src/HOL/Hilbert_Choice.thy

map_fun combinator in theory Fun

(*  Title:      HOL/Hilbert_Choice.thy
    Author:     Lawrence C Paulson, Tobias Nipkow
    Copyright   2001  University of Cambridge
*)

header {* Hilbert's Epsilon-Operator and the Axiom of Choice *}

theory Hilbert_Choice
imports Nat Wellfounded Plain
uses ("Tools/choice_specification.ML")
begin

subsection {* Hilbert's epsilon *}

axiomatization Eps :: "('a => bool) => 'a" where
  someI: "P x ==> P (Eps P)"

syntax (epsilon)
  "_Eps"        :: "[pttrn, bool] => 'a"    ("(3\<some>_./ _)" [0, 10] 10)
syntax (HOL)
  "_Eps"        :: "[pttrn, bool] => 'a"    ("(3@ _./ _)" [0, 10] 10)
syntax
  "_Eps"        :: "[pttrn, bool] => 'a"    ("(3SOME _./ _)" [0, 10] 10)
translations
  "SOME x. P" == "CONST Eps (%x. P)"

print_translation {*
  [(@{const_syntax Eps}, fn [Abs abs] =>
      let val (x, t) = atomic_abs_tr' abs
      in Syntax.const @{syntax_const "_Eps"} $ x $ t end)]
*} -- {* to avoid eta-contraction of body *}

definition inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
"inv_into A f == %x. SOME y. y : A & f y = x"

abbreviation inv :: "('a => 'b) => ('b => 'a)" where
"inv == inv_into UNIV"


subsection {*Hilbert's Epsilon-operator*}

text{*Easier to apply than @{text someI} if the witness comes from an
existential formula*}
lemma someI_ex [elim?]: "\<exists>x. P x ==> P (SOME x. P x)"
apply (erule exE)
apply (erule someI)
done

text{*Easier to apply than @{text someI} because the conclusion has only one
occurrence of @{term P}.*}
lemma someI2: "[| P a;  !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
by (blast intro: someI)

text{*Easier to apply than @{text someI2} if the witness comes from an
existential formula*}
lemma someI2_ex: "[| \<exists>a. P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
by (blast intro: someI2)

lemma some_equality [intro]:
     "[| P a;  !!x. P x ==> x=a |] ==> (SOME x. P x) = a"
by (blast intro: someI2)

lemma some1_equality: "[| EX!x. P x; P a |] ==> (SOME x. P x) = a"
by blast

lemma some_eq_ex: "P (SOME x. P x) =  (\<exists>x. P x)"
by (blast intro: someI)

lemma some_eq_trivial [simp]: "(SOME y. y=x) = x"
apply (rule some_equality)
apply (rule refl, assumption)
done

lemma some_sym_eq_trivial [simp]: "(SOME y. x=y) = x"
apply (rule some_equality)
apply (rule refl)
apply (erule sym)
done


subsection{*Axiom of Choice, Proved Using the Description Operator*}

lemma choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>f. \<forall>x. Q x (f x)"
by (fast elim: someI)

lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y ==> \<exists>f. \<forall>x\<in>S. Q x (f x)"
by (fast elim: someI)


subsection {*Function Inverse*}

lemma inv_def: "inv f = (%y. SOME x. f x = y)"
by(simp add: inv_into_def)

lemma inv_into_into: "x : f ` A ==> inv_into A f x : A"
apply (simp add: inv_into_def)
apply (fast intro: someI2)
done

lemma inv_id [simp]: "inv id = id"
by (simp add: inv_into_def id_def)

lemma inv_into_f_f [simp]:
  "[| inj_on f A;  x : A |] ==> inv_into A f (f x) = x"
apply (simp add: inv_into_def inj_on_def)
apply (blast intro: someI2)
done

lemma inv_f_f: "inj f ==> inv f (f x) = x"
by simp

lemma f_inv_into_f: "y : f`A  ==> f (inv_into A f y) = y"
apply (simp add: inv_into_def)
apply (fast intro: someI2)
done

