src/ZF/OrderArith.thy
 changeset 13140 6d97dbb189a9 parent 9883 c1c8647af477 child 13269 3ba9be497c33
```     1.1 --- a/src/ZF/OrderArith.thy	Sat May 11 20:40:31 2002 +0200
1.2 +++ b/src/ZF/OrderArith.thy	Mon May 13 09:02:13 2002 +0200
1.3 @@ -6,31 +6,507 @@
1.4  Towards ordinal arithmetic.  Also useful with wfrec.
1.5  *)
1.6
1.7 -OrderArith = Order + Sum + Ordinal +
1.8 -consts
1.9 -  radd, rmult      :: [i,i,i,i]=>i
1.10 -  rvimage          :: [i,i,i]=>i
1.11 +theory OrderArith = Order + Sum + Ordinal:
1.12 +constdefs
1.13
1.14 -defs
1.15    (*disjoint sum of two relations; underlies ordinal addition*)
1.19                  {z: (A+B) * (A+B).
1.20                      (EX x y. z = <Inl(x), Inr(y)>)   |
1.21                      (EX x' x. z = <Inl(x'), Inl(x)> & <x',x>:r)   |
1.22                      (EX y' y. z = <Inr(y'), Inr(y)> & <y',y>:s)}"
1.23
1.24    (*lexicographic product of two relations; underlies ordinal multiplication*)
1.25 -  rmult_def "rmult(A,r,B,s) ==
1.26 +  rmult   :: "[i,i,i,i]=>i"
1.27 +    "rmult(A,r,B,s) ==
1.28                  {z: (A*B) * (A*B).
1.29                      EX x' y' x y. z = <<x',y'>, <x,y>> &
1.30                         (<x',x>: r | (x'=x & <y',y>: s))}"
1.31
1.32    (*inverse image of a relation*)
1.33 -  rvimage_def "rvimage(A,f,r) == {z: A*A. EX x y. z = <x,y> & <f`x,f`y>: r}"
1.34 +  rvimage :: "[i,i,i]=>i"
1.35 +    "rvimage(A,f,r) == {z: A*A. EX x y. z = <x,y> & <f`x,f`y>: r}"
1.36 +
1.37 +  measure :: "[i, i\<Rightarrow>i] \<Rightarrow> i"
1.38 +    "measure(A,f) == {<x,y>: A*A. f(x) < f(y)}"
1.39 +
1.40 +
1.41 +(**** Addition of relations -- disjoint sum ****)
1.42 +
1.43 +(** Rewrite rules.  Can be used to obtain introduction rules **)
1.44 +
1.46 +    "<Inl(a), Inr(b)> : radd(A,r,B,s)  <->  a:A & b:B"
1.48 +apply blast
1.49 +done
1.50 +
1.52 +    "<Inl(a'), Inl(a)> : radd(A,r,B,s)  <->  a':A & a:A & <a',a>:r"
1.54 +apply blast
1.55 +done
1.56 +
1.58 +    "<Inr(b'), Inr(b)> : radd(A,r,B,s) <->  b':B & b:B & <b',b>:s"
1.60 +apply blast
1.61 +done
1.62 +
1.64 +    "<Inr(b), Inl(a)> : radd(A,r,B,s) <->  False"
1.66 +apply blast
1.67 +done
1.68 +
1.69 +(** Elimination Rule **)
1.70 +
1.72 +    "[| <p',p> : radd(A,r,B,s);
1.73 +        !!x y. [| p'=Inl(x); x:A; p=Inr(y); y:B |] ==> Q;
1.74 +        !!x' x. [| p'=Inl(x'); p=Inl(x); <x',x>: r; x':A; x:A |] ==> Q;
1.75 +        !!y' y. [| p'=Inr(y'); p=Inr(y); <y',y>: s; y':B; y:B |] ==> Q
1.76 +     |] ==> Q"
1.78 +apply (blast intro: elim:);
1.79 +done
1.80 +
1.81 +(** Type checking **)
1.82 +
1.85 +apply (rule Collect_subset)
1.86 +done
1.87 +
1.89 +
1.90 +(** Linearity **)
1.91 +
1.93 +    "[| linear(A,r);  linear(B,s) |] ==> linear(A+B,radd(A,r,B,s))"
1.94 +apply (unfold linear_def)
1.95 +apply (blast intro: elim:);
1.96 +done
1.97 +
1.98 +
1.99 +(** Well-foundedness **)
1.100 +
1.102 +apply (rule wf_onI2)
1.103 +apply (subgoal_tac "ALL x:A. Inl (x) : Ba")
