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/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab8Solution.lean | LoVe.List.length_add | [31, 1] | [48, 18] | cases hsame | x : β€
xs : List β€
hsame : List.length (x :: xs) = List.length []
β’ List.length (add (x :: xs) []) = List.length (x :: xs) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : β€
xs : List β€
hsame : List.length (x :: xs) = List.length []
β’ List.length (add (x :: xs) []) = List.length (x :: xs)
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab8Solution.lean | LoVe.List.add.comm | [64, 1] | [72, 31] | rfl | β’ add [] [] = add [] [] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
β’ add [] [] = add [] []
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab8Solution.lean | LoVe.List.add.comm | [64, 1] | [72, 31] | simp [List.add, add_comm] | x : β€
xs : List β€
y : β€
ys : List β€
β’ add (x :: xs) (y :: ys) = add (y :: ys) (x :: xs) | x : β€
xs : List β€
y : β€
ys : List β€
β’ add xs ys = add ys xs | Please generate a tactic in lean4 to solve the state.
STATE:
x : β€
xs : List β€
y : β€
ys : List β€
β’ add (x :: xs) (y :: ys) = add (y :: ys) (x :: xs)
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab8Solution.lean | LoVe.List.add.comm | [64, 1] | [72, 31] | exact List.add.comm xs ys | x : β€
xs : List β€
y : β€
ys : List β€
β’ add xs ys = add ys xs | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : β€
xs : List β€
y : β€
ys : List β€
β’ add xs ys = add ys xs
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab8Solution.lean | LoVe.List.add.comm | [64, 1] | [72, 31] | rfl | y : β€
ys : List β€
β’ add [] (y :: ys) = add (y :: ys) [] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
y : β€
ys : List β€
β’ add [] (y :: ys) = add (y :: ys) []
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab8Solution.lean | LoVe.List.add.comm | [64, 1] | [72, 31] | rfl | x : β€
xs : List β€
β’ add (x :: xs) [] = add [] (x :: xs) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : β€
xs : List β€
β’ add (x :: xs) [] = add [] (x :: xs)
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab8Solution.lean | LoVe.Vector.add.comm | [74, 1] | [79, 24] | apply Subtype.eq | n : β
u v : Vector β€ n
β’ add u v = add v u | case a
n : β
u v : Vector β€ n
β’ β(add u v) = β(add v u) | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
u v : Vector β€ n
β’ add u v = add v u
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab8Solution.lean | LoVe.Vector.add.comm | [74, 1] | [79, 24] | simp [Vector.add] | case a
n : β
u v : Vector β€ n
β’ β(add u v) = β(add v u) | case a
n : β
u v : Vector β€ n
β’ List.add βu βv = List.add βv βu | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : β
u v : Vector β€ n
β’ β(add u v) = β(add v u)
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab8Solution.lean | LoVe.Vector.add.comm | [74, 1] | [79, 24] | apply List.add.comm | case a
n : β
u v : Vector β€ n
β’ List.add βu βv = List.add βv βu | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : β
u v : Vector β€ n
β’ List.add βu βv = List.add βv βu
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab8Solution.lean | LoVe.Int.neg_eq | [109, 1] | [111, 9] | rfl | p n : β
β’ neg (Quotient.mk Setoid (p, n)) = Quotient.mk Setoid (n, p) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
p n : β
β’ neg (Quotient.mk Setoid (p, n)) = Quotient.mk Setoid (n, p)
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab8Solution.lean | LoVe.int.neg_neg | [113, 1] | [119, 35] | induction a using Quotient.inductionOn with
| h pn =>
cases pn with
| mk p n => apply Int.neg_eq | a : Int
β’ Int.neg (Int.neg a) = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : Int
β’ Int.neg (Int.neg a) = a
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab8Solution.lean | LoVe.int.neg_neg | [113, 1] | [119, 35] | cases pn with
| mk p n => apply Int.neg_eq | case h
pn : β Γ β
β’ Int.neg (Int.neg (Quotient.mk Int.Setoid pn)) = Quotient.mk Int.Setoid pn | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
pn : β Γ β
β’ Int.neg (Int.neg (Quotient.mk Int.Setoid pn)) = Quotient.mk Int.Setoid pn
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab8Solution.lean | LoVe.int.neg_neg | [113, 1] | [119, 35] | apply Int.neg_eq | case h.mk
p n : β
β’ Int.neg (Int.neg (Quotient.mk Int.Setoid (p, n))) = Quotient.mk Int.Setoid (p, n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk
p n : β
β’ Int.neg (Int.neg (Quotient.mk Int.Setoid (p, n))) = Quotient.mk Int.Setoid (p, n)
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Homework/Homework9.lean | LoVe.quarter_pos | [34, 1] | [38, 28] | have hx2 : 0 < x / 2 := half_pos hx | x : β
hx : 0 < x
β’ 0 < x / 4 | x : β
hx : 0 < x
hx2 : 0 < x / 2
β’ 0 < x / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
x : β
hx : 0 < x
β’ 0 < x / 4
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Homework/Homework9.lean | LoVe.quarter_pos | [34, 1] | [38, 28] | calc 0 < (x / 2) / 2 := half_pos hx2
