src/ZF/OrderArith.thy
author paulson
Tue Jul 02 13:28:08 2002 +0200 (2002-07-02 ago)
changeset 13269 3ba9be497c33
parent 13140 6d97dbb189a9
child 13356 c9cfe1638bf2
permissions -rw-r--r--
Tidying and introduction of various new theorems
     1 (*  Title:      ZF/OrderArith.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1994  University of Cambridge
     5 
     6 Towards ordinal arithmetic.  Also useful with wfrec.
     7 *)
     8 
     9 theory OrderArith = Order + Sum + Ordinal:
    10 constdefs
    11 
    12   (*disjoint sum of two relations; underlies ordinal addition*)
    13   radd    :: "[i,i,i,i]=>i"
    14     "radd(A,r,B,s) == 
    15                 {z: (A+B) * (A+B).  
    16                     (EX x y. z = <Inl(x), Inr(y)>)   |   
    17                     (EX x' x. z = <Inl(x'), Inl(x)> & <x',x>:r)   |      
    18                     (EX y' y. z = <Inr(y'), Inr(y)> & <y',y>:s)}"
    19 
    20   (*lexicographic product of two relations; underlies ordinal multiplication*)
    21   rmult   :: "[i,i,i,i]=>i"
    22     "rmult(A,r,B,s) == 
    23                 {z: (A*B) * (A*B).  
    24                     EX x' y' x y. z = <<x',y'>, <x,y>> &         
    25                        (<x',x>: r | (x'=x & <y',y>: s))}"
    26 
    27   (*inverse image of a relation*)
    28   rvimage :: "[i,i,i]=>i"
    29     "rvimage(A,f,r) == {z: A*A. EX x y. z = <x,y> & <f`x,f`y>: r}"
    30 
    31   measure :: "[i, i\<Rightarrow>i] \<Rightarrow> i"
    32     "measure(A,f) == {<x,y>: A*A. f(x) < f(y)}"
    33 
    34 
    35 (**** Addition of relations -- disjoint sum ****)
    36 
    37 (** Rewrite rules.  Can be used to obtain introduction rules **)
    38 
    39 lemma radd_Inl_Inr_iff [iff]: 
    40     "<Inl(a), Inr(b)> : radd(A,r,B,s)  <->  a:A & b:B"
    41 apply (unfold radd_def, blast)
    42 done
    43 
    44 lemma radd_Inl_iff [iff]: 
    45     "<Inl(a'), Inl(a)> : radd(A,r,B,s)  <->  a':A & a:A & <a',a>:r"
    46 apply (unfold radd_def, blast)
    47 done
    48 
    49 lemma radd_Inr_iff [iff]: 
    50     "<Inr(b'), Inr(b)> : radd(A,r,B,s) <->  b':B & b:B & <b',b>:s"
    51 apply (unfold radd_def, blast)
    52 done
    53 
    54 lemma radd_Inr_Inl_iff [iff]: 
    55     "<Inr(b), Inl(a)> : radd(A,r,B,s) <->  False"
    56 apply (unfold radd_def, blast)
    57 done
    58 
    59 (** Elimination Rule **)
    60 
    61 lemma raddE:
    62     "[| <p',p> : radd(A,r,B,s);                  
    63         !!x y. [| p'=Inl(x); x:A; p=Inr(y); y:B |] ==> Q;        
    64         !!x' x. [| p'=Inl(x'); p=Inl(x); <x',x>: r; x':A; x:A |] ==> Q;  
    65         !!y' y. [| p'=Inr(y'); p=Inr(y); <y',y>: s; y':B; y:B |] ==> Q   
    66      |] ==> Q"
    67 apply (unfold radd_def, blast) 
    68 done
    69 
    70 (** Type checking **)
    71 
    72 lemma radd_type: "radd(A,r,B,s) <= (A+B) * (A+B)"
    73 apply (unfold radd_def)
    74 apply (rule Collect_subset)
    75 done
    76 
    77 lemmas field_radd = radd_type [THEN field_rel_subset]
    78 
    79 (** Linearity **)
    80 
    81 lemma linear_radd: 
    82     "[| linear(A,r);  linear(B,s) |] ==> linear(A+B,radd(A,r,B,s))"
    83 apply (unfold linear_def, blast) 
