src/HOL/Binomial.thy
author haftmann
Sat, 02 Jul 2016 08:41:05 +0200
changeset 63363 bd483ddb17f2
parent 63306 ca187a9f66da
child 63366 209c4cbbc4cd
permissions -rw-r--r--
more correct comment
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
59669
de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
parents: 59667
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(*  Title       : Binomial.thy
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a3be6b3a9c0b new theories from Jacques Fleuriot
paulson
parents:
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    Author      : Jacques D. Fleuriot
a3be6b3a9c0b new theories from Jacques Fleuriot
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parents:
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    Copyright   : 1998  University of Cambridge
15094
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
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bd483ddb17f2 more correct comment
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    Various additions by Jeremy Avigad.
61554
84901b8aa4f5 added acknowledgement in Binomial.thy
eberlm
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    Additional binomial identities by Chaitanya Mangla and Manuel Eberl
12196
a3be6b3a9c0b new theories from Jacques Fleuriot
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*)
a3be6b3a9c0b new theories from Jacques Fleuriot
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d8d85a8172b5 isabelle update_cartouches;
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section\<open>Factorial Function, Binomial Coefficients and Binomial Theorem\<close>
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a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
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de7792ea4090 renaming HOL/Fact.thy -> Binomial.thy
paulson <lp15@cam.ac.uk>
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theory Binomial
33319
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imports Main
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c69542757a4d New theory header syntax.
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begin
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subsection \<open>Factorial\<close>
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paulson <lp15@cam.ac.uk>
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61533
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
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fun (in semiring_char_0) fact :: "nat \<Rightarrow> 'a"
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b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
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  where "fact 0 = 1" | "fact (Suc n) = of_nat (Suc n) * fact n"
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
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59730
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paulson <lp15@cam.ac.uk>
parents: 59669
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lemmas fact_Suc = fact.simps(2)
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paulson <lp15@cam.ac.uk>
parents: 59669
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b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
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lemma fact_1 [simp]: "fact 1 = 1"
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paulson <lp15@cam.ac.uk>
parents: 59669
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    23
  by simp
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
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b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
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lemma fact_Suc_0 [simp]: "fact (Suc 0) = Suc 0"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
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  by simp
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
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62347
2230b7047376 generalized some lemmas;
haftmann
parents: 62344
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lemma of_nat_fact [simp]:
2230b7047376 generalized some lemmas;
haftmann
parents: 62344
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  "of_nat (fact n) = fact n"
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b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
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  by (induct n) (auto simp add: algebra_simps of_nat_mult)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
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62347
2230b7047376 generalized some lemmas;
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lemma of_int_fact [simp]:
2230b7047376 generalized some lemmas;
haftmann
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  "of_int (fact n) = fact n"
2230b7047376 generalized some lemmas;
haftmann
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proof -
2230b7047376 generalized some lemmas;
haftmann
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    35
  have "of_int (of_nat (fact n)) = fact n"
2230b7047376 generalized some lemmas;
haftmann
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    36
    unfolding of_int_of_nat_eq by simp
2230b7047376 generalized some lemmas;
haftmann
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  then show ?thesis
2230b7047376 generalized some lemmas;
haftmann
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    by simp
2230b7047376 generalized some lemmas;
haftmann
parents: 62344
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qed
2230b7047376 generalized some lemmas;
haftmann
parents: 62344
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    40
59730
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paulson <lp15@cam.ac.uk>
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lemma fact_reduce: "n > 0 \<Longrightarrow> fact n = of_nat n * fact (n - 1)"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
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    42
  by (cases n) auto
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
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    43
59733
cd945dc13bec more general type class for factorial. Now allows code generation (?)
paulson <lp15@cam.ac.uk>
parents: 59730
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lemma fact_nonzero [simp]: "fact n \<noteq> (0::'a::{semiring_char_0,semiring_no_zero_divisors})"
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b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
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    45
  apply (induct n)
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    46
  apply auto
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    47
  using of_nat_eq_0_iff by fastforce
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
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    48
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
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    49
lemma fact_mono_nat: "m \<le> n \<Longrightarrow> fact m \<le> (fact n :: nat)"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    50
  by (induct n) (auto simp: le_Suc_eq)
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
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    51
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
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lemma fact_in_Nats: "fact n \<in> \<nat>" by (induction n) auto
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
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    53
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
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    54
lemma fact_in_Ints: "fact n \<in> \<int>" by (induction n) auto
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
    55
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
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context
60241
wenzelm
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    57
  assumes "SORT_CONSTRAINT('a::linordered_semidom)"
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
    58
begin
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
    59
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    60
  lemma fact_mono: "m \<le> n \<Longrightarrow> fact m \<le> (fact n :: 'a)"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    61
    by (metis of_nat_fact of_nat_le_iff fact_mono_nat)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
    62
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    63
  lemma fact_ge_1 [simp]: "fact n \<ge> (1 :: 'a)"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    64
    by (metis le0 fact.simps(1) fact_mono)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
    65
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    66
  lemma fact_gt_zero [simp]: "fact n > (0 :: 'a)"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    67
    using fact_ge_1 less_le_trans zero_less_one by blast
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
    68
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    69
  lemma fact_ge_zero [simp]: "fact n \<ge> (0 :: 'a)"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    70
    by (simp add: less_imp_le)
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
    71
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    72
  lemma fact_not_neg [simp]: "~ (fact n < (0 :: 'a))"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    73
    by (simp add: not_less_iff_gr_or_eq)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
    74
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    75
  lemma fact_le_power:
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    76
      "fact n \<le> (of_nat (n^n) ::'a)"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    77
  proof (induct n)
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    78
    case (Suc n)
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    79
    then have *: "fact n \<le> (of_nat (Suc n ^ n) ::'a)"
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61554
diff changeset
    80
      by (rule order_trans) (simp add: power_mono del: of_nat_power)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    81
    have "fact (Suc n) = (of_nat (Suc n) * fact n ::'a)"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    82
      by (simp add: algebra_simps)
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    83
    also have "... \<le> (of_nat (Suc n) * of_nat (Suc n ^ n) ::'a)"
61649
268d88ec9087 Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents: 61554
diff changeset
    84
      by (simp add: "*" ordered_comm_semiring_class.comm_mult_left_mono del: of_nat_power)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    85
    also have "... \<le> (of_nat (Suc n ^ Suc n) ::'a)"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    86
      by (metis of_nat_mult order_refl power_Suc)
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    87
    finally show ?case .
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    88
  qed simp
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
    89
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
    90
end
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
    91
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
    92
text\<open>Note that @{term "fact 0 = fact 1"}\<close>
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    93
lemma fact_less_mono_nat: "\<lbrakk>0 < m; m < n\<rbrakk> \<Longrightarrow> fact m < (fact n :: nat)"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    94
  by (induct n) (auto simp: less_Suc_eq)
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
    95
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
    96
lemma fact_less_mono:
60241
wenzelm
parents: 59867
diff changeset
    97
  "\<lbrakk>0 < m; m < n\<rbrakk> \<Longrightarrow> fact m < (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
    98
  by (metis of_nat_fact of_nat_less_iff fact_less_mono_nat)
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
    99
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   100
lemma fact_ge_Suc_0_nat [simp]: "fact n \<ge> Suc 0"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   101
  by (metis One_nat_def fact_ge_1)
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   102
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   103
lemma dvd_fact:
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   104
  shows "1 \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> m dvd fact n"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   105
  by (induct n) (auto simp: dvdI le_Suc_eq)
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   106
62128
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61799
diff changeset
   107
lemma fact_ge_self: "fact n \<ge> n"
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61799
diff changeset
   108
  by (cases "n = 0") (simp_all add: dvd_imp_le dvd_fact)
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61799
diff changeset
   109
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   110
lemma fact_altdef_nat: "fact n = \<Prod>{1..n}"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   111
  by (induct n) (auto simp: atLeastAtMostSuc_conv)
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   112
62128
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61799
diff changeset
   113
lemma fact_altdef: "fact n = (\<Prod>i=1..n. of_nat i)"
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   114
  by (induct n) (auto simp: atLeastAtMostSuc_conv)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   115
62128
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61799
diff changeset
   116
lemma fact_altdef': "fact n = of_nat (\<Prod>{1..n})"
3201ddb00097 Integrated some material from Algebraic_Numbers AFP entry to Polynomials; generalised some polynomial stuff.
eberlm
parents: 61799
diff changeset
   117
  by (subst fact_altdef_nat [symmetric]) simp
15094
a7d1a3fdc30d conversion of Hyperreal/{Fact,Filter} to Isar scripts
paulson
parents: 12196
diff changeset
   118
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   119
lemma fact_dvd: "n \<le> m \<Longrightarrow> fact n dvd (fact m :: 'a :: {semiring_div,linordered_semidom})"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   120
  by (induct m) (auto simp: le_Suc_eq)
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   121
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   122
lemma fact_mod: "m \<le> n \<Longrightarrow> fact n mod (fact m :: 'a :: {semiring_div,linordered_semidom}) = 0"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   123
  by (auto simp add: fact_dvd)
40033
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   124
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   125
lemma fact_div_fact:
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   126
  assumes "m \<ge> n"
40033
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   127
  shows "(fact m) div (fact n) = \<Prod>{n + 1..m}"
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   128
proof -
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   129
  obtain d where "d = m - n" by auto
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   130
  from assms this have "m = n + d" by auto
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   131
  have "fact (n + d) div (fact n) = \<Prod>{n + 1..n + d}"
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   132
  proof (induct d)
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   133
    case 0
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   134
    show ?case by simp
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   135
  next
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   136
    case (Suc d')
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   137
    have "fact (n + Suc d') div fact n = Suc (n + d') * fact (n + d') div fact n"
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   138
      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
   139
    also from Suc.hyps have "... = Suc (n + d') * \<Prod>{n + 1..n + d'}"
40033
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   140
      unfolding div_mult1_eq[of _ "fact (n + d')"] by (simp add: fact_mod)
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   141
    also have "... = \<Prod>{n + 1..n + Suc d'}"
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   142
      by (simp add: atLeastAtMostSuc_conv)
40033
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   143
    finally show ?case .
