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permissions  rwrr 
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theory Inner_Syntax 
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imports Base Main 
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begin 
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chapter {* Inner syntax  the term language \label{ch:innersyntax} *} 
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text {* The inner syntax of Isabelle provides concrete notation for 
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the main entities of the logical framework, notably @{text 

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"\<lambda>"}terms with types and type classes. Applications may either 

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extend existing syntactic categories by additional notation, or 

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define new sublanguages that are linked to the standard term 

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language via some explicit markers. For example @{verbatim 

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FOO}~@{text "foo"} could embed the syntax corresponding for some 

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userdefined nonterminal @{text "foo"}  within the bounds of the 

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given lexical syntax of Isabelle/Pure. 

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The most basic way to specify concrete syntax for logical entities 

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works via mixfix annotations (\secref{sec:mixfix}), which may be 

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usually given as part of the original declaration or via explicit 

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notation commands later on (\secref{sec:notation}). This already 

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covers many needs of concrete syntax without having to understand 

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the full complexity of inner syntax layers. 

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Further details of the syntax engine involves the classical 

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distinction of lexical language versus contextfree grammar (see 

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\secref{sec:puresyntax}), and various mechanisms for \emph{syntax 

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translations}  either as rewrite systems on firstorder ASTs 

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(\secref{sec:syntrans}) or ML functions on ASTs or @{text 

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"\<lambda>"}terms that represent parse trees (\secref{sec:trfuns}). 

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*} 

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section {* Printing logical entities *} 
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subsection {* Diagnostic commands \label{sec:printdiag} *} 
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text {* 

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\begin{matharray}{rcl} 

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@{command_def "typ"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ 
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@{command_def "term"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ 
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@{command_def "prop"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ 

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@{command_def "thm"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ 
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@{command_def "prf"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ 
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@{command_def "full_prf"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ 

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@{command_def "pr"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\ 
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\end{matharray} 
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These diagnostic commands assist interactive development by printing 

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internal logical entities in a humanreadable fashion. 

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@{rail " 
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@@{command typ} @{syntax modes}? @{syntax type} 
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@@{command term} @{syntax modes}? @{syntax term} 
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@@{command prop} @{syntax modes}? @{syntax prop} 
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@@{command thm} @{syntax modes}? @{syntax thmrefs} 
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( @@{command prf}  @@{command full_prf} ) @{syntax modes}? @{syntax thmrefs}? 
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@@{command pr} @{syntax modes}? @{syntax nat}? 
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; 
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@{syntax_def modes}: '(' (@{syntax name} + ) ')' 
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"} 
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\begin{description} 

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\item @{command "typ"}~@{text \<tau>} reads and prints types of the 
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metalogic according to the current theory or proof context. 
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\item @{command "term"}~@{text t} and @{command "prop"}~@{text \<phi>} 
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read, typecheck and print terms or propositions according to the 
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current theory or proof context; the inferred type of @{text t} is 
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output as well. Note that these commands are also useful in 
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inspecting the current environment of term abbreviations. 
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\item @{command "thm"}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} retrieves 

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theorems from the current theory or proof context. Note that any 

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attributes included in the theorem specifications are applied to a 

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temporary context derived from the current theory or proof; the 

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result is discarded, i.e.\ attributes involved in @{text "a\<^sub>1, 

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\<dots>, a\<^sub>n"} do not have any permanent effect. 

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\item @{command "prf"} displays the (compact) proof term of the 

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current proof state (if present), or of the given theorems. Note 

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that this requires proof terms to be switched on for the current 

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object logic (see the ``Proof terms'' section of the Isabelle 

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reference manual for information on how to do this). 

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\item @{command "full_prf"} is like @{command "prf"}, but displays 

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the full proof term, i.e.\ also displays information omitted in the 

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compact proof term, which is denoted by ``@{text _}'' placeholders 

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there. 

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\item @{command "pr"}~@{text "goals"} prints the current proof state 
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(if present), including current facts and goals. The optional limit 

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arguments affect the number of goals to be displayed, which is 

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initially 10. Omitting limit value leaves the current setting 

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unchanged. 

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\end{description} 
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All of the diagnostic commands above admit a list of @{text modes} 

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to be specified, which is appended to the current print mode; see 
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also \secref{sec:printmodes}. Thus the output behavior may be 
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modified according particular print mode features. For example, 

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@{command "pr"}~@{text "(latex xsymbols)"} would print the current 

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proof state with mathematical symbols and special characters 

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represented in {\LaTeX} source, according to the Isabelle style 

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\cite{isabellesys}. 
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Note that antiquotations (cf.\ \secref{sec:antiq}) provide a more 

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systematic way to include formal items into the printed text 

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document. 

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*} 

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subsection {* Details of printed content *} 
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text {* 
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\begin{tabular}{rcll} 
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@{attribute_def show_types} & : & @{text attribute} & default @{text false} \\ 

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@{attribute_def show_sorts} & : & @{text attribute} & default @{text false} \\ 

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@{attribute_def show_consts} & : & @{text attribute} & default @{text false} \\ 

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@{attribute_def show_abbrevs} & : & @{text attribute} & default @{text true} \\ 

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@{attribute_def show_brackets} & : & @{text attribute} & default @{text false} \\ 

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@{attribute_def names_long} & : & @{text attribute} & default @{text false} \\ 
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@{attribute_def names_short} & : & @{text attribute} & default @{text false} \\ 
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@{attribute_def names_unique} & : & @{text attribute} & default @{text true} \\ 
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@{attribute_def eta_contract} & : & @{text attribute} & default @{text true} \\ 
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@{attribute_def goals_limit} & : & @{text attribute} & default @{text 10} \\ 

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@{attribute_def show_main_goal} & : & @{text attribute} & default @{text false} \\ 

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@{attribute_def show_hyps} & : & @{text attribute} & default @{text false} \\ 

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@{attribute_def show_tags} & : & @{text attribute} & default @{text false} \\ 

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@{attribute_def show_question_marks} & : & @{text attribute} & default @{text true} \\ 

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\end{tabular} 

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\medskip 

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These configuration options control the detail of information that 
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is displayed for types, terms, theorems, goals etc. See also 

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\secref{sec:config}. 

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\begin{description} 
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\item @{attribute show_types} and @{attribute show_sorts} control 
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printing of type constraints for term variables, and sort 

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constraints for type variables. By default, neither of these are 

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shown in output. If @{attribute show_sorts} is enabled, types are 

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always shown as well. 

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Note that displaying types and sorts may explain why a polymorphic 
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inference rule fails to resolve with some goal, or why a rewrite 
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rule does not apply as expected. 
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\item @{attribute show_consts} controls printing of types of 
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constants when displaying a goal state. 

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Note that the output can be enormous, because polymorphic constants 
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often occur at several different type instances. 
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\item @{attribute show_abbrevs} controls folding of constant 
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abbreviations. 

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\item @{attribute show_brackets} controls bracketing in pretty 
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printed output. If enabled, all subexpressions of the pretty 

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printing tree will be parenthesized, even if this produces malformed 
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term syntax! This crude way of showing the internal structure of 
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pretty printed entities may occasionally help to diagnose problems 
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with operator priorities, for example. 
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\item @{attribute names_long}, @{attribute names_short}, and 
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@{attribute names_unique} control the way of printing fully 
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qualified internal names in external form. See also 
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\secref{sec:antiq} for the document antiquotation options of the 
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same names. 
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\item @{attribute eta_contract} controls @{text "\<eta>"}contracted 
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printing of terms. 

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The @{text \<eta>}contraction law asserts @{prop "(\<lambda>x. f x) \<equiv> f"}, 
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provided @{text x} is not free in @{text f}. It asserts 
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\emph{extensionality} of functions: @{prop "f \<equiv> g"} if @{prop "f x \<equiv> 
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g x"} for all @{text x}. Higherorder unification frequently puts 
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terms into a fully @{text \<eta>}expanded form. For example, if @{text 
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F} has type @{text "(\<tau> \<Rightarrow> \<tau>) \<Rightarrow> \<tau>"} then its expanded form is @{term 
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"\<lambda>h. F (\<lambda>x. h x)"}. 
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Enabling @{attribute eta_contract} makes Isabelle perform @{text 
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\<eta>}contractions before printing, so that @{term "\<lambda>h. F (\<lambda>x. h x)"} 
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appears simply as @{text F}. 
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Note that the distinction between a term and its @{text \<eta>}expanded 
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form occasionally matters. While higherorder resolution and 
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rewriting operate modulo @{text "\<alpha>\<beta>\<eta>"}conversion, some other tools 
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might look at terms more discretely. 
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\item @{attribute goals_limit} controls the maximum number of 
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subgoals to be shown in goal output. 
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\item @{attribute show_main_goal} controls whether the main result 
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to be proven should be displayed. This information might be 

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relevant for schematic goals, to inspect the current claim that has 
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been synthesized so far. 

