--- 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";
--- a/src/HOLCF/Cont.thy Wed Mar 02 23:15:16 2005 +0100
+++ b/src/HOLCF/Cont.thy Wed Mar 02 23:28:17 2005 +0100
@@ -1,11 +1,12 @@
(* Title: HOLCF/cont.thy
ID: $Id$
Author: Franz Regensburger
+ License: GPL (GNU GENERAL PUBLIC LICENSE)
Results about continuity and monotonicity
*)
-Cont = Fun3 +
+theory Cont = Fun3:
(*
@@ -15,7 +16,7 @@
*)
-default po
+defaultsort po
consts
monofun :: "('a => 'b) => bool" (* monotonicity *)
@@ -24,12 +25,12 @@
defs
-monofun "monofun(f) == ! x y. x << y --> f(x) << f(y)"
+monofun: "monofun(f) == ! x y. x << y --> f(x) << f(y)"
-contlub "contlub(f) == ! Y. chain(Y) -->
+contlub: "contlub(f) == ! Y. chain(Y) -->
f(lub(range(Y))) = lub(range(% i. f(Y(i))))"
-cont "cont(f) == ! Y. chain(Y) -->
+cont: "cont(f) == ! Y. chain(Y) -->
range(% i. f(Y(i))) <<| f(lub(range(Y)))"
(* ------------------------------------------------------------------------ *)
@@ -37,4 +38,545 @@
(* monofun(f) & contlub(f) <==> cont(f) *)
(* ------------------------------------------------------------------------ *)
+(* Title: HOLCF/Cont.ML
+ ID: $Id$
+ Author: Franz Regensburger
+ License: GPL (GNU GENERAL PUBLIC LICENSE)
+
+Results about continuity and monotonicity
+*)
+
+(* ------------------------------------------------------------------------ *)
+(* access to definition *)
+(* ------------------------------------------------------------------------ *)
+
+lemma contlubI:
+ "! Y. chain(Y) --> f(lub(range(Y))) = lub(range(%i. f(Y(i))))==>
+ contlub(f)"
+apply (unfold contlub)
+apply assumption
+done
+
+lemma contlubE:
+ " contlub(f)==>
+ ! Y. chain(Y) --> f(lub(range(Y))) = lub(range(%i. f(Y(i))))"
+apply (unfold contlub)
+apply assumption
+done
+
+
+lemma contI:
+ "! Y. chain(Y) --> range(% i. f(Y(i))) <<| f(lub(range(Y))) ==> cont(f)"
+
+apply (unfold cont)
+apply assumption
+done
+
+lemma contE:
+ "cont(f) ==> ! Y. chain(Y) --> range(% i. f(Y(i))) <<| f(lub(range(Y)))"
+apply (unfold cont)
+apply assumption
+done
+
+
+lemma monofunI:
+ "! x y. x << y --> f(x) << f(y) ==> monofun(f)"
+apply (unfold monofun)
+apply assumption
+done
+
+lemma monofunE:
+ "monofun(f) ==> ! x y. x << y --> f(x) << f(y)"
+apply (unfold monofun)
+apply assumption
+done
+
+(* ------------------------------------------------------------------------ *)
+(* the main purpose of cont.thy is to show: *)
+(* monofun(f) & contlub(f) <==> cont(f) *)
+(* ------------------------------------------------------------------------ *)
+
+(* ------------------------------------------------------------------------ *)
+(* monotone functions map chains to chains *)
+(* ------------------------------------------------------------------------ *)
+
+lemma ch2ch_monofun:
+ "[| monofun(f); chain(Y) |] ==> chain(%i. f(Y(i)))"
+apply (rule chainI)
+apply (erule monofunE [THEN spec, THEN spec, THEN mp])
+apply (erule chainE)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* monotone functions map upper bound to upper bounds *)
+(* ------------------------------------------------------------------------ *)
+
+lemma ub2ub_monofun:
+ "[| monofun(f); range(Y) <| u|] ==> range(%i. f(Y(i))) <| f(u)"
+apply (rule ub_rangeI)
+apply (erule monofunE [THEN spec, THEN spec, THEN mp])
