(* Title: HOL/Hilbert_Choice.thy Author: Lawrence C Paulson, Tobias Nipkow Copyright 2001 University of Cambridge *) section ‹Hilbert's Epsilon-Operator and the Axiom of Choice› theory Hilbert_Choice imports Wellfounded keywords "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" ("(3ϵ_./ _)" [0, 10] 10) syntax (input) "_Eps" :: "pttrn ⇒ bool ⇒ 'a" ("(3@ _./ _)" [0, 10] 10) syntax "_Eps" :: "pttrn ⇒ bool ⇒ 'a" ("(3SOME _./ _)" [0, 10] 10) translations "SOME x. P" ⇌ "CONST Eps (λx. 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 = (λx. SOME y. y ∈ A ∧ f y = x)" lemma inv_into_def2: "inv_into A f x = (SOME y. y ∈ A ∧ f y = x)" by(simp add: inv_into_def) abbreviation inv :: "('a ⇒ 'b) ⇒ ('b ⇒ 'a)" where "inv ≡ inv_into UNIV" subsection ‹Hilbert's Epsilon-operator› text ‹ Easier to apply than ‹someI› if the witness comes from an existential formula. › lemma someI_ex [elim?]: "∃x. P x ⟹ P (SOME x. P x)" apply (erule exE) apply (erule someI) done text ‹ Easier to apply than ‹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 ‹someI2› if the witness comes from an existential formula. › lemma someI2_ex: "∃a. P a ⟹ (⋀x. P x ⟹ Q x) ⟹ Q (SOME x. P x)" by (blast intro: someI2) lemma someI2_bex: "∃a∈A. P a ⟹ (⋀x. x ∈ A ∧ P x ⟹ Q x) ⟹ Q (SOME x. x ∈ A ∧ 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: "∃!x. P x ⟹ P a ⟹ (SOME x. P x) = a" by blast lemma some_eq_ex: "P (SOME x. P x) ⟷ (∃x. P x)" by (blast intro: someI) lemma some_in_eq: "(SOME x. x ∈ A) ∈ A ⟷ A ≠ {}" unfolding ex_in_conv[symmetric] by (rule some_eq_ex) lemma some_eq_trivial [simp]: "(SOME y. y = x) = x" by (rule some_equality) (rule refl) 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: "∀x. ∃y. Q x y ⟹ ∃f. ∀x. Q x (f x)" by (fast elim: someI) lemma bchoice: "∀x∈S. ∃y. Q x y ⟹ ∃f. ∀x∈S. Q x (f x)" by (fast elim: someI) lemma choice_iff: "(∀x. ∃y. Q x y) ⟷ (∃f. ∀x. Q x (f x))" by (fast elim: someI) lemma choice_iff': "(∀x. P x ⟶ (∃y. Q x y)) ⟷ (∃f. ∀x. P x ⟶ Q x (f x))" by (fast elim: someI) lemma bchoice_iff: "(∀x∈S. ∃y. Q x y) ⟷ (∃f. ∀x∈S. Q x (f x))" by (fast elim: someI) lemma bchoice_iff': "(∀x∈S. P x ⟶ (∃y. Q x y)) ⟷ (∃f. ∀x∈S. P x ⟶ Q x (f x))" by (fast elim: someI) lemma dependent_nat_choice: assumes 1: "∃x. P 0 x" and 2: "⋀x n. P n x ⟹ ∃y. P (Suc n) y ∧ Q n x y" shows "∃f. ∀n. P n (f n) ∧ Q n (f n) (f (Suc n))" proof (intro exI allI conjI) fix n define f where "f = rec_nat (SOME x. P 0 x) (λn x. SOME y. P (Suc n) y ∧ Q n x y)" then have "P 0 (f 0)" "⋀n. P n (f n) ⟹ P (Suc n) (f (Suc n)) ∧ Q n (f n) (f (Suc n))" using someI_ex[OF 1] someI_ex[OF 2] by simp_all then show "P n (f n)" "Q n (f n) (f (Suc n))" by (induct n) auto qed subsection ‹Function Inverse› lemma inv_def: "inv f = (λy. SOME x. f x = y)" by (simp add: inv_into_def) lemma inv_into_into: "x ∈ f ` A ⟹ inv_into A f x ∈ A" by (simp add: inv_into_def) (fast intro: someI2) lemma inv_identity [simp]: "inv (λa. a) = (λa. a)" by (simp add: inv_def) lemma inv_id [simp]: "inv id = id" by (simp add: id_def) lemma inv_into_f_f [simp]: "inj_on f A ⟹ x ∈ A ⟹ inv_into A f (f x) = x" by (simp add: inv_into_def inj_on_def) (blast intro: someI2) 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" by (simp add: inv_into_def) (fast intro: someI2) lemma inv_into_f_eq: "inj_on f A ⟹ x ∈ A ⟹ f x = y ⟹ inv_into A f y = x" by (erule subst) (fast intro: inv_into_f_f) 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 ⟹ ∀x. f (g x) = x ⟹ inv f = g" by (blast intro: inv_into_f_eq) text ‹But is it useful?