auto merge of #11664 : bjz/rust/identities, r=alexcrichton

`Zero` and `One` have precise definitions in mathematics as the identities of the `Add` and `Mul` operations respectively. As such, types that implement these identities are now also required to implement their respective operator traits. This should reduce their misuse whilst still enabling them to be used in generalized algebraic structures (not just numbers). Existing usages of `#[deriving(Zero)]` in client code could break under these new rules, but this is probably a sign that they should have been using something like `#[deriving(Default)]` in the first place.

For more information regarding the mathematical definitions of the additive and multiplicative identities, see the following Wikipedia articles:

- http://wikipedia.org/wiki/Additive_identity
- http://wikipedia.org/wiki/Multiplicative_identity

Note that for floating point numbers the laws specified in the doc comments of `Zero::zero` and `One::one` may not always hold. This is true however for many other traits currently implemented by floating point numbers. What traits floating point numbers should and should not implement is an open question that is beyond the scope of this pull request.

The implementation of `std::num::pow` has been made more succinct and no longer requires `Clone`. The coverage of the associated unit test has also been increased to test for more combinations of bases, exponents, and expected results.

6 files changed, 82 insertions, 98 deletions

diff --git a/src/libstd/bool.rs b/src/libstd/bool.rs
index d080262ccc7..af745f94fb5 100644
--- a/src/libstd/bool.rs
+++ b/src/libstd/bool.rs
@@ -40,7 +40,6 @@ use num::FromPrimitive;
 #[cfg(not(test))] use cmp::{Eq, Ord, TotalOrd, Ordering};
 #[cfg(not(test))] use ops::{Not, BitAnd, BitOr, BitXor};
 #[cfg(not(test))] use default::Default;
-#[cfg(not(test))] use num::Zero;
 
 /////////////////////////////////////////////////////////////////////////////
 // Freestanding functions
@@ -309,12 +308,6 @@ impl Default for bool {
     fn default() -> bool { false }
 }
 
-#[cfg(not(test))]
-impl Zero for bool {
-    fn zero() -> bool { false }
-    fn is_zero(&self) -> bool { *self == false }
-}
-
 #[cfg(test)]
 mod tests {
     use prelude::*;
diff --git a/src/libstd/char.rs b/src/libstd/char.rs
index 4e9c72de618..71a297d7176 100644
--- a/src/libstd/char.rs
+++ b/src/libstd/char.rs
@@ -22,7 +22,6 @@ use str;
 
 #[cfg(not(test))] use cmp::{Eq, Ord};
 #[cfg(not(test))] use default::Default;
-#[cfg(not(test))] use num::Zero;
 
 // UTF-8 ranges and tags for encoding characters
 static TAG_CONT: uint = 128u;
@@ -449,15 +448,6 @@ impl Default for char {
     fn default() -> char { '\x00' }
 }
 
-#[cfg(not(test))]
-impl Zero for char {
-    #[inline]
-    fn zero() -> char { '\x00' }
-
-    #[inline]
-    fn is_zero(&self) -> bool { *self == '\x00' }
-}
-
 #[test]
 fn test_is_lowercase() {
     assert!('a'.is_lowercase());
diff --git a/src/libstd/iter.rs b/src/libstd/iter.rs
index fcf0f4f2444..8081c6ed8db 100644
--- a/src/libstd/iter.rs
+++ b/src/libstd/iter.rs
@@ -2872,6 +2872,12 @@ mod tests {
             }
         }
 
+        impl Mul<Foo, Foo> for Foo {
+            fn mul(&self, _: &Foo) -> Foo {
+                Foo
+            }
+        }
+
         impl num::One for Foo {
             fn one() -> Foo {
                 Foo
diff --git a/src/libstd/num/mod.rs b/src/libstd/num/mod.rs
index c374d6c2157..34dd313d442 100644
--- a/src/libstd/num/mod.rs
+++ b/src/libstd/num/mod.rs
@@ -50,19 +50,59 @@ pub trait Orderable: Ord {
 /// Returns the number constrained within the range `mn <= self <= mx`.
 #[inline(always)] pub fn clamp<T: Orderable>(value: T, mn: T, mx: T) -> T { value.clamp(&mn, &mx) }
 
