removed obsolete Datatype_Universe.thy (cf. Datatype.thy);

--- a/src/HOL/Datatype_Universe.thy	Sun Oct 01 22:19:21 2006 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,634 +0,0 @@
-(*  Title:      HOL/Datatype_Universe.thy
-    ID:         $Id$
-    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
-    Copyright   1993  University of Cambridge
-
-Could <*> be generalized to a general summation (Sigma)?
-*)
-
-header{*Analogues of the Cartesian Product and Disjoint Sum for Datatypes*}
-
-theory Datatype_Universe
-imports NatArith Sum_Type
-begin
-
-
-typedef (Node)
-  ('a,'b) node = "{p. EX f x k. p = (f::nat=>'b+nat, x::'a+nat) & f k = Inr 0}"
-    --{*it is a subtype of @{text "(nat=>'b+nat) * ('a+nat)"}*}
-  by auto
-
-text{*Datatypes will be represented by sets of type @{text node}*}
-
-types 'a item        = "('a, unit) node set"
-      ('a, 'b) dtree = "('a, 'b) node set"
-
-consts
-  apfst     :: "['a=>'c, 'a*'b] => 'c*'b"
-  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*)
-  apfst_def:  "apfst == (%f (x,y). (f(x),y))"
-  Push_def:   "Push == (%b h. nat_case 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'))})"
-
-
-
-(** apfst -- can be used in similar type definitions **)
-
-lemma apfst_conv [simp]: "apfst f (a,b) = (f(a),b)"
-by (simp add: apfst_def)
-
-
-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 expand_fun_eq) 
-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 expand_fun_eq) 
-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 expand_fun_eq split: nat.split_asm)
-
-lemmas Abs_Node_inj = Abs_Node_inject [THEN [2] rev_iffD1, standard]
-
-
-(*** 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_Suc [THEN trans])
-done
-
-
-subsection{*Freeness: Distinctness of Constructors*}
-
-(** Scons vs Atom **)
-
-lemma Scons_not_Atom [iff]: "Scons M N \<noteq> Atom(a)"
-apply (simp add: Atom_def Scons_def Push_Node_def One_nat_def)
-apply (blast intro: Node_K0_I Rep_Node [THEN Node_Push_I] 
-         dest!: Abs_Node_inj 
-         elim!: apfst_convE sym [THEN Push_neq_K0])  
-done
-
-lemmas Atom_not_Scons = Scons_not_Atom [THEN not_sym, standard]
-declare Atom_not_Scons [iff]
-
-(*** 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, standard]
-
-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 = inj_Leaf [THEN injD, standard]
-declare Leaf_inject [dest!]
-
-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 = inj_Numb [THEN injD, standard]
-declare Numb_inject [dest!]
-
-
-(** 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'"
-apply (simp add: Scons_def One_nat_def)
-apply (blast dest!: Push_Node_inject)
-done
-
-lemma Scons_inject_lemma2: "Scons M N <= Scons M' N' ==> N<=N'"
-apply (simp add: Scons_def One_nat_def)
-apply (blast dest!: Push_Node_inject)
-done
-
-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)"
-by (simp add: Leaf_def o_def Scons_not_Atom)
-
-lemmas Leaf_not_Scons = Scons_not_Leaf [THEN not_sym, standard]
-declare Leaf_not_Scons [iff]
-
-(** Scons vs Numb **)
-
-lemma Scons_not_Numb [iff]: "Scons M N \<noteq> Numb(k)"
-by (simp add: Numb_def o_def Scons_not_Atom)
-
-lemmas Numb_not_Scons = Scons_not_Numb [THEN not_sym, standard]
-declare Numb_not_Scons [iff]
-
-
-(** Leaf vs Numb **)
-
-lemma Leaf_not_Numb [iff]: "Leaf(a) \<noteq> Numb(k)"
-by (simp add: Leaf_def Numb_def)
-
-lemmas Numb_not_Leaf = Leaf_not_Numb [THEN not_sym, standard]
-declare Numb_not_Leaf [iff]
-
-
-(*** 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:
-     "nat_case (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)"
-by (simp add: Leaf_def o_def ntrunc_Atom)
-
-lemma ntrunc_Numb [simp]: "ntrunc (Suc k) (Numb i) = Numb(i)"
-by (simp add: Numb_def o_def ntrunc_Atom)
-
-lemma ntrunc_Scons [simp]: 
-    "ntrunc (Suc k) (Scons M N) = Scons (ntrunc k M) (ntrunc k N)"
-by (auto simp add: Scons_def ntrunc_def One_nat_def 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)"
-by (auto simp add: In0_def In1_def One_nat_def)
-
-lemmas In1_not_In0 = In0_not_In1 [THEN not_sym, standard]
