src/ZF/Constructible/Rec_Separation.thy
author paulson
Thu Jul 25 10:56:35 2002 +0200 (2002-07-25)
changeset 13422 af9bc8d87a75
parent 13418 7c0ba9dba978
child 13428 99e52e78eb65
permissions -rw-r--r--
Added the assumption nth_replacement to locale M_datatypes.
Moved up its proof to make it available for the instantiation of that locale.
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header{*Separation for Facts About Recursion*}
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theory Rec_Separation = Separation + Datatype_absolute:
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text{*This theory proves all instances needed for locales @{text
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"M_trancl"}, @{text "M_wfrank"} and @{text "M_datatypes"}*}
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lemma eq_succ_imp_lt: "[|i = succ(j); Ord(i)|] ==> j<i"
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by simp 
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subsection{*The Locale @{text "M_trancl"}*}
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subsubsection{*Separation for Reflexive/Transitive Closure*}
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text{*First, The Defining Formula*}
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(* "rtran_closure_mem(M,A,r,p) ==
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      \<exists>nnat[M]. \<exists>n[M]. \<exists>n'[M]. 
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       omega(M,nnat) & n\<in>nnat & successor(M,n,n') &
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       (\<exists>f[M]. typed_function(M,n',A,f) &
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	(\<exists>x[M]. \<exists>y[M]. \<exists>zero[M]. pair(M,x,y,p) & empty(M,zero) &
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	  fun_apply(M,f,zero,x) & fun_apply(M,f,n,y)) &
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	(\<forall>j[M]. j\<in>n --> 
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	  (\<exists>fj[M]. \<exists>sj[M]. \<exists>fsj[M]. \<exists>ffp[M]. 
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	    fun_apply(M,f,j,fj) & successor(M,j,sj) &
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	    fun_apply(M,f,sj,fsj) & pair(M,fj,fsj,ffp) & ffp \<in> r)))"*)
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constdefs rtran_closure_mem_fm :: "[i,i,i]=>i"
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 "rtran_closure_mem_fm(A,r,p) == 
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   Exists(Exists(Exists(
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    And(omega_fm(2),
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     And(Member(1,2),
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      And(succ_fm(1,0),
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       Exists(And(typed_function_fm(1, A#+4, 0),
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	And(Exists(Exists(Exists(
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	      And(pair_fm(2,1,p#+7), 
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	       And(empty_fm(0),
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		And(fun_apply_fm(3,0,2), fun_apply_fm(3,5,1))))))),
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	    Forall(Implies(Member(0,3),
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	     Exists(Exists(Exists(Exists(
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	      And(fun_apply_fm(5,4,3),
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	       And(succ_fm(4,2),
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		And(fun_apply_fm(5,2,1),
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		 And(pair_fm(3,1,0), Member(0,r#+9))))))))))))))))))))"
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lemma rtran_closure_mem_type [TC]:
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 "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> rtran_closure_mem_fm(x,y,z) \<in> formula"
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by (simp add: rtran_closure_mem_fm_def) 
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lemma arity_rtran_closure_mem_fm [simp]:
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     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
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      ==> arity(rtran_closure_mem_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
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by (simp add: rtran_closure_mem_fm_def succ_Un_distrib [symmetric] Un_ac) 
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lemma sats_rtran_closure_mem_fm [simp]:
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   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
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    ==> sats(A, rtran_closure_mem_fm(x,y,z), env) <-> 
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        rtran_closure_mem(**A, nth(x,env), nth(y,env), nth(z,env))"
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by (simp add: rtran_closure_mem_fm_def rtran_closure_mem_def)
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lemma rtran_closure_mem_iff_sats:
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      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
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          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
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       ==> rtran_closure_mem(**A, x, y, z) <-> sats(A, rtran_closure_mem_fm(i,j,k), env)"
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by (simp add: sats_rtran_closure_mem_fm)
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theorem rtran_closure_mem_reflection:
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     "REFLECTS[\<lambda>x. rtran_closure_mem(L,f(x),g(x),h(x)), 
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               \<lambda>i x. rtran_closure_mem(**Lset(i),f(x),g(x),h(x))]"
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apply (simp only: rtran_closure_mem_def setclass_simps)
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apply (intro FOL_reflections function_reflections fun_plus_reflections)  
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done
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text{*Separation for @{term "rtrancl(r)"}.*}
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lemma rtrancl_separation:
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     "[| L(r); L(A) |] ==> separation (L, rtran_closure_mem(L,A,r))"
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apply (rule separation_CollectI) 
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apply (rule_tac A="{r,A,z}" in subset_LsetE, blast ) 
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apply (rule ReflectsE [OF rtran_closure_mem_reflection], assumption)
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apply (drule subset_Lset_ltD, assumption) 
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apply (erule reflection_imp_L_separation)
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  apply (simp_all add: lt_Ord2)
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apply (rule DPow_LsetI)
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apply (rename_tac u)
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apply (rule_tac env = "[u,r,A]" in rtran_closure_mem_iff_sats)
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apply (rule sep_rules | simp)+
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done
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subsubsection{*Reflexive/Transitive Closure, Internalized*}
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(*  "rtran_closure(M,r,s) == 
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        \<forall>A[M]. is_field(M,r,A) -->
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 	 (\<forall>p[M]. p \<in> s <-> rtran_closure_mem(M,A,r,p))" *)
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constdefs rtran_closure_fm :: "[i,i]=>i"
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 "rtran_closure_fm(r,s) == 
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   Forall(Implies(field_fm(succ(r),0),
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                  Forall(Iff(Member(0,succ(succ(s))),
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                             rtran_closure_mem_fm(1,succ(succ(r)),0)))))"
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lemma rtran_closure_type [TC]:
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     "[| x \<in> nat; y \<in> nat |] ==> rtran_closure_fm(x,y) \<in> formula"
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by (simp add: rtran_closure_fm_def) 
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lemma arity_rtran_closure_fm [simp]:
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     "[| x \<in> nat; y \<in> nat |] 
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      ==> arity(rtran_closure_fm(x,y)) = succ(x) \<union> succ(y)"
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by (simp add: rtran_closure_fm_def succ_Un_distrib [symmetric] Un_ac)
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lemma sats_rtran_closure_fm [simp]:
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   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
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    ==> sats(A, rtran_closure_fm(x,y), env) <-> 
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        rtran_closure(**A, nth(x,env), nth(y,env))"
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by (simp add: rtran_closure_fm_def rtran_closure_def)
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lemma rtran_closure_iff_sats:
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      "[| nth(i,env) = x; nth(j,env) = y; 
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          i \<in> nat; j \<in> nat; env \<in> list(A)|]
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       ==> rtran_closure(**A, x, y) <-> sats(A, rtran_closure_fm(i,j), env)"
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by simp
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theorem rtran_closure_reflection:
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     "REFLECTS[\<lambda>x. rtran_closure(L,f(x),g(x)), 
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               \<lambda>i x. rtran_closure(**Lset(i),f(x),g(x))]"
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apply (simp only: rtran_closure_def setclass_simps)
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apply (intro FOL_reflections function_reflections rtran_closure_mem_reflection)
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done
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subsubsection{*Transitive Closure of a Relation, Internalized*}
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(*  "tran_closure(M,r,t) ==
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         \<exists>s[M]. rtran_closure(M,r,s) & composition(M,r,s,t)" *)
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constdefs tran_closure_fm :: "[i,i]=>i"
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 "tran_closure_fm(r,s) == 
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   Exists(And(rtran_closure_fm(succ(r),0), composition_fm(succ(r),0,succ(s))))"
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lemma tran_closure_type [TC]:
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     "[| x \<in> nat; y \<in> nat |] ==> tran_closure_fm(x,y) \<in> formula"
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by (simp add: tran_closure_fm_def) 
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lemma arity_tran_closure_fm [simp]:
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     "[| x \<in> nat; y \<in> nat |] 
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      ==> arity(tran_closure_fm(x,y)) = succ(x) \<union> succ(y)"
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by (simp add: tran_closure_fm_def succ_Un_distrib [symmetric] Un_ac)
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lemma sats_tran_closure_fm [simp]:
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   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
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    ==> sats(A, tran_closure_fm(x,y), env) <-> 
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        tran_closure(**A, nth(x,env), nth(y,env))"
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by (simp add: tran_closure_fm_def tran_closure_def)
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lemma tran_closure_iff_sats:
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      "[| nth(i,env) = x; nth(j,env) = y; 
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          i \<in> nat; j \<in> nat; env \<in> list(A)|]
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       ==> tran_closure(**A, x, y) <-> sats(A, tran_closure_fm(i,j), env)"
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by simp
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theorem tran_closure_reflection:
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     "REFLECTS[\<lambda>x. tran_closure(L,f(x),g(x)), 
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               \<lambda>i x. tran_closure(**Lset(i),f(x),g(x))]"
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apply (simp only: tran_closure_def setclass_simps)
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apply (intro FOL_reflections function_reflections 
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             rtran_closure_reflection composition_reflection)
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done
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subsection{*Separation for the Proof of @{text "wellfounded_on_trancl"}*}
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lemma wellfounded_trancl_reflects:
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  "REFLECTS[\<lambda>x. \<exists>w[L]. \<exists>wx[L]. \<exists>rp[L]. 
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	         w \<in> Z & pair(L,w,x,wx) & tran_closure(L,r,rp) & wx \<in> rp,
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   \<lambda>i x. \<exists>w \<in> Lset(i). \<exists>wx \<in> Lset(i). \<exists>rp \<in> Lset(i). 
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       w \<in> Z & pair(**Lset(i),w,x,wx) & tran_closure(**Lset(i),r,rp) &
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       wx \<in> rp]"
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by (intro FOL_reflections function_reflections fun_plus_reflections 
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          tran_closure_reflection)
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lemma wellfounded_trancl_separation:
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	 "[| L(r); L(Z) |] ==> 
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	  separation (L, \<lambda>x. 
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	      \<exists>w[L]. \<exists>wx[L]. \<exists>rp[L]. 
