--- a/src/HOL/HOLCF/Map_Functions.thy Mon Jan 01 21:17:28 2018 +0100
+++ b/src/HOL/HOLCF/Map_Functions.thy Mon Jan 01 23:07:24 2018 +0100
@@ -5,28 +5,24 @@
section \<open>Map functions for various types\<close>
theory Map_Functions
-imports Deflation Sprod Ssum Sfun Up
+ imports Deflation Sprod Ssum Sfun Up
begin
subsection \<open>Map operator for continuous function space\<close>
default_sort cpo
-definition
- cfun_map :: "('b \<rightarrow> 'a) \<rightarrow> ('c \<rightarrow> 'd) \<rightarrow> ('a \<rightarrow> 'c) \<rightarrow> ('b \<rightarrow> 'd)"
-where
- "cfun_map = (\<Lambda> a b f x. b\<cdot>(f\<cdot>(a\<cdot>x)))"
+definition cfun_map :: "('b \<rightarrow> 'a) \<rightarrow> ('c \<rightarrow> 'd) \<rightarrow> ('a \<rightarrow> 'c) \<rightarrow> ('b \<rightarrow> 'd)"
+ where "cfun_map = (\<Lambda> a b f x. b\<cdot>(f\<cdot>(a\<cdot>x)))"
lemma cfun_map_beta [simp]: "cfun_map\<cdot>a\<cdot>b\<cdot>f\<cdot>x = b\<cdot>(f\<cdot>(a\<cdot>x))"
-unfolding cfun_map_def by simp
+ by (simp add: cfun_map_def)
lemma cfun_map_ID: "cfun_map\<cdot>ID\<cdot>ID = ID"
-unfolding cfun_eq_iff by simp
+ by (simp add: cfun_eq_iff)
-lemma cfun_map_map:
- "cfun_map\<cdot>f1\<cdot>g1\<cdot>(cfun_map\<cdot>f2\<cdot>g2\<cdot>p) =
- cfun_map\<cdot>(\<Lambda> x. f2\<cdot>(f1\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p"
-by (rule cfun_eqI) simp
+lemma cfun_map_map: "cfun_map\<cdot>f1\<cdot>g1\<cdot>(cfun_map\<cdot>f2\<cdot>g2\<cdot>p) = cfun_map\<cdot>(\<Lambda> x. f2\<cdot>(f1\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p"
+ by (rule cfun_eqI) simp
lemma ep_pair_cfun_map:
assumes "ep_pair e1 p1" and "ep_pair e2 p2"
@@ -34,9 +30,9 @@
proof
interpret e1p1: ep_pair e1 p1 by fact
interpret e2p2: ep_pair e2 p2 by fact
- fix f show "cfun_map\<cdot>e1\<cdot>p2\<cdot>(cfun_map\<cdot>p1\<cdot>e2\<cdot>f) = f"
+ show "cfun_map\<cdot>e1\<cdot>p2\<cdot>(cfun_map\<cdot>p1\<cdot>e2\<cdot>f) = f" for f
by (simp add: cfun_eq_iff)
- fix g show "cfun_map\<cdot>p1\<cdot>e2\<cdot>(cfun_map\<cdot>e1\<cdot>p2\<cdot>g) \<sqsubseteq> g"
+ show "cfun_map\<cdot>p1\<cdot>e2\<cdot>(cfun_map\<cdot>e1\<cdot>p2\<cdot>g) \<sqsubseteq> g" for g
apply (rule cfun_belowI, simp)
apply (rule below_trans [OF e2p2.e_p_below])
apply (rule monofun_cfun_arg)
@@ -79,13 +75,13 @@
proof (rule inj_onI, rule cfun_eqI, clarsimp)
fix x f g
assume "range (\<lambda>x. (a\<cdot>x, b\<cdot>(f\<cdot>(a\<cdot>x)))) = range (\<lambda>x. (a\<cdot>x, b\<cdot>(g\<cdot>(a\<cdot>x))))"
- hence "range (\<lambda>x. (a\<cdot>x, b\<cdot>(f\<cdot>(a\<cdot>x)))) \<subseteq> range (\<lambda>x. (a\<cdot>x, b\<cdot>(g\<cdot>(a\<cdot>x))))"
+ then have "range (\<lambda>x. (a\<cdot>x, b\<cdot>(f\<cdot>(a\<cdot>x)))) \<subseteq> range (\<lambda>x. (a\<cdot>x, b\<cdot>(g\<cdot>(a\<cdot>x))))"
by (rule equalityD1)
