src/ZF/Constructible/Separation.thy

 text{*Helps us solve for de Bruijn indices!*}
 lemma nth_ConsI: "[|nth(n,l) = x; n \<in> nat|] ==> nth(succ(n), Cons(a,l)) = x"
 by simp
 
 lemmas nth_rules = nth_0 nth_ConsI nat_0I nat_succI
-lemmas sep_rules = nth_0 nth_ConsI FOL_iff_sats function_iff_sats 
+lemmas sep_rules = nth_0 nth_ConsI FOL_iff_sats function_iff_sats
                    fun_plus_iff_sats
 
 lemma Collect_conj_in_DPow:
-     "[| {x\<in>A. P(x)} \<in> DPow(A);  {x\<in>A. Q(x)} \<in> DPow(A) |] 
+     "[| {x\<in>A. P(x)} \<in> DPow(A);  {x\<in>A. Q(x)} \<in> DPow(A) |]
       ==> {x\<in>A. P(x) & Q(x)} \<in> DPow(A)"
-by (simp add: Int_in_DPow Collect_Int_Collect_eq [symmetric]) 
+by (simp add: Int_in_DPow Collect_Int_Collect_eq [symmetric])
 
 lemma Collect_conj_in_DPow_Lset:
      "[|z \<in> Lset(j); {x \<in> Lset(j). P(x)} \<in> DPow(Lset(j))|]
       ==> {x \<in> Lset(j). x \<in> z & P(x)} \<in> DPow(Lset(j))"
 apply (frule mem_Lset_imp_subset_Lset)
-apply (simp add: Collect_conj_in_DPow Collect_mem_eq 
+apply (simp add: Collect_conj_in_DPow Collect_mem_eq
                  subset_Int_iff2 elem_subset_in_DPow)
 done
 
 lemma separation_CollectI:
      "(\<And>z. L(z) ==> L({x \<in> z . P(x)})) ==> separation(L, \<lambda>x. P(x))"
-apply (unfold separation_def, clarify) 
-apply (rule_tac x="{x\<in>z. P(x)}" in rexI) 
+apply (unfold separation_def, clarify)
+apply (rule_tac x="{x\<in>z. P(x)}" in rexI)
 apply simp_all
 done
 
 text{*Reduces the original comprehension to the reflected one*}
 lemma reflection_imp_L_separation:
       "[| \<forall>x\<in>Lset(j). P(x) <-> Q(x);
-          {x \<in> Lset(j) . Q(x)} \<in> DPow(Lset(j)); 
+          {x \<in> Lset(j) . Q(x)} \<in> DPow(Lset(j));
           Ord(j);  z \<in> Lset(j)|] ==> L({x \<in> z . P(x)})"
 apply (rule_tac i = "succ(j)" in L_I)
  prefer 2 apply simp
 apply (subgoal_tac "{x \<in> z. P(x)} = {x \<in> Lset(j). x \<in> z & (Q(x))}")
  prefer 2
- apply (blast dest: mem_Lset_imp_subset_Lset) 
+ apply (blast dest: mem_Lset_imp_subset_Lset)
 apply (simp add: Lset_succ Collect_conj_in_DPow_Lset)
 done
 
 
 subsection{*Separation for Intersection*}
 
 lemma Inter_Reflects:
-     "REFLECTS[\<lambda>x. \<forall>y[L]. y\<in>A --> x \<in> y, 
+     "REFLECTS[\<lambda>x. \<forall>y[L]. y\<in>A --> x \<in> y,
                \<lambda>i x. \<forall>y\<in>Lset(i). y\<in>A --> x \<in> y]"
-by (intro FOL_reflections)  
+by (intro FOL_reflections)
 
 lemma Inter_separation:
      "L(A) ==> separation(L, \<lambda>x. \<forall>y[L]. y\<in>A --> x\<in>y)"
-apply (rule separation_CollectI) 
-apply (rule_tac A="{A,z}" in subset_LsetE, blast ) 
+apply (rule separation_CollectI)
+apply (rule_tac A="{A,z}" in subset_LsetE, blast )
 apply (rule ReflectsE [OF Inter_Reflects], assumption)
-apply (drule subset_Lset_ltD, assumption) 
+apply (drule subset_Lset_ltD, assumption)
 apply (erule reflection_imp_L_separation)
   apply (simp_all add: lt_Ord2, clarify)
-apply (rule DPow_LsetI) 
-apply (rule ball_iff_sats) 
+apply (rule DPow_LsetI)
+apply (rule ball_iff_sats)
 apply (rule imp_iff_sats)
 apply (rule_tac [2] i=1 and j=0 and env="[y,x,A]" in mem_iff_sats)
 apply (rule_tac i=0 and j=2 in mem_iff_sats)
 apply (simp_all add: succ_Un_distrib [symmetric])
 done
 
 subsection{*Separation for Cartesian Product*}
 
 lemma cartprod_Reflects:
      "REFLECTS[\<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>B & pair(L,x,y,z)),
-                \<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). y\<in>B & 
+                \<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). y\<in>B &
                                    pair(**Lset(i),x,y,z))]"
 by (intro FOL_reflections function_reflections)
 
 lemma cartprod_separation:
-     "[| L(A); L(B) |] 
+     "[| L(A); L(B) |]
       ==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>B & pair(L,x,y,z)))"
-apply (rule separation_CollectI) 
-apply (rule_tac A="{A,B,z}" in subset_LsetE, blast ) 
+apply (rule separation_CollectI)
+apply (rule_tac A="{A,B,z}" in subset_LsetE, blast )
 apply (rule ReflectsE [OF cartprod_Reflects], assumption)
-apply (drule subset_Lset_ltD, assumption) 
-apply (erule reflection_imp_L_separation)
-  apply (simp_all add: lt_Ord2, clarify) 
-apply (rule DPow_LsetI)
-apply (rename_tac u)  
-apply (rule bex_iff_sats) 
+apply (drule subset_Lset_ltD, assumption)
+apply (erule reflection_imp_L_separation)
+  apply (simp_all add: lt_Ord2, clarify)
+apply (rule DPow_LsetI)
+apply (rename_tac u)
+apply (rule bex_iff_sats)
 apply (rule conj_iff_sats)
 apply (rule_tac i=0 and j=2 and env="[x,u,A,B]" in mem_iff_sats, simp_all)
 apply (rule sep_rules | simp)+
 done
 

