src/HOL/WF.ML
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Tue, 30 Jan 1996 15:24:36 +0100
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(*  Title:      HOL/WF.ML
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    ID:         $Id$
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    Author:     Tobias Nipkow
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    Copyright   1992  University of Cambridge
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For WF.thy.  Well-founded Recursion
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*)
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open WF;
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val H_cong = read_instantiate [("f","H")] (standard(refl RS cong RS cong));
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val H_cong1 = refl RS H_cong;
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(*Restriction to domain A.  If r is well-founded over A then wf(r)*)
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val [prem1,prem2] = goalw WF.thy [wf_def]
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 "[| r <= Sigma A (%u.A);  \
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\    !!x P. [| ! x. (! y. (y,x) : r --> P(y)) --> P(x);  x:A |] ==> P(x) |]  \
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\ ==>  wf(r)";
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by (strip_tac 1);
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by (rtac allE 1);
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by (assume_tac 1);
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by (best_tac (HOL_cs addSEs [prem1 RS subsetD RS SigmaE2] addIs [prem2]) 1);
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qed "wfI";
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val major::prems = goalw WF.thy [wf_def]
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    "[| wf(r);          \
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\       !!x.[| ! y. (y,x): r --> P(y) |] ==> P(x) \
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\    |]  ==>  P(a)";
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by (rtac (major RS spec RS mp RS spec) 1);
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by (fast_tac (HOL_cs addEs prems) 1);
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qed "wf_induct";
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(*Perform induction on i, then prove the wf(r) subgoal using prems. *)
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fun wf_ind_tac a prems i = 
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    EVERY [res_inst_tac [("a",a)] wf_induct i,
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           rename_last_tac a ["1"] (i+1),
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           ares_tac prems i];
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val prems = goal WF.thy "[| wf(r);  (a,x):r;  (x,a):r |] ==> P";
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by (subgoal_tac "! x. (a,x):r --> (x,a):r --> P" 1);
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by (fast_tac (HOL_cs addIs prems) 1);
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by (wf_ind_tac "a" prems 1);
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by (fast_tac set_cs 1);
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qed "wf_asym";
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val prems = goal WF.thy "[| wf(r);  (a,a): r |] ==> P";
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by (rtac wf_asym 1);
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by (REPEAT (resolve_tac prems 1));
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qed "wf_anti_refl";
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(*transitive closure of a WF relation is WF!*)
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val [prem] = goal WF.thy "wf(r) ==> wf(r^+)";
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by (rewtac wf_def);
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by (strip_tac 1);
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(*must retain the universal formula for later use!*)
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by (rtac allE 1 THEN assume_tac 1);
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by (etac mp 1);
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by (res_inst_tac [("a","x")] (prem RS wf_induct) 1);
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by (rtac (impI RS allI) 1);
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by (etac tranclE 1);
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by (fast_tac HOL_cs 1);
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by (fast_tac HOL_cs 1);
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qed "wf_trancl";
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(** cut **)
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(*This rewrite rule works upon formulae; thus it requires explicit use of
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  H_cong to expose the equality*)
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goalw WF.thy [cut_def]
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    "(cut f r x = cut g r x) = (!y. (y,x):r --> f(y)=g(y))";
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by(simp_tac (!simpset addsimps [expand_fun_eq]
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                        setloop (split_tac [expand_if])) 1);
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qed "cut_cut_eq";
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goalw WF.thy [cut_def] "!!x. (x,a):r ==> (cut f r a)(x) = f(x)";
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by(Asm_simp_tac 1);
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qed "cut_apply";
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(*** is_recfun ***)
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goalw WF.thy [is_recfun_def,cut_def]
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    "!!f. [| is_recfun r a H f;  ~(b,a):r |] ==> f(b) = (@z.True)";
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by (etac ssubst 1);
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by(Asm_simp_tac 1);
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qed "is_recfun_undef";
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(*eresolve_tac transD solves (a,b):r using transitivity AT MOST ONCE
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  mp amd allE  instantiate induction hypotheses*)
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fun indhyp_tac hyps =
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    ares_tac (TrueI::hyps) ORELSE' 
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    (cut_facts_tac hyps THEN'
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       DEPTH_SOLVE_1 o (ares_tac [TrueI] ORELSE'
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                        eresolve_tac [transD, mp, allE]));
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(*** NOTE! some simplifications need a different finish_tac!! ***)
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fun indhyp_tac hyps =
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    resolve_tac (TrueI::refl::hyps) ORELSE' 
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    (cut_facts_tac hyps THEN'
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       DEPTH_SOLVE_1 o (ares_tac [TrueI] ORELSE'
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                        eresolve_tac [transD, mp, allE]));
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val wf_super_ss = !simpset setsolver indhyp_tac;
