(* 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 imports "../UNITY_Main" begin
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)
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\<in> A Int lessThan k. {#if i<k then f i else g i#}) =
(\<Sum>i\<in> A Int lessThan k. {#f i#})"
by (rule setsum_cong, auto)
lemma bag_of_sublist:
"bag_of (sublist l A) =
(\<Sum>i\<in> 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\<in>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
end