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	       w \<in> Z & pair(L,w,x,wx) & tran_closure(L,r,rp) & wx \<in> rp)"
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apply (rule separation_CollectI) 
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apply (rule_tac A="{r,Z,z}" in subset_LsetE, blast ) 
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apply (rule ReflectsE [OF wellfounded_trancl_reflects], assumption)
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apply (drule subset_Lset_ltD, assumption) 
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apply (erule reflection_imp_L_separation)
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  apply (simp_all add: lt_Ord2)
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apply (rule DPow_LsetI)
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apply (rename_tac u) 
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apply (rule bex_iff_sats conj_iff_sats)+
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apply (rule_tac env = "[w,u,r,Z]" in mem_iff_sats) 
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apply (rule sep_rules tran_closure_iff_sats | simp)+
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done
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subsubsection{*Instantiating the locale @{text M_trancl}*}
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ML
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{*
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val rtrancl_separation = thm "rtrancl_separation";
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val wellfounded_trancl_separation = thm "wellfounded_trancl_separation";
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val m_trancl = [rtrancl_separation, wellfounded_trancl_separation];
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fun trancl_L th =
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    kill_flex_triv_prems (m_trancl MRS (axioms_L th));
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bind_thm ("iterates_abs", trancl_L (thm "M_trancl.iterates_abs"));
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bind_thm ("rtran_closure_rtrancl", trancl_L (thm "M_trancl.rtran_closure_rtrancl"));
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bind_thm ("rtrancl_closed", trancl_L (thm "M_trancl.rtrancl_closed"));
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bind_thm ("rtrancl_abs", trancl_L (thm "M_trancl.rtrancl_abs"));
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bind_thm ("trancl_closed", trancl_L (thm "M_trancl.trancl_closed"));
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bind_thm ("trancl_abs", trancl_L (thm "M_trancl.trancl_abs"));
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bind_thm ("wellfounded_on_trancl", trancl_L (thm "M_trancl.wellfounded_on_trancl"));
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bind_thm ("wellfounded_trancl", trancl_L (thm "M_trancl.wellfounded_trancl"));
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bind_thm ("wfrec_relativize", trancl_L (thm "M_trancl.wfrec_relativize"));
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bind_thm ("trans_wfrec_relativize", trancl_L (thm "M_trancl.trans_wfrec_relativize"));
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bind_thm ("trans_wfrec_abs", trancl_L (thm "M_trancl.trans_wfrec_abs"));
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bind_thm ("trans_eq_pair_wfrec_iff", trancl_L (thm "M_trancl.trans_eq_pair_wfrec_iff"));
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bind_thm ("eq_pair_wfrec_iff", trancl_L (thm "M_trancl.eq_pair_wfrec_iff"));
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*}
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declare rtrancl_closed [intro,simp]
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declare rtrancl_abs [simp]
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declare trancl_closed [intro,simp]
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declare trancl_abs [simp]
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subsection{*Well-Founded Recursion!*}
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(* M_is_recfun :: "[i=>o, [i,i,i]=>o, i, i, i] => o"
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   "M_is_recfun(M,MH,r,a,f) == 
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     \<forall>z[M]. z \<in> f <-> 
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            5      4       3       2       1           0
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            (\<exists>x[M]. \<exists>y[M]. \<exists>xa[M]. \<exists>sx[M]. \<exists>r_sx[M]. \<exists>f_r_sx[M]. 
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	       pair(M,x,y,z) & pair(M,x,a,xa) & upair(M,x,x,sx) &
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               pre_image(M,r,sx,r_sx) & restriction(M,f,r_sx,f_r_sx) &
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               xa \<in> r & MH(x, f_r_sx, y))"
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*)
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constdefs is_recfun_fm :: "[[i,i,i]=>i, i, i, i]=>i"
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 "is_recfun_fm(p,r,a,f) == 
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   Forall(Iff(Member(0,succ(f)),
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    Exists(Exists(Exists(Exists(Exists(Exists(
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     And(pair_fm(5,4,6),
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      And(pair_fm(5,a#+7,3),
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       And(upair_fm(5,5,2),
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        And(pre_image_fm(r#+7,2,1),
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         And(restriction_fm(f#+7,1,0),
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          And(Member(3,r#+7), p(5,0,4)))))))))))))))"
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lemma is_recfun_type_0:
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     "[| !!x y z. [| x \<in> nat; y \<in> nat; z \<in> nat |] ==> p(x,y,z) \<in> formula;  
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         x \<in> nat; y \<in> nat; z \<in> nat |] 
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      ==> is_recfun_fm(p,x,y,z) \<in> formula"
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apply (unfold is_recfun_fm_def)
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(*FIXME: FIND OUT why simp loops!*)
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apply typecheck
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by simp 
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lemma is_recfun_type [TC]:
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     "[| p(5,0,4) \<in> formula;  
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         x \<in> nat; y \<in> nat; z \<in> nat |] 
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      ==> is_recfun_fm(p,x,y,z) \<in> formula"
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by (simp add: is_recfun_fm_def) 
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lemma arity_is_recfun_fm [simp]:
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     "[| arity(p(5,0,4)) le 8; x \<in> nat; y \<in> nat; z \<in> nat |] 
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      ==> arity(is_recfun_fm(p,x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
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apply (frule lt_nat_in_nat, simp) 
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apply (simp add: is_recfun_fm_def succ_Un_distrib [symmetric] ) 
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apply (subst subset_Un_iff2 [of "arity(p(5,0,4))", THEN iffD1]) 
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apply (rule le_imp_subset) 
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apply (erule le_trans, simp) 
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apply (simp add: succ_Un_distrib [symmetric] Un_ac) 
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done
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lemma sats_is_recfun_fm:
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  assumes MH_iff_sats: 
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      "!!x y z env. 
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	 [| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
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	 ==> MH(nth(x,env), nth(y,env), nth(z,env)) <-> sats(A, p(x,y,z), env)"
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  shows 
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      "[|x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
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       ==> sats(A, is_recfun_fm(p,x,y,z), env) <-> 
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           M_is_recfun(**A, MH, nth(x,env), nth(y,env), nth(z,env))"
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by (simp add: is_recfun_fm_def M_is_recfun_def MH_iff_sats [THEN iff_sym])
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lemma is_recfun_iff_sats:
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  "[| (!!x y z env. [| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
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                    ==> MH(nth(x,env), nth(y,env), nth(z,env)) <->
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                        sats(A, p(x,y,z), env));
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      nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
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      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
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   ==> M_is_recfun(**A, MH, x, y, z) <-> sats(A, is_recfun_fm(p,i,j,k), env)" 
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by (simp add: sats_is_recfun_fm [of A MH])
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theorem is_recfun_reflection:
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  assumes MH_reflection:
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    "!!f g h. REFLECTS[\<lambda>x. MH(L, f(x), g(x), h(x)), 
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                     \<lambda>i x. MH(**Lset(i), f(x), g(x), h(x))]"
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  shows "REFLECTS[\<lambda>x. M_is_recfun(L, MH(L), f(x), g(x), h(x)), 
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               \<lambda>i x. M_is_recfun(**Lset(i), MH(**Lset(i)), f(x), g(x), h(x))]"
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apply (simp (no_asm_use) only: M_is_recfun_def setclass_simps)
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apply (intro FOL_reflections function_reflections 
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             restriction_reflection MH_reflection)  
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done
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text{*Currently, @{text sats}-theorems for higher-order operators don't seem
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useful.  Reflection theorems do work, though.  This one avoids the repetition
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of the @{text MH}-term.*}
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theorem is_wfrec_reflection:
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  assumes MH_reflection:
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    "!!f g h. REFLECTS[\<lambda>x. MH(L, f(x), g(x), h(x)), 
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                     \<lambda>i x. MH(**Lset(i), f(x), g(x), h(x))]"
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  shows "REFLECTS[\<lambda>x. is_wfrec(L, MH(L), f(x), g(x), h(x)), 
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               \<lambda>i x. is_wfrec(**Lset(i), MH(**Lset(i)), f(x), g(x), h(x))]"
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apply (simp (no_asm_use) only: is_wfrec_def setclass_simps)
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apply (intro FOL_reflections MH_reflection is_recfun_reflection)  
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done
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subsection{*The Locale @{text "M_wfrank"}*}
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subsubsection{*Separation for @{term "wfrank"}*}
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lemma wfrank_Reflects:
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 "REFLECTS[\<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
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              ~ (\<exists>f[L]. M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f)),
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      \<lambda>i x. \<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) -->
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         ~ (\<exists>f \<in> Lset(i). 
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            M_is_recfun(**Lset(i), %x f y. is_range(**Lset(i),f,y), 
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                        rplus, x, f))]"
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by (intro FOL_reflections function_reflections is_recfun_reflection tran_closure_reflection)  
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lemma wfrank_separation:
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     "L(r) ==>
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   341
      separation (L, \<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
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   342
         ~ (\<exists>f[L]. M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f)))"
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apply (rule separation_CollectI) 
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apply (rule_tac A="{r,z}" in subset_LsetE, blast ) 
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apply (rule ReflectsE [OF wfrank_Reflects], assumption)
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apply (drule subset_Lset_ltD, assumption) 
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   347
apply (erule reflection_imp_L_separation)
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  apply (simp_all add: lt_Ord2, clarify)
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apply (rule DPow_LsetI)
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apply (rename_tac u)  
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apply (rule ball_iff_sats imp_iff_sats)+
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apply (rule_tac env="[rplus,u,r]" in tran_closure_iff_sats)
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apply (rule sep_rules is_recfun_iff_sats | simp)+
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   354
done
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paulson@13348
   356
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   357
subsubsection{*Replacement for @{term "wfrank"}*}
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   359
lemma wfrank_replacement_Reflects:
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 "REFLECTS[\<lambda>z. \<exists>x[L]. x \<in> A & 
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   361
        (\<forall>rplus[L]. tran_closure(L,r,rplus) -->
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   362
         (\<exists>y[L]. \<exists>f[L]. pair(L,x,y,z)  & 
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                        M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f) &
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   364
                        is_range(L,f,y))),
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 \<lambda>i z. \<exists>x \<in> Lset(i). x \<in> A & 
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   366
      (\<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) -->
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       (\<exists>y \<in> Lset(i). \<exists>f \<in> Lset(i). pair(**Lset(i),x,y,z)  & 
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   368
         M_is_recfun(**Lset(i), %x f y. is_range(**Lset(i),f,y), rplus, x, f) &
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   369
         is_range(**Lset(i),f,y)))]"
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by (intro FOL_reflections function_reflections fun_plus_reflections
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             is_recfun_reflection tran_closure_reflection)
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   372
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   373
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   374
lemma wfrank_strong_replacement:
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     "L(r) ==>
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   376
      strong_replacement(L, \<lambda>x z. 
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   377
         \<forall>rplus[L]. tran_closure(L,r,rplus) -->
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   378
         (\<exists>y[L]. \<exists>f[L]. pair(L,x,y,z)  & 
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                        M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f) &
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   380
                        is_range(L,f,y)))"
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apply (rule strong_replacementI) 
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   382
apply (rule rallI)
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   383
apply (rename_tac B)  
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   384
apply (rule separation_CollectI) 
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   385
apply (rule_tac A="{B,r,z}" in subset_LsetE, blast ) 
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   386
apply (rule ReflectsE [OF wfrank_replacement_Reflects], assumption)
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   387
apply (drule subset_Lset_ltD, assumption) 
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   388
apply (erule reflection_imp_L_separation)
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   389
  apply (simp_all add: lt_Ord2)
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   390
apply (rule DPow_LsetI)
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   391
apply (rename_tac u) 
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   392
apply (rule bex_iff_sats ball_iff_sats conj_iff_sats)+
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   393
apply (rule_tac env = "[x,u,B,r]" in mem_iff_sats) 
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   394
apply (rule sep_rules tran_closure_iff_sats is_recfun_iff_sats | simp)+
paulson@13348
   395
done
paulson@13348
   396
paulson@13348
   397
paulson@13363
   398
subsubsection{*Separation for Proving @{text Ord_wfrank_range}*}
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   399
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   400
lemma Ord_wfrank_Reflects:
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   401
 "REFLECTS[\<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) --> 
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   402
          ~ (\<forall>f[L]. \<forall>rangef[L]. 
