
paulson@7400: (*  Title:      HOL/UNITY/Guar.thy
paulson@7400:     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
wenzelm@32960:     Author:     Sidi Ehmety
paulson@7400: 
ehmety@11190: From Chandy and Sanders, "Reasoning About Program Composition",
ehmety@11190: Technical Report 2000-003, University of Florida, 2000.
ehmety@11190: 
wenzelm@32960: Compatibility, weakest guarantees, etc.  and Weakest existential
wenzelm@32960: property, from Charpentier and Chandy "Theorems about Composition",
ehmety@11190: Fifth International Conference on Mathematics of Program, 2000.
paulson@7400: *)
paulson@7400: 
paulson@13798: header{*Guarantees Specifications*}
paulson@13798: 
haftmann@27682: theory Guar
haftmann@27682: imports Comp
haftmann@27682: begin
paulson@7400: 
wenzelm@12338: instance program :: (type) order
haftmann@27682: proof qed (auto simp add: program_less_le dest: component_antisym intro: component_refl component_trans)
paulson@7400: 
paulson@14112: text{*Existential and Universal properties.  I formalize the two-program
paulson@14112:       case, proving equivalence with Chandy and Sanders's n-ary definitions*}
paulson@7400: 
haftmann@35416: definition ex_prop :: "'a program set => bool" where
paulson@13819:    "ex_prop X == \<forall>F G. F ok G -->F \<in> X | G \<in> X --> (F\<squnion>G) \<in> X"
paulson@7400: 
haftmann@35416: definition strict_ex_prop  :: "'a program set => bool" where
paulson@13819:    "strict_ex_prop X == \<forall>F G.  F ok G --> (F \<in> X | G \<in> X) = (F\<squnion>G \<in> X)"
paulson@7400: 
haftmann@35416: definition uv_prop  :: "'a program set => bool" where
paulson@13819:    "uv_prop X == SKIP \<in> X & (\<forall>F G. F ok G --> F \<in> X & G \<in> X --> (F\<squnion>G) \<in> X)"
paulson@7400: 
haftmann@35416: definition strict_uv_prop  :: "'a program set => bool" where
paulson@13792:    "strict_uv_prop X == 
paulson@13819:       SKIP \<in> X & (\<forall>F G. F ok G --> (F \<in> X & G \<in> X) = (F\<squnion>G \<in> X))"
paulson@7400: 
paulson@14112: 
paulson@14112: text{*Guarantees properties*}
paulson@14112: 
haftmann@35416: definition guar :: "['a program set, 'a program set] => 'a program set" (infixl "guarantees" 55) where
haftmann@35416:           (*higher than membership, lower than Co*)
paulson@13819:    "X guarantees Y == {F. \<forall>G. F ok G --> F\<squnion>G \<in> X --> F\<squnion>G \<in> Y}"
ehmety@11190:   
paulson@7400: 
ehmety@11190:   (* Weakest guarantees *)
haftmann@35416: definition wg :: "['a program, 'a program set] => 'a program set" where
paulson@13805:   "wg F Y == Union({X. F \<in> X guarantees Y})"
ehmety@11190: 
ehmety@11190:    (* Weakest existential property stronger than X *)
haftmann@35416: definition wx :: "('a program) set => ('a program)set" where
paulson@13805:    "wx X == Union({Y. Y \<subseteq> X & ex_prop Y})"
ehmety@11190:   
ehmety@11190:   (*Ill-defined programs can arise through "Join"*)
haftmann@35416: definition welldef :: "'a program set" where
paulson@13805:   "welldef == {F. Init F \<noteq> {}}"
