# HG changeset patch # User paulson # Date 1390576860 0 # Node ID 70db8d380d621625fe34e2dbda7afe8fe556e1b1 # Parent 11408b65e9a6676339ed4ebe555e93ced84c9de9 Restored Suc rather than +1, and using Library/Binimial diff -r 11408b65e9a6 -r 70db8d380d62 src/HOL/Number_Theory/Binomial.thy --- a/src/HOL/Number_Theory/Binomial.thy Thu Jan 23 16:09:28 2014 +0100 +++ b/src/HOL/Number_Theory/Binomial.thy Fri Jan 24 15:21:00 2014 +0000 @@ -2,238 +2,638 @@ Authors: Lawrence C. Paulson, Jeremy Avigad, Tobias Nipkow Defines the "choose" function, and establishes basic properties. - -The original theory "Binomial" was by Lawrence C. Paulson, based on -the work of Andy Gordon and Florian Kammueller. The approach here, -which derives the definition of binomial coefficients in terms of the -factorial function, is due to Jeremy Avigad. The binomial theorem was -formalized by Tobias Nipkow. *) header {* Binomial *} theory Binomial -imports Cong Fact +imports Cong Fact Complex_Main begin -subsection {* Main definitions *} +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) + ((n - 1) choose (k - 1))" + by (metis Suc_diff_1 binomial.simps(2) nat_add_commute neq0_conv) + +lemma binomial_eq_0: "n < k \ n choose k = 0" + by (induct n arbitrary: k) auto + +declare binomial.simps [simp del] -class binomial = - fixes binomial :: "'a \ 'a \ 'a" (infixl "choose" 65) +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) -(* definitions for the natural numbers *) +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) + +(*Might be more useful if re-oriented*) +lemma Suc_times_binomial_eq: + "k \ n \ Suc n * (n choose k) = (Suc n choose Suc k) * Suc k" + apply (induct n arbitrary: k) + apply (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{*This is the well-known version, but it's harder to use because of the + need to reason about division.*} +lemma binomial_Suc_Suc_eq_times: + "k \ n \ (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) -instantiation nat :: binomial -begin +text{*Another version, with -1 instead of Suc.*} +lemma times_binomial_minus1_eq: + "k \ n \ 0 < k \ (n choose k) * 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) -fun binomial_nat :: "nat \ nat \ nat" -where - "binomial_nat n k = - (if k = 0 then 1 else - if n = 0 then 0 else - (binomial (n - 1) k) + (binomial (n - 1) (k - 1)))" +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)", standard]) + apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset]) + done + qed +qed + + +subsection {* The binomial theorem (courtesy of Tobias Nipkow): *} -instance .. - -end +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) + 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 mult_ac) + 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_addf [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 -(* definitions for the integers *) +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) + +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) -instantiation int :: binomial -begin +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 + -definition binomial_int :: "int => int \ int" where - "binomial_int n k = - (if n \ 0 \ k \ 0 then int (binomial (nat n) (nat k)) - else 0)" +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 f = 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 f=Suc]) + apply (auto simp add: fun_eq_iff) + done -instance .. +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 -end +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 -subsection {* Set up Transfer *} +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) = setprod (%i. - 1) {0 .. h}" + using setprod_constant[where A="{0 .. h}" and y="- 1 :: 'a"] + by auto + show ?thesis + unfolding Suc pochhammer_Suc_setprod eq setprod_timesf[symmetric] + apply (rule strong_setprod_reindex_cong[where f = "%i. h - i"]) + using Suc + apply (auto simp add: inj_on_def image_def of_nat_diff) + apply (metis atLeast0AtMost atMost_iff diff_diff_cancel diff_le_self) + done +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 