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stringlengths 7
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| full_name
stringlengths 1
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| start
stringlengths 6
10
| end
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https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.aux | [275, 1] | [283, 8] | have Bpos : 0 < B := lt_of_le_of_lt (abs_nonneg _) (h₀ N₀ (le_refl _)) | case intro.intro
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
N₀ : ℕ
B : ℝ
h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε | case intro.intro
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
N₀ : ℕ
B : ℝ
h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B
Bpos : 0 < B
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.aux | [275, 1] | [283, 8] | have pos₀ : ε / B > 0 := div_pos εpos Bpos | case intro.intro
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
N₀ : ℕ
B : ℝ
h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B
Bpos : 0 < B
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε | case intro.intro
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
N₀ : ℕ
B : ℝ
h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B
Bpos : 0 < B
pos₀ : ε / B > 0
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.aux | [275, 1] | [283, 8] | rcases ct _ pos₀ with ⟨N₁, h₁⟩ | case intro.intro
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
N₀ : ℕ
B : ℝ
h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B
Bpos : 0 < B
pos₀ : ε / B > 0
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε | case intro.intro.intro
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
N₀ : ℕ
B : ℝ
h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B
Bpos : 0 < B
pos₀ : ε / B > 0
N₁ : ℕ
h₁ : ∀ n ≥ N₁, |t n - 0| < ε / B
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.aux | [275, 1] | [283, 8] | sorry | case intro.intro.intro
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
N₀ : ℕ
B : ℝ
h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B
Bpos : 0 < B
pos₀ : ε / B > 0
N₁ : ℕ
h₁ : ∀ n ≥ N₁, |t n - 0| < ε / B
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.auxαα | [287, 1] | [302, 52] | intro ε εpos | s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
⊢ ConvergesTo (fun n => s n * t n) 0 | s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => s n * t n) n - 0| < ε |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.auxαα | [287, 1] | [302, 52] | dsimp | s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => s n * t n) n - 0| < ε | s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.auxαα | [287, 1] | [302, 52] | rcases exists_abs_le_of_convergesTo cs with ⟨N₀, B, h₀⟩ | s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε | case intro.intro
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
N₀ : ℕ
B : ℝ
h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.auxαα | [287, 1] | [302, 52] | have Bpos : 0 < B := lt_of_le_of_lt (abs_nonneg _) (h₀ N₀ (le_refl _)) | case intro.intro
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
N₀ : ℕ
B : ℝ
h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε | case intro.intro
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
N₀ : ℕ
B : ℝ
h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B
Bpos : 0 < B
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.auxαα | [287, 1] | [302, 52] | have pos₀ : ε / B > 0 := div_pos εpos Bpos | case intro.intro
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
N₀ : ℕ
B : ℝ
h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B
Bpos : 0 < B
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε | case intro.intro
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
N₀ : ℕ
B : ℝ
h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B
Bpos : 0 < B
pos₀ : ε / B > 0
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.auxαα | [287, 1] | [302, 52] | rcases ct _ pos₀ with ⟨N₁, h₁⟩ | case intro.intro
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
N₀ : ℕ
B : ℝ
h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B
Bpos : 0 < B
pos₀ : ε / B > 0
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε | case intro.intro.intro