ehmety@11190:   
haftmann@35416: definition refines :: "['a program, 'a program, 'a program set] => bool"
haftmann@35416:                         ("(3_ refines _ wrt _)" [10,10,10] 10) where
ehmety@11190:   "G refines F wrt X ==
paulson@14112:      \<forall>H. (F ok H & G ok H & F\<squnion>H \<in> welldef \<inter> X) --> 
paulson@13819:          (G\<squnion>H \<in> welldef \<inter> X)"
paulson@7400: 
haftmann@35416: definition iso_refines :: "['a program, 'a program, 'a program set] => bool"
haftmann@35416:                               ("(3_ iso'_refines _ wrt _)" [10,10,10] 10) where
ehmety@11190:   "G iso_refines F wrt X ==
paulson@13805:    F \<in> welldef \<inter> X --> G \<in> welldef \<inter> X"
paulson@7400: 
paulson@13792: 
paulson@13792: lemma OK_insert_iff:
paulson@13792:      "(OK (insert i I) F) = 
paulson@13805:       (if i \<in> I then OK I F else OK I F & (F i ok JOIN I F))"
paulson@13792: by (auto intro: ok_sym simp add: OK_iff_ok)
paulson@13792: 
paulson@13792: 
paulson@14112: subsection{*Existential Properties*}
paulson@14112: 
paulson@13798: lemma ex1 [rule_format]: 
paulson@13792:  "[| ex_prop X; finite GG |] ==>  
paulson@13805:      GG \<inter> X \<noteq> {}--> OK GG (%G. G) --> (\<Squnion>G \<in> GG. G) \<in> X"
paulson@13792: apply (unfold ex_prop_def)
paulson@13792: apply (erule finite_induct)
paulson@13792: apply (auto simp add: OK_insert_iff Int_insert_left)
paulson@13792: done
paulson@13792: 
paulson@13792: 
paulson@13792: lemma ex2: 
paulson@13805:      "\<forall>GG. finite GG & GG \<inter> X \<noteq> {} --> OK GG (%G. G) -->(\<Squnion>G \<in> GG. G):X 
paulson@13792:       ==> ex_prop X"
paulson@13792: apply (unfold ex_prop_def, clarify)
paulson@13792: apply (drule_tac x = "{F,G}" in spec)
paulson@13792: apply (auto dest: ok_sym simp add: OK_iff_ok)
paulson@13792: done
paulson@13792: 
paulson@13792: 
paulson@13792: (*Chandy & Sanders take this as a definition*)
paulson@13792: lemma ex_prop_finite:
paulson@13792:      "ex_prop X = 
paulson@13805:       (\<forall>GG. finite GG & GG \<inter> X \<noteq> {} & OK GG (%G. G)--> (\<Squnion>G \<in> GG. G) \<in> X)"
paulson@13792: by (blast intro: ex1 ex2)
paulson@13792: 
paulson@13792: 
paulson@13792: (*Their "equivalent definition" given at the end of section 3*)
paulson@13792: lemma ex_prop_equiv: 
paulson@13805:      "ex_prop X = (\<forall>G. G \<in> X = (\<forall>H. (G component_of H) --> H \<in> X))"
paulson@13792: apply auto
paulson@14112: apply (unfold ex_prop_def component_of_def, safe, blast, blast) 
paulson@13792: apply (subst Join_commute) 
paulson@13792: apply (drule ok_sym, blast) 
paulson@13792: done
paulson@13792: 
paulson@13792: 
paulson@14112: subsection{*Universal Properties*}
paulson@14112: 
paulson@13792: lemma uv1 [rule_format]: 
paulson@13792:      "[| uv_prop X; finite GG |] 
paulson@13805:       ==> GG \<subseteq> X & OK GG (%G. G) --> (\<Squnion>G \<in> GG. G) \<in> X"
paulson@13792: apply (unfold uv_prop_def)
paulson@13792: apply (erule finite_induct)
