--- a/src/HOL/Analysis/Gamma_Function.thy Mon Mar 30 10:35:10 2020 +0200
+++ b/src/HOL/Analysis/Gamma_Function.thy Tue Mar 31 15:51:15 2020 +0200
@@ -395,7 +395,7 @@
from d have "\<lceil>2 * (cmod z + d)\<rceil> \<ge> \<lceil>0::real\<rceil>"
by (intro ceiling_mono mult_nonneg_nonneg add_nonneg_nonneg) simp_all
hence "2 * (norm z + d) \<le> of_nat (nat \<lceil>2 * (norm z + d)\<rceil>)" unfolding N_def
- by (simp_all add: le_of_int_ceiling)
+ by (simp_all)
also have "... \<le> of_nat N" unfolding N_def
by (subst of_nat_le_iff) (rule max.coboundedI2, rule max.cobounded1)
finally have "of_nat N \<ge> 2 * (norm z + d)" .
@@ -540,7 +540,7 @@
also have "\<dots> = z * of_real (harm n) - (\<Sum>k=1..n. ln (1 + z / of_nat k))"
by (simp add: harm_def sum_subtractf sum_distrib_left divide_inverse)
also from n have "\<dots> - ?g n = 0"
- by (simp add: ln_Gamma_series_def sum_subtractf algebra_simps Ln_of_nat)
+ by (simp add: ln_Gamma_series_def sum_subtractf algebra_simps)
finally show "(\<Sum>k<n. ?f k) - ?g n = 0" .
qed
show "(\<lambda>n. (\<Sum>k<n. ?f k) - ?g n) \<longlonglongrightarrow> 0" by (subst tendsto_cong[OF A]) simp_all
@@ -583,7 +583,7 @@
with t_nz have "?f n t = 1 / (of_nat (Suc n) * (1 + of_nat (Suc n)/t))"
by (simp add: field_split_simps eq_neg_iff_add_eq_0 del: of_nat_Suc)
also have "norm \<dots> = inverse (of_nat (Suc n)) * inverse (norm (of_nat (Suc n)/t + 1))"
- by (simp add: norm_divide norm_mult field_split_simps add_ac del: of_nat_Suc)
+ by (simp add: norm_divide norm_mult field_split_simps del: of_nat_Suc)
also {
from z t_nz ball[OF t] have "of_nat (Suc n) / (4 * norm z) \<le> of_nat (Suc n) / (2 * norm t)"
by (intro divide_left_mono mult_pos_pos) simp_all
@@ -1110,7 +1110,7 @@
pochhammer z (Suc (Suc n)) / (fact n * exp (z * of_real (ln (of_nat n))))"
by (subst pochhammer_rec) (simp add: rGamma_series_def field_simps exp_add exp_of_real)
also from n have "\<dots> = ?f n * rGamma_series z n"
- by (subst pochhammer_rec') (simp_all add: field_split_simps rGamma_series_def add_ac)
+ by (subst pochhammer_rec') (simp_all add: field_split_simps rGamma_series_def)
finally show "?f n * rGamma_series z n = z * rGamma_series (z + 1) n" ..
qed
moreover have "(\<lambda>n. ?f n * rGamma_series z n) \<longlonglongrightarrow> ((z+1) * 0 + 1) * rGamma z"
@@ -1175,7 +1175,7 @@
have "1/2 > (0::real)" by simp
from tendstoD[OF assms, OF this]
show "eventually (\<lambda>n. g n / Gamma_series z n * Gamma_series z n = g n) sequentially"
- by (force elim!: eventually_mono simp: dist_real_def dist_0_norm)
+ by (force elim!: eventually_mono simp: dist_real_def)
from assms have "(\<lambda>n. g n / Gamma_series z n * Gamma_series z n) \<longlonglongrightarrow> 1 * Gamma z"
by (intro tendsto_intros)
thus "(\<lambda>n. g n / Gamma_series z n * Gamma_series z n) \<longlonglongrightarrow> Gamma z" by simp
@@ -1189,7 +1189,7 @@
proof (rule Lim_transform_eventually)
have "1/2 > (0::real)" by simp
from tendstoD[OF assms(2), OF this] show "eventually (\<lambda>n. g n * f n / f n = g n) sequentially"
- by (force elim!: eventually_mono simp: dist_real_def dist_0_norm)
+ by (force elim!: eventually_mono simp: dist_real_def)
from assms have "(\<lambda>n. g n * f n / f n) \<longlonglongrightarrow> 1 / rGamma z"
by (intro tendsto_divide assms) (simp_all add: rGamma_eq_zero_iff)
thus "(\<lambda>n. g n * f n / f n) \<longlonglongrightarrow> Gamma z" by (simp add: Gamma_def divide_inverse)
@@ -1409,13 +1409,13 @@
\<longlonglongrightarrow> d) - euler_mascheroni *\<^sub>R 1 in (\<lambda>y. (rGamma y - rGamma z +
rGamma z * d * (y - z)) /\<^sub>R cmod (y - z)) \<midarrow>z\<rightarrow> 0"
