(* Title: HOL/UNITY/UNITY
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1998 University of Cambridge
The basic UNITY theory (revised version, based upon the "co" operator)
From Misra, "A Logic for Concurrent Programming", 1994
*)
header {*The Basic UNITY Theory*}
theory UNITY imports Main begin
typedef (Program)
'a program = "{(init:: 'a set, acts :: ('a * 'a)set set,
allowed :: ('a * 'a)set set). Id \<in> acts & Id: allowed}"
by blast
constdefs
Acts :: "'a program => ('a * 'a)set set"
"Acts F == (%(init, acts, allowed). acts) (Rep_Program F)"
"constrains" :: "['a set, 'a set] => 'a program set" (infixl "co" 60)
"A co B == {F. \<forall>act \<in> Acts F. act``A \<subseteq> B}"
unless :: "['a set, 'a set] => 'a program set" (infixl "unless" 60)
"A unless B == (A-B) co (A \<union> B)"
mk_program :: "('a set * ('a * 'a)set set * ('a * 'a)set set)
=> 'a program"
"mk_program == %(init, acts, allowed).
Abs_Program (init, insert Id acts, insert Id allowed)"
Init :: "'a program => 'a set"
"Init F == (%(init, acts, allowed). init) (Rep_Program F)"
AllowedActs :: "'a program => ('a * 'a)set set"
"AllowedActs F == (%(init, acts, allowed). allowed) (Rep_Program F)"
Allowed :: "'a program => 'a program set"
"Allowed F == {G. Acts G \<subseteq> AllowedActs F}"
stable :: "'a set => 'a program set"
"stable A == A co A"
strongest_rhs :: "['a program, 'a set] => 'a set"
"strongest_rhs F A == Inter {B. F \<in> A co B}"
invariant :: "'a set => 'a program set"
"invariant A == {F. Init F \<subseteq> A} \<inter> stable A"
increasing :: "['a => 'b::{order}] => 'a program set"
--{*Polymorphic in both states and the meaning of @{text "\<le>"}*}
"increasing f == \<Inter>z. stable {s. z \<le> f s}"
text{*Perhaps HOL shouldn't add this in the first place!*}
declare image_Collect [simp del]
subsubsection{*The abstract type of programs*}
lemmas program_typedef =
Rep_Program Rep_Program_inverse Abs_Program_inverse
Program_def Init_def Acts_def AllowedActs_def mk_program_def
lemma Id_in_Acts [iff]: "Id \<in> Acts F"
apply (cut_tac x = F in Rep_Program)
apply (auto simp add: program_typedef)
done
lemma insert_Id_Acts [iff]: "insert Id (Acts F) = Acts F"
by (simp add: insert_absorb Id_in_Acts)
lemma Acts_nonempty [simp]: "Acts F \<noteq> {}"
by auto
lemma Id_in_AllowedActs [iff]: "Id \<in> AllowedActs F"
apply (cut_tac x = F in Rep_Program)
apply (auto simp add: program_typedef)
done
lemma insert_Id_AllowedActs [iff]: "insert Id (AllowedActs F) = AllowedActs F"
by (simp add: insert_absorb Id_in_AllowedActs)
subsubsection{*Inspectors for type "program"*}
lemma Init_eq [simp]: "Init (mk_program (init,acts,allowed)) = init"
by (simp add: program_typedef)
lemma Acts_eq [simp]: "Acts (mk_program (init,acts,allowed)) = insert Id acts"
by (simp add: program_typedef)
lemma AllowedActs_eq [simp]:
"AllowedActs (mk_program (init,acts,allowed)) = insert Id allowed"
by (simp add: program_typedef)
subsubsection{*Equality for UNITY programs*}
lemma surjective_mk_program [simp]:
"mk_program (Init F, Acts F, AllowedActs F) = F"
apply (cut_tac x = F in Rep_Program)
apply (auto simp add: program_typedef)
apply (drule_tac f = Abs_Program in arg_cong)+
apply (simp add: program_typedef insert_absorb)
done
