(* Author: RenĂ© Thiemann Akihisa Yamada License: BSD *) section ‹Missing Permutations› text ‹This theory provides some definitions and lemmas on permutations which we did not find in the Isabelle distribution.› theory Missing_Permutations imports Missing_Ring "HOL-Library.Permutations" begin definition signof :: "(nat ⇒ nat) ⇒ 'a :: ring_1" where "signof p = (if sign p = 1 then 1 else - 1)" lemma signof_id[simp]: "signof id = 1" "signof (λ x. x) = 1" unfolding signof_def sign_id id_def[symmetric] by auto lemma signof_inv: "finite S ⟹ p permutes S ⟹ signof (Hilbert_Choice.inv p) = signof p" unfolding signof_def using sign_inverse permutation_permutes by metis lemma signof_pm_one: "signof p ∈ {1, - 1}" unfolding signof_def by auto lemma signof_compose: assumes "p permutes {0..<(n :: nat)}" and "q permutes {0 ..<(m :: nat)}" shows "signof (p o q) = signof p * signof q" proof - from assms have pp: "permutation p" "permutation q" by (auto simp: permutation_permutes) show "signof (p o q) = signof p * signof q" unfolding signof_def sign_compose[OF pp] by (auto simp: sign_def split: if_splits) qed lemma permutes_funcset: "p permutes A ⟹ (p ` A → B) = (A → B)" by (simp add: permutes_image) context comm_monoid begin lemma finprod_permute: assumes p: "p permutes S" and f: "f ∈ S → carrier G" shows "finprod G f S = finprod G (f ∘ 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) from finprod_reindex[OF _ this, unfolded permutes_image[OF p], OF f] show ?thesis unfolding o_def . qed lemma finprod_singleton_set[simp]: assumes "f a ∈ carrier G" shows "finprod G f {a} = f a" proof - have "finprod G f {a} = f a ⊗ finprod G f {}" by (rule finprod_insert, insert assms, auto) also have "… = f a" using assms by auto finally show ?thesis . qed end lemmas (in semiring) finsum_permute = add.finprod_permute lemmas (in semiring) finsum_singleton_set = add.finprod_singleton_set lemma permutes_less[simp]: assumes p: "p permutes {0..<(n :: nat)}" shows "i < n ⟹ p i < n" "i < n ⟹ Hilbert_Choice.inv p i < n" "p (Hilbert_Choice.inv p i) = i" "Hilbert_Choice.inv p (p i) = i" proof - assume i: "i < n" show "p i < n" using permutes_in_image[OF p] i by auto let ?inv = "Hilbert_Choice.inv p" have "⋀n. ?inv (p n) = n" using permutes_inverses[OF p] by simp thus "?inv i < n" by (metis (no_types) atLeastLessThan_iff f_inv_into_f inv_into_into le0 permutes_image[OF p] i) qed (insert permutes_inverses[OF p], auto) context cring begin lemma finsum_permutations_inverse: assumes f: "f ∈ {p. p permutes S} → carrier R" shows "finsum R f {p. p permutes S} = finsum R (λp. f(Hilbert_Choice.inv p)) {p. p permutes S}" (is "?lhs = ?rhs") proof - let ?inv = "Hilbert_Choice.inv" let ?S = "{p . p permutes S}" have th0: "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 th1: "?inv ` ?S = ?S" using image_inverse_permutations by blast have th2: "?rhs = finsum R (f ∘ ?inv) ?S" by (simp add: o_def) from finsum_reindex[OF _ th0, of f] show ?thesis unfolding th1 th2 using f . qed lemma finsum_permutations_compose_right: assumes q: "q permutes S" and *: "f ∈ {p. p permutes S} → carrier R" shows "finsum