(* Title: HOLCF/sprod3.thy
ID: $Id$
Author: Franz Regensburger
Copyright 1993 Technische Universitaet Muenchen
Lemmas for Sprod3.thy
*)
open Sprod3;
(* ------------------------------------------------------------------------ *)
(* continuity of Ispair, Isfst, Issnd *)
(* ------------------------------------------------------------------------ *)
qed_goal "sprod3_lemma1" Sprod3.thy
"[| is_chain(Y); x~= UU; lub(range(Y))~= UU |] ==>\
\ Ispair (lub(range Y)) x =\
\ Ispair (lub(range(%i. Isfst(Ispair(Y i) x)))) \
\ (lub(range(%i. Issnd(Ispair(Y i) x))))"
(fn prems =>
[
(cut_facts_tac prems 1),
(res_inst_tac [("f1","Ispair")] (arg_cong RS cong) 1),
(rtac lub_equal 1),
(atac 1),
(rtac (monofun_Isfst RS ch2ch_monofun) 1),
(rtac ch2ch_fun 1),
(rtac (monofun_Ispair1 RS ch2ch_monofun) 1),
(atac 1),
(rtac allI 1),
(asm_simp_tac Sprod0_ss 1),
(rtac sym 1),
(rtac lub_chain_maxelem 1),
(res_inst_tac [("P","%j.~Y(j)=UU")] exE 1),
(rtac (not_all RS iffD1) 1),
(res_inst_tac [("Q","lub(range(Y)) = UU")] contrapos 1),
(atac 1),
(rtac chain_UU_I_inverse 1),
(atac 1),
(rtac exI 1),
(etac Issnd2 1),
(rtac allI 1),
(res_inst_tac [("Q","Y(i)=UU")] (excluded_middle RS disjE) 1),
(asm_simp_tac Sprod0_ss 1),
(rtac refl_less 1),
(res_inst_tac [("s","UU"),("t","Y(i)")] subst 1),
(etac sym 1),
(asm_simp_tac Sprod0_ss 1),
(rtac minimal 1)
]);
qed_goal "sprod3_lemma2" Sprod3.thy
"[| is_chain(Y); x ~= UU; lub(range(Y)) = UU |] ==>\
\ Ispair (lub(range Y)) x =\
\ Ispair (lub(range(%i. Isfst(Ispair(Y i) x))))\
\ (lub(range(%i. Issnd(Ispair(Y i) x))))"
(fn prems =>
[
(cut_facts_tac prems 1),
(res_inst_tac [("s","UU"),("t","lub(range(Y))")] ssubst 1),
(atac 1),
(rtac trans 1),
(rtac strict_Ispair1 1),
(rtac (strict_Ispair RS sym) 1),
(rtac disjI1 1),
(rtac chain_UU_I_inverse 1),
(rtac allI 1),
(asm_simp_tac Sprod0_ss 1),
(etac (chain_UU_I RS spec) 1),
(atac 1)
]);
qed_goal "sprod3_lemma3" Sprod3.thy
"[| is_chain(Y); x = UU |] ==>\
\ Ispair (lub(range Y)) x =\
\ Ispair (lub(range(%i. Isfst(Ispair (Y i) x))))\
\ (lub(range(%i. Issnd(Ispair (Y i) x))))"
(fn prems =>
[
(cut_facts_tac prems 1),
(res_inst_tac [("s","UU"),("t","x")] ssubst 1),
(atac 1),
(rtac trans 1),
(rtac strict_Ispair2 1),
(rtac (strict_Ispair RS sym) 1),
(rtac disjI1 1),
(rtac chain_UU_I_inverse 1),
(rtac allI 1),
(simp_tac Sprod0_ss 1)
]);
qed_goal "contlub_Ispair1" Sprod3.thy "contlub(Ispair)"
(fn prems =>
[
(rtac contlubI 1),
(strip_tac 1),
(rtac (expand_fun_eq RS iffD2) 1),
(strip_tac 1),
(stac (lub_fun RS thelubI) 1),
