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Mirrors > Home > ILE Home > Th. List > df-iseq | GIF version |
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 9222 and iseqp1 9225.
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.) |
Ref | Expression |
---|---|
df-iseq | ⊢ seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) |
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 9211 | . 2 class seq𝑀( + , 𝐹, 𝑆) |
6 | vx | . . . . 5 setvar 𝑥 | |
7 | vy | . . . . 5 setvar 𝑦 | |
8 | cuz 8473 | . . . . . 6 class ℤ≥ | |
9 | 4, 8 | cfv 4902 | . . . . 5 class (ℤ≥‘𝑀) |
10 | 6 | cv 1242 | . . . . . . 7 class 𝑥 |
11 | c1 6890 | . . . . . . 7 class 1 | |
12 | caddc 6892 | . . . . . . 7 class + | |
13 | 10, 11, 12 | co 5512 | . . . . . 6 class (𝑥 + 1) |
14 | 7 | cv 1242 | . . . . . . 7 class 𝑦 |
15 | 13, 3 | cfv 4902 | . . . . . . 7 class (𝐹‘(𝑥 + 1)) |
16 | 14, 15, 1 | co 5512 | . . . . . 6 class (𝑦 + (𝐹‘(𝑥 + 1))) |
17 | 13, 16 | cop 3378 | . . . . 5 class 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉 |
18 | 6, 7, 9, 2, 17 | cmpt2 5514 | . . . 4 class (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉) |
19 | 4, 3 | cfv 4902 | . . . . 5 class (𝐹‘𝑀) |
20 | 4, 19 | cop 3378 | . . . 4 class 〈𝑀, (𝐹‘𝑀)〉 |
21 | 18, 20 | cfrec 5977 | . . 3 class frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) |
22 | 21 | crn 4346 | . 2 class ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) |
23 | 5, 22 | wceq 1243 | 1 wff seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) |
Colors of variables: wff set class |
This definition is referenced by: iseqex 9213 iseqeq1 9214 iseqeq2 9215 iseqeq3 9216 iseqeq4 9217 nfiseq 9218 iseqval 9220 |
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