src/ZF/Constructible/Rec_Separation.thy
author wenzelm
Mon Jul 29 00:57:16 2002 +0200 (2002-07-29)
changeset 13428 99e52e78eb65
parent 13422 af9bc8d87a75
child 13429 2232810416fc
permissions -rw-r--r--
eliminate open locales and special ML code;
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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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theorem 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
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     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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  apply (rule M_trancl.intro)
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    apply (rule M_triv_axioms_L)
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   apply (rule M_axioms_axioms_L)
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  apply (rule M_trancl_axioms_L)
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  done
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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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(* M_is_recfun :: "[i=>o, [i,i,i]=>o, i, i, i] => o"
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   "M_is_recfun(M,MH,r,a,f) ==
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     \<forall>z[M]. z \<in> f <->
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            5      4       3       2       1           0
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            (\<exists>x[M]. \<exists>y[M]. \<exists>xa[M]. \<exists>sx[M]. \<exists>r_sx[M]. \<exists>f_r_sx[M].
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               pair(M,x,y,z) & pair(M,x,a,xa) & upair(M,x,x,sx) &
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               pre_image(M,r,sx,r_sx) & restriction(M,f,r_sx,f_r_sx) &
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               xa \<in> r & MH(x, f_r_sx, y))"
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*)
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constdefs is_recfun_fm :: "[[i,i,i]=>i, i, i, i]=>i"
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 "is_recfun_fm(p,r,a,f) ==
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   Forall(Iff(Member(0,succ(f)),
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    Exists(Exists(Exists(Exists(Exists(Exists(
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     And(pair_fm(5,4,6),
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      And(pair_fm(5,a#+7,3),
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       And(upair_fm(5,5,2),
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        And(pre_image_fm(r#+7,2,1),
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         And(restriction_fm(f#+7,1,0),
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          And(Member(3,r#+7), p(5,0,4)))))))))))))))"
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lemma is_recfun_type_0:
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     "[| !!x y z. [| x \<in> nat; y \<in> nat; z \<in> nat |] ==> p(x,y,z) \<in> formula;
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         x \<in> nat; y \<in> nat; z \<in> nat |]
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      ==> is_recfun_fm(p,x,y,z) \<in> formula"
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apply (unfold is_recfun_fm_def)
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(*FIXME: FIND OUT why simp loops!*)
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apply typecheck
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by simp
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lemma is_recfun_type [TC]:
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     "[| p(5,0,4) \<in> formula;
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         x \<in> nat; y \<in> nat; z \<in> nat |]
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      ==> is_recfun_fm(p,x,y,z) \<in> formula"
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by (simp add: is_recfun_fm_def)
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lemma arity_is_recfun_fm [simp]:
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     "[| arity(p(5,0,4)) le 8; x \<in> nat; y \<in> nat; z \<in> nat |]
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      ==> arity(is_recfun_fm(p,x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
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apply (frule lt_nat_in_nat, simp)
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apply (simp add: is_recfun_fm_def succ_Un_distrib [symmetric] )
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apply (subst subset_Un_iff2 [of "arity(p(5,0,4))", THEN iffD1])
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apply (rule le_imp_subset)
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apply (erule le_trans, simp)
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apply (simp add: succ_Un_distrib [symmetric] Un_ac)
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done
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lemma sats_is_recfun_fm:
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  assumes MH_iff_sats:
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      "!!x y z env.
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         [| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
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         ==> MH(nth(x,env), nth(y,env), nth(z,env)) <-> sats(A, p(x,y,z), env)"
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  shows
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      "[|x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
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       ==> sats(A, is_recfun_fm(p,x,y,z), env) <->
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           M_is_recfun(**A, MH, nth(x,env), nth(y,env), nth(z,env))"
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by (simp add: is_recfun_fm_def M_is_recfun_def MH_iff_sats [THEN iff_sym])
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   294
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lemma is_recfun_iff_sats:
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  "[| (!!x y z env. [| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
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                    ==> MH(nth(x,env), nth(y,env), nth(z,env)) <->
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                        sats(A, p(x,y,z), env));
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      nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
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      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
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   ==> M_is_recfun(**A, MH, x, y, z) <-> sats(A, is_recfun_fm(p,i,j,k), env)"
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by (simp add: sats_is_recfun_fm [of A MH])
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   303
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theorem is_recfun_reflection:
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  assumes MH_reflection:
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    "!!f g h. REFLECTS[\<lambda>x. MH(L, f(x), g(x), h(x)),
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                     \<lambda>i x. MH(**Lset(i), f(x), g(x), h(x))]"
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  shows "REFLECTS[\<lambda>x. M_is_recfun(L, MH(L), f(x), g(x), h(x)),
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   309
               \<lambda>i x. M_is_recfun(**Lset(i), MH(**Lset(i)), f(x), g(x), h(x))]"
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   310
apply (simp (no_asm_use) only: M_is_recfun_def setclass_simps)
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apply (intro FOL_reflections function_reflections
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             restriction_reflection MH_reflection)
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done
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   314
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   315
text{*Currently, @{text sats}-theorems for higher-order operators don't seem
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useful.  Reflection theorems do work, though.  This one avoids the repetition
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of the @{text MH}-term.*}
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theorem is_wfrec_reflection:
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  assumes MH_reflection:
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    "!!f g h. REFLECTS[\<lambda>x. MH(L, f(x), g(x), h(x)),
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                     \<lambda>i x. MH(**Lset(i), f(x), g(x), h(x))]"
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  shows "REFLECTS[\<lambda>x. is_wfrec(L, MH(L), f(x), g(x), h(x)),
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               \<lambda>i x. is_wfrec(**Lset(i), MH(**Lset(i)), f(x), g(x), h(x))]"
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apply (simp (no_asm_use) only: is_wfrec_def setclass_simps)
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   325
apply (intro FOL_reflections MH_reflection is_recfun_reflection)
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done
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   327
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   328
subsection{*The Locale @{text "M_wfrank"}*}
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   329
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   330
subsubsection{*Separation for @{term "wfrank"}*}
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   331
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lemma wfrank_Reflects:
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 "REFLECTS[\<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
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              ~ (\<exists>f[L]. M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f)),
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      \<lambda>i x. \<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) -->
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         ~ (\<exists>f \<in> Lset(i).
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            M_is_recfun(**Lset(i), %x f y. is_range(**Lset(i),f,y),
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                        rplus, x, f))]"
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by (intro FOL_reflections function_reflections is_recfun_reflection tran_closure_reflection)
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   340
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   341
lemma wfrank_separation:
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   342
     "L(r) ==>
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   343
      separation (L, \<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
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         ~ (\<exists>f[L]. M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f)))"
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apply (rule separation_CollectI)
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   346
apply (rule_tac A="{r,z}" in subset_LsetE, blast )
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apply (rule ReflectsE [OF wfrank_Reflects], assumption)
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   348
apply (drule subset_Lset_ltD, assumption)
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   349
apply (erule reflection_imp_L_separation)
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   350
  apply (simp_all add: lt_Ord2, clarify)
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   351
apply (rule DPow_LsetI)
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   352
apply (rename_tac u)
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   353
apply (rule ball_iff_sats imp_iff_sats)+
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apply (rule_tac env="[rplus,u,r]" in tran_closure_iff_sats)
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   355
apply (rule sep_rules is_recfun_iff_sats | simp)+
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   356
done
paulson@13348
   357
paulson@13348
   358
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   359
subsubsection{*Replacement for @{term "wfrank"}*}
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   360
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   361
lemma wfrank_replacement_Reflects:
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   362
 "REFLECTS[\<lambda>z. \<exists>x[L]. x \<in> A &
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   363
        (\<forall>rplus[L]. tran_closure(L,r,rplus) -->
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   364
         (\<exists>y[L]. \<exists>f[L]. pair(L,x,y,z)  &
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   365
                        M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f) &
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   366
                        is_range(L,f,y))),
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   367
 \<lambda>i z. \<exists>x \<in> Lset(i). x \<in> A &
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   368
      (\<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) -->
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   369
       (\<exists>y \<in> Lset(i). \<exists>f \<in> Lset(i). pair(**Lset(i),x,y,z)  &
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   370
         M_is_recfun(**Lset(i), %x f y. is_range(**Lset(i),f,y), rplus, x, f) &
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   371
         is_range(**Lset(i),f,y)))]"
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   372
by (intro FOL_reflections function_reflections fun_plus_reflections
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   373
             is_recfun_reflection tran_closure_reflection)
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   374
paulson@13348
   375
paulson@13348
   376
lemma wfrank_strong_replacement:
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     "L(r) ==>
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   378
      strong_replacement(L, \<lambda>x z.
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   379
         \<forall>rplus[L]. tran_closure(L,r,rplus) -->
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   380
         (\<exists>y[L]. \<exists>f[L]. pair(L,x,y,z)  &
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   381
                        M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f) &
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   382
                        is_range(L,f,y)))"
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   383
apply (rule strong_replacementI)
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   384
apply (rule rallI)
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   385
apply (rename_tac B)
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   386
apply (rule separation_CollectI)
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   387
apply (rule_tac A="{B,r,z}" in subset_LsetE, blast )
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   388
apply (rule ReflectsE [OF wfrank_replacement_Reflects], assumption)
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   389
apply (drule subset_Lset_ltD, assumption)
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   390
apply (erule reflection_imp_L_separation)
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   391
  apply (simp_all add: lt_Ord2)
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   392
apply (rule DPow_LsetI)
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   393
apply (rename_tac u)
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   394
apply (rule bex_iff_sats ball_iff_sats conj_iff_sats)+
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   395
apply (rule_tac env = "[x,u,B,r]" in mem_iff_sats)
paulson@13348
   396
apply (rule sep_rules tran_closure_iff_sats is_recfun_iff_sats | simp)+
paulson@13348
   397
done
paulson@13348
   398
paulson@13348
   399
paulson@13363
   400
subsubsection{*Separation for Proving @{text Ord_wfrank_range}*}
paulson@13348
   401
paulson@13348
   402
lemma Ord_wfrank_Reflects:
wenzelm@13428
   403
 "REFLECTS[\<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
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   404
          ~ (\<forall>f[L]. \<forall>rangef[L].
paulson@13348
   405
             is_range(L,f,rangef) -->
paulson@13352
   406
             M_is_recfun(L, \<lambda>x f y. is_range(L,f,y), rplus, x, f) -->
paulson@13348
   407
             ordinal(L,rangef)),
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   408
      \<lambda>i x. \<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) -->
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   409
          ~ (\<forall>f \<in> Lset(i). \<forall>rangef \<in> Lset(i).
paulson@13348
   410
             is_range(**Lset(i),f,rangef) -->
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   411
             M_is_recfun(**Lset(i), \<lambda>x f y. is_range(**Lset(i),f,y),
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   412
                         rplus, x, f) -->
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   413
             ordinal(**Lset(i),rangef))]"
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   414
by (intro FOL_reflections function_reflections is_recfun_reflection
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   415
          tran_closure_reflection ordinal_reflection)
paulson@13348
   416
paulson@13348
   417
lemma  Ord_wfrank_separation:
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   418
     "L(r) ==>
paulson@13348
   419
      separation (L, \<lambda>x.
