--- a/src/HOLCF/Bifinite.thy Mon Oct 11 07:09:42 2010 -0700
+++ b/src/HOLCF/Bifinite.thy Mon Oct 11 08:32:09 2010 -0700
@@ -18,12 +18,12 @@
class bifinite = pcpo +
fixes emb :: "'a::pcpo \<rightarrow> udom"
fixes prj :: "udom \<rightarrow> 'a::pcpo"
- fixes sfp :: "'a itself \<Rightarrow> sfp"
+ fixes defl :: "'a itself \<Rightarrow> defl"
assumes ep_pair_emb_prj: "ep_pair emb prj"
- assumes cast_SFP: "cast\<cdot>(sfp TYPE('a)) = emb oo prj"
+ assumes cast_DEFL: "cast\<cdot>(defl TYPE('a)) = emb oo prj"
-syntax "_SFP" :: "type \<Rightarrow> sfp" ("(1SFP/(1'(_')))")
-translations "SFP('t)" \<rightleftharpoons> "CONST sfp TYPE('t)"
+syntax "_DEFL" :: "type \<Rightarrow> defl" ("(1DEFL/(1'(_')))")
+translations "DEFL('t)" \<rightleftharpoons> "CONST defl TYPE('t)"
interpretation bifinite:
pcpo_ep_pair "emb :: 'a::bifinite \<rightarrow> udom" "prj :: udom \<rightarrow> 'a::bifinite"
@@ -47,24 +47,24 @@
ideal_completion below Rep_compact_basis "approximants::'a::bifinite \<Rightarrow> _"
proof -
obtain Y where Y: "\<forall>i. Y i \<sqsubseteq> Y (Suc i)"
- and SFP: "SFP('a) = (\<Squnion>i. sfp_principal (Y i))"
- by (rule sfp.obtain_principal_chain)
- def approx \<equiv> "\<lambda>i. (prj oo cast\<cdot>(sfp_principal (Y i)) oo emb) :: 'a \<rightarrow> 'a"
- interpret sfp_approx: approx_chain approx
+ and DEFL: "DEFL('a) = (\<Squnion>i. defl_principal (Y i))"
+ by (rule defl.obtain_principal_chain)
+ def approx \<equiv> "\<lambda>i. (prj oo cast\<cdot>(defl_principal (Y i)) oo emb) :: 'a \<rightarrow> 'a"
+ interpret defl_approx: approx_chain approx
proof (rule approx_chain.intro)
show "chain (\<lambda>i. approx i)"
unfolding approx_def by (simp add: Y)
show "(\<Squnion>i. approx i) = ID"
unfolding approx_def
- by (simp add: lub_distribs Y SFP [symmetric] cast_SFP expand_cfun_eq)
+ by (simp add: lub_distribs Y DEFL [symmetric] cast_DEFL expand_cfun_eq)
show "\<And>i. finite_deflation (approx i)"
unfolding approx_def
apply (rule bifinite.finite_deflation_p_d_e)
apply (rule finite_deflation_cast)
- apply (rule sfp.compact_principal)
+ apply (rule defl.compact_principal)
apply (rule below_trans [OF monofun_cfun_fun])
apply (rule is_ub_thelub, simp add: Y)
- apply (simp add: lub_distribs Y SFP [symmetric] cast_SFP)
+ apply (simp add: lub_distribs Y DEFL [symmetric] cast_DEFL)
done
qed
(* FIXME: why does show ?thesis fail here? *)
@@ -74,30 +74,30 @@
subsection {* Type combinators *}
definition
- sfp_fun1 ::
- "(nat \<Rightarrow> 'a \<rightarrow> 'a) \<Rightarrow> ((udom \<rightarrow> udom) \<rightarrow> ('a \<rightarrow> 'a)) \<Rightarrow> (sfp \<rightarrow> sfp)"
+ defl_fun1 ::
+ "(nat \<Rightarrow> 'a \<rightarrow> 'a) \<Rightarrow> ((udom \<rightarrow> udom) \<rightarrow> ('a \<rightarrow> 'a)) \<Rightarrow> (defl \<rightarrow> defl)"
where
- "sfp_fun1 approx f =
- sfp.basis_fun (\<lambda>a.
