# HG changeset patch # User paulson # Date 1426003955 0 # Node ID de7792ea409033a8c6e06d26620d3485c68c5e63 # Parent 1c937d56a70a639b55f36efc0dd731fd61cfc819 renaming HOL/Fact.thy -> Binomial.thy diff -r 1c937d56a70a -r de7792ea4090 src/HOL/Binomial.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Binomial.thy Tue Mar 10 16:12:35 2015 +0000 @@ -0,0 +1,1180 @@ +(* Title : Binomial.thy + Author : Jacques D. Fleuriot + Copyright : 1998 University of Cambridge + Conversion to Isar and new proofs by Lawrence C Paulson, 2004 + The integer version of factorial and other additions by Jeremy Avigad. +*) + +section{*Factorial Function, Binomial Coefficients and Binomial Theorem*} + +theory Binomial +imports Main +begin + +class fact = + fixes fact :: "'a \ 'a" + +instantiation nat :: fact +begin + +fun + fact_nat :: "nat \ nat" +where + fact_0_nat: "fact_nat 0 = Suc 0" +| fact_Suc: "fact_nat (Suc x) = Suc x * fact x" + +instance .. + +end + +(* definitions for the integers *) + +instantiation int :: fact + +begin + +definition + fact_int :: "int \ int" +where + "fact_int x = (if x >= 0 then int (fact (nat x)) else 0)" + +instance proof qed + +end + + +subsection {* Set up Transfer *} + +lemma transfer_nat_int_factorial: + "(x::int) >= 0 \ fact (nat x) = nat (fact x)" + unfolding fact_int_def + by auto + + +lemma transfer_nat_int_factorial_closure: + "x >= (0::int) \ fact x >= 0" + by (auto simp add: fact_int_def) + +declare transfer_morphism_nat_int[transfer add return: + transfer_nat_int_factorial transfer_nat_int_factorial_closure] + +lemma transfer_int_nat_factorial: + "fact (int x) = int (fact x)" + unfolding fact_int_def by auto + +lemma transfer_int_nat_factorial_closure: + "is_nat x \ fact x >= 0" + by (auto simp add: fact_int_def) + +declare transfer_morphism_int_nat[transfer add return: + transfer_int_nat_factorial transfer_int_nat_factorial_closure] + + +subsection {* Factorial *} + +lemma fact_0_int [simp]: "fact (0::int) = 1" + by (simp add: fact_int_def) + +lemma fact_1_nat [simp]: "fact (1::nat) = 1" + by simp + +lemma fact_Suc_0_nat [simp]: "fact (Suc 0) = Suc 0" + by simp + +lemma fact_1_int [simp]: "fact (1::int) = 1" + by (simp add: fact_int_def) + +lemma fact_plus_one_nat: "fact ((n::nat) + 1) = (n + 1) * fact n" + by simp + +lemma fact_plus_one_int: + assumes "n >= 0" + shows "fact ((n::int) + 1) = (n + 1) * fact n" + using assms unfolding fact_int_def + by (simp add: nat_add_distrib algebra_simps int_mult) + +lemma fact_reduce_nat: "(n::nat) > 0 \ fact n = n * fact (n - 1)" + apply (subgoal_tac "n = Suc (n - 1)") + apply (erule ssubst) + apply (subst fact_Suc) + apply simp_all + done + +lemma fact_reduce_int: "(n::int) > 0 \ fact n = n * fact (n - 1)" + apply (subgoal_tac "n = (n - 1) + 1") + apply (erule ssubst) + apply (subst fact_plus_one_int) + apply simp_all + done + +lemma fact_nonzero_nat [simp]: "fact (n::nat) \ 0" + apply (induct n) + apply (auto simp add: fact_plus_one_nat) + done + +lemma fact_nonzero_int [simp]: "n >= 0 \ fact (n::int) ~= 0" + by (simp add: fact_int_def) + +lemma fact_gt_zero_nat [simp]: "fact (n :: nat) > 0" + by (insert fact_nonzero_nat [of n], arith) + +lemma fact_gt_zero_int [simp]: "n >= 0 \ fact (n :: int) > 0" + by (auto simp add: fact_int_def) + +lemma fact_ge_one_nat [simp]: "fact (n :: nat) >= 1" + by (insert fact_nonzero_nat [of n], arith) + +lemma fact_ge_Suc_0_nat [simp]: "fact (n :: nat) >= Suc 0" + by (insert fact_nonzero_nat [of n], arith) + +lemma fact_ge_one_int [simp]: "n >= 0 \ fact (n :: int) >= 1" + apply (auto simp add: fact_int_def) + apply (subgoal_tac "1 = int 1") + apply (erule ssubst) + apply (subst zle_int) + apply auto + done + +lemma dvd_fact_nat [rule_format]: "1 <= m \ m <= n \ m dvd fact (n::nat)" + apply (induct n) + apply force + apply (auto simp only: fact_Suc) + apply (subgoal_tac "m = Suc n") + apply (erule ssubst) + apply (rule dvd_triv_left) + apply auto + done + +lemma dvd_fact_int [rule_format]: "1 <= m \ m <= n \ m dvd fact (n::int)" + apply (case_tac "1 <= n") + apply (induct n rule: int_ge_induct) + apply (auto simp add: fact_plus_one_int) + apply (subgoal_tac "m = i + 1") + apply auto + done + +lemma interval_plus_one_nat: "(i::nat) <= j + 1 \ + {i..j+1} = {i..j} Un {j+1}" + by auto + +lemma interval_Suc: "i <= Suc j \ {i..Suc j} = {i..j} Un {Suc j}" + by auto + +lemma interval_plus_one_int: "(i::int) <= j + 1 \ {i..j+1} = {i..j} Un {j+1}" + by auto + +lemma fact_altdef_nat: "fact (n::nat) = (PROD i:{1..n}. i)" + apply (induct n) + apply force + apply (subst fact_Suc) + apply (subst interval_Suc) + apply auto +done + +lemma fact_altdef_int: "n >= 0 \ fact (n::int) = (PROD i:{1..n}. i)" + apply (induct n rule: int_ge_induct) + apply force + apply (subst fact_plus_one_int, assumption) + apply (subst interval_plus_one_int) + apply auto +done + +lemma fact_dvd: "n \ m \ fact n dvd fact (m::nat)" + by (auto simp add: fact_altdef_nat intro!: setprod_dvd_setprod_subset) + +lemma fact_mod: "m \ (n::nat) \ fact n mod fact m = 0" + by (auto simp add: dvd_imp_mod_0 fact_dvd) + +lemma fact_div_fact: + assumes "m \ (n :: nat)" + shows "(fact m) div (fact n) = \{n + 1..m}" +proof - + obtain d where "d = m - n" by auto + from assms this have "m = n + d" by auto + have "fact (n + d) div (fact n) = \{n + 1..n + d}" + proof (induct d) + case 0 + show ?case by simp + next + case (Suc d') + have "fact (n + Suc d') div fact n = Suc (n + d') * fact (n + d') div fact n" + by simp + also from Suc.hyps have "... = Suc (n + d') * \{n + 1..n + d'}" + unfolding div_mult1_eq[of _ "fact (n + d')"] by (simp add: fact_mod) + also have "... = \{n + 1..n + Suc d'}" + by (simp add: atLeastAtMostSuc_conv setprod.insert) + finally show ?case . + qed + from this `m = n + d` show ?thesis by simp +qed + +lemma fact_mono_nat: "(m::nat) \ n \ fact m \ fact n" +apply (drule le_imp_less_or_eq) +apply (auto dest!: less_imp_Suc_add) +apply (induct_tac k, auto) +done + +lemma fact_neg_int [simp]: "m < (0::int) \ fact m = 0" + unfolding fact_int_def by auto + +lemma fact_ge_zero_int [simp]: "fact m >= (0::int)" + apply (case_tac "m >= 0") + apply auto + apply (frule fact_gt_zero_int) + apply arith +done + +lemma fact_mono_int_aux [rule_format]: "k >= (0::int) \ + fact (m + k) >= fact m" + apply (case_tac "m < 0") + apply auto + apply (induct k rule: int_ge_induct) + apply auto + apply (subst add.assoc [symmetric]) + apply (subst fact_plus_one_int) + apply auto + apply (erule order_trans) + apply (subst mult_le_cancel_right1) + apply (subgoal_tac "fact (m + i) >= 0") + apply arith + apply auto +done + +lemma fact_mono_int: "(m::int) <= n \ fact m <= fact n" + apply (insert fact_mono_int_aux [of "n - m" "m"]) + apply auto +done + +text{*Note that @{term "fact 0 = fact 1"}*} +lemma fact_less_mono_nat: "[| (0::nat) < m; m < n |] ==> fact m < fact n" +apply (drule_tac m = m in less_imp_Suc_add, auto) +apply (induct_tac k, auto) +done + +lemma fact_less_mono_int_aux: "k >= 0 \ (0::int) < m \ + fact m < fact ((m + 1) + k)" + apply (induct k rule: int_ge_induct) + apply (simp add: fact_plus_one_int) + apply (subst (2) fact_reduce_int) + apply (auto simp add: ac_simps) + apply (erule order_less_le_trans) + apply auto + done + +lemma fact_less_mono_int: "(0::int) < m \ m < n \ fact m < fact n" + apply (insert fact_less_mono_int_aux [of "n - (m + 1)" "m"]) + apply auto +done + +lemma fact_num_eq_if_nat: "fact (m::nat) = + (if m=0 then 1 else m * fact (m - 1))" +by (cases m) auto + +lemma fact_add_num_eq_if_nat: + "fact ((m::nat) + n) = (if m + n = 0 then 1 else (m + n) * fact (m + n - 1))" +by (cases "m + n") auto + +lemma fact_add_num_eq_if2_nat: + "fact ((m::nat) + n) = + (if m = 0 then fact n else (m + n) * fact ((m - 1) + n))" +by (cases m) auto + +lemma fact_le_power: "fact n \ n^n" +proof (induct n) + case (Suc n) + then have "fact n \ Suc n ^ n" by (rule le_trans) (simp add: power_mono) + then show ?case by (simp add: add_le_mono) +qed simp + +subsection {* @{term fact} and @{term of_nat} *} + +lemma of_nat_fact_not_zero [simp]: "of_nat (fact n) \ (0::'a::semiring_char_0)" +by auto + +lemma of_nat_fact_gt_zero [simp]: "(0::'a::{linordered_semidom}) < of_nat(fact n)" by auto + +lemma of_nat_fact_ge_zero [simp]: "(0::'a::linordered_semidom) \ of_nat(fact n)" +by simp + +lemma inv_of_nat_fact_gt_zero [simp]: "(0::'a::linordered_field) < inverse (of_nat (fact n))" +by (auto simp add: positive_imp_inverse_positive) + +lemma inv_of_nat_fact_ge_zero [simp]: "(0::'a::linordered_field) \ inverse (of_nat (fact n))" +by (auto intro: order_less_imp_le) + +lemma fact_eq_rev_setprod_nat: "fact (k::nat) = (\i n" shows "fact n div fact (n - r) \ n ^ r" +proof - + have "\r. r \ n \ \{n - r..n} = (n - r) * \{Suc (n - r)..n}" + by (subst setprod.insert[symmetric]) (auto simp: atLeastAtMost_insertL) + with assms show ?thesis + by (induct r rule: nat.induct) (auto simp add: fact_div_fact Suc_diff_Suc mult_le_mono) +qed + +lemma fact_numeral: --{*Evaluation for specific numerals*} + "fact (numeral k) = (numeral k) * (fact (pred_numeral k))" + by (simp add: numeral_eq_Suc) + + +text {* This development is based on the work of Andy Gordon and + Florian Kammueller. *} + +subsection {* Basic definitions and lemmas *} + +primrec binomial :: "nat \ nat \ nat" (infixl "choose" 65) +where + "0 choose k = (if k = 0 then 1 else 0)" +| "Suc n choose k = (if k = 0 then 1 else (n choose (k - 1)) + (n choose k))" + +lemma binomial_n_0 [simp]: "(n choose 0) = 1" + by (cases n) simp_all + +lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0" + by simp + +lemma binomial_Suc_Suc [simp]: "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)" + by simp + +lemma choose_reduce_nat: + "0 < (n::nat) \ 0 < k \ + (n choose k) = ((n - 1) choose (k - 1)) + ((n - 1) choose k)" + by (metis Suc_diff_1 binomial.simps(2) neq0_conv) + +lemma binomial_eq_0: "n < k \ n choose k = 0" + by (induct n arbitrary: k) auto + +declare binomial.simps [simp del] + +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 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 Suc_times_binomial_eq: + "Suc n * (n choose k) = (Suc n choose Suc k) * Suc k" + apply (induct n arbitrary: k, simp add: binomial.simps) + apply (case_tac k) + apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq binomial_eq_0) + done + +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 add: nat_diff_split) + + +subsection {* Combinatorial theorems involving @{text "choose"} *} + +text {*By Florian Kamm\"uller, tidied by LCP.*} + +lemma card_s_0_eq_empty: "finite A \ card {B. B \ A & card B = 0} = 1" + by (simp cong add: conj_cong add: finite_subset [THEN card_0_eq]) + +lemma choose_deconstruct: "finite M \ x \ M \ + {s. s \ insert x M \ card s = Suc k} = + {s. s \ M \ card s = Suc k} \ {s. \t. t \ M \ card t = k \ s = insert x t}" + apply safe + apply (auto intro: finite_subset [THEN card_insert_disjoint]) + by (metis (full_types) Diff_insert_absorb Set.set_insert Zero_neq_Suc card_Diff_singleton_if + card_eq_0_iff diff_Suc_1 in_mono subset_insert_iff) + +lemma finite_bex_subset [simp]: + assumes "finite B" + and "\A. A \ B \ finite {x. P x A}" + shows "finite {x. \A \ B. P x A}" + by (metis (no_types) assms finite_Collect_bounded_ex finite_Collect_subsets) + +text{*There are as many subsets of @{term A} having cardinality @{term k} + as there are sets obtained from the former by inserting a fixed element + @{term x} into each.