src/ZF/Constructible/Rec_Separation.thy
author paulson
Tue Jul 16 16:29:36 2002 +0200 (2002-07-16)
changeset 13363 c26eeb000470
parent 13352 3cd767f8d78b
child 13385 31df66ca0780
permissions -rw-r--r--
instantiation of locales M_trancl and M_wfrank;
proofs of list_replacement{1,2}
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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 DPowI2)
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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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apply (simp_all add: succ_Un_distrib [symmetric])
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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 DPowI2)
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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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apply (simp_all add: succ_Un_distrib [symmetric])
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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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   294
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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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   315
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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   327
paulson@13363
   328
subsection{*The Locale @{text "M_wfrank"}*}
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   330
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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   335
      \<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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   337
            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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   339
by (intro FOL_reflections function_reflections is_recfun_reflection tran_closure_reflection)  
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   340
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   341
lemma wfrank_separation:
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     "L(r) ==>
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   343
      separation (L, \<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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apply (rule separation_CollectI) 
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   346
apply (rule_tac A="{r,z}" in subset_LsetE, blast ) 
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   347
apply (rule ReflectsE [OF wfrank_Reflects], assumption)
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   348
apply (drule subset_Lset_ltD, assumption) 
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apply (erule reflection_imp_L_separation)
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   350
  apply (simp_all add: lt_Ord2, clarify)
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apply (rule DPowI2)
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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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   356
apply (simp_all add: succ_Un_distrib [symmetric])
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   357
done
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   358
paulson@13348
   359
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   360
subsubsection{*Replacement for @{term "wfrank"}*}
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   361
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   362
lemma wfrank_replacement_Reflects:
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 "REFLECTS[\<lambda>z. \<exists>x[L]. x \<in> A & 
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   364
        (\<forall>rplus[L]. tran_closure(L,r,rplus) -->
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   365
         (\<exists>y[L]. \<exists>f[L]. pair(L,x,y,z)  & 
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   366
                        M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f) &
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   367
                        is_range(L,f,y))),
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 \<lambda>i z. \<exists>x \<in> Lset(i). x \<in> A & 
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   369
      (\<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) -->
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   370
       (\<exists>y \<in> Lset(i). \<exists>f \<in> Lset(i). pair(**Lset(i),x,y,z)  & 
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   371
         M_is_recfun(**Lset(i), %x f y. is_range(**Lset(i),f,y), rplus, x, f) &
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         is_range(**Lset(i),f,y)))]"
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   373
by (intro FOL_reflections function_reflections fun_plus_reflections
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   374
             is_recfun_reflection tran_closure_reflection)
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   375
paulson@13348
   376
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   377
lemma wfrank_strong_replacement:
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   378
     "L(r) ==>
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   379
      strong_replacement(L, \<lambda>x z. 
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   380
         \<forall>rplus[L]. tran_closure(L,r,rplus) -->
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   381
         (\<exists>y[L]. \<exists>f[L]. pair(L,x,y,z)  & 
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   382
                        M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f) &
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   383
                        is_range(L,f,y)))"
paulson@13348
   384
apply (rule strong_replacementI) 
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   385
apply (rule rallI)
paulson@13348
   386
apply (rename_tac B)  
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   387
apply (rule separation_CollectI) 
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   388
apply (rule_tac A="{B,r,z}" in subset_LsetE, blast ) 
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   389
apply (rule ReflectsE [OF wfrank_replacement_Reflects], assumption)
paulson@13348
   390
apply (drule subset_Lset_ltD, assumption) 
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   391
apply (erule reflection_imp_L_separation)
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   392
  apply (simp_all add: lt_Ord2)
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   393
apply (rule DPowI2)
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   394
apply (rename_tac u) 
paulson@13348
   395
apply (rule bex_iff_sats ball_iff_sats conj_iff_sats)+
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   396
apply (rule_tac env = "[x,u,B,r]" in mem_iff_sats) 
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   397
apply (rule sep_rules tran_closure_iff_sats is_recfun_iff_sats | simp)+
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   398
apply (simp_all add: succ_Un_distrib [symmetric])
paulson@13348
   399
done
paulson@13348
   400
paulson@13348
   401
paulson@13363
   402
subsubsection{*Separation for Proving @{text Ord_wfrank_range}*}
paulson@13348
   403
paulson@13348
   404
lemma Ord_wfrank_Reflects:
paulson@13348
   405
 "REFLECTS[\<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) --> 
paulson@13348
   406
          ~ (\<forall>f[L]. \<forall>rangef[L]. 
paulson@13348
   407
             is_range(L,f,rangef) -->
paulson@13352
   408
             M_is_recfun(L, \<lambda>x f y. is_range(L,f,y), rplus, x, f) -->
paulson@13348
   409
             ordinal(L,rangef)),
paulson@13348
   410
      \<lambda>i x. \<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) --> 
paulson@13348
   411
          ~ (\<forall>f \<in> Lset(i). \<forall>rangef \<in> Lset(i). 