by (simp add: has_field_derivative_def has_derivative_def Digamma_def sums_def [abs_def]
- netlimit_at of_real_def[symmetric] suminf_def)
+ of_real_def[symmetric] suminf_def)
next
fix n :: nat
from has_field_derivative_rGamma_complex_nonpos_Int[of n]
show "let z = - of_nat n in (\<lambda>y. (rGamma y - rGamma z - (- 1) ^ n * prod of_nat {1..n} *
(y - z)) /\<^sub>R cmod (y - z)) \<midarrow>z\<rightarrow> 0"
- by (simp add: has_field_derivative_def has_derivative_def fact_prod netlimit_at Let_def)
+ by (simp add: has_field_derivative_def has_derivative_def fact_prod Let_def)
next
fix z :: complex
from rGamma_series_complex_converges[of z] have "rGamma_series z \<longlonglongrightarrow> rGamma z"
@@ -1535,7 +1535,7 @@
fix n :: nat assume n: "n > 0"
have "Re (rGamma_series (of_real x) n) =
Re (of_real (pochhammer x (Suc n)) / (fact n * exp (of_real (x * ln (real_of_nat n)))))"
- using n by (simp add: rGamma_series_def powr_def Ln_of_nat pochhammer_of_real)
+ using n by (simp add: rGamma_series_def powr_def pochhammer_of_real)
also from n have "\<dots> = Re (of_real ((pochhammer x (Suc n)) /
(fact n * (exp (x * ln (real_of_nat n))))))"
by (subst exp_of_real) simp
@@ -1569,7 +1569,7 @@
\<longlonglongrightarrow> d) - euler_mascheroni *\<^sub>R 1 in (\<lambda>y. (rGamma y - rGamma x +
rGamma x * d * (y - x)) /\<^sub>R norm (y - x)) \<midarrow>x\<rightarrow> 0"
by (simp add: has_field_derivative_def has_derivative_def Digamma_def sums_def [abs_def]
- netlimit_at of_real_def[symmetric] suminf_def)
+ of_real_def[symmetric] suminf_def)
next
fix n :: nat
have "(rGamma has_field_derivative (-1)^n * fact n) (at (- of_nat n :: real))"
@@ -1577,7 +1577,7 @@
simp: Polygamma_of_real rGamma_real_def [abs_def])
thus "let x = - of_nat n in (\<lambda>y. (rGamma y - rGamma x - (- 1) ^ n * prod of_nat {1..n} *
(y - x)) /\<^sub>R norm (y - x)) \<midarrow>x::real\<rightarrow> 0"
- by (simp add: has_field_derivative_def has_derivative_def fact_prod netlimit_at Let_def)
+ by (simp add: has_field_derivative_def has_derivative_def fact_prod Let_def)
next
fix x :: real
have "rGamma_series x \<longlonglongrightarrow> rGamma x"
@@ -1619,7 +1619,7 @@
assume xn: "x > 0" "n > 0"
have "Ln (complex_of_real x / of_nat k + 1) = of_real (ln (x / of_nat k + 1))" if "k \<ge> 1" for k
using that xn by (subst Ln_of_real [symmetric]) (auto intro!: add_nonneg_pos simp: field_simps)
- with xn show ?thesis by (simp add: ln_Gamma_series_def Ln_of_nat Ln_of_real)
+ with xn show ?thesis by (simp add: ln_Gamma_series_def Ln_of_real)
qed
lemma ln_Gamma_real_converges:
@@ -2061,7 +2061,7 @@
by (intro not_in_Ints_imp_not_in_nonpos_Ints) simp_all
with lim[of "1/2 :: 'a"] have "?h \<longlonglongrightarrow> 2 * Gamma (1/2 :: 'a)" by (simp add: exp_of_real)
from LIMSEQ_unique[OF this lim[OF assms]] z' show ?thesis
- by (simp add: field_split_simps Gamma_eq_zero_iff ring_distribs exp_diff exp_of_real ac_simps)
+ by (simp add: field_split_simps Gamma_eq_zero_iff ring_distribs exp_diff exp_of_real)
qed
text \<open>
@@ -2126,13 +2126,13 @@
have h_eq: "h t = h2 t" if "abs (Re t) < 1" for t
proof -
- from that have t: "t \<in> \<int> \<longleftrightarrow> t = 0" by (auto elim!: Ints_cases simp: dist_0_norm)
+ from that have t: "t \<in> \<int> \<longleftrightarrow> t = 0" by (auto elim!: Ints_cases)
hence "h t = a*cot (a*t) - 1/t + Digamma (1 + t) - Digamma (1 - t)"
unfolding h_def using Digamma_plus1[of t] by (force simp: field_simps a_def)
also have "a*cot (a*t) - 1/t = (F t) / (G t)"
using t by (auto simp add: divide_simps sin_eq_0 cot_def a_def F_def G_def)
also have "\<dots> = (\<Sum>n. f n * (a*t)^n) / (\<Sum>n. g n * (a*t)^n)"
- using sums[of t] that by (simp add: sums_iff dist_0_norm)
