author | paulson |
Sat, 31 Jul 2004 20:54:23 +0200 | |
changeset 15094 | a7d1a3fdc30d |
parent 15085 | 5693a977a767 |
child 15131 | c69542757a4d |
permissions | -rw-r--r-- |
10751 | 1 |
(* Title : Series.thy |
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Author : Jacques D. Fleuriot |
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Copyright : 1998 University of Cambridge |
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Converted to Isar and polished by lcp |
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*) |
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header{*Finite Summation and Infinite Series*} |
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theory Series = SEQ + Lim: |
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syntax sumr :: "[nat,nat,(nat=>real)] => real" |
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translations |
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"sumr m n f" => "setsum (f::nat=>real) (atLeastLessThan m n)" |
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constdefs |
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sums :: "[nat=>real,real] => bool" (infixr "sums" 80) |
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"f sums s == (%n. sumr 0 n f) ----> s" |
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||
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summable :: "(nat=>real) => bool" |
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"summable f == (\<exists>s. f sums s)" |
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suminf :: "(nat=>real) => real" |
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"suminf f == (@s. f sums s)" |
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lemma sumr_Suc [simp]: |
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"sumr m (Suc n) f = (if n < m then 0 else sumr m n f + f(n))" |
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by (simp add: atLeastLessThanSuc) |
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lemma sumr_add: "sumr m n f + sumr m n g = sumr m n (%n. f n + g n)" |
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by (simp add: setsum_addf) |
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lemma sumr_mult: "r * sumr m n (f::nat=>real) = sumr m n (%n. r * f n)" |
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by (simp add: setsum_mult) |
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lemma sumr_split_add [rule_format]: |
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"n < p --> sumr 0 n f + sumr n p f = sumr 0 p (f::nat=>real)" |
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apply (induct_tac "p", auto) |
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apply (rename_tac k) |
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apply (subgoal_tac "n=k", auto) |
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done |
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||
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lemma sumr_split_add_minus: |
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"n < p ==> sumr 0 p f + - sumr 0 n f = sumr n p (f::nat=>real)" |
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apply (drule_tac f1 = f in sumr_split_add [symmetric]) |
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apply (simp add: add_ac) |
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done |
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||
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lemma sumr_rabs: "abs(sumr m n (f::nat=>real)) \<le> sumr m n (%i. abs(f i))" |
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by (simp add: setsum_abs) |
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lemma sumr_rabs_ge_zero [iff]: "0 \<le> sumr m n (%n. abs (f n))" |
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by (simp add: setsum_abs_ge_zero) |
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text{*Just a congruence rule*} |
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lemma sumr_fun_eq: |
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"(\<forall>r. m \<le> r & r < n --> f r = g r) ==> sumr m n f = sumr m n g" |
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by (auto intro: setsum_cong) |
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lemma sumr_diff_mult_const: "sumr 0 n f - (real n*r) = sumr 0 n (%i. f i - r)" |
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by (simp add: diff_minus setsum_addf real_of_nat_def) |
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lemma sumr_less_bounds_zero [simp]: "n < m ==> sumr m n f = 0" |
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by (simp add: atLeastLessThan_empty) |
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lemma sumr_minus: "sumr m n (%i. - f i) = - sumr m n f" |
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by (simp add: Finite_Set.setsum_negf) |
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lemma sumr_shift_bounds: "sumr (m+k) (n+k) f = sumr m n (%i. f(i + k))" |
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by (induct_tac "n", auto) |
