By Freek Wiedijk (auth.), Freek Wiedijk (eds.)
Commemorating the fiftieth anniversary of the 1st time a mathematical theorem was once confirmed by way of a working laptop or computer approach, Freek Wiedijk initiated the current booklet in 2004 by means of inviting formalizations of an explanation of the irrationality of the sq. root of 2 from scientists utilizing a number of theorem proving structures.
The 17 structures integrated during this quantity are one of the such a lot suitable ones for the formalization of arithmetic. The structures are showcased via presentation of the formalized facts and an outline within the type of solutions to a typical questionnaire. The 17 platforms awarded are HOL, Mizar, PVS, Coq, Otter/Ivy, Isabelle/Isar, Alfa/Agda, ACL2, PhoX, IMPS, Metamath, Theorema, Leog, Nuprl, Omega, B technique, and Minlog.
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4  -d(2,m(x,y))|d(2,x)|d(2,y). 5  -d(x,a)| -d(x,b)|x=1. =1. 3] -d(2,m(x,x))|d(2,x). 13  m(x,m(y,z))=m(m(x,y),z). 1] m(m(x,y),z)=m(x,m(y,z)). 16  m(x,y)=m(y,x). 17  m(a,a)=m(2,m(b,b)). 1] m(2,m(b,b))=m(a,a). 30 [hyper,18,3] d(2,m(a,a)). =m(2,x)|m(b,b)=x. 42 [hyper,30,7] d(2,a). 46 [hyper,42,2] m(2,f(2,a))=a. 48 [ur,42,5,6] -d(2,b). 50 [ur,48,7] -d(2,m(b,b)). =m(b,b). =m(2,x). 1] m(2,m(f(2,a),x))=m(a,x). =m(2,m(2,x)). =m(a,a). 1] m(2,m(x,f(2,a)))=m(a,x). =m(a,a). 1] $F. 01 That finishes the proof of the theorem.
Pieper. A Fascinating Country in the World of Computing: Your Guide to Automated Reasoning. World Scientiﬁc, Singapore, 1999. – L. Wos. The Automation of Reasoning: An Experimenter’s Notebook with Otter Tutorial. Academic Press, New York, 1996. – A. Quaife. Automated Development of Fundamental Mathematical Theories. Kluwer Academic, 1992. What is the logic of the system? Untyped ﬁrst-order logic with equality. What is the implementation architecture of the system? Resolution, paramodulation, rewriting, with indexing.
After the script a proof term of one of the lemmas is shown. Coq 35 What needs to be explained about this specific proof ? In this proof we have decided to use as much as possible the notions that were already present in the system. The predicates even and odd are mutually defined in the theory Even. The function div2 and double are defined in Div2. The key point of the main proof (main thm) is the application of the well founded induction lt wf ind (second line of the script) whose statement is: ∀p, P .