src/ZF/OrderArith.thy
author paulson
Thu Jan 23 10:30:14 2003 +0100 (2003-01-23)
changeset 13784 b9f6154427a4
parent 13634 99a593b49b04
child 13823 d49ffd9f9662
permissions -rw-r--r--
tidying (by script)
     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 *)
     7 
     8 header{*Combining Orderings: Foundations of Ordinal Arithmetic*}
     9 
    10 theory OrderArith = Order + Sum + Ordinal:
    11 constdefs
    12 
    13   (*disjoint sum of two relations; underlies ordinal addition*)
    14   radd    :: "[i,i,i,i]=>i"
    15     "radd(A,r,B,s) == 
    16                 {z: (A+B) * (A+B).  
    17                     (EX x y. z = <Inl(x), Inr(y)>)   |   
    18                     (EX x' x. z = <Inl(x'), Inl(x)> & <x',x>:r)   |      
    19                     (EX y' y. z = <Inr(y'), Inr(y)> & <y',y>:s)}"
    20 
    21   (*lexicographic product of two relations; underlies ordinal multiplication*)
    22   rmult   :: "[i,i,i,i]=>i"
    23     "rmult(A,r,B,s) == 
    24                 {z: (A*B) * (A*B).  
    25                     EX x' y' x y. z = <<x',y'>, <x,y>> &         
    26                        (<x',x>: r | (x'=x & <y',y>: s))}"
    27 
    28   (*inverse image of a relation*)
    29   rvimage :: "[i,i,i]=>i"
    30     "rvimage(A,f,r) == {z: A*A. EX x y. z = <x,y> & <f`x,f`y>: r}"
    31 
    32   measure :: "[i, i\<Rightarrow>i] \<Rightarrow> i"
    33     "measure(A,f) == {<x,y>: A*A. f(x) < f(y)}"
    34 
    35 
    36 subsection{*Addition of Relations -- Disjoint Sum*}
    37 
    38 subsubsection{*Rewrite rules.  Can be used to obtain introduction rules*}
    39 
    40 lemma radd_Inl_Inr_iff [iff]: 
    41     "<Inl(a), Inr(b)> : radd(A,r,B,s)  <->  a:A & b:B"
    42 by (unfold radd_def, blast)
    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 by (unfold radd_def, blast)
    47 
    48 lemma radd_Inr_iff [iff]: 
    49     "<Inr(b'), Inr(b)> : radd(A,r,B,s) <->  b':B & b:B & <b',b>:s"
    50 by (unfold radd_def, blast)
    51 
    52 lemma radd_Inr_Inl_iff [iff]: 
    53     "<Inr(b), Inl(a)> : radd(A,r,B,s) <->  False"
    54 by (unfold radd_def, blast)
    55 
    56 subsubsection{*Elimination Rule*}
    57 
    58 lemma raddE:
    59     "[| <p',p> : radd(A,r,B,s);                  
    60         !!x y. [| p'=Inl(x); x:A; p=Inr(y); y:B |] ==> Q;        
    61         !!x' x. [| p'=Inl(x'); p=Inl(x); <x',x>: r; x':A; x:A |] ==> Q;  
    62         !!y' y. [| p'=Inr(y'); p=Inr(y); <y',y>: s; y':B; y:B |] ==> Q   
    63      |] ==> Q"
    64 by (unfold radd_def, blast) 
    65 
    66 subsubsection{*Type checking*}
    67 
    68 lemma radd_type: "radd(A,r,B,s) <= (A+B) * (A+B)"
    69 apply (unfold radd_def)
    70 apply (rule Collect_subset)
    71 done
    72 
    73 lemmas field_radd = radd_type [THEN field_rel_subset]
    74 
    75 subsubsection{*Linearity*}
    76 
    77 lemma linear_radd: 
    78     "[| linear(A,r);  linear(B,s) |] ==> linear(A+B,radd(A,r,B,s))"
