--- a/Univ.ML Tue Oct 24 14:59:17 1995 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,615 +0,0 @@
-(* Title: HOL/univ
- ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1991 University of Cambridge
-
-For univ.thy
-*)
-
-open Univ;
-
-(** LEAST -- the least number operator **)
-
-
-val [prem1,prem2] = goalw Univ.thy [Least_def]
- "[| P(k); !!x. x<k ==> ~P(x) |] ==> (LEAST x.P(x)) = k";
-by (rtac select_equality 1);
-by (fast_tac (HOL_cs addSIs [prem1,prem2]) 1);
-by (cut_facts_tac [less_linear] 1);
-by (fast_tac (HOL_cs addSIs [prem1] addSDs [prem2]) 1);
-qed "Least_equality";
-
-val [prem] = goal Univ.thy "P(k) ==> P(LEAST x.P(x))";
-by (rtac (prem RS rev_mp) 1);
-by (res_inst_tac [("n","k")] less_induct 1);
-by (rtac impI 1);
-by (rtac classical 1);
-by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
-by (assume_tac 1);
-by (assume_tac 2);
-by (fast_tac HOL_cs 1);
-qed "LeastI";
-
-(*Proof is almost identical to the one above!*)
-val [prem] = goal Univ.thy "P(k) ==> (LEAST x.P(x)) <= k";
-by (rtac (prem RS rev_mp) 1);
-by (res_inst_tac [("n","k")] less_induct 1);
-by (rtac impI 1);
-by (rtac classical 1);
-by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
-by (assume_tac 1);
-by (rtac le_refl 2);
-by (fast_tac (HOL_cs addIs [less_imp_le,le_trans]) 1);
-qed "Least_le";
-
-val [prem] = goal Univ.thy "k < (LEAST x.P(x)) ==> ~P(k)";
-by (rtac notI 1);
-by (etac (rewrite_rule [le_def] Least_le RS notE) 1);
-by (rtac prem 1);
-qed "not_less_Least";
-
-
-(** apfst -- can be used in similar type definitions **)
-
-goalw Univ.thy [apfst_def] "apfst(f,<a,b>) = <f(a),b>";
-by (rtac split 1);
-qed "apfst_conv";
-
-val [major,minor] = goal Univ.thy
- "[| 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 **)
-
-val [major] = goalw Univ.thy [Push_def] "Push(i,f)=Push(j,g) ==> i=j";
-by (rtac (major RS fun_cong RS box_equals RS Suc_inject) 1);
-by (rtac nat_case_0 1);
-by (rtac nat_case_0 1);
-qed "Push_inject1";
-
-val [major] = goalw Univ.thy [Push_def] "Push(i,f)=Push(j,g) ==> f=g";
-by (rtac (major RS fun_cong RS ext RS box_equals) 1);
-by (rtac (nat_case_Suc RS ext) 1);
-by (rtac (nat_case_Suc RS ext) 1);
-qed "Push_inject2";
-
-val [major,minor] = goal Univ.thy
- "[| 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";
-
-val [major] = goalw Univ.thy [Push_def] "Push(k,f)=(%z.0) ==> P";
-by (rtac (major RS fun_cong RS box_equals RS Suc_neq_Zero) 1);
-by (rtac nat_case_0 1);
-by (rtac refl 1);
-qed "Push_neq_K0";
-
-(*** Isomorphisms ***)
-
-goal Univ.thy "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 Univ.thy "inj_onto(Abs_Node,Node)";
-by (rtac inj_onto_inverseI 1);
-by (etac Abs_Node_inverse 1);
