# HG changeset patch
# User blanchet
# Date 1411051660 -7200
# Node ID bfd497f2f4c253238d3d59b77f6d05bc5702d1ae
# Parent  7f30ec82fe40a7bb79e2fff7ec938ee6991fa221
moved 'old_datatype' out of 'Main' (but put it in 'HOL-Proofs' because of the inductive realizer)
* * *
made example compile again

diff -r 7f30ec82fe40 -r bfd497f2f4c2 src/Doc/Isar_Ref/HOL_Specific.thy
--- a/src/Doc/Isar_Ref/HOL_Specific.thy	Thu Sep 18 16:47:40 2014 +0200
+++ b/src/Doc/Isar_Ref/HOL_Specific.thy	Thu Sep 18 16:47:40 2014 +0200
@@ -1,5 +1,6 @@
 theory HOL_Specific
-imports Base Main "~~/src/HOL/Library/Old_Recdef" "~~/src/Tools/Adhoc_Overloading"
+imports Base "~~/src/HOL/Library/Old_Datatype" "~~/src/HOL/Library/Old_Recdef"
+  "~~/src/Tools/Adhoc_Overloading"
 begin
 
 chapter {* Higher-Order Logic *}
diff -r 7f30ec82fe40 -r bfd497f2f4c2 src/Doc/Tutorial/ToyList/ToyList.thy
--- a/src/Doc/Tutorial/ToyList/ToyList.thy	Thu Sep 18 16:47:40 2014 +0200
+++ b/src/Doc/Tutorial/ToyList/ToyList.thy	Thu Sep 18 16:47:40 2014 +0200
@@ -1,14 +1,14 @@
 theory ToyList
-imports Old_Datatype
+imports BNF_Least_Fixpoint
 begin
 
 text{*\noindent
 HOL already has a predefined theory of lists called @{text List} ---
 @{text ToyList} is merely a small fragment of it chosen as an example. In
 contrast to what is recommended in \S\ref{sec:Basic:Theories},
-@{text ToyList} is not based on @{text Main} but on @{text Datatype}, a
-theory that contains pretty much everything but lists, thus avoiding
-ambiguities caused by defining lists twice.
+@{text ToyList} is not based on @{text Main} but on
+@{text BNF_Least_Fixpoint}, a theory that contains pretty much everything
+but lists, thus avoiding ambiguities caused by defining lists twice.
 *}
 
 datatype 'a list = Nil                          ("[]")
diff -r 7f30ec82fe40 -r bfd497f2f4c2 src/Doc/Tutorial/ToyList/ToyList1.txt
--- a/src/Doc/Tutorial/ToyList/ToyList1.txt	Thu Sep 18 16:47:40 2014 +0200
+++ b/src/Doc/Tutorial/ToyList/ToyList1.txt	Thu Sep 18 16:47:40 2014 +0200
@@ -1,5 +1,5 @@
 theory ToyList
-imports Old_Datatype
+imports BNF_Least_Fixpoint
 begin
 
 datatype 'a list = Nil                          ("[]")
diff -r 7f30ec82fe40 -r bfd497f2f4c2 src/Doc/Tutorial/ToyList/ToyList_Test.thy
--- a/src/Doc/Tutorial/ToyList/ToyList_Test.thy	Thu Sep 18 16:47:40 2014 +0200
+++ b/src/Doc/Tutorial/ToyList/ToyList_Test.thy	Thu Sep 18 16:47:40 2014 +0200
@@ -1,5 +1,5 @@
 theory ToyList_Test
-imports Old_Datatype
+imports BNF_Least_Fixpoint
 begin
 
 ML {*
diff -r 7f30ec82fe40 -r bfd497f2f4c2 src/HOL/Bali/Basis.thy
--- a/src/HOL/Bali/Basis.thy	Thu Sep 18 16:47:40 2014 +0200
+++ b/src/HOL/Bali/Basis.thy	Thu Sep 18 16:47:40 2014 +0200
@@ -145,9 +145,7 @@
 
 section "sums"
 
-hide_const In0 In1
-
-notation case_sum  (infixr "'(+')"80)
+notation case_sum  (infixr "'(+')" 80)
 
 primrec the_Inl :: "'a + 'b \<Rightarrow> 'a"
   where "the_Inl (Inl a) = a"
diff -r 7f30ec82fe40 -r bfd497f2f4c2 src/HOL/Datatype_Examples/Compat.thy
--- a/src/HOL/Datatype_Examples/Compat.thy	Thu Sep 18 16:47:40 2014 +0200
+++ b/src/HOL/Datatype_Examples/Compat.thy	Thu Sep 18 16:47:40 2014 +0200
@@ -8,7 +8,7 @@
 header {* Tests for Compatibility with the Old Datatype Package *}
 
 theory Compat
-imports Main
+imports "~~/src/HOL/Library/Old_Datatype"
 begin
 
 subsection {* Viewing and Registering New-Style Datatypes as Old-Style Ones *}
diff -r 7f30ec82fe40 -r bfd497f2f4c2 src/HOL/Induct/Sexp.thy
--- a/src/HOL/Induct/Sexp.thy	Thu Sep 18 16:47:40 2014 +0200
+++ b/src/HOL/Induct/Sexp.thy	Thu Sep 18 16:47:40 2014 +0200
@@ -7,7 +7,7 @@
 *)
 
 theory Sexp
-imports Main
+imports "~~/src/HOL/Library/Old_Datatype"
 begin
 
 type_synonym 'a item = "'a Old_Datatype.item"
diff -r 7f30ec82fe40 -r bfd497f2f4c2 src/HOL/Library/Countable.thy
--- a/src/HOL/Library/Countable.thy	Thu Sep 18 16:47:40 2014 +0200
+++ b/src/HOL/Library/Countable.thy	Thu Sep 18 16:47:40 2014 +0200
@@ -7,7 +7,7 @@
 header {* Encoding (almost) everything into natural numbers *}
 
 theory Countable
-imports Main Rat Nat_Bijection
+imports Old_Datatype Rat Nat_Bijection
 begin
 