- hence "(a\<cdot>x, b\<cdot>(f\<cdot>(a\<cdot>x))) \<in> range (\<lambda>x. (a\<cdot>x, b\<cdot>(g\<cdot>(a\<cdot>x))))"
+ then have "(a\<cdot>x, b\<cdot>(f\<cdot>(a\<cdot>x))) \<in> range (\<lambda>x. (a\<cdot>x, b\<cdot>(g\<cdot>(a\<cdot>x))))"
by (simp add: subset_eq)
then obtain y where "(a\<cdot>x, b\<cdot>(f\<cdot>(a\<cdot>x))) = (a\<cdot>y, b\<cdot>(g\<cdot>(a\<cdot>y)))"
by (rule rangeE)
- thus "b\<cdot>(f\<cdot>(a\<cdot>x)) = b\<cdot>(g\<cdot>(a\<cdot>x))"
+ then show "b\<cdot>(f\<cdot>(a\<cdot>x)) = b\<cdot>(g\<cdot>(a\<cdot>x))"
by clarsimp
qed
qed
@@ -97,41 +93,39 @@
interpret d1: finite_deflation d1 by fact
interpret d2: finite_deflation d2 by fact
have "deflation d1" and "deflation d2" by fact+
- thus "deflation (cfun_map\<cdot>d1\<cdot>d2)" by (rule deflation_cfun_map)
+ then show "deflation (cfun_map\<cdot>d1\<cdot>d2)"
+ by (rule deflation_cfun_map)
have "finite (range (\<lambda>f. cfun_map\<cdot>d1\<cdot>d2\<cdot>f))"
using d1.finite_range d2.finite_range
by (rule finite_range_cfun_map)
- thus "finite {f. cfun_map\<cdot>d1\<cdot>d2\<cdot>f = f}"
+ then show "finite {f. cfun_map\<cdot>d1\<cdot>d2\<cdot>f = f}"
by (rule finite_range_imp_finite_fixes)
qed
text \<open>Finite deflations are compact elements of the function space\<close>
lemma finite_deflation_imp_compact: "finite_deflation d \<Longrightarrow> compact d"
-apply (frule finite_deflation_imp_deflation)
-apply (subgoal_tac "compact (cfun_map\<cdot>d\<cdot>d\<cdot>d)")
-apply (simp add: cfun_map_def deflation.idem eta_cfun)
-apply (rule finite_deflation.compact)
-apply (simp only: finite_deflation_cfun_map)
-done
+ apply (frule finite_deflation_imp_deflation)
+ apply (subgoal_tac "compact (cfun_map\<cdot>d\<cdot>d\<cdot>d)")
+ apply (simp add: cfun_map_def deflation.idem eta_cfun)
+ apply (rule finite_deflation.compact)
+ apply (simp only: finite_deflation_cfun_map)
+ done
+
subsection \<open>Map operator for product type\<close>
-definition
- prod_map :: "('a \<rightarrow> 'b) \<rightarrow> ('c \<rightarrow> 'd) \<rightarrow> 'a \<times> 'c \<rightarrow> 'b \<times> 'd"
-where
- "prod_map = (\<Lambda> f g p. (f\<cdot>(fst p), g\<cdot>(snd p)))"
+definition prod_map :: "('a \<rightarrow> 'b) \<rightarrow> ('c \<rightarrow> 'd) \<rightarrow> 'a \<times> 'c \<rightarrow> 'b \<times> 'd"
+ where "prod_map = (\<Lambda> f g p. (f\<cdot>(fst p), g\<cdot>(snd p)))"
lemma prod_map_Pair [simp]: "prod_map\<cdot>f\<cdot>g\<cdot>(x, y) = (f\<cdot>x, g\<cdot>y)"
-unfolding prod_map_def by simp
+ by (simp add: prod_map_def)
lemma prod_map_ID: "prod_map\<cdot>ID\<cdot>ID = ID"
-unfolding cfun_eq_iff by auto
+ by (auto simp: cfun_eq_iff)
-lemma prod_map_map:
- "prod_map\<cdot>f1\<cdot>g1\<cdot>(prod_map\<cdot>f2\<cdot>g2\<cdot>p) =
- prod_map\<cdot>(\<Lambda> x. f1\<cdot>(f2\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p"