paulson@13348
   412
             is_range(**Lset(i),f,rangef) -->
paulson@13352
   413
             M_is_recfun(**Lset(i), \<lambda>x f y. is_range(**Lset(i),f,y), 
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   414
                         rplus, x, f) -->
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   415
             ordinal(**Lset(i),rangef))]"
paulson@13348
   416
by (intro FOL_reflections function_reflections is_recfun_reflection 
paulson@13348
   417
          tran_closure_reflection ordinal_reflection)
paulson@13348
   418
paulson@13348
   419
lemma  Ord_wfrank_separation:
paulson@13348
   420
     "L(r) ==>
paulson@13348
   421
      separation (L, \<lambda>x.
paulson@13348
   422
         \<forall>rplus[L]. tran_closure(L,r,rplus) --> 
paulson@13348
   423
          ~ (\<forall>f[L]. \<forall>rangef[L]. 
paulson@13348
   424
             is_range(L,f,rangef) -->
paulson@13352
   425
             M_is_recfun(L, \<lambda>x f y. is_range(L,f,y), rplus, x, f) -->
paulson@13348
   426
             ordinal(L,rangef)))" 
paulson@13348
   427
apply (rule separation_CollectI) 
paulson@13348
   428
apply (rule_tac A="{r,z}" in subset_LsetE, blast ) 
paulson@13348
   429
apply (rule ReflectsE [OF Ord_wfrank_Reflects], assumption)
paulson@13348
   430
apply (drule subset_Lset_ltD, assumption) 
paulson@13348
   431
apply (erule reflection_imp_L_separation)
paulson@13348
   432
  apply (simp_all add: lt_Ord2, clarify)
paulson@13348
   433
apply (rule DPowI2)
paulson@13348
   434
apply (rename_tac u)  
paulson@13348
   435
apply (rule ball_iff_sats imp_iff_sats)+
paulson@13348
   436
apply (rule_tac env="[rplus,u,r]" in tran_closure_iff_sats)
paulson@13348
   437
apply (rule sep_rules is_recfun_iff_sats | simp)+
paulson@13348
   438
apply (simp_all add: succ_Un_distrib [symmetric])
paulson@13348
   439
done
paulson@13348
   440
paulson@13348
   441
paulson@13363
   442
subsubsection{*Instantiating the locale @{text M_wfrank}*}
paulson@13363
   443
ML
paulson@13363
   444
{*
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   445
val wfrank_separation = thm "wfrank_separation";
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   446
val wfrank_strong_replacement = thm "wfrank_strong_replacement";
paulson@13363
   447
val Ord_wfrank_separation = thm "Ord_wfrank_separation";
paulson@13363
   448
paulson@13363
   449
val m_wfrank = 
paulson@13363
   450
    [wfrank_separation, wfrank_strong_replacement, Ord_wfrank_separation];
paulson@13363
   451
paulson@13363
   452
fun wfrank_L th =
paulson@13363
   453
    kill_flex_triv_prems (m_wfrank MRS (trancl_L th));
paulson@13363
   454
paulson@13363
   455
paulson@13363
   456
paulson@13363
   457
bind_thm ("iterates_closed", wfrank_L (thm "M_wfrank.iterates_closed"));
paulson@13363
   458
bind_thm ("exists_wfrank", wfrank_L (thm "M_wfrank.exists_wfrank"));
paulson@13363
   459
bind_thm ("M_wellfoundedrank", wfrank_L (thm "M_wfrank.M_wellfoundedrank"));
paulson@13363
   460
bind_thm ("Ord_wfrank_range", wfrank_L (thm "M_wfrank.Ord_wfrank_range"));
paulson@13363
   461
bind_thm ("Ord_range_wellfoundedrank", wfrank_L (thm "M_wfrank.Ord_range_wellfoundedrank"));
paulson@13363
   462
bind_thm ("function_wellfoundedrank", wfrank_L (thm "M_wfrank.function_wellfoundedrank"));
paulson@13363
   463
bind_thm ("domain_wellfoundedrank", wfrank_L (thm "M_wfrank.domain_wellfoundedrank"));
paulson@13363
   464
bind_thm ("wellfoundedrank_type", wfrank_L (thm "M_wfrank.wellfoundedrank_type"));
paulson@13363
   465
bind_thm ("Ord_wellfoundedrank", wfrank_L (thm "M_wfrank.Ord_wellfoundedrank"));
paulson@13363
   466
bind_thm ("wellfoundedrank_eq", wfrank_L (thm "M_wfrank.wellfoundedrank_eq"));
paulson@13363
   467
