--- a/CookBook/Package/Ind_Code.thy Wed Mar 18 23:52:51 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,579 +0,0 @@
-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"}
-
- So we have @{text "pred zs"} and @{text "orules[preds::=Ps]"}; 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"}.
-
- We have to show @{text "\<forall>xs. As \<longrightarrow> (\<forall>ys. Bs \<longrightarrow> pred ss)\<^isup>* \<longrightarrow> pred ts"};
- expanding the defs
-
- @{text [display]
- "\<forall>xs. As \<longrightarrow> (\<forall>ys. Bs \<longrightarrow> (\<forall>preds. orules \<longrightarrow> pred ss))\<^isup>* \<longrightarrow> (\<forall>preds. orules \<longrightarrow> pred ts"}
-
- 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 @{text "orules"} are 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