      "REFLECTS[\<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>A & pair(L,x,y,p)),
            \<lambda>i y. \<exists>p\<in>Lset(i). p\<in>r & (\<exists>x\<in>Lset(i). x\<in>A & pair(**Lset(i),x,y,p))]"
 by (intro FOL_reflections function_reflections)
 
 lemma image_separation:
-     "[| L(A); L(r) |] 
+     "[| L(A); L(r) |]
       ==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>A & pair(L,x,y,p)))"
-apply (rule separation_CollectI) 
-apply (rule_tac A="{A,r,z}" in subset_LsetE, blast ) 
+apply (rule separation_CollectI)
+apply (rule_tac A="{A,r,z}" in subset_LsetE, blast )
 apply (rule ReflectsE [OF image_Reflects], assumption)
-apply (drule subset_Lset_ltD, assumption) 
+apply (drule subset_Lset_ltD, assumption)
 apply (erule reflection_imp_L_separation)
   apply (simp_all add: lt_Ord2, clarify)
 apply (rule DPow_LsetI)
-apply (rule bex_iff_sats) 
+apply (rule bex_iff_sats)
 apply (rule conj_iff_sats)
 apply (rule_tac env="[p,y,A,r]" in mem_iff_sats)
 apply (rule sep_rules | simp)+
 done
 
 
 subsection{*Separation for Converse*}
 
 lemma converse_Reflects:
   "REFLECTS[\<lambda>z. \<exists>p[L]. p\<in>r & (\<exists>x[L]. \<exists>y[L]. pair(L,x,y,p) & pair(L,y,x,z)),
-     \<lambda>i z. \<exists>p\<in>Lset(i). p\<in>r & (\<exists>x\<in>Lset(i). \<exists>y\<in>Lset(i). 
+     \<lambda>i z. \<exists>p\<in>Lset(i). p\<in>r & (\<exists>x\<in>Lset(i). \<exists>y\<in>Lset(i).
                      pair(**Lset(i),x,y,p) & pair(**Lset(i),y,x,z))]"
 by (intro FOL_reflections function_reflections)
 
 lemma converse_separation:
-     "L(r) ==> separation(L, 
+     "L(r) ==> separation(L,
          \<lambda>z. \<exists>p[L]. p\<in>r & (\<exists>x[L]. \<exists>y[L]. pair(L,x,y,p) & pair(L,y,x,z)))"
-apply (rule separation_CollectI) 
-apply (rule_tac A="{r,z}" in subset_LsetE, blast ) 
+apply (rule separation_CollectI)
+apply (rule_tac A="{r,z}" in subset_LsetE, blast )
 apply (rule ReflectsE [OF converse_Reflects], assumption)
-apply (drule subset_Lset_ltD, assumption) 
+apply (drule subset_Lset_ltD, assumption)
 apply (erule reflection_imp_L_separation)
   apply (simp_all add: lt_Ord2, clarify)
 apply (rule DPow_LsetI)
-apply (rename_tac u) 
-apply (rule bex_iff_sats) 
+apply (rename_tac u)
+apply (rule bex_iff_sats)
 apply (rule conj_iff_sats)
 apply (rule_tac i=0 and j=2 and env="[p,u,r]" in mem_iff_sats, simp_all)
 apply (rule sep_rules | simp)+
 done
 

         \<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). pair(**Lset(i),x,y,z))]"
 by (intro FOL_reflections function_reflections)
 
 lemma restrict_separation:
    "L(A) ==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)))"
-apply (rule separation_CollectI) 
-apply (rule_tac A="{A,z}" in subset_LsetE, blast ) 
+apply (rule separation_CollectI)
+apply (rule_tac A="{A,z}" in subset_LsetE, blast )
 apply (rule ReflectsE [OF restrict_Reflects], assumption)
-apply (drule subset_Lset_ltD, assumption) 
+apply (drule subset_Lset_ltD, assumption)
 apply (erule reflection_imp_L_separation)
   apply (simp_all add: lt_Ord2, clarify)
 apply (rule DPow_LsetI)
-apply (rename_tac u) 
-apply (rule bex_iff_sats) 
+apply (rename_tac u)
+apply (rule bex_iff_sats)
 apply (rule conj_iff_sats)
 apply (rule_tac i=0 and j=2 and env="[x,u,A]" in mem_iff_sats, simp_all)
 apply (rule sep_rules | simp)+
 done
 
 
 subsection{*Separation for Composition*}
 
 lemma comp_Reflects:
-     "REFLECTS[\<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L]. 
-		  pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) & 
+     "REFLECTS[\<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L].
+                  pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) &
                   xy\<in>s & yz\<in>r,
-        \<lambda>i xz. \<exists>x\<in>Lset(i). \<exists>y\<in>Lset(i). \<exists>z\<in>Lset(i). \<exists>xy\<in>Lset(i). \<exists>yz\<in>Lset(i). 
-		  pair(**Lset(i),x,z,xz) & pair(**Lset(i),x,y,xy) & 
+        \<lambda>i xz. \<exists>x\<in>Lset(i). \<exists>y\<in>Lset(i). \<exists>z\<in>Lset(i). \<exists>xy\<in>Lset(i). \<exists>yz\<in>Lset(i).
+                  pair(**Lset(i),x,z,xz) & pair(**Lset(i),x,y,xy) &
                   pair(**Lset(i),y,z,yz) & xy\<in>s & yz\<in>r]"
 by (intro FOL_reflections function_reflections)
 
 lemma comp_separation:
      "[| L(r); L(s) |]
-      ==> separation(L, \<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L]. 
-		  pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) & 
+      ==> separation(L, \<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L].
+                  pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) &
                   xy\<in>s & yz\<in>r)"
-apply (rule separation_CollectI) 
-apply (rule_tac A="{r,s,z}" in subset_LsetE, blast ) 
+apply (rule separation_CollectI)
+apply (rule_tac A="{r,s,z}" in subset_LsetE, blast )
 apply (rule ReflectsE [OF comp_Reflects], assumption)
-apply (drule subset_Lset_ltD, assumption) 
+apply (drule subset_Lset_ltD, assumption)
 apply (erule reflection_imp_L_separation)
   apply (simp_all add: lt_Ord2, clarify)
 apply (rule DPow_LsetI)
-apply (rename_tac u) 
+apply (rename_tac u)
 apply (rule bex_iff_sats)+
-apply (rename_tac x y z)  
+apply (rename_tac x y z)
 apply (rule conj_iff_sats)
 apply (rule_tac env="[z,y,x,u,r,s]" in pair_iff_sats)
 apply (rule sep_rules | simp)+
 done
 

