(*  Title:      HOL/UNITY/SubstAx.thy

    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

    Copyright   1998  University of Cambridge

Weak LeadsTo relation (restricted to the set of reachable states)

*)

header{*Weak Progress*}

theory SubstAx imports WFair Constrains begin

definition Ensures :: "['a set, 'a set] => 'a program set" (infixl "Ensures" 60) where

    "A Ensures B == {F. F \<in> (reachable F \<inter> A) ensures B}"

definition LeadsTo :: "['a set, 'a set] => 'a program set" (infixl "LeadsTo" 60) where

    "A LeadsTo B == {F. F \<in> (reachable F \<inter> A) leadsTo B}"

notation (xsymbols)

  LeadsTo  (infixl " \<longmapsto>w " 60)

text{*Resembles the previous definition of LeadsTo*}

lemma LeadsTo_eq_leadsTo: 

     "A LeadsTo B = {F. F \<in> (reachable F \<inter> A) leadsTo (reachable F \<inter> B)}"

apply (unfold LeadsTo_def)

apply (blast dest: psp_stable2 intro: leadsTo_weaken)

done

subsection{*Specialized laws for handling invariants*}

(** Conjoining an Always property **)

lemma Always_LeadsTo_pre:

     "F \<in> Always INV ==> (F \<in> (INV \<inter> A) LeadsTo A') = (F \<in> A LeadsTo A')"

by (simp add: LeadsTo_def Always_eq_includes_reachable Int_absorb2 

              Int_assoc [symmetric])

lemma Always_LeadsTo_post:

     "F \<in> Always INV ==> (F \<in> A LeadsTo (INV \<inter> A')) = (F \<in> A LeadsTo A')"

by (simp add: LeadsTo_eq_leadsTo Always_eq_includes_reachable Int_absorb2 

              Int_assoc [symmetric])

(* [| F \<in> Always C;  F \<in> (C \<inter> A) LeadsTo A' |] ==> F \<in> A LeadsTo A' *)

lemmas Always_LeadsToI = Always_LeadsTo_pre [THEN iffD1]

(* [| F \<in> Always INV;  F \<in> A LeadsTo A' |] ==> F \<in> A LeadsTo (INV \<inter> A') *)

lemmas Always_LeadsToD = Always_LeadsTo_post [THEN iffD2]

subsection{*Introduction rules: Basis, Trans, Union*}

lemma leadsTo_imp_LeadsTo: "F \<in> A leadsTo B ==> F \<in> A LeadsTo B"

apply (simp add: LeadsTo_def)

apply (blast intro: leadsTo_weaken_L)

done

lemma LeadsTo_Trans:

     "[| F \<in> A LeadsTo B;  F \<in> B LeadsTo C |] ==> F \<in> A LeadsTo C"

apply (simp add: LeadsTo_eq_leadsTo)

apply (blast intro: leadsTo_Trans)

done

lemma LeadsTo_Union: 

     "(!!A. A \<in> S ==> F \<in> A LeadsTo B) ==> F \<in> (Union S) LeadsTo B"

apply (simp add: LeadsTo_def)

apply (subst Int_Union)

apply (blast intro: leadsTo_UN)

done

subsection{*Derived rules*}

lemma LeadsTo_UNIV [simp]: "F \<in> A LeadsTo UNIV"

by (simp add: LeadsTo_def)

text{*Useful with cancellation, disjunction*}

lemma LeadsTo_Un_duplicate:

     "F \<in> A LeadsTo (A' \<union> A') ==> F \<in> A LeadsTo A'"

by (simp add: Un_ac)

lemma LeadsTo_Un_duplicate2:

     "F \<in> A LeadsTo (A' \<union> C \<union> C) ==> F \<in> A LeadsTo (A' \<union> C)"

by (simp add: Un_ac)

lemma LeadsTo_UN: 

     "(!!i. i \<in> I ==> F \<in> (A i) LeadsTo B) ==> F \<in> (\<Union>i \<in> I. A i) LeadsTo B"

apply (unfold SUP_def)

apply (blast intro: LeadsTo_Union)

done

text{*Binary union introduction rule*}

lemma LeadsTo_Un:

     "[| F \<in> A LeadsTo C; F \<in> B LeadsTo C |] ==> F \<in> (A \<union> B) LeadsTo C"

  using LeadsTo_UN [of "{A, B}" F id C] by auto

text{*Lets us look at the starting state*}

lemma single_LeadsTo_I:

     "(!!s. s \<in> A ==> F \<in> {s} LeadsTo B) ==> F \<in> A LeadsTo B"

by (subst UN_singleton [symmetric], rule LeadsTo_UN, blast)

lemma subset_imp_LeadsTo: "A \<subseteq> B ==> F \<in> A LeadsTo B"

apply (simp add: LeadsTo_def)

apply (blast intro: subset_imp_leadsTo)

done

lemmas empty_LeadsTo = empty_subsetI [THEN subset_imp_LeadsTo, simp]

lemma LeadsTo_weaken_R:

     "[| F \<in> A LeadsTo A';  A' \<subseteq> B' |] ==> F \<in> A LeadsTo B'"

apply (simp add: LeadsTo_def)

apply (blast intro: leadsTo_weaken_R)

done

lemma LeadsTo_weaken_L:

     "[| F \<in> A LeadsTo A';  B \<subseteq> A |]   

      ==> F \<in> B LeadsTo A'"

apply (simp add: LeadsTo_def)

apply (blast intro: leadsTo_weaken_L)

done

lemma LeadsTo_weaken:

     "[| F \<in> A LeadsTo A';    

         B  \<subseteq> A;   A' \<subseteq> B' |]  

      ==> F \<in> B LeadsTo B'"

by (blast intro: LeadsTo_weaken_R LeadsTo_weaken_L LeadsTo_Trans)

lemma Always_LeadsTo_weaken:

     "[| F \<in> Always C;  F \<in> A LeadsTo A';    

         C \<inter> B \<subseteq> A;   C \<inter> A' \<subseteq> B' |]  

      ==> F \<in> B LeadsTo B'"

by (blast dest: Always_LeadsToI intro: LeadsTo_weaken intro: Always_LeadsToD)

(** Two theorems for "proof lattices" **)

lemma LeadsTo_Un_post: "F \<in> A LeadsTo B ==> F \<in> (A \<union> B) LeadsTo B"

by (blast intro: LeadsTo_Un subset_imp_LeadsTo)

lemma LeadsTo_Trans_Un:

     "[| F \<in> A LeadsTo B;  F \<in> B LeadsTo C |]  

      ==> F \<in> (A \<union> B) LeadsTo C"

by (blast intro: LeadsTo_Un subset_imp_LeadsTo LeadsTo_weaken_L LeadsTo_Trans)

(** Distributive laws **)

lemma LeadsTo_Un_distrib:

     "(F \<in> (A \<union> B) LeadsTo C)  = (F \<in> A LeadsTo C & F \<in> B LeadsTo C)"

by (blast intro: LeadsTo_Un LeadsTo_weaken_L)

lemma LeadsTo_UN_distrib:

     "(F \<in> (\<Union>i \<in> I. A i) LeadsTo B)  =  (\<forall>i \<in> I. F \<in> (A i) LeadsTo B)"

by (blast intro: LeadsTo_UN LeadsTo_weaken_L)

lemma LeadsTo_Union_distrib:

     "(F \<in> (Union S) LeadsTo B)  =  (\<forall>A \<in> S. F \<in> A LeadsTo B)"

by (blast intro: LeadsTo_Union LeadsTo_weaken_L)

(** More rules using the premise "Always INV" **)

lemma LeadsTo_Basis: "F \<in> A Ensures B ==> F \<in> A LeadsTo B"

by (simp add: Ensures_def LeadsTo_def leadsTo_Basis)

lemma EnsuresI:

     "[| F \<in> (A-B) Co (A \<union> B);  F \<in> transient (A-B) |]    

      ==> F \<in> A Ensures B"

apply (simp add: Ensures_def Constrains_eq_constrains)

apply (blast intro: ensuresI constrains_weaken transient_strengthen)

done

lemma Always_LeadsTo_Basis:

     "[| F \<in> Always INV;       

         F \<in> (INV \<inter> (A-A')) Co (A \<union> A');  

         F \<in> transient (INV \<inter> (A-A')) |]    

  ==> F \<in> A LeadsTo A'"

apply (rule Always_LeadsToI, assumption)

apply (blast intro: EnsuresI LeadsTo_Basis Always_ConstrainsD [THEN Constrains_weaken] transient_strengthen)

done

text{*Set difference: maybe combine with @{text leadsTo_weaken_L}??

