30, 31, 35, 40, 42 (#11)

030-get-positive.dfy

@@ -0,0 +1,30 @@
+
+method get_positive(l : seq<int>) returns (result : seq<int>)
+    ensures forall i : int :: i >= 0 && i < |result| ==> result[i] > 0
+    ensures |result| <= |l|
+    ensures forall i1 :: i1 >= 0 && i1 < |l| ==> (l[i1] > 0 ==> exists i2 :: i2 >= 0 && i2 < |result| && result[i2] == l[i1])
+    ensures |result| == 0 || (forall i1 : int :: i1 >= 0 && i1 < |result| ==> (exists i2 : int :: i2 >= 0 && i2 < |l| && l[i2] == result[i1]))
+{
+    result := [];
+    var i : int := 0;
+    while i < |l|
+        invariant i >= 0 && i <= |l|
+        invariant |result| <= i
+        invariant forall i1 : int :: i1 >= 0 && i1 < |result| ==> result[i1] > 0
+        invariant i > 0 ==> (l[i - 1] > 0 ==> exists i2 :: i2 >= 0 && i2 < |result| && result[i2] == l[i - 1])
+        invariant forall i1 : int :: i1 >= 0 && i1 < old(|result|) ==> old(result[i1]) == result[i1]
+        invariant forall i1 :: i1 >= 0 && i1 < i ==> (l[i1] > 0 ==> exists i2 :: i2 >= 0 && i2 < |result| && result[i2] == l[i1])
+        invariant |result| == 0 || (forall i1 : int :: i1 >= 0 && i1 < |result| ==> (exists i2 : int :: i2 >= 0 && i2 < |l| && l[i2] == result[i1]));
+    {
+        var n : int := l[i];
+        if n > 0 {
+            ghost var res_prev := result;
+            assert forall i1 :: i1 >= 0 && i1 < i ==> (l[i1] > 0 ==> exists i2 :: i2 >= 0 && i2 < |result| && result[i2] == l[i1]);
+            result := result + [n];
+            assert result[|result| - 1] == n;
+            assert forall i1 :: i1 >= 0 && i1 < |res_prev| ==> res_prev[i1] == result[i1];
+            assert forall i1 :: i1 >= 0 && i1 < i ==> (l[i1] > 0 ==> exists i2 :: i2 >= 0 && i2 < |res_prev| && res_prev[i2] == l[i1]);
+        }
+        i := i + 1;
+    }
+}  

031-is-prime.dfy

@@ -0,0 +1,19 @@
+
+method is_prime(k: int) returns (result: bool)
+  requires k >= 2
+  ensures result ==> forall i :: 2 <= i < k ==> k % i != 0
+  ensures !result ==> exists j :: 2 <= j < k && k % j == 0
+{
+  var i := 2;
+  result := true;
+  while i < k
+    invariant 2 <= i <= k
+    invariant !result ==> exists j :: 2 <= j < i && k % j == 0
+    invariant result ==> forall j :: 2 <= j < i ==> k % j != 0
+  {
+    if k % i == 0 {
+      result := false;
+    }
+    i := i + 1;
+  }
+}

035-max-element.dfy

@@ -0,0 +1,19 @@
+
+method max_element(l : seq<int>) returns (result : int)
+    requires |l| > 0
+    ensures forall i : int :: i >= 0 && i < |l| ==> l[i] <= result
+    ensures exists i : int :: i >= 0 && i < |l| && l[i] == result
+{
+    result := l[0];
+    var i : int := 1;
+    while i < |l|
+        invariant i >= 1 && i <= |l|
+        invariant forall i1 : int :: i1 >= 0 && i1 < i ==> l[i1] <= result
+        invariant exists i1 : int :: i1 >= 0 && i1 < i && l[i1] == result
+    {
+        if l[i] > result {
+            result := l[i];
+        }
+        i := i + 1;
+    }
+}

