 + 3 ^ 2 * -3 ∧ -63 = 3 ^ 2 * -7 ∧ -15 = 3 + 3 ^ 2 * c
Theorem: ℤθ.unit_pow_three_pow_1
Proof state before:
case h
a b c : ℤθ
⊢ ∃ c, -23 = 1 + 3 + 3 ^ 2 * -3 ∧ -63 = 3 ^ 2 * -7 ∧ -15 = 3 + 3 ^ 2 * c
Tactic:
use-2
Proof state after:
case h
a b c : ℤθ
⊢ -23 = 1 + 3 + 3 ^ 2 * -3 ∧ -63 = 3 ^ 2 * -7 ∧ -15 = 3 + 3 ^ 2 * -2
Theorem: ℤθ.unit_pow_three_pow_1
Proof state before:
case h
a b c : ℤθ
⊢ -23 = 1 + 3 + 3 ^ 2 * -3 ∧ -63 = 3 ^ 2 * -7 ∧ -15 = 3 + 3 ^ 2 * -2
Tactic:
norm_num
Proof state after:
no goals
Theorem: ℤθ.unit_pow_three_pow_1
Proof state before:
a b c : ℤθ
k : ℕ
hk :
  ∃ a b c,
    (↑(unit ^ 3 ^ (k + 1))).f = 1 + 3 ^ (k + 1) + 3 ^ (k + 2) * a ∧
      (↑(unit ^ 3 ^ (k + 1))).g = 3 ^ (k + 2) * b ∧ (↑(unit ^ 3 ^ (k + 1))).h = 3 ^ (k + 1) + 3 ^ (k + 2) * c
⊢ k + 1 = Nat.succ k
Tactic:
rfl
Proof state after:
no goals
Theorem: ℤθ.unit_pow_three_pow_1
Proof state before:
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
⊢ ↑(unit ^ 3 ^ Nat.succ k) =
    { f := 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p, g := 3 ^ (Nat.succ k + 1) * q,
      h := 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r }
Tactic:
ext <;> dsimp
Proof state after:
case f
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
⊢ (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p

case g
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
⊢ (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q

case h
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
⊢ (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
Theorem: ℤθ.unit_pow_three_pow_1
Proof state before:
case f
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
⊢ (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p

case g
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
⊢ (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q

case h
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
⊢ (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
Tactic:
exact_mod_cast h1
Proof state after:
case g
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
⊢ (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q

case h
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
⊢ (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
Theorem: ℤθ.unit_pow_three_pow_1
Proof state before:
case g
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
⊢ (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q

