src/HOL/Hyperreal/Series.thy
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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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declare atLeastLessThan_empty[simp];
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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. setsum f {0..<n}) ----> 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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  "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
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by (simp add: atLeastLessThanSuc add_commute)
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(*
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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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*)
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lemma sumr_split_add: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
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  setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
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by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)
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lemma sumr_split_add_minus:
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fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
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shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
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  setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
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using sumr_split_add [of m n p f,symmetric]
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apply (simp add: add_ac)
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done
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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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(*
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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_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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*)
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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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(* FIXME generalize *)
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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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(*
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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 setsum_addf setsum_negf)
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*)
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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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   155
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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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   162
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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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   169
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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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   176
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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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   184
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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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   189
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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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   195
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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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   203
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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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   217
**********************)
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   218
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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 diff_minus[symmetric], safe)
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apply (rule_tac x = n in exI)
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apply (clarsimp simp:sumr_split_add_minus)
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apply (drule sumr_interval_const2, auto)
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done
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   226
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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 setsum_mult [symmetric]
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         intro!: LIMSEQ_mult intro: LIMSEQ_const)
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   230
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   231
lemma sums_divide: "x sums x' ==> (%n. x(n)/c) sums (x'/c)"
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   232
by (simp add: real_divide_def sums_mult mult_commute [of _ "inverse c"])
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   233
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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 setsum_subtractf intro: LIMSEQ_diff)
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   236
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   237
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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   239
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   240
lemma suminf_mult2: "summable f ==> c * suminf f  = suminf(%n. c * f n)"
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   241
by (auto intro!: sums_unique sums_mult summable_sums)
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   242
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   243
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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   246
by (auto intro!: sums_diff sums_unique summable_sums)
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   247
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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: setsum_negf)
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   250
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   251
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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   253
apply (drule summable_sums)
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apply (auto simp add: sums_def LIMSEQ_def)