 subsection {* The class of countable types *}
diff -r 7f30ec82fe40 -r bfd497f2f4c2 src/HOL/Library/Old_Datatype.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Old_Datatype.thy	Thu Sep 18 16:47:40 2014 +0200
@@ -0,0 +1,527 @@
+(*  Title:      HOL/Library/Old_Datatype.thy
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
+    Author:     Stefan Berghofer and Markus Wenzel, TU Muenchen
+*)
+
+header {* Old Datatype package: constructing datatypes from Cartesian Products and Disjoint Sums *}
+
+theory Old_Datatype
+imports "../Main"
+keywords "old_datatype" :: thy_decl
+begin
+
+subsection {* The datatype universe *}
+
+definition "Node = {p. EX f x k. p = (f :: nat => 'b + nat, x ::'a + nat) & f k = Inr 0}"
+
+typedef ('a, 'b) node = "Node :: ((nat => 'b + nat) * ('a + nat)) set"
+  morphisms Rep_Node Abs_Node
+  unfolding Node_def by auto
+
+text{*Datatypes will be represented by sets of type @{text node}*}
+
+type_synonym 'a item        = "('a, unit) node set"
+type_synonym ('a, 'b) dtree = "('a, 'b) node set"
+
+consts
+  Push      :: "[('b + nat), nat => ('b + nat)] => (nat => ('b + nat))"
+
+  Push_Node :: "[('b + nat), ('a, 'b) node] => ('a, 'b) node"
+  ndepth    :: "('a, 'b) node => nat"
+
+  Atom      :: "('a + nat) => ('a, 'b) dtree"
+  Leaf      :: "'a => ('a, 'b) dtree"
+  Numb      :: "nat => ('a, 'b) dtree"
+  Scons     :: "[('a, 'b) dtree, ('a, 'b) dtree] => ('a, 'b) dtree"
+  In0       :: "('a, 'b) dtree => ('a, 'b) dtree"
+  In1       :: "('a, 'b) dtree => ('a, 'b) dtree"
+  Lim       :: "('b => ('a, 'b) dtree) => ('a, 'b) dtree"
+
+  ntrunc    :: "[nat, ('a, 'b) dtree] => ('a, 'b) dtree"
+
+  uprod     :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set"
+  usum      :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set"
+
+  Split     :: "[[('a, 'b) dtree, ('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c"
+  Case      :: "[[('a, 'b) dtree]=>'c, [('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c"
+
+  dprod     :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]
+                => (('a, 'b) dtree * ('a, 'b) dtree)set"
+  dsum      :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]
+                => (('a, 'b) dtree * ('a, 'b) dtree)set"
+
+
+defs
+
+  Push_Node_def:  "Push_Node == (%n x. Abs_Node (apfst (Push n) (Rep_Node x)))"
+
+  (*crude "lists" of nats -- needed for the constructions*)
+  Push_def:   "Push == (%b h. case_nat b h)"
+
+  (** operations on S-expressions -- sets of nodes **)
+
+  (*S-expression constructors*)
+  Atom_def:   "Atom == (%x. {Abs_Node((%k. Inr 0, x))})"
+  Scons_def:  "Scons M N == (Push_Node (Inr 1) ` M) Un (Push_Node (Inr (Suc 1)) ` N)"
+
+  (*Leaf nodes, with arbitrary or nat labels*)
+  Leaf_def:   "Leaf == Atom o Inl"
+  Numb_def:   "Numb == Atom o Inr"
+
+  (*Injections of the "disjoint sum"*)
+  In0_def:    "In0(M) == Scons (Numb 0) M"
+  In1_def:    "In1(M) == Scons (Numb 1) M"
+
+  (*Function spaces*)
+  Lim_def: "Lim f == Union {z. ? x. z = Push_Node (Inl x) ` (f x)}"
+
+  (*the set of nodes with depth less than k*)
+  ndepth_def: "ndepth(n) == (%(f,x). LEAST k. f k = Inr 0) (Rep_Node n)"
+  ntrunc_def: "ntrunc k N == {n. n:N & ndepth(n)<k}"
+
+  (*products and sums for the "universe"*)
+  uprod_def:  "uprod A B == UN x:A. UN y:B. { Scons x y }"
+  usum_def:   "usum A B == In0`A Un In1`B"
+
+  (*the corresponding eliminators*)
+  Split_def:  "Split c M == THE u. EX x y. M = Scons x y & u = c x y"
+
+  Case_def:   "Case c d M == THE u.  (EX x . M = In0(x) & u = c(x))
+                                  | (EX y . M = In1(y) & u = d(y))"
+
+
+  (** equality for the "universe" **)
+
+  dprod_def:  "dprod r s == UN (x,x'):r. UN (y,y'):s. {(Scons x y, Scons x' y')}"
+
+  dsum_def:   "dsum r s == (UN (x,x'):r. {(In0(x),In0(x'))}) Un
+                          (UN (y,y'):s. {(In1(y),In1(y'))})"
+
+
+
+lemma apfst_convE: 
+    "[| q = apfst f p;  !!x y. [| p = (x,y);  q = (f(x),y) |] ==> R  
+     |] ==> R"
+by (force simp add: apfst_def)
+
+(** Push -- an injection, analogous to Cons on lists **)
+
+lemma Push_inject1: "Push i f = Push j g  ==> i=j"
+apply (simp add: Push_def fun_eq_iff) 
+apply (drule_tac x=0 in spec, simp) 
+done
+
+lemma Push_inject2: "Push i f = Push j g  ==> f=g"
+apply (auto simp add: Push_def fun_eq_iff) 
+apply (drule_tac x="Suc x" in spec, simp) 
+done
+
+lemma Push_inject:
+    "[| Push i f =Push j g;  [| i=j;  f=g |] ==> P |] ==> P"
+by (blast dest: Push_inject1 Push_inject2) 
+
+lemma Push_neq_K0: "Push (Inr (Suc k)) f = (%z. Inr 0) ==> P"
+by (auto simp add: Push_def fun_eq_iff split: nat.split_asm)
+
+lemmas Abs_Node_inj = Abs_Node_inject [THEN [2] rev_iffD1]
+
+
+(*** Introduction rules for Node ***)
+
+lemma Node_K0_I: "(%k. Inr 0, a) : Node"
+by (simp add: Node_def)
+
+lemma Node_Push_I: "p: Node ==> apfst (Push i) p : Node"
+apply (simp add: Node_def Push_def) 
+apply (fast intro!: apfst_conv nat.case(2)[THEN trans])
+done
+
+
+subsection{*Freeness: Distinctness of Constructors*}
+
+(** Scons vs Atom **)
+
+lemma Scons_not_Atom [iff]: "Scons M N \<noteq> Atom(a)"
+unfolding Atom_def Scons_def Push_Node_def One_nat_def
+by (blast intro: Node_K0_I Rep_Node [THEN Node_Push_I] 
+         dest!: Abs_Node_inj 
+         elim!: apfst_convE sym [THEN Push_neq_K0])  
+
+lemmas Atom_not_Scons [iff] = Scons_not_Atom [THEN not_sym]
+
+
+(*** Injectiveness ***)
+
+(** Atomic nodes **)
+
+lemma inj_Atom: "inj(Atom)"
+apply (simp add: Atom_def)
+apply (blast intro!: inj_onI Node_K0_I dest!: Abs_Node_inj)
+done
+lemmas Atom_inject = inj_Atom [THEN injD]
+
+lemma Atom_Atom_eq [iff]: "(Atom(a)=Atom(b)) = (a=b)"
+by (blast dest!: Atom_inject)
+
+lemma inj_Leaf: "inj(Leaf)"
+apply (simp add: Leaf_def o_def)
+apply (rule inj_onI)
+apply (erule Atom_inject [THEN Inl_inject])
+done
+
+lemmas Leaf_inject [dest!] = inj_Leaf [THEN injD]
+
+lemma inj_Numb: "inj(Numb)"
+apply (simp add: Numb_def o_def)
+apply (rule inj_onI)
+apply (erule Atom_inject [THEN Inr_inject])
+done
+
+lemmas Numb_inject [dest!] = inj_Numb [THEN injD]
+
+
+(** Injectiveness of Push_Node **)
+
+lemma Push_Node_inject:
+    "[| Push_Node i m =Push_Node j n;  [| i=j;  m=n |] ==> P  
+     |] ==> P"
+apply (simp add: Push_Node_def)
+apply (erule Abs_Node_inj [THEN apfst_convE])
+apply (rule Rep_Node [THEN Node_Push_I])+
+apply (erule sym [THEN apfst_convE]) 
+apply (blast intro: Rep_Node_inject [THEN iffD1] trans sym elim!: Push_inject)
+done
+
+
+(** Injectiveness of Scons **)
+
+lemma Scons_inject_lemma1: "Scons M N <= Scons M' N' ==> M<=M'"
+unfolding Scons_def One_nat_def
+by (blast dest!: Push_Node_inject)
+
+lemma Scons_inject_lemma2: "Scons M N <= Scons M' N' ==> N<=N'"
+unfolding Scons_def One_nat_def
+by (blast dest!: Push_Node_inject)
+
+lemma Scons_inject1: "Scons M N = Scons M' N' ==> M=M'"
+apply (erule equalityE)
+apply (iprover intro: equalityI Scons_inject_lemma1)
+done
+
+lemma Scons_inject2: "Scons M N = Scons M' N' ==> N=N'"
+apply (erule equalityE)
+apply (iprover intro: equalityI Scons_inject_lemma2)
+done
+
+lemma Scons_inject:
+    "[| Scons M N = Scons M' N';  [| M=M';  N=N' |] ==> P |] ==> P"
+by (iprover dest: Scons_inject1 Scons_inject2)
+
+lemma Scons_Scons_eq [iff]: "(Scons M N = Scons M' N') = (M=M' & N=N')"
+by (blast elim!: Scons_inject)
+
+(*** Distinctness involving Leaf and Numb ***)
+
+(** Scons vs Leaf **)
+
+lemma Scons_not_Leaf [iff]: "Scons M N \<noteq> Leaf(a)"
+unfolding Leaf_def o_def by (rule Scons_not_Atom)
+
+lemmas Leaf_not_Scons  [iff] = Scons_not_Leaf [THEN not_sym]
+
+(** Scons vs Numb **)
+
+lemma Scons_not_Numb [iff]: "Scons M N \<noteq> Numb(k)"
+unfolding Numb_def o_def by (rule Scons_not_Atom)
+
+lemmas Numb_not_Scons [iff] = Scons_not_Numb [THEN not_sym]
+
+
+(** Leaf vs Numb **)
+
