050, 093, 109, 112, 121

050-encode_shift.dfy

@@ -0,0 +1,48 @@
+function encode_char(c: char): char
+  requires 'a' <= c <= 'z'
+  ensures 'a' <= encode_char(c) <= 'z'
+{
+  ((c as int - 'a' as int + 5) % 26 + 'a' as int) as char
+}
+
+function decode_char(c: char): char
+  requires 'a' <= c <= 'z'
+  ensures 'a' <= decode_char(c) <= 'z'
+  ensures encode_char(decode_char(c)) == c
+{
+  ((c as int - 'a' as int - 5) % 26 + 'a' as int) as char
+}
+
+method encode_shift(s: string) returns (t: string)
+  requires forall i :: 0 <= i < |s| ==> 'a' <= s[i] <= 'z'
+  ensures |s| == |t|
+  ensures forall i :: 0 <= i < |s| ==> t[i] == encode_char(s[i])
+{
+  t := "";
+  var i := 0;
+  while i < |s|
+    invariant 0 <= i <= |s|
+    invariant |t| == i
+    invariant forall j :: 0 <= j < i ==> t[j] == encode_char(s[j])
+  {
+    t := t + [encode_char(s[i])];
+    i := i + 1;
+  }
+}
+
+method decode_shift(s: string) returns (t: string)
+  requires forall i :: 0 <= i < |s| ==> 'a' <= s[i] <= 'z'
+  ensures |s| == |t|
+  ensures forall i :: 0 <= i < |s| ==> t[i] == decode_char(s[i])
+{
+  t := "";
+  var i := 0;
+  while i < |s|
+    invariant 0 <= i <= |s|
+    invariant |t| == i
+    invariant forall j :: 0 <= j < i ==> t[j] == decode_char(s[j])
+  {
+    t := t + [decode_char(s[i])];
+    i := i + 1;
+  }
+}

093-encode.dfy

@@ -0,0 +1,45 @@
+method encode(s: string) returns (t: string)
+  requires forall i :: 0 <= i < |s| ==> 'a' <= s[i] <= 'z' || 'A' <= s[i] <= 'Z'
+  ensures |s| == |t|
+  ensures forall i :: 0 <= i < |s| && is_vowel(s[i]) ==> t[i] == rot2(swap_case(s[i]))
+  ensures forall i :: 0 <= i < |s| && !is_vowel(s[i]) ==> t[i] == swap_case(s[i])
+{
+  t := "";
+  var i := 0;
+  while i < |s|
+    invariant 0 <= i <= |s|
+    invariant |t| == i
+    invariant forall j :: 0 <= j < i && is_vowel(s[j]) ==> t[j] == rot2(swap_case(s[j]))
+    invariant forall j :: 0 <= j < i && !is_vowel(s[j]) ==> t[j] == swap_case(s[j])
+  {
+    if is_vowel(s[i]) {
+      t := t + [rot2(swap_case(s[i]))];
+    } else {
+      t := t + [swap_case(s[i])];
+    }
+    i := i + 1;
+  }
+}
+
+function swap_case(c: char): char
+  requires 'a' <= c <= 'z' || 'A' <= c <= 'Z'
+  ensures 'a' <= c <= 'z' ==> 'A' <= swap_case(c) <= 'Z'
+  ensures 'A' <= c <= 'Z' ==> 'a' <= swap_case(c) <= 'z'
+  ensures is_vowel(swap_case(c)) == is_vowel(c)
+{
+  if 'a' <= c <= 'z' then
+    'A' + (c - 'a')
+  else
+    'a' + (c - 'A')
+}
+
+function rot2(c: char): char
+  requires is_vowel(c)
+{
+    (c as int + 2) as char
+}
+
+predicate is_vowel(c: char) {
+    (c == 'a' || c == 'e' || c == 'i' || c == 'o' || c == 'u')
+    || (c == 'A' || c == 'E' || c == 'I' || c == 'O' || c == 'U')
+}

