(* Title: HOL/HOLCF/Map_Functions.thy
Author: Brian Huffman
*)
section \<open>Map functions for various types\<close>
theory Map_Functions
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)))"
lemma cfun_map_beta [simp]: "cfun_map\<cdot>a\<cdot>b\<cdot>f\<cdot>x = b\<cdot>(f\<cdot>(a\<cdot>x))"
by (simp add: cfun_map_def)
lemma cfun_map_ID: "cfun_map\<cdot>ID\<cdot>ID = ID"
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 ep_pair_cfun_map:
assumes "ep_pair e1 p1" and "ep_pair e2 p2"
shows "ep_pair (cfun_map\<cdot>p1\<cdot>e2) (cfun_map\<cdot>e1\<cdot>p2)"
proof
interpret e1p1: ep_pair e1 p1 by fact
interpret e2p2: ep_pair e2 p2 by fact
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)
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)
apply (rule e1p1.e_p_below)
done
qed
lemma deflation_cfun_map:
assumes "deflation d1" and "deflation d2"
shows "deflation (cfun_map\<cdot>d1\<cdot>d2)"
proof
interpret d1: deflation d1 by fact
interpret d2: deflation d2 by fact
fix f
show "cfun_map\<cdot>d1\<cdot>d2\<cdot>(cfun_map\<cdot>d1\<cdot>d2\<cdot>f) = cfun_map\<cdot>d1\<cdot>d2\<cdot>f"
by (simp add: cfun_eq_iff d1.idem d2.idem)
show "cfun_map\<cdot>d1\<cdot>d2\<cdot>f \<sqsubseteq> f"
apply (rule cfun_belowI, simp)
apply (rule below_trans [OF d2.below])
apply (rule monofun_cfun_arg)
apply (rule d1.below)
done
qed
lemma finite_range_cfun_map:
assumes a: "finite (range (\<lambda>x. a\<cdot>x))"
assumes b: "finite (range (\<lambda>y. b\<cdot>y))"
shows "finite (range (\<lambda>f. cfun_map\<cdot>a\<cdot>b\<cdot>f))" (is "finite (range ?h)")
proof (rule finite_imageD)
let ?f = "\<lambda>g. range (\<lambda>x. (a\<cdot>x, g\<cdot>x))"
show "finite (?f ` range ?h)"
proof (rule finite_subset)
let ?B = "Pow (range (\<lambda>x. a\<cdot>x) \<times> range (\<lambda>y. b\<cdot>y))"
show "?f ` range ?h \<subseteq> ?B"
by clarsimp
show "finite ?B"
by (simp add: a b)
qed
show "inj_on ?f (range ?h)"
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))))"
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)
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)
then show "b\<cdot>(f\<cdot>(a\<cdot>x)) = b\<cdot>(g\<cdot>(a\<cdot>x))"
by clarsimp
qed
qed
lemma finite_deflation_cfun_map:
assumes "finite_deflation d1" and "finite_deflation d2"
shows "finite_deflation (cfun_map\<cdot>d1\<cdot>d2)"
proof (rule finite_deflation_intro)
interpret d1: finite_deflation d1 by fact
interpret d2: finite_deflation d2 by fact
from d1.deflation_axioms d2.deflation_axioms 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)
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
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)))"
lemma prod_map_Pair [simp]: "prod_map\<cdot>f\<cdot>g\<cdot>(x, y) = (f\<cdot>x, g\<cdot>y)"
by (simp add: prod_map_def)
lemma prod_map_ID: "prod_map\<cdot>ID\<cdot>ID = ID"
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 ep_pair_prod_map:
assumes "ep_pair e1 p1" and "ep_pair e2 p2"
shows "ep_pair (prod_map\<cdot>e1\<cdot>e2) (prod_map\<cdot>p1\<cdot>p2)"
proof
interpret e1p1: ep_pair e1 p1 by fact
interpret e2p2: ep_pair e2 p2 by fact
show "prod_map\<cdot>p1\<cdot>p2\<cdot>(prod_map\<cdot>e1\<cdot>e2\<cdot>x) = x" for x
by (induct x) simp
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
lemma deflation_prod_map:
assumes "deflation d1" and "deflation d2"
shows "deflation (prod_map\<cdot>d1\<cdot>d2)"
proof
interpret d1: deflation d1 by fact
interpret d2: deflation d2 by fact
fix x
show "prod_map\<cdot>d1\<cdot>d2\<cdot>(prod_map\<cdot>d1\<cdot>d2\<cdot>x) = prod_map\<cdot>d1\<cdot>d2\<cdot>x"
by (induct x) (simp add: d1.idem d2.idem)
show "prod_map\<cdot>d1\<cdot>d2\<cdot>x \<sqsubseteq> x"
by (induct x) (simp add: d1.below d2.below)
