Datasets:
internlm
/
Lean-Github

Dataset card Data Studio Files Community
1
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stringclasses
147 values
commit
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file_path
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7
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stringlengths
6
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6
2.09M
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab2Solution.lean
LoVe.BackwardProofs.Peirce_of_EM
[182, 1]
[197, 19]
intro a b haba
hem : βˆ€ (a : Prop), a ∨ Β¬a ⊒ βˆ€ (a b : Prop), ((a β†’ b) β†’ a) β†’ a
hem : βˆ€ (a : Prop), a ∨ Β¬a a b : Prop haba : (a β†’ b) β†’ a ⊒ a
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab2Solution.lean
LoVe.BackwardProofs.Peirce_of_EM
[182, 1]
[197, 19]
apply Or.elim (hem a)
hem : βˆ€ (a : Prop), a ∨ Β¬a a b : Prop haba : (a β†’ b) β†’ a ⊒ a
case left hem : βˆ€ (a : Prop), a ∨ Β¬a a b : Prop haba : (a β†’ b) β†’ a ⊒ a β†’ a case right hem : βˆ€ (a : Prop), a ∨ Β¬a a b : Prop haba : (a β†’ b) β†’ a ⊒ Β¬a β†’ a
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab2Solution.lean
LoVe.BackwardProofs.Peirce_of_EM
[182, 1]
[197, 19]
{ intro ha assumption }
case left hem : βˆ€ (a : Prop), a ∨ Β¬a a b : Prop haba : (a β†’ b) β†’ a ⊒ a β†’ a case right hem : βˆ€ (a : Prop), a ∨ Β¬a a b : Prop haba : (a β†’ b) β†’ a ⊒ Β¬a β†’ a
case right hem : βˆ€ (a : Prop), a ∨ Β¬a a b : Prop haba : (a β†’ b) β†’ a ⊒ Β¬a β†’ a
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab2Solution.lean
LoVe.BackwardProofs.Peirce_of_EM
[182, 1]
[197, 19]
{ intro hna apply haba intro ha apply False.elim apply hna assumption }
case right hem : βˆ€ (a : Prop), a ∨ Β¬a a b : Prop haba : (a β†’ b) β†’ a ⊒ Β¬a β†’ a
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab2Solution.lean
LoVe.BackwardProofs.Peirce_of_EM
[182, 1]
[197, 19]
intro ha
case left hem : βˆ€ (a : Prop), a ∨ Β¬a a b : Prop haba : (a β†’ b) β†’ a ⊒ a β†’ a
case left hem : βˆ€ (a : Prop), a ∨ Β¬a a b : Prop haba : (a β†’ b) β†’ a ha : a ⊒ a
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab2Solution.lean
LoVe.BackwardProofs.Peirce_of_EM
[182, 1]
[197, 19]
assumption
case left hem : βˆ€ (a : Prop), a ∨ Β¬a a b : Prop haba : (a β†’ b) β†’ a ha : a ⊒ a
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab2Solution.lean
LoVe.BackwardProofs.Peirce_of_EM
[182, 1]
[197, 19]
intro hna
case right hem : βˆ€ (a : Prop), a ∨ Β¬a a b : Prop haba : (a β†’ b) β†’ a ⊒ Β¬a β†’ a
case right hem : βˆ€ (a : Prop), a ∨ Β¬a a b : Prop haba : (a β†’ b) β†’ a hna : Β¬a ⊒ a
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab2Solution.lean
LoVe.BackwardProofs.Peirce_of_EM
[182, 1]
[197, 19]
apply haba
case right hem : βˆ€ (a : Prop), a ∨ Β¬a a b : Prop haba : (a β†’ b) β†’ a hna : Β¬a ⊒ a
case right hem : βˆ€ (a : Prop), a ∨ Β¬a a b : Prop haba : (a β†’ b) β†’ a hna : Β¬a ⊒ a β†’ b
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab2Solution.lean
LoVe.BackwardProofs.Peirce_of_EM
[182, 1]
[197, 19]
intro ha
case right hem : βˆ€ (a : Prop), a ∨ Β¬a a b : Prop haba : (a β†’ b) β†’ a hna : Β¬a ⊒ a β†’ b
case right hem : βˆ€ (a : Prop), a ∨ Β¬a a b : Prop haba : (a β†’ b) β†’ a hna : Β¬a ha : a ⊒ b
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab2Solution.lean
LoVe.BackwardProofs.Peirce_of_EM
[182, 1]
[197, 19]
apply False.elim
case right hem : βˆ€ (a : Prop), a ∨ Β¬a a b : Prop haba : (a β†’ b) β†’ a hna : Β¬a ha : a ⊒ b
case right.h hem : βˆ€ (a : Prop), a ∨ Β¬a a b : Prop haba : (a β†’ b) β†’ a hna : Β¬a ha : a ⊒ False
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab2Solution.lean
LoVe.BackwardProofs.Peirce_of_EM
[182, 1]
[197, 19]
apply hna
case right.h hem : βˆ€ (a : Prop), a ∨ Β¬a a b : Prop haba : (a β†’ b) β†’ a hna : Β¬a ha : a ⊒ False
case right.h hem : βˆ€ (a : Prop), a ∨ Β¬a a b : Prop haba : (a β†’ b) β†’ a hna : Β¬a ha : a ⊒ a
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab2Solution.lean
LoVe.BackwardProofs.Peirce_of_EM
[182, 1]
[197, 19]
assumption
case right.h hem : βˆ€ (a : Prop), a ∨ Β¬a a b : Prop haba : (a β†’ b) β†’ a hna : Β¬a ha : a ⊒ a
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab2Solution.lean
LoVe.BackwardProofs.DN_of_Peirce
[201, 1]
[211, 14]
rw [Peirce]
⊒ Peirce β†’ DoubleNegation
⊒ (βˆ€ (a b : Prop), ((a β†’ b) β†’ a) β†’ a) β†’ DoubleNegation
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab2Solution.lean
LoVe.BackwardProofs.DN_of_Peirce
[201, 1]