+ using sums[of t] that by (simp add: sums_iff)
finally show "h t = h2 t" by (simp only: h2_def)
qed
@@ -2167,7 +2167,7 @@
have d: "abs (Re d) < 1" "norm z < norm d" by (simp_all add: d_def norm_mult del: of_real_add)
have "eventually (\<lambda>z. h z = h2 z) (nhds z)"
using eventually_nhds_in_nhd[of z ?A] using h_eq z
- by (auto elim!: eventually_mono simp: dist_0_norm)
+ by (auto elim!: eventually_mono)
moreover from sums(2)[OF z] z have nz: "(\<Sum>n. g n * (a * z) ^ n) \<noteq> 0"
unfolding G_def by (auto simp: sums_iff sin_eq_0 a_def)
@@ -2326,7 +2326,7 @@
show "g z = of_real pi * Gamma (1 + z) * Gamma (1 - z) * ?t z"
proof (cases "z = 0")
assume z': "z \<noteq> 0"
- with z have z'': "z \<notin> \<int>\<^sub>\<le>\<^sub>0" "z \<notin> \<int>" by (auto elim!: Ints_cases simp: dist_0_norm)
+ with z have z'': "z \<notin> \<int>\<^sub>\<le>\<^sub>0" "z \<notin> \<int>" by (auto elim!: Ints_cases)
from Gamma_plus1[OF this(1)] have "Gamma z = Gamma (z + 1) / z" by simp
with z'' z' show ?thesis by (simp add: g_def ac_simps)
qed (simp add: g_def)
@@ -2471,7 +2471,7 @@
show "\<forall>z\<in>B. norm (h' z) \<le> M/2"
proof
fix t :: complex assume t: "t \<in> B"
- from h'_eq[of t] t have "h' t = (h' (t/2) + h' ((t+1)/2)) / 4" by (simp add: dist_0_norm)
+ from h'_eq[of t] t have "h' t = (h' (t/2) + h' ((t+1)/2)) / 4" by (simp)
also have "norm \<dots> = norm (h' (t/2) + h' ((t+1)/2)) / 4" by simp
also have "norm (h' (t/2) + h' ((t+1)/2)) \<le> norm (h' (t/2)) + norm (h' ((t+1)/2))"
by (rule norm_triangle_ineq)
@@ -2482,7 +2482,7 @@
also have "(M + M) / 4 = M / 2" by simp
finally show "norm (h' t) \<le> M/2" by - simp_all
qed
- qed (insert bdd, auto simp: cball_eq_empty)
+ qed (insert bdd, auto)
hence "M \<le> 0" by simp
finally show "h' z = 0" by simp
qed
@@ -2512,7 +2512,7 @@
with c have A: "h z * g z = 0" for z by simp
hence "(g has_field_derivative 0) (at z)" for z using g_g'[of z] by simp
hence "\<exists>c'. \<forall>z\<in>UNIV. g z = c'" by (intro has_field_derivative_zero_constant) simp_all
- then obtain c' where c: "\<And>z. g z = c'" by (force simp: dist_0_norm)
+ then obtain c' where c: "\<And>z. g z = c'" by (force)
from this[of 0] have "c' = pi" unfolding g_def by simp
with c have "g z = pi" by simp
@@ -2547,7 +2547,7 @@
lemma Gamma_reflection_complex':
"Gamma z * Gamma (- z :: complex) = - of_real pi / (z * sin (of_real pi * z))"
- using rGamma_reflection_complex'[of z] by (force simp add: Gamma_def field_split_simps mult_ac)
+ using rGamma_reflection_complex'[of z] by (force simp add: Gamma_def field_split_simps)
@@ -2675,7 +2675,7 @@
from z have z': "z \<noteq> 0" by auto
have "eventually (\<lambda>n. ?r' n = ?r n) sequentially"
- using z by (auto simp: field_split_simps Gamma_series_def ring_distribs exp_diff ln_div add_ac
+ using z by (auto simp: field_split_simps Gamma_series_def ring_distribs exp_diff ln_div
intro: eventually_mono eventually_gt_at_top[of "0::nat"] dest: pochhammer_eq_0_imp_nonpos_Int)
moreover have "?r' \<longlonglongrightarrow> exp (z * of_real (ln 1))"
by (intro tendsto_intros LIMSEQ_Suc_n_over_n) simp_all
@@ -2819,7 +2819,7 @@
have "(\<lambda>n. of_nat ((k + n) choose n) / of_real (exp (of_nat k * ln (real_of_nat n))))
\<longlonglongrightarrow> 1 / Gamma (of_nat (Suc k) :: 'a)" (is "?f \<longlonglongrightarrow> _")
using Gamma_gbinomial[of "of_nat k :: 'a"]
- by (simp add: binomial_gbinomial add_ac Gamma_def field_split_simps exp_of_real [symmetric] exp_minus)
+ by (simp add: binomial_gbinomial Gamma_def field_split_simps exp_of_real [symmetric] exp_minus)
also have "Gamma (of_nat (Suc k)) = fact k" by (simp add: Gamma_fact)
finally show "?f \<longlonglongrightarrow> 1 / fact k" .
