126, 146 (#30)

126-is_sorted.dfy

@@ -0,0 +1,65 @@
+method is_sorted(a: seq<int>) returns (f: bool)
+  ensures f == forall i, j :: 0 <= i <= j < |a| ==> a[i] <= a[j] && forall i :: 0 <= i < |a| ==> count_set(a, a[i]) <= 2
+{
+  if |a| == 0 {
+    return true;
+  }
+  var is_asc := true;
+  var i := 1;
+  while i < |a|
+    invariant 1 <= i <= |a|
+    invariant is_asc == forall j, k :: 0 <= j < k < i ==> a[j] <= a[k]
+  {
+    if a[i - 1] > a[i] {
+      is_asc := false;
+    }
+    i := i + 1;
+  }
+
+  if !is_asc {
+    return false;
+  }
+
+  i := 0;
+
+  var has_no_more_that_2 := true;
+  while i < |a|
+    invariant 0 <= i <= |a|
+    invariant has_no_more_that_2 == forall j :: 0 <= j < i ==> count_set(a, a[j]) <= 2
+  {
+    var count := count_set(a, a[i]);
+    if count > 2 {
+      has_no_more_that_2 := false;
+    }
+    i := i + 1;
+  }
+  return has_no_more_that_2;
+}
+
+
+method count_sorted(a: seq<int>, x: int, pos: int) returns (count: int)
+  requires forall i, j :: 0 <= i <= j < |a| ==> a[i] <= a[j]
+  requires 0 <= pos < |a|
+  requires a[pos] == x
+  requires pos == 0 || a[pos - 1] < x
+  ensures count == count_set(a, x)
+{
+  count := 0;
+  var i := pos;
+  ghost var positions := {};
+
+  while i < |a| && a[i] == x
+    invariant 0 <= i <= |a|
+    invariant positions == set j | 0 <= j < i && a[j] == x
+    invariant count == |positions|
+  {
+    count := count + 1;
+    positions := positions + {i};
+    i := i + 1;
+  }
+  assert positions == set j | 0 <= j < |a| && a[j] == x;
+}
+
+function count_set(a: seq<int>, x: int): int {
+  |set i | 0 <= i < |a| && a[i] == x|
+}

146_specialFilter.dfy

@@ -0,0 +1,35 @@
+function first_digit(n: int): int
+  requires n >= 0
+  ensures 0 <= first_digit(n) < 10
+{
+  if n < 10 then n else first_digit(n / 10)
+}
+
+function last_digit(n: int): int
+  requires n >= 0
+  ensures 0 <= last_digit(n) < 10
+  ensures last_digit(n) == n % 10
+{
+  n % 10
+}
+
+method specialFilter(s: seq<int>) returns (r: seq<int>)
+  ensures forall i :: 0 <= i < |r| ==> r[i] > 10
+  ensures forall x :: x in r ==> x in s
+  ensures forall i :: 0 <= i < |r| ==> first_digit(r[i]) % 2 == 1 && last_digit(r[i]) % 2 == 1
+{
+  var i := 0;
+  r := [];
+  while i < |s|
+    invariant 0 <= i <= |s|
+    invariant forall x :: x in r ==> x in s
+    invariant forall i :: 0 <= i < |r| ==> r[i] > 10
+    invariant forall x :: x in r ==> x in s
+    invariant forall i :: 0 <= i < |r| ==> first_digit(r[i]) % 2 == 1 && last_digit(r[i]) % 2 == 1
+  {
+    if s[i] > 10 && last_digit(s[i]) % 2 == 1 && first_digit(s[i]) % 2 == 1 {
+      r := r + [s[i]];
+    }
+    i := i + 1;
+  }
+}

README.md

@@ -128,7 +128,7 @@ Current status:
 - [ ] 123. get_odd_collatz - sorting
 - [ ] 124. valid_date
 - [ ] 125. split_words - complex strings
-- [ ] 126. is_sorted
+- [x] 126. is_sorted
 - [x] 127. intersection
 - [ ] 128. prod_signs
 - [ ] 129. minPath
@@ -148,7 +148,7 @@ Current status:
 - [ ] 143. words_in_sentence - complex strings
 - [ ] 144. simplify
 - [ ] 145. order_by_points - sorting
-- [ ] 146. specialFilter
+- [x] 146. specialFilter
 - [ ] 147. get_max_triples
 - [x] 148. bf
 - [x] 149. sorted_list_sum