(NOTE: This is work in progress) These axioms are used by GHC's solver to construct proofs/evidence for various predicates involving type-level naturals. The actual algorithm for constructing the evidence is implemented as set of rules (interactions) which are described separately. The "*Def" axioms bellow look a bit odd but all they are saying is that the predicates which are being defined behave like their corresponding mathematical operations. Notation: {{{ k,m,n: literals of kind Nat r,s,t: arbitrary terms of kind Nat }}} Comparison: {{{ leqDef: m <= n -- if "m <= n" leqLeast: 0 <= t leqRefl: t <= t leqTrans: (r <= s, s <= t) => r <= t leqAntiSym: (s <= t, t <= s) => s ~ t }}} Addition: {{{ addDef: m + n ~ k -- if "m + n == k" addUnit: 0 + t ~ t addAssoc: (r + s) + t ~ r + (s + t) | (r + s = u, s + t = v, r + v = w1, u + t = w2) => w1 ~ w2 addCommutes: t + s ~ s + t | (r + s ~ t) => (s + r ~ t) addCancel: (r + s ~ r + t) => s ~ t | (r + s ~ u, r + t = u) => s ~ t }}} Multiplication: {{{ mulDef: m * n ~ k -- if "m * n == k" mulUnit: 1 * t ~ t mulAssoc: (r * s) * t ~ r * (s * t) mulCommutes: t * s ~ s * t mulCancel: (r * s ~ r * t, 1 <= r) => s ~ t }}} Exponentiation: {{{ expDef: m ^ n ~ k -- if "m ^ n == k" exp0: a ^ 0 ~ 1 exp1: a ^ 1 ~ a log1: 1 ^ a ~ 1 XXX: m ^ a ~ a => m ~ 1 a ^ m ~ a => a <= 1 -- 2 <= m 0 ^ a ~ b => b <= 1 }}} Interactions: {{{ addMulDistr: r * (s + t) = (r * s) + (r * t) }}} References: * http://en.wikipedia.org/wiki/Semiring * http://en.wikipedia.org/wiki/Cancellation_property