src/ZF/Constructible/WFrec.thy

new theory of inner models

theory WFrec = Wellorderings:


(*WF.thy??*)

lemma is_recfunI:
     "f = (lam x: r-``{a}. H(x, restrict(f, r-``{x}))) ==> is_recfun(r,a,H,f)"
by (simp add: is_recfun_def) 

lemma is_recfun_imp_function: "is_recfun(r,a,H,f) ==> function(f)"
apply (simp add: is_recfun_def) 
apply (erule ssubst)
apply (rule function_lam) 
done

text{*Expresses @{text is_recfun} as a recursion equation*}
lemma is_recfun_iff_equation:
     "is_recfun(r,a,H,f) <->
	   f \<in> r -`` {a} \<rightarrow> range(f) &
	   (\<forall>x \<in> r-``{a}. f`x = H(x, restrict(f, r-``{x})))"  
apply (rule iffI) 
 apply (simp add: is_recfun_type apply_recfun Ball_def vimage_singleton_iff, 
        clarify)  
apply (simp add: is_recfun_def) 
apply (rule fun_extension) 
  apply assumption
 apply (fast intro: lam_type, simp) 
done

lemma trans_on_Int_eq [simp]:
      "[| trans[A](r); <y,x> \<in> r;  r \<subseteq> A*A |] 
       ==> r -`` {y} \<inter> r -`` {x} = r -`` {y}"
by (blast intro: trans_onD) 

lemma trans_on_Int_eq2 [simp]:
      "[| trans[A](r); <y,x> \<in> r;  r \<subseteq> A*A |] 
       ==> r -`` {x} \<inter> r -`` {y} = r -`` {y}"
by (blast intro: trans_onD) 


constdefs
   M_the_recfun :: "[i=>o, i, i, [i,i]=>i] => i"
     "M_the_recfun(M,r,a,H) == (THE f. M(f) & is_recfun(r,a,H,f))"


text{*Stated using @{term "trans[A](r)"} rather than
      @{term "transitive_rel(M,A,r)"} because the latter rewrites to
      the former anyway, by @{text transitive_rel_abs}.
      As always, theorems should be expressed in simplified form.*}
lemma (in M_axioms) is_recfun_equal [rule_format]: 
    "[|is_recfun(r,a,H,f);  is_recfun(r,b,H,g);  
       wellfounded_on(M,A,r);  trans[A](r); 
       M(A); M(f); M(g); M(a); M(b); 
       r \<subseteq> A*A;  x\<in>A |] 
     ==> <x,a> \<in> r --> <x,b> \<in> r --> f`x=g`x"
apply (frule_tac f="f" in is_recfun_type) 
apply (frule_tac f="g" in is_recfun_type) 
apply (simp add: is_recfun_def)
apply (erule wellfounded_on_induct2, assumption+) 
apply (force intro: is_recfun_separation, clarify)
apply (erule ssubst)+
apply (simp (no_asm_simp) add: vimage_singleton_iff restrict_def)
apply (rename_tac x1)
apply (rule_tac t="%z. H(x1,z)" in subst_context) 
apply (subgoal_tac "ALL y : r-``{x1}. ALL z. <y,z>:f <-> <y,z>:g")
 apply (blast intro: trans_onD) 
apply (simp add: apply_iff) 
apply (blast intro: trans_onD sym) 
done

lemma (in M_axioms) is_recfun_cut: 
    "[|is_recfun(r,a,H,f);  is_recfun(r,b,H,g);  
       wellfounded_on(M,A,r); trans[A](r); 
       M(A); M(f); M(g); M(a); M(b); 
       r \<subseteq> A*A;  <b,a>\<in>r |]   
      ==> restrict(f, r-``{b}) = g"
apply (frule_tac f="f" in is_recfun_type) 
apply (rule fun_extension) 
apply (blast intro: trans_onD restrict_type2) 
apply (erule is_recfun_type, simp) 
apply (blast intro: is_recfun_equal trans_onD) 
done

lemma (in M_axioms) is_recfun_functional:
     "[|is_recfun(r,a,H,f);  is_recfun(r,a,H,g);  
       wellfounded_on(M,A,r); trans[A](r); 
       M(A); M(f); M(g); M(a); 
       r \<subseteq> A*A |]   
      ==> f=g"
apply (rule fun_extension)
apply (erule is_recfun_type)+
apply (blast intro!: is_recfun_equal) 
done