*} +lemma constr_bij: + "finite A \ x \ A \ + card {B. \C. C \ A \ card C = k \ B = insert x C} = + card {B. B \ A & card(B) = k}" + apply (rule card_bij_eq [where f = "\s. s - {x}" and g = "insert x"]) + apply (auto elim!: equalityE simp add: inj_on_def) + apply (metis card_Diff_singleton_if finite_subset in_mono) + done + +text {* + Main theorem: combinatorial statement about number of subsets of a set. +*} + +theorem n_subsets: "finite A \ card {B. B \ A \ card B = k} = (card A choose k)" +proof (induct k arbitrary: A) + case 0 then show ?case by (simp add: card_s_0_eq_empty) +next + case (Suc k) + show ?case using `finite A` + proof (induct A) + case empty show ?case by (simp add: card_s_0_eq_empty) + next + case (insert x A) + then show ?case using Suc.hyps + apply (simp add: card_s_0_eq_empty choose_deconstruct) + apply (subst card_Un_disjoint) + prefer 4 apply (force simp add: constr_bij) + prefer 3 apply force + prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2] + finite_subset [of _ "Pow (insert x F)" for F]) + apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset]) + done + qed +qed + + +subsection {* The binomial theorem (courtesy of Tobias Nipkow): *} + +text{* Avigad's version, generalized to any commutative ring *} +theorem binomial_ring: "(a+b::'a::{comm_ring_1,power})^n = + (\k=0..n. (of_nat (n choose k)) * a^k * b^(n-k))" (is "?P n") +proof (induct n) + case 0 then show "?P 0" by simp +next + case (Suc n) + have decomp: "{0..n+1} = {0} Un {n+1} Un {1..n}" + by auto + have decomp2: "{0..n} = {0} Un {1..n}" + by auto + have "(a+b)^(n+1) = + (a+b) * (\k=0..n. of_nat (n choose k) * a^k * b^(n-k))" + using Suc.hyps by simp + also have "\ = a*(\k=0..n. of_nat (n choose k) * a^k * b^(n-k)) + + b*(\k=0..n. of_nat (n choose k) * a^k * b^(n-k))" + by (rule distrib_right) + also have "\ = (\k=0..n. of_nat (n choose k) * a^(k+1) * b^(n-k)) + + (\k=0..n. of_nat (n choose k) * a^k * b^(n-k+1))" + by (auto simp add: setsum_right_distrib ac_simps) + also have "\ = (\k=0..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:setsum_shift_bounds_cl_Suc_ivl Suc_diff_le field_simps + del:setsum_cl_ivl_Suc) + also have "\ = a^(n+1) + b^(n+1) + + (\k=1..n. of_nat (n choose (k - 1)) * a^k * b^(n+1-k)) + + (\k=1..n. of_nat (n choose k) * a^k * b^(n+1-k))" + by (simp add: decomp2) + 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 setsum.distrib [symmetric] choose_reduce_nat) + also have "\ = (\k=0..n+1. of_nat (n+1 choose k) * a^k * b^(n+1-k))" + using decomp by (simp add: field_simps) + finally show "?P (Suc n)" by simp +qed + +text{* Original version for the naturals *} +corollary binomial: "(a+b::nat)^n = (\k=0..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_setsum [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: "\mx\m. fact x * fact (m - x) * (m choose x) = + fact m" and kn: "k \ n" + let ?ths = "fact k * fact (n - k) * (n choose k) = fact n" + { assume "n=0" then have ?ths using kn by simp } + moreover + { assume "k=0" then have ?ths using kn by simp } + moreover + { assume nk: "n=k" then have ?ths by simp } + moreover + { fix m h assume n: "n = Suc m" and h: "k = Suc h" and hm: "h < m" + from n have mn: "m < n" by arith + from hm have hm': "h \ m" by arith + from hm h n kn have km: "k \ m" by arith + have "m - h = Suc (m - Suc h)" using h km hm by arith + with km h have th0: "fact (m - h) = (m - h) * fact (m - k)" + by simp + from n h th0 + 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 have ?ths using h n km by simp } + moreover have "n=0 \ k = 0 \ k = n \ (\m h. n = Suc m \ k = Suc h \ h < m)" + using kn by presburger + ultimately show ?ths by blast +qed + +lemma binomial_fact: + assumes kn: "k \ n" + shows "(of_nat (n choose k) :: 'a::{field,ring_char_0}) = + of_nat (fact n) / (of_nat (fact k) * of_nat (fact (n - k)))" + using binomial_fact_lemma[OF kn] + by (simp add: field_simps of_nat_mult [symmetric]) + +lemma choose_row_sum: "(\k=0..n. n choose k) = 2^n" + using binomial [of 1 "1" n] + by (simp add: numeral_2_eq_2) + +lemma sum_choose_lower: "(\k=0..n. (r+k) choose k) = Suc (r+n) choose n" + by (induct n) auto + +lemma sum_choose_upper: "(\k=0..n. k choose m) = Suc n choose Suc m" + by (induct n) auto + +lemma natsum_reverse_index: + fixes m::nat + shows "(\k. m \ k \ k \ n \ g k = f (m + n - k)) \ (\k=m..n. f k) = (\k=m..n. g k)" + by (rule setsum.reindex_bij_witness[where i="\k. m+n-k" and j="\k. m+n-k"]) auto + +text{*NW diagonal sum property*} +lemma sum_choose_diagonal: + assumes "m\n" shows "(\k=0..m. (n-k) choose (m-k)) = Suc n choose m" +proof - + have "(\k=0..m. (n-k) choose (m-k)) = (\k=0..m. (n-m+k) choose k)" + by (rule natsum_reverse_index) (simp add: assms) + 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{* Pochhammer's symbol : generalized rising factorial *} + +text {* See @{url "http://en.wikipedia.org/wiki/Pochhammer_symbol"} *} + +definition "pochhammer (a::'a::comm_semiring_1) n = + (if n = 0 then 1 else setprod (\n. a + of_nat n) {0 .. n - 1})" + +lemma pochhammer_0 [simp]: "pochhammer a 0 = 1" + by (simp add: pochhammer_def) + +lemma pochhammer_1 [simp]: "pochhammer a 1 = a" + by (simp add: pochhammer_def) + +lemma pochhammer_Suc0 [simp]: "pochhammer a (Suc 0) = a" + by (simp add: pochhammer_def) + +lemma pochhammer_Suc_setprod: "pochhammer a (Suc n) = setprod (\n. a + of_nat n) {0 .. n}" + by (simp add: pochhammer_def) + +lemma setprod_nat_ivl_Suc: "setprod f {0 .. Suc n} = setprod f {0..n} * f (Suc n)" +proof - + have "{0..Suc n} = {0..n} \ {Suc n}" by auto + then show ?thesis by (simp add: field_simps) +qed + +lemma setprod_nat_ivl_1_Suc: "setprod f {0 .. Suc n} = f 0 * setprod f {1.. Suc n}" +proof - + have "{0..Suc n} = {0} \ {1 .. Suc n}" by auto + then show ?thesis by simp +qed + + +lemma pochhammer_Suc: "pochhammer a (Suc n) = pochhammer a n * (a + of_nat n)" +proof (cases n) + case 0 + then show ?thesis by simp +next + case (Suc n) + show ?thesis unfolding Suc pochhammer_Suc_setprod setprod_nat_ivl_Suc .. +qed + +lemma pochhammer_rec: "pochhammer a (Suc n) = a * pochhammer (a + 1) n" +proof (cases "n = 0") + case True + then show ?thesis by (simp add: pochhammer_Suc_setprod) +next + case False + have *: "finite {1 .. n}" "0 \ {1 .. n}" by auto + have eq: "insert 0 {1 .. n} = {0..n}" by auto + have **: "(\n\{1\nat..n}. a + of_nat n) = (\n\{0\nat..n - 1}. a + 1 + of_nat n)" + apply (rule setprod.reindex_cong [where l = Suc]) + using False + apply (auto simp add: fun_eq_iff field_simps) + done + show ?thesis + apply (simp add: pochhammer_def) + unfolding setprod.insert [OF *, unfolded eq] + using ** apply (simp add: field_simps) + done +qed + +lemma pochhammer_fact: "of_nat (fact n) = pochhammer 1 n" + unfolding fact_altdef_nat + apply (cases n) + apply (simp_all add: of_nat_setprod pochhammer_Suc_setprod) + apply (rule setprod.reindex_cong [where l = Suc]) + apply (auto simp add: fun_eq_iff) + done + +lemma pochhammer_of_nat_eq_0_lemma: + assumes "k > n" + shows "pochhammer (- (of_nat n :: 'a:: idom)) k = 0" +proof (cases "n = 0") + case True + then show ?thesis + using assms by (cases k) (simp_all add: pochhammer_rec) +next + case False + from assms obtain h where "k = Suc h" by (cases k) auto + then show ?thesis + by (simp add: pochhammer_Suc_setprod) + (metis Suc_leI Suc_le_mono assms atLeastAtMost_iff less_eq_nat.simps(1)) +qed + +lemma pochhammer_of_nat_eq_0_lemma': + assumes kn: "k \ n" + shows "pochhammer (- (of_nat n :: 'a:: {idom,ring_char_0})) k \ 0" +proof (cases k) + case 0 + then show ?thesis by simp +next + case (Suc h) + then show ?thesis + apply (simp add: pochhammer_Suc_setprod) + using Suc kn apply (auto simp add: algebra_simps) + done +qed + +lemma pochhammer_of_nat_eq_0_iff: + shows "pochhammer (- (of_nat n :: 'a:: {idom,ring_char_0})) k = 0 \ k > n" + (is "?l = ?r") + using pochhammer_of_nat_eq_0_lemma[of n k, where ?'a='a] + pochhammer_of_nat_eq_0_lemma'[of k n, where ?'a = 'a] + by (auto simp add: not_le[symmetric]) + +lemma pochhammer_eq_0_iff: "pochhammer a n = (0::'a::field_char_0) \ (\k < n. a = - of_nat k)" + apply (auto simp add: pochhammer_of_nat_eq_0_iff) + apply (cases n) + apply (auto simp add: pochhammer_def algebra_simps group_add_class.eq_neg_iff_add_eq_0) + apply (metis leD not_less_eq) + done + +lemma pochhammer_eq_0_mono: + "pochhammer a n = (0::'a::field_char_0) \ m \ n \ pochhammer a m = 0" + unfolding pochhammer_eq_0_iff by auto + +lemma pochhammer_neq_0_mono: + "pochhammer a m \ (0::'a::field_char_0) \ m \ n \ pochhammer a n \ 0" + unfolding pochhammer_eq_0_iff by auto + +lemma pochhammer_minus: + assumes kn: "k \ n" + shows "pochhammer (- b) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (b - of_nat k + 1) k" +proof (cases k) + case 0 + then show ?thesis by simp +next + case (Suc h) + have eq: "((- 1) ^ Suc h :: 'a) = (\i=0..h. - 1)" + using setprod_constant[where A="{0 .. h}" and y="- 1 :: 'a"] + by auto + show ?thesis + unfolding Suc pochhammer_Suc_setprod eq setprod.distrib[symmetric] + by (rule setprod.reindex_bij_witness[where i="op - h" and j="op - h"]) + (auto simp: of_nat_diff) +qed + +lemma pochhammer_minus': + assumes kn: "k \ n" + shows "pochhammer (b - of_nat k + 1) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (- b) k" + unfolding pochhammer_minus[OF kn, where b=b] + unfolding mult.assoc[symmetric] + unfolding power_add[symmetric] + by simp + +lemma pochhammer_same: "pochhammer (- of_nat n) n = + ((- 1) ^ n :: 'a::comm_ring_1) * of_nat (fact n)" + unfolding pochhammer_minus[OF le_refl[of n]] + by (simp add: of_nat_diff pochhammer_fact) + + +subsection{* Generalized binomial coefficients *} + +definition gbinomial :: "'a::field_char_0 \ nat \ 'a" (infixl "gchoose" 65) + where "a gchoose n = + (if n = 0 then 1 else (setprod (\i. a - of_nat i) {0 .. n - 1}) / of_nat (fact n))" + +lemma gbinomial_0 [simp]: "a gchoose 0 = 1" "0 gchoose (Suc n) = 0" + apply (simp_all add: gbinomial_def) + apply (subgoal_tac "(\i\nat\{0\nat..n}. - of_nat i) = (0::'b)") + apply (simp del:setprod_zero_iff) + apply simp + done + +lemma gbinomial_pochhammer: "a gchoose n = (- 1) ^ n * pochhammer (- a) n / of_nat (fact n)" +proof (cases "n = 0") + case True + then show ?thesis by simp +next + case False + from this setprod_constant[of "{0 .. n - 1}" "- (1:: 'a)"] + have eq: "(- (1\'a)) ^ n = setprod (\i. - 1) {0 .. n - 1}" + by auto + from False show ?thesis + by (simp add: pochhammer_def gbinomial_def field_simps + eq setprod.distrib[symmetric]) +qed + +lemma binomial_gbinomial: "of_nat (n choose k) = of_nat n gchoose k" +proof - + { assume kn: "k > n" + then have ?thesis + by (subst binomial_eq_0[OF kn]) + (simp add: gbinomial_pochhammer field_simps pochhammer_of_nat_eq_0_iff) } + moreover + { assume "k=0" then have ?thesis by simp } + moreover + { assume kn: "k \ n" and k0: "k\ 0" + from k0 obtain h where h: "k = Suc h" by (cases k) auto + from h + have eq:"(- 1 :: 'a) ^ k = setprod (\i. - 1) {0..h}" + by (subst setprod_constant) auto + have eq': "(\i\{0..h}. of_nat n + - (of_nat i :: 'a)) = (\i\{n - h..n}. of_nat i)" + using h kn + by (intro setprod.reindex_bij_witness[where i="op - n" and j="op - n"]) + (auto simp: of_nat_diff) + have th0: "finite {1..n - Suc h}" "finite {n - h .. n}" + "{1..n - Suc h} \ {n - h .. n} = {}" and + eq3: "{1..n - Suc h} \ {n - h .. n} = {1..n}" + using h kn by auto + from eq[symmetric] + have ?thesis using kn + apply (simp add: binomial_fact[OF kn, where ?'a = 'a] + gbinomial_pochhammer field_simps pochhammer_Suc_setprod) + apply (simp add: pochhammer_Suc_setprod fact_altdef_nat h + of_nat_setprod setprod.distrib[symmetric] eq' del: One_nat_def power_Suc) + unfolding setprod.union_disjoint[OF th0, unfolded eq3, of "of_nat:: nat \ 'a"] eq[unfolded h] + unfolding mult.assoc[symmetric] + unfolding setprod.distrib[symmetric] + apply simp + apply (intro setprod.reindex_bij_witness[where i="op - n" and j="op - n"]) + apply (auto simp: of_nat_diff) + done + } + moreover + have "k > n \ k = 0 \ (k \ n \ k \ 0)" by arith + ultimately show ?thesis by blast +qed + +lemma gbinomial_1[simp]: "a gchoose 1 = a" + by (simp add: gbinomial_def) + +lemma gbinomial_Suc0[simp]: "a gchoose (Suc 0) = a" + by (simp add: gbinomial_def) + +lemma gbinomial_mult_1: + "a * (a gchoose n) = + of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))" (is "?l = ?r") +proof - + have "?r = ((- 1) ^n * pochhammer (- a) n / of_nat (fact n)) * (of_nat n - (- a + of_nat n))" + unfolding gbinomial_pochhammer + pochhammer_Suc fact_Suc of_nat_mult right_diff_distrib power_Suc + by (simp add: field_simps del: of_nat_Suc) + also have "\ = ?l" unfolding