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lemma sumr_minus_one_realpow_zero [simp]: "sumr 0 (2*n) (%i. (-1) ^ Suc i) = 0" |
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by (induct_tac "n", auto) |
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lemma sumr_interval_const [rule_format (no_asm)]: |
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"(\<forall>n. m \<le> Suc n --> f n = r) --> m \<le> k --> sumr m k f = (real(k-m) * r)" |
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apply (induct_tac "k", auto) |
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apply (drule_tac x = n in spec) |
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apply (auto dest!: le_imp_less_or_eq) |
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apply (simp add: left_distrib real_of_nat_Suc split: nat_diff_split) |
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done |
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lemma sumr_interval_const2 [rule_format (no_asm)]: |
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"(\<forall>n. m \<le> n --> f n = r) --> m \<le> k |
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--> sumr m k f = (real (k - m) * r)" |
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apply (induct_tac "k", auto) |
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apply (drule_tac x = n in spec) |
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apply (auto dest!: le_imp_less_or_eq) |
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apply (simp add: left_distrib real_of_nat_Suc split: nat_diff_split) |
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done |
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lemma sumr_le [rule_format (no_asm)]: |
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"(\<forall>n. m \<le> n --> 0 \<le> f n) --> m < k --> sumr 0 m f \<le> sumr 0 k f" |
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apply (induct_tac "k") |
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apply (auto simp add: less_Suc_eq_le) |
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apply (drule_tac [!] x = n in spec, safe) |
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apply (drule le_imp_less_or_eq, safe) |
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apply (arith) |
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apply (drule_tac a = "sumr 0 m f" in order_refl [THEN add_mono], auto) |
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done |
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lemma sumr_le2 [rule_format (no_asm)]: |
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"(\<forall>r. m \<le> r & r < n --> f r \<le> g r) --> sumr m n f \<le> sumr m n g" |
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apply (induct_tac "n") |
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apply (auto intro: add_mono simp add: le_def) |
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done |
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lemma sumr_ge_zero [rule_format (no_asm)]: "(\<forall>n. 0 \<le> f n) --> 0 \<le> sumr m n f" |
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apply (induct_tac "n", auto) |
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apply (drule_tac x = n in spec, arith) |
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done |
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lemma sumr_ge_zero2 [rule_format (no_asm)]: |
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"(\<forall>n. m \<le> n --> 0 \<le> f n) --> 0 \<le> sumr m n f" |
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apply (induct_tac "n", auto) |
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apply (drule_tac x = n in spec, arith) |
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done |
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lemma rabs_sumr_rabs_cancel [simp]: |
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"abs (sumr m n (%n. abs (f n))) = (sumr m n (%n. abs (f n)))" |
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apply (induct_tac "n") |
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apply (auto, arith) |
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done |
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lemma sumr_zero [rule_format]: |
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"\<forall>n. N \<le> n --> f n = 0 ==> N \<le> m --> sumr m n f = 0" |
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by (induct_tac "n", auto) |
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lemma Suc_le_imp_diff_ge2: |
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"[|\<forall>n. N \<le> n --> f (Suc n) = 0; Suc N \<le> m|] ==> sumr m n f = 0" |
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apply (rule sumr_zero) |
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apply (case_tac "n", auto) |
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done |
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lemma sumr_one_lb_realpow_zero [simp]: "sumr (Suc 0) n (%n. f(n) * 0 ^ n) = 0" |
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apply (induct_tac "n") |