text{*Tells us that is_recfun can (in principle) be relativized.*}
lemma (in M_axioms) is_recfun_relativize:
     "[| M(r); M(a); M(f); 
       \<forall>x g. M(x) & M(g) & function(g) --> M(H(x,g)) |] ==>
       is_recfun(r,a,H,f) <->
       (\<forall>z. z \<in> f <-> (\<exists>x y. M(x) & M(y) & z=<x,y> & <x,a> \<in> r & 
                              y = H(x, restrict(f, r-``{x}))))";
apply (simp add: is_recfun_def vimage_closed restrict_closed lam_def)
apply (safe intro!: equalityI) 
  apply (drule equalityD1 [THEN subsetD], assumption) 
  apply clarify 
  apply (rule_tac x=x in exI) 
  apply (blast dest: pair_components_in_M) 
 apply (blast elim!: equalityE dest: pair_components_in_M)
 apply simp  
 apply blast
 apply simp 
apply (subgoal_tac "function(f)")  
 prefer 2
 apply (simp add: function_def) 
apply (frule pair_components_in_M, assumption) 
  apply (simp add: is_recfun_imp_function function_restrictI restrict_closed vimage_closed) 
done

(* ideas for further weaking the H-closure premise:
apply (drule spec [THEN spec]) 
apply (erule mp)
apply (intro conjI)
apply (blast dest!: pair_components_in_M)
apply (blast intro!: function_restrictI dest!: pair_components_in_M)
apply (blast intro!: function_restrictI dest!: pair_components_in_M)
apply (simp only: subset_iff domain_iff restrict_iff vimage_iff) 
apply (simp add:  vimage_singleton_iff) 
apply (intro allI impI conjI)
apply (blast intro: transM dest!: pair_components_in_M)
prefer 4;apply blast 
*)

lemma (in M_axioms) is_recfun_restrict:
     "[| wellfounded_on(M,A,r); trans[A](r); is_recfun(r,x,H,f); \<langle>y,x\<rangle> \<in> r; 
       M(A); M(r); M(f); 
       \<forall>x g. M(x) & M(g) & function(g) --> M(H(x,g)); r \<subseteq> A * A |]
       ==> is_recfun(r, y, H, restrict(f, r -`` {y}))"
apply (frule pair_components_in_M, assumption, clarify) 
apply (simp (no_asm_simp) add: is_recfun_relativize vimage_closed restrict_closed
     restrict_iff)
apply safe
  apply (simp_all add: vimage_singleton_iff is_recfun_type [THEN apply_iff]) 
  apply (frule_tac x=xa in pair_components_in_M, assumption)
  apply (frule_tac x=xa in apply_recfun, blast intro: trans_onD)  
  apply (simp add: is_recfun_type [THEN apply_iff])
  (*???COMBINE*) 
  apply (simp add: is_recfun_imp_function function_restrictI restrict_closed vimage_closed) 
apply (blast intro: apply_recfun dest: trans_onD)+
done
 
lemma (in M_axioms) restrict_Y_lemma:
     "[| wellfounded_on(M,A,r); trans[A](r); M(A); M(r); r \<subseteq> A \<times> A;
       \<forall>x g. M(x) \<and> M(g) & function(g) --> M(H(x,g));  M(Y);
       \<forall>b. M(b) -->
	   b \<in> Y <->
	   (\<exists>x\<in>r -`` {a1}.
	       \<exists>y. M(y) \<and>
		   (\<exists>g. M(g) \<and> b = \<langle>x,y\<rangle> \<and> is_recfun(r,x,H,g) \<and> y = H(x,g)));
          \<langle>x,a1\<rangle> \<in> r; M(f); is_recfun(r,x,H,f) |]
       ==> restrict(Y, r -`` {x}) = f"
apply (subgoal_tac "ALL y : r-``{x}. ALL z. <y,z>:Y <-> <y,z>:f") 
apply (simp (no_asm_simp) add: restrict_def) 
apply (thin_tac "All(?P)")+  --{*essential for efficiency*}
apply (frule is_recfun_type [THEN fun_is_rel], blast)
apply (frule pair_components_in_M, assumption, clarify) 
apply (rule iffI)
 apply (frule_tac y="<y,z>" in transM, assumption )
 apply (rotate_tac -1)   
 apply (clarsimp simp add: vimage_singleton_iff is_recfun_type [THEN apply_iff]
			   apply_recfun is_recfun_cut) 
txt{*Opposite inclusion: something in f, show in Y*}
apply (frule_tac y="<y,z>" in transM, assumption, simp) 
apply (rule_tac x=y in bexI)
prefer 2 apply (blast dest: trans_onD)
apply (rule_tac x=z in exI, simp) 
apply (rule_tac x="restrict(f, r -`` {y})" in exI) 
apply (simp add: vimage_closed restrict_closed is_recfun_restrict
                 apply_recfun is_recfun_type [THEN apply_iff]) 
done

