--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/ProgTutorial/Package/Ind_Code.thy Thu Mar 19 13:28:16 2009 +0100
@@ -0,0 +1,589 @@
+theory Ind_Code
+imports "../Base" "../FirstSteps" Simple_Inductive_Package Ind_Prelims
+begin
+
+section {* Code *}
+
+text {*
+ @{text [display] "rule ::= \<And>xs. As \<Longrightarrow> (\<And>ys. Bs \<Longrightarrow> pred ss)\<^isup>* \<Longrightarrow> pred ts"}
+
+ @{text [display] "orule ::= \<forall>xs. As \<longrightarrow> (\<forall>ys. Bs \<longrightarrow> pred ss)\<^isup>* \<longrightarrow> pred ts"}
+
+ @{text [display] "def ::= pred \<equiv> \<lambda>zs. \<forall>preds. orules \<longrightarrow> pred zs"}
+
+ @{text [display] "ind ::= \<And>zs. pred zs \<Longrightarrow> rules[preds::=Ps] \<Longrightarrow> P zs"}
+
+ @{text [display] "oind ::= \<forall>zs. pred zs \<longrightarrow> orules[preds::=Ps] \<longrightarrow> P zs"}
+
+ \underline{Induction proof}
+
+ After ``objectivication'' we have
+ @{text "pred zs"} and @{text "orules[preds::=Ps]"}; and have to show
+ @{text "P zs"}. Expanding @{text "pred zs"} gives @{text "\<forall>preds. orules \<longrightarrow> pred zs"}.
+ Instantiating the @{text "preds"} with @{text "Ps"} gives
+ @{text "orules[preds::=Ps] \<longrightarrow> P zs"}. So we can conclude with @{text "P zs"}.
+
+ \underline{Intro proof}
+
+ Assume we want to prove the $i$th intro rule.
+
+ We have to show @{text "\<forall>xs. As \<longrightarrow> (\<forall>ys. Bs \<longrightarrow> pred ss)\<^isup>* \<longrightarrow> pred ts"};
+ expanding the defs, gives
+
+ @{text [display]
+ "\<forall>xs. As \<longrightarrow> (\<forall>ys. Bs \<longrightarrow> (\<forall>preds. orules \<longrightarrow> pred ss))\<^isup>* \<longrightarrow> (\<forall>preds. orules \<longrightarrow> pred ts"}
+
+ applying as many allI and impI as possible
+
+ so we have @{text "As"}, @{text "(\<forall>ys. Bs \<longrightarrow> (\<forall>preds. orules \<longrightarrow> pred ss))\<^isup>*"},
+ @{text "orules"}; and have to show @{text "pred ts"}
+
+ the $i$th @{text "orule"} is of the
+ form @{text "\<forall>xs. As \<longrightarrow> (\<forall>ys. Bs \<longrightarrow> pred ss)\<^isup>* \<longrightarrow> pred ts"}.
+
+ using the @{text "As"} we ????
+*}
+
+
+text {*
+ First we have to produce for each predicate its definitions of the form
+
+ @{text [display] "pred \<equiv> \<lambda>zs. \<forall>preds. orules \<longrightarrow> pred zs"}
+
+ In order to make definitions, we use the following wrapper for
+ @{ML LocalTheory.define}. The wrapper takes a predicate name, a syntax
+ annotation and a term representing the right-hand side of the definition.
+*}
+
+ML %linenosgray{*fun make_defs ((predname, syn), trm) lthy =
+let
+ val arg = ((predname, syn), (Attrib.empty_binding, trm))
+ val ((_, (_ , thm)), lthy') = LocalTheory.define Thm.internalK arg lthy
+in
+ (thm, lthy')
+end*}
+
+text {*
+ It returns the definition (as a theorem) and the local theory in which this definition has
+ been made. In Line 4, @{ML internalK in Thm} is a flag attached to the
+ theorem (others possibilities are @{ML definitionK in Thm} and @{ML axiomK in Thm}).
