--- a/src/ZF/Constructible/WF_absolute.thy Mon Jun 24 11:56:27 2002 +0200
+++ b/src/ZF/Constructible/WF_absolute.thy Mon Jun 24 11:57:23 2002 +0200
@@ -1,19 +1,12 @@
-theory WF_absolute = WF_extras + WFrec:
-
+theory WF_absolute = WFrec:
subsection{*Transitive closure without fixedpoints*}
-(*Ordinal.thy: just after succ_le_iff?*)
-lemma Ord_succ_mem_iff: "Ord(j) ==> succ(i) \<in> succ(j) <-> i\<in>j"
-apply (insert succ_le_iff [of i j])
-apply (simp add: lt_def)
-done
-
constdefs
rtrancl_alt :: "[i,i]=>i"
"rtrancl_alt(A,r) ==
{p \<in> A*A. \<exists>n\<in>nat. \<exists>f \<in> succ(n) -> A.
- \<exists>x y. p = <x,y> & f`0 = x & f`n = y &
+ (\<exists>x y. p = <x,y> & f`0 = x & f`n = y) &
(\<forall>i\<in>n. <f`i, f`succ(i)> \<in> r)}"
lemma alt_rtrancl_lemma1 [rule_format]:
@@ -37,8 +30,7 @@
lemma rtrancl_subset_rtrancl_alt: "r^* <= rtrancl_alt(field(r),r)"
apply (simp add: rtrancl_alt_def, clarify)
-apply (frule rtrancl_type [THEN subsetD], clarify)
-apply simp
+apply (frule rtrancl_type [THEN subsetD], clarify, simp)
apply (erule rtrancl_induct)
txt{*Base case, trivial*}
apply (rule_tac x=0 in bexI)
@@ -60,151 +52,314 @@
intro: rtrancl_alt_subset_rtrancl rtrancl_subset_rtrancl_alt)
+constdefs
+
+ rtran_closure :: "[i=>o,i,i] => o"
+ "rtran_closure(M,r,s) ==
+ \<forall>A. M(A) --> is_field(M,r,A) -->
+ (\<forall>p. M(p) -->
+ (p \<in> s <->
+ (\<exists>n\<in>nat. M(n) &
+ (\<exists>n'. M(n') & successor(M,n,n') &
+ (\<exists>f. M(f) & typed_function(M,n',A,f) &
+ (\<exists>x\<in>A. M(x) & (\<exists>y\<in>A. M(y) & pair(M,x,y,p) &
+ fun_apply(M,f,0,x) & fun_apply(M,f,n,y))) &
+ (\<forall>i\<in>n. M(i) -->
+ (\<forall>i'. M(i') --> successor(M,i,i') -->
+ (\<forall>fi. M(fi) --> fun_apply(M,f,i,fi) -->
+ (\<forall>fi'. M(fi') --> fun_apply(M,f,i',fi') -->
+ (\<forall>q. M(q) --> pair(M,fi,fi',q) --> q \<in> r))))))))))"
+
+ tran_closure :: "[i=>o,i,i] => o"
+ "tran_closure(M,r,t) ==
+ \<exists>s. M(s) & rtran_closure(M,r,s) & composition(M,r,s,t)"
+
+
+locale M_trancl = M_axioms +
+(*THEY NEED RELATIVIZATION*)
+ assumes rtrancl_separation:
+ "[| M(r); M(A) |] ==>
+ separation
+ (M, \<lambda>p. \<exists>n\<in>nat. \<exists>f \<in> succ(n) -> A.