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   420
         \<forall>rplus[L]. tran_closure(L,r,rplus) -->
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   421
          ~ (\<forall>f[L]. \<forall>rangef[L].
paulson@13348
   422
             is_range(L,f,rangef) -->
paulson@13352
   423
             M_is_recfun(L, \<lambda>x f y. is_range(L,f,y), rplus, x, f) -->
wenzelm@13428
   424
             ordinal(L,rangef)))"
wenzelm@13428
   425
apply (rule separation_CollectI)
wenzelm@13428
   426
apply (rule_tac A="{r,z}" in subset_LsetE, blast )
paulson@13348
   427
apply (rule ReflectsE [OF Ord_wfrank_Reflects], assumption)
wenzelm@13428
   428
apply (drule subset_Lset_ltD, assumption)
paulson@13348
   429
apply (erule reflection_imp_L_separation)
paulson@13348
   430
  apply (simp_all add: lt_Ord2, clarify)
paulson@13385
   431
apply (rule DPow_LsetI)
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   432
apply (rename_tac u)
paulson@13348
   433
apply (rule ball_iff_sats imp_iff_sats)+
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   434
apply (rule_tac env="[rplus,u,r]" in tran_closure_iff_sats)
paulson@13348
   435
apply (rule sep_rules is_recfun_iff_sats | simp)+
paulson@13348
   436
done
paulson@13348
   437
paulson@13348
   438
paulson@13363
   439
subsubsection{*Instantiating the locale @{text M_wfrank}*}
wenzelm@13428
   440
wenzelm@13428
   441
theorem M_wfrank_axioms_L: "M_wfrank_axioms(L)"
wenzelm@13428
   442
  apply (rule M_wfrank_axioms.intro)
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   443
  apply (assumption | rule
wenzelm@13428
   444
    wfrank_separation wfrank_strong_replacement Ord_wfrank_separation)+
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   445
  done
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   446
wenzelm@13428
   447
theorem M_wfrank_L: "PROP M_wfrank(L)"
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   448
  apply (rule M_wfrank.intro)
wenzelm@13428
   449
     apply (rule M_triv_axioms_L)
wenzelm@13428
   450
    apply (rule M_axioms_axioms_L)
wenzelm@13428
   451
   apply (rule M_trancl_axioms_L)
wenzelm@13428
   452
  apply (rule M_wfrank_axioms_L)
wenzelm@13428
   453
  done
paulson@13363
   454
wenzelm@13428
   455
lemmas iterates_closed = M_wfrank.iterates_closed [OF M_wfrank_L]
wenzelm@13428
   456
  and exists_wfrank = M_wfrank.exists_wfrank [OF M_wfrank_L]
wenzelm@13428
   457
  and M_wellfoundedrank = M_wfrank.M_wellfoundedrank [OF M_wfrank_L]
wenzelm@13428
   458
  and Ord_wfrank_range = M_wfrank.Ord_wfrank_range [OF M_wfrank_L]
wenzelm@13428
   459
  and Ord_range_wellfoundedrank = M_wfrank.Ord_range_wellfoundedrank [OF M_wfrank_L]
wenzelm@13428
   460
  and function_wellfoundedrank = M_wfrank.function_wellfoundedrank [OF M_wfrank_L]
wenzelm@13428
   461
  and domain_wellfoundedrank = M_wfrank.domain_wellfoundedrank [OF M_wfrank_L]
wenzelm@13428
   462
  and wellfoundedrank_type = M_wfrank.wellfoundedrank_type [OF M_wfrank_L]
wenzelm@13428
   463
  and Ord_wellfoundedrank = M_wfrank.Ord_wellfoundedrank [OF M_wfrank_L]
wenzelm@13428
   464
  and wellfoundedrank_eq = M_wfrank.wellfoundedrank_eq [OF M_wfrank_L]
wenzelm@13428
   465
  and wellfoundedrank_lt = M_wfrank.wellfoundedrank_lt [OF M_wfrank_L]
wenzelm@13428
   466
  and wellfounded_imp_subset_rvimage = M_wfrank.wellfounded_imp_subset_rvimage [OF M_wfrank_L]
wenzelm@13428
   467
  and wellfounded_imp_wf = M_wfrank.wellfounded_imp_wf [OF M_wfrank_L]
wenzelm@13428
   468
  and wellfounded_on_imp_wf_on = M_wfrank.wellfounded_on_imp_wf_on [OF M_wfrank_L]
wenzelm@13428
   469
  and wf_abs = M_wfrank.wf_abs [OF M_wfrank_L]
wenzelm@13428
   470
  and wf_on_abs = M_wfrank.wf_on_abs [OF M_wfrank_L]
wenzelm@13428
   471
  and wfrec_replacement_iff = M_wfrank.wfrec_replacement_iff [OF M_wfrank_L]
wenzelm@13428
   472
  and trans_wfrec_closed = M_wfrank.trans_wfrec_closed [OF M_wfrank_L]
wenzelm@13428
   473
  and wfrec_closed = M_wfrank.wfrec_closed [OF M_wfrank_L]
paulson@13363
   474
paulson@13363
   475
declare iterates_closed [intro,simp]
paulson@13363
   476
declare Ord_wfrank_range [rule_format]
paulson@13363
   477
declare wf_abs [simp]
paulson@13363
   478
declare wf_on_abs [simp]
paulson@13363
   479
paulson@13363
   480
paulson@13363
   481
subsection{*For Datatypes*}
paulson@13363
   482
paulson@13363
   483
subsubsection{*Binary Products, Internalized*}
paulson@13363
   484
paulson@13363
   485
constdefs cartprod_fm :: "[i,i,i]=>i"
wenzelm@13428
   486
(* "cartprod(M,A,B,z) ==
wenzelm@13428
   487
        \<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
   488
    "cartprod_fm(A,B,z) ==
paulson@13363
   489
       Forall(Iff(Member(0,succ(z)),
paulson@13363
   490
                  Exists(And(Member(0,succ(succ(A))),
paulson@13363
   491
                         Exists(And(Member(0,succ(succ(succ(B)))),
paulson@13363
   492
                                    pair_fm(1,0,2)))))))"
paulson@13363
   493
paulson@13363
   494
lemma cartprod_type [TC]:
paulson@13363
   495
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> cartprod_fm(x,y,z) \<in> formula"
wenzelm@13428
   496
by (simp add: cartprod_fm_def)
paulson@13363
   497
paulson@13363
   498
lemma arity_cartprod_fm [simp]:
wenzelm@13428
   499
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
paulson@13363
   500
      ==> arity(cartprod_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
wenzelm@13428
   501
by (simp add: cartprod_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13363
   502
paulson@13363
   503
lemma sats_cartprod_fm [simp]:
paulson@13363
   504
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
wenzelm@13428
   505
    ==> sats(A, cartprod_fm(x,y,z), env) <->
paulson@13363
   506
        cartprod(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13363
   507
by (simp add: cartprod_fm_def cartprod_def)
paulson@13363
   508
paulson@13363
   509
lemma cartprod_iff_sats:
wenzelm@13428
   510
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
paulson@13363
   511
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13363
   512
       ==> cartprod(**A, x, y, z) <-> sats(A, cartprod_fm(i,j,k), env)"
paulson@13363
   513
by (simp add: sats_cartprod_fm)
paulson@13363
   514
paulson@13363
   515
theorem cartprod_reflection:
wenzelm@13428
   516
     "REFLECTS[\<lambda>x. cartprod(L,f(x),g(x),h(x)),
paulson@13363
   517
               \<lambda>i x. cartprod(**Lset(i),f(x),g(x),h(x))]"
paulson@13363
   518
apply (simp only: cartprod_def setclass_simps)
wenzelm@13428
   519
apply (intro FOL_reflections pair_reflection)
paulson@13363
   520
done
paulson@13363
   521
paulson@13363
   522
paulson@13363
   523
subsubsection{*Binary Sums, Internalized*}
paulson@13363
   524
wenzelm@13428
   525
(* "is_sum(M,A,B,Z) ==
wenzelm@13428
   526
       \<exists>A0[M]. \<exists>n1[M]. \<exists>s1[M]. \<exists>B1[M].
