(* Title: HOL/Binomial.thy Author: Jacques D. Fleuriot Author: Lawrence C Paulson Author: Jeremy Avigad Author: Chaitanya Mangla Author: Manuel Eberl *) section ‹Binomial Coefficients and Binomial Theorem› theory Binomial imports Presburger Factorial begin subsection ‹Binomial coefficients› text ‹This development is based on the work of Andy Gordon and Florian Kammueller.› text ‹Combinatorial definition› definition binomial :: "nat ⇒ nat ⇒ nat" (infixl "choose" 65) where "n choose k = card {K∈Pow {0..<n}. card K = k}" theorem n_subsets: assumes "finite A" shows "card {B. B ⊆ A ∧ card B = k} = card A choose k" proof - from assms obtain f where bij: "bij_betw f {0..<card A} A" by (blast dest: ex_bij_betw_nat_finite) then have [simp]: "card (f ` C) = card C" if "C ⊆ {0..<card A}" for C by (meson bij_betw_imp_inj_on bij_betw_subset card_image that) from bij have "bij_betw (image f) (Pow {0..<card A}) (Pow A)" by (rule bij_betw_Pow) then have "inj_on (image f) (Pow {0..<card A})" by (rule bij_betw_imp_inj_on) moreover have "{K. K ⊆ {0..<card A} ∧ card K = k} ⊆ Pow {0..<card A}" by auto ultimately have "inj_on (image f) {K. K ⊆ {0..<card A} ∧ card K = k}" by (rule inj_on_subset) then have "card {K. K ⊆ {0..<card A} ∧ card K = k} = card (image f ` {K. K ⊆ {0..<card A} ∧ card K = k})" (is "_ = card ?C") by (simp add: card_image) also have "?C = {K. K ⊆ f ` {0..<card A} ∧ card K = k}" by (auto elim!: subset_imageE) also have "f ` {0..<card A} = A" by (meson bij bij_betw_def) finally show ?thesis by (simp add: binomial_def) qed text ‹Recursive characterization› lemma binomial_n_0 [simp]: "n choose 0 = 1" proof - have "{K ∈ Pow {0..<n}. card K = 0} = {{}}" by (auto dest: finite_subset) then show ?thesis by (simp add: binomial_def) qed lemma binomial_0_Suc [simp]: "0 choose Suc k = 0" by (simp add: binomial_def) lemma binomial_Suc_Suc [simp]: "Suc n choose Suc k = (n choose k) + (n choose Suc k)" proof - let ?P = "λn k. {K. K ⊆ {0..<n} ∧ card K = k}" let ?Q = "?P (Suc n) (Suc k)" have inj: "inj_on (insert n) (?P n k)" by rule (auto; metis atLeastLessThan_iff insert_iff less_irrefl subsetCE) have disjoint: "insert n ` ?P n k ∩ ?P n (Suc k) = {}" by auto have "?Q = {K∈?Q. n ∈ K} ∪ {K∈?Q. n ∉ K}" by auto also have "{K∈?Q. n ∈ K} = insert n ` ?P n k" (is "?A = ?B") proof (rule set_eqI) fix K have K_finite: "finite K" if "K ⊆ insert n {0..<n}" using that by (rule finite_subset) simp_all have Suc_card_K: "Suc (card K - Suc 0) = card K" if "n ∈ K" and "finite K" proof - from ‹n ∈ K› obtain L where "K = insert n L" and "n ∉ L" by (blast elim: Set.set_insert) with that show ?thesis by (simp add: card.insert_remove) qed show "K ∈ ?A ⟷ K ∈ ?B" by (subst in_image_insert_iff) (auto simp add: card.insert_remove subset_eq_atLeast0_lessThan_finite Diff_subset_conv K_finite Suc_card_K) qed also have "{K∈?Q. n ∉ K} = ?P n (Suc k)" by (auto simp add: atLeast0_lessThan_Suc) finally show ?thesis using inj disjoint by (simp add: binomial_def card_Un_disjoint card_image) qed lemma binomial_eq_0: "n < k ⟹ n choose k = 0" by (auto simp add: binomial_def dest: subset_eq_atLeast0_lessThan_card) lemma zero_less_binomial: "k ≤ n ⟹ n choose k > 0" by (induct n k rule: diff_induct) simp_all lemma binomial_eq_0_iff [simp]: "n choose k = 0 ⟷ n < k" by (metis binomial_eq_0 less_numeral_extra(3) not_less zero_less_binomial) lemma zero_less_binomial_iff [simp]: "n choose k > 0 ⟷ k ≤ n" by (metis binomial_eq_0_iff not_less0 not_less zero_less_binomial) lemma binomial_n_n [simp]: "n choose n = 1" by (induct n) (simp_all add: binomial_eq_0) lemma binomial_Suc_n [simp]: "Suc n choose n = Suc n" by (induct n) simp_all lemma binomial_1 [simp]: "n choose Suc 0 = n" by (induct n) simp_all lemma choose_reduce_nat: "0 < n ⟹ 0 < k ⟹ n choose k = ((n - 1) choose (k - 1)) + ((n - 1) choose k)" using binomial_Suc_Suc [of "n - 1" "k - 1"] by simp lemma Suc_times_binomial_eq: "Suc n * (n choose k) = (Suc n choose Suc k) * Suc k" proof (induction n arbitrary: k) case 0 then show ?case by auto next case (Suc n) show ?case proof (cases k) case (Suc k') then show ?thesis using Suc.IH by (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq binomial_eq_0) qed auto qed lemma binomial_le_pow2: "n choose k ≤ 2^n" proof (induction n arbitrary: k) case 0 then show ?case using le_less less_le_trans by fastforce next case (Suc n) show ?case proof (cases k) case (Suc k') then show ?thesis using Suc.IH by (simp add: add_le_mono mult_2) qed auto qed text ‹The absorption property.› lemma Suc_times_binomial: "Suc k * (Suc n choose Suc k) = Suc n * (n choose k)" using Suc_times_binomial_eq by auto text ‹This is the well-known version of absorption, but it's harder to use because of the need to reason about division.› lemma binomial_Suc_Suc_eq_times: "(Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k" by (simp add: Suc_times_binomial_eq del: mult_Suc mult_Suc_right) text ‹Another version of absorption, with ‹-1› instead of ‹Suc›.› lemma times_binomial_minus1_eq: "0 < k ⟹ k * (n choose k) = n * ((n - 1) choose (k - 1))" using Suc_times_binomial_eq [where n = "n - 1" and k = "k - 1"] by (auto split: nat_diff_split) subsection ‹The binomial theorem (courtesy of Tobias Nipkow):› text ‹Avigad's version, generalized to any commutative ring› theorem binomial_ring: "(a + b :: 'a::comm_semiring_1)^n = (∑k≤n. (of_nat (n choose k)) * a^k * b^(n-k))" proof (induct n) case 0 then show ?case by simp next case (Suc n) have decomp: "{0..n+1} = {0} ∪ {n + 1} ∪ {1..n}" by auto have decomp2: "{0..n} = {0} ∪ {1..n}" by auto have "(a + b)^(n+1) = (a + b) * (∑k≤n. of_nat (n choose k) * a^k * b^(n - k))" using Suc.hyps by simp also have "… = a * (∑k≤n. of_nat (n choose k) * a^k * b^(n-k)) + b * (∑k≤n. of_nat (n choose k) * a^k * b^(n-k))" by (rule distrib_right) also have "… = (∑k≤n. of_nat (n choose k) * a^(k+1) * b^(n-k)) + (∑k≤n. of_nat (n choose k) * a^k * b^(n - k + 1))" by (auto simp add: sum_distrib_left ac_simps) also have "… = (∑k≤n. of_nat (n choose k) * a^k * b^(n + 1 - k)) + (∑k=1..n+1. of_nat (n choose (k - 1)) * a^k * b^(n + 1 - k))" by (simp add: atMost_atLeast0 sum.shift_bounds_cl_Suc_ivl Suc_diff_le field_simps del: sum.cl_ivl_Suc) also have "… = b^(n + 1) + (∑k=1..n. of_nat (n choose k) * a^k * b^(n + 1 - k)) + (a^(n + 1) + (∑k=1..n. of_nat (n choose (k - 1)) * a^k * b^(n + 1 - k)))" using sum.nat_ivl_Suc' [of 1 n "λk. of_nat (n choose (k-1)) * a ^ k * b ^ (n + 1 - k)"] by (simp add: sum.atLeast_Suc_atMost atMost_atLeast0) also have "… = a^(n + 1) + b^(n + 1) + (∑k=1..n. of_nat (n + 1 choose k) * a^k * b^(n + 1 - k))" by (auto simp add: field_simps sum.distrib [symmetric] choose_reduce_nat) also have "… = (∑k≤n+1. of_nat (n + 1 choose k) * a^k * b^(n + 1 - k))" using decomp by (simp add: atMost_atLeast0 field_simps) finally show ?case by simp qed text ‹Original version for the naturals.