scholarly journals Structured Induction Proofs in Isabelle/Isar

2006 ◽  
pp. 17-30 ◽  
Author(s):  
Makarius Wenzel
Keyword(s):  
Induction Proofs

I.—Induction Proofs of the Relations between certain Asymptotic Expansions and Corresponding Generalised Hypergeometric Series.

1939 ◽  
Vol 58 ◽  
pp. 1-13 ◽  
Author(s):  
T. M. MacRobert
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Hypergeometric Functions ◽  
Hypergeometric Series ◽  
Multiple Integral ◽  
Contour Integral ◽  
Gamma Functions ◽  
Induction Proofs ◽  
The Subject

The subject of this paper was studied by Orr (Camb. Phil. Trans., vol. xvii, 1898, pp. 171–199; 1899, pp. 283–290), and later by Barnes (Proc. Lond. Math. Soc., ser. 2, vol. v, 1906, pp. 59–116), in whose paper a number of references to earlier work on the subject are given. Formulæ equivalent to (9), (18), and (20) below were given by, Barnes, who derived them by his well‐known method of integrating products and quotients of Gamma Functions. In this paper the formulæ are deduced by induction from simpler formulæ which are established in section 2. In order to simplify the notation, four functions, denoted by P, E, Q, and H, are introduced, the first two in section 3, the last two in section 5. The P and Q functions are merely generalised hypergeometric functions multiplied by convenient factors. The E function, which is equivalent to Barnes's contour integral, is defined as a multiple integral, and from it the asymptotic expansion, with a useful form for the remainder, is easily derived. The H function is a multiple of the E function.

scholarly journals Counterexample-Guided Prophecy for Model Checking Modulo the Theory of Arrays

2021 ◽  
pp. 113-132
Author(s):  
Makai Mann ◽  
Ahmed Irfan ◽  
Alberto Griggio ◽  
Oded Padon ◽  
Clark Barrett
Keyword(s):  
Model Checking ◽  
State Of The Art ◽  
Inductive Reasoning ◽  
Auxiliary Variables ◽  
Abstraction Refinement ◽  
Free Induction ◽  
Induction Proofs ◽  
Infinite State Systems ◽  
Infinite State

AbstractWe develop a framework for model checking infinite-state systems by automatically augmenting them with auxiliary variables, enabling quantifier-free induction proofs for systems that would otherwise require quantified invariants. We combine this mechanism with a counterexample-guided abstraction refinement scheme for the theory of arrays. Our framework can thus, in many cases, reduce inductive reasoning with quantifiers and arrays to quantifier-free and array-free reasoning. We evaluate the approach on a wide set of benchmarks from the literature. The results show that our implementation often outperforms state-of-the-art tools, demonstrating its practical potential.

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