author | paulson <lp15@cam.ac.uk> |
Mon, 30 May 2022 12:46:11 +0100 | |
changeset 75494 | eded3fe9e600 |
parent 73932 | fd21b4a93043 |
child 75856 | 4b507edcf6b5 |
permissions | -rw-r--r-- |
63466 | 1 |
(* Title: HOL/Binomial.thy |
2 |
Author: Jacques D. Fleuriot |
|
3 |
Author: Lawrence C Paulson |
|
4 |
Author: Jeremy Avigad |
|
5 |
Author: Chaitanya Mangla |
|
6 |
Author: Manuel Eberl |
|
12196 | 7 |
*) |
8 |
||
65812 | 9 |
section \<open>Binomial Coefficients and Binomial Theorem\<close> |
15094
a7d1a3fdc30d
conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents:
12196
diff
changeset
|
10 |
|
59669
de7792ea4090
renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents:
59667
diff
changeset
|
11 |
theory Binomial |
65813 | 12 |
imports Presburger Factorial |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
13 |
begin |
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62347
diff
changeset
|
14 |
|
63373 | 15 |
subsection \<open>Binomial coefficients\<close> |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
16 |
|
63466 | 17 |
text \<open>This development is based on the work of Andy Gordon and Florian Kammueller.\<close> |
18 |
||
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
19 |
text \<open>Combinatorial definition\<close> |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
20 |
|
63466 | 21 |
definition binomial :: "nat \<Rightarrow> nat \<Rightarrow> nat" (infixl "choose" 65) |
22 |
where "n choose k = card {K\<in>Pow {0..<n}. card K = k}" |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
23 |
|
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
24 |
theorem n_subsets: |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
25 |
assumes "finite A" |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
26 |
shows "card {B. B \<subseteq> A \<and> card B = k} = card A choose k" |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
27 |
proof - |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
28 |
from assms obtain f where bij: "bij_betw f {0..<card A} A" |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
29 |
by (blast dest: ex_bij_betw_nat_finite) |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
30 |
then have [simp]: "card (f ` C) = card C" if "C \<subseteq> {0..<card A}" for C |
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
31 |
by (meson bij_betw_imp_inj_on bij_betw_subset card_image that) |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
32 |
from bij have "bij_betw (image f) (Pow {0..<card A}) (Pow A)" |
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
33 |
by (rule bij_betw_Pow) |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
34 |
then have "inj_on (image f) (Pow {0..<card A})" |
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
35 |
by (rule bij_betw_imp_inj_on) |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
36 |
moreover have "{K. K \<subseteq> {0..<card A} \<and> card K = k} \<subseteq> Pow {0..<card A}" |
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
37 |
by auto |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
38 |
ultimately have "inj_on (image f) {K. K \<subseteq> {0..<card A} \<and> card K = k}" |
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
39 |
by (rule inj_on_subset) |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
40 |
then have "card {K. K \<subseteq> {0..<card A} \<and> card K = k} = |
63466 | 41 |
card (image f ` {K. K \<subseteq> {0..<card A} \<and> card K = k})" (is "_ = card ?C") |
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
42 |
by (simp add: card_image) |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
43 |
also have "?C = {K. K \<subseteq> f ` {0..<card A} \<and> card K = k}" |
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
44 |
by (auto elim!: subset_imageE) |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
45 |
also have "f ` {0..<card A} = A" |
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
46 |
by (meson bij bij_betw_def) |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
47 |
finally show ?thesis |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
48 |
by (simp add: binomial_def) |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
49 |
qed |
63466 | 50 |
|
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
51 |
text \<open>Recursive characterization\<close> |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
52 |
|
68785 | 53 |
lemma binomial_n_0 [simp]: "n choose 0 = 1" |
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
54 |
proof - |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
55 |
have "{K \<in> Pow {0..<n}. card K = 0} = {{}}" |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
56 |
by (auto dest: finite_subset) |
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
57 |
then show ?thesis |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
58 |
by (simp add: binomial_def) |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
59 |
qed |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
60 |
|
68785 | 61 |
lemma binomial_0_Suc [simp]: "0 choose Suc k = 0" |
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
62 |
by (simp add: binomial_def) |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
63 |
|
68785 | 64 |
lemma binomial_Suc_Suc [simp]: "Suc n choose Suc k = (n choose k) + (n choose Suc k)" |
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
65 |
proof - |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
66 |
let ?P = "\<lambda>n k. {K. K \<subseteq> {0..<n} \<and> card K = k}" |
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
67 |
let ?Q = "?P (Suc n) (Suc k)" |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
68 |
have inj: "inj_on (insert n) (?P n k)" |
63466 | 69 |
by rule (auto; metis atLeastLessThan_iff insert_iff less_irrefl subsetCE) |
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
70 |
have disjoint: "insert n ` ?P n k \<inter> ?P n (Suc k) = {}" |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
71 |
by auto |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
72 |
have "?Q = {K\<in>?Q. n \<in> K} \<union> {K\<in>?Q. n \<notin> K}" |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
73 |
by auto |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
74 |
also have "{K\<in>?Q. n \<in> K} = insert n ` ?P n k" (is "?A = ?B") |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
75 |
proof (rule set_eqI) |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
76 |
fix K |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
77 |
have K_finite: "finite K" if "K \<subseteq> insert n {0..<n}" |
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
78 |
using that by (rule finite_subset) simp_all |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
79 |
have Suc_card_K: "Suc (card K - Suc 0) = card K" if "n \<in> K" |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
80 |
and "finite K" |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
81 |
proof - |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
82 |
from \<open>n \<in> K\<close> obtain L where "K = insert n L" and "n \<notin> L" |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
83 |
by (blast elim: Set.set_insert) |
72302
d7d90ed4c74e
fixed some remarkably ugly proofs
paulson <lp15@cam.ac.uk>
parents:
71720
diff
changeset
|
84 |
with that show ?thesis by (simp add: card.insert_remove) |
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
85 |
qed |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
86 |
show "K \<in> ?A \<longleftrightarrow> K \<in> ?B" |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
87 |
by (subst in_image_insert_iff) |
72302
d7d90ed4c74e
fixed some remarkably ugly proofs
paulson <lp15@cam.ac.uk>
parents:
71720
diff
changeset
|
88 |
(auto simp add: card.insert_remove subset_eq_atLeast0_lessThan_finite |
63466 | 89 |
Diff_subset_conv K_finite Suc_card_K) |
90 |
qed |
|
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
91 |
also have "{K\<in>?Q. n \<notin> K} = ?P n (Suc k)" |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
92 |
by (auto simp add: atLeast0_lessThan_Suc) |
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
93 |
finally show ?thesis using inj disjoint |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
94 |
by (simp add: binomial_def card_Un_disjoint card_image) |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
95 |
qed |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
96 |
|
63466 | 97 |
lemma binomial_eq_0: "n < k \<Longrightarrow> n choose k = 0" |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
98 |
by (auto simp add: binomial_def dest: subset_eq_atLeast0_lessThan_card) |
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
99 |
|
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
100 |
lemma zero_less_binomial: "k \<le> n \<Longrightarrow> n choose k > 0" |
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
101 |
by (induct n k rule: diff_induct) simp_all |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
102 |
|
63466 | 103 |
lemma binomial_eq_0_iff [simp]: "n choose k = 0 \<longleftrightarrow> n < k" |
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
104 |
by (metis binomial_eq_0 less_numeral_extra(3) not_less zero_less_binomial) |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
105 |
|
63466 | 106 |
lemma zero_less_binomial_iff [simp]: "n choose k > 0 \<longleftrightarrow> k \<le> n" |
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
107 |
by (metis binomial_eq_0_iff not_less0 not_less zero_less_binomial) |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
108 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
109 |
lemma binomial_n_n [simp]: "n choose n = 1" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
110 |
by (induct n) (simp_all add: binomial_eq_0) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
111 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
112 |
lemma binomial_Suc_n [simp]: "Suc n choose n = Suc n" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
113 |
by (induct n) simp_all |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
114 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
115 |
lemma binomial_1 [simp]: "n choose Suc 0 = n" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
116 |
by (induct n) simp_all |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
117 |
|
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
118 |
lemma choose_reduce_nat: |
63466 | 119 |
"0 < n \<Longrightarrow> 0 < k \<Longrightarrow> |
120 |
n choose k = ((n - 1) choose (k - 1)) + ((n - 1) choose k)" |
|
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
121 |
using binomial_Suc_Suc [of "n - 1" "k - 1"] by simp |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
122 |
|
63466 | 123 |
lemma Suc_times_binomial_eq: "Suc n * (n choose k) = (Suc n choose Suc k) * Suc k" |
71699 | 124 |
proof (induction n arbitrary: k) |
125 |
case 0 |
|
126 |
then show ?case |
|
127 |
by auto |
|
128 |
next |
|
129 |
case (Suc n) |
|
130 |
show ?case |
|
131 |
proof (cases k) |
|
132 |
case (Suc k') |
|
133 |
then show ?thesis |
|
134 |
using Suc.IH |
|
135 |
by (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq binomial_eq_0) |
|
136 |
qed auto |
|
137 |
qed |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
138 |
|
60141
833adf7db7d8
New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
139 |
lemma binomial_le_pow2: "n choose k \<le> 2^n" |
71699 | 140 |
proof (induction n arbitrary: k) |
141 |
case 0 |
|
142 |
then show ?case |
|
143 |
using le_less less_le_trans by fastforce |
|
144 |
next |
|
145 |
case (Suc n) |
|
146 |
show ?case |
|
147 |
proof (cases k) |
|
148 |
case (Suc k') |
|
149 |
then show ?thesis |
|
150 |
using Suc.IH by (simp add: add_le_mono mult_2) |
|
151 |
qed auto |
|
152 |
qed |
|
60141
833adf7db7d8
New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
153 |
|
63466 | 154 |
text \<open>The absorption property.\<close> |
155 |
lemma Suc_times_binomial: "Suc k * (Suc n choose Suc k) = Suc n * (n choose k)" |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
156 |
using Suc_times_binomial_eq by auto |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
157 |
|
63466 | 158 |
text \<open>This is the well-known version of absorption, but it's harder to use |
159 |
because of the need to reason about division.\<close> |
|
160 |
lemma binomial_Suc_Suc_eq_times: "(Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k" |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
161 |
by (simp add: Suc_times_binomial_eq del: mult_Suc mult_Suc_right) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
162 |
|
63466 | 163 |
text \<open>Another version of absorption, with \<open>-1\<close> instead of \<open>Suc\<close>.\<close> |
164 |
lemma times_binomial_minus1_eq: "0 < k \<Longrightarrow> k * (n choose k) = n * ((n - 1) choose (k - 1))" |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
165 |
using Suc_times_binomial_eq [where n = "n - 1" and k = "k - 1"] |
63648 | 166 |
by (auto split: nat_diff_split) |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
167 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
168 |
|
60758 | 169 |
subsection \<open>The binomial theorem (courtesy of Tobias Nipkow):\<close> |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
170 |
|
63466 | 171 |
text \<open>Avigad's version, generalized to any commutative ring\<close> |
71351 | 172 |
theorem binomial_ring: "(a + b :: 'a::comm_semiring_1)^n = |
68077
ee8c13ae81e9
Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
173 |
(\<Sum>k\<le>n. (of_nat (n choose k)) * a^k * b^(n-k))" |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
174 |
proof (induct n) |
63466 | 175 |
case 0 |
176 |
then show ?case by simp |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
177 |
next |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
178 |
case (Suc n) |
