continuesfollows

Continuesfollows is a term used to describe a specific kind of relation between successive elements in a sequence or ordered set. It combines two intuitive ideas: continuation, where the next element persists or extends the ongoing pattern, and follows, where the next element is a logical consequence of the previous one under a defined rule. As a result, a pair (a_n, a_{n+1}) is said to be in the continuesfollows relation if a_{n+1} both continues the established pattern and follows from a_n under the governing rule.

Formal use: In a sequence over a set X with a rule R, a_{n+1} continuesfollows a_n when a_{n+1} = R(a_n) and the application of R at step n+1 preserves the same pattern observed at earlier steps. Some authors require additional constraints such as invariance of the local rule or compatibility with prior transitions.

Examples: In arithmetic with a constant difference d, a_{n+1} = a_n + d yields continuesfollows pairs. In grammar or language models, if a next token both maintains the current syntactic frame and is a natural consequence of the previous token’s role, the pair is described as continuesfollows.

Origin and status: The term has limited usage and appears mainly in theoretical discussions about sequence transitions and dependency relations. It is not widely adopted as a standard term; some sources treat it as a descriptive coinage for comparing continuation and derivation in a single relation.

See also: continuation, follows relation, dependency, Markov property, sequence modeling.