By Weidong H., Yulong M.

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**Example text**

20 J. M. Farmer, and V. Sorge various meaning functions; thus we will freely use Simple Type Theory (STT) [7, 10] as our general environment. As a general typographical convention, we will underline all the symbols when we refer to the syntactic version of a symbol we already have in our semantics. We will not however underline variables, to ease (somewhat) the readability of the results. We assume that the reader is proﬁcient enough in FOL+EQ and STT so that we do not need to repeat their syntactic deﬁnition here.

A context is a sequence x1 : A1 , . . , xn : An . The minimal type theory we use has three forms of judgements Γ A Γ M :A Γ The last judgement Γ expresses that Γ is a well-typed context. We may write J [x : A] for x : A J. The typing rules are as follows. B Γ M : A Γ N M : B[M ] A =β B We have also Γ A : Ui Γ Γ Ui : Ui+1 Γ A : Ui+1 We express ﬁnally that each universe Ui is closed under the product operation. B : Ui 42 T. Coquand and A. Spiwack These rules have an intuitive interpretation in set theory, where Ui are Grothendieck universes [2].

Implementing proof tools by orchestrating non-trivial calculations and deductions is a diﬃcult task. In providing a link to abstract theory mechanisms we Supported by BMBF project “Verisoft”. M. Kauers et al. ): MKM/Calculemus 2007, LNAI 4573, pp. 27–39, 2007. c Springer-Verlag Berlin Heidelberg 2007 28 A. Chaieb and M. Wenzel contribute to systematic development of advanced methods, being organized according to the logical context required at each stage. Thus we transfer principles of modular program and theory development towards proof procedures.

### 3-D wavelet transform for very low bit-rate video coding by Weidong H., Yulong M.

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