Datasets:
internlm
/
Lean-Github

Dataset card Data Studio Files Community
1
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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)
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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✝ ∈ ⊤
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
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
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
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
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
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
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
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
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
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
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
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
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}
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
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
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
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
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
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
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
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