1.104 +(*Proving the lemma, which is needed twice!*)
1.105 + prefer 2
1.106 + apply (erule_tac V = "y : A + B" in thin_rl)
1.107 + apply (rule_tac ballI)
1.108 + apply (erule_tac r = "r" and a = "x" in wf_on_induct, assumption)
1.109 + apply (blast intro: elim:);
1.110 +(*Returning to main part of proof*)
1.111 +apply safe
1.112 +apply blast
1.113 +apply (erule_tac r = "s" and a = "ya" in wf_on_induct , assumption)
1.114 +apply (blast intro: elim:);
1.115 +done
1.116 +
1.119 +apply (rule wf_on_subset_A [OF _ field_radd])
1.121 +done
1.122 +
1.124 +     "[| well_ord(A,r);  well_ord(B,s) |] ==> well_ord(A+B, radd(A,r,B,s))"
1.125 +apply (rule well_ordI)
1.128 +done
1.129 +
1.130 +(** An ord_iso congruence law **)
1.131
1.132 -constdefs
1.133 -   measure :: "[i, i\\<Rightarrow>i] \\<Rightarrow> i"
1.134 -   "measure(A,f) == {<x,y>: A*A. f(x) < f(y)}"
1.135 +lemma sum_bij:
1.136 +     "[| f: bij(A,C);  g: bij(B,D) |]
1.137 +      ==> (lam z:A+B. case(%x. Inl(f`x), %y. Inr(g`y), z)) : bij(A+B, C+D)"
1.138 +apply (rule_tac d = "case (%x. Inl (converse (f) `x) , %y. Inr (converse (g) `y))" in lam_bijective)
1.139 +apply (typecheck add: bij_is_inj inj_is_fun)
1.140 +apply (auto simp add: left_inverse_bij right_inverse_bij)
1.141 +done
1.142 +
1.143 +lemma sum_ord_iso_cong:
1.144 +    "[| f: ord_iso(A,r,A',r');  g: ord_iso(B,s,B',s') |] ==>
1.145 +            (lam z:A+B. case(%x. Inl(f`x), %y. Inr(g`y), z))
1.147 +apply (unfold ord_iso_def)
1.148 +apply (safe intro!: sum_bij)
1.149 +(*Do the beta-reductions now*)
1.151 +done
1.152 +
1.153 +(*Could we prove an ord_iso result?  Perhaps
1.154 +     ord_iso(A+B, radd(A,r,B,s), A Un B, r Un s) *)
1.155 +lemma sum_disjoint_bij: "A Int B = 0 ==>
1.156 +            (lam z:A+B. case(%x. x, %y. y, z)) : bij(A+B, A Un B)"
1.157 +apply (rule_tac d = "%z. if z:A then Inl (z) else Inr (z) " in lam_bijective)
1.158 +apply auto
1.159 +done
1.160 +
1.161 +(** Associativity **)
1.162 +
1.163 +lemma sum_assoc_bij:
1.164 +     "(lam z:(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z))
1.165 +      : bij((A+B)+C, A+(B+C))"
1.166 +apply (rule_tac d = "case (%x. Inl (Inl (x)), case (%x. Inl (Inr (x)), Inr))"
1.167 +       in lam_bijective)
1.168 +apply auto
1.169 +done
1.170 +
1.171 +lemma sum_assoc_ord_iso:
1.172 +     "(lam z:(A+B)+C. case(case(Inl, %y. Inr(Inl(y))), %y. Inr(Inr(y)), z))
1.175 +apply (rule sum_assoc_bij [THEN ord_isoI])
1.176 +apply auto
1.177 +done
1.178 +
1.179 +
1.180 +(**** Multiplication of relations -- lexicographic product ****)
1.181 +
1.182 +(** Rewrite rule.  Can be used to obtain introduction rules **)
1.183 +
1.184 +lemma  rmult_iff [iff]:
1.185 +    "<<a',b'>, <a,b>> : rmult(A,r,B,s) <->
1.186 +            (<a',a>: r  & a':A & a:A & b': B & b: B) |
1.187 +            (<b',b>: s  & a'=a & a:A & b': B & b: B)"
1.188 +
1.189 +apply (unfold rmult_def)
1.190 +apply blast
1.191 +done
1.192 +
1.193 +lemma rmultE:
1.194 +    "[| <<a',b'>, <a,b>> : rmult(A,r,B,s);