109-move_one_ball.dfy

@@ -0,0 +1,53 @@
+predicate is_sorted(a: seq<int>) {
+  forall i, j :: 0 <= i < j < |a| ==> a[i] <= a[j]
+}
+
+method move_one_ball(a: seq<int>) returns (can: bool)
+  requires |a| > 0
+  requires forall i, j :: 0 <= i < |a| && 0 <= j < |a| && i != j ==> a[i] != a[j]
+  ensures can <==> exists i :: 0 <= i < |a| && is_sorted(a[i..] + a[..i])
+{
+  if |a| <= 1 {
+    assert is_sorted(a[..0]);
+    return true;
+  }
+  can := false;
+  var i := 0;
+  var min_index := 0;
+  while i < |a|
+    invariant 0 <= i <= |a|
+    invariant 0 <= min_index < |a|
+    invariant forall j :: 0 <= j < i && min_index != j ==> a[min_index] < a[j]
+  {
+    if a[i] < a[min_index] {
+      min_index := i;
+    }
+    i := i + 1;
+  }
+
+  assert forall j :: (
+    0 <= j < |a| && min_index != j 
+    ==> (
+      (forall i :: j <= i < |a| ==> a[j..][i - j] == a[i])
+      && (forall i :: 0 <= i < j ==> (a[j..] + a[..j])[|a| - j + i] == a[i])
+      && (
+        if min_index < j then
+          (a[j..] + a[..j])[|a| - j + min_index] == a[min_index]
+        else
+          (a[j..] + a[..j])[min_index - j] == a[min_index]
+      )
+      && (
+        (exists p :: (
+          1 <= p < |a[j..] + a[..j]| 
+          && (a[j..] + a[..j])[p] == a[min_index] 
+          && (a[j..] + a[..j])[0] > (a[j..] + a[..j])[p]
+        )) ==> !is_sorted(a[j..] + a[..j])
+      )
+    )
+  );
+
+  assert forall j :: 0 <= j < |a| && min_index != j ==> !is_sorted(a[j..] + a[..j]);
+
+  var new_a := a[min_index..] + a[..min_index];
+  can := is_sorted(new_a);
+}

112-reverse_delete.dfy

@@ -0,0 +1,50 @@
+method reverse_delete(s: string, chars: string) returns (res: string, is_palindrome: bool)
+  ensures forall i :: 0 <= i < |res| ==> res[i] !in chars
+  ensures forall i :: 0 <= i < |res| ==> res[i] in s
+  ensures forall i :: 0 <= i < |s| && s[i] !in chars ==> s[i] in res
+  ensures is_palindrome <==> is_palindrome_pred(res)
+{
+  res := "";
+  var i := 0;
+  while i < |s|
+    invariant 0 <= i <= |s|
+    invariant forall i :: 0 <= i < |res| ==> res[i] !in chars
+    invariant forall i :: 0 <= i < |res| ==> res[i] in s
+    invariant forall j :: 0 <= j < i && s[j] !in chars ==> s[j] in res
+  {
+    if s[i] !in chars {
+      res := res + [s[i]];
+    }
+    i := i + 1;
+  }
+
+  is_palindrome := check_palindrome(res);
+}
+
+method check_palindrome(s: string) returns (result: bool)
+  ensures result <==> is_palindrome_pred(s)
+{
+  if |s| == 0 {
+    return true;
+  }
+  result := true;
+  var i := 0;
+  var j := |s| - 1;
+  while (i < j)
+    invariant 0 <= i < |s|
+    invariant 0 <= j < |s|
+    invariant j == |s| - i - 1
+    invariant forall k :: 0 <= k < i ==> s[k] == s[|s| - 1 - k]
+  {
+    if (s[i] != s[j]) {
+      result := false;
+      break;
+    }
+    i := i + 1;
+    j := j - 1;
+  }
+}
+
+predicate is_palindrome_pred(s : string) {
+  forall k :: 0 <= k < |s| ==> s[k] == s[|s| - 1 - k]
+}

