src/ZF/Constructible/Rec_Separation.thy
author paulson
Thu Aug 01 18:22:46 2002 +0200 (2002-08-01)
changeset 13441 d6d620639243
parent 13440 cdde97e1db1c
child 13493 5aa68c051725
permissions -rw-r--r--
better satisfies rules for is_recfun
A satisfies rule for is_wfrec!
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(*  Title:      ZF/Constructible/Rec_Separation.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   2002  University of Cambridge
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FIXME: define nth_fm and prove its "sats" theorem
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*)
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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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lemma M_trancl_axioms_L: "M_trancl_axioms(L)"
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  apply (rule M_trancl_axioms.intro)
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   apply (assumption | rule rtrancl_separation wellfounded_trancl_separation)+
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  done
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theorem M_trancl_L: "PROP M_trancl(L)"
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by (rule M_trancl.intro
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         [OF M_triv_axioms_L M_axioms_axioms_L M_trancl_axioms_L])
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lemmas iterates_abs = M_trancl.iterates_abs [OF M_trancl_L]
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  and rtran_closure_rtrancl = M_trancl.rtran_closure_rtrancl [OF M_trancl_L]
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  and rtrancl_closed = M_trancl.rtrancl_closed [OF M_trancl_L]
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  and rtrancl_abs = M_trancl.rtrancl_abs [OF M_trancl_L]
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  and trancl_closed = M_trancl.trancl_closed [OF M_trancl_L]
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  and trancl_abs = M_trancl.trancl_abs [OF M_trancl_L]
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  and wellfounded_on_trancl = M_trancl.wellfounded_on_trancl [OF M_trancl_L]
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  and wellfounded_trancl = M_trancl.wellfounded_trancl [OF M_trancl_L]
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  and wfrec_relativize = M_trancl.wfrec_relativize [OF M_trancl_L]
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  and trans_wfrec_relativize = M_trancl.trans_wfrec_relativize [OF M_trancl_L]
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  and trans_wfrec_abs = M_trancl.trans_wfrec_abs [OF M_trancl_L]
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  and trans_eq_pair_wfrec_iff = M_trancl.trans_eq_pair_wfrec_iff [OF M_trancl_L]
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  and eq_pair_wfrec_iff = M_trancl.eq_pair_wfrec_iff [OF M_trancl_L]
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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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text{*Alternative definition, minimizing nesting of quantifiers around MH*}
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lemma M_is_recfun_iff:
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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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     (\<exists>x[M]. \<exists>f_r_sx[M]. \<exists>y[M]. 
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             MH(x, f_r_sx, y) & pair(M,x,y,z) &
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             (\<exists>xa[M]. \<exists>sx[M]. \<exists>r_sx[M]. 
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                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)))"
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apply (simp add: M_is_recfun_def)
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apply (rule rall_cong, blast) 
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done
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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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               2      1           0
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new def     (\<exists>x[M]. \<exists>f_r_sx[M]. \<exists>y[M]. 
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             MH(x, f_r_sx, y) & pair(M,x,y,z) &
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             (\<exists>xa[M]. \<exists>sx[M]. \<exists>r_sx[M]. 
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                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)"
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*)
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text{*The three arguments of @{term p} are always 2, 1, 0 and z*}
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constdefs is_recfun_fm :: "[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(
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   273
     And(p, 
paulson@13441
   274
      And(pair_fm(2,0,3),
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   275
       Exists(Exists(Exists(
paulson@13441
   276
	And(pair_fm(5,a#+7,2),
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   277
	 And(upair_fm(5,5,1),
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   278
	  And(pre_image_fm(r#+7,1,0),
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   279
	   And(restriction_fm(f#+7,0,4), Member(2,r#+7)))))))))))))))"
paulson@13348
   280
paulson@13348
   281
lemma is_recfun_type [TC]:
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   282
     "[| p \<in> formula; x \<in> nat; y \<in> nat; z \<in> nat |] 
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   283
      ==> is_recfun_fm(p,x,y,z) \<in> formula"
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   284
by (simp add: is_recfun_fm_def)
paulson@13348
   285
paulson@13441
   286
paulson@13348
   287
lemma sats_is_recfun_fm:
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   288
  assumes MH_iff_sats: 
paulson@13441
   289
      "!!a0 a1 a2 a3. 
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   290
        [|a0\<in>A; a1\<in>A; a2\<in>A; a3\<in>A|] 
paulson@13441
   291
        ==> MH(a2, a1, a0) <-> sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,env)))))"
paulson@13434
   292
  shows 
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   293
      "[|x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
wenzelm@13428
   294
       ==> sats(A, is_recfun_fm(p,x,y,z), env) <->
paulson@13352
   295
           M_is_recfun(**A, MH, nth(x,env), nth(y,env), nth(z,env))"
paulson@13441
   296
by (simp add: is_recfun_fm_def M_is_recfun_iff MH_iff_sats [THEN iff_sym])
paulson@13348
   297
paulson@13348
   298
lemma is_recfun_iff_sats:
paulson@13434
   299
  assumes MH_iff_sats: 
paulson@13441
   300
      "!!a0 a1 a2 a3. 
paulson@13441
   301
        [|a0\<in>A; a1\<in>A; a2\<in>A; a3\<in>A|] 
paulson@13441
   302
        ==> MH(a2, a1, a0) <-> sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,env)))))"
paulson@13434
   303
  shows
paulson@13434
   304
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13348
   305
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
wenzelm@13428
   306
   ==> M_is_recfun(**A, MH, x, y, z) <-> sats(A, is_recfun_fm(p,i,j,k), env)"
paulson@13434
   307
apply (rule iff_sym) 
paulson@13434
   308
apply (rule iff_trans)
paulson@13434
   309
apply (rule sats_is_recfun_fm [of A MH]) 
paulson@13434
   310
apply (rule MH_iff_sats, simp_all) 
paulson@13434
   311
done
paulson@13434
   312
(*FIXME: surely proof can be improved?*)
paulson@13434
   313
paulson@13348
   314
paulson@13437
   315
text{*The additional variable in the premise, namely @{term f'}, is essential.
paulson@13437
   316
It lets @{term MH} depend upon @{term x}, which seems often necessary.
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   317
The same thing occurs in @{text is_wfrec_reflection}.*}
paulson@13348
   318
theorem is_recfun_reflection:
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   319
  assumes MH_reflection:
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   320
    "!!f' f g h. REFLECTS[\<lambda>x. MH(L, f'(x), f(x), g(x), h(x)), 
paulson@13437
   321
                     \<lambda>i x. MH(**Lset(i), f'(x), f(x), g(x), h(x))]"
paulson@13437
   322
  shows "REFLECTS[\<lambda>x. M_is_recfun(L, MH(L,x), f(x), g(x), h(x)), 
paulson@13437
   323
             \<lambda>i x. M_is_recfun(**Lset(i), MH(**Lset(i),x), f(x), g(x), h(x))]"
paulson@13348
   324
apply (simp (no_asm_use) only: M_is_recfun_def setclass_simps)
wenzelm@13428
   325
apply (intro FOL_reflections function_reflections
wenzelm@13428
   326
             restriction_reflection MH_reflection)
paulson@13348
   327
done
paulson@13348
   328
paulson@13441
   329
subsubsection{*The Operator @{term is_wfrec}*}
paulson@13441
   330
paulson@13441
   331
text{*The three arguments of @{term p} are always 2, 1, 0*}
paulson@13441
   332
paulson@13441
   333
(* is_wfrec :: "[i=>o, i, [i,i,i]=>o, i, i] => o"
paulson@13441
   334
    "is_wfrec(M,MH,r,a,z) == 
paulson@13441
   335
      \<exists>f[M]. M_is_recfun(M,MH,r,a,f) & MH(a,f,z)" *)
paulson@13441
   336
constdefs is_wfrec_fm :: "[i, i, i, i]=>i"
paulson@13441
   337
 "is_wfrec_fm(p,r,a,z) == 
paulson@13441
   338
    Exists(And(is_recfun_fm(p, succ(r), succ(a), 0),
paulson@13441
   339
           Exists(Exists(Exists(Exists(
paulson@13441
   340
             And(Equal(2,a#+5), And(Equal(1,4), And(Equal(0,z#+5), p)))))))))"
paulson@13441
   341
paulson@13441
   342
text{*We call @{term p} with arguments a, f, z by equating them with 
paulson@13441
   343
  the corresponding quantified variables with de Bruijn indices 2, 1, 0.*}
paulson@13441
   344
paulson@13441
   345
text{*There's an additional existential quantifier to ensure that the
paulson@13441
   346
      environments in both calls to MH have the same length.*}
paulson@13441
   347
paulson@13441
   348
lemma is_wfrec_type [TC]:
paulson@13441
   349
     "[| p \<in> formula; x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13441
   350
      ==> is_wfrec_fm(p,x,y,z) \<in> formula"
paulson@13441
   351
by (simp add: is_wfrec_fm_def) 
paulson@13441
   352
paulson@13441
   353
lemma sats_is_wfrec_fm:
paulson@13441
   354
  assumes MH_iff_sats: 
paulson@13441
   355
      "!!a0 a1 a2 a3 a4. 
paulson@13441
   356
        [|a0\<in>A; a1\<in>A; a2\<in>A; a3\<in>A; a4\<in>A|] 
paulson@13441
   357
        ==> MH(a2, a1, a0) <-> sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,Cons(a4,env))))))"
paulson@13441
   358
  shows 
paulson@13441
   359
      "[|x \<in> nat; y < length(env); z < length(env); env \<in> list(A)|]
paulson@13441
   360
       ==> sats(A, is_wfrec_fm(p,x,y,z), env) <-> 
paulson@13441
   361
           is_wfrec(**A, MH, nth(x,env), nth(y,env), nth(z,env))"
paulson@13441
   362
apply (frule_tac x=z in lt_length_in_nat, assumption)  
paulson@13441
   363
apply (frule lt_length_in_nat, assumption)  
paulson@13441
   364
apply (simp add: is_wfrec_fm_def sats_is_recfun_fm is_wfrec_def MH_iff_sats [THEN iff_sym], blast) 
paulson@13441
   365
done
paulson@13441
   366
paulson@13441
   367
paulson@13441
   368
lemma is_wfrec_iff_sats:
paulson@13441
   369
  assumes MH_iff_sats: 
paulson@13441
   370
      "!!a0 a1 a2 a3 a4. 
