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/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_4 | [79, 1] | [98, 13] | specialize hN' n (le_of_max_le_right hn) | case h
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
hs₁s₂ : ∀ (n : ℕ), s₁ n ≤ s₂ n
hs₂s₃ : ∀ (n : ℕ), s₂ n ≤ s₃ n
ε : ℝ
hε : ε > 0
N N' : ℕ
hN' : ∀ n ≥ N', |s₃ n - a| < ε
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hN : |s₁ n - a| < ε
⊢ |s₂ n - a| < ε | case h
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
hs₁s₂ : ∀ (n : ℕ), s₁ n ≤ s₂ n
hs₂s₃ : ∀ (n : ℕ), s₂ n ≤ s₃ n
ε : ℝ
hε : ε > 0
N N' : ℕ
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hN : |s₁ n - a| < ε
hN' : |s₃ n - a| < ε
⊢ |s₂ n - a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
hs₁s₂ : ∀ (n : ℕ), s₁ n ≤ s₂ n
hs₂s₃ : ∀ (n : ℕ), s₂ n ≤ s₃ n
ε : ℝ
hε : ε > 0
N N' : ℕ
hN' : ∀ n ≥ N', |s₃ n - a| < ε
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hN : |s₁ n - a| < ε
⊢ |s₂ n - a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_4 | [79, 1] | [98, 13] | specialize hs₁s₂ n | case h
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
hs₁s₂ : ∀ (n : ℕ), s₁ n ≤ s₂ n
hs₂s₃ : ∀ (n : ℕ), s₂ n ≤ s₃ n
ε : ℝ
hε : ε > 0
N N' : ℕ
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hN : |s₁ n - a| < ε
hN' : |s₃ n - a| < ε
⊢ |s₂ n - a| < ε | case h
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
hs₂s₃ : ∀ (n : ℕ), s₂ n ≤ s₃ n
ε : ℝ
hε : ε > 0
N N' : ℕ
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hN : |s₁ n - a| < ε
hN' : |s₃ n - a| < ε
hs₁s₂ : s₁ n ≤ s₂ n
⊢ |s₂ n - a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
hs₁s₂ : ∀ (n : ℕ), s₁ n ≤ s₂ n
hs₂s₃ : ∀ (n : ℕ), s₂ n ≤ s₃ n
ε : ℝ
hε : ε > 0
N N' : ℕ
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hN : |s₁ n - a| < ε
hN' : |s₃ n - a| < ε
⊢ |s₂ n - a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_4 | [79, 1] | [98, 13] | specialize hs₂s₃ n | case h
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
hs₂s₃ : ∀ (n : ℕ), s₂ n ≤ s₃ n
ε : ℝ
hε : ε > 0
N N' : ℕ
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hN : |s₁ n - a| < ε
hN' : |s₃ n - a| < ε
hs₁s₂ : s₁ n ≤ s₂ n
⊢ |s₂ n - a| < ε | case h
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
ε : ℝ
hε : ε > 0
N N' : ℕ
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hN : |s₁ n - a| < ε
hN' : |s₃ n - a| < ε
hs₁s₂ : s₁ n ≤ s₂ n
hs₂s₃ : s₂ n ≤ s₃ n
⊢ |s₂ n - a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
hs₂s₃ : ∀ (n : ℕ), s₂ n ≤ s₃ n
ε : ℝ
hε : ε > 0
N N' : ℕ
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hN : |s₁ n - a| < ε
hN' : |s₃ n - a| < ε
hs₁s₂ : s₁ n ≤ s₂ n
⊢ |s₂ n - a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_4 | [79, 1] | [98, 13] | rw [abs_lt] at hN hN' ⊢ | case h
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
ε : ℝ
hε : ε > 0
N N' : ℕ
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hN : |s₁ n - a| < ε
hN' : |s₃ n - a| < ε
hs₁s₂ : s₁ n ≤ s₂ n
hs₂s₃ : s₂ n ≤ s₃ n
⊢ |s₂ n - a| < ε | case h
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
ε : ℝ
hε : ε > 0
N N' : ℕ
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hN : -ε < s₁ n - a ∧ s₁ n - a < ε
hN' : -ε < s₃ n - a ∧ s₃ n - a < ε
hs₁s₂ : s₁ n ≤ s₂ n
hs₂s₃ : s₂ n ≤ s₃ n
⊢ -ε < s₂ n - a ∧ s₂ n - a < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
ε : ℝ
hε : ε > 0
N N' : ℕ
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hN : |s₁ n - a| < ε
hN' : |s₃ n - a| < ε
hs₁s₂ : s₁ n ≤ s₂ n
hs₂s₃ : s₂ n ≤ s₃ n
⊢ |s₂ n - a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_4 | [79, 1] | [98, 13] | obtain ⟨h₁N, h₂N⟩ := hN | case h
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
ε : ℝ
hε : ε > 0
N N' : ℕ
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hN : -ε < s₁ n - a ∧ s₁ n - a < ε
hN' : -ε < s₃ n - a ∧ s₃ n - a < ε
hs₁s₂ : s₁ n ≤ s₂ n
hs₂s₃ : s₂ n ≤ s₃ n
⊢ -ε < s₂ n - a ∧ s₂ n - a < ε | case h.intro
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
ε : ℝ
hε : ε > 0
N N' : ℕ
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hN' : -ε < s₃ n - a ∧ s₃ n - a < ε
hs₁s₂ : s₁ n ≤ s₂ n
hs₂s₃ : s₂ n ≤ s₃ n
h₁N : -ε < s₁ n - a
h₂N : s₁ n - a < ε
⊢ -ε < s₂ n - a ∧ s₂ n - a < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
ε : ℝ
hε : ε > 0
N N' : ℕ
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hN : -ε < s₁ n - a ∧ s₁ n - a < ε
hN' : -ε < s₃ n - a ∧ s₃ n - a < ε
hs₁s₂ : s₁ n ≤ s₂ n
hs₂s₃ : s₂ n ≤ s₃ n
⊢ -ε < s₂ n - a ∧ s₂ n - a < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_4 | [79, 1] | [98, 13] | obtain ⟨h₁N', h₂N'⟩ := hN' | case h.intro
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
ε : ℝ
hε : ε > 0
N N' : ℕ
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hN' : -ε < s₃ n - a ∧ s₃ n - a < ε
hs₁s₂ : s₁ n ≤ s₂ n
hs₂s₃ : s₂ n ≤ s₃ n
h₁N : -ε < s₁ n - a
h₂N : s₁ n - a < ε
⊢ -ε < s₂ n - a ∧ s₂ n - a < ε | case h.intro.intro
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
ε : ℝ
hε : ε > 0
N N' : ℕ
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hs₁s₂ : s₁ n ≤ s₂ n
