src/ZF/InfDatatype.ML

 val fun_Limit_VfromE = 
     [apply_funtype, InfCard_csucc RS InfCard_is_Limit] MRS Limit_VfromE
 	|> standard;
 
 goal InfDatatype.thy
-    "!!K. [| f: I -> Vfrom(A,csucc(K));  |I| le K;  InfCard(K)	\
-\         |] ==> EX j. f: I -> Vfrom(A,j) & j < csucc(K)";
-by (res_inst_tac [("x", "UN x:I. LEAST i. f`x : Vfrom(A,i)")] exI 1);
+    "!!K. [| f: W -> Vfrom(A,csucc(K));  |W| le K;  InfCard(K)	\
+\         |] ==> EX j. f: W -> Vfrom(A,j) & j < csucc(K)";
+by (res_inst_tac [("x", "UN w:W. LEAST i. f`w : Vfrom(A,i)")] exI 1);
 by (resolve_tac [conjI] 1);
-by (resolve_tac [ballI RSN (2,cardinal_UN_Ord_lt_csucc)] 2);
-by (eresolve_tac [fun_Limit_VfromE] 3 THEN REPEAT_SOME assume_tac);
+by (resolve_tac [le_UN_Ord_lt_csucc] 2);
+by (rtac ballI 4  THEN
+    eresolve_tac [fun_Limit_VfromE] 4 THEN REPEAT_SOME assume_tac);
 by (fast_tac (ZF_cs addEs [Least_le RS lt_trans1, ltE]) 2);
 by (resolve_tac [Pi_type] 1);
-by (rename_tac "k" 2);
+by (rename_tac "w" 2);
 by (eresolve_tac [fun_Limit_VfromE] 2 THEN REPEAT_SOME assume_tac);
-by (subgoal_tac "f`k : Vfrom(A, LEAST i. f`k : Vfrom(A,i))" 1);
+by (subgoal_tac "f`w : Vfrom(A, LEAST i. f`w : Vfrom(A,i))" 1);
 by (fast_tac (ZF_cs addEs [LeastI, ltE]) 2);
 by (eresolve_tac [[subset_refl, UN_upper] MRS Vfrom_mono RS subsetD] 1);
 by (assume_tac 1);
 val fun_Vcsucc_lemma = result();
 
 goal InfDatatype.thy
-    "!!K. [| f: K -> Vfrom(A,csucc(K));  InfCard(K)	\
-\         |] ==> EX j. f: K -> Vfrom(A,j) & j < csucc(K)";
-by (res_inst_tac [("x", "UN k:K. LEAST i. f`k : Vfrom(A,i)")] exI 1);
-by (resolve_tac [conjI] 1);
-by (resolve_tac [ballI RSN (2,cardinal_UN_Ord_lt_csucc)] 2);
-by (eresolve_tac [fun_Limit_VfromE] 3 THEN REPEAT_SOME assume_tac);
-by (fast_tac (ZF_cs addEs [Least_le RS lt_trans1, ltE]) 2);
-by (resolve_tac [Pi_type] 1);
-by (rename_tac "k" 2);
-by (eresolve_tac [fun_Limit_VfromE] 2 THEN REPEAT_SOME assume_tac);
-by (subgoal_tac "f`k : Vfrom(A, LEAST i. f`k : Vfrom(A,i))" 1);
-by (fast_tac (ZF_cs addEs [LeastI, ltE]) 2);
-by (eresolve_tac [[subset_refl, UN_upper] MRS Vfrom_mono RS subsetD] 1);
-by (assume_tac 1);
-val fun_Vcsucc_lemma = result();
+    "!!K. [| W <= Vfrom(A,csucc(K));  |W| le K;  InfCard(K)	\
+\         |] ==> EX j. W <= Vfrom(A,j) & j < csucc(K)";
+by (asm_full_simp_tac (ZF_ss addsimps [subset_iff_id, fun_Vcsucc_lemma]) 1);
+val subset_Vcsucc = result();
 
