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Definition df-seqfrec 10231
Description: Define a general-purpose operation that builds a recursive sequence (i.e., a function on an upper integer set such as or 0) 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 10246, seq3-1 10245 and seq3p1 10247. 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. seq𝑀( + , 𝐹) ⇝ 2 means "the sum of F(n) from n = M to infinity is 2." Since limits are unique (climuni 11074), by climdm 11076 the "sum of F(n) from n = 1 to infinity" can be expressed as ( ⇝ ‘seq1( + , 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, (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 NM, 18-Apr-2005.) (Revised by Jim Kingdon, 4-Nov-2022.)

Assertion
Ref Expression
df-seqfrec seq𝑀( + , 𝐹) = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
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 10230 . 2 class seq𝑀( + , 𝐹)
5 vx . . . . 5 setvar 𝑥
6 vy . . . . 5 setvar 𝑦
7 cuz 9338 . . . . . 6 class
83, 7cfv 5123 . . . . 5 class (ℤ𝑀)
9 cvv 2686 . . . . 5 class V
105cv 1330 . . . . . . 7 class 𝑥
11 c1 7633 . . . . . . 7 class 1
12 caddc 7635 . . . . . . 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 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)
193, 2cfv 5123 . . . . 5 class (𝐹𝑀)
203, 19cop 3530 . . . 4 class 𝑀, (𝐹𝑀)⟩
2118, 20cfrec 6287 . . 3 class frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
2221crn 4540 . 2 class ran frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
234, 22wceq 1331 1 wff seq𝑀( + , 𝐹) = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
Colors of variables: wff set class
This definition is referenced by:  seqex  10232  seqeq1  10233  seqeq2  10234  seqeq3  10235  nfseq  10240  seq3val  10243  seqvalcd  10244
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