(* Author: Amine Chaieb, University of Cambridge *) section ‹Permutations, both general and specifically on finite sets.› theory Permutations imports "HOL-Library.Multiset" "HOL-Library.Disjoint_Sets" Transposition begin subsection ‹Auxiliary› abbreviation (input) fixpoints :: ‹('a ⇒ 'a) ⇒ 'a set› where ‹fixpoints f ≡ {x. f x = x}› lemma inj_on_fixpoints: ‹inj_on f (fixpoints f)› by (rule inj_onI) simp lemma bij_betw_fixpoints: ‹bij_betw f (fixpoints f) (fixpoints f)› using inj_on_fixpoints by (auto simp add: bij_betw_def) subsection ‹Basic definition and consequences› definition permutes :: ‹('a ⇒ 'a) ⇒ 'a set ⇒ bool› (infixr ‹permutes› 41) where ‹p permutes S ⟷ (∀x. x ∉ S ⟶ p x = x) ∧ (∀y. ∃!x. p x = y)› lemma bij_imp_permutes: ‹p permutes S› if ‹bij_betw p S S› and stable: ‹⋀x. x ∉ S ⟹ p x = x› proof - note ‹bij_betw p S S› moreover have ‹bij_betw p (- S) (- S)› by (auto simp add: stable intro!: bij_betw_imageI inj_onI) ultimately have ‹bij_betw p (S ∪ - S) (S ∪ - S)› by (rule bij_betw_combine) simp then have ‹∃!x. p x = y› for y by (simp add: bij_iff) with stable show ?thesis by (simp add: permutes_def) qed context fixes p :: ‹'a ⇒ 'a› and S :: ‹'a set› assumes perm: ‹p permutes S› begin lemma permutes_inj: ‹inj p› using perm by (auto simp: permutes_def inj_on_def) lemma permutes_image: ‹p ` S = S› proof (rule set_eqI) fix x show ‹x ∈ p ` S ⟷ x ∈ S› proof assume ‹x ∈ p ` S› then obtain y where ‹y ∈ S› ‹p y = x› by blast with perm show ‹x ∈ S› by (cases ‹y = x›) (auto simp add: permutes_def) next assume ‹x ∈ S› with perm obtain y where ‹y ∈ S› ‹p y = x› by (metis permutes_def) then show ‹x ∈ p ` S› by blast qed qed lemma permutes_not_in: ‹x ∉ S ⟹ p x = x› using perm by (auto simp: permutes_def) lemma permutes_image_complement: ‹p ` (- S) = - S› by (auto simp add: permutes_not_in) lemma permutes_in_image: ‹p x ∈ S ⟷ x ∈ S› using permutes_image permutes_inj by (auto dest: inj_image_mem_iff) lemma permutes_surj: ‹surj p› proof - have ‹p ` (S ∪ - S) = p ` S ∪ p ` (- S)› by (rule image_Un) then show ?thesis by (simp add: permutes_image permutes_image_complement) qed lemma permutes_inv_o: shows "p ∘ inv p = id" and "inv p ∘ p = id" using permutes_inj permutes_surj unfolding inj_iff [symmetric] surj_iff [symmetric] by auto lemma permutes_inverses: shows "p (inv p x) = x" and "inv p (p x) = x" using permutes_inv_o [unfolded fun_eq_iff o_def] by auto lemma permutes_inv_eq: ‹inv p y = x ⟷ p x = y› by (auto simp add: permutes_inverses) lemma permutes_inj_on: ‹inj_on p A› by (rule inj_on_subset [of _ UNIV]) (auto intro: permutes_inj) lemma permutes_bij: ‹bij p› unfolding bij_def by (metis permutes_inj permutes_surj) lemma permutes_imp_bij: ‹bij_betw p S S› by (simp add: bij_betw_def permutes_image permutes_inj_on) lemma permutes_subset: ‹p permutes T› if ‹S ⊆ T› proof (rule bij_imp_permutes) define R where ‹R = T - S› with that have ‹T = R ∪ S› ‹R ∩ S = {}› by auto then have ‹p x = x› if ‹x ∈ R› for x using that by (auto intro: permutes_not_in) then have ‹p ` R = R› by simp with ‹T = R ∪ S› show ‹bij_betw p T T› by (simp add: bij_betw_def permutes_inj_on image_Un permutes_image) fix x assume ‹x ∉ T› with ‹T = R ∪ S› show ‹p x = x› by (simp add: permutes_not_in) qed lemma permutes_imp_permutes_insert: ‹p permutes insert x S› by (rule permutes_subset) auto end lemma permutes_id [simp]: ‹id permutes S› by (auto intro: bij_imp_permutes) lemma permutes_empty [simp]: ‹p permutes {} ⟷ p = id› proof assume ‹p permutes {}› then show ‹p = id› by (auto simp add: fun_eq_iff permutes_not_in) next assume ‹p = id› then show ‹p permutes {}› by simp qed lemma permutes_sing [simp]: ‹p permutes {a} ⟷ p = id› proof assume perm: ‹p permutes {a}› show ‹p = id› proof fix x from perm have ‹p ` {a} = {a}› by (rule permutes_image) with perm show ‹p x = id x› by (cases ‹x = a›) (auto simp add: permutes_not_in) qed next assume ‹p = id› then show ‹p permutes {a}› by simp qed lemma permutes_univ: "p permutes UNIV ⟷ (∀y. ∃!x. p x = y)" by (simp add: permutes_def) lemma permutes_swap_id: "a ∈ S ⟹ b ∈ S ⟹ transpose a b permutes S" by (rule bij_imp_permutes) (auto intro: transpose_apply_other) lemma permutes_superset: ‹p permutes T› if ‹p permutes S› ‹⋀x. x ∈ S - T ⟹ p x = x› proof - define R U where ‹R = T ∩ S› and ‹U = S - T› then have ‹T = R ∪ (T - S)› ‹S = R ∪ U› ‹R ∩ U = {}› by auto from that ‹U = S - T› have ‹p ` U = U› by simp from ‹p permutes S› have ‹bij_betw p (R ∪ U) (R ∪ U)› by (simp add: permutes_imp_bij ‹S = R ∪ U›) moreover have ‹bij_betw p U U› using that ‹U = S - T› by (simp add: bij_betw_def permutes_inj_on) ultimately have ‹bij_betw p R R› using ‹R ∩ U = {}› ‹R ∩ U = {}› by (rule bij_betw_partition) then have ‹p permutes R› proof (rule bij_imp_permutes) fix x assume ‹x ∉ R› with ‹R = T ∩ S› ‹p permutes S› show ‹p x = x› by (cases ‹x ∈ S›) (auto simp add: permutes_not_in that(2)) qed then have ‹p permutes R ∪ (T - S)› by (rule permutes_subset) simp with ‹T = R ∪ (T - S)› show ?thesis by simp qed lemma permutes_bij_inv_into: ✐‹contributor ‹Lukas Bulwahn›› fixes A :: "'a set" and B :: "'b set" assumes "p permutes A" and "bij_betw f A B" shows "(λx. if x ∈ B then f (p (inv_into A f x)) else x) permutes B" proof (rule bij_imp_permutes) from assms have "bij_betw p A A" "bij_betw f A B" "bij_betw (inv_into A f) B A" by (auto simp add: permutes_imp_bij bij_betw_inv_into) then have "bij_betw (f ∘ p ∘ inv_into A f) B B" by (simp add: bij_betw_trans) then show "bij_betw (λx. if x ∈ B then f (p (inv_into A f x)) else x) B B" by (subst bij_betw_cong[where g="f ∘ p ∘ inv_into A f"]) auto next fix x assume "x ∉ B" then show "(if x ∈ B then f (p (inv_into A f x)) else x) = x" by auto qed lemma permutes_image_mset: ✐‹contributor ‹Lukas Bulwahn›› assumes "p permutes A" shows "image_mset p (mset_set A) = mset_set A" using assms by (metis image_mset_mset_set bij_betw_imp_inj_on permutes_imp_bij permutes_image) lemma permutes_implies_image_mset_eq: ✐‹contributor ‹Lukas Bulwahn›› assumes "p permutes A" "⋀x. x ∈ A ⟹ f x = f' (p x)" shows "image_mset f' (mset_set A) = image_mset f (mset_set A)" proof - have "f x = f' (p x)" if "x ∈# mset_set A" for x using assms(2)[of x] that by (cases "finite A") auto