src/HOL/Library/Binomial.thy
 author huffman Thu Jun 11 09:03:24 2009 -0700 (2009-06-11) changeset 31563 ded2364d14d4 parent 31287 6c593b431f04 child 32042 df28ead1cf19 child 32158 4dc119d4fc8b permissions -rw-r--r--
cleaned up some proofs
1 (*  Title:      HOL/Binomial.thy
2     Author:     Lawrence C Paulson, Amine Chaieb
3     Copyright   1997  University of Cambridge
4 *)
6 header {* Binomial Coefficients *}
8 theory Binomial
9 imports Fact SetInterval Presburger Main Rational
10 begin
12 text {* This development is based on the work of Andy Gordon and
13   Florian Kammueller. *}
15 primrec binomial :: "nat \<Rightarrow> nat \<Rightarrow> nat" (infixl "choose" 65) where
16   binomial_0: "(0 choose k) = (if k = 0 then 1 else 0)"
17   | binomial_Suc: "(Suc n choose k) =
18                  (if k = 0 then 1 else (n choose (k - 1)) + (n choose k))"
20 lemma binomial_n_0 [simp]: "(n choose 0) = 1"
21 by (cases n) simp_all
23 lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0"
24 by simp
26 lemma binomial_Suc_Suc [simp]:
27   "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)"
28 by simp
30 lemma binomial_eq_0: "!!k. n < k ==> (n choose k) = 0"
31 by (induct n) auto
33 declare binomial_0 [simp del] binomial_Suc [simp del]
35 lemma binomial_n_n [simp]: "(n choose n) = 1"
36 by (induct n) (simp_all add: binomial_eq_0)
38 lemma binomial_Suc_n [simp]: "(Suc n choose n) = Suc n"
39 by (induct n) simp_all
41 lemma binomial_1 [simp]: "(n choose Suc 0) = n"
42 by (induct n) simp_all
44 lemma zero_less_binomial: "k \<le> n ==> (n choose k) > 0"
45 by (induct n k rule: diff_induct) simp_all
47 lemma binomial_eq_0_iff: "(n choose k = 0) = (n<k)"
48 apply (safe intro!: binomial_eq_0)
49 apply (erule contrapos_pp)
51 done
53 lemma zero_less_binomial_iff: "(n choose k > 0) = (k\<le>n)"
54 by(simp add: linorder_not_less binomial_eq_0_iff neq0_conv[symmetric]
55         del:neq0_conv)
57 (*Might be more useful if re-oriented*)
58 lemma Suc_times_binomial_eq:
59   "!!k. k \<le> n ==> Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
60 apply (induct n)
62 apply (case_tac k)
64     binomial_eq_0)
65 done
67 text{*This is the well-known version, but it's harder to use because of the
68   need to reason about division.*}
69 lemma binomial_Suc_Suc_eq_times:
70     "k \<le> n ==> (Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k"
71   by (simp add: Suc_times_binomial_eq div_mult_self_is_m zero_less_Suc
72     del: mult_Suc mult_Suc_right)
74 text{*Another version, with -1 instead of Suc.*}
75 lemma times_binomial_minus1_eq:
76     "[|k \<le> n;  0<k|] ==> (n choose k) * k = n * ((n - 1) choose (k - 1))"
77   apply (cut_tac n = "n - 1" and k = "k - 1" in Suc_times_binomial_eq)
78   apply (simp split add: nat_diff_split, auto)
