diff -r f556a7a9080c -r f3cd78ba687c src/ZF/ZF.thy
--- a/src/ZF/ZF.thy	Tue Apr 11 10:29:25 2017 +0200
+++ b/src/ZF/ZF.thy	Wed Apr 12 13:48:07 2017 +0200
@@ -1,650 +1,73 @@
-(*  Title:      ZF/ZF.thy
-    Author:     Lawrence C Paulson and Martin D Coen, CU Computer Laboratory
-    Copyright   1993  University of Cambridge
-*)
-
-section \<open>Zermelo-Fraenkel Set Theory\<close>
-
-theory ZF
-imports "~~/src/FOL/FOL"
-begin
-
-subsection \<open>Signature\<close>
-
-declare [[eta_contract = false]]
+section\<open>Main ZF Theory: Everything Except AC\<close>
 
-typedecl i
-instance i :: "term" ..
-
-axiomatization mem :: "[i, i] \<Rightarrow> o"  (infixl "\<in>" 50)  \<comment> \<open>membership relation\<close>
-  and zero :: "i"  ("0")  \<comment> \<open>the empty set\<close>
-  and Pow :: "i \<Rightarrow> i"  \<comment> \<open>power sets\<close>
-  and Inf :: "i"  \<comment> \<open>infinite set\<close>
-  and Union :: "i \<Rightarrow> i"  ("\<Union>_" [90] 90)
-  and PrimReplace :: "[i, [i, i] \<Rightarrow> o] \<Rightarrow> i"
-
-abbreviation not_mem :: "[i, i] \<Rightarrow> o"  (infixl "\<notin>" 50)  \<comment> \<open>negated membership relation\<close>
-  where "x \<notin> y \<equiv> \<not> (x \<in> y)"
-
-
-subsection \<open>Bounded Quantifiers\<close>
-
-definition Ball :: "[i, i \<Rightarrow> o] \<Rightarrow> o"
-  where "Ball(A, P) \<equiv> \<forall>x. x\<in>A \<longrightarrow> P(x)"
+theory ZF imports List_ZF IntDiv_ZF CardinalArith begin
 
-definition Bex :: "[i, i \<Rightarrow> o] \<Rightarrow> o"
-  where "Bex(A, P) \<equiv> \<exists>x. x\<in>A \<and> P(x)"
-
-syntax
-  "_Ball" :: "[pttrn, i, o] \<Rightarrow> o"  ("(3\<forall>_\<in>_./ _)" 10)
-  "_Bex" :: "[pttrn, i, o] \<Rightarrow> o"  ("(3\<exists>_\<in>_./ _)" 10)
-translations
-  "\<forall>x\<in>A. P" \<rightleftharpoons> "CONST Ball(A, \<lambda>x. P)"
-  "\<exists>x\<in>A. P" \<rightleftharpoons> "CONST Bex(A, \<lambda>x. P)"
-
-
-subsection \<open>Variations on Replacement\<close>
+(*The theory of "iterates" logically belongs to Nat, but can't go there because
+  primrec isn't available into after Datatype.*)
 
-(* Derived form of replacement, restricting P to its functional part.
-   The resulting set (for functional P) is the same as with
-   PrimReplace, but the rules are simpler. *)
-definition Replace :: "[i, [i, i] \<Rightarrow> o] \<Rightarrow> i"
-  where "Replace(A,P) == PrimReplace(A, %x y. (\<exists>!z. P(x,z)) & P(x,y))"
-
-syntax
-  "_Replace"  :: "[pttrn, pttrn, i, o] => i"  ("(1{_ ./ _ \<in> _, _})")
-translations
-  "{y. x\<in>A, Q}" \<rightleftharpoons> "CONST Replace(A, \<lambda>x y. Q)"
-
-
-(* Functional form of replacement -- analgous to ML's map functional *)
-
-definition RepFun :: "[i, i \<Rightarrow> i] \<Rightarrow> i"
-  where "RepFun(A,f) == {y . x\<in>A, y=f(x)}"
-
-syntax
-  "_RepFun" :: "[i, pttrn, i] => i"  ("(1{_ ./ _ \<in> _})" [51,0,51])
-translations
-  "{b. x\<in>A}" \<rightleftharpoons> "CONST RepFun(A, \<lambda>x. b)"
-
+subsection\<open>Iteration of the function @{term F}\<close>
 