› lemma inj_transfer: assumes inj: "inj f" and minor: "⋀y. y ∈ range f ⟹ P (inv f y)" shows "P x" proof - have "f x ∈ range f" by auto then have "P(inv f (f x))" by (rule minor) then show "P x" by (simp add: inv_into_f_f [OF inj]) qed lemma inj_iff: "inj f ⟷ inv f ∘ f = id" by (simp add: o_def fun_eq_iff) (blast intro: inj_on_inverseI inv_into_f_f) lemma inv_o_cancel[simp]: "inj f ⟹ inv f ∘ f = id" by (simp add: inj_iff) lemma o_inv_o_cancel[simp]: "inj f ⟹ g ∘ inv f ∘ 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 - from eq have "f (inv_into A f x) = f (inv_into A f y)" by simp with x y show ?thesis by (simp add: f_inv_into_f) 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 ∘ inv f = id" by (auto intro!: surjI simp: surj_f_inv_f fun_eq_iff[where 'b='a]) lemma surj_iff_all: "surj f ⟷ (∀x. f (inv f x) = x)" by (simp add: o_def surj_iff fun_eq_iff) lemma surj_imp_inv_eq: "surj f ⟹ ∀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" by (rule ext) (auto simp add: inv_into_def) lemma inv_inv_eq: "bij f ⟹ inv (inv f) = f" by (rule inv_equality) (auto simp add: bij_def surj_f_inv_f) text ‹ ‹bij (inv f)› implies little about ‹f›. Consider ‹f :: bool ⇒ bool› such that ‹f True = f False = True›. Then it ia 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 ∘ g) x = (inv_into A g ∘ 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 ∘ g) = inv g ∘ inv f" by (rule inv_equality) (auto simp add: bij_def surj_f_inv_f) lemma image_f_inv_f: "surj f ⟹ f ` (inv f ` A) = A" by (simp add: surj_f_inv_f image_comp comp_def) lemma image_inv_f_f: "inj f ⟹ inv f ` (f ` A) = A" by simp 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 ⇒ 'b) set)" and card: "card (UNIV :: 'b set) ≠ Suc 0" shows "finite (UNIV :: 'a set)" proof - let ?UNIV_b = "UNIV :: 'b set" from fin have "finite ?UNIV_b" by (rule finite_fun_UNIVD2) with card have "card ?UNIV_b ≥ Suc (Suc 0)" by (cases "card ?UNIV_b") (auto simp: card_eq_0_iff) then have "card ?UNIV_b = Suc (Suc (card ?UNIV_b - Suc (Suc 0)))" by simp then obtain b1 b2 :: 'b where b1b2: "b1 ≠ b2" by (auto simp: card_Suc_eq) from fin have fin': "finite (range (λf :: 'a ⇒ 'b. inv f b1))" by (rule finite_imageI) have "UNIV = range (λf :: 'a ⇒ 'b. inv f b1)" proof (rule UNIV_eq_I) fix x :: 'a from b1b2 have "x = inv (λy. if y = x then b1 else b2) b1" by (simp add: inv_into_def) then show "x ∈ range (λf::'a ⇒ 'b. inv f b1)" by blast qed with fin' show ?thesis by simp qed text ‹ Every infinite set contains a countable subset. More precisely we show that a set ‹S› is infinite if and only if there exists an injective function from the naturals into ‹S›. The ``only if'' direction is harder because it requires the construction of a sequence of pairwise different elements of an infinite set ‹S›. The idea is to construct a sequence of non-empty and infinite subsets of ‹S› obtained by successively removing elements of ‹S›. › lemma infinite_countable_subset: assumes inf: "¬ finite S" shows "∃f::nat ⇒ 'a. inj