text{*Proof of the inductive step for @{text exists_is_recfun}, since
      we must prove two versions.*}
lemma (in M_axioms) exists_is_recfun_indstep:
    "[|a1 \<in> A;  \<forall>y. \<langle>y, a1\<rangle> \<in> r --> (\<exists>f. M(f) & is_recfun(r, y, H, f)); 
       wellfounded_on(M,A,r); trans[A](r); 
       strong_replacement(M, \<lambda>x z. \<exists>y g. M(y) & M(g) &
                   pair(M,x,y,z) & is_recfun(r,x,H,g) & y = H(x,g)); 
       M(A); M(r); r \<subseteq> A * A;
       \<forall>x g. M(x) & M(g) & function(g) --> M(H(x,g))|]   
      ==> \<exists>f. M(f) & is_recfun(r,a1,H,f)"
apply (frule_tac y=a1 in transM, assumption)
apply (drule_tac A="r-``{a1}" in strong_replacementD)
  apply blast
 txt{*Discharge the "univalent" obligation of Replacement*}
 apply (clarsimp simp add: univalent_def)
 apply (blast dest!: is_recfun_functional)
txt{*Show that the constructed object satisfies @{text is_recfun}*} 
apply clarify 
apply (rule_tac x=Y in exI)  
apply (simp (no_asm_simp) add: is_recfun_relativize vimage_closed restrict_closed) 
(*Tried using is_recfun_iff2 here.  Much more simplification takes place
  because an assumption can kick in.  Not sure how to relate the new 
  proof state to the current one.*)
apply safe
txt{*Show that elements of @{term Y} are in the right relationship.*}
apply (frule_tac x=z and P="%b. M(b) --> ?Q(b)" in spec)
apply (erule impE, blast intro: transM)
txt{*We have an element of  @{term Y}, so we have x, y, z*} 
apply (frule_tac y=z in transM, assumption, clarify)
apply (simp add: vimage_closed restrict_closed restrict_Y_lemma [of A r H]) 
txt{*one more case*}
apply (simp add: vimage_closed restrict_closed )
apply (rule_tac x=x in bexI) 
prefer 2 apply blast 
apply (rule_tac x="H(x, restrict(Y, r -`` {x}))" in exI) 
apply (simp add: vimage_closed restrict_closed )
apply (drule_tac x1=x in spec [THEN mp], assumption, clarify) 
apply (rule_tac x=f in exI) 
apply (simp add: restrict_Y_lemma [of A r H]) 
done


text{*Relativized version, when we have the (currently weaker) premise
      @{term "wellfounded_on(M,A,r)"}*}
lemma (in M_axioms) wellfounded_exists_is_recfun:
    "[|wellfounded_on(M,A,r);  trans[A](r);  a\<in>A; 
       separation(M, \<lambda>x. x \<in> A --> ~ (\<exists>f. M(f) \<and> is_recfun(r, x, H, f)));
       strong_replacement(M, \<lambda>x z. \<exists>y g. M(y) & M(g) &
                   pair(M,x,y,z) & is_recfun(r,x,H,g) & y = H(x,g)); 
       M(A);  M(r);  r \<subseteq> A*A;  
       \<forall>x g. M(x) & M(g) & function(g) --> M(H(x,g)) |]   
      ==> \<exists>f. M(f) & is_recfun(r,a,H,f)"
apply (rule wellfounded_on_induct2, assumption+, clarify)
apply (rule exists_is_recfun_indstep, assumption+)
done

lemma (in M_axioms) wf_exists_is_recfun:
    "[|wf[A](r);  trans[A](r);  a\<in>A; 
       strong_replacement(M, \<lambda>x z. \<exists>y g. M(y) & M(g) &
                   pair(M,x,y,z) & is_recfun(r,x,H,g) & y = H(x,g)); 
       M(A);  M(r);  r \<subseteq> A*A;  
       \<forall>x g. M(x) & M(g) & function(g) --> M(H(x,g)) |]   
      ==> \<exists>f. M(f) & is_recfun(r,a,H,f)"
apply (rule wf_on_induct2, assumption+)
apply (frule wf_on_imp_relativized)  
apply (rule exists_is_recfun_indstep, assumption+)
done