                     \<lambda>i y. \<exists>p \<in> Lset(i). p\<in>r & pair(**Lset(i),y,x,p)]"
 by (intro FOL_reflections function_reflections)
 
 lemma pred_separation:
      "[| L(r); L(x) |] ==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & pair(L,y,x,p))"
-apply (rule separation_CollectI) 
-apply (rule_tac A="{r,x,z}" in subset_LsetE, blast ) 
+apply (rule separation_CollectI)
+apply (rule_tac A="{r,x,z}" in subset_LsetE, blast )
 apply (rule ReflectsE [OF pred_Reflects], assumption)
-apply (drule subset_Lset_ltD, assumption) 
+apply (drule subset_Lset_ltD, assumption)
 apply (erule reflection_imp_L_separation)
   apply (simp_all add: lt_Ord2, clarify)
 apply (rule DPow_LsetI)
-apply (rename_tac u) 
+apply (rename_tac u)
 apply (rule bex_iff_sats)
 apply (rule conj_iff_sats)
-apply (rule_tac env = "[p,u,r,x]" in mem_iff_sats) 
+apply (rule_tac env = "[p,u,r,x]" in mem_iff_sats)
 apply (rule sep_rules | simp)+
 done
 
 
 subsection{*Separation for the Membership Relation*}

             \<lambda>i z. \<exists>x \<in> Lset(i). \<exists>y \<in> Lset(i). pair(**Lset(i),x,y,z) & x \<in> y]"
 by (intro FOL_reflections function_reflections)
 
 lemma Memrel_separation:
      "separation(L, \<lambda>z. \<exists>x[L]. \<exists>y[L]. pair(L,x,y,z) & x \<in> y)"
-apply (rule separation_CollectI) 
-apply (rule_tac A="{z}" in subset_LsetE, blast ) 
+apply (rule separation_CollectI)
+apply (rule_tac A="{z}" in subset_LsetE, blast )
 apply (rule ReflectsE [OF Memrel_Reflects], assumption)
-apply (drule subset_Lset_ltD, assumption) 
+apply (drule subset_Lset_ltD, assumption)
 apply (erule reflection_imp_L_separation)
   apply (simp_all add: lt_Ord2)
 apply (rule DPow_LsetI)
-apply (rename_tac u) 
+apply (rename_tac u)
 apply (rule bex_iff_sats conj_iff_sats)+
-apply (rule_tac env = "[y,x,u]" in pair_iff_sats) 
+apply (rule_tac env = "[y,x,u]" in pair_iff_sats)
 apply (rule sep_rules | simp)+
 done
 
 
 subsection{*Replacement for FunSpace*}
-		
+
 lemma funspace_succ_Reflects:
- "REFLECTS[\<lambda>z. \<exists>p[L]. p\<in>A & (\<exists>f[L]. \<exists>b[L]. \<exists>nb[L]. \<exists>cnbf[L]. 
-	    pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) &
-	    upair(L,cnbf,cnbf,z)),
-	\<lambda>i z. \<exists>p \<in> Lset(i). p\<in>A & (\<exists>f \<in> Lset(i). \<exists>b \<in> Lset(i). 
-	      \<exists>nb \<in> Lset(i). \<exists>cnbf \<in> Lset(i). 
-		pair(**Lset(i),f,b,p) & pair(**Lset(i),n,b,nb) & 
-		is_cons(**Lset(i),nb,f,cnbf) & upair(**Lset(i),cnbf,cnbf,z))]"
+ "REFLECTS[\<lambda>z. \<exists>p[L]. p\<in>A & (\<exists>f[L]. \<exists>b[L]. \<exists>nb[L]. \<exists>cnbf[L].
+            pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) &
+            upair(L,cnbf,cnbf,z)),
+        \<lambda>i z. \<exists>p \<in> Lset(i). p\<in>A & (\<exists>f \<in> Lset(i). \<exists>b \<in> Lset(i).
+              \<exists>nb \<in> Lset(i). \<exists>cnbf \<in> Lset(i).
+                pair(**Lset(i),f,b,p) & pair(**Lset(i),n,b,nb) &
+                is_cons(**Lset(i),nb,f,cnbf) & upair(**Lset(i),cnbf,cnbf,z))]"
 by (intro FOL_reflections function_reflections)
 
 lemma funspace_succ_replacement:
-     "L(n) ==> 
-      strong_replacement(L, \<lambda>p z. \<exists>f[L]. \<exists>b[L]. \<exists>nb[L]. \<exists>cnbf[L]. 
+     "L(n) ==>
+      strong_replacement(L, \<lambda>p z. \<exists>f[L]. \<exists>b[L]. \<exists>nb[L]. \<exists>cnbf[L].
                 pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) &
                 upair(L,cnbf,cnbf,z))"
-apply (rule strong_replacementI) 
-apply (rule rallI) 
-apply (rule separation_CollectI) 
-apply (rule_tac A="{n,A,z}" in subset_LsetE, blast ) 
+apply (rule strong_replacementI)
+apply (rule rallI)
+apply (rule separation_CollectI)
+apply (rule_tac A="{n,A,z}" in subset_LsetE, blast )
 apply (rule ReflectsE [OF funspace_succ_Reflects], assumption)
-apply (drule subset_Lset_ltD, assumption) 
+apply (drule subset_Lset_ltD, assumption)
 apply (erule reflection_imp_L_separation)
   apply (simp_all add: lt_Ord2)
 apply (rule DPow_LsetI)
-apply (rename_tac u) 
+apply (rename_tac u)
 apply (rule bex_iff_sats)
 apply (rule conj_iff_sats)
-apply (rule_tac env = "[p,u,n,A]" in mem_iff_sats) 
+apply (rule_tac env = "[p,u,n,A]" in mem_iff_sats)
 apply (rule sep_rules | simp)+
 done
 
 
 subsection{*Separation for Order-Isomorphisms*}
 
 lemma well_ord_iso_Reflects:
-  "REFLECTS[\<lambda>x. x\<in>A --> 
+  "REFLECTS[\<lambda>x. x\<in>A -->
                 (\<exists>y[L]. \<exists>p[L]. fun_apply(L,f,x,y) & pair(L,y,x,p) & p \<in> r),
-        \<lambda>i x. x\<in>A --> (\<exists>y \<in> Lset(i). \<exists>p \<in> Lset(i). 
+        \<lambda>i x. x\<in>A --> (\<exists>y \<in> Lset(i). \<exists>p \<in> Lset(i).
                 fun_apply(**Lset(i),f,x,y) & pair(**Lset(i),y,x,p) & p \<in> r)]"
 by (intro FOL_reflections function_reflections)
 
 lemma well_ord_iso_separation:
-     "[| L(A); L(f); L(r) |] 
-      ==> separation (L, \<lambda>x. x\<in>A --> (\<exists>y[L]. (\<exists>p[L]. 
-		     fun_apply(L,f,x,y) & pair(L,y,x,p) & p \<in> r)))"
-apply (rule separation_CollectI) 
-apply (rule_tac A="{A,f,r,z}" in subset_LsetE, blast ) 
+     "[| L(A); L(f); L(r) |]
+      ==> separation (L, \<lambda>x. x\<in>A --> (\<exists>y[L]. (\<exists>p[L].
+                     fun_apply(L,f,x,y) & pair(L,y,x,p) & p \<in> r)))"
+apply (rule separation_CollectI)
+apply (rule_tac A="{A,f,r,z}" in subset_LsetE, blast )
 apply (rule ReflectsE [OF well_ord_iso_Reflects], assumption)
-apply (drule subset_Lset_ltD, assumption) 
+apply (drule subset_Lset_ltD, assumption)
 apply (erule reflection_imp_L_separation)
   apply (simp_all add: lt_Ord2)
 apply (rule DPow_LsetI)
-apply (rename_tac u) 
+apply (rename_tac u)
 apply (rule imp_iff_sats)
-apply (rule_tac env = "[u,A,f,r]" in mem_iff_sats) 
+apply (rule_tac env = "[u,A,f,r]" in mem_iff_sats)
 apply (rule sep_rules | simp)+
 done
 
