src/ZF/OrderArith.thy
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```     1 (*  Title:      ZF/OrderArith.thy
```
```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     3     Copyright   1994  University of Cambridge
```
```     4 *)
```
```     5
```
```     6 section{*Combining Orderings: Foundations of Ordinal Arithmetic*}
```
```     7
```
```     8 theory OrderArith imports Order Sum Ordinal begin
```
```     9
```
```    10 definition
```
```    11   (*disjoint sum of two relations; underlies ordinal addition*)
```
```    12   radd    :: "[i,i,i,i]=>i"  where
```
```    13     "radd(A,r,B,s) ==
```
```    14                 {z: (A+B) * (A+B).
```
```    15                     (\<exists>x y. z = <Inl(x), Inr(y)>)   |
```
```    16                     (\<exists>x' x. z = <Inl(x'), Inl(x)> & <x',x>:r)   |
```
```    17                     (\<exists>y' y. z = <Inr(y'), Inr(y)> & <y',y>:s)}"
```
```    18
```
```    19 definition
```
```    20   (*lexicographic product of two relations; underlies ordinal multiplication*)
```
```    21   rmult   :: "[i,i,i,i]=>i"  where
```
```    22     "rmult(A,r,B,s) ==
```
```    23                 {z: (A*B) * (A*B).
```
```    24                     \<exists>x' y' x y. z = <<x',y'>, <x,y>> &
```
```    25                        (<x',x>: r | (x'=x & <y',y>: s))}"
```
```    26
```
```    27 definition
```
```    28   (*inverse image of a relation*)
```
```    29   rvimage :: "[i,i,i]=>i"  where
```
```    30     "rvimage(A,f,r) == {z \<in> A*A. \<exists>x y. z = <x,y> & <f`x,f`y>: r}"
```
```    31
```
```    32 definition
```
```    33   measure :: "[i, i\<Rightarrow>i] \<Rightarrow> i"  where
```
```    34     "measure(A,f) == {<x,y>: A*A. f(x) < f(y)}"
```
```    35
```
```    36
```
```    37 subsection{*Addition of Relations -- Disjoint Sum*}
```
```    38
```
```    39 subsubsection{*Rewrite rules.  Can be used to obtain introduction rules*}
```
```    40
```
```    41 lemma radd_Inl_Inr_iff [iff]:
```
```    42     "<Inl(a), Inr(b)> \<in> radd(A,r,B,s)  \<longleftrightarrow>  a \<in> A & b \<in> B"
```
```    43 by (unfold radd_def, blast)
```
```    44
```
```    45 lemma radd_Inl_iff [iff]:
```
```    46     "<Inl(a'), Inl(a)> \<in> radd(A,r,B,s)  \<longleftrightarrow>  a':A & a \<in> A & <a',a>:r"
```
```    47 by (unfold radd_def, blast)
```
```    48
```
```    49 lemma radd_Inr_iff [iff]:
```
```    50     "<Inr(b'), Inr(b)> \<in> radd(A,r,B,s) \<longleftrightarrow>  b':B & b \<in> B & <b',b>:s"
```
```    51 by (unfold radd_def, blast)
```
```    52
```
```    53 lemma radd_Inr_Inl_iff [simp]:
```
```    54     "<Inr(b), Inl(a)> \<in> radd(A,r,B,s) \<longleftrightarrow> False"
```
```    55 by (unfold radd_def, blast)
```
```    56
```
```    57 declare radd_Inr_Inl_iff [THEN iffD1, dest!]