+ These flags just classify theorems and have no significant meaning, except
+ for tools that, for example, find theorems in the theorem database. We also
+ use @{ML empty_binding in Attrib} in Line 3, since the definition does
+ not need to have any theorem attributes. A testcase for this function is
+*}
+
+local_setup %gray {* fn lthy =>
+let
+ val arg = ((@{binding "MyTrue"}, NoSyn), @{term True})
+ val (def, lthy') = make_defs arg lthy
+in
+ warning (str_of_thm lthy' def); lthy'
+end *}
+
+text {*
+ which makes the definition @{prop "MyTrue \<equiv> True"} and then prints it out.
+ Since we are testing the function inside \isacommand{local\_setup}, i.e.~make
+ changes to the ambient theory, we can query the definition using the usual
+ command \isacommand{thm}:
+
+ \begin{isabelle}
+ \isacommand{thm}~@{text "MyTrue_def"}\\
+ @{text "> MyTrue \<equiv> True"}
+ \end{isabelle}
+
+ The next two functions construct the terms we need for the definitions for
+ our \isacommand{simple\_inductive} command. These
+ terms are of the form
+
+ @{text [display] "\<lambda>\<^raw:$zs$>. \<forall>preds. orules \<longrightarrow> pred \<^raw:$zs$>"}
+
+ The variables @{text "\<^raw:$zs$>"} need to be chosen so that they do not occur
+ in the @{text orules} and also be distinct from the @{text "preds"}.
+
+ The first function constructs the term for one particular predicate, say
+ @{text "pred"}; the number of arguments of this predicate is
+ determined by the number of argument types of @{text "arg_tys"}.
+ So it takes these two parameters as arguments. The other arguments are
+ all the @{text "preds"} and the @{text "orules"}.
+*}
+
+ML %linenosgray{*fun defs_aux lthy orules preds (pred, arg_tys) =
+let
+ fun mk_all x P = HOLogic.all_const (fastype_of x) $ lambda x P
+
+ val fresh_args =
+ arg_tys
+ |> map (pair "z")
+ |> Variable.variant_frees lthy (preds @ orules)
+ |> map Free
+in
+ list_comb (pred, fresh_args)
+ |> fold_rev (curry HOLogic.mk_imp) orules
+ |> fold_rev mk_all preds
+ |> fold_rev lambda fresh_args
+end*}
+
+text {*
+ The function in Line 3 is just a helper function for constructing universal
+ quantifications. The code in Lines 5 to 9 produces the fresh @{text
+ "\<^raw:$zs$>"}. For this it pairs every argument type with the string
+ @{text [quotes] "z"} (Line 7); then generates variants for all these strings
+ so that they are unique w.r.t.~to the @{text "orules"} and the predicates;
+ in Line 9 it generates the corresponding variable terms for the unique
+ strings.
+
+ The unique free variables are applied to the predicate (Line 11) using the
+ function @{ML list_comb}; then the @{text orules} are prefixed (Line 12); in
+ Line 13 we quantify over all predicates; and in line 14 we just abstract
+ over all the @{text "\<^raw:$zs$>"}, i.e.~the fresh arguments of the
+ predicate.