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   403
             is_range(L,f,rangef) -->
paulson@13352
   404
             M_is_recfun(L, \<lambda>x f y. is_range(L,f,y), rplus, x, f) -->
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   405
             ordinal(L,rangef)),
paulson@13348
   406
      \<lambda>i x. \<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) --> 
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   407
          ~ (\<forall>f \<in> Lset(i). \<forall>rangef \<in> Lset(i). 
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   408
             is_range(**Lset(i),f,rangef) -->
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   409
             M_is_recfun(**Lset(i), \<lambda>x f y. is_range(**Lset(i),f,y), 
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   410
                         rplus, x, f) -->
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   411
             ordinal(**Lset(i),rangef))]"
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   412
by (intro FOL_reflections function_reflections is_recfun_reflection 
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   413
          tran_closure_reflection ordinal_reflection)
paulson@13348
   414
paulson@13348
   415
lemma  Ord_wfrank_separation:
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   416
     "L(r) ==>
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   417
      separation (L, \<lambda>x.
paulson@13348
   418
         \<forall>rplus[L]. tran_closure(L,r,rplus) --> 
paulson@13348
   419
          ~ (\<forall>f[L]. \<forall>rangef[L]. 
paulson@13348
   420
             is_range(L,f,rangef) -->
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   421
             M_is_recfun(L, \<lambda>x f y. is_range(L,f,y), rplus, x, f) -->
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   422
             ordinal(L,rangef)))" 
paulson@13348
   423
apply (rule separation_CollectI) 
paulson@13348
   424
apply (rule_tac A="{r,z}" in subset_LsetE, blast ) 
paulson@13348
   425
apply (rule ReflectsE [OF Ord_wfrank_Reflects], assumption)
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   426
apply (drule subset_Lset_ltD, assumption) 
paulson@13348
   427
apply (erule reflection_imp_L_separation)
paulson@13348
   428
  apply (simp_all add: lt_Ord2, clarify)
paulson@13385
   429
apply (rule DPow_LsetI)
paulson@13348
   430
apply (rename_tac u)  
paulson@13348
   431
apply (rule ball_iff_sats imp_iff_sats)+
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   432
apply (rule_tac env="[rplus,u,r]" in tran_closure_iff_sats)
paulson@13348
   433
apply (rule sep_rules is_recfun_iff_sats | simp)+
paulson@13348
   434
done
paulson@13348
   435
paulson@13348
   436
paulson@13363
   437
subsubsection{*Instantiating the locale @{text M_wfrank}*}
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   438
ML
paulson@13363
   439
{*
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   440
val wfrank_separation = thm "wfrank_separation";
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   441
val wfrank_strong_replacement = thm "wfrank_strong_replacement";
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   442
val Ord_wfrank_separation = thm "Ord_wfrank_separation";
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   443
paulson@13363
   444
val m_wfrank = 
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   445
    [wfrank_separation, wfrank_strong_replacement, Ord_wfrank_separation];
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   446
paulson@13363
   447
fun wfrank_L th =
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   448
    kill_flex_triv_prems (m_wfrank MRS (trancl_L th));
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   449
paulson@13363
   450
paulson@13363
   451
paulson@13363
   452
bind_thm ("iterates_closed", wfrank_L (thm "M_wfrank.iterates_closed"));
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   453
bind_thm ("exists_wfrank", wfrank_L (thm "M_wfrank.exists_wfrank"));
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   454
bind_thm ("M_wellfoundedrank", wfrank_L (thm "M_wfrank.M_wellfoundedrank"));
paulson@13363
   455
bind_thm ("Ord_wfrank_range", wfrank_L (thm "M_wfrank.Ord_wfrank_range"));
paulson@13363
   456
bind_thm ("Ord_range_wellfoundedrank", wfrank_L (thm "M_wfrank.Ord_range_wellfoundedrank"));
paulson@13363
   457
bind_thm ("function_wellfoundedrank", wfrank_L (thm "M_wfrank.function_wellfoundedrank"));
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   458
bind_thm ("domain_wellfoundedrank", wfrank_L (thm "M_wfrank.domain_wellfoundedrank"));
paulson@13363
   459
bind_thm ("wellfoundedrank_type", wfrank_L (thm "M_wfrank.wellfoundedrank_type"));
paulson@13363
   460
bind_thm ("Ord_wellfoundedrank", wfrank_L (thm "M_wfrank.Ord_wellfoundedrank"));
paulson@13363
   461
bind_thm ("wellfoundedrank_eq", wfrank_L (thm "M_wfrank.wellfoundedrank_eq"));
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   462
bind_thm ("wellfoundedrank_lt", wfrank_L (thm "M_wfrank.wellfoundedrank_lt"));
paulson@13363
   463
bind_thm ("wellfounded_imp_subset_rvimage", wfrank_L (thm "M_wfrank.wellfounded_imp_subset_rvimage"));
paulson@13363
   464
bind_thm ("wellfounded_imp_wf", wfrank_L (thm "M_wfrank.wellfounded_imp_wf"));
paulson@13363
   465
bind_thm ("wellfounded_on_imp_wf_on", wfrank_L (thm "M_wfrank.wellfounded_on_imp_wf_on"));
paulson@13363
   466
bind_thm ("wf_abs", wfrank_L (thm "M_wfrank.wf_abs"));
paulson@13363
   467
bind_thm ("wf_on_abs", wfrank_L (thm "M_wfrank.wf_on_abs"));
paulson@13363
   468
bind_thm ("wfrec_replacement_iff", wfrank_L (thm "M_wfrank.wfrec_replacement_iff"));
paulson@13363
   469
bind_thm ("trans_wfrec_closed", wfrank_L (thm "M_wfrank.trans_wfrec_closed"));
paulson@13363
   470
bind_thm ("wfrec_closed", wfrank_L (thm "M_wfrank.wfrec_closed"));
paulson@13363
   471
*}
paulson@13363
   472
paulson@13363
   473
declare iterates_closed [intro,simp]
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   474
declare Ord_wfrank_range [rule_format]
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   475
declare wf_abs [simp]
paulson@13363
   476
declare wf_on_abs [simp]
paulson@13363
   477
paulson@13363
   478
paulson@13363
   479
subsection{*For Datatypes*}
paulson@13363
   480
paulson@13363
   481
subsubsection{*Binary Products, Internalized*}
paulson@13363
   482
paulson@13363
   483
constdefs cartprod_fm :: "[i,i,i]=>i"
paulson@13363
   484
(* "cartprod(M,A,B,z) == 
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   485
	\<forall>u[M]. u \<in> z <-> (\<exists>x[M]. x\<in>A & (\<exists>y[M]. y\<in>B & pair(M,x,y,u)))" *)
paulson@13363
   486
    "cartprod_fm(A,B,z) == 
paulson@13363
   487
       Forall(Iff(Member(0,succ(z)),
paulson@13363
   488
                  Exists(And(Member(0,succ(succ(A))),
paulson@13363
   489
                         Exists(And(Member(0,succ(succ(succ(B)))),
paulson@13363
   490
                                    pair_fm(1,0,2)))))))"
paulson@13363
   491
paulson@13363
   492
lemma cartprod_type [TC]:
paulson@13363
   493
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cartprod_fm(x,y,z) \<in> formula"
paulson@13363
   494
by (simp add: cartprod_fm_def) 
paulson@13363
   495
paulson@13363
   496
lemma arity_cartprod_fm [simp]:
paulson@13363
   497
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13363
   498
      ==> arity(cartprod_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13363
   499
by (simp add: cartprod_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13363
   500
paulson@13363
   501
lemma sats_cartprod_fm [simp]:
paulson@13363
   502
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13363
   503
    ==> sats(A, cartprod_fm(x,y,z), env) <-> 
paulson@13363
   504
        cartprod(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13363
   505
by (simp add: cartprod_fm_def cartprod_def)
paulson@13363
   506
paulson@13363
   507
lemma cartprod_iff_sats:
paulson@13363
   508
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13363
   509
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13363
   510
       ==> cartprod(**A, x, y, z) <-> sats(A, cartprod_fm(i,j,k), env)"
paulson@13363
   511
by (simp add: sats_cartprod_fm)
paulson@13363
   512
paulson@13363
   513
theorem cartprod_reflection:
paulson@13363
   514
     "REFLECTS[\<lambda>x. cartprod(L,f(x),g(x),h(x)), 
paulson@13363
   515
               \<lambda>i x. cartprod(**Lset(i),f(x),g(x),h(x))]"
paulson@13363
   516
apply (simp only: cartprod_def setclass_simps)
paulson@13363
   517
apply (intro FOL_reflections pair_reflection)  
paulson@13363
   518
done
paulson@13363
   519
paulson@13363
   520
paulson@13363
   521
subsubsection{*Binary Sums, Internalized*}
paulson@13363
   522
paulson@13363
   523
(* "is_sum(M,A,B,Z) == 
paulson@13363
   524
       \<exists>A0[M]. \<exists>n1[M]. \<exists>s1[M]. \<exists>B1[M]. 
