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/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet5.lean | tends_to_add | [32, 1] | [80, 7] | intro hn | case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
⊢ Nab ≤ n → |a n + b n - (t + u)| < ε | case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
⊢ |a n + b n - (t + u)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
⊢ Nab ≤ n → |a n + b n - (t + u)| < ε
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet5.lean | tends_to_add | [32, 1] | [80, 7] | have ha'' := ha' n | case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
⊢ |a n + b n - (t + u)| < ε | case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
⊢ |a n + b n - (t + u)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
⊢ |a n + b n - (t + u)| < ε
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet5.lean | tends_to_add | [32, 1] | [80, 7] | have hb'' := hb' n | case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
⊢ |a n + b n - (t + u)| < ε | case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
⊢ |a n + b n - (t + u)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
⊢ |a n + b n - (t + u)| < ε
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet5.lean | tends_to_add | [32, 1] | [80, 7] | have hNan : Na ≤ n :=
by
exact le_of_max_le_left hn | case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
⊢ |a n + b n - (t + u)| < ε | case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNan : Na ≤ n
⊢ |a n + b n - (t + u)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
⊢ |a n + b n - (t + u)| < ε
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet5.lean | tends_to_add | [32, 1] | [80, 7] | have hNbn : Nb ≤ n :=
by
exact le_of_max_le_right hn | case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNan : Na ≤ n
⊢ |a n + b n - (t + u)| < ε | case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNan : Na ≤ n
hNbn : Nb ≤ n
⊢ |a n + b n - (t + u)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNan : Na ≤ n
⊢ |a n + b n - (t + u)| < ε
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet5.lean | tends_to_add | [32, 1] | [80, 7] | apply ha'' at hNan | case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNan : Na ≤ n
hNbn : Nb ≤ n
⊢ |a n + b n - (t + u)| < ε | case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNbn : Nb ≤ n
hNan : |a n - t| < ε / 2
⊢ |a n + b n - (t + u)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNan : Na ≤ n
hNbn : Nb ≤ n
⊢ |a n + b n - (t + u)| < ε
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet5.lean | tends_to_add | [32, 1] | [80, 7] | apply hb'' at hNbn | case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNbn : Nb ≤ n
hNan : |a n - t| < ε / 2
⊢ |a n + b n - (t + u)| < ε | case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNan : |a n - t| < ε / 2
hNbn : |b n - u| < ε / 2
⊢ |a n + b n - (t + u)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNbn : Nb ≤ n
hNan : |a n - t| < ε / 2
⊢ |a n + b n - (t + u)| < ε
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet5.lean | tends_to_add | [32, 1] | [80, 7] | calc
|a n + b n - (t + u)|
= |(a n - t) + (b n - u)| :=
by
apply congrArg
ring
_ ≤ |a n - t| + |b n - u| :=
by
exact abs_add (a n - t) (b n - u)
_ < ε / 2 + ε / 2 :=
by
exact add_lt_add hNan hNbn
_ = ε :=
by
ring | case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNan : |a n - t| < ε / 2
hNbn : |b n - u| < ε / 2
⊢ |a n + b n - (t + u)| < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNan : |a n - t| < ε / 2
hNbn : |b n - u| < ε / 2
⊢ |a n + b n - (t + u)| < ε
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet5.lean | tends_to_add | [32, 1] | [80, 7] | linarith | a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
⊢ 0 < ε / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
⊢ 0 < ε / 2
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet5.lean | tends_to_add | [32, 1] | [80, 7] | linarith | a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
⊢ 0 < ε / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
⊢ 0 < ε / 2
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet5.lean | tends_to_add | [32, 1] | [80, 7] | exact le_of_max_le_left hn | a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
⊢ Na ≤ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
⊢ Na ≤ n
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet5.lean | tends_to_add | [32, 1] | [80, 7] | exact le_of_max_le_right hn | a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNan : Na ≤ n
⊢ Nb ≤ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNan : Na ≤ n
⊢ Nb ≤ n
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet5.lean | tends_to_add | [32, 1] | [80, 7] | apply congrArg | a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNan : |a n - t| < ε / 2
hNbn : |b n - u| < ε / 2
⊢ |a n + b n - (t + u)| = |a n - t + (b n - u)| | case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNan : |a n - t| < ε / 2
hNbn : |b n - u| < ε / 2
⊢ a n + b n - (t + u) = a n - t + (b n - u) | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNan : |a n - t| < ε / 2
hNbn : |b n - u| < ε / 2
⊢ |a n + b n - (t + u)| = |a n - t + (b n - u)|