-declare In1_not_In0 [iff]
-
-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: expand_fun_eq) 
-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 subset_refl Scons_mono)
-
-lemma In1_mono: "M<=N ==> In1(M) <= In1(N)"
-by (simp add: In1_def subset_refl 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, standard]
-
-(*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, standard]
-
-
-(*** Domain ***)
-
-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
-
-
-subsection {* Finishing the datatype package setup *}
-
-text {* Belongs to theory @{text Datatype_Universe}; hides popular names. *}
-hide (open) const Push Node Atom Leaf Numb Lim Split Case
-hide (open) type node item
-
-ML
-{*
-val apfst_conv = thm "apfst_conv";
-val apfst_convE = thm "apfst_convE";
-val Push_inject1 = thm "Push_inject1";
-val Push_inject2 = thm "Push_inject2";
-val Push_inject = thm "Push_inject";
-val Push_neq_K0 = thm "Push_neq_K0";
-val Abs_Node_inj = thm "Abs_Node_inj";
-val Node_K0_I = thm "Node_K0_I";
-val Node_Push_I = thm "Node_Push_I";
-val Scons_not_Atom = thm "Scons_not_Atom";
-val Atom_not_Scons = thm "Atom_not_Scons";
-val inj_Atom = thm "inj_Atom";
-val Atom_inject = thm "Atom_inject";
-val Atom_Atom_eq = thm "Atom_Atom_eq";
-val inj_Leaf = thm "inj_Leaf";
-val Leaf_inject = thm "Leaf_inject";
-val inj_Numb = thm "inj_Numb";
-val Numb_inject = thm "Numb_inject";
-val Push_Node_inject = thm "Push_Node_inject";
-val Scons_inject1 = thm "Scons_inject1";
-val Scons_inject2 = thm "Scons_inject2";
-val Scons_inject = thm "Scons_inject";
-val Scons_Scons_eq = thm "Scons_Scons_eq";
-val Scons_not_Leaf = thm "Scons_not_Leaf";
-val Leaf_not_Scons = thm "Leaf_not_Scons";
-val Scons_not_Numb = thm "Scons_not_Numb";
-val Numb_not_Scons = thm "Numb_not_Scons";
-val Leaf_not_Numb = thm "Leaf_not_Numb";
-val Numb_not_Leaf = thm "Numb_not_Leaf";
-val ndepth_K0 = thm "ndepth_K0";
-val ndepth_Push_Node_aux = thm "ndepth_Push_Node_aux";
-val ndepth_Push_Node = thm "ndepth_Push_Node";
-val ntrunc_0 = thm "ntrunc_0";
-val ntrunc_Atom = thm "ntrunc_Atom";
-val ntrunc_Leaf = thm "ntrunc_Leaf";
-val ntrunc_Numb = thm "ntrunc_Numb";
-val ntrunc_Scons = thm "ntrunc_Scons";
-val ntrunc_one_In0 = thm "ntrunc_one_In0";
-val ntrunc_In0 = thm "ntrunc_In0";
-val ntrunc_one_In1 = thm "ntrunc_one_In1";
-val ntrunc_In1 = thm "ntrunc_In1";
-val uprodI = thm "uprodI";
-val uprodE = thm "uprodE";
-val uprodE2 = thm "uprodE2";
-val usum_In0I = thm "usum_In0I";
-val usum_In1I = thm "usum_In1I";
-val usumE = thm "usumE";
-val In0_not_In1 = thm "In0_not_In1";
-val In1_not_In0 = thm "In1_not_In0";
-val In0_inject = thm "In0_inject";
-val In1_inject = thm "In1_inject";
-val In0_eq = thm "In0_eq";
-val In1_eq = thm "In1_eq";
-val inj_In0 = thm "inj_In0";
-val inj_In1 = thm "inj_In1";
-val Lim_inject = thm "Lim_inject";
-val ntrunc_subsetI = thm "ntrunc_subsetI";
-val ntrunc_subsetD = thm "ntrunc_subsetD";
-val ntrunc_equality = thm "ntrunc_equality";
-val ntrunc_o_equality = thm "ntrunc_o_equality";
-val uprod_mono = thm "uprod_mono";
-val usum_mono = thm "usum_mono";
-val Scons_mono = thm "Scons_mono";
-val In0_mono = thm "In0_mono";
-val In1_mono = thm "In1_mono";
-val Split = thm "Split";
-val Case_In0 = thm "Case_In0";
-val Case_In1 = thm "Case_In1";
-val ntrunc_UN1 = thm "ntrunc_UN1";
-val Scons_UN1_x = thm "Scons_UN1_x";
-val Scons_UN1_y = thm "Scons_UN1_y";
-val In0_UN1 = thm "In0_UN1";
-val In1_UN1 = thm "In1_UN1";
-val dprodI = thm "dprodI";
-val dprodE = thm "dprodE";
-val dsum_In0I = thm "dsum_In0I";
-val dsum_In1I = thm "dsum_In1I";
-val dsumE = thm "dsumE";
-val dprod_mono = thm "dprod_mono";
-val dsum_mono = thm "dsum_mono";
-val dprod_Sigma = thm "dprod_Sigma";
-val dprod_subset_Sigma = thm "dprod_subset_Sigma";
-val dprod_subset_Sigma2 = thm "dprod_subset_Sigma2";
-val dsum_Sigma = thm "dsum_Sigma";
-val dsum_subset_Sigma = thm "dsum_subset_Sigma";
-val Domain_dprod = thm "Domain_dprod";
-val Domain_dsum = thm "Domain_dsum";
-*}
-
-end

