--- a/src/HOLCF/Cont.ML Wed Mar 02 23:15:16 2005 +0100
+++ b/src/HOLCF/Cont.ML Wed Mar 02 23:28:17 2005 +0100
@@ -1,525 +1,48 @@
-(* Title: HOLCF/Cont.ML
- ID: $Id$
- Author: Franz Regensburger
-Results about continuity and monotonicity
-*)
-
-(* ------------------------------------------------------------------------ *)
-(* access to definition *)
-(* ------------------------------------------------------------------------ *)
-
-Goalw [contlub]
- "! Y. chain(Y) --> f(lub(range(Y))) = lub(range(%i. f(Y(i))))==>\
-\ contlub(f)";
-by (atac 1);
-qed "contlubI";
-
-Goalw [contlub]
- " contlub(f)==>\
-\ ! Y. chain(Y) --> f(lub(range(Y))) = lub(range(%i. f(Y(i))))";
-by (atac 1);
-qed "contlubE";
-
-
-Goalw [cont]
- "! Y. chain(Y) --> range(% i. f(Y(i))) <<| f(lub(range(Y))) ==> cont(f)";
-by (atac 1);
-qed "contI";
-
-Goalw [cont]
- "cont(f) ==> ! Y. chain(Y) --> range(% i. f(Y(i))) <<| f(lub(range(Y)))";
-by (atac 1);
-qed "contE";
-
-
-Goalw [monofun]
- "! x y. x << y --> f(x) << f(y) ==> monofun(f)";
-by (atac 1);
-qed "monofunI";
-
-Goalw [monofun]
- "monofun(f) ==> ! x y. x << y --> f(x) << f(y)";
-by (atac 1);
-qed "monofunE";
-
-(* ------------------------------------------------------------------------ *)
-(* the main purpose of cont.thy is to show: *)
-(* monofun(f) & contlub(f) <==> cont(f) *)
-(* ------------------------------------------------------------------------ *)
-
-(* ------------------------------------------------------------------------ *)
-(* monotone functions map chains to chains *)
-(* ------------------------------------------------------------------------ *)
-
-Goal
- "[| monofun(f); chain(Y) |] ==> chain(%i. f(Y(i)))";
-by (rtac chainI 1);
-by (etac (monofunE RS spec RS spec RS mp) 1);
-by (etac (chainE) 1);
-qed "ch2ch_monofun";
-
-(* ------------------------------------------------------------------------ *)
-(* monotone functions map upper bound to upper bounds *)
-(* ------------------------------------------------------------------------ *)
-
-Goal
- "[| monofun(f); range(Y) <| u|] ==> range(%i. f(Y(i))) <| f(u)";
-by (rtac ub_rangeI 1);
-by (etac (monofunE RS spec RS spec RS mp) 1);
-by (etac (ub_rangeD) 1);
-qed "ub2ub_monofun";
-
-(* ------------------------------------------------------------------------ *)
-(* left to right: monofun(f) & contlub(f) ==> cont(f) *)
-(* ------------------------------------------------------------------------ *)
-
-Goalw [cont]
- "[|monofun(f);contlub(f)|] ==> cont(f)";
-by (strip_tac 1);
-by (rtac thelubE 1);
-by (etac ch2ch_monofun 1);
-by (atac 1);
-by (etac (contlubE RS spec RS mp RS sym) 1);
-by (atac 1);
-qed "monocontlub2cont";
-
-(* ------------------------------------------------------------------------ *)
-(* first a lemma about binary chains *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "[| cont(f); x << y |] \
-\ ==> range(%i::nat. f(if i = 0 then x else y)) <<| f(y)";
-by (rtac subst 1);
-by (etac (contE RS spec RS mp) 2);
-by (etac bin_chain 2);
-by (res_inst_tac [("y","y")] arg_cong 1);
