# HG changeset patch # User berghofe # Date 1160756642 -7200 # Node ID d559870306f48142796155e745e759751bf46656 # Parent 3634641f940546c6f65bdfdce116c92eb5ba43fb Old version of inductive definition package (for sets). diff -r 3634641f9405 -r d559870306f4 src/HOL/Tools/old_inductive_package.ML --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Tools/old_inductive_package.ML Fri Oct 13 18:24:02 2006 +0200 @@ -0,0 +1,916 @@ +(* Title: HOL/Tools/old_inductive_package.ML + ID: $Id$ + Author: Lawrence C Paulson, Cambridge University Computer Laboratory + Author: Stefan Berghofer, TU Muenchen + Author: Markus Wenzel, TU Muenchen + +(Co)Inductive Definition module for HOL. + +Features: + * least or greatest fixedpoints + * user-specified product and sum constructions + * mutually recursive definitions + * definitions involving arbitrary monotone operators + * automatically proves introduction and elimination rules + +The recursive sets must *already* be declared as constants in the +current theory! + + Introduction rules have the form + [| ti:M(Sj), ..., P(x), ... |] ==> t: Sk + where M is some monotone operator (usually the identity) + P(x) is any side condition on the free variables + ti, t are any terms + Sj, Sk are two of the sets being defined in mutual recursion + +Sums are used only for mutual recursion. Products are used only to +derive "streamlined" induction rules for relations. +*) + +signature OLD_INDUCTIVE_PACKAGE = +sig + val quiet_mode: bool ref + val trace: bool ref + val unify_consts: theory -> term list -> term list -> term list * term list + val split_rule_vars: term list -> thm -> thm + val get_inductive: theory -> string -> ({names: string list, coind: bool} * + {defs: thm list, elims: thm list, raw_induct: thm, induct: thm, + intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm}) option + val the_mk_cases: theory -> string -> string -> thm + val print_inductives: theory -> unit + val mono_add: attribute + val mono_del: attribute + val get_monos: theory -> thm list + val inductive_forall_name: string + val inductive_forall_def: thm + val rulify: thm -> thm + val inductive_cases: ((bstring * Attrib.src list) * string list) list -> theory -> theory + val inductive_cases_i: ((bstring * attribute list) * term list) list -> theory -> theory + val add_inductive_i: bool -> bool -> bstring -> bool -> bool -> bool -> term list -> + ((bstring * term) * attribute list) list -> thm list -> theory -> theory * + {defs: thm list, elims: thm list, raw_induct: thm, induct: thm, + intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm} + val add_inductive: bool -> bool -> string list -> + ((bstring * string) * Attrib.src list) list -> (thmref * Attrib.src list) list -> + theory -> theory * + {defs: thm list, elims: thm list, raw_induct: thm, induct: thm, + intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm} + val setup: theory -> theory +end; + +structure OldInductivePackage: OLD_INDUCTIVE_PACKAGE = +struct + + +(** theory context references **) + +val mono_name = "Orderings.mono"; +val gfp_name = "FixedPoint.gfp"; +val lfp_name = "FixedPoint.lfp"; +val vimage_name = "Set.vimage"; +val Const _ $ (vimage_f $ _) $ _ = HOLogic.dest_Trueprop (Thm.concl_of vimageD); + +val inductive_forall_name = "HOL.induct_forall"; +val inductive_forall_def = thm "induct_forall_def"; +val inductive_conj_name = "HOL.induct_conj"; +val inductive_conj_def = thm "induct_conj_def"; +val inductive_conj = thms "induct_conj"; +val inductive_atomize = thms "induct_atomize"; +val inductive_rulify = thms "induct_rulify"; +val inductive_rulify_fallback = thms "induct_rulify_fallback"; + + + +(** theory data **) + +type inductive_info = + {names: string list, coind: bool} * {defs: thm list, elims: thm list, raw_induct: thm, + induct: thm, intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm}; + +structure InductiveData = TheoryDataFun +(struct + val name = "HOL/inductive"; + type T = inductive_info