src/HOLCF/Ssum.thy
changeset 15576 efb95d0d01f7
child 15577 e16da3068ad6
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOLCF/Ssum.thy	Fri Mar 04 23:12:36 2005 +0100
@@ -0,0 +1,1565 @@
+(*  Title:      HOLCF/Ssum0.thy
+    ID:         $Id$
+    Author:     Franz Regensburger
+    License:    GPL (GNU GENERAL PUBLIC LICENSE)
+
+Strict sum with typedef
+*)
+
+header {* The type of strict sums *}
+
+theory Ssum = Cfun:
+
+constdefs
+  Sinl_Rep      :: "['a,'a,'b,bool]=>bool"
+ "Sinl_Rep == (%a.%x y p. (a~=UU --> x=a & p))"
+  Sinr_Rep      :: "['b,'a,'b,bool]=>bool"
+ "Sinr_Rep == (%b.%x y p.(b~=UU --> y=b & ~p))"
+
+typedef (Ssum)  ('a, 'b) "++" (infixr 10) = 
+	"{f.(? a. f=Sinl_Rep(a::'a))|(? b. f=Sinr_Rep(b::'b))}"
+by auto
+
+syntax (xsymbols)
+  "++"		:: "[type, type] => type"	("(_ \<oplus>/ _)" [21, 20] 20)
+syntax (HTML output)
+  "++"		:: "[type, type] => type"	("(_ \<oplus>/ _)" [21, 20] 20)
+
+consts
+  Isinl         :: "'a => ('a ++ 'b)"
+  Isinr         :: "'b => ('a ++ 'b)"
+  Iwhen         :: "('a->'c)=>('b->'c)=>('a ++ 'b)=> 'c"
+
+defs   (*defining the abstract constants*)
+  Isinl_def:     "Isinl(a) == Abs_Ssum(Sinl_Rep(a))"
+  Isinr_def:     "Isinr(b) == Abs_Ssum(Sinr_Rep(b))"
+
+  Iwhen_def:     "Iwhen(f)(g)(s) == @z.
+                                    (s=Isinl(UU) --> z=UU)
+                        &(!a. a~=UU & s=Isinl(a) --> z=f$a)  
+                        &(!b. b~=UU & s=Isinr(b) --> z=g$b)"  
+
+(* ------------------------------------------------------------------------ *)
+(* A non-emptyness result for Sssum                                         *)
+(* ------------------------------------------------------------------------ *)
+
+lemma SsumIl: "Sinl_Rep(a):Ssum"
+apply (unfold Ssum_def)
+apply blast
+done
+
+lemma SsumIr: "Sinr_Rep(a):Ssum"
+apply (unfold Ssum_def)
+apply blast
+done
+
+lemma inj_on_Abs_Ssum: "inj_on Abs_Ssum Ssum"
+apply (rule inj_on_inverseI)
+apply (erule Abs_Ssum_inverse)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* Strictness of Sinr_Rep, Sinl_Rep and Isinl, Isinr                        *)
+(* ------------------------------------------------------------------------ *)
+
+lemma strict_SinlSinr_Rep: 
+ "Sinl_Rep(UU) = Sinr_Rep(UU)"
+apply (unfold Sinr_Rep_def Sinl_Rep_def)
+apply (rule ext)
+apply (rule ext)
+apply (rule ext)
+apply fast
+done
+
+lemma strict_IsinlIsinr: 
+ "Isinl(UU) = Isinr(UU)"
+apply (unfold Isinl_def Isinr_def)
+apply (rule strict_SinlSinr_Rep [THEN arg_cong])
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* distinctness of  Sinl_Rep, Sinr_Rep and Isinl, Isinr                     *)
+(* ------------------------------------------------------------------------ *)
+
+lemma noteq_SinlSinr_Rep: 
+        "(Sinl_Rep(a) = Sinr_Rep(b)) ==> a=UU & b=UU"
+apply (unfold Sinl_Rep_def Sinr_Rep_def)
+apply (blast dest!: fun_cong)
+done
+
+
+lemma noteq_IsinlIsinr: 
+        "Isinl(a)=Isinr(b) ==> a=UU & b=UU"
+apply (unfold Isinl_def Isinr_def)
+apply (rule noteq_SinlSinr_Rep)
+apply (erule inj_on_Abs_Ssum [THEN inj_onD])
+apply (rule SsumIl)
+apply (rule SsumIr)
+done
+
+
+
+(* ------------------------------------------------------------------------ *)
+(* injectivity of Sinl_Rep, Sinr_Rep and Isinl, Isinr                       *)
+(* ------------------------------------------------------------------------ *)
+
+lemma inject_Sinl_Rep1: "(Sinl_Rep(a) = Sinl_Rep(UU)) ==> a=UU"
+apply (unfold Sinl_Rep_def)
+apply (blast dest!: fun_cong)
+done
+
+lemma inject_Sinr_Rep1: "(Sinr_Rep(b) = Sinr_Rep(UU)) ==> b=UU"
+apply (unfold Sinr_Rep_def)
+apply (blast dest!: fun_cong)
+done
+
+lemma inject_Sinl_Rep2: 
+"[| a1~=UU ; a2~=UU ; Sinl_Rep(a1)=Sinl_Rep(a2) |] ==> a1=a2"
+apply (unfold Sinl_Rep_def)
+apply (blast dest!: fun_cong)
+done
+
+lemma inject_Sinr_Rep2: 
+"[|b1~=UU ; b2~=UU ; Sinr_Rep(b1)=Sinr_Rep(b2) |] ==> b1=b2"
+apply (unfold Sinr_Rep_def)
+apply (blast dest!: fun_cong)
+done
+
+lemma inject_Sinl_Rep: "Sinl_Rep(a1)=Sinl_Rep(a2) ==> a1=a2"
+apply (case_tac "a1=UU")
+apply simp
+apply (rule inject_Sinl_Rep1 [symmetric])
+apply (erule sym)
+apply (case_tac "a2=UU")
+apply simp
+apply (drule inject_Sinl_Rep1)
+apply simp
+apply (erule inject_Sinl_Rep2)
+apply assumption
+apply assumption
+done
+
+lemma inject_Sinr_Rep: "Sinr_Rep(b1)=Sinr_Rep(b2) ==> b1=b2"
+apply (case_tac "b1=UU")
+apply simp
+apply (rule inject_Sinr_Rep1 [symmetric])
+apply (erule sym)
+apply (case_tac "b2=UU")
+apply simp
+apply (drule inject_Sinr_Rep1)
+apply simp
+apply (erule inject_Sinr_Rep2)
+apply assumption
+apply assumption
+done
+
+lemma inject_Isinl: "Isinl(a1)=Isinl(a2)==> a1=a2"
+apply (unfold Isinl_def)
+apply (rule inject_Sinl_Rep)
+apply (erule inj_on_Abs_Ssum [THEN inj_onD])
+apply (rule SsumIl)
+apply (rule SsumIl)
+done
+
+lemma inject_Isinr: "Isinr(b1)=Isinr(b2) ==> b1=b2"
+apply (unfold Isinr_def)
+apply (rule inject_Sinr_Rep)
+apply (erule inj_on_Abs_Ssum [THEN inj_onD])
+apply (rule SsumIr)
+apply (rule SsumIr)
+done
+
+declare inject_Isinl [dest!] inject_Isinr [dest!]