hs₂s₃ : s₂ n ≤ s₃ n
h₁N : -ε < s₁ n - a
h₂N : s₁ n - a < ε
h₁N' : -ε < s₃ n - a
h₂N' : s₃ n - a < ε
⊢ -ε < s₂ n - a ∧ s₂ n - a < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
ε : ℝ
hε : ε > 0
N N' : ℕ
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hN' : -ε < s₃ n - a ∧ s₃ n - a < ε
hs₁s₂ : s₁ n ≤ s₂ n
hs₂s₃ : s₂ n ≤ s₃ n
h₁N : -ε < s₁ n - a
h₂N : s₁ n - a < ε
⊢ -ε < s₂ n - a ∧ s₂ n - a < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_4 | [79, 1] | [98, 13] | constructor | case h.intro.intro
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
ε : ℝ
hε : ε > 0
N N' : ℕ
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hs₁s₂ : s₁ n ≤ s₂ n
hs₂s₃ : s₂ n ≤ s₃ n
h₁N : -ε < s₁ n - a
h₂N : s₁ n - a < ε
h₁N' : -ε < s₃ n - a
h₂N' : s₃ n - a < ε
⊢ -ε < s₂ n - a ∧ s₂ n - a < ε | case h.intro.intro.left
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
ε : ℝ
hε : ε > 0
N N' : ℕ
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hs₁s₂ : s₁ n ≤ s₂ n
hs₂s₃ : s₂ n ≤ s₃ n
h₁N : -ε < s₁ n - a
h₂N : s₁ n - a < ε
h₁N' : -ε < s₃ n - a
h₂N' : s₃ n - a < ε
⊢ -ε < s₂ n - a
case h.intro.intro.right
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
ε : ℝ
hε : ε > 0
N N' : ℕ
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hs₁s₂ : s₁ n ≤ s₂ n
hs₂s₃ : s₂ n ≤ s₃ n
h₁N : -ε < s₁ n - a
h₂N : s₁ n - a < ε
h₁N' : -ε < s₃ n - a
h₂N' : s₃ n - a < ε
⊢ s₂ n - a < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
ε : ℝ
hε : ε > 0
N N' : ℕ
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hs₁s₂ : s₁ n ≤ s₂ n
hs₂s₃ : s₂ n ≤ s₃ n
h₁N : -ε < s₁ n - a
h₂N : s₁ n - a < ε
h₁N' : -ε < s₃ n - a
h₂N' : s₃ n - a < ε
⊢ -ε < s₂ n - a ∧ s₂ n - a < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_4 | [79, 1] | [98, 13] | linarith | case h.intro.intro.left
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
ε : ℝ
hε : ε > 0
N N' : ℕ
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hs₁s₂ : s₁ n ≤ s₂ n
hs₂s₃ : s₂ n ≤ s₃ n
h₁N : -ε < s₁ n - a
h₂N : s₁ n - a < ε
h₁N' : -ε < s₃ n - a
h₂N' : s₃ n - a < ε
⊢ -ε < s₂ n - a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.left
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
ε : ℝ
hε : ε > 0
N N' : ℕ
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hs₁s₂ : s₁ n ≤ s₂ n
hs₂s₃ : s₂ n ≤ s₃ n
h₁N : -ε < s₁ n - a
h₂N : s₁ n - a < ε
h₁N' : -ε < s₃ n - a
h₂N' : s₃ n - a < ε
⊢ -ε < s₂ n - a
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_4 | [79, 1] | [98, 13] | linarith | case h.intro.intro.right
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
ε : ℝ
hε : ε > 0
N N' : ℕ
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hs₁s₂ : s₁ n ≤ s₂ n
hs₂s₃ : s₂ n ≤ s₃ n
h₁N : -ε < s₁ n - a
h₂N : s₁ n - a < ε
h₁N' : -ε < s₃ n - a
h₂N' : s₃ n - a < ε
⊢ s₂ n - a < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.right
s₁ s₂ s₃ : ℕ → ℝ
a : ℝ
hs₁ : SequentialLimit s₁ a
hs₃ : SequentialLimit s₃ a
ε : ℝ
hε : ε > 0
N N' : ℕ
N'' : ℕ := max N N'
n : ℕ
hn : n ≥ N''
hs₁s₂ : s₁ n ≤ s₂ n
hs₂s₃ : s₂ n ≤ s₃ n
h₁N : -ε < s₁ n - a
h₂N : s₁ n - a < ε
h₁N' : -ε < s₃ n - a
h₂N' : s₃ n - a < ε
⊢ s₂ n - a < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_5 | [110, 1] | [123, 2] | intro ε hε | ⊢ SequentialLimit (fun n => 1 / (↑n + 1)) 0 | ε : ℝ
hε : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => 1 / (↑n + 1)) n - 0| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ SequentialLimit (fun n => 1 / (↑n + 1)) 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_5 | [110, 1] | [123, 2] | use ⌈1 / ε⌉₊ | ε : ℝ
hε : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => 1 / (↑n + 1)) n - 0| < ε | case h
ε : ℝ
hε : ε > 0
⊢ ∀ n ≥ ⌈1 / ε⌉₊, |(fun n => 1 / (↑n + 1)) n - 0| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => 1 / (↑n + 1)) n - 0| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_5 | [110, 1] | [123, 2] | intro n hn | case h
ε : ℝ
hε : ε > 0
⊢ ∀ n ≥ ⌈1 / ε⌉₊, |(fun n => 1 / (↑n + 1)) n - 0| < ε | case h
ε : ℝ
hε : ε > 0
n : ℕ
hn : n ≥ ⌈1 / ε⌉₊
⊢ |(fun n => 1 / (↑n + 1)) n - 0| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ε : ℝ
hε : ε > 0
⊢ ∀ n ≥ ⌈1 / ε⌉₊, |(fun n => 1 / (↑n + 1)) n - 0| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_5 | [110, 1] | [123, 2] | simp at hn | case h
ε : ℝ
hε : ε > 0
n : ℕ
hn : n ≥ ⌈1 / ε⌉₊
⊢ |(fun n => 1 / (↑n + 1)) n - 0| < ε | case h
ε : ℝ
hε : ε > 0
n : ℕ
hn : ε⁻¹ ≤ ↑n
⊢ |(fun n => 1 / (↑n + 1)) n - 0| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ε : ℝ
hε : ε > 0
n : ℕ
hn : n ≥ ⌈1 / ε⌉₊
⊢ |(fun n => 1 / (↑n + 1)) n - 0| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_5 | [110, 1] | [123, 2] | simp [abs_inv] | case h
ε : ℝ
hε : ε > 0
n : ℕ
hn : ε⁻¹ ≤ ↑n
⊢ |(fun n => 1 / (↑n + 1)) n - 0| < ε | case h
ε : ℝ
hε : ε > 0
n : ℕ
hn : ε⁻¹ ≤ ↑n
⊢ |↑n + 1|⁻¹ < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ε : ℝ
hε : ε > 0
n : ℕ
hn : ε⁻¹ ≤ ↑n
⊢ |(fun n => 1 / (↑n + 1)) n - 0| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_5 | [110, 1] | [123, 2] | rw [inv_lt] | case h