[211, 14]
rw [DoubleNegation]
⊒ (βˆ€ (a b : Prop), ((a β†’ b) β†’ a) β†’ a) β†’ DoubleNegation
⊒ (βˆ€ (a b : Prop), ((a β†’ b) β†’ a) β†’ a) β†’ βˆ€ (a : Prop), ¬¬a β†’ a
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab2Solution.lean
LoVe.BackwardProofs.DN_of_Peirce
[201, 1]
[211, 14]
intro hpeirce a hnna
⊒ (βˆ€ (a b : Prop), ((a β†’ b) β†’ a) β†’ a) β†’ βˆ€ (a : Prop), ¬¬a β†’ a
hpeirce : βˆ€ (a b : Prop), ((a β†’ b) β†’ a) β†’ a a : Prop hnna : ¬¬a ⊒ a
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab2Solution.lean
LoVe.BackwardProofs.DN_of_Peirce
[201, 1]
[211, 14]
apply hpeirce a False
hpeirce : βˆ€ (a b : Prop), ((a β†’ b) β†’ a) β†’ a a : Prop hnna : ¬¬a ⊒ a
hpeirce : βˆ€ (a b : Prop), ((a β†’ b) β†’ a) β†’ a a : Prop hnna : ¬¬a ⊒ (a β†’ False) β†’ a
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab2Solution.lean
LoVe.BackwardProofs.DN_of_Peirce
[201, 1]
[211, 14]
intro hna
hpeirce : βˆ€ (a b : Prop), ((a β†’ b) β†’ a) β†’ a a : Prop hnna : ¬¬a ⊒ (a β†’ False) β†’ a
hpeirce : βˆ€ (a b : Prop), ((a β†’ b) β†’ a) β†’ a a : Prop hnna : ¬¬a hna : a β†’ False ⊒ a
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab2Solution.lean
LoVe.BackwardProofs.DN_of_Peirce
[201, 1]
[211, 14]
apply False.elim
hpeirce : βˆ€ (a b : Prop), ((a β†’ b) β†’ a) β†’ a a : Prop hnna : ¬¬a hna : a β†’ False ⊒ a
case h hpeirce : βˆ€ (a b : Prop), ((a β†’ b) β†’ a) β†’ a a : Prop hnna : ¬¬a hna : a β†’ False ⊒ False
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab2Solution.lean
LoVe.BackwardProofs.DN_of_Peirce
[201, 1]
[211, 14]
apply hnna
case h hpeirce : βˆ€ (a b : Prop), ((a β†’ b) β†’ a) β†’ a a : Prop hnna : ¬¬a hna : a β†’ False ⊒ False
case h hpeirce : βˆ€ (a b : Prop), ((a β†’ b) β†’ a) β†’ a a : Prop hnna : ¬¬a hna : a β†’ False ⊒ Β¬a
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab2Solution.lean
LoVe.BackwardProofs.DN_of_Peirce
[201, 1]
[211, 14]
exact hna
case h hpeirce : βˆ€ (a b : Prop), ((a β†’ b) β†’ a) β†’ a a : Prop hnna : ¬¬a hna : a β†’ False ⊒ Β¬a
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.Setoid_Iff
[74, 1]
[76, 9]
rfl
a b : Fraction ⊒ a β‰ˆ b ↔ a.num * b.denom = b.num * a.denom
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.add_num
[102, 9]
[104, 9]
rfl
a b : Fraction ⊒ (a + b).num = a.num * b.denom + b.num * a.denom
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.add_denom
[106, 9]
[108, 9]
rfl
a b : Fraction ⊒ (a + b).denom = a.denom * b.denom
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.Setoid_add
[110, 1]
[130, 17]
simp [Setoid_Iff, add_denom, add_num] at *
a a' b b' : Fraction ha : a β‰ˆ a' hb : b β‰ˆ b' ⊒ a + b β‰ˆ a' + b'
a a' b b' : Fraction ha : a.num * a'.denom = a'.num * a.denom hb : b.num * b'.denom = b'.num * b.denom ⊒ (a.num * b.denom + b.num * a.denom) * (a'.denom * b'.denom) = (a'.num * b'.denom + b'.num * a'.denom) * (a.denom * b.denom)
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.Setoid_add
[110, 1]
[130, 17]
calc (num a * denom b + num b * denom a) * (denom a' * denom b') = num a * denom a' * denom b * denom b' + num b * denom b' * denom a * denom a' := by simp [add_mul, mul_add] ac_rfl _ = num a' * denom a * denom b * denom b' + num b' * denom b * denom a * denom a' := by simp [*] _ = (num a' * denom b' + num b' * denom a') * (denom a * denom b) := by simp [add_mul, mul_add] ac_rfl
a a' b b' : Fraction ha : a.num * a'.denom = a'.num * a.denom hb : b.num * b'.denom = b'.num * b.denom ⊒ (a.num * b.denom + b.num * a.denom) * (a'.denom * b'.denom) = (a'.num * b'.denom + b'.num * a'.denom) * (a.denom * b.denom)
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.Setoid_add
[110, 1]
[130, 17]
simp [add_mul, mul_add]
a a' b b' : Fraction ha : a.num * a'.denom = a'.num * a.denom hb : b.num * b'.denom = b'.num * b.denom ⊒ (a.num * b.denom + b.num * a.denom) * (a'.denom * b'.denom) = a.num * a'.denom * b.denom * b'.denom + b.num * b'.denom * a.denom * a'.denom
a a' b b' : Fraction ha : a.num * a'.denom = a'.num * a.denom hb : b.num * b'.denom = b'.num * b.denom ⊒ a.num * b.denom * (a'.denom * b'.denom) + b.num * a.denom * (a'.denom * b'.denom) = a.num * a'.denom * b.denom * b'.denom + b.num * b'.denom * a.denom * a'.denom
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.Setoid_add
[110, 1]
[130, 17]
ac_rfl
a a' b b' : Fraction ha : a.num * a'.denom = a'.num * a.denom hb : b.num * b'.denom = b'.num * b.denom ⊒ a.num * b.denom * (a'.denom * b'.denom) + b.num * a.denom * (a'.denom * b'.denom) = a.num * a'.denom * b.denom * b'.denom + b.num * b'.denom * a.denom * a'.denom
