lots of new results about sublist, bag_of

--- a/src/HOL/UNITY/AllocBase.ML	Fri Jun 02 18:30:38 2000 +0200
+++ b/src/HOL/UNITY/AllocBase.ML	Fri Jun 02 18:31:18 2000 +0200
@@ -5,7 +5,8 @@
 
 Basis declarations for Chandy and Charpentier's Allocator
 
-with_path "../Induct" time_use_thy "AllocBase";
+add_path "../Induct";
+time_use_thy "AllocBase";
 *)
 
 Goal "(ALL i. i<n --> f i <= (g i :: nat)) --> sum_below f n <= sum_below g n";
@@ -25,11 +26,101 @@
 by (blast_tac (claset() addIs [tokens_mono_prefix]) 1);
 qed "mono_tokens";
 
-Goal "length xs <= k --> map fst [p:zip xs [0..length xs(] . snd p < k] = xs";
-by (res_inst_tac [("xs","xs")] rev_induct 1);
+
+(*** sublist ***)
+
+Goalw [sublist_def] "sublist l {} = []";
+by Auto_tac;
+qed "sublist_empty";
+
+Goalw [sublist_def] "sublist [] A = []";
 by Auto_tac;
-qed_spec_mp "sublist_length_lemma";
+qed "sublist_nil";
+
+Goal "map fst [p:zip xs [i..i + length xs(] . snd p : A] =     \
+\     map fst [p:zip xs [0..length xs(] . snd p + i : A]";
+by (res_inst_tac [("xs","xs")] rev_induct 1);
+ by (asm_simp_tac (simpset() addsimps [add_commute]) 2);
+by (Simp_tac 1);
+qed "sublist_shift_lemma";
+
+Goalw [sublist_def]
+     "sublist (l@l') A = sublist l A @ sublist l' {j. j + length l : A}";
+by (res_inst_tac [("xs","l'")] rev_induct 1);
+by (Simp_tac 1);
+by (asm_simp_tac (simpset() addsimps [inst "i" "0" upt_add_eq_append, 
+	                              zip_append, sublist_shift_lemma]) 1);
+by (asm_simp_tac (simpset() addsimps [add_commute]) 1);
+qed "sublist_append";
+
+Addsimps [sublist_empty, sublist_nil];
+
+Goal "sublist (x#l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}";
+by (res_inst_tac [("xs","l")] rev_induct 1);
+ by (asm_simp_tac (simpset() delsimps [append_Cons]
+	 		     addsimps [append_Cons RS sym, sublist_append]) 2);
+by (simp_tac (simpset() addsimps [sublist_def]) 1);
+qed "sublist_Cons";
+
+Goal "sublist [x] A = (if 0 : A then [x] else [])";
+by (simp_tac (simpset() addsimps [sublist_Cons]) 1);
+qed "sublist_singleton";
+Addsimps [sublist_singleton];
+
+Goal "sublist l {..n(} = take n l";
+by (res_inst_tac [("xs","l")] rev_induct 1);
+ by (asm_simp_tac (simpset() addsplits [nat_diff_split]
+                             addsimps [sublist_append]) 2);
+by (Simp_tac 1);
+qed "sublist_upt_eq_take";
+Addsimps [sublist_upt_eq_take];
+
+
+(** bag_of **)
 
-Goalw [sublist_def] "sublist l {..length l(} = l";
-by (simp_tac (simpset() addsimps [sublist_length_lemma]) 1);
-qed "sublist_length";
+Goal "bag_of (l@l') = bag_of l + bag_of l'";
+by (induct_tac "l" 1);
+ by (asm_simp_tac (simpset() addsimps (thms "plus_ac0")) 2);
+by (Simp_tac 1);
+qed "bag_of_append";
+Addsimps [bag_of_append];
+
+Goal "mono (bag_of :: 'a list => ('a::order) multiset)";
+by (rtac monoI 1); 
+by (rewtac prefix_def);
+by (etac genPrefix.induct 1);
+by Auto_tac;
+by (asm_full_simp_tac (simpset() addsimps [union_le_mono]) 1); 
+by (etac order_trans 1); 
+by (rtac union_upper1 1); 
+qed "mono_bag_of";
+
+Goal "finite A ==> \
+\     setsum (%i. if i < Suc(length zs) then {#(zs @ [z]) ! i#} else {#}) A = \
+\       setsum (%i. if i < length zs then {#zs ! i#} else {#}) A + \
+\       (if length zs : A then {#z#} else {#})";
+by (etac finite_induct 1);
+by (Simp_tac 1);
+by (asm_simp_tac (simpset() addsimps (nth_append :: thms "plus_ac0")) 1);
+by (subgoal_tac "x < Suc (length zs) & ~ x < length zs --> length zs = x" 1);
+by Auto_tac;
+qed "bag_of_sublist_lemma";
+
+
+Goal "finite A \
+\     ==> bag_of (sublist l A) = \
+\         setsum (%i. if i < length l then {# nth l i #} else {#}) A";
+by (res_inst_tac [("xs","l")] rev_induct 1);
+by (Simp_tac 1);
+by (stac (symmetric Zero_def) 1);
+by (etac (setsum_0 RS sym) 1);
+by (asm_simp_tac (simpset() addsimps [sublist_append, 
+				      bag_of_sublist_lemma]) 1);
+qed "bag_of_sublist";
+
+
+Goal "[| finite A; finite B |] \
+\     ==> bag_of (sublist l (A Un B)) + bag_of (sublist l (A Int B)) = \
+\         bag_of (sublist l A) + bag_of (sublist l B)";
+by (asm_simp_tac (simpset() addsimps [bag_of_sublist, setsum_Un_Int]) 1);
+qed "bag_of_sublist_Un_Int";