src/HOL/Hyperreal/Series.thy

 (*  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
 *) 
 
 header{*Finite Summation and Infinite Series*}
 
 theory Series
 imports SEQ Lim
 begin
 
+(* FIXME why not globally? *)
 declare atLeastLessThan_empty[simp];
-
-syntax sumr :: "[nat,nat,(nat=>real)] => real"
-translations
-  "sumr m n f" => "setsum (f::nat=>real) (atLeastLessThan m n)"
+declare atLeastLessThan_iff[iff]
 
 constdefs
    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 == (@s. f sums s)"
-
-
-lemma sumr_Suc [simp]:
+   "suminf f == SOME s. f sums s"
+
+lemma setsum_Suc[simp]:
   "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
 by (simp add: atLeastLessThanSuc add_commute)
 
 (*
 lemma sumr_add: "sumr m n f + sumr m n g = sumr m n (%n. f n + g n)"

      "n < p --> sumr 0 n f + sumr n p f = sumr 0 p (f::nat=>real)"
 apply (induct "p", auto)
 apply (rename_tac k) 
 apply (subgoal_tac "n=k", auto) 
 done
-*)
 
 lemma sumr_split_add: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
   setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
 by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)
 

 shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
   setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
 using sumr_split_add [of m n p f,symmetric]
 apply (simp add: add_ac)
 done
-
-lemma sumr_diff_mult_const: "sumr 0 n f - (real n*r) = sumr 0 n (%i. f i - r)"
+*)
+
+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 sumr_rabs: "abs(sumr m n  (f::nat=>real)) \<le> sumr m n (%i. abs(f i))"
 by (simp add: setsum_abs)

 lemma sumr_less_bounds_zero [simp]: "n < m ==> sumr m n f = 0"
 by (simp add: atLeastLessThan_empty)
 
 lemma sumr_minus: "sumr m n (%i. - f i) = - sumr m n f"
 by (simp add: Finite_Set.setsum_negf)
-*)
-
-lemma sumr_shift_bounds: "sumr (m+k) (n+k) f = sumr m n (%i. f(i + k))"
+
+lemma sumr_shift_bounds:
+  "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
 by (induct "n", auto)
-
-lemma sumr_minus_one_realpow_zero [simp]: "sumr 0 (2*n) (%i. (-1) ^ Suc i) = 0"
+*)
+
+(* Generalize from real to some algebraic structure? *)
+lemma sumr_minus_one_realpow_zero [simp]:
+  "setsum (%i. (-1) ^ Suc i) {0..<2*n} = (0::real)"
 by (induct "n", auto)
 
-lemma sumr_interval_const:
-     "\<lbrakk>\<forall>n. m \<le> Suc n --> f n = r; m \<le> k\<rbrakk> \<Longrightarrow> sumr m k f = (real(k-m) * r)"
-apply (induct "k", auto) 
+(*
+lemma sumr_interval_const2:
+     "[|\<forall>n\<ge>m. f n = r; m \<le> k|]
+      ==> sumr m k f = (real (k - m) * r)"
+apply (induct "k", auto)
 apply (drule_tac x = k in spec)
 apply (auto dest!: le_imp_less_or_eq)
 apply (simp add: left_distrib real_of_nat_Suc split: nat_diff_split)
 done
-
-lemma sumr_interval_const2:
-     "[|\<forall>n\<ge>m. f n = r; m \<le> k|]
-      ==> sumr m k f = (real (k - m) * r)"
-apply (induct "k", auto) 
-apply (drule_tac x = k in spec)
-apply (auto dest!: le_imp_less_or_eq)
-apply (simp add: left_distrib real_of_nat_Suc split: nat_diff_split)
-done
-
+*)
+(* FIXME split in tow steps
+lemma setsum_nat_set_real_const:
+  "(\<And>n. n\<in>A \<Longrightarrow> f n = r) \<Longrightarrow> setsum f A = real(card A) * r" (is "PROP ?P")
+proof (cases "finite A")
+  case True
+  thus "PROP ?P"
+  proof induct
+    case empty thus ?case by simp
+  next
+    case insert thus ?case by(simp add: left_distrib real_of_nat_Suc)
+  qed
+next
+  case False thus "PROP ?P" by (simp add: setsum_def)
+qed
+ *)
+
+(*
+lemma sumr_le:
+     "[|\<forall>n\<ge>m. 0 \<le> (f n::real); m < k|] ==> setsum f {0..<m} \<le> setsum f {0..<k::nat}"
+apply (induct "k")
+apply (auto simp add: less_Suc_eq_le)
+apply (drule_tac x = k in spec, safe)
+apply (drule le_imp_less_or_eq, safe)
+apply (arith)
+apply (drule_tac a = "sumr 0 m f" in order_refl [THEN add_mono], auto)
+done
 
