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/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | calc
|c * s n - c * a| = |c| * |s n - a| := by rw [β abs_mul, mul_sub]
_ < |c| * (Ξ΅ / |c|) := (mul_lt_mul_of_pos_left (hs n ngt) acpos)
_ = Ξ΅ := mul_div_cancel' _ (ne_of_lt acpos).symm | case h
s : β β β
a c : β
cs : ConvergesTo s a
h : Β¬c = 0
acpos : 0 < |c|
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Ξ΅cpos : 0 < Ξ΅ / |c|
Ns : β
hs : β n β₯ Ns, |s n - a| < Ξ΅ / |c|
n : β
ngt : n β₯ Ns
β’ |c * s n - c * a| < Ξ΅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s : β β β
a c : β
cs : ConvergesTo s a
h : Β¬c = 0
acpos : 0 < |c|
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Ξ΅cpos : 0 < Ξ΅ / |c|
Ns : β
hs : β n β₯ Ns, |s n - a| < Ξ΅ / |c|
n : β
ngt : n β₯ Ns
β’ |c * s n - c * a| < Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | convert convergesTo_const 0 | case pos
s : β β β
a c : β
cs : ConvergesTo s a
h : c = 0
β’ ConvergesTo (fun n => c * s n) (c * a) | case h.e'_1.h
s : β β β
a c : β
cs : ConvergesTo s a
h : c = 0
xβ : β
β’ c * s xβ = 0
case h.e'_2
s : β β β
a c : β
cs : ConvergesTo s a
h : c = 0
β’ c * a = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
s : β β β
a c : β
cs : ConvergesTo s a
h : c = 0
β’ ConvergesTo (fun n => c * s n) (c * a)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | rw [h] | case h.e'_2
s : β β β
a c : β
cs : ConvergesTo s a
h : c = 0
β’ c * a = 0 | case h.e'_2
s : β β β
a c : β
cs : ConvergesTo s a
h : c = 0
β’ 0 * a = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
s : β β β
a c : β
cs : ConvergesTo s a
h : c = 0
β’ c * a = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | ring | case h.e'_2
s : β β β
a c : β
cs : ConvergesTo s a
h : c = 0
β’ 0 * a = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
s : β β β
a c : β
cs : ConvergesTo s a
h : c = 0
β’ 0 * a = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | rw [h] | case h.e'_1.h
s : β β β
a c : β
cs : ConvergesTo s a
h : c = 0
xβ : β
β’ c * s xβ = 0 | case h.e'_1.h
s : β β β
a c : β
cs : ConvergesTo s a
h : c = 0
xβ : β
β’ 0 * s xβ = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1.h
s : β β β
a c : β
cs : ConvergesTo s a
h : c = 0
xβ : β
β’ c * s xβ = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | ring | case h.e'_1.h
s : β β β
a c : β
cs : ConvergesTo s a
h : c = 0
xβ : β
β’ 0 * s xβ = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1.h
s : β β β
a c : β
cs : ConvergesTo s a
h : c = 0
xβ : β
β’ 0 * s xβ = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | apply div_pos Ξ΅pos acpos | s : β β β
a c : β
cs : ConvergesTo s a
h : Β¬c = 0
acpos : 0 < |c|
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
β’ 0 < Ξ΅ / |c| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : β β β
a c : β
cs : ConvergesTo s a
h : Β¬c = 0
acpos : 0 < |c|
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
β’ 0 < Ξ΅ / |c|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | rw [β abs_mul, mul_sub] | s : β β β
a c : β
cs : ConvergesTo s a
h : Β¬c = 0
acpos : 0 < |c|
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Ξ΅cpos : 0 < Ξ΅ / |c|
Ns : β
hs : β n β₯ Ns, |s n - a| < Ξ΅ / |c|
n : β
ngt : n β₯ Ns
β’ |c * s n - c * a| = |c| * |s n - a| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : β β β
a c : β
cs : ConvergesTo s a
h : Β¬c = 0
acpos : 0 < |c|
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Ξ΅cpos : 0 < Ξ΅ / |c|
Ns : β
hs : β n β₯ Ns, |s n - a| < Ξ΅ / |c|
n : β
ngt : n β₯ Ns
β’ |c * s n - c * a| = |c| * |s n - a|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.exists_abs_le_of_convergesTo | [56, 1] | [66, 41] | rcases cs 1 zero_lt_one with β¨N, hβ© | s : β β β
a : β
cs : ConvergesTo s a
β’ β N b, β (n : β), N β€ n β |s n| < b | case intro
s : β β β
a : β
cs : ConvergesTo s a
N : β
h : β n β₯ N, |s n - a| < 1
β’ β N b, β (n : β), N β€ n β |s n| < b | Please generate a tactic in lean4 to solve the state.