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\item @{attribute show_hyps} controls printing of implicit 
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hypotheses of local facts. Normally, only those hypotheses are 

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displayed that are \emph{not} covered by the assumptions of the 

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current context: this situation indicates a fault in some tool being 

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used. 

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By enabling @{attribute show_hyps}, output of \emph{all} hypotheses 
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can be enforced, which is occasionally useful for diagnostic 

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purposes. 

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\item @{attribute show_tags} controls printing of extra annotations 
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within theorems, such as internal position information, or the case 

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names being attached by the attribute @{attribute case_names}. 

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Note that the @{attribute tagged} and @{attribute untagged} 
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attributes provide lowlevel access to the collection of tags 
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associated with a theorem. 
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\item @{attribute show_question_marks} controls printing of question 
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marks for schematic variables, such as @{text ?x}. Only the leading 

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question mark is affected, the remaining text is unchanged 
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(including proper markup for schematic variables that might be 
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relevant for user interfaces). 
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\end{description} 
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*} 
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subsection {* Alternative print modes \label{sec:printmodes} *} 
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text {* 

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\begin{mldecls} 

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@{index_ML print_mode_value: "unit > string list"} \\ 

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@{index_ML Print_Mode.with_modes: "string list > ('a > 'b) > 'a > 'b"} \\ 

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\end{mldecls} 

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The \emph{print mode} facility allows to modify various operations 

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for printing. Commands like @{command typ}, @{command term}, 

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@{command thm} (see \secref{sec:printdiag}) take additional print 

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modes as optional argument. The underlying ML operations are as 

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follows. 

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\begin{description} 

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\item @{ML "print_mode_value ()"} yields the list of currently 

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active print mode names. This should be understood as symbolic 

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representation of certain individual features for printing (with 

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precedence from left to right). 

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\item @{ML Print_Mode.with_modes}~@{text "modes f x"} evaluates 

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@{text "f x"} in an execution context where the print mode is 

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prepended by the given @{text "modes"}. This provides a threadsafe 

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way to augment print modes. It is also monotonic in the set of mode 

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names: it retains the default print mode that certain 

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userinterfaces might have installed for their proper functioning! 

262 

263 
\end{description} 

264 

265 
\begin{warn} 

266 
The old global reference @{ML print_mode} should never be used 

267 
directly in applications. Its main reason for being publicly 

268 
accessible is to support historic versions of Proof~General. 

269 
\end{warn} 

270 

271 
\medskip The pretty printer for inner syntax maintains alternative 

272 
mixfix productions for any print mode name invented by the user, say 

273 
in commands like @{command notation} or @{command abbreviation}. 

274 
Mode names can be arbitrary, but the following ones have a specific 

275 
meaning by convention: 

276 

277 
\begin{itemize} 

278 

279 
\item @{verbatim "\"\""} (the empty string): default mode; 

280 
implicitly active as last element in the list of modes. 

281 

282 
\item @{verbatim input}: dummy print mode that is never active; may 

283 
be used to specify notation that is only available for input. 

284 

285 
\item @{verbatim internal} dummy print mode that is never active; 

286 
used internally in Isabelle/Pure. 

287 

288 
\item @{verbatim xsymbols}: enable proper mathematical symbols 

289 
instead of ASCII art.\footnote{This traditional mode name stems from 

290 
the ``XSymbol'' package for old versions Proof~General with XEmacs, 

291 
although that package has been superseded by Unicode in recent 

292 
years.} 

293 

294 
\item @{verbatim HTML}: additional mode that is active in HTML 

295 
presentation of Isabelle theory sources; allows to provide 

296 
alternative output notation. 

297 

298 
\item @{verbatim latex}: additional mode that is active in {\LaTeX} 

299 
document preparation of Isabelle theory sources; allows to provide 

300 
alternative output notation. 

301 

302 
\end{itemize} 

303 
*} 

304 

305 

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subsection {* Printing limits *} 
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307 

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text {* 
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\begin{mldecls} 
36745  310 
@{index_ML Pretty.margin_default: "int Unsynchronized.ref"} \\ 
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@{index_ML print_depth: "int > unit"} \\ 
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\end{mldecls} 
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These ML functions set limits for pretty printed text. 
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315 

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\begin{description} 
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36745  318 
\item @{ML Pretty.margin_default} indicates the global default for 
319 
the right margin of the builtin pretty printer, with initial value 

320 
76. Note that userinterfaces typically control margins 

321 
automatically when resizing windows, or even bypass the formatting 

322 
engine of Isabelle/ML altogether and do it within the front end via 

323 
Isabelle/Scala. 

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\item @{ML print_depth}~@{text n} limits the printing depth of the 
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ML toplevel pretty printer; the precise effect depends on the ML 
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compiler and runtime system. Typically @{text n} should be less 
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than 10. Bigger values such as 1001000 are useful for debugging. 
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\end{description} 
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*} 
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46282  334 
section {* Mixfix annotations \label{sec:mixfix} *} 
28762  335 

336 
text {* Mixfix annotations specify concrete \emph{inner syntax} of 

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Isabelle types and terms. Locally fixed parameters in toplevel 
46290  338 
theorem statements, locale and class specifications also admit 
339 
mixfix annotations in a fairly uniform manner. A mixfix annotation 

340 
describes describes the concrete syntax, the translation to abstract 

341 
syntax, and the pretty printing. Special case annotations provide a 

342 
simple means of specifying infix operators and binders. 

343 

344 
Isabelle mixfix syntax is inspired by {\OBJ} \cite{OBJ}. It allows 

345 
to specify any contextfree priority grammar, which is more general 

346 
than the fixity declarations of ML and Prolog. 

28762  347 

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@{rail " 
46289  349 
@{syntax_def mixfix}: '(' mfix ')' 
28762  350 
; 
46289  351 
@{syntax_def struct_mixfix}: '(' ( mfix  @'structure' ) ')' 
28762  352 
; 
353 

46290  354 
mfix: @{syntax template} prios? @{syntax nat}?  
355 
(@'infix'  @'infixl'  @'infixr') @{syntax template} @{syntax nat}  

356 
@'binder' @{syntax template} prios? @{syntax nat} 

357 
; 

358 
template: string 

46289  359 
; 
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prios: '[' (@{syntax nat} + ',') ']' 
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"} 
28762  362 

46290  363 
The string given as @{text template} may include literal text, 
364 
spacing, blocks, and arguments (denoted by ``@{text _}''); the 

365 
special symbol ``@{verbatim "\<index>"}'' (printed as ``@{text "\<index>"}'') 

366 
represents an index argument that specifies an implicit structure 

367 
reference (see also \secref{sec:locale}). Infix and binder 

368 
declarations provide common abbreviations for particular mixfix 

369 
declarations. So in practice, mixfix templates mostly degenerate to 

370 
literal text for concrete syntax, such as ``@{verbatim "++"}'' for 

371 
an infix symbol. 

372 
*} 

28762  373 

46290  374 

375 
subsection {* The general mixfix form *} 

376 

377 
text {* In full generality, mixfix declarations work as follows. 

378 
Suppose a constant @{text "c :: \<tau>\<^sub>1 \<Rightarrow> \<dots> \<tau>\<^sub>n \<Rightarrow> \<tau>"} is annotated by 

379 
@{text "(mixfix [p\<^sub>1, \<dots>, p\<^sub>n] p)"}, where @{text "mixfix"} is a string 

380 
@{text "d\<^sub>0 _ d\<^sub>1 _ \<dots> _ d\<^sub>n"} consisting of delimiters that surround 

381 
argument positions as indicated by underscores. 

