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Definition df-seqfrec 10233
Description: Define a general-purpose operation that builds a recursive sequence (i.e., a function on an upper integer set such as  NN or  NN0) 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 seqf 10248, seq3-1 10247 and seq3p1 10249. 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 11076), by climdm 11078 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 frec 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 frec is exactly the sequence we want, and we just extract the range and throw away the domain.

(Contributed by NM, 18-Apr-2005.) (Revised by Jim Kingdon, 4-Nov-2022.)

Assertion
Ref Expression
df-seqfrec  |-  seq M
(  .+  ,  F
)  =  ran frec (
( x  e.  (
ZZ>= `  M ) ,  y  e.  _V  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )
Distinct variable groups:    x,  .+ , y    x, F, y    x, M, y

Detailed syntax breakdown of Definition df-seqfrec
StepHypRef Expression
1 c.pl . . 3  class  .+
2 cF . . 3  class  F
3 cM . . 3  class  M
41, 2, 3cseq 10232 . 2  class  seq M
(  .+  ,  F
)
5 vx . . . . 5  setvar  x
6 vy . . . . 5  setvar  y
7 cuz 9340 . . . . . 6  class  ZZ>=
83, 7cfv 5123 . . . . 5  class  ( ZZ>= `  M )
9 cvv 2686 . . . . 5  class  _V
105cv 1330 . . . . . . 7  class  x
11 c1 7635 . . . . . . 7  class  1
12 caddc 7637 . . . . . . 7  class  +
1310, 11, 12co 5774 . . . . . 6  class  ( x  +  1 )
146cv 1330 . . . . . . 7  class  y
1513, 2cfv 5123 . . . . . . 7  class  ( F `
 ( x  + 
1 ) )
1614, 15, 1co 5774 . . . . . 6  class  ( y 
.+  ( F `  ( x  +  1
) ) )
1713, 16cop 3530 . . . . 5  class  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >.
185, 6, 8, 9, 17cmpo 5776 . . . 4  class  ( x  e.  ( ZZ>= `  M
) ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. )
193, 2cfv 5123 . . . . 5  class  ( F `
 M )
203, 19cop 3530 . . . 4  class  <. M , 
( F `  M
) >.
2118, 20cfrec 6287 . . 3  class frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. )
2221crn 4540 . 2  class  ran frec (
( x  e.  (
ZZ>= `  M ) ,  y  e.  _V  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )
234, 22wceq 1331 1  wff  seq M
(  .+  ,  F
)  =  ran frec (
( x  e.  (
ZZ>= `  M ) ,  y  e.  _V  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )
Colors of variables: wff set class
This definition is referenced by:  seqex  10234  seqeq1  10235  seqeq2  10236  seqeq3  10237  nfseq  10242  seq3val  10245  seqvalcd  10246
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