63466 | 179 |
have decomp: "{0..n+1} = {0} \<union> {n + 1} \<union> {1..n}" |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
180 |
by auto |
63466 | 181 |
have decomp2: "{0..n} = {0} \<union> {1..n}" |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
182 |
by auto |
68077
ee8c13ae81e9
Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
183 |
have "(a + b)^(n+1) = (a + b) * (\<Sum>k\<le>n. of_nat (n choose k) * a^k * b^(n - k))" |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
184 |
using Suc.hyps by simp |
68077
ee8c13ae81e9
Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
185 |
also have "\<dots> = a * (\<Sum>k\<le>n. of_nat (n choose k) * a^k * b^(n-k)) + |
ee8c13ae81e9
Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
186 |
b * (\<Sum>k\<le>n. of_nat (n choose k) * a^k * b^(n-k))" |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
187 |
by (rule distrib_right) |
68077
ee8c13ae81e9
Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
188 |
also have "\<dots> = (\<Sum>k\<le>n. of_nat (n choose k) * a^(k+1) * b^(n-k)) + |
ee8c13ae81e9
Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
189 |
(\<Sum>k\<le>n. of_nat (n choose k) * a^k * b^(n - k + 1))" |
64267 | 190 |
by (auto simp add: sum_distrib_left ac_simps) |
68077
ee8c13ae81e9
Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
191 |
also have "\<dots> = (\<Sum>k\<le>n. of_nat (n choose k) * a^k * b^(n + 1 - k)) + |
63466 | 192 |
(\<Sum>k=1..n+1. of_nat (n choose (k - 1)) * a^k * b^(n + 1 - k))" |
70113
c8deb8ba6d05
Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
193 |
by (simp add: atMost_atLeast0 sum.shift_bounds_cl_Suc_ivl Suc_diff_le field_simps del: sum.cl_ivl_Suc) |
71351 | 194 |
also have "\<dots> = b^(n + 1) + |
195 |
(\<Sum>k=1..n. of_nat (n choose k) * a^k * b^(n + 1 - k)) + (a^(n + 1) + |
|
196 |
(\<Sum>k=1..n. of_nat (n choose (k - 1)) * a^k * b^(n + 1 - k)))" |
|
197 |
using sum.nat_ivl_Suc' [of 1 n "\<lambda>k. of_nat (n choose (k-1)) * a ^ k * b ^ (n + 1 - k)"] |
|
198 |
by (simp add: sum.atLeast_Suc_atMost atMost_atLeast0) |
|
63466 | 199 |
also have "\<dots> = a^(n + 1) + b^(n + 1) + |
200 |
(\<Sum>k=1..n. of_nat (n + 1 choose k) * a^k * b^(n + 1 - k))" |
|
64267 | 201 |
by (auto simp add: field_simps sum.distrib [symmetric] choose_reduce_nat) |
68077
ee8c13ae81e9
Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
202 |
also have "\<dots> = (\<Sum>k\<le>n+1. of_nat (n + 1 choose k) * a^k * b^(n + 1 - k))" |
ee8c13ae81e9
Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
203 |
using decomp by (simp add: atMost_atLeast0 field_simps) |
63466 | 204 |
finally show ?case |
205 |
by simp |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
206 |
qed |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
207 |
|
63466 | 208 |
text \<open>Original version for the naturals.\<close> |
68077
ee8c13ae81e9
Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
209 |
corollary binomial: "(a + b :: nat)^n = (\<Sum>k\<le>n. (of_nat (n choose k)) * a^k * b^(n - k))" |
63466 | 210 |
using binomial_ring [of "int a" "int b" n] |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
211 |
by (simp only: of_nat_add [symmetric] of_nat_mult [symmetric] of_nat_power [symmetric] |
64267 | 212 |
of_nat_sum [symmetric] of_nat_eq_iff of_nat_id) |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
213 |
|
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
214 |
lemma binomial_fact_lemma: "k \<le> n \<Longrightarrow> fact k * fact (n - k) * (n choose k) = fact n" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
215 |
proof (induct n arbitrary: k rule: nat_less_induct) |
63466 | 216 |
fix n k |
217 |
assume H: "\<forall>m<n. \<forall>x\<le>m. fact x * fact (m - x) * (m choose x) = fact m" |
|
218 |
assume kn: "k \<le> n" |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
219 |
let ?ths = "fact k * fact (n - k) * (n choose k) = fact n" |
63466 | 220 |
consider "n = 0 \<or> k = 0 \<or> n = k" | m h where "n = Suc m" "k = Suc h" "h < m" |
221 |
using kn by atomize_elim presburger |
|
222 |
then show "fact k * fact (n - k) * (n choose k) = fact n" |
|
223 |
proof cases |
|
224 |
case 1 |
|
225 |
with kn show ?thesis by auto |
|
226 |
next |
|
227 |
case 2 |
|
228 |
note n = \<open>n = Suc m\<close> |
|
229 |
note k = \<open>k = Suc h\<close> |
|
230 |
note hm = \<open>h < m\<close> |
|
231 |
have mn: "m < n" |
|
232 |
using n by arith |
|
233 |
have hm': "h \<le> m" |
|
234 |
using hm by arith |
|
235 |
have km: "k \<le> m" |
|
236 |
using hm k n kn by arith |
|
237 |
have "m - h = Suc (m - Suc h)" |
|
238 |
using k km hm by arith |
|
239 |
with km k have "fact (m - h) = (m - h) * fact (m - k)" |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
240 |
by simp |
63466 | 241 |
with n k have "fact k * fact (n - k) * (n choose k) = |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
242 |
k * (fact h * fact (m - h) * (m choose h)) + |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
243 |
(m - h) * (fact k * fact (m - k) * (m choose k))" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
244 |
by (simp add: field_simps) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
245 |
also have "\<dots> = (k + (m - h)) * fact m" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
246 |
using H[rule_format, OF mn hm'] H[rule_format, OF mn km] |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
247 |
by (simp add: field_simps) |
63466 | 248 |
finally show ?thesis |
249 |
using k n km by simp |
|
250 |
qed |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
251 |
qed |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
252 |
|
63372
492b49535094
relating gbinomial and binomial, still using distinct definitions
haftmann
parents:
63367
diff
changeset
|
253 |
lemma binomial_fact': |
492b49535094
relating gbinomial and binomial, still using distinct definitions
haftmann
parents:
63367
diff
changeset
|
254 |
assumes "k \<le> n" |
492b49535094
relating gbinomial and binomial, still using distinct definitions
haftmann
parents:
63367
diff
changeset
|
255 |
shows "n choose k = fact n div (fact k * fact (n - k))" |
492b49535094
relating gbinomial and binomial, still using distinct definitions
haftmann
parents:
63367
diff
changeset
|
256 |
using binomial_fact_lemma [OF assms] |
64240 | 257 |
by (metis fact_nonzero mult_eq_0_iff nonzero_mult_div_cancel_left) |
63372
492b49535094
relating gbinomial and binomial, still using distinct definitions
haftmann
parents:
63367
diff
changeset
|
258 |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
259 |
lemma binomial_fact: |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
260 |
assumes kn: "k \<le> n" |
63466 | 261 |
shows "(of_nat (n choose k) :: 'a::field_char_0) = fact n / (fact k * fact (n - k))" |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
262 |
using binomial_fact_lemma[OF kn] |
71699 | 263 |
by (metis (mono_tags, lifting) fact_nonzero mult_eq_0_iff nonzero_mult_div_cancel_left of_nat_fact of_nat_mult) |
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
264 |
|
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
265 |
lemma fact_binomial: |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
266 |
assumes "k \<le> n" |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
267 |
shows "fact k * of_nat (n choose k) = (fact n / fact (n - k) :: 'a::field_char_0)" |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
268 |
unfolding binomial_fact [OF assms] by (simp add: field_simps) |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
269 |
|
63466 | 270 |
lemma choose_two: "n choose 2 = n * (n - 1) div 2" |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
271 |
proof (cases "n \<ge> 2") |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
272 |
case False |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
273 |
then have "n = 0 \<or> n = 1" |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
274 |
by auto |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
275 |
then show ?thesis by auto |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
276 |
next |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
277 |
case True |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
278 |
define m where "m = n - 2" |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
279 |
with True have "n = m + 2" |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
280 |
by simp |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
281 |
then have "fact n = n * (n - 1) * fact (n - 2)" |
64272 | 282 |
by (simp add: fact_prod_Suc atLeast0_lessThan_Suc algebra_simps) |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
283 |
with True show ?thesis |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
284 |
by (simp add: binomial_fact') |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
285 |
qed |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
286 |
|
68077
ee8c13ae81e9
Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
287 |
lemma choose_row_sum: "(\<Sum>k\<le>n. n choose k) = 2^n" |
63466 | 288 |
using binomial [of 1 "1" n] by (simp add: numeral_2_eq_2) |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
289 |
|
68077
ee8c13ae81e9
Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
290 |
lemma sum_choose_lower: "(\<Sum>k\<le>n. (r+k) choose k) = Suc (r+n) choose n" |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
291 |
by (induct n) auto |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
292 |
|
68077
ee8c13ae81e9
Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
293 |
lemma sum_choose_upper: "(\<Sum>k\<le>n. k choose m) = Suc n choose Suc m" |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
294 |
by (induct n) auto |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
295 |
|
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62347
diff
changeset
|
296 |
lemma choose_alternating_sum: |
63466 | 297 |
"n > 0 \<Longrightarrow> (\<Sum>i\<le>n. (-1)^i * of_nat (n choose i)) = (0 :: 'a::comm_ring_1)" |
298 |
using binomial_ring[of "-1 :: 'a" 1 n] |
|
299 |
by (simp add: atLeast0AtMost mult_of_nat_commute zero_power) |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
300 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
301 |
lemma choose_even_sum: |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
302 |
assumes "n > 0" |
63466 | 303 |
shows "2 * (\<Sum>i\<le>n. if even i then of_nat (n choose i) else 0) = (2 ^ n :: 'a::comm_ring_1)" |
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62347
diff
changeset
|
304 |
proof - |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
305 |
have "2 ^ n = (\<Sum>i\<le>n. of_nat (n choose i)) + (\<Sum>i\<le>n. (-1) ^ i * of_nat (n choose i) :: 'a)" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
306 |
using choose_row_sum[of n] |
64267 | 307 |
by (simp add: choose_alternating_sum assms atLeast0AtMost of_nat_sum[symmetric]) |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
308 |
also have "\<dots> = (\<Sum>i\<le>n. of_nat (n choose i) + (-1) ^ i * of_nat (n choose i))" |
64267 | 309 |
by (simp add: sum.distrib) |
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62347
diff
changeset
|
310 |
also have "\<dots> = 2 * (\<Sum>i\<le>n. if even i then of_nat (n choose i) else 0)" |
64267 | 311 |
by (subst sum_distrib_left, intro sum.cong) simp_all |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
312 |
finally show ?thesis .. |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
313 |
qed |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
314 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
315 |
lemma choose_odd_sum: |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
316 |
assumes "n > 0" |
63466 | 317 |
shows "2 * (\<Sum>i\<le>n. if odd i then of_nat (n choose i) else 0) = (2 ^ n :: 'a::comm_ring_1)" |
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62347
diff
changeset
|
318 |
proof - |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
319 |
have "2 ^ n = (\<Sum>i\<le>n. of_nat (n choose i)) - (\<Sum>i\<le>n. (-1) ^ i * of_nat (n choose i) :: 'a)" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
320 |
using choose_row_sum[of n] |
64267 | 321 |
by (simp add: choose_alternating_sum assms atLeast0AtMost of_nat_sum[symmetric]) |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
322 |
also have "\<dots> = (\<Sum>i\<le>n. of_nat (n choose i) - (-1) ^ i * of_nat (n choose i))" |
64267 | 323 |
by (simp add: sum_subtractf) |
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62347
diff
changeset
|
324 |
also have "\<dots> = 2 * (\<Sum>i\<le>n. if odd i then of_nat (n choose i) else 0)" |
64267 | 325 |
by (subst sum_distrib_left, intro sum.cong) simp_all |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
326 |
finally show ?thesis .. |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
327 |
qed |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
328 |
|
60758 | 329 |
text\<open>NW diagonal sum property\<close> |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
330 |
lemma sum_choose_diagonal: |
63466 | 331 |
assumes "m \<le> n" |
68077
ee8c13ae81e9
Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
332 |
shows "(\<Sum>k\<le>m. (n - k) choose (m - k)) = Suc n choose m" |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
333 |
proof - |
68077
ee8c13ae81e9
Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
334 |
have "(\<Sum>k\<le>m. (n-k) choose (m - k)) = (\<Sum>k\<le>m. (n - m + k) choose k)" |
67411
3f4b0c84630f
restored naming of lemmas after corresponding constants
haftmann
parents:
67399
diff
changeset
|
335 |
using sum.atLeastAtMost_rev [of "\<lambda>k. (n - k) choose (m - k)" 0 m] assms |
68077
ee8c13ae81e9
Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
336 |
by (simp add: atMost_atLeast0) |
63466 | 337 |
also have "\<dots> = Suc (n - m + m) choose m" |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
338 |
