src/HOLCF/Ssum3.ML

added local simpsets

(*  Title: 	HOLCF/ssum3.ML
    ID:         $Id$
    Author: 	Franz Regensburger
    Copyright   1993 Technische Universitaet Muenchen

Lemmas for ssum3.thy
*)

open Ssum3;

(* ------------------------------------------------------------------------ *)
(* continuity for Isinl and Isinr                                           *)
(* ------------------------------------------------------------------------ *)


qed_goal "contlub_Isinl" Ssum3.thy "contlub(Isinl)"
 (fn prems =>
	[
	(rtac contlubI 1),
	(strip_tac 1),
	(rtac trans 1),
	(rtac (thelub_ssum1a RS sym) 2),
	(rtac allI 3),
	(rtac exI 3),
	(rtac refl 3),
	(etac (monofun_Isinl RS ch2ch_monofun) 2),
	(res_inst_tac [("Q","lub(range(Y))=UU")] classical2 1),
	(res_inst_tac [("s","UU"),("t","lub(range(Y))")] ssubst 1),
	(atac 1),
	(res_inst_tac [("f","Isinl")] arg_cong  1),
	(rtac (chain_UU_I_inverse RS sym) 1),
	(rtac allI 1),
	(res_inst_tac [("s","UU"),("t","Y(i)")] ssubst 1),
	(etac (chain_UU_I RS spec ) 1),
	(atac 1),
	(rtac Iwhen1 1),
	(res_inst_tac [("f","Isinl")] arg_cong  1),
	(rtac lub_equal 1),
	(atac 1),
	(rtac (monofun_Iwhen3 RS ch2ch_monofun) 1),
	(etac (monofun_Isinl RS ch2ch_monofun) 1),
	(rtac allI 1),
	(res_inst_tac [("Q","Y(k)=UU")] classical2 1),
	(Asm_simp_tac 1),
	(Asm_simp_tac 1)
	]);

qed_goal "contlub_Isinr" Ssum3.thy "contlub(Isinr)"
 (fn prems =>
	[
	(rtac contlubI 1),
	(strip_tac 1),
	(rtac trans 1),
	(rtac (thelub_ssum1b RS sym) 2),
	(rtac allI 3),
	(rtac exI 3),
	(rtac refl 3),
	(etac (monofun_Isinr RS ch2ch_monofun) 2),
	(res_inst_tac [("Q","lub(range(Y))=UU")] classical2 1),
	(res_inst_tac [("s","UU"),("t","lub(range(Y))")] ssubst 1),
	(atac 1),
	((rtac arg_cong 1) THEN (rtac (chain_UU_I_inverse RS sym) 1)),
	(rtac allI 1),
	(res_inst_tac [("s","UU"),("t","Y(i)")] ssubst 1),
	(etac (chain_UU_I RS spec ) 1),
	(atac 1),
	(rtac (strict_IsinlIsinr RS subst) 1),
	(rtac Iwhen1 1),
	((rtac arg_cong 1) THEN (rtac lub_equal 1)),
	(atac 1),
	(rtac (monofun_Iwhen3 RS ch2ch_monofun) 1),
	(etac (monofun_Isinr RS ch2ch_monofun) 1),
	(rtac allI 1),
	(res_inst_tac [("Q","Y(k)=UU")] classical2 1),
	(Asm_simp_tac 1),
	(Asm_simp_tac 1)
	]);

qed_goal "cont_Isinl" Ssum3.thy "cont(Isinl)"
 (fn prems =>
	[
	(rtac monocontlub2cont 1),
	(rtac monofun_Isinl 1),
	(rtac contlub_Isinl 1)
	]);

qed_goal "cont_Isinr" Ssum3.thy "cont(Isinr)"
 (fn prems =>
	[
	(rtac monocontlub2cont 1),
	(rtac monofun_Isinr 1),
	(rtac contlub_Isinr 1)
	]);