    84 done
    85 
    86 
    87 (** Well-foundedness **)
    88 
    89 lemma wf_on_radd: "[| wf[A](r);  wf[B](s) |] ==> wf[A+B](radd(A,r,B,s))"
    90 apply (rule wf_onI2)
    91 apply (subgoal_tac "ALL x:A. Inl (x) : Ba")
    92 (*Proving the lemma, which is needed twice!*)
    93  prefer 2
    94  apply (erule_tac V = "y : A + B" in thin_rl)
    95  apply (rule_tac ballI)
    96  apply (erule_tac r = "r" and a = "x" in wf_on_induct, assumption)
    97  apply blast 
    98 (*Returning to main part of proof*)
    99 apply safe
   100 apply blast
   101 apply (erule_tac r = "s" and a = "ya" in wf_on_induct, assumption, blast) 
   102 done
   103 
   104 lemma wf_radd: "[| wf(r);  wf(s) |] ==> wf(radd(field(r),r,field(s),s))"
   105 apply (simp add: wf_iff_wf_on_field)
   106 apply (rule wf_on_subset_A [OF _ field_radd])
   107 apply (blast intro: wf_on_radd) 
   108 done
   109 
   110 lemma well_ord_radd:
   111      "[| well_ord(A,r);  well_ord(B,s) |] ==> well_ord(A+B, radd(A,r,B,s))"
   112 apply (rule well_ordI)
   113 apply (simp add: well_ord_def wf_on_radd)
   114 apply (simp add: well_ord_def tot_ord_def linear_radd)
   115 done
   116 
   117 (** An ord_iso congruence law **)
   118 
   119 lemma sum_bij:
   120      "[| f: bij(A,C);  g: bij(B,D) |]
   121       ==> (lam z:A+B. case(%x. Inl(f`x), %y. Inr(g`y), z)) : bij(A+B, C+D)"
   122 apply (rule_tac d = "case (%x. Inl (converse (f) `x), %y. Inr (converse (g) `y))" in lam_bijective)
   123 apply (typecheck add: bij_is_inj inj_is_fun) 
   124 apply (auto simp add: left_inverse_bij right_inverse_bij) 
   125 done
   126 
   127 lemma sum_ord_iso_cong: 
   128     "[| f: ord_iso(A,r,A',r');  g: ord_iso(B,s,B',s') |] ==>      
   129             (lam z:A+B. case(%x. Inl(f`x), %y. Inr(g`y), z))             
   130             : ord_iso(A+B, radd(A,r,B,s), A'+B', radd(A',r',B',s'))"
   131 apply (unfold ord_iso_def)
   132 apply (safe intro!: sum_bij)
   133 (*Do the beta-reductions now*)
   134 apply (auto cong add: conj_cong simp add: bij_is_fun [THEN apply_type])
   135 done
   136 
   137 (*Could we prove an ord_iso result?  Perhaps 
   138      ord_iso(A+B, radd(A,r,B,s), A Un B, r Un s) *)
   139 lemma sum_disjoint_bij: "A Int B = 0 ==>      
   140             (lam z:A+B. case(%x. x, %y. y, z)) : bij(A+B, A Un B)"
   141 apply (rule_tac d = "%z. if z:A then Inl (z) else Inr (z) " in lam_bijective)
   142 apply auto
   143 done
   144 
   145 (** Associativity **)
   146 
   147 lemma sum_assoc_bij:
   148      "(lam z:(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z))  
   149       : bij((A+B)+C, A+(B+C))"
   150 apply (rule_tac d = "case (%x. Inl (Inl (x)), case (%x. Inl (Inr (x)), Inr))" 
   151        in lam_bijective)
   152 apply auto
   153 done
   154 
   155 lemma sum_assoc_ord_iso:
   156      "(lam z:(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z))  
   157       : ord_iso((A+B)+C, radd(A+B, radd(A,r,B,s), C, t),     
   158                 A+(B+C), radd(A, r, B+C, radd(B,s,C,t)))"
   159 apply (rule sum_assoc_bij [THEN ord_isoI], auto)
   160 done
   161 
   162 
   163 (**** Multiplication of relations -- lexicographic product ****)
