--- a/src/ZF/Constructible/L_axioms.thy Thu Jul 11 10:48:30 2002 +0200
+++ b/src/ZF/Constructible/L_axioms.thy Thu Jul 11 13:43:24 2002 +0200
@@ -884,6 +884,46 @@
done
+subsubsection{*Pre-Image under a Relation, Internalized*}
+
+(* "pre_image(M,r,A,z) ==
+ \<forall>x[M]. x \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. y\<in>A & pair(M,x,y,w)))" *)
+constdefs pre_image_fm :: "[i,i,i]=>i"
+ "pre_image_fm(r,A,z) ==
+ Forall(Iff(Member(0,succ(z)),
+ Exists(And(Member(0,succ(succ(r))),
+ Exists(And(Member(0,succ(succ(succ(A)))),
+ pair_fm(2,0,1)))))))"
+
+lemma pre_image_type [TC]:
+ "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> pre_image_fm(x,y,z) \<in> formula"
+by (simp add: pre_image_fm_def)
+
+lemma arity_pre_image_fm [simp]:
+ "[| x \<in> nat; y \<in> nat; z \<in> nat |]
+ ==> arity(pre_image_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
+by (simp add: pre_image_fm_def succ_Un_distrib [symmetric] Un_ac)
+
+lemma sats_pre_image_fm [simp]:
+ "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
+ ==> sats(A, pre_image_fm(x,y,z), env) <->
+ pre_image(**A, nth(x,env), nth(y,env), nth(z,env))"
+by (simp add: pre_image_fm_def Relative.pre_image_def)
+
+lemma pre_image_iff_sats:
+ "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
+ i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
+ ==> pre_image(**A, x, y, z) <-> sats(A, pre_image_fm(i,j,k), env)"
+by (simp add: sats_pre_image_fm)
+
+theorem pre_image_reflection:
+ "REFLECTS[\<lambda>x. pre_image(L,f(x),g(x),h(x)),
+ \<lambda>i x. pre_image(**Lset(i),f(x),g(x),h(x))]"
+apply (simp only: Relative.pre_image_def setclass_simps)
+apply (intro FOL_reflections pair_reflection)
+done
+
+
subsubsection{*The Concept of Relation, Internalized*}
(* "is_relation(M,r) ==
@@ -1000,7 +1040,7 @@
fun_apply_reflection subset_reflection
transitive_set_reflection membership_reflection
pred_set_reflection domain_reflection range_reflection field_reflection
- image_reflection
+ image_reflection pre_image_reflection
is_relation_reflection is_function_reflection
lemmas function_iff_sats =
@@ -1008,7 +1048,7 @@
cons_iff_sats successor_iff_sats
fun_apply_iff_sats Memrel_iff_sats
pred_set_iff_sats domain_iff_sats range_iff_sats field_iff_sats
- image_iff_sats
+ image_iff_sats pre_image_iff_sats
relation_iff_sats function_iff_sats
@@ -1189,6 +1229,46 @@
done
+subsubsection{*Restriction of a Relation, Internalized*}
+
+
+(* "restriction(M,r,A,z) ==
+ \<forall>x[M]. x \<in> z <-> (x \<in> r & (\<exists>u[M]. u\<in>A & (\<exists>v[M]. pair(M,u,v,x))))" *)
+constdefs restriction_fm :: "[i,i,i]=>i"
+ "restriction_fm(r,A,z) ==
+ Forall(Iff(Member(0,succ(z)),
+ And(Member(0,succ(r)),
+ Exists(And(Member(0,succ(succ(A))),
+ Exists(pair_fm(1,0,2)))))))"
+
+lemma restriction_type [TC]:
+ "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> restriction_fm(x,y,z) \<in> formula"
+by (simp add: restriction_fm_def)
+
+lemma arity_restriction_fm [simp]:
+ "[| x \<in> nat; y \<in> nat; z \<in> nat |]
+ ==> arity(restriction_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
+by (simp add: restriction_fm_def succ_Un_distrib [symmetric] Un_ac)