bind_thm ("wellfoundedrank_lt", wfrank_L (thm "M_wfrank.wellfoundedrank_lt"));
paulson@13363
   468
bind_thm ("wellfounded_imp_subset_rvimage", wfrank_L (thm "M_wfrank.wellfounded_imp_subset_rvimage"));
paulson@13363
   469
bind_thm ("wellfounded_imp_wf", wfrank_L (thm "M_wfrank.wellfounded_imp_wf"));
paulson@13363
   470
bind_thm ("wellfounded_on_imp_wf_on", wfrank_L (thm "M_wfrank.wellfounded_on_imp_wf_on"));
paulson@13363
   471
bind_thm ("wf_abs", wfrank_L (thm "M_wfrank.wf_abs"));
paulson@13363
   472
bind_thm ("wf_on_abs", wfrank_L (thm "M_wfrank.wf_on_abs"));
paulson@13363
   473
bind_thm ("wfrec_replacement_iff", wfrank_L (thm "M_wfrank.wfrec_replacement_iff"));
paulson@13363
   474
bind_thm ("trans_wfrec_closed", wfrank_L (thm "M_wfrank.trans_wfrec_closed"));
paulson@13363
   475
bind_thm ("wfrec_closed", wfrank_L (thm "M_wfrank.wfrec_closed"));
paulson@13363
   476
*}
paulson@13363
   477
paulson@13363
   478
declare iterates_closed [intro,simp]
paulson@13363
   479
declare Ord_wfrank_range [rule_format]
paulson@13363
   480
declare wf_abs [simp]
paulson@13363
   481
declare wf_on_abs [simp]
paulson@13363
   482
paulson@13363
   483
paulson@13363
   484
subsection{*For Datatypes*}
paulson@13363
   485
paulson@13363
   486
subsubsection{*Binary Products, Internalized*}
paulson@13363
   487
paulson@13363
   488
constdefs cartprod_fm :: "[i,i,i]=>i"
paulson@13363
   489
(* "cartprod(M,A,B,z) == 
paulson@13363
   490
	\<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
   491
    "cartprod_fm(A,B,z) == 
paulson@13363
   492
       Forall(Iff(Member(0,succ(z)),
paulson@13363
   493
                  Exists(And(Member(0,succ(succ(A))),
paulson@13363
   494
                         Exists(And(Member(0,succ(succ(succ(B)))),
paulson@13363
   495
                                    pair_fm(1,0,2)))))))"
paulson@13363
   496
paulson@13363
   497
lemma cartprod_type [TC]:
paulson@13363
   498
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cartprod_fm(x,y,z) \<in> formula"
paulson@13363
   499
by (simp add: cartprod_fm_def) 
paulson@13363
   500
paulson@13363
   501
lemma arity_cartprod_fm [simp]:
paulson@13363
   502
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13363
   503
      ==> arity(cartprod_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13363
   504
by (simp add: cartprod_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13363
   505
paulson@13363
   506
lemma sats_cartprod_fm [simp]:
paulson@13363
   507
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13363
   508
    ==> sats(A, cartprod_fm(x,y,z), env) <-> 
paulson@13363
   509
        cartprod(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13363
   510
by (simp add: cartprod_fm_def cartprod_def)
paulson@13363
   511
paulson@13363
   512
lemma cartprod_iff_sats:
paulson@13363
   513
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13363
   514
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13363
   515
       ==> cartprod(**A, x, y, z) <-> sats(A, cartprod_fm(i,j,k), env)"
paulson@13363
   516
by (simp add: sats_cartprod_fm)
paulson@13363
   517
paulson@13363
   518
theorem cartprod_reflection:
paulson@13363
   519
     "REFLECTS[\<lambda>x. cartprod(L,f(x),g(x),h(x)), 
paulson@13363
   520
               \<lambda>i x. cartprod(**Lset(i),f(x),g(x),h(x))]"
paulson@13363
   521
apply (simp only: cartprod_def setclass_simps)
paulson@13363
   522
apply (intro FOL_reflections pair_reflection)  
paulson@13363
   523
done
paulson@13363
   524
paulson@13363
   525
paulson@13363
   526
subsubsection{*Binary Sums, Internalized*}
paulson@13363
   527
paulson@13363
   528
(* "is_sum(M,A,B,Z) == 
paulson@13363
   529
       \<exists>A0[M]. \<exists>n1[M]. \<exists>s1[M]. \<exists>B1[M]. 
