(* 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 = Main:
typedef (Program)
'a program = "{(init:: 'a set, acts :: ('a * 'a)set set,
allowed :: ('a * 'a)set set). Id:acts & Id: allowed}"
by blast
constdefs
constrains :: "['a set, 'a set] => 'a program set" (infixl "co" 60)
"A co B == {F. ALL act: Acts F. act``A <= B}"
unless :: "['a set, 'a set] => 'a program set" (infixl "unless" 60)
"A unless B == (A-B) co (A Un 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)"
Acts :: "'a program => ('a * 'a)set set"
"Acts F == (%(init, acts, allowed). acts) (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 <= 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 : A co B}"
invariant :: "'a set => 'a program set"
"invariant A == {F. Init F <= A} Int stable A"
(*Polymorphic in both states and the meaning of <= *)
increasing :: "['a => 'b::{order}] => 'a program set"
"increasing f == INT z. stable {s. z <= f s}"
(*Perhaps equalities.ML shouldn't add this in the first place!*)
declare image_Collect [simp del]
(*** 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 : 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 Id_in_AllowedActs [iff]: "Id : 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)
(** Inspectors for type "program" **)
lemma Init_eq [simp]: "Init (mk_program (init,acts,allowed)) = init"
by (auto simp add: program_typedef)
lemma Acts_eq [simp]: "Acts (mk_program (init,acts,allowed)) = insert Id acts"
by (auto simp add: program_typedef)
lemma AllowedActs_eq [simp]:
"AllowedActs (mk_program (init,acts,allowed)) = insert Id allowed"
by (auto simp add: program_typedef)
(** 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)
(*** These rules allow "lazy" definition expansion
They avoid expanding the full program, which is a large expression
***)
lemma def_prg_Init: "F == mk_program (init,acts,allowed) ==> Init F = init"
by auto
lemma def_prg_Acts:
"F == mk_program (init,acts,allowed) ==> Acts F = insert Id acts"
by auto
lemma def_prg_AllowedActs:
"F == mk_program (init,acts,allowed)
==> AllowedActs F = insert Id allowed"
by auto
(*The program is not expanded, but its Init and Acts are*)
lemma def_prg_simps:
"[| F == mk_program (init,acts,allowed) |]
==> Init F = init & Acts F = insert Id acts"
by simp
(*An action is expanded only if a pair of states is being tested against it*)
lemma def_act_simp:
"[| act == {(s,s'). P s s'} |] ==> ((s,s') : act) = P s s'"
by auto
(*A set is expanded only if an element is being tested against it*)
lemma def_set_simp: "A == B ==> (x : A) = (x : B)"
by auto
(*** co ***)
lemma constrainsI:
"(!!act s s'. [| act: Acts F; (s,s') : act; s: A |] ==> s': A')
==> F : A co A'"
by (simp add: constrains_def, blast)
lemma constrainsD:
"[| F : A co A'; act: Acts F; (s,s'): act; s: A |] ==> s': A'"
by (unfold constrains_def, blast)
lemma constrains_empty [iff]: "F : {} co B"
by (unfold constrains_def, blast)
lemma constrains_empty2 [iff]: "(F : A co {}) = (A={})"
by (unfold constrains_def, blast)
lemma constrains_UNIV [iff]: "(F : UNIV co B) = (B = UNIV)"
by (unfold constrains_def, blast)
lemma constrains_UNIV2 [iff]: "F : A co UNIV"
by (unfold constrains_def, blast)
(*monotonic in 2nd argument*)
lemma constrains_weaken_R:
"[| F : A co A'; A'<=B' |] ==> F : A co B'"
by (unfold constrains_def, blast)
(*anti-monotonic in 1st argument*)
lemma constrains_weaken_L:
"[| F : A co A'; B<=A |] ==> F : B co A'"
by (unfold constrains_def, blast)
lemma constrains_weaken:
"[| F : A co A'; B<=A; A'<=B' |] ==> F : B co B'"
by (unfold constrains_def, blast)
(** Union **)
lemma constrains_Un:
"[| F : A co A'; F : B co B' |] ==> F : (A Un B) co (A' Un B')"
by (unfold constrains_def, blast)
lemma constrains_UN:
"(!!i. i:I ==> F : (A i) co (A' i))
==> F : (UN i:I. A i) co (UN i:I. A' i)"
by (unfold constrains_def, blast)
lemma constrains_Un_distrib: "(A Un B) co C = (A co C) Int (B co C)"
by (unfold constrains_def, blast)
lemma constrains_UN_distrib: "(UN i:I. A i) co B = (INT i:I. A i co B)"
by (unfold constrains_def, blast)
lemma constrains_Int_distrib: "C co (A Int B) = (C co A) Int (C co B)"