paulson@13363
   525
         3      2       1        0
paulson@13363
   526
       number1(M,n1) & cartprod(M,n1,A,A0) & upair(M,n1,n1,s1) &
paulson@13363
   527
       cartprod(M,s1,B,B1) & union(M,A0,B1,Z)"  *)
paulson@13363
   528
constdefs sum_fm :: "[i,i,i]=>i"
paulson@13363
   529
    "sum_fm(A,B,Z) == 
paulson@13363
   530
       Exists(Exists(Exists(Exists(
paulson@13363
   531
	And(number1_fm(2),
paulson@13363
   532
            And(cartprod_fm(2,A#+4,3),
paulson@13363
   533
                And(upair_fm(2,2,1),
paulson@13363
   534
                    And(cartprod_fm(1,B#+4,0), union_fm(3,0,Z#+4)))))))))"
paulson@13363
   535
paulson@13363
   536
lemma sum_type [TC]:
paulson@13363
   537
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> sum_fm(x,y,z) \<in> formula"
paulson@13363
   538
by (simp add: sum_fm_def) 
paulson@13363
   539
paulson@13363
   540
lemma arity_sum_fm [simp]:
paulson@13363
   541
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13363
   542
      ==> arity(sum_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13363
   543
by (simp add: sum_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13363
   544
paulson@13363
   545
lemma sats_sum_fm [simp]:
paulson@13363
   546
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13363
   547
    ==> sats(A, sum_fm(x,y,z), env) <-> 
paulson@13363
   548
        is_sum(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13363
   549
by (simp add: sum_fm_def is_sum_def)
paulson@13363
   550
paulson@13363
   551
lemma sum_iff_sats:
paulson@13363
   552
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13363
   553
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13363
   554
       ==> is_sum(**A, x, y, z) <-> sats(A, sum_fm(i,j,k), env)"
paulson@13363
   555
by simp
paulson@13363
   556
paulson@13363
   557
theorem sum_reflection:
paulson@13363
   558
     "REFLECTS[\<lambda>x. is_sum(L,f(x),g(x),h(x)), 
paulson@13363
   559
               \<lambda>i x. is_sum(**Lset(i),f(x),g(x),h(x))]"
paulson@13363
   560
apply (simp only: is_sum_def setclass_simps)
paulson@13363
   561
apply (intro FOL_reflections function_reflections cartprod_reflection)  
paulson@13363
   562
done
paulson@13363
   563
paulson@13363
   564
paulson@13363
   565
subsubsection{*The Operator @{term quasinat}*}
paulson@13363
   566
paulson@13363
   567
(* "is_quasinat(M,z) == empty(M,z) | (\<exists>m[M]. successor(M,m,z))" *)
paulson@13363
   568
constdefs quasinat_fm :: "i=>i"
paulson@13363
   569
    "quasinat_fm(z) == Or(empty_fm(z), Exists(succ_fm(0,succ(z))))"
paulson@13363
   570
paulson@13363
   571
lemma quasinat_type [TC]:
paulson@13363
   572
     "x \<in> nat ==> quasinat_fm(x) \<in> formula"
paulson@13363
   573
by (simp add: quasinat_fm_def) 
paulson@13363
   574
paulson@13363
   575
lemma arity_quasinat_fm [simp]:
paulson@13363
   576
     "x \<in> nat ==> arity(quasinat_fm(x)) = succ(x)"
paulson@13363
   577
by (simp add: quasinat_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13363
   578
paulson@13363
   579
lemma sats_quasinat_fm [simp]:
paulson@13363
   580
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13363
   581
    ==> sats(A, quasinat_fm(x), env) <-> is_quasinat(**A, nth(x,env))"
paulson@13363
   582
by (simp add: quasinat_fm_def is_quasinat_def)
paulson@13363
   583
paulson@13363
   584
lemma quasinat_iff_sats:
paulson@13363
   585
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13363
   586
          i \<in> nat; env \<in> list(A)|]
paulson@13363
   587
       ==> is_quasinat(**A, x) <-> sats(A, quasinat_fm(i), env)"
paulson@13363
   588
by simp
paulson@13363
   589
paulson@13363
   590
theorem quasinat_reflection:
paulson@13363
   591
     "REFLECTS[\<lambda>x. is_quasinat(L,f(x)), 
paulson@13363
   592
               \<lambda>i x. is_quasinat(**Lset(i),f(x))]"
paulson@13363
   593
apply (simp only: is_quasinat_def setclass_simps)
paulson@13363
   594
apply (intro FOL_reflections function_reflections)  
paulson@13363
   595
done
paulson@13363
   596
paulson@13363
   597
paulson@13363
   598
subsubsection{*The Operator @{term is_nat_case}*}
paulson@13363
   599
paulson@13363
   600
(* is_nat_case :: "[i=>o, i, [i,i]=>o, i, i] => o"
paulson@13363
   601
    "is_nat_case(M, a, is_b, k, z) == 
paulson@13363
   602
       (empty(M,k) --> z=a) &
paulson@13363
   603
       (\<forall>m[M]. successor(M,m,k) --> is_b(m,z)) &
paulson@13363
   604
       (is_quasinat(M,k) | empty(M,z))" *)
paulson@13363
   605
text{*The formula @{term is_b} has free variables 1 and 0.*}
paulson@13363
   606
constdefs is_nat_case_fm :: "[i, [i,i]=>i, i, i]=>i"
paulson@13363
   607
 "is_nat_case_fm(a,is_b,k,z) == 
paulson@13363
   608
    And(Implies(empty_fm(k), Equal(z,a)),
paulson@13363
   609
        And(Forall(Implies(succ_fm(0,succ(k)), 
paulson@13363
   610
                   Forall(Implies(Equal(0,succ(succ(z))), is_b(1,0))))),
paulson@13363
   611
            Or(quasinat_fm(k), empty_fm(z))))"
paulson@13363
   612
paulson@13363
   613
lemma is_nat_case_type [TC]:
paulson@13363
   614
     "[| is_b(1,0) \<in> formula;  
paulson@13363
   615
         x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13363
   616
      ==> is_nat_case_fm(x,is_b,y,z) \<in> formula"
paulson@13363
   617
by (simp add: is_nat_case_fm_def) 
paulson@13363
   618
paulson@13363
   619
lemma arity_is_nat_case_fm [simp]:
paulson@13363
   620
     "[| is_b(1,0) \<in> formula; x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13363
   621
      ==> arity(is_nat_case_fm(x,is_b,y,z)) = 
paulson@13363
   622
          succ(x) \<union> succ(y) \<union> succ(z) \<union> (arity(is_b(1,0)) #- 2)" 
paulson@13363
   623
apply (subgoal_tac "arity(is_b(1,0)) \<in> nat")  
paulson@13363
   624
apply typecheck
paulson@13363
   625
(*FIXME: could nat_diff_split work?*)
paulson@13363
   626
apply (auto simp add: diff_def raw_diff_succ is_nat_case_fm_def nat_imp_quasinat
paulson@13363
   627
                 succ_Un_distrib [symmetric] Un_ac
paulson@13363
   628
                 split: split_nat_case) 
paulson@13363
   629
done
paulson@13363
   630
paulson@13363
   631
lemma sats_is_nat_case_fm:
paulson@13363
   632
  assumes is_b_iff_sats: 
paulson@13363
   633
      "!!a b. [| a \<in> A; b \<in> A|] 
paulson@13363
   634
              ==> is_b(a,b) <-> sats(A, p(1,0), Cons(b, Cons(a,env)))"
paulson@13363
   635
  shows 
paulson@13363
   636
      "[|x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A)|]
paulson@13363
   637
       ==> sats(A, is_nat_case_fm(x,p,y,z), env) <-> 
paulson@13363
   638
           is_nat_case(**A, nth(x,env), is_b, nth(y,env), nth(z,env))"
paulson@13363
   639
apply (frule lt_length_in_nat, assumption)  
paulson@13363
   640
apply (simp add: is_nat_case_fm_def is_nat_case_def is_b_iff_sats [THEN iff_sym])
paulson@13363
   641
done
paulson@13363
   642
paulson@13363
   643
lemma is_nat_case_iff_sats:
paulson@13363
   644
  "[| (!!a b. [| a \<in> A; b \<in> A|] 
paulson@13363
   645
              ==> is_b(a,b) <-> sats(A, p(1,0), Cons(b, Cons(a,env))));
paulson@13363
   646
      nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13363
   647
      i \<in> nat; j \<in> nat; k < length(env); env \<in> list(A)|]
paulson@13363
   648
   ==> is_nat_case(**A, x, is_b, y, z) <-> sats(A, is_nat_case_fm(i,p,j,k), env)" 
paulson@13363
   649
by (simp add: sats_is_nat_case_fm [of A is_b])
paulson@13363
   650
paulson@13363
   651
paulson@13363
   652
text{*The second argument of @{term is_b} gives it direct access to @{term x},
paulson@13363
   653
  which is essential for handling free variable references.  Without this 
paulson@13363
   654
  argument, we cannot prove reflection for @{term iterates_MH}.*}
paulson@13363
   655
theorem is_nat_case_reflection:
paulson@13363
   656
  assumes is_b_reflection:
paulson@13363
   657
    "!!h f g. REFLECTS[\<lambda>x. is_b(L, h(x), f(x), g(x)), 
paulson@13363
   658
                     \<lambda>i x. is_b(**Lset(i), h(x), f(x), g(x))]"
paulson@13363
   659
  shows "REFLECTS[\<lambda>x. is_nat_case(L, f(x), is_b(L,x), g(x), h(x)), 
paulson@13363
   660
               \<lambda>i x. is_nat_case(**Lset(i), f(x), is_b(**Lset(i), x), g(x), h(x))]"
paulson@13363
   661
apply (simp (no_asm_use) only: is_nat_case_def setclass_simps)
paulson@13363
   662
apply (intro FOL_reflections function_reflections 
paulson@13363
   663
             restriction_reflection is_b_reflection quasinat_reflection)  
paulson@13363
   664
done
paulson@13363
   665
paulson@13363
   666
paulson@13363
   667
paulson@13363
   668
subsection{*The Operator @{term iterates_MH}, Needed for Iteration*}
paulson@13363
   669
paulson@13363
   670
(*  iterates_MH :: "[i=>o, [i,i]=>o, i, i, i, i] => o"
paulson@13363
   671
   "iterates_MH(M,isF,v,n,g,z) ==
paulson@13363
   672
        is_nat_case(M, v, \<lambda>m u. \<exists>gm[M]. fun_apply(M,g,m,gm) & isF(gm,u),
paulson@13363
   673
                    n, z)" *)
paulson@13363
   674
constdefs iterates_MH_fm :: "[[i,i]=>i, i, i, i, i]=>i"
paulson@13363
   675
 "iterates_MH_fm(isF,v,n,g,z) == 
paulson@13363
   676
    is_nat_case_fm(v, 
paulson@13363
   677
      \<lambda>m u. Exists(And(fun_apply_fm(succ(succ(succ(g))),succ(m),0), 
paulson@13363
   678
                     Forall(Implies(Equal(0,succ(succ(u))), isF(1,0))))), 
paulson@13363
   679
      n, z)"
paulson@13363
   680
paulson@13363
   681
lemma iterates_MH_type [TC]:
paulson@13363
   682
     "[| p(1,0) \<in> formula;  
paulson@13363
   683
         v \<in> nat; x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13363
   684
      ==> iterates_MH_fm(p,v,x,y,z) \<in> formula"
paulson@13363
   685
by (simp add: iterates_MH_fm_def) 
paulson@13363
   686
paulson@13363
   687
paulson@13363
   688
lemma arity_iterates_MH_fm [simp]:
paulson@13363
   689
     "[| p(1,0) \<in> formula; 
paulson@13363
   690
         v \<in> nat; x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13363
   691
      ==> arity(iterates_MH_fm(p,v,x,y,z)) = 
paulson@13363
   692
          succ(v) \<union> succ(x) \<union> succ(y) \<union> succ(z) \<union> (arity(p(1,0)) #- 4)"
paulson@13363
   693
apply (subgoal_tac "arity(p(1,0)) \<in> nat")
paulson@13363
   694
apply typecheck
paulson@13363
   695
apply (simp add: nat_imp_quasinat iterates_MH_fm_def Un_ac
paulson@13363
   696
            split: split_nat_case, clarify)
paulson@13363
   697
apply (rename_tac i j)
paulson@13363
   698
apply (drule eq_succ_imp_eq_m1, simp) 
paulson@13363
   699
apply (drule eq_succ_imp_eq_m1, simp)
paulson@13363
   700
apply (simp add: diff_Un_distrib succ_Un_distrib Un_ac diff_diff_left)
paulson@13363
   701
done
paulson@13363
   702
paulson@13363
   703
lemma sats_iterates_MH_fm:
paulson@13363
   704
  assumes is_F_iff_sats: 
paulson@13363