paulson@13441
   371
        [|a0\<in>A; a1\<in>A; a2\<in>A; a3\<in>A; a4\<in>A|] 
paulson@13441
   372
        ==> MH(a2, a1, a0) <-> sats(A, p, Cons(a0,Cons(a1,Cons(a2,Cons(a3,Cons(a4,env))))))"
paulson@13441
   373
  shows
paulson@13441
   374
  "[|nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13441
   375
      i \<in> nat; j < length(env); k < length(env); env \<in> list(A)|]
paulson@13441
   376
   ==> is_wfrec(**A, MH, x, y, z) <-> sats(A, is_wfrec_fm(p,i,j,k), env)" 
paulson@13441
   377
by (simp add: sats_is_wfrec_fm [OF MH_iff_sats])
paulson@13441
   378
paulson@13363
   379
theorem is_wfrec_reflection:
paulson@13363
   380
  assumes MH_reflection:
paulson@13437
   381
    "!!f' f g h. REFLECTS[\<lambda>x. MH(L, f'(x), f(x), g(x), h(x)), 
paulson@13437
   382
                     \<lambda>i x. MH(**Lset(i), f'(x), f(x), g(x), h(x))]"
paulson@13437
   383
  shows "REFLECTS[\<lambda>x. is_wfrec(L, MH(L,x), f(x), g(x), h(x)), 
paulson@13437
   384
               \<lambda>i x. is_wfrec(**Lset(i), MH(**Lset(i),x), f(x), g(x), h(x))]"
paulson@13363
   385
apply (simp (no_asm_use) only: is_wfrec_def setclass_simps)
wenzelm@13428
   386
apply (intro FOL_reflections MH_reflection is_recfun_reflection)
paulson@13363
   387
done
paulson@13363
   388
paulson@13363
   389
subsection{*The Locale @{text "M_wfrank"}*}
paulson@13363
   390
paulson@13363
   391
subsubsection{*Separation for @{term "wfrank"}*}
paulson@13348
   392
paulson@13348
   393
lemma wfrank_Reflects:
paulson@13348
   394
 "REFLECTS[\<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
paulson@13352
   395
              ~ (\<exists>f[L]. M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f)),
paulson@13348
   396
      \<lambda>i x. \<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) -->
wenzelm@13428
   397
         ~ (\<exists>f \<in> Lset(i).
wenzelm@13428
   398
            M_is_recfun(**Lset(i), %x f y. is_range(**Lset(i),f,y),
paulson@13352
   399
                        rplus, x, f))]"
wenzelm@13428
   400
by (intro FOL_reflections function_reflections is_recfun_reflection tran_closure_reflection)
paulson@13348
   401
paulson@13348
   402
lemma wfrank_separation:
paulson@13348
   403
     "L(r) ==>
paulson@13348
   404
      separation (L, \<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
paulson@13352
   405
         ~ (\<exists>f[L]. M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f)))"
wenzelm@13428
   406
apply (rule separation_CollectI)
wenzelm@13428
   407
apply (rule_tac A="{r,z}" in subset_LsetE, blast )
paulson@13348
   408
apply (rule ReflectsE [OF wfrank_Reflects], assumption)
wenzelm@13428
   409
apply (drule subset_Lset_ltD, assumption)
paulson@13348
   410
apply (erule reflection_imp_L_separation)
paulson@13348
   411
  apply (simp_all add: lt_Ord2, clarify)
paulson@13385
   412
apply (rule DPow_LsetI)
wenzelm@13428
   413
apply (rename_tac u)
paulson@13348
   414
apply (rule ball_iff_sats imp_iff_sats)+
paulson@13348
   415
apply (rule_tac env="[rplus,u,r]" in tran_closure_iff_sats)
paulson@13441
   416
apply (rule sep_rules | simp)+
paulson@13348
   417
apply (rule sep_rules is_recfun_iff_sats | simp)+
paulson@13348
   418
done
paulson@13348
   419
paulson@13348
   420
paulson@13363
   421
subsubsection{*Replacement for @{term "wfrank"}*}
paulson@13348
   422
paulson@13348
   423
lemma wfrank_replacement_Reflects:
wenzelm@13428
   424
 "REFLECTS[\<lambda>z. \<exists>x[L]. x \<in> A &
paulson@13348
   425
        (\<forall>rplus[L]. tran_closure(L,r,rplus) -->
wenzelm@13428
   426
         (\<exists>y[L]. \<exists>f[L]. pair(L,x,y,z)  &
paulson@13352
   427
                        M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f) &
paulson@13348
   428
                        is_range(L,f,y))),
wenzelm@13428
   429
 \<lambda>i z. \<exists>x \<in> Lset(i). x \<in> A &
paulson@13348
   430
      (\<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) -->
wenzelm@13428
   431
       (\<exists>y \<in> Lset(i). \<exists>f \<in> Lset(i). pair(**Lset(i),x,y,z)  &
paulson@13352
   432
         M_is_recfun(**Lset(i), %x f y. is_range(**Lset(i),f,y), rplus, x, f) &
paulson@13348
   433
         is_range(**Lset(i),f,y)))]"
paulson@13348
   434
by (intro FOL_reflections function_reflections fun_plus_reflections
paulson@13348
   435
             is_recfun_reflection tran_closure_reflection)
paulson@13348
   436
paulson@13348
   437
paulson@13348
   438
lemma wfrank_strong_replacement:
paulson@13348
   439
     "L(r) ==>
wenzelm@13428
   440
      strong_replacement(L, \<lambda>x z.
paulson@13348
   441
         \<forall>rplus[L]. tran_closure(L,r,rplus) -->
wenzelm@13428
   442
         (\<exists>y[L]. \<exists>f[L]. pair(L,x,y,z)  &
paulson@13352
   443
                        M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f) &
paulson@13348
   444
                        is_range(L,f,y)))"
wenzelm@13428
   445
apply (rule strong_replacementI)
paulson@13348
   446
apply (rule rallI)
wenzelm@13428
   447
apply (rename_tac B)
wenzelm@13428
   448
apply (rule separation_CollectI)
wenzelm@13428
   449
apply (rule_tac A="{B,r,z}" in subset_LsetE, blast )
paulson@13348
   450
apply (rule ReflectsE [OF wfrank_replacement_Reflects], assumption)
wenzelm@13428
   451
apply (drule subset_Lset_ltD, assumption)
paulson@13348
   452
apply (erule reflection_imp_L_separation)
paulson@13348
   453
  apply (simp_all add: lt_Ord2)
paulson@13385
   454
apply (rule DPow_LsetI)
wenzelm@13428
   455
apply (rename_tac u)
paulson@13348
   456
apply (rule bex_iff_sats ball_iff_sats conj_iff_sats)+
wenzelm@13428
   457
apply (rule_tac env = "[x,u,B,r]" in mem_iff_sats)
paulson@13441
   458
apply (rule sep_rules list.intros app_type tran_closure_iff_sats is_recfun_iff_sats | simp)+
paulson@13348
   459
done
paulson@13348
   460
paulson@13348
   461
paulson@13363
   462
subsubsection{*Separation for Proving @{text Ord_wfrank_range}*}
paulson@13348
   463
paulson@13348
   464
lemma Ord_wfrank_Reflects:
wenzelm@13428
   465
 "REFLECTS[\<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
wenzelm@13428
   466
          ~ (\<forall>f[L]. \<forall>rangef[L].
paulson@13348
   467
             is_range(L,f,rangef) -->
paulson@13352
   468
             M_is_recfun(L, \<lambda>x f y. is_range(L,f,y), rplus, x, f) -->
paulson@13348
   469
             ordinal(L,rangef)),
wenzelm@13428
   470
      \<lambda>i x. \<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) -->
wenzelm@13428
   471
          ~ (\<forall>f \<in> Lset(i). \<forall>rangef \<in> Lset(i).
paulson@13348
   472
             is_range(**Lset(i),f,rangef) -->
wenzelm@13428
   473
             M_is_recfun(**Lset(i), \<lambda>x f y. is_range(**Lset(i),f,y),
paulson@13352
   474
                         rplus, x, f) -->
paulson@13348
   475
             ordinal(**Lset(i),rangef))]"
wenzelm@13428
   476
by (intro FOL_reflections function_reflections is_recfun_reflection
paulson@13348
   477
          tran_closure_reflection ordinal_reflection)
paulson@13348
   478
paulson@13348
   479
lemma  Ord_wfrank_separation:
paulson@13348
   480
     "L(r) ==>
paulson@13348
   481
      separation (L, \<lambda>x.
wenzelm@13428
   482
         \<forall>rplus[L]. tran_closure(L,r,rplus) -->
wenzelm@13428
   483
          ~ (\<forall>f[L]. \<forall>rangef[L].