+
+lemma inject_Isinl_rev: "a1~=a2 ==> Isinl(a1) ~= Isinl(a2)"
+apply blast
+done
+
+lemma inject_Isinr_rev: "b1~=b2 ==> Isinr(b1) ~= Isinr(b2)"
+apply blast
+done
+
+(* ------------------------------------------------------------------------ *)
+(* Exhaustion of the strict sum ++                                          *)
+(* choice of the bottom representation is arbitrary                         *)
+(* ------------------------------------------------------------------------ *)
+
+lemma Exh_Ssum: 
+        "z=Isinl(UU) | (? a. z=Isinl(a) & a~=UU) | (? b. z=Isinr(b) & b~=UU)"
+apply (unfold Isinl_def Isinr_def)
+apply (rule Rep_Ssum[unfolded Ssum_def, THEN CollectE])
+apply (erule disjE)
+apply (erule exE)
+apply (case_tac "z= Abs_Ssum (Sinl_Rep (UU))")
+apply (erule disjI1)
+apply (rule disjI2)
+apply (rule disjI1)
+apply (rule exI)
+apply (rule conjI)
+apply (rule Rep_Ssum_inverse [symmetric, THEN trans])
+apply (erule arg_cong)
+apply (rule_tac Q = "Sinl_Rep (a) =Sinl_Rep (UU) " in contrapos_nn)
+apply (erule_tac [2] arg_cong)
+apply (erule contrapos_nn)
+apply (rule Rep_Ssum_inverse [symmetric, THEN trans])
+apply (rule trans)
+apply (erule arg_cong)
+apply (erule arg_cong)
+apply (erule exE)
+apply (case_tac "z= Abs_Ssum (Sinl_Rep (UU))")
+apply (erule disjI1)
+apply (rule disjI2)
+apply (rule disjI2)
+apply (rule exI)
+apply (rule conjI)
+apply (rule Rep_Ssum_inverse [symmetric, THEN trans])
+apply (erule arg_cong)
+apply (rule_tac Q = "Sinr_Rep (b) =Sinl_Rep (UU) " in contrapos_nn)
+prefer 2 apply simp
+apply (rule strict_SinlSinr_Rep [symmetric])
+apply (erule contrapos_nn)
+apply (rule Rep_Ssum_inverse [symmetric, THEN trans])
+apply (rule trans)
+apply (erule arg_cong)
+apply (erule arg_cong)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* elimination rules for the strict sum ++                                  *)
+(* ------------------------------------------------------------------------ *)
+
+lemma IssumE:
+        "[|p=Isinl(UU) ==> Q ; 
+        !!x.[|p=Isinl(x); x~=UU |] ==> Q; 
+        !!y.[|p=Isinr(y); y~=UU |] ==> Q|] ==> Q"
+apply (rule Exh_Ssum [THEN disjE])
+apply auto
+done
+
+lemma IssumE2:
+"[| !!x. [| p = Isinl(x) |] ==> Q;   !!y. [| p = Isinr(y) |] ==> Q |] ==>Q"
+apply (rule IssumE)
+apply auto
+done
+
+
+
+
+(* ------------------------------------------------------------------------ *)
+(* rewrites for Iwhen                                                       *)
+(* ------------------------------------------------------------------------ *)
+
+lemma Iwhen1: 
+        "Iwhen f g (Isinl UU) = UU"
+apply (unfold Iwhen_def)
+apply (rule some_equality)
+apply (rule conjI)
+apply fast
+apply (rule conjI)
+apply (intro strip)
+apply (rule_tac P = "a=UU" in notE)
+apply fast
+apply (rule inject_Isinl)
+apply (rule sym)
+apply fast
+apply (intro strip)
+apply (rule_tac P = "b=UU" in notE)
+apply fast
+apply (rule inject_Isinr)
+apply (rule sym)
+apply (rule strict_IsinlIsinr [THEN subst])
+apply fast
+apply fast
+done
+
+
+lemma Iwhen2: 
+        "x~=UU ==> Iwhen f g (Isinl x) = f$x"
+apply (unfold Iwhen_def)
+apply (rule some_equality)
+prefer 2 apply fast
+apply (rule conjI)
+apply (intro strip)
+apply (rule_tac P = "x=UU" in notE)
+apply assumption
+apply (rule inject_Isinl)
+apply assumption
+apply (rule conjI)
+apply (intro strip)
+apply (rule cfun_arg_cong)
+apply (rule inject_Isinl)
+apply fast
+apply (intro strip)
+apply (rule_tac P = "Isinl (x) = Isinr (b) " in notE)
+prefer 2 apply fast
+apply (rule contrapos_nn)
+apply (erule_tac [2] noteq_IsinlIsinr)
+apply fast
+done
+
+lemma Iwhen3: 
+        "y~=UU ==> Iwhen f g (Isinr y) = g$y"
+apply (unfold Iwhen_def)
+apply (rule some_equality)
+prefer 2 apply fast
+apply (rule conjI)
+apply (intro strip)
+apply (rule_tac P = "y=UU" in notE)
+apply assumption
+apply (rule inject_Isinr)
+apply (rule strict_IsinlIsinr [THEN subst])
+apply assumption
+apply (rule conjI)
+apply (intro strip)
+apply (rule_tac P = "Isinr (y) = Isinl (a) " in notE)
+prefer 2 apply fast
+apply (rule contrapos_nn)
+apply (erule_tac [2] sym [THEN noteq_IsinlIsinr])
+apply fast
+apply (intro strip)
+apply (rule cfun_arg_cong)
+apply (rule inject_Isinr)
+apply fast
+done
+
+(* ------------------------------------------------------------------------ *)
+(* instantiate the simplifier                                               *)
+(* ------------------------------------------------------------------------ *)
+
+lemmas Ssum0_ss = strict_IsinlIsinr[symmetric] Iwhen1 Iwhen2 Iwhen3
+
+declare Ssum0_ss [simp]
+
+(* Partial ordering for the strict sum ++ *)
+
+instance "++"::(pcpo,pcpo)sq_ord ..
+
+defs (overloaded)
+  less_ssum_def: "(op <<) == (%s1 s2.@z.