-(* Separation and Pairing can be derived from the Replacement
-   and Powerset Axioms using the following definitions. *)
-definition Collect :: "[i, i \<Rightarrow> o] \<Rightarrow> i"
-  where "Collect(A,P) == {y . x\<in>A, x=y & P(x)}"
-
-syntax
-  "_Collect" :: "[pttrn, i, o] \<Rightarrow> i"  ("(1{_ \<in> _ ./ _})")
-translations
-  "{x\<in>A. P}" \<rightleftharpoons> "CONST Collect(A, \<lambda>x. P)"
-
-
-subsection \<open>General union and intersection\<close>
-
-definition Inter :: "i => i"  ("\<Inter>_" [90] 90)
-  where "\<Inter>(A) == { x\<in>\<Union>(A) . \<forall>y\<in>A. x\<in>y}"
-
-syntax
-  "_UNION" :: "[pttrn, i, i] => i"  ("(3\<Union>_\<in>_./ _)" 10)
-  "_INTER" :: "[pttrn, i, i] => i"  ("(3\<Inter>_\<in>_./ _)" 10)
-translations
-  "\<Union>x\<in>A. B" == "CONST Union({B. x\<in>A})"
-  "\<Inter>x\<in>A. B" == "CONST Inter({B. x\<in>A})"
-
-
-subsection \<open>Finite sets and binary operations\<close>
-
-(*Unordered pairs (Upair) express binary union/intersection and cons;
-  set enumerations translate as {a,...,z} = cons(a,...,cons(z,0)...)*)
-
-definition Upair :: "[i, i] => i"
-  where "Upair(a,b) == {y. x\<in>Pow(Pow(0)), (x=0 & y=a) | (x=Pow(0) & y=b)}"
-
-definition Subset :: "[i, i] \<Rightarrow> o"  (infixl "\<subseteq>" 50)  \<comment> \<open>subset relation\<close>
-  where subset_def: "A \<subseteq> B \<equiv> \<forall>x\<in>A. x\<in>B"
+consts  iterates :: "[i=>i,i,i] => i"   ("(_^_ '(_'))" [60,1000,1000] 60)
 
-definition Diff :: "[i, i] \<Rightarrow> i"  (infixl "-" 65)  \<comment> \<open>set difference\<close>
-  where "A - B == { x\<in>A . ~(x\<in>B) }"
-
-definition Un :: "[i, i] \<Rightarrow> i"  (infixl "\<union>" 65)  \<comment> \<open>binary union\<close>
-  where "A \<union> B == \<Union>(Upair(A,B))"
-
-definition Int :: "[i, i] \<Rightarrow> i"  (infixl "\<inter>" 70)  \<comment> \<open>binary intersection\<close>
-  where "A \<inter> B == \<Inter>(Upair(A,B))"
-
-definition cons :: "[i, i] => i"
-  where "cons(a,A) == Upair(a,a) \<union> A"
-
-definition succ :: "i => i"
-  where "succ(i) == cons(i, i)"
+primrec
+    "F^0 (x) = x"
+    "F^(succ(n)) (x) = F(F^n (x))"
 
-nonterminal "is"
-syntax
-  "" :: "i \<Rightarrow> is"  ("_")
-  "_Enum" :: "[i, is] \<Rightarrow> is"  ("_,/ _")
-  "_Finset" :: "is \<Rightarrow> i"  ("{(_)}")
-translations
-  "{x, xs}" == "CONST cons(x, {xs})"
-  "{x}" == "CONST cons(x, 0)"
-
-
-subsection \<open>Axioms\<close>
-
-(* ZF axioms -- see Suppes p.238
-   Axioms for Union, Pow and Replace state existence only,
-   uniqueness is derivable using extensionality. *)
-
-axiomatization
-where
-  extension:     "A = B \<longleftrightarrow> A \<subseteq> B \<and> B \<subseteq> A" and
-  Union_iff:     "A \<in> \<Union>(C) \<longleftrightarrow> (\<exists>B\<in>C. A\<in>B)" and
-  Pow_iff:       "A \<in> Pow(B) \<longleftrightarrow> A \<subseteq> B" and
-
-  (*We may name this set, though it is not uniquely defined.*)
-  infinity:      "0 \<in> Inf \<and> (\<forall>y\<in>Inf. succ(y) \<in> Inf)" and
+definition
+  iterates_omega :: "[i=>i,i] => i" ("(_^\<omega> '(_'))" [60,1000] 60) where
+    "F^\<omega> (x) == \<Union>n\<in>nat. F^n (x)"
 