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val prems = goalw WF.thy [is_recfun_def,cut_def]
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    "[| wf(r);  trans(r);  is_recfun r a H f;  is_recfun r b H g |] ==> \
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    \ (x,a):r --> (x,b):r --> f(x)=g(x)";
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by (cut_facts_tac prems 1);
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by (etac wf_induct 1);
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by (REPEAT (rtac impI 1 ORELSE etac ssubst 1));
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by (asm_simp_tac (wf_super_ss addcongs [if_cong]) 1);
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qed "is_recfun_equal_lemma";
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bind_thm ("is_recfun_equal", (is_recfun_equal_lemma RS mp RS mp));
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val prems as [wfr,transr,recfa,recgb,_] = goalw WF.thy [cut_def]
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    "[| wf(r);  trans(r); \
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\       is_recfun r a H f;  is_recfun r b H g;  (b,a):r |] ==> \
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\    cut f r b = g";
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val gundef = recgb RS is_recfun_undef
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and fisg   = recgb RS (recfa RS (transr RS (wfr RS is_recfun_equal)));
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by (cut_facts_tac prems 1);
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by (rtac ext 1);
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by (asm_simp_tac (wf_super_ss addsimps [gundef,fisg]
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                              setloop (split_tac [expand_if])) 1);
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qed "is_recfun_cut";
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(*** Main Existence Lemma -- Basic Properties of the_recfun ***)
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val prems = goalw WF.thy [the_recfun_def]
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    "is_recfun r a H f ==> is_recfun r a H (the_recfun r a H)";
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by (res_inst_tac [("P", "is_recfun r a H")] selectI 1);
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by (resolve_tac prems 1);
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qed "is_the_recfun";
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val prems = goal WF.thy
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    "[| wf(r);  trans(r) |] ==> is_recfun r a H (the_recfun r a H)";
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by (cut_facts_tac prems 1);
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by (wf_ind_tac "a" prems 1);
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by (res_inst_tac [("f", "cut (%y. wftrec r y H) r a1")] is_the_recfun 1);
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by (rewrite_goals_tac [is_recfun_def, wftrec_def]);
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by (rtac (cut_cut_eq RS ssubst) 1);
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(*Applying the substitution: must keep the quantified assumption!!*)
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by (EVERY1 [strip_tac, rtac H_cong1, rtac allE, atac,
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            etac (mp RS ssubst), atac]);
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by (fold_tac [is_recfun_def]);
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by (asm_simp_tac (wf_super_ss addsimps[cut_apply,is_recfun_cut,cut_cut_eq]) 1);
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qed "unfold_the_recfun";
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(*Beware incompleteness of unification!*)
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val prems = goal WF.thy
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    "[| wf(r);  trans(r);  (c,a):r;  (c,b):r |] \
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\    ==> the_recfun r a H c = the_recfun r b H c";
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by (DEPTH_SOLVE (ares_tac (prems@[is_recfun_equal,unfold_the_recfun]) 1));
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qed "the_recfun_equal";
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val prems = goal WF.thy
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    "[| wf(r); trans(r); (b,a):r |] \
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\    ==> cut (the_recfun r a H) r b = the_recfun r b H";
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by (REPEAT (ares_tac (prems@[is_recfun_cut,unfold_the_recfun]) 1));
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qed "the_recfun_cut";
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(*** Unfolding wftrec ***)
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goalw WF.thy [wftrec_def]
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    "!!r. [| wf(r);  trans(r) |] ==> \
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\    wftrec r a H = H a (cut (%x.wftrec r x H) r a)";
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by (EVERY1 [stac (rewrite_rule [is_recfun_def] unfold_the_recfun),
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            REPEAT o atac, rtac H_cong1]);
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by (asm_simp_tac (!simpset addsimps [cut_cut_eq,the_recfun_cut]) 1);
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qed "wftrec";
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(*Unused but perhaps interesting*)
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val prems = goal WF.thy
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    "[| wf(r);  trans(r);  !!f x. H x (cut f r x) = H x f |] ==> \
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\               wftrec r a H = H a (%x.wftrec r x H)";
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by (rtac (wftrec RS trans) 1);
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by (REPEAT (resolve_tac prems 1));
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qed "wftrec2";
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(** Removal of the premise trans(r) **)
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goalw WF.thy [wfrec_def]
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    "!!r. wf(r) ==> wfrec r a H = H a (cut (%x.wfrec r x H) r a)";
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by (etac (wf_trancl RS wftrec RS ssubst) 1);
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by (rtac trans_trancl 1);
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by (rtac (refl RS H_cong) 1);    (*expose the equality of cuts*)
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by (simp_tac (!simpset addsimps [cut_cut_eq, cut_apply, r_into_trancl]) 1);
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qed "wfrec";
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(*This form avoids giant explosions in proofs.  NOTE USE OF == *)
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val rew::prems = goal WF.thy
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    "[| !!x. f(x)==wfrec r x H;  wf(r) |] ==> f(a) = H a (cut (%x.f(x)) r a)";
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by (rewtac rew);
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by (REPEAT (resolve_tac (prems@[wfrec]) 1));
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qed "def_wfrec";