Added triangular numbers

--- a/src/HOL/ROOT	Tue Jan 22 15:29:22 2019 +0100
+++ b/src/HOL/ROOT	Fri Dec 14 14:33:26 2018 +0100
@@ -625,6 +625,7 @@
     Transfer_Int_Nat
     Transitive_Closure_Table_Ex
     Tree23
+    Triangular_Numbers
     Unification
     While_Combinator_Example
     Word_Type

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/Triangular_Numbers.thy	Fri Dec 14 14:33:26 2018 +0100
@@ -0,0 +1,76 @@
+(*
+  Title:     HOL/ex/Triangular_Numbers.thy
+  Author:    Manuel Eberl, TU München
+*)
+section \<open>Triangular Numbers\<close>
+theory Triangular_Numbers
+  imports Complex_Main
+begin
+
+definition triangle_num :: "nat \<Rightarrow> nat" where
+  "triangle_num n = (n * (n + 1)) div 2"
+
+lemma real_triangle_num:
+  "real (triangle_num n) = real n * (real n + 1) / 2"
+  by (simp add: triangle_num_def field_char_0_class.of_nat_div algebra_simps)
+
+lemma triangle_num_altdef: "triangle_num n = (\<Sum>k\<le>n. k)"
+  by (induction n) (auto simp: triangle_num_def)
+
+
+lemma triangle_num_ge: "triangle_num n \<ge> n"
+  unfolding triangle_num_altdef by (rule member_le_sum) auto
+
+lemma triangle_num_Suc: "triangle_num (Suc n) = triangle_num n + Suc n"
+  by (simp add: triangle_num_altdef)
+
+lemma triangle_num_0 [simp]: "triangle_num 0 = 0"
+  and triangle_num_1 [simp]: "triangle_num 1 = 1"
+  by (simp_all add: triangle_num_def)
+
+lemma triangle_num_numeral [simp]:
+  "triangle_num (numeral n) = fst (divmod (n * Num.inc n) (Num.Bit0 Num.One))"
+  unfolding fst_divmod numeral_mult numeral_inc triangle_num_def ..
+
+lemma triangle_num_eq_0_iff [simp]: "triangle_num n = 0 \<longleftrightarrow> n = 0"
+  using triangle_num_ge[of n] by auto
+
+lemma triangle_num_gt_0_iff [simp]: "triangle_num n > 0 \<longleftrightarrow> n > 0"
+  using triangle_num_eq_0_iff[of n] by linarith
+
+
+lemma strict_mono_triangle_num: "strict_mono triangle_num"
+  unfolding strict_mono_Suc_iff by (auto simp: triangle_num_altdef)
+
+lemma triangle_num_le: "m \<le> n \<Longrightarrow> triangle_num m \<le> triangle_num n"
+  using strict_mono_leD[OF strict_mono_triangle_num] .
+
+lemma triangle_num_less: "m < n \<Longrightarrow> triangle_num m < triangle_num n"
+  using strict_monoD[OF strict_mono_triangle_num] .
+
+lemma triangle_num_less_iff: "triangle_num m < triangle_num n \<longleftrightarrow> m < n"
+  using strict_mono_less[OF strict_mono_triangle_num] .
+
+lemma triangle_num_le_iff: "triangle_num m \<le> triangle_num n \<longleftrightarrow> m \<le> n"
+  using strict_mono_less_eq[OF strict_mono_triangle_num] .
+
+lemma triangle_num_eq_iff: "triangle_num m = triangle_num n \<longleftrightarrow> m = n"
+  using strict_mono_eq[OF strict_mono_triangle_num] .
+
+
+theorem inverse_triangle_num_sums: "(\<lambda>n. 1 / triangle_num (Suc n)) sums 2"
+proof -
+  have "(\<lambda>n. inverse (real (Suc n)) - inverse (real (Suc (Suc n)))) sums
+          (inverse (real (Suc 0)) - 0)"
+    by (intro telescope_sums' LIMSEQ_inverse_real_of_nat)
+  also have "(\<lambda>n. inverse (real (Suc n)) - inverse (real (Suc (Suc n)))) =
+               (\<lambda>n. 1 / real (2 * triangle_num (Suc n)))"
+    by (auto simp: field_simps triangle_num_def)
+  also have "inverse (real (Suc 0)) - 0 = 1"
+    by simp
+  finally have "(\<lambda>n. 2 * (1 / real (2 * triangle_num (Suc n)))) sums (2 * 1)"
+    by (intro sums_mult)
+  thus ?thesis by simp
+qed
+
+end
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