```
```    58
```
```    59 subsubsection{*Elimination Rule*}
```
```    60
```
```    61 lemma raddE:
```
```    62     "[| <p',p> \<in> radd(A,r,B,s);
```
```    63         !!x y. [| p'=Inl(x); x \<in> A; p=Inr(y); y \<in> B |] ==> Q;
```
```    64         !!x' x. [| p'=Inl(x'); p=Inl(x); <x',x>: r; x':A; x \<in> A |] ==> Q;
```
```    65         !!y' y. [| p'=Inr(y'); p=Inr(y); <y',y>: s; y':B; y \<in> B |] ==> Q
```
```    66      |] ==> Q"
```
```    67 by (unfold radd_def, blast)
```
```    68
```
```    69 subsubsection{*Type checking*}
```
```    70
```
```    71 lemma radd_type: "radd(A,r,B,s) \<subseteq> (A+B) * (A+B)"
```
```    72 apply (unfold radd_def)
```
```    73 apply (rule Collect_subset)
```
```    74 done
```
```    75
```
```    76 lemmas field_radd = radd_type [THEN field_rel_subset]
```
```    77
```
```    78 subsubsection{*Linearity*}
```
```    79
```
```    80 lemma linear_radd:
```
```    81     "[| linear(A,r);  linear(B,s) |] ==> linear(A+B,radd(A,r,B,s))"
```
```    82 by (unfold linear_def, blast)
```
```    83
```
```    84
```
```    85 subsubsection{*Well-foundedness*}
```
```    86
```
```    87 lemma wf_on_radd: "[| wf[A](r);  wf[B](s) |] ==> wf[A+B](radd(A,r,B,s))"
```
```    88 apply (rule wf_onI2)
```
```    89 apply (subgoal_tac "\<forall>x\<in>A. Inl (x) \<in> Ba")
```
```    90  --{*Proving the lemma, which is needed twice!*}
```
```    91  prefer 2
```
```    92  apply (erule_tac V = "y \<in> A + B" in thin_rl)
```
```    93  apply (rule_tac ballI)
```
```    94  apply (erule_tac r = r and a = x in wf_on_induct, assumption)
```
```    95  apply blast
```
```    96 txt{*Returning to main part of proof*}
```
```    97 apply safe
```
```    98 apply blast
```
```    99 apply (erule_tac r = s and a = ya in wf_on_induct, assumption, blast)
```
```   100 done
```
```   101
```
```   102 lemma wf_radd: "[| wf(r);  wf(s) |] ==> wf(radd(field(r),r,field(s),s))"
```
```   103 apply (simp add: wf_iff_wf_on_field)
```
```   104 apply (rule wf_on_subset_A [OF _ field_radd])
```
```   105 apply (blast intro: wf_on_radd)
```
```   106 done
```
```   107
```
```   108 lemma well_ord_radd:
```
```   109      "[| well_ord(A,r);  well_ord(B,s) |] ==> well_ord(A+B, radd(A,r,B,s))"
```
```   110 apply (rule well_ordI)
```
```   111 apply (simp add: well_ord_def wf_on_radd)
```
```   112 apply (simp add: well_ord_def tot_ord_def linear_radd)
```
```   113 done
```
```   114
```
```   115 subsubsection{*An @{term ord_iso} congruence law*}
```
```   116
```
```   117 lemma sum_bij:
```
```   118      "[| f \<in> bij(A,C);  g \<in> bij(B,D) |]
```
```   119       ==> (\<lambda>z\<in>A+B. case(%x. Inl(f`x), %y. Inr(g`y), z)) \<in> bij(A+B, C+D)"
```
```   120 apply (rule_tac d = "case (%x. Inl (converse(f)`x), %y. Inr(converse(g)`y))"
```
```   121        in lam_bijective)
```
```   122 apply (typecheck add: bij_is_inj inj_is_fun)
```
```   123 apply (auto simp add: left_inverse_bij right_inverse_bij)
```
```   124 done
```
```   125
```
```   126 lemma sum_ord_iso_cong:
```
```   127     "[| f \<in> ord_iso(A,r,A',r');  g \<in> ord_iso(B,s,B',s') |] ==>
```
```   128             (\<lambda>z\<in>A+B. case(%x. Inl(f`x), %y. Inr(g`y), z))
```
```   129             \<in> ord_iso(A+B, radd(A,r,B,s), A'+B', radd(A',r',B',s'))"
```
```   130 apply (unfold ord_iso_def)
```
```   131 apply (safe intro!: sum_bij)
```
```   132 (*Do the beta-reductions now*)