@@ -2862,7 +2862,7 @@
next
case False
with \<open>z = 0\<close> show ?thesis
- by (simp_all add: Beta_pole1 one_minus_of_nat_in_nonpos_Ints_iff gbinomial_1)
+ by (simp_all add: Beta_pole1 one_minus_of_nat_in_nonpos_Ints_iff)
qed
next
case False
@@ -2885,7 +2885,7 @@
have "(z gchoose n) = Gamma (z + 2) / (z + 1) / (fact n * Gamma (z - of_nat n + 1))"
by (subst gbinomial_Beta[OF assms]) (simp_all add: Beta_def Gamma_fact [symmetric] add_ac)
also from assms have "Gamma (z + 2) / (z + 1) = Gamma (z + 1)"
- using Gamma_plus1[of "z+1"] by (auto simp add: field_split_simps mult_ac add_ac)
+ using Gamma_plus1[of "z+1"] by (auto simp add: field_split_simps)
finally show ?thesis .
qed
@@ -3073,7 +3073,7 @@
have "eventually (\<lambda>n. Gamma_series z n =
of_nat n powr z * fact n / pochhammer z (n+1)) sequentially"
using eventually_gt_at_top[of "0::nat"]
- by eventually_elim (simp add: powr_def algebra_simps Ln_of_nat Gamma_series_def)
+ by eventually_elim (simp add: powr_def algebra_simps Gamma_series_def)
from this and Gamma_series_LIMSEQ[of z]
have C: "(\<lambda>k. of_nat k powr z * fact k / pochhammer z (k+1)) \<longlonglongrightarrow> Gamma z"
by (blast intro: Lim_transform_eventually)
@@ -3441,7 +3441,7 @@
fix n :: nat and x :: real assume x: "x \<in> ball 0 1"
{
fix k :: nat assume k: "k \<ge> 1"
- from x have "x^2 < 1" by (auto simp: dist_0_norm abs_square_less_1)
+ from x have "x^2 < 1" by (auto simp: abs_square_less_1)
also from k have "\<dots> \<le> of_nat k^2" by simp
finally have "(1 - x^2 / of_nat k^2) \<in> {0..1}" using k
by (simp_all add: field_simps del: of_nat_Suc)
@@ -3470,7 +3470,7 @@
by (simp add: eval_nat_numeral)
from sums_divide[OF this, of "x^3 * pi"] x
have "(\<lambda>n. - (sin_coeff (n+3) * pi^(n+2) * x^n)) sums ((1 - sin (pi*x) / (pi*x)) / x^2)"
- by (simp add: field_split_simps eval_nat_numeral power_mult_distrib mult_ac)
+ by (simp add: field_split_simps eval_nat_numeral)
with x have "(\<lambda>n. - (sin_coeff (n+3) * pi^(n+2) * x^n)) sums (g x / x^2)"
by (simp add: g_def)
hence "f' x = g x / x^2" by (simp add: sums_iff f'_def)
@@ -3485,7 +3485,7 @@
assume x: "x \<noteq> 0"
from summable_divide[OF sums_summable[OF sums_split_initial_segment[OF
sin_converges[of "pi*x"]], of 3], of "-pi*x^3"] x
- show ?thesis by (simp add: mult_ac power_mult_distrib field_split_simps eval_nat_numeral)
+ show ?thesis by (simp add: field_split_simps eval_nat_numeral)
qed (simp only: summable_0_powser)
qed
hence "f' \<midarrow> 0 \<rightarrow> f' 0" by (simp add: isCont_def)