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src/HOLCF/Ssum2.ML
changeset 243 c22b85994e17
child 892 d0dc8d057929 
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOLCF/Ssum2.ML	Wed Jan 19 17:35:01 1994 +0100
@@ -0,0 +1,419 @@
+(*  Title: 	HOLCF/ssum2.ML
+    ID:         $Id$
+    Author: 	Franz Regensburger
+    Copyright   1993 Technische Universitaet Muenchen
+
+Lemmas for ssum2.thy
+*)
+
+open Ssum2;
+
+(* ------------------------------------------------------------------------ *)
+(* access to less_ssum in class po                                          *)
+(* ------------------------------------------------------------------------ *)
+
+val less_ssum3a = prove_goal Ssum2.thy 
+	"(Isinl(x) << Isinl(y)) = (x << y)"
+ (fn prems =>
+	[
+	(rtac (inst_ssum_po RS ssubst) 1),
+	(rtac less_ssum2a 1)
+	]);
+
+val less_ssum3b = prove_goal Ssum2.thy 
+	"(Isinr(x) << Isinr(y)) = (x << y)"
+ (fn prems =>
+	[
+	(rtac (inst_ssum_po RS ssubst) 1),
+	(rtac less_ssum2b 1)
+	]);
+
+val less_ssum3c = prove_goal Ssum2.thy 
+	"(Isinl(x) << Isinr(y)) = (x = UU)"
+ (fn prems =>
+	[
+	(rtac (inst_ssum_po RS ssubst) 1),
+	(rtac less_ssum2c 1)
+	]);
+
+val less_ssum3d = prove_goal Ssum2.thy 
+	"(Isinr(x) << Isinl(y)) = (x = UU)"
+ (fn prems =>
+	[
+	(rtac (inst_ssum_po RS ssubst) 1),
+	(rtac less_ssum2d 1)
+	]);
+
+
+(* ------------------------------------------------------------------------ *)
+(* type ssum ++ is pointed                                                  *)
+(* ------------------------------------------------------------------------ *)
+
+val minimal_ssum = prove_goal Ssum2.thy "Isinl(UU) << s"
+ (fn prems =>
+	[
+	(res_inst_tac [("p","s")] IssumE2 1),
+	(hyp_subst_tac 1),
+	(rtac (less_ssum3a RS iffD2) 1),
+	(rtac minimal 1),
+	(hyp_subst_tac 1),
+	(rtac (strict_IsinlIsinr RS ssubst) 1),
+	(rtac (less_ssum3b RS iffD2) 1),
+	(rtac minimal 1)
+	]);
+
+
+(* ------------------------------------------------------------------------ *)
+(* Isinl, Isinr are monotone                                                *)
+(* ------------------------------------------------------------------------ *)
+
+val monofun_Isinl = prove_goalw Ssum2.thy [monofun] "monofun(Isinl)"
+ (fn prems =>
+	[
+	(strip_tac 1),
+	(etac (less_ssum3a RS iffD2) 1)
+	]);
+
+val monofun_Isinr = prove_goalw Ssum2.thy [monofun] "monofun(Isinr)"
+ (fn prems =>
+	[
+	(strip_tac 1),
+	(etac (less_ssum3b RS iffD2) 1)
+	]);
+
+
+(* ------------------------------------------------------------------------ *)
+(* Iwhen is monotone in all arguments                                       *)
+(* ------------------------------------------------------------------------ *)
+
+
+val monofun_Iwhen1 = prove_goalw Ssum2.thy [monofun] "monofun(Iwhen)"
+ (fn prems =>
+	[
+	(strip_tac 1),
+	(rtac (less_fun RS iffD2) 1),
+	(strip_tac 1),
+	(rtac (less_fun RS iffD2) 1),
+	(strip_tac 1),
+	(res_inst_tac [("p","xb")] IssumE 1),
+	(hyp_subst_tac 1),
+	(asm_simp_tac Ssum_ss 1),
+	(asm_simp_tac Ssum_ss 1),