+
+lemma sats_restriction_fm [simp]:
+ "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
+ ==> sats(A, restriction_fm(x,y,z), env) <->
+ restriction(**A, nth(x,env), nth(y,env), nth(z,env))"
+by (simp add: restriction_fm_def restriction_def)
+
+lemma restriction_iff_sats:
+ "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
+ i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
+ ==> restriction(**A, x, y, z) <-> sats(A, restriction_fm(i,j,k), env)"
+by simp
+
+theorem restriction_reflection:
+ "REFLECTS[\<lambda>x. restriction(L,f(x),g(x),h(x)),
+ \<lambda>i x. restriction(**Lset(i),f(x),g(x),h(x))]"
+apply (simp only: restriction_def setclass_simps)
+apply (intro FOL_reflections pair_reflection)
+done
+
subsubsection{*Order-Isomorphisms, Internalized*}
(* order_isomorphism :: "[i=>o,i,i,i,i,i] => o"
@@ -1327,12 +1407,14 @@
lemmas fun_plus_reflections =
typed_function_reflection composition_reflection
injection_reflection surjection_reflection
- bijection_reflection order_isomorphism_reflection
+ bijection_reflection restriction_reflection
+ order_isomorphism_reflection
ordinal_reflection limit_ordinal_reflection omega_reflection
lemmas fun_plus_iff_sats =
typed_function_iff_sats composition_iff_sats
- injection_iff_sats surjection_iff_sats bijection_iff_sats
+ injection_iff_sats surjection_iff_sats
+ bijection_iff_sats restriction_iff_sats
order_isomorphism_iff_sats
ordinal_iff_sats limit_ordinal_iff_sats omega_iff_sats
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/ZF/Constructible/Rec_Separation.thy Thu Jul 11 13:43:24 2002 +0200
@@ -0,0 +1,387 @@
+header{*Separation for the Absoluteness of Recursion*}
+
+theory Rec_Separation = Separation:
+
+text{*This theory proves all instances needed for locales @{text
+"M_trancl"}, @{text "M_wfrank"} and @{text "M_datatypes"}*}
+
+subsection{*The Locale @{text "M_trancl"}*}
+
+subsubsection{*Separation for Reflexive/Transitive Closure*}
+
+text{*First, The Defining Formula*}
+
+(* "rtran_closure_mem(M,A,r,p) ==
+ \<exists>nnat[M]. \<exists>n[M]. \<exists>n'[M].
+ omega(M,nnat) & n\<in>nnat & successor(M,n,n') &
+ (\<exists>f[M]. typed_function(M,n',A,f) &
+ (\<exists>x[M]. \<exists>y[M]. \<exists>zero[M]. pair(M,x,y,p) & empty(M,zero) &
+ fun_apply(M,f,zero,x) & fun_apply(M,f,n,y)) &
+ (\<forall>j[M]. j\<in>n -->
+ (\<exists>fj[M]. \<exists>sj[M]. \<exists>fsj[M]. \<exists>ffp[M].
+ fun_apply(M,f,j,fj) & successor(M,j,sj) &
+ fun_apply(M,f,sj,fsj) & pair(M,fj,fsj,ffp) & ffp \<in> r)))"*)
+constdefs rtran_closure_mem_fm :: "[i,i,i]=>i"
+ "rtran_closure_mem_fm(A,r,p) ==
+ Exists(Exists(Exists(
+ And(omega_fm(2),
+ And(Member(1,2),
+ And(succ_fm(1,0),
+ Exists(And(typed_function_fm(1, A#+4, 0),
+ And(Exists(Exists(Exists(
+ And(pair_fm(2,1,p#+7),
+ And(empty_fm(0),
+ And(fun_apply_fm(3,0,2), fun_apply_fm(3,5,1))))))),
+ Forall(Implies(Member(0,3),
+ Exists(Exists(Exists(Exists(
+ And(fun_apply_fm(5,4,3),
+ And(succ_fm(4,2),
+ And(fun_apply_fm(5,2,1),
+ And(pair_fm(3,1,0), Member(0,r#+9))))))))))))))))))))"
+
+
+lemma rtran_closure_mem_type [TC]:
+ "[| x \<in> nat; y \<in> nat; z \<in> nat |] ==> rtran_closure_mem_fm(x,y,z) \<in> formula"
+by (simp add: rtran_closure_mem_fm_def)
+
+lemma arity_rtran_closure_mem_fm [simp]:
+ "[| x \<in> nat; y \<in> nat; z \<in> nat |]
+ ==> arity(rtran_closure_mem_fm(x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
+by (simp add: rtran_closure_mem_fm_def succ_Un_distrib [symmetric] Un_ac)
+
+lemma sats_rtran_closure_mem_fm [simp]:
+ "[| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
+ ==> sats(A, rtran_closure_mem_fm(x,y,z), env) <->
+ rtran_closure_mem(**A, nth(x,env), nth(y,env), nth(z,env))"
+by (simp add: rtran_closure_mem_fm_def rtran_closure_mem_def)
+
+lemma rtran_closure_mem_iff_sats:
+ "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
+ i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
+ ==> rtran_closure_mem(**A, x, y, z) <-> sats(A, rtran_closure_mem_fm(i,j,k), env)"
+by (simp add: sats_rtran_closure_mem_fm)
+
+theorem rtran_closure_mem_reflection:
+ "REFLECTS[\<lambda>x. rtran_closure_mem(L,f(x),g(x),h(x)),
+ \<lambda>i x. rtran_closure_mem(**Lset(i),f(x),g(x),h(x))]"
+apply (simp only: rtran_closure_mem_def setclass_simps)
+apply (intro FOL_reflections function_reflections fun_plus_reflections)
+done
+
+text{*Separation for @{term "rtrancl(r)"}.*}
+lemma rtrancl_separation:
+ "[| L(r); L(A) |] ==> separation (L, rtran_closure_mem(L,A,r))"
+apply (rule separation_CollectI)
+apply (rule_tac A="{r,A,z}" in subset_LsetE, blast )
+apply (rule ReflectsE [OF rtran_closure_mem_reflection], assumption)
+apply (drule subset_Lset_ltD, assumption)
+apply (erule reflection_imp_L_separation)
+ apply (simp_all add: lt_Ord2)
+apply (rule DPowI2)
+apply (rename_tac u)
+apply (rule_tac env = "[u,r,A]" in rtran_closure_mem_iff_sats)
+apply (rule sep_rules | simp)+
+apply (simp_all add: succ_Un_distrib [symmetric])
+done
+
+
+subsubsection{*Reflexive/Transitive Closure, Internalized*}
+
+(* "rtran_closure(M,r,s) ==
+ \<forall>A[M]. is_field(M,r,A) -->
+ (\<forall>p[M]. p \<in> s <-> rtran_closure_mem(M,A,r,p))" *)
+constdefs rtran_closure_fm :: "[i,i]=>i"
+ "rtran_closure_fm(r,s) ==
+ Forall(Implies(field_fm(succ(r),0),
+ Forall(Iff(Member(0,succ(succ(s))),
+ rtran_closure_mem_fm(1,succ(succ(r)),0)))))"
+
+lemma rtran_closure_type [TC]:
+ "[| x \<in> nat; y \<in> nat |] ==> rtran_closure_fm(x,y) \<in> formula"
+by (simp add: rtran_closure_fm_def)
+
+lemma arity_rtran_closure_fm [simp]:
+ "[| x \<in> nat; y \<in> nat |]
+ ==> arity(rtran_closure_fm(x,y)) = succ(x) \<union> succ(y)"
+by (simp add: rtran_closure_fm_def succ_Un_distrib [symmetric] Un_ac)
+
+lemma sats_rtran_closure_fm [simp]:
+ "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
+ ==> sats(A, rtran_closure_fm(x,y), env) <->
+ rtran_closure(**A, nth(x,env), nth(y,env))"
+by (simp add: rtran_closure_fm_def rtran_closure_def)
+
+lemma rtran_closure_iff_sats:
+ "[| nth(i,env) = x; nth(j,env) = y;
+ i \<in> nat; j \<in> nat; env \<in> list(A)|]
+ ==> rtran_closure(**A, x, y) <-> sats(A, rtran_closure_fm(i,j), env)"
+by simp
+
+theorem rtran_closure_reflection:
+ "REFLECTS[\<lambda>x. rtran_closure(L,f(x),g(x)),
