--- a/src/HOL/Datatype_Universe.ML Wed Dec 08 15:15:49 2004 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,561 +0,0 @@
-(* Title: HOL/Datatype_Universe.ML
- ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1991 University of Cambridge
-*)
-
-(** apfst -- can be used in similar type definitions **)
-
-Goalw [apfst_def] "apfst f (a,b) = (f(a),b)";
-by (rtac split_conv 1);
-qed "apfst_conv";
-
-val [major,minor] = Goal
- "[| q = apfst f p; !!x y. [| p = (x,y); q = (f(x),y) |] ==> R \
-\ |] ==> R";
-by (rtac PairE 1);
-by (rtac minor 1);
-by (assume_tac 1);
-by (rtac (major RS trans) 1);
-by (etac ssubst 1);
-by (rtac apfst_conv 1);
-qed "apfst_convE";
-
-(** Push -- an injection, analogous to Cons on lists **)
-
-Goalw [Push_def] "Push i f = Push j g ==> i=j";
-by (etac (fun_cong RS box_equals) 1);
-by (rtac nat_case_0 1);
-by (rtac nat_case_0 1);
-qed "Push_inject1";
-
-Goalw [Push_def] "Push i f = Push j g ==> f=g";
-by (rtac (ext RS box_equals) 1);
-by (etac fun_cong 1);
-by (rtac (nat_case_Suc RS ext) 1);
-by (rtac (nat_case_Suc RS ext) 1);
-qed "Push_inject2";
-
-val [major,minor] = Goal
- "[| Push i f =Push j g; [| i=j; f=g |] ==> P \
-\ |] ==> P";
-by (rtac ((major RS Push_inject2) RS ((major RS Push_inject1) RS minor)) 1);
-qed "Push_inject";
-
-Goalw [Push_def] "Push (Inr (Suc k)) f = (%z. Inr 0) ==> P";
-by (rtac Suc_neq_Zero 1);
-by (etac (fun_cong RS box_equals RS Inr_inject) 1);
-by (rtac nat_case_0 1);
-by (rtac refl 1);
-qed "Push_neq_K0";
-
-(*** Isomorphisms ***)
-
-Goal "inj(Rep_Node)";
-by (rtac inj_inverseI 1); (*cannot combine by RS: multiple unifiers*)
-by (rtac Rep_Node_inverse 1);
-qed "inj_Rep_Node";
-
-Goal "inj_on Abs_Node Node";
-by (rtac inj_on_inverseI 1);
-by (etac Abs_Node_inverse 1);
-qed "inj_on_Abs_Node";
-
-bind_thm ("Abs_Node_inj", inj_on_Abs_Node RS inj_onD);
-
-
-(*** Introduction rules for Node ***)
-
-Goalw [Node_def] "(%k. Inr 0, a) : Node";
-by (Blast_tac 1);
-qed "Node_K0_I";
-
-Goalw [Node_def,Push_def]
- "p: Node ==> apfst (Push i) p : Node";
-by (fast_tac (claset() addSIs [apfst_conv, nat_case_Suc RS trans]) 1);
-qed "Node_Push_I";
-
-
-(*** Distinctness of constructors ***)
-
-(** Scons vs Atom **)
-
-Goalw [Atom_def,Scons_def,Push_Node_def,One_nat_def]
- "Scons M N ~= Atom(a)";
-by (rtac notI 1);
-by (etac (equalityD2 RS subsetD RS UnE) 1);
-by (rtac singletonI 1);
-by (REPEAT (eresolve_tac [imageE, Abs_Node_inj RS apfst_convE,
- Pair_inject, sym RS Push_neq_K0] 1
- ORELSE resolve_tac [Node_K0_I, Rep_Node RS Node_Push_I] 1));
-qed "Scons_not_Atom";
-bind_thm ("Atom_not_Scons", Scons_not_Atom RS not_sym);
-
-
-(*** Injectiveness ***)
-
-(** Atomic nodes **)
-
-Goalw [Atom_def] "inj(Atom)";
-by (blast_tac (claset() addSIs [injI, Node_K0_I] addSDs [Abs_Node_inj]) 1);
-qed "inj_Atom";
-bind_thm ("Atom_inject", inj_Atom RS injD);
-
-Goal "(Atom(a)=Atom(b)) = (a=b)";
-by (blast_tac (claset() addSDs [Atom_inject]) 1);
-qed "Atom_Atom_eq";
-AddIffs [Atom_Atom_eq];
-
-Goalw [Leaf_def,o_def] "inj(Leaf)";
-by (rtac injI 1);