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.Setoid_add
[110, 1]
[130, 17]
simp [*]
a a' b b' : Fraction ha : a.num * a'.denom = a'.num * a.denom hb : b.num * b'.denom = b'.num * b.denom ⊒ a.num * a'.denom * b.denom * b'.denom + b.num * b'.denom * a.denom * a'.denom = a'.num * a.denom * b.denom * b'.denom + b'.num * b.denom * a.denom * a'.denom
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.Setoid_add
[110, 1]
[130, 17]
simp [add_mul, mul_add]
a a' b b' : Fraction ha : a.num * a'.denom = a'.num * a.denom hb : b.num * b'.denom = b'.num * b.denom ⊒ a'.num * a.denom * b.denom * b'.denom + b'.num * b.denom * a.denom * a'.denom = (a'.num * b'.denom + b'.num * a'.denom) * (a.denom * b.denom)
a a' b b' : Fraction ha : a.num * a'.denom = a'.num * a.denom hb : b.num * b'.denom = b'.num * b.denom ⊒ a'.num * a.denom * b.denom * b'.denom + b'.num * b.denom * a.denom * a'.denom = a'.num * b'.denom * (a.denom * b.denom) + b'.num * a'.denom * (a.denom * b.denom)
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.Setoid_add
[110, 1]
[130, 17]
ac_rfl
a a' b b' : Fraction ha : a.num * a'.denom = a'.num * a.denom hb : b.num * b'.denom = b'.num * b.denom ⊒ a'.num * a.denom * b.denom * b'.denom + b'.num * b.denom * a.denom * a'.denom = a'.num * b'.denom * (a.denom * b.denom) + b'.num * a'.denom * (a.denom * b.denom)
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.neg_num
[137, 9]
[139, 9]
rfl
a : Fraction ⊒ (-a).num = -a.num
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.neg_denom
[141, 9]
[143, 9]
rfl
a : Fraction ⊒ (-a).denom = a.denom
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.Setoid_neg
[145, 1]
[149, 14]
simp [Setoid_Iff] at *
a a' : Fraction hab : a β‰ˆ a' ⊒ -a β‰ˆ -a'
a a' : Fraction hab : a.num * a'.denom = a'.num * a.denom ⊒ a.num * a'.denom = a'.num * a.denom
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.Setoid_neg
[145, 1]
[149, 14]
exact hab
a a' : Fraction hab : a.num * a'.denom = a'.num * a.denom ⊒ a.num * a'.denom = a'.num * a.denom
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.mul_num
[158, 9]
[160, 9]
rfl
a b : Fraction ⊒ (a * b).num = a.num * b.num
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.mul_denom
[162, 9]
[164, 9]
rfl
a b : Fraction ⊒ (a * b).denom = a.denom * b.denom
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.Setoid_mul
[166, 1]
[178, 18]
simp [Setoid_Iff] at *
a a' b b' : Fraction ha : a β‰ˆ a' hb : b β‰ˆ b' ⊒ a * b β‰ˆ a' * b'
a a' b b' : Fraction ha : a.num * a'.denom = a'.num * a.denom hb : b.num * b'.denom = b'.num * b.denom ⊒ a.num * b.num * (a'.denom * b'.denom) = a'.num * b'.num * (a.denom * b.denom)
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.Setoid_mul
[166, 1]
[178, 18]
calc num a * num b * (denom a' * denom b') = (num a * denom a') * (num b * denom b') := by ac_rfl _ = (num a' * denom a) * (num b' * denom b) := by simp [*] _ = num a' * num b' * (denom a * denom b) := by ac_rfl
a a' b b' : Fraction ha : a.num * a'.denom = a'.num * a.denom hb : b.num * b'.denom = b'.num * b.denom ⊒ a.num * b.num * (a'.denom * b'.denom) = a'.num * b'.num * (a.denom * b.denom)
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.Setoid_mul
[166, 1]
[178, 18]
ac_rfl
a a' b b' : Fraction ha : a.num * a'.denom = a'.num * a.denom hb : b.num * b'.denom = b'.num * b.denom ⊒ a.num * b.num * (a'.denom * b'.denom) = a.num * a'.denom * (b.num * b'.denom)
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.Setoid_mul
[166, 1]
[178, 18]
simp [*]
a a' b b' : Fraction ha : a.num * a'.denom = a'.num * a.denom hb : b.num * b'.denom = b'.num * b.denom ⊒ a.num * a'.denom * (b.num * b'.denom) = a'.num * a.denom * (b'.num * b.denom)
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.Setoid_mul
[166, 1]
[178, 18]
ac_rfl
a a' b b' : Fraction ha : a.num * a'.denom = a'.num * a.denom hb : b.num * b'.denom = b'.num * b.denom ⊒ a'.num * a.denom * (b'.num * b.denom) = a'.num * b'.num * (a.denom * b.denom)
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.inv_num
[200, 9]
[202, 23]
rw [inv_def a ha]
a : Fraction ha : a.num β‰  0 ⊒ a⁻¹.num = a.denom
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.inv_denom
[204, 9]
[206, 23]
rw [inv_def a ha]
a : Fraction ha : a.num β‰  0 ⊒ a⁻¹.denom = a.num
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.Setoid_inv
[208, 1]
[224, 17]