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
N₀ : ℕ
B : ℝ
h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B
Bpos : 0 < B
pos₀ : ε / B > 0
N₁ : ℕ
h₁ : ∀ n ≥ N₁, |t n - 0| < ε / B
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.auxαα | [287, 1] | [302, 52] | use max N₀ N₁ | case intro.intro.intro
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
N₀ : ℕ
B : ℝ
h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B
Bpos : 0 < B
pos₀ : ε / B > 0
N₁ : ℕ
h₁ : ∀ n ≥ N₁, |t n - 0| < ε / B
⊢ ∃ N, ∀ n ≥ N, |s n * t n - 0| < ε | case h
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
N₀ : ℕ
B : ℝ
h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B
Bpos : 0 < B
pos₀ : ε / B > 0
N₁ : ℕ
h₁ : ∀ n ≥ N₁, |t n - 0| < ε / B
⊢ ∀ n ≥ max N₀ N₁, |s n * t n - 0| < ε |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.auxαα | [287, 1] | [302, 52] | intro n ngt | case h
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
N₀ : ℕ
B : ℝ
h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B
Bpos : 0 < B
pos₀ : ε / B > 0
N₁ : ℕ
h₁ : ∀ n ≥ N₁, |t n - 0| < ε / B
⊢ ∀ n ≥ max N₀ N₁, |s n * t n - 0| < ε | case h
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
N₀ : ℕ
B : ℝ
h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B
Bpos : 0 < B
pos₀ : ε / B > 0
N₁ : ℕ
h₁ : ∀ n ≥ N₁, |t n - 0| < ε / B
n : ℕ
ngt : n ≥ max N₀ N₁
⊢ |s n * t n - 0| < ε |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.auxαα | [287, 1] | [302, 52] | have ngeN₀ : n ≥ N₀ := le_of_max_le_left ngt | case h
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
N₀ : ℕ
B : ℝ
h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B
Bpos : 0 < B
pos₀ : ε / B > 0
N₁ : ℕ
h₁ : ∀ n ≥ N₁, |t n - 0| < ε / B
n : ℕ
ngt : n ≥ max N₀ N₁
⊢ |s n * t n - 0| < ε | case h
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
N₀ : ℕ
B : ℝ
h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B
Bpos : 0 < B
pos₀ : ε / B > 0
N₁ : ℕ
h₁ : ∀ n ≥ N₁, |t n - 0| < ε / B
n : ℕ
ngt : n ≥ max N₀ N₁
ngeN₀ : n ≥ N₀
⊢ |s n * t n - 0| < ε |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.auxαα | [287, 1] | [302, 52] | have ngeN₁ : n ≥ N₁ := le_of_max_le_right ngt | case h
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
N₀ : ℕ
B : ℝ
h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B
Bpos : 0 < B
pos₀ : ε / B > 0
N₁ : ℕ
h₁ : ∀ n ≥ N₁, |t n - 0| < ε / B
n : ℕ
ngt : n ≥ max N₀ N₁
ngeN₀ : n ≥ N₀
⊢ |s n * t n - 0| < ε | case h
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
N₀ : ℕ
B : ℝ
h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B
Bpos : 0 < B
pos₀ : ε / B > 0
N₁ : ℕ
h₁ : ∀ n ≥ N₁, |t n - 0| < ε / B
n : ℕ
ngt : n ≥ max N₀ N₁
ngeN₀ : n ≥ N₀
ngeN₁ : n ≥ N₁
⊢ |s n * t n - 0| < ε |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.auxαα | [287, 1] | [302, 52] | calc
|s n * t n - 0| = |s n| * |t n - 0| := by rw [sub_zero, abs_mul, sub_zero]
_ < B * (ε / B) := (mul_lt_mul'' (h₀ n ngeN₀) (h₁ n ngeN₁) (abs_nonneg _) (abs_nonneg _))
_ = ε := mul_div_cancel₀ _ (ne_of_lt Bpos).symm | case h
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
N₀ : ℕ
B : ℝ
h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B
Bpos : 0 < B
pos₀ : ε / B > 0
N₁ : ℕ
h₁ : ∀ n ≥ N₁, |t n - 0| < ε / B
n : ℕ
ngt : n ≥ max N₀ N₁
ngeN₀ : n ≥ N₀
ngeN₁ : n ≥ N₁
⊢ |s n * t n - 0| < ε | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.auxαα | [287, 1] | [302, 52] | rw [sub_zero, abs_mul, sub_zero] | s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
ε : ℝ
εpos : ε > 0
N₀ : ℕ
B : ℝ
h₀ : ∀ (n : ℕ), N₀ ≤ n → |s n| < B
Bpos : 0 < B
pos₀ : ε / B > 0
N₁ : ℕ
h₁ : ∀ n ≥ N₁, |t n - 0| < ε / B
n : ℕ
ngt : n ≥ max N₀ N₁
ngeN₀ : n ≥ N₀
ngeN₁ : n ≥ N₁
⊢ |s n * t n - 0| = |s n| * |t n - 0| | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [311, 1] | [321, 7] | have h₁ : ConvergesTo (fun n ↦ s n * (t n + -b)) 0 := by
apply aux cs
convert convergesTo_add ct (convergesTo_const (-b))
ring | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ ConvergesTo (fun n => s n * t n) (a * b) | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
⊢ ConvergesTo (fun n => s n * t n) (a * b) |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [311, 1] | [321, 7] | have := convergesTo_add h₁ (convergesTo_mul_const b cs) | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