-by (etac (Atom_inject RS Inl_inject) 1);
-qed "inj_Leaf";
-
-bind_thm ("Leaf_inject", inj_Leaf RS injD);
-AddSDs [Leaf_inject];
-
-Goalw [Numb_def,o_def] "inj(Numb)";
-by (rtac injI 1);
-by (etac (Atom_inject RS Inr_inject) 1);
-qed "inj_Numb";
-
-bind_thm ("Numb_inject", inj_Numb RS injD);
-AddSDs [Numb_inject];
-
-(** Injectiveness of Push_Node **)
-
-val [major,minor] = Goalw [Push_Node_def]
- "[| Push_Node i m =Push_Node j n; [| i=j; m=n |] ==> P \
-\ |] ==> P";
-by (rtac (major RS Abs_Node_inj RS apfst_convE) 1);
-by (REPEAT (resolve_tac [Rep_Node RS Node_Push_I] 1));
-by (etac (sym RS apfst_convE) 1);
-by (rtac minor 1);
-by (etac Pair_inject 1);
-by (etac (Push_inject1 RS sym) 1);
-by (rtac (inj_Rep_Node RS injD) 1);
-by (etac trans 1);
-by (safe_tac (claset() addSEs [Push_inject,sym]));
-qed "Push_Node_inject";
-
-
-(** Injectiveness of Scons **)
-
-Goalw [Scons_def,One_nat_def] "Scons M N <= Scons M' N' ==> M<=M'";
-by (blast_tac (claset() addSDs [Push_Node_inject]) 1);
-qed "Scons_inject_lemma1";
-
-Goalw [Scons_def,One_nat_def] "Scons M N <= Scons M' N' ==> N<=N'";
-by (blast_tac (claset() addSDs [Push_Node_inject]) 1);
-qed "Scons_inject_lemma2";
-
-Goal "Scons M N = Scons M' N' ==> M=M'";
-by (etac equalityE 1);
-by (REPEAT (ares_tac [equalityI, Scons_inject_lemma1] 1));
-qed "Scons_inject1";
-
-Goal "Scons M N = Scons M' N' ==> N=N'";
-by (etac equalityE 1);
-by (REPEAT (ares_tac [equalityI, Scons_inject_lemma2] 1));
-qed "Scons_inject2";
-
-val [major,minor] = Goal
- "[| Scons M N = Scons M' N'; [| M=M'; N=N' |] ==> P \
-\ |] ==> P";
-by (rtac ((major RS Scons_inject2) RS ((major RS Scons_inject1) RS minor)) 1);
-qed "Scons_inject";
-
-Goal "(Scons M N = Scons M' N') = (M=M' & N=N')";
-by (blast_tac (claset() addSEs [Scons_inject]) 1);
-qed "Scons_Scons_eq";
-
-(*** Distinctness involving Leaf and Numb ***)
-
-(** Scons vs Leaf **)
-
-Goalw [Leaf_def,o_def] "Scons M N ~= Leaf(a)";
-by (rtac Scons_not_Atom 1);
-qed "Scons_not_Leaf";
-bind_thm ("Leaf_not_Scons", Scons_not_Leaf RS not_sym);
-
-AddIffs [Scons_not_Leaf, Leaf_not_Scons];
-
-
-(** Scons vs Numb **)
-
-Goalw [Numb_def,o_def] "Scons M N ~= Numb(k)";
-by (rtac Scons_not_Atom 1);
-qed "Scons_not_Numb";
-bind_thm ("Numb_not_Scons", Scons_not_Numb RS not_sym);
-
-AddIffs [Scons_not_Numb, Numb_not_Scons];
-
-
-(** Leaf vs Numb **)
-
-Goalw [Leaf_def,Numb_def] "Leaf(a) ~= Numb(k)";
-by (simp_tac (simpset() addsimps [Inl_not_Inr]) 1);
-qed "Leaf_not_Numb";
-bind_thm ("Numb_not_Leaf", Leaf_not_Numb RS not_sym);
-
-AddIffs [Leaf_not_Numb, Numb_not_Leaf];
-
-
-(*** ndepth -- the depth of a node ***)
-
-Addsimps [apfst_conv];
-AddIffs [Scons_not_Atom, Atom_not_Scons, Scons_Scons_eq];
-
-
-Goalw [ndepth_def] "ndepth (Abs_Node(%k. Inr 0, x)) = 0";
-by (EVERY1[stac (Node_K0_I RS Abs_Node_inverse), stac split_conv]);
-by (rtac Least_equality 1);
-by Auto_tac;
-qed "ndepth_K0";
-
-Goal "nat_case (Inr (Suc i)) f k = Inr 0 --> Suc(LEAST x. f x = Inr 0) <= k";
-by (induct_tac "k" 1);
-by (ALLGOALS Simp_tac);
-by (rtac impI 1);
-by (etac Least_le 1);