paulson@13348
   484
             is_range(L,f,rangef) -->
paulson@13352
   485
             M_is_recfun(L, \<lambda>x f y. is_range(L,f,y), rplus, x, f) -->
wenzelm@13428
   486
             ordinal(L,rangef)))"
wenzelm@13428
   487
apply (rule separation_CollectI)
wenzelm@13428
   488
apply (rule_tac A="{r,z}" in subset_LsetE, blast )
paulson@13348
   489
apply (rule ReflectsE [OF Ord_wfrank_Reflects], assumption)
wenzelm@13428
   490
apply (drule subset_Lset_ltD, assumption)
paulson@13348
   491
apply (erule reflection_imp_L_separation)
paulson@13348
   492
  apply (simp_all add: lt_Ord2, clarify)
paulson@13385
   493
apply (rule DPow_LsetI)
wenzelm@13428
   494
apply (rename_tac u)
paulson@13348
   495
apply (rule ball_iff_sats imp_iff_sats)+
paulson@13348
   496
apply (rule_tac env="[rplus,u,r]" in tran_closure_iff_sats)
paulson@13348
   497
apply (rule sep_rules is_recfun_iff_sats | simp)+
paulson@13348
   498
done
paulson@13348
   499
paulson@13348
   500
paulson@13363
   501
subsubsection{*Instantiating the locale @{text M_wfrank}*}
wenzelm@13428
   502
paulson@13437
   503
lemma M_wfrank_axioms_L: "M_wfrank_axioms(L)"
paulson@13437
   504
  apply (rule M_wfrank_axioms.intro)
paulson@13437
   505
   apply (assumption | rule
paulson@13437
   506
     wfrank_separation wfrank_strong_replacement Ord_wfrank_separation)+
paulson@13437
   507
  done
paulson@13437
   508
wenzelm@13428
   509
theorem M_wfrank_L: "PROP M_wfrank(L)"
wenzelm@13428
   510
  apply (rule M_wfrank.intro)
wenzelm@13429
   511
     apply (rule M_trancl.axioms [OF M_trancl_L])+
paulson@13437
   512
  apply (rule M_wfrank_axioms_L) 
wenzelm@13428
   513
  done
paulson@13363
   514
wenzelm@13428
   515
lemmas iterates_closed = M_wfrank.iterates_closed [OF M_wfrank_L]
wenzelm@13428
   516
  and exists_wfrank = M_wfrank.exists_wfrank [OF M_wfrank_L]
wenzelm@13428
   517
  and M_wellfoundedrank = M_wfrank.M_wellfoundedrank [OF M_wfrank_L]
wenzelm@13428
   518
  and Ord_wfrank_range = M_wfrank.Ord_wfrank_range [OF M_wfrank_L]
wenzelm@13428
   519
  and Ord_range_wellfoundedrank = M_wfrank.Ord_range_wellfoundedrank [OF M_wfrank_L]
wenzelm@13428
   520
  and function_wellfoundedrank = M_wfrank.function_wellfoundedrank [OF M_wfrank_L]
wenzelm@13428
   521
  and domain_wellfoundedrank = M_wfrank.domain_wellfoundedrank [OF M_wfrank_L]
wenzelm@13428
   522
  and wellfoundedrank_type = M_wfrank.wellfoundedrank_type [OF M_wfrank_L]
wenzelm@13428
   523
  and Ord_wellfoundedrank = M_wfrank.Ord_wellfoundedrank [OF M_wfrank_L]
wenzelm@13428
   524
  and wellfoundedrank_eq = M_wfrank.wellfoundedrank_eq [OF M_wfrank_L]
wenzelm@13428
   525
  and wellfoundedrank_lt = M_wfrank.wellfoundedrank_lt [OF M_wfrank_L]
wenzelm@13428
   526
  and wellfounded_imp_subset_rvimage = M_wfrank.wellfounded_imp_subset_rvimage [OF M_wfrank_L]
wenzelm@13428
   527
  and wellfounded_imp_wf = M_wfrank.wellfounded_imp_wf [OF M_wfrank_L]
wenzelm@13428
   528
  and wellfounded_on_imp_wf_on = M_wfrank.wellfounded_on_imp_wf_on [OF M_wfrank_L]
wenzelm@13428
   529
  and wf_abs = M_wfrank.wf_abs [OF M_wfrank_L]
wenzelm@13428
   530
  and wf_on_abs = M_wfrank.wf_on_abs [OF M_wfrank_L]
wenzelm@13428
   531
  and wfrec_replacement_iff = M_wfrank.wfrec_replacement_iff [OF M_wfrank_L]
wenzelm@13428
   532
  and trans_wfrec_closed = M_wfrank.trans_wfrec_closed [OF M_wfrank_L]
wenzelm@13428
   533
  and wfrec_closed = M_wfrank.wfrec_closed [OF M_wfrank_L]
paulson@13363
   534
paulson@13363
   535
declare iterates_closed [intro,simp]
paulson@13363
   536
declare Ord_wfrank_range [rule_format]
paulson@13363
   537
declare wf_abs [simp]
paulson@13363
   538
declare wf_on_abs [simp]
paulson@13363
   539
paulson@13363
   540
paulson@13363
   541
subsection{*For Datatypes*}
paulson@13363
   542
paulson@13363
   543
subsubsection{*Binary Products, Internalized*}
paulson@13363
   544
paulson@13363
   545
constdefs cartprod_fm :: "[i,i,i]=>i"
wenzelm@13428
   546
(* "cartprod(M,A,B,z) ==
wenzelm@13428
   547
        \<forall>u[M]. u \<in> z <-> (\<exists>x[M]. x\<in>A & (\<exists>y[M]. y\<in>B & pair(M,x,y,u)))" *)
wenzelm@13428
   548
    "cartprod_fm(A,B,z) ==
paulson@13363
   549
       Forall(Iff(Member(0,succ(z)),
paulson@13363
   550
                  Exists(And(Member(0,succ(succ(A))),
paulson@13363
   551
                         Exists(And(Member(0,succ(succ(succ(B)))),
paulson@13363
   552
                                    pair_fm(1,0,2)))))))"
paulson@13363
   553
paulson@13363
   554
lemma cartprod_type [TC]:
paulson@13363
   555
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cartprod_fm(x,y,z) \<in> formula"
wenzelm@13428
   556
by (simp add: cartprod_fm_def)
paulson@13363
   557
paulson@13363
   558
lemma arity_cartprod_fm [simp]:
wenzelm@13428
   559
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
paulson@13363
   560
      ==> arity(cartprod_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
wenzelm@13428
   561
by (simp add: cartprod_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13363
   562
paulson@13363
   563
lemma sats_cartprod_fm [simp]:
paulson@13363
   564
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
wenzelm@13428
   565
    ==> sats(A, cartprod_fm(x,y,z), env) <->
paulson@13363
   566
        cartprod(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13363
   567
by (simp add: cartprod_fm_def cartprod_def)
paulson@13363
   568
paulson@13363
   569
lemma cartprod_iff_sats:
wenzelm@13428
   570
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
paulson@13363
   571
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13363
   572
       ==> cartprod(**A, x, y, z) <-> sats(A, cartprod_fm(i,j,k), env)"
paulson@13363
   573
by (simp add: sats_cartprod_fm)
paulson@13363
   574
paulson@13363
   575
theorem cartprod_reflection:
wenzelm@13428
   576
     "REFLECTS[\<lambda>x. cartprod(L,f(x),g(x),h(x)),
paulson@13363
   577
               \<lambda>i x. cartprod(**Lset(i),f(x),g(x),h(x))]"
paulson@13363
   578
apply (simp only: cartprod_def setclass_simps)
wenzelm@13428
   579
apply (intro FOL_reflections pair_reflection)
paulson@13363
   580
done
paulson@13363
   581
paulson@13363
   582
paulson@13363
   583
subsubsection{*Binary Sums, Internalized*}
paulson@13363
   584
wenzelm@13428
   585
(* "is_sum(M,A,B,Z) ==
wenzelm@13428
   586
       \<exists>A0[M]. \<exists>n1[M]. \<exists>s1[M]. \<exists>B1[M].
paulson@13363
   587
         3      2       1        0
paulson@13363
   588
       number1(M,n1) & cartprod(M,n1,A,A0) & upair(M,n1,n1,s1) &
paulson@13363
   589
       cartprod(M,s1,B,B1) & union(M,A0,B1,Z)"  *)
paulson@13363
   590
constdefs sum_fm :: "[i,i,i]=>i"
wenzelm@13428
   591
    "sum_fm(A,B,Z) ==
paulson@13363
   592
       Exists(Exists(Exists(Exists(
wenzelm@13428
   593
        And(number1_fm(2),
paulson@13363
   594
            And(cartprod_fm(2,A#+4,3),
paulson@13363
   595
                And(upair_fm(2,2,1),
paulson@13363
   596
                    And(cartprod_fm(1,B#+4,0), union_fm(3,0,Z#+4)))))))))"
paulson@13363
   597
paulson@13363
   598
lemma sum_type [TC]:
paulson@13363
   599
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> sum_fm(x,y,z) \<in> formula"
wenzelm@13428
   600
by (simp add: sum_fm_def)
paulson@13363
   601
paulson@13363
   602
lemma arity_sum_fm [simp]:
wenzelm@13428
   603
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
paulson@13363
   604
      ==> arity(sum_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
wenzelm@13428
   605
by (simp add: sum_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13363
   606
paulson@13363
   607
lemma sats_sum_fm [simp]:
paulson@13363
   608
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
wenzelm@13428
   609
    ==> sats(A, sum_fm(x,y,z), env) <->
paulson@13363
   610
        is_sum(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13363
   611
by (simp add: sum_fm_def is_sum_def)
paulson@13363
   612
paulson@13363
   613
lemma sum_iff_sats:
wenzelm@13428
   614
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
paulson@13363
   615
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13363
   616
       ==> is_sum(**A, x, y, z) <-> sats(A, sum_fm(i,j,k), env)"
paulson@13363
   617
by simp
paulson@13363
   618
paulson@13363
   619
theorem sum_reflection:
wenzelm@13428
   620
     "REFLECTS[\<lambda>x. is_sum(L,f(x),g(x),h(x)),
paulson@13363
   621
               \<lambda>i x. is_sum(**Lset(i),f(x),g(x),h(x))]"
paulson@13363
   622
apply (simp only: is_sum_def setclass_simps)
wenzelm@13428
   623
apply (intro FOL_reflections function_reflections cartprod_reflection)
paulson@13363
   624
done
paulson@13363
   625
paulson@13363
   626
paulson@13363
   627
subsubsection{*The Operator @{term quasinat}*}
paulson@13363
   628
paulson@13363
   629
(* "is_quasinat(M,z) == empty(M,z) | (\<exists>m[M]. successor(M,m,z))" *)
paulson@13363
   630
constdefs quasinat_fm :: "i=>i"
paulson@13363
   631
    "quasinat_fm(z) == Or(empty_fm(z), Exists(succ_fm(0,succ(z))))"
paulson@13363
   632
paulson@13363
   633
lemma quasinat_type [TC]:
paulson@13363
   634
     "x \<in> nat ==> quasinat_fm(x) \<in> formula"
wenzelm@13428
   635
by (simp add: quasinat_fm_def)
paulson@13363
   636
paulson@13363
   637
lemma arity_quasinat_fm [simp]:
paulson@13363
   638
     "x \<in> nat ==> arity(quasinat_fm(x)) = succ(x)"
wenzelm@13428
   639
by (simp add: quasinat_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13363
   640
paulson@13363
   641
lemma sats_quasinat_fm [simp]:
paulson@13363
   642
