Program constructs and their verification rules
Sequence
S = (S1 ; S2 ) is the sequence of two programs S1 , S2 . Q, R and Q0 are logical functions over A. If
1. Q =⇒ wp(S1 , Q0 ) and
2. Q0 =⇒ wp(S2 , R)
then Q =⇒ wp(S, R)
S
S1
S2
Selection
IF = (π1 : S1 , . . . , πn : Sn ) is a branches constructed from programs S1 , . . . , Sn and logical functions
π1 , . . . , πn . Q and R are logical functions over A. If
n
W
1. Q =⇒
πi and
i=1
2. Q =⇒
n
V
(πi ∨ ¬πi ) and
i=1
3. ∀i ∈ [1..n] : Q ∧ πi =⇒ wp(Si , R)
then Q =⇒ wp(IF, R)
IF
A π1
A
· · · A πn
S1
Sn
A
···
Loop
DO = (π, S0 ) denotes the loop constructed from the program S0 and the logical function π.
Inv, Q, R are logical functions over A and t : A → Z is a function. If
1. Q =⇒ Inv and
2. Inv ∧ ¬π =⇒ R and
3. Inv =⇒ π ∨ ¬π and
4. Inv ∧ π =⇒ t > 0 and
5. Inv ∧ π ∧ t = t0 =⇒ wp(S0 , Inv ∧ t < t0 ) for any t0 integer number
then Q =⇒ wp(DO, R)
(Inv is called loop invariant, t is called variant function.)
DO
π
S0