ex/term.ML

Initial revision

(*  Title: 	HOL/ex/term
    ID:         $Id$
    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1992  University of Cambridge

For term.thy.  illustrates List functor
  (essentially the same type as in Trees & Forests)
*)

open Term;

(*** Monotonicity and unfolding of the function ***)

goal Term.thy "mono(%Z.  A <*> List(Z))";
by (REPEAT (ares_tac [monoI, subset_refl, List_mono, uprod_mono] 1));
val Term_fun_mono = result();

val Term_unfold = Term_fun_mono RS (Term_def RS def_lfp_Tarski);

(*This justifies using Term in other recursive type definitions*)
goalw Term.thy [Term_def] "!!A B. A<=B ==> Term(A) <= Term(B)";
by (REPEAT (ares_tac [lfp_mono, subset_refl, List_mono, uprod_mono] 1));
val Term_mono = result();

(** Type checking rules -- Term creates well-founded sets **)

val prems = goalw Term.thy [Term_def] "Term(Sexp) <= Sexp";
by (rtac lfp_lowerbound 1);
by (fast_tac (univ_cs addIs [Sexp_SconsI, List_Sexp RS subsetD]) 1);
val Term_Sexp = result();

(* A <= Sexp ==> Term(A) <= Sexp *)
val Term_subset_Sexp = standard
    (Term_mono RS (Term_Sexp RSN (2,subset_trans)));


(** Elimination -- structural induction on the set Term(A) **)

(*Induction for the set Term(A) *)
val [major,minor] = goal Term.thy 
    "[| M: Term(A);  \
\       !!x zs. [| x: A;  zs: List(Term(A));  zs: List({x.R(x)}) \
\               |] ==> R(x.zs)  \
\    |] ==> R(M)";
by (rtac (major RS (Term_def RS def_induct)) 1);
by (rtac Term_fun_mono 1);
by (REPEAT (eresolve_tac ([uprodE, ssubst, minor] @
 		([Int_lower1,Int_lower2] RL [List_mono RS subsetD])) 1));
val Term_induct = result();

(*Induction on Term(A) followed by induction on List *)
val major::prems = goal Term.thy
    "[| M: Term(A);  \
\       !!x.      [| x: A |] ==> R(x.NIL);  \
\       !!x z zs. [| x: A;  z: Term(A);  zs: List(Term(A));  R(x.zs)  \
\                 |] ==> R(x . CONS(z,zs))  \
\    |] ==> R(M)";
by (rtac (major RS Term_induct) 1);
by (etac List_induct 1);
by (REPEAT (ares_tac prems 1));
val Term_induct2 = result();

(*** Structural Induction on the abstract type 'a term ***)

val list_all_ss = map_ss addsimps [list_all_Nil, list_all_Cons];

val Rep_Term_in_Sexp =
    Rep_Term RS (range_Leaf_subset_Sexp RS Term_subset_Sexp RS subsetD);

(*Induction for the abstract type 'a term*)
val prems = goalw Term.thy [App_def,Rep_TList_def,Abs_TList_def]
    "[| !!x ts. list_all(R,ts) ==> R(App(x,ts))  \
\    |] ==> R(t)";
by (rtac (Rep_Term_inverse RS subst) 1);   (*types force good instantiation*)
by (res_inst_tac [("P","Rep_Term(t) : Sexp")] conjunct2 1);
by (rtac (Rep_Term RS Term_induct) 1);
by (REPEAT (ares_tac [conjI, Sexp_SconsI, Term_subset_Sexp RS 
    List_subset_Sexp,range_Leaf_subset_Sexp] 1
     ORELSE etac rev_subsetD 1));
by (eres_inst_tac [("A1","Term(?u)"), ("f1","Rep_Term"), ("g1","Abs_Term")]
    	(Abs_map_inverse RS subst) 1);
by (rtac (range_Leaf_subset_Sexp RS Term_subset_Sexp) 1);
by (etac Abs_Term_inverse 1);
by (etac rangeE 1);
by (hyp_subst_tac 1);
by (resolve_tac prems 1);
by (etac List_induct 1);
by (etac CollectE 2);
by (stac Abs_map_CONS 2);
by (etac conjunct1 2);
by (etac rev_subsetD 2);
by (rtac List_subset_Sexp 2);
by (fast_tac set_cs 2);
by (ALLGOALS (asm_simp_tac list_all_ss));
val term_induct = result();