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apply (case_tac [2] "n", auto) |
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done |
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lemma sumr_diff: "sumr m n f - sumr m n g = sumr m n (%n. f n - g n)" |
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by (simp add: diff_minus sumr_add [symmetric] sumr_minus) |
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lemma sumr_subst [rule_format (no_asm)]: |
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"(\<forall>p. m \<le> p & p < m+n --> (f p = g p)) --> sumr m n f = sumr m n g" |
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by (induct_tac "n", auto) |
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lemma sumr_bound [rule_format (no_asm)]: |
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"(\<forall>p. m \<le> p & p < m + n --> (f(p) \<le> K)) |
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--> (sumr m (m + n) f \<le> (real n * K))" |
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apply (induct_tac "n") |
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apply (auto intro: add_mono simp add: left_distrib real_of_nat_Suc) |
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done |
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lemma sumr_bound2 [rule_format (no_asm)]: |
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"(\<forall>p. 0 \<le> p & p < n --> (f(p) \<le> K)) |
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--> (sumr 0 n f \<le> (real n * K))" |
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apply (induct_tac "n") |
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apply (auto intro: add_mono simp add: left_distrib real_of_nat_Suc add_commute) |
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done |
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lemma sumr_group [simp]: |
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"sumr 0 n (%m. sumr (m * k) (m*k + k) f) = sumr 0 (n * k) f" |
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apply (subgoal_tac "k = 0 | 0 < k", auto) |
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apply (induct_tac "n") |
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apply (simp_all add: sumr_split_add add_commute) |
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done |
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subsection{* Infinite Sums, by the Properties of Limits*} |
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(*---------------------- |
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suminf is the sum |
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---------------------*) |
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lemma sums_summable: "f sums l ==> summable f" |
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by (simp add: sums_def summable_def, blast) |
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lemma summable_sums: "summable f ==> f sums (suminf f)" |
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apply (simp add: summable_def suminf_def) |
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apply (blast intro: someI2) |
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done |
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lemma summable_sumr_LIMSEQ_suminf: |
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"summable f ==> (%n. sumr 0 n f) ----> (suminf f)" |
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apply (simp add: summable_def suminf_def sums_def) |
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apply (blast intro: someI2) |
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done |
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(*------------------- |
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sum is unique |
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------------------*) |
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lemma sums_unique: "f sums s ==> (s = suminf f)" |
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apply (frule sums_summable [THEN summable_sums]) |
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apply (auto intro!: LIMSEQ_unique simp add: sums_def) |
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done |
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(* |
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Goalw [sums_def,LIMSEQ_def] |
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"(\<forall>m. n \<le> Suc m --> f(m) = 0) ==> f sums (sumr 0 n f)" |
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by safe |
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by (res_inst_tac [("x","n")] exI 1); |
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by (safe THEN ftac le_imp_less_or_eq 1) |
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by safe |
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by (dres_inst_tac [("f","f")] sumr_split_add_minus 1); |
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by (ALLGOALS (Asm_simp_tac)); |