STATE:
s : β β β
a : β
cs : ConvergesTo s a
β’ β N b, β (n : β), N β€ n β |s n| < b
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.exists_abs_le_of_convergesTo | [56, 1] | [66, 41] | use N, |a| + 1 | case intro
s : β β β
a : β
cs : ConvergesTo s a
N : β
h : β n β₯ N, |s n - a| < 1
β’ β N b, β (n : β), N β€ n β |s n| < b | case h
s : β β β
a : β
cs : ConvergesTo s a
N : β
h : β n β₯ N, |s n - a| < 1
β’ β (n : β), N β€ n β |s n| < |a| + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
s : β β β
a : β
cs : ConvergesTo s a
N : β
h : β n β₯ N, |s n - a| < 1
β’ β N b, β (n : β), N β€ n β |s n| < b
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.exists_abs_le_of_convergesTo | [56, 1] | [66, 41] | intro n ngt | case h
s : β β β
a : β
cs : ConvergesTo s a
N : β
h : β n β₯ N, |s n - a| < 1
β’ β (n : β), N β€ n β |s n| < |a| + 1 | case h
s : β β β
a : β
cs : ConvergesTo s a
N : β
h : β n β₯ N, |s n - a| < 1
n : β
ngt : N β€ n
β’ |s n| < |a| + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s : β β β
a : β
cs : ConvergesTo s a
N : β
h : β n β₯ N, |s n - a| < 1
β’ β (n : β), N β€ n β |s n| < |a| + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.exists_abs_le_of_convergesTo | [56, 1] | [66, 41] | calc
|s n| = |s n - a + a| := by
congr
abel
_ β€ |s n - a| + |a| := (abs_add _ _)
_ < |a| + 1 := by linarith [h n ngt] | case h
s : β β β
a : β
cs : ConvergesTo s a
N : β
h : β n β₯ N, |s n - a| < 1
n : β
ngt : N β€ n
β’ |s n| < |a| + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s : β β β
a : β
cs : ConvergesTo s a
N : β
h : β n β₯ N, |s n - a| < 1
n : β
ngt : N β€ n
β’ |s n| < |a| + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.exists_abs_le_of_convergesTo | [56, 1] | [66, 41] | congr | s : β β β
a : β
cs : ConvergesTo s a
N : β
h : β n β₯ N, |s n - a| < 1
n : β
ngt : N β€ n
β’ |s n| = |s n - a + a| | case e_a
s : β β β
a : β
cs : ConvergesTo s a
N : β
h : β n β₯ N, |s n - a| < 1
n : β
ngt : N β€ n
β’ s n = s n - a + a | Please generate a tactic in lean4 to solve the state.
STATE:
s : β β β
a : β
cs : ConvergesTo s a
N : β
h : β n β₯ N, |s n - a| < 1
n : β
ngt : N β€ n
β’ |s n| = |s n - a + a|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.exists_abs_le_of_convergesTo | [56, 1] | [66, 41] | abel | case e_a
s : β β β
a : β
cs : ConvergesTo s a
N : β
h : β n β₯ N, |s n - a| < 1
n : β
ngt : N β€ n
β’ s n = s n - a + a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
s : β β β
a : β
cs : ConvergesTo s a
N : β
h : β n β₯ N, |s n - a| < 1
n : β
ngt : N β€ n
β’ s n = s n - a + a
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.exists_abs_le_of_convergesTo | [56, 1] | [66, 41] | linarith [h n ngt] | s : β β β
a : β
cs : ConvergesTo s a
N : β
h : β n β₯ N, |s n - a| < 1
n : β
ngt : N β€ n
β’ |s n - a| + |a| < |a| + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : β β β
a : β
cs : ConvergesTo s a
N : β
h : β n β₯ N, |s n - a| < 1
n : β
ngt : N β€ n
β’ |s n - a| + |a| < |a| + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.aux | [68, 1] | [83, 52] | intro Ξ΅ Ξ΅pos | s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
β’ ConvergesTo (fun n => s n * t n) 0 | s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
β’ β N, β n β₯ N, |(fun n => s n * t n) n - 0| < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
β’ ConvergesTo (fun n => s n * t n) 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.aux | [68, 1] | [83, 52] | dsimp | s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
β’ β N, β n β₯ N, |(fun n => s n * t n) n - 0| < Ξ΅ | s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
β’ β N, β n β₯ N, |s n * t n - 0| < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
β’ β N, β n β₯ N, |(fun n => s n * t n) n - 0| < Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.aux | [68, 1] | [83, 52] | rcases exists_abs_le_of_convergesTo cs with β¨Nβ, B, hββ© | s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
β’ β N, β n β₯ N, |s n * t n - 0| < Ξ΅ | case intro.intro
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
β’ β N, β n β₯ N, |s n * t n - 0| < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
β’ β N, β n β₯ N, |s n * t n - 0| < Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.aux | [68, 1] | [83, 52] | have Bpos : 0 < B := lt_of_le_of_lt (abs_nonneg _) (hβ Nβ (le_refl _)) | case intro.intro
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
β’ β N, β n β₯ N, |s n * t n - 0| < Ξ΅ | case intro.intro
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
Bpos : 0 < B
β’ β N, β n β₯ N, |s n * t n - 0| < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
β’ β N, β n β₯ N, |s n * t n - 0| < Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.aux | [68, 1] | [83, 52] | have posβ : Ξ΅ / B > 0 := div_pos Ξ΅pos Bpos | case intro.intro
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
Bpos : 0 < B
β’ β N, β n β₯ N, |s n * t n - 0| < Ξ΅ | case intro.intro
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
Bpos : 0 < B
posβ : Ξ΅ / B > 0
β’ β N, β n β₯ N, |s n * t n - 0| < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
Bpos : 0 < B