-by (induct p) simp
+lemma prod_map_map: "prod_map\<cdot>f1\<cdot>g1\<cdot>(prod_map\<cdot>f2\<cdot>g2\<cdot>p) = prod_map\<cdot>(\<Lambda> x. f1\<cdot>(f2\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p"
+ by (induct p) simp
lemma ep_pair_prod_map:
assumes "ep_pair e1 p1" and "ep_pair e2 p2"
@@ -139,9 +133,9 @@
proof
interpret e1p1: ep_pair e1 p1 by fact
interpret e2p2: ep_pair e2 p2 by fact
- fix x show "prod_map\<cdot>p1\<cdot>p2\<cdot>(prod_map\<cdot>e1\<cdot>e2\<cdot>x) = x"
+ show "prod_map\<cdot>p1\<cdot>p2\<cdot>(prod_map\<cdot>e1\<cdot>e2\<cdot>x) = x" for x
by (induct x) simp
- fix y show "prod_map\<cdot>e1\<cdot>e2\<cdot>(prod_map\<cdot>p1\<cdot>p2\<cdot>y) \<sqsubseteq> y"
+ show "prod_map\<cdot>e1\<cdot>e2\<cdot>(prod_map\<cdot>p1\<cdot>p2\<cdot>y) \<sqsubseteq> y" for y
by (induct y) (simp add: e1p1.e_p_below e2p2.e_p_below)
qed
@@ -165,91 +159,90 @@
interpret d1: finite_deflation d1 by fact
interpret d2: finite_deflation d2 by fact
have "deflation d1" and "deflation d2" by fact+
- thus "deflation (prod_map\<cdot>d1\<cdot>d2)" by (rule deflation_prod_map)
+ then show "deflation (prod_map\<cdot>d1\<cdot>d2)" by (rule deflation_prod_map)
have "{p. prod_map\<cdot>d1\<cdot>d2\<cdot>p = p} \<subseteq> {x. d1\<cdot>x = x} \<times> {y. d2\<cdot>y = y}"
- by clarsimp
- thus "finite {p. prod_map\<cdot>d1\<cdot>d2\<cdot>p = p}"
+ by auto
+ then show "finite {p. prod_map\<cdot>d1\<cdot>d2\<cdot>p = p}"
by (rule finite_subset, simp add: d1.finite_fixes d2.finite_fixes)
qed
+
subsection \<open>Map function for lifted cpo\<close>
-definition
- u_map :: "('a \<rightarrow> 'b) \<rightarrow> 'a u \<rightarrow> 'b u"
-where
- "u_map = (\<Lambda> f. fup\<cdot>(up oo f))"
+definition u_map :: "('a \<rightarrow> 'b) \<rightarrow> 'a u \<rightarrow> 'b u"
+ where "u_map = (\<Lambda> f. fup\<cdot>(up oo f))"
lemma u_map_strict [simp]: "u_map\<cdot>f\<cdot>\<bottom> = \<bottom>"
-unfolding u_map_def by simp
+ by (simp add: u_map_def)
lemma u_map_up [simp]: "u_map\<cdot>f\<cdot>(up\<cdot>x) = up\<cdot>(f\<cdot>x)"
-unfolding u_map_def by simp
+ by (simp add: u_map_def)
lemma u_map_ID: "u_map\<cdot>ID = ID"
-unfolding u_map_def by (simp add: cfun_eq_iff eta_cfun)
+ by (simp add: u_map_def cfun_eq_iff eta_cfun)
lemma u_map_map: "u_map\<cdot>f\<cdot>(u_map\<cdot>g\<cdot>p) = u_map\<cdot>(\<Lambda> x. f\<cdot>(g\<cdot>x))\<cdot>p"
-by (induct p) simp_all
+ by (induct p) simp_all
lemma u_map_oo: "u_map\<cdot>(f oo g) = u_map\<cdot>f oo u_map\<cdot>g"
-by (simp add: cfcomp1 u_map_map eta_cfun)
+ by (simp add: cfcomp1 u_map_map eta_cfun)
lemma ep_pair_u_map: "ep_pair e p \<Longrightarrow> ep_pair (u_map\<cdot>e) (u_map\<cdot>p)"
-apply standard
-apply (case_tac x, simp, simp add: ep_pair.e_inverse)
-apply (case_tac y, simp, simp add: ep_pair.e_p_below)
