055, 057, 062, 063, 064, 066 (#16)

055-fib.dfy

@@ -0,0 +1,35 @@
+function fib(n: nat): nat
+  decreases n
+{
+  if n == 0 then 0
+  else if n == 1 then 1
+  else fib(n - 1) + fib(n - 2)
+}
+
+method ComputeFib(n: nat) returns (result: nat)
+  ensures result == fib(n)
+{
+  if n == 0 {
+    return 0;
+  }
+
+  if n == 1 {
+    return 1;
+  }
+
+  var a, b := 0, 1;
+  var i := 2;
+
+  while i <= n
+    invariant 2 <= i <= n + 1
+    invariant a == fib(i - 2)
+    invariant b == fib(i - 1)
+  {
+    var temp := a + b;
+    a := b;
+    b := temp;
+    i := i + 1;
+  }
+
+  return b;
+}

057-monotonic.dfy

@@ -0,0 +1,28 @@
+method monotonic(xs: seq<int>) returns (result: bool)
+  requires |xs| > 0
+  ensures result <==> (forall i, j :: 0 <= i < j < |xs| ==> xs[i] < xs[j]) || (forall i, j :: 0 <= i < j < |xs| ==> xs[i] > xs[j])  
+{
+    if |xs| == 1 {
+        return true;
+    }
+
+    var increasing := true;
+    var decreasing := true;
+    var i := 1;
+
+    while i < |xs|
+        invariant 1 <= i <= |xs|
+        invariant increasing <==> (forall j, k :: 0 <= j < k < i ==> xs[j] < xs[k])
+        invariant decreasing <==> (forall j, k :: 0 <= j < k < i ==> xs[j] > xs[k])
+    {
+        if xs[i - 1] >= xs[i] {
+            increasing := false;
+        }
+        if xs[i - 1] <= xs[i] {
+            decreasing := false;
+        }
+        i := i + 1;
+    }
+
+    result := increasing || decreasing;
+}

062-derivative.dfy

@@ -0,0 +1,16 @@
+method derivative(xs: seq<int>) returns (result: seq<int>)
+  requires |xs| > 0
+  ensures |result| == |xs| - 1
+  ensures forall i :: 0 <= i < |result| ==> result[i] == xs[i+1] * (i+1)
+{
+  result := [];
+  var i := 1;
+  while i < |xs|
+    invariant 1 <= i <= |xs|
+    invariant |result| == i - 1
+    invariant forall j :: 0 <= j < |result| ==> result[j] == xs[j+1] * (j+1)
+  {
+    result := result + [xs[i] * i];
+    i := i + 1;
+  }
+}

063-fibfib.dfy

@@ -0,0 +1,37 @@
+function fibfib(n: nat): nat
+  decreases n
+{
+  if n == 0 || n == 1 then 0
+  else if n == 2 then 1
+  else fibfib(n - 1) + fibfib(n - 2) + fibfib(n - 3)
+}
+
+method ComputeFibFib(n: nat) returns (result: nat)
+  ensures result == fibfib(n)
+{
+  if n == 0 || n == 1 {
+    return 0;
+  }
+
+  if n == 2 {
+    return 1;
+  }
+
+  var a, b, c := 0, 0, 1;
+  var i := 3;
+
+  while i <= n
+    invariant 3 <= i <= n + 1
+    invariant a == fibfib(i-3)
+    invariant b == fibfib(i-2)
+    invariant c == fibfib(i-1)
+  {
+    var temp := c + b + a;
+    a := b;
+    b := c;
+    c := temp;
+    i := i + 1;
+  }
+
+  return c;
+}

064-vowel_count.dfy

@@ -0,0 +1,28 @@
+method vowel_count(s: string) returns (count: int)
+  ensures count >= 0
+  ensures count == |(set i | 0 <= i < |s| && is_vowel(s[i]))| + if |s| > 0 && s[|s| - 1] == 'y' then 1 else 0
+{
+    ghost var found := {};
+    count := 0;
+    var i := 0;
+    while i < |s|
+      invariant 0 <= i <= |s|
+      invariant count == |found|
+      invariant forall j :: 0 <= j < i && is_vowel(s[j]) ==> j in found
+      invariant forall x :: x in found ==> x < i
+      invariant found == (set j | 0 <= j < i && is_vowel(s[j]))
+    {
+        if is_vowel(s[i]) {
+            found := found + {i};
+            count := count + 1;
+        }
+        i := i + 1;
+    }
+    count := count + if |s| > 0 && s[|s| - 1] == 'y' then 1 else 0;
+}
+
+function is_vowel(c: char): bool
+  ensures is_vowel(c) <==> c == 'a' || c == 'e' || c == 'i' || c == 'o' || c == 'u' || c == 'A' || c == 'E' || c == 'I' || c == 'O' || c == 'U'
+{
+    c == 'a' || c == 'e' || c == 'i' || c == 'o' || c == 'u' || c == 'A' || c == 'E' || c == 'I' || c == 'O' || c == 'U'
+}

066-digitSum.dfy

@@ -0,0 +1,44 @@
+function upper_sum_rec(s: string): int
+  ensures upper_sum_rec(s) >= 0
+{
+  if |s| == 0 then
+    0
+  else
+    var remaining := upper_sum_rec(s[1..]);
+    to_int(s[0]) + remaining
+}
+
+lemma upper_sum_rec_prop(s: string)
+  requires |s| > 0
+  ensures upper_sum_rec(s) == upper_sum_rec(s[..|s|-1]) + to_int(s[|s|-1])
+{
+    if (|s| > 1) {
+        assert (s[1..][..|s[1..]| - 1]) == s[1..|s| - 1];
+    }
+}
+
+function to_int(c: char): int 
+    ensures 'A' <= c <= 'Z' ==> to_int(c) == c as int
+    ensures c < 'A' || c > 'Z' ==> to_int(c) == 0
+{
+    if 'A' <= c <= 'Z' then c as int else 0
+}
+
+method upper_sum(s: string) returns (res: int)
+    ensures res == upper_sum_rec(s)
+{
+    res := 0;
+    var i := 0;
+    while (i < |s|)
+        invariant 0 <= i <= |s|
+        invariant res == upper_sum_rec(s[..i])
+    {
+        res := res + to_int(s[i]);
+        assert upper_sum_rec(s[..i + 1]) == upper_sum_rec(s[..i]) + to_int(s[i]) by {
+            assert s[..i+1][..i] == s[..i];
+            upper_sum_rec_prop(s[..i + 1]);
+        }
+        i := i + 1;
+    }
+    assert s == s[..|s|];
+}   

README.md

@@ -57,18 +57,18 @@ Current status:
 - [x] 52. below_threshold
 - [x] 53. add
 - [x] 54. same_chars
-- [ ] 55. fib
+- [x] 55. fib
 - [ ] 56. correct_bracketing
-- [ ] 57. monotonic
+- [x] 57. monotonic
 - [x] 58. common
 - [x] 59. largest_prime_factor
 - [x] 60. sum_to_n
 - [ ] 61. correct_bracketing
-- [ ] 62. derivative
-- [ ] 63. fibfib
-- [ ] 64. vowels_count
+- [x] 62. derivative
+- [x] 63. fibfib
+- [x] 64. vowels_count
 - [ ] 65. circular_shift
-- [ ] 66. digitSum
+- [x] 66. digitSum
 - [ ] 67. fruit_distribution
 - [x] 68. pluck
 - [x] 69. search