# Theory Map_Functions

```(*  Title:      HOL/HOLCF/Map_Functions.thy
Author:     Brian Huffman
*)

section ‹Map functions for various types›

theory Map_Functions
imports Deflation Sprod Ssum Sfun Up
begin

subsection ‹Map operator for continuous function space›

default_sort cpo

definition cfun_map :: "('b → 'a) → ('c → 'd) → ('a → 'c) → ('b → 'd)"
where "cfun_map = (Λ a b f x. b⋅(f⋅(a⋅x)))"

lemma cfun_map_beta [simp]: "cfun_map⋅a⋅b⋅f⋅x = b⋅(f⋅(a⋅x))"

lemma cfun_map_ID: "cfun_map⋅ID⋅ID = ID"

lemma cfun_map_map: "cfun_map⋅f1⋅g1⋅(cfun_map⋅f2⋅g2⋅p) = cfun_map⋅(Λ x. f2⋅(f1⋅x))⋅(Λ x. g1⋅(g2⋅x))⋅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⋅p1⋅e2) (cfun_map⋅e1⋅p2)"
proof
interpret e1p1: ep_pair e1 p1 by fact
interpret e2p2: ep_pair e2 p2 by fact
show "cfun_map⋅e1⋅p2⋅(cfun_map⋅p1⋅e2⋅f) = f" for f
show "cfun_map⋅p1⋅e2⋅(cfun_map⋅e1⋅p2⋅g) ⊑ 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⋅d1⋅d2)"
proof
interpret d1: deflation d1 by fact
interpret d2: deflation d2 by fact
fix f
show "cfun_map⋅d1⋅d2⋅(cfun_map⋅d1⋅d2⋅f) = cfun_map⋅d1⋅d2⋅f"
by (simp add: cfun_eq_iff d1.idem d2.idem)
show "cfun_map⋅d1⋅d2⋅f ⊑ 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 (λx. a⋅x))"
assumes b: "finite (range (λy. b⋅y))"
shows "finite (range (λf. cfun_map⋅a⋅b⋅f))"  (is "finite (range ?h)")
proof (rule finite_imageD)
let ?f = "λg. range (λx. (a⋅x, g⋅x))"
show "finite (?f ` range ?h)"
proof (rule finite_subset)
let ?B = "Pow (range (λx. a⋅x) × range (λy. b⋅y))"
show "?f ` range ?h ⊆ ?B"
by clarsimp
show "finite ?B"
qed
show "inj_on ?f (range ?h)"
proof (rule inj_onI, rule cfun_eqI, clarsimp)
fix x f g
assume "range (λx. (a⋅x, b⋅(f⋅(a⋅x)))) = range (λx. (a⋅x, b⋅(g⋅(a⋅x))))"
then have "range (λx. (a⋅x, b⋅(f⋅(a⋅x)))) ⊆ range (λx. (a⋅x, b⋅(g⋅(a⋅x))))"
by (rule equalityD1)
then have "(a⋅x, b⋅(f⋅(a⋅x))) ∈ range (λx. (a⋅x, b⋅(g⋅(a⋅x))))"
then obtain y where "(a⋅x, b⋅(f⋅(a⋅x))) = (a⋅y, b⋅(g⋅(a⋅y)))"
by (rule rangeE)
then show "b⋅(f⋅(a⋅x)) = b⋅(g⋅(a⋅x))"
by clarsimp
qed
qed

lemma finite_deflation_cfun_map:
assumes "finite_deflation d1" and "finite_deflation d2"
shows "finite_deflation (cfun_map⋅d1⋅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⋅d1⋅d2)"
by (rule deflation_cfun_map)
have "finite (range (λf. cfun_map⋅d1⋅d2⋅f))"
using d1.finite_range d2.finite_range
by (rule finite_range_cfun_map)
then show "finite {f. cfun_map⋅d1⋅d2⋅f = f}"
by (rule finite_range_imp_finite_fixes)
qed

text ‹Finite deflations are compact elements of the function space›

lemma finite_deflation_imp_compact: "finite_deflation d ⟹ compact d"
apply (frule finite_deflation_imp_deflation)
apply (subgoal_tac "compact (cfun_map⋅d⋅d⋅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 ‹Map operator for product type›

definition prod_map :: "('a → 'b) → ('c → 'd) → 'a × 'c → 'b × 'd"