case h
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
⊢ (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
Tactic:
exact_mod_cast h2
Proof state after:
case h
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
⊢ (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
Theorem: ℤθ.unit_pow_three_pow_1
Proof state before:
case h
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
⊢ (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
Tactic:
exact_mod_cast h3
Proof state after:
no goals
Theorem: ℤθ.unit_pow_three_pow_1
Proof state before:
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
t :
  ↑(unit ^ 3 ^ Nat.succ k) =
    { f := 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p, g := 3 ^ (Nat.succ k + 1) * q,
      h := 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r }
g : Nat.succ k ≤ 1 ∨ Nat.succ k > 1
⊢ 1 ≤ Nat.succ k
Tactic:
rw [← l]
Proof state after:
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
t :
  ↑(unit ^ 3 ^ Nat.succ k) =
    { f := 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p, g := 3 ^ (Nat.succ k + 1) * q,
      h := 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r }
g : Nat.succ k ≤ 1 ∨ Nat.succ k > 1
⊢ 1 ≤ k + 1
Theorem: ℤθ.unit_pow_three_pow_1
Proof state before:
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
t :
  ↑(unit ^ 3 ^ Nat.succ k) =
    { f := 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p, g := 3 ^ (Nat.succ k + 1) * q,
      h := 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r }
g : Nat.succ k ≤ 1 ∨ Nat.succ k > 1
⊢ 1 ≤ k + 1
Tactic:
simp
Proof state after:
no goals
Theorem: ℤθ.unit_pow_three_pow_1
Proof state before:
case succ.intro.intro.intro.intro.intro.inl
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
t :
  ↑(unit ^ 3 ^ Nat.succ k) =
    { f := 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p, g := 3 ^ (Nat.succ k + 1) * q,
      h := 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r }
lower : 1 ≤ Nat.succ k
g1 : Nat.succ k ≤ 1
⊢ ∃ a b c,
    (↑(unit ^ 3 ^ (Nat.succ k + 1))).f = 1 + 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * a ∧
      (↑(unit ^ 3 ^ (Nat.succ k + 1))).g = 3 ^ (Nat.succ k + 2) * b ∧
        (↑(unit ^ 3 ^ (Nat.succ k + 1))).h = 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * c
Tactic:
have upper : k.succ < 2 :=
by
  rw [← Nat.succ_le_succ_iff] at g1
  exact g1
Proof state after:
case succ.intro.intro.intro.intro.intro.inl
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
t :
  ↑(unit ^ 3 ^ Nat.succ k) =
    { f := 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p, g := 3 ^ (Nat.succ k + 1) * q,
      h := 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r }
lower : 1 ≤ Nat.succ k
g1 : Nat.succ k ≤ 1
upper : Nat.succ k < 2
⊢ ∃ a b c,
    (↑(unit ^ 3 ^ (Nat.succ k + 1))).f = 1 + 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * a ∧
      (↑(unit ^ 3 ^ (Nat.succ k + 1))).g = 3 ^ (Nat.succ k + 2) * b ∧
        (↑(unit ^ 3 ^ (Nat.succ k + 1))).h = 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * c
Theorem: ℤθ.unit_pow_three_pow_1
Proof state before:
case succ.intro.intro.intro.intro.intro.inl
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
t :
  ↑(unit ^ 3 ^ Nat.succ k) =
    { f := 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p, g := 3 ^ (Nat.succ k + 1) * q,
      h := 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r }
lower : 1 ≤ Nat.succ k
g1 : Nat.succ k ≤ 1
upper : Nat.succ k < 2
⊢ ∃ a b c,
    (↑(unit ^ 3 ^ (Nat.succ k + 1))).f = 1 + 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * a ∧
      (↑(unit ^ 3 ^ (Nat.succ k + 1))).g = 3 ^ (Nat.succ k + 2) * b ∧
        (↑(unit ^ 3 ^ (Nat.succ k + 1))).h = 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * c
Tactic:
have elem := eq_or_lt_of_le g1
Proof state after:
case succ.intro.intro.intro.intro.intro.inl
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
t :
  ↑(unit ^ 3 ^ Nat.succ k) =
    { f := 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p, g := 3 ^ (Nat.succ k + 1) * q,
      h := 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r }
lower : 1 ≤ Nat.succ k
g1 : Nat.succ k ≤ 1
upper : Nat.succ k < 2
elem : Nat.succ k = 1 ∨ Nat.succ k < 1
⊢ ∃ a b c,
    (↑(unit ^ 3 ^ (Nat.succ k + 1))).f = 1 + 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * a ∧
      (↑(unit ^ 3 ^ (Nat.succ k + 1))).g = 3 ^ (Nat.succ k + 2) * b ∧
        (↑(unit ^ 3 ^ (Nat.succ k + 1))).h = 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * c
Theorem: ℤθ.unit_pow_three_pow_1
Proof state before:
case succ.intro.intro.intro.intro.intro.inl
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
t :
  ↑(unit ^ 3 ^ Nat.succ k) =
    { f := 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p, g := 3 ^ (Nat.succ k + 1) * q,
      h := 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r }
lower : 1 ≤ Nat.succ k
g1 : Nat.succ k ≤ 1
upper : Nat.succ k < 2
elem : Nat.succ k = 1 ∨ Nat.succ k < 1
⊢ ∃ a b c,
    (↑(unit ^ 3 ^ (Nat.succ k + 1))).f = 1 + 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * a ∧
      (↑(unit ^ 3 ^ (Nat.succ k + 1))).g = 3 ^ (Nat.succ k + 2) * b ∧
        (↑(unit ^ 3 ^ (Nat.succ k + 1))).h = 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * c
Tactic:
cases' elem with f1 f2
Proof state after:
case succ.intro.intro.intro.intro.intro.inl.inl
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
t :
  ↑(unit ^ 3 ^ Nat.succ k) =
    { f := 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p, g := 3 ^ (Nat.succ k + 1) * q,
      h := 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r }
lower : 1 ≤ Nat.succ k
g1 : Nat.succ k ≤ 1
upper : Nat.succ k < 2
f1 : Nat.succ k = 1
⊢ ∃ a b c,
    (↑(unit ^ 3 ^ (Nat.succ k + 1))).f = 1 + 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * a ∧
      (↑(unit ^ 3 ^ (Nat.succ k + 1))).g = 3 ^ (Nat.succ k + 2) * b ∧
        (↑(unit ^ 3 ^ (Nat.succ k + 1))).h = 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * c

