Quelle  Main_Doc.thy

(*<*)
theory Main_Doc
imports Main
begin

setup ‹
  Document_Output.antiquotation_pretty_source 🍋‹term_type_only› (
    (fn ctxt => fn (t, T) =>
      (if fastype_of t = Sign.certify_typ (Proof_Context.theory_of ctxt) T then ()
       else error "term_type_only: type mismatch";
       Syntax.pretty_typ ctxt T))
›
setup ‹
  Document_Output.antiquotation_pretty_source 🍋‹expanded_typ› A
    Syntax.pretty_typ
›
(*>*)
text‹
 
 \begin{abstract}
 This document lists the main types, functions and syntax provided by theory 🍋‹Main›.
\end{abstract}

\section*{HOL}

The basic logic: 🍋‹x = y›, 🍋‹True›, 🍋‹False›, 🍋‹¬ P›, 🍋‹P ∧ Q›,
🍋‹P ∨ Q›, 🍋‹P ⟶ Q›, 🍋‹∀x. P›, 🍋‹∃x. P›, 🍋‹∃! x. P›,
🍋‹THE x. P›.
🪙

\begin{tabular}{@ {} l @ {~::~} l @ {}}
🍋‹HOL.undefined› & 🪙‹HOL.undefined›\java.lang.NullPointerException
🍋‹HOL.default› & 🪙‹HOL.default›\java.lang.NullPointerException
\end{tabular}

\subsubsection*{Syntax}

\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}}
🍋‹¬ (x = y)› & @{term[source]"¬ (x = y)"} & (🍋‹~=›)java.lang.NullPointerException
@{term[source]"P ⟷ Q"} & 🍋‹P ⟷ Q› java.lang.NullPointerException
🍋‹If x y z› & @{term[source]"If x y z"}java.lang.NullPointerException
🍋‹Let e🪙1 (λx. e🪙2)› & @{term[source]"Let e🪙1 (λx. e🪙2)"}java.lang.NullPointerException
\end{tabular}


\section*{Orderings}

A collection of classes defining basic orderings:
preorder, partial order, linear order, dense linear order and wellorder.
🪙

\begin{tabular}{@ {} l @ {~::~} l l @ {}}
🍋‹Orderings.less_eq› & 🪙‹Orderings.less_eq› & (🍋‹🚫›)java.lang.NullPointerException
🍋‹Orderings.less› & 🪙‹Orderings.less›\java.lang.NullPointerException
🍋‹Orderings.Least› & 🪙‹Orderings.Least›\java.lang.NullPointerException
🍋‹Orderings.Greatest› & 🪙‹Orderings.Greatest›\java.lang.NullPointerException
🍋‹Orderings.min› & 🪙‹Orderings.min›\java.lang.NullPointerException
🍋‹Orderings.max› & 🪙‹Orderings.max›\java.lang.NullPointerException
@{const[source] top} & 🪙‹Orderings.top›\java.lang.NullPointerException
@{const[source] bot} & 🪙‹Orderings.bot›\java.lang.NullPointerException
\end{tabular}

\subsubsection*{Syntax}

\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}}
@{term[source]"x ≥ y"} & 🍋‹x ≥ y› & (🍋‹>=›)java.lang.NullPointerException
@{term[source]"x > y"} & 🍋‹x > y›\java.lang.NullPointerException
🍋‹∀x≤y. P› & @{term[source]"∀x. x ≤ y ⟶ P"}java.lang.NullPointerException
🍋‹∃x≤y. P› & @{term[source]"∃x. x ≤ y ∧ P"}java.lang.NullPointerException
\multicolumn{2}{@ {}l@ {}}{Similarly for $<$, $\ge$ and $>$}java.lang.NullPointerException
🍋‹LEAST x. P› & @{term[source]"Least (λx. P)"}java.lang.NullPointerException
🍋‹GREATEST x. P› & @{term[source]"Greatest (λx. P)"}java.lang.NullPointerException
\end{tabular}


\section*{Lattices}

Classes semilattice, lattice, distributive lattice and complete lattice (the
latter in theory 🍋‹HOL.Set›).

\begin{tabular}{@ {} l @ {~::~} l @ {}}
🍋‹Lattices.inf› & 🪙‹Lattices.inf›\java.lang.NullPointerException
🍋‹Lattices.sup› & 🪙‹Lattices.sup›\java.lang.NullPointerException
🍋‹Complete_Lattices.Inf› & @{term_type_only Complete_Lattices.Inf "'a set ==> 'a::Inf"}java.lang.NullPointerException
🍋‹Complete_Lattices.Sup› & @{term_type_only Complete_Lattices.Sup "'a set ==> 'a::Sup"}java.lang.NullPointerException
\end{tabular}

