(*  Title:      HOL/UNITY/Lift_prog.thy

     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

     Copyright   1999  University of Cambridge

 lift_prog, etc: replication of components and arrays of processes. 

 *)

 header{*Replication of Components*}

 theory Lift_prog imports Rename begin

 definition insert_map :: "[nat, 'b, nat=>'b] => (nat=>'b)" where

     "insert_map i z f k == if k<i then f k

                            else if k=i then z

                            else f(k - 1)"

 definition delete_map :: "[nat, nat=>'b] => (nat=>'b)" where

     "delete_map i g k == if k<i then g k else g (Suc k)"

 definition lift_map :: "[nat, 'b * ((nat=>'b) * 'c)] => (nat=>'b) * 'c" where

     "lift_map i == %(s,(f,uu)). (insert_map i s f, uu)"

 definition drop_map :: "[nat, (nat=>'b) * 'c] => 'b * ((nat=>'b) * 'c)" where

     "drop_map i == %(g, uu). (g i, (delete_map i g, uu))"

 definition lift_set :: "[nat, ('b * ((nat=>'b) * 'c)) set] => ((nat=>'b) * 'c) set" where

     "lift_set i A == lift_map i ` A"

 definition lift :: "[nat, ('b * ((nat=>'b) * 'c)) program] => ((nat=>'b) * 'c) program" where

     "lift i == rename (lift_map i)"

   (*simplifies the expression of specifications*)

 definition sub :: "['a, 'a=>'b] => 'b" where

     "sub == %i f. f i"

 declare insert_map_def [simp] delete_map_def [simp]

 lemma insert_map_inverse: "delete_map i (insert_map i x f) = f"

 by (rule ext, simp)

 lemma insert_map_delete_map_eq: "(insert_map i x (delete_map i g)) = g(i:=x)"

 apply (rule ext)

 apply (auto split add: nat_diff_split)

 done

 subsection{*Injectiveness proof*}

 lemma insert_map_inject1: "(insert_map i x f) = (insert_map i y g) ==> x=y"

 by (drule_tac x = i in fun_cong, simp)

 lemma insert_map_inject2: "(insert_map i x f) = (insert_map i y g) ==> f=g"

 apply (drule_tac f = "delete_map i" in arg_cong)

 apply (simp add: insert_map_inverse)

 done

 lemma insert_map_inject':

      "(insert_map i x f) = (insert_map i y g) ==> x=y & f=g"

 by (blast dest: insert_map_inject1 insert_map_inject2)

 lemmas insert_map_inject = insert_map_inject' [THEN conjE, elim!]

 (*The general case: we don't assume i=i'*)

 lemma lift_map_eq_iff [iff]: 

      "(lift_map i (s,(f,uu)) = lift_map i' (s',(f',uu')))  

       = (uu = uu' & insert_map i s f = insert_map i' s' f')"

 by (unfold lift_map_def, auto)

 (*The !!s allows the automatic splitting of the bound variable*)

 lemma drop_map_lift_map_eq [simp]: "!!s. drop_map i (lift_map i s) = s"

 apply (unfold lift_map_def drop_map_def)

 apply (force intro: insert_map_inverse)

 done

 lemma inj_lift_map: "inj (lift_map i)"

 apply (unfold lift_map_def)

 apply (rule inj_onI, auto)

 done

 subsection{*Surjectiveness proof*}

 lemma lift_map_drop_map_eq [simp]: "!!s. lift_map i (drop_map i s) = s"

 apply (unfold lift_map_def drop_map_def)

 apply (force simp add: insert_map_delete_map_eq)

 done

 lemma drop_map_inject [dest!]: "(drop_map i s) = (drop_map i s') ==> s=s'"

 by (drule_tac f = "lift_map i" in arg_cong, simp)

 lemma surj_lift_map: "surj (lift_map i)"

 apply (rule surjI)

 apply (rule lift_map_drop_map_eq)

 done

 lemma bij_lift_map [iff]: "bij (lift_map i)"

 by (simp add: bij_def inj_lift_map surj_lift_map)

 lemma inv_lift_map_eq [simp]: "inv (lift_map i) = drop_map i"

 by (rule inv_equality, auto)

 lemma inv_drop_map_eq [simp]: "inv (drop_map i) = lift_map i"

 by (rule inv_equality, auto)

 lemma bij_drop_map [iff]: "bij (drop_map i)"

 by (simp del: inv_lift_map_eq add: inv_lift_map_eq [symmetric] bij_imp_bij_inv)

