1. CSE 3341.03 Winter 2008 Introduction to Program Verification February 7
prover

2. applying Leibnitz's law
Ex. 5.6: verify that
push(top(s), pop(s)) = s ?
no stack axiom covers this case
need a new inference rule:
if pop(S1) = pop(S2)
top(S1) = top(S2)
then
S1 = S2

3. stacks as lists
alternative notation: stacks as Prolog lists push(X, nil) ->> [X]. push(X,S) ->> [X | S]. pop([X | Y]) ->> Y. top([X | Y]) ->> X.
what does push(a, push(b, nil)) simplify to?

4. list computations /: theory(stack). % including rules for lists
% example: pop([X|Y]) ->> Y.
|: dup([a]). dup([a]) ->>[a,a]
|: over([a,b,c]). over([a,b,c]) ->> [b,a,b,c]

5. tracing |:plus([a, b, x]).
push(top([a, b, x])+top(pop([a, b, x])), pop(pop([a, b, x])))
[top([a, b, x])+top(pop([a, b, x]))|pop(pop([a, b, x]))]
[top([a, b, x])+top(pop([a, b, x]))|pop([b, x])]
[top([a, b, x])+top(pop([a, b, x])), x]
[top([a, b, x])+top([b, x]), x]
[top([a, b, x])+b, x]
[a+b, x]
plus([a, b, x]) ->>
[b+a, x]

6. Forth
a still useful language from the 70s (the era of the mini-computer)
scripting language for the Palace graphic-based chat group: iptscrae
used in writing device drivers by several hardware vendors.
bumper sticker from the 80s:
"Forth you love if honk then"

7. translating Forth
how can we convert a Forth expression into a corresponding stack expression?
text shows how to automate translation of Forth into stack expressions for verification

8. translating Forth expressions to stack expressions
forth(S) ->> forth1(R) :- reverse(S, R).
forth1([dup | Rest]) ->> dup(forth1(Rest)).
forth1([over | Rest]) ->> over(forth1(Rest)).
forth1(['+' | Rest]) ->> plus(forth1(Rest)).
forth1(['-' | Rest]) ->> minus(forth1(Rest)).
forth1(['*' | Rest]) ->> times(forth1(Rest)).
forth1(['/' | Rest]) ->> divide(forth1(Rest)).
forth1([X | Rest]) ->> push(X, forth1(Rest)).
forth1([]) ->> nil.

9. simplify + wang = prover
prover tool combines simplification with tautology checking, and handles identities:
example: x=3 and x=y+3 implies y=0.
How does prover establish this?
substituting 3 for x:
x=3 and 3=y+3 implies y=0
rule in equality.simp: X+Y=W ->> X=Z :- ?

10. properties of ADT functions
defined ADT functions are intended to satisfy specific properties:
E. g.
top(dup(s)) = top(pop(dup(s))) and top(dup(s)) = top(s) (p. 24)

11. proving with equalities
to verify that an ADT (e. g. stack) function satisfies a set of desired conditions,
we have to prove a proposition of the form E1 and E where the Ei are equalities.
we can use simplification to prove each equality, but we need one more step to check that the conjunction is true.
to improve the scope of automatic proof, prover has the capability to process identities (paramodulation):

12. when does paramodulation work?
example: x=0 implies x<7.
substitute 0 for x in the consequent x<7
Ch. 5, p. 21 discusses why this works
it doesn’t always, e. g. not x<7 implies x= 0 is not equivalent to not 0<7 implies x=0.
only use paramodulation when the equality occurs on the left of the sequent reduced to non-logical terms:
{ , x= E, } >> { }[E / x]

13. substitution note notation for substitution in expressions
constant minor irritation in logic & computing: name clashes
to substitute E for V and eliminate V,
E must be free of V
i. e., V doesn't occur in E

14. Exercise 4.1 with prover indigo 301 % prover
Version 1.6.6SWI, February 14, 2007
Loading /cs/dept/course/ /W/3341/arithmetic.simp
Loading /cs/dept/course/ /W/3341/equality.simp
Loading /cs/dept/course/ /W/3341/logic.simp
|:(a + b = b + a) implies a < b < a.
a+b=b+a implies a<b<a
* Cannot prove true implies b<a and a<b.

15. |:assert((A <B < A ->> false)).
|:(a + b = b + a) implies a < b < a.
a+b=b+a implies a<b<a
* Cannot prove true implies false.
|:(a + b = b + a) implies a < b < a implies (a + b = b + a).
a + b = b + a implies a < b < a implies a+b=b+a
* Valid.

16. Queue axioms exercise 5.4 p. 20 what datatypes are X and Y ?
drop(Y -- X) = if(empty(Y), nil, drop(Y) -- X)
first(Y -- X) = if(empty(Y), X, first(Y))
what datatypes are X and Y ?
queue.simp?
empty(nil) ->> true. empty(Y--X) ->> false.
what else?