   164 
   165 (** Rewrite rule.  Can be used to obtain introduction rules **)
   166 
   167 lemma  rmult_iff [iff]: 
   168     "<<a',b'>, <a,b>> : rmult(A,r,B,s) <->        
   169             (<a',a>: r  & a':A & a:A & b': B & b: B) |   
   170             (<b',b>: s  & a'=a & a:A & b': B & b: B)"
   171 
   172 apply (unfold rmult_def, blast)
   173 done
   174 
   175 lemma rmultE: 
   176     "[| <<a',b'>, <a,b>> : rmult(A,r,B,s);               
   177         [| <a',a>: r;  a':A;  a:A;  b':B;  b:B |] ==> Q;         
   178         [| <b',b>: s;  a:A;  a'=a;  b':B;  b:B |] ==> Q  
   179      |] ==> Q"
   180 apply blast 
   181 done
   182 
   183 (** Type checking **)
   184 
   185 lemma rmult_type: "rmult(A,r,B,s) <= (A*B) * (A*B)"
   186 apply (unfold rmult_def)
   187 apply (rule Collect_subset)
   188 done
   189 
   190 lemmas field_rmult = rmult_type [THEN field_rel_subset]
   191 
   192 (** Linearity **)
   193 
   194 lemma linear_rmult:
   195     "[| linear(A,r);  linear(B,s) |] ==> linear(A*B,rmult(A,r,B,s))"
   196 apply (simp add: linear_def, blast) 
   197 done
   198 
   199 (** Well-foundedness **)
   200 
   201 lemma wf_on_rmult: "[| wf[A](r);  wf[B](s) |] ==> wf[A*B](rmult(A,r,B,s))"
   202 apply (rule wf_onI2)
   203 apply (erule SigmaE)
   204 apply (erule ssubst)
   205 apply (subgoal_tac "ALL b:B. <x,b>: Ba", blast)
   206 apply (erule_tac a = "x" in wf_on_induct, assumption)
   207 apply (rule ballI)
   208 apply (erule_tac a = "b" in wf_on_induct, assumption)
   209 apply (best elim!: rmultE bspec [THEN mp])
   210 done
   211 
   212 
   213 lemma wf_rmult: "[| wf(r);  wf(s) |] ==> wf(rmult(field(r),r,field(s),s))"
   214 apply (simp add: wf_iff_wf_on_field)
   215 apply (rule wf_on_subset_A [OF _ field_rmult])
   216 apply (blast intro: wf_on_rmult) 
   217 done
   218 
   219 lemma well_ord_rmult:
   220      "[| well_ord(A,r);  well_ord(B,s) |] ==> well_ord(A*B, rmult(A,r,B,s))"
   221 apply (rule well_ordI)
   222 apply (simp add: well_ord_def wf_on_rmult)
   223 apply (simp add: well_ord_def tot_ord_def linear_rmult)
   224 done
   225 
   226 
   227 (** An ord_iso congruence law **)
   228 
   229 lemma prod_bij:
   230      "[| f: bij(A,C);  g: bij(B,D) |] 
   231       ==> (lam <x,y>:A*B. <f`x, g`y>) : bij(A*B, C*D)"
   232 apply (rule_tac d = "%<x,y>. <converse (f) `x, converse (g) `y>" 
   233        in lam_bijective)
   234 apply (typecheck add: bij_is_inj inj_is_fun) 
   235 apply (auto simp add: left_inverse_bij right_inverse_bij) 
   236 done
   237 
   238 lemma prod_ord_iso_cong: 
   239     "[| f: ord_iso(A,r,A',r');  g: ord_iso(B,s,B',s') |]      
   240      ==> (lam <x,y>:A*B. <f`x, g`y>)                                  
   241          : ord_iso(A*B, rmult(A,r,B,s), A'*B', rmult(A',r',B',s'))"
   242 apply (unfold ord_iso_def)
   243 apply (safe intro!: prod_bij)
   244 apply (simp_all add: bij_is_fun [THEN apply_type])
   245 apply (blast intro: bij_is_inj [THEN inj_apply_equality])
   246 done
   247 
   248 lemma singleton_prod_bij: "(lam z:A. <x,z>) : bij(A, {x}*A)"
   249 by (rule_tac d = "snd" in lam_bijective, auto)
   250 
