paulson@7186: (*  Title:      HOL/UNITY/Lift_prog.thy
paulson@7186:     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
paulson@7186:     Copyright   1999  University of Cambridge
paulson@7186: 
paulson@13786: lift_prog, etc: replication of components and arrays of processes. 
paulson@7186: *)
paulson@7186: 
wenzelm@58889: section{*Replication of Components*}
paulson@13798: 
haftmann@16417: theory Lift_prog imports Rename begin
paulson@7186: 
haftmann@35416: definition insert_map :: "[nat, 'b, nat=>'b] => (nat=>'b)" where
paulson@8251:     "insert_map i z f k == if k<i then f k
paulson@8251:                            else if k=i then z
wenzelm@11701:                            else f(k - 1)"
paulson@7186: 
haftmann@35416: definition delete_map :: "[nat, nat=>'b] => (nat=>'b)" where
paulson@8251:     "delete_map i g k == if k<i then g k else g (Suc k)"
paulson@7186: 
haftmann@35416: definition lift_map :: "[nat, 'b * ((nat=>'b) * 'c)] => (nat=>'b) * 'c" where
paulson@8251:     "lift_map i == %(s,(f,uu)). (insert_map i s f, uu)"
paulson@8251: 
haftmann@35416: definition drop_map :: "[nat, (nat=>'b) * 'c] => 'b * ((nat=>'b) * 'c)" where
paulson@8251:     "drop_map i == %(g, uu). (g i, (delete_map i g, uu))"
paulson@7186: 
haftmann@35416: definition lift_set :: "[nat, ('b * ((nat=>'b) * 'c)) set] => ((nat=>'b) * 'c) set" where
nipkow@10834:     "lift_set i A == lift_map i ` A"
paulson@7186: 
haftmann@35416: definition lift :: "[nat, ('b * ((nat=>'b) * 'c)) program] => ((nat=>'b) * 'c) program" where
paulson@8251:     "lift i == rename (lift_map i)"
paulson@7186: 
paulson@7186:   (*simplifies the expression of specifications*)
haftmann@35416: definition sub :: "['a, 'a=>'b] => 'b" where
paulson@13786:     "sub == %i f. f i"
paulson@13786: 
paulson@13786: 
paulson@13786: declare insert_map_def [simp] delete_map_def [simp]
paulson@13786: 
paulson@13786: lemma insert_map_inverse: "delete_map i (insert_map i x f) = f"
paulson@13798: by (rule ext, simp)
paulson@13786: 
paulson@13786: lemma insert_map_delete_map_eq: "(insert_map i x (delete_map i g)) = g(i:=x)"
paulson@13786: apply (rule ext)
paulson@13786: apply (auto split add: nat_diff_split)
paulson@13786: done
paulson@13786: 
paulson@13812: subsection{*Injectiveness proof*}
paulson@13786: 
paulson@13786: lemma insert_map_inject1: "(insert_map i x f) = (insert_map i y g) ==> x=y"
paulson@13798: by (drule_tac x = i in fun_cong, simp)
paulson@13786: 
paulson@13786: lemma insert_map_inject2: "(insert_map i x f) = (insert_map i y g) ==> f=g"
paulson@13786: apply (drule_tac f = "delete_map i" in arg_cong)
paulson@13798: apply (simp add: insert_map_inverse)
paulson@13786: done
paulson@13786: 
paulson@13786: lemma insert_map_inject':
paulson@13786:      "(insert_map i x f) = (insert_map i y g) ==> x=y & f=g"
paulson@13786: by (blast dest: insert_map_inject1 insert_map_inject2)
paulson@13786: 
paulson@13786: lemmas insert_map_inject = insert_map_inject' [THEN conjE, elim!]