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet5.lean | tends_to_add | [32, 1] | [80, 7] | ring | case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNan : |a n - t| < ε / 2
hNbn : |b n - u| < ε / 2
⊢ a n + b n - (t + u) = a n - t + (b n - u) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNan : |a n - t| < ε / 2
hNbn : |b n - u| < ε / 2
⊢ a n + b n - (t + u) = a n - t + (b n - u)
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet5.lean | tends_to_add | [32, 1] | [80, 7] | exact abs_add (a n - t) (b n - u) | a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNan : |a n - t| < ε / 2
hNbn : |b n - u| < ε / 2
⊢ |a n - t + (b n - u)| ≤ |a n - t| + |b n - u| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNan : |a n - t| < ε / 2
hNbn : |b n - u| < ε / 2
⊢ |a n - t + (b n - u)| ≤ |a n - t| + |b n - u|
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet5.lean | tends_to_add | [32, 1] | [80, 7] | exact add_lt_add hNan hNbn | a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNan : |a n - t| < ε / 2
hNbn : |b n - u| < ε / 2
⊢ |a n - t| + |b n - u| < ε / 2 + ε / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNan : |a n - t| < ε / 2
hNbn : |b n - u| < ε / 2
⊢ |a n - t| + |b n - u| < ε / 2 + ε / 2
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet5.lean | tends_to_add | [32, 1] | [80, 7] | ring | a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNan : |a n - t| < ε / 2
hNbn : |b n - u| < ε / 2
⊢ ε / 2 + ε / 2 = ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
t u : ℝ
ha : ∀ ε > 0, ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε
hb : tends_to b u
ε : ℝ
hεpos : 0 < ε
Na : ℕ
ha' : ∀ (n : ℕ), Na ≤ n → |a n - t| < ε / 2
Nb : ℕ
hb' : ∀ (n : ℕ), Nb ≤ n → |b n - u| < ε / 2
Nab : ℕ := max Na Nb
n : ℕ
hn : Nab ≤ n
ha'' : Na ≤ n → |a n - t| < ε / 2
hb'' : Nb ≤ n → |b n - u| < ε / 2
hNan : |a n - t| < ε / 2
hNbn : |b n - u| < ε / 2
⊢ ε / 2 + ε / 2 = ε
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet5.lean | tends_to_sub | [82, 1] | [88, 11] | have hb' := tends_to_neg hb | a b : ℕ → ℝ
t u : ℝ
ha : tends_to a t
hb : tends_to b u
⊢ tends_to (fun n => a n - b n) (t - u) | a b : ℕ → ℝ
t u : ℝ
ha : tends_to a t
hb : tends_to b u
hb' : tends_to (fun n => -b n) (-u)
⊢ tends_to (fun n => a n - b n) (t - u) | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
t u : ℝ
ha : tends_to a t
hb : tends_to b u
⊢ tends_to (fun n => a n - b n) (t - u)
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet5.lean | tends_to_sub | [82, 1] | [88, 11] | have h' := tends_to_add ha hb' | a b : ℕ → ℝ
t u : ℝ
ha : tends_to a t
hb : tends_to b u
hb' : tends_to (fun n => -b n) (-u)
⊢ tends_to (fun n => a n - b n) (t - u) | a b : ℕ → ℝ
t u : ℝ
ha : tends_to a t
hb : tends_to b u
hb' : tends_to (fun n => -b n) (-u)
h' : tends_to (fun n => a n + -b n) (t + -u)
⊢ tends_to (fun n => a n - b n) (t - u) | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
t u : ℝ
ha : tends_to a t
hb : tends_to b u
hb' : tends_to (fun n => -b n) (-u)
⊢ tends_to (fun n => a n - b n) (t - u)
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet5.lean | tends_to_sub | [82, 1] | [88, 11] | apply h' | a b : ℕ → ℝ
t u : ℝ
ha : tends_to a t
hb : tends_to b u
hb' : tends_to (fun n => -b n) (-u)
h' : tends_to (fun n => a n + -b n) (t + -u)
⊢ tends_to (fun n => a n - b n) (t - u) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
t u : ℝ
ha : tends_to a t
hb : tends_to b u
hb' : tends_to (fun n => -b n) (-u)
h' : tends_to (fun n => a n + -b n) (t + -u)
⊢ tends_to (fun n => a n - b n) (t - u)
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section03_functions/sheet3.lean | Yb_ne_Yc | [25, 1] | [28, 10] | intro h | ⊢ Y.b ≠ Y.c | h : Y.b = Y.c
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Y.b ≠ Y.c
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section03_functions/sheet3.lean | Yb_ne_Yc | [25, 1] | [28, 10] | cases h | h : Y.b = Y.c
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
h : Y.b = Y.c
⊢ False
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section03_functions/sheet3.lean | gYb_eq_gYc | [30, 1] | [32, 9] | rw [g] | ⊢ g Y.b = g Y.c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ g Y.b = g Y.c
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section03_functions/sheet3.lean | gf_injective | [36, 1] | [41, 8] | rw [Injective] | ⊢ Injective (g ∘ f) | ⊢ ∀ ⦃a₁ a₂ : X⦄, (g ∘ f) a₁ = (g ∘ f) a₂ → a₁ = a₂ | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Injective (g ∘ f)
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section03_functions/sheet3.lean | gf_injective | [36, 1] | [41, 8] | intro x₁ x₂ hgf | ⊢ ∀ ⦃a₁ a₂ : X⦄, (g ∘ f) a₁ = (g ∘ f) a₂ → a₁ = a₂ | x₁ x₂ : X
hgf : (g ∘ f) x₁ = (g ∘ f) x₂
⊢ x₁ = x₂ | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ ∀ ⦃a₁ a₂ : X⦄, (g ∘ f) a₁ = (g ∘ f) a₂ → a₁ = a₂
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section03_functions/sheet3.lean | gf_injective | [36, 1] | [41, 8] | cases x₁ | x₁ x₂ : X
hgf : (g ∘ f) x₁ = (g ∘ f) x₂
⊢ x₁ = x₂ | case a
x₂ : X
hgf : (g ∘ f) X.a = (g ∘ f) x₂
⊢ X.a = x₂ | Please generate a tactic in lean4 to solve the state.