- sfp_principal (Abs_fin_defl
+ "defl_fun1 approx f =
+ defl.basis_fun (\<lambda>a.
+ defl_principal (Abs_fin_defl
(udom_emb approx oo f\<cdot>(Rep_fin_defl a) oo udom_prj approx)))"
definition
- sfp_fun2 ::
+ defl_fun2 ::
"(nat \<Rightarrow> 'a \<rightarrow> 'a) \<Rightarrow> ((udom \<rightarrow> udom) \<rightarrow> (udom \<rightarrow> udom) \<rightarrow> ('a \<rightarrow> 'a))
- \<Rightarrow> (sfp \<rightarrow> sfp \<rightarrow> sfp)"
+ \<Rightarrow> (defl \<rightarrow> defl \<rightarrow> defl)"
where
- "sfp_fun2 approx f =
- sfp.basis_fun (\<lambda>a.
- sfp.basis_fun (\<lambda>b.
- sfp_principal (Abs_fin_defl
+ "defl_fun2 approx f =
+ defl.basis_fun (\<lambda>a.
+ defl.basis_fun (\<lambda>b.
+ defl_principal (Abs_fin_defl
(udom_emb approx oo
f\<cdot>(Rep_fin_defl a)\<cdot>(Rep_fin_defl b) oo udom_prj approx))))"
-lemma cast_sfp_fun1:
+lemma cast_defl_fun1:
assumes approx: "approx_chain approx"
assumes f: "\<And>a. finite_deflation a \<Longrightarrow> finite_deflation (f\<cdot>a)"
- shows "cast\<cdot>(sfp_fun1 approx f\<cdot>A) = udom_emb approx oo f\<cdot>(cast\<cdot>A) oo udom_prj approx"
+ shows "cast\<cdot>(defl_fun1 approx f\<cdot>A) = udom_emb approx oo f\<cdot>(cast\<cdot>A) oo udom_prj approx"
proof -
have 1: "\<And>a. finite_deflation
(udom_emb approx oo f\<cdot>(Rep_fin_defl a) oo udom_prj approx)"
@@ -106,23 +106,23 @@
apply (rule f, rule finite_deflation_Rep_fin_defl)
done
show ?thesis
- by (induct A rule: sfp.principal_induct, simp)
- (simp only: sfp_fun1_def
- sfp.basis_fun_principal
- sfp.basis_fun_mono
- sfp.principal_mono
+ by (induct A rule: defl.principal_induct, simp)
+ (simp only: defl_fun1_def
+ defl.basis_fun_principal
+ defl.basis_fun_mono
+ defl.principal_mono
Abs_fin_defl_mono [OF 1 1]
monofun_cfun below_refl
Rep_fin_defl_mono
- cast_sfp_principal
+ cast_defl_principal
Abs_fin_defl_inverse [unfolded mem_Collect_eq, OF 1])
qed
-lemma cast_sfp_fun2:
+lemma cast_defl_fun2:
assumes approx: "approx_chain approx"
assumes f: "\<And>a b. finite_deflation a \<Longrightarrow> finite_deflation b \<Longrightarrow>
finite_deflation (f\<cdot>a\<cdot>b)"
- shows "cast\<cdot>(sfp_fun2 approx f\<cdot>A\<cdot>B) =
+ shows "cast\<cdot>(defl_fun2 approx f\<cdot>A\<cdot>B) =
udom_emb approx oo f\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj approx"
proof -
have 1: "\<And>a b. finite_deflation (udom_emb approx oo
@@ -132,15 +132,15 @@
apply (rule f, (rule finite_deflation_Rep_fin_defl)+)
done
show ?thesis
- by (induct A B rule: sfp.principal_induct2, simp, simp)
- (simp only: sfp_fun2_def
- sfp.basis_fun_principal
- sfp.basis_fun_mono
- sfp.principal_mono
+ by (induct A B rule: defl.principal_induct2, simp, simp)
+ (simp only: defl_fun2_def
+ defl.basis_fun_principal
+ defl.basis_fun_mono
+ defl.principal_mono
Abs_fin_defl_mono [OF 1 1]
monofun_cfun below_refl
Rep_fin_defl_mono
- cast_sfp_principal
+ cast_defl_principal
Abs_fin_defl_inverse [unfolded mem_Collect_eq, OF 1])
qed