paulson@13792: apply (auto simp add: Int_insert_left OK_insert_iff)
paulson@13792: done
paulson@13792: 
paulson@13792: lemma uv2: 
paulson@13805:      "\<forall>GG. finite GG & GG \<subseteq> X & OK GG (%G. G) --> (\<Squnion>G \<in> GG. G) \<in> X  
paulson@13792:       ==> uv_prop X"
paulson@13792: apply (unfold uv_prop_def)
paulson@13792: apply (rule conjI)
paulson@13792:  apply (drule_tac x = "{}" in spec)
paulson@13792:  prefer 2
paulson@13792:  apply clarify 
paulson@13792:  apply (drule_tac x = "{F,G}" in spec)
paulson@13792: apply (auto dest: ok_sym simp add: OK_iff_ok)
paulson@13792: done
paulson@13792: 
paulson@13792: (*Chandy & Sanders take this as a definition*)
paulson@13792: lemma uv_prop_finite:
paulson@13792:      "uv_prop X = 
paulson@13805:       (\<forall>GG. finite GG & GG \<subseteq> X & OK GG (%G. G) --> (\<Squnion>G \<in> GG. G): X)"
paulson@13792: by (blast intro: uv1 uv2)
paulson@13792: 
paulson@14112: subsection{*Guarantees*}
paulson@13792: 
paulson@13792: lemma guaranteesI:
paulson@14112:      "(!!G. [| F ok G; F\<squnion>G \<in> X |] ==> F\<squnion>G \<in> Y) ==> F \<in> X guarantees Y"
paulson@13792: by (simp add: guar_def component_def)
paulson@13792: 
paulson@13792: lemma guaranteesD: 
paulson@14112:      "[| F \<in> X guarantees Y;  F ok G;  F\<squnion>G \<in> X |] ==> F\<squnion>G \<in> Y"
paulson@13792: by (unfold guar_def component_def, blast)
paulson@13792: 
paulson@13792: (*This version of guaranteesD matches more easily in the conclusion
paulson@13805:   The major premise can no longer be  F \<subseteq> H since we need to reason about G*)
paulson@13792: lemma component_guaranteesD: 
paulson@14112:      "[| F \<in> X guarantees Y;  F\<squnion>G = H;  H \<in> X;  F ok G |] ==> H \<in> Y"
paulson@13792: by (unfold guar_def, blast)
paulson@13792: 
paulson@13792: lemma guarantees_weaken: 
paulson@13805:      "[| F \<in> X guarantees X'; Y \<subseteq> X; X' \<subseteq> Y' |] ==> F \<in> Y guarantees Y'"
paulson@13792: by (unfold guar_def, blast)
paulson@13792: 
paulson@13805: lemma subset_imp_guarantees_UNIV: "X \<subseteq> Y ==> X guarantees Y = UNIV"
paulson@13792: by (unfold guar_def, blast)
paulson@13792: 
paulson@13792: (*Equivalent to subset_imp_guarantees_UNIV but more intuitive*)
paulson@13805: lemma subset_imp_guarantees: "X \<subseteq> Y ==> F \<in> X guarantees Y"
paulson@13792: by (unfold guar_def, blast)
paulson@13792: 
paulson@13792: (*Remark at end of section 4.1 *)
paulson@13792: 
paulson@13792: lemma ex_prop_imp: "ex_prop Y ==> (Y = UNIV guarantees Y)"
paulson@13792: apply (simp (no_asm_use) add: guar_def ex_prop_equiv)
paulson@13792: apply safe
paulson@13792:  apply (drule_tac x = x in spec)
paulson@13792:  apply (drule_tac [2] x = x in spec)
paulson@13792:  apply (drule_tac [2] sym)
paulson@13792: apply (auto simp add: component_of_def)
paulson@13792: done
paulson@13792: 
paulson@13792: lemma guarantees_imp: "(Y = UNIV guarantees Y) ==> ex_prop(Y)"
paulson@14112: by (auto simp add: guar_def ex_prop_equiv component_of_def dest: sym)