+
+ A testcase for this function is
+*}
+
+local_setup %gray{* fn lthy =>
+let
+ val orules = [@{prop "even 0"},
+ @{prop "\<forall>n::nat. odd n \<longrightarrow> even (Suc n)"},
+ @{prop "\<forall>n::nat. even n \<longrightarrow> odd (Suc n)"}]
+ val preds = [@{term "even::nat\<Rightarrow>bool"}, @{term "odd::nat\<Rightarrow>bool"}, @{term "z::nat"}]
+ val pred = @{term "even::nat\<Rightarrow>bool"}
+ val arg_tys = [@{typ "nat"}]
+ val def = defs_aux lthy orules preds (pred, arg_tys)
+in
+ warning (Syntax.string_of_term lthy def); lthy
+end *}
+
+text {*
+ It constructs the left-hand side for the definition of @{text "even"}. So we obtain
+ as printout the term
+
+ @{text [display]
+"\<lambda>z. \<forall>even odd. (even 0) \<longrightarrow> (\<forall>n. odd n \<longrightarrow> even (Suc n))
+ \<longrightarrow> (\<forall>n. even n \<longrightarrow> odd (Suc n)) \<longrightarrow> even z"}
+
+ The main function for the definitions now has to just iterate the function
+ @{ML defs_aux} over all predicates. The argument @{text "preds"} is again
+ the the list of predicates as @{ML_type term}s; the argument @{text
+ "prednames"} is the list of names of the predicates; @{text "arg_tyss"} is
+ the list of argument-type-lists for each predicate.
+*}
+
+ML %linenosgray{*fun definitions rules preds prednames syns arg_typss lthy =
+let
+ val thy = ProofContext.theory_of lthy
+ val orules = map (ObjectLogic.atomize_term thy) rules
+ val defs = map (defs_aux lthy orules preds) (preds ~~ arg_typss)
+in
+ fold_map make_defs (prednames ~~ syns ~~ defs) lthy
+end*}
+
+text {*
+ The user will state the introduction rules using meta-implications and
+ meta-quanti\-fications. In Line 4, we transform these introduction rules into
+ the object logic (since definitions cannot be stated with
+ meta-connectives). To do this transformation we have to obtain the theory
+ behind the local theory (Line 3); with this theory we can use the function
+ @{ML ObjectLogic.atomize_term} to make the transformation (Line 4). The call
+ to @{ML defs_aux} in Line 5 produces all left-hand sides of the
+ definitions. The actual definitions are then made in Line 7. The result
+ of the function is a list of theorems and a local theory.
+
+
+ A testcase for this function is
+*}
+
+local_setup %gray {* fn lthy =>
+let
+ val rules = [@{prop "even 0"},
+ @{prop "\<And>n::nat. odd n \<Longrightarrow> even (Suc n)"},
+ @{prop "\<And>n::nat. even n \<Longrightarrow> odd (Suc n)"}]
+ val preds = [@{term "even::nat\<Rightarrow>bool"}, @{term "odd::nat\<Rightarrow>bool"}]
+ val prednames = [@{binding "even"}, @{binding "odd"}]
+ val syns = [NoSyn, NoSyn]
+ val arg_tyss = [[@{typ "nat"}], [@{typ "nat"}]]
+ val (defs, lthy') = definitions rules preds prednames syns arg_tyss lthy
+in
+ warning (str_of_thms lthy' defs); lthy'
+end *}
+
+text {*
+ where we feed into the functions all parameters corresponding to
+ the @{text even}-@{text odd} example. The definitions we obtain
+ are:
+
+ \begin{isabelle}
+ \isacommand{thm}~@{text "even_def odd_def"}\\
+ @{text [break]
+"> even \<equiv> \<lambda>z. \<forall>even odd. (even 0) \<longrightarrow> (\<forall>n. odd n \<longrightarrow> even (Suc n))
+> \<longrightarrow> (\<forall>n. even n \<longrightarrow> odd (Suc n)) \<longrightarrow> even z,
+> odd \<equiv> \<lambda>z. \<forall>even odd. (even 0) \<longrightarrow> (\<forall>n. odd n \<longrightarrow> even (Suc n))
+> \<longrightarrow> (\<forall>n. even n \<longrightarrow> odd (Suc n)) \<longrightarrow> odd z"}
+ \end{isabelle}
+
+
+ This completes the code for making the definitions. Next we deal with
+ the induction principles. Recall that the proof of the induction principle
+ for @{text "even"} was:
+*}
+
+lemma man_ind_principle:
+assumes prems: "even n"
+shows "P 0 \<Longrightarrow> (\<And>m. Q m \<Longrightarrow> P (Suc m)) \<Longrightarrow> (\<And>m. P m \<Longrightarrow> Q (Suc m)) \<Longrightarrow> P n"
+apply(atomize (full))
+apply(cut_tac prems)
+apply(unfold even_def)
+apply(drule spec[where x=P])
+apply(drule spec[where x=Q])
+apply(assumption)
+done
+
+text {*
+ The code for such induction principles has to accomplish two tasks:
+ constructing the induction principles from the given introduction
+ rules and then automatically generating a proof of them using a tactic.