β’ β N, β n β₯ N, |s n * t n - 0| < Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.aux | [68, 1] | [83, 52] | rcases ct _ posβ with β¨Nβ, hββ© | case intro.intro
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
Bpos : 0 < B
posβ : Ξ΅ / B > 0
β’ β N, β n β₯ N, |s n * t n - 0| < Ξ΅ | case intro.intro.intro
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
Bpos : 0 < B
posβ : Ξ΅ / B > 0
Nβ : β
hβ : β n β₯ Nβ, |t n - 0| < Ξ΅ / B
β’ β N, β n β₯ N, |s n * t n - 0| < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
Bpos : 0 < B
posβ : Ξ΅ / B > 0
β’ β N, β n β₯ N, |s n * t n - 0| < Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.aux | [68, 1] | [83, 52] | use max Nβ Nβ | case intro.intro.intro
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
Bpos : 0 < B
posβ : Ξ΅ / B > 0
Nβ : β
hβ : β n β₯ Nβ, |t n - 0| < Ξ΅ / B
β’ β N, β n β₯ N, |s n * t n - 0| < Ξ΅ | case h
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
Bpos : 0 < B
posβ : Ξ΅ / B > 0
Nβ : β
hβ : β n β₯ Nβ, |t n - 0| < Ξ΅ / B
β’ β n β₯ max Nβ Nβ, |s n * t n - 0| < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
Bpos : 0 < B
posβ : Ξ΅ / B > 0
Nβ : β
hβ : β n β₯ Nβ, |t n - 0| < Ξ΅ / B
β’ β N, β n β₯ N, |s n * t n - 0| < Ξ΅
TAC... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.aux | [68, 1] | [83, 52] | intro n ngt | case h
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
Bpos : 0 < B
posβ : Ξ΅ / B > 0
Nβ : β
hβ : β n β₯ Nβ, |t n - 0| < Ξ΅ / B
β’ β n β₯ max Nβ Nβ, |s n * t n - 0| < Ξ΅ | case h
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
Bpos : 0 < B
posβ : Ξ΅ / B > 0
Nβ : β
hβ : β n β₯ Nβ, |t n - 0| < Ξ΅ / B
n : β
ngt : n β₯ max Nβ Nβ
β’ |s n * t n - 0| < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
Bpos : 0 < B
posβ : Ξ΅ / B > 0
Nβ : β
hβ : β n β₯ Nβ, |t n - 0| < Ξ΅ / B
β’ β n β₯ max Nβ Nβ, |s n * t n - 0| < Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.aux | [68, 1] | [83, 52] | have ngeNβ : n β₯ Nβ := le_of_max_le_left ngt | case h
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
Bpos : 0 < B
posβ : Ξ΅ / B > 0
Nβ : β
hβ : β n β₯ Nβ, |t n - 0| < Ξ΅ / B
n : β
ngt : n β₯ max Nβ Nβ
β’ |s n * t n - 0| < Ξ΅ | case h
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
Bpos : 0 < B
posβ : Ξ΅ / B > 0
Nβ : β
hβ : β n β₯ Nβ, |t n - 0| < Ξ΅ / B
n : β
ngt : n β₯ max Nβ Nβ
ngeNβ : n β₯ Nβ
β’ |s n * t n - 0| < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
Bpos : 0 < B
posβ : Ξ΅ / B > 0
Nβ : β
hβ : β n β₯ Nβ, |t n - 0| < Ξ΅ / B
n : β
ngt : n β₯ max Nβ Nβ
β’ |s n * t n - 0| < Ξ΅
TACTIC:... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.aux | [68, 1] | [83, 52] | have ngeNβ : n β₯ Nβ := le_of_max_le_right ngt | case h
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
Bpos : 0 < B
posβ : Ξ΅ / B > 0
Nβ : β
hβ : β n β₯ Nβ, |t n - 0| < Ξ΅ / B
n : β
ngt : n β₯ max Nβ Nβ
ngeNβ : n β₯ Nβ
β’ |s n * t n - 0| < Ξ΅ | case h
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
Bpos : 0 < B
posβ : Ξ΅ / B > 0
Nβ : β
hβ : β n β₯ Nβ, |t n - 0| < Ξ΅ / B
n : β
ngt : n β₯ max Nβ Nβ
ngeNβ : n β₯ Nβ
ngeNβ : n β₯ Nβ
β’ |s n * t n - 0| < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
Bpos : 0 < B
posβ : Ξ΅ / B > 0
Nβ : β
hβ : β n β₯ Nβ, |t n - 0| < Ξ΅ / B
n : β
ngt : n β₯ max Nβ Nβ
ngeNβ : n β₯ Nβ
β’ |s n * t n -... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.aux | [68, 1] | [83, 52] | calc
|s n * t n - 0| = |s n| * |t n - 0| := by rw [sub_zero, abs_mul, sub_zero]
_ < B * (Ξ΅ / B) := (mul_lt_mul'' (hβ n ngeNβ) (hβ n ngeNβ) (abs_nonneg _) (abs_nonneg _))
_ = Ξ΅ := mul_div_cancel' _ (ne_of_lt Bpos).symm | case h
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
Bpos : 0 < B
posβ : Ξ΅ / B > 0
Nβ : β
hβ : β n β₯ Nβ, |t n - 0| < Ξ΅ / B
n : β
ngt : n β₯ max Nβ Nβ
ngeNβ : n β₯ Nβ
ngeNβ : n β₯ Nβ
β’ |s n * t n - 0| < Ξ΅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
Bpos : 0 < B
posβ : Ξ΅ / B > 0
Nβ : β
hβ : β n β₯ Nβ, |t n - 0| < Ξ΅ / B
n : β
ngt : n β₯ max Nβ Nβ
ngeNβ : n β₯ Nβ
ngeNβ : n β₯ Nβ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.aux | [68, 1] | [83, 52] | rw [sub_zero, abs_mul, sub_zero] | s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
Bpos : 0 < B
posβ : Ξ΅ / B > 0
Nβ : β
hβ : β n β₯ Nβ, |t n - 0| < Ξ΅ / B
n : β
ngt : n β₯ max Nβ Nβ
ngeNβ : n β₯ Nβ
ngeNβ : n β₯ Nβ
β’ |s n * t n - 0| = |s n| * |t n - 0| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s t : β β β
a : β
cs : ConvergesTo s a
ct : ConvergesTo t 0
Ξ΅ : β
Ξ΅pos : Ξ΅ > 0
Nβ : β
B : β
hβ : β (n : β), Nβ β€ n β |s n| < B
Bpos : 0 < B
posβ : Ξ΅ / B > 0
Nβ : β
hβ : β n β₯ Nβ, |t n - 0| < Ξ΅ / B
n : β
ngt : n β₯ max Nβ Nβ
ngeNβ : n β₯ Nβ
ngeNβ : n β₯ Nβ
β’ |s n... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [85, 1] | [95, 7] | have hβ : ConvergesTo (fun n β¦ s n * (t n + -b)) 0 := by
apply aux cs
convert convergesTo_add ct (convergesTo_const (-b))
ring | s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesTo t b
β’ ConvergesTo (fun n => s n * t n) (a * b) | s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesTo t b
hβ : ConvergesTo (fun n => s n * (t n + -b)) 0
β’ ConvergesTo (fun n => s n * t n) (a * b) | Please generate a tactic in lean4 to solve the state.
