38,56,90,119,132 (#55)

038-encode_cyclic.dfy

@@ -0,0 +1,44 @@
+method encode_cyclic(s: seq<int>) returns (res: seq<int>) 
+    // post-conditions-start
+    ensures |s| == |res|
+    ensures forall i :: 0 <= i < |s| - |s| % 3 ==> (i % 3 == 0 ==> res[i] == s[i + 1])
+    ensures forall i :: 0 <= i < |s| - |s| % 3 ==> (i % 3 == 1 ==> res[i] == s[i + 1])
+    ensures forall i :: 0 <= i < |s| - |s| % 3 ==> (i % 3 == 2 ==> res[i] == s[i - 2])
+    ensures forall i :: |s| - |s| % 3 <= i < |s| ==> (res[i] == s[i])
+    // post-conditions-end
+{
+    // impl-start
+    res := s;
+    var i := 0;
+    while i + 2 < |s|
+        // invariants-start
+        invariant 0 <= i <= |s|
+        invariant |s| == |res|
+        invariant i % 3 == 0
+        invariant forall j :: i <= j < |s| ==> (res[j] == s[j])
+        invariant forall j :: 0 <= j < i ==> (j % 3 == 0 ==> res[j] == s[j + 1])
+        invariant forall j :: 0 <= j < i ==> (j % 3 == 1 ==> res[j] == s[j + 1])
+        invariant forall j :: 0 <= j < i ==> (j % 3 == 2 ==> res[j] == s[j - 2])
+        // invariants-end
+    {
+        res := res[i := s[i + 1]];
+        res := res[i + 1 := s[i + 2]];
+        res := res[i + 2 := s[i]];
+        i := i + 3;
+    }
+    // impl-end
+}
+
+method decode_cyclic(s: seq<int>) returns (res: seq<int>)
+    // post-conditions-start
+    ensures |s| == |res|
+    ensures forall i :: |s| - |s| % 3 <= i < |s| ==> (res[i] == s[i])
+    ensures forall i :: 0 <= i < |s| - |s| % 3 ==> (i % 3 == 0 ==> res[i] == s[i + 2])
+    ensures forall i :: 0 <= i < |s| - |s| % 3 ==> (i % 3 == 1 ==> res[i] == s[i - 1])
+    // post-conditions-end
+{
+    // impl-start
+    var intermediate := encode_cyclic(s);
+    res := encode_cyclic(intermediate);
+    // impl-end
+}

056-correct_bracketing.dfy

@@ -0,0 +1,40 @@
+function CalcBal(s: seq<int>, i: int, j: int) : int 
+    requires 0 <= i <= j <= |s|
+{
+    if i == j then 0
+    else (if s[j - 1] == 0 then 1 else -1) + CalcBal(s, i, j - 1)
+}
+
+method correct_bracketing(s: seq<int>) returns (b: bool)
+    // pre-conditions-start
+    requires forall i :: 0 <= i < |s| ==> s[i] == 0 || s[i] == 1
+    // pre-conditions-end
+    // post-conditions-start
+    ensures (forall i :: 0 <= i <= |s| ==> CalcBal(s, 0, i) >= 0) ==> b
+    ensures b ==> (forall i :: 0 <= i <= |s| ==> CalcBal(s, 0, i) >= 0)
+    // post-conditions-end
+{
+    // impl-start
+    var i := 0;
+    var depth := 0;
+    b := true;
+    while i < |s|
+        // invariants-start
+        invariant 0 <= i <= |s|
+        invariant depth == CalcBal(s, 0, i)
+        invariant (forall j :: 0 <= j <= i ==> CalcBal(s, 0, j) >= 0) ==> b
+        invariant b ==> (forall j :: 0 <= j <= i ==> CalcBal(s, 0, j) >= 0)
+        // invariants-end
+    {
+        if s[i] == 0 {
+            depth := depth + 1;
+        } else {
+            depth := depth - 1;
+        }
+        if depth < 0 {
+            b := false;
+        }
+        i := i + 1;
+    }
+    // impl-end
+}

