041, 045, 046, 135, 136, 138 (#29)

041-car_race_collision.dfy

@@ -0,0 +1,6 @@
+method car_race_collision(n: int) returns (cnt: int)
+  requires n >= 0
+  ensures cnt == n * n
+{
+  cnt := n * n;
+}

045-triangle_area.dfy

@@ -0,0 +1,6 @@
+method CalculateTriangleArea(a: real, h: real) returns (area: real)
+  requires h >= 0.0 && a >= 0.0
+  ensures area == (h * a) / 2.0
+{
+  area := (h * a) / 2.0;
+}

046-fib4.dfy

@@ -0,0 +1,39 @@
+function fib4_rec(n: nat): nat
+  decreases n
+{
+  if n == 0 || n == 1 || n == 2 then 0
+  else if n == 3 then 1
+  else fib4_rec(n - 1) + fib4_rec(n - 2) + fib4_rec(n - 3) + fib4_rec(n - 4)
+}
+
+method fib4(n: nat) returns (result: nat)
+  ensures result == fib4_rec(n)
+{
+  if n == 0 || n == 1 || n == 2 {
+    return 0;
+  }
+
+  if n == 3 {
+    return 1;
+  }
+
+  var a, b, c, d := 0, 0, 0, 1;
+  var i := 4;
+
+  while i <= n
+    invariant 4 <= i <= n + 1
+    invariant a == fib4_rec(i-4)
+    invariant b == fib4_rec(i-3)
+    invariant c == fib4_rec(i-2)
+    invariant d == fib4_rec(i-1)
+  {
+    var temp := d + c + b + a;
+    a := b;
+    b := c;
+    c := d;
+    d := temp;
+    i := i + 1;
+  }
+
+  return d;
+}

135_can_arrange.dfy

@@ -0,0 +1,21 @@
+method can_arrange(arr: seq<int>) returns (pos: int)
+  requires |arr| > 0
+  requires forall i, j :: 0 <= i < j < |arr| ==> arr[i] != arr[j]
+  ensures pos == -1 ==> forall i :: 1 <= i < |arr| ==> arr[i] >= arr[i - 1]
+  ensures pos >= 0 ==> 1 <= pos < |arr| && arr[pos] < arr[pos - 1]
+  ensures pos >= 0 ==> forall i :: pos < i < |arr| ==> arr[i] >= arr[i - 1]
+{
+  var i := 1;
+  pos := -1;
+  while i < |arr|
+    invariant 1 <= i <= |arr|
+    invariant pos == -1 ==> forall j :: 1 <= j < i ==> arr[j] >= arr[j - 1]
+    invariant pos >= 0 ==> 1 <= pos < i && arr[pos] < arr[pos - 1]
+    invariant pos >= 0 ==> forall j :: pos < j < i ==> arr[j] >= arr[j - 1]
+  {
+    if arr[i] < arr[i - 1] {
+      pos := i;
+    }
+    i := i + 1;
+  }
+}

136-largest_smallest_integers.dfy

@@ -0,0 +1,41 @@
+datatype Option<T> = None | Some(value: T)
+
+function get_value(o: Option<int>): int
+  requires o.Some?
+  ensures get_value(o) == o.value
+{
+  o.value
+}
+
+method largest_smallest_integers(arr: seq<int>) returns (a: Option<int>, b: Option<int>)
+  ensures a.None? ==> forall i :: 0 <= i < |arr| ==> arr[i] >= 0
+  ensures a.Some? ==> get_value(a) in arr && get_value(a) < 0
+  ensures a.Some? ==> forall i :: 0 <= i < |arr| && arr[i] < 0 ==> arr[i] <= get_value(a)
+
+  ensures b.None? ==> forall i :: 0 <= i < |arr| ==> arr[i] <= 0
+  ensures b.Some? ==> get_value(b) in arr && get_value(b) > 0
+  ensures b.Some? ==> forall i :: 0 <= i < |arr| && arr[i] > 0 ==> arr[i] >= get_value(b)
+{
+  var i := 0;
+  a := None;
+  b := None;
+  while i < |arr|
+    invariant 0 <= i <= |arr|
+    invariant a.None? ==> forall j :: 0 <= j < i ==> arr[j] >= 0
+    invariant a.Some? ==> get_value(a) in arr && get_value(a) < 0
+    invariant a.Some? ==> forall j :: 0 <= j < i && arr[j] < 0 ==> arr[j] <= get_value(a)
+
+    invariant b.None? ==> forall j :: 0 <= j < i ==> arr[j] <= 0
+    invariant b.Some? ==> get_value(b) in arr && get_value(b) > 0
+    invariant b.Some? ==> forall j :: 0 <= j < i && arr[j] > 0 ==> arr[j] >= get_value(b)
+  {
+    if arr[i] < 0 && (a.None? || arr[i] >= get_value(a)) {
+      a := Some(arr[i]);
+    }
+
+    if arr[i] > 0 && (b.None? || arr[i] <= get_value(b)) {
+      b := Some(arr[i]);
+    }
+    i := i + 1;
+  }
+}

138_is_equal_to_sum_even.dfy

@@ -0,0 +1,5 @@
+method is_equal_to_sum_even(n: int) returns (b : bool)
+  ensures b <==> n % 2 == 0 && n >= 8 // 2 + 2 + 2 + (n - 6)
+{
+  b := n % 2 == 0 && n >= 8;
+}

README.md

@@ -43,12 +43,12 @@ Current status:
 - [ ] 38. encode_cyclic
 - [ ] 39. ~prime_fib~ -- unbounded loop, probably impossible
 - [x] 40. triples_sum_to_zero
-- [ ] 41. car_race_collision
+- [x] 41. car_race_collision
 - [x] 42. incr_list
 - [x] 43. pairs_sum_to_zero
 - [ ] 44. change_base
-- [ ] 45. triangle_area
-- [ ] 46. fib4
+- [x] 45. triangle_area
+- [x] 46. fib4
 - [ ] 47. median
 - [x] 48. is_palindrome
 - [ ] 49. modp
@@ -137,10 +137,10 @@ Current status:
 - [ ] 132. is_nested
 - [x] 133. sum_squares
 - [ ] 134. check_if_last_char_is_a_letter
-- [ ] 135. can_arrange
-- [ ] 136. largest_smallest_integers
+- [x] 135. can_arrange
+- [x] 136. largest_smallest_integers
 - [ ] 137. compare_one - ill-typed
-- [ ] 138. is_equal_to_sum_even
+- [x] 138. is_equal_to_sum_even
 - [x] 139. special_factorial
 - [ ] 140. fix_spaces - complex strings
 - [ ] 141. file_name_check