› corollary binomial: "(a + b :: nat)^n = (∑k≤n. (of_nat (n choose k)) * a^k * b^(n - k))" using binomial_ring [of "int a" "int b" n] by (simp only: of_nat_add [symmetric] of_nat_mult [symmetric] of_nat_power [symmetric] of_nat_sum [symmetric] of_nat_eq_iff of_nat_id) lemma binomial_fact_lemma: "k ≤ n ⟹ fact k * fact (n - k) * (n choose k) = fact n" proof (induct n arbitrary: k rule: nat_less_induct) fix n k assume H: "∀m<n. ∀x≤m. fact x * fact (m - x) * (m choose x) = fact m" assume kn: "k ≤ n" let ?ths = "fact k * fact (n - k) * (n choose k) = fact n" consider "n = 0 ∨ k = 0 ∨ n = k" | m h where "n = Suc m" "k = Suc h" "h < m" using kn by atomize_elim presburger then show "fact k * fact (n - k) * (n choose k) = fact n" proof cases case 1 with kn show ?thesis by auto next case 2 note n = ‹n = Suc m› note k = ‹k = Suc h› note hm = ‹h < m› have mn: "m < n" using n by arith have hm': "h ≤ m" using hm by arith have km: "k ≤ m" using hm k n kn by arith have "m - h = Suc (m - Suc h)" using k km hm by arith with km k have "fact (m - h) = (m - h) * fact (m - k)" by simp with n k have "fact k * fact (n - k) * (n choose k) = k * (fact h * fact (m - h) * (m choose h)) + (m - h) * (fact k * fact (m - k) * (m choose k))" by (simp add: field_simps) also have "… = (k + (m - h)) * fact m" using H[rule_format, OF mn hm'] H[rule_format, OF mn km] by (simp add: field_simps) finally show ?thesis using k n km by simp qed qed lemma binomial_fact': assumes "k ≤ n" shows "n choose k = fact n div (fact k * fact (n - k))" using binomial_fact_lemma [OF assms] by (metis fact_nonzero mult_eq_0_iff nonzero_mult_div_cancel_left) lemma binomial_fact: assumes kn: "k ≤ n" shows "(of_nat (n choose k) :: 'a::field_char_0) = fact n / (fact k * fact (n - k))" using binomial_fact_lemma[OF kn] by (metis (mono_tags, lifting) fact_nonzero mult_eq_0_iff nonzero_mult_div_cancel_left of_nat_fact of_nat_mult) lemma fact_binomial: assumes "k ≤ n" shows "fact k * of_nat (n choose k) = (fact n / fact (n - k) :: 'a::field_char_0)" unfolding binomial_fact [OF assms] by (simp add: field_simps) lemma choose_two: "n choose 2 = n * (n - 1) div 2" proof (cases "n ≥ 2") case False then have "n = 0 ∨ n = 1" by auto then show ?thesis by auto next case True define m where "m = n - 2" with True have "n = m + 2" by simp then have "fact n = n * (n - 1) * fact (n - 2)" by (simp add: fact_prod_Suc atLeast0_lessThan_Suc algebra_simps) with True show ?thesis by (simp add: binomial_fact') qed lemma choose_row_sum: "(∑k≤n. n choose k) = 2^n" using binomial [of 1 "1" n] by (simp add: numeral_2_eq_2) lemma sum_choose_lower: "(∑k≤n. (r+k) choose k) = Suc (r+n) choose n" by (induct n) auto lemma sum_choose_upper: "(∑k≤n. k choose m) = Suc n choose Suc m" by (induct n) auto lemma choose_alternating_sum: "n > 0 ⟹ (∑i≤n. (-1)^i * of_nat (n choose i)) = (0 :: 'a::comm_ring_1)" using binomial_ring[of "-1 :: 'a" 1 n] by (simp add: atLeast0AtMost mult_of_nat_commute zero_power) lemma choose_even_sum: assumes "n > 0" shows "2 * (∑i≤n. if even i then of_nat (n choose i) else 0) = (2 ^ n :: 'a::comm_ring_1)" proof - have "2 ^ n = (∑i≤n. of_nat (n choose i)) + (∑i≤n. (-1) ^ i * of_nat (n choose i) :: 'a)" using choose_row_sum[of n] by (simp add: choose_alternating_sum assms atLeast0AtMost of_nat_sum[symmetric]) also have "… = (∑i≤n. of_nat (n choose i) + (-1) ^ i * of_nat (n choose i))" by (simp add: sum.distrib) also have "… = 2 * (∑i≤n. if even i then of_nat (n choose i) else 0)" by (subst sum_distrib_left, intro sum.cong) simp_all finally show ?thesis .. qed lemma choose_odd_sum: assumes "n > 0" shows "2 * (∑i≤n. if odd i then of_nat (n choose i) else 0) = (2 ^ n :: 'a::comm_ring_1)" proof - have "2 ^ n = (∑i≤n. of_nat (n choose i)) - (∑i≤n. (-1) ^ i * of_nat (n choose i) :: 'a)" using choose_row_sum[of n] by (simp add: choose_alternating_sum assms atLeast0AtMost of_nat_sum[symmetric]) also have "… = (∑i≤n. of_nat (n choose i) - (-1) ^ i * of_nat (n choose i))" by (simp add: sum_subtractf) also have "… = 2 * (∑i≤n. if odd i then of_nat (n choose i) else 0)" by (subst sum_distrib_left, intro sum.cong) simp_all finally show ?thesis .. qed text‹NW diagonal sum property› lemma sum_choose_diagonal: assumes "m ≤ n" shows "(∑k≤m. (n - k) choose (m - k)) = Suc n choose m" proof - have "(∑k≤m. (n-k) choose (m - k)) = (∑k≤m. (n - m + k) choose k)" using sum.atLeastAtMost_rev [of "λk. (n - k) choose (m - k)" 0 m] assms by (simp add: atMost_atLeast0) also have "… = Suc (n - m + m) choose m" by (rule sum_choose_lower) also have "… = Suc n choose m" using assms by simp finally show ?thesis . qed subsection ‹Generalized binomial coefficients› definition gbinomial :: "'a::{semidom_divide,semiring_char_0} ⇒ nat ⇒ 'a" (infixl "gchoose" 65) where gbinomial_prod_rev: "a gchoose k = prod (λi. a - of_nat i) {0..<k} div fact k" lemma gbinomial_0 [simp]: "a gchoose 0 = 1" "0 gchoose (Suc k) = 0" by (simp_all add: gbinomial_prod_rev prod.atLeast0_lessThan_Suc_shift del: prod.op_ivl_Suc) lemma gbinomial_Suc: "a gchoose (Suc k) = prod (λi. a - of_nat i) {0..k} div fact (Suc k)" by (simp add: gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost) lemma gbinomial_1 [simp]: "a gchoose 1 = a" by (simp add: gbinomial_prod_rev lessThan_Suc) lemma gbinomial_Suc0 [simp]: "a gchoose Suc 0 = a" by (simp add: gbinomial_prod_rev lessThan_Suc) lemma gbinomial_mult_fact: "fact k * (a gchoose k) = (∏i = 0..<k. a - of_nat i)" for a :: "'a::field_char_0" by (simp_all add: gbinomial_prod_rev field_simps) lemma gbinomial_mult_fact': "(a gchoose k) * fact k = (∏i = 0..