qed
lemma finite_deflation_prod_map:
assumes "finite_deflation d1" and "finite_deflation d2"
shows "finite_deflation (prod_map\<cdot>d1\<cdot>d2)"
proof (rule finite_deflation_intro)
interpret d1: finite_deflation d1 by fact
interpret d2: finite_deflation d2 by fact
from d1.deflation_axioms d2.deflation_axioms 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 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))"
lemma u_map_strict [simp]: "u_map\<cdot>f\<cdot>\<bottom> = \<bottom>"
by (simp add: u_map_def)
lemma u_map_up [simp]: "u_map\<cdot>f\<cdot>(up\<cdot>x) = up\<cdot>(f\<cdot>x)"
by (simp add: u_map_def)
lemma u_map_ID: "u_map\<cdot>ID = ID"
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
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)
lemma ep_pair_u_map: "ep_pair e p \<Longrightarrow> ep_pair (u_map\<cdot>e) (u_map\<cdot>p)"
apply standard
subgoal for x by (cases x) (simp_all add: ep_pair.e_inverse)
subgoal for y by (cases y) (simp_all add: ep_pair.e_p_below)
done
lemma deflation_u_map: "deflation d \<Longrightarrow> deflation (u_map\<cdot>d)"
apply standard
subgoal for x by (cases x) (simp_all add: deflation.idem)
subgoal for x by (cases x) (simp_all add: deflation.below)
done
lemma finite_deflation_u_map:
assumes "finite_deflation d"
shows "finite_deflation (u_map\<cdot>d)"
proof (rule finite_deflation_intro)
interpret d: finite_deflation d by fact
from d.deflation_axioms 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)
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:)))"
lemma sprod_map_strict [simp]: "sprod_map\<cdot>a\<cdot>b\<cdot>\<bottom> = \<bottom>"
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_ID: "sprod_map\<cdot>ID\<cdot>ID = ID"
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"
proof (induct p)
case bottom
then show ?case by simp
next
case (spair x y)
then show ?case
apply (cases "f2\<cdot>x = \<bottom>", simp)
apply (cases "g2\<cdot>y = \<bottom>", simp)
apply simp
done
qed
lemma ep_pair_sprod_map:
assumes "ep_pair e1 p1" and "ep_pair e2 p2"
shows "ep_pair (sprod_map\<cdot>e1\<cdot>e2) (sprod_map\<cdot>p1\<cdot>p2)"
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
show "sprod_map\<cdot>p1\<cdot>p2\<cdot>(sprod_map\<cdot>e1\<cdot>e2\<cdot>x) = x" for x
by (induct x) simp_all
show "sprod_map\<cdot>e1\<cdot>e2\<cdot>(sprod_map\<cdot>p1\<cdot>p2\<cdot>y) \<sqsubseteq> y" for y
proof (induct y)
case bottom
then show ?case by simp
next
case (spair x y)
then show ?case
apply simp
apply (cases "p1\<cdot>x = \<bottom>", simp, cases "p2\<cdot>y = \<bottom>", simp)
apply (simp add: monofun_cfun e1p1.e_p_below e2p2.e_p_below)
done
qed
qed
lemma deflation_sprod_map:
assumes "deflation d1" and "deflation d2"
shows "deflation (sprod_map\<cdot>d1\<cdot>d2)"
proof
interpret d1: deflation d1 by fact
interpret d2: deflation d2 by fact
fix x
show "sprod_map\<cdot>d1\<cdot>d2\<cdot>(sprod_map\<cdot>d1\<cdot>d2\<cdot>x) = sprod_map\<cdot>d1\<cdot>d2\<cdot>x"
proof (induct x)
case bottom
then show ?case by simp
next
case (spair x y)
then show ?case
apply (cases "d1\<cdot>x = \<bottom>", simp, cases "d2\<cdot>y = \<bottom>", simp)
apply (simp add: d1.idem d2.idem)
done
qed
show "sprod_map\<cdot>d1\<cdot>d2\<cdot>x \<sqsubseteq> x"
proof (induct x)
case bottom
then show ?case by simp
next
case spair
then show ?case by (simp add: monofun_cfun d1.below d2.below)
qed
qed
lemma finite_deflation_sprod_map:
assumes "finite_deflation d1" and "finite_deflation d2"
shows "finite_deflation (sprod_map\<cdot>d1\<cdot>d2)"
proof (rule finite_deflation_intro)
interpret d1: finite_deflation d1 by fact
interpret d2: finite_deflation d2 by fact
from d1.deflation_axioms d2.deflation_axioms 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)