with assms have "image_mset f (mset_set A) = image_mset (f' ∘ p) (mset_set A)" by (auto intro!: image_mset_cong) also have "… = image_mset f' (image_mset p (mset_set A))" by (simp add: image_mset.compositionality) also have "… = image_mset f' (mset_set A)" proof - from assms permutes_image_mset have "image_mset p (mset_set A) = mset_set A" by blast then show ?thesis by simp qed finally show ?thesis .. qed subsection ‹Group properties› lemma permutes_compose: "p permutes S ⟹ q permutes S ⟹ q ∘ p permutes S" unfolding permutes_def o_def by metis lemma permutes_inv: assumes "p permutes S" shows "inv p permutes S" using assms unfolding permutes_def permutes_inv_eq[OF assms] by metis lemma permutes_inv_inv: assumes "p permutes S" shows "inv (inv p) = p" unfolding fun_eq_iff permutes_inv_eq[OF assms] permutes_inv_eq[OF permutes_inv[OF assms]] by blast lemma permutes_invI: assumes perm: "p permutes S" and inv: "⋀x. x ∈ S ⟹ p' (p x) = x" and outside: "⋀x. x ∉ S ⟹ p' x = x" shows "inv p = p'" proof show "inv p x = p' x" for x proof (cases "x ∈ S") case True from assms have "p' x = p' (p (inv p x))" by (simp add: permutes_inverses) also from permutes_inv[OF perm] True have "… = inv p x" by (subst inv) (simp_all add: permutes_in_image) finally show ?thesis .. next case False with permutes_inv[OF perm] show ?thesis by (simp_all add: outside permutes_not_in) qed qed lemma permutes_vimage: "f permutes A ⟹ f -` A = A" by (simp add: bij_vimage_eq_inv_image permutes_bij permutes_image[OF permutes_inv]) subsection ‹Mapping permutations with bijections› lemma bij_betw_permutations: assumes "bij_betw f A B" shows "bij_betw (λπ x. if x ∈ B then f (π (inv_into A f x)) else x) {π. π permutes A} {π. π permutes B}" (is "bij_betw ?f _ _") proof - let ?g = "(λπ x. if x ∈ A then inv_into A f (π (f x)) else x)" show ?thesis proof (rule bij_betw_byWitness [of _ ?g], goal_cases) case 3 show ?case using permutes_bij_inv_into[OF _ assms] by auto next case 4 have bij_inv: "bij_betw (inv_into A f) B A" by (intro bij_betw_inv_into assms) { fix π assume "π permutes B" from permutes_bij_inv_into[OF this bij_inv] and assms have "(λx. if x ∈ A then inv_into A f (π (f x)) else x) permutes A" by (simp add: inv_into_inv_into_eq cong: if_cong) } from this show ?case by (auto simp: permutes_inv) next case 1 thus ?case using assms by (auto simp: fun_eq_iff permutes_not_in permutes_in_image bij_betw_inv_into_left dest: bij_betwE) next case 2 moreover have "bij_betw (inv_into A f) B A" by (intro bij_betw_inv_into assms) ultimately show ?case using assms by (auto simp: fun_eq_iff permutes_not_in permutes_in_image bij_betw_inv_into_right dest: bij_betwE) qed qed lemma bij_betw_derangements: assumes "bij_betw f A B" shows "bij_betw (λπ x. if x ∈ B then f (π (inv_into A f x)) else x) {π. π permutes A ∧ (∀x∈A. π x ≠ x)} {π. π permutes B ∧ (∀x∈B. π x ≠ x)}" (is "bij_betw ?f _ _") proof - let ?g = "(λπ x. if x ∈ A then inv_into A f (π (f x)) else x)" show ?thesis proof (rule bij_betw_byWitness [of _ ?g], goal_cases) case 3 have "?f π x ≠ x" if "π permutes A" "⋀x. x ∈ A ⟹ π x ≠ x" "x ∈ B" for π x using that and assms by (metis bij_betwE bij_betw_imp_inj_on bij_betw_imp_surj_on inv_into_f_f inv_into_into permutes_imp_bij) with permutes_bij_inv_into[OF _ assms] show ?case by auto next case 4 have bij_inv: "bij_betw (inv_into A f) B A" by (intro bij_betw_inv_into assms) have "?g π permutes A" if "π permutes B" for π using permutes_bij_inv_into[OF that bij_inv] and assms by (simp add: inv_into_inv_into_eq cong: if_cong) moreover have "?g π x ≠ x" if "π permutes B" "⋀x. x ∈ B ⟹ π x ≠ x" "x ∈ A" for π x using that and assms by (metis bij_betwE bij_betw_imp_surj_on f_inv_into_f permutes_imp_bij) ultimately show ?case by auto next case 1 thus ?case using assms by (force simp: fun_eq_iff permutes_not_in permutes_in_image bij_betw_inv_into_left dest: bij_betwE) next case 2 moreover have "bij_betw (inv_into A f) B A" by (intro bij_betw_inv_into assms) ultimately show ?case using assms by (force simp: fun_eq_iff permutes_not_in permutes_in_image bij_betw_inv_into_right dest: bij_betwE) qed qed subsection ‹The number of permutations on a finite set› lemma permutes_insert_lemma: assumes "p permutes (insert a S)" shows "transpose a (p a) ∘ p permutes S" apply (rule permutes_superset[where S = "insert a S"]) apply (rule permutes_compose[OF assms]) apply (rule permutes_swap_id, simp) using permutes_in_image[OF assms, of a] apply simp apply (auto simp add: Ball_def) done lemma permutes_insert: "{p. p permutes (insert a S)} = (λ(b, p). transpose a b ∘ p) ` {(b, p). b ∈ insert a S ∧ p ∈ {p. p permutes S}}" proof - have "p permutes insert a S ⟷ (∃b q. p = transpose a b ∘ q ∧ b ∈ insert a S ∧ q permutes S)" for p proof - have "∃b q. p = transpose a b ∘ q ∧ b ∈ insert a S ∧ q permutes S" if p: "p permutes insert a S" proof - let ?b = "p a" let ?q = "transpose a (p a) ∘ p" have *: "p = transpose a ?b ∘ ?q" by (simp add: fun_eq_iff o_assoc) have **: "?b ∈ insert a S" unfolding permutes_in_image[OF p] by simp from permutes_insert_lemma[OF p] * ** show ?thesis by blast qed moreover have "p permutes insert a S" if bq: "p = transpose a b ∘ q" "b ∈ insert a S" "q permutes S" for b q proof - from permutes_subset[OF bq(3), of "insert a S"] have q: "q permutes insert a S" by auto have a: "a ∈ insert a S" by simp from bq(1) permutes_compose[OF q permutes_swap_id[OF a bq(2)]] show ?thesis by simp qed ultimately show ?thesis by blast qed then show ?thesis by auto qed lemma card_permutations: assumes "card S = n" and "finite S" shows "card {p. p permutes S} = fact n" using assms(2,1) proof (induct arbitrary: n) case empty then show ?case by simp next case (insert x F) { fix n assume card_insert: "card (insert x F) = n" let ?xF = "{p. p permutes insert x F}" let ?pF = "{p. p permutes F}" let ?pF' = "{(b, p). b ∈ insert x F ∧ p ∈ ?pF}" let ?g = "(λ(b, p). transpose x b ∘ p)" have xfgpF': "?xF = ?g ` ?pF'" by (rule permutes_insert[of x F]) from ‹x ∉ F› ‹finite F› card_insert have Fs: "card F = n - 1" by auto from ‹finite F› insert.hyps Fs have pFs: "card ?pF = fact (n - 1)" by auto then have "finite ?pF" by (auto intro: card_ge_0_finite) with ‹finite F› card.insert_remove have pF'f: "finite ?pF'" apply (simp only: Collect_case_prod Collect_mem_eq) apply (rule finite_cartesian_product) apply simp_all done have ginj: "inj_on ?g ?pF'" proof - { fix b p c q assume bp: "(b, p) ∈ ?pF'" assume cq: "(c, q) ∈ ?pF'" assume eq: "?g (b, p) = ?g (c, q)" from bp cq have pF: "p permutes F" and qF: "q permutes F" by auto from pF ‹x ∉ F› eq have "b = ?g (b, p) x" by (auto simp: permutes_def fun_upd_def fun_eq_iff) also from qF ‹x ∉ F› eq have "… = ?g (c, q) x" by (auto simp: fun_upd_def fun_eq_iff) also from qF ‹x ∉ F› have "… = c" by (auto simp: permutes_def fun_upd_def fun_eq_iff) finally have "b = c" . then have "transpose x b = transpose x c" by simp with eq have "transpose x b ∘ p = transpose x b ∘ q" by simp then have "transpose x b ∘ (transpose x b ∘ p) = transpose x b ∘ (transpose x b ∘ q)" by simp then have "p = q" by (simp add: o_assoc) with ‹b = c› have "(b, p) = (c, q)" by simp } then show ?thesis unfolding inj_on_def by blast qed from ‹x ∉ F› ‹finite F› card_insert have "n ≠ 0" by auto then have "∃m. n = Suc m" by presburger then obtain m where n: "n = Suc m" by blast from pFs card_insert have *: "card ?xF = fact n" unfolding xfgpF' card_image[OF ginj] using ‹finite F› ‹finite ?pF› by (simp only: Collect_case_prod Collect_mem_eq card_cartesian_product) (simp add: n) from finite_imageI[OF pF'f, of ?g] have xFf: "finite ?xF" by (simp add: xfgpF' n) from * have "card ?xF = fact n" unfolding xFf by blast } with insert show ?case by simp qed lemma finite_permutations: assumes "finite S" shows "finite {p. p permutes S}" using card_permutations[OF refl assms] by (auto intro: card_ge_0_finite) subsection ‹Hence a sort of induction principle composing by swaps› lemma permutes_induct [consumes 2, case_names id swap]: ‹P p› if ‹p permutes S› ‹finite S› and id: ‹P id› and swap: ‹⋀a b p. a ∈ S ⟹ b ∈ S ⟹ p permutes S ⟹ P p ⟹ P (transpose a b ∘ p)› using ‹finite S› ‹p permutes S› swap proof (induction S arbitrary: p) case empty with id show ?case by (simp only: permutes_empty) next case (insert x S p) define q where ‹q = transpose x (p x) ∘ p› then have swap_q: ‹transpose x (p x) ∘ q = p› by (simp add: o_assoc) from ‹p permutes insert x S› have ‹q permutes S› by (simp add: q_def permutes_insert_lemma) then have ‹q permutes insert x S› by (simp add: permutes_imp_permutes_insert) from ‹q permutes S› have ‹P q› by (auto intro: insert.IH insert.prems(2) permutes_imp_permutes_insert) have ‹x ∈ insert x S› by simp moreover from ‹p permutes insert x S› have ‹p x ∈ insert x S› using permutes_in_image [of p ‹insert x S› x] by simp ultimately have ‹P (transpose x (p x) ∘ q)› using ‹q permutes insert x S› ‹P q› by (rule insert.prems(2)) then show ?case by (simp add: swap_q) qed lemma permutes_rev_induct [consumes 2, case_names id swap]: ‹P p› if ‹p permutes S› ‹finite S› and id': ‹P id› and swap': ‹⋀a b p. a ∈ S ⟹ b ∈ S ⟹ p permutes S ⟹ P p ⟹ P (p ∘ transpose a b)› using ‹p permutes S› ‹finite S› proof (induction rule: permutes_induct) case id from id' show ?case . next case (swap a b p) then have ‹bij p› using permutes_bij by blast have ‹P (p ∘ transpose (inv p a) (inv p b))› by (rule swap') (auto simp add: swap permutes_in_image permutes_inv) also have ‹p ∘ transpose (inv p a) (inv p b) = transpose a b ∘ p› using ‹bij p› by (rule transpose_comp_eq [symmetric]) finally show ?case . qed subsection ‹Permutations of index set for iterated operations› lemma (in comm_monoid_set) permute: assumes "p permutes S" shows "F g S = F (g ∘ p) S" proof - from ‹p permutes S› have "inj p" by (rule permutes_inj) then have "inj_on p S" by (auto intro: subset_inj_on) then have "F g (p ` S) = F (g ∘ p) S" by (rule reindex) moreover from ‹p permutes S› have "p ` S = S" by (rule permutes_image) ultimately show ?thesis by simp qed subsection ‹Permutations as transposition sequences› inductive swapidseq :: "nat ⇒ ('a ⇒ 'a) ⇒ bool" where id[simp]: "swapidseq 0 id" | comp_Suc: "swapidseq n p ⟹ a ≠ b ⟹ swapidseq (Suc n) (transpose a b ∘ p)" declare id[unfolded id_def, simp] definition "permutation p ⟷ (∃n. swapidseq n p)" subsection ‹Some closure properties of the set of permutations, with lengths› lemma permutation_id[simp]: "permutation id" unfolding permutation_def by (rule exI[where x=0]) simp declare permutation_id[unfolded id_def, simp] lemma swapidseq_swap: "swapidseq (if a = b then 0 else 1) (transpose a b)" apply clarsimp using comp_Suc[of 0 id a b] apply simp done lemma permutation_swap_id: "permutation (transpose a b)" proof (cases "a = b") case True then show ?thesis by simp next case False then show ?thesis unfolding permutation_def using swapidseq_swap[of a b] by blast qed lemma swapidseq_comp_add: "swapidseq n p ⟹ swapidseq m q ⟹ swapidseq (n + m) (p ∘ q)" proof (induct n p arbitrary: m q rule: swapidseq.induct) case (id m q) then show ?case by simp next case (comp_Suc n p a b m q) have eq: "Suc n + m = Suc (n + m)" by arith show ?case apply (simp only: eq comp_assoc) apply (rule swapidseq.comp_Suc) using comp_Suc.hyps(2)[OF comp_Suc.prems] comp_Suc.hyps(3) apply blast+ done qed lemma permutation_compose: "permutation p ⟹ permutation q ⟹ permutation (p ∘ q)" unfolding permutation_def using swapidseq_comp_add[of _ p _ q] by metis lemma swapidseq_endswap: "swapidseq n p ⟹ a ≠ b ⟹ swapidseq (Suc n) (p ∘ transpose a b)" by (induct n p rule: swapidseq.induct) (use swapidseq_swap[of a b] in ‹auto simp add: comp_assoc intro: swapidseq.comp_Suc›) lemma swapidseq_inverse_exists: "swapidseq n p ⟹ ∃q. swapidseq n q ∧ p ∘ q = id ∧ q ∘ p = id" proof (induct n p rule: swapidseq.induct) case id then show ?case by (rule exI[where x=id]) simp next case (comp_Suc n p a b) from comp_Suc.hyps obtain q where q: "swapidseq n q" "p ∘ q = id" "q ∘ p = id" by blast let ?q = "q ∘ transpose a b" note H = comp_Suc.hyps from swapidseq_swap[of a b] H(3) have *: "swapidseq 1 (transpose a b)" by simp from swapidseq_comp_add[OF q(1) *] have **: "swapidseq (Suc n) ?q" by simp have "transpose a b ∘ p ∘ ?q = transpose a b ∘ (p ∘ q) ∘ transpose a b" by (simp add: o_assoc) also have "… = id" by (simp add: q(2)) finally have ***: "transpose a b ∘ p ∘ ?q = id" . have "?q ∘ (transpose a b ∘ p) = q ∘ (transpose a b ∘ transpose a b) ∘ p" by (simp only: o_assoc) then have "?q ∘ (transpose a b ∘ p) = id" by (simp add: q(3)) with ** *** show ?case by blast qed lemma swapidseq_inverse: assumes "swapidseq n p" shows "swapidseq n (inv p)" using swapidseq_inverse_exists[OF assms] inv_unique_comp[of p] by auto lemma permutation_inverse: "permutation p ⟹ permutation (inv p)" using permutation_def swapidseq_inverse by blast subsection ‹Various combinations of transpositions with 2, 1 and 0 common elements› lemma swap_id_common:" a ≠ c ⟹ b ≠ c ⟹ transpose a b ∘ transpose a c = transpose b c ∘ transpose a b" by (simp add: fun_eq_iff transpose_def) lemma swap_id_common': "a ≠ b ⟹ a ≠ c ⟹ transpose a c ∘ transpose b c = transpose b c ∘ transpose a b" by (simp add: fun_eq_iff transpose_def) lemma swap_id_independent: "a ≠ c ⟹ a ≠ d ⟹ b ≠ c ⟹ b ≠ d ⟹ transpose a b ∘ transpose c d = transpose c d ∘ transpose a b" by (simp add: fun_eq_iff transpose_def) subsection ‹The identity map only has even transposition sequences› lemma symmetry_lemma: assumes "⋀a b c d. P a b c d ⟹ P a b d c" and "⋀a b c d. a ≠ b ⟹ c ≠ d ⟹ a = c ∧ b = d ∨ a = c ∧ b ≠ d ∨ a ≠ c ∧ b = d ∨ a ≠ c ∧ a ≠ d ∧ b ≠ c ∧ b ≠ d ⟹ P a b c d" shows "⋀a b c d. a ≠ b ⟶ c ≠ d ⟶ P a b c d" using assms by metis lemma swap_general: "a ≠ b ⟹ c ≠ d ⟹ transpose a b ∘ transpose c d = id ∨ (∃x y z. x ≠ a ∧ y ≠ a ∧ z ≠ a ∧ x ≠ y ∧ transpose a b ∘ transpose c d = transpose x y ∘ transpose a z)" proof - assume neq: "a ≠ b" "c ≠ d" have "a ≠ b ⟶ c ≠ d ⟶ (transpose a b ∘ transpose c d = id ∨ (∃x y z. x ≠ a ∧ y ≠ a ∧ z ≠ a ∧ x ≠ y ∧ transpose a b ∘ transpose c d = transpose x y ∘ transpose a z))" apply (rule symmetry_lemma[where a=a and b=b and c=c and d=d]) apply (simp_all only: ac_simps) apply (metis id_comp swap_id_common swap_id_common' swap_id_independent transpose_comp_involutory) done with neq show ?thesis by metis qed lemma swapidseq_id_iff[simp]: "swapidseq 0 p ⟷ p = id" using swapidseq.cases[of 0 p "p = id"] by auto lemma swapidseq_cases: "swapidseq n p ⟷ n = 0 ∧ p = id ∨ (∃a b q m. n = Suc m ∧ p = transpose a b ∘ q ∧ swapidseq m q ∧ a ≠ b)" apply (rule iffI) apply (erule swapidseq.cases[of n p]) apply simp apply (rule disjI2) apply (rule_tac x= "a" in exI) apply (rule_tac x= "b" in exI) apply (rule_tac x= "pa" in exI) apply (rule_tac x= "na" in exI) apply simp apply auto apply (rule comp_Suc, simp_all) done lemma fixing_swapidseq_decrease: assumes "swapidseq n p" and "a ≠ b" and "(transpose a b ∘ p) a = a" shows "n ≠ 0 ∧ swapidseq (n - 1) (transpose a b ∘ p)" using assms proof (induct n arbitrary: p a b) case 0 then show ?case by (auto simp add: fun_upd_def) next case (Suc n p a b) from Suc.prems(1) swapidseq_cases[of "Suc n" p] obtain c d q m where cdqm: "Suc n = Suc m" "p = transpose c d ∘ q" "swapidseq m q" "c ≠ d" "n = m" by auto consider "transpose a b ∘ transpose c d = id" | x y z where "x ≠ a" "y ≠ a" "z ≠ a" "x ≠ y" "transpose a b ∘ transpose c d = transpose x y ∘ transpose a z" using swap_general[OF Suc.prems(2) cdqm(4)] by metis then show ?case proof cases case 1 then show ?thesis by (simp only: cdqm o_assoc) (simp add: cdqm) next case prems: 2 then have az: "a ≠ z" by simp from prems have *: "(transpose x y ∘ h) a = a ⟷ h a = a" for h by (simp add: transpose_def) from cdqm(2) have "transpose a b ∘ p = transpose a b ∘ (transpose c d ∘ q)" by simp then have "transpose a b ∘ p = transpose x y ∘ (transpose a z ∘ q)" by (simp add: o_assoc prems) then have "(transpose a b ∘ p) a = (transpose x y ∘ (transpose a z ∘ q)) a" by simp then have "(transpose x y ∘ (transpose a z ∘ q)) a = a" unfolding Suc by metis then have "(transpose a z ∘ q) a = a" by (simp only: *) from Suc.hyps[OF cdqm(3)[ unfolded cdqm(5)[symmetric]] az this] have **: "swapidseq (n - 1) (transpose a z ∘ q)" "n ≠ 0" by blast+ from ‹n ≠ 0› have ***: "Suc n - 1 = Suc (n - 1)" by auto show ?thesis apply (simp only: cdqm(2) prems o_assoc ***) apply (simp only: Suc_not_Zero simp_thms comp_assoc) apply (rule comp_Suc) using ** prems apply blast+ done qed qed lemma swapidseq_identity_even: assumes "swapidseq n (id :: 'a ⇒ 'a)" shows "even n" using ‹swapidseq n id› proof (induct n rule: nat_less_induct) case H: (1 n) consider "n = 0" | a b :: 'a and q m where "n = Suc m" "id = transpose a b ∘ q" "swapidseq m q" "a ≠ b" using H(2)[unfolded swapidseq_cases[of n id]] by auto then show ?case proof cases case 1 then show ?thesis by presburger next case h: 2 from fixing_swapidseq_decrease[OF h(3,4), unfolded h(2)[symmetric]] have m: "m ≠ 0" "swapidseq (m - 1) (id :: 'a ⇒ 'a)" by auto from h m have mn: "m - 1 < n" by arith from H(1)[rule_format, OF mn m(2)] h(1) m(1) show ?thesis by presburger qed qed subsection ‹Therefore we have a welldefined notion of parity› definition "evenperm p = even (SOME n. swapidseq n p)" lemma swapidseq_even_even: assumes m: "swapidseq m p" and n: "swapidseq n p" shows "even m ⟷ even n" proof - from swapidseq_inverse_exists[OF n] obtain q where q: "swapidseq n q" "p ∘ q = id" "q ∘ p = id" by blast from swapidseq_identity_even[OF swapidseq_comp_add[OF m q(1), unfolded q]] show ?thesis by arith qed lemma evenperm_unique: assumes p: "swapidseq n p" and n:"even n = b" shows "evenperm p = b" unfolding n[symmetric] evenperm_def apply (rule swapidseq_even_even[where p = p]) apply (rule someI[where x = n]) using p apply blast+ done subsection ‹And it has the expected composition properties› lemma evenperm_id[simp]: "evenperm id = True" by (rule evenperm_unique[where n = 0]) simp_all lemma evenperm_identity [simp]: ‹evenperm (λx. x)› using evenperm_id by (simp add: id_def [abs_def]) lemma evenperm_swap: "evenperm (transpose a b) = (a = b)" by (rule evenperm_unique[where n="if a = b then 0 else 1"]) (simp_all add: swapidseq_swap) lemma evenperm_comp: assumes "permutation p" "permutation q" shows "evenperm (p ∘ q) ⟷ evenperm p = evenperm q" proof - from assms obtain n m where n: "swapidseq n p" and m: "swapidseq m q" unfolding permutation_def by blast have "even (n + m) ⟷ (even n ⟷ even m)" by arith from evenperm_unique[OF n refl] evenperm_unique[OF m refl] and evenperm_unique[OF swapidseq_comp_add[OF n m] this] show ?thesis by blast qed lemma evenperm_inv: assumes "permutation p" shows "evenperm (inv p) = evenperm p" proof - from assms obtain n where n: "swapidseq n p" unfolding permutation_def by blast show ?thesis by (rule evenperm_unique[OF swapidseq_inverse[OF n] evenperm_unique[OF n refl, symmetric]]) qed subsection ‹A more abstract characterization of permutations› lemma permutation_bijective: assumes "permutation p" shows "bij p" proof - from assms obtain n where n: "swapidseq n p" unfolding permutation_def by blast from swapidseq_inverse_exists[OF n] obtain q where q: "swapidseq n q" "p ∘ q = id" "q ∘ p = id" by blast then show ?thesis unfolding bij_iff apply (auto simp add: fun_eq_iff) apply metis done qed lemma permutation_finite_support: assumes "permutation p" shows "finite {x. p x ≠ x}" proof - from assms obtain n where "swapidseq n p" unfolding permutation_def by blast then show ?thesis proof (induct n p rule: swapidseq.induct) case id then show ?case by simp next case (comp_Suc n p a b) let ?S = "insert a (insert b {x. p x ≠ x})" from comp_Suc.hyps(2) have *: "finite ?S" by simp from ‹a ≠ b› have **: "{x. (transpose a b ∘ p) x ≠ x} ⊆ ?S" by auto show ?case by (rule finite_subset[OF ** *]) qed qed lemma permutation_lemma: assumes "finite S" and "bij p" and "∀x. x ∉ S ⟶ p x = x" shows "permutation p" using assms proof (induct S arbitrary: p rule: finite_induct) case empty then show ?case by simp next case (insert a F p) let ?r = "transpose a (p a) ∘ p" let ?q = "transpose a (p a) ∘ ?r" have *: "?r a = a" by simp from insert * have **: "∀x. x ∉ F ⟶ ?r x = x" by (metis bij_pointE comp_apply id_apply insert_iff swap_apply(3)) have "bij ?r" using insert by (simp add: bij_comp) have "permutation ?r" by (rule insert(3)[OF ‹bij ?r› **]) then have "permutation ?q" by (simp add: permutation_compose permutation_swap_id) then show ?case by (simp add: o_assoc) qed lemma permutation: "permutation p ⟷ bij p ∧ finite {x. p x ≠ x}" (is "?lhs ⟷ ?b ∧ ?f") proof assume ?lhs with permutation_bijective permutation_finite_support show "?b ∧ ?f" by auto next assume "?b ∧ ?f" then have "?f" "?b" by blast+ from permutation_lemma[OF this] show ?lhs by blast qed lemma permutation_inverse_works: assumes "permutation p" shows "inv p ∘ p = id" and "p ∘ inv p = id" using permutation_bijective [OF assms] by (auto simp: bij_def inj_iff surj_iff) lemma permutation_inverse_compose: assumes p: "permutation p" and q: "permutation q" shows "inv (p ∘ q) = inv q ∘ inv p" proof - note ps = permutation_inverse_works[OF p] note qs = permutation_inverse_works[OF q] have "p ∘ q ∘ (inv q ∘ inv p) = p ∘ (q ∘ inv q) ∘ inv p" by (simp add: o_assoc) also have "… = id" by (simp add: ps qs) finally have *: "p ∘ q ∘ (inv q ∘ inv p) = id" . have "inv q ∘ inv p ∘ (p ∘ q) = inv q ∘ (inv p ∘ p) ∘ q" by (simp add: o_assoc) also have "… = id" by (simp add: ps qs) finally have **: "inv q ∘ inv p ∘ (p ∘ q) = id" . show ?thesis by (rule inv_unique_comp[OF * **]) qed subsection ‹Relation to ‹permutes›› lemma permutes_imp_permutation: ‹permutation p› if ‹finite S› ‹p permutes S› proof - from ‹p permutes S› have ‹{x. p x ≠ x} ⊆ S› by (auto dest: permutes_not_in) then have ‹finite {x. p x ≠ x}› using ‹finite S› by (rule finite_subset) moreover from ‹p permutes S› have ‹bij p› by (auto dest: permutes_bij) ultimately show ?thesis by (simp add: permutation) qed lemma permutation_permutesE: assumes ‹permutation p› obtains S where ‹finite S› ‹p permutes S› proof - from assms have fin: ‹finite {x. p x ≠ x}› by (simp add: permutation) from assms have ‹bij p› by (simp add: permutation) also have ‹UNIV = {x. p x ≠ x} ∪ {x. p x = x}› by auto finally have ‹bij_betw p {x. p x ≠ x} {x. p x ≠ x}› by (rule bij_betw_partition) (auto simp add: bij_betw_fixpoints) then have ‹p permutes {x. p x ≠ x}› by (auto intro: bij_imp_permutes) with fin show thesis .. qed lemma permutation_permutes: "permutation p ⟷ (∃S. finite S ∧ p permutes S)" by (auto elim: permutation_permutesE intro: permutes_imp_permutation) subsection ‹Sign of a permutation as a real number› definition sign :: ‹('a ⇒ 'a) ⇒ int› ― ‹TODO: prefer less generic name› where ‹sign p = (if evenperm p then 1 else - 1)› lemma sign_cases [case_names even odd]: obtains ‹sign p = 1› | ‹sign p = - 1› by (cases ‹evenperm p›) (simp_all add: sign_def) lemma sign_nz [simp]: "sign p ≠ 0" by (cases p rule: sign_cases) simp_all lemma sign_id [simp]: "sign id = 1" by (simp add: sign_def) lemma sign_identity [simp]: ‹sign (λx. x) = 1› by (simp add: sign_def) lemma sign_inverse: "permutation p ⟹ sign (inv p) = sign p" by (simp add: sign_def evenperm_inv) lemma sign_compose: "permutation p ⟹ permutation q ⟹ sign (p ∘ q) = sign p * sign q" by (simp add: sign_def evenperm_comp) lemma sign_swap_id: "sign (transpose a b) = (if a = b then 1 else - 1)" by (simp add: sign_def evenperm_swap) lemma sign_idempotent [simp]: "sign p * sign p = 1" by (simp add: sign_def) lemma sign_left_idempotent [simp]: ‹sign p * (sign p * sign q) = sign q› by (simp add: sign_def) term "(bij, bij_betw, permutation)" subsection ‹Permuting a list› text ‹This function permutes a list by applying a permutation to the indices.› definition permute_list :: "(nat ⇒ nat) ⇒ 'a list ⇒ 'a list" where "permute_list f xs = map (λi. xs ! (f i)) [0..<length xs]" lemma permute_list_map: assumes "f permutes {..<length xs}" shows "permute_list f (map g xs) = map g (permute_list f xs)" using permutes_in_image[OF assms] by (auto simp: permute_list_def) lemma permute_list_nth: assumes "f permutes {..<length xs}" "i < length xs" shows "permute_list f xs ! i = xs ! f i" using permutes_in_image[OF assms(1)] assms(2) by (simp add: permute_list_def) lemma permute_list_Nil [simp]: "permute_list f [] = []" by (simp add: permute_list_def) lemma length_permute_list [simp]: "length (permute_list f xs) = length xs" by (simp add: permute_list_def) lemma permute_list_compose: assumes "g permutes {..<length xs}" shows "permute_list (f ∘ g) xs = permute_list g (permute_list f xs)" using assms[THEN permutes_in_image] by (auto simp add: permute_list_def) lemma permute_list_ident [simp]: "permute_list (λx. x) xs = xs" by (simp add: permute_list_def map_nth) lemma permute_list_id [simp]: "permute_list id xs = xs" by (simp add: id_def) lemma mset_permute_list [simp]: fixes xs :: "'a list" assumes "f permutes {..