src/ZF/intr_elim.ML
author clasohm
Fri Jul 01 11:03:42 1994 +0200 (1994-07-01 ago)
changeset 444 3ca9d49fd662
parent 435 ca5356bd315a
child 454 0d19ab250cc9
permissions -rw-r--r--
replaced extend_theory by new add_* functions;
changed syntax of datatype declaration
     1 (*  Title: 	ZF/intr-elim.ML
     2     ID:         $Id$
     3     Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1993  University of Cambridge
     5 
     6 Introduction/elimination rule module -- for Inductive/Coinductive Definitions
     7 
     8 Features:
     9 * least or greatest fixedpoints
    10 * user-specified product and sum constructions
    11 * mutually recursive definitions
    12 * definitions involving arbitrary monotone operators
    13 * automatically proves introduction and elimination rules
    14 
    15 The recursive sets must *already* be declared as constants in parent theory!
    16 
    17   Introduction rules have the form
    18   [| ti:M(Sj), ..., P(x), ... |] ==> t: Sk |]
    19   where M is some monotone operator (usually the identity)
    20   P(x) is any (non-conjunctive) side condition on the free variables
    21   ti, t are any terms
    22   Sj, Sk are two of the sets being defined in mutual recursion
    23 
    24 Sums are used only for mutual recursion;
    25 Products are used only to derive "streamlined" induction rules for relations
    26 *)
    27 
    28 signature FP =		(** Description of a fixed point operator **)
    29   sig
    30   val oper	: term			(*fixed point operator*)
    31   val bnd_mono	: term			(*monotonicity predicate*)
    32   val bnd_monoI	: thm			(*intro rule for bnd_mono*)
    33   val subs	: thm			(*subset theorem for fp*)
    34   val Tarski	: thm			(*Tarski's fixed point theorem*)
    35   val induct	: thm			(*induction/coinduction rule*)
    36   end;
    37 
    38 signature PR =			(** Description of a Cartesian product **)
    39   sig
    40   val sigma	: term			(*Cartesian product operator*)
    41   val pair	: term			(*pairing operator*)
    42   val split_const  : term		(*splitting operator*)
    43   val fsplit_const : term		(*splitting operator for formulae*)
    44   val pair_iff	: thm			(*injectivity of pairing, using <->*)
    45   val split_eq	: thm			(*equality rule for split*)
    46   val fsplitI	: thm			(*intro rule for fsplit*)
    47   val fsplitD	: thm			(*destruct rule for fsplit*)
    48   val fsplitE	: thm			(*elim rule for fsplit*)
    49   end;
    50 
    51 signature SU =			(** Description of a disjoint sum **)
    52   sig
    53   val sum	: term			(*disjoint sum operator*)
    54   val inl	: term			(*left injection*)
    55   val inr	: term			(*right injection*)
    56   val elim	: term			(*case operator*)
    57   val case_inl	: thm			(*inl equality rule for case*)
    58   val case_inr	: thm			(*inr equality rule for case*)
    59   val inl_iff	: thm			(*injectivity of inl, using <->*)
    60   val inr_iff	: thm			(*injectivity of inr, using <->*)
    61   val distinct	: thm			(*distinctness of inl, inr using <->*)
    62   val distinct'	: thm			(*distinctness of inr, inl using <->*)
    63   end;
    64 
    65 signature INDUCTIVE =		(** Description of a (co)inductive defn **)
    66   sig
    67   val thy        : theory		(*parent theory*)
    68   val rec_doms   : (string*string) list	(*recursion ops and their domains*)
    69   val sintrs     : string list		(*desired introduction rules*)
    70   val monos      : thm list		(*monotonicity of each M operator*)
    71   val con_defs   : thm list		(*definitions of the constructors*)
    72   val type_intrs : thm list		(*type-checking intro rules*)
    73   val type_elims : thm list		(*type-checking elim rules*)
    74   end;
    75 
    76 signature INTR_ELIM =
    77   sig
    78   val thy        : theory		(*new theory*)
    79   val defs	 : thm list		(*definitions made in thy*)
    80   val bnd_mono   : thm			(*monotonicity for the lfp definition*)
    81   val unfold     : thm			(*fixed-point equation*)
    82   val dom_subset : thm			(*inclusion of recursive set in dom*)
    83   val intrs      : thm list		(*introduction rules*)
    84   val elim       : thm			(*case analysis theorem*)
    85   val raw_induct : thm			(*raw induction rule from Fp.induct*)
    86   val mk_cases : thm list -> string -> thm	(*generates case theorems*)
    87   (*internal items...*)
    88   val big_rec_tm : term			(*the lhs of the fixedpoint defn*)