040-triples-sum-to-zero.dfy

@@ -0,0 +1,37 @@
+
+method triples_sum_to_zero(l : seq<int>) returns (result : bool)
+    ensures result ==> exists i : int, j : int, k : int :: 0 <= i < |l| && 0 <= j < |l| && 0 <= k < |l| && i != j && j != k && i != k && l[i] + l[j] + l[k] == 0
+    ensures !result ==> forall i : int, j : int, k : int :: 0 <= i < |l| && 0 <= j < |l| && 0 <= k < |l| && i != j && j != k && i != k ==> l[i] + l[j] + l[k] != 0
+{
+    result := false;
+    var i : int := 0;
+    while i < |l|
+        invariant i >= 0 && i <= |l|
+        invariant !result ==> forall i1 : int, j : int, k : int :: 0 <= i1 < i && 0 <= j < |l| && 0 <= k < |l| && i1 != j && j != k && i1 != k ==> l[i1] + l[j] + l[k] != 0
+        invariant result ==> exists i1 : int, j : int, k : int :: 0 <= i1 < i && 0 <= j < |l| && 0 <= k < |l| && i1 != j && j != k && i1 != k && l[i1] + l[j] + l[k] == 0
+    {
+        var j : int := 0;
+        while j < |l|
+            invariant j >= 0 && j <= |l|
+            invariant !result ==> forall i1 : int, j : int, k : int :: 0 <= i1 < i && 0 <= j < |l| && 0 <= k < |l| && i1 != j && j != k && i1 != k ==> l[i1] + l[j] + l[k] != 0
+            invariant !result ==> forall j1 : int, k : int :: 0 <= j1 < j && 0 <= k < |l| && i != j1 && j1 != k && i != k ==> l[i] + l[j1] + l[k] != 0
+            invariant result ==> (exists i1 : int, j : int, k : int :: 0 <= i1 < i && 0 <= j < |l| && 0 <= k < |l| && i1 != j && j != k && i1 != k && l[i1] + l[j] + l[k] == 0) || (exists j1 : int, k : int :: 0 <= j1 < j && 0 <= k < |l| && i != j1 && j1 != k && i != k && l[i] + l[j1] + l[k] == 0)
+        {
+            var k : int := 0;
+            while k < |l|
+                invariant k >= 0 && k <= |l|
+                invariant !result ==> forall i1 : int, j : int, k : int :: 0 <= i1 < i && 0 <= j < |l| && 0 <= k < |l| && i1 != j && j != k && i1 != k ==> l[i1] + l[j] + l[k] != 0
+                invariant !result ==> forall j1 : int, k : int :: 0 <= j1 < j && 0 <= k < |l| && i != j1 && j1 != k && i != k ==> l[i] + l[j1] + l[k] != 0
+                invariant !result ==> forall k1 : int :: 0 <= k1 < k && i != j && j != k1 && i != k1 ==> l[i] + l[j] + l[k1] != 0
+                invariant result ==> (exists i1 : int, j : int, k : int :: 0 <= i1 < i && 0 <= j < |l| && 0 <= k < |l| && i1 != j && j != k && i1 != k && l[i1] + l[j] + l[k] == 0) || (exists j1 : int, k : int :: 0 <= j1 < j && 0 <= k < |l| && i != j1 && j1 != k && i != k && l[i] + l[j1] + l[k] == 0) || (exists k1 : int :: 0 <= k1 < k && i != j && j != k1 && i != k1 && l[i] + l[j] + l[k1] == 0)
+            {
+                if i != j && j != k && i != k && l[i] + l[j] + l[k] == 0 {
+                    result := true;
+                }
+                k := k + 1;
+            }
+            j := j + 1;
+        }
+        i := i + 1;
+    }
+}

042-incr-list.dfy

@@ -0,0 +1,16 @@
+
+method incr_list(l: seq<int>) returns (result: seq<int>)
+    ensures |result| == |l|
+    ensures forall i :: 0 <= i < |l| ==> result[i] == l[i] + 1
+{
+    result := [];
+    var i := 0;
+    while i < |l|
+        invariant 0 <= i <= |l|
+        invariant |result| == i
+        invariant forall i1 :: 0 <= i1 < i ==> result[i1] == l[i1] + 1
+    {
+        result := result + [l[i] + 1];
+        i := i + 1;
+    }
+}

README.md

@@ -32,19 +32,19 @@ Current status:
 - [ ] 27. flip_case
 - [ ] 28. concatenate
 - [ ] 29. filter_by_prefix
-- [ ] 30. get_positive
-- [ ] 31. is_prime
+- [x] 30. get_positive
+- [x] 31. is_prime
 - [ ] 32. poly
 - [ ] 33. sort_third
 - [ ] 34. unique
-- [ ] 35. max_element
+- [x] 35. max_element
 - [ ] 36. fizz_buzz
 - [ ] 37. sort_even
 - [ ] 38. encode_cyclic
 - [ ] 39. prime_fib
-- [ ] 40. triples_sum_to_zero
+- [x] 40. triples_sum_to_zero
 - [ ] 41. car_race_collision
-- [ ] 42. incr_list
+- [x] 42. incr_list
 - [ ] 43. pairs_sum_to_zero
 - [ ] 44. change_base
 - [ ] 45. triangle_area