ε : ℝ
hε : ε > 0
n : ℕ
hn : ε⁻¹ ≤ ↑n
⊢ |↑n + 1|⁻¹ < ε | case h
ε : ℝ
hε : ε > 0
n : ℕ
hn : ε⁻¹ ≤ ↑n
⊢ ε⁻¹ < |↑n + 1|
case h.ha
ε : ℝ
hε : ε > 0
n : ℕ
hn : ε⁻¹ ≤ ↑n
⊢ 0 < |↑n + 1|
case h.hb
ε : ℝ
hε : ε > 0
n : ℕ
hn : ε⁻¹ ≤ ↑n
⊢ 0 < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ε : ℝ
hε : ε > 0
n : ℕ
hn : ε⁻¹ ≤ ↑n
⊢ |↑n + 1|⁻¹ < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_5 | [110, 1] | [123, 2] | have fact1 : (n : ℝ) + 1 ≤ |(n : ℝ) + 1| := by apply? | case h
ε : ℝ
hε : ε > 0
n : ℕ
hn : ε⁻¹ ≤ ↑n
⊢ ε⁻¹ < |↑n + 1| | case h
ε : ℝ
hε : ε > 0
n : ℕ
hn : ε⁻¹ ≤ ↑n
fact1 : ↑n + 1 ≤ |↑n + 1|
⊢ ε⁻¹ < |↑n + 1| | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ε : ℝ
hε : ε > 0
n : ℕ
hn : ε⁻¹ ≤ ↑n
⊢ ε⁻¹ < |↑n + 1|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_5 | [110, 1] | [123, 2] | linarith | case h
ε : ℝ
hε : ε > 0
n : ℕ
hn : ε⁻¹ ≤ ↑n
fact1 : ↑n + 1 ≤ |↑n + 1|
⊢ ε⁻¹ < |↑n + 1| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
ε : ℝ
hε : ε > 0
n : ℕ
hn : ε⁻¹ ≤ ↑n
fact1 : ↑n + 1 ≤ |↑n + 1|
⊢ ε⁻¹ < |↑n + 1|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_5 | [110, 1] | [123, 2] | apply? | ε : ℝ
hε : ε > 0
n : ℕ
hn : ε⁻¹ ≤ ↑n
⊢ ↑n + 1 ≤ |↑n + 1| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
ε : ℝ
hε : ε > 0
n : ℕ
hn : ε⁻¹ ≤ ↑n
⊢ ↑n + 1 ≤ |↑n + 1|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_5 | [110, 1] | [123, 2] | rw [abs_pos] | case h.ha
ε : ℝ
hε : ε > 0
n : ℕ
hn : ε⁻¹ ≤ ↑n
⊢ 0 < |↑n + 1| | case h.ha
ε : ℝ
hε : ε > 0
n : ℕ
hn : ε⁻¹ ≤ ↑n
⊢ ↑n + 1 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.ha
ε : ℝ
hε : ε > 0
n : ℕ
hn : ε⁻¹ ≤ ↑n
⊢ 0 < |↑n + 1|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_5 | [110, 1] | [123, 2] | apply? | case h.ha
ε : ℝ
hε : ε > 0
n : ℕ
hn : ε⁻¹ ≤ ↑n
⊢ ↑n + 1 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.ha
ε : ℝ
hε : ε > 0
n : ℕ
hn : ε⁻¹ ≤ ↑n
⊢ ↑n + 1 ≠ 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_5 | [110, 1] | [123, 2] | assumption | case h.hb
ε : ℝ
hε : ε > 0
n : ℕ
hn : ε⁻¹ ≤ ↑n
⊢ 0 < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hb
ε : ℝ
hε : ε > 0
n : ℕ
hn : ε⁻¹ ≤ ↑n
⊢ 0 < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | convergesTo_mul_const | [129, 1] | [142, 57] | intro ε hε | s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
⊢ SequentialLimit (fun n => c * s n) (c * a) | s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
⊢ SequentialLimit (fun n => c * s n) (c * a)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | convergesTo_mul_const | [129, 1] | [142, 57] | obtain ⟨N, hN⟩ := hs (ε / max |c| 1) (by positivity) | s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε | case intro
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N : ℕ
hN : ∀ n ≥ N, |s n - a| < ε / max |c| 1
⊢ ∃ N, ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | convergesTo_mul_const | [129, 1] | [142, 57] | use N | case intro
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N : ℕ
hN : ∀ n ≥ N, |s n - a| < ε / max |c| 1
⊢ ∃ N, ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε | case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N : ℕ
hN : ∀ n ≥ N, |s n - a| < ε / max |c| 1
⊢ ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N : ℕ
hN : ∀ n ≥ N, |s n - a| < ε / max |c| 1
⊢ ∃ N, ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | convergesTo_mul_const | [129, 1] | [142, 57] | intro n hn | case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N : ℕ
hN : ∀ n ≥ N, |s n - a| < ε / max |c| 1
⊢ ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε | case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N : ℕ
hN : ∀ n ≥ N, |s n - a| < ε / max |c| 1
n : ℕ
hn : n ≥ N
⊢ |(fun n => c * s n) n - c * a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N : ℕ
hN : ∀ n ≥ N, |s n - a| < ε / max |c| 1
⊢ ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | convergesTo_mul_const | [129, 1] | [142, 57] | specialize hN n hn | case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N : ℕ
hN : ∀ n ≥ N, |s n - a| < ε / max |c| 1
n : ℕ
hn : n ≥ N
⊢ |(fun n => c * s n) n - c * a| < ε | case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |(fun n => c * s n) n - c * a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N : ℕ
hN : ∀ n ≥ N, |s n - a| < ε / max |c| 1
n : ℕ
hn : n ≥ N
⊢ |(fun n => c * s n) n - c * a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | convergesTo_mul_const | [129, 1] | [142, 57] | simp | case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |(fun n => c * s n) n - c * a| < ε | case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |c * s n - c * a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |(fun n => c * s n) n - c * a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | convergesTo_mul_const | [129, 1] | [142, 57] | calc |c * s n - c * a|
= |c * (s n - a)| := by ring
_ = |c| * |s n - a| := by exact abs_mul c (s n - a)
_ ≤ max |c| 1 * |s n - a| := by gcongr; exact le_max_left |c| 1