lemma program_equalityI:
"[| Init F = Init G; Acts F = Acts G; AllowedActs F = AllowedActs G |]
==> F = G"
apply (rule_tac t = F in surjective_mk_program [THEN subst])
apply (rule_tac t = G in surjective_mk_program [THEN subst], simp)
done
lemma program_equalityE:
"[| F = G;
[| Init F = Init G; Acts F = Acts G; AllowedActs F = AllowedActs G |]
==> P |] ==> P"
by simp
lemma program_equality_iff:
"(F=G) =
(Init F = Init G & Acts F = Acts G &AllowedActs F = AllowedActs G)"
by (blast intro: program_equalityI program_equalityE)
subsubsection{*co*}
lemma constrainsI:
"(!!act s s'. [| act: Acts F; (s,s') \<in> act; s \<in> A |] ==> s': A')
==> F \<in> A co A'"
by (simp add: constrains_def, blast)
lemma constrainsD:
"[| F \<in> A co A'; act: Acts F; (s,s'): act; s \<in> A |] ==> s': A'"
by (unfold constrains_def, blast)
lemma constrains_empty [iff]: "F \<in> {} co B"
by (unfold constrains_def, blast)
lemma constrains_empty2 [iff]: "(F \<in> A co {}) = (A={})"
by (unfold constrains_def, blast)
lemma constrains_UNIV [iff]: "(F \<in> UNIV co B) = (B = UNIV)"
by (unfold constrains_def, blast)
lemma constrains_UNIV2 [iff]: "F \<in> A co UNIV"
by (unfold constrains_def, blast)
text{*monotonic in 2nd argument*}
lemma constrains_weaken_R:
"[| F \<in> A co A'; A'<=B' |] ==> F \<in> A co B'"
by (unfold constrains_def, blast)
text{*anti-monotonic in 1st argument*}
lemma constrains_weaken_L:
"[| F \<in> A co A'; B \<subseteq> A |] ==> F \<in> B co A'"
by (unfold constrains_def, blast)
lemma constrains_weaken:
"[| F \<in> A co A'; B \<subseteq> A; A'<=B' |] ==> F \<in> B co B'"
by (unfold constrains_def, blast)
subsubsection{*Union*}
lemma constrains_Un:
"[| F \<in> A co A'; F \<in> B co B' |] ==> F \<in> (A \<union> B) co (A' \<union> B')"
by (unfold constrains_def, blast)
lemma constrains_UN:
"(!!i. i \<in> I ==> F \<in> (A i) co (A' i))
==> F \<in> (\<Union>i \<in> I. A i) co (\<Union>i \<in> I. A' i)"
by (unfold constrains_def, blast)
lemma constrains_Un_distrib: "(A \<union> B) co C = (A co C) \<inter> (B co C)"
by (unfold constrains_def, blast)
lemma constrains_UN_distrib: "(\<Union>i \<in> I. A i) co B = (\<Inter>i \<in> I. A i co B)"
by (unfold constrains_def, blast)
lemma constrains_Int_distrib: "C co (A \<inter> B) = (C co A) \<inter> (C co B)"
by (unfold constrains_def, blast)
lemma constrains_INT_distrib: "A co (\<Inter>i \<in> I. B i) = (\<Inter>i \<in> I. A co B i)"
by (unfold constrains_def, blast)
subsubsection{*Intersection*}
lemma constrains_Int:
"[| F \<in> A co A'; F \<in> B co B' |] ==> F \<in> (A \<inter> B) co (A' \<inter> B')"
by (unfold constrains_def, blast)
lemma constrains_INT:
"(!!i. i \<in> I ==> F \<in> (A i) co (A' i))
==> F \<in> (\<Inter>i \<in> I. A i) co (\<Inter>i \<in> I. A' i)"
by (unfold constrains_def, blast)
lemma constrains_imp_subset: "F \<in> A co A' ==> A \<subseteq> A'"
by (unfold constrains_def, auto)
text{*The reasoning is by subsets since "co" refers to single actions
only. So this rule isn't that useful.*}
lemma constrains_trans:
"[| F \<in> A co B; F \<in> B co C |] ==> F \<in> A co C"
by (unfold constrains_def, blast)
lemma constrains_cancel:
"[| F \<in> A co (A' \<union> B); F \<in> B co B' |] ==> F \<in> A co (A' \<union> B')"
