^f(x) ::= x f(x)

##</code><code>##

    0   - f(x)
    1   - x f(x)
    2   - x x f(x)
    3   - x x x f(x)
    ...

##</code><code>##

    ^g(x) ::= x x

##</code><code>##

    g(f(x))

##</code><code>##

    0   - g(f(x))
    1   - f(x) f(x)
    2   - x f(x) x f(x)
    3   - x x f(x) x x f(x)
    ...

##</code><code>##

    my $x = 3;
    my $y = 3;
    
    if (3 == $x) {                  # substitutability works
        print "$x equals $y.\n";
    }
    
    $x++;
    
    if (3 == $y) {                  # substitutability fails
        print "$x equals $y.\n";
    }

##</code><code>##

    my $x = 3;      # start frame of reference for variable $x
                    # start frame of reference for value $x==3

    my $y = 3;      # start frame of reference for variable $y
                    # start frame of reference for value $y==3
    
    if (3 == $x) {
        print "$x equals $y.\n";
    }
    
                    # end frame of reference for value $x==3
    $x++;           # start frame of reference for value $x==4
    
    if (3 == $y) {
        print "$x equals $y.\n";
    }
    
                    # end frame of reference for value $x==4
                    # end frame of reference for variable $x
                    # end frame of reference for value $y==3
                    # end frame of reference for variable $y

##</code><code>##

    p1  ::= 'the morning star' equals 'the planet venus'
    p2  ::= 'the evening star' equals 'the planet venus'
    p3  ::= 'the morning star' equals 'the evening star'
    
    (+) ::= an operator that means 'two things which equal 
            the same thing, equal each other'

    =>  ::= an operator which means 'this rule produces this result'

##</code><code>##

    p1 (+) p2 => p3
    p1 (+) p3 => p2
    p2 (+) p3 => p1

##</code><code>##

    j1  ::= john knows 'the morning star' equals 'the planet venus'
    j2  ::= john knows 'the evening star' equals 'the planet venus'
    j3  ::= john knows 'the morning star' equals 'the evening star'

##</code><code>##

    j1 (+) p2 => j3
    p1 (+) j2 => j3
    j1 (+) p3 => j2
    p1 (+) j3 => j2
    j2 (+) p3 => j1
    p2 (+) j3 => j1

##</code><code>##

    sub outer_1 {
        local ($x) = 1;
        return (inner ($_[0]));
    }
    
    sub outer_2 {
        local ($x) = 2;
        return (innner ($_[0]));
    }
    
    sub inner {
        my $y = shift;
        return ($x + $y);
    }

##</code><code>##

    sub equals {
        my ($str, $num) = @_;
        
        my %lut = (
            'one'   =>  1,
            'two'   =>  2,
            'three' =>  3,
            ...
        );
        
        if ($num == $lut{ $str }) {
            return 1;
        } else {
            return 0;
        }
    }

##</code><code>##

    my $inc  = sub { return ($_[0] + 1) };
    my $dec  = sub { return ($_[0] - 1) };
    my $zero = sub { return (0) };
    
    sub apply {
        my ($val, $func, @etc) = @_;
        
        if (@etc) {
            return ($val, apply ($func->($val), @etc));
        } else {
            return ($val, $func->($val));
        }
    }
    
    sub counter {
        return apply (0, @_);
    }
    
    my @ops = (
        $inc, $inc, $inc, $dec,
        $inc, $dec, $dec, $inc,
        $zero, $inc, $inc, $inc
    );
    
    print join (' ', counter (@ops));
    
    ----
    
    output == '0 1 2 3 2 3 2 1 2 0 1 2 3'

##</code><code>##

    sub apply {
        my ($val, $func, @etc) = @_;
        
        if (@etc) {
            return (apply ($func->($val), @etc));
        } else {
            return ($func->($val));
        }
    }

##</code><code>##

    my @x = ($inc, $inc, $inc);
    my @y = ($inc, $inc, $inc);
    
    if (3 == counter (@x)) {
        printf "%d equals %d.\n", counter (@x), counter (@y);
    }
    
    do {
        my @x = (@x, $inc);
        
        if (3 == counter (@y)) {
           printf "%d equals %d.\n", counter (@x), counter (@y);
        }
    }