+
+ The tactic will use the following helper function for instantiating universal
+ quantifiers.
+*}
+
+ML{*fun inst_spec ctrm =
+ Drule.instantiate' [SOME (ctyp_of_term ctrm)] [NONE, SOME ctrm] @{thm spec}*}
+
+text {*
+ This helper function instantiates the @{text "?x"} in the theorem
+ @{thm spec} with a given @{ML_type cterm}. Together with the tactic
+*}
+
+ML{*fun inst_spec_tac ctrms =
+ EVERY' (map (dtac o inst_spec) ctrms)*}
+
+text {*
+ we can use @{ML inst_spec} in the following proof to instantiate the
+ three quantifiers in the assumption.
+*}
+
+lemma
+ fixes P::"nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"
+ shows "\<forall>x y z. P x y z \<Longrightarrow> True"
+apply (tactic {*
+ inst_spec_tac [@{cterm "a::nat"},@{cterm "b::nat"},@{cterm "c::nat"}] 1 *})
+txt {*
+ We obtain the goal state
+
+ \begin{minipage}{\textwidth}
+ @{subgoals}
+ \end{minipage}*}
+(*<*)oops(*>*)
+
+text {*
+ Now the complete tactic for proving the induction principles can
+ be implemented as follows:
+*}
+
+ML %linenosgray{*fun induction_tac defs prems insts =
+ EVERY1 [ObjectLogic.full_atomize_tac,
+ cut_facts_tac prems,
+ K (rewrite_goals_tac defs),
+ inst_spec_tac insts,
+ assume_tac]*}
+
+text {*
+ We only have to give it as arguments the definitions, the premise
+ (like @{text "even n"})
+ and the instantiations. Compare this with the manual proof given for the
+ lemma @{thm [source] man_ind_principle}.
+ A testcase for this tactic is the function
+*}
+
+ML{*fun test_tac prems =
+let
+ val defs = [@{thm even_def}, @{thm odd_def}]
+ val insts = [@{cterm "P::nat\<Rightarrow>bool"}, @{cterm "Q::nat\<Rightarrow>bool"}]
+in
+ induction_tac defs prems insts
+end*}
+
+text {*
+ which indeed proves the induction principle:
+*}
+
+lemma
+assumes prems: "even n"
+shows "P 0 \<Longrightarrow> (\<And>m. Q m \<Longrightarrow> P (Suc m)) \<Longrightarrow> (\<And>m. P m \<Longrightarrow> Q (Suc m)) \<Longrightarrow> P n"
+apply(tactic {* test_tac @{thms prems} *})
+done
+
+text {*
+ While the tactic for the induction principle is relatively simple,
+ it is a bit harder to construct the goals from the introduction
+ rules the user provides. In general we have to construct for each predicate
+ @{text "pred"} a goal of the form
+
+ @{text [display]
+ "\<And>\<^raw:$zs$>. pred \<^raw:$zs$> \<Longrightarrow> rules[preds := \<^raw:$Ps$>] \<Longrightarrow> \<^raw:$P$> \<^raw:$zs$>"}
+
+ where the given predicates @{text preds} are replaced in the introduction
+ rules by new distinct variables written @{text "\<^raw:$Ps$>"}.