_ = x / 4 := by ring | x : β
hx : 0 < x
hx2 : 0 < x / 2
β’ 0 < x / 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : β
hx : 0 < x
hx2 : 0 < x / 2
β’ 0 < x / 4
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Homework/Homework9.lean | LoVe.quarter_pos | [34, 1] | [38, 28] | ring | x : β
hx : 0 < x
hx2 : 0 < x / 2
β’ x / 2 / 2 = x / 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : β
hx : 0 < x
hx2 : 0 < x / 2
β’ x / 2 / 2 = x / 4
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/TacticStrategies.lean | normalize_correct | [62, 1] | [75, 10] | induction b with
| atom a => cases' a <;> simp [normalize, interp]
| and a b iha ihb => simp [normalize, interp, *]
| or a b iha ihb => simp [normalize, interp, *]
| not b ih =>
simp [normalize, interp]
rw [Bool.eq_false_iff]
tauto
| imp a b iha ihb =>
simp [normalize, interp, β iha, β ihb]
rw [Bool.eq_false_iff]
tauto | b : bexpr
β’ normalize b = true β interp b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
b : bexpr
β’ normalize b = true β interp b
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/TacticStrategies.lean | normalize_correct | [62, 1] | [75, 10] | cases' a <;> simp [normalize, interp] | case atom
a : Bool
β’ normalize (atom a) = true β interp (atom a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case atom
a : Bool
β’ normalize (atom a) = true β interp (atom a)
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/TacticStrategies.lean | normalize_correct | [62, 1] | [75, 10] | simp [normalize, interp, *] | case and
a b : bexpr
iha : normalize a = true β interp a
ihb : normalize b = true β interp b
β’ normalize (bexpr.and a b) = true β interp (bexpr.and a b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case and
a b : bexpr
iha : normalize a = true β interp a
ihb : normalize b = true β interp b
β’ normalize (bexpr.and a b) = true β interp (bexpr.and a b)
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/TacticStrategies.lean | normalize_correct | [62, 1] | [75, 10] | simp [normalize, interp, *] | case or
a b : bexpr
iha : normalize a = true β interp a
ihb : normalize b = true β interp b
β’ normalize (bexpr.or a b) = true β interp (bexpr.or a b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case or
a b : bexpr
iha : normalize a = true β interp a
ihb : normalize b = true β interp b
β’ normalize (bexpr.or a b) = true β interp (bexpr.or a b)
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/TacticStrategies.lean | normalize_correct | [62, 1] | [75, 10] | simp [normalize, interp] | case not
b : bexpr
ih : normalize b = true β interp b
β’ normalize (bexpr.not b) = true β interp (bexpr.not b) | case not
b : bexpr
ih : normalize b = true β interp b
β’ normalize b = false β Β¬interp b | Please generate a tactic in lean4 to solve the state.
STATE:
case not
b : bexpr
ih : normalize b = true β interp b
β’ normalize (bexpr.not b) = true β interp (bexpr.not b)
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/TacticStrategies.lean | normalize_correct | [62, 1] | [75, 10] | rw [Bool.eq_false_iff] | case not
b : bexpr
ih : normalize b = true β interp b
β’ normalize b = false β Β¬interp b | case not
b : bexpr
ih : normalize b = true β interp b
β’ normalize b β true β Β¬interp b | Please generate a tactic in lean4 to solve the state.
STATE:
case not
b : bexpr
ih : normalize b = true β interp b
β’ normalize b = false β Β¬interp b
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/TacticStrategies.lean | normalize_correct | [62, 1] | [75, 10] | tauto | case not
b : bexpr
ih : normalize b = true β interp b
β’ normalize b β true β Β¬interp b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case not
b : bexpr
ih : normalize b = true β interp b
β’ normalize b β true β Β¬interp b
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/TacticStrategies.lean | normalize_correct | [62, 1] | [75, 10] | simp [normalize, interp, β iha, β ihb] | case imp
a b : bexpr
iha : normalize a = true β interp a
ihb : normalize b = true β interp b
β’ normalize (imp a b) = true β interp (imp a b) | case imp
a b : bexpr
iha : normalize a = true β interp a
ihb : normalize b = true β interp b
β’ normalize a = false β¨ normalize b = true β normalize a = true β normalize b = true | Please generate a tactic in lean4 to solve the state.
STATE:
case imp
a b : bexpr
iha : normalize a = true β interp a
ihb : normalize b = true β interp b
β’ normalize (imp a b) = true β interp (imp a b)
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/TacticStrategies.lean | normalize_correct | [62, 1] | [75, 10] | rw [Bool.eq_false_iff] | case imp
a b : bexpr
iha : normalize a = true β interp a
ihb : normalize b = true β interp b
β’ normalize a = false β¨ normalize b = true β normalize a = true β normalize b = true | case imp
a b : bexpr
iha : normalize a = true β interp a
ihb : normalize b = true β interp b
β’ normalize a β true β¨ normalize b = true β normalize a = true β normalize b = true | Please generate a tactic in lean4 to solve the state.
