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 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, (𝐹‘𝑀)⟩, 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.) |