-done
+ apply standard
+ subgoal for x by (cases x, simp, simp add: ep_pair.e_inverse)
+ subgoal for y by (cases y, simp, simp add: ep_pair.e_p_below)
+ done
lemma deflation_u_map: "deflation d \<Longrightarrow> deflation (u_map\<cdot>d)"
-apply standard
-apply (case_tac x, simp, simp add: deflation.idem)
-apply (case_tac x, simp, simp add: deflation.below)
-done
+ apply standard
+ subgoal for x by (cases x, simp, simp add: deflation.idem)
+ subgoal for x by (cases x, simp, simp add: deflation.below)
+ done
lemma finite_deflation_u_map:
- assumes "finite_deflation d" shows "finite_deflation (u_map\<cdot>d)"
+ assumes "finite_deflation d"
+ shows "finite_deflation (u_map\<cdot>d)"
proof (rule finite_deflation_intro)
interpret d: finite_deflation d by fact
have "deflation d" by fact
- thus "deflation (u_map\<cdot>d)" by (rule deflation_u_map)
+ then show "deflation (u_map\<cdot>d)"
+ by (rule deflation_u_map)
have "{x. u_map\<cdot>d\<cdot>x = x} \<subseteq> insert \<bottom> ((\<lambda>x. up\<cdot>x) ` {x. d\<cdot>x = x})"
by (rule subsetI, case_tac x, simp_all)
- thus "finite {x. u_map\<cdot>d\<cdot>x = x}"
- by (rule finite_subset, simp add: d.finite_fixes)
+ then show "finite {x. u_map\<cdot>d\<cdot>x = x}"
+ by (rule finite_subset) (simp add: d.finite_fixes)
qed
+
subsection \<open>Map function for strict products\<close>
default_sort pcpo
-definition
- sprod_map :: "('a \<rightarrow> 'b) \<rightarrow> ('c \<rightarrow> 'd) \<rightarrow> 'a \<otimes> 'c \<rightarrow> 'b \<otimes> 'd"
-where
- "sprod_map = (\<Lambda> f g. ssplit\<cdot>(\<Lambda> x y. (:f\<cdot>x, g\<cdot>y:)))"
+definition sprod_map :: "('a \<rightarrow> 'b) \<rightarrow> ('c \<rightarrow> 'd) \<rightarrow> 'a \<otimes> 'c \<rightarrow> 'b \<otimes> 'd"
+ where "sprod_map = (\<Lambda> f g. ssplit\<cdot>(\<Lambda> x y. (:f\<cdot>x, g\<cdot>y:)))"
lemma sprod_map_strict [simp]: "sprod_map\<cdot>a\<cdot>b\<cdot>\<bottom> = \<bottom>"
-unfolding sprod_map_def by simp
+ by (simp add: sprod_map_def)
-lemma sprod_map_spair [simp]:
- "x \<noteq> \<bottom> \<Longrightarrow> y \<noteq> \<bottom> \<Longrightarrow> sprod_map\<cdot>f\<cdot>g\<cdot>(:x, y:) = (:f\<cdot>x, g\<cdot>y:)"
-by (simp add: sprod_map_def)
+lemma sprod_map_spair [simp]: "x \<noteq> \<bottom> \<Longrightarrow> y \<noteq> \<bottom> \<Longrightarrow> sprod_map\<cdot>f\<cdot>g\<cdot>(:x, y:) = (:f\<cdot>x, g\<cdot>y:)"
+ by (simp add: sprod_map_def)
-lemma sprod_map_spair':
- "f\<cdot>\<bottom> = \<bottom> \<Longrightarrow> g\<cdot>\<bottom> = \<bottom> \<Longrightarrow> sprod_map\<cdot>f\<cdot>g\<cdot>(:x, y:) = (:f\<cdot>x, g\<cdot>y:)"
-by (cases "x = \<bottom> \<or> y = \<bottom>") auto
+lemma sprod_map_spair': "f\<cdot>\<bottom> = \<bottom> \<Longrightarrow> g\<cdot>\<bottom> = \<bottom> \<Longrightarrow> sprod_map\<cdot>f\<cdot>g\<cdot>(:x, y:) = (:f\<cdot>x, g\<cdot>y:)"