 
 subsection{*Separation for @{term "obase"}*}
 
 lemma obase_reflects:
-  "REFLECTS[\<lambda>a. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. 
-	     ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) &
-	     order_isomorphism(L,par,r,x,mx,g),
-        \<lambda>i a. \<exists>x \<in> Lset(i). \<exists>g \<in> Lset(i). \<exists>mx \<in> Lset(i). \<exists>par \<in> Lset(i). 
-	     ordinal(**Lset(i),x) & membership(**Lset(i),x,mx) & pred_set(**Lset(i),A,a,r,par) &
-	     order_isomorphism(**Lset(i),par,r,x,mx,g)]"
+  "REFLECTS[\<lambda>a. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L].
+             ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) &
+             order_isomorphism(L,par,r,x,mx,g),
+        \<lambda>i a. \<exists>x \<in> Lset(i). \<exists>g \<in> Lset(i). \<exists>mx \<in> Lset(i). \<exists>par \<in> Lset(i).
+             ordinal(**Lset(i),x) & membership(**Lset(i),x,mx) & pred_set(**Lset(i),A,a,r,par) &
+             order_isomorphism(**Lset(i),par,r,x,mx,g)]"
 by (intro FOL_reflections function_reflections fun_plus_reflections)
 
 lemma obase_separation:
      --{*part of the order type formalization*}
-     "[| L(A); L(r) |] 
-      ==> separation(L, \<lambda>a. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. 
-	     ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) &
-	     order_isomorphism(L,par,r,x,mx,g))"
-apply (rule separation_CollectI) 
-apply (rule_tac A="{A,r,z}" in subset_LsetE, blast ) 
+     "[| L(A); L(r) |]
+      ==> separation(L, \<lambda>a. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L].
+             ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) &
+             order_isomorphism(L,par,r,x,mx,g))"
+apply (rule separation_CollectI)
+apply (rule_tac A="{A,r,z}" in subset_LsetE, blast )
 apply (rule ReflectsE [OF obase_reflects], assumption)
-apply (drule subset_Lset_ltD, assumption) 
+apply (drule subset_Lset_ltD, assumption)
 apply (erule reflection_imp_L_separation)
   apply (simp_all add: lt_Ord2)
 apply (rule DPow_LsetI)
-apply (rename_tac u) 
+apply (rename_tac u)
 apply (rule bex_iff_sats)
 apply (rule conj_iff_sats)
-apply (rule_tac env = "[x,u,A,r]" in ordinal_iff_sats) 
+apply (rule_tac env = "[x,u,A,r]" in ordinal_iff_sats)
 apply (rule sep_rules | simp)+
 done
 
 
 subsection{*Separation for a Theorem about @{term "obase"}*}
 
 lemma obase_equals_reflects:
-  "REFLECTS[\<lambda>x. x\<in>A --> ~(\<exists>y[L]. \<exists>g[L]. 
-		ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L]. 
-		membership(L,y,my) & pred_set(L,A,x,r,pxr) &
-		order_isomorphism(L,pxr,r,y,my,g))),
-	\<lambda>i x. x\<in>A --> ~(\<exists>y \<in> Lset(i). \<exists>g \<in> Lset(i). 
-		ordinal(**Lset(i),y) & (\<exists>my \<in> Lset(i). \<exists>pxr \<in> Lset(i). 
-		membership(**Lset(i),y,my) & pred_set(**Lset(i),A,x,r,pxr) &
-		order_isomorphism(**Lset(i),pxr,r,y,my,g)))]"
+  "REFLECTS[\<lambda>x. x\<in>A --> ~(\<exists>y[L]. \<exists>g[L].
+                ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L].
+                membership(L,y,my) & pred_set(L,A,x,r,pxr) &
+                order_isomorphism(L,pxr,r,y,my,g))),
+        \<lambda>i x. x\<in>A --> ~(\<exists>y \<in> Lset(i). \<exists>g \<in> Lset(i).
+                ordinal(**Lset(i),y) & (\<exists>my \<in> Lset(i). \<exists>pxr \<in> Lset(i).
+                membership(**Lset(i),y,my) & pred_set(**Lset(i),A,x,r,pxr) &
+                order_isomorphism(**Lset(i),pxr,r,y,my,g)))]"
 by (intro FOL_reflections function_reflections fun_plus_reflections)
 
 
 lemma obase_equals_separation:
-     "[| L(A); L(r) |] 
-      ==> separation (L, \<lambda>x. x\<in>A --> ~(\<exists>y[L]. \<exists>g[L]. 
-			      ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L]. 
-			      membership(L,y,my) & pred_set(L,A,x,r,pxr) &
-			      order_isomorphism(L,pxr,r,y,my,g))))"
-apply (rule separation_CollectI) 
-apply (rule_tac A="{A,r,z}" in subset_LsetE, blast ) 
+     "[| L(A); L(r) |]
+      ==> separation (L, \<lambda>x. x\<in>A --> ~(\<exists>y[L]. \<exists>g[L].
+                              ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L].
+                              membership(L,y,my) & pred_set(L,A,x,r,pxr) &
+                              order_isomorphism(L,pxr,r,y,my,g))))"
+apply (rule separation_CollectI)
+apply (rule_tac A="{A,r,z}" in subset_LsetE, blast )
 apply (rule ReflectsE [OF obase_equals_reflects], assumption)
-apply (drule subset_Lset_ltD, assumption) 
+apply (drule subset_Lset_ltD, assumption)
 apply (erule reflection_imp_L_separation)
   apply (simp_all add: lt_Ord2)
 apply (rule DPow_LsetI)
-apply (rename_tac u) 
+apply (rename_tac u)
 apply (rule imp_iff_sats ball_iff_sats disj_iff_sats not_iff_sats)+
-apply (rule_tac env = "[u,A,r]" in mem_iff_sats) 
+apply (rule_tac env = "[u,A,r]" in mem_iff_sats)
 apply (rule sep_rules | simp)+
 done
 
 
 subsection{*Replacement for @{term "omap"}*}
 
 lemma omap_reflects:
- "REFLECTS[\<lambda>z. \<exists>a[L]. a\<in>B & (\<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. 
-     ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) & 
+ "REFLECTS[\<lambda>z. \<exists>a[L]. a\<in>B & (\<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L].
+     ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) &
      pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g)),
- \<lambda>i z. \<exists>a \<in> Lset(i). a\<in>B & (\<exists>x \<in> Lset(i). \<exists>g \<in> Lset(i). \<exists>mx \<in> Lset(i). 
-        \<exists>par \<in> Lset(i). 
-	 ordinal(**Lset(i),x) & pair(**Lset(i),a,x,z) & 
-         membership(**Lset(i),x,mx) & pred_set(**Lset(i),A,a,r,par) & 
+ \<lambda>i z. \<exists>a \<in> Lset(i). a\<in>B & (\<exists>x \<in> Lset(i). \<exists>g \<in> Lset(i). \<exists>mx \<in> Lset(i).
+        \<exists>par \<in> Lset(i).
+         ordinal(**Lset(i),x) & pair(**Lset(i),a,x,z) &
+         membership(**Lset(i),x,mx) & pred_set(**Lset(i),A,a,r,par) &
          order_isomorphism(**Lset(i),par,r,x,mx,g))]"
 by (intro FOL_reflections function_reflections fun_plus_reflections)
 
