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
3
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
4
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
5
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
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
R+. R \plus. Transitive closure. Râ. R \star. Reflexiveâtrans. closure. Functions f (x) f(x). Function application. (λ x : T | P ⢠E). (\lambda ...) Lambda-expression. X §.
Free type definition. Ans ::= ok ¡. £ ¢ ... Domain. ranR. \ran R. Range id X. \id X. Identity relation. Q ¥ R. Q \comp R. Composition ... Domain anti-restriction. R § S.
Domain. ranR. \ran R. Range id X. \id X. Identity relation. Q £ R. Q \comp R. Composition. Q ⦠R. Q \circ R. Backwards comp. S ¨ R. S \dres R. Domain restriction.
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