Machine clients are increasingly making use of the Web to perform tasks. While Web services traditionally mimic remote procedure calling interfaces, a new generation of so-called hypermedia APIs works through hyperlinks and forms, in a way similar to how people browse the Web. This means that existing composition techniques, which determine a procedural plan upfront, are not sufficient to consume hypermedia APIs, which need to be navigated at runtime. Clients instead need a more dynamic plan that allows them to follow hyperlinks and use forms with a preset goal. Therefore, in this paper, we show how compositions of hypermedia APIs can be created by generic Semantic Web reasoners. This is achieved through the generation of a proof based on semantic descriptions of the APIs' functionality. To pragmatically verify the applicability of compositions, we introduce the notion of pre-execution and post-execution proofs. The runtime interaction between a client and a server is guided by proofs but driven by hypermedia, allowing the client to react to the application's actual state indicated by the server's response. We describe how to generate compositions from descriptions, discuss a computer-assisted process to generate descriptions, and verify reasoner performance on various composition tasks using a benchmark suite. The experimental results lead to the conclusion that proof-based consumption of hypermedia APIs is a feasible strategy at Web scale.