\subsubsection*{Syntax}

Available via 🪙‹unbundle lattice_syntax›.

\begin{supertabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}}
@{text[source]"x ⊑ y"} & 🍋‹x ≤ y›\java.lang.NullPointerException
@{text[source]"x ⊏ y"} & 🍋‹x 🚫›\java.lang.NullPointerException
@{text[source]"x ⊓ y"} & 🍋‹inf x y›\java.lang.NullPointerException
@{text[source]"x ⊔ y"} & 🍋‹sup x y›\java.lang.NullPointerException
@{text[source]"⊓A"} & 🍋‹Inf A›\java.lang.NullPointerException
@{text[source]"⊔A"} & 🍋‹Sup A›\java.lang.NullPointerException
@{text[source]"⊤"} & @{term[source] top}java.lang.NullPointerException
@{text[source]"⊥"} & @{term[source] bot}java.lang.NullPointerException
\end{supertabular}


\section*{Set}

\begin{tabular}{@ {} l @ {~::~} l l @ {}}
🍋‹Set.empty› & @{term_type_only "Set.empty" "'a set"}java.lang.NullPointerException
🍋‹Set.insert› & @{term_type_only insert "'a==>'a set==>'a set"}java.lang.NullPointerException
🍋‹Collect› & @{term_type_only Collect "('a==>bool)==>'a set"}java.lang.NullPointerException
🍋‹Set.member› & @{term_type_only Set.member "'a==>'a set==>bool"} & (🍋‹:›)java.lang.NullPointerException
🍋‹Set.union› & @{term_type_only Set.union "'a set==>'a set ==> 'a set"} & (??‹Un›)java.lang.NullPointerException
🍋‹Set.inter› & @{term_type_only Set.inter "'a set==>'a set ==> 'a set"} & (??‹Int›)java.lang.NullPointerException
🍋‹Union› & @{term_type_only Union "'a set set==>'a set"}java.lang.NullPointerException
🍋‹Inter› & @{term_type_only Inter "'a set set==>'a set"}java.lang.NullPointerException
🍋‹Pow› & @{term_type_only Pow "'a set ==>'a set set"}java.lang.NullPointerException
🍋‹UNIV› & @{term_type_only UNIV "'a set"}java.lang.NullPointerException
🍋‹image› & @{term_type_only image "('a==>'b)==>'a set==>'b set"}java.lang.NullPointerException
🍋‹Ball› & @{term_type_only Ball "'a set==>('a==>bool)==>bool"}java.lang.NullPointerException
🍋‹Bex› & @{term_type_only Bex "'a set==>('a==>bool)==>bool"}java.lang.NullPointerException
\end{tabular}

\subsubsection*{Syntax}

\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}}
‹{a🪙1,…,a🪙n}› & ‹insert a🪙1 (… (insert a🪙n {})…)›\java.lang.NullPointerException
🍋‹a ∉ A› & @{term[source]"¬(x ∈ A)"}java.lang.NullPointerException
🍋‹A ⊆ B› & @{term[source]"A ≤ B"}java.lang.NullPointerException
🍋‹A ⊂ B› & @{term[source]"A < B"}java.lang.NullPointerException
@{term[source]"A 🪙 B"} & @{term[source]"B ≤ A"}java.lang.NullPointerException
@{term[source]"A 🪙 B"} & @{term[source]"B < A"}java.lang.NullPointerException
🍋‹{x. P}› & @{term[source]"Collect (λx. P)"}java.lang.NullPointerException
‹{t | x🪙1 … x🪙n. P}› & ‹{v. ∃x🪙1 … x🪙n. v = t ∧ P}›\java.lang.NullPointerException
@{term[source]"∪x∈I. A"} & @{term[source]"∪((λx. A) ` I)"} & (\texttt{UN})java.lang.NullPointerException
@{term[source]"∪x. A"} & @{term[source]"∪((λx. A) ` UNIV)"}java.lang.NullPointerException
@{term[source]"∩x∈I. A"} & @{term[source]"∩((λx. A) ` I)"} & (\texttt{INT})java.lang.NullPointerException
@{term[source]"∩x. A"} & @{term[source]"∩((λx. A) ` UNIV)"}java.lang.NullPointerException
🍋‹∀x∈A. P› & @{term[source]"Ball A (λx. P)"}java.lang.NullPointerException
🍋‹∃x∈A. P› & @{term[source]"Bex A (λx. P)"}java.lang.NullPointerException
🍋‹range f› & @{term[source]"f ` UNIV"}java.lang.NullPointerException
\end{tabular}