1.195 +        [| <a',a>: r;  a':A;  a:A;  b':B;  b:B |] ==> Q;
1.196 +        [| <b',b>: s;  a:A;  a'=a;  b':B;  b:B |] ==> Q
1.197 +     |] ==> Q"
1.198 +apply (blast intro: elim:);
1.199 +done
1.200 +
1.201 +(** Type checking **)
1.202 +
1.203 +lemma rmult_type: "rmult(A,r,B,s) <= (A*B) * (A*B)"
1.204 +apply (unfold rmult_def)
1.205 +apply (rule Collect_subset)
1.206 +done
1.207 +
1.208 +lemmas field_rmult = rmult_type [THEN field_rel_subset]
1.209 +
1.210 +(** Linearity **)
1.211 +
1.212 +lemma linear_rmult:
1.213 +    "[| linear(A,r);  linear(B,s) |] ==> linear(A*B,rmult(A,r,B,s))"
1.215 +apply (blast intro: elim:);
1.216 +done
1.217 +
1.218 +(** Well-foundedness **)
1.219 +
1.220 +lemma wf_on_rmult: "[| wf[A](r);  wf[B](s) |] ==> wf[A*B](rmult(A,r,B,s))"
1.221 +apply (rule wf_onI2)
1.222 +apply (erule SigmaE)
1.223 +apply (erule ssubst)
1.224 +apply (subgoal_tac "ALL b:B. <x,b>: Ba")
1.225 +apply blast
1.226 +apply (erule_tac a = "x" in wf_on_induct , assumption)
1.227 +apply (rule ballI)
1.228 +apply (erule_tac a = "b" in wf_on_induct , assumption)
1.229 +apply (best elim!: rmultE bspec [THEN mp])
1.230 +done
1.231 +
1.232 +
1.233 +lemma wf_rmult: "[| wf(r);  wf(s) |] ==> wf(rmult(field(r),r,field(s),s))"
1.235 +apply (rule wf_on_subset_A [OF _ field_rmult])
1.236 +apply (blast intro: wf_on_rmult)
1.237 +done
1.238 +
1.239 +lemma well_ord_rmult:
1.240 +     "[| well_ord(A,r);  well_ord(B,s) |] ==> well_ord(A*B, rmult(A,r,B,s))"
1.241 +apply (rule well_ordI)
1.242 +apply (simp add: well_ord_def wf_on_rmult)
1.243 +apply (simp add: well_ord_def tot_ord_def linear_rmult)
1.244 +done
1.245
1.246
1.247 +(** An ord_iso congruence law **)
1.248 +
1.249 +lemma prod_bij:
1.250 +     "[| f: bij(A,C);  g: bij(B,D) |]
1.251 +      ==> (lam <x,y>:A*B. <f`x, g`y>) : bij(A*B, C*D)"
1.252 +apply (rule_tac d = "%<x,y>. <converse (f) `x, converse (g) `y>"
1.253 +       in lam_bijective)
1.254 +apply (typecheck add: bij_is_inj inj_is_fun)
1.255 +apply (auto simp add: left_inverse_bij right_inverse_bij)
1.256 +done
1.257 +
1.258 +lemma prod_ord_iso_cong:
1.259 +    "[| f: ord_iso(A,r,A',r');  g: ord_iso(B,s,B',s') |]
1.260 +     ==> (lam <x,y>:A*B. <f`x, g`y>)
1.261 +         : ord_iso(A*B, rmult(A,r,B,s), A'*B', rmult(A',r',B',s'))"
1.262 +apply (unfold ord_iso_def)
1.263 +apply (safe intro!: prod_bij)
1.264 +apply (simp_all add: bij_is_fun [THEN apply_type])
1.265 +apply (blast intro: bij_is_inj [THEN inj_apply_equality])
1.266 +done
1.267 +
1.268 +lemma singleton_prod_bij: "(lam z:A. <x,z>) : bij(A, {x}*A)"
1.269 +apply (rule_tac d = "snd" in lam_bijective)
1.270 +apply auto
1.271 +done
1.272 +
1.273 +(*Used??*)
1.274 +lemma singleton_prod_ord_iso:
1.275 +     "well_ord({x},xr) ==>
1.276 +          (lam z:A. <x,z>) : ord_iso(A, r, {x}*A, rmult({x}, xr, A, r))"
1.277 +apply (rule singleton_prod_bij [THEN ord_isoI])
1.278 +apply (simp (no_asm_simp))
1.279 +apply (blast dest: well_ord_is_wf [THEN wf_on_not_refl])
