Fixed bug in transform_rule.
(* Title: HOL/Tools/inductive_package.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Stefan Berghofer, TU Muenchen
Copyright 1994 University of Cambridge
1998 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 parent 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 INDUCTIVE_PACKAGE =
sig
val add_inductive_i : bool -> bool -> bstring -> bool -> bool -> bool -> term list ->
term list -> thm list -> thm list -> theory -> theory *
{defs:thm list, elims:thm list, raw_induct:thm, induct:thm,
intrs:thm list,
mk_cases:thm list -> string -> thm, mono:thm,
unfold:thm}
val add_inductive : bool -> bool -> string list -> string list
-> thm list -> thm list -> theory -> theory *
{defs:thm list, elims:thm list, raw_induct:thm, induct:thm,
intrs:thm list,
mk_cases:thm list -> string -> thm, mono:thm,
unfold:thm}
end;
structure InductivePackage : INDUCTIVE_PACKAGE =
struct
(*For proving monotonicity of recursion operator*)
val basic_monos = [subset_refl, imp_refl, disj_mono, conj_mono,
ex_mono, Collect_mono, in_mono, vimage_mono];
val Const _ $ (vimage_f $ _) $ _ = HOLogic.dest_Trueprop (concl_of vimageD);
(*Delete needless equality assumptions*)
val refl_thin = prove_goal HOL.thy "!!P. [| a=a; P |] ==> P"
(fn _ => [assume_tac 1]);
(*For simplifying the elimination rule*)
val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE, Pair_inject];
val vimage_name = Sign.intern_const (sign_of Vimage.thy) "op -``";
val mono_name = Sign.intern_const (sign_of Ord.thy) "mono";
(* make injections needed 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 ("Inl", T1 --> T) $ (mk_inj' T1 n2 i)
else
Const ("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;
(**************************** well-formedness checks **************************)
fun err_in_rule sign t msg = error ("Ill-formed introduction rule\n" ^
(Sign.string_of_term sign t) ^ "\n" ^ msg);
fun err_in_prem sign t p msg = error ("Ill-formed premise\n" ^
(Sign.string_of_term sign p) ^ "\nin introduction rule\n" ^
(Sign.string_of_term sign t) ^ "\n" ^ msg);
val msg1 = "Conclusion of introduction rule must have form\
\ ' t : S_i '";
val msg2 = "Premises mentioning recursive sets must have form\
\ ' t : M S_i '";
val msg3 = "Recursion term on left of member symbol";
fun check_rule sign cs r =
let
fun check_prem prem = if exists (Logic.occs o (rpair prem)) cs then
(case prem of
(Const ("op :", _) $ t $ u) =>
if exists (Logic.occs o (rpair t)) cs then
err_in_prem sign r prem msg3 else ()
| _ => err_in_prem sign r prem msg2)
else ()
in (case (HOLogic.dest_Trueprop o Logic.strip_imp_concl) r of
(Const ("op :", _) $ _ $ u) =>
if u mem cs then map (check_prem o HOLogic.dest_Trueprop)
(Logic.strip_imp_prems r)
else err_in_rule sign r msg1
| _ => err_in_rule sign r msg1)
end;
fun try' f msg sign t = (f t) handle _ => error (msg ^ Sign.string_of_term sign t);
(*********************** properties of (co)inductive sets *********************)
(***************************** elimination rules ******************************)
fun mk_elims cs cTs params intr_ts =
let
val used = foldr add_term_names (intr_ts, []);
val [aname, pname] = variantlist (["a", "P"], used);
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;
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));
in
Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (a, c)) ::
map mk_elim_prem (filter (equal c o #1) intrs), P)
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 (variantlist (if length cs < 2 then ["P"] else
map (fn i => "P" ^ string_of_int i) (1 upto length cs), used) ~~
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);
fun subst (prem as (Const ("op :", T) $ t $ u), prems) =
let val n = find_index_eq u cs in
if n >= 0 then prem :: (nth_elem (n, preds)) $ t :: prems else
(subst_free (map (fn (c, P) => (c, HOLogic.mk_binop "op Int"
(c, HOLogic.Collect_const (HOLogic.dest_setT
(fastype_of c)) $ P))) (cs ~~ preds)) prem)::prems
end
| subst (prem, prems) = prem::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 subst