(*If some f satisfies is_recfun(r,a,H,-) then so does M_the_recfun(M,r,a,H) *)
lemma (in M_axioms) M_is_the_recfun: 
    "[|is_recfun(r,a,H,f);  
       wellfounded_on(M,A,r); trans[A](r); 
       M(A); M(f); M(a); r \<subseteq> A*A |]   
     ==> M(M_the_recfun(M,r,a,H)) & 
         is_recfun(r, a, H, M_the_recfun(M,r,a,H))"
apply (unfold M_the_recfun_def)
apply (rule ex1I [THEN theI2], fast)
apply (blast intro: is_recfun_functional, blast) 
done

constdefs
   M_is_recfun :: "[i=>o, i, i, [i=>o,i,i,i]=>o, i] => o"
     "M_is_recfun(M,r,a,MH,f) == 
      \<forall>z. M(z) -->
          (z \<in> f <-> 
           (\<exists>x y xa sx r_sx f_r_sx. 
              M(x) & M(y) & M(xa) & M(sx) & M(r_sx) & M(f_r_sx) &
	      pair(M,x,y,z) & pair(M,x,a,xa) & upair(M,x,x,sx) &
              pre_image(M,r,sx,r_sx) & restriction(M,f,r_sx,f_r_sx) &
              xa \<in> r & MH(M, x, f_r_sx, y)))"

lemma (in M_axioms) is_recfun_iff_M:
     "[| M(r); M(a); M(f); \<forall>x g. M(x) & M(g) & function(g) --> M(H(x,g));
       \<forall>x g y. M(x) --> M(g) --> M(y) --> MH(M,x,g,y) <-> y = H(x,g) |] ==>
       is_recfun(r,a,H,f) <-> M_is_recfun(M,r,a,MH,f)"
apply (simp add: vimage_closed restrict_closed M_is_recfun_def is_recfun_relativize)
apply (rule all_cong, safe)
 apply (thin_tac "\<forall>x. ?P(x)")+
 apply (blast dest: transM)  (*or del: allE*)
done

lemma M_is_recfun_cong [cong]:
     "[| r = r'; a = a'; f = f'; 
       !!x g y. [| M(x); M(g); M(y) |] ==> MH(M,x,g,y) <-> MH'(M,x,g,y) |]
      ==> M_is_recfun(M,r,a,MH,f) <-> M_is_recfun(M,r',a',MH',f')"
by (simp add: M_is_recfun_def) 


constdefs
 (*This expresses ordinal addition as a formula in the LAST.  It also 
   provides an abbreviation that can be used in the instance of strong
   replacement below.  Here j is used to define the relation, namely
   Memrel(succ(j)), while x determines the domain of f.*)
 is_oadd_fun :: "[i=>o,i,i,i,i] => o"
    "is_oadd_fun(M,i,j,x,f) == 
       (\<forall>sj msj. M(sj) --> M(msj) --> 
                 successor(M,j,sj) --> membership(M,sj,msj) --> 
	         M_is_recfun(M, msj, x, 
		     %M x g y. \<exists>gx. M(gx) & image(M,g,x,gx) & union(M,i,gx,y),
		     f))"

 is_oadd :: "[i=>o,i,i,i] => o"
    "is_oadd(M,i,j,k) == 
        (~ ordinal(M,i) & ~ ordinal(M,j) & k=0) |
        (~ ordinal(M,i) & ordinal(M,j) & k=j) |
        (ordinal(M,i) & ~ ordinal(M,j) & k=i) |
        (ordinal(M,i) & ordinal(M,j) & 
	 (\<exists>f fj sj. M(f) & M(fj) & M(sj) & 
		    successor(M,j,sj) & is_oadd_fun(M,i,sj,sj,f) & 
		    fun_apply(M,f,j,fj) & fj = k))"