STATE:
s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesTo t b
β’ ConvergesTo (fun n => s n * t n) (a * b)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [85, 1] | [95, 7] | have := convergesTo_add hβ (convergesTo_mul_const b cs) | s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesTo t b
hβ : ConvergesTo (fun n => s n * (t n + -b)) 0
β’ ConvergesTo (fun n => s n * t n) (a * b) | s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesTo t b
hβ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
β’ ConvergesTo (fun n => s n * t n) (a * b) | Please generate a tactic in lean4 to solve the state.
STATE:
s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesTo t b
hβ : ConvergesTo (fun n => s n * (t n + -b)) 0
β’ ConvergesTo (fun n => s n * t n) (a * b)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [85, 1] | [95, 7] | convert convergesTo_add hβ (convergesTo_mul_const b cs) using 1 | s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesTo t b
hβ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
β’ ConvergesTo (fun n => s n * t n) (a * b) | case h.e'_1
s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesTo t b
hβ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
β’ (fun n => s n * t n) = fun n => s n * (t n + -b) + b * s n
case h.e'_2
s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesT... | Please generate a tactic in lean4 to solve the state.
STATE:
s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesTo t b
hβ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
β’ ConvergesTo (fun n => s n * t n) (a * b)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [85, 1] | [95, 7] | ring | case h.e'_2
s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesTo t b
hβ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
β’ a * b = 0 + b * a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesTo t b
hβ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
β’ a * b = 0 + b * a
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [85, 1] | [95, 7] | apply aux cs | s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesTo t b
β’ ConvergesTo (fun n => s n * (t n + -b)) 0 | s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesTo t b
β’ ConvergesTo (fun n => t n + -b) 0 | Please generate a tactic in lean4 to solve the state.
STATE:
s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesTo t b
β’ ConvergesTo (fun n => s n * (t n + -b)) 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [85, 1] | [95, 7] | convert convergesTo_add ct (convergesTo_const (-b)) | s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesTo t b
β’ ConvergesTo (fun n => t n + -b) 0 | case h.e'_2
s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesTo t b
β’ 0 = b + -b | Please generate a tactic in lean4 to solve the state.
STATE:
s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesTo t b
β’ ConvergesTo (fun n => t n + -b) 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [85, 1] | [95, 7] | ring | case h.e'_2
s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesTo t b
β’ 0 = b + -b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesTo t b
β’ 0 = b + -b
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [85, 1] | [95, 7] | ext | case h.e'_1
s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesTo t b
hβ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
β’ (fun n => s n * t n) = fun n => s n * (t n + -b) + b * s n | case h.e'_1.h
s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesTo t b
hβ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
xβ : β
β’ s xβ * t xβ = s xβ * (t xβ + -b) + b * s xβ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1
s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesTo t b
hβ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
β’ (fun n => s n * t n) = fun n => s n * (t n + -b) + b * s n
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [85, 1] | [95, 7] | ring | case h.e'_1.h
s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesTo t b
hβ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
xβ : β
β’ s xβ * t xβ = s xβ * (t xβ + -b) + b * s xβ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1.h
s t : β β β
a b : β
cs : ConvergesTo s a
ct : ConvergesTo t b
hβ : ConvergesTo (fun n => s n * (t n + -b)) 0
this : ConvergesTo (fun n => s n * (t n + -b) + b * s n) (0 + b * a)
xβ : β
β’ s xβ * t xβ = s xβ * (t xβ + -b) + b * s xβ
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | by_contra abne | s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
β’ a = b | s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
β’ a = b
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | let Ξ΅ := |a - b| / 2 | s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
β’ False | s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | have Ξ΅pos : Ξ΅ > 0 := by
change |a - b| / 2 > 0
linarith | s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
β’ False | s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | rcases sa Ξ΅ Ξ΅pos with β¨Na, hNaβ© | s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
β’ False | case intro
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | rcases sb Ξ΅ Ξ΅pos with β¨Nb, hNbβ© | case intro
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
β’ False | case intro.intro
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | let N := max Na Nb | case intro.intro
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
β’ False | case intro.intro
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | have absa : |s N - a| < Ξ΅ := by
apply hNa
apply le_max_left | case intro.intro
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
β’ False | case intro.intro
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | have absb : |s N - b| < Ξ΅ := by
apply hNb
apply le_max_right | case intro.intro
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
β’ False | case intro.intro
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
absb : |s N - b| < Ξ΅
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
β’... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | have : |a - b| < |a - b| | case intro.intro
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
absb : |s N - b| < Ξ΅
β’ False | case this
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
absb : |s N - b| < Ξ΅
β’ |a - b| < |a - b|
case intro.intro
s : β β β
... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
a... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | calc
|a - b| = |(-(s N - a)) + (s N - b)| := by
congr
ring
_ β€ |(-(s N - a))| + |s N - b| := (abs_add _ _)
_ = |s N - a| + |s N - b| := by rw [abs_neg]
_ < Ξ΅ + Ξ΅ := (add_lt_add absa absb)
_ = |a - b| := by norm_num | case this
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
absb : |s N - b| < Ξ΅
β’ |a - b| < |a - b|
case intro.intro
s : β β β
... | case intro.intro
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
thisβ : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
absb : |s N - b| < Ξ΅
this : |a - b| < |a - b|
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case this
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
absb : |... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | exact lt_irrefl _ this | case intro.intro
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
thisβ : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
absb : |s N - b| < Ξ΅
this : |a - b| < |a - b|
β’ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
thisβ : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | apply lt_of_le_of_ne | s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
β’ |a - b| > 0 | case a
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
β’ 0 β€ |a - b|
case a
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
β’ 0 β |a - b| | Please generate a tactic in lean4 to solve the state.