-qed "inj_onto_Abs_Node";
-
-val Abs_Node_inject = inj_onto_Abs_Node RS inj_ontoD;
-
-
-(*** Introduction rules for Node ***)
-
-goalw Univ.thy [Node_def] "<%k. 0,a> : Node";
-by (fast_tac set_cs 1);
-qed "Node_K0_I";
-
-goalw Univ.thy [Node_def,Push_def]
- "!!p. p: Node ==> apfst(Push(i), p) : Node";
-by (fast_tac (set_cs addSIs [apfst_conv, nat_case_Suc RS trans]) 1);
-qed "Node_Push_I";
-
-
-(*** Distinctness of constructors ***)
-
-(** Scons vs Atom **)
-
-goalw Univ.thy [Atom_def,Scons_def,Push_Node_def] "(M$N) ~= Atom(a)";
-by (rtac notI 1);
-by (etac (equalityD2 RS subsetD RS UnE) 1);
-by (rtac singletonI 1);
-by (REPEAT (eresolve_tac [imageE, Abs_Node_inject RS apfst_convE,
- Pair_inject, sym RS Push_neq_K0] 1
- ORELSE resolve_tac [Node_K0_I, Rep_Node RS Node_Push_I] 1));
-qed "Scons_not_Atom";
-bind_thm ("Atom_not_Scons", (Scons_not_Atom RS not_sym));
-
-bind_thm ("Scons_neq_Atom", (Scons_not_Atom RS notE));
-val Atom_neq_Scons = sym RS Scons_neq_Atom;
-
-(*** Injectiveness ***)
-
-(** Atomic nodes **)
-
-goalw Univ.thy [Atom_def] "inj(Atom)";
-by (rtac injI 1);
-by (etac (singleton_inject RS Abs_Node_inject RS Pair_inject) 1);
-by (REPEAT (ares_tac [Node_K0_I] 1));
-qed "inj_Atom";
-val Atom_inject = inj_Atom RS injD;
-
-goalw Univ.thy [Leaf_def,o_def] "inj(Leaf)";
-by (rtac injI 1);
-by (etac (Atom_inject RS Inl_inject) 1);
-qed "inj_Leaf";
-
-val Leaf_inject = inj_Leaf RS injD;
-
-goalw Univ.thy [Numb_def,o_def] "inj(Numb)";
-by (rtac injI 1);
-by (etac (Atom_inject RS Inr_inject) 1);
-qed "inj_Numb";
-
-val Numb_inject = inj_Numb RS injD;
-
-(** Injectiveness of Push_Node **)
-
-val [major,minor] = goalw Univ.thy [Push_Node_def]
- "[| Push_Node(i,m)=Push_Node(j,n); [| i=j; m=n |] ==> P \
-\ |] ==> P";
-by (rtac (major RS Abs_Node_inject RS apfst_convE) 1);
-by (REPEAT (resolve_tac [Rep_Node RS Node_Push_I] 1));
-by (etac (sym RS apfst_convE) 1);
-by (rtac minor 1);
-by (etac Pair_inject 1);
-by (etac (Push_inject1 RS sym) 1);
-by (rtac (inj_Rep_Node RS injD) 1);
-by (etac trans 1);
-by (safe_tac (HOL_cs addSEs [Pair_inject,Push_inject,sym]));
-qed "Push_Node_inject";
-
-
-(** Injectiveness of Scons **)
-
-val [major] = goalw Univ.thy [Scons_def] "M$N <= M'$N' ==> M<=M'";
-by (cut_facts_tac [major] 1);
-by (fast_tac (set_cs addSDs [Suc_inject]
- addSEs [Push_Node_inject, Zero_neq_Suc]) 1);
-qed "Scons_inject_lemma1";
-
-val [major] = goalw Univ.thy [Scons_def] "M$N <= M'$N' ==> N<=N'";
-by (cut_facts_tac [major] 1);
-by (fast_tac (set_cs addSDs [Suc_inject]
- addSEs [Push_Node_inject, Suc_neq_Zero]) 1);
-qed "Scons_inject_lemma2";
-
-val [major] = goal Univ.thy "M$N = M'$N' ==> M=M'";
-by (rtac (major RS equalityE) 1);
