--- a/src/ZF/Constructible/WF_absolute.thy Tue Sep 27 13:34:54 2022 +0200
+++ b/src/ZF/Constructible/WF_absolute.thy Tue Sep 27 16:51:35 2022 +0100
@@ -10,14 +10,14 @@
definition
rtrancl_alt :: "[i,i]=>i" where
- "rtrancl_alt(A,r) ==
+ "rtrancl_alt(A,r) \<equiv>
{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) &
(\<forall>i\<in>n. <f`i, f`succ(i)> \<in> r)}"
lemma alt_rtrancl_lemma1 [rule_format]:
"n \<in> nat
- ==> \<forall>f \<in> succ(n) -> field(r).
+ \<Longrightarrow> \<forall>f \<in> succ(n) -> field(r).
(\<forall>i\<in>n. \<langle>f`i, f ` succ(i)\<rangle> \<in> r) \<longrightarrow> \<langle>f`0, f`n\<rangle> \<in> r^*"
apply (induct_tac n)
apply (simp_all add: apply_funtype rtrancl_refl, clarify)
@@ -61,7 +61,7 @@
definition
rtran_closure_mem :: "[i=>o,i,i,i] => o" where
\<comment> \<open>The property of belonging to \<open>rtran_closure(r)\<close>\<close>
- "rtran_closure_mem(M,A,r,p) ==
+ "rtran_closure_mem(M,A,r,p) \<equiv>
\<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) &
@@ -74,28 +74,28 @@
definition
rtran_closure :: "[i=>o,i,i] => o" where
- "rtran_closure(M,r,s) ==
+ "rtran_closure(M,r,s) \<equiv>
\<forall>A[M]. is_field(M,r,A) \<longrightarrow>
(\<forall>p[M]. p \<in> s \<longleftrightarrow> rtran_closure_mem(M,A,r,p))"
definition
tran_closure :: "[i=>o,i,i] => o" where
- "tran_closure(M,r,t) ==
+ "tran_closure(M,r,t) \<equiv>
\<exists>s[M]. rtran_closure(M,r,s) & composition(M,r,s,t)"
locale M_trancl = M_basic +
assumes rtrancl_separation:
- "[| M(r); M(A) |] ==> separation (M, rtran_closure_mem(M,A,r))"
+ "\<lbrakk>M(r); M(A)\<rbrakk> \<Longrightarrow> separation (M, rtran_closure_mem(M,A,r))"
and wellfounded_trancl_separation:
- "[| M(r); M(Z) |] ==>
+ "\<lbrakk>M(r); M(Z)\<rbrakk> \<Longrightarrow>
separation (M, \<lambda>x.
\<exists>w[M]. \<exists>wx[M]. \<exists>rp[M].
w \<in> Z & pair(M,w,x,wx) & tran_closure(M,r,rp) & wx \<in> rp)"
and M_nat [iff] : "M(nat)"
lemma (in M_trancl) rtran_closure_mem_iff:
- "[|M(A); M(r); M(p)|]
- ==> rtran_closure_mem(M,A,r,p) \<longleftrightarrow>
+ "\<lbrakk>M(A); M(r); M(p)\<rbrakk>
+ \<Longrightarrow> rtran_closure_mem(M,A,r,p) \<longleftrightarrow>
(\<exists>n[M]. n\<in>nat &
(\<exists>f[M]. f \<in> succ(n) -> A &
(\<exists>x[M]. \<exists>y[M]. p = <x,y> & f`0 = x & f`n = y) &
@@ -104,21 +104,21 @@
done
lemma (in M_trancl) rtran_closure_rtrancl:
- "M(r) ==> rtran_closure(M,r,rtrancl(r))"
+ "M(r) \<Longrightarrow> rtran_closure(M,r,rtrancl(r))"
apply (simp add: rtran_closure_def rtran_closure_mem_iff
rtrancl_alt_eq_rtrancl [symmetric] rtrancl_alt_def)
apply (auto simp add: nat_0_le [THEN ltD] apply_funtype)
done
lemma (in M_trancl) rtrancl_closed [intro,simp]:
- "M(r) ==> M(rtrancl(r))"
+ "M(r) \<Longrightarrow> M(rtrancl(r))"
apply (insert rtrancl_separation [of r "field(r)"])
apply (simp add: rtrancl_alt_eq_rtrancl [symmetric]
rtrancl_alt_def rtran_closure_mem_iff)
done
lemma (in M_trancl) rtrancl_abs [simp]:
- "[| M(r); M(z) |] ==> rtran_closure(M,r,z) \<longleftrightarrow> z = rtrancl(r)"
+ "\<lbrakk>M(r); M(z)\<rbrakk> \<Longrightarrow> rtran_closure(M,r,z) \<longleftrightarrow> z = rtrancl(r)"
apply (rule iffI)
txt\<open>Proving the right-to-left implication\<close>
prefer 2 apply (blast intro: rtran_closure_rtrancl)
@@ -129,20 +129,20 @@
done
lemma (in M_trancl) trancl_closed [intro,simp]:
- "M(r) ==> M(trancl(r))"
+ "M(r) \<Longrightarrow> M(trancl(r))"
by (simp add: trancl_def)
lemma (in M_trancl) trancl_abs [simp]:
- "[| M(r); M(z) |] ==> tran_closure(M,r,z) \<longleftrightarrow> z = trancl(r)"
+ "\<lbrakk>M(r); M(z)\<rbrakk> \<Longrightarrow> tran_closure(M,r,z) \<longleftrightarrow> z = trancl(r)"
by (simp add: tran_closure_def trancl_def)
lemma (in M_trancl) wellfounded_trancl_separation':
- "[| M(r); M(Z) |] ==> separation (M, \<lambda>x. \<exists>w[M]. w \<in> Z & <w,x> \<in> r^+)"
+ "\<lbrakk>M(r); M(Z)\<rbrakk> \<Longrightarrow> separation (M, \<lambda>x. \<exists>w[M]. w \<in> Z & <w,x> \<in> r^+)"
by (insert wellfounded_trancl_separation [of r Z], simp)
text\<open>Alternative proof of \<open>wf_on_trancl\<close>; inspiration for the
relativized version. Original version is on theory WF.\<close>
-lemma "[| wf[A](r); r-``A \<subseteq> A |] ==> wf[A](r^+)"
+lemma "\<lbrakk>wf[A](r); r-``A \<subseteq> A\<rbrakk> \<Longrightarrow> wf[A](r^+)"
apply (simp add: wf_on_def wf_def)
apply (safe)
apply (drule_tac x = "{x\<in>A. \<exists>w. \<langle>w,x\<rangle> \<in> r^+ & w \<in> Z}" in spec)
@@ -150,8 +150,8 @@
done
lemma (in M_trancl) wellfounded_on_trancl:
- "[| wellfounded_on(M,A,r); r-``A \<subseteq> A; M(r); M(A) |]
- ==> wellfounded_on(M,A,r^+)"
+ "\<lbrakk>wellfounded_on(M,A,r); r-``A \<subseteq> A; M(r); M(A)\<rbrakk>
+ \<Longrightarrow> wellfounded_on(M,A,r^+)"
apply (simp add: wellfounded_on_def)
apply (safe intro!: equalityI)
apply (rename_tac Z x)
@@ -168,7 +168,7 @@
done
lemma (in M_trancl) wellfounded_trancl:
- "[|wellfounded(M,r); M(r)|] ==> wellfounded(M,r^+)"
+ "\<lbrakk>wellfounded(M,r); M(r)\<rbrakk> \<Longrightarrow> wellfounded(M,r^+)"
apply (simp add: wellfounded_iff_wellfounded_on_field)
apply (rule wellfounded_on_subset_A, erule wellfounded_on_trancl)
apply blast
@@ -181,13 +181,13 @@
(*first use is_recfun, then M_is_recfun*)
lemma (in M_trancl) wfrec_relativize:
- "[|wf(r); M(a); M(r);
+ "\<lbrakk>wf(r); M(a); M(r);
strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M].
pair(M,x,y,z) &
is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), g) &
y = H(x, restrict(g, r -`` {x})));
- \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))|]
- ==> wfrec(r,a,H) = z \<longleftrightarrow>
+ \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))\<rbrakk>
+ \<Longrightarrow> wfrec(r,a,H) = z \<longleftrightarrow>
(\<exists>f[M]. is_recfun(r^+, a, \<lambda>x f. H(x, restrict(f, r -`` {x})), f) &
z = H(a,restrict(f,r-``{a})))"
apply (frule wf_trancl)
@@ -204,10 +204,10 @@
The premise \<^term>\<open>relation(r)\<close> is necessary
before we can replace \<^term>\<open>r^+\<close> by \<^term>\<open>r\<close>.\<close>
theorem (in M_trancl) trans_wfrec_relativize:
- "[|wf(r); trans(r); relation(r); M(r); M(a);
+ "\<lbrakk>wf(r); trans(r); relation(r); M(r); M(a);
wfrec_replacement(M,MH,r); relation2(M,MH,H);
- \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))|]
- ==> wfrec(r,a,H) = z \<longleftrightarrow> (\<exists>f[M]. is_recfun(r,a,H,f) & z = H(a,f))"
+ \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))\<rbrakk>
+ \<Longrightarrow> wfrec(r,a,H) = z \<longleftrightarrow> (\<exists>f[M]. is_recfun(r,a,H,f) & z = H(a,f))"
apply (frule wfrec_replacement', assumption+)
apply (simp cong: is_recfun_cong
add: wfrec_relativize trancl_eq_r
@@ -215,18 +215,18 @@
done
theorem (in M_trancl) trans_wfrec_abs:
- "[|wf(r); trans(r); relation(r); M(r); M(a); M(z);
+ "\<lbrakk>wf(r); trans(r); relation(r); M(r); M(a); M(z);
wfrec_replacement(M,MH,r); relation2(M,MH,H);
- \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))|]