paulson@13363
   530
         3      2       1        0
paulson@13363
   531
       number1(M,n1) & cartprod(M,n1,A,A0) & upair(M,n1,n1,s1) &
paulson@13363
   532
       cartprod(M,s1,B,B1) & union(M,A0,B1,Z)"  *)
paulson@13363
   533
constdefs sum_fm :: "[i,i,i]=>i"
paulson@13363
   534
    "sum_fm(A,B,Z) == 
paulson@13363
   535
       Exists(Exists(Exists(Exists(
paulson@13363
   536
	And(number1_fm(2),
paulson@13363
   537
            And(cartprod_fm(2,A#+4,3),
paulson@13363
   538
                And(upair_fm(2,2,1),
paulson@13363
   539
                    And(cartprod_fm(1,B#+4,0), union_fm(3,0,Z#+4)))))))))"
paulson@13363
   540
paulson@13363
   541
lemma sum_type [TC]:
paulson@13363
   542
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> sum_fm(x,y,z) \<in> formula"
paulson@13363
   543
by (simp add: sum_fm_def) 
paulson@13363
   544
paulson@13363
   545
lemma arity_sum_fm [simp]:
paulson@13363
   546
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13363
   547
      ==> arity(sum_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13363
   548
by (simp add: sum_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13363
   549
paulson@13363
   550
lemma sats_sum_fm [simp]:
paulson@13363
   551
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13363
   552
    ==> sats(A, sum_fm(x,y,z), env) <-> 
paulson@13363
   553
        is_sum(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13363
   554
by (simp add: sum_fm_def is_sum_def)
paulson@13363
   555
paulson@13363
   556
lemma sum_iff_sats:
paulson@13363
   557
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13363
   558
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13363
   559
       ==> is_sum(**A, x, y, z) <-> sats(A, sum_fm(i,j,k), env)"
paulson@13363
   560
by simp
paulson@13363
   561
paulson@13363
   562
theorem sum_reflection:
paulson@13363
   563
     "REFLECTS[\<lambda>x. is_sum(L,f(x),g(x),h(x)), 
paulson@13363
   564
               \<lambda>i x. is_sum(**Lset(i),f(x),g(x),h(x))]"
paulson@13363
   565
apply (simp only: is_sum_def setclass_simps)
paulson@13363
   566
apply (intro FOL_reflections function_reflections cartprod_reflection)  
paulson@13363
   567
done
paulson@13363
   568
paulson@13363
   569
paulson@13363
   570
subsubsection{*The List Functor, Internalized*}
paulson@13363
   571
paulson@13363
   572
constdefs list_functor_fm :: "[i,i,i]=>i"
paulson@13363
   573
(* "is_list_functor(M,A,X,Z) == 
paulson@13363
   574
        \<exists>n1[M]. \<exists>AX[M]. 
paulson@13363
   575
         number1(M,n1) & cartprod(M,A,X,AX) & is_sum(M,n1,AX,Z)" *)
paulson@13363
   576
    "list_functor_fm(A,X,Z) == 
paulson@13363
   577
       Exists(Exists(
paulson@13363
   578
	And(number1_fm(1),
paulson@13363
   579
            And(cartprod_fm(A#+2,X#+2,0), sum_fm(1,0,Z#+2)))))"
paulson@13363
   580
paulson@13363
   581
lemma list_functor_type [TC]:
paulson@13363
   582
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> list_functor_fm(x,y,z) \<in> formula"
paulson@13363
   583
by (simp add: list_functor_fm_def) 
paulson@13363
   584
paulson@13363
   585
lemma arity_list_functor_fm [simp]:
paulson@13363
   586
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13363
   587
      ==> arity(list_functor_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
paulson@13363
   588
by (simp add: list_functor_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13363
   589
paulson@13363
   590
lemma sats_list_functor_fm [simp]:
paulson@13363
   591
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13363
   592
    ==> sats(A, list_functor_fm(x,y,z), env) <-> 
paulson@13363
   593
        is_list_functor(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13363
   594
by (simp add: list_functor_fm_def is_list_functor_def)
paulson@13363
   595
paulson@13363
   596
lemma list_functor_iff_sats:
paulson@13363
   597
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13363
   598
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13363
   599
   ==> is_list_functor(**A, x, y, z) <-> sats(A, list_functor_fm(i,j,k), env)"
paulson@13363
   600
by simp
paulson@13363
   601
paulson@13363
   602
theorem list_functor_reflection:
paulson@13363
   603
