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Definition df-seq 12615
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 12627 and seqp1 12629. 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 14073), by climdm 14075 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 12559 through om2uzf1oi 12565, originally proved by Raph Levien for use with df-exp 12674 and later generalized for arbitrary recursive sequences. Definition df-sum 14207 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 12614 . 2 class seq𝑀( + , 𝐹)
5 vx . . . . 5 setvar 𝑥
6 vy . . . . 5 setvar 𝑦
7 cvv 3168 . . . . 5 class V
85cv 1473 . . . . . . 7 class 𝑥
9 c1 9789 . . . . . . 7 class 1
10 caddc 9791 . . . . . . 7 class +
118, 9, 10co 6523 . . . . . 6 class (𝑥 + 1)
126cv 1473 . . . . . . 7 class 𝑦
1311, 2cfv 5786 . . . . . . 7 class (𝐹‘(𝑥 + 1))
1412, 13, 1co 6523 . . . . . 6 class (𝑦 + (𝐹‘(𝑥 + 1)))
1511, 14cop 4126 . . . . 5 class ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩
165, 6, 7, 7, 15cmpt2 6525 . . . 4 class (𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩)
173, 2cfv 5786 . . . . 5 class (𝐹𝑀)
183, 17cop 4126 . . . 4 class 𝑀, (𝐹𝑀)⟩
1916, 18crdg 7365 . . 3 class rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
20 com 6930 . . 3 class ω
2119, 20cima 5027 . 2 class (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
224, 21wceq 1474 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  12616  seqeq1  12617  seqeq2  12618  seqeq3  12619  nfseq  12624  seqval  12625
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