STATE:
x₁ x₂ : X
hgf : (g ∘ f) x₁ = (g ∘ f) x₂
⊢ x₁ = x₂
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section03_functions/sheet3.lean | gf_injective | [36, 1] | [41, 8] | . rfl | case a
x₂ : X
hgf : (g ∘ f) X.a = (g ∘ f) x₂
⊢ X.a = x₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
x₂ : X
hgf : (g ∘ f) X.a = (g ∘ f) x₂
⊢ X.a = x₂
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section03_functions/sheet3.lean | gf_injective | [36, 1] | [41, 8] | rfl | case a
x₂ : X
hgf : (g ∘ f) X.a = (g ∘ f) x₂
⊢ X.a = x₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
x₂ : X
hgf : (g ∘ f) X.a = (g ∘ f) x₂
⊢ X.a = x₂
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section03_functions/sheet3.lean | gf_surjective | [51, 1] | [54, 10] | intro z | ⊢ Surjective (g ∘ f) | z : Z
⊢ ∃ a, (g ∘ f) a = z | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ Surjective (g ∘ f)
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section03_functions/sheet3.lean | gf_surjective | [51, 1] | [54, 10] | use X.a | z : Z
⊢ ∃ a, (g ∘ f) a = z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z : Z
⊢ ∃ a, (g ∘ f) a = z
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section05_sets/sheet1.lean | subset_def | [10, 1] | [12, 6] | rfl | X : Type
A B C D : Set X
x y z : X
⊢ A ⊆ B ↔ ∀ x ∈ A, x ∈ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
A B C D : Set X
x y z : X
⊢ A ⊆ B ↔ ∀ x ∈ A, x ∈ B
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section05_sets/sheet1.lean | mem_union_iff | [14, 1] | [16, 6] | rfl | X : Type
A B C D : Set X
x y z : X
⊢ x ∈ A ∪ B ↔ x ∈ A ∨ x ∈ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
A B C D : Set X
x y z : X
⊢ x ∈ A ∪ B ↔ x ∈ A ∨ x ∈ B
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section05_sets/sheet1.lean | mem_inter_iff | [18, 1] | [20, 6] | rfl | X : Type
A B C D : Set X
x y z : X
⊢ x ∈ A ∩ B ↔ x ∈ A ∧ x ∈ B | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
A B C D : Set X
x y z : X
⊢ x ∈ A ∩ B ↔ x ∈ A ∧ x ∈ B
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section06_orderings_and_lattices/sheet2.lean | inf_le_inf_left' | [45, 1] | [53, 7] | apply le_inf | L : Type
inst✝ : Lattice L
a b c : L
h : b ≤ c
⊢ a ⊓ b ≤ a ⊓ c | case a
L : Type
inst✝ : Lattice L
a b c : L
h : b ≤ c
⊢ a ⊓ b ≤ a
case a
L : Type
inst✝ : Lattice L
a b c : L
h : b ≤ c
⊢ a ⊓ b ≤ c | Please generate a tactic in lean4 to solve the state.
STATE:
L : Type
inst✝ : Lattice L
a b c : L
h : b ≤ c
⊢ a ⊓ b ≤ a ⊓ c
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section06_orderings_and_lattices/sheet2.lean | inf_le_inf_left' | [45, 1] | [53, 7] | . apply inf_le_left | case a
L : Type
inst✝ : Lattice L
a b c : L
h : b ≤ c
⊢ a ⊓ b ≤ a
case a
L : Type
inst✝ : Lattice L
a b c : L
h : b ≤ c
⊢ a ⊓ b ≤ c | case a
L : Type
inst✝ : Lattice L
a b c : L
h : b ≤ c
⊢ a ⊓ b ≤ c | Please generate a tactic in lean4 to solve the state.
STATE:
case a
L : Type
inst✝ : Lattice L
a b c : L
h : b ≤ c
⊢ a ⊓ b ≤ a
case a
L : Type
inst✝ : Lattice L
a b c : L
h : b ≤ c
⊢ a ⊓ b ≤ c
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section06_orderings_and_lattices/sheet2.lean | inf_le_inf_left' | [45, 1] | [53, 7] | . have h' : a ⊓ b ≤ b :=
by
apply inf_le_right
apply le_trans h' h | case a
L : Type
inst✝ : Lattice L
a b c : L
h : b ≤ c
⊢ a ⊓ b ≤ c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
L : Type
inst✝ : Lattice L
a b c : L
h : b ≤ c
⊢ a ⊓ b ≤ c
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section06_orderings_and_lattices/sheet2.lean | inf_le_inf_left' | [45, 1] | [53, 7] | apply inf_le_left | case a
L : Type
inst✝ : Lattice L
a b c : L
h : b ≤ c
⊢ a ⊓ b ≤ a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
L : Type
inst✝ : Lattice L
a b c : L
h : b ≤ c
⊢ a ⊓ b ≤ a
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section06_orderings_and_lattices/sheet2.lean | inf_le_inf_left' | [45, 1] | [53, 7] | have h' : a ⊓ b ≤ b :=
by
apply inf_le_right | case a
L : Type
inst✝ : Lattice L
a b c : L
h : b ≤ c
⊢ a ⊓ b ≤ c | case a
L : Type
inst✝ : Lattice L
a b c : L
h : b ≤ c
h' : a ⊓ b ≤ b
⊢ a ⊓ b ≤ c | Please generate a tactic in lean4 to solve the state.
