Z Reference Card Mike Spivey

Specifications Schema box Name[Params] Declarations Predicates

Axiomatic description Declarations Predicates

Generic definition [Params] Declarations Predicates

\begin{schema}{Name}[Params] Declarations \where Predicates \end{schema}

\begin{axdef} Declarations \where Predicates \end{axdef}

\begin{gendef}[Params] Declarations \where Predicates \end{gendef}

c J. M. Spivey 1992 Copyright

\begin{zed} ... Basic type definition [NAME , DATE ]

[NAME, DATE]

Abbreviation definition DOC == seq CHAR

DOC == \seq CHAR

Constraint n disks < 5

n\_disks < 5

Schema definition Point = b [ x, y : ]

Point \defs [~x, y: \num~]

Free type definition  Ans ::= ok
| error

Ans ::= ok \ldata\num\rdata | error

... \end{zed}

Logic and schema calculus true, false ¬P P ∧Q P ∨Q P ⇒Q P ⇔Q ∀x : T | P • Q ∃x : T | P • Q ∃1 x : T | P • Q

true, false \lnot P P \land Q P \lor Q P \implies Q P \iff Q \forall ... \exists ... \exists_1 ...

Logical constants Negation Conjunction Disjunction Implication Equivalence Universal quantifier Existential quantifier Unique quantifier

Special schema operators S [y1 /x1 , y2 /x2 ] S \ (x1 , x2 ) S1  S2 pre Op Op1  Op2 Op1 >> Op2

S[y_1/x_1, y_2/x_2] S \hide (x_1, x_2) S1 \project S2 \pre Op Op1 \semi Op2 Op1 \pipe Op2

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Renaming Hiding Projection Pre-condition Sequential composition Piping

Basic expressions x =y x 6= y if P then E1 else E2 θS E .x (µ x : T | P • E ) (let x ==E 1 • E 2)

x = y x \neq y \IF P \THEN E_1 \ELSE E_2 \theta S E.x (\mu x: T | P @ E) (\LET x == E1 @ E2)

Equality Inequality Conditional Expression Theta-expression Selection Mu-expression Let-expression

x \in S x \notin S \{x_1, ..., \{~x: T | P \emptyset S \subseteq S \subset T \power S \power_1 S S \cross T (x, y, z) first~p second~p S \cup T S \cap T S \setminus \bigcup A \bigcap A \finset X \finset_1 X

Membership Non-membership Set display Set comprehension Empty set Subset relation Proper subset relation Power set Non-empty subsets Cartesian product Tuple First of pair Second of pair Set union Set intersection Set difference Generalized union Generalized intersection Finite sets Non-empty finite sets

Sets x ∈S x∈ /S {x1 , . . . , xn } {x : T | P • E } S ⊆T S ⊂T S 1S S ×T (x , y, z ) first p second p S ∪T S ∩T S S\ T TA A X 1X 









x_n\} @ E~\} T

T

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Relations X Y x 7→ y dom R ran R id X Q R Q ◦R S R R S S R R S R∼ R S Q ⊕R Rk R+ R∗












X \rel Y x \mapsto y \dom R \ran R \id X Q \comp R Q \circ R S \dres R R \rres S S \ndres R R \nrres S R \inv R \limg S\rimg Q \oplus R R^{k} R \plus R \star

Binary relations Maplet Domain Range Identity relation Composition Backwards composition Domain restriction Range restriction Domain anti-restriction Range anti-restriction Relational inverse Relational image Overriding Iteration Transitive closure Reflexive–trans. closure

f(x) (\lambda ...) X \pfun Y X \fun Y X \pinj Y X \inj Y X \psurj Y X \surj Y X \bij Y X \ffun Y X \finj Y

Function application Lambda-expression Partial functions Total functions Partial injections Total injections Partial surjections Total surjections Bijections Finite partial functions Finite partial injections

Functions f (x ) (λ x : T | P • E ) X Y X Y X Y X Y X Y X Y X Y X Y X Y 















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Numbers and arithmetic

+ − ∗ div mod <≤≥> 1

succ m ..n #S min S max S

\nat \num + - * \div \mod < \leq \geq > \nat_1 succ m \upto n \# S min~S max~S

Natural numbers Integers Arithmetic operations Arithmetic comparisons Strictly positive integers Successor function Number range Size of a set Minimum of a set Maximum of a set

\seq X \seq_1 X \iseq X \langle ... \rangle s \cat t rev~s head~s last~s tail~s front~s U \extract S s \filter V squash~f s \prefix t s \suffix t s \inseq t \dcat ss \disjoint SS SS \partition T

Finite sequences Non-empty sequences Injective sequences Sequence display Concatenation Reverse Head of sequence Last element of sequence Tail of sequence All but last element Extraction Filtering Compaction Prefix relation Suffix relation Segment relation Distributed concat. Disjointness Partition relation

Sequences seq X seq1 X iseq X hx1 , . . . , xn i s t rev s head s last s tail s front s U s s  V squash f s prefix t s suffix t s in t /ss disjoint SS SS partition T







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Bags bag X x1 , . . . , xn count B x B ]x n ⊗B x B B vC B ]C B C items s






\bag X \lbag ... \rbag count~B~x B \bcount x n \otimes B x \inbag B B \subbageq C B \uplus C B \uminus C items~s

Bags Bag display Count of an element Infix count operator Bag scaling Bag membership Sub-bag relation Bag union Bag difference Items in a sequence

f uzz flags Usage: fuzz [ -aqstv] [ -p prelude] [ file . . . ] -a -p predude -q -d -s -t -v

Don’t use type abbreviations Use prelude in place of the standard one Assume implicit quantifiers for undeclared variables Dependency analysis Syntax check only Report types of global definitions Echo formal text as it is parsed

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