src/HOLCF/Lift3.ML

adapted several proofs

(*  Title:      HOLCF/lift3.ML
    ID:         $Id$
    Author:     Franz Regensburger
    Copyright   1993 Technische Universitaet Muenchen

Lemmas for lift3.thy
*)

open Lift3;

(* -------------------------------------------------------------------------*)
(* some lemmas restated for class pcpo                                      *)
(* ------------------------------------------------------------------------ *)

qed_goal "less_lift3b" Lift3.thy "~ Iup(x) << UU"
 (fn prems =>
        [
        (rtac (inst_lift_pcpo RS ssubst) 1),
        (rtac less_lift2b 1)
        ]);

qed_goal "defined_Iup2" Lift3.thy "Iup(x) ~= UU"
 (fn prems =>
        [
        (rtac (inst_lift_pcpo RS ssubst) 1),
        (rtac defined_Iup 1)
        ]);

(* ------------------------------------------------------------------------ *)
(* continuity for Iup                                                       *)
(* ------------------------------------------------------------------------ *)

qed_goal "contlub_Iup" Lift3.thy "contlub(Iup)"
 (fn prems =>
        [
        (rtac contlubI 1),
        (strip_tac 1),
        (rtac trans 1),
        (rtac (thelub_lift1a RS sym) 2),
        (fast_tac HOL_cs 3),
        (etac (monofun_Iup RS ch2ch_monofun) 2),
        (res_inst_tac [("f","Iup")] arg_cong  1),
        (rtac lub_equal 1),
        (atac 1),
        (rtac (monofun_Ilift2 RS ch2ch_monofun) 1),
        (etac (monofun_Iup RS ch2ch_monofun) 1),
        (asm_simp_tac Lift0_ss 1)
        ]);

qed_goal "cont_Iup" Lift3.thy "cont(Iup)"
 (fn prems =>
        [
        (rtac monocontlub2cont 1),
        (rtac monofun_Iup 1),
        (rtac contlub_Iup 1)
        ]);


(* ------------------------------------------------------------------------ *)
(* continuity for Ilift                                                     *)
(* ------------------------------------------------------------------------ *)

qed_goal "contlub_Ilift1" Lift3.thy "contlub(Ilift)"
 (fn prems =>
        [
        (rtac contlubI 1),
        (strip_tac 1),
        (rtac trans 1),
        (rtac (thelub_fun RS sym) 2),
        (etac (monofun_Ilift1 RS ch2ch_monofun) 2),
        (rtac ext 1),
        (res_inst_tac [("p","x")] liftE 1),
        (asm_simp_tac Lift0_ss 1),
        (rtac (lub_const RS thelubI RS sym) 1),
        (asm_simp_tac Lift0_ss 1),
        (etac contlub_cfun_fun 1)
        ]);


qed_goal "contlub_Ilift2" Lift3.thy "contlub(Ilift(f))"
 (fn prems =>
        [
        (rtac contlubI 1),
        (strip_tac 1),
        (rtac disjE 1),
        (rtac (thelub_lift1a RS ssubst) 2),
        (atac 2),
        (atac 2),
        (asm_simp_tac Lift0_ss 2),
        (rtac (thelub_lift1b RS ssubst) 3),
        (atac 3),
        (atac 3),
        (fast_tac HOL_cs 1),
        (asm_simp_tac Lift0_ss 2),
        (rtac (chain_UU_I_inverse RS sym) 2),
        (rtac allI 2),
        (res_inst_tac [("p","Y(i)")] liftE 2),
        (asm_simp_tac Lift0_ss 2),
        (rtac notE 2),
        (dtac spec 2),
        (etac spec 2),
        (atac 2),
        (rtac (contlub_cfun_arg RS ssubst) 1),
        (etac (monofun_Ilift2 RS ch2ch_monofun) 1),
        (rtac lub_equal2 1),
        (rtac (monofun_fapp2 RS ch2ch_monofun) 2),
        (etac (monofun_Ilift2 RS ch2ch_monofun) 2),
        (etac (monofun_Ilift2 RS ch2ch_monofun) 2),
        (rtac (chain_mono2 RS exE) 1),
        (atac 2),
        (etac exE 1),
        (etac exE 1),
        (rtac exI 1),
        (res_inst_tac [("s","Iup(x)"),("t","Y(i)")] ssubst 1),
        (atac 1),
        (rtac defined_Iup2 1),
        (rtac exI 1),
        (strip_tac 1),
        (res_inst_tac [("p","Y(i)")] liftE 1),
        (asm_simp_tac Lift0_ss 2),
        (res_inst_tac [("P","Y(i) = UU")] notE 1),
        (fast_tac HOL_cs 1),
        (rtac (inst_lift_pcpo RS ssubst) 1),
        (atac 1)
        ]);

