url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | sum_squared_eq_sum_cubes | [7, 1] | [42, 21] | rw [add_sub_cancel, mul_comm] | n✝ : ℕ
ih : (∑ x in Finset.range (n✝ + 1), x) ^ 2 = ∑ x in Finset.range (n✝ + 1), x ^ 3
S : ℕ := ∑ x in Finset.range (n✝ + 1), x
hS : S = ∑ x in Finset.range (n✝ + 1), x
bigger : ∀ (n : ℕ), 1 ≤ n + 1
n : ℤ
⊢ 2 ∣ (n + 1) * (n + 1 - 1) | n✝ : ℕ
ih : (∑ x in Finset.range (n✝ + 1), x) ^ 2 = ∑ x in Finset.range (n✝ + 1), x ^ 3
S : ℕ := ∑ x in Finset.range (n✝ + 1), x
hS : S = ∑ x in Finset.range (n✝ + 1), x
bigger : ∀ (n : ℕ), 1 ≤ n + 1
n : ℤ
⊢ 2 ∣ n * (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
n✝ : ℕ
ih : (∑ x in Finset.range (n✝ + 1), x) ^ 2 = ∑ x in Finset.range (n✝ + 1), x ^ 3
S : ℕ := ∑ x in Finset.range (n✝ + 1), x
hS : S = ∑ x in Finset.range (n✝ + 1), x
bigger : ∀ (n : ℕ), 1 ≤ n + 1
n : ℤ
⊢ 2 ∣ (n + 1) * (n + 1 - 1)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | sum_squared_eq_sum_cubes | [7, 1] | [42, 21] | exact even_iff_two_dvd.1 (Int.even_mul_succ_self n) | n✝ : ℕ
ih : (∑ x in Finset.range (n✝ + 1), x) ^ 2 = ∑ x in Finset.range (n✝ + 1), x ^ 3
S : ℕ := ∑ x in Finset.range (n✝ + 1), x
hS : S = ∑ x in Finset.range (n✝ + 1), x
bigger : ∀ (n : ℕ), 1 ≤ n + 1
n : ℤ
⊢ 2 ∣ n * (n + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n✝ : ℕ
ih : (∑ x in Finset.range (n✝ + 1), x) ^ 2 = ∑ x in Finset.range (n✝ + 1), x ^ 3
S : ℕ := ∑ x in Finset.range (n✝ + 1), x
hS : S = ∑ x in Finset.range (n✝ + 1), x
bigger : ∀ (n : ℕ), 1 ≤ n + 1
n : ℤ
⊢ 2 ∣ n * (n + 1)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | have gt_pow3 (n : ℕ) : n ≤ n ^ 3 := by
apply Nat.le_self_pow
simp | n : ℕ
⊢ 6 ∣ n ^ 3 - n | n : ℕ
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
⊢ 6 ∣ n ^ 3 - n | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ 6 ∣ n ^ 3 - n
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | zify [gt_pow3 n] | n : ℕ
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
⊢ 6 ∣ n ^ 3 - n | n : ℕ
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
⊢ 6 ∣ ↑n ^ 3 - ↑n | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
⊢ 6 ∣ n ^ 3 - n
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | induction' n using Nat.caseStrongInductionOn with n ih | n : ℕ
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
⊢ 6 ∣ ↑n ^ 3 - ↑n | case zero
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
⊢ 6 ∣ ↑0 ^ 3 - ↑0
case ind
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
⊢ 6 ∣ ↑(Nat.succ n) ^ 3 - ↑(Nat.succ n) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
⊢ 6 ∣ ↑n ^ 3 - ↑n
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | apply Nat.le_self_pow | n✝ n : ℕ
⊢ n ≤ n ^ 3 | case hn
n✝ n : ℕ
⊢ 3 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
n✝ n : ℕ
⊢ n ≤ n ^ 3
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | simp | case hn
n✝ n : ℕ
⊢ 3 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hn
n✝ n : ℕ
⊢ 3 ≠ 0
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | exact dvd_zero 6 | case zero
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
⊢ 6 ∣ ↑0 ^ 3 - ↑0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
⊢ 6 ∣ ↑0 ^ 3 - ↑0
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | by_cases hp : n ≤ 3 | case ind
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
⊢ 6 ∣ ↑(Nat.succ n) ^ 3 - ↑(Nat.succ n) | case pos
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : n ≤ 3
⊢ 6 ∣ ↑(Nat.succ n) ^ 3 - ↑(Nat.succ n)
case neg
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
⊢ 6 ∣ ↑(Nat.succ n) ^ 3 - ↑(Nat.succ n) | Please generate a tactic in lean4 to solve the state.