STATE:
case imp
a b : bexpr
iha : normalize a = true β interp a
ihb : normalize b = true β interp b
β’ normalize a = false β¨ normalize b = true β normalize a = true β normalize b = true
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/TacticStrategies.lean | normalize_correct | [62, 1] | [75, 10] | tauto | case imp
a b : bexpr
iha : normalize a = true β interp a
ihb : normalize b = true β interp b
β’ normalize a β true β¨ normalize b = true β normalize a = true β normalize b = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case imp
a b : bexpr
iha : normalize a = true β interp a
ihb : normalize b = true β interp b
β’ normalize a β true β¨ normalize b = true β normalize a = true β normalize b = true
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/Complex.lean | new.to_and_from | [85, 1] | [102, 38] | cases' p with pangle pmag | p : Polar
hpmag : p.magnitude > 0
hpangle1 : -(Real.pi / 2) < p.angle
hpangle2 : p.angle < Real.pi / 2
β’ Complex.toPolar (Polar.toComplex p) = p | case mk
pangle pmag : β
hpmag : { angle := pangle, magnitude := pmag }.magnitude > 0
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
β’ Complex.toPolar (Polar.toComplex { angle := pangle, magnitude := pmag }) = { angle := pangle, magnitude := pmag } | Please generate a tactic in lean4 to solve the state.
STATE:
p : Polar
hpmag : p.magnitude > 0
hpangle1 : -(Real.pi / 2) < p.angle
hpangle2 : p.angle < Real.pi / 2
β’ Complex.toPolar (Polar.toComplex p) = p
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/Complex.lean | new.to_and_from | [85, 1] | [102, 38] | simp at hpmag | case mk
pangle pmag : β
hpmag : { angle := pangle, magnitude := pmag }.magnitude > 0
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
β’ Complex.toPolar (Polar.toComplex { angle := pangle, magnitude := pmag }) = { angle := pangle, magnitude := pmag } | case mk
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Complex.toPolar (Polar.toComplex { angle := pangle, magnitude := pmag }) = { angle := pangle, magnitude := pmag } | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
pangle pmag : β
hpmag : { angle := pangle, magnitude := pmag }.magnitude > 0
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
β’ Complex.toPolar (Polar.toComplex { angle := pangle, magnitude := pmag }) = { angle := pangle, magnitude := pmag }
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/Complex.lean | new.to_and_from | [85, 1] | [102, 38] | simp [Complex.toPolar, Polar.toComplex] | case mk
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Complex.toPolar (Polar.toComplex { angle := pangle, magnitude := pmag }) = { angle := pangle, magnitude := pmag } | case mk
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Real.arctan (pmag * Real.sin pangle / (pmag * Real.cos pangle)) = pangle β§
Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Complex.toPolar (Polar.toComplex { angle := pangle, magnitude := pmag }) = { angle := pangle, magnitude := pmag }
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/Complex.lean | new.to_and_from | [85, 1] | [102, 38] | constructor | case mk
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Real.arctan (pmag * Real.sin pangle / (pmag * Real.cos pangle)) = pangle β§
Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag | case mk.left
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Real.arctan (pmag * Real.sin pangle / (pmag * Real.cos pangle)) = pangle
case mk.right
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Real.arctan (pmag * Real.sin pangle / (pmag * Real.cos pangle)) = pangle β§
Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/Complex.lean | new.to_and_from | [85, 1] | [102, 38] | { rw [mul_div_mul_left]
rw [β Real.tan_eq_sin_div_cos]
rw [Real.arctan_tan]
all_goals aesop } | case mk.left
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Real.arctan (pmag * Real.sin pangle / (pmag * Real.cos pangle)) = pangle
case mk.right
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag | case mk.right
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.left
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Real.arctan (pmag * Real.sin pangle / (pmag * Real.cos pangle)) = pangle
case mk.right
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/Complex.lean | new.to_and_from | [85, 1] | [102, 38] | { suffices : (pmag * Real.cos pangle)^2 + (pmag * Real.sin pangle)^2 = pmag^2
rw [this]
rw [Real.sqrt_sq]
linarith
have h : (Real.sin pangle)^2 + (Real.cos pangle)^2 = 1 :=
Real.sin_sq_add_cos_sq pangle
linear_combination pmag ^ 2 * h } | case mk.right
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.right
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/Complex.lean | new.to_and_from | [85, 1] | [102, 38] | rw [mul_div_mul_left] | case mk.left
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Real.arctan (pmag * Real.sin pangle / (pmag * Real.cos pangle)) = pangle | case mk.left
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Real.arctan (Real.sin pangle / Real.cos pangle) = pangle
case mk.left.hc
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ pmag β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.left
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Real.arctan (pmag * Real.sin pangle / (pmag * Real.cos pangle)) = pangle
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/Complex.lean | new.to_and_from | [85, 1] | [102, 38] | rw [β Real.tan_eq_sin_div_cos] | case mk.left
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Real.arctan (Real.sin pangle / Real.cos pangle) = pangle
case mk.left.hc
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ pmag β 0 | case mk.left
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Real.arctan (Real.tan pangle) = pangle
case mk.left.hc
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ pmag β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.left
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Real.arctan (Real.sin pangle / Real.cos pangle) = pangle
case mk.left.hc
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ pmag β 0
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/Complex.lean | new.to_and_from | [85, 1] | [102, 38] | rw [Real.arctan_tan] | case mk.left
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Real.arctan (Real.tan pangle) = pangle
case mk.left.hc
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ pmag β 0 | case mk.left.hxβ
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ -(Real.pi / 2) < pangle
case mk.left.hxβ
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ pangle < Real.pi / 2
case mk.left.hc