+ \<lambda>i x. rtran_closure(**Lset(i),f(x),g(x))]"
+apply (simp only: rtran_closure_def setclass_simps)
+apply (intro FOL_reflections function_reflections rtran_closure_mem_reflection)
+done
+
+
+subsubsection{*Transitive Closure of a Relation, Internalized*}
+
+(* "tran_closure(M,r,t) ==
+ \<exists>s[M]. rtran_closure(M,r,s) & composition(M,r,s,t)" *)
+constdefs tran_closure_fm :: "[i,i]=>i"
+ "tran_closure_fm(r,s) ==
+ Exists(And(rtran_closure_fm(succ(r),0), composition_fm(succ(r),0,succ(s))))"
+
+lemma tran_closure_type [TC]:
+ "[| x \<in> nat; y \<in> nat |] ==> tran_closure_fm(x,y) \<in> formula"
+by (simp add: tran_closure_fm_def)
+
+lemma arity_tran_closure_fm [simp]:
+ "[| x \<in> nat; y \<in> nat |]
+ ==> arity(tran_closure_fm(x,y)) = succ(x) \<union> succ(y)"
+by (simp add: tran_closure_fm_def succ_Un_distrib [symmetric] Un_ac)
+
+lemma sats_tran_closure_fm [simp]:
+ "[| x \<in> nat; y \<in> nat; env \<in> list(A)|]
+ ==> sats(A, tran_closure_fm(x,y), env) <->
+ tran_closure(**A, nth(x,env), nth(y,env))"
+by (simp add: tran_closure_fm_def tran_closure_def)
+
+lemma tran_closure_iff_sats:
+ "[| nth(i,env) = x; nth(j,env) = y;
+ i \<in> nat; j \<in> nat; env \<in> list(A)|]
+ ==> tran_closure(**A, x, y) <-> sats(A, tran_closure_fm(i,j), env)"
+by simp
+
+theorem tran_closure_reflection:
+ "REFLECTS[\<lambda>x. tran_closure(L,f(x),g(x)),
+ \<lambda>i x. tran_closure(**Lset(i),f(x),g(x))]"
+apply (simp only: tran_closure_def setclass_simps)
+apply (intro FOL_reflections function_reflections
+ rtran_closure_reflection composition_reflection)
+done
+
+
+subsection{*Separation for the Proof of @{text "wellfounded_on_trancl"}*}
+
+lemma wellfounded_trancl_reflects:
+ "REFLECTS[\<lambda>x. \<exists>w[L]. \<exists>wx[L]. \<exists>rp[L].
+ w \<in> Z & pair(L,w,x,wx) & tran_closure(L,r,rp) & wx \<in> rp,
+ \<lambda>i x. \<exists>w \<in> Lset(i). \<exists>wx \<in> Lset(i). \<exists>rp \<in> Lset(i).
+ w \<in> Z & pair(**Lset(i),w,x,wx) & tran_closure(**Lset(i),r,rp) &
+ wx \<in> rp]"
+by (intro FOL_reflections function_reflections fun_plus_reflections
+ tran_closure_reflection)
+
+
+lemma wellfounded_trancl_separation:
+ "[| L(r); L(Z) |] ==>
+ separation (L, \<lambda>x.
+ \<exists>w[L]. \<exists>wx[L]. \<exists>rp[L].
+ w \<in> Z & pair(L,w,x,wx) & tran_closure(L,r,rp) & wx \<in> rp)"
+apply (rule separation_CollectI)
+apply (rule_tac A="{r,Z,z}" in subset_LsetE, blast )
+apply (rule ReflectsE [OF wellfounded_trancl_reflects], assumption)
+apply (drule subset_Lset_ltD, assumption)
+apply (erule reflection_imp_L_separation)
+ apply (simp_all add: lt_Ord2)
+apply (rule DPowI2)
+apply (rename_tac u)
+apply (rule bex_iff_sats conj_iff_sats)+
+apply (rule_tac env = "[w,u,r,Z]" in mem_iff_sats)
+apply (rule sep_rules tran_closure_iff_sats | simp)+
+apply (simp_all add: succ_Un_distrib [symmetric])
+done
+
+subsection{*Well-Founded Recursion!*}
+
+(* M_is_recfun :: "[i=>o, i, i, [i,i,i]=>o, i] => o"
+ "M_is_recfun(M,r,a,MH,f) ==
+ \<forall>z[M]. z \<in> f <->
+ 5 4 3 2 1 0
+ (\<exists>x[M]. \<exists>y[M]. \<exists>xa[M]. \<exists>sx[M]. \<exists>r_sx[M]. \<exists>f_r_sx[M].