Symtab.table * thm list; + + val empty = (Symtab.empty, []); + val copy = I; + val extend = I; + fun merge _ ((tab1, monos1), (tab2, monos2)) = + (Symtab.merge (K true) (tab1, tab2), Drule.merge_rules (monos1, monos2)); + + fun print thy (tab, monos) = + [Pretty.strs ("(co)inductives:" :: + map #1 (NameSpace.extern_table (Sign.const_space thy, tab))), + Pretty.big_list "monotonicity rules:" (map (Display.pretty_thm_sg thy) monos)] + |> Pretty.chunks |> Pretty.writeln; +end); + +val print_inductives = InductiveData.print; + + +(* get and put data *) + +val get_inductive = Symtab.lookup o #1 o InductiveData.get; + +fun the_inductive thy name = + (case get_inductive thy name of + NONE => error ("Unknown (co)inductive set " ^ quote name) + | SOME info => info); + +val the_mk_cases = (#mk_cases o #2) oo the_inductive; + +fun put_inductives names info = InductiveData.map (apfst (fn tab => + fold (fn name => Symtab.update_new (name, info)) names tab + handle Symtab.DUP dup => error ("Duplicate definition of (co)inductive set " ^ quote dup))); + + + +(** monotonicity rules **) + +val get_monos = #2 o InductiveData.get; +val map_monos = InductiveData.map o Library.apsnd; + +fun mk_mono thm = + let + fun eq2mono thm' = [standard (thm' RS (thm' RS eq_to_mono))] @ + (case concl_of thm of + (_ $ (_ $ (Const ("Not", _) $ _) $ _)) => [] + | _ => [standard (thm' RS (thm' RS eq_to_mono2))]); + val concl = concl_of thm + in + if can Logic.dest_equals concl then + eq2mono (thm RS meta_eq_to_obj_eq) + else if can (HOLogic.dest_eq o HOLogic.dest_Trueprop) concl then + eq2mono thm + else [thm] + end; + + +(* attributes *) + +val mono_add = Thm.declaration_attribute (fn th => + Context.mapping (map_monos (fold Drule.add_rule (mk_mono th))) I); + +val mono_del = Thm.declaration_attribute (fn th => + Context.mapping (map_monos (fold Drule.del_rule (mk_mono th))) I); + + + +(** misc utilities **) + +val quiet_mode = ref false; +val trace = ref false; (*for debugging*) +fun message s = if ! quiet_mode then () else writeln s; +fun clean_message s = if ! quick_and_dirty then () else message s; + +fun coind_prefix true = "co" + | coind_prefix false = ""; + + +(*the following code ensures that each recursive set always has the + same type in all introduction rules*) +fun unify_consts thy cs intr_ts = + (let + val add_term_consts_2 = fold_aterms (fn Const c => insert (op =) c | _ => I); + fun varify (t, (i, ts)) = + let val t' = map_types (Logic.incr_tvar (i + 1)) (#1 (Type.varify (t, []))) + in (maxidx_of_term t', t'::ts) end; + val (i, cs') = foldr varify (~1, []) cs; + val (i', intr_ts') = foldr varify (i, []) intr_ts; + val rec_consts = fold add_term_consts_2 cs' []; + val intr_consts = fold add_term_consts_2 intr_ts' []; + fun unify (cname, cT) = + let val consts = map snd (List.filter (fn c => fst c = cname) intr_consts) + in fold (Sign.typ_unify thy) ((replicate (length consts) cT) ~~ consts) end; + val (env, _) = fold unify rec_consts (Vartab.empty, i'); + val subst = Type.freeze o map_types (Envir.norm_type env) + + in (map subst cs', map subst intr_ts') + end) handle Type.TUNIFY => + (warning "Occurrences of recursive constant have non-unifiable types"; (cs, intr_ts)); + + +(*make injections used in mutually recursive definitions*) +fun mk_inj cs sumT c x = + let + fun mk_inj' T n i = + if n = 1 then x else + let val n2 = n div 2; + val Type (_, [T1, T2]) = T + in + if i <= n2 then + Const ("Sum_Type.Inl", T1 --> T) $ (mk_inj' T1 n2 i) + else + Const ("Sum_Type.Inr", T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2)) + end + in mk_inj' sumT (length cs) (1 + find_index_eq c cs) + end; + +(*make "vimage" terms for selecting out components of mutually rec.def*) +fun mk_vimage cs sumT t c = if length cs < 2 then t else + let + val cT = HOLogic.dest_setT (fastype_of c); + val vimageT = [cT --> sumT, HOLogic.mk_setT sumT] ---> HOLogic.mk_setT cT + in + Const (vimage_name, vimageT) $ + Abs ("y", cT, mk_inj