gbinomial_pochhammer + by (simp add: field_simps) + finally show ?thesis .. +qed + +lemma gbinomial_mult_1': + "(a gchoose n) * a = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))" + by (simp add: mult.commute gbinomial_mult_1) + +lemma gbinomial_Suc: + "a gchoose (Suc k) = (setprod (\i. a - of_nat i) {0 .. k}) / of_nat (fact (Suc k))" + by (simp add: gbinomial_def) + +lemma gbinomial_mult_fact: + "(of_nat (fact (Suc k)) :: 'a) * ((a::'a::field_char_0) gchoose (Suc k)) = + (setprod (\i. a - of_nat i) {0 .. k})" + by (simp_all add: gbinomial_Suc field_simps del: fact_Suc) + +lemma gbinomial_mult_fact': + "((a::'a::field_char_0) gchoose (Suc k)) * (of_nat (fact (Suc k)) :: 'a) = + (setprod (\i. a - of_nat i) {0 .. k})" + using gbinomial_mult_fact[of k a] + by (subst mult.commute) + + +lemma gbinomial_Suc_Suc: + "((a::'a::field_char_0) + 1) gchoose (Suc k) = a gchoose k + (a gchoose (Suc k))" +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)" + apply (rule setprod.reindex_cong [where l = Suc]) + using Suc + apply auto + done + have "of_nat (fact (Suc k)) * (a gchoose k + (a gchoose (Suc k))) = + ((a gchoose Suc h) * of_nat (fact (Suc h)) * of_nat (Suc k)) + (\i\{0\nat..Suc h}. a - of_nat i)" + apply (simp add: Suc field_simps del: fact_Suc) + unfolding gbinomial_mult_fact' + apply (subst fact_Suc) + unfolding of_nat_mult + apply (subst mult.commute) + unfolding mult.assoc + unfolding gbinomial_mult_fact + apply (simp add: field_simps) + done + also have "\ = (\i\{0..h}. a - of_nat i) * (a + 1)" + unfolding gbinomial_mult_fact' setprod_nat_ivl_Suc + by (simp add: field_simps Suc) + also have "\ = (\i\{0..k}. (a + 1) - of_nat i)" + using eq0 + by (simp add: Suc setprod_nat_ivl_1_Suc) + also have "\ = of_nat (fact (Suc k)) * ((a + 1) gchoose (Suc k))" + unfolding gbinomial_mult_fact .. + finally show ?thesis by (simp del: fact_Suc) +qed + +lemma gbinomial_reduce_nat: + "0 < k \ (a::'a::field_char_0) gchoose k = (a - 1) gchoose (k - 1) + ((a - 1) gchoose k)" +by (metis Suc_pred' diff_add_cancel gbinomial_Suc_Suc) + + +lemma binomial_symmetric: + assumes kn: "k \ n" + shows "n choose k = n choose (n - k)" +proof- + from kn have kn': "n - k \ n" 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 + +text{*Contributed by Manuel Eberl, generalised by LCP. + Alternative definition of the binomial coefficient as @{term "\iii = (\i x" + shows "(x / of_nat k :: 'a) ^ k \ x gchoose k" +proof - + have x: "0 \ x" + using assms of_nat_0_le_iff order_trans by blast + have "(x / of_nat k :: 'a) ^ k = (\i \ x gchoose k" + unfolding gbinomial_altdef_of_nat + proof (safe intro!: setprod_mono) + fix i :: nat + assume ik: "i < k" + from assms have "x * 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 "x * of_nat k - x * of_nat i \ x * of_nat k - of_nat (i * k)" by arith + then have "x * of_nat (k - i) \ (x - of_nat i) * of_nat k" + using ik + by (simp add: algebra_simps zero_less_mult_iff of_nat_diff of_nat_mult) + then have "x * of_nat (k - i) \ (x - of_nat i) * (of_nat k :: 'a)" + unfolding of_nat_mult[symmetric] of_nat_le_iff . + with assms show "x / of_nat k \ (x - of_nat i) / (of_nat (k - i) :: 'a)" + using `i < k` by (simp add: field_simps) + qed (simp add: x zero_le_divide_iff) + finally show ?thesis . +qed + +text{*Versions of the theorems above for the natural-number version of "choose"*} +lemma binomial_altdef_of_nat: + fixes n k :: nat + and x :: "'a :: {field_char_0,field_inverse_zero}" --{*the point is to constrain @{typ 'a}*} + assumes "k \ n" + shows "of_nat (n choose k) = (\i n" + shows "(of_nat n / of_nat k :: 'a) ^ k \ of_nat (n choose k)" +by (simp add: assms 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 `r \ n` 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::nat) \ n \ + n choose k = fact n div (fact k * fact (n - k))" + by (subst binomial_fact_lemma [symmetric]) auto + +lemma choose_dvd_nat: "(k::nat) \ n \ fact k * fact (n - k) dvd fact n" +by (metis binomial_fact_lemma dvd_def) + +lemma choose_dvd_int: + assumes "(0::int) <= k" and "k <= n" + shows "fact k * fact (n - k) dvd fact n" + apply (subst tsub_eq [symmetric], rule assms) + apply (rule choose_dvd_nat [transferred]) + using assms apply auto + done + +lemma fact_fact_dvd_fact: fixes k::nat shows "fact k * fact n dvd fact (n + k)" +by (metis add.commute add_diff_cancel_left' choose_dvd_nat 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)" +proof - + have "((m+r+k) choose (m+k)) * ((m+k) choose k) = + fact (m+r + k) div (fact (m + k) * fact (m+r - m)) * (fact (m + k) div (fact k * fact m))" + by (simp add: assms binomial_altdef_nat) + also have "... = fact (m+r+k) div (fact r * (fact k * fact m))" + apply (subst div_mult_div_if_dvd) + apply (auto simp: fact_fact_dvd_fact) + apply (metis add.assoc add.commute fact_fact_dvd_fact) + done + 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: fact_fact_dvd_fact) + apply (metis dvd_trans dvd.dual_order.refl fact_fact_dvd_fact mult_dvd_mono mult.left_commute) + done + also have "... = (fact (m+r+k) div (fact k * fact (m+r)) * (fact (m+r) div (fact r * fact m)))" + apply (subst div_mult_div_if_dvd) + apply (auto simp: fact_fact_dvd_fact) + apply(metis mult.left_commute) + done + finally show ?thesis + by (simp add: binomial_altdef_nat mult.commute) +qed + +text{*The "Subset of a Subset" identity*} +lemma choose_mult: + assumes "k\m" "m\n" + shows "(n choose m) * (m choose k) = (n choose k) * ((n-k) choose (m-k))" +using assms choose_mult_lemma [of "m-k" "n-m" k] +by simp + + +subsection {* Binomial coefficients *} + +lemma choose_one: "(n::nat) choose 1 = n" + by simp + +(*FIXME: messy and apparently unused*) +lemma binomial_induct [rule_format]: "(ALL (n::nat). P n n) \ + (ALL n. P (Suc n) 0) \ (ALL n. (ALL k < n. P n k \ P n (Suc k) \ + P (Suc n) (Suc k))) \ (ALL k <= n. P n k)" + apply (induct n) + apply auto + apply (case_tac "k = 0") + apply auto + apply (case_tac "k = Suc n") + apply auto + apply (metis Suc_le_eq fact_nat.cases le_Suc_eq le_eq_less_or_eq) + done + +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 setsum_right_distrib) simp + also have "?rhs = (\(I, _)\Sigma {I. I \ A \ I \ {}} Inter. (- 1) ^ (card I + 1))" + using assms by(subst setsum.Sigma)(auto) + also have "\ = (\(x, I)\(SIGMA x:UNIV. {I. I \ A \ I \ {} \ x \ \I}). (- 1) ^ (card I + 1))" + by (rule setsum.reindex_cong [where l = "\(x, y). (y, x)"]) (auto intro: inj_onI simp add: split_beta) + also have "\ = (\(x, I)\(SIGMA x:\A. {I. I \ A \ I \ {} \ x \ \I}). (- 1) ^ (card I + 1))" + using assms by(auto intro!: setsum.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 setsum.Sigma) auto + also have "\ = (\_\\A. 1)" (is "setsum ?lhs _ = _") + proof(rule setsum.cong[OF refl]) + fix x + assume x: "x \ \A" + def K \ "{X \ A. x \ X}" + with `finite A` have K: "finite K" by auto + let ?I = "\i. {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 setsum.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 setsum.Sigma) auto + also have "\ = (\i=1..card A. (- 1) ^ (i + 1) * (\I|I \ A \ card I = i \ x \ \I. 1))" + by(subst setsum_right_distrib) simp + also have "\ = (\i=1..card K. (- 1) ^ (i + 1) * (\I|I \ K \ card I = i. 1))" (is "_ = ?rhs") + proof(rule setsum.mono_neutral_cong_right[rule_format]) + 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}" + hence 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 setsum.cong)(auto simp add: K_def) + thus "(- 1) ^ (i + 1) * ?lhs = (- 1) ^ (i + 1) * ?rhs" by simp + qed simp + also have "{I. I \ K \ card I = 0} = {{}}" using assms + by(auto simp add: card_eq_0_iff K_def dest: finite_subset) + hence "?rhs = (\i = 0..card K. (- 1) ^ (i + 1) * (\I | I \ K \ card I = i. 1 :: int)) + 1" + by(subst (2) setsum_head_Suc)(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 setsum_right_distrib[symmetric]) simp + also have "\ = - ((-1 + 1) ^ card K) + 1" + by(subst binomial_ring)(simp add: ac_simps) + 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 @{text m} summing to @{text N} is +@{term "(N + m - 1) choose N"}: *} + +lemma card_length_listsum_rec: + assumes "m\1" + shows "card {l::nat list. length l = m \ listsum l = N} = + (card {l. length l = (m - 1) \ listsum l = N} + + card {l. length l = m \ listsum l + 1 = N})" + (is "card ?C = (card ?A + card ?B)") +proof - + let ?A'="{l. length l = m \ listsum l = N \ hd l = 0}" + let ?B'="{l. length l = m \ listsum l = N \ hd l \ 0}" + let ?f ="\ l. 0#l" + let ?g ="\ l. (hd l + 1) # tl l" + have 1: "\xs x. xs \ [] \ x = hd xs \ x # tl xs = xs" by simp + have 2: "\xs. (xs::nat list) \ [] \ listsum(tl xs) = listsum xs - hd xs" + by(auto simp add: neq_Nil_conv) + have f: "bij_betw ?f ?A ?A'" + apply(rule bij_betw_byWitness[where f' = tl]) + using assms + by (auto simp: 2 length_0_conv[symmetric] 1 simp del: length_0_conv) + have 3: "\xs:: nat list. xs \ [] \ hd xs + (listsum xs - hd xs) = listsum xs" + by (metis 1 listsum_simps(2) 2) + have g: "bij_betw ?g ?B ?B'" + apply(rule bij_betw_byWitness[where f' = "\ l. (hd l - 1) # tl l"]) + using assms + by (auto simp: 2 length_0_conv[symmetric] intro!: 3 + simp del: length_greater_0_conv length_0_conv) + { fix M N :: nat have "finite {xs. size xs = M \ set xs \ {0.. ?B'" by auto + have disj: "?A' \ ?B' = {}" by auto + 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_listsum: --"By Holden Lee, tidied by Tobias Nipkow" + "card {l::nat list. size l = m \ listsum l = N} = (N + m - 1) choose N" +proof (cases m) + case 0 then show ?thesis + by (cases N) (auto simp: 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) \ listsum l = N} = + (N + (m - 1) - 1) choose N" + proof cases + assume "m = 1" + with Suc.hyps have "N\1" by auto + with `m = 1` show ?thesis by (simp add: binomial_eq_0) + next + assume "m \ 1" thus ?thesis using Suc by fastforce + qed + + from Suc have c2: "card {l::nat list. size l = m \ listsum l + 1 = N} = + (if N>0 then ((N - 1) + m - 1) choose (N - 1) else 0)" + proof - + have aux: "\m n. n > 0 \ Suc m = n \ m = n - 1" by arith + from Suc have "N>0 \ + card {l::nat list. size l = m \ listsum l + 1 = N} = + ((N - 1) + m - 1) choose (N - 1)" by (simp add: aux) + thus ?thesis by auto + qed + + from Suc.prems have "(card {l::nat list. size l = (m - 1) \ listsum l = N} + + card {l::nat list. size l = m \ listsum 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) + thus ?case using card_length_listsum_rec[OF Suc.prems] by auto + qed +qed + +end diff -r 1c937d56a70a -r de7792ea4090 src/HOL/Fact.thy --- a/src/HOL/Fact.thy Tue Mar 10 15:21:26 2015 +0000 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,1180 +0,0 @@ -(* Title : Fact.thy - Author : Jacques D. Fleuriot - Copyright : 1998 University of Cambridge - Conversion to Isar and new proofs by Lawrence C Paulson, 2004 - The integer version of factorial and other additions by Jeremy Avigad. -*) - -section{*Factorial Function*} - -theory Fact -imports Main -begin - -class fact = - fixes fact :: "'a \ 'a" - -instantiation nat :: fact -begin - -fun - fact_nat :: "nat \ nat" -where - fact_0_nat: "fact_nat 0 = Suc 0" -| fact_Suc: "fact_nat (Suc x) = Suc x * fact x" - -instance .. - -end - -(* definitions for the integers *) - -instantiation int :: fact - -begin - -definition - fact_int :: "int \ int" -where - "fact_int x = (if x >= 0 then int (fact (nat x)) else 0)" - -instance proof qed - -end - - -subsection {* Set up Transfer *} - -lemma transfer_nat_int_factorial: - "(x::int) >= 0 \ fact (nat x) = nat (fact x)" - unfolding fact_int_def - by auto - - -lemma transfer_nat_int_factorial_closure: - "x >= (0::int) \ fact x >= 0" - by (auto simp add: fact_int_def) - -declare transfer_morphism_nat_int[transfer add return: - transfer_nat_int_factorial transfer_nat_int_factorial_closure] - -lemma transfer_int_nat_factorial: - "fact (int x) = int (fact x)" - unfolding fact_int_def by auto - -lemma transfer_int_nat_factorial_closure: - "is_nat x \ fact x >= 0" - by (auto simp add: fact_int_def) - -declare transfer_morphism_int_nat[transfer add return: - transfer_int_nat_factorial transfer_int_nat_factorial_closure] - - -subsection {* Factorial *} - -lemma fact_0_int [simp]: "fact (0::int) = 1" - by (simp add: fact_int_def) - -lemma fact_1_nat [simp]: "fact (1::nat) = 1" - by simp - -lemma fact_Suc_0_nat [simp]: "fact (Suc 0) = Suc 0" - by simp - -lemma fact_1_int [simp]: "fact (1::int) = 1" - by (simp add: fact_int_def) - -lemma fact_plus_one_nat: "fact ((n::nat) + 1) = (n + 1) * fact n" - by simp - -lemma fact_plus_one_int: - assumes "n >= 0" - shows "fact ((n::int) + 1) = (n + 1) * fact n" - using assms unfolding fact_int_def - by (simp add: nat_add_distrib algebra_simps int_mult) - -lemma fact_reduce_nat: "(n::nat) > 0 \ fact n = n * fact (n - 1)" - apply (subgoal_tac "n = Suc (n - 1)") - apply (erule ssubst) - apply (subst fact_Suc) - apply simp_all - done - -lemma fact_reduce_int: "(n::int) > 0 \ fact n = n * fact (n - 1)" - apply (subgoal_tac "n = (n - 1) + 1") - apply (erule ssubst) - apply (subst fact_plus_one_int) - apply simp_all - done - -lemma fact_nonzero_nat [simp]: "fact (n::nat) \ 0" - apply (induct n) - apply (auto simp add: fact_plus_one_nat) - done - -lemma fact_nonzero_int [simp]: "n >= 0 \ fact (n::int) ~= 0" - by (simp add: fact_int_def) - -lemma fact_gt_zero_nat [simp]: "fact (n :: nat) > 0" - by (insert fact_nonzero_nat [of n], arith) - -lemma fact_gt_zero_int [simp]: "n >= 0 \ fact (n :: int) > 0" - by (auto simp add: fact_int_def) - -lemma fact_ge_one_nat [simp]: "fact (n :: nat) >= 1" - by (insert fact_nonzero_nat [of n], arith) - -lemma fact_ge_Suc_0_nat [simp]: "fact (n :: nat) >= Suc 0" - by (insert fact_nonzero_nat [of n], arith) - -lemma fact_ge_one_int [simp]: "n >= 0 \ fact (n :: int) >= 1" - apply (auto simp add: fact_int_def) - apply (subgoal_tac "1 = int 1") - apply (erule ssubst) - apply (subst zle_int) - apply auto - done - -lemma dvd_fact_nat [rule_format]: "1 <= m \ m <= n \ m dvd fact (n::nat)" - apply (induct n) - apply force - apply (auto simp only: fact_Suc) - apply (subgoal_tac "m = Suc n") - apply (erule ssubst) - apply (rule dvd_triv_left) - apply auto - done - -lemma dvd_fact_int [rule_format]: "1 <= m \ m <= n \ m dvd fact (n::int)" - apply (case_tac "1 <= n") - apply (induct n rule: int_ge_induct) - apply (auto simp add: fact_plus_one_int) - apply (subgoal_tac "m = i + 1") - apply auto - done - -lemma interval_plus_one_nat: "(i::nat) <= j + 1 \ - {i..j+1} = {i..j} Un {j+1}" - by auto - -lemma interval_Suc: "i <= Suc j \ {i..Suc j} = {i..j} Un {Suc j}" - by auto - -lemma interval_plus_one_int: "(i::int) <= j + 1 \ {i..j+1} = {i..j} Un {j+1}" - by auto - -lemma fact_altdef_nat: "fact (n::nat) = (PROD i:{1..n}. i)" - apply (induct n) - apply force - apply (subst fact_Suc) - apply (subst interval_Suc) - apply auto -done - -lemma fact_altdef_int: "n >= 0 \ fact (n::int) = (PROD i:{1..n}. i)" - apply (induct n rule: int_ge_induct) - apply force - apply (subst fact_plus_one_int, assumption) - apply (subst interval_plus_one_int) - apply auto -done - -lemma fact_dvd: "n \ m \ fact n dvd fact (m::nat)" - by (auto simp add: fact_altdef_nat intro!: setprod_dvd_setprod_subset) - -lemma fact_mod: "m \ (n::nat) \ fact n mod fact m = 0" - by (auto simp add: dvd_imp_mod_0 fact_dvd) - -lemma fact_div_fact: - assumes "m \ (n :: nat)" - shows "(fact m) div (fact n) = \{n + 1..m}" -proof - - obtain d where "d = m - n" by auto - from assms this have "m = n + d" by auto - have "fact (n + d) div (fact n) = \{n + 1..n + d}" - proof (induct d) - case 0 - show ?case by simp - next - case (Suc d') - have "fact (n + Suc d') div fact n = Suc (n + d') * fact (n + d') div fact n" - by simp - also from Suc.hyps have "... = Suc (n + d') * \{n + 1..n + d'}" - unfolding div_mult1_eq[of _ "fact (n + d')"] by (simp add: fact_mod) - also have "... = \{n + 1..n + Suc d'}" - by (simp add: atLeastAtMostSuc_conv setprod.insert) - finally show ?case . - qed - from this `m = n + d` show ?thesis by simp -qed - -lemma fact_mono_nat: "(m::nat) \ n \ fact m \ fact n" -apply (drule le_imp_less_or_eq) -apply (auto dest!: less_imp_Suc_add) -apply (induct_tac k, auto) -done - -lemma fact_neg_int [simp]: "m < (0::int) \ fact m = 0" - unfolding fact_int_def by auto - -lemma fact_ge_zero_int [simp]: "fact m >= (0::int)" - apply (case_tac "m >= 0") - apply auto - apply (frule fact_gt_zero_int) - apply arith -done - -lemma fact_mono_int_aux [rule_format]: "k >= (0::int) \ - fact (m + k) >= fact m" - apply (case_tac "m < 0") - apply auto - apply (induct k rule: int_ge_induct) - apply auto - apply (subst add.assoc [symmetric]) - apply (subst fact_plus_one_int) - apply auto - apply (erule order_trans) - apply (subst mult_le_cancel_right1) - apply (subgoal_tac "fact (m + i) >= 0") - apply arith - apply auto -done - -lemma fact_mono_int: "(m::int) <= n \ fact m <= fact n" - apply (insert fact_mono_int_aux [of "n - m" "m"]) - apply auto -done - -text{*Note that @{term "fact 0 = fact 1"}*} -lemma fact_less_mono_nat: "[| (0::nat) < m; m < n |] ==> fact m < fact n" -apply (drule_tac m = m in less_imp_Suc_add, auto) -apply (induct_tac k, auto) -done - -lemma fact_less_mono_int_aux: "k >= 0 \ (0::int) < m \ - fact m < fact ((m + 1) + k)" - apply (induct k rule: int_ge_induct) - apply (simp add: fact_plus_one_int) - apply (subst (2) fact_reduce_int) - apply (auto simp add: ac_simps) - apply (erule order_less_le_trans) - apply auto - done - -lemma fact_less_mono_int: "(0::int) < m \ m < n \ fact m < fact n" - apply (insert fact_less_mono_int_aux [of "n - (m + 1)" "m"]) - apply auto -done - -lemma fact_num_eq_if_nat: "fact (m::nat) = - (if m=0 then 1 else m * fact (m - 1))" -by (cases m) auto - -lemma fact_add_num_eq_if_nat: - "fact ((m::nat) + n) = (if m + n = 0 then 1 else (m + n) * fact (m + n - 1))" -by (cases "m + n") auto - -lemma fact_add_num_eq_if2_nat: - "fact ((m::nat) + n) = - (if m = 0 then fact n else (m + n) * fact ((m - 1) + n))" -by (cases m) auto - -lemma fact_le_power: "fact n \ n^n" -proof (induct n) - case (Suc n) - then have "fact n \ Suc n ^ n" by (rule le_trans) (simp add: power_mono) - then show ?case by (simp add: add_le_mono) -qed simp - -subsection {* @{term fact} and @{term of_nat} *} - -lemma of_nat_fact_not_zero [simp]: "of_nat (fact n) \ (0::'a::semiring_char_0)" -by auto - -lemma of_nat_fact_gt_zero [simp]: "(0::'a::{linordered_semidom}) < of_nat(fact n)" by auto - -lemma of_nat_fact_ge_zero [simp]: "(0::'a::linordered_semidom) \ of_nat(fact n)" -by simp - -lemma inv_of_nat_fact_gt_zero [simp]: "(0::'a::linordered_field) < inverse (of_nat (fact n))" -by (auto simp add: positive_imp_inverse_positive) - -lemma inv_of_nat_fact_ge_zero [simp]: "(0::'a::linordered_field) \ inverse (of_nat (fact n))" -by (auto intro: order_less_imp_le) - -lemma fact_eq_rev_setprod_nat: "fact (k::nat) = (\i n" shows "fact n div fact (n - r) \ n ^ r" -proof - - have "\r. r \ n \ \{n - r..n} = (n - r) * \{Suc (n - r)..n}" - by (subst setprod.insert[symmetric]) (auto simp: atLeastAtMost_insertL) - with assms show ?thesis - by (induct r rule: nat.induct) (auto simp add: fact_div_fact Suc_diff_Suc mult_le_mono) -qed - -lemma fact_numeral: --{*Evaluation for specific numerals*} - "fact (numeral k) = (numeral k) * (fact (pred_numeral k))" - by (simp add: numeral_eq_Suc) - - -text {* This development is based on the work of Andy Gordon and - Florian Kammueller. *} - -subsection {* Basic definitions and lemmas *} - -primrec binomial :: "nat \ nat \ nat" (infixl "choose" 65) -where - "0 choose k = (if k = 0 then 1 else 0)" -| "Suc n choose k = (if k = 0 then 1 else (n choose (k - 1)) + (n choose k))" - -lemma binomial_n_0 [simp]: "(n choose 0) = 1" - by (cases n) simp_all - -lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0" - by simp - -lemma binomial_Suc_Suc [simp]: "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)" - by simp - -lemma choose_reduce_nat: - "0 < (n::nat) \ 0 < k \ - (n choose k) = ((n - 1) choose (k - 1)) + ((n - 1) choose k)" - by (metis Suc_diff_1 binomial.simps(2) neq0_conv) - -lemma binomial_eq_0: "n < k \ n choose k = 0" - by (induct n arbitrary: k) auto - -declare binomial.simps [simp del] - -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 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 Suc_times_binomial_eq: - "Suc n * (n choose k) = (Suc n choose Suc k) * Suc k" - apply (induct n arbitrary: k, simp add: binomial.simps) - apply (case_tac k) - apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq binomial_eq_0) - done - -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 add: nat_diff_split) - - -subsection {* Combinatorial theorems involving @{text "choose"} *} - -text {*By Florian Kamm\"uller, tidied by LCP.*} - -lemma card_s_0_eq_empty: "finite A \ card {B. B \ A & card B = 0} = 1" - by (simp cong add: conj_cong add: finite_subset [THEN card_0_eq]) - -lemma choose_deconstruct: "finite M \ x \ M \ - {s. s \ insert x M \ card s = Suc k} = - {s. s \ M \ card s = Suc k} \ {s. \t. t \ M \ card t = k \ s = insert x t}" - apply safe - apply (auto intro: finite_subset [THEN card_insert_disjoint]) - by (metis (full_types) Diff_insert_absorb Set.set_insert Zero_neq_Suc card_Diff_singleton_if - card_eq_0_iff diff_Suc_1 in_mono subset_insert_iff) - -lemma finite_bex_subset [simp]: - assumes "finite B" - and "\A. A \ B \ finite {x. P x A}" - shows "finite {x. \A \ B. P x A}" - by (metis (no_types) assms finite_Collect_bounded_ex finite_Collect_subsets) - -text{*There are as many subsets of @{term A} having cardinality @{term k} - as there are sets obtained from the former by inserting a fixed element - @{term x} into each.*} -lemma constr_bij: - "finite A \ x \ A \ - card {B. \C. C \ A \ card C = k \ B = insert x C} = - card {B. B \ A & card(B) = k}" - apply (rule card_bij_eq [where f = "\s. s - {x}" and g = "insert x"]) - apply (auto elim!: equalityE simp add: inj_on_def) - apply (metis card_Diff_singleton_if finite_subset in_mono) - done - -text {* - Main theorem: combinatorial statement about number of subsets of a set. -*} - -theorem n_subsets: "finite A \ card {B. B \ A \ card B = k} = (card A choose k)" -proof (induct k arbitrary: A) - case 0 then show ?case by (simp add: card_s_0_eq_empty) -next - case (Suc k) - show ?case using `finite A` - proof (induct A) - case empty show ?case by (simp add: card_s_0_eq_empty) - next - case (insert x A) - then show ?case using Suc.hyps - apply (simp add: card_s_0_eq_empty choose_deconstruct) - apply (subst card_Un_disjoint) - prefer 4 apply (force simp add: constr_bij) - prefer 3 apply force - prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2] - finite_subset [of _ "Pow (insert x F)" for F]) - apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset]) - done - qed -qed - - -subsection {* The binomial theorem (courtesy of Tobias Nipkow): *} - -text{* Avigad's version, generalized to any commutative ring *} -theorem binomial_ring: "(a+b::'a::{comm_ring_1,power})^n = - (\k=0..n. (of_nat (n choose k)) * a^k * b^(n-k))" (is "?P n") -proof (induct n) - case 0 then show "?P 0" by simp -next - case (Suc n) - have decomp: "{0..n+1} = {0} Un {n+1} Un {1..n}" - by auto - have decomp2: "{0..n} = {0} Un {1..n}" - by auto - have "(a+b)^(n+1) = - (a+b) * (\k=0..n. of_nat (n choose k) * a^k * b^(n-k))" - using Suc.hyps by simp - also have "\ = a*(\k=0..n. of_nat (n choose k) * a^k * b^(n-k)) + - b*(\k=0..n. of_nat (n choose k) * a^k * b^(n-k))" - by (rule distrib_right) - also have "\ = (\k=0..n. of_nat (n choose k) * a^(k+1) * b^(n-k)) + - (\k=0..n. of_nat (n choose k) * a^k * b^(n-k+1))" - by (auto simp add: setsum_right_distrib ac_simps) - also have "\ = (\k=0..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:setsum_shift_bounds_cl_Suc_ivl Suc_diff_le field_simps - del:setsum_cl_ivl_Suc) - also have "\ = a^(n+1) + b^(n+1) + - (\k=1..n. of_nat (n choose (k - 1)) * a^k * b^(n+1-k)) + - (\k=1..n. of_nat (n choose k) * a^k * b^(n+1-k))" - by (simp add: decomp2) - 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 setsum.distrib [symmetric] choose_reduce_nat) - also have "\ = (\k=0..n+1. of_nat (n+1 choose k) * a^k * b^(n+1-k))" - using decomp by (simp add: field_simps) - finally show "?P (Suc n)" by simp -qed - -text{* Original version for the naturals *} -corollary binomial: "(a+b::nat)^n = (\k=0..