   705
      "!!a b c d. [| a \<in> A; b \<in> A; c \<in> A; d \<in> A|] 
paulson@13363
   706
              ==> is_F(a,b) <->
paulson@13363
   707
                  sats(A, p(1,0), Cons(b, Cons(a, Cons(c, Cons(d,env)))))"
paulson@13363
   708
  shows 
paulson@13363
   709
      "[|v \<in> nat; x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A)|]
paulson@13363
   710
       ==> sats(A, iterates_MH_fm(p,v,x,y,z), env) <-> 
paulson@13363
   711
           iterates_MH(**A, is_F, nth(v,env), nth(x,env), nth(y,env), nth(z,env))"
paulson@13363
   712
by (simp add: iterates_MH_fm_def iterates_MH_def sats_is_nat_case_fm 
paulson@13363
   713
              is_F_iff_sats [symmetric])
paulson@13363
   714
paulson@13363
   715
lemma iterates_MH_iff_sats:
paulson@13363
   716
  "[| (!!a b c d. [| a \<in> A; b \<in> A; c \<in> A; d \<in> A|] 
paulson@13363
   717
              ==> is_F(a,b) <->
paulson@13363
   718
                  sats(A, p(1,0), Cons(b, Cons(a, Cons(c, Cons(d,env))))));
paulson@13363
   719
      nth(i',env) = v; nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13363
   720
      i' \<in> nat; i \<in> nat; j \<in> nat; k < length(env); env \<in> list(A)|]
paulson@13363
   721
   ==> iterates_MH(**A, is_F, v, x, y, z) <-> 
paulson@13363
   722
       sats(A, iterates_MH_fm(p,i',i,j,k), env)"
paulson@13363
   723
apply (rule iff_sym) 
paulson@13363
   724
apply (rule iff_trans) 
paulson@13363
   725
apply (rule sats_iterates_MH_fm [of A is_F], blast, simp_all) 
paulson@13363
   726
done
paulson@13363
   727
paulson@13363
   728
theorem iterates_MH_reflection:
paulson@13363
   729
  assumes p_reflection:
paulson@13363
   730
    "!!f g h. REFLECTS[\<lambda>x. p(L, f(x), g(x)), 
paulson@13363
   731
                     \<lambda>i x. p(**Lset(i), f(x), g(x))]"
paulson@13363
   732
 shows "REFLECTS[\<lambda>x. iterates_MH(L, p(L), e(x), f(x), g(x), h(x)), 
paulson@13363
   733
               \<lambda>i x. iterates_MH(**Lset(i), p(**Lset(i)), e(x), f(x), g(x), h(x))]"
paulson@13363
   734
apply (simp (no_asm_use) only: iterates_MH_def)
paulson@13363
   735
txt{*Must be careful: simplifying with @{text setclass_simps} above would
paulson@13363
   736
     change @{text "\<exists>gm[**Lset(i)]"} into @{text "\<exists>gm \<in> Lset(i)"}, when
paulson@13363
   737
     it would no longer match rule @{text is_nat_case_reflection}. *}
paulson@13363
   738
apply (rule is_nat_case_reflection) 
paulson@13363
   739
apply (simp (no_asm_use) only: setclass_simps)
paulson@13363
   740
apply (intro FOL_reflections function_reflections is_nat_case_reflection
paulson@13363
   741
             restriction_reflection p_reflection)  
paulson@13363
   742
done
paulson@13363
   743
paulson@13363
   744
paulson@13363
   745
paulson@13363
   746
subsection{*@{term L} is Closed Under the Operator @{term list}*} 
paulson@13363
   747
paulson@13386
   748
subsubsection{*The List Functor, Internalized*}
paulson@13386
   749
paulson@13386
   750
constdefs list_functor_fm :: "[i,i,i]=>i"
paulson@13386
   751
(* "is_list_functor(M,A,X,Z) == 
paulson@13386
   752
        \<exists>n1[M]. \<exists>AX[M]. 
paulson@13386
   753
         number1(M,n1) & cartprod(M,A,X,AX) & is_sum(M,n1,AX,Z)" *)
paulson@13386
   754
    "list_functor_fm(A,X,Z) == 
paulson@13386
   755
       Exists(Exists(
paulson@13386
   756
	And(number1_fm(1),
paulson@13386
   757
            And(cartprod_fm(A#+2,X#+2,0), sum_fm(1,0,Z#+2)))))"
paulson@13386
   758
paulson@13386
   759
lemma list_functor_type [TC]:
paulson@13386
   760
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> list_functor_fm(x,y,z) \<in> formula"
paulson@13386
   761
by (simp add: list_functor_fm_def) 
paulson@13386
   762
paulson@13386
   763
lemma arity_list_functor_fm [simp]:
paulson@13386
   764
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13386
   765
      ==> arity(list_functor_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13386
   766
by (simp add: list_functor_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13386
   767
paulson@13386
   768
lemma sats_list_functor_fm [simp]:
paulson@13386
   769
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13386
   770
    ==> sats(A, list_functor_fm(x,y,z), env) <-> 
paulson@13386
   771
        is_list_functor(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13386
   772
by (simp add: list_functor_fm_def is_list_functor_def)
paulson@13386
   773
paulson@13386
   774
lemma list_functor_iff_sats:
paulson@13386
   775
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13386
   776
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13386
   777
   ==> is_list_functor(**A, x, y, z) <-> sats(A, list_functor_fm(i,j,k), env)"
paulson@13386
   778
by simp
paulson@13386
   779
paulson@13386
   780
theorem list_functor_reflection:
paulson@13386
   781
     "REFLECTS[\<lambda>x. is_list_functor(L,f(x),g(x),h(x)), 
paulson@13386
   782
               \<lambda>i x. is_list_functor(**Lset(i),f(x),g(x),h(x))]"
paulson@13386
   783
apply (simp only: is_list_functor_def setclass_simps)
paulson@13386
   784
apply (intro FOL_reflections number1_reflection
paulson@13386
   785
             cartprod_reflection sum_reflection)  
paulson@13386
   786
done
paulson@13386
   787
paulson@13386
   788
paulson@13386
   789
subsubsection{*Instances of Replacement for Lists*}
paulson@13386
   790
paulson@13363
   791
lemma list_replacement1_Reflects:
paulson@13363
   792
 "REFLECTS
paulson@13363
   793
   [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
paulson@13363
   794
         is_wfrec(L, iterates_MH(L, is_list_functor(L,A), 0), memsn, u, y)),
paulson@13363
   795
    \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
paulson@13363
   796
         is_wfrec(**Lset(i), 
paulson@13363
   797
                  iterates_MH(**Lset(i), 
paulson@13363
   798
                          is_list_functor(**Lset(i), A), 0), memsn, u, y))]"
paulson@13363
   799
by (intro FOL_reflections function_reflections is_wfrec_reflection 
paulson@13363
   800
          iterates_MH_reflection list_functor_reflection) 
paulson@13363
   801
paulson@13363
   802
lemma list_replacement1: 
paulson@13363
   803
   "L(A) ==> iterates_replacement(L, is_list_functor(L,A), 0)"
paulson@13363
   804
apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
paulson@13363
   805
apply (rule strong_replacementI) 
paulson@13363
   806
apply (rule rallI)
paulson@13363
   807
apply (rename_tac B)   
paulson@13363
   808
apply (rule separation_CollectI) 
paulson@13363
   809
apply (insert nonempty) 
paulson@13363
   810
apply (subgoal_tac "L(Memrel(succ(n)))") 
paulson@13363
   811
apply (rule_tac A="{B,A,n,z,0,Memrel(succ(n))}" in subset_LsetE, blast ) 
paulson@13363
   812
apply (rule ReflectsE [OF list_replacement1_Reflects], assumption)
paulson@13363
   813
apply (drule subset_Lset_ltD, assumption) 
paulson@13363
   814
apply (erule reflection_imp_L_separation)
paulson@13386
   815
  apply (simp_all add: lt_Ord2 Memrel_closed)
paulson@13386
   816
apply (elim conjE) 
paulson@13385
   817
apply (rule DPow_LsetI)
paulson@13363
   818
apply (rename_tac v) 
paulson@13363
   819
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13363
   820
apply (rule_tac env = "[u,v,A,n,B,0,Memrel(succ(n))]" in mem_iff_sats)
paulson@13363
   821
apply (rule sep_rules | simp)+
paulson@13363
   822
txt{*Can't get sat rules to work for higher-order operators, so just expand them!*}
paulson@13363
   823
apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
paulson@13363
   824
apply (rule sep_rules list_functor_iff_sats quasinat_iff_sats | simp)+
paulson@13363
   825
done
paulson@13363
   826
paulson@13363
   827
paulson@13363
   828
lemma list_replacement2_Reflects:
paulson@13363
   829
 "REFLECTS
paulson@13363
   830
   [\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
paulson@13363
   831
         (\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
paulson@13363
   832
           is_wfrec (L, iterates_MH (L, is_list_functor(L, A), 0),
paulson@13363
   833
                              msn, u, x)),
paulson@13363
   834
    \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
paulson@13363
   835
         (\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i). 
paulson@13363
   836
          successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
paulson@13363
   837
           is_wfrec (**Lset(i), 
paulson@13363
   838
                 iterates_MH (**Lset(i), is_list_functor(**Lset(i), A), 0),
paulson@13363
   839
                     msn, u, x))]"
paulson@13363
   840
by (intro FOL_reflections function_reflections is_wfrec_reflection 
paulson@13363
   841
          iterates_MH_reflection list_functor_reflection) 
paulson@13363
   842
paulson@13363
   843
paulson@13363
   844
lemma list_replacement2: 
paulson@13363
   845
   "L(A) ==> strong_replacement(L, 
paulson@13363
   846
         \<lambda>n y. n\<in>nat & 
paulson@13363
   847
               (\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
paulson@13363
   848
               is_wfrec(L, iterates_MH(L,is_list_functor(L,A), 0), 
paulson@13363
   849
                        msn, n, y)))"
paulson@13363
   850
apply (rule strong_replacementI) 
paulson@13363
   851
apply (rule rallI)
paulson@13363
   852
apply (rename_tac B)   
paulson@13363
   853
apply (rule separation_CollectI) 
paulson@13363
   854
apply (insert nonempty) 
paulson@13363
   855
apply (rule_tac A="{A,B,z,0,nat}" in subset_LsetE) 
paulson@13363
   856
apply (blast intro: L_nat) 
paulson@13363
   857
apply (rule ReflectsE [OF list_replacement2_Reflects], assumption)
paulson@13363
   858
apply (drule subset_Lset_ltD, assumption) 
paulson@13363
   859
apply (erule reflection_imp_L_separation)
paulson@13363
   860
  apply (simp_all add: lt_Ord2)
paulson@13385
   861
apply (rule DPow_LsetI)
paulson@13363
   862
apply (rename_tac v) 
paulson@13363
   863
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13363
   864
apply (rule_tac env = "[u,v,A,B,0,nat]" in mem_iff_sats)
paulson@13363
   865
apply (rule sep_rules | simp)+
paulson@13363
   866
apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
paulson@13363
   867
apply (rule sep_rules list_functor_iff_sats quasinat_iff_sats | simp)+
paulson@13363
   868
done
paulson@13363
   869
paulson@13386
   870
paulson@13386
   871
subsection{*@{term L} is Closed Under the Operator @{term formula}*} 
paulson@13386
   872
paulson@13386
   873
subsubsection{*The Formula Functor, Internalized*}
paulson@13386
   874
paulson@13386
   875
constdefs formula_functor_fm :: "[i,i]=>i"
paulson@13386
   876
(*     "is_formula_functor(M,X,Z) == 
paulson@13398
   877
        \<exists>nat'[M]. \<exists>natnat[M]. \<exists>natnatsum[M]. \<exists>XX[M]. \<exists>X3[M]. 
