Some basic aspects of weakest preconditions, demostrating programs that do not use while loops. We use a simple increment program as an example.

It is instructive to look at how the proof changes as we put in different pre and post values.

Violating the post condition

If we stick in an extra increment for y after, we can easily break the postcondition:

>>> :set -XNamedFieldPuns
>>> import Control.Monad (void)
>>> void $ correctness Skip $ Assign $ \st@IncS{y} -> st{y = y+1}
Following proof obligation failed:
==================================
  Postcondition fails:
    Start: IncS {x = 0, y = 0}
    End  : IncS {x = 0, y = 2}

We're told that the program ends up in a state where x=0 and y=2, violating the requirement y=x+1, as expected.

Using assert

There are two main use cases for assert, which merely ends up being a call to Abort. One is making sure the inputs are well formed. And the other is the user putting in their own invariants into the code.

Let's assume that we only want to accept strictly positive values of x. We can try:

>>> void $ correctness (assert "x > 0" (\st@IncS{x} -> x .> 0)) Skip
Following proof obligation failed:
==================================
  Abort "x > 0" condition is satisfiable:
    Before: IncS {x = 0, y = 0}
    After : IncS {x = 0, y = 0}

Recall that our precondition (pre) required x to be non-negative. So, we can put in something weaker and it would be fine:

>>> void $ correctness (assert "x > -5" (\st@IncS{x} -> x .> -5)) Skip
Total correctness is established.

In this case the precondition to our program ensures that the assert will always be satisfied.

As another example, let us put a post assertion that y is even:

>>> void $ correctness Skip (assert "y is even" (\st@IncS{y} -> y `sMod` 2 .== 0))
Following proof obligation failed:
==================================
  Abort "y is even" condition is satisfiable:
    Before: IncS {x = 0, y = 0}
    After : IncS {x = 0, y = 1}

It is important to emphasize that you can put whatever invariant you might want:

>>> void $ correctness Skip (assert "y > x" (\st@IncS{x, y} -> y .> x))
Total correctness is established.

Violating stability

What happens if our program modifies x? After all, we can simply set x=10 and y=11 and our post condition would be satisfied:

>>> void $ correctness Skip (Assign $ \st -> st{x = 10, y = 11})
Following proof obligation failed:
==================================
  Stability fails for "x":
    Before: IncS {x = 0, y = 1}
    After : IncS {x = 10, y = 11}

So, the stability condition prevents programs from cheating!