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13363
   643
    ==> sats(A, quasinat_fm(x), env) <-> is_quasinat(**A, nth(x,env))"
paulson@13363
   644
by (simp add: quasinat_fm_def is_quasinat_def)
paulson@13363
   645
paulson@13363
   646
lemma quasinat_iff_sats:
wenzelm@13428
   647
      "[| nth(i,env) = x; nth(j,env) = y;
paulson@13363
   648
          i \<in> nat; env \<in> list(A)|]
paulson@13363
   649
       ==> is_quasinat(**A, x) <-> sats(A, quasinat_fm(i), env)"
paulson@13363
   650
by simp
paulson@13363
   651
paulson@13363
   652
theorem quasinat_reflection:
wenzelm@13428
   653
     "REFLECTS[\<lambda>x. is_quasinat(L,f(x)),
paulson@13363
   654
               \<lambda>i x. is_quasinat(**Lset(i),f(x))]"
paulson@13363
   655
apply (simp only: is_quasinat_def setclass_simps)
wenzelm@13428
   656
apply (intro FOL_reflections function_reflections)
paulson@13363
   657
done
paulson@13363
   658
paulson@13363
   659
paulson@13363
   660
subsubsection{*The Operator @{term is_nat_case}*}
paulson@13434
   661
text{*I could not get it to work with the more natural assumption that 
paulson@13434
   662
 @{term is_b} takes two arguments.  Instead it must be a formula where 1 and 0
paulson@13434
   663
 stand for @{term m} and @{term b}, respectively.*}
paulson@13363
   664
paulson@13363
   665
(* is_nat_case :: "[i=>o, i, [i,i]=>o, i, i] => o"
wenzelm@13428
   666
    "is_nat_case(M, a, is_b, k, z) ==
paulson@13363
   667
       (empty(M,k) --> z=a) &
paulson@13363
   668
       (\<forall>m[M]. successor(M,m,k) --> is_b(m,z)) &
paulson@13363
   669
       (is_quasinat(M,k) | empty(M,z))" *)
paulson@13363
   670
text{*The formula @{term is_b} has free variables 1 and 0.*}
paulson@13434
   671
constdefs is_nat_case_fm :: "[i, i, i, i]=>i"
paulson@13434
   672
 "is_nat_case_fm(a,is_b,k,z) == 
paulson@13363
   673
    And(Implies(empty_fm(k), Equal(z,a)),
paulson@13434
   674
        And(Forall(Implies(succ_fm(0,succ(k)), 
paulson@13434
   675
                   Forall(Implies(Equal(0,succ(succ(z))), is_b)))),
paulson@13363
   676
            Or(quasinat_fm(k), empty_fm(z))))"
paulson@13363
   677
paulson@13363
   678
lemma is_nat_case_type [TC]:
paulson@13434
   679
     "[| is_b \<in> formula;  
paulson@13434
   680
         x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13363
   681
      ==> is_nat_case_fm(x,is_b,y,z) \<in> formula"
wenzelm@13428
   682
by (simp add: is_nat_case_fm_def)
paulson@13363
   683
paulson@13363
   684
lemma sats_is_nat_case_fm:
paulson@13434
   685
  assumes is_b_iff_sats: 
paulson@13434
   686
      "!!a. a \<in> A ==> is_b(a,nth(z, env)) <-> 
paulson@13434
   687
                      sats(A, p, Cons(nth(z,env), Cons(a, env)))"
paulson@13434
   688
  shows 
paulson@13363
   689
      "[|x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A)|]
wenzelm@13428
   690
       ==> sats(A, is_nat_case_fm(x,p,y,z), env) <->
paulson@13363
   691
           is_nat_case(**A, nth(x,env), is_b, nth(y,env), nth(z,env))"
wenzelm@13428
   692
apply (frule lt_length_in_nat, assumption)
paulson@13363
   693
apply (simp add: is_nat_case_fm_def is_nat_case_def is_b_iff_sats [THEN iff_sym])
paulson@13363
   694
done
paulson@13363
   695
paulson@13363
   696
lemma is_nat_case_iff_sats:
paulson@13434
   697
  "[| (!!a. a \<in> A ==> is_b(a,z) <->
paulson@13434
   698
                      sats(A, p, Cons(z, Cons(a,env))));
paulson@13434
   699
      nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13363
   700
      i \<in> nat; j \<in> nat; k < length(env); env \<in> list(A)|]
wenzelm@13428
   701
   ==> is_nat_case(**A, x, is_b, y, z) <-> sats(A, is_nat_case_fm(i,p,j,k), env)"
paulson@13363
   702
by (simp add: sats_is_nat_case_fm [of A is_b])
paulson@13363
   703
paulson@13363
   704
paulson@13363
   705
text{*The second argument of @{term is_b} gives it direct access to @{term x},
wenzelm@13428
   706
  which is essential for handling free variable references.  Without this
paulson@13363
   707
  argument, we cannot prove reflection for @{term iterates_MH}.*}
paulson@13363
   708
theorem is_nat_case_reflection:
paulson@13363
   709
  assumes is_b_reflection:
wenzelm@13428
   710
    "!!h f g. REFLECTS[\<lambda>x. is_b(L, h(x), f(x), g(x)),
paulson@13363
   711
                     \<lambda>i x. is_b(**Lset(i), h(x), f(x), g(x))]"
wenzelm@13428
   712
  shows "REFLECTS[\<lambda>x. is_nat_case(L, f(x), is_b(L,x), g(x), h(x)),
paulson@13363
   713
               \<lambda>i x. is_nat_case(**Lset(i), f(x), is_b(**Lset(i), x), g(x), h(x))]"
paulson@13363
   714
apply (simp (no_asm_use) only: is_nat_case_def setclass_simps)
wenzelm@13428
   715
apply (intro FOL_reflections function_reflections
wenzelm@13428
   716
             restriction_reflection is_b_reflection quasinat_reflection)
paulson@13363
   717
done
paulson@13363
   718
paulson@13363
   719
paulson@13363
   720
paulson@13363
   721
subsection{*The Operator @{term iterates_MH}, Needed for Iteration*}
paulson@13363
   722
paulson@13363
   723
(*  iterates_MH :: "[i=>o, [i,i]=>o, i, i, i, i] => o"
paulson@13363
   724
   "iterates_MH(M,isF,v,n,g,z) ==
paulson@13363
   725
        is_nat_case(M, v, \<lambda>m u. \<exists>gm[M]. fun_apply(M,g,m,gm) & isF(gm,u),
paulson@13363
   726
                    n, z)" *)
paulson@13434
   727
constdefs iterates_MH_fm :: "[i, i, i, i, i]=>i"
paulson@13434
   728
 "iterates_MH_fm(isF,v,n,g,z) == 
paulson@13434
   729
    is_nat_case_fm(v, 
paulson@13434
   730
      Exists(And(fun_apply_fm(succ(succ(succ(g))),2,0), 
paulson@13434
   731
                     Forall(Implies(Equal(0,2), isF)))), 
paulson@13363
   732
      n, z)"
paulson@13363
   733
paulson@13363
   734
lemma iterates_MH_type [TC]:
paulson@13434
   735
     "[| p \<in> formula;  
paulson@13434
   736
         v \<in> nat; x \<in> nat; y \<in> nat; z \<in> nat |] 
paulson@13363
   737
      ==> iterates_MH_fm(p,v,x,y,z) \<in> formula"
wenzelm@13428
   738
by (simp add: iterates_MH_fm_def)
paulson@13363
   739
paulson@13363
   740
lemma sats_iterates_MH_fm:
wenzelm@13428
   741
  assumes is_F_iff_sats:
wenzelm@13428
   742
      "!!a b c d. [| a \<in> A; b \<in> A; c \<in> A; d \<in> A|]
paulson@13363
   743
              ==> is_F(a,b) <->
paulson@13434
   744
                  sats(A, p, Cons(b, Cons(a, Cons(c, Cons(d,env)))))"
paulson@13434
   745
  shows 
paulson@13363
   746
      "[|v \<in> nat; x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A)|]
wenzelm@13428
   747
       ==> sats(A, iterates_MH_fm(p,v,x,y,z), env) <->
paulson@13363
   748
           iterates_MH(**A, is_F, nth(v,env), nth(x,env), nth(y,env), nth(z,env))"
paulson@13434
   749
apply (frule lt_length_in_nat, assumption)  
paulson@13434
   750
apply (simp add: iterates_MH_fm_def iterates_MH_def sats_is_nat_case_fm 
paulson@13363
   751
              is_F_iff_sats [symmetric])
paulson@13434
   752
apply (rule is_nat_case_cong) 
paulson@13434
   753
apply (simp_all add: setclass_def)
paulson@13434
   754
done
paulson@13434
   755
paulson@13363
   756
paulson@13363
   757
lemma iterates_MH_iff_sats:
wenzelm@13428
   758
  "[| (!!a b c d. [| a \<in> A; b \<in> A; c \<in> A; d \<in> A|]
paulson@13363
   759
              ==> is_F(a,b) <->
paulson@13434
   760
                  sats(A, p, Cons(b, Cons(a, Cons(c, Cons(d,env))))));
paulson@13434
   761
      nth(i',env) = v; nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; 
paulson@13363
   762
      i' \<in> nat; i \<in> nat; j \<in> nat; k < length(env); env \<in> list(A)|]
wenzelm@13428
   763
   ==> iterates_MH(**A, is_F, v, x, y, z) <->
paulson@13363
   764
       sats(A, iterates_MH_fm(p,i',i,j,k), env)"
paulson@13434
   765
apply (rule iff_sym) 
wenzelm@13428
   766
apply (rule iff_trans)
paulson@13441
   767
apply (rule sats_iterates_MH_fm [of A is_F], blast, simp_all) 
paulson@13363
   768
done
paulson@13434
   769
(*FIXME: surely proof can be improved?*)
paulson@13434
   770
paulson@13363
   771
paulson@13363
   772
theorem iterates_MH_reflection:
paulson@13363
   773
  assumes p_reflection:
wenzelm@13428
   774
    "!!f g h. REFLECTS[\<lambda>x. p(L, f(x), g(x)),
paulson@13363
   775
                     \<lambda>i x. p(**Lset(i), f(x), g(x))]"
wenzelm@13428
   776
 shows "REFLECTS[\<lambda>x. iterates_MH(L, p(L), e(x), f(x), g(x), h(x)),
paulson@13363
   777
               \<lambda>i x. iterates_MH(**Lset(i), p(**Lset(i)), e(x), f(x), g(x), h(x))]"
paulson@13363
   778
apply (simp (no_asm_use) only: iterates_MH_def)
paulson@13363
   779
txt{*Must be careful: simplifying with @{text setclass_simps} above would
paulson@13363
   780
     change @{text "\<exists>gm[**Lset(i)]"} into @{text "\<exists>gm \<in> Lset(i)"}, when
paulson@13363
   781
     it would no longer match rule @{text is_nat_case_reflection}. *}
wenzelm@13428
   782
apply (rule is_nat_case_reflection)
paulson@13363
   783
apply (simp (no_asm_use) only: setclass_simps)
paulson@13363
   784
apply (intro FOL_reflections function_reflections is_nat_case_reflection
wenzelm@13428
   785
             restriction_reflection p_reflection)
paulson@13363
   786
done
paulson@13363
   787
paulson@13363
   788
paulson@13363
   789
wenzelm@13428
   790
subsection{*@{term L} is Closed Under the Operator @{term list}*}
paulson@13363
   791
paulson@13386
   792
subsubsection{*The List Functor, Internalized*}
paulson@13386
   793
paulson@13386
   794
constdefs list_functor_fm :: "[i,i,i]=>i"
wenzelm@13428
   795
(* "is_list_functor(M,A,X,Z) ==
wenzelm@13428
   796
        \<exists>n1[M]. \<exists>AX[M].