(map HOLogic.dest_Trueprop (Logic.strip_imp_prems r), [])),
HOLogic.mk_Trueprop (nth_elem (find_index_eq u cs, preds) $ t)))
end;
val ind_prems = map mk_ind_prem intr_ts;
(* make conclusions for induction rules *)
fun mk_ind_concl ((c, P), (ts, x)) =
let val T = HOLogic.dest_setT (fastype_of c);
val Ts = HOLogic.prodT_factors T;
val (frees, x') = foldr (fn (T', (fs, s)) =>
((Free (s, T'))::fs, bump_string s)) (Ts, ([], x));
val tuple = HOLogic.mk_tuple 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 (app HOLogic.conj)
(fst (foldr mk_ind_concl (cs ~~ preds, ([], "xa")))))
in (preds, ind_prems, mutual_ind_concl)
end;
(********************** proofs for (co)inductive sets *************************)
(**************************** prove monotonicity ******************************)
fun prove_mono setT fp_fun monos thy =
let
val _ = writeln " Proving monotonicity...";
val mono = prove_goalw_cterm [] (cterm_of (sign_of thy) (HOLogic.mk_Trueprop
(Const (mono_name, (setT --> setT) --> HOLogic.boolT) $ fp_fun)))
(fn _ => [rtac monoI 1, REPEAT (ares_tac (basic_monos @ monos) 1)])
in mono end;
(************************* prove introduction rules ***************************)
fun prove_intrs coind mono fp_def intr_ts con_defs rec_sets_defs thy =
let
val _ = writeln " Proving the introduction rules...";
val unfold = standard (mono RS (fp_def RS
(if coind then def_gfp_Tarski else def_lfp_Tarski)));
fun select_disj 1 1 = []
| select_disj _ 1 = [rtac disjI1]
| select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
val intrs = map (fn (i, intr) => prove_goalw_cterm rec_sets_defs
(cterm_of (sign_of thy) intr) (fn prems =>
[(*insert prems and underlying sets*)
cut_facts_tac prems 1,
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*)
rewrite_goals_tac con_defs,
REPEAT (rtac refl 1)])) (1 upto (length intr_ts) ~~ intr_ts)
in (intrs, unfold) end;
(*************************** prove elimination rules **************************)
fun prove_elims cs cTs params intr_ts unfold rec_sets_defs thy =
let
val _ = writeln " Proving the elimination rules...";
val rules1 = [CollectE, disjE, make_elim vimageD];
val rules2 = [exE, conjE, Inl_neq_Inr, Inr_neq_Inl] @
map make_elim [Inl_inject, Inr_inject];
val elims = map (fn t => prove_goalw_cterm rec_sets_defs
(cterm_of (sign_of thy) t) (fn prems =>
[cut_facts_tac [hd prems] 1,
dtac (unfold RS subst) 1,
REPEAT (FIRSTGOAL (eresolve_tac rules1)),
REPEAT (FIRSTGOAL (eresolve_tac rules2)),
EVERY (map (fn prem =>
DEPTH_SOLVE_1 (ares_tac [prem, conjI] 1)) (tl prems))]))
(mk_elims cs cTs params intr_ts)
in elims end;
(** derivation of simplified elimination rules **)
(*Applies freeness of the given constructors, which *must* be unfolded by
the given defs. Cannot simply use the local con_defs because con_defs=[]
for inference systems.
*)
fun con_elim_tac simps =
let val elim_tac = REPEAT o (eresolve_tac elim_rls)
in ALLGOALS(EVERY'[elim_tac,
asm_full_simp_tac (simpset_of Nat.thy addsimps simps),
elim_tac,
REPEAT o bound_hyp_subst_tac])
THEN prune_params_tac
end;
(*String s should have the form t:Si where Si is an inductive set*)
fun mk_cases elims simps s =
let val prem = assume (read_cterm (sign_of_thm (hd elims)) (s, propT));
val elims' = map (try (fn r =>
rule_by_tactic (con_elim_tac simps) (prem RS r) |> standard)) elims
in case find_first is_some elims' of
Some (Some r) => r
| None => error ("mk_cases: string '" ^ s ^ "' not of form 't : S_i'")
end;
(**************************** prove induction rule ****************************)
fun prove_indrule cs cTs sumT rec_const params intr_ts mono
fp_def rec_sets_defs thy =
let
val _ = writeln " Proving the induction rule...";
val sign = sign_of thy;
val (preds, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
(* 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 ("sum_case",
[T1 --> HOLogic.boolT, T2 --> HOLogic.boolT, T] ---> HOLogic.boolT) $
mk_ind_pred T1 (take (n, Ps)) $ mk_ind_pred T2 (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 => prove_goalw_cterm [] (cterm_of sign
