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src/HOLCF/cont.ML
changeset 243 c22b85994e17 
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOLCF/cont.ML	Wed Jan 19 17:35:01 1994 +0100
@@ -0,0 +1,670 @@
+(*  Title: 	HOLCF/cont.ML
+    ID:         $Id$
+    Author: 	Franz Regensburger
+    Copyright   1993 Technische Universitaet Muenchen
+
+Lemmas for cont.thy 
+*)
+
+open Cont;
+
+(* ------------------------------------------------------------------------ *)
+(* access to definition                                                     *)
+(* ------------------------------------------------------------------------ *)
+
+val contlubI = prove_goalw Cont.thy [contlub]
+	"! Y. is_chain(Y) --> f(lub(range(Y))) = lub(range(%i. f(Y(i))))==>\
+\	 contlub(f)"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(atac 1)
+	]);
+
+val contlubE = prove_goalw Cont.thy [contlub]
+	" contlub(f)==>\
+\         ! Y. is_chain(Y) --> f(lub(range(Y))) = lub(range(%i. f(Y(i))))"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(atac 1)
+	]);
+
+
+val contXI = prove_goalw Cont.thy [contX]
+ "! Y. is_chain(Y) --> range(% i.f(Y(i))) <<| f(lub(range(Y))) ==> contX(f)"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(atac 1)
+	]);
+
+val contXE = prove_goalw Cont.thy [contX]
+ "contX(f) ==> ! Y. is_chain(Y) --> range(% i.f(Y(i))) <<| f(lub(range(Y)))"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(atac 1)
+	]);
+
+
+val monofunI = prove_goalw Cont.thy [monofun]
+	"! x y. x << y --> f(x) << f(y) ==> monofun(f)"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(atac 1)
+	]);
+
+val monofunE = prove_goalw Cont.thy [monofun]
+	"monofun(f) ==> ! x y. x << y --> f(x) << f(y)"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(atac 1)
+	]);
+
+(* ------------------------------------------------------------------------ *)
+(* the main purpose of cont.thy is to show:                                 *)
+(*              monofun(f) & contlub(f)  <==> contX(f)                      *)
+(* ------------------------------------------------------------------------ *)
+
+(* ------------------------------------------------------------------------ *)
+(* monotone functions map chains to chains                                  *)
+(* ------------------------------------------------------------------------ *)
+
+val ch2ch_monofun= prove_goal Cont.thy 
+	"[| monofun(f); is_chain(Y) |] ==> is_chain(%i. f(Y(i)))"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(rtac is_chainI 1),
+	(rtac allI 1),
+	(etac (monofunE RS spec RS spec RS mp) 1),
+	(etac (is_chainE RS spec) 1)
+	]);
+
+(* ------------------------------------------------------------------------ *)
+(* monotone functions map upper bound to upper bounds                       *)
+(* ------------------------------------------------------------------------ *)
+
+val ub2ub_monofun = prove_goal Cont.thy 
+ "[| monofun(f); range(Y) <| u|]  ==> range(%i.f(Y(i))) <| f(u)"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(rtac ub_rangeI 1),
+	(rtac allI 1),
+	(etac (monofunE RS spec RS spec RS mp) 1),
+	(etac (ub_rangeE RS spec) 1)
+	]);
+