--- a/src/HOL/Induct/SList.thy	Sun Oct 01 22:19:21 2006 +0200
+++ b/src/HOL/Induct/SList.thy	Sun Oct 01 22:19:23 2006 +0200
@@ -56,8 +56,8 @@
   by (blast intro: list.NIL_I)
 
 abbreviation
-  "Case == Datatype_Universe.Case"
-  "Split == Datatype_Universe.Split"
+  "Case == Datatype.Case"
+  "Split == Datatype.Split"
 
 definition
   List_case :: "['b, ['a item, 'a item]=>'b, 'a item] => 'b"

--- a/src/HOL/Induct/Sexp.thy	Sun Oct 01 22:19:21 2006 +0200
+++ b/src/HOL/Induct/Sexp.thy	Sun Oct 01 22:19:23 2006 +0200
@@ -10,10 +10,10 @@
 theory Sexp imports Main begin
 
 types
-  'a item = "'a Datatype_Universe.item"
+  'a item = "'a Datatype.item"
 abbreviation
-  "Leaf == Datatype_Universe.Leaf"
-  "Numb == Datatype_Universe.Numb"
+  "Leaf == Datatype.Leaf"
+  "Numb == Datatype.Numb"
 
 consts
   sexp      :: "'a item set"

--- a/src/HOL/IsaMakefile	Sun Oct 01 22:19:21 2006 +0200
+++ b/src/HOL/IsaMakefile	Sun Oct 01 22:19:23 2006 +0200
@@ -85,7 +85,7 @@
   $(SRC)/TFL/thry.ML $(SRC)/TFL/usyntax.ML $(SRC)/TFL/utils.ML			\
   Tools/res_atpset.ML \
   Binomial.thy Datatype.ML Datatype.thy			\
-  Datatype_Universe.thy Divides.thy						\
+  Divides.thy						\
   Equiv_Relations.thy Extraction.thy Finite_Set.ML Finite_Set.thy		\
   FixedPoint.thy Fun.thy HOL.ML HOL.thy Hilbert_Choice.thy Inductive.thy	\
   Integ/IntArith.thy Integ/IntDef.thy Integ/IntDiv.thy				\