79   done
82 subsection {* Theorems about @{text "choose"} *}
84 text {*
85   \medskip Basic theorem about @{text "choose"}.  By Florian
86   Kamm\"uller, tidied by LCP.
87 *}
89 lemma card_s_0_eq_empty:
90     "finite A ==> card {B. B \<subseteq> A & card B = 0} = 1"
93 lemma choose_deconstruct: "finite M ==> x \<notin> M
94   ==> {s. s <= insert x M & card(s) = Suc k}
95        = {s. s <= M & card(s) = Suc k} Un
96          {s. EX t. t <= M & card(t) = k & s = insert x t}"
97   apply safe
98    apply (auto intro: finite_subset [THEN card_insert_disjoint])
99   apply (drule_tac x = "xa - {x}" in spec)
100   apply (subgoal_tac "x \<notin> xa", auto)
101   apply (erule rev_mp, subst card_Diff_singleton)
102   apply (auto intro: finite_subset)
103   done
104 (*
105 lemma "finite(UN y. {x. P x y})"
106 apply simp
107 lemma Collect_ex_eq
109 lemma "{x. EX y. P x y} = (UN y. {x. P x y})"
110 apply blast
111 *)
113 lemma finite_bex_subset[simp]:
114   "finite B \<Longrightarrow> (!!A. A<=B \<Longrightarrow> finite{x. P x A}) \<Longrightarrow> finite{x. EX A<=B. P x A}"
115 apply(subgoal_tac "{x. EX A<=B. P x A} = (UN A:Pow B. {x. P x A})")
116  apply simp
117 apply blast
118 done
120 text{*There are as many subsets of @{term A} having cardinality @{term k}
121  as there are sets obtained from the former by inserting a fixed element
122  @{term x} into each.*}
123 lemma constr_bij:
124    "[|finite A; x \<notin> A|] ==>
125     card {B. EX C. C <= A & card(C) = k & B = insert x C} =
126     card {B. B <= A & card(B) = k}"
127 apply (rule_tac f = "%s. s - {x}" and g = "insert x" in card_bij_eq)
128      apply (auto elim!: equalityE simp add: inj_on_def)
129 apply (subst Diff_insert0, auto)
130 done
132 text {*
133   Main theorem: combinatorial statement about number of subsets of a set.
134 *}
136 lemma n_sub_lemma:
137     "!!A. finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
138   apply (induct k)
139    apply (simp add: card_s_0_eq_empty, atomize)
140   apply (rotate_tac -1, erule finite_induct)
141    apply (simp_all (no_asm_simp) cong add: conj_cong
143   apply (subst card_Un_disjoint)
144      prefer 4 apply (force simp add: constr_bij)
145     prefer 3 apply force
146    prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2]
147      finite_subset [of _ "Pow (insert x F)", standard])
148   apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset])
149   done
151 theorem n_subsets:
152     "finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
156 text{* The binomial theorem (courtesy of Tobias Nipkow): *}
158 theorem binomial: "(a+b::nat)^n = (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"
159 proof (induct n)
160   case 0 thus ?case by simp
161 next
162   case (Suc n)
163   have decomp: "{0..n+1} = {0} \<union> {n+1} \<union> {1..n}"
164     by (auto simp add:atLeastAtMost_def atLeast_def atMost_def)
165   have decomp2: "{0..n} = {0} \<union> {1..n}"
166     by (auto simp add:atLeastAtMost_def atLeast_def atMost_def)
167   have "(a+b::nat)^(n+1) = (a+b) * (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"
168     using Suc by simp
169   also have "\<dots> =  a*(\<Sum>k=0..n. (n choose k) * a^k * b^(n-k)) +
170                    b*(\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"
171     by (rule nat_distrib)
172   also have "\<dots> = (\<Sum>k=0..n. (n choose k) * a^(k+1) * b^(n-k)) +
173                   (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k+1))"
174     by (simp add: setsum_right_distrib mult_ac)
175   also have "\<dots> = (\<Sum>k=0..n. (n choose k) * a^k * b^(n+1-k)) +
176                   (\<Sum>k=1..n+1. (n choose (k - 1)) * a^k * b^(n+1-k))"
178              del:setsum_cl_ivl_Suc)
179   also have "\<dots> = a^(n+1) + b^(n+1) +
