--- 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;
-