 (*NEEDS RELATIVIZATION*)
 omult_eqns :: "[i,i,i,i] => o"
    "omult_eqns(i,x,g,z) ==
            Ord(x) & 
	    (x=0 --> z=0) &
            (\<forall>j. x = succ(j) --> z = g`j ++ i) &
            (Limit(x) --> z = \<Union>(g``x))"

 is_omult_fun :: "[i=>o,i,i,i] => o"
    "is_omult_fun(M,i,j,f) == 
	    (\<exists>df. M(df) & is_function(M,f) & 
                  is_domain(M,f,df) & subset(M, j, df)) & 
            (\<forall>x\<in>j. omult_eqns(i,x,f,f`x))"

 is_omult :: "[i=>o,i,i,i] => o"
    "is_omult(M,i,j,k) == 
	\<exists>f fj sj. M(f) & M(fj) & M(sj) & 
                  successor(M,j,sj) & is_omult_fun(M,i,sj,f) & 
                  fun_apply(M,f,j,fj) & fj = k"


locale M_recursion = M_axioms +
  assumes oadd_strong_replacement:
   "[| M(i); M(j) |] ==>
    strong_replacement(M, 
         \<lambda>x z. \<exists>y f fx. M(y) & M(f) & M(fx) & 
		         pair(M,x,y,z) & is_oadd_fun(M,i,j,x,f) & 
		         image(M,f,x,fx) & y = i Un fx)" 
 and omult_strong_replacement':
   "[| M(i); M(j) |] ==>
    strong_replacement(M, \<lambda>x z. \<exists>y g. M(y) & M(g) &
	     pair(M,x,y,z) & 
	     is_recfun(Memrel(succ(j)),x,%x g. THE z. omult_eqns(i,x,g,z),g) & 
	     y = (THE z. omult_eqns(i, x, g, z)))" 



text{*is_oadd_fun: Relating the pure "language of set theory" to Isabelle/ZF*}
lemma (in M_recursion) is_oadd_fun_iff:
   "[| a\<le>j; M(i); M(j); M(a); M(f) |] 
    ==> is_oadd_fun(M,i,j,a,f) <->
	f \<in> a \<rightarrow> range(f) & (\<forall>x. M(x) --> x < a --> f`x = i Un f``x)"
apply (frule lt_Ord) 
apply (simp add: is_oadd_fun_def Memrel_closed Un_closed 
             is_recfun_iff_M [of concl: _ _ "%x g. i Un g``x", THEN iff_sym]
             image_closed is_recfun_iff_equation  
             Ball_def lt_trans [OF ltI, of _ a] lt_Memrel)
apply (simp add: lt_def) 
apply (blast dest: transM) 
done


lemma (in M_recursion) oadd_strong_replacement':
    "[| M(i); M(j) |] ==>
     strong_replacement(M, \<lambda>x z. \<exists>y g. M(y) & M(g) &
		  pair(M,x,y,z) & 
		  is_recfun(Memrel(succ(j)),x,%x g. i Un g``x,g) & 
		  y = i Un g``x)" 
apply (insert oadd_strong_replacement [of i j]) 
apply (simp add: Memrel_closed Un_closed image_closed is_oadd_fun_def
                 is_recfun_iff_M)  
done


lemma (in M_recursion) exists_oadd:
    "[| Ord(j);  M(i);  M(j) |]
     ==> \<exists>f. M(f) & is_recfun(Memrel(succ(j)), j, %x g. i Un g``x, f)"
apply (rule wf_exists_is_recfun) 
apply (rule wf_Memrel [THEN wf_imp_wf_on]) 
apply (rule trans_Memrel [THEN trans_imp_trans_on], simp)  
apply (rule succI1) 
apply (blast intro: oadd_strong_replacement') 
apply (simp_all add: Memrel_type Memrel_closed Un_closed image_closed)
done

lemma (in M_recursion) exists_oadd_fun:
    "[| Ord(j);  M(i);  M(j) |] 
     ==> \<exists>f. M(f) & is_oadd_fun(M,i,succ(j),succ(j),f)"
apply (rule exists_oadd [THEN exE])
apply (erule Ord_succ, assumption, simp) 
apply (rename_tac f, clarify) 
apply (frule is_recfun_type)
apply (rule_tac x=f in exI) 
apply (simp add: fun_is_function domain_of_fun lt_Memrel apply_recfun lt_def
                 is_oadd_fun_iff Ord_trans [OF _ succI1])
done