28762  382 

383 
Altogether this determines a production for a contextfree priority 

384 
grammar, where for each argument @{text "i"} the syntactic category 

46292  385 
is determined by @{text "\<tau>\<^sub>i"} (with priority @{text "p\<^sub>i"}), and the 
386 
result category is determined from @{text "\<tau>"} (with priority @{text 

387 
"p"}). Priority specifications are optional, with default 0 for 

388 
arguments and 1000 for the result.\footnote{Omitting priorities is 

389 
prone to syntactic ambiguities unless the delimiter tokens determine 

390 
fully bracketed notation, as in @{text "if _ then _ else _ fi"}.} 

28762  391 

392 
Since @{text "\<tau>"} may be again a function type, the constant 

393 
type scheme may have more argument positions than the mixfix 

394 
pattern. Printing a nested application @{text "c t\<^sub>1 \<dots> t\<^sub>m"} for 

395 
@{text "m > n"} works by attaching concrete notation only to the 

396 
innermost part, essentially by printing @{text "(c t\<^sub>1 \<dots> t\<^sub>n) \<dots> t\<^sub>m"} 

397 
instead. If a term has fewer arguments than specified in the mixfix 

398 
template, the concrete syntax is ignored. 

399 

400 
\medskip A mixfix template may also contain additional directives 

401 
for pretty printing, notably spaces, blocks, and breaks. The 

402 
general template format is a sequence over any of the following 

403 
entities. 

404 

28778  405 
\begin{description} 
28762  406 

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\item @{text "d"} is a delimiter, namely a nonempty sequence of 
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characters other than the following special characters: 
28762  409 

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\smallskip 
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\begin{tabular}{ll} 
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@{verbatim "'"} & single quote \\ 
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@{verbatim "_"} & underscore \\ 
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@{text "\<index>"} & index symbol \\ 
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@{verbatim "("} & open parenthesis \\ 
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@{verbatim ")"} & close parenthesis \\ 
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@{verbatim "/"} & slash \\ 
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\end{tabular} 
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419 
\medskip 
28762  420 

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\item @{verbatim "'"} escapes the special meaning of these 
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422 
metacharacters, producing a literal version of the following 
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423 
character, unless that is a blank. 
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424 

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A single quote followed by a blank separates delimiters, without 
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426 
affecting printing, but input tokens may have additional white space 
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427 
here. 
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428 

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\item @{verbatim "_"} is an argument position, which stands for a 
28762  430 
certain syntactic category in the underlying grammar. 
431 

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\item @{text "\<index>"} is an indexed argument position; this is the place 
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433 
where implicit structure arguments can be attached. 
28762  434 

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\item @{text "s"} is a nonempty sequence of spaces for printing. 
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This and the following specifications do not affect parsing at all. 
28762  437 

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\item @{verbatim "("}@{text n} opens a pretty printing block. The 
28762  439 
optional number specifies how much indentation to add when a line 
440 
break occurs within the block. If the parenthesis is not followed 

441 
by digits, the indentation defaults to 0. A block specified via 

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@{verbatim "(00"} is unbreakable. 
28762  443 

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\item @{verbatim ")"} closes a pretty printing block. 
28762  445 

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\item @{verbatim "//"} forces a line break. 
28762  447 

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\item @{verbatim "/"}@{text s} allows a line break. Here @{text s} 
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stands for the string of spaces (zero or more) right after the 
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slash. These spaces are printed if the break is \emph{not} taken. 
28762  451 

28778  452 
\end{description} 
28762  453 

454 
The general idea of pretty printing with blocks and breaks is also 

46286  455 
described in \cite{paulsonml2}; it goes back to \cite{Oppen:1980}. 
28762  456 
*} 
457 

458 

46290  459 
subsection {* Infixes *} 
460 

461 
text {* Infix operators are specified by convenient short forms that 

462 
abbreviate general mixfix annotations as follows: 

463 

464 
\begin{center} 

465 
\begin{tabular}{lll} 

466 

46292  467 
@{verbatim "("}@{keyword_def "infix"}~@{verbatim "\""}@{text sy}@{verbatim "\""} @{text "p"}@{verbatim ")"} 
46290  468 
& @{text "\<mapsto>"} & 
469 
@{verbatim "(\"(_ "}@{text sy}@{verbatim "/ _)\" ["}@{text "p + 1"}@{verbatim ", "}@{text "p + 1"}@{verbatim "]"}@{text " p"}@{verbatim ")"} \\ 

46292  470 
@{verbatim "("}@{keyword_def "infixl"}~@{verbatim "\""}@{text sy}@{verbatim "\""} @{text "p"}@{verbatim ")"} 
46290  471 
& @{text "\<mapsto>"} & 
472 
@{verbatim "(\"(_ "}@{text sy}@{verbatim "/ _)\" ["}@{text "p"}@{verbatim ", "}@{text "p + 1"}@{verbatim "]"}@{text " p"}@{verbatim ")"} \\ 

46292  473 
@{verbatim "("}@{keyword_def "infixr"}~@{verbatim "\""}@{text sy}@{verbatim "\""} @{text "p"}@{verbatim ")"} 
46290  474 
& @{text "\<mapsto>"} & 
475 
@{verbatim "(\"(_ "}@{text sy}@{verbatim "/ _)\" ["}@{text "p + 1"}@{verbatim ", "}@{text "p"}@{verbatim "]"}@{text " p"}@{verbatim ")"} \\ 

476 

477 
\end{tabular} 

478 
\end{center} 

479 

46292  480 
The mixfix template @{verbatim "\"(_ "}@{text sy}@{verbatim "/ _)\""} 
481 
specifies two argument positions; the delimiter is preceded by a 

482 
space and followed by a space or line break; the entire phrase is a 

483 
pretty printing block. 

46290  484 

485 
The alternative notation @{verbatim "op"}~@{text sy} is introduced 

486 
in addition. Thus any infix operator may be written in prefix form 

487 
(as in ML), independently of the number of arguments in the term. 

488 
*} 

489 

490 

491 
subsection {* Binders *} 

492 

493 
text {* A \emph{binder} is a variablebinding construct such as a 

494 
quantifier. The idea to formalize @{text "\<forall>x. b"} as @{text "All 

495 
(\<lambda>x. b)"} for @{text "All :: ('a \<Rightarrow> bool) \<Rightarrow> bool"} already goes back 

496 
to \cite{church40}. Isabelle declarations of certain higherorder 

46292  497 
operators may be annotated with @{keyword_def "binder"} annotations 
498 
as follows: 

46290  499 

500 
\begin{center} 

501 
@{text "c :: "}@{verbatim "\""}@{text "(\<tau>\<^sub>1 \<Rightarrow> \<tau>\<^sub>2) \<Rightarrow> \<tau>\<^sub>3"}@{verbatim "\" ("}@{keyword "binder"}@{verbatim " \""}@{text "sy"}@{verbatim "\" ["}@{text "p"}@{verbatim "] "}@{text "q"}@{verbatim ")"} 

502 
\end{center} 

503 

504 
This introduces concrete binder syntax @{text "sy x. b"}, where 

505 
@{text x} is a bound variable of type @{text "\<tau>\<^sub>1"}, the body @{text 

506 
b} has type @{text "\<tau>\<^sub>2"} and the whole term has type @{text "\<tau>\<^sub>3"}. 

507 
The optional integer @{text p} specifies the syntactic priority of 

508 
the body; the default is @{text "q"}, which is also the priority of 

509 
the whole construct. 

510 

511 
Internally, the binder syntax is expanded to something like this: 

512 
\begin{center} 

513 
@{text "c_binder :: "}@{verbatim "\""}@{text "idts \<Rightarrow> \<tau>\<^sub>2 \<Rightarrow> \<tau>\<^sub>3"}@{verbatim "\" (\"(3"}@{text sy}@{verbatim "_./ _)\" [0, "}@{text "p"}@{verbatim "] "}@{text "q"}@{verbatim ")"} 

514 
\end{center} 

515 

516 
Here @{syntax (inner) idts} is the nonterminal symbol for a list of 

517 
identifiers with optional type constraints (see also 

518 
\secref{sec:puregrammar}). The mixfix template @{verbatim 

519 
"\"(3"}@{text sy}@{verbatim "_./ _)\""} defines argument positions 

520 
for the bound identifiers and the body, separated by a dot with 

521 
optional line break; the entire phrase is a pretty printing block of 

522 
indentation level 3. Note that there is no extra space after @{text 

523 
"sy"}, so it needs to be included user specification if the binder 

524 
syntax ends with a token that may be continued by an identifier 

525 
token at the start of @{syntax (inner) idts}. 

526 

527 
Furthermore, a syntax translation to transforms @{text "c_binder x\<^sub>1 

528 
\<dots> x\<^sub>n b"} into iterated application @{text "c (\<lambda>x\<^sub>1. \<dots> c (\<lambda>x\<^sub>n. b)\<dots>)"}. 

