A library for reasoning on randomized algorithms in Coq Version 2
نویسنده
چکیده
3 Cpo.v: Speci cation and properties of a cpo 7 3.1 Ordered type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.1.1 Associated equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.1.2 Setoid relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.1.3 Dual order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.1.4 Order on functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2.1 De nition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2.2 Type of monotonic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2.3 Monotonicity and dual order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2.4 Monotonic functions with 2 arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3.1 Order on natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3.2 Monotonicity and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.4 Basic operators of omega-cpos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.4.1 De nition of cpos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.4.2 Least upper bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4.3 Functional cpos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.5 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.6 Cpo of monotonic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.6.1 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.7 Cpo of continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.8 Product of two cpos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.9 Indexed product of cpo's . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.9.1 Particular cases with one or two elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.10 Fixpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.10.1 Iteration of functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.10.2 Induction principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.11 Directed complete partial orders without minimal element . . . . . . . . . . . . . . . . . . . . . . 22 3.11.1 A cpo is a dcpo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.12 Setoid type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.12.1 A setoid is an ordered set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.12.2 A Type is an ordered set and a setoid with Leibniz equality . . . . . . . . . . . . . . . . . 23 3.12.3 A setoid is a dcpo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
منابع مشابه
A library for reasoning on randomized algorithms in Coq
This library forms a basis for reasoning on randomised algorithms in the proof assistant Coq [6]. The source les are available from the author homepage (http://www.lri.fr/ paulin). They can be compiled using Coq version V8.2pl1. The theoretical basis of this work is a joint work with Philippe Audebaud and is described in [1, 2]. It is developed in the framework of the project SCALP : ANR-07-SES...
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