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Definition df-iseq 8893
 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 iseq1 8902 and iseqp1 8904. 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 , 3/2, etc. In other words, transforms a sequence into an infinite series. Internally, the frec 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 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 frec
Distinct variable groups:   , ,   ,,   ,,   ,,

Detailed syntax breakdown of Definition df-iseq
StepHypRef Expression
1 c.pl . . 3
2 cS . . 3
3 cF . . 3
4 cM . . 3
51, 2, 3, 4cseq 8892 . 2
6 vx . . . . 5
7 vy . . . . 5
8 cuz 8249 . . . . . 6
94, 8cfv 4845 . . . . 5
106cv 1241 . . . . . . 7
11 c1 6712 . . . . . . 7
12 caddc 6714 . . . . . . 7
1310, 11, 12co 5455 . . . . . 6
147cv 1241 . . . . . . 7
1513, 3cfv 4845 . . . . . . 7
1614, 15, 1co 5455 . . . . . 6
1713, 16cop 3370 . . . . 5
186, 7, 9, 2, 17cmpt2 5457 . . . 4
194, 3cfv 4845 . . . . 5
204, 19cop 3370 . . . 4
2118, 20cfrec 5917 . . 3 frec
2221crn 4289 . 2 frec
235, 22wceq 1242 1 frec
 Colors of variables: wff set class This definition is referenced by:  iseqeq1  8894  iseqeq2  8895  iseqeq3  8896  iseqeq4  8897  nfiseq  8898  iseqval  8900
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