<k. a - of_nat i)" for a :: "'a::field_char_0" using gbinomial_mult_fact [of k a] by (simp add: ac_simps) lemma gbinomial_pochhammer: "a gchoose k = (- 1) ^ k * pochhammer (- a) k / fact k" for a :: "'a::field_char_0" proof (cases k) case (Suc k') then have "a gchoose k = pochhammer (a - of_nat k') (Suc k') / ((1 + of_nat k') * fact k')" by (simp add: gbinomial_prod_rev pochhammer_prod_rev atLeastLessThanSuc_atLeastAtMost prod.atLeast_Suc_atMost_Suc_shift of_nat_diff flip: power_mult_distrib prod.cl_ivl_Suc) then show ?thesis by (simp add: pochhammer_minus Suc) qed auto lemma gbinomial_pochhammer': "a gchoose k = pochhammer (a - of_nat k + 1) k / fact k" for a :: "'a::field_char_0" proof - have "a gchoose k = ((-1)^k * (-1)^k) * pochhammer (a - of_nat k + 1) k / fact k" by (simp add: gbinomial_pochhammer pochhammer_minus mult_ac) also have "(-1 :: 'a)^k * (-1)^k = 1" by (subst power_add [symmetric]) simp finally show ?thesis by simp qed lemma gbinomial_binomial: "n gchoose k = n choose k" proof (cases "k ≤ n") case False then have "n < k" by (simp add: not_le) then have "0 ∈ ((-) n) ` {0..<k}" by auto then have "prod ((-) n) {0..<k} = 0" by (auto intro: prod_zero) with ‹n < k› show ?thesis by (simp add: binomial_eq_0 gbinomial_prod_rev prod_zero) next case True from True have *: "prod ((-) n) {0..<k} = ∏{Suc (n - k)..n}" by (intro prod.reindex_bij_witness[of _ "λi. n - i" "λi. n - i"]) auto from True have "n choose k = fact n div (fact k * fact (n - k))" by (rule binomial_fact') with * show ?thesis by (simp add: gbinomial_prod_rev mult.commute [of "fact k"] div_mult2_eq fact_div_fact) qed lemma of_nat_gbinomial: "of_nat (n gchoose k) = (of_nat n gchoose k :: 'a::field_char_0)" proof (cases "k ≤ n") case False then show ?thesis by (simp add: not_le gbinomial_binomial binomial_eq_0 gbinomial_prod_rev) next case True define m where "m = n - k" with True have n: "n = m + k" by arith from n have "fact n = ((∏i = 0..<m + k. of_nat (m + k - i) ):: 'a)" by (simp add: fact_prod_rev) also have "… = ((∏i∈{0..<k} ∪ {k..<m + k}. of_nat (m + k - i)) :: 'a)" by (simp add: ivl_disj_un) finally have "fact n = (fact m * (∏i = 0..<k. of_nat m + of_nat k - of_nat i) :: 'a)" using prod.shift_bounds_nat_ivl [of "λi. of_nat (m + k - i) :: 'a" 0 k m] by (simp add: fact_prod_rev [of m] prod.union_disjoint of_nat_diff) then have "fact n / fact (n - k) = ((∏i = 0..<k. of_nat n - of_nat i) :: 'a)" by (simp add: n) with True have "fact k * of_nat (n gchoose k) = (fact k * (of_nat n gchoose k) :: 'a)" by (simp only: gbinomial_mult_fact [of k "of_nat n"] gbinomial_binomial [of n k] fact_binomial) then show ?thesis by simp qed lemma binomial_gbinomial: "of_nat (n choose k) = (of_nat n gchoose k :: 'a::field_char_0)" by (simp add: gbinomial_binomial [symmetric] of_nat_gbinomial) setup ‹Sign.add_const_constraint (\<^const_name>‹gbinomial›, SOME \<^typ>‹'a::field_char_0 ⇒ nat ⇒ 'a›)› lemma gbinomial_mult_1: fixes a :: "'a::field_char_0" shows "a * (a gchoose k) = of_nat k * (a gchoose k) + of_nat (Suc k) * (a gchoose (Suc k))" (is "?l = ?r") proof - have "?r = ((- 1) ^k * pochhammer (- a) k / fact k) * (of_nat k - (- a + of_nat k))" unfolding gbinomial_pochhammer pochhammer_Suc right_diff_distrib power_Suc by (auto simp add: field_simps simp del: of_nat_Suc) also have "… = ?l" by (simp add: field_simps gbinomial_pochhammer) finally show ?thesis .. qed lemma gbinomial_mult_1': "(a gchoose k) * a = of_nat k * (a gchoose k) + of_nat (Suc k) * (a gchoose (Suc k))" for a :: "'a::field_char_0" by (simp add: mult.commute gbinomial_mult_1) lemma gbinomial_Suc_Suc: "(a + 1) gchoose (Suc k) = a gchoose k + (a gchoose (Suc k))" for a :: "'a::field_char_0" proof (cases k) case 0 then show ?thesis by simp next case (Suc h) have eq0: "(∏i∈{1..k}. (a + 1) - of_nat i) = (∏i∈{0..h}. a - of_nat i)" proof (rule prod.reindex_cong) show "{1..k} = Suc ` {0..h}" using Suc by (auto simp add: image_Suc_atMost) qed auto have "fact (Suc k) * (a gchoose k + (a gchoose (Suc k))) = (a gchoose Suc h) * (fact (Suc (Suc h))) + (a gchoose Suc (Suc h)) * (fact (Suc (Suc h)))" by (simp add: Suc field_simps del: fact_Suc) also have "… = (a gchoose Suc h) * of_nat (Suc (Suc h) * fact (Suc h)) + (∏i=0..Suc h. a - of_nat i)" apply (simp only: gbinomial_mult_fact field_simps mult.left_commute [of _ "2"]) apply (simp del: fact_Suc add: fact_Suc [of "Suc h"] field_simps gbinomial_mult_fact mult.left_commute [of _ "2"] atLeastLessThanSuc_atLeastAtMost) done also have "… = (fact (Suc h) * (a gchoose Suc h)) * of_nat (Suc (Suc h)) + (∏i=0..Suc h. a - of_nat i)" by (simp only: fact_Suc mult.commute mult.left_commute of_nat_fact of_nat_id of_nat_mult) also have "… = of_nat (Suc (Suc h)) * (∏i=0..h. a - of_nat i) + (∏i=0..Suc h. a - of_nat i)" unfolding gbinomial_mult_fact atLeastLessThanSuc_atLeastAtMost by auto also have "… = (∏i=0..Suc h. a - of_nat i) + (of_nat h * (∏i=0..h. a - of_nat i) + 2 * (∏i=0..h. a - of_nat i))" by (simp add: field_simps) also have "… = ((a gchoose Suc h) * (fact (Suc h)) * of_nat (Suc k)) + (∏i∈{0..Suc h}. a - of_nat i)" unfolding gbinomial_mult_fact' by (simp add: comm_semiring_class.distrib field_simps Suc atLeastLessThanSuc_atLeastAtMost) also have "… = (∏i∈{0..h}. a - of_nat i) * (a + 1)" unfolding gbinomial_mult_fact' atLeast0_atMost_Suc by (simp add: field_simps Suc atLeastLessThanSuc_atLeastAtMost) also have "… = (∏i∈{0..k}. (a + 1) - of_nat i)" using eq0 by (simp add: Suc prod.atLeast0_atMost_Suc_shift del: prod.cl_ivl_Suc) also have "… = (fact (Suc k)) * ((a + 1) gchoose (Suc k))" by (simp only: gbinomial_mult_fact atLeastLessThanSuc_atLeastAtMost) finally show ?thesis using fact_nonzero [of "Suc k"] by auto qed lemma gbinomial_reduce_nat: "0 < k ⟹ a gchoose k = (a - 1) gchoose (k - 1) + ((a - 1) gchoose k)" for a :: "'a::field_char_0" by (metis Suc_pred' diff_add_cancel gbinomial_Suc_Suc) lemma gchoose_row_sum_weighted: "(∑k = 0..m. (r gchoose k) * (r/2 - of_nat k)) = of_nat(Suc m) / 2 * (r gchoose (Suc m))" for r :: "'a::field_char_0" by (induct m) (simp_all add: field_simps distrib gbinomial_mult_1) lemma binomial_symmetric: assumes kn: "k ≤ n" shows "n choose k = n choose (n - k)" proof - have kn': "n - k ≤ n" using kn by arith from binomial_fact_lemma[OF kn] binomial_fact_lemma[OF kn'] have "fact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))" by simp then show ?thesis using kn by simp qed lemma choose_rising_sum: "(∑j≤m. ((n + j) choose n)) = ((n + m + 1) choose (n + 1))" "(∑j≤m. ((n + j) choose n)) = ((n + m + 1) choose m)" proof - show "(∑j≤m. ((n + j) choose n)) = ((n + m + 1) choose (n + 1))" by (induct m) simp_all also have "… = (n + m + 1) choose m" by (subst binomial_symmetric) simp_all finally show "(∑j≤m. ((n + j) choose n)) = (n + m + 1) choose m" . qed lemma choose_linear_sum: "(∑i≤n. i * (n choose i)) = n * 2 ^ (n - 1)" proof (cases n) case 0 then show ?thesis by simp next case (Suc m) have "(∑i≤n. i * (n choose i)) = (∑i≤Suc m. i * (Suc m choose i))" by (simp add: Suc) also have "… = Suc m * 2 ^ m" unfolding sum.atMost_Suc_shift Suc_times_binomial sum_distrib_left[symmetric] by (simp add: choose_row_sum) finally show ?thesis using Suc by simp qed lemma choose_alternating_linear_sum: assumes "n ≠ 1" shows "(∑i≤n. (-1)^i * of_nat i * of_nat (n choose i) :: 'a::comm_ring_1) = 0" proof (cases n) case 0 then show ?thesis by simp next case (Suc m) with assms have "m > 0" by simp have "(∑i≤n. (-1) ^ i * of_nat i * of_nat (n choose i) :: 'a) = (∑i≤Suc m. (-1) ^ i * of_nat i * of_nat (Suc m choose i))" by (simp add: Suc) also have "… = (∑i≤m. (-1) ^ (Suc i) * of_nat (Suc i * (Suc m choose Suc i)))" by (simp only: sum.atMost_Suc_shift sum_distrib_left[symmetric] mult_ac of_nat_mult) simp also have "… = - of_nat (Suc m) * (∑i≤m. (-1) ^ i * of_nat (m choose i))" by (subst sum_distrib_left, rule sum.cong[OF refl], subst Suc_times_binomial) (simp add: algebra_simps) also have "(∑i≤m. (-1 :: 'a) ^ i * of_nat ((m choose i))) = 0" using choose_alternating_sum[OF ‹m > 0›] by simp finally show ?thesis by simp qed lemma vandermonde: "(∑k≤r. (m choose k) * (n choose (r - k))) = (m + n) choose r" proof (induct n arbitrary: r) case 0 have "(∑k≤r. (m choose k) * (0 choose (r - k))) = (∑k≤r. if k = r then (m choose k) else 0)" by (intro sum.cong) simp_all also have "… = m choose r" by simp finally show ?case by simp next case (Suc n r) show ?case by (cases r) (simp_all add: Suc [symmetric] algebra_simps sum.distrib Suc_diff_le) qed lemma choose_square_sum: "(∑k≤n. (n choose k)^2) = ((2*n) choose n)" using vandermonde[of n n n] by (simp add: power2_eq_square mult_2 binomial_symmetric [symmetric]) lemma pochhammer_binomial_sum: fixes a b :: "'a::comm_ring_1" shows "pochhammer (a + b) n = (∑k≤n. of_nat (n choose k) * pochhammer a k * pochhammer b (n - k))" proof (induction n arbitrary: a b) case 0 then show ?case by simp next case (Suc n a b) have "(∑k≤Suc n. of_nat (Suc n choose k) * pochhammer a k * pochhammer b (Suc n - k)) = (∑i≤n. of_nat (n choose i) * pochhammer a (Suc i) * pochhammer b (n - i)) + ((∑i≤n. of_nat (n choose Suc i) * pochhammer a (Suc i) * pochhammer b (n - i)) + pochhammer b (Suc n))" by (subst sum.atMost_Suc_shift) (simp add: ring_distribs sum.distrib) also have "(∑i≤n. of_nat (n choose i) * pochhammer a (Suc i) * pochhammer b (n - i)) = a * pochhammer ((a + 1) + b) n" by (subst Suc) (simp add: sum_distrib_left pochhammer_rec mult_ac) also have "(∑i≤n. of_nat (n choose Suc i) * pochhammer a (Suc i) * pochhammer b (n - i)) + pochhammer b (Suc n) = (∑i=0..Suc n. of_nat (n choose i) * pochhammer a i * pochhammer b (Suc n - i))" apply (subst sum.atLeast_Suc_atMost, simp) apply (simp add: sum.shift_bounds_cl_Suc_ivl atLeast0AtMost del: sum.cl_ivl_Suc) done also have "… = (∑i≤n. of_nat (n choose i) * pochhammer a i * pochhammer b (Suc n - i))" using Suc by (intro sum.mono_neutral_right) (auto simp: not_le binomial_eq_0) also have "… = (∑i≤n. of_nat (n choose i) * pochhammer a i * pochhammer b (Suc (n - i)))" by (intro sum.cong) (simp_all add: Suc_diff_le) also have "… = b * pochhammer (a + (b + 1)) n" by (subst Suc) (simp add: sum_distrib_left mult_ac pochhammer_rec) also have "a * pochhammer ((a + 1) + b) n + b * pochhammer (a + (b + 1)) n = pochhammer (a + b) (Suc n)" by (simp add: pochhammer_rec algebra_simps) finally show ?case .. qed text ‹Contributed by Manuel Eberl, generalised by LCP. Alternative definition of the binomial coefficient as \<^term>‹∏i<k. (n - i) / (k - i)›.› lemma gbinomial_altdef_of_nat: "a gchoose k = (∏i = 0..<k. (a - of_nat i) / of_nat (k - i) :: 'a)" for k :: nat and a :: "'a::field_char_0" by (simp add: prod_dividef gbinomial_prod_rev fact_prod_rev) lemma gbinomial_ge_n_over_k_pow_k: fixes k :: nat and a :: "'a::linordered_field" assumes "of_nat k ≤ a" shows "(a / of_nat k :: 'a) ^ k ≤ a gchoose k" proof - have x: "0 ≤ a" using assms of_nat_0_le_iff order_trans by blast have "(a / of_nat k :: 'a) ^ k = (∏i = 0..<k. a / of_nat k :: 'a)" by simp also have "… ≤ a gchoose k" proof - have "⋀i. i < k ⟹ 0 ≤ a / of_nat k" by (simp add: x zero_le_divide_iff) moreover have "a / of_nat k ≤ (a - of_nat i) / of_nat (k - i)" if "i < k" for i proof - from assms have "a * of_nat i ≥ of_nat (i * k)" by (metis mult.commute mult_le_cancel_right of_nat_less_0_iff of_nat_mult) then have "a * of_nat k - a * of_nat i ≤ a * of_nat k - of_nat (i * k)" by arith then have "a * of_nat (k - i) ≤ (a - of_nat i) * of_nat k" using ‹i < k› by (simp add: algebra_simps zero_less_mult_iff of_nat_diff) then have "a * of_nat (k - i) ≤ (a - of_nat i) * (of_nat k :: 'a)" by blast with assms show ?thesis using ‹i < k› by (simp add: field_simps) qed ultimately show ?thesis unfolding gbinomial_altdef_of_nat by (intro prod_mono) auto qed finally show ?thesis . qed lemma gbinomial_negated_upper: "(a gchoose k) = (-1) ^ k * ((of_nat k - a - 1) gchoose k)" by (simp add: gbinomial_pochhammer pochhammer_minus algebra_simps) lemma gbinomial_minus: "((-a) gchoose k) = (-1) ^ k * ((a + of_nat k - 1) gchoose k)" by (subst gbinomial_negated_upper) (simp add: add_ac) lemma Suc_times_gbinomial: "of_nat (Suc k) * ((a + 1) gchoose (Suc k)) = (a + 1) * (a gchoose k)" proof (cases k) case 0 then show ?thesis by simp next case (Suc b) then have "((a + 1) gchoose (Suc (Suc b))) = (∏i = 0..Suc b. a + (1 - of_nat i)) / fact (b + 2)" by (simp add: field_simps gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost) also have "(∏i = 0..Suc b. a + (1 - of_nat i)) = (a + 1) * (∏i = 0..b. a - of_nat i)" by (simp add: prod.atLeast0_atMost_Suc_shift del: prod.cl_ivl_Suc) also have "… / fact (b + 2) = (a + 1) / of_nat (Suc (Suc b)) * (a gchoose Suc b)" by (simp_all add: gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost) finally show ?thesis by (simp add: Suc field_simps del: of_nat_Suc) qed lemma gbinomial_factors: "((a + 1) gchoose (Suc k)) = (a + 1) / of_nat (Suc k) * (a gchoose k)" proof (cases k) case 0 then show ?thesis by simp next case (Suc b) then have "((a + 1) gchoose (Suc (Suc b))) = (∏i = 0 .. Suc b. a + (1 - of_nat i)) / fact (b + 2)" by (simp add: field_simps gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost) also have "(∏i = 0 .. Suc b. a + (1 - of_nat i)) = (a + 1) * (∏i = 0..b. a - of_nat i)" by (simp add: prod.atLeast0_atMost_Suc_shift del: prod.cl_ivl_Suc) also have "… / fact (b + 2) = (a + 1) / of_nat (Suc (Suc b)) * (a gchoose Suc b)" by (simp_all add: gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost atLeast0AtMost) finally show ?thesis by (simp add: Suc) qed lemma gbinomial_rec: "((a + 1) gchoose (Suc k)) = (a gchoose k) * ((a + 1) / of_nat (Suc k))" using gbinomial_mult_1[of a k] by (subst gbinomial_Suc_Suc) (simp add: field_simps del: of_nat_Suc, simp add: algebra_simps) lemma gbinomial_of_nat_symmetric: "k ≤ n ⟹ (of_nat n) gchoose k = (of_nat n) gchoose (n - k)" using binomial_symmetric[of k n] by (simp add: binomial_gbinomial [symmetric]) text ‹The absorption identity (equation 5.5 @{cite ‹p.~157› GKP_CM}): \[ {r \choose k} = \frac{r}{k}{r - 1 \choose k - 1},\quad \textnormal{integer } k \neq 0. \]› lemma gbinomial_absorption': "k > 0 ⟹ a gchoose k = (a / of_nat k) * (a - 1 gchoose (k - 1))" using gbinomial_rec[of "a - 1" "k - 1"] by (simp_all add: gbinomial_rec field_simps del: of_nat_Suc) text ‹The absorption identity is written in the following form to avoid division by $k$ (the lower index) and therefore remove the $k \neq 0$ restriction @{cite ‹p.~157› GKP_CM}: \[ k{r \choose k} = r{r - 1 \choose k - 1}, \quad \textnormal{integer } k. \]› lemma gbinomial_absorption: "of_nat (Suc k) * (a gchoose Suc k) = a * ((a - 1) gchoose k)" using gbinomial_absorption'[of "Suc k" a] by (simp add: field_simps del: of_nat_Suc) text ‹The absorption identity for natural number binomial coefficients:› lemma binomial_absorption: "Suc k * (n choose Suc k) = n * ((n - 1) choose k)" by (cases n) (simp_all add: binomial_eq_0 Suc_times_binomial del: binomial_Suc_Suc mult_Suc) text ‹The absorption companion identity for natural number coefficients, following the proof by GKP @{cite ‹p.~157› GKP_CM}:› lemma binomial_absorb_comp: "(n - k) * (n choose k) = n * ((n - 1) choose k)" (is "?lhs = ?rhs") proof (cases "n ≤ k") case True then show ?thesis by auto next case False then have "?rhs = Suc ((n - 1) - k) * (n choose Suc ((n - 1) - k))" using binomial_symmetric[of k "n - 1"] binomial_absorption[of "(n - 1) - k" n] by simp also have "Suc ((n - 1) - k) = n - k" using False by simp also have "n choose … = n choose k" using False by (intro binomial_symmetric [symmetric]) simp_all finally show ?thesis .. qed text ‹The generalised absorption companion identity:› lemma gbinomial_absorb_comp: "(a - of_nat k) * (a gchoose k) = a * ((a - 1) gchoose k)" using pochhammer_absorb_comp[of a k] by (simp add: gbinomial_pochhammer) lemma gbinomial_addition_formula: "a gchoose (Suc k) = ((a - 1) gchoose (Suc k)) + ((a - 1) gchoose k)" using gbinomial_Suc_Suc[of "a - 1" k] by (simp add: algebra_simps) lemma binomial_addition_formula: "0 < n ⟹ n choose (Suc k) = ((n - 1) choose (Suc k)) + ((n - 1) choose k)" by (subst choose_reduce_nat) simp_all text ‹ Equation 5.9 of the reference material @{cite ‹p.~159› GKP_CM} is a useful summation formula, operating on both indices: \[ \sum\limits_{k \leq n}{r + k \choose k} = {r + n + 1 \choose n}, \quad \textnormal{integer } n. \] › lemma gbinomial_parallel_sum: "(∑k≤n. (a + of_nat k) gchoose k) = (a + of_nat n + 1) gchoose n" proof (induct n) case 0 then show ?case by simp next case (Suc m) then show ?case using gbinomial_Suc_Suc[of "(a + of_nat m + 1)" m] by (simp add: add_ac) qed subsubsection ‹Summation on the upper index› text ‹ Another summation formula is equation 5.10 of the reference material @{cite ‹p.~160› GKP_CM}, aptly named \emph{summation on the upper index}:\[\sum_{0 \leq k \leq n} {k \choose m} = {n + 1 \choose m + 1}, \quad \textnormal{integers } m, n \geq 0.\] › lemma gbinomial_sum_up_index: "(∑j = 0..n. (of_nat j gchoose k) :: 'a::field_char_0) = (of_nat n + 1) gchoose (k + 1)" proof (induct n) case 0 show ?case using gbinomial_Suc_Suc[of 0 k] by (cases k) auto next case (Suc n) then show ?case using gbinomial_Suc_Suc[of "of_nat (Suc n) :: 'a" k] by (simp add: add_ac) qed lemma gbinomial_index_swap: "((-1) ^ k) * ((- (of_nat n) - 1) gchoose k) = ((-1) ^ n) * ((- (of_nat k) - 1) gchoose n)" (is "?lhs = ?rhs") proof - have "?lhs = (of_nat (k + n) gchoose k)" by (subst gbinomial_negated_upper) (simp add: power_mult_distrib [symmetric]) also have "… = (of_nat (k + n) gchoose n)" by (subst gbinomial_of_nat_symmetric) simp_all also have "… = ?rhs" by (subst gbinomial_negated_upper) simp finally show ?thesis . qed lemma gbinomial_sum_lower_neg: "(∑k≤m. (a gchoose k) * (- 1) ^ k) = (- 1) ^ m * (a - 1 gchoose m)" (is "?lhs = ?rhs") proof - have "?lhs = (∑k≤m. -(a + 1) + of_nat k gchoose k)" by (intro sum.cong[OF refl]) (subst gbinomial_negated_upper, simp add: power_mult_distrib) also have "… = - a + of_nat m gchoose m" by (subst gbinomial_parallel_sum) simp also have "… = ?rhs" by (subst gbinomial_negated_upper) (simp add: power_mult_distrib) finally show ?thesis . qed lemma gbinomial_partial_row_sum: "(∑k≤m. (a gchoose k) * ((a / 2) - of_nat k)) = ((of_nat m + 1)/2) * (a gchoose (m + 1))" proof (induct m) case 0 then show ?case by simp next case (Suc mm) then have "(∑k≤Suc mm. (a gchoose k) * (a / 2 - of_nat k)) = (a - of_nat (Suc mm)) * (a gchoose Suc mm) / 2" by (simp add: field_simps) also have "… = a * (a - 1 gchoose Suc mm) / 2" by (subst gbinomial_absorb_comp) (rule refl) also have "… = (of_nat (Suc mm) + 1) / 2 * (a gchoose (Suc mm + 1))" by (subst gbinomial_absorption [symmetric]) simp finally show ?case . qed lemma sum_bounds_lt_plus1: "(∑k<mm. f (Suc k)) = (∑k=1..mm. f k)" by (induct mm) simp_all lemma gbinomial_partial_sum_poly: "(∑k≤m. (of_nat m + a gchoose k) * x^k * y^(m-k)) = (∑k≤m. (-a gchoose k) * (-x)^k * (x + y)^(m-k))" (is "?lhs m = ?rhs m") proof (induction m) case 0 then show ?case by simp next case (Suc mm) define G where "G i k = (of_nat i + a gchoose k) * x^k * y^(i - k)" for i k define S where "S = ?lhs" have SG_def: "S = (λi. (∑k≤i. (G i k)))" unfolding S_def G_def .. have "S (Suc mm) = G (Suc mm) 0 + (∑k=Suc 0..Suc mm. G (Suc mm) k)" using SG_def by (simp add: sum.atLeast_Suc_atMost atLeast0AtMost [symmetric]) also have "(∑k=Suc 0..Suc mm. G (Suc mm) k) = (∑k=0..mm. G (Suc mm) (Suc k))" by (subst sum.shift_bounds_cl_Suc_ivl) simp also have "… = (∑k=0..mm. ((of_nat mm + a gchoose (Suc k)) + (of_nat mm + a gchoose k)) * x^(Suc k) * y^(mm - k))" unfolding G_def by (subst gbinomial_addition_formula) simp also have "… = (∑k=0..mm. (of_nat mm + a gchoose (Suc k)) * x^(Suc k) * y^(mm - k)) + (∑k=0..mm. (of_nat mm + a gchoose k) * x^(Suc k) * y^(mm - k))" by (subst sum.distrib [symmetric]) (simp add: algebra_simps) also have "(∑k=0..mm. (of_nat mm + a gchoose (Suc k)) * x^(Suc k) * y^(mm - k)) = (∑k<Suc mm. (of_nat mm + a gchoose (Suc k)) * x^(Suc k) * y^(mm - k))" by (simp only: atLeast0AtMost lessThan_Suc_atMost) also have "… = (∑k<mm. (of_nat mm + a gchoose Suc k) * x^(Suc k) * y^(mm-k)) + (of_nat mm + a gchoose (Suc mm)) * x^(Suc mm)" (is "_ = ?A + ?B") by (subst sum.lessThan_Suc) simp also have "?A = (∑k=1..mm. (of_nat mm + a gchoose k) * x^k * y^(mm - k + 1))" proof (subst sum_bounds_lt_plus1 [symmetric], intro sum.cong[OF refl], clarify) fix k assume "k < mm" then have "mm - k = mm - Suc k + 1" by linarith then show "(of_nat mm + a gchoose Suc k) * x ^ Suc k * y ^ (mm - k) = (of_nat mm + a gchoose Suc k) * x ^ Suc k * y ^ (mm - Suc k + 1)" by (simp only:) qed also have "… + ?B = y * (∑k=1..mm. (G mm k)) + (of_nat mm + a gchoose (Suc mm)) * x^(Suc mm)" unfolding G_def by (subst sum_distrib_left) (simp add: algebra_simps) also have "(∑k=0..mm. (of_nat mm + a gchoose k) * x^(Suc k) * y^(mm - k)) = x * (S mm)" unfolding S_def by (subst sum_distrib_left) (simp add: atLeast0AtMost algebra_simps) also have "(G (Suc mm) 0) = y * (G mm 0)" by (simp add: G_def) finally have "S (Suc mm) = y * (G mm 0 + (∑k=1..mm. (G mm k))) + (of_nat mm + a gchoose (Suc mm)) * x^(Suc mm) + x * (S mm)" by (simp add: ring_distribs) also have "G mm 0 + (∑k=1..mm. (G mm k)) = S mm" by (simp add: sum.atLeast_Suc_atMost[symmetric] SG_def atLeast0AtMost) finally have "S (Suc mm) = (x + y) * (S mm) + (of_nat mm + a gchoose (Suc mm)) * x^(Suc mm)" by (simp add: algebra_simps) also have "(of_nat mm + a gchoose (Suc mm)) = (-1) ^ (Suc mm) * (- a gchoose (Suc mm))" by (subst gbinomial_negated_upper) simp also have "(-1) ^ Suc mm * (- a gchoose Suc mm) * x ^ Suc mm = (- a gchoose (Suc mm)) * (-x) ^ Suc mm" by (simp add: power_minus[of x]) also have "(x + y) * S mm + … = (x + y) * ?rhs mm + (- a gchoose (Suc mm)) * (- x)^Suc mm" unfolding S_def by (subst Suc.IH) simp also have "(x + y) * ?rhs mm = (∑n≤mm. ((- a gchoose n) * (- x) ^ n * (x + y) ^ (Suc mm - n)))" by (subst sum_distrib_left, rule sum.cong) (simp_all add: Suc_diff_le) also have "… + (-a gchoose (Suc mm)) * (-x)^Suc mm = (∑n≤Suc mm. (- a gchoose n) * (- x) ^ n * (x + y) ^ (Suc mm - n))" by simp finally show ?case by (simp only: S_def) qed lemma gbinomial_partial_sum_poly_xpos: "(∑k≤m. (of_nat m + a gchoose k) * x^k * y^(m-k)) = (∑k≤m. (of_nat k + a - 1 gchoose k) * x^k * (x + y)^(m-k))" (is "?lhs = ?rhs") proof - have "?lhs = (∑k≤m. (- a gchoose k) * (- x) ^ k * (x + y) ^ (m - k))" by (simp add: gbinomial_partial_sum_poly) also have "... = (∑k≤m. (-1) ^ k * (of_nat k - - a - 1 gchoose k) * (- x) ^ k * (x + y) ^ (m - k))" by (metis (no_types, hide_lams) gbinomial_negated_upper) also have "... = ?rhs" by (intro sum.cong) (auto simp flip: power_mult_distrib) finally show ?thesis . qed lemma binomial_r_part_sum: "(∑k≤m. (2 * m + 1 choose k)) = 2 ^ (2 * m)" proof - have "2 * 2^(2*m) = (∑k = 0..(2 * m + 1). (2 * m + 1 choose k))" using choose_row_sum[where n="2 * m + 1"] by (simp add: atMost_atLeast0) also have "(∑k = 0..(2 * m + 1). (2 * m + 1 choose k)) = (∑k = 0..m. (2 * m + 1 choose k)) + (∑k = m+1..2*m+1. (2 * m + 1 choose k))" using sum.ub_add_nat[of 0 m "λk. 2 * m + 1 choose k" "m+1"] by (simp add: mult_2) also have "(∑k = m+1..2*m+1. (2 * m + 1 choose k)) = (∑k = 0..m. (2 * m + 1 choose (k + (m + 1))))" by (subst sum.shift_bounds_cl_nat_ivl [symmetric]) (simp add: mult_2) also have "… = (∑k = 0..m. (2 * m + 1 choose (m - k)))" by (intro sum.cong[OF refl], subst binomial_symmetric) simp_all also have "… = (∑k = 0..m. (2 * m + 1 choose k))" using sum.atLeastAtMost_rev [of "λk. 2 * m + 1 choose (m - k)" 0 m] by simp also have "… + … = 2 * …" by simp finally show ?thesis by (subst (asm) mult_cancel1) (simp add: atLeast0AtMost) qed lemma gbinomial_r_part_sum: "(∑k≤m. (2 * (of_nat m) + 1 gchoose k)) = 2 ^ (2 * m)" (is "?lhs = ?rhs") proof - have "?lhs = of_nat (∑k≤m. (2 * m + 1) choose k)" by (simp add: binomial_gbinomial add_ac) also have "… = of_nat (2 ^ (2 * m))" by (subst binomial_r_part_sum) (rule refl) finally show ?thesis by simp qed lemma gbinomial_sum_nat_pow2: "(∑k≤m. (of_nat (m + k) gchoose k :: 'a::field_char_0) / 2 ^ k) = 2 ^ m" (is "?lhs = ?rhs") proof - have "2 ^ m * 2 ^ m = (2 ^ (2*m) :: 'a)" by (induct m) simp_all also have "… = (∑k≤m. (2 * (of_nat m) + 1 gchoose k))" using gbinomial_r_part_sum .. also have "… = (∑k≤m. (of_nat (m + k) gchoose k) * 2 ^ (m - k))" using gbinomial_partial_sum_poly_xpos[where x="1" and y="1" and a="of_nat m + 1" and m="m"] by (simp add: add_ac) also have "… = 2 ^ m * (∑k≤m. (of_nat (m + k) gchoose k) / 2 ^ k)" by (subst sum_distrib_left) (simp add: algebra_simps power_diff) finally show ?thesis by (subst (asm) mult_left_cancel) simp_all qed lemma gbinomial_trinomial_revision: assumes "k ≤ m" shows "(a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))" proof - have "(a gchoose m) * (of_nat m gchoose k) = (a gchoose m) * fact m / (fact k * fact (m - k))" using assms by (simp add: binomial_gbinomial [symmetric] binomial_fact) also have "… = (a gchoose k) * (a - of_nat k gchoose (m - k))" using assms by (simp add: gbinomial_pochhammer power_diff pochhammer_product) finally show ?thesis . qed text ‹Versions of the theorems above for the natural-number version of "choose"› lemma binomial_altdef_of_nat: "k ≤ n ⟹ of_nat (n choose k) = (∏i = 0..