(* ------------------------------------------------------------------------ *)
(* continuity for Iwhen in the firts two arguments                          *)
(* ------------------------------------------------------------------------ *)

qed_goal "contlub_Iwhen1" Ssum3.thy "contlub(Iwhen)"
 (fn prems =>
	[
	(rtac contlubI 1),
	(strip_tac 1),
	(rtac trans 1),
	(rtac (thelub_fun RS sym) 2),
	(etac (monofun_Iwhen1 RS ch2ch_monofun) 2),
	(rtac (expand_fun_eq RS iffD2) 1),
	(strip_tac 1),
	(rtac trans 1),
	(rtac (thelub_fun RS sym) 2),
	(rtac ch2ch_fun 2),
	(etac (monofun_Iwhen1 RS ch2ch_monofun) 2),
	(rtac (expand_fun_eq RS iffD2) 1),
	(strip_tac 1),
	(res_inst_tac [("p","xa")] IssumE 1),
	(Asm_simp_tac 1),
	(rtac (lub_const RS thelubI RS sym) 1),
	(Asm_simp_tac 1),
	(etac contlub_cfun_fun 1),
	(Asm_simp_tac 1),
	(rtac (lub_const RS thelubI RS sym) 1)
	]);

qed_goal "contlub_Iwhen2" Ssum3.thy "contlub(Iwhen(f))"
 (fn prems =>
	[
	(rtac contlubI 1),
	(strip_tac 1),
	(rtac trans 1),
	(rtac (thelub_fun RS sym) 2),
	(etac (monofun_Iwhen2 RS ch2ch_monofun) 2),
	(rtac (expand_fun_eq RS iffD2) 1),
	(strip_tac 1),
	(res_inst_tac [("p","x")] IssumE 1),
	(Asm_simp_tac 1),
	(rtac (lub_const RS thelubI RS sym) 1),
	(Asm_simp_tac 1),
	(rtac (lub_const RS thelubI RS sym) 1),
	(Asm_simp_tac 1),
	(etac contlub_cfun_fun 1)
	]);

(* ------------------------------------------------------------------------ *)
(* continuity for Iwhen in its third argument                               *)
(* ------------------------------------------------------------------------ *)

(* ------------------------------------------------------------------------ *)
(* first 5 ugly lemmas                                                      *)
(* ------------------------------------------------------------------------ *)

qed_goal "ssum_lemma9" Ssum3.thy 
"[| is_chain(Y); lub(range(Y)) = Isinl(x)|] ==> !i.? x.Y(i)=Isinl(x)"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(strip_tac 1),
	(res_inst_tac [("p","Y(i)")] IssumE 1),
	(etac exI 1),
	(etac exI 1),
	(res_inst_tac [("P","y=UU")] notE 1),
	(atac 1),
	(rtac (less_ssum3d RS iffD1) 1),
	(etac subst 1),
	(etac subst 1),
	(etac is_ub_thelub 1)
	]);


qed_goal "ssum_lemma10" Ssum3.thy 
"[| is_chain(Y); lub(range(Y)) = Isinr(x)|] ==> !i.? x.Y(i)=Isinr(x)"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(strip_tac 1),
	(res_inst_tac [("p","Y(i)")] IssumE 1),
	(rtac exI 1),
	(etac trans 1),
	(rtac strict_IsinlIsinr 1),
	(etac exI 2),
	(res_inst_tac [("P","xa=UU")] notE 1),
	(atac 1),
	(rtac (less_ssum3c RS iffD1) 1),
	(etac subst 1),
	(etac subst 1),
	(etac is_ub_thelub 1)
	]);

qed_goal "ssum_lemma11" Ssum3.thy 
"[| is_chain(Y); lub(range(Y)) = Isinl(UU) |] ==>\
\   Iwhen f g (lub(range Y)) = lub(range(%i. Iwhen f g (Y i)))"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(Asm_simp_tac 1),
	(rtac (chain_UU_I_inverse RS sym) 1),
	(rtac allI 1),
	(res_inst_tac [("s","Isinl(UU)"),("t","Y(i)")] subst 1),
	(rtac (inst_ssum_pcpo RS subst) 1),
	(rtac (chain_UU_I RS spec RS sym) 1),
	(atac 1),
	(etac (inst_ssum_pcpo RS ssubst) 1),
	(Asm_simp_tac 1)
	]);