+apply (erule ub_rangeD)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* left to right: monofun(f) & contlub(f) ==> cont(f) *)
+(* ------------------------------------------------------------------------ *)
+
+lemma monocontlub2cont:
+ "[|monofun(f);contlub(f)|] ==> cont(f)"
+apply (unfold cont)
+apply (intro strip)
+apply (rule thelubE)
+apply (erule ch2ch_monofun)
+apply assumption
+apply (erule contlubE [THEN spec, THEN mp, symmetric])
+apply assumption
+done
+
+(* ------------------------------------------------------------------------ *)
+(* first a lemma about binary chains *)
+(* ------------------------------------------------------------------------ *)
+
+lemma binchain_cont: "[| cont(f); x << y |]
+ ==> range(%i::nat. f(if i = 0 then x else y)) <<| f(y)"
+apply (rule subst)
+prefer 2 apply (erule contE [THEN spec, THEN mp])
+apply (erule bin_chain)
+apply (rule_tac y = "y" in arg_cong)
+apply (erule lub_bin_chain [THEN thelubI])
+done
+
+(* ------------------------------------------------------------------------ *)
+(* right to left: cont(f) ==> monofun(f) & contlub(f) *)
+(* part1: cont(f) ==> monofun(f *)
+(* ------------------------------------------------------------------------ *)
+
+lemma cont2mono: "cont(f) ==> monofun(f)"
+apply (unfold monofun)
+apply (intro strip)
+apply (drule binchain_cont [THEN is_ub_lub])
+apply (auto split add: split_if_asm)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* right to left: cont(f) ==> monofun(f) & contlub(f) *)
+(* part2: cont(f) ==> contlub(f) *)
+(* ------------------------------------------------------------------------ *)
+
+lemma cont2contlub: "cont(f) ==> contlub(f)"
+apply (unfold contlub)
+apply (intro strip)
+apply (rule thelubI [symmetric])
+apply (erule contE [THEN spec, THEN mp])
+apply assumption
+done
+
+(* ------------------------------------------------------------------------ *)
+(* monotone functions map finite chains to finite chains *)
+(* ------------------------------------------------------------------------ *)
+
+lemma monofun_finch2finch:
+ "[| monofun f; finite_chain Y |] ==> finite_chain (%n. f (Y n))"
+apply (unfold finite_chain_def)
+apply (force elim!: ch2ch_monofun simp add: max_in_chain_def)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* The same holds for continuous functions *)
+(* ------------------------------------------------------------------------ *)
+
+lemmas cont_finch2finch = cont2mono [THEN monofun_finch2finch, standard]
+(* [| 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 *)
+(* ------------------------------------------------------------------------ *)
+
+lemma ch2ch_MF2L:
+"[|monofun(MF2); chain(F)|] ==> chain(%i. MF2 (F i) x)"
+apply (erule ch2ch_monofun [THEN ch2ch_fun])
+apply assumption
+done
+
+
+lemma ch2ch_MF2R:
+"[|monofun(MF2(f)); chain(Y)|] ==> chain(%i. MF2 f (Y i))"
+apply (erule ch2ch_monofun)
+apply assumption
+done
+
+lemma ch2ch_MF2LR:
+"[|monofun(MF2); !f. monofun(MF2(f)); chain(F); chain(Y)|] ==>
+ chain(%i. MF2(F(i))(Y(i)))"
+apply (rule chainI)
+apply (rule trans_less)
+apply (erule ch2ch_MF2L [THEN chainE])
+apply assumption
+apply (rule monofunE [THEN spec, THEN spec, THEN mp], erule spec)
+apply (erule chainE)
+done
+
+
+lemma ch2ch_lubMF2R:
+"[|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))))"
