(* Title: HOL/Library/Permutations.thy Author: Amine Chaieb, University of Cambridge *) section ‹Permutations, both general and specifically on finite sets.› theory Permutations imports Multiset Disjoint_Sets begin subsection ‹Transpositions› lemma swap_id_idempotent [simp]: "Fun.swap a b id ∘ Fun.swap a b id = id" by (rule ext) (auto simp add: Fun.swap_def) lemma inv_swap_id: "inv (Fun.swap a b id) = Fun.swap a b id" by (rule inv_unique_comp) simp_all lemma swap_id_eq: "Fun.swap a b id x = (if x = a then b else if x = b then a else x)" by (simp add: Fun.swap_def) lemma bij_swap_comp: assumes "bij p" shows "Fun.swap a b id ∘ p = Fun.swap (inv p a) (inv p b) p" using surj_f_inv_f[OF bij_is_surj[OF ‹bij p›]] by (simp add: fun_eq_iff Fun.swap_def bij_inv_eq_iff[OF ‹bij p›]) lemma bij_swap_compose_bij: assumes "bij p" shows "bij (Fun.swap a b id ∘ p)" by (simp only: bij_swap_comp[OF ‹bij p›] bij_swap_iff ‹bij p›) subsection ‹Basic consequences of the definition› definition permutes (infixr "permutes" 41) where "(p permutes S) ⟷ (∀x. x ∉ S ⟶ p x = x) ∧ (∀y. ∃!x. p x = y)" lemma permutes_in_image: "p permutes S ⟹ p x ∈ S ⟷ x ∈ S" unfolding permutes_def by metis lemma permutes_not_in: "f permutes S ⟹ x ∉ S ⟹ f x = x" by (auto simp: permutes_def) lemma permutes_image: "p permutes S ⟹ p ` S = S" unfolding permutes_def apply (rule set_eqI) apply (simp add: image_iff) apply metis done lemma permutes_inj: "p permutes S ⟹ inj p" unfolding permutes_def inj_def by blast lemma permutes_inj_on: "f permutes S ⟹ inj_on f A" by (auto simp: permutes_def inj_on_def) lemma permutes_surj: "p permutes s ⟹ surj p" unfolding permutes_def surj_def by metis lemma permutes_bij: "p permutes s ⟹ bij p" unfolding bij_def by (metis permutes_inj permutes_surj) lemma permutes_imp_bij: "p permutes S ⟹ bij_betw p S S" by (metis UNIV_I bij_betw_subset permutes_bij permutes_image subsetI) lemma bij_imp_permutes: "bij_betw p S S ⟹ (⋀x. x ∉ S ⟹ p x = x) ⟹ p permutes S" unfolding permutes_def bij_betw_def inj_on_def by auto (metis image_iff)+ lemma permutes_inv_o: assumes permutes: "p permutes S" shows "p ∘ inv p = id" and "inv p ∘ p = id" using permutes_inj[OF permutes] permutes_surj[OF permutes] unfolding inj_iff[symmetric] surj_iff[symmetric] by blast+ lemma permutes_inverses: fixes p :: "'a ⇒ 'a" assumes permutes: "p permutes S" shows "p (inv p x) = x" and "inv p (p x) = x" using permutes_inv_o[OF permutes, unfolded fun_eq_iff o_def] by auto lemma permutes_subset: "p permutes S ⟹ S ⊆ T ⟹ p permutes T" unfolding permutes_def by blast lemma permutes_empty[simp]: "p permutes {} ⟷ p = id" by (auto simp add: fun_eq_iff permutes_def) lemma permutes_sing[simp]: "p permutes {a} ⟷ p = id" by (simp add: fun_eq_iff permutes_def) metis (*somewhat slow*) lemma permutes_univ: "p permutes UNIV ⟷ (∀y. ∃!x. p x = y)" by (simp add: permutes_def) lemma permutes_inv_eq: "p permutes S ⟹ inv p y = x ⟷ p x = y" unfolding permutes_def inv_def apply auto apply (erule allE[where x=y]) apply (erule allE[where x=y]) apply (rule someI_ex) apply blast apply (rule some1_equality) apply blast apply blast done lemma permutes_swap_id: "a ∈ S ⟹ b ∈ S ⟹ Fun.swap a b id permutes S" unfolding permutes_def Fun.swap_def fun_upd_def by auto metis lemma permutes_superset: "p permutes S ⟹ (∀x ∈ S - T. p x = x) ⟹ p permutes T" by (simp add: Ball_def permutes_def) metis 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_id: "id permutes S" by (simp add: permutes_def) 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 "Fun.swap a (p a) id ∘ 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 Fun.swap_def) done lemma permutes_insert: "{p. p permutes (insert a S)} = (λ(b, p). Fun.swap a b id ∘ p) ` {(b, p). b ∈ insert a S ∧ p ∈ {p. p permutes S}}" proof - have "p permutes insert a S ⟷ (∃b q. p = Fun.swap a b id ∘ q ∧ b ∈ insert a S ∧ q permutes S)" for p proof - have "∃b q. p = Fun.swap a b id ∘ q ∧ b ∈ insert a S ∧ q permutes S" if p: "p permutes insert a S" proof - let ?b = "p a" let ?q = "Fun.swap a (p a) id ∘ p" have *: "p = Fun.swap a ?b id ∘ ?