529 
This works in both directions, for parsing and printing. *} 

530 

531 

46282  532 
section {* Explicit notation \label{sec:notation} *} 
28762  533 

534 
text {* 

535 
\begin{matharray}{rcll} 

35413  536 
@{command_def "type_notation"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ 
537 
@{command_def "no_type_notation"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ 

28762  538 
@{command_def "notation"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ 
539 
@{command_def "no_notation"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ 

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@{command_def "write"} & : & @{text "proof(state) \<rightarrow> proof(state)"} \\ 
28762  541 
\end{matharray} 
542 

46288  543 
Commands that introduce new logical entities (terms or types) 
544 
usually allow to provide mixfix annotations on the spot, which is 

545 
convenient for default notation. Nonetheless, the syntax may be 

546 
modified later on by declarations for explicit notation. This 

547 
allows to add or delete mixfix annotations for of existing logical 

548 
entities within the current context. 

549 

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@{rail " 
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(@@{command type_notation}  @@{command no_type_notation}) @{syntax target}? 
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@{syntax mode}? \\ (@{syntax nameref} @{syntax mixfix} + @'and') 
35413  553 
; 
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(@@{command notation}  @@{command no_notation}) @{syntax target}? @{syntax mode}? \\ 
42705  555 
(@{syntax nameref} @{syntax struct_mixfix} + @'and') 
28762  556 
; 
42705  557 
@@{command write} @{syntax mode}? (@{syntax nameref} @{syntax struct_mixfix} + @'and') 
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"} 
28762  559 

560 
\begin{description} 

561 

35413  562 
\item @{command "type_notation"}~@{text "c (mx)"} associates mixfix 
563 
syntax with an existing type constructor. The arity of the 

564 
constructor is retrieved from the context. 

46282  565 

35413  566 
\item @{command "no_type_notation"} is similar to @{command 
567 
"type_notation"}, but removes the specified syntax annotation from 

568 
the present context. 

569 

28762  570 
\item @{command "notation"}~@{text "c (mx)"} associates mixfix 
35413  571 
syntax with an existing constant or fixed variable. The type 
572 
declaration of the given entity is retrieved from the context. 

46282  573 

28762  574 
\item @{command "no_notation"} is similar to @{command "notation"}, 
575 
but removes the specified syntax annotation from the present 

576 
context. 

577 

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\item @{command "write"} is similar to @{command "notation"}, but 
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works within an Isar proof body. 
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580 

28762  581 
\end{description} 
582 
*} 

583 

28778  584 

585 
section {* The Pure syntax \label{sec:puresyntax} *} 

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586 

46282  587 
subsection {* Lexical matters \label{sec:innerlex} *} 
588 

589 
text {* The inner lexical syntax vaguely resembles the outer one 

590 
(\secref{sec:outerlex}), but some details are different. There are 

591 
two main categories of inner syntax tokens: 

592 

593 
\begin{enumerate} 

594 

595 
\item \emph{delimiters}  the literal tokens occurring in 

596 
productions of the given priority grammar (cf.\ 

597 
\secref{sec:prioritygrammar}); 

598 

599 
\item \emph{named tokens}  various categories of identifiers etc. 

600 

601 
\end{enumerate} 

602 

603 
Delimiters override named tokens and may thus render certain 

604 
identifiers inaccessible. Sometimes the logical context admits 

605 
alternative ways to refer to the same entity, potentially via 

606 
qualified names. 

607 

608 
\medskip The categories for named tokens are defined once and for 

609 
all as follows, reusing some categories of the outer token syntax 

610 
(\secref{sec:outerlex}). 

611 

612 
\begin{center} 

613 
\begin{supertabular}{rcl} 

614 
@{syntax_def (inner) id} & = & @{syntax_ref ident} \\ 

615 
@{syntax_def (inner) longid} & = & @{syntax_ref longident} \\ 

616 
@{syntax_def (inner) var} & = & @{syntax_ref var} \\ 

617 
@{syntax_def (inner) tid} & = & @{syntax_ref typefree} \\ 

618 
@{syntax_def (inner) tvar} & = & @{syntax_ref typevar} \\ 

619 
@{syntax_def (inner) num_token} & = & @{syntax_ref nat}@{text "  "}@{verbatim ""}@{syntax_ref nat} \\ 

620 
@{syntax_def (inner) float_token} & = & @{syntax_ref nat}@{verbatim "."}@{syntax_ref nat}@{text "  "}@{verbatim ""}@{syntax_ref nat}@{verbatim "."}@{syntax_ref nat} \\ 

621 
@{syntax_def (inner) xnum_token} & = & @{verbatim "#"}@{syntax_ref nat}@{text "  "}@{verbatim "#"}@{syntax_ref nat} \\ 

622 

623 
@{syntax_def (inner) xstr} & = & @{verbatim "''"} @{text "\<dots>"} @{verbatim "''"} \\ 

624 
\end{supertabular} 

625 
\end{center} 

626 

627 
The token categories @{syntax (inner) num_token}, @{syntax (inner) 

628 
float_token}, @{syntax (inner) xnum_token}, and @{syntax (inner) 

629 
xstr} are not used in Pure. Objectlogics may implement numerals 

630 
and string constants by adding appropriate syntax declarations, 

631 
together with some translation functions (e.g.\ see Isabelle/HOL). 

632 

633 
The derived categories @{syntax_def (inner) num_const}, @{syntax_def 

634 
(inner) float_const}, and @{syntax_def (inner) num_const} provide 

635 
robust access to the respective tokens: the syntax tree holds a 

636 
syntactic constant instead of a free variable. 

637 
*} 

638 

639 

28777  640 
subsection {* Priority grammars \label{sec:prioritygrammar} *} 
28769
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641 

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642 
text {* A contextfree grammar consists of a set of \emph{terminal 
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643 
symbols}, a set of \emph{nonterminal symbols} and a set of 
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644 
\emph{productions}. Productions have the form @{text "A = \<gamma>"}, 
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645 
where @{text A} is a nonterminal and @{text \<gamma>} is a string of 
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646 
terminals and nonterminals. One designated nonterminal is called 
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647 
the \emph{root symbol}. The language defined by the grammar 
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648 
consists of all strings of terminals that can be derived from the 
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649 
root symbol by applying productions as rewrite rules. 
8fc228f21861
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650 

8fc228f21861
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651 
The standard Isabelle parser for inner syntax uses a \emph{priority 
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652 
grammar}. Each nonterminal is decorated by an integer priority: 
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653 
@{text "A\<^sup>(\<^sup>p\<^sup>)"}. In a derivation, @{text "A\<^sup>(\<^sup>p\<^sup>)"} may be rewritten 
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654 
using a production @{text "A\<^sup>(\<^sup>q\<^sup>) = \<gamma>"} only if @{text "p \<le> q"}. Any 
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655 
priority grammar can be translated into a normal contextfree 
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656 
grammar by introducing new nonterminals and productions. 
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657 

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658 
\medskip Formally, a set of context free productions @{text G} 
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659 
induces a derivation relation @{text "\<longrightarrow>\<^sub>G"} as follows. Let @{text 
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660 
\<alpha>} and @{text \<beta>} denote strings of terminal or nonterminal symbols. 
28774  661 
Then @{text "\<alpha> A\<^sup>(\<^sup>p\<^sup>) \<beta> \<longrightarrow>\<^sub>G \<alpha> \<gamma> \<beta>"} holds if and only if @{text G} 
662 
contains some production @{text "A\<^sup>(\<^sup>q\<^sup>) = \<gamma>"} for @{text "p \<le> q"}. 

