author | paulson <lp15@cam.ac.uk> |
Sun, 14 Aug 2022 23:51:47 +0100 | |
changeset 75864 | 3842556b757c |
parent 70114 | 089c17514794 |
permissions | -rw-r--r-- |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
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1 |
(* Author: Lukas Bulwahn <lukas.bulwahn-at-gmail.com> *) |
61343 | 2 |
section \<open>Sum of Powers\<close> |
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|
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theory Sum_of_Powers |
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imports Complex_Main |
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begin |
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subsection \<open>Preliminaries\<close> |
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|
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lemma integrals_eq: |
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assumes "f a = g a" |
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assumes "\<And> x. ((\<lambda>x. f x - g x) has_real_derivative 0) (at x)" |
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shows "f x = g x" |
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parents:
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by (metis (no_types, lifting) DERIV_isconst_all assms(1) assms(2) eq_iff_diff_eq_0) |
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|
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lemma sum_diff: "((\<Sum>i\<le>n::nat. f (i + 1) - f i)::'a::field) = f (n + 1) - f 0" |
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parents:
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by (induct n) (auto simp add: field_simps) |
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|
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declare One_nat_def [simp del] |
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61343 | 21 |
subsection \<open>Bernoulli Numbers and Bernoulli Polynomials\<close> |
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declare sum.cong [fundef_cong] |
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|
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fun bernoulli :: "nat \<Rightarrow> real" |
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where |
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"bernoulli 0 = (1::real)" |
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| "bernoulli (Suc n) = (-1 / (n + 2)) * (\<Sum>k \<le> n. ((n + 2 choose k) * bernoulli k))" |
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|
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declare bernoulli.simps[simp del] |
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|
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definition |
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"bernpoly n = (\<lambda>x. \<Sum>k \<le> n. (n choose k) * bernoulli k * x ^ (n - k))" |
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subsection \<open>Basic Observations on Bernoulli Polynomials\<close> |
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|
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lemma bernpoly_0: "bernpoly n 0 = bernoulli n" |
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proof (cases n) |
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case 0 |
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paulson <lp15@cam.ac.uk>
parents:
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40 |
then show "bernpoly n 0 = bernoulli n" |
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unfolding bernpoly_def bernoulli.simps by auto |
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next |
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case (Suc n') |
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have "(\<Sum>k\<le>n'. real (Suc n' choose k) * bernoulli k * 0 ^ (Suc n' - k)) = 0" |
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by (rule sum.neutral) auto |
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with Suc show ?thesis |
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unfolding bernpoly_def by simp |
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qed |
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|
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lemma sum_binomial_times_bernoulli: |
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"(\<Sum>k\<le>n. ((Suc n) choose k) * bernoulli k) = (if n = 0 then 1 else 0)" |
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proof (cases n) |
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53 |
case 0 |
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Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61343
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54 |
then show ?thesis by (simp add: bernoulli.simps) |
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next |
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case Suc |
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Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61343
diff
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57 |
then show ?thesis |
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58 |
by (simp add: bernoulli.simps) |
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59 |
(simp add: field_simps add_2_eq_Suc'[symmetric] del: add_2_eq_Suc add_2_eq_Suc') |
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60 |
qed |
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61 |
|
61343 | 62 |
subsection \<open>Sum of Powers with Bernoulli Polynomials\<close> |
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|
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lemma bernpoly_derivative [derivative_intros]: |
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65 |
"(bernpoly (Suc n) has_real_derivative ((n + 1) * bernpoly n x)) (at x)" |
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66 |
proof - |
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67 |
have "(bernpoly (Suc n) has_real_derivative (\<Sum>k\<le>n. real (Suc n - k) * x ^ (n - k) * (real (Suc n choose k) * bernoulli k))) (at x)" |
