Proof Mining - Mathematics, Algorithms and Proofs

Proof Mining
THEOREM 3 (K./Paulo Oliva, APAL 2003) Let
dist1(f, Pn) := inf
p∈Pn
continuity for f.
f − p1 and ω a modulus of uniform
Ψ(ω, n, ε) := min{ cnε
8(n+1) 2 , cnε
2
cn := ⌊n/2⌋!⌈n/2⌉!
2 4n+3 (n+1) 3n+1 and
ωn(ε) := min{ω( ε
4 ),
Then ∀n ∈ IN, p1, p2 ∈ Pn
∀ε ∈ Q ∗ +(
ε
40(n+1) 4 ⌈ 1
ω(1) ⌉}.
cnε ωn( 2 )}, where
2
(f−pi1−dist1(f, Pn) ≤ Ψ(ω, n, ε)) → p1−p21 ≤ ε).
i=1
Ulrich Kohlenbach 2