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663 

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664 
\medskip The following grammar for arithmetic expressions 
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665 
demonstrates how binding power and associativity of operators can be 
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666 
enforced by priorities. 
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667 

8fc228f21861
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668 
\begin{center} 
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669 
\begin{tabular}{rclr} 
28774  670 
@{text "A\<^sup>(\<^sup>1\<^sup>0\<^sup>0\<^sup>0\<^sup>)"} & @{text "="} & @{verbatim "("} @{text "A\<^sup>(\<^sup>0\<^sup>)"} @{verbatim ")"} \\ 
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671 
@{text "A\<^sup>(\<^sup>1\<^sup>0\<^sup>0\<^sup>0\<^sup>)"} & @{text "="} & @{verbatim 0} \\ 
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672 
@{text "A\<^sup>(\<^sup>0\<^sup>)"} & @{text "="} & @{text "A\<^sup>(\<^sup>0\<^sup>)"} @{verbatim "+"} @{text "A\<^sup>(\<^sup>1\<^sup>)"} \\ 
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673 
@{text "A\<^sup>(\<^sup>2\<^sup>)"} & @{text "="} & @{text "A\<^sup>(\<^sup>3\<^sup>)"} @{verbatim "*"} @{text "A\<^sup>(\<^sup>2\<^sup>)"} \\ 
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674 
@{text "A\<^sup>(\<^sup>3\<^sup>)"} & @{text "="} & @{verbatim ""} @{text "A\<^sup>(\<^sup>3\<^sup>)"} \\ 
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675 
\end{tabular} 
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676 
\end{center} 
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677 
The choice of priorities determines that @{verbatim ""} binds 
8fc228f21861
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678 
tighter than @{verbatim "*"}, which binds tighter than @{verbatim 
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679 
"+"}. Furthermore @{verbatim "+"} associates to the left and 
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680 
@{verbatim "*"} to the right. 
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681 

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682 
\medskip For clarity, grammars obey these conventions: 
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683 
\begin{itemize} 
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684 

8fc228f21861
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685 
\item All priorities must lie between 0 and 1000. 
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diff
changeset

686 

8fc228f21861
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diff
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687 
\item Priority 0 on the righthand side and priority 1000 on the 
8fc228f21861
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688 
lefthand side may be omitted. 
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689 

8fc228f21861
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690 
\item The production @{text "A\<^sup>(\<^sup>p\<^sup>) = \<alpha>"} is written as @{text "A = \<alpha> 
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691 
(p)"}, i.e.\ the priority of the lefthand side actually appears in 
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diff
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692 
a column on the far right. 
8fc228f21861
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diff
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693 

8fc228f21861
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diff
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694 
\item Alternatives are separated by @{text ""}. 
8fc228f21861
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diff
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695 

8fc228f21861
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696 
\item Repetition is indicated by dots @{text "(\<dots>)"} in an informal 
8fc228f21861
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diff
changeset

697 
but obvious way. 
8fc228f21861
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diff
changeset

698 

8fc228f21861
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diff
changeset

699 
\end{itemize} 
8fc228f21861
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diff
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700 

8fc228f21861
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diff
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701 
Using these conventions, the example grammar specification above 
8fc228f21861
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diff
changeset

702 
takes the form: 
8fc228f21861
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diff
changeset

703 
\begin{center} 
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704 
\begin{tabular}{rclc} 
28774  705 
@{text A} & @{text "="} & @{verbatim "("} @{text A} @{verbatim ")"} \\ 
706 
& @{text ""} & @{verbatim 0} & \qquad\qquad \\ 

28769
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707 
& @{text ""} & @{text A} @{verbatim "+"} @{text "A\<^sup>(\<^sup>1\<^sup>)"} & @{text "(0)"} \\ 
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708 
& @{text ""} & @{text "A\<^sup>(\<^sup>3\<^sup>)"} @{verbatim "*"} @{text "A\<^sup>(\<^sup>2\<^sup>)"} & @{text "(2)"} \\ 
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709 
& @{text ""} & @{verbatim ""} @{text "A\<^sup>(\<^sup>3\<^sup>)"} & @{text "(3)"} \\ 
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710 
\end{tabular} 
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711 
\end{center} 
8fc228f21861
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712 
*} 
8fc228f21861
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changeset

713 

8fc228f21861
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714 

46290  715 
subsection {* The Pure grammar \label{sec:puregrammar} *} 
28770
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716 

46287  717 
text {* The priority grammar of the @{text "Pure"} theory is defined 
718 
approximately like this: 

28774  719 

28770
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720 
\begin{center} 
28773  721 
\begin{supertabular}{rclr} 
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722 

28778  723 
@{syntax_def (inner) any} & = & @{text "prop  logic"} \\\\ 
28772  724 

28778  725 
@{syntax_def (inner) prop} & = & @{verbatim "("} @{text prop} @{verbatim ")"} \\ 
28772  726 
& @{text ""} & @{text "prop\<^sup>(\<^sup>4\<^sup>)"} @{verbatim "::"} @{text type} & @{text "(3)"} \\ 
727 
& @{text ""} & @{text "any\<^sup>(\<^sup>3\<^sup>)"} @{verbatim "=="} @{text "any\<^sup>(\<^sup>2\<^sup>)"} & @{text "(2)"} \\ 

28773  728 
& @{text ""} & @{text "any\<^sup>(\<^sup>3\<^sup>)"} @{text "\<equiv>"} @{text "any\<^sup>(\<^sup>2\<^sup>)"} & @{text "(2)"} \\ 
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729 
& @{text ""} & @{text "prop\<^sup>(\<^sup>3\<^sup>)"} @{verbatim "&&&"} @{text "prop\<^sup>(\<^sup>2\<^sup>)"} & @{text "(2)"} \\ 
28772  730 
& @{text ""} & @{text "prop\<^sup>(\<^sup>2\<^sup>)"} @{verbatim "==>"} @{text "prop\<^sup>(\<^sup>1\<^sup>)"} & @{text "(1)"} \\ 
28773  731 
& @{text ""} & @{text "prop\<^sup>(\<^sup>2\<^sup>)"} @{text "\<Longrightarrow>"} @{text "prop\<^sup>(\<^sup>1\<^sup>)"} & @{text "(1)"} \\ 
28772  732 
& @{text ""} & @{verbatim "["} @{text prop} @{verbatim ";"} @{text "\<dots>"} @{verbatim ";"} @{text prop} @{verbatim "]"} @{verbatim "==>"} @{text "prop\<^sup>(\<^sup>1\<^sup>)"} & @{text "(1)"} \\ 
28773  733 
& @{text ""} & @{text "\<lbrakk>"} @{text prop} @{verbatim ";"} @{text "\<dots>"} @{verbatim ";"} @{text prop} @{text "\<rbrakk>"} @{text "\<Longrightarrow>"} @{text "prop\<^sup>(\<^sup>1\<^sup>)"} & @{text "(1)"} \\ 
28772  734 
& @{text ""} & @{verbatim "!!"} @{text idts} @{verbatim "."} @{text prop} & @{text "(0)"} \\ 
28773  735 
& @{text ""} & @{text "\<And>"} @{text idts} @{verbatim "."} @{text prop} & @{text "(0)"} \\ 
736 
& @{text ""} & @{verbatim OFCLASS} @{verbatim "("} @{text type} @{verbatim ","} @{text logic} @{verbatim ")"} \\ 

737 
& @{text ""} & @{verbatim SORT_CONSTRAINT} @{verbatim "("} @{text type} @{verbatim ")"} \\ 

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738 
& @{text ""} & @{verbatim TERM} @{text logic} \\ 
28773  739 
& @{text ""} & @{verbatim PROP} @{text aprop} \\\\ 
28772  740 

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741 
@{syntax_def (inner) aprop} & = & @{verbatim "("} @{text aprop} @{verbatim ")"} \\ 
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742 
& @{text ""} & @{text "id  longid  var  "}@{verbatim "_"}@{text "  "}@{verbatim "..."} \\ 
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743 
& @{text ""} & @{verbatim CONST} @{text "id  "}@{verbatim CONST} @{text "longid"} \\ 
46287  744 
& @{text ""} & @{verbatim XCONST} @{text "id  "}@{verbatim XCONST} @{text "longid"} \\ 
28773  745 
& @{text ""} & @{text "logic\<^sup>(\<^sup>1\<^sup>0\<^sup>0\<^sup>0\<^sup>) any\<^sup>(\<^sup>1\<^sup>0\<^sup>0\<^sup>0\<^sup>) \<dots> any\<^sup>(\<^sup>1\<^sup>0\<^sup>0\<^sup>0\<^sup>)"} & @{text "(999)"} \\\\ 
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746 