\section*{Fun}

\begin{tabular}{@ {} l @ {~::~} l l @ {}}
🍋‹Fun.id› & 🪙‹Fun.id›\java.lang.NullPointerException
🍋‹Fun.comp› & 🪙‹Fun.comp› & (\texttt{o})java.lang.NullPointerException
🍋‹Fun.inj_on› & @{term_type_only Fun.inj_on "('a==>'b)==>'a set==>bool"}java.lang.NullPointerException
🍋‹Fun.inj› & 🪙‹Fun.inj›\java.lang.NullPointerException
🍋‹Fun.surj› & 🪙‹Fun.surj›\java.lang.NullPointerException
🍋‹Fun.bij› & 🪙‹Fun.bij›\java.lang.NullPointerException
🍋‹Fun.bij_betw› & @{term_type_only Fun.bij_betw "('a==>'b)==>'a set==>'b set==>bool"}java.lang.NullPointerException
🍋‹Fun.monotone_on› & 🪙‹Fun.monotone_on›\java.lang.NullPointerException
🍋‹Fun.monotone› & 🪙‹Fun.monotone›\java.lang.NullPointerException
🍋‹Fun.mono_on› & 🪙‹Fun.mono_on›\java.lang.NullPointerException
🍋‹Fun.mono› & 🪙‹Fun.mono›\java.lang.NullPointerException
🍋‹Fun.strict_mono_on› & 🪙‹Fun.strict_mono_on›\java.lang.NullPointerException
🍋‹Fun.strict_mono› & 🪙‹Fun.strict_mono›\java.lang.NullPointerException
🍋‹Fun.antimono› & 🪙‹Fun.antimono›\java.lang.NullPointerException
🍋‹Fun.fun_upd› & 🪙‹Fun.fun_upd›\java.lang.NullPointerException
\end{tabular}

\subsubsection*{Syntax}

\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}}
🍋‹fun_upd f x y› & @{term[source]"fun_upd f x y"}java.lang.NullPointerException
‹f(x🪙1:=y🪙1,…,x🪙n:=y🪙n)› & ‹f(x🪙1:=y🪙1)…(x🪙n:=y🪙n)›\java.lang.NullPointerException
\end{tabular}


\section*{Hilbert\_Choice}

Hilbert's selection ($\varepsilon$) operator: 🍋‹SOME x. P›.
🪙

\begin{tabular}{@ {} l @ {~::~} l @ {}}
🍋‹Hilbert_Choice.inv_into› & @{term_type_only Hilbert_Choice.inv_into "'a set ==> ('a ==> 'b) ==> ('b ==> 'a)"}
\end{tabular}

\subsubsection*{Syntax}

\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}}
🍋‹inv› & @{term[source]"inv_into UNIV"}
\end{tabular}

\section*{Fixed Points}

Theory: 🍋‹HOL.Inductive›.

Least and greatest fixed points in a complete lattice 🍋‹'a›:

\begin{tabular}{@ {} l @ {~::~} l @ {}}
🍋‹Inductive.lfp› & 🪙‹Inductive.lfp›\java.lang.NullPointerException
🍋‹Inductive.gfp› & 🪙‹Inductive.gfp›\java.lang.NullPointerException
\end{tabular}

Note that in particular sets (🍋‹'a ==> bool›) are complete lattices.

\section*{Sum\_Type}

Type constructor ‹+›.

\begin{tabular}{@ {} l @ {~::~} l @ {}}
🍋‹Sum_Type.Inl› & 🪙‹Sum_Type.Inl›\java.lang.NullPointerException
🍋‹Sum_Type.Inr› & 🪙‹Sum_Type.Inr›\java.lang.NullPointerException
🍋‹Sum_Type.Plus› & @{term_type_only Sum_Type.Plus "'a set==>'b set==>('a+'b)set"}
\end{tabular}


\section*{Product\_Type}

Types 🍋‹unit› and ‹×›.

\begin{tabular}{@ {} l @ {~::~} l @ {}}
🍋‹Product_Type.Unity› & 🪙‹Product_Type.Unity›\java.lang.NullPointerException
🍋‹Pair› & 🪙‹Pair›\java.lang.NullPointerException
🍋‹fst› & 🪙‹fst›\java.lang.NullPointerException
🍋‹snd› & 🪙‹snd›\java.lang.NullPointerException
🍋‹case_prod› & 🪙‹case_prod›\java.lang.NullPointerException
🍋‹curry› & 🪙‹curry›\java.lang.NullPointerException
🍋‹Product_Type.Sigma› & @{term_type_only Product_Type.Sigma "'a set==>('a==>'b set)==>('a*'b)set"}java.lang.NullPointerException
\end{tabular}