STATE:
case ind
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
⊢ 6 ∣ ↑(Nat.succ n) ^ 3 - ↑(Nat.succ n)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | let k : ℕ := n - 3 | case neg
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
⊢ 6 ∣ ↑(Nat.succ n) ^ 3 - ↑(Nat.succ n) | case neg
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
⊢ 6 ∣ ↑(Nat.succ n) ^ 3 - ↑(Nat.succ n) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
⊢ 6 ∣ ↑(Nat.succ n) ^ 3 - ↑(Nat.succ n)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | have hk : Nat.succ n = k + 4 := by
zify [le_of_not_le hp]
ring_nf | case neg
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
⊢ 6 ∣ ↑(Nat.succ n) ^ 3 - ↑(Nat.succ n) | case neg
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
hk : Nat.succ n = k + 4
⊢ 6 ∣ ↑(Nat.succ n) ^ 3 - ↑(Nat.succ n) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
⊢ 6 ∣ ↑(Nat.succ n) ^ 3 - ↑(Nat.succ n)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | rw [hk] | case neg
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
hk : Nat.succ n = k + 4
⊢ 6 ∣ ↑(Nat.succ n) ^ 3 - ↑(Nat.succ n) | case neg
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
hk : Nat.succ n = k + 4
⊢ 6 ∣ ↑(k + 4) ^ 3 - ↑(k + 4) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
hk : Nat.succ n = k + 4
⊢ 6 ∣ ↑(Nat.succ n) ^ 3 - ↑(Nat.succ n)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | push_cast (config := { zeta := false }) | case neg
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
hk : Nat.succ n = k + 4
⊢ 6 ∣ ↑(k + 4) ^ 3 - ↑(k + 4) | case neg
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
hk : Nat.succ n = k + 4
⊢ 6 ∣ (↑k + 4) ^ 3 - (↑k + 4) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
hk : Nat.succ n = k + 4
⊢ 6 ∣ ↑(k + 4) ^ 3 - ↑(k + 4)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | have hring (k : ℤ) :
(k + 4) ^ 3 - (k + 4)
= (k ^ 3 - k) + 6 * (2 * k ^ 2 + 8 * k + 10) := by
ring_nf | case neg
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
hk : Nat.succ n = k + 4
⊢ 6 ∣ (↑k + 4) ^ 3 - (↑k + 4) | case neg
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
hk : Nat.succ n = k + 4
hring : ∀ (k : ℤ), (k + 4) ^ 3 - (k + 4) = k ^ 3 - k + 6 * (2 * k ^ 2 + 8 * k + 10)
⊢ 6 ∣ (↑k + 4) ^ 3 - (↑k + 4) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
hk : Nat.succ n = k + 4
⊢ 6 ∣ (↑k + 4) ^ 3 - (↑k + 4)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | rw [hring k] | case neg
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
hk : Nat.succ n = k + 4
hring : ∀ (k : ℤ), (k + 4) ^ 3 - (k + 4) = k ^ 3 - k + 6 * (2 * k ^ 2 + 8 * k + 10)
⊢ 6 ∣ (↑k + 4) ^ 3 - (↑k + 4) | case neg
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
hk : Nat.succ n = k + 4
hring : ∀ (k : ℤ), (k + 4) ^ 3 - (k + 4) = k ^ 3 - k + 6 * (2 * k ^ 2 + 8 * k + 10)
⊢ 6 ∣ ↑k ^ 3 - ↑k + 6 * (2 * ↑k ^ 2 + 8 * ↑k + 10) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
hk : Nat.succ n = k + 4
hring : ∀ (k : ℤ), (k + 4) ^ 3 - (k + 4) = k ^ 3 - k + 6 * (2 * k ^ 2 + 8 * k + 10)
⊢ 6 ∣ (↑k + 4) ^ 3 - (↑k + 4)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | rcases ih k (by simp) with ⟨m, hm⟩ | case neg
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
hk : Nat.succ n = k + 4
hring : ∀ (k : ℤ), (k + 4) ^ 3 - (k + 4) = k ^ 3 - k + 6 * (2 * k ^ 2 + 8 * k + 10)
⊢ 6 ∣ ↑k ^ 3 - ↑k + 6 * (2 * ↑k ^ 2 + 8 * ↑k + 10) | case neg.intro
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
hk : Nat.succ n = k + 4
hring : ∀ (k : ℤ), (k + 4) ^ 3 - (k + 4) = k ^ 3 - k + 6 * (2 * k ^ 2 + 8 * k + 10)
m : ℤ
hm : ↑k ^ 3 - ↑k = 6 * m
⊢ 6 ∣ ↑k ^ 3 - ↑k + 6 * (2 * ↑k ^ 2 + 8 * ↑k + 10) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
hk : Nat.succ n = k + 4
hring : ∀ (k : ℤ), (k + 4) ^ 3 - (k + 4) = k ^ 3 - k + 6 * (2 * k ^ 2 + 8 * k + 10)
⊢ 6 ∣ ↑k ^ 3 - ↑k + 6 * (2 * ↑k ^ 2 + 8 * ↑k + 10)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | rw [hm, ←mul_add] | case neg.intro
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
hk : Nat.succ n = k + 4
hring : ∀ (k : ℤ), (k + 4) ^ 3 - (k + 4) = k ^ 3 - k + 6 * (2 * k ^ 2 + 8 * k + 10)
m : ℤ
hm : ↑k ^ 3 - ↑k = 6 * m
⊢ 6 ∣ ↑k ^ 3 - ↑k + 6 * (2 * ↑k ^ 2 + 8 * ↑k + 10) | case neg.intro
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
hk : Nat.succ n = k + 4
hring : ∀ (k : ℤ), (k + 4) ^ 3 - (k + 4) = k ^ 3 - k + 6 * (2 * k ^ 2 + 8 * k + 10)
m : ℤ
hm : ↑k ^ 3 - ↑k = 6 * m
⊢ 6 ∣ 6 * (m + (2 * ↑k ^ 2 + 8 * ↑k + 10)) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
hk : Nat.succ n = k + 4
hring : ∀ (k : ℤ), (k + 4) ^ 3 - (k + 4) = k ^ 3 - k + 6 * (2 * k ^ 2 + 8 * k + 10)
m : ℤ
hm : ↑k ^ 3 - ↑k = 6 * m
⊢ 6 ∣ ↑k ^ 3 - ↑k + 6 * (2 * ↑k ^ 2 + 8 * ↑k + 10)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | simp only [ge_iff_le, dvd_mul_right] | case neg.intro
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
hk : Nat.succ n = k + 4
hring : ∀ (k : ℤ), (k + 4) ^ 3 - (k + 4) = k ^ 3 - k + 6 * (2 * k ^ 2 + 8 * k + 10)
m : ℤ
hm : ↑k ^ 3 - ↑k = 6 * m
⊢ 6 ∣ 6 * (m + (2 * ↑k ^ 2 + 8 * ↑k + 10)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
hk : Nat.succ n = k + 4
hring : ∀ (k : ℤ), (k + 4) ^ 3 - (k + 4) = k ^ 3 - k + 6 * (2 * k ^ 2 + 8 * k + 10)
m : ℤ
hm : ↑k ^ 3 - ↑k = 6 * m
⊢ 6 ∣ 6 * (m + (2 * ↑k ^ 2 + 8 * ↑k + 10))
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | rw [Nat.succ_eq_add_one] | case pos
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : n ≤ 3
⊢ 6 ∣ ↑(Nat.succ n) ^ 3 - ↑(Nat.succ n) | case pos
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : n ≤ 3
⊢ 6 ∣ ↑(n + 1) ^ 3 - ↑(n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : n ≤ 3
⊢ 6 ∣ ↑(Nat.succ n) ^ 3 - ↑(Nat.succ n)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | rcases hp with _ | hp | case pos
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : n ≤ 3
⊢ 6 ∣ ↑(n + 1) ^ 3 - ↑(n + 1) | case pos.refl
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 3, 6 ∣ ↑m ^ 3 - ↑m
⊢ 6 ∣ ↑(3 + 1) ^ 3 - ↑(3 + 1)
case pos.step
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : Nat.le n 2
⊢ 6 ∣ ↑(n + 1) ^ 3 - ↑(n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : n ≤ 3
⊢ 6 ∣ ↑(n + 1) ^ 3 - ↑(n + 1)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | ring_nf | case pos.refl
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 3, 6 ∣ ↑m ^ 3 - ↑m
⊢ 6 ∣ ↑(3 + 1) ^ 3 - ↑(3 + 1) | case pos.refl