180                   (\<Sum>k=1..n. (n choose (k - 1)) * a^k * b^(n+1-k)) +
181                   (\<Sum>k=1..n. (n choose k) * a^k * b^(n+1-k))"
183   also have
184       "\<dots> = a^(n+1) + b^(n+1) + (\<Sum>k=1..n. (n+1 choose k) * a^k * b^(n+1-k))"
186   also have "\<dots> = (\<Sum>k=0..n+1. (n+1 choose k) * a^k * b^(n+1-k))"
187     using decomp by simp
188   finally show ?case by simp
189 qed
191 subsection{* Pochhammer's symbol : generalized raising factorial*}
193 definition "pochhammer (a::'a::comm_semiring_1) n = (if n = 0 then 1 else setprod (\<lambda>n. a + of_nat n) {0 .. n - 1})"
195 lemma pochhammer_0[simp]: "pochhammer a 0 = 1"
198 lemma pochhammer_1[simp]: "pochhammer a 1 = a" by (simp add: pochhammer_def)
199 lemma pochhammer_Suc0[simp]: "pochhammer a (Suc 0) = a"
202 lemma pochhammer_Suc_setprod: "pochhammer a (Suc n) = setprod (\<lambda>n. a + of_nat n) {0 .. n}"
205 lemma setprod_nat_ivl_Suc: "setprod f {0 .. Suc n} = setprod f {0..n} * f (Suc n)"
206 proof-
207   have th: "finite {0..n}" "finite {Suc n}" "{0..n} \<inter> {Suc n} = {}" by auto
208   have eq: "{0..Suc n} = {0..n} \<union> {Suc n}" by auto
209   show ?thesis unfolding eq setprod_Un_disjoint[OF th] by simp
210 qed
212 lemma setprod_nat_ivl_1_Suc: "setprod f {0 .. Suc n} = f 0 * setprod f {1.. Suc n}"
213 proof-
214   have th: "finite {0}" "finite {1..Suc n}" "{0} \<inter> {1.. Suc n} = {}" by auto
215   have eq: "{0..Suc n} = {0} \<union> {1 .. Suc n}" by auto
216   show ?thesis unfolding eq setprod_Un_disjoint[OF th] by simp
217 qed
220 lemma pochhammer_Suc: "pochhammer a (Suc n) = pochhammer a n * (a + of_nat n)"
221 proof-
222   {assume "n=0" then have ?thesis by simp}
223   moreover
224   {fix m assume m: "n = Suc m"
225     have ?thesis  unfolding m pochhammer_Suc_setprod setprod_nat_ivl_Suc ..}
226   ultimately show ?thesis by (cases n, auto)
227 qed
229 lemma pochhammer_rec: "pochhammer a (Suc n) = a * pochhammer (a + 1) n"
230 proof-
231   {assume "n=0" then have ?thesis by (simp add: pochhammer_Suc_setprod)}
232   moreover
233   {assume n0: "n \<noteq> 0"
234     have th0: "finite {1 .. n}" "0 \<notin> {1 .. n}" by auto
235     have eq: "insert 0 {1 .. n} = {0..n}" by auto
236     have th1: "(\<Prod>n\<in>{1\<Colon>nat..n}. a + of_nat n) =
237       (\<Prod>n\<in>{0\<Colon>nat..n - 1}. a + 1 + of_nat n)"
238       apply (rule setprod_reindex_cong[where f = "Suc"])
239       using n0 by (auto simp add: expand_fun_eq ring_simps)
240     have ?thesis apply (simp add: pochhammer_def)
241     unfolding setprod_insert[OF th0, unfolded eq]
242     using th1 by (simp add: ring_simps)}
243 ultimately show ?thesis by blast
244 qed
246 lemma fact_setprod: "fact n = setprod id {1 .. n}"
247   apply (induct n, simp)
248   apply (simp only: fact_Suc atLeastAtMostSuc_conv)
249   apply (subst setprod_insert)
250   by simp_all
252 lemma pochhammer_fact: "of_nat (fact n) = pochhammer 1 n"
253   unfolding fact_setprod
255   apply (cases n, simp_all add: of_nat_setprod pochhammer_Suc_setprod)
256   apply (rule setprod_reindex_cong[where f=Suc])
257   by (auto simp add: expand_fun_eq)
259 lemma pochhammer_of_nat_eq_0_lemma: assumes kn: "k > n"
260   shows "pochhammer (- (of_nat n :: 'a:: idom)) k = 0"
261 proof-
262   from kn obtain h where h: "k = Suc h" by (cases k, auto)
263   {assume n0: "n=0" then have ?thesis using kn
264       by (cases k, simp_all add: pochhammer_rec del: pochhammer_Suc)}
265   moreover
266   {assume n0: "n \<noteq> 0"
267     then have ?thesis apply (simp add: h pochhammer_Suc_setprod)
268   apply (rule_tac x="n" in bexI)
269   using h kn by auto}
270 ultimately show ?thesis by blast
271 qed
273 lemma pochhammer_of_nat_eq_0_lemma': assumes kn: "k \<le> n"
274   shows "pochhammer (- (of_nat n :: 'a:: {idom, ring_char_0})) k \<noteq> 0"