_ < max |c| 1 * (ε / max |c| 1) := by gcongr
_ = ε := by refine mul_div_cancel' ε ?hb; positivity | case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |c * s n - c * a| < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |c * s n - c * a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | convergesTo_mul_const | [129, 1] | [142, 57] | positivity | s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
⊢ ε / max |c| 1 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
⊢ ε / max |c| 1 > 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | convergesTo_mul_const | [129, 1] | [142, 57] | ring | s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |c * s n - c * a| = |c * (s n - a)| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |c * s n - c * a| = |c * (s n - a)|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | convergesTo_mul_const | [129, 1] | [142, 57] | exact abs_mul c (s n - a) | s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |c * (s n - a)| = |c| * |s n - a| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |c * (s n - a)| = |c| * |s n - a|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | convergesTo_mul_const | [129, 1] | [142, 57] | gcongr | s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |c| * |s n - a| ≤ max |c| 1 * |s n - a| | case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |c| ≤ max |c| 1 | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |c| * |s n - a| ≤ max |c| 1 * |s n - a|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | convergesTo_mul_const | [129, 1] | [142, 57] | exact le_max_left |c| 1 | case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |c| ≤ max |c| 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ |c| ≤ max |c| 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | convergesTo_mul_const | [129, 1] | [142, 57] | gcongr | s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ max |c| 1 * |s n - a| < max |c| 1 * (ε / max |c| 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ max |c| 1 * |s n - a| < max |c| 1 * (ε / max |c| 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | convergesTo_mul_const | [129, 1] | [142, 57] | refine mul_div_cancel' ε ?hb | s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ max |c| 1 * (ε / max |c| 1) = ε | case hb
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ max |c| 1 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ max |c| 1 * (ε / max |c| 1) = ε
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | convergesTo_mul_const | [129, 1] | [142, 57] | positivity | case hb
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ max |c| 1 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hb
s : ℕ → ℝ
a c : ℝ
hs : SequentialLimit s a
ε : ℝ
hε : ε > 0
N n : ℕ
hn : n ≥ N
hN : |s n - a| < ε / max |c| 1
⊢ max |c| 1 ≠ 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | use_me | [144, 1] | [148, 23] | have : SequentialLimit (fun n ↦ (-1) * (1 / (n+1))) (-1 * 0) :=
convergesTo_mul_const (-1) exercise3_5 | ⊢ SequentialLimit (fun n => -1 / (↑n + 1)) 0 | this : SequentialLimit (fun n => -1 * (1 / (↑n + 1))) (-1 * 0)
⊢ SequentialLimit (fun n => -1 / (↑n + 1)) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ SequentialLimit (fun n => -1 / (↑n + 1)) 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | use_me | [144, 1] | [148, 23] | simp at this | this : SequentialLimit (fun n => -1 * (1 / (↑n + 1))) (-1 * 0)
⊢ SequentialLimit (fun n => -1 / (↑n + 1)) 0 | this : SequentialLimit (fun n => -(↑n + 1)⁻¹) 0
⊢ SequentialLimit (fun n => -1 / (↑n + 1)) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
this : SequentialLimit (fun n => -1 * (1 / (↑n + 1))) (-1 * 0)
⊢ SequentialLimit (fun n => -1 / (↑n + 1)) 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | use_me | [144, 1] | [148, 23] | simp [neg_div, this] | this : SequentialLimit (fun n => -(↑n + 1)⁻¹) 0
⊢ SequentialLimit (fun n => -1 / (↑n + 1)) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
this : SequentialLimit (fun n => -(↑n + 1)⁻¹) 0
⊢ SequentialLimit (fun n => -1 / (↑n + 1)) 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_6 | [150, 1] | [157, 11] | apply exercise3_4 use_me exercise3_5 | ⊢ SequentialLimit (fun n => sin ↑n / (↑n + 1)) 0 | case hs₁s₂
⊢ ∀ (n : ℕ), -1 / (↑n + 1) ≤ sin ↑n / (↑n + 1)
case hs₂s₃
⊢ ∀ (n : ℕ), sin ↑n / (↑n + 1) ≤ 1 / (↑n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ SequentialLimit (fun n => sin ↑n / (↑n + 1)) 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_6 | [150, 1] | [157, 11] | intro n | case hs₁s₂
⊢ ∀ (n : ℕ), -1 / (↑n + 1) ≤ sin ↑n / (↑n + 1) | case hs₁s₂
n : ℕ
⊢ -1 / (↑n + 1) ≤ sin ↑n / (↑n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case hs₁s₂
⊢ ∀ (n : ℕ), -1 / (↑n + 1) ≤ sin ↑n / (↑n + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_6 | [150, 1] | [157, 11] | gcongr | case hs₁s₂
n : ℕ
⊢ -1 / (↑n + 1) ≤ sin ↑n / (↑n + 1) | case hs₁s₂.h
n : ℕ
⊢ -1 ≤ sin ↑n | Please generate a tactic in lean4 to solve the state.