+         (! u x. s1=Isinl u & s2=Isinl x --> z = u << x)
+        &(! v y. s1=Isinr v & s2=Isinr y --> z = v << y)
+        &(! u y. s1=Isinl u & s2=Isinr y --> z = (u = UU))
+        &(! v x. s1=Isinr v & s2=Isinl x --> z = (v = UU)))"
+
+lemma less_ssum1a: 
+"[|s1=Isinl(x::'a); s2=Isinl(y::'a)|] ==> s1 << s2 = (x << y)"
+apply (unfold less_ssum_def)
+apply (rule some_equality)
+apply (drule_tac [2] conjunct1)
+apply (drule_tac [2] spec)
+apply (drule_tac [2] spec)
+apply (erule_tac [2] mp)
+prefer 2 apply fast
+apply (rule conjI)
+apply (intro strip)
+apply (erule conjE)
+apply simp
+apply (drule inject_Isinl)
+apply (drule inject_Isinl)
+apply simp
+apply (rule conjI)
+apply (intro strip)
+apply (erule conjE)
+apply simp
+apply (drule noteq_IsinlIsinr[OF sym])
+apply simp
+apply (rule conjI)
+apply (intro strip)
+apply (erule conjE)
+apply simp
+apply (drule inject_Isinl)
+apply (drule noteq_IsinlIsinr[OF sym])
+apply simp
+apply (rule eq_UU_iff[symmetric])
+apply (intro strip)
+apply (erule conjE)
+apply simp
+apply (drule noteq_IsinlIsinr[OF sym])
+apply simp
+done
+
+
+lemma less_ssum1b: 
+"[|s1=Isinr(x::'b); s2=Isinr(y::'b)|] ==> s1 << s2 = (x << y)"
+apply (unfold less_ssum_def)
+apply (rule some_equality)
+apply (drule_tac [2] conjunct2)
+apply (drule_tac [2] conjunct1)
+apply (drule_tac [2] spec)
+apply (drule_tac [2] spec)
+apply (erule_tac [2] mp)
+prefer 2 apply fast
+apply (rule conjI)
+apply (intro strip)
+apply (erule conjE)
+apply simp
+apply (drule noteq_IsinlIsinr)
+apply (drule noteq_IsinlIsinr)
+apply simp
+apply (rule conjI)
+apply (intro strip)
+apply (erule conjE)
+apply simp
+apply (drule inject_Isinr)
+apply (drule inject_Isinr)
+apply simp
+apply (rule conjI)
+apply (intro strip)
+apply (erule conjE)
+apply simp
+apply (drule noteq_IsinlIsinr)
+apply (drule inject_Isinr)
+apply simp
+apply (intro strip)
+apply (erule conjE)
+apply simp
+apply (drule inject_Isinr)
+apply (drule noteq_IsinlIsinr)
+apply simp
+apply (rule eq_UU_iff[symmetric])
+done
+
+
+lemma less_ssum1c: 
+"[|s1=Isinl(x::'a); s2=Isinr(y::'b)|] ==> s1 << s2 = ((x::'a) = UU)"
+apply (unfold less_ssum_def)
+apply (rule some_equality)
+apply (rule conjI)
+apply (intro strip)
+apply (erule conjE)
+apply simp
+apply (drule inject_Isinl)
+apply (drule noteq_IsinlIsinr)
+apply simp
+apply (rule eq_UU_iff)
+apply (rule conjI)
+apply (intro strip)
+apply (erule conjE)
+apply simp
+apply (drule noteq_IsinlIsinr[OF sym])
+apply (drule inject_Isinr)
+apply simp
+apply (rule conjI)
+apply (intro strip)
+apply (erule conjE)
+apply simp
+apply (drule inject_Isinl)
+apply (drule inject_Isinr)
+apply simp
+apply (intro strip)
+apply (erule conjE)
+apply simp
+apply (drule noteq_IsinlIsinr[OF sym])
+apply (drule noteq_IsinlIsinr)
+apply simp
+apply (drule conjunct2)
+apply (drule conjunct2)
+apply (drule conjunct1)
+apply (drule spec)
+apply (drule spec)
+apply (erule mp)
+apply fast
+done
+
+
+lemma less_ssum1d: 
+"[|s1=Isinr(x); s2=Isinl(y)|] ==> s1 << s2 = (x = UU)"
+apply (unfold less_ssum_def)
+apply (rule some_equality)
+apply (drule_tac [2] conjunct2)
+apply (drule_tac [2] conjunct2)
+apply (drule_tac [2] conjunct2)
+apply (drule_tac [2] spec)
+apply (drule_tac [2] spec)
+apply (erule_tac [2] mp)
+prefer 2 apply fast
+apply (rule conjI)
+apply (intro strip)
+apply (erule conjE)
+apply simp
+apply (drule noteq_IsinlIsinr)
+apply (drule inject_Isinl)
+apply simp
+apply (rule conjI)
+apply (intro strip)
+apply (erule conjE)
+apply simp
+apply (drule noteq_IsinlIsinr[OF sym])
+apply (drule inject_Isinr)
+apply simp
+apply (rule eq_UU_iff)
+apply (rule conjI)
+apply (intro strip)
+apply (erule conjE)
+apply simp
+apply (drule noteq_IsinlIsinr)
+apply (drule noteq_IsinlIsinr[OF sym])
+apply simp
+apply (intro strip)
+apply (erule conjE)
+apply simp
+apply (drule inject_Isinr)
+apply simp
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* optimize lemmas about less_ssum                                          *)
+(* ------------------------------------------------------------------------ *)
+
+lemma less_ssum2a: "(Isinl x) << (Isinl y) = (x << y)"
+apply (rule less_ssum1a)
+apply (rule refl)
+apply (rule refl)
+done
+
+lemma less_ssum2b: "(Isinr x) << (Isinr y) = (x << y)"
+apply (rule less_ssum1b)
+apply (rule refl)
+apply (rule refl)
+done
+
+lemma less_ssum2c: "(Isinl x) << (Isinr y) = (x = UU)"
+apply (rule less_ssum1c)
+apply (rule refl)
+apply (rule refl)
+done
+
+lemma less_ssum2d: "(Isinr x) << (Isinl y) = (x = UU)"
+apply (rule less_ssum1d)
+apply (rule refl)
+apply (rule refl)
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* less_ssum is a partial order on ++                                     *)
+(* ------------------------------------------------------------------------ *)
+
+lemma refl_less_ssum: "(p::'a++'b) << p"
+apply (rule_tac p = "p" in IssumE2)
+apply (erule ssubst)
+apply (rule less_ssum2a [THEN iffD2])
+apply (rule refl_less)
+apply (erule ssubst)
+apply (rule less_ssum2b [THEN iffD2])
+apply (rule refl_less)
+done
+
+lemma antisym_less_ssum: "[|(p1::'a++'b) << p2; p2 << p1|] ==> p1=p2"
+apply (rule_tac p = "p1" in IssumE2)
+apply simp
+apply (rule_tac p = "p2" in IssumE2)
+apply simp
+apply (rule_tac f = "Isinl" in arg_cong)
+apply (rule antisym_less)
+apply (erule less_ssum2a [THEN iffD1])
+apply (erule less_ssum2a [THEN iffD1])
+apply simp
+apply (erule less_ssum2d [THEN iffD1, THEN ssubst])
+apply (erule less_ssum2c [THEN iffD1, THEN ssubst])
+apply (rule strict_IsinlIsinr)
+apply simp
+apply (rule_tac p = "p2" in IssumE2)
+apply simp
+apply (erule less_ssum2c [THEN iffD1, THEN ssubst])
+apply (erule less_ssum2d [THEN iffD1, THEN ssubst])
+apply (rule strict_IsinlIsinr [symmetric])
+apply simp
+apply (rule_tac f = "Isinr" in arg_cong)
+apply (rule antisym_less)
+apply (erule less_ssum2b [THEN iffD1])
+apply (erule less_ssum2b [THEN iffD1])
+done
+
+lemma trans_less_ssum: "[|(p1::'a++'b) << p2; p2 << p3|] ==> p1 << p3"
+apply (rule_tac p = "p1" in IssumE2)
+apply simp
+apply (rule_tac p = "p3" in IssumE2)
+apply simp
+apply (rule less_ssum2a [THEN iffD2])
+apply (rule_tac p = "p2" in IssumE2)
+apply simp
+apply (rule trans_less)
+apply (erule less_ssum2a [THEN iffD1])
+apply (erule less_ssum2a [THEN iffD1])
+apply simp
+apply (erule less_ssum2c [THEN iffD1, THEN ssubst])
+apply (rule minimal)
+apply simp
+apply (rule less_ssum2c [THEN iffD2])
+apply (rule_tac p = "p2" in IssumE2)
+apply simp
+apply (rule UU_I)
+apply (rule trans_less)
+apply (erule less_ssum2a [THEN iffD1])
+apply (rule antisym_less_inverse [THEN conjunct1])
+apply (erule less_ssum2c [THEN iffD1])
+apply simp
+apply (erule less_ssum2c [THEN iffD1])
+apply simp
+apply (rule_tac p = "p3" in IssumE2)
+apply simp
+apply (rule less_ssum2d [THEN iffD2])
+apply (rule_tac p = "p2" in IssumE2)
+apply simp
+apply (erule less_ssum2d [THEN iffD1])
+apply simp
+apply (rule UU_I)
+apply (rule trans_less)
+apply (erule less_ssum2b [THEN iffD1])
+apply (rule antisym_less_inverse [THEN conjunct1])
+apply (erule less_ssum2d [THEN iffD1])
+apply simp
+apply (rule less_ssum2b [THEN iffD2])
+apply (rule_tac p = "p2" in IssumE2)
+apply simp
+apply (erule less_ssum2d [THEN iffD1, THEN ssubst])
+apply (rule minimal)
+apply simp
+apply (rule trans_less)
+apply (erule less_ssum2b [THEN iffD1])
+apply (erule less_ssum2b [THEN iffD1])
+done
+
+(* Class Instance ++::(pcpo,pcpo)po *)
+
+instance "++"::(pcpo,pcpo)po
+apply (intro_classes)
+apply (rule refl_less_ssum)
+apply (rule antisym_less_ssum, assumption+)
+apply (rule trans_less_ssum, assumption+)
+done
+
+(* for compatibility with old HOLCF-Version *)
+lemma inst_ssum_po: "(op <<)=(%s1 s2.@z.
