--- a/src/HOL/Library/ASeries.thy Thu Apr 06 17:29:40 2006 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,114 +0,0 @@
-(* Title: HOL/Library/ASeries.thy
- ID: $Id$
- Author: Benjamin Porter, 2006
-*)
-
-
-header {* Arithmetic Series *}
-
-theory ASeries
-imports Main
-begin
-
-section {* Abstract *}
-
-text {* The following document presents a proof of the Arithmetic
-Series Sum formalised in Isabelle/Isar.
-
-{\em Theorem:} The series $\sum_{i=1}^{n} a_i$ where $a_{i+1} = a_i +
-d$ for some constant $d$ has the sum $\frac{n}{2} (a_1 + a_n)$
-(i.e. $n$ multiplied by the arithmetic mean of the first and last
-element).
-
-{\em Informal Proof:} (from
-"http://mathworld.wolfram.com/ArithmeticSeries.html")
- The proof is a simple forward proof. Let $S$ equal the sum above and
- $a$ the first element, then we have
-\begin{tabular}{ll}
- $S$ &$= a + (a+d) + (a+2d) + ... a_n$ \\
- &$= n*a + d (0 + 1 + 2 + ... n-1)$ \\
- &$= n*a + d (\frac{1}{2} * (n-1) * n)$ ..using a simple sum identity \\
- &$= \frac{n}{2} (2a + d(n-1))$ \\
- & ..but $(a+a_n = a + (a + d(n-1)) = 2a + d(n-1))$ so \\
- $S$ &$= \frac{n}{2} (a + a_n)$
-\end{tabular}
-*}
-
-section {* Formal Proof *}
-
-text {* We present a proof for the abstract case of a commutative ring,
-we then instantiate for three common types nats, ints and reals. The
-function @{text "of_nat"} maps the natural numbers into any
-commutative ring.
-*}
-
-lemmas comm_simp [simp] = left_distrib right_distrib add_assoc mult_ac
-
-text {* Next we prove the following simple summation law $\sum_{i=1}^n
-i = \frac {n * (n+1)}{2}$. *}
-
-lemma sum_ident:
- "((1::'a::comm_semiring_1_cancel) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =
- of_nat n*((of_nat n)+1)"
-proof (induct n)
- case 0
- show ?case by simp
-next
- case (Suc n)
- then show ?case by simp
-qed
-
-text {* The abstract theorem follows. Note that $2$ is displayed as
-$1+1$ to keep the structure as abstract as possible. *}
-
-theorem arith_series_general:
- "((1::'a::comm_semiring_1_cancel) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
- of_nat n * (a + (a + of_nat(n - 1)*d))"
-proof cases
- assume ngt1: "n > 1"
- let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"
- have
- "(\<Sum>i\<in>{..<n}. a+?I i*d) =
- ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"
- by (rule setsum_addf)
- also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp
- also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"
- by (simp add: setsum_right_distrib setsum_head_upt)
- also have "(1+1)*\<dots> = (1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..<n}. ?I i)"
- by simp
- also from ngt1 have "{1..<n} = {1..n - 1}"
- by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)
- also from ngt1
- have "(1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..n - 1}. ?I i) = ((1+1)*?n*a + d*?I (n - 1)*?I n)"
- by (simp only: mult_ac sum_ident [of "n - 1"]) (simp add: of_nat_Suc [symmetric])
- finally show ?thesis by simp
-next
- assume "\<not>(n > 1)"
- hence "n = 1 \<or> n = 0" by auto
- thus ?thesis by auto
-qed
-
-subsection {* Instantiation *}
-
-lemma arith_series_nat:
- "(2::nat) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"
-proof -
- have
- "((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =
- of_nat(n) * (a + (a + of_nat(n - 1)*d))"
- by (rule arith_series_general)
- thus ?thesis by (auto simp add: of_nat_id)
-qed
-
-lemma arith_series_int:
- "(2::int) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
- of_nat n * (a + (a + of_nat(n - 1)*d))"
-proof -
- have
- "((1::int) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
- of_nat(n) * (a + (a + of_nat(n - 1)*d))"
- by (rule arith_series_general)
- thus ?thesis by simp
-qed
-
-end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Arithmetic_Series.thy Fri Apr 07 03:20:34 2006 +0200
@@ -0,0 +1,114 @@
+(* Title: HOL/Library/Arithmetic_Series.thy
+ ID: $Id$
+ Author: Benjamin Porter, 2006
+*)
+
+
+header {* Arithmetic Series *}
+
+theory Arithmetic_Series
+imports Main
+begin
+
+section {* Abstract *}
+
+text {* The following document presents a proof of the Arithmetic
+Series Sum formalised in Isabelle/Isar.
