# Theory Merge

theory Merge
imports AllocBase
```(*  Title:      ZF/UNITY/Merge.thy
Author:     Sidi O Ehmety, Cambridge University Computer Laboratory

A multiple-client allocator from a single-client allocator: Merge
specification.
*)

theory Merge imports AllocBase Follows  Guar GenPrefix begin

(** Merge specification (the number of inputs is Nclients) ***)
(** Parameter A represents the type of items to Merge **)

definition
(*spec (10)*)
merge_increasing :: "[i, i, i] =>i"  where
"merge_increasing(A, Out, iOut) == program guarantees
(lift(Out) IncreasingWrt  prefix(A)/list(A)) Int
(lift(iOut) IncreasingWrt prefix(nat)/list(nat))"

definition
(*spec (11)*)
merge_eq_Out :: "[i, i] =>i"  where
"merge_eq_Out(Out, iOut) == program guarantees
Always({s ∈ state. length(s`Out) = length(s`iOut)})"

definition
(*spec (12)*)
merge_bounded :: "i=>i"  where
"merge_bounded(iOut) == program guarantees
Always({s ∈ state. ∀elt ∈ set_of_list(s`iOut). elt<Nclients})"

definition
(*spec (13)*)
(* Parameter A represents the type of tokens *)
merge_follows :: "[i, i=>i, i, i] =>i"  where
"merge_follows(A, In, Out, iOut) ==
(⋂n ∈ Nclients. lift(In(n)) IncreasingWrt prefix(A)/list(A))
guarantees
(⋂n ∈ Nclients.
(%s. sublist(s`Out, {k ∈ nat. k < length(s`iOut) &
nth(k, s`iOut) = n})) Fols lift(In(n))
Wrt prefix(A)/list(A))"

definition
(*spec: preserves part*)
merge_preserves :: "[i=>i] =>i"  where
"merge_preserves(In) == ⋂n ∈ nat. preserves(lift(In(n)))"

definition
(* environmental constraints*)
merge_allowed_acts :: "[i, i] =>i"  where
"merge_allowed_acts(Out, iOut) ==
{F ∈ program. AllowedActs(F) =
cons(id(state), (⋃G ∈ preserves(lift(Out)) ∩
preserves(lift(iOut)). Acts(G)))}"

definition
merge_spec :: "[i, i =>i, i, i]=>i"  where
"merge_spec(A, In, Out, iOut) ==
merge_increasing(A, Out, iOut) ∩ merge_eq_Out(Out, iOut) ∩
merge_bounded(iOut) ∩  merge_follows(A, In, Out, iOut)
∩ merge_allowed_acts(Out, iOut) ∩ merge_preserves(In)"

(** State definitions.  OUTPUT variables are locals **)
locale merge =
fixes In   ―‹merge's INPUT histories: streams to merge›
and Out  ―‹merge's OUTPUT history: merged items›
and iOut ―‹merge's OUTPUT history: origins of merged items›
and A    ―‹the type of items being merged›
and M
assumes var_assumes [simp]:
"(∀n. In(n):var) & Out ∈ var & iOut ∈ var"
and all_distinct_vars:
"∀n. all_distinct([In(n), Out, iOut])"
and type_assumes [simp]:
"(∀n. type_of(In(n))=list(A)) &
type_of(Out)=list(A) &
type_of(iOut)=list(nat)"
and default_val_assumes [simp]:
"(∀n. default_val(In(n))=Nil) &
default_val(Out)=Nil &
default_val(iOut)=Nil"
and merge_spec:  "M ∈ merge_spec(A, In, Out, iOut)"

lemma (in merge) In_value_type [TC,simp]: "s ∈ state ==> s`In(n) ∈ list(A)"
apply (unfold state_def)
apply (drule_tac a = "In (n)" in apply_type)
apply auto
done

lemma (in merge) Out_value_type [TC,simp]: "s ∈ state ==> s`Out ∈ list(A)"
apply (unfold state_def)
apply (drule_tac a = Out in apply_type, auto)
done

lemma (in merge) iOut_value_type [TC,simp]: "s ∈ state ==> s`iOut ∈ list(nat)"
apply (unfold state_def)
apply (drule_tac a = iOut in apply_type, auto)
done

lemma (in merge) M_in_program [intro,simp]: "M ∈ program"
apply (cut_tac merge_spec)
