ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-seqfrec Unicode version

Definition df-seqfrec 10460
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 10475, seq3-1 10474 and seq3p1 10476. 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 11315), by climdm 11317 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 10459 . 2  class  seq M
(  .+  ,  F
)
5 vx . . . . 5  setvar  x
6 vy . . . . 5  setvar  y
7 cuz 9542 . . . . . 6  class  ZZ>=
83, 7cfv 5228 . . . . 5  class  ( ZZ>= `  M )
9 cvv 2749 . . . . 5  class  _V
105cv 1362 . . . . . . 7  class  x
11 c1 7826 . . . . . . 7  class  1
12 caddc 7828 . . . . . . 7  class  +
1310, 11, 12co 5888 . . . . . 6  class  ( x  +  1 )
146cv 1362 . . . . . . 7  class  y
1513, 2cfv 5228 . . . . . . 7  class  ( F `
 ( x  + 
1 ) )
1614, 15, 1co 5888 . . . . . 6  class  ( y 
.+  ( F `  ( x  +  1
) ) )
1713, 16cop 3607 . . . . 5  class  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >.
185, 6, 8, 9, 17cmpo 5890 . . . 4  class  ( x  e.  ( ZZ>= `  M
) ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. )
193, 2cfv 5228 . . . . 5  class  ( F `
 M )
203, 19cop 3607 . . . 4  class  <. M , 
( F `  M
) >.
2118, 20cfrec 6405 . . 3  class frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. )
2221crn 4639 . 2  class  ran frec (
( x  e.  (
ZZ>= `  M ) ,  y  e.  _V  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )
234, 22wceq 1363 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  10461  seqeq1  10462  seqeq2  10463  seqeq3  10464  nfseq  10469  seq3val  10472  seqvalcd  10473
  Copyright terms: Public domain W3C validator