STATE:
case a
L : Type
inst✝ : Lattice L
a b c : L
h : b ≤ c
⊢ a ⊓ b ≤ c
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section06_orderings_and_lattices/sheet2.lean | inf_le_inf_left' | [45, 1] | [53, 7] | apply le_trans h' h | case a
L : Type
inst✝ : Lattice L
a b c : L
h : b ≤ c
h' : a ⊓ b ≤ b
⊢ a ⊓ b ≤ c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
L : Type
inst✝ : Lattice L
a b c : L
h : b ≤ c
h' : a ⊓ b ≤ b
⊢ a ⊓ b ≤ c
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section06_orderings_and_lattices/sheet2.lean | inf_le_inf_left' | [45, 1] | [53, 7] | apply inf_le_right | L : Type
inst✝ : Lattice L
a b c : L
h : b ≤ c
⊢ a ⊓ b ≤ b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
L : Type
inst✝ : Lattice L
a b c : L
h : b ≤ c
⊢ a ⊓ b ≤ b
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section05_sets/sheet4.lean | mem_def | [3, 1] | [6, 6] | rfl | X : Type
P : X → Prop
a : X
⊢ a ∈ {x | P x} ↔ P a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
P : X → Prop
a : X
⊢ a ∈ {x | P x} ↔ P a
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet6.lean | tends_to_thirtyseven_mul | [4, 1] | [21, 7] | rw [tends_to_def] | a : ℕ → ℝ
t : ℝ
h : tends_to a t
⊢ tends_to (fun n => 37 * a n) (37 * t) | a : ℕ → ℝ
t : ℝ
h : tends_to a t
⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |37 * a n - 37 * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t : ℝ
h : tends_to a t
⊢ tends_to (fun n => 37 * a n) (37 * t)
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet6.lean | tends_to_thirtyseven_mul | [4, 1] | [21, 7] | intro ε hεpos | a : ℕ → ℝ
t : ℝ
h : tends_to a t
⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |37 * a n - 37 * t| < ε | a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |37 * a n - 37 * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t : ℝ
h : tends_to a t
⊢ ∀ (ε : ℝ), 0 < ε → ∃ B, ∀ (n : ℕ), B ≤ n → |37 * a n - 37 * t| < ε
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet6.lean | tends_to_thirtyseven_mul | [4, 1] | [21, 7] | have hεpos' : 0 < ε / 37 :=
by
apply div_pos hεpos (by norm_num) | a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |37 * a n - 37 * t| < ε | a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |37 * a n - 37 * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |37 * a n - 37 * t| < ε
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet6.lean | tends_to_thirtyseven_mul | [4, 1] | [21, 7] | have h' := h (ε / 37) hεpos' | a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |37 * a n - 37 * t| < ε | a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
h' : ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε / 37
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |37 * a n - 37 * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |37 * a n - 37 * t| < ε
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet6.lean | tends_to_thirtyseven_mul | [4, 1] | [21, 7] | obtain ⟨N, hN⟩ := h' | a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
h' : ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε / 37
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |37 * a n - 37 * t| < ε | case intro
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
N : ℕ
hN : ∀ (n : ℕ), N ≤ n → |a n - t| < ε / 37
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |37 * a n - 37 * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
h' : ∃ B, ∀ (n : ℕ), B ≤ n → |a n - t| < ε / 37
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |37 * a n - 37 * t| < ε
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet6.lean | tends_to_thirtyseven_mul | [4, 1] | [21, 7] | use N | case intro
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
N : ℕ
hN : ∀ (n : ℕ), N ≤ n → |a n - t| < ε / 37
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |37 * a n - 37 * t| < ε | case h
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
N : ℕ
hN : ∀ (n : ℕ), N ≤ n → |a n - t| < ε / 37
⊢ ∀ (n : ℕ), N ≤ n → |37 * a n - 37 * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
N : ℕ
hN : ∀ (n : ℕ), N ≤ n → |a n - t| < ε / 37
⊢ ∃ B, ∀ (n : ℕ), B ≤ n → |37 * a n - 37 * t| < ε
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet6.lean | tends_to_thirtyseven_mul | [4, 1] | [21, 7] | intro n hn | case h
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
N : ℕ
hN : ∀ (n : ℕ), N ≤ n → |a n - t| < ε / 37
⊢ ∀ (n : ℕ), N ≤ n → |37 * a n - 37 * t| < ε | case h
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
N : ℕ
hN : ∀ (n : ℕ), N ≤ n → |a n - t| < ε / 37
n : ℕ
hn : N ≤ n
⊢ |37 * a n - 37 * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
N : ℕ
hN : ∀ (n : ℕ), N ≤ n → |a n - t| < ε / 37
⊢ ∀ (n : ℕ), N ≤ n → |37 * a n - 37 * t| < ε
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet6.lean | tends_to_thirtyseven_mul | [4, 1] | [21, 7] | have h'' := hN n hn | case h
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
N : ℕ
hN : ∀ (n : ℕ), N ≤ n → |a n - t| < ε / 37
n : ℕ
hn : N ≤ n
⊢ |37 * a n - 37 * t| < ε | case h
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
N : ℕ
hN : ∀ (n : ℕ), N ≤ n → |a n - t| < ε / 37
n : ℕ
hn : N ≤ n
h'' : |a n - t| < ε / 37
⊢ |37 * a n - 37 * t| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
N : ℕ
hN : ∀ (n : ℕ), N ≤ n → |a n - t| < ε / 37
n : ℕ
hn : N ≤ n
⊢ |37 * a n - 37 * t| < ε
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet6.lean | tends_to_thirtyseven_mul | [4, 1] | [21, 7] | rw [← mul_sub, abs_mul, abs_of_nonneg] | case h
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
N : ℕ
hN : ∀ (n : ℕ), N ≤ n → |a n - t| < ε / 37
n : ℕ
hn : N ≤ n
h'' : |a n - t| < ε / 37
⊢ |37 * a n - 37 * t| < ε | case h
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
N : ℕ
hN : ∀ (n : ℕ), N ≤ n → |a n - t| < ε / 37
n : ℕ
hn : N ≤ n
h'' : |a n - t| < ε / 37
⊢ 37 * |a n - t| < ε
case h
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
N : ℕ
hN : ∀ (n : ℕ), N ≤ n → |a n - t| < ε / 37
n : ℕ
hn : N ≤ n
h'' : |a n - t| < ε / 37
⊢ 0 ≤ 37 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
N : ℕ
