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_S05_Disjunction.lean | C03S05.MyAbs.abs_add | [24, 1] | [29, 52] | rcases le_or_gt 0 (x + y) with h | h | x✝ y✝ x y : ℝ
⊢ |x + y| ≤ |x| + |y| | case inl
x✝ y✝ x y : ℝ
h : 0 ≤ x + y
⊢ |x + y| ≤ |x| + |y|
case inr
x✝ y✝ x y : ℝ
h : 0 > x + y
⊢ |x + y| ≤ |x| + |y| | Please generate a tactic in lean4 to solve the state.
STATE:
x✝ y✝ x y : ℝ
⊢ |x + y| ≤ |x| + |y|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_add | [24, 1] | [29, 52] | . rw [abs_of_neg h]
linarith [neg_le_abs_self x, neg_le_abs_self y] | case inr
x✝ y✝ x y : ℝ
h : 0 > x + y
⊢ |x + y| ≤ |x| + |y| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x✝ y✝ x y : ℝ
h : 0 > x + y
⊢ |x + y| ≤ |x| + |y|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_add | [24, 1] | [29, 52] | rw [abs_of_nonneg h] | case inl
x✝ y✝ x y : ℝ
h : 0 ≤ x + y
⊢ |x + y| ≤ |x| + |y| | case inl
x✝ y✝ x y : ℝ
h : 0 ≤ x + y
⊢ x + y ≤ |x| + |y| | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
x✝ y✝ x y : ℝ
h : 0 ≤ x + y
⊢ |x + y| ≤ |x| + |y|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_add | [24, 1] | [29, 52] | linarith [le_abs_self x, le_abs_self y] | case inl
x✝ y✝ x y : ℝ
h : 0 ≤ x + y
⊢ x + y ≤ |x| + |y| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
x✝ y✝ x y : ℝ
h : 0 ≤ x + y
⊢ x + y ≤ |x| + |y|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_add | [24, 1] | [29, 52] | rw [abs_of_neg h] | case inr
x✝ y✝ x y : ℝ
h : 0 > x + y
⊢ |x + y| ≤ |x| + |y| | case inr
x✝ y✝ x y : ℝ
h : 0 > x + y
⊢ -(x + y) ≤ |x| + |y| | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x✝ y✝ x y : ℝ
h : 0 > x + y
⊢ |x + y| ≤ |x| + |y|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_add | [24, 1] | [29, 52] | linarith [neg_le_abs_self x, neg_le_abs_self y] | case inr
x✝ y✝ x y : ℝ
h : 0 > x + y
⊢ -(x + y) ≤ |x| + |y| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x✝ y✝ x y : ℝ
h : 0 > x + y
⊢ -(x + y) ≤ |x| + |y|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | rcases le_or_gt 0 y with h | h | x y : ℝ
⊢ x < |y| ↔ x < y ∨ x < -y | case inl
x y : ℝ
h : 0 ≤ y
⊢ x < |y| ↔ x < y ∨ x < -y
case inr
x y : ℝ
h : 0 > y
⊢ x < |y| ↔ x < y ∨ x < -y | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
⊢ x < |y| ↔ x < y ∨ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | rw [abs_of_neg h] | case inr
x y : ℝ
h : 0 > y
⊢ x < |y| ↔ x < y ∨ x < -y | case inr
x y : ℝ
h : 0 > y
⊢ x < -y ↔ x < y ∨ x < -y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x y : ℝ
h : 0 > y
⊢ x < |y| ↔ x < y ∨ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | constructor | case inr
x y : ℝ
h : 0 > y
⊢ x < -y ↔ x < y ∨ x < -y | case inr.mp
x y : ℝ
h : 0 > y
⊢ x < -y → x < y ∨ x < -y
case inr.mpr
x y : ℝ
h : 0 > y
⊢ x < y ∨ x < -y → x < -y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x y : ℝ
h : 0 > y
⊢ x < -y ↔ x < y ∨ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | rw [abs_of_nonneg h] | case inl
x y : ℝ
h : 0 ≤ y
⊢ x < |y| ↔ x < y ∨ x < -y | case inl
x y : ℝ
h : 0 ≤ y
⊢ x < y ↔ x < y ∨ x < -y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
x y : ℝ
h : 0 ≤ y
⊢ x < |y| ↔ x < y ∨ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | constructor | case inl
x y : ℝ
h : 0 ≤ y
⊢ x < y ↔ x < y ∨ x < -y | case inl.mp
x y : ℝ
h : 0 ≤ y
⊢ x < y → x < y ∨ x < -y
case inl.mpr
x y : ℝ
h : 0 ≤ y
⊢ x < y ∨ x < -y → x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
x y : ℝ
h : 0 ≤ y
⊢ x < y ↔ x < y ∨ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | intro h' | case inl.mp
x y : ℝ
h : 0 ≤ y
⊢ x < y → x < y ∨ x < -y | case inl.mp
x y : ℝ
h : 0 ≤ y
h' : x < y
⊢ x < y ∨ x < -y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mp
x y : ℝ
h : 0 ≤ y
⊢ x < y → x < y ∨ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | left | case inl.mp
x y : ℝ
h : 0 ≤ y
h' : x < y
⊢ x < y ∨ x < -y | case inl.mp.h
x y : ℝ
h : 0 ≤ y
h' : x < y
⊢ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mp
x y : ℝ
h : 0 ≤ y
h' : x < y
⊢ x < y ∨ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | exact h' | case inl.mp.h