+lemma Leaf_not_Numb [iff]: "Leaf(a) \<noteq> Numb(k)"
+by (simp add: Leaf_def Numb_def)
+
+lemmas Numb_not_Leaf [iff] = Leaf_not_Numb [THEN not_sym]
+
+
+(*** ndepth -- the depth of a node ***)
+
+lemma ndepth_K0: "ndepth (Abs_Node(%k. Inr 0, x)) = 0"
+by (simp add: ndepth_def  Node_K0_I [THEN Abs_Node_inverse] Least_equality)
+
+lemma ndepth_Push_Node_aux:
+     "case_nat (Inr (Suc i)) f k = Inr 0 --> Suc(LEAST x. f x = Inr 0) <= k"
+apply (induct_tac "k", auto)
+apply (erule Least_le)
+done
+
+lemma ndepth_Push_Node: 
+    "ndepth (Push_Node (Inr (Suc i)) n) = Suc(ndepth(n))"
+apply (insert Rep_Node [of n, unfolded Node_def])
+apply (auto simp add: ndepth_def Push_Node_def
+                 Rep_Node [THEN Node_Push_I, THEN Abs_Node_inverse])
+apply (rule Least_equality)
+apply (auto simp add: Push_def ndepth_Push_Node_aux)
+apply (erule LeastI)
+done
+
+
+(*** ntrunc applied to the various node sets ***)
+
+lemma ntrunc_0 [simp]: "ntrunc 0 M = {}"
+by (simp add: ntrunc_def)
+
+lemma ntrunc_Atom [simp]: "ntrunc (Suc k) (Atom a) = Atom(a)"
+by (auto simp add: Atom_def ntrunc_def ndepth_K0)
+
+lemma ntrunc_Leaf [simp]: "ntrunc (Suc k) (Leaf a) = Leaf(a)"
+unfolding Leaf_def o_def by (rule ntrunc_Atom)
+
+lemma ntrunc_Numb [simp]: "ntrunc (Suc k) (Numb i) = Numb(i)"
+unfolding Numb_def o_def by (rule ntrunc_Atom)
+
+lemma ntrunc_Scons [simp]: 
+    "ntrunc (Suc k) (Scons M N) = Scons (ntrunc k M) (ntrunc k N)"
+unfolding Scons_def ntrunc_def One_nat_def
+by (auto simp add: ndepth_Push_Node)
+
+
+
+(** Injection nodes **)
+
+lemma ntrunc_one_In0 [simp]: "ntrunc (Suc 0) (In0 M) = {}"
+apply (simp add: In0_def)
+apply (simp add: Scons_def)
+done
+
+lemma ntrunc_In0 [simp]: "ntrunc (Suc(Suc k)) (In0 M) = In0 (ntrunc (Suc k) M)"
+by (simp add: In0_def)
+
+lemma ntrunc_one_In1 [simp]: "ntrunc (Suc 0) (In1 M) = {}"
+apply (simp add: In1_def)
+apply (simp add: Scons_def)
+done
+
+lemma ntrunc_In1 [simp]: "ntrunc (Suc(Suc k)) (In1 M) = In1 (ntrunc (Suc k) M)"
+by (simp add: In1_def)
+
+
+subsection{*Set Constructions*}
+
+
+(*** Cartesian Product ***)
+
+lemma uprodI [intro!]: "[| M:A;  N:B |] ==> Scons M N : uprod A B"
+by (simp add: uprod_def)
+
+(*The general elimination rule*)
+lemma uprodE [elim!]:
+    "[| c : uprod A B;   
+        !!x y. [| x:A;  y:B;  c = Scons x y |] ==> P  
+     |] ==> P"
+by (auto simp add: uprod_def) 
+
+
+(*Elimination of a pair -- introduces no eigenvariables*)
+lemma uprodE2: "[| Scons M N : uprod A B;  [| M:A;  N:B |] ==> P |] ==> P"
+by (auto simp add: uprod_def)
+
+
+(*** Disjoint Sum ***)
+
+lemma usum_In0I [intro]: "M:A ==> In0(M) : usum A B"
+by (simp add: usum_def)
+
+lemma usum_In1I [intro]: "N:B ==> In1(N) : usum A B"
+by (simp add: usum_def)
+
+lemma usumE [elim!]: 
+    "[| u : usum A B;   
+        !!x. [| x:A;  u=In0(x) |] ==> P;  
+        !!y. [| y:B;  u=In1(y) |] ==> P  
+     |] ==> P"
+by (auto simp add: usum_def)
+
+
+(** Injection **)
+
+lemma In0_not_In1 [iff]: "In0(M) \<noteq> In1(N)"
+unfolding In0_def In1_def One_nat_def by auto
+
+lemmas In1_not_In0 [iff] = In0_not_In1 [THEN not_sym]
+
+lemma In0_inject: "In0(M) = In0(N) ==>  M=N"
+by (simp add: In0_def)
+
+lemma In1_inject: "In1(M) = In1(N) ==>  M=N"
+by (simp add: In1_def)
+
+lemma In0_eq [iff]: "(In0 M = In0 N) = (M=N)"
+by (blast dest!: In0_inject)
+
+lemma In1_eq [iff]: "(In1 M = In1 N) = (M=N)"
+by (blast dest!: In1_inject)
+
+lemma inj_In0: "inj In0"
+by (blast intro!: inj_onI)
+
+lemma inj_In1: "inj In1"
+by (blast intro!: inj_onI)
+
+
+(*** Function spaces ***)
+
+lemma Lim_inject: "Lim f = Lim g ==> f = g"
+apply (simp add: Lim_def)
+apply (rule ext)
+apply (blast elim!: Push_Node_inject)
+done
+
+
+(*** proving equality of sets and functions using ntrunc ***)
+
+lemma ntrunc_subsetI: "ntrunc k M <= M"
+by (auto simp add: ntrunc_def)
+
+lemma ntrunc_subsetD: "(!!k. ntrunc k M <= N) ==> M<=N"
+by (auto simp add: ntrunc_def)
+
+(*A generalized form of the take-lemma*)
+lemma ntrunc_equality: "(!!k. ntrunc k M = ntrunc k N) ==> M=N"
+apply (rule equalityI)
+apply (rule_tac [!] ntrunc_subsetD)
+apply (rule_tac [!] ntrunc_subsetI [THEN [2] subset_trans], auto) 
+done
+
+lemma ntrunc_o_equality: 
+    "[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2"
+apply (rule ntrunc_equality [THEN ext])
+apply (simp add: fun_eq_iff) 
+done
+
+
+(*** Monotonicity ***)
+
+lemma uprod_mono: "[| A<=A';  B<=B' |] ==> uprod A B <= uprod A' B'"
+by (simp add: uprod_def, blast)
+
+lemma usum_mono: "[| A<=A';  B<=B' |] ==> usum A B <= usum A' B'"
+by (simp add: usum_def, blast)
+
+lemma Scons_mono: "[| M<=M';  N<=N' |] ==> Scons M N <= Scons M' N'"
+by (simp add: Scons_def, blast)
+
+lemma In0_mono: "M<=N ==> In0(M) <= In0(N)"
+by (simp add: In0_def Scons_mono)
+
+lemma In1_mono: "M<=N ==> In1(M) <= In1(N)"
+by (simp add: In1_def Scons_mono)
+
+
+(*** Split and Case ***)
+
+lemma Split [simp]: "Split c (Scons M N) = c M N"
+by (simp add: Split_def)
+
+lemma Case_In0 [simp]: "Case c d (In0 M) = c(M)"
+by (simp add: Case_def)
+
+lemma Case_In1 [simp]: "Case c d (In1 N) = d(N)"
+by (simp add: Case_def)
+
+
+
+(**** UN x. B(x) rules ****)
+
+lemma ntrunc_UN1: "ntrunc k (UN x. f(x)) = (UN x. ntrunc k (f x))"
+by (simp add: ntrunc_def, blast)
+
+lemma Scons_UN1_x: "Scons (UN x. f x) M = (UN x. Scons (f x) M)"
+by (simp add: Scons_def, blast)
+
+lemma Scons_UN1_y: "Scons M (UN x. f x) = (UN x. Scons M (f x))"
+by (simp add: Scons_def, blast)
+
+lemma In0_UN1: "In0(UN x. f(x)) = (UN x. In0(f(x)))"
+by (simp add: In0_def Scons_UN1_y)
+
+lemma In1_UN1: "In1(UN x. f(x)) = (UN x. In1(f(x)))"
+by (simp add: In1_def Scons_UN1_y)
+
+
+(*** Equality for Cartesian Product ***)
+
+lemma dprodI [intro!]: 
+    "[| (M,M'):r;  (N,N'):s |] ==> (Scons M N, Scons M' N') : dprod r s"
+by (auto simp add: dprod_def)
+
+(*The general elimination rule*)
+lemma dprodE [elim!]: 
+    "[| c : dprod r s;   
+        !!x y x' y'. [| (x,x') : r;  (y,y') : s;  
+                        c = (Scons x y, Scons x' y') |] ==> P  
+     |] ==> P"
+by (auto simp add: dprod_def)
+
+
+(*** Equality for Disjoint Sum ***)
+
+lemma dsum_In0I [intro]: "(M,M'):r ==> (In0(M), In0(M')) : dsum r s"
+by (auto simp add: dsum_def)
+
+lemma dsum_In1I [intro]: "(N,N'):s ==> (In1(N), In1(N')) : dsum r s"
+by (auto simp add: dsum_def)
+
+lemma dsumE [elim!]: 
+    "[| w : dsum r s;   
+        !!x x'. [| (x,x') : r;  w = (In0(x), In0(x')) |] ==> P;  
+        !!y y'. [| (y,y') : s;  w = (In1(y), In1(y')) |] ==> P  
+     |] ==> P"
+by (auto simp add: dsum_def)
+
+
+(*** Monotonicity ***)
+
+lemma dprod_mono: "[| r<=r';  s<=s' |] ==> dprod r s <= dprod r' s'"
+by blast
+
+lemma dsum_mono: "[| r<=r';  s<=s' |] ==> dsum r s <= dsum r' s'"
+by blast
+
+
+(*** Bounding theorems ***)
+
+lemma dprod_Sigma: "(dprod (A <*> B) (C <*> D)) <= (uprod A C) <*> (uprod B D)"
+by blast
+
+lemmas dprod_subset_Sigma = subset_trans [OF dprod_mono dprod_Sigma]
+
+(*Dependent version*)
+lemma dprod_subset_Sigma2:
+    "(dprod (Sigma A B) (Sigma C D)) <= Sigma (uprod A C) (Split (%x y. uprod (B x) (D y)))"
+by auto
+
+lemma dsum_Sigma: "(dsum (A <*> B) (C <*> D)) <= (usum A C) <*> (usum B D)"
+by blast
+
+lemmas dsum_subset_Sigma = subset_trans [OF dsum_mono dsum_Sigma]
+
+
+(*** Domain theorems ***)
+
+lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)"
+  by auto
+
+lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)"
+  by auto
+
+
+text {* hides popular names *}
+hide_type (open) node item
+hide_const (open) Push Node Atom Leaf Numb Lim Split Case
+
+ML_file "~~/src/HOL/Tools/Old_Datatype/old_datatype.ML"
+ML_file "~~/src/HOL/Tools/inductive_realizer.ML"
+
+end
diff -r 7f30ec82fe40 -r bfd497f2f4c2 src/HOL/Main.thy
--- a/src/HOL/Main.thy	Thu Sep 18 16:47:40 2014 +0200
+++ b/src/HOL/Main.thy	Thu Sep 18 16:47:40 2014 +0200
@@ -2,7 +2,7 @@
 