+	(etac monofun_cfun_fun 1),
+	(asm_simp_tac Ssum_ss 1)
+	]);
+
+val monofun_Iwhen2 = prove_goalw Ssum2.thy [monofun] "monofun(Iwhen(f))"
+ (fn prems =>
+	[
+	(strip_tac 1),
+	(rtac (less_fun RS iffD2) 1),
+	(strip_tac 1),
+	(res_inst_tac [("p","xa")] IssumE 1),
+	(hyp_subst_tac 1),
+	(asm_simp_tac Ssum_ss 1),
+	(asm_simp_tac Ssum_ss 1),
+	(asm_simp_tac Ssum_ss 1),
+	(etac monofun_cfun_fun 1)
+	]);
+
+val monofun_Iwhen3 = prove_goalw Ssum2.thy [monofun] "monofun(Iwhen(f)(g))"
+ (fn prems =>
+	[
+	(strip_tac 1),
+	(res_inst_tac [("p","x")] IssumE 1),
+	(hyp_subst_tac 1),
+	(asm_simp_tac Ssum_ss 1),
+	(hyp_subst_tac 1),
+	(res_inst_tac [("p","y")] IssumE 1),
+	(hyp_subst_tac 1),
+	(asm_simp_tac Ssum_ss 1),
+	(res_inst_tac  [("P","xa=UU")] notE 1),
+	(atac 1),
+	(rtac UU_I 1),
+	(rtac (less_ssum3a  RS iffD1) 1),
+	(atac 1),
+	(hyp_subst_tac 1),
+	(asm_simp_tac Ssum_ss 1),
+	(rtac monofun_cfun_arg 1),
+	(etac (less_ssum3a  RS iffD1) 1),
+	(hyp_subst_tac 1),
+	(res_inst_tac [("s","UU"),("t","xa")] subst 1),
+	(etac (less_ssum3c  RS iffD1 RS sym) 1),
+	(asm_simp_tac Ssum_ss 1),
+	(hyp_subst_tac 1),
+	(res_inst_tac [("p","y")] IssumE 1),
+	(hyp_subst_tac 1),
+	(res_inst_tac [("s","UU"),("t","ya")] subst 1),
+	(etac (less_ssum3d  RS iffD1 RS sym) 1),
+	(asm_simp_tac Ssum_ss 1),
+	(hyp_subst_tac 1),
+	(res_inst_tac [("s","UU"),("t","ya")] subst 1),
+	(etac (less_ssum3d  RS iffD1 RS sym) 1),
+	(asm_simp_tac Ssum_ss 1),
+	(hyp_subst_tac 1),
+	(asm_simp_tac Ssum_ss 1),
+	(rtac monofun_cfun_arg 1),
+	(etac (less_ssum3b  RS iffD1) 1)
+	]);
+
+
+
+
+(* ------------------------------------------------------------------------ *)
+(* some kind of exhaustion rules for chains in 'a ++ 'b                     *)
+(* ------------------------------------------------------------------------ *)
+
+
+val ssum_lemma1 = prove_goal Ssum2.thy 
+"[|~(!i.? x.Y(i::nat)=Isinl(x))|] ==> (? i.! x.~Y(i)=Isinl(x))"
+ (fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(fast_tac HOL_cs 1)
+	]);
+
+val ssum_lemma2 = prove_goal Ssum2.thy 
+"[|(? i.!x.~(Y::nat => 'a++'b)(i::nat)=Isinl(x::'a))|] ==>\
+\   (? i y. (Y::nat => 'a++'b)(i::nat)=Isinr(y::'b) & ~y=UU)"
+ (fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(etac exE 1),
+	(res_inst_tac [("p","Y(i)")] IssumE 1),
+	(dtac spec 1),
+	(contr_tac 1),
+	(dtac spec 1),
+	(contr_tac 1),
+	(fast_tac HOL_cs 1)
+	]);
+
+
+val ssum_lemma3 = prove_goal Ssum2.thy 
+"[|is_chain(Y);(? i x. Y(i)=Isinr(x) & ~x=UU)|] ==> (!i.? y.Y(i)=Isinr(y))"
+ (fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(etac exE 1),
+	(etac exE 1),
+	(rtac allI 1),
+	(res_inst_tac [("p","Y(ia)")] IssumE 1),
+	(rtac exI 1),
+	(rtac trans 1),
+	(rtac strict_IsinlIsinr 2),
+	(atac 1),
+	(etac exI 2),
+	(etac conjE 1),
+	(res_inst_tac [("m","i"),("n","ia")] nat_less_cases 1),
+	(hyp_subst_tac 2),
+	(etac exI 2),
+	(res_inst_tac [("P","x=UU")] notE 1),
+	(atac 1),
+	(rtac (less_ssum3d RS iffD1) 1),
+	(res_inst_tac [("s","Y(i)"),("t","Isinr(x)")] subst 1),
+	(atac 1),