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   144
  qed
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
   145
  from this \<open>m = n + d\<close> show ?thesis by simp
40033
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   146
qed
84200d970bf0 added some facts about factorial and dvd, div and mod
bulwahn
parents: 35644
diff changeset
   147
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   148
lemma fact_num_eq_if:
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   149
    "fact m = (if m=0 then 1 else of_nat m * fact (m - 1))"
32036
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   150
by (cases m) auto
8a9228872fbd Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
avigad
parents: 30242
diff changeset
   151
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   152
lemma fact_eq_rev_setprod_nat: "fact k = (\<Prod>i<k. k - i)"
50224
aacd6da09825 add binomial_ge_n_over_k_pow_k
hoelzl
parents: 46240
diff changeset
   153
  unfolding fact_altdef_nat
57129
7edb7550663e introduce more powerful reindexing rules for big operators
hoelzl
parents: 57113
diff changeset
   154
  by (rule setprod.reindex_bij_witness[where i="\<lambda>i. k - i" and j="\<lambda>i. k - i"]) auto
50224
aacd6da09825 add binomial_ge_n_over_k_pow_k
hoelzl
parents: 46240
diff changeset
   155
50240
019d642d422d add upper bounds for factorial and binomial; add equation for binomial using nat-division (both from AFP/Girth_Chromatic)
hoelzl
parents: 50224
diff changeset
   156
lemma fact_div_fact_le_pow:
019d642d422d add upper bounds for factorial and binomial; add equation for binomial using nat-division (both from AFP/Girth_Chromatic)
hoelzl
parents: 50224
diff changeset
   157
  assumes "r \<le> n" shows "fact n div fact (n - r) \<le> n ^ r"
019d642d422d add upper bounds for factorial and binomial; add equation for binomial using nat-division (both from AFP/Girth_Chromatic)
hoelzl
parents: 50224
diff changeset
   158
proof -
019d642d422d add upper bounds for factorial and binomial; add equation for binomial using nat-division (both from AFP/Girth_Chromatic)
hoelzl
parents: 50224
diff changeset
   159
  have "\<And>r. r \<le> n \<Longrightarrow> \<Prod>{n - r..n} = (n - r) * \<Prod>{Suc (n - r)..n}"
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57129
diff changeset
   160
    by (subst setprod.insert[symmetric]) (auto simp: atLeastAtMost_insertL)
50240
019d642d422d add upper bounds for factorial and binomial; add equation for binomial using nat-division (both from AFP/Girth_Chromatic)
hoelzl
parents: 50224
diff changeset
   161
  with assms show ?thesis
019d642d422d add upper bounds for factorial and binomial; add equation for binomial using nat-division (both from AFP/Girth_Chromatic)
hoelzl
parents: 50224
diff changeset
   162
    by (induct r rule: nat.induct) (auto simp add: fact_div_fact Suc_diff_Suc mult_le_mono)
019d642d422d add upper bounds for factorial and binomial; add equation for binomial using nat-division (both from AFP/Girth_Chromatic)
hoelzl
parents: 50224
diff changeset
   163
qed
019d642d422d add upper bounds for factorial and binomial; add equation for binomial using nat-division (both from AFP/Girth_Chromatic)
hoelzl
parents: 50224
diff changeset
   164
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61738
diff changeset
   165
lemma fact_numeral:  \<comment>\<open>Evaluation for specific numerals\<close>
57113
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 50240
diff changeset
   166
  "fact (numeral k) = (numeral k) * (fact (pred_numeral k))"
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   167
  by (metis fact.simps(2) numeral_eq_Suc of_nat_numeral)
57113
7e95523302e6 New theorems to enable the simplification of certain functions when applied to specific natural number constants (such as 4)
paulson <lp15@cam.ac.uk>
parents: 50240
diff changeset
   168
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
   169
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
   170
text \<open>This development is based on the work of Andy Gordon and
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
   171
  Florian Kammueller.\<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
   172
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
   173
subsection \<open>Basic definitions and lemmas\<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
   174
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
   175
primrec binomial :: "nat \<Rightarrow> nat \<Rightarrow> nat" (infixl "choose" 65)
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
   176
where
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
  "0 choose k = (if k = 0 then 1 else 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
   178
| "Suc n choose k = (if k = 0 then 1 else (n choose (k - 1)) + (n 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
   179
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
lemma binomial_n_0 [simp]: "(n choose 0) = 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
   181
  by (cases 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
   182
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
   183
lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 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
   184
  by simp
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
   185
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
   186
lemma binomial_Suc_Suc [simp]: "(Suc n choose Suc k) = (n choose k) + (n choose Suc 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
   187
  by simp
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
   188
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
   189
lemma choose_reduce_nat:
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
   190
  "0 < (n::nat) \<Longrightarrow> 0 < k \<Longrightarrow>
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
   191
    (n choose k) = ((n - 1) choose (k - 1)) + ((n - 1) 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
   192
  by (metis Suc_diff_1 binomial.simps(2) neq0_conv)
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
   193
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
   194
lemma binomial_eq_0: "n < k \<Longrightarrow> n choose k = 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
   195
  by (induct n arbitrary: k) 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
   196
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
   197
declare binomial.simps [simp del]
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
   198
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
   199
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
   200
  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
   201
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
   202
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
   203
  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
   204
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
   205
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
   206
  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
   207
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
   208
lemma zero_less_binomial: "k \<le> n \<Longrightarrow> n choose k > 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
   209
  by (induct n k rule: diff_induct) 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
   210
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
lemma binomial_eq_0_iff [simp]: "n choose k = 0 \<longleftrightarrow> n < 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
   212
  by (metis binomial_eq_0 less_numeral_extra(3) not_less zero_less_binomial)
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 zero_less_binomial_iff [simp]: "n choose k > 0 \<longleftrightarrow> k \<le> 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
  by (metis binomial_eq_0_iff not_less0 not_less zero_less_binomial)
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
   216
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
   217
lemma Suc_times_binomial_eq:
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
   218
  "Suc n * (n choose k) = (Suc n choose Suc k) * Suc 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
   219
  apply (induct n arbitrary: k, simp add: binomial.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
   220
  apply (case_tac 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
   221
   apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq 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
   222
  done
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
   223
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
   224
lemma binomial_le_pow2: "n choose k \<le> 2^n"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
   225
  apply (induction n arbitrary: k)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
   226
  apply (simp add: binomial.simps)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
   227
  apply (case_tac k)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
   228
  apply (auto simp: power_Suc)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
   229
  by (simp add: add_le_mono mult_2)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
   230
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
   231
text\<open>The absorption property\<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
   232
lemma Suc_times_binomial:
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
   233
  "Suc k * (Suc n choose Suc k) = Suc n * (n 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
   234
  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
   235
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
   236
text\<open>This is the well-known version of absorption, but it's harder to use because of the
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
   237
  need to reason about division.\<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
   238
lemma binomial_Suc_Suc_eq_times:
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
   239
    "(Suc n choose Suc k) = (Suc n * (n choose k)) div Suc 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
   240
  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
   241
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
   242
text\<open>Another version of absorption, with -1 instead of Suc.\<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
   243
lemma times_binomial_minus1_eq:
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
  "0 < k \<Longrightarrow> k * (n choose k) = n * ((n - 1) choose (k - 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
   245
  using Suc_times_binomial_eq [where n = "n - 1" and k = "k - 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
   246
  by (auto split add: nat_diff_split)
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
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
   248
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61738
diff changeset
   249
subsection \<open>Combinatorial theorems involving \<open>choose\<close>\<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
   250
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
   251
text \<open>By Florian Kamm\"uller, tidied by LCP.\<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
   252
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
   253
lemma card_s_0_eq_empty: "finite A \<Longrightarrow> card {B. B \<subseteq> A & card B = 0} = 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
   254
  by (simp cong add: conj_cong add: finite_subset [THEN card_0_eq])
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
   255
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
   256
lemma choose_deconstruct: "finite M \<Longrightarrow> x \<notin> M \<Longrightarrow>
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
   257
    {s. s \<subseteq> insert x M \<and> card s = Suc 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
   258
    {s. s \<subseteq> M \<and> card s = Suc k} \<union> {s. \<exists>t. t \<subseteq> M \<and> card t = k \<and> s = insert x t}"
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
  apply safe
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
     apply (auto intro: finite_subset [THEN card_insert_disjoint])
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
   261
  by (metis (full_types) Diff_insert_absorb Set.set_insert Zero_neq_Suc card_Diff_singleton_if
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
     card_eq_0_iff diff_Suc_1 in_mono subset_insert_iff)
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
   263
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
lemma finite_bex_subset [simp]:
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
   265
  assumes "finite B"
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
   266
    and "\<And>A. A \<subseteq> B \<Longrightarrow> finite {x. P x A}"
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
   267
  shows "finite {x. \<exists>A \<subseteq> B. P x A}"
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
   268
  by (metis (no_types) assms finite_Collect_bounded_ex finite_Collect_subsets)
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
   269
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
   270
text\<open>There are as many subsets of @{term A} having cardinality @{term 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
   271
 as there are sets obtained from the former by inserting a fixed element
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
   272
 @{term x} into each.\<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
   273
lemma constr_bij:
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
   274
   "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow>
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
   275
    card {B. \<exists>C. C \<subseteq> A \<and> card C = k \<and> B = insert x C} =
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
   276
    card {B. B \<subseteq> A & card(B) = 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
   277
  apply (rule card_bij_eq [where f = "\<lambda>s. s - {x}" and g = "insert x"])
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
   278
  apply (auto elim!: equalityE simp add: inj_on_def)
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
   279
  apply (metis card_Diff_singleton_if finite_subset in_mono)
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
   280
  done
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
   281
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
   282
text \<open>
