--- a/src/ZF/Epsilon.thy Tue Sep 27 13:34:54 2022 +0200
+++ b/src/ZF/Epsilon.thy Tue Sep 27 16:51:35 2022 +0100
@@ -9,19 +9,19 @@
definition
eclose :: "i=>i" where
- "eclose(A) == \<Union>n\<in>nat. nat_rec(n, A, %m r. \<Union>(r))"
+ "eclose(A) \<equiv> \<Union>n\<in>nat. nat_rec(n, A, %m r. \<Union>(r))"
definition
transrec :: "[i, [i,i]=>i] =>i" where
- "transrec(a,H) == wfrec(Memrel(eclose({a})), a, H)"
+ "transrec(a,H) \<equiv> wfrec(Memrel(eclose({a})), a, H)"
definition
rank :: "i=>i" where
- "rank(a) == transrec(a, %x f. \<Union>y\<in>x. succ(f`y))"
+ "rank(a) \<equiv> transrec(a, %x f. \<Union>y\<in>x. succ(f`y))"
definition
transrec2 :: "[i, i, [i,i]=>i] =>i" where
- "transrec2(k, a, b) ==
+ "transrec2(k, a, b) \<equiv>
transrec(k,
%i r. if(i=0, a,
if(\<exists>j. i=succ(j),
@@ -30,11 +30,11 @@
definition
recursor :: "[i, [i,i]=>i, i]=>i" where
- "recursor(a,b,k) == transrec(k, %n f. nat_case(a, %m. b(m, f`m), n))"
+ "recursor(a,b,k) \<equiv> transrec(k, %n f. nat_case(a, %m. b(m, f`m), n))"
definition
rec :: "[i, i, [i,i]=>i]=>i" where
- "rec(k,a,b) == recursor(a,b,k)"
+ "rec(k,a,b) \<equiv> recursor(a,b,k)"
subsection\<open>Basic Closure Properties\<close>
@@ -56,19 +56,19 @@
apply (erule UnionI, assumption)
done
-(* @{term"x \<in> eclose(A) ==> x \<subseteq> eclose(A)"} *)
+(* @{term"x \<in> eclose(A) \<Longrightarrow> x \<subseteq> eclose(A)"} *)
lemmas eclose_subset =
Transset_eclose [unfolded Transset_def, THEN bspec]
-(* @{term"[| A \<in> eclose(B); c \<in> A |] ==> c \<in> eclose(B)"} *)
+(* @{term"\<lbrakk>A \<in> eclose(B); c \<in> A\<rbrakk> \<Longrightarrow> c \<in> eclose(B)"} *)
lemmas ecloseD = eclose_subset [THEN subsetD]
lemmas arg_in_eclose_sing = arg_subset_eclose [THEN singleton_subsetD]
lemmas arg_into_eclose_sing = arg_in_eclose_sing [THEN ecloseD]
(* This is epsilon-induction for eclose(A); see also eclose_induct_down...
- [| a \<in> eclose(A); !!x. [| x \<in> eclose(A); \<forall>y\<in>x. P(y) |] ==> P(x)
- |] ==> P(a)
+ \<lbrakk>a \<in> eclose(A); \<And>x. \<lbrakk>x \<in> eclose(A); \<forall>y\<in>x. P(y)\<rbrakk> \<Longrightarrow> P(x)
+\<rbrakk> \<Longrightarrow> P(a)
*)
lemmas eclose_induct =
Transset_induct [OF _ Transset_eclose, induct set: eclose]
@@ -76,7 +76,7 @@
(*Epsilon induction*)
lemma eps_induct:
- "[| !!x. \<forall>y\<in>x. P(y) ==> P(x) |] ==> P(a)"
+ "\<lbrakk>\<And>x. \<forall>y\<in>x. P(y) \<Longrightarrow> P(x)\<rbrakk> \<Longrightarrow> P(a)"
by (rule arg_in_eclose_sing [THEN eclose_induct], blast)
@@ -85,7 +85,7 @@
(** eclose(A) is the least transitive set including A as a subset. **)
lemma eclose_least_lemma:
- "[| Transset(X); A<=X; n \<in> nat |] ==> nat_rec(n, A, %m r. \<Union>(r)) \<subseteq> X"
+ "\<lbrakk>Transset(X); A<=X; n \<in> nat\<rbrakk> \<Longrightarrow> nat_rec(n, A, %m r. \<Union>(r)) \<subseteq> X"
apply (unfold Transset_def)
apply (erule nat_induct)
apply (simp add: nat_rec_0)