+ We also need to generate fresh arguments for the predicate @{text "pred"} in
+ the premise and the @{text "\<^raw:$P$>"} in the conclusion. We achieve
+ that in two steps.
+
+ The function below expects that the introduction rules are already appropriately
+ substituted. The argument @{text "srules"} stands for these substituted
+ rules; @{text cnewpreds} are the certified terms coresponding
+ to the variables @{text "\<^raw:$Ps$>"}; @{text "pred"} is the predicate for
+ which we prove the introduction principle; @{text "newpred"} is its
+ replacement and @{text "tys"} are the argument types of this predicate.
+*}
+
+ML %linenosgray{*fun prove_induction lthy defs srules cnewpreds ((pred, newpred), tys) =
+let
+ val zs = replicate (length tys) "z"
+ val (newargnames, lthy') = Variable.variant_fixes zs lthy;
+ val newargs = map Free (newargnames ~~ tys)
+
+ val prem = HOLogic.mk_Trueprop (list_comb (pred, newargs))
+ val goal = Logic.list_implies
+ (srules, HOLogic.mk_Trueprop (list_comb (newpred, newargs)))
+in
+ Goal.prove lthy' [] [prem] goal
+ (fn {prems, ...} => induction_tac defs prems cnewpreds)
+ |> singleton (ProofContext.export lthy' lthy)
+end *}
+
+text {*
+ In Line 3 we produce names @{text "\<^raw:$zs$>"} for each type in the
+ argument type list. Line 4 makes these names unique and declares them as
+ \emph{free} (but fixed) variables in the local theory @{text "lthy'"}. In
+ Line 5 we just construct the terms corresponding to these variables.
+ The term variables are applied to the predicate in Line 7 (this corresponds
+ to the first premise @{text "pred \<^raw:$zs$>"} of the induction principle).
+ In Line 8 and 9, we first construct the term @{text "\<^raw:$P$>\<^raw:$zs$>"}
+ and then add the (substituded) introduction rules as premises. In case that
+ no introduction rules are given, the conclusion of this implication needs
+ to be wrapped inside a @{term Trueprop}, otherwise the Isabelle's goal
+ mechanism will fail.
+
+ In Line 11 we set up the goal to be proved; in the next line call the tactic
+ for proving the induction principle. This tactic expects definitions, the
+ premise and the (certified) predicates with which the introduction rules
+ have been substituted. This will return a theorem. However, it is a theorem
+ proved inside the local theory @{text "lthy'"}, where the variables @{text
+ "\<^raw:$zs$>"} are fixed, but free. By exporting this theorem from @{text
+ "lthy'"} (which contains the @{text "\<^raw:$zs$>"} as free) to @{text
+ "lthy"} (which does not), we obtain the desired quantifications @{text
+ "\<And>\<^raw:$zs$>"}.
+
+ (FIXME testcase)
+
+
+ Now it is left to produce the new predicates with which the introduction
+ rules are substituted.
+*}
+
+ML %linenosgray{*fun inductions rules defs preds arg_tyss lthy =
+let
+ val Ps = replicate (length preds) "P"
+ val (newprednames, lthy') = Variable.variant_fixes Ps lthy
+
+ val thy = ProofContext.theory_of lthy'
+
+ val tyss' = map (fn tys => tys ---> HOLogic.boolT) arg_tyss
+ val newpreds = map Free (newprednames ~~ tyss')
+ val cnewpreds = map (cterm_of thy) newpreds
+ val srules = map (subst_free (preds ~~ newpreds)) rules
+
+in
+ map (prove_induction lthy' defs srules cnewpreds)
+ (preds ~~ newpreds ~~ arg_tyss)
+ |> ProofContext.export lthy' lthy
+end*}
+
+text {*
+ In Line 3 we generate a string @{text [quotes] "P"} for each predicate.
