Theory Guar (Isabelle2005: October 2005)

# Theory Guar

Up to index of Isabelle/HOL/UNITY

theory Guar
imports Comp
begin

    (*  Title:      HOL/UNITY/Guar.thy
        ID:         $Id: Guar.thy,v 1.14 2005/07/01 12:16:32 berghofe Exp $
        Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
        Copyright   1999  University of Cambridge

    From Chandy and Sanders, "Reasoning About Program Composition",
    Technical Report 2000-003, University of Florida, 2000.

    Revised by Sidi Ehmety on January 2001

    Added: Compatibility, weakest guarantees, etc.

    and Weakest existential property,
    from Charpentier and Chandy "Theorems about Composition",
    Fifth International Conference on Mathematics of Program, 2000.

    *)

    header{*Guarantees Specifications*}

    theory Guar imports Comp begin

    instance program :: (type) order
      by (intro_classes,
          (assumption | rule component_refl component_trans component_antisym
                         program_less_le)+)

    text{*Existential and Universal properties.  I formalize the two-program
          case, proving equivalence with Chandy and Sanders's n-ary definitions*}

    constdefs

      ex_prop  :: "'a program set => bool"
       "ex_prop X == ∀F G. F ok G -->F ∈ X | G ∈ X --> (F\<squnion>G) ∈ X"

      strict_ex_prop  :: "'a program set => bool"
       "strict_ex_prop X == ∀F G.  F ok G --> (F ∈ X | G ∈ X) = (F\<squnion>G ∈ X)"

      uv_prop  :: "'a program set => bool"
       "uv_prop X == SKIP ∈ X & (∀F G. F ok G --> F ∈ X & G ∈ X --> (F\<squnion>G) ∈ X)"

      strict_uv_prop  :: "'a program set => bool"
       "strict_uv_prop X ==
          SKIP ∈ X & (∀F G. F ok G --> (F ∈ X & G ∈ X) = (F\<squnion>G ∈ X))"

    text{*Guarantees properties*}

    constdefs

      guar :: "['a program set, 'a program set] => 'a program set"
              (infixl "guarantees" 55)  (*higher than membership, lower than Co*)
       "X guarantees Y == {F. ∀G. F ok G --> F\<squnion>G ∈ X --> F\<squnion>G ∈ Y}"

      (* Weakest guarantees *)
       wg :: "['a program, 'a program set] =>  'a program set"
      "wg F Y == Union({X. F ∈ X guarantees Y})"

       (* Weakest existential property stronger than X *)
       wx :: "('a program) set => ('a program)set"
       "wx X == Union({Y. Y ⊆ X & ex_prop Y})"

      (*Ill-defined programs can arise through "Join"*)
      welldef :: "'a program set"
      "welldef == {F. Init F ≠ {}}"

      refines :: "['a program, 'a program, 'a program set] => bool"
                            ("(3_ refines _ wrt _)" [10,10,10] 10)
      "G refines F wrt X ==
         ∀H. (F ok H & G ok H & F\<squnion>H ∈ welldef ∩ X) -->
             (G\<squnion>H ∈ welldef ∩ X)"

      iso_refines :: "['a program, 'a program, 'a program set] => bool"
                                  ("(3_ iso'_refines _ wrt _)" [10,10,10] 10)
      "G iso_refines F wrt X ==
       F ∈ welldef ∩ X --> G ∈ welldef ∩ X"

    lemma OK_insert_iff:
         "(OK (insert i I) F) =
          (if i ∈ I then OK I F else OK I F & (F i ok JOIN I F))"
    by (auto intro: ok_sym simp add: OK_iff_ok)

    subsection{*Existential Properties*}

    lemma ex1 [rule_format]:
     "[| ex_prop X; finite GG |] ==>
         GG ∩ X ≠ {}--> OK GG (%G. G) --> (\<Squnion>G ∈ GG. G) ∈ X"
    apply (unfold ex_prop_def)
    apply (erule finite_induct)
    apply (auto simp add: OK_insert_iff Int_insert_left)
    done

    lemma ex2:
         "∀GG. finite GG & GG ∩ X ≠ {} --> OK GG (%G. G) -->(\<Squnion>G ∈ GG. G):X
          ==> ex_prop X"
    apply (unfold ex_prop_def, clarify)
    apply (drule_tac x = "{F,G}" in spec)
    apply (auto dest: ok_sym simp add: OK_iff_ok)
    done

