test(tests/lean/run): add more rewrite tactic examples

This commit is contained in:
Leonardo de Moura 2015-02-04 19:19:46 -08:00
parent 08dcc9e2fc
commit d0171ffe7a

View file

@ -0,0 +1,42 @@
import data.nat
open algebra
constant f {A : Type} : A → A → A
theorem test1 {A : Type} [s : comm_ring A] (a b c : A) : f (a + 0) (f (a + 0) (a + 0)) = f a (f (0 + a) a) :=
begin
rewrite
add_zero at {1 3} -- rewrite 1st and 3rd occurrences
[0 + _]add.comm -- apply commutativity to (0 + _)
end
check @mul_zero
axiom Ax {A : Type} [s₁ : has_mul A] [s₂ : has_zero A] (a : A) : f (a * 0) (a * 0) = 0
theorem test2 {A : Type} [s : comm_ring A] (a b c : A) : f 0 0 = 0 :=
begin
rewrite
-(mul_zero a) at {1 2} -- - means apply symmetry, rewrite 0 ==> a * 0 at 1st and 2nd occurrences
Ax -- use Ax as rewrite rule
end
theorem test3 {A : Type} [s : comm_ring A] (a b c : A) : a * 0 + 0 * b + c * 0 + 0 * a = 0 :=
begin
rewrite +mul_zero +zero_mul +add_zero -- in rewrite rules, + is notation for one or more
end
print definition test3
theorem test4 {A : Type} [s : comm_ring A] (a b c : A) : a * 0 + 0 * b + c * 0 + 0 * a = 0 :=
begin
rewrite *mul_zero *zero_mul *add_zero *zero_add -- in rewrite rules, * is notation for zero or more
end
theorem test5 {A : Type} [s : comm_ring A] (a b c : A) : a * 0 + 0 * b + c * 0 + 0 * a = 0 :=
begin
rewrite
2 mul_zero -- apply mul_zero exactly twice
2 zero_mul -- apply zero_mul exactly twice
5>add_zero -- apply add_zero at most 5 times
end