    79 by (unfold linear_def, blast) 
    80 
    81 
    82 subsubsection{*Well-foundedness*}
    83 
    84 lemma wf_on_radd: "[| wf[A](r);  wf[B](s) |] ==> wf[A+B](radd(A,r,B,s))"
    85 apply (rule wf_onI2)
    86 apply (subgoal_tac "ALL x:A. Inl (x) : Ba")
    87  --{*Proving the lemma, which is needed twice!*}
    88  prefer 2
    89  apply (erule_tac V = "y : A + B" in thin_rl)
    90  apply (rule_tac ballI)
    91  apply (erule_tac r = r and a = x in wf_on_induct, assumption)
    92  apply blast 
    93 txt{*Returning to main part of proof*}
    94 apply safe
    95 apply blast
    96 apply (erule_tac r = s and a = ya in wf_on_induct, assumption, blast) 
    97 done
    98 
    99 lemma wf_radd: "[| wf(r);  wf(s) |] ==> wf(radd(field(r),r,field(s),s))"
   100 apply (simp add: wf_iff_wf_on_field)
   101 apply (rule wf_on_subset_A [OF _ field_radd])
   102 apply (blast intro: wf_on_radd) 
   103 done
   104 
   105 lemma well_ord_radd:
   106      "[| well_ord(A,r);  well_ord(B,s) |] ==> well_ord(A+B, radd(A,r,B,s))"
   107 apply (rule well_ordI)
   108 apply (simp add: well_ord_def wf_on_radd)
   109 apply (simp add: well_ord_def tot_ord_def linear_radd)
   110 done
   111 
   112 subsubsection{*An @{term ord_iso} congruence law*}
   113 
   114 lemma sum_bij:
   115      "[| f: bij(A,C);  g: bij(B,D) |]
   116       ==> (lam z:A+B. case(%x. Inl(f`x), %y. Inr(g`y), z)) : bij(A+B, C+D)"
   117 apply (rule_tac d = "case (%x. Inl (converse(f)`x), %y. Inr(converse(g)`y))" 
   118        in lam_bijective)
   119 apply (typecheck add: bij_is_inj inj_is_fun) 
   120 apply (auto simp add: left_inverse_bij right_inverse_bij) 
   121 done
   122 
   123 lemma sum_ord_iso_cong: 
   124     "[| f: ord_iso(A,r,A',r');  g: ord_iso(B,s,B',s') |] ==>      
   125             (lam z:A+B. case(%x. Inl(f`x), %y. Inr(g`y), z))             
   126             : ord_iso(A+B, radd(A,r,B,s), A'+B', radd(A',r',B',s'))"
   127 apply (unfold ord_iso_def)
   128 apply (safe intro!: sum_bij)
   129 (*Do the beta-reductions now*)
   130 apply (auto cong add: conj_cong simp add: bij_is_fun [THEN apply_type])
   131 done
   132 
   133 (*Could we prove an ord_iso result?  Perhaps 
   134      ord_iso(A+B, radd(A,r,B,s), A Un B, r Un s) *)
   135 lemma sum_disjoint_bij: "A Int B = 0 ==>      
   136             (lam z:A+B. case(%x. x, %y. y, z)) : bij(A+B, A Un B)"
   137 apply (rule_tac d = "%z. if z:A then Inl (z) else Inr (z) " in lam_bijective)
   138 apply auto
   139 done
   140 
   141 subsubsection{*Associativity*}
   142 
   143 lemma sum_assoc_bij:
   144      "(lam z:(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z))  
   145       : bij((A+B)+C, A+(B+C))"
   146 apply (rule_tac d = "case (%x. Inl (Inl (x)), case (%x. Inl (Inr (x)), Inr))" 
   147        in lam_bijective)
   148 apply auto
   149 done
   150 
   151 lemma sum_assoc_ord_iso:
   152      "(lam z:(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z))  
