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stringlengths 7
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stringlengths 1
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stringlengths 6
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https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab8Solution.lean | LoVe.List.length_add | [31, 1] | [48, 18] | cases hsame | x : ℤ
xs : List ℤ
hsame : List.length (x :: xs) = List.length []
⊢ List.length (add (x :: xs) []) = List.length (x :: xs) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab8Solution.lean | LoVe.List.add.comm | [64, 1] | [72, 31] | rfl | ⊢ add [] [] = add [] [] | no goals |
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 : ℤ
xs : List ℤ
y : ℤ
ys : List ℤ
⊢ add (x :: xs) (y :: ys) = add (y :: ys) (x :: xs) | x : ℤ
xs : List ℤ
y : ℤ
ys : List ℤ
⊢ add xs ys = add ys xs |
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 : ℤ
xs : List ℤ
y : ℤ
ys : List ℤ
⊢ add xs ys = add ys xs | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab8Solution.lean | LoVe.List.add.comm | [64, 1] | [72, 31] | rfl | y : ℤ
ys : List ℤ
⊢ add [] (y :: ys) = add (y :: ys) [] | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab8Solution.lean | LoVe.List.add.comm | [64, 1] | [72, 31] | rfl | x : ℤ
xs : List ℤ
⊢ add (x :: xs) [] = add [] (x :: xs) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab8Solution.lean | LoVe.Vector.add.comm | [74, 1] | [79, 24] | apply Subtype.eq | n : ℕ
u v : Vector ℤ n
⊢ add u v = add v u | case a
n : ℕ
u v : Vector ℤ n
⊢ ↑(add u v) = ↑(add v u) |
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 : ℕ
u v : Vector ℤ n
⊢ ↑(add u v) = ↑(add v u) | case a
n : ℕ
u v : Vector ℤ n
⊢ List.add ↑u ↑v = List.add ↑v ↑u |
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 : ℕ
u v : Vector ℤ n
⊢ List.add ↑u ↑v = List.add ↑v ↑u | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab8Solution.lean | LoVe.Int.neg_eq | [109, 1] | [111, 9] | rfl | p n : ℕ
⊢ neg (Quotient.mk Setoid (p, n)) = Quotient.mk Setoid (n, p) | no goals |
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
⊢ Int.neg (Int.neg a) = a | no goals |
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 : ℕ × ℕ
⊢ Int.neg (Int.neg (Quotient.mk Int.Setoid pn)) = Quotient.mk Int.Setoid pn | no goals |
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 : ℕ
⊢ Int.neg (Int.neg (Quotient.mk Int.Setoid (p, n))) = Quotient.mk Int.Setoid (p, n) | no goals |
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 : ℚ
hx : 0 < x
⊢ 0 < x / 4 | x : ℚ
hx : 0 < x
hx2 : 0 < x / 2
⊢ 0 < x / 4 |
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 : ℚ
hx : 0 < x
hx2 : 0 < x / 2
⊢ 0 < x / 4 | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Homework/Homework9.lean | LoVe.quarter_pos | [34, 1] | [38, 28] | ring | x : ℚ
hx : 0 < x
hx2 : 0 < x / 2
⊢ x / 2 / 2 = x / 4 | no goals |
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, ← iha, ← ihb]
rw [Bool.eq_false_iff]
tauto | b : bexpr
⊢ normalize b = true ↔ interp b | no goals |
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
⊢ normalize (atom a) = true ↔ interp (atom a) | no goals |
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 ↔ interp a
ihb : normalize b = true ↔ interp b
⊢ normalize (bexpr.and a b) = true ↔ interp (bexpr.and a b) | no goals |
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 ↔ interp a
ihb : normalize b = true ↔ interp b
⊢ normalize (bexpr.or a b) = true ↔ interp (bexpr.or a b) | no goals |
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 ↔ interp b
⊢ normalize (bexpr.not b) = true ↔ interp (bexpr.not b) | case not
b : bexpr
ih : normalize b = true ↔ interp b
⊢ normalize b = false ↔ ¬interp b |
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 ↔ interp b
⊢ normalize b = false ↔ ¬interp b | case not
b : bexpr
ih : normalize b = true ↔ interp b
⊢ normalize b ≠ true ↔ ¬interp b |
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 ↔ interp b
⊢ normalize b ≠ true ↔ ¬interp b | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/TacticStrategies.lean | normalize_correct | [62, 1] | [75, 10] | simp [normalize, interp, ← iha, ← ihb] | case imp