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
   283
  Main theorem: combinatorial statement about number of subsets of a set.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
   284
\<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
   285
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
   286
theorem n_subsets: "finite A \<Longrightarrow> card {B. B \<subseteq> A \<and> card B = k} = (card A 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
   287
proof (induct k arbitrary: A)
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
   288
  case 0 then show ?case by (simp add: card_s_0_eq_empty)
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
   289
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
   290
  case (Suc k)
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
   291
  show ?case using \<open>finite A\<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
   292
  proof (induct A)
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
   293
    case empty show ?case by (simp add: card_s_0_eq_empty)
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
   294
  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
   295
    case (insert x A)
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
   296
    then show ?case using Suc.hyps
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
   297
      apply (simp add: card_s_0_eq_empty choose_deconstruct)
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
   298
      apply (subst card_Un_disjoint)
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
   299
         prefer 4 apply (force simp add: constr_bij)
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
   300
        prefer 3 apply force
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
   301
       prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2]
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
   302
         finite_subset [of _ "Pow (insert x F)" for F])
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
   303
      apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset])
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
   304
      done
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
   305
  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
   306
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
   307
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
   308
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
   309
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
   310
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
   311
text\<open>Avigad's version, generalized to any commutative ring\<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
   312
theorem binomial_ring: "(a+b::'a::{comm_ring_1,power})^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
   313
  (\<Sum>k=0..n. (of_nat (n choose k)) * a^k * b^(n-k))" (is "?P 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
   314
proof (induct 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
   315
  case 0 then show "?P 0" by simp
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
   316
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
   317
  case (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
   318
  have decomp: "{0..n+1} = {0} Un {n+1} Un {1..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
   319
    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
   320
  have decomp2: "{0..n} = {0} Un {1..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
   321
    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
   322
  have "(a+b)^(n+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
   323
      (a+b) * (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-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
   324
    using Suc.hyps by simp
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
   325
  also have "\<dots> = a*(\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-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
   326
                   b*(\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-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
   327
    by (rule distrib_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
   328
  also have "\<dots> = (\<Sum>k=0..n. of_nat (n choose k) * a^(k+1) * b^(n-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
   329
                  (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n-k+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
   330
    by (auto simp add: setsum_right_distrib ac_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
   331
  also have "\<dots> = (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n+1-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
   332
                  (\<Sum>k=1..n+1. of_nat (n choose (k - 1)) * a^k * b^(n+1-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
   333
    by (simp add:setsum_shift_bounds_cl_Suc_ivl Suc_diff_le field_simps
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
   334
        del:setsum_cl_ivl_Suc)
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
   335
  also have "\<dots> = a^(n+1) + b^(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
   336
                  (\<Sum>k=1..n. of_nat (n choose (k - 1)) * a^k * b^(n+1-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
   337
                  (\<Sum>k=1..n. of_nat (n choose k) * a^k * b^(n+1-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
   338
    by (simp add: decomp2)
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
   339
  also have
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
   340
      "\<dots> = a^(n+1) + b^(n+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
   341
            (\<Sum>k=1..n. of_nat(n+1 choose k) * a^k * b^(n+1-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
   342
    by (auto simp add: field_simps setsum.distrib [symmetric] choose_reduce_nat)
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
   343
  also have "\<dots> = (\<Sum>k=0..n+1. of_nat (n+1 choose k) * a^k * b^(n+1-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
   344
    using decomp 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
   345
  finally show "?P (Suc n)" by simp
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
   346
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
   347
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
   348
text\<open>Original version for the naturals\<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
   349
corollary binomial: "(a+b::nat)^n = (\<Sum>k=0..n. (of_nat (n choose k)) * a^k * b^(n-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
   350
    using binomial_ring [of "int a" "int b" 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
   351
  by (simp only: of_nat_add [symmetric] of_nat_mult [symmetric] of_nat_power [symmetric]
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
   352
           of_nat_setsum [symmetric]
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
   353
           of_nat_eq_iff of_nat_id)
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
   354
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
   355
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
   356
proof (induct n arbitrary: k rule: nat_less_induct)
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
   357
  fix n k assume H: "\<forall>m<n. \<forall>x\<le>m. fact x * fact (m - x) * (m choose x) =
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
   358
                      fact m" and kn: "k \<le> 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
   359
  let ?ths = "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
   360
  { assume "n=0" then have ?ths using kn by simp }
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
   361
  moreover
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
   362
  { assume "k=0" then have ?ths using kn by simp }
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
   363
  moreover
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
   364
  { assume nk: "n=k" then have ?ths by simp }
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
   365
  moreover
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
   366
  { fix m h assume n: "n = Suc m" and h: "k = Suc h" and hm: "h < 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
   367
    from n have mn: "m < n" by arith
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
   368
    from hm have hm': "h \<le> m" by arith
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
   369
    from hm h n kn have km: "k \<le> m" by arith
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
   370
    have "m - h = Suc (m - Suc h)" using  h km hm by arith
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
   371
    with km h have th0: "fact (m - h) = (m - h) * fact (m - 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
   372
      by simp
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
   373
    from n h th0
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
   374
    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
   375
        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
   376
        (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
   377
      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
   378
    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
   379
      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
   380
      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
   381
    finally have ?ths using h n km by simp }
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
   382
  moreover have "n=0 \<or> k = 0 \<or> k = n \<or> (\<exists>m h. n = Suc m \<and> k = Suc h \<and> h < 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
   383
    using kn by presburger
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
   384
  ultimately show ?ths by blast
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
   385
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
   386
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
   387
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
   388
  assumes kn: "k \<le> n"
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   389
  shows "(of_nat (n choose k) :: 'a::field_char_0) =
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   390
         (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
   391
  using binomial_fact_lemma[OF kn]
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   392
  apply (simp add: field_simps)
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   393
  by (metis mult.commute 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
   394
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
   395
lemma choose_row_sum: "(\<Sum>k=0..n. n choose k) = 2^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
   396
  using binomial [of 1 "1" 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
   397
  by (simp add: numeral_2_eq_2)
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
   398
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
   399
lemma sum_choose_lower: "(\<Sum>k=0..n. (r+k) choose k) = Suc (r+n) choose 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
   400
  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
   401
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
   402
lemma sum_choose_upper: "(\<Sum>k=0..n. k choose m) = Suc n choose Suc m"
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
   403
  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
   404
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   405
lemma choose_alternating_sum:
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   406
  "n > 0 \<Longrightarrow> (\<Sum>i\<le>n. (-1)^i * of_nat (n choose i)) = (0 :: 'a :: comm_ring_1)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   407
  using binomial_ring[of "-1 :: 'a" 1 n] by (simp add: atLeast0AtMost mult_of_nat_commute zero_power)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   408
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   409
lemma choose_even_sum:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   410
  assumes "n > 0"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   411
  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
   412
proof -
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   413
  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
   414
    using choose_row_sum[of n]
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   415
    by (simp add: choose_alternating_sum assms atLeast0AtMost of_nat_setsum[symmetric] of_nat_power)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   416
  also have "\<dots> = (\<Sum>i\<le>n. of_nat (n choose i) + (-1) ^ i * of_nat (n choose i))"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   417
    by (simp add: setsum.distrib)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   418
  also have "\<dots> = 2 * (\<Sum>i\<le>n. if even i then of_nat (n choose i) else 0)"
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   419
    by (subst setsum_right_distrib, intro setsum.cong) simp_all
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   420
  finally show ?thesis ..
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   421
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   422
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   423
lemma choose_odd_sum:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   424
  assumes "n > 0"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   425
  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
   426
proof -
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   427
  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
   428
    using choose_row_sum[of n]
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   429
    by (simp add: choose_alternating_sum assms atLeast0AtMost of_nat_setsum[symmetric] of_nat_power)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   430
  also have "\<dots> = (\<Sum>i\<le>n. of_nat (n choose i) - (-1) ^ i * of_nat (n choose i))"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   431
    by (simp add: setsum_subtractf)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   432
  also have "\<dots> = 2 * (\<Sum>i\<le>n. if odd i then of_nat (n choose i) else 0)"
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   433
    by (subst setsum_right_distrib, intro setsum.cong) simp_all
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   434
  finally show ?thesis ..