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_setsum [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: "\mx\m. fact x * fact (m - x) * (m choose x) = - fact m" and kn: "k \ n" - let ?ths = "fact k * fact (n - k) * (n choose k) = fact n" - { assume "n=0" then have ?ths using kn by simp } - moreover - { assume "k=0" then have ?ths using kn by simp } - moreover - { assume nk: "n=k" then have ?ths by simp } - moreover - { fix m h assume n: "n = Suc m" and h: "k = Suc h" and hm: "h < m" - from n have mn: "m < n" by arith - from hm have hm': "h \ m" by arith - from hm h n kn have km: "k \ m" by arith - have "m - h = Suc (m - Suc h)" using h km hm by arith - with km h have th0: "fact (m - h) = (m - h) * fact (m - k)" - by simp - from n h th0 - 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 have ?ths using h n km by simp } - moreover have "n=0 \ k = 0 \ k = n \ (\m h. n = Suc m \ k = Suc h \ h < m)" - using kn by presburger - ultimately show ?ths by blast -qed - -lemma binomial_fact: - assumes kn: "k \ n" - shows "(of_nat (n choose k) :: 'a::{field,ring_char_0}) = - of_nat (fact n) / (of_nat (fact k) * of_nat (fact (n - k)))" - using binomial_fact_lemma[OF kn] - by (simp add: field_simps of_nat_mult [symmetric]) - -lemma choose_row_sum: "(\k=0..n. n choose k) = 2^n" - using binomial [of 1 "1" n] - by (simp add: numeral_2_eq_2) - -lemma sum_choose_lower: "(\k=0..n. (r+k) choose k) = Suc (r+n) choose n" - by (induct n) auto - -lemma sum_choose_upper: "(\k=0..n. k choose m) = Suc n choose Suc m" - by (induct n) auto - -lemma natsum_reverse_index: - fixes m::nat - shows "(\k. m \ k \ k \ n \ g k = f (m + n - k)) \ (\k=m..n. f k) = (\k=m..n. g k)" - by (rule setsum.reindex_bij_witness[where i="\k. m+n-k" and j="\k. m+n-k"]) auto - -text{*NW diagonal sum property*} -lemma sum_choose_diagonal: - assumes "m\n" shows "(\k=0..m. (n-k) choose (m-k)) = Suc n choose m" -proof - - have "(\k=0..m. (n-k) choose (m-k)) = (\k=0..m. (n-m+k) choose k)" - by (rule natsum_reverse_index) (simp add: assms) - 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{* Pochhammer's symbol : generalized rising factorial *} - -text {* See @{url "http://en.wikipedia.org/wiki/Pochhammer_symbol"} *} - -definition "pochhammer (a::'a::comm_semiring_1) n = - (if n = 0 then 1 else setprod (\n. a + of_nat n) {0 .. n - 1})" - -lemma pochhammer_0 [simp]: "pochhammer a 0 = 1" - by (simp add: pochhammer_def) - -lemma pochhammer_1 [simp]: "pochhammer a 1 = a" - by (simp add: pochhammer_def) - -lemma pochhammer_Suc0 [simp]: "pochhammer a (Suc 0) = a" - by (simp add: pochhammer_def) - -lemma pochhammer_Suc_setprod: "pochhammer a (Suc n) = setprod (\n. a + of_nat n) {0 .. n}" - by (simp add: pochhammer_def) - -lemma setprod_nat_ivl_Suc: "setprod f {0 .. Suc n} = setprod f {0..n} * f (Suc n)" -proof - - have "{0..Suc n} = {0..n} \ {Suc n}" by auto - then show ?thesis by (simp add: field_simps) -qed - -lemma setprod_nat_ivl_1_Suc: "setprod f {0 .. Suc n} = f 0 * setprod f {1.. Suc n}" -proof - - have "{0..Suc n} = {0} \ {1 .. Suc n}" by auto - then show ?thesis by simp -qed - - -lemma pochhammer_Suc: "pochhammer a (Suc n) = pochhammer a n * (a + of_nat n)" -proof (cases n) - case 0 - then show ?thesis by simp -next - case (Suc n) - show ?thesis unfolding Suc pochhammer_Suc_setprod setprod_nat_ivl_Suc .. -qed - -lemma pochhammer_rec: "pochhammer a (Suc n) = a * pochhammer (a + 1) n" -proof (cases "n = 0") - case True - then show ?thesis by (simp add: pochhammer_Suc_setprod) -next - case False - have *: "finite {1 .. n}" "0 \ {1 .. n}" by auto - have eq: "insert 0 {1 .. n} = {0..n}" by auto - have **: "(\n\{1\nat..n}. a + of_nat n) = (\n\{0\nat..n - 1}. a + 1 + of_nat n)" - apply (rule setprod.reindex_cong [where l = Suc]) - using False - apply (auto simp add: fun_eq_iff field_simps) - done - show ?thesis - apply (simp add: pochhammer_def) - unfolding setprod.insert [OF *, unfolded eq] - using ** apply (simp add: field_simps) - done -qed - -lemma pochhammer_fact: "of_nat (fact n) = pochhammer 1 n" - unfolding fact_altdef_nat - apply (cases n) - apply (simp_all add: of_nat_setprod pochhammer_Suc_setprod) - apply (rule setprod.reindex_cong [where l = Suc]) - apply (auto simp add: fun_eq_iff) - done - -lemma pochhammer_of_nat_eq_0_lemma: - assumes "k > n" - shows "pochhammer (- (of_nat n :: 'a:: idom)) k = 0" -proof (cases "n = 0") - case True - then show ?thesis - using assms by (cases k) (simp_all add: pochhammer_rec) -next - case False - from assms obtain h where "k = Suc h" by (cases k) auto - then show ?thesis - by (simp add: pochhammer_Suc_setprod) - (metis Suc_leI Suc_le_mono assms atLeastAtMost_iff less_eq_nat.simps(1)) -qed - -lemma pochhammer_of_nat_eq_0_lemma': - assumes kn: "k \ n" - shows "pochhammer (- (of_nat n :: 'a:: {idom,ring_char_0})) k \ 0" -proof (cases k) - case 0 - then show ?thesis by simp -next - case (Suc h) - then show ?thesis - apply (simp add: pochhammer_Suc_setprod) - using Suc kn apply (auto simp add: algebra_simps) - done -qed - -lemma pochhammer_of_nat_eq_0_iff: - shows "pochhammer (- (of_nat n :: 'a:: {idom,ring_char_0})) k = 0 \ k > n" - (is "?l = ?r") - using pochhammer_of_nat_eq_0_lemma[of n k, where ?'a='a] - pochhammer_of_nat_eq_0_lemma'[of k n, where ?'a = 'a] - by (auto simp add: not_le[symmetric]) - -lemma pochhammer_eq_0_iff: "pochhammer a n = (0::'a::field_char_0) \ (\k < n. a = - of_nat k)" - apply (auto simp add: pochhammer_of_nat_eq_0_iff) - apply (cases n) - apply (auto simp add: pochhammer_def algebra_simps group_add_class.eq_neg_iff_add_eq_0) - apply (metis leD not_less_eq) - done - -lemma pochhammer_eq_0_mono: - "pochhammer a n = (0::'a::field_char_0) \ m \ n \ pochhammer a m = 0" - unfolding pochhammer_eq_0_iff by auto - -lemma pochhammer_neq_0_mono: - "pochhammer a m \ (0::'a::field_char_0) \ m \ n \ pochhammer a n \ 0" - unfolding pochhammer_eq_0_iff by auto - -lemma pochhammer_minus: - assumes kn: "k \ n" - shows "pochhammer (- b) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (b - of_nat k + 1) k" -proof (cases k) - case 0 - then show ?thesis by simp -next - case (Suc h) - have eq: "((- 1) ^ Suc h :: 'a) = (\i=0..h. - 1)" - using setprod_constant[where A="{0 .. h}" and y="- 1 :: 'a"] - by auto - show ?thesis - unfolding Suc pochhammer_Suc_setprod eq setprod.distrib[symmetric] - by (rule setprod.reindex_bij_witness[where i="op - h" and j="op - h"]) - (auto simp: of_nat_diff) -qed - -lemma pochhammer_minus': - assumes kn: "k \ n" - shows "pochhammer (b - of_nat k + 1) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (- b) k" - unfolding pochhammer_minus[OF kn, where b=b] - unfolding mult.assoc[symmetric] - unfolding power_add[symmetric] - by simp - -lemma pochhammer_same: "pochhammer (- of_nat n) n = - ((- 1) ^ n :: 'a::comm_ring_1) * of_nat (fact n)" - unfolding pochhammer_minus[OF le_refl[of n]] - by (simp add: of_nat_diff pochhammer_fact) - - -subsection{* Generalized binomial coefficients *} - -definition gbinomial :: "'a::field_char_0 \ nat \ 'a" (infixl "gchoose" 65) - where "a gchoose n = - (if n = 0 then 1 else (setprod (\i. a - of_nat i) {0 .. n - 1}) / of_nat (fact n))" - -lemma gbinomial_0 [simp]: "a gchoose 0 = 1" "0 gchoose (Suc n) = 0" - apply (simp_all add: gbinomial_def) - apply (subgoal_tac "(\i\nat\{0\nat..n}. - of_nat i) = (0::'b)") - apply (simp del:setprod_zero_iff) - apply simp - done - -lemma gbinomial_pochhammer: "a gchoose n = (- 1) ^ n * pochhammer (- a) n / of_nat (fact n)" -proof (cases "n = 0") - case True - then show ?thesis by simp -next - case False - from this setprod_constant[of "{0 .. n - 1}" "- (1:: 'a)"] - have eq: "(- (1\'a)) ^ n = setprod (\i. - 1) {0 .. n - 1}" - by auto - from False show ?thesis - by (simp add: pochhammer_def gbinomial_def field_simps - eq setprod.distrib[symmetric]) -qed - -lemma binomial_gbinomial: "of_nat (n choose k) = of_nat n gchoose k" -proof - - { assume kn: "k > n" - then have ?thesis - by (subst binomial_eq_0[OF kn]) - (simp add: gbinomial_pochhammer field_simps pochhammer_of_nat_eq_0_iff) } - moreover - { assume "k=0" then have ?thesis by simp } - moreover - { assume kn: "k \ n" and k0: "k\ 0" - from k0 obtain h where h: "k = Suc h" by (cases k) auto - from h - have eq:"(- 1 :: 'a) ^ k = setprod (\i. - 1) {0..h}" - by (subst setprod_constant) auto - have eq': "(\i\{0..h}. of_nat n + - (of_nat i :: 'a)) = (\i\{n - h..n}. of_nat i)" - using h kn - by (intro setprod.reindex_bij_witness[where i="op - n" and j="op - n"]) - (auto simp: of_nat_diff) - have th0: "finite {1..n - Suc h}" "finite {n - h .. n}" - "{1..n - Suc h} \ {n - h .. n} = {}" and - eq3: "{1..n - Suc h} \ {n - h .. n} = {1..n}" - using h kn by auto - from eq[symmetric] - have ?thesis using kn - apply (simp add: binomial_fact[OF kn, where ?'a = 'a] - gbinomial_pochhammer field_simps pochhammer_Suc_setprod) - apply (simp add: pochhammer_Suc_setprod fact_altdef_nat h - of_nat_setprod setprod.distrib[symmetric] eq' del: One_nat_def power_Suc) - unfolding setprod.union_disjoint[OF th0, unfolded eq3, of "of_nat:: nat \ 'a"] eq[unfolded h] - unfolding mult.assoc[symmetric] - unfolding setprod.distrib[symmetric] - apply simp - apply (intro setprod.reindex_bij_witness[where i="op - n" and j="op - n"]) - apply (auto simp: of_nat_diff) - done - } - moreover - have "k > n \ k = 0 \ (k \ n \ k \ 0)" by arith - ultimately show ?thesis by blast -qed - -lemma gbinomial_1[simp]: "a gchoose 1 = a" - by (simp add: gbinomial_def) - -lemma gbinomial_Suc0[simp]: "a gchoose (Suc 0) = a" - by (simp add: gbinomial_def) - -lemma gbinomial_mult_1: - "a * (a gchoose n) = - of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))" (is "?l = ?r") -proof - - have "?r = ((- 1) ^n * pochhammer (- a) n / of_nat (fact n)) * (of_nat n - (- a + of_nat n))" - unfolding gbinomial_pochhammer - pochhammer_Suc fact_Suc of_nat_mult right_diff_distrib power_Suc - by (simp add: field_simps del: of_nat_Suc) - also have "\ = ?l" unfolding gbinomial_pochhammer - by (simp add: field_simps) - finally show ?thesis .. -qed - -lemma gbinomial_mult_1': - "(a gchoose n) * a = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))" - by (simp add: mult.commute gbinomial_mult_1) - -lemma gbinomial_Suc: - "a gchoose (Suc k) = (setprod (\i. a - of_nat i) {0 .. k}) / of_nat (fact (Suc k))" - by (simp add: gbinomial_def) - -lemma gbinomial_mult_fact: - "(of_nat (fact (Suc k)) :: 'a) * ((a::'a::field_char_0) gchoose (Suc k)) = - (setprod (\i. a - of_nat i) {0 .. k})" - by (simp_all add: gbinomial_Suc field_simps del: fact_Suc) - -lemma gbinomial_mult_fact': - "((a::'a::field_char_0) gchoose (Suc k)) * (of_nat (fact (Suc k)) :: 'a) = - (setprod (\i. a - of_nat i) {0 .. k})" - using gbinomial_mult_fact[of k a] - by (subst mult.commute) - - -lemma gbinomial_Suc_Suc: - "((a::'a::field_char_0) + 1) gchoose (Suc k) = a gchoose k + (a gchoose (Suc k))" -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)" - apply (rule setprod.reindex_cong [where l = Suc]) - using Suc - apply auto - done - have "of_nat (fact (Suc k)) * (a gchoose k + (a gchoose (Suc k))) = - ((a gchoose Suc h) * of_nat (fact (Suc h)) * of_nat (Suc k)) + (\i\{0\nat..Suc h}. a - of_nat i)" - apply (simp add: Suc field_simps del: fact_Suc) - unfolding gbinomial_mult_fact' - apply (subst fact_Suc) - unfolding of_nat_mult - apply (subst mult.commute) - unfolding mult.assoc - unfolding gbinomial_mult_fact - apply (simp add: field_simps) - done - also have "\ = (\i\{0..h}. a - of_nat i) * (a + 1)" - unfolding gbinomial_mult_fact' setprod_nat_ivl_Suc - by (simp add: field_simps Suc) - also have "\ = (\i\{0..k}. (a + 1) - of_nat i)" - using eq0 - by (simp add: Suc setprod_nat_ivl_1_Suc) - also have "\ = of_nat (fact (Suc k)) * ((a + 1) gchoose (Suc k))" - unfolding gbinomial_mult_fact .. - finally show ?thesis by (simp del: fact_Suc) -qed - -lemma gbinomial_reduce_nat: - "0 < k \ (a::'a::field_char_0) gchoose k = (a - 1) gchoose (k - 1) + ((a - 1) gchoose k)" -by (metis Suc_pred' diff_add_cancel gbinomial_Suc_Suc) - - -lemma binomial_symmetric: - assumes kn: "k \ n" - shows "n choose k = n choose (n - k)" -proof- - from kn have kn': "n - k \ n" 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 - -text{*Contributed by Manuel Eberl, generalised by LCP. - Alternative definition of the binomial coefficient as @{term "\iii = (\i x" - shows "(x / of_nat k :: 'a) ^ k \ x gchoose k" -proof - - have x: "0 \ x" - using assms of_nat_0_le_iff order_trans by blast - have "(x / of_nat k :: 'a) ^ k = (\i \ x gchoose k" - unfolding gbinomial_altdef_of_nat - proof (safe intro!: setprod_mono) - fix i :: nat - assume ik: "i < k" - from assms have "x * 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 "x * of_nat k - x * of_nat i \ x * of_nat k - of_nat (i * k)" by arith - then have "x * of_nat (k - i) \ (x - of_nat i) * of_nat k" - using ik - by (simp add: algebra_simps zero_less_mult_iff of_nat_diff of_nat_mult) - then have "x * of_nat (k - i) \ (x - of_nat i) * (of_nat k :: 'a)" - unfolding of_nat_mult[symmetric] of_nat_le_iff . - with assms show "x / of_nat k \ (x - of_nat i) / (of_nat (k - i) :: 'a)" - using `i < k` by (simp add: field_simps) - qed (simp add: x zero_le_divide_iff) - finally show ?thesis . -qed - -text{*Versions of the theorems above for the natural-number version of "choose"*} -lemma binomial_altdef_of_nat: - fixes n k :: nat - and x :: "'a :: {field_char_0,field_inverse_zero}" --{*the point is to constrain @{typ 'a}*} - assumes "k \ n" - shows "of_nat (n choose k) = (\i n" - shows "(of_nat n / of_nat k :: 'a) ^ k \ of_nat (n choose k)" -by (simp add: assms 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 `r \ n` 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::nat) \ n \ - n choose k = fact n div (fact k * fact (n - k))" - by (subst binomial_fact_lemma [symmetric]) auto - -lemma choose_dvd_nat: "(k::nat) \ n \ fact k * fact (n - k) dvd fact n" -by (metis binomial_fact_lemma dvd_def) - -lemma choose_dvd_int: - assumes "(0::int) <= k" and "k <= n" - shows "fact k * fact (n - k) dvd fact n" - apply (subst tsub_eq [symmetric], rule assms) - apply (rule choose_dvd_nat [transferred]) - using assms apply auto - done - -lemma fact_fact_dvd_fact: fixes k::nat shows "fact k * fact n dvd fact (n + k)" -by (metis add.commute add_diff_cancel_left' choose_dvd_nat 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)" -proof - - have "((m+r+k) choose (m+k)) * ((m+k) choose k) = - fact (m+r + k) div (fact (m + k) * fact (m+r - m)) * (fact (m + k) div (fact k * fact m))" - by (simp add: assms binomial_altdef_nat) - also have "... = fact (m+r+k) div (fact r * (fact k * fact m))" - apply (subst div_mult_div_if_dvd) - apply (auto simp: fact_fact_dvd_fact) - apply (metis add.assoc add.commute fact_fact_dvd_fact) - done - 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: fact_fact_dvd_fact) - apply (metis dvd_trans dvd.dual_order.refl fact_fact_dvd_fact mult_dvd_mono mult.left_commute) - done - also have "... = (fact (m+r+k) div (fact k * fact (m+r)) * (fact (m+r) div (fact r * fact m)))" - apply (subst div_mult_div_if_dvd) - apply (auto simp: fact_fact_dvd_fact) - apply(metis mult.left_commute) - done - finally show ?thesis - by (simp add: binomial_altdef_nat mult.commute) -qed - -text{*The "Subset of a Subset" identity*} -lemma choose_mult: - assumes "k\m" "m\n" - shows "(n choose m) * (m choose k) = (n choose k) * ((n-k) choose (m-k))" -using assms choose_mult_lemma [of "m-k" "n-m" k] -by simp - - -subsection {* Binomial coefficients *} - -lemma choose_one: "(n::nat) choose 1 = n" - by simp - -(*FIXME: messy and apparently unused*) -lemma binomial_induct [rule_format]: "(ALL (n::nat). P n n) \ - (ALL n. P (Suc n) 0) \ (ALL n. (ALL k < n. P n k \ P n (Suc k) \ - P (Suc n) (Suc k))) \ (ALL k <= n. P n k)" - apply (induct n) - apply auto - apply (case_tac "k = 0") - apply auto - apply (case_tac "k = Suc n") - apply auto - apply (metis Suc_le_eq fact_nat.cases le_Suc_eq le_eq_less_or_eq) - done - -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 setsum_right_distrib) simp - also have "?rhs = (\(I, _)\Sigma {I. I \ A \ I \ {}} Inter. (- 1) ^ (card I + 1))" - using assms by(subst setsum.Sigma)(auto) - also have "\ = (\(x, I)\(SIGMA x:UNIV. {I. I \ A \ I \ {} \ x \ \I}). (- 1) ^ (card I + 1))" - by (rule setsum.reindex_cong [where l = "\(x, y). (y, x)"]) (auto intro: inj_onI simp add: split_beta) - also have "\ = (\(x, I)\(SIGMA x:\A. {I. I \ A \ I \ {} \ x \ \I}). (- 1) ^ (card I + 1))" - using assms by(auto intro!: setsum.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 setsum.Sigma) auto - also have "\ = (\_\\A. 1)" (is "setsum ?lhs _ = _") - proof(rule setsum.cong[OF refl]) - fix x - assume x: "x \ \A" - def K \ "{X \ A. x \ X}" - with `finite A` have K: "finite K" by auto - let ?I = "\i. {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 setsum.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 setsum.Sigma) auto - also have "\ = (\i=1..card A. (- 1) ^ (i + 1) * (\I|I \ A \ card I = i \ x \ \I. 1))" - by(subst setsum_right_distrib) simp - also have "\ = (\i=1..card K. (- 1) ^ (i + 1) * (\I|I \ K \ card I = i. 1))" (is "_ = ?rhs") - proof(rule setsum.mono_neutral_cong_right[rule_format]) - 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}" - hence 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 setsum.cong)(auto simp add: K_def) - thus "(- 1) ^ (i + 1) * ?lhs = (- 1) ^ (i + 1) * ?rhs" by simp - qed simp - also have "{I. I \ K \ card I = 0} = {{}}" using assms - by(auto simp add: card_eq_0_iff K_def dest: finite_subset) - hence "?rhs = (\i = 0..card K. (- 1) ^ (i + 1) * (\I | I \ K \ card I = i. 1 :: int)) + 1" - by(subst (2) setsum_head_Suc)(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 setsum_right_distrib[symmetric]) simp - also have "\ = - ((-1 + 1) ^ card K) + 1" - by(subst binomial_ring)(simp add: ac_simps) - 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 @{text m} summing to @{text N} is -@{term "(N + m - 1) choose N"}: *} - -lemma card_length_listsum_rec: - assumes "m\1" - shows "card {l::nat list. length l = m \ listsum l = N} = - (card {l. length l = (m - 1) \ listsum l = N} + - card {l. length l = m \ listsum l + 1 = N})" - (is "card ?C = (card ?A + card ?B)") -proof - - let ?A'="{l. length l = m \ listsum l = N \ hd l = 0}" - let ?B'="{l. length l = m \ listsum l = N \ hd l \ 0}" - let ?f ="\ l. 0#l" - let ?g ="\ l. (hd l + 1) # tl l" - have 1: "\xs x. xs \ [] \ x = hd xs \ x # tl xs = xs" by simp - have 2: "\xs. (xs::nat list) \ [] \ listsum(tl xs) = listsum xs - hd xs" - by(auto simp add: neq_Nil_conv) - have f: "bij_betw ?f ?A ?A'" - apply(rule bij_betw_byWitness[where f' = tl]) - using assms - by (auto simp: 2 length_0_conv[symmetric] 1 simp del: length_0_conv) - have 3: "\xs:: nat list. xs \ [] \ hd xs + (listsum xs - hd xs) = listsum xs" - by (metis 1 listsum_simps(2) 2) - have g: "bij_betw ?g ?B ?B'" - apply(rule bij_betw_byWitness[where f' = "\ l. (hd l - 1) # tl l"]) - using assms - by (auto simp: 2 length_0_conv[symmetric] intro!: 3 - simp del: length_greater_0_conv length_0_conv) - { fix M N :: nat have "finite {xs. size xs = M \ set xs \ {0.. ?B'" by auto - have disj: "?A' \ ?B' = {}" by auto - 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_listsum: --"By Holden Lee, tidied by Tobias Nipkow" - "card {l::nat list. size l = m \ listsum l = N} = (N + m - 1) choose N" -proof (cases m) - case 0 then show ?thesis - by (cases N) (auto simp: 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) \ listsum l = N} = - (N + (m - 1) - 1) choose N" - proof cases - assume "m = 1" - with Suc.hyps have "N\1" by auto - with `m = 1` show ?thesis by (simp add: binomial_eq_0) - next - assume "m \ 1" thus ?thesis using Suc by fastforce - qed - - from Suc have c2: "card {l::nat list. size l = m \ listsum l + 1 = N} = - (if N>0 then ((N - 1) + m - 1) choose (N - 1) else 0)" - proof - - have aux: "\m n. n > 0 \ Suc m = n \ m = n - 1" by arith - from Suc have "N>0 \ - card {l::nat list. size l = m \ listsum l + 1 = N} = - ((N - 1) + m - 1) choose (N - 1)" by (simp add: aux) - thus ?thesis by auto - qed - - from Suc.prems have "(card {l::nat list. size l = (m - 1) \ listsum l = N} + - card {l::nat list. size l = m \ listsum 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) - thus ?case using card_length_listsum_rec[OF Suc.prems] by auto - qed -qed - -end diff -r 1c937d56a70a -r de7792ea4090 src/HOL/Import/Import_Setup.thy --- a/src/HOL/Import/Import_Setup.thy Tue Mar 10 15:21:26 2015 +0000 +++ b/src/HOL/Import/Import_Setup.thy Tue Mar 10 16:12:35 2015 +0000 @@ -6,7 +6,7 @@ section {* Importer machinery and required theorems *} theory Import_Setup -imports Main "~~/src/HOL/Fact" +imports Main "~~/src/HOL/Binomial" keywords "import_type_map" "import_const_map" "import_file" :: thy_decl begin diff -r 1c937d56a70a -r de7792ea4090 src/HOL/Library/Permutations.thy --- a/src/HOL/Library/Permutations.thy Tue Mar 10 15:21:26 2015 +0000 +++ b/src/HOL/Library/Permutations.thy Tue Mar 10 16:12:35 2015 +0000 @@ -5,7 +5,7 @@ section {* Permutations, both general and specifically on finite sets.*} theory Permutations -imports Fact +imports Binomial begin subsection {* Transpositions *} @@ -46,7 +46,7 @@ lemma permutes_imp_bij: "p permutes S \ bij_betw p S S" by (metis UNIV_I bij_betw_def permutes_image permutes_inj subsetI subset_inj_on) - + 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)+ diff -r 1c937d56a70a -r de7792ea4090 src/HOL/Number_Theory/Primes.thy --- a/src/HOL/Number_Theory/Primes.thy Tue Mar 10 15:21:26 2015 +0000 +++ b/src/HOL/Number_Theory/Primes.thy Tue Mar 10 16:12:35 2015 +0000 @@ -28,7 +28,7 @@ section {* Primes *} theory Primes -imports "~~/src/HOL/GCD" "~~/src/HOL/Fact" +imports "~~/src/HOL/GCD" "~~/src/HOL/Binomial" begin declare [[coercion int]] @@ -72,7 +72,7 @@ apply (metis gcd_dvd1_nat gcd_dvd2_nat) done -lemma prime_int_altdef: +lemma prime_int_altdef: "prime p = (1 < p \ (\m::int. m \ 0 \ m dvd p \ m = 1 \ m = p))" apply (simp add: prime_def) @@ -90,7 +90,7 @@ lemma prime_dvd_mult_nat: "prime p \ p dvd m * n \ p dvd m \ p dvd n" by (blast intro: coprime_dvd_mult_nat prime_imp_coprime_nat) -lemma prime_dvd_mult_int: +lemma prime_dvd_mult_int: fixes n::int shows "prime p \ p dvd m * n \ p dvd m \ p dvd n" by (blast intro: coprime_dvd_mult_int prime_imp_coprime_int) @@ -99,7 +99,7 @@ by (rule iffI, rule prime_dvd_mult_nat, auto) lemma prime_dvd_mult_eq_int [simp]: - fixes n::int + fixes n::int shows "prime p \ p dvd m * n = (p dvd m \ p dvd n)" by (rule iffI, rule prime_dvd_mult_int, auto) @@ -121,7 +121,7 @@ by (cases n) (auto elim: prime_dvd_power_nat) lemma prime_dvd_power_int_iff: - fixes x::int + fixes x::int shows "prime p \ n > 0 \ p dvd x^n \ p dvd x" by (cases n) (auto elim: prime_dvd_power_int) @@ -226,14 +226,14 @@ lemma next_prime_bound: "\p. prime p \ n < p \ p <= fact n + 1" proof- - have f1: "fact n + 1 \ 1" using fact_ge_one_nat [of n] by arith + have f1: "fact n + 1 \ 1" using fact_ge_one_nat [of n] by arith from prime_factor_nat [OF f1] obtain p where "prime p" and "p dvd fact n + 1" by auto then have "p \ fact n + 1" apply (intro dvd_imp_le) apply auto done { assume "p \ n" - from `prime p` have "p \ 1" + from `prime p` have "p \ 1" by (cases p, simp_all) - with `p <= n` have "p dvd fact n" + with `p <= n` have "p dvd fact n" by (intro dvd_fact_nat) with `p dvd fact n + 1` have "p dvd fact n + 1 - fact n" by (rule dvd_diff_nat) @@ -245,7 +245,7 @@ with `prime p` and `p <= fact n + 1` show ?thesis by auto qed -lemma bigger_prime: "\p. prime p \ p > (n::nat)" +lemma bigger_prime: "\p. prime p \ p > (n::nat)" using next_prime_bound by auto lemma primes_infinite: "\ (finite {(p::nat). prime p})" @@ -263,12 +263,12 @@ text{*Versions for type nat only*} -lemma