| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/Isabelle/Archive-of-Formal-Proofs/thys/Stable_Matching/ (Archive of formal Proofs Version 2026-5©) Datei vom 29.4.2026 mit Größe 66 kB |
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Quelle COP.thy Sprache: Isabelle |
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(*<*) theory COP imports Contracts begin (*>*) section‹ 🚫‹"HatfieldKojima:2010"›: Substitutes and stability for matching with text‹ 🚫‹"HatfieldKojima:2010"› set about weakening @{const "substitutes"} therefore making the @{emph ‹cumulative offer processes›} (COPs, S\ref{sec:contracts-cop}) applicable to more matching problems. In so they lose the lattice structure of the stable matches, which redeveloping the results of \S\ref{sec:contracts}. contrast to the COP of \S\ref{sec:contracts-cop}, 🚫‹"HatfieldKojima:2010"› develop and analyze a @{emph ‹single-offer›} variant, where only one doctor (who has held contract) proposes per round. The order of doctors making is not specified. We persist with the simultaneous-offer COP as is deterministic. See 🚫‹"HirataKasuya:2014"› for equivalence . begin with some observations due to citeauthor{AygunSonmez:2012-WP}. Firstly, as for the -with-contracts setting of \S\ref{sec:contracts}, 🚫‹"AygunSonmez:2012-WP"› demonstrate that these results depend on preferences satisfying @{const "irc"}. We do not formalize examples. Secondly, an alternative to hospitals having choice (as we have up to now) is for the hospitals to have orders over sets, which is suggested by both 🚫‹"HatfieldMilgrom:2005"› (weakly) and 🚫‹"HatfieldKojima:2010"›. 🚫‹‹\S2› in "AygunSonmez:2012-WP"› argue that this approach is -specified and propose to define ‹Ch› as choosing maximal elements of some non-strict preference order (i.e., indifference). They then claim that this is equivalent to ‹Ch› as primitive, and so we continue down that . › subsection‹ Theorem~1: the COP yields a stable match under @{emph ‹bilateral sub text‹ weakest replacement condition suggested by 🚫‹‹\S1› in "HatfieldKojima:2010"› for the @{const "substitutes"} on hospital choice functions is termed @{emph ‹bilateral ›}: begin{quote} are @{emph ‹bilateral substitutes›} for a hospital if there are two contracts ‹x› and ‹z› and a set of ‹Y› with other doctors than those associated ‹x› and ‹z› such that the hospital that ‹Y› as available wants to sign ‹z› and only if ‹x› becomes available. In other words, are bilateral substitutes when any hospital, presented with offer from a doctor he does not currently employ, never wishes to hire another doctor he does not currently employ at a contract he rejected. end{quote} that this constraint is specific to this matching-with-contracts , unlike those of \S\ref{sec:cf}. › context Contracts begin definition bilateral_substitutes_on :: "'x set → 'x cfun → bool" where "bilateral_substitutes_on A f ⟷ ¬(∃B⊆A. ∃a b. {a, b} ⊆ A ∧ Xd a ∉ Xd ` B ∧ Xd b ∉ Xd ` B ∧ b ∉ f (B ∪ {b}) ∧ b ∈ f (B ∪ {a, b}))" abbreviation bilateral_substitutes :: "'x cfun → bool" where "bilateral_substitutes ≡ bilateral_substitutes_on UNIV" lemma bilateral_substitutes_on_def2: "bilateral_substitutes_on A f ⟷ (∀B⊆A. ∀a∈A. ∀b∈A. Xd a ∉ Xd ` B ∧ Xd b ∉ Xd ` B ∧ b ∉ f (B ∪ {b}) ⟶ b ∉ f (B (*<*) (is "?lhs ⟷ ?rhs") proof %invisible (rule iffI, clarsimp) fix B a b assume lhs: ?lhs and XXX: "B ⊆ A" "a ∈ A" "b ∈ A" "Xd a ∉ Xd ` B" "Xd b ∉ Xd ` B" "b ∉ f ( show False proof(cases "a ∈ B") case True with XXX show ?thesis by (simp add: insert_absorb) next case False with lhs XXX show ?thesis unfolding bilateral_substitutes_on_def by (cases "b ∈ B") (auto simp: insert_commute insert_absorb) qed qed (fastforce simp: bilateral_substitutes_on_def) lemmas bilateral_substitutes_onI = iffD2[OF bilateral_substitutes_on_def2, rule_forma lemmas bilateral_substitutes_onD = iffD1[OF bilateral_substitutes_on_def2, rule_forma lemmas bilateral_substitutes_def = bilateral_substitutes_on_def2[where A=UNIV, simpli lemmas bilateral_substitutesI = iffD2[OF bilateral_substitutes_def, rule_format] lemmas bilateral_substitutesD = iffD1[OF bilateral_substitutes_def, rule_format, simpl (*>*) text‹› lemma substitutes_on_bilateral_substitutes_on: assumes "substitutes_on A f" shows "bilateral_substitutes_on A f" using %invisible assms by (auto intro!: bilateral_substitutes_onI dest: substitutes_onD[ text‹ 🚫‹‹\S4, Definition~5› in "AygunSonmez:2012-WP"› give the following definition: › lemma bilateral_substitutes_on_def3: "bilateral_substitutes_on A f ⟷ (∀B⊆A. ∀a∈A. ∀b∈A. b ∉ f (B ∪ {b}) ∧ b ∈ f (B ∪ {a, b}) ⟶ Xd a ∈ Xd ` B ∨ Xd b unfolding %invisible bilateral_substitutes_on_def2 by blast end text‹ before, we define a series of locales that capture the relevant about hospital choice functions. › locale ContractsWithBilateralSubstitutes = Contracts + assumes Ch_bilateral_substitutes: "∀h. bilateral_substitutes (Ch h)" sublocale ContractsWithSubstitutes < ContractsWithBilateralSubstitutes using %invisible Ch_substitutes by unfold_locales (blast intro: substitutes_on_bilatera locale ContractsWithBilateralSubstitutesAndIRC = ContractsWithBilateralSubstitutes + ContractsWithIRC sublocale ContractsWithSubstitutesAndIRC < ContractsWithBilateralSubstitutesAndIRC by %invisible unfold_locales context ContractsWithBilateralSubstitutesAndIRC begin text‹ key difficulty in showing the stability of the result of the COP this condition 🚫‹‹Theorem~1› in "HatfieldKojima:2010"› is in that it ensures we get an @{const "allocation"}; the remainder the proof of \S\ref{sec:contracts-cop} (for a single hospital, this property is trivial) goes through unchanged. We avail of 🚫‹‹Lemma› in "HirataKasuya:2014"›, which they say is a of the proof of 🚫‹‹Theorem~1› in "HatfieldKojima:2010"›. See also 🚫‹‹Appendix~A› in "AygunSonmez:2012-WP"›. › lemma bilateral_substitutes_lemma: assumes "Xd x ∉ Xd ` Ch h X" assumes "d ∉ Xd ` Ch h X" assumes "d ≠Xd x" shows "d ∉ Xd ` Ch h (insert x X)" proof(rule notI) assume "d ∈ Xd ` Ch h (insert x X)" then obtain x' where x': "x' ∈ Ch h (insert x X)" "Xd x' = d" by blast with Ch_irc ‹d ∉ Xd ` Ch h X› have "x ∈ Ch h (insert x X)" unfolding irc_def by blast let ?X' = "{y ∈ X. Xd y ∉ {Xd x, d}}" from Ch_range ‹Xd x ∉ Xd ` Ch h X› ‹d ∉ Xd ` Ch h X› ‹d ≠Xd x› x' have "Ch h (insert x' ?X') = Ch h X" using consistencyD[OF Ch_consistency[where h=h], where B="X" and C="insert x' ?X'"] by (fastforce iff: image_iff) (* slow *) moreover from Ch_range Ch_singular ‹d ∉ Xd ` Ch h X› x' ‹x ∈ Ch h (insert x X)› have "Ch h (insert x (insert x' ?X')) = Ch h (insert x X)" using consistencyD[OF Ch_consistency[where h=h], where B="insert x X" and C="insert x ( by (clarsimp simp: insert_commute) (blast dest: inj_onD) moreover note ‹d ∉ Xd ` Ch h X› x' ultimately show False using bilateral_substitutesD[OF spec[OF Ch_bilateral_substitutes, of h], where a=x and b= qed text ‹ proof essentially adds the inductive details these earlier efforts over. It is somewhat complicated by our use of the -offer COP. › lemma bilateral_substitutes_lemma_union: assumes "Xd ` Ch h X ∩ Xd ` Y = {}" assumes "d ∉ Xd ` Ch h X" assumes "d ∉ Xd ` Y" assumes "allocation Y" shows "d ∉ Xd ` Ch h (X ∪ Y)" using %invisible finite[of Y] assms by (induct arbitrary: d) (simp_all add: bilateral_subst lemma cop_F_CH_CD_on_disjoint: assumes "cop_F_closed_inv ds fp" assumes "cop_F_range_inv ds fp" shows "Xd ` CH fp ∩ Xd ` (CD_on ds (- RH fp) - fp) = {}" using %invisible assms CH_range unfolding cop_F_range_inv_def cop_F_closed_inv_def abov by (fastforce simp: set_eq_iff mem_CD_on_Cd Cd_greatest greatest_def) text‹ key lemma shows that we effectively have @{const "substitutes"} rejected contracts, provided the relevant doctor does not have a held with the relevant hospital. Note the similarity to ~4 (\S\ref{sec:cop-theorem-4}). › lemma cop_F_RH_mono: assumes "cop_F_closed_inv ds fp" assumes "cop_F_range_inv ds fp" assumes "Xd x ∉ Xd ` Ch (Xh x) fp" assumes "x ∈ RH fp" shows "x ∈ RH (cop_F ds fp)" proof(safe) from ‹x ∈ RH fp› show "x ∈ cop_F ds fp" using cop_F_increasing by blast next assume "x ∈ CH (cop_F ds fp)" from Ch_singular ‹x ∈ CH (cop_F ds fp)› ‹x ∈ RH fp› have "Ch (Xh x) (cop_F ds fp) = Ch (Xh x) (fp ∪ (CD_on ds (-RH fp) - fp - {z. Xd unfolding cop_F_def by - (rule consistencyD[OF Ch_consistency], auto simp: mem_CH_Ch dest: Ch_range' inj_onD) with cop_F_CH_CD_on_disjoint[OF ‹cop_F_closed_inv ds fp› ‹cop_F_range_inv ds fp›] have "Xd x ∉ Xd ` Ch (Xh x) (cop_F ds fp)" by simp (rule bilateral_substitutes_lemma_union[OF _ ‹Xd x ∉ Xd ` Ch (Xh x) fp›], auto simp: CH_def CD_on_inj_on_Xd inj_on_diff) with ‹x ∈ CH (cop_F ds fp)› show False by (simp add: mem_CH_Ch) qed lemma cop_F_allocation_inv: "valid ds (λds fp. cop_F_range_inv ds fp ∧ cop_F_closed_inv ds fp) (λds fp. alloca proof(induct rule: validI) case base show ?case by (simp add: CH_simps) next case (step fp) then have "allocation (CH fp)" and "cop_F_closed_inv ds fp" and "cop_F_range_inv ds fp" by blast+ note cop_F_CH_CD_on_disjoint = cop_F_CH_CD_on_disjoint[OF ‹cop_F_closed_inv ds fp› ‹ note cop_F_RH_mono = cop_F_RH_mono[OF ‹cop_F_closed_inv ds fp› ‹cop_F_range_inv ds f show ?case proof(rule inj_onI) fix x y assume "x ∈ CH (cop_F ds fp)" and "y ∈ CH (cop_F ds fp)" and "Xd x = Xd y" show "x = y" proof(cases "Xh y = Xh x") case True with Ch_singular ‹x ∈ CH (cop_F ds fp)› ‹y ∈ CH (cop_F ds fp)› ‹Xd x = Xd show ?thesis by (fastforce simp: mem_CH_Ch dest: inj_onD) next case False note ‹Xh y ≠Xh x› from ‹x ∈ CH (cop_F ds fp)› show ?thesis proof(cases x rule: CH_cop_F_cases) case CH note ‹x ∈ CH fp› from ‹y ∈ CH (cop_F ds fp)› show ?thesis proof(cases y rule: CH_cop_F_cases) case CH note ‹y ∈ CH fp› with ‹allocation (CH fp)› ‹Xd x = Xd y› ‹x ∈ CH fp› show ?thesis by (blast dest: inj_onD) next case RH_fp note ‹y ∈ RH fp› from ‹allocation (CH fp)› ‹Xd x = Xd y› ‹Xh y ≠Xh x› ‹x ∈ CH fp› have "Xd y ∉ Xd by clarsimp (metis Ch_CH_irc_idem Ch_range' inj_on_contraD) with ‹y ∈ CH (cop_F ds fp)› ‹y ∈ RH fp› cop_F_RH_mono show ?thesis by blast next case CD_on note y' = ‹y ∈ CD_on ds (- RH fp) - fp› with cop_F_CH_CD_on_disjoint ‹Xd x = Xd y› ‹x ∈ CH fp› show ?thesis by blast qed next case RH_fp note ‹x ∈ RH fp› from ‹y ∈ CH (cop_F ds fp)› show ?thesis proof(cases y rule: CH_cop_F_cases) case CH note ‹y ∈ CH fp› from ‹allocation (CH fp)› \<open>Xd x = Xd y\<close> \<open>Xh y \<noteq> Xh x\<close> \<open>y \<in> CH fp\< by clarsimp (metis Ch_CH_irc_idem Ch_range' inj_on_contraD) with \<open>x \<in> CH (cop_F ds fp)\<close> \<open>x \<in> RH fp\<close> cop_F_RH_mono show ?thesis by bla next case RH_fp note \<open>y \<in> RH fp\<close> show ?thesis proof(cases "Xd x \<in> Xd ` Ch (Xh x) fp") case True with \<open>allocation (CH fp)\<close> \<open>Xd x = Xd y\<close> \<open>Xh y \<noteq> Xh x\<close> have "Xd y by clarsimp (metis Ch_range' inj_onD mem_CH_Ch) with \<open>y \<in> CH (cop_F ds fp)\<close> \<open>y \<in> RH fp\<close> cop_F_RH_mono show ?thesis by bla next case False note \<open>Xd x \<notin> Xd ` Ch (Xh x) fp\<close> with \<open>x \<in> CH (cop_F ds fp)\<close> \<open>x \<in> RH fp\<close> cop_F_RH_mono show ?thesis by bla qed next case CD_on note \<open>y \<in> CD_on ds (- RH fp) - fp\<close> from cop_F_CH_CD_on_disjoint \<open>Xd x = Xd y\<close> \<open>y \<in> CD_on ds (- RH fp) - fp\<close> have "Xd x \<notin> Xd ` Ch (Xh x) fp" by (auto simp: CH_def dest: Ch_range') with \<open>x \<in> CH (cop_F ds fp)\<close> \<open>x \<in> RH fp\<close> cop_F_RH_mono show ?thesis by bla qed next case CD_on note \<open>x \<in> CD_on ds (- RH fp) - fp\<close> from \<open>y \<in> CH (cop_F ds fp)\<close> show ?thesis proof(cases y rule: CH_cop_F_cases) case CH note \<open>y \<in> CH fp\<close> with cop_F_CH_CD_on_disjoint \<open>Xd x = Xd y\<close> \<open>x \<in> CD_on ds (- RH fp) - fp\<close> sho next case RH_fp note \<open>y \<in> RH fp\<close> from cop_F_CH_CD_on_disjoint \<open>Xd x = Xd y\<close> \<open>x \<in> CD_on ds (- RH fp) - fp\<close> have "Xd y \<notin> Xd ` Ch (Xh y) fp" unfolding CH_def by clarsimp (blast dest: Ch_range') with \<open>y \<in> CH (cop_F ds fp)\<close> \<open>y \<in> RH fp\<close> cop_F_RH_mono show ?thesis by bla next case CD_on note \<open>y \<in> CD_on ds (- RH fp) - fp\<close> with \<open>Xd x = Xd y\<close> \<open>x \<in> CD_on ds (- RH fp) - fp\<close> show ?thesis by (meson CD_on_inj_on_Xd DiffD1 inj_on_eq_iff) qed qed qed qed qed lemma fp_cop_F_allocation: shows "allocation (cop ds)" proof %invisible - have "invariant ds (\<lambda>ds fp. cop_F_range_inv ds fp \<and> cop_F_closed_inv ds fp \<and> allo using cop_F_range_inv cop_F_closed_inv cop_F_allocation_inv by - (rule valid_conj | blast | rule valid_pre)+ then show ?thesis by (blast dest: invariantD) qed theorem Theorem_1: shows "stable_on ds (cop ds)" proof %invisible (rule stable_onI) show "individually_rational_on ds (cop ds)" proof(rule individually_rational_onI) from fp_cop_F_allocation show "CD_on ds (cop ds) = cop ds" by (rule CD_on_closed) (blast dest: fp_cop_F_range_inv' CH_range') show "CH (cop ds) = cop ds" by (simp add: CH_irc_idem) qed show "stable_no_blocking_on ds (cop ds)" by (rule cop_stable_no_blocking_on) qed end text (in ContractsWithBilateralSubstitutesAndIRC) \<open> \<^citet>\<open>\<open>\S3.1\<close> in "HatfieldKojima:2010"\<close> provide an example that sho the traditional optimality and strategic results do not hold under @{const "bilateral_substitutes"}, which motivates looking for a stronger condition that remains weaker than @{const "substitutes"}. Their example involves two doctors, two hospitals, and five contracts. \<close> datatype X5 = Xd1 | Xd1' | Xd2 | Xd2' | Xd2'' primrec X5d :: "X5 \<Rightarrow> D2" where "X5d Xd1 = D1" | "X5d Xd1' = D1" | "X5d Xd2 = D2" | "X5d Xd2' = D2" | "X5d Xd2'' = D2" primrec X5h :: "X5 \<Rightarrow> H2" where "X5h Xd1 = H1" | "X5h Xd1' = H1" | "X5h Xd2 = H1" | "X5h Xd2' = H2" | "X5h Xd2'' = H1" primrec PX5d :: "D2 \<Rightarrow> X5 rel" where "PX5d D1 = linord_of_list [Xd1, Xd1']" | "PX5d D2 = linord_of_list [Xd2, Xd2', Xd2'']" primrec CX5h :: "H2 \<Rightarrow> X5 cfun" where "CX5h H1 A = (if {Xd1', Xd2} \<subseteq> A then {Xd1', Xd2} else if {Xd2''} \<subseteq> A then {Xd2''} else if {Xd1} \<subseteq> A then {Xd1} else if {Xd1'} \<subseteq> A then {Xd1'} else if {Xd2} \<subseteq> A then {Xd2} else {})" | "CX5h H2 A = { x . x \<in> A \<and> x = Xd2' }" (*<*) lemma X5_UNIV: shows "UNIV = set [Xd1, Xd1', Xd2, Xd2', Xd2'']" using X5.exhaust by auto lemmas X5_pow = subset_subseqs[OF subset_trans[OF subset_UNIV Set.equalityD1[OF X5_UNIV instance X5 :: finite by standard (simp add: X5_UNIV) lemma X5_ALL: shows "(\<forall>X''. P X'') \<longleftrightarrow> (\<forall>X''\<in>set ` set (subseqs [Xd1, Xd1' using X5_pow by blast lemma PX5d_linear: shows "Linear_order (PX5d d)" by (cases d) (simp_all add: linord_of_list_Linear_order) lemma PX5d_range: shows "Field (PX5d d) \<subseteq> {x. X5d x = d}" by (cases d) simp_all lemma CX5h_range: shows "CX5h h X \<subseteq> {x \<in> X. X5h x = h}" by (cases h) auto lemma CX5h_singular: shows "inj_on X5d (CX5h h X)" by (cases h) (auto intro: inj_onI) (*>*) text\<open>\<close> interpretation BSI: Contracts X5d X5h PX5d CX5h using %invisible PX5d_linear PX5d_range CX5h_range CX5h_singular by unfold_locales blast lemma CX5h_bilateral_substitutes: shows "BSI.bilateral_substitutes (CX5h h)" unfolding BSI.bilateral_substitutes_def by (cases h) (auto simp: X5_ALL) lemma CX5h_irc: shows "irc (CX5h h)" unfolding irc_def by (cases h) (auto simp: X5_ALL) interpretation BSI: ContractsWithBilateralSubstitutesAndIRC X5d X5h PX5d CX5h using %invisible CX5h_bilateral_substitutes CX5h_irc by unfold_locales blast+ text\<open> There are two stable matches in this model. \<close> (*<*) lemma Xd1_Xd2'_stable: shows "BSI.stable {Xd1, Xd2'}" proof(rule BSI.stable_onI) note image_cong_simp [cong del] note INF_cong_simp [cong] note SUP_cong_simp [cong] show "BSI.individually_rational {Xd1, Xd2'}" unfolding BSI.individually_rational_on_def BSI.CD_on_def BSI.CH_def by (auto simp: insert_commute D2_UNION H2_UNION) show "BSI.stable_no_blocking {Xd1, Xd2'}" unfolding BSI.stable_no_blocking_on_def by (auto simp: X5_ALL BSI.blocking_on_def BSI.mem_CD_on_Cd BSI.maxR_def linord_of_list_ qed lemma Xd1'_Xd2_stable: shows "BSI.stable {Xd1', Xd2}" proof(rule BSI.stable_onI) note image_cong_simp [cong del] note INF_cong_simp [cong] note SUP_cong_simp [cong] show "BSI.individually_rational {Xd1', Xd2}" unfolding BSI.individually_rational_on_def BSI.CD_on_def BSI.CH_def by (auto simp: insert_commute D2_UNION H2_UNION) show "BSI.stable_no_blocking {Xd1', Xd2}" unfolding BSI.stable_no_blocking_on_def by (auto simp: X5_ALL BSI.blocking_on_def BSI.mem_CD_on_Cd BSI.maxR_def linord_of_list_ qed (*>*) text\<open>\<close> lemma BSI_stable: shows "BSI.stable X \<longleftrightarrow> X = {Xd1, Xd2'} \<or> X = {Xd1', Xd2}" (*<*) (is "?lhs = ?rhs") proof(rule iffI) assume ?lhs then show ?rhs using X5_pow[where X=X] BSI.stable_on_allocation[OF \<open>?lhs\<close>] apply clarsimp apply (elim disjE; simp add: insert_eq_iff) apply (simp_all only: BSI.stable_on_def BSI.individually_rational_on_def BSI.stable_n apply (auto 0 1 simp: D2_UNION H2_UNION X5_ALL BSI.mem_CD_on_Cd BSI.maxR_def linord_of_lis done next assume ?rhs then show ?lhs using Xd1'_Xd2_stable Xd1_Xd2'_stable by blast qed (*>*) text (in Contracts) \<open> Therefore there is no doctor-optimal match under these preferences: \<close> lemma "\<not>(\<exists> (Y::X5 set). BSI.doctor_optimal_match UNIV Y)" unfolding BSI.doctor_optimal_match_def BSI_stable apply clarsimp apply (cut_tac X=Y in X5_pow) apply clarsimp apply (elim disjE; simp add: insert_eq_iff; simp add: X5_ALL linord_of_list_linord_of_lis done subsection\<open> Theorem~3: @{emph \<open>pareto separability\<close>} relates @{emph \<open>u text\<open> \<^citet>\<open>\<open>\S4\<close> in "HatfieldKojima:2010"\<close> proceed to define @{emph \<ope substitutes\<close>}: \begin{quote} [P]references satisfy @{emph \<open>unilateral substitutes\<close>} if whenever a hospital rejects the contract @{term "z"} when that is the only contract with @{term "Xd z"} available, it still rejects the contract @{term "z"} when the choice set expands. \end{quote} \<close> context Contracts begin definition unilateral_substitutes_on :: "'x set \<Rightarrow> 'x cfun \<Rightarrow> bool" wher "unilateral_substitutes_on A f \<longleftrightarrow> \<not>(\<exists>B\<subseteq>A. \<exists>a b. {a, b} \<subseteq> A \<and> Xd b \<noti abbreviation unilateral_substitutes :: "'x cfun \<Rightarrow> bool" where "unilateral_substitutes \<equiv> unilateral_substitutes_on UNIV" lemma unilateral_substitutes_on_def2: "unilateral_substitutes_on A f \<longleftrightarrow> (\<forall>B\<subseteq>A. \<forall>a\<in>A. \<forall>b\<in>A. Xd b \<notin> Xd ` B \<a (*<*) (is "?lhs \<longleftrightarrow> ?rhs") proof %invisible (rule iffI, clarsimp) fix B a b assume lhs: ?lhs and XXX: "B \<subseteq> A" "a \<in> A" "b \<in> A" "Xd b \<notin> Xd ` B" "b \<notin> f (insert b show False proof(cases "a \<in> B") case True with XXX show ?thesis by (simp add: insert_absorb) next case False with lhs XXX show ?thesis unfolding unilateral_substitutes_on_def by (cases "b \<in> B") (auto simp: insert_commute insert_absorb) qed qed (fastforce simp: unilateral_substitutes_on_def) lemmas %invisible unilateral_substitutes_onI = iffD2[OF unilateral_substitutes_on_def lemmas %invisible unilateral_substitutes_onD = iffD1[OF unilateral_substitutes_on_def lemmas %invisible unilateral_substitutes_def = unilateral_substitutes_on_def2[where A lemmas %invisible unilateral_substitutesI = iffD2[OF unilateral_substitutes_def, rule_ lemmas %invisible unilateral_substitutesD = iffD1[OF unilateral_substitutes_def, rule_ (*>*) text\<open> \<^citet>\<open>\<open>\S4, Definition~6\<close> in "AygunSonmez:2012-WP"\<close> give the follo \<close> lemma unilateral_substitutes_on_def3: "unilateral_substitutes_on A f \<longleftrightarrow> (\<forall>B\<subseteq>A. \<forall>a\<in>A. \<forall>b\<in>A. b \<notin> f (B \<unio (*<*) unfolding %invisible unilateral_substitutes_on_def2 by blast lemmas %invisible unilateral_substitutes_def3 = unilateral_substitutes_on_def3[where lemmas %invisible unilateral_substitutesD3 = iffD1[OF unilateral_substitutes_def3, rul (*>*) text\<open>\<close> lemma substitutes_on_unilateral_substitutes_on: assumes "substitutes_on A f" shows "unilateral_substitutes_on A f" using %invisible assms by (auto intro!: unilateral_substitutes_onI dest: substitutes_onD lemma unilateral_substitutes_on_bilateral_substitutes_on: assumes "unilateral_substitutes_on A f" shows "bilateral_substitutes_on A f" using %invisible assms by (auto intro!: bilateral_substitutes_onI dest: unilateral_subst text\<open> The following defines locales for the @{const "unilateral_substitutes"} hypothesis, and inserts these between those for @{const "substitutes"} and @{const "bilateral_substitutes"}. \<close> end locale ContractsWithUnilateralSubstitutes = Contracts + assumes Ch_unilateral_substitutes: "\<forall>h. unilateral_substitutes (Ch h)" sublocale ContractsWithUnilateralSubstitutes < ContractsWithBilateralSubstitutes using %invisible Ch_unilateral_substitutes by unfold_locales (blast intro: unilateral_s sublocale ContractsWithSubstitutes < ContractsWithUnilateralSubstitutes using %invisible Ch_substitutes by unfold_locales (blast intro: substitutes_on_unilater locale ContractsWithUnilateralSubstitutesAndIRC = ContractsWithUnilateralSubstitutes + ContractsWithIRC sublocale ContractsWithUnilateralSubstitutesAndIRC < ContractsWithBilateralSubstitut by %invisible unfold_locales sublocale ContractsWithSubstitutesAndIRC < ContractsWithUnilateralSubstitutesAndIRC by %invisible unfold_locales text (in Contracts) \<open> \<^citet>\<open>\<open>Theorem~3\<close> in "HatfieldKojima:2010"\<close> relate @{const "unilateral_substitutes"} to @{const "substitutes"} using @{emph \<open>Pareto separability\<close>}: \begin{quote} Preferences are @{emph \<open>Pareto separable\<close>} for a hospital if the hospital's choice between \<open>x\<close> and \<open>x'\<close>, two [distinct] contracts with the same doctor, does not depend on what other contracts the hospital has access to. \end{quote} This result also depends on @{const "irc"}. \<close> context Contracts begin definition pareto_separable_on :: "'x set \<Rightarrow> bool" where "pareto_separable_on A \<longleftrightarrow> (\<forall>B\<subseteq>A. \<forall>C\<subseteq>A. \<forall>a b. {a, b} \<subset \<and> a \<in> Ch (Xh b) (B \<union> {a, b}) \<longrightarrow> b \<notin> Ch (Xh b) (C \<union> {a, b}))" abbreviation pareto_separable :: "bool" where "pareto_separable \<equiv> pareto_separable_on UNIV" lemmas %invisible pareto_separable_def = pareto_separable_on_def[where A=UNIV, simplif lemmas %invisible pareto_separable_onI = iffD2[OF pareto_separable_on_def, rule_format lemmas %invisible pareto_separable_onD = iffD1[OF pareto_separable_on_def, rule_format lemma substitutes_on_pareto_separable_on: assumes "\<forall>h. substitutes_on A (Ch h)" shows "pareto_separable_on A" proof(rule pareto_separable_onI) fix B C a b assume XXX: "B \<subseteq> A" "C \<subseteq> A" "a \<in> A" "b \<in> A" "a \<noteq> b" "Xd a = Xd b" "Xh a = Xh b" "a note Ch_iiaD = iia_onD[OF iffD1[OF substitutes_iia spec[OF \<open>\<forall>h. substitutes_on from XXX have "a \<in> Ch (Xh b) {a, b}" by (fast elim: Ch_iiaD) with XXX have "b \<notin> Ch (Xh b) {a, b}" by (meson Ch_singular inj_on_eq_iff) with XXX have "b \<notin> Ch (Xh b) (C \<union> {a, b})" by (blast dest: Ch_iiaD) with XXX show "b \<notin> Ch (Xh b) (insert a (insert b C))" by simp qed lemma unilateral_substitutes_on_pareto_separable_on_substitutes_on: assumes "\<forall>h. unilateral_substitutes_on A (Ch h)" assumes "\<forall>h. irc_on A (Ch h)" assumes "pareto_separable_on A" shows "substitutes_on A (Ch h)" proof(rule substitutes_onI) fix B a b assume XXX: "B \<subseteq> A" "a \<in> A" "b \<in> A" "b \<notin> Ch h (insert b B)" show "b \<notin> Ch h (insert a (insert b B))" proof(cases "Xd b \<in> Xd ` B") case True show ?thesis proof(cases "Xd b \<in> Xd ` Ch h (insert b B)") case True then obtain x where "x \<in> Ch h (insert b B)" "Xd x = Xd b" by force moreover with XXX have "x \<in> B" "x \<noteq> b" using Ch_range by blast+ moreover note \<open>pareto_separable_on A\<close> XXX ultimately show ?thesis using pareto_separable_onD[where A=A and B="B - {x}" and a=x and b=b and C="insert a (B - {x})"] by (cases "Xh b = h") (auto simp: insert_commute insert_absorb) next case False let ?B' = "{x \<in> B . Xd x \<noteq> Xd b}" from False have "b \<notin> Ch h (insert b B)" by blast with \<open>\<forall>h. irc_on A (Ch h)\<close> XXX False have "b \<notin> Ch h (insert b ?B')" using consistency_onD[OF irc_on_consistency_on[where f="Ch h"], where B="insert b B" and C by (fastforce iff: image_iff) with \<open>\<forall>h. unilateral_substitutes_on A (Ch h)\<close> XXX False have "b \<notin> Ch h (i using unilateral_substitutes_onD[where f="Ch h" and B="?B'"] by (fastforce iff: image_iff) with \<open>\<forall>h. irc_on A (Ch h)\<close> XXX False show ?thesis using consistency_onD[OF irc_on_consistency_on[where f="Ch h"], where A=A and B="insert a (insert b B)" and C="insert a (insert b ?B')"] Ch_range'[of _ h "insert a (insert b B)"] Ch_singular by simp (blast dest: inj_on_contraD) qed next case False with \<open>\<forall>h. unilateral_substitutes_on A (Ch h)\<close> XXX show ?thesis by (blast des qed qed theorem Theorem_3: assumes "\<forall>h. irc_on A (Ch h)" shows "(\<forall>h. substitutes_on A (Ch h)) \<longleftrightarrow> (\<forall>h. unilateral_sub using %invisible assms unilateral_substitutes_on_pareto_separable_on_substitutes_on substitutes_on_paret substitutes_on_unilateral_substitutes_on by blast end subsubsection\<open> \<^citet>\<open>"AfacanTurhan:2015"\<close>: @{emph \<open>doctor separab context Contracts begin text \<open> \<^citet>\<open>\<open>Theorem~1\<close> in "AfacanTurhan:2015"\<close> relate @{const "bilateral_substitutes"} and @{const "unilateral_substitutes"} using @{emph \<open>doctor separability\<close>}: \begin{quote} @{emph \<open>[Doctor separability (DS)]\<close>} says that if a doctor is not chosen from a set of contracts in the sense that no contract of him is selected, then that doctor should still not be chosen unless a contract of a new doctor (that is, doctor having no contract in the given set of contracts) becomes available. For practical purposes, we can consider DS as capturing contracts where certain groups of doctors are substitutes. [footnote: If \<open>Xd x \<notin> Xd ` Ch h (Y \<union> {x, z})\<close>, then doctor \<open>Xd x\<close> is not chosen. And under DS, he continues not to be chosen unless a new doctor comes. Hence, we can interpret it as the doctors in the given set of contracts are substitutes.] \end{quote} \<close> definition doctor_separable_on :: "'x set \<Rightarrow> 'x cfun \<Rightarrow> bool" where "doctor_separable_on A f \<longleftrightarrow> (\<forall>B\<subseteq>A. \<forall>a b c. {a, b, c} \<subseteq> A \<and> Xd a \<noteq> \<longrightarrow> Xd a \<notin> Xd ` f (B \<union> {a, b, c}))" abbreviation doctor_separable :: "'x cfun \<Rightarrow> bool" where "doctor_separable \<equiv> doctor_separable_on UNIV" (*<*) lemmas doctor_separable_def = doctor_separable_on_def[where A=UNIV, simplified] lemmas doctor_separable_onI = iffD2[OF doctor_separable_on_def, rule_format, simplifie lemmas doctor_separable_onD = iffD1[OF doctor_separable_on_def, rule_format, simplifie (*>*) lemma unilateral_substitutes_on_doctor_separable_on: assumes "unilateral_substitutes_on A f" assumes "irc_on A f" assumes "\<forall>B\<subseteq>A. allocation (f B)" assumes "f_range_on A f" shows "doctor_separable_on A f" proof(rule doctor_separable_onI) fix B a b c assume XXX: "B \<subseteq> A" "a \<in> A" "b \<in> A" "c \<in> A" "Xd a \<noteq> Xd b" "Xd b = Xd c" "Xd a \<notin> Xd ` have "a \<notin> f (insert a (insert b (insert c B)))" proof(rule notI) assume a: "a \<in> f (insert a (insert b (insert c B)))" let ?C = "{x \<in> B . Xd x \<noteq> Xd a \<or> x = a}" from \<open>irc_on A f\<close> \<open>f_range_on A f\<close> XXX(1-3,7) have "f (insert a (insert b B)) = f (insert a (insert b ?C))" by - (rule consistency_onD[OF irc_on_consistency_on[where A=A and f=f]]; fastforce dest: f_range_onD[where A=A and f=f and B="insert a (insert b B)"] simp: rev_image_ with \<open>unilateral_substitutes_on A f\<close> XXX have abcC: "a \<notin> f (insert a (insert b (insert c ?C)))" using unilateral_substitutes_onD[where A=A and f=f and a=c and b=a and B="insert b ?C - {a}"] by (force simp: insert_commute) from \<open>irc_on A f\<close> \<open>\<forall>B\<subseteq>A. allocation (f B)\<close> \<open>f_range_ have "f (insert a (insert b (insert c B))) = f (insert a (insert b (insert c ?C)))" by - (rule consistency_onD[OF irc_on_consistency_on[where A=A and f=f]], (auto)[4], clarsimp, rule conjI, blast dest!: f_range_onD'[where A=A], metis inj_on_contraD insert_s with a abcC show False by simp qed moreover have "a' \<notin> f (insert a (insert b (insert c B)))" if a': "a' \<in> B" "Xd a' = Xd a" for a' proof(rule notI) assume a'X: "a' \<in> f (insert a (insert b (insert c B)))" let ?B = "insert a B - {a'}" from XXX a' have XXX_7': "Xd a \<notin> Xd ` f (insert a' (insert b ?B))" by clarsimp (metis imageI insert_Diff_single insert_absorb insert_commute) let ?C = "{x \<in> ?B . Xd x \<noteq> Xd a \<or> x = a'}" from \<open>irc_on A f\<close> \<open>f_range_on A f\<close> XXX(1-3) a' XXX_7' have "f (insert a' (insert b ?B)) = f (insert a' (insert b ?C))" by - (rule consistency_onD[OF irc_on_consistency_on[where A=A and f=f]]; fastforce dest: f_range_onD[where A=A and f=f and B="insert a' (insert b ?B)"] simp: rev_imag with \<open>unilateral_substitutes_on A f\<close> XXX(1-6) XXX_7' a' have abcC: "a' \<notin> f (insert a' (insert b (insert c ?C)))" using unilateral_substitutes_onD[where A=A and f=f and a=c and b=a' and B="insert b ?C - {a'}" by (force simp: insert_commute rev_image_eqI) have "f (insert a' (insert b (insert c ?B))) = f (insert a' (insert b (insert c ?C)))" proof(rule consistency_onD[OF irc_on_consistency_on[where A=A and f=f]]) from a' have "insert a' (insert b (insert c ?B)) = insert a (insert b (insert c B))" by blast with \<open>\<forall>B\<subseteq>A. allocation (f B)\<close> \<open>f_range_on A f\<close> XXX(1-4) a show "f (insert a' (insert b (insert c ?B))) \<subseteq> insert a' (insert b (insert c {x \<in> ?B. Xd x by clarsimp (rule conjI, blast dest!: f_range_onD'[where A=A], metis inj_on_contraD insert qed (use \<open>irc_on A f\<close> XXX(1-4) a' in auto) with a' a'X abcC show False by simp (metis insert_Diff insert_Diff_single insert_commute) qed moreover note \<open>f_range_on A f\<close> XXX ultimately show "Xd a \<notin> Xd ` f (insert a (insert b (insert c B)))" by (fastforce dest: f_range_onD[where B="insert a (insert b (insert c B))"]) qed lemma bilateral_substitutes_on_doctor_separable_on_unilateral_substitutes_on: assumes "bilateral_substitutes_on A f" assumes "doctor_separable_on A f" assumes "f_range_on A f" shows "unilateral_substitutes_on A f" proof(rule unilateral_substitutes_onI) fix B a b assume XXX: "B \<subseteq> A" "a \<in> A" "b \<in> A" "Xd b \<notin> Xd ` B" "b \<notin> f (insert b B)" show "b \<notin> f (insert a (insert b B))" proof(cases "Xd a \<in> Xd ` B") case True then obtain C c where Cc: "B = insert c C" "c \<notin> C" "Xd c = Xd a" by (metis Set.set_insert image_i from \<open>b \<notin> f (insert b B)\<close> Cc have "b \<notin> f (insert b (insert c C))" by simp with \<open>f_range_on A f\<close> XXX Cc have "Xd b \<notin> Xd ` f (insert b (insert c C))" by clarsimp (metis f_range_onD' image_eqI insertE insert_subset) with \<open>doctor_separable_on A f\<close> XXX Cc show ?thesis by (auto simp: insert_commute dest: doctor_separable_onD) qed (use \<open>bilateral_substitutes_on A f\<close> XXX in \<open>simp add: bilateral_substitu qed theorem unilateral_substitutes_on_doctor_separable_on_bilateral_substitutes_on: assumes "irc_on A f" assumes "\<forall>B\<subseteq>A. allocation (f B)" \<comment> \<open>A rephrasing of @{thm [source assumes "f_range_on A f" shows "unilateral_substitutes_on A f \<longleftrightarrow> bilateral_substitutes_on A f \<a using %invisible assms unilateral_substitutes_on_bilateral_substitutes_on unilateral_substitutes_on_doctor_separable_on bilateral_substitutes_on_doctor_separable_on_unilateral_substitutes_on by metis text\<open> \<^citet>\<open>\<open>Remark~2\<close> in "AfacanTurhan:2015"\<close> observe the independence @{const "doctor_separable"}, @{const "pareto_separable"} and @{const "bilateral_substitutes"} conditions. \<close> end subsection\<open> Theorems~4 and 5: Doctor optimality \<close> context ContractsWithUnilateralSubstitutesAndIRC begin text \<open> We return to analyzing the COP following \<^citet>\<open>"HatfieldKojima:2010"\<close>. The next goal is to establish a doctor-optimality result for it in the spirit of \S\ref{sec:contracts-optimality}. We first show that, with hospital choice functions satisfying @{const "unilateral_substitutes"}, we effectively have the @{const "substitutes"} condition for all contracts that have been rejected. In other words, hospitals never renegotiate with doctors. The proof is by induction over the finite set @{term "Y"}. \<close> lemma assumes "Xd x \<notin> Xd ` Ch h X" assumes "x \<in> X" shows no_renegotiation_union: "x \<notin> Ch h (X \<union> Y)" and "x \<notin> Ch h (insert x ((X \<union> Y) - {z. Xd z = Xd x}))" using %invisible finite[of Y] proof induct case empty { case 1 from \<open>Xd x \<notin> Xd ` Ch h X\<close> show ?case by clarsimp } { case 2 from assms show ?case by - (clarsimp, drule subsetD[OF equalityD2[OF consistencyD[OF Ch_consistency[of h], wher next case (insert y Y) { case 1 show ?case proof(cases "y \<in> Ch h (insert y (X \<union> Y))") case True note \<open>y \<in> Ch h (insert y (X \<union> Y))\<close> show ?thesis proof(cases "Xd y = Xd x") case True with \<open>x \<in> X\<close> \<open>x \<notin> Ch h (X \<union> Y)\<close> \<open>y \<in> Ch h (insert y ( by clarsimp (metis Ch_singular Un_iff inj_onD insert_absorb) next case False note xy = \<open>Xd y \<noteq> Xd x\<close> show ?thesis proof(rule notI) assume "x \<in> Ch h (X \<union> insert y Y)" with \<open>x \<in> X\<close> have "x \<in> Ch h (insert y (insert x ((X \<union> Y) - {z. Xd z = Xd x})))" by - (rule subsetD[OF equalityD1[OF consistencyD[OF Ch_consistency[of h]]], rotated -1], assumption, clarsimp+, metis Ch_range' Ch_singular Un_iff inj_onD insertE) with \<open>x \<notin> Ch h (insert x (X \<union> Y - {z. Xd z = Xd x}))\<close> show False by (force dest: unilateral_substitutesD3[OF spec[OF Ch_unilateral_substitutes, of h], ro qed qed next case False with \<open>x \<notin> Ch h (X \<union> Y)\<close> show ?thesis using Ch_irc unfolding irc_def by simp qed } { case 2 from insert(4) show ?case by (auto simp: insert_commute insert_Diff_if split: if_splits dest: unilateral_substitutesD3[OF spec[OF Ch_unilateral_substitutes, of h], where a=y]) qed text\<open> To discharge the first antecedent of this lemma, we need an invariant for the COP that asserts that, for each doctor @{term "d"}, there is a subset of the contracts currently offered by @{term "d"} that was previously uniformly rejected by the COP, for each contract that is rejected at the current step. To support a later theorem (see \S\ref{sec:cop-worst}) we require these subsets to be upwards-closed with respect to the doctor's preferences. \<close> definition cop_F_rejected_inv :: "'b set \<Rightarrow> 'a set \<Rightarrow> bool" where "cop_F_rejected_inv ds fp \<longleftrightarrow> (\<forall>x\<in>RH fp. \<exists>fp'\<subseteq>fp (*<*) lemmas cop_F_rejected_invI = iffD2[OF cop_F_rejected_inv_def, rule_format, simplified, (*>*) lemma cop_F_rejected_inv: shows "valid ds (\<lambda>ds fp. cop_F_range_inv ds fp \<and> cop_F_closed_inv ds fp \<and> allocat proof %invisible (induct rule: validI) case base show ?case unfolding cop_F_rejected_inv_def by simp next case (step fp) then have "cop_F_closed_inv ds fp" and "cop_F_range_inv ds fp" and "allocation (CH fp)" and "cop_F_rejected_inv ds fp" by blast+ show ?case proof(rule cop_F_rejected_invI) fix x assume x: "x \<in> cop_F ds fp" "x \<notin> CH (cop_F ds fp)" show "\<exists>fp'\<subseteq>cop_F ds fp. x \<in> fp' \<and> above (Pd (Xd x)) x \<subseteq> fp' \<and> Xd x \< proof(cases "Xd x \<in> Xd ` CH (cop_F ds fp)") case True with \<open>x \<notin> CH (cop_F ds fp)\<close> obtain y where y: "Xd y = Xd x" "y \<in> CH (cop_F ds fp)" "x \<noteq> y" by (metis imageE) from \<open>x \<in> cop_F ds fp\<close> show ?thesis proof(cases x rule: cop_F_cases) case fp note \<open>x \<in> fp\<close> from \<open>y \<in> CH (cop_F ds fp)\<close> show ?thesis proof(cases y rule: CH_cop_F_cases) case CH note \<open>y \<in> CH fp\<close> with \<open>allocation (CH fp)\<close> y(1,3) have "x \<notin> CH fp" by (metis inj_on_eq_iff) with \<open>cop_F_rejected_inv ds fp\<close> \<open>x \<in> fp\<close> show ?thesis unfolding cop_F_rejected_inv_def by (metis Diff_iff Un_upper1 cop_F_def subset_refl subs next case RH_fp note \<open>y \<in> RH fp\<close> with \<open>cop_F_rejected_inv ds fp\<close> obtain fp' where "fp' \<subseteq> fp \<and> y \<in> fp' \<and> above (Pd (Xd y)) y \<subseteq> fp' \<and> Xd y \<n unfolding cop_F_rejected_inv_def by (fastforce simp: mem_CH_Ch) with \<open>y \<in> CH (cop_F ds fp)\<close> show ?thesis using no_renegotiation_union[where x=y and X="fp'" and Y="cop_F ds fp"] cop_F_increasing by (clarsimp simp: mem_Ch_CH image_iff) (metis Un_absorb1 mem_CH_Ch subset_trans) next case CD_on note \<open>y \<in> CD_on ds (- RH fp) - fp\<close> with cop_F_increasing \<open>cop_F_closed_inv ds fp\<close> cop_F_CH_CD_on_disjoint[OF \<op unfolding cop_F_closed_inv_def by - (rule exI[where x=fp]; clarsimp; blast) qed next case CD_on { fix z assume z: "z \<in> CH (cop_F ds fp)" "Xd z = Xd x" from \<open>allocation (CH fp)\<close> x(2) z have "z \<noteq> x" by blast from z(1) have False proof(cases z rule: CH_cop_F_cases) case CH with \<open>cop_F_range_inv ds fp\<close> \<open>cop_F_closed_inv ds fp\<close> \<open>x \<in> CD_on d show False using cop_F_CH_CD_on_disjoint by blast next case RH_fp note \<open>z \<in> RH fp\<close> with \<open>cop_F_rejected_inv ds fp\<close> obtain fp' where "fp' \<subseteq> fp \<and> z \<in> fp' \<and> above (Pd (Xd z)) z \<subseteq> fp' \<and> Xd z \<n unfolding cop_F_rejected_inv_def by (fastforce simp: mem_CH_Ch) with \<open>z \<in> CH (cop_F ds fp)\<close> show ?thesis using no_renegotiation_union[where x=z and X="fp'" and Y="cop_F ds fp"] cop_F_increasing by (clarsimp simp: mem_Ch_CH image_iff) (metis Un_absorb1 mem_CH_Ch subset_trans) next case CD_on note \<open>z \<in> CD_on ds (- RH fp) - fp\<close> with z(2) \<open>x \<in> CD_on ds (- RH fp) - fp\<close> \<open>z \<noteq> x\<close> show False by (meson CD_on_inj_on_Xd DiffD1 inj_onD) qed } with True have False by clarsimp blast then show ?thesis .. qed next case False note \<open>Xd x \<notin> Xd ` CH (cop_F ds fp)\<close> with cop_F_closed_inv \<open>cop_F_closed_inv ds fp\<close> \<open>x \<in> cop_F ds fp\<close> show ?thesis by (metis cop_F_closed_inv_def subsetI valid_def) qed qed qed lemma fp_cop_F_rejected_inv: shows "cop_F_rejected_inv ds (fp_cop_F ds)" proof %invisible - have "invariant ds (\<lambda>ds fp. cop_F_range_inv ds fp \<and> cop_F_closed_inv ds fp \<and> allo using cop_F_range_inv cop_F_closed_inv cop_F_allocation_inv cop_F_rejected_inv by - (rule valid_conj | blast | rule valid_pre)+ then show ?thesis by (blast dest: invariantD) qed text\<open> \label{sec:cop-theorem-4} \<^citet>\<open>\<open>Theorem~4\<close> in "HatfieldKojima:2010"\<close> assert that we effecti recover @{const "substitutes"} for the contracts relevant to the COP. We cannot adopt their phrasing as it talks about the execution traces of the COP, and not just its final state. Instead we present the result we use, which relates two consecutive states in an execution trace of the COP: \<close> theorem Theorem_4: assumes "cop_F_rejected_inv ds fp" assumes "x \<in> RH fp" shows "x \<in> RH (cop_F ds fp)" using %invisible bspec[OF assms[unfolded cop_F_rejected_inv_def]] cop_F_increasing[of no_renegotiation_union[where x=x and h="Xh x" and Y="cop_F ds fp"] by (auto simp: mem_CH_Ch image_iff Ch_CH_irc_idem) (metis le_iff_sup mem_Ch_CH sup.coboun text\<open> \label{sec:cop-worst} Another way to interpret @{const "cop_F_rejected_inv"} is to observe that the doctor-optimal match contains the least preferred of the contracts that the doctors have offered. \<close> corollary fp_cop_F_worst: assumes "x \<in> cop ds" assumes "y \<in> fp_cop_F ds" assumes "Xd y = Xd x" shows "(x, y) \<in> Pd (Xd x)" proof %invisible (rule ccontr) assume "(x, y) \<notin> Pd (Xd x)" with assms have "(y, x) \<in> Pd (Xd x) \<and> x \<noteq> y" by (metis CH_range' Pd_linear eq_iff fp_cop_F_range_inv' order_on_defs(3) total_on_def u with assms have "y \<notin> (cop ds)" using fp_cop_F_allocation by (blast dest: inj_onD) with fp_cop_F_rejected_inv[of ds] \<open>y \<in> fp_cop_F ds\<close> obtain fp' where "fp' \<subseteq> fp_cop_F ds \<and> y \<in> fp' \<and> above (Pd (Xd y)) y \<subseteq> fp' \< unfolding cop_F_rejected_inv_def by fastforce with \<open>(y, x) \<in> Pd (Xd x) \<and> x \<noteq> y\<close> assms show False using no_renegotiation_union[where x=x and h="Xh x" and X=fp' and Y="fp_cop_F ds"] unfolding above_def by (clarsimp simp: mem_CH_Ch image_iff) (metis contra_subsetD le_iff_sup mem_Ch_CH mem_Co qed text\<open> The doctor optimality result, Theorem~5, hinges on showing that no contract in any stable match is ever rejected. \<close> definition theorem_5_inv :: "'b set \<Rightarrow> 'a set \<Rightarrow> bool" where "theorem_5_inv ds fp \<longleftrightarrow> RH fp \<inter> \<Union>{X. stable_on ds X} = {}" (*<*) lemma theorem_5_invI: assumes "\<And>z X. \<lbrakk>z \<in> RH fp; z \<in> X; stable_on ds X\<rbrakk> \<Longrightarrow> False" shows "theorem_5_inv ds fp" unfolding theorem_5_inv_def using assms by blast (*>*) lemma theorem_5_inv: shows "valid ds (\<lambda>ds fp. cop_F_range_inv ds fp \<and> cop_F_closed_inv ds fp \<and> allocation (CH fp) \<and> cop_F_rejected_inv ds fp) theorem_5_inv" proof(induct rule: validI) case base show ?case unfolding theorem_5_inv_def by simp next case (step fp) then have "cop_F_range_inv ds fp" and "cop_F_closed_inv ds fp" and "allocation (CH fp)" and "cop_F_rejected_inv ds fp" and "theorem_5_inv ds fp" by blast+ show ?case proof(rule theorem_5_invI) fix z X assume z: "z \<in> RH (cop_F ds fp)" and "z \<in> X" and "stable_on ds X" from \<open>theorem_5_inv ds fp\<close> \<open>z \<in> X\<close> \<open>stable_on ds X\<close> have z': "z \<notin> RH fp" unfolding theorem_5_inv_def by blast define Y where "Y \<equiv> Ch (Xh z) (cop_F ds fp)" from z have YYY: "z \<notin> Ch (Xh z) (insert z Y)" using consistencyD[OF Ch_consistency] by (simp add: mem_CH_Ch Y_def) (metis Ch_f_range f_range_on_def insert_subset subset_insertI top_greatest) have yRx: "(x, y) \<in> Pd (Xd y)" if "x \<in> X" and "y \<in> Y" and "Xd y = Xd x" for x y proof(rule ccontr) assume "(x, y) \<notin> Pd (Xd y)" with Pd_linear \<open>cop_F_range_inv ds fp\<close> \<open>stable_on ds X\<close> that have BBB: "(y, x) \<in> Pd (Xd y) \<and> x \<noteq> y" unfolding Y_def cop_F_def cop_F_range_inv_def order_on_defs total_on_def by (clarsimp simp: mem_CD_on_Cd dest!: Ch_range') (metis Cd_range' Int_iff refl_onD stable from \<open>stable_on