-by (etac (lub_bin_chain RS thelubI) 1);
-qed "binchain_cont";
-
-(* ------------------------------------------------------------------------ *)
-(* right to left: cont(f) ==> monofun(f) & contlub(f) *)
-(* part1: cont(f) ==> monofun(f *)
-(* ------------------------------------------------------------------------ *)
-
-Goalw [monofun] "cont(f) ==> monofun(f)";
-by (strip_tac 1);
-by (dtac (binchain_cont RS is_ub_lub) 1);
-by (auto_tac (claset(), simpset() addsplits [split_if_asm]));
-qed "cont2mono";
-
-(* ------------------------------------------------------------------------ *)
-(* right to left: cont(f) ==> monofun(f) & contlub(f) *)
-(* part2: cont(f) ==> contlub(f) *)
-(* ------------------------------------------------------------------------ *)
-
-Goalw [contlub] "cont(f) ==> contlub(f)";
-by (strip_tac 1);
-by (rtac (thelubI RS sym) 1);
-by (etac (contE RS spec RS mp) 1);
-by (atac 1);
-qed "cont2contlub";
-
-(* ------------------------------------------------------------------------ *)
-(* monotone functions map finite chains to finite chains *)
-(* ------------------------------------------------------------------------ *)
-
-Goalw [finite_chain_def]
- "[| monofun f; finite_chain Y |] ==> finite_chain (%n. f (Y n))";
-by (force_tac (claset() addSEs [ch2ch_monofun],
- simpset() addsimps [max_in_chain_def]) 1);
-qed "monofun_finch2finch";
-
-(* ------------------------------------------------------------------------ *)
-(* The same holds for continuous functions *)
-(* ------------------------------------------------------------------------ *)
-
-bind_thm ("cont_finch2finch", cont2mono RS monofun_finch2finch);
-(* [| cont ?f; finite_chain ?Y |] ==> finite_chain (%n. ?f (?Y n)) *)
-
-
-(* ------------------------------------------------------------------------ *)
-(* The following results are about a curried function that is monotone *)
-(* in both arguments *)
-(* ------------------------------------------------------------------------ *)
-
-Goal
-"[|monofun(MF2); chain(F)|] ==> chain(%i. MF2 (F i) x)";
-by (etac (ch2ch_monofun RS ch2ch_fun) 1);
-by (atac 1);
-qed "ch2ch_MF2L";
-
-
-Goal
-"[|monofun(MF2(f)); chain(Y)|] ==> chain(%i. MF2 f (Y i))";
-by (etac ch2ch_monofun 1);
-by (atac 1);
-qed "ch2ch_MF2R";
-
-Goal
-"[|monofun(MF2); !f. monofun(MF2(f)); chain(F); chain(Y)|] ==> \
-\ chain(%i. MF2(F(i))(Y(i)))";
-by (rtac chainI 1);
-by (rtac trans_less 1);
-by (etac (ch2ch_MF2L RS chainE) 1);
-by (atac 1);
-by ((rtac (monofunE RS spec RS spec RS mp) 1) THEN (etac spec 1));
-by (etac (chainE) 1);
-qed "ch2ch_MF2LR";
-
-
-Goal
-"[|monofun(MF2::('a::po=>'b::po=>'c::cpo));\
-\ !f. monofun(MF2(f)::('b::po=>'c::cpo));\
-\ chain(F);chain(Y)|] ==> \
-\ chain(%j. lub(range(%i. MF2 (F j) (Y i))))";
-by (rtac (lub_mono RS chainI) 1);
-by ((rtac ch2ch_MF2R 1) THEN (etac spec 1));
-by (atac 1);
-by ((rtac ch2ch_MF2R 1) THEN (etac spec 1));
-by (atac 1);
-by (strip_tac 1);
-by (rtac (chainE) 1);
-by (etac ch2ch_MF2L 1);
-by (atac 1);
-qed "ch2ch_lubMF2R";
-
-
-Goal
-"[|monofun(MF2::('a::po=>'b::po=>'c::cpo));\