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   255
apply (drule_tac x = r in spec, safe)
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   256
apply (rule_tac x = no in exI, safe)
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   257
apply (drule_tac x = "n*k" in spec)
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   258
apply (auto dest!: not_leE)
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   259
apply (drule_tac j = no in less_le_trans, auto)
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done
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   261
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   262
lemma sumr_pos_lt_pair_lemma:
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   263
     "[|\<forall>d. - f (n + (d + d)) < f (Suc (n + (d + d)))|]
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   264
      ==> 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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   266
apply (drule_tac x = "Suc no" in spec)
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   267
apply (simp add: add_ac) 
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   268
done
10751
a81ea5d3dd41 separation of HOL-Hyperreal from HOL-Real
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parents:
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   269
a81ea5d3dd41 separation of HOL-Hyperreal from HOL-Real
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parents:
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   270
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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)
15085
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apply (auto iff: real_0_less_add_iff)
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   279
   --{*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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   281
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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   283
apply (rotate_tac 1, drule_tac x = "Suc (Suc 0) * (Suc no) + n" in spec)
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   284
apply (safe, simp)
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   285
apply (subgoal_tac "suminf f + (f (n) + f (n + 1)) \<le> sumr 0 (Suc (Suc 0) * (Suc no) + n) f")
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   286
apply (rule_tac [2] y = "sumr 0 (n+ Suc (Suc 0)) f" in order_trans)
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   287
prefer 3 apply assumption
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   288
apply (rule_tac [2] y = "sumr 0 n f + (f (n) + f (n + 1))" in order_trans)
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   289
apply simp_all 
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   290
apply (subgoal_tac "suminf f \<le> sumr 0 (Suc (Suc 0) * (Suc no) + n) f")
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   291
apply (rule_tac [2] y = "suminf f + (f (n) + f (n + 1))" in order_trans)
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   292
prefer 3 apply simp 
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   293
apply (drule_tac [2] x = 0 in spec)
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   294
 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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parents: 12018
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   296
apply (simp add: abs_if) 
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parents: 12018
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   297
apply (auto simp add: linorder_not_less [symmetric])
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parents: 12018
diff changeset
   298
done
1f256287d4f0 converted Hyperreal/Series to Isar script
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   299
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   300
text{*A summable series of positive terms has limit that is at least as
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
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   301
great as any partial sum.*}
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   302
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   303
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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parents: 12018
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   305
apply (drule summable_sums)
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parents: 12018
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   306
apply (simp add: sums_def)
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parents: 12018
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   307
apply (cut_tac k = "sumr 0 n f" in LIMSEQ_const)
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parents: 12018
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   308
apply (erule LIMSEQ_le, blast) 
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parents: 12018
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   309
apply (rule_tac x = n in exI, clarify) 
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parents: 12018
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   310
apply (drule le_imp_less_or_eq)
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parents: 12018
diff changeset
   311
apply (auto intro: sumr_le)
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parents: 12018
diff changeset
   312
done
1f256287d4f0 converted Hyperreal/Series to Isar script
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diff changeset
   313
1f256287d4f0 converted Hyperreal/Series to Isar script
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   314
lemma series_pos_less:
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diff changeset
   315
     "[| summable f; \<forall>m \<ge> n. 0 < f(m) |] ==> sumr 0 n f < suminf f"
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parents: 12018
diff changeset
   316
apply (rule_tac y = "sumr 0 (Suc n) f" in order_less_le_trans)
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parents: 12018
diff changeset
   317
apply (rule_tac [2] series_pos_le, auto)
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parents: 12018
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   318
apply (drule_tac x = m in spec, auto)
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parents: 12018
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   319
done
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   320
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   321
text{*Sum of a geometric progression.*}
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parents: 12018
diff changeset
   322
1f256287d4f0 converted Hyperreal/Series to Isar script
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   323