 theory Main
 imports Predicate_Compile Quickcheck_Narrowing Extraction Lifting_Sum Coinduction Nitpick
-  Basic_BNF_Least_Fixpoints BNF_Greatest_Fixpoint Old_Datatype
+  Basic_BNF_Least_Fixpoints BNF_Greatest_Fixpoint
 begin
 
 text {*
diff -r 7f30ec82fe40 -r bfd497f2f4c2 src/HOL/Nominal/Nominal.thy
--- a/src/HOL/Nominal/Nominal.thy	Thu Sep 18 16:47:40 2014 +0200
+++ b/src/HOL/Nominal/Nominal.thy	Thu Sep 18 16:47:40 2014 +0200
@@ -1,5 +1,5 @@
 theory Nominal 
-imports Main "~~/src/HOL/Library/Infinite_Set"
+imports "~~/src/HOL/Library/Infinite_Set" "~~/src/HOL/Library/Old_Datatype"
 keywords
   "atom_decl" "nominal_datatype" "equivariance" :: thy_decl and
   "nominal_primrec" "nominal_inductive" "nominal_inductive2" :: thy_goal and
diff -r 7f30ec82fe40 -r bfd497f2f4c2 src/HOL/Nominal/nominal_datatype.ML
--- a/src/HOL/Nominal/nominal_datatype.ML	Thu Sep 18 16:47:40 2014 +0200
+++ b/src/HOL/Nominal/nominal_datatype.ML	Thu Sep 18 16:47:40 2014 +0200
@@ -752,7 +752,7 @@
                   Old_Datatype_Aux.DtRec k => if k < length new_type_names then
                       Const (nth rep_names k, Old_Datatype_Aux.typ_of_dtyp descr'' dt -->
                         Old_Datatype_Aux.typ_of_dtyp descr dt) $ x
-                    else error "nested recursion not (yet) supported"
+                    else error "nested recursion not supported"
                 | _ => x) :: r_args)
           end
 
