Theory FuncSet

(*  Title:      HOL/Library/FuncSet.thy
    Author:     Florian Kammueller and Lawrence C Paulson, Lukas Bulwahn
*)

section Pi and Function Sets

theory FuncSet
  imports Main
  abbrevs PiE = "PiE"
    and PIE = "ΠE"
begin

definition Pi :: "'a set  ('a  'b set)  ('a  'b) set"
  where "Pi A B = {f. x. x  A  f x  B x}"

definition extensional :: "'a set  ('a  'b) set"
  where "extensional A = {f. x. x  A  f x = undefined}"

definition "restrict" :: "('a  'b)  'a set  'a  'b"
  where "restrict f A = (λx. if x  A then f x else undefined)"

abbreviation funcset :: "'a set  'b set  ('a  'b) set"  (infixr "" 60)
  where "A  B  Pi A (λ_. B)"

syntax
  "_Pi" :: "pttrn  'a set  'b set  ('a  'b) set"  ("(3Π __./ _)"   10)
  "_lam" :: "pttrn  'a set  ('a  'b)  ('a  'b)"  ("(3λ__./ _)" [0,0,3] 3)
translations
  "Π xA. B"  "CONST Pi A (λx. B)"
  "λxA. f"  "CONST restrict (λx. f) A"

definition "compose" :: "'a set  ('b  'c)  ('a  'b)  ('a  'c)"
  where "compose A g f = (λxA. g (f x))"


subsection Basic Properties of termPi

lemma Pi_I[intro!]: "(x. x  A  f x  B x)  f  Pi A B"
  by (simp add: Pi_def)

lemma Pi_I'[simp]: "(x. x  A  f x  B x)  f  Pi A B"
  by (simp add:Pi_def)

lemma funcsetI: "(x. x  A  f x  B)  f  A  B"
  by (simp add: Pi_def)

lemma Pi_mem: "f  Pi A B  x  A  f x  B x"
  by (simp add: Pi_def)

lemma Pi_iff: "f  Pi I X  (iI. f i  X i)"
  unfolding Pi_def by auto

lemma PiE [elim]: "f  Pi A B  (f x  B x  Q)  (x  A  Q)  Q"
  by (auto simp: Pi_def)

lemma Pi_cong: "(w. w  A  f w = g w)  f  Pi A B  g  Pi A B"
  by (auto simp: Pi_def)

lemma funcset_id [simp]: "(λx. x)  A  A"
  by auto

lemma funcset_mem: "f  A  B  x  A  f x  B"
  by (simp add: Pi_def)

lemma funcset_image: "f  A  B  f ` A  B"
  by auto

lemma image_subset_iff_funcset: "F ` A  B  F  A  B"
  by auto

lemma funcset_to_empty_iff: "A  {} = (if A={} then UNIV else {})"
  by auto

lemma Pi_eq_empty[simp]: "(Π x  A. B x) = {}  (xA. B x = {})"
proof -
  have "xA. B x = {}" if "f. y. y  A  f y  B y"
    using that [of "λu. SOME y. y  B u"] some_in_eq by blast
  then show ?thesis
    by force
qed

lemma Pi_empty [simp]: "Pi {} B = UNIV"
  by (simp add: Pi_def)

lemma Pi_Int: "Pi I E  Pi I F = (Π iI. E i  F i)"
  by auto

lemma Pi_UN:
  fixes A :: "nat  'i  'a set"
  assumes "finite I"
    and mono: "i n m. i  I  n  m  A n i  A m i"
  shows "(n. Pi I (A n)) = (Π iI. n. A n i)"
proof (intro set_eqI iffI)
  fix f
  assume "f  (Π iI. n. A n i)"
  then have "iI. n. f i  A n i"
    by auto
  from bchoice[OF this] obtain n where n: "f i  A (n i) i" if "i  I" for i
    by auto
  obtain k where k: "n i  k" if "i  I" for i
    using finite I finite_nat_set_iff_bounded_le[of "n`I"] by auto
  have "f  Pi I (A k)"
  proof (intro Pi_I)
    fix i
    assume "i  I"
    from mono[OF this, of "n i" k] k[OF this] n[OF this]
    show "f i  A k i" by auto
  qed
  then show "f  (n. Pi I (A n))"
    by auto
qed auto