 lemma sumr_le:
      "[|\<forall>n\<ge>m. 0 \<le> f n; m < k|] ==> sumr 0 m f \<le> sumr 0 k f"
 apply (induct "k")
 apply (auto simp add: less_Suc_eq_le)

 lemma sumr_le2 [rule_format (no_asm)]:
      "(\<forall>r. m \<le> r & r < n --> f r \<le> g r) --> sumr m n f \<le> sumr m n g"
 apply (induct "n")
 apply (auto intro: add_mono simp add: le_def)
 done
-
+*)
+
+(*
 lemma sumr_ge_zero: "(\<forall>n\<ge>m. 0 \<le> f n) --> 0 \<le> sumr m n f"
 apply (induct "n", auto)
 apply (drule_tac x = n in spec, arith)
 done
-
-(* FIXME generalize *)
+*)
+
+(*
 lemma rabs_sumr_rabs_cancel [simp]:
      "abs (sumr m n (%k. abs (f k))) = (sumr m n (%k. abs (f k)))"
 by (induct "n", simp_all add: add_increasing)
 
 lemma sumr_zero [rule_format]:
      "\<forall>n \<ge> N. f n = 0 ==> N \<le> m --> sumr m n f = 0"
 by (induct "n", auto)
+*)
 
 lemma Suc_le_imp_diff_ge2:
-     "[|\<forall>n \<ge> N. f (Suc n) = 0; Suc N \<le> m|] ==> sumr m n f = 0"
-apply (rule sumr_zero) 
+     "[|\<forall>n \<ge> N. f (Suc n) = 0; Suc N \<le> m|] ==> setsum f {m..<n} = 0"
+apply (rule setsum_0')
 apply (case_tac "n", auto)
-done
-
-lemma sumr_one_lb_realpow_zero [simp]: "sumr (Suc 0) n (%n. f(n) * 0 ^ n) = 0"
+apply(erule_tac x = "a - 1" in allE)
+apply (simp split:nat_diff_split)
+done
+
+(* 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_diff: "sumr m n f - sumr m n g = sumr m n (%n. f n - g n)"
 by (simp add: diff_minus setsum_addf setsum_negf)
 *)
+(*
 lemma sumr_subst [rule_format (no_asm)]:
      "(\<forall>p. m \<le> p & p < m+n --> (f p = g p)) --> sumr m n f = sumr m n g"
 by (induct "n", auto)
-
+*)
+
+lemma setsum_bounded:
+  assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{comm_semiring_1_cancel, pordered_ab_semigroup_add})"
+  shows "setsum f A \<le> of_nat(card A) * K"
+proof (cases "finite A")
+  case True
+  thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
+next
+  case False thus ?thesis by (simp add: setsum_def)
+qed
+(*
 lemma sumr_bound [rule_format (no_asm)]:
      "(\<forall>p. m \<le> p & p < m + n --> (f(p) \<le> K))  
-      --> (sumr m (m + n) f \<le> (real n * K))"
+      --> setsum f {m..<m+n::nat} \<le> (real n * K)"
 apply (induct "n")
 apply (auto intro: add_mono simp add: left_distrib real_of_nat_Suc)
 done
-
+*)
+(* FIXME should be generalized
 lemma sumr_bound2 [rule_format (no_asm)]:
      "(\<forall>p. 0 \<le> p & p < n --> (f(p) \<le> K))  
-      --> (sumr 0 n f \<le> (real n * K))"
+      --> setsum f {0..<n::nat} \<le> (real n * K)"
 apply (induct "n")
 apply (auto intro: add_mono simp add: left_distrib real_of_nat_Suc add_commute)
 done
-
+ *)
+(* FIXME a bit specialized for [simp]! *)
 lemma sumr_group [simp]:
-     "sumr 0 n (%m. sumr (m * k) (m*k + k) f) = sumr 0 (n * k) f"
-apply (subgoal_tac "k = 0 | 0 < k", auto)
+     "(\<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 simp:setsum_0')
 apply (induct "n")
-apply (simp_all add: sumr_split_add add_commute)
-done
+apply (simp_all add: setsum_add_nat_ivl add_commute)
+done
+(* FIXME setsum_0[simp] *)
+
 