   251 (*Used??*)
   252 lemma singleton_prod_ord_iso:
   253      "well_ord({x},xr) ==>   
   254           (lam z:A. <x,z>) : ord_iso(A, r, {x}*A, rmult({x}, xr, A, r))"
   255 apply (rule singleton_prod_bij [THEN ord_isoI])
   256 apply (simp (no_asm_simp))
   257 apply (blast dest: well_ord_is_wf [THEN wf_on_not_refl])
   258 done
   259 
   260 (*Here we build a complicated function term, then simplify it using
   261   case_cong, id_conv, comp_lam, case_case.*)
   262 lemma prod_sum_singleton_bij:
   263      "a~:C ==>  
   264        (lam x:C*B + D. case(%x. x, %y.<a,y>, x))  
   265        : bij(C*B + D, C*B Un {a}*D)"
   266 apply (rule subst_elem)
   267 apply (rule id_bij [THEN sum_bij, THEN comp_bij])
   268 apply (rule singleton_prod_bij)
   269 apply (rule sum_disjoint_bij, blast)
   270 apply (simp (no_asm_simp) cong add: case_cong)
   271 apply (rule comp_lam [THEN trans, symmetric])
   272 apply (fast elim!: case_type)
   273 apply (simp (no_asm_simp) add: case_case)
   274 done
   275 
   276 lemma prod_sum_singleton_ord_iso:
   277  "[| a:A;  well_ord(A,r) |] ==>  
   278     (lam x:pred(A,a,r)*B + pred(B,b,s). case(%x. x, %y.<a,y>, x))  
   279     : ord_iso(pred(A,a,r)*B + pred(B,b,s),               
   280                   radd(A*B, rmult(A,r,B,s), B, s),       
   281               pred(A,a,r)*B Un {a}*pred(B,b,s), rmult(A,r,B,s))"
   282 apply (rule prod_sum_singleton_bij [THEN ord_isoI])
   283 apply (simp (no_asm_simp) add: pred_iff well_ord_is_wf [THEN wf_on_not_refl])
   284 apply (auto elim!: well_ord_is_wf [THEN wf_on_asym] predE)
   285 done
   286 
   287 (** Distributive law **)
   288 
   289 lemma sum_prod_distrib_bij:
   290      "(lam <x,z>:(A+B)*C. case(%y. Inl(<y,z>), %y. Inr(<y,z>), x))  
   291       : bij((A+B)*C, (A*C)+(B*C))"
   292 apply (rule_tac d = "case (%<x,y>.<Inl (x),y>, %<x,y>.<Inr (x),y>) " 
   293        in lam_bijective)
   294 apply auto
   295 done
   296 
   297 lemma sum_prod_distrib_ord_iso:
   298  "(lam <x,z>:(A+B)*C. case(%y. Inl(<y,z>), %y. Inr(<y,z>), x))  
   299   : ord_iso((A+B)*C, rmult(A+B, radd(A,r,B,s), C, t),  
   300             (A*C)+(B*C), radd(A*C, rmult(A,r,C,t), B*C, rmult(B,s,C,t)))"
   301 apply (rule sum_prod_distrib_bij [THEN ord_isoI], auto)
   302 done
   303 
   304 (** Associativity **)
   305 
   306 lemma prod_assoc_bij:
   307      "(lam <<x,y>, z>:(A*B)*C. <x,<y,z>>) : bij((A*B)*C, A*(B*C))"
   308 apply (rule_tac d = "%<x, <y,z>>. <<x,y>, z>" in lam_bijective, auto)
   309 done
   310 
   311 lemma prod_assoc_ord_iso:
   312  "(lam <<x,y>, z>:(A*B)*C. <x,<y,z>>)                    
   313   : ord_iso((A*B)*C, rmult(A*B, rmult(A,r,B,s), C, t),   
   314             A*(B*C), rmult(A, r, B*C, rmult(B,s,C,t)))"
   315 apply (rule prod_assoc_bij [THEN ord_isoI], auto)
   316 done
   317 
   318 (**** Inverse image of a relation ****)
   319 
   320 (** Rewrite rule **)
   321 
   322 lemma rvimage_iff: "<a,b> : rvimage(A,f,r)  <->  <f`a,f`b>: r & a:A & b:A"
   323 by (unfold rvimage_def, blast)
   324 
   325 (** Type checking **)
   326 
   327 lemma rvimage_type: "rvimage(A,f,r) <= A*A"
   328 apply (unfold rvimage_def)
   329 apply (rule Collect_subset)