diff -r 7f30ec82fe40 -r bfd497f2f4c2 src/HOL/Old_Datatype.thy
--- a/src/HOL/Old_Datatype.thy	Thu Sep 18 16:47:40 2014 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,529 +0,0 @@
-(*  Title:      HOL/Old_Datatype.thy
-    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
-    Author:     Stefan Berghofer and Markus Wenzel, TU Muenchen
-*)
-
-header {* Old Datatype package: constructing datatypes from Cartesian Products and Disjoint Sums *}
-
-theory Old_Datatype
-imports Power
-keywords "old_datatype" :: thy_decl
-begin
-
-subsection {* The datatype universe *}
-
-definition "Node = {p. EX f x k. p = (f :: nat => 'b + nat, x ::'a + nat) & f k = Inr 0}"
-
-typedef ('a, 'b) node = "Node :: ((nat => 'b + nat) * ('a + nat)) set"
-  morphisms Rep_Node Abs_Node
-  unfolding Node_def by auto
-
-text{*Datatypes will be represented by sets of type @{text node}*}
-
-type_synonym 'a item        = "('a, unit) node set"
-type_synonym ('a, 'b) dtree = "('a, 'b) node set"
-
-consts
-  Push      :: "[('b + nat), nat => ('b + nat)] => (nat => ('b + nat))"
-
-  Push_Node :: "[('b + nat), ('a, 'b) node] => ('a, 'b) node"
-  ndepth    :: "('a, 'b) node => nat"
-
-  Atom      :: "('a + nat) => ('a, 'b) dtree"
-  Leaf      :: "'a => ('a, 'b) dtree"
-  Numb      :: "nat => ('a, 'b) dtree"
-  Scons     :: "[('a, 'b) dtree, ('a, 'b) dtree] => ('a, 'b) dtree"
-  In0       :: "('a, 'b) dtree => ('a, 'b) dtree"
-  In1       :: "('a, 'b) dtree => ('a, 'b) dtree"
-  Lim       :: "('b => ('a, 'b) dtree) => ('a, 'b) dtree"
-
-  ntrunc    :: "[nat, ('a, 'b) dtree] => ('a, 'b) dtree"
-
-  uprod     :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set"
-  usum      :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set"
-
-  Split     :: "[[('a, 'b) dtree, ('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c"
-  Case      :: "[[('a, 'b) dtree]=>'c, [('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c"
-
-  dprod     :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]
-                => (('a, 'b) dtree * ('a, 'b) dtree)set"
-  dsum      :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set]
-                => (('a, 'b) dtree * ('a, 'b) dtree)set"
-
-
-defs
-
-  Push_Node_def:  "Push_Node == (%n x. Abs_Node (apfst (Push n) (Rep_Node x)))"
-
-  (*crude "lists" of nats -- needed for the constructions*)
-  Push_def:   "Push == (%b h. case_nat b h)"
-
-  (** operations on S-expressions -- sets of nodes **)
-
-  (*S-expression constructors*)
-  Atom_def:   "Atom == (%x. {Abs_Node((%k. Inr 0, x))})"
-  Scons_def:  "Scons M N == (Push_Node (Inr 1) ` M) Un (Push_Node (Inr (Suc 1)) ` N)"
-
-  (*Leaf nodes, with arbitrary or nat labels*)
-  Leaf_def:   "Leaf == Atom o Inl"
-  Numb_def:   "Numb == Atom o Inr"
-
-  (*Injections of the "disjoint sum"*)
-  In0_def:    "In0(M) == Scons (Numb 0) M"
-  In1_def:    "In1(M) == Scons (Numb 1) M"
-
-  (*Function spaces*)
-  Lim_def: "Lim f == Union {z. ? x. z = Push_Node (Inl x) ` (f x)}"
-
-  (*the set of nodes with depth less than k*)
-  ndepth_def: "ndepth(n) == (%(f,x). LEAST k. f k = Inr 0) (Rep_Node n)"
-  ntrunc_def: "ntrunc k N == {n. n:N & ndepth(n)<k}"
-
-  (*products and sums for the "universe"*)
-  uprod_def:  "uprod A B == UN x:A. UN y:B. { Scons x y }"
-  usum_def:   "usum A B == In0`A Un In1`B"
-
-  (*the corresponding eliminators*)
-  Split_def:  "Split c M == THE u. EX x y. M = Scons x y & u = c x y"
-
-  Case_def:   "Case c d M == THE u.  (EX x . M = In0(x) & u = c(x))
-                                  | (EX y . M = In1(y) & u = d(y))"
-
-
-  (** equality for the "universe" **)
-
-  dprod_def:  "dprod r s == UN (x,x'):r. UN (y,y'):s. {(Scons x y, Scons x' y')}"
-
-  dsum_def:   "dsum r s == (UN (x,x'):r. {(In0(x),In0(x'))}) Un
-                          (UN (y,y'):s. {(In1(y),In1(y'))})"
-
-
-
-lemma apfst_convE: 
-    "[| q = apfst f p;  !!x y. [| p = (x,y);  q = (f(x),y) |] ==> R  
-     |] ==> R"
-by (force simp add: apfst_def)
-
-(** Push -- an injection, analogous to Cons on lists **)
-
-lemma Push_inject1: "Push i f = Push j g  ==> i=j"
-apply (simp add: Push_def fun_eq_iff) 
-apply (drule_tac x=0 in spec, simp) 
-done
-
-lemma Push_inject2: "Push i f = Push j g  ==> f=g"
-apply (auto simp add: Push_def fun_eq_iff) 
-apply (drule_tac x="Suc x" in spec, simp) 
-done
-
-lemma Push_inject:
-    "[| Push i f =Push j g;  [| i=j;  f=g |] ==> P |] ==> P"
-by (blast dest: Push_inject1 Push_inject2) 
-
-lemma Push_neq_K0: "Push (Inr (Suc k)) f = (%z. Inr 0) ==> P"
-by (auto simp add: Push_def fun_eq_iff split: nat.split_asm)
-
-lemmas Abs_Node_inj = Abs_Node_inject [THEN [2] rev_iffD1]
-
-
-(*** Introduction rules for Node ***)
-
-lemma Node_K0_I: "(%k. Inr 0, a) : Node"
-by (simp add: Node_def)
-
-lemma Node_Push_I: "p: Node ==> apfst (Push i) p : Node"
-apply (simp add: Node_def Push_def) 
-apply (fast intro!: apfst_conv nat.case(2)[THEN trans])
-done
-
-
-subsection{*Freeness: Distinctness of Constructors*}
-
-(** Scons vs Atom **)
-
-lemma Scons_not_Atom [iff]: "Scons M N \<noteq> Atom(a)"
-unfolding Atom_def Scons_def Push_Node_def One_nat_def
-by (blast intro: Node_K0_I Rep_Node [THEN Node_Push_I] 
-         dest!: Abs_Node_inj 