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by (dtac (conjI RS sumr_interval_const) 1); |
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by Auto_tac |
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qed "series_zero"; |
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next one was called series_zero2 |
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**********************) |
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lemma series_zero: |
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"(\<forall>m. n \<le> m --> f(m) = 0) ==> f sums (sumr 0 n f)" |
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apply (simp add: sums_def LIMSEQ_def, safe) |
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apply (rule_tac x = n in exI) |
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apply (safe, frule le_imp_less_or_eq, safe) |
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apply (drule_tac f = f in sumr_split_add_minus, simp_all) |
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apply (drule sumr_interval_const2, auto) |
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done |
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lemma sums_mult: "x sums x0 ==> (%n. c * x(n)) sums (c * x0)" |
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by (auto simp add: sums_def sumr_mult [symmetric] |
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intro!: LIMSEQ_mult intro: LIMSEQ_const) |
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lemma sums_divide: "x sums x' ==> (%n. x(n)/c) sums (x'/c)" |
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by (simp add: real_divide_def sums_mult mult_commute [of _ "inverse c"]) |
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lemma sums_diff: "[| x sums x0; y sums y0 |] ==> (%n. x n - y n) sums (x0-y0)" |
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by (auto simp add: sums_def sumr_diff [symmetric] intro: LIMSEQ_diff) |
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lemma suminf_mult: "summable f ==> suminf f * c = suminf(%n. f n * c)" |
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by (auto intro!: sums_unique sums_mult summable_sums simp add: mult_commute) |
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lemma suminf_mult2: "summable f ==> c * suminf f = suminf(%n. c * f n)" |
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by (auto intro!: sums_unique sums_mult summable_sums) |
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lemma suminf_diff: |
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"[| summable f; summable g |] |
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==> suminf f - suminf g = suminf(%n. f n - g n)" |
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by (auto intro!: sums_diff sums_unique summable_sums) |
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lemma sums_minus: "x sums x0 ==> (%n. - x n) sums - x0" |
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by (auto simp add: sums_def intro!: LIMSEQ_minus simp add: sumr_minus) |
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lemma sums_group: |
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"[|summable f; 0 < k |] ==> (%n. sumr (n*k) (n*k + k) f) sums (suminf f)" |
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apply (drule summable_sums) |
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apply (auto simp add: sums_def LIMSEQ_def) |
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apply (drule_tac x = r in spec, safe) |
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apply (rule_tac x = no in exI, safe) |
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apply (drule_tac x = "n*k" in spec) |
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apply (auto dest!: not_leE) |
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apply (drule_tac j = no in less_le_trans, auto) |
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done |
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lemma sumr_pos_lt_pair_lemma: |
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"[|\<forall>d. - f (n + (d + d)) < f (Suc (n + (d + d)))|] |
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==> sumr 0 (n + Suc (Suc 0)) f \<le> sumr 0 (Suc (Suc 0) * Suc no + n) f" |
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apply (induct_tac "no", simp) |
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apply (rule_tac y = "sumr 0 (Suc (Suc 0) * (Suc na) +n) f" in order_trans) |
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apply assumption |
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apply (drule_tac x = "Suc na" in spec) |
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apply (simp add: add_ac) |
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done |
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lemma sumr_pos_lt_pair: |
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"[|summable f; \<forall>d. 0 < (f(n + (Suc(Suc 0) * d))) + f(n + ((Suc(Suc 0) * d) + 1))|] |
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==> sumr 0 n f < suminf f" |
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apply (drule summable_sums) |