prime_product: +lemma prime_product: fixes p::nat assumes "prime (p * q)" shows "p = 1 \ q = 1" proof - - from assms have + from assms have "1 < p * q" and P: "\m. m dvd p * q \ m = 1 \ m = p * q" unfolding prime_nat_def by auto from `1 < p * q` have "p \ 0" by (cases p) auto @@ -278,7 +278,7 @@ then show ?thesis by (simp add: Q) qed -lemma prime_exp: +lemma prime_exp: fixes p::nat shows "prime (p^n) \ prime p \ n = 1" proof(induct n) @@ -301,7 +301,7 @@ ultimately show ?case by blast qed -lemma prime_power_mult: +lemma prime_power_mult: fixes p::nat assumes p: "prime p" and xy: "x * y = p ^ k" shows "\i j. x = p ^i \ y = p^ j" @@ -312,28 +312,28 @@ case (Suc k x y) from Suc.prems have pxy: "p dvd x*y" by auto from Primes.prime_dvd_mult_nat [OF p pxy] have pxyc: "p dvd x \ p dvd y" . - from p have p0: "p \ 0" by - (rule ccontr, simp) + from p have p0: "p \ 0" by - (rule ccontr, simp) {assume px: "p dvd x" then obtain d where d: "x = p*d" unfolding dvd_def by blast from Suc.prems d have "p*d*y = p^Suc k" by simp hence th: "d*y = p^k" using p0 by simp from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "y = p^j" by blast - with d have "x = p^Suc i" by simp + with d have "x = p^Suc i" by simp with ij(2) have ?case by blast} - moreover + moreover {assume px: "p dvd y" then obtain d where d: "y = p*d" unfolding dvd_def by blast from Suc.prems d have "p*d*x = p^Suc k" by (simp add: mult.commute) hence th: "d*x = p^k" using p0 by simp from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "x = p^j" by blast - with d have "y = p^Suc i" by simp + with d have "y = p^Suc i" by simp with ij(2) have ?case by blast} ultimately show ?case using pxyc by blast qed -lemma prime_power_exp: +lemma prime_power_exp: fixes p::nat - assumes p: "prime p" and n: "n \ 0" + assumes p: "prime p" and n: "n \ 0" and xn: "x^n = p^k" shows "\i. x = p^i" using n xn proof(induct n arbitrary: k) @@ -343,7 +343,7 @@ {assume "n = 0" with Suc have ?case by simp (rule exI[where x="k"], simp)} moreover {assume n: "n \ 0" - from prime_power_mult[OF p th] + from prime_power_mult[OF p th] obtain i j where ij: "x = p^i" "x^n = p^j"by blast from Suc.hyps[OF n ij(2)] have ?case .} ultimately show ?case by blast @@ -351,14 +351,14 @@ lemma divides_primepow: fixes p::nat - assumes p: "prime p" + assumes p: "prime p" shows "d dvd p^k \ (\ i. i \ k \ d = p ^i)" proof - assume H: "d dvd p^k" then obtain e where e: "d*e = p^k" + assume H: "d dvd p^k" then obtain e where e: "d*e = p^k" unfolding dvd_def apply (auto simp add: mult.commute) by blast from prime_power_mult[OF p e] obtain i j where ij: "d = p^i" "e=p^j" by blast from e ij have "p^(i + j) = p^k" by (simp add: power_add) - hence "i + j = k" using p prime_gt_1_nat power_inject_exp[of p "i+j" k] by simp + hence "i + j = k" using p prime_gt_1_nat power_inject_exp[of p "i+j" k] by simp hence "i \ k" by arith with ij(1) show "\i\k. d = p ^ i" by blast next @@ -375,16 +375,16 @@ lemma bezout_gcd_nat: fixes a::nat shows "\x y. a * x - b * y = gcd a b \ b * x - a * y = gcd a b" using bezout_nat[of a b] -by (metis bezout_nat diff_add_inverse gcd_add_mult_nat gcd_nat.commute - gcd_nat.right_neutral mult_0) +by (metis bezout_nat diff_add_inverse gcd_add_mult_nat gcd_nat.commute + gcd_nat.right_neutral mult_0) lemma gcd_bezout_sum_nat: - fixes a::nat - assumes "a * x + b * y = d" + fixes a::nat + assumes "a * x + b * y = d" shows "gcd a b dvd d" proof- let ?g = "gcd a b" - have dv: "?g dvd a*x" "?g dvd b * y" + have dv: "?g dvd a*x" "?g dvd b * y" by simp_all from dvd_add[OF dv] assms show ?thesis by auto @@ -393,19 +393,19 @@ text {* A binary form of the Chinese Remainder Theorem. *} -lemma chinese_remainder: +lemma chinese_remainder: fixes a::nat assumes ab: "coprime a b" and a: "a \ 0" and b: "b \ 0" shows "\x q1 q2. x = u + q1 * a \ x = v + q2 * b" proof- from bezout_add_strong_nat[OF a, of b] bezout_add_strong_nat[OF b, of a] - obtain d1 x1 y1 d2 x2 y2 where dxy1: "d1 dvd a" "d1 dvd b" "a * x1 = b * y1 + d1" + obtain d1 x1 y1 d2 x2 y2 where dxy1: "d1 dvd a" "d1 dvd b" "a * x1 = b * y1 + d1" and dxy2: "d2 dvd b" "d2 dvd a" "b * x2 = a * y2 + d2" by blast then have d12: "d1 = 1" "d2 =1" by (metis ab coprime_nat)+ let ?x = "v * a * x1 + u * b * x2" let ?q1 = "v * x1 + u * y2" let ?q2 = "v * y1 + u * x2" - from dxy2(3)[simplified d12] dxy1(3)[simplified d12] + from dxy2(3)[simplified d12] dxy1(3)[simplified d12] have "?x = u + ?q1 * a" "?x = v + ?q2 * b" by algebra+ thus ?thesis by blast @@ -418,14 +418,14 @@ shows "\x y. a * x = b * y + 1" by (metis assms bezout_nat gcd_nat.left_neutral) -lemma bezout_prime: +lemma bezout_prime: assumes p: "prime p" and pa: "\ p dvd a" shows "\x y. a*x = Suc (p*y)" proof- have ap: "coprime a p" - by (metis gcd_nat.commute p pa prime_imp_coprime_nat) + by (metis gcd_nat.commute p pa prime_imp_coprime_nat) from coprime_bezout_strong[OF ap] show ?thesis - by (metis Suc_eq_plus1 gcd_lcm_complete_lattice_nat.bot.extremum pa) + by (metis Suc_eq_plus1 gcd_lcm_complete_lattice_nat.bot.extremum pa) qed end diff -r 1c937d56a70a -r de7792ea4090 src/HOL/Transcendental.thy --- a/src/HOL/Transcendental.thy Tue Mar 10 15:21:26 2015 +0000 +++ b/src/HOL/Transcendental.thy Tue Mar 10 16:12:35 2015 +0000 @@ -7,7 +7,7 @@ section{*Power Series, Transcendental Functions etc.*} theory Transcendental -imports Fact Series Deriv NthRoot +imports Binomial Series Deriv NthRoot begin lemma root_test_convergence: @@ -81,13 +81,13 @@ lemma power_diff_1_eq: fixes x :: "'a::{comm_ring,monoid_mult}" shows "n \ 0 \ x^n - 1 = (x - 1) * (\i 0 \ 1 - x^n = (1 - x) * (\iexp x::real\ = exp x" by simp -(*FIXME: superseded by exp_of_nat_mult*) -lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n" +(*FIXME: superseded by exp_of_nat_mult*) +lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n" by (induct n) (auto simp add: real_of_nat_Suc distrib_left exp_add mult.commute) - + text {* Strict monotonicity of exponential. *} -lemma exp_ge_add_one_self_aux: +lemma exp_ge_add_one_self_aux: assumes "0 \ (x::real)" shows "1+x \ exp(x)" using order_le_imp_less_or_eq [OF assms] -proof +proof assume "0 < x" have "1+x \ (\n<2. inverse (real (fact n)) * x ^ n)" by (auto simp add: numeral_2_eq_2) @@ -1189,7 +1189,7 @@ using `0 < x` apply (auto simp add: zero_le_mult_iff) done - finally show "1+x \ exp x" + finally show "1+x \ exp x" by (simp add: exp_def) next assume "0 = x" @@ -1443,7 +1443,7 @@ proof - have "exp x = suminf (\n. inverse(fact n) * (x ^ n))" by (simp add: exp_def) - also from summable_exp have "... = (\ n. inverse(fact(n+2)) * (x ^ (n+2))) + + also from summable_exp have "... = (\ n. inverse(fact(n+2)) * (x ^ (n+2))) + (\ n::nat<2. inverse(fact n) * (x ^ n))" (is "_ = _ + ?a") by (rule suminf_split_initial_segment) also have "?a = 1 + x" @@ -1536,7 +1536,7 @@ ultimately have "1 - x <= 1 / (1 + x + x\<^sup>2)" by (elim mult_imp_le_div_pos) also have "... <= 1 / exp x" - by (metis a abs_one b exp_bound exp_gt_zero frac_le less_eq_real_def real_sqrt_abs + by (metis a abs_one b exp_bound exp_gt_zero frac_le less_eq_real_def real_sqrt_abs real_sqrt_pow2_iff real_sqrt_power) also have "... = exp (-x)" by (auto simp add: exp_minus divide_inverse) @@ -1584,7 +1584,7 @@ qed finally have "exp (x - x\<^sup>2) <= exp (ln (1 + x))" . thus ?thesis - by (metis exp_le_cancel_iff) + by (metis exp_le_cancel_iff) qed lemma ln_one_minus_pos_lower_bound: @@ -1690,7 +1690,7 @@ also have "... = 1 + (y - x) / x" using x a by (simp add: field_simps) also have "x * ln(1 + (y - x) / x) <= x * ((y - x) / x)" - using x a + using x a by (intro mult_left_mono ln_add_one_self_le_self) simp_all also have "... = y - x" using a by simp also have "... = (y - x) * ln (exp 1)" by simp @@ -2204,7 +2204,7 @@ unfolding powr_def exp_inj_iff by simp lemma ln_powr_bound: "1 <= x ==> 0 < a ==> ln x <= (x powr a) / a" - by (metis less_eq_real_def ln_less_self mult_imp_le_div_pos ln_powr mult.commute + by (metis less_eq_real_def ln_less_self mult_imp_le_div_pos ln_powr mult.commute order.strict_trans2 powr_gt_zero zero_less_one) lemma ln_powr_bound2: @@ -2302,7 +2302,7 @@ have "((\y. ln (1 + x * y)::real) has_real_derivative 1 * x) (at 0)" by (auto intro!: derivative_eq_intros) then have "((\y. ln (1 + x * y) / y) ---> x) (at 0)" - by (auto simp add: has_field_derivative_def field_has_derivative_at) + by (auto simp add: has_field_derivative_def field_has_derivative_at) then have *: "((\y. exp (ln (1 + x * y) / y)) ---> exp x) (at 0)" by (rule tendsto_intros) then show ?thesis @@ -2367,15 +2367,15 @@ unfolding cos_coeff_def sin_coeff_def by (simp del: mult_Suc) (auto elim: oddE) -lemma summable_norm_sin: +lemma summable_norm_sin: fixes x :: "'a::{real_normed_algebra_1,banach}" shows "summable (\n. norm (sin_coeff n *\<^sub>R x^n))" - unfolding sin_coeff_def + unfolding sin_coeff_def apply (rule summable_comparison_test [OF _ summable_norm_exp [where x=x]]) apply (auto simp: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff) done -lemma summable_norm_cos: +lemma summable_norm_cos: fixes x :: "'a::{real_normed_algebra_1,banach}" shows "summable (\n. norm (cos_coeff n *\<^sub>R x ^ n))" unfolding cos_coeff_def @@ -2405,7 +2405,7 @@ by (rule sin_converges) finally have "(\n. of_real (sin_coeff n *\<^sub>R x^n)) sums (sin (of_real x))" . then show ?thesis - using sums_unique2 sums_of_real [OF sin_converges] + using sums_unique2 sums_of_real [OF sin_converges] by blast qed @@ -2423,7 +2423,7 @@ by (rule cos_converges) finally have "(\n. of_real (cos_coeff n *\<^sub>R x^n)) sums (cos (of_real x))" . then show ?thesis - using sums_unique2 sums_of_real [OF cos_converges] + using sums_unique2 sums_of_real [OF cos_converges] by blast qed @@ -2441,22 +2441,22 @@ unfolding sin_def cos_def scaleR_conv_of_real apply (rule DERIV_cong) apply (rule termdiffs [where K="of_real (norm x) + 1 :: 'a"]) - apply (simp_all add: norm_less_p1 diffs_of_real diffs_sin_coeff diffs_cos_coeff + apply (simp_all add: norm_less_p1 diffs_of_real diffs_sin_coeff diffs_cos_coeff summable_minus_iff scaleR_conv_of_real [symmetric] summable_norm_sin [THEN summable_norm_cancel] summable_norm_cos [THEN summable_norm_cancel]) done - + declare DERIV_sin[THEN DERIV_chain2, derivative_intros] -lemma DERIV_cos [simp]: +lemma DERIV_cos [simp]: fixes x :: "'a::{real_normed_field,banach}" shows "DERIV cos x :> -sin(x)" unfolding sin_def cos_def scaleR_conv_of_real apply (rule DERIV_cong) apply (rule termdiffs [where K="of_real (norm x) + 1 :: 'a"]) - apply (simp_all add: norm_less_p1 diffs_of_real diffs_minus suminf_minus - diffs_sin_coeff diffs_cos_coeff + apply (simp_all add: norm_less_p1 diffs_of_real diffs_minus suminf_minus + diffs_sin_coeff diffs_cos_coeff summable_minus_iff scaleR_conv_of_real [symmetric] summable_norm_sin [THEN summable_norm_cancel] summable_norm_cos [THEN summable_norm_cancel]) @@ -2469,7 +2469,7 @@ shows "isCont sin x" by (rule DERIV_sin [THEN DERIV_isCont]) -lemma isCont_cos: +lemma isCont_cos: fixes x :: "'a::{real_normed_field,banach}" shows "isCont cos x" by (rule DERIV_cos [THEN DERIV_isCont]) @@ -2481,7 +2481,7 @@ (*FIXME A CONTEXT FOR F WOULD BE BETTER*) -lemma isCont_cos' [simp]: +lemma isCont_cos' [simp]: fixes f:: "_ \ 'a::{real_normed_field,banach}" shows "isCont f a \ isCont (\x. cos (f x)) a" by (rule isCont_o2 [OF _ isCont_cos]) @@ -2545,23 +2545,23 @@ subsection {*Deriving the Addition Formulas*} text{*The the product of two cosine series*} -lemma cos_x_cos_y: +lemma cos_x_cos_y: fixes x :: "'a::{real_normed_field,banach}" - shows "(\p. \n\p. - if even p \ even n - then ((-1) ^ (p div 2) * (p choose n) / of_nat (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) + shows "(\p. \n\p. + if even p \ even n + then ((-1) ^ (p div 2) * (p choose n) / of_nat (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) sums (cos x * cos y)" proof - { fix n p::nat assume "n\p" then have *: "even n \ even p \ (-1) ^ (n div 2) * (-1) ^ ((p - n) div 2) = (-1 :: real) ^ (p div 2)" by (metis div_add power_add le_add_diff_inverse