+         (! u x. s1=Isinl u & s2=Isinl x --> z = u << x)
+        &(! v y. s1=Isinr v & s2=Isinr y --> z = v << y)
+        &(! u y. s1=Isinl u & s2=Isinr y --> z = (u = UU))
+        &(! v x. s1=Isinr v & s2=Isinl x --> z = (v = UU)))"
+apply (fold less_ssum_def)
+apply (rule refl)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* access to less_ssum in class po                                          *)
+(* ------------------------------------------------------------------------ *)
+
+lemma less_ssum3a: "Isinl x << Isinl y = x << y"
+apply (simp (no_asm) add: less_ssum2a)
+done
+
+lemma less_ssum3b: "Isinr x << Isinr y = x << y"
+apply (simp (no_asm) add: less_ssum2b)
+done
+
+lemma less_ssum3c: "Isinl x << Isinr y = (x = UU)"
+apply (simp (no_asm) add: less_ssum2c)
+done
+
+lemma less_ssum3d: "Isinr x << Isinl y = (x = UU)"
+apply (simp (no_asm) add: less_ssum2d)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* type ssum ++ is pointed                                                  *)
+(* ------------------------------------------------------------------------ *)
+
+lemma minimal_ssum: "Isinl UU << s"
+apply (rule_tac p = "s" in IssumE2)
+apply simp
+apply (rule less_ssum3a [THEN iffD2])
+apply (rule minimal)
+apply simp
+apply (subst strict_IsinlIsinr)
+apply (rule less_ssum3b [THEN iffD2])
+apply (rule minimal)
+done
+
+lemmas UU_ssum_def = minimal_ssum [THEN minimal2UU, symmetric, standard]
+
+lemma least_ssum: "? x::'a++'b.!y. x<<y"
+apply (rule_tac x = "Isinl UU" in exI)
+apply (rule minimal_ssum [THEN allI])
+done
+
+(* ------------------------------------------------------------------------ *)
+(* Isinl, Isinr are monotone                                                *)
+(* ------------------------------------------------------------------------ *)
+
+lemma monofun_Isinl: "monofun(Isinl)"
+apply (unfold monofun)
+apply (intro strip)
+apply (erule less_ssum3a [THEN iffD2])
+done
+
+lemma monofun_Isinr: "monofun(Isinr)"
+apply (unfold monofun)
+apply (intro strip)
+apply (erule less_ssum3b [THEN iffD2])
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* Iwhen is monotone in all arguments                                       *)
+(* ------------------------------------------------------------------------ *)
+
+
+lemma monofun_Iwhen1: "monofun(Iwhen)"
+apply (unfold monofun)
+apply (intro strip)
+apply (rule less_fun [THEN iffD2])
+apply (intro strip)
+apply (rule less_fun [THEN iffD2])
+apply (intro strip)
+apply (rule_tac p = "xb" in IssumE)
+apply simp
+apply simp
+apply (erule monofun_cfun_fun)
+apply simp
+done
+
+lemma monofun_Iwhen2: "monofun(Iwhen(f))"
+apply (unfold monofun)
+apply (intro strip)
+apply (rule less_fun [THEN iffD2])
+apply (intro strip)
+apply (rule_tac p = "xa" in IssumE)
+apply simp
+apply simp
+apply simp
+apply (erule monofun_cfun_fun)
+done
+
+lemma monofun_Iwhen3: "monofun(Iwhen(f)(g))"
+apply (unfold monofun)
+apply (intro strip)
+apply (rule_tac p = "x" in IssumE)
+apply simp
+apply (rule_tac p = "y" in IssumE)
+apply simp
+apply (rule_tac P = "xa=UU" in notE)
+apply assumption
+apply (rule UU_I)
+apply (rule less_ssum3a [THEN iffD1])
+apply assumption
+apply simp
+apply (rule monofun_cfun_arg)
+apply (erule less_ssum3a [THEN iffD1])
+apply (simp del: Iwhen2)
+apply (rule_tac s = "UU" and t = "xa" in subst)
+apply (erule less_ssum3c [THEN iffD1, symmetric])
+apply simp
+apply (rule_tac p = "y" in IssumE)
+apply simp
+apply (simp only: less_ssum3d)
+apply (simp only: less_ssum3d)
+apply simp
+apply (rule monofun_cfun_arg)
+apply (erule less_ssum3b [THEN iffD1])
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* some kind of exhaustion rules for chains in 'a ++ 'b                     *)
+(* ------------------------------------------------------------------------ *)
+
+lemma ssum_lemma1: "[|~(!i.? x. Y(i::nat)=Isinl(x))|] ==> (? i.! x. Y(i)~=Isinl(x))"
+apply fast
+done
+
+lemma ssum_lemma2: "[|(? i.!x.(Y::nat => 'a++'b)(i::nat)~=Isinl(x::'a))|]   
+      ==> (? i y. (Y::nat => 'a++'b)(i::nat)=Isinr(y::'b) & y~=UU)"
+apply (erule exE)
+apply (rule_tac p = "Y (i) " in IssumE)
+apply (drule spec)
+apply (erule notE, assumption)
+apply (drule spec)
+apply (erule notE, assumption)
+apply fast
+done
+
+
+lemma ssum_lemma3: "[|chain(Y);(? i x. Y(i)=Isinr(x::'b) & (x::'b)~=UU)|]  
+      ==> (!i.? y. Y(i)=Isinr(y))"
+apply (erule exE)
+apply (erule exE)
+apply (rule allI)
+apply (rule_tac p = "Y (ia) " in IssumE)
+apply (rule exI)
+apply (rule trans)
+apply (rule_tac [2] strict_IsinlIsinr)
+apply assumption
+apply (erule_tac [2] exI)
+apply (erule conjE)
+apply (rule_tac m = "i" and n = "ia" in nat_less_cases)
+prefer 2 apply simp
+apply (rule exI, rule refl)
+apply (erule_tac P = "x=UU" in notE)
+apply (rule less_ssum3d [THEN iffD1])
+apply (erule_tac s = "Y (i) " and t = "Isinr (x) ::'a++'b" in subst)
+apply (erule_tac s = "Y (ia) " and t = "Isinl (xa) ::'a++'b" in subst)
+apply (erule chain_mono)
+apply assumption
+apply (erule_tac P = "xa=UU" in notE)
+apply (rule less_ssum3c [THEN iffD1])
+apply (erule_tac s = "Y (i) " and t = "Isinr (x) ::'a++'b" in subst)
+apply (erule_tac s = "Y (ia) " and t = "Isinl (xa) ::'a++'b" in subst)
+apply (erule chain_mono)
