Definition df-iseq 9432

Description: Define a general-purpose operation that builds a recursive sequence (i.e. a function on the positive integers ℕ 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 9442 and iseqp1 9445. 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, ℚ) with values 1, 3/2, 7/4, 15/8,.., so that (seq1( + , 𝐹, ℚ)‘1) = 1, (seq1( + , 𝐹, ℚ)‘2) = 3/2, etc. In other words, seq𝑀( + , 𝐹, ℚ) 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	⊢ seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 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

Step	Hyp	Ref	Expression
1		c.pl	. . 3 class +
2		cS	. . 3 class 𝑆
3		cF	. . 3 class 𝐹
4		cM	. . 3 class 𝑀
5	1, 2, 3, 4	cseq 9431	. 2 class seq𝑀( + , 𝐹, 𝑆)
6		vx	. . . . 5 setvar 𝑥
7		vy	. . . . 5 setvar 𝑦
8		cuz 8619	. . . . . 6 class ℤ≥
9	4, 8	cfv 4922	. . . . 5 class (ℤ≥‘𝑀)
10	6	cv 1283	. . . . . . 7 class 𝑥
11		c1 6982	. . . . . . 7 class 1
12		caddc 6984	. . . . . . 7 class +
13	10, 11, 12	co 5532	. . . . . 6 class (𝑥 + 1)
14	7	cv 1283	. . . . . . 7 class 𝑦
15	13, 3	cfv 4922	. . . . . . 7 class (𝐹‘(𝑥 + 1))
16	14, 15, 1	co 5532	. . . . . 6 class (𝑦 + (𝐹‘(𝑥 + 1)))
17	13, 16	cop 3401	. . . . 5 class ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩
18	6, 7, 9, 2, 17	cmpt2 5534	. . . 4 class (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)
19	4, 3	cfv 4922	. . . . 5 class (𝐹‘𝑀)
20	4, 19	cop 3401	. . . 4 class ⟨𝑀, (𝐹‘𝑀)⟩
21	18, 20	cfrec 6000	. . 3 class frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
22	21	crn 4364	. 2 class ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)
23	5, 22	wceq 1284	1 wff seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩)

This definition is referenced by:  iseqex  9433  iseqeq1  9434  iseqeq2  9435  iseqeq3  9436  iseqeq4  9437  nfiseq  9438  iseqval  9440