   153       : ord_iso((A+B)+C, radd(A+B, radd(A,r,B,s), C, t),     
   154                 A+(B+C), radd(A, r, B+C, radd(B,s,C,t)))"
   155 by (rule sum_assoc_bij [THEN ord_isoI], auto)
   156 
   157 
   158 subsection{*Multiplication of Relations -- Lexicographic Product*}
   159 
   160 subsubsection{*Rewrite rule.  Can be used to obtain introduction rules*}
   161 
   162 lemma  rmult_iff [iff]: 
   163     "<<a',b'>, <a,b>> : rmult(A,r,B,s) <->        
   164             (<a',a>: r  & a':A & a:A & b': B & b: B) |   
   165             (<b',b>: s  & a'=a & a:A & b': B & b: B)"
   166 
   167 by (unfold rmult_def, blast)
   168 
   169 lemma rmultE: 
   170     "[| <<a',b'>, <a,b>> : rmult(A,r,B,s);               
   171         [| <a',a>: r;  a':A;  a:A;  b':B;  b:B |] ==> Q;         
   172         [| <b',b>: s;  a:A;  a'=a;  b':B;  b:B |] ==> Q  
   173      |] ==> Q"
   174 by blast 
   175 
   176 subsubsection{*Type checking*}
   177 
   178 lemma rmult_type: "rmult(A,r,B,s) <= (A*B) * (A*B)"
   179 by (unfold rmult_def, rule Collect_subset)
   180 
   181 lemmas field_rmult = rmult_type [THEN field_rel_subset]
   182 
   183 subsubsection{*Linearity*}
   184 
   185 lemma linear_rmult:
   186     "[| linear(A,r);  linear(B,s) |] ==> linear(A*B,rmult(A,r,B,s))"
   187 by (simp add: linear_def, blast) 
   188 
   189 subsubsection{*Well-foundedness*}
   190 
   191 lemma wf_on_rmult: "[| wf[A](r);  wf[B](s) |] ==> wf[A*B](rmult(A,r,B,s))"
   192 apply (rule wf_onI2)
   193 apply (erule SigmaE)
   194 apply (erule ssubst)
   195 apply (subgoal_tac "ALL b:B. <x,b>: Ba", blast)
   196 apply (erule_tac a = x in wf_on_induct, assumption)
   197 apply (rule ballI)
   198 apply (erule_tac a = b in wf_on_induct, assumption)
   199 apply (best elim!: rmultE bspec [THEN mp])
   200 done
   201 
   202 
   203 lemma wf_rmult: "[| wf(r);  wf(s) |] ==> wf(rmult(field(r),r,field(s),s))"
   204 apply (simp add: wf_iff_wf_on_field)
   205 apply (rule wf_on_subset_A [OF _ field_rmult])
   206 apply (blast intro: wf_on_rmult) 
   207 done
   208 
   209 lemma well_ord_rmult:
   210      "[| well_ord(A,r);  well_ord(B,s) |] ==> well_ord(A*B, rmult(A,r,B,s))"
   211 apply (rule well_ordI)
   212 apply (simp add: well_ord_def wf_on_rmult)
   213 apply (simp add: well_ord_def tot_ord_def linear_rmult)
   214 done
   215 
   216 
   217 subsubsection{*An @{term ord_iso} congruence law*}
   218 
   219 lemma prod_bij:
   220      "[| f: bij(A,C);  g: bij(B,D) |] 
   221       ==> (lam <x,y>:A*B. <f`x, g`y>) : bij(A*B, C*D)"
   222 apply (rule_tac d = "%<x,y>. <converse (f) `x, converse (g) `y>" 
   223        in lam_bijective)
   224 apply (typecheck add: bij_is_inj inj_is_fun) 
   225 apply (auto simp add: left_inverse_bij right_inverse_bij) 
   226 done
   227 
   228 lemma prod_ord_iso_cong: 
   229     "[| f: ord_iso(A,r,A',r');  g: ord_iso(B,s,B',s') |]      
   230      ==> (lam <x,y>:A*B. <f`x, g`y>)                                  