     "REFLECTS[\<lambda>x. is_list_functor(L,f(x),g(x),h(x)), 
paulson@13363
   604
               \<lambda>i x. is_list_functor(**Lset(i),f(x),g(x),h(x))]"
paulson@13363
   605
apply (simp only: is_list_functor_def setclass_simps)
paulson@13363
   606
apply (intro FOL_reflections number1_reflection
paulson@13363
   607
             cartprod_reflection sum_reflection)  
paulson@13363
   608
done
paulson@13363
   609
paulson@13363
   610
subsubsection{*The Operator @{term quasinat}*}
paulson@13363
   611
paulson@13363
   612
(* "is_quasinat(M,z) == empty(M,z) | (\<exists>m[M]. successor(M,m,z))" *)
paulson@13363
   613
constdefs quasinat_fm :: "i=>i"
paulson@13363
   614
    "quasinat_fm(z) == Or(empty_fm(z), Exists(succ_fm(0,succ(z))))"
paulson@13363
   615
paulson@13363
   616
lemma quasinat_type [TC]:
paulson@13363
   617
     "x \<in> nat ==> quasinat_fm(x) \<in> formula"
paulson@13363
   618
by (simp add: quasinat_fm_def) 
paulson@13363
   619
paulson@13363
   620
lemma arity_quasinat_fm [simp]:
paulson@13363
   621
     "x \<in> nat ==> arity(quasinat_fm(x)) = succ(x)"
paulson@13363
   622
by (simp add: quasinat_fm_def succ_Un_distrib [symmetric] Un_ac) 
paulson@13363
   623
paulson@13363
   624
lemma sats_quasinat_fm [simp]:
paulson@13363
   625
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13363
   626
    ==> sats(A, quasinat_fm(x), env) <-> is_quasinat(**A, nth(x,env))"
paulson@13363
   627
by (simp add: quasinat_fm_def is_quasinat_def)
paulson@13363
   628
paulson@13363
   629
lemma quasinat_iff_sats:
paulson@13363
   630
      "[| nth(i,env) = x; nth(j,env) = y; 
paulson@13363
   631
          i \<in> nat; env \<in> list(A)|]
paulson@13363
   632
       ==> is_quasinat(**A, x) <-> sats(A, quasinat_fm(i), env)"
paulson@13363
   633
by simp
paulson@13363
   634
paulson@13363
   635
theorem quasinat_reflection:
paulson@13363
   636
     "REFLECTS[\<lambda>x. is_quasinat(L,f(x)), 
paulson@13363
   637
               \<lambda>i x. is_quasinat(**Lset(i),f(x))]"
paulson@13363
   638
apply (simp only: is_quasinat_def setclass_simps)
paulson@13363
   639
apply (intro FOL_reflections function_reflections)  
paulson@13363
   640
done
paulson@13363
   641
paulson@13363
   642
paulson@13363
   643
subsubsection{*The Operator @{term is_nat_case}*}
paulson@13363
   644
paulson@13363
   645
(* is_nat_case :: "[i=>o, i, [i,i]=>o, i, i] => o"
paulson@13363
   646
    "is_nat_case(M, a, is_b, k, z) == 
paulson@13363
   647
       (empty(M,k) --> z=a) &
paulson@13363
   648
       (\<forall>m[M]. successor(M,m,k) --> is_b(m,z)) &
paulson@13363
   649
       (is_quasinat(M,k) | empty(M,z))" *)
paulson@13363
   650
text{*The formula @{term is_b} has free variables 1 and 0.*}
paulson@13363
   651
constdefs is_nat_case_fm :: "[i, [i,i]=>i, i, i]=>i"
paulson@13363
   652
 "is_nat_case_fm(a,is_b,k,z) == 
paulson@13363
   653
    And(Implies(empty_fm(k), Equal(z,a)),
paulson@13363
   654
        And(Forall(Implies(succ_fm(0,succ(k)), 
paulson@13363
   655
                   Forall(Implies(Equal(0,succ(succ(z))), is_b(1,0))))),
paulson@13363
   656
            Or(quasinat_fm(k), empty_fm(z))))"
paulson@13363
   657
paulson@13363
   658
lemma is_nat_case_type [TC]:
paulson@13363
   659
     "[| is_b(1,0) \<in> formula;  
paulson@13363
   660
         x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13363
   661
      ==> is_nat_case_fm(x,is_b,y,z) \<in> formula"
paulson@13363
   662
by (simp add: is_nat_case_fm_def) 
paulson@13363
   663
paulson@13363
   664
lemma arity_is_nat_case_fm [simp]:
paulson@13363
   665
     "[| is_b(1,0) \<in> formula; x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13363
   666
      ==> arity(is_nat_case_fm(x,is_b,y,z)) = 
paulson@13363
   667
          succ(x) \<union> succ(y) \<union> succ(z) \<union> (arity(is_b(1,0)) #- 2)" 
paulson@13363
   668
apply (subgoal_tac "arity(is_b(1,0)) \<in> nat")  
paulson@13363
   669
apply typecheck
paulson@13363
   670
(*FIXME: could nat_diff_split work?*)
paulson@13363
   671
apply (auto simp add: diff_def raw_diff_succ is_nat_case_fm_def nat_imp_quasinat
paulson@13363
   672
                 succ_Un_distrib [symmetric] Un_ac
paulson@13363
   673
                 split: split_nat_case) 
paulson@13363