⊢ ConvergesTo (fun n => s n * t n) (a * b) | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
⊢ ConvergesTo (fun n => s n * t n) (a * b) |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [311, 1] | [321, 7] | convert convergesTo_add h₁ (convergesTo_mul_const b cs) using 1 | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
⊢ ConvergesTo (fun n => s n * t n) (a * b) | case h.e'_1
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
⊢ (fun n => s n * t n) = fun n => s n * (t n + -b) + b * s n
case h.e'_2
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
⊢ a * b = 0 + b * a |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [311, 1] | [321, 7] | ring | case h.e'_2
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
⊢ a * b = 0 + b * a | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [311, 1] | [321, 7] | apply aux cs | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ ConvergesTo (fun n => s n * (t n + -b)) 0 | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ ConvergesTo (fun n => t n + -b) 0 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [311, 1] | [321, 7] | convert convergesTo_add ct (convergesTo_const (-b)) | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ ConvergesTo (fun n => t n + -b) 0 | case h.e'_2
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ 0 = b + -b |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [311, 1] | [321, 7] | ring | case h.e'_2
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ 0 = b + -b | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [311, 1] | [321, 7] | ext | case h.e'_1
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
⊢ (fun n => s n * t n) = fun n => s n * (t n + -b) + b * s n | case h.e'_1.h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
x✝ : ℕ
⊢ s x✝ * t x✝ = s x✝ * (t x✝ + -b) + b * s x✝ |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [311, 1] | [321, 7] | ring | case h.e'_1.h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
x✝ : ℕ
⊢ s x✝ * t x✝ = s x✝ * (t x✝ + -b) + b * s x✝ | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [332, 1] | [347, 25] | by_contra abne | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
⊢ a = b | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [332, 1] | [347, 25] | have : |a - b| > 0 := by sorry | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
⊢ False | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [332, 1] | [347, 25] | let ε := |a - b| / 2 | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
⊢ False | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [332, 1] | [347, 25] | have εpos : ε > 0 := by
change |a - b| / 2 > 0
linarith | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
⊢ False | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [332, 1] | [347, 25] | rcases sa ε εpos with ⟨Na, hNa⟩ | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
⊢ False | case intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [332, 1] | [347, 25] | rcases sb ε εpos with ⟨Nb, hNb⟩ | case intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
⊢ False | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [332, 1] | [347, 25] | let N := max Na Nb | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
⊢ False | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [332, 1] | [347, 25] | have absa : |s N - a| < ε := by sorry | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
⊢ False | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [332, 1] | [347, 25] | have absb : |s N - b| < ε := by sorry | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
⊢ False | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [332, 1] | [347, 25] | have : |a - b| < |a - b| := by sorry | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
⊢ False | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this✝ : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
this : |a - b| < |a - b|
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [332, 1] | [347, 25] | exact lt_irrefl _ this | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this✝ : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