+ pair(M,x,y,z) & pair(M,x,a,xa) & upair(M,x,x,sx) &
+ pre_image(M,r,sx,r_sx) & restriction(M,f,r_sx,f_r_sx) &
+ xa \<in> r & MH(x, f_r_sx, y))"
+*)
+
+constdefs is_recfun_fm :: "[[i,i,i]=>i, i, i, i]=>i"
+ "is_recfun_fm(p,r,a,f) ==
+ Forall(Iff(Member(0,succ(f)),
+ Exists(Exists(Exists(Exists(Exists(Exists(
+ And(pair_fm(5,4,6),
+ And(pair_fm(5,a#+7,3),
+ And(upair_fm(5,5,2),
+ And(pre_image_fm(r#+7,2,1),
+ And(restriction_fm(f#+7,1,0),
+ And(Member(3,r#+7), p(5,0,4)))))))))))))))"
+
+
+lemma is_recfun_type_0:
+ "[| !!x y z. [| x \<in> nat; y \<in> nat; z \<in> nat |] ==> p(x,y,z) \<in> formula;
+ x \<in> nat; y \<in> nat; z \<in> nat |]
+ ==> is_recfun_fm(p,x,y,z) \<in> formula"
+apply (unfold is_recfun_fm_def)
+(*FIXME: FIND OUT why simp loops!*)
+apply typecheck
+by simp
+
+lemma is_recfun_type [TC]:
+ "[| p(5,0,4) \<in> formula;
+ x \<in> nat; y \<in> nat; z \<in> nat |]
+ ==> is_recfun_fm(p,x,y,z) \<in> formula"
+by (simp add: is_recfun_fm_def)
+
+lemma arity_is_recfun_fm [simp]:
+ "[| arity(p(5,0,4)) le 8; x \<in> nat; y \<in> nat; z \<in> nat |]
+ ==> arity(is_recfun_fm(p,x,y,z)) = succ(x) \<union> succ(y) \<union> succ(z)"
+apply (frule lt_nat_in_nat, simp)
+apply (simp add: is_recfun_fm_def succ_Un_distrib [symmetric] )
+apply (subst subset_Un_iff2 [of "arity(p(5,0,4))", THEN iffD1])
+apply (rule le_imp_subset)
+apply (erule le_trans, simp)
+apply (simp add: succ_Un_distrib [symmetric] Un_ac)
+done
+
+lemma sats_is_recfun_fm:
+ assumes MH_iff_sats:
+ "!!x y z env.
+ [| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
+ ==> MH(nth(x,env), nth(y,env), nth(z,env)) <-> sats(A, p(x,y,z), env)"
+ shows
+ "[|x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
+ ==> sats(A, is_recfun_fm(p,x,y,z), env) <->
+ M_is_recfun(**A, nth(x,env), nth(y,env), MH, nth(z,env))"
+by (simp add: is_recfun_fm_def M_is_recfun_def MH_iff_sats [THEN iff_sym])
+
+lemma is_recfun_iff_sats:
+ "[| (!!x y z env. [| x \<in> nat; y \<in> nat; z \<in> nat; env \<in> list(A)|]
+ ==> MH(nth(x,env), nth(y,env), nth(z,env)) <->
+ sats(A, p(x,y,z), env));
+ nth(i,env) = x; nth(j,env) = y; nth(k,env) = z;
+ i \<in> nat; j \<in> nat; k \<in> nat; env \<in> list(A)|]
+ ==> M_is_recfun(**A, x, y, MH, z) <-> sats(A, is_recfun_fm(p,i,j,k), env)"
+by (simp add: sats_is_recfun_fm [of A MH])
+
+theorem is_recfun_reflection:
+ assumes MH_reflection:
+ "!!f g h. REFLECTS[\<lambda>x. MH(L, f(x), g(x), h(x)),
+ \<lambda>i x. MH(**Lset(i), f(x), g(x), h(x))]"
+ shows "REFLECTS[\<lambda>x. M_is_recfun(L, f(x), g(x), MH(L), h(x)),
+ \<lambda>i x. M_is_recfun(**Lset(i), f(x), g(x), MH(**Lset(i)), h(x))]"
+apply (simp (no_asm_use) only: M_is_recfun_def setclass_simps)
+apply (intro FOL_reflections function_reflections
+ restriction_reflection MH_reflection)
+done
+
+subsection{*Separation for @{term "wfrank"}*}
+
+lemma wfrank_Reflects:
+ "REFLECTS[\<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
+ ~ (\<exists>f[L]. M_is_recfun(L, rplus, x, %x f y. is_range(L,f,y), f)),
+ \<lambda>i x. \<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) -->
+ ~ (\<exists>f \<in> Lset(i). M_is_recfun(**Lset(i), rplus, x, %x f y. is_range(**Lset(i),f,y), f))]"