+	(res_inst_tac [("s","Y(ia)"),("t","Isinl(xa)")] subst 1),
+	(atac 1),
+	(etac (chain_mono RS mp) 1),
+	(atac 1),
+	(res_inst_tac [("P","xa=UU")] notE 1),
+	(atac 1),
+	(rtac (less_ssum3c RS iffD1) 1),
+	(res_inst_tac [("s","Y(i)"),("t","Isinr(x)")] subst 1),
+	(atac 1),
+	(res_inst_tac [("s","Y(ia)"),("t","Isinl(xa)")] subst 1),
+	(atac 1),
+	(etac (chain_mono RS mp) 1),
+	(atac 1)
+	]);
+
+val ssum_lemma4 = prove_goal Ssum2.thy 
+"is_chain(Y) ==> (!i.? x.Y(i)=Isinl(x))|(!i.? y.Y(i)=Isinr(y))"
+ (fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(rtac classical2 1),
+	(etac disjI1 1),
+	(rtac disjI2 1),
+	(etac ssum_lemma3 1),
+	(rtac ssum_lemma2 1),
+	(etac ssum_lemma1 1)
+	]);
+
+
+(* ------------------------------------------------------------------------ *)
+(* restricted surjectivity of Isinl                                         *)
+(* ------------------------------------------------------------------------ *)
+
+val ssum_lemma5 = prove_goal Ssum2.thy 
+"z=Isinl(x)==> Isinl((Iwhen (LAM x.x) (LAM y.UU))(z)) = z"
+ (fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(hyp_subst_tac 1),
+	(res_inst_tac [("Q","x=UU")] classical2 1),
+	(asm_simp_tac Ssum_ss 1),
+	(asm_simp_tac Ssum_ss 1)
+	]);
+
+(* ------------------------------------------------------------------------ *)
+(* restricted surjectivity of Isinr                                         *)
+(* ------------------------------------------------------------------------ *)
+
+val ssum_lemma6 = prove_goal Ssum2.thy 
+"z=Isinr(x)==> Isinr((Iwhen (LAM y.UU) (LAM x.x))(z)) = z"
+ (fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(hyp_subst_tac 1),
+	(res_inst_tac [("Q","x=UU")] classical2 1),
+	(asm_simp_tac Ssum_ss 1),
+	(asm_simp_tac Ssum_ss 1)
+	]);
+
+(* ------------------------------------------------------------------------ *)
+(* technical lemmas                                                         *)
+(* ------------------------------------------------------------------------ *)
+
+val ssum_lemma7 = prove_goal Ssum2.thy 
+"[|Isinl(x) << z; ~x=UU|] ==> ? y.z=Isinl(y) & ~y=UU"
+ (fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(res_inst_tac [("p","z")] IssumE 1),
+	(hyp_subst_tac 1),
+	(etac notE 1),
+	(rtac antisym_less 1),
+	(etac (less_ssum3a RS iffD1) 1),
+	(rtac minimal 1),
+	(fast_tac HOL_cs 1),
+	(hyp_subst_tac 1),
+	(rtac notE 1),
+	(etac (less_ssum3c RS iffD1) 2),
+	(atac 1)
+	]);
+
+val ssum_lemma8 = prove_goal Ssum2.thy 
+"[|Isinr(x) << z; ~x=UU|] ==> ? y.z=Isinr(y) & ~y=UU"
+ (fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(res_inst_tac [("p","z")] IssumE 1),
+	(hyp_subst_tac 1),
+	(etac notE 1),
+	(etac (less_ssum3d RS iffD1) 1),
+	(hyp_subst_tac 1),
+	(rtac notE 1),
+	(etac (less_ssum3d RS iffD1) 2),
+	(atac 1),
+	(fast_tac HOL_cs 1)
+	]);
+
+(* ------------------------------------------------------------------------ *)
+(* the type 'a ++ 'b is a cpo in three steps                                *)
+(* ------------------------------------------------------------------------ *)