paulson@13363
   527
         3      2       1        0
paulson@13363
   528
       number1(M,n1) & cartprod(M,n1,A,A0) & upair(M,n1,n1,s1) &
paulson@13363
   529
       cartprod(M,s1,B,B1) & union(M,A0,B1,Z)"  *)
paulson@13363
   530
constdefs sum_fm :: "[i,i,i]=>i"
wenzelm@13428
   531
    "sum_fm(A,B,Z) ==
paulson@13363
   532
       Exists(Exists(Exists(Exists(
wenzelm@13428
   533
        And(number1_fm(2),
paulson@13363
   534
            And(cartprod_fm(2,A#+4,3),
paulson@13363
   535
                And(upair_fm(2,2,1),
paulson@13363
   536
                    And(cartprod_fm(1,B#+4,0), union_fm(3,0,Z#+4)))))))))"
paulson@13363
   537
paulson@13363
   538
lemma sum_type [TC]:
paulson@13363
   539
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> sum_fm(x,y,z) \<in> formula"
wenzelm@13428
   540
by (simp add: sum_fm_def)
paulson@13363
   541
paulson@13363
   542
lemma arity_sum_fm [simp]:
wenzelm@13428
   543
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
paulson@13363
   544
      ==> arity(sum_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
wenzelm@13428
   545
by (simp add: sum_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13363
   546
paulson@13363
   547
lemma sats_sum_fm [simp]:
paulson@13363
   548
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
wenzelm@13428
   549
    ==> sats(A, sum_fm(x,y,z), env) <->
paulson@13363
   550
        is_sum(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13363
   551
by (simp add: sum_fm_def is_sum_def)
paulson@13363
   552
paulson@13363
   553
lemma sum_iff_sats:
wenzelm@13428
   554
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
paulson@13363
   555
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13363
   556
       ==> is_sum(**A, x, y, z) <-> sats(A, sum_fm(i,j,k), env)"
paulson@13363
   557
by simp
paulson@13363
   558
paulson@13363
   559
theorem sum_reflection:
wenzelm@13428
   560
     "REFLECTS[\<lambda>x. is_sum(L,f(x),g(x),h(x)),
paulson@13363
   561
               \<lambda>i x. is_sum(**Lset(i),f(x),g(x),h(x))]"
paulson@13363
   562
apply (simp only: is_sum_def setclass_simps)
wenzelm@13428
   563
apply (intro FOL_reflections function_reflections cartprod_reflection)
paulson@13363
   564
done
paulson@13363
   565
paulson@13363
   566
paulson@13363
   567
subsubsection{*The Operator @{term quasinat}*}
paulson@13363
   568
paulson@13363
   569
(* "is_quasinat(M,z) == empty(M,z) | (\<exists>m[M]. successor(M,m,z))" *)
paulson@13363
   570
constdefs quasinat_fm :: "i=>i"
paulson@13363
   571
    "quasinat_fm(z) == Or(empty_fm(z), Exists(succ_fm(0,succ(z))))"
paulson@13363
   572
paulson@13363
   573
lemma quasinat_type [TC]:
paulson@13363
   574
     "x \<in> nat ==> quasinat_fm(x) \<in> formula"
wenzelm@13428
   575
by (simp add: quasinat_fm_def)
paulson@13363
   576
paulson@13363
   577
lemma arity_quasinat_fm [simp]:
paulson@13363
   578
     "x \<in> nat ==> arity(quasinat_fm(x)) = succ(x)"
wenzelm@13428
   579
by (simp add: quasinat_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13363
   580
paulson@13363
   581
lemma sats_quasinat_fm [simp]:
paulson@13363
   582
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13363
   583
    ==> sats(A, quasinat_fm(x), env) <-> is_quasinat(**A, nth(x,env))"
paulson@13363
   584
by (simp add: quasinat_fm_def is_quasinat_def)
paulson@13363
   585
paulson@13363
   586
lemma quasinat_iff_sats:
wenzelm@13428
   587
      "[| nth(i,env) = x; nth(j,env) = y;
paulson@13363
   588
          i \<in> nat; env \<in> list(A)|]
paulson@13363
   589
       ==> is_quasinat(**A, x) <-> sats(A, quasinat_fm(i), env)"
paulson@13363
   590
by simp
paulson@13363
   591
paulson@13363
   592
theorem quasinat_reflection:
wenzelm@13428
   593
     "REFLECTS[\<lambda>x. is_quasinat(L,f(x)),
paulson@13363
   594
               \<lambda>i x. is_quasinat(**Lset(i),f(x))]"
paulson@13363
   595
apply (simp only: is_quasinat_def setclass_simps)
wenzelm@13428
   596
apply (intro FOL_reflections function_reflections)
paulson@13363
   597
done
paulson@13363
   598
paulson@13363
   599
paulson@13363
   600
subsubsection{*The Operator @{term is_nat_case}*}
paulson@13363
   601
paulson@13363
   602
(* is_nat_case :: "[i=>o, i, [i,i]=>o, i, i] => o"
wenzelm@13428
   603
    "is_nat_case(M, a, is_b, k, z) ==
paulson@13363
   604
       (empty(M,k) --> z=a) &
paulson@13363
   605
       (\<forall>m[M]. successor(M,m,k) --> is_b(m,z)) &
paulson@13363
   606
       (is_quasinat(M,k) | empty(M,z))" *)
paulson@13363
   607
text{*The formula @{term is_b} has free variables 1 and 0.*}
paulson@13363
   608
constdefs is_nat_case_fm :: "[i, [i,i]=>i, i, i]=>i"
wenzelm@13428
   609
 "is_nat_case_fm(a,is_b,k,z) ==
paulson@13363
   610
    And(Implies(empty_fm(k), Equal(z,a)),
wenzelm@13428
   611
        And(Forall(Implies(succ_fm(0,succ(k)),
paulson@13363
   612
                   Forall(Implies(Equal(0,succ(succ(z))), is_b(1,0))))),
paulson@13363
   613
            Or(quasinat_fm(k), empty_fm(z))))"
paulson@13363
   614
paulson@13363
   615
lemma is_nat_case_type [TC]:
wenzelm@13428
   616
     "[| is_b(1,0) \<in> formula;
wenzelm@13428
   617
         x \<in> nat; y \<in> nat; z \<in> nat |]
paulson@13363
   618
      ==> is_nat_case_fm(x,is_b,y,z) \<in> formula"
wenzelm@13428
   619
by (simp add: is_nat_case_fm_def)
paulson@13363
   620
paulson@13363
   621
lemma arity_is_nat_case_fm [simp]:
wenzelm@13428
   622
     "[| is_b(1,0) \<in> formula; x \<in> nat; y \<in> nat; z \<in> nat |]
wenzelm@13428
   623
      ==> arity(is_nat_case_fm(x,is_b,y,z)) =
wenzelm@13428
   624
          succ(x) \<union> succ(y) \<union> succ(z) \<union> (arity(is_b(1,0)) #- 2)"
wenzelm@13428
   625
apply (subgoal_tac "arity(is_b(1,0)) \<in> nat")
paulson@13363
   626
apply typecheck
paulson@13363
   627
(*FIXME: could nat_diff_split work?*)
paulson@13363
   628
apply (auto simp add: diff_def raw_diff_succ is_nat_case_fm_def nat_imp_quasinat
paulson@13363
   629
                 succ_Un_distrib [symmetric] Un_ac
wenzelm@13428
   630
                 split: split_nat_case)
paulson@13363
   631
done
paulson@13363
   632
paulson@13363
   633
lemma sats_is_nat_case_fm:
wenzelm@13428
   634
  assumes is_b_iff_sats:
wenzelm@13428
   635
      "!!a b. [| a \<in> A; b \<in> A|]
paulson@13363
   636
              ==> is_b(a,b) <-> sats(A, p(1,0), Cons(b, Cons(a,env)))"
wenzelm@13428
   637
  shows
paulson@13363
   638
      "[|x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A)|]
wenzelm@13428
   639
       ==> sats(A, is_nat_case_fm(x,p,y,z), env) <->
paulson@13363
   640
           is_nat_case(**A, nth(x,env), is_b, nth(y,env), nth(z,env))"
wenzelm@13428
   641
apply (frule lt_length_in_nat, assumption)
paulson@13363
   642
apply (simp add: is_nat_case_fm_def is_nat_case_def is_b_iff_sats [THEN iff_sym])
paulson@13363
   643
done
paulson@13363
   644
paulson@13363
   645
lemma is_nat_case_iff_sats:
wenzelm@13428
   646
  "[| (!!a b. [| a \<in> A; b \<in> A|]
paulson@13363
   647
              ==> is_b(a,b) <-> sats(A, p(1,0), Cons(b, Cons(a,env))));
wenzelm@13428
   648
      nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
paulson@13363
   649
      i \<in> nat; j \<in> nat; k < length(env); env \<in> list(A)|]
wenzelm@13428
   650
   ==> is_nat_case(**A, x, is_b, y, z) <-> sats(A, is_nat_case_fm(i,p,j,k), env)"
paulson@13363
   651
by (simp add: sats_is_nat_case_fm [of A is_b])
paulson@13363
   652
paulson@13363
   653
paulson@13363
   654
text{*The second argument of @{term is_b} gives it direct access to @{term x},
wenzelm@13428
   655
  which is essential for handling free variable references.  Without this
paulson@13363
   656
  argument, we cannot prove reflection for @{term iterates_MH}.*}
paulson@13363
   657
theorem is_nat_case_reflection:
paulson@13363
   658
  assumes is_b_reflection:
wenzelm@13428
   659
    "!!h f g. REFLECTS[\<lambda>x. is_b(L, h(x), f(x), g(x)),
paulson@13363
   660
                     \<lambda>i x. is_b(**Lset(i), h(x), f(x), g(x))]"
wenzelm@13428
   661
  shows "REFLECTS[\<lambda>x. is_nat_case(L, f(x), is_b(L,x), g(x), h(x)),
paulson@13363
   662
               \<lambda>i x. is_nat_case(**Lset(i), f(x), is_b(**Lset(i), x), g(x), h(x))]"
paulson@13363
   663
apply (simp (no_asm_use) only: is_nat_case_def setclass_simps)
wenzelm@13428
   664
apply (intro FOL_reflections function_reflections
wenzelm@13428
   665
             restriction_reflection is_b_reflection quasinat_reflection)
paulson@13363
   666
done
paulson@13363
   667
paulson@13363
   668
paulson@13363
   669
paulson@13363
   670
subsection{*The Operator @{term iterates_MH}, Needed for Iteration*}
paulson@13363
   671
paulson@13363
   672
(*  iterates_MH :: "[i=>o, [i,i]=>o, i, i, i, i] => o"
paulson@13363
   673
   "iterates_MH(M,isF,v,n,g,z) ==
paulson@13363
   674
        is_nat_case(M, v, \<lambda>m u. \<exists>gm[M]. fun_apply(M,g,m,gm) & isF(gm,u),
paulson@13363
   675
                    n, z)" *)
paulson@13363
   676
constdefs iterates_MH_fm :: "[[i,i]=>i, i, i, i, i]=>i"
wenzelm@13428
   677
 "iterates_MH_fm(isF,v,n,g,z) ==
wenzelm@13428
   678
    is_nat_case_fm(v,
wenzelm@13428
   679
      \<lambda>m u. Exists(And(fun_apply_fm(succ(succ(succ(g))),succ(m),0),
wenzelm@13428
   680
                     Forall(Implies(Equal(0,succ(succ(u))), isF(1,0))))),
paulson@13363
   681
      n, z)"
paulson@13363
   682
paulson@13363
   683
lemma iterates_MH_type [TC]:
wenzelm@13428
   684
     "[| p(1,0) \<in> formula;
wenzelm@13428
   685