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apply (auto simp add: sums_def LIMSEQ_def) |
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apply (drule_tac x = "f (n) + f (n + 1) " in spec) |
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apply (auto iff: real_0_less_add_iff) |
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--{*legacy proof: not necessarily better!*} |
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apply (rule_tac [2] ccontr, drule_tac [2] linorder_not_less [THEN iffD1]) |
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apply (frule_tac [2] no=no in sumr_pos_lt_pair_lemma) |
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apply (drule_tac x = 0 in spec, simp) |
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apply (rotate_tac 1, drule_tac x = "Suc (Suc 0) * (Suc no) + n" in spec) |
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apply (safe, simp) |
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apply (subgoal_tac "suminf f + (f (n) + f (n + 1)) \<le> sumr 0 (Suc (Suc 0) * (Suc no) + n) f") |
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apply (rule_tac [2] y = "sumr 0 (n+ Suc (Suc 0)) f" in order_trans) |
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prefer 3 apply assumption |
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apply (rule_tac [2] y = "sumr 0 n f + (f (n) + f (n + 1))" in order_trans) |
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apply simp_all |
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apply (subgoal_tac "suminf f \<le> sumr 0 (Suc (Suc 0) * (Suc no) + n) f") |
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apply (rule_tac [2] y = "suminf f + (f (n) + f (n + 1))" in order_trans) |
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prefer 3 apply simp |
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apply (drule_tac [2] x = 0 in spec) |
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prefer 2 apply simp |
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apply (subgoal_tac "0 \<le> sumr 0 (Suc (Suc 0) * Suc no + n) f + - suminf f") |
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apply (simp add: abs_if) |
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apply (auto simp add: linorder_not_less [symmetric]) |
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done |
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15085
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text{*A summable series of positive terms has limit that is at least as |
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great as any partial sum.*} |
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lemma series_pos_le: |
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"[| summable f; \<forall>m. n \<le> m --> 0 \<le> f(m) |] ==> sumr 0 n f \<le> suminf f" |
|
300 |
apply (drule summable_sums) |
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apply (simp add: sums_def) |
|
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apply (cut_tac k = "sumr 0 n f" in LIMSEQ_const) |
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apply (erule LIMSEQ_le, blast) |
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apply (rule_tac x = n in exI, clarify) |
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apply (drule le_imp_less_or_eq) |
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apply (auto intro: sumr_le) |
|
307 |
done |
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lemma series_pos_less: |
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"[| summable f; \<forall>m. n \<le> m --> 0 < f(m) |] ==> sumr 0 n f < suminf f" |
|
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apply (rule_tac y = "sumr 0 (Suc n) f" in order_less_le_trans) |
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apply (rule_tac [2] series_pos_le, auto) |
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apply (drule_tac x = m in spec, auto) |
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done |
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text{*Sum of a geometric progression.*} |
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lemma sumr_geometric: "x ~= 1 ==> sumr 0 n (%n. x ^ n) = (x ^ n - 1) / (x - 1)" |
|
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apply (induct_tac "n", auto) |
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apply (rule_tac c1 = "x - 1" in real_mult_right_cancel [THEN iffD1]) |
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apply (auto simp add: real_mult_assoc left_distrib) |
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apply (simp add: right_distrib real_diff_def real_mult_commute) |
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done |
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324 |
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lemma geometric_sums: "abs(x) < 1 ==> (%n. x ^ n) sums (1/(1 - x))" |
|
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apply (case_tac "x = 1") |
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apply (auto dest!: LIMSEQ_rabs_realpow_zero2 simp add: sumr_geometric sums_def real_diff_def add_divide_distrib) |