--- a/src/HOL/Library/Coinductive_List.thy	Sun Oct 01 22:19:21 2006 +0200
+++ b/src/HOL/Library/Coinductive_List.thy	Sun Oct 01 22:19:23 2006 +0200
@@ -12,8 +12,8 @@
 subsection {* List constructors over the datatype universe *}
 
 definition
-  "NIL = Datatype_Universe.In0 (Datatype_Universe.Numb 0)"
-  "CONS M N = Datatype_Universe.In1 (Datatype_Universe.Scons M N)"
+  "NIL = Datatype.In0 (Datatype.Numb 0)"
+  "CONS M N = Datatype.In1 (Datatype.Scons M N)"
 
 lemma CONS_not_NIL [iff]: "CONS M N \<noteq> NIL"
   and NIL_not_CONS [iff]: "NIL \<noteq> CONS M N"
@@ -28,7 +28,7 @@
   by (simp add: CONS_def In1_UN1 Scons_UN1_y)
 
 definition
-  "List_case c h = Datatype_Universe.Case (\<lambda>_. c) (Datatype_Universe.Split h)"
+  "List_case c h = Datatype.Case (\<lambda>_. c) (Datatype.Split h)"
 
 lemma List_case_NIL [simp]: "List_case c h NIL = c"
   and List_case_CONS [simp]: "List_case c h (CONS M N) = h M N"
@@ -38,7 +38,7 @@
 subsection {* Corecursive lists *}
 
 consts
-  LList  :: "'a Datatype_Universe.item set \<Rightarrow> 'a Datatype_Universe.item set"
+  LList  :: "'a Datatype.item set \<Rightarrow> 'a Datatype.item set"
 
 coinductive "LList A"
   intros
@@ -50,8 +50,8 @@
   unfolding LList.defs by (blast intro!: gfp_mono)
 
 consts
-  LList_corec_aux :: "nat \<Rightarrow> ('a \<Rightarrow> ('b Datatype_Universe.item \<times> 'a) option) \<Rightarrow>
-    'a \<Rightarrow> 'b Datatype_Universe.item"
+  LList_corec_aux :: "nat \<Rightarrow> ('a \<Rightarrow> ('b Datatype.item \<times> 'a) option) \<Rightarrow>
+    'a \<Rightarrow> 'b Datatype.item"
 primrec
   "LList_corec_aux 0 f x = {}"
   "LList_corec_aux (Suc k) f x =
@@ -117,7 +117,7 @@
 subsection {* Abstract type definition *}
 
 typedef 'a llist =
-  "LList (range Datatype_Universe.Leaf) :: 'a Datatype_Universe.item set"
+  "LList (range Datatype.Leaf) :: 'a Datatype.item set"
 proof
   show "NIL \<in> ?llist" ..
 qed
@@ -125,20 +125,20 @@
 lemma NIL_type: "NIL \<in> llist"
   unfolding llist_def by (rule LList.NIL)
 
-lemma CONS_type: "a \<in> range Datatype_Universe.Leaf \<Longrightarrow>
+lemma CONS_type: "a \<in> range Datatype.Leaf \<Longrightarrow>
     M \<in> llist \<Longrightarrow> CONS a M \<in> llist"
   unfolding llist_def by (rule LList.CONS)
 
-lemma llistI: "x \<in> LList (range Datatype_Universe.Leaf) \<Longrightarrow> x \<in> llist"
+lemma llistI: "x \<in> LList (range Datatype.Leaf) \<Longrightarrow> x \<in> llist"
   by (simp add: llist_def)
 
-lemma llistD: "x \<in> llist \<Longrightarrow> x \<in> LList (range Datatype_Universe.Leaf)"
+lemma llistD: "x \<in> llist \<Longrightarrow> x \<in> LList (range Datatype.Leaf)"
   by (simp add: llist_def)
 
 lemma Rep_llist_UNIV: "Rep_llist x \<in> LList UNIV"
 proof -
   have "Rep_llist x \<in> llist" by (rule Rep_llist)
-  then have "Rep_llist x \<in> LList (range Datatype_Universe.Leaf)"
+  then have "Rep_llist x \<in> LList (range Datatype.Leaf)"
     by (simp add: llist_def)
   also have "\<dots> \<subseteq> LList UNIV" by (rule LList_mono) simp
   finally show ?thesis .
@@ -146,7 +146,7 @@
 