where "prod_map = (Λ f g p. (f⋅(fst p), g⋅(snd p)))"

lemma prod_map_Pair [simp]: "prod_map⋅f⋅g⋅(x, y) = (f⋅x, g⋅y)"

lemma prod_map_ID: "prod_map⋅ID⋅ID = ID"
by (auto simp: cfun_eq_iff)

lemma prod_map_map: "prod_map⋅f1⋅g1⋅(prod_map⋅f2⋅g2⋅p) = prod_map⋅(Λ x. f1⋅(f2⋅x))⋅(Λ x. g1⋅(g2⋅x))⋅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⋅e1⋅e2) (prod_map⋅p1⋅p2)"
proof
interpret e1p1: ep_pair e1 p1 by fact
interpret e2p2: ep_pair e2 p2 by fact
show "prod_map⋅p1⋅p2⋅(prod_map⋅e1⋅e2⋅x) = x" for x
by (induct x) simp
show "prod_map⋅e1⋅e2⋅(prod_map⋅p1⋅p2⋅y) ⊑ 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⋅d1⋅d2)"
proof
interpret d1: deflation d1 by fact
interpret d2: deflation d2 by fact
fix x
show "prod_map⋅d1⋅d2⋅(prod_map⋅d1⋅d2⋅x) = prod_map⋅d1⋅d2⋅x"
by (induct x) (simp add: d1.idem d2.idem)
show "prod_map⋅d1⋅d2⋅x ⊑ 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⋅d1⋅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⋅d1⋅d2)"
by (rule deflation_prod_map)
have "{p. prod_map⋅d1⋅d2⋅p = p} ⊆ {x. d1⋅x = x} × {y. d2⋅y = y}"
by auto
then show "finite {p. prod_map⋅d1⋅d2⋅p = p}"
by (rule finite_subset, simp add: d1.finite_fixes d2.finite_fixes)
qed

subsection ‹Map function for lifted cpo›

definition u_map :: "('a → 'b) → 'a u → 'b u"
where "u_map = (Λ f. fup⋅(up oo f))"

lemma u_map_strict [simp]: "u_map⋅f⋅⊥ = ⊥"

lemma u_map_up [simp]: "u_map⋅f⋅(up⋅x) = up⋅(f⋅x)"

lemma u_map_ID: "u_map⋅ID = ID"
by (simp add: u_map_def cfun_eq_iff eta_cfun)

lemma u_map_map: "u_map⋅f⋅(u_map⋅g⋅p) = u_map⋅(Λ x. f⋅(g⋅x))⋅p"
by (induct p) simp_all

lemma u_map_oo: "u_map⋅(f oo g) = u_map⋅f oo u_map⋅g"
by (simp add: cfcomp1 u_map_map eta_cfun)

lemma ep_pair_u_map: "ep_pair e p ⟹ ep_pair (u_map⋅e) (u_map⋅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 ⟹ deflation (u_map⋅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⋅d)"
proof (rule finite_deflation_intro)
interpret d: finite_deflation d by fact
from d.deflation_axioms show "deflation (u_map⋅d)"
by (rule deflation_u_map)
have "{x. u_map⋅d⋅x = x} ⊆ insert ⊥ ((λx. up⋅x) ` {x. d⋅x = x})"
by (rule subsetI, case_tac x, simp_all)
then show "finite {x. u_map⋅d⋅x = x}"
by (rule finite_subset) (simp add: d.finite_fixes)
qed

subsection ‹Map function for strict products›

default_sort pcpo

definition sprod_map :: "('a → 'b) → ('c → 'd) → 'a ⊗ 'c → 'b ⊗ 'd"
where "sprod_map = (Λ f g. ssplit⋅(Λ x y. (:f⋅x, g⋅y:)))"

lemma sprod_map_strict [simp]: "sprod_map⋅a⋅b⋅⊥ = ⊥"

lemma sprod_map_spair [simp]: "x ≠ ⊥ ⟹ y ≠ ⊥ ⟹ sprod_map⋅f⋅g⋅(:x, y:) = (:f⋅x, g⋅y:)"

lemma sprod_map_spair': "f⋅⊥ = ⊥ ⟹ g⋅⊥ = ⊥ ⟹ sprod_map⋅f⋅g⋅(:x, y:) = (:f⋅x, g⋅y:)"
by (cases "x = ⊥ ∨ y = ⊥") auto

lemma sprod_map_ID: "sprod_map⋅ID⋅ID = ID"
by (simp add: sprod_map_def cfun_eq_iff eta_cfun)

