(*  Title       : Series.thy

     Author      : Jacques D. Fleuriot

     Copyright   : 1998  University of Cambridge

 Converted to Isar and polished by lcp

 Converted to setsum and polished yet more by TNN

 Additional contributions by Jeremy Avigad

 *) 

 header{*Finite Summation and Infinite Series*}

 theory Series

 imports SEQ Lim

 begin

 declare atLeastLessThan_iff[iff]

 declare setsum_op_ivl_Suc[simp]

 definition

    sums  :: "(nat => real) => real => bool"     (infixr "sums" 80)

    "f sums s = (%n. setsum f {0..<n}) ----> s"

    summable :: "(nat=>real) => bool"

    "summable f = (\<exists>s. f sums s)"

    suminf   :: "(nat=>real) => real"

    "suminf f = (SOME s. f sums s)"

 syntax

   "_suminf" :: "idt => real => real"    ("\<Sum>_. _" [0, 10] 10)

 translations

   "\<Sum>i. b" == "suminf (%i. b)"

 lemma sumr_diff_mult_const:

  "setsum f {0..<n} - (real n*r) = setsum (%i. f i - r) {0..<n::nat}"

 by (simp add: diff_minus setsum_addf real_of_nat_def)

 lemma real_setsum_nat_ivl_bounded:

      "(!!p. p < n \<Longrightarrow> f(p) \<le> K)

       \<Longrightarrow> setsum f {0..<n::nat} \<le> real n * K"

 using setsum_bounded[where A = "{0..<n}"]

 by (auto simp:real_of_nat_def)

 (* Generalize from real to some algebraic structure? *)

 lemma sumr_minus_one_realpow_zero [simp]:

   "(\<Sum>i=0..<2*n. (-1) ^ Suc i) = (0::real)"

 by (induct "n", auto)

 (* FIXME this is an awful lemma! *)

 lemma sumr_one_lb_realpow_zero [simp]:

   "(\<Sum>n=Suc 0..<n. f(n) * (0::real) ^ n) = 0"

 apply (induct "n")

 apply (case_tac [2] "n", auto)

 done

 lemma sumr_group:

      "(\<Sum>m=0..<n::nat. setsum f {m * k ..< m*k + k}) = setsum f {0 ..< n * k}"

 apply (subgoal_tac "k = 0 | 0 < k", auto)

 apply (induct "n")

 apply (simp_all add: setsum_add_nat_ivl add_commute)

 done

 (* FIXME generalize? *)

 lemma sumr_offset:

  "(\<Sum>m=0..<n::nat. f(m+k)::real) = setsum f {0..<n+k} - setsum f {0..<k}"

 by (induct "n", auto)

 lemma sumr_offset2:

  "\<forall>f. (\<Sum>m=0..<n::nat. f(m+k)::real) = setsum f {0..<n+k} - setsum f {0..<k}"

 by (induct "n", auto)

 lemma sumr_offset3:

   "setsum f {0::nat..<n+k} = (\<Sum>m=0..<n. f (m+k)::real) + setsum f {0..<k}"

 by (simp  add: sumr_offset)

 lemma sumr_offset4:

  "\<forall>n f. setsum f {0::nat..<n+k} =

         (\<Sum>m=0..<n. f (m+k)::real) + setsum f {0..<k}"

 by (simp add: sumr_offset)

 (*

 lemma sumr_from_1_from_0: "0 < n ==>

       (\<Sum>n=Suc 0 ..< Suc n. if even(n) then 0 else

              ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n =

       (\<Sum>n=0..<Suc n. if even(n) then 0 else

              ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n"

 by (rule_tac n1 = 1 in sumr_split_add [THEN subst], auto)

 *)

 subsection{* Infinite Sums, by the Properties of Limits*}

 (*----------------------

    suminf is the sum   

  ---------------------*)

 lemma sums_summable: "f sums l ==> summable f"

 by (simp add: sums_def summable_def, blast)

 lemma summable_sums: "summable f ==> f sums (suminf f)"

 apply (simp add: summable_def suminf_def)

 apply (blast intro: someI2)

 done

 lemma summable_sumr_LIMSEQ_suminf: 

