src/HOLCF/cont.ML

PureThy.smart_store_thms;

(*  Title: 	HOLCF/cont.ML
    ID:         $Id$
    Author: 	Franz Regensburger
    Copyright   1993 Technische Universitaet Muenchen

Lemmas for cont.thy 
*)

open Cont;

(* ------------------------------------------------------------------------ *)
(* access to definition                                                     *)
(* ------------------------------------------------------------------------ *)

val contlubI = prove_goalw Cont.thy [contlub]
	"! Y. is_chain(Y) --> f(lub(range(Y))) = lub(range(%i. f(Y(i))))==>\
\	 contlub(f)"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(atac 1)
	]);

val contlubE = prove_goalw Cont.thy [contlub]
	" contlub(f)==>\
\         ! Y. is_chain(Y) --> f(lub(range(Y))) = lub(range(%i. f(Y(i))))"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(atac 1)
	]);


val contXI = prove_goalw Cont.thy [contX]
 "! Y. is_chain(Y) --> range(% i.f(Y(i))) <<| f(lub(range(Y))) ==> contX(f)"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(atac 1)
	]);

val contXE = prove_goalw Cont.thy [contX]
 "contX(f) ==> ! Y. is_chain(Y) --> range(% i.f(Y(i))) <<| f(lub(range(Y)))"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(atac 1)
	]);


val monofunI = prove_goalw Cont.thy [monofun]
	"! x y. x << y --> f(x) << f(y) ==> monofun(f)"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(atac 1)
	]);

val monofunE = prove_goalw Cont.thy [monofun]
	"monofun(f) ==> ! x y. x << y --> f(x) << f(y)"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(atac 1)
	]);

(* ------------------------------------------------------------------------ *)
(* the main purpose of cont.thy is to show:                                 *)
(*              monofun(f) & contlub(f)  <==> contX(f)                      *)
(* ------------------------------------------------------------------------ *)

(* ------------------------------------------------------------------------ *)
(* monotone functions map chains to chains                                  *)
(* ------------------------------------------------------------------------ *)

val ch2ch_monofun= prove_goal Cont.thy 
	"[| monofun(f); is_chain(Y) |] ==> is_chain(%i. f(Y(i)))"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac is_chainI 1),
	(rtac allI 1),
	(etac (monofunE RS spec RS spec RS mp) 1),
	(etac (is_chainE RS spec) 1)
	]);

(* ------------------------------------------------------------------------ *)
(* monotone functions map upper bound to upper bounds                       *)
(* ------------------------------------------------------------------------ *)

val ub2ub_monofun = prove_goal Cont.thy 
 "[| monofun(f); range(Y) <| u|]  ==> range(%i.f(Y(i))) <| f(u)"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac ub_rangeI 1),
	(rtac allI 1),
	(etac (monofunE RS spec RS spec RS mp) 1),
	(etac (ub_rangeE RS spec) 1)
	]);

(* ------------------------------------------------------------------------ *)
(* left to right: monofun(f) & contlub(f)  ==> contX(f)                     *)
(* ------------------------------------------------------------------------ *)

val monocontlub2contX = prove_goalw Cont.thy [contX]
	"[|monofun(f);contlub(f)|] ==> contX(f)"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(strip_tac 1),
	(rtac thelubE 1),
	(etac ch2ch_monofun 1),
	(atac 1),
	(etac (contlubE RS spec RS mp RS sym) 1),
	(atac 1)
	]);

(* ------------------------------------------------------------------------ *)
(* first a lemma about binary chains                                        *)
(* ------------------------------------------------------------------------ *)

val binchain_contX =  prove_goal Cont.thy
"[| contX(f); x << y |]  ==> range(%i. f(if(i = 0,x,y))) <<| f(y)"
(fn prems => 
	[
	(cut_facts_tac prems 1),
	(rtac subst 1), 
	(etac (contXE RS spec RS mp) 2),
	(etac bin_chain 2),
	(res_inst_tac [("y","y")] arg_cong 1),
	(etac (lub_bin_chain RS thelubI) 1)
	]);

(* ------------------------------------------------------------------------ *)
(* right to left: contX(f) ==> monofun(f) & contlub(f)                      *)
(* part1:         contX(f) ==> monofun(f                                    *)
(* ------------------------------------------------------------------------ *)

val contX2mono =  prove_goalw Cont.thy [monofun]
	"contX(f) ==> monofun(f)"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(strip_tac 1),
	(res_inst_tac [("s","if(0 = 0,x,y)")] subst 1),
	(rtac (binchain_contX RS is_ub_lub) 2),
	(atac 2),
	(atac 2),
	(simp_tac nat_ss 1)
	]);

