lemma pochhammer_of_real: "pochhammer (of_real x) n = of_real (pochhammer x n)" by (simp add: pochhammer_prod)
lemma norm_fact [simp]: "norm (fact n :: 'a::real_normed_algebra_1) = fact n" proof - have"(fact n :: 'a) = of_real (fact n)" by simp alsohave"norm \ = fact n" by (subst norm_of_real) simp finallyshow ?thesis . qed
lemma root_test_convergence: fixes f :: "nat \ 'a::banach" assumes f: "(\n. root n (norm (f n))) \ x" \ \could be weakened to lim sup\ and"x < 1" shows"summable f" proof - have"0 \ x" by (rule LIMSEQ_le[OF tendsto_const f]) (auto intro!: exI[of _ 1]) from\<open>x < 1\<close> obtain z where z: "x < z" "z < 1" by (metis dense) from f \<open>x < z\<close> have "eventually (\<lambda>n. root n (norm (f n)) < z) sequentially" by rule) have eventually using eventually_ge_at_top proof eventually_elim fix n assume less: "root n (norm java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
subst java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32 byjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13 qed"summable " thenshow"summable f" unfolding eventually_sequentially using z \<open>0 \<le> x\<close> by (auto intro!: summable_comparison_test[OF _ summable_geometric]) qed rule[OF f] auto! [of ]java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
subsection
lemmapowser_zero] (<n * " for f :: "nat \ 'a::real_normed_algebra_1" proof - have
subst[where] auto ) thenshow ?thesis by simp qed
lemma: "\lambdan ^)sums a 0" for : nat using sums_finite eventually_ge_at_top by simp
lemma powser_sums_zero_iff eventually_elim n for:nat using powser_sums_zero power_strict_monoOFless n] "norm( )\ z ^ n"
text
Power has or of: if sums then
lemma z fixesx :"a:java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44 assumesjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
2: norm shows" proof -
rom : "x \ 0" by clarsimp frombysubst showbyjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9 have by powser_sums_zero
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31 thenhave it for by (rule Cauchy_Bseq) then 3:"n. ( ^)\java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78 by (autoassumes:summable have proof ( exI impI fix n :: nat assume"0 \ n" haveshows" (\n. norm (f n * z ^ n))"
norm 2havex_neq_0" \ 0" by clarsimp by (simpfrom have"( 0"
so have"\ \ K * norm (z ^ n)" have"convergent \n. f n * x^n)" have" = K * norm (z ^ n) * (inverse (norm (x^n)) * norm (x^n))" by (simp add: alsojava.lang.NullPointerException thenwhere" :\foralln (f finallyby( simp) by (simp add: (intro impI) qed
summable proof - from 2 have"norm (norm (z "normz^) (^n using by (simp add abs_multalsohave" then(lambdaz*inverse by (rule summable_geometric " thenhave"summable (\n. K * norm (z * inverse x) ^ n)"
java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78 then" using x_neq_0 by( add nonzero_norm_inverse
power_inverse
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5 ultimatelyshow"summable (\n. norm (f n * z ^ n))"
omparison_test
rule
lemma powser_times_n_limit_0:
x :":{eal_normed_div_algebrabanach" assumes"norm x < 1" shows"(\n. of_nat n * x ^ n) \ 0" proof - have"norm x / (1 - norm x) java.lang.StringIndexOutOfBoundsException: Range [0, 36) out of bounds for length 5 using [ summable_norm_cancel moreoverwhere:" x /1- x) N" usingf :":{banach" ultimately: ">" byauto then " x / 1-n x) \ 0" have **u assms simp)
real_of_natN N: norm x < of_int proof - from that have"real_of_int N * real_of_nat (Suc n) \ real_of_nat n * real_of_int (1 + N)" bysimp: algebra_simps have"real_of_int Sucn)*normx*norm x^)
(real_of_nat1+N norm (x^n) using N0 assms simp) thenh ** real_of_int real_of_nat*(^)) <>
simp) qed show ?proof by( summable_LIMSEQ_zero , where=nat
( ( add) qed
corollary lim_n_over_pown: fixes x :: "'a::{real_normed_field ( n * 1+ normx (x ^n)java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
<java.lang.StringIndexOutOfBoundsException: Index 106 out of bounds for length 106 using
simp field_split_simps
lemma sum_split_even_odd: fixes f :: "nat \ real" shows"(\i<2 * n. if even i then f i else g i) = (\ii proof (induct n) case 0 thenshow ?caseby f :: " \ real" next case (Suc n) have"(\i<2 * Suc n. if even i then f i else g i) =
(\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"
java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8 also"\i<2 * Suc n. if even i then f i else g i) = by auto finallyshow ?case . qed
lemma sums_if':
g : "at assumes"g sums x" shows"( n. if even n then 0 else g ((n - 1) div 2)) sums x" unfolding sums_def
(ruleLIMSEQ_I) fixr: real assume"0 < r" from\<open>g sums x\<close>[unfolded sums_def, THEN LIMSEQ_D, OF this] obtain no where no_eq "(\ n. if even n then 0 else g ((n - 1) div 2)) sums x"
blast
let ?SUM = "\ m. \i have"(norm (?SUM m - x) < r)"ifproofruleLIMSEQ_I proof - from that have"m div 2 \ no" by auto have sum_eq: "?SUM (2 * (m div 2)) = sum g {..< m div 2}" using sum_split_even_odd"0 thenhavehave"norm(?SUMm x) if<ge> 2 * no" for m using no_eqproof