```
```   133 apply (auto cong add: conj_cong simp add: bij_is_fun [THEN apply_type])
```
```   134 done
```
```   135
```
```   136 (*Could we prove an ord_iso result?  Perhaps
```
```   137      ord_iso(A+B, radd(A,r,B,s), A \<union> B, r \<union> s) *)
```
```   138 lemma sum_disjoint_bij: "A \<inter> B = 0 ==>
```
```   139             (\<lambda>z\<in>A+B. case(%x. x, %y. y, z)) \<in> bij(A+B, A \<union> B)"
```
```   140 apply (rule_tac d = "%z. if z \<in> A then Inl (z) else Inr (z) " in lam_bijective)
```
```   141 apply auto
```
```   142 done
```
```   143
```
```   144 subsubsection{*Associativity*}
```
```   145
```
```   146 lemma sum_assoc_bij:
```
```   147      "(\<lambda>z\<in>(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z))
```
```   148       \<in> bij((A+B)+C, A+(B+C))"
```
```   149 apply (rule_tac d = "case (%x. Inl (Inl (x)), case (%x. Inl (Inr (x)), Inr))"
```
```   150        in lam_bijective)
```
```   151 apply auto
```
```   152 done
```
```   153
```
```   154 lemma sum_assoc_ord_iso:
```
```   155      "(\<lambda>z\<in>(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z))
```
```   156       \<in> ord_iso((A+B)+C, radd(A+B, radd(A,r,B,s), C, t),
```
```   157                 A+(B+C), radd(A, r, B+C, radd(B,s,C,t)))"
```
```   158 by (rule sum_assoc_bij [THEN ord_isoI], auto)
```
```   159
```
```   160
```
```   161 subsection{*Multiplication of Relations -- Lexicographic Product*}
```
```   162
```
```   163 subsubsection{*Rewrite rule.  Can be used to obtain introduction rules*}
```
```   164
```
```   165 lemma  rmult_iff [iff]:
```
```   166     "<<a',b'>, <a,b>> \<in> rmult(A,r,B,s) \<longleftrightarrow>
```
```   167             (<a',a>: r  & a':A & a \<in> A & b': B & b \<in> B) |
```
```   168             (<b',b>: s  & a'=a & a \<in> A & b': B & b \<in> B)"
```
```   169
```
```   170 by (unfold rmult_def, blast)
```
```   171
```
```   172 lemma rmultE:
```
```   173     "[| <<a',b'>, <a,b>> \<in> rmult(A,r,B,s);
```
```   174         [| <a',a>: r;  a':A;  a \<in> A;  b':B;  b \<in> B |] ==> Q;
```
```   175         [| <b',b>: s;  a \<in> A;  a'=a;  b':B;  b \<in> B |] ==> Q
```
```   176      |] ==> Q"
```
```   177 by blast
```
```   178
```
```   179 subsubsection{*Type checking*}
```
```   180
```
```   181 lemma rmult_type: "rmult(A,r,B,s) \<subseteq> (A*B) * (A*B)"
```
```   182 by (unfold rmult_def, rule Collect_subset)
```
```   183
```
```   184 lemmas field_rmult = rmult_type [THEN field_rel_subset]
```
```   185
```
```   186 subsubsection{*Linearity*}
```
```   187
```
```   188 lemma linear_rmult:
```
```   189     "[| linear(A,r);  linear(B,s) |] ==> linear(A*B,rmult(A,r,B,s))"
```
```   190 by (simp add: linear_def, blast)
```
```   191
```
```   192 subsubsection{*Well-foundedness*}
```
```   193
```
```   194 lemma wf_on_rmult: "[| wf[A](r);  wf[B](s) |] ==> wf[A*B](rmult(A,r,B,s))"
```
```   195 apply (rule wf_onI2)
```
```   196 apply (erule SigmaE)
```
```   197 apply (erule ssubst)
```
```   198 apply (subgoal_tac "\<forall>b\<in>B. <x,b>: Ba", blast)
```
```   199 apply (erule_tac a = x in wf_on_induct, assumption)
```
```   200 apply (rule ballI)
```
```   201 apply (erule_tac a = b in wf_on_induct, assumption)
```
```   202 apply (best elim!: rmultE bspec [THEN mp])
```
```   203 done
```
```   204
```
```   205
```
