--- layout: fc_discuss_archives title: Message 17 from Frama-C-discuss on February 2015 ---
Several mistakes in your code: - loop assigns are incomplete : missing `rev`. Indeed, assigns must be completely proved _before_ any attempt to prove another property (otherwise, proof-obligations are inconsistent) ; - you need to specify that, during the loop, indices are monotonic. With proper renaming of variables, during loop: * P elements have been found with the desired property, and Q without. * 0 <= P && 0 <= Q && P+Q <= length * first indices ind[0..P-1] are all in range (0..P+Q-1) * last indices ind[length-Q..length-1] are all in range (0..P+Q-1) * ind[rev[k]] == k for all k in (0..P+Q-1) The monotonicity of indices is necessary to apply the induction hypothesis ind[ref[k]]==k for small enough k?s. L. > Le 25 f?vr. 2015 ? 13:49, Marko Sch?tz Schmuck <MarkoSchuetz at web.de> a ?crit : > > At Mon, 16 Feb 2015 11:29:02 +0100, > Lo?c Correnson wrote: >> >> Hi, >> Instead of an axiomatic definition of the permutation, you should introduce the inverse indices array as a ghost variable, say ?rev?, maintain this array during the loop with ghost code and prove that rev[ind[i]]==i in suitable ranges. >> We already used such a technique to prove a bubble-sort algorithm with permutation indices bookkeeping. >> This is much easier than converting C-arrays to abstract lists : I?m afraid you will need, at some point, some frame lemmas to compare `arrayToList{A}` and `arrayToList{B}` when you have memory updates between points `A` and `B`. >> L. > > thanks for the recommendation. I pursued this a bit, but got stuck > again. The following PO is not discharged: > > Goal Assigns (file partition-mine-momomorphic-ghost.c, line 17) in 'partition' (4/4): > Effect at line 21 > partition-mine-momomorphic-ghost.c:21: warning from Typed Model: > - Warning: Call assigns everything, looking for context inconsistency > Reason: Cast with incompatible pointers types (source: sint8*) (target: sint32*) > Let x_0 = 4*(to_uint32 length_4). > Let a_0 = (shift arr_6 0). > Let a_1 = (shift ind_7 0). > Assume { > (* Domain *) > Type: (is_sint32 length_4). > (* Heap *) > Have: (linked Malloc_4). > (* Pre-condition (file partition-mine-momomorphic-ghost.c, line 11) in 'partition' *) > (* Pre-condition: *) > Have: (0<=length_4) /\ (valid_rw Malloc_4 a_0 length_4) > /\ (valid_rw Malloc_4 a_1 length_4). > (* Pre-condition (file partition-mine-momomorphic-ghost.c, line 12) in 'partition' *) > (* Pre-condition: *) > Have: (separated a_0 length_4 a_1 length_4). > (* Pre-condition (file partition-mine-momomorphic-ghost.c, line 13) in 'partition' *) > (* Pre-condition: *) > Have: (4*length_4)<=4294967295. > (* Assertion 'rte,signed_overflow' (file partition-mine-momomorphic-ghost.c, line 20) *) > (* partition-mine-momomorphic-ghost.c:20: Assertion 'rte,signed_overflow': *) > Have: -2147483647<=length_4. > (* Assertion 'rte,unsigned_overflow' (file partition-mine-momomorphic-ghost.c, line 21) *) > (* partition-mine-momomorphic-ghost.c:21: Assertion 'rte,unsigned_overflow': *) > Have: 0<=x_0. > (* Assertion 'rte,unsigned_overflow' (file partition-mine-momomorphic-ghost.c, line 21) *) > (* partition-mine-momomorphic-ghost.c:21: Assertion 'rte,unsigned_overflow': *) > Have: x_0<=4294967295. > } > Prove: false. > Prover Alt-Ergo returns Unknown > > I suspect it is related to the implicit use of __builtin_alloca, but > could not yet find or come up with a way to handle it. > > Any hints, ideas? > > Thanks in advance, > > Marko > >>> Le 12 f?vr. 2015 ? 18:48, Marko Sch?tz Schmuck <MarkoSchuetz at web.de> a ?crit : >>> >>> Dear All, >>> >>> verifying an implementation of partition I have something like >>> >>> /*@ axiomatic List { >>> @ type listInt; >>> @ logic listInt nil; >>> @ logic listInt cons(integer x, listInt xs); >>> @ logic listInt append(listInt xs, listInt ys); >>> @ logic listInt arrayToList{L}(int *arr, integer length); >>> @ logic boolean member(integer elem, listInt xs); >>> @ axiom appendNil: >>> @ \forall listInt ys; append(nil, ys) == ys; >>> @ axiom appendCons: >>> @ \forall listInt xs, ys; \forall integer x; append(cons(x, xs), ys) == cons(x, append(xs, ys)); >>> @ axiom arrayToListNull{L}: >>> @ \forall int *arr; \forall integer i; i == 0 ==> arrayToList{L}(arr, i) == nil; >>> @ axiom arrayToListN{L}: >>> @ \forall int *arr; \forall integer length, newLength; length > 0 && newLength == length-1 >>> @ ==> arrayToList{L}(arr, length) == cons(\at(arr[0], L), arrayToList{L}(arr+1, newLength)); >>> @ axiom memberNil: >>> @ \forall integer elem; !member(elem, nil); >>> @ axiom memberConsHead: >>> @ \forall integer elem; \forall listInt xs; member(elem, cons(elem, xs)); >>> @ axiom memberConsTail: >>> @ \forall integer elem, x; \forall listInt xs; member(elem, xs) ==> member(elem, cons(x, xs)); >>> @ predicate permutationLists(listInt a, listInt b); >>> @ axiom permutationListsNil: >>> @ permutationLists(nil, nil); >>> @ axiom permutationListsCons: >>> @ \forall listInt a1, a2, a3, ta, b1, b2, b3, tb; ta == append(a1, append(a2, a3)) >>> @ && tb == append(b1, append(b2, b3)) && a2 != nil && a2 == b2 >>> @ && permutationLists(append(a1, a3), append(b1, b3)) >>> @ ==> permutationLists(ta, b1); >>> @} >>> @*/ >>> >>> /*@ >>> @predicate permutation{L1, L2}(int *a, int *b, integer count) = >>> @ permutationLists(arrayToList{L1}(a, count), arrayToList{L2}(b, count)); >>> @ >>> @predicate property(integer x); >>> @*/ >>> >>> /*@ assigns \nothing; >>> @ ensures \result == \true <==> property(x); >>> @*/ >>> int property(int x); >>> >>> /*@ requires length >= 0 && \valid(arr+(0..length-1)) && \valid(ind+(0..length-1)); >>> @ ensures 0 <= \result < length; >>> @ ensures \forall int i; 0 <= i <= \result ==> property(arr[ind[i]]); >>> @ ensures \forall int i; \result + 1 <= i < length ==> !property(arr[ind[i]]); >>> @ assigns ind[0..length-1]; >>> @*/ >>> int partition(int *arr, int *ind, int length) { >>> int gr = 0, j = length-1; >>> /*@ loop invariant 0 <= gr <= j+1 <= length ; >>> @ loop invariant \forall integer i; 0 <= i < gr ==> property(arr[ind[i]]); >>> @ loop invariant \forall integer i; j+1 <= i < length ==> !property(arr[ind[i]]); >>> @ loop invariant permutationLists(append(arrayToList{Here}(ind, gr), arrayToList{Here}(ind+j+1, length-j)), arrayToList{Here}(arr, gr + length - j)); >>> @ loop variant j - gr; >>> @*/ >>> while (gr <= j) { >>> if (property(arr[gr + length - 1 - j])) { >>> ind[gr] = gr + length - 1 - j; >>> //@ assert ind[gr] == gr + length - 1 - j && property(arr[gr + length - 1 - j]) && property(arr[ind[gr]]); >>> gr++; >>> } else { >>> ind[j] = gr + length - 1 - j; >>> j--; >>> } >>> } >>> return gr-1; >>> } >>> >>> Frama-C Neon/WP failed to discharge the second part of the invariant, >>> so I started experimenting with some assertions. When I write the >>> assertion as above the PO is not discharged. When I turn on 'split' >>> the first of the three generated POs is discharged. When I write the >>> three assertions as separate assertions: >>> >>> //@ assert ind[gr] == gr + length - 1 - j; >>> //@ assert property(arr[gr + length - 1 - j]); >>> //@ assert property(arr[ind[gr]]); >>> >>> then the first and the last are discharged. >>> >>> Any hints on how I should deal with this and how I could get the >>> invariant proved? >>> >>> Thanks and best regards, >>> >>> Marko_______________________________________________ >>> Frama-c-discuss mailing list >>> Frama-c-discuss at lists.gforge.inria.fr >>> http://lists.gforge.inria.fr/cgi-bin/mailman/listinfo/frama-c-discuss >> >> _______________________________________________ >> Frama-c-discuss mailing list >> Frama-c-discuss at lists.gforge.inria.fr <mailto:Frama-c-discuss at lists.gforge.inria.fr> >> http://lists.gforge.inria.fr/cgi-bin/mailman/listinfo/frama-c-discuss <http://lists.gforge.inria.fr/cgi-bin/mailman/listinfo/frama-c-discuss> > <partition-mine-momomorphic-ghost.c>_______________________________________________ > Frama-c-discuss mailing list > Frama-c-discuss at lists.gforge.inria.fr <mailto:Frama-c-discuss at lists.gforge.inria.fr> > http://lists.gforge.inria.fr/cgi-bin/mailman/listinfo/frama-c-discuss <http://lists.gforge.inria.fr/cgi-bin/mailman/listinfo/frama-c-discuss> -------------- next part -------------- An HTML attachment was scrubbed... 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