paulson@13398
   878
           4           3               2       1       0
paulson@13386
   879
          omega(M,nat') & cartprod(M,nat',nat',natnat) & 
paulson@13386
   880
          is_sum(M,natnat,natnat,natnatsum) &
paulson@13398
   881
          cartprod(M,X,X,XX) & is_sum(M,XX,X,X3) & 
paulson@13398
   882
          is_sum(M,natnatsum,X3,Z)" *) 
paulson@13386
   883
    "formula_functor_fm(X,Z) == 
paulson@13398
   884
       Exists(Exists(Exists(Exists(Exists(
paulson@13398
   885
	And(omega_fm(4),
paulson@13398
   886
         And(cartprod_fm(4,4,3),
paulson@13398
   887
          And(sum_fm(3,3,2),
paulson@13398
   888
           And(cartprod_fm(X#+5,X#+5,1),
paulson@13398
   889
            And(sum_fm(1,X#+5,0), sum_fm(2,0,Z#+5)))))))))))"
paulson@13386
   890
paulson@13386
   891
lemma formula_functor_type [TC]:
paulson@13386
   892
     "[| x \<in> nat; y \<in> nat |] ==> formula_functor_fm(x,y) \<in> formula"
paulson@13386
   893
by (simp add: formula_functor_fm_def) 
paulson@13386
   894
paulson@13386
   895
lemma sats_formula_functor_fm [simp]:
paulson@13386
   896
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
paulson@13386
   897
    ==> sats(A, formula_functor_fm(x,y), env) <-> 
paulson@13386
   898
        is_formula_functor(**A, nth(x,env), nth(y,env))"
paulson@13386
   899
by (simp add: formula_functor_fm_def is_formula_functor_def)
paulson@13386
   900
paulson@13386
   901
lemma formula_functor_iff_sats:
paulson@13386
   902
  "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13386
   903
      i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13386
   904
   ==> is_formula_functor(**A, x, y) <-> sats(A, formula_functor_fm(i,j), env)"
paulson@13386
   905
by simp
paulson@13386
   906
paulson@13386
   907
theorem formula_functor_reflection:
paulson@13386
   908
     "REFLECTS[\<lambda>x. is_formula_functor(L,f(x),g(x)), 
paulson@13386
   909
               \<lambda>i x. is_formula_functor(**Lset(i),f(x),g(x))]"
paulson@13386
   910
apply (simp only: is_formula_functor_def setclass_simps)
paulson@13386
   911
apply (intro FOL_reflections omega_reflection
paulson@13386
   912
             cartprod_reflection sum_reflection)  
paulson@13386
   913
done
paulson@13386
   914
paulson@13386
   915
subsubsection{*Instances of Replacement for Formulas*}
paulson@13386
   916
paulson@13386
   917
lemma formula_replacement1_Reflects:
paulson@13386
   918
 "REFLECTS
paulson@13386
   919
   [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
paulson@13386
   920
         is_wfrec(L, iterates_MH(L, is_formula_functor(L), 0), memsn, u, y)),
paulson@13386
   921
    \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
paulson@13386
   922
         is_wfrec(**Lset(i), 
paulson@13386
   923
                  iterates_MH(**Lset(i), 
paulson@13386
   924
                          is_formula_functor(**Lset(i)), 0), memsn, u, y))]"
paulson@13386
   925
by (intro FOL_reflections function_reflections is_wfrec_reflection 
paulson@13386
   926
          iterates_MH_reflection formula_functor_reflection) 
paulson@13386
   927
paulson@13386
   928
lemma formula_replacement1: 
paulson@13386
   929
   "iterates_replacement(L, is_formula_functor(L), 0)"
paulson@13386
   930
apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
paulson@13386
   931
apply (rule strong_replacementI) 
paulson@13386
   932
apply (rule rallI)
paulson@13386
   933
apply (rename_tac B)   
paulson@13386
   934
apply (rule separation_CollectI) 
paulson@13386
   935
apply (insert nonempty) 
paulson@13386
   936
apply (subgoal_tac "L(Memrel(succ(n)))") 
paulson@13386
   937
apply (rule_tac A="{B,n,z,0,Memrel(succ(n))}" in subset_LsetE, blast ) 
paulson@13386
   938
apply (rule ReflectsE [OF formula_replacement1_Reflects], assumption)
paulson@13386
   939
apply (drule subset_Lset_ltD, assumption) 
paulson@13386
   940
apply (erule reflection_imp_L_separation)
paulson@13386
   941
  apply (simp_all add: lt_Ord2 Memrel_closed)
paulson@13386
   942
apply (rule DPow_LsetI)
paulson@13386
   943
apply (rename_tac v) 
paulson@13386
   944
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13386
   945
apply (rule_tac env = "[u,v,n,B,0,Memrel(succ(n))]" in mem_iff_sats)
paulson@13386
   946
apply (rule sep_rules | simp)+
paulson@13386
   947
txt{*Can't get sat rules to work for higher-order operators, so just expand them!*}
paulson@13386
   948
apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
paulson@13386
   949
apply (rule sep_rules formula_functor_iff_sats quasinat_iff_sats | simp)+
paulson@13386
   950
txt{*SLOW: like 40 seconds!*}
paulson@13386
   951
done
paulson@13386
   952
paulson@13386
   953
lemma formula_replacement2_Reflects:
paulson@13386
   954
 "REFLECTS
paulson@13386
   955
   [\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
paulson@13386
   956
         (\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
paulson@13386
   957
           is_wfrec (L, iterates_MH (L, is_formula_functor(L), 0),
paulson@13386
   958
                              msn, u, x)),
paulson@13386
   959
    \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
paulson@13386
   960
         (\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i). 
paulson@13386
   961
          successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
paulson@13386
   962
           is_wfrec (**Lset(i), 
paulson@13386
   963
                 iterates_MH (**Lset(i), is_formula_functor(**Lset(i)), 0),
paulson@13386
   964
                     msn, u, x))]"
paulson@13386
   965
by (intro FOL_reflections function_reflections is_wfrec_reflection 
paulson@13386
   966
          iterates_MH_reflection formula_functor_reflection) 
paulson@13386
   967
paulson@13386
   968
paulson@13386
   969
lemma formula_replacement2: 
paulson@13386
   970
   "strong_replacement(L, 
paulson@13386
   971
         \<lambda>n y. n\<in>nat & 
paulson@13386
   972
               (\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
paulson@13386
   973
               is_wfrec(L, iterates_MH(L,is_formula_functor(L), 0), 
paulson@13386
   974
                        msn, n, y)))"
paulson@13386
   975
apply (rule strong_replacementI) 
paulson@13386
   976
apply (rule rallI)
paulson@13386
   977
apply (rename_tac B)   
paulson@13386
   978
apply (rule separation_CollectI) 
paulson@13386
   979
apply (insert nonempty) 
paulson@13386
   980
apply (rule_tac A="{B,z,0,nat}" in subset_LsetE) 
paulson@13386
   981
apply (blast intro: L_nat) 
paulson@13386
   982
apply (rule ReflectsE [OF formula_replacement2_Reflects], assumption)
paulson@13386
   983
apply (drule subset_Lset_ltD, assumption) 
paulson@13386
   984
apply (erule reflection_imp_L_separation)
paulson@13386
   985
  apply (simp_all add: lt_Ord2)
paulson@13386
   986
apply (rule DPow_LsetI)
paulson@13386
   987
apply (rename_tac v) 
paulson@13386
   988
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13386
   989
apply (rule_tac env = "[u,v,B,0,nat]" in mem_iff_sats)
paulson@13386
   990
apply (rule sep_rules | simp)+
paulson@13386
   991
apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
paulson@13386
   992
apply (rule sep_rules formula_functor_iff_sats quasinat_iff_sats | simp)+
paulson@13386
   993
done
paulson@13386
   994
paulson@13386
   995
text{*NB The proofs for type @{term formula} are virtually identical to those
paulson@13386
   996
for @{term "list(A)"}.  It was a cut-and-paste job! *}
paulson@13386
   997
paulson@13387
   998
paulson@13409
   999
subsection{*Internalized Forms of Data Structuring Operators*}
paulson@13409
  1000
paulson@13409
  1001
subsubsection{*The Formula @{term is_Inl}, Internalized*}
paulson@13409
  1002
paulson@13409
  1003
(*  is_Inl(M,a,z) == \<exists>zero[M]. empty(M,zero) & pair(M,zero,a,z) *)
paulson@13409
  1004
constdefs Inl_fm :: "[i,i]=>i"
paulson@13409
  1005
    "Inl_fm(a,z) == Exists(And(empty_fm(0), pair_fm(0,succ(a),succ(z))))"
paulson@13409
  1006
paulson@13409
  1007
lemma Inl_type [TC]:
paulson@13409
  1008
     "[| x \<in> nat; z \<in> nat |] ==> Inl_fm(x,z) \<in> formula"
paulson@13409
  1009
by (simp add: Inl_fm_def) 
paulson@13409
  1010
paulson@13409
  1011
lemma sats_Inl_fm [simp]:
paulson@13409
  1012
   "[| x \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13409
  1013
    ==> sats(A, Inl_fm(x,z), env) <-> is_Inl(**A, nth(x,env), nth(z,env))"
paulson@13409
  1014
by (simp add: Inl_fm_def is_Inl_def)
paulson@13409
  1015
paulson@13409
  1016
lemma Inl_iff_sats:
paulson@13409
  1017
      "[| nth(i,env) = x; nth(k,env) = z; 
paulson@13409
  1018
          i \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13409
  1019
       ==> is_Inl(**A, x, z) <-> sats(A, Inl_fm(i,k), env)"
paulson@13409
  1020
by simp
paulson@13409
  1021
paulson@13409
  1022
theorem Inl_reflection:
paulson@13409
  1023
     "REFLECTS[\<lambda>x. is_Inl(L,f(x),h(x)), 
paulson@13409
  1024
               \<lambda>i x. is_Inl(**Lset(i),f(x),h(x))]"
paulson@13409
  1025
apply (simp only: is_Inl_def setclass_simps)
paulson@13409
  1026
apply (intro FOL_reflections function_reflections)  
paulson@13409
  1027
done
paulson@13409
