--- a/src/HOLCF/Cont.ML Tue Jul 04 14:58:40 2000 +0200
+++ b/src/HOLCF/Cont.ML Tue Jul 04 15:58:11 2000 +0200
@@ -10,57 +10,45 @@
(* access to definition *)
(* ------------------------------------------------------------------------ *)
-qed_goalw "contlubI" thy [contlub]
+val prems = goalw thy [contlub]
"! Y. chain(Y) --> f(lub(range(Y))) = lub(range(%i. f(Y(i))))==>\
-\ contlub(f)"
-(fn prems =>
- [
- (cut_facts_tac prems 1),
- (atac 1)
- ]);
+\ contlub(f)";
+by (cut_facts_tac prems 1);
+by (atac 1);
+qed "contlubI";
-qed_goalw "contlubE" thy [contlub]
+val prems = goalw thy [contlub]
" contlub(f)==>\
-\ ! Y. chain(Y) --> f(lub(range(Y))) = lub(range(%i. f(Y(i))))"
-(fn prems =>
- [
- (cut_facts_tac prems 1),
- (atac 1)
- ]);
+\ ! Y. chain(Y) --> f(lub(range(Y))) = lub(range(%i. f(Y(i))))";
+by (cut_facts_tac prems 1);
+by (atac 1);
+qed "contlubE";
-qed_goalw "contI" thy [cont]
- "! Y. chain(Y) --> range(% i. f(Y(i))) <<| f(lub(range(Y))) ==> cont(f)"
-(fn prems =>
- [
- (cut_facts_tac prems 1),
- (atac 1)
- ]);
+val prems = goalw thy [cont]
+ "! Y. chain(Y) --> range(% i. f(Y(i))) <<| f(lub(range(Y))) ==> cont(f)";
+by (cut_facts_tac prems 1);
+by (atac 1);
+qed "contI";
-qed_goalw "contE" thy [cont]
- "cont(f) ==> ! Y. chain(Y) --> range(% i. f(Y(i))) <<| f(lub(range(Y)))"
-(fn prems =>
- [
- (cut_facts_tac prems 1),
- (atac 1)
- ]);
+val prems = goalw thy [cont]
+ "cont(f) ==> ! Y. chain(Y) --> range(% i. f(Y(i))) <<| f(lub(range(Y)))";
+by (cut_facts_tac prems 1);
+by (atac 1);
+qed "contE";
-qed_goalw "monofunI" thy [monofun]
- "! x y. x << y --> f(x) << f(y) ==> monofun(f)"
-(fn prems =>
- [
- (cut_facts_tac prems 1),
- (atac 1)
- ]);
+val prems = goalw thy [monofun]
+ "! x y. x << y --> f(x) << f(y) ==> monofun(f)";
+by (cut_facts_tac prems 1);
+by (atac 1);
+qed "monofunI";
-qed_goalw "monofunE" thy [monofun]
- "monofun(f) ==> ! x y. x << y --> f(x) << f(y)"
-(fn prems =>
- [
- (cut_facts_tac prems 1),
- (atac 1)
- ]);
+val prems = goalw thy [monofun]
+ "monofun(f) ==> ! x y. x << y --> f(x) << f(y)";
+by (cut_facts_tac prems 1);
+by (atac 1);
+qed "monofunE";
(* ------------------------------------------------------------------------ *)
(* the main purpose of cont.thy is to show: *)
@@ -71,115 +59,91 @@
(* monotone functions map chains to chains *)
(* ------------------------------------------------------------------------ *)
-qed_goal "ch2ch_monofun" thy
- "[| monofun(f); chain(Y) |] ==> chain(%i. f(Y(i)))"
-(fn prems =>
- [
- (cut_facts_tac prems 1),
- (rtac chainI 1),
- (rtac allI 1),
- (etac (monofunE RS spec RS spec RS mp) 1),
- (etac (chainE RS spec) 1)
- ]);
+val prems = goal thy
+ "[| monofun(f); chain(Y) |] ==> chain(%i. f(Y(i)))";
+by (cut_facts_tac prems 1);
+by (rtac chainI 1);
+by (rtac allI 1);
+by (etac (monofunE RS spec RS spec RS mp) 1);
+by (etac (chainE RS spec) 1);
+qed "ch2ch_monofun";
(* ------------------------------------------------------------------------ *)
(* monotone functions map upper bound to upper bounds *)
(* ------------------------------------------------------------------------ *)
-qed_goal "ub2ub_monofun" 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)
- ]);
+val prems = goal thy
+ "[| monofun(f); range(Y) <| u|] ==> range(%i. f(Y(i))) <| f(u)";
+by (cut_facts_tac prems 1);
+by (rtac ub_rangeI 1);
+by (rtac allI 1);
+by (etac (monofunE RS spec RS spec RS mp) 1);