R f {p. p permutes S} = finsum R (λp. f(p ∘ q)) {p. p permutes S}" (is "?lhs = ?rhs") proof - let ?S = "{p. p permutes S}" let ?inv = "Hilbert_Choice.inv" have th0: "?rhs = finsum R (f ∘ (λp. p ∘ q)) ?S" by (simp add: o_def) have th1: "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 have th3: "(λp. p ∘ q) ` ?S = ?S" using image_compose_permutations_right[OF q] by auto from finsum_reindex[OF _ th1, of f] show ?thesis unfolding th0 th1 th3 using * . qed end text ‹The following lemma is slightly generalized from Determinants.thy in HMA.› lemma finite_bounded_functions: assumes fS: "finite S" shows "finite T ⟹ finite {f. (∀i ∈ T. f i ∈ S) ∧ (∀i. i ∉ T ⟶ f i = i)}" proof (induct T rule: finite_induct) case empty have th: "{f. ∀i. f i = i} = {id}" by auto show ?case by (auto simp add: th) next case (insert a T) let ?f = "λ(y,g) i. if i = a then y else g i" let ?S = "?f ` (S × {f. (∀i∈T. f i ∈ S) ∧ (∀i. i ∉ T ⟶ f i = i)})" have "?S = {f. (∀i∈ insert a T. f i ∈ S) ∧ (∀i. i ∉ insert a T ⟶ f i = i)}" apply (auto simp add: image_iff) apply (rule_tac x="x a" in bexI) apply (rule_tac x = "λi. if i = a then i else x i" in exI) apply (insert insert, auto) done with finite_imageI[OF finite_cartesian_product[OF fS insert.hyps(3)], of ?f] show ?case by metis qed lemma finite_bounded_functions': assumes fS: "finite S" shows "finite T ⟹ finite {f. (∀i ∈ T. f i ∈ S) ∧ (∀i. i ∉ T ⟶ f i = j)}" proof (induct T rule: finite_induct) case empty have th: "{f. ∀i. f i = j} = {(λ x. j)}" by auto show ?case by (auto simp add: th) next case (insert a T) let ?f = "λ(y,g) i. if i = a then y else g i" let ?S = "?f ` (S × {f. (∀i∈T. f i ∈ S) ∧ (∀i. i ∉ T ⟶ f i = j)})" have "?S = {f. (∀i∈ insert a T. f i ∈ S) ∧ (∀i. i ∉ insert a T ⟶ f i = j)}" apply (auto simp add: image_iff) apply (rule_tac x="x a" in bexI) apply (rule_tac x = "λi. if i = a then j else x i" in exI) apply (insert insert, auto) done with finite_imageI[OF finite_cartesian_product[OF fS insert.hyps(3)], of ?f] show ?case by metis qed context fixes A :: "'a set" and B :: "'b set" and a_to_b :: "'a ⇒ 'b" and b_to_a :: "'b ⇒ 'a" assumes ab: "⋀ a. a ∈ A ⟹ a_to_b a ∈ B" and ba: "⋀ b. b ∈ B ⟹ b_to_a b ∈ A" and ab_ba: "⋀ a. a ∈ A ⟹ b_to_a (a_to_b a) = a" and ba_ab: "⋀ b. b ∈ B ⟹ a_to_b (b_to_a b) = b" begin qualified lemma permutes_memb: fixes p :: "'b ⇒ 'b" assumes p: "p permutes B" and a: "a ∈ A" defines "ip ≡ Hilbert_Choice.inv p" shows "a ∈ A" "a_to_b a ∈ B" "ip (a_to_b a) ∈ B" "p (a_to_b a) ∈ B" "b_to_a (p (a_to_b a)) ∈ A" "b_to_a (ip (a_to_b a)) ∈ A" proof - let ?b = "a_to_b a" from p have ip: "ip permutes B" unfolding ip_def by (rule permutes_inv) note in_ip = permutes_in_image[OF ip] note in_p = permutes_in_image[OF p] show a: "a ∈ A" by fact show b: "?b ∈ B" by (rule ab[OF a]) show pb: "p ?b ∈ B" unfolding in_p by (rule b) show ipb: "ip ?b ∈ B" unfolding in_ip by (rule b) show "b_to_a (p ?b) ∈ A" by (rule ba[OF pb]) show "b_to_a (ip ?b) ∈ A" by (rule ba[OF ipb]) qed lemma permutes_bij_main: "{p . p permutes A} ⊇ (λ p a. if a ∈ A then b_to_a (p (a_to_b a)) else a) ` {p . p permutes B}" (is "?A ⊇ ?f ` ?B") proof