@@ -156,22 +156,22 @@
"prj = (ID :: udom \<rightarrow> udom)"
definition
- "sfp (t::udom itself) = (\<Squnion>i. sfp_principal (Abs_fin_defl (udom_approx i)))"
+ "defl (t::udom itself) = (\<Squnion>i. defl_principal (Abs_fin_defl (udom_approx i)))"
instance proof
show "ep_pair emb (prj :: udom \<rightarrow> udom)"
by (simp add: ep_pair.intro)
next
- show "cast\<cdot>SFP(udom) = emb oo (prj :: udom \<rightarrow> udom)"
- unfolding sfp_udom_def
+ show "cast\<cdot>DEFL(udom) = emb oo (prj :: udom \<rightarrow> udom)"
+ unfolding defl_udom_def
apply (subst contlub_cfun_arg)
apply (rule chainI)
- apply (rule sfp.principal_mono)
+ apply (rule defl.principal_mono)
apply (simp add: below_fin_defl_def)
apply (simp add: Abs_fin_defl_inverse finite_deflation_udom_approx)
apply (rule chainE)
apply (rule chain_udom_approx)
- apply (subst cast_sfp_principal)
+ apply (subst cast_defl_principal)
apply (simp add: Abs_fin_defl_inverse finite_deflation_udom_approx)
done
qed
@@ -197,14 +197,14 @@
by (intro finite_deflation_cfun_map finite_deflation_udom_approx)
qed
-definition cfun_sfp :: "sfp \<rightarrow> sfp \<rightarrow> sfp"
-where "cfun_sfp = sfp_fun2 cfun_approx cfun_map"
+definition cfun_defl :: "defl \<rightarrow> defl \<rightarrow> defl"
+where "cfun_defl = defl_fun2 cfun_approx cfun_map"
-lemma cast_cfun_sfp:
- "cast\<cdot>(cfun_sfp\<cdot>A\<cdot>B) =
+lemma cast_cfun_defl:
+ "cast\<cdot>(cfun_defl\<cdot>A\<cdot>B) =
udom_emb cfun_approx oo cfun_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj cfun_approx"
-unfolding cfun_sfp_def
-apply (rule cast_sfp_fun2 [OF cfun_approx])
+unfolding cfun_defl_def
+apply (rule cast_defl_fun2 [OF cfun_approx])
apply (erule (1) finite_deflation_cfun_map)
done
@@ -218,7 +218,7 @@
"prj = cfun_map\<cdot>emb\<cdot>prj oo udom_prj cfun_approx"
definition
- "sfp (t::('a \<rightarrow> 'b) itself) = cfun_sfp\<cdot>SFP('a)\<cdot>SFP('b)"
+ "defl (t::('a \<rightarrow> 'b) itself) = cfun_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
instance proof
show "ep_pair emb (prj :: udom \<rightarrow> 'a \<rightarrow> 'b)"
@@ -226,16 +226,16 @@
using ep_pair_udom [OF cfun_approx]
by (intro ep_pair_comp ep_pair_cfun_map ep_pair_emb_prj)
next
- show "cast\<cdot>SFP('a \<rightarrow> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<rightarrow> 'b)"
- unfolding emb_cfun_def prj_cfun_def sfp_cfun_def cast_cfun_sfp
- by (simp add: cast_SFP oo_def expand_cfun_eq cfun_map_map)
+ show "cast\<cdot>DEFL('a \<rightarrow> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<rightarrow> 'b)"
+ unfolding emb_cfun_def prj_cfun_def defl_cfun_def cast_cfun_defl
+ by (simp add: cast_DEFL oo_def expand_cfun_eq cfun_map_map)
qed
end
-lemma SFP_cfun:
- "SFP('a::bifinite \<rightarrow> 'b::bifinite) = cfun_sfp\<cdot>SFP('a)\<cdot>SFP('b)"
-by (rule sfp_cfun_def)
+lemma DEFL_cfun:
+ "DEFL('a::bifinite \<rightarrow> 'b::bifinite) = cfun_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