 (*sub's main property!*)

 lemma sub_apply [simp]: "sub i f = f i"

 by (simp add: sub_def)

 lemma all_total_lift: "all_total F ==> all_total (lift i F)"

 by (simp add: lift_def rename_def Extend.all_total_extend)

 lemma insert_map_upd_same: "(insert_map i t f)(i := s) = insert_map i s f"

 by (rule ext, auto)

 lemma insert_map_upd:

      "(insert_map j t f)(i := s) =  

       (if i=j then insert_map i s f  

        else if i<j then insert_map j t (f(i:=s))  

        else insert_map j t (f(i - Suc 0 := s)))"

 apply (rule ext) 

 apply (simp split add: nat_diff_split)

  txt{*This simplification is VERY slow*}

 done

 lemma insert_map_eq_diff:

      "[| insert_map i s f = insert_map j t g;  i\<noteq>j |]  

       ==> \<exists>g'. insert_map i s' f = insert_map j t g'"

 apply (subst insert_map_upd_same [symmetric])

 apply (erule ssubst)

 apply (simp only: insert_map_upd if_False split: split_if, blast)

 done

 lemma lift_map_eq_diff: 

      "[| lift_map i (s,(f,uu)) = lift_map j (t,(g,vv));  i\<noteq>j |]  

       ==> \<exists>g'. lift_map i (s',(f,uu)) = lift_map j (t,(g',vv))"

 apply (unfold lift_map_def, auto)

 apply (blast dest: insert_map_eq_diff)

 done

 subsection{*The Operator @{term lift_set}*}

 lemma lift_set_empty [simp]: "lift_set i {} = {}"

 by (unfold lift_set_def, auto)

 lemma lift_set_iff: "(lift_map i x \<in> lift_set i A) = (x \<in> A)"

 apply (unfold lift_set_def)

 apply (rule inj_lift_map [THEN inj_image_mem_iff])

 done

 (*Do we really need both this one and its predecessor?*)

 lemma lift_set_iff2 [iff]:

      "((f,uu) \<in> lift_set i A) = ((f i, (delete_map i f, uu)) \<in> A)"

 by (simp add: lift_set_def mem_rename_set_iff drop_map_def)

 lemma lift_set_mono: "A \<subseteq> B ==> lift_set i A \<subseteq> lift_set i B"

 apply (unfold lift_set_def)

 apply (erule image_mono)

 done

 lemma lift_set_Un_distrib: "lift_set i (A \<union> B) = lift_set i A \<union> lift_set i B"

 by (simp add: lift_set_def image_Un)

 lemma lift_set_Diff_distrib: "lift_set i (A-B) = lift_set i A - lift_set i B"

 apply (unfold lift_set_def)

 apply (rule inj_lift_map [THEN image_set_diff])

 done

 subsection{*The Lattice Operations*}

 lemma bij_lift [iff]: "bij (lift i)"

 by (simp add: lift_def)

 lemma lift_SKIP [simp]: "lift i SKIP = SKIP"

 by (simp add: lift_def)

 lemma lift_Join [simp]: "lift i (F Join G) = lift i F Join lift i G"

 by (simp add: lift_def)

 lemma lift_JN [simp]: "lift j (JOIN I F) = (\<Squnion>i \<in> I. lift j (F i))"

 by (simp add: lift_def)

 subsection{*Safety: constrains, stable, invariant*}

 lemma lift_constrains: 

      "(lift i F \<in> (lift_set i A) co (lift_set i B)) = (F \<in> A co B)"

 by (simp add: lift_def lift_set_def rename_constrains)

 lemma lift_stable: 

      "(lift i F \<in> stable (lift_set i A)) = (F \<in> stable A)"

 by (simp add: lift_def lift_set_def rename_stable)

 lemma lift_invariant: 

      "(lift i F \<in> invariant (lift_set i A)) = (F \<in> invariant A)"

 by (simp add: lift_def lift_set_def rename_invariant)

 lemma lift_Constrains: 

      "(lift i F \<in> (lift_set i A) Co (lift_set i B)) = (F \<in> A Co B)"

 by (simp add: lift_def lift_set_def rename_Constrains)

 lemma lift_Stable: 

      "(lift i F \<in> Stable (lift_set i A)) = (F \<in> Stable A)"

 by (simp add: lift_def lift_set_def rename_Stable)

 lemma lift_Always: 

      "(lift i F \<in> Always (lift_set i A)) = (F \<in> Always A)"

 by (simp add: lift_def lift_set_def rename_Always)

 subsection{*Progress: transient, ensures*}

 lemma lift_transient: 

      "(lift i F \<in> transient (lift_set i A)) = (F \<in> transient A)"

 by (simp add: lift_def lift_set_def rename_transient)

 lemma lift_ensures: 