28778  747 
@{syntax_def (inner) logic} & = & @{verbatim "("} @{text logic} @{verbatim ")"} \\ 
28772  748 
& @{text ""} & @{text "logic\<^sup>(\<^sup>4\<^sup>)"} @{verbatim "::"} @{text type} & @{text "(3)"} \\ 
28773  749 
& @{text ""} & @{text "id  longid  var  "}@{verbatim "_"}@{text "  "}@{verbatim "..."} \\ 
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750 
& @{text ""} & @{verbatim CONST} @{text "id  "}@{verbatim CONST} @{text "longid"} \\ 
46287  751 
& @{text ""} & @{verbatim XCONST} @{text "id  "}@{verbatim XCONST} @{text "longid"} \\ 
28773  752 
& @{text ""} & @{text "logic\<^sup>(\<^sup>1\<^sup>0\<^sup>0\<^sup>0\<^sup>) any\<^sup>(\<^sup>1\<^sup>0\<^sup>0\<^sup>0\<^sup>) \<dots> any\<^sup>(\<^sup>1\<^sup>0\<^sup>0\<^sup>0\<^sup>)"} & @{text "(999)"} \\ 
46287  753 
& @{text ""} & @{text "\<struct> index\<^sup>(\<^sup>1\<^sup>0\<^sup>0\<^sup>0\<^sup>)"} \\ 
28772  754 
& @{text ""} & @{verbatim "%"} @{text pttrns} @{verbatim "."} @{text "any\<^sup>(\<^sup>3\<^sup>)"} & @{text "(3)"} \\ 
28773  755 
& @{text ""} & @{text \<lambda>} @{text pttrns} @{verbatim "."} @{text "any\<^sup>(\<^sup>3\<^sup>)"} & @{text "(3)"} \\ 
46287  756 
& @{text ""} & @{verbatim op} @{verbatim "=="}@{text "  "}@{verbatim op} @{text "\<equiv>"}@{text "  "}@{verbatim op} @{verbatim "&&&"} \\ 
757 
& @{text ""} & @{verbatim op} @{verbatim "==>"}@{text "  "}@{verbatim op} @{text "\<Longrightarrow>"} \\ 

28772  758 
& @{text ""} & @{verbatim TYPE} @{verbatim "("} @{text type} @{verbatim ")"} \\\\ 
759 

28778  760 
@{syntax_def (inner) idt} & = & @{verbatim "("} @{text idt} @{verbatim ")"}@{text "  id  "}@{verbatim "_"} \\ 
28773  761 
& @{text ""} & @{text id} @{verbatim "::"} @{text type} & @{text "(0)"} \\ 
762 
& @{text ""} & @{verbatim "_"} @{verbatim "::"} @{text type} & @{text "(0)"} \\\\ 

28772  763 

46287  764 
@{syntax_def (inner) index} & = & @{verbatim "\<^bsub>"} @{text "logic\<^sup>(\<^sup>0\<^sup>)"} @{verbatim "\<^esub>"}@{text "   \<index>"} \\\\ 
765 

28778  766 
@{syntax_def (inner) idts} & = & @{text "idt  idt\<^sup>(\<^sup>1\<^sup>) idts"} & @{text "(0)"} \\\\ 
28772  767 

28778  768 
@{syntax_def (inner) pttrn} & = & @{text idt} \\\\ 
28772  769 

28778  770 
@{syntax_def (inner) pttrns} & = & @{text "pttrn  pttrn\<^sup>(\<^sup>1\<^sup>) pttrns"} & @{text "(0)"} \\\\ 
28774  771 

28778  772 
@{syntax_def (inner) type} & = & @{verbatim "("} @{text type} @{verbatim ")"} \\ 
28773  773 
& @{text ""} & @{text "tid  tvar  "}@{verbatim "_"} \\ 
774 
& @{text ""} & @{text "tid"} @{verbatim "::"} @{text "sort  tvar "}@{verbatim "::"} @{text "sort  "}@{verbatim "_"} @{verbatim "::"} @{text "sort"} \\ 

46287  775 
& @{text ""} & @{text "type_name  type\<^sup>(\<^sup>1\<^sup>0\<^sup>0\<^sup>0\<^sup>) type_name"} \\ 
776 
& @{text ""} & @{verbatim "("} @{text type} @{verbatim ","} @{text "\<dots>"} @{verbatim ","} @{text type} @{verbatim ")"} @{text type_name} \\ 

28772  777 
& @{text ""} & @{text "type\<^sup>(\<^sup>1\<^sup>)"} @{verbatim "=>"} @{text type} & @{text "(0)"} \\ 
28773  778 
& @{text ""} & @{text "type\<^sup>(\<^sup>1\<^sup>)"} @{text "\<Rightarrow>"} @{text type} & @{text "(0)"} \\ 
779 
& @{text ""} & @{verbatim "["} @{text type} @{verbatim ","} @{text "\<dots>"} @{verbatim ","} @{text type} @{verbatim "]"} @{verbatim "=>"} @{text type} & @{text "(0)"} \\ 

46287  780 
& @{text ""} & @{verbatim "["} @{text type} @{verbatim ","} @{text "\<dots>"} @{verbatim ","} @{text type} @{verbatim "]"} @{text "\<Rightarrow>"} @{text type} & @{text "(0)"} \\ 
781 
@{syntax_def (inner) type_name} & = & @{text "id  longid"} \\\\ 

28772  782 

46287  783 
@{syntax_def (inner) sort} & = & @{syntax class_name}~@{text "  "}@{verbatim "{}"} \\ 
784 
& @{text ""} & @{verbatim "{"} @{syntax class_name} @{verbatim ","} @{text "\<dots>"} @{verbatim ","} @{syntax class_name} @{verbatim "}"} \\ 

785 
@{syntax_def (inner) class_name} & = & @{text "id  longid"} \\ 

28773  786 
\end{supertabular} 
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787 
\end{center} 
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788 

28774  789 
\medskip Here literal terminals are printed @{verbatim "verbatim"}; 
790 
see also \secref{sec:innerlex} for further token categories of the 

791 
inner syntax. The meaning of the nonterminals defined by the above 

792 
grammar is as follows: 

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793 

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794 
\begin{description} 
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795 

28778  796 
\item @{syntax_ref (inner) any} denotes any term. 
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797 

28778  798 
\item @{syntax_ref (inner) prop} denotes metalevel propositions, 
799 
which are terms of type @{typ prop}. The syntax of such formulae of 

800 
the metalogic is carefully distinguished from usual conventions for 

801 
objectlogics. In particular, plain @{text "\<lambda>"}term notation is 

802 
\emph{not} recognized as @{syntax (inner) prop}. 

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803 

28778  804 
\item @{syntax_ref (inner) aprop} denotes atomic propositions, which 
805 
are embedded into regular @{syntax (inner) prop} by means of an 

806 
explicit @{verbatim PROP} token. 

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807 

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808 
Terms of type @{typ prop} with nonconstant head, e.g.\ a plain 
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809 
variable, are printed in this form. Constants that yield type @{typ 
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810 
prop} are expected to provide their own concrete syntax; otherwise 
28778  811 
the printed version will appear like @{syntax (inner) logic} and 
812 
cannot be parsed again as @{syntax (inner) prop}. 

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813 

28778  814 
\item @{syntax_ref (inner) logic} denotes arbitrary terms of a 
815 
logical type, excluding type @{typ prop}. This is the main 

816 
syntactic category of objectlogic entities, covering plain @{text 

817 
\<lambda>}term notation (variables, abstraction, application), plus 

818 
anything defined by the user. 

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819 

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820 
When specifying notation for logical entities, all logical types 
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821 
(excluding @{typ prop}) are \emph{collapsed} to this single category 
28778  822 
of @{syntax (inner) logic}. 
28770
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823 

46287  824 
\item @{syntax_ref (inner) index} denotes an optional index term for 
825 
indexed syntax. If omitted, it refers to the first @{keyword 

826 
"structure"} variable in the context. The special dummy ``@{text 

827 
"\<index>"}'' serves as pattern variable in mixfix annotations that 

828 
introduce indexed notation. 

829 

28778  830 
\item @{syntax_ref (inner) idt} denotes identifiers, possibly 
831 
constrained by types. 

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832 

28778  833 
\item @{syntax_ref (inner) idts} denotes a sequence of @{syntax_ref 
834 
(inner) idt}. This is the most basic category for variables in 

835 
iterated binders, such as @{text "\<lambda>"} or @{text "\<And>"}. 

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836 

28778  837 
\item @{syntax_ref (inner) pttrn} and @{syntax_ref (inner) pttrns} 
838 
denote patterns for abstraction, cases bindings etc. In Pure, these 

839 
categories start as a merely copy of @{syntax (inner) idt} and 

840 
@{syntax (inner) idts}, respectively. Objectlogics may add 

841 
additional productions for binding forms. 

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842 

28778  843 
\item @{syntax_ref (inner) type} denotes types of the metalogic. 
28770
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844 

28778  845 
\item @{syntax_ref (inner) sort} denotes metalevel sorts. 
28770
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846 

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847 
\end{description} 
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848 

28774  849 
Here are some further explanations of certain syntax features. 
28773  850 

851 
\begin{itemize} 

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852 

28778  853 
\item In @{syntax (inner) idts}, note that @{text "x :: nat y"} is 
854 
parsed as @{text "x :: (nat y)"}, treating @{text y} like a type 

855 
constructor applied to @{text nat}. To avoid this interpretation, 

856 
write @{text "(x :: nat) y"} with explicit parentheses. 