lemma sumr_geometric: "x ~= 1 ==> sumr 0 n (%n. x ^ n) = (x ^ n - 1) / (x - 1)"
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parents: 15234
diff changeset
   324
apply (induct "n", auto)
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parents: 12018
diff changeset
   325
apply (rule_tac c1 = "x - 1" in real_mult_right_cancel [THEN iffD1])
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   326
apply (auto simp add: mult_assoc left_distrib  times_divide_eq)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   327
apply (simp add: right_distrib diff_minus mult_commute)
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parents: 12018
diff changeset
   328
done
1f256287d4f0 converted Hyperreal/Series to Isar script
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parents: 12018
diff changeset
   329
1f256287d4f0 converted Hyperreal/Series to Isar script
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parents: 12018
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   330
lemma geometric_sums: "abs(x) < 1 ==> (%n. x ^ n) sums (1/(1 - x))"
1f256287d4f0 converted Hyperreal/Series to Isar script
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parents: 12018
diff changeset
   331
apply (case_tac "x = 1")
15234
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parents: 15229
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   332
apply (auto dest!: LIMSEQ_rabs_realpow_zero2 
ec91a90c604e simplification tweaks for better arithmetic reasoning
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parents: 15229
diff changeset
   333
        simp add: sumr_geometric sums_def diff_minus add_divide_distrib)
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1f256287d4f0 converted Hyperreal/Series to Isar script
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parents: 12018
diff changeset
   334
apply (subgoal_tac "1 / (1 + -x) = 0/ (x - 1) + - 1/ (x - 1) ")
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   335
apply (erule ssubst)
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parents: 12018
diff changeset
   336
apply (rule LIMSEQ_add, rule LIMSEQ_divide)
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parents: 15229
diff changeset
   337
apply (auto intro: LIMSEQ_const simp add: diff_minus minus_divide_right LIMSEQ_rabs_realpow_zero2)
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paulson
parents: 12018
diff changeset
   338
done
1f256287d4f0 converted Hyperreal/Series to Isar script
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parents: 12018
diff changeset
   339
15085
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   340
text{*Cauchy-type criterion for convergence of series (c.f. Harrison)*}
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
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parents: 15053
diff changeset
   341
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diff changeset
   342
lemma summable_convergent_sumr_iff: "summable f = convergent (%n. sumr 0 n f)"
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diff changeset
   343
by (simp add: summable_def sums_def convergent_def)
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parents: 12018
diff changeset
   344
1f256287d4f0 converted Hyperreal/Series to Isar script
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diff changeset
   345
lemma summable_Cauchy:
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parents: 12018
diff changeset
   346
     "summable f =  
15360
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parents: 15251
diff changeset
   347
      (\<forall>e > 0. \<exists>N. \<forall>m \<ge> N. \<forall>n. abs(sumr m n f) < e)"
15537
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parents: 15536
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   348
apply (auto simp add: summable_convergent_sumr_iff Cauchy_convergent_iff [symmetric] Cauchy_def diff_minus[symmetric])
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parents: 12018
diff changeset
   349
apply (drule_tac [!] spec, auto) 
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paulson
parents: 12018
diff changeset
   350
apply (rule_tac x = M in exI)
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paulson
parents: 12018
diff changeset
   351
apply (rule_tac [2] x = N in exI, auto)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   352
apply (cut_tac [!] m = m and n = n in less_linear, auto)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   353
apply (frule le_less_trans [THEN less_imp_le], assumption)
15360
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nipkow
parents: 15251
diff changeset
   354
apply (drule_tac x = n in spec, simp)
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paulson
parents: 12018
diff changeset
   355
apply (drule_tac x = m in spec)
15537
5538d3244b4d continued eliminating sumr
nipkow
parents: 15536
diff changeset
   356
apply(simp add: sumr_split_add_minus)
5538d3244b4d continued eliminating sumr
nipkow
parents: 15536
diff changeset
   357
apply(subst abs_minus_commute)
5538d3244b4d continued eliminating sumr
nipkow
parents: 15536
diff changeset
   358
apply(simp add: sumr_split_add_minus)
5538d3244b4d continued eliminating sumr
nipkow
parents: 15536
diff changeset
   359
apply (simp add: sumr_split_add_minus)
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paulson
parents: 12018
diff changeset
   360
done
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   361
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
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diff changeset
   362
text{*Comparison test*}
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   363
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parents: 12018
diff changeset
   364
lemma summable_comparison_test:
15360
300e09825d8b Added "ALL x > y" and relatives.
nipkow
parents: 15251
diff changeset
   365
     "[| \<exists>N. \<forall>n \<ge> N. abs(f n) \<le> g n; summable g |] ==> summable f"
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   366
apply (auto simp add: summable_Cauchy)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   367
apply (drule spec, auto)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   368
apply (rule_tac x = "N + Na" in exI, auto)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   369
apply (rotate_tac 2)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   370
apply (drule_tac x = m in spec)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   371
apply (auto, rotate_tac 2, drule_tac x = n in spec)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   372
apply (rule_tac y = "sumr m n (%k. abs (f k))" in order_le_less_trans)
15536
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nipkow
parents: 15360
diff changeset
   373
apply (rule setsum_abs)
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1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   374
