univ.ML

changed "." to "$" and Cons to infix "#" to eliminate ambiguity

(*  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);
val Least_equality = result();

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);
val LeastI = result();

(*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);
val Least_le = result();

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);
val not_less_Least = result();


(** apfst -- can be used in similar type definitions **)

goalw Univ.thy [apfst_def] "apfst(f,<a,b>) = <f(a),b>";
by (rtac split 1);
val apfst = result();

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 1);
val apfstE = result();

(** 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);
val Push_inject1 = result();

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);
val Push_inject2 = result();

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);
val Push_inject = result();

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);
val Push_neq_K0 = result();

(*** Isomorphisms ***)

goal Univ.thy "inj(Rep_Node)";
by (rtac inj_inverseI 1);	(*cannot combine by RS: multiple unifiers*)
by (rtac Rep_Node_inverse 1);
val inj_Rep_Node = result();

goal Univ.thy "inj_onto(Abs_Node,Node)";
by (rtac inj_onto_inverseI 1);
by (etac Abs_Node_inverse 1);
val inj_onto_Abs_Node = result();

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);
val Node_K0_I = result();

goalw Univ.thy [Node_def,Push_def]
    "!!p. p: Node ==> apfst(Push(i), p) : Node";
by (fast_tac (set_cs addSIs [apfst, nat_case_Suc RS trans]) 1);
val Node_Push_I = result();


(*** 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 apfstE, 
			  Pair_inject, sym RS Push_neq_K0] 1
     ORELSE resolve_tac [Node_K0_I, Rep_Node RS Node_Push_I] 1));
val Scons_not_Atom = result();
val Atom_not_Scons = standard (Scons_not_Atom RS not_sym);

val Scons_neq_Atom = standard (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));
val inj_Atom = result();
val Atom_inject = inj_Atom RS injD;

goalw Univ.thy [Leaf_def] "inj(Leaf)";
by (stac o_def 1);
by (rtac injI 1);
by (etac (Atom_inject RS Inl_inject) 1);
val inj_Leaf = result();

val Leaf_inject = inj_Leaf RS injD;

goalw Univ.thy [Numb_def] "inj(Numb)";
by (stac o_def 1);
by (rtac injI 1);
by (etac (Atom_inject RS Inr_inject) 1);
val inj_Numb = result();

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 apfstE) 1);
by (REPEAT (resolve_tac [Rep_Node RS Node_Push_I] 1));
by (etac (sym RS apfstE) 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]));
val Push_Node_inject = result();


(** 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);
val Scons_inject_lemma1 = result();

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);
val Scons_inject_lemma2 = result();

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));
val Scons_inject1 = result();

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));
val Scons_inject2 = result();

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);
val Scons_inject = result();

(*rewrite rules*)
goal Univ.thy "(Atom(a)=Atom(b)) = (a=b)";
by (fast_tac (HOL_cs addSEs [Atom_inject]) 1);
val Atom_Atom_eq = result();

goal Univ.thy "(M$N = M'$N') = (M=M' & N=N')";
by (fast_tac (HOL_cs addSEs [Scons_inject]) 1);
val Scons_Scons_eq = result();

(*** Distinctness involving Leaf and Numb ***)

(** Scons vs Leaf **)

goalw Univ.thy [Leaf_def] "(M$N) ~= Leaf(a)";
by (stac o_def 1);
by (rtac Scons_not_Atom 1);
val Scons_not_Leaf = result();
val Leaf_not_Scons = standard (Scons_not_Leaf RS not_sym);

val Scons_neq_Leaf = standard (Scons_not_Leaf RS notE);
val Leaf_neq_Scons = sym RS Scons_neq_Leaf;

(** Scons vs Numb **)

goalw Univ.thy [Numb_def] "(M$N) ~= Numb(k)";
by (stac o_def 1);
by (rtac Scons_not_Atom 1);
val Scons_not_Numb = result();
val Numb_not_Scons = standard (Scons_not_Numb RS not_sym);

val Scons_neq_Numb = standard (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);
val Leaf_not_Numb = result();
val Numb_not_Leaf = standard (Leaf_not_Numb RS not_sym);

val Leaf_neq_Numb = standard (Leaf_not_Numb RS notE);
val Numb_neq_Leaf = sym RS Leaf_neq_Numb;


(*** ndepth -- the depth of a node ***)

val univ_simps = [apfst,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);
val ndepth_K0 = result();

goal Univ.thy "k < Suc(LEAST x. f(x)=0) --> nat_case(k, Suc(i), f) ~= 0";
by (nat_ind_tac "k" 1);
by (ALLGOALS (simp_tac nat_ss));
by (rtac impI 1);
by (etac not_less_Least 1);
val ndepth_Push_lemma = result();

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);
val ndepth_Push_Node = result();


(*** 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]));
val ntrunc_0 = result();

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);
val ntrunc_Atom = result();

goalw Univ.thy [Leaf_def] "ntrunc(Suc(k), Leaf(a)) = Leaf(a)";
by (stac o_def 1);
by (rtac ntrunc_Atom 1);
val ntrunc_Leaf = result();

goalw Univ.thy [Numb_def] "ntrunc(Suc(k), Numb(i)) = Numb(i)";
by (stac o_def 1);
by (rtac ntrunc_Atom 1);
val ntrunc_Numb = result();