121-solution.dfy

@@ -0,0 +1,49 @@
+method solution(numbers: seq<int>) returns (s: int)
+  ensures s == sum(numbers, seq(|numbers|, i requires 0 <= i < |numbers| => i % 2 == 0 && numbers[i] % 2 == 1)) 
+{
+  var i := 0;
+  s := 0;
+  ghost var p := [];
+  while i < |numbers|
+    invariant 0 <= i <= |numbers|
+    invariant |p| == i
+    invariant s == sum(numbers[..i], p[..i])
+    invariant p == seq(i, j requires 0 <= j < i => j % 2 == 0 && numbers[j] % 2 == 1)
+  {
+    ghost var old_p := p;
+    p := p + [i % 2 == 0 && numbers[i] % 2 == 1];
+
+    assert sum(numbers[..i], p[..i]) == sum(numbers[..i], old_p[..i]) by {
+      assert p[..i] == old_p[..i];
+    }
+    assert sum(numbers[..i + 1], p[..i + 1]) == sum(numbers[..i], p[..i]) + (if p[i] then numbers[i] else 0) by {
+      assert numbers[..i+1][..i] == numbers[..i];
+      assert p[..i + 1][..i] == p[..i] by {
+        assert forall j :: 0 <= j < i ==> p[..i + 1][j] == p[..i][j];
+      }
+      sum_prop(numbers[..i + 1], p[..i + 1]); 
+    }
+    s := s + (if i % 2 == 0 && numbers[i] % 2 == 1 then numbers[i] else 0);
+
+    i := i + 1;
+  }
+  assert numbers[..|numbers|] == numbers;
+  assert p[..|p|] == p;
+}
+
+function sum(s: seq<int>, p: seq<bool>) : int 
+  requires |s| == |p|
+{
+  if |s| == 0 then 0 else (if p[0] then s[0] else 0) + sum(s[1..], p[1..])
+}
+
+lemma sum_prop(s: seq<int>, p: seq<bool>)
+  requires |s| > 0
+  requires |p| == |s|
+  ensures sum(s, p) == sum(s[..|s| - 1], p[..|s| - 1]) + (if p[|s| - 1] then s[|s| - 1] else 0)
+{
+  if (|s| > 1) {
+      assert (s[1..][..|s[1..]| - 1]) == s[1..|s| - 1];
+      assert (p[1..][..|p[1..]| - 1]) == p[1..|p| - 1];
+  }
+}

README.md

@@ -52,7 +52,7 @@ Current status:
 - [ ] 47. median
 - [x] 48. is_palindrome
 - [x] 49. modp
-- [ ] 50. encode_shift
+- [x] 50. encode_shift
 - [x] 51. remove_vowels
 - [x] 52. below_threshold
 - [x] 53. add
@@ -95,7 +95,7 @@ Current status:
 - [ ] 90. next_smallest - nullable
 - [ ] 91. is_bored - complex strings
 - [x] 92. any_int
-- [ ] 93. encode
+- [x] 93. encode
 - [x] 94. skjkasdkd
 - [x] 95. check_dict_case
 - [x] 96. count_up_to
@@ -111,19 +111,19 @@ Current status:
 - [x] 106. f
 - [ ] 107. even_odd_palindrome
 - [x] 108. count_nums
-- [ ] 109. move_one_ball
+- [x] 109. move_one_ball
 - [x] 110. exchange
 - [ ] 111. histogram
-- [ ] 112. reverse_delete
+- [x] 112. reverse_delete
 - [ ] 113. odd_count
-- [ ] 114. minSubArraySum
+- [x] 114. minSubArraySum
 - [x] 115. max_fill
 - [x] 116. sort_array - sorting
 - [ ] 117. select_words - complex strings
 - [x] 118. get_closest_vowel
 - [ ] 119. match_parens
 - [ ] 120. maximum - sorting
-- [ ] 121. solution - sum
+- [x] 121. solution - sum
 - [x] 122. add_elements
 - [x] 123. get_odd_collatz - sorting
 - [ ] 124. valid_date

REMAINING.md

@@ -2,30 +2,25 @@
 
 Current status:
 
+## Priority
 - [ ] 25. factorize
-- [ ] 28. concatenate
-- [ ] 32. poly
 - [ ] 36. fizz_buzz
+
+## Other
+
+- [ ] 32. poly
 - [ ] 38. encode_cyclic
 - [ ] 44. change_base
 - [ ] 47. median
-- [ ] 50. encode_shift
-- [ ] 56. correct_bracketing
-- [ ] 61. correct_bracketing
 - [ ] 65. circular_shift
 - [ ] 79. decimal_to_binary
 - [ ] 84. solve
 - [ ] 90. next_smallest - nullable
-- [ ] 93. encode
 - [ ] 103. rounded_avg
 - [ ] 107. even_odd_palindrome
-- [ ] 109. move_one_ball
 - [ ] 111. histogram
-- [ ] 112. reverse_delete
 - [ ] 113. odd_count
-- [ ] 114. minSubArraySum
 - [ ] 119. match_parens
-- [ ] 121. solution - sum
 - [ ] 124. valid_date
 - [ ] 129. minPath
 - [ ] 131. digits
@@ -38,11 +33,15 @@ Current status:
 - [ ] 161. solve
 - [ ] 162. string_to_md5 - needs hash
 
+
 ## Complex strings
 - [ ] 15. string_sequence
 - [ ] 17. parse_music
 - [ ] 18. how_many_times
 - [ ] 19. sort_numbers
+- [ ] 28. concatenate
+- [ ] 56. correct_bracketing
+- [ ] 61. correct_bracketing
 - [ ] 67. fruit_distribution
 - [ ] 86. anti_shuffle
 - [ ] 91. is_bored