   674
done
paulson@13363
   675
paulson@13363
   676
lemma sats_is_nat_case_fm:
paulson@13363
   677
  assumes is_b_iff_sats: 
paulson@13363
   678
      "!!a b. [| a \<in> A; b \<in> A|] 
paulson@13363
   679
              ==> is_b(a,b) <-> sats(A, p(1,0), Cons(b, Cons(a,env)))"
paulson@13363
   680
  shows 
paulson@13363
   681
      "[|x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A)|]
paulson@13363
   682
       ==> sats(A, is_nat_case_fm(x,p,y,z), env) <-> 
paulson@13363
   683
           is_nat_case(**A, nth(x,env), is_b, nth(y,env), nth(z,env))"
paulson@13363
   684
apply (frule lt_length_in_nat, assumption)  
paulson@13363
   685
apply (simp add: is_nat_case_fm_def is_nat_case_def is_b_iff_sats [THEN iff_sym])
paulson@13363
   686
done
paulson@13363
   687
paulson@13363
   688
lemma is_nat_case_iff_sats:
paulson@13363
   689
  "[| (!!a b. [| a \<in> A; b \<in> A|] 
paulson@13363
   690
              ==> is_b(a,b) <-> sats(A, p(1,0), Cons(b, Cons(a,env))));
paulson@13363
   691
      nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13363
   692
      i \<in> nat; j \<in> nat; k < length(env); env \<in> list(A)|]
paulson@13363
   693
   ==> is_nat_case(**A, x, is_b, y, z) <-> sats(A, is_nat_case_fm(i,p,j,k), env)" 
paulson@13363
   694
by (simp add: sats_is_nat_case_fm [of A is_b])
paulson@13363
   695
paulson@13363
   696
paulson@13363
   697
text{*The second argument of @{term is_b} gives it direct access to @{term x},
paulson@13363
   698
  which is essential for handling free variable references.  Without this 
paulson@13363
   699
  argument, we cannot prove reflection for @{term iterates_MH}.*}
paulson@13363
   700
theorem is_nat_case_reflection:
paulson@13363
   701
  assumes is_b_reflection:
paulson@13363
   702
    "!!h f g. REFLECTS[\<lambda>x. is_b(L, h(x), f(x), g(x)), 
paulson@13363
   703
                     \<lambda>i x. is_b(**Lset(i), h(x), f(x), g(x))]"
paulson@13363
   704
  shows "REFLECTS[\<lambda>x. is_nat_case(L, f(x), is_b(L,x), g(x), h(x)), 
paulson@13363
   705
               \<lambda>i x. is_nat_case(**Lset(i), f(x), is_b(**Lset(i), x), g(x), h(x))]"
paulson@13363
   706
apply (simp (no_asm_use) only: is_nat_case_def setclass_simps)
paulson@13363
   707
apply (intro FOL_reflections function_reflections 
paulson@13363
   708
             restriction_reflection is_b_reflection quasinat_reflection)  
paulson@13363
   709
done
paulson@13363
   710
paulson@13363
   711
paulson@13363
   712
paulson@13363
   713
subsection{*The Operator @{term iterates_MH}, Needed for Iteration*}
paulson@13363
   714
paulson@13363
   715
(*  iterates_MH :: "[i=>o, [i,i]=>o, i, i, i, i] => o"
paulson@13363
   716
   "iterates_MH(M,isF,v,n,g,z) ==
paulson@13363
   717
        is_nat_case(M, v, \<lambda>m u. \<exists>gm[M]. fun_apply(M,g,m,gm) & isF(gm,u),
paulson@13363
   718
                    n, z)" *)
paulson@13363
   719
constdefs iterates_MH_fm :: "[[i,i]=>i, i, i, i, i]=>i"
paulson@13363
   720
 "iterates_MH_fm(isF,v,n,g,z) == 
paulson@13363
   721
    is_nat_case_fm(v, 
paulson@13363
   722
      \<lambda>m u. Exists(And(fun_apply_fm(succ(succ(succ(g))),succ(m),0), 
paulson@13363
   723
                     Forall(Implies(Equal(0,succ(succ(u))), isF(1,0))))), 
paulson@13363
   724
      n, z)"
paulson@13363
   725
paulson@13363
   726
lemma iterates_MH_type [TC]:
paulson@13363
   727
     "[| p(1,0) \<in> formula;  
paulson@13363
   728
         v \<in> nat; x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13363
   729
      ==> iterates_MH_fm(p,v,x,y,z) \<in> formula"
paulson@13363
   730
by (simp add: iterates_MH_fm_def) 
paulson@13363
   731
paulson@13363
   732
paulson@13363
   733
lemma arity_iterates_MH_fm [simp]:
paulson@13363
   734
     "[| p(1,0) \<in> formula; 
paulson@13363
   735
         v \<in> nat; x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13363
   736
      ==> arity(iterates_MH_fm(p,v,x,y,z)) = 
paulson@13363
   737
          succ(v) \<union> succ(x) \<union> succ(y) \<union> succ(z) \<union> (arity(p(1,0)) #- 4)"
paulson@13363
   738
apply (subgoal_tac "arity(p(1,0)) \<in> nat")
paulson@13363
   739
apply typecheck
paulson@13363
   740
apply (simp add: nat_imp_quasinat iterates_MH_fm_def Un_ac