by (unfold constrains_def, blast)
lemma constrains_INT_distrib: "A co (INT i:I. B i) = (INT i:I. A co B i)"
by (unfold constrains_def, blast)
(** Intersection **)
lemma constrains_Int:
"[| F : A co A'; F : B co B' |] ==> F : (A Int B) co (A' Int B')"
by (unfold constrains_def, blast)
lemma constrains_INT:
"(!!i. i:I ==> F : (A i) co (A' i))
==> F : (INT i:I. A i) co (INT i:I. A' i)"
by (unfold constrains_def, blast)
lemma constrains_imp_subset: "F : A co A' ==> A <= A'"
by (unfold constrains_def, auto)
(*The reasoning is by subsets since "co" refers to single actions
only. So this rule isn't that useful.*)
lemma constrains_trans:
"[| F : A co B; F : B co C |] ==> F : A co C"
by (unfold constrains_def, blast)
lemma constrains_cancel:
"[| F : A co (A' Un B); F : B co B' |] ==> F : A co (A' Un B')"
by (unfold constrains_def, clarify, blast)
(*** unless ***)
lemma unlessI: "F : (A-B) co (A Un B) ==> F : A unless B"
by (unfold unless_def, assumption)
lemma unlessD: "F : A unless B ==> F : (A-B) co (A Un B)"
by (unfold unless_def, assumption)
(*** stable ***)
lemma stableI: "F : A co A ==> F : stable A"
by (unfold stable_def, assumption)
lemma stableD: "F : stable A ==> F : A co A"
by (unfold stable_def, assumption)
lemma stable_UNIV [simp]: "stable UNIV = UNIV"
by (unfold stable_def constrains_def, auto)
(** Union **)
lemma stable_Un:
"[| F : stable A; F : stable A' |] ==> F : stable (A Un A')"
apply (unfold stable_def)
apply (blast intro: constrains_Un)
done
lemma stable_UN:
"(!!i. i:I ==> F : stable (A i)) ==> F : stable (UN i:I. A i)"
apply (unfold stable_def)
apply (blast intro: constrains_UN)
done
(** Intersection **)
lemma stable_Int:
"[| F : stable A; F : stable A' |] ==> F : stable (A Int A')"
apply (unfold stable_def)
apply (blast intro: constrains_Int)
done
lemma stable_INT:
"(!!i. i:I ==> F : stable (A i)) ==> F : stable (INT i:I. A i)"
apply (unfold stable_def)
apply (blast intro: constrains_INT)
done
lemma stable_constrains_Un:
"[| F : stable C; F : A co (C Un A') |] ==> F : (C Un A) co (C Un A')"
by (unfold stable_def constrains_def, blast)
lemma stable_constrains_Int:
"[| F : stable C; F : (C Int A) co A' |] ==> F : (C Int A) co (C Int A')"
by (unfold stable_def constrains_def, blast)
(*[| F : stable C; F : (C Int A) co A |] ==> F : stable (C Int A) *)
lemmas stable_constrains_stable = stable_constrains_Int [THEN stableI, standard]
(*** invariant ***)
lemma invariantI: "[| Init F<=A; F: stable A |] ==> F : invariant A"
by (simp add: invariant_def)
(*Could also say "invariant A Int invariant B <= invariant (A Int B)"*)
lemma invariant_Int:
"[| F : invariant A; F : invariant B |] ==> F : invariant (A Int B)"
by (auto simp add: invariant_def stable_Int)
(*** increasing ***)
lemma increasingD:
"F : increasing f ==> F : stable {s. z <= f s}"
by (unfold increasing_def, blast)
lemma increasing_constant [iff]: "F : increasing (%s. c)"
by (unfold increasing_def stable_def, auto)
lemma mono_increasing_o:
"mono g ==> increasing f <= 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 <= n) = (m < n) *)
lemma strict_increasingD:
"!!z::nat. F : increasing f ==> F: 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 ALL m ? Would make it harder to use
in forward proof. **)
lemma elimination:
"[| ALL m:M. F : {s. s x = m} co (B m) |]
==> F : {s. s x : M} co (UN m:M. B m)"
by (unfold constrains_def, blast)
(*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:
"(ALL m:M. F : {m} co (B m)) ==> F : M co (UN m:M. B m)"
by (unfold constrains_def, blast)
(*** Theoretical Results from Section 6 ***)
lemma constrains_strongest_rhs:
"F : A co (strongest_rhs F A )"
by (unfold constrains_def strongest_rhs_def, blast)
lemma strongest_rhs_is_strongest:
"F : A co B ==> strongest_rhs F A <= B"
by (unfold constrains_def strongest_rhs_def, blast)
(** Ad-hoc set-theory rules **)
lemma Un_Diff_Diff [simp]: "A Un B - (A - B) = B"
by blast
lemma Int_Union_Union: "Union(B) Int A = Union((%C. C Int A)`B)"
by blast
(** Needed for WF reasoning in WFair.ML **)
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
end