+by (rule defl_cfun_def)
subsection {* Cartesian product is a bifinite domain *}
@@ -256,14 +256,14 @@
by (intro finite_deflation_cprod_map finite_deflation_udom_approx)
qed
-definition prod_sfp :: "sfp \<rightarrow> sfp \<rightarrow> sfp"
-where "prod_sfp = sfp_fun2 prod_approx cprod_map"
+definition prod_defl :: "defl \<rightarrow> defl \<rightarrow> defl"
+where "prod_defl = defl_fun2 prod_approx cprod_map"
-lemma cast_prod_sfp:
- "cast\<cdot>(prod_sfp\<cdot>A\<cdot>B) = udom_emb prod_approx oo
+lemma cast_prod_defl:
+ "cast\<cdot>(prod_defl\<cdot>A\<cdot>B) = udom_emb prod_approx oo
cprod_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj prod_approx"
-unfolding prod_sfp_def
-apply (rule cast_sfp_fun2 [OF prod_approx])
+unfolding prod_defl_def
+apply (rule cast_defl_fun2 [OF prod_approx])
apply (erule (1) finite_deflation_cprod_map)
done
@@ -277,7 +277,7 @@
"prj = cprod_map\<cdot>prj\<cdot>prj oo udom_prj prod_approx"
definition
- "sfp (t::('a \<times> 'b) itself) = prod_sfp\<cdot>SFP('a)\<cdot>SFP('b)"
+ "defl (t::('a \<times> 'b) itself) = prod_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
instance proof
show "ep_pair emb (prj :: udom \<rightarrow> 'a \<times> 'b)"
@@ -285,16 +285,16 @@
using ep_pair_udom [OF prod_approx]
by (intro ep_pair_comp ep_pair_cprod_map ep_pair_emb_prj)
next
- show "cast\<cdot>SFP('a \<times> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<times> 'b)"
- unfolding emb_prod_def prj_prod_def sfp_prod_def cast_prod_sfp
- by (simp add: cast_SFP oo_def expand_cfun_eq cprod_map_map)
+ show "cast\<cdot>DEFL('a \<times> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<times> 'b)"
+ unfolding emb_prod_def prj_prod_def defl_prod_def cast_prod_defl
+ by (simp add: cast_DEFL oo_def expand_cfun_eq cprod_map_map)
qed
end
-lemma SFP_prod:
- "SFP('a::bifinite \<times> 'b::bifinite) = prod_sfp\<cdot>SFP('a)\<cdot>SFP('b)"
-by (rule sfp_prod_def)
+lemma DEFL_prod:
+ "DEFL('a::bifinite \<times> 'b::bifinite) = prod_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
+by (rule defl_prod_def)
subsection {* Strict product is a bifinite domain *}
@@ -315,16 +315,16 @@
by (intro finite_deflation_sprod_map finite_deflation_udom_approx)
qed
-definition sprod_sfp :: "sfp \<rightarrow> sfp \<rightarrow> sfp"
-where "sprod_sfp = sfp_fun2 sprod_approx sprod_map"
+definition sprod_defl :: "defl \<rightarrow> defl \<rightarrow> defl"
+where "sprod_defl = defl_fun2 sprod_approx sprod_map"
-lemma cast_sprod_sfp:
- "cast\<cdot>(sprod_sfp\<cdot>A\<cdot>B) =
+lemma cast_sprod_defl:
+ "cast\<cdot>(sprod_defl\<cdot>A\<cdot>B) =
udom_emb sprod_approx oo
sprod_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo
udom_prj sprod_approx"
-unfolding sprod_sfp_def
-apply (rule cast_sfp_fun2 [OF sprod_approx])
+unfolding sprod_defl_def
+apply (rule cast_defl_fun2 [OF sprod_approx])
apply (erule (1) finite_deflation_sprod_map)
done
@@ -338,7 +338,7 @@
"prj = sprod_map\<cdot>prj\<cdot>prj oo udom_prj sprod_approx"
