src/HOL/Hyperreal/Series.thy
author nipkow
Thu Dec 02 11:42:01 2004 +0100 (2004-12-02)
changeset 15360 300e09825d8b
parent 15251 bb6f072c8d10
child 15536 3ce1cb7a24f0
permissions -rw-r--r--
Added "ALL x > y" and relatives.
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(*  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
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imports SEQ Lim
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begin
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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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   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 "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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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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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 "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 "n", auto)
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lemma sumr_interval_const:
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     "\<lbrakk>\<forall>n. m \<le> Suc n --> f n = r; m \<le> k\<rbrakk> \<Longrightarrow> sumr m k f = (real(k-m) * r)"
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apply (induct "k", auto) 
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apply (drule_tac x = k 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:
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     "[|\<forall>n\<ge>m. f n = r; m \<le> k|]
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      ==> sumr m k f = (real (k - m) * r)"
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apply (induct "k", auto) 
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apply (drule_tac x = k 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:
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     "[|\<forall>n\<ge>m. 0 \<le> f n; m < k|] ==> sumr 0 m f \<le> sumr 0 k f"
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apply (induct "k")
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apply (auto simp add: less_Suc_eq_le)
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apply (drule_tac x = k 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 "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: "(\<forall>n\<ge>m. 0 \<le> f n) --> 0 \<le> sumr m n f"
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apply (induct "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 (%k. abs (f k))) = (sumr m n (%k. abs (f k)))"
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by (induct "n", simp_all add: add_increasing)
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lemma sumr_zero [rule_format]:
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     "\<forall>n \<ge> N. f n = 0 ==> N \<le> m --> sumr m n f = 0"
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by (induct "n", auto)
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lemma Suc_le_imp_diff_ge2:
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     "[|\<forall>n \<ge> 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 "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 "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 "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 "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 "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 "no", auto)
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apply (drule_tac x = "Suc no" 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; 
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        \<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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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 \<ge> n. 0 \<le> f(m) |] ==> sumr 0 n f \<le> suminf f"
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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)
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done
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lemma series_pos_less:
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     "[| summable f; \<forall>m \<ge> n. 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 "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: mult_assoc left_distrib  times_divide_eq)
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   315
apply (simp add: right_distrib diff_minus mult_commute)
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   316
done
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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 
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        simp add: sumr_geometric sums_def diff_minus 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: diff_minus minus_divide_right LIMSEQ_rabs_realpow_zero2)
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   326
done
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   328
text{*Cauchy-type criterion for convergence of series (c.f. Harrison)*}
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lemma summable_convergent_sumr_iff: "summable f = convergent (%n. sumr 0 n f)"
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by (simp add: summable_def sums_def convergent_def)
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   333
lemma summable_Cauchy:
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     "summable f =  
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      (\<forall>e > 0. \<exists>N. \<forall>m \<ge> N. \<forall>n. abs(sumr m n f) < e)"
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   336
apply (auto simp add: summable_convergent_sumr_iff Cauchy_convergent_iff [symmetric] Cauchy_def)
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apply (drule_tac [!] spec, auto) 
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apply (rule_tac x = M in exI)
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apply (rule_tac [2] x = N in exI, auto)
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   340
apply (cut_tac [!] m = m and n = n in less_linear, auto)
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apply (frule le_less_trans [THEN less_imp_le], assumption)
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apply (drule_tac x = n in spec, simp)
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   343
apply (drule_tac x = m in spec)
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apply (auto intro: abs_minus_add_cancel [THEN subst]
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   345
            simp add: sumr_split_add_minus abs_minus_add_cancel)
