  We study pro excision in algebraic K-theory, following Suslin--Wodzicki,
Cuntz--Quillen, Corti\~nas, and Geisser--Hesselholt, as well as Artin--Rees and
continuity properties of Andr\'e--Quillen, Hochschild, and cyclic homology. Our
key tool is to first establish the equivalence of various pro Tor vanishing
conditions which appear in the literature. Using this we prove that all ideals
of commutative, Noetherian rings are pro unital in a certain sense, and show
that such ideals satisfy pro excision in $K$-theory as well as in cyclic and
topological cyclic homology. In addition, our techniques yield a strong form of
the pro Hochschild--Kostant--Rosenberg theorem, an extension to general base
rings of the Cuntz--Quillen excision theorem in periodic cyclic homology, and a
generalisation of the Fe\u{\i}gin--Tsygan theorem.