apply (rule_tac y = "sumr m n g" in order_le_less_trans)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   375
apply (auto intro: sumr_le2 simp add: abs_interval_iff)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   376
done
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   377
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   378
lemma summable_rabs_comparison_test:
15360
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nipkow
parents: 15251
diff changeset
   379
     "[| \<exists>N. \<forall>n \<ge> N. abs(f n) \<le> g n; summable g |] 
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   380
      ==> summable (%k. abs (f k))"
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   381
apply (rule summable_comparison_test)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   382
apply (auto simp add: abs_idempotent)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   383
done
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   384
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   385
text{*Limit comparison property for series (c.f. jrh)*}
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   386
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   387
lemma summable_le:
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   388
     "[|\<forall>n. f n \<le> g n; summable f; summable g |] ==> suminf f \<le> suminf g"
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   389
apply (drule summable_sums)+
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   390
apply (auto intro!: LIMSEQ_le simp add: sums_def)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   391
apply (rule exI)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   392
apply (auto intro!: sumr_le2)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   393
done
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   394
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   395
lemma summable_le2:
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   396
     "[|\<forall>n. abs(f n) \<le> g n; summable g |]  
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   397
      ==> summable f & suminf f \<le> suminf g"
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   398
apply (auto intro: summable_comparison_test intro!: summable_le)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   399
apply (simp add: abs_le_interval_iff)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   400
done
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   401
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   402
text{*Absolute convergence imples normal convergence*}
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   403
lemma summable_rabs_cancel: "summable (%n. abs (f n)) ==> summable f"
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15360
diff changeset
   404
apply (auto simp add: summable_Cauchy)
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   405
apply (drule spec, auto)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   406
apply (rule_tac x = N in exI, auto)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   407
apply (drule spec, auto)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   408
apply (rule_tac y = "sumr m n (%n. abs (f n))" in order_le_less_trans)
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15360
diff changeset
   409
apply (auto)
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   410
done
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   411
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   412
text{*Absolute convergence of series*}
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   413
lemma summable_rabs:
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   414
     "summable (%n. abs (f n)) ==> abs(suminf f) \<le> suminf (%n. abs(f n))"
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15360
diff changeset
   415
by (auto intro: LIMSEQ_le LIMSEQ_imp_rabs summable_rabs_cancel summable_sumr_LIMSEQ_suminf)
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   417
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   418
subsection{* The Ratio Test*}
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   419
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   420
lemma rabs_ratiotest_lemma: "[| c \<le> 0; abs x \<le> c * abs y |] ==> x = (0::real)"
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   421
apply (drule order_le_imp_less_or_eq, auto)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   422
apply (subgoal_tac "0 \<le> c * abs y")
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   423
apply (simp add: zero_le_mult_iff, arith)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   424
done
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   425
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   426
lemma le_Suc_ex: "(k::nat) \<le> l ==> (\<exists>n. l = k + n)"
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   427
apply (drule le_imp_less_or_eq)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   428
apply (auto dest: less_imp_Suc_add)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   429
done
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   430
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   431
lemma le_Suc_ex_iff: "((k::nat) \<le> l) = (\<exists>n. l = k + n)"
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   432
by (auto simp add: le_Suc_ex)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   433
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   434
(*All this trouble just to get 0<c *)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   435
lemma ratio_test_lemma2:
15360
300e09825d8b Added "ALL x > y" and relatives.
nipkow
parents: 15251
diff changeset
   436
     "[| \<forall>n \<ge> N. abs(f(Suc n)) \<le> c*abs(f n) |]  
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   437
      ==> 0 < c | summable f"
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   438
apply (simp (no_asm) add: linorder_not_le [symmetric])
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   439
apply (simp add: summable_Cauchy)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   440
apply (safe, subgoal_tac "\<forall>n. N \<le> n --> f (Suc n) = 0")
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   441
prefer 2 apply (blast intro: rabs_ratiotest_lemma)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   442
apply (rule_tac x = "Suc N" in exI, clarify)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   443
apply (drule_tac n=n in Suc_le_imp_diff_ge2, auto) 
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   444
done
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   445
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   446
lemma ratio_test:
15360
300e09825d8b Added "ALL x > y" and relatives.