-pub trait Zero {
-    fn zero() -> Self;      // FIXME (#5527): This should be an associated constant
+/// Defines an additive identity element for `Self`.
+///
+/// # Deriving
+///
+/// This trait can be automatically be derived using `#[deriving(Zero)]`
+/// attribute. If you choose to use this, make sure that the laws outlined in
+/// the documentation for `Zero::zero` still hold.
+pub trait Zero: Add<Self, Self> {
+    /// Returns the additive identity element of `Self`, `0`.
+    ///
+    /// # Laws
+    ///
+    /// ~~~
+    /// a + 0 = a       ∀ a ∈ Self
+    /// 0 + a = a       ∀ a ∈ Self
+    /// ~~~
+    ///
+    /// # Purity
+    ///
+    /// This function should return the same result at all times regardless of
+    /// external mutable state, for example values stored in TLS or in
+    /// `static mut`s.
+    // FIXME (#5527): This should be an associated constant
+    fn zero() -> Self;
+
+    /// Returns `true` if `self` is equal to the additive identity.
     fn is_zero(&self) -> bool;
 }
 
-/// Returns `0` of appropriate type.
+/// Returns the additive identity, `0`.
 #[inline(always)] pub fn zero<T: Zero>() -> T { Zero::zero() }
 
-pub trait One {
-    fn one() -> Self;       // FIXME (#5527): This should be an associated constant
+/// Defines a multiplicative identity element for `Self`.
+pub trait One: Mul<Self, Self> {
+    /// Returns the multiplicative identity element of `Self`, `1`.
+    ///
+    /// # Laws
+    ///
+    /// ~~~
+    /// a * 1 = a       ∀ a ∈ Self
+    /// 1 * a = a       ∀ a ∈ Self
+    /// ~~~
+    ///
+    /// # Purity
+    ///
+    /// This function should return the same result at all times regardless of
+    /// external mutable state, for example values stored in TLS or in
+    /// `static mut`s.
+    // FIXME (#5527): This should be an associated constant
+    fn one() -> Self;
 }
 
-/// Returns `1` of appropriate type.
+/// Returns the multiplicative identity, `1`.
 #[inline(always)] pub fn one<T: One>() -> T { One::one() }
 
 pub trait Signed: Num
@@ -264,48 +304,29 @@ pub trait Real: Signed
     fn to_radians(&self) -> Self;
 }
 
-/// Raises a value to the power of exp, using
-/// exponentiation by squaring.
+/// Raises a value to the power of exp, using exponentiation by squaring.
 ///
 /// # Example
 ///
 /// ```rust
 /// use std::num;
 ///
-/// let sixteen = num::pow(2, 4u);
-/// assert_eq!(sixteen, 16);
+/// assert_eq!(num::pow(2, 4), 16);
 /// ```
 #[inline]
-pub fn pow<T: Clone+One+Mul<T, T>>(num: T, exp: uint) -> T {
-    let one: uint = One::one();
-    let num_one: T = One::one();
-
-    if exp.is_zero() { return num_one; }
-    if exp == one { return num.clone(); }
-
-    let mut i: uint = exp;
-    let mut v: T;
-    let mut r: T = num_one;
-
-    // This if is to avoid cloning self.
-    if (i & one) == one {
-        r = r * num;
-        i = i - one;
-    }
-
-    i = i >> one;
-    v = num * num;
-
-    while !i.is_zero() {
-        if (i & one) == one {
-            r = r * v;
-            i = i - one;
+pub fn pow<T: One + Mul<T, T>>(mut base: T, mut exp: uint) -> T {
+    if exp == 1 { base }
+    else {
+        let mut acc = one::<T>();
+        while exp > 0 {
+            if (exp & 1) == 1 {
+                acc = acc * base;
+            }
+            base = base * base;
+            exp = exp >> 1;
         }
-        i = i >> one;
-        v = v * v;
+        acc
     }
-
-    r
 }
 