+apply assumption
+done
+
+lemma ssum_lemma4: "chain(Y) ==> (!i.? x. Y(i)=Isinl(x))|(!i.? y. Y(i)=Isinr(y))"
+apply (rule case_split_thm)
+apply (erule disjI1)
+apply (rule disjI2)
+apply (erule ssum_lemma3)
+apply (rule ssum_lemma2)
+apply (erule ssum_lemma1)
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* restricted surjectivity of Isinl                                         *)
+(* ------------------------------------------------------------------------ *)
+
+lemma ssum_lemma5: "z=Isinl(x)==> Isinl((Iwhen (LAM x. x) (LAM y. UU))(z)) = z"
+apply simp
+apply (case_tac "x=UU")
+apply simp
+apply simp
+done
+
+(* ------------------------------------------------------------------------ *)
+(* restricted surjectivity of Isinr                                         *)
+(* ------------------------------------------------------------------------ *)
+
+lemma ssum_lemma6: "z=Isinr(x)==> Isinr((Iwhen (LAM y. UU) (LAM x. x))(z)) = z"
+apply simp
+apply (case_tac "x=UU")
+apply simp
+apply simp
+done
+
+(* ------------------------------------------------------------------------ *)
+(* technical lemmas                                                         *)
+(* ------------------------------------------------------------------------ *)
+
+lemma ssum_lemma7: "[|Isinl(x) << z; x~=UU|] ==> ? y. z=Isinl(y) & y~=UU"
+apply (rule_tac p = "z" in IssumE)
+apply simp
+apply (erule notE)
+apply (rule antisym_less)
+apply (erule less_ssum3a [THEN iffD1])
+apply (rule minimal)
+apply fast
+apply simp
+apply (rule notE)
+apply (erule_tac [2] less_ssum3c [THEN iffD1])
+apply assumption
+done
+
+lemma ssum_lemma8: "[|Isinr(x) << z; x~=UU|] ==> ? y. z=Isinr(y) & y~=UU"
+apply (rule_tac p = "z" in IssumE)
+apply simp
+apply (erule notE)
+apply (erule less_ssum3d [THEN iffD1])
+apply simp
+apply (rule notE)
+apply (erule_tac [2] less_ssum3d [THEN iffD1])
+apply assumption
+apply fast
+done
+
+(* ------------------------------------------------------------------------ *)
+(* the type 'a ++ 'b is a cpo in three steps                                *)
+(* ------------------------------------------------------------------------ *)
+
+lemma lub_ssum1a: "[|chain(Y);(!i.? x. Y(i)=Isinl(x))|] ==> 
+      range(Y) <<| Isinl(lub(range(%i.(Iwhen (LAM x. x) (LAM y. UU))(Y i))))"
+apply (rule is_lubI)
+apply (rule ub_rangeI)
+apply (erule allE)
+apply (erule exE)
+apply (rule_tac t = "Y (i) " in ssum_lemma5 [THEN subst])
+apply assumption
+apply (rule monofun_Isinl [THEN monofunE, THEN spec, THEN spec, THEN mp])
+apply (rule is_ub_thelub)
+apply (erule monofun_Iwhen3 [THEN ch2ch_monofun])
+apply (rule_tac p = "u" in IssumE2)
+apply (rule_tac t = "u" in ssum_lemma5 [THEN subst])
+apply assumption
+apply (rule monofun_Isinl [THEN monofunE, THEN spec, THEN spec, THEN mp])
+apply (rule is_lub_thelub)
+apply (erule monofun_Iwhen3 [THEN ch2ch_monofun])
+apply (erule monofun_Iwhen3 [THEN ub2ub_monofun])
+apply simp
+apply (rule less_ssum3c [THEN iffD2])
+apply (rule chain_UU_I_inverse)
+apply (rule allI)
+apply (rule_tac p = "Y (i) " in IssumE)
+apply simp
+apply simp
+apply (erule notE)
+apply (rule less_ssum3c [THEN iffD1])
+apply (rule_tac t = "Isinl (x) " in subst)
+apply assumption
+apply (erule ub_rangeD)
+apply simp
+done
+
+
+lemma lub_ssum1b: "[|chain(Y);(!i.? x. Y(i)=Isinr(x))|] ==> 
+      range(Y) <<| Isinr(lub(range(%i.(Iwhen (LAM y. UU) (LAM x. x))(Y i))))"
+apply (rule is_lubI)
+apply (rule ub_rangeI)
+apply (erule allE)
+apply (erule exE)
+apply (rule_tac t = "Y (i) " in ssum_lemma6 [THEN subst])
+apply assumption
+apply (rule monofun_Isinr [THEN monofunE, THEN spec, THEN spec, THEN mp])
+apply (rule is_ub_thelub)
+apply (erule monofun_Iwhen3 [THEN ch2ch_monofun])
+apply (rule_tac p = "u" in IssumE2)
+apply simp
+apply (rule less_ssum3d [THEN iffD2])
+apply (rule chain_UU_I_inverse)
+apply (rule allI)
+apply (rule_tac p = "Y (i) " in IssumE)
+apply simp
+apply simp
+apply (erule notE)
+apply (rule less_ssum3d [THEN iffD1])
+apply (rule_tac t = "Isinr (y) " in subst)
+apply assumption
+apply (erule ub_rangeD)
+apply (rule_tac t = "u" in ssum_lemma6 [THEN subst])
+apply assumption
+apply (rule monofun_Isinr [THEN monofunE, THEN spec, THEN spec, THEN mp])
+apply (rule is_lub_thelub)
+apply (erule monofun_Iwhen3 [THEN ch2ch_monofun])
+apply (erule monofun_Iwhen3 [THEN ub2ub_monofun])
+done
+
+
+lemmas thelub_ssum1a = lub_ssum1a [THEN thelubI, standard]
+(*
+[| chain ?Y1; ! i. ? x. ?Y1 i = Isinl x |] ==>
+ lub (range ?Y1) = Isinl
+ (lub (range (%i. Iwhen (LAM x. x) (LAM y. UU) (?Y1 i))))
+*)
+
+lemmas thelub_ssum1b = lub_ssum1b [THEN thelubI, standard]
+(*
+[| chain ?Y1; ! i. ? x. ?Y1 i = Isinr x |] ==>
+ lub (range ?Y1) = Isinr
+ (lub (range (%i. Iwhen (LAM y. UU) (LAM x. x) (?Y1 i))))
+*)
+
+lemma cpo_ssum: "chain(Y::nat=>'a ++'b) ==> ? x. range(Y) <<|x"
+apply (rule ssum_lemma4 [THEN disjE])
+apply assumption
+apply (rule exI)
+apply (erule lub_ssum1a)
+apply assumption
+apply (rule exI)
+apply (erule lub_ssum1b)
+apply assumption
+done
+
+(* Class instance of  ++ for class pcpo *)
+
+instance "++" :: (pcpo,pcpo)pcpo
+apply (intro_classes)
+apply (erule cpo_ssum)
+apply (rule least_ssum)
+done
+
+consts  
+        sinl    :: "'a -> ('a++'b)" 
