(* 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 Big_Operators
keywords "specification" "ax_specification" :: thy_goal
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 _ => fn [Abs abs] =>
let val (x, t) = Syntax_Trans.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)
lemma choice_iff: "(\<forall>x. \<exists>y. Q x y) \<longleftrightarrow> (\<exists>f. \<forall>x. Q x (f x))"
by (fast elim: someI)
lemma choice_iff': "(\<forall>x. P x \<longrightarrow> (\<exists>y. Q x y)) \<longleftrightarrow> (\<exists>f. \<forall>x. P x \<longrightarrow> Q x (f x))"
by (fast elim: someI)
lemma bchoice_iff: "(\<forall>x\<in>S. \<exists>y. Q x y) \<longleftrightarrow> (\<exists>f. \<forall>x\<in>S. Q x (f x))"
by (fast elim: someI)
lemma bchoice_iff': "(\<forall>x\<in>S. P x \<longrightarrow> (\<exists>y. Q x y)) \<longleftrightarrow> (\<exists>f. \<forall>x\<in>S. P x \<longrightarrow> 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: 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: comp_assoc)
lemma inv_into_image_cancel[simp]:
"inj_on f A ==> S <= A ==> inv_into A f ` f ` S = S"
by(fastforce 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)
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)
lemma surj_iff: "(surj f) = (f o inv f = id)"
by (auto intro!: surjI simp: surj_f_inv_f fun_eq_iff[where 'b='a])
lemma surj_iff_all: "surj f \<longleftrightarrow> (\<forall>x. f (inv f x) = x)"
unfolding surj_iff by (simp add: o_def fun_eq_iff)
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
lemma image_inv_into_cancel:
assumes SURJ: "f`A=A'" and SUB: "B' \<le> A'"
shows "f `((inv_into A f)`B') = B'"
using assms
proof (auto simp add: f_inv_into_f)
let ?f' = "(inv_into A f)"
fix a' assume *: "a' \<in> B'"
then have "a' \<in> A'" using SUB by auto
then have "a' = f (?f' a')"
using SURJ by (auto simp add: f_inv_into_f)
then show "a' \<in> f ` (?f' ` B')" using * by blast
qed
lemma inv_into_inv_into_eq:
assumes "bij_betw f A A'" "a \<in> A"
shows "inv_into A' (inv_into A f) a = f a"
proof -
let ?f' = "inv_into A f" let ?f'' = "inv_into A' ?f'"
have 1: "bij_betw ?f' A' A" using assms
by (auto simp add: bij_betw_inv_into)
obtain a' where 2: "a' \<in> A'" and 3: "?f' a' = a"
using 1 `a \<in> A` unfolding bij_betw_def by force
hence "?f'' a = a'"
using `a \<in> A` 1 3 by (auto simp add: f_inv_into_f bij_betw_def)
moreover have "f a = a'" using assms 2 3
by (auto simp add: bij_betw_def)
ultimately show "?f'' a = f a" by simp
qed
lemma inj_on_iff_surj:
assumes "A \<noteq> {}"
shows "(\<exists>f. inj_on f A \<and> f ` A \<le> A') \<longleftrightarrow> (\<exists>g. g ` A' = A)"
proof safe
fix f assume INJ: "inj_on f A" and INCL: "f ` A \<le> A'"
let ?phi = "\<lambda>a' a. a \<in> A \<and> f a = a'" let ?csi = "\<lambda>a. a \<in> A"
let ?g = "\<lambda>a'. if a' \<in> f ` A then (SOME a. ?phi a' a) else (SOME a. ?csi a)"
have "?g ` A' = A"
proof
show "?g ` A' \<le> A"
proof clarify
fix a' assume *: "a' \<in> A'"
show "?g a' \<in> A"
proof cases
assume Case1: "a' \<in> f ` A"
then obtain a where "?phi a' a" by blast