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   346
done
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   347
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   348
text{*Comparison test*}
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   349
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lemma summable_comparison_test:
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     "[| \<exists>N. \<forall>n \<ge> N. abs(f n) \<le> g n; summable g |] ==> summable f"
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apply (auto simp add: summable_Cauchy)
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   353
apply (drule spec, auto)
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   354
apply (rule_tac x = "N + Na" in exI, auto)
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   355
apply (rotate_tac 2)
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   356
apply (drule_tac x = m in spec)
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   357
apply (auto, rotate_tac 2, drule_tac x = n in spec)
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   358
apply (rule_tac y = "sumr m n (%k. abs (f k))" in order_le_less_trans)
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   359
apply (rule sumr_rabs)
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   360
apply (rule_tac y = "sumr m n g" in order_le_less_trans)
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   361
apply (auto intro: sumr_le2 simp add: abs_interval_iff)
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   362
done
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   363
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   364
lemma summable_rabs_comparison_test:
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     "[| \<exists>N. \<forall>n \<ge> N. abs(f n) \<le> g n; summable g |] 
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      ==> summable (%k. abs (f k))"
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   367
apply (rule summable_comparison_test)
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apply (auto simp add: abs_idempotent)
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   369
done
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   370
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   371
text{*Limit comparison property for series (c.f. jrh)*}
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   372
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   373
lemma summable_le:
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     "[|\<forall>n. f n \<le> g n; summable f; summable g |] ==> suminf f \<le> suminf g"
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   375
apply (drule summable_sums)+
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   376
apply (auto intro!: LIMSEQ_le simp add: sums_def)
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   377
apply (rule exI)
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   378
apply (auto intro!: sumr_le2)
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   379
done
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   380
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   381
lemma summable_le2:
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     "[|\<forall>n. abs(f n) \<le> g n; summable g |]  
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   383
      ==> summable f & suminf f \<le> suminf g"
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   384
apply (auto intro: summable_comparison_test intro!: summable_le)
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   385
apply (simp add: abs_le_interval_iff)
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   386
done
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   387
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   388
text{*Absolute convergence imples normal convergence*}
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   389
lemma summable_rabs_cancel: "summable (%n. abs (f n)) ==> summable f"
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   390
apply (auto simp add: sumr_rabs summable_Cauchy)
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   391
apply (drule spec, auto)
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   392
apply (rule_tac x = N in exI, auto)
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   393
apply (drule spec, auto)
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   394
apply (rule_tac y = "sumr m n (%n. abs (f n))" in order_le_less_trans)
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   395
apply (auto intro: sumr_rabs)
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   396
done
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   397
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   398
text{*Absolute convergence of series*}
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   399
lemma summable_rabs:
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     "summable (%n. abs (f n)) ==> abs(suminf f) \<le> suminf (%n. abs(f n))"
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   401
by (auto intro: LIMSEQ_le LIMSEQ_imp_rabs summable_rabs_cancel summable_sumr_LIMSEQ_suminf sumr_rabs)
paulson@14416
   402
paulson@14416
   403
paulson@14416
   404
subsection{* The Ratio Test*}
paulson@14416
   405
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   406
lemma rabs_ratiotest_lemma: "[| c \<le> 0; abs x \<le> c * abs y |] ==> x = (0::real)"
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   407
apply (drule order_le_imp_less_or_eq, auto)
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   408
apply (subgoal_tac "0 \<le> c * abs y")
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   409
apply (simp add: zero_le_mult_iff, arith)
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   410
done
paulson@14416
   411
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   412
lemma le_Suc_ex: "(k::nat) \<le> l ==> (\<exists>n. l = k + n)"
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   413
apply (drule le_imp_less_or_eq)
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   414
apply (auto dest: less_imp_Suc_add)
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   415
done
paulson@14416
   416
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   417
lemma le_Suc_ex_iff: "((k::nat) \<le> l) = (\<exists>n. l = k + n)"
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   418
by (auto simp add: le_Suc_ex)
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   419
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   420
(*All this trouble just to get 0<c *)
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   421