 definition
   "LNil = Abs_llist NIL"
-  "LCons x xs = Abs_llist (CONS (Datatype_Universe.Leaf x) (Rep_llist xs))"
+  "LCons x xs = Abs_llist (CONS (Datatype.Leaf x) (Rep_llist xs))"
 
 lemma LCons_not_LNil [iff]: "LCons x xs \<noteq> LNil"
   apply (simp add: LNil_def LCons_def)
@@ -167,7 +167,7 @@
   by (simp add: LNil_def add: Abs_llist_inverse NIL_type)
 
 lemma Rep_llist_LCons: "Rep_llist (LCons x l) =
-    CONS (Datatype_Universe.Leaf x) (Rep_llist l)"
+    CONS (Datatype.Leaf x) (Rep_llist l)"
   by (simp add: LCons_def Abs_llist_inverse CONS_type Rep_llist)
 
 lemma llist_cases [cases type: llist]:
@@ -176,7 +176,7 @@
   | (LCons) x l' where "l = LCons x l'"
 proof (cases l)
   case (Abs_llist L)
-  from `L \<in> llist` have "L \<in> LList (range Datatype_Universe.Leaf)" by (rule llistD)
+  from `L \<in> llist` have "L \<in> LList (range Datatype.Leaf)" by (rule llistD)
   then show ?thesis
   proof cases
     case NIL
@@ -195,7 +195,7 @@
 
 definition
   "llist_case c d l =
-    List_case c (\<lambda>x y. d (inv Datatype_Universe.Leaf x) (Abs_llist y)) (Rep_llist l)"
+    List_case c (\<lambda>x y. d (inv Datatype.Leaf x) (Abs_llist y)) (Rep_llist l)"
 
 syntax  (* FIXME? *)
   LNil :: logic
@@ -217,17 +217,17 @@
     Abs_llist (LList_corec a
       (\<lambda>z.
         case f z of None \<Rightarrow> None
-        | Some (v, w) \<Rightarrow> Some (Datatype_Universe.Leaf v, w)))"
+        | Some (v, w) \<Rightarrow> Some (Datatype.Leaf v, w)))"
 
 lemma LList_corec_type2:
   "LList_corec a
     (\<lambda>z. case f z of None \<Rightarrow> None
-      | Some (v, w) \<Rightarrow> Some (Datatype_Universe.Leaf v, w)) \<in> llist"
+      | Some (v, w) \<Rightarrow> Some (Datatype.Leaf v, w)) \<in> llist"
   (is "?corec a \<in> _")
 proof (unfold llist_def)
   let "LList_corec a ?g" = "?corec a"
   have "?corec a \<in> {?corec x | x. True}" by blast
-  then show "?corec a \<in> LList (range Datatype_Universe.Leaf)"
+  then show "?corec a \<in> LList (range Datatype.Leaf)"
   proof coinduct
     case (LList L)
     then obtain x where L: "L = ?corec x" by blast
@@ -241,7 +241,7 @@
     next
       case (Some p)
       then have "?corec x =
-          CONS (Datatype_Universe.Leaf (fst p)) (?corec (snd p))"
+          CONS (Datatype.Leaf (fst p)) (?corec (snd p))"
         by (simp add: split_def LList_corec)
       with L have ?CONS by auto
       then show ?thesis ..
@@ -263,12 +263,12 @@
   let "?rep_corec a" =
     "LList_corec a
       (\<lambda>z. case f z of None \<Rightarrow> None
-        | Some (v, w) \<Rightarrow> Some (Datatype_Universe.Leaf v, w))"
+        | Some (v, w) \<Rightarrow> Some (Datatype.Leaf v, w))"
 
   have "?corec a = Abs_llist (?rep_corec a)"
     by (simp only: llist_corec_def)
   also from Some have "?rep_corec a =
-      CONS (Datatype_Universe.Leaf (fst p)) (?rep_corec (snd p))"
+      CONS (Datatype.Leaf (fst p)) (?rep_corec (snd p))"
     by (simp add: split_def LList_corec)
   also have "?rep_corec (snd p) = Rep_llist (?corec (snd p))"
     by (simp only: llist_corec_def Abs_llist_inverse LList_corec_type2)
@@ -281,8 +281,8 @@
 subsection {* Equality as greatest fixed-point; the bisimulation principle. *}
 