```   206 lemma wf_rmult: "[| wf(r);  wf(s) |] ==> wf(rmult(field(r),r,field(s),s))"
```
```   207 apply (simp add: wf_iff_wf_on_field)
```
```   208 apply (rule wf_on_subset_A [OF _ field_rmult])
```
```   209 apply (blast intro: wf_on_rmult)
```
```   210 done
```
```   211
```
```   212 lemma well_ord_rmult:
```
```   213      "[| well_ord(A,r);  well_ord(B,s) |] ==> well_ord(A*B, rmult(A,r,B,s))"
```
```   214 apply (rule well_ordI)
```
```   215 apply (simp add: well_ord_def wf_on_rmult)
```
```   216 apply (simp add: well_ord_def tot_ord_def linear_rmult)
```
```   217 done
```
```   218
```
```   219
```
```   220 subsubsection{*An @{term ord_iso} congruence law*}
```
```   221
```
```   222 lemma prod_bij:
```
```   223      "[| f \<in> bij(A,C);  g \<in> bij(B,D) |]
```
```   224       ==> (lam <x,y>:A*B. <f`x, g`y>) \<in> bij(A*B, C*D)"
```
```   225 apply (rule_tac d = "%<x,y>. <converse (f) `x, converse (g) `y>"
```
```   226        in lam_bijective)
```
```   227 apply (typecheck add: bij_is_inj inj_is_fun)
```
```   228 apply (auto simp add: left_inverse_bij right_inverse_bij)
```
```   229 done
```
```   230
```
```   231 lemma prod_ord_iso_cong:
```
```   232     "[| f \<in> ord_iso(A,r,A',r');  g \<in> ord_iso(B,s,B',s') |]
```
```   233      ==> (lam <x,y>:A*B. <f`x, g`y>)
```
```   234          \<in> ord_iso(A*B, rmult(A,r,B,s), A'*B', rmult(A',r',B',s'))"
```
```   235 apply (unfold ord_iso_def)
```
```   236 apply (safe intro!: prod_bij)
```
```   237 apply (simp_all add: bij_is_fun [THEN apply_type])
```
```   238 apply (blast intro: bij_is_inj [THEN inj_apply_equality])
```
```   239 done
```
```   240
```
```   241 lemma singleton_prod_bij: "(\<lambda>z\<in>A. <x,z>) \<in> bij(A, {x}*A)"
```
```   242 by (rule_tac d = snd in lam_bijective, auto)
```
```   243
```
```   244 (*Used??*)
```
```   245 lemma singleton_prod_ord_iso:
```
```   246      "well_ord({x},xr) ==>
```
```   247           (\<lambda>z\<in>A. <x,z>) \<in> ord_iso(A, r, {x}*A, rmult({x}, xr, A, r))"
```
```   248 apply (rule singleton_prod_bij [THEN ord_isoI])
```
```   249 apply (simp (no_asm_simp))
```
```   250 apply (blast dest: well_ord_is_wf [THEN wf_on_not_refl])
```
```   251 done
```
```   252
```
```   253 (*Here we build a complicated function term, then simplify it using
```
```   254   case_cong, id_conv, comp_lam, case_case.*)
```
```   255 lemma prod_sum_singleton_bij:
```
```   256      "a\<notin>C ==>
```
```   257        (\<lambda>x\<in>C*B + D. case(%x. x, %y.<a,y>, x))
```
```   258        \<in> bij(C*B + D, C*B \<union> {a}*D)"
```
```   259 apply (rule subst_elem)
```
```   260 apply (rule id_bij [THEN sum_bij, THEN comp_bij])
```
```   261 apply (rule singleton_prod_bij)
```
```   262 apply (rule sum_disjoint_bij, blast)
```
```   263 apply (simp (no_asm_simp) cong add: case_cong)
```
```   264 apply (rule comp_lam [THEN trans, symmetric])
```
```   265 apply (fast elim!: case_type)
```
```   266 apply (simp (no_asm_simp) add: case_case)
```
```   267 done
```
```   268
```
```   269 lemma prod_sum_singleton_ord_iso:
```
```   270  "[| a \<in> A;  well_ord(A,r) |] ==>
```
```   271     (\<lambda>x\<in>pred(A,a,r)*B + pred(B,b,s). case(%x. x, %y.<a,y>, x))
```
```   272     \<in> ord_iso(pred(A,a,r)*B + pred(B,b,s),
```
```   273                   radd(A*B, rmult(A,r,B,s), B, s),
```
```   274               pred(A,a,r)*B \<union> {a}*pred(B,b,s), rmult(A,r,B,s))"
```