 /// Raise a number to a power.
@@ -993,16 +1014,6 @@ pub fn from_str_radix<T: FromStrRadix>(str: &str, radix: uint) -> Option<T> {
     FromStrRadix::from_str_radix(str, radix)
 }
 
-impl<T: Zero + 'static> Zero for @T {
-    fn zero() -> @T { @Zero::zero() }
-    fn is_zero(&self) -> bool { (**self).is_zero() }
-}
-
-impl<T: Zero> Zero for ~T {
-    fn zero() -> ~T { ~Zero::zero() }
-    fn is_zero(&self) -> bool { (**self).is_zero() }
-}
-
 /// Saturating math operations
 pub trait Saturating {
     /// Saturating addition operator.
@@ -1640,17 +1651,24 @@ mod tests {
 
     #[test]
     fn test_pow() {
-        fn assert_pow<T: Eq+Clone+One+Mul<T, T>>(num: T, exp: uint) -> () {
-            assert_eq!(num::pow(num.clone(), exp),
-                       range(1u, exp).fold(num.clone(), |acc, _| acc * num));
+        fn naive_pow<T: One + Mul<T, T>>(base: T, exp: uint) -> T {
+            range(0, exp).fold(one::<T>(), |acc, _| acc * base)
         }
-
-        assert_eq!(num::pow(3, 0), 1);
-        assert_eq!(num::pow(5, 1), 5);
-        assert_pow(-4, 2);
-        assert_pow(8, 3);
-        assert_pow(8, 5);
-        assert_pow(2u64, 50);
+        macro_rules! assert_pow(
+            (($num:expr, $exp:expr) => $expected:expr) => {{
+                let result = pow($num, $exp);
+                assert_eq!(result, $expected);
+                assert_eq!(result, naive_pow($num, $exp));
+            }}
+        )
+        assert_pow!((3,    0 ) => 1);
+        assert_pow!((5,    1 ) => 5);
+        assert_pow!((-4,   2 ) => 16);
+        assert_pow!((0.5,  5 ) => 0.03125);
+        assert_pow!((8,    3 ) => 512);
+        assert_pow!((8.0,  5 ) => 32768.0);
+        assert_pow!((8.5,  5 ) => 44370.53125);
+        assert_pow!((2u64, 50) => 1125899906842624);
     }
 }
 
diff --git a/src/libstd/tuple.rs b/src/libstd/tuple.rs
index 313fd9c79b4..8e278aeb2ea 100644
--- a/src/libstd/tuple.rs
+++ b/src/libstd/tuple.rs
@@ -15,7 +15,6 @@
 use clone::Clone;
 #[cfg(not(test))] use cmp::*;
 #[cfg(not(test))] use default::Default;
-#[cfg(not(test))] use num::Zero;
 
 /// Method extensions to pairs where both types satisfy the `Clone` bound
 pub trait CopyableTuple<T, U> {
@@ -177,18 +176,6 @@ macro_rules! tuple_impls {
                     ($({ let x: $T = Default::default(); x},)+)
                 }
             }
-
-            #[cfg(not(test))]
-            impl<$($T:Zero),+> Zero for ($($T,)+) {
-                #[inline]
-                fn zero() -> ($($T,)+) {
-                    ($({ let x: $T = Zero::zero(); x},)+)
-                }
-                #[inline]
-                fn is_zero(&self) -> bool {
-                    $(self.$get_ref_fn().is_zero())&&+
-                }
-            }
         )+
     }
 }
diff --git a/src/libstd/unit.rs b/src/libstd/unit.rs
index c27f6e3d086..786a7f42bb3 100644
--- a/src/libstd/unit.rs
+++ b/src/libstd/unit.rs
@@ -12,8 +12,6 @@
 
 #[cfg(not(test))]
 use prelude::*;
-#[cfg(not(test))]
-use num::Zero;
 
 #[cfg(not(test))]
 impl Eq for () {
@@ -46,11 +44,3 @@ impl Default for () {
     #[inline]
     fn default() -> () { () }
 }
-
-#[cfg(not(test))]
-impl Zero for () {
-    #[inline]
-    fn zero() -> () { () }
-    #[inline]
-    fn is_zero(&self) -> bool { true }
-}