src/HOLCF/Sprod2.ML

Changes of HOLCF from Oscar Slotosch:

1. axclass instead of class
* less instead of
less_fun,
less_cfun,
less_sprod,
less_cprod,
less_ssum,
less_up,
less_lift
* @x.!y.x<<y instead of UUU instead of
UU_fun, UU_cfun, ...
* no witness type void needed (eliminated Void.thy.Void.ML)
* inst_<typ>_<class> derived as theorems

2. improved some proves on less_sprod and less_cprod
* eliminated the following theorems
Sprod1.ML: less_sprod1a
Sprod1.ML: less_sprod1b
Sprod1.ML: less_sprod2a
Sprod1.ML: less_sprod2b
Sprod1.ML: less_sprod2c
Sprod2.ML: less_sprod3a
Sprod2.ML: less_sprod3b
Sprod2.ML: less_sprod4b
Sprod2.ML: less_sprod4c
Sprod3.ML: less_sprod5b
Sprod3.ML: less_sprod5c
Cprod1.ML: less_cprod1b
Cprod1.ML: less_cprod2a
Cprod1.ML: less_cprod2b
Cprod1.ML: less_cprod2c
Cprod2.ML: less_cprod3a
Cprod2.ML: less_cprod3b

3. new classes:
* cpo<po,
* chfin<pcpo,
* flat<pcpo,
* derived: flat<chfin
to do: show instances for lift

4. Data Type One
* Used lift for the definition: one = unit lift
* Changed the constant one into ONE

5. Data Type Tr
* Used lift for the definition: tr = bool lift
* adopted definitions of if,andalso,orelse,neg
* only one theory Tr.thy,Tr.ML instead of
Tr1.thy,Tr1.ML, Tr2.thy,Tr2.ML
* reintroduced ceils for =TT,=FF

6. typedef
* Using typedef instead of faking type definitions
to do: change fapp, fabs from Cfun1 to Rep_Cfun, Abs_Cfun

7. adopted examples and domain construct to theses changes

These changes eliminated all rules and arities from HOLCF

(*  Title:      HOLCF/Sprod2.ML
    ID:         $Id$
    Author:     Franz Regensburger
    Copyright   1993 Technische Universitaet Muenchen

Lemmas for Sprod2.thy
*)

open Sprod2;

(* for compatibility with old HOLCF-Version *)
qed_goal "inst_sprod_po" thy "(op <<)=(%x y.Isfst x<<Isfst y&Issnd x<<Issnd y)"
 (fn prems => 
        [
	(fold_goals_tac [po_def,less_sprod_def]),
	(rtac refl 1)
        ]);

(* ------------------------------------------------------------------------ *)
(* type sprod is pointed                                                    *)
(* ------------------------------------------------------------------------ *)

qed_goal "minimal_sprod" thy "Ispair UU UU << p"
(fn prems =>
        [
        (simp_tac(Sprod0_ss addsimps[inst_sprod_po,minimal])1)
        ]);

bind_thm ("UU_sprod_def",minimal_sprod RS minimal2UU RS sym);

qed_goal "least_sprod" thy "? x::'a**'b.!y.x<<y"
(fn prems =>
        [
        (res_inst_tac [("x","Ispair UU UU")] exI 1),
        (rtac (minimal_sprod RS allI) 1)
        ]);

(* ------------------------------------------------------------------------ *)
(* Ispair is monotone in both arguments                                     *)
(* ------------------------------------------------------------------------ *)

qed_goalw "monofun_Ispair1" Sprod2.thy [monofun] "monofun(Ispair)"
(fn prems =>
        [
        (strip_tac 1),
        (rtac (less_fun RS iffD2) 1),
        (strip_tac 1),
        (res_inst_tac [("Q","xa=UU")] (excluded_middle RS disjE) 1),
        (res_inst_tac [("Q","x=UU")] (excluded_middle RS disjE) 1),
        (forward_tac [notUU_I] 1),
        (atac 1),
        (REPEAT(asm_simp_tac(Sprod0_ss 
                addsimps[inst_sprod_po,refl_less,minimal]) 1))
        ]);

qed_goalw "monofun_Ispair2" Sprod2.thy [monofun] "monofun(Ispair(x))"
(fn prems =>
        [
        (strip_tac 1),
        (res_inst_tac [("Q","x=UU")] (excluded_middle RS disjE) 1),
        (res_inst_tac [("Q","xa=UU")] (excluded_middle RS disjE) 1),
        (forward_tac [notUU_I] 1),
        (atac 1),
        (REPEAT(asm_simp_tac(Sprod0_ss 
                addsimps[inst_sprod_po,refl_less,minimal]) 1))
        ]);


qed_goal " monofun_Ispair" Sprod2.thy 
 "[|x1<<x2; y1<<y2|] ==> Ispair x1 y1 << Ispair x2 y2"
(fn prems =>
        [
        (cut_facts_tac prems 1),
        (rtac trans_less 1),
        (rtac (monofun_Ispair1 RS monofunE RS spec RS spec RS mp RS 
        (less_fun RS iffD1 RS spec)) 1),
        (rtac (monofun_Ispair2 RS monofunE RS spec RS spec RS mp) 2),
        (atac 1),
        (atac 1)
        ]);


(* ------------------------------------------------------------------------ *)
(* Isfst and Issnd are monotone                                             *)
(* ------------------------------------------------------------------------ *)

qed_goalw "monofun_Isfst" Sprod2.thy [monofun] "monofun(Isfst)"
(fn prems => [(simp_tac (HOL_ss addsimps [inst_sprod_po]) 1)]);

qed_goalw "monofun_Issnd" Sprod2.thy [monofun] "monofun(Issnd)"
(fn prems => [(simp_tac (HOL_ss addsimps [inst_sprod_po]) 1)]);

(* ------------------------------------------------------------------------ *)
(* the type 'a ** 'b is a cpo                                               *)
(* ------------------------------------------------------------------------ *)

qed_goal "lub_sprod" Sprod2.thy 
"[|is_chain(S)|] ==> range(S) <<| \
\ Ispair (lub(range(%i.Isfst(S i)))) (lub(range(%i.Issnd(S i))))"
(fn prems =>
        [
        (cut_facts_tac prems 1),
        (rtac (conjI RS is_lubI) 1),
        (rtac (allI RS ub_rangeI) 1),
        (res_inst_tac [("t","S(i)")] (surjective_pairing_Sprod RS ssubst) 1),
        (rtac monofun_Ispair 1),
        (rtac is_ub_thelub 1),
        (etac (monofun_Isfst RS ch2ch_monofun) 1),
        (rtac is_ub_thelub 1),
        (etac (monofun_Issnd RS ch2ch_monofun) 1),
        (strip_tac 1),
        (res_inst_tac [("t","u")] (surjective_pairing_Sprod RS ssubst) 1),
        (rtac monofun_Ispair 1),
        (rtac is_lub_thelub 1),
        (etac (monofun_Isfst RS ch2ch_monofun) 1),
        (etac (monofun_Isfst RS ub2ub_monofun) 1),
        (rtac is_lub_thelub 1),
        (etac (monofun_Issnd RS ch2ch_monofun) 1),
        (etac (monofun_Issnd RS ub2ub_monofun) 1)
        ]);

bind_thm ("thelub_sprod", lub_sprod RS thelubI);


qed_goal "cpo_sprod" Sprod2.thy 
        "is_chain(S::nat=>'a**'b)==>? x.range(S)<<| x"
(fn prems =>
        [
        (cut_facts_tac prems 1),
        (rtac exI 1),
        (etac lub_sprod 1)
        ]);