+ by (cases "x = \<bottom> \<or> y = \<bottom>") auto
lemma sprod_map_ID: "sprod_map\<cdot>ID\<cdot>ID = ID"
-unfolding sprod_map_def by (simp add: cfun_eq_iff eta_cfun)
+ by (simp add: sprod_map_def cfun_eq_iff eta_cfun)
lemma sprod_map_map:
"\<lbrakk>f1\<cdot>\<bottom> = \<bottom>; g1\<cdot>\<bottom> = \<bottom>\<rbrakk> \<Longrightarrow>
sprod_map\<cdot>f1\<cdot>g1\<cdot>(sprod_map\<cdot>f2\<cdot>g2\<cdot>p) =
sprod_map\<cdot>(\<Lambda> x. f1\<cdot>(f2\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p"
-apply (induct p, simp)
-apply (case_tac "f2\<cdot>x = \<bottom>", simp)
-apply (case_tac "g2\<cdot>y = \<bottom>", simp)
-apply simp
-done
+ apply (induct p)
+ apply simp
+ apply (case_tac "f2\<cdot>x = \<bottom>", simp)
+ apply (case_tac "g2\<cdot>y = \<bottom>", simp)
+ apply simp
+ done
lemma ep_pair_sprod_map:
assumes "ep_pair e1 p1" and "ep_pair e2 p2"
@@ -257,10 +250,11 @@
proof
interpret e1p1: pcpo_ep_pair e1 p1 unfolding pcpo_ep_pair_def by fact
interpret e2p2: pcpo_ep_pair e2 p2 unfolding pcpo_ep_pair_def by fact
- fix x show "sprod_map\<cdot>p1\<cdot>p2\<cdot>(sprod_map\<cdot>e1\<cdot>e2\<cdot>x) = x"
+ show "sprod_map\<cdot>p1\<cdot>p2\<cdot>(sprod_map\<cdot>e1\<cdot>e2\<cdot>x) = x" for x
by (induct x) simp_all
- fix y show "sprod_map\<cdot>e1\<cdot>e2\<cdot>(sprod_map\<cdot>p1\<cdot>p2\<cdot>y) \<sqsubseteq> y"
- apply (induct y, simp)
+ show "sprod_map\<cdot>e1\<cdot>e2\<cdot>(sprod_map\<cdot>p1\<cdot>p2\<cdot>y) \<sqsubseteq> y" for y
+ apply (induct y)
+ apply simp
apply (case_tac "p1\<cdot>x = \<bottom>", simp, case_tac "p2\<cdot>y = \<bottom>", simp)
apply (simp add: monofun_cfun e1p1.e_p_below e2p2.e_p_below)
done
@@ -291,47 +285,48 @@
interpret d1: finite_deflation d1 by fact
interpret d2: finite_deflation d2 by fact
have "deflation d1" and "deflation d2" by fact+
- thus "deflation (sprod_map\<cdot>d1\<cdot>d2)" by (rule deflation_sprod_map)
- have "{x. sprod_map\<cdot>d1\<cdot>d2\<cdot>x = x} \<subseteq> insert \<bottom>
- ((\<lambda>(x, y). (:x, y:)) ` ({x. d1\<cdot>x = x} \<times> {y. d2\<cdot>y = y}))"
+ then show "deflation (sprod_map\<cdot>d1\<cdot>d2)"
+ by (rule deflation_sprod_map)
+ have "{x. sprod_map\<cdot>d1\<cdot>d2\<cdot>x = x} \<subseteq>
+ insert \<bottom> ((\<lambda>(x, y). (:x, y:)) ` ({x. d1\<cdot>x = x} \<times> {y. d2\<cdot>y = y}))"
by (rule subsetI, case_tac x, auto simp add: spair_eq_iff)
- thus "finite {x. sprod_map\<cdot>d1\<cdot>d2\<cdot>x = x}"
- by (rule finite_subset, simp add: d1.finite_fixes d2.finite_fixes)
+ then show "finite {x. sprod_map\<cdot>d1\<cdot>d2\<cdot>x = x}"
+ by (rule finite_subset) (simp add: d1.finite_fixes d2.finite_fixes)
qed
+
subsection \<open>Map function for strict sums\<close>
-definition