lemma Pi_UNIV [simp]: "A  UNIV = UNIV"
  by (simp add: Pi_def)

text Covariance of Pi-sets in their second argument
lemma Pi_mono: "(x. x  A  B x  C x)  Pi A B  Pi A C"
  by auto

text Contravariance of Pi-sets in their first argument
lemma Pi_anti_mono: "A'  A  Pi A B  Pi A' B"
  by auto

lemma prod_final:
  assumes 1: "fst  f  Pi A B"
    and 2: "snd  f  Pi A C"
  shows "f  (Π z  A. B z × C z)"
proof (rule Pi_I)
  fix z
  assume z: "z  A"
  have "f z = (fst (f z), snd (f z))"
    by simp
  also have "  B z × C z"
    by (metis SigmaI PiE o_apply 1 2 z)
  finally show "f z  B z × C z" .
qed

lemma Pi_split_domain[simp]: "x  Pi (I  J) X  x  Pi I X  x  Pi J X"
  by (auto simp: Pi_def)

lemma Pi_split_insert_domain[simp]: "x  Pi (insert i I) X  x  Pi I X  x i  X i"
  by (auto simp: Pi_def)

lemma Pi_cancel_fupd_range[simp]: "i  I  x  Pi I (B(i := b))  x  Pi I B"
  by (auto simp: Pi_def)

lemma Pi_cancel_fupd[simp]: "i  I  x(i := a)  Pi I B  x  Pi I B"
  by (auto simp: Pi_def)

lemma Pi_fupd_iff: "i  I  f  Pi I (B(i := A))  f  Pi (I - {i}) B  f i  A"
  apply auto
  apply (metis PiE fun_upd_apply)
  by force


subsection Composition With a Restricted Domain: termcompose

lemma funcset_compose: "f  A  B  g  B  C  compose A g f  A  C"
  by (simp add: Pi_def compose_def restrict_def)

lemma compose_assoc:
  assumes "f  A  B"
  shows "compose A h (compose A g f) = compose A (compose B h g) f"
  using assms by (simp add: fun_eq_iff Pi_def compose_def restrict_def)

lemma compose_eq: "x  A  compose A g f x = g (f x)"
  by (simp add: compose_def restrict_def)

lemma surj_compose: "f ` A = B  g ` B = C  compose A g f ` A = C"
  by (auto simp add: image_def compose_eq)


subsection Bounded Abstraction: termrestrict

lemma restrict_cong: "I = J  (i. i  J =simp=> f i = g i)  restrict f I = restrict g J"
  by (auto simp: restrict_def fun_eq_iff simp_implies_def)

lemma restrictI[intro!]: "(x. x  A  f x  B x)  (λxA. f x)  Pi A B"
  by (simp add: Pi_def restrict_def)

lemma restrict_apply[simp]: "(λyA. f y) x = (if x  A then f x else undefined)"
  by (simp add: restrict_def)

lemma restrict_apply': "x  A  (λyA. f y) x = f x"
  by simp

lemma restrict_ext: "(x. x  A  f x = g x)  (λxA. f x) = (λxA. g x)"
  by (simp add: fun_eq_iff Pi_def restrict_def)

lemma restrict_UNIV: "restrict f UNIV = f"
  by (simp add: restrict_def)

lemma inj_on_restrict_eq [simp]: "inj_on (restrict f A) A  inj_on f A"
  by (simp add: inj_on_def restrict_def)

lemma inj_on_restrict_iff: "A  B  inj_on (restrict f B) A  inj_on f A"
  by (metis inj_on_cong restrict_def subset_iff)

lemma Id_compose: "f  A  B  f  extensional A  compose A (λyB. y) f = f"
  by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def)

lemma compose_Id: "g  A  B  g  extensional A  compose A g (λxA. x) = g"
  by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def)

lemma image_restrict_eq [simp]: "(restrict f A) ` A = f ` A"
  by (auto simp add: restrict_def)

lemma restrict_restrict[simp]: "restrict (restrict f A) B = restrict f (A  B)"
  unfolding restrict_def by (simp add: fun_eq_iff)