(etac (monofun_Ispair1 RS ch2ch_monofun) 1),
(rtac trans 1),
(rtac (thelub_sprod RS sym) 2),
(rtac ch2ch_fun 2),
(etac (monofun_Ispair1 RS ch2ch_monofun) 2),
(res_inst_tac [("Q","x=UU")] (excluded_middle RS disjE) 1),
(res_inst_tac
[("Q","lub(range(Y))=UU")] (excluded_middle RS disjE) 1),
(etac sprod3_lemma1 1),
(atac 1),
(atac 1),
(etac sprod3_lemma2 1),
(atac 1),
(atac 1),
(etac sprod3_lemma3 1),
(atac 1)
]);
qed_goal "sprod3_lemma4" Sprod3.thy
"[| is_chain(Y); x ~= UU; lub(range(Y)) ~= UU |] ==>\
\ Ispair x (lub(range Y)) =\
\ Ispair (lub(range(%i. Isfst (Ispair x (Y i)))))\
\ (lub(range(%i. Issnd (Ispair x (Y i)))))"
(fn prems =>
[
(cut_facts_tac prems 1),
(res_inst_tac [("f1","Ispair")] (arg_cong RS cong) 1),
(rtac sym 1),
(rtac lub_chain_maxelem 1),
(res_inst_tac [("P","%j.Y(j)~=UU")] exE 1),
(rtac (not_all RS iffD1) 1),
(res_inst_tac [("Q","lub(range(Y)) = UU")] contrapos 1),
(atac 1),
(rtac chain_UU_I_inverse 1),
(atac 1),
(rtac exI 1),
(etac Isfst2 1),
(rtac allI 1),
(res_inst_tac [("Q","Y(i)=UU")] (excluded_middle RS disjE) 1),
(asm_simp_tac Sprod0_ss 1),
(rtac refl_less 1),
(res_inst_tac [("s","UU"),("t","Y(i)")] subst 1),
(etac sym 1),
(asm_simp_tac Sprod0_ss 1),
(rtac minimal 1),
(rtac lub_equal 1),
(atac 1),
(rtac (monofun_Issnd RS ch2ch_monofun) 1),
(rtac (monofun_Ispair2 RS ch2ch_monofun) 1),
(atac 1),
(rtac allI 1),
(asm_simp_tac Sprod0_ss 1)
]);
qed_goal "sprod3_lemma5" Sprod3.thy
"[| is_chain(Y); x ~= UU; lub(range(Y)) = UU |] ==>\
\ Ispair x (lub(range Y)) =\
\ Ispair (lub(range(%i. Isfst(Ispair x (Y i)))))\
\ (lub(range(%i. Issnd(Ispair x (Y i)))))"
(fn prems =>
[
(cut_facts_tac prems 1),
(res_inst_tac [("s","UU"),("t","lub(range(Y))")] ssubst 1),
(atac 1),
(rtac trans 1),
(rtac strict_Ispair2 1),
(rtac (strict_Ispair RS sym) 1),
(rtac disjI2 1),
(rtac chain_UU_I_inverse 1),
(rtac allI 1),
(asm_simp_tac Sprod0_ss 1),
(etac (chain_UU_I RS spec) 1),
(atac 1)
]);
qed_goal "sprod3_lemma6" Sprod3.thy
"[| is_chain(Y); x = UU |] ==>\
\ Ispair x (lub(range Y)) =\
\ Ispair (lub(range(%i. Isfst (Ispair x (Y i)))))\
\ (lub(range(%i. Issnd (Ispair x (Y i)))))"
(fn prems =>
[
(cut_facts_tac prems 1),
(res_inst_tac [("s","UU"),("t","x")] ssubst 1),
(atac 1),
(rtac trans 1),
(rtac strict_Ispair1 1),
(rtac (strict_Ispair RS sym) 1),
(rtac disjI1 1),
(rtac chain_UU_I_inverse 1),
(rtac allI 1),
(simp_tac Sprod0_ss 1)
]);
qed_goal "contlub_Ispair2" Sprod3.thy "contlub(Ispair(x))"
(fn prems =>
[
(rtac contlubI 1),
(strip_tac 1),
(rtac trans 1),