  This is the most useful form of the "disjunction" rule*}

lemma LeadsTo_Diff:

     "[| F \<in> (A-B) LeadsTo C;  F \<in> (A \<inter> B) LeadsTo C |]  

      ==> F \<in> A LeadsTo C"

by (blast intro: LeadsTo_Un LeadsTo_weaken)

lemma LeadsTo_UN_UN: 

     "(!! i. i \<in> I ==> F \<in> (A i) LeadsTo (A' i))  

      ==> F \<in> (\<Union>i \<in> I. A i) LeadsTo (\<Union>i \<in> I. A' i)"

apply (simp only: Union_image_eq [symmetric])

apply (blast intro: LeadsTo_Union LeadsTo_weaken_R)

done

text{*Version with no index set*}

lemma LeadsTo_UN_UN_noindex: 

     "(!!i. F \<in> (A i) LeadsTo (A' i)) ==> F \<in> (\<Union>i. A i) LeadsTo (\<Union>i. A' i)"

by (blast intro: LeadsTo_UN_UN)

text{*Version with no index set*}

lemma all_LeadsTo_UN_UN:

     "\<forall>i. F \<in> (A i) LeadsTo (A' i)  

      ==> F \<in> (\<Union>i. A i) LeadsTo (\<Union>i. A' i)"

by (blast intro: LeadsTo_UN_UN)

text{*Binary union version*}

lemma LeadsTo_Un_Un:

     "[| F \<in> A LeadsTo A'; F \<in> B LeadsTo B' |]  

            ==> F \<in> (A \<union> B) LeadsTo (A' \<union> B')"

by (blast intro: LeadsTo_Un LeadsTo_weaken_R)

(** The cancellation law **)

lemma LeadsTo_cancel2:

     "[| F \<in> A LeadsTo (A' \<union> B); F \<in> B LeadsTo B' |]     

      ==> F \<in> A LeadsTo (A' \<union> B')"

by (blast intro: LeadsTo_Un_Un subset_imp_LeadsTo LeadsTo_Trans)

lemma LeadsTo_cancel_Diff2:

     "[| F \<in> A LeadsTo (A' \<union> B); F \<in> (B-A') LeadsTo B' |]  

      ==> F \<in> A LeadsTo (A' \<union> B')"

apply (rule LeadsTo_cancel2)

prefer 2 apply assumption

apply (simp_all (no_asm_simp))

done

lemma LeadsTo_cancel1:

     "[| F \<in> A LeadsTo (B \<union> A'); F \<in> B LeadsTo B' |]  

      ==> F \<in> A LeadsTo (B' \<union> A')"

apply (simp add: Un_commute)

apply (blast intro!: LeadsTo_cancel2)

done

lemma LeadsTo_cancel_Diff1:

     "[| F \<in> A LeadsTo (B \<union> A'); F \<in> (B-A') LeadsTo B' |]  

      ==> F \<in> A LeadsTo (B' \<union> A')"

apply (rule LeadsTo_cancel1)

prefer 2 apply assumption

apply (simp_all (no_asm_simp))

done

text{*The impossibility law*}

text{*The set "A" may be non-empty, but it contains no reachable states*}

lemma LeadsTo_empty: "[|F \<in> A LeadsTo {}; all_total F|] ==> F \<in> Always (-A)"

apply (simp add: LeadsTo_def Always_eq_includes_reachable)

apply (drule leadsTo_empty, auto)

done

subsection{*PSP: Progress-Safety-Progress*}

text{*Special case of PSP: Misra's "stable conjunction"*}

lemma PSP_Stable:

     "[| F \<in> A LeadsTo A';  F \<in> Stable B |]  

      ==> F \<in> (A \<inter> B) LeadsTo (A' \<inter> B)"

apply (simp add: LeadsTo_eq_leadsTo Stable_eq_stable)

apply (drule psp_stable, assumption)

apply (simp add: Int_ac)

done

lemma PSP_Stable2:

     "[| F \<in> A LeadsTo A'; F \<in> Stable B |]  

      ==> F \<in> (B \<inter> A) LeadsTo (B \<inter> A')"

by (simp add: PSP_Stable Int_ac)

lemma PSP:

     "[| F \<in> A LeadsTo A'; F \<in> B Co B' |]  

      ==> F \<in> (A \<inter> B') LeadsTo ((A' \<inter> B) \<union> (B' - B))"

apply (simp add: LeadsTo_def Constrains_eq_constrains)

apply (blast dest: psp intro: leadsTo_weaken)

done

lemma PSP2:

     "[| F \<in> A LeadsTo A'; F \<in> B Co B' |]  

      ==> F \<in> (B' \<inter> A) LeadsTo ((B \<inter> A') \<union> (B' - B))"

by (simp add: PSP Int_ac)

lemma PSP_Unless: 

     "[| F \<in> A LeadsTo A'; F \<in> B Unless B' |]  

      ==> F \<in> (A \<inter> B) LeadsTo ((A' \<inter> B) \<union> B')"

apply (unfold Unless_def)

apply (drule PSP, assumption)

apply (blast intro: LeadsTo_Diff LeadsTo_weaken subset_imp_LeadsTo)

done

lemma Stable_transient_Always_LeadsTo:

     "[| F \<in> Stable A;  F \<in> transient C;   

         F \<in> Always (-A \<union> B \<union> C) |] ==> F \<in> A LeadsTo B"

apply (erule Always_LeadsTo_weaken)

apply (rule LeadsTo_Diff)

   prefer 2

   apply (erule

          transient_imp_leadsTo [THEN leadsTo_imp_LeadsTo, THEN PSP_Stable2])

   apply (blast intro: subset_imp_LeadsTo)+

done

subsection{*Induction rules*}

(** Meta or object quantifier ????? **)

lemma LeadsTo_wf_induct:

     "[| wf r;      

         \<forall>m. F \<in> (A \<inter> f-`{m}) LeadsTo                      

                    ((A \<inter> f-`(r^-1 `` {m})) \<union> B) |]  

      ==> F \<in> A LeadsTo B"

apply (simp add: LeadsTo_eq_leadsTo)

apply (erule leadsTo_wf_induct)

apply (blast intro: leadsTo_weaken)

done

lemma Bounded_induct:

     "[| wf r;      

         \<forall>m \<in> I. F \<in> (A \<inter> f-`{m}) LeadsTo                    

                      ((A \<inter> f-`(r^-1 `` {m})) \<union> B) |]  

      ==> F \<in> A LeadsTo ((A - (f-`I)) \<union> B)"

apply (erule LeadsTo_wf_induct, safe)

apply (case_tac "m \<in> I")

apply (blast intro: LeadsTo_weaken)

apply (blast intro: subset_imp_LeadsTo)

done

lemma LessThan_induct:

     "(!!m::nat. F \<in> (A \<inter> f-`{m}) LeadsTo ((A \<inter> f-`(lessThan m)) \<union> B))

      ==> F \<in> A LeadsTo B"

by (rule wf_less_than [THEN LeadsTo_wf_induct], auto)

text{*Integer version.  Could generalize from 0 to any lower bound*}

lemma integ_0_le_induct:

     "[| F \<in> Always {s. (0::int) \<le> f s};   

         !! z. F \<in> (A \<inter> {s. f s = z}) LeadsTo                      

                   ((A \<inter> {s. f s < z}) \<union> B) |]  

      ==> F \<in> A LeadsTo B"

apply (rule_tac f = "nat o f" in LessThan_induct)

apply (simp add: vimage_def)

apply (rule Always_LeadsTo_weaken, assumption+)

apply (auto simp add: nat_eq_iff nat_less_iff)

done

lemma LessThan_bounded_induct:

     "!!l::nat. \<forall>m \<in> greaterThan l. 