-         elim!: apfst_convE sym [THEN Push_neq_K0])  
-
-lemmas Atom_not_Scons [iff] = Scons_not_Atom [THEN not_sym]
-
-
-(*** Injectiveness ***)
-
-(** Atomic nodes **)
-
-lemma inj_Atom: "inj(Atom)"
-apply (simp add: Atom_def)
-apply (blast intro!: inj_onI Node_K0_I dest!: Abs_Node_inj)
-done
-lemmas Atom_inject = inj_Atom [THEN injD]
-
-lemma Atom_Atom_eq [iff]: "(Atom(a)=Atom(b)) = (a=b)"
-by (blast dest!: Atom_inject)
-
-lemma inj_Leaf: "inj(Leaf)"
-apply (simp add: Leaf_def o_def)
-apply (rule inj_onI)
-apply (erule Atom_inject [THEN Inl_inject])
-done
-
-lemmas Leaf_inject [dest!] = inj_Leaf [THEN injD]
-
-lemma inj_Numb: "inj(Numb)"
-apply (simp add: Numb_def o_def)
-apply (rule inj_onI)
-apply (erule Atom_inject [THEN Inr_inject])
-done
-
-lemmas Numb_inject [dest!] = inj_Numb [THEN injD]
-
-
-(** Injectiveness of Push_Node **)
-
-lemma Push_Node_inject:
-    "[| Push_Node i m =Push_Node j n;  [| i=j;  m=n |] ==> P  
-     |] ==> P"
-apply (simp add: Push_Node_def)
-apply (erule Abs_Node_inj [THEN apfst_convE])
-apply (rule Rep_Node [THEN Node_Push_I])+
-apply (erule sym [THEN apfst_convE]) 
-apply (blast intro: Rep_Node_inject [THEN iffD1] trans sym elim!: Push_inject)
-done
-
-
-(** Injectiveness of Scons **)
-
-lemma Scons_inject_lemma1: "Scons M N <= Scons M' N' ==> M<=M'"
-unfolding Scons_def One_nat_def
-by (blast dest!: Push_Node_inject)
-
-lemma Scons_inject_lemma2: "Scons M N <= Scons M' N' ==> N<=N'"
-unfolding Scons_def One_nat_def
-by (blast dest!: Push_Node_inject)
-
-lemma Scons_inject1: "Scons M N = Scons M' N' ==> M=M'"
-apply (erule equalityE)
-apply (iprover intro: equalityI Scons_inject_lemma1)
-done
-
-lemma Scons_inject2: "Scons M N = Scons M' N' ==> N=N'"
-apply (erule equalityE)
-apply (iprover intro: equalityI Scons_inject_lemma2)
-done
-
-lemma Scons_inject:
-    "[| Scons M N = Scons M' N';  [| M=M';  N=N' |] ==> P |] ==> P"
-by (iprover dest: Scons_inject1 Scons_inject2)
-
-lemma Scons_Scons_eq [iff]: "(Scons M N = Scons M' N') = (M=M' & N=N')"
-by (blast elim!: Scons_inject)
-
-(*** Distinctness involving Leaf and Numb ***)
-
-(** Scons vs Leaf **)
-
-lemma Scons_not_Leaf [iff]: "Scons M N \<noteq> Leaf(a)"
-unfolding Leaf_def o_def by (rule Scons_not_Atom)
-
-lemmas Leaf_not_Scons  [iff] = Scons_not_Leaf [THEN not_sym]
-
-(** Scons vs Numb **)
-
-lemma Scons_not_Numb [iff]: "Scons M N \<noteq> Numb(k)"
-unfolding Numb_def o_def by (rule Scons_not_Atom)
-
-lemmas Numb_not_Scons [iff] = Scons_not_Numb [THEN not_sym]
-
-
-(** Leaf vs Numb **)
-
-lemma Leaf_not_Numb [iff]: "Leaf(a) \<noteq> Numb(k)"
-by (simp add: Leaf_def Numb_def)
-
-lemmas Numb_not_Leaf [iff] = Leaf_not_Numb [THEN not_sym]
-
-
-(*** ndepth -- the depth of a node ***)
-
-lemma ndepth_K0: "ndepth (Abs_Node(%k. Inr 0, x)) = 0"
-by (simp add: ndepth_def  Node_K0_I [THEN Abs_Node_inverse] Least_equality)
-
-lemma ndepth_Push_Node_aux:
-     "case_nat (Inr (Suc i)) f k = Inr 0 --> Suc(LEAST x. f x = Inr 0) <= k"
-apply (induct_tac "k", auto)
-apply (erule Least_le)
-done
-
-lemma ndepth_Push_Node: 
-    "ndepth (Push_Node (Inr (Suc i)) n) = Suc(ndepth(n))"
-apply (insert Rep_Node [of n, unfolded Node_def])
-apply (auto simp add: ndepth_def Push_Node_def
-                 Rep_Node [THEN Node_Push_I, THEN Abs_Node_inverse])
-apply (rule Least_equality)
-apply (auto simp add: Push_def ndepth_Push_Node_aux)
-apply (erule LeastI)
-done
-
-
-(*** ntrunc applied to the various node sets ***)
-
-lemma ntrunc_0 [simp]: "ntrunc 0 M = {}"
-by (simp add: ntrunc_def)
-
-lemma ntrunc_Atom [simp]: "ntrunc (Suc k) (Atom a) = Atom(a)"
-by (auto simp add: Atom_def ntrunc_def ndepth_K0)
-
-lemma ntrunc_Leaf [simp]: "ntrunc (Suc k) (Leaf a) = Leaf(a)"
-unfolding Leaf_def o_def by (rule ntrunc_Atom)
-
-lemma ntrunc_Numb [simp]: "ntrunc (Suc k) (Numb i) = Numb(i)"
-unfolding Numb_def o_def by (rule ntrunc_Atom)
-
-lemma ntrunc_Scons [simp]: 
-    "ntrunc (Suc k) (Scons M N) = Scons (ntrunc k M) (ntrunc k N)"
-unfolding Scons_def ntrunc_def One_nat_def
-by (auto simp add: ndepth_Push_Node)
-
-
-
-(** Injection nodes **)
-
-lemma ntrunc_one_In0 [simp]: "ntrunc (Suc 0) (In0 M) = {}"
-apply (simp add: In0_def)
-apply (simp add: Scons_def)
-done
-
-lemma ntrunc_In0 [simp]: "ntrunc (Suc(Suc k)) (In0 M) = In0 (ntrunc (Suc k) M)"
-by (simp add: In0_def)
-
-lemma ntrunc_one_In1 [simp]: "ntrunc (Suc 0) (In1 M) = {}"
-apply (simp add: In1_def)
-apply (simp add: Scons_def)
-done
-
-lemma ntrunc_In1 [simp]: "ntrunc (Suc(Suc k)) (In1 M) = In1 (ntrunc (Suc k) M)"
-by (simp add: In1_def)
-
-
-subsection{*Set Constructions*}
-
-
-(*** Cartesian Product ***)
-
-lemma uprodI [intro!]: "[| M:A;  N:B |] ==> Scons M N : uprod A B"
-by (simp add: uprod_def)
-
-(*The general elimination rule*)
-lemma uprodE [elim!]:
-    "[| c : uprod A B;   
-        !!x y. [| x:A;  y:B;  c = Scons x y |] ==> P  
-     |] ==> P"
-by (auto simp add: uprod_def) 
-
-
-(*Elimination of a pair -- introduces no eigenvariables*)
-lemma uprodE2: "[| Scons M N : uprod A B;  [| M:A;  N:B |] ==> P |] ==> P"
-by (auto simp add: uprod_def)
-
-
-(*** Disjoint Sum ***)
-
-lemma usum_In0I [intro]: "M:A ==> In0(M) : usum A B"
-by (simp add: usum_def)
-
-lemma usum_In1I [intro]: "N:B ==> In1(N) : usum A B"