```   275 apply (rule prod_sum_singleton_bij [THEN ord_isoI])
```
```   276 apply (simp (no_asm_simp) add: pred_iff well_ord_is_wf [THEN wf_on_not_refl])
```
```   277 apply (auto elim!: well_ord_is_wf [THEN wf_on_asym] predE)
```
```   278 done
```
```   279
```
```   280 subsubsection{*Distributive law*}
```
```   281
```
```   282 lemma sum_prod_distrib_bij:
```
```   283      "(lam <x,z>:(A+B)*C. case(%y. Inl(<y,z>), %y. Inr(<y,z>), x))
```
```   284       \<in> bij((A+B)*C, (A*C)+(B*C))"
```
```   285 by (rule_tac d = "case (%<x,y>.<Inl (x),y>, %<x,y>.<Inr (x),y>) "
```
```   286     in lam_bijective, auto)
```
```   287
```
```   288 lemma sum_prod_distrib_ord_iso:
```
```   289  "(lam <x,z>:(A+B)*C. case(%y. Inl(<y,z>), %y. Inr(<y,z>), x))
```
```   290   \<in> ord_iso((A+B)*C, rmult(A+B, radd(A,r,B,s), C, t),
```
```   291             (A*C)+(B*C), radd(A*C, rmult(A,r,C,t), B*C, rmult(B,s,C,t)))"
```
```   292 by (rule sum_prod_distrib_bij [THEN ord_isoI], auto)
```
```   293
```
```   294 subsubsection{*Associativity*}
```
```   295
```
```   296 lemma prod_assoc_bij:
```
```   297      "(lam <<x,y>, z>:(A*B)*C. <x,<y,z>>) \<in> bij((A*B)*C, A*(B*C))"
```
```   298 by (rule_tac d = "%<x, <y,z>>. <<x,y>, z>" in lam_bijective, auto)
```
```   299
```
```   300 lemma prod_assoc_ord_iso:
```
```   301  "(lam <<x,y>, z>:(A*B)*C. <x,<y,z>>)
```
```   302   \<in> ord_iso((A*B)*C, rmult(A*B, rmult(A,r,B,s), C, t),
```
```   303             A*(B*C), rmult(A, r, B*C, rmult(B,s,C,t)))"
```
```   304 by (rule prod_assoc_bij [THEN ord_isoI], auto)
```
```   305
```
```   306 subsection{*Inverse Image of a Relation*}
```
```   307
```
```   308 subsubsection{*Rewrite rule*}
```
```   309
```
```   310 lemma rvimage_iff: "<a,b> \<in> rvimage(A,f,r)  \<longleftrightarrow>  <f`a,f`b>: r & a \<in> A & b \<in> A"
```
```   311 by (unfold rvimage_def, blast)
```
```   312
```
```   313 subsubsection{*Type checking*}
```
```   314
```
```   315 lemma rvimage_type: "rvimage(A,f,r) \<subseteq> A*A"
```
```   316 by (unfold rvimage_def, rule Collect_subset)
```
```   317
```
```   318 lemmas field_rvimage = rvimage_type [THEN field_rel_subset]
```
```   319
```
```   320 lemma rvimage_converse: "rvimage(A,f, converse(r)) = converse(rvimage(A,f,r))"
```
```   321 by (unfold rvimage_def, blast)
```
```   322
```
```   323
```
```   324 subsubsection{*Partial Ordering Properties*}
```
```   325
```
```   326 lemma irrefl_rvimage:
```
```   327     "[| f \<in> inj(A,B);  irrefl(B,r) |] ==> irrefl(A, rvimage(A,f,r))"
```
```   328 apply (unfold irrefl_def rvimage_def)
```
```   329 apply (blast intro: inj_is_fun [THEN apply_type])
```
```   330 done
```
```   331
```
```   332 lemma trans_on_rvimage:
```
```   333     "[| f \<in> inj(A,B);  trans[B](r) |] ==> trans[A](rvimage(A,f,r))"
```
```   334 apply (unfold trans_on_def rvimage_def)
```
```   335 apply (blast intro: inj_is_fun [THEN apply_type])
```
```   336 done
```
```   337
```
```   338 lemma part_ord_rvimage:
```
```   339     "[| f \<in> inj(A,B);  part_ord(B,r) |] ==> part_ord(A, rvimage(A,f,r))"
```
```   340 apply (unfold part_ord_def)
```
```   341 apply (blast intro!: irrefl_rvimage trans_on_rvimage)
```
```   342 done
```
```   343
```
```   344 subsubsection{*Linearity*}
```
```   345
```
```   346 lemma linear_rvimage:
```
```   347     "[| f \<in> inj(A,B);  linear(B,r) |] ==> linear(A,rvimage(A,f,r))"
```