f ∧ range f ⊆ S" ― ‹Courtesy of Stephan Merz› proof - define Sseq where "Sseq = rec_nat S (λn T. T - {SOME e. e ∈ T})" define pick where "pick n = (SOME e. e ∈ Sseq n)" for n have *: "Sseq n ⊆ S" "¬ finite (Sseq n)" for n by (induct n) (auto simp: Sseq_def inf) then have **: "⋀n. pick n ∈ Sseq n" unfolding pick_def by (subst (asm) finite.simps) (auto simp add: ex_in_conv intro: someI_ex) with * have "range pick ⊆ S" by auto moreover have "pick n ≠ pick (n + Suc m)" for m n proof - have "pick n ∉ Sseq (n + Suc m)" by (induct m) (auto simp add: Sseq_def pick_def) with ** show ?thesis by auto qed then have "inj pick" by (intro linorder_injI) (auto simp add: less_iff_Suc_add) ultimately show ?thesis by blast qed lemma infinite_iff_countable_subset: "¬ finite S ⟷ (∃f::nat ⇒ 'a. inj f ∧ range f ⊆ S)" ― ‹Courtesy of Stephan Merz› using finite_imageD finite_subset infinite_UNIV_char_0 infinite_countable_subset by auto lemma image_inv_into_cancel: assumes surj: "f`A = A'" and sub: "B' ⊆ A'" shows "f `((inv_into A f)`B') = B'" using assms proof (auto simp: f_inv_into_f) let ?f' = "inv_into A f" fix a' assume *: "a' ∈ B'" with sub have "a' ∈ A'" by auto with surj have "a' = f (?f' a')" by (auto simp: f_inv_into_f) with * show "a' ∈ f ` (?f' ` B')" by blast qed lemma inv_into_inv_into_eq: assumes "bij_betw f A A'" and a: "a ∈ 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'" from assms have *: "bij_betw ?f' A' A" by (auto simp: bij_betw_inv_into) with a obtain a' where a': "a' ∈ A'" "?f' a' = a" unfolding bij_betw_def by force with a * have "?f'' a = a'" by (auto simp: f_inv_into_f bij_betw_def) moreover from assms a' have "f a = a'" by (auto simp: bij_betw_def) ultimately show "?f'' a = f a" by simp qed lemma inj_on_iff_surj: assumes "A ≠ {}" shows "(∃f. inj_on f A ∧ f ` A ⊆ A') ⟷ (∃g. g ` A' = A)" proof safe fix f assume inj: "inj_on f A" and incl: "f ` A ⊆ A'" let ?phi = "λa' a. a ∈ A ∧ f a = a'" let ?csi = "λa. a ∈ A" let ?g = "λa'. if a' ∈ f ` A then (SOME a. ?phi a' a) else (SOME a. ?csi a)" have "?g ` A' = A" proof show "?g ` A' ⊆ A" proof clarify fix a' assume *: "a' ∈ A'" show "?g a' ∈ A" proof (cases "a' ∈ f ` A") case True then obtain a where "?phi a' a" by blast then have "?phi a' (SOME a. ?phi a' a)" using someI[of "?phi a'" a] by blast with True show ?thesis by auto next case False with assms have "?csi (SOME a. ?csi a)" using someI_ex[of ?csi] by blast with False show ?thesis by auto qed qed next show "A ⊆ ?g ` A'" proof - have "?g (f a) = a ∧ f a ∈ A'" if a: "a ∈ A" for a proof - let ?b = "SOME aa. ?phi (f a) aa" from a have "?phi (f a) a" by auto then have *: "?phi (f a) ?b" using someI[of "?phi(f a)" a] by blast then have "?g (f a) = ?b" using a by auto moreover from inj * a have "a = ?b" by (auto simp add: inj_on_def) ultimately have "?g(f a) = a" by simp with incl a show ?thesis by auto qed then show ?thesis by force qed qed then show "∃g. g ` A' = A" by blast next fix g let ?f = "inv_into A' g" have "inj_on ?f (g ` A')" by (auto simp: inj_on_inv_into) moreover have "?f (g a') ∈ A'" if a': "a' ∈ A'" for a' proof - let ?phi = "λ b'. b' ∈ A' ∧ g b' = g a'" from a' have "?phi a'" by auto then have "?phi (SOME b'. ?phi b')" using someI[of ?phi] by blast then show ?thesis by (auto simp: inv_into_def) qed ultimately show "∃f. inj_on f (g ` A') ∧ f ` g ` A' ⊆ A'" by auto qed lemma Ex_inj_on_UNION_Sigma: "∃f. (inj_on f (⋃i ∈ I. A i) ∧ f ` (⋃i ∈ I. A i) ⊆ (SIGMA i : I. A i))" proof let ?phi = "λa i. i ∈ I ∧ a ∈ A i" let ?sm = "λa. SOME i. ?phi a i" let ?f = "λa. (?sm a, a)" have "inj_on ?f (⋃i ∈ I. A i)" by (auto simp: inj_on_def) moreover have "?sm a ∈ I ∧ a ∈ A(?sm a)" if "i ∈ I" and "a ∈ A i" for i a using that someI[of "?phi a" i] by auto then have "?f ` (⋃i ∈ I. A i) ⊆ (SIGMA i : I. A i)" by auto ultimately show "inj_on ?f (⋃i ∈ I. A i) ∧ ?f ` (⋃i ∈ I. A i) ⊆ (SIGMA i : I. A i)" by auto qed lemma inv_unique_comp: assumes fg: "f ∘ g = id" and gf: "g ∘ f = id" shows "inv f = g" using fg gf inv_equality[of g f] by (auto simp add: fun_eq_iff) 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_case_prod: "Eps (case_prod P) = (SOME xy. P (fst xy) (snd xy))" by (simp add: split_def) lemma Eps_case_prod_eq [simp]: "(SOME (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 ⟷ (∄f. ∀i. (f (Suc i), f i) ∈ r)" (is "_ ⟷ ¬ ?ex") proof assume "wf r" show "¬ ?ex" proof assume ?ex then obtain f where f: "(f (Suc i), f i) ∈ r" for i by blast from ‹wf r› have minimal: "x ∈ Q ⟹ ∃z∈Q. ∀y. (y, z) ∈ r ⟶ y ∉ Q" for x Q by (auto simp: wf_eq_minimal) let ?Q = "{w. ∃i. w = f i}" fix n have "f n ∈ ?Q" by blast from minimal [OF this] obtain j where "(y, f j) ∈ r ⟹ y ∉ ?Q" for y by blast with this [OF ‹(f (Suc j), f j) ∈ r›] have "f (Suc j) ∉ ?Q" by simp then show False by blast qed next assume "¬ ?ex" then show "wf r" proof (rule contrapos_np) assume "¬ wf r" then obtain Q x where x: "x ∈ Q" and rec: "z ∈ Q ⟹ ∃y. (y, z) ∈ r ∧ y ∈ Q" for z by (auto simp add: wf_eq_minimal) obtain descend :: "nat ⇒ 'a" where descend_0: "descend 0 = x" and descend_Suc: "descend (Suc n) = (SOME y. y ∈ Q ∧ (y, descend n) ∈ r)" for n by (rule that [of "rec_nat x (λ_ rec. (SOME y. y ∈ Q ∧ (y, rec) ∈ r))"]) simp_all have descend_Q: "descend n ∈ Q" for n proof (induct n) case 0 with x show ?case by (simp only: descend_0) next case Suc then show ?case by (simp only: descend_Suc) (rule someI2_ex; use rec in blast) qed have "(descend (Suc i), descend i) ∈ r" for i by (simp only: descend_Suc) (rule someI2_ex; use descend_Q rec in blast) then show "∃f. ∀i. (f (Suc i), f i) ∈ r" by blast qed qed lemma wf_no_infinite_down_chainE: assumes "wf r" obtains k where "(f (Suc k), f k) ∉ r" using assms wf_iff_no_infinite_down_chain[of r] by blast text ‹A dynamically-scoped fact for TFL› lemma tfl_some: "∀P x. P x ⟶ P (Eps P)" by (blast intro: someI) subsection ‹An aside: bounded accessible part› text ‹Finite monotone eventually stable sequences› lemma finite_mono_remains_stable_implies_strict_prefix: fixes f :: "nat ⇒ 'a::order" assumes S: "finite (range f)" "mono f" and eq: "∀n. f n = f (Suc n) ⟶ f (Suc n) = f (Suc (Suc n))" shows "∃N. (∀n≤N. ∀m≤N. m < n ⟶ f m < f n) ∧ (∀n≥N. f N = f n)" using assms proof - have "∃n. f n = f (Suc n)" proof (rule ccontr) assume "¬ ?thesis" then have "⋀n. f n ≠ f (Suc n)" by auto with ‹mono f› have "⋀n. f n < f (Suc n)" by (auto simp: le_less mono_iff_le_Suc) with lift_Suc_mono_less_iff[of f] have *: "⋀n m. n < m ⟹ f n < f m" by auto have "inj f" proof (intro injI) fix x