a b : bexpr
iha : normalize a = true ↔ interp a
ihb : normalize b = true ↔ interp b
⊢ normalize (imp a b) = true ↔ interp (imp a b) | case imp
a b : bexpr
iha : normalize a = true ↔ interp a
ihb : normalize b = true ↔ interp b
⊢ normalize a = false ∨ normalize b = true ↔ normalize a = true → normalize b = true |
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 ↔ interp a
ihb : normalize b = true ↔ interp b
⊢ normalize a = false ∨ normalize b = true ↔ normalize a = true → normalize b = true | case imp
a b : bexpr
iha : normalize a = true ↔ interp a
ihb : normalize b = true ↔ interp b
⊢ normalize a ≠ true ∨ normalize b = true ↔ normalize a = true → normalize b = true |
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 ↔ interp a
ihb : normalize b = true ↔ interp b
⊢ normalize a ≠ true ∨ normalize b = true ↔ normalize a = true → normalize b = true | no goals |
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
⊢ Complex.toPolar (Polar.toComplex p) = p | case mk
pangle pmag : ℝ
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
⊢ Complex.toPolar (Polar.toComplex { angle := pangle, magnitude := pmag }) = { angle := pangle, magnitude := pmag } |
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 : ℝ
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
⊢ Complex.toPolar (Polar.toComplex { angle := pangle, magnitude := pmag }) = { angle := pangle, magnitude := pmag } | case mk
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ Complex.toPolar (Polar.toComplex { angle := pangle, magnitude := pmag }) = { angle := pangle, magnitude := pmag } |
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 : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ Complex.toPolar (Polar.toComplex { angle := pangle, magnitude := pmag }) = { angle := pangle, magnitude := pmag } | case mk
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ Real.arctan (pmag * Real.sin pangle / (pmag * Real.cos pangle)) = pangle ∧
Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/Complex.lean | new.to_and_from | [85, 1] | [102, 38] | constructor | case mk
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ Real.arctan (pmag * Real.sin pangle / (pmag * Real.cos pangle)) = pangle ∧
Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag | case mk.left
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ Real.arctan (pmag * Real.sin pangle / (pmag * Real.cos pangle)) = pangle
case mk.right
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag |
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 [← Real.tan_eq_sin_div_cos]
rw [Real.arctan_tan]
all_goals aesop } | case mk.left
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ Real.arctan (pmag * Real.sin pangle / (pmag * Real.cos pangle)) = pangle
case mk.right
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag | case mk.right
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag |
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 : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag | no goals |
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 : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ Real.arctan (pmag * Real.sin pangle / (pmag * Real.cos pangle)) = pangle | case mk.left
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ Real.arctan (Real.sin pangle / Real.cos pangle) = pangle
case mk.left.hc
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ pmag ≠ 0 |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/Complex.lean | new.to_and_from | [85, 1] | [102, 38] | rw [← Real.tan_eq_sin_div_cos] | case mk.left
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ Real.arctan (Real.sin pangle / Real.cos pangle) = pangle
case mk.left.hc
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ pmag ≠ 0 | case mk.left
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ Real.arctan (Real.tan pangle) = pangle
case mk.left.hc
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ pmag ≠ 0 |
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 : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ Real.arctan (Real.tan pangle) = pangle
case mk.left.hc
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ pmag ≠ 0 | case mk.left.hx₁