paulson@13386
   797
         number1(M,n1) & cartprod(M,A,X,AX) & is_sum(M,n1,AX,Z)" *)
wenzelm@13428
   798
    "list_functor_fm(A,X,Z) ==
paulson@13386
   799
       Exists(Exists(
wenzelm@13428
   800
        And(number1_fm(1),
paulson@13386
   801
            And(cartprod_fm(A#+2,X#+2,0), sum_fm(1,0,Z#+2)))))"
paulson@13386
   802
paulson@13386
   803
lemma list_functor_type [TC]:
paulson@13386
   804
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> list_functor_fm(x,y,z) \<in> formula"
wenzelm@13428
   805
by (simp add: list_functor_fm_def)
paulson@13386
   806
paulson@13386
   807
lemma arity_list_functor_fm [simp]:
wenzelm@13428
   808
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
paulson@13386
   809
      ==> arity(list_functor_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
wenzelm@13428
   810
by (simp add: list_functor_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13386
   811
paulson@13386
   812
lemma sats_list_functor_fm [simp]:
paulson@13386
   813
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
wenzelm@13428
   814
    ==> sats(A, list_functor_fm(x,y,z), env) <->
paulson@13386
   815
        is_list_functor(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13386
   816
by (simp add: list_functor_fm_def is_list_functor_def)
paulson@13386
   817
paulson@13386
   818
lemma list_functor_iff_sats:
wenzelm@13428
   819
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
paulson@13386
   820
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13386
   821
   ==> is_list_functor(**A, x, y, z) <-> sats(A, list_functor_fm(i,j,k), env)"
paulson@13386
   822
by simp
paulson@13386
   823
paulson@13386
   824
theorem list_functor_reflection:
wenzelm@13428
   825
     "REFLECTS[\<lambda>x. is_list_functor(L,f(x),g(x),h(x)),
paulson@13386
   826
               \<lambda>i x. is_list_functor(**Lset(i),f(x),g(x),h(x))]"
paulson@13386
   827
apply (simp only: is_list_functor_def setclass_simps)
paulson@13386
   828
apply (intro FOL_reflections number1_reflection
wenzelm@13428
   829
             cartprod_reflection sum_reflection)
paulson@13386
   830
done
paulson@13386
   831
paulson@13386
   832
paulson@13386
   833
subsubsection{*Instances of Replacement for Lists*}
paulson@13386
   834
paulson@13363
   835
lemma list_replacement1_Reflects:
paulson@13363
   836
 "REFLECTS
paulson@13363
   837
   [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
paulson@13363
   838
         is_wfrec(L, iterates_MH(L, is_list_functor(L,A), 0), memsn, u, y)),
paulson@13363
   839
    \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
wenzelm@13428
   840
         is_wfrec(**Lset(i),
wenzelm@13428
   841
                  iterates_MH(**Lset(i),
paulson@13363
   842
                          is_list_functor(**Lset(i), A), 0), memsn, u, y))]"
wenzelm@13428
   843
by (intro FOL_reflections function_reflections is_wfrec_reflection
wenzelm@13428
   844
          iterates_MH_reflection list_functor_reflection)
paulson@13363
   845
paulson@13441
   846
wenzelm@13428
   847
lemma list_replacement1:
paulson@13363
   848
   "L(A) ==> iterates_replacement(L, is_list_functor(L,A), 0)"
paulson@13363
   849
apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
wenzelm@13428
   850
apply (rule strong_replacementI)
paulson@13363
   851
apply (rule rallI)
wenzelm@13428
   852
apply (rename_tac B)
wenzelm@13428
   853
apply (rule separation_CollectI)
wenzelm@13428
   854
apply (insert nonempty)
wenzelm@13428
   855
apply (subgoal_tac "L(Memrel(succ(n)))")
wenzelm@13428
   856
apply (rule_tac A="{B,A,n,z,0,Memrel(succ(n))}" in subset_LsetE, blast )
paulson@13363
   857
apply (rule ReflectsE [OF list_replacement1_Reflects], assumption)
wenzelm@13428
   858
apply (drule subset_Lset_ltD, assumption)
paulson@13363
   859
apply (erule reflection_imp_L_separation)
paulson@13386
   860
  apply (simp_all add: lt_Ord2 Memrel_closed)
wenzelm@13428
   861
apply (elim conjE)
paulson@13385
   862
apply (rule DPow_LsetI)
wenzelm@13428
   863
apply (rename_tac v)
paulson@13363
   864
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13363
   865
apply (rule_tac env = "[u,v,A,n,B,0,Memrel(succ(n))]" in mem_iff_sats)
paulson@13434
   866
apply (rule sep_rules is_nat_case_iff_sats list_functor_iff_sats
paulson@13441
   867
            is_wfrec_iff_sats iterates_MH_iff_sats quasinat_iff_sats | simp)+
paulson@13363
   868
done
paulson@13363
   869
paulson@13441
   870
paulson@13363
   871
lemma list_replacement2_Reflects:
paulson@13363
   872
 "REFLECTS
paulson@13363
   873
   [\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
paulson@13363
   874
         (\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
paulson@13363
   875
           is_wfrec (L, iterates_MH (L, is_list_functor(L, A), 0),
paulson@13363
   876
                              msn, u, x)),
paulson@13363
   877
    \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
wenzelm@13428
   878
         (\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i).
paulson@13363
   879
          successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
wenzelm@13428
   880
           is_wfrec (**Lset(i),
paulson@13363
   881
                 iterates_MH (**Lset(i), is_list_functor(**Lset(i), A), 0),
paulson@13363
   882
                     msn, u, x))]"
wenzelm@13428
   883
by (intro FOL_reflections function_reflections is_wfrec_reflection
wenzelm@13428
   884
          iterates_MH_reflection list_functor_reflection)
paulson@13363
   885
paulson@13363
   886
wenzelm@13428
   887
lemma list_replacement2:
wenzelm@13428
   888
   "L(A) ==> strong_replacement(L,
wenzelm@13428
   889
         \<lambda>n y. n\<in>nat &
paulson@13363
   890
               (\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
wenzelm@13428
   891
               is_wfrec(L, iterates_MH(L,is_list_functor(L,A), 0),
paulson@13363
   892
                        msn, n, y)))"
wenzelm@13428
   893
apply (rule strong_replacementI)
paulson@13363
   894
apply (rule rallI)
wenzelm@13428
   895
apply (rename_tac B)
wenzelm@13428
   896
apply (rule separation_CollectI)
wenzelm@13428
   897
apply (insert nonempty)
wenzelm@13428
   898
apply (rule_tac A="{A,B,z,0,nat}" in subset_LsetE)
wenzelm@13428
   899
apply (blast intro: L_nat)
paulson@13363
   900
apply (rule ReflectsE [OF list_replacement2_Reflects], assumption)
wenzelm@13428
   901
apply (drule subset_Lset_ltD, assumption)
paulson@13363
   902
apply (erule reflection_imp_L_separation)
paulson@13363
   903
  apply (simp_all add: lt_Ord2)
paulson@13385
   904
apply (rule DPow_LsetI)
wenzelm@13428
   905
apply (rename_tac v)
paulson@13363
   906
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13363
   907
apply (rule_tac env = "[u,v,A,B,0,nat]" in mem_iff_sats)
paulson@13434
   908
apply (rule sep_rules is_nat_case_iff_sats list_functor_iff_sats
paulson@13441
   909
            is_wfrec_iff_sats iterates_MH_iff_sats quasinat_iff_sats | simp)+
paulson@13363
   910
done
paulson@13363
   911
paulson@13386
   912
wenzelm@13428
   913
subsection{*@{term L} is Closed Under the Operator @{term formula}*}
paulson@13386
   914
paulson@13386
   915
subsubsection{*The Formula Functor, Internalized*}
paulson@13386
   916
paulson@13386
   917
constdefs formula_functor_fm :: "[i,i]=>i"
wenzelm@13428
   918
(*     "is_formula_functor(M,X,Z) ==
wenzelm@13428
   919
        \<exists>nat'[M]. \<exists>natnat[M]. \<exists>natnatsum[M]. \<exists>XX[M]. \<exists>X3[M].
paulson@13398
   920
           4           3               2       1       0
wenzelm@13428
   921
          omega(M,nat') & cartprod(M,nat',nat',natnat) &
paulson@13386
   922
          is_sum(M,natnat,natnat,natnatsum) &
wenzelm@13428
   923
          cartprod(M,X,X,XX) & is_sum(M,XX,X,X3) &
wenzelm@13428
   924
          is_sum(M,natnatsum,X3,Z)" *)
wenzelm@13428
   925
    "formula_functor_fm(X,Z) ==
paulson@13398
   926
       Exists(Exists(Exists(Exists(Exists(
wenzelm@13428
   927
        And(omega_fm(4),
paulson@13398
   928
         And(cartprod_fm(4,4,3),
paulson@13398
   929
          And(sum_fm(3,3,2),
paulson@13398
   930
           And(cartprod_fm(X#+5,X#+5,1),
paulson@13398
   931
            And(sum_fm(1,X#+5,0), sum_fm(2,0,Z#+5)))))))))))"
paulson@13386
   932
paulson@13386
   933
lemma formula_functor_type [TC]:
paulson@13386
   934
     "[| x \<in> nat; y \<in> nat |] ==> formula_functor_fm(x,y) \<in> formula"
wenzelm@13428
   935
by (simp add: formula_functor_fm_def)
paulson@13386
   936
paulson@13386
   937
lemma sats_formula_functor_fm [simp]:
paulson@13386
   938
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
wenzelm@13428
   939
    ==> sats(A, formula_functor_fm(x,y), env) <->
paulson@13386
   940
        is_formula_functor(**A, nth(x,env), nth(y,env))"
paulson@13386
   941
by (simp add: formula_functor_fm_def is_formula_functor_def)
paulson@13386
   942
paulson@13386
   943
lemma formula_functor_iff_sats:
wenzelm@13428
   944
  "[| nth(i,env) = x; nth(j,env) = y;
paulson@13386
   945
      i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13386
   946
   ==> is_formula_functor(**A, x, y) <-> sats(A, formula_functor_fm(i,j), env)"
paulson@13386
   947
by simp
paulson@13386
   948
paulson@13386
   949
theorem formula_functor_reflection:
wenzelm@13428
   950
     "REFLECTS[\<lambda>x. is_formula_functor(L,f(x),g(x)),
paulson@13386
   951
               \<lambda>i x. is_formula_functor(**Lset(i),f(x),g(x))]"
paulson@13386
   952
apply (simp only: is_formula_functor_def setclass_simps)
paulson@13386
   953
apply (intro FOL_reflections omega_reflection
wenzelm@13428
   954
             cartprod_reflection sum_reflection)
paulson@13386
   955
done
paulson@13386
   956
paulson@13386
   957
subsubsection{*Instances of Replacement for Formulas*}
paulson@13386
   958
paulson@13386
   959
lemma formula_replacement1_Reflects:
paulson@13386
   960
 "REFLECTS
paulson@13386
   961
   [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
paulson@13386
   962
         is_wfrec(L, iterates_MH(L, is_formula_functor(L), 0), memsn, u, y)),
paulson@13386
   963
    \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
wenzelm@13428
   964
         is_wfrec(**Lset(i),
wenzelm@13428
   965
                  iterates_MH(**Lset(i),
paulson@13386
   966
                          is_formula_functor(**Lset(i)), 0), memsn, u, y))]"
wenzelm@13428
   967
by (intro FOL_reflections function_reflections is_wfrec_reflection
wenzelm@13428
   968
          iterates_MH_reflection formula_functor_reflection)
paulson@13386
   969
wenzelm@13428
   970
lemma formula_replacement1:
paulson@13386
   971
   "iterates_replacement(L, is_formula_functor(L), 0)"
paulson@13386
   972
apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
wenzelm@13428
   973
apply (rule strong_replacementI)
paulson@13386
   974
apply (rule rallI)
wenzelm@13428
   975
apply (rename_tac B)
wenzelm@13428
   976
apply (rule separation_CollectI)
wenzelm@13428
   977
apply (insert nonempty)
wenzelm@13428
   978
apply (subgoal_tac "L(Memrel(succ(n)))")
wenzelm@13428
   979
apply (rule_tac A="{B,n,z,0,Memrel(succ(n))}" in subset_LsetE, blast )
paulson@13386
   980
apply (rule ReflectsE [OF formula_replacement1_Reflects], assumption)
wenzelm@13428
   981
apply (drule subset_Lset_ltD, assumption)
paulson@13386
   982
apply (erule reflection_imp_L_separation)
paulson@13386
   983
  apply (simp_all add: lt_Ord2 Memrel_closed)
paulson@13386
   984
apply (rule DPow_LsetI)
wenzelm@13428
   985
apply (rename_tac v)
paulson@13386
   986
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13386
   987
apply (rule_tac env = "[u,v,n,B,0,Memrel(succ(n))]" in mem_iff_sats)
paulson@13434
   988
apply (rule sep_rules is_nat_case_iff_sats formula_functor_iff_sats
paulson@13441
   989
            is_wfrec_iff_sats iterates_MH_iff_sats quasinat_iff_sats | simp)+
paulson@13386
   990
done
paulson@13386
   991
paulson@13386
   992
lemma formula_replacement2_Reflects:
paulson@13386
   993
 "REFLECTS
paulson@13386
   994
   [\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
paulson@13386
   995
         (\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
paulson@13386
   996
           is_wfrec (L, iterates_MH (L, is_formula_functor(L), 0),
paulson@13386
   997
                              msn, u, x)),
paulson@13386
   998
    \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
wenzelm@13428
   999
         (\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i).
