author | haftmann |
Tue, 13 Oct 2015 09:21:15 +0200 | |
changeset 61424 | c3658c18b7bc |
parent 60352 | d46de31a50c4 |
child 61943 | 7fba644ed827 |
permissions | -rw-r--r-- |
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(*<*) |
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theory Main_Doc |
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imports Main |
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begin |
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setup {* |
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let |
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fun pretty_term_type_only ctxt (t, T) = |
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(if fastype_of t = Sign.certify_typ (Proof_Context.theory_of ctxt) T then () |
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else error "term_type_only: type mismatch"; |
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Syntax.pretty_typ ctxt T) |
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in |
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Thy_Output.antiquotation @{binding term_type_only} |
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(Args.term -- Args.typ_abbrev) |
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(fn {source, context = ctxt, ...} => fn arg => |
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Thy_Output.output ctxt |
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(Thy_Output.maybe_pretty_source pretty_term_type_only ctxt source [arg])) |
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end |
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*} |
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setup {* |
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Thy_Output.antiquotation @{binding expanded_typ} (Args.typ >> single) |
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(fn {source, context, ...} => Thy_Output.output context o |
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Thy_Output.maybe_pretty_source Syntax.pretty_typ context source) |
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*} |
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(*>*) |
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text{* |
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\begin{abstract} |
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This document lists the main types, functions and syntax provided by theory @{theory Main}. It is meant as a quick overview of what is available. For infix operators and their precedences see the final section. The sophisticated class structure is only hinted at. For details see @{url "http://isabelle.in.tum.de/library/HOL/"}. |
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\end{abstract} |
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\section*{HOL} |
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The basic logic: @{prop "x = y"}, @{const True}, @{const False}, @{prop"Not P"}, @{prop"P & Q"}, @{prop "P | Q"}, @{prop "P --> Q"}, @{prop"ALL x. P"}, @{prop"EX x. P"}, @{prop"EX! x. P"}, @{term"THE x. P"}. |
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\smallskip |
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\begin{tabular}{@ {} l @ {~::~} l @ {}} |
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@{const HOL.undefined} & @{typeof HOL.undefined}\\ |
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@{const HOL.default} & @{typeof HOL.default}\\ |
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\end{tabular} |
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\subsubsection*{Syntax} |
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\begin{supertabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}} |
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@{term"~(x = y)"} & @{term[source]"\<not> (x = y)"} & (\verb$~=$)\\ |
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@{term[source]"P \<longleftrightarrow> Q"} & @{term"P \<longleftrightarrow> Q"} \\ |
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@{term"If x y z"} & @{term[source]"If x y z"}\\ |
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@{term"Let e\<^sub>1 (%x. e\<^sub>2)"} & @{term[source]"Let e\<^sub>1 (\<lambda>x. e\<^sub>2)"}\\ |
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\end{supertabular} |
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\section*{Orderings} |
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A collection of classes defining basic orderings: |
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preorder, partial order, linear order, dense linear order and wellorder. |
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\smallskip |
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\begin{supertabular}{@ {} l @ {~::~} l l @ {}} |
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@{const Orderings.less_eq} & @{typeof Orderings.less_eq} & (\verb$<=$)\\ |
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@{const Orderings.less} & @{typeof Orderings.less}\\ |
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@{const Orderings.Least} & @{typeof Orderings.Least}\\ |
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@{const Orderings.min} & @{typeof Orderings.min}\\ |
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@{const Orderings.max} & @{typeof Orderings.max}\\ |
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@{const[source] top} & @{typeof Orderings.top}\\ |
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@{const[source] bot} & @{typeof Orderings.bot}\\ |
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@{const Orderings.mono} & @{typeof Orderings.mono}\\ |