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ -(Real.pi / 2) < pangle
case mk.left.hx₂
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ pangle < Real.pi / 2
case mk.left.hc
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ pmag ≠ 0 |
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.hx₁
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ -(Real.pi / 2) < pangle
case mk.left.hx₂
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ pangle < Real.pi / 2
case mk.left.hc
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ pmag ≠ 0 | no goals |
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 : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ pmag ≠ 0 | no goals |
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 : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag | case mk.right
pangle pmag : ℝ
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
⊢ Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag
case this
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2 |
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 : ℝ
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
⊢ Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag
case this
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2 | case mk.right
pangle pmag : ℝ
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
⊢ Real.sqrt (pmag ^ 2) = pmag
case this
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2 |
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 : ℝ
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
⊢ Real.sqrt (pmag ^ 2) = pmag
case this
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2 | case mk.right
pangle pmag : ℝ
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
⊢ 0 ≤ pmag
case this
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2 |
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 : ℝ
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
⊢ 0 ≤ pmag
case this
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2 | case this
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2 |
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 : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2 | case this
pangle pmag : ℝ
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
⊢ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2 |
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 : ℝ
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
⊢ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2 | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.I | [24, 1] | [28, 13] | intro ha | a : Prop
⊢ a → a | a : Prop
ha : a
⊢ a |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.I | [24, 1] | [28, 13] | exact ha | a : Prop
ha : a
⊢ a | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.K | [30, 1] | [34, 13] | intro ha hb | a b : Prop
⊢ a → b → b | a b : Prop
ha : a
hb : b
⊢ b |
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
⊢ b | no goals |
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
⊢ (a → b → c) → b → a → c | a b c : Prop
hg : a → b → c
hb : b
ha : a
⊢ c |
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 → b → c
hb : b
ha : a
⊢ c | case a
a b c : Prop
hg : a → b → c
hb : b
ha : a
⊢ a
case a
a b c : Prop
hg : a → b → c
hb : b
ha : a
⊢ b |
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 → b → c
hb : b
ha : a
⊢ a
case a
a b c : Prop
hg : a → b → c
hb : b
ha : a
⊢ b | case a
a b c : Prop
hg : a → b → c
hb : b
ha : a
⊢ b |
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 → b → c
hb : b
ha : a
⊢ b | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.proj_fst | [44, 1] | [48, 13] | intro ha ha' | a : Prop
⊢ a → a → a | a : Prop
ha ha' : a
⊢ a |
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
⊢ a | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.proj_snd | [52, 1] | [56, 14] | intro ha ha' | a : Prop
⊢ a → a → a | a : Prop
ha ha' : a
⊢ a |
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
⊢ a | no goals |
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
⊢ (a → b → c) → a → (a → c) → b → c | a b c : Prop
hg : a → b → c
ha : a
hf : a → c
hb : b
⊢ c |
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 → b → c
