1,341 research outputs found
A Vernacular for Coherent Logic
We propose a simple, yet expressive proof representation from which proofs
for different proof assistants can easily be generated. The representation uses
only a few inference rules and is based on a frag- ment of first-order logic
called coherent logic. Coherent logic has been recognized by a number of
researchers as a suitable logic for many ev- eryday mathematical developments.
The proposed proof representation is accompanied by a corresponding XML format
and by a suite of XSL transformations for generating formal proofs for
Isabelle/Isar and Coq, as well as proofs expressed in a natural language form
(formatted in LATEX or in HTML). Also, our automated theorem prover for
coherent logic exports proofs in the proposed XML format. All tools are
publicly available, along with a set of sample theorems.Comment: CICM 2014 - Conferences on Intelligent Computer Mathematics (2014
The computability path ordering
This paper aims at carrying out termination proofs for simply typed
higher-order calculi automatically by using ordering comparisons. To this end,
we introduce the computability path ordering (CPO), a recursive relation on
terms obtained by lifting a precedence on function symbols. A first version,
core CPO, is essentially obtained from the higher-order recursive path ordering
(HORPO) by eliminating type checks from some recursive calls and by
incorporating the treatment of bound variables as in the com-putability
closure. The well-foundedness proof shows that core CPO captures the essence of
computability arguments \'a la Tait and Girard, therefore explaining its name.
We further show that no further type check can be eliminated from its recursive
calls without loosing well-foundedness, but for one for which we found no
counterexample yet. Two extensions of core CPO are then introduced which allow
one to consider: the first, higher-order inductive types; the second, a
precedence in which some function symbols are smaller than application and
abstraction
Uniform Substitution for Differential Game Logic
This paper presents a uniform substitution calculus for differential game
logic (dGL). Church's uniform substitutions substitute a term or formula for a
function or predicate symbol everywhere. After generalizing them to
differential game logic and allowing for the substitution of hybrid games for
game symbols, uniform substitutions make it possible to only use axioms instead
of axiom schemata, thereby substantially simplifying implementations. Instead
of subtle schema variables and soundness-critical side conditions on the
occurrence patterns of logical variables to restrict infinitely many axiom
schema instances to sound ones, the resulting axiomatization adopts only a
finite number of ordinary dGL formulas as axioms, which uniform substitutions
instantiate soundly. This paper proves soundness and completeness of uniform
substitutions for the monotone modal logic dGL. The resulting axiomatization
admits a straightforward modular implementation of dGL in theorem provers
Sharing a Library between Proof Assistants: Reaching out to the HOL Family
We observe today a large diversity of proof systems. This diversity has the
negative consequence that a lot of theorems are proved many times. Unlike
programming languages, it is difficult for these systems to co-operate because
they do not implement the same logic. Logical frameworks are a class of theorem
provers that overcome this issue by their capacity of implementing various
logics. In this work, we study the STTforall logic, an extension of Simple Type
Theory that has been encoded in the logical framework Dedukti. We present a
translation from this logic to OpenTheory, a proof system and interoperability
tool between provers of the HOL family. We have used this translation to export
an arithmetic library containing Fermat's little theorem to OpenTheory and to
two other proof systems that are Coq and Matita.Comment: In Proceedings LFMTP 2018, arXiv:1807.0135
Probabilistic Interval Temporal Logic and Duration Calculus with Infinite Intervals: Complete Proof Systems
The paper presents probabilistic extensions of interval temporal logic (ITL)
and duration calculus (DC) with infinite intervals and complete Hilbert-style
proof systems for them. The completeness results are a strong completeness
theorem for the system of probabilistic ITL with respect to an abstract
semantics and a relative completeness theorem for the system of probabilistic
DC with respect to real-time semantics. The proposed systems subsume
probabilistic real-time DC as known from the literature. A correspondence
between the proposed systems and a system of probabilistic interval temporal
logic with finite intervals and expanding modalities is established too.Comment: 43 page
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