28773  857 

858 
\item Similarly, @{text "x :: nat y :: nat"} is parsed as @{text "x :: 

28770
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859 
(nat y :: nat)"}. The correct form is @{text "(x :: nat) (y :: 
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860 
nat)"}, or @{text "(x :: nat) y :: nat"} if @{text y} is last in the 
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861 
sequence of identifiers. 
28773  862 

863 
\item Type constraints for terms bind very weakly. For example, 

864 
@{text "x < y :: nat"} is normally parsed as @{text "(x < y) :: 

865 
nat"}, unless @{text "<"} has a very low priority, in which case the 

866 
input is likely to be ambiguous. The correct form is @{text "x < (y 

867 
:: nat)"}. 

868 

869 
\item Constraints may be either written with two literal colons 

870 
``@{verbatim "::"}'' or the doublecolon symbol @{verbatim "\<Colon>"}, 

28774  871 
which actually looks exactly the same in some {\LaTeX} styles. 
28773  872 

28774  873 
\item Dummy variables (written as underscore) may occur in different 
874 
roles. 

28773  875 

876 
\begin{description} 

877 

28774  878 
\item A type ``@{text "_"}'' or ``@{text "_ :: sort"}'' acts like an 
879 
anonymous inference parameter, which is filledin according to the 

880 
most general type produced by the typechecking phase. 

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881 

28774  882 
\item A bound ``@{text "_"}'' refers to a vacuous abstraction, where 
883 
the body does not refer to the binding introduced here. As in the 

884 
term @{term "\<lambda>x _. x"}, which is @{text "\<alpha>"}equivalent to @{text 

885 
"\<lambda>x y. x"}. 

28773  886 

28774  887 
\item A free ``@{text "_"}'' refers to an implicit outer binding. 
888 
Higher definitional packages usually allow forms like @{text "f x _ 

889 
= x"}. 

28773  890 

28774  891 
\item A schematic ``@{text "_"}'' (within a term pattern, see 
892 
\secref{sec:termdecls}) refers to an anonymous variable that is 

893 
implicitly abstracted over its context of locally bound variables. 

894 
For example, this allows pattern matching of @{text "{x. f x = g 

895 
x}"} against @{text "{x. _ = _}"}, or even @{text "{_. _ = _}"} by 

896 
using both bound and schematic dummies. 

28773  897 

898 
\end{description} 

899 

28774  900 
\item The three literal dots ``@{verbatim "..."}'' may be also 
901 
written as ellipsis symbol @{verbatim "\<dots>"}. In both cases this 

902 
refers to a special schematic variable, which is bound in the 

903 
context. This special term abbreviation works nicely with 

904 
calculational reasoning (\secref{sec:calculation}). 

905 

46287  906 
\item @{verbatim CONST} ensures that the given identifier is treated 
907 
as constant term, and passed through the parse tree in fully 

908 
internalized form. This is particularly relevant for translation 

909 
rules (\secref{sec:syntrans}), notably on the RHS. 

910 

911 
\item @{verbatim XCONST} is similar to @{verbatim CONST}, but 

912 
retains the constant name as given. This is only relevant to 

913 
translation rules (\secref{sec:syntrans}), notably on the LHS. 

914 

28773  915 
\end{itemize} 
28770
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916 
*} 
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917 

28777  918 

46282  919 
subsection {* Inspecting the syntax *} 
28777  920 

46282  921 
text {* 
922 
\begin{matharray}{rcl} 

923 
@{command_def "print_syntax"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\ 

924 
\end{matharray} 

28777  925 

46282  926 
\begin{description} 
927 

928 
\item @{command "print_syntax"} prints the inner syntax of the 

929 
current context. The output can be quite large; the most important 

930 
sections are explained below. 

28777  931 

46282  932 
\begin{description} 
28777  933 

46282  934 
\item @{text "lexicon"} lists the delimiters of the inner token 
935 
language; see \secref{sec:innerlex}. 

28777  936 

46282  937 
\item @{text "prods"} lists the productions of the underlying 
938 
priority grammar; see \secref{sec:prioritygrammar}. 

28777  939 

46282  940 
The nonterminal @{text "A\<^sup>(\<^sup>p\<^sup>)"} is rendered in plain text as @{text 
941 
"A[p]"}; delimiters are quoted. Many productions have an extra 

942 
@{text "\<dots> => name"}. These names later become the heads of parse 

943 
trees; they also guide the pretty printer. 

28777  944 

46282  945 
Productions without such parse tree names are called \emph{copy 
946 
productions}. Their righthand side must have exactly one 

947 
nonterminal symbol (or named token). The parser does not create a 

948 
new parse tree node for copy productions, but simply returns the 

949 
parse tree of the righthand symbol. 

950 

951 
If the righthand side of a copy production consists of a single 

952 
nonterminal without any delimiters, then it is called a \emph{chain 

953 
production}. Chain productions act as abbreviations: conceptually, 

954 
they are removed from the grammar by adding new productions. 

955 
Priority information attached to chain productions is ignored; only 

956 
the dummy value @{text "1"} is displayed. 

957 

958 
\item @{text "print modes"} lists the alternative print modes 

959 
provided by this grammar; see \secref{sec:printmodes}. 

28777  960 

46282  961 
\item @{text "parse_rules"} and @{text "print_rules"} relate to 
962 
syntax translations (macros); see \secref{sec:syntrans}. 

963 

964 
\item @{text "parse_ast_translation"} and @{text 

965 
"print_ast_translation"} list sets of constants that invoke 

966 
translation functions for abstract syntax trees, which are only 

967 
required in very special situations; see \secref{sec:trfuns}. 

28777  968 

46282  969 
\item @{text "parse_translation"} and @{text "print_translation"} 
970 
list the sets of constants that invoke regular translation 

971 
functions; see \secref{sec:trfuns}. 

29157  972 

46282  973 
\end{description} 
974 

975 
\end{description} 

28777  976 
*} 
28774  977 

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978 

46291  979 
subsection {* Ambiguity of parsed expressions *} 
980 

981 
text {* 

982 
\begin{tabular}{rcll} 

983 
@{attribute_def syntax_ambiguity_level} & : & @{text attribute} & default @{text 1} \\ 

984 
\end{tabular} 

985 

986 
\begin{mldecls} 

987 
@{index_ML Syntax.ambiguity_limit: "int Config.T"} \\ %FIXME attribute 

988 
\end{mldecls} 

989 

990 
Depending on the grammar and the given input, parsing may be 

991 
ambiguous. Isabelle lets the Earley parser enumerate all possible 

992 
parse trees, and then tries to make the best out of the situation. 

993 
Terms that cannot be typechecked are filtered out, which often 

994 
leads to a unique result in the end. Unlike regular type 

995 
reconstruction, which is applied to the whole collection of input 

996 
terms simultaneously, the filtering stage only treats each given 

997 
term in isolation. Filtering is also not attempted for individual 

998 
types or raw ASTs (as required for @{command translations}). 

999 

1000 
Certain warning or error messages are printed, depending on the 

1001 
situation and the given configuration options. Parsing ultimately 

1002 
fails, if multiple results remain after the filtering phase. 

1003 

1004 
\begin{description} 

1005 

1006 
\item @{attribute syntax_ambiguity_level} determines the number of 

1007 
parser results that are tolerated without printing a detailed 

1008 
message. 

1009 

1010 
\item @{ML Syntax.ambiguity_limit} determines the number of 

1011 
resulting parse trees that are shown as part of the printed message 

1012 
in case of an ambiguity. 

1013 

1014 
\end{description} 

1015 
*} 

1016 

1017 

46282  1018 
section {* Raw syntax and translations \label{sec:syntrans} *} 
28762  1019 

1020 
text {* 

1021 
\begin{matharray}{rcl} 

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1022 
@{command_def "nonterminal"} & : & @{text "theory \<rightarrow> theory"} \\ 
28762  1023 
@{command_def "syntax"} & : & @{text "theory \<rightarrow> theory"} \\ 
1024 
@{command_def "no_syntax"} & : & @{text "theory \<rightarrow> theory"} \\ 

1025 
@{command_def "translations"} & : & @{text "theory \<rightarrow> theory"} \\ 

1026 
@{command_def "no_translations"} & : & @{text "theory \<rightarrow> theory"} \\ 

1027 
\end{matharray} 

1028 

46292  1029 
Unlike mixfix notation for existing formal entities 
1030 
(\secref{sec:notation}), raw syntax declarations provide full access 

1031 
to the priority grammar of the inner syntax. This includes 

1032 
additional syntactic categories (via @{command nonterminal}) and 

1033 
freeform grammar productions (via @{command syntax}). Additional 

1034 
syntax translations (or macros, via @{command translations}) are 

1035 
required to turn resulting parse trees into proper representations 

1036 
of formal entities again. 

