src/HOL/WF.ML

Removed LK since Sequents contains everything in it

(*  Title:      HOL/wf.ML
    ID:         $Id$
    Author:     Tobias Nipkow, with minor changes by Konrad Slind
    Copyright   1992  University of Cambridge/1995 TU Munich

For WF.thy.  Wellfoundedness, induction, and  recursion
*)

open WF;

val H_cong = read_instantiate [("f","H")] (standard(refl RS cong RS cong));
val H_cong1 = refl RS H_cong;

(*Restriction to domain A.  If r is well-founded over A then wf(r)*)
val [prem1,prem2] = goalw WF.thy [wf_def]
 "[| r <= A Times A;  \
\    !!x P. [| ! x. (! y. (y,x) : r --> P(y)) --> P(x);  x:A |] ==> P(x) |]  \
\ ==>  wf(r)";
by (strip_tac 1);
by (rtac allE 1);
by (assume_tac 1);
by (best_tac (!claset addSEs [prem1 RS subsetD RS SigmaE2] addIs [prem2]) 1);
qed "wfI";

val major::prems = goalw WF.thy [wf_def]
    "[| wf(r);          \
\       !!x.[| ! y. (y,x): r --> P(y) |] ==> P(x) \
\    |]  ==>  P(a)";
by (rtac (major RS spec RS mp RS spec) 1);
by (fast_tac (!claset addEs prems) 1);
qed "wf_induct";

(*Perform induction on i, then prove the wf(r) subgoal using prems. *)
fun wf_ind_tac a prems i = 
    EVERY [res_inst_tac [("a",a)] wf_induct i,
           rename_last_tac a ["1"] (i+1),
           ares_tac prems i];

val prems = goal WF.thy "[| wf(r);  (a,x):r;  (x,a):r |] ==> P";
by (subgoal_tac "! x. (a,x):r --> (x,a):r --> P" 1);
by (fast_tac (!claset addIs prems) 1);
by (wf_ind_tac "a" prems 1);
by (Fast_tac 1);
qed "wf_asym";

val prems = goal WF.thy "[| wf(r);  (a,a): r |] ==> P";
by (rtac wf_asym 1);
by (REPEAT (resolve_tac prems 1));
qed "wf_irrefl";

(*transitive closure of a wf relation is wf! *)
val [prem] = goal WF.thy "wf(r) ==> wf(r^+)";
by (rewtac wf_def);
by (strip_tac 1);
(*must retain the universal formula for later use!*)
by (rtac allE 1 THEN assume_tac 1);
by (etac mp 1);
by (res_inst_tac [("a","x")] (prem RS wf_induct) 1);
by (rtac (impI RS allI) 1);
by (etac tranclE 1);
by (Fast_tac 1);
by (Fast_tac 1);
qed "wf_trancl";


(** cut **)

(*This rewrite rule works upon formulae; thus it requires explicit use of
  H_cong to expose the equality*)
goalw WF.thy [cut_def]
    "(cut f r x = cut g r x) = (!y. (y,x):r --> f(y)=g(y))";
by (simp_tac (HOL_ss addsimps [expand_fun_eq]
                    setloop (split_tac [expand_if])) 1);
qed "cuts_eq";

goalw WF.thy [cut_def] "!!x. (x,a):r ==> (cut f r a)(x) = f(x)";
by (asm_simp_tac HOL_ss 1);
qed "cut_apply";

(*** is_recfun ***)

goalw WF.thy [is_recfun_def,cut_def]