         v \<in> nat; x \<in> nat; y \<in> nat; z \<in> nat |]
paulson@13363
   686
      ==> iterates_MH_fm(p,v,x,y,z) \<in> formula"
wenzelm@13428
   687
by (simp add: iterates_MH_fm_def)
paulson@13363
   688
paulson@13363
   689
paulson@13363
   690
lemma arity_iterates_MH_fm [simp]:
wenzelm@13428
   691
     "[| p(1,0) \<in> formula;
wenzelm@13428
   692
         v \<in> nat; x \<in> nat; y \<in> nat; z \<in> nat |]
wenzelm@13428
   693
      ==> arity(iterates_MH_fm(p,v,x,y,z)) =
paulson@13363
   694
          succ(v) \<union> succ(x) \<union> succ(y) \<union> succ(z) \<union> (arity(p(1,0)) #- 4)"
paulson@13363
   695
apply (subgoal_tac "arity(p(1,0)) \<in> nat")
paulson@13363
   696
apply typecheck
paulson@13363
   697
apply (simp add: nat_imp_quasinat iterates_MH_fm_def Un_ac
paulson@13363
   698
            split: split_nat_case, clarify)
paulson@13363
   699
apply (rename_tac i j)
wenzelm@13428
   700
apply (drule eq_succ_imp_eq_m1, simp)
paulson@13363
   701
apply (drule eq_succ_imp_eq_m1, simp)
paulson@13363
   702
apply (simp add: diff_Un_distrib succ_Un_distrib Un_ac diff_diff_left)
paulson@13363
   703
done
paulson@13363
   704
paulson@13363
   705
lemma sats_iterates_MH_fm:
wenzelm@13428
   706
  assumes is_F_iff_sats:
wenzelm@13428
   707
      "!!a b c d. [| a \<in> A; b \<in> A; c \<in> A; d \<in> A|]
paulson@13363
   708
              ==> is_F(a,b) <->
paulson@13363
   709
                  sats(A, p(1,0), Cons(b, Cons(a, Cons(c, Cons(d,env)))))"
wenzelm@13428
   710
  shows
paulson@13363
   711
      "[|v \<in> nat; x \<in> nat; y \<in> nat; z < length(env); env \<in> list(A)|]
wenzelm@13428
   712
       ==> sats(A, iterates_MH_fm(p,v,x,y,z), env) <->
paulson@13363
   713
           iterates_MH(**A, is_F, nth(v,env), nth(x,env), nth(y,env), nth(z,env))"
wenzelm@13428
   714
by (simp add: iterates_MH_fm_def iterates_MH_def sats_is_nat_case_fm
paulson@13363
   715
              is_F_iff_sats [symmetric])
paulson@13363
   716
paulson@13363
   717
lemma iterates_MH_iff_sats:
wenzelm@13428
   718
  "[| (!!a b c d. [| a \<in> A; b \<in> A; c \<in> A; d \<in> A|]
paulson@13363
   719
              ==> is_F(a,b) <->
paulson@13363
   720
                  sats(A, p(1,0), Cons(b, Cons(a, Cons(c, Cons(d,env))))));
wenzelm@13428
   721
      nth(i',env) = v; nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
paulson@13363
   722
      i' \<in> nat; i \<in> nat; j \<in> nat; k < length(env); env \<in> list(A)|]
wenzelm@13428
   723
   ==> iterates_MH(**A, is_F, v, x, y, z) <->
paulson@13363
   724
       sats(A, iterates_MH_fm(p,i',i,j,k), env)"
wenzelm@13428
   725
apply (rule iff_sym)
wenzelm@13428
   726
apply (rule iff_trans)
wenzelm@13428
   727
apply (rule sats_iterates_MH_fm [of A is_F], blast, simp_all)
paulson@13363
   728
done
paulson@13363
   729
paulson@13363
   730
theorem iterates_MH_reflection:
paulson@13363
   731
  assumes p_reflection:
wenzelm@13428
   732
    "!!f g h. REFLECTS[\<lambda>x. p(L, f(x), g(x)),
paulson@13363
   733
                     \<lambda>i x. p(**Lset(i), f(x), g(x))]"
wenzelm@13428
   734
 shows "REFLECTS[\<lambda>x. iterates_MH(L, p(L), e(x), f(x), g(x), h(x)),
paulson@13363
   735
               \<lambda>i x. iterates_MH(**Lset(i), p(**Lset(i)), e(x), f(x), g(x), h(x))]"
paulson@13363
   736
apply (simp (no_asm_use) only: iterates_MH_def)
paulson@13363
   737
txt{*Must be careful: simplifying with @{text setclass_simps} above would
paulson@13363
   738
     change @{text "\<exists>gm[**Lset(i)]"} into @{text "\<exists>gm \<in> Lset(i)"}, when
paulson@13363
   739
     it would no longer match rule @{text is_nat_case_reflection}. *}
wenzelm@13428
   740
apply (rule is_nat_case_reflection)
paulson@13363
   741
apply (simp (no_asm_use) only: setclass_simps)
paulson@13363
   742
apply (intro FOL_reflections function_reflections is_nat_case_reflection
wenzelm@13428
   743
             restriction_reflection p_reflection)
paulson@13363
   744
done
paulson@13363
   745
paulson@13363
   746
paulson@13363
   747
wenzelm@13428
   748
subsection{*@{term L} is Closed Under the Operator @{term list}*}
paulson@13363
   749
paulson@13386
   750
subsubsection{*The List Functor, Internalized*}
paulson@13386
   751
paulson@13386
   752
constdefs list_functor_fm :: "[i,i,i]=>i"
wenzelm@13428
   753
(* "is_list_functor(M,A,X,Z) ==
wenzelm@13428
   754
        \<exists>n1[M]. \<exists>AX[M].
paulson@13386
   755
         number1(M,n1) & cartprod(M,A,X,AX) & is_sum(M,n1,AX,Z)" *)
wenzelm@13428
   756
    "list_functor_fm(A,X,Z) ==
paulson@13386
   757
       Exists(Exists(
wenzelm@13428
   758
        And(number1_fm(1),
paulson@13386
   759
            And(cartprod_fm(A#+2,X#+2,0), sum_fm(1,0,Z#+2)))))"
paulson@13386
   760
paulson@13386
   761
lemma list_functor_type [TC]:
paulson@13386
   762
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> list_functor_fm(x,y,z) \<in> formula"
wenzelm@13428
   763
by (simp add: list_functor_fm_def)
paulson@13386
   764
paulson@13386
   765
lemma arity_list_functor_fm [simp]:
wenzelm@13428
   766
     "[| x \<in> nat; y \<in> nat; z \<in> nat |]
paulson@13386
   767
      ==> arity(list_functor_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
wenzelm@13428
   768
by (simp add: list_functor_fm_def succ_Un_distrib [symmetric] Un_ac)
paulson@13386
   769
paulson@13386
   770
lemma sats_list_functor_fm [simp]:
paulson@13386
   771
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
wenzelm@13428
   772
    ==> sats(A, list_functor_fm(x,y,z), env) <->
paulson@13386
   773
        is_list_functor(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13386
   774
by (simp add: list_functor_fm_def is_list_functor_def)
paulson@13386
   775
paulson@13386
   776
lemma list_functor_iff_sats:
wenzelm@13428
   777
  "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
paulson@13386
   778
      i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13386
   779
   ==> is_list_functor(**A, x, y, z) <-> sats(A, list_functor_fm(i,j,k), env)"
paulson@13386
   780
by simp
paulson@13386
   781
paulson@13386
   782
theorem list_functor_reflection:
wenzelm@13428
   783
     "REFLECTS[\<lambda>x. is_list_functor(L,f(x),g(x),h(x)),
paulson@13386
   784
               \<lambda>i x. is_list_functor(**Lset(i),f(x),g(x),h(x))]"
paulson@13386
   785
apply (simp only: is_list_functor_def setclass_simps)
paulson@13386
   786
apply (intro FOL_reflections number1_reflection
wenzelm@13428
   787
             cartprod_reflection sum_reflection)
paulson@13386
   788
done
paulson@13386
   789
paulson@13386
   790
paulson@13386
   791
subsubsection{*Instances of Replacement for Lists*}
paulson@13386
   792
paulson@13363
   793
lemma list_replacement1_Reflects:
paulson@13363
   794
 "REFLECTS
paulson@13363
   795
   [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
paulson@13363
   796
         is_wfrec(L, iterates_MH(L, is_list_functor(L,A), 0), memsn, u, y)),
paulson@13363
   797
    \<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
   798
         is_wfrec(**Lset(i),
wenzelm@13428
   799
                  iterates_MH(**Lset(i),
paulson@13363
   800
                          is_list_functor(**Lset(i), A), 0), memsn, u, y))]"
wenzelm@13428
   801
by (intro FOL_reflections function_reflections is_wfrec_reflection
wenzelm@13428
   802
          iterates_MH_reflection list_functor_reflection)
paulson@13363
   803
wenzelm@13428
   804
lemma list_replacement1:
paulson@13363
   805
   "L(A) ==> iterates_replacement(L, is_list_functor(L,A), 0)"
paulson@13363
   806
apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
wenzelm@13428
   807
apply (rule strong_replacementI)
paulson@13363
   808
apply (rule rallI)
wenzelm@13428
   809
apply (rename_tac B)
wenzelm@13428
   810
apply (rule separation_CollectI)
wenzelm@13428
   811
apply (insert nonempty)
wenzelm@13428
   812
apply (subgoal_tac "L(Memrel(succ(n)))")
wenzelm@13428
   813
apply (rule_tac A="{B,A,n,z,0,Memrel(succ(n))}" in subset_LsetE, blast )
paulson@13363
   814
apply (rule ReflectsE [OF list_replacement1_Reflects], assumption)
wenzelm@13428
   815
apply (drule subset_Lset_ltD, assumption)
paulson@13363
   816
apply (erule reflection_imp_L_separation)
paulson@13386
   817
  apply (simp_all add: lt_Ord2 Memrel_closed)
wenzelm@13428
   818
apply (elim conjE)
paulson@13385
   819
apply (rule DPow_LsetI)
wenzelm@13428
   820
apply (rename_tac v)
paulson@13363
   821
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13363
   822
apply (rule_tac env = "[u,v,A,n,B,0,Memrel(succ(n))]" in mem_iff_sats)
paulson@13363
   823
apply (rule sep_rules | simp)+
paulson@13363
   824
txt{*Can't get sat rules to work for higher-order operators, so just expand them!*}
paulson@13363
   825
apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
paulson@13363
   826
apply (rule sep_rules list_functor_iff_sats quasinat_iff_sats | simp)+
paulson@13363
   827
done
paulson@13363
   828
paulson@13363
   829
paulson@13363
   830
lemma list_replacement2_Reflects:
paulson@13363
   831
 "REFLECTS
paulson@13363
   832
   [\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
paulson@13363
   833
         (\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
paulson@13363
   834
           is_wfrec (L, iterates_MH (L, is_list_functor(L, A), 0),
paulson@13363
   835
                              msn, u, x)),
paulson@13363
   836
    \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
wenzelm@13428
   837
         (\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i).