\subsubsection*{Syntax}

\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} ll @ {}}
🍋‹Pair a b› & @{term[source]"Pair a b"}java.lang.NullPointerException
🍋‹case_prod (λx y. t)› & @{term[source]"case_prod (λx y. t)"}java.lang.NullPointerException
🍋‹A × B› &  ‹Sigma A (λ🍋‹\_›. B)›
\end{tabular}

Pairs may be nested. Nesting to the right is printed as a tuple,
e.g.\ \mbox{🍋‹(a,b,c)›} is really \mbox{‹(a, (b, c))›.}
Pattern matching with pairs and tuples extends to all binders,
e.g.\ \mbox{🍋‹∀(x,y)∈A. P›,} 🍋‹{(x,y). P}›, etc.


\section*{Relation}

\begin{supertabular}{@ {} l @ {~::~} l @ {}}
🍋‹Relation.converse› & @{term_type_only Relation.converse "('a * 'b)set ==> ('b*'a)set"}java.lang.NullPointerException
🍋‹Relation.relcomp› & @{term_type_only Relation.relcomp "('a*'b)set==>('b*'c)set==>('a*'c)set"}java.lang.NullPointerException
🍋‹Relation.Image› & @{term_type_only Relation.Image "('a*'b)set==>'a set==>'b set"}java.lang.NullPointerException
🍋‹Relation.inv_image› & @{term_type_only Relation.inv_image "('a*'a)set==>('b==>'a)==>('b*'b)set"}java.lang.NullPointerException
🍋‹Relation.Id_on› & @{term_type_only Relation.Id_on "'a set==>('a*'a)set"}java.lang.NullPointerException
🍋‹Relation.Id› & @{term_type_only Relation.Id "('a*'a)set"}java.lang.NullPointerException
🍋‹Relation.Domain› & @{term_type_only Relation.Domain "('a*'b)set==>'a set"}java.lang.NullPointerException
🍋‹Relation.Range› & @{term_type_only Relation.Range "('a*'b)set==>'b set"}java.lang.NullPointerException
🍋‹Relation.Field› & @{term_type_only Relation.Field "('a*'a)set==>'a set"}java.lang.NullPointerException
🍋‹Relation.refl_on› & @{term_type_only Relation.refl_on "'a set==>('a*'a)set==>bool"}java.lang.NullPointerException
🍋‹Relation.refl› & @{term_type_only Relation.refl "('a*'a)set==>bool"}java.lang.NullPointerException
🍋‹Relation.sym› & @{term_type_only Relation.sym "('a*'a)set==>bool"}java.lang.NullPointerException
🍋‹Relation.antisym› & @{term_type_only Relation.antisym "('a*'a)set==>bool"}java.lang.NullPointerException
🍋‹Relation.trans› & @{term_type_only Relation.trans "('a*'a)set==>bool"}java.lang.NullPointerException
🍋‹Relation.irrefl› & @{term_type_only Relation.irrefl "('a*'a)set==>bool"}java.lang.NullPointerException
🍋‹Relation.total_on› & @{term_type_only Relation.total_on "'a set==>('a*'a)set==>bool"}java.lang.NullPointerException
🍋‹Relation.total› & @{term_type_only Relation.total "('a*'a)set==>bool"}java.lang.NullPointerException
\end{supertabular}

\subsubsection*{Syntax}

\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}}
🍋‹converse r› & @{term[source]"converse r"} & (🍋‹^-1›)
\end{tabular}
🪙

\noindent
Type synonym \ 🍋‹'a rel› ‹=› @{expanded_typ "'a rel"}

\section*{Equiv\_Relations}

\begin{tabular}{@ {} l @ {~::~} l @ {}}
🍋‹Equiv_Relations.equiv› & @{term_type_only Equiv_Relations.equiv "'a set ==> ('a*'a)set==>bool"}java.lang.NullPointerException
🍋‹Equiv_Relations.quotient› & @{term_type_only Equiv_Relations.quotient "'a set ==> ('a × 'a) set ==> 'a set set"}java.lang.NullPointerException
🍋‹Equiv_Relations.congruent› & @{term_type_only Equiv_Relations.congruent "('a*'a)set==>('a==>'b)==>bool"}java.lang.NullPointerException
🍋‹Equiv_Relations.congruent2› & @{term_type_only Equiv_Relations.congruent2 "('a*'a)set==>('b*'b)set==>('a==>'b==>'c)==>bool"}java.lang.NullPointerException
%@ {const Equiv_Relations.} & @ {term_type_only Equiv_Relations. ""}java.lang.NullPointerException
\end{tabular}