note d = permutes_def let ?g = "λ q b. if b ∈ B then a_to_b (q (b_to_a b)) else b" let ?inv = "Hilbert_Choice.inv" fix p assume p: "p ∈ ?f ` ?B" then obtain q where q: "q permutes B" and p: "p = ?f q" by auto let ?iq = "?inv q" from q have iq: "?iq permutes B" by (rule permutes_inv) note in_iq = permutes_in_image[OF iq] note in_q = permutes_in_image[OF q] have qiB: "⋀ b. b ∈ B ⟹ q (?iq b) = b" using q by (rule permutes_inverses) have iqB: "⋀ b. b ∈ B ⟹ ?iq (q b) = b" using q by (rule permutes_inverses) from q[unfolded d] have q1: "⋀ b. b ∉ B ⟹ q b = b" and q2: "⋀ b. ∃!b'. q b' = b" by auto note memb = permutes_memb[OF q] show "p ∈ ?A" unfolding p d proof (rule, intro conjI impI allI, force) fix a show "∃!a'. ?f q a' = a" proof (cases "a ∈ A") case True note a = memb[OF True] let ?a = "b_to_a (?iq (a_to_b a))" show ?thesis proof show "?f q ?a = a" using a by (simp add: ba_ab qiB ab_ba) next fix a' assume id: "?f q a' = a" show "a' = ?a" proof (cases "a' ∈ A") case False thus ?thesis using id a by auto next case True note a' = memb[OF this] from id True have "b_to_a (q (a_to_b a')) = a" by simp from arg_cong[OF this, of "a_to_b"] a' a have "q (a_to_b a') = a_to_b a" by (simp add: ba_ab) from arg_cong[OF this, of ?iq] have "a_to_b a' = ?iq (a_to_b a)" unfolding iqB[OF a'(2)] . from arg_cong[OF this, of b_to_a] show ?thesis unfolding ab_ba[OF True] . qed qed next case False note a = this show ?thesis proof show "?f q a = a" using a by simp next fix a' assume id: "?f q a' = a" show "a' = a" proof (cases "a' ∈ A") case False with id show ?thesis by simp next case True note a' = memb[OF True] with id False show ?thesis by auto qed qed qed qed qed end lemma permutes_bij': assumes ab: "⋀ a. a ∈ A ⟹ a_to_b a ∈ B" and ba: "⋀ b. b ∈ B ⟹ b_to_a b ∈ A" and ab_ba: "⋀ a. a ∈ A ⟹ b_to_a (a_to_b a) = a" and ba_ab: "⋀ b. b ∈ B ⟹ a_to_b (b_to_a b) = b" shows "{p . p permutes A} = (λ p a. if a ∈ A then b_to_a (p (a_to_b a)) else a) ` {p . p permutes B}" (is "?A = ?f ` ?B") proof - note one_dir = ab ba ab_ba ba_ab note other_dir = ba ab ba_ab ab_ba let ?g = "(λ p b. if b ∈ B then a_to_b (p (b_to_a b)) else b)" define PA where "PA = ?A" define f where "f = ?f" define g where "g = ?g" { fix p assume "p ∈ PA" hence p: "p permutes A" unfolding PA_def by simp from p[unfolded permutes_def] have pnA: "⋀ a. a ∉ A ⟹ p a = a" by auto have "?f (?g p) = p" proof (rule ext) fix a show "?f (?g p) a = p a" proof (cases "a ∈ A") case False thus ?thesis by (simp add: pnA) next case True note a = this hence "p a ∈ A" unfolding permutes_in_image[OF p] . thus ?thesis using a by (simp add: ab_ba ba_ab ab) qed qed hence "f (g p) = p" unfolding f_def g_def . } hence "f ` g ` PA = PA" by force hence id: "?f ` ?g ` ?A = ?A" unfolding PA_def f_def g_def . have "?f ` ?B ⊆ ?A" by (rule permutes_bij_main[OF one_dir]) moreover have "?g ` ?A ⊆ ?B" by (rule permutes_bij_main[OF ba ab ba_ab ab_ba]) hence "?f ` ?g ` ?A ⊆ ?f ` ?B" by auto hence "?A ⊆ ?f ` ?B" unfolding id . ultimately show ?thesis by blast qed lemma inj_on_nat_permutes: assumes i: "inj_on f (S :: nat set)" and fS: "f ∈ S → S" and