lemma inv_into_f_eq: "[| inj_on f A; x : A; f x = y |] ==> inv_into A f y = x"
apply (erule subst)
apply (fast intro: inv_into_f_f)
done

lemma inv_f_eq: "[| inj f; f x = y |] ==> inv f y = x"
by (simp add:inv_into_f_eq)

lemma inj_imp_inv_eq: "[| inj f; ALL x. f(g x) = x |] ==> inv f = g"
by (blast intro: ext inv_into_f_eq)

text{*But is it useful?*}
lemma inj_transfer:
  assumes injf: "inj f" and minor: "!!y. y \<in> range(f) ==> P(inv f y)"
  shows "P x"
proof -
  have "f x \<in> range f" by auto
  hence "P(inv f (f x))" by (rule minor)
  thus "P x" by (simp add: inv_into_f_f [OF injf])
qed

lemma inj_iff: "(inj f) = (inv f o f = id)"
apply (simp add: o_def fun_eq_iff)
apply (blast intro: inj_on_inverseI inv_into_f_f)
done

lemma inv_o_cancel[simp]: "inj f ==> inv f o f = id"
by (simp add: inj_iff)

lemma o_inv_o_cancel[simp]: "inj f ==> g o inv f o f = g"
by (simp add: o_assoc[symmetric])

lemma inv_into_image_cancel[simp]:
  "inj_on f A ==> S <= A ==> inv_into A f ` f ` S = S"
by(fastsimp simp: image_def)

lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"
by (blast intro: surjI inv_into_f_f)

lemma surj_f_inv_f: "surj f ==> f(inv f y) = y"
by (simp add: f_inv_into_f surj_range)

lemma inv_into_injective:
  assumes eq: "inv_into A f x = inv_into A f y"
      and x: "x: f`A"
      and y: "y: f`A"
  shows "x=y"
proof -
  have "f (inv_into A f x) = f (inv_into A f y)" using eq by simp
  thus ?thesis by (simp add: f_inv_into_f x y)
qed

lemma inj_on_inv_into: "B <= f`A ==> inj_on (inv_into A f) B"
by (blast intro: inj_onI dest: inv_into_injective injD)

lemma bij_betw_inv_into: "bij_betw f A B ==> bij_betw (inv_into A f) B A"
by (auto simp add: bij_betw_def inj_on_inv_into)

lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"
by (simp add: inj_on_inv_into surj_range)

lemma surj_iff: "(surj f) = (f o inv f = id)"
apply (simp add: o_def fun_eq_iff)
apply (blast intro: surjI surj_f_inv_f)
done

lemma surj_imp_inv_eq: "[| surj f; \<forall>x. g(f x) = x |] ==> inv f = g"
apply (rule ext)
apply (drule_tac x = "inv f x" in spec)
apply (simp add: surj_f_inv_f)
done

lemma bij_imp_bij_inv: "bij f ==> bij (inv f)"
by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv)

lemma inv_equality: "[| !!x. g (f x) = x;  !!y. f (g y) = y |] ==> inv f = g"
apply (rule ext)
apply (auto simp add: inv_into_def)
done

lemma inv_inv_eq: "bij f ==> inv (inv f) = f"
apply (rule inv_equality)
apply (auto simp add: bij_def surj_f_inv_f)
done

(** bij(inv f) implies little about f.  Consider f::bool=>bool such that
    f(True)=f(False)=True.  Then it's consistent with axiom someI that
    inv f could be any function at all, including the identity function.
    If inv f=id then inv f is a bijection, but inj f, surj(f) and
    inv(inv f)=f all fail.
**)

lemma inv_into_comp:
  "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
  inv_into A (f o g) x = (inv_into A g o inv_into (g ` A) f) x"
apply (rule inv_into_f_eq)
  apply (fast intro: comp_inj_on)
 apply (simp add: inv_into_into)
apply (simp add: f_inv_into_f inv_into_into)
done

lemma o_inv_distrib: "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f"
apply (rule inv_equality)
apply (auto simp add: bij_def surj_f_inv_f)
done

lemma image_surj_f_inv_f: "surj f ==> f ` (inv f ` A) = A"
by (simp add: image_eq_UN surj_f_inv_f)