(rtac (thelub_sprod RS sym) 2),
(etac (monofun_Ispair2 RS ch2ch_monofun) 2),
(res_inst_tac [("Q","x=UU")] (excluded_middle RS disjE) 1),
(res_inst_tac [("Q","lub(range(Y))=UU")]
(excluded_middle RS disjE) 1),
(etac sprod3_lemma4 1),
(atac 1),
(atac 1),
(etac sprod3_lemma5 1),
(atac 1),
(atac 1),
(etac sprod3_lemma6 1),
(atac 1)
]);
qed_goal "cont_Ispair1" Sprod3.thy "cont(Ispair)"
(fn prems =>
[
(rtac monocontlub2cont 1),
(rtac monofun_Ispair1 1),
(rtac contlub_Ispair1 1)
]);
qed_goal "cont_Ispair2" Sprod3.thy "cont(Ispair(x))"
(fn prems =>
[
(rtac monocontlub2cont 1),
(rtac monofun_Ispair2 1),
(rtac contlub_Ispair2 1)
]);
qed_goal "contlub_Isfst" Sprod3.thy "contlub(Isfst)"
(fn prems =>
[
(rtac contlubI 1),
(strip_tac 1),
(stac (lub_sprod RS thelubI) 1),
(atac 1),
(res_inst_tac [("Q","lub(range(%i. Issnd(Y(i))))=UU")]
(excluded_middle RS disjE) 1),
(asm_simp_tac Sprod0_ss 1),
(res_inst_tac [("s","UU"),("t","lub(range(%i. Issnd(Y(i))))")]
ssubst 1),
(atac 1),
(rtac trans 1),
(asm_simp_tac Sprod0_ss 1),
(rtac sym 1),
(rtac chain_UU_I_inverse 1),
(rtac allI 1),
(rtac strict_Isfst 1),
(rtac swap 1),
(etac (defined_IsfstIssnd RS conjunct2) 2),
(fast_tac (HOL_cs addSDs [monofun_Issnd RS ch2ch_monofun RS
chain_UU_I RS spec]) 1)
]);
qed_goal "contlub_Issnd" Sprod3.thy "contlub(Issnd)"
(fn prems =>
[
(rtac contlubI 1),
(strip_tac 1),
(stac (lub_sprod RS thelubI) 1),
(atac 1),
(res_inst_tac [("Q","lub(range(%i. Isfst(Y(i))))=UU")]
(excluded_middle RS disjE) 1),
(asm_simp_tac Sprod0_ss 1),
(res_inst_tac [("s","UU"),("t","lub(range(%i. Isfst(Y(i))))")]
ssubst 1),
(atac 1),
(asm_simp_tac Sprod0_ss 1),
(rtac sym 1),
(rtac chain_UU_I_inverse 1),
(rtac allI 1),
(rtac strict_Issnd 1),
(rtac swap 1),
(etac (defined_IsfstIssnd RS conjunct1) 2),
(fast_tac (HOL_cs addSDs [monofun_Isfst RS ch2ch_monofun RS
chain_UU_I RS spec]) 1)
]);
qed_goal "cont_Isfst" Sprod3.thy "cont(Isfst)"
(fn prems =>
[
(rtac monocontlub2cont 1),
(rtac monofun_Isfst 1),
(rtac contlub_Isfst 1)
]);
qed_goal "cont_Issnd" Sprod3.thy "cont(Issnd)"
(fn prems =>
[
(rtac monocontlub2cont 1),
(rtac monofun_Issnd 1),
(rtac contlub_Issnd 1)
]);
(*
--------------------------------------------------------------------------
more lemmas for Sprod3.thy
--------------------------------------------------------------------------
*)
qed_goal "spair_eq" Sprod3.thy "[|x1=x2;y1=y2|] ==> (|x1,y1|) = (|x2,y2|)"
(fn prems =>
[
(cut_facts_tac prems 1),
(fast_tac HOL_cs 1)
]);
(* ------------------------------------------------------------------------ *)