-  (*This formulation facilitates case analysis on A.*)
-  foundation:    "A = 0 \<or> (\<exists>x\<in>A. \<forall>y\<in>x. y\<notin>A)" and
-
-  (*Schema axiom since predicate P is a higher-order variable*)
-  replacement:   "(\<forall>x\<in>A. \<forall>y z. P(x,y) \<and> P(x,z) \<longrightarrow> y = z) \<Longrightarrow>
-                         b \<in> PrimReplace(A,P) \<longleftrightarrow> (\<exists>x\<in>A. P(x,b))"
-
-
-subsection \<open>Definite descriptions -- via Replace over the set "1"\<close>
-
-definition The :: "(i \<Rightarrow> o) \<Rightarrow> i"  (binder "THE " 10)
-  where the_def: "The(P)    == \<Union>({y . x \<in> {0}, P(y)})"
-
-definition If :: "[o, i, i] \<Rightarrow> i"  ("(if (_)/ then (_)/ else (_))" [10] 10)
-  where if_def: "if P then a else b == THE z. P & z=a | ~P & z=b"
-
-abbreviation (input)
-  old_if :: "[o, i, i] => i"  ("if '(_,_,_')")
-  where "if(P,a,b) == If(P,a,b)"
-
-
-subsection \<open>Ordered Pairing\<close>
-
-(* this "symmetric" definition works better than {{a}, {a,b}} *)
-definition Pair :: "[i, i] => i"
-  where "Pair(a,b) == {{a,a}, {a,b}}"
-
-definition fst :: "i \<Rightarrow> i"
-  where "fst(p) == THE a. \<exists>b. p = Pair(a, b)"
-
-definition snd :: "i \<Rightarrow> i"
-  where "snd(p) == THE b. \<exists>a. p = Pair(a, b)"
+lemma iterates_triv:
+     "[| n\<in>nat;  F(x) = x |] ==> F^n (x) = x"
+by (induct n rule: nat_induct, simp_all)
 
-definition split :: "[[i, i] \<Rightarrow> 'a, i] \<Rightarrow> 'a::{}"  \<comment> \<open>for pattern-matching\<close>
-  where "split(c) == \<lambda>p. c(fst(p), snd(p))"
-
-(* Patterns -- extends pre-defined type "pttrn" used in abstractions *)
-nonterminal patterns
-syntax
-  "_pattern"  :: "patterns => pttrn"         ("\<langle>_\<rangle>")
-  ""          :: "pttrn => patterns"         ("_")
-  "_patterns" :: "[pttrn, patterns] => patterns"  ("_,/_")
-  "_Tuple"    :: "[i, is] => i"              ("\<langle>(_,/ _)\<rangle>")
-translations
-  "\<langle>x, y, z\<rangle>"   == "\<langle>x, \<langle>y, z\<rangle>\<rangle>"
-  "\<langle>x, y\<rangle>"      == "CONST Pair(x, y)"
-  "\<lambda>\<langle>x,y,zs\<rangle>.b" == "CONST split(\<lambda>x \<langle>y,zs\<rangle>.b)"
-  "\<lambda>\<langle>x,y\<rangle>.b"    == "CONST split(\<lambda>x y. b)"
-
-definition Sigma :: "[i, i \<Rightarrow> i] \<Rightarrow> i"
-  where "Sigma(A,B) == \<Union>x\<in>A. \<Union>y\<in>B(x). {\<langle>x,y\<rangle>}"
-
-abbreviation cart_prod :: "[i, i] => i"  (infixr "\<times>" 80)  \<comment> \<open>Cartesian product\<close>
-  where "A \<times> B \<equiv> Sigma(A, \<lambda>_. B)"
-
-
-subsection \<open>Relations and Functions\<close>
-
-(*converse of relation r, inverse of function*)
-definition converse :: "i \<Rightarrow> i"
-  where "converse(r) == {z. w\<in>r, \<exists>x y. w=\<langle>x,y\<rangle> \<and> z=\<langle>y,x\<rangle>}"
-
-definition domain :: "i \<Rightarrow> i"
-  where "domain(r) == {x. w\<in>r, \<exists>y. w=\<langle>x,y\<rangle>}"
-
-definition range :: "i \<Rightarrow> i"
-  where "range(r) == domain(converse(r))"
-
-definition field :: "i \<Rightarrow> i"
-  where "field(r) == domain(r) \<union> range(r)"
+lemma iterates_type [TC]:
+     "[| n \<in> nat;  a \<in> A; !!x. x \<in> A ==> F(x) \<in> A |]
+      ==> F^n (a) \<in> A"
+by (induct n rule: nat_induct, simp_all)
 