+        sinr    :: "'b -> ('a++'b)" 
+        sscase  :: "('a->'c)->('b->'c)->('a ++ 'b)-> 'c"
+
+defs
+
+sinl_def:        "sinl   == (LAM x. Isinl(x))"
+sinr_def:        "sinr   == (LAM x. Isinr(x))"
+sscase_def:      "sscase   == (LAM f g s. Iwhen(f)(g)(s))"
+
+translations
+"case s of sinl$x => t1 | sinr$y => t2" == "sscase$(LAM x. t1)$(LAM y. t2)$s"
+
+(* for compatibility with old HOLCF-Version *)
+lemma inst_ssum_pcpo: "UU = Isinl UU"
+apply (simp add: UU_def UU_ssum_def)
+done
+
+declare inst_ssum_pcpo [symmetric, simp]
+
+(* ------------------------------------------------------------------------ *)
+(* continuity for Isinl and Isinr                                           *)
+(* ------------------------------------------------------------------------ *)
+
+lemma contlub_Isinl: "contlub(Isinl)"
+apply (rule contlubI)
+apply (intro strip)
+apply (rule trans)
+apply (rule_tac [2] thelub_ssum1a [symmetric])
+apply (rule_tac [3] allI)
+apply (rule_tac [3] exI)
+apply (rule_tac [3] refl)
+apply (erule_tac [2] monofun_Isinl [THEN ch2ch_monofun])
+apply (case_tac "lub (range (Y))=UU")
+apply (rule_tac s = "UU" and t = "lub (range (Y))" in ssubst)
+apply assumption
+apply (rule_tac f = "Isinl" in arg_cong)
+apply (rule chain_UU_I_inverse [symmetric])
+apply (rule allI)
+apply (rule_tac s = "UU" and t = "Y (i) " in ssubst)
+apply (erule chain_UU_I [THEN spec])
+apply assumption
+apply (rule Iwhen1)
+apply (rule_tac f = "Isinl" in arg_cong)
+apply (rule lub_equal)
+apply assumption
+apply (rule monofun_Iwhen3 [THEN ch2ch_monofun])
+apply (erule monofun_Isinl [THEN ch2ch_monofun])
+apply (rule allI)
+apply (case_tac "Y (k) =UU")
+apply (erule ssubst)
+apply (rule Iwhen1[symmetric])
+apply simp
+done
+
+lemma contlub_Isinr: "contlub(Isinr)"
+apply (rule contlubI)
+apply (intro strip)
+apply (rule trans)
+apply (rule_tac [2] thelub_ssum1b [symmetric])
+apply (rule_tac [3] allI)
+apply (rule_tac [3] exI)
+apply (rule_tac [3] refl)
+apply (erule_tac [2] monofun_Isinr [THEN ch2ch_monofun])
+apply (case_tac "lub (range (Y))=UU")
+apply (rule_tac s = "UU" and t = "lub (range (Y))" in ssubst)
+apply assumption
+apply (rule arg_cong, rule chain_UU_I_inverse [symmetric])
+apply (rule allI)
+apply (rule_tac s = "UU" and t = "Y (i) " in ssubst)
+apply (erule chain_UU_I [THEN spec])
+apply assumption
+apply (rule strict_IsinlIsinr [THEN subst])
+apply (rule Iwhen1)
+apply (rule arg_cong, rule lub_equal)
+apply assumption
+apply (rule monofun_Iwhen3 [THEN ch2ch_monofun])
+apply (erule monofun_Isinr [THEN ch2ch_monofun])
+apply (rule allI)
+apply (case_tac "Y (k) =UU")
+apply (simp only: Ssum0_ss)
+apply simp
+done
+
+lemma cont_Isinl: "cont(Isinl)"
+apply (rule monocontlub2cont)
+apply (rule monofun_Isinl)
+apply (rule contlub_Isinl)
+done
+
+lemma cont_Isinr: "cont(Isinr)"
+apply (rule monocontlub2cont)
+apply (rule monofun_Isinr)
+apply (rule contlub_Isinr)
+done
+
+declare cont_Isinl [iff] cont_Isinr [iff]
+
+
+(* ------------------------------------------------------------------------ *)
+(* continuity for Iwhen in the firts two arguments                          *)
+(* ------------------------------------------------------------------------ *)
+
+lemma contlub_Iwhen1: "contlub(Iwhen)"
+apply (rule contlubI)
+apply (intro strip)
+apply (rule trans)
+apply (rule_tac [2] thelub_fun [symmetric])
+apply (erule_tac [2] monofun_Iwhen1 [THEN ch2ch_monofun])
+apply (rule expand_fun_eq [THEN iffD2])
+apply (intro strip)
+apply (rule trans)
+apply (rule_tac [2] thelub_fun [symmetric])
+apply (rule_tac [2] ch2ch_fun)
+apply (erule_tac [2] monofun_Iwhen1 [THEN ch2ch_monofun])
+apply (rule expand_fun_eq [THEN iffD2])
+apply (intro strip)
+apply (rule_tac p = "xa" in IssumE)
+apply (simp only: Ssum0_ss)
+apply (rule lub_const [THEN thelubI, symmetric])
+apply simp
+apply (erule contlub_cfun_fun)
+apply simp
+apply (rule lub_const [THEN thelubI, symmetric])
+done
+
+lemma contlub_Iwhen2: "contlub(Iwhen(f))"
+apply (rule contlubI)
+apply (intro strip)
+apply (rule trans)
+apply (rule_tac [2] thelub_fun [symmetric])
+apply (erule_tac [2] monofun_Iwhen2 [THEN ch2ch_monofun])
+apply (rule expand_fun_eq [THEN iffD2])
+apply (intro strip)
+apply (rule_tac p = "x" in IssumE)
+apply (simp only: Ssum0_ss)
+apply (rule lub_const [THEN thelubI, symmetric])
+apply simp
+apply (rule lub_const [THEN thelubI, symmetric])
+apply simp
+apply (erule contlub_cfun_fun)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* continuity for Iwhen in its third argument                               *)
+(* ------------------------------------------------------------------------ *)
+
+(* ------------------------------------------------------------------------ *)
+(* first 5 ugly lemmas                                                      *)
+(* ------------------------------------------------------------------------ *)
+
+lemma ssum_lemma9: "[| chain(Y); lub(range(Y)) = Isinl(x)|] ==> !i.? x. Y(i)=Isinl(x)"
+apply (intro strip)
+apply (rule_tac p = "Y (i) " in IssumE)
+apply (erule exI)
+apply (erule exI)
+apply (rule_tac P = "y=UU" in notE)
+apply assumption
+apply (rule less_ssum3d [THEN iffD1])
+apply (erule subst)
+apply (erule subst)
