src/HOLCF/Up1.ML

Removal of ex/LexProd

(*  Title:      HOLCF/Up1.ML
    ID:         $Id$
    Author:     Franz Regensburger
    Copyright   1993  Technische Universitaet Muenchen
*)

open Up1;

qed_goal "Abs_Up_inverse2" thy "Rep_Up (Abs_Up y) = y"
 (fn prems =>
        [
	(simp_tac (!simpset addsimps [Up_def,Abs_Up_inverse]) 1)
	]);

qed_goalw "Exh_Up" thy [Iup_def ]
        "z = Abs_Up(Inl ()) | (? x. z = Iup x)"
 (fn prems =>
        [
        (rtac (Rep_Up_inverse RS subst) 1),
        (res_inst_tac [("s","Rep_Up z")] sumE 1),
        (rtac disjI1 1),
        (res_inst_tac [("f","Abs_Up")] arg_cong 1),
        (rtac (unit_eq RS subst) 1),
        (atac 1),
        (rtac disjI2 1),
        (rtac exI 1),
        (res_inst_tac [("f","Abs_Up")] arg_cong 1),
        (atac 1)
        ]);

qed_goal "inj_Abs_Up" thy "inj(Abs_Up)"
 (fn prems =>
        [
        (rtac inj_inverseI 1),
        (rtac Abs_Up_inverse2 1)
        ]);

qed_goal "inj_Rep_Up" thy "inj(Rep_Up)"
 (fn prems =>
        [
        (rtac inj_inverseI 1),
        (rtac Rep_Up_inverse 1)
        ]);

qed_goalw "inject_Iup" thy [Iup_def] "Iup x=Iup y ==> x=y"
 (fn prems =>
        [
        (cut_facts_tac prems 1),
        (rtac (inj_Inr RS injD) 1),
        (rtac (inj_Abs_Up RS injD) 1),
        (atac 1)
        ]);

qed_goalw "defined_Iup" thy [Iup_def] "Iup x~=Abs_Up(Inl ())"
 (fn prems =>
        [
        (rtac notI 1),
        (rtac notE 1),
        (rtac Inl_not_Inr 1),
        (rtac sym 1),
        (etac (inj_Abs_Up RS  injD) 1)
        ]);


qed_goal "upE"  thy
        "[| p=Abs_Up(Inl ()) ==> Q; !!x. p=Iup(x)==>Q|] ==>Q"
 (fn prems =>
        [
        (rtac (Exh_Up RS disjE) 1),
        (eresolve_tac prems 1),
        (etac exE 1),
        (eresolve_tac prems 1)
        ]);

qed_goalw "Ifup1"  thy [Ifup_def]
        "Ifup(f)(Abs_Up(Inl ()))=UU"
 (fn prems =>
        [
        (stac Abs_Up_inverse2 1),
        (stac sum_case_Inl 1),
        (rtac refl 1)
        ]);

qed_goalw "Ifup2"  thy [Ifup_def,Iup_def]
        "Ifup(f)(Iup(x))=f`x"
 (fn prems =>
        [
        (stac Abs_Up_inverse2 1),
        (stac sum_case_Inr 1),
        (rtac refl 1)
        ]);

val Up0_ss = (simpset_of "Cfun3") addsimps [Ifup1,Ifup2];

qed_goalw "less_up1a"  thy [less_up_def]
        "less(Abs_Up(Inl ()))(z)"
 (fn prems =>
        [
        (stac Abs_Up_inverse2 1),
        (stac sum_case_Inl 1),
        (rtac TrueI 1)
        ]);

qed_goalw "less_up1b" thy [Iup_def,less_up_def]
        "~less (Iup x) (Abs_Up(Inl ()))"
 (fn prems =>
        [
        (rtac notI 1),
        (rtac iffD1 1),
        (atac 2),
        (stac Abs_Up_inverse2 1),
        (stac Abs_Up_inverse2 1),
        (stac sum_case_Inr 1),
        (stac sum_case_Inl 1),
        (rtac refl 1)
        ]);

qed_goalw "less_up1c"  thy [Iup_def,less_up_def]
        "less (Iup x) (Iup y)=(x<<y)"
 (fn prems =>
        [
        (stac Abs_Up_inverse2 1),
        (stac Abs_Up_inverse2 1),
        (stac sum_case_Inr 1),
        (stac sum_case_Inr 1),
        (rtac refl 1)
        ]);


qed_goal "refl_less_up"  thy "less (p::'a u) p"
 (fn prems =>
        [
        (res_inst_tac [("p","p")] upE 1),
        (hyp_subst_tac 1),
        (rtac less_up1a 1),
        (hyp_subst_tac 1),
        (rtac (less_up1c RS iffD2) 1),
        (rtac refl_less 1)
        ]);

qed_goal "antisym_less_up"  thy 
        "!!p1.[|less(p1::'a u) p2;less p2 p1|] ==> p1=p2"
 (fn _ =>
        [
        (res_inst_tac [("p","p1")] upE 1),
        (hyp_subst_tac 1),
        (res_inst_tac [("p","p2")] upE 1),
        (etac sym 1),
        (hyp_subst_tac 1),
        (res_inst_tac [("P","less (Iup x) (Abs_Up(Inl ()))")] notE 1),
        (rtac less_up1b 1),
        (atac 1),
        (hyp_subst_tac 1),
        (res_inst_tac [("p","p2")] upE 1),
        (hyp_subst_tac 1),
        (res_inst_tac [("P","less (Iup x) (Abs_Up(Inl ()))")] notE 1),
        (rtac less_up1b 1),
        (atac 1),
        (hyp_subst_tac 1),
        (rtac arg_cong 1),
        (rtac antisym_less 1),
        (etac (less_up1c RS iffD1) 1),
        (etac (less_up1c RS iffD1) 1)
        ]);

qed_goal "trans_less_up"  thy 
        "[|less (p1::'a u) p2;less p2 p3|] ==> less p1 p3"
 (fn prems =>
        [
        (cut_facts_tac prems 1),
        (res_inst_tac [("p","p1")] upE 1),
        (hyp_subst_tac 1),
        (rtac less_up1a 1),
        (hyp_subst_tac 1),
        (res_inst_tac [("p","p2")] upE 1),
        (hyp_subst_tac 1),
        (rtac notE 1),
        (rtac less_up1b 1),
        (atac 1),
        (hyp_subst_tac 1),
        (res_inst_tac [("p","p3")] upE 1),
        (hyp_subst_tac 1),
        (rtac notE 1),
        (rtac less_up1b 1),
        (atac 1),
        (hyp_subst_tac 1),
        (rtac (less_up1c RS iffD2) 1),
        (rtac trans_less 1),
        (etac (less_up1c RS iffD1) 1),
        (etac (less_up1c RS iffD1) 1)
        ]);