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Definition df-iseq 9780
Description: Define a general-purpose operation that builds a recursive sequence (i.e. a function on the positive integers  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 iseq1 9791 and iseqp1 9796. 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 ,  QQ ) with values 1, 3/2, 7/4, 15/8,.., so that  (  seq 1
(  +  ,  F ,  QQ ) `  1
)  =  1,  (  seq 1
(  +  ,  F ,  QQ ) `  2
)  = 3/2, etc. In other words,  seq M (  +  ,  F ,  QQ ) transforms a sequence  F into an infinite series.

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 Jim Kingdon, 29-May-2020.)

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

Detailed syntax breakdown of Definition df-iseq
StepHypRef Expression
1 c.pl . . 3  class  .+
2 cS . . 3  class  S
3 cF . . 3  class  F
4 cM . . 3  class  M
51, 2, 3, 4cseq 9779 . 2  class  seq M
(  .+  ,  F ,  S )
6 vx . . . . 5  setvar  x
7 vy . . . . 5  setvar  y
8 cuz 8951 . . . . . 6  class  ZZ>=
94, 8cfv 4981 . . . . 5  class  ( ZZ>= `  M )
106cv 1286 . . . . . . 7  class  x
11 c1 7295 . . . . . . 7  class  1
12 caddc 7297 . . . . . . 7  class  +
1310, 11, 12co 5613 . . . . . 6  class  ( x  +  1 )
147cv 1286 . . . . . . 7  class  y
1513, 3cfv 4981 . . . . . . 7  class  ( F `
 ( x  + 
1 ) )
1614, 15, 1co 5613 . . . . . 6  class  ( y 
.+  ( F `  ( x  +  1
) ) )
1713, 16cop 3434 . . . . 5  class  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >.
186, 7, 9, 2, 17cmpt2 5615 . . . 4  class  ( x  e.  ( ZZ>= `  M
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. )
194, 3cfv 4981 . . . . 5  class  ( F `
 M )
204, 19cop 3434 . . . 4  class  <. M , 
( F `  M
) >.
2118, 20cfrec 6109 . . 3  class frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. )
2221crn 4412 . 2  class  ran frec (
( x  e.  (
ZZ>= `  M ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )
235, 22wceq 1287 1  wff  seq M
(  .+  ,  F ,  S )  =  ran frec ( ( x  e.  (
ZZ>= `  M ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )
Colors of variables: wff set class
This definition is referenced by:  iseqex  9781  iseqeq1  9782  iseqeq2  9783  iseqeq3  9784  iseqeq4  9785  nfiseq  9786  iseqval  9788  iseqvalt  9790
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