by (unfold constrains_def, clarify, blast)
subsubsection{*unless*}
lemma unlessI: "F \<in> (A-B) co (A \<union> B) ==> F \<in> A unless B"
by (unfold unless_def, assumption)
lemma unlessD: "F \<in> A unless B ==> F \<in> (A-B) co (A \<union> B)"
by (unfold unless_def, assumption)
subsubsection{*stable*}
lemma stableI: "F \<in> A co A ==> F \<in> stable A"
by (unfold stable_def, assumption)
lemma stableD: "F \<in> stable A ==> F \<in> A co A"
by (unfold stable_def, assumption)
lemma stable_UNIV [simp]: "stable UNIV = UNIV"
by (unfold stable_def constrains_def, auto)
subsubsection{*Union*}
lemma stable_Un:
"[| F \<in> stable A; F \<in> stable A' |] ==> F \<in> stable (A \<union> A')"
apply (unfold stable_def)
apply (blast intro: constrains_Un)
done
lemma stable_UN:
"(!!i. i \<in> I ==> F \<in> stable (A i)) ==> F \<in> stable (\<Union>i \<in> I. A i)"
apply (unfold stable_def)
apply (blast intro: constrains_UN)
done
lemma stable_Union:
"(!!A. A \<in> X ==> F \<in> stable A) ==> F \<in> stable (\<Union>X)"
by (unfold stable_def constrains_def, blast)
subsubsection{*Intersection*}
lemma stable_Int:
"[| F \<in> stable A; F \<in> stable A' |] ==> F \<in> stable (A \<inter> A')"
apply (unfold stable_def)
apply (blast intro: constrains_Int)
done
lemma stable_INT:
"(!!i. i \<in> I ==> F \<in> stable (A i)) ==> F \<in> stable (\<Inter>i \<in> I. A i)"
apply (unfold stable_def)
apply (blast intro: constrains_INT)
done
lemma stable_Inter:
"(!!A. A \<in> X ==> F \<in> stable A) ==> F \<in> stable (\<Inter>X)"
by (unfold stable_def constrains_def, blast)
lemma stable_constrains_Un:
"[| F \<in> stable C; F \<in> A co (C \<union> A') |] ==> F \<in> (C \<union> A) co (C \<union> A')"
by (unfold stable_def constrains_def, blast)
lemma stable_constrains_Int:
"[| F \<in> stable C; F \<in> (C \<inter> A) co A' |] ==> F \<in> (C \<inter> A) co (C \<inter> A')"
by (unfold stable_def constrains_def, blast)
(*[| F \<in> stable C; F \<in> (C \<inter> A) co A |] ==> F \<in> stable (C \<inter> A) *)
lemmas stable_constrains_stable = stable_constrains_Int[THEN stableI, standard]
subsubsection{*invariant*}
lemma invariantI: "[| Init F \<subseteq> A; F \<in> stable A |] ==> F \<in> invariant A"
by (simp add: invariant_def)
text{*Could also say @{term "invariant A \<inter> invariant B \<subseteq> invariant(A \<inter> B)"}*}
lemma invariant_Int:
"[| F \<in> invariant A; F \<in> invariant B |] ==> F \<in> invariant (A \<inter> B)"
by (auto simp add: invariant_def stable_Int)
subsubsection{*increasing*}
lemma increasingD:
"F \<in> increasing f ==> F \<in> stable {s. z \<subseteq> f s}"
by (unfold increasing_def, blast)
lemma increasing_constant [iff]: "F \<in> increasing (%s. c)"
by (unfold increasing_def stable_def, auto)
lemma mono_increasing_o:
"mono g ==> increasing f \<subseteq> increasing (g o f)"
apply (unfold increasing_def stable_def constrains_def, auto)
apply (blast intro: monoD order_trans)
done
(*Holds by the theorem (Suc m \<subseteq> n) = (m < n) *)
lemma strict_increasingD:
"!!z::nat. F \<in> increasing f ==> F \<in> stable {s. z < f s}"
by (simp add: increasing_def Suc_le_eq [symmetric])
(** The Elimination Theorem. The "free" m has become universally quantified!