(* ------------------------------------------------------------------------ *)
(* right to left: contX(f) ==> monofun(f) & contlub(f)                      *)
(* part2:         contX(f) ==>              contlub(f)                      *)
(* ------------------------------------------------------------------------ *)

val contX2contlub = prove_goalw Cont.thy [contlub]
	"contX(f) ==> contlub(f)"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(strip_tac 1),
	(rtac (thelubI RS sym) 1),
	(etac (contXE RS spec RS mp) 1),
	(atac 1)
	]);

(* ------------------------------------------------------------------------ *)
(* The following results are about a curried function that is monotone      *)
(* in both arguments                                                        *)
(* ------------------------------------------------------------------------ *)

val ch2ch_MF2L = prove_goal Cont.thy 
"[|monofun(MF2::('a::po=>'b::po=>'c::po));\
\	is_chain(F)|] ==> is_chain(%i. MF2(F(i),x))"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(etac (ch2ch_monofun RS ch2ch_fun) 1),
	(atac 1)
	]);


val ch2ch_MF2R = prove_goal Cont.thy "[|monofun(MF2(f)::('b::po=>'c::po));\
\	is_chain(Y)|] ==> is_chain(%i. MF2(f,Y(i)))"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(etac ch2ch_monofun 1),
	(atac 1)
	]);

val ch2ch_MF2LR = prove_goal Cont.thy 
"[|monofun(MF2::('a::po=>'b::po=>'c::pcpo));\
\  !f.monofun(MF2(f)::('b::po=>'c::pcpo));\
\	is_chain(F); is_chain(Y)|] ==> \
\  is_chain(%i. MF2(F(i))(Y(i)))"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac is_chainI 1),
	(strip_tac 1 ),
	(rtac trans_less 1),
	(etac (ch2ch_MF2L RS is_chainE RS spec) 1),
	(atac 1),
	((rtac (monofunE RS spec RS spec RS mp) 1) THEN (etac spec 1)),
	(etac (is_chainE RS spec) 1)
	]);

val ch2ch_lubMF2R = prove_goal Cont.thy 
"[|monofun(MF2::('a::po=>'b::po=>'c::pcpo));\
\  !f.monofun(MF2(f)::('b::po=>'c::pcpo));\
\	is_chain(F);is_chain(Y)|] ==> \
\	is_chain(%j. lub(range(%i. MF2(F(j),Y(i)))))"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac (lub_mono RS allI RS is_chainI) 1),
	((rtac ch2ch_MF2R 1) THEN (etac spec 1)),
	(atac 1),
	((rtac ch2ch_MF2R 1) THEN (etac spec 1)),
	(atac 1),
	(strip_tac 1),
	(rtac (is_chainE RS spec) 1),
	(etac ch2ch_MF2L 1),
	(atac 1)
	]);


val ch2ch_lubMF2L = prove_goal Cont.thy 
"[|monofun(MF2::('a::po=>'b::po=>'c::pcpo));\
\  !f.monofun(MF2(f)::('b::po=>'c::pcpo));\
\	is_chain(F);is_chain(Y)|] ==> \
\	is_chain(%i. lub(range(%j. MF2(F(j),Y(i)))))"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac (lub_mono RS allI RS is_chainI) 1),
	(etac ch2ch_MF2L 1),
	(atac 1),
	(etac ch2ch_MF2L 1),
	(atac 1),
	(strip_tac 1),
	(rtac (is_chainE RS spec) 1),
	((rtac ch2ch_MF2R 1) THEN (etac spec 1)),
	(atac 1)
	]);


val lub_MF2_mono = prove_goal Cont.thy 
"[|monofun(MF2::('a::po=>'b::po=>'c::pcpo));\
\  !f.monofun(MF2(f)::('b::po=>'c::pcpo));\
\	is_chain(F)|] ==> \
\	monofun(% x.lub(range(% j.MF2(F(j),x))))"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac monofunI 1),
	(strip_tac 1),
	(rtac lub_mono 1),
	(etac ch2ch_MF2L 1),
	(atac 1),
	(etac ch2ch_MF2L 1),
	(atac 1),
	(strip_tac 1),
	((rtac (monofunE RS spec RS spec RS mp) 1) THEN (etac spec 1)),
	(atac 1)
	]);


