<html><head><meta http-equiv="Content-Type" content="text/html charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><blockquote type="cite" class="">On Jan 16, 2017, at 3:25 AM, Xiaodi Wu via swift-evolution <<a href="mailto:swift-evolution@swift.org" class="">swift-evolution@swift.org</a>> wrote:<br class=""></blockquote><div><blockquote type="cite" class=""><div class=""><div style="font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px;" class=""><br class="Apple-interchange-newline">Unless I'm mistaken, after removing division, models of SignedArithmetic would have the mathematical properties of a ring. For every element a in ring R, there must exist an additive inverse -a in R such that a + (-a) = 0. Models of Arithmetic alone would not necessarily have that property.</div></div></blockquote></div><br class=""><div class="">Closure under the arithmetic operations is a sticky point for all the finite integer models vs. the actual ring axioms. No finite [non-modulo] integer type is closed, because of overflow. Similarly, additive inverses don’t exist for the most negative value of a signed type, or for any non-zero value of an unsigned type. The obvious way around this is to say that types conforming to Arithmetic model a subset of a ring that need not be closed under the operations.</div><div class=""><br class=""></div><div class="">– Steve</div></body></html>