 lemma omap_replacement:
-     "[| L(A); L(r) |] 
+     "[| L(A); L(r) |]
       ==> strong_replacement(L,
-             \<lambda>a z. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. 
-	     ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) & 
-	     pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g))"
-apply (rule strong_replacementI) 
+             \<lambda>a z. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L].
+             ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) &
+             pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g))"
+apply (rule strong_replacementI)
 apply (rule rallI)
-apply (rename_tac B)  
-apply (rule separation_CollectI) 
-apply (rule_tac A="{A,B,r,z}" in subset_LsetE, blast ) 
+apply (rename_tac B)
+apply (rule separation_CollectI)
+apply (rule_tac A="{A,B,r,z}" in subset_LsetE, blast )
 apply (rule ReflectsE [OF omap_reflects], assumption)
-apply (drule subset_Lset_ltD, assumption) 
+apply (drule subset_Lset_ltD, assumption)
 apply (erule reflection_imp_L_separation)
   apply (simp_all add: lt_Ord2)
 apply (rule DPow_LsetI)
-apply (rename_tac u) 
+apply (rename_tac u)
 apply (rule bex_iff_sats conj_iff_sats)+
-apply (rule_tac env = "[a,u,A,B,r]" in mem_iff_sats) 
+apply (rule_tac env = "[a,u,A,B,r]" in mem_iff_sats)
 apply (rule sep_rules | simp)+
 done
 
 
 subsection{*Separation for a Theorem about @{term "obase"}*}
 
 lemma is_recfun_reflects:
-  "REFLECTS[\<lambda>x. \<exists>xa[L]. \<exists>xb[L]. 
-                pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r & 
-                (\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) & 
+  "REFLECTS[\<lambda>x. \<exists>xa[L]. \<exists>xb[L].
+                pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r &
+                (\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) &
                                    fx \<noteq> gx),
-   \<lambda>i x. \<exists>xa \<in> Lset(i). \<exists>xb \<in> Lset(i). 
+   \<lambda>i x. \<exists>xa \<in> Lset(i). \<exists>xb \<in> Lset(i).
           pair(**Lset(i),x,a,xa) & xa \<in> r & pair(**Lset(i),x,b,xb) & xb \<in> r &
-                (\<exists>fx \<in> Lset(i). \<exists>gx \<in> Lset(i). fun_apply(**Lset(i),f,x,fx) & 
+                (\<exists>fx \<in> Lset(i). \<exists>gx \<in> Lset(i). fun_apply(**Lset(i),f,x,fx) &
                   fun_apply(**Lset(i),g,x,gx) & fx \<noteq> gx)]"
 by (intro FOL_reflections function_reflections fun_plus_reflections)
 
 lemma is_recfun_separation:
      --{*for well-founded recursion*}
-     "[| L(r); L(f); L(g); L(a); L(b) |] 
-     ==> separation(L, 
-            \<lambda>x. \<exists>xa[L]. \<exists>xb[L]. 
-                pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r & 
-                (\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) & 
+     "[| L(r); L(f); L(g); L(a); L(b) |]
+     ==> separation(L,
+            \<lambda>x. \<exists>xa[L]. \<exists>xb[L].
+                pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r &
+                (\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) &
                                    fx \<noteq> gx))"
-apply (rule separation_CollectI) 
-apply (rule_tac A="{r,f,g,a,b,z}" in subset_LsetE, blast ) 
+apply (rule separation_CollectI)
+apply (rule_tac A="{r,f,g,a,b,z}" in subset_LsetE, blast )
 apply (rule ReflectsE [OF is_recfun_reflects], assumption)
-apply (drule subset_Lset_ltD, assumption) 
+apply (drule subset_Lset_ltD, assumption)
 apply (erule reflection_imp_L_separation)
   apply (simp_all add: lt_Ord2)
 apply (rule DPow_LsetI)
-apply (rename_tac u) 
+apply (rename_tac u)
 apply (rule bex_iff_sats conj_iff_sats)+
-apply (rule_tac env = "[xa,u,r,f,g,a,b]" in pair_iff_sats) 
+apply (rule_tac env = "[xa,u,r,f,g,a,b]" in pair_iff_sats)
 apply (rule sep_rules | simp)+
 done
 
 
 subsection{*Instantiating the locale @{text M_axioms}*}
 text{*Separation (and Strong Replacement) for basic set-theoretic constructions
 such as intersection, Cartesian Product and image.*}
 