   231          : ord_iso(A*B, rmult(A,r,B,s), A'*B', rmult(A',r',B',s'))"
   232 apply (unfold ord_iso_def)
   233 apply (safe intro!: prod_bij)
   234 apply (simp_all add: bij_is_fun [THEN apply_type])
   235 apply (blast intro: bij_is_inj [THEN inj_apply_equality])
   236 done
   237 
   238 lemma singleton_prod_bij: "(lam z:A. <x,z>) : bij(A, {x}*A)"
   239 by (rule_tac d = snd in lam_bijective, auto)
   240 
   241 (*Used??*)
   242 lemma singleton_prod_ord_iso:
   243      "well_ord({x},xr) ==>   
   244           (lam z:A. <x,z>) : ord_iso(A, r, {x}*A, rmult({x}, xr, A, r))"
   245 apply (rule singleton_prod_bij [THEN ord_isoI])
   246 apply (simp (no_asm_simp))
   247 apply (blast dest: well_ord_is_wf [THEN wf_on_not_refl])
   248 done
   249 
   250 (*Here we build a complicated function term, then simplify it using
   251   case_cong, id_conv, comp_lam, case_case.*)
   252 lemma prod_sum_singleton_bij:
   253      "a~:C ==>  
   254        (lam x:C*B + D. case(%x. x, %y.<a,y>, x))  
   255        : bij(C*B + D, C*B Un {a}*D)"
   256 apply (rule subst_elem)
   257 apply (rule id_bij [THEN sum_bij, THEN comp_bij])
   258 apply (rule singleton_prod_bij)
   259 apply (rule sum_disjoint_bij, blast)
   260 apply (simp (no_asm_simp) cong add: case_cong)
   261 apply (rule comp_lam [THEN trans, symmetric])
   262 apply (fast elim!: case_type)
   263 apply (simp (no_asm_simp) add: case_case)
   264 done
   265 
   266 lemma prod_sum_singleton_ord_iso:
   267  "[| a:A;  well_ord(A,r) |] ==>  
   268     (lam x:pred(A,a,r)*B + pred(B,b,s). case(%x. x, %y.<a,y>, x))  
   269     : ord_iso(pred(A,a,r)*B + pred(B,b,s),               
   270                   radd(A*B, rmult(A,r,B,s), B, s),       
   271               pred(A,a,r)*B Un {a}*pred(B,b,s), rmult(A,r,B,s))"
   272 apply (rule prod_sum_singleton_bij [THEN ord_isoI])
   273 apply (simp (no_asm_simp) add: pred_iff well_ord_is_wf [THEN wf_on_not_refl])
   274 apply (auto elim!: well_ord_is_wf [THEN wf_on_asym] predE)
   275 done
   276 
   277 subsubsection{*Distributive law*}
   278 
   279 lemma sum_prod_distrib_bij:
   280      "(lam <x,z>:(A+B)*C. case(%y. Inl(<y,z>), %y. Inr(<y,z>), x))  
   281       : bij((A+B)*C, (A*C)+(B*C))"
   282 by (rule_tac d = "case (%<x,y>.<Inl (x),y>, %<x,y>.<Inr (x),y>) " 
   283     in lam_bijective, auto)
   284 
   285 lemma sum_prod_distrib_ord_iso:
   286  "(lam <x,z>:(A+B)*C. case(%y. Inl(<y,z>), %y. Inr(<y,z>), x))  
   287   : ord_iso((A+B)*C, rmult(A+B, radd(A,r,B,s), C, t),  
   288             (A*C)+(B*C), radd(A*C, rmult(A,r,C,t), B*C, rmult(B,s,C,t)))"
   289 by (rule sum_prod_distrib_bij [THEN ord_isoI], auto)
   290 
   291 subsubsection{*Associativity*}
   292 
   293 lemma prod_assoc_bij:
   294      "(lam <<x,y>, z>:(A*B)*C. <x,<y,z>>) : bij((A*B)*C, A*(B*C))"
   295 by (rule_tac d = "%<x, <y,z>>. <<x,y>, z>" in lam_bijective, auto)
   296 
   297 lemma prod_assoc_ord_iso:
   298  "(lam <<x,y>, z>:(A*B)*C. <x,<y,z>>)                    