ds X\<close> \<open>cop_F_closed_inv ds fp\<close> \<open>theorem_5_inv ds fp unfolding cop_F_def cop_F_closed_inv_def theorem_5_inv_def above_def Y_def by (fastforce simp: mem_CD_on_Cd dest: Ch_range' Cd_preferred) with \<open>stable_on ds X\<close> \<open>theorem_5_inv ds fp\<close> \<open>x \<in> X\<close> have "x \<in> unfolding theorem_5_inv_def by (force simp: mem_CH_Ch) with \<open>allocation (CH fp)\<close> \<open>Xd y = Xd x\<close> BBB have "y \<notin> Ch (Xh z) fp" by (metis Ch_range' inj_onD mem_CH_Ch) with \<open>y \<in> Y\<close> \<open>x \<in> fp \<and> y \<in> fp\<close> show False unfolding Y_def using Theorem_4[OF \<open>cop_F_rejected_inv ds fp\<close>, where x=y] by (metis Ch_range' Diff_iff mem_CH_Ch) qed have "Xd z \<notin> Xd ` Y" proof(safe) fix w assume w: "Xd z = Xd w" "w \<in> Y" show False proof(cases "z \<in> fp") case True note \<open>z \<in> fp\<close> show False proof(cases "w \<in> fp") case True note \<open>w \<in> fp\<close> from \<open>Xd z = Xd w\<close> \<open>w \<in> Y\<close> z' \<open>z \<in> fp\<close> have "w \<notin> CH fp" by (metis Ch_irc_idem DiffI YYY Y_def \<open>allocation (CH fp)\<close> inj_on_eq_iff insert_a with Theorem_4[OF \<open>cop_F_rejected_inv ds fp\<close>, where x=w] \<open>w \<in> Y\<close> \<open> unfolding Y_def CH_def by simp next case False note \<open>w \<notin> fp\<close> with \<open>w \<in> Y\<close> have "w \<notin> Ch (Xh z) fp \<and> w \<in> cop_F ds fp \<and> Xh w = Xh z" unfolding Y_def by (blast dest: Ch_range') with \<open>cop_F_closed_inv ds fp\<close> \<open>cop_F_range_inv ds fp\<close> \<open>z \<notin> RH f show False unfolding cop_F_closed_inv_def cop_F_range_inv_def above_def by (fastforce simp: cop_F_def mem_CD_on_Cd Cd_greatest greatest_def) qed next case False note \<open>z \<notin> fp\<close> from \<open>cop_F_range_inv ds fp\<close> \<open>cop_F_closed_inv ds fp\<close> z \<open>z \<notin> fp unfolding cop_F_range_inv_def cop_F_closed_inv_def above_def by (clarsimp simp: mem_CD_on_Cd Cd_greatest greatest_def dest!: mem_Ch_CH elim!: cop_F_ca (blast dest: CH_range') with w \<open>z \<in> RH (cop_F ds fp)\<close> \<open>z \<notin> fp\<close> show False by (clarsimp simp: Y_def cop_F_def mem_CH_Ch) (metis CD_on_inj_on_Xd Ch_range' Un_iff inj_onD no_renegotiation_union) qed qed show False proof(cases "z \<in> Ch (Xh z) (X \<union> Y)") case True note \<open>z \<in> Ch (Xh z) (X \<union> Y)\<close> with \<open>z \<in> X\<close> have "Xd z \<in> Xd ` Ch (Xh z) (insert z Y)" using no_renegotiation_union[where X="insert z Y" and Y="X - {z}" and x=z and h="Xh z"] by clarsimp (metis Un_insert_right insert_Diff Un_commute) with \<open>Xd z \<notin> Xd ` Y\<close> \<open>z \<notin> Ch (Xh z) (insert z Y)\<close> show False by (blast next case False note \<open>z \<notin> Ch (Xh z) (X \<union> Y)\<close> have "\<not>stable_on ds X" proof(rule blocking_on_imp_not_stable[OF blocking_onI]) from False \<open>z \<in> X\<close> \<open>stable_on ds X\<close> show "Ch (Xh z) (X \<union> Y) \<noteq> Ch (Xh z) X" using mem_CH_Ch stable_on_CH by blast show "Ch (Xh z) (X \<union> Y) = Ch (Xh z) (X \<union> Ch (Xh z) (X \<union> Y))" using Ch_range' by (blast intro!: consistencyD[OF Ch_consistency]) next fix x assume "x \<in> Ch (Xh z) (X \<union> Y)" with Ch_singular'[of "Xh z" "X \<union> Ch (Xh z) (cop_F ds fp)"] invariant_cop_FD[OF cop_F_range_inv \<open>cop_F_range_inv ds fp\<close>] stable_on_allocation[OF \<open>stable_on ds X\<close>] stable_on_Xd[OF \<open>stable_on ds X\< stable_on_range'[OF \<open>stable_on ds X\<close>] show "x \<in> CD_on ds (X \<union> Ch (Xh z) (X \<union> Y))" unfolding cop_F_range_inv_def by (clarsimp simp: mem_CD_on_Cd Cd_greatest greatest_def) (metis Ch_range' IntE Pd_range' Pd_refl Un_iff Y_def inj_onD yRx) qed with \<open>stable_on ds X\<close> show False by blast qed qed qed lemma fp_cop_F_theorem_5_inv: shows "theorem_5_inv ds (fp_cop_F ds)" proof %invisible - have "invariant ds (\<lambda>ds fp. cop_F_range_inv ds fp \<and> cop_F_closed_inv ds fp \<and> allo using cop_F_range_inv cop_F_closed_inv cop_F_allocation_inv cop_F_rejected_inv theore by - (rule valid_conj | blast | rule valid_pre)+ then show ?thesis by (blast dest: invariantD) qed theorem Theorem_5: assumes "stable_on ds X" assumes "x \<in> X" shows "\<exists>y \<in> cop ds. (x, y) \<in> Pd (Xd x)" proof - from fp_cop_F_theorem_5_inv assms have x: "x \<notin> RH (fp_cop_F ds)" unfolding theorem_5_inv_def by blast show ?thesis proof(cases "Xd x \<in> Xd ` cop ds") case True then obtain z where z: "z \<in> cop ds" "Xd z = Xd x" by auto show ?thesis proof(cases "(x, z) \<in> Pd (Xd x)") case True with z show ?thesis by blast next case False with Pd_linear'[where d="Xd x"] fp_cop_F_range_inv'[of z ds] assms z have "(z, x) \<in> Pd (Xd x)" unfolding order_on_defs total_on_def by (metis CH_range' refl_onD stable_on_range') with fp_cop_F_closed_inv'[of z ds x] x z have "x \<in> fp_cop_F ds" unfolding above_def by (force simp: mem_CH_Ch dest: Ch_range') with fp_cop_F_allocation x z have "z = x" by (fastforce dest: inj_onD) with Pd_linear assms z show ?thesis by (meson equalityD2 stable_on_range' underS_incl_iff) qed next case False note \<open>Xd x \<notin> Xd ` cop ds\<close> with assms x show ?thesis by (metis DiffI Diff_eq_empty_iff fp_cop_F_all emptyE imageI stable_on_Xd stable_on_rang qed qed theorem fp_cop_F_doctor_optimal_match: shows "doctor_optimal_match ds (cop ds)" by %invisible (rule doctor_optimal_matchI[OF Theorem_1 Theorem_5]) auto end text\<open> \label{sec:cop-opposition-of-interests} The next lemma demonstrates the @{emph \<open>opposition of interests\<close>} of doctors and hospitals: if all doctors weakly prefer one stable match to another, then the hospitals weakly prefer the converse. As we do not have linear preferences for hospitals, we use revealed preference and hence assume @{const "irc"} holds of hospital choice functions. Our definition of the doctor-preferred ordering @{term "dpref"} follows the Isabelle/HOL convention of putting the larger (more preferred) element on the right, and takes care with unemployment. \<close> context Contracts begin definition dpref :: "'x set \<Rightarrow> 'x set \<Rightarrow> bool" where "dpref X Y = (\<forall>x\<in>X. \<exists>y\<in>Y. (x, y) \<in> Pd (Xd x))" lemmas %invisible dprefI = iffD2[OF dpref_def, rule_format] end context ContractsWithIRC begin theorem Lemma_1: assumes "stable_on ds Y" assumes "stable_on ds Z" assumes "dpref Z Y" assumes "x \<in> Ch h Z" shows "x \<in> Ch h (Y \<union> Z)" proof(rule ccontr) assume "x \<notin> Ch h (Y \<union> Z)" from \<open>x \<in> Ch h Z\<close> \<open>x \<notin> Ch h (Y \<union> Z)\<close> have "Ch h (Y \<union> Z) \<noteq> Ch h Z" by blast moreover have "Ch h (Y \<union> Z) = Ch h (Z \<union> Ch h (Y \<union> Z))" by (rule consistency_onD[OF Ch_consistency]; auto dest: Ch_range') moreover have "y \<in> CD_on ds (Z \<union> Ch h (Y \<union> Z))" if "y \<in> Ch h (Y \<union> Z)" for y proof - from \<open>stable_on ds Y\<close> \<open>stable_on ds Z\<close> that have "Xd y \<in> ds \<and> y \<in> Field (Pd (Xd y))" using stable_on_Xd stable_on_range' Ch_range' by (meson Un_iff) with Pd_linear'[of "Xd y"] Ch_singular \<open>stable_on ds Y\<close> \<open>stable_on ds Z\<close> unfolding