lemma (in M_recursion) is_oadd_fun_apply:
    "[| x < j; M(i); M(j); M(f); is_oadd_fun(M,i,j,j,f) |] 
     ==> f`x = i Un (\<Union>k\<in>x. {f ` k})"
apply (simp add: is_oadd_fun_iff lt_Ord2, clarify) 
apply (frule lt_closed, simp)
apply (frule leI [THEN le_imp_subset])  
apply (simp add: image_fun, blast) 
done

lemma (in M_recursion) is_oadd_fun_iff_oadd [rule_format]:
    "[| is_oadd_fun(M,i,J,J,f); M(i); M(J); M(f); Ord(i); Ord(j) |] 
     ==> j<J --> f`j = i++j"
apply (erule_tac i=j in trans_induct, clarify) 
apply (subgoal_tac "\<forall>k\<in>x. k<J")
 apply (simp (no_asm_simp) add: is_oadd_def oadd_unfold is_oadd_fun_apply)
apply (blast intro: lt_trans ltI lt_Ord) 
done

lemma (in M_recursion) oadd_abs_fun_apply_iff:
    "[| M(i); M(J); M(f); M(k); j<J; is_oadd_fun(M,i,J,J,f) |] 
     ==> fun_apply(M,f,j,k) <-> f`j = k"
by (force simp add: lt_def is_oadd_fun_iff subsetD typed_apply_abs) 

lemma (in M_recursion) Ord_oadd_abs:
    "[| M(i); M(j); M(k); Ord(i); Ord(j) |] ==> is_oadd(M,i,j,k) <-> k = i++j"
apply (simp add: is_oadd_def oadd_abs_fun_apply_iff is_oadd_fun_iff_oadd)
apply (frule exists_oadd_fun [of j i], blast+)
done

lemma (in M_recursion) oadd_abs:
    "[| M(i); M(j); M(k) |] ==> is_oadd(M,i,j,k) <-> k = i++j"
apply (case_tac "Ord(i) & Ord(j)")
 apply (simp add: Ord_oadd_abs)
apply (auto simp add: is_oadd_def oadd_eq_if_raw_oadd)
done

lemma (in M_recursion) oadd_closed [intro]:
    "[| M(i); M(j) |] ==> M(i++j)"
apply (simp add: oadd_eq_if_raw_oadd, clarify) 
apply (simp add: raw_oadd_eq_oadd) 
apply (frule exists_oadd_fun [of j i], auto)
apply (simp add: apply_closed is_oadd_fun_iff_oadd [symmetric]) 
done


text{*Ordinal Multiplication*}

lemma omult_eqns_unique:
     "[| omult_eqns(i,x,g,z); omult_eqns(i,x,g,z') |] ==> z=z'";
apply (simp add: omult_eqns_def, clarify) 
apply (erule Ord_cases, simp_all) 
done

lemma omult_eqns_0: "omult_eqns(i,0,g,z) <-> z=0"
by (simp add: omult_eqns_def)

lemma the_omult_eqns_0: "(THE z. omult_eqns(i,0,g,z)) = 0"
by (simp add: omult_eqns_0)

lemma omult_eqns_succ: "omult_eqns(i,succ(j),g,z) <-> Ord(j) & z = g`j ++ i"
by (simp add: omult_eqns_def)

lemma the_omult_eqns_succ:
     "Ord(j) ==> (THE z. omult_eqns(i,succ(j),g,z)) = g`j ++ i"
by (simp add: omult_eqns_succ) 

lemma omult_eqns_Limit:
     "Limit(x) ==> omult_eqns(i,x,g,z) <-> z = \<Union>(g``x)"
apply (simp add: omult_eqns_def) 
apply (blast intro: Limit_is_Ord) 
done

lemma the_omult_eqns_Limit:
     "Limit(x) ==> (THE z. omult_eqns(i,x,g,z)) = \<Union>(g``x)"
by (simp add: omult_eqns_Limit)

lemma omult_eqns_Not: "~ Ord(x) ==> ~ omult_eqns(i,x,g,z)"
by (simp add: omult_eqns_def)