+(*Version for arbitrary index sets*)
 goal InfDatatype.thy
-    "!!K. InfCard(K) ==> K -> Vfrom(A,csucc(K)) <= Vfrom(A,csucc(K))";
-by (safe_tac (ZF_cs addSDs [fun_Vcsucc_lemma]));
+    "!!K. [| |W| le K;  W <= Vfrom(A,csucc(K));  InfCard(K) |] ==> \
+\         W -> Vfrom(A,csucc(K)) <= Vfrom(A,csucc(K))";
+by (safe_tac (ZF_cs addSDs [fun_Vcsucc_lemma, subset_Vcsucc]));
 by (resolve_tac [Vfrom RS ssubst] 1);
 by (eresolve_tac [PiE] 1);
 (*This level includes the function, and is below csucc(K)*)
-by (res_inst_tac [("a1", "succ(succ(K Un j))")] (UN_I RS UnI2) 1);
+by (res_inst_tac [("a1", "succ(succ(j Un ja))")] (UN_I RS UnI2) 1);
 by (eresolve_tac [subset_trans RS PowI] 2);
-by (safe_tac (ZF_cs addSIs [Pair_in_Vfrom]));
-by (fast_tac (ZF_cs addIs [i_subset_Vfrom RS subsetD]) 2);
-by (eresolve_tac [[subset_refl, Un_upper2] MRS Vfrom_mono RS subsetD] 2);
+by (fast_tac (ZF_cs addIs [Pair_in_Vfrom, Vfrom_UnI1, Vfrom_UnI2]) 2);
+
 by (REPEAT (ares_tac [ltD, InfCard_csucc, InfCard_is_Limit, 
 		      Limit_has_succ, Un_least_lt] 1));
-by (eresolve_tac [InfCard_is_Card RS Card_is_Ord RS lt_csucc] 1);
-by (assume_tac 1);
 val fun_Vcsucc = result();
+
+goal InfDatatype.thy
+    "!!K. [| f: W -> Vfrom(A, csucc(K));  |W| le K;  InfCard(K);	\
+\            W <= Vfrom(A,csucc(K)) 					\
+\         |] ==> f: Vfrom(A,csucc(K))";
+by (REPEAT (ares_tac [fun_Vcsucc RS subsetD] 1));
+val fun_in_Vcsucc = result();
+
+goal InfDatatype.thy
+    "!!K. [| W <= Vfrom(A,csucc(K));  B <= Vfrom(A,csucc(K));	\
+\            |W| le K;  InfCard(K)				\
+\         |] ==> W -> B <= Vfrom(A, csucc(K))";
+by (REPEAT (ares_tac [[Pi_mono, fun_Vcsucc] MRS subset_trans] 1));
+val fun_subset_Vcsucc = result();
+
+goal InfDatatype.thy
+    "!!f. [| f: W -> B;  W <= Vfrom(A,csucc(K));  B <= Vfrom(A,csucc(K)); \
+\            |W| le K;  InfCard(K) 					  \
+\         |] ==> f: Vfrom(A,csucc(K))";
+by (DEPTH_SOLVE (ares_tac [fun_subset_Vcsucc RS subsetD] 1));
+val fun_into_Vcsucc = result();
+
+(*Version where K itself is the index set*)
+goal InfDatatype.thy
+    "!!K. InfCard(K) ==> K -> Vfrom(A,csucc(K)) <= Vfrom(A,csucc(K))";
+by (forward_tac [InfCard_is_Card RS Card_is_Ord] 1);
+by (REPEAT (ares_tac [fun_Vcsucc, Ord_cardinal_le,
+		      i_subset_Vfrom,
+		      lt_csucc RS leI RS le_imp_subset RS subset_trans] 1));
+val Card_fun_Vcsucc = result();
 
 goal InfDatatype.thy
     "!!K. [| f: K -> Vfrom(A, csucc(K));  InfCard(K) \
 \         |] ==> f: Vfrom(A,csucc(K))";
-by (REPEAT (ares_tac [fun_Vcsucc RS subsetD] 1));
-val fun_in_Vcsucc = result();
+by (REPEAT (ares_tac [Card_fun_Vcsucc RS subsetD] 1));
+val Card_fun_in_Vcsucc = result();
 
-val fun_subset_Vcsucc = 
-	[Pi_mono, fun_Vcsucc] MRS subset_trans |> standard;
+val Card_fun_subset_Vcsucc = 
+	[Pi_mono, Card_fun_Vcsucc] MRS subset_trans |> standard;
 
 goal InfDatatype.thy
     "!!f. [| f: K -> B;  B <= Vfrom(A,csucc(K));  InfCard(K) \
 \         |] ==> f: Vfrom(A,csucc(K))";
-by (REPEAT (ares_tac [fun_subset_Vcsucc RS subsetD] 1));
-val fun_into_Vcsucc = result();
-
-val Limit_csucc = InfCard_csucc RS InfCard_is_Limit |> standard;
+by (REPEAT (ares_tac [Card_fun_subset_Vcsucc RS subsetD] 1));
+val Card_fun_into_Vcsucc = result();
 
 val Pair_in_Vcsucc = Limit_csucc RSN (3, Pair_in_VLimit) |> standard;
 val Inl_in_Vcsucc  = Limit_csucc RSN (2, Inl_in_VLimit) |> standard;
 val Inr_in_Vcsucc  = Limit_csucc RSN (2, Inr_in_VLimit) |> standard;
 val zero_in_Vcsucc = Limit_csucc RS zero_in_VLimit |> standard;
 val nat_into_Vcsucc = Limit_csucc RSN (2, nat_into_VLimit) |> standard;
 
 (*For most K-branching datatypes with domain Vfrom(A, csucc(K)) *)
 val inf_datatype_intrs =  
-    [fun_in_Vcsucc, InfCard_nat, Pair_in_Vcsucc, 
+    [Card_fun_in_Vcsucc, fun_in_Vcsucc, InfCard_nat, Pair_in_Vcsucc, 
      Inl_in_Vcsucc, Inr_in_Vcsucc, 
      zero_in_Vcsucc, A_into_Vfrom, nat_into_Vcsucc] @ datatype_intrs;