x y : ℝ
h : 0 ≤ y
h' : x < y
⊢ x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mp.h
x y : ℝ
h : 0 ≤ y
h' : x < y
⊢ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | intro h' | case inl.mpr
x y : ℝ
h : 0 ≤ y
⊢ x < y ∨ x < -y → x < y | case inl.mpr
x y : ℝ
h : 0 ≤ y
h' : x < y ∨ x < -y
⊢ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mpr
x y : ℝ
h : 0 ≤ y
⊢ x < y ∨ x < -y → x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | rcases h' with h' | h' | case inl.mpr
x y : ℝ
h : 0 ≤ y
h' : x < y ∨ x < -y
⊢ x < y | case inl.mpr.inl
x y : ℝ
h : 0 ≤ y
h' : x < y
⊢ x < y
case inl.mpr.inr
x y : ℝ
h : 0 ≤ y
h' : x < -y
⊢ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mpr
x y : ℝ
h : 0 ≤ y
h' : x < y ∨ x < -y
⊢ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | . linarith | case inl.mpr.inr
x y : ℝ
h : 0 ≤ y
h' : x < -y
⊢ x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mpr.inr
x y : ℝ
h : 0 ≤ y
h' : x < -y
⊢ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | exact h' | case inl.mpr.inl
x y : ℝ
h : 0 ≤ y
h' : x < y
⊢ x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mpr.inl
x y : ℝ
h : 0 ≤ y
h' : x < y
⊢ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | linarith | case inl.mpr.inr
x y : ℝ
h : 0 ≤ y
h' : x < -y
⊢ x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mpr.inr
x y : ℝ
h : 0 ≤ y
h' : x < -y
⊢ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | intro h' | case inr.mp
x y : ℝ
h : 0 > y
⊢ x < -y → x < y ∨ x < -y | case inr.mp
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < y ∨ x < -y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mp
x y : ℝ
h : 0 > y
⊢ x < -y → x < y ∨ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | right | case inr.mp
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < y ∨ x < -y | case inr.mp.h
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < -y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mp
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < y ∨ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | exact h' | case inr.mp.h
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < -y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mp.h
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | intro h' | case inr.mpr
x y : ℝ
h : 0 > y
⊢ x < y ∨ x < -y → x < -y | case inr.mpr
x y : ℝ
h : 0 > y
h' : x < y ∨ x < -y
⊢ x < -y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mpr
x y : ℝ
h : 0 > y
⊢ x < y ∨ x < -y → x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | rcases h' with h' | h' | case inr.mpr
x y : ℝ
h : 0 > y
h' : x < y ∨ x < -y
⊢ x < -y | case inr.mpr.inl
x y : ℝ
h : 0 > y
h' : x < y
⊢ x < -y
case inr.mpr.inr
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < -y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mpr
x y : ℝ
h : 0 > y
h' : x < y ∨ x < -y
⊢ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | . exact h' | case inr.mpr.inr
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < -y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mpr.inr
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | linarith | case inr.mpr.inl
x y : ℝ
h : 0 > y
h' : x < y
⊢ x < -y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mpr.inl
x y : ℝ
h : 0 > y
h' : x < y
⊢ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | exact h' | case inr.mpr.inr
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < -y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mpr.inr
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | rcases le_or_gt 0 x with h | h | x y : ℝ
⊢ |x| < y ↔ -y < x ∧ x < y | case inl
x y : ℝ
h : 0 ≤ x
⊢ |x| < y ↔ -y < x ∧ x < y
case inr
x y : ℝ
h : 0 > x
⊢ |x| < y ↔ -y < x ∧ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
⊢ |x| < y ↔ -y < x ∧ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | rw [abs_of_nonneg h] | case inl
x y : ℝ
h : 0 ≤ x
⊢ |x| < y ↔ -y < x ∧ x < y | case inl
x y : ℝ
h : 0 ≤ x
⊢ x < y ↔ -y < x ∧ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
x y : ℝ
h : 0 ≤ x
