(*  Title:      HOL/UNITY/AllocBase.thy

    ID:         $Id$

    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

    Copyright   1998  University of Cambridge

*)

header{*Common Declarations for Chandy and Charpentier's Allocator*}

theory AllocBase = UNITY_Main:

consts

  NbT      :: nat       (*Number of tokens in system*)

  Nclients :: nat       (*Number of clients*)

axioms

  NbT_pos:  "0 < NbT"

(*This function merely sums the elements of a list*)

consts tokens     :: "nat list => nat"

primrec 

  "tokens [] = 0"

  "tokens (x#xs) = x + tokens xs"

consts

  bag_of :: "'a list => 'a multiset"

primrec

  "bag_of []     = {#}"

  "bag_of (x#xs) = {#x#} + bag_of xs"

lemma setsum_fun_mono [rule_format]:

     "!!f :: nat=>nat.  

      (ALL i. i<n --> f i <= g i) -->  

      setsum f (lessThan n) <= setsum g (lessThan n)"

apply (induct_tac "n")

apply (auto simp add: lessThan_Suc)

apply (drule_tac x = n in spec, arith)

done

lemma tokens_mono_prefix [rule_format]:

     "ALL xs. xs <= ys --> tokens xs <= tokens ys"

apply (induct_tac "ys")

apply (auto simp add: prefix_Cons)

done

lemma mono_tokens: "mono tokens"

apply (unfold mono_def)

apply (blast intro: tokens_mono_prefix)

done

(** bag_of **)

lemma bag_of_append [simp]: "bag_of (l@l') = bag_of l + bag_of l'"

apply (induct_tac "l", simp)

 apply (simp add: add_ac)

done

lemma mono_bag_of: "mono (bag_of :: 'a list => ('a::order) multiset)"

apply (rule monoI)

apply (unfold prefix_def)

apply (erule genPrefix.induct, auto)

apply (simp add: union_le_mono)

apply (erule order_trans)

apply (rule union_upper1)

done

(** setsum **)

declare setsum_cong [cong]

lemma bag_of_sublist_lemma:

     "(\<Sum>i: A Int lessThan k. {#if i<k then f i else g i#}) =  

      (\<Sum>i: A Int lessThan k. {#f i#})"

by (rule setsum_cong, auto)

lemma bag_of_sublist:

     "bag_of (sublist l A) =  

      (\<Sum>i: A Int lessThan (length l). {# l!i #})"

apply (rule_tac xs = l in rev_induct, simp)

apply (simp add: sublist_append Int_insert_right lessThan_Suc nth_append 

                 bag_of_sublist_lemma add_ac)

done

lemma bag_of_sublist_Un_Int:

     "bag_of (sublist l (A Un B)) + bag_of (sublist l (A Int B)) =  

      bag_of (sublist l A) + bag_of (sublist l B)"

apply (subgoal_tac "A Int B Int {..length l (} =

	            (A Int {..length l (}) Int (B Int {..length l (}) ")

apply (simp add: bag_of_sublist Int_Un_distrib2 setsum_Un_Int, blast)

done

lemma bag_of_sublist_Un_disjoint:

     "A Int B = {}  

      ==> bag_of (sublist l (A Un B)) =  

          bag_of (sublist l A) + bag_of (sublist l B)"

by (simp add: bag_of_sublist_Un_Int [symmetric])

lemma bag_of_sublist_UN_disjoint [rule_format]:

     "[| finite I; ALL i:I. ALL j:I. i~=j --> A i Int A j = {} |]  

      ==> bag_of (sublist l (UNION I A)) =   

          (\<Sum>i:I. bag_of (sublist l (A i)))"

apply (simp del: UN_simps 

            add: UN_simps [symmetric] add: bag_of_sublist)

apply (subst setsum_UN_disjoint, auto)

done

ML

{*

val NbT_pos = thm "NbT_pos";

val setsum_fun_mono = thm "setsum_fun_mono";

val tokens_mono_prefix = thm "tokens_mono_prefix";

val mono_tokens = thm "mono_tokens";

val bag_of_append = thm "bag_of_append";

val mono_bag_of = thm "mono_bag_of";

val bag_of_sublist_lemma = thm "bag_of_sublist_lemma";

val bag_of_sublist = thm "bag_of_sublist";

val bag_of_sublist_Un_Int = thm "bag_of_sublist_Un_Int";

val bag_of_sublist_Un_disjoint = thm "bag_of_sublist_Un_disjoint";

val bag_of_sublist_UN_disjoint = thm "bag_of_sublist_UN_disjoint";

*}

end