 consts
-  EqLList :: "('a Datatype_Universe.item \<times> 'a Datatype_Universe.item) set \<Rightarrow>
-    ('a Datatype_Universe.item \<times> 'a Datatype_Universe.item) set"
+  EqLList :: "('a Datatype.item \<times> 'a Datatype.item) set \<Rightarrow>
+    ('a Datatype.item \<times> 'a Datatype.item) set"
 
 coinductive "EqLList r"
   intros
@@ -291,7 +291,7 @@
       (CONS a M, CONS b N) \<in> EqLList r"
 
 lemma EqLList_unfold:
-    "EqLList r = dsum (diag {Datatype_Universe.Numb 0}) (dprod r (EqLList r))"
+    "EqLList r = dsum (diag {Datatype.Numb 0}) (dprod r (EqLList r))"
   by (fast intro!: EqLList.intros [unfolded NIL_def CONS_def]
            elim: EqLList.cases [unfolded NIL_def CONS_def])
 
@@ -612,7 +612,7 @@
   have "(?lhs M, ?rhs M) \<in> {(?lhs N, ?rhs N) | N. N \<in> LList A}"
     using M by blast
   then show ?thesis
-  proof (coinduct taking: "range (\<lambda>N :: 'a Datatype_Universe.item. N)"
+  proof (coinduct taking: "range (\<lambda>N :: 'a Datatype.item. N)"
       rule: LList_equalityI)
     case (EqLList q)
     then obtain N where q: "q = (?lhs N, ?rhs N)" and N: "N \<in> LList A" by blast
@@ -635,7 +635,7 @@
 proof -
   have "(?lmap M, M) \<in> {(?lmap N, N) | N. N \<in> LList A}" using M by blast
   then show ?thesis
-  proof (coinduct taking: "range (\<lambda>N :: 'a Datatype_Universe.item. N)"
+  proof (coinduct taking: "range (\<lambda>N :: 'a Datatype.item. N)"
       rule: LList_equalityI)
     case (EqLList q)
     then obtain N where q: "q = (?lmap N, N)" and N: "N \<in> LList A" by blast

--- a/src/HOL/Tools/datatype_package.ML	Sun Oct 01 22:19:21 2006 +0200
+++ b/src/HOL/Tools/datatype_package.ML	Sun Oct 01 22:19:23 2006 +0200
@@ -927,7 +927,7 @@
 
 fun gen_add_datatype prep_typ err flat_names new_type_names dts thy =
   let
-    val _ = Theory.requires thy "Datatype_Universe" "datatype definitions";
+    val _ = Theory.requires thy "Datatype" "datatype definitions";
 
     (* this theory is used just for parsing *)
 

--- a/src/HOL/Tools/datatype_rep_proofs.ML	Sun Oct 01 22:19:21 2006 +0200
+++ b/src/HOL/Tools/datatype_rep_proofs.ML	Sun Oct 01 22:19:23 2006 +0200
@@ -43,13 +43,13 @@
       new_type_names descr sorts types_syntax constr_syntax case_names_induct thy =
   let
     val Datatype_thy = ThyInfo.the_theory "Datatype" thy;
-    val node_name = "Datatype_Universe.node";
-    val In0_name = "Datatype_Universe.In0";
-    val In1_name = "Datatype_Universe.In1";
-    val Scons_name = "Datatype_Universe.Scons";
-    val Leaf_name = "Datatype_Universe.Leaf";
-    val Numb_name = "Datatype_Universe.Numb";
-    val Lim_name = "Datatype_Universe.Lim";
+    val node_name = "Datatype.node";
+    val In0_name = "Datatype.In0";
+    val In1_name = "Datatype.In1";
+    val Scons_name = "Datatype.Scons";
+    val Leaf_name = "Datatype.Leaf";
+    val Numb_name = "Datatype.Numb";
+    val Lim_name = "Datatype.Lim";
     val Suml_name = "Datatype.Suml";
     val Sumr_name = "Datatype.Sumr";