transfer_nat_int_binomial: - "(n::int) >= 0 \ k >= 0 \ binomial (nat n) (nat k) = - nat (binomial n k)" - unfolding binomial_int_def - by auto +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_timesf[symmetric]) +qed -lemma transfer_nat_int_binomial_closure: - "n >= (0::int) \ k >= 0 \ binomial n k >= 0" - by (auto simp add: binomial_int_def) +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_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]) -declare transfer_morphism_nat_int[transfer add return: - transfer_nat_int_binomial transfer_nat_int_binomial_closure] +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)" + apply (rule strong_setprod_reindex_cong[where f="op - n"]) + using h kn + apply (simp_all add: inj_on_def image_iff Bex_def set_eq_iff) + apply clarsimp + apply presburger + apply presburger + apply (simp add: fun_eq_iff field_simps of_nat_add[symmetric] del: of_nat_add) + done + 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_timesf[symmetric] eq' del: One_nat_def power_Suc) + unfolding setprod_Un_disjoint[OF th0, unfolded eq3, of "of_nat:: nat \ 'a"] eq[unfolded h] + unfolding mult_assoc[symmetric] + unfolding setprod_timesf[symmetric] + apply simp + apply (rule strong_setprod_reindex_cong[where f= "op - n"]) + apply (auto simp add: inj_on_def image_iff Bex_def) + apply presburger + apply (subgoal_tac "(of_nat (n - x) :: 'a) = of_nat n - of_nat x") + apply simp + apply (rule of_nat_diff) + apply simp + done + } + moreover + have "k > n \ k = 0 \ (k \ n \ k \ 0)" by arith + ultimately show ?thesis by blast +qed -lemma transfer_int_nat_binomial: - "binomial (int n) (int k) = int (binomial n k)" - unfolding fact_int_def binomial_int_def by auto +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 transfer_int_nat_binomial_closure: - "is_nat n \ is_nat k \ binomial n k >= 0" - by (auto simp add: binomial_int_def) +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 strong_setprod_reindex_cong[where f = 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 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 -declare transfer_morphism_int_nat[transfer add return: - transfer_int_nat_binomial transfer_int_nat_binomial_closure] +(* Contributed by Manuel Eberl *) +(* Alternative definition of the binomial coefficient as \i n" + shows "of_nat (n choose k) = (\ii = (\i n` unfolding fact_eq_rev_setprod_nat of_nat_setprod + by (auto simp add: setprod_dividef intro!: setprod_cong of_nat_diff[symmetric]) + finally show ?thesis . +next + case False + then show ?thesis by simp +qed + +lemma binomial_ge_n_over_k_pow_k: + fixes k n :: nat + and x :: "'a :: linordered_field_inverse_zero" + assumes "0 < k" + and "k \ n" + shows "(of_nat n / of_nat k :: 'a) ^ k \ of_nat (n choose k)" +proof - + have "(of_nat n / of_nat k :: 'a) ^ k = (\i \ of_nat (n choose k)" + unfolding binomial_altdef_of_nat[OF `k\n`] + proof (safe intro!: setprod_mono) + fix i :: nat + assume "i < k" + from assms have "n * i \ i * k" by simp + then have "n * k - n * i \ n * k - i * k" by arith + then have "n * (k - i) \ (n - i) * k" + by (simp add: diff_mult_distrib2 nat_mult_commute) + then have "of_nat n * of_nat (k - i) \ of_nat (n - i) * (of_nat k :: 'a)" + unfolding of_nat_mult[symmetric] of_nat_le_iff . + with assms show "of_nat n / of_nat k \ of_nat (n - i) / (of_nat (k - i) :: 'a)" + using `i < k` by (simp add: field_simps) + qed (simp add: zero_le_divide_iff) + finally show ?thesis . +qed + +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 + subsection {* Binomial coefficients *} -lemma choose_zero_nat [simp]: "(n::nat) choose 0 = 1" - by simp - -lemma choose_zero_int [simp]: "n \ 0 \ (n::int) choose 0 = 1" - by (simp add: binomial_int_def) - -lemma zero_choose_nat [rule_format,simp]: "ALL (k::nat) > n. n choose k = 0" - by (induct n rule: induct'_nat, auto) +lemma choose_plus_one_nat: + "((n::nat) + 1) choose (k + 1) =(n choose (k + 1)) + (n choose k)" + by (simp add: choose_reduce_nat) -lemma zero_choose_int [rule_format,simp]: "(k::int) > n \ n choose k = 0" - unfolding