  1028
paulson@13409
  1029
paulson@13409
  1030
subsubsection{*The Formula @{term is_Inr}, Internalized*}
paulson@13409
  1031
paulson@13409
  1032
(*  is_Inr(M,a,z) == \<exists>n1[M]. number1(M,n1) & pair(M,n1,a,z) *)
paulson@13409
  1033
constdefs Inr_fm :: "[i,i]=>i"
paulson@13409
  1034
    "Inr_fm(a,z) == Exists(And(number1_fm(0), pair_fm(0,succ(a),succ(z))))"
paulson@13409
  1035
paulson@13409
  1036
lemma Inr_type [TC]:
paulson@13409
  1037
     "[| x \<in> nat; z \<in> nat |] ==> Inr_fm(x,z) \<in> formula"
paulson@13409
  1038
by (simp add: Inr_fm_def) 
paulson@13409
  1039
paulson@13409
  1040
lemma sats_Inr_fm [simp]:
paulson@13409
  1041
   "[| x \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13409
  1042
    ==> sats(A, Inr_fm(x,z), env) <-> is_Inr(**A, nth(x,env), nth(z,env))"
paulson@13409
  1043
by (simp add: Inr_fm_def is_Inr_def)
paulson@13409
  1044
paulson@13409
  1045
lemma Inr_iff_sats:
paulson@13409
  1046
      "[| nth(i,env) = x; nth(k,env) = z; 
paulson@13409
  1047
          i \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13409
  1048
       ==> is_Inr(**A, x, z) <-> sats(A, Inr_fm(i,k), env)"
paulson@13409
  1049
by simp
paulson@13409
  1050
paulson@13409
  1051
theorem Inr_reflection:
paulson@13409
  1052
     "REFLECTS[\<lambda>x. is_Inr(L,f(x),h(x)), 
paulson@13409
  1053
               \<lambda>i x. is_Inr(**Lset(i),f(x),h(x))]"
paulson@13409
  1054
apply (simp only: is_Inr_def setclass_simps)
paulson@13409
  1055
apply (intro FOL_reflections function_reflections)  
paulson@13409
  1056
done
paulson@13409
  1057
paulson@13409
  1058
paulson@13409
  1059
subsubsection{*The Formula @{term is_Nil}, Internalized*}
paulson@13409
  1060
paulson@13409
  1061
(* is_Nil(M,xs) == \<exists>zero[M]. empty(M,zero) & is_Inl(M,zero,xs) *)
paulson@13409
  1062
paulson@13409
  1063
constdefs Nil_fm :: "i=>i"
paulson@13409
  1064
    "Nil_fm(x) == Exists(And(empty_fm(0), Inl_fm(0,succ(x))))"
paulson@13409
  1065
 
paulson@13409
  1066
lemma Nil_type [TC]: "x \<in> nat ==> Nil_fm(x) \<in> formula"
paulson@13409
  1067
by (simp add: Nil_fm_def) 
paulson@13409
  1068
paulson@13409
  1069
lemma sats_Nil_fm [simp]:
paulson@13409
  1070
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13409
  1071
    ==> sats(A, Nil_fm(x), env) <-> is_Nil(**A, nth(x,env))"
paulson@13409
  1072
by (simp add: Nil_fm_def is_Nil_def)
paulson@13409
  1073
paulson@13409
  1074
lemma Nil_iff_sats:
paulson@13409
  1075
      "[| nth(i,env) = x; i \<in> nat; env \<in> list(A)|]
paulson@13409
  1076
       ==> is_Nil(**A, x) <-> sats(A, Nil_fm(i), env)"
paulson@13409
  1077
by simp
paulson@13409
  1078
paulson@13409
  1079
theorem Nil_reflection:
paulson@13409
  1080
     "REFLECTS[\<lambda>x. is_Nil(L,f(x)), 
paulson@13409
  1081
               \<lambda>i x. is_Nil(**Lset(i),f(x))]"
paulson@13409
  1082
apply (simp only: is_Nil_def setclass_simps)
paulson@13409
  1083
apply (intro FOL_reflections function_reflections Inl_reflection)  
paulson@13409
  1084
done
paulson@13409
  1085
paulson@13409
  1086
paulson@13422
  1087
subsubsection{*The Formula @{term is_Cons}, Internalized*}
paulson@13395
  1088
paulson@13387
  1089
paulson@13409
  1090
(*  "is_Cons(M,a,l,Z) == \<exists>p[M]. pair(M,a,l,p) & is_Inr(M,p,Z)" *)
paulson@13409
  1091
constdefs Cons_fm :: "[i,i,i]=>i"
paulson@13409
  1092
    "Cons_fm(a,l,Z) == 
paulson@13409
  1093
       Exists(And(pair_fm(succ(a),succ(l),0), Inr_fm(0,succ(Z))))"
paulson@13409
  1094
paulson@13409
  1095
lemma Cons_type [TC]:
paulson@13409
  1096
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> Cons_fm(x,y,z) \<in> formula"
paulson@13409
  1097
by (simp add: Cons_fm_def) 
paulson@13409
  1098
paulson@13409
  1099
lemma sats_Cons_fm [simp]:
paulson@13409
  1100
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13409
  1101
    ==> sats(A, Cons_fm(x,y,z), env) <-> 
paulson@13409
  1102
       is_Cons(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13409
  1103
by (simp add: Cons_fm_def is_Cons_def)
paulson@13409
  1104
paulson@13409
  1105
lemma Cons_iff_sats:
paulson@13409
  1106
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13409
  1107
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13409
  1108
       ==>is_Cons(**A, x, y, z) <-> sats(A, Cons_fm(i,j,k), env)"
paulson@13409
  1109
by simp
paulson@13409
  1110
paulson@13409
  1111
theorem Cons_reflection:
paulson@13409
  1112
     "REFLECTS[\<lambda>x. is_Cons(L,f(x),g(x),h(x)), 
paulson@13409
  1113
               \<lambda>i x. is_Cons(**Lset(i),f(x),g(x),h(x))]"
paulson@13409
  1114
apply (simp only: is_Cons_def setclass_simps)
paulson@13409
  1115
apply (intro FOL_reflections pair_reflection Inr_reflection)  
paulson@13409
  1116
done
paulson@13409
  1117
paulson@13409
  1118
subsubsection{*The Formula @{term is_quasilist}, Internalized*}
paulson@13409
  1119
paulson@13409
  1120
(* is_quasilist(M,xs) == is_Nil(M,z) | (\<exists>x[M]. \<exists>l[M]. is_Cons(M,x,l,z))" *)
paulson@13409
  1121
paulson@13409
  1122
constdefs quasilist_fm :: "i=>i"
paulson@13409
  1123
    "quasilist_fm(x) == 
paulson@13409
  1124
       Or(Nil_fm(x), Exists(Exists(Cons_fm(1,0,succ(succ(x))))))"
paulson@13409
  1125
 
paulson@13409
  1126
lemma quasilist_type [TC]: "x \<in> nat ==> quasilist_fm(x) \<in> formula"
paulson@13409
  1127
by (simp add: quasilist_fm_def) 
paulson@13409
  1128
paulson@13409
  1129
lemma sats_quasilist_fm [simp]:
paulson@13409
  1130
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13409
  1131
    ==> sats(A, quasilist_fm(x), env) <-> is_quasilist(**A, nth(x,env))"
paulson@13409
  1132
by (simp add: quasilist_fm_def is_quasilist_def)
paulson@13409
  1133
paulson@13409
  1134
lemma quasilist_iff_sats:
paulson@13409
  1135
      "[| nth(i,env) = x; i \<in> nat; env \<in> list(A)|]
paulson@13409
  1136
       ==> is_quasilist(**A, x) <-> sats(A, quasilist_fm(i), env)"
paulson@13409
  1137
by simp
paulson@13409
  1138
paulson@13409
  1139
theorem quasilist_reflection:
paulson@13409
  1140
     "REFLECTS[\<lambda>x. is_quasilist(L,f(x)), 
paulson@13409
  1141
               \<lambda>i x. is_quasilist(**Lset(i),f(x))]"
paulson@13409
  1142
apply (simp only: is_quasilist_def setclass_simps)
paulson@13409
  1143
apply (intro FOL_reflections Nil_reflection Cons_reflection)  
paulson@13409
  1144
done
paulson@13409
  1145
paulson@13409
  1146
paulson@13409
  1147
subsection{*Absoluteness for the Function @{term nth}*}
paulson@13409
  1148
paulson@13409
  1149
paulson@13409
  1150
subsubsection{*The Formula @{term is_tl}, Internalized*}
paulson@13409
  1151
paulson@13409
  1152
(*     "is_tl(M,xs,T) == 
paulson@13409
  1153
       (is_Nil(M,xs) --> T=xs) &
paulson@13409
  1154
       (\<forall>x[M]. \<forall>l[M]. ~ is_Cons(M,x,l,xs) | T=l) &
paulson@13409
  1155
       (is_quasilist(M,xs) | empty(M,T))" *)
paulson@13409
  1156
constdefs tl_fm :: "[i,i]=>i"
paulson@13409
  1157
    "tl_fm(xs,T) == 
paulson@13409
  1158
       And(Implies(Nil_fm(xs), Equal(T,xs)),
paulson@13409
  1159
           And(Forall(Forall(Or(Neg(Cons_fm(1,0,xs#+2)), Equal(T#+2,0)))),
paulson@13409
  1160
               Or(quasilist_fm(xs), empty_fm(T))))"
paulson@13409
  1161
paulson@13409
  1162
lemma tl_type [TC]:
paulson@13409
  1163
     "[| x \<in> nat; y \<in> nat |] ==> tl_fm(x,y) \<in> formula"
paulson@13409
  1164
by (simp add: tl_fm_def) 
paulson@13409
  1165
paulson@13409
  1166
lemma sats_tl_fm [simp]:
paulson@13409
  1167
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
paulson@13409
  1168
    ==> sats(A, tl_fm(x,y), env) <-> is_tl(**A, nth(x,env), nth(y,env))"
paulson@13409
  1169
by (simp add: tl_fm_def is_tl_def)
paulson@13409
  1170
paulson@13409
  1171
lemma tl_iff_sats:
paulson@13409
  1172
      "[| nth(i,env) = x; nth(j,env) = y;
paulson@13409
  1173
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13409
  1174
       ==> is_tl(**A, x, y) <-> sats(A, tl_fm(i,j), env)"
paulson@13409
  1175
by simp
paulson@13409
  1176
paulson@13409
  1177
theorem tl_reflection:
paulson@13409
  1178
     "REFLECTS[\<lambda>x. is_tl(L,f(x),g(x)), 
paulson@13409
  1179
               \<lambda>i x. is_tl(**Lset(i),f(x),g(x))]"
paulson@13409
  1180
apply (simp only: is_tl_def setclass_simps)
paulson@13409
  1181
apply (intro FOL_reflections Nil_reflection Cons_reflection
paulson@13409
  1182
             quasilist_reflection empty_reflection)  
paulson@13409
  1183
done
paulson@13409
  1184
paulson@13409
  1185
paulson@13409
  1186
subsubsection{*An Instance of Replacement for @{term nth}*}
paulson@13409
  1187
paulson@13409
  1188
lemma nth_replacement_Reflects:
paulson@13409
  1189
 "REFLECTS
paulson@13409
  1190
   [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
paulson@13409
  1191
         is_wfrec(L, iterates_MH(L, is_tl(L), z), memsn, u, y)),
paulson@13409
  1192
    \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
paulson@13409
  1193
         is_wfrec(**Lset(i), 
paulson@13409
  1194
                  iterates_MH(**Lset(i), 
paulson@13409
  1195
                          is_tl(**Lset(i)), z), memsn, u, y))]"
paulson@13409
  1196