this : |a - b| < |a - b|
⊢ False | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [332, 1] | [347, 25] | sorry | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
⊢ |a - b| > 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [332, 1] | [347, 25] | change |a - b| / 2 > 0 | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
⊢ ε > 0 | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
⊢ |a - b| / 2 > 0 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [332, 1] | [347, 25] | linarith | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
⊢ |a - b| / 2 > 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [332, 1] | [347, 25] | sorry | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
⊢ |s N - a| < ε | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [332, 1] | [347, 25] | sorry | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
⊢ |s N - b| < ε | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [332, 1] | [347, 25] | sorry | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
⊢ |a - b| < |a - b| | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | by_contra abne | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
⊢ a = b | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | let ε := |a - b| / 2 | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
⊢ False | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | have εpos : ε > 0 := by
change |a - b| / 2 > 0
linarith | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
⊢ False | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | rcases sa ε εpos with ⟨Na, hNa⟩ | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
⊢ False | case intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | rcases sb ε εpos with ⟨Nb, hNb⟩ | case intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
⊢ False | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | let N := max Na Nb | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
⊢ False | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | have absa : |s N - a| < ε := by
apply hNa
apply le_max_left | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
⊢ False | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | have absb : |s N - b| < ε := by
apply hNb
apply le_max_right | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
⊢ False | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | have : |a - b| < |a - b| | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
⊢ False | case this
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
⊢ |a - b| < |a - b|
case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this✝ : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
this : |a - b| < |a - b|
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | calc
|a - b| = |(-(s N - a)) + (s N - b)| := by
congr
ring
_ ≤ |(-(s N - a))| + |s N - b| := (abs_add _ _)
_ = |s N - a| + |s N - b| := by rw [abs_neg]
_ < ε + ε := (add_lt_add absa absb)
_ = |a - b| := by norm_num [ε] | case this
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
⊢ |a - b| < |a - b|
case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this✝ : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
this : |a - b| < |a - b|
⊢ False | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this✝ : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
this : |a - b| < |a - b|
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | exact lt_irrefl _ this | case intro.intro
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this✝ : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
this : |a - b| < |a - b|
⊢ False | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | apply lt_of_le_of_ne | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
⊢ |a - b| > 0 | case a
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
⊢ 0 ≤ |a - b|
case a
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
⊢ 0 ≠ |a - b| |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | intro h'' | case a
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
⊢ 0 ≠ |a - b| | case a
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
h'' : 0 = |a - b|