hN : ∀ (n : ℕ), N ≤ n → |a n - t| < ε / 37
n : ℕ
hn : N ≤ n
h'' : |a n - t| < ε / 37
⊢ |37 * a n - 37 * t| < ε
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet6.lean | tends_to_thirtyseven_mul | [4, 1] | [21, 7] | linarith | case h
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
N : ℕ
hN : ∀ (n : ℕ), N ≤ n → |a n - t| < ε / 37
n : ℕ
hn : N ≤ n
h'' : |a n - t| < ε / 37
⊢ 37 * |a n - t| < ε
case h
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
N : ℕ
hN : ∀ (n : ℕ), N ≤ n → |a n - t| < ε / 37
n : ℕ
hn : N ≤ n
h'' : |a n - t| < ε / 37
⊢ 0 ≤ 37 | case h
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
N : ℕ
hN : ∀ (n : ℕ), N ≤ n → |a n - t| < ε / 37
n : ℕ
hn : N ≤ n
h'' : |a n - t| < ε / 37
⊢ 0 ≤ 37 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
N : ℕ
hN : ∀ (n : ℕ), N ≤ n → |a n - t| < ε / 37
n : ℕ
hn : N ≤ n
h'' : |a n - t| < ε / 37
⊢ 37 * |a n - t| < ε
case h
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
N : ℕ
hN : ∀ (n : ℕ), N ≤ n → |a n - t| < ε / 37
n : ℕ
hn : N ≤ n
h'' : |a n - t| < ε / 37
⊢ 0 ≤ 37
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet6.lean | tends_to_thirtyseven_mul | [4, 1] | [21, 7] | norm_num | case h
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
N : ℕ
hN : ∀ (n : ℕ), N ≤ n → |a n - t| < ε / 37
n : ℕ
hn : N ≤ n
h'' : |a n - t| < ε / 37
⊢ 0 ≤ 37 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
hεpos' : 0 < ε / 37
N : ℕ
hN : ∀ (n : ℕ), N ≤ n → |a n - t| < ε / 37
n : ℕ
hn : N ≤ n
h'' : |a n - t| < ε / 37
⊢ 0 ≤ 37
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet6.lean | tends_to_thirtyseven_mul | [4, 1] | [21, 7] | apply div_pos hεpos (by norm_num) | a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
⊢ 0 < ε / 37 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
⊢ 0 < ε / 37
TACTIC:
|
https://github.com/rikitoro/FM2023_exercise.git | 5f189bdf83b1e5fba19d25a36272bd87dfcdcc55 | FM2023Exrcise/section02_reals/sheet6.lean | tends_to_thirtyseven_mul | [4, 1] | [21, 7] | norm_num | a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
⊢ 0 < 37 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t : ℝ
h : tends_to a t
ε : ℝ
hεpos : 0 < ε
⊢ 0 < 37
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.coe_inj | [42, 1] | [45, 75] | ext x | α : Type u_1
x y : α
f✝ g✝ f g : Finperm α
h : f.toPerm = g.toPerm
⊢ f = g | case toPerm.H
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : f.toPerm = g.toPerm
x : α
⊢ f.toPerm x = g.toPerm x
case support.a
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : f.toPerm = g.toPerm
x : α
⊢ x ∈ f.support ↔ x ∈ g.support | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ g✝ f g : Finperm α
h : f.toPerm = g.toPerm
⊢ f = g
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.coe_inj | [42, 1] | [45, 75] | rw [mem_support_iff, mem_support_iff, ← toPerm_eq_coe, h, toPerm_eq_coe] | case support.a
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : f.toPerm = g.toPerm
x : α
⊢ x ∈ f.support ↔ x ∈ g.support | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case support.a
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : f.toPerm = g.toPerm
x : α
⊢ x ∈ f.support ↔ x ∈ g.support
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.coe_inj | [42, 1] | [45, 75] | rw [h] | case toPerm.H
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : f.toPerm = g.toPerm
x : α
⊢ f.toPerm x = g.toPerm x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case toPerm.H
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : f.toPerm = g.toPerm
x : α
⊢ f.toPerm x = g.toPerm x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.funext | [47, 1] | [50, 12] | apply coe_inj | α : Type u_1
x y : α
f✝ g✝ f g : Finperm α
h : ∀ (x : α), f x = g x
⊢ f = g | case h
α : Type u_1
x y : α
f✝ g✝ f g : Finperm α
h : ∀ (x : α), f x = g x
⊢ f.toPerm = g.toPerm | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ g✝ f g : Finperm α
h : ∀ (x : α), f x = g x
⊢ f = g
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.funext | [47, 1] | [50, 12] | ext x | case h
α : Type u_1
x y : α
f✝ g✝ f g : Finperm α
h : ∀ (x : α), f x = g x
⊢ f.toPerm = g.toPerm | case h.H
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : ∀ (x : α), f x = g x
x : α
⊢ f.toPerm x = g.toPerm x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
x y : α
f✝ g✝ f g : Finperm α
h : ∀ (x : α), f x = g x
⊢ f.toPerm = g.toPerm
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.funext | [47, 1] | [50, 12] | exact h x | case h.H
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : ∀ (x : α), f x = g x
x : α
⊢ f.toPerm x = g.toPerm x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.H
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : ∀ (x : α), f x = g x
x : α
⊢ f.toPerm x = g.toPerm x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.funext_support | [55, 1] | [63, 15] | refine funext <| fun x ↦ ?_ | α : Type u_1
x y : α
f✝ g✝ f g : Finperm α
h : f.support = g.support
h' : ∀ i ∈ f.support, f i = g i
⊢ f = g | α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : f.support = g.support
h' : ∀ i ∈ f.support, f i = g i
x : α
⊢ f x = g x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ g✝ f g : Finperm α
h : f.support = g.support
h' : ∀ i ∈ f.support, f i = g i
⊢ f = g
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.funext_support | [55, 1] | [63, 15] | obtain (hx | hx) := em (x ∈ f.support) | α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : f.support = g.support
h' : ∀ i ∈ f.support, f i = g i
x : α
⊢ f x = g x | case inl
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : f.support = g.support
h' : ∀ i ∈ f.support, f i = g i
x : α
hx : x ∈ f.support
⊢ f x = g x
case inr
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : f.support = g.support
h' : ∀ i ∈ f.support, f i = g i
x : α
hx : x ∉ f.support
⊢ f x = g x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : f.support = g.support
h' : ∀ i ∈ f.support, f i = g i
x : α
⊢ f x = g x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.funext_support | [55, 1] | [63, 15] | have hx' := hx | case inr