by (intro FOL_reflections function_reflections is_wfrec_reflection 
paulson@13409
  1197
          iterates_MH_reflection list_functor_reflection tl_reflection) 
paulson@13409
  1198
paulson@13409
  1199
lemma nth_replacement: 
paulson@13409
  1200
   "L(w) ==> iterates_replacement(L, %l t. is_tl(L,l,t), w)"
paulson@13409
  1201
apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
paulson@13409
  1202
apply (rule strong_replacementI) 
paulson@13409
  1203
apply (rule rallI)   
paulson@13409
  1204
apply (rule separation_CollectI) 
paulson@13409
  1205
apply (subgoal_tac "L(Memrel(succ(n)))") 
paulson@13409
  1206
apply (rule_tac A="{A,n,w,z,Memrel(succ(n))}" in subset_LsetE, blast ) 
paulson@13409
  1207
apply (rule ReflectsE [OF nth_replacement_Reflects], assumption)
paulson@13409
  1208
apply (drule subset_Lset_ltD, assumption) 
paulson@13409
  1209
apply (erule reflection_imp_L_separation)
paulson@13409
  1210
  apply (simp_all add: lt_Ord2 Memrel_closed)
paulson@13409
  1211
apply (elim conjE) 
paulson@13409
  1212
apply (rule DPow_LsetI)
paulson@13409
  1213
apply (rename_tac v) 
paulson@13409
  1214
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13409
  1215
apply (rule_tac env = "[u,v,A,z,w,Memrel(succ(n))]" in mem_iff_sats)
paulson@13409
  1216
apply (rule sep_rules | simp)+
paulson@13409
  1217
apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
paulson@13409
  1218
apply (rule sep_rules quasinat_iff_sats tl_iff_sats | simp)+
paulson@13409
  1219
done
paulson@13409
  1220
paulson@13422
  1221
paulson@13422
  1222
paulson@13422
  1223
subsubsection{*Instantiating the locale @{text M_datatypes}*}
paulson@13409
  1224
ML
paulson@13409
  1225
{*
paulson@13422
  1226
val list_replacement1 = thm "list_replacement1"; 
paulson@13422
  1227
val list_replacement2 = thm "list_replacement2";
paulson@13422
  1228
val formula_replacement1 = thm "formula_replacement1";
paulson@13422
  1229
val formula_replacement2 = thm "formula_replacement2";
paulson@13422
  1230
val nth_replacement = thm "nth_replacement";
paulson@13422
  1231
paulson@13422
  1232
val m_datatypes = [list_replacement1, list_replacement2, 
paulson@13422
  1233
                   formula_replacement1, formula_replacement2, 
paulson@13422
  1234
                   nth_replacement];
paulson@13422
  1235
paulson@13422
  1236
fun datatypes_L th =
paulson@13422
  1237
    kill_flex_triv_prems (m_datatypes MRS (wfrank_L th));
paulson@13422
  1238
paulson@13422
  1239
bind_thm ("list_closed", datatypes_L (thm "M_datatypes.list_closed"));
paulson@13422
  1240
bind_thm ("formula_closed", datatypes_L (thm "M_datatypes.formula_closed"));
paulson@13422
  1241
bind_thm ("list_abs", datatypes_L (thm "M_datatypes.list_abs"));
paulson@13422
  1242
bind_thm ("formula_abs", datatypes_L (thm "M_datatypes.formula_abs"));
paulson@13422
  1243
bind_thm ("nth_abs", datatypes_L (thm "M_datatypes.nth_abs"));
paulson@13409
  1244
*}
paulson@13409
  1245
paulson@13422
  1246
declare list_closed [intro,simp]
paulson@13422
  1247
declare formula_closed [intro,simp]
paulson@13422
  1248
declare list_abs [simp]
paulson@13422
  1249
declare formula_abs [simp]
paulson@13422
  1250
declare nth_abs [simp]
paulson@13422
  1251
paulson@13422
  1252
paulson@13422
  1253
paulson@13422
  1254
subsection{*@{term L} is Closed Under the Operator @{term eclose}*} 
paulson@13422
  1255
paulson@13422
  1256
subsubsection{*Instances of Replacement for @{term eclose}*}
paulson@13422
  1257
paulson@13422
  1258
lemma eclose_replacement1_Reflects:
paulson@13422
  1259
 "REFLECTS
paulson@13422
  1260
   [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
paulson@13422
  1261
         is_wfrec(L, iterates_MH(L, big_union(L), A), memsn, u, y)),
paulson@13422
  1262
    \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
paulson@13422
  1263
         is_wfrec(**Lset(i), 
paulson@13422
  1264
                  iterates_MH(**Lset(i), big_union(**Lset(i)), A), 
paulson@13422
  1265
                  memsn, u, y))]"
paulson@13422
  1266
by (intro FOL_reflections function_reflections is_wfrec_reflection 
paulson@13422
  1267
          iterates_MH_reflection) 
paulson@13422
  1268
paulson@13422
  1269
lemma eclose_replacement1: 
paulson@13422
  1270
   "L(A) ==> iterates_replacement(L, big_union(L), A)"
paulson@13422
  1271
apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
paulson@13422
  1272
apply (rule strong_replacementI) 
paulson@13422
  1273
apply (rule rallI)
paulson@13422
  1274
apply (rename_tac B)   
paulson@13422
  1275
apply (rule separation_CollectI) 
paulson@13422
  1276
apply (subgoal_tac "L(Memrel(succ(n)))") 
paulson@13422
  1277
apply (rule_tac A="{B,A,n,z,Memrel(succ(n))}" in subset_LsetE, blast ) 
paulson@13422
  1278
apply (rule ReflectsE [OF eclose_replacement1_Reflects], assumption)
paulson@13422
  1279
apply (drule subset_Lset_ltD, assumption) 
paulson@13422
  1280
apply (erule reflection_imp_L_separation)
paulson@13422
  1281
  apply (simp_all add: lt_Ord2 Memrel_closed)
paulson@13422
  1282
apply (elim conjE) 
paulson@13422
  1283
apply (rule DPow_LsetI)
paulson@13422
  1284
apply (rename_tac v) 
paulson@13422
  1285
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13422
  1286
apply (rule_tac env = "[u,v,A,n,B,Memrel(succ(n))]" in mem_iff_sats)
paulson@13422
  1287
apply (rule sep_rules | simp)+
paulson@13422
  1288
txt{*Can't get sat rules to work for higher-order operators, so just expand them!*}
paulson@13422
  1289
apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
paulson@13422
  1290
apply (rule sep_rules big_union_iff_sats quasinat_iff_sats | simp)+
paulson@13409
  1291
done
paulson@13409
  1292
paulson@13422
  1293
paulson@13422
  1294
lemma eclose_replacement2_Reflects:
paulson@13422
  1295
 "REFLECTS
paulson@13422
  1296
   [\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
paulson@13422
  1297
         (\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
paulson@13422
  1298
           is_wfrec (L, iterates_MH (L, big_union(L), A),
paulson@13422
  1299
                              msn, u, x)),
paulson@13422
  1300
    \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
paulson@13422
  1301
         (\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i). 
paulson@13422
  1302
          successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
paulson@13422
  1303
           is_wfrec (**Lset(i), 
paulson@13422
  1304
                 iterates_MH (**Lset(i), big_union(**Lset(i)), A),
paulson@13422
  1305
                     msn, u, x))]"
paulson@13422
  1306
by (intro FOL_reflections function_reflections is_wfrec_reflection 
paulson@13422
  1307
          iterates_MH_reflection) 
paulson@13422
  1308
paulson@13422
  1309
paulson@13422
  1310
lemma eclose_replacement2: 
paulson@13422
  1311
   "L(A) ==> strong_replacement(L, 
paulson@13422
  1312
         \<lambda>n y. n\<in>nat & 
paulson@13422
  1313
               (\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
paulson@13422
  1314
               is_wfrec(L, iterates_MH(L,big_union(L), A), 
paulson@13422
  1315
                        msn, n, y)))"
paulson@13422
  1316
apply (rule strong_replacementI) 
paulson@13422
  1317
apply (rule rallI)
paulson@13422
  1318
apply (rename_tac B)   
paulson@13422
  1319
apply (rule separation_CollectI) 
paulson@13422
  1320
apply (rule_tac A="{A,B,z,nat}" in subset_LsetE) 
paulson@13422
  1321
apply (blast intro: L_nat) 
paulson@13422
  1322
apply (rule ReflectsE [OF eclose_replacement2_Reflects], assumption)
paulson@13422
  1323
apply (drule subset_Lset_ltD, assumption) 
paulson@13422
  1324
apply (erule reflection_imp_L_separation)
paulson@13422
  1325
  apply (simp_all add: lt_Ord2)
paulson@13422
  1326
apply (rule DPow_LsetI)
paulson@13422
  1327
apply (rename_tac v) 
paulson@13422
  1328
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13422
  1329
apply (rule_tac env = "[u,v,A,B,nat]" in mem_iff_sats)
paulson@13422
  1330
apply (rule sep_rules | simp)+
paulson@13422
  1331
apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
paulson@13422
  1332
apply (rule sep_rules big_union_iff_sats quasinat_iff_sats | simp)+
paulson@13422
  1333
done
paulson@13422
  1334
paulson@13422
  1335
paulson@13422
  1336
subsubsection{*Instantiating the locale @{text M_eclose}*}
paulson@13422
  1337
ML
paulson@13422
  1338
{*
paulson@13422
  1339
val eclose_replacement1 = thm "eclose_replacement1"; 
paulson@13422
  1340
val eclose_replacement2 = thm "eclose_replacement2";
paulson@13422
  1341
paulson@13422
  1342
val m_eclose = [eclose_replacement1, eclose_replacement2];
paulson@13422
  1343
paulson@13422
  1344
fun eclose_L th =
paulson@13422
  1345
    kill_flex_triv_prems (m_eclose MRS (datatypes_L th));
paulson@13422
  1346
paulson@13422
  1347
bind_thm ("eclose_closed", eclose_L (thm "M_eclose.eclose_closed"));
paulson@13422
  1348
bind_thm ("eclose_abs", eclose_L (thm "M_eclose.eclose_abs"));
paulson@13422
  1349
*}
paulson@13422
  1350
paulson@13422
  1351
declare eclose_closed [intro,simp]
paulson@13422
  1352
declare eclose_abs [intro,simp]
paulson@13422
  1353
paulson@13422
  1354
paulson@13348
  1355
end