+apply (rule lub_mono [THEN chainI])
+apply (rule ch2ch_MF2R, erule spec)
+apply assumption
+apply (rule ch2ch_MF2R, erule spec)
+apply assumption
+apply (intro strip)
+apply (rule chainE)
+apply (erule ch2ch_MF2L)
+apply assumption
+done
+
+
+lemma ch2ch_lubMF2L:
+"[|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))))"
+apply (rule lub_mono [THEN chainI])
+apply (erule ch2ch_MF2L)
+apply assumption
+apply (erule ch2ch_MF2L)
+apply assumption
+apply (intro strip)
+apply (rule chainE)
+apply (rule ch2ch_MF2R, erule spec)
+apply assumption
+done
+
+
+lemma lub_MF2_mono:
+"[|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))))"
+apply (rule monofunI)
+apply (intro strip)
+apply (rule lub_mono)
+apply (erule ch2ch_MF2L)
+apply assumption
+apply (erule ch2ch_MF2L)
+apply assumption
+apply (intro strip)
+apply (rule monofunE [THEN spec, THEN spec, THEN mp], erule spec)
+apply assumption
+done
+
+lemma ex_lubMF2:
+"[|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)))))"
+apply (rule antisym_less)
+apply (rule is_lub_thelub[OF _ ub_rangeI])
+apply (erule ch2ch_lubMF2R)
+apply (assumption+)
+apply (rule lub_mono)
+apply (rule ch2ch_MF2R, erule spec)
+apply assumption
+apply (erule ch2ch_lubMF2L)
+apply (assumption+)
+apply (intro strip)
+apply (rule is_ub_thelub)
+apply (erule ch2ch_MF2L)
+apply assumption
+apply (rule is_lub_thelub[OF _ ub_rangeI])
+apply (erule ch2ch_lubMF2L)
+apply (assumption+)
+apply (rule lub_mono)
+apply (erule ch2ch_MF2L)
+apply assumption
+apply (erule ch2ch_lubMF2R)
+apply (assumption+)
+apply (intro strip)
+apply (rule is_ub_thelub)
+apply (rule ch2ch_MF2R, erule spec)
+apply assumption
+done
+
+
+lemma diag_lubMF2_1:
+"[|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))))"
+apply (rule antisym_less)
+apply (rule is_lub_thelub[OF _ ub_rangeI])
+apply (erule ch2ch_lubMF2L)
+apply (assumption+)
+apply (rule lub_mono3)
+apply (erule ch2ch_MF2L)
+apply (assumption+)
+apply (erule ch2ch_MF2LR)
+apply (assumption+)
+apply (rule allI)
+apply (rule_tac m = "i" and n = "ia" in nat_less_cases)
+apply (rule_tac x = "ia" in exI)
+apply (rule chain_mono)
+apply (erule allE)
+apply (erule ch2ch_MF2R)
+apply (assumption+)
+apply (erule ssubst)
+apply (rule_tac x = "ia" in exI)
+apply (rule refl_less)
+apply (rule_tac x = "i" in exI)
+apply (rule chain_mono)
+apply (erule ch2ch_MF2L)
+apply (assumption+)
+apply (rule lub_mono)
+apply (erule ch2ch_MF2LR)
+apply (assumption+)
+apply (erule ch2ch_lubMF2L)
+apply (assumption+)
+apply (intro strip)
+apply (rule is_ub_thelub)
+apply (erule ch2ch_MF2L)
+apply assumption
+done
+
+lemma diag_lubMF2_2:
+"[|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))))"
+apply (rule trans)
+apply (rule ex_lubMF2)
+apply (assumption+)
+apply (erule diag_lubMF2_1)
+apply (assumption+)
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* The following results are about a curried function that is continuous *)
+(* in both arguments *)
+(* ------------------------------------------------------------------------ *)
+
+lemma contlub_CF2:
+assumes prem1: "cont(CF2)"
+assumes prem2: "!f. cont(CF2(f))"
+assumes prem3: "chain(FY)"
+assumes prem4: "chain(TY)"
+shows "CF2(lub(range(FY)))(lub(range(TY))) = lub(range(%i. CF2(FY(i))(TY(i))))"