STATE:
case hs₁s₂
n : ℕ
⊢ -1 / (↑n + 1) ≤ sin ↑n / (↑n + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_6 | [150, 1] | [157, 11] | apply? | case hs₁s₂.h
n : ℕ
⊢ -1 ≤ sin ↑n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hs₁s₂.h
n : ℕ
⊢ -1 ≤ sin ↑n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_6 | [150, 1] | [157, 11] | intro n | case hs₂s₃
⊢ ∀ (n : ℕ), sin ↑n / (↑n + 1) ≤ 1 / (↑n + 1) | case hs₂s₃
n : ℕ
⊢ sin ↑n / (↑n + 1) ≤ 1 / (↑n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case hs₂s₃
⊢ ∀ (n : ℕ), sin ↑n / (↑n + 1) ≤ 1 / (↑n + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_6 | [150, 1] | [157, 11] | gcongr | case hs₂s₃
n : ℕ
⊢ sin ↑n / (↑n + 1) ≤ 1 / (↑n + 1) | case hs₂s₃.h
n : ℕ
⊢ sin ↑n ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case hs₂s₃
n : ℕ
⊢ sin ↑n / (↑n + 1) ≤ 1 / (↑n + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_6 | [150, 1] | [157, 11] | apply? | case hs₂s₃.h
n : ℕ
⊢ sin ↑n ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hs₂s₃.h
n : ℕ
⊢ sin ↑n ≤ 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_7 | [164, 1] | [177, 13] | intro n | u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
⊢ ∀ (n : ℕ), u n ≤ l | u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
⊢ u n ≤ l | Please generate a tactic in lean4 to solve the state.
STATE:
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
⊢ ∀ (n : ℕ), u n ≤ l
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_7 | [164, 1] | [177, 13] | by_contra hn | u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
⊢ u n ≤ l | u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
⊢ u n ≤ l
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_7 | [164, 1] | [177, 13] | have : u n > l := by exact not_le.mp hn | u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
⊢ False | u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this : u n > l
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_7 | [164, 1] | [177, 13] | obtain ⟨N, hN⟩ := h1 (u n - l) (by linarith) | u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this : u n > l
⊢ False | case intro
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this : u n > l
N : ℕ
hN : ∀ n_1 ≥ N, |u n_1 - l| < u n - l
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this : u n > l
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_7 | [164, 1] | [177, 13] | specialize hN (max n N) (by apply?) | case intro
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this : u n > l
N : ℕ
hN : ∀ n_1 ≥ N, |u n_1 - l| < u n - l
⊢ False | case intro
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this : u n > l
N : ℕ
hN : |u (max n N) - l| < u n - l
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this : u n > l
N : ℕ
hN : ∀ n_1 ≥ N, |u n_1 - l| < u n - l
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_7 | [164, 1] | [177, 13] | rw [← not_le] at hN | case intro
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this : u n > l
N : ℕ
hN : |u (max n N) - l| < u n - l
⊢ False | case intro
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this : u n > l
N : ℕ
hN : ¬u n - l ≤ |u (max n N) - l|
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this : u n > l
N : ℕ
hN : |u (max n N) - l| < u n - l
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_7 | [164, 1] | [177, 13] | apply hN | case intro
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this : u n > l
N : ℕ
hN : ¬u n - l ≤ |u (max n N) - l|
⊢ False | case intro
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this : u n > l
N : ℕ
hN : ¬u n - l ≤ |u (max n N) - l|
⊢ u n - l ≤ |u (max n N) - l| | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this : u n > l
N : ℕ
hN : ¬u n - l ≤ |u (max n N) - l|
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_7 | [164, 1] | [177, 13] | have : u n ≤ u (max n N) := h2 _ _ (by apply?) | case intro
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this : u n > l
N : ℕ
hN : ¬u n - l ≤ |u (max n N) - l|
⊢ u n - l ≤ |u (max n N) - l| | case intro
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this✝ : u n > l
N : ℕ
hN : ¬u n - l ≤ |u (max n N) - l|
this : u n ≤ u (max n N)
⊢ u n - l ≤ |u (max n N) - l| | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this : u n > l
N : ℕ
hN : ¬u n - l ≤ |u (max n N) - l|
⊢ u n - l ≤ |u (max n N) - l|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_7 | [164, 1] | [177, 13] | rw [abs_eq_self.2] | case intro
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this✝ : u n > l
N : ℕ
hN : ¬u n - l ≤ |u (max n N) - l|
this : u n ≤ u (max n N)
⊢ u n - l ≤ |u (max n N) - l| | case intro
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this✝ : u n > l
N : ℕ
hN : ¬u n - l ≤ |u (max n N) - l|
this : u n ≤ u (max n N)
⊢ u n - l ≤ u (max n N) - l
case intro
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this✝ : u n > l
N : ℕ
hN : ¬u n - l ≤ |u (max n N) - l|
this : u n ≤ u (max n N)
⊢ 0 ≤ u (max n N) - l | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this✝ : u n > l
N : ℕ
hN : ¬u n - l ≤ |u (max n N) - l|
this : u n ≤ u (max n N)
⊢ u n - l ≤ |u (max n N) - l|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_7 | [164, 1] | [177, 13] | exact not_le.mp hn | u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
⊢ u n > l | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
⊢ u n > l