1.280 +done
1.281 +
1.282 +(*Here we build a complicated function term, then simplify it using
1.283 +  case_cong, id_conv, comp_lam, case_case.*)
1.284 +lemma prod_sum_singleton_bij:
1.285 +     "a~:C ==>
1.286 +       (lam x:C*B + D. case(%x. x, %y.<a,y>, x))
1.287 +       : bij(C*B + D, C*B Un {a}*D)"
1.288 +apply (rule subst_elem)
1.289 +apply (rule id_bij [THEN sum_bij, THEN comp_bij])
1.290 +apply (rule singleton_prod_bij)
1.291 +apply (rule sum_disjoint_bij)
1.292 +apply blast
1.293 +apply (simp (no_asm_simp) cong add: case_cong)
1.294 +apply (rule comp_lam [THEN trans, symmetric])
1.295 +apply (fast elim!: case_type)
1.296 +apply (simp (no_asm_simp) add: case_case)
1.297 +done
1.298 +
1.299 +lemma prod_sum_singleton_ord_iso:
1.300 + "[| a:A;  well_ord(A,r) |] ==>
1.301 +    (lam x:pred(A,a,r)*B + pred(B,b,s). case(%x. x, %y.<a,y>, x))
1.302 +    : ord_iso(pred(A,a,r)*B + pred(B,b,s),
1.303 +                  radd(A*B, rmult(A,r,B,s), B, s),
1.304 +              pred(A,a,r)*B Un {a}*pred(B,b,s), rmult(A,r,B,s))"
1.305 +apply (rule prod_sum_singleton_bij [THEN ord_isoI])
1.306 +apply (simp (no_asm_simp) add: pred_iff well_ord_is_wf [THEN wf_on_not_refl])
1.307 +apply (auto elim!: well_ord_is_wf [THEN wf_on_asym] predE)
1.308 +done
1.309 +
1.310 +(** Distributive law **)
1.311 +
1.312 +lemma sum_prod_distrib_bij:
1.313 +     "(lam <x,z>:(A+B)*C. case(%y. Inl(<y,z>), %y. Inr(<y,z>), x))
1.314 +      : bij((A+B)*C, (A*C)+(B*C))"
1.315 +apply (rule_tac d = "case (%<x,y>.<Inl (x) ,y>, %<x,y>.<Inr (x) ,y>) "
1.316 +       in lam_bijective)
1.317 +apply auto
1.318 +done
1.319 +
1.320 +lemma sum_prod_distrib_ord_iso:
1.321 + "(lam <x,z>:(A+B)*C. case(%y. Inl(<y,z>), %y. Inr(<y,z>), x))
1.322 +  : ord_iso((A+B)*C, rmult(A+B, radd(A,r,B,s), C, t),
1.323 +            (A*C)+(B*C), radd(A*C, rmult(A,r,C,t), B*C, rmult(B,s,C,t)))"
1.324 +apply (rule sum_prod_distrib_bij [THEN ord_isoI])
1.325 +apply auto
1.326 +done
1.327 +
1.328 +(** Associativity **)
1.329 +
1.330 +lemma prod_assoc_bij:
1.331 +     "(lam <<x,y>, z>:(A*B)*C. <x,<y,z>>) : bij((A*B)*C, A*(B*C))"
1.332 +apply (rule_tac d = "%<x, <y,z>>. <<x,y>, z>" in lam_bijective)
1.333 +apply auto
1.334 +done
1.335 +
1.336 +lemma prod_assoc_ord_iso:
1.337 + "(lam <<x,y>, z>:(A*B)*C. <x,<y,z>>)
1.338 +  : ord_iso((A*B)*C, rmult(A*B, rmult(A,r,B,s), C, t),
1.339 +            A*(B*C), rmult(A, r, B*C, rmult(B,s,C,t)))"
1.340 +apply (rule prod_assoc_bij [THEN ord_isoI])
1.341 +apply auto
1.342 +done
1.343 +
1.344 +(**** Inverse image of a relation ****)
1.345 +
1.346 +(** Rewrite rule **)
1.347 +
1.348 +lemma rvimage_iff: "<a,b> : rvimage(A,f,r)  <->  <f`a,f`b>: r & a:A & b:A"
1.349 +apply (unfold rvimage_def)
1.350 +apply blast
1.351 +done
1.352 +
1.353 +(** Type checking **)
1.354 +
1.355 +lemma rvimage_type: "rvimage(A,f,r) <= A*A"
1.356 +apply (unfold rvimage_def)
1.357 +apply (rule Collect_subset)
1.358 +done
1.359 +