-\ !f. monofun(MF2(f)::('b::po=>'c::cpo));\
-\ chain(F);chain(Y)|] ==> \
-\ chain(%i. lub(range(%j. MF2 (F j) (Y i))))";
-by (rtac (lub_mono RS chainI) 1);
-by (etac ch2ch_MF2L 1);
-by (atac 1);
-by (etac ch2ch_MF2L 1);
-by (atac 1);
-by (strip_tac 1);
-by (rtac (chainE) 1);
-by ((rtac ch2ch_MF2R 1) THEN (etac spec 1));
-by (atac 1);
-qed "ch2ch_lubMF2L";
-
-
-Goal
-"[|monofun(MF2::('a::po=>'b::po=>'c::cpo));\
-\ !f. monofun(MF2(f)::('b::po=>'c::cpo));\
-\ chain(F)|] ==> \
-\ monofun(% x. lub(range(% j. MF2 (F j) (x))))";
-by (rtac monofunI 1);
-by (strip_tac 1);
-by (rtac lub_mono 1);
-by (etac ch2ch_MF2L 1);
-by (atac 1);
-by (etac ch2ch_MF2L 1);
-by (atac 1);
-by (strip_tac 1);
-by ((rtac (monofunE RS spec RS spec RS mp) 1) THEN (etac spec 1));
-by (atac 1);
-qed "lub_MF2_mono";
-
-Goal
-"[|monofun(MF2::('a::po=>'b::po=>'c::cpo));\
-\ !f. monofun(MF2(f)::('b::po=>'c::cpo));\
-\ chain(F); chain(Y)|] ==> \
-\ lub(range(%j. lub(range(%i. MF2(F j) (Y i))))) =\
-\ lub(range(%i. lub(range(%j. MF2(F j) (Y i)))))";
-by (rtac antisym_less 1);
-by (rtac (ub_rangeI RSN (2,is_lub_thelub)) 1);
-by (etac ch2ch_lubMF2R 1);
-by (REPEAT (atac 1));
-by (strip_tac 1);
-by (rtac lub_mono 1);
-by ((rtac ch2ch_MF2R 1) THEN (etac spec 1));
-by (atac 1);
-by (etac ch2ch_lubMF2L 1);
-by (REPEAT (atac 1));
-by (strip_tac 1);
-by (rtac is_ub_thelub 1);
-by (etac ch2ch_MF2L 1);
-by (atac 1);
-by (rtac (ub_rangeI RSN (2,is_lub_thelub)) 1);
-by (etac ch2ch_lubMF2L 1);
-by (REPEAT (atac 1));
-by (strip_tac 1);
-by (rtac lub_mono 1);
-by (etac ch2ch_MF2L 1);
-by (atac 1);
-by (etac ch2ch_lubMF2R 1);
-by (REPEAT (atac 1));
-by (strip_tac 1);
-by (rtac is_ub_thelub 1);
-by ((rtac ch2ch_MF2R 1) THEN (etac spec 1));
-by (atac 1);
-qed "ex_lubMF2";
-
+(* legacy ML bindings *)
-Goal
-"[|monofun(MF2::('a::po=>'b::po=>'c::cpo));\
-\ !f. monofun(MF2(f)::('b::po=>'c::cpo));\
-\ chain(FY);chain(TY)|] ==>\
-\ lub(range(%i. lub(range(%j. MF2(FY(j))(TY(i)))))) =\
-\ lub(range(%i. MF2(FY(i))(TY(i))))";
-by (rtac antisym_less 1);
-by (rtac (ub_rangeI RSN (2,is_lub_thelub)) 1);
-by (etac ch2ch_lubMF2L 1);
-by (REPEAT (atac 1));
-by (strip_tac 1 );
-by (rtac lub_mono3 1);
-by (etac ch2ch_MF2L 1);
-by (REPEAT (atac 1));
-by (etac ch2ch_MF2LR 1);
-by (REPEAT (atac 1));
-by (rtac allI 1);
-by (res_inst_tac [("m","i"),("n","ia")] nat_less_cases 1);
-by (res_inst_tac [("x","ia")] exI 1);
-by (rtac (chain_mono) 1);
-by (etac allE 1);
-by (etac ch2ch_MF2R 1);
-by (REPEAT (atac 1));
-by (hyp_subst_tac 1);
-by (res_inst_tac [("x","ia")] exI 1);
-by (rtac refl_less 1);
-by (res_inst_tac [("x","i")] exI 1);
-by (rtac (chain_mono) 1);
-by (etac ch2ch_MF2L 1);
-by (REPEAT (atac 1));
-by (rtac lub_mono 1);
-by (etac ch2ch_MF2LR 1);
-by (REPEAT(atac 1));
-by (etac ch2ch_lubMF2L 1);
-by (REPEAT (atac 1));
-by (strip_tac 1 );
-by (rtac is_ub_thelub 1);
-by (etac ch2ch_MF2L 1);
-by (atac 1);
-qed "diag_lubMF2_1";
-
-Goal
-"[|monofun(MF2::('a::po=>'b::po=>'c::cpo));\
-\ !f. monofun(MF2(f)::('b::po=>'c::cpo));\