+by (etac (ub_rangeE RS spec) 1);
+qed "ub2ub_monofun";
(* ------------------------------------------------------------------------ *)
(* left to right: monofun(f) & contlub(f) ==> cont(f) *)
(* ------------------------------------------------------------------------ *)
-qed_goalw "monocontlub2cont" thy [cont]
- "[|monofun(f);contlub(f)|] ==> cont(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)
- ]);
+val prems = goalw thy [cont]
+ "[|monofun(f);contlub(f)|] ==> cont(f)";
+by (cut_facts_tac prems 1);
+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 *)
(* ------------------------------------------------------------------------ *)
-qed_goal "binchain_cont" thy
-"[| cont(f); x << y |] ==> range(%i::nat. f(if i = 0 then x else y)) <<| f(y)"
-(fn prems =>
- [
- (cut_facts_tac prems 1),
- (rtac subst 1),
- (etac (contE RS spec RS mp) 2),
- (etac bin_chain 2),
- (res_inst_tac [("y","y")] arg_cong 1),
- (etac (lub_bin_chain RS thelubI) 1)
- ]);
+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 *)
(* ------------------------------------------------------------------------ *)
-qed_goalw "cont2mono" thy [monofun]
- "cont(f) ==> monofun(f)"
-(fn prems =>
- [
- (cut_facts_tac prems 1),
- (strip_tac 1),
- (res_inst_tac [("s","if 0 = 0 then x else y")] subst 1),
- (rtac (binchain_cont RS is_ub_lub) 2),
- (atac 2),
- (atac 2),
- (Simp_tac 1)
- ]);
+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) *)
(* ------------------------------------------------------------------------ *)
-qed_goalw "cont2contlub" thy [contlub]
- "cont(f) ==> contlub(f)"
-(fn prems =>
- [
- (cut_facts_tac prems 1),
- (strip_tac 1),
- (rtac (thelubI RS sym) 1),
- (etac (contE RS spec RS mp) 1),
- (atac 1)
- ]);
+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 *)
+(* monotone functions map finite chains to finite chains *)
(* ------------------------------------------------------------------------ *)
-qed_goalw "monofun_finch2finch" thy [finite_chain_def]
- "[| monofun f; finite_chain Y |] ==> finite_chain (%n. f (Y n))"
-(fn prems =>
- [
- cut_facts_tac prems 1,
- safe_tac HOL_cs,
- fast_tac (HOL_cs addSEs [ch2ch_monofun]) 1,
- fast_tac (HOL_cs addss (HOL_ss addsimps [max_in_chain_def])) 1
- ]);
+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 *)
+(* The same holds for continuous functions *)
(* ------------------------------------------------------------------------ *)
bind_thm ("cont_finch2finch", cont2mono RS monofun_finch2finch);
@@ -191,199 +155,181 @@
(* in both arguments *)
(* ------------------------------------------------------------------------ *)
-qed_goal "ch2ch_MF2L" thy
-"[|monofun(MF2); chain(F)|] ==> chain(%i. MF2 (F i) x)"
-(fn prems =>
- [
- (cut_facts_tac prems 1),
- (etac (ch2ch_monofun RS ch2ch_fun) 1),
- (atac 1)
- ]);
+val prems = goal thy
+"[|monofun(MF2); chain(F)|] ==> chain(%i. MF2 (F i) x)";
+by (cut_facts_tac prems 1);
+by (etac (ch2ch_monofun RS ch2ch_fun) 1);
+by (atac 1);
+qed "ch2ch_MF2L";
-qed_goal "ch2ch_MF2R" thy
-"[|monofun(MF2(f)); chain(Y)|] ==> chain(%i. MF2 f (Y i))"
-(fn prems =>
- [
- (cut_facts_tac prems 1),
- (etac ch2ch_monofun 1),
- (atac 1)
- ]);
+val prems = goal thy
+"[|monofun(MF2(f)); chain(Y)|] ==> chain(%i. MF2 f (Y i))";
+by (cut_facts_tac prems 1);
+by (etac ch2ch_monofun 1);
+by (atac 1);
+qed "ch2ch_MF2R";
-qed_goal "ch2ch_MF2LR" thy
+val prems = goal thy
