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Definition df-seq 11042
Description: Define a general-purpose operation that builds an recursive sequence (i.e. a function on the natural numbers  NN 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 11054 and seqp1 11056. 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  F with values 1, 1/2, 1/4, 1/8,... would be transformed into the output sequence  seq  1 (  +  ,  F ) with values 1, 3/2, 7/4, 15/8,.., so that  (  seq  1
(  +  ,  F
) `  1 )  =  1,  (  seq  1 (  +  ,  F ) `  2
)  = 3/2, etc. In other words, 
seq  M (  +  ,  F ) transforms a sequence  F into an infinite series.  seq  M (  +  ,  F )  ~~>  2 means "the sum of F(n) from n = M to infinity is 2." Since limits are unique (climuni 12021), by climdm 12023 the "sum of F(n) from n = 1 to infinity" can be expressed as  (  ~~>  `  seq  1
(  +  ,  F
) ) (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 
<. M ,  ( F `
 M ) >., 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 11005 through om2uzf1oi 11011, originally proved by Raph Levien for use with df-exp 11100 and later generalized for arbitrary recursive sequences. Definition df-sum 12154 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  |-  seq  M
(  .+  ,  F
)  =  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )
Distinct variable groups:    x,  .+ , y    x, F, y    x, M, y

Detailed syntax breakdown of Definition df-seq
StepHypRef Expression
1 c.pl . . 3  class  .+
2 cF . . 3  class  F
3 cM . . 3  class  M
41, 2, 3cseq 11041 . 2  class  seq  M
(  .+  ,  F
)
5 vx . . . . 5  set  x
6 vy . . . . 5  set  y
7 cvv 2790 . . . . 5  class  _V
85cv 1623 . . . . . . 7  class  x
9 c1 8734 . . . . . . 7  class  1
10 caddc 8736 . . . . . . 7  class  +
118, 9, 10co 5820 . . . . . 6  class  ( x  +  1 )
126cv 1623 . . . . . . 7  class  y
1311, 2cfv 5222 . . . . . . 7  class  ( F `
 ( x  + 
1 ) )
1412, 13, 1co 5820 . . . . . 6  class  ( y 
.+  ( F `  ( x  +  1
) ) )
1511, 14cop 3645 . . . . 5  class  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >.
165, 6, 7, 7, 15cmpt2 5822 . . . 4  class  ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. )
173, 2cfv 5222 . . . . 5  class  ( F `
 M )
183, 17cop 3645 . . . 4  class  <. M , 
( F `  M
) >.
1916, 18crdg 6418 . . 3  class  rec (
( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )
20 com 4656 . . 3  class  om
2119, 20cima 4692 . 2  class  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )
224, 21wceq 1624 1  wff  seq  M
(  .+  ,  F
)  =  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )
Colors of variables: wff set class
This definition is referenced by:  seqex  11043  seqeq1  11044  seqeq2  11045  seqeq3  11046  nfseq  11051  seqval  11052
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