+ In Line 4, we use the same trick as in the previous function, that is making the
+ @{text "\<^raw:$Ps$>"} fresh and declaring them as fixed but free in
+ the new local theory @{text "lthy'"}. From the local theory we extract
+ the ambient theory in Line 6. We need this theory in order to certify
+ the new predicates. In Line 8 we calculate the types of these new predicates
+ using the argument types. Next we turn them into terms and subsequently
+ certify them. We can now produce the substituted introduction rules
+ (Line 11). Line 14 and 15 just iterate the proofs for all predicates.
+ From this we obtain a list of theorems. Finally we need to export the
+ fixed variables @{text "\<^raw:$Ps$>"} to obtain the correct quantification
+ (Line 16).
+
+ A testcase for this function is
+*}
+
+local_setup %gray {* fn lthy =>
+let
+ val rules = [@{prop "even (0::nat)"},
+ @{prop "\<And>n::nat. odd n \<Longrightarrow> even (Suc n)"},
+ @{prop "\<And>n::nat. even n \<Longrightarrow> odd (Suc n)"}]
+ val defs = [@{thm even_def}, @{thm odd_def}]
+ val preds = [@{term "even::nat\<Rightarrow>bool"}, @{term "odd::nat\<Rightarrow>bool"}]
+ val tyss = [[@{typ "nat"}], [@{typ "nat"}]]
+ val ind_thms = inductions rules defs preds tyss lthy
+in
+ warning (str_of_thms lthy ind_thms); lthy
+end
+*}
+
+
+text {*
+ which prints out
+
+@{text [display]
+"> even z \<Longrightarrow>
+> P 0 \<Longrightarrow> (\<And>m. Pa m \<Longrightarrow> P (Suc m)) \<Longrightarrow> (\<And>m. P m \<Longrightarrow> Pa (Suc m)) \<Longrightarrow> P z,
+> odd z \<Longrightarrow>
+> P 0 \<Longrightarrow> (\<And>m. Pa m \<Longrightarrow> P (Suc m)) \<Longrightarrow> (\<And>m. P m \<Longrightarrow> Pa (Suc m)) \<Longrightarrow> Pa z"}
+
+
+ This completes the code for the induction principles. Finally we can
+ prove the introduction rules.
+
+*}
+
+ML {* ObjectLogic.rulify *}
+
+
+ML{*val all_elims = fold (fn ct => fn th => th RS inst_spec ct)
+val imp_elims = fold (fn th => fn th' => [th', th] MRS @{thm mp})*}
+
+ML{*fun subproof2 prem params2 prems2 =
+ SUBPROOF (fn {prems, ...} =>
+ let
+ val prem' = prems MRS prem;
+ val prem'' =
+ case prop_of prem' of
+ _ $ (Const (@{const_name All}, _) $ _) =>
+ prem' |> all_elims params2
+ |> imp_elims prems2
+ | _ => prem';
+ in
+ rtac prem'' 1
+ end)*}
+
+ML{*fun subproof1 rules preds i =
+ SUBPROOF (fn {params, prems, context = ctxt', ...} =>
+ let
+ val (prems1, prems2) = chop (length prems - length rules) prems;
+ val (params1, params2) = chop (length params - length preds) params;
+ in
+ rtac (ObjectLogic.rulify (all_elims params1 (nth prems2 i))) 1
+ THEN
+ EVERY1 (map (fn prem => subproof2 prem params2 prems2 ctxt') prems1)
+ end)*}
+
+ML{*
+fun introductions_tac defs rules preds i ctxt =
+ EVERY1 [ObjectLogic.rulify_tac,
+ K (rewrite_goals_tac defs),
+ REPEAT o (resolve_tac [@{thm allI}, @{thm impI}]),
+ subproof1 rules preds i ctxt]*}
+
+lemma evenS:
+ shows "odd m \<Longrightarrow> even (Suc m)"
+apply(tactic {*
+let
+ val rules = [@{prop "even (0::nat)"},