-val lemma = result();
-
-Goalw [ndepth_def,Push_Node_def]
- "ndepth (Push_Node (Inr (Suc i)) n) = Suc(ndepth(n))";
-by (stac (Rep_Node RS Node_Push_I RS Abs_Node_inverse) 1);
-by (cut_facts_tac [rewrite_rule [Node_def] Rep_Node] 1);
-by Safe_tac;
-by (etac ssubst 1); (*instantiates type variables!*)
-by (Simp_tac 1);
-by (rtac Least_equality 1);
-by (rewtac Push_def);
-by (auto_tac (claset(), simpset() addsimps [lemma]));
-by (etac LeastI 1);
-qed "ndepth_Push_Node";
-
-
-(*** ntrunc applied to the various node sets ***)
-
-Goalw [ntrunc_def] "ntrunc 0 M = {}";
-by (Blast_tac 1);
-qed "ntrunc_0";
-
-Goalw [Atom_def,ntrunc_def] "ntrunc (Suc k) (Atom a) = Atom(a)";
-by (fast_tac (claset() addss (simpset() addsimps [ndepth_K0])) 1);
-qed "ntrunc_Atom";
-
-Goalw [Leaf_def,o_def] "ntrunc (Suc k) (Leaf a) = Leaf(a)";
-by (rtac ntrunc_Atom 1);
-qed "ntrunc_Leaf";
-
-Goalw [Numb_def,o_def] "ntrunc (Suc k) (Numb i) = Numb(i)";
-by (rtac ntrunc_Atom 1);
-qed "ntrunc_Numb";
-
-Goalw [Scons_def,ntrunc_def,One_nat_def]
- "ntrunc (Suc k) (Scons M N) = Scons (ntrunc k M) (ntrunc k N)";
-by (safe_tac (claset() addSIs [imageI]));
-by (REPEAT (stac ndepth_Push_Node 3 THEN etac Suc_mono 3));
-by (REPEAT (rtac Suc_less_SucD 1 THEN
- rtac (ndepth_Push_Node RS subst) 1 THEN
- assume_tac 1));
-qed "ntrunc_Scons";
-
-Addsimps [ntrunc_0, ntrunc_Atom, ntrunc_Leaf, ntrunc_Numb, ntrunc_Scons];
-
-
-(** Injection nodes **)
-
-Goalw [In0_def] "ntrunc (Suc 0) (In0 M) = {}";
-by (Simp_tac 1);
-by (rewtac Scons_def);
-by (Blast_tac 1);
-qed "ntrunc_one_In0";
-
-Goalw [In0_def]
- "ntrunc (Suc (Suc k)) (In0 M) = In0 (ntrunc (Suc k) M)";
-by (Simp_tac 1);
-qed "ntrunc_In0";
-
-Goalw [In1_def] "ntrunc (Suc 0) (In1 M) = {}";
-by (Simp_tac 1);
-by (rewtac Scons_def);
-by (Blast_tac 1);
-qed "ntrunc_one_In1";
-
-Goalw [In1_def]
- "ntrunc (Suc (Suc k)) (In1 M) = In1 (ntrunc (Suc k) M)";
-by (Simp_tac 1);
-qed "ntrunc_In1";
-
-Addsimps [ntrunc_one_In0, ntrunc_In0, ntrunc_one_In1, ntrunc_In1];
-
-
-(*** Cartesian Product ***)
-
-Goalw [uprod_def] "[| M:A; N:B |] ==> Scons M N : uprod A B";
-by (REPEAT (ares_tac [singletonI,UN_I] 1));
-qed "uprodI";
-
-(*The general elimination rule*)
-val major::prems = Goalw [uprod_def]
- "[| c : uprod A B; \
-\ !!x y. [| x:A; y:B; c = Scons x y |] ==> P \
-\ |] ==> P";
-by (cut_facts_tac [major] 1);
-by (REPEAT (eresolve_tac [asm_rl,singletonE,UN_E] 1
- ORELSE resolve_tac prems 1));
-qed "uprodE";
-
-(*Elimination of a pair -- introduces no eigenvariables*)
-val prems = Goal
- "[| Scons M N : uprod A B; [| M:A; N:B |] ==> P \
-\ |] ==> P";
-by (rtac uprodE 1);
-by (REPEAT (ares_tac prems 1 ORELSE eresolve_tac [Scons_inject,ssubst] 1));
-qed "uprodE2";
-
-
-(*** Disjoint Sum ***)
-
-Goalw [usum_def] "M:A ==> In0(M) : usum A B";
-by (Blast_tac 1);
-qed "usum_In0I";
-
-Goalw [usum_def] "N:B ==> In1(N) : usum A B";
-by (Blast_tac 1);
-qed "usum_In1I";
-
-val major::prems = Goalw [usum_def]
- "[| u : usum A B; \
-\ !!x. [| x:A; u=In0(x) |] ==> P; \
-\ !!y. [| y:B; u=In1(y) |] ==> P \
