src/HOL/WF_Rel.ML

 
 (*----------------------------------------------------------------------------
  * "Less than" on the natural numbers
  *---------------------------------------------------------------------------*)
 
-goalw thy [less_than_def] "wf less_than"; 
+Goalw [less_than_def] "wf less_than"; 
 by (rtac (wf_pred_nat RS wf_trancl) 1);
 qed "wf_less_than";
 AddIffs [wf_less_than];
 
-goalw thy [less_than_def] "trans less_than"; 
+Goalw [less_than_def] "trans less_than"; 
 by (rtac trans_trancl 1);
 qed "trans_less_than";
 AddIffs [trans_less_than];
 
-goalw thy [less_than_def, less_def] "((x,y): less_than) = (x<y)"; 
+Goalw [less_than_def, less_def] "((x,y): less_than) = (x<y)"; 
 by (Simp_tac 1);
 qed "less_than_iff";
 AddIffs [less_than_iff];
 
 (*----------------------------------------------------------------------------
  * The inverse image into a wellfounded relation is wellfounded.
  *---------------------------------------------------------------------------*)
 
-goal thy "!!r. wf(r) ==> wf(inv_image r (f::'a=>'b))"; 
+Goal "!!r. wf(r) ==> wf(inv_image r (f::'a=>'b))"; 
 by (full_simp_tac (simpset() addsimps [inv_image_def, wf_eq_minimal]) 1);
 by (Clarify_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];
 
-goalw thy [trans_def,inv_image_def]
+Goalw [trans_def,inv_image_def]
     "!!r. trans r ==> trans (inv_image r f)";
 by (Simp_tac 1);
 by (Blast_tac 1);
 qed "trans_inv_image";
 
 
 (*----------------------------------------------------------------------------
  * All measures are wellfounded.
  *---------------------------------------------------------------------------*)
 
-goalw thy [measure_def] "wf (measure f)";
+Goalw [measure_def] "wf (measure f)";
 by (rtac (wf_less_than RS wf_inv_image) 1);
 qed "wf_measure";
 AddIffs [wf_measure];
 
 val measure_induct = standard

 AddSIs [wf_lex_prod];
 
 (*---------------------------------------------------------------------------
  * Transitivity of WF combinators.
  *---------------------------------------------------------------------------*)
-goalw thy [trans_def, lex_prod_def]
+Goalw [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];
 
 
 (*---------------------------------------------------------------------------
  * Wellfoundedness of proper subset on finite sets.
  *---------------------------------------------------------------------------*)
-goalw thy [finite_psubset_def] "wf(finite_psubset)";
+Goalw [finite_psubset_def] "wf(finite_psubset)";
 by (rtac (wf_measure RS wf_subset) 1);
 by (simp_tac (simpset() addsimps [measure_def, inv_image_def, less_than_def,
 				 symmetric less_def])1);
 by (fast_tac (claset() addSIs [psubset_card]) 1);
 qed "wf_finite_psubset";
 
-goalw thy [finite_psubset_def, trans_def] "trans finite_psubset";
+Goalw [finite_psubset_def, trans_def] "trans finite_psubset";
 by (simp_tac (simpset() addsimps [psubset_def]) 1);
 by (Blast_tac 1);
 qed "trans_finite_psubset";
 
 (*---------------------------------------------------------------------------
  * Wellfoundedness of finite acyclic relations
  * Cannot go into WF because it needs Finite
  *---------------------------------------------------------------------------*)
 
-goal thy "!!r. finite r ==> acyclic r --> wf r";
+Goal "!!r. finite r ==> acyclic r --> wf r";
 by (etac finite_induct 1);
  by (Blast_tac 1);
 by (split_all_tac 1);
 by (Asm_full_simp_tac 1);
 qed_spec_mp "finite_acyclic_wf";

 qed_goal "finite_acyclic_wf_converse" thy 
  "!!X. [|finite r; acyclic r|] ==> wf (r^-1)" (K [
 	etac (finite_converse RS iffD2 RS finite_acyclic_wf) 1,
 	etac (acyclic_converse RS iffD2) 1]);
 
-goal thy "!!r. finite r ==> wf r = acyclic r";
+Goal "!!r. finite r ==> wf r = acyclic r";
 by (blast_tac (claset() addIs [finite_acyclic_wf,wf_acyclic]) 1);
 qed "wf_iff_acyclic_if_finite";
 
 
 (*---------------------------------------------------------------------------
  * A relation is wellfounded iff it has no infinite descending chain
  *---------------------------------------------------------------------------*)
 
-goalw thy [wf_eq_minimal RS eq_reflection]
+Goalw [wf_eq_minimal RS eq_reflection]
   "wf r = (~(? f. !i. (f(Suc i),f i) : r))";
 by (rtac iffI 1);
  by (rtac notI 1);
  by (etac exE 1);
  by (eres_inst_tac [("x","{w. ? i. w=f i}")] allE 1);