qed_goal "cont_Ilift1" Lift3.thy "cont(Ilift)"
 (fn prems =>
        [
        (rtac monocontlub2cont 1),
        (rtac monofun_Ilift1 1),
        (rtac contlub_Ilift1 1)
        ]);

qed_goal "cont_Ilift2" Lift3.thy "cont(Ilift(f))"
 (fn prems =>
        [
        (rtac monocontlub2cont 1),
        (rtac monofun_Ilift2 1),
        (rtac contlub_Ilift2 1)
        ]);


(* ------------------------------------------------------------------------ *)
(* continuous versions of lemmas for ('a)u                                  *)
(* ------------------------------------------------------------------------ *)

qed_goalw "Exh_Lift1" Lift3.thy [up_def] "z = UU | (? x. z = up`x)"
 (fn prems =>
        [
        (simp_tac (Lift0_ss addsimps [cont_Iup]) 1),
        (rtac (inst_lift_pcpo RS ssubst) 1),
        (rtac Exh_Lift 1)
        ]);

qed_goalw "inject_up" Lift3.thy [up_def] "up`x=up`y ==> x=y"
 (fn prems =>
        [
        (cut_facts_tac prems 1),
        (rtac inject_Iup 1),
        (etac box_equals 1),
        (simp_tac (Lift0_ss addsimps [cont_Iup]) 1),
        (simp_tac (Lift0_ss addsimps [cont_Iup]) 1)
        ]);

qed_goalw "defined_up" Lift3.thy [up_def] " up`x ~= UU"
 (fn prems =>
        [
        (simp_tac (Lift0_ss addsimps [cont_Iup]) 1),
        (rtac defined_Iup2 1)
        ]);

qed_goalw "liftE1" Lift3.thy [up_def] 
        "[| p=UU ==> Q; !!x. p=up`x==>Q|] ==>Q"
 (fn prems =>
        [
        (rtac liftE 1),
        (resolve_tac prems 1),
        (etac (inst_lift_pcpo RS ssubst) 1),
        (resolve_tac (tl prems) 1),
        (asm_simp_tac (Lift0_ss addsimps [cont_Iup]) 1)
        ]);


qed_goalw "lift1" Lift3.thy [up_def,lift_def] "lift`f`UU=UU"
 (fn prems =>
        [
        (rtac (inst_lift_pcpo RS ssubst) 1),
        (rtac (beta_cfun RS ssubst) 1),
        (REPEAT (resolve_tac (cont_lemmas @ [cont_Iup,cont_Ilift1,
                cont_Ilift2,cont2cont_CF1L]) 1)),
        (rtac (beta_cfun RS ssubst) 1),
        (REPEAT (resolve_tac (cont_lemmas @ [cont_Iup,cont_Ilift1,
                cont_Ilift2,cont2cont_CF1L]) 1)),
        (simp_tac (Lift0_ss addsimps [cont_Iup,cont_Ilift1,cont_Ilift2]) 1)
        ]);

qed_goalw "lift2" Lift3.thy [up_def,lift_def] "lift`f`(up`x)=f`x"
 (fn prems =>
        [
        (rtac (beta_cfun RS ssubst) 1),
        (rtac cont_Iup 1),
        (rtac (beta_cfun RS ssubst) 1),
        (REPEAT (resolve_tac (cont_lemmas @ [cont_Iup,cont_Ilift1,
                cont_Ilift2,cont2cont_CF1L]) 1)),
        (rtac (beta_cfun RS ssubst) 1),
        (rtac cont_Ilift2 1),
        (simp_tac (Lift0_ss addsimps [cont_Iup,cont_Ilift1,cont_Ilift2]) 1)
        ]);

qed_goalw "less_lift4b" Lift3.thy [up_def,lift_def] "~ up`x << UU"
 (fn prems =>
        [
        (simp_tac (Lift0_ss addsimps [cont_Iup]) 1),
        (rtac less_lift3b 1)
        ]);

qed_goalw "less_lift4c" Lift3.thy [up_def,lift_def]
         "(up`x << up`y) = (x<<y)"
 (fn prems =>
        [
        (simp_tac (Lift0_ss addsimps [cont_Iup]) 1),
        (rtac less_lift2c 1)
        ]);

qed_goalw "thelub_lift2a" Lift3.thy [up_def,lift_def] 
"[| is_chain(Y); ? i x. Y(i) = up`x |] ==>\
\      lub(range(Y)) = up`(lub(range(%i. lift`(LAM x. x)`(Y i))))"
 (fn prems =>
        [
        (cut_facts_tac prems 1),
        (rtac (beta_cfun RS ssubst) 1),
        (REPEAT (resolve_tac (cont_lemmas @ [cont_Iup,cont_Ilift1,
                cont_Ilift2,cont2cont_CF1L]) 1)),
        (rtac (beta_cfun RS ssubst) 1),
        (REPEAT (resolve_tac (cont_lemmas @ [cont_Iup,cont_Ilift1,
                cont_Ilift2,cont2cont_CF1L]) 1)),