paulson@13363
   741
            split: split_nat_case, clarify)
paulson@13363
   742
apply (rename_tac i j)
paulson@13363
   743
apply (drule eq_succ_imp_eq_m1, simp) 
paulson@13363
   744
apply (drule eq_succ_imp_eq_m1, simp)
paulson@13363
   745
apply (simp add: diff_Un_distrib succ_Un_distrib Un_ac diff_diff_left)
paulson@13363
   746
done
paulson@13363
   747
paulson@13363
   748
lemma sats_iterates_MH_fm:
paulson@13363
   749
  assumes is_F_iff_sats: 
paulson@13363
   750
      "!!a b c d. [| a \<in> A; b \<in> A; c \<in> A; d \<in> A|] 
paulson@13363
   751
              ==> is_F(a,b) <->
paulson@13363
   752
                  sats(A, p(1,0), Cons(b, Cons(a, Cons(c, Cons(d,env)))))"
paulson@13363
   753
  shows 
paulson@13363
   754
      "[|v \<in> nat; x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A)|]
paulson@13363
   755
       ==> sats(A, iterates_MH_fm(p,v,x,y,z), env) <-> 
paulson@13363
   756
           iterates_MH(**A, is_F, nth(v,env), nth(x,env), nth(y,env), nth(z,env))"
paulson@13363
   757
by (simp add: iterates_MH_fm_def iterates_MH_def sats_is_nat_case_fm 
paulson@13363
   758
              is_F_iff_sats [symmetric])
paulson@13363
   759
paulson@13363
   760
lemma iterates_MH_iff_sats:
paulson@13363
   761
  "[| (!!a b c d. [| a \<in> A; b \<in> A; c \<in> A; d \<in> A|] 
paulson@13363
   762
              ==> is_F(a,b) <->
paulson@13363
   763
                  sats(A, p(1,0), Cons(b, Cons(a, Cons(c, Cons(d,env))))));
paulson@13363
   764
      nth(i',env) = v; nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13363
   765
      i' \<in> nat; i \<in> nat; j \<in> nat; k < length(env); env \<in> list(A)|]
paulson@13363
   766
   ==> iterates_MH(**A, is_F, v, x, y, z) <-> 
paulson@13363
   767
       sats(A, iterates_MH_fm(p,i',i,j,k), env)"
paulson@13363
   768
apply (rule iff_sym) 
paulson@13363
   769
apply (rule iff_trans) 
paulson@13363
   770
apply (rule sats_iterates_MH_fm [of A is_F], blast, simp_all) 
paulson@13363
   771
done
paulson@13363
   772
paulson@13363
   773
theorem iterates_MH_reflection:
paulson@13363
   774
  assumes p_reflection:
paulson@13363
   775
    "!!f g h. REFLECTS[\<lambda>x. p(L, f(x), g(x)), 
paulson@13363
   776
                     \<lambda>i x. p(**Lset(i), f(x), g(x))]"
paulson@13363
   777
 shows "REFLECTS[\<lambda>x. iterates_MH(L, p(L), e(x), f(x), g(x), h(x)), 
paulson@13363
   778
               \<lambda>i x. iterates_MH(**Lset(i), p(**Lset(i)), e(x), f(x), g(x), h(x))]"
paulson@13363
   779
apply (simp (no_asm_use) only: iterates_MH_def)
paulson@13363
   780
txt{*Must be careful: simplifying with @{text setclass_simps} above would
paulson@13363
   781
     change @{text "\<exists>gm[**Lset(i)]"} into @{text "\<exists>gm \<in> Lset(i)"}, when
paulson@13363
   782
     it would no longer match rule @{text is_nat_case_reflection}. *}
paulson@13363
   783
apply (rule is_nat_case_reflection) 
paulson@13363
   784
apply (simp (no_asm_use) only: setclass_simps)
paulson@13363
   785
apply (intro FOL_reflections function_reflections is_nat_case_reflection
paulson@13363
   786
             restriction_reflection p_reflection)  
paulson@13363
   787
done
paulson@13363
   788
paulson@13363
   789
paulson@13363
   790
paulson@13363
   791
subsection{*@{term L} is Closed Under the Operator @{term list}*} 
paulson@13363
   792
paulson@13363
   793
paulson@13363
   794
lemma list_replacement1_Reflects:
paulson@13363
   795
 "REFLECTS
paulson@13363
   796
   [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
paulson@13363
   797
         is_wfrec(L, iterates_MH(L, is_list_functor(L,A), 0), memsn, u, y)),
paulson@13363
   798
    \<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
   799
         is_wfrec(**Lset(i), 
paulson@13363
   800
                  iterates_MH(**Lset(i), 
paulson@13363
   801
                          is_list_functor(**Lset(i), A), 0), memsn, u, y))]"
paulson@13363
   802
by (intro FOL_reflections function_reflections is_wfrec_reflection 
paulson@13363
   803
          iterates_MH_reflection list_functor_reflection) 
paulson@13363
   804
paulson@13363
   805
lemma list_replacement1: 
paulson@13363
   806
   "L(A) ==> iterates_replacement(L, is_list_functor(L,A), 0)"