lemma image_inv_f_f: "inj f ==> (inv f) ` (f ` A) = A"
by (simp add: image_eq_UN)

lemma inv_image_comp: "inj f ==> inv f ` (f`X) = X"
by (auto simp add: image_def)

lemma bij_image_Collect_eq: "bij f ==> f ` Collect P = {y. P (inv f y)}"
apply auto
apply (force simp add: bij_is_inj)
apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric])
done

lemma bij_vimage_eq_inv_image: "bij f ==> f -` A = inv f ` A" 
apply (auto simp add: bij_is_surj [THEN surj_f_inv_f])
apply (blast intro: bij_is_inj [THEN inv_into_f_f, symmetric])
done

lemma finite_fun_UNIVD1:
  assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
  and card: "card (UNIV :: 'b set) \<noteq> Suc 0"
  shows "finite (UNIV :: 'a set)"
proof -
  from fin have finb: "finite (UNIV :: 'b set)" by (rule finite_fun_UNIVD2)
  with card have "card (UNIV :: 'b set) \<ge> Suc (Suc 0)"
    by (cases "card (UNIV :: 'b set)") (auto simp add: card_eq_0_iff)
  then obtain n where "card (UNIV :: 'b set) = Suc (Suc n)" "n = card (UNIV :: 'b set) - Suc (Suc 0)" by auto
  then obtain b1 b2 where b1b2: "(b1 :: 'b) \<noteq> (b2 :: 'b)" by (auto simp add: card_Suc_eq)
  from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1))" by (rule finite_imageI)
  moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1)"
  proof (rule UNIV_eq_I)
    fix x :: 'a
    from b1b2 have "x = inv (\<lambda>y. if y = x then b1 else b2) b1" by (simp add: inv_into_def)
    thus "x \<in> range (\<lambda>f\<Colon>'a \<Rightarrow> 'b. inv f b1)" by blast
  qed
  ultimately show "finite (UNIV :: 'a set)" by simp
qed


subsection {*Other Consequences of Hilbert's Epsilon*}

text {*Hilbert's Epsilon and the @{term split} Operator*}

text{*Looping simprule*}
lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))"
  by simp

lemma Eps_split: "Eps (split P) = (SOME xy. P (fst xy) (snd xy))"
  by (simp add: split_def)

lemma Eps_split_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)"
  by blast


text{*A relation is wellfounded iff it has no infinite descending chain*}
lemma wf_iff_no_infinite_down_chain:
  "wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))"
apply (simp only: wf_eq_minimal)
apply (rule iffI)
 apply (rule notI)
 apply (erule exE)
 apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast)
apply (erule contrapos_np, simp, clarify)
apply (subgoal_tac "\<forall>n. nat_rec x (%i y. @z. z:Q & (z,y) :r) n \<in> Q")
 apply (rule_tac x = "nat_rec x (%i y. @z. z:Q & (z,y) :r)" in exI)
 apply (rule allI, simp)
 apply (rule someI2_ex, blast, blast)
apply (rule allI)
apply (induct_tac "n", simp_all)
apply (rule someI2_ex, blast+)
done

lemma wf_no_infinite_down_chainE:
  assumes "wf r" obtains k where "(f (Suc k), f k) \<notin> r"
using `wf r` wf_iff_no_infinite_down_chain[of r] by blast


text{*A dynamically-scoped fact for TFL *}
lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
  by (blast intro: someI)


subsection {* Least value operator *}

definition
  LeastM :: "['a => 'b::ord, 'a => bool] => 'a" where
  "LeastM m P == SOME x. P x & (\<forall>y. P y --> m x <= m y)"

syntax
  "_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"    ("LEAST _ WRT _. _" [0, 4, 10] 10)
translations
  "LEAST x WRT m. P" == "CONST LeastM m (%x. P)"

lemma LeastMI2:
  "P x ==> (!!y. P y ==> m x <= m y)
    ==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x)
    ==> Q (LeastM m P)"
  apply (simp add: LeastM_def)
  apply (rule someI2_ex, blast, blast)
  done