by (rule sum_choose_lower) |
63466 | 339 |
also have "\<dots> = Suc n choose m" |
340 |
using assms by simp |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
341 |
finally show ?thesis . |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
342 |
qed |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
343 |
|
63373 | 344 |
|
63372
492b49535094
relating gbinomial and binomial, still using distinct definitions
haftmann
parents:
63367
diff
changeset
|
345 |
subsection \<open>Generalized binomial coefficients\<close> |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
346 |
|
63466 | 347 |
definition gbinomial :: "'a::{semidom_divide,semiring_char_0} \<Rightarrow> nat \<Rightarrow> 'a" (infixl "gchoose" 65) |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
348 |
where gbinomial_prod_rev: "a gchoose k = prod (\<lambda>i. a - of_nat i) {0..<k} div fact k" |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
349 |
|
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
350 |
lemma gbinomial_0 [simp]: |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
351 |
"a gchoose 0 = 1" |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
352 |
"0 gchoose (Suc k) = 0" |
70113
c8deb8ba6d05
Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
353 |
by (simp_all add: gbinomial_prod_rev prod.atLeast0_lessThan_Suc_shift del: prod.op_ivl_Suc) |
63367
6c731c8b7f03
simplified definitions of combinatorial functions
haftmann
parents:
63366
diff
changeset
|
354 |
|
64272 | 355 |
lemma gbinomial_Suc: "a gchoose (Suc k) = prod (\<lambda>i. a - of_nat i) {0..k} div fact (Suc k)" |
356 |
by (simp add: gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost) |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
357 |
|
68786 | 358 |
lemma gbinomial_1 [simp]: "a gchoose 1 = a" |
359 |
by (simp add: gbinomial_prod_rev lessThan_Suc) |
|
360 |
||
361 |
lemma gbinomial_Suc0 [simp]: "a gchoose Suc 0 = a" |
|
362 |
by (simp add: gbinomial_prod_rev lessThan_Suc) |
|
363 |
||
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
364 |
lemma gbinomial_mult_fact: "fact k * (a gchoose k) = (\<Prod>i = 0..<k. a - of_nat i)" |
63466 | 365 |
for a :: "'a::field_char_0" |
64272 | 366 |
by (simp_all add: gbinomial_prod_rev field_simps) |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
367 |
|
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
368 |
lemma gbinomial_mult_fact': "(a gchoose k) * fact k = (\<Prod>i = 0..<k. a - of_nat i)" |
63466 | 369 |
for a :: "'a::field_char_0" |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
370 |
using gbinomial_mult_fact [of k a] by (simp add: ac_simps) |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
371 |
|
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
372 |
lemma gbinomial_pochhammer: "a gchoose k = (- 1) ^ k * pochhammer (- a) k / fact k" |
63466 | 373 |
for a :: "'a::field_char_0" |
70113
c8deb8ba6d05
Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
374 |
proof (cases k) |
c8deb8ba6d05
Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
375 |
case (Suc k') |
71699 | 376 |
then have "a gchoose k = pochhammer (a - of_nat k') (Suc k') / ((1 + of_nat k') * fact k')" |
377 |
by (simp add: gbinomial_prod_rev pochhammer_prod_rev atLeastLessThanSuc_atLeastAtMost |
|
378 |
prod.atLeast_Suc_atMost_Suc_shift of_nat_diff flip: power_mult_distrib prod.cl_ivl_Suc) |
|
70113
c8deb8ba6d05
Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
379 |
then show ?thesis |
71699 | 380 |
by (simp add: pochhammer_minus Suc) |
70113
c8deb8ba6d05
Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
381 |
qed auto |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
382 |
|
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
383 |
lemma gbinomial_pochhammer': "a gchoose k = pochhammer (a - of_nat k + 1) k / fact k" |
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
384 |
for a :: "'a::field_char_0" |
61552
980dd46a03fb
Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents:
61531
diff
changeset
|
385 |
proof - |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
386 |
have "a gchoose k = ((-1)^k * (-1)^k) * pochhammer (a - of_nat k + 1) k / fact k" |
61552
980dd46a03fb
Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents:
61531
diff
changeset
|
387 |
by (simp add: gbinomial_pochhammer pochhammer_minus mult_ac) |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
388 |
also have "(-1 :: 'a)^k * (-1)^k = 1" |
63466 | 389 |
by (subst power_add [symmetric]) simp |
390 |
finally show ?thesis |
|
391 |
by simp |
|
61552
980dd46a03fb
Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents:
61531
diff
changeset
|
392 |
qed |
980dd46a03fb
Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents:
61531
diff
changeset
|
393 |
|
63466 | 394 |
lemma gbinomial_binomial: "n gchoose k = n choose k" |
63372
492b49535094
relating gbinomial and binomial, still using distinct definitions
haftmann
parents:
63367
diff
changeset
|
395 |
proof (cases "k \<le> n") |
492b49535094
relating gbinomial and binomial, still using distinct definitions
haftmann
parents:
63367
diff
changeset
|
396 |
case False |
63466 | 397 |
then have "n < k" |
398 |
by (simp add: not_le) |
|
67399 | 399 |
then have "0 \<in> ((-) n) ` {0..<k}" |
63372
492b49535094
relating gbinomial and binomial, still using distinct definitions
haftmann
parents:
63367
diff
changeset
|
400 |
by auto |
67399 | 401 |
then have "prod ((-) n) {0..<k} = 0" |
64272 | 402 |
by (auto intro: prod_zero) |
63372
492b49535094
relating gbinomial and binomial, still using distinct definitions
haftmann
parents:
63367
diff
changeset
|
403 |
with \<open>n < k\<close> show ?thesis |
64272 | 404 |
by (simp add: binomial_eq_0 gbinomial_prod_rev prod_zero) |
63372
492b49535094
relating gbinomial and binomial, still using distinct definitions
haftmann
parents:
63367
diff
changeset
|
405 |
next |
492b49535094
relating gbinomial and binomial, still using distinct definitions
haftmann
parents:
63367
diff
changeset
|
406 |
case True |
67399 | 407 |
from True have *: "prod ((-) n) {0..<k} = \<Prod>{Suc (n - k)..n}" |
65350
b149abe619f7
added shuffle product to HOL/List
eberlm <eberlm@in.tum.de>
parents:
64272
diff
changeset
|
408 |
by (intro prod.reindex_bij_witness[of _ "\<lambda>i. n - i" "\<lambda>i. n - i"]) auto |
63466 | 409 |
from True have "n choose k = fact n div (fact k * fact (n - k))" |
63372
492b49535094
relating gbinomial and binomial, still using distinct definitions
haftmann
parents:
63367
diff
changeset
|
410 |
by (rule binomial_fact') |
492b49535094
relating gbinomial and binomial, still using distinct definitions
haftmann
parents:
63367
diff
changeset
|
411 |
with * show ?thesis |
64272 | 412 |
by (simp add: gbinomial_prod_rev mult.commute [of "fact k"] div_mult2_eq fact_div_fact) |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
413 |
qed |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
414 |
|
63466 | 415 |
lemma of_nat_gbinomial: "of_nat (n gchoose k) = (of_nat n gchoose k :: 'a::field_char_0)" |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
416 |
proof (cases "k \<le> n") |
63466 | 417 |
case False |
418 |
then show ?thesis |
|
64272 | 419 |
by (simp add: not_le gbinomial_binomial binomial_eq_0 gbinomial_prod_rev) |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
420 |
next |
63466 | 421 |
case True |
422 |
define m where "m = n - k" |
|
423 |
with True have n: "n = m + k" |
|
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
424 |
by arith |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
425 |
from n have "fact n = ((\<Prod>i = 0..<m + k. of_nat (m + k - i) ):: 'a)" |
64272 | 426 |
by (simp add: fact_prod_rev) |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
427 |
also have "\<dots> = ((\<Prod>i\<in>{0..<k} \<union> {k..<m + k}. of_nat (m + k - i)) :: 'a)" |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
428 |
by (simp add: ivl_disj_un) |
63466 | 429 |
finally have "fact n = (fact m * (\<Prod>i = 0..<k. of_nat m + of_nat k - of_nat i) :: 'a)" |
70113
c8deb8ba6d05
Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
430 |
using prod.shift_bounds_nat_ivl [of "\<lambda>i. of_nat (m + k - i) :: 'a" 0 k m] |
64272 | 431 |
by (simp add: fact_prod_rev [of m] prod.union_disjoint of_nat_diff) |
63466 | 432 |
then have "fact n / fact (n - k) = ((\<Prod>i = 0..<k. of_nat n - of_nat i) :: 'a)" |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
433 |
by (simp add: n) |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
434 |
with True have "fact k * of_nat (n gchoose k) = (fact k * (of_nat n gchoose k) :: 'a)" |
63466 | 435 |
by (simp only: gbinomial_mult_fact [of k "of_nat n"] gbinomial_binomial [of n k] fact_binomial) |
436 |
then show ?thesis |
|
437 |
by simp |
|
63372
492b49535094
relating gbinomial and binomial, still using distinct definitions
haftmann
parents:
63367
diff
changeset
|
438 |
qed |
492b49535094
relating gbinomial and binomial, still using distinct definitions
haftmann
parents:
63367
diff
changeset
|
439 |
|
63466 | 440 |
lemma binomial_gbinomial: "of_nat (n choose k) = (of_nat n gchoose k :: 'a::field_char_0)" |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
441 |
by (simp add: gbinomial_binomial [symmetric] of_nat_gbinomial) |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
442 |
|
63466 | 443 |
setup |
69593 | 444 |
\<open>Sign.add_const_constraint (\<^const_name>\<open>gbinomial\<close>, SOME \<^typ>\<open>'a::field_char_0 \<Rightarrow> nat \<Rightarrow> 'a\<close>)\<close> |
63372
492b49535094
relating gbinomial and binomial, still using distinct definitions
haftmann
parents:
63367
diff
changeset
|
445 |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
446 |
lemma gbinomial_mult_1: |
63466 | 447 |
fixes a :: "'a::field_char_0" |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
448 |
shows "a * (a gchoose k) = of_nat k * (a gchoose k) + of_nat (Suc k) * (a gchoose (Suc k))" |
63466 | 449 |
(is "?l = ?r") |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
450 |
proof - |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
451 |
have "?r = ((- 1) ^k * pochhammer (- a) k / fact k) * (of_nat k - (- a + of_nat k))" |
71699 | 452 |
unfolding gbinomial_pochhammer pochhammer_Suc right_diff_distrib power_Suc |
453 |
by (auto simp add: field_simps simp del: of_nat_Suc) |
|
63466 | 454 |
also have "\<dots> = ?l" |
455 |
by (simp add: field_simps gbinomial_pochhammer) |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
456 |
finally show ?thesis .. |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
457 |
qed |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
458 |
|
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
459 |
lemma gbinomial_mult_1': |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
460 |
"(a gchoose k) * a = of_nat k * (a gchoose k) + of_nat (Suc k) * (a gchoose (Suc k))" |
63466 | 461 |
for a :: "'a::field_char_0" |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
462 |
by (simp add: mult.commute gbinomial_mult_1) |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
463 |
|
63466 | 464 |
lemma gbinomial_Suc_Suc: "(a + 1) gchoose (Suc k) = a gchoose k + (a gchoose (Suc k))" |
465 |
for a :: "'a::field_char_0" |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
466 |
proof (cases k) |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
467 |
case 0 |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
468 |
then show ?thesis by simp |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
469 |
next |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
470 |
case (Suc h) |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
471 |
have eq0: "(\<Prod>i\<in>{1..k}. (a + 1) - of_nat i) = (\<Prod>i\<in>{0..h}. a - of_nat i)" |
71699 | 472 |
proof (rule prod.reindex_cong) |
473 |
show "{1..k} = Suc ` {0..h}" |
|
474 |
using Suc by (auto simp add: image_Suc_atMost) |
|
475 |
qed auto |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
476 |
have "fact (Suc k) * (a gchoose k + (a gchoose (Suc k))) = |
63466 | 477 |
(a gchoose Suc h) * (fact (Suc (Suc h))) + |
478 |
(a gchoose Suc (Suc h)) * (fact (Suc (Suc h)))" |
|
63367
6c731c8b7f03
simplified definitions of combinatorial functions
haftmann
parents:
63366
diff
changeset
|
479 |
by (simp add: Suc field_simps del: fact_Suc) |
63466 | 480 |
also have "\<dots> = |
481 |
(a gchoose Suc h) * of_nat (Suc (Suc h) * fact (Suc h)) + (\<Prod>i=0..Suc h. a - of_nat i)" |
|
71699 | 482 |
apply (simp only: gbinomial_mult_fact field_simps mult.left_commute [of _ "2"]) |
63466 | 483 |
apply (simp del: fact_Suc add: fact_Suc [of "Suc h"] field_simps gbinomial_mult_fact |
484 |
mult.left_commute [of _ "2"] atLeastLessThanSuc_atLeastAtMost) |
|
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
485 |
done |
63466 | 486 |
also have "\<dots> = |
487 |
(fact (Suc h) * (a gchoose Suc h)) * of_nat (Suc (Suc h)) + (\<Prod>i=0..Suc h. a - of_nat i)" |
|
63367
6c731c8b7f03
simplified definitions of combinatorial functions
haftmann
parents:
63366
diff
changeset
|
488 |
by (simp only: fact_Suc mult.commute mult.left_commute of_nat_fact of_nat_id of_nat_mult) |
63466 | 489 |
also have "\<dots> = |
490 |
of_nat (Suc (Suc h)) * (\<Prod>i=0..h. a - of_nat i) + (\<Prod>i=0..Suc h. a - of_nat i)" |
|
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
491 |
unfolding gbinomial_mult_fact atLeastLessThanSuc_atLeastAtMost by auto |
63466 | 492 |
also have "\<dots> = |
493 |
(\<Prod>i=0..Suc h. a - of_nat i) + (of_nat h * (\<Prod>i=0..h. a - of_nat i) + 2 * (\<Prod>i=0..h. a - of_nat i))" |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
494 |
by (simp add: field_simps) |
63466 | 495 |
also have "\<dots> = |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
496 |
((a gchoose Suc h) * (fact (Suc h)) * of_nat (Suc k)) + (\<Prod>i\<in>{0..Suc h}. a - of_nat i)" |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
497 |