lemma sprod_map_map:
"⟦f1⋅⊥ = ⊥; g1⋅⊥ = ⊥⟧ ⟹
sprod_map⋅f1⋅g1⋅(sprod_map⋅f2⋅g2⋅p) =
sprod_map⋅(Λ x. f1⋅(f2⋅x))⋅(Λ x. g1⋅(g2⋅x))⋅p"
proof (induct p)
case bottom
then show ?case by simp
next
case (spair x y)
then show ?case
apply (cases "f2⋅x = ⊥", simp)
apply (cases "g2⋅y = ⊥", 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⋅e1⋅e2) (sprod_map⋅p1⋅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⋅p1⋅p2⋅(sprod_map⋅e1⋅e2⋅x) = x" for x
by (induct x) simp_all
show "sprod_map⋅e1⋅e2⋅(sprod_map⋅p1⋅p2⋅y) ⊑ 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⋅x = ⊥", simp, cases "p2⋅y = ⊥", 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⋅d1⋅d2)"
proof
interpret d1: deflation d1 by fact
interpret d2: deflation d2 by fact
fix x
show "sprod_map⋅d1⋅d2⋅(sprod_map⋅d1⋅d2⋅x) = sprod_map⋅d1⋅d2⋅x"
proof (induct x)
case bottom
then show ?case by simp
next
case (spair x y)
then show ?case
apply (cases "d1⋅x = ⊥", simp, cases "d2⋅y = ⊥", simp)
done
qed
show "sprod_map⋅d1⋅d2⋅x ⊑ 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⋅d1⋅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⋅d1⋅d2)"
by (rule deflation_sprod_map)
have "{x. sprod_map⋅d1⋅d2⋅x = x} ⊆
insert ⊥ ((λ(x, y). (:x, y:)) ` ({x. d1⋅x = x} × {y. d2⋅y = y}))"
by (rule subsetI, case_tac x, auto simp add: spair_eq_iff)
then show "finite {x. sprod_map⋅d1⋅d2⋅x = x}"
by (rule finite_subset) (simp add: d1.finite_fixes d2.finite_fixes)
qed

subsection ‹Map function for strict sums›

definition ssum_map :: "('a → 'b) → ('c → 'd) → 'a ⊕ 'c → 'b ⊕ 'd"
where "ssum_map = (Λ f g. sscase⋅(sinl oo f)⋅(sinr oo g))"

lemma ssum_map_strict [simp]: "ssum_map⋅f⋅g⋅⊥ = ⊥"

lemma ssum_map_sinl [simp]: "x ≠ ⊥ ⟹ ssum_map⋅f⋅g⋅(sinl⋅x) = sinl⋅(f⋅x)"

lemma ssum_map_sinr [simp]: "x ≠ ⊥ ⟹ ssum_map⋅f⋅g⋅(sinr⋅x) = sinr⋅(g⋅x)"

lemma ssum_map_sinl': "f⋅⊥ = ⊥ ⟹ ssum_map⋅f⋅g⋅(sinl⋅x) = sinl⋅(f⋅x)"
by (cases "x = ⊥") simp_all

lemma ssum_map_sinr': "g⋅⊥ = ⊥ ⟹ ssum_map⋅f⋅g⋅(sinr⋅x) = sinr⋅(g⋅x)"
by (cases "x = ⊥") simp_all

lemma ssum_map_ID: "ssum_map⋅ID⋅ID = ID"
by (simp add: ssum_map_def cfun_eq_iff eta_cfun)

lemma ssum_map_map:
"⟦f1⋅⊥ = ⊥; g1⋅⊥ = ⊥⟧ ⟹
ssum_map⋅f1⋅g1⋅(ssum_map⋅f2⋅g2⋅p) =
ssum_map⋅(Λ x. f1⋅(f2⋅x))⋅(Λ x. g1⋅(g2⋅x))⋅p"
proof (induct p)
case bottom
then show ?case by simp
next
case (sinl x)
then show ?case by (cases "f2⋅x = ⊥") simp_all
next
case (sinr y)
then show ?case by (cases "g2⋅y = ⊥") simp_all
qed

lemma ep_pair_ssum_map:
assumes "ep_pair e1 p1" and "ep_pair e2 p2"
shows "ep_pair (ssum_map⋅e1⋅e2) (ssum_map⋅p1⋅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⋅p1⋅p2⋅(ssum_map⋅e1⋅e2⋅x) = x" for x
by (induct x) simp_all
show "ssum_map⋅e1⋅e2⋅(ssum_map⋅p1⋅p2⋅y) ⊑ y" for y
proof (induct y)
case bottom
then show ?case by simp
next
case (sinl x)