y assume "f x = f y" then show "x = y" by (cases x y rule: linorder_cases) (auto dest: *) qed 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)" .. define N where "N = (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 ≤ n" then have "⋀m. N ≤ m ⟹ m ≤ n ⟹ f m = f N" proof (induct rule: dec_induct) case base then show ?case by simp next case (step n) then show ?case using eq [rule_format, of "n - 1"] N by (cases n) (auto simp add: le_Suc_eq) qed from this[of n] ‹N ≤ n› show "f N = f n" by auto next fix n m :: nat assume "m < n" "n ≤ N" then show "f m < f n" proof (induct rule: less_Suc_induct) case (1 i) then have "i < N" by simp then have "f i ≠ 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) next case 2 then show ?case by simp qed qed qed lemma finite_mono_strict_prefix_implies_finite_fixpoint: fixes f :: "nat ⇒ 'a set" assumes S: "⋀i. f i ⊆ S" "finite S" and ex: "∃N. (∀n≤N. ∀m≤N. m < n ⟶ f m ⊂ f n) ∧ (∀n≥N. f N = f n)" shows "f (card S) = (⋃n. f n)" proof - from ex obtain N where inj: "⋀n m. n ≤ N ⟹ m ≤ N ⟹ m < n ⟹ f m ⊂ f n" and eq: "∀n≥N. f N = f n" by atomize auto have "i ≤ N ⟹ i ≤ card (f i)" for i proof (induct i) case 0 then show ?case by simp next case (Suc i) with inj [of "Suc i" i] have "(f i) ⊂ (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 inj show ?case by auto qed then have "N ≤ card (f N)" by simp also have "… ≤ 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 [of N] apply auto apply (case_tac "n < N") apply (auto simp: not_less) done qed subsection ‹More on injections, bijections, and inverses› locale bijection = fixes f :: "'a ⇒ 'a" assumes bij: "bij f" begin lemma bij_inv: "bij (inv f)" using bij by (rule bij_imp_bij_inv) lemma surj [simp]: "surj f" using bij by (rule bij_is_surj) lemma inj: "inj f" using bij by (rule bij_is_inj) lemma surj_inv [simp]: "surj (inv f)" using inj by (rule inj_imp_surj_inv) lemma inj_inv: "inj (inv f)" using surj by (rule surj_imp_inj_inv) lemma eqI: "f a = f b ⟹ a = b" using inj by (rule injD) lemma eq_iff [simp]: "f a = f b ⟷ a = b" by (auto intro: eqI) lemma eq_invI: "inv f a = inv f b ⟹ a = b" using inj_inv by (rule injD) lemma eq_inv_iff [simp]: "inv f a = inv f b ⟷ a = b" by (auto intro: eq_invI) lemma inv_left [simp]: "inv f (f a) = a" using inj by (simp add: inv_f_eq) lemma inv_comp_left [simp]: "inv f ∘ f = id" by (simp add: fun_eq_iff) lemma inv_right [simp]: "f (inv f a) = a" using surj by (simp add: surj_f_inv_f) lemma inv_comp_right [simp]: "f ∘ inv f = id" by (simp add: fun_eq_iff) lemma inv_left_eq_iff [simp]: "inv f a = b ⟷ f b = a" by auto lemma inv_right_eq_iff [simp]: "b = inv f a ⟷ f b = a" by auto end lemma infinite_imp_bij_betw: assumes infinite: "¬ finite A" shows "∃h. bij_betw h A (A - {a})" proof (cases "a ∈ A") case False then have "A - {a} = A" by blast then show ?thesis using bij_betw_id[of A] by auto next case True with infinite have "¬ finite (A - {a})" by auto with infinite_iff_countable_subset[of "A - {a}"] obtain f :: "nat ⇒ 'a" where 1: "inj f" and 2: "f ` UNIV ⊆ A - {a}" by blast define g where "g n = (if n = 0 then a else f (Suc n))" for n define A' where "A' = g ` UNIV" have *: "∀y. f y ≠ a" using 2 by blast have 3: "inj_on g UNIV ∧ g ` UNIV ⊆ A ∧ a ∈ g ` UNIV" apply (auto simp add: True g_def [abs_def]) apply (unfold inj_on_def) apply (intro ballI impI) apply (case_tac "x = 0") apply (auto simp add: 2) proof - fix y assume "a = (if y = 0 then a else f (Suc y))" then show "y = 0" by (cases "y = 0") (use * in auto) next fix x y assume "f (Suc x) = (if y = 0 then a else f (Suc y))" with 1 * show "x = y" by (cases "y = 0") (auto simp: inj_on_def) next fix n from 2 show "f (Suc n) ∈ A" by blast qed then have 4: "bij_betw g UNIV A' ∧ a ∈ A' ∧ A' ⊆ A" using inj_on_imp_bij_betw[of g] by (auto simp: A'_def) then have 5: "bij_betw (inv g) A' UNIV" by (auto simp add: bij_betw_inv_into) from 3 obtain n where n: "g n = a" by auto have 6: "bij_betw g (UNIV - {n}) (A' - {a})" by (rule bij_betw_subset) (use 3 4 n in ‹auto simp: image_set_diff A'_def›) define v where "v m = (if m < n then m else Suc m)" for m have 7: "bij_betw v UNIV (UNIV - {n})" proof (unfold bij_betw_def inj_on_def, intro conjI, clarify) fix m1 m2 assume "v m1 = v m2" then show "m1 = m2" apply (cases "m1 < n") apply (cases "m2 < n") apply (auto simp: inj_on_def v_def [abs_def]) apply (cases "m2 < n") apply auto done next show "v ` UNIV = UNIV - {n}" proof (auto simp: v_def [abs_def]) fix m assume "m ≠ n" assume *: "m ∉ Suc ` {m'. ¬ m' < n}" have False if "n ≤ m" proof - from ‹m ≠ n› that have **: "Suc n ≤ m" by auto from Suc_le_D [OF this] obtain m' where m': "m = Suc m'" .. with ** have "n ≤ m'" by auto with m' * show ?thesis by auto qed then show "m < n" by force qed qed define h' where "h' = g ∘ v ∘ (inv g)" with 5 6 7 have 8: "bij_betw h' A' (A' - {a})" by (auto simp add: bij_betw_trans) define h where "h b = (if b ∈ A' then h' b else b)" for b then have "∀b ∈ A'. h b = h' b" by simp with 8 have "bij_betw h A' (A' - {a})" using bij_betw_cong[of A' h] by auto moreover have "∀b ∈ A - A'. h b = b" by (auto simp: h_def) then have "bij_betw h (A - A') (A - A')" using bij_betw_cong[of "A - A'" h id] bij_betw_id[of "A - A'"] by auto moreover from 4 have "(A' ∩ (A - A') = {} ∧ A' ∪ (A - A') = A) ∧ ((A' - {a}) ∩ (A - A') = {} ∧ (A' - {a}) ∪ (A - A') = A - {a})" by blast ultimately have "bij_betw h A (A - {a})" using bij_betw_combine[of h A' "A' - {a}" "A - A'" "A - A'"] by simp then show ?thesis by blast qed lemma infinite_imp_bij_betw2: assumes "¬ finite A" shows "∃h. bij_betw h A (A ∪ {a})" proof (cases "a ∈ A") case True then have "A ∪ {a} = A" by blast then show ?thesis using bij_betw_id[of A] by auto next case False let ?A' = "A ∪ {a}" from False have "A = ?A' - {a}" by blast moreover from assms have "¬ finite ?A'" by auto ultimately obtain f where "bij_betw f ?A' A" using infinite_imp_bij_betw[of ?A' a] by auto then have "bij_betw (inv_into ?A' f) A ?A'" by (rule bij_betw_inv_into) then show ?thesis by auto qed lemma bij_betw_inv_into_left: "bij_betw f A A' ⟹ a ∈ A ⟹ inv_into A f (f a) = a" unfolding bij_betw_def by clarify (rule inv_into_f_f) lemma bij_betw_inv_into_right: "bij_betw f A A' ⟹ a' ∈ A' ⟹ f (inv_into A f a') = a'" unfolding bij_betw_def using f_inv_into_f by force lemma bij_betw_inv_into_subset: "bij_betw f A A' ⟹ B ⊆ A ⟹ f ` B = B' ⟹ bij_betw (inv_into A f) B' B" by (auto simp: bij_betw_def intro: inj_on_inv_into) 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