paulson@13363
   838
          successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
wenzelm@13428
   839
           is_wfrec (**Lset(i),
paulson@13363
   840
                 iterates_MH (**Lset(i), is_list_functor(**Lset(i), A), 0),
paulson@13363
   841
                     msn, u, x))]"
wenzelm@13428
   842
by (intro FOL_reflections function_reflections is_wfrec_reflection
wenzelm@13428
   843
          iterates_MH_reflection list_functor_reflection)
paulson@13363
   844
paulson@13363
   845
wenzelm@13428
   846
lemma list_replacement2:
wenzelm@13428
   847
   "L(A) ==> strong_replacement(L,
wenzelm@13428
   848
         \<lambda>n y. n\<in>nat &
paulson@13363
   849
               (\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
wenzelm@13428
   850
               is_wfrec(L, iterates_MH(L,is_list_functor(L,A), 0),
paulson@13363
   851
                        msn, n, y)))"
wenzelm@13428
   852
apply (rule strong_replacementI)
paulson@13363
   853
apply (rule rallI)
wenzelm@13428
   854
apply (rename_tac B)
wenzelm@13428
   855
apply (rule separation_CollectI)
wenzelm@13428
   856
apply (insert nonempty)
wenzelm@13428
   857
apply (rule_tac A="{A,B,z,0,nat}" in subset_LsetE)
wenzelm@13428
   858
apply (blast intro: L_nat)
paulson@13363
   859
apply (rule ReflectsE [OF list_replacement2_Reflects], assumption)
wenzelm@13428
   860
apply (drule subset_Lset_ltD, assumption)
paulson@13363
   861
apply (erule reflection_imp_L_separation)
paulson@13363
   862
  apply (simp_all add: lt_Ord2)
paulson@13385
   863
apply (rule DPow_LsetI)
wenzelm@13428
   864
apply (rename_tac v)
paulson@13363
   865
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13363
   866
apply (rule_tac env = "[u,v,A,B,0,nat]" in mem_iff_sats)
paulson@13363
   867
apply (rule sep_rules | simp)+
paulson@13363
   868
apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
paulson@13363
   869
apply (rule sep_rules list_functor_iff_sats quasinat_iff_sats | simp)+
paulson@13363
   870
done
paulson@13363
   871
paulson@13386
   872
wenzelm@13428
   873
subsection{*@{term L} is Closed Under the Operator @{term formula}*}
paulson@13386
   874
paulson@13386
   875
subsubsection{*The Formula Functor, Internalized*}
paulson@13386
   876
paulson@13386
   877
constdefs formula_functor_fm :: "[i,i]=>i"
wenzelm@13428
   878
(*     "is_formula_functor(M,X,Z) ==
wenzelm@13428
   879
        \<exists>nat'[M]. \<exists>natnat[M]. \<exists>natnatsum[M]. \<exists>XX[M]. \<exists>X3[M].
paulson@13398
   880
           4           3               2       1       0
wenzelm@13428
   881
          omega(M,nat') & cartprod(M,nat',nat',natnat) &
paulson@13386
   882
          is_sum(M,natnat,natnat,natnatsum) &
wenzelm@13428
   883
          cartprod(M,X,X,XX) & is_sum(M,XX,X,X3) &
wenzelm@13428
   884
          is_sum(M,natnatsum,X3,Z)" *)
wenzelm@13428
   885
    "formula_functor_fm(X,Z) ==
paulson@13398
   886
       Exists(Exists(Exists(Exists(Exists(
wenzelm@13428
   887
        And(omega_fm(4),
paulson@13398
   888
         And(cartprod_fm(4,4,3),
paulson@13398
   889
          And(sum_fm(3,3,2),
paulson@13398
   890
           And(cartprod_fm(X#+5,X#+5,1),
paulson@13398
   891
            And(sum_fm(1,X#+5,0), sum_fm(2,0,Z#+5)))))))))))"
paulson@13386
   892
paulson@13386
   893
lemma formula_functor_type [TC]:
paulson@13386
   894
     "[| x \<in> nat; y \<in> nat |] ==> formula_functor_fm(x,y) \<in> formula"
wenzelm@13428
   895
by (simp add: formula_functor_fm_def)
paulson@13386
   896
paulson@13386
   897
lemma sats_formula_functor_fm [simp]:
paulson@13386
   898
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
wenzelm@13428
   899
    ==> sats(A, formula_functor_fm(x,y), env) <->
paulson@13386
   900
        is_formula_functor(**A, nth(x,env), nth(y,env))"
paulson@13386
   901
by (simp add: formula_functor_fm_def is_formula_functor_def)
paulson@13386
   902
paulson@13386
   903
lemma formula_functor_iff_sats:
wenzelm@13428
   904
  "[| nth(i,env) = x; nth(j,env) = y;
paulson@13386
   905
      i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13386
   906
   ==> is_formula_functor(**A, x, y) <-> sats(A, formula_functor_fm(i,j), env)"
paulson@13386
   907
by simp
paulson@13386
   908
paulson@13386
   909
theorem formula_functor_reflection:
wenzelm@13428
   910
     "REFLECTS[\<lambda>x. is_formula_functor(L,f(x),g(x)),
paulson@13386
   911
               \<lambda>i x. is_formula_functor(**Lset(i),f(x),g(x))]"
paulson@13386
   912
apply (simp only: is_formula_functor_def setclass_simps)
paulson@13386
   913
apply (intro FOL_reflections omega_reflection
wenzelm@13428
   914
             cartprod_reflection sum_reflection)
paulson@13386
   915
done
paulson@13386
   916
paulson@13386
   917
subsubsection{*Instances of Replacement for Formulas*}
paulson@13386
   918
paulson@13386
   919
lemma formula_replacement1_Reflects:
paulson@13386
   920
 "REFLECTS
paulson@13386
   921
   [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
paulson@13386
   922
         is_wfrec(L, iterates_MH(L, is_formula_functor(L), 0), memsn, u, y)),
paulson@13386
   923
    \<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
   924
         is_wfrec(**Lset(i),
wenzelm@13428
   925
                  iterates_MH(**Lset(i),
paulson@13386
   926
                          is_formula_functor(**Lset(i)), 0), memsn, u, y))]"
wenzelm@13428
   927
by (intro FOL_reflections function_reflections is_wfrec_reflection
wenzelm@13428
   928
          iterates_MH_reflection formula_functor_reflection)
paulson@13386
   929
wenzelm@13428
   930
lemma formula_replacement1:
paulson@13386
   931
   "iterates_replacement(L, is_formula_functor(L), 0)"
paulson@13386
   932
apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
wenzelm@13428
   933
apply (rule strong_replacementI)
paulson@13386
   934
apply (rule rallI)
wenzelm@13428
   935
apply (rename_tac B)
wenzelm@13428
   936
apply (rule separation_CollectI)
wenzelm@13428
   937
apply (insert nonempty)
wenzelm@13428
   938
apply (subgoal_tac "L(Memrel(succ(n)))")
wenzelm@13428
   939
apply (rule_tac A="{B,n,z,0,Memrel(succ(n))}" in subset_LsetE, blast )
paulson@13386
   940
apply (rule ReflectsE [OF formula_replacement1_Reflects], assumption)
wenzelm@13428
   941
apply (drule subset_Lset_ltD, assumption)
paulson@13386
   942
apply (erule reflection_imp_L_separation)
paulson@13386
   943
  apply (simp_all add: lt_Ord2 Memrel_closed)
paulson@13386
   944
apply (rule DPow_LsetI)
wenzelm@13428
   945
apply (rename_tac v)
paulson@13386
   946
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13386
   947
apply (rule_tac env = "[u,v,n,B,0,Memrel(succ(n))]" in mem_iff_sats)
paulson@13386
   948
apply (rule sep_rules | simp)+
paulson@13386
   949
txt{*Can't get sat rules to work for higher-order operators, so just expand them!*}
paulson@13386
   950
apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
paulson@13386
   951
apply (rule sep_rules formula_functor_iff_sats quasinat_iff_sats | simp)+
paulson@13386
   952
txt{*SLOW: like 40 seconds!*}
paulson@13386
   953
done
paulson@13386
   954
paulson@13386
   955
lemma formula_replacement2_Reflects:
paulson@13386
   956
 "REFLECTS
paulson@13386
   957
   [\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
paulson@13386
   958
         (\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
paulson@13386
   959
           is_wfrec (L, iterates_MH (L, is_formula_functor(L), 0),
paulson@13386
   960
                              msn, u, x)),
paulson@13386
   961
    \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
wenzelm@13428
   962
         (\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i).