paulson@13386
  1000
          successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
wenzelm@13428
  1001
           is_wfrec (**Lset(i),
paulson@13386
  1002
                 iterates_MH (**Lset(i), is_formula_functor(**Lset(i)), 0),
paulson@13386
  1003
                     msn, u, x))]"
wenzelm@13428
  1004
by (intro FOL_reflections function_reflections is_wfrec_reflection
wenzelm@13428
  1005
          iterates_MH_reflection formula_functor_reflection)
paulson@13386
  1006
paulson@13386
  1007
wenzelm@13428
  1008
lemma formula_replacement2:
wenzelm@13428
  1009
   "strong_replacement(L,
wenzelm@13428
  1010
         \<lambda>n y. n\<in>nat &
paulson@13386
  1011
               (\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
wenzelm@13428
  1012
               is_wfrec(L, iterates_MH(L,is_formula_functor(L), 0),
paulson@13386
  1013
                        msn, n, y)))"
wenzelm@13428
  1014
apply (rule strong_replacementI)
paulson@13386
  1015
apply (rule rallI)
wenzelm@13428
  1016
apply (rename_tac B)
wenzelm@13428
  1017
apply (rule separation_CollectI)
wenzelm@13428
  1018
apply (insert nonempty)
wenzelm@13428
  1019
apply (rule_tac A="{B,z,0,nat}" in subset_LsetE)
wenzelm@13428
  1020
apply (blast intro: L_nat)
paulson@13386
  1021
apply (rule ReflectsE [OF formula_replacement2_Reflects], assumption)
wenzelm@13428
  1022
apply (drule subset_Lset_ltD, assumption)
paulson@13386
  1023
apply (erule reflection_imp_L_separation)
paulson@13386
  1024
  apply (simp_all add: lt_Ord2)
paulson@13386
  1025
apply (rule DPow_LsetI)
wenzelm@13428
  1026
apply (rename_tac v)
paulson@13386
  1027
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13386
  1028
apply (rule_tac env = "[u,v,B,0,nat]" in mem_iff_sats)
paulson@13434
  1029
apply (rule sep_rules is_nat_case_iff_sats formula_functor_iff_sats
paulson@13441
  1030
            is_wfrec_iff_sats iterates_MH_iff_sats quasinat_iff_sats | simp)+
paulson@13386
  1031
done
paulson@13386
  1032
paulson@13386
  1033
text{*NB The proofs for type @{term formula} are virtually identical to those
paulson@13386
  1034
for @{term "list(A)"}.  It was a cut-and-paste job! *}
paulson@13386
  1035
paulson@13387
  1036
paulson@13409
  1037
subsection{*Internalized Forms of Data Structuring Operators*}
paulson@13409
  1038
paulson@13409
  1039
subsubsection{*The Formula @{term is_Inl}, Internalized*}
paulson@13409
  1040
paulson@13409
  1041
(*  is_Inl(M,a,z) == \<exists>zero[M]. empty(M,zero) & pair(M,zero,a,z) *)
paulson@13409
  1042
constdefs Inl_fm :: "[i,i]=>i"
paulson@13409
  1043
    "Inl_fm(a,z) == Exists(And(empty_fm(0), pair_fm(0,succ(a),succ(z))))"
paulson@13409
  1044
paulson@13409
  1045
lemma Inl_type [TC]:
paulson@13409
  1046
     "[| x \<in> nat; z \<in> nat |] ==> Inl_fm(x,z) \<in> formula"
wenzelm@13428
  1047
by (simp add: Inl_fm_def)
paulson@13409
  1048
paulson@13409
  1049
lemma sats_Inl_fm [simp]:
paulson@13409
  1050
   "[| x \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13409
  1051
    ==> sats(A, Inl_fm(x,z), env) <-> is_Inl(**A, nth(x,env), nth(z,env))"
paulson@13409
  1052
by (simp add: Inl_fm_def is_Inl_def)
paulson@13409
  1053
paulson@13409
  1054
lemma Inl_iff_sats:
wenzelm@13428
  1055
      "[| nth(i,env) = x; nth(k,env) = z;
paulson@13409
  1056
          i \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13409
  1057
       ==> is_Inl(**A, x, z) <-> sats(A, Inl_fm(i,k), env)"
paulson@13409
  1058
by simp
paulson@13409
  1059
paulson@13409
  1060
theorem Inl_reflection:
wenzelm@13428
  1061
     "REFLECTS[\<lambda>x. is_Inl(L,f(x),h(x)),
paulson@13409
  1062
               \<lambda>i x. is_Inl(**Lset(i),f(x),h(x))]"
paulson@13409
  1063
apply (simp only: is_Inl_def setclass_simps)
wenzelm@13428
  1064
apply (intro FOL_reflections function_reflections)
paulson@13409
  1065
done
paulson@13409
  1066
paulson@13409
  1067
paulson@13409
  1068
subsubsection{*The Formula @{term is_Inr}, Internalized*}
paulson@13409
  1069
paulson@13409
  1070
(*  is_Inr(M,a,z) == \<exists>n1[M]. number1(M,n1) & pair(M,n1,a,z) *)
paulson@13409
  1071
constdefs Inr_fm :: "[i,i]=>i"
paulson@13409
  1072
    "Inr_fm(a,z) == Exists(And(number1_fm(0), pair_fm(0,succ(a),succ(z))))"
paulson@13409
  1073
paulson@13409
  1074
lemma Inr_type [TC]:
paulson@13409
  1075
     "[| x \<in> nat; z \<in> nat |] ==> Inr_fm(x,z) \<in> formula"
wenzelm@13428
  1076
by (simp add: Inr_fm_def)
paulson@13409
  1077
paulson@13409
  1078
lemma sats_Inr_fm [simp]:
paulson@13409
  1079
   "[| x \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13409
  1080
    ==> sats(A, Inr_fm(x,z), env) <-> is_Inr(**A, nth(x,env), nth(z,env))"
paulson@13409
  1081
by (simp add: Inr_fm_def is_Inr_def)
paulson@13409
  1082
paulson@13409
  1083
lemma Inr_iff_sats:
wenzelm@13428
  1084
      "[| nth(i,env) = x; nth(k,env) = z;
paulson@13409
  1085
          i \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13409
  1086
       ==> is_Inr(**A, x, z) <-> sats(A, Inr_fm(i,k), env)"
paulson@13409
  1087
by simp
paulson@13409
  1088
paulson@13409
  1089
theorem Inr_reflection:
wenzelm@13428
  1090
     "REFLECTS[\<lambda>x. is_Inr(L,f(x),h(x)),
paulson@13409
  1091
               \<lambda>i x. is_Inr(**Lset(i),f(x),h(x))]"
paulson@13409
  1092
apply (simp only: is_Inr_def setclass_simps)
wenzelm@13428
  1093
apply (intro FOL_reflections function_reflections)
paulson@13409
  1094
done
paulson@13409
  1095
paulson@13409
  1096
paulson@13409
  1097
subsubsection{*The Formula @{term is_Nil}, Internalized*}
paulson@13409
  1098
paulson@13409
  1099
(* is_Nil(M,xs) == \<exists>zero[M]. empty(M,zero) & is_Inl(M,zero,xs) *)
paulson@13409
  1100
paulson@13409
  1101
constdefs Nil_fm :: "i=>i"
paulson@13409
  1102
    "Nil_fm(x) == Exists(And(empty_fm(0), Inl_fm(0,succ(x))))"
wenzelm@13428
  1103
paulson@13409
  1104
lemma Nil_type [TC]: "x \<in> nat ==> Nil_fm(x) \<in> formula"
wenzelm@13428
  1105
by (simp add: Nil_fm_def)
paulson@13409
  1106
paulson@13409
  1107
lemma sats_Nil_fm [simp]:
paulson@13409
  1108
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13409
  1109
    ==> sats(A, Nil_fm(x), env) <-> is_Nil(**A, nth(x,env))"
paulson@13409
  1110
by (simp add: Nil_fm_def is_Nil_def)
paulson@13409
  1111
paulson@13409
  1112
lemma Nil_iff_sats:
paulson@13409
  1113
      "[| nth(i,env) = x; i \<in> nat; env \<in> list(A)|]
paulson@13409
  1114
       ==> is_Nil(**A, x) <-> sats(A, Nil_fm(i), env)"
paulson@13409
  1115
by simp
paulson@13409
  1116
paulson@13409
  1117
theorem Nil_reflection:
wenzelm@13428
  1118
     "REFLECTS[\<lambda>x. is_Nil(L,f(x)),
paulson@13409
  1119
               \<lambda>i x. is_Nil(**Lset(i),f(x))]"
paulson@13409
  1120
apply (simp only: is_Nil_def setclass_simps)
wenzelm@13428
  1121
apply (intro FOL_reflections function_reflections Inl_reflection)
paulson@13409
  1122
done
paulson@13409
  1123
paulson@13409
  1124
paulson@13422
  1125
subsubsection{*The Formula @{term is_Cons}, Internalized*}
paulson@13395
  1126
paulson@13387
  1127
paulson@13409
  1128
(*  "is_Cons(M,a,l,Z) == \<exists>p[M]. pair(M,a,l,p) & is_Inr(M,p,Z)" *)
paulson@13409
  1129
constdefs Cons_fm :: "[i,i,i]=>i"
wenzelm@13428
  1130
    "Cons_fm(a,l,Z) ==
paulson@13409
  1131