+apply (subst prem1 [THEN cont2contlub, THEN contlubE, THEN spec, THEN mp])
+apply assumption
+apply (subst thelub_fun)
+apply (rule prem1 [THEN cont2mono [THEN ch2ch_monofun]])
+apply assumption
+apply (rule trans)
+apply (rule prem2 [THEN spec, THEN cont2contlub, THEN contlubE, THEN spec, THEN mp, THEN ext, THEN arg_cong, THEN arg_cong])
+apply (rule prem4)
+apply (rule diag_lubMF2_2)
+apply (auto simp add: cont2mono prems)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* The following results are about application for functions in 'a=>'b *)
+(* ------------------------------------------------------------------------ *)
+
+lemma monofun_fun_fun: "f1 << f2 ==> f1(x) << f2(x)"
+apply (erule less_fun [THEN iffD1, THEN spec])
+done
+
+lemma monofun_fun_arg: "[|monofun(f); x1 << x2|] ==> f(x1) << f(x2)"
+apply (erule monofunE [THEN spec, THEN spec, THEN mp])
+apply assumption
+done
+
+lemma monofun_fun: "[|monofun(f1); monofun(f2); f1 << f2; x1 << x2|] ==> f1(x1) << f2(x2)"
+apply (rule trans_less)
+apply (erule monofun_fun_arg)
+apply assumption
+apply (erule monofun_fun_fun)
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* The following results are about the propagation of monotonicity and *)
+(* continuity *)
+(* ------------------------------------------------------------------------ *)
+
+lemma mono2mono_MF1L: "[|monofun(c1)|] ==> monofun(%x. c1 x y)"
+apply (rule monofunI)
+apply (intro strip)
+apply (erule monofun_fun_arg [THEN monofun_fun_fun])
+apply assumption
+done
+
+lemma cont2cont_CF1L: "[|cont(c1)|] ==> cont(%x. c1 x y)"
+apply (rule monocontlub2cont)
+apply (erule cont2mono [THEN mono2mono_MF1L])
+apply (rule contlubI)
+apply (intro strip)
+apply (frule asm_rl)
+apply (erule cont2contlub [THEN contlubE, THEN spec, THEN mp, THEN ssubst])
+apply assumption
+apply (subst thelub_fun)
+apply (rule ch2ch_monofun)
+apply (erule cont2mono)
+apply assumption
+apply (rule refl)
+done
+
+(********* Note "(%x.%y.c1 x y) = c1" ***********)
+
+lemma mono2mono_MF1L_rev: "!y. monofun(%x. c1 x y) ==> monofun(c1)"
+apply (rule monofunI)
+apply (intro strip)
+apply (rule less_fun [THEN iffD2])
+apply (blast dest: monofunE)
+done
+
+lemma cont2cont_CF1L_rev: "!y. cont(%x. c1 x y) ==> cont(c1)"
+apply (rule monocontlub2cont)
+apply (rule cont2mono [THEN allI, THEN mono2mono_MF1L_rev])
+apply (erule spec)
+apply (rule contlubI)
+apply (intro strip)
+apply (rule ext)
+apply (subst thelub_fun)
+apply (rule cont2mono [THEN allI, THEN mono2mono_MF1L_rev, THEN ch2ch_monofun])
+apply (erule spec)
+apply assumption
+apply (blast dest: cont2contlub [THEN contlubE])
+done
+
+(* ------------------------------------------------------------------------ *)
+(* What D.A.Schmidt calls continuity of abstraction *)
+(* never used here *)
+(* ------------------------------------------------------------------------ *)
+
+lemma contlub_abstraction:
+"[|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)))"
+apply (rule trans)
+prefer 2 apply (rule cont2contlub [THEN contlubE, THEN spec, THEN mp])
+prefer 2 apply (assumption)
+apply (erule cont2cont_CF1L_rev)
+apply (rule ext)
+apply (rule cont2contlub [THEN contlubE, THEN spec, THEN mp, symmetric])
+apply (erule spec)
+apply assumption
+done
+