unfolding gbinomial_mult_fact' |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
498 |
by (simp add: comm_semiring_class.distrib field_simps Suc atLeastLessThanSuc_atLeastAtMost) |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
499 |
also have "\<dots> = (\<Prod>i\<in>{0..h}. a - of_nat i) * (a + 1)" |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
500 |
unfolding gbinomial_mult_fact' atLeast0_atMost_Suc |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
501 |
by (simp add: field_simps Suc atLeastLessThanSuc_atLeastAtMost) |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
502 |
also have "\<dots> = (\<Prod>i\<in>{0..k}. (a + 1) - of_nat i)" |
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
503 |
using eq0 |
70113
c8deb8ba6d05
Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
504 |
by (simp add: Suc prod.atLeast0_atMost_Suc_shift del: prod.cl_ivl_Suc) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
505 |
also have "\<dots> = (fact (Suc k)) * ((a + 1) gchoose (Suc k))" |
63466 | 506 |
by (simp only: gbinomial_mult_fact atLeastLessThanSuc_atLeastAtMost) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
507 |
finally show ?thesis |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
508 |
using fact_nonzero [of "Suc k"] by auto |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
509 |
qed |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
510 |
|
63466 | 511 |
lemma gbinomial_reduce_nat: "0 < k \<Longrightarrow> a gchoose k = (a - 1) gchoose (k - 1) + ((a - 1) gchoose k)" |
512 |
for a :: "'a::field_char_0" |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
513 |
by (metis Suc_pred' diff_add_cancel gbinomial_Suc_Suc) |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
514 |
|
60141
833adf7db7d8
New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
515 |
lemma gchoose_row_sum_weighted: |
63466 | 516 |
"(\<Sum>k = 0..m. (r gchoose k) * (r/2 - of_nat k)) = of_nat(Suc m) / 2 * (r gchoose (Suc m))" |
517 |
for r :: "'a::field_char_0" |
|
518 |
by (induct m) (simp_all add: field_simps distrib gbinomial_mult_1) |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
519 |
|
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
520 |
lemma binomial_symmetric: |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
521 |
assumes kn: "k \<le> n" |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
522 |
shows "n choose k = n choose (n - k)" |
63466 | 523 |
proof - |
524 |
have kn': "n - k \<le> n" |
|
525 |
using kn by arith |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
526 |
from binomial_fact_lemma[OF kn] binomial_fact_lemma[OF kn'] |
63466 | 527 |
have "fact k * fact (n - k) * (n choose k) = fact (n - k) * fact (n - (n - k)) * (n choose (n - k))" |
528 |
by simp |
|
529 |
then show ?thesis |
|
530 |
using kn by simp |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
531 |
qed |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
532 |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
533 |
lemma choose_rising_sum: |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
534 |
"(\<Sum>j\<le>m. ((n + j) choose n)) = ((n + m + 1) choose (n + 1))" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
535 |
"(\<Sum>j\<le>m. ((n + j) choose n)) = ((n + m + 1) choose m)" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
536 |
proof - |
63466 | 537 |
show "(\<Sum>j\<le>m. ((n + j) choose n)) = ((n + m + 1) choose (n + 1))" |
538 |
by (induct m) simp_all |
|
539 |
also have "\<dots> = (n + m + 1) choose m" |
|
540 |
by (subst binomial_symmetric) simp_all |
|
541 |
finally show "(\<Sum>j\<le>m. ((n + j) choose n)) = (n + m + 1) choose m" . |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
542 |
qed |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
543 |
|
63466 | 544 |
lemma choose_linear_sum: "(\<Sum>i\<le>n. i * (n choose i)) = n * 2 ^ (n - 1)" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
545 |
proof (cases n) |
63466 | 546 |
case 0 |
547 |
then show ?thesis by simp |
|
548 |
next |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
549 |
case (Suc m) |
63466 | 550 |
have "(\<Sum>i\<le>n. i * (n choose i)) = (\<Sum>i\<le>Suc m. i * (Suc m choose i))" |
551 |
by (simp add: Suc) |
|
552 |
also have "\<dots> = Suc m * 2 ^ m" |
|
70113
c8deb8ba6d05
Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
553 |
unfolding sum.atMost_Suc_shift Suc_times_binomial sum_distrib_left[symmetric] |
68077
ee8c13ae81e9
Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
554 |
by (simp add: choose_row_sum) |
63466 | 555 |
finally show ?thesis |
556 |
using Suc by simp |
|
557 |
qed |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
558 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
559 |
lemma choose_alternating_linear_sum: |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
560 |
assumes "n \<noteq> 1" |
63466 | 561 |
shows "(\<Sum>i\<le>n. (-1)^i * of_nat i * of_nat (n choose i) :: 'a::comm_ring_1) = 0" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
562 |
proof (cases n) |
63466 | 563 |
case 0 |
564 |
then show ?thesis by simp |
|
565 |
next |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
566 |
case (Suc m) |
63466 | 567 |
with assms have "m > 0" |
568 |
by simp |
|
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62347
diff
changeset
|
569 |
have "(\<Sum>i\<le>n. (-1) ^ i * of_nat i * of_nat (n choose i) :: 'a) = |
63466 | 570 |
(\<Sum>i\<le>Suc m. (-1) ^ i * of_nat i * of_nat (Suc m choose i))" |
571 |
by (simp add: Suc) |
|
572 |
also have "\<dots> = (\<Sum>i\<le>m. (-1) ^ (Suc i) * of_nat (Suc i * (Suc m choose Suc i)))" |
|
70113
c8deb8ba6d05
Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
573 |
by (simp only: sum.atMost_Suc_shift sum_distrib_left[symmetric] mult_ac of_nat_mult) simp |
63466 | 574 |
also have "\<dots> = - of_nat (Suc m) * (\<Sum>i\<le>m. (-1) ^ i * of_nat (m choose i))" |
64267 | 575 |
by (subst sum_distrib_left, rule sum.cong[OF refl], subst Suc_times_binomial) |
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
576 |
(simp add: algebra_simps) |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
577 |
also have "(\<Sum>i\<le>m. (-1 :: 'a) ^ i * of_nat ((m choose i))) = 0" |
61799 | 578 |
using choose_alternating_sum[OF \<open>m > 0\<close>] by simp |
63466 | 579 |
finally show ?thesis |
580 |
by simp |
|
581 |
qed |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
582 |
|
63466 | 583 |
lemma vandermonde: "(\<Sum>k\<le>r. (m choose k) * (n choose (r - k))) = (m + n) choose r" |
584 |
proof (induct n arbitrary: r) |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
585 |
case 0 |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
586 |
have "(\<Sum>k\<le>r. (m choose k) * (0 choose (r - k))) = (\<Sum>k\<le>r. if k = r then (m choose k) else 0)" |
64267 | 587 |
by (intro sum.cong) simp_all |
63466 | 588 |
also have "\<dots> = m choose r" |
68784 | 589 |
by simp |
63466 | 590 |
finally show ?case |
591 |
by simp |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
592 |
next |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
593 |
case (Suc n r) |
63466 | 594 |
show ?case |
64267 | 595 |
by (cases r) (simp_all add: Suc [symmetric] algebra_simps sum.distrib Suc_diff_le) |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
596 |
qed |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
597 |
|
63466 | 598 |
lemma choose_square_sum: "(\<Sum>k\<le>n. (n choose k)^2) = ((2*n) choose n)" |
599 |
using vandermonde[of n n n] |
|
600 |
by (simp add: power2_eq_square mult_2 binomial_symmetric [symmetric]) |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
601 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
602 |
lemma pochhammer_binomial_sum: |
63466 | 603 |
fixes a b :: "'a::comm_ring_1" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
604 |
shows "pochhammer (a + b) n = (\<Sum>k\<le>n. of_nat (n choose k) * pochhammer a k * pochhammer b (n - k))" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
605 |
proof (induction n arbitrary: a b) |
63466 | 606 |
case 0 |
607 |
then show ?case by simp |
|
608 |
next |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
609 |
case (Suc n a b) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
610 |
have "(\<Sum>k\<le>Suc n. of_nat (Suc n choose k) * pochhammer a k * pochhammer b (Suc n - k)) = |
63466 | 611 |
(\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a (Suc i) * pochhammer b (n - i)) + |
612 |
((\<Sum>i\<le>n. of_nat (n choose Suc i) * pochhammer a (Suc i) * pochhammer b (n - i)) + |
|
613 |
pochhammer b (Suc n))" |
|
70113
c8deb8ba6d05
Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
614 |
by (subst sum.atMost_Suc_shift) (simp add: ring_distribs sum.distrib) |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
615 |
also have "(\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a (Suc i) * pochhammer b (n - i)) = |
63466 | 616 |
a * pochhammer ((a + 1) + b) n" |
64267 | 617 |
by (subst Suc) (simp add: sum_distrib_left pochhammer_rec mult_ac) |
63466 | 618 |
also have "(\<Sum>i\<le>n. of_nat (n choose Suc i) * pochhammer a (Suc i) * pochhammer b (n - i)) + |
619 |
pochhammer b (Suc n) = |
|
620 |
(\<Sum>i=0..Suc n. of_nat (n choose i) * pochhammer a i * pochhammer b (Suc n - i))" |
|
71699 | 621 |
apply (subst sum.atLeast_Suc_atMost, simp) |
622 |
apply (simp add: sum.shift_bounds_cl_Suc_ivl atLeast0AtMost del: sum.cl_ivl_Suc) |
|
63466 | 623 |
done |
624 |
also have "\<dots> = (\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a i * pochhammer b (Suc n - i))" |
|
64267 | 625 |
using Suc by (intro sum.mono_neutral_right) (auto simp: not_le binomial_eq_0) |
63466 | 626 |
also have "\<dots> = (\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a i * pochhammer b (Suc (n - i)))" |
64267 | 627 |
by (intro sum.cong) (simp_all add: Suc_diff_le) |
63466 | 628 |
also have "\<dots> = b * pochhammer (a + (b + 1)) n" |
64267 | 629 |
by (subst Suc) (simp add: sum_distrib_left mult_ac pochhammer_rec) |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
630 |
also have "a * pochhammer ((a + 1) + b) n + b * pochhammer (a + (b + 1)) n = |
63466 | 631 |
pochhammer (a + b) (Suc n)" |
632 |
by (simp add: pochhammer_rec algebra_simps) |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
633 |
finally show ?case .. |
63466 | 634 |
qed |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
635 |
|
63466 | 636 |
text \<open>Contributed by Manuel Eberl, generalised by LCP. |
69593 | 637 |
Alternative definition of the binomial coefficient as \<^term>\<open>\<Prod>i<k. (n - i) / (k - i)\<close>.\<close> |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
638 |
lemma gbinomial_altdef_of_nat: "a gchoose k = (\<Prod>i = 0..<k. (a - of_nat i) / of_nat (k - i) :: 'a)" |
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
639 |
for k :: nat and a :: "'a::field_char_0" |
64272 | 640 |
by (simp add: prod_dividef gbinomial_prod_rev fact_prod_rev) |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
641 |
|
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
642 |
lemma gbinomial_ge_n_over_k_pow_k: |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
643 |
fixes k :: nat |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
644 |
and a :: "'a::linordered_field" |
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
645 |
assumes "of_nat k \<le> a" |
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
646 |
shows "(a / of_nat k :: 'a) ^ k \<le> a gchoose k" |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
647 |
proof - |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
648 |
have x: "0 \<le> a" |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
649 |
using assms of_nat_0_le_iff order_trans by blast |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
650 |
have "(a / of_nat k :: 'a) ^ k = (\<Prod>i = 0..<k. a / of_nat k :: 'a)" |
68784 | 651 |
by simp |
71699 | 652 |
also have "\<dots> \<le> a gchoose k" |
653 |
proof - |
|
654 |
have "\<And>i. i < k \<Longrightarrow> 0 \<le> a / of_nat k" |
|
655 |
by (simp add: x zero_le_divide_iff) |
|
656 |
moreover have "a / of_nat k \<le> (a - of_nat i) / of_nat (k - i)" if "i < k" for i |
|
63466 | 657 |
proof - |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
658 |
from assms have "a * of_nat i \<ge> of_nat (i * k)" |
63466 | 659 |
by (metis mult.commute mult_le_cancel_right of_nat_less_0_iff of_nat_mult) |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
660 |
then have "a * of_nat k - a * of_nat i \<le> a * of_nat k - of_nat (i * k)" |
63466 | 661 |
by arith |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
662 |
then have "a * of_nat (k - i) \<le> (a - of_nat i) * of_nat k" |
63466 | 663 |
using \<open>i < k\<close> by (simp add: algebra_simps zero_less_mult_iff of_nat_diff) |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
664 |
then have "a * of_nat (k - i) \<le> (a - of_nat i) * (of_nat k :: 'a)" |
71699 | 665 |
by blast |
63466 | 666 |
with assms show ?thesis |
667 |
using \<open>i < k\<close> by (simp add: field_simps) |
|
668 |
qed |
|
71699 | 669 |
ultimately show ?thesis |
670 |
unfolding gbinomial_altdef_of_nat |
|
671 |
by (intro prod_mono) auto |
|
672 |
qed |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
673 |
finally show ?thesis . |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
674 |
qed |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
675 |
|
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
676 |
lemma gbinomial_negated_upper: "(a gchoose k) = (-1) ^ k * ((of_nat k - a - 1) gchoose k)" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
677 |
by (simp add: gbinomial_pochhammer pochhammer_minus algebra_simps) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
678 |
|
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
679 |
lemma gbinomial_minus: "((-a) gchoose k) = (-1) ^ k * ((a + of_nat k - 1) gchoose k)" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