paulson@13792: 
paulson@13792: lemma ex_prop_equiv2: "(ex_prop Y) = (Y = UNIV guarantees Y)"
paulson@13792: apply (rule iffI)
paulson@13792: apply (rule ex_prop_imp)
paulson@13792: apply (auto simp add: guarantees_imp) 
paulson@13792: done
paulson@13792: 
paulson@13792: 
paulson@14112: subsection{*Distributive Laws.  Re-Orient to Perform Miniscoping*}
paulson@13792: 
paulson@13792: lemma guarantees_UN_left: 
paulson@13805:      "(\<Union>i \<in> I. X i) guarantees Y = (\<Inter>i \<in> I. X i guarantees Y)"
paulson@13792: by (unfold guar_def, blast)
paulson@13792: 
paulson@13792: lemma guarantees_Un_left: 
paulson@13805:      "(X \<union> Y) guarantees Z = (X guarantees Z) \<inter> (Y guarantees Z)"
paulson@13792: by (unfold guar_def, blast)
paulson@13792: 
paulson@13792: lemma guarantees_INT_right: 
paulson@13805:      "X guarantees (\<Inter>i \<in> I. Y i) = (\<Inter>i \<in> I. X guarantees Y i)"
paulson@13792: by (unfold guar_def, blast)
paulson@13792: 
paulson@13792: lemma guarantees_Int_right: 
paulson@13805:      "Z guarantees (X \<inter> Y) = (Z guarantees X) \<inter> (Z guarantees Y)"
paulson@13792: by (unfold guar_def, blast)
paulson@13792: 
paulson@13792: lemma guarantees_Int_right_I:
paulson@13805:      "[| F \<in> Z guarantees X;  F \<in> Z guarantees Y |]  
paulson@13805:      ==> F \<in> Z guarantees (X \<inter> Y)"
paulson@13792: by (simp add: guarantees_Int_right)
paulson@13792: 
paulson@13792: lemma guarantees_INT_right_iff:
paulson@13805:      "(F \<in> X guarantees (INTER I Y)) = (\<forall>i\<in>I. F \<in> X guarantees (Y i))"
paulson@13792: by (simp add: guarantees_INT_right)
paulson@13792: 
paulson@13805: lemma shunting: "(X guarantees Y) = (UNIV guarantees (-X \<union> Y))"
paulson@13792: by (unfold guar_def, blast)
paulson@13792: 
paulson@13792: lemma contrapositive: "(X guarantees Y) = -Y guarantees -X"
paulson@13792: by (unfold guar_def, blast)
paulson@13792: 
paulson@13792: (** The following two can be expressed using intersection and subset, which
paulson@13792:     is more faithful to the text but looks cryptic.
paulson@13792: **)
paulson@13792: 
paulson@13792: lemma combining1: 
paulson@13805:     "[| F \<in> V guarantees X;  F \<in> (X \<inter> Y) guarantees Z |] 
paulson@13805:      ==> F \<in> (V \<inter> Y) guarantees Z"
paulson@13792: by (unfold guar_def, blast)
paulson@13792: 
paulson@13792: lemma combining2: 
paulson@13805:     "[| F \<in> V guarantees (X \<union> Y);  F \<in> Y guarantees Z |] 
paulson@13805:      ==> F \<in> V guarantees (X \<union> Z)"
paulson@13792: by (unfold guar_def, blast)
paulson@13792: 
paulson@13792: (** The following two follow Chandy-Sanders, but the use of object-quantifiers
paulson@13792:     does not suit Isabelle... **)
paulson@13792: 
paulson@13805: (*Premise should be (!!i. i \<in> I ==> F \<in> X guarantees Y i) *)
paulson@13792: lemma all_guarantees: 
paulson@13805:      "\<forall>i\<in>I. F \<in> X guarantees (Y i) ==> F \<in> X guarantees (\<Inter>i \<in> I. Y i)"
