Proof Mining - Mathematics, Algorithms and Proofs

Proof Mining
3. Attempt: Monotone functional interpretation
(K.96)
Definition 1 (Howard) (x ∗ majorizes x):
x ∗ maj0 x :≡ x ∗ ≥ x,
x ∗ majρ→τ x :≡ ∀y ∗ , y(y ∗ majρy → x ∗ y ∗ majτ xy).
Extract Φ ∗ , Ω ∗ with
∃Φ
∃Ω
Φ ∗ maj Φ ∧ ∀x, gAqf (x, Φ(x, g), g(Φ(x, g)))
Ω ∗ maj Ω ∧ ∀u, Y ( . . . → Bqf (u, Ω(u, Y )))
Define F ∗ (u) := Ω ∗ (u, Φ ∗ ). Then
∀u∃v ≤ F ∗ (u) Bqf (u, v).
.
and
Ulrich Kohlenbach 9