-definition relation :: "i \<Rightarrow> o"  \<comment> \<open>recognizes sets of pairs\<close>
-  where "relation(r) == \<forall>z\<in>r. \<exists>x y. z = \<langle>x,y\<rangle>"
-
-definition "function" :: "i \<Rightarrow> o"  \<comment> \<open>recognizes functions; can have non-pairs\<close>
-  where "function(r) == \<forall>x y. \<langle>x,y\<rangle> \<in> r \<longrightarrow> (\<forall>y'. \<langle>x,y'\<rangle> \<in> r \<longrightarrow> y = y')"
-
-definition Image :: "[i, i] \<Rightarrow> i"  (infixl "``" 90)  \<comment> \<open>image\<close>
-  where image_def: "r `` A  == {y \<in> range(r). \<exists>x\<in>A. \<langle>x,y\<rangle> \<in> r}"
-
-definition vimage :: "[i, i] \<Rightarrow> i"  (infixl "-``" 90)  \<comment> \<open>inverse image\<close>
-  where vimage_def: "r -`` A == converse(r)``A"
-
-(* Restrict the relation r to the domain A *)
-definition restrict :: "[i, i] \<Rightarrow> i"
-  where "restrict(r,A) == {z \<in> r. \<exists>x\<in>A. \<exists>y. z = \<langle>x,y\<rangle>}"
-
-
-(* Abstraction, application and Cartesian product of a family of sets *)
-
-definition Lambda :: "[i, i \<Rightarrow> i] \<Rightarrow> i"
-  where lam_def: "Lambda(A,b) == {\<langle>x,b(x)\<rangle>. x\<in>A}"
-
-definition "apply" :: "[i, i] \<Rightarrow> i"  (infixl "`" 90)  \<comment> \<open>function application\<close>
-  where "f`a == \<Union>(f``{a})"
-
-definition Pi :: "[i, i \<Rightarrow> i] \<Rightarrow> i"
-  where "Pi(A,B) == {f\<in>Pow(Sigma(A,B)). A\<subseteq>domain(f) & function(f)}"
-
-abbreviation function_space :: "[i, i] \<Rightarrow> i"  (infixr "->" 60)  \<comment> \<open>function space\<close>
-  where "A -> B \<equiv> Pi(A, \<lambda>_. B)"
-
-
-(* binder syntax *)
+lemma iterates_omega_triv:
+    "F(x) = x ==> F^\<omega> (x) = x"
+by (simp add: iterates_omega_def iterates_triv)
 
-syntax
-  "_PROD"     :: "[pttrn, i, i] => i"        ("(3\<Prod>_\<in>_./ _)" 10)
-  "_SUM"      :: "[pttrn, i, i] => i"        ("(3\<Sum>_\<in>_./ _)" 10)
-  "_lam"      :: "[pttrn, i, i] => i"        ("(3\<lambda>_\<in>_./ _)" 10)
-translations
-  "\<Prod>x\<in>A. B"   == "CONST Pi(A, \<lambda>x. B)"
-  "\<Sum>x\<in>A. B"   == "CONST Sigma(A, \<lambda>x. B)"
-  "\<lambda>x\<in>A. f"    == "CONST Lambda(A, \<lambda>x. f)"
-
-
-subsection \<open>ASCII syntax\<close>
-
-notation (ASCII)
-  cart_prod       (infixr "*" 80) and
-  Int             (infixl "Int" 70) and
-  Un              (infixl "Un" 65) and
-  function_space  (infixr "\<rightarrow>" 60) and
-  Subset          (infixl "<=" 50) and
-  mem             (infixl ":" 50) and
-  not_mem         (infixl "~:" 50)
+lemma Ord_iterates [simp]:
+     "[| n\<in>nat;  !!i. Ord(i) ==> Ord(F(i));  Ord(x) |]
+      ==> Ord(F^n (x))"
+by (induct n rule: nat_induct, simp_all)
 
-syntax (ASCII)
-  "_Ball"     :: "[pttrn, i, o] => o"        ("(3ALL _:_./ _)" 10)
-  "_Bex"      :: "[pttrn, i, o] => o"        ("(3EX _:_./ _)" 10)
-  "_Collect"  :: "[pttrn, i, o] => i"        ("(1{_: _ ./ _})")
-  "_Replace"  :: "[pttrn, pttrn, i, o] => i" ("(1{_ ./ _: _, _})")
-  "_RepFun"   :: "[i, pttrn, i] => i"        ("(1{_ ./ _: _})" [51,0,51])
-  "_UNION"    :: "[pttrn, i, i] => i"        ("(3UN _:_./ _)" 10)
-  "_INTER"    :: "[pttrn, i, i] => i"        ("(3INT _:_./ _)" 10)
-  "_PROD"     :: "[pttrn, i, i] => i"        ("(3PROD _:_./ _)" 10)
-  "_SUM"      :: "[pttrn, i, i] => i"        ("(3SUM _:_./ _)" 10)
-  "_lam"      :: "[pttrn, i, i] => i"        ("(3lam _:_./ _)" 10)
-  "_Tuple"    :: "[i, is] => i"              ("<(_,/ _)>")
-  "_pattern"  :: "patterns => pttrn"         ("<_>")
-
-
-subsection \<open>Substitution\<close>
-
-(*Useful examples:  singletonI RS subst_elem,  subst_elem RSN (2,IntI) *)
-lemma subst_elem: "[| b\<in>A;  a=b |] ==> a\<in>A"
-by (erule ssubst, assumption)
+lemma iterates_commute: "n \<in> nat ==> F(F^n (x)) = F^n (F(x))"
+by (induct_tac n, simp_all)
 