                   F \<in> (A \<inter> f-`{m}) LeadsTo ((A \<inter> f-`(lessThan m)) \<union> B)

            ==> F \<in> A LeadsTo ((A \<inter> (f-`(atMost l))) \<union> B)"

apply (simp only: Diff_eq [symmetric] vimage_Compl 

                  Compl_greaterThan [symmetric])

apply (rule wf_less_than [THEN Bounded_induct], simp)

done

lemma GreaterThan_bounded_induct:

     "!!l::nat. \<forall>m \<in> lessThan l. 

                 F \<in> (A \<inter> f-`{m}) LeadsTo ((A \<inter> f-`(greaterThan m)) \<union> B)

      ==> F \<in> A LeadsTo ((A \<inter> (f-`(atLeast l))) \<union> B)"

apply (rule_tac f = f and f1 = "%k. l - k" 

       in wf_less_than [THEN wf_inv_image, THEN LeadsTo_wf_induct])

apply (simp add: Image_singleton, clarify)

apply (case_tac "m<l")

 apply (blast intro: LeadsTo_weaken_R diff_less_mono2)

apply (blast intro: not_leE subset_imp_LeadsTo)

done

subsection{*Completion: Binary and General Finite versions*}

lemma Completion:

     "[| F \<in> A LeadsTo (A' \<union> C);  F \<in> A' Co (A' \<union> C);  

         F \<in> B LeadsTo (B' \<union> C);  F \<in> B' Co (B' \<union> C) |]  

      ==> F \<in> (A \<inter> B) LeadsTo ((A' \<inter> B') \<union> C)"

apply (simp add: LeadsTo_eq_leadsTo Constrains_eq_constrains Int_Un_distrib)

apply (blast intro: completion leadsTo_weaken)

done

lemma Finite_completion_lemma:

     "finite I  

      ==> (\<forall>i \<in> I. F \<in> (A i) LeadsTo (A' i \<union> C)) -->   

          (\<forall>i \<in> I. F \<in> (A' i) Co (A' i \<union> C)) -->  

          F \<in> (\<Inter>i \<in> I. A i) LeadsTo ((\<Inter>i \<in> I. A' i) \<union> C)"

apply (erule finite_induct, auto)

apply (rule Completion)

   prefer 4

   apply (simp only: INT_simps [symmetric])

   apply (rule Constrains_INT, auto)

done

lemma Finite_completion: 

     "[| finite I;   

         !!i. i \<in> I ==> F \<in> (A i) LeadsTo (A' i \<union> C);  

         !!i. i \<in> I ==> F \<in> (A' i) Co (A' i \<union> C) |]    

      ==> F \<in> (\<Inter>i \<in> I. A i) LeadsTo ((\<Inter>i \<in> I. A' i) \<union> C)"

by (blast intro: Finite_completion_lemma [THEN mp, THEN mp])

lemma Stable_completion: 

     "[| F \<in> A LeadsTo A';  F \<in> Stable A';    

         F \<in> B LeadsTo B';  F \<in> Stable B' |]  

      ==> F \<in> (A \<inter> B) LeadsTo (A' \<inter> B')"

apply (unfold Stable_def)

apply (rule_tac C1 = "{}" in Completion [THEN LeadsTo_weaken_R])

apply (force+)

done

lemma Finite_stable_completion: 

     "[| finite I;   

         !!i. i \<in> I ==> F \<in> (A i) LeadsTo (A' i);  

         !!i. i \<in> I ==> F \<in> Stable (A' i) |]    

      ==> F \<in> (\<Inter>i \<in> I. A i) LeadsTo (\<Inter>i \<in> I. A' i)"

apply (unfold Stable_def)

apply (rule_tac C1 = "{}" in Finite_completion [THEN LeadsTo_weaken_R])

apply (simp_all, blast+)

done

end