-\ |] ==> P";
-by (rtac (major RS UnE) 1);
-by (REPEAT (rtac refl 1
- ORELSE eresolve_tac (prems@[imageE,ssubst]) 1));
-qed "usumE";
-
-
-(** Injection **)
-
-Goalw [In0_def,In1_def,One_nat_def] "In0(M) ~= In1(N)";
-by (rtac notI 1);
-by (etac (Scons_inject1 RS Numb_inject RS Zero_neq_Suc) 1);
-qed "In0_not_In1";
-
-bind_thm ("In1_not_In0", In0_not_In1 RS not_sym);
-
-AddIffs [In0_not_In1, In1_not_In0];
-
-Goalw [In0_def] "In0(M) = In0(N) ==> M=N";
-by (etac (Scons_inject2) 1);
-qed "In0_inject";
-
-Goalw [In1_def] "In1(M) = In1(N) ==> M=N";
-by (etac (Scons_inject2) 1);
-qed "In1_inject";
-
-Goal "(In0 M = In0 N) = (M=N)";
-by (blast_tac (claset() addSDs [In0_inject]) 1);
-qed "In0_eq";
-
-Goal "(In1 M = In1 N) = (M=N)";
-by (blast_tac (claset() addSDs [In1_inject]) 1);
-qed "In1_eq";
-
-AddIffs [In0_eq, In1_eq];
-
-Goal "inj In0";
-by (blast_tac (claset() addSIs [injI]) 1);
-qed "inj_In0";
-
-Goal "inj In1";
-by (blast_tac (claset() addSIs [injI]) 1);
-qed "inj_In1";
-
-
-(*** Function spaces ***)
-
-Goalw [Lim_def] "Lim f = Lim g ==> f = g";
-by (rtac ext 1);
-by (blast_tac (claset() addSEs [Push_Node_inject]) 1);
-qed "Lim_inject";
-
-
-(*** proving equality of sets and functions using ntrunc ***)
-
-Goalw [ntrunc_def] "ntrunc k M <= M";
-by (Blast_tac 1);
-qed "ntrunc_subsetI";
-
-val [major] = Goalw [ntrunc_def] "(!!k. ntrunc k M <= N) ==> M<=N";
-by (blast_tac (claset() addIs [less_add_Suc1, less_add_Suc2,
- major RS subsetD]) 1);
-qed "ntrunc_subsetD";
-
-(*A generalized form of the take-lemma*)
-val [major] = Goal "(!!k. ntrunc k M = ntrunc k N) ==> M=N";
-by (rtac equalityI 1);
-by (ALLGOALS (rtac ntrunc_subsetD));
-by (ALLGOALS (rtac (ntrunc_subsetI RSN (2, subset_trans))));
-by (rtac (major RS equalityD1) 1);
-by (rtac (major RS equalityD2) 1);
-qed "ntrunc_equality";
-
-val [major] = Goalw [o_def]
- "[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2";
-by (rtac (ntrunc_equality RS ext) 1);
-by (rtac (major RS fun_cong) 1);
-qed "ntrunc_o_equality";
-
-(*** Monotonicity ***)
-
-Goalw [uprod_def] "[| A<=A'; B<=B' |] ==> uprod A B <= uprod A' B'";
-by (Blast_tac 1);
-qed "uprod_mono";
-
-Goalw [usum_def] "[| A<=A'; B<=B' |] ==> usum A B <= usum A' B'";
-by (Blast_tac 1);
-qed "usum_mono";
-
-Goalw [Scons_def] "[| M<=M'; N<=N' |] ==> Scons M N <= Scons M' N'";
-by (Blast_tac 1);
-qed "Scons_mono";
-
-Goalw [In0_def] "M<=N ==> In0(M) <= In0(N)";
-by (REPEAT (ares_tac [subset_refl,Scons_mono] 1));
-qed "In0_mono";
-
-Goalw [In1_def] "M<=N ==> In1(M) <= In1(N)";
-by (REPEAT (ares_tac [subset_refl,Scons_mono] 1));
-qed "In1_mono";
-
-
-(*** Split and Case ***)
-
-Goalw [Split_def] "Split c (Scons M N) = c M N";
-by (Blast_tac 1);
-qed "Split";
-
-Goalw [Case_def] "Case c d (In0 M) = c(M)";
-by (Blast_tac 1);
-qed "Case_In0";
-
-Goalw [Case_def] "Case c d (In1 N) = d(N)";
-by (Blast_tac 1);
-qed "Case_In1";
-
-Addsimps [Split, Case_In0, Case_In1];
-
-
-(**** UN x. B(x) rules ****)
-
-Goalw [ntrunc_def] "ntrunc k (UN x. f(x)) = (UN x. ntrunc k (f x))";
-by (Blast_tac 1);