val ex_lubMF2 = prove_goal Cont.thy 
"[|monofun(MF2::('a::po=>'b::po=>'c::pcpo));\
\  !f.monofun(MF2(f)::('b::po=>'c::pcpo));\
\	is_chain(F); is_chain(Y)|] ==> \
\		lub(range(%j. lub(range(%i. MF2(F(j),Y(i)))))) =\
\		lub(range(%i. lub(range(%j. MF2(F(j),Y(i))))))"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac antisym_less 1),
	(rtac is_lub_thelub 1),
	(etac ch2ch_lubMF2R 1),
	(atac 1),(atac 1),(atac 1),
	(rtac ub_rangeI 1),
	(strip_tac 1),
	(rtac lub_mono 1),
	((rtac ch2ch_MF2R 1) THEN (etac spec 1)),
	(atac 1),
	(etac ch2ch_lubMF2L 1),
	(atac 1),(atac 1),(atac 1),
	(strip_tac 1),
	(rtac is_ub_thelub 1),
	(etac ch2ch_MF2L 1),(atac 1),
	(rtac is_lub_thelub 1),
	(etac ch2ch_lubMF2L 1),
	(atac 1),(atac 1),(atac 1),
	(rtac ub_rangeI 1),
	(strip_tac 1),
	(rtac lub_mono 1),
	(etac ch2ch_MF2L 1),(atac 1),
	(etac ch2ch_lubMF2R 1),
	(atac 1),(atac 1),(atac 1),
	(strip_tac 1),
	(rtac is_ub_thelub 1),
	((rtac ch2ch_MF2R 1) THEN (etac spec 1)),
	(atac 1)
	]);

(* ------------------------------------------------------------------------ *)
(* The following results are about a curried function that is continuous    *)
(* in both arguments                                                        *)
(* ------------------------------------------------------------------------ *)

val diag_lubCF2_1 = prove_goal Cont.thy 
"[|contX(CF2);!f.contX(CF2(f));is_chain(FY);is_chain(TY)|] ==>\
\ lub(range(%i. lub(range(%j. CF2(FY(j))(TY(i)))))) =\
\ lub(range(%i. CF2(FY(i))(TY(i))))"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac antisym_less 1),
	(rtac is_lub_thelub 1),
	(rtac ch2ch_lubMF2L 1),
	(rtac contX2mono 1),
	(atac 1),
	(rtac allI 1),
	(rtac contX2mono 1),
	(etac spec 1),
	(atac 1),
	(atac 1),
	(rtac ub_rangeI 1),
	(strip_tac 1 ),
	(rtac is_lub_thelub 1),
	((rtac ch2ch_MF2L 1) THEN (rtac contX2mono 1) THEN (atac 1)),
	(atac 1),
	(rtac ub_rangeI 1),
	(strip_tac 1 ),
	(res_inst_tac [("m","i"),("n","ia")] nat_less_cases 1),
	(rtac trans_less 1),
	(rtac is_ub_thelub 2),
	(rtac (chain_mono RS mp) 1),
	((rtac ch2ch_MF2R 1) THEN (rtac contX2mono 1) THEN (etac spec 1)),
	(atac 1),
	(atac 1),
	((rtac ch2ch_MF2LR 1) THEN (etac contX2mono 1)),
	(rtac allI 1),
	((rtac contX2mono 1) THEN (etac spec 1)),
	(atac 1),
	(atac 1),
	(hyp_subst_tac 1),
	(rtac is_ub_thelub 1),
	((rtac ch2ch_MF2LR 1) THEN (etac contX2mono 1)),
	(rtac allI 1),
	((rtac contX2mono 1) THEN (etac spec 1)),
	(atac 1),
	(atac 1),
	(rtac trans_less 1),
	(rtac is_ub_thelub 2),
	(res_inst_tac [("x1","ia")] (chain_mono RS mp) 1),
	((rtac ch2ch_MF2L 1) THEN (etac contX2mono 1)),
	(atac 1),
	(atac 1),
	((rtac ch2ch_MF2LR 1) THEN (etac contX2mono 1)),
	(rtac allI 1),
	((rtac contX2mono 1) THEN (etac spec 1)),
	(atac 1),
	(atac 1),
	(rtac lub_mono 1),
	((rtac ch2ch_MF2LR 1) THEN (etac contX2mono 1)),
	(rtac allI 1),
	((rtac contX2mono 1) THEN (etac spec 1)),
	(atac 1),
	(atac 1),
	(rtac ch2ch_lubMF2L 1),
	(rtac contX2mono 1),
	(atac 1),
	(rtac allI 1),
	((rtac contX2mono 1) THEN (etac spec 1)),
	(atac 1),
	(atac 1),
	(strip_tac 1 ),
	(rtac is_ub_thelub 1),
	((rtac ch2ch_MF2L 1) THEN (etac contX2mono 1)),
	(atac 1)
	]);


