MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-seq Unicode version

Definition df-seq 12108
Description: Define a general-purpose operation that builds a recursive sequence (i.e. a function on the positive integers or some other upper integer set) whose value at an index is a function of its previous value and the value of an input sequence at that index. This definition is complicated, but fortunately it is not intended to be used directly. Instead, the only purpose of this definition is to provide us with an object that has the properties expressed by seq1 12120 and seqp1 12122. Typically, those are the main theorems that would be used in practice.

The first operand in the parentheses is the operation that is applied to the previous value and the value of the input sequence (second operand). The operand to the left of the parenthesis is the integer to start from. For example, for the operation , an input sequence with values 1, 1/2, 1/4, 1/8,... would be transformed into the output sequence with values 1, 3/2, 7/4, 15/8,.., so that , (seq1( , )`2)= 3/2, etc. In other words, transforms a sequence into an infinite series. means "the sum of F(n) from n = M to infinity is 2." Since limits are unique (climuni 13375), by climdm 13377 the "sum of F(n) from n = 1 to infinity" can be expressed as (provided the sequence converges) and evaluates to 2 in this example.

Internally, the rec function generates as its values a set of ordered pairs starting at , with the first member of each pair incremented by one in each successive value. So, the range of rec is exactly the sequence we want, and we just extract the range (restricted to omega) and throw away the domain.

This definition has its roots in a series of theorems from om2uz0i 12058 through om2uzf1oi 12064, originally proved by Raph Levien for use with df-exp 12167 and later generalized for arbitrary recursive sequences. Definition df-sum 13509 extracts the summation values from partial (finite) and complete (infinite) series. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 4-Sep-2013.)

Assertion
Ref Expression
df-seq
Distinct variable groups:   , ,   , ,   ,M,

Detailed syntax breakdown of Definition df-seq
StepHypRef Expression
1 c.pl . . 3
2 cF . . 3
3 cM . . 3
41, 2, 3cseq 12107 . 2
5 vx . . . . 5
6 vy . . . . 5
7 cvv 3109 . . . . 5
85cv 1394 . . . . . . 7
9 c1 9514 . . . . . . 7
10 caddc 9516 . . . . . . 7
118, 9, 10co 6296 . . . . . 6
126cv 1394 . . . . . . 7
1311, 2cfv 5593 . . . . . . 7
1412, 13, 1co 6296 . . . . . 6
1511, 14cop 4035 . . . . 5
165, 6, 7, 7, 15cmpt2 6298 . . . 4
173, 2cfv 5593 . . . . 5
183, 17cop 4035 . . . 4
1916, 18crdg 7094 . . 3
20 com 6700 . . 3
2119, 20cima 5007 . 2
224, 21wceq 1395 1
Colors of variables: wff setvar class
This definition is referenced by:  seqex  12109  seqeq1  12110  seqeq2  12111  seqeq3  12112  nfseq  12117  seqval  12118
  Copyright terms: Public domain W3C validator