Should the premise be !!m instead of \<forall>m ? Would make it harder to use
in forward proof. **)
lemma elimination:
"[| \<forall>m \<in> M. F \<in> {s. s x = m} co (B m) |]
==> F \<in> {s. s x \<in> M} co (\<Union>m \<in> M. B m)"
by (unfold constrains_def, blast)
text{*As above, but for the trivial case of a one-variable state, in which the
state is identified with its one variable.*}
lemma elimination_sing:
"(\<forall>m \<in> M. F \<in> {m} co (B m)) ==> F \<in> M co (\<Union>m \<in> M. B m)"
by (unfold constrains_def, blast)
subsubsection{*Theoretical Results from Section 6*}
lemma constrains_strongest_rhs:
"F \<in> A co (strongest_rhs F A )"
by (unfold constrains_def strongest_rhs_def, blast)
lemma strongest_rhs_is_strongest:
"F \<in> A co B ==> strongest_rhs F A \<subseteq> B"
by (unfold constrains_def strongest_rhs_def, blast)
subsubsection{*Ad-hoc set-theory rules*}
lemma Un_Diff_Diff [simp]: "A \<union> B - (A - B) = B"
by blast
lemma Int_Union_Union: "Union(B) \<inter> A = Union((%C. C \<inter> A)`B)"
by blast
text{*Needed for WF reasoning in WFair.thy*}
lemma Image_less_than [simp]: "less_than `` {k} = greaterThan k"
by blast
lemma Image_inverse_less_than [simp]: "less_than^-1 `` {k} = lessThan k"
by blast
subsection{*Partial versus Total Transitions*}
constdefs
totalize_act :: "('a * 'a)set => ('a * 'a)set"
"totalize_act act == act \<union> diag (-(Domain act))"
totalize :: "'a program => 'a program"
"totalize F == mk_program (Init F,
totalize_act ` Acts F,
AllowedActs F)"
mk_total_program :: "('a set * ('a * 'a)set set * ('a * 'a)set set)
=> 'a program"
"mk_total_program args == totalize (mk_program args)"
all_total :: "'a program => bool"
"all_total F == \<forall>act \<in> Acts F. Domain act = UNIV"
lemma insert_Id_image_Acts: "f Id = Id ==> insert Id (f`Acts F) = f ` Acts F"
by (blast intro: sym [THEN image_eqI])
subsubsection{*Basic properties*}
lemma totalize_act_Id [simp]: "totalize_act Id = Id"
by (simp add: totalize_act_def)
lemma Domain_totalize_act [simp]: "Domain (totalize_act act) = UNIV"
by (auto simp add: totalize_act_def)
lemma Init_totalize [simp]: "Init (totalize F) = Init F"
by (unfold totalize_def, auto)
lemma Acts_totalize [simp]: "Acts (totalize F) = (totalize_act ` Acts F)"
by (simp add: totalize_def insert_Id_image_Acts)
lemma AllowedActs_totalize [simp]: "AllowedActs (totalize F) = AllowedActs F"
by (simp add: totalize_def)
lemma totalize_constrains_iff [simp]: "(totalize F \<in> A co B) = (F \<in> A co B)"
by (simp add: totalize_def totalize_act_def constrains_def, blast)
lemma totalize_stable_iff [simp]: "(totalize F \<in> stable A) = (F \<in> stable A)"
by (simp add: stable_def)
lemma totalize_invariant_iff [simp]:
"(totalize F \<in> invariant A) = (F \<in> invariant A)"
by (simp add: invariant_def)
lemma all_total_totalize: "all_total (totalize F)"
by (simp add: totalize_def all_total_def)
lemma Domain_iff_totalize_act: "(Domain act = UNIV) = (totalize_act act = act)"
by (force simp add: totalize_act_def)
lemma all_total_imp_totalize: "all_total F ==> (totalize F = F)"
apply (simp add: all_total_def totalize_def)
apply (rule program_equalityI)
apply (simp_all add: Domain_iff_totalize_act image_def)
done
lemma all_total_iff_totalize: "all_total F = (totalize F = F)"
apply (rule iffI)
apply (erule all_total_imp_totalize)
apply (erule subst)
apply (rule all_total_totalize)
done
lemma mk_total_program_constrains_iff [simp]:
"(mk_total_program args \<in> A co B) = (mk_program args \<in> A co B)"
by (simp add: mk_total_program_def)
subsection{*Rules for Lazy Definition Expansion*}
text{*They avoid expanding the full program, which is a large expression*}
lemma def_prg_Init:
"F == mk_total_program (init,acts,allowed) ==> Init F = init"
by (simp add: mk_total_program_def)
lemma def_prg_Acts:
"F == mk_total_program (init,acts,allowed)
==> Acts F = insert Id (totalize_act ` acts)"
by (simp add: mk_total_program_def)
lemma def_prg_AllowedActs:
"F == mk_total_program (init,acts,allowed)
==> AllowedActs F = insert Id allowed"
by (simp add: mk_total_program_def)
text{*An action is expanded if a pair of states is being tested against it*}
lemma def_act_simp:
"act == {(s,s'). P s s'} ==> ((s,s') \<in> act) = P s s'"
by (simp add: mk_total_program_def)
text{*A set is expanded only if an element is being tested against it*}
lemma def_set_simp: "A == B ==> (x \<in> A) = (x \<in> B)"
by (simp add: mk_total_program_def)
subsubsection{*Inspectors for type "program"*}
lemma Init_total_eq [simp]:
"Init (mk_total_program (init,acts,allowed)) = init"
by (simp add: mk_total_program_def)
lemma Acts_total_eq [simp]:
"Acts(mk_total_program(init,acts,allowed)) = insert Id (totalize_act`acts)"
by (simp add: mk_total_program_def)
lemma AllowedActs_total_eq [simp]:
"AllowedActs (mk_total_program (init,acts,allowed)) = insert Id allowed"
by (auto simp add: mk_total_program_def)
end