680 |
by (subst gbinomial_negated_upper) (simp add: add_ac) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
681 |
|
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
682 |
lemma Suc_times_gbinomial: "of_nat (Suc k) * ((a + 1) gchoose (Suc k)) = (a + 1) * (a gchoose k)" |
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
683 |
proof (cases k) |
63466 | 684 |
case 0 |
685 |
then show ?thesis by simp |
|
686 |
next |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
687 |
case (Suc b) |
63466 | 688 |
then have "((a + 1) gchoose (Suc (Suc b))) = (\<Prod>i = 0..Suc b. a + (1 - of_nat i)) / fact (b + 2)" |
64272 | 689 |
by (simp add: field_simps gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost) |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
690 |
also have "(\<Prod>i = 0..Suc b. a + (1 - of_nat i)) = (a + 1) * (\<Prod>i = 0..b. a - of_nat i)" |
70113
c8deb8ba6d05
Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
691 |
by (simp add: prod.atLeast0_atMost_Suc_shift del: prod.cl_ivl_Suc) |
63466 | 692 |
also have "\<dots> / fact (b + 2) = (a + 1) / of_nat (Suc (Suc b)) * (a gchoose Suc b)" |
64272 | 693 |
by (simp_all add: gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost) |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
694 |
finally show ?thesis by (simp add: Suc field_simps del: of_nat_Suc) |
63466 | 695 |
qed |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
696 |
|
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
697 |
lemma gbinomial_factors: "((a + 1) gchoose (Suc k)) = (a + 1) / of_nat (Suc k) * (a gchoose k)" |
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
698 |
proof (cases k) |
63466 | 699 |
case 0 |
700 |
then show ?thesis by simp |
|
701 |
next |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
702 |
case (Suc b) |
63466 | 703 |
then have "((a + 1) gchoose (Suc (Suc b))) = (\<Prod>i = 0 .. Suc b. a + (1 - of_nat i)) / fact (b + 2)" |
64272 | 704 |
by (simp add: field_simps gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost) |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
705 |
also have "(\<Prod>i = 0 .. Suc b. a + (1 - of_nat i)) = (a + 1) * (\<Prod>i = 0..b. a - of_nat i)" |
70113
c8deb8ba6d05
Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
706 |
by (simp add: prod.atLeast0_atMost_Suc_shift del: prod.cl_ivl_Suc) |
63466 | 707 |
also have "\<dots> / fact (b + 2) = (a + 1) / of_nat (Suc (Suc b)) * (a gchoose Suc b)" |
64272 | 708 |
by (simp_all add: gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost atLeast0AtMost) |
63466 | 709 |
finally show ?thesis |
710 |
by (simp add: Suc) |
|
711 |
qed |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
712 |
|
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
713 |
lemma gbinomial_rec: "((a + 1) gchoose (Suc k)) = (a gchoose k) * ((a + 1) / of_nat (Suc k))" |
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
714 |
using gbinomial_mult_1[of a k] |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
715 |
by (subst gbinomial_Suc_Suc) (simp add: field_simps del: of_nat_Suc, simp add: algebra_simps) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
716 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
717 |
lemma gbinomial_of_nat_symmetric: "k \<le> n \<Longrightarrow> (of_nat n) gchoose k = (of_nat n) gchoose (n - k)" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
718 |
using binomial_symmetric[of k n] by (simp add: binomial_gbinomial [symmetric]) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
719 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
720 |
|
67299 | 721 |
text \<open>The absorption identity (equation 5.5 @{cite \<open>p.~157\<close> GKP_CM}): |
63466 | 722 |
\[ |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
723 |
{r \choose k} = \frac{r}{k}{r - 1 \choose k - 1},\quad \textnormal{integer } k \neq 0. |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
724 |
\]\<close> |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
725 |
lemma gbinomial_absorption': "k > 0 \<Longrightarrow> a gchoose k = (a / of_nat k) * (a - 1 gchoose (k - 1))" |
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
726 |
using gbinomial_rec[of "a - 1" "k - 1"] |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
727 |
by (simp_all add: gbinomial_rec field_simps del: of_nat_Suc) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
728 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
729 |
text \<open>The absorption identity is written in the following form to avoid |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
730 |
division by $k$ (the lower index) and therefore remove the $k \neq 0$ |
67299 | 731 |
restriction @{cite \<open>p.~157\<close> GKP_CM}: |
63466 | 732 |
\[ |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
733 |
k{r \choose k} = r{r - 1 \choose k - 1}, \quad \textnormal{integer } k. |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
734 |
\]\<close> |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
735 |
lemma gbinomial_absorption: "of_nat (Suc k) * (a gchoose Suc k) = a * ((a - 1) gchoose k)" |
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
736 |
using gbinomial_absorption'[of "Suc k" a] by (simp add: field_simps del: of_nat_Suc) |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
737 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
738 |
text \<open>The absorption identity for natural number binomial coefficients:\<close> |
63466 | 739 |
lemma binomial_absorption: "Suc k * (n choose Suc k) = n * ((n - 1) choose k)" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
740 |
by (cases n) (simp_all add: binomial_eq_0 Suc_times_binomial del: binomial_Suc_Suc mult_Suc) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
741 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
742 |
text \<open>The absorption companion identity for natural number coefficients, |
67299 | 743 |
following the proof by GKP @{cite \<open>p.~157\<close> GKP_CM}:\<close> |
63466 | 744 |
lemma binomial_absorb_comp: "(n - k) * (n choose k) = n * ((n - 1) choose k)" |
745 |
(is "?lhs = ?rhs") |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
746 |
proof (cases "n \<le> k") |
63466 | 747 |
case True |
748 |
then show ?thesis by auto |
|
749 |
next |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
750 |
case False |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
751 |
then have "?rhs = Suc ((n - 1) - k) * (n choose Suc ((n - 1) - k))" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
752 |
using binomial_symmetric[of k "n - 1"] binomial_absorption[of "(n - 1) - k" n] |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
753 |
by simp |
63466 | 754 |
also have "Suc ((n - 1) - k) = n - k" |
755 |
using False by simp |
|
756 |
also have "n choose \<dots> = n choose k" |
|
757 |
using False by (intro binomial_symmetric [symmetric]) simp_all |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
758 |
finally show ?thesis .. |
63466 | 759 |
qed |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
760 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
761 |
text \<open>The generalised absorption companion identity:\<close> |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
762 |
lemma gbinomial_absorb_comp: "(a - of_nat k) * (a gchoose k) = a * ((a - 1) gchoose k)" |
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
763 |
using pochhammer_absorb_comp[of a k] by (simp add: gbinomial_pochhammer) |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
764 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
765 |
lemma gbinomial_addition_formula: |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
766 |
"a gchoose (Suc k) = ((a - 1) gchoose (Suc k)) + ((a - 1) gchoose k)" |
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
767 |
using gbinomial_Suc_Suc[of "a - 1" k] by (simp add: algebra_simps) |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
768 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
769 |
lemma binomial_addition_formula: |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
770 |
"0 < n \<Longrightarrow> n choose (Suc k) = ((n - 1) choose (Suc k)) + ((n - 1) choose k)" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
771 |
by (subst choose_reduce_nat) simp_all |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
772 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
773 |
text \<open> |
67299 | 774 |
Equation 5.9 of the reference material @{cite \<open>p.~159\<close> GKP_CM} is a useful |
63466 | 775 |
summation formula, operating on both indices: |
776 |
\[ |
|
777 |
\sum\limits_{k \leq n}{r + k \choose k} = {r + n + 1 \choose n}, |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
778 |
\quad \textnormal{integer } n. |
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62347
diff
changeset
|
779 |
\] |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
780 |
\<close> |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
781 |
lemma gbinomial_parallel_sum: "(\<Sum>k\<le>n. (a + of_nat k) gchoose k) = (a + of_nat n + 1) gchoose n" |
63466 | 782 |
proof (induct n) |
783 |
case 0 |
|
784 |
then show ?case by simp |
|
785 |
next |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
786 |
case (Suc m) |
63466 | 787 |
then show ?case |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
788 |
using gbinomial_Suc_Suc[of "(a + of_nat m + 1)" m] |
63466 | 789 |
by (simp add: add_ac) |
790 |
qed |
|
791 |
||
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
792 |
|
63373 | 793 |
subsubsection \<open>Summation on the upper index\<close> |
63466 | 794 |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
795 |
text \<open> |
67299 | 796 |
Another summation formula is equation 5.10 of the reference material @{cite \<open>p.~160\<close> GKP_CM}, |
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62347
diff
changeset
|
797 |
aptly named \emph{summation on the upper index}:\[\sum_{0 \leq k \leq n} {k \choose m} = |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
798 |
{n + 1 \choose m + 1}, \quad \textnormal{integers } m, n \geq 0.\] |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
799 |
\<close> |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
800 |
lemma gbinomial_sum_up_index: |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
801 |
"(\<Sum>j = 0..n. (of_nat j gchoose k) :: 'a::field_char_0) = (of_nat n + 1) gchoose (k + 1)" |
63466 | 802 |
proof (induct n) |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
803 |
case 0 |
63466 | 804 |
show ?case |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
805 |
using gbinomial_Suc_Suc[of 0 k] |
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
806 |
by (cases k) auto |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
807 |
next |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
808 |
case (Suc n) |
63466 | 809 |
then show ?case |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
810 |
using gbinomial_Suc_Suc[of "of_nat (Suc n) :: 'a" k] |
63466 | 811 |
by (simp add: add_ac) |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
812 |
qed |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
813 |
|
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62347
diff
changeset
|
814 |
lemma gbinomial_index_swap: |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
815 |
"((-1) ^ k) * ((- (of_nat n) - 1) gchoose k) = ((-1) ^ n) * ((- (of_nat k) - 1) gchoose n)" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
816 |
(is "?lhs = ?rhs") |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
817 |
proof - |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
818 |
have "?lhs = (of_nat (k + n) gchoose k)" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
819 |
by (subst gbinomial_negated_upper) (simp add: power_mult_distrib [symmetric]) |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
820 |
also have "\<dots> = (of_nat (k + n) gchoose n)" |
63466 | 821 |
by (subst gbinomial_of_nat_symmetric) simp_all |
822 |
also have "\<dots> = ?rhs" |
|
823 |
by (subst gbinomial_negated_upper) simp |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
824 |
finally show ?thesis . |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
825 |
qed |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
826 |
|
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
827 |
lemma gbinomial_sum_lower_neg: "(\<Sum>k\<le>m. (a gchoose k) * (- 1) ^ k) = (- 1) ^ m * (a - 1 gchoose m)" |
63466 | 828 |
(is "?lhs = ?rhs") |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
829 |
proof - |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
830 |
have "?lhs = (\<Sum>k\<le>m. -(a + 1) + of_nat k gchoose k)" |
64267 | 831 |
by (intro sum.cong[OF refl]) (subst gbinomial_negated_upper, simp add: power_mult_distrib) |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
832 |
also have "\<dots> = - a + of_nat m gchoose m" |
63466 | 833 |
by (subst gbinomial_parallel_sum) simp |
834 |
also have "\<dots> = ?rhs" |
|
835 |
by (subst gbinomial_negated_upper) (simp add: power_mult_distrib) |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
836 |
finally show ?thesis . |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
837 |
qed |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
838 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
839 |
lemma gbinomial_partial_row_sum: |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
840 |
"(\<Sum>k\<le>m. (a gchoose k) * ((a / 2) - of_nat k)) = ((of_nat m + 1)/2) * (a gchoose (m + 1))" |
63466 | 841 |
proof (induct m) |
842 |
case 0 |
|
843 |
then show ?case by simp |
|
844 |
next |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
845 |
case (Suc mm) |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
846 |
then have "(\<Sum>k\<le>Suc mm. (a gchoose k) * (a / 2 - of_nat k)) = |
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