+by (intro FOL_reflections function_reflections is_recfun_reflection tran_closure_reflection)
+
+lemma wfrank_separation:
+ "L(r) ==>
+ separation (L, \<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
+ ~ (\<exists>f[L]. M_is_recfun(L, rplus, x, %x f y. is_range(L,f,y), f)))"
+apply (rule separation_CollectI)
+apply (rule_tac A="{r,z}" in subset_LsetE, blast )
+apply (rule ReflectsE [OF wfrank_Reflects], assumption)
+apply (drule subset_Lset_ltD, assumption)
+apply (erule reflection_imp_L_separation)
+ apply (simp_all add: lt_Ord2, clarify)
+apply (rule DPowI2)
+apply (rename_tac u)
+apply (rule ball_iff_sats imp_iff_sats)+
+apply (rule_tac env="[rplus,u,r]" in tran_closure_iff_sats)
+apply (rule sep_rules is_recfun_iff_sats | simp)+
+apply (simp_all add: succ_Un_distrib [symmetric])
+done
+
+
+subsection{*Replacement for @{term "wfrank"}*}
+
+lemma wfrank_replacement_Reflects:
+ "REFLECTS[\<lambda>z. \<exists>x[L]. x \<in> A &
+ (\<forall>rplus[L]. tran_closure(L,r,rplus) -->
+ (\<exists>y[L]. \<exists>f[L]. pair(L,x,y,z) &
+ M_is_recfun(L, rplus, x, %x f y. is_range(L,f,y), f) &
+ is_range(L,f,y))),
+ \<lambda>i z. \<exists>x \<in> Lset(i). x \<in> A &
+ (\<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) -->
+ (\<exists>y \<in> Lset(i). \<exists>f \<in> Lset(i). pair(**Lset(i),x,y,z) &
+ M_is_recfun(**Lset(i), rplus, x, %x f y. is_range(**Lset(i),f,y), f) &
+ is_range(**Lset(i),f,y)))]"
+by (intro FOL_reflections function_reflections fun_plus_reflections
+ is_recfun_reflection tran_closure_reflection)
+
+
+lemma wfrank_strong_replacement:
+ "L(r) ==>
+ strong_replacement(L, \<lambda>x z.
+ \<forall>rplus[L]. tran_closure(L,r,rplus) -->
+ (\<exists>y[L]. \<exists>f[L]. pair(L,x,y,z) &
+ M_is_recfun(L, rplus, x, %x f y. is_range(L,f,y), f) &
+ is_range(L,f,y)))"
+apply (rule strong_replacementI)
+apply (rule rallI)
+apply (rename_tac B)
+apply (rule separation_CollectI)
+apply (rule_tac A="{B,r,z}" in subset_LsetE, blast )
+apply (rule ReflectsE [OF wfrank_replacement_Reflects], assumption)
+apply (drule subset_Lset_ltD, assumption)
+apply (erule reflection_imp_L_separation)
+ apply (simp_all add: lt_Ord2)
+apply (rule DPowI2)
+apply (rename_tac u)
+apply (rule bex_iff_sats ball_iff_sats conj_iff_sats)+
+apply (rule_tac env = "[x,u,B,r]" in mem_iff_sats)
+apply (rule sep_rules tran_closure_iff_sats is_recfun_iff_sats | simp)+
+apply (simp_all add: succ_Un_distrib [symmetric])
+done
+
+
+subsection{*Separation for @{term "wfrank"}*}
+
+lemma Ord_wfrank_Reflects:
+ "REFLECTS[\<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
+ ~ (\<forall>f[L]. \<forall>rangef[L].
+ is_range(L,f,rangef) -->
+ M_is_recfun(L, rplus, x, \<lambda>x f y. is_range(L,f,y), f) -->
+ ordinal(L,rangef)),
+ \<lambda>i x. \<forall>rplus \<in> Lset(i). tran_closure(**Lset(i),r,rplus) -->
+ ~ (\<forall>f \<in> Lset(i). \<forall>rangef \<in> Lset(i).
+ is_range(**Lset(i),f,rangef) -->
+ M_is_recfun(**Lset(i), rplus, x, \<lambda>x f y. is_range(**Lset(i),f,y), f) -->
+ ordinal(**Lset(i),rangef))]"
+by (intro FOL_reflections function_reflections is_recfun_reflection
+ tran_closure_reflection ordinal_reflection)
+
+lemma Ord_wfrank_separation:
+ "L(r) ==>
+ separation (L, \<lambda>x.