- ssum_map :: "('a \<rightarrow> 'b) \<rightarrow> ('c \<rightarrow> 'd) \<rightarrow> 'a \<oplus> 'c \<rightarrow> 'b \<oplus> 'd"
-where
- "ssum_map = (\<Lambda> f g. sscase\<cdot>(sinl oo f)\<cdot>(sinr oo g))"
+definition ssum_map :: "('a \<rightarrow> 'b) \<rightarrow> ('c \<rightarrow> 'd) \<rightarrow> 'a \<oplus> 'c \<rightarrow> 'b \<oplus> 'd"
+ where "ssum_map = (\<Lambda> f g. sscase\<cdot>(sinl oo f)\<cdot>(sinr oo g))"
lemma ssum_map_strict [simp]: "ssum_map\<cdot>f\<cdot>g\<cdot>\<bottom> = \<bottom>"
-unfolding ssum_map_def by simp
+ by (simp add: ssum_map_def)
lemma ssum_map_sinl [simp]: "x \<noteq> \<bottom> \<Longrightarrow> ssum_map\<cdot>f\<cdot>g\<cdot>(sinl\<cdot>x) = sinl\<cdot>(f\<cdot>x)"
-unfolding ssum_map_def by simp
+ by (simp add: ssum_map_def)
lemma ssum_map_sinr [simp]: "x \<noteq> \<bottom> \<Longrightarrow> ssum_map\<cdot>f\<cdot>g\<cdot>(sinr\<cdot>x) = sinr\<cdot>(g\<cdot>x)"
-unfolding ssum_map_def by simp
+ by (simp add: ssum_map_def)
lemma ssum_map_sinl': "f\<cdot>\<bottom> = \<bottom> \<Longrightarrow> ssum_map\<cdot>f\<cdot>g\<cdot>(sinl\<cdot>x) = sinl\<cdot>(f\<cdot>x)"
-by (cases "x = \<bottom>") simp_all
+ by (cases "x = \<bottom>") simp_all
lemma ssum_map_sinr': "g\<cdot>\<bottom> = \<bottom> \<Longrightarrow> ssum_map\<cdot>f\<cdot>g\<cdot>(sinr\<cdot>x) = sinr\<cdot>(g\<cdot>x)"
-by (cases "x = \<bottom>") simp_all
+ by (cases "x = \<bottom>") simp_all
lemma ssum_map_ID: "ssum_map\<cdot>ID\<cdot>ID = ID"
-unfolding ssum_map_def by (simp add: cfun_eq_iff eta_cfun)
+ by (simp add: ssum_map_def cfun_eq_iff eta_cfun)
lemma ssum_map_map:
"\<lbrakk>f1\<cdot>\<bottom> = \<bottom>; g1\<cdot>\<bottom> = \<bottom>\<rbrakk> \<Longrightarrow>
ssum_map\<cdot>f1\<cdot>g1\<cdot>(ssum_map\<cdot>f2\<cdot>g2\<cdot>p) =
ssum_map\<cdot>(\<Lambda> x. f1\<cdot>(f2\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p"
-apply (induct p, simp)
-apply (case_tac "f2\<cdot>x = \<bottom>", simp, simp)
-apply (case_tac "g2\<cdot>y = \<bottom>", simp, simp)
-done
+ apply (induct p)
+ apply simp
+ apply (case_tac "f2\<cdot>x = \<bottom>", simp, simp)
+ apply (case_tac "g2\<cdot>y = \<bottom>", simp, simp)
+ done
lemma ep_pair_ssum_map:
assumes "ep_pair e1 p1" and "ep_pair e2 p2"
@@ -339,11 +334,12 @@
proof
interpret e1p1: pcpo_ep_pair e1 p1 unfolding pcpo_ep_pair_def by fact
interpret e2p2: pcpo_ep_pair e2 p2 unfolding pcpo_ep_pair_def by fact
- fix x show "ssum_map\<cdot>p1\<cdot>p2\<cdot>(ssum_map\<cdot>e1\<cdot>e2\<cdot>x) = x"
+ show "ssum_map\<cdot>p1\<cdot>p2\<cdot>(ssum_map\<cdot>e1\<cdot>e2\<cdot>x) = x" for x
by (induct x) simp_all
- fix y show "ssum_map\<cdot>e1\<cdot>e2\<cdot>(ssum_map\<cdot>p1\<cdot>p2\<cdot>y) \<sqsubseteq> y"
- apply (induct y, simp)
- apply (case_tac "p1\<cdot>x = \<bottom>", simp, simp add: e1p1.e_p_below)