hence "?phi a' (SOME a. ?phi a' a)" using someI[of "?phi a'" a] by blast
with Case1 show ?thesis by auto
next
assume Case2: "a' \<notin> f ` A"
hence "?csi (SOME a. ?csi a)" using assms someI_ex[of ?csi] by blast
with Case2 show ?thesis by auto
qed
qed
next
show "A \<le> ?g ` A'"
proof-
{fix a assume *: "a \<in> A"
let ?b = "SOME aa. ?phi (f a) aa"
have "?phi (f a) a" using * by auto
hence 1: "?phi (f a) ?b" using someI[of "?phi(f a)" a] by blast
hence "?g(f a) = ?b" using * by auto
moreover have "a = ?b" using 1 INJ * by (auto simp add: inj_on_def)
ultimately have "?g(f a) = a" by simp
with INCL * have "?g(f a) = a \<and> f a \<in> A'" by auto
}
thus ?thesis by force
qed
qed
thus "\<exists>g. g ` A' = A" by blast
next
fix g let ?f = "inv_into A' g"
have "inj_on ?f (g ` A')"
by (auto simp add: inj_on_inv_into)
moreover
{fix a' assume *: "a' \<in> A'"
let ?phi = "\<lambda> b'. b' \<in> A' \<and> g b' = g a'"
have "?phi a'" using * by auto
hence "?phi(SOME b'. ?phi b')" using someI[of ?phi] by blast
hence "?f(g a') \<in> A'" unfolding inv_into_def by auto
}
ultimately show "\<exists>f. inj_on f (g ` A') \<and> f ` g ` A' \<subseteq> A'" by auto
qed
lemma Ex_inj_on_UNION_Sigma:
"\<exists>f. (inj_on f (\<Union> i \<in> I. A i) \<and> f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i))"
proof
let ?phi = "\<lambda> a i. i \<in> I \<and> a \<in> A i"
let ?sm = "\<lambda> a. SOME i. ?phi a i"
let ?f = "\<lambda>a. (?sm a, a)"
have "inj_on ?f (\<Union> i \<in> I. A i)" unfolding inj_on_def by auto
moreover
{ { fix i a assume "i \<in> I" and "a \<in> A i"
hence "?sm a \<in> I \<and> a \<in> A(?sm a)" using someI[of "?phi a" i] by auto
}
hence "?f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i)" by auto
}
ultimately
show "inj_on ?f (\<Union> i \<in> I. A i) \<and> ?f ` (\<Union> i \<in> I. A i) \<le> (SIGMA i : I. A i)"
by auto
qed
subsection {* The Cantor-Bernstein Theorem *}
lemma Cantor_Bernstein_aux:
shows "\<exists>A' h. A' \<le> A \<and>
(\<forall>a \<in> A'. a \<notin> g`(B - f ` A')) \<and>
(\<forall>a \<in> A'. h a = f a) \<and>
(\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a))"
proof-
obtain H where H_def: "H = (\<lambda> A'. A - (g`(B - (f ` A'))))" by blast
have 0: "mono H" unfolding mono_def H_def by blast
then obtain A' where 1: "H A' = A'" using lfp_unfold by blast
hence 2: "A' = A - (g`(B - (f ` A')))" unfolding H_def by simp
hence 3: "A' \<le> A" by blast
have 4: "\<forall>a \<in> A'. a \<notin> g`(B - f ` A')"
using 2 by blast
have 5: "\<forall>a \<in> A - A'. \<exists>b \<in> B - (f ` A'). a = g b"
using 2 by blast
(* *)
obtain h where h_def:
"h = (\<lambda> a. if a \<in> A' then f a else (SOME b. b \<in> B - (f ` A') \<and> a = g b))" by blast
hence "\<forall>a \<in> A'. h a = f a" by auto
moreover
have "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
proof
fix a assume *: "a \<in> A - A'"
let ?phi = "\<lambda> b. b \<in> B - (f ` A') \<and> a = g b"
have "h a = (SOME b. ?phi b)" using h_def * by auto
moreover have "\<exists>b. ?phi b" using 5 * by auto
ultimately show "?phi (h a)" using someI_ex[of ?phi] by auto
qed