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : f.support = g.support
h' : ∀ i ∈ f.support, f i = g i
x : α
hx : x ∉ f.support
⊢ f x = g x | case inr
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : f.support = g.support
h' : ∀ i ∈ f.support, f i = g i
x : α
hx hx' : x ∉ f.support
⊢ f x = g x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : f.support = g.support
h' : ∀ i ∈ f.support, f i = g i
x : α
hx : x ∉ f.support
⊢ f x = g x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.funext_support | [55, 1] | [63, 15] | rw [h] at hx' | case inr
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : f.support = g.support
h' : ∀ i ∈ f.support, f i = g i
x : α
hx hx' : x ∉ f.support
⊢ f x = g x | case inr
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : f.support = g.support
h' : ∀ i ∈ f.support, f i = g i
x : α
hx : x ∉ f.support
hx' : x ∉ g.support
⊢ f x = g x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : f.support = g.support
h' : ∀ i ∈ f.support, f i = g i
x : α
hx hx' : x ∉ f.support
⊢ f x = g x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.funext_support | [55, 1] | [63, 15] | rw [mem_support_iff, not_not] at hx hx' | case inr
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : f.support = g.support
h' : ∀ i ∈ f.support, f i = g i
x : α
hx : x ∉ f.support
hx' : x ∉ g.support
⊢ f x = g x | case inr
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : f.support = g.support
h' : ∀ i ∈ f.support, f i = g i
x : α
hx : f x = x
hx' : g x = x
⊢ f x = g x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : f.support = g.support
h' : ∀ i ∈ f.support, f i = g i
x : α
hx : x ∉ f.support
hx' : x ∉ g.support
⊢ f x = g x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.funext_support | [55, 1] | [63, 15] | rw [hx, hx'] | case inr
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : f.support = g.support
h' : ∀ i ∈ f.support, f i = g i
x : α
hx : f x = x
hx' : g x = x
⊢ f x = g x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : f.support = g.support
h' : ∀ i ∈ f.support, f i = g i
x : α
hx : f x = x
hx' : g x = x
⊢ f x = g x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.funext_support | [55, 1] | [63, 15] | rw [h' x hx] | case inl
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : f.support = g.support
h' : ∀ i ∈ f.support, f i = g i
x : α
hx : x ∈ f.support
⊢ f x = g x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
x✝ y : α
f✝ g✝ f g : Finperm α
h : f.support = g.support
h' : ∀ i ∈ f.support, f i = g i
x : α
hx : x ∈ f.support
⊢ f x = g x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.funext_support_subset | [65, 1] | [71, 54] | refine funext <| fun x ↦ ?_ | α : Type u_1
x y : α
f✝ g✝ : Finperm α
s : Finset α
f g : Finperm α
hf : f.support ⊆ s
hg : g.support ⊆ s
h : ∀ i ∈ s, f i = g i
⊢ f = g | α : Type u_1
x✝ y : α
f✝ g✝ : Finperm α
s : Finset α
f g : Finperm α
hf : f.support ⊆ s
hg : g.support ⊆ s
h : ∀ i ∈ s, f i = g i
x : α
⊢ f x = g x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ g✝ : Finperm α
s : Finset α
f g : Finperm α
hf : f.support ⊆ s
hg : g.support ⊆ s
h : ∀ i ∈ s, f i = g i
⊢ f = g
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.funext_support_subset | [65, 1] | [71, 54] | obtain (hx | hx) := em (x ∈ s) | α : Type u_1
x✝ y : α
f✝ g✝ : Finperm α
s : Finset α
f g : Finperm α
hf : f.support ⊆ s
hg : g.support ⊆ s
h : ∀ i ∈ s, f i = g i
x : α
⊢ f x = g x | case inl
α : Type u_1
x✝ y : α
f✝ g✝ : Finperm α
s : Finset α
f g : Finperm α
hf : f.support ⊆ s
hg : g.support ⊆ s
h : ∀ i ∈ s, f i = g i
x : α
hx : x ∈ s
⊢ f x = g x
case inr
α : Type u_1
x✝ y : α
f✝ g✝ : Finperm α
s : Finset α
f g : Finperm α
hf : f.support ⊆ s
hg : g.support ⊆ s
h : ∀ i ∈ s, f i = g i
x : α
hx : x ∉ s
⊢ f x = g x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x✝ y : α
f✝ g✝ : Finperm α
s : Finset α
f g : Finperm α
hf : f.support ⊆ s
hg : g.support ⊆ s
h : ∀ i ∈ s, f i = g i
x : α
⊢ f x = g x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.funext_support_subset | [65, 1] | [71, 54] | rw [(show f x = x by simpa using not_mem_mono hf hx),
(show g x = x by simpa using not_mem_mono hg hx)] | case inr
α : Type u_1
x✝ y : α
f✝ g✝ : Finperm α
s : Finset α
f g : Finperm α
hf : f.support ⊆ s
hg : g.support ⊆ s
h : ∀ i ∈ s, f i = g i
x : α
hx : x ∉ s
⊢ f x = g x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
x✝ y : α
f✝ g✝ : Finperm α
s : Finset α
f g : Finperm α
hf : f.support ⊆ s
hg : g.support ⊆ s
h : ∀ i ∈ s, f i = g i
x : α
hx : x ∉ s
⊢ f x = g x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.funext_support_subset | [65, 1] | [71, 54] | exact h _ hx | case inl
α : Type u_1
x✝ y : α
f✝ g✝ : Finperm α
s : Finset α
f g : Finperm α
hf : f.support ⊆ s
hg : g.support ⊆ s
h : ∀ i ∈ s, f i = g i
x : α
hx : x ∈ s
⊢ f x = g x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
x✝ y : α
f✝ g✝ : Finperm α
s : Finset α
f g : Finperm α
hf : f.support ⊆ s
hg : g.support ⊆ s
h : ∀ i ∈ s, f i = g i
x : α
hx : x ∈ s
⊢ f x = g x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.funext_support_subset | [65, 1] | [71, 54] | simpa using not_mem_mono hf hx | α : Type u_1
x✝ y : α
f✝ g✝ : Finperm α
s : Finset α
f g : Finperm α
hf : f.support ⊆ s
hg : g.support ⊆ s
h : ∀ i ∈ s, f i = g i
x : α
hx : x ∉ s
⊢ f x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x✝ y : α
f✝ g✝ : Finperm α
s : Finset α
f g : Finperm α
hf : f.support ⊆ s
hg : g.support ⊆ s
h : ∀ i ∈ s, f i = g i
x : α
hx : x ∉ s
⊢ f x = x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.funext_support_subset | [65, 1] | [71, 54] | simpa using not_mem_mono hg hx | α : Type u_1
x✝ y : α
f✝ g✝ : Finperm α
s : Finset α
f g : Finperm α
hf : f.support ⊆ s
hg : g.support ⊆ s
h : ∀ i ∈ s, f i = g i
x : α
hx : x ∉ s
⊢ g x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x✝ y : α
f✝ g✝ : Finperm α
s : Finset α
f g : Finperm α
hf : f.support ⊆ s
hg : g.support ⊆ s
h : ∀ i ∈ s, f i = g i
x : α
hx : x ∉ s
