# Theory Datatype_Universe

Up to index of Isabelle/HOL

theory Datatype_Universe
imports NatArith Sum_Type
begin

    (*  Title:      HOL/Datatype_Universe.thy
        ID:         $Id: Datatype_Universe.thy,v 1.12 2005/09/22 21:56:15 nipkow Exp $
        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 ≠ 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 ≠ 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 ≠ 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) ≠ 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) ≠ 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

    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

lemma apfst_conv:

      apfst f (a, b) = (f a, b)

lemma apfst_convE:

      [| q = apfst f p; !!x y. [| p = (x, y); q = (f x, y) |] ==> R |] ==> R

lemma Push_inject1:

      Push i f = Push j g ==> i = j

lemma Push_inject2:

      Push i f = Push j g ==> f = g

lemma Push_inject:

      [| Push i f = Push j g; [| i = j; f = g |] ==> P |] ==> P

lemma Push_neq_K0:

      Push (Inr (Suc k)) f = (%z. Inr 0) ==> P

lemmas Abs_Node_inj:

      [| Abs_Node x = Abs_Node y; x ∈ Node; y ∈ Node |] ==> x = y

lemmas Abs_Node_inj:

      [| Abs_Node x = Abs_Node y; x ∈ Node; y ∈ Node |] ==> x = y

lemma Node_K0_I:

      (%k. Inr 0, a) ∈ Node

lemma Node_Push_I:

      p ∈ Node ==> apfst (Push i) p ∈ Node

### Freeness: Distinctness of Constructors

lemma Scons_not_Atom:

      Scons M N ≠ Atom a

lemmas Atom_not_Scons:

      Atom a ≠ Scons M N

lemmas Atom_not_Scons:

      Atom a ≠ Scons M N

lemma inj_Atom:

      inj Atom

lemmas Atom_inject:

      Atom x = Atom y ==> x = y

lemmas Atom_inject:

      Atom x = Atom y ==> x = y

lemma Atom_Atom_eq:

      (Atom a = Atom b) = (a = b)

lemma inj_Leaf:

      inj Leaf

lemmas Leaf_inject:

      Leaf x = Leaf y ==> x = y

lemmas Leaf_inject:

      Leaf x = Leaf y ==> x = y

lemma inj_Numb:

      inj Numb

lemmas Numb_inject:

      Numb x = Numb y ==> x = y

lemmas Numb_inject:

      Numb x = Numb y ==> x = y

lemma Push_Node_inject:

      [| Push_Node i m = Push_Node j n; [| i = j; m = n |] ==> P |] ==> P

lemma Scons_inject_lemma1:

      Scons M N ⊆ Scons M' N' ==> M ⊆ M'

lemma Scons_inject_lemma2:

      Scons M N ⊆ Scons M' N' ==> N ⊆ N'

lemma Scons_inject1:

      Scons M N = Scons M' N' ==> M = M'

lemma Scons_inject2:

      Scons M N = Scons M' N' ==> N = N'

lemma Scons_inject:

      [| Scons M N = Scons M' N'; [| M = M'; N = N' |] ==> P |] ==> P

lemma Scons_Scons_eq:

      (Scons M N = Scons M' N') = (M = M' ∧ N = N')

lemma Scons_not_Leaf:

      Scons M N ≠ Leaf a

lemmas Leaf_not_Scons:

      Leaf a ≠ Scons M N

lemmas Leaf_not_Scons:

      Leaf a ≠ Scons M N

lemma Scons_not_Numb:

      Scons M N ≠ Numb k

lemmas Numb_not_Scons:

      Numb k ≠ Scons M N

lemmas Numb_not_Scons:

      Numb k ≠ Scons M N

lemma Leaf_not_Numb:

      Leaf a ≠ Numb k

lemmas Numb_not_Leaf:

      Numb k ≠ Leaf a

lemmas Numb_not_Leaf:

      Numb k ≠ Leaf a

lemma ndepth_K0:

      ndepth (Abs_Node (%k. Inr 0, x)) = 0

lemma ndepth_Push_Node_aux:

      nat_case (Inr (Suc i)) f k = Inr 0 --> Suc (LEAST x. f x = Inr 0) ≤ k

lemma ndepth_Push_Node:

      ndepth (Push_Node (Inr (Suc i)) n) = Suc (ndepth n)

lemma ntrunc_0:

      ntrunc 0 M = {}

lemma ntrunc_Atom:

      ntrunc (Suc k) (Atom a) = Atom a

lemma ntrunc_Leaf:

      ntrunc (Suc k) (Leaf a) = Leaf a

lemma ntrunc_Numb:

      ntrunc (Suc k) (Numb i) = Numb i

lemma ntrunc_Scons:

      ntrunc (Suc k) (Scons M N) = Scons (ntrunc k M) (ntrunc k N)

lemma ntrunc_one_In0:

      ntrunc (Suc 0) (In0 M) = {}

lemma ntrunc_In0:

      ntrunc (Suc (Suc k)) (In0 M) = In0 (ntrunc (Suc k) M)

lemma ntrunc_one_In1:

      ntrunc (Suc 0) (In1 M) = {}

lemma ntrunc_In1:

      ntrunc (Suc (Suc k)) (In1 M) = In1 (ntrunc (Suc k) M)

### Set Constructions

lemma uprodI:

      [| M ∈ A; N ∈ B |] ==> Scons M N ∈ uprod A B

lemma uprodE:

      [| c ∈ uprod A B; !!x y. [| x ∈ A; y ∈ B; c = Scons x y |] ==> P |] ==> P

lemma uprodE2:

      [| Scons M N ∈ uprod A B; [| M ∈ A; N ∈ B |] ==> P |] ==> P

lemma usum_In0I:

      M ∈ A ==> In0 M ∈ usum A B

lemma usum_In1I:

      N ∈ B ==> In1 N ∈ usum A B

lemma usumE:

      [| u ∈ usum A B; !!x. [| x ∈ A; u = In0 x |] ==> P;
         !!y. [| y ∈ B; u = In1 y |] ==> P |]
      ==> P

lemma In0_not_In1:

      In0 M ≠ In1 N

lemmas In1_not_In0:

      In1 N ≠ In0 M

lemmas In1_not_In0:

      In1 N ≠ In0 M

lemma In0_inject:

      In0 M = In0 N ==> M = N

lemma In1_inject:

      In1 M = In1 N ==> M = N

lemma In0_eq:

      (In0 M = In0 N) = (M = N)

lemma In1_eq:

      (In1 M = In1 N) = (M = N)

lemma inj_In0:

      inj In0

lemma inj_In1:

      inj In1

lemma Lim_inject:

      Lim f = Lim g ==> f = g

lemma ntrunc_subsetI:

      ntrunc k M ⊆ M

lemma ntrunc_subsetD:

      (!!k. ntrunc k M ⊆ N) ==> M ⊆ N

lemma ntrunc_equality:

      (!!k. ntrunc k M = ntrunc k N) ==> M = N

lemma ntrunc_o_equality:

      (!!k. ntrunc k o h1.0 = ntrunc k o h2.0) ==> h1.0 = h2.0

lemma uprod_mono:

      [| A ⊆ A'; B ⊆ B' |] ==> uprod A B ⊆ uprod A' B'

lemma usum_mono:

      [| A ⊆ A'; B ⊆ B' |] ==> usum A B ⊆ usum A' B'

lemma Scons_mono:

      [| M ⊆ M'; N ⊆ N' |] ==> Scons M N ⊆ Scons M' N'

lemma In0_mono:

      M ⊆ N ==> In0 M ⊆ In0 N

lemma In1_mono:

      M ⊆ N ==> In1 M ⊆ In1 N

lemma Split:

      Split c (Scons M N) = c M N

lemma Case_In0:

      Case c d (In0 M) = c M

lemma Case_In1:

      Case c d (In1 N) = d N

lemma ntrunc_UN1:

      ntrunc k (UN x. f x) = (UN x. ntrunc k (f x))

lemma Scons_UN1_x:

      Scons (UN x. f x) M = (UN x. Scons (f x) M)

lemma Scons_UN1_y:

      Scons M (UN x. f x) = (UN x. Scons M (f x))

lemma In0_UN1:

      In0 (UN x. f x) = (UN x. In0 (f x))

lemma In1_UN1:

      In1 (UN x. f x) = (UN x. In1 (f x))

lemma dprodI:

      [| (M, M') ∈ r; (N, N') ∈ s |] ==> (Scons M N, Scons M' N') ∈ dprod r s

lemma dprodE:

      [| c ∈ dprod r s;
         !!x y x' y'.
            [| (x, x') ∈ r; (y, y') ∈ s; c = (Scons x y, Scons x' y') |] ==> P |]
      ==> P

lemma dsum_In0I:

      (M, M') ∈ r ==> (In0 M, In0 M') ∈ dsum r s

lemma dsum_In1I:

      (N, N') ∈ s ==> (In1 N, In1 N') ∈ dsum r s

lemma dsumE:

      [| 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

lemma dprod_mono:

      [| r ⊆ r'; s ⊆ s' |] ==> dprod r s ⊆ dprod r' s'

lemma dsum_mono:

      [| r ⊆ r'; s ⊆ s' |] ==> dsum r s ⊆ dsum r' s'

lemma dprod_Sigma:

      dprod (A × B) (C × D) ⊆ uprod A C × uprod B D

lemmas dprod_subset_Sigma:

      [| r ⊆ A × B; s ⊆ C × D |] ==> dprod r s ⊆ uprod A C × uprod B D

lemmas dprod_subset_Sigma:

      [| r ⊆ A × B; s ⊆ C × D |] ==> dprod r s ⊆ uprod A C × uprod B D

lemma dprod_subset_Sigma2:

      dprod (Sigma A B) (Sigma C D)
      ⊆ Sigma (uprod A C) (Split (%x y. uprod (B x) (D y)))

lemma dsum_Sigma:

      dsum (A × B) (C × D) ⊆ usum A C × usum B D

lemmas dsum_subset_Sigma:

      [| r ⊆ A × B; s ⊆ C × D |] ==> dsum r s ⊆ usum A C × usum B D

lemmas dsum_subset_Sigma:

      [| r ⊆ A × B; s ⊆ C × D |] ==> dsum r s ⊆ usum A C × usum B D

lemma Domain_dprod:

      Domain (dprod r s) = uprod (Domain r) (Domain s)

lemma Domain_dsum:

      Domain (dsum r s) = usum (Domain r) (Domain s)