diff --git a/thys/PAPP_Impossibility/#Anonymous_PAPP_Lowering.thy# b/thys/PAPP_Impossibility/#Anonymous_PAPP_Lowering.thy# new file mode 100644 --- /dev/null +++ b/thys/PAPP_Impossibility/#Anonymous_PAPP_Lowering.thy# @@ -0,0 +1,1309 @@ +(* + File: Anonymous_PAPP_Lowering.thy + Author: Manuel Eberl, University of Innsbruck +*) +section \Lowering P-APP Rules to Smaller Settings\ +theory Anonymous_PAPP_Lowering + imports Anonymous_PAPP +begin + +text \ + In this section, we prove a number of lemmas (corresponding to Lemma~1 in the paper) that allow + us to take an anonymous P-APP rule with some additional properties (typically + Cardinality-Strategyproofness and Weak Representation or Weak Proportional Representation) and + construct from it an anonymous P-APP rule for a different setting, i.e.\ different number of + voters, parties, and/or result committee size. + + In the reverse direction, this also allows us to lift impossibility results from one setting + to another. +\ + +subsection \Preliminary Lemmas\ + +context card_stratproof_anon_papp +begin + +text \ + The following lemma is obtained by applying Strategyproofness repeatedly. It shows that + if we have \l\ voters with identical approval lists, then this entire group of voters has no + incentive to submit wrong preferences. That is, the outcome they obtain by submitting their + genuine approval lists is weakly preferred by them over all outcomes obtained where these \l\ + voters submit any other preferences (and the remaining $n - l$ voters submit the same preferences + as before). + + This is stronger than regular Strategyproofness, where we only demand that no voter has an + incentive to submit wrong preferences \<^emph>\unilaterally\ (and everyone else keeps the same + preferences). Here we know that the entire group of \l\ voters has no incentive to submit + wrong preferences in coordination with one another. +\ +lemma proposition2: + assumes "size B = l" "size A + l = n_voters" + assumes "X \ {}" "X \ parties" "{} \# A+B" "\X'\#A+B. X' \ parties" + shows "r (replicate_mset l X + A) \[Comm(X)] r (B + A)" + using assms +proof (induction l arbitrary: A B) + case 0 + thus ?case + by simp +next + case (Suc l A B) + from Suc.prems have "set_mset B \ {}" + by auto + then obtain Y where Y: "Y \# B" + by blast + define B' where "B' = B - {#Y#}" + define A' where "A' = A + {#Y#}" + have [simp]: "size B' = l" + using Suc.prems Y by (simp add: B'_def size_Diff_singleton) + have [simp]: "size A' = n_voters - l" + using Suc.prems Y by (simp add: A'_def) + + have "r (B' + A') \[Comm(X)] r (replicate_mset l X + A')" + by (rule Suc.IH) (use Suc.prems Y in \auto simp: A'_def B'_def size_Diff_singleton\) + also have "B' + A' = B + A" + using Y by (simp add: B'_def A'_def) + also have "r (replicate_mset l X + A') \[Comm(X)] r (replicate_mset (Suc l) X + A)" + proof (rule not_manipulable'') + show "replicate_mset (Suc l) X + A + {#Y#} = replicate_mset l X + A' + {#X#}" + by (simp add: A'_def) + next + show "is_pref_profile (replicate_mset (Suc l) X + A)" + using Suc.prems by unfold_locales (auto split: if_splits) + next + show "is_pref_profile (replicate_mset l X + A')" + using Suc.prems Y by unfold_locales (auto split: if_splits simp: A'_def) + qed + finally show ?case . +qed + +end + + +context card_stratproof_weak_rep_anon_papp +begin + +text \ + In a setting with Weak Representation and Cardinality-Strategyproofness, Proposition 2 allows + us to strengthen Weak Representation in the following way: Suppose we at least + $l \lfloor n/k\rfloor$ voters with the same approval list \X\, and \X\ consists of at least \l\ + parties. Then at least \l\ of the members of the result committee are in \X\. +\ +lemma proposition3: + assumes "is_pref_profile A" "X \ parties" "card X \ l" + assumes "committee_size > 0" + assumes "count A X \ l * \n_voters / committee_size\" + shows "size {# x\# r A. x \ X #} \ l" + using assms +proof (induction l arbitrary: A X rule: less_induct) + case (less l A X) + interpret A: anon_papp_profile n_voters parties committee_size A + by fact + consider "l = 0" | "l = 1" | "l > 1" + by force + thus ?case + proof cases + assume "l = 0" + thus ?thesis by simp + next + assume [simp]: "l = 1" + define n where "n = count A X" + with less.prems have "X \ {}" + by auto + then obtain x where x: "x \ X" + by blast + have "n \ size A" + unfolding n_def by (rule count_le_size) + hence "n \ n_voters" + by (simp add: A.size_A) + + have "count A X > 0" + by (rule Nat.gr0I) (use n_voters_pos less.prems in \auto simp: field_simps\) + hence "X \# A" + by force + have [simp]: "replicate_mset n X \# A" + by (simp add: n_def flip: count_le_replicate_mset_subset_eq) + + define A'' where "A'' = A - replicate_mset n X" + define A' where "A' = A'' + replicate_mset n {x}" + interpret A': anon_papp_profile n_voters parties committee_size A' + using A.A_nonempty A.A_subset A.size_A x \X \# A\ + by unfold_locales + (fastforce simp: A'_def A''_def size_Diff_submset subset_mset.add_increasing2 + split: if_splits dest!: in_diffD)+ + + have "x \# r A'" + proof (rule weak_representation) + show "is_pref_profile A'" + by (fact A'.anon_papp_profile_axioms) + next + have "n_voters \ committee_size * n" + using less.prems by (simp add: n_def ceiling_le_iff field_simps flip: of_nat_mult) + also have "n \ count A' {x}" + by (simp add: A'_def) + finally show "n_voters \ committee_size * count A' {x}" + by simp + qed + hence "1 \ count (r A') x" + by simp + also have "\ = size {# y \# r A'. y = x #}" + by simp + also have "\ \ size {# y \# r A'. y \ X #}" + by (intro size_mset_mono multiset_filter_mono') (use x in auto) + + also have "r A' \[Comm(X)] r A" + proof - + have "r (replicate_mset n {x} + A'') \[Comm(X)] r (replicate_mset n X + A'')" + proof (rule proposition2) + show "{} \# A'' + replicate_mset n {x}" + using A'.A_nonempty by (auto simp: A'_def) + show "\X'\#A'' + replicate_mset n {x}. X' \ parties" + using A'.A_subset x by (auto simp: A'_def dest: in_diffD) + show "size A'' + n = n_voters" + using \n \ n_voters\ by (auto simp: A''_def size_Diff_submset A.size_A) + qed (use less.prems in auto) + also have "replicate_mset n X + A'' = A" + by (simp add: A''_def n_def flip: count_le_replicate_mset_subset_eq) + finally show ?thesis + by (simp add: A'_def add_ac) + qed + hence "size {# y \# r A'. y \ X #} \ size {# y \# r A. y \ X #}" + by (simp add: committee_preference_def) + + finally show ?thesis + by simp + + next + assume l: "l > 1" + + define n where "n = count A X" + have "n \ size A" + unfolding n_def by (rule count_le_size) + hence "n \ n_voters" + by (simp add: A.size_A) + + define m where "m = nat (ceiling (n_voters / committee_size))" + have "n_voters / committee_size \ m" + unfolding m_def by linarith + hence m: "n_voters \ committee_size * m" + using \committee_size > 0\ by (simp add: field_simps flip: of_nat_mult) + have "real n_voters / real committee_size > 0" + using n_voters_pos less.prems by auto + hence m': "\real n_voters / real committee_size\ = int m" + by (simp add: m_def) + have "1 * m \ l * m" + using l by (intro mult_right_mono) auto + also have "l * m \ n" + using less.prems by (simp add: m' n_def flip: of_nat_mult) + finally have "m \ n" + by simp + + with less.prems l have "X \ {}" + by auto + then obtain x where x: "x \ X" + by blast + have "card (X - {x}) > 0" + using less.prems x l by simp + hence "X - {x} \ {}" + by force + + have "count A X > 0" + by (rule Nat.gr0I) (use n_voters_pos less.prems l in \auto simp: field_simps mult_le_0_iff\) + hence "X \# A" + by force + have [simp]: "replicate_mset n X \# A" + by (simp add: n_def flip: count_le_replicate_mset_subset_eq) + + define A'' where "A'' = A - replicate_mset n X" + define A' where "A' = A'' + replicate_mset m {x} + replicate_mset (n - m) (X - {x})" + interpret A': anon_papp_profile n_voters parties committee_size A' + proof + show "Y \ parties" if "Y \# A'" for Y + using that A.A_subset x \X \# A\ + by (fastforce simp: A'_def A''_def dest!: in_diffD split: if_splits) + next + show "{} \# A'" + using A.A_nonempty x \X \# A\ \X - {x} \ {}\ + by (auto simp: A'_def A''_def dest!: in_diffD split: if_splits) + next + show "size A' = n_voters" + using \m \ n\ + by (auto simp: A'_def A''_def A.size_A subset_mset.add_increasing2 size_Diff_submset) + qed + + have "x \# r A'" + proof (rule weak_representation) + show "is_pref_profile A'" + by (fact A'.anon_papp_profile_axioms) + next + have "n_voters \ committee_size * m" + by (fact m) + also have "m \ count A' {x}" + by (simp add: A'_def) + finally show "n_voters \ committee_size * count A' {x}" + by simp + qed + hence "1 \ count (r A') x" + by simp + also have "\ = size {# y \# r A'. y = x #}" + by simp + finally have 1: "size {#y \# r A'. y = x#} \ 1" . + + have 2: "size {# y \# r A'. y \ X - {x} #} \ l - 1" + proof (rule less.IH) + have "int (l - 1) * \real n_voters / real committee_size\ = int ((l - 1) * m)" + by (auto simp add: m' not_less) + also have "(l - 1) * m = l * m - m" + by (simp add: algebra_simps) + also have "l * m \ n" + using less.prems by (simp add: m' n_def flip: of_nat_mult) + hence "l * m - m \ n - m" + by (meson diff_le_mono) + also have "n - m \ count A' (X - {x})" + by (simp add: A'_def A''_def) + finally show "int (l - 1) * \real n_voters / real committee_size\ \ int (count A' (X - {x}))" + by simp + qed (use l A'.anon_papp_profile_axioms x less.prems in \auto\) + + have "1 + (l - 1) \ size {#y \# r A'. y = x#} + size {#y \# r A'. y \ X - {x}#}" + by (intro add_mono 1 2) + also have "\ = size ({#y \# r A'. y = x#} + {#y \# r A'. y \ X - {x}#})" + by simp + also have "{#y \# r A'. y = x#} + {#y \# r A'. y \ X - {x}#} = + {#y \# r A'. y = x \ y \ X - {x}#}" + by (rule filter_mset_disjunction [symmetric]) auto + also have "(\y. y = x \ y \ X - {x}) = (\y. y \ X)" + using x by auto + also have "1 + (l - 1) = l" + using l by simp + + also have "r A' \[Comm(X)] r A" + proof - + have "r (replicate_mset m {x} + replicate_mset (n - m) (X - {x}) + A'') \[Comm(X)] + r (replicate_mset n X + A'')" + proof (rule proposition2) + show " {} \# A'' + (replicate_mset m {x} + replicate_mset (n - m) (X - {x}))" + using A'.A_nonempty by (auto simp: A'_def) + show "\X'\# A'' + (replicate_mset m {x} + replicate_mset (n - m) (X - {x})). X' \ parties" + using A'.A_subset x by (auto simp: A'_def dest: in_diffD) + show "size A'' + n = n_voters" + using \n \ n_voters\ by (auto simp: A''_def size_Diff_submset A.size_A) + qed (use less.prems l \m \ n\ in auto) + also have "replicate_mset n X + A'' = A" + by (simp add: A''_def n_def flip: count_le_replicate_mset_subset_eq) + finally show ?thesis + by (simp add: A'_def add_ac) + qed + hence "size {# y \# r A'. y \ X #} \ size {# y \# r A. y \ X #}" + by (simp add: committee_preference_def) + + finally show ?thesis + by simp + qed +qed + +end + + +subsection \Dividing the number of voters\ + +text \ + If we have a PAPP rule that satisfies weak representation and cardinality strategyproofness, + for $ln$ voters, we can turn it into one for \n\ voters. + This is done by simply cloning each voter \l\ times. + + Consequently, if we have an impossibility result for $n$ voters, it also holds for any + integer multiple of \n\. +\ +locale divide_voters_card_stratproof_weak_rep_anon_papp = + card_stratproof_weak_rep_anon_papp "l * n_voters" parties committee_size r + for l n_voters parties committee_size r +begin + +definition lift_profile :: "'a set multiset \ 'a set multiset" where + "lift_profile A = (\X\#A. replicate_mset l X)" + +sublocale lowered: anon_papp_election n_voters parties + by standard (use n_voters_pos in auto) + +lemma l_pos: "l > 0" + using n_voters_pos by auto + +lemma empty_in_lift_profile_iff [simp]: "{} \# lift_profile A \ {} \# A" + using l_pos by (auto simp: lift_profile_def) + +lemma set_mset_lift_profile [simp]: "set_mset (lift_profile A) = set_mset A" + using l_pos by (auto simp: lift_profile_def) + +lemma size_lift_profile: "size (lift_profile A) = l * size A" + by (simp add: size_mset_sum_mset lift_profile_def image_mset.compositionality o_def) + +lemma count_lift_profile [simp]: "count (lift_profile A) x = l * count A x" + unfolding lift_profile_def by (induction A) auto + +lemma is_pref_profile_lift_profile [intro]: + assumes "lowered.is_pref_profile A" + shows "is_pref_profile (lift_profile A)" +proof - + interpret anon_papp_profile n_voters parties committee_size A + by fact + show ?thesis + using A_nonempty A_subset size_A + by unfold_locales + (auto simp: lift_profile_def size_mset_sum_mset image_mset.compositionality o_def) +qed + +sublocale lowered: anon_papp n_voters parties committee_size "r \ lift_profile" +proof + fix A assume "lowered.is_pref_profile A" + hence "is_pref_profile (lift_profile A)" + by blast + hence "is_committee (r (lift_profile A))" + using rule_wf by blast + thus "lowered.is_committee ((r \ lift_profile) A)" + by simp +qed + +sublocale lowered: weak_rep_anon_papp n_voters parties committee_size "r \ lift_profile" +proof + fix A x + assume A: "lowered.is_pref_profile A" and x: "n_voters \ committee_size * count A {x}" + from A have A': "is_pref_profile (lift_profile A)" + by blast + from x have "l * n_voters \ l * (committee_size * count A {x})" + by (rule mult_left_mono) auto + also have "\ = committee_size * count (lift_profile A) {x}" + by simp + finally have "x \# r (lift_profile A)" + by (intro weak_representation A') + thus "x \# (r \ lift_profile) A" + by simp +qed + +sublocale lowered: card_stratproof_anon_papp n_voters parties committee_size "r \ lift_profile" +proof + fix A X Y + show "\lowered.card_manipulable A X Y" + unfolding lowered.card_manipulable_def + proof (rule notI, elim conjE) + assume A: "lowered.is_pref_profile A" and XY: "X \# A" "Y \ {}" "Y \ parties" + assume *: "(r \ lift_profile) A \[lowered.committee_preference X] + (r \ lift_profile) (A - {#X#} + {#Y#})" + interpret anon_papp_profile n_voters parties committee_size A + by fact + have X: "X \ {}" "X \ parties" + using XY A_nonempty A_subset by auto + + define A' where "A' = A - {#X#}" + have A': "A = A' + {#X#}" + using XY by (simp add: A'_def) + + have "r (lift_profile A) \[committee_preference X] + r (lift_profile (A - {#X#} + {#Y#}))" + using * by simp + also have "r (lift_profile (A - {#X#} + {#Y#})) \[committee_preference X] + r (lift_profile A - {#X#} + {#Y#})" + proof - + have "r (replicate_mset (l - 1) Y + (lift_profile A' + {#Y#})) \[committee_preference X] + r (replicate_mset (l - 1) X + (lift_profile A' + {#Y#}))" + proof (rule proposition2) + show "size (lift_profile A' + {#Y#}) + (l - 1) = l * n_voters" + using XY l_pos n_voters_pos + by (simp add: A'_def size_lift_profile size_Diff_singleton + algebra_simps Suc_diff_le size_A) + next + show "{} \# lift_profile A' + {#Y#} + replicate_mset (l - 1) Y" + using XY A_nonempty by (auto simp: A'_def dest: in_diffD) + next + show "\X'\#lift_profile A' + {#Y#} + replicate_mset (l - 1) Y. X' \ parties" + using XY A_subset by (auto simp: A'_def dest: in_diffD) + qed (use X in auto) + thus ?thesis + by (simp add: A' replicate_mset_rec l_pos lift_profile_def) + qed + finally have "card_manipulable (lift_profile A) X Y" + unfolding card_manipulable_def using XY A by auto + with not_manipulable show False + by blast + qed +qed + +sublocale lowered: card_stratproof_weak_rep_anon_papp n_voters parties committee_size "r \ lift_profile" + .. + +end + + +locale divide_voters_card_stratproof_weak_prop_rep_anon_papp = + card_stratproof_weak_prop_rep_anon_papp "l * n_voters" parties committee_size r + for l n_voters parties committee_size r +begin + +sublocale divide_voters_card_stratproof_weak_rep_anon_papp .. + +sublocale lowered: card_stratproof_weak_prop_rep_anon_papp + n_voters parties committee_size "r \ lift_profile" +proof + fix A x l' + assume A: "lowered.is_pref_profile A" and x: "l' * n_voters \ committee_size * count A {x}" + from A have A': "is_pref_profile (lift_profile A)" + by blast + from x have "l * (l' * n_voters) \ l * (committee_size * count A {x})" + by (rule mult_left_mono) auto + also have "\ = committee_size * count (lift_profile A) {x}" + by simp + also have "l * (l' * n_voters) = l' * (l * n_voters)" + by (simp add: algebra_simps) + finally have "count (r (lift_profile A)) x \ l'" + by (intro weak_proportional_representation A') + thus "count ((r \ lift_profile) A) x \ l'" + by simp +qed + +end + + + +subsection \Decreasing the number of parties\ + +text \ + If we have a PAPP rule that satisfies weak representation and cardinality strategyproofness, + for $m$ parties, we can turn it into one for $m-1$ parties. This is done by simply + duplicating one particular party (say \x\) in the preference profile, i.e.\ whenever + \x\ is part of an approval list, we add a clone of \x\ (say \y\) as well. Should \y\ then end up + in the committee, we simply replace it with \x\. + + Consequently, if we have an impossibility result for $k$ parties, it also holds for + $\geq m$ parties. +\ +locale remove_alt_card_stratproof_weak_rep_anon_papp = + card_stratproof_weak_rep_anon_papp n_voters parties committee_size r + for n_voters and parties :: "'a set" and committee_size r + + fixes x y :: 'a + assumes xy: "x \ parties" "y \ parties" "x \ y" +begin + +definition lift_applist :: "'a set \ 'a set" where + "lift_applist X = (if x \ X then insert y X else X)" + +definition lift_profile :: "'a set multiset \ 'a set multiset" where + "lift_profile A = image_mset lift_applist A" + +definition lower_result where "lower_result C = image_mset (\z. if z = y then x else z) C" + +definition lowered where "lowered = lower_result \ r \ lift_profile" + + +lemma lift_profile_empty [simp]: "lift_profile {#} = {#}" + by (simp add: lift_profile_def) + +lemma lift_profile_add_mset [simp]: + "lift_profile (add_mset X A) = add_mset (lift_applist X) (lift_profile A)" + by (simp add: lift_profile_def) + +lemma empty_in_lift_profile_iff [simp]: "{} \# lift_profile A \ {} \# A" + by (auto simp: lift_applist_def lift_profile_def) + +lemma size_lift_profile [simp]: "size (lift_profile A) = size A" + by (simp add: lift_profile_def) + +lemma lift_applist_eq_self_iff [simp]: "lift_applist X = X \ x \ X \ y \ X" + by (auto simp: lift_applist_def) + +lemma lift_applist_eq_self_iff' [simp]: "lift_applist (X - {y}) = X \ (x \ X \ y \ X)" + by (cases "y \ X") (auto simp: lift_applist_def xy) + +lemma in_lift_applist_iff: "z \ lift_applist X \ z \ X \ (z = y \ x \ X)" + by (auto simp: lift_applist_def) + +lemma count_lift_profile: + assumes "\Y\#A. y \ Y" + shows "count (lift_profile A) X = (if x \ X \ y \ X then count A (X - {y}) else 0)" + using assms xy by (induction A) (auto simp: lift_applist_def) + +lemma y_notin_lower_result [simp]: "y \# lower_result C" + using xy by (auto simp: lower_result_def) + +lemma lower_result_subset: "set_mset (lower_result C) \ insert x (set_mset C - {y})" + using xy by (auto simp: lower_result_def) + +lemma lower_result_subset': "set_mset C \ parties \ set_mset (lower_result C) \ parties" + using xy by (auto simp: lower_result_def) + +lemma size_lower_result [simp]: "size (lower_result C) = size C" + by (simp add: lower_result_def) + +lemma count_lower_result: + "count (lower_result C) z = + (if z = y then 0 + else if z = x then count C x + count C y + else count C z)" + using xy by (induction C) (auto simp: lower_result_def) + +lemma in_lower_result_iff: + "z \# lower_result C \ z \ y \ (z \# C \ (z = x \ y \# C))" + unfolding lower_result_def using xy by (induction C) auto + + +sublocale lowered: anon_papp_election n_voters "parties - {y}" + by standard (use n_voters_pos xy in auto) + +lemma is_pref_profile_lift_profile [intro]: + assumes "lowered.is_pref_profile A" + shows "is_pref_profile (lift_profile A)" +proof - + interpret anon_papp_profile n_voters "parties - {y}" committee_size A + by fact + show ?thesis + using A_nonempty A_subset size_A + by unfold_locales + (auto simp: lift_profile_def lift_applist_def xy + size_mset_sum_mset image_mset.compositionality o_def) +qed + +sublocale lowered: anon_papp n_voters "parties - {y}" committee_size lowered +proof + fix A assume "lowered.is_pref_profile A" + hence "is_pref_profile (lift_profile A)" + by blast + hence "is_committee (r (lift_profile A))" + using rule_wf by blast + thus "lowered.is_committee (lowered A)" + unfolding lowered.is_committee_def is_committee_def lowered_def + using lower_result_subset'[of "r (lift_profile A)"] by auto +qed + +sublocale lowered: weak_rep_anon_papp n_voters "parties - {y}" committee_size lowered +proof + fix A z + assume A: "lowered.is_pref_profile A" and z: "n_voters \ committee_size * count A {z}" + interpret A: anon_papp_profile n_voters "parties - {y}" committee_size A + by fact + have "committee_size > 0" + using z n_voters_pos by (intro Nat.gr0I) auto + + from A have A': "is_pref_profile (lift_profile A)" + by blast + have "count A {z} > 0" + using z n_voters_pos by (intro Nat.gr0I) auto + hence "{z} \# A" + by simp + hence z': "z \ parties - {y}" + using A.A_subset z by auto + + define C where "C = r (lift_profile A)" + + show "z \# lowered A" + proof (cases "z = x") + case False + have "n_voters \ committee_size * count A {z}" + by fact + also have "count A {z} \ count (lift_profile A) {z}" + using A.A_subset z' False by (subst count_lift_profile) auto + hence "committee_size * count A {z} \ committee_size * count (lift_profile A) {z}" + by (intro mult_left_mono) auto + finally have "z \# r (lift_profile A)" + by (intro weak_representation A') + thus "z \# lowered A" + using False z' by (simp add: lowered_def in_lower_result_iff) + next + case [simp]: True + have "1 \ size {#z \# C. z \ {x, y}#}" + unfolding C_def + proof (rule proposition3) + have [simp]: "{x, y} - {y} = {x}" + using xy by auto + hence "n_voters \ committee_size * count (lift_profile A) {x, y}" + using xy A.A_subset z by (subst count_lift_profile) auto + thus "int 1 * \real n_voters / real committee_size\ \ int (count (lift_profile A) {x, y})" + using \committee_size > 0\ + by (auto simp: ceiling_le_iff field_simps simp flip: of_nat_mult) + qed (use A' xy \committee_size > 0\ in auto) + also have "\ = count C x + count