lemma (in M_recursion) the_omult_eqns_closed:
    "[| M(i); M(x); M(g); function(g) |] 
     ==> M(THE z. omult_eqns(i, x, g, z))"
apply (case_tac "Ord(x)")
 prefer 2 apply (simp add: omult_eqns_Not) --{*trivial, non-Ord case*}
apply (erule Ord_cases) 
  apply (simp add: omult_eqns_0)
 apply (simp add: omult_eqns_succ apply_closed oadd_closed) 
apply (simp add: omult_eqns_Limit) 
apply (simp add: Union_closed image_closed) 
done

lemma (in M_recursion) exists_omult:
    "[| Ord(j);  M(i);  M(j) |]
     ==> \<exists>f. M(f) & is_recfun(Memrel(succ(j)), j, %x g. THE z. omult_eqns(i,x,g,z), f)"
apply (rule wf_exists_is_recfun) 
apply (rule wf_Memrel [THEN wf_imp_wf_on]) 
apply (rule trans_Memrel [THEN trans_imp_trans_on], simp)  
apply (rule succI1) 
apply (blast intro: omult_strong_replacement') 
apply (simp_all add: Memrel_type Memrel_closed Un_closed image_closed)
apply (blast intro: the_omult_eqns_closed) 
done

lemma (in M_recursion) exists_omult_fun:
    "[| Ord(j);  M(i);  M(j) |] ==> \<exists>f. M(f) & is_omult_fun(M,i,succ(j),f)"
apply (rule exists_omult [THEN exE])
apply (erule Ord_succ, assumption, simp) 
apply (rename_tac f, clarify) 
apply (frule is_recfun_type)
apply (rule_tac x=f in exI) 
apply (simp add: fun_is_function domain_of_fun lt_Memrel apply_recfun lt_def
                 is_omult_fun_def Ord_trans [OF _ succI1])
apply (force dest: Ord_in_Ord' 
             simp add: omult_eqns_def the_omult_eqns_0 the_omult_eqns_succ
                       the_omult_eqns_Limit) 
done

lemma (in M_recursion) is_omult_fun_apply_0:
    "[| 0 < j; is_omult_fun(M,i,j,f) |] ==> f`0 = 0"
by (simp add: is_omult_fun_def omult_eqns_def lt_def ball_conj_distrib)

lemma (in M_recursion) is_omult_fun_apply_succ:
    "[| succ(x) < j; is_omult_fun(M,i,j,f) |] ==> f`succ(x) = f`x ++ i"
by (simp add: is_omult_fun_def omult_eqns_def lt_def, blast) 

lemma (in M_recursion) is_omult_fun_apply_Limit:
    "[| x < j; Limit(x); M(j); M(f); is_omult_fun(M,i,j,f) |] 
     ==> f ` x = (\<Union>y\<in>x. f`y)"
apply (simp add: is_omult_fun_def omult_eqns_def domain_closed lt_def, clarify)
apply (drule subset_trans [OF OrdmemD], assumption+)  
apply (simp add: ball_conj_distrib omult_Limit image_function)
done

lemma (in M_recursion) is_omult_fun_eq_omult:
    "[| is_omult_fun(M,i,J,f); M(J); M(f); Ord(i); Ord(j) |] 
     ==> j<J --> f`j = i**j"
apply (erule_tac i=j in trans_induct3)
apply (safe del: impCE)
  apply (simp add: is_omult_fun_apply_0) 
 apply (subgoal_tac "x<J") 
  apply (simp add: is_omult_fun_apply_succ omult_succ)  
 apply (blast intro: lt_trans) 
apply (subgoal_tac "\<forall>k\<in>x. k<J")
 apply (simp add: is_omult_fun_apply_Limit omult_Limit) 
apply (blast intro: lt_trans ltI lt_Ord) 
done

lemma (in M_recursion) omult_abs_fun_apply_iff:
    "[| M(i); M(J); M(f); M(k); j<J; is_omult_fun(M,i,J,f) |] 
     ==> fun_apply(M,f,j,k) <-> f`j = k"
by (auto simp add: lt_def is_omult_fun_def subsetD apply_abs) 

lemma (in M_recursion) omult_abs:
    "[| M(i); M(j); M(k); Ord(i); Ord(j) |] ==> is_omult(M,i,j,k) <-> k = i**j"
apply (simp add: is_omult_def omult_abs_fun_apply_iff is_omult_fun_eq_omult)
apply (frule exists_omult_fun [of j i], blast+)
done

end