<length xs}" shows "mset (permute_list f xs) = mset xs" proof (rule multiset_eqI) fix y :: 'a from assms have [simp]: "f x < length xs ⟷ x < length xs" for x using permutes_in_image[OF assms] by auto have "count (mset (permute_list f xs)) y = card ((λi. xs ! f i) -` {y} ∩ {..<length xs})" by (simp add: permute_list_def count_image_mset atLeast0LessThan) also have "(λi. xs ! f i) -` {y} ∩ {..<length xs} = f -` {i. i < length xs ∧ y = xs ! i}" by auto also from assms have "card … = card {i. i < length xs ∧ y = xs ! i}" by (intro card_vimage_inj) (auto simp: permutes_inj permutes_surj) also have "… = count (mset xs) y" by (simp add: count_mset length_filter_conv_card) finally show "count (mset (permute_list f xs)) y = count (mset xs) y" by simp qed lemma set_permute_list [simp]: assumes "f permutes {..<length xs}" shows "set (permute_list f xs) = set xs" by (rule mset_eq_setD[OF mset_permute_list]) fact lemma distinct_permute_list [simp]: assumes "f permutes {..<length xs}" shows "distinct (permute_list f xs) = distinct xs" by (simp add: distinct_count_atmost_1 assms) lemma permute_list_zip: assumes "f permutes A" "A = {..<length xs}" assumes [simp]: "length xs = length ys" shows "permute_list f (zip xs ys) = zip (permute_list f xs) (permute_list f ys)" proof - from permutes_in_image[OF assms(1)] assms(2) have *: "f i < length ys ⟷ i < length ys" for i by simp have "permute_list f (zip xs ys) = map (λi. zip xs ys ! f i) [0..<length ys]" by (simp_all add: permute_list_def zip_map_map) also have "… = map (λ(x, y). (xs ! f x, ys ! f y)) (zip [0..<length ys] [0..<length ys])" by (intro nth_equalityI) (simp_all add: *) also have "… = zip (permute_list f xs) (permute_list f ys)" by (simp_all add: permute_list_def zip_map_map) finally show ?thesis . qed lemma map_of_permute: assumes "σ permutes fst ` set xs" shows "map_of xs ∘ σ = map_of (map (λ(x,y). (inv σ x, y)) xs)" (is "_ = map_of (map ?f _)") proof from assms have "inj σ" "surj σ" by (simp_all add: permutes_inj permutes_surj) then show "(map_of xs ∘ σ) x = map_of (map ?f xs) x" for x by (induct xs) (auto simp: inv_f_f surj_f_inv_f) qed lemma list_all2_permute_list_iff: ‹list_all2 P (permute_list p xs) (permute_list p ys) ⟷ list_all2 P xs ys› if ‹p permutes {..<length xs}› using that by (auto simp add: list_all2_iff simp flip: permute_list_zip) subsection ‹More lemmas about permutations› lemma permutes_in_funpow_image: ✐‹contributor ‹Lars Noschinski›› assumes "f permutes S" "x ∈ S" shows "(f ^^ n) x ∈ S" using assms by (induction n) (auto simp: permutes_in_image) lemma permutation_self: ✐‹contributor ‹Lars Noschinski›› assumes ‹permutation p› obtains n where ‹n > 0› ‹(p ^^ n) x = x› proof (cases ‹p x = x›) case True with that [of 1] show thesis by simp next case False from ‹permutation p› have ‹inj p› by (intro permutation_bijective bij_is_inj) moreover from ‹p x ≠ x› have ‹(p ^^ Suc n) x ≠ (p ^^ n) x› for n proof (induction n arbitrary: x) case 0 then show ?case by simp next case (Suc n) have "p (p x) ≠ p x" proof (rule notI) assume "p (p x) = p x" then show False using ‹p x ≠ x› ‹inj p› by (simp add: inj_eq) qed have "(p ^^ Suc (Suc n)) x = (p ^^ Suc n) (p x)" by (simp add: funpow_swap1) also have "… ≠ (p ^^ n) (p x)" by (rule Suc) fact also have "(p ^^ n) (p x) = (p ^^ Suc n) x" by (simp add: funpow_swap1) finally show ?case by simp qed then have "{y. ∃n. y = (p ^^ n) x} ⊆ {x. p x ≠ x}" by auto then have "finite {y. ∃n. y = (p ^^ n) x}" using permutation_finite_support[OF assms] by (rule finite_subset) ultimately obtain n where ‹n > 0› ‹(p ^^ n) x = x› by (rule funpow_inj_finite) with that [of n] show thesis by blast qed text ‹The following few lemmas were contributed by Lukas Bulwahn.› lemma count_image_mset_eq_card_vimage: assumes "finite A" shows "count (image_mset f (mset_set A)) b = card {a ∈ A. f a = b}" using assms proof (induct A) case empty show ?case by simp next case (insert x F) show ?case proof (cases "f x = b") case True with insert.hyps have "count (image_mset f (mset_set (insert x F))) b = Suc (card {a ∈ F. f a = f x})" by auto also from insert.hyps(1,2) have "… = card (insert x {a ∈ F. f a = f x})" by simp also from ‹f x = b› have "card (insert x {a ∈ F. f a = f x}) = card {a ∈ insert x F. f a = b}" by (auto intro: arg_cong[where f="card"]) finally show ?thesis using insert by auto next case False then have "{a ∈ F. f a = b} = {a ∈ insert x F. f a = b}" by auto with insert False show ?thesis by simp qed qed ― ‹Prove ‹image_mset_eq_implies_permutes› ...› lemma image_mset_eq_implies_permutes: fixes f :: "'a ⇒ 'b" assumes "finite A" and mset_eq: "image_mset f (mset_set A) = image_mset f' (mset_set A)" obtains p where "p permutes A" and "∀x∈A. f x = f' (p x)" proof - from ‹finite A› have [simp]: "finite {a ∈ A. f a = (b::'b)}" for f b by auto have "f ` A = f' ` A" proof - from ‹finite A› have "f ` A = f ` (set_mset (mset_set A))" by simp also have "… = f' ` set_mset (mset_set A)" by (metis mset_eq multiset.set_map) also from ‹finite A› have "… = f' ` A" by simp finally show ?thesis . qed have "∀b∈(f ` A). ∃p. bij_betw p {a ∈ A. f a = b} {a ∈ A. f' a = b}" proof fix b from mset_eq have "count (image_mset f (mset_set A)) b = count (image_mset f' (mset_set A)) b" by simp with ‹finite A› have "card {a ∈ A. f a = b} = card {a ∈ A. f' a = b}" by (simp add: count_image_mset_eq_card_vimage) then show "∃p. bij_betw p {a∈A. f a = b} {a ∈ A. f' a = b}" by (intro finite_same_card_bij) simp_all qed then have "∃p. ∀b∈f ` A. bij_betw (p b) {a ∈ A. f a = b} {a ∈ A. f' a = b}" by (rule bchoice) then obtain p where p: "∀b∈f ` A. bij_betw (p b) {a ∈ A. f a = b} {a ∈ A. f' a = b}" .. define p' where "p' = (λa. if a ∈ A then p (f a) a else a)" have "p' permutes A" proof (rule bij_imp_permutes) have "disjoint_family_on (λi. {a ∈ A. f' a = i}) (f ` A)" by (auto simp: disjoint_family_on_def) moreover have "bij_betw (λa. p (f a) a) {a ∈ A. f a = b} {a ∈ A. f' a = b}" if "b ∈ f ` A" for b using p that by (subst bij_betw_cong[where g="p b"]) auto ultimately have "bij_betw (λa. p (f a) a) (⋃b∈f ` A. {a ∈ A. f a = b}) (⋃b∈f ` A. {a ∈ A. f' a = b})" by (rule bij_betw_UNION_disjoint) moreover have "(⋃b∈f ` A. {a ∈ A. f a = b}) = A" by auto moreover from ‹f ` A = f' ` A› have "(⋃b∈f ` A. {a ∈ A. f' a = b}) = A" by auto ultimately show "bij_betw p' A A" unfolding p'_def by (subst bij_betw_cong[where g="(λa. p (f a) a)"]) auto next show "⋀x. x ∉ A ⟹ p' x = x" by (simp add: p'_def) qed moreover from p have "∀x∈A. f x = f' (p' x)" unfolding p'_def using bij_betwE by fastforce ultimately show ?thesis .. qed ― ‹... and derive the existing property:› lemma mset_eq_permutation: fixes xs ys :: "'a list" assumes mset_eq: "mset xs = mset ys" obtains p where "p permutes {..