    (*Chandy & Sanders take this as a definition*)
    lemma ex_prop_finite:
         "ex_prop X =
          (∀GG. finite GG & GG ∩ X ≠ {} & OK GG (%G. G)--> (\<Squnion>G ∈ GG. G) ∈ X)"
    by (blast intro: ex1 ex2)

    (*Their "equivalent definition" given at the end of section 3*)
    lemma ex_prop_equiv:
         "ex_prop X = (∀G. G ∈ X = (∀H. (G component_of H) --> H ∈ X))"
    apply auto
    apply (unfold ex_prop_def component_of_def, safe, blast, blast)
    apply (subst Join_commute)
    apply (drule ok_sym, blast)
    done

    subsection{*Universal Properties*}

    lemma uv1 [rule_format]:
         "[| uv_prop X; finite GG |]
          ==> GG ⊆ X & OK GG (%G. G) --> (\<Squnion>G ∈ GG. G) ∈ X"
    apply (unfold uv_prop_def)
    apply (erule finite_induct)
    apply (auto simp add: Int_insert_left OK_insert_iff)
    done

    lemma uv2:
         "∀GG. finite GG & GG ⊆ X & OK GG (%G. G) --> (\<Squnion>G ∈ GG. G) ∈ X
          ==> uv_prop X"
    apply (unfold uv_prop_def)
    apply (rule conjI)
     apply (drule_tac x = "{}" in spec)
     prefer 2
     apply clarify
     apply (drule_tac x = "{F,G}" in spec)
    apply (auto dest: ok_sym simp add: OK_iff_ok)
    done

    (*Chandy & Sanders take this as a definition*)
    lemma uv_prop_finite:
         "uv_prop X =
          (∀GG. finite GG & GG ⊆ X & OK GG (%G. G) --> (\<Squnion>G ∈ GG. G): X)"
    by (blast intro: uv1 uv2)

    subsection{*Guarantees*}

    lemma guaranteesI:
         "(!!G. [| F ok G; F\<squnion>G ∈ X |] ==> F\<squnion>G ∈ Y) ==> F ∈ X guarantees Y"
    by (simp add: guar_def component_def)

    lemma guaranteesD:
         "[| F ∈ X guarantees Y;  F ok G;  F\<squnion>G ∈ X |] ==> F\<squnion>G ∈ Y"
    by (unfold guar_def component_def, blast)

    (*This version of guaranteesD matches more easily in the conclusion
      The major premise can no longer be  F ⊆ H since we need to reason about G*)
    lemma component_guaranteesD:
         "[| F ∈ X guarantees Y;  F\<squnion>G = H;  H ∈ X;  F ok G |] ==> H ∈ Y"
    by (unfold guar_def, blast)

    lemma guarantees_weaken:
         "[| F ∈ X guarantees X'; Y ⊆ X; X' ⊆ Y' |] ==> F ∈ Y guarantees Y'"
    by (unfold guar_def, blast)

    lemma subset_imp_guarantees_UNIV: "X ⊆ Y ==> X guarantees Y = UNIV"
    by (unfold guar_def, blast)

    (*Equivalent to subset_imp_guarantees_UNIV but more intuitive*)
    lemma subset_imp_guarantees: "X ⊆ Y ==> F ∈ X guarantees Y"
    by (unfold guar_def, blast)