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@{const Orderings.strict_mono} & @{typeof Orderings.strict_mono}\\ |
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\end{supertabular} |
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\subsubsection*{Syntax} |
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\begin{supertabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}} |
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@{term[source]"x \<ge> y"} & @{term"x \<ge> y"} & (\verb$>=$)\\ |
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@{term[source]"x > y"} & @{term"x > y"}\\ |
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@{term"ALL x<=y. P"} & @{term[source]"\<forall>x. x \<le> y \<longrightarrow> P"}\\ |
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@{term"EX x<=y. P"} & @{term[source]"\<exists>x. x \<le> y \<and> P"}\\ |
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\multicolumn{2}{@ {}l@ {}}{Similarly for $<$, $\ge$ and $>$}\\ |
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@{term"LEAST x. P"} & @{term[source]"Least (\<lambda>x. P)"}\\ |
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\end{supertabular} |
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\section*{Lattices} |
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Classes semilattice, lattice, distributive lattice and complete lattice (the |
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latter in theory @{theory Set}). |
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\begin{tabular}{@ {} l @ {~::~} l @ {}} |
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@{const Lattices.inf} & @{typeof Lattices.inf}\\ |
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@{const Lattices.sup} & @{typeof Lattices.sup}\\ |
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@{const Complete_Lattices.Inf} & @{term_type_only Complete_Lattices.Inf "'a set \<Rightarrow> 'a::Inf"}\\ |
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@{const Complete_Lattices.Sup} & @{term_type_only Complete_Lattices.Sup "'a set \<Rightarrow> 'a::Sup"}\\ |
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\end{tabular} |
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\subsubsection*{Syntax} |
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Available by loading theory @{text Lattice_Syntax} in directory @{text |
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Library}. |
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\begin{supertabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}} |
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@{text[source]"x \<sqsubseteq> y"} & @{term"x \<le> y"}\\ |
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@{text[source]"x \<sqsubset> y"} & @{term"x < y"}\\ |
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@{text[source]"x \<sqinter> y"} & @{term"inf x y"}\\ |
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@{text[source]"x \<squnion> y"} & @{term"sup x y"}\\ |
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@{text[source]"\<Sqinter> A"} & @{term"Sup A"}\\ |
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@{text[source]"\<Squnion> A"} & @{term"Inf A"}\\ |
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@{text[source]"\<top>"} & @{term[source] top}\\ |
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@{text[source]"\<bottom>"} & @{term[source] bot}\\ |
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\end{supertabular} |
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\section*{Set} |
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\begin{supertabular}{@ {} l @ {~::~} l l @ {}} |
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@{const Set.empty} & @{term_type_only "Set.empty" "'a set"}\\ |
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@{const Set.insert} & @{term_type_only insert "'a\<Rightarrow>'a set\<Rightarrow>'a set"}\\ |
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@{const Collect} & @{term_type_only Collect "('a\<Rightarrow>bool)\<Rightarrow>'a set"}\\ |
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@{const Set.member} & @{term_type_only Set.member "'a\<Rightarrow>'a set\<Rightarrow>bool"} & (\texttt{:})\\ |
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@{const Set.union} & @{term_type_only Set.union "'a set\<Rightarrow>'a set \<Rightarrow> 'a set"} & (\texttt{Un})\\ |
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@{const Set.inter} & @{term_type_only Set.inter "'a set\<Rightarrow>'a set \<Rightarrow> 'a set"} & (\texttt{Int})\\ |
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@{const UNION} & @{term_type_only UNION "'a set\<Rightarrow>('a \<Rightarrow> 'b set) \<Rightarrow> 'b set"}\\ |
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@{const INTER} & @{term_type_only INTER "'a set\<Rightarrow>('a \<Rightarrow> 'b set) \<Rightarrow> 'b set"}\\ |
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@{const Union} & @{term_type_only Union "'a set set\<Rightarrow>'a set"}\\ |
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@{const Inter} & @{term_type_only Inter "'a set set\<Rightarrow>'a set"}\\ |
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@{const Pow} & @{term_type_only Pow "'a set \<Rightarrow>'a set set"}\\ |
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@{const UNIV} & @{term_type_only UNIV "'a set"}\\ |
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@{const image} & @{term_type_only image "('a\<Rightarrow>'b)\<Rightarrow>'a set\<Rightarrow>'b set"}\\ |