090-next_smallest.dfy

@@ -0,0 +1,57 @@
+datatype Option<T> = None | Some(T)
+
+
+function getVal(mx : Option<int>) : int
+    requires mx != None
+{
+    match mx {
+        case Some(n) => n
+    }
+}
+
+method rolling_max(s: seq<int>) returns (res: Option<int>) 
+    // post-conditions-start
+    ensures res == None <==> |s| < 2
+    ensures res != None ==> exists x :: 0 <= x < |s| && s[x] == getVal(res)
+    ensures forall x, y :: 0 <= x < y < |s| ==> s[x] <= getVal(res) || s[y] <= getVal(res)
+    // post-conditions-end
+{
+    // impl-start
+    if |s| < 2 {
+        return None;
+    }
+
+    res := None;
+    var mx := s[0];
+    var i := 1;
+
+    while i < |s| 
+        // invariants-start
+        invariant 0 <= i <= |s|
+        invariant i > 1 ==> res != None
+        invariant exists x :: 0 <= x < |s| && s[x] == mx
+        invariant res != None ==> exists x :: 0 <= x < |s| && s[x] == getVal(res)
+        invariant i == 1 ==> res == None
+        invariant i > 1 ==> getVal(res) <= mx
+        invariant forall x :: 0 <= x < i ==> s[x] == mx || (res != None && s[x] <= getVal(res))
+        invariant i > 1 ==> forall x, y :: 0 <= x < y < i ==> s[x] <= getVal(res) || s[y] <= getVal(res)
+        // invariants-end
+    {
+        if s[i] > mx {
+            res := Some(mx);
+            mx := s[i];
+        } else {
+            match res {
+                case None => res := Some(s[i]);
+                case Some(n1) =>
+                {
+                    if (s[i] > n1) {
+                        res := Some(s[i]);
+                    }
+                }
+            }
+        }
+        i := i + 1;
+    }
+    // impl-end
+}

119-match_parens.dfy

@@ -0,0 +1,67 @@
+function CalcBal(s: seq<int>, i: int, j: int, acc: int) : int 
+    requires 0 <= i <= j <= |s|
+{
+    if i == j then acc
+    else (if s[j - 1] == 0 then 1 else -1) + CalcBal(s, i, j - 1, acc)
+}
+
+method checkFixed(s1: seq<int>, s2: seq<int>) returns (b: bool) 
+    // pre-conditions-start
+    requires forall i :: 0 <= i < |s1| ==> s1[i] == 0 || s1[i] == 1
+    requires forall i :: 0 <= i < |s2| ==> s2[i] == 0 || s2[i] == 1
+    // pre-conditions-end
+    // post-conditions-start
+    ensures b ==> forall i :: 0 <= i <= |s1| ==> CalcBal(s1, 0, i, 0) >= 0
+    ensures b ==> forall i :: 0 <= i <= |s2| ==> CalcBal(s1, 0, |s1|, 0) + CalcBal(s2, 0, i, 0) >= 0
+    ensures !b ==> (exists i :: 0 <= i <= |s1| && CalcBal(s1, 0, i, 0) < 0) || (exists i :: 0 <= i <= |s2| && CalcBal(s1, 0, |s1|, 0) + CalcBal(s2, 0, i, 0) < 0)
+    // post-conditions-end
+{
+    // impl-start
+    var bal := 0;
+    var i := 0;
+    while i < |s1|
+        // invariants-start
+        invariant 0 <= i <= |s1|
+        invariant bal == CalcBal(s1, 0, i, 0)
+        invariant forall j :: 0 <= j < i ==> CalcBal(s1, 0, j, 0) >= 0
+        // invariants-end
+    {
+        if s1[i] == 0 {
+            bal := bal + 1;
+        } else {
+            bal := bal - 1;
+        }
+        if bal < 0 {
+            // assert-start
+            assert CalcBal(s1, 0, i + 1, 0) < 0;
+            // assert-end
+            return false;
+        }
+        i := i + 1;
+    }
+
+    i := 0;
+    while i < |s2| 
+        // invariants-start
+        invariant 0 <= i <= |s2|
+        invariant bal == CalcBal(s1, 0, |s1|, 0) + CalcBal(s2, 0, i, 0) 
+        invariant forall j :: 0 <= j < i ==> CalcBal(s1, 0, |s1|, 0) + CalcBal(s2, 0, j, 0) >= 0
+        // invariants-end
+    {
+        if s2[i] == 0 {
+            bal := bal + 1;
+        } else {
+            bal := bal - 1;
+        }
+        if bal < 0 {
+            // assert-start
+            assert CalcBal(s1, 0, |s1|, 0) + CalcBal(s2, 0, i + 1, 0) < 0;
+            assert exists i :: 0 <= i <= |s2| && CalcBal(s1, 0, |s1|, 0) + CalcBal(s2, 0, i, 0) < 0;
+            // assert-end
+            return false;
+        }
+        i := i + 1;
+    }
+    return true;
+    // impl-end
+}