cs sumT c (Bound 0)) $ t + end; + +(** proper splitting **) + +fun prod_factors p (Const ("Pair", _) $ t $ u) = + p :: prod_factors (1::p) t @ prod_factors (2::p) u + | prod_factors p _ = []; + +fun mg_prod_factors ts (t $ u) fs = if t mem ts then + let val f = prod_factors [] u + in AList.update (op =) (t, f inter (AList.lookup (op =) fs t) |> the_default f) fs end + else mg_prod_factors ts u (mg_prod_factors ts t fs) + | mg_prod_factors ts (Abs (_, _, t)) fs = mg_prod_factors ts t fs + | mg_prod_factors ts _ fs = fs; + +fun prodT_factors p ps (T as Type ("*", [T1, T2])) = + if p mem ps then prodT_factors (1::p) ps T1 @ prodT_factors (2::p) ps T2 + else [T] + | prodT_factors _ _ T = [T]; + +fun ap_split p ps (Type ("*", [T1, T2])) T3 u = + if p mem ps then HOLogic.split_const (T1, T2, T3) $ + Abs ("v", T1, ap_split (2::p) ps T2 T3 (ap_split (1::p) ps T1 + (prodT_factors (2::p) ps T2 ---> T3) (incr_boundvars 1 u) $ Bound 0)) + else u + | ap_split _ _ _ _ u = u; + +fun mk_tuple p ps (Type ("*", [T1, T2])) (tms as t::_) = + if p mem ps then HOLogic.mk_prod (mk_tuple (1::p) ps T1 tms, + mk_tuple (2::p) ps T2 (Library.drop (length (prodT_factors (1::p) ps T1), tms))) + else t + | mk_tuple _ _ _ (t::_) = t; + +fun split_rule_var' ((t as Var (v, Type ("fun", [T1, T2])), ps), rl) = + let val T' = prodT_factors [] ps T1 ---> T2 + val newt = ap_split [] ps T1 T2 (Var (v, T')) + val cterm = Thm.cterm_of (Thm.theory_of_thm rl) + in + instantiate ([], [(cterm t, cterm newt)]) rl + end + | split_rule_var' (_, rl) = rl; + +val remove_split = rewrite_rule [split_conv RS eq_reflection]; + +fun split_rule_vars vs rl = standard (remove_split (foldr split_rule_var' + rl (mg_prod_factors vs (Thm.prop_of rl) []))); + +fun split_rule vs rl = standard (remove_split (foldr split_rule_var' + rl (List.mapPartial (fn (t as Var ((a, _), _)) => + Option.map (pair t) (AList.lookup (op =) vs a)) (term_vars (Thm.prop_of rl))))); + + +(** process rules **) + +local + +fun err_in_rule thy name t msg = + error (cat_lines ["Ill-formed introduction rule " ^ quote name, + Sign.string_of_term thy t, msg]); + +fun err_in_prem thy name t p msg = + error (cat_lines ["Ill-formed premise", Sign.string_of_term thy p, + "in introduction rule " ^ quote name, Sign.string_of_term thy t, msg]); + +val bad_concl = "Conclusion of introduction rule must have form \"t : S_i\""; + +val all_not_allowed = + "Introduction rule must not have a leading \"!!\" quantifier"; + +fun atomize_term thy = MetaSimplifier.rewrite_term thy inductive_atomize []; + +in + +fun check_rule thy cs ((name, rule), att) = + let + val concl = Logic.strip_imp_concl rule; + val prems = Logic.strip_imp_prems rule; + val aprems = map (atomize_term thy) prems; + val arule = Logic.list_implies (aprems, concl); + + fun check_prem (prem, aprem) = + if can HOLogic.dest_Trueprop aprem then () + else err_in_prem thy name rule prem "Non-atomic premise"; + in + (case concl of + Const ("Trueprop", _) $ (Const ("op :", _) $ t $ u) => + if u mem cs then + if exists (Logic.occs o rpair t) cs then + err_in_rule thy name rule "Recursion term on left of member symbol" + else List.app check_prem (prems ~~ aprems) + else err_in_rule thy name rule bad_concl + | Const ("all", _) $ _ => err_in_rule thy name rule all_not_allowed + | _ => err_in_rule thy name rule bad_concl); + ((name, arule), att) + end; + +val rulify = (* FIXME norm_hhf *) + hol_simplify inductive_conj + #> hol_simplify inductive_rulify + #> hol_simplify inductive_rulify_fallback + #> standard; + +end; + + + +(** properties of (co)inductive sets **) + +(* elimination rules *) + +fun mk_elims cs cTs params intr_ts intr_names = + let + val used = foldr add_term_names [] intr_ts; + val [aname, pname] = Name.variant_list used ["a", "P"]; + val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT)); + + fun dest_intr r = + let val Const ("op :", _) $ t $ u = + HOLogic.dest_Trueprop (Logic.strip_imp_concl r) + in (u, t, Logic.strip_imp_prems r) end; + + val intrs = map dest_intr intr_ts ~~ intr_names; + + fun mk_elim (c, T) = + let + val a = Free (aname, T); + + fun mk_elim_prem (_, t, ts) = + list_all_free (map dest_Free ((foldr add_term_frees [] (t::ts)) \\ params), + Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (a, t)) :: ts, P)); + val c_intrs = (List.filter (equal c o #1 o #1) intrs); + in + (Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (a, c)) :: + map mk_elim_prem (map #1 c_intrs), P), map #2 c_intrs) + end + in + map mk_elim (cs ~~ cTs) + end; + + +(* premises and conclusions of induction rules *) + +fun mk_indrule cs cTs params intr_ts = + let + val used = foldr add_term_names [] intr_ts; + + (* predicates for induction rule *) + + val preds = map Free (Name.variant_list used (if length cs < 2 then ["P"] else + map (fn i => "P" ^ string_of_int i) (1 upto length cs)) ~~ + map (fn T => T --> HOLogic.boolT) cTs); + + (* transform an introduction rule into a premise for induction rule *) + + fun mk_ind_prem r = + let + val frees = map dest_Free ((add_term_frees (r, [])) \\ params); + + val pred_of = AList.lookup (op aconv) (cs ~~ preds); + + fun subst (s as ((m as Const ("op :", T)) $ t $ u)) = + (case pred_of u of + NONE => (m $ fst (subst t) $ fst (subst u), NONE) + | SOME P => (HOLogic.mk_binop inductive_conj_name (s, P $ t), SOME (s, P $ t))) + | subst s = + (case pred_of s of + SOME P => (HOLogic.mk_binop "op Int" + (s, HOLogic.Collect_const (HOLogic.dest_setT + (fastype_of s)) $ P), NONE) + | NONE => (case s of + (t $ u) => (fst (subst t) $ fst (subst u), NONE) + | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), NONE) + | _ => (s, NONE))); + + fun mk_prem (s, prems) = (case subst s of + (_, SOME (t, u)) => t :: u :: prems + | (t, _) => t :: prems); + + val Const ("op :", _) $ t $ u = + HOLogic.dest_Trueprop (Logic.strip_imp_concl r) + + in list_all_free (frees, + Logic.list_implies (map HOLogic.mk_Trueprop (foldr mk_prem + [] (map HOLogic.dest_Trueprop (Logic.strip_imp_prems r))), + HOLogic.mk_Trueprop (valOf (pred_of u) $ t))) + end; + + val ind_prems = map mk_ind_prem intr_ts; + + val factors = Library.fold (mg_prod_factors preds) ind_prems []; + + (* make conclusions for induction rules *) + + fun mk_ind_concl ((c, P), (ts, x)) = + let val T = HOLogic.dest_setT (fastype_of c); + val ps = AList.lookup (op =) factors P |> the_default []; + val Ts = prodT_factors [] ps T; + val (frees, x') = foldr (fn (T', (fs, s)) => + ((Free (s, T'))::fs, Symbol.bump_string s)) ([], x) Ts; + val tuple = mk_tuple [] ps T frees; + in ((HOLogic.mk_binop "op -->" + (HOLogic.mk_mem (tuple, c), P $ tuple))::ts, x') + end; + + val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj + (fst (foldr mk_ind_concl ([], "xa") (cs ~~ preds)))) + + in (preds, ind_prems, mutual_ind_concl, + map (apfst (fst o dest_Free)) factors) + end; + + +(* prepare cases and induct rules *) + +fun add_cases_induct no_elim no_induct coind names elims induct = + let + fun cases_spec name elim thy = + thy + |> Theory.parent_path + |> Theory.add_path (Sign.base_name name) + |> PureThy.add_thms [(("cases", elim), [InductAttrib.cases_set name])] |> snd + |> Theory.restore_naming thy; + val cases_specs = if no_elim then [] else map2 cases_spec names elims; + + val induct_att = if coind then InductAttrib.coinduct_set else InductAttrib.induct_set; + fun induct_specs thy = + if no_induct then thy + else + let + val ctxt = ProofContext.init thy; + val rules = names ~~ ProjectRule.projects ctxt (1 upto length names) induct; + val inducts = map (RuleCases.save induct o standard o #2) rules; + in + thy + |> PureThy.add_thms (rules |> map (fn (name, th) => + (("", th), [RuleCases.consumes 1, induct_att name]))) |> snd + |> PureThy.add_thmss + [((coind_prefix coind ^ "inducts", inducts), [RuleCases.consumes 1])] |> snd + end; + in Library.apply cases_specs #> induct_specs end; + + + +(** proofs for (co)inductive sets **) + +(* prove monotonicity -- NOT subject to quick_and_dirty! *) + +fun prove_mono setT fp_fun monos thy = + (message " Proving monotonicity ..."; + Goal.prove_global thy [] [] (*NO quick_and_dirty here!