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ pmag β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.left
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Real.arctan (Real.tan pangle) = pangle
case mk.left.hc
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ pmag β 0
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/Complex.lean | new.to_and_from | [85, 1] | [102, 38] | all_goals aesop | case mk.left.hxβ
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ -(Real.pi / 2) < pangle
case mk.left.hxβ
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ pangle < Real.pi / 2
case mk.left.hc
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ pmag β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.left.hxβ
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ -(Real.pi / 2) < pangle
case mk.left.hxβ
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ pangle < Real.pi / 2
case mk.left.hc
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ pmag β 0
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/Complex.lean | new.to_and_from | [85, 1] | [102, 38] | aesop | case mk.left.hc
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ pmag β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.left.hc
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ pmag β 0
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/Complex.lean | new.to_and_from | [85, 1] | [102, 38] | suffices : (pmag * Real.cos pangle)^2 + (pmag * Real.sin pangle)^2 = pmag^2 | case mk.right
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag | case mk.right
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
this : (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2
β’ Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag
case this
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.right
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/Complex.lean | new.to_and_from | [85, 1] | [102, 38] | rw [this] | case mk.right
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
this : (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2
β’ Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag
case this
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2 | case mk.right
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
this : (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2
β’ Real.sqrt (pmag ^ 2) = pmag
case this
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.right
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
this : (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2
β’ Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag
case this
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/Complex.lean | new.to_and_from | [85, 1] | [102, 38] | rw [Real.sqrt_sq] | case mk.right
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
this : (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2
β’ Real.sqrt (pmag ^ 2) = pmag
case this
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2 | case mk.right
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
this : (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2
β’ 0 β€ pmag
case this
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.right
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
this : (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2
β’ Real.sqrt (pmag ^ 2) = pmag
case this
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/Complex.lean | new.to_and_from | [85, 1] | [102, 38] | linarith | case mk.right
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
this : (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2
β’ 0 β€ pmag
case this
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2 | case this
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.right
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
this : (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2
β’ 0 β€ pmag
case this
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/Complex.lean | new.to_and_from | [85, 1] | [102, 38] | have h : (Real.sin pangle)^2 + (Real.cos pangle)^2 = 1 :=
Real.sin_sq_add_cos_sq pangle | case this
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2 | case this
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
h : Real.sin pangle ^ 2 + Real.cos pangle ^ 2 = 1
β’ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case this
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
β’ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/Complex.lean | new.to_and_from | [85, 1] | [102, 38] | linear_combination pmag ^ 2 * h | case this
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
h : Real.sin pangle ^ 2 + Real.cos pangle ^ 2 = 1
β’ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this
pangle pmag : β
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
h : Real.sin pangle ^ 2 + Real.cos pangle ^ 2 = 1
β’ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.I | [24, 1] | [28, 13] | intro ha | a : Prop
β’ a β a | a : Prop
ha : a
β’ a | Please generate a tactic in lean4 to solve the state.
STATE:
a : Prop
β’ a β a
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.I | [24, 1] | [28, 13] | exact ha | a : Prop
ha : a
β’ a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : Prop
ha : a
β’ a
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.K | [30, 1] | [34, 13] | intro ha hb | a b : Prop
β’ a β b β b | a b : Prop
ha : a
hb : b
β’ b | Please generate a tactic in lean4 to solve the state.
STATE:
a b : Prop
β’ a β b β b
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.K | [30, 1] | [34, 13] | exact hb | a b : Prop
ha : a
hb : b
β’ b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : Prop
ha : a
hb : b
β’ b
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.C | [36, 1] | [42, 13] | intro hg hb ha | a b c : Prop
β’ (a β b β c) β b β a β c | a b c : Prop
hg : a β b β c
hb : b
ha : a
β’ c | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : Prop
β’ (a β b β c) β b β a β c
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.C | [36, 1] | [42, 13] | apply hg | a b c : Prop
hg : a β b β c
hb : b
ha : a
β’ c | case a
a b c : Prop
hg : a β b β c
hb : b
ha : a
β’ a
case a
a b c : Prop
hg : a β b β c
hb : b
ha : a
β’ b | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : Prop
hg : a β b β c
hb : b
ha : a
β’ c
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.C | [36, 1] | [42, 13] | exact ha | case a
a b c : Prop
hg : a β b β c
hb : b
ha : a
β’ a
case a
a b c : Prop
hg : a β b β c
hb : b
ha : a
β’ b | case a
a b c : Prop
hg : a β b β c
hb : b
ha : a
β’ b | Please generate a tactic in lean4 to solve the state.