+(* ------------------------------------------------------------------------ *)
+(* left to right: monofun(f) & contlub(f)  ==> contX(f)                     *)
+(* ------------------------------------------------------------------------ *)
+
+val monocontlub2contX = prove_goalw Cont.thy [contX]
+	"[|monofun(f);contlub(f)|] ==> contX(f)"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(strip_tac 1),
+	(rtac thelubE 1),
+	(etac ch2ch_monofun 1),
+	(atac 1),
+	(etac (contlubE RS spec RS mp RS sym) 1),
+	(atac 1)
+	]);
+
+(* ------------------------------------------------------------------------ *)
+(* first a lemma about binary chains                                        *)
+(* ------------------------------------------------------------------------ *)
+
+val binchain_contX =  prove_goal Cont.thy
+"[| contX(f); x << y |]  ==> range(%i. f(if(i = 0,x,y))) <<| f(y)"
+(fn prems => 
+	[
+	(cut_facts_tac prems 1),
+	(rtac subst 1), 
+	(etac (contXE RS spec RS mp) 2),
+	(etac bin_chain 2),
+	(res_inst_tac [("y","y")] arg_cong 1),
+	(etac (lub_bin_chain RS thelubI) 1)
+	]);
+
+(* ------------------------------------------------------------------------ *)
+(* right to left: contX(f) ==> monofun(f) & contlub(f)                      *)
+(* part1:         contX(f) ==> monofun(f                                    *)
+(* ------------------------------------------------------------------------ *)
+
+val contX2mono =  prove_goalw Cont.thy [monofun]
+	"contX(f) ==> monofun(f)"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(strip_tac 1),
+	(res_inst_tac [("s","if(0 = 0,x,y)")] subst 1),
+	(rtac (binchain_contX RS is_ub_lub) 2),
+	(atac 2),
+	(atac 2),
+	(simp_tac nat_ss 1)
+	]);
+
+(* ------------------------------------------------------------------------ *)
+(* right to left: contX(f) ==> monofun(f) & contlub(f)                      *)
+(* part2:         contX(f) ==>              contlub(f)                      *)
+(* ------------------------------------------------------------------------ *)
+
+val contX2contlub = prove_goalw Cont.thy [contlub]
+	"contX(f) ==> contlub(f)"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(strip_tac 1),
+	(rtac (thelubI RS sym) 1),
+	(etac (contXE RS spec RS mp) 1),
+	(atac 1)
+	]);
+
+(* ------------------------------------------------------------------------ *)
+(* The following results are about a curried function that is monotone      *)
+(* in both arguments                                                        *)
+(* ------------------------------------------------------------------------ *)
+
+val ch2ch_MF2L = prove_goal Cont.thy 
+"[|monofun(MF2::('a::po=>'b::po=>'c::po));\
+\	is_chain(F)|] ==> is_chain(%i. MF2(F(i),x))"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(etac (ch2ch_monofun RS ch2ch_fun) 1),
+	(atac 1)
+	]);
+
+
+val ch2ch_MF2R = prove_goal Cont.thy "[|monofun(MF2(f)::('b::po=>'c::po));\
+\	is_chain(Y)|] ==> is_chain(%i. MF2(f,Y(i)))"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(etac ch2ch_monofun 1),
+	(atac 1)
+	]);
+
+val ch2ch_MF2LR = prove_goal Cont.thy 
+"[|monofun(MF2::('a::po=>'b::po=>'c::pcpo));\
+\  !f.monofun(MF2(f)::('b::po=>'c::pcpo));\
+\	is_chain(F); is_chain(Y)|] ==> \