dpref_def by (clarsimp simp: mem_CD_on_Cd Cd_greatest greatest_def) (metis Ch_range' Pd_Xd Un_iff eq_iff inj_on_contraD stable_on_allocation underS_incl_if qed ultimately show False by (blast dest: stable_on_blocking_onD[OF \<open>stable_on ds Z\<close> qed end text (in Contracts) \<open> \<^citet>\<open>\<open>Corollary~1 (of Theorem~5 and Lemma~1)\<close> in "HatfieldKojima:2010" @{const "unilateral_substitutes"} implies there is a hospital-pessimal match, which is indeed the doctor-optimal one. \<close> context ContractsWithUnilateralSubstitutesAndIRC begin theorem Corollary_1: assumes "stable_on ds Z" shows "dpref Z (cop ds)" and "x \<in> Z \<Longrightarrow> x \<in> Ch (Xh x) (cop ds \<union> Z)" proof - show "dpref Z (cop ds)" by (rule dprefI[OF Theorem_5[OF \<open>stable_on ds Z\<close>]]) fix x assume "x \<in> Z" with assms show "x \<in> Ch (Xh x) (cop ds \<union> Z)" using Lemma_1[OF Theorem_1 assms \<open>dpref Z (cop ds)\<close>] stable_on_CH by (fastforce simp: mem_CH_Ch) qed text\<open> \<^citet>\<open>\<open>p1717\<close> in "HatfieldKojima:2010"\<close> show that there is not alway hospital-optimal/doctor-pessimal match when hospital preferences satisfy @{const "unilateral_substitutes"}, in contrast to the situation under @{const "substitutes"} (see \S\ref{sec:contracts-optimality}). This reflects the loss of the lattice structure. \<close> end subsection\<open> Theorem~6: A ``rural hospitals'' theorem \label{sec:cop-rh} \<close> text (in Contracts) \<open> \<^citet>\<open>\<open>Theorem~6\<close> in "HatfieldKojima:2010"\<close> demonstrates a ``rura hospitals'' theorem for the COP assuming hospital choice functions satisfy @{const "unilateral_substitutes"} and @{const "lad"}, as for \S\ref{sec:contracts-rh}. However \<^citet>\<open>\<open>\S4, Example~1\<close> in "AygunSonmez:2012-WP"\<close> observe that @{thm [source] "lad_on_substitutes_on_irc_on"} does not hold with @{const "bilateral_substitutes"} instead of @{const "substitutes"}, and their Example~3 similarly for @{const "unilateral_substitutes"}. Moreover @{const "fp_cop_F"} can yield an unstable allocation with just these two hypotheses. Ergo we need to assume @{const "irc"} even when we have @{const "lad"}, unlike before (see \S\ref{sec:contracts-rh}). This theorem is the foundation for all later strategic results. \<close> locale ContractsWithUnilateralSubstitutesAndIRCAndLAD = ContractsWithUnilateralSubs sublocale ContractsWithSubstitutesAndLAD < ContractsWithUnilateralSubstitutesAndIRCA using %invisible Ch_lad by unfold_locales context ContractsWithUnilateralSubstitutesAndIRCAndLAD begin context fixes ds :: "'b set" fixes X :: "'a set" assumes "stable_on ds X" java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5 text\<open) \ proofs these lemmasprovided by \<in definition of the function @{term "A"} as we \S\ref{sec:contracts-t1-converse}. \<close> lemma RHT_Cd_card: assumes "d \<in> ds" shows "card (Cd d X) \<le> card proof java.lang.StringIndexOutOfBoundsException: Range [9, 10) out of bounds for case True then obtain x where "x \<in> X" "Xd x = d" by blast with \<open>stable_on ds X\<close> have "Cd d X = {x}" using Cd_singleton mem_CD_on_Cd stable_on_CD_on by blast from Theorem_5[OF \ret < 0) \ using Cd_single Cd_singleton FieldI2 \<open>Xd x = d\<close> fp_cop_F_allocation by metis java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 33 next case False then have "Cd d X = return (,java.lang.StringIndexOutOfBoundsException: Index 48 out o using ' then show ?thesis by (bits>64 qed lemma shows "card (Ch java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds f proof - define A where "A \<equiv> \<lambda>X. {y |y. Xd y \<in> ds \<and> y \<in> Field (Pd (Xd y)) \<and> (\<forall>x \<in> have "A (cop ds) = java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds proof(rule |) fixxassume> ?lhs" ret = bch2_varint_dec(, (casesx \<in>") case True with \: -BCH_ERR_ENOENT_inode subvol_inum inum, next \open <n lhs<lose>fp_cop_F_alljava.lang.StringIndexOutOfBoundsException: Index 75 out unfolding A_def bch2_inode_find_by_inum(bch_fsinum, qed next fix x assume "x \<in> ?rhs" with fp_cop_F_worst show "x \<in> ? bch_inode_unpacked *inodejava.lang.StringIndexOutOfBo bch2_btree_insert_nonextent(trans, BTREE_ID_inodes, qed = commit_dotransNULL, BCH_TRANS_COMMIT_no_enospc, moreover int ret; proof(rule ccontr) A)\ " then have "CH java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for thenhXXX(rans, have "\<not>stable_on ds X" proof(rule blocking_on_imp_not_stable[OF blocking_onI]) show "Ch h (A X) \<noteq> Ch h X" by fact from Pd_linear \<open>stable_on ds X\<close> show "Ch unfolding A_def by - java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length next fix x assume "x \<in> Ch h (A X)" with java.lang.StringIndexOutOfBoundsException: Range [32, 22) out of bounds for unfolding A_def by (auto 9 3 simp: mem_CD_on_Cd Cd_greatest greatest_def struct btree_iter java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of intro: FieldI1) qed with \<open>stable_on ds X\<close>bch2_inode_pack(node_p qed moreover from Pd_linear Theorem_5[OF \<open> prt_printf(out, "bi_version=%llu\n", inode-); unfolding then have "card (Ch h (A (cop ds))) \<le> card (Ch h (A X))" by (fastforce intro: ladD[OFjava.lang.StringIndexOutOfBoundsException: Index 1 out ultimately show ?thesis by (metis (no_types, lifting) java.lang.StringIndexOutOfBoundsE qed text\<open> The top-level proof is the same as in \ \<close> lemma Theorem_6_fp_cop_Fs->c; shows>\> card( java.lang.StringIndexOutOfBoundsException: Range [49, 48) out of boun and " proof - java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length let ? :cnn; } have ifjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for leng using Theorem_1 stable_on_CD_on CD_on_card[symmetric] CH_card[symmetric] by simp also have "\<dots> \<le> ?Sum_Ch_X" usingRHT_Ch_card java.lang.StringIndexOutOfBoundsException: Range [36, 30) out of also have "\<dots> = ?Sum_Cd_X" using] using \<open>stable_on ds caseKEY_TYPE_inodejava.lang.StringIndexOutOfBoundsException java.lang.StringIndexOutOfBoundsException: Range [17, 9) out of bounds for lengt variatePolynomialByCoefficients(fam, with java.lang.StringIndexOutOfBoundsException: Range [26, 18) out of bounds for by (fastforce elim: sum_mono_inv) have "?Sum_Ch_X = ?Sum_Cd_X" also using RHT_Cd_card by (simp add: sum_mono) also have "… return ret; using CD_on_cardsymmetric] using Theorem_1 stable_on_CD_on stable_on_CH by auto finally have "?Sum_Ch_COP = ?Sum_Ch_X" using RHT_Ch_card by (simp add: eq_iff sum_mono) with RHT_Ch_card by (fastforce elim: sym[OF sum_mono_inv]) qed end theorem Theorem_6: assumes "stable_on ds X = assumes "stable_on ds Y" shows "d ∈ ds ==> card (Cd d X) = card (Cd d Y)" and using(parent java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds fo end subsection‹ Concluding remarks › text‹ next discuss a kind of interference between doctors termed {emph ‹bossiness›} in \S\ref{sec:bossiness}. This has some implications the strategic issues we discuss in \S\ref{sec:strategic}. › (*<*) end (*>*)
¤ Dauer der Verarbeitung: 0.21 Sekunden (vorverarbeitet am 2026-11-04) ¤*© Formatika GbR, Deutschland |
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2026-11-01