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_7 | [164, 1] | [177, 13] | linarith | u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this : u n > l
⊢ u n - l > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this : u n > l
⊢ u n - l > 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_7 | [164, 1] | [177, 13] | apply? | u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this : u n > l
N : ℕ
hN : ∀ n_1 ≥ N, |u n_1 - l| < u n - l
⊢ max n N ≥ N | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this : u n > l
N : ℕ
hN : ∀ n_1 ≥ N, |u n_1 - l| < u n - l
⊢ max n N ≥ N
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_7 | [164, 1] | [177, 13] | apply? | u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this : u n > l
N : ℕ
hN : ¬u n - l ≤ |u (max n N) - l|
⊢ n ≤ max n N | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this : u n > l
N : ℕ
hN : ¬u n - l ≤ |u (max n N) - l|
⊢ n ≤ max n N
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_7 | [164, 1] | [177, 13] | gcongr | case intro
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this✝ : u n > l
N : ℕ
hN : ¬u n - l ≤ |u (max n N) - l|
this : u n ≤ u (max n N)
⊢ u n - l ≤ u (max n N) - l | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this✝ : u n > l
N : ℕ
hN : ¬u n - l ≤ |u (max n N) - l|
this : u n ≤ u (max n N)
⊢ u n - l ≤ u (max n N) - l
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_7 | [164, 1] | [177, 13] | rw [sub_nonneg] | case intro
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this✝ : u n > l
N : ℕ
hN : ¬u n - l ≤ |u (max n N) - l|
this : u n ≤ u (max n N)
⊢ 0 ≤ u (max n N) - l | case intro
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this✝ : u n > l
N : ℕ
hN : ¬u n - l ≤ |u (max n N) - l|
this : u n ≤ u (max n N)
⊢ l ≤ u (max n N) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this✝ : u n > l
N : ℕ
hN : ¬u n - l ≤ |u (max n N) - l|
this : u n ≤ u (max n N)
⊢ 0 ≤ u (max n N) - l
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_7 | [164, 1] | [177, 13] | linarith | case intro
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this✝ : u n > l
N : ℕ
hN : ¬u n - l ≤ |u (max n N) - l|
this : u n ≤ u (max n N)
⊢ l ≤ u (max n N) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
u : ℕ → ℝ
l : ℝ
h1 : SequentialLimit u l
h2 : NondecreasingSequence u
n : ℕ
hn : ¬u n ≤ l
this✝ : u n > l
N : ℕ
hN : ¬u n - l ≤ |u (max n N) - l|
this : u n ≤ u (max n N)
⊢ l ≤ u (max n N)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_8 | [184, 1] | [190, 30] | ext y | α : Type u_1
β : Type u_2
f : α → β
s : Set α
t : Set β
⊢ f '' s ∩ t = f '' (s ∩ f ⁻¹' t) | case h
α : Type u_1
β : Type u_2
f : α → β
s : Set α
t : Set β
y : β
⊢ y ∈ f '' s ∩ t ↔ y ∈ f '' (s ∩ f ⁻¹' t) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
f : α → β
s : Set α
t : Set β
⊢ f '' s ∩ t = f '' (s ∩ f ⁻¹' t)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_8 | [184, 1] | [190, 30] | constructor | case h
α : Type u_1
β : Type u_2
f : α → β
s : Set α
t : Set β
y : β
⊢ y ∈ f '' s ∩ t ↔ y ∈ f '' (s ∩ f ⁻¹' t) | case h.mp
α : Type u_1
β : Type u_2
f : α → β
s : Set α
t : Set β
y : β
⊢ y ∈ f '' s ∩ t → y ∈ f '' (s ∩ f ⁻¹' t)
case h.mpr
α : Type u_1
β : Type u_2
f : α → β
s : Set α
t : Set β
y : β
⊢ y ∈ f '' (s ∩ f ⁻¹' t) → y ∈ f '' s ∩ t | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
β : Type u_2
f : α → β
s : Set α
t : Set β
y : β
⊢ y ∈ f '' s ∩ t ↔ y ∈ f '' (s ∩ f ⁻¹' t)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_8 | [184, 1] | [190, 30] | rintro ⟨⟨x, xs, rfl⟩, fxv⟩ | case h.mp
α : Type u_1
β : Type u_2
f : α → β
s : Set α
t : Set β
y : β
⊢ y ∈ f '' s ∩ t → y ∈ f '' (s ∩ f ⁻¹' t) | case h.mp.intro.intro.intro
α : Type u_1
β : Type u_2
f : α → β
s : Set α
t : Set β
x : α
xs : x ∈ s
fxv : f x ∈ t
⊢ f x ∈ f '' (s ∩ f ⁻¹' t) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
α : Type u_1
β : Type u_2
f : α → β
s : Set α
t : Set β
y : β
⊢ y ∈ f '' s ∩ t → y ∈ f '' (s ∩ f ⁻¹' t)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_8 | [184, 1] | [190, 30] | use x, ⟨xs, fxv⟩ | case h.mp.intro.intro.intro
α : Type u_1
β : Type u_2
f : α → β
s : Set α
t : Set β
x : α
xs : x ∈ s
fxv : f x ∈ t
⊢ f x ∈ f '' (s ∩ f ⁻¹' t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro.intro.intro
α : Type u_1
β : Type u_2
f : α → β
s : Set α
t : Set β
x : α
xs : x ∈ s
fxv : f x ∈ t
⊢ f x ∈ f '' (s ∩ f ⁻¹' t)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_8 | [184, 1] | [190, 30] | rintro ⟨x, ⟨⟨xs, fxv⟩, rfl⟩⟩ | case h.mpr
α : Type u_1
β : Type u_2
f : α → β
s : Set α
t : Set β
y : β
⊢ y ∈ f '' (s ∩ f ⁻¹' t) → y ∈ f '' s ∩ t | case h.mpr.intro.intro.intro
α : Type u_1
β : Type u_2
f : α → β
s : Set α
t : Set β
x : α
xs : x ∈ s
fxv : x ∈ f ⁻¹' t
⊢ f x ∈ f '' s ∩ t | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
α : Type u_1
β : Type u_2
f : α → β
s : Set α
t : Set β
y : β
⊢ y ∈ f '' (s ∩ f ⁻¹' t) → y ∈ f '' s ∩ t
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_8 | [184, 1] | [190, 30] | exact ⟨⟨x, xs, rfl⟩, fxv⟩ | case h.mpr.intro.intro.intro
α : Type u_1
β : Type u_2
f : α → β
s : Set α
t : Set β
x : α
xs : x ∈ s
fxv : x ∈ f ⁻¹' t
⊢ f x ∈ f '' s ∩ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro.intro.intro
α : Type u_1
β : Type u_2
f : α → β
s : Set α
t : Set β
x : α
xs : x ∈ s
fxv : x ∈ f ⁻¹' t
⊢ f x ∈ f '' s ∩ t