ultimately show ?thesis using 3 4 by blast
qed
theorem Cantor_Bernstein:
assumes INJ1: "inj_on f A" and SUB1: "f ` A \<le> B" and
INJ2: "inj_on g B" and SUB2: "g ` B \<le> A"
shows "\<exists>h. bij_betw h A B"
proof-
obtain A' and h where 0: "A' \<le> A" and
1: "\<forall>a \<in> A'. a \<notin> g`(B - f ` A')" and
2: "\<forall>a \<in> A'. h a = f a" and
3: "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
using Cantor_Bernstein_aux[of A g B f] by blast
have "inj_on h A"
proof (intro inj_onI)
fix a1 a2
assume 4: "a1 \<in> A" and 5: "a2 \<in> A" and 6: "h a1 = h a2"
show "a1 = a2"
proof(cases "a1 \<in> A'")
assume Case1: "a1 \<in> A'"
show ?thesis
proof(cases "a2 \<in> A'")
assume Case11: "a2 \<in> A'"
hence "f a1 = f a2" using Case1 2 6 by auto
thus ?thesis using INJ1 Case1 Case11 0
unfolding inj_on_def by blast
next
assume Case12: "a2 \<notin> A'"
hence False using 3 5 2 6 Case1 by force
thus ?thesis by simp
qed
next
assume Case2: "a1 \<notin> A'"
show ?thesis
proof(cases "a2 \<in> A'")
assume Case21: "a2 \<in> A'"
hence False using 3 4 2 6 Case2 by auto
thus ?thesis by simp
next
assume Case22: "a2 \<notin> A'"
hence "a1 = g(h a1) \<and> a2 = g(h a2)" using Case2 4 5 3 by auto
thus ?thesis using 6 by simp
qed
qed
qed
(* *)
moreover
have "h ` A = B"
proof safe
fix a assume "a \<in> A"
thus "h a \<in> B" using SUB1 2 3 by (cases "a \<in> A'") auto
next
fix b assume *: "b \<in> B"
show "b \<in> h ` A"
proof(cases "b \<in> f ` A'")
assume Case1: "b \<in> f ` A'"
then obtain a where "a \<in> A' \<and> b = f a" by blast
thus ?thesis using 2 0 by force
next
assume Case2: "b \<notin> f ` A'"
hence "g b \<notin> A'" using 1 * by auto
hence 4: "g b \<in> A - A'" using * SUB2 by auto
hence "h(g b) \<in> B \<and> g(h(g b)) = g b"
using 3 by auto
hence "h(g b) = b" using * INJ2 unfolding inj_on_def by auto
thus ?thesis using 4 by force
qed
qed
(* *)
ultimately show ?thesis unfolding bij_betw_def by auto
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]
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]
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 {* An aside: bounded accessible part *}
text {* Finite monotone eventually stable sequences *}
lemma finite_mono_remains_stable_implies_strict_prefix:
fixes f :: "nat \<Rightarrow> 'a::order"
assumes S: "finite (range f)" "mono f" and eq: "\<forall>n. f n = f (Suc n) \<longrightarrow> f (Suc n) = f (Suc (Suc n))"
shows "\<exists>N. (\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m < f n) \<and> (\<forall>n\<ge>N. f N = f n)"
using assms
proof -
have "\<exists>n. f n = f (Suc n)"
proof (rule ccontr)
assume "\<not> ?thesis"
then have "\<And>n. f n \<noteq> f (Suc n)" by auto
then have "\<And>n. f n < f (Suc n)"
using `mono f` by (auto simp: le_less mono_iff_le_Suc)
with lift_Suc_mono_less_iff[of f]
have "\<And>n m. n < m \<Longrightarrow> f n < f m" by auto
then have "inj f"
by (auto simp: inj_on_def) (metis linorder_less_linear order_less_imp_not_eq)
with `finite (range f)` have "finite (UNIV::nat set)"
by (rule finite_imageD)
then show False by simp
qed
then obtain n where n: "f n = f (Suc n)" ..
