src/ZF/AC/recfunAC16.ML

-(*  Title: 	ZF/AC/recfunAC16.ML
+(*  Title:      ZF/AC/recfunAC16.ML
     ID:         $Id$
-    Author: 	Krzysztof Grabczewski
+    Author:     Krzysztof Grabczewski
 
 Properties of the recursive definition used in the proof of WO2 ==> AC16
 *)
 
 open recfunAC16;
 
 (* ********************************************************************** *)
-(* Basic properties of recfunAC16					  *)
+(* Basic properties of recfunAC16                                         *)
 (* ********************************************************************** *)
 
 goalw thy [recfunAC16_def] "recfunAC16(f,fa,0,a) = 0";
 by (rtac transrec2_0 1);
 val recfunAC16_0 = result();
 
 goalw thy [recfunAC16_def] "recfunAC16(f,fa,succ(i),a) =  \
 \       if (EX y:recfunAC16(f,fa,i,a). fa ` i <= y, recfunAC16(f,fa,i,a), \
 \       recfunAC16(f,fa,i,a) Un {f ` (LEAST j. fa ` i <= f ` j &  \
-\	(ALL b<a. (fa`b <= f`j  \
-\	--> (ALL t:recfunAC16(f,fa,i,a). ~ fa`b <= t))))})";
+\       (ALL b<a. (fa`b <= f`j  \
+\       --> (ALL t:recfunAC16(f,fa,i,a). ~ fa`b <= t))))})";
 by (rtac transrec2_succ 1);
 val recfunAC16_succ = result();
 
 goalw thy [recfunAC16_def] "!!i. Limit(i)  \
-\	==> recfunAC16(f,fa,i,a) = (UN j<i. recfunAC16(f,fa,j,a))";
+\       ==> recfunAC16(f,fa,i,a) = (UN j<i. recfunAC16(f,fa,j,a))";
 by (etac transrec2_Limit 1);
 val recfunAC16_Limit = result();
 
 (* ********************************************************************** *)
-(* Monotonicity of transrec2						  *)
+(* Monotonicity of transrec2                                              *)
 (* ********************************************************************** *)
 
 val [prem1, prem2] = goal thy 
     "[| !!g r. r <= B(g,r); Ord(i) |]  \
-\	==> j<i --> transrec2(j, 0, B) <= transrec2(i, 0, B)";
+\       ==> j<i --> transrec2(j, 0, B) <= transrec2(i, 0, B)";
 by (resolve_tac [prem2 RS trans_induct] 1);
 by (rtac Ord_cases 1 THEN (REPEAT (assume_tac 1)));
 by (fast_tac lt_cs 1);
 by (asm_simp_tac (AC_ss addsimps [transrec2_succ]) 1);
 by (fast_tac (FOL_cs addSIs [succI1, prem1]
-	addSEs [ballE, leE, prem1 RSN (2, subset_trans)]) 1);
+        addSEs [ballE, leE, prem1 RSN (2, subset_trans)]) 1);
 by (fast_tac (AC_cs addIs [OUN_I, ltI]
-	addSEs [Limit_has_succ RS ltE, succI1 RSN (2, Ord_in_Ord) RS le_refl,
-		transrec2_Limit RS ssubst]) 1);
+        addSEs [Limit_has_succ RS ltE, succI1 RSN (2, Ord_in_Ord) RS le_refl,
+                transrec2_Limit RS ssubst]) 1);
 val transrec2_mono_lemma = result();
 
 val [prem1, prem2] = goal thy "[| !!g r. r <= B(g,r); j le i |]  \
-\	==> transrec2(j, 0, B) <= transrec2(i, 0, B)";
+\       ==> transrec2(j, 0, B) <= transrec2(i, 0, B)";
 by (resolve_tac [prem2 RS leE] 1);
 by (resolve_tac [transrec2_mono_lemma RS impE] 1);
 by (TRYALL (fast_tac (AC_cs addSIs [prem1, prem2, lt_Ord2])));
 val transrec2_mono = result();
 
 (* ********************************************************************** *)
-(* Monotonicity of recfunAC16						  *)
+(* Monotonicity of recfunAC16                                             *)
 (* ********************************************************************** *)
 
 goalw thy [recfunAC16_def]
-	"!!i. i le j ==> recfunAC16(f, g, i, a) <= recfunAC16(f, g, j, a)";
+        "!!i. i le j ==> recfunAC16(f, g, i, a) <= recfunAC16(f, g, j, a)";
 by (rtac transrec2_mono 1);
 by (REPEAT (fast_tac (AC_cs addIs [expand_if RS iffD2]) 1));
 val recfunAC16_mono = result();