<length ys}" "permute_list p ys = xs" proof - from mset_eq have length_eq: "length xs = length ys" by (rule mset_eq_length) have "mset_set {..<length ys} = mset [0..<length ys]" by (rule mset_set_upto_eq_mset_upto) with mset_eq length_eq have "image_mset (λi. xs ! i) (mset_set {..<length ys}) = image_mset (λi. ys ! i) (mset_set {..<length ys})" by (metis map_nth mset_map) from image_mset_eq_implies_permutes[OF _ this] obtain p where p: "p permutes {..<length ys}" and "∀i∈{..<length ys}. xs ! i = ys ! (p i)" by auto with length_eq have "permute_list p ys = xs" by (auto intro!: nth_equalityI simp: permute_list_nth) with p show thesis .. qed lemma permutes_natset_le: fixes S :: "'a::wellorder set" assumes "p permutes S" and "∀i ∈ S. p i ≤ i" shows "p = id" proof - have "p n = n" for n using assms proof (induct n arbitrary: S rule: less_induct) case (less n) show ?case proof (cases "n ∈ S") case False with less(2) show ?thesis unfolding permutes_def by metis next case True with less(3) have "p n < n ∨ p n = n" by auto then show ?thesis proof assume "p n < n" with less have "p (p n) = p n" by metis with permutes_inj[OF less(2)] have "p n = n" unfolding inj_def by blast with ‹p n < n› have False by simp then show ?thesis .. qed qed qed then show ?thesis by (auto simp: fun_eq_iff) qed lemma permutes_natset_ge: fixes S :: "'a::wellorder set" assumes p: "p permutes S" and le: "∀i ∈ S. p i ≥ i" shows "p = id" proof - have "i ≥ inv p i" if "i ∈ S" for i proof - from that permutes_in_image[OF permutes_inv[OF p]] have "inv p i ∈ S" by simp with le have "p (inv p i) ≥ inv p i" by blast with permutes_inverses[OF p] show ?thesis by simp qed then have "∀i∈S. inv p i ≤ i" by blast from permutes_natset_le[OF permutes_inv[OF p] this] have "inv p = inv id" by simp then show ?thesis apply (subst permutes_inv_inv[OF p, symmetric]) apply (rule inv_unique_comp) apply simp_all done qed lemma image_inverse_permutations: "{inv p |p. p permutes S} = {p. p permutes S}" apply (rule set_eqI) apply auto using permutes_inv_inv permutes_inv apply auto apply (rule_tac x="inv x" in exI) apply auto done lemma image_compose_permutations_left: assumes "q permutes S" shows "{q ∘ p |p. p permutes S} = {p. p permutes S}" apply (rule set_eqI) apply auto apply (rule permutes_compose) using assms apply auto apply (rule_tac x = "inv q ∘ x" in exI) apply (simp add: o_assoc permutes_inv permutes_compose permutes_inv_o) done lemma image_compose_permutations_right: assumes "q permutes S" shows "{p ∘ q | p. p permutes S} = {p . p permutes S}" apply (rule set_eqI) apply auto apply (rule permutes_compose) using assms apply auto apply (rule_tac x = "x ∘ inv q" in exI) apply (simp add: o_assoc permutes_inv permutes_compose permutes_inv_o comp_assoc) done lemma permutes_in_seg: "p permutes {1 ..n} ⟹ i ∈ {1..n} ⟹ 1 ≤ p i ∧ p i ≤ n" by (simp add: permutes_def) metis lemma sum_permutations_inverse: "sum f {p. p permutes S} = sum (λp. f(inv p)) {p. p permutes S}" (is "?lhs = ?rhs") proof - let ?S = "{p . p permutes S}" have *: "inj_on inv ?S" proof (auto simp add: inj_on_def) fix q r assume q: "q permutes S" and r: "r permutes S" and qr: "inv q = inv r" then have "inv (inv q) = inv (inv r)" by simp with permutes_inv_inv[OF q] permutes_inv_inv[OF r] show "q = r" by metis qed have **: "inv ` ?S = ?S" using image_inverse_permutations by blast have ***: "?rhs = sum (f ∘ inv) ?S" by (simp add: o_def) from sum.reindex[OF *, of f] show ?thesis by (simp only: ** ***) qed lemma setum_permutations_compose_left: assumes q: "q permutes S" shows "sum f {p. p permutes S} = sum (λp. f(q ∘ p)) {p. p permutes S}" (is "?lhs = ?rhs") proof - let ?S = "{p. p permutes S}" have *: "?rhs = sum (f ∘ ((∘) q)) ?S" by (simp add: o_def) have **: "inj_on ((∘) q) ?S" proof (auto simp add: inj_on_def) fix p r assume "p permutes S" and r: "r permutes S" and rp: "q ∘ p = q ∘ r" then have "inv q ∘ q ∘ p = inv q ∘ q ∘ r" by (simp add: comp_assoc) with permutes_inj[OF q, unfolded inj_iff] show "p = r" by simp qed have "((∘) q) ` ?S = ?S" using image_compose_permutations_left[OF q] by auto with * sum.reindex[OF **, of f] show ?thesis by (simp only:) qed lemma sum_permutations_compose_right: assumes q: "q permutes S" shows "sum f {p. p permutes S} = sum (λp. f(p ∘ q)) {p. p permutes S}" (is "?lhs = ?rhs") proof - let ?S = "{p. p permutes S}" have *: "?rhs = sum (f ∘ (λp. p ∘ q)) ?S" by (simp add: o_def) have **: "inj_on (λp. p ∘ q) ?S" proof (auto simp add: inj_on_def) fix p r assume "p permutes S" and r: "r permutes S" and rp: "p ∘ q = r ∘ q" then have "p ∘ (q ∘ inv q) = r ∘ (q ∘ inv q)" by (simp add: o_assoc) with permutes_surj[OF q, unfolded surj_iff] show "p = r" by simp qed from image_compose_permutations_right[OF q] have "(λp. p ∘ q) ` ?S = ?S" by auto with * sum.reindex[OF **, of f] show ?thesis by (simp only:) qed lemma inv_inj_on_permutes: ‹inj_on inv {p. p permutes S}› proof (intro inj_onI, unfold mem_Collect_eq) fix p q assume p: "p permutes S" and q: "q permutes S" and eq: "inv p = inv q" have "inv (inv p) = inv (inv q)" using eq by simp thus "p = q" using inv_inv_eq[OF permutes_bij] p q by metis qed lemma permutes_pair_eq: ‹{(p s, s) |s. s ∈ S} = {(s, inv p s) |s. s ∈ S}› (is ‹?L = ?R›) if ‹p permutes S› proof show "?L ⊆ ?R" proof fix x assume "x ∈ ?L" then obtain s where x: "x = (p s, s)" and s: "s ∈ S" by auto note x also have "(p s, s) = (p s, Hilbert_Choice.inv p (p s))" using permutes_inj [OF that] inv_f_f by auto also have "... ∈ ?R" using s permutes_in_image[OF that] by auto finally show "x ∈ ?R". qed show "?R ⊆ ?L" proof fix x assume "x ∈ ?R" then obtain s where x: "x = (s, Hilbert_Choice.inv p s)" (is "_ = (s, ?ips)") and s: "s ∈ S" by auto note x also have "(s, ?ips) = (p ?ips, ?ips)" using inv_f_f[OF permutes_inj[OF permutes_inv[OF that]]] using inv_inv_eq[OF permutes_bij[OF that]] by auto also have "... ∈ ?L" using s permutes_in_image[OF permutes_inv[OF that]] by auto finally show "x ∈ ?L". qed qed context fixes p and n i :: nat assumes p: ‹p permutes {0..