C y" + using xy by (induction C) auto + also have "\ = count (lowered A) x" + using xy by (simp add: lowered_def count_lower_result C_def) + finally show "z \# lowered A" + by simp + qed +qed + +lemma filter_lower_result_eq: + "y \ X \ {#x \# lower_result C. x \ X#} = lower_result {#x \# C. x \ lift_applist X#}" + by (induction C) (auto simp: lower_result_def lift_applist_def) + +sublocale lowered: card_stratproof_anon_papp n_voters "parties - {y}" committee_size lowered +proof + fix A X Y + show "\lowered.card_manipulable A X Y" + unfolding lowered.card_manipulable_def + proof (rule notI, elim conjE) + assume A: "lowered.is_pref_profile A" and XY: "X \# A" "Y \ {}" "Y \ parties - {y}" + assume *: "lowered A \[lowered.committee_preference X] lowered (A - {#X#} + {#Y#})" + interpret anon_papp_profile n_voters "parties - {y}" committee_size A + by fact + have X: "X \ {}" "X \ parties - {y}" + using XY A_nonempty A_subset by auto + define A' where "A' = A - {#X#}" + have A': "A = A' + {#X#}" + using XY by (simp add: A'_def) + + from * have "size {#x \# lower_result (r (lift_profile A' + {#lift_applist X#})). x \ X#} < + size {#x \# lower_result (r (lift_profile A' + {#lift_applist Y#})). x \ X#}" + by (simp add: lowered_def A' lowered.strong_committee_preference_iff) + also have "{#x \# lower_result (r (lift_profile A' + {#lift_applist X#})). x \ X#} = + lower_result {#x \# r (lift_profile A' + {#lift_applist X#}). x \ lift_applist X#}" + using X by (subst filter_lower_result_eq) auto + also have "{#x \# lower_result (r (lift_profile A' + {#lift_applist Y#})). x \ X#} = + lower_result {#x \# r (lift_profile A' + {#lift_applist Y#}). x \ lift_applist X#}" + using X by (subst filter_lower_result_eq) auto + finally have "size {#x \# r (lift_profile A' + {#lift_applist X#}). x \ lift_applist X#} < + size {#x \# r (lift_profile A' + {#lift_applist Y#}). x \ lift_applist X#}" + by simp + hence "r (lift_profile A' + {#lift_applist X#}) \[committee_preference (lift_applist X)] + r (lift_profile A' + {#lift_applist Y#})" + by (simp add: strong_committee_preference_iff) + moreover have "\r (lift_profile A' + {#lift_applist X#}) \[committee_preference (lift_applist X)] + r (lift_profile A' + {#lift_applist Y#})" + proof (rule not_manipulable' [where Y = "lift_applist Y"]) + have "is_pref_profile (lift_profile A)" + using A by blast + thus "is_pref_profile (lift_profile A' + {#lift_applist X#})" + using A by (simp add: A') + next + have "is_pref_profile (lift_profile (A - {#X#} + {#Y#}))" + using A XY lowered.is_pref_profile_replace by blast + thus "is_pref_profile (lift_profile A' + {#lift_applist Y#})" + by (simp add: A') + qed auto + ultimately show False + by contradiction + qed +qed + +sublocale lowered: card_stratproof_weak_rep_anon_papp n_voters "parties - {y}" committee_size lowered + .. + +end + + +text \ + The following lemma is now simply an iterated application of the above. This allows us to + restrict a P-APP rule to any non-empty subset of parties. +\ +lemma card_stratproof_weak_rep_anon_papp_restrict_parties: + assumes "card_stratproof_weak_rep_anon_papp n parties k r" "parties' \ parties" "parties' \ {}" + shows "\r. card_stratproof_weak_rep_anon_papp n parties' k r" +proof - + have "finite parties" + proof - + interpret card_stratproof_weak_rep_anon_papp n parties k r + by fact + show ?thesis + by (rule finite_parties) + qed + thus ?thesis + using assms + proof (induction parties arbitrary: r rule: finite_psubset_induct) + case (psubset parties r) + show ?thesis + proof (cases "parties = parties'") + case True + thus ?thesis + using psubset.prems by auto + next + case False + obtain x where x: "x \ parties'" + using psubset.prems by blast + from False and psubset.prems have "parties - parties' \ {}" + by auto + then obtain y where y: "y \ parties - parties'" + by blast + + interpret card_stratproof_weak_rep_anon_papp n parties k r + by fact + interpret remove_alt_card_stratproof_weak_rep_anon_papp n parties k r x y + by standard (use x y psubset.prems in auto) + show ?thesis + proof (rule psubset.IH) + show "parties - {y} \ parties" and "parties' \ parties - {y}" "parties' \ {}" + using x y psubset.prems by auto + next + show "card_stratproof_weak_rep_anon_papp n (parties - {y}) k lowered" + using lowered.card_stratproof_weak_rep_anon_papp_axioms . + qed + qed + qed +qed + + + +subsection \Decreasing the committee size\ + +text \ + If we have a PAPP rule that satisfies weak representation and cardinality strategyproofness, + for $l(k+1)$ voters, $m+1$ parties, and a committee size of $k+1$, we can turn it into + one for $lk$ voters, $m$ parties, and a committee size of $k$. + + This is done by again cloning a party \x\ into a new party \y\ and additionally + adding \l\ new voters whose preferences are \{x, y}\. We again replace any \y\ occuring in the + output committee with \x\. Weak representation then ensures that \x\ occurs in the output + at least once, and we then simply remove one \x\ from the committee to obtain an output + committee of size \k - 1\. + + Consequently, if we have an impossibility result for a committee size of $m$, we can + extend it to a larger committee size, but at the cost of introducing a new party + and new voters, and with a restriction on the number of voters. +\ +(* n / k = l *) +locale decrease_committee_card_stratproof_weak_rep_anon_papp = + card_stratproof_weak_rep_anon_papp "l * (k + 1)" "insert y parties" "k + 1" r + for l k y and parties :: "'a set" and r + + fixes x :: 'a + assumes xy: "x \ parties" "y \ parties" + assumes k: "k > 0" +begin + +definition lift_applist :: "'a set \ 'a set" where + "lift_applist X = (if x \ X then insert y X else X)" + +definition lift_profile :: "'a set multiset \ 'a set multiset" where + "lift_profile A = image_mset lift_applist A + replicate_mset l {x, y}" + +definition lower_result + where "lower_result C = image_mset (\z. if z = y then x else z) C - {#x#}" + +definition lowered where "lowered = lower_result \ r \ lift_profile" + + +lemma l: "l > 0" + using n_voters_pos by auto + +lemma x_neq_y [simp]: "x \ y" "y \ x" + using xy by auto + +lemma lift_profile_empty [simp]: "lift_profile {#} = replicate_mset l {x, y}" + by (simp add: lift_profile_def) + +lemma lift_profile_add_mset [simp]: + "lift_profile (add_mset X A) = add_mset (lift_applist X) (lift_profile A)" + by (simp add: lift_profile_def) + +lemma empty_in_lift_profile_iff [simp]: "{} \# lift_profile A \ {} \# A" + by (auto simp: lift_applist_def lift_profile_def) + +lemma size_lift_profile [simp]: "size (lift_profile A) = size A + l" + by (simp add: lift_profile_def) + +lemma lift_applist_eq_self_iff [simp]: "lift_applist X = X \ x \ X \ y \ X" + by (auto simp: lift_applist_def) + +lemma lift_applist_eq_self_iff' [simp]: "lift_applist (X - {y}) = X \ (x \ X \ y \ X)" + by (cases "y \ X") (auto simp: lift_applist_def xy) + +lemma in_lift_applist_iff: "z \ lift_applist X \ z \ X \ (z = y \ x \ X)" + by (auto simp: lift_applist_def) + +lemma count_lift_profile: + assumes "\Y\#A. y \ Y" + shows "count (lift_profile A) X = + (if x \ X \ y \ X then count A (X - {y}) else 0) + + (if X = {x, y} then l else 0)" + using assms xy by (induction A) (auto simp: lift_applist_def) + + + +lemma y_notin_lower_result [simp]: "y \# lower_result C" + using xy by (auto simp: lower_result_def dest: in_diffD) + +lemma lower_result_subset: "set_mset (lower_result C) \ insert x (set_mset C - {y})" + using xy by (auto simp: lower_result_def dest: in_diffD) + +lemma lower_result_subset': "set_mset C \ insert y parties \ set_mset (lower_result C) \ parties" + by (rule order.trans[OF lower_result_subset]) (use xy in auto) + +lemma size_lower_result [simp]: + assumes "x \# C \ y \# C" + shows "size (lower_result C) = size C - 1" + using assms by (auto simp: lower_result_def size_Diff_singleton) + +lemma size_lower_result': "size (lower_result C) = size C - (if x \# C \ y \# C then 1 else 0)" +proof - + define f where "f = (\C. image_mset (\z. if z = y then x else z) C)" + have "size (lower_result C) = size (f C - {#x#})" + by (simp add: lower_result_def f_def) + also have "\ = size (f C) - (if x \# f C then 1 else 0)" + by (simp add: diff_single_trivial size_Diff_singleton) + finally show ?thesis + by (auto simp: f_def) +qed + +lemma count_lower_result: + "count (lower_result C) z = + (if z = y then 0 + else if z = x then count C x + count C y - 1 + else count C z)" (is "_ = ?rhs") +proof - + define f where "f = (\C. image_mset (\z. if z = y then x else z) C)" + have "count (lower_result C) z = count (f C - {#x#}) z" + by (simp add: lower_result_def f_def) + also have "\ = count (f C) z - (if z = x then 1 else 0)" + by auto + also have "count (f C) z = (if z = y then 0 else if z = x then count C x + count C y else count C z)" + using xy by (induction C) (auto simp: f_def) + also have "\ - (if z = x then 1 else 0) = ?rhs" + by auto + finally show ?thesis . +qed + +lemma in_lower_resultD: + "z \# lower_result C \ z = x \ z \# C" + using xy by (auto simp: lower_result_def dest!: in_diffD) + +lemma in_lower_result_iff: + "z \# lower_result C \ z \ y \ (if z = x then count C x + count C y > 1 else z \# C)" + (is "_ = ?rhs") +proof - + have "z \# lower_result C \ count (lower_result C) z > 0" + by auto + also have "\ \ ?rhs" + by (subst count_lower_result) auto + finally show ?thesis . +qed + +lemma filter_lower_result_eq: + assumes "y \ X" + shows "{#z \# lower_result C. z \ X#} = lower_result {#z \# C. z \ lift_applist X#}" +proof - + define f where "f = (\C. {#if z = y then x else z. z \# C#})" + have "lower_result {#z \# C. z \ lift_applist X#} = f {#z \# C. z \ lift_applist X#} - {#x#}" + by (simp add: f_def lower_result_def) + also have "f {#z \# C. z \ lift_applist X#} = {#z \# f C. z \ X#}" + using assms by (induction C) (auto simp: f_def lift_applist_def) + also have "\ - {#x#} = {#z \# f C - {#x#}. z \ X#}" + by (subst filter_diff_mset') auto + also have "f C - {#x#} = lower_result C" + by (simp add: f_def lower_result_def) + finally show ?thesis .. +qed + + +sublocale lowered: anon_papp_election "l * k" parties k + by standard (use n_voters_pos xy finite_parties k in auto) + +lemma is_pref_profile_lift_profile [intro]: + assumes "lowered.is_pref_profile A" + shows "is_pref_profile (lift_profile A)" +proof - + interpret A: anon_papp_profile "l * k" parties k A + by fact + show ?thesis + using A.A_nonempty A.A_subset A.size_A l + by unfold_locales + (auto simp: lift_profile_def lift_applist_def xy + size_mset_sum_mset image_mset.compositionality o_def) +qed + +lemma is_committee_lower_result: + assumes "is_committee C" "x \# C \ y \# C" + shows "lowered.is_committee (lower_result C)" + using assms unfolding is_committee_def lowered.is_committee_def + using lower_result_subset'[of C] by auto + +lemma x_or_y_in_r_lift_profile: + assumes "lowered.is_pref_profile A" + shows "x \# r (lift_profile A) \ y \# r (lift_profile A)" +proof - + interpret A: anon_papp_profile "l * k" parties k A + by fact + have "size {#z \# r (lift_profile A). z \ {x, y}#} \ 1" + proof (rule proposition3) + have "real (l * (k + 1)) / real (k + 1) = real l" + by (simp add: field_simps) + also have "int 1 * \\\ = int l" + by simp + also have "l \ count (lift_profile A) {x, y}" + using xy A.A_subset by (subst count_lift_profile) auto + finally show "int 1 * \real (l * (k + 1)) / real (k + 1)\ \ int (count (lift_profile A) {x, y})" + by simp + next + show "is_pref_profile (lift_profile A)" + by (intro is_pref_profile_lift_profile) fact + qed (use xy in auto) + hence "{#z \# r (lift_profile A). z \ {x, y}#} \ {#}" + by force + thus ?thesis + by auto +qed + +sublocale lowered: anon_papp "l * k" parties k lowered +proof + fix A assume A: "lowered.is_pref_profile A" + hence "is_pref_profile (lift_profile A)" + by blast + hence "is_committee (r (lift_profile A))" + using rule_wf by blast + thus "lowered.is_committee (lowered A)" + unfolding lowered_def o_def using x_or_y_in_r_lift_profile[of A] A + by (intro is_committee_lower_result) auto +qed + +sublocale lowered: weak_rep_anon_papp "l * k" parties k lowered +proof + fix A z + assume A: "lowered.is_pref_profile A" and z: "l * k \ k * count A {z}" + interpret A: anon_papp_profile "l * k" "parties" k A + by fact + + from A have A': "is_pref_profile (lift_profile A)" + by blast + have "count A {z} > 0" + using z k n_voters_pos by (intro Nat.gr0I) auto + hence "{z} \# A" + by simp + hence z': "z \ parties" + using A.A_subset z by auto + hence [simp]: "y \ z" "z \ y" + using xy by auto + + define C where "C = r (lift_profile A)" + + show "z \# lowered A" + proof (cases "z = x") + case False + have "l \ count A {z}" + using z k by (simp add: algebra_simps) + hence "l * (k + 1) \ (k + 1) * count A {z}" + by (subst mult.commute, intro mult_right_mono) auto + also have "count A {z} = count (lift_profile A) {z}" + using False A.A_subset xy by (subst count_lift_profile) auto + finally have "z \# r (lift_profile A)" + by (intro weak_representation A') + thus "z \# lowered A" + using False by (auto simp: lowered_def in_lower_result_iff) + next + case [simp]: True + from xy have [simp]: "{x, y} - {y} = {x}" + by auto + + have "size {#z \# C. z \ {x, y}#} \ 2" + unfolding C_def + proof (rule proposition3) + have "real (l * (k + 1)) / real (k + 1) = l" + unfolding of_nat_mult using k by (simp add: divide_simps) + also have "int 2 * \\\ = int (2 * l)" + by simp + also have "\ \ count (lift_profile A) {x, y}" + using z k xy A.A_subset by (subst count_lift_profile) auto + finally show "int 2 * \real (l * (k + 1)) / real (k + 1)\ \ \" . + qed (use A' xy in auto) + also have "size {#z \# C. z \ {x, y}#} = count C x + count C y" + by (induction C) auto + finally have "x \# lower_result C" + by (subst in_lower_result_iff) auto + thus "z \# lowered A" + by (simp add: lowered_def C_def) + qed +qed + +sublocale lowered: card_stratproof_anon_papp "l * k" parties k lowered +proof + fix A X Y + show "\lowered.card_manipulable A X Y" + unfolding lowered.card_manipulable_def + proof (rule notI, elim conjE) + assume A: "lowered.is_pref_profile A" and XY: "X \# A" "Y \ {}" "Y \ parties" + assume *: "lowered A \[lowered.committee_preference X] lowered (A - {#X#} + {#Y#})" + interpret anon_papp_profile "l * k" "parties" k A + by fact + have X: "X \ {}" "X \ parties" + using XY A_nonempty A_subset by auto + define A' where "A' = A - {#X#}" + have A': "A = A' + {#X#}" + using XY by (simp add: A'_def) + from xy X XY have [simp]: "y \ X" "y \ Y" + by auto + + define Al1 where "Al1 = lift_profile A" + define Al2 where "Al2 = lift_profile (A' + {#Y#})" + have A'_plus_Y: "lowered.is_pref_profile (A' + {#Y#})" + unfolding A'_def using A XY lowered.is_pref_profile_replace by blast + have Al1: "is_pref_profile Al1" + unfolding Al1_def using A by blast + have Al2: "is_pref_profile Al2" + unfolding Al2_def unfolding A'_def using A XY lowered.is_pref_profile_replace by blast + + have size_aux: "size (lower_result {#x \# r (lift_profile A). x \ lift_applist X#}) = + size {#x \# r (lift_profile A). x \ lift_applist X#} - (if x \ X then 1 else 0)" + if A: "lowered.is_pref_profile A" for A + using x_or_y_in_r_lift_profile[OF A] + by (subst size_lower_result') (auto simp: lift_applist_def) + + from * have "size {#x \# lower_result (r Al1). x \ X#} < + size {#x \# lower_result (r Al2). x \ X#}" + by (simp add: Al1_def Al2_def lowered_def A' lowered.strong_committee_preference_iff) + also have "{#x \# lower_result (r Al1). x \ X#} = lower_result {#x \# r Al1. x \ lift_applist X#}" + using X xy by (subst filter_lower_result_eq) auto + also have "{#x \# lower_result (r Al2). x \ X#} = lower_result {#x \# r Al2. x \ lift_applist X#}" + using X xy by (subst filter_lower_result_eq) auto + also have "size (lower_result {#x \# r Al1. x \ lift_applist X#}) = + size {#x \# r Al1. x \ lift_applist X#} - (if x \ X then 1 else 0)" + unfolding Al1_def by (rule size_aux) fact + also have "size (lower_result {#x \# r Al2. x \ lift_applist X#}) = + size {#x \# r Al2. x \ lift_applist X#} - (if x \ X then 1 else 0)" + unfolding Al2_def by (rule size_aux) fact + finally have "size {#x \# r Al1. x \ lift_applist X#} < size {#x \# r Al2. x \ lift_applist X#}" + by auto + hence "r Al1 \[committee_preference (lift_applist X)] r Al2" + by (simp add: strong_committee_preference_iff) + + moreover have "\r Al1 \[committee_preference (lift_applist X)] r Al2" + by (rule not_manipulable' [where Y = "lift_applist Y"]) + (use Al1 Al2 in \auto simp: Al1_def Al2_def A'\) + + ultimately show False + by contradiction + qed +qed + +sublocale lowered: card_stratproof_weak_rep_anon_papp "l * k" parties k lowered + .. + +end + + +text \ + For Weak \<^emph>\Proportional\ Representation, the lowering argument to decrease the committee size + is somewhat easier since it does not involve adding a new party; instead, we simply + add \l\ new voters whose preferences are \{x}\. +\ +locale decrease_committee_card_stratproof_weak_prop_rep_anon_papp = + card_stratproof_weak_prop_rep_anon_papp "l * (k + 1)" parties "k + 1" r + for l k and parties :: "'a set" and r + + fixes x :: 'a + assumes x: "x \ parties" + assumes k: "k > 0" +begin + +definition lift_profile :: "'a set multiset \ 'a set multiset" where + "lift_profile A = A + replicate_mset l {x}" + +definition lower_result + where "lower_result C = C - {#x#}" + +definition lowered where "lowered = lower_result \ r \ lift_profile" + + +lemma l: "l > 0" + using n_voters_pos by auto + +lemma lift_profile_empty [simp]: "lift_profile {#} = replicate_mset l {x}" + by (simp add: lift_profile_def) + +lemma lift_profile_add_mset [simp]: + "lift_profile (add_mset X A) = add_mset X (lift_profile A)" + by (simp add: lift_profile_def) + +lemma empty_in_lift_profile_iff [simp]: "{} \# lift_profile A \ {} \# A" + by (auto simp: lift_profile_def) + +lemma size_lift_profile [simp]: "size (lift_profile A) = size A + l" + by (simp add: lift_profile_def) + +lemma count_lift_profile: + "count (lift_profile A) X = count A X + (if X = {x} then l else 0)" + by (auto simp: lift_profile_def) + + +lemma size_lower_result [simp]: + assumes "x \# C" + shows "size (lower_result C) = size C - 1" + using assms by (auto simp: lower_result_def size_Diff_singleton) + +lemma size_lower_result': "size (lower_result C) = size C - (if x \# C then 1 else 0)" + by (induction C) (auto simp: lower_result_def size_Diff_singleton) + +lemma count_lower_result: + "count (lower_result C) z = count C z - (if z = x then 1 else 0)" + by (auto simp: lower_result_def) + +lemma in_lower_resultD: + "z \# lower_result C \ z \# C" + by (auto simp: lower_result_def dest!: in_diffD) + +lemma in_lower_result_iff: + "z \# lower_result C \ (if z = x then count C x > 1 else z \# C)" + (is "_ = ?rhs") +proof - + have "z \# lower_result C \ count (lower_result C) z > 0" + by auto + also have "\ \ ?rhs" + by (subst count_lower_result) auto + finally show ?thesis . +qed + + +sublocale lowered: anon_papp_election "l * k" parties k + by standard (use n_voters_pos finite_parties k in auto) + +lemma is_pref_profile_lift_profile [intro]: + assumes "lowered.is_pref_profile A" + shows "is_pref_profile (lift_profile A)" +proof - + interpret A: anon_papp_profile "l * k" parties k A + by fact + show ?thesis + using A.A_nonempty A.A_subset A.size_A l + by unfold_locales + (auto simp: lift_profile_def x size_mset_sum_mset image_mset.compositionality o_def) +qed + +lemma is_committee_lower_result: + assumes "is_committee C" "x \# C" + shows "lowered.is_committee (lower_result C)" + using assms unfolding is_committee_def lowered.is_committee_def + by (auto simp: lower_result_def size_Diff_singleton dest: in_diffD) + +lemma x_in_r_lift_profile: + assumes "lowered.is_pref_profile A" + shows "x \# r (lift_profile A)" +proof (rule weak_representation) + show "is_pref_profile (lift_profile A)" + using assms by blast +next + have "(k + 1) * l \ (k + 1) * count (lift_profile A) {x}" + by (intro mult_left_mono) (auto simp: count_lift_profile) + thus "l * (k + 1) \ (k + 1) * count (lift_profile A) {x}" + by (simp add: algebra_simps) +qed + +sublocale lowered: anon_papp "l * k" parties k lowered +proof + fix A assume A: "lowered.is_pref_profile A" + hence "is_pref_profile (lift_profile A)" + by blast + hence "is_committee (r (lift_profile A))" + using rule_wf by blast + thus "lowered.is_committee (lowered A)" + unfolding lowered_def o_def using x_in_r_lift_profile[of A] A + by (intro is_committee_lower_result) auto +qed + +sublocale lowered: weak_prop_rep_anon_papp "l * k" parties k lowered +proof + fix A z l' + assume A: "lowered.is_pref_profile A" and z: "l' * (l * k) \ k * count A {z}" + interpret A: anon_papp_profile "l * k" "parties" k A + by fact + + show "count (lowered A) z \ l'" + proof (cases "l' > 0") + case False + thus ?thesis by auto + next + case l: True + + from A have A': "is_pref_profile (lift_profile A)" + by blast + have "count A {z} > 0" + using z k n_voters_pos l by (intro Nat.gr0I) auto + hence "{z} \# A" + by simp + hence z': "z \ parties" + using A.A_subset z by auto + + define C where "C = r (lift_profile A)" + + show "count (lowered A) z \ l'" + proof (cases "z = x") + case False + have "l' * l \ count A {z}" + using z k by (simp add: algebra_simps) + hence "l' * l * (k + 1) \ (k + 1) * count A {z}" + by (subst mult.commute, intro mult_right_mono) auto + also have "count A {z} = count (lift_profile A) {z}" + using False A.A_subset by (subst count_lift_profile) auto + finally have "count (r (lift_profile A)) z \ l'" + by (intro weak_proportional_representation A') (auto simp: algebra_simps) + thus "l' \ count (lowered A) z" + using False by (simp add: lowered_def lower_result_def) + next + case [simp]: True + have "l' * l \ count A {x}" + using z k by (simp add: algebra_simps) + hence "l' * l * (k + 1) \ (k + 1) * count A {x}" + by (subst mult.commute, intro mult_right_mono) auto + also have "\ + (k + 1) * l = (k + 1) * count (lift_profile A) {x}" + by (simp add: count_lift_profile algebra_simps) + finally have "(l' + 1) * (l * (k + 1)) \ (k + 1) * count (lift_profile A) {x}" + by (simp add: algebra_simps) + hence "count (r (lift_profile A)) x \ l' + 1" + by (intro weak_proportional_representation A') + thus "l' \ count (lowered A) z" + by (simp add: lowered_def lower_result_def) + qed + qed +qed + +sublocale lowered: card_stratproof_anon_papp "l * k" parties k lowered +proof + fix A X Y + show "\lowered.card_manipulable A X Y" + unfolding lowered.card_manipulable_def + proof (rule notI, elim conjE) + assume A: "lowered.is_pref_profile A" and XY: "X \# A" "Y \ {}" "Y \ parties" + assume *: "lowered A \[lowered.committee_preference X] lowered (A - {#X#} + {#Y#})" + interpret anon_papp_profile "l * k" "parties" k A + by fact + have X: "X \ {}" "X \ parties" + using XY A_nonempty A_subset by auto + define A' where "A' = A - {#X#}" + have A': "A = A' + {#X#}" + using XY by (simp add: A'_def) + + define Al1 where "Al1 = lift_profile A" + define Al2 where "Al2 = lift_profile (A' + {#Y#})" + have A'_plus_Y: "lowered.is_pref_profile (A' + {#Y#})" + unfolding A'_def using A XY lowered.is_pref_profile_replace by blast + have Al1: "is_pref_profile Al1" + unfolding Al1_def using A by blast + have Al2: "is_pref_profile Al2" + unfolding Al2_def unfolding A'_def using A XY lowered.is_pref_profile_replace by blast + + have size_aux: "size (lower_result {#x \# r (lift_profile A). x \ X#}) = + size {#x \# r (lift_profile A). x \ X#} - (if x \ X then 1 else 0)" + if A: "lowered.is_pref_profile A" for A + using x_in_r_lift_profile[OF A] by (subst size_lower_result') auto + + from * have "size {#x \# lower_result (r Al1). x \ X#} < + size {#x \# lower_result (r Al2). x \ X#}" + by (simp add: Al1_def Al2_def lowered_def A' lowered.strong_committee_preference_iff) + also have "{#x \# lower_result (r Al1). x \ X#} = lower_result {#x \# r Al1. x \ X#}" + using X x unfolding lower_result_def by (subst filter_diff_mset') auto + also have "{#x \# lower_result (r Al2). x \ X#} = lower_result {#x \# r Al2. x \ X#}" + using X x unfolding lower_result_def by (subst filter_diff_mset') auto + also have "size (lower_result {#x \# r Al1. x \ X#}) = + size {#x \# r Al1. x \ X#} - (if x \ X then 1 else 0)" + unfolding Al1_def by (rule size_aux) fact + also have "size (lower_result {#x \# r Al2. x \ X#}) = + size {#x \# r Al2. x \ X#} - (if x \ X then 1 else 0)" + unfolding Al2_def by (rule size_aux) fact + finally have "size {#x \# r Al1. x \ X#} < size {#x \# r Al2. x \ X#}" + by auto + hence "r Al1 \[committee_preference X] r Al2" + by (simp add: strong_committee_preference_iff) + + moreover have "\r Al1 \[committee_preference X] r Al2" + by (rule not_manipulable' [where Y = "Y"]) + (use Al1 Al2 in \auto simp: Al1_def Al2_def A'\) + + ultimately show False + by contradiction + qed +qed + +sublocale lowered: card_stratproof_weak_prop_rep_anon_papp "l * k" parties k lowered + .. + +end + +end diff --git a/thys/PAPP_Impossibility/#ROOT# b/thys/PAPP_Impossibility/#ROOT# new file mode 100644 --- /dev/null +++ b/thys/PAPP_Impossibility/#ROOT# @@ -0,0 +1,14 @@ +chapter AFP + +session PAPP_Impossibility (AFP) = "HOL-Library" + + options [timeout = 900] + sessions + Randomised_Social_Choice + theories [document=false] + SAT_Replay + theories + PAPP_Impossibility + document_files + root.tex + root.bib + diff --git a/thys/PAPP_Impossibility/Anonymous_PAPP.thy b/thys/PAPP_Impossibility/Anonymous_PAPP.thy new file mode 100644 --- /dev/null +++ b/thys/PAPP_Impossibility/Anonymous_PAPP.thy @@ -0,0 +1,452 @@ +(* + File: Anonymous_PAPP.thy + Author: Manuel Eberl, University of Innsbruck +*) +section \Anonymous Party Approval Rules\ +theory Anonymous_PAPP + imports Complex_Main "Randomised_Social_Choice.Order_Predicates" PAPP_Multiset_Extras +begin + +text \ + In this section we will define (anonymous) P-APP rules and some basic desirable properties + of P-APP rules. +\ + +subsection \Definition of the General Setting\ + +text \ + The following locale encapsulates an anonymous \<^emph>\party approval election\; that is: + + \<^item> a number of voters + + \<^item> a set of parties + + \<^item> the size of the desired committee + + The number of parties and voters is assumed to be finite and non-zero. + As a modelling choice, we do not distinguish the voters at all; there is no explicit set + of voters. We only care about their number. +\ +locale anon_papp_election = + fixes n_voters :: nat and parties :: "'a set" and committee_size :: nat + assumes finite_parties [simp, intro]: "finite parties" + assumes n_voters_pos: "n_voters > 0" + assumes nonempty_parties [simp]: "parties \ {}" +begin + +text \ + The result of a P-APP election is a committee, i.e.\ a multiset of parties with + the desired size. +\ +definition is_committee :: "'a multiset \ bool" where + "is_committee W \ set_mset W \ parties \ size W = committee_size" + +end + + +text \ + A \<^emph>\preference profile\ for a P-APP collection consists of one approval list (i.e.\ a set of + approved parties) for each voter. Since we are in an anonymous setting, this means that + we have a \<^emph>\multiset\ consisting of \n\ sets of parties (where \n\ is the number of voters). + + Moreover, we make the usual assumption that the approval lists must be non-empty. +\ +locale anon_papp_profile = anon_papp_election + + fixes A :: "'a set multiset" + assumes A_subset: "\X. X \# A \ X \ parties" + assumes A_nonempty: "{} \# A" + assumes size_A: "size A = n_voters" +begin + +lemma A_nonempty': "A \ {#}" + using size_A n_voters_pos by auto + +end + + +context anon_papp_election +begin + +abbreviation + is_pref_profile where "is_pref_profile \ anon_papp_profile n_voters parties" + +lemma is_pref_profile_iff: + "is_pref_profile A \ set_mset A \ Pow parties - {{}} \ size A = n_voters" + unfolding anon_papp_profile_def anon_papp_profile_axioms_def + using anon_papp_election_axioms by auto + + +lemma not_is_pref_profile_empty [simp]: "\is_pref_profile {#}" + using anon_papp_profile.A_nonempty'[of n_voters] + by auto + + + +text \ + The following relation is a key definition: it takes an approval list \A\ and turns it + into a preference relation on committees. A committee is to be at least as good as another + if the number of approved parties in it is at least as big. + + This relation is a reflexive, transitive, and total. +\ +definition committee_preference :: "'a set \ 'a multiset relation" ("Comm") where + "W1 \[Comm(A)] W2 \ size {# x\#W1. x \ A #} \ size {# x\#W2. x \ A #}" + +lemma not_strict_Comm [simp]: "\(W1 \[Comm(A)] W2) \ W1 \[Comm(A)] W2" + by (auto simp: committee_preference_def strongly_preferred_def) + +lemma not_weak_Comm [simp]: "\(W1 \[Comm(A)] W2) \ W1 \[Comm(A)] W2" + by (auto simp: committee_preference_def strongly_preferred_def) + +sublocale Comm: preorder "Comm(A)" "\x y. x \[Comm(A)] y" + by standard (auto simp: committee_preference_def strongly_preferred_def) + +lemma strong_committee_preference_iff: + "W1 \[Comm(A)] W2 \ size {# x\#W1. x \ A #} < size {# x\#W2. x \ A #}" + by (auto simp: committee_preference_def strongly_preferred_def) + + +text \ + We also define the Pareto ordering on parties induced by a given preference profile: + One party is at least as good (in the Pareto relation) as another if all voters agree that + it is at least as good. That is, $y \succeq x$ in the Pareto ordering if all voters who + approve $x$ also approve $y$. + + This relation is also reflexive and transitive. +\ +definition Pareto :: "'a set multiset \ 'a relation" where + "x \[Pareto(A)] y \ x \ parties \ y \ parties \ (\X\#A. x \ X \ y \ X)" + +sublocale Pareto: preorder_on parties "Pareto A" + by standard (auto simp: Pareto_def) + + +text \ + Pareto losers are parties that are (strictly) Pareto-dominated, i.e.\ there + exists some other party that all voters consider to be at least as good and at least + one voter considers it to be strictly better. +\ +definition pareto_losers :: "'a set multiset \ 'a set" where + "pareto_losers A = {x. \y. y \[Pareto(A)] x}" + +end + + + +subsection \P-APP rules and Desirable Properties\ + +text \ + The following locale describes a P-APP rule. This is simply a function that maps every + preference profile to a committee of the desired size. + + Note that in our setting, a P-APP rule has a fixed number of voters, a fixed set of parties, + and a fixed desired committee size. +\ +locale anon_papp = anon_papp_election + + fixes r :: "'a set multiset \ 'a multiset" + assumes rule_wf: "is_pref_profile A \ is_committee (r A)" + + + +subsection \Efficiency\ + +text \ + Efficiency is a common notion in Social Choice Theory. The idea is that if a party is + ``obviously bad'', then it should not be chosen. What ``obviously bad'' means depends on the + precise notion of Efficiency that is used. We will talk about two notions: Weak Efficiency and + Pareto Efficiency. + + A P-APP rule is \<^emph>\weakly efficient\ if a party that is approved by no one is never + part of the output committee. + + Note that approval lists must be non-empty, so there is always at least one party that + is approved by at least one voter. +\ +locale weakly_efficient_anon_papp = anon_papp + + assumes weakly_efficient: "is_pref_profile A \ \X\#A. x \ X \ x \# r A" + + +text \ + A P-APP rule is \<^emph>\Pareto-efficient\ if a Pareto-dominated party is never part of + the output committee. +\ +locale pareto_optimal_anon_papp = anon_papp + + assumes pareto_optimal: "is_pref_profile A \ x \ pareto_losers A \ x \# r A" +begin + +text \ + Pareto-efficiency implies weak efficiency: +\ +sublocale weakly_efficient_anon_papp +proof + fix A x + assume A: "is_pref_profile A" and x: "\X\#A. x \ X" + interpret anon_papp_profile n_voters parties committee_size A + by fact + have "A \ {#}" + using A_nonempty' . + then obtain X where X: "X \# A" + by auto + with A_nonempty have "X \ {}" + by auto + then obtain y where y: "y \ X" + by auto + show "x \# r A" + proof (cases "x \ parties") + case False + thus ?thesis + using rule_wf[OF A] by (auto simp: is_committee_def) + next + case True + have "y \[Pareto(A)] x" + unfolding Pareto_def using X x y True A_subset[of X] + by (auto simp: strongly_preferred_def) + hence "x \ pareto_losers A" + by (auto simp: pareto_losers_def) + thus ?thesis + using pareto_optimal[OF A] by auto + qed +qed + +end + + +subsection \Strategyproofness\ + +text \ + Strategyproofness is another common notion in Social Choice Theory that generally encapsulates + the notion that an voter should not be able to manipulate the outcome of an election in their + favour by (unilaterally) submitting fake preferences; i.e.\ reporting one's preferences + truthfully should always be the optimal choice. + + A P-APP rule is called \<^emph>\cardinality-strategyproof\ if an voter cannot obtain a better committee + (i.e.\ one that contains strictly more of their approved parties) by submitting an approval + list that is different from their real approval list. +\ + +text \ + To make the definition simpler, we first define the notion of \<^emph>\manipulability\: in the context + of a particular P-APP rule \r\, a preference profile \A\ is said to be manipulable by the voter + \i\ with the fake preference list \Y\ if \r(A(i := Y))\ contains strictly more parties + approved by \i\ than \r(A)\. + + Since we have anonymous profiles and do not talk about particular voters, we replace \i\ with + their approval list \X\. Since \A\ is a multiset, the definition of manipulability + becomes $r(A-\{X\}+\{Y\}) \succ_{X} r(A)$. +\ +definition (in anon_papp) card_manipulable where + "card_manipulable A X Y \ + is_pref_profile A \ X \# A \ Y \ {} \ Y \ parties \ r (A - {#X#} + {#Y#}) \[Comm(X)] r A" + +text \ + A technical (and fairly obvious) lemma: replacing an voter's approval list with a different + approval list again yields a valid preference profile. +\ +lemma (in anon_papp) is_pref_profile_replace: + assumes "is_pref_profile A" and "X \# A" and "Y \ {}" and "Y \ parties" + shows "is_pref_profile (A - {#X#} + {#Y#})" +proof - + interpret anon_papp_profile n_voters parties committee_size A + by fact + show ?thesis + using assms A_subset A_nonempty unfolding is_pref_profile_iff + by (auto dest: in_diffD simp: size_Suc_Diff1) +qed + +locale card_stratproof_anon_papp = anon_papp + + assumes not_manipulable: "\card_manipulable A X Y" +begin + +text \ + The two following alternative versions of non-manipulability are somewhat nicer to use + in practice. +\ +lemma not_manipulable': + assumes "is_pref_profile A" "is_pref_profile A'" "A + {#Y#} = A' + {#X#}" + shows "\(r A' \[Comm(X)] r A)" +proof (cases "X = Y") + case True + thus ?thesis + using assms by (simp add: strongly_preferred_def) +next + case False + interpret A: anon_papp_profile n_voters parties committee_size A + by fact + interpret A': anon_papp_profile n_voters parties committee_size A' + by fact + from assms(3) False have *: "Y \# A'" "X \# A" + by (metis add_mset_add_single insert_noteq_member)+ + + have "\card_manipulable A X Y" + by (intro not_manipulable) + hence "\r (A - {#X#} + {#Y#}) \[Comm(X)] r A" + using assms * A.A_subset A'.A_subset A.A_nonempty A'.A_nonempty + by (auto simp: card_manipulable_def) + also have "A - {#X#} + {#Y#} = A'" + using assms(3) False by (metis add_eq_conv_diff add_mset_add_single) + finally show ?thesis . +qed + +lemma not_manipulable'': + assumes "is_pref_profile A" "is_pref_profile A'" "A + {#Y#} = A' + {#X#}" + shows "r A' \[Comm(X)] r A" + using not_manipulable'[OF assms] by simp + +end + + +subsection \Representation\ + +text \ + \<^emph>\Representation\ properties are in a sense the opposite of Efficiency properties: if a + sufficiently high voters agree that certain parties are good, then these should, to some + extent, be present in the result. For instance, if we have 20 voters and 5 of them approve + parties \A\ and \B\, then if the output committee has size 4, we would expect either + \A\ or \B\ to be in the committee to ensure that these voters' preferences are represented fairly. + + Weak representation is a particularly weak variant of this that states that if at least one $k$-th + of the voters (where $k$ is the size of the output committee) approve only a single + party \x\, then \x\ should be in the committee at least once: +\ + +locale weak_rep_anon_papp = + anon_papp n_voters parties committee_size r + for n_voters and parties :: "'alt set" and committee_size :: nat and r + + assumes weak_representation: + "is_pref_profile A \ committee_size * count A {x} \ n_voters \ x \# r A" + +text \ + The following alternative definition of Weak Representation is a bit closer to the definition + given in the paper. +\ +lemma weak_rep_anon_papp_altdef: + "weak_rep_anon_papp n_voters parties committee_size r \ + anon_papp n_voters parties committee_size r \ (committee_size = 0 \ + (\A x. anon_papp_profile n_voters parties A \ + count A {x} \ n_voters / committee_size \ x \# r A))" + by (cases "committee_size = 0") + (auto simp: field_simps weak_rep_anon_papp_def + weak_rep_anon_papp_axioms_def + anon_papp_def anon_papp_axioms_def anon_papp_election_def + simp flip: of_nat_mult) + + +text \ + \<^emph>\Justified Representation\ is a stronger notion which demands that if there is a subgroup of + voters that comprises at least one $k$-th of all voters and for which the intersection of their + approval lists is some nonempty set \X\, then at least one of the parties approved by at least + one voter in that subgroup must be in the result committee. +\ +locale justified_rep_anon_papp = + anon_papp n_voters parties committee_size r + for n_voters and parties :: "'alt set" and committee_size :: nat and r + + assumes justified_representation: + "is_pref_profile A \ G \# A \ committee_size * size G \ n_voters \ + (\X\set_mset G. X) \ {} \ \X x. X \# G \ x \ X \ x \# r A" +begin + +text \ + Any rule that satisfies Justified Representation also satisfies Weak Representation +\ +sublocale weak_rep_anon_papp +proof + fix A x + assume *: "is_pref_profile A" "n_voters \ committee_size * count A {x}" + define G where "G = replicate_mset (count A {x}) {x}" + have [simp]: "size G = count A {x}" + by (auto simp: G_def) + have **: "set_mset G \ {{x}}" + by (auto simp: G_def) + have ***: "G \# A" + unfolding G_def by (meson count_le_replicate_mset_subset_eq order_refl) + have "\X x. X \# G \ x \ X \ x \# r A" + by (rule justified_representation) (use * ** *** in auto) + thus "x \# r A" + using ** by auto +qed + +end + + +locale card_stratproof_weak_rep_anon_papp = + card_stratproof_anon_papp + weak_rep_anon_papp + + +subsection \Proportional Representation\ + +text \ + The notions of Representation we have seen so far are fairly week in that they only demand + that certain parties be in the committee at least once if enough voters approve them. Notions of + Proportional Representation strengthen this by demanding that if a sufficiently large subgroup of + voters approve some parties, then these voters must be represented in the result committe not + just once, but to a degree proportional to the size of that subgroup of voters. + + For Weak Representation, the proportional generalization is fairly simple: if a fraction of + at least $\frac{ln}{k}$ of the voters uniquely approve a party \x\, then \x\ must be in the + committee at least \l\ times. +\ + +locale weak_prop_rep_anon_papp = + anon_papp n_voters parties committee_size r + for n_voters and parties :: "'alt set" and committee_size :: nat and r + + assumes weak_proportional_representation: + "is_pref_profile A \ committee_size * count A {x} \ l * n_voters \ count (r A) x \ l" +begin + +sublocale weak_rep_anon_papp +proof + fix A x + assume "is_pref_profile A" "n_voters \ committee_size * count A {x}" + thus "x \# r A" + using weak_proportional_representation[of A 1] by auto +qed + +end + + +text \ + Similarly, Justified \<^emph>\Proportional\ Representation demands that if the approval lists of + a subgroup of at least $\frac{ln}{k}$ voters have a non-empty intersection, then at least \l\ + parties in the result committee are each approved by at least one of the voters in the subgroup. +\ +locale justified_prop_rep_anon_papp = + anon_papp n_voters parties committee_size r + for n_voters and parties :: "'alt set" and committee_size :: nat and r + + assumes justified_proportional_representation: + "is_pref_profile A \ G \# A \ committee_size * size G \ l * n_voters \ + (\X\set_mset G. X) \ {} \ size {# x \# r A. x \ (\X\set_mset G. X) #} \ l" +begin + +sublocale justified_rep_anon_papp +proof + fix A G + assume "is_pref_profile A" "G \# A" "n_voters \ committee_size * size G" + "(\X\set_mset G. X) \ {}" + hence "size {#x \# r A. \X\#G. x \ X#} \ 1" + using justified_proportional_representation[of A G 1] by auto + hence "{#x \# r A. \X\#G. x \ X#} \ {#}" + by auto + thus "\X x. X \# G \ x \ X \ x \# r A" + by fastforce +qed + +sublocale weak_prop_rep_anon_papp +proof + fix A l x + assume *: "is_pref_profile A" "l * n_voters \ committee_size * count A {x}" + define G where "G = replicate_mset (count A {x}) {x}" + from * have "size {#x \# r A. x \ (\X\set_mset G. X)#} \ l" + by (intro justified_proportional_representation) + (auto simp: G_def simp flip: count_le_replicate_mset_subset_eq) + also have "size {#x \# r A. x \ (\X\set_mset G. X)#} \ count (r A) x" + by (auto simp: G_def) + finally show "count (r A) x \ l" . +qed + +end + + +locale card_stratproof_weak_prop_rep_anon_papp = + card_stratproof_anon_papp + weak_prop_rep_anon_papp + +end diff --git a/thys/PAPP_Impossibility/Anonymous_PAPP_Lowering.thy b/thys/PAPP_Impossibility/Anonymous_PAPP_Lowering.thy new file mode 100644 --- /dev/null +++ b/thys/PAPP_Impossibility/Anonymous_PAPP_Lowering.thy @@ -0,0 +1,1374 @@ +(* + File: Anonymous_PAPP_Lowering.thy + Author: Manuel Eberl, University of Innsbruck +*) +section \Lowering P-APP Rules to Smaller Settings\ +theory Anonymous_PAPP_Lowering + imports Anonymous_PAPP +begin + +text \ + In this section, we prove a number of lemmas (corresponding to Lemma~1 in the paper) that allow + us to take an anonymous P-APP rule with some additional properties (typically + Cardinality-Strategyproofness and Weak Representation or Weak Proportional Representation) and + construct from it an anonymous P-APP rule for a different setting, i.e.