val diag_lubCF2_2 = prove_goal Cont.thy 
"[|contX(CF2);!f.contX(CF2(f));is_chain(FY);is_chain(TY)|] ==>\
\ lub(range(%j. lub(range(%i. CF2(FY(j))(TY(i)))))) =\
\ lub(range(%i. CF2(FY(i))(TY(i))))"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac trans 1),
	(rtac ex_lubMF2 1),
	(rtac ((hd prems) RS contX2mono) 1), 
	(rtac allI 1),
	(rtac (((hd (tl prems)) RS spec RS contX2mono)) 1),
	(atac 1),
	(atac 1),
	(rtac diag_lubCF2_1 1),
	(atac 1),
	(atac 1),
	(atac 1),
	(atac 1)
	]);


val contlub_CF2 = prove_goal Cont.thy 
"[|contX(CF2);!f.contX(CF2(f));is_chain(FY);is_chain(TY)|] ==>\
\ CF2(lub(range(FY)))(lub(range(TY))) = lub(range(%i.CF2(FY(i))(TY(i))))"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac ((hd prems) RS contX2contlub RS contlubE RS 
		spec RS mp RS ssubst) 1),
	(atac 1),
	(rtac (thelub_fun RS ssubst) 1),
	(rtac ((hd prems) RS contX2mono RS ch2ch_monofun) 1), 
	(atac 1),
	(rtac trans 1),
	(rtac (((hd (tl prems)) RS spec RS contX2contlub) RS 
	contlubE RS spec RS mp RS ext RS arg_cong RS arg_cong) 1),
	(atac 1),
	(rtac diag_lubCF2_2 1),
	(atac 1),
	(atac 1),
	(atac 1),
	(atac 1)
	]);

(* ------------------------------------------------------------------------ *)
(* The following results are about application for functions in 'a=>'b      *)
(* ------------------------------------------------------------------------ *)

val monofun_fun_fun = prove_goal Cont.thy 
	"f1 << f2 ==> f1(x) << f2(x)"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(etac (less_fun RS iffD1 RS spec) 1)
	]);

val monofun_fun_arg = prove_goal Cont.thy 
	"[|monofun(f); x1 << x2|] ==> f(x1) << f(x2)"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(etac (monofunE RS spec RS spec RS mp) 1),
	(atac 1)
	]);

val monofun_fun = prove_goal Cont.thy 
"[|monofun(f1); monofun(f2); f1 << f2; x1 << x2|] ==> f1(x1) << f2(x2)"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac trans_less 1),
	(etac monofun_fun_arg 1),
	(atac 1),
	(etac monofun_fun_fun 1)
	]);


(* ------------------------------------------------------------------------ *)
(* The following results are about the propagation of monotonicity and      *)
(* continuity                                                               *)
(* ------------------------------------------------------------------------ *)

val mono2mono_MF1L = prove_goal Cont.thy 
	"[|monofun(c1)|] ==> monofun(%x. c1(x,y))"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac monofunI 1),
	(strip_tac 1),
	(etac (monofun_fun_arg RS monofun_fun_fun) 1),
	(atac 1)
	]);

val contX2contX_CF1L = prove_goal Cont.thy 
	"[|contX(c1)|] ==> contX(%x. c1(x,y))"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac monocontlub2contX 1),
	(etac (contX2mono RS mono2mono_MF1L) 1),
	(rtac contlubI 1),
	(strip_tac 1),
	(rtac ((hd prems) RS contX2contlub RS 
		contlubE RS spec RS mp RS ssubst) 1),
	(atac 1),
	(rtac (thelub_fun RS ssubst) 1),
	(rtac ch2ch_monofun 1),
	(etac contX2mono 1),
	(atac 1),
	(rtac refl 1)
	]);

(*********  Note "(%x.%y.c1(x,y)) = c1" ***********)

val mono2mono_MF1L_rev = prove_goal Cont.thy
	"!y.monofun(%x.c1(x,y)) ==> monofun(c1)"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac monofunI 1),
	(strip_tac 1),
	(rtac (less_fun RS iffD2) 1),
	(strip_tac 1),
	(rtac ((hd prems) RS spec RS monofunE RS spec RS spec RS mp) 1),
	(atac 1)
	]);

val contX2contX_CF1L_rev = prove_goal Cont.thy
	"!y.contX(%x.c1(x,y)) ==> contX(c1)"
(fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac monocontlub2contX 1),
	(rtac (contX2mono RS allI RS mono2mono_MF1L_rev ) 1),
	(etac spec 1),
	(rtac contlubI 1),
	(strip_tac 1),
	(rtac ext 1),
	(rtac (thelub_fun RS ssubst) 1),
	(rtac (contX2mono RS allI RS mono2mono_MF1L_rev RS ch2ch_monofun) 1),
	(etac spec 1),
	(atac 1),
	(rtac 
	((hd prems) RS spec RS contX2contlub RS contlubE RS spec RS mp) 1),
	(atac 1)
	]);


