
haftmann@37613: theory Inductive_Predicate
haftmann@37613: imports Setup
haftmann@37613: begin
haftmann@37613: 
haftmann@37613: (*<*)
haftmann@38441: hide_const %invisible append
haftmann@37613: 
haftmann@38441: inductive %invisible append where
haftmann@37613:   "append [] ys ys"
haftmann@38811: | "append xs ys zs \<Longrightarrow> append (x # xs) ys (x # zs)"
haftmann@38441: 
haftmann@38441: lemma %invisible append: "append xs ys zs = (xs @ ys = zs)"
haftmann@38441:   by (induct xs arbitrary: ys zs) (auto elim: append.cases intro: append.intros)
haftmann@46515: 
haftmann@46515: lemmas lexordp_def = 
haftmann@46515:   lexordp_def [unfolded lexord_def mem_Collect_eq split]
haftmann@37613: (*>*)
haftmann@38441: 
haftmann@59377: section \<open>Inductive Predicates \label{sec:inductive}\<close>
haftmann@38441: 
haftmann@59377: text \<open>
haftmann@38508:   The @{text "predicate compiler"} is an extension of the code generator
haftmann@38441:   which turns inductive specifications into equational ones, from
haftmann@38441:   which in turn executable code can be generated.  The mechanisms of
haftmann@38441:   this compiler are described in detail in
wenzelm@58620:   @{cite "Berghofer-Bulwahn-Haftmann:2009:TPHOL"}.
haftmann@38441: 
haftmann@38441:   Consider the simple predicate @{const append} given by these two
haftmann@38441:   introduction rules:
haftmann@59377: \<close>
haftmann@38441: 
haftmann@59377: text %quote \<open>
haftmann@38441:   @{thm append.intros(1)[of ys]} \\
haftmann@38441:   @{thm append.intros(2)[of xs ys zs x]}
haftmann@59377: \<close>
haftmann@38441: 
haftmann@59377: text \<open>
haftmann@38441:   \noindent To invoke the compiler, simply use @{command_def "code_pred"}:
haftmann@59377: \<close>
haftmann@37613: 
haftmann@38441: code_pred %quote append .
haftmann@38441: 
haftmann@59377: text \<open>
haftmann@38441:   \noindent The @{command "code_pred"} command takes the name of the
haftmann@38441:   inductive predicate and then you put a period to discharge a trivial
haftmann@38441:   correctness proof.  The compiler infers possible modes for the
haftmann@38441:   predicate and produces the derived code equations.  Modes annotate
haftmann@38441:   which (parts of the) arguments are to be taken as input, and which
haftmann@38441:   output. Modes are similar to types, but use the notation @{text "i"}
haftmann@38441:   for input and @{text "o"} for output.
haftmann@38441:  
haftmann@38441:   For @{term "append"}, the compiler can infer the following modes:
haftmann@38441:   \begin{itemize}
haftmann@38441:     \item @{text "i \<Rightarrow> i \<Rightarrow> i \<Rightarrow> bool"}
haftmann@38441:     \item @{text "i \<Rightarrow> i \<Rightarrow> o \<Rightarrow> bool"}
haftmann@38441:     \item @{text "o \<Rightarrow> o \<Rightarrow> i \<Rightarrow> bool"}
haftmann@38441:   \end{itemize}
haftmann@38441:   You can compute sets of predicates using @{command_def "values"}:
haftmann@59377: \<close>
haftmann@38441: 
haftmann@38441: values %quote "{zs. append [(1::nat),2,3] [4,5] zs}"
haftmann@38441: 
haftmann@59377: text \<open>\noindent outputs @{text "{[1, 2, 3, 4, 5]}"}, and\<close>
haftmann@38441: 
haftmann@38441: values %quote "{(xs, ys). append xs ys [(2::nat),3]}"
haftmann@38441: 
haftmann@59377: text \<open>\noindent outputs @{text "{([], [2, 3]), ([2], [3]), ([2, 3], [])}"}.\<close>
haftmann@37613: 
haftmann@59377: text \<open>
haftmann@38441:   \noindent If you are only interested in the first elements of the
haftmann@38441:   set comprehension (with respect to a depth-first search on the
haftmann@38441:   introduction rules), you can pass an argument to @{command "values"}
haftmann@38441:   to specify the number of elements you want:
haftmann@59377: \<close>
haftmann@37613: 
haftmann@38441: values %quote 1 "{(xs, ys). append xs ys [(1::nat), 2, 3, 4]}"
haftmann@38441: values %quote 3 "{(xs, ys). append xs ys [(1::nat), 2, 3, 4]}"
haftmann@38441: 
haftmann@59377: text \<open>
haftmann@38441:   \noindent The @{command "values"} command can only compute set
haftmann@38441:   comprehensions for which a mode has been inferred.