+
+val lub_ssum1a = prove_goal Ssum2.thy 
+"[|is_chain(Y);(!i.? x.Y(i)=Isinl(x))|] ==>\
+\ range(Y) <<|\
+\ Isinl(lub(range(%i.(Iwhen (LAM x.x) (LAM y.UU))(Y(i)))))"
+ (fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(rtac is_lubI 1),
+	(rtac conjI 1),
+	(rtac ub_rangeI 1),
+	(rtac allI 1),
+	(etac allE 1),
+	(etac exE 1),
+	(res_inst_tac [("t","Y(i)")] (ssum_lemma5 RS subst) 1),
+	(atac 1),
+	(rtac (monofun_Isinl RS monofunE RS spec RS spec RS mp) 1),
+	(rtac is_ub_thelub 1),
+	(etac (monofun_Iwhen3 RS ch2ch_monofun) 1),
+	(strip_tac 1),
+	(res_inst_tac [("p","u")] IssumE2 1),
+	(res_inst_tac [("t","u")] (ssum_lemma5 RS subst) 1),
+	(atac 1),
+	(rtac (monofun_Isinl RS monofunE RS spec RS spec RS mp) 1),
+	(rtac is_lub_thelub 1),
+	(etac (monofun_Iwhen3 RS ch2ch_monofun) 1),
+	(etac (monofun_Iwhen3 RS ub2ub_monofun) 1),
+	(hyp_subst_tac 1),
+	(rtac (less_ssum3c RS iffD2) 1),
+	(rtac chain_UU_I_inverse 1),
+	(rtac allI 1),
+	(res_inst_tac [("p","Y(i)")] IssumE 1),
+	(asm_simp_tac Ssum_ss 1),
+	(asm_simp_tac Ssum_ss 2),
+	(etac notE 1),
+	(rtac (less_ssum3c RS iffD1) 1),
+	(res_inst_tac [("t","Isinl(x)")] subst 1),
+	(atac 1),
+	(etac (ub_rangeE RS spec) 1)
+	]);
+
+
+val lub_ssum1b = prove_goal Ssum2.thy 
+"[|is_chain(Y);(!i.? x.Y(i)=Isinr(x))|] ==>\
+\ range(Y) <<|\
+\ Isinr(lub(range(%i.(Iwhen (LAM y.UU) (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),
+	(etac allE 1),
+	(etac exE 1),
+	(res_inst_tac [("t","Y(i)")] (ssum_lemma6 RS subst) 1),
+	(atac 1),
+	(rtac (monofun_Isinr RS monofunE RS spec RS spec RS mp) 1),
+	(rtac is_ub_thelub 1),
+	(etac (monofun_Iwhen3 RS ch2ch_monofun) 1),
+	(strip_tac 1),
+	(res_inst_tac [("p","u")] IssumE2 1),
+	(hyp_subst_tac 1),
+	(rtac (less_ssum3d RS iffD2) 1),
+	(rtac chain_UU_I_inverse 1),
+	(rtac allI 1),
+	(res_inst_tac [("p","Y(i)")] IssumE 1),
+	(asm_simp_tac Ssum_ss 1),
+	(asm_simp_tac Ssum_ss 1),
+	(etac notE 1),
+	(rtac (less_ssum3d RS iffD1) 1),
+	(res_inst_tac [("t","Isinr(y)")] subst 1),
+	(atac 1),
+	(etac (ub_rangeE RS spec) 1),
+	(res_inst_tac [("t","u")] (ssum_lemma6 RS subst) 1),
+	(atac 1),
+	(rtac (monofun_Isinr RS monofunE RS spec RS spec RS mp) 1),
+	(rtac is_lub_thelub 1),
+	(etac (monofun_Iwhen3 RS ch2ch_monofun) 1),
+	(etac (monofun_Iwhen3 RS ub2ub_monofun) 1)
+	]);
+
+
+val thelub_ssum1a = lub_ssum1a RS thelubI;
+(* [| is_chain(?Y1); ! i. ? x. ?Y1(i) = Isinl(x) |] ==>                     *)
+(* lub(range(?Y1)) = Isinl(lub(range(%i. Iwhen(LAM x. x,LAM y. UU,?Y1(i)))))*)
+
+val thelub_ssum1b = lub_ssum1b RS thelubI;
+(* [| is_chain(?Y1); ! i. ? x. ?Y1(i) = Isinr(x) |] ==>                     *)
+(* lub(range(?Y1)) = Isinr(lub(range(%i. Iwhen(LAM y. UU,LAM x. x,?Y1(i)))))*)
+
+val cpo_ssum = prove_goal Ssum2.thy 
+	"is_chain(Y::nat=>'a ++'b) ==> ? x.range(Y) <<|x"
+ (fn prems =>
+	[
+	(cut_facts_tac prems 1),
+	(rtac (ssum_lemma4 RS disjE) 1),
+	(atac 1),
+	(rtac exI 1),
+	(etac lub_ssum1a 1),
+	(atac 1),
+	(rtac exI 1),
+	(etac lub_ssum1b 1),
+	(atac 1)
+	]);
isabelle