paulson@13386
   963
          successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
wenzelm@13428
   964
           is_wfrec (**Lset(i),
paulson@13386
   965
                 iterates_MH (**Lset(i), is_formula_functor(**Lset(i)), 0),
paulson@13386
   966
                     msn, u, x))]"
wenzelm@13428
   967
by (intro FOL_reflections function_reflections is_wfrec_reflection
wenzelm@13428
   968
          iterates_MH_reflection formula_functor_reflection)
paulson@13386
   969
paulson@13386
   970
wenzelm@13428
   971
lemma formula_replacement2:
wenzelm@13428
   972
   "strong_replacement(L,
wenzelm@13428
   973
         \<lambda>n y. n\<in>nat &
paulson@13386
   974
               (\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
wenzelm@13428
   975
               is_wfrec(L, iterates_MH(L,is_formula_functor(L), 0),
paulson@13386
   976
                        msn, n, y)))"
wenzelm@13428
   977
apply (rule strong_replacementI)
paulson@13386
   978
apply (rule rallI)
wenzelm@13428
   979
apply (rename_tac B)
wenzelm@13428
   980
apply (rule separation_CollectI)
wenzelm@13428
   981
apply (insert nonempty)
wenzelm@13428
   982
apply (rule_tac A="{B,z,0,nat}" in subset_LsetE)
wenzelm@13428
   983
apply (blast intro: L_nat)
paulson@13386
   984
apply (rule ReflectsE [OF formula_replacement2_Reflects], assumption)
wenzelm@13428
   985
apply (drule subset_Lset_ltD, assumption)
paulson@13386
   986
apply (erule reflection_imp_L_separation)
paulson@13386
   987
  apply (simp_all add: lt_Ord2)
paulson@13386
   988
apply (rule DPow_LsetI)
wenzelm@13428
   989
apply (rename_tac v)
paulson@13386
   990
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13386
   991
apply (rule_tac env = "[u,v,B,0,nat]" in mem_iff_sats)
paulson@13386
   992
apply (rule sep_rules | simp)+
paulson@13386
   993
apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
paulson@13386
   994
apply (rule sep_rules formula_functor_iff_sats quasinat_iff_sats | simp)+
paulson@13386
   995
done
paulson@13386
   996
paulson@13386
   997
text{*NB The proofs for type @{term formula} are virtually identical to those
paulson@13386
   998
for @{term "list(A)"}.  It was a cut-and-paste job! *}
paulson@13386
   999
paulson@13387
  1000
paulson@13409
  1001
subsection{*Internalized Forms of Data Structuring Operators*}
paulson@13409
  1002
paulson@13409
  1003
subsubsection{*The Formula @{term is_Inl}, Internalized*}
paulson@13409
  1004
paulson@13409
  1005
(*  is_Inl(M,a,z) == \<exists>zero[M]. empty(M,zero) & pair(M,zero,a,z) *)
paulson@13409
  1006
constdefs Inl_fm :: "[i,i]=>i"
paulson@13409
  1007
    "Inl_fm(a,z) == Exists(And(empty_fm(0), pair_fm(0,succ(a),succ(z))))"
paulson@13409
  1008
paulson@13409
  1009
lemma Inl_type [TC]:
paulson@13409
  1010
     "[| x \<in> nat; z \<in> nat |] ==> Inl_fm(x,z) \<in> formula"
wenzelm@13428
  1011
by (simp add: Inl_fm_def)
paulson@13409
  1012
paulson@13409
  1013
lemma sats_Inl_fm [simp]:
paulson@13409
  1014
   "[| x \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13409
  1015
    ==> sats(A, Inl_fm(x,z), env) <-> is_Inl(**A, nth(x,env), nth(z,env))"
paulson@13409
  1016
by (simp add: Inl_fm_def is_Inl_def)
paulson@13409
  1017
paulson@13409
  1018
lemma Inl_iff_sats:
wenzelm@13428
  1019
      "[| nth(i,env) = x; nth(k,env) = z;
paulson@13409
  1020
          i \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13409
  1021
       ==> is_Inl(**A, x, z) <-> sats(A, Inl_fm(i,k), env)"
paulson@13409
  1022
by simp
paulson@13409
  1023
paulson@13409
  1024
theorem Inl_reflection:
wenzelm@13428
  1025
     "REFLECTS[\<lambda>x. is_Inl(L,f(x),h(x)),
paulson@13409
  1026
               \<lambda>i x. is_Inl(**Lset(i),f(x),h(x))]"
paulson@13409
  1027
apply (simp only: is_Inl_def setclass_simps)
wenzelm@13428
  1028
apply (intro FOL_reflections function_reflections)
paulson@13409
  1029
done
paulson@13409
  1030
paulson@13409
  1031
paulson@13409
  1032
subsubsection{*The Formula @{term is_Inr}, Internalized*}
paulson@13409
  1033
paulson@13409
  1034
(*  is_Inr(M,a,z) == \<exists>n1[M]. number1(M,n1) & pair(M,n1,a,z) *)
paulson@13409
  1035
constdefs Inr_fm :: "[i,i]=>i"
paulson@13409
  1036
    "Inr_fm(a,z) == Exists(And(number1_fm(0), pair_fm(0,succ(a),succ(z))))"
paulson@13409
  1037
paulson@13409
  1038
lemma Inr_type [TC]:
paulson@13409
  1039
     "[| x \<in> nat; z \<in> nat |] ==> Inr_fm(x,z) \<in> formula"
wenzelm@13428
  1040
by (simp add: Inr_fm_def)
paulson@13409
  1041
paulson@13409
  1042
lemma sats_Inr_fm [simp]:
paulson@13409
  1043
   "[| x \<in> nat; z \<in> nat; env \<in> list(A)|]
paulson@13409
  1044
    ==> sats(A, Inr_fm(x,z), env) <-> is_Inr(**A, nth(x,env), nth(z,env))"
paulson@13409
  1045
by (simp add: Inr_fm_def is_Inr_def)
paulson@13409
  1046
paulson@13409
  1047
lemma Inr_iff_sats:
wenzelm@13428
  1048
      "[| nth(i,env) = x; nth(k,env) = z;
paulson@13409
  1049
          i \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13409
  1050
       ==> is_Inr(**A, x, z) <-> sats(A, Inr_fm(i,k), env)"
paulson@13409
  1051
by simp
paulson@13409
  1052
paulson@13409
  1053
theorem Inr_reflection:
wenzelm@13428
  1054
     "REFLECTS[\<lambda>x. is_Inr(L,f(x),h(x)),
paulson@13409
  1055
               \<lambda>i x. is_Inr(**Lset(i),f(x),h(x))]"
paulson@13409
  1056
apply (simp only: is_Inr_def setclass_simps)
wenzelm@13428
  1057
apply (intro FOL_reflections function_reflections)
paulson@13409
  1058
done
paulson@13409
  1059
paulson@13409
  1060
paulson@13409
  1061
subsubsection{*The Formula @{term is_Nil}, Internalized*}
paulson@13409
  1062
paulson@13409
  1063
(* is_Nil(M,xs) == \<exists>zero[M]. empty(M,zero) & is_Inl(M,zero,xs) *)
paulson@13409
  1064
paulson@13409
  1065
constdefs Nil_fm :: "i=>i"
paulson@13409
  1066
    "Nil_fm(x) == Exists(And(empty_fm(0), Inl_fm(0,succ(x))))"
wenzelm@13428
  1067
paulson@13409
  1068
lemma Nil_type [TC]: "x \<in> nat ==> Nil_fm(x) \<in> formula"
wenzelm@13428
  1069
by (simp add: Nil_fm_def)
paulson@13409
  1070
paulson@13409
  1071
lemma sats_Nil_fm [simp]:
paulson@13409
  1072
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13409
  1073
    ==> sats(A, Nil_fm(x), env) <-> is_Nil(**A, nth(x,env))"
paulson@13409
  1074
by (simp add: Nil_fm_def is_Nil_def)
paulson@13409
  1075
paulson@13409
  1076
lemma Nil_iff_sats:
paulson@13409
  1077
      "[| nth(i,env) = x; i \<in> nat; env \<in> list(A)|]
paulson@13409
  1078
       ==> is_Nil(**A, x) <-> sats(A, Nil_fm(i), env)"
paulson@13409
  1079
by simp
paulson@13409
  1080
paulson@13409
  1081
theorem Nil_reflection:
wenzelm@13428
  1082
     "REFLECTS[\<lambda>x. is_Nil(L,f(x)),
paulson@13409
  1083
               \<lambda>i x. is_Nil(**Lset(i),f(x))]"
paulson@13409
  1084
apply (simp only: is_Nil_def setclass_simps)
wenzelm@13428
  1085
apply (intro FOL_reflections function_reflections Inl_reflection)
paulson@13409
  1086
done
paulson@13409
  1087
paulson@13409
  1088
paulson@13422
  1089
subsubsection{*The Formula @{term is_Cons}, Internalized*}
paulson@13395
  1090
paulson@13387
  1091
paulson@13409
  1092
(*  "is_Cons(M,a,l,Z) == \<exists>p[M]. pair(M,a,l,p) & is_Inr(M,p,Z)" *)
paulson@13409
  1093
constdefs Cons_fm :: "[i,i,i]=>i"
wenzelm@13428
  1094
    "Cons_fm(a,l,Z) ==
paulson@13409
  1095
       Exists(And(pair_fm(succ(a),succ(l),0), Inr_fm(0,succ(Z))))"
paulson@13409
  1096
paulson@13409
  1097
lemma Cons_type [TC]:
paulson@13409
  1098
     "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> Cons_fm(x,y,z) \<in> formula"