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   435
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   436
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   437
lemma choose_row_sum': "(\<Sum>k\<le>n. (n choose k)) = 2 ^ n"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   438
  using choose_row_sum[of n] by (simp add: atLeast0AtMost)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   439
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
   440
lemma natsum_reverse_index:
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
   441
  fixes m::nat
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
  shows "(\<And>k. m \<le> k \<Longrightarrow> k \<le> n \<Longrightarrow> g k = f (m + n - k)) \<Longrightarrow> (\<Sum>k=m..n. f k) = (\<Sum>k=m..n. g 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
   443
  by (rule setsum.reindex_bij_witness[where i="\<lambda>k. m+n-k" and j="\<lambda>k. m+n-k"]) 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
   444
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
   445
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
   446
lemma sum_choose_diagonal:
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
   447
  assumes "m\<le>n" shows "(\<Sum>k=0..m. (n-k) choose (m-k)) = Suc n choose m"
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
   448
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
   449
  have "(\<Sum>k=0..m. (n-k) choose (m-k)) = (\<Sum>k=0..m. (n-m+k) choose 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
   450
    by (rule natsum_reverse_index) (simp add: assms)
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
   451
  also have "... = Suc (n-m+m) choose m"
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
   452
    by (rule sum_choose_lower)
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
   453
  also have "... = Suc n choose m" using assms
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
   454
    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
   455
  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
   456
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
   457
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
   458
subsection\<open>Pochhammer's symbol : generalized rising factorial\<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
   459
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
   460
text \<open>See @{url "http://en.wikipedia.org/wiki/Pochhammer_symbol"}\<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
   461
61533
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   462
definition (in comm_semiring_1) "pochhammer (a :: 'a) 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
   463
  (if n = 0 then 1 else setprod (\<lambda>n. a + of_nat n) {0 .. n - 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
   464
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
   465
lemma pochhammer_0 [simp]: "pochhammer a 0 = 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
   466
  by (simp add: pochhammer_def)
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
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
lemma pochhammer_1 [simp]: "pochhammer a 1 = a"
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
  by (simp add: pochhammer_def)
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
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
   471
lemma pochhammer_Suc0 [simp]: "pochhammer a (Suc 0) = a"
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
   472
  by (simp add: pochhammer_def)
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
   473
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
   474
lemma pochhammer_Suc_setprod: "pochhammer a (Suc n) = setprod (\<lambda>n. a + of_nat n) {0 .. 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
   475
  by (simp add: pochhammer_def)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   476
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   477
lemma pochhammer_of_nat: "pochhammer (of_nat x) n = of_nat (pochhammer x n)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   478
  by (simp add: pochhammer_def of_nat_setprod)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   479
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   480
lemma pochhammer_of_int: "pochhammer (of_int x) n = of_int (pochhammer x n)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   481
  by (simp add: pochhammer_def of_int_setprod)
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
   482
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
   483
lemma setprod_nat_ivl_Suc: "setprod f {0 .. Suc n} = setprod f {0..n} * f (Suc 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
   484
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
   485
  have "{0..Suc n} = {0..n} \<union> {Suc n}" by 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
   486
  then show ?thesis by (simp add: field_simps)
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
   487
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
   488
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
   489
lemma setprod_nat_ivl_1_Suc: "setprod f {0 .. Suc n} = f 0 * setprod f {1.. Suc 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
   490
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
   491
  have "{0..Suc n} = {0} \<union> {1 .. Suc n}" by 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
   492
  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
   493
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
   494
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
   495
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
   496
lemma pochhammer_Suc: "pochhammer a (Suc n) = pochhammer a n * (a + of_nat 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
   497
proof (cases 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
   498
  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
   499
  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
   500
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
   501
  case (Suc 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
   502
  show ?thesis unfolding Suc pochhammer_Suc_setprod setprod_nat_ivl_Suc ..
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
   503
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
   504
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
   505
lemma pochhammer_rec: "pochhammer a (Suc n) = a * pochhammer (a + 1) 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
   506
proof (cases "n = 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
   507
  case True
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
   508
  then show ?thesis by (simp add: pochhammer_Suc_setprod)
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
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
   510
  case False
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
   511
  have *: "finite {1 .. n}" "0 \<notin> {1 .. n}" by 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
   512
  have eq: "insert 0 {1 .. n} = {0..n}" by auto
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60758
diff changeset
   513
  have **: "(\<Prod>n\<in>{1::nat..n}. a + of_nat n) = (\<Prod>n\<in>{0::nat..n - 1}. a + 1 + of_nat 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
   514
    apply (rule setprod.reindex_cong [where l = Suc])
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
   515
    using False
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
   516
    apply (auto simp add: fun_eq_iff field_simps)
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
   517
    done
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
   518
  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
   519
    apply (simp add: pochhammer_def)
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
    unfolding setprod.insert [OF *, unfolded 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
   521
    using ** apply (simp add: field_simps)
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
    done
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
   523
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
   524
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   525
lemma pochhammer_rec': "pochhammer z (Suc n) = (z + of_nat n) * pochhammer z n"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   526
proof (induction n arbitrary: z)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   527
  case (Suc n z)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   528
  have "pochhammer z (Suc (Suc n)) = z * pochhammer (z + 1) (Suc n)"
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   529
    by (simp add: pochhammer_rec)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   530
  also note Suc
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   531
  also have "z * ((z + 1 + of_nat n) * pochhammer (z + 1) n) =
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   532
               (z + of_nat (Suc n)) * pochhammer z (Suc n)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   533
    by (simp_all add: pochhammer_rec algebra_simps)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   534
  finally show ?case .
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   535
qed simp_all
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   536
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   537
lemma pochhammer_fact: "fact n = pochhammer 1 n"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   538
  unfolding fact_altdef
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
   539
  apply (cases 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
   540
   apply (simp_all add: of_nat_setprod pochhammer_Suc_setprod)
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
   541
  apply (rule setprod.reindex_cong [where l = Suc])
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
   542
    apply (auto simp add: fun_eq_iff)
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
   543
  done
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
   544
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
   545
lemma pochhammer_of_nat_eq_0_lemma:
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
   546
  assumes "k > 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
   547
  shows "pochhammer (- (of_nat n :: 'a:: idom)) k = 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
   548
proof (cases "n = 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
   549
  case True
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
   550
  then 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
   551
    using assms by (cases k) (simp_all add: pochhammer_rec)
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
   552
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
   553
  case False
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
   554
  from assms obtain h where "k = Suc h" by (cases k) 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
   555
  then 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
   556
    by (simp add: pochhammer_Suc_setprod)
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
   557
       (metis Suc_leI Suc_le_mono assms atLeastAtMost_iff less_eq_nat.simps(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
   558
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
   559
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
   560
lemma pochhammer_of_nat_eq_0_lemma':
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
   561
  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
   562
  shows "pochhammer (- (of_nat n :: 'a:: {idom,ring_char_0})) k \<noteq> 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
   563
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
   564
  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
   565
  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
   566
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
   567
  case (Suc h)
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
   568
  then 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
   569
    apply (simp add: pochhammer_Suc_setprod)
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
   570
    using Suc kn apply (auto simp add: algebra_simps)
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
   571
    done
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
   572
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
   573
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
   574
lemma pochhammer_of_nat_eq_0_iff:
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
   575
  shows "pochhammer (- (of_nat n :: 'a:: {idom,ring_char_0})) k = 0 \<longleftrightarrow> k > 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
   576
  (is "?l = ?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
   577
  using pochhammer_of_nat_eq_0_lemma[of n k, where ?'a='a]
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
   578
    pochhammer_of_nat_eq_0_lemma'[of k n, where ?'a = 'a]
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
   579
  by (auto simp add: not_le[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
   580
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
   581
lemma pochhammer_eq_0_iff: "pochhammer a n = (0::'a::field_char_0) \<longleftrightarrow> (\<exists>k < n. a = - of_nat 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
   582
  apply (auto simp add: pochhammer_of_nat_eq_0_iff)
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
   583
  apply (cases 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
   584
   apply (auto simp add: pochhammer_def algebra_simps group_add_class.eq_neg_iff_add_eq_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
   585
  apply (metis leD not_less_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
   586
  done
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
   587
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
   588
lemma pochhammer_eq_0_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
   589
  "pochhammer a n = (0::'a::field_char_0) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a m = 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
   590
  unfolding pochhammer_eq_0_iff by 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
   591
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
   592
lemma pochhammer_neq_0_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
   593
  "pochhammer a m \<noteq> (0::'a::field_char_0) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a n \<noteq> 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
   594
  unfolding pochhammer_eq_0_iff by 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
   595
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
   596
lemma pochhammer_minus:
59862
44b3f4fa33ca New material and binomial fix
paulson <lp15@cam.ac.uk>
parents: 59733
diff changeset
   597
    "pochhammer (- b) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (b - of_nat k + 1) 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
   598
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
   599
  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
   600
  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
   601
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
   602
  case (Suc h)
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
   603
  have eq: "((- 1) ^ Suc h :: 'a) = (\<Prod>i=0..h. - 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
   604
    using setprod_constant[where A="{0 .. h}" and y="- 1 :: 'a"]
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
   605
    by 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
   606
  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
   607
    unfolding Suc pochhammer_Suc_setprod eq setprod.distrib[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
   608
    by (rule setprod.reindex_bij_witness[where i="op - h" and j="op - h"])
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
   609
       (auto simp: of_nat_diff)
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
   610
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
   611
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
   612
lemma pochhammer_minus':
59862
44b3f4fa33ca New material and binomial fix
paulson <lp15@cam.ac.uk>
parents: 59733
diff changeset
   613
    "pochhammer (b - of_nat k + 1) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (- b) k"
44b3f4fa33ca New material and binomial fix
paulson <lp15@cam.ac.uk>
parents: 59733
diff changeset
   614
  unfolding pochhammer_minus[where b=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
   615
  unfolding mult.assoc[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
   616
  unfolding power_add[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
   617
  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
   618
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
   619
lemma pochhammer_same: "pochhammer (- of_nat n) n =
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   620
    ((- 1) ^ n :: 'a::{semiring_char_0,comm_ring_1,semiring_no_zero_divisors}) * (fact n)"
59862