 
-subsection\<open>Bounded universal quantifier\<close>
-
-lemma ballI [intro!]: "[| !!x. x\<in>A ==> P(x) |] ==> \<forall>x\<in>A. P(x)"
-by (simp add: Ball_def)
-
-lemmas strip = impI allI ballI
-
-lemma bspec [dest?]: "[| \<forall>x\<in>A. P(x);  x: A |] ==> P(x)"
-by (simp add: Ball_def)
-
-(*Instantiates x first: better for automatic theorem proving?*)
-lemma rev_ballE [elim]:
-    "[| \<forall>x\<in>A. P(x);  x\<notin>A ==> Q;  P(x) ==> Q |] ==> Q"
-by (simp add: Ball_def, blast)
-
-lemma ballE: "[| \<forall>x\<in>A. P(x);  P(x) ==> Q;  x\<notin>A ==> Q |] ==> Q"
-by blast
-
-(*Used in the datatype package*)
-lemma rev_bspec: "[| x: A;  \<forall>x\<in>A. P(x) |] ==> P(x)"
-by (simp add: Ball_def)
-
-(*Trival rewrite rule;   @{term"(\<forall>x\<in>A.P)<->P"} holds only if A is nonempty!*)
-lemma ball_triv [simp]: "(\<forall>x\<in>A. P) <-> ((\<exists>x. x\<in>A) \<longrightarrow> P)"
-by (simp add: Ball_def)
-
-(*Congruence rule for rewriting*)
-lemma ball_cong [cong]:
-    "[| A=A';  !!x. x\<in>A' ==> P(x) <-> P'(x) |] ==> (\<forall>x\<in>A. P(x)) <-> (\<forall>x\<in>A'. P'(x))"
-by (simp add: Ball_def)
-
-lemma atomize_ball:
-    "(!!x. x \<in> A ==> P(x)) == Trueprop (\<forall>x\<in>A. P(x))"
-  by (simp only: Ball_def atomize_all atomize_imp)
-
-lemmas [symmetric, rulify] = atomize_ball
-  and [symmetric, defn] = atomize_ball
-
-
-subsection\<open>Bounded existential quantifier\<close>
+subsection\<open>Transfinite Recursion\<close>
 
-lemma bexI [intro]: "[| P(x);  x: A |] ==> \<exists>x\<in>A. P(x)"
-by (simp add: Bex_def, blast)
-
-(*The best argument order when there is only one @{term"x\<in>A"}*)
-lemma rev_bexI: "[| x\<in>A;  P(x) |] ==> \<exists>x\<in>A. P(x)"
-by blast
-
-(*Not of the general form for such rules. The existential quanitifer becomes universal. *)
-lemma bexCI: "[| \<forall>x\<in>A. ~P(x) ==> P(a);  a: A |] ==> \<exists>x\<in>A. P(x)"
-by blast
-
-lemma bexE [elim!]: "[| \<exists>x\<in>A. P(x);  !!x. [| x\<in>A; P(x) |] ==> Q |] ==> Q"
-by (simp add: Bex_def, blast)
-
-(*We do not even have @{term"(\<exists>x\<in>A. True) <-> True"} unless @{term"A" is nonempty!!*)
-lemma bex_triv [simp]: "(\<exists>x\<in>A. P) <-> ((\<exists>x. x\<in>A) & P)"
-by (simp add: Bex_def)
-
-lemma bex_cong [cong]:
-    "[| A=A';  !!x. x\<in>A' ==> P(x) <-> P'(x) |]
-     ==> (\<exists>x\<in>A. P(x)) <-> (\<exists>x\<in>A'. P'(x))"
-by (simp add: Bex_def cong: conj_cong)
-
-
-
-subsection\<open>Rules for subsets\<close>
-
-lemma subsetI [intro!]:
-    "(!!x. x\<in>A ==> x\<in>B) ==> A \<subseteq> B"
-by (simp add: subset_def)
-
-(*Rule in Modus Ponens style [was called subsetE] *)
-lemma subsetD [elim]: "[| A \<subseteq> B;  c\<in>A |] ==> c\<in>B"
-apply (unfold subset_def)
-apply (erule bspec, assumption)
-done
-
-(*Classical elimination rule*)
-lemma subsetCE [elim]:
-    "[| A \<subseteq> B;  c\<notin>A ==> P;  c\<in>B ==> P |] ==> P"
-by (simp add: subset_def, blast)
+text\<open>Transfinite recursion for definitions based on the
+    three cases of ordinals\<close>
 