+ (\<exists>x y. p = <x,y> & f`0 = x & f`n = y) &
+ (\<forall>i\<in>n. <f`i, f`succ(i)> \<in> r))"
+ and wellfounded_trancl_separation:
+ "[| M(r); M(Z) |] ==> separation (M, \<lambda>x. \<exists>w. M(w) & \<langle>w,x\<rangle> \<in> r^+ & w \<in> Z)"
+
+
+lemma (in M_trancl) rtran_closure_rtrancl:
+ "M(r) ==> rtran_closure(M,r,rtrancl(r))"
+apply (simp add: rtran_closure_def rtrancl_alt_eq_rtrancl [symmetric]
+ rtrancl_alt_def field_closed typed_apply_abs apply_closed
+ Ord_succ_mem_iff M_nat nat_0_le [THEN ltD], clarify)
+apply (rule iffI)
+ apply clarify
+ apply simp
+ apply (rename_tac n f)
+ apply (rule_tac x=n in bexI)
+ apply (rule_tac x=f in exI)
+ apply simp
+ apply (blast dest: finite_fun_closed dest: transM)
+ apply assumption
+apply clarify
+apply (simp add: nat_0_le [THEN ltD] apply_funtype, blast)
+done
+
+lemma (in M_trancl) rtrancl_closed [intro,simp]:
+ "M(r) ==> M(rtrancl(r))"
+apply (insert rtrancl_separation [of r "field(r)"])
+apply (simp add: rtrancl_alt_eq_rtrancl [symmetric]
+ rtrancl_alt_def field_closed typed_apply_abs apply_closed
+ Ord_succ_mem_iff M_nat
+ nat_0_le [THEN ltD] leI [THEN ltD] ltI apply_funtype)
+done
+
+lemma (in M_trancl) rtrancl_abs [simp]:
+ "[| M(r); M(z) |] ==> rtran_closure(M,r,z) <-> z = rtrancl(r)"
+apply (rule iffI)
+ txt{*Proving the right-to-left implication*}
+ prefer 2 apply (blast intro: rtran_closure_rtrancl)
+apply (rule M_equalityI)
+apply (simp add: rtran_closure_def rtrancl_alt_eq_rtrancl [symmetric]
+ rtrancl_alt_def field_closed typed_apply_abs apply_closed
+ Ord_succ_mem_iff M_nat
+ nat_0_le [THEN ltD] leI [THEN ltD] ltI apply_funtype)
+ prefer 2 apply assumption
+ prefer 2 apply blast
+apply (rule iffI, clarify)
+apply (simp add: nat_0_le [THEN ltD] apply_funtype, blast, clarify)
+apply simp
+ apply (rename_tac n f)
+ apply (rule_tac x=n in bexI)
+ apply (rule_tac x=f in exI)
+ apply (blast dest!: finite_fun_closed, assumption)
+done
+
+
+lemma (in M_trancl) trancl_closed [intro,simp]:
+ "M(r) ==> M(trancl(r))"
+by (simp add: trancl_def comp_closed rtrancl_closed)
+
+lemma (in M_trancl) trancl_abs [simp]:
+ "[| M(r); M(z) |] ==> tran_closure(M,r,z) <-> z = trancl(r)"
+by (simp add: tran_closure_def trancl_def)
+
+
+text{*Alternative proof of @{text wf_on_trancl}; inspiration for the
+ relativized version. Original version is on theory WF.*}
+lemma "[| wf[A](r); r-``A <= A |] ==> wf[A](r^+)"
+apply (simp add: wf_on_def wf_def)
+apply (safe intro!: equalityI)
+apply (drule_tac x = "{x\<in>A. \<exists>w. \<langle>w,x\<rangle> \<in> r^+ & w \<in> Z}" in spec)
+apply (blast elim: tranclE)
+done
+
+
+lemma (in M_trancl) wellfounded_on_trancl:
+ "[| wellfounded_on(M,A,r); r-``A <= A; M(r); M(A) |]
+ ==> wellfounded_on(M,A,r^+)"
+apply (simp add: wellfounded_on_def)
+apply (safe intro!: equalityI)
+apply (rename_tac Z x)
+apply (subgoal_tac "M({x\<in>A. \<exists>w. M(w) & \<langle>w,x\<rangle> \<in> r^+ & w \<in> Z})")
+ prefer 2
+ apply (simp add: wellfounded_trancl_separation)
