# HG changeset patch # User paulson # Date 1426000840 0 # Node ID 651ea265d56811ec3b7ed20be0ee282dcae52d0a # Parent 37adca7fd48ff367a116f2ddaac77a619f055c56 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy diff -r 37adca7fd48f -r 651ea265d568 src/Doc/Tutorial/Sets/Examples.thy --- a/src/Doc/Tutorial/Sets/Examples.thy Tue Mar 10 11:56:32 2015 +0100 +++ b/src/Doc/Tutorial/Sets/Examples.thy Tue Mar 10 15:20:40 2015 +0000 @@ -1,4 +1,4 @@ -theory Examples imports "~~/src/HOL/Number_Theory/Binomial" begin +theory Examples imports Complex_Main begin declare [[eta_contract = false]] diff -r 37adca7fd48f -r 651ea265d568 src/HOL/Algebra/Exponent.thy --- a/src/HOL/Algebra/Exponent.thy Tue Mar 10 11:56:32 2015 +0100 +++ b/src/HOL/Algebra/Exponent.thy Tue Mar 10 15:20:40 2015 +0000 @@ -6,7 +6,7 @@ *) theory Exponent -imports Main "~~/src/HOL/Number_Theory/Primes" "~~/src/HOL/Number_Theory/Binomial" +imports Main "~~/src/HOL/Number_Theory/Primes" begin section {*Sylow's Theorem*} @@ -35,7 +35,7 @@ lemma prime_dvd_cases: fixes p::nat - shows "[| p*k dvd m*n; prime p |] + shows "[| p*k dvd m*n; prime p |] ==> (\x. k dvd x*n & m = p*x) | (\y. k dvd m*y & n = p*y)" apply (simp add: prime_iff) apply (frule dvd_mult_left) @@ -48,10 +48,10 @@ done -lemma prime_power_dvd_cases [rule_format (no_asm)]: +lemma prime_power_dvd_cases [rule_format (no_asm)]: fixes p::nat shows "prime p - ==> \m n. p^c dvd m*n --> + ==> \m n. p^c dvd m*n --> (\a b. a+b = Suc c --> p^a dvd m | p^b dvd n)" apply (induct c) apply (metis dvd_1_left nat_power_eq_Suc_0_iff one_is_add) @@ -119,7 +119,7 @@ lemma power_Suc_exponent_Not_dvd: "[|(p * p ^ exponent p s) dvd s; prime p |] ==> s=0" apply (subgoal_tac "p ^ Suc (exponent p s) dvd s") - prefer 2 apply simp + prefer 2 apply simp apply (rule ccontr) apply (drule exponent_ge, auto) done @@ -147,7 +147,7 @@ by (metis mult_dvd_mono power_exponent_dvd) (* exponent_mult_add, opposite inclusion *) -lemma exponent_mult_add2: "[| a > 0; b > 0 |] +lemma exponent_mult_add2: "[| a > 0; b > 0 |] ==> exponent p (a * b) <= (exponent p a) + (exponent p b)" apply (case_tac "prime p") apply (rule leI, clarify) @@ -155,7 +155,7 @@ apply (subgoal_tac "p ^ (Suc (exponent p a + exponent p b)) dvd a * b") apply (rule_tac [2] le_imp_power_dvd [THEN dvd_trans]) prefer 3 apply assumption - prefer 2 apply simp + prefer 2 apply simp apply (frule_tac a = "Suc (exponent p a) " and b = "Suc (exponent p b) " in prime_power_dvd_cases) apply (assumption, force, simp) apply (blast dest: power_Suc_exponent_Not_dvd) @@ -185,7 +185,7 @@ text{*Main Combinatorial Argument*} lemma gcd_mult': fixes a::nat shows "gcd b (a * b) = b" -by (simp add: mult.commute[of a b]) +by (simp add: mult.commute[of a b]) lemma le_extend_mult: "[| c > 0; a <= b |] ==> a <= b * (c::nat)" apply (rule_tac P = "%x. x <= b * c" in subst) @@ -204,7 +204,7 @@ apply (metis diff_is_0_eq dvd_diffD1 gcd_dvd2_nat gcd_mult' gr0I le_extend_mult less_diff_conv nat_dvd_not_less mult.commute not_add_less2 xt1(10)) done -lemma p_fac_forw: "[| (m::nat) > 0; k>0; k < p^a; (p^r) dvd (p^a)* m - k |] +lemma p_fac_forw: "[| (m::nat) > 0; k>0; k < p^a; (p^r) dvd (p^a)* m - k |] ==> (p^r) dvd (p^a) - k" apply (frule p_fac_forw_lemma [THEN le_imp_power_dvd, of _ k p], auto) apply (subgoal_tac "p^r dvd p^a*m") @@ -220,7 +220,7 @@ "[| (k::nat) > 0; k < p^a; p>0; (p^r) dvd (p^a) - k |] ==> r <= a" by (rule_tac m = "Suc 0" in p_fac_forw_lemma, auto) -lemma p_fac_backw: "[| m>0; k>0; (p::nat)\0; k < p^a; (p^r) dvd p^a - k |] +lemma p_fac_backw: "[| m>0; k>0; (p::nat)\0; k < p^a; (p^r) dvd p^a - k |] ==> (p^r) dvd (p^a)*m - k" apply (frule_tac k1 = k and p1 = p in r_le_a_forw [THEN le_imp_power_dvd], auto) apply (subgoal_tac "p^r dvd p^a*m") @@ -231,7 +231,7 @@ apply (drule less_imp_Suc_add, auto) done -lemma exponent_p_a_m_k_equation: "[| m>0; k>0; (p::nat)\0; k < p^a |] +lemma exponent_p_a_m_k_equation: "[| m>0; k>0; (p::nat)\0; k < p^a |] ==> exponent p (p^a * m - k) = exponent p (p^a - k)" apply (blast intro: exponent_equalityI p_fac_forw p_fac_backw) done @@ -241,16 +241,16 @@ (*The bound K is needed; otherwise it's too weak to be used.*) lemma p_not_div_choose_lemma [rule_format]: - "[| \i. Suc i < K --> exponent p (Suc i) = exponent p (Suc(j+i))|] + "[| \i. Suc i < K --> exponent p (Suc i) = exponent p (Suc(j+i))|] ==> k exponent p ((j+k) choose k) = 0" apply (cases "prime p") - prefer 2 apply simp + prefer 2 apply simp apply (induct k) apply (simp (no_asm)) (*induction step*) apply (subgoal_tac "(Suc (j+k) choose Suc k) > 0") prefer 2 apply (simp, clarify) -apply (subgoal_tac "exponent p ((Suc (j+k) choose Suc k) * Suc k) = +apply (subgoal_tac "exponent p ((Suc (j+k) choose Suc k) * Suc k) = exponent p (Suc k)") txt{*First, use the assumed equation. We simplify the LHS to @{term "exponent p (Suc (j + k) choose Suc k) + exponent p (Suc k)"} @@ -276,7 +276,7 @@ lemma const_p_fac_right: "m>0 ==> exponent p ((p^a * m - Suc 0) choose (p^a - Suc 0)) = 0" apply (case_tac "prime p") - prefer 2 apply simp + prefer 2 apply simp apply (frule_tac a = a in zero_less_prime_power) apply (rule_tac K = "p^a" in p_not_div_choose) apply simp @@ -294,14 +294,14 @@ lemma const_p_fac: "m>0 ==> exponent p (((p^a) * m) choose p^a) = exponent p m" apply (case_tac "prime p") - prefer 2 apply simp + prefer 2 apply simp apply (subgoal_tac "0 < p^a * m & p^a <= p^a * m") prefer 2 apply (force simp add: prime_iff) txt{*A similar trick to the one used in @{text p_not_div_choose_lemma}: insert an equation; use @{text exponent_mult_add} on the LHS; on the RHS, first transform the binomial coefficient, then use @{text exponent_mult_add}.*} -apply (subgoal_tac "exponent p ((( (p^a) * m) choose p^a) * p^a) = +apply (subgoal_tac "exponent p ((( (p^a) * m) choose p^a) * p^a) = a + exponent p m") apply (simp add: exponent_mult_add) txt{*one subgoal left!*} diff -r 37adca7fd48f -r 651ea265d568 src/HOL/Fact.thy --- a/src/HOL/Fact.thy Tue Mar 10 11:56:32 2015 +0100 +++ b/src/HOL/Fact.thy Tue Mar 10 15:20:40 2015 +0000 @@ -15,7 +15,7 @@ fixes fact :: "'a \ 'a" instantiation nat :: fact -begin +begin fun fact_nat :: "nat \ nat" @@ -31,11 +31,11 @@ instantiation int :: fact -begin +begin definition fact_int :: "int \ int" -where +where "fact_int x = (if x >= 0 then int (fact (nat x)) else 0)" instance proof qed @@ -55,7 +55,7 @@ "x >= (0::int) \ fact x >= 0" by (auto simp add: fact_int_def) -declare transfer_morphism_nat_int[transfer add return: +declare transfer_morphism_nat_int[transfer add return: transfer_nat_int_factorial