1.360 +lemmas field_rvimage = rvimage_type [THEN field_rel_subset]
1.361 +
1.362 +lemma rvimage_converse: "rvimage(A,f, converse(r)) = converse(rvimage(A,f,r))"
1.363 +apply (unfold rvimage_def)
1.364 +apply blast
1.365 +done
1.366 +
1.367 +
1.368 +(** Partial Ordering Properties **)
1.369 +
1.370 +lemma irrefl_rvimage:
1.371 +    "[| f: inj(A,B);  irrefl(B,r) |] ==> irrefl(A, rvimage(A,f,r))"
1.372 +apply (unfold irrefl_def rvimage_def)
1.373 +apply (blast intro: inj_is_fun [THEN apply_type])
1.374 +done
1.375 +
1.376 +lemma trans_on_rvimage:
1.377 +    "[| f: inj(A,B);  trans[B](r) |] ==> trans[A](rvimage(A,f,r))"
1.378 +apply (unfold trans_on_def rvimage_def)
1.379 +apply (blast intro: inj_is_fun [THEN apply_type])
1.380 +done
1.381 +
1.382 +lemma part_ord_rvimage:
1.383 +    "[| f: inj(A,B);  part_ord(B,r) |] ==> part_ord(A, rvimage(A,f,r))"
1.384 +apply (unfold part_ord_def)
1.385 +apply (blast intro!: irrefl_rvimage trans_on_rvimage)
1.386 +done
1.387 +
1.388 +(** Linearity **)
1.389 +
1.390 +lemma linear_rvimage:
1.391 +    "[| f: inj(A,B);  linear(B,r) |] ==> linear(A,rvimage(A,f,r))"
1.392 +apply (simp add: inj_def linear_def rvimage_iff)
1.393 +apply (blast intro: apply_funtype);
1.394 +done
1.395 +
1.396 +lemma tot_ord_rvimage:
1.397 +    "[| f: inj(A,B);  tot_ord(B,r) |] ==> tot_ord(A, rvimage(A,f,r))"
1.398 +apply (unfold tot_ord_def)
1.399 +apply (blast intro!: part_ord_rvimage linear_rvimage)
1.400 +done
1.401 +
1.402 +
1.403 +(** Well-foundedness **)
1.404 +
1.405 +(*Not sure if wf_on_rvimage could be proved from this!*)
1.406 +lemma wf_rvimage [intro!]: "wf(r) ==> wf(rvimage(A,f,r))"
1.407 +apply (simp (no_asm_use) add: rvimage_def wf_eq_minimal)
1.408 +apply clarify
1.409 +apply (subgoal_tac "EX w. w : {w: {f`x. x:Q}. EX x. x: Q & (f`x = w) }")
1.410 + apply (erule allE)
1.411 + apply (erule impE)
1.412 + apply assumption;
1.413 + apply blast
1.414 +apply (blast intro: elim:);
1.415 +done
1.416 +
1.417 +lemma wf_on_rvimage: "[| f: A->B;  wf[B](r) |] ==> wf[A](rvimage(A,f,r))"
1.418 +apply (rule wf_onI2)
1.419 +apply (subgoal_tac "ALL z:A. f`z=f`y --> z: Ba")
1.420 + apply blast
1.421 +apply (erule_tac a = "f`y" in wf_on_induct)
1.422 + apply (blast intro!: apply_funtype)
1.423 +apply (blast intro!: apply_funtype dest!: rvimage_iff [THEN iffD1])
1.424 +done
1.425 +
1.426 +(*Note that we need only wf[A](...) and linear(A,...) to get the result!*)
1.427 +lemma well_ord_rvimage:
1.428 +     "[| f: inj(A,B);  well_ord(B,r) |] ==> well_ord(A, rvimage(A,f,r))"
1.429 +apply (rule well_ordI)
1.430 +apply (unfold well_ord_def tot_ord_def)
1.431 +apply (blast intro!: wf_on_rvimage inj_is_fun)
1.432 +apply (blast intro!: linear_rvimage)
1.433 +done
1.434 +
1.435 +lemma ord_iso_rvimage:
1.436 +    "f: bij(A,B) ==> f: ord_iso(A, rvimage(A,f,s), B, s)"
1.437 +apply (unfold ord_iso_def)
1.439 +done
1.440 +
1.441 +lemma ord_iso_rvimage_eq:
1.442 +    "f: ord_iso(A,r, B,s) ==> rvimage(A,f,s) = r Int A*A"
