58 lines
2.2 KiB
Text
58 lines
2.2 KiB
Text
variable N : Type
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variable h : N -> N -> N
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theorem congrH {a1 a2 b1 b2 : N} (H1 : a1 = b1) (H2 : a2 = b2) : (h a1 a2) = (h b1 b2) :=
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congr (congr (refl h) H1) H2
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-- Display the theorem showing implicit arguments
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set::option lean::pp::implicit true
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print environment 2
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-- Display the theorem hiding implicit arguments
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set::option lean::pp::implicit false
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print environment 2
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theorem Example1 (a b c d : N) (H: (a = b ∧ b = c) ∨ (a = d ∧ d = c)) : (h a b) = (h c b) :=
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or::elim H
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(fun H1 : a = b ∧ b = c,
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congrH (trans (and::eliml H1) (and::elimr H1)) (refl b))
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(fun H1 : a = d ∧ d = c,
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congrH (trans (and::eliml H1) (and::elimr H1)) (refl b))
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-- print proof of the last theorem with all implicit arguments
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set::option lean::pp::implicit true
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print environment 1
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-- Using placeholders to hide the type of H1
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theorem Example2 (a b c d : N) (H: (a = b ∧ b = c) ∨ (a = d ∧ d = c)) : (h a b) = (h c b) :=
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or::elim H
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(fun H1 : _,
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congrH (trans (and::eliml H1) (and::elimr H1)) (refl b))
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(fun H1 : _,
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congrH (trans (and::eliml H1) (and::elimr H1)) (refl b))
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set::option lean::pp::implicit true
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print environment 1
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-- Same example but the first conjuct has unnecessary stuff
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theorem Example3 (a b c d e : N) (H: (a = b ∧ b = e ∧ b = c) ∨ (a = d ∧ d = c)) : (h a b) = (h c b) :=
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or::elim H
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(fun H1 : _,
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congrH (trans (and::eliml H1) (and::elimr (and::elimr H1))) (refl b))
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(fun H1 : _,
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congrH (trans (and::eliml H1) (and::elimr H1)) (refl b))
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set::option lean::pp::implicit false
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print environment 1
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theorem Example4 (a b c d e : N) (H: (a = b ∧ b = e ∧ b = c) ∨ (a = d ∧ d = c)) : (h a c) = (h c a) :=
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or::elim H
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(fun H1 : _,
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let AeqC := trans (and::eliml H1) (and::elimr (and::elimr H1))
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in congrH AeqC (symm AeqC))
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(fun H1 : _,
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let AeqC := trans (and::eliml H1) (and::elimr H1)
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in congrH AeqC (symm AeqC))
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set::option lean::pp::implicit false
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print environment 1
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