lemma restrict_fupd[simp]: "i  I  restrict (f (i := x)) I = restrict f I"
  by (auto simp: restrict_def)

lemma restrict_upd[simp]: "i  I  (restrict f I)(i := y) = restrict (f(i := y)) (insert i I)"
  by (auto simp: fun_eq_iff)

lemma restrict_Pi_cancel: "restrict x I  Pi I A  x  Pi I A"
  by (auto simp: restrict_def Pi_def)

lemma sum_restrict' [simp]: "sum' (λiI. g i) I = sum' (λi. g i) I"
  by (simp add: sum.G_def conj_commute cong: conj_cong)

lemma prod_restrict' [simp]: "prod' (λiI. g i) I = prod' (λi. g i) I"
  by (simp add: prod.G_def conj_commute cong: conj_cong)


subsection Bijections Between Sets

text The definition of constbij_betw is in Fun.thy›, but most of
the theorems belong here, or need at least termHilbert_Choice.

lemma bij_betwI:
  assumes "f  A  B"
    and "g  B  A"
    and g_f: "x. xA  g (f x) = x"
    and f_g: "y. yB  f (g y) = y"
  shows "bij_betw f A B"
  unfolding bij_betw_def
proof
  show "inj_on f A"
    by (metis g_f inj_on_def)
  have "f ` A  B"
    using f  A  B by auto
  moreover
  have "B  f ` A"
    by auto (metis Pi_mem g  B  A f_g image_iff)
  ultimately show "f ` A = B"
    by blast
qed

lemma bij_betw_imp_funcset: "bij_betw f A B  f  A  B"
  by (auto simp add: bij_betw_def)

lemma inj_on_compose: "bij_betw f A B  inj_on g B  inj_on (compose A g f) A"
  by (auto simp add: bij_betw_def inj_on_def compose_eq)

lemma bij_betw_compose: "bij_betw f A B  bij_betw g B C  bij_betw (compose A g f) A C"
  apply (simp add: bij_betw_def compose_eq inj_on_compose)
  apply (auto simp add: compose_def image_def)
  done

lemma bij_betw_restrict_eq [simp]: "bij_betw (restrict f A) A B = bij_betw f A B"
  by (simp add: bij_betw_def)


subsection Extensionality

lemma extensional_empty[simp]: "extensional {} = {λx. undefined}"
  unfolding extensional_def by auto

lemma extensional_arb: "f  extensional A  x  A  f x = undefined"
  by (simp add: extensional_def)

lemma restrict_extensional [simp]: "restrict f A  extensional A"
  by (simp add: restrict_def extensional_def)

lemma compose_extensional [simp]: "compose A f g  extensional A"
  by (simp add: compose_def)

lemma extensionalityI:
  assumes "f  extensional A"
    and "g  extensional A"
    and "x. x  A  f x = g x"
  shows "f = g"
  using assms by (force simp add: fun_eq_iff extensional_def)

lemma extensional_restrict:  "f  extensional A  restrict f A = f"
  by (rule extensionalityI[OF restrict_extensional]) auto

lemma extensional_subset: "f  extensional A  A  B  f  extensional B"
  unfolding extensional_def by auto

lemma inv_into_funcset: "f ` A = B  (λxB. inv_into A f x)  B  A"
  by (unfold inv_into_def) (fast intro: someI2)

lemma compose_inv_into_id: "bij_betw f A B  compose A (λyB. inv_into A f y) f = (λxA. x)"
  apply (simp add: bij_betw_def compose_def)
  apply (rule restrict_ext, auto)
  done

lemma compose_id_inv_into: "f ` A = B  compose B f (λyB. inv_into A f y) = (λxB. x)"
  apply (simp add: compose_def)
  apply (rule restrict_ext)
  apply (simp add: f_inv_into_f)
  done

lemma extensional_insert[intro, simp]:
  assumes "a  extensional (insert i I)"
  shows "a(i := b)  extensional (insert i I)"
  using assms unfolding extensional_def by auto

lemma extensional_Int[simp]: "extensional I  extensional I' = extensional (I  I')"
  unfolding extensional_def by auto