lemma ratio_test_lemma2:
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   422
     "[| \<forall>n \<ge> N. abs(f(Suc n)) \<le> c*abs(f n) |]  
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   423
      ==> 0 < c | summable f"
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   424
apply (simp (no_asm) add: linorder_not_le [symmetric])
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   425
apply (simp add: summable_Cauchy)
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   426
apply (safe, subgoal_tac "\<forall>n. N \<le> n --> f (Suc n) = 0")
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   427
prefer 2 apply (blast intro: rabs_ratiotest_lemma)
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   428
apply (rule_tac x = "Suc N" in exI, clarify)
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   429
apply (drule_tac n=n in Suc_le_imp_diff_ge2, auto) 
paulson@14416
   430
done
paulson@14416
   431
paulson@14416
   432
lemma ratio_test:
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   433
     "[| c < 1; \<forall>n \<ge> N. abs(f(Suc n)) \<le> c*abs(f n) |]  
paulson@14416
   434
      ==> summable f"
paulson@14416
   435
apply (frule ratio_test_lemma2, auto)
paulson@15234
   436
apply (rule_tac g = "%n. (abs (f N) / (c ^ N))*c ^ n" 
paulson@15234
   437
       in summable_comparison_test)
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   438
apply (rule_tac x = N in exI, safe)
paulson@14416
   439
apply (drule le_Suc_ex_iff [THEN iffD1])
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   440
apply (auto simp add: power_add realpow_not_zero)
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   441
apply (induct_tac "na", auto simp add: times_divide_eq)
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   442
apply (rule_tac y = "c*abs (f (N + n))" in order_trans)
paulson@14416
   443
apply (auto intro: mult_right_mono simp add: summable_def)
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   444
apply (simp add: mult_ac)
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   445
apply (rule_tac x = "abs (f N) * (1/ (1 - c)) / (c ^ N)" in exI)
paulson@15234
   446
apply (rule sums_divide) 
paulson@15234
   447
apply (rule sums_mult) 
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   448
apply (auto intro!: geometric_sums)
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   449
done
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   450
paulson@14416
   451
paulson@15085
   452
text{*Differentiation of finite sum*}
paulson@14416
   453
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   454
lemma DERIV_sumr [rule_format (no_asm)]:
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   455
     "(\<forall>r. m \<le> r & r < (m + n) --> DERIV (%x. f r x) x :> (f' r x))  
paulson@14416
   456
      --> DERIV (%x. sumr m n (%n. f n x)) x :> sumr m n (%r. f' r x)"
paulson@15251
   457
apply (induct "n")
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   458
apply (auto intro: DERIV_add)
paulson@14416
   459
done
paulson@14416
   460
paulson@14416
   461
ML
paulson@14416
   462
{*
paulson@14416
   463
val sumr_Suc = thm"sumr_Suc";
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   464
val sums_def = thm"sums_def";
paulson@14416
   465
val summable_def = thm"summable_def";
paulson@14416
   466
val suminf_def = thm"suminf_def";
paulson@14416
   467
paulson@14416
   468
val sumr_add = thm "sumr_add";
paulson@14416
   469
val sumr_mult = thm "sumr_mult";
paulson@14416
   470
val sumr_split_add = thm "sumr_split_add";
paulson@14416
   471
val sumr_rabs = thm "sumr_rabs";
paulson@14416
   472
val sumr_fun_eq = thm "sumr_fun_eq";
paulson@14416
   473
val sumr_diff_mult_const = thm "sumr_diff_mult_const";
paulson@14416
   474
val sumr_minus_one_realpow_zero = thm "sumr_minus_one_realpow_zero";
paulson@14416
   475
val sumr_le2 = thm "sumr_le2";
paulson@14416
   476
val rabs_sumr_rabs_cancel = thm "rabs_sumr_rabs_cancel";
paulson@14416
   477
val sumr_zero = thm "sumr_zero";
paulson@14416
   478
val Suc_le_imp_diff_ge2 = thm "Suc_le_imp_diff_ge2";
paulson@14416
   479
val sumr_one_lb_realpow_zero = thm "sumr_one_lb_realpow_zero";
paulson@14416
   480
val sumr_diff = thm "sumr_diff";
paulson@14416
   481
val sumr_subst = thm "sumr_subst";
paulson@14416
   482
val sumr_bound = thm "sumr_bound";
paulson@14416
   483
val sumr_bound2 = thm "sumr_bound2";
paulson@14416
   484
val sumr_group = thm "sumr_group";
paulson@14416
   485
val sums_summable = thm "sums_summable";
paulson@14416
   486
val summable_sums = thm "summable_sums";
paulson@14416
   487
val summable_sumr_LIMSEQ_suminf = thm "summable_sumr_LIMSEQ_suminf";
paulson@14416
   488
val sums_unique = thm "sums_unique";
paulson@14416
   489
val series_zero = thm "series_zero";
paulson@14416
   490
val sums_mult = thm "sums_mult";
paulson@14416
   491
val sums_divide = thm "sums_divide";
paulson@14416
   492
val sums_diff = thm "sums_diff";
paulson@14416
   493
val suminf_mult = thm "suminf_mult";
paulson@14416
   494
val suminf_mult2 = thm "suminf_mult2";
paulson@14416
   495
val suminf_diff = thm "suminf_diff";
paulson@14416
   496
val sums_minus = thm "sums_minus";
paulson@14416
   497
val sums_group = thm "sums_group";
paulson@14416
   498
val sumr_pos_lt_pair_lemma = thm "sumr_pos_lt_pair_lemma";
paulson@14416
   499
val sumr_pos_lt_pair = thm "sumr_pos_lt_pair";
paulson@14416
   500
val series_pos_le = thm "series_pos_le";
paulson@14416
   501
val series_pos_less = thm "series_pos_less";
paulson@14416
   502
val sumr_geometric = thm "sumr_geometric";
paulson@14416
   503
val geometric_sums = thm "geometric_sums";
paulson@14416
   504
val summable_convergent_sumr_iff = thm "summable_convergent_sumr_iff";
paulson@14416
   505
val summable_Cauchy = thm "summable_Cauchy";
paulson@14416
   506
val summable_comparison_test = thm "summable_comparison_test";
paulson@14416
   507
val summable_rabs_comparison_test = thm "summable_rabs_comparison_test";
paulson@14416
   508
val summable_le = thm "summable_le";
paulson@14416
   509
val summable_le2 = thm "summable_le2";
paulson@14416
   510
val summable_rabs_cancel = thm "summable_rabs_cancel";
paulson@14416
   511
val summable_rabs = thm "summable_rabs";
paulson@14416
   512
val rabs_ratiotest_lemma = thm "rabs_ratiotest_lemma";
paulson@14416
   513
val le_Suc_ex = thm "le_Suc_ex";
paulson@14416
   514
val le_Suc_ex_iff = thm "le_Suc_ex_iff";
paulson@14416
   515
val ratio_test_lemma2 = thm "ratio_test_lemma2";
paulson@14416
   516
val ratio_test = thm "ratio_test";
paulson@14416
   517
val DERIV_sumr = thm "DERIV_sumr";
paulson@14416
   518
*}
paulson@14416
   519
paulson@14416
   520
end