STATE:
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
β’ |a - b| > 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | intro h'' | case a
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
β’ 0 β |a - b| | case a
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
h'' : 0 = |a - b|
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case a
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
β’ 0 β |a - b|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | apply abne | case a
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
h'' : 0 = |a - b|
β’ False | case a
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
h'' : 0 = |a - b|
β’ a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case a
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
h'' : 0 = |a - b|
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | apply eq_of_abs_sub_eq_zero h''.symm | case a
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
h'' : 0 = |a - b|
β’ a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
h'' : 0 = |a - b|
β’ a = b
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | apply abs_nonneg | case a
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
β’ 0 β€ |a - b| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
β’ 0 β€ |a - b|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | change |a - b| / 2 > 0 | s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
β’ Ξ΅ > 0 | s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
β’ |a - b| / 2 > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
β’ Ξ΅ > 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | linarith | s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
β’ |a - b| / 2 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
β’ |a - b| / 2 > 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | apply hNa | s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
β’ |s N - a| < Ξ΅ | case a
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
β’ N β₯ Na | Please generate a tactic in lean4 to solve the state.
STATE:
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
β’ |s N - a| < Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | apply le_max_left | case a
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
β’ N β₯ Na | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
β’ N β₯ Na
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | apply hNb | s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
β’ |s N - b| < Ξ΅ | case a
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
β’ N β₯ Nb | Please generate a tactic in lean4 to solve the state.
STATE:
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
β’ |s N - b| < Ξ΅
TA... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | apply le_max_right | case a
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
β’ N β₯ Nb | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
β’ N β₯ Nb
TA... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | congr | s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
absb : |s N - b| < Ξ΅
β’ |a - b| = |-(s N - a) + (s N - b)| | case e_a
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
absb : |s N - b| < Ξ΅
β’ a - b = -(s N - a) + (s N - b) | Please generate a tactic in lean4 to solve the state.
STATE:
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
absb : |s N - b| <... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | ring | case e_a
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
absb : |s N - b| < Ξ΅
β’ a - b = -(s N - a) + (s N - b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
absb : |s... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | rw [abs_neg] | s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
absb : |s N - b| < Ξ΅
β’ |(-(s N - a))| + |s N - b| = |s N - a| + |s N - b| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
absb : |s N - b| <... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [97, 1] | [130, 25] | norm_num | s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
absb : |s N - b| < Ξ΅
β’ Ξ΅ + Ξ΅ = |a - b| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s : β β β
a b : β
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : Β¬a = b
this : |a - b| > 0
Ξ΅ : β := |a - b| / 2
Ξ΅pos : Ξ΅ > 0
Na : β
hNa : β n β₯ Na, |s n - a| < Ξ΅
Nb : β
hNb : β n β₯ Nb, |s n - b| < Ξ΅
N : β := max Na Nb
absa : |s N - a| < Ξ΅
absb : |s N - b| <... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S5_Topology/S02_Metric_Spaces.lean | cauchySeq_of_le_geometric_two' | [146, 1] | [160, 19] | rw [Metric.cauchySeq_iff'] | X : Type u_1
instβ : MetricSpace X
a b c : X
r : β
u : β β X
hu : β (n : β), dist (u n) (u (n + 1)) β€ (1 / 2) ^ n
β’ CauchySeq u | X : Type u_1
instβ : MetricSpace X
a b c : X
r : β
u : β β X
hu : β (n : β), dist (u n) (u (n + 1)) β€ (1 / 2) ^ n
β’ β Ξ΅ > 0, β N, β n β₯ N, dist (u n) (u N) < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
instβ : MetricSpace X
a b c : X
r : β
u : β β X
hu : β (n : β), dist (u n) (u (n + 1)) β€ (1 / 2) ^ n
β’ CauchySeq u