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 3, 6 ∣ ↑m ^ 3 - ↑m
⊢ 6 ∣ 60 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.refl
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 3, 6 ∣ ↑m ^ 3 - ↑m
⊢ 6 ∣ ↑(3 + 1) ^ 3 - ↑(3 + 1)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | have : (60 : ℤ) = 6 * 10 := by ring | case pos.refl
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 3, 6 ∣ ↑m ^ 3 - ↑m
⊢ 6 ∣ 60 | case pos.refl
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 3, 6 ∣ ↑m ^ 3 - ↑m
this : 60 = 6 * 10
⊢ 6 ∣ 60 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.refl
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 3, 6 ∣ ↑m ^ 3 - ↑m
⊢ 6 ∣ 60
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | simp_rw [this, dvd_mul_right] | case pos.refl
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 3, 6 ∣ ↑m ^ 3 - ↑m
this : 60 = 6 * 10
⊢ 6 ∣ 60 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.refl
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 3, 6 ∣ ↑m ^ 3 - ↑m
this : 60 = 6 * 10
⊢ 6 ∣ 60
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | ring | gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 3, 6 ∣ ↑m ^ 3 - ↑m
⊢ 60 = 6 * 10 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 3, 6 ∣ ↑m ^ 3 - ↑m
⊢ 60 = 6 * 10
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | rcases hp with _ | hp | case pos.step
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : Nat.le n 2
⊢ 6 ∣ ↑(n + 1) ^ 3 - ↑(n + 1) | case pos.step.refl
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 2, 6 ∣ ↑m ^ 3 - ↑m
⊢ 6 ∣ ↑(2 + 1) ^ 3 - ↑(2 + 1)
case pos.step.step
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : Nat.le n 1
⊢ 6 ∣ ↑(n + 1) ^ 3 - ↑(n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.step
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : Nat.le n 2
⊢ 6 ∣ ↑(n + 1) ^ 3 - ↑(n + 1)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | ring_nf | case pos.step.refl
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 2, 6 ∣ ↑m ^ 3 - ↑m
⊢ 6 ∣ ↑(2 + 1) ^ 3 - ↑(2 + 1) | case pos.step.refl
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 2, 6 ∣ ↑m ^ 3 - ↑m
⊢ 6 ∣ 24 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.step.refl
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 2, 6 ∣ ↑m ^ 3 - ↑m
⊢ 6 ∣ ↑(2 + 1) ^ 3 - ↑(2 + 1)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | have : (24 : ℤ) = 6 * 4 := by ring | case pos.step.refl
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 2, 6 ∣ ↑m ^ 3 - ↑m
⊢ 6 ∣ 24 | case pos.step.refl
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 2, 6 ∣ ↑m ^ 3 - ↑m
this : 24 = 6 * 4
⊢ 6 ∣ 24 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.step.refl
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 2, 6 ∣ ↑m ^ 3 - ↑m
⊢ 6 ∣ 24
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | simp_rw [this, dvd_mul_right] | case pos.step.refl
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 2, 6 ∣ ↑m ^ 3 - ↑m
this : 24 = 6 * 4
⊢ 6 ∣ 24 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.step.refl
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 2, 6 ∣ ↑m ^ 3 - ↑m
this : 24 = 6 * 4
⊢ 6 ∣ 24
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | ring | gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 2, 6 ∣ ↑m ^ 3 - ↑m
⊢ 24 = 6 * 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 2, 6 ∣ ↑m ^ 3 - ↑m
⊢ 24 = 6 * 4
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | rcases hp with _ | hp | case pos.step.step
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : Nat.le n 1
⊢ 6 ∣ ↑(n + 1) ^ 3 - ↑(n + 1) | case pos.step.step.refl
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 1, 6 ∣ ↑m ^ 3 - ↑m
⊢ 6 ∣ ↑(1 + 1) ^ 3 - ↑(1 + 1)
case pos.step.step.step
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : Nat.le n 0
⊢ 6 ∣ ↑(n + 1) ^ 3 - ↑(n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.step.step
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : Nat.le n 1
⊢ 6 ∣ ↑(n + 1) ^ 3 - ↑(n + 1)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | ring_nf | case pos.step.step.refl
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 1, 6 ∣ ↑m ^ 3 - ↑m
⊢ 6 ∣ ↑(1 + 1) ^ 3 - ↑(1 + 1) | case pos.step.step.refl
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 1, 6 ∣ ↑m ^ 3 - ↑m
⊢ 6 ∣ 6 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.step.step.refl
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 1, 6 ∣ ↑m ^ 3 - ↑m
⊢ 6 ∣ ↑(1 + 1) ^ 3 - ↑(1 + 1)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | exact Int.dvd_refl 6 | case pos.step.step.refl
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 1, 6 ∣ ↑m ^ 3 - ↑m
⊢ 6 ∣ 6 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.step.step.refl
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
ih : ∀ m ≤ 1, 6 ∣ ↑m ^ 3 - ↑m
⊢ 6 ∣ 6
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | simp at hp | case pos.step.step.step
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : Nat.le n 0
⊢ 6 ∣ ↑(n + 1) ^ 3 - ↑(n + 1) | case pos.step.step.step
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : n = 0
⊢ 6 ∣ ↑(n + 1) ^ 3 - ↑(n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.step.step.step
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : Nat.le n 0
⊢ 6 ∣ ↑(n + 1) ^ 3 - ↑(n + 1)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | rw [hp] | case pos.step.step.step
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : n = 0
⊢ 6 ∣ ↑(n + 1) ^ 3 - ↑(n + 1) | case pos.step.step.step
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : n = 0
⊢ 6 ∣ ↑(0 + 1) ^ 3 - ↑(0 + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.step.step.step
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : n = 0
⊢ 6 ∣ ↑(n + 1) ^ 3 - ↑(n + 1)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | exact dvd_zero 6 | case pos.step.step.step
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : n = 0
⊢ 6 ∣ ↑(0 + 1) ^ 3 - ↑(0 + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.step.step.step
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : n = 0
⊢ 6 ∣ ↑(0 + 1) ^ 3 - ↑(0 + 1)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | zify [le_of_not_le hp] | gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
⊢ Nat.succ n = k + 4 | gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
⊢ ↑n + 1 = ↑n - 3 + 4 | Please generate a tactic in lean4 to solve the state.