<n}› and i: ‹i < n› begin lemma permutes_nat_less: ‹p i < n› proof - have ‹?thesis ⟷ p i ∈ {0..<n}› by simp also from p have ‹p i ∈ {0..<n} ⟷ i ∈ {0..<n}› by (rule permutes_in_image) finally show ?thesis using i by simp qed lemma permutes_nat_inv_less: ‹inv p i < n› proof - from p have ‹inv p permutes {0..<n}› by (rule permutes_inv) then show ?thesis using i by (rule Permutations.permutes_nat_less) qed end context comm_monoid_set begin lemma permutes_inv: ‹F (λs. g (p s) s) S = F (λs. g s (inv p s)) S› (is ‹?l = ?r›) if ‹p permutes S› proof - let ?g = "λ(x, y). g x y" let ?ps = "λs. (p s, s)" let ?ips = "λs. (s, inv p s)" have inj1: "inj_on ?ps S" by (rule inj_onI) auto have inj2: "inj_on ?ips S" by (rule inj_onI) auto have "?l = F ?g (?ps ` S)" using reindex [OF inj1, of ?g] by simp also have "?ps ` S = {(p s, s) |s. s ∈ S}" by auto also have "... = {(s, inv p s) |s. s ∈ S}" unfolding permutes_pair_eq [OF that] by simp also have "... = ?ips ` S" by auto also have "F ?g ... = ?r" using reindex [OF inj2, of ?g] by simp finally show ?thesis. qed end subsection ‹Sum over a set of permutations (could generalize to iteration)› lemma sum_over_permutations_insert: assumes fS: "finite S" and aS: "a ∉ S" shows "sum f {p. p permutes (insert a S)} = sum (λb. sum (λq. f (transpose a b ∘ q)) {p. p permutes S}) (insert a S)" proof - have *: "⋀f a b. (λ(b, p). f (transpose a b ∘ p)) = f ∘ (λ(b,p). transpose a b ∘ p)" by (simp add: fun_eq_iff) have **: "⋀P Q. {(a, b). a ∈ P ∧ b ∈ Q} = P × Q" by blast show ?thesis unfolding * ** sum.cartesian_product permutes_insert proof (rule sum.reindex) let ?f = "(λ(b, y). transpose a b ∘ y)" let ?P = "{p. p permutes S}" { fix b c p q assume b: "b ∈ insert a S" assume c: "c ∈ insert a S" assume p: "p permutes S" assume q: "q permutes S" assume eq: "transpose a b ∘ p = transpose a c ∘ q" from p q aS have pa: "p a = a" and qa: "q a = a" unfolding permutes_def by metis+ from eq have "(transpose a b ∘ p) a = (transpose a c ∘ q) a" by simp then have bc: "b = c" by (simp add: permutes_def pa qa o_def fun_upd_def id_def cong del: if_weak_cong split: if_split_asm) from eq[unfolded bc] have "(λp. transpose a c ∘ p) (transpose a c ∘ p) = (λp. transpose a c ∘ p) (transpose a c ∘ q)" by simp then have "p = q" unfolding o_assoc swap_id_idempotent by simp with bc have "b = c ∧ p = q" by blast } then show "inj_on ?f (insert a S × ?P)" unfolding inj_on_def by clarify metis qed qed subsection ‹Constructing permutations from association lists› definition list_permutes :: "('a × 'a) list ⇒ 'a set ⇒ bool" where "list_permutes xs A ⟷ set (map fst xs) ⊆ A ∧ set (map snd xs) = set (map fst xs) ∧ distinct (map fst xs) ∧ distinct (map snd xs)" lemma list_permutesI [simp]: assumes "set (map fst xs) ⊆ A" "set (map snd xs) = set (map fst xs)" "distinct (map fst xs)" shows "list_permutes xs A" proof - from assms(2,3) have "distinct (map snd xs)" by (intro card_distinct) (simp_all add: distinct_card del: set_map) with assms show ?thesis by (simp add: list_permutes_def) qed definition permutation_of_list :: "('a × 'a) list ⇒ 'a ⇒ 'a" where "permutation_of_list xs x = (case map_of xs x of None ⇒ x | Some y ⇒ y)" lemma permutation_of_list_Cons: "permutation_of_list ((x, y) # xs) x' = (if x = x' then y else permutation_of_list xs x')" by (simp add: permutation_of_list_def) fun inverse_permutation_of_list :: "('a × 'a) list ⇒ 'a ⇒ 'a" where "inverse_permutation_of_list [] x = x" | "inverse_permutation_of_list ((y, x') # xs) x = (if x = x' then y else inverse_permutation_of_list xs x)" declare inverse_permutation_of_list.simps [simp del] lemma inj_on_map_of: assumes "distinct (map snd xs)" shows "inj_on (map_of xs) (set (map fst xs))" proof (rule inj_onI) fix x y assume xy: "x ∈ set (map fst xs)" "y ∈ set (map fst xs)" assume eq: "map_of xs x = map_of xs y" from xy obtain x' y' where x'y': "map_of xs x = Some x'" "map_of xs y = Some y'" by (cases "map_of xs x"; cases "map_of xs y") (simp_all add: map_of_eq_None_iff) moreover from x'y' have *: "(x, x') ∈ set xs" "(y, y') ∈ set xs" by (force dest: map_of_SomeD)+ moreover from * eq x'y' have "x' = y'" by simp ultimately show "x = y" using assms by (force simp: distinct_map dest: inj_onD[of _ _ "(x,x')" "(y,y')"]) qed lemma inj_on_the: "None ∉ A ⟹ inj_on the A" by (auto simp: inj_on_def option.the_def split: option.splits) lemma inj_on_map_of': assumes "distinct (map snd xs)" shows "inj_on (the ∘ map_of xs) (set (map fst xs))" by (intro comp_inj_on inj_on_map_of assms inj_on_the) (force simp: eq_commute[of None] map_of_eq_None_iff) lemma image_map_of: assumes "distinct (map fst xs)" shows "map_of xs ` set (map fst xs) = Some ` set (map snd xs)" using assms by (auto simp: rev_image_eqI) lemma the_Some_image [simp]: "the ` Some ` A = A" by (subst image_image) simp lemma image_map_of': assumes "distinct (map fst xs)" shows "(the ∘ map_of xs) ` set (map fst xs) = set (map snd xs)" by (simp only: image_comp [symmetric] image_map_of assms the_Some_image) lemma permutation_of_list_permutes [simp]: assumes "list_permutes xs A" shows "permutation_of_list xs permutes A" (is "?f permutes _") proof (rule permutes_subset[OF bij_imp_permutes]) from assms show "set (map fst xs) ⊆ A" by (simp add: list_permutes_def) from assms have "inj_on (the ∘ map_of xs) (set (map fst xs))" (is ?P) by (intro inj_on_map_of') (simp_all add: list_permutes_def) also have "?P ⟷ inj_on ?f (set (map fst xs))" by (intro inj_on_cong) (auto simp: permutation_of_list_def map_of_eq_None_iff split: option.splits) finally have "bij_betw ?f (set (map fst xs)) (?f ` set (map fst xs))" by (rule inj_on_imp_bij_betw) also from assms have "?f ` set (