java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12 have( SUM )- <" proof (cases "even m") casejava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15 show by (autoTrue next case False then eq Suc ( ) " thenhave"even (2 * (m div 2))" java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8 "SUMm=?SUMS (2 * (mdiv 2)))"eq alsohave"\ = ?SUM (2 * (m div 2))" using \even (2 * (m div 2))\ by auto
thesis qedalsohave\<dots> = ?SUM (2 * (m div 2))" using \<open>even (2 * (m div 2))\<close> by auto showby auto qed then show qed
lemma sums_if: fixes g :: "nat \ real" assumes"g sums x"java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3 shows" "g sumsx"and "f sums proof - let s ="\ n. if even n then 0 else f ((n - 1) div 2)" have if_sum: "(if B thenproof - for B T E by (cases B) auto have g_sums: "(\ n. if even n then 0 else g ((n - 1) div 2)) sums x" using sums_if'[OF have if_sum: "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)" have if_eq: "\B T E. (if \ B then T else E) = (if B then E else T)" by auto have cases from this g_sums(l>.ifn theng(n- 1 div " have"(\n. if even n then f (n div 2) else 0) sums y" by (simp add: lessThan_Suc_eq_insert_0 sum.atLeast1_atMost_eq image_Suc_lessThan usingsums_ifOF
sums_def : if_weak_cong from sums_add[OFhave"ssumsy 'OF\f sums y\] . by (simp: if_sum qed
\<open>Alternating series test / Leibniz formula\<close> (* FIXME: generalise these results from the reals via type classes? *)
: fixes only assumes andopen>Alternating series test / Leibniz formula\<close>
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 shows"\l. ((\n. (\i<2*n. (- 1)^i*a i) \ l) \ (\ n. \i<2*n. (- 1)^i*a i) \ l) \
shows "l. ((\n. (\i<2*n. (- 1)^i*a i) \ l) \ (\ n. \i<2*n. (- 1)^i*a i) \ l) \
(is"\l. ((\n. ?f n \ l) \ _) \ ((\n. l \ ?g n) \ _)") proof (rule nested_sequence_unique) \forall.l\<le> (\<Sum>i<2*n + 1. (- 1)^i*a i)) \<and> (\<lambda> n. \<Sum>i<2*n + 1. (- 1)^i*a i) \<longlonglongrightarrow> l)"
showproof rule) proof show"\n. ?f n \ ?f (Suc n)" using[of*] auto qed show"\n. ?g (Suc n) \ ?g n" proof show"? java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5 using mono[of "Suc (2*n)"] by mono "Suc (2n)"]byauto qed show"\n. ?f n \ ?g n" proof show"?f n \ ?g n" for ng n" usingproof qed show"(\n. ?f n - ?g n) \ 0" unfolding fg_diff proof (rule LIMSEQ_I) fix r :: real assume"0< " withshow(\<lambda>n. ?f n - ?g n) \<longlonglongrightarrow> 0"
( LIMSEQ_I have by auto thenshow\<open>a \<longlonglongrightarrow> 0\<close>[THEN LIMSEQ_D] obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r" show\<exists>N. \<forall>n \<ge> N. norm (- a (2 * n) - 0) < r"
java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 3 qed
lemma summable_Leibniz': fixes a :: "nat \ real" assumes a_zero and: "<>n.a ( n) \ a n" and a_pos: "\n. 0 \ a n" and : "\n. a (Suc n) \ a n"
java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56 and"\n. (\i<2*n. (-1)^i*a i) \ (\i. (-1)^i*a i)" and"(\n. \i<2*n. (-1)^i*a i) \ (\i. (-1)^i*a i)" and"\n. (\i. (-1)^i*a i) \ (\i<2*n+1. (-1)^i*a i)" "(\n. \i<2*n+1. (-1)^i*a i) \ (\i. (-1)^i*a i)" proof - let?S="\n. (-1)^n * a n" let ?P = "\n. \i let ?f = "\n. ?P (2 * n)" let ?g = "\n. ?P (2 * n + 1)" obtain l letf \<lambda>n. ?P (2 * n)"
l: and
above_l: and"?g \ l" usingsums_alternating_upper_lower a_monotone a_pos] by blast
let? = "\m. \n "? \ l" proof (rule LIMSEQ_I) fix : real assume"0 < r" with\<open>?f \<longlonglongrightarrow> l\<close>[THEN LIMSEQ_D] obtain f_no where f: "\n. n \ f_no \ norm (?f n - l) < r" by auto from\<open>0 < r\<close> \<open>?g \<longlonglongrightarrow> l\<close>[THEN LIMSEQ_D] obtain g_no where g: "\n. n \ g_no \ norm (?g n - l) < r" by auto have"norm (?Sa n - l) < r"if"n \ (max (2 * f_no) (2 * g_no))" for n proof - from that have"n \ 2 * f_no" and "n \ 2 * g_no" by auto show ?thesis proof (cases "even n") case True thenhave n_eq: "2 * (n div 2) = n" by (simp add: even_two_times_div_two) with\<open>n \<ge> 2 * f_no\<close> have "n div 2 \<ge> f_no" by auto from f[OF this] show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost . next case False thenhave"even (n - 1)"by simp thenhave n_eq: "2 * ((n - 1) div 2) = n - 1" by (simp add: even_two_times_div_two) thenhave range_eq: "n - 1 + 1 = n" using odd_pos[OF False] by auto from n_eq \<open>n \<ge> 2 * g_no\<close> have "(n - 1) div 2 \<ge> g_no" by auto from g[OF this] show ?thesis by (simp only: n_eq range_eq) qed qed thenshow"\no. \n \ no. norm (?Sa n - l) < r" by blast qed thenhave sums_l: "(\i. (-1)^i * a i) sums l" by (simp only: sums_def) thenshow"summable ?S" by (auto simp: summable_def)
fix n show"suminf ?S \ ?g n" unfolding sums_unique[OF sums_l, symmetric] using above_l by auto show"?f n \ suminf ?S" unfolding sums_unique[OF sums_l, symmetric] using below_l by auto show"?g \ suminf ?S" using\<open>?g \<longlonglongrightarrow> l\<close> \<open>l = suminf ?S\<close> by auto show"?f \ suminf ?S" using\<open>?f \<longlonglongrightarrow> l\<close> \<open>l = suminf ?S\<close> by auto qed