STATE:
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
⊢ Nat.succ n = k + 4
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | ring_nf | gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
⊢ ↑n + 1 = ↑n - 3 + 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
⊢ ↑n + 1 = ↑n - 3 + 4
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | ring_nf | gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k✝ : ℕ := n - 3
hk : Nat.succ n = k✝ + 4
k : ℤ
⊢ (k + 4) ^ 3 - (k + 4) = k ^ 3 - k + 6 * (2 * k ^ 2 + 8 * k + 10) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k✝ : ℕ := n - 3
hk : Nat.succ n = k✝ + 4
k : ℤ
⊢ (k + 4) ^ 3 - (k + 4) = k ^ 3 - k + 6 * (2 * k ^ 2 + 8 * k + 10)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | six_dvd_n_cubed_minus_n | [45, 1] | [88, 41] | simp | gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
hk : Nat.succ n = k + 4
hring : ∀ (k : ℤ), (k + 4) ^ 3 - (k + 4) = k ^ 3 - k + 6 * (2 * k ^ 2 + 8 * k + 10)
⊢ k ≤ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
gt_pow3 : ∀ (n : ℕ), n ≤ n ^ 3
n : ℕ
ih : ∀ m ≤ n, 6 ∣ ↑m ^ 3 - ↑m
hp : ¬n ≤ 3
k : ℕ := n - 3
hk : Nat.succ n = k + 4
hring : ∀ (k : ℤ), (k + 4) ^ 3 - (k + 4) = k ^ 3 - k + 6 * (2 * k ^ 2 + 8 * k + 10)
⊢ k ≤ n
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | binomial_theorem | [91, 1] | [112, 36] | induction' n with n ih | x y n : ℕ
⊢ (x + y) ^ n = ∑ k in Finset.range (n + 1), Nat.choose n k * x ^ (n - k) * y ^ k | case zero
x y : ℕ
⊢ (x + y) ^ Nat.zero = ∑ k in Finset.range (Nat.zero + 1), Nat.choose Nat.zero k * x ^ (Nat.zero - k) * y ^ k
case succ
x y n : ℕ
ih : (x + y) ^ n = ∑ k in Finset.range (n + 1), Nat.choose n k * x ^ (n - k) * y ^ k
⊢ (x + y) ^ Nat.succ n = ∑ k in Finset.range (Nat.succ n + 1), Nat.choose (Nat.succ n) k * x ^ (Nat.succ n - k) * y ^ k | Please generate a tactic in lean4 to solve the state.
STATE:
x y n : ℕ
⊢ (x + y) ^ n = ∑ k in Finset.range (n + 1), Nat.choose n k * x ^ (n - k) * y ^ k
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | binomial_theorem | [91, 1] | [112, 36] | simp | case zero
x y : ℕ
⊢ (x + y) ^ Nat.zero = ∑ k in Finset.range (Nat.zero + 1), Nat.choose Nat.zero k * x ^ (Nat.zero - k) * y ^ k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
x y : ℕ
⊢ (x + y) ^ Nat.zero = ∑ k in Finset.range (Nat.zero + 1), Nat.choose Nat.zero k * x ^ (Nat.zero - k) * y ^ k
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | binomial_theorem | [91, 1] | [112, 36] | rw [Finset.sum_range_succ,
Nat.succ_eq_add_one,
Nat.choose_self,
Nat.sub_self,
pow_zero,
mul_one,
one_mul,
Finset.sum_range_succ',
Nat.choose_zero_right,
one_mul,
Nat.sub_zero,
pow_zero,
mul_one] | case succ
x y n : ℕ
ih : (x + y) ^ n = ∑ k in Finset.range (n + 1), Nat.choose n k * x ^ (n - k) * y ^ k
⊢ (x + y) ^ Nat.succ n = ∑ k in Finset.range (Nat.succ n + 1), Nat.choose (Nat.succ n) k * x ^ (Nat.succ n - k) * y ^ k | case succ
x y n : ℕ
ih : (x + y) ^ n = ∑ k in Finset.range (n + 1), Nat.choose n k * x ^ (n - k) * y ^ k
⊢ (x + y) ^ (n + 1) =
∑ k in Finset.range n, Nat.choose (n + 1) (k + 1) * x ^ (n + 1 - (k + 1)) * y ^ (k + 1) + x ^ (n + 1) + y ^ (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
x y n : ℕ
ih : (x + y) ^ n = ∑ k in Finset.range (n + 1), Nat.choose n k * x ^ (n - k) * y ^ k
⊢ (x + y) ^ Nat.succ n = ∑ k in Finset.range (Nat.succ n + 1), Nat.choose (Nat.succ n) k * x ^ (Nat.succ n - k) * y ^ k
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | binomial_theorem | [91, 1] | [112, 36] | simp_rw [Nat.choose_succ_succ'] | case succ
x y n : ℕ
ih : (x + y) ^ n = ∑ k in Finset.range (n + 1), Nat.choose n k * x ^ (n - k) * y ^ k
⊢ (x + y) ^ (n + 1) =
∑ k in Finset.range n, Nat.choose (n + 1) (k + 1) * x ^ (n + 1 - (k + 1)) * y ^ (k + 1) + x ^ (n + 1) + y ^ (n + 1) | case succ
x y n : ℕ
ih : (x + y) ^ n = ∑ k in Finset.range (n + 1), Nat.choose n k * x ^ (n - k) * y ^ k
⊢ (x + y) ^ (n + 1) =
∑ x_1 in Finset.range n, (Nat.choose n x_1 + Nat.choose n (x_1 + 1)) * x ^ (n + 1 - (x_1 + 1)) * y ^ (x_1 + 1) +
x ^ (n + 1) +
y ^ (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
x y n : ℕ
ih : (x + y) ^ n = ∑ k in Finset.range (n + 1), Nat.choose n k * x ^ (n - k) * y ^ k
⊢ (x + y) ^ (n + 1) =
∑ k in Finset.range n, Nat.choose (n + 1) (k + 1) * x ^ (n + 1 - (k + 1)) * y ^ (k + 1) + x ^ (n + 1) + y ^ (n + 1)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | cons_le_is_ordered | [25, 1] | [34, 17] | simp only [IsOrdered, List.pairwise_cons] at * | α : Type
inst✝ : LE α
a : α
xs : List α
h : IsOrdered xs
hle : ∀ x ∈ xs, a ≤ x
⊢ IsOrdered (a :: xs) | α : Type
inst✝ : LE α
a : α
xs : List α
h : List.Pairwise LE.le xs
hle : ∀ x ∈ xs, a ≤ x
⊢ (∀ a' ∈ xs, a ≤ a') ∧ List.Pairwise LE.le xs | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : LE α
a : α
xs : List α
h : IsOrdered xs
hle : ∀ x ∈ xs, a ≤ x
⊢ IsOrdered (a :: xs)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | cons_le_is_ordered | [25, 1] | [34, 17] | exact ⟨hle, h⟩ | α : Type
inst✝ : LE α
a : α
xs : List α
h : List.Pairwise LE.le xs
hle : ∀ x ∈ xs, a ≤ x
⊢ (∀ a' ∈ xs, a ≤ a') ∧ List.Pairwise LE.le xs | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : LE α
a : α
xs : List α
h : List.Pairwise LE.le xs
hle : ∀ x ∈ xs, a ≤ x
⊢ (∀ a' ∈ xs, a ≤ a') ∧ List.Pairwise LE.le xs
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | constructor | α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
l : List α
⊢ x ∈ insert' a l ↔ x = a ∨ x ∈ l | case mp
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
l : List α
⊢ x ∈ insert' a l → x = a ∨ x ∈ l
case mpr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
l : List α
⊢ x = a ∨ x ∈ l → x ∈ insert' a l | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
l : List α
⊢ x ∈ insert' a l ↔ x = a ∨ x ∈ l
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | intro hel | case mp
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
l : List α
⊢ x ∈ insert' a l → x = a ∨ x ∈ l | case mp
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
l : List α
hel : x ∈ insert' a l
⊢ x = a ∨ x ∈ l | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
l : List α
⊢ x ∈ insert' a l → x = a ∨ x ∈ l