(* ------------------------------------------------------------------------ *)
(* What D.A.Schmidt calls continuity of abstraction                         *)
(* never used here                                                          *)
(* ------------------------------------------------------------------------ *)

val contlub_abstraction = prove_goal Cont.thy
"[|is_chain(Y::nat=>'a);!y.contX(%x.(c::'a=>'b=>'c)(x,y))|] ==>\
\ (%y.lub(range(%i.c(Y(i),y)))) = (lub(range(%i.%y.c(Y(i),y))))"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac trans 1),
	(rtac (contX2contlub RS contlubE RS spec RS mp) 2),
	(atac 3),
	(etac contX2contX_CF1L_rev 2),
	(rtac ext 1), 
	(rtac (contX2contlub RS contlubE RS spec RS mp RS sym) 1),
	(etac spec 1),
	(atac 1)
	]);


val mono2mono_app = prove_goal Cont.thy 
"[|monofun(ft);!x.monofun(ft(x));monofun(tt)|] ==>\
\	 monofun(%x.(ft(x))(tt(x)))"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac monofunI 1),
	(strip_tac 1),
	(res_inst_tac [("f1.0","ft(x)"),("f2.0","ft(y)")] monofun_fun 1),
	(etac spec 1),
	(etac spec 1),
	(etac (monofunE RS spec RS spec RS mp) 1),
	(atac 1),
	(etac (monofunE RS spec RS spec RS mp) 1),
	(atac 1)
	]);

val contX2contlub_app = prove_goal Cont.thy 
"[|contX(ft);!x.contX(ft(x));contX(tt)|] ==>\
\	 contlub(%x.(ft(x))(tt(x)))"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac contlubI 1),
	(strip_tac 1),
	(res_inst_tac [("f3","tt")] (contlubE RS spec RS mp RS ssubst) 1),
	(rtac contX2contlub 1),
	(resolve_tac prems 1),
	(atac 1),
	(rtac contlub_CF2 1),
	(resolve_tac prems 1),
	(resolve_tac prems 1),
	(atac 1),
	(rtac (contX2mono RS ch2ch_monofun) 1),
	(resolve_tac prems 1),
	(atac 1)
	]);


val contX2contX_app = prove_goal Cont.thy 
"[|contX(ft);!x.contX(ft(x));contX(tt)|] ==>\
\	 contX(%x.(ft(x))(tt(x)))"
 (fn prems =>
	[
	(rtac monocontlub2contX 1),
	(rtac mono2mono_app 1),
	(rtac contX2mono 1),
	(resolve_tac prems 1),
	(strip_tac 1),
	(rtac contX2mono 1),
	(cut_facts_tac prems 1),
	(etac spec 1),
	(rtac contX2mono 1),
	(resolve_tac prems 1),
	(rtac contX2contlub_app 1),
	(resolve_tac prems 1),
	(resolve_tac prems 1),
	(resolve_tac prems 1)
	]);


val contX2contX_app2 = (allI RSN (2,contX2contX_app));
(*  [| contX(?ft); !!x. contX(?ft(x)); contX(?tt) |] ==>                 *)
(*                                      contX(%x. ?ft(x,?tt(x)))         *)


(* ------------------------------------------------------------------------ *)
(* The identity function is continuous                                      *)
(* ------------------------------------------------------------------------ *)

val contX_id = prove_goal Cont.thy "contX(% x.x)"
 (fn prems =>
	[
	(rtac contXI 1),
	(strip_tac 1),
	(etac thelubE 1),
	(rtac refl 1)
	]);



(* ------------------------------------------------------------------------ *)
(* constant functions are continuous                                        *)
(* ------------------------------------------------------------------------ *)

val contX_const = prove_goalw Cont.thy [contX] "contX(%x.c)"
 (fn prems =>
	[
	(strip_tac 1),
	(rtac is_lubI 1),
	(rtac conjI 1),
	(rtac ub_rangeI 1),
	(strip_tac 1),
	(rtac refl_less 1),
	(strip_tac 1),
	(dtac ub_rangeE 1),
	(etac spec 1)
	]);


val contX2contX_app3 = prove_goal Cont.thy 
 "[|contX(f);contX(t) |] ==> contX(%x. f(t(x)))"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac contX2contX_app2 1),
	(rtac contX_const 1),
	(atac 1),
	(atac 1)
	]);