-by (REPEAT (ares_tac [equalityI, Scons_inject_lemma1] 1));
-qed "Scons_inject1";
-
-val [major] = goal Univ.thy "M$N = M'$N' ==> N=N'";
-by (rtac (major RS equalityE) 1);
-by (REPEAT (ares_tac [equalityI, Scons_inject_lemma2] 1));
-qed "Scons_inject2";
-
-val [major,minor] = goal Univ.thy
- "[| M$N = M'$N'; [| M=M'; N=N' |] ==> P \
-\ |] ==> P";
-by (rtac ((major RS Scons_inject2) RS ((major RS Scons_inject1) RS minor)) 1);
-qed "Scons_inject";
-
-(*rewrite rules*)
-goal Univ.thy "(Atom(a)=Atom(b)) = (a=b)";
-by (fast_tac (HOL_cs addSEs [Atom_inject]) 1);
-qed "Atom_Atom_eq";
-
-goal Univ.thy "(M$N = M'$N') = (M=M' & N=N')";
-by (fast_tac (HOL_cs addSEs [Scons_inject]) 1);
-qed "Scons_Scons_eq";
-
-(*** Distinctness involving Leaf and Numb ***)
-
-(** Scons vs Leaf **)
-
-goalw Univ.thy [Leaf_def,o_def] "(M$N) ~= Leaf(a)";
-by (rtac Scons_not_Atom 1);
-qed "Scons_not_Leaf";
-bind_thm ("Leaf_not_Scons", (Scons_not_Leaf RS not_sym));
-
-bind_thm ("Scons_neq_Leaf", (Scons_not_Leaf RS notE));
-val Leaf_neq_Scons = sym RS Scons_neq_Leaf;
-
-(** Scons vs Numb **)
-
-goalw Univ.thy [Numb_def,o_def] "(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));
-
-bind_thm ("Scons_neq_Numb", (Scons_not_Numb RS notE));
-val Numb_neq_Scons = sym RS Scons_neq_Numb;
-
-(** Leaf vs Numb **)
-
-goalw Univ.thy [Leaf_def,Numb_def] "Leaf(a) ~= Numb(k)";
-by (simp_tac (HOL_ss addsimps [Atom_Atom_eq,Inl_not_Inr]) 1);
-qed "Leaf_not_Numb";
-bind_thm ("Numb_not_Leaf", (Leaf_not_Numb RS not_sym));
-
-bind_thm ("Leaf_neq_Numb", (Leaf_not_Numb RS notE));
-val Numb_neq_Leaf = sym RS Leaf_neq_Numb;
-
-
-(*** ndepth -- the depth of a node ***)
-
-val univ_simps = [apfst_conv,Scons_not_Atom,Atom_not_Scons,Scons_Scons_eq];
-val univ_ss = nat_ss addsimps univ_simps;
-
-
-goalw Univ.thy [ndepth_def] "ndepth (Abs_Node(<%k.0, x>)) = 0";
-by (sstac [Node_K0_I RS Abs_Node_inverse, split] 1);
-by (rtac Least_equality 1);
-by (rtac refl 1);
-by (etac less_zeroE 1);
-qed "ndepth_K0";
-
-goal Univ.thy "k < Suc(LEAST x. f(x)=0) --> nat_case(Suc(i), f, k) ~= 0";
-by (nat_ind_tac "k" 1);
-by (ALLGOALS (simp_tac nat_ss));
-by (rtac impI 1);
-by (etac not_less_Least 1);
-qed "ndepth_Push_lemma";
-
-goalw Univ.thy [ndepth_def,Push_Node_def]
- "ndepth (Push_Node(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 set_cs);
-be ssubst 1; (*instantiates type variables!*)
-by (simp_tac univ_ss 1);
-by (rtac Least_equality 1);
-by (rewtac Push_def);
-by (rtac (nat_case_Suc RS trans) 1);
-by (etac LeastI 1);
-by (etac (ndepth_Push_lemma RS mp) 1);
-qed "ndepth_Push_Node";
-
-
-(*** ntrunc applied to the various node sets ***)
-
-goalw Univ.thy [ntrunc_def] "ntrunc(0, M) = {}";