+\  is_chain(%i. MF2(F(i))(Y(i)))"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(rtac is_chainI 1),
+	(strip_tac 1 ),
+	(rtac trans_less 1),
+	(etac (ch2ch_MF2L RS is_chainE RS spec) 1),
+	(atac 1),
+	((rtac (monofunE RS spec RS spec RS mp) 1) THEN (etac spec 1)),
+	(etac (is_chainE RS spec) 1)
+	]);
+
+val ch2ch_lubMF2R = prove_goal Cont.thy 
+"[|monofun(MF2::('a::po=>'b::po=>'c::pcpo));\
+\  !f.monofun(MF2(f)::('b::po=>'c::pcpo));\
+\	is_chain(F);is_chain(Y)|] ==> \
+\	is_chain(%j. lub(range(%i. MF2(F(j),Y(i)))))"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(rtac (lub_mono RS allI RS is_chainI) 1),
+	((rtac ch2ch_MF2R 1) THEN (etac spec 1)),
+	(atac 1),
+	((rtac ch2ch_MF2R 1) THEN (etac spec 1)),
+	(atac 1),
+	(strip_tac 1),
+	(rtac (is_chainE RS spec) 1),
+	(etac ch2ch_MF2L 1),
+	(atac 1)
+	]);
+
+
+val ch2ch_lubMF2L = prove_goal Cont.thy 
+"[|monofun(MF2::('a::po=>'b::po=>'c::pcpo));\
+\  !f.monofun(MF2(f)::('b::po=>'c::pcpo));\
+\	is_chain(F);is_chain(Y)|] ==> \
+\	is_chain(%i. lub(range(%j. MF2(F(j),Y(i)))))"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(rtac (lub_mono RS allI RS is_chainI) 1),
+	(etac ch2ch_MF2L 1),
+	(atac 1),
+	(etac ch2ch_MF2L 1),
+	(atac 1),
+	(strip_tac 1),
+	(rtac (is_chainE RS spec) 1),
+	((rtac ch2ch_MF2R 1) THEN (etac spec 1)),
+	(atac 1)
+	]);
+
+
+val lub_MF2_mono = prove_goal Cont.thy 
+"[|monofun(MF2::('a::po=>'b::po=>'c::pcpo));\
+\  !f.monofun(MF2(f)::('b::po=>'c::pcpo));\
+\	is_chain(F)|] ==> \
+\	monofun(% x.lub(range(% j.MF2(F(j),x))))"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(rtac monofunI 1),
+	(strip_tac 1),
+	(rtac lub_mono 1),
+	(etac ch2ch_MF2L 1),
+	(atac 1),
+	(etac ch2ch_MF2L 1),
+	(atac 1),
+	(strip_tac 1),
+	((rtac (monofunE RS spec RS spec RS mp) 1) THEN (etac spec 1)),
+	(atac 1)
+	]);
+
+
+val ex_lubMF2 = prove_goal Cont.thy 
+"[|monofun(MF2::('a::po=>'b::po=>'c::pcpo));\
+\  !f.monofun(MF2(f)::('b::po=>'c::pcpo));\
+\	is_chain(F); is_chain(Y)|] ==> \
+\		lub(range(%j. lub(range(%i. MF2(F(j),Y(i)))))) =\
+\		lub(range(%i. lub(range(%j. MF2(F(j),Y(i))))))"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(rtac antisym_less 1),
+	(rtac is_lub_thelub 1),
+	(etac ch2ch_lubMF2R 1),
+	(atac 1),(atac 1),(atac 1),
+	(rtac ub_rangeI 1),
+	(strip_tac 1),
+	(rtac lub_mono 1),
+	((rtac ch2ch_MF2R 1) THEN (etac spec 1)),
+	(atac 1),
+	(etac ch2ch_lubMF2L 1),
+	(atac 1),(atac 1),(atac 1),
+	(strip_tac 1),
+	(rtac is_ub_thelub 1),
+	(etac ch2ch_MF2L 1),(atac 1),
+	(rtac is_lub_thelub 1),
+	(etac ch2ch_lubMF2L 1),
+	(atac 1),(atac 1),(atac 1),
+	(rtac ub_rangeI 1),
+	(strip_tac 1),
+	(rtac lub_mono 1),
+	(etac ch2ch_MF2L 1),(atac 1),
+	(etac ch2ch_lubMF2R 1),
+	(atac 1),(atac 1),(atac 1),
+	(strip_tac 1),
+	(rtac is_ub_thelub 1),
+	((rtac ch2ch_MF2R 1) THEN (etac spec 1)),
+	(atac 1)
+	]);
+
+(* ------------------------------------------------------------------------ *)
+(* The following results are about a curried function that is continuous    *)
+(* in both arguments                                                        *)
+(* ------------------------------------------------------------------------ *)
+