paulson@13786: 
paulson@13786: (*The general case: we don't assume i=i'*)
paulson@13786: lemma lift_map_eq_iff [iff]: 
paulson@13786:      "(lift_map i (s,(f,uu)) = lift_map i' (s',(f',uu')))  
paulson@13786:       = (uu = uu' & insert_map i s f = insert_map i' s' f')"
paulson@13798: by (unfold lift_map_def, auto)
paulson@13786: 
paulson@13786: (*The !!s allows the automatic splitting of the bound variable*)
paulson@13786: lemma drop_map_lift_map_eq [simp]: "!!s. drop_map i (lift_map i s) = s"
paulson@13786: apply (unfold lift_map_def drop_map_def)
paulson@13786: apply (force intro: insert_map_inverse)
paulson@13786: done
paulson@13786: 
paulson@13786: lemma inj_lift_map: "inj (lift_map i)"
paulson@13786: apply (unfold lift_map_def)
paulson@13786: apply (rule inj_onI, auto)
paulson@13786: done
paulson@13786: 
paulson@13812: subsection{*Surjectiveness proof*}
paulson@13786: 
paulson@13786: lemma lift_map_drop_map_eq [simp]: "!!s. lift_map i (drop_map i s) = s"
paulson@13786: apply (unfold lift_map_def drop_map_def)
paulson@13786: apply (force simp add: insert_map_delete_map_eq)
paulson@13786: done
paulson@13786: 
paulson@13786: lemma drop_map_inject [dest!]: "(drop_map i s) = (drop_map i s') ==> s=s'"
paulson@13798: by (drule_tac f = "lift_map i" in arg_cong, simp)
paulson@13786: 
paulson@13786: lemma surj_lift_map: "surj (lift_map i)"
paulson@13786: apply (rule surjI)
paulson@13786: apply (rule lift_map_drop_map_eq)
paulson@13786: done
paulson@13786: 
paulson@13786: lemma bij_lift_map [iff]: "bij (lift_map i)"
paulson@13798: by (simp add: bij_def inj_lift_map surj_lift_map)
paulson@13786: 
paulson@13786: lemma inv_lift_map_eq [simp]: "inv (lift_map i) = drop_map i"
paulson@13786: by (rule inv_equality, auto)
paulson@13786: 
paulson@13786: lemma inv_drop_map_eq [simp]: "inv (drop_map i) = lift_map i"
paulson@13786: by (rule inv_equality, auto)
paulson@13786: 
paulson@13786: lemma bij_drop_map [iff]: "bij (drop_map i)"
paulson@13786: by (simp del: inv_lift_map_eq add: inv_lift_map_eq [symmetric] bij_imp_bij_inv)
paulson@13786: 
paulson@13786: (*sub's main property!*)
paulson@13786: lemma sub_apply [simp]: "sub i f = f i"
paulson@13798: by (simp add: sub_def)
paulson@13786: 
paulson@13812: lemma all_total_lift: "all_total F ==> all_total (lift i F)"
paulson@13812: by (simp add: lift_def rename_def Extend.all_total_extend)
paulson@13812: 
paulson@13836: lemma insert_map_upd_same: "(insert_map i t f)(i := s) = insert_map i s f"
paulson@13836: by (rule ext, auto)
paulson@13836: 
paulson@13836: lemma insert_map_upd:
paulson@13836:      "(insert_map j t f)(i := s) =  
paulson@13836:       (if i=j then insert_map i s f  
paulson@13836:        else if i<j then insert_map j t (f(i:=s))  
paulson@13836:        else insert_map j t (f(i - Suc 0 := s)))"
paulson@13836: apply (rule ext) 
wenzelm@46577: apply (simp split add: nat_diff_split)
paulson@13836:  txt{*This simplification is VERY slow*}
paulson@13836: done
paulson@13836: 
paulson@13836: lemma insert_map_eq_diff:
paulson@13836:      "[| insert_map i s f = insert_map j t g;  i\<noteq>j |]  
paulson@13836:       ==> \<exists>g'. insert_map i s' f = insert_map j t g'"
paulson@13836: apply (subst insert_map_upd_same [symmetric])
paulson@13836: apply (erule ssubst)