cases Classical.em (num a = 0) with | inl ha0 => cases Classical.em (num a' = 0) with | inl ha'0 => simp [ha0, ha'0, inv_zero] | inr ha'0 => simp [ha0, ha'0, Setoid_Iff, denom_Neq_zero] at ha | inr ha0 => cases Classical.em (num a' = 0) with | inl ha'0 => simp [ha0, ha'0, Setoid_Iff, denom_Neq_zero] at ha | inr ha'0 => simp [Setoid_Iff, ha0, ha'0] at * linarith
a a' : Fraction ha : a β‰ˆ a' ⊒ a⁻¹ β‰ˆ a'⁻¹
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.Setoid_inv
[208, 1]
[224, 17]
cases Classical.em (num a' = 0) with | inl ha'0 => simp [ha0, ha'0, inv_zero] | inr ha'0 => simp [ha0, ha'0, Setoid_Iff, denom_Neq_zero] at ha
case inl a a' : Fraction ha : a β‰ˆ a' ha0 : a.num = 0 ⊒ a⁻¹ β‰ˆ a'⁻¹
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.Setoid_inv
[208, 1]
[224, 17]
simp [ha0, ha'0, inv_zero]
case inl.inl a a' : Fraction ha : a β‰ˆ a' ha0 : a.num = 0 ha'0 : a'.num = 0 ⊒ a⁻¹ β‰ˆ a'⁻¹
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.Setoid_inv
[208, 1]
[224, 17]
simp [ha0, ha'0, Setoid_Iff, denom_Neq_zero] at ha
case inl.inr a a' : Fraction ha : a β‰ˆ a' ha0 : a.num = 0 ha'0 : Β¬a'.num = 0 ⊒ a⁻¹ β‰ˆ a'⁻¹
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.Setoid_inv
[208, 1]
[224, 17]
cases Classical.em (num a' = 0) with | inl ha'0 => simp [ha0, ha'0, Setoid_Iff, denom_Neq_zero] at ha | inr ha'0 => simp [Setoid_Iff, ha0, ha'0] at * linarith
case inr a a' : Fraction ha : a β‰ˆ a' ha0 : Β¬a.num = 0 ⊒ a⁻¹ β‰ˆ a'⁻¹
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.Setoid_inv
[208, 1]
[224, 17]
simp [ha0, ha'0, Setoid_Iff, denom_Neq_zero] at ha
case inr.inl a a' : Fraction ha : a β‰ˆ a' ha0 : Β¬a.num = 0 ha'0 : a'.num = 0 ⊒ a⁻¹ β‰ˆ a'⁻¹
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.Setoid_inv
[208, 1]
[224, 17]
simp [Setoid_Iff, ha0, ha'0] at *
case inr.inr a a' : Fraction ha : a β‰ˆ a' ha0 : Β¬a.num = 0 ha'0 : Β¬a'.num = 0 ⊒ a⁻¹ β‰ˆ a'⁻¹
case inr.inr a a' : Fraction ha : a.num * a'.denom = a'.num * a.denom ha0 ha'0 : True ⊒ a.denom * a'.num = a'.denom * a.num
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.Fraction.Setoid_inv
[208, 1]
[224, 17]
linarith
case inr.inr a a' : Fraction ha : a.num * a'.denom = a'.num * a.denom ha0 ha'0 : True ⊒ a.denom * a'.num = a'.denom * a.num
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.id_Not_CauchySeq
[343, 1]
[352, 34]
rw [IsCauchySeq]
⊒ Β¬IsCauchySeq fun n => ↑n
⊒ Β¬βˆ€ (Ξ΅ : β„š), Ξ΅ > 0 β†’ βˆƒ N, βˆ€ (m : β„•), m β‰₯ N β†’ |↑N - ↑m| < Ξ΅
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.id_Not_CauchySeq
[343, 1]
[352, 34]
intro h
⊒ Β¬βˆ€ (Ξ΅ : β„š), Ξ΅ > 0 β†’ βˆƒ N, βˆ€ (m : β„•), m β‰₯ N β†’ |↑N - ↑m| < Ξ΅
h : βˆ€ (Ξ΅ : β„š), Ξ΅ > 0 β†’ βˆƒ N, βˆ€ (m : β„•), m β‰₯ N β†’ |↑N - ↑m| < Ξ΅ ⊒ False
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.id_Not_CauchySeq
[343, 1]
[352, 34]
cases h 1 zero_lt_one with | intro i hi => have hi_succi := hi (i + 1) (by simp) simp [←sub_sub] at hi_succi
h : βˆ€ (Ξ΅ : β„š), Ξ΅ > 0 β†’ βˆƒ N, βˆ€ (m : β„•), m β‰₯ N β†’ |↑N - ↑m| < Ξ΅ ⊒ False
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.id_Not_CauchySeq
[343, 1]
[352, 34]
have hi_succi := hi (i + 1) (by simp)
case intro h : βˆ€ (Ξ΅ : β„š), Ξ΅ > 0 β†’ βˆƒ N, βˆ€ (m : β„•), m β‰₯ N β†’ |↑N - ↑m| < Ξ΅ i : β„• hi : βˆ€ (m : β„•), m β‰₯ i β†’ |↑i - ↑m| < 1 ⊒ False
case intro h : βˆ€ (Ξ΅ : β„š), Ξ΅ > 0 β†’ βˆƒ N, βˆ€ (m : β„•), m β‰₯ N β†’ |↑N - ↑m| < Ξ΅ i : β„• hi : βˆ€ (m : β„•), m β‰₯ i β†’ |↑i - ↑m| < 1 hi_succi : |↑i - ↑(i + 1)| < 1 ⊒ False
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.id_Not_CauchySeq
[343, 1]
[352, 34]
simp [←sub_sub] at hi_succi
case intro h : βˆ€ (Ξ΅ : β„š), Ξ΅ > 0 β†’ βˆƒ N, βˆ€ (m : β„•), m β‰₯ N β†’ |↑N - ↑m| < Ξ΅ i : β„• hi : βˆ€ (m : β„•), m β‰₯ i β†’ |↑i - ↑m| < 1 hi_succi : |↑i - ↑(i + 1)| < 1 ⊒ False
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.id_Not_CauchySeq
[343, 1]
[352, 34]
simp
h : βˆ€ (Ξ΅ : β„š), Ξ΅ > 0 β†’ βˆƒ N, βˆ€ (m : β„•), m β‰₯ N β†’ |↑N - ↑m| < Ξ΅ i : β„• hi : βˆ€ (m : β„•), m β‰₯ i β†’ |↑i - ↑m| < 1 ⊒ i + 1 β‰₯ i
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.CauchySeq.Setoid_iff
[413, 1]
[416, 9]
rfl
f g : CauchySeq ⊒ f β‰ˆ g ↔ βˆ€ (Ξ΅ : β„š), Ξ΅ > 0 β†’ βˆƒ N, βˆ€ (m : β„•), m β‰₯ N β†’ |seqOf f m - seqOf g m| < Ξ΅
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.CauchySeq.Setoid_add
[440, 1]
[474, 20]
intro Ξ΅β‚€ hΞ΅β‚€
f f' g g' : CauchySeq hf : f β‰ˆ f' hg : g β‰ˆ g' ⊒ f + g β‰ˆ f' + g'
f f' g g' : CauchySeq hf : f β‰ˆ f' hg : g β‰ˆ g' Ξ΅β‚€ : β„š hΞ΅β‚€ : Ξ΅β‚€ > 0 ⊒ βˆƒ N, βˆ€ (m : β„•), m β‰₯ N β†’ |seqOf (f + g) m - seqOf (f' + g') m| < Ξ΅β‚€