binomial_int_def - apply (cases "n < 0") - apply force - apply (simp del: binomial_nat.simps) - done - -lemma choose_reduce_nat: "(n::nat) > 0 \ 0 < k \ - (n choose k) = ((n - 1) choose k) + ((n - 1) choose (k - 1))" - by simp - -lemma choose_reduce_int: "(n::int) > 0 \ 0 < k \ - (n choose k) = ((n - 1) choose k) + ((n - 1) choose (k - 1))" - unfolding binomial_int_def - apply (subst choose_reduce_nat) - apply (auto simp del: binomial_nat.simps simp add: nat_diff_distrib) - done - -lemma choose_plus_one_nat: "((n::nat) + 1) choose (k + 1) = - (n choose (k + 1)) + (n choose k)" +lemma choose_Suc_nat: + "(Suc n) choose (Suc k) = (n choose (Suc k)) + (n choose k)" by (simp add: choose_reduce_nat) -lemma choose_Suc_nat: "(Suc n) choose (Suc k) = - (n choose (Suc k)) + (n choose k)" - by (simp add: choose_reduce_nat One_nat_def) - -lemma choose_plus_one_int: "n \ 0 \ k \ 0 \ ((n::int) + 1) choose (k + 1) = - (n choose (k + 1)) + (n choose k)" - by (simp add: binomial_int_def choose_plus_one_nat nat_add_distrib del: binomial_nat.simps) - -declare binomial_nat.simps [simp del] - -lemma choose_self_nat [simp]: "((n::nat) choose n) = 1" - by (induct n rule: induct'_nat) (auto simp add: choose_plus_one_nat) - -lemma choose_self_int [simp]: "n \ 0 \ ((n::int) choose n) = 1" - by (auto simp add: binomial_int_def) - -lemma choose_one_nat [simp]: "(n::nat) choose 1 = n" - by (induct n rule: induct'_nat) (auto simp add: choose_reduce_nat) - -lemma choose_one_int [simp]: "n \ 0 \ (n::int) choose 1 = n" - by (auto simp add: binomial_int_def) - -lemma plus_one_choose_self_nat [simp]: "(n::nat) + 1 choose n = n + 1" - apply (induct n rule: induct'_nat, force) - apply (case_tac "n = 0") - apply auto - apply (subst choose_reduce_nat) - apply (auto simp add: One_nat_def) - (* natdiff_cancel_numerals introduces Suc *) -done - -lemma Suc_choose_self_nat [simp]: "(Suc n) choose n = Suc n" - using plus_one_choose_self_nat by (simp add: One_nat_def) - -lemma plus_one_choose_self_int [rule_format, simp]: - "(n::int) \ 0 \ n + 1 choose n = n + 1" - by (auto simp add: binomial_int_def nat_add_distrib) - -(* bounded quantification doesn't work with the unicode characters? *) -lemma choose_pos_nat [rule_format]: "ALL k <= (n::nat). - ((n::nat) choose k) > 0" - apply (induct n rule: induct'_nat) - apply force - apply clarify - apply (case_tac "k = 0") - apply force - apply (subst choose_reduce_nat) - apply auto - done - -lemma choose_pos_int: "n \ 0 \ k >= 0 \ k \ n \ - ((n::int) choose k) > 0" - by (auto simp add: binomial_int_def choose_pos_nat) +lemma choose_one: "(n::nat) choose 1 = n" + by simp lemma binomial_induct [rule_format]: "(ALL (n::nat). P n n) \ - (ALL n. P (n + 1) 0) \ (ALL n. (ALL k < n. P n k \ P n (k + 1) \ - P (n + 1) (k + 1))) \ (ALL k <= n. P n k)" - apply (induct n rule: induct'_nat) + (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 = n + 1") - apply auto - apply (drule_tac x = n in spec) back back - apply (drule_tac x = "k - 1" in spec) back back back - apply auto - done - -lemma choose_altdef_aux_nat: "(k::nat) \ n \ - fact k * fact (n - k) * (n choose k) = fact n" - apply (rule binomial_induct [of _ k n]) + apply (case_tac "k = Suc n") apply auto -proof - - fix k :: nat and n - assume less: "k < n" - assume ih1: "fact k * fact (n - k) * (n choose k) = fact n" - then have one: "fact (k + 1) * fact (n - k) * (n choose k) = (k + 1) * fact n" - by (subst fact_plus_one_nat, auto) - assume ih2: "fact (k + 1) * fact (n - (k + 1)) * (n choose (k + 1)) = fact n" - with less have "fact (k + 1) * fact ((n - (k + 1)) + 1) * - (n choose (k + 1)) = (n - k) * fact n" - by (subst (2) fact_plus_one_nat, auto) - with less have two: "fact (k + 1) * fact (n - k) * (n choose (k + 1)) = - (n - k) * fact n" by simp - have "fact (k + 1) * fact (n - k) * (n + 1 choose (k + 1)) = - fact (k + 1) * fact (n - k) * (n choose (k + 1)) + - fact (k + 1) * fact (n - k) * (n choose k)" - by (subst choose_reduce_nat, auto simp add: field_simps) - also note one - also note two - also from less have "(n - k) * fact n + (k + 1) * fact n= fact (n + 1)" - apply (subst fact_plus_one_nat) - apply (subst distrib_right [symmetric]) - apply simp - done - finally show "fact (k + 1) * fact (n - k) * (n + 1 choose (k + 1)) = - fact (n + 1)" . -qed - -lemma choose_altdef_nat: "(k::nat) \ n \ - n choose k = fact n div (fact k * fact (n - k))" - apply (frule choose_altdef_aux_nat) - apply (erule subst) - apply (simp add: mult_ac) - done - - -lemma choose_altdef_int: - assumes "(0::int) <= k" and "k <= n" - shows "n choose k = fact n div (fact k * fact (n - k))" - apply (subst tsub_eq [symmetric], rule assms) - apply (rule choose_altdef_nat [transferred]) - using assms apply auto + apply (metis Suc_le_eq fact_nat.cases le_Suc_eq le_eq_less_or_eq) done lemma choose_dvd_nat: "(k::nat) \ n \ fact k * fact (n - k) dvd fact n" - unfolding dvd_def apply (frule choose_altdef_aux_nat) - (* why don't blast and auto get this??? *) - apply (rule exI) - apply (erule sym) - done +by (metis binomial_fact_lemma dvd_def) lemma choose_dvd_int: assumes "(0::int) <= k" and "k <= n" @@ -243,125 +643,6 @@ using assms apply auto done -(* generalizes Tobias Nipkow's proof to any commutative semiring *) -theorem binomial: "(a+b::'a::{comm_ring_1,power})^n = - (SUM k=0..n. (of_nat (n choose k)) * a^k * b^(n-k))" (is "?P n") -proof (induct n rule: induct'_nat) - show "?P 0" by simp -next - fix n - assume ih: "?P 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 decomp3: "{1..n+1} = {n+1} Un {1..n}" - by auto - have "(a+b)^(n+1) = - (a+b) * (SUM k=0..n. of_nat (n choose k) * a^k * b^(n-k))" - using ih by simp - also have "... = a*(SUM k=0..n. of_nat (n choose k) * a^k * b^(n-k)) + - b*(SUM k=0..n. of_nat (n choose k) * a^k * b^(n-k))" - by (rule distrib) - also have "... = (SUM k=0..n. of_nat (n choose k) * a^(k+1) * b^(n-k)) + - (SUM k=0..n. of_nat (n choose k) * a^k * b^(n-k+1))" - by (subst (1 2) power_plus_one, simp add: setsum_right_distrib mult_ac) - also have "... = (SUM k=0..n. of_nat (n choose k) * a^k * b^(n+1-k)) + - (SUM 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 One_nat_def del:setsum_cl_ivl_Suc) - also have "... = a^(n+1) + b^(n+1) + - (SUM k=1..n. of_nat (n choose (k - 1)) * a^k * b^(n+1-k)) + - (SUM k=1..n. of_nat (n choose k) * a^k * b^(n+1-k))" - by (simp add: decomp2 decomp3) - also have - "... = a^(n+1) + b^(n+1) + - (SUM k=1..n. of_nat(n+1 choose k) * a^k * b^(n+1-k))" - by (auto simp add: field_simps setsum_addf [symmetric] - choose_reduce_nat) - also have "... = (SUM 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 (n + 1)" by simp -qed - -lemma card_subsets_nat: - fixes S :: "'a set" - shows "finite S \ card {T. T \ S \ card T = k} = card S choose k" -proof (induct arbitrary: k set: finite) - case empty - show ?case by (auto simp add: Collect_conv_if) -next - case (insert x F) - note iassms = insert(1,2) - note ih = insert(3) - show ?case - proof (induct k rule: induct'_nat) - case zero - from iassms have "{T. T \ (insert x F) \ card T = 0} = {{}}" - by (auto simp: finite_subset) - then show ?case by auto - next - case (plus1 k) - from iassms have fin: "finite (insert x F)" by auto - then have "{ T. T \ insert x F \ card T = k + 1} = - {T. T \ F & card T = k + 1} Un - {T. T \ insert x F & x : T & card T = k + 1}" - by auto - with iassms fin have "card ({T. T \ insert x F \ card T = k + 1}) = - card ({T. T \ F \ card T = k + 1}) + - card ({T. T \ insert x F \ x : T \ card T = k + 1})" - apply (subst card_Un_disjoint [symmetric]) - apply auto - (* note: nice! Didn't have to say anything here *) - done - also from ih have "card ({T. T \ F \ card T = k + 1}) = - card F choose (k+1)" by auto - also have "card ({T. T \ insert x F \ x : T \ card T = k + 1}) = - card ({T. T <= F & card T = k})" - proof - - let ?f = "%T. T Un {x}" - from iassms have "inj_on ?f {T. T <= F & card T = k}" - unfolding inj_on_def by auto - then have "card ({T. T <= F & card T = k}) = - card(?f ` {T. T <= F & card T = k})" - by (rule card_image [symmetric]) - also have "?f ` {T. T <= F & card T = k} = - {T. T \ insert x F \ x : T \ card T = k + 1}" (is "?L=?R") - proof- - { fix S assume "S \ F" - then have "card(insert x S) = card S +1" - using iassms by (auto simp: finite_subset) } - moreover - { fix T assume 1: "T \ insert x F" "x : T" "card T = k+1" - let ?S = "T - {x}" - have "?S <= F & card ?S = k \ T = insert x ?S" - using 1 fin by (auto simp: finite_subset) } - ultimately show ?thesis by(auto simp: image_def) - qed - finally show ?thesis by (rule sym) - qed - also from ih have "card ({T. T <= F & card T = k}) = card F choose k" - by auto - finally have "card ({T. T \ insert x F \ card T = k + 1}) = - card F choose (k + 1) + (card F choose k)". - with iassms choose_plus_one_nat show ?case - by (auto simp del: card.insert) - qed -qed - -lemma choose_le_pow: - assumes "r \ n" shows "n choose r \ n ^ r" -proof - - have X: "\r. r \ n \ \{n - r..n} = (n - r) * \{Suc (n - r)..n}" - by (subst setprod_insert[symmetric]) (auto simp: atLeastAtMost_insertL) - have "n choose r \ fact n div fact (n - r)" - using `r \ n` by (simp add: choose_altdef_nat div_le_mono2) - also have "\ \ n ^ r" using `r \ n` - by (induct r rule: nat.induct) - (auto simp add: fact_div_fact Suc_diff_Suc X One_nat_def mult_le_mono) - finally show ?thesis . -qed - 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)))" @@ -384,12 +665,11 @@ 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)", standard] simp add: card_gt_0_iff[folded Suc_le_eq] One_nat_def dest: finite_subset intro: card_mono) + using assms by(auto intro!: rev_image_eqI[where x="(card a, a)", standard] 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 f=snd]) fastforce also have "\ = (\i=1..card A. (\I|I \ A \ card I = i \ x \ \I. -1 ^ (i + 1)))" @@ -420,13 +700,13 @@ 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 add: One_nat_def) + 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 card_subsets_nat[symmetric]) simp_all + 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)(simp add: mult_ac) + by(subst binomial_ring)(simp add: mult_ac) also have "\ = 1" using x K by(auto simp add: K_def card_gt_0_iff) finally show "?lhs x = 1" . qed diff -r 11408b65e9a6 -r 70db8d380d62 src/HOL/Number_Theory/Cong.thy --- a/src/HOL/Number_Theory/Cong.thy Thu Jan 23 16:09:28 2014 +0100 +++ b/src/HOL/Number_Theory/Cong.thy Fri Jan 24 15:21:00 2014 +0000 @@ -31,34 +31,9 @@ subsection {* Turn off @{text One_nat_def} *} -lemma induct'_nat [case_names zero plus1, induct type: nat]: - "P (0::nat) \ (\n. P n \ P (n + 1)) \ P n" - by (induct n) (simp_all add: One_nat_def) - -lemma cases_nat [case_names zero plus1, cases type: nat]: - "P (0::nat) \ (\n. P (n + 1)) \ P n" - by (rule induct'_nat) - -lemma power_plus_one [simp]: "(x::'a::power)^(n + 1) = x * x^n" - by (simp add: One_nat_def) - lemma power_eq_one_eq_nat [simp]: "((x::nat)^m = 1) = (m = 0 | x = 1)" by (induct m) auto -lemma card_insert_if' [simp]: "finite A \ - card (insert x A) = (if x \ A then (card A) else (card A) + 1)" - by (auto simp add: insert_absorb) - -lemma nat_1' [simp]: "nat 1 = 1" - by simp - -(* For those annoying moments where Suc reappears, use Suc_eq_plus1 *) - -declare nat_1 [simp del] -declare add_2_eq_Suc [simp del] -declare add_2_eq_Suc' [simp del] - - declare mod_pos_pos_trivial [simp] @@ -106,11 +81,8 @@ "(x::int) >= 0 \ y >= 0 \ m >= 0 \ ([(nat x) = (nat y)] (mod (nat m))) = ([x = y] (mod m))" unfolding cong_int_def cong_nat_def - apply (auto simp add: nat_mod_distrib [symmetric]) - apply (subst (asm) eq_nat_nat_iff) - apply (cases "m = 0", force, rule pos_mod_sign, force)+ - apply assumption - done + by (metis Divides.transfer_int_nat_functions(2) nat_0_le nat_mod_distrib) + declare transfer_morphism_nat_int[transfer add return: transfer_nat_int_cong] @@ -138,7 +110,7 @@ unfolding cong_nat_def by auto lemma cong_Suc_0_nat [simp, presburger]: "[(a::nat) = b] (mod Suc 0)" - unfolding cong_nat_def by (auto simp add: One_nat_def) + unfolding cong_nat_def by auto lemma cong_1_int [simp, presburger]: "[(a::int) = b] (mod 1)" unfolding cong_int_def by auto @@ -171,32 +143,20 @@ lemma cong_add_nat: "[(a::nat) = b] (mod m) \ [c = d] (mod m) \ [a + c = b + d] (mod m)" - apply (unfold cong_nat_def) - apply (subst (1 2) mod_add_eq) - apply simp - done + unfolding cong_nat_def by (metis mod_add_cong) lemma cong_add_int: "[(a::int) = b] (mod m) \ [c = d] (mod m) \ [a + c = b + d] (mod m)" - apply (unfold cong_int_def) - apply (subst (1 2) mod_add_left_eq) - apply (subst (1 2) mod_add_right_eq) - apply simp - done + unfolding cong_int_def by (metis mod_add_cong) lemma cong_diff_int: "[(a::int) = b] (mod m) \ [c = d] (mod m) \ [a - c = b - d] (mod m)" - apply (unfold cong_int_def) - apply (subst (1 2) mod_diff_eq) - apply simp - done + unfolding cong_int_def by (metis mod_diff_cong) lemma cong_diff_aux_int: "(a::int) >= c \ b >= d \ [(a::int) = b] (mod m) \ [c = d] (mod m) \ [tsub a c = tsub b d] (mod m)" - apply (subst (1 2) tsub_eq) - apply (auto intro: cong_diff_int) - done + by (metis cong_diff_int tsub_eq) lemma cong_diff_nat: assumes "(a::nat) >= c" and "b >= d" and "[a = b] (mod m)" and @@ -206,19 +166,11 @@ lemma cong_mult_nat: "[(a::nat) = b] (mod m) \ [c = d] (mod m) \ [a * c = b * d] (mod m)" - apply (unfold cong_nat_def) - apply (subst (1 2) mod_mult_eq) - apply simp - done + unfolding cong_nat_def by (metis mod_mult_cong) lemma cong_mult_int: "[(a::int) = b] (mod m) \ [c = d] (mod m) \ [a * c = b * d] (mod m)" - apply (unfold cong_int_def) - apply (subst (1 2) mod_mult_right_eq) - apply (subst (1 2) mult_commute) - apply (subst (1 2) mod_mult_right_eq) - apply simp - done + unfolding cong_int_def by (metis mod_mult_cong) lemma cong_exp_nat: "[(x::nat) = y] (mod n) \ [x^k = y^k] (mod n)" by (induct k) (auto simp add: cong_mult_nat) @@ -277,10 +229,7 @@ unfolding cong_int_def by auto lemma cong_eq_diff_cong_0_int: "[(a::int) = b] (mod m) = [a - b = 0] (mod m)" - apply (rule iffI) - apply (erule cong_diff_int [of a b m b b, simplified]) - apply (erule cong_add_int [of "a - b" 0 m b b, simplified]) - done + by (metis cong_add_int cong_diff_int cong_refl_int diff_add_cancel diff_self) lemma cong_eq_diff_cong_0_aux_int: "a >= b \ [(a::int) = b] (mod m) = [tsub a b = 0] (mod m)" @@ -294,14 +243,7 @@ lemma cong_diff_cong_0'_nat: "[(x::nat) = y] (mod n) \ (if x <= y then [y - x = 0] (mod n) else [x - y = 0] (mod n))" - apply (cases "y <= x") - apply (frule cong_eq_diff_cong_0_nat [where m = n]) - apply auto [1] - apply (subgoal_tac "x <= y") - apply (frule cong_eq_diff_cong_0_nat [where m = n]) - apply (subst cong_sym_eq_nat) - apply auto - done + by (metis cong_eq_diff_cong_0_nat cong_sym_nat nat_le_linear) lemma cong_altdef_nat: "(a::nat) >= b \ [a = b] (mod m) = (m dvd (a - b))" apply (subst cong_eq_diff_cong_0_nat, assumption) @@ -447,12 +389,6 @@ apply auto done -lemma cong_zero_nat: "[(a::nat) = b] (mod 0) = (a = b)" - by auto - -lemma cong_zero_int: "[(a::int) = b] (mod 0) = (a = b)" - by auto - (* lemma mod_dvd_mod_int: "0 < (m::int) \ m dvd b \ (a mod b mod m) = (a mod m)" @@ -552,6 +488,9 @@ apply auto done +lemma cong_0_1_nat': "[(0::nat) = Suc 0] (mod n) = (n = Suc 0)" + unfolding cong_nat_def by auto + lemma cong_0_1_nat: "[(0::nat) = 1] (mod n) = (n = 1)" unfolding cong_nat_def by auto @@ -565,7 +504,7 @@ apply (drule_tac x = "a - 1" in spec) apply force apply (cases "a = 0") - apply (auto simp add: cong_0_1_nat) [1] + apply (auto simp add: cong_0_1_nat') [1] apply (rule iffI) apply (drule cong_to_1_nat) apply (unfold dvd_def) @@ -667,9 +606,10 @@ apply auto done -lemma coprime_iff_invertible_nat: "m > (1::nat) \ coprime a m = (EX x. [a * x = 1] (mod m))" +lemma coprime_iff_invertible_nat: "m > Suc 0 \ coprime a m = (EX x. [a * x = Suc 0] (mod m))" apply (auto intro: cong_solve_coprime_nat) - apply (unfold cong_nat_def, auto intro: invertible_coprime_nat) + apply (metis cong_solve_nat gcd_nat.left_neutral nat_neq_iff) + apply (unfold cong_nat_def, auto intro: invertible_coprime_nat [unfolded One_nat_def]) done lemma coprime_iff_invertible_int: "m > (1::int) \ coprime a m = (EX x. [a * x = 1] (mod m))" diff -r 