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | apply abne | case a
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
h'' : 0 = |a - b|
⊢ False | case a
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
h'' : 0 = |a - b|
⊢ a = b |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | apply eq_of_abs_sub_eq_zero h''.symm | case a
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
h'' : 0 = |a - b|
⊢ a = b | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | apply abs_nonneg | case a
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
⊢ 0 ≤ |a - b| | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | change |a - b| / 2 > 0 | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
⊢ ε > 0 | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
⊢ |a - b| / 2 > 0 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | linarith | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
⊢ |a - b| / 2 > 0 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | apply hNa | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
⊢ |s N - a| < ε | case a
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
⊢ N ≥ Na |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | apply le_max_left | case a
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
⊢ N ≥ Na | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | apply hNb | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
⊢ |s N - b| < ε | case a
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
⊢ N ≥ Nb |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | apply le_max_right | case a
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
⊢ N ≥ Nb | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | congr | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
⊢ |a - b| = |-(s N - a) + (s N - b)| | case e_a
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
⊢ a - b = -(s N - a) + (s N - b) |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | ring | case e_a
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
⊢ a - b = -(s N - a) + (s N - b) | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | rw [abs_neg] | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
⊢ |(-(s N - a))| + |s N - b| = |s N - a| + |s N - b| | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C03_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_uniqueαα | [351, 1] | [384, 25] | norm_num [ε] | s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
ε : ℝ := |a - b| / 2
εpos : ε > 0
Na : ℕ
hNa : ∀ n ≥ Na, |s n - a| < ε
Nb : ℕ
hNb : ∀ n ≥ Nb, |s n - b| < ε
N : ℕ := max Na Nb
absa : |s N - a| < ε
absb : |s N - b| < ε
⊢ ε + ε = |a - b| | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C08_Groups_and_Rings/S01_Groups.lean | eq_bot_iff_card | [453, 1] | [465, 41] | suffices (∀ x ∈ H, x = 1) ↔ ∃ x ∈ H, ∀ a ∈ H, a = x by
simpa [eq_bot_iff_forall, card_eq_one_iff] | G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype ↥H
⊢ H = ⊥ ↔ card ↥H = 1 | G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype ↥H
⊢ (∀ x ∈ H, x = 1) ↔ ∃ x ∈ H, ∀ a ∈ H, a = x |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C08_Groups_and_Rings/S01_Groups.lean | eq_bot_iff_card | [453, 1] | [465, 41] | constructor | G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype ↥H
⊢ (∀ x ∈ H, x = 1) ↔ ∃ x ∈ H, ∀ a ∈ H, a = x | case mp
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype ↥H
⊢ (∀ x ∈ H, x = 1) → ∃ x ∈ H, ∀ a ∈ H, a = x
case mpr
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype ↥H
⊢ (∃ x ∈ H, ∀ a ∈ H, a = x) → ∀ x ∈ H, x = 1 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C08_Groups_and_Rings/S01_Groups.lean | eq_bot_iff_card | [453, 1] | [465, 41] | simpa [eq_bot_iff_forall, card_eq_one_iff] | G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype ↥H
this : (∀ x ∈ H, x = 1) ↔ ∃ x ∈ H, ∀ a ∈ H, a = x
⊢ H = ⊥ ↔ card ↥H = 1 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C08_Groups_and_Rings/S01_Groups.lean | eq_bot_iff_card | [453, 1] | [465, 41] | intro h | case mp
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype ↥H
⊢ (∀ x ∈ H, x = 1) → ∃ x ∈ H, ∀ a ∈ H, a = x | case mp
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype ↥H
h : ∀ x ∈ H, x = 1