-(*Sometimes useful with premises in this order*)
-lemma rev_subsetD: "[| c\<in>A; A<=B |] ==> c\<in>B"
-by blast
-
-lemma contra_subsetD: "[| A \<subseteq> B; c \<notin> B |] ==> c \<notin> A"
-by blast
-
-lemma rev_contra_subsetD: "[| c \<notin> B;  A \<subseteq> B |] ==> c \<notin> A"
-by blast
-
-lemma subset_refl [simp]: "A \<subseteq> A"
-by blast
-
-lemma subset_trans: "[| A<=B;  B<=C |] ==> A<=C"
-by blast
-
-(*Useful for proving A<=B by rewriting in some cases*)
-lemma subset_iff:
-     "A<=B <-> (\<forall>x. x\<in>A \<longrightarrow> x\<in>B)"
-apply (unfold subset_def Ball_def)
-apply (rule iff_refl)
-done
-
-text\<open>For calculations\<close>
-declare subsetD [trans] rev_subsetD [trans] subset_trans [trans]
-
-
-subsection\<open>Rules for equality\<close>
-
-(*Anti-symmetry of the subset relation*)
-lemma equalityI [intro]: "[| A \<subseteq> B;  B \<subseteq> A |] ==> A = B"
-by (rule extension [THEN iffD2], rule conjI)
-
-
-lemma equality_iffI: "(!!x. x\<in>A <-> x\<in>B) ==> A = B"
-by (rule equalityI, blast+)
-
-lemmas equalityD1 = extension [THEN iffD1, THEN conjunct1]
-lemmas equalityD2 = extension [THEN iffD1, THEN conjunct2]
-
-lemma equalityE: "[| A = B;  [| A<=B; B<=A |] ==> P |]  ==>  P"
-by (blast dest: equalityD1 equalityD2)
+definition
+  transrec3 :: "[i, i, [i,i]=>i, [i,i]=>i] =>i" where
+    "transrec3(k, a, b, c) ==
+       transrec(k, \<lambda>x r.
+         if x=0 then a
+         else if Limit(x) then c(x, \<lambda>y\<in>x. r`y)
+         else b(Arith.pred(x), r ` Arith.pred(x)))"
 
-lemma equalityCE:
-    "[| A = B;  [| c\<in>A; c\<in>B |] ==> P;  [| c\<notin>A; c\<notin>B |] ==> P |]  ==>  P"
-by (erule equalityE, blast)
-
-lemma equality_iffD:
-  "A = B ==> (!!x. x \<in> A <-> x \<in> B)"
-  by auto
-
-
-subsection\<open>Rules for Replace -- the derived form of replacement\<close>
-
-lemma Replace_iff:
-    "b \<in> {y. x\<in>A, P(x,y)}  <->  (\<exists>x\<in>A. P(x,b) & (\<forall>y. P(x,y) \<longrightarrow> y=b))"
-apply (unfold Replace_def)
-apply (rule replacement [THEN iff_trans], blast+)
-done
+lemma transrec3_0 [simp]: "transrec3(0,a,b,c) = a"
+by (rule transrec3_def [THEN def_transrec, THEN trans], simp)
 
-(*Introduction; there must be a unique y such that P(x,y), namely y=b. *)
-lemma ReplaceI [intro]:
-    "[| P(x,b);  x: A;  !!y. P(x,y) ==> y=b |] ==>
-     b \<in> {y. x\<in>A, P(x,y)}"
-by (rule Replace_iff [THEN iffD2], blast)
-
-(*Elimination; may asssume there is a unique y such that P(x,y), namely y=b. *)
-lemma ReplaceE:
-    "[| b \<in> {y. x\<in>A, P(x,y)};
-        !!x. [| x: A;  P(x,b);  \<forall>y. P(x,y)\<longrightarrow>y=b |] ==> R
-     |] ==> R"
-by (rule Replace_iff [THEN iffD1, THEN bexE], simp+)
+lemma transrec3_succ [simp]:
+     "transrec3(succ(i),a,b,c) = b(i, transrec3(i,a,b,c))"
+by (rule transrec3_def [THEN def_transrec, THEN trans], simp)
 
-(*As above but without the (generally useless) 3rd assumption*)
-lemma ReplaceE2 [elim!]:
-    "[| b \<in> {y. x\<in>A, P(x,y)};
-        !!x. [| x: A;  P(x,b) |] ==> R
-     |] ==> R"
-by (erule ReplaceE, blast)
-
-lemma Replace_cong [cong]:
-    "[| A=B;  !!x y. x\<in>B ==> P(x,y) <-> Q(x,y) |] ==>
-     Replace(A,P) = Replace(B,Q)"
-apply (rule equality_iffI)
-apply (simp add: Replace_iff)
-done
+lemma transrec3_Limit:
+     "Limit(i) ==>
+      transrec3(i,a,b,c) = c(i, \<lambda>j\<in>i. transrec3(j,a,b,c))"
+by (rule transrec3_def [THEN def_transrec, THEN trans], force)
 