definition
- "sfp (t::('a \<otimes> 'b) itself) = sprod_sfp\<cdot>SFP('a)\<cdot>SFP('b)"
+ "defl (t::('a \<otimes> 'b) itself) = sprod_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
instance proof
show "ep_pair emb (prj :: udom \<rightarrow> 'a \<otimes> 'b)"
@@ -346,16 +346,16 @@
using ep_pair_udom [OF sprod_approx]
by (intro ep_pair_comp ep_pair_sprod_map ep_pair_emb_prj)
next
- show "cast\<cdot>SFP('a \<otimes> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<otimes> 'b)"
- unfolding emb_sprod_def prj_sprod_def sfp_sprod_def cast_sprod_sfp
- by (simp add: cast_SFP oo_def expand_cfun_eq sprod_map_map)
+ show "cast\<cdot>DEFL('a \<otimes> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<otimes> 'b)"
+ unfolding emb_sprod_def prj_sprod_def defl_sprod_def cast_sprod_defl
+ by (simp add: cast_DEFL oo_def expand_cfun_eq sprod_map_map)
qed
end
-lemma SFP_sprod:
- "SFP('a::bifinite \<otimes> 'b::bifinite) = sprod_sfp\<cdot>SFP('a)\<cdot>SFP('b)"
-by (rule sfp_sprod_def)
+lemma DEFL_sprod:
+ "DEFL('a::bifinite \<otimes> 'b::bifinite) = sprod_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
+by (rule defl_sprod_def)
subsection {* Lifted cpo is a bifinite domain *}
@@ -374,14 +374,14 @@
by (intro finite_deflation_u_map finite_deflation_udom_approx)
qed
-definition u_sfp :: "sfp \<rightarrow> sfp"
-where "u_sfp = sfp_fun1 u_approx u_map"
+definition u_defl :: "defl \<rightarrow> defl"
+where "u_defl = defl_fun1 u_approx u_map"
-lemma cast_u_sfp:
- "cast\<cdot>(u_sfp\<cdot>A) =
+lemma cast_u_defl:
+ "cast\<cdot>(u_defl\<cdot>A) =
udom_emb u_approx oo u_map\<cdot>(cast\<cdot>A) oo udom_prj u_approx"
-unfolding u_sfp_def
-apply (rule cast_sfp_fun1 [OF u_approx])
+unfolding u_defl_def
+apply (rule cast_defl_fun1 [OF u_approx])
apply (erule finite_deflation_u_map)
done
@@ -395,7 +395,7 @@
"prj = u_map\<cdot>prj oo udom_prj u_approx"
definition
- "sfp (t::'a u itself) = u_sfp\<cdot>SFP('a)"
+ "defl (t::'a u itself) = u_defl\<cdot>DEFL('a)"
instance proof
show "ep_pair emb (prj :: udom \<rightarrow> 'a u)"
@@ -403,15 +403,15 @@
using ep_pair_udom [OF u_approx]
by (intro ep_pair_comp ep_pair_u_map ep_pair_emb_prj)
next
- show "cast\<cdot>SFP('a u) = emb oo (prj :: udom \<rightarrow> 'a u)"
- unfolding emb_u_def prj_u_def sfp_u_def cast_u_sfp
- by (simp add: cast_SFP oo_def expand_cfun_eq u_map_map)
+ show "cast\<cdot>DEFL('a u) = emb oo (prj :: udom \<rightarrow> 'a u)"
+ unfolding emb_u_def prj_u_def defl_u_def cast_u_defl
+ by (simp add: cast_DEFL oo_def expand_cfun_eq u_map_map)
qed
end
-lemma SFP_u: "SFP('a::bifinite u) = u_sfp\<cdot>SFP('a)"
-by (rule sfp_u_def)
+lemma DEFL_u: "DEFL('a::bifinite u) = u_defl\<cdot>DEFL('a)"
+by (rule defl_u_def)
subsection {* Lifted countable types are bifinite domains *}
@@ -472,25 +472,25 @@
"prj = udom_prj lift_approx"
definition
- "sfp (t::'a lift itself) =
- (\<Squnion>i. sfp_principal (Abs_fin_defl (emb oo lift_approx i oo prj)))"