⊢ ∃ x ∈ H, ∀ a ∈ H, a = x |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C08_Groups_and_Rings/S01_Groups.lean | eq_bot_iff_card | [453, 1] | [465, 41] | use 1, H.one_mem | case mp
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype ↥H
h : ∀ x ∈ H, x = 1
⊢ ∃ x ∈ H, ∀ a ∈ H, a = x | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C08_Groups_and_Rings/S01_Groups.lean | eq_bot_iff_card | [453, 1] | [465, 41] | rintro ⟨y, -, hy'⟩ x hx | case mpr
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype ↥H
⊢ (∃ x ∈ H, ∀ a ∈ H, a = x) → ∀ x ∈ H, x = 1 | case mpr.intro.intro
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype ↥H
y : G
hy' : ∀ a ∈ H, a = y
x : G
hx : x ∈ H
⊢ x = 1 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C08_Groups_and_Rings/S01_Groups.lean | eq_bot_iff_card | [453, 1] | [465, 41] | calc x = y := hy' x hx
_ = 1 := (hy' 1 H.one_mem).symm | case mpr.intro.intro
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Fintype ↥H
y : G
hy' : ∀ a ∈ H, a = y
x : G
hx : x ∈ H
⊢ x = 1 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C08_Groups_and_Rings/S01_Groups.lean | inf_bot_of_coprime | [471, 1] | [478, 63] | have D₁ : card (H ⊓ K : Subgroup G) ∣ card H := card_dvd_of_le inf_le_left | G : Type u_1
inst✝² : Group G
H K : Subgroup G
inst✝¹ : Fintype ↥H
inst✝ : Fintype ↥K
h : (card ↥H).Coprime (card ↥K)
⊢ H ⊓ K = ⊥ | G : Type u_1
inst✝² : Group G
H K : Subgroup G
inst✝¹ : Fintype ↥H
inst✝ : Fintype ↥K
h : (card ↥H).Coprime (card ↥K)
D₁ : card ↥(H ⊓ K) ∣ card ↥H
⊢ H ⊓ K = ⊥ |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C08_Groups_and_Rings/S01_Groups.lean | inf_bot_of_coprime | [471, 1] | [478, 63] | have D₂ : card (H ⊓ K : Subgroup G) ∣ card K := card_dvd_of_le inf_le_right | G : Type u_1
inst✝² : Group G
H K : Subgroup G
inst✝¹ : Fintype ↥H
inst✝ : Fintype ↥K
h : (card ↥H).Coprime (card ↥K)
D₁ : card ↥(H ⊓ K) ∣ card ↥H
⊢ H ⊓ K = ⊥ | G : Type u_1
inst✝² : Group G
H K : Subgroup G
inst✝¹ : Fintype ↥H
inst✝ : Fintype ↥K
h : (card ↥H).Coprime (card ↥K)
D₁ : card ↥(H ⊓ K) ∣ card ↥H
D₂ : card ↥(H ⊓ K) ∣ card ↥K
⊢ H ⊓ K = ⊥ |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C08_Groups_and_Rings/S01_Groups.lean | inf_bot_of_coprime | [471, 1] | [478, 63] | exact eq_bot_iff_card.2 (Nat.eq_one_of_dvd_coprimes h D₁ D₂) | G : Type u_1
inst✝² : Group G
H K : Subgroup G
inst✝¹ : Fintype ↥H
inst✝ : Fintype ↥K
h : (card ↥H).Coprime (card ↥K)
D₁ : card ↥(H ⊓ K) ∣ card ↥H
D₂ : card ↥(H ⊓ K) ∣ card ↥K
⊢ H ⊓ K = ⊥ | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C08_Groups_and_Rings/S01_Groups.lean | compat_myMap | [568, 1] | [572, 9] | rintro _ rfl | ⊢ ∀ r ∈ {FreeGroup.of () ^ 3}, (FreeGroup.lift myMap) r = 1 | ⊢ (FreeGroup.lift myMap) (FreeGroup.of () ^ 3) = 1 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C08_Groups_and_Rings/S01_Groups.lean | compat_myMap | [568, 1] | [572, 9] | simp | ⊢ (FreeGroup.lift myMap) (FreeGroup.of () ^ 3) = 1 | ⊢ myMap () ^ 3 = 1 |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C08_Groups_and_Rings/S01_Groups.lean | compat_myMap | [568, 1] | [572, 9] | decide | ⊢ myMap () ^ 3 = 1 | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C08_Groups_and_Rings/S01_Groups.lean | conjugate_one | [698, 1] | [703, 19] | ext x | G : Type u_1
inst✝ : Group G
H : Subgroup G
⊢ conjugate 1 H = H | case h
G : Type u_1
inst✝ : Group G
H : Subgroup G
x : G
⊢ x ∈ conjugate 1 H ↔ x ∈ H |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C08_Groups_and_Rings/S01_Groups.lean | conjugate_one | [698, 1] | [703, 19] | simp [conjugate] | case h
G : Type u_1
inst✝ : Group G
H : Subgroup G
x : G
⊢ x ∈ conjugate 1 H ↔ x ∈ H | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C08_Groups_and_Rings/S01_Groups.lean | aux_card_eq | [832, 1] | [840, 49] | have := calc
card (G ⧸ H) * card H = card G := by rw [← H.index_eq_card, H.index_mul_card]
_ = card K * card H := by rw [h', mul_comm] | G : Type u_1
inst✝¹ : Group G
H K : Subgroup G
inst✝ : Fintype G
h' : card G = card ↥H * card ↥K
⊢ card (G ⧸ H) = card ↥K | G : Type u_1
inst✝¹ : Group G
H K : Subgroup G
inst✝ : Fintype G
h' : card G = card ↥H * card ↥K
this : card (G ⧸ H) * card ↥H = card ↥K * card ↥H