44b3f4fa33ca New material and binomial fix
paulson <lp15@cam.ac.uk>
parents: 59733
diff changeset
   621
  unfolding pochhammer_minus
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
   622
  by (simp add: of_nat_diff pochhammer_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
   623
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   624
lemma pochhammer_product':
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   625
  "pochhammer z (n + m) = pochhammer z n * pochhammer (z + of_nat n) m"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   626
proof (induction n arbitrary: z)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   627
  case (Suc n z)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   628
  have "pochhammer z (Suc n) * pochhammer (z + of_nat (Suc n)) m =
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   629
            z * (pochhammer (z + 1) n * pochhammer (z + 1 + of_nat n) m)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   630
    by (simp add: pochhammer_rec ac_simps)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   631
  also note Suc[symmetric]
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   632
  also have "z * pochhammer (z + 1) (n + m) = pochhammer z (Suc (n + m))"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   633
    by (subst pochhammer_rec) simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   634
  finally show ?case by simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   635
qed simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   636
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   637
lemma pochhammer_product:
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   638
  "m \<le> n \<Longrightarrow> pochhammer z n = pochhammer z m * pochhammer (z + of_nat m) (n - m)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   639
  using pochhammer_product'[of z m "n - m"] by simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   640
61533
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   641
lemma pochhammer_times_pochhammer_half:
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   642
  fixes z :: "'a :: field_char_0"
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   643
  shows "pochhammer z (Suc n) * pochhammer (z + 1/2) (Suc n) = (\<Prod>k=0..2*n+1. z + of_nat k / 2)"
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   644
proof (induction n)
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   645
  case (Suc n)
63044
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62481
diff changeset
   646
  define n' where "n' = Suc n"
61533
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   647
  have "pochhammer z (Suc n') * pochhammer (z + 1 / 2) (Suc n') =
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   648
          (pochhammer z n' * pochhammer (z + 1 / 2) n') *
61533
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   649
          ((z + of_nat n') * (z + 1/2 + of_nat n'))" (is "_ = _ * ?A")
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   650
     by (simp_all add: pochhammer_rec' mult_ac)
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   651
  also have "?A = (z + of_nat (Suc (2 * n + 1)) / 2) * (z + of_nat (Suc (Suc (2 * n + 1))) / 2)"
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   652
    (is "_ = ?A") by (simp add: field_simps n'_def of_nat_mult)
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   653
  also note Suc[folded n'_def]
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   654
  also have "(\<Prod>k = 0..2 * n + 1. z + of_nat k / 2) * ?A = (\<Prod>k = 0..2 * Suc n + 1. z + of_nat k / 2)"
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   655
    by (simp add: setprod_nat_ivl_Suc)
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   656
  finally show ?case by (simp add: n'_def)
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   657
qed (simp add: setprod_nat_ivl_Suc)
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   658
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   659
lemma pochhammer_double:
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   660
  fixes z :: "'a :: field_char_0"
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   661
  shows "pochhammer (2 * z) (2 * n) = of_nat (2^(2*n)) * pochhammer z n * pochhammer (z+1/2) n"
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   662
proof (induction n)
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   663
  case (Suc n)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   664
  have "pochhammer (2 * z) (2 * (Suc n)) = pochhammer (2 * z) (2 * n) *
61533
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   665
          (2 * (z + of_nat n)) * (2 * (z + of_nat n) + 1)"
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   666
    by (simp add: pochhammer_rec' ac_simps of_nat_mult)
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   667
  also note Suc
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   668
  also have "of_nat (2 ^ (2 * n)) * pochhammer z n * pochhammer (z + 1/2) n *
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   669
                 (2 * (z + of_nat n)) * (2 * (z + of_nat n) + 1) =
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   670
             of_nat (2 ^ (2 * (Suc n))) * pochhammer z (Suc n) * pochhammer (z + 1/2) (Suc n)"
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   671
    by (simp add: of_nat_mult field_simps pochhammer_rec')
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   672
  finally show ?case .
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   673
qed simp
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   674
63306
ca187a9f66da Various additions to polynomials, FPSs, Gamma function
eberlm
parents: 63092
diff changeset
   675
lemma fact_double:
ca187a9f66da Various additions to polynomials, FPSs, Gamma function
eberlm
parents: 63092
diff changeset
   676
  "fact (2 * n) = (2 ^ (2 * n) * pochhammer (1 / 2) n * fact n :: 'a :: field_char_0)"
ca187a9f66da Various additions to polynomials, FPSs, Gamma function
eberlm
parents: 63092
diff changeset
   677
  using pochhammer_double[of "1/2::'a" n] by (simp add: pochhammer_fact)
ca187a9f66da Various additions to polynomials, FPSs, Gamma function
eberlm
parents: 63092
diff changeset
   678
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   679
lemma pochhammer_absorb_comp:
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   680
  "((r :: 'a :: comm_ring_1) - of_nat k) * pochhammer (- r) k = r * pochhammer (-r + 1) k"
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   681
  (is "?lhs = ?rhs")
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   682
proof -
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   683
  have "?lhs = -pochhammer (-r) (Suc k)" by (subst pochhammer_rec') (simp add: algebra_simps)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   684
  also have "\<dots> = ?rhs" by (subst pochhammer_rec) simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   685
  finally show ?thesis .
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   686
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   687
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
   688
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
   689
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
   690
61533
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   691
definition (in field_char_0) gbinomial :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixl "gchoose" 65)
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
   692
  where "a gchoose n =
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   693
    (if n = 0 then 1 else (setprod (\<lambda>i. a - of_nat i) {0 .. n - 1}) / (fact 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
   694
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
   695
lemma gbinomial_0 [simp]: "a gchoose 0 = 1" "0 gchoose (Suc n) = 0"
59867
58043346ca64 given up separate type classes demanding `inverse 0 = 0`
haftmann
parents: 59862
diff changeset
   696
  by (simp_all add: gbinomial_def)
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
   697
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   698
lemma gbinomial_pochhammer: "a gchoose n = (- 1) ^ n * pochhammer (- a) n / (fact 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
   699
proof (cases "n = 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
   700
  case True
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
   701
  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
   702
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
   703
  case False
62481
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
   704
  then have eq: "(- 1) ^ n = (\<Prod>i = 0..n - 1. - 1)"
b5d8e57826df tuned bootstrap order to provide type classes in a more sensible order
haftmann
parents: 62378
diff changeset
   705
    by (auto simp add: setprod_constant)
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
   706
  from False 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
   707
    by (simp add: pochhammer_def gbinomial_def field_simps
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
   708
      eq setprod.distrib[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
   709
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
   710
61533
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   711
lemma gbinomial_pochhammer':
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   712
  "(s :: 'a :: field_char_0) gchoose n = pochhammer (s - of_nat n + 1) n / fact n"
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   713
proof -
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   714
  have "s gchoose n = ((-1)^n * (-1)^n) * pochhammer (s - of_nat n + 1) n / fact n"
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   715
    by (simp add: gbinomial_pochhammer pochhammer_minus mult_ac)
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   716
  also have "(-1 :: 'a)^n * (-1)^n = 1" by (subst power_add [symmetric]) simp
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   717
  finally show ?thesis by simp
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   718
qed
980dd46a03fb Added binomial identities to CONTRIBUTORS; small lemmas on of_int/pochhammer
eberlm
parents: 61525
diff changeset
   719
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   720
lemma binomial_gbinomial:
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   721
    "of_nat (n choose k) = (of_nat n gchoose k :: '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
   722
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
   723
  { assume kn: "k > 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
   724
    then have ?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
   725
      by (subst binomial_eq_0[OF kn])
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
   726
         (simp add: gbinomial_pochhammer field_simps  pochhammer_of_nat_eq_0_iff) }
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
   727
  moreover
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
   728
  { assume "k=0" then have ?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
   729
  moreover
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
   730
  { assume kn: "k \<le> n" and k0: "k\<noteq> 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
   731
    from k0 obtain h where h: "k = Suc h" by (cases k) 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
   732
    from h
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
   733
    have eq:"(- 1 :: 'a) ^ k = setprod (\<lambda>i. - 1) {0..h}"
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
   734
      by (subst setprod_constant) 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
   735
    have eq': "(\<Prod>i\<in>{0..h}. of_nat n + - (of_nat i :: 'a)) = (\<Prod>i\<in>{n - h..n}. of_nat 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
   736
        using h kn
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
   737
      by (intro setprod.reindex_bij_witness[where i="op - n" and j="op - 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
   738
         (auto simp: of_nat_diff)
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
   739
    have th0: "finite {1..n - Suc h}" "finite {n - h .. 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
   740
        "{1..n - Suc h} \<inter> {n - h .. n} = {}" and
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
   741
        eq3: "{1..n - Suc h} \<union> {n - h .. n} = {1..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
   742
      using h kn by 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
   743
    from eq[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
   744
    have ?thesis using kn
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
   745
      apply (simp add: binomial_fact[OF kn, where ?'a = 'a]
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
   746
        gbinomial_pochhammer field_simps pochhammer_Suc_setprod)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   747
      apply (simp add: pochhammer_Suc_setprod fact_altdef h
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
   748
        of_nat_setprod setprod.distrib[symmetric] eq' del: One_nat_def power_Suc)
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
   749
      unfolding setprod.union_disjoint[OF th0, unfolded eq3, of "of_nat:: nat \<Rightarrow> 'a"] eq[unfolded h]
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   750
      unfolding mult.assoc
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
   751
      unfolding setprod.distrib[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
   752
      apply 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
   753
      apply (intro setprod.reindex_bij_witness[where i="op - n" and j="op - 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
   754
      apply (auto simp: of_nat_diff)
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
   755
      done
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
   756
  }
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
   757
  moreover
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
   758
  have "k > n \<or> k = 0 \<or> (k \<le> n \<and> k \<noteq> 0)" by arith
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
   759
  ultimately show ?thesis by blast
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
   760
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
   761
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
   762
lemma gbinomial_1[simp]: "a gchoose 1 = a"
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
   763
  by (simp add: gbinomial_def)
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
   764
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
   765
lemma gbinomial_Suc0[simp]: "a gchoose (Suc 0) = a"
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
   766
  by (simp add: gbinomial_def)
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
   767
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
   768
lemma gbinomial_mult_1:
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   769
  fixes a :: "'a :: field_char_0"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   770
  shows "a * (a gchoose 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
   771
    of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"  (is "?l = ?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
   772
proof -
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   773
  have "?r = ((- 1) ^n * pochhammer (- a) n / (fact n)) * (of_nat n - (- a + of_nat 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
   774
    unfolding gbinomial_pochhammer
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   775
      pochhammer_Suc of_nat_mult right_diff_distrib power_Suc
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   776
    apply (simp del: of_nat_Suc fact.simps)
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   777
    apply (auto simp add: field_simps simp del: of_nat_Suc)
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   778
    done
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
   779
  also have "\<dots> = ?l" unfolding gbinomial_pochhammer
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
   780
    by (simp add: field_simps)
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
   781
  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
   782
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
   783
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
   784
lemma gbinomial_mult_1':
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   785
  fixes a :: "'a :: field_char_0"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   786
  shows "(a gchoose n) * a = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc 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
   787
  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
   788
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
   789
lemma gbinomial_Suc:
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   790