⊢ |x| < y ↔ -y < x ∧ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | constructor | case inl
x y : ℝ
h : 0 ≤ x
⊢ x < y ↔ -y < x ∧ x < y | case inl.mp
x y : ℝ
h : 0 ≤ x
⊢ x < y → -y < x ∧ x < y
case inl.mpr
x y : ℝ
h : 0 ≤ x
⊢ -y < x ∧ x < y → x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
x y : ℝ
h : 0 ≤ x
⊢ x < y ↔ -y < x ∧ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | . intro h'
rcases h' with ⟨h1, h2⟩
exact h2 | case inl.mpr
x y : ℝ
h : 0 ≤ x
⊢ -y < x ∧ x < y → x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mpr
x y : ℝ
h : 0 ≤ x
⊢ -y < x ∧ x < y → x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | intro h' | case inl.mp
x y : ℝ
h : 0 ≤ x
⊢ x < y → -y < x ∧ x < y | case inl.mp
x y : ℝ
h : 0 ≤ x
h' : x < y
⊢ -y < x ∧ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mp
x y : ℝ
h : 0 ≤ x
⊢ x < y → -y < x ∧ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | constructor | case inl.mp
x y : ℝ
h : 0 ≤ x
h' : x < y
⊢ -y < x ∧ x < y | case inl.mp.left
x y : ℝ
h : 0 ≤ x
h' : x < y
⊢ -y < x
case inl.mp.right
x y : ℝ
h : 0 ≤ x
h' : x < y
⊢ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mp
x y : ℝ
h : 0 ≤ x
h' : x < y
⊢ -y < x ∧ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | exact h' | case inl.mp.right
x y : ℝ
h : 0 ≤ x
h' : x < y
⊢ x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mp.right
x y : ℝ
h : 0 ≤ x
h' : x < y
⊢ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | linarith | case inl.mp.left
x y : ℝ
h : 0 ≤ x
h' : x < y
⊢ -y < x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mp.left
x y : ℝ
h : 0 ≤ x
h' : x < y
⊢ -y < x
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | intro h' | case inl.mpr
x y : ℝ
h : 0 ≤ x
⊢ -y < x ∧ x < y → x < y | case inl.mpr
x y : ℝ
h : 0 ≤ x
h' : -y < x ∧ x < y
⊢ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mpr
x y : ℝ
h : 0 ≤ x
⊢ -y < x ∧ x < y → x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | rcases h' with ⟨h1, h2⟩ | case inl.mpr
x y : ℝ
h : 0 ≤ x
h' : -y < x ∧ x < y
⊢ x < y | case inl.mpr.intro
x y : ℝ
h : 0 ≤ x
h1 : -y < x
h2 : x < y
⊢ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mpr
x y : ℝ
h : 0 ≤ x
h' : -y < x ∧ x < y
⊢ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | exact h2 | case inl.mpr.intro
x y : ℝ
h : 0 ≤ x
h1 : -y < x
h2 : x < y
⊢ x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mpr.intro
x y : ℝ
h : 0 ≤ x
h1 : -y < x
h2 : x < y
⊢ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | rw [abs_of_neg h] | case inr
x y : ℝ
h : 0 > x
⊢ |x| < y ↔ -y < x ∧ x < y | case inr
x y : ℝ
h : 0 > x
⊢ -x < y ↔ -y < x ∧ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x y : ℝ
h : 0 > x
⊢ |x| < y ↔ -y < x ∧ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | constructor | case inr
x y : ℝ
h : 0 > x
⊢ -x < y ↔ -y < x ∧ x < y | case inr.mp
x y : ℝ
h : 0 > x
⊢ -x < y → -y < x ∧ x < y
case inr.mpr
x y : ℝ
h : 0 > x
⊢ -y < x ∧ x < y → -x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x y : ℝ
h : 0 > x
⊢ -x < y ↔ -y < x ∧ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | . intro h'
linarith | case inr.mpr
x y : ℝ
h : 0 > x
⊢ -y < x ∧ x < y → -x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mpr
x y : ℝ
h : 0 > x
⊢ -y < x ∧ x < y → -x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | intro h' | case inr.mp
x y : ℝ
h : 0 > x
⊢ -x < y → -y < x ∧ x < y | case inr.mp
x y : ℝ
h : 0 > x
h' : -x < y
⊢ -y < x ∧ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mp
x y : ℝ
h : 0 > x
⊢ -x < y → -y < x ∧ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | constructor | case inr.mp
x y : ℝ
h : 0 > x
h' : -x < y
⊢ -y < x ∧ x < y | case inr.mp.left
x y : ℝ
h : 0 > x
h' : -x < y
⊢ -y < x
case inr.mp.right
x y : ℝ
h : 0 > x
h' : -x < y
⊢ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mp
x y : ℝ
h : 0 > x
h' : -x < y
⊢ -y < x ∧ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | . linarith | case inr.mp.right
x y : ℝ
h : 0 > x
h' : -x < y
⊢ x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mp.right
x y : ℝ