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));
val ntrunc_Scons = result();

(** 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]));
val ntrunc_one_In0 = result();

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);
val ntrunc_In0 = result();

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]));
val ntrunc_one_In1 = result();

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);
val ntrunc_In1 = result();


(*** 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));
val uprodI = result();

(*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));
val uprodE = result();

(*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));
val uprodE2 = result();


(*** Disjoint Sum ***)

goalw Univ.thy [usum_def] "!!M. M:A ==> In0(M) : A<+>B";
by (fast_tac set_cs 1);
val usum_In0I = result();

goalw Univ.thy [usum_def] "!!N. N:B ==> In1(N) : A<+>B";
by (fast_tac set_cs 1);
val usum_In1I = result();

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));
val usumE = result();


(** 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);
val In0_not_In1 = result();

val In1_not_In0 = standard (In0_not_In1 RS not_sym);
val In0_neq_In1 = standard (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);
val In0_inject = result();

val [major] = goalw Univ.thy [In1_def] "In1(M) = In1(N) ==>  M=N";
by (rtac (major RS Scons_inject2) 1);
val In1_inject = result();


(*** proving equality of sets and functions using ntrunc ***)

goalw Univ.thy [ntrunc_def] "ntrunc(k,M) <= M";
by (fast_tac set_cs 1);
val ntrunc_subsetI = result();

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);
val ntrunc_subsetD = result();

(*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);
val ntrunc_equality = result();

val [major] = goal Univ.thy
    "[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2";
by (rtac (ntrunc_equality RS ext) 1);
by (resolve_tac ([major RS fun_cong] RL [o_def RS subst]) 1);
val ntrunc_o_equality = result();

(*** Monotonicity ***)

goalw Univ.thy [uprod_def] "!!A B. [| A<=A';  B<=B' |] ==> A<*>B <= A'<*>B'";
by (fast_tac set_cs 1);
val uprod_mono = result();

goalw Univ.thy [usum_def] "!!A B. [| A<=A';  B<=B' |] ==> A<+>B <= A'<+>B'";
by (fast_tac set_cs 1);
val usum_mono = result();

goalw Univ.thy [Scons_def] "!!M N. [| M<=M';  N<=N' |] ==> M$N <= M'$N'";
by (fast_tac set_cs 1);
val Scons_mono = result();

goalw Univ.thy [In0_def] "!!M N. M<=N ==> In0(M) <= In0(N)";
by (REPEAT (ares_tac [subset_refl,Scons_mono] 1));
val In0_mono = result();

goalw Univ.thy [In1_def] "!!M N. M<=N ==> In1(M) <= In1(N)";
by (REPEAT (ares_tac [subset_refl,Scons_mono] 1));
val In1_mono = result();


(*** Split and Case ***)

goalw Univ.thy [Split_def] "Split(M$N, c) = c(M,N)";
by (fast_tac (set_cs addIs [select_equality] addEs [Scons_inject]) 1);
val Split = result();

goalw Univ.thy [Case_def] "Case(In0(M), c, d) = c(M)";
by (fast_tac (set_cs addIs [select_equality] 
		     addEs [make_elim In0_inject, In0_neq_In1]) 1);
val Case_In0 = result();

goalw Univ.thy [Case_def] "Case(In1(N), c, d) = d(N)";
by (fast_tac (set_cs addIs [select_equality] 
		     addEs [make_elim In1_inject, In1_neq_In0]) 1);
val Case_In1 = result();

(**** 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);
val ntrunc_UN1 = result();

goalw Univ.thy [Scons_def] "(UN x.f(x)) $ M = (UN x. f(x) $ M)";
by (fast_tac (set_cs addIs [equalityI]) 1);
val Scons_UN1_x = result();

goalw Univ.thy [Scons_def] "M $ (UN x.f(x)) = (UN x. M $ f(x))";
by (fast_tac (set_cs addIs [equalityI]) 1);
val Scons_UN1_y = result();

goalw Univ.thy [In0_def] "In0(UN x.f(x)) = (UN x. In0(f(x)))";
br Scons_UN1_y 1;
val In0_UN1 = result();

goalw Univ.thy [In1_def] "In1(UN x.f(x)) = (UN x. In1(f(x)))";
br Scons_UN1_y 1;
val In1_UN1 = result();


(*** Equality : the diagonal relation ***)

goalw Univ.thy [diag_def] "!!a A. a:A ==> <a,a> : diag(A)";
by (REPEAT (ares_tac [singletonI,UN_I] 1));
val diagI = result();

(*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));
val diagE = result();

(*** Equality for Cartesian Product ***)

goal Univ.thy
 "split(<M,M'>, %x x'. split(<N,N'>, %y y'. {<x$y,x'$y'>})) = {<M$N, M'$N'>}";
by (simp_tac univ_ss 1);
val dprod_lemma = result();

goalw Univ.thy [dprod_def]
    "!!r s. [| <M,M'>:r;  <N,N'>:s |] ==> <M$N, M'$N'> : r<**>s";
by (REPEAT (ares_tac [UN_I] 1));
by (rtac (singletonI RS (dprod_lemma RS equalityD2 RS subsetD)) 1);
val dprodI = result();