    "a gchoose (Suc k) = (setprod (\<lambda>i. a - of_nat i) {0 .. k}) / (fact (Suc 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
   791
  by (simp add: gbinomial_def)
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
   792
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
   793
lemma gbinomial_mult_fact:
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   794
  fixes a :: "'a::field_char_0"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   795
  shows
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   796
   "fact (Suc k) * (a gchoose (Suc 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
   797
    (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   798
  by (simp_all add: gbinomial_Suc field_simps del: fact.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
   799
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
   800
lemma gbinomial_mult_fact':
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   801
  fixes a :: "'a::field_char_0"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   802
  shows "(a gchoose (Suc k)) * fact (Suc k) = (setprod (\<lambda>i. a - of_nat i) {0 .. 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
   803
  using gbinomial_mult_fact[of k a]
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
   804
  by (subst 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
   805
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
   806
lemma gbinomial_Suc_Suc:
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   807
  fixes a :: "'a :: field_char_0"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   808
  shows "(a + 1) gchoose (Suc k) = a gchoose k + (a gchoose (Suc 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
   809
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
   810
  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
   811
  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
   812
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
   813
  case (Suc h)
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
   814
  have eq0: "(\<Prod>i\<in>{1..k}. (a + 1) - of_nat i) = (\<Prod>i\<in>{0..h}. a - of_nat 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
   815
    apply (rule setprod.reindex_cong [where l = Suc])
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
   816
      using Suc
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
   817
      apply 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
   818
    done
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   819
  have "fact (Suc k) * (a gchoose k + (a gchoose (Suc k))) =
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   820
        (a gchoose Suc h) * (fact (Suc (Suc h))) +
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   821
        (a gchoose Suc (Suc h)) * (fact (Suc (Suc h)))"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   822
    by (simp add: Suc field_simps del: fact.simps)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   823
  also have "... = (a gchoose Suc h) * of_nat (Suc (Suc h) * fact (Suc h)) +
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   824
                   (\<Prod>i = 0..Suc h. a - of_nat i)"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   825
    by (metis fact.simps(2) gbinomial_mult_fact' of_nat_fact of_nat_id)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   826
  also have "... = (fact (Suc h) * (a gchoose Suc h)) * of_nat (Suc (Suc h)) +
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   827
                   (\<Prod>i = 0..Suc h. a - of_nat i)"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   828
    by (simp only: fact.simps(2) mult.commute mult.left_commute of_nat_fact of_nat_id of_nat_mult)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   829
  also have "... =  of_nat (Suc (Suc h)) * (\<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
   830
                    (\<Prod>i = 0..Suc h. a - of_nat i)"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   831
    by (metis gbinomial_mult_fact mult.commute)
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   832
  also have "... = (\<Prod>i = 0..Suc h. a - of_nat i) +
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   833
                   (of_nat h * (\<Prod>i = 0..h. a - of_nat i) + 2 * (\<Prod>i = 0..h. a - of_nat i))"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   834
    by (simp add: field_simps)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   835
  also have "... =
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 60758
diff changeset
   836
    ((a gchoose Suc h) * (fact (Suc h)) * of_nat (Suc k)) + (\<Prod>i\<in>{0::nat..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
   837
    unfolding gbinomial_mult_fact'
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   838
    by (simp add: comm_semiring_class.distrib field_simps 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
   839
  also have "\<dots> = (\<Prod>i\<in>{0..h}. a - of_nat i) * (a + 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
   840
    unfolding gbinomial_mult_fact' setprod_nat_ivl_Suc
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
   841
    by (simp add: field_simps Suc)
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
   842
  also have "\<dots> = (\<Prod>i\<in>{0..k}. (a + 1) - of_nat 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
   843
    using eq0
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
   844
    by (simp add: Suc setprod_nat_ivl_1_Suc)
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   845
  also have "\<dots> = (fact (Suc k)) * ((a + 1) gchoose (Suc 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
   846
    unfolding gbinomial_mult_fact ..
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   847
  finally show ?thesis
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   848
    by (metis fact_nonzero mult_cancel_left)
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
   849
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
   850
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
   851
lemma gbinomial_reduce_nat:
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   852
  fixes a :: "'a :: field_char_0"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   853
  shows "0 < k \<Longrightarrow> a gchoose k = (a - 1) gchoose (k - 1) + ((a - 1) gchoose k)"
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   854
  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
   855
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
   856
lemma gchoose_row_sum_weighted:
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
   857
  fixes r :: "'a::field_char_0"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
   858
  shows "(\<Sum>k = 0..m. (r gchoose k) * (r/2 - of_nat k)) = of_nat(Suc m) / 2 * (r gchoose (Suc m))"
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
   859
proof (induct m)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
   860
  case 0 show ?case by simp
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
   861
next
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
   862
  case (Suc m)
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
   863
  from Suc show ?case
61738
c4f6031f1310 New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
   864
    by (simp add: field_simps distrib gbinomial_mult_1)
60141
833adf7db7d8 New material, mostly about limits. Consolidation.
paulson <lp15@cam.ac.uk>
parents: 59867
diff changeset
   865
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
   866
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
   867
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
   868
  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
   869
  shows "n choose k = n choose (n - 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
   870
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
   871
  from kn have kn': "n - k \<le> n" by arith
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
   872
  from binomial_fact_lemma[OF kn] binomial_fact_lemma[OF kn']
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
   873
  have "fact k * fact (n - k) * (n choose 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
   874
    fact (n - k) * fact (n - (n - k)) * (n choose (n - k))" 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
   875
  then show ?thesis using kn 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
   876
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
   877
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   878
lemma choose_rising_sum:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   879
  "(\<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
   880
  "(\<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
   881
proof -
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   882
  show "(\<Sum>j\<le>m. ((n + j) choose n)) = ((n + m + 1) choose (n + 1))" by (induction m) simp_all
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   883
  also have "... = ((n + m + 1) choose m)" by (subst binomial_symmetric) simp_all
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   884
  finally show "(\<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
   885
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   886
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   887
lemma choose_linear_sum:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   888
  "(\<Sum>i\<le>n. i * (n choose i)) = n * 2 ^ (n - 1)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   889
proof (cases n)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   890
  case (Suc m)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   891
  have "(\<Sum>i\<le>n. i * (n choose i)) = (\<Sum>i\<le>Suc m. i * (Suc m choose i))" by (simp add: Suc)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   892
  also have "... = Suc m * 2 ^ m"
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   893
    by (simp only: setsum_atMost_Suc_shift Suc_times_binomial setsum_right_distrib[symmetric])
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   894
       (simp add: choose_row_sum')
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   895
  finally show ?thesis using Suc by simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   896
qed simp_all
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   897
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   898
lemma choose_alternating_linear_sum:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   899
  assumes "n \<noteq> 1"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   900
  shows   "(\<Sum>i\<le>n. (-1)^i * of_nat i * of_nat (n choose i) :: 'a :: comm_ring_1) = 0"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   901
proof (cases n)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   902
  case (Suc m)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   903
  with assms have "m > 0" by simp
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   904
  have "(\<Sum>i\<le>n. (-1) ^ i * of_nat i * of_nat (n choose i) :: 'a) =
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   905
            (\<Sum>i\<le>Suc m. (-1) ^ i * of_nat i * of_nat (Suc m choose i))" by (simp add: Suc)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   906
  also have "... = (\<Sum>i\<le>m. (-1) ^ (Suc i) * of_nat (Suc i * (Suc m choose Suc i)))"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   907
    by (simp only: setsum_atMost_Suc_shift setsum_right_distrib[symmetric] of_nat_mult mult_ac) simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   908
  also have "... = -of_nat (Suc m) * (\<Sum>i\<le>m. (-1) ^ i * of_nat ((m choose i)))"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   909
    by (subst setsum_right_distrib, rule setsum.cong[OF refl], subst Suc_times_binomial)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   910
       (simp add: algebra_simps of_nat_mult)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   911
  also have "(\<Sum>i\<le>m. (-1 :: 'a) ^ i * of_nat ((m choose i))) = 0"
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61738
diff changeset
   912
    using choose_alternating_sum[OF \<open>m > 0\<close>] by simp
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   913
  finally show ?thesis by simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   914
qed simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   915
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   916
lemma vandermonde:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   917
  "(\<Sum>k\<le>r. (m choose k) * (n choose (r - k))) = (m + n) choose r"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   918
proof (induction n arbitrary: r)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   919
  case 0
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   920
  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)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   921
    by (intro setsum.cong) simp_all
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   922
  also have "... = m choose r" by (simp add: setsum.delta)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   923
  finally show ?case by simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   924
next
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   925
  case (Suc n r)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   926
  show ?case by (cases r) (simp_all add: Suc [symmetric] algebra_simps setsum.distrib Suc_diff_le)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   927
qed
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 choose_square_sum:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   930
  "(\<Sum>k\<le>n. (n choose k)^2) = ((2*n) choose n)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   931
  using vandermonde[of n n n] by (simp add: power2_eq_square mult_2 binomial_symmetric [symmetric])
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   932
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   933
lemma pochhammer_binomial_sum:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   934
  fixes a b :: "'a :: comm_ring_1"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   935
  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
   936
proof (induction n arbitrary: a b)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   937
  case (Suc n a b)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   938
  have "(\<Sum>k\<le>Suc n. of_nat (Suc n choose k) * pochhammer a k * pochhammer b (Suc n - k)) =
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   939
            (\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a (Suc i) * pochhammer b (n - i)) +
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   940
            ((\<Sum>i\<le>n. of_nat (n choose Suc i) * pochhammer a (Suc i) * pochhammer b (n - i)) +
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   941
            pochhammer b (Suc n))"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   942
    by (subst setsum_atMost_Suc_shift) (simp add: ring_distribs setsum.distrib)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   943
  also have "(\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a (Suc i) * pochhammer b (n - i)) =
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   944
               a * pochhammer ((a + 1) + b) n"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   945
    by (subst Suc) (simp add: setsum_right_distrib pochhammer_rec mult_ac)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
   946
  also have "(\<Sum>i\<le>n. of_nat (n choose Suc i) * pochhammer a (Suc i) * pochhammer b (n - i)) + pochhammer b (Suc n) =
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   947
               (\<Sum>i=0..Suc n. of_nat (n choose i) * pochhammer a i * pochhammer b (Suc n - i))"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   948
    by (subst setsum_head_Suc, simp, subst setsum_shift_bounds_cl_Suc_ivl) (simp add: atLeast0AtMost)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   949
  also have "... = (\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a i * pochhammer b (Suc n - i))"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   950
    using Suc by (intro setsum.mono_neutral_right) (auto simp: not_le binomial_eq_0)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   951
  also have "... = (\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a i * pochhammer b (Suc (n - i)))"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   952
    by (intro setsum.cong) (simp_all add: Suc_diff_le)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   953
  also have "... = b * pochhammer (a + (b + 1)) n"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   954
    by (subst Suc) (simp add: setsum_right_distrib mult_ac pochhammer_rec)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   955
  also have "a * pochhammer ((a + 1) + b) n + b * pochhammer (a + (b + 1)) n =
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   956
               pochhammer (a + b) (Suc n)" by (simp add: pochhammer_rec algebra_simps)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   957
  finally show ?case ..