-qed "ntrunc_UN1";
-
-Goalw [Scons_def] "Scons (UN x. f x) M = (UN x. Scons (f x) M)";
-by (Blast_tac 1);
-qed "Scons_UN1_x";
-
-Goalw [Scons_def] "Scons M (UN x. f x) = (UN x. Scons M (f x))";
-by (Blast_tac 1);
-qed "Scons_UN1_y";
-
-Goalw [In0_def] "In0(UN x. f(x)) = (UN x. In0(f(x)))";
-by (rtac Scons_UN1_y 1);
-qed "In0_UN1";
-
-Goalw [In1_def] "In1(UN x. f(x)) = (UN x. In1(f(x)))";
-by (rtac Scons_UN1_y 1);
-qed "In1_UN1";
-
-
-(*** Equality for Cartesian Product ***)
-
-Goalw [dprod_def]
- "[| (M,M'):r; (N,N'):s |] ==> (Scons M N, Scons M' N') : dprod r s";
-by (Blast_tac 1);
-qed "dprodI";
-
-(*The general elimination rule*)
-val major::prems = Goalw [dprod_def]
- "[| c : dprod r s; \
-\ !!x y x' y'. [| (x,x') : r; (y,y') : s; c = (Scons x y, Scons x' y') |] ==> P \
-\ |] ==> P";
-by (cut_facts_tac [major] 1);
-by (REPEAT_FIRST (eresolve_tac [asm_rl, UN_E, mem_splitE, singletonE]));
-by (REPEAT (ares_tac prems 1 ORELSE hyp_subst_tac 1));
-qed "dprodE";
-
-
-(*** Equality for Disjoint Sum ***)
-
-Goalw [dsum_def] "(M,M'):r ==> (In0(M), In0(M')) : dsum r s";
-by (Blast_tac 1);
-qed "dsum_In0I";
-
-Goalw [dsum_def] "(N,N'):s ==> (In1(N), In1(N')) : dsum r s";
-by (Blast_tac 1);
-qed "dsum_In1I";
-
-val major::prems = Goalw [dsum_def]
- "[| 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 (cut_facts_tac [major] 1);
-by (REPEAT_FIRST (eresolve_tac [asm_rl, UN_E, UnE, mem_splitE, singletonE]));
-by (DEPTH_SOLVE (ares_tac prems 1 ORELSE hyp_subst_tac 1));
-qed "dsumE";
-
-AddSIs [uprodI, dprodI];
-AddIs [usum_In0I, usum_In1I, dsum_In0I, dsum_In1I];
-AddSEs [uprodE, dprodE, usumE, dsumE];
-
-
-(*** Monotonicity ***)
-
-Goal "[| r<=r'; s<=s' |] ==> dprod r s <= dprod r' s'";
-by (Blast_tac 1);
-qed "dprod_mono";
-
-Goal "[| r<=r'; s<=s' |] ==> dsum r s <= dsum r' s'";
-by (Blast_tac 1);
-qed "dsum_mono";
-
-
-(*** Bounding theorems ***)
-
-Goal "(dprod (A <*> B) (C <*> D)) <= (uprod A C) <*> (uprod B D)";
-by (Blast_tac 1);
-qed "dprod_Sigma";
-
-bind_thm ("dprod_subset_Sigma", [dprod_mono, dprod_Sigma] MRS subset_trans |> standard);
-
-(*Dependent version*)
-Goal "(dprod (Sigma A B) (Sigma C D)) <= Sigma (uprod A C) (Split (%x y. uprod (B x) (D y)))";
-by Safe_tac;
-by (stac Split 1);
-by (Blast_tac 1);
-qed "dprod_subset_Sigma2";
-
-Goal "(dsum (A <*> B) (C <*> D)) <= (usum A C) <*> (usum B D)";
-by (Blast_tac 1);
-qed "dsum_Sigma";
-
-bind_thm ("dsum_subset_Sigma", [dsum_mono, dsum_Sigma] MRS subset_trans |> standard);
-
-
-(*** Domain ***)
-
-Goal "Domain (dprod r s) = uprod (Domain r) (Domain s)";
-by Auto_tac;
-qed "Domain_dprod";
-
-Goal "Domain (dsum r s) = usum (Domain r) (Domain s)";
-by Auto_tac;
-qed "Domain_dsum";
-
-Addsimps [Domain_dprod, Domain_dsum];
--- a/src/HOL/Datatype_Universe.thy Wed Dec 08 15:15:49 2004 +0100
+++ b/src/HOL/Datatype_Universe.thy Thu Dec 09 12:03:06 2004 +0100
@@ -3,97 +3,637 @@
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
-Declares the type ('a, 'b) node, a subtype of (nat=>'b+nat) * ('a+nat)
-
-Defines "Cartesian Product" and "Disjoint Sum" as set operations.