case succ.intro.intro.intro.intro.intro.inl.inr
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
t :
  ↑(unit ^ 3 ^ Nat.succ k) =
    { f := 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p, g := 3 ^ (Nat.succ k + 1) * q,
      h := 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r }
lower : 1 ≤ Nat.succ k
g1 : Nat.succ k ≤ 1
upper : Nat.succ k < 2
f2 : Nat.succ k < 1
⊢ ∃ a b c,
    (↑(unit ^ 3 ^ (Nat.succ k + 1))).f = 1 + 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * a ∧
      (↑(unit ^ 3 ^ (Nat.succ k + 1))).g = 3 ^ (Nat.succ k + 2) * b ∧
        (↑(unit ^ 3 ^ (Nat.succ k + 1))).h = 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * c
Theorem: ℤθ.unit_pow_three_pow_1
Proof state before:
case succ.intro.intro.intro.intro.intro.inl.inl
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
t :
  ↑(unit ^ 3 ^ Nat.succ k) =
    { f := 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p, g := 3 ^ (Nat.succ k + 1) * q,
      h := 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r }
lower : 1 ≤ Nat.succ k
g1 : Nat.succ k ≤ 1
upper : Nat.succ k < 2
f1 : Nat.succ k = 1
⊢ (∃ x, ((-1 - 3 * θ - θ ^ 2) ^ 9).f = 10 + 27 * x) ∧
    (∃ x, ((-1 - 3 * θ - θ ^ 2) ^ 9).g = 27 * x) ∧ ∃ x, ((-1 - 3 * θ - θ ^ 2) ^ 9).h = 9 + 27 * x