-by (safe_tac (set_cs addSIs [equalityI] addSEs [less_zeroE]));
-qed "ntrunc_0";
-
-goalw Univ.thy [Atom_def,ntrunc_def] "ntrunc(Suc(k), Atom(a)) = Atom(a)";
-by (safe_tac (set_cs addSIs [equalityI]));
-by (stac ndepth_K0 1);
-by (rtac zero_less_Suc 1);
-qed "ntrunc_Atom";
-
-goalw Univ.thy [Leaf_def,o_def] "ntrunc(Suc(k), Leaf(a)) = Leaf(a)";
-by (rtac ntrunc_Atom 1);
-qed "ntrunc_Leaf";
-
-goalw Univ.thy [Numb_def,o_def] "ntrunc(Suc(k), Numb(i)) = Numb(i)";
-by (rtac ntrunc_Atom 1);
-qed "ntrunc_Numb";
-
-goalw Univ.thy [Scons_def,ntrunc_def]
- "ntrunc(Suc(k), M$N) = ntrunc(k,M) $ ntrunc(k,N)";
-by (safe_tac (set_cs addSIs [equalityI,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";
-
-(** Injection nodes **)
-
-goalw Univ.thy [In0_def] "ntrunc(Suc(0), In0(M)) = {}";
-by (simp_tac (univ_ss addsimps [ntrunc_Scons,ntrunc_0]) 1);
-by (rewtac Scons_def);
-by (safe_tac (set_cs addSIs [equalityI]));
-qed "ntrunc_one_In0";
-
-goalw Univ.thy [In0_def]
- "ntrunc(Suc(Suc(k)), In0(M)) = In0 (ntrunc(Suc(k),M))";
-by (simp_tac (univ_ss addsimps [ntrunc_Scons,ntrunc_Numb]) 1);
-qed "ntrunc_In0";
-
-goalw Univ.thy [In1_def] "ntrunc(Suc(0), In1(M)) = {}";
-by (simp_tac (univ_ss addsimps [ntrunc_Scons,ntrunc_0]) 1);
-by (rewtac Scons_def);
-by (safe_tac (set_cs addSIs [equalityI]));
-qed "ntrunc_one_In1";
-
-goalw Univ.thy [In1_def]
- "ntrunc(Suc(Suc(k)), In1(M)) = In1 (ntrunc(Suc(k),M))";
-by (simp_tac (univ_ss addsimps [ntrunc_Scons,ntrunc_Numb]) 1);
-qed "ntrunc_In1";
-
-
-(*** Cartesian Product ***)
-
-goalw Univ.thy [uprod_def] "!!M N. [| M:A; N:B |] ==> (M$N) : A<*>B";
-by (REPEAT (ares_tac [singletonI,UN_I] 1));
-qed "uprodI";
-
-(*The general elimination rule*)
-val major::prems = goalw Univ.thy [uprod_def]
- "[| c : A<*>B; \
-\ !!x y. [| x:A; y:B; c=x$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 Univ.thy
- "[| (M$N) : 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 Univ.thy [usum_def] "!!M. M:A ==> In0(M) : A<+>B";
-by (fast_tac set_cs 1);
-qed "usum_In0I";
-
-goalw Univ.thy [usum_def] "!!N. N:B ==> In1(N) : A<+>B";
-by (fast_tac set_cs 1);
-qed "usum_In1I";
-
-val major::prems = goalw Univ.thy [usum_def]
- "[| u : 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 Univ.thy [In0_def,In1_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));
-bind_thm ("In0_neq_In1", (In0_not_In1 RS notE));
-val In1_neq_In0 = sym RS In0_neq_In1;
-
-val [major] = goalw Univ.thy [In0_def] "In0(M) = In0(N) ==> M=N";
-by (rtac (major RS Scons_inject2) 1);
-qed "In0_inject";
-
-val [major] = goalw Univ.thy [In1_def] "In1(M) = In1(N) ==> M=N";
-by (rtac (major RS Scons_inject2) 1);
-qed "In1_inject";
-
-