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.CauchySeq.Setoid_add
[440, 1]
[474, 20]
simp [Setoid_iff]
f f' g g' : CauchySeq hf : f β‰ˆ f' hg : g β‰ˆ g' Ξ΅β‚€ : β„š hΞ΅β‚€ : Ξ΅β‚€ > 0 ⊒ βˆƒ N, βˆ€ (m : β„•), m β‰₯ N β†’ |seqOf (f + g) m - seqOf (f' + g') m| < Ξ΅β‚€
f f' g g' : CauchySeq hf : f β‰ˆ f' hg : g β‰ˆ g' Ξ΅β‚€ : β„š hΞ΅β‚€ : Ξ΅β‚€ > 0 ⊒ βˆƒ N, βˆ€ (m : β„•), N ≀ m β†’ |seqOf (f + g) m - seqOf (f' + g') m| < Ξ΅β‚€
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.CauchySeq.Setoid_add
[440, 1]
[474, 20]
cases hf (Ξ΅β‚€ / 2) (half_pos hΞ΅β‚€) with | intro Nf hNf => cases hg (Ξ΅β‚€ / 2) (half_pos hΞ΅β‚€) with | intro Ng hNg => apply Exists.intro (max Nf Ng) intro m hm calc abs (seqOf (f + g) m - seqOf (f' + g') m) = abs ((seqOf f m + seqOf g m) - (seqOf f' m + seqOf g' m)) := by rfl _ = abs ((seqOf f m - seqOf f' m) + (seqOf g m - seqOf g' m)) := by have arg_eq : seqOf f m + seqOf g m - (seqOf f' m + seqOf g' m) = seqOf f m - seqOf f' m + (seqOf g m - seqOf g' m) := by linarith rw [arg_eq] _ ≀ abs (seqOf f m - seqOf f' m) + abs (seqOf g m - seqOf g' m) := by apply abs_add _ < Ξ΅β‚€ / 2 + Ξ΅β‚€ / 2 := add_lt_add (hNf m (le_of_max_le_left hm)) (hNg m (le_of_max_le_right hm)) _ = Ξ΅β‚€ := by simp
f f' g g' : CauchySeq hf : f β‰ˆ f' hg : g β‰ˆ g' Ξ΅β‚€ : β„š hΞ΅β‚€ : Ξ΅β‚€ > 0 ⊒ βˆƒ N, βˆ€ (m : β„•), N ≀ m β†’ |seqOf (f + g) m - seqOf (f' + g') m| < Ξ΅β‚€
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.CauchySeq.Setoid_add
[440, 1]
[474, 20]
cases hg (Ξ΅β‚€ / 2) (half_pos hΞ΅β‚€) with | intro Ng hNg => apply Exists.intro (max Nf Ng) intro m hm calc abs (seqOf (f + g) m - seqOf (f' + g') m) = abs ((seqOf f m + seqOf g m) - (seqOf f' m + seqOf g' m)) := by rfl _ = abs ((seqOf f m - seqOf f' m) + (seqOf g m - seqOf g' m)) := by have arg_eq : seqOf f m + seqOf g m - (seqOf f' m + seqOf g' m) = seqOf f m - seqOf f' m + (seqOf g m - seqOf g' m) := by linarith rw [arg_eq] _ ≀ abs (seqOf f m - seqOf f' m) + abs (seqOf g m - seqOf g' m) := by apply abs_add _ < Ξ΅β‚€ / 2 + Ξ΅β‚€ / 2 := add_lt_add (hNf m (le_of_max_le_left hm)) (hNg m (le_of_max_le_right hm)) _ = Ξ΅β‚€ := by simp
case intro f f' g g' : CauchySeq hf : f β‰ˆ f' hg : g β‰ˆ g' Ξ΅β‚€ : β„š hΞ΅β‚€ : Ξ΅β‚€ > 0 Nf : β„• hNf : βˆ€ (m : β„•), m β‰₯ Nf β†’ |seqOf f m - seqOf f' m| < Ξ΅β‚€ / 2 ⊒ βˆƒ N, βˆ€ (m : β„•), N ≀ m β†’ |seqOf (f + g) m - seqOf (f' + g') m| < Ξ΅β‚€
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.CauchySeq.Setoid_add
[440, 1]
[474, 20]
apply Exists.intro (max Nf Ng)
case intro.intro f f' g g' : CauchySeq hf : f β‰ˆ f' hg : g β‰ˆ g' Ξ΅β‚€ : β„š hΞ΅β‚€ : Ξ΅β‚€ > 0 Nf : β„• hNf : βˆ€ (m : β„•), m β‰₯ Nf β†’ |seqOf f m - seqOf f' m| < Ξ΅β‚€ / 2 Ng : β„• hNg : βˆ€ (m : β„•), m β‰₯ Ng β†’ |seqOf g m - seqOf g' m| < Ξ΅β‚€ / 2 ⊒ βˆƒ N, βˆ€ (m : β„•), N ≀ m β†’ |seqOf (f + g) m - seqOf (f' + g') m| < Ξ΅β‚€
case intro.intro f f' g g' : CauchySeq hf : f β‰ˆ f' hg : g β‰ˆ g' Ξ΅β‚€ : β„š hΞ΅β‚€ : Ξ΅β‚€ > 0 Nf : β„• hNf : βˆ€ (m : β„•), m β‰₯ Nf β†’ |seqOf f m - seqOf f' m| < Ξ΅β‚€ / 2 Ng : β„• hNg : βˆ€ (m : β„•), m β‰₯ Ng β†’ |seqOf g m - seqOf g' m| < Ξ΅β‚€ / 2 ⊒ βˆ€ (m : β„•), max Nf Ng ≀ m β†’ |seqOf (f + g) m - seqOf (f' + g') m| < Ξ΅β‚€
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.CauchySeq.Setoid_add
[440, 1]
[474, 20]
intro m hm
case intro.intro f f' g g' : CauchySeq hf : f β‰ˆ f' hg : g β‰ˆ g' Ξ΅β‚€ : β„š hΞ΅β‚€ : Ξ΅β‚€ > 0 Nf : β„• hNf : βˆ€ (m : β„•), m β‰₯ Nf β†’ |seqOf f m - seqOf f' m| < Ξ΅β‚€ / 2 Ng : β„• hNg : βˆ€ (m : β„•), m β‰₯ Ng β†’ |seqOf g m - seqOf g' m| < Ξ΅β‚€ / 2 ⊒ βˆ€ (m : β„•), max Nf Ng ≀ m β†’ |seqOf (f + g) m - seqOf (f' + g') m| < Ξ΅β‚€
case intro.intro f f' g g' : CauchySeq hf : f β‰ˆ f' hg : g β‰ˆ g' Ξ΅β‚€ : β„š hΞ΅β‚€ : Ξ΅β‚€ > 0 Nf : β„• hNf : βˆ€ (m : β„•), m β‰₯ Nf β†’ |seqOf f m - seqOf f' m| < Ξ΅β‚€ / 2 Ng : β„• hNg : βˆ€ (m : β„•), m β‰₯ Ng β†’ |seqOf g m - seqOf g' m| < Ξ΅β‚€ / 2 m : β„• hm : max Nf Ng ≀ m ⊒ |seqOf (f + g) m - seqOf (f' + g') m| < Ξ΅β‚€
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.CauchySeq.Setoid_add
[440, 1]
[474, 20]
calc abs (seqOf (f + g) m - seqOf (f' + g') m) = abs ((seqOf f m + seqOf g m) - (seqOf f' m + seqOf g' m)) := by rfl _ = abs ((seqOf f m - seqOf f' m) + (seqOf g m - seqOf g' m)) := by have arg_eq : seqOf f m + seqOf g m - (seqOf f' m + seqOf g' m) = seqOf f m - seqOf f' m + (seqOf g m - seqOf g' m) := by linarith rw [arg_eq] _ ≀ abs (seqOf f m - seqOf f' m) + abs (seqOf g m - seqOf g' m) := by apply abs_add _ < Ξ΅β‚€ / 2 + Ξ΅β‚€ / 2 := add_lt_add (hNf m (le_of_max_le_left hm)) (hNg m (le_of_max_le_right hm)) _ = Ξ΅β‚€ := by simp