1.443 +apply (unfold ord_iso_def rvimage_def)
1.444 +apply blast
1.445 +done
1.446 +
1.447 +
1.448 +(** The "measure" relation is useful with wfrec **)
1.449 +
1.450 +lemma measure_eq_rvimage_Memrel:
1.451 +     "measure(A,f) = rvimage(A,Lambda(A,f),Memrel(Collect(RepFun(A,f),Ord)))"
1.452 +apply (simp (no_asm) add: measure_def rvimage_def Memrel_iff)
1.453 +apply (rule equalityI)
1.454 +apply auto
1.455 +apply (auto intro: Ord_in_Ord simp add: lt_def)
1.456 +done
1.457 +
1.458 +lemma wf_measure [iff]: "wf(measure(A,f))"
1.459 +apply (simp (no_asm) add: measure_eq_rvimage_Memrel wf_Memrel wf_rvimage)
1.460 +done
1.461 +
1.462 +lemma measure_iff [iff]: "<x,y> : measure(A,f) <-> x:A & y:A & f(x)<f(y)"
1.463 +apply (simp (no_asm) add: measure_def)
1.464 +done
1.465 +
1.466 +ML {*
1.467 +val measure_def = thm "measure_def";
1.479 +val sum_bij = thm "sum_bij";
1.480 +val sum_ord_iso_cong = thm "sum_ord_iso_cong";
1.481 +val sum_disjoint_bij = thm "sum_disjoint_bij";
1.482 +val sum_assoc_bij = thm "sum_assoc_bij";
1.483 +val sum_assoc_ord_iso = thm "sum_assoc_ord_iso";
1.484 +val rmult_iff = thm "rmult_iff";
1.485 +val rmultE = thm "rmultE";
1.486 +val rmult_type = thm "rmult_type";
1.487 +val field_rmult = thm "field_rmult";
1.488 +val linear_rmult = thm "linear_rmult";
1.489 +val wf_on_rmult = thm "wf_on_rmult";
1.490 +val wf_rmult = thm "wf_rmult";
1.491 +val well_ord_rmult = thm "well_ord_rmult";
1.492 +val prod_bij = thm "prod_bij";
1.493 +val prod_ord_iso_cong = thm "prod_ord_iso_cong";
1.494 +val singleton_prod_bij = thm "singleton_prod_bij";
1.495 +val singleton_prod_ord_iso = thm "singleton_prod_ord_iso";
1.496 +val prod_sum_singleton_bij = thm "prod_sum_singleton_bij";
1.497 +val prod_sum_singleton_ord_iso = thm "prod_sum_singleton_ord_iso";
1.498 +val sum_prod_distrib_bij = thm "sum_prod_distrib_bij";
1.499 +val sum_prod_distrib_ord_iso = thm "sum_prod_distrib_ord_iso";
1.500 +val prod_assoc_bij = thm "prod_assoc_bij";
1.501 +val prod_assoc_ord_iso = thm "prod_assoc_ord_iso";
1.502 +val rvimage_iff = thm "rvimage_iff";
1.503 +val rvimage_type = thm "rvimage_type";
1.504 +val field_rvimage = thm "field_rvimage";
1.505 +val rvimage_converse = thm "rvimage_converse";
1.506 +val irrefl_rvimage = thm "irrefl_rvimage";
1.507 +val trans_on_rvimage = thm "trans_on_rvimage";
1.508 +val part_ord_rvimage = thm "part_ord_rvimage";
1.509 +val linear_rvimage = thm "linear_rvimage";
1.510 +val tot_ord_rvimage = thm "tot_ord_rvimage";
1.511 +val wf_rvimage = thm "wf_rvimage";
1.512 +val wf_on_rvimage = thm "wf_on_rvimage";
1.513 +val well_ord_rvimage = thm "well_ord_rvimage";
1.514 +val ord_iso_rvimage = thm "ord_iso_rvimage";
1.515 +val ord_iso_rvimage_eq = thm "ord_iso_rvimage_eq";
1.516 +val measure_eq_rvimage_Memrel = thm "measure_eq_rvimage_Memrel";
1.517 +val wf_measure = thm "wf_measure";
1.518 +val measure_iff = thm "measure_iff";
1.519 +*}
1.520 +
1.521  end
```