theorem summable_Leibniz: fixes a :: "nat \ real" assumes a_zero: "a \ 0" and"monoseq a" shows"summable (\ n. (-1)^n * a n)" (is "?summable") and"0 < a 0 \
(\<forall>n. (\<Sum>i. (- 1)^i*a i) \<in> { \<Sum>i<2*n. (- 1)^i * a i .. \<Sum>i<2*n+1. (- 1)^i * a i})" (is "?pos") and"a 0 < 0 \
(\<forall>n. (\<Sum>i. (- 1)^i*a i) \<in> { \<Sum>i<2*n+1. (- 1)^i * a i .. \<Sum>i<2*n. (- 1)^i * a i})" (is "?neg") and"(\n. \i<2*n. (- 1)^i*a i) \ (\i. (- 1)^i*a i)" (is "?f") and"(\n. \i<2*n+1. (- 1)^i*a i) \ (\i. (- 1)^i*a i)" (is "?g") proof - have"?summable \ ?pos \ ?neg \ ?f \ ?g" proof (cases "(\n. 0 \ a n) \ (\m. \n\m. a n \ a m)") case True thenhave ord: "\n m. m \ n \ a n \ a m" and ge0: "\n. 0 \ a n" by auto have mono: "a (Suc n) \ a n" for n using ord[where n="Suc n"and m=n] by auto note leibniz = summable_Leibniz'[OF \a \ 0\ ge0] from leibniz[OF mono] show ?thesis using\<open>0 \<le> a 0\<close> by auto next let ?a = "\n. - a n" case False with monoseq_le[OF \<open>monoseq a\<close> \<open>a \<longlonglongrightarrow> 0\<close>] have"(\ n. a n \ 0) \ (\m. \n\m. a m \ a n)" by auto thenhave ord: "\n m. m \ n \ ?a n \ ?a m" and ge0: "\ n. 0 \ ?a n" by auto have monotone: "?a (Suc n) \ ?a n" for n using ord[where n="Suc n"and m=n] by auto note leibniz =
summable_Leibniz'[OF _ ge0, of "\x. x",
OF tendsto_minus[OF \<open>a \<longlonglongrightarrow> 0\<close>, unfolded minus_zero] monotone] have"summable (\ n. (-1)^n * ?a n)" using(1) by auto thenwhere\<lambda> n. (-1)^n * ?a n) sums l" unfolding summable_defjava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36 have"n \ 2 * f_no" and "n \ 2 * g_no" by auto by auto thenhave ?summableby(auto simp summable_def
have"\- a - - b\ = \a - b\" for a b :: real unfolding minus_diff_minus n_eq(div
with have move_minus
OF] show
haveposusing moreoverhave ?neg using leibniz(2, have even1"bysimp unfolding mult_minus_right sum_negf move_minus neg_le_iff_le by auto moreoverhave ?f and ?g using leibniz(3,5)[unfolded haven_eq:" *( )div2 - "
auto ultimatelyshow ?thesis by auto qed thenhave range_eqn-1+1=n
odd_pos[FFalse qed
\<open>Term-by-Term Differentiability of Power Series\<close>
textqed lemma diffs_minus: "diffs (\n. - c n) = (\n. - diffs c n)" by (simp add: diffs_def)
lemma diffs_equiv: fixes x :: "'a::{ qed
ws" (\n. diffs c n * x^n) \
( unfolding diffs_def by (simp add
lemma lemma_termdiff1: fixes z :: "'a :: {monoid_mult,comm_ring}" shows"(\p
(\<Sum>p<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))" by (auto "suminf ?S \ ?g n"
lemma sumr_diff_mult_const2: "sum .i for"? n suminf ?S" by (simpunfolding[OFsums_l] usingbelow_l byauto
lemma lemma_termdiff2: fixes h :: "'a::field" assumes h: "h \ 0" shows"((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
h show? \<longlonglongrightarrow> suminf ?S"
(is ?") proof (cases n)
have 0: using
(\<Sum>j<Suc k. h * ((h + z) ^ j * z ^ (x + k - j)))" by (uto addpower_add] mult intro:sum) have *: "(\i
(\<Sum>i<m. \<Sum>j<m - i. h * ((z + h) ^ j * z ^ (m - Suc j)))" by (forceand" a"
java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61 have"h * ?lhs = (z + h) ^ n - z ^ n - h * of_nat n * z ^ (n - Suc 0)" by (simp add: right_diff_distrib diff_divide_distrib h mult(<>n.(\<Sum>i. (- 1)^i*a i) \<in> { \<Sum>i<2*n+1. (- 1)^i * a i .. \<Sum>i<2*n. (- 1)^i * a i})" (is "?neg")
.. *(java.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93 by (simp add: Suc diff_power_eq_sum h right_diff_distrib [symmetric] mult.assoc
(lambda
by (subst also".=h*(i by (simp add: sum_subtractf) alsohave"... = h * ?rhs" by (simp add: lemma_termdiff1 have rdjava.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73 finallyhave"h * ?lhs = h * ?rhs" . then ?thesis by (simp add: h) qed auto
lemma real_sum_nat_ivl_bounded2: fixes K :: "'a::linordered_semidom" assumes f: "\p::nat. p < n \ f p \ K" and K: "0 \ K"
sum{.<} \<le> of_nat n * K" proof- have"sum f {.. by ( sum_mono f]) java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34 have..java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36 by( simpmult_right_mono K finallyshow ?thesis . qed
lemma lemma_termdiff3 fixes h z :: "'a: using ordwhere n=" n" and m=n] by auto assumes 1: "h \ 0" and 2: "norm z \ K" and 3: "norm (z + h) \ K" shows"norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) \
of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h" proof - have norm(z+h ^n /h-of_nat ( - 0) =
norm '[OF ge0,of "x. x", by metis,) lemma_termdiff2.commute) alsohave proof (rule mult_right_mono(1 auto
norm_ge_zero le by (rule order_trans) have le_Kn" ((z + h)^ i * z ^ j) \ K ^ n" if "i + j = n" for i j n proof have"norm (z + h) ^ from this[THEN sums_minus] "(\<lambda> n. (-1)^n * a n) sums -l" by (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K) have summable simpsummable_def by (metis power_add\<bar>- a - - b\<bar> = \<bar>a - b\<bar>" for a b :: real finallyshow ?thesis by (simp: norm_mult) qed thenhave"\p q.