"[|monofun(MF2); !f. monofun(MF2(f)); chain(F); chain(Y)|] ==> \
-\ chain(%i. MF2(F(i))(Y(i)))"
- (fn prems =>
- [
- (cut_facts_tac prems 1),
- (rtac chainI 1),
- (strip_tac 1 ),
- (rtac trans_less 1),
- (etac (ch2ch_MF2L RS chainE RS spec) 1),
- (atac 1),
- ((rtac (monofunE RS spec RS spec RS mp) 1) THEN (etac spec 1)),
- (etac (chainE RS spec) 1)
- ]);
+\ chain(%i. MF2(F(i))(Y(i)))";
+by (cut_facts_tac prems 1);
+by (rtac chainI 1);
+by (strip_tac 1 );
+by (rtac trans_less 1);
+by (etac (ch2ch_MF2L RS chainE RS spec) 1);
+by (atac 1);
+by ((rtac (monofunE RS spec RS spec RS mp) 1) THEN (etac spec 1));
+by (etac (chainE RS spec) 1);
+qed "ch2ch_MF2LR";
-qed_goal "ch2ch_lubMF2R" thy
+val prems = goal thy
"[|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))))"
-(fn prems =>
- [
- (cut_facts_tac prems 1),
- (rtac (lub_mono RS allI RS 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 (chainE RS spec) 1),
- (etac ch2ch_MF2L 1),
- (atac 1)
- ]);
+\ chain(%j. lub(range(%i. MF2 (F j) (Y i))))";
+by (cut_facts_tac prems 1);
+by (rtac (lub_mono RS allI 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 RS spec) 1);
+by (etac ch2ch_MF2L 1);
+by (atac 1);
+qed "ch2ch_lubMF2R";
-qed_goal "ch2ch_lubMF2L" thy
+val prems = goal thy
"[|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))))"
-(fn prems =>
- [
- (cut_facts_tac prems 1),
- (rtac (lub_mono RS allI RS chainI) 1),
- (etac ch2ch_MF2L 1),
- (atac 1),
- (etac ch2ch_MF2L 1),
- (atac 1),
- (strip_tac 1),
- (rtac (chainE RS spec) 1),
- ((rtac ch2ch_MF2R 1) THEN (etac spec 1)),
- (atac 1)
- ]);
+\ chain(%i. lub(range(%j. MF2 (F j) (Y i))))";
+by (cut_facts_tac prems 1);
+by (rtac (lub_mono RS allI 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 RS spec) 1);
+by ((rtac ch2ch_MF2R 1) THEN (etac spec 1));
+by (atac 1);
+qed "ch2ch_lubMF2L";
-qed_goal "lub_MF2_mono" thy
+val prems = goal thy
"[|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))))"
-(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)
- ]);
+\ monofun(% x. lub(range(% j. MF2 (F j) (x))))";
+by (cut_facts_tac prems 1);
+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";
-qed_goal "ex_lubMF2" thy
+val prems = goal thy
"[|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)))))"
- (fn prems =>
- [
- (cut_facts_tac prems 1),
- (rtac antisym_less 1),
- (rtac (ub_rangeI RSN (2,is_lub_thelub)) 1),
- (etac ch2ch_lubMF2R 1),
- (REPEAT (atac 1)),
- (strip_tac 1),
- (rtac lub_mono 1),
- ((rtac ch2ch_MF2R 1) THEN (etac spec 1)),
- (atac 1),
- (etac ch2ch_lubMF2L 1),
- (REPEAT (atac 1)),
- (strip_tac 1),
- (rtac is_ub_thelub 1),
- (etac ch2ch_MF2L 1),
- (atac 1),
- (rtac (ub_rangeI RSN (2,is_lub_thelub)) 1),
- (etac ch2ch_lubMF2L 1),
- (REPEAT (atac 1)),
- (strip_tac 1),
- (rtac lub_mono 1),
- (etac ch2ch_MF2L 1),
- (atac 1),
- (etac ch2ch_lubMF2R 1),
- (REPEAT (atac 1)),
- (strip_tac 1),
- (rtac is_ub_thelub 1),
- ((rtac ch2ch_MF2R 1) THEN (etac spec 1)),
- (atac 1)
- ]);
+\ lub(range(%i. lub(range(%j. MF2(F j) (Y i)))))";
+by (cut_facts_tac prems 1);
+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";
-qed_goal "diag_lubMF2_1" thy
+val prems = goal thy
"[|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))))"
- (fn prems =>
- [
- (cut_facts_tac prems 1),
- (rtac antisym_less 1),
- (rtac (ub_rangeI RSN (2,is_lub_thelub)) 1),
- (etac ch2ch_lubMF2L 1),
- (REPEAT (atac 1)),
- (strip_tac 1 ),
- (rtac lub_mono3 1),