\subsubsection*{Syntax}

\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}}
🍋‹congruent r f› & @{term[source]"congruent r f"}java.lang.NullPointerException
🍋‹congruent2 r r f› & @{term[source]"congruent2 r r f"}java.lang.NullPointerException
\end{tabular}


\section*{Transitive\_Closure}

\begin{tabular}{@ {} l @ {~::~} l @ {}}
🍋‹Transitive_Closure.rtrancl› & @{term_type_only Transitive_Closure.rtrancl "('a*'a)set==>('a*'a)set"}java.lang.NullPointerException
🍋‹Transitive_Closure.trancl› & @{term_type_only Transitive_Closure.trancl "('a*'a)set==>('a*'a)set"}java.lang.NullPointerException
🍋‹Transitive_Closure.reflcl› & @{term_type_only Transitive_Closure.reflcl "('a*'a)set==>('a*'a)set"}java.lang.NullPointerException
🍋‹Transitive_Closure.acyclic› & @{term_type_only Transitive_Closure.acyclic "('a*'a)set==>bool"}java.lang.NullPointerException
🍋‹compower› & @{term_type_only "(^^) :: ('a*'a)set==>nat==>('a*'a)set" "('a*'a)set==>nat==>('a*'a)set"}java.lang.NullPointerException
\end{tabular}

\subsubsection*{Syntax}

\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}}
🍋‹rtrancl r› & @{term[source]"rtrancl r"} & (🍋‹^*›)java.lang.NullPointerException
🍋‹trancl r› & @{term[source]"trancl r"} & (🍋‹^+›)java.lang.NullPointerException
🍋‹reflcl r› & @{term[source]"reflcl r"} & (🍋‹^=›)
\end{tabular}


\section*{Algebra}

Theories 🍋‹HOL.Groups›, 🍋‹HOL.Rings›,
🍋‹HOL.Euclidean_Rings› and 🍋‹HOL.Fields›
define a large collection of classes describing common algebraic
structures from semigroups up to fields. Everything is done in terms of
overloaded operators:

\begin{tabular}{@ {} l @ {~::~} l l @ {}}
‹0› & 🪙‹zero›\java.lang.NullPointerException
‹1› & 🪙‹one›\java.lang.NullPointerException
🍋‹plus› & 🪙‹plus›\java.lang.NullPointerException
🍋‹minus› & 🪙‹minus›\java.lang.NullPointerException
🍋‹uminus› & 🪙‹uminus› & (🍋‹-›)java.lang.NullPointerException
🍋‹times› & 🪙‹times›\java.lang.NullPointerException
🍋‹inverse› & 🪙‹inverse›\java.lang.NullPointerException
🍋‹divide› & 🪙‹divide›\java.lang.NullPointerException
🍋‹abs› & 🪙‹abs›\java.lang.NullPointerException
🍋‹sgn› & 🪙‹sgn›\java.lang.NullPointerException
🍋‹Rings.dvd› & 🪙‹Rings.dvd›\java.lang.NullPointerException
🍋‹divide› & 🪙‹divide›\java.lang.NullPointerException
🍋‹modulo› & 🪙‹modulo›\java.lang.NullPointerException
\end{tabular}

\subsubsection*{Syntax}

\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}}
🍋‹∣x∣› & @{term[source] "abs x"}
\end{tabular}


\section*{Nat}

🪙‹nat›
🪙

\begin{tabular}{@ {} lllllll @ {}}
🍋‹(+) :: nat ==> nat ==> nat› &
🍋‹(-) :: nat ==> nat ==> nat› &
🍋‹(*) :: nat \ nat \ nat\ &
 🍋‹(^) :: nat ==> nat ==> nat› &
🍋‹(div) :: nat ==> nat ==> nat›&
🍋‹(mod) :: nat ==> nat ==> nat›&
🍋‹(dvd) :: nat ==> nat ==> bool›\java.lang.NullPointerException
🍋‹(≤) :: nat ==> nat ==> bool› &
🍋‹(🚫:: nat ==> nat ==> bool› &
🍋‹min :: nat ==> nat ==> nat› &
🍋‹max :: nat ==> nat ==> nat› &
🍋‹Min :: nat set ==> nat› &
🍋‹Max :: nat set ==> nat›\java.lang.NullPointerException
\end{tabular}

\begin{tabular}{@ {} l @ {~::~} l @ {}}
🍋‹Nat.of_nat› & 🪙‹Nat.of_nat›\java.lang.NullPointerException
🍋‹(^^) :: ('a ==> 'a) ==> nat ==> 'a ==> 'a› &
  @{term_type_only "(^^) :: ('a ==> 'a) ==> nat ==> 'a ==> 'a" "('a ==> 'a) ==> nat ==> 'a ==> 'a"}
\end{tabular}

