Substructural Prolog?
Logic programming with substructural logics has been studied, starting in the second half of the 90's. I am not an expert, but I can probably provide enough references to get you going.

There is Dale Miller's Lolli, a programming language for linear logic programming.

Major research was carried out by Frank Pfenning and his coworkers and students. For example:


Frank Pfenning: lecture notes on Linear logic programming, part of a course on linear logic (the web site also has some software)
Iliano Cervesato, Frank Pfenning: A Linear Logical Framework
Jeff Polakow, Frank Pfenning: Properties of Terms in Continuation-Passing Style in an Ordered Logical Framework
Frank Pfenning, Robert J. Simmons: Substructural Operational Semantics as Ordered Logic Programming


I am pretty sure someone implemented logic programming for ordered linear logic, poke around the work done by Jeff Polakow and Frank Pfenning. I don't seem to be able to find it right now.