Theory UNITY

(*  Title:      HOL/UNITY/UNITY.thy
    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.
*)

section ‹The Basic UNITY Theory›

theory UNITY imports Main begin

definition
  "Program =
    {(init:: 'a set, acts :: ('a * 'a)set set,
      allowed :: ('a * 'a)set set). Id ∈ acts & Id ∈ allowed}"

typedef 'a program = "Program :: ('a set * ('a * 'a) set set * ('a * 'a) set set) set"
  morphisms Rep_Program Abs_Program
  unfolding Program_def by blast

definition Acts :: "'a program => ('a * 'a)set set" where
    "Acts F == (%(init, acts, allowed). acts) (Rep_Program F)"

definition "constrains" :: "['a set, 'a set] => 'a program set"  (infixl "co"     60) where
    "A co B == {F. ∀act ∈ Acts F. act``A ⊆ B}"

definition unless  :: "['a set, 'a set] => 'a program set"  (infixl "unless" 60)  where
    "A unless B == (A-B) co (A ∪ B)"

definition mk_program :: "('a set * ('a * 'a)set set * ('a * 'a)set set)
                   => 'a program" where
    "mk_program == %(init, acts, allowed).
                      Abs_Program (init, insert Id acts, insert Id allowed)"

definition Init :: "'a program => 'a set" where
    "Init F == (%(init, acts, allowed). init) (Rep_Program F)"

definition AllowedActs :: "'a program => ('a * 'a)set set" where
    "AllowedActs F == (%(init, acts, allowed). allowed) (Rep_Program F)"

definition Allowed :: "'a program => 'a program set" where
    "Allowed F == {G. Acts G ⊆ AllowedActs F}"

definition stable     :: "'a set => 'a program set" where
    "stable A == A co A"

definition strongest_rhs :: "['a program, 'a set] => 'a set" where
    "strongest_rhs F A == ⋂{B. F ∈ A co B}"

definition invariant :: "'a set => 'a program set" where
    "invariant A == {F. Init F ⊆ A} ∩ stable A"

definition increasing :: "['a => 'b::{order}] => 'a program set" where
    ― ‹Polymorphic in both states and the meaning of ‹≤››
    "increasing f == ⋂z. stable {s. z ≤ f s}"


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 ∈ 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)

lemma Acts_nonempty [simp]: "Acts F ≠ {}"
by auto

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)

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') ∈ 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)

text‹monotonic in 2nd argument›
lemma constrains_weaken_R: 
    "[| F ∈ A co A'; A'<=B' |] ==> F ∈ A co B'"
by (unfold constrains_def, blast)

text‹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)

subsubsection‹Union›

lemma constrains_Un: 
    "[| F ∈ A co A'; F ∈ B co B' |] ==> F ∈ (A ∪ B) co (A' ∪ B')"
by (unfold constrains_def, blast)

lemma constrains_UN: 
    "(!!i. i ∈ I ==> F ∈ (A i) co (A' i)) 
     ==> F ∈ (⋃i ∈ I. A i) co (⋃i ∈ I. A' i)"
by (unfold constrains_def, blast)

lemma constrains_Un_distrib: "(A ∪ B) co C = (A co C) ∩ (B co C)"
by (unfold constrains_def, blast)

lemma constrains_UN_distrib: "(⋃i ∈ I. A i) co B = (⋂i ∈ I. A i co B)"
by (unfold constrains_def, blast)

lemma constrains_Int_distrib: "C co (A ∩ B) = (C co A) ∩ (C co B)"
by (unfold constrains_def, blast)

lemma constrains_INT_distrib: "A co (⋂i ∈ I. B i) = (⋂i ∈ I. A co B i)"
by (unfold constrains_def, blast)

subsubsection‹Intersection›

lemma constrains_Int: 
    "[| F ∈ A co A'; F ∈ B co B' |] ==> F ∈ (A ∩ B) co (A' ∩ B')"
by (unfold constrains_def, blast)

lemma constrains_INT: 
    "(!!i. i ∈ I ==> F ∈ (A i) co (A' i)) 
     ==> F ∈ (⋂i ∈ I. A i) co (⋂i ∈ I. A' i)"
by (unfold constrains_def, blast)

lemma constrains_imp_subset: "F ∈ A co A' ==> A ⊆ 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 ∈ A co B; F ∈ B co C |] ==> F ∈ A co C"
by (unfold constrains_def, blast)

lemma constrains_cancel: 
   "[| F ∈ A co (A' ∪ B); F ∈ B co B' |] ==> F ∈ A co (A' ∪ B')"
by (unfold constrains_def, clarify, blast)