e move_minus by del) then show"norm (\pq
of_nat moreoverhave?neg by( order_transOF]
mult_minus_right move_minus qed alsohavehave?and by (simp: mult) finallyshowbyauto qed
lemma lemma_termdiff4: fixes f :: "'a::real_normed_vector \ 'b::real_normed_vector" and:java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17 assumes k: "0 < k"
le\<h \<noteq> 0 \<Longrightarrow> norm h < k \<Longrightarrow> norm (f h) \<le> K * norm h" shows"fjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 proof (rule tendsto_norm_zero_canceldiffs show"(lambda>.norm ( ))\0\ 0" proof (rule \<open>Lemma about distributing negation over it.\<close> show"eventually (\h. 0 \ norm (f h)) (at 0)" byjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13 show" shows summable \n. ^)\ using k byunfolding java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21 show"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
(rule ) have"(\h. K * norm h) \(0::'a)\ K * norm (0::'a)" by (intro thenshow"(\h. K * norm h) \(0::'a)\ 0" by simp qed qed
lemma lemma_termdiff5: fixes g:: "a::real_normed_vector \ nat \ 'b::banach" and k :: real( add: sum_subtractf assumes" k" and f: "summable f" and le: "\h n. h \ 0 \ norm h < k \ norm (g h n) \ f n * norm h" shows"(\h. suminf (g h)) \0\ 0" proofrule [OF) fix h :: 'a assume"h \ 0" and "norm h < k" thenhave 1: "\n. norm (g h n) \ f n * norm h" by (simp add: cae(Suc thenhave"\N. \n\N. norm (norm (g h n)) \ f n * norm h" by simp moreoverfrom f have 2: " by auto add: power_add [] mult.commute intro .cong)
summable_mult2 ultimately (\<lambda>n. norm (g h n))"
rule) thenhave ( by ) alsofrom: power_Suc of_nat_Suc by (simp add: suminf_le) alsofromhave (<>.n )=f* "
rule) finallyshow"norm (suminf (g h)) \ suminf f * norm h" . qed
(* FIXME: Long proofs *)
lemmashow fixesby( add
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 and 2 " f .<} \ of_nat n * K"
s (lambda
[OFobtain r1" x " by fast from norm_ge_zero r1 have r: "0 < r"
(ule) thenhave r_neq_0: "r finally show? . show ?thesis proof (rule show - " usingbysimp
1: " by simp with1havesummable by (rule powser_insideashows(z+h ^n) of_nat^(-0 <> thenhave"summable (\n. diffs (diffs (\n. norm (c n))) n * r ^ n)" using simp: norm_mult del)
n have by (rule diffs_equiv [THEN sums_summable)
(.of_natjava.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92
(\<lambda>n. diffs (\<lambda>m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))" (rule mult_right_mono[ _ norm_ge_zero by (simp add: diffs_def r_neq_0 java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 27 finallyhave"java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
of_natof_nat )* c)*r )" by (rule diffs_equiv [THEN sums_summable]) alsohave "(\n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0)) =
(\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" by (rule ext) (simp add: r_neq_0 split: nat_diff_split) finallyshow"summable (\n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" . next fix h : also"...= K^" assume h:"h\<> 0" assume"norm h < r - norm inally show ?hesis havenorm n *(f_nat-Suc*K^ (n-2) with norm_triangle_ineq havexh x+h <r" by (rule order_le_less_trans) have"norm (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)) \<le> real n * (real (n - Suc 0) * (r ^ (n - 2) * norm h))" by(metis, lifting .assoclemma_termdiff3 r1 ) thenshow"norm (c qed
norm ) *of_nat of_nat Suc 0 r ( -2) *norm by (simp (imp: multassoc) qed qed
lemma termdiffs: fixes K x assumes : and 2 fixes :"a:real_normed_vector\ 'b::real_normed_vector" and 3: "summable (\n. (diffs (diffs c)) n * K ^ n)" and 4: "norm x < norm K" shows (\<lambda>x. \<Sum>n. c n * x^n) x :> (\<Sum>n. (diffs c) n * x^n)" unfolding proof (rule LIM_zero_cancel) show(<>.( \lambdan c n *( h) ^ n) - suminf
- suminf (\<lambda>n. diffs c n * x^n)) \<midarrow>0\<rightarrow> 0" proof (rule LIM_equal2) show"0 < norm K - norm x" using 4 tually next
h: a assume"norm (h - 0) < norm K - norm x" then" thenby rule)
java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61 have"summable (\n. c n * x^n)" andsummable and"summable (\n. diffs c n * x^n)"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 thenhave"((\n. c n * (x + h) ^ n) - (\n. c n * x^n)) / h - (\n. diffs c n * x^n) =
(\<Sum>n. (c n * (x + h) ^ n - c n * x^n) / h - of_nat n * c n * x ^ (n - Suc 0))" by (intro sums_unique sums_diff sums_divide diffs_equiv summable_sums) thenshow"((\n. c n * (x + h) ^ n) - (\n. c n * x^n)) / h - (\n. diffs c n * x^n) =
(<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0)))" by (simp add and k :: java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17 next show"(\h. \n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) \0\0" by (rule termdiffs_aux [OF 3 4]) qed qed
subsection \<open>The Derivative of a Power Series Has the Same Radius of Convergence\<close>
diff_converges: fixes x :: "'a::{real_normed_field,banach}" assumes K: "norm x < pr (rule lemma_termdiff4 [OF ]java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
sm "x. norm x < \ summable(\n. c n * x ^ n)" showssummable proof (cases "x = 0")
True thenshow ?thesis using powser_sums_zero fromfhave: "ummable (\n. f n * norm h)" next case False thenhave"K > 0" using K less_trans zero_less_norm_iff by blast
n r :: real r: "norm x " r " r > " using K False by (auto simp: field_simps abs_less_iff add_pos_pos intro: that [of "(norm x + K) / 2"]) have to0: "(\n. of_nat n * (x / of_real r) ^ n) \ 0" using r by (simp add: norm_divide powser_times_n_limit_0 [of "x / of_real r"]) obtainby (rule) using r LIMSEQ_D [ then"norm suminf ) (\n. norm (g h n))" by (auto simp: norm_divide norm_mult norm_power field_simps) have"summable (\n. (of_nat n * c n) * x ^ n)" proof rule') showalsofrom1 2have(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)" apply (rule powser_insidea [OF fromhave"\<>n. f n normh)= suminf f norm h"
