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))"
  by (simp add: cfun_map_def)

lemma cfun_map_ID: "cfun_map⋅ID⋅ID = ID"
  by (simp add: cfun_eq_iff)

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
    by (simp add: cfun_eq_iff)
  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"
      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 (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))))"
      by (simp add: subset_eq)
    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)"
  by (simp add: prod_map_def)

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⋅⊥ = ⊥"
  by (simp add: u_map_def)

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

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⋅⊥ = ⊥"
  by (simp add: sprod_map_def)

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

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)
      apply (simp add: d1.idem d2.idem)
      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⋅⊥ = ⊥"
  by (simp add: ssum_map_def)

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

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

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 (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⋅p1⋅e2⋅(sfun_map⋅e1⋅p2⋅g) ⊑ 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⋅d1⋅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⋅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