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Definition df-iseq 8873
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 8882 and at successors. 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 seq1( + , 𝐹) with values 1, 3/2, 7/4, 15/8,.., so that (seq1( + , 𝐹)‘1) = 1, (seq1( + , 𝐹)‘2) = 3/2, etc. In other words, seq𝑀( + , 𝐹) transforms a sequence 𝐹 into an infinite series.

Internally, the frec function generates as its values a set of ordered pairs starting at 𝑀, (𝐹𝑀)⟩, 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 seq𝑀( + , 𝐹, 𝑆) = ran frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
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 8872 . 2 class seq𝑀( + , 𝐹, 𝑆)
6 vx . . . . 5 setvar x
7 vy . . . . 5 setvar y
8 cuz 8229 . . . . . 6 class
94, 8cfv 4845 . . . . 5 class (ℤ𝑀)
106cv 1241 . . . . . . 7 class x
11 c1 6692 . . . . . . 7 class 1
12 caddc 6694 . . . . . . 7 class +
1310, 11, 12co 5455 . . . . . 6 class (x + 1)
147cv 1241 . . . . . . 7 class y
1513, 3cfv 4845 . . . . . . 7 class (𝐹‘(x + 1))
1614, 15, 1co 5455 . . . . . 6 class (y + (𝐹‘(x + 1)))
1713, 16cop 3370 . . . . 5 class ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩
186, 7, 9, 2, 17cmpt2 5457 . . . 4 class (x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩)
194, 3cfv 4845 . . . . 5 class (𝐹𝑀)
204, 19cop 3370 . . . 4 class 𝑀, (𝐹𝑀)⟩
2118, 20cfrec 5917 . . 3 class frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
2221crn 4289 . 2 class ran frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
235, 22wceq 1242 1 wff seq𝑀( + , 𝐹, 𝑆) = ran frec((x (ℤ𝑀), y 𝑆 ↦ ⟨(x + 1), (y + (𝐹‘(x + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
Colors of variables: wff set class
This definition is referenced by:  iseqeq1  8874  iseqeq2  8875  iseqeq3  8876  iseqeq4  8877  nfiseq  8878  iseqval  8880
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