Functions

front s U s s V squash f s prefix t s suffix t s in t /ss disjoint SS SS partition T

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
















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

Application Lambda-expression Partial functions Total functions Partial injections Total injections Partial surjections Total surjections Bijections Finite functions Finite injections







+ − ∗ div mod <≤≥>

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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 Operations Comparisons Integers > 0 Successor function Number range Size of a set Minimum of a set Maximum of a set

All but last element Extraction Filtering Compaction Prefix relation Suffix relation Segment relation Distributed concat. Disjointness Partition relation

\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 element Infix count operator Bag scaling Bag membership Sub-bag relation Bag union Bag difference Items in a sequence

Z Reference Card Mike Spivey The Spivey Partnership

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

Numbers and arithmetic

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







f uzz flags Usage: fuzz [ -aqstv] [ -p prelude] [ file . . . ]

Sequences seq X seq1 X iseq X hx1 , . . . , xn i s t rev s head s last s tail s 

\seq X \seq_1 X \iseq X \langle \rangle s \cat t rev~s head~s last~s tail~s

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Finite sequences Sequences 6= hi Injective sequences Sequence display Concatenation Reverse Head of sequence Last element Tail of sequence

-a -p predude -q -d -s -t -v

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

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c J. M. Spivey 1992 Copyright

Specifications Schema box Name[Params] Declarations Predicates

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

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

Generic definition [Params] Declarations Predicates

{x1 , . . . , xn } { x :T | P • E }

Logic and schema calculus

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

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 quant. Unique quantifier

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

S[y1/x1, y2/x2] S \hide (x1, x2) S \project T \pre Op Op1 \semi Op2 Op1 \pipe Op2

Renaming Hiding Projection Pre-condition Sequential comp. Piping

Basic expressions \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 : ]

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

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

Membership Non-membership

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

Free type definition 
Ans ::= ok | error Ans ::= ok \ldata\num\rdata | error

... \end{zed}

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Sets x ∈S x∈ /S

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S ⊆T S ⊂T   S 1S S ×T (x , y, z ) first p second p S ∪T S ∩T S \T S TA  A  X 1X

\{x_1,...,x_n\} \{~x:T | P @ E~\} \emptyset S \subseteq T 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 T \bigcup A \bigcap A \finset X \finset_1 X

Set display Set comprehension Empty set Subset relation Proper subset Power set Non-empty subsets Cartesian product Tuple First of pair Second of pair Set union Set intersection Set difference Generalized union Gen. intersection Finite sets  Finite sets 6=

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 comp. Domain restriction Range restriction Domain anti-res. Range anti-restrict. Relational inverse Relational image Overriding Iteration Transitive closure Refl.–trans. closure

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∗

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