+apply (drule_tac x = "{x\<in>A. \<exists>w. M(w) & \<langle>w,x\<rangle> \<in> r^+ & w \<in> Z}" in spec)
+apply safe
+apply (blast dest: transM, simp)
+apply (rename_tac y w)
+apply (drule_tac x=w in bspec, assumption, clarify)
+apply (erule tranclE)
+ apply (blast dest: transM) (*transM is needed to prove M(xa)*)
+ apply blast
+done
+
+
text{*Relativized to M: Every well-founded relation is a subset of some
inverse image of an ordinal. Key step is the construction (in M) of a
rank function.*}
(*NEEDS RELATIVIZATION*)
-locale M_recursion = M_axioms +
+locale M_recursion = M_trancl +
assumes wfrank_separation':
- "[| M(r); M(a); r \<subseteq> A*A |] ==>
+ "[| M(r); M(A) |] ==>
separation
(M, \<lambda>x. x \<in> A -->
- ~(\<exists>f. M(f) \<and>
- is_recfun(r, x, %x f. \<Union>y \<in> r-``{x}. succ(f`y), f)))"
+ ~(\<exists>f. M(f) \<and> is_recfun(r^+, x, %x f. range(f), f)))"
and wfrank_strong_replacement':
- "[| M(r); M(a); r \<subseteq> A*A |] ==>
- strong_replacement(M, \<lambda>x z. \<exists>y g. M(y) & M(g) &
- pair(M,x,y,z) &
- is_recfun(r, x, %x f. \<Union>y \<in> r-``{x}. succ(f`y), f) &
- y = (\<Union>y \<in> r-``{x}. succ(g`y)))"
+ "M(r) ==>
+ strong_replacement(M, \<lambda>x z. \<exists>y f. M(y) & M(f) &
+ pair(M,x,y,z) & is_recfun(r^+, x, %x f. range(f), f) &
+ y = range(f))"
+ and Ord_wfrank_separation:
+ "[| M(r); M(A) |] ==>
+ separation (M, \<lambda>x. x \<in> A \<longrightarrow>
+ \<not> (\<forall>f. M(f) \<longrightarrow>
+ is_recfun(r^+, x, \<lambda>x. range, f) \<longrightarrow> Ord(range(f))))"
-
-constdefs (*????????????????NEEDED?*)
- is_wfrank_fun :: "[i=>o,i,i,i] => o"
- "is_wfrank_fun(M,r,a,f) ==
- function(f) & domain(f) = r-``{a} &
- (\<forall>x. M(x) --> <x,a> \<in> r --> f`x = (\<Union>y \<in> r-``{x}. succ(f`y)))"
-
-
-
+constdefs
+ wellfoundedrank :: "[i=>o,i,i] => i"
+ "wellfoundedrank(M,r,A) ==
+ {p. x\<in>A, \<exists>y f. M(y) & M(f) &
+ p = <x,y> & is_recfun(r^+, x, %x f. range(f), f) &
+ y = range(f)}"
lemma (in M_recursion) exists_wfrank:
"[| wellfounded(M,r); r \<subseteq> A*A; a\<in>A; M(r); M(A) |]
- ==> \<exists>f. M(f) & is_recfun(r, a, %x g. (\<Union>y \<in> r-``{x}. succ(g`y)), f)"
-apply (rule exists_is_recfun [of A r])
-apply (erule wellfounded_imp_wellfounded_on)
-apply assumption;
-apply (rule trans_Memrel [THEN trans_imp_trans_on], simp)
-apply (rule succI1)
-apply (blast intro: wfrank_separation')
-apply (blast intro: wfrank_strong_replacement')
-apply (simp_all add: Memrel_type Memrel_closed Un_closed image_closed)
+ ==> \<exists>f. M(f) & is_recfun(r^+, a, %x f. range(f), f)"
+apply (rule wellfounded_exists_is_recfun [of A])
+apply (blast intro: wellfounded_on_trancl wellfounded_imp_wellfounded_on)
+apply (rule trans_trancl [THEN trans_imp_trans_on], assumption+)
+apply (simp_all add: trancl_subset_times)
done
-lemma (in M_recursion) exists_wfrank_fun:
- "[| Ord(j); M(i); M(j) |] ==> \<exists>f. M(f) & is_wfrank_fun(M,i,succ(j),f)"
-apply (rule exists_wfrank [THEN exE])