   299   : ord_iso((A*B)*C, rmult(A*B, rmult(A,r,B,s), C, t),   
   300             A*(B*C), rmult(A, r, B*C, rmult(B,s,C,t)))"
   301 by (rule prod_assoc_bij [THEN ord_isoI], auto)
   302 
   303 subsection{*Inverse Image of a Relation*}
   304 
   305 subsubsection{*Rewrite rule*}
   306 
   307 lemma rvimage_iff: "<a,b> : rvimage(A,f,r)  <->  <f`a,f`b>: r & a:A & b:A"
   308 by (unfold rvimage_def, blast)
   309 
   310 subsubsection{*Type checking*}
   311 
   312 lemma rvimage_type: "rvimage(A,f,r) <= A*A"
   313 by (unfold rvimage_def, rule Collect_subset)
   314 
   315 lemmas field_rvimage = rvimage_type [THEN field_rel_subset]
   316 
   317 lemma rvimage_converse: "rvimage(A,f, converse(r)) = converse(rvimage(A,f,r))"
   318 by (unfold rvimage_def, blast)
   319 
   320 
   321 subsubsection{*Partial Ordering Properties*}
   322 
   323 lemma irrefl_rvimage: 
   324     "[| f: inj(A,B);  irrefl(B,r) |] ==> irrefl(A, rvimage(A,f,r))"
   325 apply (unfold irrefl_def rvimage_def)
   326 apply (blast intro: inj_is_fun [THEN apply_type])
   327 done
   328 
   329 lemma trans_on_rvimage: 
   330     "[| f: inj(A,B);  trans[B](r) |] ==> trans[A](rvimage(A,f,r))"
   331 apply (unfold trans_on_def rvimage_def)
   332 apply (blast intro: inj_is_fun [THEN apply_type])
   333 done
   334 
   335 lemma part_ord_rvimage: 
   336     "[| f: inj(A,B);  part_ord(B,r) |] ==> part_ord(A, rvimage(A,f,r))"
   337 apply (unfold part_ord_def)
   338 apply (blast intro!: irrefl_rvimage trans_on_rvimage)
   339 done
   340 
   341 subsubsection{*Linearity*}
   342 
   343 lemma linear_rvimage:
   344     "[| f: inj(A,B);  linear(B,r) |] ==> linear(A,rvimage(A,f,r))"
   345 apply (simp add: inj_def linear_def rvimage_iff) 
   346 apply (blast intro: apply_funtype) 
   347 done
   348 
   349 lemma tot_ord_rvimage: 
   350     "[| f: inj(A,B);  tot_ord(B,r) |] ==> tot_ord(A, rvimage(A,f,r))"
   351 apply (unfold tot_ord_def)
   352 apply (blast intro!: part_ord_rvimage linear_rvimage)
   353 done
   354 
   355 
   356 subsubsection{*Well-foundedness*}
   357 
   358 lemma wf_rvimage [intro!]: "wf(r) ==> wf(rvimage(A,f,r))"
   359 apply (simp (no_asm_use) add: rvimage_def wf_eq_minimal)
   360 apply clarify
   361 apply (subgoal_tac "EX w. w : {w: {f`x. x:Q}. EX x. x: Q & (f`x = w) }")
   362  apply (erule allE)
   363  apply (erule impE)
   364  apply assumption
   365  apply blast
   366 apply blast 
   367 done
   368 
   369 text{*But note that the combination of @{text wf_imp_wf_on} and
   370  @{text wf_rvimage} gives @{term "wf(r) ==> wf[C](rvimage(A,f,r))"}*}
   371 lemma wf_on_rvimage: "[| f: A->B;  wf[B](r) |] ==> wf[A](rvimage(A,f,r))"
   372 apply (rule wf_onI2)
   373 apply (subgoal_tac "ALL z:A. f`z=f`y --> z: Ba")
   374  apply blast
   375 apply (erule_tac a = "f`y" in wf_on_induct)
   376  apply (blast intro!: apply_funtype)
   377 apply (blast intro!: apply_funtype dest!: rvimage_iff [THEN iffD1])
   378 done
   379 
