--- a/doc-src/Codegen/Thy/Program.thy Tue Jun 29 11:25:03 2010 +0200
+++ b/doc-src/Codegen/Thy/Program.thy Tue Jun 29 11:25:04 2010 +0200
@@ -174,7 +174,7 @@
*}
lemma %quote [code_unfold]:
- "x \<in> set xs \<longleftrightarrow> member xs x" by (fact in_set_code)
+ "x \<in> set xs \<longleftrightarrow> List.member xs x" by (fact in_set_member)
text {*
\item eliminating superfluous constants:
@@ -188,7 +188,7 @@
*}
lemma %quote [code_unfold]:
- "xs = [] \<longleftrightarrow> List.null xs" by (fact empty_null)
+ "xs = [] \<longleftrightarrow> List.null xs" by (fact eq_Nil_null)
text_raw {*
\end{itemize}
@@ -339,7 +339,7 @@
else collect_duplicates (z#xs) (z#ys) zs)"
text {*
- \noindent The membership test during preprocessing is rewritten,
+ \noindent During preprocessing, the membership test is rewritten,
resulting in @{const List.member}, which itself
performs an explicit equality check.
*}
@@ -429,219 +429,4 @@
likely to be used in such situations.
*}
-subsection {* Inductive Predicates *}
-(*<*)
-hide_const append
-
-inductive append
-where
- "append [] ys ys"
-| "append xs ys zs ==> append (x # xs) ys (x # zs)"
-(*>*)
-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 .
-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.
-*}
-subsubsection {* 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 => i => o => bool as concat,
- o => o => i => bool as split,
- i => o => i => bool as suffix) append .
-
-subsubsection {* 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]}\\
-\noindent @{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 append: "append xs ys zs = (xs @ ys = zs)"
-by (induct xs arbitrary: ys zs) (auto elim: append.cases intro: append.intros)
-(*>*)
-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
-(*>*)
-subsubsection {* 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 succ :: "nat \<Rightarrow> nat \<Rightarrow> bool"
-where
- "succ 0 (Suc 0)"
-| "succ x y \<Longrightarrow> succ (Suc x) (Suc y)"
-
-code_pred 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 => 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 *}
-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}
-*}
-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.
-*}
end