-apply (erule Ord_succ, assumption, simp)
-apply (rename_tac f, clarify)
-apply (frule is_recfun_type)
-apply (rule_tac x=f in exI)
-apply (simp add: fun_is_function domain_of_fun lt_Memrel apply_recfun lt_def
- is_wfrank_fun_eq Ord_trans [OF _ succI1])
+lemma (in M_recursion) M_wellfoundedrank:
+ "[| wellfounded(M,r); r \<subseteq> A*A; M(r); M(A) |]
+ ==> M(wellfoundedrank(M,r,A))"
+apply (insert wfrank_strong_replacement' [of r])
+apply (simp add: wellfoundedrank_def)
+apply (rule strong_replacement_closed)
+ apply assumption+
+ apply (rule univalent_is_recfun)
+ apply (blast intro: wellfounded_on_trancl wellfounded_imp_wellfounded_on)
+ apply (rule trans_on_trancl)
+ apply (simp add: trancl_subset_times)
+ apply blast+
done
-lemma (in M_recursion) is_wfrank_fun_apply:
- "[| x < j; M(i); M(j); M(f); is_wfrank_fun(M,r,a,f) |]
- ==> f`x = i Un (\<Union>k\<in>x. {f ` k})"
-apply (simp add: is_wfrank_fun_eq lt_Ord2)
-apply (frule lt_closed, simp)
-apply (subgoal_tac "x <= domain(f)")
- apply (simp add: Ord_trans [OF _ succI1] image_function)
- apply (blast intro: elim:);
-apply (blast intro: dest!: leI [THEN le_imp_subset] )
-done
-
-lemma (in M_recursion) is_wfrank_fun_eq_wfrank [rule_format]:
- "[| is_wfrank_fun(M,i,J,f); M(i); M(J); M(f); Ord(i); Ord(j) |]
- ==> j<J --> f`j = i++j"
-apply (erule_tac i=j in trans_induct, clarify)
-apply (subgoal_tac "\<forall>k\<in>x. k<J")
- apply (simp (no_asm_simp) add: is_wfrank_def wfrank_unfold is_wfrank_fun_apply)
-apply (blast intro: lt_trans ltI lt_Ord)
-done
-
-lemma (in M_recursion) wfrank_abs_fun_apply_iff:
- "[| M(i); M(J); M(f); M(k); j<J; is_wfrank_fun(M,i,J,f) |]
- ==> fun_apply(M,f,j,k) <-> f`j = k"
-by (auto simp add: lt_def is_wfrank_fun_eq subsetD apply_abs)
-
-lemma (in M_recursion) Ord_wfrank_abs:
- "[| M(i); M(j); M(k); Ord(i); Ord(j) |] ==> is_wfrank(M,r,a,k) <-> k = i++j"
-apply (simp add: is_wfrank_def wfrank_abs_fun_apply_iff is_wfrank_fun_eq_wfrank)
-apply (frule exists_wfrank_fun [of j i], blast+)
+lemma (in M_recursion) Ord_wfrank_range [rule_format]:
+ "[| wellfounded(M,r); r \<subseteq> A*A; a\<in>A; M(r); M(A) |]
+ ==> \<forall>f. M(f) --> is_recfun(r^+, a, %x f. range(f), f) --> Ord(range(f))"
+apply (subgoal_tac "wellfounded_on(M, A, r^+)")
+ prefer 2
+ apply (blast intro: wellfounded_on_trancl wellfounded_imp_wellfounded_on)
+apply (erule wellfounded_on_induct2, assumption+)
+apply (simp add: trancl_subset_times)
+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})"}. *}
+apply (rule OrdI [OF _ Ord_is_Transset])
+ txt{*An ordinal is a transitive set...*}
+ apply (simp add: Transset_def)
+ apply clarify
+ apply (frule apply_recfun2, assumption)
+ apply (force simp add: restrict_iff)
+txt{*...of ordinals. This second case requires the induction hyp.*}
+apply clarify
+apply (rename_tac i y)
+apply (frule apply_recfun2, assumption)
+apply (frule is_recfun_imp_in_r, assumption)
+apply (frule is_recfun_restrict)
+ (*simp_all won't work*)