\section*{Int}

Type 🍋‹int›
🪙

\begin{tabular}{@ {} llllllll @ {}}
🍋‹(+) :: int ==> int ==> int› &
🍋‹(-) :: int ==> int ==> int› &
🍋‹uminus :: int ==> int› &
🍋‹(*) :: int \ int \ int\ &
 🍋‹(^) :: int ==> nat ==> int› &
 🍋‹(div) :: int ==> int ==> int›&
 🍋‹(mod) :: int ==> int ==> int›&
java.lang.NullPointerException
 🍋‹(≤) :: int ==> int ==> bool› &
 🍋‹(🚫:: int ==> int ==> bool› &
 🍋‹min :: int ==> int ==> int› &
 🍋‹max :: int ==> int ==> int› &
 🍋‹Min :: int set ==> int› &
java.lang.NullPointerException
 🍋‹abs :: int ==> int› &
java.lang.NullPointerException
 \end{tabular}
 
 \begin{tabular}{@ {} l @ {~::~} l l @ {}}
java.lang.NullPointerException
java.lang.NullPointerException
 🍋‹Int.Ints› & @{term_type_only Int.Ints "'a::ring_1 set"} & (🍋‹Ints›)
 \end{tabular}
 
 \subsubsection*{Syntax}
 
 \begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}}
 🍋‹of_nat::nat==>int› & @{term[source]"of_nat"}\\
 \end{tabular}
 
 
 \section*{Finite\_Set}
 
 \begin{tabular}{@ {} l @ {~::~} l @ {}}
 🍋‹Finite_Set.finite› & @{term_type_only Finite_Set.finite "'a set==>bool"}\\
 🍋‹Finite_Set.card› & @{term_type_only Finite_Set.card "'a set ==> nat"}\\
 🍋‹Finite_Set.fold› & @{term_type_only Finite_Set.fold "('a ==> 'b ==> 'b) ==> 'b ==> 'a set ==> 'b"}\\
 \end{tabular}
 
 
 \section*{Lattices\_Big}
 
 \begin{tabular}{@ {} l @ {~::~} l l @ {}}
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
 \end{tabular}
 
 \subsubsection*{Syntax}
 
 \begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}}
 🍋‹ARG_MIN f x. P› & @{term[source]"arg_min f (λx. P)"}\\
 🍋‹ARG_MAX f x. P› & @{term[source]"arg_max f (λx. P)"}\\
 \end{tabular}
 
 
 \section*{Groups\_Big}
 
 \begin{tabular}{@ {} l @ {~::~} l @ {}}
 🍋‹Groups_Big.sum› & @{term_type_only Groups_Big.sum "('a ==> 'b) ==> 'a set ==> 'b::comm_monoid_add"}\\
 🍋‹Groups_Big.prod› & @{term_type_only Groups_Big.prod "('a ==> 'b) ==> 'a set ==> 'b::comm_monoid_mult"}\\
 \end{tabular}
 
 
 \subsubsection*{Syntax}
 
 \begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}}
 🍋‹sum (λx. x) A› & @{term[source]"sum (λx. x) A"} & (🍋‹SUM›)\\
 🍋‹sum (λx. t) A› & @{term[source]"sum (λx. t) A"}\\
java.lang.NullPointerException
 \multicolumn{2}{@ {}l@ {}}{Similarly for ‹∏› instead of ‹∑›} & (🍋‹PROD›)\\
 \end{tabular}
 
 
 \section*{Wellfounded}
 
 \begin{tabular}{@ {} l @ {~::~} l @ {}}
 🍋‹Wellfounded.wf› & @{term_type_only Wellfounded.wf "('a*'a)set==>bool"}\\
 🍋‹Wellfounded.acc› & @{term_type_only Wellfounded.acc "('a*'a)set==>'a set"}\\
 🍋‹Wellfounded.measure› & @{term_type_only Wellfounded.measure "('a==>nat)==>('a*'a)set"}\\
 🍋‹Wellfounded.lex_prod› & @{term_type_only Wellfounded.lex_prod "('a*'a)set==>('b*'b)set==>(('a*'b)*('a*'b))set"}\\
 🍋‹Wellfounded.mlex_prod› & @{term_type_only Wellfounded.mlex_prod "('a==>nat)==>('a*'a)set==>('a*'a)set"}\\
 🍋‹Wellfounded.less_than› & @{term_type_only Wellfounded.less_than "(nat*nat)set"}\\
 🍋‹Wellfounded.pred_nat› & @{term_type_only Wellfounded.pred_nat "(nat*nat)set"}\\
 \end{tabular}
 