    (*Remark at end of section 4.1 *)

    lemma ex_prop_imp: "ex_prop Y ==> (Y = UNIV guarantees Y)"
    apply (simp (no_asm_use) add: guar_def ex_prop_equiv)
    apply safe
     apply (drule_tac x = x in spec)
     apply (drule_tac [2] x = x in spec)
     apply (drule_tac [2] sym)
    apply (auto simp add: component_of_def)
    done

    lemma guarantees_imp: "(Y = UNIV guarantees Y) ==> ex_prop(Y)"
    by (auto simp add: guar_def ex_prop_equiv component_of_def dest: sym)

    lemma ex_prop_equiv2: "(ex_prop Y) = (Y = UNIV guarantees Y)"
    apply (rule iffI)
    apply (rule ex_prop_imp)
    apply (auto simp add: guarantees_imp)
    done

    subsection{*Distributive Laws.  Re-Orient to Perform Miniscoping*}

    lemma guarantees_UN_left:
         "(\<Union>i ∈ I. X i) guarantees Y = (\<Inter>i ∈ I. X i guarantees Y)"
    by (unfold guar_def, blast)

    lemma guarantees_Un_left:
         "(X ∪ Y) guarantees Z = (X guarantees Z) ∩ (Y guarantees Z)"
    by (unfold guar_def, blast)

    lemma guarantees_INT_right:
         "X guarantees (\<Inter>i ∈ I. Y i) = (\<Inter>i ∈ I. X guarantees Y i)"
    by (unfold guar_def, blast)

    lemma guarantees_Int_right:
         "Z guarantees (X ∩ Y) = (Z guarantees X) ∩ (Z guarantees Y)"
    by (unfold guar_def, blast)

    lemma guarantees_Int_right_I:
         "[| F ∈ Z guarantees X;  F ∈ Z guarantees Y |]
         ==> F ∈ Z guarantees (X ∩ Y)"
    by (simp add: guarantees_Int_right)

    lemma guarantees_INT_right_iff:
         "(F ∈ X guarantees (INTER I Y)) = (∀i∈I. F ∈ X guarantees (Y i))"
    by (simp add: guarantees_INT_right)

    lemma shunting: "(X guarantees Y) = (UNIV guarantees (-X ∪ Y))"
    by (unfold guar_def, blast)

    lemma contrapositive: "(X guarantees Y) = -Y guarantees -X"
    by (unfold guar_def, blast)

    (** The following two can be expressed using intersection and subset, which
        is more faithful to the text but looks cryptic.
    **)

    lemma combining1:
        "[| F ∈ V guarantees X;  F ∈ (X ∩ Y) guarantees Z |]
         ==> F ∈ (V ∩ Y) guarantees Z"
    by (unfold guar_def, blast)

    lemma combining2:
        "[| F ∈ V guarantees (X ∪ Y);  F ∈ Y guarantees Z |]
         ==> F ∈ V guarantees (X ∪ Z)"
    by (unfold guar_def, blast)

    (** The following two follow Chandy-Sanders, but the use of object-quantifiers
        does not suit Isabelle... **)

    (*Premise should be (!!i. i ∈ I ==> F ∈ X guarantees Y i) *)
    lemma all_guarantees:
         "∀i∈I. F ∈ X guarantees (Y i) ==> F ∈ X guarantees (\<Inter>i ∈ I. Y i)"
    by (unfold guar_def, blast)

    (*Premises should be [| F ∈ X guarantees Y i; i ∈ I |] *)
    lemma ex_guarantees:
         "∃i∈I. F ∈ X guarantees (Y i) ==> F ∈ X guarantees (\<Union>i ∈ I. Y i)"
    by (unfold guar_def, blast)

    subsection{*Guarantees: Additional Laws (by lcp)*}

    lemma guarantees_Join_Int:
        "[| F ∈ U guarantees V;  G ∈ X guarantees Y; F ok G |]
         ==> F\<squnion>G ∈ (U ∩ X) guarantees (V ∩ Y)"
    apply (simp add: guar_def, safe)
     apply (simp add: Join_assoc)
    apply (subgoal_tac "F\<squnion>G\<squnion>Ga = G\<squnion>(F\<squnion>Ga) ")
     apply (simp add: ok_commute)
    apply (simp add: Join_ac)
    done