Could <*> be generalized to a general summation (Sigma)?
*)
-Datatype_Universe = NatArith + Sum_Type +
+header{*Analogues of the Cartesian Product and Disjoint Sum for Datatypes*}
+theory Datatype_Universe
+imports NatArith Sum_Type
+begin
-(** lists, trees will be sets of nodes **)
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
-types
- 'a item = ('a, unit) node set
- ('a, 'b) dtree = ('a, 'b) node set
+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
+ 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,In1 :: ('a, 'b) dtree => ('a, 'b) dtree
- Lim :: ('b => ('a, 'b) dtree) => ('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
+ 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
+ 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
+ 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]
+ 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]
+ 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)))"
+ 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)"
+ 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)"
+ 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"
+ 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"
+ 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)}"
+ 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}"
+ 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"
+ 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"
+ 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))
+ 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')}"
+ 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
+ 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)
+
+(*** Isomorphisms ***)
+
+lemma inj_Rep_Node: "inj(Rep_Node)"
+apply (rule inj_on_inverseI)
+apply (rule Rep_Node_inverse)
+done
+
+lemma inj_on_Abs_Node: "inj_on Abs_Node Node"
+apply (rule inj_on_inverseI)
+apply (erule Abs_Node_inverse)
+done
+
+lemmas Abs_Node_inj = inj_on_Abs_Node [THEN inj_onD, 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
+
+
+subsubsection{*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, 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, 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, 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: inj_Rep_Node [THEN injD] 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 (rules intro: equalityI Scons_inject_lemma1)
+done
+
+lemma Scons_inject2: "Scons M N = Scons M' N' ==> N=N'"
+apply (erule equalityE)
+apply (rules intro: equalityI Scons_inject_lemma2)
+done
+
+lemma Scons_inject:
+ "[| Scons M N = Scons M' N'; [| M=M'; N=N' |] ==> P |] ==> P"
+by (rules 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, 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, 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, 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, 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 inj_Rep_Node = thm "inj_Rep_Node";
+val inj_on_Abs_Node = thm "inj_on_Abs_Node";
+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