 subsection{* Infinite Sums, by the Properties of Limits*}
 
 (*----------------------
    suminf is the sum   

 apply (simp add: summable_def suminf_def)
 apply (blast intro: someI2)
 done
 
 lemma summable_sumr_LIMSEQ_suminf: 
-     "summable f ==> (%n. sumr 0 n f) ----> (suminf f)"
+     "summable f ==> (%n. setsum f {0..<n}) ----> (suminf f)"
 apply (simp add: summable_def suminf_def sums_def)
 apply (blast intro: someI2)
 done
 
 (*-------------------

 by Auto_tac
 qed "series_zero";
 next one was called series_zero2
 **********************)
 
+lemma ivl_subset[simp]:
+ "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
+apply(auto simp:linorder_not_le)
+apply(rule ccontr)
+apply(frule subsetCE[where c = n])
+apply(auto simp:linorder_not_le)
+done
+
+lemma ivl_diff[simp]:
+ "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
+by(auto)
+
+
+(* FIXME the last step should work w/o Ball_def, ideally just with
+   setsum_0 and setsum_cong. Currently the simplifier does not simplify
+   the premise x:{i..<j} that ball_cong (or a modified version of setsum_0')
+   generates. FIX simplifier??? *)
 lemma series_zero: 
-     "(\<forall>m. n \<le> m --> f(m) = 0) ==> f sums (sumr 0 n f)"
+     "(\<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:sumr_split_add_minus)
-apply (drule sumr_interval_const2, auto)
-done
+apply (clarsimp simp add:setsum_diff[symmetric] setsum_0' Ball_def)
+done
+
 
 lemma sums_mult: "x sums x0 ==> (%n. c * x(n)) sums (c * x0)"
 by (auto simp add: sums_def setsum_mult [symmetric]
          intro!: LIMSEQ_mult intro: LIMSEQ_const)
 

 
 lemma sums_minus: "x sums x0 ==> (%n. - x n) sums - x0"
 by (auto simp add: sums_def intro!: LIMSEQ_minus simp add: setsum_negf)
 
 lemma sums_group:
-     "[|summable f; 0 < k |] ==> (%n. sumr (n*k) (n*k + k) f) sums (suminf f)"
+     "[|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)
 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)))|]
-      ==> sumr 0 (n + Suc (Suc 0)) f \<le> sumr 0 (Suc (Suc 0) * Suc no + n) f"
+  "[|\<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) 
+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))|]  
-      ==> sumr 0 n f < suminf f"
+      ==> 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> sumr 0 (Suc (Suc 0) * (Suc no) + n) f")
-apply (rule_tac [2] y = "sumr 0 (n+ Suc (Suc 0)) f" in order_trans)
+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 = "sumr 0 n f + (f (n) + f (n + 1))" in order_trans)
+apply (rule_tac [2] y = "setsum f {0..<n} + (f (n) + f (n + 1))" in order_trans)
 apply simp_all 
-apply (subgoal_tac "suminf f \<le> sumr 0 (Suc (Suc 0) * (Suc no) + n) f")
+apply (subgoal_tac "suminf f \<le> setsum f {0..< Suc (Suc 0) * (Suc no) + n}")
 apply (rule_tac [2] y = "suminf f + (f (n) + f (n + 1))" in order_trans)
-prefer 3 apply simp 
+prefer 3 apply simp
 apply (drule_tac [2] x = 0 in spec)
  prefer 2 apply simp 
-apply (subgoal_tac "0 \<le> sumr 0 (Suc (Suc 0) * Suc no + n) f + - suminf f")
-apply (simp add: abs_if) 
+apply (subgoal_tac "0 \<le> setsum f {0 ..< Suc (Suc 0) * Suc no + n} + - suminf f")
+apply (simp add: abs_if)
 apply (auto simp add: linorder_not_less [symmetric])
 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) |] ==> sumr 0 n f \<le> suminf f"
+     "[| 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 = "sumr 0 n f" in LIMSEQ_const)
-apply (erule LIMSEQ_le, blast) 
-apply (rule_tac x = n in exI, clarify) 
-apply (drule le_imp_less_or_eq)
-apply (auto intro: sumr_le)
+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) |] ==> sumr 0 n f < suminf f"
-apply (rule_tac y = "sumr 0 (Suc n) f" in order_less_le_trans)
+     "[| 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.*}
 