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S5_Topology/S02_Metric_Spaces.lean | cauchySeq_of_le_geometric_two' | [146, 1] | [160, 19] | intro Ξ΅ Ξ΅_pos | X : Type u_1
instβ : MetricSpace X
a b c : X
r : β
u : β β X
hu : β (n : β), dist (u n) (u (n + 1)) β€ (1 / 2) ^ n
β’ β Ξ΅ > 0, β N, β n β₯ N, dist (u n) (u N) < Ξ΅ | X : Type u_1
instβ : MetricSpace X
a b c : X
r : β
u : β β X
hu : β (n : β), dist (u n) (u (n + 1)) β€ (1 / 2) ^ n
Ξ΅ : β
Ξ΅_pos : Ξ΅ > 0
β’ β N, β n β₯ N, dist (u n) (u N) < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
instβ : MetricSpace X
a b c : X
r : β
u : β β X
hu : β (n : β), dist (u n) (u (n + 1)) β€ (1 / 2) ^ n
β’ β Ξ΅ > 0, β N, β n β₯ N, dist (u n) (u N) < Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S5_Topology/S02_Metric_Spaces.lean | cauchySeq_of_le_geometric_two' | [146, 1] | [160, 19] | obtain β¨N, hNβ© : β N : β, 1 / 2 ^ N * 2 < Ξ΅ := by sorry | X : Type u_1
instβ : MetricSpace X
a b c : X
r : β
u : β β X
hu : β (n : β), dist (u n) (u (n + 1)) β€ (1 / 2) ^ n
Ξ΅ : β
Ξ΅_pos : Ξ΅ > 0
β’ β N, β n β₯ N, dist (u n) (u N) < Ξ΅ | case intro
X : Type u_1
instβ : MetricSpace X
a b c : X
r : β
u : β β X
hu : β (n : β), dist (u n) (u (n + 1)) β€ (1 / 2) ^ n
Ξ΅ : β
Ξ΅_pos : Ξ΅ > 0
N : β
hN : 1 / 2 ^ N * 2 < Ξ΅
β’ β N, β n β₯ N, dist (u n) (u N) < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
instβ : MetricSpace X
a b c : X
r : β
u : β β X
hu : β (n : β), dist (u n) (u (n + 1)) β€ (1 / 2) ^ n
Ξ΅ : β
Ξ΅_pos : Ξ΅ > 0
β’ β N, β n β₯ N, dist (u n) (u N) < Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S5_Topology/S02_Metric_Spaces.lean | cauchySeq_of_le_geometric_two' | [146, 1] | [160, 19] | use N | case intro
X : Type u_1
instβ : MetricSpace X
a b c : X
r : β
u : β β X
hu : β (n : β), dist (u n) (u (n + 1)) β€ (1 / 2) ^ n
Ξ΅ : β
Ξ΅_pos : Ξ΅ > 0
N : β
hN : 1 / 2 ^ N * 2 < Ξ΅
β’ β N, β n β₯ N, dist (u n) (u N) < Ξ΅ | case h
X : Type u_1
instβ : MetricSpace X
a b c : X
r : β
u : β β X
hu : β (n : β), dist (u n) (u (n + 1)) β€ (1 / 2) ^ n
Ξ΅ : β
Ξ΅_pos : Ξ΅ > 0
N : β
hN : 1 / 2 ^ N * 2 < Ξ΅
β’ β n β₯ N, dist (u n) (u N) < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type u_1
instβ : MetricSpace X
a b c : X
r : β
u : β β X
hu : β (n : β), dist (u n) (u (n + 1)) β€ (1 / 2) ^ n
Ξ΅ : β
Ξ΅_pos : Ξ΅ > 0
N : β
hN : 1 / 2 ^ N * 2 < Ξ΅
β’ β N, β n β₯ N, dist (u n) (u N) < Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S5_Topology/S02_Metric_Spaces.lean | cauchySeq_of_le_geometric_two' | [146, 1] | [160, 19] | intro n hn | case h
X : Type u_1
instβ : MetricSpace X
a b c : X
r : β
u : β β X
hu : β (n : β), dist (u n) (u (n + 1)) β€ (1 / 2) ^ n
Ξ΅ : β
Ξ΅_pos : Ξ΅ > 0
N : β
hN : 1 / 2 ^ N * 2 < Ξ΅
β’ β n β₯ N, dist (u n) (u N) < Ξ΅ | case h
X : Type u_1
instβ : MetricSpace X
a b c : X
r : β
u : β β X
hu : β (n : β), dist (u n) (u (n + 1)) β€ (1 / 2) ^ n
Ξ΅ : β
Ξ΅_pos : Ξ΅ > 0
N : β
hN : 1 / 2 ^ N * 2 < Ξ΅
n : β
hn : n β₯ N
β’ dist (u n) (u N) < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type u_1
instβ : MetricSpace X
a b c : X
r : β
u : β β X
hu : β (n : β), dist (u n) (u (n + 1)) β€ (1 / 2) ^ n
Ξ΅ : β
Ξ΅_pos : Ξ΅ > 0
N : β
hN : 1 / 2 ^ N * 2 < Ξ΅
β’ β n β₯ N, dist (u n) (u N) < Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S5_Topology/S02_Metric_Spaces.lean | cauchySeq_of_le_geometric_two' | [146, 1] | [160, 19] | obtain β¨k, rfl : n = N + kβ© := le_iff_exists_add.mp hn | case h
X : Type u_1
instβ : MetricSpace X
a b c : X
r : β
u : β β X
hu : β (n : β), dist (u n) (u (n + 1)) β€ (1 / 2) ^ n
Ξ΅ : β
Ξ΅_pos : Ξ΅ > 0
N : β
hN : 1 / 2 ^ N * 2 < Ξ΅
n : β
hn : n β₯ N
β’ dist (u n) (u N) < Ξ΅ | case h.intro
X : Type u_1
instβ : MetricSpace X
a b c : X
r : β
u : β β X
hu : β (n : β), dist (u n) (u (n + 1)) β€ (1 / 2) ^ n
Ξ΅ : β
Ξ΅_pos : Ξ΅ > 0
N : β
hN : 1 / 2 ^ N * 2 < Ξ΅
k : β
hn : N + k β₯ N
β’ dist (u (N + k)) (u N) < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type u_1
instβ : MetricSpace X
a b c : X
r : β
u : β β X
hu : β (n : β), dist (u n) (u (n + 1)) β€ (1 / 2) ^ n
Ξ΅ : β
Ξ΅_pos : Ξ΅ > 0
N : β
hN : 1 / 2 ^ N * 2 < Ξ΅
n : β
hn : n β₯ N
β’ dist (u n) (u N) < Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S5_Topology/S02_Metric_Spaces.lean | cauchySeq_of_le_geometric_two' | [146, 1] | [160, 19] | calc
dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := sorry
_ β€ β i in range k, dist (u (N + i)) (u (N + (i + 1))) := sorry
_ β€ β i in range k, (1 / 2 : β) ^ (N + i) := sorry
_ = 1 / 2 ^ N * β i in range k, (1 / 2 : β) ^ i := sorry
_ β€ 1 / 2 ^ N * 2 := sorry
_ < Ξ΅ := sorry | case h.intro
X : Type u_1
instβ : MetricSpace X
a b c : X
r : β
u : β β X
hu : β (n : β), dist (u n) (u (n + 1)) β€ (1 / 2) ^ n
Ξ΅ : β
Ξ΅_pos : Ξ΅ > 0
N : β
hN : 1 / 2 ^ N * 2 < Ξ΅
k : β
hn : N + k β₯ N
β’ dist (u (N + k)) (u N) < Ξ΅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro
X : Type u_1
instβ : MetricSpace X
a b c : X
r : β
u : β β X
hu : β (n : β), dist (u n) (u (n + 1)) β€ (1 / 2) ^ n
Ξ΅ : β
Ξ΅_pos : Ξ΅ > 0
N : β
hN : 1 / 2 ^ N * 2 < Ξ΅
k : β
hn : N + k β₯ N