h : 0 > x
h' : -x < y
⊢ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | linarith | case inr.mp.left
x y : ℝ
h : 0 > x
h' : -x < y
⊢ -y < x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mp.left
x y : ℝ
h : 0 > x
h' : -x < y
⊢ -y < x
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | linarith | case inr.mp.right
x y : ℝ
h : 0 > x
h' : -x < y
⊢ x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mp.right
x y : ℝ
h : 0 > x
h' : -x < y
⊢ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | intro h' | case inr.mpr
x y : ℝ
h : 0 > x
⊢ -y < x ∧ x < y → -x < y | case inr.mpr
x y : ℝ
h : 0 > x
h' : -y < x ∧ x < y
⊢ -x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mpr
x y : ℝ
h : 0 > x
⊢ -y < x ∧ x < y → -x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | linarith | case inr.mpr
x y : ℝ
h : 0 > x
h' : -y < x ∧ x < y
⊢ -x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mpr
x y : ℝ
h : 0 > x
h' : -y < x ∧ x < y
⊢ -x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S5_Topology/solutions/Solutions_S03_Topological_Spaces.lean | aux | [112, 1] | [116, 89] | simpa [and_assoc] using ((nhds_basis_opens' x).comap c).tendsto_left_iff.mp h V' V'_in | X✝ : Type u_1
Y✝ : Type u_2
X : Type u_3
Y : Type u_4
A : Type u_5
inst✝ : TopologicalSpace X
c : A → X
f : A → Y
x : X
F : Filter Y
h : Tendsto f (comap c (𝓝 x)) F
V' : Set Y
V'_in : V' ∈ F
⊢ ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ : Type u_1
Y✝ : Type u_2
X : Type u_3
Y : Type u_4
A : Type u_5
inst✝ : TopologicalSpace X
c : A → X
f : A → Y
x : X
F : Filter Y
h : Tendsto f (comap c (𝓝 x)) F
V' : Set Y
V'_in : V' ∈ F
⊢ ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V'
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_const | [22, 1] | [27, 13] | intro ε εpos | a : ℝ
⊢ ConvergesTo (fun x => a) a | a ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun x => a) n - a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℝ
⊢ ConvergesTo (fun x => a) a
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_const | [22, 1] | [27, 13] | use 0 | a ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun x => a) n - a| < ε | case h
a ε : ℝ
εpos : ε > 0
⊢ ∀ n ≥ 0, |(fun x => a) n - a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
a ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun x => a) n - a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_const | [22, 1] | [27, 13] | intro n nge | case h
a ε : ℝ
εpos : ε > 0
⊢ ∀ n ≥ 0, |(fun x => a) n - a| < ε | case h
a ε : ℝ
εpos : ε > 0
n : ℕ
nge : n ≥ 0
⊢ |(fun x => a) n - a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a ε : ℝ
εpos : ε > 0
⊢ ∀ n ≥ 0, |(fun x => a) n - a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_const | [22, 1] | [27, 13] | rw [sub_self, abs_zero] | case h
a ε : ℝ
εpos : ε > 0
n : ℕ
nge : n ≥ 0
⊢ |(fun x => a) n - a| < ε | case h
a ε : ℝ
εpos : ε > 0
n : ℕ
nge : n ≥ 0
⊢ 0 < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a ε : ℝ
εpos : ε > 0
n : ℕ
nge : n ≥ 0
⊢ |(fun x => a) n - a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_const | [22, 1] | [27, 13] | apply εpos | case h
a ε : ℝ
εpos : ε > 0
n : ℕ
nge : n ≥ 0
⊢ 0 < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a ε : ℝ
εpos : ε > 0
n : ℕ
nge : n ≥ 0
⊢ 0 < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [29, 1] | [38, 8] | intro ε εpos | 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
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => s n + t n) n - (a + b)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ ConvergesTo (fun n => s n + t n) (a + b)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [29, 1] | [38, 8] | dsimp | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => s n + t n) n - (a + b)| < ε | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => s n + t n) n - (a + b)| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [29, 1] | [38, 8] | have ε2pos : 0 < ε / 2 := by linarith | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [29, 1] | [38, 8] | rcases cs (ε / 2) ε2pos with ⟨Ns, hs⟩ | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε | case intro
