Classical tactics now use default claset.
(* Title: HOL/ex/Term
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
Terms over a given alphabet -- function applications; illustrates list functor
(essentially the same type as in Trees & Forests)
*)
open Term;
(*** Monotonicity and unfolding of the function ***)
goal Term.thy "term(A) = A <*> list(term(A))";
by (fast_tac (!claset addSIs (equalityI :: term.intrs)
addEs [term.elim]) 1);
qed "term_unfold";
(*This justifies using term in other recursive type definitions*)
goalw Term.thy term.defs "!!A B. A<=B ==> term(A) <= term(B)";
by (REPEAT (ares_tac ([lfp_mono, list_mono] @ basic_monos) 1));
qed "term_mono";
(** Type checking -- term creates well-founded sets **)
goalw Term.thy term.defs "term(sexp) <= sexp";
by (rtac lfp_lowerbound 1);
by (fast_tac (!claset addIs [sexp.SconsI, list_sexp RS subsetD]) 1);
qed "term_sexp";
(* A <= sexp ==> term(A) <= sexp *)
bind_thm ("term_subset_sexp", ([term_mono, term_sexp] MRS 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$zs) \
\ |] ==> R(M)";
by (rtac (major RS term.induct) 1);
by (REPEAT (eresolve_tac ([minor] @
([Int_lower1,Int_lower2] RL [list_mono RS subsetD])) 1));
(*Proof could also use mono_Int RS subsetD RS IntE *)
qed "Term_induct";
(*Induction on term(A) followed by induction on list *)
val major::prems = goal Term.thy
"[| M: term(A); \
\ !!x. [| x: A |] ==> R(x$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));
qed "Term_induct2";
(*** Structural Induction on the abstract type 'a term ***)
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. (ALL t: setOfList ts. R t) ==> 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 (ALLGOALS Asm_simp_tac);
by (ALLGOALS Fast_tac);
qed "term_induct";
(*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 (t#ts)) \
\ |] ==> R(t)";
by (rtac term_induct 1); (*types force good instantiation*)
by (etac rev_mp 1);
by (rtac list_induct2 1); (*types force good instantiation*)
by (ALLGOALS (asm_simp_tac (!simpset addsimps prems)));
qed "term_induct2";
(*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)];
(*** Term_rec -- by wf recursion on pred_sexp ***)
goal Term.thy
"(%M. Term_rec M d) = wfrec (trancl pred_sexp) \
\ (%g. Split(%x y. d x y (Abs_map g y)))";
by (simp_tac (HOL_ss addsimps [Term_rec_def]) 1);
bind_thm("Term_rec_unfold", (wf_pred_sexp RS wf_trancl) RS
((result() RS eq_reflection) RS def_wfrec));
(*---------------------------------------------------------------------------
* Old:
* 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 1);
by (strip_tac 1);
by (etac (pred_sexp_CONS_D RS conjE) 1);
by (asm_simp_tac (!simpset addsimps [trancl_pred_sexpD1]) 1);
qed "Abs_map_lemma";
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 (simp_tac (HOL_ss addsimps
[Split,
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);
qed "Term_rec";
(*** 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 term.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_ident2, 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;