Cumulative versus Noncumulative Ramified Types

Feasible functionals and intersection of ramified types

We show that the basic feasible functions of Cook and Urquhart’s BFF [8,9] are precisely the functionals definable in a natural system of ramified recurrence that uses type intersection (for tier-variants of a common type). This further confirms the stability of BFF as a notion of computational feasibility in higher type. It also suggests the potential usefulness of type-intersection restricted...

Cumulative Inductive Types In Coq

In order to avoid well-know paradoxes associated with self-referential definitions, higher-order dependent type theories stratify the theory using a countably infinite hierarchy of universes (also known as sorts), Type0 : Type1 : · · · . Such type systems are called cumulative if for any type A we have that A : Typei implies A : Typei+1. The Predicative Calculus of Inductive Constructions (pCIC...

Decomposition types in minimally tamely ramified extensions of

Implant failure: regional versus cumulative evaluation.

In this paper the success rate of implant therapy in various bone regions is discussed. The objective is to determine whether differences existed in success rates of cylinder implants placed in different areas in the both maxilla and mandible. Forty four patients have been treated and reviewed five years after the placement of the fixed prosthetic restoration. The patients were provided with a ...

Ramified Frege Arithmetic

Frege’s definitions of zero, predecession, and natural number will be explained below. As for second-order Dedekind-Peano arithmetic, the axiomatization most convenient for our purposes is the following: (1) N0 (2) Nx∧Pxy→ Ny (3) ∀x∀y∀z(Nx∧Pxy∧Pxz→ y = z) (4) ∀x∀y∀z(Nx∧Ny∧Pxz∧Pyz→ x = y) (5) ¬∃x(Nx∧Px0) (6) ∀x(Nx→∃y(Pxy)) (7) ∀F(F0∧∀x∀y(Nx∧Fx∧Pxy→ Fy)→∀x(Nx→ Fx) If (slightly non-standardly) we ...