A datatype defining rewrite system (DDRS) is an algebraic (equational) specification intended to specify a datatype. When interpreting the equations from left-to-right, a DDRS defines a term rewriting system that must be ground-complete. First we define two DDRSs for the ring of integers, each comprising twelve rewrite rules, and prove their ground-completeness. Then we introduce natural number and integer arithmetic specified according to unary view, that is, arithmetic based on a postfix unary append constructor (a form of tallying). Next we specify arithmetic based on two other views: binary and decimal notation. The binary and decimal view have as their characteristic that each normal form resembles common number notation, that is, either a digit, or a string of digits without leading zero, or the negated versions of the latter. Integer arithmetic in binary and decimal notation is based on (postfix) digit append functions. For each view we define a DDRS, and in each case the resulting datatype is a canonical term algebra that extends a corresponding canonical term algebra for natural numbers. Then, for each view, we consider an alternative DDRS based on tree constructors that yields comparable normal forms, which for that view admits expressions that are algorithmically more involved. For all DDRSs considered, ground-completeness is proven.

翻译：数据类型定义重写系统( DDRS) 是用于指定数据类型的代数( equation) 。 在解释左对右方方的方程式时, 一个 DDS 定义了一个术语重写系统, 它必须是地面完成的。 首先, 我们为整数环定义了两个 DDS, 每个整数环由12 重写规则组成, 并证明它们的地面完整性。 然后, 我们根据单词视图, 即根据后缀单子附加器( 一种计算形式) 引入自然数和整数计算。 下一步, 我们根据后缀附加器( 后缀) 进行算术( 一种计算形式) 。 我们根据另外两种观点定义了解算术: 二进制和小数标记。 二进制和小数视图的特性是, 每个正常形式都类似于通用的编号, 即数字, 或数字, 或数字, 或数字, 或数, 或数, 或数数, 或数, 以( 后缀) 数字为( 后缀) 数附加功能附加功能功能。 对于每个视图, 我们定义定义的代变数, 变数为一个 Campal- realbra 。