haftmann@38441: 
haftmann@38441:   The code equations for a predicate are made available as theorems with
haftmann@38441:   the suffix @{text "equation"}, and can be inspected with:
haftmann@59377: \<close>
haftmann@38441: 
haftmann@38441: thm %quote append.equation
haftmann@38441: 
haftmann@59377: text \<open>
haftmann@38441:   \noindent More advanced options are described in the following subsections.
haftmann@59377: \<close>
haftmann@38441: 
haftmann@59377: subsection \<open>Alternative names for functions\<close>
haftmann@38441: 
haftmann@59377: text \<open>
haftmann@38441:   By default, the functions generated from a predicate are named after
haftmann@38441:   the predicate with the mode mangled into the name (e.g., @{text
haftmann@38441:   "append_i_i_o"}).  You can specify your own names as follows:
haftmann@59377: \<close>
haftmann@38441: 
haftmann@38811: code_pred %quote (modes: i \<Rightarrow> i \<Rightarrow> o \<Rightarrow> bool as concat,
haftmann@38811:   o \<Rightarrow> o \<Rightarrow> i \<Rightarrow> bool as split,
haftmann@38811:   i \<Rightarrow> o \<Rightarrow> i \<Rightarrow> bool as suffix) append .
haftmann@37613: 
haftmann@59377: subsection \<open>Alternative introduction rules\<close>
haftmann@38441: 
haftmann@59377: text \<open>
haftmann@38441:   Sometimes the introduction rules of an predicate are not executable
haftmann@38441:   because they contain non-executable constants or specific modes
haftmann@38441:   could not be inferred.  It is also possible that the introduction
haftmann@38441:   rules yield a function that loops forever due to the execution in a
haftmann@38441:   depth-first search manner.  Therefore, you can declare alternative
haftmann@38441:   introduction rules for predicates with the attribute @{attribute
haftmann@38441:   "code_pred_intro"}.  For example, the transitive closure is defined
haftmann@38441:   by:
haftmann@59377: \<close>
haftmann@38441: 
haftmann@59377: text %quote \<open>
haftmann@39682:   @{lemma [source] "r a b \<Longrightarrow> tranclp r a b" by (fact tranclp.intros(1))}\\
haftmann@39682:   @{lemma [source] "tranclp r a b \<Longrightarrow> r b c \<Longrightarrow> tranclp r a c" by (fact tranclp.intros(2))}
haftmann@59377: \<close>
haftmann@38441: 
haftmann@59377: text \<open>
haftmann@38441:   \noindent These rules do not suit well for executing the transitive
haftmann@38441:   closure with the mode @{text "(i \<Rightarrow> o \<Rightarrow> bool) \<Rightarrow> i \<Rightarrow> o \<Rightarrow> bool"}, as
haftmann@38441:   the second rule will cause an infinite loop in the recursive call.