      "summable f ==> (%n. setsum f {0..<n}) ----> (suminf f)"

 apply (simp add: summable_def suminf_def sums_def)

 apply (blast intro: someI2)

 done

 (*-------------------

     sum is unique                    

  ------------------*)

 lemma sums_unique: "f sums s ==> (s = suminf f)"

 apply (frule sums_summable [THEN summable_sums])

 apply (auto intro!: LIMSEQ_unique simp add: sums_def)

 done

 lemma sums_split_initial_segment: "f sums s ==> 

   (%n. f(n + k)) sums (s - (SUM i = 0..< k. f i))"

   apply (unfold sums_def);

   apply (simp add: sumr_offset); 

   apply (rule LIMSEQ_diff_const)

   apply (rule LIMSEQ_ignore_initial_segment)

   apply assumption

 done

 lemma summable_ignore_initial_segment: "summable f ==> 

     summable (%n. f(n + k))"

   apply (unfold summable_def)

   apply (auto intro: sums_split_initial_segment)

 done

 lemma suminf_minus_initial_segment: "summable f ==>

     suminf f = s ==> suminf (%n. f(n + k)) = s - (SUM i = 0..< k. f i)"

   apply (frule summable_ignore_initial_segment)

   apply (rule sums_unique [THEN sym])

   apply (frule summable_sums)

   apply (rule sums_split_initial_segment)

   apply auto

 done

 lemma suminf_split_initial_segment: "summable f ==> 

     suminf f = (SUM i = 0..< k. f i) + suminf (%n. f(n + k))"

 by (auto simp add: suminf_minus_initial_segment)

 lemma series_zero: 

      "(\<forall>m. n \<le> m --> f(m) = 0) ==> f sums (setsum f {0..<n})"

 apply (simp add: sums_def LIMSEQ_def diff_minus[symmetric], safe)

 apply (rule_tac x = n in exI)

 apply (clarsimp simp add:setsum_diff[symmetric] cong:setsum_ivl_cong)

 done

 lemma sums_zero: "(%n. 0) sums 0";

   apply (unfold sums_def);

   apply simp;

   apply (rule LIMSEQ_const);

 done;

 lemma summable_zero: "summable (%n. 0)";

   apply (rule sums_summable);

   apply (rule sums_zero);

 done;

 lemma suminf_zero: "suminf (%n. 0) = 0";

   apply (rule sym);

   apply (rule sums_unique);

   apply (rule sums_zero);

 done;

 lemma sums_mult: "f sums a ==> (%n. c * f n) sums (c * a)"

 by (auto simp add: sums_def setsum_right_distrib [symmetric]

          intro!: LIMSEQ_mult intro: LIMSEQ_const)

 lemma summable_mult: "summable f ==> summable (%n. c * f n)";

   apply (unfold summable_def);

   apply (auto intro: sums_mult);

 done;

 lemma suminf_mult: "summable f ==> suminf (%n. c * f n) = c * suminf f";

   apply (rule sym);

   apply (rule sums_unique);

   apply (rule sums_mult);

   apply (erule summable_sums);

 done;

 lemma sums_mult2: "f sums a ==> (%n. f n * c) sums (a * c)"

 apply (subst mult_commute)

 apply (subst mult_commute);back;

 apply (erule sums_mult)

 done

 lemma summable_mult2: "summable f ==> summable (%n. f n * c)"

   apply (unfold summable_def)

   apply (auto intro: sums_mult2)

 done

 lemma suminf_mult2: "summable f ==> suminf f * c = (\<Sum>n. f n * c)"

 by (auto intro!: sums_unique sums_mult summable_sums simp add: mult_commute)

 lemma sums_divide: "f sums a ==> (%n. (f n)/c) sums (a/c)"

 by (simp add: real_divide_def sums_mult mult_commute [of _ "inverse c"])

 lemma summable_divide: "summable f ==> summable (%n. (f n) / c)";

   apply (unfold summable_def);

   apply (auto intro: sums_divide);

 done;

 lemma suminf_divide: "summable f ==> suminf (%n. (f n) / c) = (suminf f) / c";

   apply (rule sym);

   apply (rule sums_unique);

   apply (rule sums_divide);

   apply (erule summable_sums);

 done;

 lemma sums_add: "[| x sums x0; y sums y0 |] ==> (%n. x n + y n) sums (x0+y0)"

 by (auto simp add: sums_def setsum_addf intro: LIMSEQ_add)

 lemma summable_add: "summable f ==> summable g ==> summable (%x. f x + g x)";

   apply (unfold summable_def);

   apply clarify;

   apply (rule exI);

   apply (erule sums_add);

   apply assumption;

 done;

 lemma suminf_add:

      "[| summable f; summable g |]   

       ==> suminf f + suminf g  = (\<Sum>n. f n + g n)"

 by (auto intro!: sums_add sums_unique summable_sums)

 lemma sums_diff: "[| x sums x0; y sums y0 |] ==> (%n. x n - y n) sums (x0-y0)"

 by (auto simp add: sums_def setsum_subtractf intro: LIMSEQ_diff)

 lemma summable_diff: "summable f ==> summable g ==> summable (%x. f x - g x)";

   apply (unfold summable_def);

   apply clarify;

   apply (rule exI);

   apply (erule sums_diff);

   apply assumption;

 done;

 lemma suminf_diff:

      "[| summable f; summable g |]   

       ==> suminf f - suminf g  = (\<Sum>n. f n - g n)"

 by (auto intro!: sums_diff sums_unique summable_sums)

 lemma sums_minus: "f sums s ==> (%x. - f x) sums (- s)";

   by (simp add: sums_def setsum_negf LIMSEQ_minus);

 lemma summable_minus: "summable f ==> summable (%x. - f x)";

   by (auto simp add: summable_def intro: sums_minus);

 lemma suminf_minus: "summable f ==> suminf (%x. - f x) = - (suminf f)";

   apply (rule sym);

   apply (rule sums_unique);

   apply (rule sums_minus);

   apply (erule summable_sums);

 done;

 lemma sums_group:

      "[|summable f; 0 < k |] ==> (%n. setsum f {n*k..<n*k+k}) sums (suminf f)"

 apply (drule summable_sums)

 apply (auto simp add: sums_def LIMSEQ_def sumr_group)

 apply (drule_tac x = r in spec, safe)

 apply (rule_tac x = no in exI, safe)

 apply (drule_tac x = "n*k" in spec)

 apply (auto dest!: not_leE)

 apply (drule_tac j = no in less_le_trans, auto)

 done

 lemma sumr_pos_lt_pair_lemma:

   "[|\<forall>d. - f (n + (d + d)) < (f (Suc (n + (d + d))) :: real) |]

    ==> setsum f {0..<n+Suc(Suc 0)} \<le> setsum f {0..<Suc(Suc 0) * Suc no + n}"

 apply (induct "no", auto)

 apply (drule_tac x = "Suc no" in spec)

 apply (simp add: add_ac)

 done

 lemma sumr_pos_lt_pair:

      "[|summable f; 

         \<forall>d. 0 < (f(n + (Suc(Suc 0) * d))) + f(n + ((Suc(Suc 0) * d) + 1))|]  

       ==> setsum f {0..<n} < suminf f"

 apply (drule summable_sums)

 apply (auto simp add: sums_def LIMSEQ_def)

 apply (drule_tac x = "f (n) + f (n + 1)" in spec)

 apply (auto iff: real_0_less_add_iff)

    --{*legacy proof: not necessarily better!*}

 apply (rule_tac [2] ccontr, drule_tac [2] linorder_not_less [THEN iffD1])

 apply (frule_tac [2] no=no in sumr_pos_lt_pair_lemma) 

 apply (drule_tac x = 0 in spec, simp)

 apply (rotate_tac 1, drule_tac x = "Suc (Suc 0) * (Suc no) + n" in spec)

 apply (safe, simp)

 apply (subgoal_tac "suminf f + (f (n) + f (n + 1)) \<le>

  setsum f {0 ..< Suc (Suc 0) * (Suc no) + n}")

 apply (rule_tac [2] y = "setsum f {0..<n+ Suc (Suc 0)}" in order_trans)

 prefer 3 apply assumption

 apply (rule_tac [2] y = "setsum f {0..<n} + (f (n) + f (n + 1))" in order_trans)

 apply simp_all

 done

 text{*A summable series of positive terms has limit that is at least as

 great as any partial sum.*}

 lemma series_pos_le: 

      "[| summable f; \<forall>m \<ge> n. 0 \<le> f(m) |] ==> setsum f {0..<n} \<le> suminf f"

 apply (drule summable_sums)

 apply (simp add: sums_def)

 apply (cut_tac k = "setsum f {0..<n}" in LIMSEQ_const)

 apply (erule LIMSEQ_le, blast)

 apply (rule_tac x = n in exI, clarify)

 apply (rule setsum_mono2)

 apply auto

 done

 lemma series_pos_less:

      "[| summable f; \<forall>m \<ge> n. 0 < f(m) |] ==> setsum f {0..<n} < suminf f"

 apply (rule_tac y = "setsum f {0..<Suc n}" in order_less_le_trans)

 apply (rule_tac [2] series_pos_le, auto)

 apply (drule_tac x = m in spec, auto)

 done

 text{*Sum of a geometric progression.*}

 lemmas sumr_geometric = geometric_sum [where 'a = real]

 lemma geometric_sums: "abs(x) < 1 ==> (%n. x ^ n) sums (1/(1 - x))"

 apply (case_tac "x = 1")

 apply (auto dest!: LIMSEQ_rabs_realpow_zero2 

         simp add: sumr_geometric sums_def diff_minus add_divide_distrib)

 apply (subgoal_tac "1 / (1 + -x) = 0/ (x - 1) + - 1/ (x - 1) ")

 apply (erule ssubst)

 apply (rule LIMSEQ_add, rule LIMSEQ_divide)

 apply (auto intro: LIMSEQ_const simp add: diff_minus minus_divide_right LIMSEQ_rabs_realpow_zero2)

 done

 text{*Cauchy-type criterion for convergence of series (c.f. Harrison)*}

 lemma summable_convergent_sumr_iff:

  "summable f = convergent (%n. setsum f {0..<n})"

 by (simp add: summable_def sums_def convergent_def)

 lemma summable_Cauchy:

      "summable f =  

       (\<forall>e > 0. \<exists>N. \<forall>m \<ge> N. \<forall>n. abs(setsum f {m..<n}) < e)"

 apply (auto simp add: summable_convergent_sumr_iff Cauchy_convergent_iff [symmetric] Cauchy_def diff_minus[symmetric])

 apply (drule_tac [!] spec, auto)

 apply (rule_tac x = M in exI)

 apply (rule_tac [2] x = N in exI, auto)

 apply (cut_tac [!] m = m and n = n in less_linear, auto)

 apply (frule le_less_trans [THEN less_imp_le], assumption)

 apply (drule_tac x = n in spec, simp)

 apply (drule_tac x = m in spec)

 apply(simp add: setsum_diff[symmetric])

 apply(subst abs_minus_commute)

 apply(simp add: setsum_diff[symmetric])

 apply(simp add: setsum_diff[symmetric])

 done

 text{*Comparison test*}

 lemma summable_comparison_test:

      "[| \<exists>N. \<forall>n \<ge> N. abs(f n) \<le> g n; summable g |] ==> summable f"

 apply (auto simp add: summable_Cauchy)

 apply (drule spec, auto)

 apply (rule_tac x = "N + Na" in exI, auto)

 apply (rotate_tac 2)

 apply (drule_tac x = m in spec)

 apply (auto, rotate_tac 2, drule_tac x = n in spec)

 apply (rule_tac y = "\<Sum>k=m..<n. abs(f k)" in order_le_less_trans)

 apply (rule setsum_abs)

 apply (rule_tac y = "setsum g {m..<n}" in order_le_less_trans)

 apply (auto intro: setsum_mono simp add: abs_interval_iff)

 done

 lemma summable_rabs_comparison_test:

      "[| \<exists>N. \<forall>n \<ge> N. abs(f n) \<le> g n; summable g |] 

       ==> summable (%k. abs (f k))"

 apply (rule summable_comparison_test)

 apply (auto)

 done

 text{*Limit comparison property for series (c.f. jrh)*}

 lemma summable_le:

      "[|\<forall>n. f n \<le> g n; summable f; summable g |] ==> suminf f \<le> suminf g"

 apply (drule summable_sums)+

 apply (auto intro!: LIMSEQ_le simp add: sums_def)

 apply (rule exI)

 apply (auto intro!: setsum_mono)

 done

 lemma summable_le2:

      "[|\<forall>n. abs(f n) \<le> g n; summable g |]  