+ show "ssum_map\<cdot>e1\<cdot>e2\<cdot>(ssum_map\<cdot>p1\<cdot>p2\<cdot>y) \<sqsubseteq> y" for y
+ apply (induct y)
+ apply simp
+ apply (case_tac "p1\<cdot>x = \<bottom>", simp, simp add: e1p1.e_p_below)
apply (case_tac "p2\<cdot>y = \<bottom>", simp, simp add: e2p2.e_p_below)
done
qed
@@ -374,32 +370,30 @@
interpret d1: finite_deflation d1 by fact
interpret d2: finite_deflation d2 by fact
have "deflation d1" and "deflation d2" by fact+
- thus "deflation (ssum_map\<cdot>d1\<cdot>d2)" by (rule deflation_ssum_map)
+ then show "deflation (ssum_map\<cdot>d1\<cdot>d2)"
+ by (rule deflation_ssum_map)
have "{x. ssum_map\<cdot>d1\<cdot>d2\<cdot>x = x} \<subseteq>
(\<lambda>x. sinl\<cdot>x) ` {x. d1\<cdot>x = x} \<union>
(\<lambda>x. sinr\<cdot>x) ` {x. d2\<cdot>x = x} \<union> {\<bottom>}"
by (rule subsetI, case_tac x, simp_all)
- thus "finite {x. ssum_map\<cdot>d1\<cdot>d2\<cdot>x = x}"
+ then show "finite {x. ssum_map\<cdot>d1\<cdot>d2\<cdot>x = x}"
by (rule finite_subset, simp add: d1.finite_fixes d2.finite_fixes)
qed
+
subsection \<open>Map operator for strict function space\<close>
-definition
- sfun_map :: "('b \<rightarrow> 'a) \<rightarrow> ('c \<rightarrow> 'd) \<rightarrow> ('a \<rightarrow>! 'c) \<rightarrow> ('b \<rightarrow>! 'd)"
-where
- "sfun_map = (\<Lambda> a b. sfun_abs oo cfun_map\<cdot>a\<cdot>b oo sfun_rep)"
+definition sfun_map :: "('b \<rightarrow> 'a) \<rightarrow> ('c \<rightarrow> 'd) \<rightarrow> ('a \<rightarrow>! 'c) \<rightarrow> ('b \<rightarrow>! 'd)"
+ where "sfun_map = (\<Lambda> a b. sfun_abs oo cfun_map\<cdot>a\<cdot>b oo sfun_rep)"
lemma sfun_map_ID: "sfun_map\<cdot>ID\<cdot>ID = ID"
- unfolding sfun_map_def
- by (simp add: cfun_map_ID cfun_eq_iff)
+ by (simp add: sfun_map_def cfun_map_ID cfun_eq_iff)
lemma sfun_map_map:
- assumes "f2\<cdot>\<bottom> = \<bottom>" and "g2\<cdot>\<bottom> = \<bottom>" shows
- "sfun_map\<cdot>f1\<cdot>g1\<cdot>(sfun_map\<cdot>f2\<cdot>g2\<cdot>p) =
+ assumes "f2\<cdot>\<bottom> = \<bottom>" and "g2\<cdot>\<bottom> = \<bottom>"
+ shows "sfun_map\<cdot>f1\<cdot>g1\<cdot>(sfun_map\<cdot>f2\<cdot>g2\<cdot>p) =
sfun_map\<cdot>(\<Lambda> x. f2\<cdot>(f1\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p"
-unfolding sfun_map_def
-by (simp add: cfun_eq_iff strictify_cancel assms cfun_map_map)
+ by (simp add: sfun_map_def cfun_eq_iff strictify_cancel assms cfun_map_map)
lemma ep_pair_sfun_map:
assumes 1: "ep_pair e1 p1"
@@ -410,13 +404,13 @@
unfolding pcpo_ep_pair_def by fact
interpret e2p2: pcpo_ep_pair e2 p2
unfolding pcpo_ep_pair_def by fact
- fix f show "sfun_map\<cdot>e1\<cdot>p2\<cdot>(sfun_map\<cdot>p1\<cdot>e2\<cdot>f) = f"
+ show "sfun_map\<cdot>e1\<cdot>p2\<cdot>(sfun_map\<cdot>p1\<cdot>e2\<cdot>f) = f" for f
unfolding sfun_map_def
apply (simp add: sfun_eq_iff strictify_cancel)
apply (rule ep_pair.e_inverse)