+val diag_lubCF2_1 = prove_goal Cont.thy 
+"[|contX(CF2);!f.contX(CF2(f));is_chain(FY);is_chain(TY)|] ==>\
+\ lub(range(%i. lub(range(%j. CF2(FY(j))(TY(i)))))) =\
+\ lub(range(%i. CF2(FY(i))(TY(i))))"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(rtac antisym_less 1),
+	(rtac is_lub_thelub 1),
+	(rtac ch2ch_lubMF2L 1),
+	(rtac contX2mono 1),
+	(atac 1),
+	(rtac allI 1),
+	(rtac contX2mono 1),
+	(etac spec 1),
+	(atac 1),
+	(atac 1),
+	(rtac ub_rangeI 1),
+	(strip_tac 1 ),
+	(rtac is_lub_thelub 1),
+	((rtac ch2ch_MF2L 1) THEN (rtac contX2mono 1) THEN (atac 1)),
+	(atac 1),
+	(rtac ub_rangeI 1),
+	(strip_tac 1 ),
+	(res_inst_tac [("m","i"),("n","ia")] nat_less_cases 1),
+	(rtac trans_less 1),
+	(rtac is_ub_thelub 2),
+	(rtac (chain_mono RS mp) 1),
+	((rtac ch2ch_MF2R 1) THEN (rtac contX2mono 1) THEN (etac spec 1)),
+	(atac 1),
+	(atac 1),
+	((rtac ch2ch_MF2LR 1) THEN (etac contX2mono 1)),
+	(rtac allI 1),
+	((rtac contX2mono 1) THEN (etac spec 1)),
+	(atac 1),
+	(atac 1),
+	(hyp_subst_tac 1),
+	(rtac is_ub_thelub 1),
+	((rtac ch2ch_MF2LR 1) THEN (etac contX2mono 1)),
+	(rtac allI 1),
+	((rtac contX2mono 1) THEN (etac spec 1)),
+	(atac 1),
+	(atac 1),
+	(rtac trans_less 1),
+	(rtac is_ub_thelub 2),
+	(res_inst_tac [("x1","ia")] (chain_mono RS mp) 1),
+	((rtac ch2ch_MF2L 1) THEN (etac contX2mono 1)),
+	(atac 1),
+	(atac 1),
+	((rtac ch2ch_MF2LR 1) THEN (etac contX2mono 1)),
+	(rtac allI 1),
+	((rtac contX2mono 1) THEN (etac spec 1)),
+	(atac 1),
+	(atac 1),
+	(rtac lub_mono 1),
+	((rtac ch2ch_MF2LR 1) THEN (etac contX2mono 1)),
+	(rtac allI 1),
+	((rtac contX2mono 1) THEN (etac spec 1)),
+	(atac 1),
+	(atac 1),
+	(rtac ch2ch_lubMF2L 1),
+	(rtac contX2mono 1),
+	(atac 1),
+	(rtac allI 1),
+	((rtac contX2mono 1) THEN (etac spec 1)),
+	(atac 1),
+	(atac 1),
+	(strip_tac 1 ),
+	(rtac is_ub_thelub 1),
+	((rtac ch2ch_MF2L 1) THEN (etac contX2mono 1)),
+	(atac 1)
+	]);
+
+
+val diag_lubCF2_2 = prove_goal Cont.thy 
+"[|contX(CF2);!f.contX(CF2(f));is_chain(FY);is_chain(TY)|] ==>\
+\ lub(range(%j. lub(range(%i. CF2(FY(j))(TY(i)))))) =\
+\ lub(range(%i. CF2(FY(i))(TY(i))))"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(rtac trans 1),
+	(rtac ex_lubMF2 1),
+	(rtac ((hd prems) RS contX2mono) 1), 
+	(rtac allI 1),
+	(rtac (((hd (tl prems)) RS spec RS contX2mono)) 1),
+	(atac 1),
+	(atac 1),
+	(rtac diag_lubCF2_1 1),
+	(atac 1),
+	(atac 1),
+	(atac 1),
+	(atac 1)
+	]);
+
+
+val contlub_CF2 = prove_goal Cont.thy 
+"[|contX(CF2);!f.contX(CF2(f));is_chain(FY);is_chain(TY)|] ==>\
+\ CF2(lub(range(FY)))(lub(range(TY))) = lub(range(%i.CF2(FY(i))(TY(i))))"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(rtac ((hd prems) RS contX2contlub RS contlubE RS 
+		spec RS mp RS ssubst) 1),
+	(atac 1),
+	(rtac (thelub_fun RS ssubst) 1),
+	(rtac ((hd prems) RS contX2mono RS ch2ch_monofun) 1), 
+	(atac 1),
+	(rtac trans 1),
+	(rtac (((hd (tl prems)) RS spec RS contX2contlub) RS 
+	contlubE RS spec RS mp RS ext RS arg_cong RS arg_cong) 1),
+	(atac 1),
+	(rtac diag_lubCF2_2 1),
+	(atac 1),
+	(atac 1),
+	(atac 1),
+	(atac 1)
+	]);
+
+(* ------------------------------------------------------------------------ *)