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_9 | [192, 1] | [195, 61] | ext x | ⊢ (fun x => x ^ 2) ⁻¹' {y | y ≥ 4} = {x | x ≤ -2} ∪ {x | x ≥ 2} | case h
x : ℝ
⊢ x ∈ (fun x => x ^ 2) ⁻¹' {y | y ≥ 4} ↔ x ∈ {x | x ≤ -2} ∪ {x | x ≥ 2} | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ (fun x => x ^ 2) ⁻¹' {y | y ≥ 4} = {x | x ≤ -2} ∪ {x | x ≥ 2}
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_9 | [192, 1] | [195, 61] | simp [show (4 : ℝ) = 2 ^ 2 by norm_num, sq_le_sq, le_abs'] | case h
x : ℝ
⊢ x ∈ (fun x => x ^ 2) ⁻¹' {y | y ≥ 4} ↔ x ∈ {x | x ≤ -2} ∪ {x | x ≥ 2} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
x : ℝ
⊢ x ∈ (fun x => x ^ 2) ⁻¹' {y | y ≥ 4} ↔ x ∈ {x | x ≤ -2} ∪ {x | x ≥ 2}
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_9 | [192, 1] | [195, 61] | norm_num | x : ℝ
⊢ 4 = 2 ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : ℝ
⊢ 4 = 2 ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_10 | [210, 1] | [237, 28] | have h1' : ∀ x y, f x ≠ g y | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | case h1'
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
⊢ ∀ (x : α) (y : β), f x ≠ g y
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_10 | [210, 1] | [237, 28] | have h1'' : ∀ y x, g y ≠ f x | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | case h1''
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
⊢ ∀ (y : β) (x : α), g y ≠ f x
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_10 | [210, 1] | [237, 28] | have h2' : ∀ x, x ∈ range f ∪ range g := eq_univ_iff_forall.1 h2 | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_10 | [210, 1] | [237, 28] | have hf' : ∀ x x', f x = f x' ↔ x = x' := fun x x' ↦ hf.eq_iff | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_10 | [210, 1] | [237, 28] | let L : Set α × Set β → Set γ :=
fun (s, t) ↦ f '' s ∪ g '' t | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_10 | [210, 1] | [237, 28] | let R : Set γ → Set α × Set β :=
fun s ↦ (f ⁻¹' s, g ⁻¹' s) | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_10 | [210, 1] | [237, 28] | use L | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id | case h
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
⊢ ∃ R, L ∘ R = id ∧ R ∘ L = id | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
⊢ ∃ L R, L ∘ R = id ∧ R ∘ L = id
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_10 | [210, 1] | [237, 28] | use R | case h
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
⊢ ∃ R, L ∘ R = id ∧ R ∘ L = id | case h
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
⊢ L ∘ R = id ∧ R ∘ L = id | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
⊢ ∃ R, L ∘ R = id ∧ R ∘ L = id
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_10 | [210, 1] | [237, 28] | constructor | case h
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
⊢ L ∘ R = id ∧ R ∘ L = id | case h.left
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
⊢ L ∘ R = id
case h.right
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
⊢ R ∘ L = id | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
⊢ L ∘ R = id ∧ R ∘ L = id
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_10 | [210, 1] | [237, 28] | intro x y h | case h1'
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
⊢ ∀ (x : α) (y : β), f x ≠ g y | case h1'
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
x : α
y : β
h : f x = g y
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h1'
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
⊢ ∀ (x : α) (y : β), f x ≠ g y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_10 | [210, 1] | [237, 28] | apply h1.subset | case h1'
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
x : α
y : β
h : f x = g y
⊢ False | case h1'.a
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
x : α
y : β
h : f x = g y
⊢ ?h1'.a ∈ range f ∩ range g
case h1'.a
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
x : α
y : β
h : f x = g y
⊢ γ | Please generate a tactic in lean4 to solve the state.
STATE:
case h1'
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
x : α
y : β
h : f x = g y
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_10 | [210, 1] | [237, 28] | exact ⟨⟨x, h⟩, ⟨y, rfl⟩⟩ | case h1'.a
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
x : α
y : β
h : f x = g y
⊢ ?h1'.a ∈ range f ∩ range g
case h1'.a
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
x : α
y : β
h : f x = g y
⊢ γ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h1'.a
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
x : α
y : β
h : f x = g y
⊢ ?h1'.a ∈ range f ∩ range g
case h1'.a
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
x : α
y : β
h : f x = g y
⊢ γ
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_10 | [210, 1] | [237, 28] | intro x y | case h1''
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
⊢ ∀ (y : β) (x : α), g y ≠ f x | case h1''
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
x : β
y : α
⊢ g x ≠ f y | Please generate a tactic in lean4 to solve the state.
STATE:
case h1''
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
⊢ ∀ (y : β) (x : α), g y ≠ f x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_10 | [210, 1] | [237, 28] | symm | case h1''
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
x : β
y : α
⊢ g x ≠ f y | case h1''
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
x : β
y : α
⊢ f y ≠ g x | Please generate a tactic in lean4 to solve the state.