11408b65e9a6 -r 70db8d380d62 src/HOL/Number_Theory/Eratosthenes.thy --- a/src/HOL/Number_Theory/Eratosthenes.thy Thu Jan 23 16:09:28 2014 +0100 +++ b/src/HOL/Number_Theory/Eratosthenes.thy Fri Jan 24 15:21:00 2014 +0000 @@ -257,7 +257,7 @@ proof (cases "n > 1") case False then have "n = 0 \ n = 1" by arith then show ?thesis - by (auto simp add: numbers_of_marks_sieve One_nat_def numeral_2_eq_2 set_primes_upto dest: prime_gt_Suc_0_nat) + by (auto simp add: numbers_of_marks_sieve numeral_2_eq_2 set_primes_upto dest: prime_gt_Suc_0_nat) next { fix m q assume "Suc (Suc 0) \ q" @@ -280,9 +280,9 @@ \m\{Suc (Suc 0).. q dvd m \ m dvd q \ m \ q \ m = 1" by auto case True then show ?thesis - apply (auto simp add: numbers_of_marks_sieve One_nat_def numeral_2_eq_2 set_primes_upto dest: prime_gt_Suc_0_nat) - apply (simp add: prime_nat_def dvd_def) - apply (auto simp add: prime_nat_def aux) + apply (auto simp add: numbers_of_marks_sieve numeral_2_eq_2 set_primes_upto dest: prime_gt_Suc_0_nat) + apply (metis One_nat_def Suc_le_eq less_not_refl prime_nat_def) + apply (metis One_nat_def Suc_le_eq aux prime_nat_def) done qed diff -r 11408b65e9a6 -r 70db8d380d62 src/HOL/Number_Theory/Primes.thy --- a/src/HOL/Number_Theory/Primes.thy Thu Jan 23 16:09:28 2014 +0100 +++ b/src/HOL/Number_Theory/Primes.thy Fri Jan 24 15:21:00 2014 +0000 @@ -31,6 +31,8 @@ imports "~~/src/HOL/GCD" begin +declare One_nat_def [simp] + class prime = one + fixes prime :: "'a \ bool" @@ -172,10 +174,7 @@ by (induct n) auto lemma prime_dvd_power_int: "prime (p::int) \ p dvd x^n \ p dvd x" - apply (induct n) - apply (frule prime_ge_0_int) - apply auto - done + by (induct n) (auto simp: prime_ge_0_int) lemma prime_dvd_power_nat_iff: "prime (p::nat) \ n > 0 \ p dvd x^n \ p dvd x" @@ -198,7 +197,7 @@ by (simp add: prime_nat_def) lemma Suc_0_not_prime_nat [simp]: "~prime (Suc 0)" - by (simp add: prime_nat_def One_nat_def) + by (simp add: prime_nat_def) lemma one_not_prime_int [simp]: "~prime (1::int)" by (simp add: prime_int_def) @@ -206,7 +205,7 @@ lemma prime_nat_code [code]: "prime (p::nat) \ p > 1 \ (\n \ {1<.. ~ p dvd a \ coprime a (p^m)" - apply (rule coprime_exp_nat) - apply (subst gcd_commute_nat) - apply (erule (1) prime_imp_coprime_nat) - done + by (metis coprime_exp_nat gcd_nat.commute prime_imp_coprime_nat) lemma prime_imp_power_coprime_int: "prime (p::int) \ ~ p dvd a \ coprime a (p^m)" - apply (rule coprime_exp_int) - apply (subst gcd_commute_int) - apply (erule (1) prime_imp_coprime_int) - done + by (metis coprime_exp_int gcd_int.commute prime_imp_coprime_int) lemma primes_coprime_nat: "prime (p::nat) \ prime q \ p \ q \ coprime p q" - apply (rule prime_imp_coprime_nat, assumption) - apply (unfold prime_nat_def) - apply auto - done + by (metis gcd_nat.absorb1 gcd_nat.absorb2 prime_imp_coprime_nat) lemma primes_coprime_int: "prime (p::int) \ prime q \ p \ q \ coprime p q" - apply (rule prime_imp_coprime_int, assumption) - apply (unfold prime_int_altdef) - apply (metis int_one_le_iff_zero_less less_le) - done + by (metis leD linear prime_gt_0_int prime_imp_coprime_int prime_int_altdef) lemma primes_imp_powers_coprime_nat: "prime (p::nat) \ prime q \ p ~= q \ coprime (p^m) (q^n)" @@ -286,46 +273,6 @@ nat_dvd_not_less neq0_conv prime_nat_def) done -(* An Isar version: - -lemma prime_factor_b_nat: - fixes n :: nat - assumes "n \ 1" - shows "\p. prime p \ p dvd n" - -using `n ~= 1` -proof (induct n rule: less_induct_nat) - fix n :: nat - assume "n ~= 1" and - ih: "\m 1 \ (\p. prime p \ p dvd m)" - then show "\p. prime p \ p dvd n" - proof - - { - assume "n = 0" - moreover note two_is_prime_nat - ultimately have ?thesis - by (auto simp del: two_is_prime_nat) - } - moreover - { - assume "prime n" - then have ?thesis by auto - } - moreover - { - assume "n ~= 0" and "~ prime n" - with `n ~= 1` have "n > 1" by auto - with `~ prime n` and not_prime_eq_prod_nat obtain m k where - "n = m * k" and "1 < m" and "m < n" by blast - with ih obtain p where "prime p" and "p dvd m" by blast - with `n = m * k` have ?thesis by auto - } - ultimately show ?thesis by blast - qed -qed - -*) - text {* One property of coprimality is easier to prove via prime factors. *} lemma prime_divprod_pow_nat: diff -r 11408b65e9a6 -r 70db8d380d62 src/HOL/Number_Theory/UniqueFactorization.thy --- a/src/HOL/Number_Theory/UniqueFactorization.thy Thu Jan 23 16:09:28 2014 +0100 +++ b/src/HOL/Number_Theory/UniqueFactorization.thy Fri Jan 24 15:21:00 2014 +0000 @@ -14,9 +14,6 @@ imports Cong "~~/src/HOL/Library/Multiset" begin -(* inherited from Multiset *) -declare One_nat_def [simp del] - (* As a simp or intro rule, prime p \ p > 0 @@ -290,9 +287,6 @@ using assms apply auto done -lemma neq_zero_eq_gt_zero_nat: "((x::nat) ~= 0) = (x > 0)" - by auto - lemma prime_factorization_unique_nat: fixes f :: "nat \ _" assumes S_eq: "S = {p. 0 < f p}" and "finite S" @@ -412,18 +406,19 @@ lemma multiplicity_zero_int [simp]: "multiplicity (p::int) 0 = 0" by (simp add: multiplicity_int_def) -lemma multiplicity_one_nat [simp]: "multiplicity p (1::nat) = 0" +lemma multiplicity_one_nat': "multiplicity p (1::nat) = 0" by (subst multiplicity_characterization_nat [where f = "%x. 0"], auto) +lemma multiplicity_one_nat [simp]: "multiplicity p (Suc 0) = 0" + by (metis One_nat_def multiplicity_one_nat') + lemma multiplicity_one_int [simp]: "multiplicity p (1::int) = 0" - by (simp add: multiplicity_int_def) + by (metis multiplicity_int_def multiplicity_one_nat' transfer_nat_int_numerals(2)) lemma multiplicity_prime_nat [simp]: "prime (p::nat) \ multiplicity p p = 1" apply (subst multiplicity_characterization_nat [where f = "(%q. if q = p then 1 else 0)"]) apply auto - apply (case_tac "x = p") - apply auto - done + by (metis (full_types) less_not_refl) lemma multiplicity_prime_int [simp]: "prime (p::int) \ multiplicity p p = 1" unfolding prime_int_def multiplicity_int_def by auto @@ -433,9 +428,7 @@ apply auto apply (subst multiplicity_characterization_nat [where f = "(%q. if q = p then n else 0)"]) apply auto - apply (case_tac "x = p") - apply auto - done + by (metis (full_types) less_not_refl) lemma multiplicity_prime_power_int [simp]: "prime (p::int) \ multiplicity p (p^n) = n" apply (frule prime_ge_0_int) @@ -464,6 +457,7 @@ apply auto done +(*FIXME: messy*) lemma multiplicity_product_aux_nat: "(k::nat) > 0 \ l > 0 \ (prime_factors k) Un (prime_factors l) = prime_factors (k * l) & (ALL p. multiplicity p k + multiplicity p l = multiplicity p (k * l))" @@ -472,13 +466,13 @@ apply auto apply (subst power_add) apply (subst setprod_timesf) - apply (rule arg_cong2)back back + apply (rule arg_cong2 [where f = "\x y. x*y"]) apply (subgoal_tac "prime_factors k Un prime_factors l = prime_factors k Un (prime_factors l - prime_factors k)") apply (erule ssubst) apply (subst setprod_Un_disjoint) apply auto - apply(simp add: prime_factorization_nat) + apply (metis One_nat_def nat_mult_1_right prime_factorization_nat setprod.neutral_const) apply (subgoal_tac "prime_factors k Un prime_factors l = prime_factors l Un (prime_factors k - prime_factors l)") apply (erule ssubst) @@ -486,8 +480,8 @@ apply auto apply (subgoal_tac "(\p\prime_factors k - prime_factors l. p ^ multiplicity p l) = (\p\prime_factors k - prime_factors l. 1)") - apply (simp add: prime_factorization_nat) - apply (rule setprod_cong, auto) + apply auto + apply (metis One_nat_def nat_mult_1_right prime_factorization_nat setprod.neutral_const) done (* transfer doesn't have the same problem here with the right @@ -639,13 +633,13 @@ "0 < (y::nat) \ x dvd y \ prime_factors x <= prime_factors y" apply (simp only: prime_factors_altdef_nat) apply auto - apply (metis dvd_multiplicity_nat le_0_eq neq_zero_eq_gt_zero_nat) + apply (metis dvd_multiplicity_nat le_0_eq neq0_conv) done lemma dvd_prime_factors_int [intro]: "0 < (y::int) \ 0 <= x \ x dvd y \ prime_factors x <= prime_factors y" apply (auto simp add: prime_factors_altdef_int) - apply (metis dvd_multiplicity_int le_0_eq neq_zero_eq_gt_zero_nat) + apply (metis dvd_multiplicity_int le_0_eq neq0_conv) done lemma multiplicity_dvd_nat: "0 < (x::nat) \ 0 < y \