# HG changeset patch # User berghofe # Date 1184146890 -7200 # Node ID b136b53fcd2ae580b5e6f68c3bcdd863219f106e # Parent 24eef53a9ad31c07a82646cc9726929ae3f97b16 Old (co)inductive command is now replaced by (co)inductive_set. diff -r 24eef53a9ad3 -r b136b53fcd2a src/HOL/Tools/old_inductive_package.ML --- a/src/HOL/Tools/old_inductive_package.ML Wed Jul 11 11:39:59 2007 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,901 +0,0 @@ -(* 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 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 -( - 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)); -); - -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)) (snd (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 (filter (fn (c, _) => 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 = 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 (raw_props, ctxt) = - 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 = BalancedTree.make (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 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 |> IsarCmd.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 = - 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.$$$ "intros" |-- - P.!!! (Scan.repeat (SpecParse.opt_thm_name ":" -- P.prop))) -- - Scan.optional (P.$$$ "monos" |-- P.!!! SpecParse.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 (SpecParse.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; -