transfer_nat_int_factorial_closure] lemma transfer_int_nat_factorial: @@ -66,7 +66,7 @@ "is_nat x \ fact x >= 0" by (auto simp add: fact_int_def) -declare transfer_morphism_int_nat[transfer add return: +declare transfer_morphism_int_nat[transfer add return: transfer_int_nat_factorial transfer_int_nat_factorial_closure] @@ -87,10 +87,10 @@ lemma fact_plus_one_nat: "fact ((n::nat) + 1) = (n + 1) * fact n" by simp -lemma fact_plus_one_int: +lemma fact_plus_one_int: assumes "n >= 0" shows "fact ((n::int) + 1) = (n + 1) * fact n" - using assms unfolding fact_int_def + 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)" @@ -153,7 +153,7 @@ apply auto done -lemma interval_plus_one_nat: "(i::nat) <= j + 1 \ +lemma interval_plus_one_nat: "(i::nat) <= j + 1 \ {i..j+1} = {i..j} Un {j+1}" by auto @@ -199,7 +199,7 @@ 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'}" + 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) @@ -224,7 +224,7 @@ apply arith done -lemma fact_mono_int_aux [rule_format]: "k >= (0::int) \ +lemma fact_mono_int_aux [rule_format]: "k >= (0::int) \ fact (m + k) >= fact m" apply (case_tac "m < 0") apply auto @@ -266,7 +266,7 @@ apply auto done -lemma fact_num_eq_if_nat: "fact (m::nat) = +lemma fact_num_eq_if_nat: "fact (m::nat) = (if m=0 then 1 else m * fact (m - 1))" by (cases m) auto @@ -275,7 +275,7 @@ by (cases "m + n") auto lemma fact_add_num_eq_if2_nat: - "fact ((m::nat) + n) = + "fact ((m::nat) + n) = (if m = 0 then fact n else (m + n) * fact ((m - 1) + n))" by (cases m) auto @@ -339,7 +339,7 @@ lemma binomial_Suc_Suc [simp]: "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)" by simp -lemma choose_reduce_nat: +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) @@ -404,7 +404,7 @@ {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 + 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]: @@ -455,7 +455,7 @@ 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 = +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 @@ -465,7 +465,7 @@ by auto have decomp2: "{0..n} = {0} Un {1..n}" by auto - have "(a+b)^(n+1) = + 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)) + @@ -476,14 +476,14 @@ 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 + 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) + + "\ = 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))" @@ -518,7 +518,7 @@ by simp from n h th0 have "fact k * fact (n - k) * (n choose k) = - k * (fact h * fact (m - h) * (m choose h)) + + 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" @@ -537,4 +537,644 @@ 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 37adca7fd48f -r 651ea265d568 src/HOL/Fields.thy --- a/src/HOL/Fields.thy Tue Mar 10 11:56:32 2015 +0100 +++ b/src/HOL/Fields.thy Tue Mar 10 15:20:40 2015 +0000 @@ -91,7 +91,7 @@ apply auto done -lemma inverse_unique: +lemma inverse_unique: assumes ab: "a * b = 1" shows "inverse a = b" proof - @@ -121,7 +121,7 @@ lemma inverse_1 [simp]: "inverse 1 = 1" by (rule inverse_unique) simp -lemma nonzero_inverse_mult_distrib: +lemma nonzero_inverse_mult_distrib: assumes "a \ 0" and "b \ 0" shows "inverse (a * b) = inverse b * inverse a" proof - @@ -199,7 +199,7 @@ proof - assume [simp]: "c \ 0" have "b / c = a \ (b / c) * c = a * c" by simp - also have "... \ b = a * c" by (simp add: divide_inverse mult.assoc) + also have "... \ b = a * c" by (simp add: divide_inverse mult.assoc) finally show ?thesis . qed @@ -262,7 +262,7 @@ proof cases assume "a=0" thus ?thesis by simp next - assume "a\0" + assume "a\0" thus ?thesis by (simp add: nonzero_inverse_minus_eq) qed @@ -271,7 +271,7 @@ proof cases assume "a=0" thus ?thesis by simp next - assume "a\0" + assume "a\0" thus ?thesis by (simp add: nonzero_inverse_inverse_eq) qed @@ -394,7 +394,7 @@ lemma divide_minus1 [simp]: "x / - 1 = - x" using nonzero_minus_divide_right [of "1" x] by simp - + end class field_inverse_zero = field + @@ -409,10 +409,10 @@ lemma inverse_mult_distrib [simp]: "inverse (a * b) = inverse a * inverse b" proof cases - assume "a \ 0 & b \ 0" + assume "a \ 0 & b \ 0" thus ?thesis by (simp add: nonzero_inverse_mult_distrib ac_simps) next - assume "~ (a \ 0 & b \ 0)" + assume "~ (a \ 0 & b \ 0)" thus ?thesis by force qed @@ -470,13 +470,13 @@ lemma minus_divide_divide: "(- a) / (- b) = a / b" -apply (cases "b=0", simp) -apply (simp add: nonzero_minus_divide_divide) +apply (cases "b=0", simp) +apply (simp add: nonzero_minus_divide_divide) done lemma inverse_eq_1_iff [simp]: "inverse x = 1 \ x = 1" - by (insert inverse_eq_iff_eq [of x 1], simp) + by (insert inverse_eq_iff_eq [of x 1], simp) lemma divide_eq_0_iff [simp]: "a / b = 0 \ a = 0 \ b = 0" @@ -489,7 +489,7 @@ done lemma divide_cancel_left [simp]: - "c / a = c / b \ c = 0 \ a = b" + "c / a = c / b \ c = 0 \ a = b" apply (cases "c=0", simp) apply (simp add: divide_inverse) done @@ -524,19 +524,19 @@ class linordered_field = field + linordered_idom begin -lemma positive_imp_inverse_positive: - assumes a_gt_0: "0 < a" +lemma positive_imp_inverse_positive: + assumes a_gt_0: "0 < a" shows "0 < inverse a" proof - - have "0 < a * inverse a" + have "0 < a * inverse a" by (simp add: a_gt_0 [THEN less_imp_not_eq2]) - thus "0 < inverse a" + thus "0 < inverse a" by (simp add: a_gt_0 [THEN less_not_sym] zero_less_mult_iff) qed lemma negative_imp_inverse_negative: "a < 0 \ inverse a < 0" - by (insert positive_imp_inverse_positive [of "-a"], + by (insert positive_imp_inverse_positive [of "-a"], simp add: nonzero_inverse_minus_eq less_imp_not_eq) lemma inverse_le_imp_le: @@ -577,7 +577,7 @@ proof fix x::'a have m1: "- (1::'a) < 0" by simp - from add_strict_right_mono[OF m1, where c=x] + from add_strict_right_mono[OF m1, where c=x] have "(- 1) + x < x" by simp thus "\y. y < x" by blast qed @@ -587,7 +587,7 @@ proof fix x::'a have m1: " (1::'a) > 0" by simp - from add_strict_right_mono[OF m1, where c=x] + from add_strict_right_mono[OF m1, where c=x] have "1 + x > x" by simp thus "\y. y > x" by blast qed @@ -606,13 +606,13 @@ lemma inverse_less_imp_less: "inverse a < inverse b \ 0 < a \ b < a" apply (simp add: less_le [of "inverse a"] less_le [of "b"]) -apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) +apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) done text{*Both premises are essential. Consider -1 and 1.*} lemma inverse_less_iff_less [simp]: "0 < a \ 0 < b \ inverse a < inverse b \ b < a" - by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) + by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) lemma le_imp_inverse_le: "a \ b \ 0 < a \ inverse b \