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@{const Ball} & @{term_type_only Ball "'a set\<Rightarrow>('a\<Rightarrow>bool)\<Rightarrow>bool"}\\ |
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@{const Bex} & @{term_type_only Bex "'a set\<Rightarrow>('a\<Rightarrow>bool)\<Rightarrow>bool"}\\ |
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\end{supertabular} |
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\subsubsection*{Syntax} |
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\begin{supertabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}} |
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@{text"{a\<^sub>1,\<dots>,a\<^sub>n}"} & @{text"insert a\<^sub>1 (\<dots> (insert a\<^sub>n {})\<dots>)"}\\ |
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@{term"a ~: A"} & @{term[source]"\<not>(x \<in> A)"}\\ |
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@{term"A \<subseteq> B"} & @{term[source]"A \<le> B"}\\ |
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@{term"A \<subset> B"} & @{term[source]"A < B"}\\ |
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@{term[source]"A \<supseteq> B"} & @{term[source]"B \<le> A"}\\ |
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@{term[source]"A \<supset> B"} & @{term[source]"B < A"}\\ |
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@{term"{x. P}"} & @{term[source]"Collect (\<lambda>x. P)"}\\ |
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@{text"{t | x\<^sub>1 \<dots> x\<^sub>n. P}"} & @{text"{v. \<exists>x\<^sub>1 \<dots> x\<^sub>n. v = t \<and> P}"}\\ |
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@{term[mode=xsymbols]"UN x:I. A"} & @{term[source]"UNION I (\<lambda>x. A)"} & (\texttt{UN})\\ |
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@{term[mode=xsymbols]"UN x. A"} & @{term[source]"UNION UNIV (\<lambda>x. A)"}\\ |
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@{term[mode=xsymbols]"INT x:I. A"} & @{term[source]"INTER I (\<lambda>x. A)"} & (\texttt{INT})\\ |
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@{term[mode=xsymbols]"INT x. A"} & @{term[source]"INTER UNIV (\<lambda>x. A)"}\\ |
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@{term"ALL x:A. P"} & @{term[source]"Ball A (\<lambda>x. P)"}\\ |
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@{term"EX x:A. P"} & @{term[source]"Bex A (\<lambda>x. P)"}\\ |
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@{term"range f"} & @{term[source]"f ` UNIV"}\\ |
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\end{supertabular} |
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\section*{Fun} |
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\begin{supertabular}{@ {} l @ {~::~} l l @ {}} |
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@{const "Fun.id"} & @{typeof Fun.id}\\ |
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@{const "Fun.comp"} & @{typeof Fun.comp} & (\texttt{o})\\ |
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@{const "Fun.inj_on"} & @{term_type_only Fun.inj_on "('a\<Rightarrow>'b)\<Rightarrow>'a set\<Rightarrow>bool"}\\ |
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@{const "Fun.inj"} & @{typeof Fun.inj}\\ |
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@{const "Fun.surj"} & @{typeof Fun.surj}\\ |
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@{const "Fun.bij"} & @{typeof Fun.bij}\\ |
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@{const "Fun.bij_betw"} & @{term_type_only Fun.bij_betw "('a\<Rightarrow>'b)\<Rightarrow>'a set\<Rightarrow>'b set\<Rightarrow>bool"}\\ |
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@{const "Fun.fun_upd"} & @{typeof Fun.fun_upd}\\ |
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\end{supertabular} |
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\subsubsection*{Syntax} |
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\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}} |
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@{term"fun_upd f x y"} & @{term[source]"fun_upd f x y"}\\ |
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@{text"f(x\<^sub>1:=y\<^sub>1,\<dots>,x\<^sub>n:=y\<^sub>n)"} & @{text"f(x\<^sub>1:=y\<^sub>1)\<dots>(x\<^sub>n:=y\<^sub>n)"}\\ |
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\end{tabular} |
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\section*{Hilbert\_Choice} |
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Hilbert's selection ($\varepsilon$) operator: @{term"SOME x. P"}. |
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\smallskip |
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\begin{tabular}{@ {} l @ {~::~} l @ {}} |
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@{const Hilbert_Choice.inv_into} & @{term_type_only Hilbert_Choice.inv_into "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)"} |
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\end{tabular} |
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\subsubsection*{Syntax} |
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\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}} |
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@{term inv} & @{term[source]"inv_into UNIV"} |
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\end{tabular} |
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\section*{Fixed Points} |
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Theory: @{theory Inductive}. |
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Least and greatest fixed points in a complete lattice @{typ 'a}: |
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\begin{tabular}{@ {} l @ {~::~} l @ {}} |