```   348 apply (simp add: inj_def linear_def rvimage_iff)
```
```   349 apply (blast intro: apply_funtype)
```
```   350 done
```
```   351
```
```   352 lemma tot_ord_rvimage:
```
```   353     "[| f \<in> inj(A,B);  tot_ord(B,r) |] ==> tot_ord(A, rvimage(A,f,r))"
```
```   354 apply (unfold tot_ord_def)
```
```   355 apply (blast intro!: part_ord_rvimage linear_rvimage)
```
```   356 done
```
```   357
```
```   358
```
```   359 subsubsection{*Well-foundedness*}
```
```   360
```
```   361 lemma wf_rvimage [intro!]: "wf(r) ==> wf(rvimage(A,f,r))"
```
```   362 apply (simp (no_asm_use) add: rvimage_def wf_eq_minimal)
```
```   363 apply clarify
```
```   364 apply (subgoal_tac "\<exists>w. w \<in> {w: {f`x. x \<in> Q}. \<exists>x. x \<in> Q & (f`x = w) }")
```
```   365  apply (erule allE)
```
```   366  apply (erule impE)
```
```   367  apply assumption
```
```   368  apply blast
```
```   369 apply blast
```
```   370 done
```
```   371
```
```   372 text{*But note that the combination of @{text wf_imp_wf_on} and
```
```   373  @{text wf_rvimage} gives @{prop "wf(r) ==> wf[C](rvimage(A,f,r))"}*}
```
```   374 lemma wf_on_rvimage: "[| f \<in> A->B;  wf[B](r) |] ==> wf[A](rvimage(A,f,r))"
```
```   375 apply (rule wf_onI2)
```
```   376 apply (subgoal_tac "\<forall>z\<in>A. f`z=f`y \<longrightarrow> z \<in> Ba")
```
```   377  apply blast
```
```   378 apply (erule_tac a = "f`y" in wf_on_induct)
```
```   379  apply (blast intro!: apply_funtype)
```
```   380 apply (blast intro!: apply_funtype dest!: rvimage_iff [THEN iffD1])
```
```   381 done
```
```   382
```
```   383 (*Note that we need only wf[A](...) and linear(A,...) to get the result!*)
```
```   384 lemma well_ord_rvimage:
```
```   385      "[| f \<in> inj(A,B);  well_ord(B,r) |] ==> well_ord(A, rvimage(A,f,r))"
```
```   386 apply (rule well_ordI)
```
```   387 apply (unfold well_ord_def tot_ord_def)
```
```   388 apply (blast intro!: wf_on_rvimage inj_is_fun)
```
```   389 apply (blast intro!: linear_rvimage)
```
```   390 done
```
```   391
```
```   392 lemma ord_iso_rvimage:
```
```   393     "f \<in> bij(A,B) ==> f \<in> ord_iso(A, rvimage(A,f,s), B, s)"
```
```   394 apply (unfold ord_iso_def)
```
```   395 apply (simp add: rvimage_iff)
```
```   396 done
```
```   397
```
```   398 lemma ord_iso_rvimage_eq:
```
```   399     "f \<in> ord_iso(A,r, B,s) ==> rvimage(A,f,s) = r \<inter> A*A"
```
```   400 by (unfold ord_iso_def rvimage_def, blast)
```
```   401
```
```   402
```
```   403 subsection{*Every well-founded relation is a subset of some inverse image of
```
```   404       an ordinal*}
```
```   405
```
```   406 lemma wf_rvimage_Ord: "Ord(i) \<Longrightarrow> wf(rvimage(A, f, Memrel(i)))"
```
```   407 by (blast intro: wf_rvimage wf_Memrel)
```
```   408
```
```   409
```
```   410 definition
```
```   411   wfrank :: "[i,i]=>i"  where
```
```   412     "wfrank(r,a) == wfrec(r, a, %x f. \<Union>y \<in> r-``{x}. succ(f`y))"
```
```   413
```
```   414 definition
```
```   415   wftype :: "i=>i"  where
```
```   416     "wftype(r) == \<Union>y \<in> range(r). succ(wfrank(r,y))"
```
```   417
```
```   418 lemma wfrank: "wf(r) ==> wfrank(r,a) = (\<Union>y \<in> r-``{a}. succ(wfrank(r,y)))"
```
```   419 by (subst wfrank_def [THEN def_wfrec], simp_all)
```
```   420
```
```   421 lemma Ord_wfrank: "wf(r) ==> Ord(wfrank(r,a))"
```
```   422 apply (rule_tac a=a in wf_induct, assumption)