paulson@13792: by (unfold guar_def, blast)
paulson@13792: 
paulson@13805: (*Premises should be [| F \<in> X guarantees Y i; i \<in> I |] *)
paulson@13792: lemma ex_guarantees: 
paulson@13805:      "\<exists>i\<in>I. F \<in> X guarantees (Y i) ==> F \<in> X guarantees (\<Union>i \<in> I. Y i)"
paulson@13792: by (unfold guar_def, blast)
paulson@13792: 
paulson@13792: 
paulson@14112: subsection{*Guarantees: Additional Laws (by lcp)*}
paulson@13792: 
paulson@13792: lemma guarantees_Join_Int: 
paulson@13805:     "[| F \<in> U guarantees V;  G \<in> X guarantees Y; F ok G |]  
paulson@13819:      ==> F\<squnion>G \<in> (U \<inter> X) guarantees (V \<inter> Y)"
paulson@14112: apply (simp add: guar_def, safe)
paulson@14112:  apply (simp add: Join_assoc)
paulson@13819: apply (subgoal_tac "F\<squnion>G\<squnion>Ga = G\<squnion>(F\<squnion>Ga) ")
paulson@14112:  apply (simp add: ok_commute)
paulson@14112: apply (simp add: Join_ac)
paulson@13792: done
paulson@13792: 
paulson@13792: lemma guarantees_Join_Un: 
paulson@13805:     "[| F \<in> U guarantees V;  G \<in> X guarantees Y; F ok G |]   
paulson@13819:      ==> F\<squnion>G \<in> (U \<union> X) guarantees (V \<union> Y)"
paulson@14112: apply (simp add: guar_def, safe)
paulson@14112:  apply (simp add: Join_assoc)
paulson@13819: apply (subgoal_tac "F\<squnion>G\<squnion>Ga = G\<squnion>(F\<squnion>Ga) ")
paulson@14112:  apply (simp add: ok_commute)
paulson@14112: apply (simp add: Join_ac)
paulson@13792: done
paulson@13792: 
paulson@13792: lemma guarantees_JN_INT: 
paulson@13805:      "[| \<forall>i\<in>I. F i \<in> X i guarantees Y i;  OK I F |]  
paulson@13805:       ==> (JOIN I F) \<in> (INTER I X) guarantees (INTER I Y)"
paulson@13792: apply (unfold guar_def, auto)
paulson@13792: apply (drule bspec, assumption)
paulson@13792: apply (rename_tac "i")
paulson@13819: apply (drule_tac x = "JOIN (I-{i}) F\<squnion>G" in spec)
paulson@13792: apply (auto intro: OK_imp_ok
paulson@13792:             simp add: Join_assoc [symmetric] JN_Join_diff JN_absorb)
paulson@13792: done
paulson@13792: 
paulson@13792: lemma guarantees_JN_UN: 
paulson@13805:     "[| \<forall>i\<in>I. F i \<in> X i guarantees Y i;  OK I F |]  
paulson@13805:      ==> (JOIN I F) \<in> (UNION I X) guarantees (UNION I Y)"
paulson@13792: apply (unfold guar_def, auto)
paulson@13792: apply (drule bspec, assumption)
paulson@13792: apply (rename_tac "i")
paulson@13819: apply (drule_tac x = "JOIN (I-{i}) F\<squnion>G" in spec)
paulson@13792: apply (auto intro: OK_imp_ok
paulson@13792:             simp add: Join_assoc [symmetric] JN_Join_diff JN_absorb)
paulson@13792: done
paulson@13792: 
paulson@13792: 
paulson@14112: subsection{*Guarantees Laws for Breaking Down the Program (by lcp)*}
paulson@13792: 
paulson@13792: lemma guarantees_Join_I1: 
paulson@13819:      "[| F \<in> X guarantees Y;  F ok G |] ==> F\<squnion>G \<in> X guarantees Y"
paulson@14112: by (simp add: guar_def Join_assoc)
paulson@13792: 
paulson@14112: lemma guarantees_Join_I2:         