*) + (HOLogic.mk_Trueprop + (Const (mono_name, (setT --> setT) --> HOLogic.boolT) $ fp_fun)) + (fn _ => EVERY [rtac monoI 1, + REPEAT (ares_tac (List.concat (map mk_mono monos) @ get_monos thy) 1)])); + + +(* prove introduction rules *) + +fun prove_intrs coind mono fp_def intr_ts rec_sets_defs ctxt = + let + val _ = clean_message " Proving the introduction rules ..."; + + val unfold = standard' (mono RS (fp_def RS + (if coind then def_gfp_unfold else def_lfp_unfold))); + + fun select_disj 1 1 = [] + | select_disj _ 1 = [rtac disjI1] + | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1)); + + val intrs = (1 upto (length intr_ts) ~~ intr_ts) |> map (fn (i, intr) => + rulify (SkipProof.prove ctxt [] [] intr (fn _ => EVERY + [rewrite_goals_tac rec_sets_defs, + stac unfold 1, + REPEAT (resolve_tac [vimageI2, CollectI] 1), + (*Now 1-2 subgoals: the disjunction, perhaps equality.*) + EVERY1 (select_disj (length intr_ts) i), + (*Not ares_tac, since refl must be tried before any equality assumptions; + backtracking may occur if the premises have extra variables!*) + DEPTH_SOLVE_1 (resolve_tac [refl, exI, conjI] 1 APPEND assume_tac 1), + (*Now solve the equations like Inl 0 = Inl ?b2*) + REPEAT (rtac refl 1)]))) + + in (intrs, unfold) end; + + +(* prove elimination rules *) + +fun prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs ctxt = + let + val _ = clean_message " Proving the elimination rules ..."; + + val rules1 = [CollectE, disjE, make_elim vimageD, exE, FalseE]; + val rules2 = [conjE, Inl_neq_Inr, Inr_neq_Inl] @ map make_elim [Inl_inject, Inr_inject]; + in + mk_elims cs cTs params intr_ts intr_names |> map (fn (t, cases) => + SkipProof.prove ctxt [] (Logic.strip_imp_prems t) (Logic.strip_imp_concl t) + (fn {prems, ...} => EVERY + [cut_facts_tac [hd prems] 1, + rewrite_goals_tac rec_sets_defs, + dtac (unfold RS subst) 1, + REPEAT (FIRSTGOAL (eresolve_tac rules1)), + REPEAT (FIRSTGOAL (eresolve_tac rules2)), + EVERY (map (fn prem => + DEPTH_SOLVE_1 (ares_tac [rewrite_rule rec_sets_defs prem, conjI] 1)) (tl prems))]) + |> rulify + |> RuleCases.name cases) + end; + + +(* derivation of simplified elimination rules *) + +local + +(*cprop should have the form t:Si where Si is an inductive set*) +val mk_cases_err = "mk_cases: proposition not of form \"t : S_i\""; + +(*delete needless equality assumptions*) +val refl_thin = prove_goal HOL.thy "!!P. a = a ==> P ==> P" (fn _ => [assume_tac 1]); +val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE, Pair_inject]; +val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls; + +fun simp_case_tac solved ss i = + EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i + THEN_MAYBE (if solved then no_tac else all_tac); + +in + +fun mk_cases_i elims ss cprop = + let + val prem = Thm.assume cprop; + val tac = ALLGOALS (simp_case_tac false ss) THEN prune_params_tac; + fun mk_elim rl = Drule.standard (Tactic.rule_by_tactic tac (prem RS rl)); + in + (case get_first (try mk_elim) elims of + SOME r => r + | NONE => error (Pretty.string_of (Pretty.block + [Pretty.str mk_cases_err, Pretty.fbrk, Display.pretty_cterm cprop]))) + end; + +fun mk_cases elims s = + mk_cases_i elims (simpset()) (Thm.read_cterm (Thm.theory_of_thm (hd elims)) (s, propT)); + +fun smart_mk_cases thy ss cprop = + let + val c = #1 (Term.dest_Const (Term.head_of (#2 (HOLogic.dest_mem (HOLogic.dest_Trueprop + (Logic.strip_imp_concl (Thm.term_of cprop))))))) handle TERM _ => error mk_cases_err; + val (_, {elims, ...