77, 78, 82, 83 (#12)

077-is_cube.dfy

@@ -0,0 +1,30 @@
+method cube_root(N: nat) returns (r: nat)
+  ensures cube(r) <= N < cube(r + 1)
+  ensures r <= N
+{
+  r := 0;
+  while cube(r + 1) <= N
+    invariant cube(r) <= N
+  {
+    r := r + 1;
+  }
+}
+
+method is_cube(n: nat) returns (r: bool)
+  ensures r ==> exists r :: 0 <= r <= n && n == cube(r)
+  ensures !r ==> forall r :: 0 <= r <= n ==> n != cube(r)
+{
+    var root := cube_root(n);
+    if cube(root) == n {
+        r := true;
+    } else {
+        cube_of_larger_is_larger();
+        r := false;
+    }
+}
+
+function cube(n: int): int { n * n * n }
+
+lemma cube_of_larger_is_larger()
+    ensures forall smaller : int, larger : int :: smaller <= larger ==> cube(smaller) <= cube(larger)
+{}

078-hex_key.dfy

@@ -0,0 +1,45 @@
+ function IsPrimeHexDigit(c: char): bool
+{
+  c == '2' || c == '3' || c == '5' || c == '7' || c == 'B' || c == 'D'
+}
+
+function count_prime_hex_digits_rec(num: seq<char>) : (count : int)
+  ensures 0 <= count <= |num|
+{
+  if |num| == 0 then 0
+  else (if IsPrimeHexDigit(num[0]) then 1 else 0) + count_prime_hex_digits_rec(num[1..])
+}
+
+lemma count_prop(s: seq<char>) 
+    requires |s| > 0
+    ensures count_prime_hex_digits_rec(s) == count_prime_hex_digits_rec(s[..|s| - 1]) + (
+        if IsPrimeHexDigit(s[ |s| - 1 ]) then 1 else 0
+    )
+{
+    if (|s| > 1) {
+        assert (s[1..][..|s[1..]| - 1]) == s[1..|s| - 1];
+    }
+}
+
+method count_prime_hex_digits(s: seq<char>) returns (count : int)
+    ensures count == count_prime_hex_digits_rec(s)
+    ensures 0 <= count <= |s|
+{
+    count := 0;
+    var i := 0;
+    while(i < |s|) 
+        invariant 0 <= i <= |s|
+        invariant count == count_prime_hex_digits_rec(s[..i])
+    {
+        assert count_prime_hex_digits_rec(s[..i + 1]) == count_prime_hex_digits_rec(s[..i]) + (
+            if IsPrimeHexDigit(s[ i ]) then 1 else 0
+        ) by {
+            assert s[..i+1][..i] == s[..i];
+            count_prop(s[..i + 1]);
+        }
+        count := count + if IsPrimeHexDigit(s[i]) then 1 else 0;
+        i := i + 1;
+    }
+    assert s[..i] == s;
+    return count;
+}

082-prime_length.dfy

@@ -0,0 +1,11 @@
+predicate IsPrime(n: int)
+{
+  n > 1 &&
+  forall k :: 2 <= k < n ==> n % k != 0
+}
+
+function PrimeLength(s: string): bool
+  ensures PrimeLength(s) <==> IsPrime(|s|)
+{
+  IsPrime(|s|)
+}

083-starts_one_ends.dfy

@@ -0,0 +1,18 @@
+method CountNumbersStartingOrEndingWithOne(n: nat) returns (count: nat)
+    requires n > 0
+    ensures n == 1 ==> count == 1
+    ensures n > 1 ==> count == 18 * Pow(10, n - 2)
+{
+    if n == 1 {
+        count := 1;
+    } else {
+        count := 18 * Pow(10, n - 2);
+    }
+}
+
+function Pow(base: nat, exponent: nat): nat
+    ensures exponent == 0 ==> Pow(base, exponent) == 1
+    ensures exponent > 0 ==> Pow(base, exponent) == base * Pow(base, exponent-1)
+{
+    if exponent == 0 then 1 else base * Pow(base, exponent-1)
+}

README.md

@@ -79,13 +79,13 @@ Current status:
 - [ ] 74. total_match
 - [ ] 75. is_multiply_prime
 - [ ] 76. is_simple_power
-- [ ] 77. iscube
-- [ ] 78. hex_key
+- [x] 77. iscube
+- [x] 78. hex_key
 - [ ] 79. decimal_to_binary
 - [ ] 80. is_happy
 - [ ] 81. numerical_letter_grade
-- [ ] 82. prime_length
-- [ ] 83. starts_one_ends
+- [x] 82. prime_length
+- [x] 83. starts_one_ends
 - [ ] 84. solve
 - [x] 85. add
 - [ ] 86. anti_shuffle