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68 |
unfolding bernpoly_def by (rule DERIV_cong) (fast intro!: derivative_intros, simp) |
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69 |
moreover have "(\<Sum>k\<le>n. real (Suc n - k) * x ^ (n - k) * (real (Suc n choose k) * bernoulli k)) = (n + 1) * bernpoly n x" |
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parents:
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|
70 |
unfolding bernpoly_def |
64267 | 71 |
by (auto intro: sum.cong simp add: sum_distrib_left real_binomial_eq_mult_binomial_Suc[of _ n] Suc_eq_plus1 of_nat_diff) |
60603
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72 |
ultimately show ?thesis by auto |
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parents:
diff
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|
73 |
qed |
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diff
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74 |
|
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75 |
lemma diff_bernpoly: |
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76 |
"bernpoly n (x + 1) - bernpoly n x = n * x ^ (n - 1)" |
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77 |
proof (induct n arbitrary: x) |
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78 |
case 0 |
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79 |
show ?case unfolding bernpoly_def by auto |
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80 |
next |
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81 |
case (Suc n) |
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|
82 |
have "bernpoly (Suc n) (0 + 1) - bernpoly (Suc n) 0 = (Suc n) * 0 ^ n" |
64267 | 83 |
unfolding bernpoly_0 unfolding bernpoly_def by (simp add: sum_binomial_times_bernoulli zero_power) |
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Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61343
diff
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|
84 |
then have const: "bernpoly (Suc n) (0 + 1) - bernpoly (Suc n) 0 = real (Suc n) * 0 ^ n" by (simp add: power_0_left) |
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parents:
diff
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|
85 |
have hyps': "\<And>x. (real n + 1) * bernpoly n (x + 1) - (real n + 1) * bernpoly n x = real n * x ^ (n - Suc 0) * real (Suc n)" |
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parents:
diff
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|
86 |
unfolding right_diff_distrib[symmetric] by (simp add: Suc.hyps One_nat_def) |
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parents:
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87 |
note [derivative_intros] = DERIV_chain'[where f = "\<lambda>x::real. x + 1" and g = "bernpoly (Suc n)" and s="UNIV"] |
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parents:
diff
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|
88 |
have derivative: "\<And>x. ((%x. bernpoly (Suc n) (x + 1) - bernpoly (Suc n) x - real (Suc n) * x ^ n) has_real_derivative 0) (at x)" |
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parents:
diff
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|
89 |
by (rule DERIV_cong) (fast intro!: derivative_intros, simp add: hyps') |
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parents:
diff
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|
90 |
from integrals_eq[OF const derivative] show ?case by simp |
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parents:
diff
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|
91 |
qed |
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bulwahn
parents:
diff
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|
92 |
|
09ecbd791d4a
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|
93 |
lemma sum_of_powers: "(\<Sum>k\<le>n::nat. (real k) ^ m) = (bernpoly (Suc m) (n + 1) - bernpoly (Suc m) 0) / (m + 1)" |
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parents:
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|
94 |
proof - |
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parents:
diff
changeset
|
95 |
from diff_bernpoly[of "Suc m", simplified] have "(m + (1::real)) * (\<Sum>k\<le>n. (real k) ^ m) = (\<Sum>k\<le>n. bernpoly (Suc m) (real k + 1) - bernpoly (Suc m) (real k))" |
64267 | 96 |
by (auto simp add: sum_distrib_left intro!: sum.cong) |
60603
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parents:
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|
97 |
also have "... = (\<Sum>k\<le>n. bernpoly (Suc m) (real (k + 1)) - bernpoly (Suc m) (real k))" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61343
diff
changeset
|
98 |
by simp |
60603
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parents:
diff
changeset
|
99 |
also have "... = bernpoly (Suc m) (n + 1) - bernpoly (Suc m) 0" |
64267 | 100 |
by (simp only: sum_diff[where f="\<lambda>k. bernpoly (Suc m) (real k)"]) simp |
60603
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|
101 |
finally show ?thesis by (auto simp add: field_simps intro!: eq_divide_imp) |
09ecbd791d4a
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parents:
diff
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|
102 |
qed |
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parents:
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|
103 |
|
61343 | 104 |
subsection \<open>Instances for Square And Cubic Numbers\<close> |
60603
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changeset
|
105 |
|
09ecbd791d4a
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parents:
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changeset
|
106 |
lemma binomial_unroll: |
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parents:
diff
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|
107 |
"n > 0 \<Longrightarrow> (n choose k) = (if k = 0 then 1 else (n - 1) choose (k - 1) + ((n - 1) choose k))" |
63367
6c731c8b7f03
simplified definitions of combinatorial functions
haftmann
parents:
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|