⊢ g x = x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.funext_support_iff | [77, 1] | [79, 56] | simp [h] | α : Type u_1
x y : α
f✝ g✝ f g : Finperm α
h : f = g
⊢ f.support = g.support ∧ ∀ i ∈ f.support, f i = g i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ g✝ f g : Finperm α
h : f = g
⊢ f.support = g.support ∧ ∀ i ∈ f.support, f i = g i
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.symm_symm | [92, 9] | [93, 22] | apply coe_inj | α : Type u_1
x y : α
f✝ g f : Finperm α
⊢ f.symm.symm = f | case h
α : Type u_1
x y : α
f✝ g f : Finperm α
⊢ f.symm.symm.toPerm = f.toPerm | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ g f : Finperm α
⊢ f.symm.symm = f
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.symm_symm | [92, 9] | [93, 22] | simp | case h
α : Type u_1
x y : α
f✝ g f : Finperm α
⊢ f.symm.symm.toPerm = f.toPerm | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
x y : α
f✝ g f : Finperm α
⊢ f.symm.symm.toPerm = f.toPerm
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_eq_empty_iff | [109, 9] | [110, 72] | simp [h] | α : Type u_1
x y : α
f✝ g f : Finperm α
h : f.support = ∅
⊢ f.support = refl.support | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ g f : Finperm α
h : f.support = ∅
⊢ f.support = refl.support
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_eq_empty_iff | [109, 9] | [110, 72] | simp [h] | α : Type u_1
x y : α
f✝ g f : Finperm α
h : f.support = ∅
⊢ ∀ i ∈ f.support, f i = refl i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ g f : Finperm α
h : f.support = ∅
⊢ ∀ i ∈ f.support, f i = refl i
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.restrict_univ | [193, 9] | [195, 12] | ext | α✝¹ : Type u_1
x y : α✝¹
f✝ g : Finperm α✝¹
α✝ : Type u_2
β : Type u_3
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq β
f : Finperm α✝
α : Type u_4
inst✝ : DecidableEq α
⊢ restrict Set.univ = ⊤ | case h
α✝¹ : Type u_1
x y : α✝¹
f✝ g : Finperm α✝¹
α✝ : Type u_2
β : Type u_3
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq β
f : Finperm α✝
α : Type u_4
inst✝ : DecidableEq α
x✝ : Finperm α
⊢ x✝ ∈ restrict Set.univ ↔ x✝ ∈ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝¹ : Type u_1
x y : α✝¹
f✝ g : Finperm α✝¹
α✝ : Type u_2
β : Type u_3
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq β
f : Finperm α✝
α : Type u_4
inst✝ : DecidableEq α
⊢ restrict Set.univ = ⊤
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.restrict_univ | [193, 9] | [195, 12] | simp | case h
α✝¹ : Type u_1
x y : α✝¹
f✝ g : Finperm α✝¹
α✝ : Type u_2
β : Type u_3
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq β
f : Finperm α✝
α : Type u_4
inst✝ : DecidableEq α
x✝ : Finperm α
⊢ x✝ ∈ restrict Set.univ ↔ x✝ ∈ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α✝¹ : Type u_1
x y : α✝¹
f✝ g : Finperm α✝¹
α✝ : Type u_2
β : Type u_3
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq β
f : Finperm α✝
α : Type u_4
inst✝ : DecidableEq α
x✝ : Finperm α
⊢ x✝ ∈ restrict Set.univ ↔ x✝ ∈ ⊤
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.mem_restrict_support | [197, 1] | [198, 7] | simp | α✝ : Type u_1
x y : α✝
f✝¹ g : Finperm α✝
α : Type u_2
β : Type u_3
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f✝ f : Finperm α
⊢ f ∈ restrict ↑f.support | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
x y : α✝
f✝¹ g : Finperm α✝
α : Type u_2
β : Type u_3
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f✝ f : Finperm α
⊢ f ∈ restrict ↑f.support
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_self | [243, 9] | [246, 6] | apply coe_inj | α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
⊢ swap x x = refl | case h
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
⊢ (swap x x).toPerm = refl.toPerm | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
⊢ swap x x = refl
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_self | [243, 9] | [246, 6] | simp only [swap_toPerm, Equiv.swap_self, refl_toPerm] | case h
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
⊢ (swap x x).toPerm = refl.toPerm | case h
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
⊢ Equiv.refl α = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
⊢ (swap x x).toPerm = refl.toPerm
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_self | [243, 9] | [246, 6] | rfl | case h
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
⊢ Equiv.refl α = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
⊢ Equiv.refl α = 1
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_support_of_ne | [257, 1] | [258, 19] | simp [swap, hxy] | α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
hxy : x ≠ y
⊢ (swap x y).support = {x, y} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
hxy : x ≠ y
⊢ (swap x y).support = {x, y}
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_comm | [260, 1] | [261, 33] | rw [funext_support_iff] | α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a b : α
⊢ swap a b = swap b a | α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a b : α
⊢ (swap a b).support = (swap b a).support ∧ ∀ i ∈ (swap a b).support, (swap a b) i = (swap b a) i | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a b : α
⊢ swap a b = swap b a
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_comm | [260, 1] | [261, 33] | aesop | α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a b : α
⊢ (swap a b).support = (swap b a).support ∧ ∀ i ∈ (swap a b).support, (swap a b) i = (swap b a) i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a b : α
⊢ (swap a b).support = (swap b a).support ∧ ∀ i ∈ (swap a b).support, (swap a b) i = (swap b a) i
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_mul_swap | [263, 1] | [270, 7] | obtain (rfl | hne) := eq_or_ne a b | α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a b : α
⊢ swap a b * swap a b = 1 | case inl
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a : α
⊢ swap a a * swap a a = 1
case inr
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a b : α
hne : a ≠ b
⊢ swap a b * swap a b = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a b : α