-lemma sumr_geometric: "x ~= 1 ==> sumr 0 n (%n. x ^ n) = (x ^ n - 1) / (x - 1)"
+lemma sumr_geometric:
+ "x ~= 1 ==> (\<Sum>i=0..<n. x ^ i) = (x ^ n - 1) / (x - 1::real)"
 apply (induct "n", auto)
 apply (rule_tac c1 = "x - 1" in real_mult_right_cancel [THEN iffD1])
-apply (auto simp add: mult_assoc left_distrib  times_divide_eq)
+apply (auto simp add: mult_assoc left_distrib)
 apply (simp add: right_distrib diff_minus mult_commute)
 done
 
 lemma geometric_sums: "abs(x) < 1 ==> (%n. x ^ n) sums (1/(1 - x))"
 apply (case_tac "x = 1")

 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. sumr 0 n f)"
+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(sumr m n f) < e)"
+      (\<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 (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: sumr_split_add_minus)
+apply(simp add: setsum_diff[symmetric])
 apply(subst abs_minus_commute)
-apply(simp add: sumr_split_add_minus)
-apply (simp add: sumr_split_add_minus)
-done
+apply(simp add: setsum_diff[symmetric])
+apply(simp add: setsum_diff[symmetric])
+done
+(* FIXME move ivl_ lemmas out of this theory *)
 
 text{*Comparison test*}
 
 lemma summable_comparison_test:
      "[| \<exists>N. \<forall>n \<ge> N. abs(f n) \<le> g n; summable g |] ==> summable f"

 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 = "sumr m n (%k. abs (f k))" in order_le_less_trans)
+apply (rule_tac y = "\<Sum>k=m..<n. abs(f k)" in order_le_less_trans)
 apply (rule setsum_abs)
-apply (rule_tac y = "sumr m n g" in order_le_less_trans)
-apply (auto intro: sumr_le2 simp add: abs_interval_iff)
+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))"

 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!: sumr_le2)
+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"

 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 = "sumr m n (%n. abs (f n))" in order_le_less_trans)
+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:

 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 simp add: times_divide_eq)
+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) 

 
 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. sumr m n (%n. f n x)) x :> sumr m n (%r. 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
 
 ML
 {*
-val sumr_Suc = thm"sumr_Suc";
 val sums_def = thm"sums_def";
 val summable_def = thm"summable_def";
 val suminf_def = thm"suminf_def";
 
-val sumr_split_add = thm "sumr_split_add";
 val sumr_minus_one_realpow_zero = thm "sumr_minus_one_realpow_zero";
-val sumr_le2 = thm "sumr_le2";
-val rabs_sumr_rabs_cancel = thm "rabs_sumr_rabs_cancel";
-val sumr_zero = thm "sumr_zero";
 val Suc_le_imp_diff_ge2 = thm "Suc_le_imp_diff_ge2";
 val sumr_one_lb_realpow_zero = thm "sumr_one_lb_realpow_zero";
-val sumr_subst = thm "sumr_subst";
-val sumr_bound = thm "sumr_bound";
-val sumr_bound2 = thm "sumr_bound2";
 val sumr_group = thm "sumr_group";
 val sums_summable = thm "sums_summable";
 val summable_sums = thm "summable_sums";
 val summable_sumr_LIMSEQ_suminf = thm "summable_sumr_LIMSEQ_suminf";
 val sums_unique = thm "sums_unique";