paulson@13819:      "[| G \<in> X guarantees Y;  F ok G |] ==> F\<squnion>G \<in> X guarantees Y"
paulson@13792: apply (simp add: Join_commute [of _ G] ok_commute [of _ G])
paulson@13792: apply (blast intro: guarantees_Join_I1)
paulson@13792: done
paulson@13792: 
paulson@13792: lemma guarantees_JN_I: 
paulson@13805:      "[| i \<in> I;  F i \<in> X guarantees Y;  OK I F |]  
paulson@13805:       ==> (\<Squnion>i \<in> I. (F i)) \<in> X guarantees Y"
paulson@13792: apply (unfold guar_def, clarify)
paulson@13819: apply (drule_tac x = "JOIN (I-{i}) F\<squnion>G" in spec)
paulson@14112: apply (auto intro: OK_imp_ok 
paulson@14112:             simp add: JN_Join_diff JN_Join_diff Join_assoc [symmetric])
paulson@13792: done
paulson@13792: 
paulson@13792: 
paulson@13792: (*** well-definedness ***)
paulson@13792: 
paulson@13819: lemma Join_welldef_D1: "F\<squnion>G \<in> welldef ==> F \<in> welldef"
paulson@13792: by (unfold welldef_def, auto)
paulson@13792: 
paulson@13819: lemma Join_welldef_D2: "F\<squnion>G \<in> welldef ==> G \<in> welldef"
paulson@13792: by (unfold welldef_def, auto)
paulson@13792: 
paulson@13792: (*** refinement ***)
paulson@13792: 
paulson@13792: lemma refines_refl: "F refines F wrt X"
paulson@13792: by (unfold refines_def, blast)
paulson@13792: 
paulson@14112: (*We'd like transitivity, but how do we get it?*)
paulson@14112: lemma refines_trans:
paulson@13792:      "[| H refines G wrt X;  G refines F wrt X |] ==> H refines F wrt X"
paulson@14112: apply (simp add: refines_def) 
paulson@14112: oops
paulson@13792: 
paulson@13792: 
paulson@13792: lemma strict_ex_refine_lemma: 
paulson@13792:      "strict_ex_prop X  
paulson@13819:       ==> (\<forall>H. F ok H & G ok H & F\<squnion>H \<in> X --> G\<squnion>H \<in> X)  
paulson@13805:               = (F \<in> X --> G \<in> X)"
paulson@13792: by (unfold strict_ex_prop_def, auto)
paulson@13792: 
paulson@13792: lemma strict_ex_refine_lemma_v: 
paulson@13792:      "strict_ex_prop X  
paulson@13819:       ==> (\<forall>H. F ok H & G ok H & F\<squnion>H \<in> welldef & F\<squnion>H \<in> X --> G\<squnion>H \<in> X) =  
paulson@13805:           (F \<in> welldef \<inter> X --> G \<in> X)"
paulson@13792: apply (unfold strict_ex_prop_def, safe)
paulson@13792: apply (erule_tac x = SKIP and P = "%H. ?PP H --> ?RR H" in allE)
paulson@13792: apply (auto dest: Join_welldef_D1 Join_welldef_D2)
paulson@13792: done
paulson@13792: 
paulson@13792: lemma ex_refinement_thm:
paulson@13792:      "[| strict_ex_prop X;   
paulson@13819:          \<forall>H. F ok H & G ok H & F\<squnion>H \<in> welldef \<inter> X --> G\<squnion>H \<in> welldef |]  
paulson@13792:       ==> (G refines F wrt X) = (G iso_refines F wrt X)"
paulson@13792: apply (rule_tac x = SKIP in allE, assumption)
paulson@13792: apply (simp add: refines_def iso_refines_def strict_ex_refine_lemma_v)
paulson@13792: done
paulson@13792: 
paulson@13792: 
paulson@13792: lemma strict_uv_refine_lemma: 
paulson@13792:      "strict_uv_prop X ==> 