haftmann@38441:   This can be avoided using the following alternative rules which are
haftmann@38441:   declared to the predicate compiler by the attribute @{attribute
haftmann@38441:   "code_pred_intro"}:
haftmann@59377: \<close>
haftmann@38441: 
haftmann@37613: lemma %quote [code_pred_intro]:
haftmann@39682:   "r a b \<Longrightarrow> tranclp r a b"
haftmann@39682:   "r a b \<Longrightarrow> tranclp r b c \<Longrightarrow> tranclp r a c"
haftmann@37613: by auto
haftmann@38441: 
haftmann@59377: text \<open>
haftmann@38441:   \noindent After declaring all alternative rules for the transitive
haftmann@38441:   closure, you invoke @{command "code_pred"} as usual.  As you have
haftmann@38441:   declared alternative rules for the predicate, you are urged to prove
haftmann@38441:   that these introduction rules are complete, i.e., that you can
haftmann@38441:   derive an elimination rule for the alternative rules:
haftmann@59377: \<close>
haftmann@38441: 
haftmann@37613: code_pred %quote tranclp
haftmann@37613: proof -
haftmann@37613:   case tranclp
haftmann@39682:   from this converse_tranclpE [OF tranclp.prems] show thesis by metis
haftmann@37613: qed
haftmann@38441: 
haftmann@59377: text \<open>
haftmann@38441:   \noindent Alternative rules can also be used for constants that have
haftmann@38441:   not been defined inductively. For example, the lexicographic order
haftmann@38441:   which is defined as:
haftmann@59377: \<close>
haftmann@38441: 
haftmann@59377: text %quote \<open>
haftmann@46515:   @{thm [display] lexordp_def [of r]}
haftmann@59377: \<close>
haftmann@38441: 
haftmann@59377: text \<open>
haftmann@38441:   \noindent To make it executable, you can derive the following two
haftmann@38441:   rules and prove the elimination rule:
haftmann@59377: \<close>
haftmann@38441: 
haftmann@37613: lemma %quote [code_pred_intro]:
haftmann@46515:   "append xs (a # v) ys \<Longrightarrow> lexordp r xs ys"
haftmann@46515: (*<*)unfolding lexordp_def by (auto simp add: append)(*>*)
haftmann@37613: 
haftmann@37613: lemma %quote [code_pred_intro]:
haftmann@46515:   "append u (a # v) xs \<Longrightarrow> append u (b # w) ys \<Longrightarrow> r a b
haftmann@46515:   \<Longrightarrow> lexordp r xs ys"
haftmann@46515: (*<*)unfolding lexordp_def append apply simp
haftmann@38441: apply (rule disjI2) by auto(*>*)
haftmann@37613: 
haftmann@46515: code_pred %quote lexordp
haftmann@38441: (*<*)proof -
haftmann@37613:   fix r xs ys
haftmann@46515:   assume lexord: "lexordp r xs ys"
haftmann@46515:   assume 1: "\<And>r' xs' ys' a v. r = r' \<Longrightarrow> xs = xs' \<Longrightarrow> ys = ys'
haftmann@46515:     \<Longrightarrow> append xs' (a # v) ys' \<Longrightarrow> thesis"
haftmann@46515:   assume 2: "\<And>r' xs' ys' u a v b w. r = r' \<Longrightarrow> xs = xs' \<Longrightarrow> ys = ys'
haftmann@46515:     \<Longrightarrow> append u (a # v) xs' \<Longrightarrow> append u (b # w) ys' \<Longrightarrow> r' a b \<Longrightarrow> thesis"
haftmann@37613:   {
haftmann@37613:     assume "\<exists>a v. ys = xs @ a # v"
haftmann@37613:     from this 1 have thesis
haftmann@46515:         by (fastforce simp add: append)
haftmann@37613:   } moreover
haftmann@37613:   {
haftmann@46515:     assume "\<exists>u a b v w. r a b \<and> xs = u @ a # v \<and> ys = u @ b # w"
haftmann@46515:     from this 2 have thesis by (fastforce simp add: append)
haftmann@37613:   } moreover
haftmann@37613:   note lexord
haftmann@37613:   ultimately show thesis
haftmann@46515:     unfolding lexordp_def
haftmann@46515:     by fastforce
haftmann@38441: qed(*>*)
haftmann@38441: 
haftmann@38441: 
haftmann@59377: subsection \<open>Options for values\<close>
haftmann@38441: 
haftmann@59377: text \<open>
haftmann@38441:   In the presence of higher-order predicates, multiple modes for some
haftmann@38441:   predicate could be inferred that are not disambiguated by the
haftmann@38441:   pattern of the set comprehension.  To disambiguate the modes for the
haftmann@38441:   arguments of a predicate, you can state the modes explicitly in the
haftmann@38441:   @{command "values"} command.  Consider the simple predicate @{term
haftmann@38441:   "succ"}:
haftmann@59377: \<close>
haftmann@38441: 
haftmann@38441: inductive %quote succ :: "nat \<Rightarrow> nat \<Rightarrow> bool" where
haftmann@37613:   "succ 0 (Suc 0)"
haftmann@37613: | "succ x y \<Longrightarrow> succ (Suc x) (Suc y)"
haftmann@37613: 
haftmann@38441: code_pred %quote succ .