lemma extensional_UNIV[simp]: "extensional UNIV = UNIV"
  by (auto simp: extensional_def)

lemma restrict_extensional_sub[intro]: "A  B  restrict f A  extensional B"
  unfolding restrict_def extensional_def by auto

lemma extensional_insert_undefined[intro, simp]:
  "a  extensional (insert i I)  a(i := undefined)  extensional I"
  unfolding extensional_def by auto

lemma extensional_insert_cancel[intro, simp]:
  "a  extensional I  a  extensional (insert i I)"
  unfolding extensional_def by auto


subsection Cardinality

lemma card_inj: "f  A  B  inj_on f A  finite B  card A  card B"
  by (rule card_inj_on_le) auto

lemma card_bij:
  assumes "f  A  B" "inj_on f A"
    and "g  B  A" "inj_on g B"
    and "finite A" "finite B"
  shows "card A = card B"
  using assms by (blast intro: card_inj order_antisym)


subsection Extensional Function Spaces

definition PiE :: "'a set  ('a  'b set)  ('a  'b) set"
  where "PiE S T = Pi S T  extensional S"

abbreviation "PiE A B  PiE A B"

syntax
  "_PiE" :: "pttrn  'a set  'b set  ('a  'b) set"  ("(3ΠE __./ _)" 10)
translations
  "ΠE xA. B"  "CONST PiE A (λx. B)"

abbreviation extensional_funcset :: "'a set  'b set  ('a  'b) set" (infixr "E" 60)
  where "A E B  (ΠE iA. B)"

lemma extensional_funcset_def: "extensional_funcset S T = (S  T)  extensional S"
  by (simp add: PiE_def)

lemma PiE_empty_domain[simp]: "PiE {} T = {λx. undefined}"
  unfolding PiE_def by simp

lemma PiE_UNIV_domain: "PiE UNIV T = Pi UNIV T"
  unfolding PiE_def by simp

lemma PiE_empty_range[simp]: "i  I  F i = {}  (ΠE iI. F i) = {}"
  unfolding PiE_def by auto

lemma PiE_eq_empty_iff: "PiE I F = {}  (iI. F i = {})"
proof
  assume "PiE I F = {}"
  show "iI. F i = {}"
  proof (rule ccontr)
    assume "¬ ?thesis"
    then have "i. y. (i  I  y  F i)  (i  I  y = undefined)"
      by auto
    from choice[OF this]
    obtain f where " x. (x  I  f x  F x)  (x  I  f x = undefined)" ..
    then have "f  PiE I F"
      by (auto simp: extensional_def PiE_def)
    with PiE I F = {} show False
      by auto
  qed
qed (auto simp: PiE_def)

lemma PiE_arb: "f  PiE S T  x  S  f x = undefined"
  unfolding PiE_def by auto (auto dest!: extensional_arb)

lemma PiE_mem: "f  PiE S T  x  S  f x  T x"
  unfolding PiE_def by auto

lemma PiE_fun_upd: "y  T x  f  PiE S T  f(x := y)  PiE (insert x S) T"
  unfolding PiE_def extensional_def by auto

lemma fun_upd_in_PiE: "x  S  f  PiE (insert x S) T  f(x := undefined)  PiE S T"
  unfolding PiE_def extensional_def by auto

lemma PiE_insert_eq: "PiE (insert x S) T = (λ(y, g). g(x := y)) ` (T x × PiE S T)"
proof -
  {
    fix f assume "f  PiE (insert x S) T" "x  S"
    then have "f  (λ(y, g). g(x := y)) ` (T x × PiE S T)"
      by (auto intro!: image_eqI[where x="(f x, f(x := undefined))"] intro: fun_upd_in_PiE PiE_mem)
  }
  moreover
  {
    fix f assume "f  PiE (insert x S) T" "x  S"
    then have "f  (λ(y, g). g(x := y)) ` (T x × PiE S T)"
      by (auto intro!: image_eqI[where x="(f x, f)"] intro: fun_upd_in_PiE PiE_mem simp: insert_absorb)
  }
  ultimately show ?thesis
    by (auto intro: PiE_fun_upd)
qed

lemma PiE_Int: "PiE I A  PiE I B = PiE I (λx. A x  B x)"
  by (auto simp: PiE_def)