STATE:
case a
a b c : Prop
hg : a β b β c
hb : b
ha : a
β’ a
case a
a b c : Prop
hg : a β b β c
hb : b
ha : a
β’ b
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.C | [36, 1] | [42, 13] | exact hb | case a
a b c : Prop
hg : a β b β c
hb : b
ha : a
β’ b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
a b c : Prop
hg : a β b β c
hb : b
ha : a
β’ b
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.proj_fst | [44, 1] | [48, 13] | intro ha ha' | a : Prop
β’ a β a β a | a : Prop
ha ha' : a
β’ a | Please generate a tactic in lean4 to solve the state.
STATE:
a : Prop
β’ a β a β a
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.proj_fst | [44, 1] | [48, 13] | exact ha | a : Prop
ha ha' : a
β’ a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : Prop
ha ha' : a
β’ a
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.proj_snd | [52, 1] | [56, 14] | intro ha ha' | a : Prop
β’ a β a β a | a : Prop
ha ha' : a
β’ a | Please generate a tactic in lean4 to solve the state.
STATE:
a : Prop
β’ a β a β a
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.proj_snd | [52, 1] | [56, 14] | exact ha' | a : Prop
ha ha' : a
β’ a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : Prop
ha ha' : a
β’ a
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.some_nonsense | [58, 1] | [64, 13] | intro hg ha hf hb | a b c : Prop
β’ (a β b β c) β a β (a β c) β b β c | a b c : Prop
hg : a β b β c
ha : a
hf : a β c
hb : b
β’ c | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : Prop
β’ (a β b β c) β a β (a β c) β b β c
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.some_nonsense | [58, 1] | [64, 13] | apply hg | a b c : Prop
hg : a β b β c
ha : a
hf : a β c
hb : b
β’ c | case a
a b c : Prop
hg : a β b β c
ha : a
hf : a β c
hb : b
β’ a
case a
a b c : Prop
hg : a β b β c
ha : a
hf : a β c
hb : b
β’ b | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : Prop
hg : a β b β c
ha : a
hf : a β c
hb : b
β’ c
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.some_nonsense | [58, 1] | [64, 13] | exact ha | case a
a b c : Prop
hg : a β b β c
ha : a
hf : a β c
hb : b
β’ a
case a
a b c : Prop
hg : a β b β c
ha : a
hf : a β c
hb : b
β’ b | case a
a b c : Prop
hg : a β b β c
ha : a
hf : a β c
hb : b
β’ b | Please generate a tactic in lean4 to solve the state.
STATE:
case a
a b c : Prop
hg : a β b β c
ha : a
hf : a β c
hb : b
β’ a
case a
a b c : Prop
hg : a β b β c
ha : a
hf : a β c
hb : b
β’ b
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.some_nonsense | [58, 1] | [64, 13] | exact hb | case a
a b c : Prop
hg : a β b β c
ha : a
hf : a β c
hb : b
β’ b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
a b c : Prop
hg : a β b β c
ha : a
hf : a β c
hb : b
β’ b
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.contrapositive | [68, 1] | [74, 13] | intro hab hnb ha | a b : Prop
β’ (a β b) β Β¬b β Β¬a | a b : Prop
hab : a β b
hnb : Β¬b
ha : a
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
a b : Prop
β’ (a β b) β Β¬b β Β¬a
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.contrapositive | [68, 1] | [74, 13] | apply hnb | a b : Prop
hab : a β b
hnb : Β¬b
ha : a
β’ False | a b : Prop
hab : a β b
hnb : Β¬b
ha : a
β’ b | Please generate a tactic in lean4 to solve the state.
STATE:
a b : Prop
hab : a β b
hnb : Β¬b
ha : a
β’ False
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.contrapositive | [68, 1] | [74, 13] | apply hab | a b : Prop
hab : a β b
hnb : Β¬b
ha : a
β’ b | a b : Prop
hab : a β b
hnb : Β¬b
ha : a
β’ a | Please generate a tactic in lean4 to solve the state.