\ different number of + voters, parties, and/or result committee size. + + In the reverse direction, this also allows us to lift impossibility results from one setting + to another. +\ + +subsection \Preliminary Lemmas\ + +context card_stratproof_anon_papp +begin + +text \ + The following lemma is obtained by applying Strategyproofness repeatedly. It shows that + if we have \l\ voters with identical approval lists, then this entire group of voters has no + incentive to submit wrong preferences. That is, the outcome they obtain by submitting their + genuine approval lists is weakly preferred by them over all outcomes obtained where these \l\ + voters submit any other preferences (and the remaining $n - l$ voters submit the same preferences + as before). + + This is stronger than regular Strategyproofness, where we only demand that no voter has an + incentive to submit wrong preferences \<^emph>\unilaterally\ (and everyone else keeps the same + preferences). Here we know that the entire group of \l\ voters has no incentive to submit + wrong preferences in coordination with one another. +\ +lemma proposition2: + assumes "size B = l" "size A + l = n_voters" + assumes "X \ {}" "X \ parties" "{} \# A+B" "\X'\#A+B. X' \ parties" + shows "r (replicate_mset l X + A) \[Comm(X)] r (B + A)" + using assms +proof (induction l arbitrary: A B) + case 0 + thus ?case + by simp +next + case (Suc l A B) + from Suc.prems have "set_mset B \ {}" + by auto + then obtain Y where Y: "Y \# B" + by blast + define B' where "B' = B - {#Y#}" + define A' where "A' = A + {#Y#}" + have [simp]: "size B' = l" + using Suc.prems Y by (simp add: B'_def size_Diff_singleton) + have [simp]: "size A' = n_voters - l" + using Suc.prems Y by (simp add: A'_def) + + have "r (B' + A') \[Comm(X)] r (replicate_mset l X + A')" + by (rule Suc.IH) (use Suc.prems Y in \auto simp: A'_def B'_def size_Diff_singleton\) + also have "B' + A' = B + A" + using Y by (simp add: B'_def A'_def) + also have "r (replicate_mset l X + A') \[Comm(X)] r (replicate_mset (Suc l) X + A)" + proof (rule not_manipulable'') + show "replicate_mset (Suc l) X + A + {#Y#} = replicate_mset l X + A' + {#X#}" + by (simp add: A'_def) + next + show "is_pref_profile (replicate_mset (Suc l) X + A)" + using Suc.prems by unfold_locales (auto split: if_splits) + next + show "is_pref_profile (replicate_mset l X + A')" + using Suc.prems Y by unfold_locales (auto split: if_splits simp: A'_def) + qed + finally show ?case . +qed + +end + + +context card_stratproof_weak_rep_anon_papp +begin + +text \ + In a setting with Weak Representation and Cardinality-Strategyproofness, Proposition 2 allows + us to strengthen Weak Representation in the following way: Suppose we at least + $l \lfloor n/k\rfloor$ voters with the same approval list \X\, and \X\ consists of at least \l\ + parties. Then at least \l\ of the members of the result committee are in \X\. +\ +lemma proposition3: + assumes "is_pref_profile A" "X \ parties" "card X \ l" + assumes "committee_size > 0" + assumes "count A X \ l * \n_voters / committee_size\" + shows "size {# x\# r A. x \ X #} \ l" + using assms +proof (induction l arbitrary: A X rule: less_induct) + case (less l A X) + interpret A: anon_papp_profile n_voters parties committee_size A + by fact + consider "l = 0" | "l = 1" | "l > 1" + by force + thus ?case + proof cases + assume "l = 0" + thus ?thesis by simp + next + assume [simp]: "l = 1" + define n where "n = count A X" + with less.prems have "X \ {}" + by auto + then obtain x where x: "x \ X" + by blast + have "n \ size A" + unfolding n_def by (rule count_le_size) + hence "n \ n_voters" + by (simp add: A.size_A) + + have "count A X > 0" + by (rule Nat.gr0I) (use n_voters_pos less.prems in \auto simp: field_simps\) + hence "X \# A" + by force + have [simp]: "replicate_mset n X \# A" + by (simp add: n_def flip: count_le_replicate_mset_subset_eq) + + define A'' where "A'' = A - replicate_mset n X" + define A' where "A' = A'' + replicate_mset n {x}" + interpret A': anon_papp_profile n_voters parties committee_size A' + using A.A_nonempty A.A_subset A.size_A x \X \# A\ + by unfold_locales + (fastforce simp: A'_def A''_def size_Diff_submset subset_mset.add_increasing2 + split: if_splits dest!: in_diffD)+ + + have "x \# r A'" + proof (rule weak_representation) + show "is_pref_profile A'" + by (fact A'.anon_papp_profile_axioms) + next + have "n_voters \ committee_size * n" + using less.prems by (simp add: n_def ceiling_le_iff field_simps flip: of_nat_mult) + also have "n \ count A' {x}" + by (simp add: A'_def) + finally show "n_voters \ committee_size * count A' {x}" + by simp + qed + hence "1 \ count (r A') x" + by simp + also have "\ = size {# y \# r A'. y = x #}" + by simp + also have "\ \ size {# y \# r A'. y \ X #}" + by (intro size_mset_mono multiset_filter_mono') (use x in auto) + + also have "r A' \[Comm(X)] r A" + proof - + have "r (replicate_mset n {x} + A'') \[Comm(X)] r (replicate_mset n X + A'')" + proof (rule proposition2) + show "{} \# A'' + replicate_mset n {x}" + using A'.A_nonempty by (auto simp: A'_def) + show "\X'\#A'' + replicate_mset n {x}. X' \ parties" + using A'.A_subset x by (auto simp: A'_def dest: in_diffD) + show "size A'' + n = n_voters" + using \n \ n_voters\ by (auto simp: A''_def size_Diff_submset A.size_A) + qed (use less.prems in auto) + also have "replicate_mset n X + A'' = A" + by (simp add: A''_def n_def flip: count_le_replicate_mset_subset_eq) + finally show ?thesis + by (simp add: A'_def add_ac) + qed + hence "size {# y \# r A'. y \ X #} \ size {# y \# r A. y \ X #}" + by (simp add: committee_preference_def) + + finally show ?thesis + by simp + + next + assume l: "l > 1" + + define n where "n = count A X" + have "n \ size A" + unfolding n_def by (rule count_le_size) + hence "n \ n_voters" + by (simp add: A.size_A) + + define m where "m = nat (ceiling (n_voters / committee_size))" + have "n_voters / committee_size \ m" + unfolding m_def by linarith + hence m: "n_voters \ committee_size * m" + using \committee_size > 0\ by (simp add: field_simps flip: of_nat_mult) + have "real n_voters / real committee_size > 0" + using n_voters_pos less.prems by auto + hence m': "\real n_voters / real committee_size\ = int m" + by (simp add: m_def) + have "1 * m \ l * m" + using l by (intro mult_right_mono) auto + also have "l * m \ n" + using less.prems by (simp add: m' n_def flip: of_nat_mult) + finally have "m \ n" + by simp + + with less.prems l have "X \ {}" + by auto + then obtain x where x: "x \ X" + by blast + have "card (X - {x}) > 0" + using less.prems x l by simp + hence "X - {x} \ {}" + by force + + have "count A X > 0" + by (rule Nat.gr0I) (use n_voters_pos less.prems l in \auto simp: field_simps mult_le_0_iff\) + hence "X \# A" + by force + have [simp]: "replicate_mset n X \# A" + by (simp add: n_def flip: count_le_replicate_mset_subset_eq) + + define A'' where "A'' = A - replicate_mset n X" + define A' where "A' = A'' + replicate_mset m {x} + replicate_mset (n - m) (X - {x})" + interpret A': anon_papp_profile n_voters parties committee_size A' + proof + show "Y \ parties" if "Y \# A'" for Y + using that A.A_subset x \X \# A\ + by (fastforce simp: A'_def A''_def dest!: in_diffD split: if_splits) + next + show "{} \# A'" + using A.A_nonempty x \X \# A\ \X - {x} \ {}\ + by (auto simp: A'_def A''_def dest!: in_diffD split: if_splits) + next + show "size A' = n_voters" + using \m \ n\ + by (auto simp: A'_def A''_def A.size_A subset_mset.add_increasing2 size_Diff_submset) + qed + + have "x \# r A'" + proof (rule weak_representation) + show "is_pref_profile A'" + by (fact A'.anon_papp_profile_axioms) + next + have "n_voters \ committee_size * m" + by (fact m) + also have "m \ count A' {x}" + by (simp add: A'_def) + finally show "n_voters \ committee_size * count A' {x}" + by simp + qed + hence "1 \ count (r A') x" + by simp + also have "\ = size {# y \# r A'. y = x #}" + by simp + finally have 1: "size {#y \# r A'. y = x#} \ 1" . + + have 2: "size {# y \# r A'. y \ X - {x} #} \ l - 1" + proof (rule less.IH) + have "int (l - 1) * \real n_voters / real committee_size\ = int ((l - 1) * m)" + by (auto simp add: m' not_less) + also have "(l - 1) * m = l * m - m" + by (simp add: algebra_simps) + also have "l * m \ n" + using less.prems by (simp add: m' n_def flip: of_nat_mult) + hence "l * m - m \ n - m" + by (meson diff_le_mono) + also have "n - m \ count A' (X - {x})" + by (simp add: A'_def A''_def) + finally show "int (l - 1) * \real n_voters / real committee_size\ \ int (count A' (X - {x}))" + by simp + qed (use l A'.anon_papp_profile_axioms x less.prems in \auto\) + + have "1 + (l - 1) \ size {#y \# r A'. y = x#} + size {#y \# r A'. y \ X - {x}#}" + by (intro add_mono 1 2) + also have "\ = size ({#y \# r A'. y = x#} + {#y \# r A'. y \ X - {x}#})" + by simp + also have "{#y \# r A'. y = x#} + {#y \# r A'. y \ X - {x}#} = + {#y \# r A'. y = x \ y \ X - {x}#}" + by (rule filter_mset_disjunction [symmetric]) auto + also have "(\y. y = x \ y \ X - {x}) = (\y. y \ X)" + using x by auto + also have "1 + (l - 1) = l" + using l by simp + + also have "r A' \[Comm(X)] r A" + proof - + have "r (replicate_mset m {x} + replicate_mset (n - m) (X - {x}) + A'') \[Comm(X)] + r (replicate_mset n X + A'')" + proof (rule proposition2) + show " {} \# A'' + (replicate_mset m {x} + replicate_mset (n - m) (X - {x}))" + using A'.A_nonempty by (auto simp: A'_def) + show "\X'\# A'' + (replicate_mset m {x} + replicate_mset (n - m) (X - {x})). X' \ parties" + using A'.A_subset x by (auto simp: A'_def dest: in_diffD) + show "size A'' + n = n_voters" + using \n \ n_voters\ by (auto simp: A''_def size_Diff_submset A.size_A) + qed (use less.prems l \m \ n\ in auto) + also have "replicate_mset n X + A'' = A" + by (simp add: A''_def n_def flip: count_le_replicate_mset_subset_eq) + finally show ?thesis + by (simp add: A'_def add_ac) + qed + hence "size {# y \# r A'. y \ X #} \ size {# y \# r A. y \ X #}" + by (simp add: committee_preference_def) + + finally show ?thesis + by simp + qed +qed + +end + + +subsection \Dividing the number of voters\ + +text \ + If we have a PAPP rule that satisfies weak representation and cardinality strategyproofness, + for $ln$ voters, we can turn it into one for \n\ voters. + This is done by simply cloning each voter \l\ times. + + Consequently, if we have an impossibility result for $n$ voters, it also holds for any + integer multiple of \n\. +\ +locale divide_voters_card_stratproof_weak_rep_anon_papp = + card_stratproof_weak_rep_anon_papp "l * n_voters" parties committee_size r + for l n_voters parties committee_size r +begin + +definition lift_profile :: "'a set multiset \ 'a set multiset" where + "lift_profile A = (\X\#A. replicate_mset l X)" + +sublocale lowered: anon_papp_election n_voters parties + by standard (use n_voters_pos in auto) + +lemma l_pos: "l > 0" + using n_voters_pos by auto + +lemma empty_in_lift_profile_iff [simp]: "{} \# lift_profile A \ {} \# A" + using l_pos by (auto simp: lift_profile_def) + +lemma set_mset_lift_profile [simp]: "set_mset (lift_profile A) = set_mset A" + using l_pos by (auto simp: lift_profile_def) + +lemma size_lift_profile: "size (lift_profile A) = l * size A" + by (simp add: size_mset_sum_mset lift_profile_def image_mset.compositionality o_def) + +lemma count_lift_profile [simp]: "count (lift_profile A) x = l * count A x" + unfolding lift_profile_def by (induction A) auto + +lemma is_pref_profile_lift_profile [intro]: + assumes "lowered.is_pref_profile A" + shows "is_pref_profile (lift_profile A)" +proof - + interpret anon_papp_profile n_voters parties committee_size A + by fact + show ?thesis + using A_nonempty A_subset size_A + by unfold_locales + (auto simp: lift_profile_def size_mset_sum_mset image_mset.compositionality o_def) +qed + +sublocale lowered: anon_papp n_voters parties committee_size "r \ lift_profile" +proof + fix A assume "lowered.is_pref_profile A" + hence "is_pref_profile (lift_profile A)" + by blast + hence "is_committee (r (lift_profile A))" + using rule_wf by blast + thus "lowered.is_committee ((r \ lift_profile) A)" + by simp +qed + +sublocale lowered: weak_rep_anon_papp n_voters parties committee_size "r \ lift_profile" +proof + fix A x + assume A: "lowered.is_pref_profile A" and x: "n_voters \ committee_size * count A {x}" + from A have A': "is_pref_profile (lift_profile A)" + by blast + from x have "l * n_voters \ l * (committee_size * count A {x})" + by (rule mult_left_mono) auto + also have "\ = committee_size * count (lift_profile A) {x}" + by simp + finally have "x \# r (lift_profile A)" + by (intro weak_representation A') + thus "x \# (r \ lift_profile) A" + by simp +qed + +sublocale lowered: card_stratproof_anon_papp n_voters parties committee_size "r \ lift_profile" +proof + fix A X Y + show "\lowered.card_manipulable A X Y" + unfolding lowered.card_manipulable_def + proof (rule notI, elim conjE) + assume A: "lowered.is_pref_profile A" and XY: "X \# A" "Y \ {}" "Y \ parties" + assume *: "(r \ lift_profile) A \[lowered.committee_preference X] + (r \ lift_profile) (A - {#X#} + {#Y#})" + interpret anon_papp_profile n_voters parties committee_size A + by fact + have X: "X \ {}" "X \ parties" + using XY A_nonempty A_subset by auto + + define A' where "A' = A - {#X#}" + have A': "A = A' + {#X#}" + using XY by (simp add: A'_def) + + have "r (lift_profile A) \[committee_preference X] + r (lift_profile (A - {#X#} + {#Y#}))" + using * by simp + also have "r (lift_profile (A - {#X#} + {#Y#})) \[committee_preference X] + r (lift_profile A - {#X#} + {#Y#})" + proof - + have "r (replicate_mset (l - 1) Y + (lift_profile A' + {#Y#})) \[committee_preference X] + r (replicate_mset (l - 1) X + (lift_profile A' + {#Y#}))" + proof (rule proposition2) + show "size (lift_profile A' + {#Y#}) + (l - 1) = l * n_voters" + using XY l_pos n_voters_pos + by (simp add: A'_def size_lift_profile size_Diff_singleton + algebra_simps Suc_diff_le size_A) + next + show "{} \# lift_profile A' + {#Y#} + replicate_mset (l - 1) Y" + using XY A_nonempty by (auto simp: A'_def dest: in_diffD) + next + show "\X'\#lift_profile A' + {#Y#} + replicate_mset (l - 1) Y. X' \ parties" + using XY A_subset by (auto simp: A'_def dest: in_diffD) + qed (use X in auto) + thus ?thesis + by (simp add: A' replicate_mset_rec l_pos lift_profile_def) + qed + finally have "card_manipulable (lift_profile A) X Y" + unfolding card_manipulable_def using XY A by auto + with not_manipulable show False + by blast + qed +qed + +sublocale lowered: card_stratproof_weak_rep_anon_papp n_voters parties committee_size "r \ lift_profile" + .. + +end + + +locale divide_voters_card_stratproof_weak_prop_rep_anon_papp = + card_stratproof_weak_prop_rep_anon_papp "l * n_voters" parties committee_size r + for l n_voters parties committee_size r +begin + +sublocale divide_voters_card_stratproof_weak_rep_anon_papp .. + +sublocale lowered: card_stratproof_weak_prop_rep_anon_papp + n_voters parties committee_size "r \ lift_profile" +proof + fix A x l' + assume A: "lowered.is_pref_profile A" and x: "l' * n_voters \ committee_size * count A {x}" + from A have A': "is_pref_profile (lift_profile A)" + by blast + from x have "l * (l' * n_voters) \ l * (committee_size * count A {x})" + by (rule mult_left_mono) auto + also have "\ = committee_size * count (lift_profile A) {x}" + by simp + also have "l * (l' * n_voters) = l' * (l * n_voters)" + by (simp add: algebra_simps) + finally have "count (r (lift_profile A)) x \ l'" + by (intro weak_proportional_representation A') + thus "count ((r \ lift_profile) A) x \ l'" + by simp +qed + +end + + + +subsection \Decreasing the number of parties\ + +text \ + If we have a PAPP rule that satisfies weak representation and cardinality strategyproofness, + for $m$ parties, we can turn it into one for $m-1$ parties. This is done by simply + duplicating one particular party (say \x\) in the preference profile, i.e.\ whenever + \x\ is part of an approval list, we add a clone of \x\ (say \y\) as well. Should \y\ then end up + in the committee, we simply replace it with \x\. + + Consequently, if we have an impossibility result for $k$ parties, it also holds for + $\geq m$ parties. +\ +locale remove_alt_card_stratproof_weak_rep_anon_papp = + card_stratproof_weak_rep_anon_papp n_voters parties committee_size r + for n_voters and parties :: "'a set" and committee_size r + + fixes x y :: 'a + assumes xy: "x \ parties" "y \ parties" "x \ y" +begin + +definition lift_applist :: "'a set \ 'a set" where + "lift_applist X = (if x \ X then insert y X else X)" + +definition lift_profile :: "'a set multiset \ 'a set multiset" where + "lift_profile A = image_mset lift_applist A" + +definition lower_result where "lower_result C = image_mset (\z. if z = y then x else z) C" + +definition lowered where "lowered = lower_result \ r \ lift_profile" + + +lemma lift_profile_empty [simp]: "lift_profile {#} = {#}" + by (simp add: lift_profile_def) + +lemma lift_profile_add_mset [simp]: + "lift_profile (add_mset X A) = add_mset (lift_applist X) (lift_profile A)" + by (simp add: lift_profile_def) + +lemma empty_in_lift_profile_iff [simp]: "{} \# lift_profile A \ {} \# A" + by (auto simp: lift_applist_def lift_profile_def) + +lemma size_lift_profile [simp]: "size (lift_profile A) = size A" + by (simp add: lift_profile_def) + +lemma lift_applist_eq_self_iff [simp]: "lift_applist X = X \ x \ X \ y \ X" + by (auto simp: lift_applist_def) + +lemma lift_applist_eq_self_iff' [simp]: "lift_applist (X - {y}) = X \ (x \ X \ y \ X)" + by (cases "y \ X") (auto simp: lift_applist_def xy) + +lemma in_lift_applist_iff: "z \ lift_applist X \ z \ X \ (z = y \ x \ X)" + by (auto simp: lift_applist_def) + +lemma count_lift_profile: + assumes "\Y\#A. y \ Y" + shows "count (lift_profile A) X = (if x \ X \ y \ X then count A (X - {y}) else 0)" + using assms xy by (induction A) (auto simp: lift_applist_def) + +lemma y_notin_lower_result [simp]: "y \# lower_result C" + using xy by (auto simp: lower_result_def) + +lemma lower_result_subset: "set_mset (lower_result C) \ insert x (set_mset C - {y})" + using xy by (auto simp: lower_result_def) + +lemma lower_result_subset': "set_mset C \ parties \ set_mset (lower_result C) \ parties" + using xy by (auto simp: lower_result_def) + +lemma size_lower_result [simp]: "size (lower_result C) = size C" + by (simp add: lower_result_def) + +lemma count_lower_result: + "count (lower_result C) z = + (if z = y then 0 + else if z = x then count C x + count C y + else count C z)" + using xy by (induction C) (auto simp: lower_result_def) + +lemma in_lower_result_iff: + "z \# lower_result C \ z \ y \ (z \# C \ (z = x \ y \# C))" + unfolding lower_result_def using xy by (induction C) auto + + +sublocale lowered: anon_papp_election n_voters "parties - {y}" + by standard (use n_voters_pos xy in auto) + +lemma is_pref_profile_lift_profile [intro]: + assumes "lowered.is_pref_profile A" + shows "is_pref_profile (lift_profile A)" +proof - + interpret anon_papp_profile n_voters "parties - {y}" committee_size A + by fact + show ?thesis + using A_nonempty A_subset size_A + by unfold_locales + (auto simp: lift_profile_def lift_applist_def xy + size_mset_sum_mset image_mset.compositionality o_def) +qed + +sublocale lowered: anon_papp n_voters "parties - {y}" committee_size lowered +proof + fix A assume "lowered.is_pref_profile A" + hence "is_pref_profile (lift_profile A)" + by blast + hence "is_committee (r (lift_profile A))" + using rule_wf by blast + thus "lowered.is_committee (lowered A)" + unfolding lowered.is_committee_def is_committee_def lowered_def + using lower_result_subset'[of "r (lift_profile A)"] by auto +qed + +sublocale lowered: weak_rep_anon_papp n_voters "parties - {y}" committee_size lowered +proof + fix A z + assume A: "lowered.is_pref_profile A" and z: "n_voters \ committee_size * count A {z}" + interpret A: anon_papp_profile n_voters "parties - {y}" committee_size A + by fact + have "committee_size > 0" + using z n_voters_pos by (intro Nat.gr0I) auto + + from A have A': "is_pref_profile (lift_profile A)" + by blast + have "count A {z} > 0" + using z n_voters_pos by (intro Nat.gr0I) auto + hence "{z} \# A" + by simp + hence z': "z \ parties - {y}" + using A.A_subset z by auto + + define C where "C = r (lift_profile A)" + + show "z \# lowered A" + proof (cases "z = x") + case False + have "n_voters \ committee_size * count A {z}" + by fact + also have "count A {z} \ count (lift_profile A) {z}" + using A.A_subset z' False by (subst count_lift_profile) auto + hence "committee_size * count A {z} \ committee_size * count (lift_profile A) {z}" + by (intro mult_left_mono) auto + finally have "z \# r (lift_profile A)" + by (intro weak_representation A') + thus "z \# lowered A" + using False z' by (simp add: lowered_def in_lower_result_iff) + next + case [simp]: True + have "1 \ size {#z \# C. z \ {x, y}#}" + unfolding C_def + proof (rule proposition3) + have [simp]: "{x, y} - {y} = {x}" + using xy by auto + hence "n_voters \ committee_size * count (lift_profile A) {x, y}" + using xy A.A_subset z by (subst count_lift_profile) auto + thus "int 1 * \real n_voters / real committee_size\ \ int (count (lift_profile A) {x, y})" + using \committee_size > 0\ + by (auto simp: ceiling_le_iff field_simps simp flip: of_nat_mult) + qed (use A' xy \committee_size > 0\ in auto) + also have "\ = count C x + count C y" + using xy by (induction C) auto + also have "\ = count (lowered A) x" + using xy by (simp add: lowered_def count_lower_result C_def) + finally show "z \# lowered A" + by simp + qed +qed + +lemma filter_lower_result_eq: + "y \ X \ {#x \# lower_result C. x \ X#} = lower_result {#x \# C. x \ lift_applist X#}" + by (induction C) (auto simp: lower_result_def lift_applist_def) + +sublocale lowered: card_stratproof_anon_papp n_voters "parties - {y}" committee_size lowered +proof + fix A X Y + show "\lowered.card_manipulable A X Y" + unfolding lowered.card_manipulable_def + proof (rule notI, elim conjE) + assume A: "lowered.is_pref_profile A" and XY: "X \# A" "Y \ {}" "Y \ parties - {y}" + assume *: "lowered A \[lowered.committee_preference X] lowered (A - {#X#} + {#Y#})" + interpret anon_papp_profile n_voters "parties - {y}" committee_size A + by fact + have X: "X \ {}" "X \ parties - {y}" + using XY A_nonempty A_subset by auto + define A' where "A' = A - {#X#}" + have A': "A = A' + {#X#}" + using XY by (simp add: A'_def) + + from * have "size {#x \# lower_result (r (lift_profile A' + {#lift_applist X#})). x \ X#} < + size {#x \# lower_result (r (lift_profile A' + {#lift_applist Y#})). x \ X#}" + by (simp add: lowered_def A' lowered.strong_committee_preference_iff) + also have "{#x \# lower_result (r (lift_profile A' + {#lift_applist X#})). x \ X#} = + lower_result {#x \# r (lift_profile A' + {#lift_applist X#}). x \ lift_applist X#}" + using X by (subst filter_lower_result_eq) auto + also have "{#x \# lower_result (r (lift_profile A' + {#lift_applist Y#})). x \ X#} = + lower_result {#x \# r (lift_profile A' + {#lift_applist Y#}). x \ lift_applist X#}" + using X by (subst filter_lower_result_eq) auto + finally have "size {#x \# r (lift_profile A' + {#lift_applist X#}). x \ lift_applist X#} < + size {#x \# r (lift_profile A' + {#lift_applist Y#}). x \ lift_applist X#}" + by simp + hence "r (lift_profile A' + {#lift_applist X#}) \[committee_preference (lift_applist X)] + r (lift_profile A' + {#lift_applist Y#})" + by (simp add: strong_committee_preference_iff) + moreover have "\r (lift_profile A' + {#lift_applist X#}) \[committee_preference (lift_applist X)] + r (lift_profile A' + {#lift_applist Y#})" + proof (rule not_manipulable' [where Y = "lift_applist Y"]) + have "is_pref_profile (lift_profile A)" + using A by blast + thus "is_pref_profile (lift_profile A' + {#lift_applist X#})" + using A by (simp add: A') + next + have "is_pref_profile (lift_profile (A - {#X#} + {#Y#}))" + using A XY lowered.is_pref_profile_replace by blast + thus "is_pref_profile (lift_profile A' + {#lift_applist Y#})" + by (simp add: A') + qed auto + ultimately show False + by contradiction + qed +qed + +sublocale lowered: card_stratproof_weak_rep_anon_papp n_voters "parties - {y}" committee_size lowered + .. + +end + + +(* +locale remove_alt_card_stratproof_weak_prop_rep_anon_papp = + card_stratproof_weak_prop_rep_anon_papp n_voters parties committee_size r + for n_voters and parties :: "'a set" and committee_size r + + fixes x y :: 'a + assumes xy: "x \ parties" "y \ parties" "x \ y" +begin + +sublocale remove_alt_card_stratproof_weak_rep_anon_papp + by standard (use xy in auto) + +sublocale lowered: weak_prop_rep_anon_papp n_voters "parties - {y}" committee_size lowered +proof + fix A z l + assume A: "lowered.is_pref_profile A" and z: "l * n_voters \ committee_size * count A {z}" + interpret A: anon_papp_profile n_voters "parties - {y}" committee_size A + by fact + + show "count (lowered A) z \ l" + proof (cases "l = 0") + case True + thus ?thesis + by simp + next + case l: False + + have "committee_size > 0" + using z n_voters_pos l by (intro Nat.gr0I) auto + + from A have A': "is_pref_profile (lift_profile A)" + by blast + have "count A {z} > 0" + using z n_voters_pos l by (intro Nat.gr0I) auto + hence "{z} \# A" + by simp + hence z': "z \ parties - {y}" + using A.A_subset z by auto + + define C where "C = r (lift_profile A)" + + show ?thesis + proof (cases "z = x") + case False + have "l * n_voters \ committee_size * count A {z}" + by fact + also have "count A {z} \ count (lift_profile A) {z}" + using A.A_subset z' False by (subst count_lift_profile) auto + hence "committee_size * count A {z} \ committee_size * count (lift_profile A) {z}" + by (intro mult_left_mono) auto + finally have "count (r (lift_profile A)) z \ l" + by (intro weak_proportional_representation A') + thus ?thesis + using False z' by (simp add: lowered_def count_lower_result) + next + case [simp]: True + show ?thesis + sorry + qed + qed +qed + +end +*) + + +text \ + The following lemma is now simply an iterated application of the above. This allows us to + restrict a P-APP rule to any non-empty subset of parties. +\ +lemma card_stratproof_weak_rep_anon_papp_restrict_parties: + assumes "card_stratproof_weak_rep_anon_papp n parties k r" "parties' \ parties" "parties' \ {}" + shows "\r. card_stratproof_weak_rep_anon_papp n parties' k r" +proof - + have "finite parties" + proof - + interpret card_stratproof_weak_rep_anon_papp n parties k r + by fact + show ?thesis + by (rule finite_parties) + qed + thus ?thesis + using assms + proof (induction parties arbitrary: r rule: finite_psubset_induct) + case (psubset parties r) + show ?thesis + proof (cases "parties = parties'") + case True + thus ?thesis + using psubset.prems by auto + next + case False + obtain x where x: "x \ parties'" + using psubset.prems by blast + from False and psubset.prems have "parties - parties' \ {}" + by auto + then obtain y where y: "y \ parties - parties'" + by blast + + interpret card_stratproof_weak_rep_anon_papp n parties k r + by fact + interpret remove_alt_card_stratproof_weak_rep_anon_papp n parties k r x y + by standard (use x y psubset.prems in auto) + show ?thesis + proof (rule psubset.IH) + show "parties - {y} \ parties" and "parties' \ parties - {y}" "parties' \ {}" + using x y psubset.prems by auto + next + show "card_stratproof_weak_rep_anon_papp n (parties - {y}) k lowered" + using lowered.card_stratproof_weak_rep_anon_papp_axioms . + qed + qed + qed +qed + + + +subsection \Decreasing the committee size\ + +text \ + If we have a PAPP rule that satisfies weak representation and cardinality strategyproofness, + for $l(k+1)$ voters, $m+1$ parties, and a committee size of $k+1$, we can turn it into + one for $lk$ voters, $m$ parties, and a committee size of $k$. + + This is done by again cloning a party \x\ into a new party \y\ and additionally + adding \l\ new voters whose preferences are \{x, y}\. We again replace any \y\ occuring in the + output committee with \x\. Weak representation then ensures that \x\ occurs in the output + at least once, and we then simply remove one \x\ from the committee to obtain an output + committee of size \k - 1\. + + Consequently, if we have an impossibility result for a committee size of $m$, we can + extend it to a larger committee size, but at the cost of introducing a new party + and new voters, and with a restriction on the number of voters. +\ +(* n / k = l *) +locale decrease_committee_card_stratproof_weak_rep_anon_papp = + card_stratproof_weak_rep_anon_papp "l * (k + 1)" "insert y parties" "k + 1" r + for l k y and parties :: "'a set" and r + + fixes x :: 'a + assumes xy: "x \ parties" "y \ parties" + assumes k: "k > 0" +begin + +definition lift_applist :: "'a set \ 'a set" where + "lift_applist X = (if x \ X then insert y X else X)" + +definition lift_profile :: "'a set multiset \ 'a set multiset" where + "lift_profile A = image_mset lift_applist A + replicate_mset l {x, y}" + +definition lower_result + where "lower_result C = image_mset (\z. if z = y then x else z) C - {#x#}" + +definition lowered where "lowered = lower_result \ r \ lift_profile" + + +lemma l: "l > 0" + using n_voters_pos by auto + +lemma x_neq_y [simp]: "x \ y" "y \ x" + using xy by auto + +lemma lift_profile_empty [simp]: "lift_profile {#} = replicate_mset l {x, y}" + by (simp add: lift_profile_def) + +lemma lift_profile_add_mset [simp]: + "lift_profile (add_mset X A) = add_mset (lift_applist X) (lift_profile A)" + by (simp add: lift_profile_def) + +lemma empty_in_lift_profile_iff [simp]: "{} \# lift_profile A \ {} \# A" + by (auto simp: lift_applist_def lift_profile_def) + +lemma size_lift_profile [simp]: "size (lift_profile A) = size A + l" + by (simp add: lift_profile_def) + +lemma lift_applist_eq_self_iff [simp]: "lift_applist X = X \ x \ X \ y \ X" + by (auto simp: lift_applist_def) + +lemma lift_applist_eq_self_iff' [simp]: "lift_applist (X - {y}) = X \ (x \ X \ y \ X)" + by (cases "y \ X") (auto simp: lift_applist_def xy) + +lemma in_lift_applist_iff: "z \ lift_applist X \ z \ X \ (z = y \ x \ X)" + by (auto simp: lift_applist_def) + +lemma count_lift_profile: + assumes "\Y\#A. y \ Y" + shows "count (lift_profile A) X = + (if x \ X \ y \ X then count A (X - {y}) else 0) + + (if X = {x, y} then l else 0)" + using assms xy by (induction A) (auto simp: lift_applist_def) + + + +lemma y_notin_lower_result [simp]: "y \# lower_result C" + using xy by (auto simp: lower_result_def dest: in_diffD) + +lemma lower_result_subset: "set_mset (lower_result C) \ insert x (set_mset C - {y})" + using xy by (auto simp: lower_result_def dest: in_diffD) + +lemma lower_result_subset': "set_mset C \ insert y parties \ set_mset (lower_result C) \ parties" + by (rule order.trans[OF lower_result_subset]) (use xy in auto) + +lemma size_lower_result [simp]: + assumes "x \# C \ y \# C" + shows "size (lower_result C) = size C - 1" + using assms by (auto simp: lower_result_def size_Diff_singleton) + +lemma size_lower_result': "size (lower_result C) = size C - (if x \# C \ y \# C then 1 else 0)" +proof - + define f where "f = (\C. image_mset (\z. if z = y then x else z) C)" + have "size (lower_result C) = size (f C - {#x#})" + by (simp add: lower_result_def f_def) + also have "\ = size (f C) - (if x \# f C then 1 else 0)" + by (simp add: diff_single_trivial size_Diff_singleton) + finally show ?thesis + by (auto simp: f_def) +qed + +lemma count_lower_result: + "count (lower_result C) z = + (if z = y then 0 + else if z = x then count C x + count C y - 1 + else count C z)" (is "_ = ?rhs") +proof - + define f where "f = (\C. image_mset (\z. if z = y then x else z) C)" + have "count (lower_result C) z = count (f C - {#x#}) z" + by (simp add: lower_result_def f_def) + also have "\ = count (f C) z - (if z = x then 1 else 0)" + by auto + also have "count (f C) z = (if z = y then 0 else if z = x then count C x + count C y else count C z)" + using xy by (induction C) (auto simp: f_def) + also have "\ - (if z = x then 1 else 0) = ?rhs" + by auto + finally show ?thesis . +qed + +lemma in_lower_resultD: + "z \# lower_result C \ z = x \ z \# C" + using xy by (auto simp: lower_result_def dest!: in_diffD) + +lemma in_lower_result_iff: + "z \# lower_result C \ z \ y \ (if z = x then count C x + count C y > 1 else z \# C)" + (is "_ = ?rhs") +proof - + have "z \# lower_result C \ count (lower_result C) z > 0" + by auto + also have "\ \ ?rhs" + by (subst count_lower_result) auto + finally show ?thesis . +qed + +lemma filter_lower_result_eq: + assumes "y \ X" + shows "{#z \# lower_result C. z \ X#} = lower_result {#z \# C. z \ lift_applist X#}" +proof - + define f where "f = (\C. {#if z = y then x else z. z \# C#})" + have "lower_result {#z \# C. z \ lift_applist X#} = f {#z \# C. z \ lift_applist X#} - {#x#}" + by (simp add: f_def lower_result_def) + also have "f {#z \# C. z \ lift_applist X#} = {#z \# f C. z \ X#}" + using assms by (induction C) (auto simp: f_def lift_applist_def) + also have "\ - {#x#} = {#z \# f C - {#x#}. z \ X#}" + by (subst filter_diff_mset') auto + also have "f C - {#x#} = lower_result C" + by (simp add: f_def lower_result_def) + finally show ?thesis .. +qed + + +sublocale lowered: anon_papp_election "l * k" parties k + by standard (use n_voters_pos xy finite_parties k in auto) + +lemma is_pref_profile_lift_profile [intro]: + assumes "lowered.is_pref_profile A" + shows "is_pref_profile (lift_profile A)" +proof - + interpret A: anon_papp_profile "l * k" parties k A + by fact + show ?thesis + using A.A_nonempty A.A_subset A.size_A l + by unfold_locales + (auto simp: lift_profile_def lift_applist_def xy + size_mset_sum_mset image_mset.compositionality o_def) +qed + +lemma is_committee_lower_result: + assumes "is_committee C" "x \# C \ y \# C" + shows "lowered.is_committee (lower_result C)" + using assms unfolding is_committee_def lowered.is_committee_def + using lower_result_subset'[of C] by auto + +lemma x_or_y_in_r_lift_profile: + assumes "lowered.is_pref_profile A" + shows "x \# r (lift_profile A) \ y \# r (lift_profile A)" +proof - + interpret A: anon_papp_profile "l * k" parties k A + by fact + have "size {#z \# r (lift_profile A). z \ {x, y}#} \ 1" + proof (rule proposition3) + have "real (l * (k + 1)) / real (k + 1) = real l" + by (simp add: field_simps) + also have "int 1 * \\\ = int l" + by simp + also have "l \ count (lift_profile A) {x, y}" + using xy A.A_subset by (subst count_lift_profile) auto + finally show "int 1 * \real (l * (k + 1)) / real (k + 1)\ \ int (count (lift_profile A) {x, y})" + by simp + next + show "is_pref_profile (lift_profile A)" + by (intro is_pref_profile_lift_profile) fact + qed (use xy in auto) + hence "{#z \# r (lift_profile A). z \ {x, y}#} \ {#}" + by force + thus ?thesis + by auto +qed + +sublocale lowered: anon_papp "l * k" parties k lowered +proof + fix A assume A: "lowered.is_pref_profile A" + hence "is_pref_profile (lift_profile A)" + by blast + hence "is_committee (r (lift_profile A))" + using rule_wf by blast + thus "lowered.is_committee (lowered A)" + unfolding lowered_def o_def using x_or_y_in_r_lift_profile[of A] A + by (intro is_committee_lower_result) auto +qed + +sublocale lowered: weak_rep_anon_papp "l * k" parties k lowered +proof + fix A z + assume A: "lowered.is_pref_profile A" and z: "l * k \ k * count A {z}" + interpret A: anon_papp_profile "l * k" "parties" k A + by fact + + from A have A': "is_pref_profile (lift_profile A)" + by blast + have "count A {z} > 0" + using z k n_voters_pos by (intro Nat.gr0I) auto + hence "{z} \# A" + by simp + hence z': "z \ parties" + using A.A_subset z by auto + hence [simp]: "y \ z" "z \ y" + using xy by auto + + define C where "C = r (lift_profile A)" + + show "z \# lowered A" + proof (cases "z = x") + case False + have "l \ count A {z}" + using z k by (simp add: algebra_simps) + hence "l * (k + 1) \ (k + 1) * count A {z}" + by (subst mult.commute, intro mult_right_mono) auto + also have "count A {z} = count (lift_profile A) {z}" + using False A.A_subset xy by (subst count_lift_profile) auto + finally have "z \# r (lift_profile A)" + by (intro weak_representation A') + thus "z \# lowered A" + using False by (auto simp: lowered_def in_lower_result_iff) + next + case [simp]: True + from xy have [simp]: "{x, y} - {y} = {x}" + by auto + + have "size {#z \# C. z \ {x, y}#} \ 2" + unfolding C_def + proof (rule proposition3) + have "real (l * (k + 1)) / real (k + 1) = l" + unfolding of_nat_mult using k by (simp add: divide_simps) + also have "int 2 * \\\ = int (2 * l)" + by simp + also have "\ \ count (lift_profile A) {x, y}" + using z k xy A.A_subset by (subst count_lift_profile) auto + finally show "int 2 * \real (l * (k + 1)) / real (k + 1)\ \ \" . + qed (use A' xy in auto) + also have "size {#z \# C. z \ {x, y}#} = count C x + count C y" + by (induction C) auto + finally have "x \# lower_result C" + by (subst in_lower_result_iff) auto + thus "z \# lowered A" + by (simp add: lowered_def C_def) + qed +qed + +sublocale lowered: card_stratproof_anon_papp "l * k" parties k lowered +proof + fix A X Y + show "\lowered.card_manipulable A X Y" + unfolding lowered.card_manipulable_def + proof (rule notI, elim conjE) + assume A: "lowered.is_pref_profile A" and XY: "X \# A" "Y \ {}" "Y \ parties" + assume *: "lowered A \[lowered.committee_preference X] lowered (A - {#X#} + {#Y#})" + interpret anon_papp_profile "l * k" "parties" k A + by fact + have X: "X \ {}" "X \ parties" + using XY A_nonempty A_subset by auto + define A' where "A' = A - {#X#}" + have A': "A = A' + {#X#}" + using XY by (simp add: A'_def) + from xy X XY have [simp]: "y \ X" "y \ Y" + by auto + + define Al1 where "Al1 = lift_profile A" + define Al2 where "Al2 = lift_profile (A' + {#Y#})" + have A'_plus_Y: "lowered.is_pref_profile (A' + {#Y#})" + unfolding A'_def using A XY lowered.is_pref_profile_replace by blast + have Al1: "is_pref_profile Al1" + unfolding Al1_def using A by blast + have Al2: "is_pref_profile Al2" + unfolding Al2_def unfolding A'_def using A XY lowered.is_pref_profile_replace by blast + + have size_aux: "size (lower_result {#x \# r (lift_profile A). x \ lift_applist X#}) = + size {#x \# r (lift_profile A). x \ lift_applist X#} - (if x \ X then 1 else 0)" + if A: "lowered.is_pref_profile A" for A + using x_or_y_in_r_lift_profile[OF A] + by (subst size_lower_result') (auto simp: lift_applist_def) + + from * have "size {#x \# lower_result (r Al1). x \ X#} < + size {#x \# lower_result (r Al2). x \ X#}" + by (simp add: Al1_def Al2_def lowered_def A' lowered.strong_committee_preference_iff) + also have "{#x \# lower_result (r Al1). x \ X#} = lower_result {#x \# r Al1. x \ lift_applist X#}" + using X xy by (subst filter_lower_result_eq) auto + also have "{#x \# lower_result (r Al2). x \ X#} = lower_result {#x \# r Al2. x \ lift_applist X#}" + using X xy by (subst filter_lower_result_eq) auto + also have "size (lower_result {#x \# r Al1. x \ lift_applist X#}) = + size {#x \# r Al1. x \ lift_applist X#} - (if x \ X then 1 else 0)" + unfolding Al1_def by (rule size_aux) fact + also have "size (lower_result {#x \# r Al2. x \ lift_applist X#}) = + size {#x \# r Al2. x \ lift_applist X#} - (if x \ X then 1 else 0)" + unfolding Al2_def by (rule size_aux) fact + finally have "size {#x \# r Al1. x \ lift_applist X#} < size {#x \# r Al2. x \ lift_applist X#}" + by auto + hence "r Al1 \[committee_preference (lift_applist X)] r Al2" + by (simp add: strong_committee_preference_iff) + + moreover have "\r Al1 \[committee_preference (lift_applist X)] r Al2" + by (rule not_manipulable' [where Y = "lift_applist Y"]) + (use Al1 Al2 in \auto simp: Al1_def Al2_def A'\) + + ultimately show False + by contradiction + qed +qed + +sublocale lowered: card_stratproof_weak_rep_anon_papp "l * k" parties k lowered + .. + +end + + +text \ + For Weak \<^emph>\Proportional\ Representation, the lowering argument to decrease the committee size + is somewhat easier since it does not involve adding a new party; instead, we simply + add \l\ new voters whose preferences are \{x}\. +\ +locale decrease_committee_card_stratproof_weak_prop_rep_anon_papp = + card_stratproof_weak_prop_rep_anon_papp "l * (k + 1)" parties "k + 1" r + for l k and parties :: "'a set" and r + + fixes x :: 'a + assumes x: "x \ parties" + assumes k: "k > 0" +begin + +definition lift_profile :: "'a set multiset \ 'a set multiset" where + "lift_profile A = A + replicate_mset l {x}" + +definition lower_result + where "lower_result C = C - {#x#}" + +definition lowered where "lowered = lower_result \ r \ lift_profile" + + +lemma l: "l > 0" + using n_voters_pos by auto + +lemma lift_profile_empty [simp]: "lift_profile {#} = replicate_mset l {x}" + by (simp add: lift_profile_def) + +lemma lift_profile_add_mset [simp]: + "lift_profile (add_mset X A) = add_mset X (lift_profile A)" + by (simp add: lift_profile_def) + +lemma empty_in_lift_profile_iff [simp]: "{} \# lift_profile A \ {} \# A" + by (auto simp: lift_profile_def) + +lemma size_lift_profile [simp]: "size (lift_profile A) = size A + l" + by (simp add: lift_profile_def) + +lemma count_lift_profile: + "count (lift_profile A) X = count A X + (if X = {x} then l else 0)" + by (auto simp: lift_profile_def) + + +lemma size_lower_result [simp]: + assumes "x \# C" + shows "size (lower_result C) = size C - 1" + using assms by (auto simp: lower_result_def size_Diff_singleton) + +lemma size_lower_result': "size (lower_result C) = size C - (if x \# C then 1 else 0)" + by (induction C) (auto simp: lower_result_def size_Diff_singleton) + +lemma count_lower_result: + "count (lower_result C) z = count C z - (if z = x then 1 else 0)" + by (auto simp: lower_result_def) + +lemma in_lower_resultD: + "z \# lower_result C \ z \# C" + by (auto simp: lower_result_def dest!: in_diffD) + +lemma in_lower_result_iff: + "z \# lower_result C \ (if z = x then count C x > 1 else z \# C)" + (is "_ = ?rhs") +proof - + have "z \# lower_result C \ count (lower_result C) z > 0" + by auto + also have "\ \ ?rhs" + by (subst count_lower_result) auto + finally show ?thesis . +qed + + +sublocale lowered: anon_papp_election "l * k" parties k + by standard (use n_voters_pos finite_parties k in auto) + +lemma is_pref_profile_lift_profile [intro]: + assumes "lowered.is_pref_profile A" + shows "is_pref_profile (lift_profile A)" +proof - + interpret A: anon_papp_profile "l * k" parties k A + by fact + show ?thesis + using A.A_nonempty A.A_subset A.size_A l + by unfold_locales + (auto simp: lift_profile_def x size_mset_sum_mset image_mset.compositionality o_def) +qed + +lemma is_committee_lower_result: + assumes "is_committee C" "x \# C" + shows "lowered.is_committee (lower_result C)" + using assms unfolding is_committee_def lowered.is_committee_def + by (auto simp: lower_result_def size_Diff_singleton dest: in_diffD) + +lemma x_in_r_lift_profile: + assumes "lowered.is_pref_profile A" + shows "x \# r (lift_profile A)" +proof (rule weak_representation) + show "is_pref_profile (lift_profile A)" + using assms by blast +next + have "(k + 1) * l \ (k + 1) * count (lift_profile A) {x}" + by (intro mult_left_mono) (auto simp: count_lift_profile) + thus "l * (k + 1) \ (k + 1) * count (lift_profile A) {x}" + by (simp add: algebra_simps) +qed + +sublocale lowered: anon_papp "l * k" parties k lowered +proof + fix A assume A: "lowered.is_pref_profile A" + hence "is_pref_profile (lift_profile A)" + by blast + hence "is_committee (r (lift_profile A))" + using rule_wf by blast + thus "lowered.is_committee (lowered A)" + unfolding lowered_def o_def using x_in_r_lift_profile[of A] A + by (intro is_committee_lower_result) auto +qed + +sublocale lowered: weak_prop_rep_anon_papp "l * k" parties k lowered +proof + fix A z l' + assume A: "lowered.is_pref_profile A" and z: "l' * (l * k) \ k * count A {z}" + interpret A: anon_papp_profile "l * k" "parties" k A + by fact + + show "count (lowered A) z \ l'" + proof (cases "l' > 0") + case False + thus ?thesis by auto + next + case l: True + + from A have A': "is_pref_profile (lift_profile A)" + by blast + have "count A {z} > 0" + using z k n_voters_pos l by (intro Nat.gr0I) auto + hence "{z} \# A" + by simp + hence z': "z \ parties" + using A.A_subset z by auto + + define C where "C = r (lift_profile A)" + + show "count (lowered A) z \ l'" + proof (cases "z = x") + case False + have "l' * l \ count A {z}" + using z k by (simp add: algebra_simps) + hence "l' * l * (k + 1) \ (k + 1) * count A {z}" + by (subst mult.commute, intro mult_right_mono) auto + also have "count A {z} = count (lift_profile A) {z}" + using False A.A_subset by (subst count_lift_profile) auto + finally have "count (r (lift_profile A)) z \ l'" + by (intro weak_proportional_representation A') (auto simp: algebra_simps) + thus "l' \ count (lowered A) z" + using False by (simp add: lowered_def lower_result_def) + next + case [simp]: True + have "l' * l \ count A {x}" + using z k by (simp add: algebra_simps) + hence "l' * l * (k + 1) \ (k + 1) * count A {x}" + by (subst mult.commute, intro mult_right_mono) auto + also have "\ + (k + 1) * l = (k + 1) * count (lift_profile A) {x}" + by (simp add: count_lift_profile algebra_simps) + finally have "(l' + 1) * (l * (k + 1)) \ (k + 1) * count (lift_profile A) {x}" + by (simp add: algebra_simps) + hence "count (r (lift_profile A)) x \ l' + 1" + by (intro weak_proportional_representation A') + thus "l' \ count (lowered A) z" + by (simp add: lowered_def lower_result_def) + qed + qed +qed + +sublocale lowered: card_stratproof_anon_papp "l * k" parties k lowered +proof + fix A X Y + show "\lowered.card_manipulable A X Y" + unfolding lowered.card_manipulable_def + proof (rule notI, elim conjE) + assume A: "lowered.is_pref_profile A" and XY: "X \# A" "Y \ {}" "Y \ parties" + assume *: "lowered A \[lowered.committee_preference X] lowered (A - {#X#} + {#Y#})" + interpret anon_papp_profile "l * k" "parties" k A + by fact + have X: "X \ {}" "X \ parties" + using XY A_nonempty A_subset by auto + define A' where "A' = A - {#X#}" + have A': "A = A' + {#X#}" + using XY by (simp add: A'_def) + + define Al1 where "Al1 = lift_profile A" + define Al2 where "Al2 = lift_profile (A' + {#Y#})" + have A'_plus_Y: "lowered.is_pref_profile (A' + {#Y#})" + unfolding A'_def using A XY lowered.is_pref_profile_replace by blast + have Al1: "is_pref_profile Al1" + unfolding Al1_def using A by blast + have Al2: "is_pref_profile Al2" + unfolding