src/HOL/WF_Rel.ML

little changes

(*  Title: 	HOL/WF_Rel
    ID:         $Id$
    Author: 	Konrad Slind
    Copyright   1996  TU Munich

Derived wellfounded relations: inverse image, relational product, measure, ...
*)

open WF_Rel;


(*----------------------------------------------------------------------------
 * The inverse image into a wellfounded relation is wellfounded.
 *---------------------------------------------------------------------------*)

goal thy "!!r. wf(r) ==> wf(inv_image r (f::'a=>'b))"; 
by (full_simp_tac (!simpset addsimps [inv_image_def, wf_eq_minimal]) 1);
by (Step_tac 1);
by (subgoal_tac "? (w::'b). w : {w. ? (x::'a). x: Q & (f x = w)}" 1);
by (blast_tac (!claset delrules [allE]) 2);
by (etac allE 1);
by (mp_tac 1);
by (Blast_tac 1);
qed "wf_inv_image";
AddSIs [wf_inv_image];

(*----------------------------------------------------------------------------
 * All measures are wellfounded.
 *---------------------------------------------------------------------------*)

goalw thy [measure_def] "wf (measure f)";
by (rtac wf_inv_image 1);
by (rtac wf_trancl 1);
by (rtac wf_pred_nat 1);
qed "wf_measure";
AddIffs [wf_measure];

(*----------------------------------------------------------------------------
 * Wellfoundedness of lexicographic combinations
 *---------------------------------------------------------------------------*)

goal Prod.thy "!!P. !a b. P((a,b)) ==> !p. P(p)";
by (rtac allI 1);
by (rtac (surjective_pairing RS ssubst) 1);
by (Blast_tac 1);
qed "split_all_pair";

val [wfa,wfb] = goalw thy [wf_def,lex_prod_def]
 "[| wf(ra); wf(rb) |] ==> wf(ra**rb)";
by (EVERY1 [rtac allI,rtac impI, rtac split_all_pair]);
by (rtac (wfa RS spec RS mp) 1);
by (EVERY1 [rtac allI,rtac impI]);
by (rtac (wfb RS spec RS mp) 1);
by (Blast_tac 1);
qed "wf_lex_prod";
AddSIs [wf_lex_prod];

(*----------------------------------------------------------------------------
 * Wellfoundedness of relational product
 *---------------------------------------------------------------------------*)
val [wfa,wfb] = goalw thy [wf_def,rprod_def]
 "[| wf(ra); wf(rb) |] ==> wf(rprod ra rb)";
by (EVERY1 [rtac allI,rtac impI, rtac split_all_pair]);
by (rtac (wfa RS spec RS mp) 1);
by (EVERY1 [rtac allI,rtac impI]);
by (rtac (wfb RS spec RS mp) 1);
by (Blast_tac 1);
qed "wf_rel_prod";
AddSIs [wf_rel_prod];


(*---------------------------------------------------------------------------
 * Wellfoundedness of subsets
 *---------------------------------------------------------------------------*)

goal thy "!!r. [| wf(r);  p<=r |] ==> wf(p)";
by (full_simp_tac (!simpset addsimps [wf_eq_minimal]) 1);
by (Fast_tac 1);
qed "wf_subset";

(*---------------------------------------------------------------------------
 * Wellfoundedness of the empty relation.
 *---------------------------------------------------------------------------*)

goal thy "wf({})";
by (simp_tac (!simpset addsimps [wf_def]) 1);
qed "wf_empty";
AddSIs [wf_empty];


(*---------------------------------------------------------------------------
 * Transitivity of WF combinators.
 *---------------------------------------------------------------------------*)
goalw thy [trans_def, lex_prod_def]
    "!!R1 R2. [| trans R1; trans R2 |] ==> trans (R1 ** R2)";
by (Simp_tac 1);
by (Blast_tac 1);
qed "trans_lex_prod";
AddSIs [trans_lex_prod];

goalw thy [trans_def, rprod_def]
    "!!R1 R2. [| trans R1; trans R2 |] ==> trans (rprod R1 R2)";
by (Simp_tac 1);
by (Blast_tac 1);
qed "trans_rprod";
AddSIs [trans_rprod];


(*---------------------------------------------------------------------------
 * Wellfoundedness of proper subset on finite sets.
 *---------------------------------------------------------------------------*)
goalw thy [finite_psubset_def] "wf(finite_psubset)";
by (rtac (wf_measure RS wf_subset) 1);
by (simp_tac (!simpset addsimps[measure_def, inv_image_def,
				symmetric less_def])1);
by (fast_tac (!claset addIs [psubset_card]) 1);
qed "wf_finite_psubset";