src/HOLCF/cprod3.ML

ISABELLE_TMP_PREFIX;

(*  Title: 	HOLCF/cprod3.ML
    ID:         $Id$
    Author: 	Franz Regensburger
    Copyright   1993 Technische Universitaet Muenchen

Lemmas for Cprod3.thy 
*)

open Cprod3;

(* ------------------------------------------------------------------------ *)
(* continuity of <_,_> , fst, snd                                           *)
(* ------------------------------------------------------------------------ *)

val Cprod3_lemma1 = prove_goal Cprod3.thy 
"is_chain(Y::(nat=>'a)) ==>\
\ <lub(range(Y)),(x::'b)> =\
\ <lub(range(%i. fst(<Y(i),x>))),lub(range(%i. snd(<Y(i),x>)))>"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(res_inst_tac [("f1","Pair")] (arg_cong RS cong) 1),
	(rtac lub_equal 1),
	(atac 1),
	(rtac (monofun_fst RS ch2ch_monofun) 1),
	(rtac ch2ch_fun 1),
	(rtac (monofun_pair1 RS ch2ch_monofun) 1),
	(atac 1),
	(rtac allI 1),
	(simp_tac pair_ss 1),
	(rtac sym 1),
	(simp_tac pair_ss 1),
	(rtac (lub_const RS thelubI) 1)
	]);

val contlub_pair1 = prove_goal Cprod3.thy "contlub(Pair)"
 (fn prems =>
	[
	(rtac contlubI 1),
	(strip_tac 1),
	(rtac (expand_fun_eq RS iffD2) 1),
	(strip_tac 1),
	(rtac (lub_fun RS thelubI RS ssubst) 1),
	(etac (monofun_pair1 RS ch2ch_monofun) 1),
	(rtac trans 1),
	(rtac (thelub_cprod RS sym) 2),
	(rtac ch2ch_fun 2),
	(etac (monofun_pair1 RS ch2ch_monofun) 2),
	(etac Cprod3_lemma1 1)
	]);

val Cprod3_lemma2 = prove_goal Cprod3.thy 
"is_chain(Y::(nat=>'a)) ==>\
\ <(x::'b),lub(range(Y))> =\
\ <lub(range(%i. fst(<x,Y(i)>))),lub(range(%i. snd(<x,Y(i)>)))>"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(res_inst_tac [("f1","Pair")] (arg_cong RS cong) 1),
	(rtac sym 1),
	(simp_tac pair_ss 1),
	(rtac (lub_const RS thelubI) 1),
	(rtac lub_equal 1),
	(atac 1),
	(rtac (monofun_snd RS ch2ch_monofun) 1),
	(rtac (monofun_pair2 RS ch2ch_monofun) 1),
	(atac 1),
	(rtac allI 1),
	(simp_tac pair_ss 1)
	]);

val contlub_pair2 = prove_goal Cprod3.thy "contlub(Pair(x))"
 (fn prems =>
	[
	(rtac contlubI 1),
	(strip_tac 1),
	(rtac trans 1),
	(rtac (thelub_cprod RS sym) 2),
	(etac (monofun_pair2 RS ch2ch_monofun) 2),
	(etac Cprod3_lemma2 1)
	]);

val contX_pair1 = prove_goal Cprod3.thy "contX(Pair)"
(fn prems =>
	[
	(rtac monocontlub2contX 1),
	(rtac monofun_pair1 1),
	(rtac contlub_pair1 1)
	]);

val contX_pair2 = prove_goal Cprod3.thy "contX(Pair(x))"
(fn prems =>
	[
	(rtac monocontlub2contX 1),
	(rtac monofun_pair2 1),
	(rtac contlub_pair2 1)
	]);

val contlub_fst = prove_goal Cprod3.thy "contlub(fst)"
 (fn prems =>
	[
	(rtac contlubI 1),
	(strip_tac 1),
	(rtac (lub_cprod RS thelubI RS ssubst) 1),
	(atac 1),
	(simp_tac pair_ss 1)
	]);

val contlub_snd = prove_goal Cprod3.thy "contlub(snd)"
 (fn prems =>
	[
	(rtac contlubI 1),
	(strip_tac 1),
	(rtac (lub_cprod RS thelubI RS ssubst) 1),
	(atac 1),
	(simp_tac pair_ss 1)
	]);

val contX_fst = prove_goal Cprod3.thy "contX(fst)"
(fn prems =>
	[
	(rtac monocontlub2contX 1),
	(rtac monofun_fst 1),
	(rtac contlub_fst 1)
	]);

val contX_snd = prove_goal Cprod3.thy "contX(snd)"
(fn prems =>
	[
	(rtac monocontlub2contX 1),
	(rtac monofun_snd 1),
	(rtac contlub_snd 1)
	]);

(* 
 -------------------------------------------------------------------------- 
 more lemmas for Cprod3.thy 
 
 -------------------------------------------------------------------------- 
*)