+(* The following results are about application for functions in 'a=>'b      *)
+(* ------------------------------------------------------------------------ *)
+
+val monofun_fun_fun = prove_goal Cont.thy 
+	"f1 << f2 ==> f1(x) << f2(x)"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(etac (less_fun RS iffD1 RS spec) 1)
+	]);
+
+val monofun_fun_arg = prove_goal Cont.thy 
+	"[|monofun(f); x1 << x2|] ==> f(x1) << f(x2)"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(etac (monofunE RS spec RS spec RS mp) 1),
+	(atac 1)
+	]);
+
+val monofun_fun = prove_goal Cont.thy 
+"[|monofun(f1); monofun(f2); f1 << f2; x1 << x2|] ==> f1(x1) << f2(x2)"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(rtac trans_less 1),
+	(etac monofun_fun_arg 1),
+	(atac 1),
+	(etac monofun_fun_fun 1)
+	]);
+
+
+(* ------------------------------------------------------------------------ *)
+(* The following results are about the propagation of monotonicity and      *)
+(* continuity                                                               *)
+(* ------------------------------------------------------------------------ *)
+
+val mono2mono_MF1L = prove_goal Cont.thy 
+	"[|monofun(c1)|] ==> monofun(%x. c1(x,y))"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(rtac monofunI 1),
+	(strip_tac 1),
+	(etac (monofun_fun_arg RS monofun_fun_fun) 1),
+	(atac 1)
+	]);
+
+val contX2contX_CF1L = prove_goal Cont.thy 
+	"[|contX(c1)|] ==> contX(%x. c1(x,y))"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(rtac monocontlub2contX 1),
+	(etac (contX2mono RS mono2mono_MF1L) 1),
+	(rtac contlubI 1),
+	(strip_tac 1),
+	(rtac ((hd prems) RS contX2contlub RS 
+		contlubE RS spec RS mp RS ssubst) 1),
+	(atac 1),
+	(rtac (thelub_fun RS ssubst) 1),
+	(rtac ch2ch_monofun 1),
+	(etac contX2mono 1),
+	(atac 1),
+	(rtac refl 1)
+	]);
+
+(*********  Note "(%x.%y.c1(x,y)) = c1" ***********)
+
+val mono2mono_MF1L_rev = prove_goal Cont.thy
+	"!y.monofun(%x.c1(x,y)) ==> monofun(c1)"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(rtac monofunI 1),
+	(strip_tac 1),
+	(rtac (less_fun RS iffD2) 1),
+	(strip_tac 1),
+	(rtac ((hd prems) RS spec RS monofunE RS spec RS spec RS mp) 1),
+	(atac 1)
+	]);
+
+val contX2contX_CF1L_rev = prove_goal Cont.thy
+	"!y.contX(%x.c1(x,y)) ==> contX(c1)"
+(fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(rtac monocontlub2contX 1),
+	(rtac (contX2mono RS allI RS mono2mono_MF1L_rev ) 1),
+	(etac spec 1),
+	(rtac contlubI 1),
+	(strip_tac 1),
+	(rtac ext 1),
+	(rtac (thelub_fun RS ssubst) 1),
+	(rtac (contX2mono RS allI RS mono2mono_MF1L_rev RS ch2ch_monofun) 1),
+	(etac spec 1),
+	(atac 1),
+	(rtac 
+	((hd prems) RS spec RS contX2contlub RS contlubE RS spec RS mp) 1),
+	(atac 1)
+	]);
+
+
+(* ------------------------------------------------------------------------ *)
+(* What D.A.Schmidt calls continuity of abstraction                         *)
+(* never used here                                                          *)
+(* ------------------------------------------------------------------------ *)
+
+val contlub_abstraction = prove_goal Cont.thy
+"[|is_chain(Y::nat=>'a);!y.contX(%x.(c::'a=>'b=>'c)(x,y))|] ==>\