⊢ swap a b * swap a b = 1
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_mul_swap | [263, 1] | [270, 7] | apply funext_support_subset (s := {a,b}) | case inr
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a b : α
hne : a ≠ b
⊢ swap a b * swap a b = 1 | case inr.hf
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a b : α
hne : a ≠ b
⊢ (swap a b * swap a b).support ⊆ {a, b}
case inr.hg
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a b : α
hne : a ≠ b
⊢ 1.support ⊆ {a, b}
case inr.h
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a b : α
hne : a ≠ b
⊢ ∀ i ∈ {a, b}, (swap a b * swap a b) i = 1 i | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a b : α
hne : a ≠ b
⊢ swap a b * swap a b = 1
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_mul_swap | [263, 1] | [270, 7] | simp | case inr.h
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a b : α
hne : a ≠ b
⊢ ∀ i ∈ {a, b}, (swap a b * swap a b) i = 1 i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a b : α
hne : a ≠ b
⊢ ∀ i ∈ {a, b}, (swap a b * swap a b) i = 1 i
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_mul_swap | [263, 1] | [270, 7] | simp [one_def] | case inl
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a : α
⊢ swap a a * swap a a = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a : α
⊢ swap a a * swap a a = 1
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_mul_swap | [263, 1] | [270, 7] | refine (mul_support_subset _ _).trans ?_ | case inr.hf
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a b : α
hne : a ≠ b
⊢ (swap a b * swap a b).support ⊆ {a, b} | case inr.hf
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a b : α
hne : a ≠ b
⊢ (swap a b).support ∪ (swap a b).support ⊆ {a, b} | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.hf
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a b : α
hne : a ≠ b
⊢ (swap a b * swap a b).support ⊆ {a, b}
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_mul_swap | [263, 1] | [270, 7] | rw [Finset.union_self, swap_support_of_ne hne] | case inr.hf
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a b : α
hne : a ≠ b
⊢ (swap a b).support ∪ (swap a b).support ⊆ {a, b} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.hf
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a b : α
hne : a ≠ b
⊢ (swap a b).support ∪ (swap a b).support ⊆ {a, b}
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_mul_swap | [263, 1] | [270, 7] | simp | case inr.hg
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a b : α
hne : a ≠ b
⊢ 1.support ⊆ {a, b} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.hg
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
a b : α
hne : a ≠ b
⊢ 1.support ⊆ {a, b}
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_conj_eq | [272, 1] | [286, 7] | obtain (rfl | hxy) := eq_or_ne x y | α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
⊢ swap x y * swap y z * swap x y = swap x z | case inl
α : Type u_1
x : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz hyz : x ≠ z
⊢ swap x x * swap x z * swap x x = swap x z
case inr
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ swap x y * swap y z * swap x y = swap x z | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
⊢ swap x y * swap y z * swap x y = swap x z
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_conj_eq | [272, 1] | [286, 7] | apply funext_support_subset (s := {x,y,z}) | case inr
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ swap x y * swap y z * swap x y = swap x z | case inr.hf
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ (swap x y * swap y z * swap x y).support ⊆ {x, y, z}
case inr.hg
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ (swap x z).support ⊆ {x, y, z}
case inr.h
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ ∀ i ∈ {x, y, z}, (swap x y * swap y z * swap x y) i = (swap x z) i | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ swap x y * swap y z * swap x y = swap x z
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_conj_eq | [272, 1] | [286, 7] | simp only [mem_insert, mem_singleton, mul_apply, forall_eq_or_imp, swap_apply_left,
swap_apply_right, forall_eq] | case inr.h
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ ∀ i ∈ {x, y, z}, (swap x y * swap y z * swap x y) i = (swap x z) i | case inr.h
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ (swap x y) z = z ∧ (swap x y) ((swap y z) x) = (swap x z) y ∧ (swap x y) ((swap y z) ((swap x y) z)) = x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ ∀ i ∈ {x, y, z}, (swap x y * swap y z * swap x y) i = (swap x z) i
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_conj_eq | [272, 1] | [286, 7] | rw [swap_apply_of_ne_of_ne hxz.symm hyz.symm, swap_apply_of_ne_of_ne hxy hxz,
swap_apply_of_ne_of_ne hxy.symm hyz] | case inr.h
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ (swap x y) z = z ∧ (swap x y) ((swap y z) x) = (swap x z) y ∧ (swap x y) ((swap y z) ((swap x y) z)) = x | case inr.h
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ z = z ∧ (swap x y) x = y ∧ (swap x y) ((swap y z) z) = x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ (swap x y) z = z ∧ (swap x y) ((swap y z) x) = (swap x z) y ∧ (swap x y) ((swap y z) ((swap x y) z)) = x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_conj_eq | [272, 1] | [286, 7] | simp | case inr.h
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ z = z ∧ (swap x y) x = y ∧ (swap x y) ((swap y z) z) = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ z = z ∧ (swap x y) x = y ∧ (swap x y) ((swap y z) z) = x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_conj_eq | [272, 1] | [286, 7] | simp | case inl
α : Type u_1
x : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz hyz : x ≠ z
⊢ swap x x * swap x z * swap x x = swap x z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
x : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz hyz : x ≠ z
⊢ swap x x * swap x z * swap x x = swap x z
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.