nipkow
parents: 15251
diff changeset
   447
     "[| c < 1; \<forall>n \<ge> N. abs(f(Suc n)) \<le> c*abs(f n) |]  
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   448
      ==> summable f"
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   449
apply (frule ratio_test_lemma2, auto)
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   450
apply (rule_tac g = "%n. (abs (f N) / (c ^ N))*c ^ n" 
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   451
       in summable_comparison_test)
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   452
apply (rule_tac x = N in exI, safe)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   453
apply (drule le_Suc_ex_iff [THEN iffD1])
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   454
apply (auto simp add: power_add realpow_not_zero)
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   455
apply (induct_tac "na", auto simp add: times_divide_eq)
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   456
apply (rule_tac y = "c*abs (f (N + n))" in order_trans)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   457
apply (auto intro: mult_right_mono simp add: summable_def)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   458
apply (simp add: mult_ac)
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   459
apply (rule_tac x = "abs (f N) * (1/ (1 - c)) / (c ^ N)" in exI)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   460
apply (rule sums_divide) 
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   461
apply (rule sums_mult) 
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   462
apply (auto intro!: geometric_sums)
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   463
done
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   464
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   465
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   466
text{*Differentiation of finite sum*}
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   467
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   468
lemma DERIV_sumr [rule_format (no_asm)]:
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   469
     "(\<forall>r. m \<le> r & r < (m + n) --> DERIV (%x. f r x) x :> (f' r x))  
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   470
      --> DERIV (%x. sumr m n (%n. f n x)) x :> sumr m n (%r. f' r x)"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15234
diff changeset
   471
apply (induct "n")
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   472
apply (auto intro: DERIV_add)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   473
done
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   474
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   475
ML
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   476
{*
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   477
val sumr_Suc = thm"sumr_Suc";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   478
val sums_def = thm"sums_def";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   479
val summable_def = thm"summable_def";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   480
val suminf_def = thm"suminf_def";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   481
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   482
val sumr_split_add = thm "sumr_split_add";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   483
val sumr_minus_one_realpow_zero = thm "sumr_minus_one_realpow_zero";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   484
val sumr_le2 = thm "sumr_le2";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   485
val rabs_sumr_rabs_cancel = thm "rabs_sumr_rabs_cancel";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   486
val sumr_zero = thm "sumr_zero";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   487
val Suc_le_imp_diff_ge2 = thm "Suc_le_imp_diff_ge2";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   488
val sumr_one_lb_realpow_zero = thm "sumr_one_lb_realpow_zero";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   489
val sumr_subst = thm "sumr_subst";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   490
val sumr_bound = thm "sumr_bound";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   491
val sumr_bound2 = thm "sumr_bound2";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   492
val sumr_group = thm "sumr_group";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   493
val sums_summable = thm "sums_summable";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   494
val summable_sums = thm "summable_sums";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   495
val summable_sumr_LIMSEQ_suminf = thm "summable_sumr_LIMSEQ_suminf";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   496
val sums_unique = thm "sums_unique";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   497
val series_zero = thm "series_zero";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   498
val sums_mult = thm "sums_mult";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   499
val sums_divide = thm "sums_divide";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   500
val sums_diff = thm "sums_diff";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   501
val suminf_mult = thm "suminf_mult";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   502
val suminf_mult2 = thm "suminf_mult2";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   503
val suminf_diff = thm "suminf_diff";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   504
val sums_minus = thm "sums_minus";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   505
val sums_group = thm "sums_group";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   506
val sumr_pos_lt_pair_lemma = thm "sumr_pos_lt_pair_lemma";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   507
val sumr_pos_lt_pair = thm "sumr_pos_lt_pair";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   508
val series_pos_le = thm "series_pos_le";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   509
val series_pos_less = thm "series_pos_less";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   510
val sumr_geometric = thm "sumr_geometric";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   511
val geometric_sums = thm "geometric_sums";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   512
val summable_convergent_sumr_iff = thm "summable_convergent_sumr_iff";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   513
val summable_Cauchy = thm "summable_Cauchy";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   514
val summable_comparison_test = thm "summable_comparison_test";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   515
val summable_rabs_comparison_test = thm "summable_rabs_comparison_test";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   516
val summable_le = thm "summable_le";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   517
val summable_le2 = thm "summable_le2";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   518
val summable_rabs_cancel = thm "summable_rabs_cancel";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   519
val summable_rabs = thm "summable_rabs";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   520
val rabs_ratiotest_lemma = thm "rabs_ratiotest_lemma";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   521
val le_Suc_ex = thm "le_Suc_ex";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   522
val le_Suc_ex_iff = thm "le_Suc_ex_iff";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   523
val ratio_test_lemma2 = thm "ratio_test_lemma2";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   524
val ratio_test = thm "ratio_test";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   525
val DERIV_sumr = thm "DERIV_sumr";
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   526
*}
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   527
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   528
end