then show ?case by (cases "p1⋅x = ⊥") (simp_all add: e1p1.e_p_below)
next
case (sinr y)
then show ?case by (cases "p2⋅y = ⊥") (simp_all add: e2p2.e_p_below)
qed
qed

lemma deflation_ssum_map:
assumes "deflation d1" and "deflation d2"
shows "deflation (ssum_map⋅d1⋅d2)"
proof
interpret d1: deflation d1 by fact
interpret d2: deflation d2 by fact
fix x
show "ssum_map⋅d1⋅d2⋅(ssum_map⋅d1⋅d2⋅x) = ssum_map⋅d1⋅d2⋅x"
proof (induct x)
case bottom
then show ?case by simp
next
case (sinl x)
then show ?case by (cases "d1⋅x = ⊥") (simp_all add: d1.idem)
next
case (sinr y)
then show ?case by (cases "d2⋅y = ⊥") (simp_all add: d2.idem)
qed
show "ssum_map⋅d1⋅d2⋅x ⊑ x"
proof (induct x)
case bottom
then show ?case by simp
next
case (sinl x)
then show ?case by (cases "d1⋅x = ⊥") (simp_all add: d1.below)
next
case (sinr y)
then show ?case by (cases "d2⋅y = ⊥") (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⋅d1⋅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⋅d1⋅d2)"
by (rule deflation_ssum_map)
have "{x. ssum_map⋅d1⋅d2⋅x = x} ⊆
(λx. sinl⋅x) ` {x. d1⋅x = x} ∪
(λx. sinr⋅x) ` {x. d2⋅x = x} ∪ {⊥}"
by (rule subsetI, case_tac x, simp_all)
then show "finite {x. ssum_map⋅d1⋅d2⋅x = x}"
by (rule finite_subset, simp add: d1.finite_fixes d2.finite_fixes)
qed

subsection ‹Map operator for strict function space›

definition sfun_map :: "('b → 'a) → ('c → 'd) → ('a →! 'c) → ('b →! 'd)"
where "sfun_map = (Λ a b. sfun_abs oo cfun_map⋅a⋅b oo sfun_rep)"

lemma sfun_map_ID: "sfun_map⋅ID⋅ID = ID"
by (simp add: sfun_map_def cfun_map_ID cfun_eq_iff)

lemma sfun_map_map:
assumes "f2⋅⊥ = ⊥" and "g2⋅⊥ = ⊥"
shows "sfun_map⋅f1⋅g1⋅(sfun_map⋅f2⋅g2⋅p) =
sfun_map⋅(Λ x. f2⋅(f1⋅x))⋅(Λ x. g1⋅(g2⋅x))⋅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⋅p1⋅e2) (sfun_map⋅e1⋅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⋅e1⋅p2⋅(sfun_map⋅p1⋅e2⋅f) = f" for f
unfolding sfun_map_def
apply (rule ep_pair.e_inverse)
apply (rule ep_pair_cfun_map [OF 1 2])
done
show "sfun_map⋅p1⋅e2⋅(sfun_map⋅e1⋅p2⋅g) ⊑ g" for g
unfolding sfun_map_def
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⋅d1⋅d2)"
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 (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⋅d1⋅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⋅d1⋅d2)"
by (rule deflation_sfun_map)
from assms have "finite_deflation (cfun_map⋅d1⋅d2)"
by (rule finite_deflation_cfun_map)
then have "finite {f. cfun_map⋅d1⋅d2⋅f = f}"
by (rule finite_deflation.finite_fixes)
moreover have "inj (λf. sfun_rep⋅f)"
by (rule inj_onI) (simp add: sfun_eq_iff)
ultimately have "finite ((λf. sfun_rep⋅f) -` {f. cfun_map⋅d1⋅d2⋅f = f})"
by (rule finite_vimageI)
with ‹deflation d1› ‹deflation d2› show "finite {f. sfun_map⋅d1⋅d2⋅f = f}"
by (simp add: sfun_map_def sfun_eq_iff strictify_cancel deflation_strict)
qed

end
```