paulson@13836: apply (simp only: insert_map_upd if_False split: split_if, blast)
paulson@13836: done
paulson@13836: 
paulson@13836: lemma lift_map_eq_diff: 
paulson@13836:      "[| lift_map i (s,(f,uu)) = lift_map j (t,(g,vv));  i\<noteq>j |]  
paulson@13836:       ==> \<exists>g'. lift_map i (s',(f,uu)) = lift_map j (t,(g',vv))"
paulson@13836: apply (unfold lift_map_def, auto)
paulson@13836: apply (blast dest: insert_map_eq_diff)
paulson@13836: done
paulson@13836: 
paulson@13812: 
paulson@13812: subsection{*The Operator @{term lift_set}*}
paulson@13786: 
paulson@13786: lemma lift_set_empty [simp]: "lift_set i {} = {}"
paulson@13786: by (unfold lift_set_def, auto)
paulson@13786: 
paulson@13805: lemma lift_set_iff: "(lift_map i x \<in> lift_set i A) = (x \<in> A)"
paulson@13786: apply (unfold lift_set_def)
paulson@13786: apply (rule inj_lift_map [THEN inj_image_mem_iff])
paulson@13786: done
paulson@13786: 
paulson@13786: (*Do we really need both this one and its predecessor?*)
paulson@13786: lemma lift_set_iff2 [iff]:
paulson@13805:      "((f,uu) \<in> lift_set i A) = ((f i, (delete_map i f, uu)) \<in> A)"
paulson@13798: by (simp add: lift_set_def mem_rename_set_iff drop_map_def)
paulson@13786: 
paulson@13786: 
paulson@13805: lemma lift_set_mono: "A \<subseteq> B ==> lift_set i A \<subseteq> lift_set i B"
paulson@13786: apply (unfold lift_set_def)
paulson@13786: apply (erule image_mono)
paulson@13786: done
paulson@13786: 
paulson@13805: lemma lift_set_Un_distrib: "lift_set i (A \<union> B) = lift_set i A \<union> lift_set i B"
paulson@13812: by (simp add: lift_set_def image_Un)
paulson@13786: 
paulson@13786: lemma lift_set_Diff_distrib: "lift_set i (A-B) = lift_set i A - lift_set i B"
paulson@13786: apply (unfold lift_set_def)
paulson@13786: apply (rule inj_lift_map [THEN image_set_diff])
paulson@13786: done
paulson@13786: 
paulson@13786: 
paulson@13812: subsection{*The Lattice Operations*}
paulson@13786: 
paulson@13786: lemma bij_lift [iff]: "bij (lift i)"
paulson@13798: by (simp add: lift_def)
paulson@13786: 
paulson@13786: lemma lift_SKIP [simp]: "lift i SKIP = SKIP"
paulson@13798: by (simp add: lift_def)
paulson@13786: 
wenzelm@60773: lemma lift_Join [simp]: "lift i (F \<squnion> G) = lift i F \<squnion> lift i G"
paulson@13798: by (simp add: lift_def)
paulson@13786: 
paulson@13805: lemma lift_JN [simp]: "lift j (JOIN I F) = (\<Squnion>i \<in> I. lift j (F i))"
paulson@13798: by (simp add: lift_def)
paulson@13786: 
paulson@13812: subsection{*Safety: constrains, stable, invariant*}
paulson@13786: 
paulson@13786: lemma lift_constrains: 
paulson@13805:      "(lift i F \<in> (lift_set i A) co (lift_set i B)) = (F \<in> A co B)"
paulson@13798: by (simp add: lift_def lift_set_def rename_constrains)
paulson@13786: 
paulson@13786: lemma lift_stable: 
paulson@13805:      "(lift i F \<in> stable (lift_set i A)) = (F \<in> stable A)"
paulson@13798: by (simp add: lift_def lift_set_def rename_stable)
paulson@13786: 
paulson@13786: lemma lift_invariant: 
paulson@13805:      "(lift i F \<in> invariant (lift_set i A)) = (F \<in> invariant A)"
paulson@13812: by (simp add: lift_def lift_set_def rename_invariant)
paulson@13786: 
paulson@13786: lemma lift_Constrains: 
paulson@13805:      "(lift i F \<in> (lift_set i A) Co (lift_set i B)) = (F \<in> A Co B)"