-(*** proving equality of sets and functions using ntrunc ***)
-
-goalw Univ.thy [ntrunc_def] "ntrunc(k,M) <= M";
-by (fast_tac set_cs 1);
-qed "ntrunc_subsetI";
-
-val [major] = goalw Univ.thy [ntrunc_def]
- "(!!k. ntrunc(k,M) <= N) ==> M<=N";
-by (fast_tac (set_cs addIs [less_add_Suc1, less_add_Suc2,
- major RS subsetD]) 1);
-qed "ntrunc_subsetD";
-
-(*A generalized form of the take-lemma*)
-val [major] = goal Univ.thy "(!!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 Univ.thy [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 Univ.thy [uprod_def] "!!A B. [| A<=A'; B<=B' |] ==> A<*>B <= A'<*>B'";
-by (fast_tac set_cs 1);
-qed "uprod_mono";
-
-goalw Univ.thy [usum_def] "!!A B. [| A<=A'; B<=B' |] ==> A<+>B <= A'<+>B'";
-by (fast_tac set_cs 1);
-qed "usum_mono";
-
-goalw Univ.thy [Scons_def] "!!M N. [| M<=M'; N<=N' |] ==> M$N <= M'$N'";
-by (fast_tac set_cs 1);
-qed "Scons_mono";
-
-goalw Univ.thy [In0_def] "!!M N. M<=N ==> In0(M) <= In0(N)";
-by (REPEAT (ares_tac [subset_refl,Scons_mono] 1));
-qed "In0_mono";
-
-goalw Univ.thy [In1_def] "!!M N. M<=N ==> In1(M) <= In1(N)";
-by (REPEAT (ares_tac [subset_refl,Scons_mono] 1));
-qed "In1_mono";
-
-
-(*** Split and Case ***)
-
-goalw Univ.thy [Split_def] "Split(c, M$N) = c(M,N)";
-by (fast_tac (set_cs addIs [select_equality] addEs [Scons_inject]) 1);
-qed "Split";
-
-goalw Univ.thy [Case_def] "Case(c, d, In0(M)) = c(M)";
-by (fast_tac (set_cs addIs [select_equality]
- addEs [make_elim In0_inject, In0_neq_In1]) 1);
-qed "Case_In0";
-
-goalw Univ.thy [Case_def] "Case(c, d, In1(N)) = d(N)";
-by (fast_tac (set_cs addIs [select_equality]
- addEs [make_elim In1_inject, In1_neq_In0]) 1);
-qed "Case_In1";
-
-(**** UN x. B(x) rules ****)
-
-goalw Univ.thy [ntrunc_def] "ntrunc(k, UN x.f(x)) = (UN x. ntrunc(k, f(x)))";
-by (fast_tac (set_cs addIs [equalityI]) 1);
-qed "ntrunc_UN1";
-
-goalw Univ.thy [Scons_def] "(UN x.f(x)) $ M = (UN x. f(x) $ M)";
-by (fast_tac (set_cs addIs [equalityI]) 1);
-qed "Scons_UN1_x";
-
-goalw Univ.thy [Scons_def] "M $ (UN x.f(x)) = (UN x. M $ f(x))";
-by (fast_tac (set_cs addIs [equalityI]) 1);
-qed "Scons_UN1_y";
-
-goalw Univ.thy [In0_def] "In0(UN x.f(x)) = (UN x. In0(f(x)))";
-br Scons_UN1_y 1;
-qed "In0_UN1";
-
-goalw Univ.thy [In1_def] "In1(UN x.f(x)) = (UN x. In1(f(x)))";
-br Scons_UN1_y 1;
-qed "In1_UN1";
-
-
-(*** Equality : the diagonal relation ***)
-
-goalw Univ.thy [diag_def] "!!a A. [| a=b; a:A |] ==> <a,b> : diag(A)";
-by (fast_tac set_cs 1);
-qed "diag_eqI";
-
-val diagI = refl RS diag_eqI |> standard;
-
-(*The general elimination rule*)
-val major::prems = goalw Univ.thy [diag_def]
- "[| c : diag(A); \
-\ !!x y. [| x:A; c = <x,x> |] ==> P \
-\ |] ==> P";