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@{const Inductive.lfp} & @{typeof Inductive.lfp}\\ |
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@{const Inductive.gfp} & @{typeof Inductive.gfp}\\ |
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\end{tabular} |
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Note that in particular sets (@{typ"'a \<Rightarrow> bool"}) are complete lattices. |
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\section*{Sum\_Type} |
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Type constructor @{text"+"}. |
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\begin{tabular}{@ {} l @ {~::~} l @ {}} |
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@{const Sum_Type.Inl} & @{typeof Sum_Type.Inl}\\ |
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@{const Sum_Type.Inr} & @{typeof Sum_Type.Inr}\\ |
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@{const Sum_Type.Plus} & @{term_type_only Sum_Type.Plus "'a set\<Rightarrow>'b set\<Rightarrow>('a+'b)set"} |
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\end{tabular} |
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\section*{Product\_Type} |
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Types @{typ unit} and @{text"\<times>"}. |
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\begin{supertabular}{@ {} l @ {~::~} l @ {}} |
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@{const Product_Type.Unity} & @{typeof Product_Type.Unity}\\ |
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@{const Pair} & @{typeof Pair}\\ |
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@{const fst} & @{typeof fst}\\ |
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@{const snd} & @{typeof snd}\\ |
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@{const case_prod} & @{typeof case_prod}\\ |
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@{const curry} & @{typeof curry}\\ |
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@{const Product_Type.Sigma} & @{term_type_only Product_Type.Sigma "'a set\<Rightarrow>('a\<Rightarrow>'b set)\<Rightarrow>('a*'b)set"}\\ |
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\end{supertabular} |
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\subsubsection*{Syntax} |
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\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} ll @ {}} |
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@{term"Pair a b"} & @{term[source]"Pair a b"}\\ |
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@{term"case_prod (\<lambda>x y. t)"} & @{term[source]"case_prod (\<lambda>x y. t)"}\\ |
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@{term"A <*> B"} & @{text"Sigma A (\<lambda>\<^raw:\_>. B)"} & (\verb$<*>$) |
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\end{tabular} |
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Pairs may be nested. Nesting to the right is printed as a tuple, |
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e.g.\ \mbox{@{term"(a,b,c)"}} is really \mbox{@{text"(a, (b, c))"}.} |
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Pattern matching with pairs and tuples extends to all binders, |
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e.g.\ \mbox{@{prop"ALL (x,y):A. P"},} @{term"{(x,y). P}"}, etc. |
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\section*{Relation} |
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\begin{tabular}{@ {} l @ {~::~} l @ {}} |
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@{const Relation.converse} & @{term_type_only Relation.converse "('a * 'b)set \<Rightarrow> ('b*'a)set"}\\ |
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@{const Relation.relcomp} & @{term_type_only Relation.relcomp "('a*'b)set\<Rightarrow>('b*'c)set\<Rightarrow>('a*'c)set"}\\ |
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@{const Relation.Image} & @{term_type_only Relation.Image "('a*'b)set\<Rightarrow>'a set\<Rightarrow>'b set"}\\ |
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@{const Relation.inv_image} & @{term_type_only Relation.inv_image "('a*'a)set\<Rightarrow>('b\<Rightarrow>'a)\<Rightarrow>('b*'b)set"}\\ |
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@{const Relation.Id_on} & @{term_type_only Relation.Id_on "'a set\<Rightarrow>('a*'a)set"}\\ |
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@{const Relation.Id} & @{term_type_only Relation.Id "('a*'a)set"}\\ |
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@{const Relation.Domain} & @{term_type_only Relation.Domain "('a*'b)set\<Rightarrow>'a set"}\\ |
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@{const Relation.Range} & @{term_type_only Relation.Range "('a*'b)set\<Rightarrow>'b set"}\\ |
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@{const Relation.Field} & @{term_type_only Relation.Field "('a*'a)set\<Rightarrow>'a set"}\\ |
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@{const Relation.refl_on} & @{term_type_only Relation.refl_on "'a set\<Rightarrow>('a*'a)set\<Rightarrow>bool"}\\ |
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@{const Relation.refl} & @{term_type_only Relation.refl "('a*'a)set\<Rightarrow>bool"}\\ |
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@{const Relation.sym} & @{term_type_only Relation.sym "('a*'a)set\<Rightarrow>bool"}\\ |
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@{const Relation.antisym} & @{term_type_only Relation.antisym "('a*'a)set\<Rightarrow>bool"}\\ |
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@{const Relation.trans} & @{term_type_only Relation.trans "('a*'a)set\<Rightarrow>bool"}\\ |