-ML
-{*
-val Inter_separation = thm "Inter_separation";
-val cartprod_separation = thm "cartprod_separation";
-val image_separation = thm "image_separation";
-val converse_separation = thm "converse_separation";
-val restrict_separation = thm "restrict_separation";
-val comp_separation = thm "comp_separation";
-val pred_separation = thm "pred_separation";
-val Memrel_separation = thm "Memrel_separation";
-val funspace_succ_replacement = thm "funspace_succ_replacement";
-val well_ord_iso_separation = thm "well_ord_iso_separation";
-val obase_separation = thm "obase_separation";
-val obase_equals_separation = thm "obase_equals_separation";
-val omap_replacement = thm "omap_replacement";
-val is_recfun_separation = thm "is_recfun_separation";
-
-val m_axioms = 
-    [Inter_separation, cartprod_separation, image_separation, 
-     converse_separation, restrict_separation, comp_separation, 
-     pred_separation, Memrel_separation, funspace_succ_replacement, 
-     well_ord_iso_separation, obase_separation,
-     obase_equals_separation, omap_replacement, is_recfun_separation]
-
-fun axioms_L th =
-    kill_flex_triv_prems (m_axioms MRS (triv_axioms_L th));
-
-bind_thm ("cartprod_iff", axioms_L (thm "M_axioms.cartprod_iff"));
-bind_thm ("cartprod_closed", axioms_L (thm "M_axioms.cartprod_closed"));
-bind_thm ("sum_closed", axioms_L (thm "M_axioms.sum_closed"));
-bind_thm ("M_converse_iff", axioms_L (thm "M_axioms.M_converse_iff"));
-bind_thm ("converse_closed", axioms_L (thm "M_axioms.converse_closed"));
-bind_thm ("converse_abs", axioms_L (thm "M_axioms.converse_abs"));
-bind_thm ("image_closed", axioms_L (thm "M_axioms.image_closed"));
-bind_thm ("vimage_abs", axioms_L (thm "M_axioms.vimage_abs"));
-bind_thm ("vimage_closed", axioms_L (thm "M_axioms.vimage_closed"));
-bind_thm ("domain_abs", axioms_L (thm "M_axioms.domain_abs"));
-bind_thm ("domain_closed", axioms_L (thm "M_axioms.domain_closed"));
-bind_thm ("range_abs", axioms_L (thm "M_axioms.range_abs"));
-bind_thm ("range_closed", axioms_L (thm "M_axioms.range_closed"));
-bind_thm ("field_abs", axioms_L (thm "M_axioms.field_abs"));
-bind_thm ("field_closed", axioms_L (thm "M_axioms.field_closed"));
-bind_thm ("relation_abs", axioms_L (thm "M_axioms.relation_abs"));
-bind_thm ("function_abs", axioms_L (thm "M_axioms.function_abs"));
-bind_thm ("apply_closed", axioms_L (thm "M_axioms.apply_closed"));
-bind_thm ("apply_abs", axioms_L (thm "M_axioms.apply_abs"));
-bind_thm ("typed_function_abs", axioms_L (thm "M_axioms.typed_function_abs"));
-bind_thm ("injection_abs", axioms_L (thm "M_axioms.injection_abs"));
-bind_thm ("surjection_abs", axioms_L (thm "M_axioms.surjection_abs"));
-bind_thm ("bijection_abs", axioms_L (thm "M_axioms.bijection_abs"));
-bind_thm ("M_comp_iff", axioms_L (thm "M_axioms.M_comp_iff"));
-bind_thm ("comp_closed", axioms_L (thm "M_axioms.comp_closed"));
-bind_thm ("composition_abs", axioms_L (thm "M_axioms.composition_abs"));
-bind_thm ("restriction_is_function", axioms_L (thm "M_axioms.restriction_is_function"));
-bind_thm ("restriction_abs", axioms_L (thm "M_axioms.restriction_abs"));
-bind_thm ("M_restrict_iff", axioms_L (thm "M_axioms.M_restrict_iff"));
-bind_thm ("restrict_closed", axioms_L (thm "M_axioms.restrict_closed"));
-bind_thm ("Inter_abs", axioms_L (thm "M_axioms.Inter_abs"));
-bind_thm ("Inter_closed", axioms_L (thm "M_axioms.Inter_closed"));
-bind_thm ("Int_closed", axioms_L (thm "M_axioms.Int_closed"));
-bind_thm ("finite_fun_closed", axioms_L (thm "M_axioms.finite_fun_closed"));
-bind_thm ("is_funspace_abs", axioms_L (thm "M_axioms.is_funspace_abs"));
-bind_thm ("succ_fun_eq2", axioms_L (thm "M_axioms.succ_fun_eq2"));
-bind_thm ("funspace_succ", axioms_L (thm "M_axioms.funspace_succ"));
-bind_thm ("finite_funspace_closed", axioms_L (thm "M_axioms.finite_funspace_closed"));
-*}
-
-ML
-{*
-bind_thm ("is_recfun_equal", axioms_L (thm "M_axioms.is_recfun_equal"));  
-bind_thm ("is_recfun_cut", axioms_L (thm "M_axioms.is_recfun_cut")); 
-bind_thm ("is_recfun_functional", axioms_L (thm "M_axioms.is_recfun_functional"));
-bind_thm ("is_recfun_relativize", axioms_L (thm "M_axioms.is_recfun_relativize"));
-bind_thm ("is_recfun_restrict", axioms_L (thm "M_axioms.is_recfun_restrict"));
-bind_thm ("univalent_is_recfun", axioms_L (thm "M_axioms.univalent_is_recfun"));
-bind_thm ("exists_is_recfun_indstep", axioms_L (thm "M_axioms.exists_is_recfun_indstep"));
-bind_thm ("wellfounded_exists_is_recfun", axioms_L (thm "M_axioms.wellfounded_exists_is_recfun"));
-bind_thm ("wf_exists_is_recfun", axioms_L (thm "M_axioms.wf_exists_is_recfun")); 
-bind_thm ("is_recfun_abs", axioms_L (thm "M_axioms.is_recfun_abs"));
-bind_thm ("irreflexive_abs", axioms_L (thm "M_axioms.irreflexive_abs"));  
-bind_thm ("transitive_rel_abs", axioms_L (thm "M_axioms.transitive_rel_abs"));  
-bind_thm ("linear_rel_abs", axioms_L (thm "M_axioms.linear_rel_abs"));  
-bind_thm ("wellordered_is_trans_on", axioms_L (thm "M_axioms.wellordered_is_trans_on")); 
-bind_thm ("wellordered_is_linear", axioms_L (thm "M_axioms.wellordered_is_linear")); 
-bind_thm ("wellordered_is_wellfounded_on", axioms_L (thm "M_axioms.wellordered_is_wellfounded_on")); 
-bind_thm ("wellfounded_imp_wellfounded_on", axioms_L (thm "M_axioms.wellfounded_imp_wellfounded_on")); 
-bind_thm ("wellfounded_on_subset_A", axioms_L (thm "M_axioms.wellfounded_on_subset_A"));
-bind_thm ("wellfounded_on_iff_wellfounded", axioms_L (thm "M_axioms.wellfounded_on_iff_wellfounded"));
-bind_thm ("wellfounded_on_imp_wellfounded", axioms_L (thm "M_axioms.wellfounded_on_imp_wellfounded"));
-bind_thm ("wellfounded_on_field_imp_wellfounded", axioms_L (thm "M_axioms.wellfounded_on_field_imp_wellfounded"));