```
```   423 apply (subst wfrank, assumption)
```
```   424 apply (rule Ord_succ [THEN Ord_UN], blast)
```
```   425 done
```
```   426
```
```   427 lemma wfrank_lt: "[|wf(r); <a,b> \<in> r|] ==> wfrank(r,a) < wfrank(r,b)"
```
```   428 apply (rule_tac a1 = b in wfrank [THEN ssubst], assumption)
```
```   429 apply (rule UN_I [THEN ltI])
```
```   430 apply (simp add: Ord_wfrank vimage_iff)+
```
```   431 done
```
```   432
```
```   433 lemma Ord_wftype: "wf(r) ==> Ord(wftype(r))"
```
```   434 by (simp add: wftype_def Ord_wfrank)
```
```   435
```
```   436 lemma wftypeI: "\<lbrakk>wf(r);  x \<in> field(r)\<rbrakk> \<Longrightarrow> wfrank(r,x) \<in> wftype(r)"
```
```   437 apply (simp add: wftype_def)
```
```   438 apply (blast intro: wfrank_lt [THEN ltD])
```
```   439 done
```
```   440
```
```   441
```
```   442 lemma wf_imp_subset_rvimage:
```
```   443      "[|wf(r); r \<subseteq> A*A|] ==> \<exists>i f. Ord(i) & r \<subseteq> rvimage(A, f, Memrel(i))"
```
```   444 apply (rule_tac x="wftype(r)" in exI)
```
```   445 apply (rule_tac x="\<lambda>x\<in>A. wfrank(r,x)" in exI)
```
```   446 apply (simp add: Ord_wftype, clarify)
```
```   447 apply (frule subsetD, assumption, clarify)
```
```   448 apply (simp add: rvimage_iff wfrank_lt [THEN ltD])
```
```   449 apply (blast intro: wftypeI)
```
```   450 done
```
```   451
```
```   452 theorem wf_iff_subset_rvimage:
```
```   453   "relation(r) ==> wf(r) \<longleftrightarrow> (\<exists>i f A. Ord(i) & r \<subseteq> rvimage(A, f, Memrel(i)))"
```
```   454 by (blast dest!: relation_field_times_field wf_imp_subset_rvimage
```
```   455           intro: wf_rvimage_Ord [THEN wf_subset])
```
```   456
```
```   457
```
```   458 subsection{*Other Results*}
```
```   459
```
```   460 lemma wf_times: "A \<inter> B = 0 ==> wf(A*B)"
```
```   461 by (simp add: wf_def, blast)
```
```   462
```
```   463 text{*Could also be used to prove @{text wf_radd}*}
```
```   464 lemma wf_Un:
```
```   465      "[| range(r) \<inter> domain(s) = 0; wf(r);  wf(s) |] ==> wf(r \<union> s)"
```
```   466 apply (simp add: wf_def, clarify)
```
```   467 apply (rule equalityI)
```
```   468  prefer 2 apply blast
```
```   469 apply clarify
```
```   470 apply (drule_tac x=Z in spec)
```
```   471 apply (drule_tac x="Z \<inter> domain(s)" in spec)
```
```   472 apply simp
```
```   473 apply (blast intro: elim: equalityE)
```
```   474 done
```
```   475
```
```   476 subsubsection{*The Empty Relation*}
```
```   477
```
```   478 lemma wf0: "wf(0)"
```
```   479 by (simp add: wf_def, blast)
```
```   480
```
```   481 lemma linear0: "linear(0,0)"
```
```   482 by (simp add: linear_def)
```
```   483
```
```   484 lemma well_ord0: "well_ord(0,0)"
```
```   485 by (blast intro: wf_imp_wf_on well_ordI wf0 linear0)
```
```   486
```
```   487 subsubsection{*The "measure" relation is useful with wfrec*}
```
```   488
```
```   489 lemma measure_eq_rvimage_Memrel:
```
```   490      "measure(A,f) = rvimage(A,Lambda(A,f),Memrel(Collect(RepFun(A,f),Ord)))"
```
```   491 apply (simp (no_asm) add: measure_def rvimage_def Memrel_iff)
```
```   492 apply (rule equalityI, auto)
```
```   493 apply (auto intro: Ord_in_Ord simp add: lt_def)
```
```   494 done
```
```   495
```
```   496 lemma wf_measure [iff]: "wf(measure(A,f))"
```
```   497 by (simp (no_asm) add: measure_eq_rvimage_Memrel wf_Memrel wf_rvimage)
```
```   498