+lemma mono2mono_app: "[|monofun(ft);!x. monofun(ft(x));monofun(tt)|] ==>
+ monofun(%x.(ft(x))(tt(x)))"
+apply (rule monofunI)
+apply (intro strip)
+apply (rule_tac ?f1.0 = "ft(x)" and ?f2.0 = "ft(y)" in monofun_fun)
+apply (erule spec)
+apply (erule spec)
+apply (erule monofunE [THEN spec, THEN spec, THEN mp])
+apply assumption
+apply (erule monofunE [THEN spec, THEN spec, THEN mp])
+apply assumption
+done
+
+
+lemma cont2contlub_app: "[|cont(ft);!x. cont(ft(x));cont(tt)|] ==> contlub(%x.(ft(x))(tt(x)))"
+apply (rule contlubI)
+apply (intro strip)
+apply (rule_tac f3 = "tt" in contlubE [THEN spec, THEN mp, THEN ssubst])
+apply (erule cont2contlub)
+apply assumption
+apply (rule contlub_CF2)
+apply (assumption+)
+apply (erule cont2mono [THEN ch2ch_monofun])
+apply assumption
+done
+
+
+lemma cont2cont_app: "[|cont(ft); !x. cont(ft(x)); cont(tt)|] ==> cont(%x.(ft(x))(tt(x)))"
+apply (blast intro: monocontlub2cont mono2mono_app cont2mono cont2contlub_app)
+done
+
+
+lemmas cont2cont_app2 = cont2cont_app[OF _ allI]
+(* [| cont ?ft; !!x. cont (?ft x); cont ?tt |] ==> *)
+(* cont (%x. ?ft x (?tt x)) *)
+
+
+(* ------------------------------------------------------------------------ *)
+(* The identity function is continuous *)
+(* ------------------------------------------------------------------------ *)
+
+lemma cont_id: "cont(% x. x)"
+apply (rule contI)
+apply (intro strip)
+apply (erule thelubE)
+apply (rule refl)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* constant functions are continuous *)
+(* ------------------------------------------------------------------------ *)
+
+lemma cont_const: "cont(%x. c)"
+apply (unfold cont)
+apply (intro strip)
+apply (blast intro: is_lubI ub_rangeI dest: ub_rangeD)
+done
+
+
+lemma cont2cont_app3: "[|cont(f); cont(t) |] ==> cont(%x. f(t(x)))"
+apply (best intro: cont2cont_app2 cont_const)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* A non-emptyness result for Cfun *)
+(* ------------------------------------------------------------------------ *)
+
+lemma CfunI: "?x:Collect cont"
+apply (rule CollectI)
+apply (rule cont_const)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* some properties of flat *)
+(* ------------------------------------------------------------------------ *)
+
+lemma flatdom2monofun: "f UU = UU ==> monofun (f::'a::flat=>'b::pcpo)"
+
+apply (unfold monofun)
+apply (intro strip)
+apply (drule ax_flat [THEN spec, THEN spec, THEN mp])
+apply auto
+done
+
+declare range_composition [simp del]
+lemma chfindom_monofun2cont: "monofun f ==> cont(f::'a::chfin=>'b::pcpo)"
+apply (rule monocontlub2cont)
+apply assumption
+apply (rule contlubI)
+apply (intro strip)
+apply (frule chfin2finch)
+apply (rule antisym_less)
+apply (clarsimp simp add: finite_chain_def maxinch_is_thelub)
+apply (rule is_ub_thelub)
+apply (erule ch2ch_monofun)
+apply assumption
+apply (drule monofun_finch2finch[COMP swap_prems_rl])
+apply assumption
+apply (simp add: finite_chain_def)
+apply (erule conjE)
+apply (erule exE)
+apply (simp add: maxinch_is_thelub)
+apply (erule monofunE [THEN spec, THEN spec, THEN mp])
+apply (erule is_ub_thelub)
+done
+
+lemmas flatdom_strict2cont = flatdom2monofun [THEN chfindom_monofun2cont, standard]
+(* f UU = UU ==> cont (f::'a=>'b::pcpo)" *)
+
end