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [29, 1] | [38, 8] | rcases ct (ε / 2) ε2pos with ⟨Nt, ht⟩ | case intro
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε | case intro.intro
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [29, 1] | [38, 8] | use max Ns Nt | case intro.intro
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε | case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
⊢ ∀ n ≥ max Ns Nt, |s n + t n - (a + b)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [29, 1] | [38, 8] | sorry | case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
⊢ ∀ n ≥ max Ns Nt, |s n + t n - (a + b)| < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
⊢ ∀ n ≥ max Ns Nt, |s n + t n - (a + b)| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [29, 1] | [38, 8] | linarith | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
⊢ 0 < ε / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
⊢ 0 < ε / 2
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [40, 1] | [49, 8] | by_cases h : c = 0 | s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
⊢ ConvergesTo (fun n => c * s n) (c * a) | case pos
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
⊢ ConvergesTo (fun n => c * s n) (c * a)
case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
⊢ ConvergesTo (fun n => c * s n) (c * a) | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
⊢ ConvergesTo (fun n => c * s n) (c * a)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [40, 1] | [49, 8] | have acpos : 0 < |c| := abs_pos.mpr h | case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
⊢ ConvergesTo (fun n => c * s n) (c * a) | case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
⊢ ConvergesTo (fun n => c * s n) (c * a) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
⊢ ConvergesTo (fun n => c * s n) (c * a)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [40, 1] | [49, 8] | sorry | case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
⊢ ConvergesTo (fun n => c * s n) (c * a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
⊢ ConvergesTo (fun n => c * s n) (c * a)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [40, 1] | [49, 8] | convert convergesTo_const 0 | case pos
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
⊢ ConvergesTo (fun n => c * s n) (c * a) | case h.e'_1.h
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
x✝ : ℕ
⊢ c * s x✝ = 0
case h.e'_2
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
⊢ c * a = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
⊢ ConvergesTo (fun n => c * s n) (c * a)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [40, 1] | [49, 8] | rw [h] | case h.e'_2
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
⊢ c * a = 0 | case h.e'_2
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
⊢ 0 * a = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
⊢ c * a = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [40, 1] | [49, 8] | ring | case h.e'_2
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
⊢ 0 * a = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
⊢ 0 * a = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [40, 1] | [49, 8] | rw [h] | case h.e'_1.h
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
x✝ : ℕ
⊢ c * s x✝ = 0 | case h.e'_1.h
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
x✝ : ℕ
⊢ 0 * s x✝ = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1.h
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
x✝ : ℕ
⊢ c * s x✝ = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [40, 1] | [49, 8] | ring | case h.e'_1.h
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
x✝ : ℕ
⊢ 0 * s x✝ = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_1.h
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
x✝ : ℕ
⊢ 0 * s x✝ = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.exists_abs_le_of_convergesTo | [51, 1] | [55, 8] | rcases cs 1 zero_lt_one with ⟨N, h⟩ | s : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
