Many automatic theorem provers are restricted to untyped logics, and existing translations from typed logics are bulky or unsound. Recent research proposes monotonicity as a means to remove some clutter when translating monomorphic to untyped first-order logic. Here we pursue this approach systematically, analysing formally a variety of encodings that further improve on efficiency while retaining soundness and completeness. We extend the approach to rank-1 polymorphism and present alternative schemes that lighten the translation of polymorphic symbols based on the novel notion of "cover". The new encodings are implemented in Isabelle/HOL as part of the Sledgehammer tool. We include informal proofs of soundness and correctness, and have formalised the monomorphic part of this work in Isabelle/HOL. Our evaluation finds the new encodings vastly superior to previous schemes.

@article{BlanchetteBPS13,
  title = {Encoding Monomorphic and Polymorphic Types},
  author = {Jasmin Christian Blanchette and Sascha Böhme and Andrei Popescu and Nicholas Smallbone},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 12, Issue 4},
  year = {2017},
  doi = {10.2168/lmcs-12(4:13)2016}
}