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Verified Over-Approximation of the Diameter of Propositionally Factored Transition Systems Read chapter Read first chapter

2015 | OriginalPaper | Chapter

Proof-Producing Reflection for HOL

With an Application to Model Polymorphism

Authors : Benja Fallenstein, Ramana Kumar

Published in: Interactive Theorem Proving

Publisher: Springer International Publishing

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Abstract

We present a reflection principle of the form “If \(\ulcorner \varphi \urcorner \) is provable, then \(\varphi \)” implemented in the HOL4 theorem prover, assuming the existence of a large cardinal. We use the large-cardinal assumption to construct a model of HOL within HOL, and show how to ensure \(\varphi \) has the same meaning both inside and outside of this model. Soundness of HOL implies that if \(\ulcorner \varphi \urcorner \) is provable, then it is true in this model, and hence \(\varphi \) holds. We additionally show how this reflection principle can be extended, assuming an infinite hierarchy of large cardinals, to implement model polymorphism, a technique designed for verifying systems with self-replacement functionality.

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previous chapter Formalizing Size-Optimal Sorting Networks: Extracting a Certified Proof Checker
next chapter Improved Tool Support for Machine-Code Decompilation in HOL4
Footnotes
1
\(\ulcorner \varphi \urcorner \) refers to \(\varphi \) as a syntactic object, represented e.g. by its Gödel number or by an abstract syntax tree.
 
2
Including, as in Gordon [9], polymorphism (type variables and polymorphic constants) and defined constants and type operators.
 
3
This is akin to working within an Isabelle [23] locale which fixes \(\mu \) and \({ {mem}}\) and assumes \(\mathsf{is\_set\_theory }\;{ {mem}}\). (We assume infinity only when necessary).
 
4
The abstract syntax here is the inner HOL rendition of our example term. \(\mathsf{Bool }\) and \(\mathsf{Fun }\;{ {a}}\;{ {b}}\) are abbreviations for \(\mathsf{Tyapp }\;{\text {``bool''}}\;\mathsf{[\,] }\) and \(\mathsf{Tyapp }\;{\text {``fun''}}\;[{ {a}};\;{ {b}}]\).
 
5
To understand the three turnstiles: \(\vdash \) denotes provability in outer HOL (and applies to the whole formula), infix |- denotes provability in inner HOL, and infix \(\models \) states that a sequent is valid according to the semantics of inner HOL.
 
6
To be pedantic, \(\mathsf{to\_inner }\) also depends on the pervasive \({ {mem}}\) relation of the set theory.
 
7
We follow the convention of treating predicates in outer HOL as sets. To be clear \(\mathcal {U}({:}\alpha )\) is a term of type https://static-content.springer.com/image/chp%3A10.1007%2F978-3-319-22102-1_11/340828_1_En_11_IEq84_HTML.gif , and \(\left\{ \right. {}{ {a}}\;|{}\;{ {a}}\;\mathsf{\lessdot }\;{ {x}}\left. \right\} {}\) is a term of type https://static-content.springer.com/image/chp%3A10.1007%2F978-3-319-22102-1_11/340828_1_En_11_IEq86_HTML.gif . These sets-as-predicates are distinct from the Zermelo sets, i.e., the terms of type \(\mu \).
 
8
Also known as indefinite choice, the Hilbert choice principle in HOL provides a constant \((\mathsf{\varepsilon {} })\), usually written as a binder, together with the axiom \((\mathsf{\exists \,{} }{ {x}}.\;{ {P}}\;{ {x}})\;\mathsf{\Rightarrow {} }\;{ {P}}\;(\mathsf{\varepsilon {} }{ {x}}.\;{ {P}}\;{ {x}})\) which holds for any predicate \({ {P}}\).
 
9
\(\mathsf{Num }\) is an abbreviation for \(\mathsf{Tyapp }\;{\text {``num''}}\;\mathsf{[\,] }\).
 
10
When a new constant or type operator is defined, theorems produced by the definition are considered axioms of the resulting theory. The other source of axioms is new-axiom updates, which we do not support since they are not sound in general.
 
11
Any valuation would do for this input since it has no free type or term variables.
 
12
Our proof constructs a potentially slightly different model of inner HOL for each value of t; this is, roughly, the origin of the term model polymorphism.
 
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Metadata
Title
Proof-Producing Reflection for HOL
Authors
Benja Fallenstein
Ramana Kumar
Copyright Year
2015
Book
Interactive Theorem Proving

Print ISBN: 978-3-319-22101-4

Electronic ISBN: 978-3-319-22102-1

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-3-319-22102-1_11

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