@@ -93,17 +93,17 @@
done
lemma eclose_least:
- "[| Transset(X); A<=X |] ==> eclose(A) \<subseteq> X"
+ "\<lbrakk>Transset(X); A<=X\<rbrakk> \<Longrightarrow> eclose(A) \<subseteq> X"
apply (unfold eclose_def)
apply (rule eclose_least_lemma [THEN UN_least], assumption+)
done
-(*COMPLETELY DIFFERENT induction principle from eclose_induct!!*)
+(*COMPLETELY DIFFERENT induction principle from eclose_induct\<And>*)
lemma eclose_induct_down [consumes 1]:
- "[| a \<in> eclose(b);
- !!y. [| y \<in> b |] ==> P(y);
- !!y z. [| y \<in> eclose(b); P(y); z \<in> y |] ==> P(z)
- |] ==> P(a)"
+ "\<lbrakk>a \<in> eclose(b);
+ \<And>y. \<lbrakk>y \<in> b\<rbrakk> \<Longrightarrow> P(y);
+ \<And>y z. \<lbrakk>y \<in> eclose(b); P(y); z \<in> y\<rbrakk> \<Longrightarrow> P(z)
+\<rbrakk> \<Longrightarrow> P(a)"
apply (rule eclose_least [THEN subsetD, THEN CollectD2, of "eclose(b)"])
prefer 3 apply assumption
apply (unfold Transset_def)
@@ -111,49 +111,49 @@
apply (blast intro: arg_subset_eclose [THEN subsetD])
done
-lemma Transset_eclose_eq_arg: "Transset(X) ==> eclose(X) = X"
+lemma Transset_eclose_eq_arg: "Transset(X) \<Longrightarrow> eclose(X) = X"
apply (erule equalityI [OF eclose_least arg_subset_eclose])
apply (rule subset_refl)
done
text\<open>A transitive set either is empty or contains the empty set.\<close>
-lemma Transset_0_lemma [rule_format]: "Transset(A) ==> x\<in>A \<longrightarrow> 0\<in>A"
+lemma Transset_0_lemma [rule_format]: "Transset(A) \<Longrightarrow> x\<in>A \<longrightarrow> 0\<in>A"
apply (simp add: Transset_def)
apply (rule_tac a=x in eps_induct, clarify)
apply (drule bspec, assumption)
apply (case_tac "x=0", auto)
done
-lemma Transset_0_disj: "Transset(A) ==> A=0 | 0\<in>A"
+lemma Transset_0_disj: "Transset(A) \<Longrightarrow> A=0 | 0\<in>A"
by (blast dest: Transset_0_lemma)
subsection\<open>Epsilon Recursion\<close>
(*Unused...*)
-lemma mem_eclose_trans: "[| A \<in> eclose(B); B \<in> eclose(C) |] ==> A \<in> eclose(C)"
+lemma mem_eclose_trans: "\<lbrakk>A \<in> eclose(B); B \<in> eclose(C)\<rbrakk> \<Longrightarrow> A \<in> eclose(C)"
by (rule eclose_least [OF Transset_eclose eclose_subset, THEN subsetD],
assumption+)
(*Variant of the previous lemma in a useable form for the sequel*)
lemma mem_eclose_sing_trans:
- "[| A \<in> eclose({B}); B \<in> eclose({C}) |] ==> A \<in> eclose({C})"
+ "\<lbrakk>A \<in> eclose({B}); B \<in> eclose({C})\<rbrakk> \<Longrightarrow> A \<in> eclose({C})"
by (rule eclose_least [OF Transset_eclose singleton_subsetI, THEN subsetD],
assumption+)
-lemma under_Memrel: "[| Transset(i); j \<in> i |] ==> Memrel(i)-``{j} = j"
+lemma under_Memrel: "\<lbrakk>Transset(i); j \<in> i\<rbrakk> \<Longrightarrow> Memrel(i)-``{j} = j"
by (unfold Transset_def, blast)
-lemma lt_Memrel: "j < i ==> Memrel(i) -`` {j} = j"
+lemma lt_Memrel: "j < i \<Longrightarrow> Memrel(i) -`` {j} = j"
by (simp add: lt_def Ord_def under_Memrel)