- (etac ch2ch_MF2L 1),
- (REPEAT (atac 1)),
- (etac ch2ch_MF2LR 1),
- (REPEAT (atac 1)),
- (rtac allI 1),
- (res_inst_tac [("m","i"),("n","ia")] nat_less_cases 1),
- (res_inst_tac [("x","ia")] exI 1),
- (rtac (chain_mono RS mp) 1),
- (etac allE 1),
- (etac ch2ch_MF2R 1),
- (REPEAT (atac 1)),
- (hyp_subst_tac 1),
- (res_inst_tac [("x","ia")] exI 1),
- (rtac refl_less 1),
- (res_inst_tac [("x","i")] exI 1),
- (rtac (chain_mono RS mp) 1),
- (etac ch2ch_MF2L 1),
- (REPEAT (atac 1)),
- (rtac lub_mono 1),
- (etac ch2ch_MF2LR 1),
- (REPEAT(atac 1)),
- (etac ch2ch_lubMF2L 1),
- (REPEAT (atac 1)),
- (strip_tac 1 ),
- (rtac is_ub_thelub 1),
- (etac ch2ch_MF2L 1),
- (atac 1)
- ]);
+\ lub(range(%i. MF2(FY(i))(TY(i))))";
+by (cut_facts_tac prems 1);
+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 RS mp) 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 RS mp) 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";
-qed_goal "diag_lubMF2_2" thy
+val prems = goal thy
"[|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))))"
- (fn prems =>
- [
- (cut_facts_tac prems 1),
- (rtac trans 1),
- (rtac ex_lubMF2 1),
- (REPEAT (atac 1)),
- (etac diag_lubMF2_1 1),
- (REPEAT (atac 1))
- ]);
+\ lub(range(%i. MF2(FY(i))(TY(i))))";
+by (cut_facts_tac prems 1);
+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";
(* ------------------------------------------------------------------------ *)
@@ -391,59 +337,51 @@
(* in both arguments *)
(* ------------------------------------------------------------------------ *)
-qed_goal "contlub_CF2" thy
+val prems = 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))))"
- (fn prems =>
- [
- (cut_facts_tac prems 1),
- (stac ((hd prems) RS cont2contlub RS contlubE RS spec RS mp) 1),
- (atac 1),
- (stac thelub_fun 1),
- (rtac ((hd prems) RS cont2mono RS ch2ch_monofun) 1),
- (atac 1),
- (rtac trans 1),
- (rtac (((hd (tl prems)) RS spec RS cont2contlub) RS contlubE RS spec RS mp RS ext RS arg_cong RS arg_cong) 1),
- (atac 1),
- (rtac diag_lubMF2_2 1),
- (etac cont2mono 1),
- (rtac allI 1),
- (etac allE 1),
- (etac cont2mono 1),
- (REPEAT (atac 1))
- ]);
+\ CF2(lub(range(FY)))(lub(range(TY))) = lub(range(%i. CF2(FY(i))(TY(i))))";
+by (cut_facts_tac prems 1);
+by (stac ((hd prems) RS cont2contlub RS contlubE RS spec RS mp) 1);
+by (atac 1);
+by (stac thelub_fun 1);
+by (rtac ((hd prems) RS cont2mono RS ch2ch_monofun) 1);
+by (atac 1);
+by (rtac trans 1);
+by (rtac (((hd (tl prems)) RS spec RS cont2contlub) RS contlubE RS spec RS mp RS ext RS arg_cong RS arg_cong) 1);
+by (atac 1);
+by (rtac diag_lubMF2_2 1);
+by (etac cont2mono 1);
+by (rtac allI 1);
+by (etac allE 1);
+by (etac cont2mono 1);
+by (REPEAT (atac 1));
+qed "contlub_CF2";
(* ------------------------------------------------------------------------ *)
(* The following results are about application for functions in 'a=>'b *)
(* ------------------------------------------------------------------------ *)
-qed_goal "monofun_fun_fun" thy
- "f1 << f2 ==> f1(x) << f2(x)"
-(fn prems =>
- [
- (cut_facts_tac prems 1),
- (etac (less_fun RS iffD1 RS spec) 1)
- ]);
+val prems = goal thy
+ "f1 << f2 ==> f1(x) << f2(x)";
+by (cut_facts_tac prems 1);
+by (etac (less_fun RS iffD1 RS spec) 1);
+qed "monofun_fun_fun";
-qed_goal "monofun_fun_arg" 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 prems = goal thy
+ "[|monofun(f); x1 << x2|] ==> f(x1) << f(x2)";