subsubsection‹unless›

lemma unlessI: "F ∈ (A-B) co (A ∪ B) ==> F ∈ A unless B"
by (unfold unless_def, assumption)

lemma unlessD: "F ∈ A unless B ==> F ∈ (A-B) co (A ∪ B)"
by (unfold unless_def, assumption)


subsubsection‹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)

subsubsection‹Union›

lemma stable_Un: 
    "[| F ∈ stable A; F ∈ stable A' |] ==> F ∈ stable (A ∪ A')"

apply (unfold stable_def)
apply (blast intro: constrains_Un)
done

lemma stable_UN: 
    "(!!i. i ∈ I ==> F ∈ stable (A i)) ==> F ∈ stable (⋃i ∈ I. A i)"
apply (unfold stable_def)
apply (blast intro: constrains_UN)
done

lemma stable_Union: 
    "(!!A. A ∈ X ==> F ∈ stable A) ==> F ∈ stable (⋃X)"
by (unfold stable_def constrains_def, blast)

subsubsection‹Intersection›

lemma stable_Int: 
    "[| F ∈ stable A;  F ∈ stable A' |] ==> F ∈ stable (A ∩ A')"
apply (unfold stable_def)
apply (blast intro: constrains_Int)
done

lemma stable_INT: 
    "(!!i. i ∈ I ==> F ∈ stable (A i)) ==> F ∈ stable (⋂i ∈ I. A i)"
apply (unfold stable_def)
apply (blast intro: constrains_INT)
done

lemma stable_Inter: 
    "(!!A. A ∈ X ==> F ∈ stable A) ==> F ∈ stable (⋂X)"
by (unfold stable_def constrains_def, blast)

lemma stable_constrains_Un: 
    "[| F ∈ stable C; F ∈ A co (C ∪ A') |] ==> F ∈ (C ∪ A) co (C ∪ A')"
by (unfold stable_def constrains_def, blast)

lemma stable_constrains_Int: 
  "[| F ∈ stable C; F ∈  (C ∩ A) co A' |] ==> F ∈ (C ∩ A) co (C ∩ A')"
by (unfold stable_def constrains_def, blast)

(*[| F ∈ stable C; F ∈  (C ∩ A) co A |] ==> F ∈ stable (C ∩ A) *)
lemmas stable_constrains_stable = stable_constrains_Int[THEN stableI]


subsubsection‹invariant›

lemma invariantI: "[| Init F ⊆ A;  F ∈ stable A |] ==> F ∈ invariant A"
by (simp add: invariant_def)

text‹Could also say \<^term>‹invariant A ∩ invariant B ⊆ invariant(A ∩ B)››
lemma invariant_Int:
     "[| F ∈ invariant A;  F ∈ invariant B |] ==> F ∈ invariant (A ∩ B)"
by (auto simp add: invariant_def stable_Int)


subsubsection‹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 ∀m ?  Would make it harder to use
    in forward proof. **)

lemma elimination: 
    "[| ∀m ∈ M. F ∈ {s. s x = m} co (B m) |]  
     ==> F ∈ {s. s x ∈ M} co (⋃m ∈ 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: 
    "(∀m ∈ M. F ∈ {m} co (B m)) ==> F ∈ M co (⋃m ∈ M. B m)"
by (unfold constrains_def, blast)



subsubsection‹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)


subsubsection‹Ad-hoc set-theory rules›

lemma Un_Diff_Diff [simp]: "A ∪ B - (A - B) = B"
by blast

lemma Int_Union_Union: "⋃B ∩ A = ⋃((%C. C ∩ 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¯ `` {k} = lessThan k"
by blast


subsection‹Partial versus Total Transitions›

definition totalize_act :: "('a * 'a)set => ('a * 'a)set" where
    "totalize_act act == act ∪ Id_on (-(Domain act))"

definition totalize :: "'a program => 'a program" where
    "totalize F == mk_program (Init F,
                               totalize_act ` Acts F,
                               AllowedActs F)"

definition mk_total_program :: "('a set * ('a * 'a)set set * ('a * 'a)set set)
                   => 'a program" where
    "mk_total_program args == totalize (mk_program args)"

definition all_total :: "'a program => bool" where
    "all_total F == ∀act ∈ 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 ∈ A co B) = (F ∈ A co B)"
by (simp add: totalize_def totalize_act_def constrains_def, blast)

lemma totalize_stable_iff [simp]: "(totalize F ∈ stable A) = (F ∈ stable A)"
by (simp add: stable_def)

lemma totalize_invariant_iff [simp]:
     "(totalize F ∈ invariant A) = (F ∈ 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 ∈ A co B) = (mk_program args ∈ 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') ∈ 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 ∈ A) = (x ∈ 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