apply (rule ep_pair_cfun_map [OF 1 2])
done
- fix g show "sfun_map\<cdot>p1\<cdot>e2\<cdot>(sfun_map\<cdot>e1\<cdot>p2\<cdot>g) \<sqsubseteq> g"
+ show "sfun_map\<cdot>p1\<cdot>e2\<cdot>(sfun_map\<cdot>e1\<cdot>p2\<cdot>g) \<sqsubseteq> g" for g
unfolding sfun_map_def
apply (simp add: sfun_below_iff strictify_cancel)
apply (rule ep_pair.e_p_below)
@@ -428,40 +422,39 @@
assumes 1: "deflation d1"
assumes 2: "deflation d2"
shows "deflation (sfun_map\<cdot>d1\<cdot>d2)"
-apply (simp add: sfun_map_def)
-apply (rule deflation.intro)
-apply simp
-apply (subst strictify_cancel)
-apply (simp add: cfun_map_def deflation_strict 1 2)
-apply (simp add: cfun_map_def deflation.idem 1 2)
-apply (simp add: sfun_below_iff)
-apply (subst strictify_cancel)
-apply (simp add: cfun_map_def deflation_strict 1 2)
-apply (rule deflation.below)
-apply (rule deflation_cfun_map [OF 1 2])
-done
+ apply (simp add: sfun_map_def)
+ apply (rule deflation.intro)
+ apply simp
+ apply (subst strictify_cancel)
+ apply (simp add: cfun_map_def deflation_strict 1 2)
+ apply (simp add: cfun_map_def deflation.idem 1 2)
+ apply (simp add: sfun_below_iff)
+ apply (subst strictify_cancel)
+ apply (simp add: cfun_map_def deflation_strict 1 2)
+ apply (rule deflation.below)
+ apply (rule deflation_cfun_map [OF 1 2])
+ done
lemma finite_deflation_sfun_map:
- assumes 1: "finite_deflation d1"
- assumes 2: "finite_deflation d2"
+ assumes "finite_deflation d1"
+ and "finite_deflation d2"
shows "finite_deflation (sfun_map\<cdot>d1\<cdot>d2)"
proof (intro finite_deflation_intro)
interpret d1: finite_deflation d1 by fact
interpret d2: finite_deflation d2 by fact
have "deflation d1" and "deflation d2" by fact+
- thus "deflation (sfun_map\<cdot>d1\<cdot>d2)" by (rule deflation_sfun_map)
- from 1 2 have "finite_deflation (cfun_map\<cdot>d1\<cdot>d2)"
+ then show "deflation (sfun_map\<cdot>d1\<cdot>d2)"
+ by (rule deflation_sfun_map)
+ from assms have "finite_deflation (cfun_map\<cdot>d1\<cdot>d2)"
by (rule finite_deflation_cfun_map)
then have "finite {f. cfun_map\<cdot>d1\<cdot>d2\<cdot>f = f}"
by (rule finite_deflation.finite_fixes)
moreover have "inj (\<lambda>f. sfun_rep\<cdot>f)"
- by (rule inj_onI, simp add: sfun_eq_iff)
+ by (rule inj_onI) (simp add: sfun_eq_iff)
ultimately have "finite ((\<lambda>f. sfun_rep\<cdot>f) -` {f. cfun_map\<cdot>d1\<cdot>d2\<cdot>f = f})"
by (rule finite_vimageI)
- then show "finite {f. sfun_map\<cdot>d1\<cdot>d2\<cdot>f = f}"
- unfolding sfun_map_def sfun_eq_iff
- by (simp add: strictify_cancel
- deflation_strict \<open>deflation d1\<close> \<open>deflation d2\<close>)
+ with \<open>deflation d1\<close> \<open>deflation d2\<close> show "finite {f. sfun_map\<cdot>d1\<cdot>d2\<cdot>f = f}"
+ by (simp add: sfun_map_def sfun_eq_iff strictify_cancel deflation_strict)
qed
end