275 proof-
276   {assume "k=0" then have ?thesis by simp}
277   moreover
278   {fix h assume h: "k = Suc h"
279     then have ?thesis apply (simp add: pochhammer_Suc_setprod)
280       using h kn by (auto simp add: algebra_simps)}
281   ultimately show ?thesis by (cases k, auto)
282 qed
284 lemma pochhammer_of_nat_eq_0_iff:
285   shows "pochhammer (- (of_nat n :: 'a:: {idom, ring_char_0})) k = 0 \<longleftrightarrow> k > n"
286   (is "?l = ?r")
287   using pochhammer_of_nat_eq_0_lemma[of n k, where ?'a='a]
288     pochhammer_of_nat_eq_0_lemma'[of k n, where ?'a = 'a]
289   by (auto simp add: not_le[symmetric])
291 subsection{* Generalized binomial coefficients *}
293 definition gbinomial :: "'a::field_char_0 \<Rightarrow> nat \<Rightarrow> 'a" (infixl "gchoose" 65)
294   where "a gchoose n = (if n = 0 then 1 else (setprod (\<lambda>i. a - of_nat i) {0 .. n - 1}) / of_nat (fact n))"
296 lemma gbinomial_0[simp]: "a gchoose 0 = 1" "0 gchoose (Suc n) = 0"
298 apply (subgoal_tac "(\<Prod>i\<Colon>nat\<in>{0\<Colon>nat..n}. - of_nat i) = (0::'b)")
299  apply (simp del:setprod_zero_iff)
300 apply simp
301 done
303 lemma gbinomial_pochhammer: "a gchoose n = (- 1) ^ n * pochhammer (- a) n / of_nat (fact n)"
304 proof-
305   {assume "n=0" then have ?thesis by simp}
306   moreover
307   {assume n0: "n\<noteq>0"
308     from n0 setprod_constant[of "{0 .. n - 1}" "- (1:: 'a)"]
309     have eq: "(- (1\<Colon>'a)) ^ n = setprod (\<lambda>i. - 1) {0 .. n - 1}"
310       by auto
311     from n0 have ?thesis
312       by (simp add: pochhammer_def gbinomial_def field_simps eq setprod_timesf[symmetric])}
313   ultimately show ?thesis by blast
314 qed
316 lemma binomial_fact_lemma:
317   "k \<le> n \<Longrightarrow> fact k * fact (n - k) * (n choose k) = fact n"
318 proof(induct n arbitrary: k rule: nat_less_induct)
319   fix n k assume H: "\<forall>m<n. \<forall>x\<le>m. fact x * fact (m - x) * (m choose x) =
320                       fact m" and kn: "k \<le> n"
321     let ?ths = "fact k * fact (n - k) * (n choose k) = fact n"
322   {assume "n=0" then have ?ths using kn by simp}
323   moreover
324   {assume "k=0" then have ?ths using kn by simp}
325   moreover
326   {assume nk: "n=k" then have ?ths by simp}
327   moreover
328   {fix m h assume n: "n = Suc m" and h: "k = Suc h" and hm: "h < m"
329     from n have mn: "m < n" by arith
330     from hm have hm': "h \<le> m" by arith
331     from hm h n kn have km: "k \<le> m" by arith
332     have "m - h = Suc (m - Suc h)" using  h km hm by arith
333     with km h have th0: "fact (m - h) = (m - h) * fact (m - k)"
334       by simp
335     from n h th0
336     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))"
338     also have "\<dots> = (k + (m - h)) * fact m"
339       using H[rule_format, OF mn hm'] H[rule_format, OF mn km]
341     finally have ?ths using h n km by simp}
342   moreover have "n=0 \<or> k = 0 \<or> k = n \<or> (EX m h. n=Suc m \<and> k = Suc h \<and> h < m)" using kn by presburger
343   ultimately show ?ths by blast
344 qed
346 lemma binomial_fact:
347   assumes kn: "k \<le> n"
348   shows "(of_nat (n choose k) :: 'a::field_char_0) = of_nat (fact n) / (of_nat (fact k) * of_nat (fact (n - k)))"
349   using binomial_fact_lemma[OF kn]
350   by (simp add: field_simps fact_not_eq_zero of_nat_mult[symmetric])
352 lemma binomial_gbinomial: "of_nat (n choose k) = of_nat n gchoose k"
353 proof-
354   {assume kn: "k > n"
355     from kn binomial_eq_0[OF kn] have ?thesis
356       by (simp add: gbinomial_pochhammer field_simps
357 	pochhammer_of_nat_eq_0_iff)}
358   moreover
359   {assume "k=0" then have ?thesis by simp}
360   moreover
361   {assume kn: "k \<le> n" and k0: "k\<noteq> 0"
362     from k0 obtain h where h: "k = Suc h" by (cases k, auto)