ha : a
hf : a → c
hb : b
⊢ c | case a
a b c : Prop
hg : a → b → c
ha : a
hf : a → c
hb : b
⊢ a
case a
a b c : Prop
hg : a → b → c
ha : a
hf : a → c
hb : b
⊢ b |
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 → b → c
ha : a
hf : a → c
hb : b
⊢ a
case a
a b c : Prop
hg : a → b → c
ha : a
hf : a → c
hb : b
⊢ b | case a
a b c : Prop
hg : a → b → c
ha : a
hf : a → c
hb : b
⊢ b |
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 → b → c
ha : a
hf : a → c
hb : b
⊢ b | no goals |
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
⊢ (a → b) → ¬b → ¬a | a b : Prop
hab : a → b
hnb : ¬b
ha : a
⊢ False |
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 → b
hnb : ¬b
ha : a
⊢ False | a b : Prop
hab : a → b
hnb : ¬b
ha : a
⊢ b |
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 → b
hnb : ¬b
ha : a
⊢ b | a b : Prop
hab : a → b
hnb : ¬b
ha : a
⊢ a |
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 → b
hnb : ¬b
ha : a
⊢ a | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | apply Iff.intro | α : Type
p q : α → Prop
⊢ (∀ (x : α), p x ∧ q x) ↔ (∀ (x : α), p x) ∧ ∀ (x : α), q x | case mp
α : Type
p q : α → Prop
⊢ (∀ (x : α), p x ∧ q x) → (∀ (x : α), p x) ∧ ∀ (x : α), q x
case mpr
α : Type
p q : α → Prop
⊢ ((∀ (x : α), p x) ∧ ∀ (x : α), q x) → ∀ (x : α), p x ∧ q x |
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
α : Type
p q : α → Prop
⊢ (∀ (x : α), p x ∧ q x) → (∀ (x : α), p x) ∧ ∀ (x : α), q x
case mpr
α : Type
p q : α → Prop
⊢ ((∀ (x : α), p x) ∧ ∀ (x : α), q x) → ∀ (x : α), p x ∧ q x | case mpr
α : Type
p q : α → Prop
⊢ ((∀ (x : α), p x) ∧ ∀ (x : α), q x) → ∀ (x : α), p x ∧ q x |
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
α : Type
p q : α → Prop
⊢ ((∀ (x : α), p x) ∧ ∀ (x : α), q x) → ∀ (x : α), p x ∧ q x | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | intro h | case mp
α : Type
p q : α → Prop
⊢ (∀ (x : α), p x ∧ q x) → (∀ (x : α), p x) ∧ ∀ (x : α), q x | case mp
α : Type
p q : α → Prop
h : ∀ (x : α), p x ∧ q x
⊢ (∀ (x : α), p x) ∧ ∀ (x : α), q x |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | apply And.intro | case mp
α : Type
p q : α → Prop
h : ∀ (x : α), p x ∧ q x
⊢ (∀ (x : α), p x) ∧ ∀ (x : α), q x | case mp.left
α : Type
p q : α → Prop
h : ∀ (x : α), p x ∧ q x
⊢ ∀ (x : α), p x
case mp.right
α : Type
p q : α → Prop
h : ∀ (x : α), p x ∧ q x
⊢ ∀ (x : α), q x |
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
α : Type
p q : α → Prop
h : ∀ (x : α), p x ∧ q x
⊢ ∀ (x : α), p x
case mp.right
α : Type
p q : α → Prop
h : ∀ (x : α), p x ∧ q x
⊢ ∀ (x : α), q x | case mp.right
α : Type
p q : α → Prop
h : ∀ (x : α), p x ∧ q x
⊢ ∀ (x : α), q x |
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
α : Type
p q : α → Prop
h : ∀ (x : α), p x ∧ q x
⊢ ∀ (x : α), q x | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | intro x | case mp.left
α : Type
p q : α → Prop
h : ∀ (x : α), p x ∧ q x
⊢ ∀ (x : α), p x | case mp.left
α : Type
p q : α → Prop
h : ∀ (x : α), p x ∧ q x
x : α
⊢ p x |
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
α : Type
p q : α → Prop
h : ∀ (x : α), p x ∧ q x
x : α
⊢ p x | case mp.left.self
α : Type
p q : α → Prop
h : ∀ (x : α), p x ∧ q x
x : α
⊢ p x ∧ ?mp.left.b
case mp.left.b
α : Type
p q : α → Prop
h : ∀ (x : α), p x ∧ q x
x : α
⊢ Prop |
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
α : Type
p q : α → Prop
h : ∀ (x : α), p x ∧ q x
x : α
⊢ p x ∧ ?mp.left.b
case mp.left.b
α : Type
p q : α → Prop
h : ∀ (x : α), p x ∧ q x
x : α
⊢ Prop | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | intro x | case mp.right
α : Type
p q : α → Prop
h : ∀ (x : α), p x ∧ q x
⊢ ∀ (x : α), q x | case mp.right
α : Type
p q : α → Prop
h : ∀ (x : α), p x ∧ q x
x : α
⊢ q x |
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
α : Type
p q : α → Prop
h : ∀ (x : α), p x ∧ q x
x : α
⊢ q x | case mp.right.self
α : Type
p q : α → Prop
h : ∀ (x : α), p x ∧ q x
x : α
⊢ ?mp.right.a ∧ q x
case mp.right.a
α : Type
p q : α → Prop
h : ∀ (x : α), p x ∧ q x
x : α
⊢ Prop |
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
α : Type
p q : α → Prop
h : ∀ (x : α), p x ∧ q x
x : α
⊢ ?mp.right.a ∧ q x
case mp.right.a
α : Type
p q : α → Prop
