--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOLCF/Lift2.ML Wed Jan 19 17:35:01 1994 +0100
@@ -0,0 +1,182 @@
+(* Title: HOLCF/lift2.ML
+ ID: $Id$
+ Author: Franz Regensburger
+ Copyright 1993 Technische Universitaet Muenchen
+
+Lemmas for lift2.thy
+*)
+
+open Lift2;
+
+(* -------------------------------------------------------------------------*)
+(* type ('a)u is pointed *)
+(* ------------------------------------------------------------------------ *)
+
+val minimal_lift = prove_goal Lift2.thy "UU_lift << z"
+ (fn prems =>
+ [
+ (rtac (inst_lift_po RS ssubst) 1),
+ (rtac less_lift1a 1)
+ ]);
+
+(* -------------------------------------------------------------------------*)
+(* access to less_lift in class po *)
+(* ------------------------------------------------------------------------ *)
+
+val less_lift2b = prove_goal Lift2.thy "~ Iup(x) << UU_lift"
+ (fn prems =>
+ [
+ (rtac (inst_lift_po RS ssubst) 1),
+ (rtac less_lift1b 1)
+ ]);
+
+val less_lift2c = prove_goal Lift2.thy "(Iup(x)<<Iup(y)) = (x<<y)"
+ (fn prems =>
+ [
+ (rtac (inst_lift_po RS ssubst) 1),
+ (rtac less_lift1c 1)
+ ]);
+
+(* ------------------------------------------------------------------------ *)
+(* Iup and Ilift are monotone *)
+(* ------------------------------------------------------------------------ *)
+
+val monofun_Iup = prove_goalw Lift2.thy [monofun] "monofun(Iup)"
+ (fn prems =>
+ [
+ (strip_tac 1),
+ (etac (less_lift2c RS iffD2) 1)
+ ]);
+
+val monofun_Ilift1 = prove_goalw Lift2.thy [monofun] "monofun(Ilift)"
+ (fn prems =>
+ [
+ (strip_tac 1),
+ (rtac (less_fun RS iffD2) 1),
+ (strip_tac 1),
+ (res_inst_tac [("p","xa")] liftE 1),
+ (asm_simp_tac Lift_ss 1),
+ (asm_simp_tac Lift_ss 1),
+ (etac monofun_cfun_fun 1)
+ ]);
+
+val monofun_Ilift2 = prove_goalw Lift2.thy [monofun] "monofun(Ilift(f))"
+ (fn prems =>
+ [
+ (strip_tac 1),
+ (res_inst_tac [("p","x")] liftE 1),
+ (asm_simp_tac Lift_ss 1),
+ (asm_simp_tac Lift_ss 1),
+ (res_inst_tac [("p","y")] liftE 1),
+ (hyp_subst_tac 1),
+ (hyp_subst_tac 1),
+ (rtac notE 1),
+ (rtac less_lift2b 1),
+ (atac 1),
+ (asm_simp_tac Lift_ss 1),
+ (rtac monofun_cfun_arg 1),
+ (hyp_subst_tac 1),
+ (hyp_subst_tac 1),
+ (etac (less_lift2c RS iffD1) 1)
+ ]);
+
+(* ------------------------------------------------------------------------ *)
+(* Some kind of surjectivity lemma *)
+(* ------------------------------------------------------------------------ *)
+
+
+val lift_lemma1 = prove_goal Lift2.thy "z=Iup(x) ==> Iup(Ilift(LAM x.x)(z)) = z"
+ (fn prems =>
+ [
+ (cut_facts_tac prems 1),
+ (asm_simp_tac Lift_ss 1)
+ ]);
+
+(* ------------------------------------------------------------------------ *)
+(* ('a)u is a cpo *)
+(* ------------------------------------------------------------------------ *)
+
+val lub_lift1a = prove_goal Lift2.thy
+"[|is_chain(Y);? i x.Y(i)=Iup(x)|] ==>\
+\ range(Y) <<| Iup(lub(range(%i.(Ilift (LAM x.x) (Y(i))))))"
+ (fn prems =>
+ [
+ (cut_facts_tac prems 1),
+ (rtac is_lubI 1),
+ (rtac conjI 1),
+ (rtac ub_rangeI 1),
+ (rtac allI 1),
+ (res_inst_tac [("p","Y(i)")] liftE 1),
+ (res_inst_tac [("s","UU_lift"),("t","Y(i)")] subst 1),
+ (etac sym 1),
+ (rtac minimal_lift 1),
+ (res_inst_tac [("t","Y(i)")] (lift_lemma1 RS subst) 1),
+ (atac 1),
+ (rtac (less_lift2c RS iffD2) 1),
+ (rtac is_ub_thelub 1),
+ (etac (monofun_Ilift2 RS ch2ch_monofun) 1),
+ (strip_tac 1),
+ (res_inst_tac [("p","u")] liftE 1),
+ (etac exE 1),
+ (etac exE 1),
+ (res_inst_tac [("P","Y(i)<<UU_lift")] notE 1),
+ (res_inst_tac [("s","Iup(x)"),("t","Y(i)")] ssubst 1),
+ (atac 1),
+ (rtac less_lift2b 1),
+ (hyp_subst_tac 1),
+ (etac (ub_rangeE RS spec) 1),
+ (res_inst_tac [("t","u")] (lift_lemma1 RS subst) 1),
+ (atac 1),
+ (rtac (less_lift2c RS iffD2) 1),
+ (rtac is_lub_thelub 1),
+ (etac (monofun_Ilift2 RS ch2ch_monofun) 1),
+ (etac (monofun_Ilift2 RS ub2ub_monofun) 1)
+ ]);
+
+val lub_lift1b = prove_goal Lift2.thy
+"[|is_chain(Y);!i x.~Y(i)=Iup(x)|] ==>\
+\ range(Y) <<| UU_lift"
+ (fn prems =>
+ [
+ (cut_facts_tac prems 1),
+ (rtac is_lubI 1),
+ (rtac conjI 1),
+ (rtac ub_rangeI 1),
+ (rtac allI 1),
+ (res_inst_tac [("p","Y(i)")] liftE 1),
+ (res_inst_tac [("s","UU_lift"),("t","Y(i)")] ssubst 1),
+ (atac 1),
+ (rtac refl_less 1),
+ (rtac notE 1),
+ (dtac spec 1),
+ (dtac spec 1),
+ (atac 1),
+ (atac 1),
+ (strip_tac 1),
+ (rtac minimal_lift 1)
+ ]);
+
+val thelub_lift1a = lub_lift1a RS thelubI;
+(* [| is_chain(?Y1); ? i x. ?Y1(i) = Iup(x) |] ==> *)
+(* lub(range(?Y1)) = Iup(lub(range(%i. Ilift(LAM x. x,?Y1(i))))) *)
+
+val thelub_lift1b = lub_lift1b RS thelubI;
+(* [| is_chain(?Y1); ! i x. ~ ?Y1(i) = Iup(x) |] ==> *)
+(* lub(range(?Y1)) = UU_lift *)
+
+
+val cpo_lift = prove_goal Lift2.thy
+ "is_chain(Y::nat=>('a)u) ==> ? x.range(Y) <<|x"
+ (fn prems =>
+ [
+ (cut_facts_tac prems 1),
+ (rtac disjE 1),
+ (rtac exI 2),
+ (etac lub_lift1a 2),
+ (atac 2),
+ (rtac exI 2),
+ (etac lub_lift1b 2),
+ (atac 2),
+ (fast_tac HOL_cs 1)
+ ]);
+