(* convert all lemmas to the continuous versions *)
(* ------------------------------------------------------------------------ *)
qed_goalw "beta_cfun_sprod" Sprod3.thy [spair_def]
"(LAM x y.Ispair x y)`a`b = Ispair a b"
(fn prems =>
[
(stac beta_cfun 1),
(cont_tac 1),
(rtac cont_Ispair2 1),
(rtac cont2cont_CF1L 1),
(rtac cont_Ispair1 1),
(stac beta_cfun 1),
(rtac cont_Ispair2 1),
(rtac refl 1)
]);
qed_goalw "inject_spair" Sprod3.thy [spair_def]
"[| aa~=UU ; ba~=UU ; (|a,b|)=(|aa,ba|) |] ==> a=aa & b=ba"
(fn prems =>
[
(cut_facts_tac prems 1),
(etac inject_Ispair 1),
(atac 1),
(etac box_equals 1),
(rtac beta_cfun_sprod 1),
(rtac beta_cfun_sprod 1)
]);
qed_goalw "inst_sprod_pcpo2" Sprod3.thy [spair_def] "UU = (|UU,UU|)"
(fn prems =>
[
(rtac sym 1),
(rtac trans 1),
(rtac beta_cfun_sprod 1),
(rtac sym 1),
(rtac inst_sprod_pcpo 1)
]);
qed_goalw "strict_spair" Sprod3.thy [spair_def]
"(a=UU | b=UU) ==> (|a,b|)=UU"
(fn prems =>
[
(cut_facts_tac prems 1),
(rtac trans 1),
(rtac beta_cfun_sprod 1),
(rtac trans 1),
(rtac (inst_sprod_pcpo RS sym) 2),
(etac strict_Ispair 1)
]);
qed_goalw "strict_spair1" Sprod3.thy [spair_def] "(|UU,b|) = UU"
(fn prems =>
[
(stac beta_cfun_sprod 1),
(rtac trans 1),
(rtac (inst_sprod_pcpo RS sym) 2),
(rtac strict_Ispair1 1)
]);
qed_goalw "strict_spair2" Sprod3.thy [spair_def] "(|a,UU|) = UU"
(fn prems =>
[
(stac beta_cfun_sprod 1),
(rtac trans 1),
(rtac (inst_sprod_pcpo RS sym) 2),
(rtac strict_Ispair2 1)
]);
qed_goalw "strict_spair_rev" Sprod3.thy [spair_def]
"(|x,y|)~=UU ==> ~x=UU & ~y=UU"
(fn prems =>
[
(cut_facts_tac prems 1),
(rtac strict_Ispair_rev 1),
(rtac (beta_cfun_sprod RS subst) 1),
(rtac (inst_sprod_pcpo RS subst) 1),
(atac 1)
]);
qed_goalw "defined_spair_rev" Sprod3.thy [spair_def]
"(|a,b|) = UU ==> (a = UU | b = UU)"
(fn prems =>
[
(cut_facts_tac prems 1),
(rtac defined_Ispair_rev 1),
(rtac (beta_cfun_sprod RS subst) 1),
(rtac (inst_sprod_pcpo RS subst) 1),
(atac 1)
]);
qed_goalw "defined_spair" Sprod3.thy [spair_def]
"[|a~=UU; b~=UU|] ==> (|a,b|) ~= UU"
(fn prems =>
[
(cut_facts_tac prems 1),
(stac beta_cfun_sprod 1),
(stac inst_sprod_pcpo 1),
(etac defined_Ispair 1),
(atac 1)
]);
qed_goalw "Exh_Sprod2" Sprod3.thy [spair_def]
"z=UU | (? a b. z=(|a,b|) & a~=UU & b~=UU)"
(fn prems =>
[
(rtac (Exh_Sprod RS disjE) 1),
(rtac disjI1 1),
(stac inst_sprod_pcpo 1),
(atac 1),
(rtac disjI2 1),
(etac exE 1),
(etac exE 1),
(rtac exI 1),
(rtac exI 1),
(rtac conjI 1),
(stac beta_cfun_sprod 1),
(fast_tac HOL_cs 1),
(fast_tac HOL_cs 1)
