30 research outputs found
Efficiently Integrating Boolean Reasoning and Mathematical Solving
Many real-world problems require the ability of reasoning efficiently on formulae which are boolean combinations of boolean and unquantified mathematical propositions. This task requires a fruitful combination of efficient boolean reasoning and mathematical solving capabilities. SAT tools and mathematical reasoners are respectively very effective on one of these activities each, but not on both. In this paper we present a formal framework, a generalized algorithm and architecture for integrating boolean reasoners and mathematical solvers so that they can efficiently solve boolean combinations of boolean and unquantified mathematical propositions. We describe many techniques to optimize this integration, and highlight the main requirements for SAT tools and mathematicalsolvers to maximize the benefits of their integration
A Synthesis of the Procedural and Declarative Styles of Interactive Theorem Proving
We propose a synthesis of the two proof styles of interactive theorem
proving: the procedural style (where proofs are scripts of commands, like in
Coq) and the declarative style (where proofs are texts in a controlled natural
language, like in Isabelle/Isar). Our approach combines the advantages of the
declarative style - the possibility to write formal proofs like normal
mathematical text - and the procedural style - strong automation and help with
shaping the proofs, including determining the statements of intermediate steps.
Our approach is new, and differs significantly from the ways in which the
procedural and declarative proof styles have been combined before in the
Isabelle, Ssreflect and Matita systems. Our approach is generic and can be
implemented on top of any procedural interactive theorem prover, regardless of
its architecture and logical foundations. To show the viability of our proposed
approach, we fully implemented it as a proof interface called miz3, on top of
the HOL Light interactive theorem prover. The declarative language that this
interface uses is a slight variant of the language of the Mizar system, and can
be used for any interactive theorem prover regardless of its logical
foundations. The miz3 interface allows easy access to the full set of tactics
and formal libraries of HOL Light, and as such has "industrial strength". Our
approach gives a way to automatically convert any procedural proof to a
declarative counterpart, where the converted proof is similar in size to the
original. As all declarative systems have essentially the same proof language,
this gives a straightforward way to port proofs between interactive theorem
provers
More on the Continuity of Real Functions
In this article we demonstrate basic properties of the continuous functions from R to Rn which correspond to state space equations in control engineering.Narita Keiko - Hirosaki-city, Aomori, JapanKornilowicz Artur - Institute of Informatics, University of BiaĆystok, Sosnowa 64, 15-887 BiaĆystok, PolandShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.CzesĆaw ByliĆski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.CzesĆaw ByliĆski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.CzesĆaw ByliĆski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.CzesĆaw ByliĆski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata DarmochwaĆ. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577-580, 2005.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces Rn. Formalized Mathematics, 15(2):65-72, 2007, doi:10.2478/v10037-007-0008-5.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Artur KorniĆowicz. Arithmetic operations on functions from sets into functional sets. Formalized Mathematics, 17(1):43-60, 2009, doi:10.2478/v10037-009-0005-y.Keiichi Miyajima and Yasunari Shidama. Riemann integral of functions from R into Rn. Formalized Mathematics, 17(2):179-185, 2009, doi: 10.2478/v10037-009-0021-y.Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004.Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. More on continuous functions on normed linear spaces. Formalized Mathematics, 19(1):45-49, 2011, doi: 10.2478/v10037-011-0008-3.Beata Padlewska and Agata DarmochwaĆ. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Jan PopioĆek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Konrad Raczkowski and PaweĆ Sadowski. Real function continuity. Formalized Mathematics, 1(4):787-791, 1990.Konrad Raczkowski and PaweĆ Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992
On the Closure Operator and the Closure System of Many Sorted Sets
this paper definitions of many sorted closure system and many sorted closure operator are introduced. These notations are also introduced in [10], but in another meaning. In this article closure system is absolutely multiplicative subset family of many sorted sets and in [10] is many sorted absolutely multiplicative subset family of many sorted sets. Analogously, closure operator is function between many sorted sets and in [10] is many sorted function from a many sorted set into a many sorted set. MML Identifier: CLOSURE