60, 85, 133, 142, 151 - generalized sum method (#10)

060-sum_to_n.dfy

@@ -0,0 +1,40 @@
+function sum(s: seq<int>) : int
+    {
+        if |s| == 0 then 0 else s[0] + sum(s[1..])
+    }
+
+lemma sum_prop(s: seq<int>) 
+    requires |s| > 0
+    ensures sum(s) == sum(s[..|s| - 1]) + s[ |s| - 1 ]
+{
+    if (|s| > 1) {
+        assert (s[1..][..|s[1..]| - 1]) == s[1..|s| - 1];
+    }
+}
+
+function number_seq(n: int) : (r : seq<int>) 
+        requires n >= 0
+        ensures |r| == n
+    {
+        seq(n, i => i + 1)
+    }
+
+method sum_squares(n: int) returns (r : int)
+    requires n >= 1
+    ensures r == sum(number_seq(n))
+    {
+        r := 0;
+        var k := 0;
+        ghost var v := number_seq(n);
+        while(k < n) 
+            invariant 0 <= k <= |v|
+            invariant r == sum(v[..k])
+        {
+            assert v[..k + 1][..k] == v[..k];
+            r := r + (k + 1);
+            k := k + 1;
+            assert sum(v[..k]) == r by { sum_prop(v[..k]); }
+        }
+        assert v[..k] == v;
+        return r;
+    }

085-add.dfy

@@ -0,0 +1,45 @@
+function sumc(s: seq<int>, p: seq<bool>) : int 
+    requires |s| == |p|
+    {
+        if |s| == 0 then 0 else (if p[0] then s[0] else 0) + sumc(s[1..], p[1..])
+    }
+
+lemma sum_prop(s: seq<int>, p: seq<bool>) 
+    requires |s| > 0
+    requires |s| == |p|
+    ensures sumc(s, p) == sumc(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..][..|s[1..]| - 1]) == p[1..|s| - 1];
+    }
+}
+
+function add_conditon(lst: seq<int>) : (p : seq<bool>)
+    ensures |lst| == |p|
+    {
+        seq(|lst|, i requires 0 <= i < |lst| => i % 2 == 1 && lst[i] % 2 == 0)
+    }
+
+method add(v: seq<int>) returns (r : int)
+    ensures r == sumc(v, add_conditon(v))
+    {
+        r := 0;
+        var k := 0;
+        var p := add_conditon(v);
+        while(k < |v|) 
+            invariant 0 <= k <= |v|
+            invariant r == sumc(v[..k], p[..k])
+        {
+            assert v[..k + 1][..k] == v[..k];
+            assert p[..k + 1][..k] == p[..k];
+            r := r + if p[k] then v[k] else 0;
+            k := k + 1;
+            assert sumc(v[..k], p[..k]) == r by {
+                sum_prop(v[..k], p[..k]);
+            }
+        }
+        assert v[..k] == v;
+        assert p[..k] == p;
+        return r;
+    }

133-sum_squares.dfy

@@ -0,0 +1,43 @@
+function sum(s: seq<int>) : int
+    {
+        if |s| == 0 then 0 else s[0] + sum(s[1..])
+    }
+
+lemma sum_prop(s: seq<int>) 
+    requires |s| > 0
+    ensures sum(s) == sum(s[..|s| - 1]) + s[ |s| - 1 ]
+{
+    if (|s| > 1) {
+        assert (s[1..][..|s[1..]| - 1]) == s[1..|s| - 1];
+    }
+}
+
+function ceil(f: real) : (c : int)
+    {
+        (f + 1.0).Floor
+    }
+
+function square_seq(lst: seq<real>) : (sq : seq<int>) 
+        ensures |sq| == |lst|
+    {
+        seq(|lst|, i requires 0 <= i < |lst| => ceil(lst[i]) * ceil(lst[i]))
+    }
+
+method sum_squares(lst: seq<real>) returns (r : int)
+    ensures r == sum(square_seq(lst))
+    {
+        r := 0;
+        var k := 0;
+        var v := square_seq(lst);
+        while(k < |lst|) 
+            invariant 0 <= k <= |lst|
+            invariant r == sum(v[..k])
+        {
+            assert v[..k + 1][..k] == v[..k];
+            r := r + v[k];
+            k := k + 1;
+            assert sum(v[..k]) == r by { sum_prop(v[..k]); }
+        }
+        assert v[..k] == v;
+        return r;
+    }