paulson@13812: by (simp add: lift_def lift_set_def rename_Constrains)
paulson@13786: 
paulson@13786: lemma lift_Stable: 
paulson@13805:      "(lift i F \<in> Stable (lift_set i A)) = (F \<in> Stable A)"
paulson@13812: by (simp add: lift_def lift_set_def rename_Stable)
paulson@13786: 
paulson@13786: lemma lift_Always: 
paulson@13805:      "(lift i F \<in> Always (lift_set i A)) = (F \<in> Always A)"
paulson@13812: by (simp add: lift_def lift_set_def rename_Always)
paulson@13786: 
paulson@13812: subsection{*Progress: transient, ensures*}
paulson@13786: 
paulson@13786: lemma lift_transient: 
paulson@13805:      "(lift i F \<in> transient (lift_set i A)) = (F \<in> transient A)"
paulson@13798: by (simp add: lift_def lift_set_def rename_transient)
paulson@13786: 
paulson@13786: lemma lift_ensures: 
paulson@13805:      "(lift i F \<in> (lift_set i A) ensures (lift_set i B)) =  
paulson@13805:       (F \<in> A ensures B)"
paulson@13798: by (simp add: lift_def lift_set_def rename_ensures)
paulson@13786: 
paulson@13786: lemma lift_leadsTo: 
paulson@13805:      "(lift i F \<in> (lift_set i A) leadsTo (lift_set i B)) =  
paulson@13805:       (F \<in> A leadsTo B)"
paulson@13798: by (simp add: lift_def lift_set_def rename_leadsTo)
paulson@13786: 
paulson@13786: lemma lift_LeadsTo: 
paulson@13805:      "(lift i F \<in> (lift_set i A) LeadsTo (lift_set i B)) =   
paulson@13805:       (F \<in> A LeadsTo B)"
paulson@13798: by (simp add: lift_def lift_set_def rename_LeadsTo)
paulson@13786: 
paulson@13786: 
paulson@13786: (** guarantees **)
paulson@13786: 
paulson@13786: lemma lift_lift_guarantees_eq: 
paulson@13805:      "(lift i F \<in> (lift i ` X) guarantees (lift i ` Y)) =  
paulson@13805:       (F \<in> X guarantees Y)"
paulson@13786: apply (unfold lift_def)
paulson@13786: apply (subst bij_lift_map [THEN rename_rename_guarantees_eq, symmetric])
paulson@13798: apply (simp add: o_def)
paulson@13786: done
paulson@13786: 
paulson@13812: lemma lift_guarantees_eq_lift_inv:
paulson@13812:      "(lift i F \<in> X guarantees Y) =  
paulson@13805:       (F \<in> (rename (drop_map i) ` X) guarantees (rename (drop_map i) ` Y))"
paulson@13786: by (simp add: bij_lift_map [THEN rename_guarantees_eq_rename_inv] lift_def)
paulson@7186: 
paulson@7186: 
paulson@13786: (*To preserve snd means that the second component is there just to allow
paulson@13786:   guarantees properties to be stated.  Converse fails, for lift i F can 
paulson@13786:   change function components other than i*)
paulson@13805: lemma lift_preserves_snd_I: "F \<in> preserves snd ==> lift i F \<in> preserves snd"
paulson@13786: apply (drule_tac w1=snd in subset_preserves_o [THEN subsetD])
paulson@13798: apply (simp add: lift_def rename_preserves)
wenzelm@46577: apply (simp add: lift_map_def o_def split_def)
paulson@13786: done
paulson@13786: 
paulson@13786: lemma delete_map_eqE':
paulson@13805:      "(delete_map i g) = (delete_map i g') ==> \<exists>x. g = g'(i:=x)"
paulson@13786: apply (drule_tac f = "insert_map i (g i) " in arg_cong)
paulson@13798: apply (simp add: insert_map_delete_map_eq)
paulson@13786: apply (erule exI)
paulson@13786: done
paulson@13786: 
paulson@13786: lemmas delete_map_eqE = delete_map_eqE' [THEN exE, elim!]