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   958
qed simp_all
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   959
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
   960
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
   961
text\<open>Contributed by Manuel Eberl, generalised by LCP.
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
   962
  Alternative definition of the binomial coefficient as @{term "\<Prod>i<k. (n - i) / (k - i)"}\<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
   963
lemma gbinomial_altdef_of_nat:
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
   964
  fixes k :: nat
59867
58043346ca64 given up separate type classes demanding `inverse 0 = 0`
haftmann
parents: 59862
diff changeset
   965
    and x :: "'a :: {field_char_0,field}"
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
   966
  shows "x gchoose k = (\<Prod>i<k. (x - of_nat i) / of_nat (k - i) :: 'a)"
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
   967
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
   968
  have "(x gchoose k) = (\<Prod>i<k. x - of_nat i) / of_nat (fact 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
   969
    unfolding gbinomial_def
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
   970
    by (auto simp: gr0_conv_Suc lessThan_Suc_atMost atLeast0AtMost)
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
   971
  also have "\<dots> = (\<Prod>i<k. (x - of_nat i) / of_nat (k - i) :: 'a)"
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
   972
    unfolding fact_eq_rev_setprod_nat of_nat_setprod
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
   973
    by (auto simp add: setprod_dividef intro!: setprod.cong of_nat_diff[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
   974
  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
   975
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
   976
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
   977
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
   978
  fixes k :: nat
59867
58043346ca64 given up separate type classes demanding `inverse 0 = 0`
haftmann
parents: 59862
diff changeset
   979
    and x :: "'a :: linordered_field"
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
   980
  assumes "of_nat k \<le> 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
   981
  shows "(x / of_nat k :: 'a) ^ k \<le> x gchoose 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
   982
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
   983
  have x: "0 \<le> 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
   984
    using assms of_nat_0_le_iff order_trans by blast
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
   985
  have "(x / of_nat k :: 'a) ^ k = (\<Prod>i<k. x / of_nat k :: 'a)"
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
   986
    by (simp add: setprod_constant)
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
   987
  also have "\<dots> \<le> x gchoose 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
   988
    unfolding gbinomial_altdef_of_nat
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
   989
  proof (safe intro!: setprod_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
   990
    fix i :: nat
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
   991
    assume ik: "i < 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
   992
    from assms have "x * of_nat i \<ge> of_nat (i * 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
   993
      by (metis mult.commute mult_le_cancel_right of_nat_less_0_iff of_nat_mult)
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
   994
    then have "x * of_nat k - x * of_nat i \<le> x * of_nat k - of_nat (i * k)" by arith
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
   995
    then have "x * of_nat (k - i) \<le> (x - of_nat i) * of_nat 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
   996
      using ik
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
   997
      by (simp add: algebra_simps zero_less_mult_iff of_nat_diff of_nat_mult)
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
   998
    then have "x * of_nat (k - i) \<le> (x - of_nat i) * (of_nat k :: 'a)"
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
   999
      unfolding of_nat_mult[symmetric] of_nat_le_iff .
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
  1000
    with assms show "x / of_nat k \<le> (x - of_nat i) / (of_nat (k - i) :: 'a)"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60604
diff changeset
  1001
      using \<open>i < k\<close> by (simp add: field_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
  1002
  qed (simp add: x zero_le_divide_iff)
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
  1003
  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
  1004
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
  1005
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1006
lemma gbinomial_negated_upper: "(a gchoose b) = (-1) ^ b * ((of_nat b - a - 1) gchoose b)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1007
  by (simp add: gbinomial_pochhammer pochhammer_minus algebra_simps)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1008
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1009
lemma gbinomial_minus: "((-a) gchoose b) = (-1) ^ b * ((a + of_nat b - 1) gchoose b)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1010
  by (subst gbinomial_negated_upper) (simp add: add_ac)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1011
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1012
lemma Suc_times_gbinomial:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1013
  "of_nat (Suc b) * ((a + 1) gchoose (Suc b)) = (a + 1) * (a gchoose b)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1014
proof (cases b)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1015
  case (Suc b)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
  1016
  hence "((a + 1) gchoose (Suc (Suc b))) =
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1017
             (\<Prod>i = 0..Suc b. a + (1 - of_nat i)) / fact (b + 2)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1018
    by (simp add: field_simps gbinomial_def)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1019
  also have "(\<Prod>i = 0..Suc b. a + (1 - of_nat i)) = (a + 1) * (\<Prod>i = 0..b. a - of_nat i)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1020
    by (simp add: setprod_nat_ivl_1_Suc setprod_shift_bounds_cl_Suc_ivl)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1021
  also have "... / fact (b + 2) = (a + 1) / of_nat (Suc (Suc b)) * (a gchoose Suc b)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1022
    by (simp_all add: gbinomial_def setprod_nat_ivl_1_Suc setprod_shift_bounds_cl_Suc_ivl)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1023
  finally show ?thesis by (simp add: Suc field_simps del: of_nat_Suc)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1024
qed simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1025
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1026
lemma gbinomial_factors:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1027
  "((a + 1) gchoose (Suc b)) = (a + 1) / of_nat (Suc b) * (a gchoose b)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1028
proof (cases b)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1029
  case (Suc b)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
  1030
  hence "((a + 1) gchoose (Suc (Suc b))) =
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1031
             (\<Prod>i = 0..Suc b. a + (1 - of_nat i)) / fact (b + 2)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1032
    by (simp add: field_simps gbinomial_def)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1033
  also have "(\<Prod>i = 0..Suc b. a + (1 - of_nat i)) = (a + 1) * (\<Prod>i = 0..b. a - of_nat i)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1034
    by (simp add: setprod_nat_ivl_1_Suc setprod_shift_bounds_cl_Suc_ivl)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1035
  also have "... / fact (b + 2) = (a + 1) / of_nat (Suc (Suc b)) * (a gchoose Suc b)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1036
    by (simp_all add: gbinomial_def setprod_nat_ivl_1_Suc setprod_shift_bounds_cl_Suc_ivl)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1037
  finally show ?thesis by (simp add: Suc)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1038
qed simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1039
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1040
lemma gbinomial_rec: "((r + 1) gchoose (Suc k)) = (r gchoose k) * ((r + 1) / of_nat (Suc k))"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1041
  using gbinomial_mult_1[of r k]
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1042
  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
  1043
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1044
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
  1045
  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
  1046
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1047
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1048
text \<open>The absorption identity (equation 5.5 \cite[p.~157]{GKP}):\[
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1049
{r \choose k} = \frac{r}{k}{r - 1 \choose k - 1},\quad \textnormal{integer } k \neq 0.
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1050
\]\<close>
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
  1051
lemma gbinomial_absorption':
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1052
  "k > 0 \<Longrightarrow> (r gchoose k) = (r / of_nat(k)) * (r - 1 gchoose (k - 1))"
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
  1053
  using gbinomial_rec[of "r - 1" "k - 1"]
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1054
  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
  1055
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1056
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
  1057
division by $k$ (the lower index) and therefore remove the $k \neq 0$
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1058
restriction\cite[p.~157]{GKP}:\[
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1059
k{r \choose k} = r{r - 1 \choose k - 1}, \quad \textnormal{integer } k.