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | induction' l with y ys ih | case mp
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
l : List α
hel : x ∈ insert' a l
⊢ x = a ∨ x ∈ l | case mp.nil
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
hel : x ∈ insert' a []
⊢ x = a ∨ x ∈ []
case mp.cons
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
hel : x ∈ insert' a (y :: ys)
⊢ x = a ∨ x ∈ y :: ys | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
l : List α
hel : x ∈ insert' a l
⊢ x = a ∨ x ∈ l
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | rw [insert', List.mem_singleton] at hel | case mp.nil
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
hel : x ∈ insert' a []
⊢ x = a ∨ x ∈ [] | case mp.nil
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
hel : x = a
⊢ x = a ∨ x ∈ [] | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.nil
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
hel : x ∈ insert' a []
⊢ x = a ∨ x ∈ []
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | left | case mp.nil
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
hel : x = a
⊢ x = a ∨ x ∈ [] | case mp.nil.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
hel : x = a
⊢ x = a | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.nil
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
hel : x = a
⊢ x = a ∨ x ∈ []
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | exact hel | case mp.nil.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
hel : x = a
⊢ x = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.nil.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
hel : x = a
⊢ x = a
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | rw [insert'] at hel | case mp.cons
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
hel : x ∈ insert' a (y :: ys)
⊢ x = a ∨ x ∈ y :: ys | case mp.cons
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
hel : x ∈ if a ≤ y then a :: y :: ys else y :: insert' a ys
⊢ x = a ∨ x ∈ y :: ys | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.cons
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
hel : x ∈ insert' a (y :: ys)
⊢ x = a ∨ x ∈ y :: ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | split_ifs at hel | case mp.cons
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
hel : x ∈ if a ≤ y then a :: y :: ys else y :: insert' a ys
⊢ x = a ∨ x ∈ y :: ys | case pos
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : a ≤ y
hel : x ∈ a :: y :: ys
⊢ x = a ∨ x ∈ y :: ys
case neg
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hel : x ∈ y :: insert' a ys
⊢ x = a ∨ x ∈ y :: ys | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.cons
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
hel : x ∈ if a ≤ y then a :: y :: ys else y :: insert' a ys
⊢ x = a ∨ x ∈ y :: ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | simp only [List.mem_cons] at hel | case pos
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : a ≤ y
hel : x ∈ a :: y :: ys
⊢ x = a ∨ x ∈ y :: ys | case pos
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : a ≤ y
hel : x = a ∨ x = y ∨ x ∈ ys
⊢ x = a ∨ x ∈ y :: ys | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : a ≤ y
hel : x ∈ a :: y :: ys
⊢ x = a ∨ x ∈ y :: ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | simp only [List.mem_cons] | case pos
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : a ≤ y
hel : x = a ∨ x = y ∨ x ∈ ys
⊢ x = a ∨ x ∈ y :: ys | case pos
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : a ≤ y
hel : x = a ∨ x = y ∨ x ∈ ys
⊢ x = a ∨ x = y ∨ x ∈ ys | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : a ≤ y
hel : x = a ∨ x = y ∨ x ∈ ys
⊢ x = a ∨ x ∈ y :: ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | exact hel | case pos
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : a ≤ y
hel : x = a ∨ x = y ∨ x ∈ ys
⊢ x = a ∨ x = y ∨ x ∈ ys | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : a ≤ y
hel : x = a ∨ x = y ∨ x ∈ ys
⊢ x = a ∨ x = y ∨ x ∈ ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | simp only [List.mem_cons] at hel | case neg
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hel : x ∈ y :: insert' a ys
⊢ x = a ∨ x ∈ y :: ys | case neg
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hel : x = y ∨ x ∈ insert' a ys
⊢ x = a ∨ x ∈ y :: ys | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hel : x ∈ y :: insert' a ys
⊢ x = a ∨ x ∈ y :: ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | simp only [List.mem_cons] | case neg
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hel : x = y ∨ x ∈ insert' a ys
⊢ x = a ∨ x ∈ y :: ys | case neg
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hel : x = y ∨ x ∈ insert' a ys
⊢ x = a ∨ x = y ∨ x ∈ ys | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hel : x = y ∨ x ∈ insert' a ys
⊢ x = a ∨ x ∈ y :: ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | rcases hel with (heq | hmem) | case neg
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hel : x = y ∨ x ∈ insert' a ys
⊢ x = a ∨ x = y ∨ x ∈ ys | case neg.inl
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
heq : x = y
⊢ x = a ∨ x = y ∨ x ∈ ys
case neg.inr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hmem : x ∈ insert' a ys
⊢ x = a ∨ x = y ∨ x ∈ ys | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hel : x = y ∨ x ∈ insert' a ys
⊢ x = a ∨ x = y ∨ x ∈ ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | right | case neg.inl
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
heq : x = y
⊢ x = a ∨ x = y ∨ x ∈ ys | case neg.inl.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
heq : x = y
⊢ x = y ∨ x ∈ ys | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inl
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
heq : x = y
⊢ x = a ∨ x = y ∨ x ∈ ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | left | case neg.inl.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
heq : x = y
⊢ x = y ∨ x ∈ ys | case neg.inl.h.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
heq : x = y
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inl.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
heq : x = y