β’ dist (u (N + k)) (u N) < Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S5_Topology/S02_Metric_Spaces.lean | cauchySeq_of_le_geometric_two' | [146, 1] | [160, 19] | sorry | X : Type u_1
instβ : MetricSpace X
a b c : X
r : β
u : β β X
hu : β (n : β), dist (u n) (u (n + 1)) β€ (1 / 2) ^ n
Ξ΅ : β
Ξ΅_pos : Ξ΅ > 0
β’ β N, 1 / 2 ^ N * 2 < Ξ΅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
instβ : MetricSpace X
a b c : X
r : β
u : β β X
hu : β (n : β), dist (u n) (u (n + 1)) β€ (1 / 2) ^ n
Ξ΅ : β
Ξ΅_pos : Ξ΅ > 0
β’ β N, 1 / 2 ^ N * 2 < Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma3 | [36, 1] | [39, 8] | intro x y Ξ΅ epos ele1 xlt ylt | β’ β {x y Ξ΅ : β}, 0 < Ξ΅ β Ξ΅ β€ 1 β |x| < Ξ΅ β |y| < Ξ΅ β |x * y| < Ξ΅ | x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |x * y| < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
β’ β {x y Ξ΅ : β}, 0 < Ξ΅ β Ξ΅ β€ 1 β |x| < Ξ΅ β |y| < Ξ΅ β |x * y| < Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma3 | [36, 1] | [39, 8] | sorry | x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |x * y| < Ξ΅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |x * y| < Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [41, 1] | [48, 19] | intro x y Ξ΅ epos ele1 xlt ylt | β’ β {x y Ξ΅ : β}, 0 < Ξ΅ β Ξ΅ β€ 1 β |x| < Ξ΅ β |y| < Ξ΅ β |x * y| < Ξ΅ | x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |x * y| < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
β’ β {x y Ξ΅ : β}, 0 < Ξ΅ β Ξ΅ β€ 1 β |x| < Ξ΅ β |y| < Ξ΅ β |x * y| < Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [41, 1] | [48, 19] | calc
|x * y| = |x| * |y| := sorry
_ β€ |x| * Ξ΅ := sorry
_ < 1 * Ξ΅ := sorry
_ = Ξ΅ := sorry | x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |x * y| < Ξ΅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |x * y| < Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S01_Implication_and_the_Universal_Quantifier.lean | C03S01.Subset.trans | [146, 1] | [147, 8] | sorry | Ξ± : Type u_1
r s t : Set Ξ±
β’ r β s β s β t β r β t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type u_1
r s t : Set Ξ±
β’ r β s β s β t β r β t
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [6, 1] | [13, 30] | intro x y Ξ΅ epos ele1 xlt ylt | β’ β {x y Ξ΅ : β}, 0 < Ξ΅ β Ξ΅ β€ 1 β |x| < Ξ΅ β |y| < Ξ΅ β |x * y| < Ξ΅ | x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |x * y| < Ξ΅ | Please generate a tactic in lean4 to solve the state.
STATE:
β’ β {x y Ξ΅ : β}, 0 < Ξ΅ β Ξ΅ β€ 1 β |x| < Ξ΅ β |y| < Ξ΅ β |x * y| < Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [6, 1] | [13, 30] | calc
|x * y| = |x| * |y| := by apply abs_mul
_ β€ |x| * Ξ΅ := by apply mul_le_mul; linarith; linarith; apply abs_nonneg; apply abs_nonneg;
_ < 1 * Ξ΅ := by rw [mul_lt_mul_right epos]; linarith
_ = Ξ΅ := by apply one_mul | x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |x * y| < Ξ΅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |x * y| < Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [6, 1] | [13, 30] | apply abs_mul | x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |x * y| = |x| * |y| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |x * y| = |x| * |y|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [6, 1] | [13, 30] | apply mul_le_mul | x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |x| * |y| β€ |x| * Ξ΅ | case hβ
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |x| β€ |x|
case hβ
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |y| β€ Ξ΅
case c0
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ 0 β€ |y|
case b0
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| <... | Please generate a tactic in lean4 to solve the state.
STATE:
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |x| * |y| β€ |x| * Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [6, 1] | [13, 30] | linarith | case hβ
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |x| β€ |x|
case hβ
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |y| β€ Ξ΅
case c0
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ 0 β€ |y|
case b0
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| <... | case hβ
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |y| β€ Ξ΅
case c0
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ 0 β€ |y|
case b0
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ 0 β€ |x| | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |x| β€ |x|
case hβ
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |y| β€ Ξ΅
case c0
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ 0 β€ |y|
case b0
... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [6, 1] | [13, 30] | linarith | case hβ
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |y| β€ Ξ΅
case c0
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ 0 β€ |y|
case b0
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ 0 β€ |x| | case c0
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ 0 β€ |y|
case b0
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ 0 β€ |x| | Please generate a tactic in lean4 to solve the state.