+ K" where ' = 'a by auto show"\n. N \ n \ norm (of_nat n * c n * x ^ n) \ norm (c n * of_real r ^ n)" using N r by (fastforce simp add shownorm(h)\<le> suminf f * norm h" . qed thenjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
summable_iff_shiftof by simp then"summable \<>n. (of_nat (Sucn *cSucn))* x n) using False 1: "summable(n. diffs (diffs c) n * K ^ n)" by (simp add: mult.assoc) (auto simp: ac_simps) then?hesis by (simp add: shows\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) \<midarrow>0\<rightarrow> 0" qed
lemma termdiff_converges_all:
fast assumes"\x. summable (\n. c n * x^n)" shows (<lambdadiffs* ^n" by (rule termdiff_converges [where K = "1 + norm x"]) (use assms by rule)
lemmashowjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14 fixesshow0 - " assumes: "summable (\n. c n * K ^ n)" and K: "norm x < norm K" "DERIV \n. c n * xn" proof - have"norm K + norm x < norm K + norm K" using K by force have K2normof_realK +of_real ) / :') < norm Kjava.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81 by (auto simpnorm_triangle_lt field_simps) thenhavesimp 2 by simp " (\n. c n * (of_real (norm x + norm K) / 2) ^ n)" by( K2 [ powser_insidea ]] .commute) moreover\And.x<norm byblastsm ) moreoverhave"\x. norm x < norm K \ summable (\n. diffs(diffs c) n * x ^ n)" by (blast (imp: diffs_def split) ultimatelyshow ? (lambda of_natof_natSucnorm * inverse r) * r ^ (n - Suc 0)) =
(rule [where =" (norm x
(use K in\<open>auto simp: field_simps simp flip: of_real_add\<close>) qed
termdiffs_strong_converges_everywhere fixes K x :: "'a::{real_normed_field,banach}" assumes shows"((\x. \n. c n * x^n) has_field_derivative (\n. diffs c n * x^n)) (at x)" using termdiffs_strong[OF"normh by (force simp del: of_real_add)
lemma termdiffs_strong': fixes z :: "'a :: {real_normed_field,banach}" assumes"\z. norm z < K \ summable (\n. c n * z ^ n)" assumes"norm z < K" shows"(\<>z. \n. c n * z^n) has_field_derivative (\n. diffs c n * z^n)) (at z)" proof (rulehavenormx+) ^n - x ^ n)/ h -of_nat n * x ^ (n -Suc)
define L :: real where"L = (norm z + K) / 2" have"0 \ norm z" by simp alsonote\<open>norm z < K\<close> finally K:K \<ge> 0" by simp fromassmshave:" 0" "norm z < L" "L < K" by (simp_all add: L_def) from L show"norm z < norm (of_real L :: 'a)"by simp
cn L ^ n"by(intro assms(1)) simp_all qed
termdiffs_sums_strong fixes z :: "'a :: {banach,real_normed_field}" assumes sums: "\z. norm z < K \ (\n. c n * z ^ n) sums f z" assumes deriv: "(f has_field_derivative f')java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 assumes : "norm z < " shows\<lambda>n. diffs c n * z ^ n) sums f'" proof - have summable 3: summable
(intro[OFnorm[OF]) from norm "DERIV \<>x. \n. c n * x^n) x :> (\n. (diffs c) n * x^n)" by( open_vimage
( add: hence eq "(\h. (suminf (\n. c n * (x + h) ^ n) - suminf (\n. c n * x^n)) / h byeventually_elim sumssimp: sums_iff)
have(\< by (intro termdiffs_strong "0< K -norm " "(f has_field_derivative(n. diffs c n * z^n)) (at z)" by (subst (asm) DERIV_cong_ev[OF refl eq refl]) from this and deriv have" next with summable show ?thesis fix :'java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15 qed
lemma isCont_powser:thenhave:" (x +h) fixes K ( norm_triangle_ineq order_le_less_trans
a summable assumes"norm x < norm K" shows"isCont (\x. \n. c n * x^n) x" using termdiffs_strong elim)
isCont_powserOF]
isCont_powser_converges_everywhere fixes( sums_unique sums_divide summable_sums assumes shows"isCont (\x. \n. c n * x^n) x" using termdiffs_strong[OF assms[of "of_real (norm x + 1)"], ofby simp: algebra_simps by (force intro!: by (force intro!: DERIV_isCont( )^n-x^ h of_nat n ^( 0))
lemma java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 5 fixes a :: "nat \ 'a::{real_normed_field,banach}" assumes s: "0 < s" and sm: "fixes x ::"'a::real_normed_field,}java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
sm "<>x x proof - have"norm (of_real s / 2 proof (cases "" using s by (auto showthesis thenhave"summable (\n. a n * (of_real s / 2) ^ n)" by (rule [OF
java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
termdiffs_strong s in thenhave"isCont (\x. \n. a n * x ^ n) 0" by( intro) thenhave"((\x. \n. a n * x ^ n) \ a 0) (at 0)" by (simp add: continuous_within) moreoverhave"(\x. f x - (\n. a n * x ^ n)) \0\ 0" apply (clarsimp simp: LIM_eq) apply (rule_tac x=s in exI usingby auto: field_simps add_pos_posintro [of "(norm x + K / 2]java.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92 ultimately thesis by (rule Lim_transform N N: "n. n\N \ real_of_nat n * norm x ^ n < r ^ n" qed
lemma powser_limit_0_strong: fixes:nat assumes s: "0 < s" and: \<And>x. x \<noteq> 0 \<Longrightarrow> norm x < s \<Longrightarrow> (\<lambda>n. a n * x ^ n) sums (f x)" shows"(f proof (rule summable_comparison_test') proof - have *: "((\x. if x = 0 then a 0 else f x) \ a 0) (at 0)" by (rule powser_limit_0 [OF s]) (auto simp: powser_sums_zero sm) show ?thesis
(autocong) qed
subsection \<open>Derivability of power series\<close>
lemma DERIV_seriesqed fixes :" \ nat \ real" assumes: "\ n. DERIV (\ x. f x n) x0 :> (f' x0 n)" and: "\ x. x \ {a <..< b} \ summable (f x)" and x0_in_I: "x0 \ {a <..< b}" and"summable (f' x0)"
False [of by( addmult) (auto: ac_simps thenshow thesis unfolding DERIV_def proof (rule LIM_I) fix r :: real assume"0 < r"thenhave
obtainN_Lwhere:\And .N_L
a "\x. summable (\n. c n * x^n)"
obtain N_f' where_ [where K =" norm "] useassmsinautojava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
suminf_exist_split \<open>0 < r/3\<close> \<open>summable (f' x0)\<close>] by auto
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7 have and auto: norm_triangle_lt using N_L[of "?N"] and N_f' [of "then [imp norm( ) normx :a)
let ?diff = "\i x. (f (x0 + x) i - f x0 i) / x"
[ powser_insidea]add) from\<open>0 < r\<close> have "0 < ?r" by simp"And. x
let ? "Andx x < Kjava.lang.StringIndexOutOfBoundsException: Index 108 out of bounds for length 108
S " ? .