-bind_thm ("wellfounded_iff_wellfounded_on_field", axioms_L (thm "M_axioms.wellfounded_iff_wellfounded_on_field"));
-bind_thm ("wellfounded_induct", axioms_L (thm "M_axioms.wellfounded_induct")); 
-bind_thm ("wellfounded_on_induct", axioms_L (thm "M_axioms.wellfounded_on_induct")); 
-bind_thm ("wellfounded_on_induct2", axioms_L (thm "M_axioms.wellfounded_on_induct2")); 
-bind_thm ("linear_imp_relativized", axioms_L (thm "M_axioms.linear_imp_relativized")); 
-bind_thm ("trans_on_imp_relativized", axioms_L (thm "M_axioms.trans_on_imp_relativized")); 
-bind_thm ("wf_on_imp_relativized", axioms_L (thm "M_axioms.wf_on_imp_relativized")); 
-bind_thm ("wf_imp_relativized", axioms_L (thm "M_axioms.wf_imp_relativized")); 
-bind_thm ("well_ord_imp_relativized", axioms_L (thm "M_axioms.well_ord_imp_relativized")); 
-bind_thm ("order_isomorphism_abs", axioms_L (thm "M_axioms.order_isomorphism_abs"));  
-bind_thm ("pred_set_abs", axioms_L (thm "M_axioms.pred_set_abs"));  
-*}
-
-ML
-{*
-bind_thm ("pred_closed", axioms_L (thm "M_axioms.pred_closed"));  
-bind_thm ("membership_abs", axioms_L (thm "M_axioms.membership_abs"));  
-bind_thm ("M_Memrel_iff", axioms_L (thm "M_axioms.M_Memrel_iff"));
-bind_thm ("Memrel_closed", axioms_L (thm "M_axioms.Memrel_closed"));  
-bind_thm ("wellordered_iso_predD", axioms_L (thm "M_axioms.wellordered_iso_predD"));
-bind_thm ("wellordered_iso_pred_eq", axioms_L (thm "M_axioms.wellordered_iso_pred_eq"));
-bind_thm ("wellfounded_on_asym", axioms_L (thm "M_axioms.wellfounded_on_asym"));
-bind_thm ("wellordered_asym", axioms_L (thm "M_axioms.wellordered_asym"));
-bind_thm ("ord_iso_pred_imp_lt", axioms_L (thm "M_axioms.ord_iso_pred_imp_lt"));
-bind_thm ("obase_iff", axioms_L (thm "M_axioms.obase_iff"));
-bind_thm ("omap_iff", axioms_L (thm "M_axioms.omap_iff"));
-bind_thm ("omap_unique", axioms_L (thm "M_axioms.omap_unique"));
-bind_thm ("omap_yields_Ord", axioms_L (thm "M_axioms.omap_yields_Ord"));
-bind_thm ("otype_iff", axioms_L (thm "M_axioms.otype_iff"));
-bind_thm ("otype_eq_range", axioms_L (thm "M_axioms.otype_eq_range"));
-bind_thm ("Ord_otype", axioms_L (thm "M_axioms.Ord_otype"));
-bind_thm ("domain_omap", axioms_L (thm "M_axioms.domain_omap"));
-bind_thm ("omap_subset", axioms_L (thm "M_axioms.omap_subset")); 
-bind_thm ("omap_funtype", axioms_L (thm "M_axioms.omap_funtype")); 
-bind_thm ("wellordered_omap_bij", axioms_L (thm "M_axioms.wellordered_omap_bij"));
-bind_thm ("omap_ord_iso", axioms_L (thm "M_axioms.omap_ord_iso"));
-bind_thm ("Ord_omap_image_pred", axioms_L (thm "M_axioms.Ord_omap_image_pred"));
-bind_thm ("restrict_omap_ord_iso", axioms_L (thm "M_axioms.restrict_omap_ord_iso"));
-bind_thm ("obase_equals", axioms_L (thm "M_axioms.obase_equals")); 
-bind_thm ("omap_ord_iso_otype", axioms_L (thm "M_axioms.omap_ord_iso_otype"));
-bind_thm ("obase_exists", axioms_L (thm "M_axioms.obase_exists"));
-bind_thm ("omap_exists", axioms_L (thm "M_axioms.omap_exists"));
-bind_thm ("otype_exists", axioms_L (thm "M_axioms.otype_exists"));
-bind_thm ("omap_ord_iso_otype", axioms_L (thm "M_axioms.omap_ord_iso_otype"));
-bind_thm ("ordertype_exists", axioms_L (thm "M_axioms.ordertype_exists"));
-bind_thm ("relativized_imp_well_ord", axioms_L (thm "M_axioms.relativized_imp_well_ord")); 
-bind_thm ("well_ord_abs", axioms_L (thm "M_axioms.well_ord_abs"));  
-*}
+theorem M_axioms_axioms_L: "M_axioms_axioms(L)"
+  apply (rule M_axioms_axioms.intro)
+               apply (assumption | rule
+                 Inter_separation cartprod_separation image_separation
+                 converse_separation restrict_separation
+                 comp_separation pred_separation Memrel_separation
+                 funspace_succ_replacement well_ord_iso_separation
+                 obase_separation obase_equals_separation
+                 omap_replacement is_recfun_separation)+
+  done
+  
+theorem M_axioms_L: "PROP M_axioms(L)"
+  apply (rule M_axioms.intro)
+   apply (rule M_triv_axioms_L)
+  apply (rule M_axioms_axioms_L)
+  done
+
+lemmas cartprod_iff = M_axioms.cartprod_iff [OF M_axioms_L]
+  and cartprod_closed = M_axioms.cartprod_closed [OF M_axioms_L]
+  and sum_closed = M_axioms.sum_closed [OF M_axioms_L]
+  and M_converse_iff = M_axioms.M_converse_iff [OF M_axioms_L]
+  and converse_closed = M_axioms.converse_closed [OF M_axioms_L]
+  and converse_abs = M_axioms.converse_abs [OF M_axioms_L]
+  and image_closed = M_axioms.image_closed [OF M_axioms_L]
+  and vimage_abs = M_axioms.vimage_abs [OF M_axioms_L]
+  and vimage_closed = M_axioms.vimage_closed [OF M_axioms_L]
+  and domain_abs = M_axioms.domain_abs [OF M_axioms_L]
+  and domain_closed = M_axioms.domain_closed [OF M_axioms_L]
+  and range_abs = M_axioms.range_abs [OF M_axioms_L]
+  and range_closed = M_axioms.range_closed [OF M_axioms_L]
+  and field_abs = M_axioms.field_abs [OF M_axioms_L]
+  and field_closed = M_axioms.field_closed [OF M_axioms_L]
+  and relation_abs = M_axioms.relation_abs [OF M_axioms_L]
+  and function_abs = M_axioms.function_abs [OF M_axioms_L]
+  and apply_closed = M_axioms.apply_closed [OF M_axioms_L]
+  and apply_abs = M_axioms.apply_abs [OF M_axioms_L]
+  and typed_function_abs = M_axioms.typed_function_abs [OF M_axioms_L]
+  and injection_abs = M_axioms.injection_abs [OF M_axioms_L]
+  and surjection_abs = M_axioms.surjection_abs [OF M_axioms_L]
+  and bijection_abs = M_axioms.bijection_abs [OF M_axioms_L]
+  and M_comp_iff = M_axioms.M_comp_iff [OF M_axioms_L]
+  and comp_closed = M_axioms.comp_closed [OF M_axioms_L]
+  and composition_abs = M_axioms.composition_abs [OF M_axioms_L]
+  and restriction_is_function = M_axioms.restriction_is_function [OF M_axioms_L]
+  and restriction_abs = M_axioms.restriction_abs [OF M_axioms_L]
