src/HOLCF/Lift2.ML

apply. -> by

(*  Title:      HOLCF/Lift2.ML
    ID:         $Id$
    Author:     Olaf Mueller
    Copyright   1996 Technische Universitaet Muenchen

Class Instance lift::(term)po
*)

(* for compatibility with old HOLCF-Version *)
Goal "(op <<)=(%x y. x=y|x=Undef)";
by (fold_goals_tac [less_lift_def]);
by (rtac refl 1);
qed "inst_lift_po";

(* -------------------------------------------------------------------------*)
(* type ('a)lift is pointed                                                *)
(* ------------------------------------------------------------------------ *)

Goal  "Undef << x";
by (simp_tac (simpset() addsimps [inst_lift_po]) 1);
qed"minimal_lift";

bind_thm ("UU_lift_def",minimal_lift RS minimal2UU RS sym);

AddIffs [minimal_lift];

Goal "EX x::'a lift. ALL y. x<<y";
by (res_inst_tac [("x","Undef")] exI 1);
by (rtac (minimal_lift RS allI) 1);
qed "least_lift";

(* ------------------------------------------------------------------------ *)
(* ('a)lift is a cpo                                                       *)
(* ------------------------------------------------------------------------ *)

(* The following Lemmata have already been proved in Pcpo.ML and Fix.ML, 
   but there class pcpo is assumed, although only po is necessary and a
   least element. Therefore they are redone here for the po lift with 
   least element Undef.   *)

(* Tailoring notUU_I of Pcpo.ML to Undef *)

Goal "[| x<<y; ~x=Undef |] ==> ~y=Undef";
by (blast_tac (claset() addIs [antisym_less]) 1);
qed"notUndef_I";


(* Tailoring chain_mono2 of Pcpo.ML to Undef *)

Goal "[| EX j.~Y(j)=Undef; chain(Y::nat=>('a)lift) |] \
\        ==> EX j. ALL i. j<i-->~Y(i)=Undef";
by Safe_tac;
by (Step_tac 1);
by (strip_tac 1);
by (rtac notUndef_I 1);
by (atac 2);
by (etac (chain_mono) 1);
by (atac 1);
qed"chain_mono2_po";


(* Tailoring flat_imp_chfin of Fix.ML to lift *)

Goal "(ALL Y. chain(Y::nat=>('a)lift)-->(EX n. max_in_chain n Y))";
by (rewtac max_in_chain_def);  
by (strip_tac 1);
by (res_inst_tac [("P","ALL i. Y(i)=Undef")] case_split_thm  1);
by (res_inst_tac [("x","0")] exI 1);
by (strip_tac 1);
by (rtac trans 1);
by (etac spec 1);
by (rtac sym 1);
by (etac spec 1); 
by (subgoal_tac "ALL x y. x<<(y::('a)lift) --> x=Undef | x=y" 1);
by (simp_tac (simpset() addsimps [inst_lift_po]) 2);
by (rtac (chain_mono2_po RS exE) 1); 
by (Fast_tac 1); 
by (atac 1); 
by (res_inst_tac [("x","Suc(x)")] exI 1); 
by (strip_tac 1); 
by (rtac disjE 1); 
by (atac 3); 
by (rtac mp 1); 
by (dtac spec 1);
by (etac spec 1); 
by (etac (le_imp_less_or_eq RS disjE) 1); 
by (etac (chain_mono) 1);
by Auto_tac;   
qed"flat_imp_chfin_poo";


(* Main Lemma: cpo_lift *)

Goal "chain(Y::nat=>('a)lift) ==> EX x. range(Y) <<|x";
by (cut_inst_tac [] flat_imp_chfin_poo 1);
by (blast_tac (claset() addIs [lub_finch1]) 1);
qed"cpo_lift";