paulson@13786: 
paulson@13786: lemma delete_map_neq_apply:
paulson@13805:      "[| delete_map j g = delete_map j g';  i\<noteq>j |] ==> g i = g' i"
paulson@13786: by force
paulson@13786: 
wenzelm@61943: (*A set of the form (A \<times> UNIV) ignores the second (dummy) state component*)
paulson@13786: 
wenzelm@61943: lemma vimage_o_fst_eq [simp]: "(f o fst) -` A = (f-`A) \<times> UNIV"
paulson@13786: by auto
paulson@13786: 
paulson@13786: lemma vimage_sub_eq_lift_set [simp]:
wenzelm@61943:      "(sub i -`A) \<times> UNIV = lift_set i (A \<times> UNIV)"
paulson@13786: by auto
paulson@13786: 
paulson@13786: lemma mem_lift_act_iff [iff]: 
paulson@13805:      "((s,s') \<in> extend_act (%(x,u::unit). lift_map i x) act) =  
paulson@13805:       ((drop_map i s, drop_map i s') \<in> act)"
paulson@13786: apply (unfold extend_act_def, auto)
paulson@13786: apply (rule bexI, auto)
paulson@13786: done
paulson@13786: 
paulson@13786: lemma preserves_snd_lift_stable:
paulson@13805:      "[| F \<in> preserves snd;  i\<noteq>j |]  
wenzelm@61943:       ==> lift j F \<in> stable (lift_set i (A \<times> UNIV))"
paulson@13798: apply (auto simp add: lift_def lift_set_def stable_def constrains_def 
paulson@13798:                       rename_def extend_def mem_rename_set_iff)
paulson@13786: apply (auto dest!: preserves_imp_eq simp add: lift_map_def drop_map_def)
paulson@13786: apply (drule_tac x = i in fun_cong, auto)
paulson@13786: done
paulson@13786: 
paulson@13805: (*If i\<noteq>j then lift j F  does nothing to lift_set i, and the 
paulson@13805:   premise ensures A \<subseteq> B.*)
paulson@13786: lemma constrains_imp_lift_constrains:
wenzelm@61943:     "[| F i \<in> (A \<times> UNIV) co (B \<times> UNIV);   
paulson@13805:         F j \<in> preserves snd |]   
wenzelm@61943:      ==> lift j (F j) \<in> (lift_set i (A \<times> UNIV)) co (lift_set i (B \<times> UNIV))"
wenzelm@46911: apply (cases "i=j")
paulson@13786: apply (simp add: lift_def lift_set_def rename_constrains)
paulson@13798: apply (erule preserves_snd_lift_stable[THEN stableD, THEN constrains_weaken_R],
paulson@13798:        assumption)
paulson@13786: apply (erule constrains_imp_subset [THEN lift_set_mono])
paulson@13786: done
paulson@13786: 
paulson@13786: (*USELESS??*)
paulson@13812: lemma lift_map_image_Times:
wenzelm@61943:      "lift_map i ` (A \<times> UNIV) =  
wenzelm@61943:       (\<Union>s \<in> A. \<Union>f. {insert_map i s f}) \<times> UNIV"
paulson@13786: apply (auto intro!: bexI image_eqI simp add: lift_map_def)
paulson@13786: apply (rule split_conv [symmetric])
paulson@13786: done
paulson@13786: 
paulson@13786: lemma lift_preserves_eq:
paulson@13805:      "(lift i F \<in> preserves v) = (F \<in> preserves (v o lift_map i))"
paulson@13786: by (simp add: lift_def rename_preserves)
paulson@13786: 
paulson@13786: (*A useful rewrite.  If o, sub have been rewritten out already then can also
paulson@13786:   use it as   rewrite_rule [sub_def, o_def] lift_preserves_sub*)
paulson@13786: lemma lift_preserves_sub:
paulson@13805:      "F \<in> preserves snd  
paulson@13805:       ==> lift i F \<in> preserves (v o sub j o fst) =  
paulson@13805:           (if i=j then F \<in> preserves (v o fst) else True)"
paulson@13786: apply (drule subset_preserves_o [THEN subsetD])
wenzelm@46577: apply (simp add: lift_preserves_eq o_def)
paulson@13798: apply (auto cong del: if_weak_cong 
wenzelm@46577:        simp add: lift_map_def eq_commute split_def o_def)
paulson@13786: done
paulson@13786: 
paulson@13786: 
paulson@13812: subsection{*Lemmas to Handle Function Composition (o) More Consistently*}
paulson@13786: 
paulson@13786: (*Lets us prove one version of a theorem and store others*)
paulson@13786: lemma o_equiv_assoc: "f o g = h ==> f' o f o g = f' o h"
nipkow@39302: by (simp add: fun_eq_iff o_def)
paulson@13786: 
