Syntropy Type view Object diagrams/mechanisms Statecharts Logic Value types Glossary

# Logic

## Definitions and declarations

 Definition of LHS as syntactically equivalent to RHS. x : T Declaration of identifier x to stand for a member of the set T (which may be a type name or any expression yielding a set). x, y : T Equivalent to x : T, y : T ( ) Groups terms in expressions

## Logic

 true, false Logical constants not P Negation: ‘not P’. Conjunction: ‘P and Q’ Disjunction: ‘P or Q’ Implication: ‘P implies Q’ or ‘if P then Q’ Equivalence: ‘P is logically equivalent to Q’ or ‘P if and only if Q’ Conditional: ‘if P then Q else R’. Universal quantification: ‘for all x in set S, P holds’. Existential quantification: ‘there exists an x in S such that P holds’. Unique existence: ‘there exists a unique x in S such that P holds’. Equality between terms.

## Sets

 Set membership: ‘t is a member of S’. * not (t * S) Set inclusion: ‘every member of S is also in T’. { } The empty set. The set containing the terms t1 through tn #S Size of the set S set of S Powerset: set of all subsets of S. The set containing exactly those x in S for which P holds. Given declarations D, the set of t’s for which P holds. Given declarations D, the set of t’s. Ordered tuple of elements Set union. Set difference. Set intersection. Distributed set union. Given SS is a set of sets with members taken from S, ‘the union of all the members of all the sets’ Cartesian product: The set of all 2-tuples such that the first component is a member of S and the second a member of T. sum S The numerical sum all of the elements of the set S. Also defined over sequences and bags. min S Minimum of a set (or sequence or bag) max S Maximum of a set (or sequence or bag)

## Functions

 The set of total functions from S to T.

## Bags

Mathematically, a bag is treated as a function mapping elements of the bag to positive integers, representing the number of times the element appears in the bag.

 bag of T The set of bags whose elements are drawn from set T. #X The number of elements in bag X The empty bag The bag containing x1, x2...,xn with the frequency in which they occur in the list. members X The set formed from the elements of bag X.

## Sequences

Mathematically, a sequence is treated as a function mapping positive integers, representing position in the sequence, to elements of the sequence.

 seq of T The set of sequences whose elements are drawn from set T. #A The length of sequence A The empty sequence The sequence containing a1, a2..., and an The sequence formed by concatenating the sequence A with the sequence B. A(n) The nth element of sequence A. members A The set formed from the elements of A. items A The bag of items contained in the sequence A. head A The first element of a sequence or nil if the sequence is empty. last A The last element of a sequence or nil if the sequence is empty. tail A All but the head of a sequence. front A All but the last of a sequence.

## Sorted sequences

 The sorted sequence formed by inserting element e into the sorted sequence S, following the sort rule for S.

## Objects

 a in Q True if the object a is in state Q, false otherwise.