(* Title: HOLCF/Cont.ML
ID: $Id$
Author: Franz Regensburger
Copyright 1993 Technische Universitaet Muenchen
Lemmas for Cont.thy
*)
open Cont;
(* ------------------------------------------------------------------------ *)
(* access to definition *)
(* ------------------------------------------------------------------------ *)
qed_goalw "contlubI" 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)
]);
qed_goalw "contlubE" 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)
]);
qed_goalw "contI" thy [cont]
"! Y. is_chain(Y) --> range(% i. f(Y(i))) <<| f(lub(range(Y))) ==> cont(f)"
(fn prems =>
[
(cut_facts_tac prems 1),
(atac 1)
]);
qed_goalw "contE" thy [cont]
"cont(f) ==> ! Y. is_chain(Y) --> range(% i. f(Y(i))) <<| f(lub(range(Y)))"
(fn prems =>
[
(cut_facts_tac prems 1),
(atac 1)
]);
qed_goalw "monofunI" thy [monofun]
"! x y. x << y --> f(x) << f(y) ==> monofun(f)"
(fn prems =>
[
(cut_facts_tac prems 1),
(atac 1)
]);
qed_goalw "monofunE" 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) <==> cont(f) *)
(* ------------------------------------------------------------------------ *)
(* ------------------------------------------------------------------------ *)
(* monotone functions map chains to chains *)
(* ------------------------------------------------------------------------ *)
qed_goal "ch2ch_monofun" 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 *)
(* ------------------------------------------------------------------------ *)
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)
]);
(* ------------------------------------------------------------------------ *)
(* 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)
]);
(* ------------------------------------------------------------------------ *)
(* first a lemma about binary chains *)
(* ------------------------------------------------------------------------ *)
qed_goal "binchain_cont" thy
"[| cont(f); x << y |] ==> range(%i. 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)
]);
(* ------------------------------------------------------------------------ *)
(* 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)
]);
(* ------------------------------------------------------------------------ *)
(* 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)
]);
(* ------------------------------------------------------------------------ *)
(* 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
]);
(* ------------------------------------------------------------------------ *)
(* 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 *)
(* ------------------------------------------------------------------------ *)
qed_goal "ch2ch_MF2L" thy
"[|monofun(MF2); 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)
]);
qed_goal "ch2ch_MF2R" thy
"[|monofun(MF2(f)); is_chain(Y)|] ==> is_chain(%i. MF2 f (Y i))"
(fn prems =>
[
(cut_facts_tac prems 1),
(etac ch2ch_monofun 1),
(atac 1)
]);
qed_goal "ch2ch_MF2LR" thy
"[|monofun(MF2); !f. monofun(MF2(f)); 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)
]);
qed_goal "ch2ch_lubMF2R" thy
"[|monofun(MF2::('a::po=>'b::po=>'c::cpo));\
\ !f. monofun(MF2(f)::('b::po=>'c::cpo));\
\ 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)
]);
qed_goal "ch2ch_lubMF2L" thy
"[|monofun(MF2::('a::po=>'b::po=>'c::cpo));\
\ !f. monofun(MF2(f)::('b::po=>'c::cpo));\
\ 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)
]);
qed_goal "lub_MF2_mono" thy
"[|monofun(MF2::('a::po=>'b::po=>'c::cpo));\
\ !f. monofun(MF2(f)::('b::po=>'c::cpo));\
\ 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)
]);
qed_goal "ex_lubMF2" thy
"[|monofun(MF2::('a::po=>'b::po=>'c::cpo));\
\ !f. monofun(MF2(f)::('b::po=>'c::cpo));\
\ 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 (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)
]);
qed_goal "diag_lubMF2_1" thy
"[|monofun(MF2::('a::po=>'b::po=>'c::cpo));\
\ !f. monofun(MF2(f)::('b::po=>'c::cpo));\
\ is_chain(FY);is_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)
]);
qed_goal "diag_lubMF2_2" thy
"[|monofun(MF2::('a::po=>'b::po=>'c::cpo));\
\ !f. monofun(MF2(f)::('b::po=>'c::cpo));\
\ is_chain(FY);is_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))
]);
(* ------------------------------------------------------------------------ *)
(* The following results are about a curried function that is continuous *)
(* in both arguments *)
(* ------------------------------------------------------------------------ *)
qed_goal "contlub_CF2" thy
"[|cont(CF2);!f. cont(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),
(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))
]);
(* ------------------------------------------------------------------------ *)
(* 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)
]);
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)
]);
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)
]);
(* ------------------------------------------------------------------------ *)
(* The following results are about the propagation of monotonicity and *)
(* 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)
]);
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)
]);
(********* 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)
]);
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)
]);
(* ------------------------------------------------------------------------ *)
(* What D.A.Schmidt calls continuity of abstraction *)
(* never used here *)
(* ------------------------------------------------------------------------ *)
qed_goal "contlub_abstraction" thy
"[|is_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)
]);
qed_goal "mono2mono_app" 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)
]);
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)
]);
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)
]);
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 *)
(* ------------------------------------------------------------------------ *)
qed_goal "cont_id" thy "cont(% x. x)"
(fn prems =>
[
(rtac contI 1),
(strip_tac 1),
(etac thelubE 1),
(rtac refl 1)
]);
(* ------------------------------------------------------------------------ *)
(* 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)
]);
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)
]);
(* ------------------------------------------------------------------------ *)
(* A non-emptyness result for Cfun *)
(* ------------------------------------------------------------------------ *)
qed_goal "CfunI" thy "?x:Collect cont"
(fn prems =>
[
(rtac CollectI 1),
(rtac cont_const 1)
]);
(* ------------------------------------------------------------------------ *)
(* 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
]);
qed_goal "chfindom_monofun2cont" thy "monofun f ==> cont(f::'a::chfin=>'b::pcpo)"
(fn prems =>
[
cut_facts_tac prems 1,
rtac monocontlub2cont 1,
atac 1,
rtac contlubI 1,
strip_tac 1,
forward_tac [chfin2finch] 1,
rtac antisym_less 1,
fast_tac ((HOL_cs addIs [is_ub_thelub,ch2ch_monofun])
addss (HOL_ss addsimps [finite_chain_def,maxinch_is_thelub])) 1,
dtac (monofun_finch2finch COMP swap_prems_rl) 1,
atac 1,
asm_full_simp_tac (HOL_ss addsimps [finite_chain_def]) 1,
etac conjE 1, etac exE 1,
asm_full_simp_tac (HOL_ss addsimps [maxinch_is_thelub]) 1,
etac (monofunE RS spec RS spec RS mp) 1,
etac is_ub_thelub 1
]);
bind_thm ("flatdom_strict2cont",flatdom2monofun RS chfindom_monofun2cont);
(* f UU = UU ==> cont (f::'a=>'b::pcpo)" *)