149,158 (#19)

149-sorted_list_sum.dfy

@@ -0,0 +1,116 @@
+function comparison(a : string, b : string, i : int): bool
+    requires 0 <= i <= |a| && 0 <= i <= |b|
+    decreases |a| - i
+    decreases |b| - i
+    ensures (a == b) ==> comparison(a, b, i)
+{
+    if (i < |a| && i < |b|) then
+        if a[i] < b[i] then
+            true
+        else if a[i] > b[i] then
+            false
+        else
+            comparison(a, b, i + 1)
+    else 
+        if |a| <= |b| then
+            true
+        else
+            false
+}
+
+method sort_strings(list: seq<string>) returns (sorted: seq<string>)
+    ensures |sorted| == |list|
+    ensures multiset(sorted) == multiset(list)
+{
+    sorted := list;
+    var i := 0;
+    while i < |list|
+        invariant 0 <= i <= |list|
+        invariant |sorted| == |list|
+        invariant multiset(sorted) == multiset(list)
+    {
+        var j := i + 1;
+        var min := i;
+        while j < |list|
+            invariant 0 <= i <= j <= |list|
+            invariant 0 <= min < j
+        {
+            if !comparison(sorted[min], sorted[j], 0) {
+                min := j;
+            } 
+            j := j + 1;
+        }
+        var temp := sorted[i];
+        sorted := sorted[i := sorted[min]][min := temp];
+        i := i + 1;
+    }
+}
+
+method sort_lengths(list: seq<string>) returns (sorted: seq<string>)
+    requires forall i : int :: 0 <= i < |list| ==> |list[i]| % 2 == 0
+    ensures forall i : int :: 0 <= i < |sorted| ==> |sorted[i]| % 2 == 0
+    ensures |sorted| == |list|
+    ensures multiset(sorted) == multiset(list)
+    ensures forall x : int, y : int :: 0 <= x < y < |sorted| ==> |sorted[x]| <= |sorted[y]|
+{
+    sorted := list;
+    var i := 0;
+    while i < |list|
+        invariant 0 <= i <= |list|
+        invariant |sorted| == |list|
+        invariant multiset(sorted) == multiset(list)
+        invariant forall x :: 0 <= x < i ==> forall y :: i <= y < |sorted| ==> |sorted[x]| <= |sorted[y]|
+        invariant forall x : int, y : int :: 0 <= x < y < i ==> |sorted[x]| <= |sorted[y]|
+        invariant forall i : int :: 0 <= i < |sorted| ==> |sorted[i]| % 2 == 0
+    {
+        var j := i + 1;
+        var min := i;
+        while j < |list|
+            invariant 0 <= i < j <= |list|
+            invariant i <= min < j
+            invariant forall x :: i <= x < j ==> |sorted[min]| <= |sorted[x]|
+        {
+            if |sorted[j]| < |sorted[min]| {
+                min := j;
+            }
+            j := j + 1;
+        }
+        var temp := sorted[i];
+        sorted := sorted[i := sorted[min]][min := temp];
+        i := i + 1;
+    }
+}
+
+method sorted_list_sum(list: seq<string>) returns (sorted: seq<string>)
+    requires |list| > 0
+    ensures |sorted| <= |list|
+    ensures forall i : int :: 0 <= i < |sorted| ==> |sorted[i]| % 2 == 0
+    ensures forall x : int, y : int :: 0 <= x < y < |sorted| ==> |sorted[x]| <= |sorted[y]|
+    ensures multiset(sorted) <= multiset(list)
+{
+    var init := sort_strings(list);
+
+    sorted := [];
+    var i := 0;
+
+    while i < |init|
+        invariant |init| > 0
+        invariant multiset(init) == multiset(list)
+        invariant 0 <= i <= |init|
+        invariant |sorted| <= i
+        invariant forall i : int :: 0 <= i < |sorted| ==> |sorted[i]| % 2 == 0
+        invariant multiset(sorted) <= multiset(init[0..i])
+    {
+        if |init[i]| % 2 == 0 {
+            sorted := sorted + [init[i]];
+            assert multiset(sorted) <= multiset(init[0..i + 1]);
+        } else {
+            assert init[0..i + 1] == init[0..i] + [init[i]];
+        }
+        i := i + 1;
+    }
+    assert multiset(sorted) <= multiset(init[0..|init|]);
+    assert init[0..|init|] == init;
+
+    sorted := sort_lengths(sorted);
+}

158-find_max.dfy

@@ -0,0 +1,19 @@
+
+method find_max(strings : seq<string>) returns (s : string) 
+    requires |strings| > 0
+    ensures s in strings
+    ensures forall i : int :: 0 <= i < |strings| ==> |set c | c in s| >= |set c | c in strings[i]|
+{
+    s := strings[0];
+    var i := 1;
+    while i < |strings|
+        invariant 0 <= i <= |strings|
+        invariant s in strings[..i]
+        invariant forall j : int :: 0 <= j < i ==> |set c | c in s| >= |set c | c in strings[j]|
+    {
+        if (|set c | c in s| < |set c | c in strings[i]|) {
+            s := strings[i];
+        }
+        i := i + 1;
+    }
+}