odd_add) - have "(cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)) = + have "(cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)) = (if even p \ even n then ((-1) ^ (p div 2) * (p choose n) / of_nat (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)" using `n\p` by (auto simp: * algebra_simps cos_coeff_def binomial_fact real_of_nat_def) - } - then have "(\p. \n\p. if even p \ even n + } + then have "(\p. \n\p. if even p \ even n then ((-1) ^ (p div 2) * (p choose n) / of_nat (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) = (\p. \n\p. (cos_coeff n * cos_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)))" by simp @@ -2574,11 +2574,11 @@ qed text{*The product of two sine series*} -lemma sin_x_sin_y: +lemma sin_x_sin_y: fixes x :: "'a::{real_normed_field,banach}" - shows "(\p. \n\p. - if even p \ odd n - then - ((-1) ^ (p div 2) * (p choose n) / of_nat (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) + shows "(\p. \n\p. + if even p \ odd n + then - ((-1) ^ (p div 2) * (p choose n) / of_nat (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) sums (sin x * sin y)" proof - { fix n p::nat @@ -2594,13 +2594,13 @@ apply (metis (no_types) One_nat_def Suc_1 le_div_geq minus_minus mult.left_neutral mult_minus_left power.simps(2) zero_less_Suc) done } then - have "(sin_coeff n * sin_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)) = - (if even p \ odd n + have "(sin_coeff n * sin_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)) = + (if even p \ odd n then -((-1) ^ (p div 2) * (p choose n) / of_nat (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)" using `n\p` by (auto simp: algebra_simps sin_coeff_def binomial_fact real_of_nat_def) - } - then have "(\p. \n\p. if even p \ odd n + } + then have "(\p. \n\p. if even p \ odd n then - ((-1) ^ (p div 2) * (p choose n) / of_nat (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) = (\p. \n\p. (sin_coeff n * sin_coeff (p - n)) *\<^sub>R (x^n * y^(p-n)))" by simp @@ -2612,18 +2612,18 @@ finally show ?thesis . qed -lemma sums_cos_x_plus_y: +lemma sums_cos_x_plus_y: fixes x :: "'a::{real_normed_field,banach}" shows - "(\p. \n\p. if even p - then ((-1) ^ (p div 2) * (p choose n) / of_nat (fact p)) *\<^sub>R (x^n) * y^(p-n) - else 0) + "(\p. \n\p. if even p + then ((-1) ^ (p div 2) * (p choose n) / of_nat (fact p)) *\<^sub>R (x^n) * y^(p-n) + else 0) sums cos (x + y)" proof - { fix p::nat have "(\n\p. if even p then ((-1) ^ (p div 2) * (p choose n) / of_nat (fact p)) *\<^sub>R (x^n) * y^(p-n) - else 0) = + else 0) = (if even p then \n\p. ((-1) ^ (p div 2) * (p choose n) / of_nat (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)" @@ -2637,11 +2637,11 @@ finally have "(\n\p. if even p then ((-1) ^ (p div 2) * (p choose n) / of_nat (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) = cos_coeff p *\<^sub>R ((x + y) ^ p)" . - } - then have "(\p. \n\p. - if even p - then ((-1) ^ (p div 2) * (p choose n) / of_nat (fact p)) *\<^sub>R (x^n) * y^(p-n) - else 0) + } + then have "(\p. \n\p. + if even p + then ((-1) ^ (p div 2) * (p choose n) / of_nat (fact p)) *\<^sub>R (x^n) * y^(p-n) + else 0) = (\p. cos_coeff p *\<^sub>R ((x+y)^p))" by simp also have "... sums cos (x + y)" @@ -2649,22 +2649,22 @@ finally show ?thesis . qed -theorem cos_add: +theorem cos_add: fixes x :: "'a::{real_normed_field,banach}" shows "cos (x + y) = cos x * cos y - sin x * sin y" proof - { fix n p::nat assume "n\p" - then have "(if even p \ even n + then have "(if even p \ even n then ((- 1) ^ (p div 2) * int (p choose n) / of_nat (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) - - (if even p \ odd n + (if even p \ odd n then - ((- 1) ^ (p div 2) * int (p choose n) / of_nat (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0) - = (if even p + = (if even p then ((-1) ^ (p div 2) * (p choose n) / of_nat (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)" by simp - } - then have "(\p. \n\p. (if even p - then ((-1) ^ (p div 2) * (p choose n) / of_nat (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)) + } + then have "(\p. \n\p. (if even p + then ((-1) ^ (p div 2) * (p choose n) / of_nat (fact p)) *\<^sub>R (x^n) * y^(p-n) else 0)) sums (cos x * cos y - sin x * sin y)" using sums_diff [OF cos_x_cos_y [of x y] sin_x_sin_y [of x y]] by (simp add: setsum_subtractf [symmetric]) @@ -2683,7 +2683,7 @@ lemma sin_minus [simp]: fixes x :: "'a::{real_normed_algebra_1,banach}" shows "sin (-x) = -sin(x)" -using sin_minus_converges [of x] +using sin_minus_converges [of x] by (auto simp: sin_def summable_norm_sin [THEN summable_norm_cancel] suminf_minus sums_iff equation_minus_iff) lemma cos_minus_converges: "(\n. (cos_coeff n *\<^sub>R (-x)^n)) sums cos(x)" @@ -2698,72 +2698,72 @@ fixes x :: "'a::{real_normed_algebra_1,banach}" shows "cos (-x) = cos(x)" using cos_minus_converges [of x] -by (simp add: cos_def summable_norm_cos [THEN summable_norm_cancel] +by (simp add: cos_def summable_norm_cos [THEN summable_norm_cancel] suminf_minus sums_iff equation_minus_iff) - -lemma sin_cos_squared_add [simp]: + +lemma sin_cos_squared_add [simp]: fixes x :: "'a::{real_normed_field,banach}" shows "(sin x)\<^sup>2 + (cos x)\<^sup>2 = 1" using cos_add [of x "-x"] by (simp add: power2_eq_square algebra_simps) -lemma sin_cos_squared_add2 [simp]: +lemma sin_cos_squared_add2 [simp]: fixes x :: "'a::{real_normed_field,banach}" shows "(cos x)\<^sup>2 + (sin x)\<^sup>2 = 1" by (subst add.commute, rule sin_cos_squared_add) -lemma sin_cos_squared_add3 [simp]: +lemma sin_cos_squared_add3 [simp]: fixes x :: "'a::{real_normed_field,banach}" shows "cos x * cos x + sin x * sin x = 1" using sin_cos_squared_add2 [unfolded power2_eq_square] . -lemma sin_squared_eq: +lemma sin_squared_eq: fixes x :: "'a::{real_normed_field,banach}" shows "(sin x)\<^sup>2 = 1 - (cos x)\<^sup>2" unfolding eq_diff_eq by (rule sin_cos_squared_add) -lemma cos_squared_eq: +lemma cos_squared_eq: fixes x :: "'a::{real_normed_field,banach}" shows "(cos x)\<^sup>2 = 1 - (sin x)\<^sup>2" unfolding eq_diff_eq by (rule sin_cos_squared_add2) -lemma abs_sin_le_one [simp]: +lemma abs_sin_le_one [simp]: fixes x :: real shows "\sin x\ \ 1" by (rule power2_le_imp_le, simp_all add: sin_squared_eq) -lemma sin_ge_minus_one [simp]: +lemma sin_ge_minus_one [simp]: fixes x :: real shows "-1 \ sin x" using abs_sin_le_one [of x] unfolding abs_le_iff by simp -lemma sin_le_one [simp]: +lemma sin_le_one [simp]: fixes x :: real shows "sin x \ 1" using abs_sin_le_one [of x] unfolding abs_le_iff by simp -lemma abs_cos_le_one [simp]: +lemma abs_cos_le_one [simp]: fixes x :: real shows "\cos x\ \ 1" by (rule power2_le_imp_le, simp_all add: cos_squared_eq) -lemma cos_ge_minus_one [simp]: +lemma cos_ge_minus_one [simp]: fixes x :: real shows "-1 \ cos x" using abs_cos_le_one [of x] unfolding abs_le_iff by simp -lemma cos_le_one [simp]: +lemma cos_le_one [simp]: fixes x :: real shows "cos x \ 1" using abs_cos_le_one [of x] unfolding abs_le_iff by simp -lemma cos_diff: +lemma cos_diff: fixes x :: "'a::{real_normed_field,banach}" shows "cos (x - y) = cos x * cos y + sin x * sin y" using cos_add [of x "- y"] by simp -lemma cos_double: +lemma cos_double: fixes x :: "'a::{real_normed_field,banach}" shows "cos(2*x) = (cos x)\<^sup>2 - (sin x)\<^sup>2" using cos_add [where x=x and y=x] @@ -2786,7 +2786,7 @@ hence define pi.*} lemma sin_paired: - fixes x :: real + fixes x :: real shows "(\n. (- 1) ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1)) sums sin x" proof - have "(\n. \k = n*2.. x < 2 \ cos (2 * x) < 1" using sin_gt_zero_02 [where x = x] by (auto simp: cos_squared_eq cos_double) lemma cos_paired: - fixes x :: real + fixes x :: real shows "(\n. (- 1) ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x" proof - have "(\n. \k = n * 2.. (\n::int. even n & (x = real n * (pi/2)))" proof safe assume "sin x = 0" @@ -3261,7 +3261,7 @@ done next fix n::int - assume "even n" + assume "even n" then show "sin (real n * (pi / 2)) = 0" apply (simp add: sin_zero_iff) apply (case_tac n rule: int_cases2, simp) @@ -3271,8 +3271,8 @@ qed lemma sin_zero_iff_int2: "sin x = 0 \ (\n::int. x = real n * pi)" - apply (simp only: sin_zero_iff_int) - apply (safe elim!: evenE) + apply (simp only: sin_zero_iff_int) + apply (safe elim!: evenE) apply (simp_all add: field_simps) using dvd_triv_left by fastforce @@ -3337,7 +3337,7 @@ using pi_ge_two and assms by auto from cos_monotone_0_pi'[OF this] show ?thesis unfolding minus_sin_cos_eq[symmetric] - by (metis minus_sin_cos_eq mult.right_neutral neg_le_iff_le of_real_def real_scaleR_def) + by (metis minus_sin_cos_eq mult.right_neutral neg_le_iff_le of_real_def real_scaleR_def) qed lemma sin_x_le_x: @@ -3401,14 +3401,14 @@ lemma tan_add: fixes x :: "'a::{real_normed_field,banach}" - shows + shows "\cos x \ 0; cos y \ 0; cos (x + y) \ 0\ \ tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))" by (simp add: add_tan_eq lemma_tan_add1 field_simps) (simp add: tan_def) lemma tan_double: fixes x :: "'a::{real_normed_field,banach}" - shows + shows "\cos x \ 0; cos (2 * x) \ 0\ \ tan (2 * x) = (2 * tan x) / (1 - (tan x)\<^sup>2)" using tan_add [of x x] by (simp add: power2_eq_square) @@ -3463,7 +3463,7 @@ lemma continuous_within_tan [continuous_intros]: fixes f :: "'a \ 'a::{real_normed_field,banach}" - shows + shows "continuous (at x within s) f \ cos (f x) \ 0 \ continuous (at x within s) (\x. tan (f x))" unfolding continuous_within by (rule tendsto_tan) @@ -4200,7 +4200,7 @@ shows "x\<^sup>2 < 1" proof - have "\x\<^sup>2\ < 1" - by (metis abs_power2 assms pos2 power2_abs power_0 power_strict_decreasing zero_eq_power2 zero_less_abs_iff) + by (metis abs_power2 assms pos2 power2_abs power_0 power_strict_decreasing zero_eq_power2 zero_less_abs_iff) thus ?thesis using zero_le_power2 by auto qed @@ -4594,7 +4594,7 @@ done show ?thesis proof (cases "0::real" y rule: linorder_cases) - case less + case less then show ?thesis by (rule polar_ex1) next case equal @@ -4602,7 +4602,7 @@ by (force simp add: intro!: cos_zero sin_zero) next case greater - then show ?thesis + then show ?thesis using polar_ex1 [where y="-y"] by auto (metis cos_minus minus_minus minus_mult_right sin_minus) qed diff -r 1c937d56a70a -r de7792ea4090 src/HOL/ex/Birthday_Paradox.thy --- a/src/HOL/ex/Birthday_Paradox.thy Tue Mar 10 15:21:26 2015 +0000 +++ b/src/HOL/ex/Birthday_Paradox.thy Tue Mar 10 16:12:35 2015 +0000 @@ -5,14 +5,14 @@ section {* A Formulation of the Birthday Paradox *} theory Birthday_Paradox -imports Main "~~/src/HOL/Fact" "~~/src/HOL/Library/FuncSet" +imports Main "~~/src/HOL/Binomial" "~~/src/HOL/Library/FuncSet" begin section {* Cardinality *} lemma card_product_dependent: assumes "finite S" - assumes "\x \ S. finite (T x)" + assumes "\x \ S. finite (T x)" shows "card {(x, y). x \ S \ y \ T x} = (\x \ S. card (T x))" using card_SigmaI[OF assms, symmetric] by (auto intro!: arg_cong[where f=card] simp add: Sigma_def) @@ -30,7 +30,7 @@ from `finite S` this have "finite (extensional_funcset S (T - {x}))" by (rule finite_PiE) moreover - have "{f : extensional_funcset S (T - {x}). inj_on f S} \ (extensional_funcset S (T - {x}))" by auto + have "{f : extensional_funcset S (T - {x}). inj_on f S} \ (extensional_funcset S (T - {x}))" by auto ultimately have "finite {f : extensional_funcset S (T - {x}). inj_on f S}" by (auto intro: finite_subset) } note finite_delete = this @@ -62,7 +62,7 @@ have subset: "{f : extensional_funcset S T. inj_on f S} <= extensional_funcset S T" by auto from finite_subset[OF subset] assms have finite: "finite {f : extensional_funcset S T. inj_on f S}" by (auto intro!: finite_PiE) - have "{f \ extensional_funcset S T. \ inj_on f S} = extensional_funcset S T - {f \ extensional_funcset S T. inj_on f S}" by auto + have "{f \ extensional_funcset S T. \ inj_on f S} = extensional_funcset S T - {f \ extensional_funcset S T. inj_on f S}" by auto from assms this finite subset show ?thesis by (simp add: card_Diff_subset card_PiE card_extensional_funcset_inj_on setprod_constant) qed