case intro.intro f f' g g' : CauchySeq hf : f β‰ˆ f' hg : g β‰ˆ g' Ξ΅β‚€ : β„š hΞ΅β‚€ : Ξ΅β‚€ > 0 Nf : β„• hNf : βˆ€ (m : β„•), m β‰₯ Nf β†’ |seqOf f m - seqOf f' m| < Ξ΅β‚€ / 2 Ng : β„• hNg : βˆ€ (m : β„•), m β‰₯ Ng β†’ |seqOf g m - seqOf g' m| < Ξ΅β‚€ / 2 m : β„• hm : max Nf Ng ≀ m ⊒ |seqOf (f + g) m - seqOf (f' + g') m| < Ξ΅β‚€
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.CauchySeq.Setoid_add
[440, 1]
[474, 20]
rfl
f f' g g' : CauchySeq hf : f β‰ˆ f' hg : g β‰ˆ g' Ξ΅β‚€ : β„š hΞ΅β‚€ : Ξ΅β‚€ > 0 Nf : β„• hNf : βˆ€ (m : β„•), m β‰₯ Nf β†’ |seqOf f m - seqOf f' m| < Ξ΅β‚€ / 2 Ng : β„• hNg : βˆ€ (m : β„•), m β‰₯ Ng β†’ |seqOf g m - seqOf g' m| < Ξ΅β‚€ / 2 m : β„• hm : max Nf Ng ≀ m ⊒ |seqOf (f + g) m - seqOf (f' + g') m| = |seqOf f m + seqOf g m - (seqOf f' m + seqOf g' m)|
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.CauchySeq.Setoid_add
[440, 1]
[474, 20]
have arg_eq : seqOf f m + seqOf g m - (seqOf f' m + seqOf g' m) = seqOf f m - seqOf f' m + (seqOf g m - seqOf g' m) := by linarith
f f' g g' : CauchySeq hf : f β‰ˆ f' hg : g β‰ˆ g' Ξ΅β‚€ : β„š hΞ΅β‚€ : Ξ΅β‚€ > 0 Nf : β„• hNf : βˆ€ (m : β„•), m β‰₯ Nf β†’ |seqOf f m - seqOf f' m| < Ξ΅β‚€ / 2 Ng : β„• hNg : βˆ€ (m : β„•), m β‰₯ Ng β†’ |seqOf g m - seqOf g' m| < Ξ΅β‚€ / 2 m : β„• hm : max Nf Ng ≀ m ⊒ |seqOf f m + seqOf g m - (seqOf f' m + seqOf g' m)| = |seqOf f m - seqOf f' m + (seqOf g m - seqOf g' m)|
f f' g g' : CauchySeq hf : f β‰ˆ f' hg : g β‰ˆ g' Ξ΅β‚€ : β„š hΞ΅β‚€ : Ξ΅β‚€ > 0 Nf : β„• hNf : βˆ€ (m : β„•), m β‰₯ Nf β†’ |seqOf f m - seqOf f' m| < Ξ΅β‚€ / 2 Ng : β„• hNg : βˆ€ (m : β„•), m β‰₯ Ng β†’ |seqOf g m - seqOf g' m| < Ξ΅β‚€ / 2 m : β„• hm : max Nf Ng ≀ m arg_eq : seqOf f m + seqOf g m - (seqOf f' m + seqOf g' m) = seqOf f m - seqOf f' m + (seqOf g m - seqOf g' m) ⊒ |seqOf f m + seqOf g m - (seqOf f' m + seqOf g' m)| = |seqOf f m - seqOf f' m + (seqOf g m - seqOf g' m)|
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.CauchySeq.Setoid_add
[440, 1]
[474, 20]
rw [arg_eq]
f f' g g' : CauchySeq hf : f β‰ˆ f' hg : g β‰ˆ g' Ξ΅β‚€ : β„š hΞ΅β‚€ : Ξ΅β‚€ > 0 Nf : β„• hNf : βˆ€ (m : β„•), m β‰₯ Nf β†’ |seqOf f m - seqOf f' m| < Ξ΅β‚€ / 2 Ng : β„• hNg : βˆ€ (m : β„•), m β‰₯ Ng β†’ |seqOf g m - seqOf g' m| < Ξ΅β‚€ / 2 m : β„• hm : max Nf Ng ≀ m arg_eq : seqOf f m + seqOf g m - (seqOf f' m + seqOf g' m) = seqOf f m - seqOf f' m + (seqOf g m - seqOf g' m) ⊒ |seqOf f m + seqOf g m - (seqOf f' m + seqOf g' m)| = |seqOf f m - seqOf f' m + (seqOf g m - seqOf g' m)|
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.CauchySeq.Setoid_add
[440, 1]
[474, 20]
linarith
f f' g g' : CauchySeq hf : f β‰ˆ f' hg : g β‰ˆ g' Ξ΅β‚€ : β„š hΞ΅β‚€ : Ξ΅β‚€ > 0 Nf : β„• hNf : βˆ€ (m : β„•), m β‰₯ Nf β†’ |seqOf f m - seqOf f' m| < Ξ΅β‚€ / 2 Ng : β„• hNg : βˆ€ (m : β„•), m β‰₯ Ng β†’ |seqOf g m - seqOf g' m| < Ξ΅β‚€ / 2 m : β„• hm : max Nf Ng ≀ m ⊒ seqOf f m + seqOf g m - (seqOf f' m + seqOf g' m) = seqOf f m - seqOf f' m + (seqOf g m - seqOf g' m)
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.CauchySeq.Setoid_add
[440, 1]
[474, 20]
apply abs_add
f f' g g' : CauchySeq hf : f β‰ˆ f' hg : g β‰ˆ g' Ξ΅β‚€ : β„š hΞ΅β‚€ : Ξ΅β‚€ > 0 Nf : β„• hNf : βˆ€ (m : β„•), m β‰₯ Nf β†’ |seqOf f m - seqOf f' m| < Ξ΅β‚€ / 2 Ng : β„• hNg : βˆ€ (m : β„•), m β‰₯ Ng β†’ |seqOf g m - seqOf g' m| < Ξ΅β‚€ / 2 m : β„• hm : max Nf Ng ≀ m ⊒ |seqOf f m - seqOf f' m + (seqOf g m - seqOf g' m)| ≀ |seqOf f m - seqOf f' m| + |seqOf g m - seqOf g' m|
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Lectures/LoVe14_RationalAndRealNumbers_Demo.lean
LoVe.CauchySeq.Setoid_add
[440, 1]
[474, 20]
simp
f f' g g' : CauchySeq hf : f β‰ˆ f' hg : g β‰ˆ g' Ξ΅β‚€ : β„š hΞ΅β‚€ : Ξ΅β‚€ > 0 Nf : β„• hNf : βˆ€ (m : β„•), m β‰₯ Nf β†’ |seqOf f m - seqOf f' m| < Ξ΅β‚€ / 2 Ng : β„• hNg : βˆ€ (m : β„•), m β‰₯ Ng β†’ |seqOf g m - seqOf g' m| < Ξ΅β‚€ / 2 m : β„• hm : max Nf Ng ≀ m ⊒ Ξ΅β‚€ / 2 + Ξ΅β‚€ / 2 = Ξ΅β‚€
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.BTree.graft_assoc
[37, 1]
[42, 61]
induction t with | empty => rfl | node a l r ih_l ih_r => simp [BTree.graft, ih_l, ih_r]
α : Type t u v : BTree α ⊒ graft (graft t u) v = graft t (graft u v)
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.BTree.graft_assoc
[37, 1]
[42, 61]
rfl
case empty α : Type u v : BTree α ⊒ graft (graft empty u) v = graft empty (graft u v)