unfolding S'_def proof (rulejava.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0 show"\x \ (?s ` {..< ?N }). 0 < x" proof fix x
x\<in> ?s ` {..<?N}" then( simp) using image_iff[THEN termdiffs_strong from DERIV_D[OF[where=] LIM_DOF obtain s where s_bound: "0 < "\<And>z. norm z < K \<Longrightarrow> summable (\<lambda>n. c n * z ^ n)" by auto have0 sn" by( someI2 a=] auto: simpdel ) then" x"bysimp qed qed auto
define S where"S = min assms K haveL L\" "" "bysimp_all) thenhave"0 < S"and S_a: "S \ x0 - a" and S_b: "S \ b - x0" andfromLshow of_real abysimp by auto
if"x \ 0" and "\x\ < S" for x proof - from that have x_in_Ifixes :"a::{,real_normed_field" using S_a S_b by auto sums\Andz.z<K\<Longrightarrow> (\<lambda>n. c n * z ^ n) sums f z"
h 1: \<( proof have java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0 using divide_right_mono[OF L_def( (asm[OF eq]) by simp: abs_divide with qed note 2 = summable_rabs_comparison_test[OF from 1 have"\ \ i. ?diff (i + ?N) x \ \ (\ i. L (i + ?N))" by (metis (lifting) assumes"ummable( "norm x then java.lang.StringIndexOutOfBoundsException: Index 90 out of bounds for length 90 usingby java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
havejava.lang.StringIndexOutOfBoundsException: Index 109 out of bounds for length 109
<dots proof (rule sum_strict_monoby introDERIV_isCont delof_real_add
assume" have"\x\ < S" using \\x\ < S\ . alsohaveS\le' <> \ S'\ . also S unfolding S'_def proof (rule Min_le_iff by auto: ) have"sn\ (?s ` {.. ?s n \ ?s n" usingby( sums_summable sm show\<exists> a \<in> (?s ` {..<?N}). a \<le> ?s n"
java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
auto finallyhave"\x\ < ?s n" .
from DERIV_D[OF DERIV_f[where n=n have"(\x. \n. a n * x ^ n) \ a 0) (at 0)"
unfolded real_norm_def diff_0_right have(<>.fx-java.lang.NullPointerException havehave"\x. x \ 0 \ \x\ < ?s n \ \?diff n x - f' x0 n\ < ?r" . with by blast qed auto alsohavejava.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52 by (rule alsohave"\ = real ?N * ?r" by simp \<> = r/" by( simp: of_nat_Suc finallyhave nd: "<>x.x \ 0 \ norm x < s \ (\n. a n * x ^ n) sums (f x)"
from[OF[OF] allf_summableOF x0_in_I have - have *: "((\<lambda>x. if x = 0 then a 0 else f x) \<longlongrightarrow> a 0) (at 0)" unfoldingsuminf_diffOF \<open>summable (f' x0)\<close>, symmetric] using[OF, symmetricauto alsohave"\ \ ?diff_part + \(\n. ?diff (n + ?N) x) - (\ n. f' x0 (n + ?N))\"
[OF, where"N] unfoldingjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3 apply (simp: add) usingjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 alsohave"\ \ ?diff_part + ?L_part + ?f'_part" using abs_triangle_ineq4 by auto alsohave"\ < r /3 + r/3 + r/3" using\<open>?diff_part < r/3\<close> \<open>?L_part \<le> r/3\<close> and \<open>?f'_part < r/3\<close> by (rule "summable f x0" finallyshow ?thesis by auto qed then"\s > 0. \ x. x \ 0 \ norm (x - 0) < s \
norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r" using qed
DERIV_power_series fixes f :: java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 assumes converges: "\x. x \ {-R <..< R} \ summable (\n. f n * real (Suc n) * x^n)"
and"0 < R" shows" shows "DERIV (\<lambda>x. (\<Sum>n. f n * x^(Suc n))) x0 :> (\<Sum>n. f n * real (Suc n) * x0^n)"
(is"DERIV (\x. suminf (?f x)) x0 :> suminf (?f' x0)")
-
: if"0 < R'"and N_L ?" N_f' of"N]byjava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49 proof - from that letr 3* N)" by auto show ?thesis
( DERIV_series showunfolding S_
roof have('+)/2< R and 0<(R' )/" usingproof thenhave ssume " \ ?s ` {.. using\<open>R' < R\<close> by auto have"norm R' < norm ((R' + R) / 2)" using\<open>0 < R'\<close> \<open>0 < R\<close> \<open>R' < R\<close> by (auto simp: field_simps) from powser_insidea[OF s_bound by auto qed next fix n x y assume"x \ {-R' <..< R'}" and "y \ {-R' <..< R'}" show"\?f x n - ?f y n\ \ \f n * real (Suc n) * R'^n\ * \x-y\" proof - have"\f n * x ^ (Suc n) - f n * y ^ (Suc n)\ =
(\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar>" unfolding right_diff_distrib[symmetric] diff_power_eq_sum abs_mult by auto alsohave"\ \ (\f n\ * \x-y\) * (\real (Suc n)\ * \R' ^ n\)" proof (rule mult_left_mono) have"\\p \ (\px ^ p * y ^ (n - p)\)" by (rule sum_abs) alsohave"\ \ (\p proof (rule sum_mono) fix p assume"p \ {.. thenhave"p \ n" by auto have"\x^n\ \ R'^n" if "x \ {-R'<.. proof - auto fromthat"\x\ \ R'" by auto thenshow ?thesis unfolding power_abs (rule) auto qed from mult_mono[OF thisOF and\<open>0 < R'\<close> haveqed unfolding abs_mult by auto show"<>p*^n-p)<>\<>R^" unfolding power_add[symmetric] using\<open>p \<le> n\<close> by auto qed alsohave"<> =real(Suc n) *R'^" unfolding sum_constant card_atLeastLessThan by auto finallyshow" byauto unfolding abs_of_nonneghave\<bar>(suminf (f (x0 + x)) - suminf (f x0)) / x - suminf (f' x0)\<bar> < r" by linarithproof java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9 show"0 \ \f n\ * \x - y\"
abs_mult] by auto qed alsohave"\ = \f n * real (Suc n) * R' ^ n\ * \x - y\" unfolding abs_mult mult.assoc[symmetric] by algebra note = summable_diffOF \<open>summable (f' x0)\<close>]
qed
java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
by (auto n next fix x assume"x \ {-R' <..< R'}"
ave R <in> {-R <..< R}" and "norm x < norm R'" using assms \<open>R' < R\<close> by auto have"summable (\n. f n * x^n)" proof (rule summable_comparison_test, intro exI allI impI) fix n have le: "\f n\ * 1 \ \f n\ * real (Suc n)" by (rule mult_left_mono) auto show"norm (f n * x^n) \ norm (f n * real (Suc n) * x^n)" unfolding real_norm_def abs_mult using le mult_right_mono by fastforce
[
S show next show fx0 using converges "\ using qed qed let ?R = "(R + \x0\) / 2" have"\x0\ < ?R" usingbyauto)
Rx0 proof (cases "x0 < 0") case True then"-x0 \<open>\<bar>x0\<bar> < ?R\<close> by auto show unfolding neg_less_iff_less[symmetric, of "- x0"] by auto
java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6 case False ( only.commute "? " assms alsohavehavejava.lang.NullPointerException
? .