+
+{\em Theorem:} The series $\sum_{i=1}^{n} a_i$ where $a_{i+1} = a_i +
+d$ for some constant $d$ has the sum $\frac{n}{2} (a_1 + a_n)$
+(i.e. $n$ multiplied by the arithmetic mean of the first and last
+element).
+
+{\em Informal Proof:} (from
+"http://mathworld.wolfram.com/ArithmeticSeries.html")
+ The proof is a simple forward proof. Let $S$ equal the sum above and
+ $a$ the first element, then we have
+\begin{tabular}{ll}
+ $S$ &$= a + (a+d) + (a+2d) + ... a_n$ \\
+ &$= n*a + d (0 + 1 + 2 + ... n-1)$ \\
+ &$= n*a + d (\frac{1}{2} * (n-1) * n)$ ..using a simple sum identity \\
+ &$= \frac{n}{2} (2a + d(n-1))$ \\
+ & ..but $(a+a_n = a + (a + d(n-1)) = 2a + d(n-1))$ so \\
+ $S$ &$= \frac{n}{2} (a + a_n)$
+\end{tabular}
+*}
+
+section {* Formal Proof *}
+
+text {* We present a proof for the abstract case of a commutative ring,
+we then instantiate for three common types nats, ints and reals. The
+function @{text "of_nat"} maps the natural numbers into any
+commutative ring.
+*}
+
+lemmas comm_simp [simp] = left_distrib right_distrib add_assoc mult_ac
+
+text {* Next we prove the following simple summation law $\sum_{i=1}^n
+i = \frac {n * (n+1)}{2}$. *}
+
+lemma sum_ident:
+ "((1::'a::comm_semiring_1_cancel) + 1)*(\<Sum>i\<in>{1..n}. of_nat i) =
+ of_nat n*((of_nat n)+1)"
+proof (induct n)
+ case 0
+ show ?case by simp
+next
+ case (Suc n)
+ then show ?case by simp
+qed
+
+text {* The abstract theorem follows. Note that $2$ is displayed as
+$1+1$ to keep the structure as abstract as possible. *}
+
+theorem arith_series_general:
+ "((1::'a::comm_semiring_1_cancel) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
+ of_nat n * (a + (a + of_nat(n - 1)*d))"
+proof cases
+ assume ngt1: "n > 1"
+ let ?I = "\<lambda>i. of_nat i" and ?n = "of_nat n"
+ have
+ "(\<Sum>i\<in>{..<n}. a+?I i*d) =
+ ((\<Sum>i\<in>{..<n}. a) + (\<Sum>i\<in>{..<n}. ?I i*d))"
+ by (rule setsum_addf)
+ also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp
+ also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"
+ by (simp add: setsum_right_distrib setsum_head_upt)
+ also have "(1+1)*\<dots> = (1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..<n}. ?I i)"
+ by simp
+ also from ngt1 have "{1..<n} = {1..n - 1}"
+ by (cases n) (auto simp: atLeastLessThanSuc_atLeastAtMost)
+ also from ngt1
+ have "(1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..n - 1}. ?I i) = ((1+1)*?n*a + d*?I (n - 1)*?I n)"
+ by (simp only: mult_ac sum_ident [of "n - 1"]) (simp add: of_nat_Suc [symmetric])
+ finally show ?thesis by simp
+next
+ assume "\<not>(n > 1)"
+ hence "n = 1 \<or> n = 0" by auto
+ thus ?thesis by auto
+qed
+
+subsection {* Instantiation *}
+
+lemma arith_series_nat:
+ "(2::nat) * (\<Sum>i\<in>{..<n}. a+i*d) = n * (a + (a+(n - 1)*d))"
+proof -
+ have
+ "((1::nat) + 1) * (\<Sum>i\<in>{..<n::nat}. a + of_nat(i)*d) =
+ of_nat(n) * (a + (a + of_nat(n - 1)*d))"
+ by (rule arith_series_general)
+ thus ?thesis by (auto simp add: of_nat_id)
+qed
+
+lemma arith_series_int:
+ "(2::int) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
+ of_nat n * (a + (a + of_nat(n - 1)*d))"
+proof -
+ have
+ "((1::int) + 1) * (\<Sum>i\<in>{..<n}. a + of_nat i * d) =
+ of_nat(n) * (a + (a + of_nat(n - 1)*d))"
+ by (rule arith_series_general)
+ thus ?thesis by simp
+qed
+
+end
--- a/src/HOL/Library/Library.thy Thu Apr 06 17:29:40 2006 +0200
+++ b/src/HOL/Library/Library.thy Fri Apr 07 03:20:34 2006 +0200
@@ -21,7 +21,7 @@
Char_ord
Commutative_Ring
Coinductive_List
- ASeries
+ Arithmetic_Series
AssocList
begin
end