qed_goal "ssum_lemma12" Ssum3.thy 
"[| is_chain(Y); lub(range(Y)) = Isinl(x); x ~= UU |] ==>\
\   Iwhen f g (lub(range Y)) = lub(range(%i. Iwhen f g (Y i)))"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(Asm_simp_tac 1),
	(res_inst_tac [("t","x")] subst 1),
	(rtac inject_Isinl 1),
	(rtac trans 1),
	(atac 2),
	(rtac (thelub_ssum1a RS sym) 1),
	(atac 1),
	(etac ssum_lemma9 1),
	(atac 1),
	(rtac trans 1),
	(rtac contlub_cfun_arg 1),
	(rtac (monofun_Iwhen3 RS ch2ch_monofun) 1),
	(atac 1),
	(rtac lub_equal2 1),
	(rtac (chain_mono2 RS exE) 1),
	(atac 2),
	(rtac chain_UU_I_inverse2 1),
	(rtac (inst_ssum_pcpo RS ssubst) 1),
	(etac swap 1),
	(rtac inject_Isinl 1),
	(rtac trans 1),
	(etac sym 1),
	(etac notnotD 1),
	(rtac exI 1),
	(strip_tac 1),
	(rtac (ssum_lemma9 RS spec RS exE) 1),
	(atac 1),
	(atac 1),
	(res_inst_tac [("t","Y(i)")] ssubst 1),
	(atac 1),
	(rtac trans 1),
	(rtac cfun_arg_cong 1),
	(rtac Iwhen2 1),
	(res_inst_tac [("Pa","Y(i)=UU")] swap 1),
	(fast_tac HOL_cs 1),
	(rtac (inst_ssum_pcpo RS ssubst) 1),
	(res_inst_tac [("t","Y(i)")] ssubst 1),
	(atac 1),
	(fast_tac HOL_cs 1),
	(rtac (Iwhen2 RS ssubst) 1),
	(res_inst_tac [("Pa","Y(i)=UU")] swap 1),
	(fast_tac HOL_cs 1),
	(rtac (inst_ssum_pcpo RS ssubst) 1),
	(res_inst_tac [("t","Y(i)")] ssubst 1),
	(atac 1),
	(fast_tac HOL_cs 1),
	(Simp_tac 1),
	(rtac (monofun_fapp2 RS ch2ch_monofun) 1),
	(etac (monofun_Iwhen3 RS ch2ch_monofun) 1),
	(etac (monofun_Iwhen3 RS ch2ch_monofun) 1)
	]);


qed_goal "ssum_lemma13" Ssum3.thy 
"[| is_chain(Y); lub(range(Y)) = Isinr(x); x ~= UU |] ==>\
\   Iwhen f g (lub(range Y)) = lub(range(%i. Iwhen f g (Y i)))"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(Asm_simp_tac 1),
	(res_inst_tac [("t","x")] subst 1),
	(rtac inject_Isinr 1),
	(rtac trans 1),
	(atac 2),
	(rtac (thelub_ssum1b RS sym) 1),
	(atac 1),
	(etac ssum_lemma10 1),
	(atac 1),
	(rtac trans 1),
	(rtac contlub_cfun_arg 1),
	(rtac (monofun_Iwhen3 RS ch2ch_monofun) 1),
	(atac 1),
	(rtac lub_equal2 1),
	(rtac (chain_mono2 RS exE) 1),
	(atac 2),
	(rtac chain_UU_I_inverse2 1),
	(rtac (inst_ssum_pcpo RS ssubst) 1),
	(etac swap 1),
	(rtac inject_Isinr 1),
	(rtac trans 1),
	(etac sym 1),
	(rtac (strict_IsinlIsinr RS subst) 1),
	(etac notnotD 1),
	(rtac exI 1),
	(strip_tac 1),
	(rtac (ssum_lemma10 RS spec RS exE) 1),
	(atac 1),
	(atac 1),
	(res_inst_tac [("t","Y(i)")] ssubst 1),
	(atac 1),
	(rtac trans 1),
	(rtac cfun_arg_cong 1),
	(rtac Iwhen3 1),
	(res_inst_tac [("Pa","Y(i)=UU")] swap 1),
	(fast_tac HOL_cs 1),
	(dtac notnotD 1),
	(rtac (inst_ssum_pcpo RS ssubst) 1),
	(rtac (strict_IsinlIsinr RS ssubst) 1),
	(res_inst_tac [("t","Y(i)")] ssubst 1),
	(atac 1),
	(fast_tac HOL_cs 1),
	(rtac (Iwhen3 RS ssubst) 1),
	(res_inst_tac [("Pa","Y(i)=UU")] swap 1),
	(fast_tac HOL_cs 1),
	(dtac notnotD 1),
	(rtac (inst_ssum_pcpo RS ssubst) 1),
	(rtac (strict_IsinlIsinr RS ssubst) 1),
	(res_inst_tac [("t","Y(i)")] ssubst 1),
	(atac 1),
	(fast_tac HOL_cs 1),
	(Simp_tac 1),
	(rtac (monofun_fapp2 RS ch2ch_monofun) 1),
	(etac (monofun_Iwhen3 RS ch2ch_monofun) 1),
	(etac (monofun_Iwhen3 RS ch2ch_monofun) 1)
	]);