       Exists(And(pair_fm(succ(a),succ(l),0), Inr_fm(0,succ(Z))))"
paulson@13409
  1132
paulson@13409
  1133
lemma Cons_type [TC]:
paulson@13409
  1134
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> Cons_fm(x,y,z) \<in> formula"
wenzelm@13428
  1135
by (simp add: Cons_fm_def)
paulson@13409
  1136
paulson@13409
  1137
lemma sats_Cons_fm [simp]:
paulson@13409
  1138
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
wenzelm@13428
  1139
    ==> sats(A, Cons_fm(x,y,z), env) <->
paulson@13409
  1140
       is_Cons(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13409
  1141
by (simp add: Cons_fm_def is_Cons_def)
paulson@13409
  1142
paulson@13409
  1143
lemma Cons_iff_sats:
wenzelm@13428
  1144
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
paulson@13409
  1145
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13409
  1146
       ==>is_Cons(**A, x, y, z) <-> sats(A, Cons_fm(i,j,k), env)"
paulson@13409
  1147
by simp
paulson@13409
  1148
paulson@13409
  1149
theorem Cons_reflection:
wenzelm@13428
  1150
     "REFLECTS[\<lambda>x. is_Cons(L,f(x),g(x),h(x)),
paulson@13409
  1151
               \<lambda>i x. is_Cons(**Lset(i),f(x),g(x),h(x))]"
paulson@13409
  1152
apply (simp only: is_Cons_def setclass_simps)
wenzelm@13428
  1153
apply (intro FOL_reflections pair_reflection Inr_reflection)
paulson@13409
  1154
done
paulson@13409
  1155
paulson@13409
  1156
subsubsection{*The Formula @{term is_quasilist}, Internalized*}
paulson@13409
  1157
paulson@13409
  1158
(* is_quasilist(M,xs) == is_Nil(M,z) | (\<exists>x[M]. \<exists>l[M]. is_Cons(M,x,l,z))" *)
paulson@13409
  1159
paulson@13409
  1160
constdefs quasilist_fm :: "i=>i"
wenzelm@13428
  1161
    "quasilist_fm(x) ==
paulson@13409
  1162
       Or(Nil_fm(x), Exists(Exists(Cons_fm(1,0,succ(succ(x))))))"
wenzelm@13428
  1163
paulson@13409
  1164
lemma quasilist_type [TC]: "x \<in> nat ==> quasilist_fm(x) \<in> formula"
wenzelm@13428
  1165
by (simp add: quasilist_fm_def)
paulson@13409
  1166
paulson@13409
  1167
lemma sats_quasilist_fm [simp]:
paulson@13409
  1168
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13409
  1169
    ==> sats(A, quasilist_fm(x), env) <-> is_quasilist(**A, nth(x,env))"
paulson@13409
  1170
by (simp add: quasilist_fm_def is_quasilist_def)
paulson@13409
  1171
paulson@13409
  1172
lemma quasilist_iff_sats:
paulson@13409
  1173
      "[| nth(i,env) = x; i \<in> nat; env \<in> list(A)|]
paulson@13409
  1174
       ==> is_quasilist(**A, x) <-> sats(A, quasilist_fm(i), env)"
paulson@13409
  1175
by simp
paulson@13409
  1176
paulson@13409
  1177
theorem quasilist_reflection:
wenzelm@13428
  1178
     "REFLECTS[\<lambda>x. is_quasilist(L,f(x)),
paulson@13409
  1179
               \<lambda>i x. is_quasilist(**Lset(i),f(x))]"
paulson@13409
  1180
apply (simp only: is_quasilist_def setclass_simps)
wenzelm@13428
  1181
apply (intro FOL_reflections Nil_reflection Cons_reflection)
paulson@13409
  1182
done
paulson@13409
  1183
paulson@13409
  1184
paulson@13409
  1185
subsection{*Absoluteness for the Function @{term nth}*}
paulson@13409
  1186
paulson@13409
  1187
paulson@13437
  1188
subsubsection{*The Formula @{term is_hd}, Internalized*}
paulson@13437
  1189
paulson@13437
  1190
(*   "is_hd(M,xs,H) == 
paulson@13437
  1191
       (is_Nil(M,xs) --> empty(M,H)) &
paulson@13437
  1192
       (\<forall>x[M]. \<forall>l[M]. ~ is_Cons(M,x,l,xs) | H=x) &
paulson@13437
  1193
       (is_quasilist(M,xs) | empty(M,H))" *)
paulson@13437
  1194
constdefs hd_fm :: "[i,i]=>i"
paulson@13437
  1195
    "hd_fm(xs,H) == 
paulson@13437
  1196
       And(Implies(Nil_fm(xs), empty_fm(H)),
paulson@13437
  1197
           And(Forall(Forall(Or(Neg(Cons_fm(1,0,xs#+2)), Equal(H#+2,1)))),
paulson@13437
  1198
               Or(quasilist_fm(xs), empty_fm(H))))"
paulson@13437
  1199
paulson@13437
  1200
lemma hd_type [TC]:
paulson@13437
  1201
     "[| x \<in> nat; y \<in> nat |] ==> hd_fm(x,y) \<in> formula"
paulson@13437
  1202
by (simp add: hd_fm_def) 
paulson@13437
  1203
paulson@13437
  1204
lemma sats_hd_fm [simp]:
paulson@13437
  1205
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
paulson@13437
  1206
    ==> sats(A, hd_fm(x,y), env) <-> is_hd(**A, nth(x,env), nth(y,env))"
paulson@13437
  1207
by (simp add: hd_fm_def is_hd_def)
paulson@13437
  1208
paulson@13437
  1209
lemma hd_iff_sats:
paulson@13437
  1210
      "[| nth(i,env) = x; nth(j,env) = y;
paulson@13437
  1211
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13437
  1212
       ==> is_hd(**A, x, y) <-> sats(A, hd_fm(i,j), env)"
paulson@13437
  1213
by simp
paulson@13437
  1214
paulson@13437
  1215
theorem hd_reflection:
paulson@13437
  1216
     "REFLECTS[\<lambda>x. is_hd(L,f(x),g(x)), 
paulson@13437
  1217
               \<lambda>i x. is_hd(**Lset(i),f(x),g(x))]"
paulson@13437
  1218
apply (simp only: is_hd_def setclass_simps)
paulson@13437
  1219
apply (intro FOL_reflections Nil_reflection Cons_reflection
paulson@13437
  1220
             quasilist_reflection empty_reflection)  
paulson@13437
  1221
done
paulson@13437
  1222
paulson@13437
  1223
paulson@13409
  1224
subsubsection{*The Formula @{term is_tl}, Internalized*}
paulson@13409
  1225
wenzelm@13428
  1226
(*     "is_tl(M,xs,T) ==
paulson@13409
  1227
       (is_Nil(M,xs) --> T=xs) &
paulson@13409
  1228
       (\<forall>x[M]. \<forall>l[M]. ~ is_Cons(M,x,l,xs) | T=l) &
paulson@13409
  1229
       (is_quasilist(M,xs) | empty(M,T))" *)
paulson@13409
  1230
constdefs tl_fm :: "[i,i]=>i"
wenzelm@13428
  1231
    "tl_fm(xs,T) ==
paulson@13409
  1232
       And(Implies(Nil_fm(xs), Equal(T,xs)),
paulson@13409
  1233
           And(Forall(Forall(Or(Neg(Cons_fm(1,0,xs#+2)), Equal(T#+2,0)))),
paulson@13409
  1234
               Or(quasilist_fm(xs), empty_fm(T))))"
paulson@13409
  1235
paulson@13409
  1236
lemma tl_type [TC]:
paulson@13409
  1237
     "[| x \<in> nat; y \<in> nat |] ==> tl_fm(x,y) \<in> formula"
wenzelm@13428
  1238
by (simp add: tl_fm_def)
paulson@13409
  1239
paulson@13409
  1240
lemma sats_tl_fm [simp]:
paulson@13409
  1241
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
paulson@13409
  1242
    ==> sats(A, tl_fm(x,y), env) <-> is_tl(**A, nth(x,env), nth(y,env))"
paulson@13409
  1243
by (simp add: tl_fm_def is_tl_def)
paulson@13409
  1244
paulson@13409
  1245
lemma tl_iff_sats:
paulson@13409
  1246
      "[| nth(i,env) = x; nth(j,env) = y;
paulson@13409
  1247
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13409
  1248
       ==> is_tl(**A, x, y) <-> sats(A, tl_fm(i,j), env)"
paulson@13409
  1249
by simp
paulson@13409
  1250
paulson@13409
  1251
theorem tl_reflection:
wenzelm@13428
  1252
     "REFLECTS[\<lambda>x. is_tl(L,f(x),g(x)),
paulson@13409
  1253
               \<lambda>i x. is_tl(**Lset(i),f(x),g(x))]"
paulson@13409
  1254
apply (simp only: is_tl_def setclass_simps)
paulson@13409
  1255
apply (intro FOL_reflections Nil_reflection Cons_reflection
wenzelm@13428
  1256
             quasilist_reflection empty_reflection)
paulson@13409
  1257
done
paulson@13409
  1258
paulson@13409
  1259
paulson@13437
  1260
subsubsection{*The Formula @{term is_nth}, Internalized*}
paulson@13437
  1261
paulson@13437
  1262
(* "is_nth(M,n,l,Z) == 
paulson@13437
  1263
      \<exists>X[M]. \<exists>sn[M]. \<exists>msn[M]. 