STATE:
a b : Prop
hab : a β b
hnb : Β¬b
ha : a
β’ b
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.contrapositive | [68, 1] | [74, 13] | apply ha | a b : Prop
hab : a β b
hnb : Β¬b
ha : a
β’ a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : Prop
hab : a β b
hnb : Β¬b
ha : a
β’ a
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | apply Iff.intro | Ξ± : Type
p q : Ξ± β Prop
β’ (β (x : Ξ±), p x β§ q x) β (β (x : Ξ±), p x) β§ β (x : Ξ±), q x | case mp
Ξ± : Type
p q : Ξ± β Prop
β’ (β (x : Ξ±), p x β§ q x) β (β (x : Ξ±), p x) β§ β (x : Ξ±), q x
case mpr
Ξ± : Type
p q : Ξ± β Prop
β’ ((β (x : Ξ±), p x) β§ β (x : Ξ±), q x) β β (x : Ξ±), p x β§ q x | Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type
p q : Ξ± β Prop
β’ (β (x : Ξ±), p x β§ q x) β (β (x : Ξ±), p x) β§ β (x : Ξ±), q x
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | { intro h
apply And.intro
{ intro x
apply And.left
apply h }
{ intro x
apply And.right
apply h } } | case mp
Ξ± : Type
p q : Ξ± β Prop
β’ (β (x : Ξ±), p x β§ q x) β (β (x : Ξ±), p x) β§ β (x : Ξ±), q x
case mpr
Ξ± : Type
p q : Ξ± β Prop
β’ ((β (x : Ξ±), p x) β§ β (x : Ξ±), q x) β β (x : Ξ±), p x β§ q x | case mpr
Ξ± : Type
p q : Ξ± β Prop
β’ ((β (x : Ξ±), p x) β§ β (x : Ξ±), q x) β β (x : Ξ±), p x β§ q x | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
Ξ± : Type
p q : Ξ± β Prop
β’ (β (x : Ξ±), p x β§ q x) β (β (x : Ξ±), p x) β§ β (x : Ξ±), q x
case mpr
Ξ± : Type
p q : Ξ± β Prop
β’ ((β (x : Ξ±), p x) β§ β (x : Ξ±), q x) β β (x : Ξ±), p x β§ q x
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | { intro h x
apply And.intro
{ apply And.left h }
{ apply And.right h } } | case mpr
Ξ± : Type
p q : Ξ± β Prop
β’ ((β (x : Ξ±), p x) β§ β (x : Ξ±), q x) β β (x : Ξ±), p x β§ q x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
Ξ± : Type
p q : Ξ± β Prop
β’ ((β (x : Ξ±), p x) β§ β (x : Ξ±), q x) β β (x : Ξ±), p x β§ q x
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | intro h | case mp
Ξ± : Type
p q : Ξ± β Prop
β’ (β (x : Ξ±), p x β§ q x) β (β (x : Ξ±), p x) β§ β (x : Ξ±), q x | case mp
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
β’ (β (x : Ξ±), p x) β§ β (x : Ξ±), q x | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
Ξ± : Type
p q : Ξ± β Prop
β’ (β (x : Ξ±), p x β§ q x) β (β (x : Ξ±), p x) β§ β (x : Ξ±), q x
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | apply And.intro | case mp
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
β’ (β (x : Ξ±), p x) β§ β (x : Ξ±), q x | case mp.left
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
β’ β (x : Ξ±), p x
case mp.right
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
β’ β (x : Ξ±), q x | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
β’ (β (x : Ξ±), p x) β§ β (x : Ξ±), q x
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | { intro x
apply And.left
apply h } | case mp.left
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
β’ β (x : Ξ±), p x
case mp.right
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
β’ β (x : Ξ±), q x | case mp.right
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
β’ β (x : Ξ±), q x | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.left
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
β’ β (x : Ξ±), p x
case mp.right
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
β’ β (x : Ξ±), q x
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | { intro x
apply And.right
apply h } | case mp.right
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
β’ β (x : Ξ±), q x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
β’ β (x : Ξ±), q x
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | intro x | case mp.left
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
β’ β (x : Ξ±), p x | case mp.left
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
x : Ξ±
β’ p x | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.left
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
β’ β (x : Ξ±), p x
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | apply And.left | case mp.left
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
x : Ξ±
β’ p x | case mp.left.self
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
x : Ξ±
β’ p x β§ ?mp.left.b
case mp.left.b
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
x : Ξ±
β’ Prop | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.left
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
x : Ξ±
β’ p x
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | apply h | case mp.left.self
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
x : Ξ±
β’ p x β§ ?mp.left.b
case mp.left.b
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
x : Ξ±
β’ Prop | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.left.self