   380 (*Note that we need only wf[A](...) and linear(A,...) to get the result!*)
   381 lemma well_ord_rvimage:
   382      "[| f: inj(A,B);  well_ord(B,r) |] ==> well_ord(A, rvimage(A,f,r))"
   383 apply (rule well_ordI)
   384 apply (unfold well_ord_def tot_ord_def)
   385 apply (blast intro!: wf_on_rvimage inj_is_fun)
   386 apply (blast intro!: linear_rvimage)
   387 done
   388 
   389 lemma ord_iso_rvimage: 
   390     "f: bij(A,B) ==> f: ord_iso(A, rvimage(A,f,s), B, s)"
   391 apply (unfold ord_iso_def)
   392 apply (simp add: rvimage_iff)
   393 done
   394 
   395 lemma ord_iso_rvimage_eq: 
   396     "f: ord_iso(A,r, B,s) ==> rvimage(A,f,s) = r Int A*A"
   397 by (unfold ord_iso_def rvimage_def, blast)
   398 
   399 
   400 subsection{*Every well-founded relation is a subset of some inverse image of
   401       an ordinal*}
   402 
   403 lemma wf_rvimage_Ord: "Ord(i) \<Longrightarrow> wf(rvimage(A, f, Memrel(i)))"
   404 by (blast intro: wf_rvimage wf_Memrel)
   405 
   406 
   407 constdefs
   408   wfrank :: "[i,i]=>i"
   409     "wfrank(r,a) == wfrec(r, a, %x f. \<Union>y \<in> r-``{x}. succ(f`y))"
   410 
   411 constdefs
   412   wftype :: "i=>i"
   413     "wftype(r) == \<Union>y \<in> range(r). succ(wfrank(r,y))"
   414 
   415 lemma wfrank: "wf(r) ==> wfrank(r,a) = (\<Union>y \<in> r-``{a}. succ(wfrank(r,y)))"
   416 by (subst wfrank_def [THEN def_wfrec], simp_all)
   417 
   418 lemma Ord_wfrank: "wf(r) ==> Ord(wfrank(r,a))"
   419 apply (rule_tac a=a in wf_induct, assumption)
   420 apply (subst wfrank, assumption)
   421 apply (rule Ord_succ [THEN Ord_UN], blast)
   422 done
   423 
   424 lemma wfrank_lt: "[|wf(r); <a,b> \<in> r|] ==> wfrank(r,a) < wfrank(r,b)"
   425 apply (rule_tac a1 = b in wfrank [THEN ssubst], assumption)
   426 apply (rule UN_I [THEN ltI])
   427 apply (simp add: Ord_wfrank vimage_iff)+
   428 done
   429 
   430 lemma Ord_wftype: "wf(r) ==> Ord(wftype(r))"
   431 by (simp add: wftype_def Ord_wfrank)
   432 
   433 lemma wftypeI: "\<lbrakk>wf(r);  x \<in> field(r)\<rbrakk> \<Longrightarrow> wfrank(r,x) \<in> wftype(r)"
   434 apply (simp add: wftype_def)
   435 apply (blast intro: wfrank_lt [THEN ltD])
   436 done
   437 
   438 
   439 lemma wf_imp_subset_rvimage:
   440      "[|wf(r); r \<subseteq> A*A|] ==> \<exists>i f. Ord(i) & r <= rvimage(A, f, Memrel(i))"
   441 apply (rule_tac x="wftype(r)" in exI)
   442 apply (rule_tac x="\<lambda>x\<in>A. wfrank(r,x)" in exI)
   443 apply (simp add: Ord_wftype, clarify)
   444 apply (frule subsetD, assumption, clarify)
   445 apply (simp add: rvimage_iff wfrank_lt [THEN ltD])
   446 apply (blast intro: wftypeI)
   447 done
   448 
   449 theorem wf_iff_subset_rvimage:
   450   "relation(r) ==> wf(r) <-> (\<exists>i f A. Ord(i) & r <= rvimage(A, f, Memrel(i)))"
   451 by (blast dest!: relation_field_times_field wf_imp_subset_rvimage
   452           intro: wf_rvimage_Ord [THEN wf_subset])
   453 
   454 
   455 subsection{*Other Results*}