    89   val rec_tms    : term list		(*the mutually recursive sets*)
    90   val domts      : term list		(*domains of the recursive sets*)
    91   val intr_tms   : term list		(*terms for the introduction rules*)
    92   val rec_params : term list		(*parameters of the recursion*)
    93   val sumprod_free_SEs : thm list       (*destruct rules for Su and Pr*)
    94   end;
    95 
    96 
    97 functor Intr_elim_Fun (structure Ind: INDUCTIVE and 
    98 		       Fp: FP and Pr : PR and Su : SU) : INTR_ELIM =
    99 struct
   100 open Logic Ind;
   101 
   102 (*** Extract basic information from arguments ***)
   103 
   104 val sign = sign_of Ind.thy;
   105 
   106 fun rd T a = 
   107     read_cterm sign (a,T)
   108     handle ERROR => error ("The error above occurred for " ^ a);
   109 
   110 val rec_names = map #1 rec_doms
   111 and domts     = map (term_of o rd iT o #2) rec_doms;
   112 
   113 val dummy = assert_all Syntax.is_identifier rec_names
   114    (fn a => "Name of recursive set not an identifier: " ^ a);
   115 
   116 val dummy = assert_all (is_some o lookup_const sign) rec_names
   117    (fn a => "Name of recursive set not declared as constant: " ^ a);
   118 
   119 val intr_tms = map (term_of o rd propT) sintrs;
   120 
   121 local (*Checking the introduction rules*)
   122   val intr_sets = map (#2 o rule_concl_msg sign) intr_tms;
   123 
   124   fun intr_ok set =
   125       case head_of set of Const(a,recT) => a mem rec_names | _ => false;
   126 
   127   val dummy =  assert_all intr_ok intr_sets
   128      (fn t => "Conclusion of rule does not name a recursive set: " ^ 
   129 	      Sign.string_of_term sign t);
   130 in
   131 val (Const(_,recT),rec_params) = strip_comb (hd intr_sets)
   132 end;
   133 
   134 val rec_hds = map (fn a=> Const(a,recT)) rec_names;
   135 val rec_tms = map (fn rec_hd=> list_comb(rec_hd,rec_params)) rec_hds;
   136 
   137 val dummy = assert_all is_Free rec_params
   138     (fn t => "Param in recursion term not a free variable: " ^
   139              Sign.string_of_term sign t);
   140 
   141 (*** Construct the lfp definition ***)
   142 
   143 val mk_variant = variant (foldr add_term_names (intr_tms,[]));
   144 
   145 val z' = mk_variant"z"
   146 and X' = mk_variant"X"
   147 and w' = mk_variant"w";
   148 
   149 (*simple error-checking in the premises*)
   150 fun chk_prem rec_hd (Const("op &",_) $ _ $ _) =
   151 	error"Premises may not be conjuctive"
   152   | chk_prem rec_hd (Const("op :",_) $ t $ X) = 
   153 	deny (rec_hd occs t) "Recursion term on left of member symbol"
   154   | chk_prem rec_hd t = 
   155 	deny (rec_hd occs t) "Recursion term in side formula";
   156 
   157 fun dest_tprop (Const("Trueprop",_) $ P) = P
   158   | dest_tprop Q = error ("Ill-formed premise of introduction rule: " ^ 
   159 			  Sign.string_of_term sign Q);
   160 
   161 (*Makes a disjunct from an introduction rule*)
   162 fun lfp_part intr = (*quantify over rule's free vars except parameters*)
   163   let val prems = map dest_tprop (strip_imp_prems intr)
   164       val dummy = seq (fn rec_hd => seq (chk_prem rec_hd) prems) rec_hds
   165       val exfrees = term_frees intr \\ rec_params
   166       val zeq = eq_const $ (Free(z',iT)) $ (#1 (rule_concl intr))
   167   in foldr mk_exists (exfrees, fold_bal (app conj) (zeq::prems)) end;
   168 
   169 val dom_sum = fold_bal (app Su.sum) domts;
   170 
   171 (*The Part(A,h) terms -- compose injections to make h*)
   172 fun mk_Part (Bound 0) = Free(X',iT)	(*no mutual rec, no Part needed*)
   173   | mk_Part h         = Part_const $ Free(X',iT) $ Abs(w',iT,h);
   174 
   175 (*Access to balanced disjoint sums via injections*)
   176 val parts = 
   177     map mk_Part (accesses_bal (ap Su.inl, ap Su.inr, Bound 0) 
   178 		              (length rec_doms));
   179 
   180 (*replace each set by the corresponding Part(A,h)*)
   181 val part_intrs = map (subst_free (rec_tms ~~ parts) o lfp_part) intr_tms;
   182 
   183 val lfp_abs = absfree(X', iT, 
   184 	         mk_Collect(z', dom_sum, fold_bal (app disj) part_intrs));
   185 
   186 val lfp_rhs = Fp.oper $ dom_sum $ lfp_abs
   187 
   188 val dummy = seq (fn rec_hd => deny (rec_hd occs lfp_rhs) 
   189 			   "Illegal occurrence of recursion operator")
   190 	 rec_hds;
   191 
   192 (*** Make the new theory ***)
   193 
   194 (*A key definition:
   195   If no mutual recursion then it equals the one recursive set.
   196   If mutual recursion then it differs from all the recursive sets. *)
   197 val big_rec_name = space_implode "_" rec_names;
   198 
   199 (*Big_rec... is the union of the mutually recursive sets*)
   200 val big_rec_tm = list_comb(Const(big_rec_name,recT), rec_params);
   201 
   202 (*The individual sets must already be declared*)
   203 val axpairs = map (mk_defpair sign) 
   204       ((big_rec_tm, lfp_rhs) ::
   205        (case parts of 
   206 	   [_] => [] 			(*no mutual recursion*)
   207 	 | _ => rec_tms ~~		(*define the sets as Parts*)
   208 		map (subst_atomic [(Free(X',iT),big_rec_tm)]) parts));
   209 
   210 val thy = 
   211   Ind.thy
   212   |> add_axioms axpairs
   213   |> add_thyname (big_rec_name ^ "_Inductive");
   214 
   215 val defs = map (get_axiom thy o #1) axpairs;
   216 
   217 val big_rec_def::part_rec_defs = defs;
   218 
   219 val prove = prove_term (sign_of thy);
   220 
   221 (********)
   222 val dummy = writeln "Proving monotonicity...";
   223 
   224 val bnd_mono = 
   225     prove [] (mk_tprop (Fp.bnd_mono $ dom_sum $ lfp_abs), 
   226        fn _ =>
   227        [rtac (Collect_subset RS bnd_monoI) 1,
   228 	REPEAT (ares_tac (basic_monos @ monos) 1)]);
   229 
   230 val dom_subset = standard (big_rec_def RS Fp.subs);
   231 
   232 val unfold = standard (bnd_mono RS (big_rec_def RS Fp.Tarski));
   233 
   234 (********)
   235 val dummy = writeln "Proving the introduction rules...";
   236 
   237 (*Mutual recursion: Needs subset rules for the individual sets???*)
   238 val rec_typechecks = [dom_subset] RL (asm_rl::monos) RL [subsetD];
   239 
   240 (*Type-checking is hardest aspect of proof;
   241   disjIn selects the correct disjunct after unfolding*)
   242 fun intro_tacsf disjIn prems = 
   243   [(*insert prems and underlying sets*)
   244    cut_facts_tac prems 1,
   245    rtac (unfold RS ssubst) 1,
   246    REPEAT (resolve_tac [Part_eqI,CollectI] 1),
   247    (*Now 2-3 subgoals: typechecking, the disjunction, perhaps equality.*)
   248    rtac disjIn 2,
   249    REPEAT (ares_tac [refl,exI,conjI] 2),
   250    rewrite_goals_tac con_defs,
   251    (*Now can solve the trivial equation*)
   252    REPEAT (rtac refl 2),
   253    REPEAT (FIRSTGOAL (eresolve_tac (asm_rl::PartE::type_elims)
   254 		      ORELSE' hyp_subst_tac
   255 		      ORELSE' dresolve_tac rec_typechecks)),
   256    DEPTH_SOLVE (swap_res_tac type_intrs 1)];
   257 
   258 (*combines disjI1 and disjI2 to access the corresponding nested disjunct...*)
   259 val mk_disj_rls = 
   260     let fun f rl = rl RS disjI1
   261         and g rl = rl RS disjI2
   262     in  accesses_bal(f, g, asm_rl)  end;
   263 
   264 val intrs = map (prove part_rec_defs) 
   265 	       (intr_tms ~~ map intro_tacsf (mk_disj_rls(length intr_tms)));
   266 
   267 (********)
   268 val dummy = writeln "Proving the elimination rule...";
   269 
   270 (*Includes rules for succ and Pair since they are common constructions*)
   271 val elim_rls = [asm_rl, FalseE, succ_neq_0, sym RS succ_neq_0, 
   272 		Pair_neq_0, sym RS Pair_neq_0, make_elim succ_inject, 
   273 		refl_thin, conjE, exE, disjE];
   274 
   275 val sumprod_free_SEs = 
   276     map (gen_make_elim [conjE,FalseE])
   277         ([Su.distinct, Su.distinct', Su.inl_iff, Su.inr_iff, Pr.pair_iff] 
   278 	 RL [iffD1]);
   279 
   280 (*Breaks down logical connectives in the monotonic function*)
   281 val basic_elim_tac =
   282     REPEAT (SOMEGOAL (eresolve_tac (elim_rls@sumprod_free_SEs)
   283               ORELSE' bound_hyp_subst_tac))
   284     THEN prune_params_tac;
   285 
   286 val elim = rule_by_tactic basic_elim_tac (unfold RS equals_CollectD);
   287 
   288 (*Applies freeness of the given constructors, which *must* be unfolded by
   289   the given defs.  Cannot simply use the local con_defs because con_defs=[] 
   290   for inference systems. *)
   291 fun con_elim_tac defs =
   292     rewrite_goals_tac defs THEN basic_elim_tac THEN fold_tac defs;
   293 
   294 (*String s should have the form t:Si where Si is an inductive set*)
   295 fun mk_cases defs s = 
   296     rule_by_tactic (con_elim_tac defs)
   297       (assume_read thy s  RS  elim);
   298 
   299 val defs = big_rec_def::part_rec_defs;
   300 
   301 val raw_induct = standard ([big_rec_def, bnd_mono] MRS Fp.induct);
   302 
   303 end;