--- a/src/HOLCF/ssum2.ML Sat Apr 05 17:03:38 2003 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,419 +0,0 @@
-(* 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)
- ]);