   456 
   457 lemma wf_times: "A Int B = 0 ==> wf(A*B)"
   458 by (simp add: wf_def, blast)
   459 
   460 text{*Could also be used to prove @{text wf_radd}*}
   461 lemma wf_Un:
   462      "[| range(r) Int domain(s) = 0; wf(r);  wf(s) |] ==> wf(r Un s)"
   463 apply (simp add: wf_def, clarify) 
   464 apply (rule equalityI) 
   465  prefer 2 apply blast 
   466 apply clarify 
   467 apply (drule_tac x=Z in spec)
   468 apply (drule_tac x="Z Int domain(s)" in spec)
   469 apply simp 
   470 apply (blast intro: elim: equalityE) 
   471 done
   472 
   473 subsubsection{*The Empty Relation*}
   474 
   475 lemma wf0: "wf(0)"
   476 by (simp add: wf_def, blast)
   477 
   478 lemma linear0: "linear(0,0)"
   479 by (simp add: linear_def)
   480 
   481 lemma well_ord0: "well_ord(0,0)"
   482 by (blast intro: wf_imp_wf_on well_ordI wf0 linear0)
   483 
   484 subsubsection{*The "measure" relation is useful with wfrec*}
   485 
   486 lemma measure_eq_rvimage_Memrel:
   487      "measure(A,f) = rvimage(A,Lambda(A,f),Memrel(Collect(RepFun(A,f),Ord)))"
   488 apply (simp (no_asm) add: measure_def rvimage_def Memrel_iff)
   489 apply (rule equalityI, auto)
   490 apply (auto intro: Ord_in_Ord simp add: lt_def)
   491 done
   492 
   493 lemma wf_measure [iff]: "wf(measure(A,f))"
   494 by (simp (no_asm) add: measure_eq_rvimage_Memrel wf_Memrel wf_rvimage)
   495 
   496 lemma measure_iff [iff]: "<x,y> : measure(A,f) <-> x:A & y:A & f(x)<f(y)"
   497 by (simp (no_asm) add: measure_def)
   498 
   499 lemma linear_measure: 
   500  assumes Ordf: "!!x. x \<in> A ==> Ord(f(x))"
   501      and inj:  "!!x y. [|x \<in> A; y \<in> A; f(x) = f(y) |] ==> x=y"
   502  shows "linear(A, measure(A,f))"
   503 apply (auto simp add: linear_def) 
   504 apply (rule_tac i="f(x)" and j="f(y)" in Ord_linear_lt) 
   505     apply (simp_all add: Ordf) 
   506 apply (blast intro: inj) 
   507 done
   508 
   509 lemma wf_on_measure: "wf[B](measure(A,f))"
   510 by (rule wf_imp_wf_on [OF wf_measure])
   511 
   512 lemma well_ord_measure: 
   513  assumes Ordf: "!!x. x \<in> A ==> Ord(f(x))"
   514      and inj:  "!!x y. [|x \<in> A; y \<in> A; f(x) = f(y) |] ==> x=y"
   515  shows "well_ord(A, measure(A,f))"
   516 apply (rule well_ordI)
   517 apply (rule wf_on_measure) 
   518 apply (blast intro: linear_measure Ordf inj) 
   519 done
   520 
   521 lemma measure_type: "measure(A,f) <= A*A"
   522 by (auto simp add: measure_def)
   523 
   524 subsubsection{*Well-foundedness of Unions*}
   525 
   526 lemma wf_on_Union:
   527  assumes wfA: "wf[A](r)"
   528      and wfB: "!!a. a\<in>A ==> wf[B(a)](s)"
   529      and ok: "!!a u v. [|<u,v> \<in> s; v \<in> B(a); a \<in> A|] 
   530                        ==> (\<exists>a'\<in>A. <a',a> \<in> r & u \<in> B(a')) | u \<in> B(a)"
   531  shows "wf[\<Union>a\<in>A. B(a)](s)"
   532 apply (rule wf_onI2)
   533 apply (erule UN_E)
   534 apply (subgoal_tac "\<forall>z \<in> B(a). z \<in> Ba", blast)
