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Definition df-seq 13016
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 seq1 13028 and seqp1 13030. 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 14502), by climdm 14504 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 rec 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 rec is exactly the sequence we want, and we just extract the range (restricted to omega) and throw away the domain.

This definition has its roots in a series of theorems from om2uz0i 12960 through om2uzf1oi 12966, originally proved by Raph Levien for use with df-exp 13075 and later generalized for arbitrary recursive sequences. Definition df-sum 14636 extracts the summation values from partial (finite) and complete (infinite) series. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 4-Sep-2013.)

Assertion
Ref Expression
df-seq seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
Distinct variable groups:   𝑥, + ,𝑦   𝑥,𝐹,𝑦   𝑥,𝑀,𝑦

Detailed syntax breakdown of Definition df-seq
StepHypRef Expression
1 c.pl . . 3 class +
2 cF . . 3 class 𝐹
3 cM . . 3 class 𝑀
41, 2, 3cseq 13015 . 2 class seq𝑀( + , 𝐹)
5 vx . . . . 5 setvar 𝑥
6 vy . . . . 5 setvar 𝑦
7 cvv 3340 . . . . 5 class V
85cv 1631 . . . . . . 7 class 𝑥
9 c1 10149 . . . . . . 7 class 1
10 caddc 10151 . . . . . . 7 class +
118, 9, 10co 6814 . . . . . 6 class (𝑥 + 1)
126cv 1631 . . . . . . 7 class 𝑦
1311, 2cfv 6049 . . . . . . 7 class (𝐹‘(𝑥 + 1))
1412, 13, 1co 6814 . . . . . 6 class (𝑦 + (𝐹‘(𝑥 + 1)))
1511, 14cop 4327 . . . . 5 class ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩
165, 6, 7, 7, 15cmpt2 6816 . . . 4 class (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)
173, 2cfv 6049 . . . . 5 class (𝐹𝑀)
183, 17cop 4327 . . . 4 class 𝑀, (𝐹𝑀)⟩
1916, 18crdg 7675 . . 3 class rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
20 com 7231 . . 3 class ω
2119, 20cima 5269 . 2 class (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
224, 21wceq 1632 1 wff seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
Colors of variables: wff setvar class
This definition is referenced by:  seqex  13017  seqeq1  13018  seqeq2  13019  seqeq3  13020  nfseq  13025  seqval  13026
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