(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)) $
nth_elem (find_index_eq c cs, preds)))))
(fn _ => [rtac vimage_Collect 1, rewrite_goals_tac
(map mk_meta_eq [sum_case_Inl, sum_case_Inr]),
rtac refl 1])) cs;
val induct = prove_goalw_cterm [] (cterm_of sign
(Logic.list_implies (ind_prems, ind_concl))) (fn prems =>
[rtac (impI RS allI) 1,
DETERM (etac (mono RS (fp_def RS def_induct)) 1),
rewrite_goals_tac (map mk_meta_eq (vimage_Int::vimage_simps)),
fold_goals_tac rec_sets_defs,
(*This CollectE and disjE separates out the introduction rules*)
REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE])),
(*Now break down the individual cases. No disjE here in case
some premise involves disjunction.*)
REPEAT (FIRSTGOAL (eresolve_tac [IntE, CollectE, exE, conjE]
ORELSE' hyp_subst_tac)),
rewrite_goals_tac (map mk_meta_eq [sum_case_Inl, sum_case_Inr]),
EVERY (map (fn prem =>
DEPTH_SOLVE_1 (ares_tac [prem, conjI, refl] 1)) prems)]);
val lemma = prove_goalw_cterm rec_sets_defs (cterm_of sign
(Logic.mk_implies (ind_concl, mutual_ind_concl))) (fn prems =>
[cut_facts_tac prems 1,
REPEAT (EVERY
[REPEAT (resolve_tac [conjI, impI] 1),
TRY (dtac vimageD 1), etac allE 1, dtac mp 1, atac 1,
rewrite_goals_tac (map mk_meta_eq [sum_case_Inl, sum_case_Inr]),
atac 1])])
in standard (split_rule (induct RS lemma))
end;
(*************** definitional introduction of (co)inductive sets **************)
fun add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs
intr_ts monos con_defs thy params paramTs cTs cnames =
let
val _ = if verbose then writeln ("Proofs for " ^
(if coind then "co" else "") ^ "inductive set(s) " ^ commas cnames) else ();
val sumT = fold_bal (fn (T, U) => Type ("+", [T, U])) cTs;
val setT = HOLogic.mk_setT sumT;
val fp_name = if coind then Sign.intern_const (sign_of Gfp.thy) "gfp"
else Sign.intern_const (sign_of Lfp.thy) "lfp";
val used = foldr add_term_names (intr_ts, []);
val [sname, xname] = variantlist (["S", "x"], used);
(* 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))
(frees, foldr1 (app HOLogic.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))))
end
(* make a disjunction of all introduction rules *)
val fp_fun = absfree (sname, setT, (HOLogic.Collect_const sumT) $
absfree (xname, sumT, foldr1 (app HOLogic.disj) (map transform_rule intr_ts)));
(* add definiton of recursive sets to theory *)
val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;
val full_rec_name = Sign.full_name (sign_of 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 thy' = thy |>
(if declare_consts then
Theory.add_consts_i (map (fn (c, n) =>
(n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
else I) |>
(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 [(("defs", def_terms), [])];
(* get definitions from theory *)
val fp_def::rec_sets_defs = get_thms thy' "defs";
(* prove and store theorems *)
val mono = prove_mono setT fp_fun monos thy';
val (intrs, unfold) = prove_intrs coind mono fp_def intr_ts con_defs
rec_sets_defs thy';
val elims = if no_elim then [] else
prove_elims cs cTs params intr_ts unfold rec_sets_defs thy';
val raw_induct = if no_ind then TrueI 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 thy';
val induct = if coind orelse no_ind orelse length cs > 1 then raw_induct
else standard (raw_induct RSN (2, rev_mp));
val thy'' = thy' |>
PureThy.add_tthmss [(("intrs", map Attribute.tthm_of intrs), [])] |>
(if no_elim then I else PureThy.add_tthmss
[(("elims", map Attribute.tthm_of elims), [])]) |>
(if no_ind then I else PureThy.add_tthms
[(((if coind then "co" else "") ^ "induct",
Attribute.tthm_of induct), [])]) |>
Theory.parent_path;
in (thy'',
{defs = fp_def::rec_sets_defs,
mono = mono,
unfold = unfold,
intrs = intrs,
elims = elims,
mk_cases = mk_cases elims,
raw_induct = raw_induct,
induct = induct})
end;
(***************** axiomatic introduction of (co)inductive sets ***************)
fun add_ind_axm verbose declare_consts alt_name coind no_elim no_ind cs
