diff --git a/include/data/encoding/halves.hpp b/include/data/encoding/halves.hpp
index 2facdffb..86411bea 100644
--- a/include/data/encoding/halves.hpp
+++ b/include/data/encoding/halves.hpp
@@ -173,17 +173,22 @@ namespace data::encoding {
     };
     
     template <typename half>
-    typename twice<half>::type combine(half greater, half lesser) {
+    typename twice<half>::type inline combine(half greater, half lesser) {
         return ((typename twice<half>::type)(greater) << count_digits<half>::value * 8) + lesser;
     };
     
     template <typename half>
-    typename twice<half>::type multiply(half a, half b, half r) {
+    typename twice<half>::type inline multiply(half a, half b, half r) {
         return (typename twice<half>::type)(a) * (typename twice<half>::type)(b) + (typename twice<half>::type)(r);
     };
     
     template <typename half>
-    typename twice<half>::type add(half a, half b, half r) {
+    typename twice<half>::type inline subtract(half a, half b, half r) {
+        return (typename twice<half>::type)(a) - b + r;
+    };
+    
+    template <typename half>
+    typename twice<half>::type inline add(half a, half b, half r) {
         return (typename twice<half>::type)(a) + b + r;
     };
     
diff --git a/include/data/encoding/words.hpp b/include/data/encoding/words.hpp
index 18ab220f..c0986f0b 100644
--- a/include/data/encoding/words.hpp
+++ b/include/data/encoding/words.hpp
@@ -163,13 +163,51 @@ namespace data::arithmetic {
     template <typename digit, typename sen, typename ita, typename itb>
     requires std::input_iterator<itb> && std::output_iterator<ita, digit> && std::sentinel_for<sen, ita>
     digit minus(sen z, ita a, itb b, itb i) {
-        throw method::unimplemented{"arithmetic::minus"};
+        using two_digits = typename encoding::twice<digit>::type;
+        
+        digit remainder = 0;
+        
+        while (i != z) {
+            two_digits result = encoding::subtract<digit>(*a, *b, remainder);
+            remainder = encoding::greater_half(result);
+            *i = encoding::lesser_half(result);
+            i++;
+            a++;
+            b++;
+        }
+        
+        return remainder;
     }
     
     template <typename digit, typename sen, typename ita, typename itb>
     requires std::input_iterator<itb> && std::output_iterator<ita, digit> && std::sentinel_for<sen, ita>
     digit times(sen z, ita a, itb b, itb i) {
-        throw method::unimplemented{"arithmetic::minus"};
+        using two_digits = typename encoding::twice<digit>::type;
+        
+        digit remainder = 0;
+        auto ab = a;
+        while (i != z) {
+            auto ax = ab;
+            auto bx = b;
+            
+            two_digits result = remainder;
+            
+            while (true) {
+                result += two_digits(*ax) * two_digits(*bx);
+                
+                if (ax == a) break;
+                ax++;
+                bx--;
+            }
+            
+            remainder = encoding::greater_half(result);
+            *i = encoding::lesser_half(result);
+            i++;
+            a++;
+            b++;
+        }
+        
+        return remainder;
     }
     
     // bit shift operations are defined in terms of big-endian numbers. 
diff --git a/include/data/tools/rb_map.hpp b/include/data/tools/rb_map.hpp
index 379c2219..39030f1a 100644
--- a/include/data/tools/rb_map.hpp
+++ b/include/data/tools/rb_map.hpp
@@ -41,7 +41,16 @@ namespace data::tool {
         rb_map remove(const K& k) const;
         rb_map remove(const entry& e) const;
         
-        bool valid() const;
+        bool valid() const {
+            try {
+                milewski::okasaki::forEach(Map, [](const K &k, const V &v) -> void {
+                    if (!data::valid(k) || !data::valid(v)) throw 0;
+                });
+            } catch (int) {
+                return false;
+            }
+            return true;
+        }
         
         bool empty() const;
         size_t size() const;