+ \<forall>rplus[L]. tran_closure(L,r,rplus) -->
+ ~ (\<forall>f[L]. \<forall>rangef[L].
+ is_range(L,f,rangef) -->
+ M_is_recfun(L, rplus, x, \<lambda>x f y. is_range(L,f,y), f) -->
+ ordinal(L,rangef)))"
+apply (rule separation_CollectI)
+apply (rule_tac A="{r,z}" in subset_LsetE, blast )
+apply (rule ReflectsE [OF Ord_wfrank_Reflects], assumption)
+apply (drule subset_Lset_ltD, assumption)
+apply (erule reflection_imp_L_separation)
+ apply (simp_all add: lt_Ord2, clarify)
+apply (rule DPowI2)
+apply (rename_tac u)
+apply (rule ball_iff_sats imp_iff_sats)+
+apply (rule_tac env="[rplus,u,r]" in tran_closure_iff_sats)
+apply (rule sep_rules is_recfun_iff_sats | simp)+
+apply (simp_all add: succ_Un_distrib [symmetric])
+done
+
+
+end
--- a/src/ZF/Constructible/WF_absolute.thy Thu Jul 11 10:48:30 2002 +0200
+++ b/src/ZF/Constructible/WF_absolute.thy Thu Jul 11 13:43:24 2002 +0200
@@ -232,21 +232,30 @@
rank function.*}
-(*NEEDS RELATIVIZATION*)
locale M_wfrank = M_trancl +
assumes wfrank_separation:
"M(r) ==>
separation (M, \<lambda>x.
- ~ (\<exists>f[M]. M_is_recfun(M, r^+, x, %mm x f y. y = range(f), f)))"
- and wfrank_strong_replacement':
+ \<forall>rplus[M]. tran_closure(M,r,rplus) -->
+ ~ (\<exists>f[M]. M_is_recfun(M, rplus, x, %x f y. is_range(M,f,y), f)))"
+ and wfrank_strong_replacement:
"M(r) ==>
- strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>f[M].
- pair(M,x,y,z) & is_recfun(r^+, x, %x f. range(f), f) &
- y = range(f))"
+ strong_replacement(M, \<lambda>x z.
+ \<forall>rplus[M]. tran_closure(M,r,rplus) -->
+ (\<exists>y[M]. \<exists>f[M]. pair(M,x,y,z) &
+ M_is_recfun(M, rplus, x, %x f y. is_range(M,f,y), f) &
+ is_range(M,f,y)))"
and Ord_wfrank_separation:
"M(r) ==>
- separation (M, \<lambda>x. ~ (\<forall>f. M(f) \<longrightarrow>
- is_recfun(r^+, x, \<lambda>x. range, f) \<longrightarrow> Ord(range(f))))"
+ separation (M, \<lambda>x.
+ \<forall>rplus[M]. tran_closure(M,r,rplus) -->
+ ~ (\<forall>f[M]. \<forall>rangef[M].
+ is_range(M,f,rangef) -->
+ M_is_recfun(M, rplus, x, \<lambda>x f y. is_range(M,f,y), f) -->
+ ordinal(M,rangef)))"
+
+text{*Proving that the relativized instances of Separation or Replacement
+agree with the "real" ones.*}
lemma (in M_wfrank) wfrank_separation':
"M(r) ==>
@@ -256,6 +265,23 @@
apply (simp add: is_recfun_iff_M [of concl: _ _ "%x. range", THEN iff_sym])
done
+lemma (in M_wfrank) wfrank_strong_replacement':
+ "M(r) ==>
+ strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>f[M].
+ pair(M,x,y,z) & is_recfun(r^+, x, %x f. range(f), f) &
+ y = range(f))"
+apply (insert wfrank_strong_replacement [of r])
+apply (simp add: is_recfun_iff_M [of concl: _ _ "%x. range", THEN iff_sym])
+done
+
+lemma (in M_wfrank) Ord_wfrank_separation':
+ "M(r) ==>
+ separation (M, \<lambda>x.
