A Mechanised Semantics for HOL with Ad-hoc Overloading

Abstract

Isabelle/HOL augments classical higher-order logic with ad-hoc overloading of constant definitions— that is, one constant may have several definitions for non-overlapping types. In this paper, we present a mechanised proof that HOL with ad-hoc overloading is consistent. All our results have been formalised in the HOL4 theorem prover.

Mechanisation of Model-theoretic Conservative Extension for HOL with Ad-hoc Overloading

“…• Our work generalises and replaces the earlier monolithic model construction of Åman Pohjola and Gengelbach [17], and obtains consistency of HOL with ad-hoc overloading as a corollary.…”

“…With overloading, model-theoretic conservativity holds for symbols that are independent of new definitions, as Gengelbach and Weber prove [4]. However, their proof was based on inherited wrong assumptions from Kunčar and Popescu [12], that Åman Pohjola and Gengelbach in their mechanised model construction [17] uncover and correct. Additionally, the mechanisation supports theory extension by the more expressive constant specification [2], which is a definitional mechanism also used in the theorem provers ProofPower and HOL4 to simultaneously introduce several new constants that satisfy some property.…”

“…This can be proved by an argument based on standard semantics [16], where Booleans, function types and the equality constant are interpreted as expected, and type variables are interpreted as elements of a fixed universe of sets. The consistency story for HOL with overloading is a long one [20,15,12,17]. Åman Pohjola and Gengelbach [17] prove HOL with overloading consistent in a machine-checked proof, by constructing models of theories of definitions through a construction that originates from Kunčar and Popescu [12].…”

“…The consistency story for HOL with overloading is a long one [20,15,12,17]. Åman Pohjola and Gengelbach [17] prove HOL with overloading consistent in a machine-checked proof, by constructing models of theories of definitions through a construction that originates from Kunčar and Popescu [12]. Kunčar and Popescu introduce a dependency relation between the symbols of a theory to track dependencies of defined symbols on their definiens.…”

“…• We use the independent fragment to prove a notion of model-theoretic conservativity [4] in a new setting for the lazy ground semantics [17], which delays type variable instantiation and does not instantiate the type of term variables. To the best of our knowledge, this is the first mechanised conservativity result for a logic with overloaded definitions.…”