065, 079

065-circular_shift.dfy

@@ -0,0 +1,92 @@
+method circular_shift(a: nat, shift: nat) returns (shifted: string)
+  // post-conditions-start
+  ensures |shifted| == |natToString(a)|
+  ensures var s := natToString(a); shift > |s| ==> forall i :: 0 <= i < |s| ==> shifted[i] == s[|s| - 1 - i]
+  ensures var s := natToString(a); shift <= |s| ==> shifted == s[|s| - shift..] + s[..|s| - shift]
+  // post-conditions-end
+{
+  // impl-start
+  var s := natToString(a);
+  if shift > |s| {
+    s := reverse(s);
+    return s;
+  }
+
+  shifted := "";
+  var i := 0;
+  var start := |s| - shift;
+  while i < shift
+    // invariants-start
+    invariant 0 <= i <= shift
+    invariant shifted == s[|s| - shift..][..i]
+    // invariants-end
+  {
+    shifted := shifted + [s[start + i]];
+    i := i + 1;
+  }
+  var added := "";
+  i := 0;
+  while i < |s| - shift
+    // invariants-start
+    invariant 0 <= i <= |s| - shift
+    invariant added == s[..|s| - shift][..i]
+    // invariants-end
+  {
+    added := added + [s[i]];
+    i := i + 1;
+  }
+  assert shifted == s[|s| - shift..]; // assert-line
+  assert added == s[..|s| - shift]; // assert-line
+  shifted := shifted + added;
+  // impl-end
+}
+
+type stringNat = s: string |
+  |s| > 0 && (|s| > 1 ==> s[0] != '0') &&
+  forall i | 0 <= i < |s| :: s[i] in "0123456789"
+  witness "1"
+
+function natToString(n: nat): stringNat {
+  match n
+    case 0 => "0" case 1 => "1" case 2 => "2" case 3 => "3" case 4 => "4"
+    case 5 => "5" case 6 => "6" case 7 => "7" case 8 => "8" case 9 => "9"
+    case _ => natToString(n / 10) + natToString(n % 10)
+}
+
+function stringToNat(s: stringNat): nat
+  decreases |s|
+{
+  if |s| == 1 then
+      match s[0]
+      case '0' => 0 case '1' => 1 case '2' => 2 case '3' => 3 case '4' => 4
+      case '5' => 5 case '6' => 6 case '7' => 7 case '8' => 8 case '9' => 9
+  else
+      stringToNat(s[..|s|-1])*10 + stringToNat(s[|s|-1..|s|])
+}
+
+lemma natToStringThenStringToNatIdem(n: nat)
+  ensures stringToNat(natToString(n)) == n
+{
+}
+
+method reverse(str: string) returns (rev: string)
+    // post-conditions-start
+    ensures |rev| == |str|
+    ensures forall k :: 0 <= k < |str| ==> rev[k] == str[|str| - 1 - k]
+    // post-conditions-end
+{
+    // impl-start
+    rev := "";
+    var i := 0;
+    while (i < |str|)
+        // invariants-start
+        invariant i >= 0 && i <= |str|
+        invariant |rev| == i
+        invariant forall k :: 0 <= k < i ==> rev[k] == str[|str| - 1 - k]
+        // invariants-end
+    {
+        rev := rev + [str[|str| - i - 1]];
+        i := i + 1;
+    }
+    // impl-end
+}

079-decimal_to_binary.dfy

@@ -0,0 +1,39 @@
+type stringBin = s: string |
+  |s| > 0 && (|s| > 1 ==> s[0] != '0') &&
+  forall i | 0 <= i < |s| :: s[i] in "01"
+  witness "1"
+
+function decimal_to_binary(n: nat): (s: string)
+  // post-conditions-start
+  ensures |s| == |decimal_to_binary_helper(n)| + 4
+  ensures s[..2] == "db"
+  ensures s[|s| - 2..] == "db"
+  ensures s[2..|s| - 2] == decimal_to_binary_helper(n)
+  // post-conditions-end
+{
+  // impl-start
+  "db" + decimal_to_binary_helper(n) + "db"
+  // impl-end
+}
+
+function decimal_to_binary_helper(n: nat): stringBin 
+  // post-conditions-start
+  ensures binary_to_decimal(decimal_to_binary_helper(n)) == n
+  // post-conditions-end
+{
+  // impl-start
+  match n
+    case 0 => "0" case 1 => "1"
+    case _ => decimal_to_binary_helper(n / 2) + decimal_to_binary_helper(n % 2)
+  // impl-end
+}
+
+function binary_to_decimal(s: stringBin): nat
+  decreases |s|
+{
+  if |s| == 1 then
+    match s[0]
+    case '0' => 0 case '1' => 1
+  else
+    binary_to_decimal(s[..|s|-1])*2 + binary_to_decimal(s[|s|-1..|s|])
+}

README.md

@@ -67,7 +67,7 @@ Current status:
 - [x] 62. derivative
 - [x] 63. fibfib
 - [x] 64. vowels_count
-- [ ] 65. circular_shift
+- [x] 65. circular_shift
 - [x] 66. digitSum
 - [ ] 67. fruit_distribution - complex strings, possibly doable
 - [x] 68. pluck
@@ -81,7 +81,7 @@ Current status:
 - [x] 76. is_simple_power
 - [x] 77. iscube
 - [x] 78. hex_key
-- [ ] 79. decimal_to_binary
+- [x] 79. decimal_to_binary
 - [x] 80. is_happy
 - [x] 81. numerical_letter_grade
 - [x] 82. prime_length

REMAINING.md

@@ -10,8 +10,6 @@ Current status:
 - [ ] 32. poly
 - [ ] 38. encode_cyclic
 - [ ] 47. median
-- [ ] 65. circular_shift
-- [ ] 79. decimal_to_binary
 - [ ] 90. next_smallest - nullable
 - [ ] 103. rounded_avg
 - [ ] 113. odd_count