lemma PiE_cong: "(i. iI  A i = B i)  PiE I A = PiE I B"
  unfolding PiE_def by (auto simp: Pi_cong)

lemma PiE_E [elim]:
  assumes "f  PiE A B"
  obtains "x  A" and "f x  B x"
    | "x  A" and "f x = undefined"
  using assms by (auto simp: Pi_def PiE_def extensional_def)

lemma PiE_I[intro!]:
  "(x. x  A  f x  B x)  (x. x  A  f x = undefined)  f  PiE A B"
  by (simp add: PiE_def extensional_def)

lemma PiE_mono: "(x. x  A  B x  C x)  PiE A B  PiE A C"
  by auto

lemma PiE_iff: "f  PiE I X  (iI. f i  X i)  f  extensional I"
  by (simp add: PiE_def Pi_iff)

lemma restrict_PiE_iff: "restrict f I  PiE I X  (i  I. f i  X i)"
  by (simp add: PiE_iff)

lemma ext_funcset_to_sing_iff [simp]: "A E {a} = {λxA. a}"
  by (auto simp: PiE_def Pi_iff extensionalityI)

lemma PiE_restrict[simp]:  "f  PiE A B  restrict f A = f"
  by (simp add: extensional_restrict PiE_def)

lemma restrict_PiE[simp]: "restrict f I  PiE I S  f  Pi I S"
  by (auto simp: PiE_iff)

lemma PiE_eq_subset:
  assumes ne: "i. i  I  F i  {}" "i. i  I  F' i  {}"
    and eq: "PiE I F = PiE I F'"
    and "i  I"
  shows "F i  F' i"
proof
  fix x
  assume "x  F i"
  with ne have "j. y. (j  I  y  F j  (i = j  x = y))  (j  I  y = undefined)"
    by auto
  from choice[OF this] obtain f
    where f: " j. (j  I  f j  F j  (i = j  x = f j))  (j  I  f j = undefined)" ..
  then have "f  PiE I F"
    by (auto simp: extensional_def PiE_def)
  then have "f  PiE I F'"
    using assms by simp
  then show "x  F' i"
    using f i  I by (auto simp: PiE_def)
qed

lemma PiE_eq_iff_not_empty:
  assumes ne: "i. i  I  F i  {}" "i. i  I  F' i  {}"
  shows "PiE I F = PiE I F'  (iI. F i = F' i)"
proof (intro iffI ballI)
  fix i
  assume eq: "PiE I F = PiE I F'"
  assume i: "i  I"
  show "F i = F' i"
    using PiE_eq_subset[of I F F', OF ne eq i]
    using PiE_eq_subset[of I F' F, OF ne(2,1) eq[symmetric] i]
    by auto
qed (auto simp: PiE_def)

lemma PiE_eq_iff:
  "PiE I F = PiE I F'  (iI. F i = F' i)  ((iI. F i = {})  (iI. F' i = {}))"
proof (intro iffI disjCI)
  assume eq[simp]: "PiE I F = PiE I F'"
  assume "¬ ((iI. F i = {})  (iI. F' i = {}))"
  then have "(iI. F i  {})  (iI. F' i  {})"
    using PiE_eq_empty_iff[of I F] PiE_eq_empty_iff[of I F'] by auto
  with PiE_eq_iff_not_empty[of I F F'] show "iI. F i = F' i"
    by auto
next
  assume "(iI. F i = F' i)  (iI. F i = {})  (iI. F' i = {})"
  then show "PiE I F = PiE I F'"
    using PiE_eq_empty_iff[of I F] PiE_eq_empty_iff[of I F'] by (auto simp: PiE_def)
qed

lemma extensional_funcset_fun_upd_restricts_rangeI:
  "y  S. f x  f y  f  (insert x S) E T  f(x := undefined)  S E (T - {f x})"
  unfolding extensional_funcset_def extensional_def
  by (auto split: if_split_asm)

lemma extensional_funcset_fun_upd_extends_rangeI:
  assumes "a  T" "f  S E (T - {a})"
  shows "f(x := a)  insert x S E  T"
  using assms unfolding extensional_funcset_def extensional_def by auto