⊢ x = y ∨ x ∈ ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | exact heq | case neg.inl.h.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
heq : x = y
⊢ x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inl.h.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
heq : x = y
⊢ x = y
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | rcases ih hmem with (heq' | hmem') | case neg.inr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hmem : x ∈ insert' a ys
⊢ x = a ∨ x = y ∨ x ∈ ys | case neg.inr.inl
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hmem : x ∈ insert' a ys
heq' : x = a
⊢ x = a ∨ x = y ∨ x ∈ ys
case neg.inr.inr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hmem : x ∈ insert' a ys
hmem' : x ∈ ys
⊢ x = a ∨ x = y ∨ x ∈ ys | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hmem : x ∈ insert' a ys
⊢ x = a ∨ x = y ∨ x ∈ ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | left | case neg.inr.inl
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hmem : x ∈ insert' a ys
heq' : x = a
⊢ x = a ∨ x = y ∨ x ∈ ys | case neg.inr.inl.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hmem : x ∈ insert' a ys
heq' : x = a
⊢ x = a | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.inl
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hmem : x ∈ insert' a ys
heq' : x = a
⊢ x = a ∨ x = y ∨ x ∈ ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | exact heq' | case neg.inr.inl.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hmem : x ∈ insert' a ys
heq' : x = a
⊢ x = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.inl.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hmem : x ∈ insert' a ys
heq' : x = a
⊢ x = a
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | right | case neg.inr.inr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hmem : x ∈ insert' a ys
hmem' : x ∈ ys
⊢ x = a ∨ x = y ∨ x ∈ ys | case neg.inr.inr.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hmem : x ∈ insert' a ys
hmem' : x ∈ ys
⊢ x = y ∨ x ∈ ys | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.inr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hmem : x ∈ insert' a ys
hmem' : x ∈ ys
⊢ x = a ∨ x = y ∨ x ∈ ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | right | case neg.inr.inr.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hmem : x ∈ insert' a ys
hmem' : x ∈ ys
⊢ x = y ∨ x ∈ ys | case neg.inr.inr.h.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hmem : x ∈ insert' a ys
hmem' : x ∈ ys
⊢ x ∈ ys | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.inr.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hmem : x ∈ insert' a ys
hmem' : x ∈ ys
⊢ x = y ∨ x ∈ ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | exact hmem' | case neg.inr.inr.h.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hmem : x ∈ insert' a ys
hmem' : x ∈ ys
⊢ x ∈ ys | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.inr.h.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x ∈ insert' a ys → x = a ∨ x ∈ ys
h✝ : ¬a ≤ y
hmem : x ∈ insert' a ys
hmem' : x ∈ ys
⊢ x ∈ ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | intro hor | case mpr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
l : List α
⊢ x = a ∨ x ∈ l → x ∈ insert' a l | case mpr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
l : List α
hor : x = a ∨ x ∈ l
⊢ x ∈ insert' a l | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
l : List α
⊢ x = a ∨ x ∈ l → x ∈ insert' a l
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | induction' l with y ys ih | case mpr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
l : List α
hor : x = a ∨ x ∈ l
⊢ x ∈ insert' a l | case mpr.nil
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
hor : x = a ∨ x ∈ []
⊢ x ∈ insert' a []
case mpr.cons
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
hor : x = a ∨ x ∈ y :: ys
⊢ x ∈ insert' a (y :: ys) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
l : List α
hor : x = a ∨ x ∈ l
⊢ x ∈ insert' a l
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | rw [insert', List.mem_singleton] | case mpr.nil
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
hor : x = a ∨ x ∈ []
⊢ x ∈ insert' a [] | case mpr.nil
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
hor : x = a ∨ x ∈ []
⊢ x = a | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.nil
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
hor : x = a ∨ x ∈ []
⊢ x ∈ insert' a []
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | rcases hor with (heq | hmem) | case mpr.nil
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
hor : x = a ∨ x ∈ []
⊢ x = a | case mpr.nil.inl
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
heq : x = a
⊢ x = a
case mpr.nil.inr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
hmem : x ∈ []
⊢ x = a | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.nil
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
hor : x = a ∨ x ∈ []
⊢ x = a
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | exact heq | case mpr.nil.inl
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
heq : x = a
⊢ x = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.nil.inl
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
heq : x = a
⊢ x = a
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | simp only [List.not_mem_nil] at hmem | case mpr.nil.inr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
hmem : x ∈ []
⊢ x = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.nil.inr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a : α
hmem : x ∈ []
⊢ x = a
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | rw [insert'] | case mpr.cons
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
hor : x = a ∨ x ∈ y :: ys
⊢ x ∈ insert' a (y :: ys) | case mpr.cons
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
hor : x = a ∨ x ∈ y :: ys
⊢ x ∈ if a ≤ y then a :: y :: ys else y :: insert' a ys | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.cons
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
hor : x = a ∨ x ∈ y :: ys
⊢ x ∈ insert' a (y :: ys)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | split_ifs | case mpr.cons
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