wenzelm@13428
  1099
by (simp add: Cons_fm_def)
paulson@13409
  1100
paulson@13409
  1101
lemma sats_Cons_fm [simp]:
paulson@13409
  1102
   "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
wenzelm@13428
  1103
    ==> sats(A, Cons_fm(x,y,z), env) <->
paulson@13409
  1104
       is_Cons(**A, nth(x,env), nth(y,env), nth(z,env))"
paulson@13409
  1105
by (simp add: Cons_fm_def is_Cons_def)
paulson@13409
  1106
paulson@13409
  1107
lemma Cons_iff_sats:
wenzelm@13428
  1108
      "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
paulson@13409
  1109
          i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
paulson@13409
  1110
       ==>is_Cons(**A, x, y, z) <-> sats(A, Cons_fm(i,j,k), env)"
paulson@13409
  1111
by simp
paulson@13409
  1112
paulson@13409
  1113
theorem Cons_reflection:
wenzelm@13428
  1114
     "REFLECTS[\<lambda>x. is_Cons(L,f(x),g(x),h(x)),
paulson@13409
  1115
               \<lambda>i x. is_Cons(**Lset(i),f(x),g(x),h(x))]"
paulson@13409
  1116
apply (simp only: is_Cons_def setclass_simps)
wenzelm@13428
  1117
apply (intro FOL_reflections pair_reflection Inr_reflection)
paulson@13409
  1118
done
paulson@13409
  1119
paulson@13409
  1120
subsubsection{*The Formula @{term is_quasilist}, Internalized*}
paulson@13409
  1121
paulson@13409
  1122
(* is_quasilist(M,xs) == is_Nil(M,z) | (\<exists>x[M]. \<exists>l[M]. is_Cons(M,x,l,z))" *)
paulson@13409
  1123
paulson@13409
  1124
constdefs quasilist_fm :: "i=>i"
wenzelm@13428
  1125
    "quasilist_fm(x) ==
paulson@13409
  1126
       Or(Nil_fm(x), Exists(Exists(Cons_fm(1,0,succ(succ(x))))))"
wenzelm@13428
  1127
paulson@13409
  1128
lemma quasilist_type [TC]: "x \<in> nat ==> quasilist_fm(x) \<in> formula"
wenzelm@13428
  1129
by (simp add: quasilist_fm_def)
paulson@13409
  1130
paulson@13409
  1131
lemma sats_quasilist_fm [simp]:
paulson@13409
  1132
   "[| x \<in> nat; env \<in> list(A)|]
paulson@13409
  1133
    ==> sats(A, quasilist_fm(x), env) <-> is_quasilist(**A, nth(x,env))"
paulson@13409
  1134
by (simp add: quasilist_fm_def is_quasilist_def)
paulson@13409
  1135
paulson@13409
  1136
lemma quasilist_iff_sats:
paulson@13409
  1137
      "[| nth(i,env) = x; i \<in> nat; env \<in> list(A)|]
paulson@13409
  1138
       ==> is_quasilist(**A, x) <-> sats(A, quasilist_fm(i), env)"
paulson@13409
  1139
by simp
paulson@13409
  1140
paulson@13409
  1141
theorem quasilist_reflection:
wenzelm@13428
  1142
     "REFLECTS[\<lambda>x. is_quasilist(L,f(x)),
paulson@13409
  1143
               \<lambda>i x. is_quasilist(**Lset(i),f(x))]"
paulson@13409
  1144
apply (simp only: is_quasilist_def setclass_simps)
wenzelm@13428
  1145
apply (intro FOL_reflections Nil_reflection Cons_reflection)
paulson@13409
  1146
done
paulson@13409
  1147
paulson@13409
  1148
paulson@13409
  1149
subsection{*Absoluteness for the Function @{term nth}*}
paulson@13409
  1150
paulson@13409
  1151
paulson@13409
  1152
subsubsection{*The Formula @{term is_tl}, Internalized*}
paulson@13409
  1153
wenzelm@13428
  1154
(*     "is_tl(M,xs,T) ==
paulson@13409
  1155
       (is_Nil(M,xs) --> T=xs) &
paulson@13409
  1156
       (\<forall>x[M]. \<forall>l[M]. ~ is_Cons(M,x,l,xs) | T=l) &
paulson@13409
  1157
       (is_quasilist(M,xs) | empty(M,T))" *)
paulson@13409
  1158
constdefs tl_fm :: "[i,i]=>i"
wenzelm@13428
  1159
    "tl_fm(xs,T) ==
paulson@13409
  1160
       And(Implies(Nil_fm(xs), Equal(T,xs)),
paulson@13409
  1161
           And(Forall(Forall(Or(Neg(Cons_fm(1,0,xs#+2)), Equal(T#+2,0)))),
paulson@13409
  1162
               Or(quasilist_fm(xs), empty_fm(T))))"
paulson@13409
  1163
paulson@13409
  1164
lemma tl_type [TC]:
paulson@13409
  1165
     "[| x \<in> nat; y \<in> nat |] ==> tl_fm(x,y) \<in> formula"
wenzelm@13428
  1166
by (simp add: tl_fm_def)
paulson@13409
  1167
paulson@13409
  1168
lemma sats_tl_fm [simp]:
paulson@13409
  1169
   "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
paulson@13409
  1170
    ==> sats(A, tl_fm(x,y), env) <-> is_tl(**A, nth(x,env), nth(y,env))"
paulson@13409
  1171
by (simp add: tl_fm_def is_tl_def)
paulson@13409
  1172
paulson@13409
  1173
lemma tl_iff_sats:
paulson@13409
  1174
      "[| nth(i,env) = x; nth(j,env) = y;
paulson@13409
  1175
          i \<in> nat; j \<in> nat; env \<in> list(A)|]
paulson@13409
  1176
       ==> is_tl(**A, x, y) <-> sats(A, tl_fm(i,j), env)"
paulson@13409
  1177
by simp
paulson@13409
  1178
paulson@13409
  1179
theorem tl_reflection:
wenzelm@13428
  1180
     "REFLECTS[\<lambda>x. is_tl(L,f(x),g(x)),
paulson@13409
  1181
               \<lambda>i x. is_tl(**Lset(i),f(x),g(x))]"
paulson@13409
  1182
apply (simp only: is_tl_def setclass_simps)
paulson@13409
  1183
apply (intro FOL_reflections Nil_reflection Cons_reflection
wenzelm@13428
  1184
             quasilist_reflection empty_reflection)
paulson@13409
  1185
done
paulson@13409
  1186
paulson@13409
  1187
paulson@13409
  1188
subsubsection{*An Instance of Replacement for @{term nth}*}
paulson@13409
  1189
paulson@13409
  1190
lemma nth_replacement_Reflects:
paulson@13409
  1191
 "REFLECTS
paulson@13409
  1192
   [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
paulson@13409
  1193
         is_wfrec(L, iterates_MH(L, is_tl(L), z), memsn, u, y)),
paulson@13409
  1194
    \<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
  1195
         is_wfrec(**Lset(i),
wenzelm@13428
  1196
                  iterates_MH(**Lset(i),
paulson@13409
  1197
                          is_tl(**Lset(i)), z), memsn, u, y))]"
wenzelm@13428
  1198
by (intro FOL_reflections function_reflections is_wfrec_reflection
wenzelm@13428
  1199
          iterates_MH_reflection list_functor_reflection tl_reflection)
paulson@13409
  1200
wenzelm@13428
  1201
lemma nth_replacement:
paulson@13409
  1202
   "L(w) ==> iterates_replacement(L, %l t. is_tl(L,l,t), w)"
paulson@13409
  1203
apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
wenzelm@13428
  1204
apply (rule strong_replacementI)
wenzelm@13428
  1205
apply (rule rallI)
wenzelm@13428
  1206
apply (rule separation_CollectI)
wenzelm@13428
  1207
apply (subgoal_tac "L(Memrel(succ(n)))")
wenzelm@13428
  1208
apply (rule_tac A="{A,n,w,z,Memrel(succ(n))}" in subset_LsetE, blast )
paulson@13409
  1209
apply (rule ReflectsE [OF nth_replacement_Reflects], assumption)
wenzelm@13428
  1210
apply (drule subset_Lset_ltD, assumption)
paulson@13409
  1211
apply (erule reflection_imp_L_separation)
paulson@13409
  1212
  apply (simp_all add: lt_Ord2 Memrel_closed)
wenzelm@13428
  1213
apply (elim conjE)
paulson@13409
  1214
apply (rule DPow_LsetI)
wenzelm@13428
  1215
apply (rename_tac v)
paulson@13409
  1216
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13409
  1217
apply (rule_tac env = "[u,v,A,z,w,Memrel(succ(n))]" in mem_iff_sats)
paulson@13409
  1218
apply (rule sep_rules | simp)+
paulson@13409
  1219
apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
paulson@13409
  1220
apply (rule sep_rules quasinat_iff_sats tl_iff_sats | simp)+
paulson@13409
  1221
done
paulson@13409
  1222
paulson@13422
  1223
paulson@13422
  1224
paulson@13422
  1225
subsubsection{*Instantiating the locale @{text M_datatypes}*}
wenzelm@13428
  1226
wenzelm@13428
  1227
theorem M_datatypes_axioms_L: "M_datatypes_axioms(L)"
wenzelm@13428
  1228
  apply (rule M_datatypes_axioms.intro)
wenzelm@13428
  1229
      apply (assumption | rule
wenzelm@13428
  1230
        list_replacement1 list_replacement2
wenzelm@13428
  1231
