SE 504
Sample Proof of Correctness of a Selection Command

Prove the following Hoare triple: 

{P}
if k > 0 ⟶ k := k+1
[] k < 0 ⟶ k := k-1
fi
{Q}
where both P and Q are k≠0.

By the Hoare Triple Selection Law (HTSL), to prove this Hoare Triple it suffices to prove each of the following:

  1. [P ⟹ k>0 ∨ k<0]
  2. {P ∧ k>0} k:=k+1 {Q}
  3. {P ∧ k<0} k:=k-1 {Q}

As for the first, which is an implication, we employ the "Assume the Antecedant" approach: 

Assume P (i.e., k≠0)
   k>0 ∨ k<0

=    < number theory: x>y ∨ x<y ≡ x≠y >

   k≠0

=    < assumption P >

   true

As for the second (i.e., {P ∧ k>0} k:=k+1 {Q}), by the relationship between Hoare Triples and the wp function, it is equivalent to [P ∧ k>0 ⟹ wp.(k:=k+1).Q], so we prove it employing the Assume the Antecedant approach: 

Assume P and k>0.

    wp.(k:=k+1).Q

=      < defn. of Q >

    wp.(k:=k+1).(k≠0)

=      < wpAL (wp Assignment law) >

    (k≠0)(k:=k+1)

=      < textual substitution >

    k+1 ≠ 0

=      < algebra >

    k ≠-1 

=      < assumption k>0 >

    true

As for the third (i.e., {P ∧ k<0} k:=k-1 {Q}), the proof of its validity is very similar to the proof just presented, so it is omitted.