lemma LeastM_equality:
  "P k ==> (!!x. P x ==> m k <= m x)
    ==> m (LEAST x WRT m. P x) = (m k::'a::order)"
  apply (rule LeastMI2, assumption, blast)
  apply (blast intro!: order_antisym)
  done

lemma wf_linord_ex_has_least:
  "wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
    ==> \<exists>x. P x & (!y. P y --> (m x,m y):r^*)"
  apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
  apply (drule_tac x = "m`Collect P" in spec, force)
  done

lemma ex_has_least_nat:
    "P k ==> \<exists>x. P x & (\<forall>y. P y --> m x <= (m y::nat))"
  apply (simp only: pred_nat_trancl_eq_le [symmetric])
  apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
   apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le, assumption)
  done

lemma LeastM_nat_lemma:
    "P k ==> P (LeastM m P) & (\<forall>y. P y --> m (LeastM m P) <= (m y::nat))"
  apply (simp add: LeastM_def)
  apply (rule someI_ex)
  apply (erule ex_has_least_nat)
  done

lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1, standard]

lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)"
by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption)


subsection {* Greatest value operator *}

definition
  GreatestM :: "['a => 'b::ord, 'a => bool] => 'a" where
  "GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)"

definition
  Greatest :: "('a::ord => bool) => 'a" (binder "GREATEST " 10) where
  "Greatest == GreatestM (%x. x)"

syntax
  "_GreatestM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"
      ("GREATEST _ WRT _. _" [0, 4, 10] 10)
translations
  "GREATEST x WRT m. P" == "CONST GreatestM m (%x. P)"

lemma GreatestMI2:
  "P x ==> (!!y. P y ==> m y <= m x)
    ==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x)
    ==> Q (GreatestM m P)"
  apply (simp add: GreatestM_def)
  apply (rule someI2_ex, blast, blast)
  done

lemma GreatestM_equality:
 "P k ==> (!!x. P x ==> m x <= m k)
    ==> m (GREATEST x WRT m. P x) = (m k::'a::order)"
  apply (rule_tac m = m in GreatestMI2, assumption, blast)
  apply (blast intro!: order_antisym)
  done

lemma Greatest_equality:
  "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k"
  apply (simp add: Greatest_def)
  apply (erule GreatestM_equality, blast)
  done

lemma ex_has_greatest_nat_lemma:
  "P k ==> \<forall>x. P x --> (\<exists>y. P y & ~ ((m y::nat) <= m x))
    ==> \<exists>y. P y & ~ (m y < m k + n)"
  apply (induct n, force)
  apply (force simp add: le_Suc_eq)
  done

lemma ex_has_greatest_nat:
  "P k ==> \<forall>y. P y --> m y < b
    ==> \<exists>x. P x & (\<forall>y. P y --> (m y::nat) <= m x)"
  apply (rule ccontr)
  apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
    apply (subgoal_tac [3] "m k <= b", auto)
  done

lemma GreatestM_nat_lemma:
  "P k ==> \<forall>y. P y --> m y < b
    ==> P (GreatestM m P) & (\<forall>y. P y --> (m y::nat) <= m (GreatestM m P))"
  apply (simp add: GreatestM_def)
  apply (rule someI_ex)
  apply (erule ex_has_greatest_nat, assumption)
  done

lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1, standard]

lemma GreatestM_nat_le:
  "P x ==> \<forall>y. P y --> m y < b
    ==> (m x::nat) <= m (GreatestM m P)"
  apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P])
  done


text {* \medskip Specialization to @{text GREATEST}. *}

lemma GreatestI: "P (k::nat) ==> \<forall>y. P y --> y < b ==> P (GREATEST x. P x)"
  apply (simp add: Greatest_def)
  apply (rule GreatestM_natI, auto)
  done

lemma Greatest_le:
    "P x ==> \<forall>y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
  apply (simp add: Greatest_def)
  apply (rule GreatestM_nat_le, auto)
  done


subsection {* Specification package -- Hilbertized version *}

lemma exE_some: "[| Ex P ; c == Eps P |] ==> P c"
  by (simp only: someI_ex)

use "Tools/choice_specification.ML"

end