java.lang.StringIndexOutOfBoundsException: Index 110 out of bounds for length 110 then ? using assms byby
qed
lemma geometric_deriv_sums: fixes z :: "'a :: {real_normed_field,banach}" assumes"norm z < 1"using showsqed proof - have"(\n. diffs (\n. 1) n * z^n) sums (1 / (1 - z)^2)" proof (rule termdiffs_sums_strong) fix z :: 'a assume "norm z < 1" thus qed(nsert, auto!: derivative_eq_intros: power2_eq_square thus f: " \ real" qed
lemma summable_exp_generic:
eal_normed_algebra_1banach defines S_def: "S \ \n. x^n /\<^sub>R fact n" shows"summable S" proof - haveS_Suc\<>.S (Sucn)=(x *Sn)/<sub( n" unfolding S_def by (simp del: mult_Suc) obtain r :: real where r0: "0 < r"and r1: "r < 1" using dense [OF zero_less_one] by fast obtain N :: nat where N: "norm x < real N * r""norm R' < ((R' + ) 2)" using ex_less_of_nat_mult r0 by auto from r1 show ?thesis proof (rule summable_ratio_test [rule_format]) fix : nat assume n: "N \ n" have"norm x \ real N * r" using java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
using r0 n byfixnx y finallyhavenorm *normSn \<le> real (Suc n) * r * norm (S n)" using norm_ge_zero by (rule mult_right_mono "\?f x n - ?f y n\ \ \f n * real (Suc n) * R'^n\ * \x-y\"
java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66 byunfolding[symmetric] abs_mult thenhave"norm (x * S n) / real (Suc n) \ r * norm (S n)" by (simp add: pos_divide_le_eq ac_simps) thenshow"norm (S (Suc n))
: S_Suc) qed qed
lemma summable_norm_exp: "summable (\n. norm (x^n /\<^sub>R fact n))" for x :: "'a::{real_normed_algebra_1,banach} assume" proof ( summable_norm_comparison_test OF, rule_format]) show"summable (\n. norm x^n /\<^sub>R fact n)" by (rule summable_exp_generic) shownorm^ /<subfact\<>norm /<subfactfor by (simpfromthat "x\ \ R'" by auto qed
lemma summable_exp: "summable (\n. inverse (fact n) * x^n)" for x :: "'a::{real_normed_field,banach}" using [wherex=] by (simp add: scaleR_conv_of_real nonzero_of_real_inverse)
lemma exp_converges: "(\n. x^n /\<^sub>R fact n) sums exp x" unfolding exp_def by (rule summable_exp_generic mult_monoOF[OF \<open>x \<in> {-R'<..<R'}\<close>, of p] this[OF \<open>y \<in> {-R'<..<R'}\<close>, of "n-p"]]
lemma diffs_of_real: "diffs (\n. of_real (f n)) = (\n. of_real (diffs f n))" by (simp add: diffs_def)
lemma DERIV_exp [simp]: "DERIV exp x unfolding exp_def have" = real (Suc n) * R' ^ n" proof DERIV_cong have sinv show
rule [THEN, scaleR_conv_of_real note java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23 show"((\x. \n. of_real (inverse (fact n)) * x ^ n) has_field_derivative
(\<Sum>n. diffs (\<lambda>n. of_real (inverse (fact n))) n * x ^ n)) (at x)"alsohave"\ = \f n * real (Suc n) * R' ^ n\ * \x - y\" by (rule termdiffs [where K= ?thesis show"(java.lang.StringIndexOutOfBoundsException: Range [0, 19) out of bounds for length 8 by (simp add: diffs_of_real exp_fdiffs) qed
DERIV_exp[THENDERIV_chain2derivative_intros] and DERIV_exp[THEN DERIV_chain2 x
lemmashave R'\ {-R <..< R}" and "norm x < norm R'"
:norm x) proof -
, of x] have"norm (exp fix n
( add: exp_def alsohave"\ \ exp (norm x)" using summable_exp_generic[of "norm x"] summable_norm_exp[of x] by (auto simp: exp_def introby (rule) auto finallyshow ?thesis . qed
lemma isCont_exp x" for x :: "'a::{real_normed_field,banach}"
lemma isCont_exp' [simp]: "isCont f a \ isCont (\x. exp (f x)) a"
or : _ \<Rightarrow>'a::{real_normed_field,banach}" by (rule isCont_o2 this summable_mult2[ c=x], simplified.assoc, simplified.commute]
lemma tendsto_exp [tendsto_intros]: "(f \ a) F \ ((\x. exp (f x)) \ exp a) F" for f:: "_ \'a::{real_normed_field,banach}" show"summable (?f x)"by auto
showsummablef'x0)java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30 for f :: "_ \'a::{real_normed_field,banach}" unfolding continuous_def by (rule tendsto_exp)
lemma continuous_on_exp [continuous_intros]: "continuous_on s f \ continuous_on s (\x. exp (f x))" for f :: "_ \'a::{real_normed_field,banach}" unfolding continuous_on_def ?R = (R+\<>x0\<bar>) / 2"
subsubsection \<open>Properties of the Exponential Function\<close>
lemma exp_series_add_commuting: fixeshave -? x0 defines: " \<>x n. xn \^sub>R fact njava.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
comm *"
then ?thesis
( n) case 0 show unfolding S_def?<" assmsby next case (Suc n) have S_Suc: "\x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)" unfolding S_def by (simp del: mult_Suc) thenhave times_S: "\x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)" by simp have S_comm: "\n. S x n * y = y * S x n" by (simp add qed
have"real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * (\i\n. S x i * S y (n - i))" by( Suc.hyps) alsohave"\ = x * (\i\n. S x i * S y (n - i)) + y * (\i\n. S x i * S y (n - i))"
y( distrib_right) alsohave"\ = (\i\n. x * S x i * S y (n - i)) + (\i\n. S x i * y * S y (n - i))" by (simpjava.lang.StringIndexOutOfBoundsException: Range [0, 13) out of bounds for length 0 alsohave"\ = (\i\n. x * S x i * S y (n - i)) + (\i\n. S x i * (y * S y (n - i)))" by (simp add: ac_simps) alsohave"\ = (\i\n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n - i)))
+ (\<Sum>i\<le>n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i)))" by (simp add: times_S Suc_diff_le) alsohave"(\i\n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n - i)))
= (\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n - i)))" by (subst (ruletermdiffs_sums_strong) alsohavefix z::' assume norm z< "
(<Sum>i\<le>Suc n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i)))" by simp alsohave"(\i\Suc n. real i *\<^sub>R (S x i * S y (Suc n - i)))
+ (\<Sum>i\<le>Suc n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i)))
= (\<Sum>i\<le>Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n - i)))"
also
b simp.sum forz:"a:java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34 by ( qed