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1060
\]\<close>
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1061
lemma gbinomial_absorption:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1062
  "of_nat (Suc k) * (r gchoose Suc k) = r * ((r - 1) gchoose k)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1063
  using gbinomial_absorption'[of "Suc k" r] by (simp add: field_simps del: of_nat_Suc)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1064
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1065
text \<open>The absorption identity for natural number binomial coefficients:\<close>
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1066
lemma binomial_absorption:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1067
  "Suc k * (n choose Suc k) = n * ((n - 1) choose k)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1068
  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
  1069
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1070
text \<open>The absorption companion identity for natural number coefficients,
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1071
following the proof by GKP \cite[p.~157]{GKP}:\<close>
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1072
lemma binomial_absorb_comp:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1073
  "(n - k) * (n choose k) = n * ((n - 1) choose k)" (is "?lhs = ?rhs")
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1074
proof (cases "n \<le> k")
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1075
  case False
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1076
  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
  1077
    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
  1078
    by simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1079
  also from False have "Suc ((n - 1) - k) = n - k" by simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1080
  also from False have "n choose \<dots> = n choose k" by (intro binomial_symmetric [symmetric]) simp_all
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1081
  finally show ?thesis ..
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1082
qed auto
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1083
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1084
text \<open>The generalised absorption companion identity:\<close>
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1085
lemma gbinomial_absorb_comp: "(r - of_nat k) * (r gchoose k) = r * ((r - 1) gchoose k)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1086
  using pochhammer_absorb_comp[of r k] by (simp add: gbinomial_pochhammer)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1087
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1088
lemma gbinomial_addition_formula:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1089
  "r gchoose (Suc k) = ((r - 1) gchoose (Suc k)) + ((r - 1) gchoose k)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1090
  using gbinomial_Suc_Suc[of "r - 1" k] by (simp add: algebra_simps)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1091
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1092
lemma binomial_addition_formula:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1093
  "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
  1094
  by (subst choose_reduce_nat) simp_all
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1095
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1096
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1097
text \<open>
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1098
  Equation 5.9 of the reference material \cite[p.~159]{GKP} is a useful
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1099
  summation formula, operating on both indices:\[
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1100
  \sum\limits_{k \leq n}{r + k \choose k} = {r + n + 1 \choose n},
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1101
   \quad \textnormal{integer } n.
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
  1102
  \]
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1103
\<close>
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1104
lemma gbinomial_parallel_sum:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1105
"(\<Sum>k\<le>n. (r + of_nat k) gchoose k) = (r + of_nat n + 1) gchoose n"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1106
proof (induction n)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1107
  case (Suc m)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1108
  thus ?case using gbinomial_Suc_Suc[of "(r + of_nat m + 1)" m] by (simp add: add_ac)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1109
qed auto
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1110
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1111
subsection \<open>Summation on the upper index\<close>
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1112
text \<open>
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1113
  Another summation formula is equation 5.10 of the reference material \cite[p.~160]{GKP},
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
  1114
  aptly named \emph{summation on the upper index}:\[\sum_{0 \leq k \leq n} {k \choose m} =
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1115
  {n + 1 \choose m + 1}, \quad \textnormal{integers } m, n \geq 0.\]
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1116
\<close>
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1117
lemma gbinomial_sum_up_index:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1118
  "(\<Sum>k = 0..n. (of_nat k gchoose m) :: 'a :: field_char_0) = (of_nat n + 1) gchoose (m + 1)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1119
proof (induction n)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1120
  case 0
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1121
  show ?case using gbinomial_Suc_Suc[of 0 m] by (cases m) auto
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1122
next
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1123
  case (Suc n)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1124
  thus ?case using gbinomial_Suc_Suc[of "of_nat (Suc n) :: 'a" m] by (simp add: add_ac)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1125
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1126
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
  1127
lemma gbinomial_index_swap:
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1128
  "((-1) ^ m) * ((- (of_nat n) - 1) gchoose m) = ((-1) ^ n) * ((- (of_nat m) - 1) gchoose n)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1129
  (is "?lhs = ?rhs")
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1130
proof -
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1131
  have "?lhs = (of_nat (m + n) gchoose m)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1132
    by (subst gbinomial_negated_upper) (simp add: power_mult_distrib [symmetric])
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1133
  also have "\<dots> = (of_nat (m + n) gchoose n)" by (subst gbinomial_of_nat_symmetric) simp_all
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1134
  also have "\<dots> = ?rhs" by (subst gbinomial_negated_upper) simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1135
  finally show ?thesis .
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1136
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1137
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
  1138
lemma gbinomial_sum_lower_neg:
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1139
  "(\<Sum>k\<le>m. (r gchoose k) * (- 1) ^ k) = (- 1) ^ m * (r - 1 gchoose m)" (is "?lhs = ?rhs")
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1140
proof -
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1141
  have "?lhs = (\<Sum>k\<le>m. -(r + 1) + of_nat k gchoose k)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1142
    by (intro setsum.cong[OF refl]) (subst gbinomial_negated_upper, simp add: power_mult_distrib)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1143
  also have "\<dots>  = -r + of_nat m gchoose m" by (subst gbinomial_parallel_sum) simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1144
  also have "\<dots> = ?rhs" by (subst gbinomial_negated_upper) (simp add: power_mult_distrib)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1145
  finally show ?thesis .
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1146
qed
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1147
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1148
lemma gbinomial_partial_row_sum:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1149
"(\<Sum>k\<le>m. (r gchoose k) * ((r / 2) - of_nat k)) = ((of_nat m + 1)/2) * (r gchoose (m + 1))"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1150
proof (induction m)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1151
  case (Suc mm)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
  1152
  hence "(\<Sum>k\<le>Suc mm. (r gchoose k) * (r / 2 - of_nat k)) =
61738
c4f6031f1310 New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents: 61649
diff changeset
  1153
             (r - of_nat (Suc mm)) * (r gchoose Suc mm) / 2" by (simp add: field_simps)
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1154
  also have "\<dots> = r * (r - 1 gchoose Suc mm) / 2" by (subst gbinomial_absorb_comp) (rule refl)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1155
  also have "\<dots> = (of_nat (Suc mm) + 1) / 2 * (r gchoose (Suc mm + 1))"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1156
    by (subst gbinomial_absorption [symmetric]) simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1157
  finally show ?case .
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1158
qed simp_all
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1159
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1160
lemma setsum_bounds_lt_plus1: "(\<Sum>k<mm. f (Suc k)) = (\<Sum>k=1..mm. f k)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1161
  by (induction mm) simp_all
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1162
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1163
lemma gbinomial_partial_sum_poly:
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1164
  "(\<Sum>k\<le>m. (of_nat m + r gchoose k) * x^k * y^(m-k)) =
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1165
       (\<Sum>k\<le>m. (-r gchoose k) * (-x)^k * (x + y)^(m-k))" (is "?lhs m = ?rhs m")
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1166
proof (induction m)
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1167
  case (Suc mm)
63044
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62481
diff changeset
  1168
  define G where "G i k = (of_nat i + r gchoose k) * x^k * y^(i-k)" for i k
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62481
diff changeset
  1169
  define S where "S = ?lhs"
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1170
  have SG_def: "S = (\<lambda>i. (\<Sum>k\<le>i. (G i k)))" unfolding S_def G_def ..
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1171
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1172
  have "S (Suc mm) = G (Suc mm) 0 + (\<Sum>k=Suc 0..Suc mm. G (Suc mm) k)"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1173
    using SG_def by (simp add: setsum_head_Suc atLeast0AtMost [symmetric])
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1174
  also have "(\<Sum>k=Suc 0..Suc mm. G (Suc mm) k) = (\<Sum>k=0..mm. G (Suc mm) (Suc k))"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1175
    by (subst setsum_shift_bounds_cl_Suc_ivl) simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1176
  also have "\<dots> = (\<Sum>k=0..mm. ((of_nat mm + r gchoose (Suc k))
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1177
                    + (of_nat mm + r gchoose k)) * x^(Suc k) * y^(mm - k))"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1178
    unfolding G_def by (subst gbinomial_addition_formula) simp
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1179
  also have "\<dots> = (\<Sum>k=0..mm. (of_nat mm + r gchoose (Suc k)) * x^(Suc k) * y^(mm - k))
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1180
                  + (\<Sum>k=0..mm. (of_nat mm + r gchoose k) * x^(Suc k) * y^(mm - k))"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1181
    by (subst setsum.distrib [symmetric]) (simp add: algebra_simps)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
  1182
  also have "(\<Sum>k=0..mm. (of_nat mm + r gchoose (Suc k)) * x^(Suc k) * y^(mm - k)) =
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1183
               (\<Sum>k<Suc mm. (of_nat mm + r gchoose (Suc k)) * x^(Suc k) * y^(mm - k))"
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1184
    by (simp only: atLeast0AtMost lessThan_Suc_atMost)
62378
85ed00c1fe7c generalize more theorems to support enat and ennreal
hoelzl
parents: 62347
diff changeset
  1185
  also have "\<dots> = (\<Sum>k<mm. (of_nat mm + r gchoose Suc k) * x^(Suc k) * y^(mm-k))
61525
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1186
                      + (of_nat mm + r gchoose (Suc mm)) * x^(Suc mm)" (is "_ = ?A + ?B")
ab2e862263e7 Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents: 61076
diff changeset
  1187
    by (subst setsum_lessThan_Suc) simp