STATE:
case h1''
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
x : β
y : α
⊢ g x ≠ f y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_10 | [210, 1] | [237, 28] | apply h1' | case h1''
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
x : β
y : α
⊢ f y ≠ g x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h1''
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
x : β
y : α
⊢ f y ≠ g x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_10 | [210, 1] | [237, 28] | ext s x | case h.left
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
⊢ L ∘ R = id | case h.left.h.h
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
s : Set γ
x : γ
⊢ x ∈ (L ∘ R) s ↔ x ∈ id s | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
⊢ L ∘ R = id
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_10 | [210, 1] | [237, 28] | obtain ⟨x, rfl⟩|⟨x, rfl⟩ := h2' x | case h.left.h.h
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
s : Set γ
x : γ
⊢ x ∈ (L ∘ R) s ↔ x ∈ id s | case h.left.h.h.inl.intro
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
s : Set γ
x : α
⊢ f x ∈ (L ∘ R) s ↔ f x ∈ id s
case h.left.h.h.inr.intro
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
s : Set γ
x : β
⊢ g x ∈ (L ∘ R) s ↔ g x ∈ id s | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.h.h
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
s : Set γ
x : γ
⊢ x ∈ (L ∘ R) s ↔ x ∈ id s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_10 | [210, 1] | [237, 28] | simp [h1'', hf.eq_iff] | case h.left.h.h.inl.intro
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
s : Set γ
x : α
⊢ f x ∈ (L ∘ R) s ↔ f x ∈ id s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.h.h.inl.intro
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
s : Set γ
x : α
⊢ f x ∈ (L ∘ R) s ↔ f x ∈ id s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_10 | [210, 1] | [237, 28] | simp [h1', hg.eq_iff] | case h.left.h.h.inr.intro
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
s : Set γ
x : β
⊢ g x ∈ (L ∘ R) s ↔ g x ∈ id s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.h.h.inr.intro
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
s : Set γ
x : β
⊢ g x ∈ (L ∘ R) s ↔ g x ∈ id s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_10 | [210, 1] | [237, 28] | ext s x | case h.right
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
⊢ R ∘ L = id | case h.right.h.h₁.h
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
s : Set α × Set β
x : α
⊢ x ∈ ((R ∘ L) s).1 ↔ x ∈ (id s).1
case h.right.h.h₂.h
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
s : Set α × Set β
x : β
⊢ x ∈ ((R ∘ L) s).2 ↔ x ∈ (id s).2 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
⊢ R ∘ L = id
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_10 | [210, 1] | [237, 28] | simp [hf.eq_iff, h1''] | case h.right.h.h₁.h
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
s : Set α × Set β
x : α
⊢ x ∈ ((R ∘ L) s).1 ↔ x ∈ (id s).1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.h.h₁.h
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
s : Set α × Set β
x : α
⊢ x ∈ ((R ∘ L) s).1 ↔ x ∈ (id s).1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions3.lean | exercise3_10 | [210, 1] | [237, 28] | simp [hg.eq_iff, h1'] | case h.right.h.h₂.h
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
s : Set α × Set β
x : β
⊢ x ∈ ((R ∘ L) s).2 ↔ x ∈ (id s).2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right.h.h₂.h
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → γ
g : β → γ
hf : Injective f
hg : Injective g
h1 : range f ∩ range g = ∅
h2 : range f ∪ range g = univ
h1' : ∀ (x : α) (y : β), f x ≠ g y
h1'' : ∀ (y : β) (x : α), g y ≠ f x
h2' : ∀ (x : γ), x ∈ range f ∪ range g
hf' : ∀ (x x' : α), f x = f x' ↔ x = x'
L : Set α × Set β → Set γ :=
fun x =>
match x with
| (s, t) => f '' s ∪ g '' t
R : Set γ → Set α × Set β := fun s => (f ⁻¹' s, g ⁻¹' s)
s : Set α × Set β
x : β
⊢ x ∈ ((R ∘ L) s).2 ↔ x ∈ (id s).2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment2.lean | exercise2_1 | [21, 1] | [23, 8] | sorry | α : Type u_1
p q : α → Prop
⊢ (∃ x, p x ∨ q x) ↔ (∃ x, p x) ∨ ∃ x, q x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
p q : α → Prop
⊢ (∃ x, p x ∨ q x) ↔ (∃ x, p x) ∨ ∃ x, q x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment2.lean | exercise2_2 | [38, 1] | [39, 8] | sorry | f g : ℝ → ℝ
x : ℝ
h : SurjectiveFunction (g ∘ f)
⊢ SurjectiveFunction g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℝ → ℝ
x : ℝ
h : SurjectiveFunction (g ∘ f)
⊢ SurjectiveFunction g
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment2.lean | exercise2_3 | [42, 1] | [44, 8] | sorry | f g : ℝ → ℝ
x : ℝ
hf : SurjectiveFunction f
⊢ SurjectiveFunction (g ∘ f) ↔ SurjectiveFunction g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℝ → ℝ
x : ℝ
hf : SurjectiveFunction f
⊢ SurjectiveFunction (g ∘ f) ↔ SurjectiveFunction g
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment2.lean | exercise2_4 | [48, 1] | [50, 8] | sorry | f g : ℝ → ℝ
x : ℝ
hf : SurjectiveFunction f
⊢ SurjectiveFunction fun x => 2 * f (3 * (x + 4)) + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f g : ℝ → ℝ
x : ℝ
hf : SurjectiveFunction f
⊢ SurjectiveFunction fun x => 2 * f (3 * (x + 4)) + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment2.lean | exercise2_5 | [71, 1] | [72, 8] | sorry | s t : ℕ → ℕ
k : ℕ
⊢ EventuallyGrowsFaster (fun n => n * s n) s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℕ
k : ℕ
⊢ EventuallyGrowsFaster (fun n => n * s n) s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment2.lean | useful_fact | [76, 1] | [76, 71] | simp | s t : ℕ → ℕ
k a b c : ℕ
⊢ c ≥ max a b ↔ c ≥ a ∧ c ≥ b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℕ
k a b c : ℕ
⊢ c ≥ max a b ↔ c ≥ a ∧ c ≥ b
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment2.lean | exercise2_6 | [78, 1] | [81, 8] | sorry | s t : ℕ → ℕ
k : ℕ
s₁ s₂ t₁ t₂ : ℕ → ℕ
h₁ : EventuallyGrowsFaster s₁ t₁
h₂ : EventuallyGrowsFaster s₂ t₂
⊢ EventuallyGrowsFaster (s₁ + s₂) (t₁ + t₂) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℕ
k : ℕ
s₁ s₂ t₁ t₂ : ℕ → ℕ
h₁ : EventuallyGrowsFaster s₁ t₁
h₂ : EventuallyGrowsFaster s₂ t₂
⊢ EventuallyGrowsFaster (s₁ + s₂) (t₁ + t₂)
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.