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 = Fun.swap a b id ∘ 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). Fun.swap x b id ∘ 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 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.swap_def fun_upd_def fun_eq_iff) also from qF ‹x ∉ F› eq have "… = ?g (c, q) x" by (auto simp: swap_def fun_upd_def fun_eq_iff) also from qF ‹x ∉ F› have "… = c" by (auto simp: permutes_def Fun.swap_def fun_upd_def fun_eq_iff) finally have "b = c" . then have "Fun.swap x b id = Fun.swap x c id" by simp with eq have "Fun.swap x b id ∘ p = Fun.swap x b id ∘ q" by simp then have "Fun.swap x b id ∘ (Fun.swap x b id ∘ p) = Fun.swap x b id ∘ (Fun.swap x b id ∘ 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 ‹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 ‹Various combinations of transpositions with 2, 1 and 0 common elements› lemma swap_id_common:" a ≠ c ⟹ b ≠ c ⟹ Fun.swap a b id ∘ Fun.swap a c id = Fun.swap b c id ∘ Fun.swap a b id" by (simp add: fun_eq_iff Fun.swap_def) lemma swap_id_common': "a ≠ b ⟹ a ≠ c ⟹ Fun.swap a c id ∘ Fun.swap b c id = Fun.swap b c id ∘ Fun.swap a b id" by (simp add: fun_eq_iff Fun.swap_def) lemma swap_id_independent: "a ≠ c ⟹ a ≠ d ⟹ b ≠ c ⟹ b ≠ d ⟹ Fun.swap a b id ∘ Fun.swap c d id = Fun.swap c d id ∘ Fun.swap a b id" by (simp add: fun_eq_iff Fun.swap_def) 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) (Fun.swap a b id ∘ 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) (Fun.swap a b id)" apply clarsimp using comp_Suc[of 0 id a b] apply simp done lemma permutation_swap_id: "permutation (Fun.swap a b id)" 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 ∘ Fun.swap a b id)" 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 ∘ Fun.swap a b id" note H = comp_Suc.hyps from swapidseq_swap[of a b] H(3) have *: "swapidseq 1 (Fun.swap a b id)" by simp from swapidseq_comp_add[OF q(1) *] have **: "swapidseq (Suc n) ?q" by simp have "Fun.swap a b id ∘ p ∘ ?q = Fun.swap a b id ∘ (p ∘ q) ∘ Fun.swap a b id" by (simp add: o_assoc) also have "… = id" by (simp add: q(2)) finally have ***: "Fun.swap a b id ∘ p ∘ ?q = id" . have "?q ∘ (Fun.swap a b id ∘ p) = q ∘ (Fun.swap a b id ∘ Fun.swap a b id) ∘ p" by (simp only: o_assoc) then have "?q ∘ (Fun.swap a b id ∘ 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 ‹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 ⟹ Fun.swap a b id ∘ Fun.swap c d id = id ∨ (∃x y z. x ≠ a ∧ y ≠ a ∧ z ≠ a ∧ x ≠ y ∧ Fun.swap a b id ∘ Fun.swap c d id = Fun.swap x y id ∘ Fun.swap a z id)" proof - assume neq: "a ≠ b" "c ≠ d" have "a ≠ b ⟶ c ≠ d ⟶ (Fun.swap a b id ∘ Fun.swap c d id = id ∨ (∃x y z. x ≠ a ∧ y ≠ a ∧ z ≠ a ∧ x ≠ y ∧ Fun.swap a b id ∘ Fun.swap c d id = Fun.swap x y id ∘ Fun.swap a z id))" apply (rule symmetry_lemma[where a=a and b=b and c=c and d=d]) apply (simp_all only: swap_commute) apply (case_tac "a = c ∧ b = d") apply (clarsimp simp only: swap_commute swap_id_idempotent) apply (case_tac "a = c ∧ b ≠ d") apply (rule disjI2) apply (rule_tac x="b" in exI) apply (rule_tac x="d" in exI) apply (rule_tac x="b" in exI) apply (clarsimp simp add: fun_eq_iff Fun.swap_def) apply (case_tac "a ≠ c ∧ b = d") apply (rule disjI2) apply (rule_tac x="c" in exI) apply (rule_tac x="d" in exI) apply (rule_tac x="c" in exI) apply (clarsimp simp add: fun_eq_iff Fun.swap_def) apply (rule disjI2) apply (rule_tac