+apply (erule is_ub_thelub)
+done
+
+
+lemma ssum_lemma10: "[| chain(Y); lub(range(Y)) = Isinr(x)|] ==> !i.? x. Y(i)=Isinr(x)"
+apply (intro strip)
+apply (rule_tac p = "Y (i) " in IssumE)
+apply (rule exI)
+apply (erule trans)
+apply (rule strict_IsinlIsinr)
+apply (erule_tac [2] exI)
+apply (rule_tac P = "xa=UU" in notE)
+apply assumption
+apply (rule less_ssum3c [THEN iffD1])
+apply (erule subst)
+apply (erule subst)
+apply (erule is_ub_thelub)
+done
+
+lemma ssum_lemma11: "[| chain(Y); lub(range(Y)) = Isinl(UU) |] ==> 
+    Iwhen f g (lub(range Y)) = lub(range(%i. Iwhen f g (Y i)))"
+apply (simp only: Ssum0_ss)
+apply (rule chain_UU_I_inverse [symmetric])
+apply (rule allI)
+apply (rule_tac s = "Isinl (UU) " and t = "Y (i) " in subst)
+apply (rule inst_ssum_pcpo [THEN subst])
+apply (rule chain_UU_I [THEN spec, symmetric])
+apply assumption
+apply (erule inst_ssum_pcpo [THEN ssubst])
+apply (simp only: Ssum0_ss)
+done
+
+lemma ssum_lemma12: "[| chain(Y); lub(range(Y)) = Isinl(x); x ~= UU |] ==> 
+    Iwhen f g (lub(range Y)) = lub(range(%i. Iwhen f g (Y i)))"
+apply simp
+apply (rule_tac t = "x" in subst)
+apply (rule inject_Isinl)
+apply (rule trans)
+prefer 2 apply (assumption)
+apply (rule thelub_ssum1a [symmetric])
+apply assumption
+apply (erule ssum_lemma9)
+apply assumption
+apply (rule trans)
+apply (rule contlub_cfun_arg)
+apply (rule monofun_Iwhen3 [THEN ch2ch_monofun])
+apply assumption
+apply (rule lub_equal2)
+apply (rule chain_mono2 [THEN exE])
+prefer 2 apply (assumption)
+apply (rule chain_UU_I_inverse2)
+apply (subst inst_ssum_pcpo)
+apply (erule contrapos_np)
+apply (rule inject_Isinl)
+apply (rule trans)
+apply (erule sym)
+apply (erule notnotD)
+apply (rule exI)
+apply (intro strip)
+apply (rule ssum_lemma9 [THEN spec, THEN exE])
+apply assumption
+apply assumption
+apply (rule_tac t = "Y (i) " in ssubst)
+apply assumption
+apply (rule trans)
+apply (rule cfun_arg_cong)
+apply (rule Iwhen2)
+apply force
+apply (rule_tac t = "Y (i) " in ssubst)
+apply assumption
+apply auto
+apply (subst Iwhen2)
+apply force
+apply (rule refl)
+apply (rule monofun_Rep_CFun2 [THEN ch2ch_monofun])
+apply (erule monofun_Iwhen3 [THEN ch2ch_monofun])
+apply (erule monofun_Iwhen3 [THEN ch2ch_monofun])
+done
+
+
+lemma ssum_lemma13: "[| chain(Y); lub(range(Y)) = Isinr(x); x ~= UU |] ==> 
+    Iwhen f g (lub(range Y)) = lub(range(%i. Iwhen f g (Y i)))"
+apply simp
+apply (rule_tac t = "x" in subst)
+apply (rule inject_Isinr)
+apply (rule trans)
+prefer 2 apply (assumption)
+apply (rule thelub_ssum1b [symmetric])
+apply assumption
+apply (erule ssum_lemma10)
+apply assumption
+apply (rule trans)
+apply (rule contlub_cfun_arg)
+apply (rule monofun_Iwhen3 [THEN ch2ch_monofun])
+apply assumption
+apply (rule lub_equal2)
+apply (rule chain_mono2 [THEN exE])
+prefer 2 apply (assumption)
+apply (rule chain_UU_I_inverse2)
+apply (subst inst_ssum_pcpo)
+apply (erule contrapos_np)
+apply (rule inject_Isinr)
+apply (rule trans)
+apply (erule sym)
+apply (rule strict_IsinlIsinr [THEN subst])
+apply (erule notnotD)
+apply (rule exI)
+apply (intro strip)
+apply (rule ssum_lemma10 [THEN spec, THEN exE])
+apply assumption
+apply assumption
+apply (rule_tac t = "Y (i) " in ssubst)
+apply assumption
+apply (rule trans)
+apply (rule cfun_arg_cong)
+apply (rule Iwhen3)
+apply force
+apply (rule_tac t = "Y (i) " in ssubst)
+apply assumption
+apply (subst Iwhen3)
+apply force
+apply (rule_tac t = "Y (i) " in ssubst)
+apply assumption
+apply simp
+apply (rule monofun_Rep_CFun2 [THEN ch2ch_monofun])
+apply (erule monofun_Iwhen3 [THEN ch2ch_monofun])
+apply (erule monofun_Iwhen3 [THEN ch2ch_monofun])
+done
+
+
+lemma contlub_Iwhen3: "contlub(Iwhen(f)(g))"
+apply (rule contlubI)
+apply (intro strip)
+apply (rule_tac p = "lub (range (Y))" in IssumE)
+apply (erule ssum_lemma11)
+apply assumption
+apply (erule ssum_lemma12)
+apply assumption
+apply assumption
+apply (erule ssum_lemma13)
+apply assumption
+apply assumption
+done
+
+lemma cont_Iwhen1: "cont(Iwhen)"
+apply (rule monocontlub2cont)
+apply (rule monofun_Iwhen1)
+apply (rule contlub_Iwhen1)
+done
+
+lemma cont_Iwhen2: "cont(Iwhen(f))"
+apply (rule monocontlub2cont)
+apply (rule monofun_Iwhen2)
+apply (rule contlub_Iwhen2)
+done
+
+lemma cont_Iwhen3: "cont(Iwhen(f)(g))"
+apply (rule monocontlub2cont)
+apply (rule monofun_Iwhen3)
+apply (rule contlub_Iwhen3)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* continuous versions of lemmas for 'a ++ 'b                               *)
+(* ------------------------------------------------------------------------ *)
+
+lemma strict_sinl [simp]: "sinl$UU =UU"
+apply (unfold sinl_def)
+apply (simp add: cont_Isinl)
+done
+
+lemma strict_sinr [simp]: "sinr$UU=UU"
+apply (unfold sinr_def)
+apply (simp add: cont_Isinr)
+done
+
+lemma noteq_sinlsinr: 
+        "sinl$a=sinr$b ==> a=UU & b=UU"
+apply (unfold sinl_def sinr_def)
+apply (auto dest!: noteq_IsinlIsinr)
+done
+
+lemma inject_sinl: 
+        "sinl$a1=sinl$a2==> a1=a2"
+apply (unfold sinl_def sinr_def)
+apply auto
+done
+
+lemma inject_sinr: 
+        "sinr$a1=sinr$a2==> a1=a2"
+apply (unfold sinl_def sinr_def)
+apply auto
+done
+
+declare inject_sinl [dest!] inject_sinr [dest!]