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apply (subgoal_tac "1 / (1 + -x) = 0/ (x - 1) + - 1/ (x - 1) ") |
|
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apply (erule ssubst) |
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apply (rule LIMSEQ_add, rule LIMSEQ_divide) |
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apply (auto intro: LIMSEQ_const simp add: real_diff_def minus_divide_right LIMSEQ_rabs_realpow_zero2) |
|
332 |
done |
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15085
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text{*Cauchy-type criterion for convergence of series (c.f. Harrison)*} |
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|
14416 | 336 |
lemma summable_convergent_sumr_iff: "summable f = convergent (%n. sumr 0 n f)" |
337 |
by (simp add: summable_def sums_def convergent_def) |
|
338 |
||
339 |
lemma summable_Cauchy: |
|
340 |
"summable f = |
|
341 |
(\<forall>e. 0 < e --> (\<exists>N. \<forall>m n. N \<le> m --> abs(sumr m n f) < e))" |
|
342 |
apply (auto simp add: summable_convergent_sumr_iff Cauchy_convergent_iff [symmetric] Cauchy_def) |
|
343 |
apply (drule_tac [!] spec, auto) |
|
344 |
apply (rule_tac x = M in exI) |
|
345 |
apply (rule_tac [2] x = N in exI, auto) |
|
346 |
apply (cut_tac [!] m = m and n = n in less_linear, auto) |
|
347 |
apply (frule le_less_trans [THEN less_imp_le], assumption) |
|
348 |
apply (drule_tac x = n in spec) |
|
349 |
apply (drule_tac x = m in spec) |
|
350 |
apply (auto intro: abs_minus_add_cancel [THEN subst] |
|
351 |
simp add: sumr_split_add_minus abs_minus_add_cancel) |
|
352 |
done |
|
353 |
||
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|
354 |
text{*Comparison test*} |
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|
355 |
|
14416 | 356 |
lemma summable_comparison_test: |
357 |
"[| \<exists>N. \<forall>n. N \<le> n --> abs(f n) \<le> g n; summable g |] ==> summable f" |
|
358 |
apply (auto simp add: summable_Cauchy) |
|
359 |
apply (drule spec, auto) |
|
360 |
apply (rule_tac x = "N + Na" in exI, auto) |
|
361 |
apply (rotate_tac 2) |
|
362 |
apply (drule_tac x = m in spec) |
|
363 |
apply (auto, rotate_tac 2, drule_tac x = n in spec) |
|
364 |
apply (rule_tac y = "sumr m n (%k. abs (f k))" in order_le_less_trans) |
|
365 |
apply (rule sumr_rabs) |
|
366 |
apply (rule_tac y = "sumr m n g" in order_le_less_trans) |
|
367 |
apply (auto intro: sumr_le2 simp add: abs_interval_iff) |
|
368 |
done |
|
369 |
||
370 |
lemma summable_rabs_comparison_test: |
|
371 |
"[| \<exists>N. \<forall>n. N \<le> n --> abs(f n) \<le> g n; summable g |] |
|
372 |
==> summable (%k. abs (f k))" |
|
373 |
apply (rule summable_comparison_test) |
|
374 |
apply (auto simp add: abs_idempotent) |
|
375 |
done |
|
376 |
||
15085
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|
377 |
text{*Limit comparison property for series (c.f. jrh)*} |
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|
378 |
|
14416 | 379 |
lemma summable_le: |
380 |
"[|\<forall>n. f n \<le> g n; summable f; summable g |] ==> suminf f \<le> suminf g" |
|
381 |
apply (drule summable_sums)+ |
|
382 |
apply (auto intro!: LIMSEQ_le simp add: sums_def) |
|
383 |
apply (rule exI) |
|
384 |
apply (auto intro!: sumr_le2) |
|
385 |
done |
|
386 |
||
387 |
lemma summable_le2: |
|
388 |
"[|\<forall>n. abs(f n) \<le> g n; summable g |] |
|
389 |
==> summable f & suminf f \<le> suminf g" |
|
390 |
apply (auto intro: summable_comparison_test intro!: summable_le) |
|
391 |
apply (simp add: abs_le_interval_iff) |
|
392 |
done |
|
393 |
||
15085
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15053
diff
changeset
|
394 |
text{*Absolute convergence imples normal convergence*} |
14416 | 395 |
lemma summable_rabs_cancel: "summable (%n. abs (f n)) ==> summable f" |
396 |
apply (auto simp add: sumr_rabs summable_Cauchy) |
|
397 |
apply (drule spec, auto) |
|
398 |
apply (rule_tac x = N in exI, auto) |
|
399 |
apply (drule spec, auto) |
|
400 |
apply (rule_tac y = "sumr m n (%n. abs (f n))" in order_le_less_trans) |
|
401 |
apply (auto intro: sumr_rabs) |
|
402 |
done |
|
403 |
||
15085
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removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset
|
404 |
text{*Absolute convergence of series*} |
14416 | 405 |
lemma summable_rabs: |
406 |
"summable (%n. abs (f n)) ==> abs(suminf f) \<le> suminf (%n. abs(f n))" |
|
407 |
by (auto intro: LIMSEQ_le LIMSEQ_imp_rabs summable_rabs_cancel summable_sumr_LIMSEQ_suminf sumr_rabs) |
|
408 |
||
409 |
||
410 |
subsection{* The Ratio Test*} |
|
411 |
||
412 |
lemma rabs_ratiotest_lemma: "[| c \<le> 0; abs x \<le> c * abs y |] ==> x = (0::real)" |
|
413 |
apply (drule order_le_imp_less_or_eq, auto) |
|
414 |
apply (subgoal_tac "0 \<le> c * abs y") |
|
415 |
apply (simp add: zero_le_mult_iff, arith) |
|
416 |
done |
|
417 |
||
418 |
lemma le_Suc_ex: "(k::nat) \<le> l ==> (\<exists>n. l = k + n)" |
|
419 |
apply (drule le_imp_less_or_eq) |
|
420 |
apply (auto dest: less_imp_Suc_add) |
|
421 |
done |
|
422 |
||
423 |