(* ------------------------------------------------------------------------ *)
(* convert all lemmas to the continuous versions                            *)
(* ------------------------------------------------------------------------ *)

val beta_cfun_cprod = prove_goalw Cprod3.thy [cpair_def]
	"(LAM x y.<x,y>)[a][b] = <a,b>"
 (fn prems =>
	[
	(rtac (beta_cfun RS ssubst) 1),
	(contX_tac 1),
	(rtac contX_pair2 1),
	(rtac contX2contX_CF1L 1),
	(rtac contX_pair1 1),
	(rtac (beta_cfun RS ssubst) 1),
	(rtac contX_pair2 1),
	(rtac refl 1)
	]);

val inject_cpair = prove_goalw Cprod3.thy [cpair_def]
	" (a#b)=(aa#ba)  ==> a=aa & b=ba"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(dtac (beta_cfun_cprod RS subst) 1),
	(dtac (beta_cfun_cprod RS subst) 1),
	(etac Pair_inject 1),
	(fast_tac HOL_cs 1)
	]);

val inst_cprod_pcpo2 = prove_goalw Cprod3.thy [cpair_def] "UU = (UU#UU)"
 (fn prems =>
	[
	(rtac sym 1),
	(rtac trans 1),
	(rtac beta_cfun_cprod 1),
	(rtac sym 1),
	(rtac inst_cprod_pcpo 1)
	]);

val defined_cpair_rev = prove_goal Cprod3.thy
 "(a#b) = UU ==> a = UU & b = UU"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(dtac (inst_cprod_pcpo2 RS subst) 1),
	(etac inject_cpair 1)
	]);

val Exh_Cprod2 = prove_goalw Cprod3.thy [cpair_def]
	"? a b. z=(a#b) "
 (fn prems =>
	[
	(rtac PairE 1),
	(rtac exI 1),
	(rtac exI 1),
	(etac (beta_cfun_cprod RS ssubst) 1)
	]);

val cprodE = prove_goalw Cprod3.thy [cpair_def]
"[|!!x y. [|p=(x#y) |] ==> Q|] ==> Q"
 (fn prems =>
	[
	(rtac PairE 1),
	(resolve_tac prems 1),
	(etac (beta_cfun_cprod RS ssubst) 1)
	]);

val cfst2 = prove_goalw Cprod3.thy [cfst_def,cpair_def] 
	"cfst[x#y]=x"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac (beta_cfun_cprod RS ssubst) 1),
	(rtac (beta_cfun RS ssubst) 1),
	(rtac contX_fst 1),
	(simp_tac pair_ss  1)
	]);

val csnd2 = prove_goalw Cprod3.thy [csnd_def,cpair_def] 
	"csnd[x#y]=y"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac (beta_cfun_cprod RS ssubst) 1),
	(rtac (beta_cfun RS ssubst) 1),
	(rtac contX_snd 1),
	(simp_tac pair_ss  1)
	]);

val surjective_pairing_Cprod2 = prove_goalw Cprod3.thy 
	[cfst_def,csnd_def,cpair_def] "(cfst[p] # csnd[p]) = p"
 (fn prems =>
	[
	(rtac (beta_cfun_cprod RS ssubst) 1),
	(rtac (beta_cfun RS ssubst) 1),
	(rtac contX_snd 1),
	(rtac (beta_cfun RS ssubst) 1),
	(rtac contX_fst 1),
	(rtac (surjective_pairing RS sym) 1)
	]);


val less_cprod5b = prove_goalw Cprod3.thy [cfst_def,csnd_def,cpair_def]
 " (p1 << p2) = (cfst[p1]<<cfst[p2] & csnd[p1]<<csnd[p2])"
 (fn prems =>
	[
	(rtac (beta_cfun RS ssubst) 1),
	(rtac contX_snd 1),
	(rtac (beta_cfun RS ssubst) 1),
	(rtac contX_snd 1),
	(rtac (beta_cfun RS ssubst) 1),
	(rtac contX_fst 1),
	(rtac (beta_cfun RS ssubst) 1),
	(rtac contX_fst 1),
	(rtac less_cprod3b 1)
	]);

val less_cprod5c = prove_goalw Cprod3.thy [cfst_def,csnd_def,cpair_def]
 "xa#ya << x#y ==>xa<<x & ya << y"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac less_cprod4c 1),
	(dtac (beta_cfun_cprod RS subst) 1),
	(dtac (beta_cfun_cprod RS subst) 1),
	(atac 1)
	]);


val lub_cprod2 = prove_goalw Cprod3.thy [cfst_def,csnd_def,cpair_def]
"[|is_chain(S)|] ==> range(S) <<| \
\ (lub(range(%i.cfst[S(i)])) # lub(range(%i.csnd[S(i)])))"
 (fn prems =>
	[
	(cut_facts_tac prems 1),
	(rtac (beta_cfun_cprod RS ssubst) 1),
	(rtac (beta_cfun RS ext RS ssubst) 1),
	(rtac contX_snd 1),
	(rtac (beta_cfun RS ext RS ssubst) 1),
	(rtac contX_fst 1),
	(rtac lub_cprod 1),
	(atac 1)
	]);

val thelub_cprod2 = (lub_cprod2 RS thelubI);
(*  "is_chain(?S1) ==> lub(range(?S1)) =                         *)
(*    lub(range(%i. cfst[?S1(i)]))#lub(range(%i. csnd[?S1(i)]))" *)

val csplit2 = prove_goalw Cprod3.thy [csplit_def]
	"csplit[f][x#y]=f[x][y]"
 (fn prems =>
	[
	(rtac (beta_cfun RS ssubst) 1),
	(contX_tacR 1),
	(simp_tac Cfun_ss 1),
	(simp_tac (Cfun_ss addsimps [cfst2,csnd2]) 1)
	]);

val csplit3 = prove_goalw Cprod3.thy [csplit_def]
  "csplit[cpair][z]=z"
 (fn prems =>
	[
	(rtac (beta_cfun RS ssubst) 1),
	(contX_tacR 1),
	(simp_tac (Cfun_ss addsimps [surjective_pairing_Cprod2]) 1)
	]);

(* ------------------------------------------------------------------------ *)
(* install simplifier for Cprod                                             *)
(* ------------------------------------------------------------------------ *)

val Cprod_rews = [cfst2,csnd2,csplit2];

val Cprod_ss = Cfun_ss addsimps Cprod_rews;