+ apply (simp add: trans_on_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 (simp add: function_apply_equality [OF _ is_recfun_imp_function])
+apply (blast dest: pair_components_in_M)
done
-lemma (in M_recursion) wfrank_abs:
- "[| M(i); M(j); M(k) |] ==> is_wfrank(M,r,a,k) <-> k = i++j"
-apply (case_tac "Ord(i) & Ord(j)")
- apply (simp add: Ord_wfrank_abs)
-apply (auto simp add: is_wfrank_def wfrank_eq_if_raw_wfrank)
+lemma (in M_recursion) Ord_range_wellfoundedrank:
+ "[| wellfounded(M,r); r \<subseteq> A*A; M(r); M(A) |]
+ ==> Ord (range(wellfoundedrank(M,r,A)))"
+apply (subgoal_tac "wellfounded_on(M, A, r^+)")
+ prefer 2
+ apply (blast intro: wellfounded_on_trancl wellfounded_imp_wellfounded_on)
+apply (frule trancl_subset_times)
+apply (simp add: wellfoundedrank_def)
+apply (rule OrdI [OF _ Ord_is_Transset])
+ prefer 2
+ txt{*by our previous result the range consists of ordinals.*}
+ apply (blast intro: Ord_wfrank_range)
+txt{*We still must show that the range is a transitive set.*}
+apply (simp add: Transset_def, clarify)
+apply simp
+apply (rename_tac x i f u)
+apply (frule is_recfun_imp_in_r, assumption)
+apply (subgoal_tac "M(u) & M(i) & M(x)")
+ prefer 2 apply (blast dest: transM, clarify)
+apply (rule_tac a=u in rangeI)
+apply (rule ReplaceI)
+ apply (rule_tac x=i in exI, simp)
+ apply (rule_tac x="restrict(f, r^+ -`` {u})" in exI)
+ apply (blast intro: is_recfun_restrict trans_on_trancl dest: apply_recfun2)
+ apply blast
+txt{*Unicity requirement of Replacement*}
+apply clarify
+apply (frule apply_recfun2, assumption)
+apply (simp add: trans_on_trancl is_recfun_cut)+
done
-lemma (in M_recursion) wfrank_closed [intro]:
- "[| M(i); M(j) |] ==> M(i++j)"
-apply (simp add: wfrank_eq_if_raw_wfrank, clarify)
-apply (simp add: raw_wfrank_eq_wfrank)
-apply (frule exists_wfrank_fun [of j i], auto)
-apply (simp add: apply_closed is_wfrank_fun_eq_wfrank [symmetric])
+lemma (in M_recursion) function_wellfoundedrank:
+ "[| wellfounded(M,r); r \<subseteq> A*A; M(r); M(A)|]
+ ==> function(wellfoundedrank(M,r,A))"
+apply (simp add: wellfoundedrank_def function_def, clarify)
+txt{*Uniqueness: repeated below!*}
+apply (drule is_recfun_functional, assumption)
+ apply (blast intro: wellfounded_on_trancl wellfounded_imp_wellfounded_on)
+ apply (simp_all add: trancl_subset_times
+ trans_trancl [THEN trans_imp_trans_on])
+apply (blast dest: transM)
done
-
-
-constdefs
- wfrank :: "[i,i]=>i"
- "wfrank(r,a) == wfrec(r, a, %x f. \<Union>y \<in> r-``{x}. succ(f`y))"
-
-constdefs
- wftype :: "i=>i"
- "wftype(r) == \<Union>y \<in> range(r). succ(wfrank(r,y))"
-
-lemma (in M_axioms) wfrank: "wellfounded(M,r) ==> wfrank(r,a) = (\<Union>y \<in> r-``{a}. succ(wfrank(r,y)))"
-by (subst wfrank_def [THEN def_wfrec], simp_all)
-
-lemma (in M_axioms) Ord_wfrank: "wellfounded(M,r) ==> Ord(wfrank(r,a))"
-apply (rule_tac a="a" in wf_induct, assumption)
-apply (subst wfrank, assumption)
-apply (rule Ord_succ [THEN Ord_UN], blast)
+lemma (in M_recursion) domain_wellfoundedrank:
+ "[| wellfounded(M,r); r \<subseteq> A*A; M(r); M(A)|]