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@{const Relation.irrefl} & @{term_type_only Relation.irrefl "('a*'a)set\<Rightarrow>bool"}\\ |
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@{const Relation.total_on} & @{term_type_only Relation.total_on "'a set\<Rightarrow>('a*'a)set\<Rightarrow>bool"}\\ |
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@{const Relation.total} & @{term_type_only Relation.total "('a*'a)set\<Rightarrow>bool"}\\ |
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\end{tabular} |
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\subsubsection*{Syntax} |
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\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}} |
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@{term"converse r"} & @{term[source]"converse r"} & (\verb$^-1$) |
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\end{tabular} |
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\medskip |
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\noindent |
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Type synonym \ @{typ"'a rel"} @{text"="} @{expanded_typ "'a rel"} |
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\section*{Equiv\_Relations} |
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|
274 |
\begin{supertabular}{@ {} l @ {~::~} l @ {}} |
|
275 |
@{const Equiv_Relations.equiv} & @{term_type_only Equiv_Relations.equiv "'a set \<Rightarrow> ('a*'a)set\<Rightarrow>bool"}\\ |
|
276 |
@{const Equiv_Relations.quotient} & @{term_type_only Equiv_Relations.quotient "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> 'a set set"}\\ |
|
277 |
@{const Equiv_Relations.congruent} & @{term_type_only Equiv_Relations.congruent "('a*'a)set\<Rightarrow>('a\<Rightarrow>'b)\<Rightarrow>bool"}\\ |
|
278 |
@{const Equiv_Relations.congruent2} & @{term_type_only Equiv_Relations.congruent2 "('a*'a)set\<Rightarrow>('b*'b)set\<Rightarrow>('a\<Rightarrow>'b\<Rightarrow>'c)\<Rightarrow>bool"}\\ |
|
279 |
%@ {const Equiv_Relations.} & @ {term_type_only Equiv_Relations. ""}\\ |
|
280 |
\end{supertabular} |
|
281 |
||
282 |
\subsubsection*{Syntax} |
|
283 |
||
284 |
\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}} |
|
285 |
@{term"congruent r f"} & @{term[source]"congruent r f"}\\ |
|
286 |
@{term"congruent2 r r f"} & @{term[source]"congruent2 r r f"}\\ |
|
287 |
\end{tabular} |
|
288 |
||
289 |
||
50581 | 290 |
\section*{Transitive\_Closure} |
30293 | 291 |
|
292 |
\begin{tabular}{@ {} l @ {~::~} l @ {}} |
|
293 |
@{const Transitive_Closure.rtrancl} & @{term_type_only Transitive_Closure.rtrancl "('a*'a)set\<Rightarrow>('a*'a)set"}\\ |
|
294 |
@{const Transitive_Closure.trancl} & @{term_type_only Transitive_Closure.trancl "('a*'a)set\<Rightarrow>('a*'a)set"}\\ |
|
295 |
@{const Transitive_Closure.reflcl} & @{term_type_only Transitive_Closure.reflcl "('a*'a)set\<Rightarrow>('a*'a)set"}\\ |
|
45618 | 296 |
@{const Transitive_Closure.acyclic} & @{term_type_only Transitive_Closure.acyclic "('a*'a)set\<Rightarrow>bool"}\\ |
30988 | 297 |
@{const compower} & @{term_type_only "op ^^ :: ('a*'a)set\<Rightarrow>nat\<Rightarrow>('a*'a)set" "('a*'a)set\<Rightarrow>nat\<Rightarrow>('a*'a)set"}\\ |
30293 | 298 |
\end{tabular} |
299 |
||
300 |
\subsubsection*{Syntax} |
|
301 |
||
30440 | 302 |
\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}} |
303 |
@{term"rtrancl r"} & @{term[source]"rtrancl r"} & (\verb$^*$)\\ |
|
304 |
@{term"trancl r"} & @{term[source]"trancl r"} & (\verb$^+$)\\ |
|
305 |
@{term"reflcl r"} & @{term[source]"reflcl r"} & (\verb$^=$) |
|
30293 | 306 |
\end{tabular} |
307 |
||
308 |
||
50581 | 309 |
\section*{Algebra} |
30293 | 310 |
|
35061 | 311 |
Theories @{theory Groups}, @{theory Rings}, @{theory Fields} and @{theory |
30440 | 312 |
Divides} define a large collection of classes describing common algebraic |
313 |
structures from semigroups up to fields. Everything is done in terms of |
|
314 |
overloaded operators: |
|
315 |
||
316 |
\begin{supertabular}{@ {} l @ {~::~} l l @ {}} |
|
317 |
@{text "0"} & @{typeof zero}\\ |
|
318 |
@{text "1"} & @{typeof one}\\ |
|
319 |
@{const plus} & @{typeof plus}\\ |
|
320 |
@{const minus} & @{typeof minus}\\ |
|
321 |
@{const uminus} & @{typeof uminus} & (\verb$-$)\\ |
|
322 |
@{const times} & @{typeof times}\\ |
|
323 |
@{const inverse} & @{typeof inverse}\\ |
|
324 |
@{const divide} & @{typeof divide}\\ |
|
325 |
@{const abs} & @{typeof abs}\\ |
|
326 |
@{const sgn} & @{typeof sgn}\\ |
|
327 |
@{const dvd_class.dvd} & @{typeof "dvd_class.dvd"}\\ |
|
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328 |
@{const Rings.divide} & @{typeof Rings.divide}\\ |
30440 | 329 |
@{const div_class.mod} & @{typeof "div_class.mod"}\\ |
330 |
\end{supertabular} |
|
331 |
||
332 |
\subsubsection*{Syntax} |
|
333 |
||
334 |
\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}} |
|
335 |
@{term"abs x"} & @{term[source]"abs x"} |
|
336 |
\end{tabular} |
|
30293 | 337 |
|
338 |
||
50581 | 339 |
\section*{Nat} |
30293 | 340 |
|
341 |
@{datatype nat} |
|
342 |
\bigskip |
|
343 |
||
344 |
\begin{tabular}{@ {} lllllll @ {}} |
|
345 |
@{term "op + :: nat \<Rightarrow> nat \<Rightarrow> nat"} & |
|
346 |
@{term "op - :: nat \<Rightarrow> nat \<Rightarrow> nat"} & |
|
347 |
@{term "op * :: nat \<Rightarrow> nat \<Rightarrow> nat"} & |
|
47187 | 348 |
@{term "op ^ :: nat \<Rightarrow> nat \<Rightarrow> nat"} & |
30293 | 349 |
@{term "op div :: nat \<Rightarrow> nat \<Rightarrow> nat"}& |
350 |
@{term "op mod :: nat \<Rightarrow> nat \<Rightarrow> nat"}& |
|
351 |
@{term "op dvd :: nat \<Rightarrow> nat \<Rightarrow> bool"}\\ |
|
352 |