}) = the_inductive thy c; + in mk_cases_i elims ss cprop end; + +end; + + +(* inductive_cases(_i) *) + +fun gen_inductive_cases prep_att prep_prop args thy = + let + val cert_prop = Thm.cterm_of thy o prep_prop (ProofContext.init thy); + val mk_cases = smart_mk_cases thy (Simplifier.simpset_of thy) o cert_prop; + + val facts = args |> map (fn ((a, atts), props) => + ((a, map (prep_att thy) atts), map (Thm.no_attributes o single o mk_cases) props)); + in thy |> PureThy.note_thmss_i "" facts |> snd end; + +val inductive_cases = gen_inductive_cases Attrib.attribute ProofContext.read_prop; +val inductive_cases_i = gen_inductive_cases (K I) ProofContext.cert_prop; + + +(* mk_cases_meth *) + +fun mk_cases_meth (ctxt, raw_props) = + let + val thy = ProofContext.theory_of ctxt; + val ss = local_simpset_of ctxt; + val cprops = map (Thm.cterm_of thy o ProofContext.read_prop ctxt) raw_props; + in Method.erule 0 (map (smart_mk_cases thy ss) cprops) end; + +val mk_cases_args = Method.syntax (Scan.lift (Scan.repeat1 Args.name)); + + +(* prove induction rule *) + +fun prove_indrule cs cTs sumT rec_const params intr_ts mono + fp_def rec_sets_defs ctxt = + let + val _ = clean_message " Proving the induction rule ..."; + val thy = ProofContext.theory_of ctxt; + + val sum_case_rewrites = + (if Context.theory_name thy = "Datatype" then + PureThy.get_thms thy (Name "sum.cases") + else + (case ThyInfo.lookup_theory "Datatype" of + NONE => [] + | SOME thy' => + if Theory.subthy (thy', thy) then + PureThy.get_thms thy' (Name "sum.cases") + else [])) + |> map mk_meta_eq; + + val (preds, ind_prems, mutual_ind_concl, factors) = + mk_indrule cs cTs params intr_ts; + + val dummy = if !trace then + (writeln "ind_prems = "; + List.app (writeln o Sign.string_of_term thy) ind_prems) + else (); + + (* make predicate for instantiation of abstract induction rule *) + + fun mk_ind_pred _ [P] = P + | mk_ind_pred T Ps = + let val n = (length Ps) div 2; + val Type (_, [T1, T2]) = T + in Const ("Datatype.sum.sum_case", + [T1 --> HOLogic.boolT, T2 --> HOLogic.boolT, T] ---> HOLogic.boolT) $ + mk_ind_pred T1 (Library.take (n, Ps)) $ mk_ind_pred T2 (Library.drop (n, Ps)) + end; + + val ind_pred = mk_ind_pred sumT preds; + + val ind_concl = HOLogic.mk_Trueprop + (HOLogic.all_const sumT $ Abs ("x", sumT, HOLogic.mk_binop "op -->" + (HOLogic.mk_mem (Bound 0, rec_const), ind_pred $ Bound 0))); + + (* simplification rules for vimage and Collect *) + + val vimage_simps = if length cs < 2 then [] else + map (fn c => standard (SkipProof.prove ctxt [] [] + (HOLogic.mk_Trueprop (HOLogic.mk_eq + (mk_vimage cs sumT (HOLogic.Collect_const sumT $ ind_pred) c, + HOLogic.Collect_const (HOLogic.dest_setT (fastype_of c)) $ + List.nth (preds, find_index_eq c cs)))) + (fn _ => EVERY + [rtac vimage_Collect 1, rewrite_goals_tac sum_case_rewrites, rtac refl 1]))) cs; + + val raw_fp_induct = (mono RS (fp_def RS def_lfp_induct_set)); + + val dummy = if !trace then + (writeln "raw_fp_induct = "; print_thm raw_fp_induct) + else (); + + val induct = standard (SkipProof.prove ctxt [] ind_prems ind_concl + (fn {prems, ...} => EVERY + [rewrite_goals_tac [inductive_conj_def], + rtac (impI RS allI) 1, + DETERM (etac raw_fp_induct 1), + rewrite_goals_tac (map mk_meta_eq (vimage_Int::Int_Collect::vimage_simps)), + fold_goals_tac rec_sets_defs, + (*This CollectE and disjE separates out the introduction rules*) + REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE, exE, FalseE])), + (*Now break down the individual cases. No disjE here in case + some premise involves disjunction.*) + REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac)), + rewrite_goals_tac sum_case_rewrites, + EVERY (map (fn prem => + DEPTH_SOLVE_1 (ares_tac [rewrite_rule [inductive_conj_def] prem, conjI, refl] 1)) prems)])); + + val lemma = standard (SkipProof.prove ctxt [] [] + (Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn _ => EVERY + [rewrite_goals_tac rec_sets_defs, + REPEAT (EVERY + [REPEAT (resolve_tac [conjI, impI] 1), + TRY (dtac vimageD 1), etac allE 1, dtac mp 1, atac 1, + rewrite_goals_tac sum_case_rewrites, + atac 1])])) + + in standard (split_rule factors (induct RS lemma)) end; + + + +(** specification of (co)inductive