Al2_def unfolding A'_def using A XY lowered.is_pref_profile_replace by blast + + have size_aux: "size (lower_result {#x \# r (lift_profile A). x \ X#}) = + size {#x \# r (lift_profile A). x \ X#} - (if x \ X then 1 else 0)" + if A: "lowered.is_pref_profile A" for A + using x_in_r_lift_profile[OF A] by (subst size_lower_result') auto + + from * have "size {#x \# lower_result (r Al1). x \ X#} < + size {#x \# lower_result (r Al2). x \ X#}" + by (simp add: Al1_def Al2_def lowered_def A' lowered.strong_committee_preference_iff) + also have "{#x \# lower_result (r Al1). x \ X#} = lower_result {#x \# r Al1. x \ X#}" + using X x unfolding lower_result_def by (subst filter_diff_mset') auto + also have "{#x \# lower_result (r Al2). x \ X#} = lower_result {#x \# r Al2. x \ X#}" + using X x unfolding lower_result_def by (subst filter_diff_mset') auto + also have "size (lower_result {#x \# r Al1. x \ X#}) = + size {#x \# r Al1. x \ X#} - (if x \ X then 1 else 0)" + unfolding Al1_def by (rule size_aux) fact + also have "size (lower_result {#x \# r Al2. x \ X#}) = + size {#x \# r Al2. x \ X#} - (if x \ X then 1 else 0)" + unfolding Al2_def by (rule size_aux) fact + finally have "size {#x \# r Al1. x \ X#} < size {#x \# r Al2. x \ X#}" + by auto + hence "r Al1 \[committee_preference X] r Al2" + by (simp add: strong_committee_preference_iff) + + moreover have "\r Al1 \[committee_preference X] r Al2" + by (rule not_manipulable' [where Y = "Y"]) + (use Al1 Al2 in \auto simp: Al1_def Al2_def A'\) + + ultimately show False + by contradiction + qed +qed + +sublocale lowered: card_stratproof_weak_prop_rep_anon_papp "l * k" parties k lowered + .. + +end + +end diff --git a/thys/PAPP_Impossibility/PAPP_Impossibility.thy b/thys/PAPP_Impossibility/PAPP_Impossibility.thy new file mode 100644 --- /dev/null +++ b/thys/PAPP_Impossibility/PAPP_Impossibility.thy @@ -0,0 +1,179 @@ +(* + File: PAPP_Impossibility.thy + Author: Manuel Eberl, University of Innsbruck +*) +section \Lifting the Impossibility Result to Larger Settings\ +theory PAPP_Impossibility + imports PAPP_Impossibility_Base_Case Anonymous_PAPP_Lowering +begin + +text \ + In this section, we now prove the main results of this work by combining the base case + with the lifting arguments formalized earlier. + + First, we prove the following very simple technical lemma: a set that is infinite or finite with + cardinality at least 2 contains two different elements \x\ and \y\. +\ + +lemma obtain_2_elements: + assumes "infinite X \ card X \ 2" + obtains x y where "x \ X" "y \ X" "x \ y" +proof - + from assms have "X \ {}" + by auto + then obtain x where "x \ X" + by blast + with assms have "infinite X \ card (X - {x}) > 0" + by (subst card_Diff_subset) auto + hence "X - {x} \ {}" + by (metis card_gt_0_iff finite.emptyI infinite_remove) + then obtain y where "y \ X - {x}" + by blast + with \x \ X\ show ?thesis + using that[of x y] by blast +qed + +text \ + We now have all the ingredients to formalise the first main impossibility result: There is + no P-APP rule that satisfies Anonymity, Cardinality-Strategyproofness, and Weak Representation + if \k \ 3\ and \m \ k + 1\ and \n\ is a multiple of \2k\. + + The proof simply uses the lowering lemmas we proved earlier to first reduce the committee size + to 3, then reduce the voters to 6, and finally restrict the parties to 4. At that point, + the base case we proved with SAT solving earlier kicks in. + + This corresponds to Theorem~1 in the paper. +\ +theorem papp_impossibility1: + assumes "k \ 3" and "card parties \ k + 1" and "finite parties" + shows "\card_stratproof_weak_rep_anon_papp (2 * k * l) parties k r" + using assms +proof (induction k arbitrary: parties r rule: less_induct) + case (less k parties r) + show ?case + proof (cases "k = 3") + assume [simp]: "k = 3" + text \ + If the committee size is 3, we first use our voter-division lemma to + go from a P-APP rule for $6l$ voters to one with just 6 voters. Next, we choose 4 arbitrary + parties and use our party-restriction lemma to obtain a P-APP rule for just 4 parties. + + But this is exactly our base case, which we already know to be infeasible. + \ + show ?thesis + proof + assume "card_stratproof_weak_rep_anon_papp (2 * k * l) parties k r" + then interpret card_stratproof_weak_rep_anon_papp "l * 6" parties 3 r + by (simp add: mult_ac) + interpret divide_voters_card_stratproof_weak_rep_anon_papp l 6 parties 3 r .. + + have "card parties \ 4" + using less.prems by auto + then obtain parties' where parties': "parties' \ parties" "card parties' = 4" + by (metis obtain_subset_with_card_n) + have "\r. card_stratproof_weak_rep_anon_papp 6 parties' 3 r" + proof (rule card_stratproof_weak_rep_anon_papp_restrict_parties) + show "card_stratproof_weak_rep_anon_papp 6 parties 3 (r \ lift_profile)" + by (rule lowered.card_stratproof_weak_rep_anon_papp_axioms) + qed (use parties' in auto) + thus False + using papp_impossibility_base_case[OF parties'(2)] by blast + qed + next + assume "k \ 3" + text \ + If the committee size is greater than 3, we use our other lowering lemma to reduce the + committee size by 1 (while also reducing the number of voters by $2l$ and the number of + parties by 1). + \ + with less.prems have "k > 3" + by simp + + obtain x y where xy: "x \ parties" "y \ parties" "x \ y" + using obtain_2_elements[of parties] less.prems by auto + define parties' where "parties' = parties - {y}" + have [simp]: "card parties' = card parties - 1" + unfolding parties'_def using xy by (subst card_Diff_subset) auto + + show ?thesis + proof + assume "card_stratproof_weak_rep_anon_papp (2 * k * l) parties k r" + then interpret card_stratproof_weak_rep_anon_papp + "2 * l * (k - 1 + 1)" "insert y parties'" "k - 1 + 1" r + using \k > 3\ xy by (simp add: parties'_def insert_absorb mult_ac) + interpret decrease_committee_card_stratproof_weak_rep_anon_papp "2 * l" "k - 1" y parties' r x + by unfold_locales (use \k > 3\ xy in \auto simp: parties'_def\) + have "\card_stratproof_weak_rep_anon_papp (2 * (k - 1) * l) parties' (k - 1) lowered" + by (rule less.IH) (use \k > 3\ xy less.prems in auto) + with lowered.card_stratproof_weak_rep_anon_papp_axioms show False + by (simp add: mult_ac) + qed + qed +qed + +text \ + If Weak Representation is replaced with Weak Proportional Representation, we can strengthen + the impossibility result by relaxing the conditions on the number of parties to \m \ 4\. + + This works because with Weak Proportional Representation, we can reduce the size of the committee + without changing the number of parties. We use this to again bring $k$ down to $3$ without + changing $m$, at which point we can simply apply our previous impossibility result for + Weak Representation. + + This corresponds to Theorem~2 in the paper. +\ +corollary papp_impossibility2: + assumes "k \ 3" and "card parties \ 4" and "finite parties" + shows "\card_stratproof_weak_prop_rep_anon_papp (2 * k * l) parties k r" + using assms +proof (induction k arbitrary: parties r rule: less_induct) + case (less k parties r) + show ?case + proof (cases "k = 3") + assume [simp]: "k = 3" + text \ + For committee size 3, we simply employ our previous impossibility result: + \ + show ?thesis + proof + assume "card_stratproof_weak_prop_rep_anon_papp (2 * k * l) parties k r" + then interpret card_stratproof_weak_prop_rep_anon_papp "l * 6" parties 3 r + by (simp add: mult_ac) + have "card_stratproof_weak_rep_anon_papp (l * 6) parties 3 r" .. + moreover have "\card_stratproof_weak_rep_anon_papp (l * 6) parties 3 r" + using papp_impossibility1[of 3 parties l r] less.prems by (simp add: mult_ac) + ultimately show False + by contradiction + qed + next + assume "k \ 3" + text \ + If the committee size is greater than 3, we use our other lowering lemma to reduce the + committee size by 1 (while also reducing the number of voters by $2l$). + \ + + with less.prems have "k > 3" + by simp + + have "parties \ {}" + using less.prems by auto + then obtain x where x: "x \ parties" + by blast + + show ?thesis + proof + assume "card_stratproof_weak_prop_rep_anon_papp (2 * k * l) parties k r" + then interpret card_stratproof_weak_prop_rep_anon_papp + "2 * l * (k - 1 + 1)" parties "k - 1 + 1" r + using \k > 3\ by (simp add: mult_ac) + interpret decrease_committee_card_stratproof_weak_prop_rep_anon_papp "2 * l" "k - 1" parties r x + by unfold_locales (use \k > 3\ x in auto) + have "\card_stratproof_weak_prop_rep_anon_papp (2 * (k - 1) * l) parties (k - 1) lowered" + by (rule less.IH) (use \k > 3\ less.prems in auto) + with lowered.card_stratproof_weak_prop_rep_anon_papp_axioms show False + by (simp add: mult_ac) + qed + qed +qed + +end diff --git a/thys/PAPP_Impossibility/PAPP_Impossibility_Base_Case.thy b/thys/PAPP_Impossibility/PAPP_Impossibility_Base_Case.thy new file mode 100644 --- /dev/null +++ b/thys/PAPP_Impossibility/PAPP_Impossibility_Base_Case.thy @@ -0,0 +1,714 @@ +(* + File: PAPP_Impossibility_Base_Case.thy + Author: Manuel Eberl, University of Innsbruck +*) +section \The Base Case of the Impossibility\ +theory PAPP_Impossibility_Base_Case + imports Anonymous_PAPP SAT_Replay +begin + +text \ + In this section, we will prove the base case of our P-APP impossibility result, namely that + there exists no anonymous P-APP rule \f\ for 6 voters, 4 parties, and committee size 3 that + satisfies Weak Representation and Cardinality Strategyproofness. + + The proof works by looking at some (comparatively small) set of preference profiles and the + set of all 20 possible output committees. Each proposition $f(A) = C$ (where \A\ is a profile + from our set and \C\ is one of the 20 possible output committees) is considered as a Boolean + variable. + + All the conditions arising on these variables based on the fact that \f\ is a function + and the additional properties (Representation, Strategyproofness) are encoded as SAT clauses. + This SAT problem is then proven unsatisfiable by an external SAT solver and the resulting + proof re-imported into Isabelle/HOL. +\ + +subsection \Auxiliary Material\ + +text \ + We define the set of committees of the given size \k\ for a given set of parties \P\. +\ +definition committees :: "nat \ 'a set \ 'a multiset set" where + "committees k P = {W. set_mset W \ P \ size W = k}" + +text \ + We now prove a recurrence for this set so that we can more easily compute the set of all + possible committees: +\ +lemma committees_0 [simp]: "committees 0 P = {{#}}" + by (auto simp: committees_def) + +lemma committees_Suc: + "committees (Suc n) P = (\x\P. \W\committees n P. {{#x#} + W})" +proof safe + fix C assume C: "C \ committees (Suc n) P" + hence "size C = Suc n" + by (auto simp: committees_def) + hence "C \ {#}" + by auto + then obtain x where x: "x \# C" + by auto + define C' where "C' = C - {#x#}" + have "C = {#x#} + C'" "x \ P" "C' \ committees n P" + using C x by (auto simp: committees_def C'_def size_Diff_singleton dest: in_diffD) + thus "C \ (\x\P. \W\committees n P. {{#x#} + W})" + by blast +qed (auto simp: committees_def) + +text \ + The following function takes a list $[a_1, \ldots, a_n]$ and computes the list of all pairs + of the form $(a_i, a_j)$ with $i < j$: +\ +fun pairs :: "'a list \ ('a \ 'a) list" where + "pairs [] = []" +| "pairs (x # xs) = map (\y. (x, y)) xs @ pairs xs" + +lemma distinct_conv_pairs: "distinct xs \ list_all (\(x,y). x \ y) (pairs xs)" + by (induction xs) (auto simp: list_all_iff) + +lemma list_ex_unfold: "list_ex P (x # y # xs) \ P x \ list_ex P (y # xs)" "list_ex P [x] \ P x" + by simp_all + +lemma list_all_unfold: "list_all P (x # y # xs) \ P x \ list_all P (y # xs)" "list_all P [x] \ P x" + by simp_all + + +subsection \Setup for the Base Case\ + +text \ + We define a locale for an anonymous P-APP rule for 6 voters, 4 parties, and committee size 3 + that satisfies weak representation and cardinality strategyproofness. Our goal is to prove + the theorem \<^term>\False\ inside this locale. +\ + +locale papp_impossibility_base_case = + card_stratproof_weak_rep_anon_papp 6 parties 3 r + for parties :: "'a set" and r + + assumes card_parties: "card parties = 4" +begin + +text \ + A slightly more convenient version of Weak Representation: +\ +lemma weak_representation': + assumes "is_pref_profile A" "A' \ A" "\z\Z. count A {z} \ 2" "\Z \ set_mset W" + shows "r A' \ W" + using weak_representation[OF assms(1)] assms(2-4) by auto + +text \ + The following lemma (Lemma~2 in the appendix of the paper) is a strengthening of Weak + Representation and Strategyproofness in our concrete setting: + + Let \A\ be a preference profile containing approval lists \X\ and + let \Z\ be a set of parties such that each element of \Z\ is uniquely approved by at least two + voters in \A\. Due to Weak Representation, at least \|X \ Z|\ members of the committee are then + approved by \X\. + + What the lemma now says is that if there exists another voter with approval list \Y \ X\ and + $Y \nsubseteq Z$, then there is an additional committee member that is approved by \X\. + + This lemma will be used both in our symmetry-breaking argument and as a means to add more + clauses to the SAT instance. Since these clauses are logical consequences of Strategyproofness + and Weak Representation, they are technically redundant -- but their presence allows us to + use consider a smaller set of profiles and still get a contradiction. Without using the lemma, + we would need to feed more profiles to the SAT solver to obtain the same information. +\ +lemma lemma2: + assumes A: "is_pref_profile A" + assumes "X \# A" and "Y \# A - {#X#}" and "Y \ X" and "\Y \ Z" + assumes Z: "\z\Z. count A {z} \ 2" + shows "size (filter_mset (\x. x \ X) (r A)) > card (X \ Z)" +proof (rule ccontr) + text \ + For the sake of contradiction, suppose the number of elements approved by \X\ were + no larger than \|X \ Z|\. + \ + assume "\size (filter_mset (\x. x \ X) (r A)) > card (X \ Z)" + hence le: "size (filter_mset (\x. x \ X) (r A)) \ card (X \ Z)" + by linarith + interpret anon_papp_profile 6 parties 3 A + by fact + have "Z \ parties" + using assms(1,6) by (meson is_committee_def order.trans rule_wf weak_representation') + have [simp]: "finite Z" + by (rule finite_subset[OF _ finite_parties]) fact + + text \ + Due to Weak Representation, each member of \X \ Z\ must be chosen at least once. But due + to the above, it cannot be chosen more than once. So it has to be chosen exactly once. + \ + have X_approved_A_eq: "filter_mset (\x. x \ X) (r A) = mset_set (X \ Z)" + proof - + have "mset_set Z \# r A" + using Z weak_representation[OF A] by (subst mset_set_subset_iff) auto + hence "size (filter_mset (\x. x \ X) (mset_set Z)) \ size (filter_mset (\x. x \ X) (r A))" + by (intro size_mset_mono multiset_filter_mono) + also have "filter_mset (\x. x \ X) (mset_set Z) = mset_set {x\Z. x \ X}" + by simp + also have "{x\Z. x \ X} = X \ Z" + by auto + also have "size (mset_set (X \ Z)) = card (X \ Z)" + by simp + finally have "size (filter_mset (\x. x \ X) (r A)) = card (X \ Z)" + using le by linarith + moreover have "mset_set (X \ Z) \# filter_mset (\x. x \ X) (r A)" + using Z weak_representation[OF A] by (subst mset_set_subset_iff) auto + ultimately show "filter_mset (\x. x \ X) (r A) = mset_set (X \ Z)" + by (intro mset_subset_size_ge_imp_eq [symmetric]) auto + qed + + have count_eq_1: "count (r A) x = 1" if "x \ X \ Z" for x + using that X_approved_A_eq + by (metis \finite Z\ count_filter_mset count_mset_set' diff_is_0_eq diff_zero + finite_subset inf_le2 not_one_le_zero) + + text \ + Let \x\ be some element of \Y\ that is not in \Z\. + \ + obtain x where x: "x \ Y - Z" + using \\Y \ Z\ by blast + with assms have x': "x \ X - Z" + by auto + have [simp]: "x \ parties" + using A_subset assms(2) x' by blast + + text \ + Let \A'\ be the preference profile obtained by having voter \X\ lying and pretending + she only approves \x\. + \ + define A' where "A' = A - {#X#} + {#{x}#}" + have A': "is_pref_profile A'" + using is_pref_profile_replace[OF A \X \# A\, of "{x}"] by (auto simp: A'_def) + + text \ + We now show that even with this manipulated profile, the committee members approved by \X\ + are exactly the same as before: + \ + have X_approved_A'_eq: "filter_mset (\x. x \ X) (r A') = mset_set (X \ Z)" + proof - + text \ + Every element of \Z\ must still be in the result committee due to Weak Representation. + \ + have "mset_set Z \# r A'" + proof (subst mset_set_subset_iff) + show "Z \ set_mset (r A')" + proof + fix z assume z: "z \ Z" + from x' z have [simp]: "x \ z" + by auto + have [simp]: "X \ {z}" + using x' by auto + show "z \# r A'" + using Z weak_representation[OF A', of z] z x x' by (auto simp: A'_def) + qed + qed auto + + text \ + Thus the parties in \X \ Z\ must be in the committee (and they are approved by \X\). + \ + have "mset_set (X \ Z) \# filter_mset (\x. x \ X) (r A')" + proof - + have "filter_mset (\x. x \ X) (mset_set Z) \# filter_mset (\x. x \ X) (r A')" + using \mset_set Z \# r A'\ by (intro multiset_filter_mono) auto + also have "filter_mset (\x. x \ X) (mset_set Z) = mset_set (X \ Z)" + by auto + finally show "mset_set (X \ Z) \# filter_mset (\x. x \ X) (r A')" . + qed + + text \ + Due to Strategyproofness, no additional committee members can be approved by \X\, + so indeed only \X \ Z\ is approved by \X\, and they each occur only once. + \ + moreover have "\card_manipulable A X {x}" + using not_manipulable by blast + hence "size (mset_set (X \ Z)) \ size (filter_mset (\x. x \ X) (r A'))" using assms + by (simp add: card_manipulable_def A'_def strong_committee_preference_iff not_less + X_approved_A_eq) + ultimately show "filter_mset (\x. x \ X) (r A') = mset_set (X \ Z)" + by (metis mset_subset_size_ge_imp_eq) + qed + + text \ + Next, we show that the set of committee members approved by \Y\ in the committee returned for + the manipulated profile is exactly \Y \ Z\ (and again, each party only occurs once). + \ + have Y_approved_A'_eq: "filter_mset (\x. x \ Y) (r A') = mset_set (Y \ Z)" + proof - + have "filter_mset (\x. x \ Y) (filter_mset (\x. x \ X) (r A')) = + filter_mset (\x. x \ Y) (mset_set (X \ Z))" + by (simp only: X_approved_A'_eq) + also have "filter_mset (\x. x \ Y) (filter_mset (\x. x \ X) (r A')) = + filter_mset (\x. x \ Y \ x \ X) (r A')" + by (simp add: filter_filter_mset conj_commute) + also have "(\x. x \ Y \ x \ X) = (\x. x \ Y)" + using assms by auto + also have "filter_mset (\x. x \ Y) (mset_set (X \ Z)) = mset_set (Y \ Z)" + using assms by auto + finally show ?thesis . + qed + + text \ + Next, define the profile \A''\ obtained from \A'\ by also having \Y\ pretend to approve + only \x\. + \ + define A'' where "A'' = A' - {#Y#} + {#{x}#}" + have "Y \# A'" + using assms by (auto simp: A'_def) + hence A'': "is_pref_profile A''" + using is_pref_profile_replace[OF A', of Y "{x}"] by (auto simp: A''_def) + + text \ + Again, the elements of \Z\ must be chosen due to Weak Representation. + \ + have "Z \ set_mset (r A'')" + proof + fix z assume z: "z \ Z" + from x' z have [simp]: "x \ z" + by auto + have [simp]: "X \ {z}" "Y \ {z}" + using x x' by auto + show "z \# r A''" + using Z weak_representation[OF A'', of z] z x x' + by (auto simp: A''_def A'_def) + qed + + text \ + But now additionally, \x\ must be chosen, since both \X\ and \Y\ uniquely approve it. + \ + moreover have "x \# r A''" + using x x' \Y \# A - {#X#}\ by (intro weak_representation A'') (auto simp: A''_def A'_def) + ultimately have "insert x (Y \ Z) \ set_mset (r A'') \ Y" + using x by blast + + text \ + Now we have a contradiction due to Strategyproofness, since \Y\ can force the additional + member \x\ into the committee by lying. + \ + hence "mset_set (insert x (Y \ Z)) \# filter_mset (\w. w \ Y) (r A'')" + by (subst mset_set_subset_iff) auto + hence "size (mset_set (insert x (Y \ Z))) \ size (filter_mset (\w. w \ Y) (r A''))" + by (rule size_mset_mono) + hence "size (filter_mset (\x. x \ Y) (r A'')) > size (filter_mset (\x. x \ Y) (r A'))" + using x by (simp add: Y_approved_A'_eq) + hence "card_manipulable A' Y {x}" + using A' x \Y \# A'\ + unfolding card_manipulable_def strong_committee_preference_iff A''_def by auto + thus False + using not_manipulable by blast +qed + +text \ + The following are merely reformulation of the above lemma for technical reasons. +\ +lemma lemma2': + assumes "is_pref_profile A" + assumes "\z\Z. count A {z} \ 2" + assumes "X \# A \ (\Y. Y \# A - {#X#} \ Y \ X \ \Y \ Z)" + shows "\filter_mset (\x. x \ X) (r A) \# mset_set (X \ Z)" +proof + assume subset: "filter_mset (\x. x \ X) (r A) \# mset_set (X \ Z)" + from assms(3) obtain Y where Y: "X \# A" "Y \# A - {#X#}" "Y \ X" "\Y \ Z" + by blast + have "card (X \ Z) < size {#x \# r A. x \ X#}" + by (rule lemma2[where Y = Y]) (use Y assms(1,2) in auto) + with size_mset_mono[OF subset] show False + by simp +qed + +lemma lemma2'': + assumes "is_pref_profile A" + assumes "A' \ A" + assumes "\z\Z. count A {z} \ 2" + assumes "X \# A \ (\Y\set_mset (A - {#X#}). Y \ X \ \Y \ Z)" + assumes "filter_mset (\x. x \ X) W \# mset_set (X \ Z)" + shows "r A' \ W" + using lemma2'[of A Z X] assms by auto + + +subsection \Symmetry Breaking\ + +text \ + In the following, we formalize the symmetry-breaking argument that shows that we can + reorder the four alternatives $C_1$ to $C_4$ in such a way that the preference profile + \[ \{C_1\}\ \ \{C_2\}\ \ \{C_1, C_2\}\ \ \{C_3\}\ \ \{C_3\}\ \ \{C_3, C_4\} \] + is mapped to one of the committees $[C_1, C_1, C_3]$ or $[C_1, C_2, C_3]$. + + We start with a simple technical lemma that states that if we have a multiset $A$ of size 3 + consisting of the elements $x$ and $y$ and $x$ occurs at least as often as $y$, then + $A = [x, x, y]$. +\ + +lemma papp_multiset_3_aux: + assumes "size A = 3" "x \# A" "y \# A" "set_mset A \ {x, y}" "x \ y" "count A x \ count A y" + shows "A = {#x, x, y#}" +proof - + have "count A x > 0" + using assms by force + have "size A = (\z\set_mset A. count A z)" + by (rule size_multiset_overloaded_eq) + also have "set_mset A = {x, y}" + using assms by auto + also have "(\z\\. count A z) = count A x + count A y" + using assms by auto + finally have "count A x + count A y = 3" + by (simp add: assms(1)) + moreover from assms have "count A x > 0" "count A y > 0" + by auto + ultimately have *: "count A x = 2 \ count A y = 1" + using \count A x \ count A y\ by linarith + show ?thesis + proof (rule multiset_eqI) + fix z show "count A z = count {#x, x, y#} z" + proof (cases "z \ {x, y}") + case False + with assms have "z \ set_mset A" + by auto + hence "count A z = 0" + by (simp add: Multiset.not_in_iff) + thus ?thesis + using False by auto + qed (use * in auto) + qed +qed + +text \ + The following is the main symmetry-breaking result. It shows that we can find parties + $C_1$ to $C_4$ with the desired property. + + This is a somewhat ad-hoc argument; in the appendix of the paper this is done more + systematically in Lemma~3. +\ +lemma symmetry_break_aux: + obtains C1 C2 C3 C4 where + "parties = {C1, C2, C3, C4}" "distinct [C1, C2, C3, C4]" + "r ({#{C1}, {C2}, {C1, C2}, {C3}, {C4}, {C3, C4}#}) \ {{#C1, C1, C3#}, {#C1, C2, C3#}}" +proof - + note I = that + have "\xs. set xs = parties \ distinct xs" + using finite_distinct_list[of parties] by blast + then obtain xs where xs: "set xs = parties" "distinct xs" + by blast + from xs have "length xs = 4" + using card_parties distinct_card[of xs] by auto + then obtain C1 C2 C3 C4 where xs_eq: "xs = [C1, C2, C3, C4]" + by (auto simp: eval_nat_numeral length_Suc_conv) + have parties_eq: "parties = {C1, C2, C3, C4}" + by (subst xs(1) [symmetric], subst xs_eq) auto + have [simp]: + "C1 \ C2" "C1 \ C3" "C1 \ C4" + "C2 \ C1" "C2 \ C3" "C2 \ C4" + "C3 \ C1" "C3 \ C2" "C3 \ C4" + "C4 \ C1" "C4 \ C2" "C4 \ C3" + using \distinct xs\ unfolding xs_eq by auto + + define A where "A = {#{C1}, {C2}, {C1, C2}, {C3}, {C4}, {C3, C4}#}" + define m where "m = Max (count (r A) ` parties)" + + have A: "is_pref_profile A" + unfolding A_def is_pref_profile_iff by (simp add: parties_eq) + hence "is_committee (r A)" + by (rule rule_wf) + hence rA: "size (r A) = 3" "set_mset (r A) \ parties" + unfolding is_committee_def by auto + define X where "X = set_mset (r A)" + have "X \ {}" "X \ parties" + using rA by (auto simp: X_def) + + have "m > 0" + proof - + obtain x where "x \ X" + using \X \ {}\ by blast + with \X \ parties\ have "C1 \ X \ C2 \ X \ C3 \ X \ C4 \ X" + unfolding parties_eq by blast + thus ?thesis + unfolding m_def X_def by (subst Max_gr_iff) (auto simp: parties_eq) + qed + + have "m \ 3" + proof - + have "m \ size (r A)" + unfolding m_def by (subst Max_le_iff) (auto simp: count_le_size) + also have "\ = 3" + by fact + finally show ?thesis . + qed + + have "m \ (count (r A) ` parties)" + unfolding m_def by (intro Max_in) auto + then obtain C1' where C1': "count (r A) C1' = m" "C1' \ parties" + by blast + have "C1' \# r A" + using \m > 0\ C1'(1) by auto + + have "\C2'\parties-{C1'}. {C1', C2'} \# A" + using C1' unfolding A_def parties_eq + by (elim insertE; simp add: insert_Diff_if insert_commute) + then obtain C2' where C2': "C2' \ parties - {C1'}" "{C1', C2'} \# A" + by blast + have [simp]: "C1' \ C2'" "C2' \ C1'" + using C2' by auto + have disj: "C1' = C1 \ C2' = C2 \ C1' = C2 \ C2' = C1 \ C1' = C3 \ C2' = C4 \ C1' = C4 \ C2' = C3" + using C1'(2) C2' unfolding A_def parties_eq + by (elim insertE; force simp: insert_commute) + + obtain C3' where C3': "C3' \ parties-{C1', C2'}" + using C1'(2) C2' unfolding parties_eq by (fastforce simp: insert_Diff_if) + obtain C4' where C4': "C4' \ parties-{C1', C2', C3'}" + using C1'(2) C2' C3' unfolding parties_eq by (fastforce simp: insert_Diff_if) + have A_eq: "A = {#{C1'}, {C2'}, {C1', C2'}, {C3'}, {C4'}, {C3', C4'}#}" + using disj C3' C4' + by (elim disjE) (auto simp: A_def parties_eq insert_commute) + have distinct: + "C1' \ C2'" "C1' \ C3'" "C1' \ C4'" + "C2' \ C1'" "C2' \ C3'" "C2' \ C4'" + "C3' \ C1'" "C3' \ C2'" "C3' \ C4'" + "C4' \ C1'" "C4' \ C2'" "C4' \ C3'" + using C1' C2' C3' C4' by blast+ + have parties_eq': "parties = {C1', C2', C3', C4'}" + using C1'(2) C2'(1) C3' C4' distinct unfolding parties_eq by (elim insertE) auto + + have "\{#x \# r A. x \ {C3', C4'}#} \# mset_set ({C3', C4'} \ {})" + by (rule lemma2'[OF A]) (auto simp: A_eq) + hence C34': "C3' \# r A \ C4' \# r A" + by auto + then consider "C3' \# r A" "C4' \# r A" | "C3' \# r A" "C4' \# r A" | "C3' \# r A" "C4' \# r A" + by blast + + thus ?thesis + proof cases + assume *: "C3' \# r A" "C4' \# r A" + have "r A = {#C3', C4', C1'#}" + by (rule sym, rule mset_subset_size_ge_imp_eq) + (use * \C1' \# r A\ distinct in + \auto simp: \size (r A) = 3\ Multiset.insert_subset_eq_iff in_diff_multiset_absorb2\) + thus ?thesis using distinct + by (intro that[of C3' C4' C1' C2']) + (auto simp: parties_eq' A_eq add_mset_commute insert_commute) + + next + + assume *: "C3' \# r A" "C4' \# r A" + show ?thesis + proof (cases "C2' \# r A") + case True + have "r A = {#C1', C2', C3'#}" + by (rule sym, rule mset_subset_size_ge_imp_eq) + (use * \C1' \# r A\ distinct True in + \auto simp: \size (r A) = 3\ Multiset.insert_subset_eq_iff in_diff_multiset_absorb2\) + thus ?thesis using distinct + by (intro that[of C1' C2' C3' C4']) + (auto simp: parties_eq' A_eq add_mset_commute insert_commute) + next + case False + have "r A = {#C1', C1', C3'#}" + proof (rule papp_multiset_3_aux) + show "set_mset (r A) \ {C1', C3'}" + using \set_mset (r A) \ _\ * False unfolding parties_eq' by auto + next + have "count (r A) C3' \ m" + unfolding m_def by (subst Max_ge_iff) (auto simp: parties_eq') + also have "m = count (r A) C1'" + by (simp add: C1') + finally show "count (r A) C3' \ count (r A) C1'" . + qed (use C1' * False \C1' \# r A\ distinct in \auto simp: \size (r A) = 3\\) + thus ?thesis using distinct + by (intro that[of C1' C2' C3' C4']) + (auto simp: parties_eq' insert_commute add_mset_commute A_eq) + qed + + next + + assume *: "C3' \# r A" "C4' \# r A" + show ?thesis + proof (cases "C2' \# r A") + case True + have "r A = {#C1', C2', C4'#}" + by (rule sym, rule mset_subset_size_ge_imp_eq) + (use * \C1' \# r A\ distinct True in + \auto simp: \size (r A) = 3\ Multiset.insert_subset_eq_iff in_diff_multiset_absorb2\) + thus ?thesis using distinct + by (intro that[of C1' C2' C4' C3']) + (auto simp: parties_eq' A_eq add_mset_commute insert_commute) + next + case False + have "r A = {#C1', C1', C4'#}" + proof (rule papp_multiset_3_aux) + show "set_mset (r A) \ {C1', C4'}" + using \set_mset (r A) \ _\ * False unfolding parties_eq' by auto + next + have "count (r A) C4' \ m" + unfolding m_def by (subst Max_ge_iff) (auto simp: parties_eq') + also have "m = count (r A) C1'" + by (simp add: C1') + finally show "count (r A) C4' \ count (r A) C1'" . + qed (use C1' * False \C1' \# r A\ distinct in \auto simp: \size (r A) = 3\\) + thus ?thesis using distinct + by (intro that[of C1' C2' C4' C3']) + (auto simp: parties_eq' insert_commute add_mset_commute A_eq) + qed + qed +qed + +text \ + We now use the choice operator to get our hands on such values $C_1$ to $C_4$. +\ +definition C1234 where + "C1234 = (SOME xs. set xs = parties \ distinct xs \ + (case xs of [C1, C2, C3, C4] \ + r ({#{C1}, {C2}, {C1, C2}, {C3}, {C4}, {C3, C4}#}) \ {{#C1, C1, C3#}, {#C1, C2, C3#}}))" + +definition C1 where "C1 = C1234 ! 0" +definition C2 where "C2 = C1234 ! 1" +definition C3 where "C3 = C1234 ! 2" +definition C4 where "C4 = C1234 ! 3" + +lemma distinct: "distinct [C1, C2, C3, C4]" + and parties_eq: "parties = {C1, C2, C3, C4}" + and symmetry_break: + "r ({#{C1}, {C2}, {C1, C2}, {C3}, {C4}, {C3, C4}#}) \ {{#C1, C1, C3#}, {#C1, C2, C3#}}" +proof - + have C1234: + "set C1234 = parties \ distinct C1234 \ + (case C1234 of [C1', C2', C3', C4'] \ + r ({#{C1'}, {C2'}, {C1', C2'}, {C3'}, {C4'}, {C3', C4'}#}) \ + {{#C1', C1', C3'#}, {#C1', C2', C3'#}})" + unfolding C1234_def + proof (rule someI_ex) + obtain C1' C2' C3' C4' where *: + "parties = {C1', C2', C3', C4'}" "distinct [C1', C2', C3', C4']" + "r ({#{C1'}, {C2'}, {C1', C2'}, {C3'}, {C4'}, {C3', C4'}#}) \ + {{#C1', C1', C3'#}, {#C1', C2', C3'#}}" + using symmetry_break_aux by blast + show "\xs. set xs = parties \ distinct xs \ + (case xs of [C1', C2', C3', C4'] \ + r ({#{C1'}, {C2'}, {C1', C2'}, {C3'}, {C4'}, {C3', C4'}#}) \ + {{#C1', C1', C3'#}, {#C1', C2', C3'#}})" + by (intro exI[of _ "[C1', C2', C3', C4']"]) (use * in auto) + qed + + have "length C1234 = 4" + using C1234 card_parties distinct_card[of C1234] by simp + then obtain C1' C2' C3' C4' where C1234_eq: "C1234 = [C1', C2', C3', C4']" + by (auto simp: eval_nat_numeral length_Suc_conv) + show "distinct [C1, C2, C3, C4]" "parties = {C1, C2, C3, C4}" + "r ({#{C1}, {C2}, {C1, C2}, {C3}, {C4}, {C3, C4}#}) \ {{#C1, C1, C3#}, {#C1, C2, C3#}}" + using C1234 by (simp_all add: C1234_eq C1_def C2_def C3_def C4_def) +qed + +lemma distinct' [simp]: + "C1 \ C2" "C1 \ C3" "C1 \ C4" "C2 \ C1" "C2 \ C3" "C2 \ C4" + "C3 \ C1" "C3 \ C2" "C3 \ C4" "C4 \ C1" "C4 \ C2" "C4 \ C3" + using distinct by auto + +lemma in_parties [simp]: "C1 \ parties" "C2 \ parties" "C3 \ parties" "C4 \ parties" + by (subst (2) parties_eq; simp; fail)+ + + +subsection \The Set of Possible Committees\ + +text \ + Next, we compute the set of the 20 possible committees. +\ + +abbreviation COM where "COM \ committees 3 parties" + +definition COM' where "COM' = + [{#C1, C1, C1#}, {#C1, C1, C2#}, {#C1, C1, C3#}, {#C1, C1, C4#}, + {#C1, C2, C2#}, {#C1, C2, C3#}, {#C1, C2, C4#}, {#C1, C3, C3#}, + {#C1, C3, C4#}, {#C1, C4, C4#}, {#C2, C2, C2#}, {#C2, C2, C3#}, + {#C2, C2, C4#}, {#C2, C3, C3#}, {#C2, C3, C4#}, {#C2, C4, C4#}, + {#C3, C3, C3#}, {#C3, C3, C4#}, {#C3, C4, C4#}, + {#C4, C4, C4#}]" + +lemma distinct_COM': "distinct COM'" + by (simp add: COM'_def add_mset_neq) + +lemma COM_eq: "COM = set COM'" + by (subst parties_eq) + (simp_all add: COM'_def numeral_3_eq_3 committees_Suc add_ac insert_commute add_mset_commute) + +lemma r_in_COM: + assumes "is_pref_profile A" + shows "r A \ COM" + using rule_wf[OF assms] unfolding committees_def is_committee_def by auto + +lemma r_in_COM': + assumes "is_pref_profile A" "A' \ A" + shows "list_ex (\W. r A' = W) COM'" + using r_in_COM[OF assms(1)] assms(2) by (auto simp: list_ex_iff COM_eq) + +lemma r_right_unique: + "list_all (\(W1,W2). r A \ W1 \ r A \ W2) (pairs COM')" +proof - + have "list_all (\(W1,W2). W1 \ W2) (pairs COM')" + using distinct_COM' unfolding distinct_conv_pairs by blast + thus ?thesis + unfolding list_all_iff by blast +qed + +end + + + +subsection \Generating Clauses and Replaying the SAT Proof\ + +text \ + We now employ some custom-written ML code to generate all the SAT clauses arising from the + given profiles (read from an external file) as Isabelle/HOL theorems. From these, we then + derive \<^term>\False\ by replaying an externally found SAT proof (also written from an external + file). + + The proof was found with the glucose SAT solver, which outputs proofs in the DRUP format + (a subset of the more powerful DRAT format). We then used the \<^emph>\DRAT-trim\ tool by + Wetzler et al.~\cite{wetzler_drat_trim} to make the proof smaller. This was done repeatedly + until the proof size did not decrease any longer. Then, the proof was converted into the \<^emph>\GRAT\ + format introduced by Lammich~\cite{lammich_grat}, which is easier to check (or in our case + replay) than the less explicit DRAT (or DRUP) format. +\ + +external_file "sat_data/profiles" +external_file "sat_data/papp_impossibility.grat.xz" + +context papp_impossibility_base_case +begin + +ML_file \papp_impossibility.ML\ + +text \ + This invocation proves a theorem called \<^emph>\contradiction\ whose statement is \<^term>\False\. + Note that the DIMACS version of the SAT file that is being generated can be viewed by + clicking on ``See theory exports'' in the messages output by the invocation below. + + On a 2021 desktop PC with 12 cores, proving all the clauses takes 8.4\,s (multithreaded; + CPU time 55\,s). Replaying the proof takes 130\,s (singlethreaded). +\ + +local_setup \fn lthy => + let + val thm = + PAPP_Impossibility.derive_false lthy + (\<^master_dir> + \<^path>\sat_data/profiles\) + (\<^master_dir> + \<^path>\sat_data/papp_impossibility.grat.xz\) + in + Local_Theory.note ((\<^binding>\contradiction\, []), [thm]) lthy |> snd + end +\ + +end + +text \ + With this, we can now prove the impossibility result: +\ +lemma papp_impossibility_base_case: + assumes "card parties = 4" + shows "\card_stratproof_weak_rep_anon_papp 6 parties 3 r" +proof + assume "card_stratproof_weak_rep_anon_papp 6 parties 3 r" + then interpret card_stratproof_weak_rep_anon_papp 6 parties 3 r . + interpret papp_impossibility_base_case parties r + by unfold_locales fact+ + show False + by (rule contradiction) +qed + +end diff --git a/thys/PAPP_Impossibility/PAPP_Multiset_Extras.thy b/thys/PAPP_Impossibility/PAPP_Multiset_Extras.thy new file mode 100644 --- /dev/null +++ b/thys/PAPP_Impossibility/PAPP_Multiset_Extras.thy @@ -0,0 +1,86 @@ +(* + File: PAPP_Multiset_Extras.thy + Author: Manuel Eberl, University of Innsbruck +*) +section \Auxiliary Facts About Multisets\ +theory PAPP_Multiset_Extras + imports "HOL-Library.Multiset" +begin + +text \ + This section contains a number of not particularly interesting small facts about multisets. +\ + +lemma mset_set_subset_iff: "finite A \ mset_set A \# B \ A \ set_mset B" + by (metis finite_set_mset finite_set_mset_mset_set mset_set_set_mset_msubset + msubset_mset_set_iff set_mset_mono subset_mset.trans) + +lemma mset_subset_size_ge_imp_eq: + assumes "A \# B" "size A \ size B" + shows "A = B" + using assms +proof (induction A arbitrary: B) + case empty + thus ?case by auto +next + case (add x A B) + have [simp]: "x \# B" + using add.prems by (simp add: insert_subset_eq_iff) + define B' where "B' = B - {#x#}" + have B_eq: "B = add_mset x B'" + using add.prems unfolding B'_def by (auto simp: add_mset_remove_trivial_If) + have "A = B'" + using add.prems by (intro add.IH) (auto simp: B_eq) + thus ?case + by (auto simp: B_eq) +qed + +lemma mset_psubset_iff: + "X \# Y \ X \# Y \ (\x. count X x < count Y x)" + by (meson less_le_not_le subset_mset.less_le_not_le subseteq_mset_def) + +lemma count_le_size: "count A x \ size A" + by (induction A) auto + +lemma size_filter_eq_conv_count [simp]: "size (filter_mset (\y. y = x) A) = count A x" + by (induction A) auto + +lemma multiset_filter_mono': + assumes "\x. x \# A \ P x \ Q x" + shows "filter_mset P A \# filter_mset Q A" + using assms by (induction A) (auto simp: subset_mset.absorb_iff1 add_mset_union) + +lemma multiset_filter_mono'': + assumes "A \# B" "\x. x \# A \ P x \ Q x" + shows "filter_mset P A \# filter_mset Q B" + using assms multiset_filter_mono multiset_filter_mono' + by (metis subset_mset.order_trans) + +lemma filter_mset_disjunction: + assumes "\x. x \# X \ P x \ Q x \ False" + shows "filter_mset (\x. P x \ Q x) X = filter_mset P X + filter_mset Q X" + using assms by (induction X) auto + +lemma size_mset_sum_mset: "size (sum_mset X) = (\x\#X. size (x :: 'a multiset))" + by (induction X) auto + +lemma count_sum_mset: "count (sum_mset X) x = (\Y\#X. count Y x)" + by (induction X) auto + +lemma replicate_mset_rec: "n > 0 \ replicate_mset n x = add_mset x (replicate_mset (n - 1) x)" + by (cases n) auto + +lemma add_mset_neq: "x \# B \ add_mset x A \ B" + by force + +lemma filter_replicate_mset: + "filter_mset P (replicate_mset n x) = (if P x then replicate_mset n x else {#})" + by (induction n) auto + +lemma filter_diff_mset': "filter_mset P (X - Y) = filter_mset P X - Y" + by (rule multiset_eqI) auto + +lemma in_diff_multiset_absorb2: "x \# B \ x \# A - B \ x \# A" + by (metis count_greater_zero_iff count_inI in_diff_count) + +end diff --git a/thys/PAPP_Impossibility/ROOT b/thys/PAPP_Impossibility/ROOT new file mode 100644 --- /dev/null +++ b/thys/PAPP_Impossibility/ROOT @@ -0,0 +1,15 @@ +chapter AFP + +session PAPP_Impossibility (AFP) = "HOL" + + options [timeout = 900] + sessions + "HOL-Library" + Randomised_Social_Choice + theories [document=false] + SAT_Replay + theories + PAPP_Impossibility + document_files + root.tex + root.bib + diff --git a/thys/PAPP_Impossibility/SAT_Replay.thy b/thys/PAPP_Impossibility/SAT_Replay.thy new file mode 100644 --- /dev/null +++ b/thys/PAPP_Impossibility/SAT_Replay.thy @@ -0,0 +1,18 @@ +(* + File: SAT_Replay.thy + Author: Manuel Eberl, University of Innsbruck +*) +section \Replaying SAT Proofs\ +theory SAT_Replay + imports Main +begin + +text \ + This file implements replaying SAT proofs in the DRUP format that were found by an external + SAT solver and preprocessed with Peter Lammich's GRAT tool.~\cite{lammich_grat}. +\ + +ML_file \array_map.ML\ +ML_file \sat_replay.ML\ + +end diff --git a/thys/PAPP_Impossibility/array_map.ML b/thys/PAPP_Impossibility/array_map.ML new file mode 100644 --- /dev/null +++ b/thys/PAPP_Impossibility/array_map.ML @@ -0,0 +1,83 @@ +(* + File: array_map.ML + Author: Manuel Eberl, University of Innsbruck + + This file implements array maps, i.e. logically a partial function from the integers + to some values. Note that the array in the background grows whenever it needs to, but it + never shrinks. +*) + +signature ARRAY_MAP = +sig + +type 'a T + +val capacity : 'a T -> int +val empty : int -> 'a T +val sub : 'a T * int -> 'a option +val update : 'a T * int * 'a option -> unit +val tabulate : int * int * (int -> 'a) -> 'a T +val fromArray : 'a option array -> 'a T +val clone : 'a T -> 'a T + +end + + +structure Array_Map : ARRAY_MAP = struct + +fun growth_factor n = (3 * n) div 2 + +type 'a T = 'a option array Unsynchronized.ref + +open Unsynchronized + +fun capacity a = Array.length (!a) + +fun empty cap = ref (Array.array (cap, NONE)) + +fun sub (a, i) = + if i < 0 orelse i >= capacity a then NONE else Array.sub (!a, i) + +fun grow a i = + if i < capacity a then () + else + let + val cap' = growth_factor i + 1 + val a' = Array.array (cap', NONE) + val _ = Array.copy {src = !a, dst = a', di = 0} + val _ = a := a' + in + () + end + +fun update (a, i, NONE) = + if i >= 0 andalso i < capacity a then Array.update (!a, i, NONE) else () + | update (a, i, x) = + if i < 0 then + raise Subscript + else + let + val _ = grow a i + in + Array.update (!a, i, x) + end + +fun tabulate (cap, n, f) = + let + fun g i = if i >= n then NONE else SOME (f i) + val cap = if cap <= 0 then growth_factor n else cap + in + ref (Array.tabulate (Int.max (cap, n), g)) + end + +fun fromArray a = + let + val a' = empty (growth_factor (Array.length a)) + val _ = Array.copy {di = 0, dst = !a', src = a} + in + a' + end + +fun clone a = fromArray (!a) + +end diff --git a/thys/PAPP_Impossibility/document/root.bib b/thys/PAPP_Impossibility/document/root.bib new file mode 100644 --- /dev/null +++ b/thys/PAPP_Impossibility/document/root.bib @@ -0,0 +1,25 @@ +@inproceedings{lammich_grat, + author = {Peter Lammich}, + editor = {Serge Gaspers and Toby Walsh}, + title = {The {GRAT} Tool Chain -- Efficient {(UN)SAT} Certificate Checking with Formal Correctness Guarantees}, + booktitle = {Theory and Applications of Satisfiability Testing -- {SAT} 2017, Proceedings}, + series = {Lecture Notes in Computer Science}, + volume = {10491}, + pages = {457--463}, + publisher = {Springer}, + year = {2017}, + doi = {10.1007/978-3-319-66263-3_29} +} + +@inproceedings{wetzler_drat_trim, + author = {Nathan Wetzler and Marijn Heule and Warren A. Hunt Jr.}, + editor = {Carsten Sinz and Uwe Egly}, + title = {{DRAT-trim}: Efficient Checking and Trimming Using Expressive Clausal Proofs}, + booktitle = {Theory and Applications of Satisfiability Testing -- {SAT} 2014, Proceedings}, + series = {Lecture Notes in Computer Science}, + volume = {8561}, + pages = {422--429}, + publisher = {Springer}, + year = {2014}, + doi = {10.1007/978-3-319-09284-3_31} +} diff --git a/thys/PAPP_Impossibility/document/root.tex b/thys/PAPP_Impossibility/document/root.tex new file mode 100644 --- /dev/null +++ b/thys/PAPP_Impossibility/document/root.tex @@ -0,0 +1,63 @@ +\documentclass[11pt,a4paper]{scrartcl} +\usepackage[T1]{fontenc} +\usepackage{isabelle,isabellesym} +\usepackage{amsfonts,amsmath,amssymb} +\usepackage{pgfplots} + +% this should be the last package used +\usepackage{pdfsetup} + +% urls in roman style, theory text in math-similar italics +\urlstyle{rm} +\isabellestyle{it} + +\begin{document} + +\title{The Incompatibility of Strategy-Proofness and Representation in Party-Approval Multi-Winner Elections} +\author{Th\'eo Delemazure\\ +Tom Demeulemeester\\ +Manuel Eberl\\ +Jonas Israel\\ +Patrick Lederer} +\maketitle + +\begin{abstract} +In party-approval multi-winner elections, the goal is to allocate the seats of a fixed-size +committee to parties based on approval ballots of the voters over the parties. In particular, each +voter can approve multiple parties and each party can be assigned multiple seats. + +Three central requirements in this settings are: +\begin{description} +\item[Anonymity:] The result is invariant under renaming the voters. +\item[Representation:] Every sufficiently large group of voters with similar +preferences is represented by some committee members. +\item[Strategy-proofness:] No voter can benefit by misreporting her true preferences. +\end{description} +We show that these three basic axioms are incompatible for party-approval multi-winner voting rules, +thus proving a far-reaching impossibility theorem. + +The proof of this result is obtained by +formulating the problem in propositional logic and then letting a SAT solver show that the +formula is unsatisfiable. The DRUP proof output by the SAT solver is then converted into +Lammich's GRAT format and imported into Isabelle/HOL with some custom-written ML code. + +This transformation is proof-producing, so the final Isabelle/HOL theorem does not rely on any oracles +or other trusted external trusted components. +\end{abstract} + +\newpage +\tableofcontents +\newpage +\parindent 0pt\parskip 0.5ex + +\input{session} + +\bibliographystyle{abbrv} +\bibliography{root} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: t +%%% End: diff --git a/thys/PAPP_Impossibility/papp_impossibility.ML b/thys/PAPP_Impossibility/papp_impossibility.ML new file mode 100644 --- /dev/null +++ b/thys/PAPP_Impossibility/papp_impossibility.ML @@ -0,0 +1,741 @@ +(* + File: papp_impossibility.ML + Author: Manuel Eberl, University of Innsbruck + + This file implements the generation of a SAT problem in DIMACS format from the situation we + have in the base case of the P-APP impossibility, and the replaying of an externally found + GRAT proof to prove that impossibility. +*) + +signature PAPP_IMPOSSIBILITY = +sig + +type clause = int list * thm +type sat_problem = (int * cterm option array) * (int * clause Array_Map.T) * Proof.context + +val mk_sat_problem : Proof.context -> Path.T -> sat_problem +val mk_dimacs : sat_problem -> string +val write_dimacs : sat_problem -> Path.T -> unit +val replay_grat : Proof.context -> sat_problem -> Path.T -> thm +val derive_false : Proof.context -> Path.T -> Path.T -> thm + +end + + +structure PAPP_Impossibility : PAPP_IMPOSSIBILITY = +struct + +type clause = int list * thm +type sat_problem = (int * cterm option array) * (int * clause Array_Map.T) * Proof.context + +exception CLAUSE of cterm * thm + +fun decode_approval_list i = + let + val i' = i + 1 + in + filter (fn j => IntInf.andb (i', IntInf.