- ==> is_wfrec(M,MH,r,a,z) \<longleftrightarrow> z=wfrec(r,a,H)"
+ \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))\<rbrakk>
+ \<Longrightarrow> is_wfrec(M,MH,r,a,z) \<longleftrightarrow> z=wfrec(r,a,H)"
by (simp add: trans_wfrec_relativize [THEN iff_sym] is_wfrec_abs, blast)
lemma (in M_trancl) trans_eq_pair_wfrec_iff:
- "[|wf(r); trans(r); relation(r); M(r); M(y);
+ "\<lbrakk>wf(r); trans(r); relation(r); M(r); M(y);
wfrec_replacement(M,MH,r); relation2(M,MH,H);
- \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))|]
- ==> y = <x, wfrec(r, x, H)> \<longleftrightarrow>
+ \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))\<rbrakk>
+ \<Longrightarrow> y = <x, wfrec(r, x, H)> \<longleftrightarrow>
(\<exists>f[M]. is_recfun(r,x,H,f) & y = <x, H(x,f)>)"
apply safe
apply (simp add: trans_wfrec_relativize [THEN iff_sym, of concl: _ x])
@@ -240,10 +240,10 @@
text\<open>Lemma with the awkward premise mentioning \<open>wfrec\<close>.\<close>
lemma (in M_trancl) wfrec_closed_lemma [rule_format]:
- "[|wf(r); M(r);
+ "\<lbrakk>wf(r); M(r);
strong_replacement(M, \<lambda>x y. y = \<langle>x, wfrec(r, x, H)\<rangle>);
- \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g)) |]
- ==> M(a) \<longrightarrow> M(wfrec(r,a,H))"
+ \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))\<rbrakk>
+ \<Longrightarrow> M(a) \<longrightarrow> M(wfrec(r,a,H))"
apply (rule_tac a=a in wf_induct, assumption+)
apply (subst wfrec, assumption, clarify)
apply (drule_tac x1=x and x="\<lambda>x\<in>r -`` {x}. wfrec(r, x, H)"
@@ -264,10 +264,10 @@
text\<open>Useful version for transitive relations\<close>
theorem (in M_trancl) trans_wfrec_closed:
- "[|wf(r); trans(r); relation(r); M(r); M(a);
+ "\<lbrakk>wf(r); trans(r); relation(r); M(r); M(a);
wfrec_replacement(M,MH,r); relation2(M,MH,H);
- \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g)) |]
- ==> M(wfrec(r,a,H))"
+ \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))\<rbrakk>
+ \<Longrightarrow> M(wfrec(r,a,H))"
apply (frule wfrec_replacement', assumption+)
apply (frule wfrec_replacement_iff [THEN iffD1])
apply (rule wfrec_closed_lemma, assumption+)
@@ -276,13 +276,13 @@
subsection\<open>Absoluteness without assuming transitivity\<close>
lemma (in M_trancl) eq_pair_wfrec_iff:
- "[|wf(r); M(r); M(y);
+ "\<lbrakk>wf(r); M(r); M(y);
strong_replacement(M, \<lambda>x z. \<exists>y[M]. \<exists>g[M].
pair(M,x,y,z) &
is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), g) &
y = H(x, restrict(g, r -`` {x})));
- \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))|]
- ==> y = <x, wfrec(r, x, H)> \<longleftrightarrow>
+ \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))\<rbrakk>
+ \<Longrightarrow> y = <x, wfrec(r, x, H)> \<longleftrightarrow>
(\<exists>f[M]. is_recfun(r^+, x, \<lambda>x f. H(x, restrict(f, r -`` {x})), f) &
y = <x, H(x,restrict(f,r-``{x}))>)"
apply safe
@@ -294,11 +294,11 @@
text\<open>Full version not assuming transitivity, but maybe not very useful.\<close>
theorem (in M_trancl) wfrec_closed:
- "[|wf(r); M(r); M(a);
+ "\<lbrakk>wf(r); M(r); M(a);
wfrec_replacement(M,MH,r^+);
relation2(M,MH, \<lambda>x f. H(x, restrict(f, r -`` {x})));
- \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g)) |]
- ==> M(wfrec(r,a,H))"
+ \<forall>x[M]. \<forall>g[M]. function(g) \<longrightarrow> M(H(x,g))\<rbrakk>
+ \<Longrightarrow> M(wfrec(r,a,H))"
apply (frule wfrec_replacement'
[of MH "r^+" "\<lambda>x f. H(x, restrict(f, r -`` {x}))"])
prefer 4