      "(lift i F \<in> (lift_set i A) ensures (lift_set i B)) =  

       (F \<in> A ensures B)"

 by (simp add: lift_def lift_set_def rename_ensures)

 lemma lift_leadsTo: 

      "(lift i F \<in> (lift_set i A) leadsTo (lift_set i B)) =  

       (F \<in> A leadsTo B)"

 by (simp add: lift_def lift_set_def rename_leadsTo)

 lemma lift_LeadsTo: 

      "(lift i F \<in> (lift_set i A) LeadsTo (lift_set i B)) =   

       (F \<in> A LeadsTo B)"

 by (simp add: lift_def lift_set_def rename_LeadsTo)

 (** guarantees **)

 lemma lift_lift_guarantees_eq: 

      "(lift i F \<in> (lift i ` X) guarantees (lift i ` Y)) =  

       (F \<in> X guarantees Y)"

 apply (unfold lift_def)

 apply (subst bij_lift_map [THEN rename_rename_guarantees_eq, symmetric])

 apply (simp add: o_def)

 done

 lemma lift_guarantees_eq_lift_inv:

      "(lift i F \<in> X guarantees Y) =  

       (F \<in> (rename (drop_map i) ` X) guarantees (rename (drop_map i) ` Y))"

 by (simp add: bij_lift_map [THEN rename_guarantees_eq_rename_inv] lift_def)

 (*To preserve snd means that the second component is there just to allow

   guarantees properties to be stated.  Converse fails, for lift i F can 

   change function components other than i*)

 lemma lift_preserves_snd_I: "F \<in> preserves snd ==> lift i F \<in> preserves snd"

 apply (drule_tac w1=snd in subset_preserves_o [THEN subsetD])

 apply (simp add: lift_def rename_preserves)

 apply (simp add: lift_map_def o_def split_def)

 done

 lemma delete_map_eqE':

      "(delete_map i g) = (delete_map i g') ==> \<exists>x. g = g'(i:=x)"

 apply (drule_tac f = "insert_map i (g i) " in arg_cong)

 apply (simp add: insert_map_delete_map_eq)

 apply (erule exI)

 done

 lemmas delete_map_eqE = delete_map_eqE' [THEN exE, elim!]

 lemma delete_map_neq_apply:

      "[| delete_map j g = delete_map j g';  i\<noteq>j |] ==> g i = g' i"

 by force

 (*A set of the form (A <*> UNIV) ignores the second (dummy) state component*)

 lemma vimage_o_fst_eq [simp]: "(f o fst) -` A = (f-`A) <*> UNIV"

 by auto

 lemma vimage_sub_eq_lift_set [simp]:

      "(sub i -`A) <*> UNIV = lift_set i (A <*> UNIV)"

 by auto

 lemma mem_lift_act_iff [iff]: 

      "((s,s') \<in> extend_act (%(x,u::unit). lift_map i x) act) =  

       ((drop_map i s, drop_map i s') \<in> act)"

 apply (unfold extend_act_def, auto)

 apply (rule bexI, auto)

 done

 lemma preserves_snd_lift_stable:

      "[| F \<in> preserves snd;  i\<noteq>j |]  

       ==> lift j F \<in> stable (lift_set i (A <*> UNIV))"

 apply (auto simp add: lift_def lift_set_def stable_def constrains_def 

                       rename_def extend_def mem_rename_set_iff)

 apply (auto dest!: preserves_imp_eq simp add: lift_map_def drop_map_def)

 apply (drule_tac x = i in fun_cong, auto)

 done

 (*If i\<noteq>j then lift j F  does nothing to lift_set i, and the 

   premise ensures A \<subseteq> B.*)

 lemma constrains_imp_lift_constrains:

     "[| F i \<in> (A <*> UNIV) co (B <*> UNIV);   

         F j \<in> preserves snd |]   

      ==> lift j (F j) \<in> (lift_set i (A <*> UNIV)) co (lift_set i (B <*> UNIV))"

 apply (case_tac "i=j")

 apply (simp add: lift_def lift_set_def rename_constrains)

 apply (erule preserves_snd_lift_stable[THEN stableD, THEN constrains_weaken_R],

        assumption)

 apply (erule constrains_imp_subset [THEN lift_set_mono])

 done

 (*USELESS??*)

 lemma lift_map_image_Times:

      "lift_map i ` (A <*> UNIV) =  

       (\<Union>s \<in> A. \<Union>f. {insert_map i s f}) <*> UNIV"

 apply (auto intro!: bexI image_eqI simp add: lift_map_def)

 apply (rule split_conv [symmetric])

 done

 lemma lift_preserves_eq:

      "(lift i F \<in> preserves v) = (F \<in> preserves (v o lift_map i))"

 by (simp add: lift_def rename_preserves)

 (*A useful rewrite.  If o, sub have been rewritten out already then can also

   use it as   rewrite_rule [sub_def, o_def] lift_preserves_sub*)

 lemma lift_preserves_sub:

      "F \<in> preserves snd  

       ==> lift i F \<in> preserves (v o sub j o fst) =  

           (if i=j then F \<in> preserves (v o fst) else True)"

 apply (drule subset_preserves_o [THEN subsetD])

 apply (simp add: lift_preserves_eq o_def)

 apply (auto cong del: if_weak_cong 

        simp add: lift_map_def eq_commute split_def o_def)

 done

 subsection{*Lemmas to Handle Function Composition (o) More Consistently*}

 (*Lets us prove one version of a theorem and store others*)

 lemma o_equiv_assoc: "f o g = h ==> f' o f o g = f' o h"

 by (simp add: fun_eq_iff o_def)

 lemma o_equiv_apply: "f o g = h ==> \<forall>x. f(g x) = h x"

 by (simp add: fun_eq_iff o_def)

 lemma fst_o_lift_map: "sub i o fst o lift_map i = fst"

 apply (rule ext)

 apply (auto simp add: o_def lift_map_def sub_def)

 done

 lemma snd_o_lift_map: "snd o lift_map i = snd o snd"

 apply (rule ext)

 apply (auto simp add: o_def lift_map_def)

 done

 subsection{*More lemmas about extend and project*}

 text{*They could be moved to theory Extend or Project*}

 lemma extend_act_extend_act:

      "extend_act h' (extend_act h act) =  

       extend_act (%(x,(y,y')). h'(h(x,y),y')) act"

 apply (auto elim!: rev_bexI simp add: extend_act_def, blast) 

 done

 lemma project_act_project_act:

      "project_act h (project_act h' act) =  

       project_act (%(x,(y,y')). h'(h(x,y),y')) act"

 by (auto elim!: rev_bexI simp add: project_act_def)

 lemma project_act_extend_act:

      "project_act h (extend_act h' act) =  

         {(x,x'). \<exists>s s' y y' z. (s,s') \<in> act &  

                  h(x,y) = h'(s,z) & h(x',y') = h'(s',z)}"

 by (simp add: extend_act_def project_act_def, blast)

 subsection{*OK and "lift"*}

 lemma act_in_UNION_preserves_fst:

      "act \<subseteq> {(x,x'). fst x = fst x'} ==> act \<in> UNION (preserves fst) Acts"

 apply (rule_tac a = "mk_program (UNIV,{act},UNIV) " in UN_I)

 apply (auto simp add: preserves_def stable_def constrains_def)

 done

 lemma UNION_OK_lift_I:

      "[| \<forall>i \<in> I. F i \<in> preserves snd;   

          \<forall>i \<in> I. UNION (preserves fst) Acts \<subseteq> AllowedActs (F i) |]  

       ==> OK I (%i. lift i (F i))"

 apply (auto simp add: OK_def lift_def rename_def Extend.Acts_extend)

 apply (simp add: Extend.AllowedActs_extend project_act_extend_act)

 apply (rename_tac "act")

 apply (subgoal_tac

        "{(x, x'). \<exists>s f u s' f' u'. 

                     ((s, f, u), s', f', u') \<in> act & 

                     lift_map j x = lift_map i (s, f, u) & 

                     lift_map j x' = lift_map i (s', f', u') } 

                 \<subseteq> { (x,x') . fst x = fst x'}")

 apply (blast intro: act_in_UNION_preserves_fst, clarify)

 apply (drule_tac x = j in fun_cong)+

 apply (drule_tac x = i in bspec, assumption)

 apply (frule preserves_imp_eq, auto)

 done

 lemma OK_lift_I:

      "[| \<forall>i \<in> I. F i \<in> preserves snd;   

          \<forall>i \<in> I. preserves fst \<subseteq> Allowed (F i) |]  

       ==> OK I (%i. lift i (F i))"

 by (simp add: safety_prop_AllowedActs_iff_Allowed UNION_OK_lift_I)

 lemma Allowed_lift [simp]: "Allowed (lift i F) = lift i ` (Allowed F)"

 by (simp add: lift_def)

 lemma lift_image_preserves:

      "lift i ` preserves v = preserves (v o drop_map i)"

 by (simp add: rename_image_preserves lift_def)

 end