<< (1, Word.fromInt j)) <> 0) (0 upto 3) + end + +fun mk_committees k n = + let + fun go (lo, hi, n) = + if hi <= lo orelse n < 0 then [] + else if n = 0 then [[]] + else map (fn xs => lo :: xs) (go (lo, hi, n-1)) @ go (lo+1, hi, n) + in + go (0, k, n) + end + +fun ccand 0 = \<^cterm>\C1\ + | ccand 1 = \<^cterm>\C2\ + | ccand 2 = \<^cterm>\C3\ + | ccand 3 = \<^cterm>\C4\ + | ccand _ = raise Match + +val cand = Thm.term_of o ccand + +val candT = \<^typ>\'a\ +val candsetT = HOLogic.mk_setT candT +fun mk_msetT T = Type (\<^type_name>\Multiset.multiset\, [T]) +fun mk_empty_mset T = Const (\<^const_name>\Groups.zero\, mk_msetT T) +fun mk_add_mset T = + let + val T' = mk_msetT T + in + Const (\<^const_name>\Multiset.add_mset\, T --> T' --> T') + end +val add_mset_cterm_candset' = \<^cterm>\add_mset :: 'a set \ _\ + +val profileT = mk_msetT candsetT +val commT = mk_msetT candT + +fun mk_mset T ts = + let + val empty = mk_empty_mset T + val add = mk_add_mset T + fun add' a b = add $ a $ b + in + fold_rev add' ts empty + end + +type profile = {id : int, profile : int list, cterm : cterm, wf_thm : thm} + +val is_pref_profile_iff_thm = @{thm eq_reflection[OF is_pref_profile_iff]} + +val profile_cterm = (#cterm : profile -> cterm) +val profile_term = Thm.term_of o profile_cterm + +val eq_trueD = @{lemma "PROP P \ PROP Trueprop True \ PROP P" by simp} + +fun simp_prove ctxt ct = + let + val thm = Simplifier.rewrite ctxt ct + val rhs = thm |> Thm.cconcl_of |> Thm.dest_equals_rhs + in + if Thm.term_of rhs = \<^prop>\True\ then + Thm.implies_elim + (Thm.instantiate (TVars.empty, Vars.make [((("P", 0), propT), ct)]) eq_trueD) thm + else + raise CTERM ("simp_prove: Failed to finish proof", [rhs]) + end + +fun simp_discharge ctxt thm = + let + val prem = Thm.cprem_of thm 1 + in + Thm.implies_elim thm (simp_prove ctxt prem) + end + +fun simp_discharge_all ctxt thm = + if Thm.nprems_of thm = 0 then thm else simp_discharge_all ctxt (simp_discharge ctxt thm) + +fun mp thm thm' = Thm.implies_elim thm' thm + +fun elim_conjunctions_parallel 0 thm = Conjunction.elim_conjunctions thm + | elim_conjunctions_parallel n thm = + case try Conjunction.elim thm of + NONE => [thm] + | SOME (thm1, thm2) => Par_List.map (elim_conjunctions_parallel (n - 1)) [thm1, thm2] |> flat + +(* make sure that all theorems in the list have pointer-equal hyps *) +fun consolidate_hyps ctxt thms = + thms + |> chop_groups 20000 + |> Par_List.map Conjunction.intr_balanced + |> Conjunction.intr_balanced + |> Thm.solve_constraints + |> Thm.strip_shyps + |> Thm.adjust_maxidx_thm (~1) + |> Thm.transfer' ctxt + |> elim_conjunctions_parallel 4 + +val mset_swap_conv = + Conv.rewr_conv @{thm eq_reflection[OF add_mset_commute]} + +fun norm_profile_conv [] ct = Conv.all_conv ct + | norm_profile_conv [_] ct = Conv.all_conv ct + | norm_profile_conv (x :: y :: xs) ct = + if x > y then + (mset_swap_conv + then_conv Conv.arg_conv (norm_profile_conv (x :: xs))) ct + else + Conv.all_conv ct + +fun committee_score app (cs : int list) = length (filter (member op= app) cs) +fun committee_less app cs1 (cs2 : int list) = committee_score app cs1 < committee_score app cs2 + +fun group [] = [] + | group (x :: xs) = + let + fun go (x, n) [] acc = rev ((x, n) :: acc) + | go (x, n) (y :: ys) acc = + if x = y then go (x, n+1) ys acc else go (y, 1) ys ((x, n) :: acc) + in + go (x, 1) xs [] + end + +fun pairs [] = [] + | pairs (x :: xs) = map (fn y => (x, y)) xs @ pairs xs + +fun encode_var (i, j) = i * 20 + j + 1 +fun encode_lit (b, i, j) = + let val v = encode_var (i, j) in if b then ~v else v end +fun decode_var v = IntInf.divMod (v-1, 20) +fun decode_lit l = case decode_var (abs l) of (i, j) => (l < 0, i, j) + +fun mk_disjs [] = raise Empty + | mk_disjs [x] = x + | mk_disjs (x :: xs) = HOLogic.mk_disj (x, mk_disjs xs) + +fun rev_maps_pairs f xs = + let + fun go [] acc = acc + | go (x :: xs) acc = + let + fun go' [] acc = acc + | go' (y :: ys) acc = go' ys (f (x, y) @ acc) + in + go xs (go' xs acc) + end + in + go xs [] + end + +fun get_approval_list ct 0 = Thm.dest_arg1 ct + | get_approval_list ct i = get_approval_list (Thm.dest_arg ct) (i - 1) + +val not_manipulable_thm' = + @{lemma "is_pref_profile A \ is_pref_profile B \ A' \ A \ B' \ B \ add_mset Y A \ add_mset X B \ + \r A \[Comm(X)] r B" + by (rule not_manipulable'[of A B Y X]) auto} + +val stratproof_aux_thm = + @{lemma "\r B \[Comm(X)] r A \ A' \ A \ B' \ B \ W \[Comm(X)] W' \ r A' \ W \ r B' \ W'" by blast} + +fun add_mset_cterm_candset ct1 profile = + Thm.apply (Thm.apply add_mset_cterm_candset' ct1) (profile_cterm profile) + +fun prove_add_mset_eq (xs, ct1) (ys, ct2) = + let + val thm1 = norm_profile_conv xs ct1 + val thm2 = norm_profile_conv ys ct2 + in + Thm.transitive thm1 (Thm.symmetric thm2) + end + +fun is_subset [] _ = true + | is_subset _ [] = false + | is_subset (x :: xs) (y :: ys) = (x = y andalso is_subset xs ys) orelse is_subset (x :: xs) ys + +fun exists_index p xs = + let + fun go _ [] = false + | go i (x :: xs) = p (i, x) orelse go (i + 1) xs + in + go 0 xs + end + + + +fun mk_sat_problem ctxt profiles_file = +let + val committee_list = mk_committees 4 3 + val committees = Array.fromList committee_list + fun committee i = Array.sub (committees, i) + + val approval_list_cache' = + Array.tabulate (14, decode_approval_list #> map cand #> HOLogic.mk_set candT) + + val approval_list_cache = + Array.tabulate (14, fn i => Thm.cterm_of ctxt (Array.sub (approval_list_cache', i))) + + fun holify_approval_list i = Array.sub (approval_list_cache, i) + fun holify_approval_list' i = Array.sub (approval_list_cache', i) + + + val holify_profile' = + map holify_approval_list' #> mk_mset candsetT + + fun mk_profile ctxt id profile = + let + val term = holify_profile' profile + val goal = \<^instantiate>\A = "term" in prop \is_pref_profile A\\ + val thm = + Goal.prove ctxt [] [] goal + (fn {context, ...} => + HEADGOAL (EqSubst.eqsubst_tac ctxt [0] [is_pref_profile_iff_thm] + THEN_ALL_NEW Simplifier.simp_tac context)) + val cterm = thm |> Thm.cconcl_of |> Thm.dest_arg |> Thm.dest_arg + in + {id = id, profile = profile, cterm = cterm, wf_thm = thm} : profile + end + + fun read_profiles ctxt path = + let + fun process_line (id, l) = + l + |> space_explode " " + |> map (the o Int.fromString) + |> mk_profile ctxt id + in + File.read_lines path + |> filter (fn l => l <> "") + |> map_index I + |> chop_groups 50 + |> Par_List.map (map process_line) + |> flat + end + + val profs = read_profiles ctxt profiles_file + val profile_array = Array.fromList profs + + local + fun mk_def prof = + ((Binding.make ("prof" ^ Int.toString (#id prof), \<^here>), NoSyn), + (Binding.empty_atts, profile_term prof)) + val xs = map mk_def profs + val (aux, ctxt) = Local_Defs.define xs ctxt + in + val ctxt' = ctxt + val profile_consts = Array.fromList (map fst aux) + val profile_defs = Array.fromList (map (snd o snd) aux) + fun profile_const (prof : profile) = Array.sub (profile_consts, #id prof) + fun profile_cconst (prof : profile) = Thm.cterm_of ctxt (profile_const prof) + fun profile_def (prof : profile) = Array.sub (profile_defs, #id prof) + end + + local + fun mk_committee i = + committee i + |> map cand + |> mk_mset candT + fun mk_def i = + ((Binding.make ("W" ^ Int.toString i, \<^here>), NoSyn), + (Binding.empty_atts, mk_committee i)) + val xs = map_range mk_def 20 + val (aux, ctxt) = Local_Defs.define xs ctxt' + in + val ctxt' = ctxt + val w_consts = Array.fromList (map (Thm.cterm_of ctxt' o fst) aux) + val w_def_list = map (snd o snd) aux + val w_def_list_sym = map Thm.symmetric w_def_list + end + + fun holify_committee i = Array.sub (w_consts, i) + val holify_committee' = holify_committee #> Thm.term_of + + fun mk_committee_pref_thms app = + let + val ctxt'' = ctxt' addsimps (w_def_list @ @{thms strongly_preferred_def committee_preference_def}) + val n = Array.length committees + val app' = decode_approval_list app + fun mk' (c1, c2) = + let + val goal = + \<^instantiate>\X = "holify_committee' c1" and Y = "holify_committee' c2" + and A = "holify_approval_list' app" + in prop \X \[Comm(A)] Y\\ + fun tac ctxt = Simplifier.simp_tac ctxt + in + (c1, c2, Goal.prove ctxt'' [] [] goal (fn {context, ...} => HEADGOAL (tac context))) + end + fun mk (c1, c2) acc = + if committee_less app' (committee c1) (committee c2) then + mk' (c1, c2) :: acc + else + acc + in + fold_range (fn i => fold_range (fn j => mk (i, j)) n) n [] |> rev + end + + val committee_pref_cache = + (0 upto 14) |> Par_List.map mk_committee_pref_thms |> Array.fromList + + fun weak_rep ctxt (profiles : profile list) = + let + val n_committees = Array.length committees + val subtract = fold (remove1 op=) + datatype elim = Weak_Rep | Lemma of int + fun sing_alts (profile : profile) = + #profile profile + |> map_filter (fn i => case decode_approval_list i of [x] => SOME x | _ => NONE) + |> sort int_ord + |> group + |> filter (fn (_, n) => n >= 2) + |> map fst + val ctxt'' = ctxt addsimps w_def_list + + (* + Returns a list of all committees that are forbidden for the given profile due to + weak representation, and why they are forbidden (basic WP or the Lemma). + Additionally, the set of candidates forced into the committee by WP is also returned. + *) + fun elims profile = + let + val sings = sing_alts profile + val ws = 0 upto (n_committees - 1) + + fun upd (idx, i) (ws, acc)= + let + val xs = decode_approval_list i + fun suitable (idx', j) = + idx <> idx' andalso + let + val ys = decode_approval_list j + in + subset op= (ys, xs) andalso + not (subset op= (ys, sings)) + end + in + if exists_index suitable (#profile profile) then + let + val es = + filter_out (fn w => exists (member op= xs) (subtract sings (committee w))) ws + in + (subtract es ws, if null es then acc else (es, Lemma i) :: acc) + end + else + (ws, acc) + end + + val es = filter_out (fn w => is_subset sings (committee w)) ws + val state = (subtract es ws, if null es then [] else [(es, Weak_Rep)]) + val state = fold_index upd (#profile profile) state + in + (sings, fst state, snd state) + end + + fun mk_clauses (profile, (sings, remaining, justifications)) = + let + val sings_cterm = Thm.cterm_of ctxt (HOLogic.mk_set candT (map cand sings)) + val goal = + \<^instantiate>\A = "profile_cterm profile" and Z = sings_cterm + in cprop \\x\Z. count A {x :: 'a} \ 2\\ + val sing_thm = simp_prove ctxt'' goal + + fun mk_clauses' (ws, Weak_Rep) = + let + fun go w = + let + val insts = [((("W", 0), commT), holify_committee w)] + val rule = Thm.instantiate (TVars.empty, Vars.make insts) @{thm weak_representation'} + val thm = + (rule OF [#wf_thm profile, profile_def profile, sing_thm]) + |> simp_discharge_all ctxt'' + in + thm + end + in + map go ws + end + | mk_clauses' (ws, Lemma i) = + let + fun go w = + let + val insts = + [((("X", 0), candsetT), holify_approval_list i), + ((("W", 0), commT), holify_committee w)] + val rule = Thm.instantiate (TVars.empty, Vars.make insts) @{thm lemma2''} + val thm = + (rule OF [#wf_thm profile, profile_def profile, sing_thm]) + |> simp_discharge_all ctxt'' + in + thm + end + in + map go ws + end + in + (remaining, maps mk_clauses' justifications) + end + + in + profiles |> chop_groups 10 |> Par_List.map (map (fn profile => mk_clauses (profile, elims profile))) |> flat + end + + local + val _ = writeln "Generating weak representation facts..." + val start = Timing.start () + val wrs = weak_rep ctxt profs + val _ = writeln (Timing.message (Timing.result start)) + in + val possible_committee_table = Array.fromList (map (Inttab.make_set o fst) wrs) + val wr_thm_table = Array.fromList (map snd wrs) + fun is_possible_committee (p : profile, i : int) = + Inttab.defined (Array.sub (possible_committee_table, #id p)) i + end + + val n_vars = Array.length committees * length profs + val vars = + Array.tabulate (n_vars + 1, + fn v => if v = 0 then NONE else SOME ( + let val (i, j) = decode_var v + in \<^instantiate>\ + A = "profile_const (Array.sub (profile_array, i))" and W = "holify_committee' j" + in term \r A = W\\ |> Thm.cterm_of ctxt + end)) + fun get_var (profile : profile, i) = + the (Array.sub (vars, encode_var (#id profile, i))) + val get_var' = Thm.term_of o get_var + + fun total_conv wr_thms = + let + val neg_imp_eq_False_thm = @{lemma "\P \ P \ False" by simp} + val reduce_thms = @{lemma "False \ P \ P" and "P \ False \ P" by simp_all} + val thms = reduce_thms @ map (fn thm => thm RS neg_imp_eq_False_thm) wr_thms + in + Conv.bottom_conv (K (Conv.repeat_conv (Conv.rewrs_conv thms))) ctxt' + end + + fun functional profiles = + let + val unfold_thms = map (fn thm => thm RS @{thm eq_reflection}) + @{thms COM'_def pairs.simps list_ex_unfold list_all_unfold list.map append.simps prod.case} + val r_in_com_thm = + @{thm r_in_COM'} + |> Local_Defs.unfold ctxt' (unfold_thms @ w_def_list_sym) + val r_right_unique_thm = + @{thm r_right_unique} + |> Local_Defs.unfold ctxt' (unfold_thms @ w_def_list_sym) + val n = Array.length committees + val committee_pairs = pairs (0 upto (n - 1)) + fun mk_clause (profile : profile) = + let + val id = #id profile + val insts = [((("A", 0), profileT), profile_cterm profile), + ((("A'", 0), profileT), Thm.cterm_of ctxt' (profile_const profile))] |> Vars.make + val inst = Thm.instantiate (TVars.empty, insts) + val insts' = [((("A", 0), profileT), Thm.cterm_of ctxt' (profile_const profile))] |> Vars.make + val inst' = Thm.instantiate (TVars.empty, insts') + + val possible = + Inttab.keys (Array.sub (possible_committee_table, id)) + val total_clause = map (fn i => (false, id, i)) possible + val wr_thms = Array.sub (wr_thm_table, id) + val total_thm = + inst r_in_com_thm + |> mp (#wf_thm profile) + |> mp (profile_def profile) + |> Conv.fconv_rule (Conv.arg_conv (total_conv wr_thms)) + val right_unique_thms = inst' r_right_unique_thm |> HOLogic.conj_elims ctxt + val right_unique_clauses = + (committee_pairs ~~ right_unique_thms) + |> filter (fn ((i, j), _) => is_possible_committee (profile, i) + andalso is_possible_committee (profile, j)) + |> map (apfst (fn (i, j) => [(true, id, i), (true, id, j)])) + in + (total_clause, total_thm) :: right_unique_clauses + end + in + profiles |> chop_groups 200 |> Par_List.map (map mk_clause) |> flat |> flat + end + + val _ = writeln "Generating totality and right-uniqueness facts..." + val start = Timing.start () + val functional_clauses = functional profs + val _ = writeln (Timing.message (Timing.result start)) + + val symm_break_thm = + @{lemma "r ({#{C1}, {C2}, {C1, C2}, {C3}, {C4}, {C3, C4}#}) = {#C1, C1, C3#} \ + r ({#{C1}, {C2}, {C1, C2}, {C3}, {C4}, {C3, C4}#}) = {#C1, C2, C3#}" + by (use symmetry_break in simp)} + |> Local_Defs.fold ctxt' (Array.sub (profile_defs, 0) :: w_def_list) + val symm_break_clauses = [([(false, 0, 2), (false, 0, 5)], symm_break_thm)] + + fun stratproof (profiles : profile list) = + let + val subtract = fold (remove1 op=) + fun analyze_pair (p1, p2) = + let + val (xs, ys) = apply2 #profile (p1, p2) + in + case (subtract ys xs, subtract xs ys) of + ([x], [y]) => [(p1, x, p2, y), (p2, y, p1, x)] + | _ => [] + end + val ps = + profiles + |> rev_maps_pairs analyze_pair + fun mk_clauses (p1 : profile, x : int, p2 : profile, y : int) = + let + val x_idx = find_index (fn u => u = x) (#profile p1) + val y_idx = find_index (fn u => u = y) (#profile p2) + val x_cterm = get_approval_list (#cterm p1) x_idx + val y_cterm = get_approval_list (#cterm p2) y_idx + val insts = + [("A", profileT, #cterm p1), ("B", profileT, #cterm p2), + ("A'", profileT, profile_cconst p1), ("B'", profileT, profile_cconst p2), + ("X", candsetT, x_cterm), ("Y", candsetT, y_cterm)] + val vars = + insts + |> map (fn (s, T, t) => (((s, 0), T), t)) + |> Vars.make + val eq_thm = + prove_add_mset_eq + (y :: #profile p1, add_mset_cterm_candset y_cterm p1) + (x :: #profile p2, add_mset_cterm_candset x_cterm p2) + val not_less_thm = + not_manipulable_thm' + |> Thm.instantiate (TVars.empty, vars) + |> mp (#wf_thm p1) + |> mp (#wf_thm p2) + |> mp (profile_def p1) + |> mp (profile_def p2) + |> mp eq_thm + val comms = Array.sub (committee_pref_cache, x) + fun mk_clause (c1, c2, less_thm) = + if is_possible_committee (p1, c1) andalso is_possible_committee (p2, c2) then + let + val insts = + insts @ [("W", commT, holify_committee c1), ("W'", commT, holify_committee c2)] + val vars = + insts + |> map (fn (s, T, t) => (((s, 0), T), t)) + |> Vars.make + val thm = + stratproof_aux_thm + |> Thm.instantiate (TVars.empty, vars) + |> mp not_less_thm + |> mp (profile_def p1) + |> mp (profile_def p2) + |> mp less_thm + in + [([(true, #id p1, c1), (true, #id p2, c2)], thm)] + end + else + [] + in + maps mk_clause comms + end + in + ps + |> chop_groups 1000 + |> Par_List.map (map mk_clauses) + |> flat + |> flat + end + + val _ = writeln "Generating strategyproofness facts..." + val start = Timing.start () + val sp_clauses = stratproof profs + val _ = writeln (Timing.message (Timing.result start)) + + fun prep (cs, thm) = + (map encode_lit cs, Thm.transfer' ctxt' thm) + + val clauses = Array_Map.empty 1000000 : (int list * thm) Array_Map.T + val add_clauses = + fold (fn x => fn i => + let val _ = Array_Map.update (clauses, i + 1, SOME (prep x)) in i + 1 end) +(* val n_clauses = fold add_clauses [symm_break_clauses, functional_clauses, sp_clauses] 0*) + + val _ = writeln "Consolidating theorems..." + val start = Timing.start () + val clauses1 = flat (map (map fst) [symm_break_clauses, functional_clauses, sp_clauses]) + val clauses2 = + flat (map (map snd) [symm_break_clauses, functional_clauses, sp_clauses]) + |> consolidate_hyps ctxt' + val n_clauses = add_clauses (clauses1 ~~ clauses2) 0 + val _ = writeln (Timing.message (Timing.result start)) + + local + fun term_of_clause' cl = + let + fun term_of_var x = the (Array.sub (vars, x)) |> Thm.term_of + fun term_of_lit x = + if x < 0 then \<^const>\HOL.Not\ $ term_of_var (abs x) else term_of_var (abs x) + fun go [] = raise Empty + | go [x] = term_of_lit x + | go (x :: xs) = HOLogic.mk_disj (term_of_lit x, go xs) + in + go cl |> HOLogic.mk_Trueprop + end + + fun check_clause' (cl, thm) = + if Thm.prop_of thm = term_of_clause' cl then + () + else + raise CLAUSE (Thm.cterm_of ctxt (term_of_clause' cl), thm) + + fun check i = if i > n_clauses then () else + let val _ = check_clause' (the (Array_Map.sub (clauses, i))) in check (i+1) end + in + val _ = writeln "Checking correctness of clauses..." + val start = Timing.start () + val _ = check 1 + val _ = writeln (Timing.message (Timing.result start)) + end + + val n_vars = + fold_range (fn i => fn a => + case Array_Map.sub (clauses, i) of + SOME cl => fold (fn a => fn b => Int.max (abs a, b)) (fst cl) a + | NONE => a) (n_clauses + 1) 0 + +in + ((n_vars, vars), (n_clauses, clauses), ctxt') +end + + +fun dimacs_gen st put ((n_vars, _), (n_clauses, clauses), _) = + let + fun write_clause st (cs, _) = + put st (space_implode " " (map signed_string_of_int cs) ^ " 0") + val st = put st ("p cnf " ^ string_of_int n_vars ^ " " ^ string_of_int n_clauses ^ "") + fun write_clauses st i = if i > n_clauses then st else + let + val st = write_clause st (the (Array_Map.sub (clauses, i))) + in + write_clauses st (i + 1) + end + val st = write_clauses st 1 + in + st + end + +fun write_dimacs sat dimacs_file = + let + val f = TextIO.openOut (Path.implode dimacs_file) + val _ = dimacs_gen () (fn _ => fn s => TextIO.output (f, s ^ "\n")) sat + val _ = TextIO.closeOut f + in + () + end + +fun mk_dimacs sat = + dimacs_gen [] (fn xs => fn s => s :: xs) sat + |> rev + |> space_implode "\n" + +fun replay_grat ctxt ((n_vars, vars), (n_clauses, clauses), ctxt') grat_file = + let + val is_xz = + case Path.split_ext grat_file of + (_, "xz") => true + | _ => false + val rup_input = + { + ctxt = ctxt', + clauses = clauses, + vars = vars, + n_vars = n_vars, + tracing = false + } : Replay_RUP.rup_input + in + Replay_RUP.replay_rup_file rup_input (grat_file, is_xz) + |> Local_Defs.export ctxt' ctxt + |> snd + end + +fun derive_false ctxt profiles_file grat_file = + let + fun mk_thm () = + let + val thy = Proof_Context.theory_of ctxt + val _ = writeln "Generating SAT problem:" + val sat = mk_sat_problem ctxt profiles_file + val (n_vars, n_clauses) = case sat of ((n_vars, _), (n_clauses, _), _) => (n_vars, n_clauses) + val _ = + writeln ("Generated SAT problems has " ^ Int.toString n_vars ^ " variables, " ^ + Int.toString n_clauses ^ " clauses") + val path = Path.basic "papp.cnf" + val _ = + Export.export thy (Path.binding (path, \<^here>)) [XML.Text (mk_dimacs sat)] + val _ = writeln (Export.message thy path) + val _ = writeln " " + + val _ = writeln "Replaying DRUP proof..." + val start = Timing.start () + val thm = replay_grat ctxt sat grat_file + val _ = writeln (Timing.message (Timing.result start)) + in + thm + end + fun tac {context, ...} = HEADGOAL (resolve_tac context [mk_thm ()]) + val thm = Goal.prove_future ctxt [] [] \<^prop>\False\ tac + in + thm + end + +end diff --git a/thys/PAPP_Impossibility/sat_data/papp_impossibility.grat.xz b/thys/PAPP_Impossibility/sat_data/papp_impossibility.grat.xz new file 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