Quotients revisited for Isabelle/HOL

Abstract

Higher-Order Logic (HOL) is based on a small logic kernel, whose only mechanism for extension is the introduction of safe definitions and of non-empty types. Both extensions are often performed in quotient constructions. To ease the work involved with such quotient constructions, we re-implemented in the Isabelle/HOL theorem prover the quotient package by Homeier. In doing so we extended his work in order to deal with compositions of quotients and also specified completely the procedure of lifting theorems from the raw level to the quotient level. The importance for theorem proving is that many formal verifications, in order to be feasible, require a convenient reasoning infrastructure for quotient constructions.

Program Analysis and Verification Based on Kleene Algebra in Isabelle/HOL

“…Isabelle's quotient package [17] now allows us to formally take the quotient of SKAT expressions with respect to skat-cong. The SKAT axioms then become available for reasoning about SKAT expressions.…”

“…For reasoning about assignments and assertions, we have added first-order syntax and axioms to KAT, following Angus and Kozen's approach to schematic Kleene algebras with tests (SKAT) [2]. Program syntax is defined as syntactic sugar on SKAT expressions; axiomatic algebraic reasoning about these expressions is implemented by using Isabelle's quotient package [16,17].…”

Nonfree Datatypes in Isabelle/HOL

“…This complication, of having to discharge goals that imply well-definedness of a function definition, is inherent in the nature of quotiented types and is shared with the quotient and nominal packages [11,13,14]. For this paper's examples, the conditions are easy to discharge by simplification (but this cannot be guaranteed in general).…”

“…Parts (3)(4)(5) and (8,9) of this process can be eased by existing Isabelle quotient/lifting/transfer packages [12,14].…”

“…The quotient/lifting/transfer packages of Isabelle [12,14] overlap in functionality with our tool for isomorphic transfer. The novelty here is its realization inside a general infrastructure and the possibility of ad hoc higher-order transformations such as currying of functions on finite products.…”