src/ZF/intr_elim.ML

New file of just the theory primitives

(*  Title:      ZF/intr_elim.ML
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1994  University of Cambridge

Introduction/elimination rule module -- for Inductive/Coinductive Definitions
*)

signature INDUCTIVE_ARG =       (** Description of a (co)inductive def **)
  sig
  val thy        : theory               (*new theory with inductive defs*)
  val monos      : thm list             (*monotonicity of each M operator*)
  val con_defs   : thm list             (*definitions of the constructors*)
  val type_intrs : thm list             (*type-checking intro rules*)
  val type_elims : thm list             (*type-checking elim rules*)
  end;


signature INDUCTIVE_I =         (** Terms read from the theory section **)
  sig
  val rec_tms    : term list            (*the recursive sets*)
  val dom_sum    : term                 (*their common domain*)
  val intr_tms   : term list            (*terms for the introduction rules*)
  end;

signature INTR_ELIM =
  sig
  val thy        : theory               (*copy of input theory*)
  val defs       : thm list             (*definitions made in thy*)
  val bnd_mono   : thm                  (*monotonicity for the lfp definition*)
  val dom_subset : thm                  (*inclusion of recursive set in dom*)
  val intrs      : thm list             (*introduction rules*)
  val elim       : thm                  (*case analysis theorem*)
  val mk_cases   : thm list -> string -> thm    (*generates case theorems*)
  end;

signature INTR_ELIM_AUX =       (** Used to make induction rules **)
  sig
  val raw_induct : thm                  (*raw induction rule from Fp.induct*)
  val rec_names  : string list          (*names of recursive sets*)
  end;

(*prove intr/elim rules for a fixedpoint definition*)
functor Intr_elim_Fun
    (structure Inductive: sig include INDUCTIVE_ARG INDUCTIVE_I end  
     and Fp: FP and Pr : PR and Su : SU) 
    : sig include INTR_ELIM INTR_ELIM_AUX end =
let

val rec_names = map (#1 o dest_Const o head_of) Inductive.rec_tms;
val big_rec_name = space_implode "_" rec_names;

val _ = deny (big_rec_name  mem  map ! (stamps_of_thy Inductive.thy))
             ("Definition " ^ big_rec_name ^ 
              " would clash with the theory of the same name!");

(*fetch fp definitions from the theory*)
val big_rec_def::part_rec_defs = 
  map (get_def Inductive.thy)
      (case rec_names of [_] => rec_names | _ => big_rec_name::rec_names);


val sign = sign_of Inductive.thy;

(********)
val _ = writeln "  Proving monotonicity...";

val Const("==",_) $ _ $ (_ $ dom_sum $ fp_abs) =
    big_rec_def |> rep_thm |> #prop |> Logic.unvarify;

val bnd_mono = 
    prove_goalw_cterm [] 
      (cterm_of sign (Ind_Syntax.mk_tprop (Fp.bnd_mono $ dom_sum $ fp_abs)))
      (fn _ =>
       [rtac (Collect_subset RS bnd_monoI) 1,
        REPEAT (ares_tac (basic_monos @ Inductive.monos) 1)]);

val dom_subset = standard (big_rec_def RS Fp.subs);

val unfold = standard ([big_rec_def, bnd_mono] MRS Fp.Tarski);

(********)
val _ = writeln "  Proving the introduction rules...";

(*Mutual recursion?  Helps to derive subset rules for the individual sets.*)
val Part_trans =
    case rec_names of
         [_] => asm_rl
       | _   => standard (Part_subset RS subset_trans);

(*To type-check recursive occurrences of the inductive sets, possibly
  enclosed in some monotonic operator M.*)
val rec_typechecks = 
   [dom_subset] RL (asm_rl :: ([Part_trans] RL Inductive.monos)) RL [subsetD];

(*Type-checking is hardest aspect of proof;
  disjIn selects the correct disjunct after unfolding*)
fun intro_tacsf disjIn prems = 
  [(*insert prems and underlying sets*)
   cut_facts_tac prems 1,
   DETERM (rtac (unfold RS ssubst) 1),
   REPEAT (resolve_tac [Part_eqI,CollectI] 1),
   (*Now 2-3 subgoals: typechecking, the disjunction, perhaps equality.*)
   rtac disjIn 2,
   (*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] 2 ORELSE assume_tac 2),
   (*Now solve the equations like Tcons(a,f) = Inl(?b4)*)
   rewrite_goals_tac Inductive.con_defs,
   REPEAT (rtac refl 2),
   (*Typechecking; this can fail*)
   REPEAT (FIRSTGOAL (        dresolve_tac rec_typechecks
                      ORELSE' eresolve_tac (asm_rl::PartE::SigmaE2::
                                            Inductive.type_elims)
                      ORELSE' hyp_subst_tac)),
   DEPTH_SOLVE (swap_res_tac (SigmaI::subsetI::Inductive.type_intrs) 1)];

(*combines disjI1 and disjI2 to access the corresponding nested disjunct...*)
val mk_disj_rls = 
    let fun f rl = rl RS disjI1
        and g rl = rl RS disjI2
    in  accesses_bal(f, g, asm_rl)  end;

val intrs = map (uncurry (prove_goalw_cterm part_rec_defs))
            (map (cterm_of sign) Inductive.intr_tms ~~ 
             map intro_tacsf (mk_disj_rls(length Inductive.intr_tms)));

(********)
val _ = writeln "  Proving the elimination rule...";

(*Breaks down logical connectives in the monotonic function*)
val basic_elim_tac =
    REPEAT (SOMEGOAL (eresolve_tac (Ind_Syntax.elim_rls @ Su.free_SEs)
              ORELSE' bound_hyp_subst_tac))
    THEN prune_params_tac
        (*Mutual recursion: collapse references to Part(D,h)*)
    THEN fold_tac part_rec_defs;

in
  struct
  val thy = Inductive.thy
  and defs = big_rec_def :: part_rec_defs
  and bnd_mono   = bnd_mono
  and dom_subset = dom_subset
  and intrs      = intrs;

  val elim = rule_by_tactic basic_elim_tac 
                  (unfold RS Ind_Syntax.equals_CollectD);

  (*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. 
    String s should have the form t:Si where Si is an inductive set*)
  fun mk_cases defs s = 
      rule_by_tactic (rewrite_goals_tac defs THEN 
                      basic_elim_tac THEN
                      fold_tac defs)
        (assume_read Inductive.thy s  RS  elim);

  val raw_induct = standard ([big_rec_def, bnd_mono] MRS Fp.induct)
  and rec_names = rec_names
  end
end;