    lemma guarantees_Join_Un:
        "[| F ∈ U guarantees V;  G ∈ X guarantees Y; F ok G |]
         ==> F\<squnion>G ∈ (U ∪ X) guarantees (V ∪ Y)"
    apply (simp add: guar_def, safe)
     apply (simp add: Join_assoc)
    apply (subgoal_tac "F\<squnion>G\<squnion>Ga = G\<squnion>(F\<squnion>Ga) ")
     apply (simp add: ok_commute)
    apply (simp add: Join_ac)
    done

    lemma guarantees_JN_INT:
         "[| ∀i∈I. F i ∈ X i guarantees Y i;  OK I F |]
          ==> (JOIN I F) ∈ (INTER I X) guarantees (INTER I Y)"
    apply (unfold guar_def, auto)
    apply (drule bspec, assumption)
    apply (rename_tac "i")
    apply (drule_tac x = "JOIN (I-{i}) F\<squnion>G" in spec)
    apply (auto intro: OK_imp_ok
                simp add: Join_assoc [symmetric] JN_Join_diff JN_absorb)
    done

    lemma guarantees_JN_UN:
        "[| ∀i∈I. F i ∈ X i guarantees Y i;  OK I F |]
         ==> (JOIN I F) ∈ (UNION I X) guarantees (UNION I Y)"
    apply (unfold guar_def, auto)
    apply (drule bspec, assumption)
    apply (rename_tac "i")
    apply (drule_tac x = "JOIN (I-{i}) F\<squnion>G" in spec)
    apply (auto intro: OK_imp_ok
                simp add: Join_assoc [symmetric] JN_Join_diff JN_absorb)
    done

    subsection{*Guarantees Laws for Breaking Down the Program (by lcp)*}

    lemma guarantees_Join_I1:
         "[| F ∈ X guarantees Y;  F ok G |] ==> F\<squnion>G ∈ X guarantees Y"
    by (simp add: guar_def Join_assoc)

    lemma guarantees_Join_I2:
         "[| G ∈ X guarantees Y;  F ok G |] ==> F\<squnion>G ∈ X guarantees Y"
    apply (simp add: Join_commute [of _ G] ok_commute [of _ G])
    apply (blast intro: guarantees_Join_I1)
    done

    lemma guarantees_JN_I:
         "[| i ∈ I;  F i ∈ X guarantees Y;  OK I F |]
          ==> (\<Squnion>i ∈ I. (F i)) ∈ X guarantees Y"
    apply (unfold guar_def, clarify)
    apply (drule_tac x = "JOIN (I-{i}) F\<squnion>G" in spec)
    apply (auto intro: OK_imp_ok
                simp add: JN_Join_diff JN_Join_diff Join_assoc [symmetric])
    done

    (*** well-definedness ***)

    lemma Join_welldef_D1: "F\<squnion>G ∈ welldef ==> F ∈ welldef"
    by (unfold welldef_def, auto)

    lemma Join_welldef_D2: "F\<squnion>G ∈ welldef ==> G ∈ welldef"
    by (unfold welldef_def, auto)

    (*** refinement ***)

    lemma refines_refl: "F refines F wrt X"
    by (unfold refines_def, blast)

    (*We'd like transitivity, but how do we get it?*)
    lemma refines_trans:
         "[| H refines G wrt X;  G refines F wrt X |] ==> H refines F wrt X"
    apply (simp add: refines_def)
    oops

    lemma strict_ex_refine_lemma:
         "strict_ex_prop X
          ==> (∀H. F ok H & G ok H & F\<squnion>H ∈ X --> G\<squnion>H ∈ X)
                  = (F ∈ X --> G ∈ X)"
    by (unfold strict_ex_prop_def, auto)

