lean2/tests/lean/run/rewriter2.lean

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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:A) :=
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
reveal test3
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