@{term "op \<le> :: nat \<Rightarrow> nat \<Rightarrow> bool"} & |
|
353 |
@{term "op < :: nat \<Rightarrow> nat \<Rightarrow> bool"} & |
|
354 |
@{term "min :: nat \<Rightarrow> nat \<Rightarrow> nat"} & |
|
355 |
@{term "max :: nat \<Rightarrow> nat \<Rightarrow> nat"} & |
|
356 |
@{term "Min :: nat set \<Rightarrow> nat"} & |
|
357 |
@{term "Max :: nat set \<Rightarrow> nat"}\\ |
|
358 |
\end{tabular} |
|
359 |
||
360 |
\begin{tabular}{@ {} l @ {~::~} l @ {}} |
|
30988 | 361 |
@{const Nat.of_nat} & @{typeof Nat.of_nat}\\ |
362 |
@{term "op ^^ :: ('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"} & |
|
363 |
@{term_type_only "op ^^ :: ('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"} |
|
30293 | 364 |
\end{tabular} |
365 |
||
50581 | 366 |
\section*{Int} |
30293 | 367 |
|
368 |
Type @{typ int} |
|
369 |
\bigskip |
|
370 |
||
371 |
\begin{tabular}{@ {} llllllll @ {}} |
|
372 |
@{term "op + :: int \<Rightarrow> int \<Rightarrow> int"} & |
|
373 |
@{term "op - :: int \<Rightarrow> int \<Rightarrow> int"} & |
|
374 |
@{term "uminus :: int \<Rightarrow> int"} & |
|
375 |
@{term "op * :: int \<Rightarrow> int \<Rightarrow> int"} & |
|
376 |
@{term "op ^ :: int \<Rightarrow> nat \<Rightarrow> int"} & |
|
377 |
@{term "op div :: int \<Rightarrow> int \<Rightarrow> int"}& |
|
378 |
@{term "op mod :: int \<Rightarrow> int \<Rightarrow> int"}& |
|
379 |
@{term "op dvd :: int \<Rightarrow> int \<Rightarrow> bool"}\\ |
|
380 |
@{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"} & |
|
381 |
@{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"} & |
|
382 |
@{term "min :: int \<Rightarrow> int \<Rightarrow> int"} & |
|
383 |
@{term "max :: int \<Rightarrow> int \<Rightarrow> int"} & |
|
384 |
@{term "Min :: int set \<Rightarrow> int"} & |
|
385 |
@{term "Max :: int set \<Rightarrow> int"}\\ |
|
386 |
@{term "abs :: int \<Rightarrow> int"} & |
|
387 |
@{term "sgn :: int \<Rightarrow> int"}\\ |
|
388 |
\end{tabular} |
|
389 |
||
30440 | 390 |
\begin{tabular}{@ {} l @ {~::~} l l @ {}} |
30293 | 391 |
@{const Int.nat} & @{typeof Int.nat}\\ |
392 |
@{const Int.of_int} & @{typeof Int.of_int}\\ |
|
30440 | 393 |
@{const Int.Ints} & @{term_type_only Int.Ints "'a::ring_1 set"} & (\verb$Ints$) |
30293 | 394 |
\end{tabular} |
395 |
||
396 |
\subsubsection*{Syntax} |
|
397 |
||
398 |
\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}} |
|
399 |
@{term"of_nat::nat\<Rightarrow>int"} & @{term[source]"of_nat"}\\ |
|
400 |
\end{tabular} |
|
401 |
||
402 |
||
50581 | 403 |
\section*{Finite\_Set} |
30401 | 404 |
|
405 |
||
406 |
\begin{supertabular}{@ {} l @ {~::~} l @ {}} |
|
407 |
@{const Finite_Set.finite} & @{term_type_only Finite_Set.finite "'a set\<Rightarrow>bool"}\\ |
|
408 |
@{const Finite_Set.card} & @{term_type_only Finite_Set.card "'a set => nat"}\\ |
|
409 |
@{const Finite_Set.fold} & @{term_type_only Finite_Set.fold "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"}\\ |
|
54744
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
54703
diff
changeset
|
410 |
@{const Groups_Big.setsum} & @{term_type_only Groups_Big.setsum "('a => 'b) => 'a set => 'b::comm_monoid_add"}\\ |
1e7f2d296e19
more algebraic terminology for theories about big operators
haftmann
parents:
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diff
changeset
|
411 |
@{const Groups_Big.setprod} & @{term_type_only Groups_Big.setprod "('a => 'b) => 'a set => 'b::comm_monoid_mult"}\\ |
30401 | 412 |
\end{supertabular} |
413 |
||
414 |
||
415 |
\subsubsection*{Syntax} |
|
416 |
||
30440 | 417 |
\begin{supertabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}} |
418 |
@{term"setsum (%x. x) A"} & @{term[source]"setsum (\<lambda>x. x) A"} & (\verb$SUM$)\\ |
|
30401 | 419 |
@{term"setsum (%x. t) A"} & @{term[source]"setsum (\<lambda>x. t) A"}\\ |
420 |
@{term[source]"\<Sum>x|P. t"} & @{term"\<Sum>x|P. t"}\\ |
|
30440 | 421 |
\multicolumn{2}{@ {}l@ {}}{Similarly for @{text"\<Prod>"} instead of @{text"\<Sum>"}} & (\verb$PROD$)\\ |
30401 | 422 |
\end{supertabular} |
423 |
||
424 |
||
50581 | 425 |
\section*{Wellfounded} |
30293 | 426 |
|
427 |
\begin{supertabular}{@ {} l @ {~::~} l @ {}} |
|
428 |
@{const Wellfounded.wf} & @{term_type_only Wellfounded.wf "('a*'a)set\<Rightarrow>bool"}\\ |
|
429 |
@{const Wellfounded.acc} & @{term_type_only Wellfounded.acc "('a*'a)set\<Rightarrow>'a set"}\\ |
|
430 |
@{const Wellfounded.measure} & @{term_type_only Wellfounded.measure "('a\<Rightarrow>nat)\<Rightarrow>('a*'a)set"}\\ |
|
431 |
@{const Wellfounded.lex_prod} & @{term_type_only Wellfounded.lex_prod "('a*'a)set\<Rightarrow>('b*'b)set\<Rightarrow>(('a*'b)*('a*'b))set"}\\ |
|
432 |
@{const Wellfounded.mlex_prod} & @{term_type_only Wellfounded.mlex_prod "('a\<Rightarrow>nat)\<Rightarrow>('a*'a)set\<Rightarrow>('a*'a)set"}\\ |
|
433 |
@{const Wellfounded.less_than} & @{term_type_only Wellfounded.less_than "(nat*nat)set"}\\ |
|
434 |
@{const Wellfounded.pred_nat} & @{term_type_only Wellfounded.pred_nat "(nat*nat)set"}\\ |
|
435 |
\end{supertabular} |
|
436 |
||
437 |
||
50581 | 438 |
\section*{Set\_Interval} % @{theory Set_Interval} |
30321 | 439 |
|
440 |
\begin{supertabular}{@ {} l @ {~::~} l @ {}} |
|
30370 | 441 |
@{const lessThan} & @{term_type_only lessThan "'a::ord \<Rightarrow> 'a set"}\\ |
442 |
@{const atMost} & @{term_type_only atMost "'a::ord \<Rightarrow> 'a set"}\\ |
|
443 |
@{const greaterThan} & @{term_type_only greaterThan "'a::ord \<Rightarrow> 'a set"}\\ |
|
444 |
@{const atLeast} & @{term_type_only atLeast "'a::ord \<Rightarrow> 'a set"}\\ |
|
445 |
@{const greaterThanLessThan} & @{term_type_only greaterThanLessThan "'a::ord \<Rightarrow> 'a \<Rightarrow> 'a set"}\\ |