-(* @{term"j \<in> eclose(A) ==> Memrel(eclose(A)) -`` j = j"} *)
+(* @{term"j \<in> eclose(A) \<Longrightarrow> Memrel(eclose(A)) -`` j = j"} *)
lemmas under_Memrel_eclose = Transset_eclose [THEN under_Memrel]
lemmas wfrec_ssubst = wf_Memrel [THEN wfrec, THEN ssubst]
lemma wfrec_eclose_eq:
- "[| k \<in> eclose({j}); j \<in> eclose({i}) |] ==>
+ "\<lbrakk>k \<in> eclose({j}); j \<in> eclose({i})\<rbrakk> \<Longrightarrow>
wfrec(Memrel(eclose({i})), k, H) = wfrec(Memrel(eclose({j})), k, H)"
apply (erule eclose_induct)
apply (rule wfrec_ssubst)
@@ -162,7 +162,7 @@
done
lemma wfrec_eclose_eq2:
- "k \<in> i ==> wfrec(Memrel(eclose({i})),k,H) = wfrec(Memrel(eclose({k})),k,H)"
+ "k \<in> i \<Longrightarrow> wfrec(Memrel(eclose({i})),k,H) = wfrec(Memrel(eclose({k})),k,H)"
apply (rule arg_in_eclose_sing [THEN wfrec_eclose_eq])
apply (erule arg_into_eclose_sing)
done
@@ -175,20 +175,20 @@
(*Avoids explosions in proofs; resolve it with a meta-level definition.*)
lemma def_transrec:
- "[| !!x. f(x)==transrec(x,H) |] ==> f(a) = H(a, \<lambda>x\<in>a. f(x))"
+ "\<lbrakk>\<And>x. f(x)\<equiv>transrec(x,H)\<rbrakk> \<Longrightarrow> f(a) = H(a, \<lambda>x\<in>a. f(x))"
apply simp
apply (rule transrec)
done
lemma transrec_type:
- "[| !!x u. [| x \<in> eclose({a}); u \<in> Pi(x,B) |] ==> H(x,u) \<in> B(x) |]
- ==> transrec(a,H) \<in> B(a)"
+ "\<lbrakk>\<And>x u. \<lbrakk>x \<in> eclose({a}); u \<in> Pi(x,B)\<rbrakk> \<Longrightarrow> H(x,u) \<in> B(x)\<rbrakk>
+ \<Longrightarrow> transrec(a,H) \<in> B(a)"
apply (rule_tac i = a in arg_in_eclose_sing [THEN eclose_induct])
apply (subst transrec)
apply (simp add: lam_type)
done
-lemma eclose_sing_Ord: "Ord(i) ==> eclose({i}) \<subseteq> succ(i)"
+lemma eclose_sing_Ord: "Ord(i) \<Longrightarrow> eclose({i}) \<subseteq> succ(i)"
apply (erule Ord_is_Transset [THEN Transset_succ, THEN eclose_least])
apply (rule succI1 [THEN singleton_subsetI])
done
@@ -198,7 +198,7 @@
apply (frule eclose_subset, blast)
done
-lemma eclose_sing_Ord_eq: "Ord(i) ==> eclose({i}) = succ(i)"
+lemma eclose_sing_Ord_eq: "Ord(i) \<Longrightarrow> eclose({i}) = succ(i)"
apply (rule equalityI)
apply (erule eclose_sing_Ord)
apply (rule succ_subset_eclose_sing)
@@ -207,7 +207,7 @@
lemma Ord_transrec_type:
assumes jini: "j \<in> i"
and ordi: "Ord(i)"
- and minor: " !!x u. [| x \<in> i; u \<in> Pi(x,B) |] ==> H(x,u) \<in> B(x)"
+ and minor: " \<And>x u. \<lbrakk>x \<in> i; u \<in> Pi(x,B)\<rbrakk> \<Longrightarrow> H(x,u) \<in> B(x)"
shows "transrec(j,H) \<in> B(j)"
apply (rule transrec_type)
apply (insert jini ordi)
@@ -229,25 +229,25 @@
apply (erule bspec, assumption)
done
-lemma rank_of_Ord: "Ord(i) ==> rank(i) = i"
+lemma rank_of_Ord: "Ord(i) \<Longrightarrow> rank(i) = i"
apply (erule trans_induct)
apply (subst rank)
apply (simp add: Ord_equality)
done
-lemma rank_lt: "a \<in> b ==> rank(a) < rank(b)"
+lemma rank_lt: "a \<in> b \<Longrightarrow> rank(a) < rank(b)"
