src/Doc/Codegen/Inductive_Predicate.thy

 
 lemmas lexordp_def = 
   lexordp_def [unfolded lexord_def mem_Collect_eq split]
 (*>*)
 
-section {* Inductive Predicates \label{sec:inductive} *}
-
-text {*
+section \<open>Inductive Predicates \label{sec:inductive}\<close>
+
+text \<open>
   The @{text "predicate compiler"} is an extension of the code generator
   which turns inductive specifications into equational ones, from
   which in turn executable code can be generated.  The mechanisms of
   this compiler are described in detail in
   @{cite "Berghofer-Bulwahn-Haftmann:2009:TPHOL"}.
 
   Consider the simple predicate @{const append} given by these two
   introduction rules:
-*}
-
-text %quote {*
+\<close>
+
+text %quote \<open>
   @{thm append.intros(1)[of ys]} \\
   @{thm append.intros(2)[of xs ys zs x]}
-*}
-
-text {*
+\<close>
+
+text \<open>
   \noindent To invoke the compiler, simply use @{command_def "code_pred"}:
-*}
+\<close>
 
 code_pred %quote append .
 
-text {*
+text \<open>
   \noindent The @{command "code_pred"} command takes the name of the
   inductive predicate and then you put a period to discharge a trivial
   correctness proof.  The compiler infers possible modes for the
   predicate and produces the derived code equations.  Modes annotate
   which (parts of the) arguments are to be taken as input, and which

     \item @{text "i \<Rightarrow> i \<Rightarrow> i \<Rightarrow> bool"}
     \item @{text "i \<Rightarrow> i \<Rightarrow> o \<Rightarrow> bool"}
     \item @{text "o \<Rightarrow> o \<Rightarrow> i \<Rightarrow> bool"}
   \end{itemize}
   You can compute sets of predicates using @{command_def "values"}:
-*}
+\<close>
 
 values %quote "{zs. append [(1::nat),2,3] [4,5] zs}"
 
-text {* \noindent outputs @{text "{[1, 2, 3, 4, 5]}"}, and *}
+text \<open>\noindent outputs @{text "{[1, 2, 3, 4, 5]}"}, and\<close>
 
 values %quote "{(xs, ys). append xs ys [(2::nat),3]}"
 
-text {* \noindent outputs @{text "{([], [2, 3]), ([2], [3]), ([2, 3], [])}"}. *}
-
-text {*
+text \<open>\noindent outputs @{text "{([], [2, 3]), ([2], [3]), ([2, 3], [])}"}.\<close>
+
+text \<open>
   \noindent If you are only interested in the first elements of the
   set comprehension (with respect to a depth-first search on the
   introduction rules), you can pass an argument to @{command "values"}
   to specify the number of elements you want:
-*}
+\<close>
 
 values %quote 1 "{(xs, ys). append xs ys [(1::nat), 2, 3, 4]}"
 values %quote 3 "{(xs, ys). append xs ys [(1::nat), 2, 3, 4]}"
 
-text {*
+text \<open>
   \noindent The @{command "values"} command can only compute set
   comprehensions for which a mode has been inferred.
 
   The code equations for a predicate are made available as theorems with
   the suffix @{text "equation"}, and can be inspected with:
-*}
+\<close>
 
 thm %quote append.equation
 
-text {*
+text \<open>
   \noindent More advanced options are described in the following subsections.
-*}
-
-subsection {* Alternative names for functions *}
-
-text {* 
+\<close>
+
+subsection \<open>Alternative names for functions\<close>
+
+text \<open>
   By default, the functions generated from a predicate are named after
   the predicate with the mode mangled into the name (e.g., @{text
   "append_i_i_o"}).  You can specify your own names as follows:
-*}
+\<close>
 
 code_pred %quote (modes: i \<Rightarrow> i \<Rightarrow> o \<Rightarrow> bool as concat,
   o \<Rightarrow> o \<Rightarrow> i \<Rightarrow> bool as split,
   i \<Rightarrow> o \<Rightarrow> i \<Rightarrow> bool as suffix) append .
 
-subsection {* Alternative introduction rules *}
-
-text {*
+subsection \<open>Alternative introduction rules\<close>
+
+text \<open>
   Sometimes the introduction rules of an predicate are not executable
   because they contain non-executable constants or specific modes
   could not be inferred.  It is also possible that the introduction
   rules yield a function that loops forever due to the execution in a
   depth-first search manner.  Therefore, you can declare alternative
   introduction rules for predicates with the attribute @{attribute
   "code_pred_intro"}.  For example, the transitive closure is defined
   by:
-*}
-
-text %quote {*
+\<close>
+
+text %quote \<open>
   @{lemma [source] "r a b \<Longrightarrow> tranclp r a b" by (fact tranclp.intros(1))}\\
   @{lemma [source] "tranclp r a b \<Longrightarrow> r b c \<Longrightarrow> tranclp r a c" by (fact tranclp.intros(2))}
-*}
-
-text {*
+\<close>
+
+text \<open>
   \noindent These rules do not suit well for executing the transitive
   closure with the mode @{text "(i \<Rightarrow> o \<Rightarrow> bool) \<Rightarrow> i \<Rightarrow> o \<Rightarrow> bool"}, as
   the second rule will cause an infinite loop in the recursive call.
   This can be avoided using the following alternative rules which are
   declared to the predicate compiler by the attribute @{attribute
   "code_pred_intro"}:
-*}
+\<close>
 