+\ (%y.lub(range(%i.c(Y(i),y)))) = (lub(range(%i.%y.c(Y(i),y))))"
+ (fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(rtac trans 1),
+	(rtac (contX2contlub RS contlubE RS spec RS mp) 2),
+	(atac 3),
+	(etac contX2contX_CF1L_rev 2),
+	(rtac ext 1), 
+	(rtac (contX2contlub RS contlubE RS spec RS mp RS sym) 1),
+	(etac spec 1),
+	(atac 1)
+	]);
+
+
+val mono2mono_app = prove_goal Cont.thy 
+"[|monofun(ft);!x.monofun(ft(x));monofun(tt)|] ==>\
+\	 monofun(%x.(ft(x))(tt(x)))"
+ (fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(rtac monofunI 1),
+	(strip_tac 1),
+	(res_inst_tac [("f1.0","ft(x)"),("f2.0","ft(y)")] monofun_fun 1),
+	(etac spec 1),
+	(etac spec 1),
+	(etac (monofunE RS spec RS spec RS mp) 1),
+	(atac 1),
+	(etac (monofunE RS spec RS spec RS mp) 1),
+	(atac 1)
+	]);
+
+val contX2contlub_app = prove_goal Cont.thy 
+"[|contX(ft);!x.contX(ft(x));contX(tt)|] ==>\
+\	 contlub(%x.(ft(x))(tt(x)))"
+ (fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(rtac contlubI 1),
+	(strip_tac 1),
+	(res_inst_tac [("f3","tt")] (contlubE RS spec RS mp RS ssubst) 1),
+	(rtac contX2contlub 1),
+	(resolve_tac prems 1),
+	(atac 1),
+	(rtac contlub_CF2 1),
+	(resolve_tac prems 1),
+	(resolve_tac prems 1),
+	(atac 1),
+	(rtac (contX2mono RS ch2ch_monofun) 1),
+	(resolve_tac prems 1),
+	(atac 1)
+	]);
+
+
+val contX2contX_app = prove_goal Cont.thy 
+"[|contX(ft);!x.contX(ft(x));contX(tt)|] ==>\
+\	 contX(%x.(ft(x))(tt(x)))"
+ (fn prems =>
+	[
+	(rtac monocontlub2contX 1),
+	(rtac mono2mono_app 1),
+	(rtac contX2mono 1),
+	(resolve_tac prems 1),
+	(strip_tac 1),
+	(rtac contX2mono 1),
+	(cut_facts_tac prems 1),
+	(etac spec 1),
+	(rtac contX2mono 1),
+	(resolve_tac prems 1),
+	(rtac contX2contlub_app 1),
+	(resolve_tac prems 1),
+	(resolve_tac prems 1),
+	(resolve_tac prems 1)
+	]);
+
+
+val contX2contX_app2 = (allI RSN (2,contX2contX_app));
+(*  [| contX(?ft); !!x. contX(?ft(x)); contX(?tt) |] ==>                 *)
+(*                                      contX(%x. ?ft(x,?tt(x)))         *)
+
+
+(* ------------------------------------------------------------------------ *)
+(* The identity function is continuous                                      *)
+(* ------------------------------------------------------------------------ *)
+
+val contX_id = prove_goal Cont.thy "contX(% x.x)"
+ (fn prems =>
+	[
+	(rtac contXI 1),
+	(strip_tac 1),
+	(etac thelubE 1),
+	(rtac refl 1)
+	]);
+
+
+
+(* ------------------------------------------------------------------------ *)
+(* constant functions are continuous                                        *)
+(* ------------------------------------------------------------------------ *)
+
+val contX_const = prove_goalw Cont.thy [contX] "contX(%x.c)"
+ (fn prems =>
+	[
+	(strip_tac 1),
+	(rtac is_lubI 1),
+	(rtac conjI 1),
+	(rtac ub_rangeI 1),
+	(strip_tac 1),
+	(rtac refl_less 1),
+	(strip_tac 1),
+	(dtac ub_rangeE 1),
+	(etac spec 1)
+	]);
+
+
+val contX2contX_app3 = prove_goal Cont.thy 
+ "[|contX(f);contX(t) |] ==> contX(%x. f(t(x)))"
+ (fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(rtac contX2contX_app2 1),
+	(rtac contX_const 1),
+	(atac 1),
+	(atac 1)
+	]);
+
isabelle