   330 done
   331 
   332 lemmas field_rvimage = rvimage_type [THEN field_rel_subset]
   333 
   334 lemma rvimage_converse: "rvimage(A,f, converse(r)) = converse(rvimage(A,f,r))"
   335 by (unfold rvimage_def, blast)
   336 
   337 
   338 (** Partial Ordering Properties **)
   339 
   340 lemma irrefl_rvimage: 
   341     "[| f: inj(A,B);  irrefl(B,r) |] ==> irrefl(A, rvimage(A,f,r))"
   342 apply (unfold irrefl_def rvimage_def)
   343 apply (blast intro: inj_is_fun [THEN apply_type])
   344 done
   345 
   346 lemma trans_on_rvimage: 
   347     "[| f: inj(A,B);  trans[B](r) |] ==> trans[A](rvimage(A,f,r))"
   348 apply (unfold trans_on_def rvimage_def)
   349 apply (blast intro: inj_is_fun [THEN apply_type])
   350 done
   351 
   352 lemma part_ord_rvimage: 
   353     "[| f: inj(A,B);  part_ord(B,r) |] ==> part_ord(A, rvimage(A,f,r))"
   354 apply (unfold part_ord_def)
   355 apply (blast intro!: irrefl_rvimage trans_on_rvimage)
   356 done
   357 
   358 (** Linearity **)
   359 
   360 lemma linear_rvimage:
   361     "[| f: inj(A,B);  linear(B,r) |] ==> linear(A,rvimage(A,f,r))"
   362 apply (simp add: inj_def linear_def rvimage_iff) 
   363 apply (blast intro: apply_funtype) 
   364 done
   365 
   366 lemma tot_ord_rvimage: 
   367     "[| f: inj(A,B);  tot_ord(B,r) |] ==> tot_ord(A, rvimage(A,f,r))"
   368 apply (unfold tot_ord_def)
   369 apply (blast intro!: part_ord_rvimage linear_rvimage)
   370 done
   371 
   372 
   373 (** Well-foundedness **)
   374 
   375 (*Not sure if wf_on_rvimage could be proved from this!*)
   376 lemma wf_rvimage [intro!]: "wf(r) ==> wf(rvimage(A,f,r))"
   377 apply (simp (no_asm_use) add: rvimage_def wf_eq_minimal)
   378 apply clarify
   379 apply (subgoal_tac "EX w. w : {w: {f`x. x:Q}. EX x. x: Q & (f`x = w) }")
   380  apply (erule allE)
   381  apply (erule impE)
   382  apply assumption
   383  apply blast
   384 apply blast 
   385 done
   386 
   387 lemma wf_on_rvimage: "[| f: A->B;  wf[B](r) |] ==> wf[A](rvimage(A,f,r))"
   388 apply (rule wf_onI2)
   389 apply (subgoal_tac "ALL z:A. f`z=f`y --> z: Ba")
   390  apply blast
   391 apply (erule_tac a = "f`y" in wf_on_induct)
   392  apply (blast intro!: apply_funtype)
   393 apply (blast intro!: apply_funtype dest!: rvimage_iff [THEN iffD1])
   394 done
   395 
   396 (*Note that we need only wf[A](...) and linear(A,...) to get the result!*)
   397 lemma well_ord_rvimage:
   398      "[| f: inj(A,B);  well_ord(B,r) |] ==> well_ord(A, rvimage(A,f,r))"
   399 apply (rule well_ordI)
   400 apply (unfold well_ord_def tot_ord_def)
   401 apply (blast intro!: wf_on_rvimage inj_is_fun)
   402 apply (blast intro!: linear_rvimage)
   403 done
   404 
   405 lemma ord_iso_rvimage: 
   406     "f: bij(A,B) ==> f: ord_iso(A, rvimage(A,f,s), B, s)"
   407 apply (unfold ord_iso_def)
   408 apply (simp add: rvimage_iff)
   409 done
   410 
   411 lemma ord_iso_rvimage_eq: 
   412     "f: ord_iso(A,r, B,s) ==> rvimage(A,f,s) = r Int A*A"
   413 apply (unfold ord_iso_def rvimage_def, blast)
   414 done
   415 
   416 
   417 (** The "measure" relation is useful with wfrec **)
   418 
   419 lemma measure_eq_rvimage_Memrel:
   420      "measure(A,f) = rvimage(A,Lambda(A,f),Memrel(Collect(RepFun(A,f),Ord)))"
   421 apply (simp (no_asm) add: measure_def rvimage_def Memrel_iff)
   422 apply (rule equalityI, auto)
   423 apply (auto intro: Ord_in_Ord simp add: lt_def)
   424 done
   425 
   426 lemma wf_measure [iff]: "wf(measure(A,f))"
   427 apply (simp (no_asm) add: measure_eq_rvimage_Memrel wf_Memrel wf_rvimage)
   428 done
   429 
   430 lemma measure_iff [iff]: "<x,y> : measure(A,f) <-> x:A & y:A & f(x)<f(y)"
   431 apply (simp (no_asm) add: measure_def)
   432 done
   433 
   434 ML {*
   435 val measure_def = thm "measure_def";
   436 val radd_Inl_Inr_iff = thm "radd_Inl_Inr_iff";
   437 val radd_Inl_iff = thm "radd_Inl_iff";
   438 val radd_Inr_iff = thm "radd_Inr_iff";
   439 val radd_Inr_Inl_iff = thm "radd_Inr_Inl_iff";
   440 val raddE = thm "raddE";
   441 val radd_type = thm "radd_type";
   442 val field_radd = thm "field_radd";
   443 val linear_radd = thm "linear_radd";
   444 val wf_on_radd = thm "wf_on_radd";
   445 val wf_radd = thm "wf_radd";
   446 val well_ord_radd = thm "well_ord_radd";
   447 val sum_bij = thm "sum_bij";
   448 val sum_ord_iso_cong = thm "sum_ord_iso_cong";
   449 val sum_disjoint_bij = thm "sum_disjoint_bij";
   450 val sum_assoc_bij = thm "sum_assoc_bij";
   451 val sum_assoc_ord_iso = thm "sum_assoc_ord_iso";
   452 val rmult_iff = thm "rmult_iff";
   453 val rmultE = thm "rmultE";
   454 val rmult_type = thm "rmult_type";
   455 val field_rmult = thm "field_rmult";
   456 val linear_rmult = thm "linear_rmult";
   457 val wf_on_rmult = thm "wf_on_rmult";
   458 val wf_rmult = thm "wf_rmult";
   459 val well_ord_rmult = thm "well_ord_rmult";
   460 val prod_bij = thm "prod_bij";
   461 val prod_ord_iso_cong = thm "prod_ord_iso_cong";
   462 val singleton_prod_bij = thm "singleton_prod_bij";
   463 val singleton_prod_ord_iso = thm "singleton_prod_ord_iso";
   464 val prod_sum_singleton_bij = thm "prod_sum_singleton_bij";
   465 val prod_sum_singleton_ord_iso = thm "prod_sum_singleton_ord_iso";
   466 val sum_prod_distrib_bij = thm "sum_prod_distrib_bij";
   467 val sum_prod_distrib_ord_iso = thm "sum_prod_distrib_ord_iso";
   468 val prod_assoc_bij = thm "prod_assoc_bij";
   469 val prod_assoc_ord_iso = thm "prod_assoc_ord_iso";
   470 val rvimage_iff = thm "rvimage_iff";
   471 val rvimage_type = thm "rvimage_type";
   472 val field_rvimage = thm "field_rvimage";
   473 val rvimage_converse = thm "rvimage_converse";
   474 val irrefl_rvimage = thm "irrefl_rvimage";
   475 val trans_on_rvimage = thm "trans_on_rvimage";
   476 val part_ord_rvimage = thm "part_ord_rvimage";
   477 val linear_rvimage = thm "linear_rvimage";
   478 val tot_ord_rvimage = thm "tot_ord_rvimage";
   479 val wf_rvimage = thm "wf_rvimage";
   480 val wf_on_rvimage = thm "wf_on_rvimage";
   481 val well_ord_rvimage = thm "well_ord_rvimage";
   482 val ord_iso_rvimage = thm "ord_iso_rvimage";
   483 val ord_iso_rvimage_eq = thm "ord_iso_rvimage_eq";
   484 val measure_eq_rvimage_Memrel = thm "measure_eq_rvimage_Memrel";
   485 val wf_measure = thm "wf_measure";
   486 val measure_iff = thm "measure_iff";
   487 *}
   488 
   489 end