lemma subset_PiE:
   "PiE I S  PiE I T  PiE I S = {}  (i  I. S i  T i)" (is "?lhs  _  ?rhs")
proof (cases "PiE I S = {}")
  case False
  moreover have "?lhs = ?rhs"
  proof
    assume L: ?lhs
    have "i. iI  S i  {}"
      using False PiE_eq_empty_iff by blast
    with L show ?rhs
      by (simp add: PiE_Int PiE_eq_iff inf.absorb_iff2)
  qed auto
  ultimately show ?thesis
    by simp
qed simp

lemma PiE_eq:
   "PiE I S = PiE I T  PiE I S = {}  PiE I T = {}  (i  I. S i = T i)"
  by (auto simp: PiE_eq_iff PiE_eq_empty_iff)

lemma PiE_UNIV [simp]: "PiE UNIV (λi. UNIV) = UNIV"
  by blast

lemma image_projection_PiE:
  "(λf. f i) ` (PiE I S) = (if PiE I S = {} then {} else if i  I then S i else {undefined})"
proof -
  have "(λf. f i) ` PiE I S = S i" if "i  I" "f  PiE I S" for f
    using that apply auto
    by (rule_tac x="(λk. if k=i then x else f k)" in image_eqI) auto
  moreover have "(λf. f i) ` PiE I S = {undefined}" if "f  PiE I S" "i  I" for f
    using that by (blast intro: PiE_arb [OF that, symmetric])
  ultimately show ?thesis
    by auto
qed

lemma PiE_singleton:
  assumes "f  extensional A"
  shows   "PiE A (λx. {f x}) = {f}"
proof -
  {
    fix g assume "g  PiE A (λx. {f x})"
    hence "g x = f x" for x
      using assms by (cases "x  A") (auto simp: extensional_def)
    hence "g = f" by (simp add: fun_eq_iff)
  }
  thus ?thesis using assms by (auto simp: extensional_def)
qed

lemma PiE_eq_singleton: "(ΠE iI. S i) = {λiI. f i}  (iI. S i = {f i})"
  by (metis (mono_tags, lifting) PiE_eq PiE_singleton insert_not_empty restrict_apply' restrict_extensional)

lemma PiE_over_singleton_iff: "(ΠE x{a}. B x) = (b  B a. {λx  {a}. b})"
  apply (auto simp: PiE_iff split: if_split_asm)
  apply (metis (no_types, lifting) extensionalityI restrict_apply' restrict_extensional singletonD)
  done

lemma all_PiE_elements:
   "(z  PiE I S. i  I. P i (z i))  PiE I S = {}  (i  I. x  S i. P i x)" (is "?lhs = ?rhs")
proof (cases "PiE I S = {}")
  case False
  then obtain f where f: "i. i  I  f i  S i"
    by fastforce
  show ?thesis
  proof
    assume L: ?lhs
    have "P i x"
      if "i  I" "x  S i" for i x
    proof -
      have "(λj  I. if j=i then x else f j)  PiE I S"
        by (simp add: f that(2))
      then have "P i ((λj  I. if j=i then x else f j) i)"
        using L that(1) by blast
      with that show ?thesis
        by simp
    qed
    then show ?rhs
      by (simp add: False)
  qed fastforce
qed simp

lemma PiE_ext: "x  PiE k s; y  PiE k s; i. i  k  x i = y i  x = y"
  by (metis ext PiE_E)


subsubsection Injective Extensional Function Spaces

lemma extensional_funcset_fun_upd_inj_onI:
  assumes "f  S E (T - {a})"
    and "inj_on f S"
  shows "inj_on (f(x := a)) S"
  using assms
  unfolding extensional_funcset_def by (auto intro!: inj_on_fun_updI)