hor : x = a ∨ x ∈ y :: ys
⊢ x ∈ if a ≤ y then a :: y :: ys else y :: insert' a ys | case pos
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
hor : x = a ∨ x ∈ y :: ys
h✝ : a ≤ y
⊢ x ∈ a :: y :: ys
case neg
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
hor : x = a ∨ x ∈ y :: ys
h✝ : ¬a ≤ y
⊢ x ∈ y :: insert' a ys | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.cons
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
hor : x = a ∨ x ∈ y :: ys
⊢ x ∈ if a ≤ y then a :: y :: ys else y :: insert' a ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | simp only [List.mem_cons] | case pos
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
hor : x = a ∨ x ∈ y :: ys
h✝ : a ≤ y
⊢ x ∈ a :: y :: ys | case pos
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
hor : x = a ∨ x ∈ y :: ys
h✝ : a ≤ y
⊢ x = a ∨ x = y ∨ x ∈ ys | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
hor : x = a ∨ x ∈ y :: ys
h✝ : a ≤ y
⊢ x ∈ a :: y :: ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | rcases hor with (heq | hmem) | case pos
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
hor : x = a ∨ x ∈ y :: ys
h✝ : a ≤ y
⊢ x = a ∨ x = y ∨ x ∈ ys | case pos.inl
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : a ≤ y
heq : x = a
⊢ x = a ∨ x = y ∨ x ∈ ys
case pos.inr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : a ≤ y
hmem : x ∈ y :: ys
⊢ x = a ∨ x = y ∨ x ∈ ys | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
hor : x = a ∨ x ∈ y :: ys
h✝ : a ≤ y
⊢ x = a ∨ x = y ∨ x ∈ ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | left | case pos.inl
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : a ≤ y
heq : x = a
⊢ x = a ∨ x = y ∨ x ∈ ys | case pos.inl.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : a ≤ y
heq : x = a
⊢ x = a | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.inl
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : a ≤ y
heq : x = a
⊢ x = a ∨ x = y ∨ x ∈ ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | exact heq | case pos.inl.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : a ≤ y
heq : x = a
⊢ x = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.inl.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : a ≤ y
heq : x = a
⊢ x = a
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | right | case pos.inr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : a ≤ y
hmem : x ∈ y :: ys
⊢ x = a ∨ x = y ∨ x ∈ ys | case pos.inr.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : a ≤ y
hmem : x ∈ y :: ys
⊢ x = y ∨ x ∈ ys | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.inr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : a ≤ y
hmem : x ∈ y :: ys
⊢ x = a ∨ x = y ∨ x ∈ ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | simp only [List.mem_cons] at hmem | case pos.inr.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : a ≤ y
hmem : x ∈ y :: ys
⊢ x = y ∨ x ∈ ys | case pos.inr.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : a ≤ y
hmem : x = y ∨ x ∈ ys
⊢ x = y ∨ x ∈ ys | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.inr.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : a ≤ y
hmem : x ∈ y :: ys
⊢ x = y ∨ x ∈ ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | exact hmem | case pos.inr.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : a ≤ y
hmem : x = y ∨ x ∈ ys
⊢ x = y ∨ x ∈ ys | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.inr.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : a ≤ y
hmem : x = y ∨ x ∈ ys
⊢ x = y ∨ x ∈ ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | simp only [List.mem_cons] | case neg
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
hor : x = a ∨ x ∈ y :: ys
h✝ : ¬a ≤ y
⊢ x ∈ y :: insert' a ys | case neg
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
hor : x = a ∨ x ∈ y :: ys
h✝ : ¬a ≤ y
⊢ x = y ∨ x ∈ insert' a ys | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
hor : x = a ∨ x ∈ y :: ys
h✝ : ¬a ≤ y
⊢ x ∈ y :: insert' a ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | rcases hor with (heq | hmem) | case neg
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
hor : x = a ∨ x ∈ y :: ys
h✝ : ¬a ≤ y
⊢ x = y ∨ x ∈ insert' a ys | case neg.inl
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : ¬a ≤ y
heq : x = a
⊢ x = y ∨ x ∈ insert' a ys
case neg.inr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : ¬a ≤ y
hmem : x ∈ y :: ys
⊢ x = y ∨ x ∈ insert' a ys | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
hor : x = a ∨ x ∈ y :: ys
h✝ : ¬a ≤ y
⊢ x = y ∨ x ∈ insert' a ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | right | case neg.inl
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : ¬a ≤ y
heq : x = a
⊢ x = y ∨ x ∈ insert' a ys | case neg.inl.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : ¬a ≤ y
heq : x = a
⊢ x ∈ insert' a ys | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inl
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : ¬a ≤ y
heq : x = a
⊢ x = y ∨ x ∈ insert' a ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | exact ih (Or.inl heq) | case neg.inl.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : ¬a ≤ y
heq : x = a
⊢ x ∈ insert' a ys | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inl.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : ¬a ≤ y
heq : x = a
⊢ x ∈ insert' a ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | simp only [List.mem_cons] at hmem | case neg.inr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : ¬a ≤ y
hmem : x ∈ y :: ys
⊢ x = y ∨ x ∈ insert' a ys | case neg.inr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : ¬a ≤ y
hmem : x = y ∨ x ∈ ys
⊢ x = y ∨ x ∈ insert' a ys | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : ¬a ≤ y
hmem : x ∈ y :: ys
⊢ x = y ∨ x ∈ insert' a ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | rcases hmem with (heq' | hmem') | case neg.inr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : ¬a ≤ y
hmem : x = y ∨ x ∈ ys
⊢ x = y ∨ x ∈ insert' a ys | case neg.inr.inl
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : ¬a ≤ y
heq' : x = y
⊢ x = y ∨ x ∈ insert' a ys
case neg.inr.inr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : ¬a ≤ y