|
446 |
@{const atLeastLessThan} & @{term_type_only atLeastLessThan "'a::ord \<Rightarrow> 'a \<Rightarrow> 'a set"}\\ |
|
447 |
@{const greaterThanAtMost} & @{term_type_only greaterThanAtMost "'a::ord \<Rightarrow> 'a \<Rightarrow> 'a set"}\\ |
|
448 |
@{const atLeastAtMost} & @{term_type_only atLeastAtMost "'a::ord \<Rightarrow> 'a \<Rightarrow> 'a set"}\\ |
|
30321 | 449 |
\end{supertabular} |
450 |
||
451 |
\subsubsection*{Syntax} |
|
452 |
||
453 |
\begin{supertabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}} |
|
454 |
@{term "lessThan y"} & @{term[source] "lessThan y"}\\ |
|
455 |
@{term "atMost y"} & @{term[source] "atMost y"}\\ |
|
456 |
@{term "greaterThan x"} & @{term[source] "greaterThan x"}\\ |
|
457 |
@{term "atLeast x"} & @{term[source] "atLeast x"}\\ |
|
458 |
@{term "greaterThanLessThan x y"} & @{term[source] "greaterThanLessThan x y"}\\ |
|
459 |
@{term "atLeastLessThan x y"} & @{term[source] "atLeastLessThan x y"}\\ |
|
460 |
@{term "greaterThanAtMost x y"} & @{term[source] "greaterThanAtMost x y"}\\ |
|
461 |
@{term "atLeastAtMost x y"} & @{term[source] "atLeastAtMost x y"}\\ |
|
30370 | 462 |
@{term[mode=xsymbols] "UN i:{..n}. A"} & @{term[source] "\<Union> i \<in> {..n}. A"}\\ |
463 |
@{term[mode=xsymbols] "UN i:{..<n}. A"} & @{term[source] "\<Union> i \<in> {..<n}. A"}\\ |
|
464 |
\multicolumn{2}{@ {}l@ {}}{Similarly for @{text"\<Inter>"} instead of @{text"\<Union>"}}\\ |
|
30321 | 465 |
@{term "setsum (%x. t) {a..b}"} & @{term[source] "setsum (\<lambda>x. t) {a..b}"}\\ |
30370 | 466 |
@{term "setsum (%x. t) {a..<b}"} & @{term[source] "setsum (\<lambda>x. t) {a..<b}"}\\ |
30386 | 467 |
@{term "setsum (%x. t) {..b}"} & @{term[source] "setsum (\<lambda>x. t) {..b}"}\\ |
468 |
@{term "setsum (%x. t) {..<b}"} & @{term[source] "setsum (\<lambda>x. t) {..<b}"}\\ |
|
30372 | 469 |
\multicolumn{2}{@ {}l@ {}}{Similarly for @{text"\<Prod>"} instead of @{text"\<Sum>"}}\\ |
30321 | 470 |
\end{supertabular} |
471 |
||
472 |
||
50581 | 473 |
\section*{Power} |
30293 | 474 |
|
475 |
\begin{tabular}{@ {} l @ {~::~} l @ {}} |
|
476 |
@{const Power.power} & @{typeof Power.power} |
|
477 |
\end{tabular} |
|
478 |
||
479 |
||
50581 | 480 |
\section*{Option} |
30293 | 481 |
|
482 |
@{datatype option} |
|
483 |
\bigskip |
|
484 |
||
485 |
\begin{tabular}{@ {} l @ {~::~} l @ {}} |
|
486 |
@{const Option.the} & @{typeof Option.the}\\ |
|
55466 | 487 |
@{const map_option} & @{typ[source]"('a \<Rightarrow> 'b) \<Rightarrow> 'a option \<Rightarrow> 'b option"}\\ |
55518
1ddb2edf5ceb
folded 'Option.set' into BNF-generated 'set_option'
blanchet
parents:
55466
diff
changeset
|
488 |
@{const set_option} & @{term_type_only set_option "'a option \<Rightarrow> 'a set"}\\ |
41532 | 489 |
@{const Option.bind} & @{term_type_only Option.bind "'a option \<Rightarrow> ('a \<Rightarrow> 'b option) \<Rightarrow> 'b option"} |
30293 | 490 |
\end{tabular} |
491 |
||
50581 | 492 |
\section*{List} |
30293 | 493 |
|
494 |
@{datatype list} |
|
495 |
\bigskip |
|
496 |
||
497 |
\begin{supertabular}{@ {} l @ {~::~} l @ {}} |
|
498 |
@{const List.append} & @{typeof List.append}\\ |
|
499 |
@{const List.butlast} & @{typeof List.butlast}\\ |
|
500 |
@{const List.concat} & @{typeof List.concat}\\ |
|
501 |
@{const List.distinct} & @{typeof List.distinct}\\ |
|
502 |
@{const List.drop} & @{typeof List.drop}\\ |
|
503 |
@{const List.dropWhile} & @{typeof List.dropWhile}\\ |
|
504 |
@{const List.filter} & @{typeof List.filter}\\ |
|
47187 | 505 |
@{const List.find} & @{typeof List.find}\\ |
46133
d9fe85d3d2cd
incorporated canonical fold combinator on lists into body of List theory; refactored passages on List.fold(l/r)
haftmann
parents:
45618
diff
changeset
|
506 |
@{const List.fold} & @{typeof List.fold}\\ |
d9fe85d3d2cd
incorporated canonical fold combinator on lists into body of List theory; refactored passages on List.fold(l/r)
haftmann
parents:
45618
diff
changeset
|
507 |
@{const List.foldr} & @{typeof List.foldr}\\ |
30293 | 508 |
@{const List.foldl} & @{typeof List.foldl}\\ |
509 |
@{const List.hd} & @{typeof List.hd}\\ |
|
510 |
@{const List.last} & @{typeof List.last}\\ |
|
511 |
@{const List.length} & @{typeof List.length}\\ |
|
512 |
@{const List.lenlex} & @{term_type_only List.lenlex "('a*'a)set\<Rightarrow>('a list * 'a list)set"}\\ |
|
513 |
@{const List.lex} & @{term_type_only List.lex "('a*'a)set\<Rightarrow>('a list * 'a list)set"}\\ |
|
514 |
@{const List.lexn} & @{term_type_only List.lexn "('a*'a)set\<Rightarrow>nat\<Rightarrow>('a list * 'a list)set"}\\ |
|
515 |
@{const List.lexord} & @{term_type_only List.lexord "('a*'a)set\<Rightarrow>('a list * 'a list)set"}\\ |
|
46488 | 516 |
@{const List.listrel} & @{term_type_only List.listrel "('a*'b)set\<Rightarrow>('a list * 'b list)set"}\\ |
40272 | 517 |
@{const List.listrel1} & @{term_type_only List.listrel1 "('a*'a)set\<Rightarrow>('a list * 'a list)set"}\\ |
30293 | 518 |
@{const List.lists} & @{term_type_only List.lists "'a set\<Rightarrow>'a list set"}\\ |
519 |
@{const List.listset} & @{term_type_only List.listset "'a set list \<Rightarrow> 'a list set"}\\ |
|
58101 | 520 |
@{const Groups_List.listsum} & @{typeof Groups_List.listsum}\\ |
30293 | 521 |
@{const List.list_all2} & @{typeof List.list_all2}\\ |
522 |
@{const List.list_update} & @{typeof List.list_update}\\ |
|
523 |
@{const List.map} & @{typeof List.map}\\ |
|
524 |
@{const List.measures} & @{term_type_only List.measures "('a\<Rightarrow>nat)list\<Rightarrow>('a*'a)set"}\\ |
|
32933 | 525 |
@{const List.nth} & @{typeof List.nth}\\ |
30293 | 526 |
@{const List.remdups} & @{typeof List.remdups}\\ |
527 |
@{const List.removeAll} & @{typeof List.removeAll}\\ |