]);
qed_goalw "sprodE" Sprod3.thy [spair_def]
"[|p=UU ==> Q;!!x y. [|p=(|x,y|);x~=UU ; y~=UU|] ==> Q|] ==> Q"
(fn prems =>
[
(rtac IsprodE 1),
(resolve_tac prems 1),
(stac inst_sprod_pcpo 1),
(atac 1),
(resolve_tac prems 1),
(atac 2),
(atac 2),
(stac beta_cfun_sprod 1),
(atac 1)
]);
qed_goalw "strict_sfst" Sprod3.thy [sfst_def]
"p=UU==>sfst`p=UU"
(fn prems =>
[
(cut_facts_tac prems 1),
(stac beta_cfun 1),
(rtac cont_Isfst 1),
(rtac strict_Isfst 1),
(rtac (inst_sprod_pcpo RS subst) 1),
(atac 1)
]);
qed_goalw "strict_sfst1" Sprod3.thy [sfst_def,spair_def]
"sfst`(|UU,y|) = UU"
(fn prems =>
[
(stac beta_cfun_sprod 1),
(stac beta_cfun 1),
(rtac cont_Isfst 1),
(rtac strict_Isfst1 1)
]);
qed_goalw "strict_sfst2" Sprod3.thy [sfst_def,spair_def]
"sfst`(|x,UU|) = UU"
(fn prems =>
[
(stac beta_cfun_sprod 1),
(stac beta_cfun 1),
(rtac cont_Isfst 1),
(rtac strict_Isfst2 1)
]);
qed_goalw "strict_ssnd" Sprod3.thy [ssnd_def]
"p=UU==>ssnd`p=UU"
(fn prems =>
[
(cut_facts_tac prems 1),
(stac beta_cfun 1),
(rtac cont_Issnd 1),
(rtac strict_Issnd 1),
(rtac (inst_sprod_pcpo RS subst) 1),
(atac 1)
]);
qed_goalw "strict_ssnd1" Sprod3.thy [ssnd_def,spair_def]
"ssnd`(|UU,y|) = UU"
(fn prems =>
[
(stac beta_cfun_sprod 1),
(stac beta_cfun 1),
(rtac cont_Issnd 1),
(rtac strict_Issnd1 1)
]);
qed_goalw "strict_ssnd2" Sprod3.thy [ssnd_def,spair_def]
"ssnd`(|x,UU|) = UU"
(fn prems =>
[
(stac beta_cfun_sprod 1),
(stac beta_cfun 1),
(rtac cont_Issnd 1),
(rtac strict_Issnd2 1)
]);
qed_goalw "sfst2" Sprod3.thy [sfst_def,spair_def]
"y~=UU ==>sfst`(|x,y|)=x"
(fn prems =>
[
(cut_facts_tac prems 1),
(stac beta_cfun_sprod 1),
(stac beta_cfun 1),
(rtac cont_Isfst 1),
(etac Isfst2 1)
]);
qed_goalw "ssnd2" Sprod3.thy [ssnd_def,spair_def]
"x~=UU ==>ssnd`(|x,y|)=y"
(fn prems =>
[
(cut_facts_tac prems 1),
(stac beta_cfun_sprod 1),
(stac beta_cfun 1),
(rtac cont_Issnd 1),
(etac Issnd2 1)
]);
qed_goalw "defined_sfstssnd" Sprod3.thy [sfst_def,ssnd_def,spair_def]
"p~=UU ==> sfst`p ~=UU & ssnd`p ~=UU"
(fn prems =>
[
(cut_facts_tac prems 1),
(stac beta_cfun 1),
(rtac cont_Issnd 1),
(stac beta_cfun 1),
(rtac cont_Isfst 1),
(rtac defined_IsfstIssnd 1),
(rtac (inst_sprod_pcpo RS subst) 1),
(atac 1)
]);
qed_goalw "surjective_pairing_Sprod2" Sprod3.thy
[sfst_def,ssnd_def,spair_def] "(|sfst`p , ssnd`p|) = p"
(fn prems =>
[
(stac beta_cfun_sprod 1),
(stac beta_cfun 1),
(rtac cont_Issnd 1),
(stac beta_cfun 1),
(rtac cont_Isfst 1),
(rtac (surjective_pairing_Sprod RS sym) 1)
]);