132-is_nested.dfy

@@ -0,0 +1,49 @@
+method is_nested(s: seq<int>) returns (res: bool) 
+    // post-conditions-start
+    ensures res == exists x, y, z, w :: 0 <= x < y < z < w < |s| && s[x] == 0 && s[y] == 0 && s[z] == 1 && s[w] == 1
+    // post-conditions-end
+{
+    // impl-start
+    var bal := 0;
+    var i := 0;
+
+    while i < |s| && bal < 2
+        // invariants-start
+        invariant 0 <= i <= |s|
+        invariant (bal >= 1) == (exists x :: 0 <= x < i && s[x] == 0)
+        invariant (bal >= 2) == (exists x, y :: 0 <= x < y < i && s[x] == 0 && s[y] == 0)
+        invariant !(exists x, y, z :: 0 <= x < y < z < i && s[x] == 0 && s[y] == 0 && s[z] == 1)
+        // invariants-end
+    {
+        if s[i] == 0 {
+            bal := bal + 1;
+        }
+        i := i + 1;
+    }
+
+    if bal < 2 {
+        return false;
+    }
+
+    while i < |s| && bal > 0 
+        // invariants-start
+        invariant 0 <= i <= |s|
+        invariant 0 <= bal <= 2
+        invariant exists x, y :: 0 <= x && x < y && y < i && s[x] == 0 && s[y] == 0
+        invariant (bal <= 1) == (exists x, y, z :: 0 <= x < y < z < i && s[x] == 0 && s[y] == 0 && s[z] == 1)
+        invariant (bal == 0) == (exists x, y, z, w :: 0 <= x < y < z < w < i && s[x] == 0 && s[y] == 0 && s[z] == 1 && s[w] == 1)
+        // invariants-end
+    {
+        if s[i] == 1 {
+            bal := bal - 1;
+        }
+        i := i + 1;
+    }
+
+    if bal > 0 {
+        return false;
+    }
+
+    return true;
+    // impl-end
+}

README.md

@@ -40,7 +40,7 @@ Current status:
 - [x] 35. max_element
 - [x] 36. fizz_buzz
 - [x] 37. sort_even - sorting
-- [ ] 38. encode_cyclic
+- [x] 38. encode_cyclic
 - [ ] 39. ~prime_fib~ -- unbounded loop, probably impossible
 - [x] 40. triples_sum_to_zero
 - [x] 41. car_race_collision
@@ -58,7 +58,7 @@ Current status:
 - [x] 53. add
 - [x] 54. same_chars
 - [x] 55. fib
-- [ ] 56. correct_bracketing
+- [x] 56. correct_bracketing
 - [x] 57. monotonic
 - [x] 58. common
 - [x] 59. largest_prime_factor
@@ -92,7 +92,7 @@ Current status:
 - [x] 87. get_row - sorting
 - [x] 88. sort_array - sorting
 - [x] 89. encrypt
-- [ ] 90. next_smallest - nullable
+- [x] 90. next_smallest - nullable
 - [ ] 91. is_bored - complex strings
 - [x] 92. any_int
 - [x] 93. encode
@@ -121,7 +121,7 @@ Current status:
 - [x] 116. sort_array - sorting
 - [ ] 117. select_words - complex strings
 - [x] 118. get_closest_vowel
-- [ ] 119. match_parens
+- [x] 119. match_parens
 - [x] 120. maximum - sorting
 - [x] 121. solution - sum
 - [x] 122. add_elements