def N \<equiv> "LEAST n. f n = f (Suc n)"
have N: "f N = f (Suc N)"
unfolding N_def using n by (rule LeastI)
show ?thesis
proof (intro exI[of _ N] conjI allI impI)
fix n assume "N \<le> n"
then have "\<And>m. N \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m = f N"
proof (induct rule: dec_induct)
case (step n) then show ?case
using eq[rule_format, of "n - 1"] N
by (cases n) (auto simp add: le_Suc_eq)
qed simp
from this[of n] `N \<le> n` show "f N = f n" by auto
next
fix n m :: nat assume "m < n" "n \<le> N"
then show "f m < f n"
proof (induct rule: less_Suc_induct[consumes 1])
case (1 i)
then have "i < N" by simp
then have "f i \<noteq> f (Suc i)"
unfolding N_def by (rule not_less_Least)
with `mono f` show ?case by (simp add: mono_iff_le_Suc less_le)
qed auto
qed
qed
lemma finite_mono_strict_prefix_implies_finite_fixpoint:
fixes f :: "nat \<Rightarrow> 'a set"
assumes S: "\<And>i. f i \<subseteq> S" "finite S"
and inj: "\<exists>N. (\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m \<subset> f n) \<and> (\<forall>n\<ge>N. f N = f n)"
shows "f (card S) = (\<Union>n. f n)"
proof -
from inj obtain N where inj: "(\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m \<subset> f n)" and eq: "(\<forall>n\<ge>N. f N = f n)" by auto
{ fix i have "i \<le> N \<Longrightarrow> i \<le> card (f i)"
proof (induct i)
case 0 then show ?case by simp
next
case (Suc i)
with inj[rule_format, of "Suc i" i]
have "(f i) \<subset> (f (Suc i))" by auto
moreover have "finite (f (Suc i))" using S by (rule finite_subset)
ultimately have "card (f i) < card (f (Suc i))" by (intro psubset_card_mono)
with Suc show ?case using inj by auto
qed
}
then have "N \<le> card (f N)" by simp
also have "\<dots> \<le> card S" using S by (intro card_mono)
finally have "f (card S) = f N" using eq by auto
then show ?thesis using eq inj[rule_format, of N]
apply auto
apply (case_tac "n < N")
apply (auto simp: not_less)
done
qed
primrec bacc :: "('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> 'a set"
where
"bacc r 0 = {x. \<forall> y. (y, x) \<notin> r}"
| "bacc r (Suc n) = (bacc r n \<union> {x. \<forall>y. (y, x) \<in> r \<longrightarrow> y \<in> bacc r n})"
lemma bacc_subseteq_acc:
"bacc r n \<subseteq> acc r"
by (induct n) (auto intro: acc.intros)
lemma bacc_mono:
"n \<le> m \<Longrightarrow> bacc r n \<subseteq> bacc r m"
by (induct rule: dec_induct) auto
lemma bacc_upper_bound:
"bacc (r :: ('a \<times> 'a) set) (card (UNIV :: 'a::finite set)) = (\<Union>n. bacc r n)"
proof -
have "mono (bacc r)" unfolding mono_def by (simp add: bacc_mono)
moreover have "\<forall>n. bacc r n = bacc r (Suc n) \<longrightarrow> bacc r (Suc n) = bacc r (Suc (Suc n))" by auto
moreover have "finite (range (bacc r))" by auto
ultimately show ?thesis
by (intro finite_mono_strict_prefix_implies_finite_fixpoint)
(auto intro: finite_mono_remains_stable_implies_strict_prefix)
qed
lemma acc_subseteq_bacc:
assumes "finite r"
shows "acc r \<subseteq> (\<Union>n. bacc r n)"
proof
fix x
assume "x : acc r"
then have "\<exists> n. x : bacc r n"
proof (induct x arbitrary: rule: acc.induct)
case (accI x)
then have "\<forall>y. \<exists> n. (y, x) \<in> r --> y : bacc r n" by simp
from choice[OF this] obtain n where n: "\<forall>y. (y, x) \<in> r \<longrightarrow> y \<in> bacc r (n y)" ..
obtain n where "\<And>y. (y, x) : r \<Longrightarrow> y : bacc r n"
proof
fix y assume y: "(y, x) : r"
with n have "y : bacc r (n y)" by auto
moreover have "n y <= Max ((%(y, x). n y) ` r)"
using y `finite r` by (auto intro!: Max_ge)
note bacc_mono[OF this, of r]
ultimately show "y : bacc r (Max ((%(y, x). n y) ` r))" by auto
qed
then show ?case
by (auto simp add: Let_def intro!: exI[of _ "Suc n"])
qed
then show "x : (UN n. bacc r n)" by auto
qed
lemma acc_bacc_eq:
fixes A :: "('a :: finite \<times> 'a) set"
assumes "finite A"
shows "acc A = bacc A (card (UNIV :: 'a set))"
using assms by (metis acc_subseteq_bacc bacc_subseteq_acc bacc_upper_bound order_eq_iff)
subsection {* Specification package -- Hilbertized version *}
lemma exE_some: "[| Ex P ; c == Eps P |] ==> P c"
by (simp only: someI_ex)
ML_file "Tools/choice_specification.ML"
end