paulson@13819:       (\<forall>H. F ok H & G ok H & F\<squnion>H \<in> X --> G\<squnion>H \<in> X) = (F \<in> X --> G \<in> X)"
paulson@13792: by (unfold strict_uv_prop_def, blast)
paulson@13792: 
paulson@13792: lemma strict_uv_refine_lemma_v: 
paulson@13792:      "strict_uv_prop X  
paulson@13819:       ==> (\<forall>H. F ok H & G ok H & F\<squnion>H \<in> welldef & F\<squnion>H \<in> X --> G\<squnion>H \<in> X) =  
paulson@13805:           (F \<in> welldef \<inter> X --> G \<in> X)"
paulson@13792: apply (unfold strict_uv_prop_def, safe)
paulson@13792: apply (erule_tac x = SKIP and P = "%H. ?PP H --> ?RR H" in allE)
paulson@13792: apply (auto dest: Join_welldef_D1 Join_welldef_D2)
paulson@13792: done
paulson@13792: 
paulson@13792: lemma uv_refinement_thm:
paulson@13792:      "[| strict_uv_prop X;   
paulson@13819:          \<forall>H. F ok H & G ok H & F\<squnion>H \<in> welldef \<inter> X --> 
paulson@13819:              G\<squnion>H \<in> welldef |]  
paulson@13792:       ==> (G refines F wrt X) = (G iso_refines F wrt X)"
paulson@13792: apply (rule_tac x = SKIP in allE, assumption)
paulson@13792: apply (simp add: refines_def iso_refines_def strict_uv_refine_lemma_v)
paulson@13792: done
paulson@13792: 
paulson@13792: (* Added by Sidi Ehmety from Chandy & Sander, section 6 *)
paulson@13792: lemma guarantees_equiv: 
paulson@13805:     "(F \<in> X guarantees Y) = (\<forall>H. H \<in> X \<longrightarrow> (F component_of H \<longrightarrow> H \<in> Y))"
paulson@13792: by (unfold guar_def component_of_def, auto)
paulson@13792: 
paulson@14112: lemma wg_weakest: "!!X. F\<in> (X guarantees Y) ==> X \<subseteq> (wg F Y)"
paulson@13792: by (unfold wg_def, auto)
paulson@13792: 
paulson@14112: lemma wg_guarantees: "F\<in> ((wg F Y) guarantees Y)"
paulson@13792: by (unfold wg_def guar_def, blast)
paulson@13792: 
paulson@14112: lemma wg_equiv: "(H \<in> wg F X) = (F component_of H --> H \<in> X)"
paulson@14112: by (simp add: guarantees_equiv wg_def, blast)
paulson@13792: 
paulson@13805: lemma component_of_wg: "F component_of H ==> (H \<in> wg F X) = (H \<in> X)"
paulson@13792: by (simp add: wg_equiv)
paulson@13792: 
paulson@13792: lemma wg_finite: 
paulson@13805:     "\<forall>FF. finite FF & FF \<inter> X \<noteq> {} --> OK FF (%F. F)  
paulson@13805:           --> (\<forall>F\<in>FF. ((\<Squnion>F \<in> FF. F): wg F X) = ((\<Squnion>F \<in> FF. F):X))"
paulson@13792: apply clarify
paulson@13805: apply (subgoal_tac "F component_of (\<Squnion>F \<in> FF. F) ")
paulson@13792: apply (drule_tac X = X in component_of_wg, simp)
paulson@13792: apply (simp add: component_of_def)
paulson@13805: apply (rule_tac x = "\<Squnion>F \<in> (FF-{F}) . F" in exI)
paulson@13792: apply (auto intro: JN_Join_diff dest: ok_sym simp add: OK_iff_ok)
paulson@13792: done
paulson@13792: 
paulson@13805: lemma wg_ex_prop: "ex_prop X ==> (F \<in> X) = (\<forall>H. H \<in> wg F X)"
paulson@13792: apply (simp (no_asm_use) add: ex_prop_equiv wg_equiv)
paulson@13792: apply blast
paulson@13792: done
paulson@13792: 