   535 apply (rule_tac a = a in wf_on_induct [OF wfA], assumption)
   536 apply (rule ballI)
   537 apply (rule_tac a = z in wf_on_induct [OF wfB], assumption, assumption)
   538 apply (rename_tac u) 
   539 apply (drule_tac x=u in bspec, blast) 
   540 apply (erule mp, clarify)
   541 apply (frule ok, assumption+, blast) 
   542 done
   543 
   544 
   545 ML {*
   546 val measure_def = thm "measure_def";
   547 val radd_Inl_Inr_iff = thm "radd_Inl_Inr_iff";
   548 val radd_Inl_iff = thm "radd_Inl_iff";
   549 val radd_Inr_iff = thm "radd_Inr_iff";
   550 val radd_Inr_Inl_iff = thm "radd_Inr_Inl_iff";
   551 val raddE = thm "raddE";
   552 val radd_type = thm "radd_type";
   553 val field_radd = thm "field_radd";
   554 val linear_radd = thm "linear_radd";
   555 val wf_on_radd = thm "wf_on_radd";
   556 val wf_radd = thm "wf_radd";
   557 val well_ord_radd = thm "well_ord_radd";
   558 val sum_bij = thm "sum_bij";
   559 val sum_ord_iso_cong = thm "sum_ord_iso_cong";
   560 val sum_disjoint_bij = thm "sum_disjoint_bij";
   561 val sum_assoc_bij = thm "sum_assoc_bij";
   562 val sum_assoc_ord_iso = thm "sum_assoc_ord_iso";
   563 val rmult_iff = thm "rmult_iff";
   564 val rmultE = thm "rmultE";
   565 val rmult_type = thm "rmult_type";
   566 val field_rmult = thm "field_rmult";
   567 val linear_rmult = thm "linear_rmult";
   568 val wf_on_rmult = thm "wf_on_rmult";
   569 val wf_rmult = thm "wf_rmult";
   570 val well_ord_rmult = thm "well_ord_rmult";
   571 val prod_bij = thm "prod_bij";
   572 val prod_ord_iso_cong = thm "prod_ord_iso_cong";
   573 val singleton_prod_bij = thm "singleton_prod_bij";
   574 val singleton_prod_ord_iso = thm "singleton_prod_ord_iso";
   575 val prod_sum_singleton_bij = thm "prod_sum_singleton_bij";
   576 val prod_sum_singleton_ord_iso = thm "prod_sum_singleton_ord_iso";
   577 val sum_prod_distrib_bij = thm "sum_prod_distrib_bij";
   578 val sum_prod_distrib_ord_iso = thm "sum_prod_distrib_ord_iso";
   579 val prod_assoc_bij = thm "prod_assoc_bij";
   580 val prod_assoc_ord_iso = thm "prod_assoc_ord_iso";
   581 val rvimage_iff = thm "rvimage_iff";
   582 val rvimage_type = thm "rvimage_type";
   583 val field_rvimage = thm "field_rvimage";
   584 val rvimage_converse = thm "rvimage_converse";
   585 val irrefl_rvimage = thm "irrefl_rvimage";
   586 val trans_on_rvimage = thm "trans_on_rvimage";
   587 val part_ord_rvimage = thm "part_ord_rvimage";
   588 val linear_rvimage = thm "linear_rvimage";
   589 val tot_ord_rvimage = thm "tot_ord_rvimage";
   590 val wf_rvimage = thm "wf_rvimage";
   591 val wf_on_rvimage = thm "wf_on_rvimage";
   592 val well_ord_rvimage = thm "well_ord_rvimage";
   593 val ord_iso_rvimage = thm "ord_iso_rvimage";
   594 val ord_iso_rvimage_eq = thm "ord_iso_rvimage_eq";
   595 val measure_eq_rvimage_Memrel = thm "measure_eq_rvimage_Memrel";
   596 val wf_measure = thm "wf_measure";
   597 val measure_iff = thm "measure_iff";
   598 *}
   599 
   600 end