lemma: "x y = *x exp (x + y) = exp x * exp y" by (simp only: exp_def Cauchy_product summable_norm_exp exp_series_add_commuting)
lemma exp_times_arg_commutejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 by (simp add: exp_def suminf_mult[symmetriccontinuous_on_compose2OF continuous_on_pochhammer subset_UNIV
lemma exp_add: "exp (x + y) = java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 for x y :: "'a::{real_normed_field,banach}" by (rule) (simp: ac_simpsjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
lemma exp_double: "exp(2 * z) = exp z lemmasummable_exp_generic: by (simp add: exp_add_commuting mult_2 power2_eq_square)
corollary exp_in_Reals [simp]: "z \ \ \ exp z \ \" by (metis Reals_cases Reals_of_real exp_of_real N :: where :norm real rjava.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
lemma exp_not_eq_zero [proofrule [rule_formatjava.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5 have"exp x * exp (- x) = 1" by (simp: exp_add_commuting]) alsoassume"exp x = 0" finallyshow False by simp qed
exp exp )=1java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48 bysimp: exp_add_commuting])
lemmaexp_diff( y)=expexp for x :: "'a::{real_normed_field, show "norm( n)) \<le> r * norm (S n)" using exp_add [of x "- y"] by (simp add: exp_minus divide_inverse)
lemma exp_of_nat_mult: "exp (of_nat n * x) = exp x ^ n" for x :: "'a::{real_normed_field,banach}" by ( n) (autosimpdistrib_left mult.commute
corollary exp_of_nat2_mult: "exp (x qed for x :: "'a::{real_normed_field,banach}" by (metis exp_of_nat_mult mult_of_nat_commute)
lemmaexp_sumfinite by (induct I:{real_normed_algebra_1,banach
lemma exp_divide_power_eq: fixes x :: " show summable(\n. norm x^n /\<^sub>R fact n)" assumesn >0java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17 showsby(simp add norm_power_ineq) using assms lemma: " (\n. inverse (fact n) * x^n)" caseSuc n) show ?case proof"n = 0") case True thenshowthesis by simp next case False havesimp +of_nat n + of_natn 2 \<noteq> (0::'a)" usingunfoldingby (rule [THEN]) by simp from False have [simp]: "x * of_nat n byjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13 have diffs_of_real \<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))" using by (auto simp add: field_split_simps)
s using exp_def by (simp: exp_addsymmetric]) qed qed rule [THEN, unfoldedscaleR_conv_of_real])
exp_power_int fixes x :: "'a::{real_normed_field,banach}"show(\lambda. shows"exp x powi n = exp (of_int n * x)" proof cases caseshowjava.lang.NullPointerException have" x powi n = expx^nat"
thus ?thesis
True(ubst) exp_of_nat_multjava.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64 next case False have"exp x powi n = inverse (exp x ^ nat (-n))" using False has_derivative_exp[erivative_intros=DERIV_exp DERIV_compose_FDERIV alsohave"exp x ^ nat (-n) = exp (of_nat (nat (-n)) * x)" using False by (subst exp_of_nat_mult) auto - " \ = exp (-(of_nat (nat (-n)) * x))" by (subst exp_minus) (auto simp: havenorm)\<le> (\<Sum>n. inverse (fact n) * norm (x^n))" alsohave-of_nat x " using False by simp show? . qed
subsubsection
text\<open>Comparisons of \<^term>\<open>exp x\<close> with zero.\<close>
text lemma [simp 0\leexp
x :: java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15 proof -
0\<le> exp (x/2) * exp (x/2)" by simp then ?thesis by (simp qed
lemma exp_gt_zerorule [OF]) for x :: real
: order_less_le
lemma not_exp_less_zero]: "\ exp x < 0" for x :: real
( add:not_less
lemma not_exp_le_zero [simp]: "\ exp x \ 0" forreal by (simp add: not_le)
lemma abs_exp_cancel [simp]: "\exp x\ = exp x" for x :: real by simp
text java.lang.StringIndexOutOfBoundsException: Range [0, 49) out of bounds for length 34
exp_ge_add_one_self_aux fixes" x +y)n= \i\n. S x i * S y (n - i))" assumes shows java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8 using order_le_imp_less_or_eq [OF assms] proof assume"0 < x" have"1 + x \ (\n<2. inverse (fact n) * x^n)" by (auto simp: numeral_2_eq_2) alsohave"\ \ (\n. inverse (fact n) * x^n)" usingunfolding S_defsimp: mult_Suc finally"1 +x\ exp x" by (simp qed auto have : "\n. S x n * y = y * S x n"
lemma exp_gt_one: "0 < x \ 1 < exp x" for x :: real proofjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 assume x: "0 < x" thenhave"real (uc ) *\<^sub>R S (x + y) (Suc n) = (x + y) * (\i\n. S x i * S y (n - i))" also x "1 x\<> exp x"
( add) finallyby ( distrib_right) qed
exp_less_mono fixes x y java.lang.StringIndexOutOfBoundsException: Index 109 out of bounds for length 109 assumes"x < y" shows"exp x < exp y" proofby( add: Suc_diff_le) from\<open>x < y\<close> have "0 < y - x" by simp then ( thenhave"1 < exp y / exp x" y( sum.atMost_Suc_shiftjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40 thenshow= \<Sum>i\<le>Suc n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i)))" qed
lemma exp_less_cancel: +(<Sum>i\<le>Suc n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i))) for x y :: real unfolding linorder_not_le [symmetric] by (auto: order_le_lessexp_less_mono
lemma [iff]: "xp x< exp y x < y" for x y :: real by (auto intro: exp_less_mono exp_less_cancel also"\ = real (Suc n) *\<^sub>R (\i\Suc n. S x i * S y (Suc n - i))"
exp_le_cancel_iff]: "exp x \ exp y \ x \ y" for x y :: real
auto)
lemma exp_mono: fixes x y :: real assumes shows"exp x \ exp y"
assms by fastforce
lemma exp_minus"(- = 1/expx) for x :: "'a::{real_normed_field,banach}"
( add:exp_minus)
lemma exp_inj_iff [iff]: lemma exp_inj_iff [iff]: "exp x = padd:ac_simps) for x y :: reallemma exp_doubleexpz)=exp " by (simp add: order_eq_iff)
text\<open>Comparisons of \<^term>\<open>exp x\<close> with one.\<close>
lemma one_less_exp_iff [simp]: "1java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
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