lemma extensional_funcset_extend_domain_inj_on_eq:
  assumes "x  S"
  shows "{f. f  (insert x S) E T  inj_on f (insert x S)} =
    (λ(y, g). g(x:=y)) ` {(y, g). y  T  g  S E (T - {y})  inj_on g S}"
  using assms
  apply (auto del: PiE_I PiE_E)
  apply (auto intro: extensional_funcset_fun_upd_inj_onI
    extensional_funcset_fun_upd_extends_rangeI del: PiE_I PiE_E)
  apply (auto simp add: image_iff inj_on_def)
  apply (rule_tac x="xa x" in exI)
  apply (auto intro: PiE_mem del: PiE_I PiE_E)
  apply (rule_tac x="xa(x := undefined)" in exI)
  apply (auto intro!: extensional_funcset_fun_upd_restricts_rangeI)
  apply (auto dest!: PiE_mem split: if_split_asm)
  done

lemma extensional_funcset_extend_domain_inj_onI:
  assumes "x  S"
  shows "inj_on (λ(y, g). g(x := y)) {(y, g). y  T  g  S E (T - {y})  inj_on g S}"
  using assms
  apply (auto intro!: inj_onI)
  apply (metis fun_upd_same)
  apply (metis assms PiE_arb fun_upd_triv fun_upd_upd)
  done


subsubsection Misc properties of functions, composition and restriction from HOL Light

lemma function_factors_left_gen:
  "(x y. P x  P y  g x = g y  f x = f y)  (h. x. P x  f x = h(g x))"
  (is "?lhs = ?rhs")
proof
  assume L: ?lhs
  then show ?rhs
    apply (rule_tac x="f  inv_into (Collect P) g" in exI)
    unfolding o_def
    by (metis (mono_tags, opaque_lifting) f_inv_into_f imageI inv_into_into mem_Collect_eq)
qed auto

lemma function_factors_left:
  "(x y. (g x = g y)  (f x = f y))  (h. f = h  g)"
  using function_factors_left_gen [of "λx. True" g f] unfolding o_def by blast

lemma function_factors_right_gen:
  "(x. P x  (y. g y = f x))  (h. x. P x  f x = g(h x))"
  by metis

lemma function_factors_right:
  "(x. y. g y = f x)  (h. f = g  h)"
  unfolding o_def by metis

lemma restrict_compose_right:
   "restrict (g  restrict f S) S = restrict (g  f) S"
  by auto

lemma restrict_compose_left:
   "f ` S  T  restrict (restrict g T  f) S = restrict (g  f) S"
  by fastforce


subsubsection Cardinality

lemma finite_PiE: "finite S  (i. i  S  finite (T i))  finite (ΠE i  S. T i)"
  by (induct S arbitrary: T rule: finite_induct) (simp_all add: PiE_insert_eq)

lemma inj_combinator: "x  S  inj_on (λ(y, g). g(x := y)) (T x × PiE S T)"
proof (safe intro!: inj_onI ext)
  fix f y g z
  assume "x  S"
  assume fg: "f  PiE S T" "g  PiE S T"
  assume "f(x := y) = g(x := z)"
  then have *: "i. (f(x := y)) i = (g(x := z)) i"
    unfolding fun_eq_iff by auto
  from this[of x] show "y = z" by simp
  fix i from *[of i] x  S fg show "f i = g i"
    by (auto split: if_split_asm simp: PiE_def extensional_def)
qed

lemma card_PiE: "finite S  card (ΠE i  S. T i) = ( iS. card (T i))"
proof (induct rule: finite_induct)
  case empty
  then show ?case by auto
next
  case (insert x S)
  then show ?case
    by (simp add: PiE_insert_eq inj_combinator card_image card_cartesian_product)
qed

lemma card_funcsetE: "finite A  card (A E B) = card B ^ card A" 
  by (subst card_PiE, auto)