intr_ts monos con_defs thy params paramTs cTs cnames =
let
val _ = if verbose then writeln ("Adding axioms for " ^
(if coind then "co" else "") ^ "inductive set(s) " ^ commas cnames) else ();
val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;
val elim_ts = mk_elims cs cTs params intr_ts;
val (_, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
val ind_t = Logic.list_implies (ind_prems, mutual_ind_concl);
val thy' = thy |>
(if declare_consts then
Theory.add_consts_i (map (fn (c, n) =>
(n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
else I) |>
Theory.add_path rec_name |>
PureThy.add_axiomss_i [(("intrs", intr_ts), []), (("elims", elim_ts), [])] |>
(if coind then I
else PureThy.add_axioms_i [(("internal_induct", ind_t), [])]);
val intrs = get_thms thy' "intrs";
val elims = get_thms thy' "elims";
val raw_induct = if coind then TrueI else
standard (split_rule (get_thm thy' "internal_induct"));
val induct = if coind orelse length cs > 1 then raw_induct
else standard (raw_induct RSN (2, rev_mp));
val thy'' = thy' |>
(if coind then I
else PureThy.add_tthms [(("induct", Attribute.tthm_of induct), [])]) |>
Theory.parent_path
in (thy'',
{defs = [],
mono = TrueI,
unfold = TrueI,
intrs = intrs,
elims = elims,
mk_cases = mk_cases elims,
raw_induct = raw_induct,
induct = induct})
end;
(********************** introduction of (co)inductive sets ********************)
fun add_inductive_i verbose declare_consts alt_name coind no_elim no_ind cs
intr_ts monos con_defs thy =
let
val _ = Theory.requires thy "Inductive"
((if coind then "co" else "") ^ "inductive definitions");
val sign = sign_of thy;
(*parameters should agree for all mutually recursive components*)
val (_, params) = strip_comb (hd cs);
val paramTs = map (try' (snd o dest_Free) "Parameter in recursive\
\ component is not a free variable: " sign) params;
val cTs = map (try' (HOLogic.dest_setT o fastype_of)
"Recursive component not of type set: " sign) cs;
val cnames = map (try' (Sign.base_name o fst o dest_Const o head_of)
"Recursive set not previously declared as constant: " sign) cs;
val _ = assert_all Syntax.is_identifier cnames
(fn a => "Base name of recursive set not an identifier: " ^ a);
val _ = map (check_rule sign cs) intr_ts;
in
(if !quick_and_dirty then add_ind_axm else add_ind_def)
verbose declare_consts alt_name coind no_elim no_ind cs intr_ts monos
con_defs thy params paramTs cTs cnames
end;
(***************************** external interface *****************************)
fun add_inductive verbose coind c_strings intr_strings monos con_defs thy =
let
val sign = sign_of thy;
val cs = map (readtm (sign_of thy) HOLogic.termTVar) c_strings;
val intr_ts = map (readtm (sign_of thy) propT) intr_strings;
(* the following code ensures that each recursive set *)
(* always has the same type in all introduction rules *)
val {tsig, ...} = Sign.rep_sg sign;
val add_term_consts_2 =
foldl_aterms (fn (cs, Const c) => c ins cs | (cs, _) => cs);
fun varify (t, (i, ts)) =
let val t' = map_term_types (incr_tvar (i + 1)) (Type.varify (t, []))
in (maxidx_of_term t', t'::ts) end;
val (i, cs') = foldr varify (cs, (~1, []));
val (i', intr_ts') = foldr varify (intr_ts, (i, []));
val rec_consts = foldl add_term_consts_2 ([], cs');
val intr_consts = foldl add_term_consts_2 ([], intr_ts');
fun unify (env, (cname, cT)) =
let val consts = map snd (filter (fn c => fst c = cname) intr_consts)
in (foldl (fn ((env', j'), Tp) => Type.unify tsig j' env' Tp)
(env, (replicate (length consts) cT) ~~ consts)) handle _ =>
error ("Occurrences of constant '" ^ cname ^
"' have incompatible types")
end;
val (env, _) = foldl unify (([], i'), rec_consts);
fun typ_subst_TVars_2 env T = let val T' = typ_subst_TVars env T
in if T = T' then T else typ_subst_TVars_2 env T' end;
val subst = fst o Type.freeze_thaw o
(map_term_types (typ_subst_TVars_2 env));
val cs'' = map subst cs';
val intr_ts'' = map subst intr_ts';
in add_inductive_i verbose false "" coind false false cs'' intr_ts''
monos con_defs thy
end;
end;