108 |
by (auto simp add: gr0_conv_Suc) |
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109 |
|
64267 | 110 |
lemma sum_unroll: |
60603
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|
111 |
"(\<Sum>k\<le>n::nat. f k) = (if n = 0 then f 0 else f n + (\<Sum>k\<le>n - 1. f k))" |
70097
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
69597
diff
changeset
|
112 |
by auto (metis One_nat_def Suc_pred add.commute sum.atMost_Suc) |
60603
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changeset
|
113 |
|
09ecbd791d4a
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|
114 |
lemma bernoulli_unroll: |
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parents:
diff
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|
115 |
"n > 0 \<Longrightarrow> bernoulli n = - 1 / (real n + 1) * (\<Sum>k\<le>n - 1. real (n + 1 choose k) * bernoulli k)" |
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parents:
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116 |
by (cases n) (simp add: bernoulli.simps One_nat_def)+ |
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parents:
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|
117 |
|
63367
6c731c8b7f03
simplified definitions of combinatorial functions
haftmann
parents:
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diff
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|
118 |
lemmas unroll = binomial_unroll |
64267 | 119 |
bernoulli.simps(1) bernoulli_unroll sum_unroll bernpoly_def |
60603
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
120 |
|
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
121 |
lemma sum_of_squares: "(\<Sum>k\<le>n::nat. k ^ 2) = (2 * n ^ 3 + 3 * n ^ 2 + n) / 6" |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
122 |
proof - |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
123 |
have "real (\<Sum>k\<le>n::nat. k ^ 2) = (\<Sum>k\<le>n::nat. (real k) ^ 2)" by simp |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
124 |
also have "... = (bernpoly 3 (real (n + 1)) - bernpoly 3 0) / real (3 :: nat)" |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
125 |
by (auto simp add: sum_of_powers) |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
126 |
also have "... = (2 * n ^ 3 + 3 * n ^ 2 + n) / 6" |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
127 |
by (simp add: unroll algebra_simps power2_eq_square power3_eq_cube One_nat_def[symmetric]) |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
128 |
finally show ?thesis by simp |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
129 |
qed |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
130 |
|
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
131 |
lemma sum_of_squares_nat: "(\<Sum>k\<le>n::nat. k ^ 2) = (2 * n ^ 3 + 3 * n ^ 2 + n) div 6" |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
132 |
proof - |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
133 |
from sum_of_squares have "real (6 * (\<Sum>k\<le>n. k ^ 2)) = real (2 * n ^ 3 + 3 * n ^ 2 + n)" |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
134 |
by (auto simp add: field_simps) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61343
diff
changeset
|
135 |
then have "6 * (\<Sum>k\<le>n. k ^ 2) = 2 * n ^ 3 + 3 * n ^ 2 + n" |
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
136 |
using of_nat_eq_iff by blast |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61343
diff
changeset
|
137 |
then show ?thesis by auto |
60603
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
138 |
qed |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
139 |
|
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
140 |
lemma sum_of_cubes: "(\<Sum>k\<le>n::nat. k ^ 3) = (n ^ 2 + n) ^ 2 / 4" |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
141 |
proof - |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
142 |
have two_plus_two: "2 + 2 = 4" by simp |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
143 |
have power4_eq: "\<And>x::real. x ^ 4 = x * x * x * x" |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
144 |
by (simp only: two_plus_two[symmetric] power_add power2_eq_square) |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
145 |
have "real (\<Sum>k\<le>n::nat. k ^ 3) = (\<Sum>k\<le>n::nat. (real k) ^ 3)" by simp |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
146 |
also have "... = ((bernpoly 4 (n + 1) - bernpoly 4 0)) / (real (4 :: nat))" |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
147 |
by (auto simp add: sum_of_powers) |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
148 |
also have "... = ((n ^ 2 + n) / 2) ^ 2" |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
149 |
by (simp add: unroll algebra_simps power2_eq_square power4_eq power3_eq_cube) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61649
diff
changeset
|
150 |
finally show ?thesis by (simp add: power_divide) |
60603
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
151 |
qed |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61343
diff
changeset
|
152 |
|
60603
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
153 |
lemma sum_of_cubes_nat: "(\<Sum>k\<le>n::nat. k ^ 3) = (n ^ 2 + n) ^ 2 div 4" |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
154 |
proof - |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
155 |
from sum_of_cubes have "real (4 * (\<Sum>k\<le>n. k ^ 3)) = real ((n ^ 2 + n) ^ 2)" |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
156 |
by (auto simp add: field_simps) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61343
diff
changeset
|
157 |
then have "4 * (\<Sum>k\<le>n. k ^ 3) = (n ^ 2 + n) ^ 2" |
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
158 |
using of_nat_eq_iff by blast |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61343
diff
changeset
|
159 |
then show ?thesis by auto |
60603
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
160 |
qed |
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
161 |
|
09ecbd791d4a
add examples from Freek's top 100 theorems (thms 30, 73, 77)
bulwahn
parents:
diff
changeset
|
162 |
end |