        formula_replacement1 formula_replacement2
wenzelm@13428
  1232
        nth_replacement)+
wenzelm@13428
  1233
  done
paulson@13422
  1234
wenzelm@13428
  1235
theorem M_datatypes_L: "PROP M_datatypes(L)"
wenzelm@13428
  1236
  apply (rule M_datatypes.intro)
wenzelm@13428
  1237
      apply (rule M_triv_axioms_L)
wenzelm@13428
  1238
     apply (rule M_axioms_axioms_L)
wenzelm@13428
  1239
    apply (rule M_trancl_axioms_L)
wenzelm@13428
  1240
   apply (rule M_wfrank_axioms_L)
wenzelm@13428
  1241
  apply (rule M_datatypes_axioms_L)
wenzelm@13428
  1242
  done
paulson@13422
  1243
wenzelm@13428
  1244
lemmas list_closed = M_datatypes.list_closed [OF M_datatypes_L]
wenzelm@13428
  1245
  and formula_closed = M_datatypes.formula_closed [OF M_datatypes_L]
wenzelm@13428
  1246
  and list_abs = M_datatypes.list_abs [OF M_datatypes_L]
wenzelm@13428
  1247
  and formula_abs = M_datatypes.formula_abs [OF M_datatypes_L]
wenzelm@13428
  1248
  and nth_abs = M_datatypes.nth_abs [OF M_datatypes_L]
paulson@13409
  1249
paulson@13422
  1250
declare list_closed [intro,simp]
paulson@13422
  1251
declare formula_closed [intro,simp]
paulson@13422
  1252
declare list_abs [simp]
paulson@13422
  1253
declare formula_abs [simp]
paulson@13422
  1254
declare nth_abs [simp]
paulson@13422
  1255
paulson@13422
  1256
wenzelm@13428
  1257
subsection{*@{term L} is Closed Under the Operator @{term eclose}*}
paulson@13422
  1258
paulson@13422
  1259
subsubsection{*Instances of Replacement for @{term eclose}*}
paulson@13422
  1260
paulson@13422
  1261
lemma eclose_replacement1_Reflects:
paulson@13422
  1262
 "REFLECTS
paulson@13422
  1263
   [\<lambda>x. \<exists>u[L]. u \<in> B \<and> (\<exists>y[L]. pair(L,u,y,x) \<and>
paulson@13422
  1264
         is_wfrec(L, iterates_MH(L, big_union(L), A), memsn, u, y)),
paulson@13422
  1265
    \<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
  1266
         is_wfrec(**Lset(i),
wenzelm@13428
  1267
                  iterates_MH(**Lset(i), big_union(**Lset(i)), A),
paulson@13422
  1268
                  memsn, u, y))]"
wenzelm@13428
  1269
by (intro FOL_reflections function_reflections is_wfrec_reflection
wenzelm@13428
  1270
          iterates_MH_reflection)
paulson@13422
  1271
wenzelm@13428
  1272
lemma eclose_replacement1:
paulson@13422
  1273
   "L(A) ==> iterates_replacement(L, big_union(L), A)"
paulson@13422
  1274
apply (unfold iterates_replacement_def wfrec_replacement_def, clarify)
wenzelm@13428
  1275
apply (rule strong_replacementI)
paulson@13422
  1276
apply (rule rallI)
wenzelm@13428
  1277
apply (rename_tac B)
wenzelm@13428
  1278
apply (rule separation_CollectI)
wenzelm@13428
  1279
apply (subgoal_tac "L(Memrel(succ(n)))")
wenzelm@13428
  1280
apply (rule_tac A="{B,A,n,z,Memrel(succ(n))}" in subset_LsetE, blast )
paulson@13422
  1281
apply (rule ReflectsE [OF eclose_replacement1_Reflects], assumption)
wenzelm@13428
  1282
apply (drule subset_Lset_ltD, assumption)
paulson@13422
  1283
apply (erule reflection_imp_L_separation)
paulson@13422
  1284
  apply (simp_all add: lt_Ord2 Memrel_closed)
wenzelm@13428
  1285
apply (elim conjE)
paulson@13422
  1286
apply (rule DPow_LsetI)
wenzelm@13428
  1287
apply (rename_tac v)
paulson@13422
  1288
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13422
  1289
apply (rule_tac env = "[u,v,A,n,B,Memrel(succ(n))]" in mem_iff_sats)
paulson@13422
  1290
apply (rule sep_rules | simp)+
paulson@13422
  1291
txt{*Can't get sat rules to work for higher-order operators, so just expand them!*}
paulson@13422
  1292
apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
paulson@13422
  1293
apply (rule sep_rules big_union_iff_sats quasinat_iff_sats | simp)+
paulson@13409
  1294
done
paulson@13409
  1295
paulson@13422
  1296
paulson@13422
  1297
lemma eclose_replacement2_Reflects:
paulson@13422
  1298
 "REFLECTS
paulson@13422
  1299
   [\<lambda>x. \<exists>u[L]. u \<in> B \<and> u \<in> nat \<and>
paulson@13422
  1300
         (\<exists>sn[L]. \<exists>msn[L]. successor(L, u, sn) \<and> membership(L, sn, msn) \<and>
paulson@13422
  1301
           is_wfrec (L, iterates_MH (L, big_union(L), A),
paulson@13422
  1302
                              msn, u, x)),
paulson@13422
  1303
    \<lambda>i x. \<exists>u \<in> Lset(i). u \<in> B \<and> u \<in> nat \<and>
wenzelm@13428
  1304
         (\<exists>sn \<in> Lset(i). \<exists>msn \<in> Lset(i).
paulson@13422
  1305
          successor(**Lset(i), u, sn) \<and> membership(**Lset(i), sn, msn) \<and>
wenzelm@13428
  1306
           is_wfrec (**Lset(i),
paulson@13422
  1307
                 iterates_MH (**Lset(i), big_union(**Lset(i)), A),
paulson@13422
  1308
                     msn, u, x))]"
wenzelm@13428
  1309
by (intro FOL_reflections function_reflections is_wfrec_reflection
wenzelm@13428
  1310
          iterates_MH_reflection)
paulson@13422
  1311
paulson@13422
  1312
wenzelm@13428
  1313
lemma eclose_replacement2:
wenzelm@13428
  1314
   "L(A) ==> strong_replacement(L,
wenzelm@13428
  1315
         \<lambda>n y. n\<in>nat &
paulson@13422
  1316
               (\<exists>sn[L]. \<exists>msn[L]. successor(L,n,sn) & membership(L,sn,msn) &
wenzelm@13428
  1317
               is_wfrec(L, iterates_MH(L,big_union(L), A),
paulson@13422
  1318
                        msn, n, y)))"
wenzelm@13428
  1319
apply (rule strong_replacementI)
paulson@13422
  1320
apply (rule rallI)
wenzelm@13428
  1321
apply (rename_tac B)
wenzelm@13428
  1322
apply (rule separation_CollectI)
wenzelm@13428
  1323
apply (rule_tac A="{A,B,z,nat}" in subset_LsetE)
wenzelm@13428
  1324
apply (blast intro: L_nat)
paulson@13422
  1325
apply (rule ReflectsE [OF eclose_replacement2_Reflects], assumption)
wenzelm@13428
  1326
apply (drule subset_Lset_ltD, assumption)
paulson@13422
  1327
apply (erule reflection_imp_L_separation)
paulson@13422
  1328
  apply (simp_all add: lt_Ord2)
paulson@13422
  1329
apply (rule DPow_LsetI)
wenzelm@13428
  1330
apply (rename_tac v)
paulson@13422
  1331
apply (rule bex_iff_sats conj_iff_sats)+
paulson@13422
  1332
apply (rule_tac env = "[u,v,A,B,nat]" in mem_iff_sats)
paulson@13422
  1333
apply (rule sep_rules | simp)+
paulson@13422
  1334
apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def)
paulson@13422
  1335
apply (rule sep_rules big_union_iff_sats quasinat_iff_sats | simp)+
paulson@13422
  1336
done
paulson@13422
  1337
paulson@13422
  1338
paulson@13422
  1339
subsubsection{*Instantiating the locale @{text M_eclose}*}
paulson@13422
  1340
wenzelm@13428
  1341
theorem M_eclose_axioms_L: "M_eclose_axioms(L)"
wenzelm@13428
  1342
  apply (rule M_eclose_axioms.intro)
wenzelm@13428
  1343
   apply (assumption | rule eclose_replacement1 eclose_replacement2)+
wenzelm@13428
  1344
  done
paulson@13422
  1345
wenzelm@13428
  1346
theorem M_eclose_L: "PROP M_eclose(L)"
wenzelm@13428
  1347
  apply (rule M_eclose.intro)
wenzelm@13428
  1348
       apply (rule M_triv_axioms_L)
wenzelm@13428
  1349
      apply (rule M_axioms_axioms_L)
wenzelm@13428
  1350
     apply (rule M_trancl_axioms_L)
wenzelm@13428
  1351
    apply (rule M_wfrank_axioms_L)
wenzelm@13428
  1352
   apply (rule M_datatypes_axioms_L)
wenzelm@13428
  1353
  apply (rule M_eclose_axioms_L)
wenzelm@13428
  1354
  done
paulson@13422
  1355
wenzelm@13428
  1356
lemmas eclose_closed [intro, simp] = M_eclose.eclose_closed [OF M_eclose_L]
wenzelm@13428
  1357
  and eclose_abs [intro, simp] = M_eclose.eclose_abs [OF M_eclose_L]
paulson@13422
  1358
paulson@13348
  1359
end