    lemma strict_ex_refine_lemma_v:
         "strict_ex_prop X
          ==> (∀H. F ok H & G ok H & F\<squnion>H ∈ welldef & F\<squnion>H ∈ X --> G\<squnion>H ∈ X) =
              (F ∈ welldef ∩ X --> G ∈ X)"
    apply (unfold strict_ex_prop_def, safe)
    apply (erule_tac x = SKIP and P = "%H. ?PP H --> ?RR H" in allE)
    apply (auto dest: Join_welldef_D1 Join_welldef_D2)
    done

    lemma ex_refinement_thm:
         "[| strict_ex_prop X;
             ∀H. F ok H & G ok H & F\<squnion>H ∈ welldef ∩ X --> G\<squnion>H ∈ welldef |]
          ==> (G refines F wrt X) = (G iso_refines F wrt X)"
    apply (rule_tac x = SKIP in allE, assumption)
    apply (simp add: refines_def iso_refines_def strict_ex_refine_lemma_v)
    done

    lemma strict_uv_refine_lemma:
         "strict_uv_prop X ==>
          (∀H. F ok H & G ok H & F\<squnion>H ∈ X --> G\<squnion>H ∈ X) = (F ∈ X --> G ∈ X)"
    by (unfold strict_uv_prop_def, blast)

    lemma strict_uv_refine_lemma_v:
         "strict_uv_prop X
          ==> (∀H. F ok H & G ok H & F\<squnion>H ∈ welldef & F\<squnion>H ∈ X --> G\<squnion>H ∈ X) =
              (F ∈ welldef ∩ X --> G ∈ X)"
    apply (unfold strict_uv_prop_def, safe)
    apply (erule_tac x = SKIP and P = "%H. ?PP H --> ?RR H" in allE)
    apply (auto dest: Join_welldef_D1 Join_welldef_D2)
    done

    lemma uv_refinement_thm:
         "[| strict_uv_prop X;
             ∀H. F ok H & G ok H & F\<squnion>H ∈ welldef ∩ X -->
                 G\<squnion>H ∈ welldef |]
          ==> (G refines F wrt X) = (G iso_refines F wrt X)"
    apply (rule_tac x = SKIP in allE, assumption)
    apply (simp add: refines_def iso_refines_def strict_uv_refine_lemma_v)
    done

    (* Added by Sidi Ehmety from Chandy & Sander, section 6 *)
    lemma guarantees_equiv:
        "(F ∈ X guarantees Y) = (∀H. H ∈ X --> (F component_of H --> H ∈ Y))"
    by (unfold guar_def component_of_def, auto)

    lemma wg_weakest: "!!X. F∈ (X guarantees Y) ==> X ⊆ (wg F Y)"
    by (unfold wg_def, auto)

    lemma wg_guarantees: "F∈ ((wg F Y) guarantees Y)"
    by (unfold wg_def guar_def, blast)

    lemma wg_equiv: "(H ∈ wg F X) = (F component_of H --> H ∈ X)"
    by (simp add: guarantees_equiv wg_def, blast)

    lemma component_of_wg: "F component_of H ==> (H ∈ wg F X) = (H ∈ X)"
    by (simp add: wg_equiv)

    lemma wg_finite:
        "∀FF. finite FF & FF ∩ X ≠ {} --> OK FF (%F. F)
              --> (∀F∈FF. ((\<Squnion>F ∈ FF. F): wg F X) = ((\<Squnion>F ∈ FF. F):X))"
    apply clarify
    apply (subgoal_tac "F component_of (\<Squnion>F ∈ FF. F) ")
    apply (drule_tac X = X in component_of_wg, simp)
    apply (simp add: component_of_def)
    apply (rule_tac x = "\<Squnion>F ∈ (FF-{F}) . F" in exI)
    apply (auto intro: JN_Join_diff dest: ok_sym simp add: OK_iff_ok)
    done

    lemma wg_ex_prop: "ex_prop X ==> (F ∈ X) = (∀H. H ∈ wg F X)"
    apply (simp (no_asm_use) add: ex_prop_equiv wg_equiv)
    apply blast
    done