lemma le_Suc_ex_iff: "((k::nat) \<le> l) = (\<exists>n. l = k + n)" |
|
424 |
by (auto simp add: le_Suc_ex) |
|
425 |
||
426 |
(*All this trouble just to get 0<c *) |
|
427 |
lemma ratio_test_lemma2: |
|
428 |
"[| \<forall>n. N \<le> n --> abs(f(Suc n)) \<le> c*abs(f n) |] |
|
429 |
==> 0 < c | summable f" |
|
430 |
apply (simp (no_asm) add: linorder_not_le [symmetric]) |
|
431 |
apply (simp add: summable_Cauchy) |
|
432 |
apply (safe, subgoal_tac "\<forall>n. N \<le> n --> f (Suc n) = 0") |
|
433 |
prefer 2 apply (blast intro: rabs_ratiotest_lemma) |
|
434 |
apply (rule_tac x = "Suc N" in exI, clarify) |
|
435 |
apply (drule_tac n=n in Suc_le_imp_diff_ge2, auto) |
|
436 |
done |
|
437 |
||
438 |
lemma ratio_test: |
|
439 |
"[| c < 1; \<forall>n. N \<le> n --> abs(f(Suc n)) \<le> c*abs(f n) |] |
|
440 |
==> summable f" |
|
441 |
apply (frule ratio_test_lemma2, auto) |
|
442 |
apply (rule_tac g = "%n. (abs (f N) / (c ^ N))*c ^ n" in summable_comparison_test) |
|
443 |
apply (rule_tac x = N in exI, safe) |
|
444 |
apply (drule le_Suc_ex_iff [THEN iffD1]) |
|
445 |
apply (auto simp add: power_add realpow_not_zero) |
|
446 |
apply (induct_tac "na", auto) |
|
447 |
apply (rule_tac y = "c*abs (f (N + n))" in order_trans) |
|
448 |
apply (auto intro: mult_right_mono simp add: summable_def) |
|
449 |
apply (simp add: mult_ac) |
|
450 |
apply (rule_tac x = "abs (f N) * (1/ (1 - c)) / (c ^ N) " in exI) |
|
451 |
apply (rule sums_divide) |
|
452 |
apply (rule sums_mult) |
|
15003 | 453 |
apply (auto intro!: sums_mult geometric_sums simp add: abs_if) |
14416 | 454 |
done |
455 |
||
456 |
||
15085
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removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset
|
457 |
text{*Differentiation of finite sum*} |
14416 | 458 |
|
459 |
lemma DERIV_sumr [rule_format (no_asm)]: |
|
460 |
"(\<forall>r. m \<le> r & r < (m + n) --> DERIV (%x. f r x) x :> (f' r x)) |
|
461 |
--> DERIV (%x. sumr m n (%n. f n x)) x :> sumr m n (%r. f' r x)" |
|
462 |
apply (induct_tac "n") |
|
463 |
apply (auto intro: DERIV_add) |
|
464 |
done |
|
465 |
||
466 |
ML |
|
467 |
{* |
|
468 |
val sumr_Suc = thm"sumr_Suc"; |
|
469 |
val sums_def = thm"sums_def"; |
|
470 |
val summable_def = thm"summable_def"; |
|
471 |
val suminf_def = thm"suminf_def"; |
|
472 |
||
473 |
val sumr_add = thm "sumr_add"; |
|
474 |
val sumr_mult = thm "sumr_mult"; |
|
475 |
val sumr_split_add = thm "sumr_split_add"; |
|
476 |
val sumr_rabs = thm "sumr_rabs"; |
|
477 |
val sumr_fun_eq = thm "sumr_fun_eq"; |
|
478 |
val sumr_diff_mult_const = thm "sumr_diff_mult_const"; |
|
479 |
val sumr_minus_one_realpow_zero = thm "sumr_minus_one_realpow_zero"; |
|
480 |
val sumr_le2 = thm "sumr_le2"; |
|
481 |
val sumr_ge_zero = thm "sumr_ge_zero"; |
|
482 |
val sumr_ge_zero2 = thm "sumr_ge_zero2"; |
|
483 |
val sumr_rabs_ge_zero = thm "sumr_rabs_ge_zero"; |
|
484 |
val rabs_sumr_rabs_cancel = thm "rabs_sumr_rabs_cancel"; |
|
485 |
val sumr_zero = thm "sumr_zero"; |
|
486 |
val Suc_le_imp_diff_ge2 = thm "Suc_le_imp_diff_ge2"; |
|
487 |
val sumr_one_lb_realpow_zero = thm "sumr_one_lb_realpow_zero"; |
|
488 |
val sumr_diff = thm "sumr_diff"; |
|
489 |
val sumr_subst = thm "sumr_subst"; |
|
490 |
val sumr_bound = thm "sumr_bound"; |
|
491 |
val sumr_bound2 = thm "sumr_bound2"; |
|
492 |
val sumr_group = thm "sumr_group"; |
|
493 |
val sums_summable = thm "sums_summable"; |
|
494 |
val summable_sums = thm "summable_sums"; |
|
495 |
val summable_sumr_LIMSEQ_suminf = thm "summable_sumr_LIMSEQ_suminf"; |
|
496 |
val sums_unique = thm "sums_unique"; |
|
497 |
val series_zero = thm "series_zero"; |
|
498 |
val sums_mult = thm "sums_mult"; |
|
499 |
val sums_divide = thm "sums_divide"; |
|
500 |
val sums_diff = thm "sums_diff"; |
|
501 |
val suminf_mult = thm "suminf_mult"; |
|
502 |
val suminf_mult2 = thm "suminf_mult2"; |
|
503 |
val suminf_diff = thm "suminf_diff"; |
|
504 |
val sums_minus = thm "sums_minus"; |
|
505 |
val sums_group = thm "sums_group"; |
|
506 |
val sumr_pos_lt_pair_lemma = thm "sumr_pos_lt_pair_lemma"; |
|
507 |
val sumr_pos_lt_pair = thm "sumr_pos_lt_pair"; |
|
508 |
val series_pos_le = thm "series_pos_le"; |
|
509 |
val series_pos_less = thm "series_pos_less"; |
|
510 |
val sumr_geometric = thm "sumr_geometric"; |
|
511 |
val geometric_sums = thm "geometric_sums"; |
|
512 |
val summable_convergent_sumr_iff = thm "summable_convergent_sumr_iff"; |
|
513 |
val summable_Cauchy = thm "summable_Cauchy"; |
|
514 |
val summable_comparison_test = thm "summable_comparison_test"; |
|
515 |
val summable_rabs_comparison_test = thm "summable_rabs_comparison_test"; |
|
516 |
val summable_le = thm "summable_le"; |
|
517 |
val summable_le2 = thm "summable_le2"; |
|
518 |
val summable_rabs_cancel = thm "summable_rabs_cancel"; |
|
519 |
val summable_rabs = thm "summable_rabs"; |
|
520 |
val rabs_ratiotest_lemma = thm "rabs_ratiotest_lemma"; |
|
521 |
val le_Suc_ex = thm "le_Suc_ex"; |
|
522 |
val le_Suc_ex_iff = thm "le_Suc_ex_iff"; |
|
523 |
val ratio_test_lemma2 = thm "ratio_test_lemma2"; |
|
524 |
val ratio_test = thm "ratio_test"; |
|
525 |
val DERIV_sumr = thm "DERIV_sumr"; |
|
526 |
*} |
|
527 |
||
528 |
end |