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))"
lemma ssum_map_strict [simp]: "ssum_map\<cdot>f\<cdot>g\<cdot>\<bottom> = \<bottom>"
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)"
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)"
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
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
lemma ssum_map_ID: "ssum_map\<cdot>ID\<cdot>ID = ID"
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"
proof (induct p)
case bottom
then show ?case by simp
next
case (sinl x)
then show ?case by (cases "f2\<cdot>x = \<bottom>") simp_all
next
case (sinr y)
then show ?case by (cases "g2\<cdot>y = \<bottom>") simp_all
qed
lemma ep_pair_ssum_map:
assumes "ep_pair e1 p1" and "ep_pair e2 p2"
shows "ep_pair (ssum_map\<cdot>e1\<cdot>e2) (ssum_map\<cdot>p1\<cdot>p2)"
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
show "ssum_map\<cdot>p1\<cdot>p2\<cdot>(ssum_map\<cdot>e1\<cdot>e2\<cdot>x) = x" for x
by (induct x) simp_all
show "ssum_map\<cdot>e1\<cdot>e2\<cdot>(ssum_map\<cdot>p1\<cdot>p2\<cdot>y) \<sqsubseteq> y" for y
proof (induct y)
case bottom
then show ?case by simp
next
case (sinl x)
then show ?case by (cases "p1\<cdot>x = \<bottom>") (simp_all add: e1p1.e_p_below)
next
case (sinr y)
then show ?case by (cases "p2\<cdot>y = \<bottom>") (simp_all add: e2p2.e_p_below)
qed
qed
lemma deflation_ssum_map:
assumes "deflation d1" and "deflation d2"
shows "deflation (ssum_map\<cdot>d1\<cdot>d2)"
proof
interpret d1: deflation d1 by fact
interpret d2: deflation d2 by fact
fix x
show "ssum_map\<cdot>d1\<cdot>d2\<cdot>(ssum_map\<cdot>d1\<cdot>d2\<cdot>x) = ssum_map\<cdot>d1\<cdot>d2\<cdot>x"
proof (induct x)
case bottom
then show ?case by simp
next
case (sinl x)
then show ?case by (cases "d1\<cdot>x = \<bottom>") (simp_all add: d1.idem)
next
case (sinr y)
then show ?case by (cases "d2\<cdot>y = \<bottom>") (simp_all add: d2.idem)
qed
show "ssum_map\<cdot>d1\<cdot>d2\<cdot>x \<sqsubseteq> x"
proof (induct x)
case bottom
then show ?case by simp
next
case (sinl x)
then show ?case by (cases "d1\<cdot>x = \<bottom>") (simp_all add: d1.below)
next
case (sinr y)
then show ?case by (cases "d2\<cdot>y = \<bottom>") (simp_all add: d2.below)
qed
qed
lemma finite_deflation_ssum_map:
assumes "finite_deflation d1" and "finite_deflation d2"
shows "finite_deflation (ssum_map\<cdot>d1\<cdot>d2)"
proof (rule finite_deflation_intro)
interpret d1: finite_deflation d1 by fact
interpret d2: finite_deflation d2 by fact
from d1.deflation_axioms d2.deflation_axioms 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)
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)"
lemma sfun_map_ID: "sfun_map\<cdot>ID\<cdot>ID = ID"
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) =
sfun_map\<cdot>(\<Lambda> x. f2\<cdot>(f1\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p"
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"
assumes 2: "ep_pair e2 p2"
shows "ep_pair (sfun_map\<cdot>p1\<cdot>e2) (sfun_map\<cdot>e1\<cdot>p2)"
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
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
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)
apply (rule ep_pair_cfun_map [OF 1 2])
done
qed
lemma deflation_sfun_map:
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
lemma finite_deflation_sfun_map:
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
from d1.deflation_axioms d2.deflation_axioms 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)
ultimately have "finite ((\<lambda>f. sfun_rep\<cdot>f) -` {f. cfun_map\<cdot>d1\<cdot>d2\<cdot>f = f})"
by (rule finite_vimageI)
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