x="c" in exI) apply (rule_tac x="d" in exI) apply (rule_tac x="b" in exI) apply (clarsimp simp add: fun_eq_iff Fun.swap_def) 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 = Fun.swap a b id ∘ 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 "(Fun.swap a b id ∘ p) a = a" shows "n ≠ 0 ∧ swapidseq (n - 1) (Fun.swap a b id ∘ p)" using assms proof (induct n arbitrary: p a b) case 0 then show ?case by (auto simp add: Fun.swap_def 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 = Fun.swap c d id ∘ q" "swapidseq m q" "c ≠ d" "n = m" by auto consider "Fun.swap a b id ∘ Fun.swap c d id = id" | x y z where "x ≠ a" "y ≠ a" "z ≠ a" "x ≠ y" "Fun.swap a b id ∘ Fun.swap c d id = Fun.swap x y id ∘ Fun.swap a z id" 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 *: "(Fun.swap x y id ∘ h) a = a ⟷ h a = a" for h by (simp add: Fun.swap_def) from cdqm(2) have "Fun.swap a b id ∘ p = Fun.swap a b id ∘ (Fun.swap c d id ∘ q)" by simp then have "Fun.swap a b id ∘ p = Fun.swap x y id ∘ (Fun.swap a z id ∘ q)" by (simp add: o_assoc prems) then have "(Fun.swap a b id ∘ p) a = (Fun.swap x y id ∘ (Fun.swap a z id ∘ q)) a" by simp then have "(Fun.swap x y id ∘ (Fun.swap a z id ∘ q)) a = a" unfolding Suc by metis then have "(Fun.swap a z id ∘ q) a = a" by (simp only: *) from Suc.hyps[OF cdqm(3)[ unfolded cdqm(5)[symmetric]] az this] have **: "swapidseq (n - 1) (Fun.swap a z id ∘ 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 = Fun.swap a b id ∘ 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_swap: "evenperm (Fun.swap a b id) = (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 bij_iff: "bij f ⟷ (∀x. ∃!y. f y = x)" unfolding bij_def inj_def surj_def apply auto apply metis apply metis done 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. (Fun.swap a b id ∘ p) x ≠ x} ⊆ ?S" by (auto simp: Fun.swap_def) 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 = "Fun.swap a (p a) id ∘ p" let ?q = "Fun.swap a (p a) id ∘ ?r" have *: "?r a = a" by (simp add: Fun.swap_def) 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" by (rule bij_swap_compose_bij[OF insert(4)]) 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 permutation_permutes: "permutation p ⟷ (∃S. finite S ∧ p permutes S)" unfolding permutation permutes_def bij_iff[symmetric] apply (rule iffI, clarify) apply (rule exI[where x="{x. p x ≠ x}"]) apply simp apply clarsimp apply (rule_tac B="S" in finite_subset) apply auto done subsection ‹Hence a sort of induction principle composing by swaps› lemma permutes_induct: "finite S ⟹ P id ⟹ (⋀a b p. a ∈ S ⟹ b ∈ S ⟹ P p ⟹ P p ⟹ permutation p ⟹ P (Fun.swap a b id ∘ p)) ⟹ (⋀p. p permutes S ⟹ P p)" proof (induct S rule: finite_induct) case empty then show ?case by auto next case (insert x F p) let ?r = "Fun.swap x (p x) id ∘ p" let ?q = "Fun.swap x (p x) id ∘ ?r" have qp: "?q = p" by (simp add: o_assoc) from permutes_insert_lemma[OF insert.prems(3)] insert have Pr: "P ?r" by blast from permutes_in_image[OF insert.prems(3), of x] have pxF: "p x ∈ insert x F" by simp have xF: "x ∈ insert x F" by simp have rp: "permutation ?r" unfolding permutation_permutes using insert.hyps(1) permutes_insert_lemma[OF insert.prems(3)] by blast from insert.prems(2)[OF xF pxF Pr Pr rp] qp show ?case by (simp only:) qed subsection ‹Sign of a permutation as a real number› definition "sign p = (if evenperm p then (1::int) else -1)" lemma sign_nz: "sign p ≠ 0" by (simp add: sign_def) lemma sign_id: "sign id = 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 (Fun.swap a b id) = (if a = b then 1 else -1)" by (simp add: sign_def evenperm_swap) lemma sign_idempotent: "sign p * sign p = 1" by (simp add: sign_def) 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 subsection ‹More lemmas about permutations› 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 lemma mset_set_upto_eq_mset_upto: "mset_set {..