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
x : Ξ±
β’ p x β§ ?mp.left.b
case mp.left.b
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
x : Ξ±
β’ Prop
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | intro x | case mp.right
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
β’ β (x : Ξ±), q x | case mp.right
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
x : Ξ±
β’ q x | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
β’ β (x : Ξ±), q x
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | apply And.right | case mp.right
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
x : Ξ±
β’ q x | case mp.right.self
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
x : Ξ±
β’ ?mp.right.a β§ q x
case mp.right.a
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
x : Ξ±
β’ Prop | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
x : Ξ±
β’ q x
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | apply h | case mp.right.self
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
x : Ξ±
β’ ?mp.right.a β§ q x
case mp.right.a
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
x : Ξ±
β’ Prop | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right.self
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
x : Ξ±
β’ ?mp.right.a β§ q x
case mp.right.a
Ξ± : Type
p q : Ξ± β Prop
h : β (x : Ξ±), p x β§ q x
x : Ξ±
β’ Prop
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | intro h x | case mpr
Ξ± : Type
p q : Ξ± β Prop
β’ ((β (x : Ξ±), p x) β§ β (x : Ξ±), q x) β β (x : Ξ±), p x β§ q x | case mpr
Ξ± : Type
p q : Ξ± β Prop
h : (β (x : Ξ±), p x) β§ β (x : Ξ±), q x
x : Ξ±
β’ p x β§ q x | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
Ξ± : Type
p q : Ξ± β Prop
β’ ((β (x : Ξ±), p x) β§ β (x : Ξ±), q x) β β (x : Ξ±), p x β§ q x
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | apply And.intro | case mpr
Ξ± : Type
p q : Ξ± β Prop
h : (β (x : Ξ±), p x) β§ β (x : Ξ±), q x
x : Ξ±
β’ p x β§ q x | case mpr.left
Ξ± : Type
p q : Ξ± β Prop
h : (β (x : Ξ±), p x) β§ β (x : Ξ±), q x
x : Ξ±
β’ p x
case mpr.right
Ξ± : Type
p q : Ξ± β Prop
h : (β (x : Ξ±), p x) β§ β (x : Ξ±), q x
x : Ξ±
β’ q x | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
Ξ± : Type
p q : Ξ± β Prop
h : (β (x : Ξ±), p x) β§ β (x : Ξ±), q x
x : Ξ±
β’ p x β§ q x
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | { apply And.left h } | case mpr.left
Ξ± : Type
p q : Ξ± β Prop
h : (β (x : Ξ±), p x) β§ β (x : Ξ±), q x
x : Ξ±
β’ p x
case mpr.right
Ξ± : Type
p q : Ξ± β Prop
h : (β (x : Ξ±), p x) β§ β (x : Ξ±), q x
x : Ξ±
β’ q x | case mpr.right
Ξ± : Type
p q : Ξ± β Prop
h : (β (x : Ξ±), p x) β§ β (x : Ξ±), q x
x : Ξ±
β’ q x | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.left
Ξ± : Type
p q : Ξ± β Prop
h : (β (x : Ξ±), p x) β§ β (x : Ξ±), q x
x : Ξ±
β’ p x
case mpr.right
Ξ± : Type
p q : Ξ± β Prop
h : (β (x : Ξ±), p x) β§ β (x : Ξ±), q x
x : Ξ±
β’ q x
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | { apply And.right h } | case mpr.right
Ξ± : Type
p q : Ξ± β Prop
h : (β (x : Ξ±), p x) β§ β (x : Ξ±), q x
x : Ξ±
β’ q x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.right
Ξ± : Type
p q : Ξ± β Prop
h : (β (x : Ξ±), p x) β§ β (x : Ξ±), q x
x : Ξ±
β’ q x
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | apply And.left h | case mpr.left
Ξ± : Type
p q : Ξ± β Prop
h : (β (x : Ξ±), p x) β§ β (x : Ξ±), q x
x : Ξ±
β’ p x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.left
Ξ± : Type
p q : Ξ± β Prop
h : (β (x : Ξ±), p x) β§ β (x : Ξ±), q x
x : Ξ±
β’ p x
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | apply And.right h | case mpr.right
Ξ± : Type
p q : Ξ± β Prop
h : (β (x : Ξ±), p x) β§ β (x : Ξ±), q x
x : Ξ±
β’ q x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.right
Ξ± : Type
p q : Ξ± β Prop
h : (β (x : Ξ±), p x) β§ β (x : Ξ±), q x
x : Ξ±
β’ q x
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.mul_zero | [107, 1] | [112, 40] | induction n with
| zero => rfl
| succ n' ih => simp only [mul, ih] | n : β
β’ mul 0 n = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
β’ mul 0 n = 0
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.mul_zero | [107, 1] | [112, 40] | rfl | case zero
β’ mul 0 Nat.zero = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
β’ mul 0 Nat.zero = 0
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.mul_zero | [107, 1] | [112, 40] | simp only [mul, ih] | case succ
n' : β
ih : mul 0 n' = 0
β’ mul 0 (Nat.succ n') = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n' : β
ih : mul 0 n' = 0
β’ mul 0 (Nat.succ n') = 0
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.mul_succ | [115, 1] | [120, 66] | induction n with
| zero => rfl
| succ n' ih => simp only [add, add_succ, add_assoc, mul, ih] | m n : β