 
-subsection\<open>Rules for RepFun\<close>
-
-lemma RepFunI: "a \<in> A ==> f(a) \<in> {f(x). x\<in>A}"
-by (simp add: RepFun_def Replace_iff, blast)
-
-(*Useful for coinduction proofs*)
-lemma RepFun_eqI [intro]: "[| b=f(a);  a \<in> A |] ==> b \<in> {f(x). x\<in>A}"
-apply (erule ssubst)
-apply (erule RepFunI)
-done
-
-lemma RepFunE [elim!]:
-    "[| b \<in> {f(x). x\<in>A};
-        !!x.[| x\<in>A;  b=f(x) |] ==> P |] ==>
-     P"
-by (simp add: RepFun_def Replace_iff, blast)
-
-lemma RepFun_cong [cong]:
-    "[| A=B;  !!x. x\<in>B ==> f(x)=g(x) |] ==> RepFun(A,f) = RepFun(B,g)"
-by (simp add: RepFun_def)
-
-lemma RepFun_iff [simp]: "b \<in> {f(x). x\<in>A} <-> (\<exists>x\<in>A. b=f(x))"
-by (unfold Bex_def, blast)
-
-lemma triv_RepFun [simp]: "{x. x\<in>A} = A"
-by blast
-
-
-subsection\<open>Rules for Collect -- forming a subset by separation\<close>
-
-(*Separation is derivable from Replacement*)
-lemma separation [simp]: "a \<in> {x\<in>A. P(x)} <-> a\<in>A & P(a)"
-by (unfold Collect_def, blast)
-
-lemma CollectI [intro!]: "[| a\<in>A;  P(a) |] ==> a \<in> {x\<in>A. P(x)}"
-by simp
-
-lemma CollectE [elim!]: "[| a \<in> {x\<in>A. P(x)};  [| a\<in>A; P(a) |] ==> R |] ==> R"
-by simp
-
-lemma CollectD1: "a \<in> {x\<in>A. P(x)} ==> a\<in>A"
-by (erule CollectE, assumption)
-
-lemma CollectD2: "a \<in> {x\<in>A. P(x)} ==> P(a)"
-by (erule CollectE, assumption)
-
-lemma Collect_cong [cong]:
-    "[| A=B;  !!x. x\<in>B ==> P(x) <-> Q(x) |]
-     ==> Collect(A, %x. P(x)) = Collect(B, %x. Q(x))"
-by (simp add: Collect_def)
-
-
-subsection\<open>Rules for Unions\<close>
-
-declare Union_iff [simp]
-
-(*The order of the premises presupposes that C is rigid; A may be flexible*)
-lemma UnionI [intro]: "[| B: C;  A: B |] ==> A: \<Union>(C)"
-by (simp, blast)
-
-lemma UnionE [elim!]: "[| A \<in> \<Union>(C);  !!B.[| A: B;  B: C |] ==> R |] ==> R"
-by (simp, blast)
-
-
-subsection\<open>Rules for Unions of families\<close>
-(* @{term"\<Union>x\<in>A. B(x)"} abbreviates @{term"\<Union>({B(x). x\<in>A})"} *)
-
-lemma UN_iff [simp]: "b \<in> (\<Union>x\<in>A. B(x)) <-> (\<exists>x\<in>A. b \<in> B(x))"
-by (simp add: Bex_def, blast)
-
-(*The order of the premises presupposes that A is rigid; b may be flexible*)
-lemma UN_I: "[| a: A;  b: B(a) |] ==> b: (\<Union>x\<in>A. B(x))"
-by (simp, blast)
-
-
-lemma UN_E [elim!]:
-    "[| b \<in> (\<Union>x\<in>A. B(x));  !!x.[| x: A;  b: B(x) |] ==> R |] ==> R"
-by blast
-
-lemma UN_cong:
-    "[| A=B;  !!x. x\<in>B ==> C(x)=D(x) |] ==> (\<Union>x\<in>A. C(x)) = (\<Union>x\<in>B. D(x))"
-by simp
-
-
-(*No "Addcongs [UN_cong]" because @{term\<Union>} is a combination of constants*)
-
-(* UN_E appears before UnionE so that it is tried first, to avoid expensive
-  calls to hyp_subst_tac.  Cannot include UN_I as it is unsafe: would enlarge
-  the search space.*)
-
-
-subsection\<open>Rules for the empty set\<close>
-
-(*The set @{term"{x\<in>0. False}"} is empty; by foundation it equals 0
-  See Suppes, page 21.*)
-lemma not_mem_empty [simp]: "a \<notin> 0"