(*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 (eresolve_tac [asm_rl,singletonE,UN_E] 1));
by (res_inst_tac [("p","u")] PairE 1);
by (res_inst_tac [("p","v")] PairE 1);
by (safe_tac HOL_cs);
by (REPEAT (ares_tac prems 1));
by (safe_tac (set_cs addSDs [dprod_lemma RS equalityD1 RS subsetD]));
val dprodE = result();


(*** Equality for Disjoint Sum ***)

goalw Univ.thy [dsum_def]  "!!r. <M,M'>:r ==> <In0(M), In0(M')> : r<++>s";
by (fast_tac (set_cs addSIs [split RS equalityD2 RS subsetD]) 1);
val dsum_In0I = result();

goalw Univ.thy [dsum_def]  "!!r. <N,N'>:s ==> <In1(N), In1(N')> : r<++>s";
by (fast_tac (set_cs addSIs [split RS equalityD2 RS subsetD]) 1);
val dsum_In1I = result();

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 (rtac (major RS UnE) 1);
by (safe_tac set_cs);
by (res_inst_tac [("p","u")] PairE 1);
by (res_inst_tac [("p","v")] PairE 2);
by (safe_tac (set_cs addSEs prems 
                     addSDs [split RS equalityD1 RS subsetD]));
val dsumE = result();


(*** Monotonicity ***)

goalw Univ.thy [dprod_def] "!!r s. [| r<=r';  s<=s' |] ==> r<**>s <= r'<**>s'";
by (fast_tac set_cs 1);
val dprod_mono = result();

goalw Univ.thy [dsum_def] "!!r s. [| r<=r';  s<=s' |] ==> r<++>s <= r'<++>s'";
by (fast_tac set_cs 1);
val dsum_mono = result();


(*** Bounding theorems ***)

goal Univ.thy "diag(A) <= Sigma(A,%x.A)";
by (fast_tac (set_cs addIs [SigmaI] addSEs [diagE]) 1);
val diag_subset_Sigma = result();

val prems = goal Univ.thy
    "[| r <= Sigma(A,%x.B);  s <= Sigma(C,%x.D) |] ==> \
\    (r<**>s) <= Sigma(A<*>C, %z. B<*>D)";
by (cut_facts_tac prems 1);
by (fast_tac (set_cs addSIs [SigmaI,uprodI]
                     addSEs [dprodE,SigmaE2]) 1);
val dprod_subset_Sigma = result();

goal Univ.thy
    "!!r s. [| r <= Sigma(A,B);  s <= Sigma(C,D) |] ==> \
\           (r<**>s) <= Sigma(A<*>C, %z. Split(z, %x y. B(x)<*>D(y)))";
by (safe_tac (set_cs addSIs [SigmaI,uprodI] addSEs [dprodE]));
by (stac Split 3);
by (ALLGOALS (fast_tac (set_cs addSIs [uprodI] addSEs [SigmaE2])));
val dprod_subset_Sigma2 = result();

goal Univ.thy
    "!!r s. [| r <= Sigma(A,%x.B);  s <= Sigma(C,%x.D) |] ==> \
\           (r<++>s) <= Sigma(A<+>C, %z. B<+>D)";
by (fast_tac (set_cs addSIs [SigmaI,usum_In0I,usum_In1I]
                     addSEs [dsumE,SigmaE2]) 1);
val dsum_subset_Sigma = result();


(*** Domain ***)

goal Univ.thy "fst `` diag(A) = A";
by (fast_tac (set_cs addIs [equalityI, fst_imageI, diagI]
                     addSEs [fst_imageE, Pair_inject, diagE]) 1);
val fst_image_diag = result();

goal Univ.thy "fst `` (r<**>s) = (fst``r) <*> (fst``s)";
by (fast_tac (set_cs addIs [equalityI, fst_imageI, uprodI, dprodI]
                     addSEs [fst_imageE, Pair_inject, uprodE, dprodE]) 1);
val fst_image_dprod = result();

goal Univ.thy "fst `` (r<++>s) = (fst``r) <+> (fst``s)";
by (fast_tac (set_cs addIs [equalityI, fst_imageI, usum_In0I, usum_In1I, 
			     dsum_In0I, dsum_In1I]
                     addSEs [fst_imageE, Pair_inject, usumE, dsumE]) 1);
val fst_image_dsum = result();

val fst_image_simps = [fst_image_diag, fst_image_dprod, fst_image_dsum];
val fst_image_ss = univ_ss addsimps fst_image_simps;

val univ_cs =
    set_cs addSIs [SigmaI,uprodI,dprodI]
           addIs [usum_In0I,usum_In1I,dsum_In0I,dsum_In1I]
           addSEs [diagE,uprodE,dprodE,usumE,dsumE,SigmaE2,Pair_inject];