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doc-src/Codegen/Thy/Inductive_Predicate.thy

author | haftmann |

Thu, 02 Sep 2010 16:41:42 +0200 | |

changeset 39065 | 16a2ed09217a |

parent 38811 | c3511b112595 |

child 39682 | 066e2d4d0de8 |

permissions | -rw-r--r-- |

set depth to 1

theory Inductive_Predicate imports Setup begin (*<*) hide_const %invisible append inductive %invisible append where "append [] ys ys" | "append xs ys zs \<Longrightarrow> 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) (*>*) section {* Inductive Predicates \label{sec:inductive} *} text {* 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 {* @{thm append.intros(1)[of ys]} \\ @{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 . 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 \<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"}: *} 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 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: *} 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 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: *} 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: *} 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 {* 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]} \\ @{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 \<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"}: *} lemma %quote [code_pred_intro]: "r a b \<Longrightarrow> r\<^sup>+\<^sup>+ a b" "r a b \<Longrightarrow> r\<^sup>+\<^sup>+ b c \<Longrightarrow> 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: *} code_pred %quote tranclp proof - case tranclp 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"]} *} 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 \<Longrightarrow> lexord r (xs, ys)" (*<*)unfolding lexord_def Collect_def by (auto simp add: append)(*>*) lemma %quote [code_pred_intro]: "append u (a # v) xs \<Longrightarrow> append u (b # w) ys \<Longrightarrow> r (a, b) \<Longrightarrow> lexord r (xs, ys)" (*<*)unfolding lexord_def Collect_def append mem_def apply simp apply (rule disjI2) by auto(*>*) code_pred %quote lexord (*<*)proof - fix r xs ys assume lexord: "lexord r (xs, ys)" assume 1: "\<And> r' xs' ys' a v. r = r' \<Longrightarrow> (xs, ys) = (xs', ys') \<Longrightarrow> append xs' (a # v) ys' \<Longrightarrow> thesis" assume 2: "\<And> r' xs' ys' u a v b w. r = r' \<Longrightarrow> (xs, ys) = (xs', ys') \<Longrightarrow> append u (a # v) xs' \<Longrightarrow> append u (b # w) ys' \<Longrightarrow> r' (a, b) \<Longrightarrow> thesis" { assume "\<exists>a v. ys = xs @ a # v" from this 1 have thesis by (fastsimp simp add: append) } moreover { assume "\<exists>u a b v w. (a, b) \<in> r \<and> xs = u @ a # v \<and> ys = u @ b # w" from this 2 have thesis by (fastsimp simp add: append mem_def) } moreover note lexord ultimately show thesis unfolding lexord_def by (fastsimp simp add: Collect_def) 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"}: *} 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 {* \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. *} 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 {* 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} *} 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