|
528 |
@{const List.remove1} & @{typeof List.remove1}\\ |
|
529 |
@{const List.replicate} & @{typeof List.replicate}\\ |
|
530 |
@{const List.rev} & @{typeof List.rev}\\ |
|
531 |
@{const List.rotate} & @{typeof List.rotate}\\ |
|
532 |
@{const List.rotate1} & @{typeof List.rotate1}\\ |
|
533 |
@{const List.set} & @{term_type_only List.set "'a list \<Rightarrow> 'a set"}\\ |
|
534 |
@{const List.sort} & @{typeof List.sort}\\ |
|
535 |
@{const List.sorted} & @{typeof List.sorted}\\ |
|
536 |
@{const List.splice} & @{typeof List.splice}\\ |
|
537 |
@{const List.sublist} & @{typeof List.sublist}\\ |
|
538 |
@{const List.take} & @{typeof List.take}\\ |
|
539 |
@{const List.takeWhile} & @{typeof List.takeWhile}\\ |
|
540 |
@{const List.tl} & @{typeof List.tl}\\ |
|
541 |
@{const List.upt} & @{typeof List.upt}\\ |
|
542 |
@{const List.upto} & @{typeof List.upto}\\ |
|
543 |
@{const List.zip} & @{typeof List.zip}\\ |
|
544 |
\end{supertabular} |
|
545 |
||
546 |
\subsubsection*{Syntax} |
|
547 |
||
548 |
\begin{supertabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}} |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51489
diff
changeset
|
549 |
@{text"[x\<^sub>1,\<dots>,x\<^sub>n]"} & @{text"x\<^sub>1 # \<dots> # x\<^sub>n # []"}\\ |
30293 | 550 |
@{term"[m..<n]"} & @{term[source]"upt m n"}\\ |
551 |
@{term"[i..j]"} & @{term[source]"upto i j"}\\ |
|
552 |
@{text"[e. x \<leftarrow> xs]"} & @{term"map (%x. e) xs"}\\ |
|
553 |
@{term"[x \<leftarrow> xs. b]"} & @{term[source]"filter (\<lambda>x. b) xs"} \\ |
|
554 |
@{term"xs[n := x]"} & @{term[source]"list_update xs n x"}\\ |
|
555 |
@{term"\<Sum>x\<leftarrow>xs. e"} & @{term[source]"listsum (map (\<lambda>x. e) xs)"}\\ |
|
556 |
\end{supertabular} |
|
557 |
\medskip |
|
558 |
||
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51489
diff
changeset
|
559 |
List comprehension: @{text"[e. q\<^sub>1, \<dots>, q\<^sub>n]"} where each |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51489
diff
changeset
|
560 |
qualifier @{text q\<^sub>i} is either a generator \mbox{@{text"pat \<leftarrow> e"}} or a |
30293 | 561 |
guard, i.e.\ boolean expression. |
562 |
||
50581 | 563 |
\section*{Map} |
30293 | 564 |
|
565 |
Maps model partial functions and are often used as finite tables. However, |
|
566 |
the domain of a map may be infinite. |
|
567 |
||
568 |
\begin{supertabular}{@ {} l @ {~::~} l @ {}} |
|
569 |
@{const Map.empty} & @{typeof Map.empty}\\ |
|
570 |
@{const Map.map_add} & @{typeof Map.map_add}\\ |
|
571 |
@{const Map.map_comp} & @{typeof Map.map_comp}\\ |
|
572 |
@{const Map.restrict_map} & @{term_type_only Map.restrict_map "('a\<Rightarrow>'b option)\<Rightarrow>'a set\<Rightarrow>('a\<Rightarrow>'b option)"}\\ |
|
573 |
@{const Map.dom} & @{term_type_only Map.dom "('a\<Rightarrow>'b option)\<Rightarrow>'a set"}\\ |
|
574 |
@{const Map.ran} & @{term_type_only Map.ran "('a\<Rightarrow>'b option)\<Rightarrow>'b set"}\\ |
|
575 |
@{const Map.map_le} & @{typeof Map.map_le}\\ |
|
576 |
@{const Map.map_of} & @{typeof Map.map_of}\\ |
|
577 |
@{const Map.map_upds} & @{typeof Map.map_upds}\\ |
|
578 |
\end{supertabular} |
|
579 |
||
580 |
\subsubsection*{Syntax} |
|
581 |
||
582 |
\begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}} |
|
30403 | 583 |
@{term"Map.empty"} & @{term"\<lambda>x. None"}\\ |
30293 | 584 |
@{term"m(x:=Some y)"} & @{term[source]"m(x:=Some y)"}\\ |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51489
diff
changeset
|
585 |
@{text"m(x\<^sub>1\<mapsto>y\<^sub>1,\<dots>,x\<^sub>n\<mapsto>y\<^sub>n)"} & @{text[source]"m(x\<^sub>1\<mapsto>y\<^sub>1)\<dots>(x\<^sub>n\<mapsto>y\<^sub>n)"}\\ |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51489
diff
changeset
|
586 |
@{text"[x\<^sub>1\<mapsto>y\<^sub>1,\<dots>,x\<^sub>n\<mapsto>y\<^sub>n]"} & @{text[source]"Map.empty(x\<^sub>1\<mapsto>y\<^sub>1,\<dots>,x\<^sub>n\<mapsto>y\<^sub>n)"}\\ |
30293 | 587 |
@{term"map_upds m xs ys"} & @{term[source]"map_upds m xs ys"}\\ |
588 |
\end{tabular} |
|
589 |
||
50581 | 590 |
\section*{Infix operators in Main} % @{theory Main} |
591 |
||
592 |
\begin{center} |
|
50605 | 593 |
\begin{tabular}{llll} |
594 |
& Operator & precedence & associativity \\ |
|
595 |
\hline |
|
596 |
Meta-logic & @{text"\<Longrightarrow>"} & 1 & right \\ |
|
597 |
& @{text"\<equiv>"} & 2 \\ |
|
598 |
\hline |
|
599 |
Logic & @{text"\<and>"} & 35 & right \\ |
|
600 |
&@{text"\<or>"} & 30 & right \\ |
|
601 |
&@{text"\<longrightarrow>"}, @{text"\<longleftrightarrow>"} & 25 & right\\ |
|
602 |
&@{text"="}, @{text"\<noteq>"} & 50 & left\\ |
|
603 |
\hline |
|
604 |
Orderings & @{text"\<le>"}, @{text"<"}, @{text"\<ge>"}, @{text">"} & 50 \\ |
|
605 |
\hline |
|
606 |
Sets & @{text"\<subseteq>"}, @{text"\<subset>"}, @{text"\<supseteq>"}, @{text"\<supset>"} & 50 \\ |
|
607 |
&@{text"\<in>"}, @{text"\<notin>"} & 50 \\ |
|
608 |
&@{text"\<inter>"} & 70 & left \\ |
|
609 |
&@{text"\<union>"} & 65 & left \\ |
|
610 |
\hline |
|
611 |
Functions and Relations & @{text"\<circ>"} & 55 & left\\ |
|
612 |
&@{text"`"} & 90 & right\\ |
|
613 |
&@{text"O"} & 75 & right\\ |
|
614 |
&@{text"``"} & 90 & right\\ |
|
57570 | 615 |
&@{text"^^"} & 80 & right\\ |
50605 | 616 |
\hline |
617 |
Numbers & @{text"+"}, @{text"-"} & 65 & left \\ |
|
618 |
&@{text"*"}, @{text"/"} & 70 & left \\ |
|
619 |
&@{text"div"}, @{text"mod"} & 70 & left\\ |
|
620 |
&@{text"^"} & 80 & right\\ |
|
621 |
&@{text"dvd"} & 50 \\ |
|
622 |
\hline |
|
623 |
Lists & @{text"#"}, @{text"@"} & 65 & right\\ |
|
624 |
&@{text"!"} & 100 & left |
|
50581 | 625 |
\end{tabular} |
626 |
\end{center} |
|
30293 | 627 |
*} |
628 |
(*<*) |
|
629 |
end |
|
630 |
(*>*) |