+  and M_restrict_iff = M_axioms.M_restrict_iff [OF M_axioms_L]
+  and restrict_closed = M_axioms.restrict_closed [OF M_axioms_L]
+  and Inter_abs = M_axioms.Inter_abs [OF M_axioms_L]
+  and Inter_closed = M_axioms.Inter_closed [OF M_axioms_L]
+  and Int_closed = M_axioms.Int_closed [OF M_axioms_L]
+  and finite_fun_closed = M_axioms.finite_fun_closed [OF M_axioms_L]
+  and is_funspace_abs = M_axioms.is_funspace_abs [OF M_axioms_L]
+  and succ_fun_eq2 = M_axioms.succ_fun_eq2 [OF M_axioms_L]
+  and funspace_succ = M_axioms.funspace_succ [OF M_axioms_L]
+  and finite_funspace_closed = M_axioms.finite_funspace_closed [OF M_axioms_L]
+
+lemmas is_recfun_equal = M_axioms.is_recfun_equal [OF M_axioms_L]
+  and is_recfun_cut = M_axioms.is_recfun_cut [OF M_axioms_L]
+  and is_recfun_functional = M_axioms.is_recfun_functional [OF M_axioms_L]
+  and is_recfun_relativize = M_axioms.is_recfun_relativize [OF M_axioms_L]
+  and is_recfun_restrict = M_axioms.is_recfun_restrict [OF M_axioms_L]
+  and univalent_is_recfun = M_axioms.univalent_is_recfun [OF M_axioms_L]
+  and exists_is_recfun_indstep = M_axioms.exists_is_recfun_indstep [OF M_axioms_L]
+  and wellfounded_exists_is_recfun = M_axioms.wellfounded_exists_is_recfun [OF M_axioms_L]
+  and wf_exists_is_recfun = M_axioms.wf_exists_is_recfun [OF M_axioms_L]
+  and is_recfun_abs = M_axioms.is_recfun_abs [OF M_axioms_L]
+  and irreflexive_abs = M_axioms.irreflexive_abs [OF M_axioms_L]
+  and transitive_rel_abs = M_axioms.transitive_rel_abs [OF M_axioms_L]
+  and linear_rel_abs = M_axioms.linear_rel_abs [OF M_axioms_L]
+  and wellordered_is_trans_on = M_axioms.wellordered_is_trans_on [OF M_axioms_L]
+  and wellordered_is_linear = M_axioms.wellordered_is_linear [OF M_axioms_L]
+  and wellordered_is_wellfounded_on = M_axioms.wellordered_is_wellfounded_on [OF M_axioms_L]
+  and wellfounded_imp_wellfounded_on = M_axioms.wellfounded_imp_wellfounded_on [OF M_axioms_L]
+  and wellfounded_on_subset_A = M_axioms.wellfounded_on_subset_A [OF M_axioms_L]
+  and wellfounded_on_iff_wellfounded = M_axioms.wellfounded_on_iff_wellfounded [OF M_axioms_L]
+  and wellfounded_on_imp_wellfounded = M_axioms.wellfounded_on_imp_wellfounded [OF M_axioms_L]
+  and wellfounded_on_field_imp_wellfounded = M_axioms.wellfounded_on_field_imp_wellfounded [OF M_axioms_L]
+  and wellfounded_iff_wellfounded_on_field = M_axioms.wellfounded_iff_wellfounded_on_field [OF M_axioms_L]
+  and wellfounded_induct = M_axioms.wellfounded_induct [OF M_axioms_L]
+  and wellfounded_on_induct = M_axioms.wellfounded_on_induct [OF M_axioms_L]
+  and wellfounded_on_induct2 = M_axioms.wellfounded_on_induct2 [OF M_axioms_L]
+  and linear_imp_relativized = M_axioms.linear_imp_relativized [OF M_axioms_L]
+  and trans_on_imp_relativized = M_axioms.trans_on_imp_relativized [OF M_axioms_L]
+  and wf_on_imp_relativized = M_axioms.wf_on_imp_relativized [OF M_axioms_L]
+  and wf_imp_relativized = M_axioms.wf_imp_relativized [OF M_axioms_L]
+  and well_ord_imp_relativized = M_axioms.well_ord_imp_relativized [OF M_axioms_L]
+  and order_isomorphism_abs = M_axioms.order_isomorphism_abs [OF M_axioms_L]
+  and pred_set_abs = M_axioms.pred_set_abs [OF M_axioms_L]
+
+lemmas pred_closed = M_axioms.pred_closed [OF M_axioms_L]
+  and membership_abs = M_axioms.membership_abs [OF M_axioms_L]
+  and M_Memrel_iff = M_axioms.M_Memrel_iff [OF M_axioms_L]
+  and Memrel_closed = M_axioms.Memrel_closed [OF M_axioms_L]
+  and wellordered_iso_predD = M_axioms.wellordered_iso_predD [OF M_axioms_L]
+  and wellordered_iso_pred_eq = M_axioms.wellordered_iso_pred_eq [OF M_axioms_L]
+  and wellfounded_on_asym = M_axioms.wellfounded_on_asym [OF M_axioms_L]
+  and wellordered_asym = M_axioms.wellordered_asym [OF M_axioms_L]
+  and ord_iso_pred_imp_lt = M_axioms.ord_iso_pred_imp_lt [OF M_axioms_L]
+  and obase_iff = M_axioms.obase_iff [OF M_axioms_L]
+  and omap_iff = M_axioms.omap_iff [OF M_axioms_L]
+  and omap_unique = M_axioms.omap_unique [OF M_axioms_L]
+  and omap_yields_Ord = M_axioms.omap_yields_Ord [OF M_axioms_L]
+  and otype_iff = M_axioms.otype_iff [OF M_axioms_L]
+  and otype_eq_range = M_axioms.otype_eq_range [OF M_axioms_L]
+  and Ord_otype = M_axioms.Ord_otype [OF M_axioms_L]
+  and domain_omap = M_axioms.domain_omap [OF M_axioms_L]
+  and omap_subset = M_axioms.omap_subset [OF M_axioms_L]
+  and omap_funtype = M_axioms.omap_funtype [OF M_axioms_L]
+  and wellordered_omap_bij = M_axioms.wellordered_omap_bij [OF M_axioms_L]
+  and omap_ord_iso = M_axioms.omap_ord_iso [OF M_axioms_L]
+  and Ord_omap_image_pred = M_axioms.Ord_omap_image_pred [OF M_axioms_L]
+  and restrict_omap_ord_iso = M_axioms.restrict_omap_ord_iso [OF M_axioms_L]
+  and obase_equals = M_axioms.obase_equals [OF M_axioms_L]
+  and omap_ord_iso_otype = M_axioms.omap_ord_iso_otype [OF M_axioms_L]
+  and obase_exists = M_axioms.obase_exists [OF M_axioms_L]
+  and omap_exists = M_axioms.omap_exists [OF M_axioms_L]
+  and otype_exists = M_axioms.otype_exists [OF M_axioms_L]
+  and omap_ord_iso_otype' = M_axioms.omap_ord_iso_otype' [OF M_axioms_L]
+  and ordertype_exists = M_axioms.ordertype_exists [OF M_axioms_L]
+  and relativized_imp_well_ord = M_axioms.relativized_imp_well_ord [OF M_axioms_L]
+  and well_ord_abs = M_axioms.well_ord_abs [OF M_axioms_L]
+
 
 declare cartprod_closed [intro,simp]
 declare sum_closed [intro,simp]
 declare converse_closed [intro,simp]
 declare converse_abs [simp]

 declare restriction_abs [simp]
 declare restrict_closed [intro,simp]
 declare Inter_abs [simp]
 declare Inter_closed [intro,simp]
 declare Int_closed [intro,simp]
-declare finite_fun_closed [rule_format]
 declare is_funspace_abs [simp]
 declare finite_funspace_closed [intro,simp]
 
 end