case succ.intro.intro.intro.intro.intro.inl.inr
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
t :
  ↑(unit ^ 3 ^ Nat.succ k) =
    { f := 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p, g := 3 ^ (Nat.succ k + 1) * q,
      h := 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r }
lower : 1 ≤ Nat.succ k
g1 : Nat.succ k ≤ 1
upper : Nat.succ k < 2
f2 : Nat.succ k < 1
⊢ ∃ a b c,
    (↑(unit ^ 3 ^ (Nat.succ k + 1))).f = 1 + 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * a ∧
      (↑(unit ^ 3 ^ (Nat.succ k + 1))).g = 3 ^ (Nat.succ k + 2) * b ∧
        (↑(unit ^ 3 ^ (Nat.succ k + 1))).h = 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * c
Tactic:
sorry
Proof state after:
case succ.intro.intro.intro.intro.intro.inl.inr
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
t :
  ↑(unit ^ 3 ^ Nat.succ k) =
    { f := 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p, g := 3 ^ (Nat.succ k + 1) * q,
      h := 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r }
lower : 1 ≤ Nat.succ k
g1 : Nat.succ k ≤ 1
upper : Nat.succ k < 2
f2 : Nat.succ k < 1
⊢ ∃ a b c,
    (↑(unit ^ 3 ^ (Nat.succ k + 1))).f = 1 + 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * a ∧
      (↑(unit ^ 3 ^ (Nat.succ k + 1))).g = 3 ^ (Nat.succ k + 2) * b ∧
        (↑(unit ^ 3 ^ (Nat.succ k + 1))).h = 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * c
Theorem: ℤθ.unit_pow_three_pow_1
Proof state before:
case succ.intro.intro.intro.intro.intro.inl.inr
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
t :
  ↑(unit ^ 3 ^ Nat.succ k) =
    { f := 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p, g := 3 ^ (Nat.succ k + 1) * q,
      h := 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r }
lower : 1 ≤ Nat.succ k
g1 : Nat.succ k ≤ 1
upper : Nat.succ k < 2
f2 : Nat.succ k < 1
⊢ ∃ a b c,
    (↑(unit ^ 3 ^ (Nat.succ k + 1))).f = 1 + 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * a ∧
      (↑(unit ^ 3 ^ (Nat.succ k + 1))).g = 3 ^ (Nat.succ k + 2) * b ∧
        (↑(unit ^ 3 ^ (Nat.succ k + 1))).h = 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * c
Tactic:
contradiction
Proof state after:
no goals
Theorem: ℤθ.unit_pow_three_pow_1
Proof state before:
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
t :
  ↑(unit ^ 3 ^ Nat.succ k) =
    { f := 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p, g := 3 ^ (Nat.succ k + 1) * q,
      h := 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r }
lower : 1 ≤ Nat.succ k
g1 : Nat.succ k ≤ 1
⊢ Nat.succ k < 2
Tactic:
rw [← Nat.succ_le_succ_iff] at g1
Proof state after:
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
t :
  ↑(unit ^ 3 ^ Nat.succ k) =
    { f := 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p, g := 3 ^ (Nat.succ k + 1) * q,
      h := 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r }
lower : 1 ≤ Nat.succ k
g1 : Nat.succ (Nat.succ k) ≤ Nat.succ 1
⊢ Nat.succ k < 2
Theorem: ℤθ.unit_pow_three_pow_1
Proof state before:
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
t :
  ↑(unit ^ 3 ^ Nat.succ k) =
    { f := 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p, g := 3 ^ (Nat.succ k + 1) * q,
      h := 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r }
lower : 1 ≤ Nat.succ k
g1 : Nat.succ (Nat.succ k) ≤ Nat.succ 1
⊢ Nat.succ k < 2
Tactic:
exact g1
Proof state after:
no goals
Theorem: ℤθ.unit_pow_three_pow_1
Proof state before:
case succ.intro.intro.intro.intro.intro.inr
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
t :
  ↑(unit ^ 3 ^ Nat.succ k) =
    { f := 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p, g := 3 ^ (Nat.succ k + 1) * q,
      h := 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r }
lower : 1 ≤ Nat.succ k
g2 : Nat.succ k > 1
⊢ ∃ a b c,
    (↑(unit ^ 3 ^ (Nat.succ k + 1))).f = 1 + 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * a ∧
      (↑(unit ^ 3 ^ (Nat.succ k + 1))).g = 3 ^ (Nat.succ k + 2) * b ∧
        (↑(unit ^ 3 ^ (Nat.succ k + 1))).h = 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * c
Tactic:
change 2 ≤ k.succ  at g2
Proof state after:
case succ.intro.intro.intro.intro.intro.inr
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
t :
  ↑(unit ^ 3 ^ Nat.succ k) =
    { f := 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p, g := 3 ^ (Nat.succ k + 1) * q,
      h := 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r }
lower : 1 ≤ Nat.succ k
g2 : 2 ≤ Nat.succ k
⊢ ∃ a b c,
    (↑(unit ^ 3 ^ (Nat.succ k + 1))).f = 1 + 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * a ∧
      (↑(unit ^ 3 ^ (Nat.succ k + 1))).g = 3 ^ (Nat.succ k + 2) * b ∧
        (↑(unit ^ 3 ^ (Nat.succ k + 1))).h = 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * c
Theorem: ℤθ.unit_pow_three_pow_1
Proof state before:
case succ.intro.intro.intro.intro.intro.inr
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
t :
  ↑(unit ^ 3 ^ Nat.succ k) =
    { f := 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p, g := 3 ^ (Nat.succ k + 1) * q,
      h := 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r }
lower : 1 ≤ Nat.succ k
g2 : 2 ≤ Nat.succ k
⊢ ∃ a b c,
    (↑(unit ^ 3 ^ (Nat.succ k + 1))).f = 1 + 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * a ∧
      (↑(unit ^ 3 ^ (Nat.succ k + 1))).g = 3 ^ (Nat.succ k + 2) * b ∧
        (↑(unit ^ 3 ^ (Nat.succ k + 1))).h = 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * c
Tactic:
clear lower
Proof state after:
case succ.intro.intro.intro.intro.intro.inr
a b c : ℤθ
k : ℕ
l : k + 1 = Nat.succ k
p q r : ℤ
h1 : (↑(unit ^ 3 ^ Nat.succ k)).f = 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p
h2 : (↑(unit ^ 3 ^ Nat.succ k)).g = 3 ^ (Nat.succ k + 1) * q
h3 : (↑(unit ^ 3 ^ Nat.succ k)).h = 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r
t :
  ↑(unit ^ 3 ^ Nat.succ k) =
    { f := 1 + 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * p, g := 3 ^ (Nat.succ k + 1) * q,
      h := 3 ^ Nat.succ k + 3 ^ (Nat.succ k + 1) * r }
g2 : 2 ≤ Nat.succ k
⊢ ∃ a b c,
    (↑(unit ^ 3 ^ (Nat.succ k + 1))).f = 1 + 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * a ∧
      (↑(unit ^ 3 ^ (Nat.succ k + 1))).g = 3 ^ (Nat.succ k + 2) * b ∧
        (↑(unit ^ 3 ^ (Nat.succ k + 1))).h = 3 ^ (Nat.succ k + 1) + 3 ^ (Nat.succ k + 2) * c