apply (rule_tac a1 = b in rank [THEN ssubst])
apply (erule UN_I [THEN ltI])
apply (rule_tac [2] Ord_UN, auto)
done
-lemma eclose_rank_lt: "a \<in> eclose(b) ==> rank(a) < rank(b)"
+lemma eclose_rank_lt: "a \<in> eclose(b) \<Longrightarrow> rank(a) < rank(b)"
apply (erule eclose_induct_down)
apply (erule rank_lt)
apply (erule rank_lt [THEN lt_trans], assumption)
done
-lemma rank_mono: "a<=b ==> rank(a) \<le> rank(b)"
+lemma rank_mono: "a<=b \<Longrightarrow> rank(a) \<le> rank(b)"
apply (rule subset_imp_le)
apply (auto simp add: rank [of a] rank [of b])
done
@@ -305,14 +305,14 @@
(*Not clear how to remove the P(a) condition, since the "then" part
must refer to "a"*)
lemma the_equality_if:
- "P(a) ==> (THE x. P(x)) = (if (\<exists>!x. P(x)) then a else 0)"
+ "P(a) \<Longrightarrow> (THE x. P(x)) = (if (\<exists>!x. P(x)) then a else 0)"
by (simp add: the_0 the_equality2)
(*The first premise not only fixs i but ensures @{term"f\<noteq>0"}.
The second premise is now essential. Consider otherwise the relation
r = {<0,0>,<0,1>,<0,2>,...}. Then f`0 = \<Union>(f``{0}) = \<Union>(nat) = nat,
whose rank equals that of r.*)
-lemma rank_apply: "[|i \<in> domain(f); function(f)|] ==> rank(f`i) < rank(f)"
+lemma rank_apply: "\<lbrakk>i \<in> domain(f); function(f)\<rbrakk> \<Longrightarrow> rank(f`i) < rank(f)"
apply clarify
apply (simp add: function_apply_equality)
apply (blast intro: lt_trans rank_lt rank_pair2)
@@ -321,12 +321,12 @@
subsection\<open>Corollaries of Leastness\<close>
-lemma mem_eclose_subset: "A \<in> B ==> eclose(A)<=eclose(B)"
+lemma mem_eclose_subset: "A \<in> B \<Longrightarrow> eclose(A)<=eclose(B)"
apply (rule Transset_eclose [THEN eclose_least])
apply (erule arg_into_eclose [THEN eclose_subset])
done
-lemma eclose_mono: "A<=B ==> eclose(A) \<subseteq> eclose(B)"
+lemma eclose_mono: "A<=B \<Longrightarrow> eclose(A) \<subseteq> eclose(B)"
apply (rule Transset_eclose [THEN eclose_least])
apply (erule subset_trans)
apply (rule arg_subset_eclose)
@@ -352,14 +352,14 @@
done
lemma transrec2_Limit:
- "Limit(i) ==> transrec2(i,a,b) = (\<Union>j<i. transrec2(j,a,b))"
+ "Limit(i) \<Longrightarrow> transrec2(i,a,b) = (\<Union>j<i. transrec2(j,a,b))"
apply (rule transrec2_def [THEN def_transrec, THEN trans])
apply (auto simp add: OUnion_def)
done
lemma def_transrec2:
- "(!!x. f(x)==transrec2(x,a,b))
- ==> f(0) = a &
+ "(\<And>x. f(x)\<equiv>transrec2(x,a,b))
+ \<Longrightarrow> f(0) = a &
f(succ(i)) = b(i, f(i)) &
(Limit(K) \<longrightarrow> f(K) = (\<Union>j<K. f(j)))"
by (simp add: transrec2_Limit)
@@ -390,10 +390,10 @@
done
lemma rec_type:
- "[| n \<in> nat;
+ "\<lbrakk>n \<in> nat;
a \<in> C(0);
- !!m z. [| m \<in> nat; z \<in> C(m) |] ==> b(m,z): C(succ(m)) |]
- ==> rec(n,a,b) \<in> C(n)"
+ \<And>m z. \<lbrakk>m \<in> nat; z \<in> C(m)\<rbrakk> \<Longrightarrow> b(m,z): C(succ(m))\<rbrakk>
+ \<Longrightarrow> rec(n,a,b) \<in> C(n)"
by (erule nat_induct, auto)
end