847 |
(a - of_nat (Suc mm)) * (a gchoose Suc mm) / 2" |
63466 | 848 |
by (simp add: field_simps) |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
849 |
also have "\<dots> = a * (a - 1 gchoose Suc mm) / 2" |
63466 | 850 |
by (subst gbinomial_absorb_comp) (rule refl) |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
851 |
also have "\<dots> = (of_nat (Suc mm) + 1) / 2 * (a gchoose (Suc mm + 1))" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
852 |
by (subst gbinomial_absorption [symmetric]) simp |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
853 |
finally show ?case . |
63466 | 854 |
qed |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
855 |
|
64267 | 856 |
lemma sum_bounds_lt_plus1: "(\<Sum>k<mm. f (Suc k)) = (\<Sum>k=1..mm. f k)" |
63466 | 857 |
by (induct mm) simp_all |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
858 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
859 |
lemma gbinomial_partial_sum_poly: |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
860 |
"(\<Sum>k\<le>m. (of_nat m + a gchoose k) * x^k * y^(m-k)) = |
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
861 |
(\<Sum>k\<le>m. (-a gchoose k) * (-x)^k * (x + y)^(m-k))" |
63466 | 862 |
(is "?lhs m = ?rhs m") |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
863 |
proof (induction m) |
63466 | 864 |
case 0 |
865 |
then show ?case by simp |
|
866 |
next |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
867 |
case (Suc mm) |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
868 |
define G where "G i k = (of_nat i + a gchoose k) * x^k * y^(i - k)" for i k |
63040 | 869 |
define S where "S = ?lhs" |
63466 | 870 |
have SG_def: "S = (\<lambda>i. (\<Sum>k\<le>i. (G i k)))" |
871 |
unfolding S_def G_def .. |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
872 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
873 |
have "S (Suc mm) = G (Suc mm) 0 + (\<Sum>k=Suc 0..Suc mm. G (Suc mm) k)" |
70097
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69768
diff
changeset
|
874 |
using SG_def by (simp add: sum.atLeast_Suc_atMost atLeast0AtMost [symmetric]) |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
875 |
also have "(\<Sum>k=Suc 0..Suc mm. G (Suc mm) k) = (\<Sum>k=0..mm. G (Suc mm) (Suc k))" |
70113
c8deb8ba6d05
Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
876 |
by (subst sum.shift_bounds_cl_Suc_ivl) simp |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
877 |
also have "\<dots> = (\<Sum>k=0..mm. ((of_nat mm + a gchoose (Suc k)) + |
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
878 |
(of_nat mm + a gchoose k)) * x^(Suc k) * y^(mm - k))" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
879 |
unfolding G_def by (subst gbinomial_addition_formula) simp |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
880 |
also have "\<dots> = (\<Sum>k=0..mm. (of_nat mm + a gchoose (Suc k)) * x^(Suc k) * y^(mm - k)) + |
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
881 |
(\<Sum>k=0..mm. (of_nat mm + a gchoose k) * x^(Suc k) * y^(mm - k))" |
64267 | 882 |
by (subst sum.distrib [symmetric]) (simp add: algebra_simps) |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
883 |
also have "(\<Sum>k=0..mm. (of_nat mm + a gchoose (Suc k)) * x^(Suc k) * y^(mm - k)) = |
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
884 |
(\<Sum>k<Suc mm. (of_nat mm + a gchoose (Suc k)) * x^(Suc k) * y^(mm - k))" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
885 |
by (simp only: atLeast0AtMost lessThan_Suc_atMost) |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
886 |
also have "\<dots> = (\<Sum>k<mm. (of_nat mm + a gchoose Suc k) * x^(Suc k) * y^(mm-k)) + |
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
887 |
(of_nat mm + a gchoose (Suc mm)) * x^(Suc mm)" |
63466 | 888 |
(is "_ = ?A + ?B") |
70097
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69768
diff
changeset
|
889 |
by (subst sum.lessThan_Suc) simp |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
890 |
also have "?A = (\<Sum>k=1..mm. (of_nat mm + a gchoose k) * x^k * y^(mm - k + 1))" |
64267 | 891 |
proof (subst sum_bounds_lt_plus1 [symmetric], intro sum.cong[OF refl], clarify) |
63466 | 892 |
fix k |
893 |
assume "k < mm" |
|
894 |
then have "mm - k = mm - Suc k + 1" |
|
895 |
by linarith |
|
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
896 |
then show "(of_nat mm + a gchoose Suc k) * x ^ Suc k * y ^ (mm - k) = |
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
897 |
(of_nat mm + a gchoose Suc k) * x ^ Suc k * y ^ (mm - Suc k + 1)" |
63466 | 898 |
by (simp only:) |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
899 |
qed |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
900 |
also have "\<dots> + ?B = y * (\<Sum>k=1..mm. (G mm k)) + (of_nat mm + a gchoose (Suc mm)) * x^(Suc mm)" |
64267 | 901 |
unfolding G_def by (subst sum_distrib_left) (simp add: algebra_simps) |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
902 |
also have "(\<Sum>k=0..mm. (of_nat mm + a gchoose k) * x^(Suc k) * y^(mm - k)) = x * (S mm)" |
64267 | 903 |
unfolding S_def by (subst sum_distrib_left) (simp add: atLeast0AtMost algebra_simps) |
63466 | 904 |
also have "(G (Suc mm) 0) = y * (G mm 0)" |
905 |
by (simp add: G_def) |
|
906 |
finally have "S (Suc mm) = |
|
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
907 |
y * (G mm 0 + (\<Sum>k=1..mm. (G mm k))) + (of_nat mm + a gchoose (Suc mm)) * x^(Suc mm) + x * (S mm)" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
908 |
by (simp add: ring_distribs) |
63466 | 909 |
also have "G mm 0 + (\<Sum>k=1..mm. (G mm k)) = S mm" |
70097
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69768
diff
changeset
|
910 |
by (simp add: sum.atLeast_Suc_atMost[symmetric] SG_def atLeast0AtMost) |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
911 |
finally have "S (Suc mm) = (x + y) * (S mm) + (of_nat mm + a gchoose (Suc mm)) * x^(Suc mm)" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
912 |
by (simp add: algebra_simps) |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
913 |
also have "(of_nat mm + a gchoose (Suc mm)) = (-1) ^ (Suc mm) * (- a gchoose (Suc mm))" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
914 |
by (subst gbinomial_negated_upper) simp |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
915 |
also have "(-1) ^ Suc mm * (- a gchoose Suc mm) * x ^ Suc mm = |
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
916 |
(- a gchoose (Suc mm)) * (-x) ^ Suc mm" |
63466 | 917 |
by (simp add: power_minus[of x]) |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
918 |
also have "(x + y) * S mm + \<dots> = (x + y) * ?rhs mm + (- a gchoose (Suc mm)) * (- x)^Suc mm" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
919 |
unfolding S_def by (subst Suc.IH) simp |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
920 |
also have "(x + y) * ?rhs mm = (\<Sum>n\<le>mm. ((- a gchoose n) * (- x) ^ n * (x + y) ^ (Suc mm - n)))" |
64267 | 921 |
by (subst sum_distrib_left, rule sum.cong) (simp_all add: Suc_diff_le) |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
922 |
also have "\<dots> + (-a gchoose (Suc mm)) * (-x)^Suc mm = |
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
923 |
(\<Sum>n\<le>Suc mm. (- a gchoose n) * (- x) ^ n * (x + y) ^ (Suc mm - n))" |
63466 | 924 |
by simp |
925 |
finally show ?case |
|
926 |
by (simp only: S_def) |
|
927 |
qed |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
928 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
929 |
lemma gbinomial_partial_sum_poly_xpos: |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
930 |
"(\<Sum>k\<le>m. (of_nat m + a gchoose k) * x^k * y^(m-k)) = |
71699 | 931 |
(\<Sum>k\<le>m. (of_nat k + a - 1 gchoose k) * x^k * (x + y)^(m-k))" (is "?lhs = ?rhs") |
932 |
proof - |
|
933 |
have "?lhs = (\<Sum>k\<le>m. (- a gchoose k) * (- x) ^ k * (x + y) ^ (m - k))" |
|
934 |
by (simp add: gbinomial_partial_sum_poly) |
|
935 |
also have "... = (\<Sum>k\<le>m. (-1) ^ k * (of_nat k - - a - 1 gchoose k) * (- x) ^ k * (x + y) ^ (m - k))" |
|
73932
fd21b4a93043
added opaque_combs and renamed hide_lams to opaque_lifting
desharna
parents:
72302
diff
changeset
|
936 |
by (metis (no_types, opaque_lifting) gbinomial_negated_upper) |
71699 | 937 |
also have "... = ?rhs" |
938 |
by (intro sum.cong) (auto simp flip: power_mult_distrib) |
|
939 |
finally show ?thesis . |
|
940 |
qed |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
941 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
942 |
lemma binomial_r_part_sum: "(\<Sum>k\<le>m. (2 * m + 1 choose k)) = 2 ^ (2 * m)" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
943 |
proof - |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
944 |
have "2 * 2^(2*m) = (\<Sum>k = 0..(2 * m + 1). (2 * m + 1 choose k))" |
68077
ee8c13ae81e9
Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
945 |
using choose_row_sum[where n="2 * m + 1"] by (simp add: atMost_atLeast0) |
63466 | 946 |
also have "(\<Sum>k = 0..(2 * m + 1). (2 * m + 1 choose k)) = |
947 |
(\<Sum>k = 0..m. (2 * m + 1 choose k)) + |
|
948 |
(\<Sum>k = m+1..2*m+1. (2 * m + 1 choose k))" |
|
70113
c8deb8ba6d05
Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
949 |
using sum.ub_add_nat[of 0 m "\<lambda>k. 2 * m + 1 choose k" "m+1"] |
63466 | 950 |
by (simp add: mult_2) |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
951 |
also have "(\<Sum>k = m+1..2*m+1. (2 * m + 1 choose k)) = |
63466 | 952 |
(\<Sum>k = 0..m. (2 * m + 1 choose (k + (m + 1))))" |
70113
c8deb8ba6d05
Fixing the main Homology theory; also moving a lot of sum/prod lemmas into their generic context
paulson <lp15@cam.ac.uk>
parents:
70097
diff
changeset
|
953 |
by (subst sum.shift_bounds_cl_nat_ivl [symmetric]) (simp add: mult_2) |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
954 |
also have "\<dots> = (\<Sum>k = 0..m. (2 * m + 1 choose (m - k)))" |
64267 | 955 |
by (intro sum.cong[OF refl], subst binomial_symmetric) simp_all |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
956 |
also have "\<dots> = (\<Sum>k = 0..m. (2 * m + 1 choose k))" |
67411
3f4b0c84630f
restored naming of lemmas after corresponding constants
haftmann
parents:
67399
diff
changeset
|
957 |
using sum.atLeastAtMost_rev [of "\<lambda>k. 2 * m + 1 choose (m - k)" 0 m] |
63417
c184ec919c70
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
haftmann
parents:
63373
diff
changeset
|
958 |
by simp |
63466 | 959 |
also have "\<dots> + \<dots> = 2 * \<dots>" |
960 |
by simp |
|
961 |
finally show ?thesis |
|
962 |
by (subst (asm) mult_cancel1) (simp add: atLeast0AtMost) |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
963 |
qed |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
964 |
|
63466 | 965 |
lemma gbinomial_r_part_sum: "(\<Sum>k\<le>m. (2 * (of_nat m) + 1 gchoose k)) = 2 ^ (2 * m)" |
966 |
(is "?lhs = ?rhs") |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
967 |
proof - |
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62347
diff
changeset
|
968 |
have "?lhs = of_nat (\<Sum>k\<le>m. (2 * m + 1) choose k)" |
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
969 |
by (simp add: binomial_gbinomial add_ac) |
63466 | 970 |
also have "\<dots> = of_nat (2 ^ (2 * m))" |
971 |
by (subst binomial_r_part_sum) (rule refl) |
|
63366
209c4cbbc4cd
define binomial coefficents directly via combinatorial definition
haftmann
parents:
63363
diff
changeset
|
972 |
finally show ?thesis by simp |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
973 |
qed |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
974 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
975 |
lemma gbinomial_sum_nat_pow2: |
63466 | 976 |
"(\<Sum>k\<le>m. (of_nat (m + k) gchoose k :: 'a::field_char_0) / 2 ^ k) = 2 ^ m" |
977 |
(is "?lhs = ?rhs") |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
978 |
proof - |
63466 | 979 |
have "2 ^ m * 2 ^ m = (2 ^ (2*m) :: 'a)" |
980 |
by (induct m) simp_all |
|
981 |
also have "\<dots> = (\<Sum>k\<le>m. (2 * (of_nat m) + 1 gchoose k))" |
|
982 |
using gbinomial_r_part_sum .. |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
983 |
also have "\<dots> = (\<Sum>k\<le>m. (of_nat (m + k) gchoose k) * 2 ^ (m - k))" |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
984 |
using gbinomial_partial_sum_poly_xpos[where x="1" and y="1" and a="of_nat m + 1" and m="m"] |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
985 |
by (simp add: add_ac) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
986 |
also have "\<dots> = 2 ^ m * (\<Sum>k\<le>m. (of_nat (m + k) gchoose k) / 2 ^ k)" |
64267 | 987 |
by (subst sum_distrib_left) (simp add: algebra_simps power_diff) |
63466 | 988 |
finally show ?thesis |
989 |
by (subst (asm) mult_left_cancel) simp_all |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
990 |
qed |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
991 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
992 |
lemma gbinomial_trinomial_revision: |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
993 |
assumes "k \<le> m" |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
994 |
shows "(a gchoose m) * (of_nat m gchoose k) = (a gchoose k) * (a - of_nat k gchoose (m - k))" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
995 |
proof - |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
996 |
have "(a gchoose m) * (of_nat m gchoose k) = (a gchoose m) * fact m / (fact k * fact (m - k))" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
997 |
using assms by (simp add: binomial_gbinomial [symmetric] binomial_fact) |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
998 |
also have "\<dots> = (a gchoose k) * (a - of_nat k gchoose (m - k))" |
63466 | 999 |
using assms by (simp add: gbinomial_pochhammer power_diff pochhammer_product) |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
1000 |
