Normal form (abstract rewriting)

In abstract rewriting, an object is in normal form if it cannot be rewritten any further, i.e. it is irreducible. Depending on the rewriting system, an object may rewrite to several normal forms or none at all. Many properties of rewriting systems relate to normal forms. Stated formally, if (A,→) is an abstract rewriting system, x∈A is in normal form if no y∈A exists such that x→y, i.e. x is an irreducible term. An object a is weakly normalizing if there exists at least one particular sequence of rewrites starting from a that eventually yields a normal form. A rewriting system has the weak normalization property or is (weakly) normalizing (WN) if every object is weakly normalizing. An object a is strongly normalizing if every sequence of rewrites starting from a eventually terminates with a normal form. An abstract rewriting system is strongly normalizing, terminating, noetherian, or has the (strong) normalization property (SN), if each of its objects is strongly normalizing. A rewriting system has the normal form property (NF) if for all objects a and normal forms b, b can be reached from a by a series of rewrites and inverse rewrites only if a reduces to b. A rewriting system has the unique normal form property (UN) if for all normal forms a, b ∈ S, a can be reached from b by a series of rewrites and inverse rewrites only if a is equal to b. A rewriting system has the unique normal form property with respect to reduction (UN→) if for every term reducing to normal forms a and b, a is equal to b. This section presents some well known results. First, SN implies WN. Confluence (abbreviated CR) implies NF implies UN implies UN→. The reverse implications do not generally hold. {a→b,a→c,c→c,d→c,d→e} is UN→ but not UN as b=e and b,e are normal forms. {a→b,a→c,b→b} is UN but not NF as b=c, c is a normal form, and b does not reduce to c. {a→b,a→c,b→b,c→c} is NF as there are no normal forms, but not CR as a reduces to b and c, and b,c have no common reduct. WN and UN→ imply confluence. Hence CR, NF, UN, and UN→ coincide if WN holds.