        (rtac (beta_cfun RS ext RS ssubst) 1),
        (REPEAT (resolve_tac (cont_lemmas @ [cont_Iup,cont_Ilift1,
                cont_Ilift2,cont2cont_CF1L]) 1)),
        (rtac thelub_lift1a 1),
        (atac 1),
        (etac exE 1),
        (etac exE 1),
        (rtac exI 1),
        (rtac exI 1),
        (etac box_equals 1),
        (rtac refl 1),
        (simp_tac (Lift0_ss addsimps [cont_Iup]) 1)
        ]);



qed_goalw "thelub_lift2b" Lift3.thy [up_def,lift_def] 
"[| is_chain(Y); ! i x. Y(i) ~= up`x |] ==> lub(range(Y)) = UU"
 (fn prems =>
        [
        (cut_facts_tac prems 1),
        (rtac (inst_lift_pcpo RS ssubst) 1),
        (rtac thelub_lift1b 1),
        (atac 1),
        (strip_tac 1),
        (dtac spec 1),
        (dtac spec 1),
        (rtac swap 1),
        (atac 1),
        (dtac notnotD 1),
        (etac box_equals 1),
        (rtac refl 1),
        (simp_tac (Lift0_ss addsimps [cont_Iup]) 1)
        ]);


qed_goal "lift_lemma2" Lift3.thy  " (? x.z = up`x) = (z~=UU)"
 (fn prems =>
        [
        (rtac iffI 1),
        (etac exE 1),
        (hyp_subst_tac 1),
        (rtac defined_up 1),
        (res_inst_tac [("p","z")] liftE1 1),
        (etac notE 1),
        (atac 1),
        (etac exI 1)
        ]);


qed_goal "thelub_lift2a_rev" Lift3.thy  
"[| is_chain(Y); lub(range(Y)) = up`x |] ==> ? i x. Y(i) = up`x"
 (fn prems =>
        [
        (cut_facts_tac prems 1),
        (rtac exE 1),
        (rtac chain_UU_I_inverse2 1),
        (rtac (lift_lemma2 RS iffD1) 1),
        (etac exI 1),
        (rtac exI 1),
        (rtac (lift_lemma2 RS iffD2) 1),
        (atac 1)
        ]);

qed_goal "thelub_lift2b_rev" Lift3.thy  
"[| is_chain(Y); lub(range(Y)) = UU |] ==> ! i x.  Y(i) ~= up`x"
 (fn prems =>
        [
        (cut_facts_tac prems 1),
        (rtac allI 1),
        (rtac (not_ex RS iffD1) 1),
        (rtac contrapos 1),
        (etac (lift_lemma2 RS iffD1) 2),
        (fast_tac (HOL_cs addSDs [chain_UU_I RS spec]) 1)
        ]);


qed_goal "thelub_lift3" Lift3.thy  
"is_chain(Y) ==> lub(range(Y)) = UU |\
\                lub(range(Y)) = up`(lub(range(%i. lift`(LAM x.x)`(Y i))))"
 (fn prems =>
        [
        (cut_facts_tac prems 1),
        (rtac disjE 1),
        (rtac disjI1 2),
        (rtac thelub_lift2b 2),
        (atac 2),
        (atac 2),
        (rtac disjI2 2),
        (rtac thelub_lift2a 2),
        (atac 2),
        (atac 2),
        (fast_tac HOL_cs 1)
        ]);

qed_goal "lift3" Lift3.thy "lift`up`x=x"
 (fn prems =>
        [
        (res_inst_tac [("p","x")] liftE1 1),
        (asm_simp_tac ((simpset_of "Cfun3") addsimps [lift1,lift2]) 1),
        (asm_simp_tac ((simpset_of "Cfun3") addsimps [lift1,lift2]) 1)
        ]);

(* ------------------------------------------------------------------------ *)
(* install simplifier for ('a)u                                             *)
(* ------------------------------------------------------------------------ *)

val lift_rews = [lift1,lift2,defined_up];