haftmann@37613: 
haftmann@59377: text \<open>
haftmann@38441:   \noindent For this, the predicate compiler can infer modes @{text "o
haftmann@38441:   \<Rightarrow> o \<Rightarrow> bool"}, @{text "i \<Rightarrow> o \<Rightarrow> bool"}, @{text "o \<Rightarrow> i \<Rightarrow> bool"} and
haftmann@38441:   @{text "i \<Rightarrow> i \<Rightarrow> bool"}.  The invocation of @{command "values"}
haftmann@38441:   @{text "{n. tranclp succ 10 n}"} loops, as multiple modes for the
haftmann@38441:   predicate @{text "succ"} are possible and here the first mode @{text
haftmann@38441:   "o \<Rightarrow> o \<Rightarrow> bool"} is chosen. To choose another mode for the argument,
haftmann@38441:   you can declare the mode for the argument between the @{command
haftmann@38441:   "values"} and the number of elements.
haftmann@59377: \<close>
haftmann@38441: 
wenzelm@61498: values %quote [mode: i \<Rightarrow> o \<Rightarrow> bool] 1 "{n. tranclp succ 10 n}" (*FIXME does not terminate for n\<ge>1*)
haftmann@39065: values %quote [mode: o \<Rightarrow> i \<Rightarrow> bool] 1 "{n. tranclp succ n 10}"
haftmann@37613: 
haftmann@38441: 
haftmann@59377: subsection \<open>Embedding into functional code within Isabelle/HOL\<close>
haftmann@38441: 
haftmann@59377: text \<open>
haftmann@38441:   To embed the computation of an inductive predicate into functions
haftmann@38441:   that are defined in Isabelle/HOL, you have a number of options:
haftmann@38441: 
haftmann@38441:   \begin{itemize}
haftmann@38441: 
wenzelm@61498:   \item You want to use the first-order predicate with the mode
wenzelm@61498:     where all arguments are input. Then you can use the predicate directly, e.g.
haftmann@38441: 
wenzelm@61498:     @{text [display]
wenzelm@61498: \<open>valid_suffix ys zs =
wenzelm@61498:   (if append [Suc 0, 2] ys zs then Some ys else None)\<close>}
haftmann@38441: 
wenzelm@61498:   \item If you know that the execution returns only one value (it is
wenzelm@61498:     deterministic), then you can use the combinator @{term
wenzelm@61498:     "Predicate.the"}, e.g., a functional concatenation of lists is
wenzelm@61498:     defined with
haftmann@38441: 
wenzelm@61498:     @{term [display] "functional_concat xs ys = Predicate.the (append_i_i_o xs ys)"}
haftmann@38441: 
wenzelm@61498:     Note that if the evaluation does not return a unique value, it
wenzelm@61498:     raises a run-time error @{term "not_unique"}.
haftmann@38441: 
haftmann@38441:   \end{itemize}
haftmann@59377: \<close>
haftmann@38441: 
haftmann@38441: 
haftmann@59377: subsection \<open>Further Examples\<close>
haftmann@38441: 
haftmann@59377: text \<open>
haftmann@38441:   Further examples for compiling inductive predicates can be found in
wenzelm@63680:   \<^file>\<open>~~/src/HOL/Predicate_Compile_Examples/Examples.thy\<close>.  There are
haftmann@38441:   also some examples in the Archive of Formal Proofs, notably in the
haftmann@38441:   @{text "POPLmark-deBruijn"} and the @{text "FeatherweightJava"}
haftmann@38441:   sessions.
haftmann@59377: \<close>
haftmann@38441: 
haftmann@37613: end
haftmann@46515: 