+ ~ (\<forall>f[M]. is_recfun(r^+, x, \<lambda>x. range, f) --> Ord(range(f))))"
+apply (insert Ord_wfrank_separation [of r])
+apply (simp add: is_recfun_iff_M [of concl: _ _ "%x. range", THEN iff_sym])
+done
+
text{*This function, defined using replacement, is a rank function for
well-founded relations within the class M.*}
constdefs
@@ -290,11 +316,11 @@
lemma (in M_wfrank) Ord_wfrank_range [rule_format]:
"[| wellfounded(M,r); a\<in>A; M(r); M(A) |]
- ==> \<forall>f. M(f) --> is_recfun(r^+, a, %x f. range(f), f) --> Ord(range(f))"
+ ==> \<forall>f[M]. is_recfun(r^+, a, %x f. range(f), f) --> Ord(range(f))"
apply (drule wellfounded_trancl, assumption)
apply (rule wellfounded_induct, assumption+)
apply simp
- apply (blast intro: Ord_wfrank_separation, clarify)
+ apply (blast intro: Ord_wfrank_separation', clarify)
txt{*The reasoning in both cases is that we get @{term y} such that
@{term "\<langle>y, x\<rangle> \<in> r^+"}. We find that
@{term "f`y = restrict(f, r^+ -`` {y})"}. *}
@@ -314,7 +340,8 @@
apply (simp add: trans_trancl trancl_subset_times)+
apply (drule spec [THEN mp], assumption)
apply (subgoal_tac "M(restrict(f, r^+ -`` {y}))")
- apply (drule_tac x="restrict(f, r^+ -`` {y})" in spec)
+ apply (drule_tac x="restrict(f, r^+ -`` {y})" in rspec)
+apply assumption
apply (simp add: function_apply_equality [OF _ is_recfun_imp_function])
apply (blast dest: pair_components_in_M)
done
--- a/src/ZF/Constructible/WFrec.thy Thu Jul 11 10:48:30 2002 +0200
+++ b/src/ZF/Constructible/WFrec.thy Thu Jul 11 13:43:24 2002 +0200
@@ -275,17 +275,17 @@
done
constdefs
- M_is_recfun :: "[i=>o, i, i, [i=>o,i,i,i]=>o, i] => o"
+ M_is_recfun :: "[i=>o, i, i, [i,i,i]=>o, i] => o"
"M_is_recfun(M,r,a,MH,f) ==
\<forall>z[M]. z \<in> f <->
(\<exists>x[M]. \<exists>y[M]. \<exists>xa[M]. \<exists>sx[M]. \<exists>r_sx[M]. \<exists>f_r_sx[M].
pair(M,x,y,z) & pair(M,x,a,xa) & upair(M,x,x,sx) &
pre_image(M,r,sx,r_sx) & restriction(M,f,r_sx,f_r_sx) &
- xa \<in> r & MH(M, x, f_r_sx, y))"
+ xa \<in> r & MH(x, f_r_sx, y))"
lemma (in M_axioms) is_recfun_iff_M:
"[| M(r); M(a); M(f); \<forall>x[M]. \<forall>g[M]. function(g) --> M(H(x,g));
- \<forall>x g y. M(x) --> M(g) --> M(y) --> MH(M,x,g,y) <-> y = H(x,g) |] ==>
+ \<forall>x g y. M(x) --> M(g) --> M(y) --> MH(x,g,y) <-> y = H(x,g) |] ==>
is_recfun(r,a,H,f) <-> M_is_recfun(M,r,a,MH,f)"
apply (simp add: M_is_recfun_def is_recfun_relativize)
apply (rule rall_cong)
@@ -294,7 +294,7 @@
lemma M_is_recfun_cong [cong]:
"[| r = r'; a = a'; f = f';
- !!x g y. [| M(x); M(g); M(y) |] ==> MH(M,x,g,y) <-> MH'(M,x,g,y) |]
+ !!x g y. [| M(x); M(g); M(y) |] ==> MH(x,g,y) <-> MH'(x,g,y) |]
==> M_is_recfun(M,r,a,MH,f) <-> M_is_recfun(M,r',a',MH',f')"
by (simp add: M_is_recfun_def)
@@ -309,7 +309,7 @@
(\<forall>sj msj. M(sj) --> M(msj) -->
successor(M,j,sj) --> membership(M,sj,msj) -->
M_is_recfun(M, msj, x,
- %M x g y. \<exists>gx. M(gx) & image(M,g,x,gx) & union(M,i,gx,y),
+ %x g y. \<exists>gx[M]. image(M,g,x,gx) & union(M,i,gx,y),
f))"
is_oadd :: "[i=>o,i,i,i] => o"