src/HOLCF/Lift2.ML

isatool fixclasimp;

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

Theorems for Lift2.thy
*)

open Lift2;

(* for compatibility with old HOLCF-Version *)
qed_goal "inst_lift_po" thy "(op <<)=(%x y. x=y|x=Undef)"
 (fn prems => 
        [
        (fold_goals_tac [less_lift_def]),
        (rtac refl 1)
        ]);

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

goal Lift2.thy  "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);

qed_goal "least_lift" thy "? x::'a lift.!y. x<<y"
(fn prems =>
        [
        (res_inst_tac [("x","Undef")] exI 1),
        (rtac (minimal_lift RS allI) 1)
        ]);

(* ------------------------------------------------------------------------ *)
(* ('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 Lift2.thy "!!x. [| x<<y; ~x=Undef |] ==> ~y=Undef";
by (etac contrapos 1);
by (hyp_subst_tac 1);
by (rtac antisym_less 1);
by (atac 1);
by (rtac minimal_lift 1);
qed"notUndef_I";


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

goal Lift2.thy
"!!Y. [|? j.~Y(j)=Undef;is_chain(Y::nat=>('a)lift)|] \
\ ==> ? j.!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 RS mp) 1);
by (atac 1);
qed"chain_mono2_po";


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

goal Lift2.thy
        "(! Y. is_chain(Y::nat=>('a)lift)-->(? n. max_in_chain n Y))";
by (rewtac max_in_chain_def);  
by (strip_tac 1);
by (res_inst_tac [("P","!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 "!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 RS mp) 1); 
by (atac 1); 
by (hyp_subst_tac 1); 
by (rtac refl_less 1); 
by (res_inst_tac [("P","Y(Suc(x)) = Undef")] notE 1); 
by (atac 2); 
by (rtac mp 1); 
by (etac spec 1); 
by (Asm_simp_tac 1);
qed"flat_imp_chain_finite_poo";


(* Main Lemma: cpo_lift *)

goal Lift2.thy  
  "!!Y. is_chain(Y::nat=>('a)lift) ==> ? x. range(Y) <<|x";
by (cut_inst_tac [] flat_imp_chain_finite_poo 1);
by (Step_tac 1);
by Safe_tac;
by (Step_tac 1); 
by (rtac exI 1);
by (rtac lub_finch1 1);
by (atac 1);
by (atac 1);
qed"cpo_lift";