Is the operation "false < true" well defined?

Question

Does the C++ specification define:

1. the existence of the 'less than' operator for boolean parameters, and if so,
2. the result of the 4 parameter permutations?

In other words, are the results from the following operations defined by the specification?

false < false
false < true
true < false
true < true

On my setup (Centos 7, gcc 4.8.2) , the code below spits out what I'd expect (given C's history of representing false as 0 and true as 1):

false < false = false
false < true = true
true < false = false
true < true = false

Whilst I'm pretty sure most (all?) compilers will give the same output, is this legislated by the C++ specification? Or is an obfuscating, but specification-compliant compiler allowed to decide that true is less than false?

#include <iostream>

const char * s(bool a)
{
  return (a ? "true" : "false");
}

void test(bool a, bool b)
{
  std::cout << s(a) << " < " << s(b) << " = " << s(a < b) << std::endl;
}

int main(int argc, char* argv[])
{
  test(false, false);
  test(false, true);
  test(true, false);
  test(true, true);
  return 0;
}

Answer 1

TL;DR:

The operations are well defined according to the draft C++ standard.

Details

We can see that by going to the draft C++ standard section 5.9 Relational operators which says (emphasis mine going forward):

The operands shall have arithmetic, enumeration, or pointer type, or type std::nullptr_t. The operators < (less than), > (greater than), <= (less than or equal to), and >= (greater than or equal to) all yield false or true. The type of the result is bool

and bools are arithematic types from 3.9.1 Fundamental types

Types bool, char, char16_t, char32_t, wchar_t, and the signed and unsigned integer types are collectively called integral types.

and

Integral and floating types are collectively called arithmetic types.

and true and false are boolean literals from 2.14.6 Boolean literals:

boolean-literal:
    false
    true

Going back to section 5.9 to see the mechanics of the relational operators further, it says:

The usual arithmetic conversions are performed on operands of arithmetic or enumeration type.

the usual arithmetic conversions are covered in section 5 which says:

Otherwise, the integral promotions (4.5) shall be performed on both operands

and section 4.5 says:

A prvalue of type bool can be converted to a prvalue of type int, with false becoming zero and true becoming one.

and so the expressions:

false < false
false < true
true < false
true < true

using these rules become:

0 < 0
0 < 1
1 < 0
1 < 1

Answer 2

Boolean values are subject to the usual integer promotions, with false defined as 0 and true defined as 1. That makes all the comparisons well defined.

Answer 3

According to the C++ Standard (5.9 Relational operators)

2 The usual arithmetic conversions are performed on operands of arithmetic or enumeration type.

and

1...The type of the result is bool.

and (3.9.1 Fundamental types)

6 Values of type bool are either true or false.49 [ Note: There are no signed, unsigned, short, or long bool types or values. —end note ] Values of type bool participate in integral promotions (4.5).

and (4.5 Integral promotions)

6 A prvalue of type bool can be converted to a prvalue of type int, with false becoming zero and true becoming one.

So in all your examples true is converted to int 1 and false is converted to int 0

These expressions

false < false
false < true
true < false
true < true

are entirely equivalent to

0 < 0
0 < 1
1 < 0
1 < 1

Answer 4

Boolean false is equivalent to int 0, and boolean true is equivalent to int 1. So this explains why the expression false < true => 0 < 1 is the only one which returns true.