+
+lemma defined_sinl [simp]: "x~=UU ==> sinl$x ~= UU"
+apply (erule contrapos_nn)
+apply (rule inject_sinl)
+apply auto
+done
+
+lemma defined_sinr [simp]: "x~=UU ==> sinr$x ~= UU"
+apply (erule contrapos_nn)
+apply (rule inject_sinr)
+apply auto
+done
+
+lemma Exh_Ssum1: 
+        "z=UU | (? a. z=sinl$a & a~=UU) | (? b. z=sinr$b & b~=UU)"
+apply (unfold sinl_def sinr_def)
+apply simp
+apply (subst inst_ssum_pcpo)
+apply (rule Exh_Ssum)
+done
+
+
+lemma ssumE:
+assumes major: "p=UU ==> Q"
+assumes prem2: "!!x.[|p=sinl$x; x~=UU |] ==> Q"
+assumes prem3: "!!y.[|p=sinr$y; y~=UU |] ==> Q"
+shows "Q"
+apply (rule major [THEN IssumE])
+apply (subst inst_ssum_pcpo)
+apply assumption
+apply (rule prem2)
+prefer 2 apply (assumption)
+apply (simp add: sinl_def)
+apply (rule prem3)
+prefer 2 apply (assumption)
+apply (simp add: sinr_def)
+done
+
+
+lemma ssumE2:
+assumes preml: "!!x.[|p=sinl$x|] ==> Q"
+assumes premr: "!!y.[|p=sinr$y|] ==> Q"
+shows "Q"
+apply (rule IssumE2)
+apply (rule preml)
+apply (rule_tac [2] premr)
+apply (unfold sinl_def sinr_def)
+apply auto
+done
+
+lemmas ssum_conts = cont_lemmas1 cont_Iwhen1 cont_Iwhen2
+                cont_Iwhen3 cont2cont_CF1L
+
+lemma sscase1 [simp]: 
+        "sscase$f$g$UU = UU"
+apply (unfold sscase_def sinl_def sinr_def)
+apply (subst inst_ssum_pcpo)
+apply (subst beta_cfun)
+apply (intro ssum_conts)
+apply (subst beta_cfun)
+apply (intro ssum_conts)
+apply (subst beta_cfun)
+apply (intro ssum_conts)
+apply (simp only: Ssum0_ss)
+done
+
+lemma sscase2 [simp]: 
+        "x~=UU==> sscase$f$g$(sinl$x) = f$x"
+apply (unfold sscase_def sinl_def sinr_def)
+apply (simplesubst beta_cfun)
+apply (rule cont_Isinl)
+apply (subst beta_cfun)
+apply (intro ssum_conts)
+apply (subst beta_cfun)
+apply (intro ssum_conts)
+apply (subst beta_cfun)
+apply (intro ssum_conts)
+apply simp
+done
+
+lemma sscase3 [simp]: 
+        "x~=UU==> sscase$f$g$(sinr$x) = g$x"
+apply (unfold sscase_def sinl_def sinr_def)
+apply (simplesubst beta_cfun)
+apply (rule cont_Isinr)
+apply (subst beta_cfun)
+apply (intro ssum_conts)
+apply (subst beta_cfun)
+apply (intro ssum_conts)
+apply (subst beta_cfun)
+apply (intro ssum_conts)
+apply simp
+done
+
+lemma less_ssum4a: 
+        "(sinl$x << sinl$y) = (x << y)"
+apply (unfold sinl_def sinr_def)
+apply (subst beta_cfun)
+apply (rule cont_Isinl)
+apply (subst beta_cfun)
+apply (rule cont_Isinl)
+apply (rule less_ssum3a)
+done
+
+lemma less_ssum4b: 
+        "(sinr$x << sinr$y) = (x << y)"
+apply (unfold sinl_def sinr_def)
+apply (subst beta_cfun)
+apply (rule cont_Isinr)
+apply (subst beta_cfun)
+apply (rule cont_Isinr)
+apply (rule less_ssum3b)
+done
+
+lemma less_ssum4c: 
+        "(sinl$x << sinr$y) = (x = UU)"
+apply (unfold sinl_def sinr_def)
+apply (simplesubst beta_cfun)
+apply (rule cont_Isinr)
+apply (subst beta_cfun)
+apply (rule cont_Isinl)
+apply (rule less_ssum3c)
+done
+
+lemma less_ssum4d: 
+        "(sinr$x << sinl$y) = (x = UU)"
+apply (unfold sinl_def sinr_def)
+apply (simplesubst beta_cfun)
+apply (rule cont_Isinl)
+apply (subst beta_cfun)
+apply (rule cont_Isinr)
+apply (rule less_ssum3d)
+done
+
+lemma ssum_chainE: 
+        "chain(Y) ==> (!i.? x.(Y i)=sinl$x)|(!i.? y.(Y i)=sinr$y)"
+apply (unfold sinl_def sinr_def)
+apply simp
+apply (erule ssum_lemma4)
+done
+
+lemma thelub_ssum2a: 
+"[| chain(Y); !i.? x. Y(i) = sinl$x |] ==>  
+    lub(range(Y)) = sinl$(lub(range(%i. sscase$(LAM x. x)$(LAM y. UU)$(Y i))))"
+apply (unfold sinl_def sinr_def sscase_def)
+apply (subst beta_cfun)
+apply (rule cont_Isinl)
+apply (subst beta_cfun)
+apply (intro ssum_conts)
+apply (subst beta_cfun)
+apply (intro ssum_conts)
+apply (subst beta_cfun [THEN ext])
+apply (intro ssum_conts)
+apply (rule thelub_ssum1a)
+apply assumption
+apply (rule allI)
+apply (erule allE)
+apply (erule exE)
+apply (rule exI)
+apply (erule box_equals)
+apply (rule refl)
+apply simp
+done
+
+lemma thelub_ssum2b: 
+"[| chain(Y); !i.? x. Y(i) = sinr$x |] ==>  
+    lub(range(Y)) = sinr$(lub(range(%i. sscase$(LAM y. UU)$(LAM x. x)$(Y i))))"
+apply (unfold sinl_def sinr_def sscase_def)
+apply (subst beta_cfun)
+apply (rule cont_Isinr)
+apply (subst beta_cfun)
+apply (intro ssum_conts)
+apply (subst beta_cfun)
+apply (intro ssum_conts)
+apply (subst beta_cfun [THEN ext])
+apply (intro ssum_conts)
+apply (rule thelub_ssum1b)
+apply assumption
+apply (rule allI)
+apply (erule allE)
+apply (erule exE)
+apply (rule exI)
+apply (erule box_equals)
+apply (rule refl)
+apply simp
+done
+
+lemma thelub_ssum2a_rev: 
+        "[| chain(Y); lub(range(Y)) = sinl$x|] ==> !i.? x. Y(i)=sinl$x"
+apply (unfold sinl_def sinr_def)
+apply simp
+apply (erule ssum_lemma9)
+apply simp
+done
+
+lemma thelub_ssum2b_rev: 
+     "[| chain(Y); lub(range(Y)) = sinr$x|] ==> !i.? x. Y(i)=sinr$x"
+apply (unfold sinl_def sinr_def)
+apply simp
+apply (erule ssum_lemma10)
+apply simp
+done
+
+lemma thelub_ssum3: "chain(Y) ==>  
+    lub(range(Y)) = sinl$(lub(range(%i. sscase$(LAM x. x)$(LAM y. UU)$(Y i)))) 
+  | lub(range(Y)) = sinr$(lub(range(%i. sscase$(LAM y. UU)$(LAM x. x)$(Y i))))"
+apply (rule ssum_chainE [THEN disjE])
+apply assumption
+apply (rule disjI1)
+apply (erule thelub_ssum2a)
+apply assumption
+apply (rule disjI2)
+apply (erule thelub_ssum2b)
+apply assumption
+done
+
+lemma sscase4: "sscase$sinl$sinr$z=z"
+apply (rule_tac p = "z" in ssumE)
+apply auto
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* install simplifier for Ssum                                              *)
+(* ------------------------------------------------------------------------ *)
+
+lemmas Ssum_rews = strict_sinl strict_sinr defined_sinl defined_sinr
+                sscase1 sscase2 sscase3
+
+end