(*Induction for the abstract type 'a term*)
val prems = goal Term.thy 
    "[| !!x. R(App(x,Nil));  \
\       !!x t ts. R(App(x,ts)) ==> R(App(x, Cons(t,ts)))  \
\    |] ==> R(t)";
by (rtac term_induct 1);  (*types force good instantiation*)
by (etac rev_mp 1);
by (rtac list_induct 1);  (*types force good instantiation*)
by (ALLGOALS (asm_simp_tac (list_all_ss addsimps prems)));
val term_induct2 = result();

(*Perform induction on xs. *)
fun term_ind2_tac a i = 
    EVERY [res_inst_tac [("t",a)] term_induct2 i,
	   rename_last_tac a ["1","s"] (i+1)];


(** Introduction rule for Term **)

(* c : A <*> List(Term(A)) ==> c : Term(A) *)
val TermI = standard (Term_unfold RS equalityD2 RS subsetD);

(*The constant APP is not declared; it is simply . *)
val prems = goal Term.thy "[| M: A;  N : List(Term(A)) |] ==> M.N : Term(A)";
by (REPEAT (resolve_tac (prems@[TermI, ListI, uprodI]) 1));
val APP_I = result();


(*** Term_rec -- by wf recursion on pred_Sexp ***)

val Term_rec_unfold =
    wf_pred_Sexp RS wf_trancl RS (Term_rec_def RS def_wfrec);

(** conversion rules **)

val [prem] = goal Term.thy
    "N: List(Term(A)) ==>  \
\    !M. <N,M>: pred_Sexp^+ --> \
\        Abs_map(cut(h, pred_Sexp^+, M), N) = \
\        Abs_map(h,N)";
by (rtac (prem RS List_induct) 1);
by (simp_tac list_all_ss 1);
by (strip_tac 1);
by (etac (pred_Sexp_CONS_D RS conjE) 1);
by (asm_simp_tac (list_all_ss addsimps [trancl_pred_SexpD1, cut_apply]) 1);
val Abs_map_lemma = result();

val [prem1,prem2,A_subset_Sexp] = goal Term.thy
    "[| M: Sexp;  N: List(Term(A));  A<=Sexp |] ==> \
\    Term_rec(M.N, d) = d(M, N, Abs_map(%Z. Term_rec(Z,d), N))";
by (rtac (Term_rec_unfold RS trans) 1);
by (rtac (Split RS trans) 1);
by (simp_tac (HOL_ss addsimps
      [prem2 RS Abs_map_lemma RS spec RS mp, pred_SexpI2 RS r_into_trancl,
       prem1, prem2 RS rev_subsetD, List_subset_Sexp,
       Term_subset_Sexp, A_subset_Sexp])1);
val Term_rec = result();

(*** term_rec -- by Term_rec ***)

local
  val Rep_map_type1 = read_instantiate_sg (sign_of Term.thy)
                        [("f","Rep_Term")] Rep_map_type;
  val Rep_TList = Rep_Term RS Rep_map_type1;
  val Rep_Term_rec = range_Leaf_subset_Sexp RSN (2,Rep_TList RSN(2,Term_rec));

  (*Now avoids conditional rewriting with the premise N: List(Term(A)),
    since A will be uninstantiated and will cause rewriting to fail. *)
  val term_rec_ss = HOL_ss 
      addsimps [Rep_TList RS (rangeI RS APP_I RS Abs_Term_inverse),  
	       Rep_Term_in_Sexp, Rep_Term_rec, Rep_Term_inverse,
	       inj_Leaf, Inv_f_f,
	       Abs_Rep_map, map_ident, Sexp_LeafI]
in

val term_rec = prove_goalw Term.thy
	 [term_rec_def, App_def, Rep_TList_def, Abs_TList_def]
    "term_rec(App(f,ts), d) = d(f, ts, map (%t. term_rec(t,d), ts))"
 (fn _ => [simp_tac term_rec_ss 1])

end;