-\ chain(FY);chain(TY)|] ==>\
-\ lub(range(%j. lub(range(%i. MF2(FY(j))(TY(i)))))) =\
-\ lub(range(%i. MF2(FY(i))(TY(i))))";
-by (rtac trans 1);
-by (rtac ex_lubMF2 1);
-by (REPEAT (atac 1));
-by (etac diag_lubMF2_1 1);
-by (REPEAT (atac 1));
-qed "diag_lubMF2_2";
-
-
-(* ------------------------------------------------------------------------ *)
-(* The following results are about a curried function that is continuous *)
-(* in both arguments *)
-(* ------------------------------------------------------------------------ *)
-
-val [prem1,prem2,prem3,prem4] = goal thy
-"[| cont(CF2); !f. cont(CF2(f)); chain(FY); chain(TY)|] ==>\
-\ CF2(lub(range(FY)))(lub(range(TY))) = lub(range(%i. CF2(FY(i))(TY(i))))";
-by (cut_facts_tac [prem1,prem2,prem3, prem4] 1);
-by (stac (prem1 RS cont2contlub RS contlubE RS spec RS mp) 1);
-by (assume_tac 1);
-by (stac thelub_fun 1);
-by (rtac (prem1 RS (cont2mono RS ch2ch_monofun)) 1);
-by (assume_tac 1);
-by (rtac trans 1);
-by (rtac ((prem2 RS spec RS cont2contlub) RS contlubE RS spec RS mp RS ext RS arg_cong RS arg_cong) 1);
-by (rtac prem4 1);
-by (blast_tac (claset() addIs [diag_lubMF2_2, cont2mono]) 1);
-qed "contlub_CF2";
-
-(* ------------------------------------------------------------------------ *)
-(* The following results are about application for functions in 'a=>'b *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "f1 << f2 ==> f1(x) << f2(x)";
-by (etac (less_fun RS iffD1 RS spec) 1);
-qed "monofun_fun_fun";
-
-Goal "[|monofun(f); x1 << x2|] ==> f(x1) << f(x2)";
-by (etac (monofunE RS spec RS spec RS mp) 1);
-by (atac 1);
-qed "monofun_fun_arg";
-
-Goal "[|monofun(f1); monofun(f2); f1 << f2; x1 << x2|] ==> f1(x1) << f2(x2)";
-by (rtac trans_less 1);
-by (etac monofun_fun_arg 1);
-by (atac 1);
-by (etac monofun_fun_fun 1);
-qed "monofun_fun";
-
-
-(* ------------------------------------------------------------------------ *)
-(* The following results are about the propagation of monotonicity and *)
-(* continuity *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "[|monofun(c1)|] ==> monofun(%x. c1 x y)";
-by (rtac monofunI 1);
-by (strip_tac 1);
-by (etac (monofun_fun_arg RS monofun_fun_fun) 1);
-by (atac 1);
-qed "mono2mono_MF1L";
-
-Goal "[|cont(c1)|] ==> cont(%x. c1 x y)";
-by (rtac monocontlub2cont 1);
-by (etac (cont2mono RS mono2mono_MF1L) 1);
-by (rtac contlubI 1);
-by (strip_tac 1);
-by (ftac asm_rl 1);
-by (etac (cont2contlub RS contlubE RS spec RS mp RS ssubst) 1);
-by (atac 1);
-by (stac thelub_fun 1);
-by (rtac ch2ch_monofun 1);
-by (etac cont2mono 1);
-by (atac 1);
-by (rtac refl 1);
-qed "cont2cont_CF1L";
-
-(********* Note "(%x.%y.c1 x y) = c1" ***********)
-
-Goal "!y. monofun(%x. c1 x y) ==> monofun(c1)";
-by (rtac monofunI 1);
-by (strip_tac 1);
-by (rtac (less_fun RS iffD2) 1);
-by (blast_tac (claset() addDs [monofunE]) 1);
-qed "mono2mono_MF1L_rev";
-
-Goal "!y. cont(%x. c1 x y) ==> cont(c1)";
-by (rtac monocontlub2cont 1);
-by (rtac (cont2mono RS allI RS mono2mono_MF1L_rev ) 1);
-by (etac spec 1);
-by (rtac contlubI 1);