fin: "finite S" and f: "⋀ i. i ∉ S ⟹ f i = i" shows "f permutes S" unfolding permutes_def proof (intro conjI allI impI, rule f) fix y from endo_inj_surj[OF fin _ i] fS have fs: "f ` S = S" by auto show "∃!x. f x = y" proof (cases "y ∈ S") case False thus ?thesis by (intro ex1I[of _ y], insert fS f, auto) next case True with fs obtain x where x: "x ∈ S" and fx: "f x = y" by force show ?thesis proof (rule ex1I, rule fx) fix x' assume fx': "f x' = y" with True f[of x'] have "x' ∈ S" by metis from inj_onD[OF i fx[folded fx'] x this] show "x' = x" by simp qed qed qed lemma permutes_pair_eq: assumes p: "p permutes S" shows "{ (p s, s) | s. s ∈ S } = { (s, Hilbert_Choice.inv p s) | s. s ∈ S }" (is "?L = ?R") 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 p] inv_f_f by auto also have "... ∈ ?R" using s permutes_in_image[OF p] 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 p]]] using inv_inv_eq[OF permutes_bij[OF p]] by auto also have "... ∈ ?L" using s permutes_in_image[OF permutes_inv[OF p]] by auto finally show "x ∈ ?L". qed qed lemma inj_on_finite[simp]: assumes inj: "inj_on f A" shows "finite (f ` A) = finite A" proof assume fin: "finite (f ` A)" show "finite A" proof (cases "card (f ` A) = 0") case True thus ?thesis using fin by auto next case False hence "card A > 0" unfolding card_image[OF inj] by auto thus ?thesis using card_infinite by force qed qed auto lemma permutes_prod: assumes p: "p permutes S" shows "(∏s∈S. f (p s) s) = (∏s∈S. f s (Hilbert_Choice.inv p s))" (is "?l = ?r") proof - let ?f = "λ(x,y). f x y" let ?ps = "λs. (p s, s)" let ?ips = "λs. (s, Hilbert_Choice.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 = prod ?f (?ps ` S)" using prod.reindex[OF inj1, of ?f] by simp also have "?ps ` S = {(p s, s) |s. s ∈ S}" by auto also have "... = {(s, Hilbert_Choice.inv p s) | s. s ∈ S}" unfolding permutes_pair_eq[OF p] by simp also have "... = ?ips ` S" by auto also have "prod ?f ... = ?r" using prod.reindex[OF inj2, of ?f] by simp finally show ?thesis. qed lemma permutes_sum: assumes p: "p permutes S" shows "(∑s∈S. f (p s) s) = (∑s∈S. f s (Hilbert_Choice.inv p s))" (is "?l = ?r") proof - let ?f = "λ(x,y). f x y" let ?ps = "λs. (p s, s)" let ?ips = "λs. (s, Hilbert_Choice.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 = sum ?f (?ps ` S)" using sum.reindex[OF inj1, of ?f] by simp also have "?ps ` S = {(p s, s) |s. s ∈ S}" by auto also have "... = {(s, Hilbert_Choice.inv p s) | s. s ∈ S}" unfolding permutes_pair_eq[OF p] by simp also have "... = ?ips ` S" by auto also have "sum ?f ... = ?r" using sum.reindex[OF inj2, of ?f] by simp finally show ?thesis. qed lemma inv_inj_on_permutes: "inj_on Hilbert_Choice.inv { p. p permutes S }" proof (intro inj_onI, unfold mem_Collect_eq) let ?i = "Hilbert_Choice.inv" fix p q assume p: "p permutes S" and q: "q permutes S" and eq: "?i p = ?i q" have "?i (?i p) = ?i (?i q)" using eq by simp thus "p = q" using inv_inv_eq[OF permutes_bij] p q by metis qed lemma permutes_others: assumes p: "p permutes S" and x: "x ∉ S" shows "p x = x" using p unfolding permutes_def using x by simp end