    (** From Charpentier and Chandy "Theorems About Composition" **)
    (* Proposition 2 *)
    lemma wx_subset: "(wx X)<=X"
    by (unfold wx_def, auto)

    lemma wx_ex_prop: "ex_prop (wx X)"
    apply (simp add: wx_def ex_prop_equiv cong: bex_cong, safe, blast)
    apply force
    done

    lemma wx_weakest: "∀Z. Z<= X --> ex_prop Z --> Z ⊆ wx X"
    by (auto simp add: wx_def)

    (* Proposition 6 *)
    lemma wx'_ex_prop: "ex_prop({F. ∀G. F ok G --> F\<squnion>G ∈ X})"
    apply (unfold ex_prop_def, safe)
     apply (drule_tac x = "G\<squnion>Ga" in spec)
     apply (force simp add: ok_Join_iff1 Join_assoc)
    apply (drule_tac x = "F\<squnion>Ga" in spec)
    apply (simp add: ok_Join_iff1 ok_commute  Join_ac)
    done

    text{* Equivalence with the other definition of wx *}

    lemma wx_equiv: "wx X = {F. ∀G. F ok G --> (F\<squnion>G) ∈ X}"
    apply (unfold wx_def, safe)
     apply (simp add: ex_prop_def, blast)
    apply (simp (no_asm))
    apply (rule_tac x = "{F. ∀G. F ok G --> F\<squnion>G ∈ X}" in exI, safe)
    apply (rule_tac [2] wx'_ex_prop)
    apply (drule_tac x = SKIP in spec)+
    apply auto
    done

    text{* Propositions 7 to 11 are about this second definition of wx.
       They are the same as the ones proved for the first definition of wx,
     by equivalence *}

    (* Proposition 12 *)
    (* Main result of the paper *)
    lemma guarantees_wx_eq: "(X guarantees Y) = wx(-X ∪ Y)"
    by (simp add: guar_def wx_equiv)

    (* Rules given in section 7 of Chandy and Sander's
        Reasoning About Program composition paper *)
    lemma stable_guarantees_Always:
         "Init F ⊆ A ==> F ∈ (stable A) guarantees (Always A)"
    apply (rule guaranteesI)
    apply (simp add: Join_commute)
    apply (rule stable_Join_Always1)
     apply (simp_all add: invariant_def Join_stable)
    done

    lemma constrains_guarantees_leadsTo:
         "F ∈ transient A ==> F ∈ (A co A ∪ B) guarantees (A leadsTo (B-A))"
    apply (rule guaranteesI)
    apply (rule leadsTo_Basis')
     apply (drule constrains_weaken_R)
      prefer 2 apply assumption
     apply blast
    apply (blast intro: Join_transient_I1)
    done

    end

lemma OK_insert_iff:

      OK (insert i I) F = (if i ∈ I then OK I F else OK I F ∧ F i ok JOIN I F)

### Existential Properties

lemma ex1:

      [| ex_prop X; finite GG; GG ∩ X ≠ {}; OK GG (%G. G) |] ==> (JN G:GG. G) ∈ X

lemma ex2:

      ∀GG. finite GG ∧ GG ∩ X ≠ {} --> OK GG (%G. G) --> (JN G:GG. G) ∈ X
      ==> ex_prop X

lemma ex_prop_finite:

      ex_prop X = (∀GG. finite GG ∧ GG ∩ X ≠ {} ∧ OK GG (%G. G) --> (JN G:GG. G) ∈ X)

lemma ex_prop_equiv:

      ex_prop X = (∀G. (G ∈ X) = (∀H. G component_of H --> H ∈ X))

### Universal Properties

lemma uv1:

      [| uv_prop X; finite GG; GG ⊆ X ∧ OK GG (%G. G) |] ==> (JN G:GG. G) ∈ X

lemma uv2:

      ∀GG. finite GG ∧ GG ⊆ X ∧ OK GG (%G. G) --> (JN G:GG. G) ∈ X ==> uv_prop X

lemma uv_prop_finite:

      uv_prop X = (∀GG. finite GG ∧ GG ⊆ X ∧ OK GG (%G. G) --> (JN G:GG. G) ∈ X)

### Guarantees

lemma guaranteesI:

      (!!G. [| F ok G; F Join G ∈ X |] ==> F Join G ∈ Y) ==> F ∈ X guarantees Y

lemma guaranteesD:

      [| F ∈ X guarantees Y; F ok G; F Join G ∈ X |] ==> F Join G ∈ Y

lemma component_guaranteesD:

      [| F ∈ X guarantees Y; F Join G = H; H ∈ X; F ok G |] ==> H ∈ Y

lemma guarantees_weaken:

      [| F ∈ X guarantees X'; Y ⊆ X; X' ⊆ Y' |] ==> F ∈ Y guarantees Y'

lemma subset_imp_guarantees_UNIV:

      X ⊆ Y ==> X guarantees Y = UNIV

lemma subset_imp_guarantees:

      X ⊆ Y ==> F ∈ X guarantees Y

lemma ex_prop_imp:

      ex_prop Y ==> Y = UNIV guarantees Y

lemma guarantees_imp:

      Y = UNIV guarantees Y ==> ex_prop Y

lemma ex_prop_equiv2:

      ex_prop Y = (Y = UNIV guarantees Y)

### Distributive Laws. Re-Orient to Perform Miniscoping

lemma guarantees_UN_left:

      (UN i:I. X i) guarantees Y = (INT i:I. X i guarantees Y)

lemma guarantees_Un_left:

      X ∪ Y guarantees Z = (X guarantees Z) ∩ (Y guarantees Z)

lemma guarantees_INT_right:

      X guarantees (INT i:I. Y i) = (INT i:I. X guarantees Y i)

lemma guarantees_Int_right:

      Z guarantees X ∩ Y = (Z guarantees X) ∩ (Z guarantees Y)

lemma guarantees_Int_right_I:

      [| F ∈ Z guarantees X; F ∈ Z guarantees Y |] ==> F ∈ Z guarantees X ∩ Y

lemma guarantees_INT_right_iff:

      (F ∈ X guarantees INTER I Y) = (∀i∈I. F ∈ X guarantees Y i)

lemma shunting:

      X guarantees Y = UNIV guarantees - X ∪ Y

lemma contrapositive:

      X guarantees Y = - Y guarantees - X

lemma combining1:

      [| F ∈ V guarantees X; F ∈ X ∩ Y guarantees Z |] ==> F ∈ V ∩ Y guarantees Z

lemma combining2:

      [| F ∈ V guarantees X ∪ Y; F ∈ Y guarantees Z |] ==> F ∈ V guarantees X ∪ Z

lemma all_guarantees:

      ∀i∈I. F ∈ X guarantees Y i ==> F ∈ X guarantees (INT i:I. Y i)

lemma ex_guarantees:

      ∃i∈I. F ∈ X guarantees Y i ==> F ∈ X guarantees (UN i:I. Y i)

### Guarantees: Additional Laws (by lcp)

lemma guarantees_Join_Int:

      [| F ∈ U guarantees V; G ∈ X guarantees Y; F ok G |]
      ==> F Join G ∈ U ∩ X guarantees V ∩ Y

lemma guarantees_Join_Un:

      [| F ∈ U guarantees V; G ∈ X guarantees Y; F ok G |]
      ==> F Join G ∈ U ∪ X guarantees V ∪ Y

lemma guarantees_JN_INT:

      [| ∀i∈I. F i ∈ X i guarantees Y i; OK I F |]
      ==> JOIN I F ∈ INTER I X guarantees INTER I Y

lemma guarantees_JN_UN:

      [| ∀i∈I. F i ∈ X i guarantees Y i; OK I F |]
      ==> JOIN I F ∈ UNION I X guarantees UNION I Y

### Guarantees Laws for Breaking Down the Program (by lcp)

lemma guarantees_Join_I1:

      [| F ∈ X guarantees Y; F ok G |] ==> F Join G ∈ X guarantees Y

lemma guarantees_Join_I2:

      [| G ∈ X guarantees Y; F ok G |] ==> F Join G ∈ X guarantees Y

lemma guarantees_JN_I:

      [| i ∈ I; F i ∈ X guarantees Y; OK I F |] ==> JOIN I F ∈ X guarantees Y

lemma Join_welldef_D1:

      F Join G ∈ welldef ==> F ∈ welldef

lemma Join_welldef_D2:

      F Join G ∈ welldef ==> G ∈ welldef

lemma refines_refl:

      F refines F wrt X

lemma strict_ex_refine_lemma:

      strict_ex_prop X
      ==> (∀H. F ok H ∧ G ok H ∧ F Join H ∈ X --> G Join H ∈ X) = (F ∈ X --> G ∈ X)

lemma strict_ex_refine_lemma_v:

      strict_ex_prop X
      ==> (∀H. F ok H ∧ G ok H ∧ F Join H ∈ welldef ∧ F Join H ∈ X --> G Join H ∈ X) =
          (F ∈ welldef ∩ X --> G ∈ X)

lemma ex_refinement_thm:

      [| strict_ex_prop X;
         ∀H. F ok H ∧ G ok H ∧ F Join H ∈ welldef ∩ X --> G Join H ∈ welldef |]
      ==> (G refines F wrt X) = (G iso_refines F wrt X)

lemma strict_uv_refine_lemma:

      strict_uv_prop X
      ==> (∀H. F ok H ∧ G ok H ∧ F Join H ∈ X --> G Join H ∈ X) = (F ∈ X --> G ∈ X)

lemma strict_uv_refine_lemma_v:

      strict_uv_prop X
      ==> (∀H. F ok H ∧ G ok H ∧ F Join H ∈ welldef ∧ F Join H ∈ X --> G Join H ∈ X) =
          (F ∈ welldef ∩ X --> G ∈ X)

lemma uv_refinement_thm:

      [| strict_uv_prop X;
         ∀H. F ok H ∧ G ok H ∧ F Join H ∈ welldef ∩ X --> G Join H ∈ welldef |]
      ==> (G refines F wrt X) = (G iso_refines F wrt X)

lemma guarantees_equiv:

      (F ∈ X guarantees Y) = (∀H. H ∈ X --> F component_of H --> H ∈ Y)

lemma wg_weakest:

      F ∈ X guarantees Y ==> X ⊆ wg F Y

lemma wg_guarantees:

      F ∈ wg F Y guarantees Y

lemma wg_equiv:

      (H ∈ wg F X) = (F component_of H --> H ∈ X)

lemma component_of_wg:

      F component_of H ==> (H ∈ wg F X) = (H ∈ X)

lemma wg_finite:

      ∀FF. finite FF ∧ FF ∩ X ≠ {} -->
           OK FF (%F. F) --> (∀F∈FF. ((JN F:FF. F) ∈ wg F X) = ((JN F:FF. F) ∈ X))

lemma wg_ex_prop:

      ex_prop X ==> (F ∈ X) = (∀H. H ∈ wg F X)

lemma wx_subset:

      wx X ⊆ X

lemma wx_ex_prop:

      ex_prop (wx X)

lemma wx_weakest:

      ∀Z⊆X. ex_prop Z --> Z ⊆ wx X

lemma wx'_ex_prop:

      ex_prop {F. ∀G. F ok G --> F Join G ∈ X}

lemma wx_equiv:

      wx X = {F. ∀G. F ok G --> F Join G ∈ X}

lemma guarantees_wx_eq:

      X guarantees Y = wx (- X ∪ Y)

lemma stable_guarantees_Always:

      Init F ⊆ A ==> F ∈ stable A guarantees Always A

lemma constrains_guarantees_leadsTo:

      F ∈ transient A ==> F ∈ A co A ∪ B guarantees A leadsTo B - A