sorting - dafny implementation of insertionsort -


i new in dafny , have problems verifying insertionsort-implementation. dafny tells me the multiset invariants not holding, else working fine. after hours of searching mistake need :) nice if tell me trick!

my code:

predicate sorted(a:array<int>, min:int, max:int) requires != null; requires 0<= min <= max <= a.length; reads a; {   forall j, k :: min <= j < k < max ==> a[j] <= a[k] }   /*  *    */   method insertionsort(a: array<int>) requires != null; requires a.length > 0; ensures sorted(a, 0, a.length); ensures multiset(a[..]) == multiset(old(a[..])); modifies a;  {   var := 1;     while(i < a.length)   invariant 1 <= <= a.length;   invariant sorted(a, 0, i);    invariant != null;   invariant multiset(old(a[..])) == multiset(a[..]);   decreases a.length-i;   {     var j := - 1;     var key := a[i];     while(j >= 0 && key < a[j])    invariant -1 <= j <= - 1 <= a.length;    invariant (j == i-1 && sorted(a, 0, i)) || (sorted(a, 0, i+1));    invariant forall k :: j < k < ==> a[k] >= key;     invariant -1 < j == - 1   ==> multiset(old(a[..])) == multiset(a[..]);    invariant |multiset(old(a[..]))| == |multiset(a[..])|;    invariant -1 < j < - 1 && key < a[j] ==> multiset(old(a[..]))  ==  multiset(a[..]) - multiset({a[j+1]}) + multiset({key});    invariant -1 == j ==> multiset(old(a[..]))  ==  multiset(a[..]) + multiset({key}) - multiset({a[j+1]});    decreases j;                                             {       a[j + 1] := a[j];       j := j - 1;     }     a[j + 1] := key;    := + 1;     } }

it produces

1   loop invariant might not maintained loop.    29,38 2   loop invariant might not maintained loop.    42,73 3   loop invariant might not maintained loop.    43,52

link: http://rise4fun.com/dafny/3r5

here modified algorithm verify. shifts elements doing swap. think bit more work adapted algorithm. needs more complex multiset invariant (which require lemma adding , removing things multisets).

predicate sorted(a:array<int>, min:int, max:int) requires != null; requires 0<= min <= max <= a.length; reads a; {   forall j, k :: min <= j < k < max ==> a[j] <= a[k] }  predicate sortedseq(a:seq<int>) {   forall j, k :: 0 <= j < k < |a| ==> a[j] <= a[k] }  lemma sortedseqsubsequencesorted(a:seq<int>, min:int, max:int) requires 0<= min <= max <= |a| requires sortedseq(a) ensures sortedseq(a[min .. max]) { }  method swap(a: array<int>, i:int, j:int)   modifies a;   requires != null && 0 <= < j < a.length   requires + 1 == j   ensures a[..i] == old(a[..i])   ensures a[j+1..] == old(a[j+1..])   ensures a[j] == old(a[i])   ensures a[i] == old(a[j])   ensures multiset(a[..]) == multiset(old(a[..])) {    var tmp := a[i];    a[i] := a[j];    a[j] := tmp;   }   method insertionsort(a: array<int>) modifies a; requires != null; requires a.length > 0; ensures sorted(a, 0, a.length); ensures multiset(a[..]) == multiset(old(a[..]));  {   var := 0;    while(i < a.length)      invariant 0 <= <= a.length      invariant sorted(a, 0, i)       invariant multiset(old(a[..])) == multiset(a[..]);   {      var key := a[i];       var j := - 1;       ghost var a' := a[..];      assert sortedseq(a'[0..i]);      while(j >= 0 && key < a[j])         invariant -1 <= j <= - 1         invariant a[0 .. j+1] == a'[0 .. j+1]         invariant sorted(a, 0, j+1);         invariant a[j+1] == key == a'[i];         invariant a[j+2 .. i+1] == a'[j+1 .. i]         invariant sorted(a, j+2, i+1);          invariant multiset(old(a[..])) == multiset(a[..])         invariant forall k :: j+1 < k <= ==> key < a[k]                                           {        ghost var a'' := a[..];        swap(a, j, j+1);        assert a[0..j] == a''[0..j];        assert a[0..j] == a'[0..j];        assert a[j] == a''[j+1] == a'[i] == key;        assert a[j+2..] == a''[j+2..];        assert a[j+2..i+1] == a''[j+2..i+1] == a'[j+1..i];         j := j - 1;         sortedseqsubsequencesorted(a'[0..i], j+1, i);        assert sortedseq(a'[j+1..i]);        assert a[j+2 .. i+1] == a'[j+1 .. i];        assert sortedseq(a[j+2..i+1]);      }      := + 1;    } }

http://rise4fun.com/dafny/gplux


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