       ==> summable f & suminf f \<le> suminf g"

 apply (auto intro: summable_comparison_test intro!: summable_le)

 apply (simp add: abs_le_interval_iff)

 done

 (* specialisation for the common 0 case *)

 lemma suminf_0_le:

   fixes f::"nat\<Rightarrow>real"

   assumes gt0: "\<forall>n. 0 \<le> f n" and sm: "summable f"

   shows "0 \<le> suminf f"

 proof -

   let ?g = "(\<lambda>n. (0::real))"

   from gt0 have "\<forall>n. ?g n \<le> f n" by simp

   moreover have "summable ?g" by (rule summable_zero)

   moreover from sm have "summable f" .

   ultimately have "suminf ?g \<le> suminf f" by (rule summable_le)

   then show "0 \<le> suminf f" by (simp add: suminf_zero)

 qed 

 text{*Absolute convergence imples normal convergence*}

 lemma summable_rabs_cancel: "summable (%n. abs (f n)) ==> summable f"

 apply (auto simp add: summable_Cauchy)

 apply (drule spec, auto)

 apply (rule_tac x = N in exI, auto)

 apply (drule spec, auto)

 apply (rule_tac y = "\<Sum>n=m..<n. abs(f n)" in order_le_less_trans)

 apply (auto)

 done

 text{*Absolute convergence of series*}

 lemma summable_rabs:

      "summable (%n. abs (f n)) ==> abs(suminf f) \<le> (\<Sum>n. abs(f n))"

 by (auto intro: LIMSEQ_le LIMSEQ_imp_rabs summable_rabs_cancel summable_sumr_LIMSEQ_suminf)

 subsection{* The Ratio Test*}

 lemma rabs_ratiotest_lemma: "[| c \<le> 0; abs x \<le> c * abs y |] ==> x = (0::real)"

 apply (drule order_le_imp_less_or_eq, auto)

 apply (subgoal_tac "0 \<le> c * abs y")

 apply (simp add: zero_le_mult_iff, arith)

 done

 lemma le_Suc_ex: "(k::nat) \<le> l ==> (\<exists>n. l = k + n)"

 apply (drule le_imp_less_or_eq)

 apply (auto dest: less_imp_Suc_add)

 done

 lemma le_Suc_ex_iff: "((k::nat) \<le> l) = (\<exists>n. l = k + n)"

 by (auto simp add: le_Suc_ex)

 (*All this trouble just to get 0<c *)

 lemma ratio_test_lemma2:

      "[| \<forall>n \<ge> N. abs(f(Suc n)) \<le> c*abs(f n) |]  

       ==> 0 < c | summable f"

 apply (simp (no_asm) add: linorder_not_le [symmetric])

 apply (simp add: summable_Cauchy)

 apply (safe, subgoal_tac "\<forall>n. N < n --> f (n) = 0")

  prefer 2

  apply clarify

  apply(erule_tac x = "n - 1" in allE)

  apply (simp add:diff_Suc split:nat.splits)

  apply (blast intro: rabs_ratiotest_lemma)

 apply (rule_tac x = "Suc N" in exI, clarify)

 apply(simp cong:setsum_ivl_cong)

 done

 lemma ratio_test:

      "[| c < 1; \<forall>n \<ge> N. abs(f(Suc n)) \<le> c*abs(f n) |]  

       ==> summable f"

 apply (frule ratio_test_lemma2, auto)

 apply (rule_tac g = "%n. (abs (f N) / (c ^ N))*c ^ n" 

        in summable_comparison_test)

 apply (rule_tac x = N in exI, safe)

 apply (drule le_Suc_ex_iff [THEN iffD1])

 apply (auto simp add: power_add realpow_not_zero)

 apply (induct_tac "na", auto)

 apply (rule_tac y = "c*abs (f (N + n))" in order_trans)

 apply (auto intro: mult_right_mono simp add: summable_def)

 apply (simp add: mult_ac)

 apply (rule_tac x = "abs (f N) * (1/ (1 - c)) / (c ^ N)" in exI)

 apply (rule sums_divide) 

 apply (rule sums_mult) 

 apply (auto intro!: geometric_sums)

 done

 text{*Differentiation of finite sum*}

 lemma DERIV_sumr [rule_format (no_asm)]:

      "(\<forall>r. m \<le> r & r < (m + n) --> DERIV (%x. f r x) x :> (f' r x))  

       --> DERIV (%x. \<Sum>n=m..<n::nat. f n x) x :> (\<Sum>r=m..<n. f' r x)"

 apply (induct "n")

 apply (auto intro: DERIV_add)

 done

 end