⊢ ∃ N b, ∀ (n : ℕ), N ≤ n → |s n| < b | case intro
s : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
N : ℕ
h : ∀ n ≥ N, |s n - a| < 1
⊢ ∃ N b, ∀ (n : ℕ), N ≤ n → |s n| < b | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
⊢ ∃ N b, ∀ (n : ℕ), N ≤ n → |s n| < b
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.exists_abs_le_of_convergesTo | [51, 1] | [55, 8] | use N, |a| + 1 | case intro
s : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
N : ℕ
h : ∀ n ≥ N, |s n - a| < 1
⊢ ∃ N b, ∀ (n : ℕ), N ≤ n → |s n| < b | case h
s : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
N : ℕ
h : ∀ n ≥ N, |s n - a| < 1
⊢ ∀ (n : ℕ), N ≤ n → |s n| < |a| + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
s : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
N : ℕ
h : ∀ n ≥ N, |s n - a| < 1
⊢ ∃ N b, ∀ (n : ℕ), N ≤ n → |s n| < b
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.exists_abs_le_of_convergesTo | [51, 1] | [55, 8] | sorry | case h
s : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
N : ℕ
h : ∀ n ≥ N, |s n - a| < 1
⊢ ∀ (n : ℕ), N ≤ n → |s n| < |a| + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
N : ℕ
h : ∀ n ≥ N, |s n - a| < 1
⊢ ∀ (n : ℕ), N ≤ n → |s n| < |a| + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.aux | [57, 1] | [65, 8] | 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 | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t 0
⊢ ConvergesTo (fun n => s n * t n) 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.aux | [57, 1] | [65, 8] | 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.aux | [57, 1] | [65, 8] | 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.aux | [57, 1] | [65, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.aux | [57, 1] | [65, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.aux | [57, 1] | [65, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.aux | [57, 1] | [65, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TAC... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [67, 1] | [77, 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) | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ ConvergesTo (fun n => s n * t n) (a * b)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [67, 1] | [77, 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) | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [67, 1] | [77, 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 : ConvergesT... | Please generate a tactic in lean4 to solve the state.
STATE:
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)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [67, 1] | [77, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [67, 1] | [77, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ ConvergesTo (fun n => s n * (t n + -b)) 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [67, 1] | [77, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ ConvergesTo (fun n => t n + -b) 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [67, 1] | [77, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ 0 = b + -b
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [67, 1] | [77, 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✝ | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul | [67, 1] | [77, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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✝
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
⊢ a = b
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a b : ℝ
sa : ConvergesTo s a
sb : ConvergesTo s b
abne : ¬a = b
this : |a - b| > 0
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
⊢... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
a... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S06_Sequences_and_Convergence.lean | C03S06.convergesTo_unique | [79, 1] | [94, 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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
... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.