<k. of_nat (n - i) / of_nat (k - i) :: 'a)" for n k :: nat and x :: "'a::field_char_0" by (simp add: gbinomial_altdef_of_nat binomial_gbinomial of_nat_diff) lemma binomial_ge_n_over_k_pow_k: "k ≤ n ⟹ (of_nat n / of_nat k :: 'a) ^ k ≤ of_nat (n choose k)" for k n :: nat and x :: "'a::linordered_field" by (simp add: gbinomial_ge_n_over_k_pow_k binomial_gbinomial of_nat_diff) lemma binomial_le_pow: assumes "r ≤ n" shows "n choose r ≤ n ^ r" proof - have "n choose r ≤ fact n div fact (n - r)" using assms by (subst binomial_fact_lemma[symmetric]) auto with fact_div_fact_le_pow [OF assms] show ?thesis by auto qed lemma binomial_altdef_nat: "k ≤ n ⟹ n choose k = fact n div (fact k * fact (n - k))" for k n :: nat by (subst binomial_fact_lemma [symmetric]) auto lemma choose_dvd: assumes "k ≤ n" shows "fact k * fact (n - k) dvd (fact n :: 'a::linordered_semidom)" unfolding dvd_def proof show "fact n = fact k * fact (n - k) * of_nat (n choose k)" by (metis assms binomial_fact_lemma of_nat_fact of_nat_mult) qed lemma fact_fact_dvd_fact: "fact k * fact n dvd (fact (k + n) :: 'a::linordered_semidom)" by (metis add.commute add_diff_cancel_left' choose_dvd le_add2) lemma choose_mult_lemma: "((m + r + k) choose (m + k)) * ((m + k) choose k) = ((m + r + k) choose k) * ((m + r) choose m)" (is "?lhs = _") proof - have "?lhs = fact (m + r + k) div (fact (m + k) * fact (m + r - m)) * (fact (m + k) div (fact k * fact m))" by (simp add: binomial_altdef_nat) also have "... = fact (m + r + k) * fact (m + k) div (fact (m + k) * fact (m + r - m) * (fact k * fact m))" apply (subst div_mult_div_if_dvd) apply (auto simp: algebra_simps fact_fact_dvd_fact) apply (metis add.assoc add.commute fact_fact_dvd_fact) done also have "… = fact (m + r + k) div (fact r * (fact k * fact m))" by (auto simp: algebra_simps fact_fact_dvd_fact) also have "… = (fact (m + r + k) * fact (m + r)) div (fact r * (fact k * fact m) * fact (m + r))" apply (subst div_mult_div_if_dvd [symmetric]) apply (auto simp add: algebra_simps) apply (metis fact_fact_dvd_fact dvd_trans nat_mult_dvd_cancel_disj) done also have "… = (fact (m + r + k) div (fact k * fact (m + r)) * (fact (m + r) div (fact r * fact m)))" by (auto simp: div_mult_div_if_dvd fact_fact_dvd_fact algebra_simps) finally show ?thesis by (simp add: binomial_altdef_nat mult.commute) qed text ‹The "Subset of a Subset" identity.› lemma choose_mult: "k ≤ m ⟹ m ≤ n ⟹ (n choose m) * (m choose k) = (n choose k) * ((n - k) choose (m - k))" using choose_mult_lemma [of "m-k" "n-m" k] by simp subsection ‹More on Binomial Coefficients› lemma choose_one: "n choose 1 = n" for n :: nat by simp lemma card_UNION: assumes "finite A" and "∀k ∈ A. finite k" shows "card (⋃A) = nat (∑I | I ⊆ A ∧ I ≠ {}. (- 1) ^ (card I + 1) * int (card (⋂I)))" (is "?lhs = ?rhs") proof - have "?rhs = nat (∑I | I ⊆ A ∧ I ≠ {}. (- 1) ^ (card I + 1) * (∑_∈⋂I. 1))" by simp also have "… = nat (∑I | I ⊆ A ∧ I ≠ {}. (∑_∈⋂I. (- 1) ^ (card I + 1)))" (is "_ = nat ?rhs") by (subst sum_distrib_left) simp also have "?rhs = (∑(I, _)∈Sigma {I. I ⊆ A ∧ I ≠ {}} Inter. (- 1) ^ (card I + 1))" using assms by (subst sum.Sigma) auto also have "… = (∑(x, I)∈(SIGMA x:UNIV. {I. I ⊆ A ∧ I ≠ {} ∧ x ∈ ⋂I}). (- 1) ^ (card I + 1))" by (rule sum.reindex_cong [where l = "λ(x, y). (y, x)"]) (auto intro: inj_onI) also have "… = (∑(x, I)∈(SIGMA x:⋃A. {I. I ⊆ A ∧ I ≠ {} ∧ x ∈ ⋂I}). (- 1) ^ (card I + 1))" using assms by (auto intro!: sum.mono_neutral_cong_right finite_SigmaI2 intro: finite_subset[where B="⋃A"]) also have "… = (∑x∈⋃A. (∑I|I ⊆ A ∧ I ≠ {} ∧ x ∈ ⋂I. (- 1) ^ (card I + 1)))" using assms by (subst sum.Sigma) auto also have "… = (∑_∈⋃A. 1)" (is "sum ?lhs _ = _") proof (rule sum.cong[OF refl]) fix x assume x: "x ∈ ⋃A" define K where "K = {X ∈ A. x ∈ X}" with ‹finite A› have K: "finite K" by auto let ?I = "λi. {I. I ⊆ A ∧ card I = i ∧ x ∈ ⋂I}" have "inj_on snd (SIGMA i:{1..card A}. ?I i)" using assms by (auto intro!: inj_onI) moreover have [symmetric]: "snd ` (SIGMA i:{1..card A}. ?I i) = {I. I ⊆ A ∧ I ≠ {} ∧ x ∈ ⋂I}" using assms by (auto intro!: rev_image_eqI[where x="(card a, a)" for a] simp add: card_gt_0_iff[folded Suc_le_eq] dest: finite_subset intro: card_mono) ultimately have "?lhs x = (∑(i, I)∈(SIGMA i:{1..card A}. ?I i). (- 1) ^ (i + 1))" by (rule sum.reindex_cong [where l = snd]) fastforce also have "… = (∑i=1..card A. (∑I|I ⊆ A ∧ card I = i ∧ x ∈ ⋂I. (- 1) ^ (i + 1)))" using assms by (subst sum.Sigma) auto also have "… = (∑i=1..card A. (- 1) ^ (i + 1) * (∑I|I ⊆ A ∧ card I = i ∧ x ∈ ⋂I. 1))" by (subst sum_distrib_left) simp also have "… = (∑i=1..card K. (- 1) ^ (i + 1) * (∑I|I ⊆ K ∧ card I = i. 1))" (is "_ = ?rhs") proof (rule sum.mono_neutral_cong_right[rule_format]) show "finite {1..card A}" by simp show "{1..card K} ⊆ {1..card A}" using ‹finite A› by (auto simp add: K_def intro: card_mono) next fix i assume "i ∈ {1..card A} - {1..card K}" then have i: "i ≤ card A" "card K < i" by auto have "{I. I ⊆ A ∧ card I = i ∧ x ∈ ⋂I} = {I. I ⊆ K ∧ card I = i}" by (auto simp add: K_def) also have "… = {}" using ‹finite A› i by (auto simp add: K_def dest: card_mono[rotated 1]) finally show "(- 1) ^ (i + 1) * (∑I | I ⊆ A ∧ card I = i ∧ x ∈ ⋂I. 1 :: int) = 0" by (simp only:) simp next fix i have "(∑I | I ⊆ A ∧ card I = i ∧ x ∈ ⋂I. 1) = (∑I | I ⊆ K ∧ card I = i. 1 :: int)" (is "?lhs = ?rhs") by (rule sum.cong) (auto simp add: K_def) then show "(- 1) ^ (i + 1) * ?lhs = (- 1) ^ (i + 1) * ?rhs" by simp qed also have "{I. I ⊆ K ∧ card I = 0} = {{}}" using assms by (auto simp add: card_eq_0_iff K_def dest: finite_subset) then have "?rhs = (∑i = 0..card K. (- 1) ^ (i + 1) * (∑I | I ⊆ K ∧ card I = i. 1 :: int)) + 1" by (subst (2) sum.atLeast_Suc_atMost) simp_all also have "… = (∑i = 0..card K. (- 1) * ((- 1) ^ i * int (card K choose i))) + 1" using K by (subst n_subsets[symmetric]) simp_all also have "… = - (∑i = 0..card K. (- 1) ^ i * int (card K choose i)) + 1" by (subst sum_distrib_left[symmetric]) simp also have "… = - ((-1 + 1) ^ card K) + 1" by (subst binomial_ring) (simp add: ac_simps atMost_atLeast0) also have "… = 1" using x K by (auto simp add: K_def card_gt_0_iff) finally show "?lhs x = 1" . qed also have "nat … = card (⋃A)" by simp finally show ?thesis .. qed text ‹The number of nat lists of length ‹m› summing to ‹N› is \<^term>‹(N + m - 1) choose N›:› lemma card_length_sum_list_rec: assumes "m ≥ 1" shows "card {l::nat list. length l = m ∧ sum_list l = N} = card {l. length l = (m - 1) ∧ sum_list l = N} + card {l. length l = m ∧ sum_list l + 1 = N}" (is "card ?C = card ?A + card ?B") proof - let ?A' = "{l. length l = m ∧ sum_list l = N ∧ hd l = 0}" let ?B' = "{l. length l = m ∧ sum_list l = N ∧ hd l ≠ 0}" let ?f = "λl. 0 # l" let ?g = "λl. (hd l + 1) # tl l" have 1: "xs ≠ [] ⟹ x = hd xs ⟹ x # tl xs = xs" for x :: nat and xs by simp have 2: "xs ≠ [] ⟹ sum_list(tl xs) = sum_list xs - hd xs" for xs :: "nat list" by (auto simp add: neq_Nil_conv) have f: "bij_betw ?f ?A ?A'" by (rule bij_betw_byWitness[where f' = tl]) (use assms in ‹auto simp: 2 1 simp flip: length_0_conv›) have 3: "xs ≠ [] ⟹ hd xs + (sum_list xs - hd xs) = sum_list xs" for xs :: "nat list" by (metis 1 sum_list_simps(2) 2) have g: "bij_betw ?g ?B ?B'" apply (rule bij_betw_byWitness[where f' = "λl. (hd l - 1) # tl l"]) using assms by (auto simp: 2 simp flip: length_0_conv intro!: 3) have fin: "finite {xs. size xs = M ∧ set xs ⊆ {0..<N}}" for M N :: nat using finite_lists_length_eq[OF finite_atLeastLessThan] conj_commute by auto have fin_A: "finite ?A" using fin[of _ "N+1"] by (intro finite_subset[where ?A = "?A" and ?B = "{xs. size xs = m - 1 ∧ set xs ⊆ {0..<N+1}}"]) (auto simp: member_le_sum_list less_Suc_eq_le) have fin_B: "finite ?B" by (intro finite_subset[where ?A = "?B" and ?B = "{xs. size xs = m ∧ set xs ⊆ {0..<N}}"]) (auto simp: member_le_sum_list less_Suc_eq_le fin) have uni: "?C = ?A' ∪ ?B'" by auto have disj: "?A' ∩ ?B' = {}" by blast have "card ?C = card(?A' ∪ ?B')" using uni by simp also have "… = card ?A + card ?B" using card_Un_disjoint[OF _ _ disj] bij_betw_finite[OF f] bij_betw_finite[OF g] bij_betw_same_card[OF f] bij_betw_same_card[OF g] fin_A fin_B by presburger finally show ?thesis . qed lemma card_length_sum_list: "card {l::nat list. size l = m ∧ sum_list l = N} = (N + m - 1) choose N" ― ‹by Holden Lee, tidied by Tobias Nipkow› proof (cases m) case 0 then show ?thesis by (cases N) (auto cong: conj_cong) next case (Suc m') have m: "m ≥ 1" by (simp add: Suc) then show ?thesis proof (induct "N + m - 1" arbitrary: N m) case 0 ― ‹In the base case, the only solution is [0].› have [simp]: "{l::nat list. length l = Suc 0 ∧ (∀n∈set l. n = 0)} = {[0]}" by (auto simp: length_Suc_conv) have "m = 1 ∧ N = 0" using 0 by linarith then show ?case by simp next case (Suc k) have c1: "card {l::nat list. size l = (m - 1) ∧ sum_list l = N} = (N + (m - 1) - 1) choose N" proof (cases "m = 1") case True with Suc.hyps have "N ≥ 1" by auto with True show ?thesis by (simp add: binomial_eq_0) next case False then show ?thesis using Suc by fastforce qed from Suc have c2: "card {l::nat list. size l = m ∧ sum_list l + 1 = N} = (if N > 0 then ((N - 1) + m - 1) choose (N - 1) else 0)" proof - have *: "n > 0 ⟹ Suc m = n ⟷ m = n - 1" for m n by arith from Suc have "N > 0 ⟹ card {l::nat list. size l = m ∧ sum_list l + 1 = N} = ((N - 1) + m - 1) choose (N - 1)" by (simp add: *) then show ?thesis by auto qed from Suc.prems have "(card {l::nat list. size l = (m - 1) ∧ sum_list l = N} + card {l::nat list. size l = m ∧ sum_list l + 1 = N}) = (N + m - 1) choose N" by (auto simp: c1 c2 choose_reduce_nat[of "N + m - 1" N] simp del: One_nat_def) then show ?case using card_length_sum_list_rec[OF Suc.prems] by auto qed qed lemma card_disjoint_shuffles: assumes "set xs ∩ set ys = {}" shows "card (shuffles xs ys) = (length xs + length ys) choose length xs" using assms proof (induction xs ys rule: shuffles.induct) case (3 x xs y ys) have "shuffles (x # xs) (y # ys) = (#) x ` shuffles xs (y # ys) ∪ (#) y ` shuffles (x # xs) ys" by (rule shuffles.simps) also have "card … = card ((#) x ` shuffles xs (y # ys)) + card ((#) y ` shuffles (x # xs) ys)" by (rule card_Un_disjoint) (insert "3.prems", auto) also have "card ((#) x ` shuffles xs (y # ys)) = card (shuffles xs (y # ys))" by (rule card_image) auto also have "… = (length xs + length (y # ys)) choose length xs" using "3.prems" by (intro "3.IH") auto also have "card ((#) y ` shuffles (x # xs) ys) = card (shuffles (x # xs) ys)" by (rule card_image) auto also have "… = (length (x # xs) + length ys) choose length (x # xs)" using "3.prems" by (intro "3.IH") auto also have "length xs + length (y # ys) choose length xs + … = (length (x # xs) + length (y # ys)) choose length (x # xs)" by simp finally show ?case . qed auto lemma Suc_times_binomial_add: "Suc a * (Suc (a + b) choose Suc a) = Suc b * (Suc (a + b) choose a)" ― ‹by Lukas Bulwahn› proof - have dvd: "Suc a * (fact a * fact b) dvd fact (Suc (a + b))" for a b using fact_fact_dvd_fact[of "Suc a" "b", where 'a=nat] by (simp only: fact_Suc add_Suc[symmetric] of_nat_id mult.assoc) have "Suc a * (fact (Suc (a + b)) div (Suc a * fact a * fact b)) = Suc a * fact (Suc (a + b)) div (Suc a * (fact a * fact b))" by (subst div_mult_swap[symmetric]; simp only: mult.assoc dvd) also have "… = Suc b * fact (Suc (a + b)) div (Suc b * (fact a * fact b))" by (simp only: div_mult_mult1) also have "… = Suc b * (fact (Suc (a + b)) div (Suc b * (fact a * fact b)))" using dvd[of b a] by (subst div_mult_swap[symmetric]; simp only: ac_simps dvd) finally show ?thesis by (subst (1 2) binomial_altdef_nat) (simp_all only: ac_simps diff_Suc_Suc Suc_diff_le diff_add_inverse fact_Suc of_nat_id) qed subsection ‹Executable code› lemma gbinomial_code [code]: "a gchoose k = (if k = 0 then 1 else fold_atLeastAtMost_nat (λk acc. (a - of_nat k) * acc) 0 (k - 1) 1 / fact k)" by (cases k) (simp_all add: gbinomial_prod_rev prod_atLeastAtMost_code [symmetric] atLeastLessThanSuc_atLeastAtMost) lemma binomial_code [code]: "n choose k = (if k > n then 0 else if 2 * k > n then n choose (n - k) else (fold_atLeastAtMost_nat (*) (n - k + 1) n 1 div fact k))" proof - { assume "k ≤ n" then have "{1..n} = {1..n-k} ∪ {n-k+1..n}" by auto then have "(fact n :: nat) = fact (n-k) * ∏{n-k+1..n}" by (simp add: prod.union_disjoint fact_prod) } then show ?thesis by (auto simp: binomial_altdef_nat mult_ac prod_atLeastAtMost_code) qed end