finally show ?thesis . |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
1001 |
qed |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61076
diff
changeset
|
1002 |
|
63466 | 1003 |
text \<open>Versions of the theorems above for the natural-number version of "choose"\<close> |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1004 |
lemma binomial_altdef_of_nat: |
63466 | 1005 |
"k \<le> n \<Longrightarrow> of_nat (n choose k) = (\<Prod>i = 0..<k. of_nat (n - i) / of_nat (k - i) :: 'a)" |
1006 |
for n k :: nat and x :: "'a::field_char_0" |
|
1007 |
by (simp add: gbinomial_altdef_of_nat binomial_gbinomial of_nat_diff) |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1008 |
|
63466 | 1009 |
lemma binomial_ge_n_over_k_pow_k: "k \<le> n \<Longrightarrow> (of_nat n / of_nat k :: 'a) ^ k \<le> of_nat (n choose k)" |
1010 |
for k n :: nat and x :: "'a::linordered_field" |
|
1011 |
by (simp add: gbinomial_ge_n_over_k_pow_k binomial_gbinomial of_nat_diff) |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1012 |
|
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1013 |
lemma binomial_le_pow: |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1014 |
assumes "r \<le> n" |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1015 |
shows "n choose r \<le> n ^ r" |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1016 |
proof - |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1017 |
have "n choose r \<le> fact n div fact (n - r)" |
63466 | 1018 |
using assms by (subst binomial_fact_lemma[symmetric]) auto |
1019 |
with fact_div_fact_le_pow [OF assms] show ?thesis |
|
1020 |
by auto |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1021 |
qed |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1022 |
|
63466 | 1023 |
lemma binomial_altdef_nat: "k \<le> n \<Longrightarrow> n choose k = fact n div (fact k * fact (n - k))" |
1024 |
for k n :: nat |
|
1025 |
by (subst binomial_fact_lemma [symmetric]) auto |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1026 |
|
63466 | 1027 |
lemma choose_dvd: |
71699 | 1028 |
assumes "k \<le> n" shows "fact k * fact (n - k) dvd (fact n :: 'a::linordered_semidom)" |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
1029 |
unfolding dvd_def |
71699 | 1030 |
proof |
1031 |
show "fact n = fact k * fact (n - k) * of_nat (n choose k)" |
|
1032 |
by (metis assms binomial_fact_lemma of_nat_fact of_nat_mult) |
|
1033 |
qed |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1034 |
|
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62347
diff
changeset
|
1035 |
lemma fact_fact_dvd_fact: |
66806
a4e82b58d833
abolished (semi)ring_div in favour of euclidean_(semi)ring_cancel
haftmann
parents:
66311
diff
changeset
|
1036 |
"fact k * fact n dvd (fact (k + n) :: 'a::linordered_semidom)" |
63466 | 1037 |
by (metis add.commute add_diff_cancel_left' choose_dvd le_add2) |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1038 |
|
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1039 |
lemma choose_mult_lemma: |
63466 | 1040 |
"((m + r + k) choose (m + k)) * ((m + k) choose k) = ((m + r + k) choose k) * ((m + r) choose m)" |
1041 |
(is "?lhs = _") |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1042 |
proof - |
63466 | 1043 |
have "?lhs = |
1044 |
fact (m + r + k) div (fact (m + k) * fact (m + r - m)) * (fact (m + k) div (fact k * fact m))" |
|
63092 | 1045 |
by (simp add: binomial_altdef_nat) |
71699 | 1046 |
also have "... = fact (m + r + k) * fact (m + k) div |
1047 |
(fact (m + k) * fact (m + r - m) * (fact k * fact m))" |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1048 |
apply (subst div_mult_div_if_dvd) |
71699 | 1049 |
apply (auto simp: algebra_simps fact_fact_dvd_fact) |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1050 |
apply (metis add.assoc add.commute fact_fact_dvd_fact) |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1051 |
done |
71699 | 1052 |
also have "\<dots> = fact (m + r + k) div (fact r * (fact k * fact m))" |
1053 |
by (auto simp: algebra_simps fact_fact_dvd_fact) |
|
63466 | 1054 |
also have "\<dots> = (fact (m + r + k) * fact (m + r)) div (fact r * (fact k * fact m) * fact (m + r))" |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1055 |
apply (subst div_mult_div_if_dvd [symmetric]) |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
1056 |
apply (auto simp add: algebra_simps) |
62344
759d684c0e60
pulled out legacy aliasses and infamous dvd interpretations into theory appendix
haftmann
parents:
62142
diff
changeset
|
1057 |
apply (metis fact_fact_dvd_fact dvd_trans nat_mult_dvd_cancel_disj) |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1058 |
done |
63466 | 1059 |
also have "\<dots> = |
1060 |
(fact (m + r + k) div (fact k * fact (m + r)) * (fact (m + r) div (fact r * fact m)))" |
|
71720 | 1061 |
by (auto simp: div_mult_div_if_dvd fact_fact_dvd_fact algebra_simps) |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1062 |
finally show ?thesis |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1063 |
by (simp add: binomial_altdef_nat mult.commute) |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1064 |
qed |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1065 |
|
63466 | 1066 |
text \<open>The "Subset of a Subset" identity.\<close> |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1067 |
lemma choose_mult: |
63466 | 1068 |
"k \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> (n choose m) * (m choose k) = (n choose k) * ((n - k) choose (m - k))" |
1069 |
using choose_mult_lemma [of "m-k" "n-m" k] by simp |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1070 |
|
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1071 |
|
63373 | 1072 |
subsection \<open>More on Binomial Coefficients\<close> |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1073 |
|
63466 | 1074 |
lemma choose_one: "n choose 1 = n" for n :: nat |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1075 |
by simp |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1076 |
|
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1077 |
lemma card_UNION: |
63466 | 1078 |
assumes "finite A" |
1079 |
and "\<forall>k \<in> A. finite k" |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1080 |
shows "card (\<Union>A) = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (- 1) ^ (card I + 1) * int (card (\<Inter>I)))" |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1081 |
(is "?lhs = ?rhs") |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1082 |
proof - |
63466 | 1083 |
have "?rhs = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (- 1) ^ (card I + 1) * (\<Sum>_\<in>\<Inter>I. 1))" |
1084 |
by simp |
|
1085 |
also have "\<dots> = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (\<Sum>_\<in>\<Inter>I. (- 1) ^ (card I + 1)))" |
|
1086 |
(is "_ = nat ?rhs") |
|
64267 | 1087 |
by (subst sum_distrib_left) simp |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1088 |
also have "?rhs = (\<Sum>(I, _)\<in>Sigma {I. I \<subseteq> A \<and> I \<noteq> {}} Inter. (- 1) ^ (card I + 1))" |
64267 | 1089 |
using assms by (subst sum.Sigma) auto |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1090 |
also have "\<dots> = (\<Sum>(x, I)\<in>(SIGMA x:UNIV. {I. I \<subseteq> A \<and> I \<noteq> {} \<and> x \<in> \<Inter>I}). (- 1) ^ (card I + 1))" |
69768 | 1091 |
by (rule sum.reindex_cong [where l = "\<lambda>(x, y). (y, x)"]) (auto intro: inj_onI) |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1092 |
also have "\<dots> = (\<Sum>(x, I)\<in>(SIGMA x:\<Union>A. {I. I \<subseteq> A \<and> I \<noteq> {} \<and> x \<in> \<Inter>I}). (- 1) ^ (card I + 1))" |
63466 | 1093 |
using assms |
64267 | 1094 |
by (auto intro!: sum.mono_neutral_cong_right finite_SigmaI2 intro: finite_subset[where B="\<Union>A"]) |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1095 |
also have "\<dots> = (\<Sum>x\<in>\<Union>A. (\<Sum>I|I \<subseteq> A \<and> I \<noteq> {} \<and> x \<in> \<Inter>I. (- 1) ^ (card I + 1)))" |
64267 | 1096 |
using assms by (subst sum.Sigma) auto |
1097 |
also have "\<dots> = (\<Sum>_\<in>\<Union>A. 1)" (is "sum ?lhs _ = _") |
|
1098 |
proof (rule sum.cong[OF refl]) |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1099 |
fix x |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1100 |
assume x: "x \<in> \<Union>A" |
63040 | 1101 |
define K where "K = {X \<in> A. x \<in> X}" |
63466 | 1102 |
with \<open>finite A\<close> have K: "finite K" |
1103 |
by auto |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1104 |
let ?I = "\<lambda>i. {I. I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I}" |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1105 |
have "inj_on snd (SIGMA i:{1..card A}. ?I i)" |
63466 | 1106 |
using assms by (auto intro!: inj_onI) |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1107 |
moreover have [symmetric]: "snd ` (SIGMA i:{1..card A}. ?I i) = {I. I \<subseteq> A \<and> I \<noteq> {} \<and> x \<in> \<Inter>I}" |
63466 | 1108 |
using assms |
1109 |
by (auto intro!: rev_image_eqI[where x="(card a, a)" for a] |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1110 |
simp add: card_gt_0_iff[folded Suc_le_eq] |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1111 |
dest: finite_subset intro: card_mono) |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1112 |
ultimately have "?lhs x = (\<Sum>(i, I)\<in>(SIGMA i:{1..card A}. ?I i). (- 1) ^ (i + 1))" |
64267 | 1113 |
by (rule sum.reindex_cong [where l = snd]) fastforce |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1114 |
also have "\<dots> = (\<Sum>i=1..card A. (\<Sum>I|I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I. (- 1) ^ (i + 1)))" |
64267 | 1115 |
using assms by (subst sum.Sigma) auto |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1116 |
also have "\<dots> = (\<Sum>i=1..card A. (- 1) ^ (i + 1) * (\<Sum>I|I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I. 1))" |
64267 | 1117 |
by (subst sum_distrib_left) simp |
63466 | 1118 |
also have "\<dots> = (\<Sum>i=1..card K. (- 1) ^ (i + 1) * (\<Sum>I|I \<subseteq> K \<and> card I = i. 1))" |
1119 |
(is "_ = ?rhs") |
|
64267 | 1120 |
proof (rule sum.mono_neutral_cong_right[rule_format]) |
63466 | 1121 |
show "finite {1..card A}" |
1122 |
by simp |
|
1123 |
show "{1..card K} \<subseteq> {1..card A}" |
|
1124 |
using \<open>finite A\<close> by (auto simp add: K_def intro: card_mono) |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1125 |
next |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1126 |
fix i |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1127 |
assume "i \<in> {1..card A} - {1..card K}" |
63466 | 1128 |
then have i: "i \<le> card A" "card K < i" |
1129 |
by auto |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1130 |
have "{I. I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I} = {I. I \<subseteq> K \<and> card I = i}" |
63466 | 1131 |
by (auto simp add: K_def) |
1132 |
also have "\<dots> = {}" |
|
1133 |
using \<open>finite A\<close> i by (auto simp add: K_def dest: card_mono[rotated 1]) |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1134 |
finally show "(- 1) ^ (i + 1) * (\<Sum>I | I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I. 1 :: int) = 0" |
63466 | 1135 |
by (simp only:) simp |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1136 |
next |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1137 |
fix i |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1138 |
have "(\<Sum>I | I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I. 1) = (\<Sum>I | I \<subseteq> K \<and> card I = i. 1 :: int)" |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1139 |
(is "?lhs = ?rhs") |
64267 | 1140 |
by (rule sum.cong) (auto simp add: K_def) |
63466 | 1141 |
then show "(- 1) ^ (i + 1) * ?lhs = (- 1) ^ (i + 1) * ?rhs" |
1142 |
by simp |
|
1143 |
qed |
|
1144 |
also have "{I. I \<subseteq> K \<and> card I = 0} = {{}}" |
|
1145 |
using assms by (auto simp add: card_eq_0_iff K_def dest: finite_subset) |
|
1146 |
then have "?rhs = (\<Sum>i = 0..card K. (- 1) ^ (i + 1) * (\<Sum>I | I \<subseteq> K \<and> card I = i. 1 :: int)) + 1" |
|
70097
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69768
diff
changeset
|
1147 |
by (subst (2) sum.atLeast_Suc_atMost) simp_all |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1148 |
also have "\<dots> = (\<Sum>i = 0..card K. (- 1) * ((- 1) ^ i * int (card K choose i))) + 1" |
63466 | 1149 |
using K by (subst n_subsets[symmetric]) simp_all |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1150 |
also have "\<dots> = - (\<Sum>i = 0..card K. (- 1) ^ i * int (card K choose i)) + 1" |
64267 | 1151 |
by (subst sum_distrib_left[symmetric]) simp |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1152 |
also have "\<dots> = - ((-1 + 1) ^ card K) + 1" |
68077
ee8c13ae81e9
Some tidying up (mostly regarding summations from 0)
paulson <lp15@cam.ac.uk>
parents:
67443
diff
changeset
|
1153 |
by (subst binomial_ring) (simp add: ac_simps atMost_atLeast0) |
63466 | 1154 |
also have "\<dots> = 1" |
1155 |
using x K by (auto simp add: K_def card_gt_0_iff) |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1156 |
finally show "?lhs x = 1" . |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1157 |
qed |
63466 | 1158 |
also have "nat \<dots> = card (\<Union>A)" |
1159 |
by simp |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1160 |
finally show ?thesis .. |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1161 |
qed |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1162 |
|
69593 | 1163 |
text \<open>The number of nat lists of length \<open>m\<close> summing to \<open>N\<close> is \<^term>\<open>(N + m - 1) choose N\<close>:\<close> |
63882
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63725
diff
changeset
|
1164 |
lemma card_length_sum_list_rec: |
63466 | 1165 |
assumes "m \<ge> 1" |
63882
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63725
diff
changeset
|
1166 |