lemma card_inj_on_subset_funcset: assumes finB: "finite B"
  and finC: "finite C" 
  and AB: "A  B" 
shows "card {f  B E C. inj_on f A} = 
  card C^(card B - card A) * prod ((-) (card C)) {0 ..< card A}"
proof -
  define D where "D = B - A" 
  from AB have B: "B = A  D" and disj: "A  D = {}" unfolding D_def by auto
  have sub: "card B - card A = card D" unfolding D_def using finB AB
    by (metis card_Diff_subset finite_subset)
  have "finite A" "finite D" using finB unfolding B by auto
  thus ?thesis unfolding sub unfolding B using disj
  proof (induct A rule: finite_induct)
    case empty
    from card_funcsetE[OF this(1), of C] show ?case by auto
  next
    case (insert a A)
    have "{f. f  insert a A  D E C  inj_on f (insert a A)}
      = {f(a := c) | f c. f  A  D E C  inj_on f A  c  C - f ` A}" 
      (is "?l = ?r")
    proof
      show "?r  ?l" 
        by (auto intro: inj_on_fun_updI split: if_splits) 
      {
        fix f
        assume f: "f  ?l" 
        let ?g = "f(a := undefined)" 
        let ?h = "?g(a := f a)" 
        have mem: "f a  C - ?g ` A" using insert(1,2,4,5) f by auto
        from f have f: "f  insert a A  D E C" "inj_on f (insert a A)" by auto
        hence "?g  A  D E C" "inj_on ?g A" using a  A insert a A  D = {}
          by (auto split: if_splits simp: inj_on_def)
        with mem have "?h  ?r" by blast
        also have "?h = f" by auto
        finally have "f  ?r" .
      }
      thus "?l  ?r" by auto
    qed
    also have " = (λ (f, c). f (a := c)) ` 
         (Sigma {f . f  A  D E C  inj_on f A} (λ f. C - f ` A))"
      by auto
    also have "card (...) = card (Sigma {f . f  A  D E C  inj_on f A} (λ f. C - f ` A))" 
    proof (rule card_image, intro inj_onI, clarsimp, goal_cases) 
      case (1 f c g d)
      let ?f = "f(a := c, a := undefined)" 
      let ?g = "g(a := d, a := undefined)" 
      from 1 have id: "f(a := c) = g(a := d)" by auto
      from fun_upd_eqD[OF id] 
      have cd: "c = d" by auto
      from id have "?f = ?g" by auto
      also have "?f = f" using `f  A  D E C` insert(1,2,4,5) 
        by (intro ext, auto)
      also have "?g = g" using `g  A  D E C` insert(1,2,4,5) 
        by (intro ext, auto)
      finally show "f = g  c = d" using cd by auto
    qed
    also have " = (f{f  A  D E C. inj_on f A}. card (C - f ` A))" 
      by (rule card_SigmaI, rule finite_subset[of _ "A  D E C"],
          insert finite C finite D finite A, auto intro!: finite_PiE)
    also have " = (f{f  A  D E C. inj_on f A}. card C - card A)"
      by (rule sum.cong[OF refl], subst card_Diff_subset, insert finite A, auto simp: card_image)
    also have " = (card C - card A) * card {f  A  D E C. inj_on f A}" 
      by simp
    also have " = card C ^ card D * ((card C - card A) * prod ((-) (card C)) {0..<card A})" 
      using insert by (auto simp: ac_simps)
    also have "(card C - card A) * prod ((-) (card C)) {0..<card A} =
      prod ((-) (card C)) {0..<Suc (card A)}" by simp
    also have "Suc (card A) = card (insert a A)" using insert by auto
    finally show ?case .
  qed
qed


subsection The pigeonhole principle

text 
  An alternative formulation of this is that for a function mapping a finite set A› of
  cardinality m› to a finite set B› of cardinality n›, there exists an element y ∈ B› that
  is hit at least $\lceil \frac{m}{n}\rceil$ times. However, since we do not have real numbers
  or rounding yet, we state it in the following equivalent form:

lemma pigeonhole_card:
  assumes "f  A  B" "finite A" "finite B" "B  {}"
  shows   "yB. card (f -` {y}  A) * card B  card A"
proof -
  from assms have "card B > 0"
    by auto
  define M where "M = Max ((λy. card (f -` {y}  A)) ` B)"
  have "A = (yB. f -` {y}  A)"
    using assms by auto
  also have "card  = (iB. card (f -` {i}  A))"
    using assms by (subst card_UN_disjoint) auto
  also have "  (iB. M)"
    unfolding M_def using assms by (intro sum_mono Max.coboundedI) auto
  also have " = card B * M"
    by simp
  finally have "M * card B  card A"
    by (simp add: mult_ac)
  moreover have "M  (λy. card (f -` {y}  A)) ` B"
    unfolding M_def using assms B  {} by (intro Max_in) auto
  ultimately show ?thesis
    by blast
qed

end