STATE:
case hβ
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |y| β€ Ξ΅
case c0
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ 0 β€ |y|
case b0
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ 0 β€ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [6, 1] | [13, 30] | apply abs_nonneg | case c0
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ 0 β€ |y|
case b0
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ 0 β€ |x| | case b0
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ 0 β€ |x| | Please generate a tactic in lean4 to solve the state.
STATE:
case c0
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ 0 β€ |y|
case b0
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ 0 β€ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [6, 1] | [13, 30] | apply abs_nonneg | case b0
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ 0 β€ |x| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case b0
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ 0 β€ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [6, 1] | [13, 30] | rw [mul_lt_mul_right epos] | x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |x| * Ξ΅ < 1 * Ξ΅ | x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |x| < 1 | Please generate a tactic in lean4 to solve the state.
STATE:
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |x| * Ξ΅ < 1 * Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [6, 1] | [13, 30] | linarith | x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |x| < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ |x| < 1
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean | C03S01.my_lemma4 | [6, 1] | [13, 30] | apply one_mul | x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ 1 * Ξ΅ = Ξ΅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y Ξ΅ : β
epos : 0 < Ξ΅
ele1 : Ξ΅ β€ 1
xlt : |x| < Ξ΅
ylt : |y| < Ξ΅
β’ 1 * Ξ΅ = Ξ΅
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S04_Conjunction_and_Iff.lean | C03S04.aux | [30, 1] | [32, 17] | linarith [pow_two_nonneg x, pow_two_nonneg y] | x y : β
h : x ^ 2 + y ^ 2 = 0
β’ x ^ 2 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : β
h : x ^ 2 + y ^ 2 = 0
β’ x ^ 2 = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S04_Conjunction_and_Iff.lean | C03S04.not_monotone_iff | [44, 1] | [47, 6] | rw [Monotone] | f : β β β
β’ Β¬Monotone f β β x y, x β€ y β§ f x > f y | f : β β β
β’ (Β¬β β¦a b : ββ¦, a β€ b β f a β€ f b) β β x y, x β€ y β§ f x > f y | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
β’ Β¬Monotone f β β x y, x β€ y β§ f x > f y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S04_Conjunction_and_Iff.lean | C03S04.not_monotone_iff | [44, 1] | [47, 6] | push_neg | f : β β β
β’ (Β¬β β¦a b : ββ¦, a β€ b β f a β€ f b) β β x y, x β€ y β§ f x > f y | f : β β β
β’ (β a b, a β€ b β§ f b < f a) β β x y, x β€ y β§ f x > f y | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
β’ (Β¬β β¦a b : ββ¦, a β€ b β f a β€ f b) β β x y, x β€ y β§ f x > f y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S04_Conjunction_and_Iff.lean | C03S04.not_monotone_iff | [44, 1] | [47, 6] | rfl | f : β β β
β’ (β a b, a β€ b β§ f b < f a) β β x y, x β€ y β§ f x > f y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
β’ (β a b, a β€ b β§ f b < f a) β β x y, x β€ y β§ f x > f y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.le_abs_self | [12, 1] | [16, 13] | rcases le_or_gt 0 x with h | h | xβ y x : β
β’ x β€ |x| | case inl
xβ y x : β
h : 0 β€ x
β’ x β€ |x|
case inr
xβ y x : β
h : 0 > x
β’ x β€ |x| | Please generate a tactic in lean4 to solve the state.
STATE:
xβ y x : β
β’ x β€ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.le_abs_self | [12, 1] | [16, 13] | . rw [abs_of_neg h]
linarith | case inr
xβ y x : β
h : 0 > x
β’ x β€ |x| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
xβ y x : β
h : 0 > x
β’ x β€ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.le_abs_self | [12, 1] | [16, 13] | rw [abs_of_nonneg h] | case inl
xβ y x : β
h : 0 β€ x
β’ x β€ |x| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
xβ y x : β
h : 0 β€ x
β’ x β€ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.le_abs_self | [12, 1] | [16, 13] | rw [abs_of_neg h] | case inr
xβ y x : β
h : 0 > x
β’ x β€ |x| | case inr
xβ y x : β
h : 0 > x
β’ x β€ -x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
xβ y x : β
h : 0 > x
β’ x β€ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.le_abs_self | [12, 1] | [16, 13] | linarith | case inr
xβ y x : β
h : 0 > x
β’ x β€ -x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
xβ y x : β
h : 0 > x
β’ x β€ -x
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.neg_le_abs_self | [18, 1] | [22, 22] | rcases le_or_gt 0 x with h | h | xβ y x : β
β’ -x β€ |x| | case inl
xβ y x : β
h : 0 β€ x
β’ -x β€ |x|
case inr
xβ y x : β
h : 0 > x
β’ -x β€ |x| | Please generate a tactic in lean4 to solve the state.
STATE:
xβ y x : β
β’ -x β€ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.neg_le_abs_self | [18, 1] | [22, 22] | . rw [abs_of_neg h] | case inr
xβ y x : β
h : 0 > x
β’ -x β€ |x| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
xβ y x : β
h : 0 > x
β’ -x β€ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.neg_le_abs_self | [18, 1] | [22, 22] | rw [abs_of_nonneg h] | case inl
xβ y x : β
h : 0 β€ x
β’ -x β€ |x| | case inl
xβ y x : β
h : 0 β€ x
β’ -x β€ x | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
xβ y x : β
h : 0 β€ x
β’ -x β€ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.neg_le_abs_self | [18, 1] | [22, 22] | linarith | case inl
xβ y x : β
h : 0 β€ x
β’ -x β€ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
xβ y x : β
h : 0 β€ x
β’ -x β€ x
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.neg_le_abs_self | [18, 1] | [22, 22] | rw [abs_of_neg h] | case inr
xβ y x : β
h : 0 > x
β’ -x β€ |x| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
xβ y x : β
h : 0 > x
β’ -x β€ |x|
TACTIC:
|
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