apply (auto dest: guarantees_type [THEN subsetD]
done

lemma (in merge) merge_Allowed:
"Allowed(M) = (preserves(lift(Out)) ∩ preserves(lift(iOut)))"
apply (insert merge_spec preserves_type [of "lift (Out)"])
apply (auto simp add: merge_spec_def merge_allowed_acts_def Allowed_def safety_prop_Acts_iff)
done

lemma (in merge) M_ok_iff:
"G ∈ program ==>
M ok G ⟷ (G ∈ preserves(lift(Out)) &
G ∈ preserves(lift(iOut)) & M ∈ Allowed(G))"
apply (cut_tac merge_spec)
apply (auto simp add: merge_Allowed ok_iff_Allowed)
done

lemma (in merge) merge_Always_Out_eq_iOut:
"[| G ∈ preserves(lift(Out)); G ∈ preserves(lift(iOut));
M ∈ Allowed(G) |]
==> M ⊔ G ∈ Always({s ∈ state. length(s`Out)=length(s`iOut)})"
apply (frule preserves_type [THEN subsetD])
apply (subgoal_tac "G ∈ program")
prefer 2 apply assumption
apply (frule M_ok_iff)
apply (cut_tac merge_spec)
apply (force dest: guaranteesD simp add: merge_spec_def merge_eq_Out_def)
done

lemma (in merge) merge_Bounded:
"[| G ∈ preserves(lift(iOut)); G ∈ preserves(lift(Out));
M ∈ Allowed(G) |] ==>
M ⊔ G: Always({s ∈ state. ∀elt ∈ set_of_list(s`iOut). elt<Nclients})"
apply (frule preserves_type [THEN subsetD])
apply (frule M_ok_iff)
apply (cut_tac merge_spec)
apply (force dest: guaranteesD simp add: merge_spec_def merge_bounded_def)
done

lemma (in merge) merge_bag_Follows_lemma:
"[| G ∈ preserves(lift(iOut));
G: preserves(lift(Out)); M ∈ Allowed(G) |]
==> M ⊔ G ∈ Always
({s ∈ state. msetsum(%i. bag_of(sublist(s`Out,
{k ∈ nat. k < length(s`iOut) & nth(k, s`iOut)=i})),
Nclients, A) = bag_of(s`Out)})"
apply (rule Always_Diff_Un_eq [THEN iffD1])
apply (rule_tac [2] state_AlwaysI [THEN Always_weaken])
apply (rule Always_Int_I [OF merge_Always_Out_eq_iOut merge_Bounded], auto)
apply (subst bag_of_sublist_UN_disjoint [symmetric])
apply (auto simp add: nat_into_Finite set_of_list_conv_nth  [OF iOut_value_type])
apply (subgoal_tac " (⋃i ∈ Nclients. {k ∈ nat. k < length (x`iOut) & nth (k, x`iOut) = i}) = length (x`iOut) ")
apply (auto simp add: sublist_upt_eq_take [OF Out_value_type]
length_type  [OF iOut_value_type]
take_all [OF _ Out_value_type]
length_type [OF iOut_value_type])
apply (rule equalityI)
apply (blast dest: ltD, clarify)
apply (subgoal_tac "length (x ` iOut) ∈ nat")
prefer 2 apply (simp add: length_type [OF iOut_value_type])
apply (subgoal_tac "xa ∈ nat")
prefer 2 apply (blast intro: lt_trans)
apply (drule_tac x = "nth (xa, x`iOut)" and P = "%elt. X (elt) ⟶ elt<Nclients" for X in bspec)
apply (blast dest: ltD)
done

theorem (in merge) merge_bag_Follows:
"M ∈ (⋂n ∈ Nclients. lift(In(n)) IncreasingWrt prefix(A)/list(A))
guarantees
(%s. bag_of(s`Out)) Fols
(%s. msetsum(%i. bag_of(s`In(i)),Nclients, A)) Wrt MultLe(A, r)/Mult(A)"
apply (cut_tac merge_spec)
apply (rule merge_bag_Follows_lemma [THEN Always_Follows1, THEN guaranteesI])
apply (rule Follows_state_ofD1 [OF Follows_msetsum_UN])
apply (simp_all add: nat_into_Finite bag_of_multiset [of _ A])
apply (simp add: INT_iff merge_spec_def merge_follows_def, clarify)
apply (cut_tac merge_spec)
apply (subgoal_tac "M ok G")
prefer 2 apply (force intro: M_ok_iff [THEN iffD2])
apply (drule guaranteesD, assumption)
apply (simp add: merge_spec_def merge_follows_def, blast)