-by (simp add: usum_def)
-
-lemma usumE [elim!]: 
-    "[| u : usum A B;   
-        !!x. [| x:A;  u=In0(x) |] ==> P;  
-        !!y. [| y:B;  u=In1(y) |] ==> P  
-     |] ==> P"
-by (auto simp add: usum_def)
-
-
-(** Injection **)
-
-lemma In0_not_In1 [iff]: "In0(M) \<noteq> In1(N)"
-unfolding In0_def In1_def One_nat_def by auto
-
-lemmas In1_not_In0 [iff] = In0_not_In1 [THEN not_sym]
-
-lemma In0_inject: "In0(M) = In0(N) ==>  M=N"
-by (simp add: In0_def)
-
-lemma In1_inject: "In1(M) = In1(N) ==>  M=N"
-by (simp add: In1_def)
-
-lemma In0_eq [iff]: "(In0 M = In0 N) = (M=N)"
-by (blast dest!: In0_inject)
-
-lemma In1_eq [iff]: "(In1 M = In1 N) = (M=N)"
-by (blast dest!: In1_inject)
-
-lemma inj_In0: "inj In0"
-by (blast intro!: inj_onI)
-
-lemma inj_In1: "inj In1"
-by (blast intro!: inj_onI)
-
-
-(*** Function spaces ***)
-
-lemma Lim_inject: "Lim f = Lim g ==> f = g"
-apply (simp add: Lim_def)
-apply (rule ext)
-apply (blast elim!: Push_Node_inject)
-done
-
-
-(*** proving equality of sets and functions using ntrunc ***)
-
-lemma ntrunc_subsetI: "ntrunc k M <= M"
-by (auto simp add: ntrunc_def)
-
-lemma ntrunc_subsetD: "(!!k. ntrunc k M <= N) ==> M<=N"
-by (auto simp add: ntrunc_def)
-
-(*A generalized form of the take-lemma*)
-lemma ntrunc_equality: "(!!k. ntrunc k M = ntrunc k N) ==> M=N"
-apply (rule equalityI)
-apply (rule_tac [!] ntrunc_subsetD)
-apply (rule_tac [!] ntrunc_subsetI [THEN [2] subset_trans], auto) 
-done
-
-lemma ntrunc_o_equality: 
-    "[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2"
-apply (rule ntrunc_equality [THEN ext])
-apply (simp add: fun_eq_iff) 
-done
-
-
-(*** Monotonicity ***)
-
-lemma uprod_mono: "[| A<=A';  B<=B' |] ==> uprod A B <= uprod A' B'"
-by (simp add: uprod_def, blast)
-
-lemma usum_mono: "[| A<=A';  B<=B' |] ==> usum A B <= usum A' B'"
-by (simp add: usum_def, blast)
-
-lemma Scons_mono: "[| M<=M';  N<=N' |] ==> Scons M N <= Scons M' N'"
-by (simp add: Scons_def, blast)
-
-lemma In0_mono: "M<=N ==> In0(M) <= In0(N)"
-by (simp add: In0_def Scons_mono)
-
-lemma In1_mono: "M<=N ==> In1(M) <= In1(N)"
-by (simp add: In1_def Scons_mono)
-
-
-(*** Split and Case ***)
-
-lemma Split [simp]: "Split c (Scons M N) = c M N"
-by (simp add: Split_def)
-
-lemma Case_In0 [simp]: "Case c d (In0 M) = c(M)"
-by (simp add: Case_def)
-
-lemma Case_In1 [simp]: "Case c d (In1 N) = d(N)"
-by (simp add: Case_def)
-
-
-
-(**** UN x. B(x) rules ****)
-
-lemma ntrunc_UN1: "ntrunc k (UN x. f(x)) = (UN x. ntrunc k (f x))"
-by (simp add: ntrunc_def, blast)
-
-lemma Scons_UN1_x: "Scons (UN x. f x) M = (UN x. Scons (f x) M)"
-by (simp add: Scons_def, blast)
-
-lemma Scons_UN1_y: "Scons M (UN x. f x) = (UN x. Scons M (f x))"
-by (simp add: Scons_def, blast)
-
-lemma In0_UN1: "In0(UN x. f(x)) = (UN x. In0(f(x)))"
-by (simp add: In0_def Scons_UN1_y)
-
-lemma In1_UN1: "In1(UN x. f(x)) = (UN x. In1(f(x)))"
-by (simp add: In1_def Scons_UN1_y)
-
-
-(*** Equality for Cartesian Product ***)
-
-lemma dprodI [intro!]: 
-    "[| (M,M'):r;  (N,N'):s |] ==> (Scons M N, Scons M' N') : dprod r s"
-by (auto simp add: dprod_def)
-
-(*The general elimination rule*)
-lemma dprodE [elim!]: 
-    "[| c : dprod r s;   
-        !!x y x' y'. [| (x,x') : r;  (y,y') : s;  
-                        c = (Scons x y, Scons x' y') |] ==> P  
-     |] ==> P"
-by (auto simp add: dprod_def)
-
-
-(*** Equality for Disjoint Sum ***)
-
-lemma dsum_In0I [intro]: "(M,M'):r ==> (In0(M), In0(M')) : dsum r s"
-by (auto simp add: dsum_def)
-
-lemma dsum_In1I [intro]: "(N,N'):s ==> (In1(N), In1(N')) : dsum r s"
-by (auto simp add: dsum_def)
-
-lemma dsumE [elim!]: 
-    "[| w : dsum r s;   
-        !!x x'. [| (x,x') : r;  w = (In0(x), In0(x')) |] ==> P;  
-        !!y y'. [| (y,y') : s;  w = (In1(y), In1(y')) |] ==> P  
-     |] ==> P"
-by (auto simp add: dsum_def)
-
-
-(*** Monotonicity ***)
-
-lemma dprod_mono: "[| r<=r';  s<=s' |] ==> dprod r s <= dprod r' s'"
-by blast
-
-lemma dsum_mono: "[| r<=r';  s<=s' |] ==> dsum r s <= dsum r' s'"
-by blast
-
-
-(*** Bounding theorems ***)
-
-lemma dprod_Sigma: "(dprod (A <*> B) (C <*> D)) <= (uprod A C) <*> (uprod B D)"
-by blast
-
-lemmas dprod_subset_Sigma = subset_trans [OF dprod_mono dprod_Sigma]
-
-(*Dependent version*)
-lemma dprod_subset_Sigma2:
-    "(dprod (Sigma A B) (Sigma C D)) <= Sigma (uprod A C) (Split (%x y. uprod (B x) (D y)))"
-by auto
-
-lemma dsum_Sigma: "(dsum (A <*> B) (C <*> D)) <= (usum A C) <*> (usum B D)"
-by blast
-
-lemmas dsum_subset_Sigma = subset_trans [OF dsum_mono dsum_Sigma]
-
-
-(*** Domain theorems ***)
-
-lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)"
-  by auto
-
-lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)"
-  by auto
-
-
-text {* hides popular names *}
-hide_type (open) node item
-hide_const (open) Push Node Atom Leaf Numb Lim Split Case
-
-ML_file "Tools/Old_Datatype/old_datatype.ML"
-
-ML_file "Tools/inductive_realizer.ML"
-setup InductiveRealizer.setup
-
-end
diff -r 7f30ec82fe40 -r bfd497f2f4c2 src/HOL/Proofs/Extraction/Higman.thy
--- a/src/HOL/Proofs/Extraction/Higman.thy	Thu Sep 18 16:47:40 2014 +0200
+++ b/src/HOL/Proofs/Extraction/Higman.thy	Thu Sep 18 16:47:40 2014 +0200
@@ -6,7 +6,7 @@
 header {* Higman's lemma *}
 