paulson@13437
  1264
       2       1       0
paulson@13437
  1265
       successor(M,n,sn) & membership(M,sn,msn) &
paulson@13437
  1266
       is_wfrec(M, iterates_MH(M, is_tl(M), l), msn, n, X) &
paulson@13437
  1267
       is_hd(M,X,Z)"
paulson@13437
  1268
constdefs nth_fm :: "[i,i,i]=>i"
paulson@13437
  1269
    "nth_fm(n,l,Z) == 
paulson@13437
  1270
       Exists(Exists(Exists(
paulson@13437
  1271
         And(successor_fm(n#+3,1),
paulson@13437
  1272
          And(membership_fm(1,0),
paulson@13437
  1273
           And(
paulson@13437
  1274
 *)
paulson@13437
  1275
paulson@13437
  1276
theorem nth_reflection:
paulson@13437
  1277
     "REFLECTS[\<lambda>x. is_nth(L, f(x), g(x), h(x)),  
paulson@13437
  1278
               \<lambda>i x. is_nth(**Lset(i), f(x), g(x), h(x))]"
paulson@13437
  1279
apply (simp only: is_nth_def setclass_simps)
paulson@13437
  1280
apply (intro FOL_reflections function_reflections is_wfrec_reflection 
paulson@13437
  1281
             iterates_MH_reflection hd_reflection tl_reflection) 
paulson@13437
  1282
done
paulson@13437
  1283
paulson@13437
  1284
theorem bool_of_o_reflection:
paulson@13440
  1285
     "REFLECTS [P(L), \<lambda>i. P(**Lset(i))] ==>
paulson@13440
  1286
      REFLECTS[\<lambda>x. is_bool_of_o(L, P(L,x), f(x)),  
paulson@13440
  1287
               \<lambda>i x. is_bool_of_o(**Lset(i), P(**Lset(i),x), f(x))]"
paulson@13440
  1288
apply (simp (no_asm) only: is_bool_of_o_def setclass_simps)
paulson@13441
  1289
apply (intro FOL_reflections function_reflections, assumption+)
paulson@13437
  1290
done
paulson@13437
  1291
paulson@13437
  1292
paulson@13409
  1293
subsubsection{*An Instance of Replacement for @{term nth}*}
paulson@13409
  1294
paulson@13409
  1295
lemma nth_replacement_Reflects:
paulson@13409
  1296
 "REFLECTS
paulson@13409
  1297
   [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
paulson@13409
  1298
         is_wfrec(L, iterates_MH(L, is_tl(L), z), memsn, u, y)),
paulson@13409
  1299
    \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
wenzelm@13428
  1300
         is_wfrec(**Lset(i),
wenzelm@13428
  1301
                  iterates_MH(**Lset(i),
paulson@13409
  1302
                          is_tl(**Lset(i)), z), memsn, u, y))]"
wenzelm@13428
  1303
by (intro FOL_reflections function_reflections is_wfrec_reflection
wenzelm@13428
  1304
          iterates_MH_reflection list_functor_reflection tl_reflection)
paulson@13409
  1305
wenzelm@13428
  1306
lemma nth_replacement:
paulson@13409
  1307
   "L(w) ==> iterates_replacement(L, %l t. is_tl(L,l,t), w)"
paulson@13409
  1308
apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
wenzelm@13428
  1309
apply (rule strong_replacementI)
wenzelm@13428
  1310
apply (rule rallI)
wenzelm@13428
  1311
apply (rule separation_CollectI)
wenzelm@13428
  1312
apply (subgoal_tac "L(Memrel(succ(n)))")
wenzelm@13428
  1313
apply (rule_tac A="{A,n,w,z,Memrel(succ(n))}" in subset_LsetE, blast )
paulson@13409
  1314
apply (rule ReflectsE [OF nth_replacement_Reflects], assumption)
wenzelm@13428
  1315
apply (drule subset_Lset_ltD, assumption)
paulson@13409
  1316
apply (erule reflection_imp_L_separation)
paulson@13409
  1317
  apply (simp_all add: lt_Ord2 Memrel_closed)
wenzelm@13428
  1318
apply (elim conjE)
paulson@13409
  1319
apply (rule DPow_LsetI)
wenzelm@13428
  1320
apply (rename_tac v)
paulson@13409
  1321
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13409
  1322
apply (rule_tac env = "[u,v,A,z,w,Memrel(succ(n))]" in mem_iff_sats)
paulson@13434
  1323
apply (rule sep_rules is_nat_case_iff_sats tl_iff_sats
paulson@13441
  1324
            is_wfrec_iff_sats iterates_MH_iff_sats quasinat_iff_sats | simp)+
paulson@13409
  1325
done
paulson@13409
  1326
paulson@13422
  1327
paulson@13422
  1328
paulson@13422
  1329
subsubsection{*Instantiating the locale @{text M_datatypes}*}
wenzelm@13428
  1330
paulson@13437
  1331
lemma M_datatypes_axioms_L: "M_datatypes_axioms(L)"
wenzelm@13428
  1332
  apply (rule M_datatypes_axioms.intro)
wenzelm@13428
  1333
      apply (assumption | rule
wenzelm@13428
  1334
        list_replacement1 list_replacement2
wenzelm@13428
  1335
        formula_replacement1 formula_replacement2
wenzelm@13428
  1336
        nth_replacement)+
wenzelm@13428
  1337
  done
paulson@13422
  1338
paulson@13437
  1339
theorem M_datatypes_L: "PROP M_datatypes(L)"
paulson@13437
  1340
  apply (rule M_datatypes.intro)
paulson@13437
  1341
      apply (rule M_wfrank.axioms [OF M_wfrank_L])+
paulson@13441
  1342
 apply (rule M_datatypes_axioms_L) 
paulson@13437
  1343
 done
paulson@13437
  1344
wenzelm@13428
  1345
lemmas list_closed = M_datatypes.list_closed [OF M_datatypes_L]
wenzelm@13428
  1346
  and formula_closed = M_datatypes.formula_closed [OF M_datatypes_L]
wenzelm@13428
  1347
  and list_abs = M_datatypes.list_abs [OF M_datatypes_L]
wenzelm@13428
  1348
  and formula_abs = M_datatypes.formula_abs [OF M_datatypes_L]
wenzelm@13428
  1349
  and nth_abs = M_datatypes.nth_abs [OF M_datatypes_L]
paulson@13409
  1350
paulson@13422
  1351
declare list_closed [intro,simp]
paulson@13422
  1352
declare formula_closed [intro,simp]
paulson@13422
  1353
declare list_abs [simp]
paulson@13422
  1354
declare formula_abs [simp]
paulson@13422
  1355
declare nth_abs [simp]
paulson@13422
  1356
paulson@13422
  1357
wenzelm@13428
  1358
subsection{*@{term L} is Closed Under the Operator @{term eclose}*}
paulson@13422
  1359
paulson@13422
  1360
subsubsection{*Instances of Replacement for @{term eclose}*}
paulson@13422
  1361
paulson@13422
  1362
lemma eclose_replacement1_Reflects:
paulson@13422
  1363
 "REFLECTS
paulson@13422
  1364
   [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
paulson@13422
  1365
         is_wfrec(L, iterates_MH(L, big_union(L), A), memsn, u, y)),
paulson@13422
  1366
    \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> (\<exists>y \<in> Lset(i). pair(**Lset(i), u, y, x) \<and>
wenzelm@13428
  1367
         is_wfrec(**Lset(i),
wenzelm@13428
  1368
                  iterates_MH(**Lset(i), big_union(**Lset(i)), A),
paulson@13422
  1369
                  memsn, u, y))]"
wenzelm@13428
  1370
by (intro FOL_reflections function_reflections is_wfrec_reflection
wenzelm@13428
  1371
          iterates_MH_reflection)
paulson@13422
  1372
wenzelm@13428
  1373
lemma eclose_replacement1:
paulson@13422
  1374
   "L(A) ==> iterates_replacement(L, big_union(L), A)"
paulson@13422
  1375
apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
wenzelm@13428
  1376
apply (rule strong_replacementI)
paulson@13422
  1377
apply (rule rallI)
wenzelm@13428
  1378
apply (rename_tac B)
wenzelm@13428
  1379
apply (rule separation_CollectI)
wenzelm@13428
  1380
apply (subgoal_tac "L(Memrel(succ(n)))")
wenzelm@13428
  1381
apply (rule_tac A="{B,A,n,z,Memrel(succ(n))}" in subset_LsetE, blast )
paulson@13422
  1382
apply (rule ReflectsE [OF eclose_replacement1_Reflects], assumption)
wenzelm@13428
  1383
apply (drule subset_Lset_ltD, assumption)
paulson@13422
  1384
apply (erule reflection_imp_L_separation)
paulson@13422
  1385
  apply (simp_all add: lt_Ord2 Memrel_closed)
wenzelm@13428
  1386
apply (elim conjE)
paulson@13422
  1387
apply (rule DPow_LsetI)
wenzelm@13428
  1388
apply (rename_tac v)
paulson@13422
  1389
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13422
  1390
apply (rule_tac env = "[u,v,A,n,B,Memrel(succ(n))]" in mem_iff_sats)
paulson@13434
  1391
apply (rule sep_rules iterates_MH_iff_sats is_nat_case_iff_sats
paulson@13441
  1392
             is_wfrec_iff_sats big_union_iff_sats quasinat_iff_sats | simp)+
paulson@13409
  1393
done
paulson@13409
  1394
paulson@13422
  1395
paulson@13422
  1396
lemma eclose_replacement2_Reflects:
paulson@13422
  1397
 "REFLECTS
paulson@13422
  1398
   [\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
paulson@13422
  1399
         (\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
paulson@13422
  1400
           is_wfrec (L, iterates_MH (L, big_union(L), A),
paulson@13422
  1401
                              msn, u, x)),
paulson@13422
  1402
    \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
wenzelm@13428
  1403
         (\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i).
paulson@13422
  1404
          successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
wenzelm@13428
  1405
           is_wfrec (**Lset(i),
paulson@13422
  1406
                 iterates_MH (**Lset(i), big_union(**Lset(i)), A),
paulson@13422
  1407
                     msn, u, x))]"
wenzelm@13428
  1408
by (intro FOL_reflections function_reflections is_wfrec_reflection
wenzelm@13428
  1409
          iterates_MH_reflection)
paulson@13422
  1410
paulson@13422
  1411
wenzelm@13428
  1412
lemma eclose_replacement2:
wenzelm@13428
  1413
   "L(A) ==> strong_replacement(L,
wenzelm@13428
  1414
         \<lambda>n y. n\<in>nat &
paulson@13422
  1415
               (\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
wenzelm@13428
  1416
               is_wfrec(L, iterates_MH(L,big_union(L), A),
paulson@13422
  1417
                        msn, n, y)))"
wenzelm@13428
  1418
apply (rule strong_replacementI)
paulson@13422
  1419
apply (rule rallI)
wenzelm@13428
  1420
apply (rename_tac B)
wenzelm@13428
  1421
apply (rule separation_CollectI)
wenzelm@13428
  1422
apply (rule_tac A="{A,B,z,nat}" in subset_LsetE)
wenzelm@13428
  1423
apply (blast intro: L_nat)
paulson@13422
  1424
apply (rule ReflectsE [OF eclose_replacement2_Reflects], assumption)
wenzelm@13428
  1425
apply (drule subset_Lset_ltD, assumption)
paulson@13422
  1426
apply (erule reflection_imp_L_separation)
paulson@13422
  1427
  apply (simp_all add: lt_Ord2)
paulson@13422
  1428
apply (rule DPow_LsetI)
wenzelm@13428
  1429
apply (rename_tac v)
paulson@13422
  1430
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13422
  1431
apply (rule_tac env = "[u,v,A,B,nat]" in mem_iff_sats)
paulson@13434
  1432
apply (rule sep_rules is_nat_case_iff_sats iterates_MH_iff_sats
paulson@13441
  1433
              is_wfrec_iff_sats big_union_iff_sats quasinat_iff_sats | simp)+
paulson@13422
  1434
done
paulson@13422
  1435
paulson@13422
  1436
paulson@13422
  1437
subsubsection{*Instantiating the locale @{text M_eclose}*}
paulson@13422
  1438
paulson@13437
  1439
lemma M_eclose_axioms_L: "M_eclose_axioms(L)"
paulson@13437
  1440
  apply (rule M_eclose_axioms.intro)
paulson@13437
  1441
   apply (assumption | rule eclose_replacement1 eclose_replacement2)+
paulson@13437
  1442
  done
paulson@13437
  1443
wenzelm@13428
  1444
theorem M_eclose_L: "PROP M_eclose(L)"
wenzelm@13428
  1445
  apply (rule M_eclose.intro)
wenzelm@13429
  1446
       apply (rule M_datatypes.axioms [OF M_datatypes_L])+
paulson@13437
  1447
  apply (rule M_eclose_axioms_L)
wenzelm@13428
  1448
  done
paulson@13422
  1449
wenzelm@13428
  1450
lemmas eclose_closed [intro, simp] = M_eclose.eclose_closed [OF M_eclose_L]
wenzelm@13428
  1451
  and eclose_abs [intro, simp] = M_eclose.eclose_abs [OF M_eclose_L]
paulson@13440
  1452
  and transrec_replacementI = M_eclose.transrec_replacementI [OF M_eclose_L]
paulson@13422
  1453
paulson@13348
  1454
end