diff -r 6c0d02f416ba -r 2fffd5ac487f doc-src/Codegen/Thy/Inductive_Predicate.thy --- a/doc-src/Codegen/Thy/Inductive_Predicate.thy Mon Aug 16 10:54:08 2010 +0200 +++ b/doc-src/Codegen/Thy/Inductive_Predicate.thy Mon Aug 16 11:18:28 2010 +0200 @@ -2,135 +2,164 @@ imports Setup begin -section {* Inductive Predicates \label{sec:inductive} *} (*<*) -hide_const append +hide_const %invisible append -inductive append -where +inductive %invisible append where "append [] ys ys" | "append xs ys zs ==> append (x # xs) ys (x # zs)" + +lemma %invisible append: "append xs ys zs = (xs @ ys = zs)" + by (induct xs arbitrary: ys zs) (auto elim: append.cases intro: append.intros) (*>*) -text {* -To execute inductive predicates, a special preprocessor, the predicate - compiler, generates code equations from the introduction rules of the predicates. -The mechanisms of this compiler are described in \cite{Berghofer-Bulwahn-Haftmann:2009:TPHOL}. -Consider the simple predicate @{const append} given by these two -introduction rules: -*} -text %quote {* -@{thm append.intros(1)[of ys]}\\ -\noindent@{thm append.intros(2)[of xs ys zs x]} -*} -text {* -\noindent To invoke the compiler, simply use @{command_def "code_pred"}: -*} -code_pred %quote append . + +section {* Inductive Predicates \label{sec:inductive} *} + text {* -\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 output. Modes are similar to types, but use the notation @{text "i"} -for input and @{text "o"} for output. - -For @{term "append"}, the compiler can infer the following modes: -\begin{itemize} -\item @{text "i \ i \ i \ bool"} -\item @{text "i \ i \ o \ bool"} -\item @{text "o \ o \ i \ bool"} -\end{itemize} -You can compute sets of predicates using @{command_def "values"}: + 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: *} -values %quote "{zs. append [(1::nat),2,3] [4,5] zs}" -text {* \noindent outputs @{text "{[1, 2, 3, 4, 5]}"}, and *} -values %quote "{(xs, ys). append xs ys [(2::nat),3]}" -text {* \noindent outputs @{text "{([], [2, 3]), ([2], [3]), ([2, 3], [])}"}. *} + +text %quote {* + @{thm append.intros(1)[of ys]} \\ + @{thm append.intros(2)[of xs ys zs x]} +*} + text {* -\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: + \noindent To invoke the compiler, simply use @{command_def "code_pred"}: *} -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]}" +code_pred %quote append . + +text {* + \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 + output. Modes are similar to types, but use the notation @{text "i"} + for input and @{text "o"} for output. + + For @{term "append"}, the compiler can infer the following modes: + \begin{itemize} + \item @{text "i \ i \ i \ bool"} + \item @{text "i \ i \ o \ bool"} + \item @{text "o \ o \ i \ bool"} + \end{itemize} + You can compute sets of predicates using @{command_def "values"}: +*} + +values %quote "{zs. append [(1::nat),2,3] [4,5] zs}" + +text {* \noindent outputs @{text "{[1, 2, 3, 4, 5]}"}, and *} + +values %quote "{(xs, ys). append xs ys [(2::nat),3]}" + +text {* \noindent outputs @{text "{([], [2, 3]), ([2], [3]), ([2, 3], [])}"}. *} text {* -\noindent The @{command "values"} command can only compute set - comprehensions for which a mode has been inferred. + \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: +*} -The code equations for a predicate are made available as theorems with - the suffix @{text "equation"}, and can be inspected with: -*} -thm %quote append.equation +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 {* -\noindent More advanced options are described in the following subsections. + \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: *} -subsubsection {* Alternative names for functions *} + +thm %quote append.equation + +text {* + \noindent More advanced options are described in the following subsections. +*} + +subsection {* Alternative names for functions *} + text {* -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: + 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: *} + code_pred %quote (modes: i => i => o => bool as concat, o => o => i => bool as split, i => o => i => bool as suffix) append . -subsubsection {* Alternative introduction rules *} +subsection {* Alternative introduction rules *} + text {* -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: + 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 {* -@{thm tranclp.intros(1)[of r a b]}\\ -\noindent @{thm tranclp.intros(2)[of r a b c]} + @{thm tranclp.intros(1)[of r a b]} \\ + @{thm tranclp.intros(2)[of r a b c]} *} + text {* -\noindent These rules do not suit well for executing the transitive closure -with the mode @{text "(i \ o \ bool) \ i \ o \ 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"}: + \noindent These rules do not suit well for executing the transitive + closure with the mode @{text "(i \ o \ bool) \ i \ o \ 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"}: *} + lemma %quote [code_pred_intro]: "r a b \ r\<^sup>+\<^sup>+ a b" "r a b \ r\<^sup>+\<^sup>+ b c \ r\<^sup>+\<^sup>+ a c" by auto + text {* -\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: + \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: *} + code_pred %quote tranclp proof - case tranclp - from this converse_tranclpE[OF this(1)] show thesis by metis + from this converse_tranclpE [OF this(1)] show thesis by metis qed + text {* -\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 {* -@{thm [display] lexord_def[of "r"]} + \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 {* -\noindent To make it executable, you can derive the following two rules and prove the -elimination rule: + +text %quote {* + @{thm [display] lexord_def[of "r"]} *} -(*<*) -lemma append: "append xs ys zs = (xs @ ys = zs)" -by (induct xs arbitrary: ys zs) (auto elim: append.cases intro: append.intros) -(*>*) + +text {* + \noindent To make it executable, you can derive the following two + rules and prove the elimination rule: +*} + lemma %quote [code_pred_intro]: "append xs (a # v) ys \ lexord r (xs, ys)" (*<*)unfolding lexord_def Collect_def by (auto simp add: append)(*>*) @@ -139,12 +168,10 @@ "append u (a # v) xs \ append u (b # w) ys \ r (a, b) \ lexord r (xs, ys)" (*<*)unfolding lexord_def Collect_def append mem_def apply simp -apply (rule disjI2) by auto -(*>*) +apply (rule disjI2) by auto(*>*) code_pred %quote lexord -(*<*) -proof - +(*<*)proof - fix r xs ys assume lexord: "lexord r (xs, ys)" assume 1: "\ r' xs' ys' a v. r = r' \ (xs, ys) = (xs', ys') \ append xs' (a # v) ys' \ thesis" @@ -162,59 +189,81 @@ ultimately show thesis unfolding lexord_def by (fastsimp simp add: Collect_def) -qed -(*>*) -subsubsection {* Options for values *} +qed(*>*) + + +subsection {* Options for values *} + text {* -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"}: + 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"}: *} -inductive succ :: "nat \ nat \ bool" -where + +inductive %quote succ :: "nat \ nat \ bool" where "succ 0 (Suc 0)" | "succ x y \ succ (Suc x) (Suc y)" -code_pred succ . +code_pred %quote succ . text {* -\noindent For this, the predicate compiler can infer modes @{text "o \ o \ bool"}, @{text "i \ o \ bool"}, - @{text "o \ i \ bool"} and @{text "i \ i \ 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 \ o \ 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. + \noindent For this, the predicate compiler can infer modes @{text "o + \ o \ bool"}, @{text "i \ o \ bool"}, @{text "o \ i \ bool"} and + @{text "i \ i \ 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 \ o \ 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. *} + values %quote [mode: i => o => bool] 20 "{n. tranclp succ 10 n}" values %quote [mode: o => i => bool] 10 "{n. tranclp succ n 10}" -subsubsection {* Embedding into functional code within Isabelle/HOL *} + +subsection {* Embedding into functional code within Isabelle/HOL *} + text {* -To embed the computation of an inductive predicate into functions that are defined in Isabelle/HOL, -you have a number of options: -\begin{itemize} -\item You want to use the first-order predicate with the mode - where all arguments are input. Then you can use the predicate directly, e.g. -\begin{quote} - @{text "valid_suffix ys zs = "}\\ - @{text "(if append [Suc 0, 2] ys zs then Some ys else None)"} -\end{quote} -\item If you know that the execution returns only one value (it is deterministic), then you can - use the combinator @{term "Predicate.the"}, e.g., a functional concatenation of lists - is defined with -\begin{quote} - @{term "functional_concat xs ys = Predicate.the (append_i_i_o xs ys)"} -\end{quote} - Note that if the evaluation does not return a unique value, it raises a run-time error - @{term "not_unique"}. -\end{itemize} + To embed the computation of an inductive predicate into functions + that are defined in Isabelle/HOL, you have a number of options: + + \begin{itemize} + + \item You want to use the first-order predicate with the mode + where all arguments are input. Then you can use the predicate directly, e.g. + + \begin{quote} + @{text "valid_suffix ys zs = "} \\ + @{text "(if append [Suc 0, 2] ys zs then Some ys else None)"} + \end{quote} + + \item If you know that the execution returns only one value (it is + deterministic), then you can use the combinator @{term + "Predicate.the"}, e.g., a functional concatenation of lists is + defined with + + \begin{quote} + @{term "functional_concat xs ys = Predicate.the (append_i_i_o xs ys)"} + \end{quote} + + Note that if the evaluation does not return a unique value, it + raises a run-time error @{term "not_unique"}. + + \end{itemize} *} -subsubsection {* Further Examples *} -text {* Further examples for compiling inductive predicates can be found in -the @{text "HOL/ex/Predicate_Compile_ex"} theory file. -There are also some examples in the Archive of Formal Proofs, notably -in the @{text "POPLmark-deBruijn"} and the @{text "FeatherweightJava"} sessions. + + +subsection {* Further Examples *} + +text {* + Further examples for compiling inductive predicates can be found in + the @{text "HOL/ex/Predicate_Compile_ex,thy"} theory file. There are + also some examples in the Archive of Formal Proofs, notably in the + @{text "POPLmark-deBruijn"} and the @{text "FeatherweightJava"} + sessions. *} + end