 theory Higman
-imports Main
+imports Old_Datatype
 begin
 
 text {*
diff -r 7f30ec82fe40 -r bfd497f2f4c2 src/HOL/Proofs/Extraction/Util.thy
--- a/src/HOL/Proofs/Extraction/Util.thy	Thu Sep 18 16:47:40 2014 +0200
+++ b/src/HOL/Proofs/Extraction/Util.thy	Thu Sep 18 16:47:40 2014 +0200
@@ -5,7 +5,7 @@
 header {* Auxiliary lemmas used in program extraction examples *}
 
 theory Util
-imports Main
+imports Old_Datatype
 begin
 
 text {*
diff -r 7f30ec82fe40 -r bfd497f2f4c2 src/HOL/Proofs/Extraction/Warshall.thy
--- a/src/HOL/Proofs/Extraction/Warshall.thy	Thu Sep 18 16:47:40 2014 +0200
+++ b/src/HOL/Proofs/Extraction/Warshall.thy	Thu Sep 18 16:47:40 2014 +0200
@@ -5,7 +5,7 @@
 header {* Warshall's algorithm *}
 
 theory Warshall
-imports Main
+imports Old_Datatype
 begin
 
 text {*
diff -r 7f30ec82fe40 -r bfd497f2f4c2 src/HOL/Proofs/Lambda/Commutation.thy
--- a/src/HOL/Proofs/Lambda/Commutation.thy	Thu Sep 18 16:47:40 2014 +0200
+++ b/src/HOL/Proofs/Lambda/Commutation.thy	Thu Sep 18 16:47:40 2014 +0200
@@ -5,7 +5,9 @@
 
 header {* Abstract commutation and confluence notions *}
 
-theory Commutation imports Main begin
+theory Commutation
+imports Old_Datatype
+begin
 
 declare [[syntax_ambiguity_warning = false]]
 
diff -r 7f30ec82fe40 -r bfd497f2f4c2 src/HOL/Proofs/Lambda/Lambda.thy
--- a/src/HOL/Proofs/Lambda/Lambda.thy	Thu Sep 18 16:47:40 2014 +0200
+++ b/src/HOL/Proofs/Lambda/Lambda.thy	Thu Sep 18 16:47:40 2014 +0200
@@ -5,7 +5,9 @@
 
 header {* Basic definitions of Lambda-calculus *}
 
-theory Lambda imports Main begin
+theory Lambda
+imports Old_Datatype
+begin
 
 declare [[syntax_ambiguity_warning = false]]
 
diff -r 7f30ec82fe40 -r bfd497f2f4c2 src/HOL/Proofs/Lambda/ListOrder.thy
--- a/src/HOL/Proofs/Lambda/ListOrder.thy	Thu Sep 18 16:47:40 2014 +0200
+++ b/src/HOL/Proofs/Lambda/ListOrder.thy	Thu Sep 18 16:47:40 2014 +0200
@@ -5,7 +5,9 @@
 
 header {* Lifting an order to lists of elements *}
 
-theory ListOrder imports Main begin
+theory ListOrder
+imports Old_Datatype
+begin
 
 declare [[syntax_ambiguity_warning = false]]
 
diff -r 7f30ec82fe40 -r bfd497f2f4c2 src/HOL/Proofs/ex/XML_Data.thy
--- a/src/HOL/Proofs/ex/XML_Data.thy	Thu Sep 18 16:47:40 2014 +0200
+++ b/src/HOL/Proofs/ex/XML_Data.thy	Thu Sep 18 16:47:40 2014 +0200
@@ -63,4 +63,3 @@
 *}
 
 end
-
diff -r 7f30ec82fe40 -r bfd497f2f4c2 src/HOL/ROOT
--- a/src/HOL/ROOT	Thu Sep 18 16:47:40 2014 +0200
+++ b/src/HOL/ROOT	Thu Sep 18 16:47:40 2014 +0200
@@ -21,7 +21,7 @@
   *}
   options [timeout = 5400, document = false]
   theories Proofs (*sequential change of global flag!*)
-  theories Main
+  theories "~~/src/HOL/Library/Old_Datatype"
   files
     "Tools/Quickcheck/Narrowing_Engine.hs"
     "Tools/Quickcheck/PNF_Narrowing_Engine.hs"
@@ -51,6 +51,7 @@
     RBT_Set
     (*legacy tools*)
     Refute
+    Old_Datatype
     Old_Recdef
     Old_SMT
   theories [condition = ISABELLE_FULL_TEST]
@@ -91,11 +92,13 @@
 
     PropLog proves the completeness of a formalization of propositional logic
     (see
-    HREF="http://www.cl.cam.ac.uk/Research/Reports/TR312-lcp-set-II.ps.gz).
+    http://www.cl.cam.ac.uk/Research/Reports/TR312-lcp-set-II.ps.gz).
 
     Exp demonstrates the use of iterated inductive definitions to reason about
     mutually recursive relations.
   *}
+  theories [document = false]
+    "~~/src/HOL/Library/Old_Datatype"
   theories [quick_and_dirty]
     Common_Patterns
   theories
@@ -741,6 +744,7 @@
   *}
   options [document = false]
   theories
+    "~~/src/HOL/Library/Old_Datatype"
     Compat
     Lambda_Term
     Process
diff -r 7f30ec82fe40 -r bfd497f2f4c2 src/HOL/Tools/inductive_realizer.ML
--- a/src/HOL/Tools/inductive_realizer.ML	Thu Sep 18 16:47:40 2014 +0200
+++ b/src/HOL/Tools/inductive_realizer.ML	Thu Sep 18 16:47:40 2014 +0200
@@ -8,7 +8,6 @@
 signature INDUCTIVE_REALIZER =
 sig
   val add_ind_realizers: string -> string list -> theory -> theory
-  val setup: theory -> theory
 end;
 
 structure InductiveRealizer : INDUCTIVE_REALIZER =
@@ -513,11 +512,10 @@
       | SOME (SOME sets') => subtract (op =) sets' sets)
   end I);
 
-val setup =
-  Attrib.setup @{binding ind_realizer}
-    ((Scan.option (Scan.lift (Args.$$$ "irrelevant") |--
-      Scan.option (Scan.lift (Args.colon) |--
-        Scan.repeat1 (Args.const {proper = true, strict = true})))) >> rlz_attrib)
-    "add realizers for inductive set";
+val _ = Theory.setup (Attrib.setup @{binding ind_realizer}
+  ((Scan.option (Scan.lift (Args.$$$ "irrelevant") |--
+    Scan.option (Scan.lift (Args.colon) |--
+      Scan.repeat1 (Args.const {proper = true, strict = true})))) >> rlz_attrib)
+  "add realizers for inductive set");
 
 end;