-by (strip_tac 1);
-by (rtac ext 1);
-by (stac thelub_fun 1);
-by (rtac (cont2mono RS allI RS mono2mono_MF1L_rev RS ch2ch_monofun) 1);
-by (etac spec 1);
-by (atac 1);
-by (blast_tac (claset() addDs [ cont2contlub RS contlubE]) 1);
-qed "cont2cont_CF1L_rev";
-
-(* ------------------------------------------------------------------------ *)
-(* What D.A.Schmidt calls continuity of abstraction *)
-(* never used here *)
-(* ------------------------------------------------------------------------ *)
-
-Goal
-"[|chain(Y::nat=>'a);!y. cont(%x.(c::'a::cpo=>'b::cpo=>'c::cpo) x y)|] ==>\
-\ (%y. lub(range(%i. c (Y i) y))) = (lub(range(%i.%y. c (Y i) y)))";
-by (rtac trans 1);
-by (rtac (cont2contlub RS contlubE RS spec RS mp) 2);
-by (atac 3);
-by (etac cont2cont_CF1L_rev 2);
-by (rtac ext 1);
-by (rtac (cont2contlub RS contlubE RS spec RS mp RS sym) 1);
-by (etac spec 1);
-by (atac 1);
-qed "contlub_abstraction";
-
-Goal "[|monofun(ft);!x. monofun(ft(x));monofun(tt)|] ==>\
-\ monofun(%x.(ft(x))(tt(x)))";
-by (rtac monofunI 1);
-by (strip_tac 1);
-by (res_inst_tac [("f1.0","ft(x)"),("f2.0","ft(y)")] monofun_fun 1);
-by (etac spec 1);
-by (etac spec 1);
-by (etac (monofunE RS spec RS spec RS mp) 1);
-by (atac 1);
-by (etac (monofunE RS spec RS spec RS mp) 1);
-by (atac 1);
-qed "mono2mono_app";
-
-
-Goal "[|cont(ft);!x. cont(ft(x));cont(tt)|] ==> contlub(%x.(ft(x))(tt(x)))";
-by (rtac contlubI 1);
-by (strip_tac 1);
-by (res_inst_tac [("f3","tt")] (contlubE RS spec RS mp RS ssubst) 1);
-by (etac cont2contlub 1);
-by (atac 1);
-by (rtac contlub_CF2 1);
-by (REPEAT (atac 1));
-by (etac (cont2mono RS ch2ch_monofun) 1);
-by (atac 1);
-qed "cont2contlub_app";
-
-
-Goal "[|cont(ft); !x. cont(ft(x)); cont(tt)|] ==> cont(%x.(ft(x))(tt(x)))";
-by (blast_tac (claset() addIs [monocontlub2cont, mono2mono_app, cont2mono,
- cont2contlub_app]) 1);
-qed "cont2cont_app";
-
-
-bind_thm ("cont2cont_app2", allI RSN (2,cont2cont_app));
-(* [| cont ?ft; !!x. cont (?ft x); cont ?tt |] ==> *)
-(* cont (%x. ?ft x (?tt x)) *)
-
-
-(* ------------------------------------------------------------------------ *)
-(* The identity function is continuous *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "cont(% x. x)";
-by (rtac contI 1);
-by (strip_tac 1);
-by (etac thelubE 1);
-by (rtac refl 1);
-qed "cont_id";
-
-(* ------------------------------------------------------------------------ *)
-(* constant functions are continuous *)
-(* ------------------------------------------------------------------------ *)
-
-Goalw [cont] "cont(%x. c)";
-by (strip_tac 1);
-by (blast_tac (claset() addIs [is_lubI, ub_rangeI] addDs [ub_rangeD]) 1);
-qed "cont_const";
-
-
-Goal "[|cont(f); cont(t) |] ==> cont(%x. f(t(x)))";
-by (best_tac (claset() addIs [ cont2cont_app2, cont_const]) 1);
-qed "cont2cont_app3";
-
-(* ------------------------------------------------------------------------ *)
-(* A non-emptyness result for Cfun *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "?x:Collect cont";
-by (rtac CollectI 1);
-by (rtac cont_const 1);
