How to Believe a Twelf Proof

Logical systems are represented in LF by giving a full and faithful (adequate) embedding of the deductive apparatus of the logic as canonical forms of certain types and kinds in LF in specified contexts. The collection of contexts over which the representation is adequate is called a world, because it provides generators for the canonical forms in question. Transferring adequacy from one world to another relies on the concept of subordination, which expresses the irrelevance of any “extra” variables in the target world. Given such a representation any metatheoretic property of the logic may be stated and proved in terms of its representation as certain canonical forms. The Twelf meta-theorem prover for LF supports mechanical verification of Π2 (∀∃) sentences that quantify over canonical forms of specified types. The proof of such a Π2 proposition may be seen as a totality proof for an associated relation, with the universal variables designated as “inputs” and the existential variables designated as “outputs”. Such totality proofs often take the form of a lexicographic induction over the structure of various canonical forms. The representation of a logic typically involves higher types of LF; this is called higher-order abstract syntax. Exploiting the LF type structure typically streamlines the presentation, but does not in any way inhibit the expressive power of the Twelf theorem prover. This is because the theorem prover works at the meta-level of LF by induction over the structure of canonical forms of higher type. It is frequently alleged that logical relations arguments cannot be formalized in LF. This is not so; LF is capable of encoding nearly any logic in which a logic relations argument might be expressed. What is true, however, is that the Twelf meta-logic, being limited to Π2 sentences, cannot express meta-theorems whose proof proceeds by logical relations. The reason is that the core idea of logical relations is that the logical complexity of the theorem varies in proportion to