-by (rtac (major RS UN_E) 1);
-by (REPEAT (eresolve_tac [asm_rl,singletonE] 1 ORELSE resolve_tac prems 1));
-qed "diagE";
-
-(*** Equality for Cartesian Product ***)
-
-goalw Univ.thy [dprod_def]
- "!!r s. [| <M,M'>:r; <N,N'>:s |] ==> <M$N, M'$N'> : r<**>s";
-by (fast_tac prod_cs 1);
-qed "dprodI";
-
-(*The general elimination rule*)
-val major::prems = goalw Univ.thy [dprod_def]
- "[| c : r<**>s; \
-\ !!x y x' y'. [| <x,x'> : r; <y,y'> : s; c = <x$y,x'$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 Univ.thy [dsum_def] "!!r. <M,M'>:r ==> <In0(M), In0(M')> : r<++>s";
-by (fast_tac prod_cs 1);
-qed "dsum_In0I";
-
-goalw Univ.thy [dsum_def] "!!r. <N,N'>:s ==> <In1(N), In1(N')> : r<++>s";
-by (fast_tac prod_cs 1);
-qed "dsum_In1I";
-
-val major::prems = goalw Univ.thy [dsum_def]
- "[| w : 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";
-
-
-val univ_cs =
- prod_cs addSIs [diagI, uprodI, dprodI]
- addIs [usum_In0I, usum_In1I, dsum_In0I, dsum_In1I]
- addSEs [diagE, uprodE, dprodE, usumE, dsumE];
-
-
-(*** Monotonicity ***)
-
-goal Univ.thy "!!r s. [| r<=r'; s<=s' |] ==> r<**>s <= r'<**>s'";
-by (fast_tac univ_cs 1);
-qed "dprod_mono";
-
-goal Univ.thy "!!r s. [| r<=r'; s<=s' |] ==> r<++>s <= r'<++>s'";
-by (fast_tac univ_cs 1);
-qed "dsum_mono";
-
-
-(*** Bounding theorems ***)
-
-goal Univ.thy "diag(A) <= Sigma(A,%x.A)";
-by (fast_tac univ_cs 1);
-qed "diag_subset_Sigma";
-
-goal Univ.thy "(Sigma(A,%x.B) <**> Sigma(C,%x.D)) <= Sigma(A<*>C, %z. B<*>D)";
-by (fast_tac univ_cs 1);
-qed "dprod_Sigma";
-
-val dprod_subset_Sigma = [dprod_mono, dprod_Sigma] MRS subset_trans |>standard;
-
-(*Dependent version*)
-goal Univ.thy
- "(Sigma(A,B) <**> Sigma(C,D)) <= Sigma(A<*>C, Split(%x y. B(x)<*>D(y)))";
-by (safe_tac univ_cs);
-by (stac Split 1);
-by (fast_tac univ_cs 1);
-qed "dprod_subset_Sigma2";
-
-goal Univ.thy "(Sigma(A,%x.B) <++> Sigma(C,%x.D)) <= Sigma(A<+>C, %z. B<+>D)";
-by (fast_tac univ_cs 1);
-qed "dsum_Sigma";
-
-val dsum_subset_Sigma = [dsum_mono, dsum_Sigma] MRS subset_trans |> standard;
-
-
-(*** Domain ***)
-
-goal Univ.thy "fst `` diag(A) = A";
-by (fast_tac (prod_cs addIs [equalityI, diagI] addSEs [diagE]) 1);
-qed "fst_image_diag";
-
-goal Univ.thy "fst `` (r<**>s) = (fst``r) <*> (fst``s)";
-by (fast_tac (prod_cs addIs [equalityI, uprodI, dprodI]
- addSEs [uprodE, dprodE]) 1);
-qed "fst_image_dprod";
-
-goal Univ.thy "fst `` (r<++>s) = (fst``r) <+> (fst``s)";
-by (fast_tac (prod_cs addIs [equalityI, usum_In0I, usum_In1I,
- dsum_In0I, dsum_In1I]
- addSEs [usumE, dsumE]) 1);
-qed "fst_image_dsum";
-
-val fst_image_simps = [fst_image_diag, fst_image_dprod, fst_image_dsum];
-val fst_image_ss = univ_ss addsimps fst_image_simps;