-apply (cut_tac foundation)
-apply (best dest: equalityD2)
-done
-
-lemmas emptyE [elim!] = not_mem_empty [THEN notE]
-
-
-lemma empty_subsetI [simp]: "0 \<subseteq> A"
-by blast
-
-lemma equals0I: "[| !!y. y\<in>A ==> False |] ==> A=0"
-by blast
-
-lemma equals0D [dest]: "A=0 ==> a \<notin> A"
-by blast
-
-declare sym [THEN equals0D, dest]
-
-lemma not_emptyI: "a\<in>A ==> A \<noteq> 0"
-by blast
-
-lemma not_emptyE:  "[| A \<noteq> 0;  !!x. x\<in>A ==> R |] ==> R"
-by blast
-
-
-subsection\<open>Rules for Inter\<close>
-
-(*Not obviously useful for proving InterI, InterD, InterE*)
-lemma Inter_iff: "A \<in> \<Inter>(C) <-> (\<forall>x\<in>C. A: x) & C\<noteq>0"
-by (simp add: Inter_def Ball_def, blast)
-
-(* Intersection is well-behaved only if the family is non-empty! *)
-lemma InterI [intro!]:
-    "[| !!x. x: C ==> A: x;  C\<noteq>0 |] ==> A \<in> \<Inter>(C)"
-by (simp add: Inter_iff)
-
-(*A "destruct" rule -- every B in C contains A as an element, but
-  A\<in>B can hold when B\<in>C does not!  This rule is analogous to "spec". *)
-lemma InterD [elim, Pure.elim]: "[| A \<in> \<Inter>(C);  B \<in> C |] ==> A \<in> B"
-by (unfold Inter_def, blast)
-
-(*"Classical" elimination rule -- does not require exhibiting @{term"B\<in>C"} *)
-lemma InterE [elim]:
-    "[| A \<in> \<Inter>(C);  B\<notin>C ==> R;  A\<in>B ==> R |] ==> R"
-by (simp add: Inter_def, blast)
-
-
-subsection\<open>Rules for Intersections of families\<close>
-
-(* @{term"\<Inter>x\<in>A. B(x)"} abbreviates @{term"\<Inter>({B(x). x\<in>A})"} *)
-
-lemma INT_iff: "b \<in> (\<Inter>x\<in>A. B(x)) <-> (\<forall>x\<in>A. b \<in> B(x)) & A\<noteq>0"
-by (force simp add: Inter_def)
-
-lemma INT_I: "[| !!x. x: A ==> b: B(x);  A\<noteq>0 |] ==> b: (\<Inter>x\<in>A. B(x))"
-by blast
-
-lemma INT_E: "[| b \<in> (\<Inter>x\<in>A. B(x));  a: A |] ==> b \<in> B(a)"
-by blast
-
-lemma INT_cong:
-    "[| A=B;  !!x. x\<in>B ==> C(x)=D(x) |] ==> (\<Inter>x\<in>A. C(x)) = (\<Inter>x\<in>B. D(x))"
-by simp
-
-(*No "Addcongs [INT_cong]" because @{term\<Inter>} is a combination of constants*)
-
-
-subsection\<open>Rules for Powersets\<close>
-
-lemma PowI: "A \<subseteq> B ==> A \<in> Pow(B)"
-by (erule Pow_iff [THEN iffD2])
-
-lemma PowD: "A \<in> Pow(B)  ==>  A<=B"
-by (erule Pow_iff [THEN iffD1])
-
-declare Pow_iff [iff]
-
-lemmas Pow_bottom = empty_subsetI [THEN PowI]    \<comment>\<open>@{term"0 \<in> Pow(B)"}\<close>
-lemmas Pow_top = subset_refl [THEN PowI]         \<comment>\<open>@{term"A \<in> Pow(A)"}\<close>
-
-
-subsection\<open>Cantor's Theorem: There is no surjection from a set to its powerset.\<close>
-
-(*The search is undirected.  Allowing redundant introduction rules may
-  make it diverge.  Variable b represents ANY map, such as
-  (lam x\<in>A.b(x)): A->Pow(A). *)
-lemma cantor: "\<exists>S \<in> Pow(A). \<forall>x\<in>A. b(x) \<noteq> S"
-by (best elim!: equalityCE del: ReplaceI RepFun_eqI)
+declaration \<open>fn _ =>
+  Simplifier.map_ss (Simplifier.set_mksimps (fn ctxt =>
+    map mk_eq o Ord_atomize o Variable.gen_all ctxt))
+\<close>
 
 end