paulson@13805: lemma o_equiv_apply: "f o g = h ==> \<forall>x. f(g x) = h x"
nipkow@39302: by (simp add: fun_eq_iff o_def)
paulson@13786: 
paulson@13786: lemma fst_o_lift_map: "sub i o fst o lift_map i = fst"
paulson@13786: apply (rule ext)
paulson@13786: apply (auto simp add: o_def lift_map_def sub_def)
paulson@13786: done
paulson@13786: 
paulson@13786: lemma snd_o_lift_map: "snd o lift_map i = snd o snd"
paulson@13786: apply (rule ext)
paulson@13786: apply (auto simp add: o_def lift_map_def)
paulson@13786: done
paulson@13786: 
paulson@13786: 
paulson@13812: subsection{*More lemmas about extend and project*}
paulson@13786: 
paulson@13812: text{*They could be moved to theory Extend or Project*}
paulson@13812: 
paulson@13812: lemma extend_act_extend_act:
paulson@13812:      "extend_act h' (extend_act h act) =  
paulson@13786:       extend_act (%(x,(y,y')). h'(h(x,y),y')) act"
paulson@13786: apply (auto elim!: rev_bexI simp add: extend_act_def, blast) 
paulson@13786: done
paulson@13786: 
paulson@13812: lemma project_act_project_act:
paulson@13812:      "project_act h (project_act h' act) =  
paulson@13786:       project_act (%(x,(y,y')). h'(h(x,y),y')) act"
paulson@13786: by (auto elim!: rev_bexI simp add: project_act_def)
paulson@13786: 
paulson@13786: lemma project_act_extend_act:
paulson@13786:      "project_act h (extend_act h' act) =  
paulson@13805:         {(x,x'). \<exists>s s' y y' z. (s,s') \<in> act &  
paulson@13786:                  h(x,y) = h'(s,z) & h(x',y') = h'(s',z)}"
paulson@13786: by (simp add: extend_act_def project_act_def, blast)
paulson@13786: 
paulson@13786: 
paulson@13812: subsection{*OK and "lift"*}
paulson@13786: 
paulson@13786: lemma act_in_UNION_preserves_fst:
paulson@13805:      "act \<subseteq> {(x,x'). fst x = fst x'} ==> act \<in> UNION (preserves fst) Acts"
paulson@13786: apply (rule_tac a = "mk_program (UNIV,{act},UNIV) " in UN_I)
paulson@13786: apply (auto simp add: preserves_def stable_def constrains_def)
paulson@13786: done
paulson@13786: 
paulson@13786: lemma UNION_OK_lift_I:
paulson@13805:      "[| \<forall>i \<in> I. F i \<in> preserves snd;   
paulson@13805:          \<forall>i \<in> I. UNION (preserves fst) Acts \<subseteq> AllowedActs (F i) |]  
paulson@13786:       ==> OK I (%i. lift i (F i))"
paulson@13790: apply (auto simp add: OK_def lift_def rename_def Extend.Acts_extend)
paulson@13798: apply (simp add: Extend.AllowedActs_extend project_act_extend_act)
paulson@13786: apply (rename_tac "act")
paulson@13786: apply (subgoal_tac
paulson@13786:        "{(x, x'). \<exists>s f u s' f' u'. 
paulson@13805:                     ((s, f, u), s', f', u') \<in> act & 
paulson@13786:                     lift_map j x = lift_map i (s, f, u) & 
paulson@13786:                     lift_map j x' = lift_map i (s', f', u') } 
paulson@13805:                 \<subseteq> { (x,x') . fst x = fst x'}")
paulson@13786: apply (blast intro: act_in_UNION_preserves_fst, clarify)
paulson@13786: apply (drule_tac x = j in fun_cong)+
paulson@13786: apply (drule_tac x = i in bspec, assumption)
paulson@13786: apply (frule preserves_imp_eq, auto)
paulson@13786: done
paulson@13786: 
paulson@13786: lemma OK_lift_I:
paulson@13805:      "[| \<forall>i \<in> I. F i \<in> preserves snd;   
paulson@13805:          \<forall>i \<in> I. preserves fst \<subseteq> Allowed (F i) |]  
paulson@13786:       ==> OK I (%i. lift i (F i))"
paulson@13786: by (simp add: safety_prop_AllowedActs_iff_Allowed UNION_OK_lift_I)
paulson@13786: 
paulson@13786: lemma Allowed_lift [simp]: "Allowed (lift i F) = lift i ` (Allowed F)"
wenzelm@46577: by (simp add: lift_def)
paulson@13786: 
paulson@13790: lemma lift_image_preserves:
paulson@13790:      "lift i ` preserves v = preserves (v o drop_map i)"
wenzelm@46577: by (simp add: rename_image_preserves lift_def)
paulson@13786: 
paulson@7186: end