# Relation COMPOSITION-OF

## Slots on this relation:

Documentation:
A binary relation R is a COMPOSITION-OF a sequence of binary relations R_1, R_2, ... R_N iff there exists a relation R' that is a COMPOSITION-OF the sequence R_1 ... R_{N-1}, and R is the (COMPOSITION R_1 R').

Relations are composed right to left. For example, if (composition-of R (listof a b c d)) then R = (composition d (composition c (composition b a))).

When the relations are unary functions, the sequence corresponds to nested parentheses in functional notation. For example, if F is composition-of functions a, b, and c, (COMPOSITION-OF F (listof a b c)) means (f ?x) is equal to (a (b (c ?x))).

Arity: 2
Domain: Binary-relation
Range: List

## Axioms:

```(<=> (Composition-Of ?Binary-Relation ?List-Of-Relations)
(And (Binary-Relation ?Binary-Relation)
(List ?List-Of-Relations)
(Not (Null ?List-Of-Relations))
(=> (Item ?R ?List-Of-Relations) (Binary-Relation ?R))
(Or (And (Single ?List-Of-Relations)
(= ?Binary-Relation (First ?List-Of-Relations)))
(And (Double ?List-Of-Relations)
(= ?Binary-Relation
(Composition (First (Rest ?List-Of-Relations))
(First ?List-Of-Relations))))
(Exists (?Left-Sub-Relation)
(And (= ?Binary-Relation
(Composition (Last ?List-Of-Relations)
?Left-Sub-Relation))
(Composition-Of ?Left-Sub-Relation
(Butlast ?List-Of-Relations)))))))

```

## Notes:

• Example: (=> (composition-of r (listof a b c d)) (= r (composition d (composition c (composition b a)))))