shows "card {l::nat list. length l = m \<and> sum_list l = N} = |
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63725
diff
changeset
|
1167 |
card {l. length l = (m - 1) \<and> sum_list l = N} + |
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63725
diff
changeset
|
1168 |
card {l. length l = m \<and> sum_list l + 1 = N}" |
63466 | 1169 |
(is "card ?C = card ?A + card ?B") |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1170 |
proof - |
63882
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63725
diff
changeset
|
1171 |
let ?A' = "{l. length l = m \<and> sum_list l = N \<and> hd l = 0}" |
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63725
diff
changeset
|
1172 |
let ?B' = "{l. length l = m \<and> sum_list l = N \<and> hd l \<noteq> 0}" |
63466 | 1173 |
let ?f = "\<lambda>l. 0 # l" |
1174 |
let ?g = "\<lambda>l. (hd l + 1) # tl l" |
|
65812 | 1175 |
have 1: "xs \<noteq> [] \<Longrightarrow> x = hd xs \<Longrightarrow> x # tl xs = xs" for x :: nat and xs |
63466 | 1176 |
by simp |
63882
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63725
diff
changeset
|
1177 |
have 2: "xs \<noteq> [] \<Longrightarrow> sum_list(tl xs) = sum_list xs - hd xs" for xs :: "nat list" |
63466 | 1178 |
by (auto simp add: neq_Nil_conv) |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1179 |
have f: "bij_betw ?f ?A ?A'" |
71720 | 1180 |
by (rule bij_betw_byWitness[where f' = tl]) (use assms in \<open>auto simp: 2 1 simp flip: length_0_conv\<close>) |
63882
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63725
diff
changeset
|
1181 |
have 3: "xs \<noteq> [] \<Longrightarrow> hd xs + (sum_list xs - hd xs) = sum_list xs" for xs :: "nat list" |
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63725
diff
changeset
|
1182 |
by (metis 1 sum_list_simps(2) 2) |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1183 |
have g: "bij_betw ?g ?B ?B'" |
63466 | 1184 |
apply (rule bij_betw_byWitness[where f' = "\<lambda>l. (hd l - 1) # tl l"]) |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1185 |
using assms |
71720 | 1186 |
by (auto simp: 2 simp flip: length_0_conv intro!: 3) |
63466 | 1187 |
have fin: "finite {xs. size xs = M \<and> set xs \<subseteq> {0..<N}}" for M N :: nat |
1188 |
using finite_lists_length_eq[OF finite_atLeastLessThan] conj_commute by auto |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1189 |
have fin_A: "finite ?A" using fin[of _ "N+1"] |
63466 | 1190 |
by (intro finite_subset[where ?A = "?A" and ?B = "{xs. size xs = m - 1 \<and> set xs \<subseteq> {0..<N+1}}"]) |
66311 | 1191 |
(auto simp: member_le_sum_list less_Suc_eq_le) |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1192 |
have fin_B: "finite ?B" |
63466 | 1193 |
by (intro finite_subset[where ?A = "?B" and ?B = "{xs. size xs = m \<and> set xs \<subseteq> {0..<N}}"]) |
66311 | 1194 |
(auto simp: member_le_sum_list less_Suc_eq_le fin) |
63466 | 1195 |
have uni: "?C = ?A' \<union> ?B'" |
1196 |
by auto |
|
65350
b149abe619f7
added shuffle product to HOL/List
eberlm <eberlm@in.tum.de>
parents:
64272
diff
changeset
|
1197 |
have disj: "?A' \<inter> ?B' = {}" by blast |
63466 | 1198 |
have "card ?C = card(?A' \<union> ?B')" |
1199 |
using uni by simp |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1200 |
also have "\<dots> = card ?A + card ?B" |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1201 |
using card_Un_disjoint[OF _ _ disj] bij_betw_finite[OF f] bij_betw_finite[OF g] |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1202 |
bij_betw_same_card[OF f] bij_betw_same_card[OF g] fin_A fin_B |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1203 |
by presburger |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1204 |
finally show ?thesis . |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1205 |
qed |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1206 |
|
63882
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63725
diff
changeset
|
1207 |
lemma card_length_sum_list: "card {l::nat list. size l = m \<and> sum_list l = N} = (N + m - 1) choose N" |
67443
3abf6a722518
standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents:
67411
diff
changeset
|
1208 |
\<comment> \<open>by Holden Lee, tidied by Tobias Nipkow\<close> |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1209 |
proof (cases m) |
63466 | 1210 |
case 0 |
1211 |
then show ?thesis |
|
1212 |
by (cases N) (auto cong: conj_cong) |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1213 |
next |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1214 |
case (Suc m') |
63466 | 1215 |
have m: "m \<ge> 1" |
1216 |
by (simp add: Suc) |
|
1217 |
then show ?thesis |
|
1218 |
proof (induct "N + m - 1" arbitrary: N m) |
|
67443
3abf6a722518
standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents:
67411
diff
changeset
|
1219 |
case 0 \<comment> \<open>In the base case, the only solution is [0].\<close> |
63466 | 1220 |
have [simp]: "{l::nat list. length l = Suc 0 \<and> (\<forall>n\<in>set l. n = 0)} = {[0]}" |
1221 |
by (auto simp: length_Suc_conv) |
|
1222 |
have "m = 1 \<and> N = 0" |
|
1223 |
using 0 by linarith |
|
1224 |
then show ?case |
|
1225 |
by simp |
|
1226 |
next |
|
1227 |
case (Suc k) |
|
63882
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63725
diff
changeset
|
1228 |
have c1: "card {l::nat list. size l = (m - 1) \<and> sum_list l = N} = (N + (m - 1) - 1) choose N" |
63466 | 1229 |
proof (cases "m = 1") |
1230 |
case True |
|
1231 |
with Suc.hyps have "N \<ge> 1" |
|
1232 |
by auto |
|
1233 |
with True show ?thesis |
|
1234 |
by (simp add: binomial_eq_0) |
|
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1235 |
next |
63466 | 1236 |
case False |
1237 |
then show ?thesis |
|
1238 |
using Suc by fastforce |
|
1239 |
qed |
|
63882
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63725
diff
changeset
|
1240 |
from Suc have c2: "card {l::nat list. size l = m \<and> sum_list l + 1 = N} = |
63466 | 1241 |
(if N > 0 then ((N - 1) + m - 1) choose (N - 1) else 0)" |
1242 |
proof - |
|
1243 |
have *: "n > 0 \<Longrightarrow> Suc m = n \<longleftrightarrow> m = n - 1" for m n |
|
1244 |
by arith |
|
1245 |
from Suc have "N > 0 \<Longrightarrow> |
|
63882
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63725
diff
changeset
|
1246 |
card {l::nat list. size l = m \<and> sum_list l + 1 = N} = |
63466 | 1247 |
((N - 1) + m - 1) choose (N - 1)" |
1248 |
by (simp add: *) |
|
1249 |
then show ?thesis |
|
1250 |
by auto |
|
1251 |
qed |
|
63882
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63725
diff
changeset
|
1252 |
from Suc.prems have "(card {l::nat list. size l = (m - 1) \<and> sum_list l = N} + |
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63725
diff
changeset
|
1253 |
card {l::nat list. size l = m \<and> sum_list l + 1 = N}) = (N + m - 1) choose N" |
63466 | 1254 |
by (auto simp: c1 c2 choose_reduce_nat[of "N + m - 1" N] simp del: One_nat_def) |
1255 |
then show ?case |
|
63882
018998c00003
renamed listsum -> sum_list, listprod ~> prod_list
nipkow
parents:
63725
diff
changeset
|
1256 |
using card_length_sum_list_rec[OF Suc.prems] by auto |
63466 | 1257 |
qed |
59667
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1258 |
qed |
651ea265d568
Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents:
59658
diff
changeset
|
1259 |
|
69107 | 1260 |
lemma card_disjoint_shuffles: |
65350
b149abe619f7
added shuffle product to HOL/List
eberlm <eberlm@in.tum.de>
parents:
64272
diff
changeset
|
1261 |
assumes "set xs \<inter> set ys = {}" |
69107 | 1262 |
shows "card (shuffles xs ys) = (length xs + length ys) choose length xs" |
65350
b149abe619f7
added shuffle product to HOL/List
eberlm <eberlm@in.tum.de>
parents:
64272
diff
changeset
|
1263 |
using assms |
69107 | 1264 |
proof (induction xs ys rule: shuffles.induct) |
65350
b149abe619f7
added shuffle product to HOL/List
eberlm <eberlm@in.tum.de>
parents:
64272
diff
changeset
|
1265 |
case (3 x xs y ys) |
69107 | 1266 |
have "shuffles (x # xs) (y # ys) = (#) x ` shuffles xs (y # ys) \<union> (#) y ` shuffles (x # xs) ys" |
1267 |
by (rule shuffles.simps) |
|
1268 |
also have "card \<dots> = card ((#) x ` shuffles xs (y # ys)) + card ((#) y ` shuffles (x # xs) ys)" |
|
65350
b149abe619f7
added shuffle product to HOL/List
eberlm <eberlm@in.tum.de>
parents:
64272
diff
changeset
|
1269 |
by (rule card_Un_disjoint) (insert "3.prems", auto) |
69107 | 1270 |
also have "card ((#) x ` shuffles xs (y # ys)) = card (shuffles xs (y # ys))" |
65350
b149abe619f7
added shuffle product to HOL/List
eberlm <eberlm@in.tum.de>
parents:
64272
diff
changeset
|
1271 |
by (rule card_image) auto |
b149abe619f7
added shuffle product to HOL/List
eberlm <eberlm@in.tum.de>
parents:
64272
diff
changeset
|
1272 |
also have "\<dots> = (length xs + length (y # ys)) choose length xs" |
b149abe619f7
added shuffle product to HOL/List
eberlm <eberlm@in.tum.de>
parents:
64272
diff
changeset
|
1273 |
using "3.prems" by (intro "3.IH") auto |
69107 | 1274 |
also have "card ((#) y ` shuffles (x # xs) ys) = card (shuffles (x # xs) ys)" |
65350
b149abe619f7
added shuffle product to HOL/List
eberlm <eberlm@in.tum.de>
parents:
64272
diff
changeset
|
1275 |
by (rule card_image) auto |
b149abe619f7
added shuffle product to HOL/List
eberlm <eberlm@in.tum.de>
parents:
64272
diff
changeset
|
1276 |
also have "\<dots> = (length (x # xs) + length ys) choose length (x # xs)" |
b149abe619f7
added shuffle product to HOL/List
eberlm <eberlm@in.tum.de>
parents:
64272
diff
changeset
|
1277 |
using "3.prems" by (intro "3.IH") auto |
65552
f533820e7248
theories "GCD" and "Binomial" are already included in "Main": this avoids improper imports in applications;
wenzelm
parents:
65350
diff
changeset
|
1278 |
also have "length xs + length (y # ys) choose length xs + \<dots> = |
65350
b149abe619f7
added shuffle product to HOL/List
eberlm <eberlm@in.tum.de>
parents:
64272
diff
changeset
|
1279 |
(length (x # xs) + length (y # ys)) choose length (x # xs)" by simp |
b149abe619f7
added shuffle product to HOL/List
eberlm <eberlm@in.tum.de>
parents:
64272
diff
changeset
|
1280 |
finally show ?case . |
b149abe619f7
added shuffle product to HOL/List
eberlm <eberlm@in.tum.de>
parents:
64272
diff
changeset
|
1281 |
qed auto |
b149abe619f7
added shuffle product to HOL/List
eberlm <eberlm@in.tum.de>
parents:
64272
diff
changeset
|
1282 |
|
63466 | 1283 |
lemma Suc_times_binomial_add: "Suc a * (Suc (a + b) choose Suc a) = Suc b * (Suc (a + b) choose a)" |
1284 |
\<comment> \<open>by Lukas Bulwahn\<close> |
|
60604 | 1285 |
proof - |
1286 |
have dvd: "Suc a * (fact a * fact b) dvd fact (Suc (a + b))" for a b |
|
1287 |
using fact_fact_dvd_fact[of "Suc a" "b", where 'a=nat] |
|
1288 |
by (simp only: fact_Suc add_Suc[symmetric] of_nat_id mult.assoc) |
|
1289 |
have "Suc a * (fact (Suc (a + b)) div (Suc a * fact a * fact b)) = |
|
1290 |
Suc a * fact (Suc (a + b)) div (Suc a * (fact a * fact b))" |
|
1291 |
by (subst div_mult_swap[symmetric]; simp only: mult.assoc dvd) |
|
1292 |
also have "\<dots> = Suc b * fact (Suc (a + b)) div (Suc b * (fact a * fact b))" |
|
1293 |
by (simp only: div_mult_mult1) |
|
1294 |
also have "\<dots> = Suc b * (fact (Suc (a + b)) div (Suc b * (fact a * fact b)))" |
|
1295 |
using dvd[of b a] by (subst div_mult_swap[symmetric]; simp only: ac_simps dvd) |
|
1296 |
finally show ?thesis |
|
1297 |
by (subst (1 2) binomial_altdef_nat) |
|
63466 | 1298 |
(simp_all only: ac_simps diff_Suc_Suc Suc_diff_le diff_add_inverse fact_Suc of_nat_id) |
60604 | 1299 |
qed |
1300 |
||
63373 | 1301 |
|
68785 | 1302 |
subsection \<open>Executable code\<close> |
63373 | 1303 |
|
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
61799
diff
changeset
|
1304 |
lemma gbinomial_code [code]: |
68787
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
1305 |
"a gchoose k = |
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
1306 |
(if k = 0 then 1 |
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
1307 |
else fold_atLeastAtMost_nat (\<lambda>k acc. (a - of_nat k) * acc) 0 (k - 1) 1 / fact k)" |
b129052644e9
more uniform parameter naming convention for choose and gchoose
haftmann
parents:
68786
diff
changeset
|
1308 |
by (cases k) |
64272 | 1309 |
(simp_all add: gbinomial_prod_rev prod_atLeastAtMost_code [symmetric] |
63466 | 1310 |
atLeastLessThanSuc_atLeastAtMost) |
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
61799
diff
changeset
|
1311 |
|
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
61799
diff
changeset
|
1312 |
lemma binomial_code [code]: |
68785 | 1313 |
"n choose k = |
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
61799
diff
changeset
|
1314 |
(if k > n then 0 |
68785 | 1315 |
else if 2 * k > n then n choose (n - k) |
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68787
diff
changeset
|
1316 |
else (fold_atLeastAtMost_nat (*) (n - k + 1) n 1 div fact k))" |
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
61799
diff
changeset
|
1317 |
proof - |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
61799
diff
changeset
|
1318 |
{ |
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
61799
diff
changeset
|
1319 |
assume "k \<le> n" |
63466 | 1320 |
then have "{1..n} = {1..n-k} \<union> {n-k+1..n}" by auto |
1321 |
then have "(fact n :: nat) = fact (n-k) * \<Prod>{n-k+1..n}" |
|
65581
baf96277ee76
better code equation for binomial
eberlm <eberlm@in.tum.de>
parents:
65552
diff
changeset
|
1322 |
by (simp add: prod.union_disjoint fact_prod) |
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
61799
diff
changeset
|
1323 |
} |
64272 | 1324 |
then show ?thesis by (auto simp: binomial_altdef_nat mult_ac prod_atLeastAtMost_code) |
62378
85ed00c1fe7c
generalize more theorems to support enat and ennreal
hoelzl
parents:
62347
diff
changeset
|
1325 |
qed |
62128
3201ddb00097
Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents:
61799
diff
changeset
|
1326 |
|
15131 | 1327 |
end |