 
 \section*{Set\_Interval} % 🍋‹HOL.Set_Interval›
 
 \begin{tabular}{@ {} l @ {~::~} l @ {}}
 🍋‹lessThan› & @{term_type_only lessThan "'a::ord ==> 'a set"}\\
 🍋‹atMost› & @{term_type_only atMost "'a::ord ==> 'a set"}\\
 🍋‹greaterThan› & @{term_type_only greaterThan "'a::ord ==> 'a set"}\\
 🍋‹atLeast› & @{term_type_only atLeast "'a::ord ==> 'a set"}\\
 🍋‹greaterThanLessThan› & @{term_type_only greaterThanLessThan "'a::ord ==> 'a ==> 'a set"}\\
 🍋‹atLeastLessThan› & @{term_type_only atLeastLessThan "'a::ord ==> 'a ==> 'a set"}\\
 🍋‹greaterThanAtMost› & @{term_type_only greaterThanAtMost "'a::ord ==> 'a ==> 'a set"}\\
 🍋‹atLeastAtMost› & @{term_type_only atLeastAtMost "'a::ord ==> 'a ==> 'a set"}\\
 \end{tabular}
 
 \subsubsection*{Syntax}
 
 \begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}}
 🍋‹lessThan y› & @{term[source] "lessThan y"}\\
 🍋‹atMost y› & @{term[source] "atMost y"}\\
 🍋‹greaterThan x› & @{term[source] "greaterThan x"}\\
 🍋‹atLeast x› & @{term[source] "atLeast x"}\\
 🍋‹greaterThanLessThan x y› & @{term[source] "greaterThanLessThan x y"}\\
 🍋‹atLeastLessThan x y› & @{term[source] "atLeastLessThan x y"}\\
 🍋‹greaterThanAtMost x y› & @{term[source] "greaterThanAtMost x y"}\\
 🍋‹atLeastAtMost x y› & @{term[source] "atLeastAtMost x y"}\\
 @{term[source] "∪i≤n. A"} & @{term[source] "∪i ∈ {..n}. A"}\\
 @{term[source] "∪i🚫 A"} & @{term[source] "∪i ∈ {..🚫. A"}\\
 \multicolumn{2}{@ {}l@ {}}{Similarly for ‹∩› instead of ‹∪›}\\
 🍋‹sum (λx. t) {a..b}› & @{term[source] "sum (λx. t) {a..b}"}\\
 🍋‹sum (λx. t) {a..🚫› & @{term[source] "sum (λx. t) {a..🚫"}\\
 🍋‹sum (λx. t) {..b}› & @{term[source] "sum (λx. t) {..b}"}\\
 🍋‹sum (λx. t) {..🚫› & @{term[source] "sum (λx. t) {..🚫"}\\
 \multicolumn{2}{@ {}l@ {}}{Similarly for ‹∏› instead of ‹∑›}\\
 \end{tabular}
 
 
 \section*{Power}
 
 \begin{tabular}{@ {} l @ {~::~} l @ {}}
 🍋‹Power.power› & 🪙‹Power.power›
 \end{tabular}
 
 
 \section*{Option}
 
 🪙‹option›
 🪙
 
 \begin{tabular}{@ {} l @ {~::~} l @ {}}
java.lang.NullPointerException
 🍋‹map_option› & @{typ[source]"('a ==> 'b) ==> 'a option ==> 'b option"}\\
 🍋‹set_option› & @{term_type_only set_option "'a option ==> 'a set"}\\
 🍋‹Option.bind› & @{term_type_only Option.bind "'a option ==> ('a ==> 'b option) ==> 'b option"}
 \end{tabular}
 
 \section*{List}
 
 🪙‹list›
 🪙
 
 \begin{supertabular}{@ {} l @ {~::~} l @ {}}
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
 🍋‹List.lenlex› & @{term_type_only List.lenlex "('a*'a)set==>('a list * 'a list)set"}\\
 🍋‹List.lex› & @{term_type_only List.lex "('a*'a)set==>('a list * 'a list)set"}\\
 🍋‹List.lexn› & @{term_type_only List.lexn "('a*'a)set==>nat==>('a list * 'a list)set"}\\
 🍋‹List.lexord› & @{term_type_only List.lexord "('a*'a)set==>('a list * 'a list)set"}\\
 🍋‹List.listrel› & @{term_type_only List.listrel "('a*'b)set==>('a list * 'b list)set"}\\
 🍋‹List.listrel1› & @{term_type_only List.listrel1 "('a*'a)set==>('a list * 'a list)set"}\\
 🍋‹List.lists› & @{term_type_only List.lists "'a set==>'a list set"}\\
 🍋‹List.listset› & @{term_type_only List.listset "'a set list ==> 'a list set"}\\
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
 🍋‹List.measures› & @{term_type_only List.measures "('a==>nat)list==>('a*'a)set"}\\
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
 🍋‹List.set› & @{term_type_only List.set "'a list ==> 'a set"}\\
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
 \end{supertabular}
 