+ @{prop "\<And>n::nat. odd n \<Longrightarrow> even (Suc n)"},
+ @{prop "\<And>n::nat. even n \<Longrightarrow> odd (Suc n)"}]
+ val defs = [@{thm even_def}, @{thm odd_def}]
+ val preds = [@{term "even::nat\<Rightarrow>bool"}, @{term "odd::nat\<Rightarrow>bool"}]
+in
+ introductions_tac defs rules preds 1 @{context}
+end *})
+done
+
+ML{*fun introductions rules preds defs lthy =
+let
+ fun prove_intro (i, goal) =
+ Goal.prove lthy [] [] goal
+ (fn {context, ...} => introductions_tac defs rules preds i context)
+in
+ map_index prove_intro rules
+end*}
+
+text {* main internal function *}
+
+ML %linenosgray{*fun add_inductive pred_specs rule_specs lthy =
+let
+ val syns = map snd pred_specs
+ val pred_specs' = map fst pred_specs
+ val prednames = map fst pred_specs'
+ val preds = map (fn (p, ty) => Free (Binding.name_of p, ty)) pred_specs'
+
+ val tyss = map (binder_types o fastype_of) preds
+ val (attrs, rules) = split_list rule_specs
+
+ val (defs, lthy') = definitions rules preds prednames syns tyss lthy
+ val ind_rules = inductions rules defs preds tyss lthy'
+ val intro_rules = introductions rules preds defs lthy'
+
+ val mut_name = space_implode "_" (map Binding.name_of prednames)
+ val case_names = map (Binding.name_of o fst) attrs
+in
+ lthy'
+ |> LocalTheory.notes Thm.theoremK (map (fn (((a, atts), _), th) =>
+ ((Binding.qualify false mut_name a, atts), [([th], [])])) (rule_specs ~~ intro_rules))
+ |-> (fn intross => LocalTheory.note Thm.theoremK
+ ((Binding.qualify false mut_name (@{binding "intros"}), []), maps snd intross))
+ |>> snd
+ ||>> (LocalTheory.notes Thm.theoremK (map (fn (((R, _), _), th) =>
+ ((Binding.qualify false (Binding.name_of R) (@{binding "induct"}),
+ [Attrib.internal (K (RuleCases.case_names case_names)),
+ Attrib.internal (K (RuleCases.consumes 1)),
+ Attrib.internal (K (Induct.induct_pred ""))]), [([th], [])]))
+ (pred_specs ~~ ind_rules)) #>> maps snd)
+ |> snd
+end*}
+
+ML{*fun add_inductive_cmd pred_specs rule_specs lthy =
+let
+ val ((pred_specs', rule_specs'), _) =
+ Specification.read_spec pred_specs rule_specs lthy
+in
+ add_inductive pred_specs' rule_specs' lthy
+end*}
+
+ML{*val spec_parser =
+ OuterParse.fixes --
+ Scan.optional
+ (OuterParse.$$$ "where" |--
+ OuterParse.!!!
+ (OuterParse.enum1 "|"
+ (SpecParse.opt_thm_name ":" -- OuterParse.prop))) []*}
+
+ML{*val specification =
+ spec_parser >>
+ (fn ((pred_specs), rule_specs) => add_inductive_cmd pred_specs rule_specs)*}
+
+ML{*val _ = OuterSyntax.local_theory "simple_inductive"
+ "define inductive predicates"
+ OuterKeyword.thy_decl specification*}
+
+text {*
+ Things to include at the end:
+
+ \begin{itemize}
+ \item say something about add-inductive-i to return
+ the rules
+ \item say that the induction principle is weaker (weaker than
+ what the standard inductive package generates)
+ \end{itemize}
+
+*}
+
+simple_inductive
+ Even and Odd
+where
+ Even0: "Even 0"
+| EvenS: "Odd n \<Longrightarrow> Even (Suc n)"
+| OddS: "Even n \<Longrightarrow> Odd (Suc n)"
+
+end