(* 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]
"!!f. [| is_recfun r H a f; ~(b,a):r |] ==> f(b) = (@z.True)";
by (etac ssubst 1);
by (asm_simp_tac HOL_ss 1);
qed "is_recfun_undef";
(*** NOTE! some simplifications need a different finish_tac!! ***)
fun indhyp_tac hyps =
(cut_facts_tac hyps THEN'
DEPTH_SOLVE_1 o (ares_tac [TrueI] ORELSE'
eresolve_tac [transD, mp, allE]));
val wf_super_ss = HOL_ss addsolver indhyp_tac;
val prems = goalw WF.thy [is_recfun_def,cut_def]
"[| wf(r); trans(r); is_recfun r H a f; is_recfun r H b 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);
qed_spec_mp "is_recfun_equal";
val prems as [wfr,transr,recfa,recgb,_] = goalw WF.thy [cut_def]
"[| wf(r); trans(r); \
\ is_recfun r H a f; is_recfun r H b 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);
qed "is_recfun_cut";
(*** Main Existence Lemma -- Basic Properties of the_recfun ***)
val prems = goalw WF.thy [the_recfun_def]
"is_recfun r H a f ==> is_recfun r H a (the_recfun r H a)";
by (res_inst_tac [("P", "is_recfun r H a")] selectI 1);
by (resolve_tac prems 1);
qed "is_the_recfun";
val prems = goal WF.thy
"[| wf(r); trans(r) |] ==> is_recfun r H a (the_recfun r H a)";
by (cut_facts_tac prems 1);
by (wf_ind_tac "a" prems 1);
by (res_inst_tac [("f","cut (%y. H (the_recfun r H y) y) r a1")]
is_the_recfun 1);
by (rewtac is_recfun_def);
by (stac cuts_eq 1);
by (rtac allI 1);
by (rtac impI 1);
by (res_inst_tac [("f1","H"),("x","y")](arg_cong RS fun_cong) 1);
by (subgoal_tac
"the_recfun r H y = cut(%x. H(cut(the_recfun r H y) r x) x) r y" 1);
by (etac allE 2);
by (dtac impE 2);
by (atac 2);
by (atac 3);
by (atac 2);
by (etac ssubst 1);
by (simp_tac (HOL_ss addsimps [cuts_eq]) 1);
by (rtac allI 1);
by (rtac impI 1);
by (asm_simp_tac (wf_super_ss addsimps[cut_apply,is_recfun_cut,cuts_eq]) 1);
by (res_inst_tac [("f1","H"),("x","ya")](arg_cong RS fun_cong) 1);
by (fold_tac [is_recfun_def]);
by (asm_simp_tac (wf_super_ss addsimps[cut_apply,is_recfun_cut,cuts_eq]) 1);
qed "unfold_the_recfun";
val unwind1_the_recfun = rewrite_rule[is_recfun_def] unfold_the_recfun;
(*--------------Old proof-----------------------------------------------------
val prems = goal WF.thy
"[| wf(r); trans(r) |] ==> is_recfun r H a (the_recfun r H a)";
by (cut_facts_tac prems 1);
by (wf_ind_tac "a" prems 1);
by (res_inst_tac [("f", "cut (%y. wftrec r H y) r a1")] is_the_recfun 1);
by (rewrite_goals_tac [is_recfun_def, wftrec_def]);
by (stac cuts_eq 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,cuts_eq]) 1);
qed "unfold_the_recfun";
---------------------------------------------------------------------------*)
(** Removal of the premise trans(r) **)
val th = rewrite_rule[is_recfun_def]
(trans_trancl RSN (2,(wf_trancl RS unfold_the_recfun)));
goalw WF.thy [wfrec_def]
"!!r. wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a";
by (rtac H_cong 1);
by (rtac refl 2);
by (simp_tac (HOL_ss addsimps [cuts_eq]) 1);
by (rtac allI 1);
by (rtac impI 1);
by (simp_tac(HOL_ss addsimps [wfrec_def]) 1);
by (res_inst_tac [("a1","a")] (th RS ssubst) 1);
by (atac 1);
by (forward_tac[wf_trancl] 1);
by (forward_tac[r_into_trancl] 1);
by (asm_simp_tac (HOL_ss addsimps [cut_apply]) 1);
by (rtac H_cong 1); (*expose the equality of cuts*)
by (rtac refl 2);
by (simp_tac (HOL_ss addsimps [cuts_eq, cut_apply, r_into_trancl]) 1);
by (strip_tac 1);
by (res_inst_tac [("r","r^+")] is_recfun_equal 1);
by (atac 1);
by (rtac trans_trancl 1);
by (rtac unfold_the_recfun 1);
by (atac 1);
by (rtac trans_trancl 1);
by (rtac unfold_the_recfun 1);
by (atac 1);
by (rtac trans_trancl 1);
by (rtac transD 1);
by (rtac trans_trancl 1);
by (forw_inst_tac [("a","ya")] r_into_trancl 1);
by (atac 1);
by (atac 1);
by (forw_inst_tac [("a","ya")] r_into_trancl 1);
by (atac 1);
qed "wfrec";
(*--------------Old proof-----------------------------------------------------
goalw WF.thy [wfrec_def]
"!!r. wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) 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 [cuts_eq, cut_apply, r_into_trancl]) 1);
qed "wfrec";
---------------------------------------------------------------------------*)
(*---------------------------------------------------------------------------
* This form avoids giant explosions in proofs. NOTE USE OF ==
*---------------------------------------------------------------------------*)
val rew::prems = goal WF.thy
"[| f==wfrec r H; wf(r) |] ==> f(a) = H (cut f r a) a";
by (rewtac rew);
by (REPEAT (resolve_tac (prems@[wfrec]) 1));
qed "def_wfrec";