hmem' : x ∈ ys
⊢ x = y ∨ x ∈ insert' a ys | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : ¬a ≤ y
hmem : x = y ∨ x ∈ ys
⊢ x = y ∨ x ∈ insert' a ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | left | case neg.inr.inl
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : ¬a ≤ y
heq' : x = y
⊢ x = y ∨ x ∈ insert' a ys | case neg.inr.inl.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : ¬a ≤ y
heq' : x = y
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.inl
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : ¬a ≤ y
heq' : x = y
⊢ x = y ∨ x ∈ insert' a ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | exact heq' | case neg.inr.inl.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : ¬a ≤ y
heq' : x = y
⊢ x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.inl.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : ¬a ≤ y
heq' : x = y
⊢ x = y
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | right | case neg.inr.inr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : ¬a ≤ y
hmem' : x ∈ ys
⊢ x = y ∨ x ∈ insert' a ys | case neg.inr.inr.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : ¬a ≤ y
hmem' : x ∈ ys
⊢ x ∈ insert' a ys | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.inr
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : ¬a ≤ y
hmem' : x ∈ ys
⊢ x = y ∨ x ∈ insert' a ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | insert_mem | [61, 1] | [115, 36] | exact ih (Or.inr hmem') | case neg.inr.inr.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : ¬a ≤ y
hmem' : x ∈ ys
⊢ x ∈ insert' a ys | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.inr.inr.h
α : Type
hp : LE α
inst✝ : DecidableRel LE.le
x a y : α
ys : List α
ih : x = a ∨ x ∈ ys → x ∈ insert' a ys
h✝ : ¬a ≤ y
hmem' : x ∈ ys
⊢ x ∈ insert' a ys
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | ordered_insert_is_ordered | [117, 1] | [155, 27] | simp at * | α : Type
hl : LinearOrder α
inst✝ : DecidableRel LE.le
a : α
l : List α
ordered : IsOrdered l
⊢ IsOrdered (insert' a l) | α : Type
hl : LinearOrder α
inst✝ : DecidableRel LE.le
a : α
l : List α
ordered : List.Pairwise LE.le l
⊢ List.Pairwise LE.le (insert' a l) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
hl : LinearOrder α
inst✝ : DecidableRel LE.le
a : α
l : List α
ordered : IsOrdered l
⊢ IsOrdered (insert' a l)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | ordered_insert_is_ordered | [117, 1] | [155, 27] | induction' l with x xs ih | α : Type
hl : LinearOrder α
inst✝ : DecidableRel LE.le
a : α
l : List α
ordered : List.Pairwise LE.le l
⊢ List.Pairwise LE.le (insert' a l) | case nil
α : Type
hl : LinearOrder α
inst✝ : DecidableRel LE.le
a : α
ordered : List.Pairwise LE.le []
⊢ List.Pairwise LE.le (insert' a [])
case cons
α : Type
hl : LinearOrder α
inst✝ : DecidableRel LE.le
a x : α
xs : List α
ih : List.Pairwise LE.le xs → List.Pairwise LE.le (insert' a xs)
ordered : List.Pairwise LE.le (x :: xs)
⊢ List.Pairwise LE.le (insert' a (x :: xs)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
hl : LinearOrder α
inst✝ : DecidableRel LE.le
a : α
l : List α
ordered : List.Pairwise LE.le l
⊢ List.Pairwise LE.le (insert' a l)
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | ordered_insert_is_ordered | [117, 1] | [155, 27] | rw [insert', List.pairwise_cons] | case nil
α : Type
hl : LinearOrder α
inst✝ : DecidableRel LE.le
a : α
ordered : List.Pairwise LE.le []
⊢ List.Pairwise LE.le (insert' a []) | case nil
α : Type
hl : LinearOrder α
inst✝ : DecidableRel LE.le
a : α
ordered : List.Pairwise LE.le []
⊢ (∀ a' ∈ [], a ≤ a') ∧ List.Pairwise LE.le [] | Please generate a tactic in lean4 to solve the state.
STATE:
case nil
α : Type
hl : LinearOrder α
inst✝ : DecidableRel LE.le
a : α
ordered : List.Pairwise LE.le []
⊢ List.Pairwise LE.le (insert' a [])
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | ordered_insert_is_ordered | [117, 1] | [155, 27] | constructor | case nil
α : Type
hl : LinearOrder α
inst✝ : DecidableRel LE.le
a : α
ordered : List.Pairwise LE.le []
⊢ (∀ a' ∈ [], a ≤ a') ∧ List.Pairwise LE.le [] | case nil.left
α : Type
hl : LinearOrder α
inst✝ : DecidableRel LE.le
a : α
ordered : List.Pairwise LE.le []
⊢ ∀ a' ∈ [], a ≤ a'
case nil.right
α : Type
hl : LinearOrder α
inst✝ : DecidableRel LE.le
a : α
ordered : List.Pairwise LE.le []
⊢ List.Pairwise LE.le [] | Please generate a tactic in lean4 to solve the state.
STATE:
case nil
α : Type
hl : LinearOrder α
inst✝ : DecidableRel LE.le
a : α
ordered : List.Pairwise LE.le []
⊢ (∀ a' ∈ [], a ≤ a') ∧ List.Pairwise LE.le []
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | ordered_insert_is_ordered | [117, 1] | [155, 27] | intros y hy | case nil.left
α : Type
hl : LinearOrder α
inst✝ : DecidableRel LE.le
a : α
ordered : List.Pairwise LE.le []
⊢ ∀ a' ∈ [], a ≤ a' | case nil.left
α : Type
hl : LinearOrder α
inst✝ : DecidableRel LE.le
a : α
ordered : List.Pairwise LE.le []
y : α
hy : y ∈ []
⊢ a ≤ y | Please generate a tactic in lean4 to solve the state.
STATE:
case nil.left
α : Type
hl : LinearOrder α
inst✝ : DecidableRel LE.le
a : α
ordered : List.Pairwise LE.le []
⊢ ∀ a' ∈ [], a ≤ a'
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | ordered_insert_is_ordered | [117, 1] | [155, 27] | simp only [List.find?_nil, List.not_mem_nil] at hy | case nil.left
α : Type
hl : LinearOrder α
inst✝ : DecidableRel LE.le
a : α
ordered : List.Pairwise LE.le []
y : α
hy : y ∈ []
⊢ a ≤ y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nil.left
α : Type
hl : LinearOrder α
inst✝ : DecidableRel LE.le
a : α
ordered : List.Pairwise LE.le []
y : α
hy : y ∈ []
⊢ a ≤ y
TACTIC:
|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | Dictionary.lean | ordered_insert_is_ordered | [117, 1] | [155, 27] | exact ordered | case nil.right
α : Type
hl : LinearOrder α
inst✝ : DecidableRel LE.le
a : α
ordered : List.Pairwise LE.le []
⊢ List.Pairwise LE.le [] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nil.right
α : Type
hl : LinearOrder α
inst✝ : DecidableRel LE.le
a : α
ordered : List.Pairwise LE.le []
⊢ List.Pairwise LE.le []
TACTIC:
|
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