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.BTree.graft_assoc
[37, 1]
[42, 61]
simp [BTree.graft, ih_l, ih_r]
case node α : Type u v : BTree α a : α l r : BTree α ih_l : graft (graft l u) v = graft l (graft u v) ih_r : graft (graft r u) v = graft r (graft u v) ⊒ graft (graft (node a l r) u) v = graft (node a l r) (graft u v)
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.BTree.graft_empty
[44, 1]
[49, 61]
induction t with | empty => rfl | node a l r ih_l ih_r => simp [BTree.graft, ih_l, ih_r]
α : Type t : BTree α ⊒ graft t empty = t
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.BTree.graft_empty
[44, 1]
[49, 61]
rfl
case empty α : Type ⊒ graft empty empty = empty
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.BTree.graft_empty
[44, 1]
[49, 61]
simp [BTree.graft, ih_l, ih_r]
case node α : Type a : α l r : BTree α ih_l : graft l empty = l ih_r : graft r empty = r ⊒ graft (node a l r) empty = node a l r
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.BTree.add_left_neg_counterexample
[87, 1]
[94, 30]
apply Exists.intro (BTree.node 0 BTree.empty BTree.empty)
⊒ βˆƒ x, βˆ€ (y : BTree β„•), graft y x β‰  empty
⊒ βˆ€ (y : BTree β„•), graft y (node 0 empty empty) β‰  empty
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.BTree.add_left_neg_counterexample
[87, 1]
[94, 30]
intro y hy
⊒ βˆ€ (y : BTree β„•), graft y (node 0 empty empty) β‰  empty
y : BTree β„• hy : graft y (node 0 empty empty) = empty ⊒ False
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.BTree.add_left_neg_counterexample
[87, 1]
[94, 30]
cases y with | empty => cases hy | node a l r => cases' hy
y : BTree β„• hy : graft y (node 0 empty empty) = empty ⊒ False
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.BTree.add_left_neg_counterexample
[87, 1]
[94, 30]
cases hy
case empty hy : graft empty (node 0 empty empty) = empty ⊒ False
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.BTree.add_left_neg_counterexample
[87, 1]
[94, 30]
cases' hy
case node a : β„• l r : BTree β„• hy : graft (node a l r) (node 0 empty empty) = empty ⊒ False
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.Multiset.elems_mirror
[114, 1]
[128, 33]
induction t with | empty => rfl | node a l r ih_l ih_r => rw [Multiset.elems] rw [mirror] rw [←ih_l] rw [←ih_r] rw [Multiset.elems] rw [Multiset.union_assoc, Multiset.union_comm (elems (mirror r)), ←Multiset.union_assoc]
t : BTree β„• ⊒ elems (mirror t) = elems t
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.Multiset.elems_mirror
[114, 1]
[128, 33]
rfl
case empty ⊒ elems (mirror BTree.empty) = elems BTree.empty
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.Multiset.elems_mirror
[114, 1]
[128, 33]
rw [Multiset.elems]
case node a : β„• l r : BTree β„• ih_l : elems (mirror l) = elems l ih_r : elems (mirror r) = elems r ⊒ elems (mirror (BTree.node a l r)) = elems (BTree.node a l r)
case node a : β„• l r : BTree β„• ih_l : elems (mirror l) = elems l ih_r : elems (mirror r) = elems r ⊒ elems (mirror (BTree.node a l r)) = {a} βˆͺ elems l βˆͺ elems r
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.Multiset.elems_mirror
[114, 1]
[128, 33]
rw [mirror]
case node a : β„• l r : BTree β„• ih_l : elems (mirror l) = elems l ih_r : elems (mirror r) = elems r ⊒ elems (mirror (BTree.node a l r)) = {a} βˆͺ elems l βˆͺ elems r
case node a : β„• l r : BTree β„• ih_l : elems (mirror l) = elems l ih_r : elems (mirror r) = elems r ⊒ elems (BTree.node a (mirror r) (mirror l)) = {a} βˆͺ elems l βˆͺ elems r
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.Multiset.elems_mirror
[114, 1]
[128, 33]
rw [←ih_l]
case node a : β„• l r : BTree β„• ih_l : elems (mirror l) = elems l ih_r : elems (mirror r) = elems r ⊒ elems (BTree.node a (mirror r) (mirror l)) = {a} βˆͺ elems l βˆͺ elems r
case node a : β„• l r : BTree β„• ih_l : elems (mirror l) = elems l ih_r : elems (mirror r) = elems r ⊒ elems (BTree.node a (mirror r) (mirror l)) = {a} βˆͺ elems (mirror l) βˆͺ elems r
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.Multiset.elems_mirror
[114, 1]
[128, 33]
rw [←ih_r]
case node a : β„• l r : BTree β„• ih_l : elems (mirror l) = elems l ih_r : elems (mirror r) = elems r ⊒ elems (BTree.node a (mirror r) (mirror l)) = {a} βˆͺ elems (mirror l) βˆͺ elems r
case node a : β„• l r : BTree β„• ih_l : elems (mirror l) = elems l ih_r : elems (mirror r) = elems r ⊒ elems (BTree.node a (mirror r) (mirror l)) = {a} βˆͺ elems (mirror l) βˆͺ elems (mirror r)