qed_goalw "less_sprod5b" Sprod3.thy [sfst_def,ssnd_def,spair_def]
"p1~=UU ==> (p1<<p2) = (sfst`p1<<sfst`p2 & ssnd`p1<<ssnd`p2)"
(fn prems =>
[
(cut_facts_tac prems 1),
(stac beta_cfun 1),
(rtac cont_Issnd 1),
(stac beta_cfun 1),
(rtac cont_Issnd 1),
(stac beta_cfun 1),
(rtac cont_Isfst 1),
(stac beta_cfun 1),
(rtac cont_Isfst 1),
(rtac less_sprod3b 1),
(rtac (inst_sprod_pcpo RS subst) 1),
(atac 1)
]);
qed_goalw "less_sprod5c" Sprod3.thy [sfst_def,ssnd_def,spair_def]
"[|(|xa,ya|) << (|x,y|);xa~=UU;ya~=UU;x~=UU;y~=UU|] ==>xa<<x & ya << y"
(fn prems =>
[
(cut_facts_tac prems 1),
(rtac less_sprod4c 1),
(REPEAT (atac 2)),
(rtac (beta_cfun_sprod RS subst) 1),
(rtac (beta_cfun_sprod RS subst) 1),
(atac 1)
]);
qed_goalw "lub_sprod2" Sprod3.thy [sfst_def,ssnd_def,spair_def]
"[|is_chain(S)|] ==> range(S) <<| \
\ (| lub(range(%i.sfst`(S i))), lub(range(%i.ssnd`(S i))) |)"
(fn prems =>
[
(cut_facts_tac prems 1),
(stac beta_cfun_sprod 1),
(stac (beta_cfun RS ext) 1),
(rtac cont_Issnd 1),
(stac (beta_cfun RS ext) 1),
(rtac cont_Isfst 1),
(rtac lub_sprod 1),
(resolve_tac prems 1)
]);
bind_thm ("thelub_sprod2", lub_sprod2 RS thelubI);
(*
"is_chain ?S1 ==>
lub (range ?S1) =
(|lub (range (%i. sfst`(?S1 i))), lub (range (%i. ssnd`(?S1 i)))|)" : thm
*)
qed_goalw "ssplit1" Sprod3.thy [ssplit_def]
"ssplit`f`UU=UU"
(fn prems =>
[
(stac beta_cfun 1),
(cont_tacR 1),
(stac strictify1 1),
(rtac refl 1)
]);
qed_goalw "ssplit2" Sprod3.thy [ssplit_def]
"[|x~=UU;y~=UU|] ==> ssplit`f`(|x,y|)= f`x`y"
(fn prems =>
[
(stac beta_cfun 1),
(cont_tacR 1),
(stac strictify2 1),
(rtac defined_spair 1),
(resolve_tac prems 1),
(resolve_tac prems 1),
(stac beta_cfun 1),
(cont_tacR 1),
(stac sfst2 1),
(resolve_tac prems 1),
(stac ssnd2 1),
(resolve_tac prems 1),
(rtac refl 1)
]);
qed_goalw "ssplit3" Sprod3.thy [ssplit_def]
"ssplit`spair`z=z"
(fn prems =>
[
(stac beta_cfun 1),
(cont_tacR 1),
(case_tac "z=UU" 1),
(hyp_subst_tac 1),
(rtac strictify1 1),
(rtac trans 1),
(rtac strictify2 1),
(atac 1),
(stac beta_cfun 1),
(cont_tacR 1),
(rtac surjective_pairing_Sprod2 1)
]);
(* ------------------------------------------------------------------------ *)
(* install simplifier for Sprod *)
(* ------------------------------------------------------------------------ *)
val Sprod_rews = [strict_spair1,strict_spair2,strict_sfst1,strict_sfst2,
strict_ssnd1,strict_ssnd2,sfst2,ssnd2,defined_spair,
ssplit1,ssplit2];
Addsimps [strict_spair1,strict_spair2,strict_sfst1,strict_sfst2,
strict_ssnd1,strict_ssnd2,sfst2,ssnd2,defined_spair,
ssplit1,ssplit2];