⊢ card (G ⧸ H) = card ↥K |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C08_Groups_and_Rings/S01_Groups.lean | aux_card_eq | [832, 1] | [840, 49] | exact Nat.eq_of_mul_eq_mul_right card_pos this | G : Type u_1
inst✝¹ : Group G
H K : Subgroup G
inst✝ : Fintype G
h' : card G = card ↥H * card ↥K
this : card (G ⧸ H) * card ↥H = card ↥K * card ↥H
⊢ card (G ⧸ H) = card ↥K | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C08_Groups_and_Rings/S01_Groups.lean | aux_card_eq | [832, 1] | [840, 49] | rw [← H.index_eq_card, H.index_mul_card] | G : Type u_1
inst✝¹ : Group G
H K : Subgroup G
inst✝ : Fintype G
h' : card G = card ↥H * card ↥K
⊢ card (G ⧸ H) * card ↥H = card G | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C08_Groups_and_Rings/S01_Groups.lean | aux_card_eq | [832, 1] | [840, 49] | rw [h', mul_comm] | G : Type u_1
inst✝¹ : Group G
H K : Subgroup G
inst✝ : Fintype G
h' : card G = card ↥H * card ↥K
⊢ card G = card ↥K * card ↥H | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | inverse_spec | [551, 1] | [553, 32] | rw [inverse, dif_pos h] | α : Type u_1
β : Type u_2
inst✝ : Inhabited α
P : α → Prop
h✝ : ∃ x, P x
f : α → β
y : β
h : ∃ x, f x = y
⊢ f (inverse f y) = y | α : Type u_1
β : Type u_2
inst✝ : Inhabited α
P : α → Prop
h✝ : ∃ x, P x
f : α → β
y : β
h : ∃ x, f x = y
⊢ f (choose h) = y |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | inverse_spec | [551, 1] | [553, 32] | exact Classical.choose_spec h | α : Type u_1
β : Type u_2
inst✝ : Inhabited α
P : α → Prop
h✝ : ∃ x, P x
f : α → β
y : β
h : ∃ x, f x = y
⊢ f (choose h) = y | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [671, 1] | [683, 16] | intro f surjf | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
⊢ ∀ (f : α → Set α), ¬Surjective f | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [671, 1] | [683, 16] | let S := { i | i ∉ f i } | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
⊢ False | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [671, 1] | [683, 16] | rcases surjf S with ⟨j, h⟩ | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
⊢ False | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [671, 1] | [683, 16] | have h₁ : j ∉ f j := by
intro h'
have : j ∉ f j := by rwa [h] at h'
contradiction | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
⊢ False | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [671, 1] | [683, 16] | have h₂ : j ∈ S | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
⊢ False | case h₂
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
⊢ j ∈ S
case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [671, 1] | [683, 16] | sorry | case h₂
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
⊢ j ∈ S
case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
⊢ False | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [671, 1] | [683, 16] | have h₃ : j ∉ S | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
⊢ False | case h₃
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
⊢ j ∉ S
case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
h₃ : j ∉ S
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [671, 1] | [683, 16] | sorry | case h₃
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
⊢ j ∉ S
case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
h₃ : j ∉ S
⊢ False | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
h₃ : j ∉ S
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [671, 1] | [683, 16] | contradiction | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
h₃ : j ∉ S
⊢ False | no goals |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [671, 1] | [683, 16] | intro h' | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
⊢ j ∉ f j | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
⊢ False |
https://github.com/avigad/mathematics_in_lean_source.git | 3fa84e6b3135a3ae41edc6ca195abf0fb1ae3ac3 | MIL/C04_Sets_and_Functions/S02_Functions.lean | Cantor | [671, 1] | [683, 16] | have : j ∉ f j := by rwa [h] at h' | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
⊢ False | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
this : j ∉ f j
⊢ False |
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