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.Multiset.elems_mirror
[114, 1]
[128, 33]
rw [Multiset.elems]
case node a : β„• l r : BTree β„• ih_l : elems (mirror l) = elems l ih_r : elems (mirror r) = elems r ⊒ elems (BTree.node a (mirror r) (mirror l)) = {a} βˆͺ elems (mirror l) βˆͺ elems (mirror r)
case node a : β„• l r : BTree β„• ih_l : elems (mirror l) = elems l ih_r : elems (mirror r) = elems r ⊒ {a} βˆͺ elems (mirror r) βˆͺ elems (mirror l) = {a} βˆͺ elems (mirror l) βˆͺ elems (mirror r)
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.Multiset.elems_mirror
[114, 1]
[128, 33]
rw [Multiset.union_assoc, Multiset.union_comm (elems (mirror r)), ←Multiset.union_assoc]
case node a : β„• l r : BTree β„• ih_l : elems (mirror l) = elems l ih_r : elems (mirror r) = elems r ⊒ {a} βˆͺ elems (mirror r) βˆͺ elems (mirror l) = {a} βˆͺ elems (mirror l) βˆͺ elems (mirror r)
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.Finset.elems_mirror
[133, 1]
[145, 13]
induction t with | empty => rfl | node a l r ih_l ih_r => rw [Finset.elems] rw [mirror] rw [←ih_l] rw [←ih_r] rw [Finset.elems] ac_rfl
t : BTree β„• ⊒ elems (mirror t) = elems t
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.Finset.elems_mirror
[133, 1]
[145, 13]
rfl
case empty ⊒ elems (mirror BTree.empty) = elems BTree.empty
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.Finset.elems_mirror
[133, 1]
[145, 13]
rw [Finset.elems]
case node a : β„• l r : BTree β„• ih_l : elems (mirror l) = elems l ih_r : elems (mirror r) = elems r ⊒ elems (mirror (BTree.node a l r)) = elems (BTree.node a l r)
case node a : β„• l r : BTree β„• ih_l : elems (mirror l) = elems l ih_r : elems (mirror r) = elems r ⊒ elems (mirror (BTree.node a l r)) = {a} βˆͺ elems l βˆͺ elems r
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.Finset.elems_mirror
[133, 1]
[145, 13]
rw [mirror]
case node a : β„• l r : BTree β„• ih_l : elems (mirror l) = elems l ih_r : elems (mirror r) = elems r ⊒ elems (mirror (BTree.node a l r)) = {a} βˆͺ elems l βˆͺ elems r
case node a : β„• l r : BTree β„• ih_l : elems (mirror l) = elems l ih_r : elems (mirror r) = elems r ⊒ elems (BTree.node a (mirror r) (mirror l)) = {a} βˆͺ elems l βˆͺ elems r
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.Finset.elems_mirror
[133, 1]
[145, 13]
rw [←ih_l]
case node a : β„• l r : BTree β„• ih_l : elems (mirror l) = elems l ih_r : elems (mirror r) = elems r ⊒ elems (BTree.node a (mirror r) (mirror l)) = {a} βˆͺ elems l βˆͺ elems r
case node a : β„• l r : BTree β„• ih_l : elems (mirror l) = elems l ih_r : elems (mirror r) = elems r ⊒ elems (BTree.node a (mirror r) (mirror l)) = {a} βˆͺ elems (mirror l) βˆͺ elems r
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.Finset.elems_mirror
[133, 1]
[145, 13]
rw [←ih_r]
case node a : β„• l r : BTree β„• ih_l : elems (mirror l) = elems l ih_r : elems (mirror r) = elems r ⊒ elems (BTree.node a (mirror r) (mirror l)) = {a} βˆͺ elems (mirror l) βˆͺ elems r
case node a : β„• l r : BTree β„• ih_l : elems (mirror l) = elems l ih_r : elems (mirror r) = elems r ⊒ elems (BTree.node a (mirror r) (mirror l)) = {a} βˆͺ elems (mirror l) βˆͺ elems (mirror r)
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.Finset.elems_mirror
[133, 1]
[145, 13]
rw [Finset.elems]
case node a : β„• l r : BTree β„• ih_l : elems (mirror l) = elems l ih_r : elems (mirror r) = elems r ⊒ elems (BTree.node a (mirror r) (mirror l)) = {a} βˆͺ elems (mirror l) βˆͺ elems (mirror r)
case node a : β„• l r : BTree β„• ih_l : elems (mirror l) = elems l ih_r : elems (mirror r) = elems r ⊒ {a} βˆͺ elems (mirror r) βˆͺ elems (mirror l) = {a} βˆͺ elems (mirror l) βˆͺ elems (mirror r)
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.Finset.elems_mirror
[133, 1]
[145, 13]
ac_rfl
case node a : β„• l r : BTree β„• ih_l : elems (mirror l) = elems l ih_r : elems (mirror r) = elems r ⊒ {a} βˆͺ elems (mirror r) βˆͺ elems (mirror l) = {a} βˆͺ elems (mirror l) βˆͺ elems (mirror r)
no goals
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.List.elems_mirror_counterexample
[159, 1]
[163, 22]
apply Exists.intro badTree
⊒ βˆƒ t, elems t β‰  elems (mirror t)
⊒ elems badTree β‰  elems (mirror badTree)
https://github.com/BrownCS1951x/fpv2023.git
9aaf6b5c454aa9a70fc4e6807adf3123b001ea66
LoVe/Labs/Lab7Solution.lean
LoVe.List.elems_mirror_counterexample
[159, 1]
[163, 22]
simp [List.elems]
⊒ elems badTree β‰  elems (mirror badTree)
no goals