+ "defl (t::'a lift itself) =
+ (\<Squnion>i. defl_principal (Abs_fin_defl (emb oo lift_approx i oo prj)))"
instance proof
show ep: "ep_pair emb (prj :: udom \<rightarrow> 'a lift)"
unfolding emb_lift_def prj_lift_def
by (rule ep_pair_udom [OF lift_approx])
- show "cast\<cdot>SFP('a lift) = emb oo (prj :: udom \<rightarrow> 'a lift)"
- unfolding sfp_lift_def
+ show "cast\<cdot>DEFL('a lift) = emb oo (prj :: udom \<rightarrow> 'a lift)"
+ unfolding defl_lift_def
apply (subst contlub_cfun_arg)
apply (rule chainI)
- apply (rule sfp.principal_mono)
+ apply (rule defl.principal_mono)
apply (simp add: below_fin_defl_def)
apply (simp add: Abs_fin_defl_inverse finite_deflation_lift_approx
ep_pair.finite_deflation_e_d_p [OF ep])
apply (intro monofun_cfun below_refl)
apply (rule chainE)
apply (rule chain_lift_approx)
- apply (subst cast_sfp_principal)
+ apply (subst cast_defl_principal)
apply (simp add: Abs_fin_defl_inverse finite_deflation_lift_approx
ep_pair.finite_deflation_e_d_p [OF ep] lub_distribs)
done
@@ -517,14 +517,14 @@
by (intro finite_deflation_ssum_map finite_deflation_udom_approx)
qed
-definition ssum_sfp :: "sfp \<rightarrow> sfp \<rightarrow> sfp"
-where "ssum_sfp = sfp_fun2 ssum_approx ssum_map"
+definition ssum_defl :: "defl \<rightarrow> defl \<rightarrow> defl"
+where "ssum_defl = defl_fun2 ssum_approx ssum_map"
-lemma cast_ssum_sfp:
- "cast\<cdot>(ssum_sfp\<cdot>A\<cdot>B) =
+lemma cast_ssum_defl:
+ "cast\<cdot>(ssum_defl\<cdot>A\<cdot>B) =
udom_emb ssum_approx oo ssum_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj ssum_approx"
-unfolding ssum_sfp_def
-apply (rule cast_sfp_fun2 [OF ssum_approx])
+unfolding ssum_defl_def
+apply (rule cast_defl_fun2 [OF ssum_approx])
apply (erule (1) finite_deflation_ssum_map)
done
@@ -538,7 +538,7 @@
"prj = ssum_map\<cdot>prj\<cdot>prj oo udom_prj ssum_approx"
definition
- "sfp (t::('a \<oplus> 'b) itself) = ssum_sfp\<cdot>SFP('a)\<cdot>SFP('b)"
+ "defl (t::('a \<oplus> 'b) itself) = ssum_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
instance proof
show "ep_pair emb (prj :: udom \<rightarrow> 'a \<oplus> 'b)"
@@ -546,15 +546,15 @@
using ep_pair_udom [OF ssum_approx]
by (intro ep_pair_comp ep_pair_ssum_map ep_pair_emb_prj)
next
- show "cast\<cdot>SFP('a \<oplus> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<oplus> 'b)"
- unfolding emb_ssum_def prj_ssum_def sfp_ssum_def cast_ssum_sfp
- by (simp add: cast_SFP oo_def expand_cfun_eq ssum_map_map)
+ show "cast\<cdot>DEFL('a \<oplus> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<oplus> 'b)"
+ unfolding emb_ssum_def prj_ssum_def defl_ssum_def cast_ssum_defl
+ by (simp add: cast_DEFL oo_def expand_cfun_eq ssum_map_map)
qed
end
-lemma SFP_ssum:
- "SFP('a::bifinite \<oplus> 'b::bifinite) = ssum_sfp\<cdot>SFP('a)\<cdot>SFP('b)"
-by (rule sfp_ssum_def)
+lemma DEFL_ssum:
+ "DEFL('a::bifinite \<oplus> 'b::bifinite) = ssum_defl\<cdot>DEFL('a)\<cdot>DEFL('b)"
+by (rule defl_ssum_def)
end