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