--- layout: fc_discuss_archives title: Message 15 from Frama-C-discuss on February 2015 ---
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. > 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