paulson@13363
   807
apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
paulson@13363
   808
apply (rule strong_replacementI) 
paulson@13363
   809
apply (rule rallI)
paulson@13363
   810
apply (rename_tac B)   
paulson@13363
   811
apply (rule separation_CollectI) 
paulson@13363
   812
apply (insert nonempty) 
paulson@13363
   813
apply (subgoal_tac "L(Memrel(succ(n)))") 
paulson@13363
   814
apply (rule_tac A="{B,A,n,z,0,Memrel(succ(n))}" in subset_LsetE, blast ) 
paulson@13363
   815
apply (rule ReflectsE [OF list_replacement1_Reflects], assumption)
paulson@13363
   816
apply (drule subset_Lset_ltD, assumption) 
paulson@13363
   817
apply (erule reflection_imp_L_separation)
paulson@13363
   818
  apply (simp_all add: lt_Ord2)
paulson@13363
   819
apply (rule DPowI2)
paulson@13363
   820
apply (rename_tac v) 
paulson@13363
   821
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13363
   822
apply (rule_tac env = "[u,v,A,n,B,0,Memrel(succ(n))]" in mem_iff_sats)
paulson@13363
   823
apply (rule sep_rules | simp)+
paulson@13363
   824
txt{*Can't get sat rules to work for higher-order operators, so just expand them!*}
paulson@13363
   825
apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
paulson@13363
   826
apply (rule sep_rules list_functor_iff_sats quasinat_iff_sats | simp)+
paulson@13363
   827
apply (simp_all add: succ_Un_distrib [symmetric] Memrel_closed)
paulson@13363
   828
done
paulson@13363
   829
paulson@13363
   830
paulson@13363
   831
lemma list_replacement2_Reflects:
paulson@13363
   832
 "REFLECTS
paulson@13363
   833
   [\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
paulson@13363
   834
         (\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
paulson@13363
   835
           is_wfrec (L, iterates_MH (L, is_list_functor(L, A), 0),
paulson@13363
   836
                              msn, u, x)),
paulson@13363
   837
    \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
paulson@13363
   838
         (\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i). 
paulson@13363
   839
          successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
paulson@13363
   840
           is_wfrec (**Lset(i), 
paulson@13363
   841
                 iterates_MH (**Lset(i), is_list_functor(**Lset(i), A), 0),
paulson@13363
   842
                     msn, u, x))]"
paulson@13363
   843
by (intro FOL_reflections function_reflections is_wfrec_reflection 
paulson@13363
   844
          iterates_MH_reflection list_functor_reflection) 
paulson@13363
   845
paulson@13363
   846
paulson@13363
   847
lemma list_replacement2: 
paulson@13363
   848
   "L(A) ==> strong_replacement(L, 
paulson@13363
   849
         \<lambda>n y. n\<in>nat & 
paulson@13363
   850
               (\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
paulson@13363
   851
               is_wfrec(L, iterates_MH(L,is_list_functor(L,A), 0), 
paulson@13363
   852
                        msn, n, y)))"
paulson@13363
   853
apply (rule strong_replacementI) 
paulson@13363
   854
apply (rule rallI)
paulson@13363
   855
apply (rename_tac B)   
paulson@13363
   856
apply (rule separation_CollectI) 
paulson@13363
   857
apply (insert nonempty) 
paulson@13363
   858
apply (rule_tac A="{A,B,z,0,nat}" in subset_LsetE) 
paulson@13363
   859
apply (blast intro: L_nat) 
paulson@13363
   860
apply (rule ReflectsE [OF list_replacement2_Reflects], assumption)
paulson@13363
   861
apply (drule subset_Lset_ltD, assumption) 
paulson@13363
   862
apply (erule reflection_imp_L_separation)
paulson@13363
   863
  apply (simp_all add: lt_Ord2)
paulson@13363
   864
apply (rule DPowI2)
paulson@13363
   865
apply (rename_tac v) 
paulson@13363
   866
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13363
   867
apply (rule_tac env = "[u,v,A,B,0,nat]" in mem_iff_sats)
paulson@13363
   868
apply (rule sep_rules | simp)+
paulson@13363
   869
apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
paulson@13363
   870
apply (rule sep_rules list_functor_iff_sats quasinat_iff_sats | simp)+
paulson@13363
   871
apply (simp_all add: succ_Un_distrib [symmetric] Memrel_closed)
paulson@13363
   872
done
paulson@13363
   873
paulson@13363
   874
paulson@13363
   875
paulson@13348
   876
end