-qed "CfunI";
-
-(* ------------------------------------------------------------------------ *)
-(* some properties of flat *)
-(* ------------------------------------------------------------------------ *)
-
-Goalw [monofun] "f UU = UU ==> monofun (f::'a::flat=>'b::pcpo)";
-by (strip_tac 1);
-by (dtac (ax_flat RS spec RS spec RS mp) 1);
-by (fast_tac ((HOL_cs addss (simpset() addsimps [minimal]))) 1);
-qed "flatdom2monofun";
-
-
-Goal "monofun f ==> cont(f::'a::chfin=>'b::pcpo)";
-by (rtac monocontlub2cont 1);
-by ( atac 1);
-by (rtac contlubI 1);
-by (strip_tac 1);
-by (ftac chfin2finch 1);
-by (rtac antisym_less 1);
-by ( force_tac (HOL_cs addIs [is_ub_thelub,ch2ch_monofun],
- HOL_ss addsimps [finite_chain_def,maxinch_is_thelub]) 1);
-by (dtac (monofun_finch2finch COMP swap_prems_rl) 1);
-by ( atac 1);
-by (asm_full_simp_tac (HOL_ss addsimps [finite_chain_def]) 1);
-by (etac conjE 1);
-by (etac exE 1);
-by (asm_full_simp_tac (HOL_ss addsimps [maxinch_is_thelub]) 1);
-by (etac (monofunE RS spec RS spec RS mp) 1);
-by (etac is_ub_thelub 1);
-qed "chfindom_monofun2cont";
-
-bind_thm ("flatdom_strict2cont",flatdom2monofun RS chfindom_monofun2cont);
-(* f UU = UU ==> cont (f::'a=>'b::pcpo)" *)
+val monofun = thm "monofun";
+val contlub = thm "contlub";
+val cont = thm "cont";
+val contlubI = thm "contlubI";
+val contlubE = thm "contlubE";
+val contI = thm "contI";
+val contE = thm "contE";
+val monofunI = thm "monofunI";
+val monofunE = thm "monofunE";
+val ch2ch_monofun = thm "ch2ch_monofun";
+val ub2ub_monofun = thm "ub2ub_monofun";
+val monocontlub2cont = thm "monocontlub2cont";
+val binchain_cont = thm "binchain_cont";
+val cont2mono = thm "cont2mono";
+val cont2contlub = thm "cont2contlub";
+val monofun_finch2finch = thm "monofun_finch2finch";
+val cont_finch2finch = thm "cont_finch2finch";
+val ch2ch_MF2L = thm "ch2ch_MF2L";
+val ch2ch_MF2R = thm "ch2ch_MF2R";
+val ch2ch_MF2LR = thm "ch2ch_MF2LR";
+val ch2ch_lubMF2R = thm "ch2ch_lubMF2R";
+val ch2ch_lubMF2L = thm "ch2ch_lubMF2L";
+val lub_MF2_mono = thm "lub_MF2_mono";
+val ex_lubMF2 = thm "ex_lubMF2";
+val diag_lubMF2_1 = thm "diag_lubMF2_1";
+val diag_lubMF2_2 = thm "diag_lubMF2_2";
+val contlub_CF2 = thm "contlub_CF2";
+val monofun_fun_fun = thm "monofun_fun_fun";
+val monofun_fun_arg = thm "monofun_fun_arg";
+val mono2mono_MF1L = thm "mono2mono_MF1L";
+val cont2cont_CF1L = thm "cont2cont_CF1L";
+val mono2mono_MF1L_rev = thm "mono2mono_MF1L_rev";
+val cont2cont_CF1L_rev = thm "cont2cont_CF1L_rev";
+val contlub_abstraction = thm "contlub_abstraction";
+val mono2mono_app = thm "mono2mono_app";
+val cont2contlub_app = thm "cont2contlub_app";
+val cont2cont_app = thm "cont2cont_app";
+val cont2cont_app2 = thm "cont2cont_app2";
+val cont_id = thm "cont_id";
+val cont_const = thm "cont_const";
+val cont2cont_app3 = thm "cont2cont_app3";
+val CfunI = thm "CfunI";
+val flatdom2monofun = thm "flatdom2monofun";
+val chfindom_monofun2cont = thm "chfindom_monofun2cont";
+val flatdom_strict2cont = thm "flatdom_strict2cont";