paulson@13792: (** From Charpentier and Chandy "Theorems About Composition" **)
paulson@13792: (* Proposition 2 *)
paulson@13792: lemma wx_subset: "(wx X)<=X"
paulson@13792: by (unfold wx_def, auto)
paulson@13792: 
paulson@13792: lemma wx_ex_prop: "ex_prop (wx X)"
berghofe@16647: apply (simp add: wx_def ex_prop_equiv cong: bex_cong, safe, blast)
paulson@14112: apply force 
paulson@13792: done
paulson@13792: 
paulson@13805: lemma wx_weakest: "\<forall>Z. Z<= X --> ex_prop Z --> Z \<subseteq> wx X"
paulson@14112: by (auto simp add: wx_def)
paulson@13792: 
paulson@13792: (* Proposition 6 *)
paulson@13819: lemma wx'_ex_prop: "ex_prop({F. \<forall>G. F ok G --> F\<squnion>G \<in> X})"
paulson@13792: apply (unfold ex_prop_def, safe)
paulson@14112:  apply (drule_tac x = "G\<squnion>Ga" in spec)
paulson@14112:  apply (force simp add: ok_Join_iff1 Join_assoc)
paulson@13819: apply (drule_tac x = "F\<squnion>Ga" in spec)
paulson@14112: apply (simp add: ok_Join_iff1 ok_commute  Join_ac) 
paulson@13792: done
paulson@13792: 
paulson@14112: text{* Equivalence with the other definition of wx *}
paulson@13792: 
paulson@14112: lemma wx_equiv: "wx X = {F. \<forall>G. F ok G --> (F\<squnion>G) \<in> X}"
paulson@13792: apply (unfold wx_def, safe)
paulson@14112:  apply (simp add: ex_prop_def, blast) 
paulson@13792: apply (simp (no_asm))
paulson@13819: apply (rule_tac x = "{F. \<forall>G. F ok G --> F\<squnion>G \<in> X}" in exI, safe)
paulson@13792: apply (rule_tac [2] wx'_ex_prop)
paulson@14112: apply (drule_tac x = SKIP in spec)+
paulson@14112: apply auto 
paulson@13792: done
paulson@13792: 
paulson@13792: 
paulson@14112: text{* Propositions 7 to 11 are about this second definition of wx. 
paulson@14112:    They are the same as the ones proved for the first definition of wx,
paulson@14112:  by equivalence *}
paulson@13792:    
paulson@13792: (* Proposition 12 *)
paulson@13792: (* Main result of the paper *)
paulson@14112: lemma guarantees_wx_eq: "(X guarantees Y) = wx(-X \<union> Y)"
paulson@14112: by (simp add: guar_def wx_equiv)
paulson@13792: 
paulson@13792: 
paulson@13792: (* Rules given in section 7 of Chandy and Sander's
paulson@13792:     Reasoning About Program composition paper *)
paulson@13792: lemma stable_guarantees_Always:
paulson@14112:      "Init F \<subseteq> A ==> F \<in> (stable A) guarantees (Always A)"
paulson@13792: apply (rule guaranteesI)
paulson@14112: apply (simp add: Join_commute)
paulson@13792: apply (rule stable_Join_Always1)
paulson@14112:  apply (simp_all add: invariant_def Join_stable)
paulson@13792: done
paulson@13792: 
paulson@13792: lemma constrains_guarantees_leadsTo:
paulson@13805:      "F \<in> transient A ==> F \<in> (A co A \<union> B) guarantees (A leadsTo (B-A))"
paulson@13792: apply (rule guaranteesI)
paulson@13792: apply (rule leadsTo_Basis')
paulson@14112:  apply (drule constrains_weaken_R)
paulson@14112:   prefer 2 apply assumption
paulson@14112:  apply blast
paulson@13792: apply (blast intro: Join_transient_I1)
paulson@13792: done
paulson@13792: 
paulson@7400: end