+by (cut_facts_tac prems 1);
+by (etac (monofunE RS spec RS spec RS mp) 1);
+by (atac 1);
+qed "monofun_fun_arg";
-qed_goal "monofun_fun" 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)
- ]);
+val prems = goal thy
+"[|monofun(f1); monofun(f2); f1 << f2; x1 << x2|] ==> f1(x1) << f2(x2)";
+by (cut_facts_tac prems 1);
+by (rtac trans_less 1);
+by (etac monofun_fun_arg 1);
+by (atac 1);
+by (etac monofun_fun_fun 1);
+qed "monofun_fun";
(* ------------------------------------------------------------------------ *)
@@ -451,147 +389,115 @@
(* continuity *)
(* ------------------------------------------------------------------------ *)
-qed_goal "mono2mono_MF1L" 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 prems = goal thy
+ "[|monofun(c1)|] ==> monofun(%x. c1 x y)";
+by (cut_facts_tac prems 1);
+by (rtac monofunI 1);
+by (strip_tac 1);
+by (etac (monofun_fun_arg RS monofun_fun_fun) 1);
+by (atac 1);
+qed "mono2mono_MF1L";
-qed_goal "cont2cont_CF1L" thy
- "[|cont(c1)|] ==> cont(%x. c1 x y)"
-(fn prems =>
- [
- (cut_facts_tac prems 1),
- (rtac monocontlub2cont 1),
- (etac (cont2mono RS mono2mono_MF1L) 1),
- (rtac contlubI 1),
- (strip_tac 1),
- (rtac ((hd prems) RS cont2contlub RS
- contlubE RS spec RS mp RS ssubst) 1),
- (atac 1),
- (stac thelub_fun 1),
- (rtac ch2ch_monofun 1),
- (etac cont2mono 1),
- (atac 1),
- (rtac refl 1)
- ]);
+val prems = goal thy
+ "[|cont(c1)|] ==> cont(%x. c1 x y)";
+by (cut_facts_tac prems 1);
+by (rtac monocontlub2cont 1);
+by (etac (cont2mono RS mono2mono_MF1L) 1);
+by (rtac contlubI 1);
+by (strip_tac 1);
+by (rtac ((hd prems) RS 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" ***********)
-qed_goal "mono2mono_MF1L_rev" 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 prems = goal thy
+ "!y. monofun(%x. c1 x y) ==> monofun(c1)";
+by (cut_facts_tac prems 1);
+by (rtac monofunI 1);
+by (strip_tac 1);
+by (rtac (less_fun RS iffD2) 1);
+by (strip_tac 1);
+by (rtac ((hd prems) RS spec RS monofunE RS spec RS spec RS mp) 1);
+by (atac 1);
+qed "mono2mono_MF1L_rev";
-qed_goal "cont2cont_CF1L_rev" thy
- "!y. cont(%x. c1 x y) ==> cont(c1)"
-(fn prems =>
- [
- (cut_facts_tac prems 1),
- (rtac monocontlub2cont 1),
- (rtac (cont2mono RS allI RS mono2mono_MF1L_rev ) 1),
- (etac spec 1),
- (rtac contlubI 1),
- (strip_tac 1),
- (rtac ext 1),
- (stac thelub_fun 1),
- (rtac (cont2mono RS allI RS mono2mono_MF1L_rev RS ch2ch_monofun) 1),
- (etac spec 1),
- (atac 1),
- (rtac
- ((hd prems) RS spec RS cont2contlub RS contlubE RS spec RS mp) 1),
- (atac 1)
- ]);
+val prems = goal thy
+ "!y. cont(%x. c1 x y) ==> cont(c1)";
+by (cut_facts_tac prems 1);
+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 (rtac ((hd prems) RS spec RS cont2contlub RS contlubE RS spec RS mp) 1);
+by (atac 1);
+qed "cont2cont_CF1L_rev";
(* ------------------------------------------------------------------------ *)
(* What D.A.Schmidt calls continuity of abstraction *)
(* never used here *)
(* ------------------------------------------------------------------------ *)
-qed_goal "contlub_abstraction" thy
+val prems = goal thy
"[|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)))"
- (fn prems =>
- [
- (cut_facts_tac prems 1),
- (rtac trans 1),
- (rtac (cont2contlub RS contlubE RS spec RS mp) 2),
- (atac 3),
- (etac cont2cont_CF1L_rev 2),
- (rtac ext 1),
- (rtac (cont2contlub RS contlubE RS spec RS mp RS sym) 1),
