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Definition df-seq 12797
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 12809 and seqp1 12811. 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 14277), by climdm 14279 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 12741 through om2uzf1oi 12747, originally proved by Raph Levien for use with df-exp 12856 and later generalized for arbitrary recursive sequences. Definition df-sum 14411 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 12796 . 2 class seq𝑀( + , 𝐹)
5 vx . . . . 5 setvar 𝑥
6 vy . . . . 5 setvar 𝑦
7 cvv 3198 . . . . 5 class V
85cv 1481 . . . . . . 7 class 𝑥
9 c1 9934 . . . . . . 7 class 1
10 caddc 9936 . . . . . . 7 class +
118, 9, 10co 6647 . . . . . 6 class (𝑥 + 1)
126cv 1481 . . . . . . 7 class 𝑦
1311, 2cfv 5886 . . . . . . 7 class (𝐹‘(𝑥 + 1))
1412, 13, 1co 6647 . . . . . 6 class (𝑦 + (𝐹‘(𝑥 + 1)))
1511, 14cop 4181 . . . . 5 class ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩
165, 6, 7, 7, 15cmpt2 6649 . . . 4 class (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)
173, 2cfv 5886 . . . . 5 class (𝐹𝑀)
183, 17cop 4181 . . . 4 class 𝑀, (𝐹𝑀)⟩
1916, 18crdg 7502 . . 3 class rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
20 com 7062 . . 3 class ω
2119, 20cima 5115 . 2 class (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
224, 21wceq 1482 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  12798  seqeq1  12799  seqeq2  12800  seqeq3  12801  nfseq  12806  seqval  12807
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