<n} = mset [0..<n]" by (induct n) (auto simp: add.commute lessThan_Suc) ― ‹... 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 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 (Fun.swap a b id ∘ q)) {p. p permutes S}) (insert a S)" proof - have *: "⋀f a b. (λ(b, p). f (Fun.swap a b id ∘ p)) = f ∘ (λ(b,p). Fun.swap a b id ∘ 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). Fun.swap a b id ∘ 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: "Fun.swap a b id ∘ p = Fun.swap a c id ∘ q" from p q aS have pa: "p a = a" and qa: "q a = a" unfolding permutes_def by metis+ from eq have "(Fun.swap a b id ∘ p) a = (Fun.swap a c id ∘ q) a" by simp then have bc: "b = c" by (simp add: permutes_def pa qa o_def fun_upd_def Fun.swap_def id_def cong del: if_weak_cong split: if_split_asm) from eq[unfolded bc] have "(λp. Fun.swap a c id ∘ p) (Fun.swap a c id ∘ p) = (λp. Fun.swap a c id ∘ p) (Fun.swap a c id ∘ 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 (map fst xs) = (the ∘ map_of xs) ` set (map fst xs)" by (intro image_cong refl) (auto simp: permutation_of_list_def map_of_eq_None_iff split: option.splits) also from assms have "… = set (map fst xs)" by (subst image_map_of') (simp_all add: list_permutes_def) finally show "bij_betw ?f (set (map fst xs)) (set (map fst xs))" . qed (force simp: permutation_of_list_def dest!: map_of_SomeD split: option.splits)+ lemma eval_permutation_of_list [simp]: "permutation_of_list [] x = x" "x = x' ⟹ permutation_of_list ((x',y)#xs) x = y" "x ≠ x' ⟹ permutation_of_list ((x',y')#xs) x = permutation_of_list xs x" by (simp_all add: permutation_of_list_def) lemma eval_inverse_permutation_of_list [simp]: "inverse_permutation_of_list [] x = x" "x = x' ⟹ inverse_permutation_of_list ((y,x')#xs) x = y" "x ≠ x' ⟹ inverse_permutation_of_list ((y',x')#xs) x = inverse_permutation_of_list xs x" by (simp_all add: inverse_permutation_of_list.simps) lemma permutation_of_list_id: "x ∉ set (map fst xs) ⟹ permutation_of_list xs x = x" by (induct xs) (auto simp: permutation_of_list_Cons) lemma permutation_of_list_unique': "distinct (map fst xs) ⟹ (x, y) ∈ set xs ⟹ permutation_of_list xs x = y" by (induct xs) (force simp: permutation_of_list_Cons)+ lemma permutation_of_list_unique: "list_permutes xs A ⟹ (x, y) ∈ set xs ⟹ permutation_of_list xs x = y" by (intro permutation_of_list_unique') (simp_all add: list_permutes_def) lemma inverse_permutation_of_list_id: "x ∉ set (map snd xs) ⟹ inverse_permutation_of_list xs x = x" by (induct xs) auto lemma inverse_permutation_of_list_unique': "distinct (map snd xs) ⟹ (x, y) ∈ set xs ⟹ inverse_permutation_of_list xs y = x" by (induct xs) (force simp: inverse_permutation_of_list.simps)+ lemma inverse_permutation_of_list_unique: "list_permutes xs A ⟹ (x,y) ∈ set xs ⟹ inverse_permutation_of_list xs y = x" by (intro inverse_permutation_of_list_unique') (simp_all add: list_permutes_def) lemma inverse_permutation_of_list_correct: fixes A :: "'a set" assumes "list_permutes xs A" shows "inverse_permutation_of_list xs = inv (permutation_of_list xs)" proof (rule ext, rule sym, subst permutes_inv_eq) from assms show "permutation_of_list xs permutes A" by simp show "permutation_of_list xs (inverse_permutation_of_list xs x) = x" for x proof (cases "x ∈ set (map snd xs)") case True then obtain y where "(y, x) ∈ set xs" by auto with assms show ?thesis by (simp add: inverse_permutation_of_list_unique permutation_of_list_unique) next case False with assms show ?thesis by (auto simp: list_permutes_def inverse_permutation_of_list_id permutation_of_list_id) qed qed end