363     from h
364     have eq:"(- 1 :: 'a) ^ k = setprod (\<lambda>i. - 1) {0..h}"
365       by (subst setprod_constant, auto)
366     have eq': "(\<Prod>i\<in>{0..h}. of_nat n + - (of_nat i :: 'a)) = (\<Prod>i\<in>{n - h..n}. of_nat i)"
367       apply (rule strong_setprod_reindex_cong[where f="op - n"])
368       using h kn
369       apply (simp_all add: inj_on_def image_iff Bex_def expand_set_eq)
370       apply clarsimp
371       apply (presburger)
372       apply presburger
374     have th0: "finite {1..n - Suc h}" "finite {n - h .. n}"
375 "{1..n - Suc h} \<inter> {n - h .. n} = {}" and eq3: "{1..n - Suc h} \<union> {n - h .. n} = {1..n}" using h kn by auto
376     from eq[symmetric]
377     have ?thesis using kn
378       apply (simp add: binomial_fact[OF kn, where ?'a = 'a]
379 	gbinomial_pochhammer field_simps pochhammer_Suc_setprod)
380       apply (simp add: pochhammer_Suc_setprod fact_setprod h of_nat_setprod setprod_timesf[symmetric] eq' del: One_nat_def power_Suc)
381       unfolding setprod_Un_disjoint[OF th0, unfolded eq3, of "of_nat:: nat \<Rightarrow> 'a"] eq[unfolded h]
382       unfolding mult_assoc[symmetric]
383       unfolding setprod_timesf[symmetric]
384       apply simp
385       apply (rule strong_setprod_reindex_cong[where f= "op - n"])
386       apply (auto simp add: inj_on_def image_iff Bex_def)
387       apply presburger
388       apply (subgoal_tac "(of_nat (n - x) :: 'a) = of_nat n - of_nat x")
389       apply simp
390       by (rule of_nat_diff, simp)
391   }
392   moreover
393   have "k > n \<or> k = 0 \<or> (k \<le> n \<and> k \<noteq> 0)" by arith
394   ultimately show ?thesis by blast
395 qed
397 lemma gbinomial_1[simp]: "a gchoose 1 = a"
400 lemma gbinomial_Suc0[simp]: "a gchoose (Suc 0) = a"
403 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")
404 proof-
405   have "?r = ((- 1) ^n * pochhammer (- a) n / of_nat (fact n)) * (of_nat n - (- a + of_nat n))"
406     unfolding gbinomial_pochhammer
407     pochhammer_Suc fact_Suc of_nat_mult right_diff_distrib power_Suc
408     by (simp add:  field_simps del: of_nat_Suc)
409   also have "\<dots> = ?l" unfolding gbinomial_pochhammer
411   finally show ?thesis ..
412 qed
414 lemma gbinomial_mult_1': "(a gchoose n) * a = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"
415   by (simp add: mult_commute gbinomial_mult_1)
417 lemma gbinomial_Suc: "a gchoose (Suc k) = (setprod (\<lambda>i. a - of_nat i) {0 .. k}) / of_nat (fact (Suc k))"
420 lemma gbinomial_mult_fact:
421   "(of_nat (fact (Suc k)) :: 'a) * ((a::'a::field_char_0) gchoose (Suc k)) = (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
422   unfolding gbinomial_Suc
423   by (simp_all add: field_simps del: fact_Suc)
425 lemma gbinomial_mult_fact':
426   "((a::'a::field_char_0) gchoose (Suc k)) * (of_nat (fact (Suc k)) :: 'a) = (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
427   using gbinomial_mult_fact[of k a]
428   apply (subst mult_commute) .
430 lemma gbinomial_Suc_Suc: "((a::'a::field_char_0) + 1) gchoose (Suc k) = a gchoose k + (a gchoose (Suc k))"
431 proof-
432   {assume "k = 0" then have ?thesis by simp}
433   moreover
434   {fix h assume h: "k = Suc h"
435    have eq0: "(\<Prod>i\<in>{1..k}. (a + 1) - of_nat i) = (\<Prod>i\<in>{0..h}. a - of_nat i)"
436      apply (rule strong_setprod_reindex_cong[where f = Suc])
437      using h by auto
439     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)) + (\<Prod>i\<in>{0\<Colon>nat..Suc h}. a - of_nat i)"
440       unfolding h
441       apply (simp add: ring_simps del: fact_Suc)
442       unfolding gbinomial_mult_fact'
443       apply (subst fact_Suc)
444       unfolding of_nat_mult
445       apply (subst mult_commute)
446       unfolding mult_assoc
447       unfolding gbinomial_mult_fact