 lemma %quote [code_pred_intro]:
   "r a b \<Longrightarrow> tranclp r a b"
   "r a b \<Longrightarrow> tranclp r b c \<Longrightarrow> tranclp r a c"
 by auto
 
-text {*
+text \<open>
   \noindent After declaring all alternative rules for the transitive
   closure, you invoke @{command "code_pred"} as usual.  As you have
   declared alternative rules for the predicate, you are urged to prove
   that these introduction rules are complete, i.e., that you can
   derive an elimination rule for the alternative rules:
-*}
+\<close>
 
 code_pred %quote tranclp
 proof -
   case tranclp
   from this converse_tranclpE [OF tranclp.prems] show thesis by metis
 qed
 
-text {*
+text \<open>
   \noindent Alternative rules can also be used for constants that have
   not been defined inductively. For example, the lexicographic order
   which is defined as:
-*}
-
-text %quote {*
+\<close>
+
+text %quote \<open>
   @{thm [display] lexordp_def [of r]}
-*}
-
-text {*
+\<close>
+
+text \<open>
   \noindent To make it executable, you can derive the following two
   rules and prove the elimination rule:
-*}
+\<close>
 
 lemma %quote [code_pred_intro]:
   "append xs (a # v) ys \<Longrightarrow> lexordp r xs ys"
 (*<*)unfolding lexordp_def by (auto simp add: append)(*>*)
 

     unfolding lexordp_def
     by fastforce
 qed(*>*)
 
 
-subsection {* Options for values *}
-
-text {*
+subsection \<open>Options for values\<close>
+
+text \<open>
   In the presence of higher-order predicates, multiple modes for some
   predicate could be inferred that are not disambiguated by the
   pattern of the set comprehension.  To disambiguate the modes for the
   arguments of a predicate, you can state the modes explicitly in the
   @{command "values"} command.  Consider the simple predicate @{term
   "succ"}:
-*}
+\<close>
 
 inductive %quote succ :: "nat \<Rightarrow> nat \<Rightarrow> bool" where
   "succ 0 (Suc 0)"
 | "succ x y \<Longrightarrow> succ (Suc x) (Suc y)"
 
 code_pred %quote succ .
 
-text {*
+text \<open>
   \noindent For this, the predicate compiler can infer modes @{text "o
   \<Rightarrow> o \<Rightarrow> bool"}, @{text "i \<Rightarrow> o \<Rightarrow> bool"}, @{text "o \<Rightarrow> i \<Rightarrow> bool"} and
   @{text "i \<Rightarrow> i \<Rightarrow> bool"}.  The invocation of @{command "values"}
   @{text "{n. tranclp succ 10 n}"} loops, as multiple modes for the
   predicate @{text "succ"} are possible and here the first mode @{text
   "o \<Rightarrow> o \<Rightarrow> bool"} is chosen. To choose another mode for the argument,
   you can declare the mode for the argument between the @{command
   "values"} and the number of elements.
-*}
+\<close>
 
 values %quote [mode: i \<Rightarrow> o \<Rightarrow> bool] 1 "{n. tranclp succ 10 n}" (*FIMXE does not terminate for n\<ge>1*)
 values %quote [mode: o \<Rightarrow> i \<Rightarrow> bool] 1 "{n. tranclp succ n 10}"
 
 
-subsection {* Embedding into functional code within Isabelle/HOL *}
-
-text {*
+subsection \<open>Embedding into functional code within Isabelle/HOL\<close>
+
+text \<open>
   To embed the computation of an inductive predicate into functions
   that are defined in Isabelle/HOL, you have a number of options:
 
   \begin{itemize}
 

 
       Note that if the evaluation does not return a unique value, it
       raises a run-time error @{term "not_unique"}.
 
   \end{itemize}
-*}
-
-
-subsection {* Further Examples *}
-
-text {*
+\<close>
+
+
+subsection \<open>Further Examples\<close>
+
+text \<open>
   Further examples for compiling inductive predicates can be found in
   @{file "~~/src/HOL/Predicate_Compile_Examples/Examples.thy"}.  There are
   also some examples in the Archive of Formal Proofs, notably in the
   @{text "POPLmark-deBruijn"} and the @{text "FeatherweightJava"}
   sessions.
-*}
+\<close>
 
 end