+ ==> domain(wellfoundedrank(M,r,A)) = A"
+apply (simp add: wellfoundedrank_def function_def)
+apply (rule equalityI, auto)
+apply (frule transM, assumption)
+apply (frule exists_wfrank, assumption+)
+apply clarify
+apply (rule domainI)
+apply (rule ReplaceI)
+apply (rule_tac x="range(f)" in exI)
+apply simp
+apply (rule_tac x=f in exI, blast)
+apply assumption
+txt{*Uniqueness (for Replacement): repeated above!*}
+apply clarify
+apply (drule is_recfun_functional, assumption)
+ apply (blast intro: wellfounded_on_trancl wellfounded_imp_wellfounded_on)
+ apply (simp_all add: trancl_subset_times
+ trans_trancl [THEN trans_imp_trans_on])
done
-lemma (in M_axioms) wfrank_lt: "[|wellfounded(M,r); <a,b> \<in> r|] ==> wfrank(r,a) < wfrank(r,b)"
-apply (rule_tac a1 = "b" in wfrank [THEN ssubst], assumption)
-apply (rule UN_I [THEN ltI])
-apply (simp add: Ord_wfrank vimage_iff)+
-done
-
-lemma (in M_axioms) Ord_wftype: "wellfounded(M,r) ==> Ord(wftype(r))"
-by (simp add: wftype_def Ord_wfrank)
-
-lemma (in M_axioms) wftypeI: "\<lbrakk>wellfounded(M,r); x \<in> field(r)\<rbrakk> \<Longrightarrow> wfrank(r,x) \<in> wftype(r)"
-apply (simp add: wftype_def)
-apply (blast intro: wfrank_lt [THEN ltD])
+lemma (in M_recursion) wellfoundedrank_type:
+ "[| wellfounded(M,r); r \<subseteq> A*A; M(r); M(A)|]
+ ==> wellfoundedrank(M,r,A) \<in> A -> range(wellfoundedrank(M,r,A))"
+apply (frule function_wellfoundedrank, assumption+)
+apply (frule function_imp_Pi)
+ apply (simp add: wellfoundedrank_def relation_def)
+ apply blast
+apply (simp add: domain_wellfoundedrank)
done
+lemma (in M_recursion) Ord_wellfoundedrank:
+ "[| wellfounded(M,r); a \<in> A; r \<subseteq> A*A; M(r); M(A) |]
+ ==> Ord(wellfoundedrank(M,r,A) ` a)"
+by (blast intro: apply_funtype [OF wellfoundedrank_type]
+ Ord_in_Ord [OF Ord_range_wellfoundedrank])
-lemma (in M_axioms) wf_imp_subset_rvimage:
- "[|wellfounded(M,r); r \<subseteq> A*A|] ==> \<exists>i f. Ord(i) & r <= rvimage(A, f, Memrel(i))"
-apply (rule_tac x="wftype(r)" in exI)
-apply (rule_tac x="\<lambda>x\<in>A. wfrank(r,x)" in exI)
-apply (simp add: Ord_wftype, clarify)
-apply (frule subsetD, assumption, clarify)
-apply (simp add: rvimage_iff wfrank_lt [THEN ltD])
-apply (blast intro: wftypeI )
+lemma (in M_recursion) wellfoundedrank_eq:
+ "[| is_recfun(r^+, a, %x. range, f);
+ wellfounded(M,r); a \<in> A; r \<subseteq> A*A; M(f); M(r); M(A)|]
+ ==> wellfoundedrank(M,r,A) ` a = range(f)"
+apply (rule apply_equality)
+ prefer 2 apply (blast intro: wellfoundedrank_type )
+apply (simp add: wellfoundedrank_def)
+apply (rule ReplaceI)
+ apply (rule_tac x="range(f)" in exI)
+ apply blast
+ apply assumption
+txt{*Unicity requirement of Replacement*}
+apply clarify
+apply (drule is_recfun_functional, assumption)
+ apply (blast intro: wellfounded_on_trancl wellfounded_imp_wellfounded_on)
+ apply (simp_all add: trancl_subset_times
+ trans_trancl [THEN trans_imp_trans_on])
+apply (blast dest: transM)
done
-
-
-
end