qed_goal "contlub_Iwhen3" Ssum3.thy "contlub(Iwhen(f)(g))"
 (fn prems =>
	[
	(rtac contlubI 1),
	(strip_tac 1),
	(res_inst_tac [("p","lub(range(Y))")] IssumE 1),
	(etac ssum_lemma11 1),
	(atac 1),
	(etac ssum_lemma12 1),
	(atac 1),
	(atac 1),
	(etac ssum_lemma13 1),
	(atac 1),
	(atac 1)
	]);

qed_goal "cont_Iwhen1" Ssum3.thy "cont(Iwhen)"
 (fn prems =>
	[
	(rtac monocontlub2cont 1),
	(rtac monofun_Iwhen1 1),
	(rtac contlub_Iwhen1 1)
	]);

qed_goal "cont_Iwhen2" Ssum3.thy "cont(Iwhen(f))"
 (fn prems =>
	[
	(rtac monocontlub2cont 1),
	(rtac monofun_Iwhen2 1),
	(rtac contlub_Iwhen2 1)
	]);

qed_goal "cont_Iwhen3" Ssum3.thy "cont(Iwhen(f)(g))"
 (fn prems =>
	[
	(rtac monocontlub2cont 1),
	(rtac monofun_Iwhen3 1),
	(rtac contlub_Iwhen3 1)
	]);

(* ------------------------------------------------------------------------ *)
(* continuous versions of lemmas for 'a ++ 'b                               *)
(* ------------------------------------------------------------------------ *)

qed_goalw "strict_sinl" Ssum3.thy [sinl_def] "sinl`UU =UU"
 (fn prems =>
	[
	(simp_tac (!simpset addsimps [cont_Isinl]) 1),
	(rtac (inst_ssum_pcpo RS sym) 1)
	]);

qed_goalw "strict_sinr" Ssum3.thy [sinr_def] "sinr`UU=UU"
 (fn prems =>
	[
	(simp_tac (!simpset addsimps [cont_Isinr]) 1),
	(rtac (inst_ssum_pcpo RS sym) 1)
	]);

qed_goalw "noteq_sinlsinr" Ssum3.thy [sinl_def,sinr_def] 
	"sinl`a=sinr`b ==> a=UU & b=UU"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac noteq_IsinlIsinr 1),
	(etac box_equals 1),
	(asm_simp_tac (!simpset addsimps [cont_Isinr,cont_Isinl]) 1),
	(asm_simp_tac (!simpset addsimps [cont_Isinr,cont_Isinl]) 1)
	]);

qed_goalw "inject_sinl" Ssum3.thy [sinl_def,sinr_def] 
	"sinl`a1=sinl`a2==> a1=a2"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac inject_Isinl 1),
	(etac box_equals 1),
	(asm_simp_tac (!simpset addsimps [cont_Isinr,cont_Isinl]) 1),
	(asm_simp_tac (!simpset addsimps [cont_Isinr,cont_Isinl]) 1)
	]);

qed_goalw "inject_sinr" Ssum3.thy [sinl_def,sinr_def] 
	"sinr`a1=sinr`a2==> a1=a2"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac inject_Isinr 1),
	(etac box_equals 1),
	(asm_simp_tac (!simpset addsimps [cont_Isinr,cont_Isinl]) 1),
	(asm_simp_tac (!simpset addsimps [cont_Isinr,cont_Isinl]) 1)
	]);


qed_goal "defined_sinl" Ssum3.thy  
	"x~=UU ==> sinl`x ~= UU"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(etac swap 1),
	(rtac inject_sinl 1),
	(rtac (strict_sinl RS ssubst) 1),
	(etac notnotD 1)
	]);

qed_goal "defined_sinr" Ssum3.thy  
	"x~=UU ==> sinr`x ~= UU"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(etac swap 1),
	(rtac inject_sinr 1),
	(rtac (strict_sinr RS ssubst) 1),
	(etac notnotD 1)
	]);

qed_goalw "Exh_Ssum1" Ssum3.thy [sinl_def,sinr_def] 
	"z=UU | (? a. z=sinl`a & a~=UU) | (? b. z=sinr`b & b~=UU)"
 (fn prems =>
	[
	(asm_simp_tac (!simpset addsimps [cont_Isinr,cont_Isinl]) 1),
	(rtac (inst_ssum_pcpo RS ssubst) 1),
	(rtac Exh_Ssum 1)
	]);


