wf.ML

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(*  Title: 	HOL/wf.ML
    ID:         $Id$
    Author: 	Tobias Nipkow
    Copyright   1992  University of Cambridge

For wf.thy.  Well-founded Recursion
*)

open WF;

val H_cong = read_instantiate [("f","H::[?'a, ?'a=>?'b]=>?'b")]
               (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 <= Sigma(A, %u.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 (HOL_cs addSEs [prem1 RS subsetD RS SigmaE2] addIs [prem2]) 1);
val wfI = result();

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 (HOL_cs addEs prems) 1);
val wf_induct = result();

(*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 (HOL_cs addIs prems) 1);
by (wf_ind_tac "a" prems 1);
by (fast_tac set_cs 1);
val wf_anti_sym = result();

val prems = goal WF.thy "[| wf(r);  <a,a>: r |] ==> P";
by (rtac wf_anti_sym 1);
by (REPEAT (resolve_tac prems 1));
val wf_anti_refl = result();

(*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 HOL_cs 1);
by (fast_tac HOL_cs 1);
val wf_trancl = result();


(** 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);
val cut_cut_eq = result();

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


(*** is_recfun ***)

goalw WF.thy [is_recfun_def,cut_def]
    "!!f. [| is_recfun(r,a,H,f);  ~<b,a>:r |] ==> f(b) = (@z.True)";
by (etac ssubst 1);
by(asm_simp_tac HOL_ss 1);
val is_recfun_undef = result();

(*eresolve_tac transD solves <a,b>:r using transitivity AT MOST ONCE
  mp amd allE  instantiate induction hypotheses*)
fun indhyp_tac hyps =
    ares_tac (TrueI::hyps) ORELSE' 
    (cut_facts_tac hyps THEN'
       DEPTH_SOLVE_1 o (ares_tac [TrueI] ORELSE'
		        eresolve_tac [transD, mp, allE]));

(*** NOTE! some simplifications need a different finish_tac!! ***)
fun indhyp_tac hyps =
    resolve_tac (TrueI::refl::hyps) ORELSE' 
    (cut_facts_tac hyps THEN'
       DEPTH_SOLVE_1 o (ares_tac [TrueI] ORELSE'
		        eresolve_tac [transD, mp, allE]));
val wf_super_ss = HOL_ss setsolver indhyp_tac;

val prems = goalw WF.thy [is_recfun_def,cut_def]
    "[| wf(r);  trans(r);  is_recfun(r,a,H,f);  is_recfun(r,b,H,g) |] ==> \
    \ <x,a>:r --> <x,b>:r --> f(x)=g(x)";
by (cut_facts_tac prems 1);
by (etac wf_induct 1);
by (REPEAT (rtac impI 1 ORELSE etac ssubst 1));
by (asm_simp_tac (wf_super_ss addcongs [if_cong]) 1);
val is_recfun_equal_lemma = result();
val is_recfun_equal = standard (is_recfun_equal_lemma RS mp RS mp);


val prems as [wfr,transr,recfa,recgb,_] = goalw WF.thy [cut_def]
    "[| wf(r);  trans(r); \
\       is_recfun(r,a,H,f);  is_recfun(r,b,H,g);  <b,a>:r |] ==> \
\    cut(f,r,b) = g";
val gundef = recgb RS is_recfun_undef
and fisg   = recgb RS (recfa RS (transr RS (wfr RS is_recfun_equal)));
by (cut_facts_tac prems 1);
by (rtac ext 1);
by (asm_simp_tac (wf_super_ss addsimps [gundef,fisg]
                              setloop (split_tac [expand_if])) 1);
val is_recfun_cut = result();

(*** Main Existence Lemma -- Basic Properties of the_recfun ***)

val prems = goalw WF.thy [the_recfun_def]
    "is_recfun(r,a,H,f) ==> is_recfun(r, a, H, the_recfun(r,a,H))";
by (res_inst_tac [("P", "is_recfun(r,a,H)")] selectI 1);
by (resolve_tac prems 1);
val is_the_recfun = result();

val prems = goal WF.thy
    "[| wf(r);  trans(r) |] ==> is_recfun(r, a, H, the_recfun(r,a,H))";
by (cut_facts_tac prems 1);
by (wf_ind_tac "a" prems 1);
by (res_inst_tac [("f", "cut(%y. wftrec(r,y,H), r, a1)")] is_the_recfun 1);
by (rewrite_goals_tac [is_recfun_def, wftrec_def]);
by (rtac (cut_cut_eq RS ssubst) 1);
(*Applying the substitution: must keep the quantified assumption!!*)
by (EVERY1 [strip_tac, rtac H_cong1, rtac allE, atac,
            etac (mp RS ssubst), atac]);
by (fold_tac [is_recfun_def]);
by (asm_simp_tac (wf_super_ss addsimps[cut_apply,is_recfun_cut,cut_cut_eq]) 1);
val unfold_the_recfun = result();


(*Beware incompleteness of unification!*)
val prems = goal WF.thy
    "[| wf(r);  trans(r);  <c,a>:r;  <c,b>:r |] \
\    ==> the_recfun(r,a,H,c) = the_recfun(r,b,H,c)";
by (DEPTH_SOLVE (ares_tac (prems@[is_recfun_equal,unfold_the_recfun]) 1));
val the_recfun_equal = result();

val prems = goal WF.thy
    "[| wf(r); trans(r); <b,a>:r |] \
\    ==> cut(the_recfun(r,a,H),r,b) = the_recfun(r,b,H)";
by (REPEAT (ares_tac (prems@[is_recfun_cut,unfold_the_recfun]) 1));
val the_recfun_cut = result();

(*** Unfolding wftrec ***)

goalw WF.thy [wftrec_def]
    "!!r. [| wf(r);  trans(r) |] ==> \
\    wftrec(r,a,H) = H(a, cut(%x.wftrec(r,x,H), r, a))";
by (EVERY1 [stac (rewrite_rule [is_recfun_def] unfold_the_recfun),
	    REPEAT o atac, rtac H_cong1]);
by (asm_simp_tac (HOL_ss addsimps [cut_cut_eq,the_recfun_cut]) 1);
val wftrec = result();

(*Unused but perhaps interesting*)
val prems = goal WF.thy
    "[| wf(r);  trans(r);  !!f x. H(x, cut(f,r,x)) = H(x,f) |] ==> \
\		wftrec(r,a,H) = H(a, %x.wftrec(r,x,H))";
by (rtac (wftrec RS trans) 1);
by (REPEAT (resolve_tac prems 1));
val wftrec2 = result();

(** Removal of the premise trans(r) **)

goalw WF.thy [wfrec_def]
    "!!r. wf(r) ==> wfrec(r,a,H) = H(a, cut(%x.wfrec(r,x,H), r, a))";
by (etac (wf_trancl RS wftrec RS ssubst) 1);
by (rtac trans_trancl 1);
by (rtac (refl RS H_cong) 1);    (*expose the equality of cuts*)
by (simp_tac (HOL_ss addsimps [cut_cut_eq, cut_apply, r_into_trancl]) 1);
val wfrec = result();

(*This form avoids giant explosions in proofs.  NOTE USE OF == *)
val rew::prems = goal WF.thy
    "[| !!x. f(x)==wfrec(r,x,H);  wf(r) |] ==> f(a) = H(a, cut(%x.f(x),r,a))";
by (rewtac rew);
by (REPEAT (resolve_tac (prems@[wfrec]) 1));
val def_wfrec = result();