h : ∀ (x : α), p x ∧ q x
x : α
⊢ Prop | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | intro h x | case mpr
α : Type
p q : α → Prop
⊢ ((∀ (x : α), p x) ∧ ∀ (x : α), q x) → ∀ (x : α), p x ∧ q x | case mpr
α : Type
p q : α → Prop
h : (∀ (x : α), p x) ∧ ∀ (x : α), q x
x : α
⊢ p x ∧ q x |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.forall_and | [82, 1] | [97, 30] | apply And.intro | case mpr
α : Type
p q : α → Prop
h : (∀ (x : α), p x) ∧ ∀ (x : α), q x
x : α
⊢ p x ∧ q x | case mpr.left
α : Type
p q : α → Prop
h : (∀ (x : α), p x) ∧ ∀ (x : α), q x
x : α
⊢ p x
case mpr.right
α : Type
p q : α → Prop
h : (∀ (x : α), p x) ∧ ∀ (x : α), q x
x : α
⊢ q x |
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
α : Type
p q : α → Prop
h : (∀ (x : α), p x) ∧ ∀ (x : α), q x
x : α
⊢ p x
case mpr.right
α : Type
p q : α → Prop
h : (∀ (x : α), p x) ∧ ∀ (x : α), q x
x : α
⊢ q x | case mpr.right
α : Type
p q : α → Prop
h : (∀ (x : α), p x) ∧ ∀ (x : α), q x
x : α
⊢ q x |
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
α : Type
p q : α → Prop
h : (∀ (x : α), p x) ∧ ∀ (x : α), q x
x : α
⊢ q x | no goals |
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
α : Type
p q : α → Prop
h : (∀ (x : α), p x) ∧ ∀ (x : α), q x
x : α
⊢ p x | no goals |
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
α : Type
p q : α → Prop
h : (∀ (x : α), p x) ∧ ∀ (x : α), q x
x : α
⊢ q x | no goals |
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 : ℕ
⊢ mul 0 n = 0 | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.mul_zero | [107, 1] | [112, 40] | rfl | case zero
⊢ mul 0 Nat.zero = 0 | no goals |
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' : ℕ
ih : mul 0 n' = 0
⊢ mul 0 (Nat.succ n') = 0 | no goals |
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 : ℕ
⊢ mul (Nat.succ m) n = add (mul m n) n | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.mul_succ | [115, 1] | [120, 66] | rfl | case zero
m : ℕ
⊢ mul (Nat.succ m) Nat.zero = add (mul m Nat.zero) Nat.zero | no goals |
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' : ℕ
ih : mul (Nat.succ m) n' = add (mul m n') n'
⊢ mul (Nat.succ m) (Nat.succ n') = add (mul m (Nat.succ n')) (Nat.succ n') | no goals |
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 : ℕ
⊢ mul m n = mul n m | no goals |
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 : ℕ
⊢ mul Nat.zero n = mul n Nat.zero | no goals |
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' : ℕ
ih : mul m' n = mul n m'
⊢ mul (Nat.succ m') n = mul n (Nat.succ m') | case succ
n m' : ℕ
ih : mul m' n = mul n m'
⊢ add (mul n m') n = mul n (Nat.succ m') |
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' : ℕ
ih : mul m' n = mul n m'
⊢ add (mul n m') n = mul n (Nat.succ m') | no goals |
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 : ℕ
⊢ mul (mul l m) n = mul l (mul m n) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.mul_assoc | [134, 1] | [139, 49] | rfl | case zero
l m : ℕ
⊢ mul (mul l m) Nat.zero = mul l (mul m Nat.zero) | no goals |
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' : ℕ
ih : mul (mul l m) n' = mul l (mul m n')
⊢ mul (mul l m) (Nat.succ n') = mul l (mul m (Nat.succ n')) | no goals |
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 : ℕ
⊢ mul (add l m) n = add (mul n l) (mul n m) | l m n : ℕ
⊢ mul n (add l m) = add (mul n l) (mul n m) |
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 : ℕ
⊢ mul n (add l m) = add (mul n l) (mul n m) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.Peirce_of_EM | [182, 1] | [197, 19] | rw [ExcludedMiddle] | ⊢ ExcludedMiddle → Peirce | ⊢ (∀ (a : Prop), a ∨ ¬a) → Peirce |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.Peirce_of_EM | [182, 1] | [197, 19] | rw [Peirce] | ⊢ (∀ (a : Prop), a ∨ ¬a) → Peirce | ⊢ (∀ (a : Prop), a ∨ ¬a) → ∀ (a b : Prop), ((a → b) → a) → a |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Labs/Lab2Solution.lean | LoVe.BackwardProofs.Peirce_of_EM | [182, 1] | [197, 19] | intro hem | ⊢ (∀ (a : Prop), a ∨ ¬a) → ∀ (a b : Prop), ((a → b) → a) → a | hem : ∀ (a : Prop), a ∨ ¬a
⊢ ∀ (a b : Prop), ((a → b) → a) → a |
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