 \subsubsection*{Syntax}
 
 \begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}}
java.lang.NullPointerException
 🍋‹[m..🚫› & @{term[source]"upt m n"}\\
 🍋‹[i..j]› & @{term[source]"upto i j"}\\
 🍋‹xs[n := x]› & @{term[source]"list_update xs n x"}\\
 🍋‹∑x←xs. e› & @{term[source]"listsum (map (λx. e) xs)"}\\
 \end{tabular}
 🪙
 
 Filter input syntax ‹[pat ← e. b]›, where
 ‹pat› is a tuple pattern, which stands for 🍋‹filter (λpat. b) e›.
 
 List comprehension input syntax: ‹[e. q🪙1, …, q🪙n]› where each
 qualifier ‹q🪙i› is either a generator \mbox{‹pat ← e›} or a
 guard, i.e.\ boolean expression.
 
 \section*{Map}
 
 Maps model partial functions and are often used as finite tables. However,
 the domain of a map may be infinite.
 
 \begin{tabular}{@ {} l @ {~::~} l @ {}}
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
 🍋‹Map.restrict_map› & @{term_type_only Map.restrict_map "('a==>'b option)==>'a set==>('a==>'b option)"}\\
 🍋‹Map.dom› & @{term_type_only Map.dom "('a==>'b option)==>'a set"}\\
 🍋‹Map.ran› & @{term_type_only Map.ran "('a==>'b option)==>'b set"}\\
java.lang.NullPointerException
java.lang.NullPointerException
java.lang.NullPointerException
 \end{tabular}
 
 \subsubsection*{Syntax}
 
 \begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}}
 🍋‹Map.empty› & @{term[source] "λ_. None"}\\
 🍋‹m(x:=Some y)› & @{term[source]"m(x:=Some y)"}\\
 ‹m(x🪙1↦y🪙1,…,x🪙n↦y🪙n)› & @{text[source]"m(xlick='alert("unbekannte/s Formatierung/Symbol sub");' ontouchend='alert("unbekannte/s Formatierung/Symbol sub");' >🪙1↦y🪙1)…(x🪙n↦y🪙n)"}\\
 ‹[x🪙1↦y🪙1,…,x🪙n↦y🪙n]› & @{text[source]"Map.empty(xick='alert("unbekannte/s Formatierung/Symbol sub");' ontouchend='alert("unbekannte/s Formatierung/Symbol sub");' >🪙1↦y🪙1,…,x🪙n↦y🪙n)"}\\
 🍋‹map_upds m xs ys› & @{term[source]"map_upds m xs ys"}\\
 \end{tabular}
 
 \section*{Infix operators in Main} % 🍋‹Main›
 
 \begin{center}
 \begin{tabular}{llll}
  & Operator & precedence & associativity \\
 \hline
 Meta-logic & ‹==>› & 1 & right \\
 & ‹≡› & 2 \\
 \hline
 Logic & ‹∧› & 35 & right \\
 &‹∨› & 30 & right \\
 &‹⟶›, ‹⟷› & 25 & right\\
 &‹=›, ‹≠› & 50 & left\\
 \hline
 Orderings & ‹≤›, ‹🚫close>, ‹≥›, ‹>› & 50 \\
 \hline
 Sets & ‹⊆›, ‹⊂›, ‹🪙›, ‹🪙› & 50 \\
 &‹∈›, ‹∉› & 50 \\
 &‹∩› & 70 & left \\
 &‹∪› & 65 & left \\
 \hline
 Functions and Relations & ‹∘› & 55 & left\\
 &‹`› & 90 & right\\
 &‹O› & 75 & right\\
 &‹``› & 90 & right\\
 &‹^^› & 80 & right\\
 \hline
 Numbers & ‹+›, ‹-› & 65 & left \\
 &‹*›, ‹/› & 70 & left \\
 &‹div›, ‹mod› & 70 & left\\
 &‹^› & 80 & right\\
 &‹dvd› & 50 \\
 \hline
 Lists & ‹#›, ‹@› & 65 & right\\
 &‹!› & 100 & left
 \end{tabular}
 \end{center}
 ›
(*<*)
end
(*>*)