```
```   499 lemma measure_iff [iff]: "<x,y> \<in> measure(A,f) \<longleftrightarrow> x \<in> A & y \<in> A & f(x)<f(y)"
```
```   500 by (simp (no_asm) add: measure_def)
```
```   501
```
```   502 lemma linear_measure:
```
```   503  assumes Ordf: "!!x. x \<in> A ==> Ord(f(x))"
```
```   504      and inj:  "!!x y. [|x \<in> A; y \<in> A; f(x) = f(y) |] ==> x=y"
```
```   505  shows "linear(A, measure(A,f))"
```
```   506 apply (auto simp add: linear_def)
```
```   507 apply (rule_tac i="f(x)" and j="f(y)" in Ord_linear_lt)
```
```   508     apply (simp_all add: Ordf)
```
```   509 apply (blast intro: inj)
```
```   510 done
```
```   511
```
```   512 lemma wf_on_measure: "wf[B](measure(A,f))"
```
```   513 by (rule wf_imp_wf_on [OF wf_measure])
```
```   514
```
```   515 lemma well_ord_measure:
```
```   516  assumes Ordf: "!!x. x \<in> A ==> Ord(f(x))"
```
```   517      and inj:  "!!x y. [|x \<in> A; y \<in> A; f(x) = f(y) |] ==> x=y"
```
```   518  shows "well_ord(A, measure(A,f))"
```
```   519 apply (rule well_ordI)
```
```   520 apply (rule wf_on_measure)
```
```   521 apply (blast intro: linear_measure Ordf inj)
```
```   522 done
```
```   523
```
```   524 lemma measure_type: "measure(A,f) \<subseteq> A*A"
```
```   525 by (auto simp add: measure_def)
```
```   526
```
```   527 subsubsection{*Well-foundedness of Unions*}
```
```   528
```
```   529 lemma wf_on_Union:
```
```   530  assumes wfA: "wf[A](r)"
```
```   531      and wfB: "!!a. a\<in>A ==> wf[B(a)](s)"
```
```   532      and ok: "!!a u v. [|<u,v> \<in> s; v \<in> B(a); a \<in> A|]
```
```   533                        ==> (\<exists>a'\<in>A. <a',a> \<in> r & u \<in> B(a')) | u \<in> B(a)"
```
```   534  shows "wf[\<Union>a\<in>A. B(a)](s)"
```
```   535 apply (rule wf_onI2)
```
```   536 apply (erule UN_E)
```
```   537 apply (subgoal_tac "\<forall>z \<in> B(a). z \<in> Ba", blast)
```
```   538 apply (rule_tac a = a in wf_on_induct [OF wfA], assumption)
```
```   539 apply (rule ballI)
```
```   540 apply (rule_tac a = z in wf_on_induct [OF wfB], assumption, assumption)
```
```   541 apply (rename_tac u)
```
```   542 apply (drule_tac x=u in bspec, blast)
```
```   543 apply (erule mp, clarify)
```
```   544 apply (frule ok, assumption+, blast)
```
```   545 done
```
```   546
```
```   547 subsubsection{*Bijections involving Powersets*}
```
```   548
```
```   549 lemma Pow_sum_bij:
```
```   550     "(\<lambda>Z \<in> Pow(A+B). <{x \<in> A. Inl(x) \<in> Z}, {y \<in> B. Inr(y) \<in> Z}>)
```
```   551      \<in> bij(Pow(A+B), Pow(A)*Pow(B))"
```
```   552 apply (rule_tac d = "%<X,Y>. {Inl (x). x \<in> X} \<union> {Inr (y). y \<in> Y}"
```
```   553        in lam_bijective)
```
```   554 apply force+
```
```   555 done
```
```   556
```
```   557 text{*As a special case, we have @{term "bij(Pow(A*B), A -> Pow(B))"} *}
```
```   558 lemma Pow_Sigma_bij:
```
```   559     "(\<lambda>r \<in> Pow(Sigma(A,B)). \<lambda>x \<in> A. r``{x})
```
```   560      \<in> bij(Pow(Sigma(A,B)), \<Pi> x \<in> A. Pow(B(x)))"
```
```   561 apply (rule_tac d = "%f. \<Union>x \<in> A. \<Union>y \<in> f`x. {<x,y>}" in lam_bijective)
```
```   562 apply (blast intro: lam_type)
```
```   563 apply (blast dest: apply_type, simp_all)
```
```   564 apply fast (*strange, but blast can't do it*)
```
```   565 apply (rule fun_extension, auto)
```
```   566 by blast
```
```   567
```
```   568 end
```