- (etac spec 1),
- (atac 1)
- ]);
+\ (%y. lub(range(%i. c (Y i) y))) = (lub(range(%i.%y. c (Y i) y)))";
+by (cut_facts_tac prems 1);
+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";
-qed_goal "mono2mono_app" thy
+val prems = goal 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)
- ]);
+\ monofun(%x.(ft(x))(tt(x)))";
+by (cut_facts_tac prems 1);
+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";
-qed_goal "cont2contlub_app" thy
-"[|cont(ft);!x. cont(ft(x));cont(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),
- (etac cont2contlub 1),
- (atac 1),
- (rtac contlub_CF2 1),
- (REPEAT (atac 1)),
- (etac (cont2mono RS ch2ch_monofun) 1),
- (atac 1)
- ]);
+val prems = goal thy
+"[|cont(ft);!x. cont(ft(x));cont(tt)|] ==> contlub(%x.(ft(x))(tt(x)))";
+by (cut_facts_tac prems 1);
+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";
-qed_goal "cont2cont_app" thy
-"[|cont(ft);!x. cont(ft(x));cont(tt)|] ==>\
-\ cont(%x.(ft(x))(tt(x)))"
- (fn prems =>
- [
- (rtac monocontlub2cont 1),
- (rtac mono2mono_app 1),
- (rtac cont2mono 1),
- (resolve_tac prems 1),
- (strip_tac 1),
- (rtac cont2mono 1),
- (cut_facts_tac prems 1),
- (etac spec 1),
- (rtac cont2mono 1),
- (resolve_tac prems 1),
- (rtac cont2contlub_app 1),
- (resolve_tac prems 1),
- (resolve_tac prems 1),
- (resolve_tac prems 1)
- ]);
+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));
@@ -603,69 +509,54 @@
(* The identity function is continuous *)
(* ------------------------------------------------------------------------ *)
-qed_goal "cont_id" thy "cont(% x. x)"
- (fn prems =>
- [
- (rtac contI 1),
- (strip_tac 1),
- (etac thelubE 1),
- (rtac refl 1)
- ]);
+val prems = goal thy "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 *)
(* ------------------------------------------------------------------------ *)
-qed_goalw "cont_const" thy [cont] "cont(%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 prems = goalw thy [cont] "cont(%x. c)";
+by (strip_tac 1);
+by (rtac is_lubI 1);
+by (rtac conjI 1);
+by (rtac ub_rangeI 1);
+by (strip_tac 1);
+by (rtac refl_less 1);
+by (strip_tac 1);
+by (dtac ub_rangeE 1);
+by (etac spec 1);
+qed "cont_const";
-qed_goal "cont2cont_app3" thy
- "[|cont(f);cont(t) |] ==> cont(%x. f(t(x)))"
- (fn prems =>
- [
- (cut_facts_tac prems 1),
- (rtac cont2cont_app2 1),
- (rtac cont_const 1),
- (atac 1),
- (atac 1)
- ]);
+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 *)
(* ------------------------------------------------------------------------ *)
-qed_goal "CfunI" thy "?x:Collect cont"
- (fn prems =>
- [
- (rtac CollectI 1),
- (rtac cont_const 1)
- ]);
+val prems = goal thy "?x:Collect cont";
+by (rtac CollectI 1);
+by (rtac cont_const 1);
+qed "CfunI";
(* ------------------------------------------------------------------------ *)
-(* some properties of flat *)
+(* some properties of flat *)
(* ------------------------------------------------------------------------ *)
-qed_goalw "flatdom2monofun" thy [monofun]
- "f UU = UU ==> monofun (f::'a::flat=>'b::pcpo)"
-(fn prems =>
- [
- cut_facts_tac prems 1,
- strip_tac 1,
- dtac (ax_flat RS spec RS spec RS mp) 1,
- fast_tac ((HOL_cs addss (simpset() addsimps [minimal]))) 1
- ]);
+val prems = goalw thy [monofun]
+ "f UU = UU ==> monofun (f::'a::flat=>'b::pcpo)";
+by (cut_facts_tac prems 1);
+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)";