β’ mul (Nat.succ m) n = add (mul m n) n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : β
β’ mul (Nat.succ m) n = add (mul m n) n
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.mul_succ | [115, 1] | [120, 66] | rfl | case zero
m : β
β’ mul (Nat.succ m) Nat.zero = add (mul m Nat.zero) Nat.zero | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
m : β
β’ mul (Nat.succ m) Nat.zero = add (mul m Nat.zero) Nat.zero
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.mul_succ | [115, 1] | [120, 66] | simp only [add, add_succ, add_assoc, mul, ih] | case succ
m n' : β
ih : mul (Nat.succ m) n' = add (mul m n') n'
β’ mul (Nat.succ m) (Nat.succ n') = add (mul m (Nat.succ n')) (Nat.succ n') | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n' : β
ih : mul (Nat.succ m) n' = add (mul m n') n'
β’ mul (Nat.succ m) (Nat.succ n') = add (mul m (Nat.succ n')) (Nat.succ n')
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.mul_comm | [125, 1] | [132, 13] | induction m with
| zero => simp only [mul, mul_zero]
| succ m' ih =>
simp only [mul_succ, ih]
ac_rfl | m n : β
β’ mul m n = mul n m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : β
β’ mul m n = mul n m
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.mul_comm | [125, 1] | [132, 13] | simp only [mul, mul_zero] | case zero
n : β
β’ mul Nat.zero n = mul n Nat.zero | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
n : β
β’ mul Nat.zero n = mul n Nat.zero
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.mul_comm | [125, 1] | [132, 13] | simp only [mul_succ, ih] | case succ
n m' : β
ih : mul m' n = mul n m'
β’ mul (Nat.succ m') n = mul n (Nat.succ m') | case succ
n m' : β
ih : mul m' n = mul n m'
β’ add (mul n m') n = mul n (Nat.succ m') | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n m' : β
ih : mul m' n = mul n m'
β’ mul (Nat.succ m') n = mul n (Nat.succ m')
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.mul_comm | [125, 1] | [132, 13] | ac_rfl | case succ
n m' : β
ih : mul m' n = mul n m'
β’ add (mul n m') n = mul n (Nat.succ m') | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n m' : β
ih : mul m' n = mul n m'
β’ add (mul n m') n = mul n (Nat.succ m')
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.mul_assoc | [134, 1] | [139, 49] | induction n with
| zero => rfl
| succ n' ih => simp only [mul, mul_add, ih] | l m n : β
β’ mul (mul l m) n = mul l (mul m n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
l m n : β
β’ mul (mul l m) n = mul l (mul m n)
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.mul_assoc | [134, 1] | [139, 49] | rfl | case zero
l m : β
β’ mul (mul l m) Nat.zero = mul l (mul m Nat.zero) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
l m : β
β’ mul (mul l m) Nat.zero = mul l (mul m Nat.zero)
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.mul_assoc | [134, 1] | [139, 49] | simp only [mul, mul_add, ih] | case succ
l m n' : β
ih : mul (mul l m) n' = mul l (mul m n')
β’ mul (mul l m) (Nat.succ n') = mul l (mul m (Nat.succ n')) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
l m n' : β
ih : mul (mul l m) n' = mul l (mul m n')
β’ mul (mul l m) (Nat.succ n') = mul l (mul m (Nat.succ n'))
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.add_mul | [145, 1] | [149, 17] | rw [mul_comm _ n] | l m n : β
β’ mul (add l m) n = add (mul n l) (mul n m) | l m n : β
β’ mul n (add l m) = add (mul n l) (mul n m) | Please generate a tactic in lean4 to solve the state.
STATE:
l m n : β
β’ mul (add l m) n = add (mul n l) (mul n m)
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.add_mul | [145, 1] | [149, 17] | rw [mul_add] | l m n : β
β’ mul n (add l m) = add (mul n l) (mul n m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
l m n : β
β’ mul n (add l m) = add (mul n l) (mul n m)
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.Peirce_of_EM | [182, 1] | [197, 19] | rw [ExcludedMiddle] | β’ ExcludedMiddle β Peirce | β’ (β (a : Prop), a β¨ Β¬a) β Peirce | Please generate a tactic in lean4 to solve the state.
STATE:
β’ ExcludedMiddle β Peirce
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.Peirce_of_EM | [182, 1] | [197, 19] | rw [Peirce] | β’ (β (a : Prop), a β¨ Β¬a) β Peirce | β’ (β (a : Prop), a β¨ Β¬a) β β (a b : Prop), ((a β b) β a) β a | Please generate a tactic in lean4 to solve the state.
STATE:
β’ (β (a : Prop), a β¨ Β¬a) β Peirce
TACTIC:
|
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.Peirce_of_EM | [182, 1] | [197, 19] | intro hem | β’ (β (a : Prop), a β¨ Β¬a) β β (a b : Prop), ((a β b) β a) β a | hem : β (a : Prop), a β¨ Β¬a
β’ β (a b : Prop), ((a β b) β a) β a | Please generate a tactic in lean4 to solve the state.
STATE:
β’ (β (a : Prop), a β¨ Β¬a) β β (a b : Prop), ((a β b) β a) β a
TACTIC:
|
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