sets **) + +fun cond_declare_consts declare_consts cs paramTs cnames = + if declare_consts then + Theory.add_consts_i (map (fn (c, n) => (Sign.base_name n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames)) + else I; + +fun mk_ind_def declare_consts alt_name coind cs intr_ts monos thy + params paramTs cTs cnames = + let + val sumT = fold_bal (fn (T, U) => Type ("+", [T, U])) cTs; + val setT = HOLogic.mk_setT sumT; + + val fp_name = if coind then gfp_name else lfp_name; + + val used = foldr add_term_names [] intr_ts; + val [sname, xname] = Name.variant_list used ["S", "x"]; + + (* transform an introduction rule into a conjunction *) + (* [| t : ... S_i ... ; ... |] ==> u : S_j *) + (* is transformed into *) + (* x = Inj_j u & t : ... Inj_i -`` S ... & ... *) + + fun transform_rule r = + let + val frees = map dest_Free ((add_term_frees (r, [])) \\ params); + val subst = subst_free + (cs ~~ (map (mk_vimage cs sumT (Free (sname, setT))) cs)); + val Const ("op :", _) $ t $ u = + HOLogic.dest_Trueprop (Logic.strip_imp_concl r) + + in foldr (fn ((x, T), P) => HOLogic.mk_exists (x, T, P)) + (foldr1 HOLogic.mk_conj + (((HOLogic.eq_const sumT) $ Free (xname, sumT) $ (mk_inj cs sumT u t)):: + (map (subst o HOLogic.dest_Trueprop) + (Logic.strip_imp_prems r)))) frees + end + + (* make a disjunction of all introduction rules *) + + val fp_fun = absfree (sname, setT, (HOLogic.Collect_const sumT) $ + absfree (xname, sumT, if null intr_ts then HOLogic.false_const + else foldr1 HOLogic.mk_disj (map transform_rule intr_ts))); + + (* add definiton of recursive sets to theory *) + + val rec_name = if alt_name = "" then + space_implode "_" (map Sign.base_name cnames) else alt_name; + val full_rec_name = if length cs < 2 then hd cnames + else Sign.full_name thy rec_name; + + val rec_const = list_comb + (Const (full_rec_name, paramTs ---> setT), params); + + val fp_def_term = Logic.mk_equals (rec_const, + Const (fp_name, (setT --> setT) --> setT) $ fp_fun); + + val def_terms = fp_def_term :: (if length cs < 2 then [] else + map (fn c => Logic.mk_equals (c, mk_vimage cs sumT rec_const c)) cs); + + val ([fp_def :: rec_sets_defs], thy') = + thy + |> cond_declare_consts declare_consts cs paramTs cnames + |> (if length cs < 2 then I + else Theory.add_consts_i [(rec_name, paramTs ---> setT, NoSyn)]) + |> Theory.add_path rec_name + |> PureThy.add_defss_i false [(("defs", def_terms), [])]; + + val mono = prove_mono setT fp_fun monos thy' + + in (thy', rec_name, mono, fp_def, rec_sets_defs, rec_const, sumT) end; + +fun add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs + intros monos thy params paramTs cTs cnames induct_cases = + let + val _ = + if verbose then message ("Proofs for " ^ coind_prefix coind ^ "inductive set(s) " ^ + commas_quote (map Sign.base_name cnames)) else (); + + val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros); + + val (thy1, rec_name, mono, fp_def, rec_sets_defs, rec_const, sumT) = + mk_ind_def declare_consts alt_name coind cs intr_ts monos thy + params paramTs cTs cnames; + val ctxt1 = ProofContext.init thy1; + + val (intrs, unfold) = prove_intrs coind mono fp_def intr_ts rec_sets_defs ctxt1; + val elims = if no_elim then [] else + prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs ctxt1; + val raw_induct = if no_ind then Drule.asm_rl else + if coind then standard (rule_by_tactic + (rewrite_tac [mk_meta_eq vimage_Un] THEN + fold_tac rec_sets_defs) (mono RS (fp_def RS def_Collect_coinduct))) + else + prove_indrule cs cTs sumT rec_const params intr_ts mono fp_def + rec_sets_defs ctxt1; + val induct = + if coind then + (raw_induct, [RuleCases.case_names [rec_name], + RuleCases.case_conclusion (rec_name, induct_cases), + RuleCases.consumes 1]) + else if no_ind orelse length cs > 1 then + (raw_induct, [RuleCases.case_names induct_cases, RuleCases.consumes 0]) + else (raw_induct RSN (2, rev_mp), [RuleCases.case_names induct_cases, RuleCases.consumes 1]); + + val (intrs', thy2) = + thy1 + |> PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts); + val (([_, elims'], [induct']), thy3) = + thy2 + |> PureThy.add_thmss + [(("intros", intrs'), []), + (("elims", elims), [RuleCases.consumes 1])] + ||>> PureThy.add_thms + [((coind_prefix coind ^ "induct", rulify (#1 induct)), #2 induct)]; + in (thy3, + {defs = fp_def :: rec_sets_defs, + mono = mono, + unfold = unfold, + intrs = intrs', + elims = elims', + mk_cases = mk_cases elims', + raw_induct = rulify raw_induct, + induct = induct'}) + end; + + +(* external interfaces *) + +fun try_term f msg thy t = + (case Library.try f t of + SOME x => x + | NONE => error (msg ^ Sign.string_of_term thy t)); + +fun add_inductive_i verbose declare_consts alt_name coind no_elim no_ind cs pre_intros monos thy = + let + val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions"); + + (*parameters should agree for all mutually recursive components*) + val (_, params) = strip_comb (hd cs); + val paramTs = map (try_term (snd o dest_Free) "Parameter in recursive\ + \ component is not a free variable: " thy) params; + + val cTs = map (try_term (HOLogic.dest_setT o fastype_of) + "Recursive component not of type set: " thy) cs; + + val cnames = map (try_term (fst o dest_Const o head_of) + "Recursive set not previously declared as constant: " thy) cs; + + val save_thy = thy + |> Theory.copy |> cond_declare_consts declare_consts cs paramTs cnames; + val intros = map (check_rule save_thy cs) pre_intros; + val induct_cases = map (#1 o #1) intros; + + val (thy1, result as {elims, induct, ...}) = + add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs intros monos + thy params paramTs cTs cnames induct_cases; + val thy2 = thy1 + |> put_inductives cnames ({names = cnames, coind = coind}, result) + |> add_cases_induct no_elim no_ind coind cnames elims induct + |> Theory.parent_path; + in (thy2, result) end; + +fun add_inductive verbose coind c_strings intro_srcs raw_monos thy = + let + val cs = map (Sign.read_term thy) c_strings; + + val intr_names = map (fst o fst) intro_srcs; + fun read_rule s = Thm.read_cterm thy (s, propT) + handle ERROR msg => cat_error msg ("The error(s) above occurred for " ^ s); + val intr_ts = map (Thm.term_of o read_rule o snd o fst) intro_srcs; + val intr_atts = map (map (Attrib.attribute thy) o snd) intro_srcs; + val (cs', intr_ts') = unify_consts thy cs intr_ts; + + val (monos, thy') = thy |> IsarThy.apply_theorems raw_monos; + in + add_inductive_i verbose false "" coind false false cs' + ((intr_names ~~ intr_ts') ~~ intr_atts) monos thy' + end; + + + +(** package setup **) + +(* setup theory *) + +val setup = + InductiveData.init #> + Method.add_methods [("ind_cases", mk_cases_meth oo mk_cases_args, + "dynamic case analysis on sets")] #> + Attrib.add_attributes [("mono", Attrib.add_del_args mono_add mono_del, + "declaration of monotonicity rule")]; + + +(* outer syntax *) + +local structure P = OuterParse and K = OuterKeyword in + +fun mk_ind coind ((sets, intrs), monos) = + #1 o add_inductive true coind sets (map P.triple_swap intrs) monos; + +fun ind_decl coind = + Scan.repeat1 P.term -- + (P.$$$ "intros" |-- + P.!!! (Scan.repeat (P.opt_thm_name ":" -- P.prop))) -- + Scan.optional (P.$$$ "monos" |-- P.!!! P.xthms1) [] + >> (Toplevel.theory o mk_ind coind); + +val inductiveP = + OuterSyntax.command "inductive" "define inductive sets" K.thy_decl (ind_decl false); + +val coinductiveP = + OuterSyntax.command "coinductive" "define coinductive sets" K.thy_decl (ind_decl true); + + +val ind_cases = + P.and_list1 (P.opt_thm_name ":" -- Scan.repeat1 P.prop) + >> (Toplevel.theory o inductive_cases); + +val inductive_casesP = + OuterSyntax.command "inductive_cases" + "create simplified instances of elimination rules (improper)" K.thy_script ind_cases; + +val _ = OuterSyntax.add_keywords ["intros", "monos"]; +val _ = OuterSyntax.add_parsers [inductiveP, coinductiveP, inductive_casesP]; + +end; + +end; +