1037 

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1038 
@{rail " 
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1039 
@@{command nonterminal} (@{syntax name} + @'and') 
28762  1040 
; 
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1041 
(@@{command syntax}  @@{command no_syntax}) @{syntax mode}? (@{syntax constdecl} +) 
28762  1042 
; 
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1043 
(@@{command translations}  @@{command no_translations}) 
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1044 
(transpat ('=='  '=>'  '<='  '\<rightleftharpoons>'  '\<rightharpoonup>'  '\<leftharpoondown>') transpat +) 
28762  1045 
; 
1046 

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1047 
mode: ('(' ( @{syntax name}  @'output'  @{syntax name} @'output' ) ')') 
28762  1048 
; 
42596
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1049 
transpat: ('(' @{syntax nameref} ')')? @{syntax string} 
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1050 
"} 
28762  1051 

1052 
\begin{description} 

46282  1053 

41229
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1054 
\item @{command "nonterminal"}~@{text c} declares a type 
28762  1055 
constructor @{text c} (without arguments) to act as purely syntactic 
1056 
type: a nonterminal symbol of the inner syntax. 

1057 

46292  1058 
\item @{command "syntax"}~@{text "(mode) c :: \<sigma> (mx)"} augments the 
1059 
priority grammar and the pretty printer table for the given print 

1060 
mode (default @{verbatim "\"\""}). An optional keyword @{keyword_ref 

1061 
"output"} means that only the pretty printer table is affected. 

1062 

1063 
Following \secref{sec:mixfix}, the mixfix annotation @{text "mx = 

1064 
template ps q"} together with type @{text "\<sigma> = \<tau>\<^sub>1 \<Rightarrow> \<dots> \<tau>\<^sub>n \<Rightarrow> \<tau>"} and 

1065 
specify a grammar production. The @{text template} contains 

1066 
delimiter tokens that surround @{text "n"} argument positions 

1067 
(@{verbatim "_"}). The latter correspond to nonterminal symbols 

1068 
@{text "A\<^sub>i"} derived from the argument types @{text "\<tau>\<^sub>i"} as 

1069 
follows: 

1070 
\begin{itemize} 

1071 

1072 
\item @{text "prop"} if @{text "\<tau>\<^sub>i = prop"} 

1073 

1074 
\item @{text "logic"} if @{text "\<tau>\<^sub>i = (\<dots>)\<kappa>"} for logical type 

1075 
constructor @{text "\<kappa> \<noteq> prop"} 

1076 

1077 
\item @{text any} if @{text "\<tau>\<^sub>i = \<alpha>"} for type variables 

1078 

1079 
\item @{text "\<kappa>"} if @{text "\<tau>\<^sub>i = \<kappa>"} for nonterminal @{text "\<kappa>"} 

1080 
(syntactic type constructor) 

1081 

1082 
\end{itemize} 

1083 

1084 
Each @{text "A\<^sub>i"} is decorated by priority @{text "p\<^sub>i"} from the 

1085 
given list @{text "ps"}; misssing priorities default to 0. 

1086 

1087 
The resulting nonterminal of the production is determined similarly 

1088 
from type @{text "\<tau>"}, with priority @{text "q"} and default 1000. 

1089 

1090 
\medskip Parsing via this production produces parse trees @{text 

1091 
"t\<^sub>1, \<dots>, t\<^sub>n"} for the argument slots. The resulting parse tree is 

1092 
composed as @{text "c t\<^sub>1 \<dots> t\<^sub>n"}, by using the syntax constant @{text 

1093 
"c"} of the syntax declaration. 

1094 

1095 
Such syntactic constants are invented on the spot, without formal 

1096 
check wrt.\ existing declarations. It is conventional to use plain 

1097 
identifiers prefixed by a single underscore (e.g.\ @{text 

1098 
"_foobar"}). Names should be chosen with care, to avoid clashes 

1099 
with unrelated syntax declarations. 

1100 

1101 
\medskip The special case of copy production is specified by @{text 

1102 
"c = "}@{verbatim "\"\""} (empty string). It means that the 

1103 
resulting parse tree @{text "t"} is copied directly, without any 

1104 
further decoration. 

46282  1105 

28762  1106 
\item @{command "no_syntax"}~@{text "(mode) decls"} removes grammar 
1107 
declarations (and translations) resulting from @{text decls}, which 

1108 
are interpreted in the same manner as for @{command "syntax"} above. 

46282  1109 

28762  1110 
\item @{command "translations"}~@{text rules} specifies syntactic 
1111 
translation rules (i.e.\ macros): parse~/ print rules (@{text "\<rightleftharpoons>"}), 

1112 
parse rules (@{text "\<rightharpoonup>"}), or print rules (@{text "\<leftharpoondown>"}). 

1113 
Translation patterns may be prefixed by the syntactic category to be 

1114 
used for parsing; the default is @{text logic}. 

46282  1115 

28762  1116 
\item @{command "no_translations"}~@{text rules} removes syntactic 
1117 
translation rules, which are interpreted in the same manner as for 

1118 
@{command "translations"} above. 

1119 

1120 
\end{description} 

1121 
*} 

1122 

1123 

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1124 
section {* Syntax translation functions \label{sec:trfuns} *} 
28762  1125 

1126 
text {* 

1127 
\begin{matharray}{rcl} 

1128 
@{command_def "parse_ast_translation"} & : & @{text "theory \<rightarrow> theory"} \\ 

1129 
@{command_def "parse_translation"} & : & @{text "theory \<rightarrow> theory"} \\ 

1130 
@{command_def "print_translation"} & : & @{text "theory \<rightarrow> theory"} \\ 

1131 
@{command_def "typed_print_translation"} & : & @{text "theory \<rightarrow> theory"} \\ 

1132 
@{command_def "print_ast_translation"} & : & @{text "theory \<rightarrow> theory"} \\ 

1133 
\end{matharray} 

1134 

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1135 
@{rail " 
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1136 
( @@{command parse_ast_translation}  @@{command parse_translation}  
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1137 
@@{command print_translation}  @@{command typed_print_translation}  
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1138 
@@{command print_ast_translation}) ('(' @'advanced' ')')? @{syntax text} 
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1139 
"} 
28762  1140 

1141 
Syntax translation functions written in ML admit almost arbitrary 

1142 
manipulations of Isabelle's inner syntax. Any of the above commands 

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1143 
have a single @{syntax text} argument that refers to an ML 
28762  1144 
expression of appropriate type, which are as follows by default: 
1145 

1146 
%FIXME proper antiquotations 

1147 
\begin{ttbox} 

1148 
val parse_ast_translation : (string * (ast list > ast)) list 

1149 
val parse_translation : (string * (term list > term)) list 

1150 
val print_translation : (string * (term list > term)) list 

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1151 
val typed_print_translation : (string * (typ > term list > term)) list 
28762  1152 
val print_ast_translation : (string * (ast list > ast)) list 
1153 
\end{ttbox} 

1154 

1155 
If the @{text "(advanced)"} option is given, the corresponding 

1156 
translation functions may depend on the current theory or proof 

1157 
context. This allows to implement advanced syntax mechanisms, as 

1158 
translations functions may refer to specific theory declarations or 

1159 
auxiliary proof data. 

1160 

30397  1161 
See also \cite{isabelleref} for more information on the general 
1162 
concept of syntax transformations in Isabelle. 

28762  1163 

1164 
%FIXME proper antiquotations 

1165 
\begin{ttbox} 

1166 
val parse_ast_translation: 

1167 
(string * (Proof.context > ast list > ast)) list 

1168 
val parse_translation: 

1169 
(string * (Proof.context > term list > term)) list 

1170 
val print_translation: 

1171 
(string * (Proof.context > term list > term)) list 

1172 
val typed_print_translation: 

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typed_print_translation: discontinued show_sorts argument;
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1173 
(string * (Proof.context > typ > term list > term)) list 
28762  1174 
val print_ast_translation: 
1175 
(string * (Proof.context > ast list > ast)) list 

1176 
\end{ttbox} 

1177 
*} 

1178 

1179 
end 