qed_goalw "ssumE" Ssum3.thy [sinl_def,sinr_def] 
	"[|p=UU ==> Q ;\
\	!!x.[|p=sinl`x; x~=UU |] ==> Q;\
\	!!y.[|p=sinr`y; y~=UU |] ==> Q|] ==> Q"
 (fn prems =>
	[
	(rtac IssumE 1),
	(resolve_tac prems 1),
	(rtac (inst_ssum_pcpo RS ssubst) 1),
	(atac 1),
	(resolve_tac prems 1),
	(atac 2),
	(asm_simp_tac (!simpset addsimps [cont_Isinr,cont_Isinl]) 1),
	(resolve_tac prems 1),
	(atac 2),
	(asm_simp_tac (!simpset addsimps [cont_Isinr,cont_Isinl]) 1)
	]);


qed_goalw "ssumE2" Ssum3.thy [sinl_def,sinr_def] 
      "[|!!x.[|p=sinl`x|] ==> Q;\
\	 !!y.[|p=sinr`y|] ==> Q|] ==> Q"
 (fn prems =>
	[
	(rtac IssumE2 1),
	(resolve_tac prems 1),
	(rtac (beta_cfun RS ssubst) 1),
	(rtac cont_Isinl 1),
	(atac 1),
	(resolve_tac prems 1),
	(rtac (beta_cfun RS ssubst) 1),
	(rtac cont_Isinr 1),
	(atac 1)
	]);

qed_goalw "sswhen1" Ssum3.thy [sswhen_def,sinl_def,sinr_def] 
	"sswhen`f`g`UU = UU"
 (fn prems =>
	[
	(rtac (inst_ssum_pcpo RS ssubst) 1),
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,
		cont_Iwhen3,cont2cont_CF1L]) 1)),
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,
		cont_Iwhen3,cont2cont_CF1L]) 1)),
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,
		cont_Iwhen3,cont2cont_CF1L]) 1)),
	(Simp_tac 1)
	]);

qed_goalw "sswhen2" Ssum3.thy [sswhen_def,sinl_def,sinr_def] 
	"x~=UU==> sswhen`f`g`(sinl`x) = f`x"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,
		cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1)),
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,
		cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1)),
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,
		cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1)),
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,
		cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1)),
	(Asm_simp_tac  1)
	]);



qed_goalw "sswhen3" Ssum3.thy [sswhen_def,sinl_def,sinr_def] 
	"x~=UU==> sswhen`f`g`(sinr`x) = g`x"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,
		cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1)),
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,
		cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1)),
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,
		cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1)),
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,
		cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1)),
	(Asm_simp_tac 1)
	]);


qed_goalw "less_ssum4a" Ssum3.thy [sinl_def,sinr_def] 
	"(sinl`x << sinl`y) = (x << y)"
 (fn prems =>
	[
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,
		cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1)),
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,
		cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1)),
	(rtac less_ssum3a 1)
	]);

qed_goalw "less_ssum4b" Ssum3.thy [sinl_def,sinr_def] 
	"(sinr`x << sinr`y) = (x << y)"
 (fn prems =>
	[
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,
		cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1)),
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,
		cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1)),
	(rtac less_ssum3b 1)
	]);

qed_goalw "less_ssum4c" Ssum3.thy [sinl_def,sinr_def] 
	"(sinl`x << sinr`y) = (x = UU)"
 (fn prems =>
	[
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,
		cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1)),
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,
		cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1)),
	(rtac less_ssum3c 1)
	]);

qed_goalw "less_ssum4d" Ssum3.thy [sinl_def,sinr_def] 
	"(sinr`x << sinl`y) = (x = UU)"
 (fn prems =>
	[
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,
		cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1)),
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,
		cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1)),
	(rtac less_ssum3d 1)
	]);

qed_goalw "ssum_chainE" Ssum3.thy [sinl_def,sinr_def] 
	"is_chain(Y) ==> (!i.? x.(Y i)=sinl`x)|(!i.? y.(Y i)=sinr`y)"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(asm_simp_tac (!simpset addsimps [cont_Isinr,cont_Isinl]) 1),
	(etac ssum_lemma4 1)
	]);