142-sum_squares.dfy

@@ -1,23 +1,38 @@
-method sum_squares(a : seq<int>) returns (result : seq<int>)
-  ensures |result| == |a|
-  ensures forall i : int :: i >= 0 && i < |result| && i % 3 == 0 ==> result[i] == a[i] * a[i]
-  ensures forall i : int :: i >= 0 && i < |result| && i % 4 == 0 && i % 3 != 0 ==> result[i] == a[i] * a[i] * a[i]
+function sum(s: seq<int>) : int
+    {
+        if |s| == 0 then 0 else s[0] + sum(s[1..])
+    }
+
+lemma sum_prop(s: seq<int>) 
+    requires |s| > 0
+    ensures sum(s) == sum(s[..|s| - 1]) + s[ |s| - 1 ]
 {
-  result := [];
-  var i := 0;
-  while i < |a|
-    invariant i >= 0 && i <= |a|
-    invariant |result| == i
-    invariant forall j : int :: j >= 0 && j < |result| && j % 3 == 0 ==> result[j] == a[j] * a[j]
-    invariant forall j : int :: j >= 0 && j < |result| && j % 4 == 0 && j % 3 != 0 ==> result[j] == a[j] * a[j] * a[j]
-  {
-    if i % 3 == 0 {
-      result := result + [a[i] * a[i]];
-    } else if i % 4 == 0 && i % 3 != 0 {
-      result := result + [a[i] * a[i] * a[i]];
-    } else {
-      result := result + [a[i]];
+    if (|s| > 1) {
+        assert (s[1..][..|s[1..]| - 1]) == s[1..|s| - 1];
     }
-    i := i + 1;
-  }
 }
+
+function square_seq(lst: seq<int>) : (sq : seq<int>) 
+        ensures |sq| == |lst|
+    {
+        seq(|lst|, i requires 0 <= i < |lst| => if i % 3 == 0 then lst[i] * lst[i] else (if i % 4 == 0 then lst[i] * lst[i] * lst[i] else lst[i]))
+    }
+
+method sum_squares(lst: seq<int>) returns (r : int)
+    ensures r == sum(square_seq(lst))
+    {
+        r := 0;
+        var k := 0;
+        var v := square_seq(lst);
+        while(k < |lst|) 
+            invariant 0 <= k <= |lst|
+            invariant r == sum(v[..k])
+        {
+            assert v[..k + 1][..k] == v[..k];
+            r := r + v[k];
+            k := k + 1;
+            assert sum(v[..k]) == r by { sum_prop(v[..k]); }
+        }
+        assert v[..k] == v;
+        return r;
+    }

151-double_the_difference.dfy

@@ -0,0 +1,53 @@
+function sumc(s: seq<int>, p: seq<bool>) : int 
+    requires |s| == |p|
+    {
+        if |s| == 0 then 0 else (if p[0] then s[0] else 0) + sumc(s[1..], p[1..])
+    }
+
+lemma sum_prop(s: seq<int>, p: seq<bool>) 
+    requires |s| > 0
+    requires |s| == |p|
+    ensures sumc(s, p) == sumc(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..][..|s[1..]| - 1]) == p[1..|s| - 1];
+    }
+}
+
+function add_conditon(lst: seq<int>) : (p : seq<bool>)
+    ensures |lst| == |p|
+    {
+        seq(|lst|, i requires 0 <= i < |lst| => lst[i] >= 0 && lst[i] % 2 == 1)
+    }
+
+function square_seq(lst: seq<int>) : (sq : seq<int>) 
+        ensures |sq| == |lst|
+        ensures forall i :: 0 <= i < |lst| ==> sq[i] == lst[i] * lst[i]
+    {
+        seq(|lst|, i requires 0 <= i < |lst| => lst[i] * lst[i])
+    }
+
+method double_the_difference(lst: seq<int>) returns (r : int)
+    ensures r == sumc(square_seq(lst), add_conditon(lst))
+    {
+        r := 0;
+        var k := 0;
+        var v := square_seq(lst);
+        var p := add_conditon(lst);
+        while(k < |lst|) 
+            invariant 0 <= k <= |lst|
+            invariant r == sumc(v[..k], p[..k])
+        {
+            assert v[..k + 1][..k] == v[..k];
+            assert p[..k + 1][..k] == p[..k];
+            r := r + if p[k] then v[k] else 0;
+            k := k + 1;
+            assert sumc(v[..k], p[..k]) == r by {
+                sum_prop(v[..k], p[..k]);
+            }
+        }
+        assert v[..k] == v;
+        assert p[..k] == p;
+        return r;
+    }

README.md

@@ -62,7 +62,7 @@ Current status:
 - [ ] 57. monotonic
 - [x] 58. common
 - [x] 59. largest_prime_factor
-- [ ] 60. sum_to_n
+- [x] 60. sum_to_n
 - [ ] 61. correct_bracketing
 - [ ] 62. derivative
 - [ ] 63. fibfib
@@ -87,7 +87,7 @@ Current status:
 - [ ] 82. prime_length
 - [ ] 83. starts_one_ends
 - [ ] 84. solve
-- [ ] 85. add
+- [x] 85. add
 - [ ] 86. anti_shuffle
 - [ ] 87. get_row
 - [ ] 88. sort_array
@@ -135,7 +135,7 @@ Current status:
 - [x] 130. tri
 - [ ] 131. digits
 - [ ] 132. is_nested
-- [ ] 133. sum_squares
+- [x] 133. sum_squares
 - [ ] 134. check_if_last_char_is_a_letter
 - [ ] 135. can_arrange
 - [ ] 136. largest_smallest_integers
@@ -144,7 +144,7 @@ Current status:
 - [x] 139. special_factorial
 - [ ] 140. fix_spaces
 - [ ] 141. file_name_check
-- [ ] 142. sum_squares
+- [x] 142. sum_squares
 - [ ] 143. words_in_sentence
 - [ ] 144. simplify
 - [ ] 145. order_by_points
@@ -153,7 +153,7 @@ Current status:
 - [x] 148. bf
 - [ ] 149. sorted_list_sum
 - [x] 150. x_or_y
-- [ ] 151. double_the_difference
+- [x] 151. double_the_difference
 - [x] 152. compare
 - [x] 153. Strongest_Extension
 - [x] 154. cycpattern_check