qed_goalw "thelub_ssum2a" Ssum3.thy [sinl_def,sinr_def,sswhen_def] 
"[| is_chain(Y); !i.? x. Y(i) = sinl`x |] ==>\ 
\   lub(range(Y)) = sinl`(lub(range(%i. sswhen`(LAM x. x)`(LAM y. UU)`(Y i))))"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,		cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1)),
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,		cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1)),
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,		cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1)),
	(rtac (beta_cfun RS ext RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,		cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1)),
	(rtac thelub_ssum1a 1),
	(atac 1),
	(rtac allI 1),
	(etac allE 1),
	(etac exE 1),
	(rtac exI 1),
	(etac box_equals 1),
	(rtac refl 1),
	(asm_simp_tac (!simpset addsimps [cont_Isinl]) 1)
	]);

qed_goalw "thelub_ssum2b" Ssum3.thy [sinl_def,sinr_def,sswhen_def] 
"[| is_chain(Y); !i.? x. Y(i) = sinr`x |] ==>\ 
\   lub(range(Y)) = sinr`(lub(range(%i. sswhen`(LAM y. UU)`(LAM x.x)`(Y i))))"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,
		cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1)),
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,
		cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1)),
	(rtac (beta_cfun RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,
		cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1)),
	(rtac (beta_cfun RS ext RS ssubst) 1),
	(REPEAT (resolve_tac (cont_lemmas @ [cont_Iwhen1,cont_Iwhen2,
		cont_Iwhen3,cont_Isinl,cont_Isinr,cont2cont_CF1L]) 1)),
	(rtac thelub_ssum1b 1),
	(atac 1),
	(rtac allI 1),
	(etac allE 1),
	(etac exE 1),
	(rtac exI 1),
	(etac box_equals 1),
	(rtac refl 1),
	(asm_simp_tac (!simpset addsimps 
	[cont_Isinr,cont_Isinl,cont_Iwhen1,cont_Iwhen2,
	cont_Iwhen3]) 1)
	]);

qed_goalw "thelub_ssum2a_rev" Ssum3.thy [sinl_def,sinr_def] 
	"[| is_chain(Y); lub(range(Y)) = sinl`x|] ==> !i.? x.Y(i)=sinl`x"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(asm_simp_tac (!simpset addsimps 
	[cont_Isinr,cont_Isinl,cont_Iwhen1,cont_Iwhen2,
	cont_Iwhen3]) 1),
	(etac ssum_lemma9 1),
	(asm_simp_tac (!simpset addsimps 
	[cont_Isinr,cont_Isinl,cont_Iwhen1,cont_Iwhen2,
	cont_Iwhen3]) 1)
	]);

qed_goalw "thelub_ssum2b_rev" Ssum3.thy [sinl_def,sinr_def] 
	"[| is_chain(Y); lub(range(Y)) = sinr`x|] ==> !i.? x.Y(i)=sinr`x"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(asm_simp_tac (!simpset addsimps 
	[cont_Isinr,cont_Isinl,cont_Iwhen1,cont_Iwhen2,
	cont_Iwhen3]) 1),
	(etac ssum_lemma10 1),
	(asm_simp_tac (!simpset addsimps 
	[cont_Isinr,cont_Isinl,cont_Iwhen1,cont_Iwhen2,
	cont_Iwhen3]) 1)
	]);

qed_goal "thelub_ssum3" Ssum3.thy  
"is_chain(Y) ==>\ 
\   lub(range(Y)) = sinl`(lub(range(%i. sswhen`(LAM x. x)`(LAM y.UU)`(Y i))))\
\ | lub(range(Y)) = sinr`(lub(range(%i. sswhen`(LAM y. UU)`(LAM x.x)`(Y i))))"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac (ssum_chainE RS disjE) 1),
	(atac 1),
	(rtac disjI1 1),
	(etac thelub_ssum2a 1),
	(atac 1),
	(rtac disjI2 1),
	(etac thelub_ssum2b 1),
	(atac 1)
	]);


qed_goal "sswhen4" Ssum3.thy  
	"sswhen`sinl`sinr`z=z"
 (fn prems =>
	[
	(res_inst_tac [("p","z")] ssumE 1),
	(asm_simp_tac (!simpset addsimps [sswhen1,sswhen2,sswhen3]) 1),
	(asm_simp_tac (!simpset addsimps [sswhen1,sswhen2,sswhen3]) 1),
	(asm_simp_tac (!simpset addsimps [sswhen1,sswhen2,sswhen3]) 1)
	]);


(* ------------------------------------------------------------------------ *)
(* install simplifier for Ssum                                              *)
(* ------------------------------------------------------------------------ *)

Addsimps [strict_sinl,strict_sinr,sswhen1,sswhen2,sswhen3];