By Cook R. J., Kalbfleisch J. D.

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**Extra info for A generalized moverstayer model for panel data**

**Example text**

If there are inﬁnitely many quantiﬁers then the function associating the type si with each i is recursive. A predicate language with equality contains an additional binary predicate = (the equality predicate) distinct from P1 , . . , Pn . Formulae are deﬁned inductively as usual: • • • Each expression Pi (u1 , . . , uti ) where u1 , . . , uti are variables is an atomic formula (and u1 = u2 is an atomic formula). Atomic formula is a formula, 0 and 1 are formulae. If ϕ and ψ are formulae, then ¬ϕ, ϕ ∧ ψ, ϕ ∨ ψ, ϕ → ψ and ϕ ↔ ψ are formulae.

In: Fayyad UM, et al (eds). Advances in Knowledge Discovery and Data Mining. AAAI Press, Menlo Park California 2. H´ ajek P, Havr´ anek T (1978) Mechanising Hypothesis Formation - Mathematical Foundations for a General Theory. cz/~hajek/guhabook/) 3. H´ ajek P, Havr´ anek T, Chytil M (1983) GUHA Method. Academia, Prague (in Czech) 4. Rauch J (1986) Logical Foundations of Hypothesis Formation from Databases. PhD Thesis, Mathematical Institute of the Czechoslovak Academy of Sciences, Prague (in Czech) 5.

Thus there is p ∈ {1, . . , K} such that ||ψp ||M = 1 that implies TM ∈ Ip and also TM ∈ K j=1 Ij . According to the supposition it means Asf∼ (TM ) = 1 and thus ||(∼ x)(P1 (x), P2 (x))||M = 1. The quantiﬁer ∃k is for each k > 0 deﬁnable in the extension of P = by adding ∃k (see lemma 3). It means that there is a formula Φ∗ of the calculus P = that is logically equivalent to the formula Φ. It also means that Φ∗ is logically equivalent to the formula (∼ x)(P1 (x), P2 (x)). It ﬁnishes the proof.

### A generalized moverstayer model for panel data by Cook R. J., Kalbfleisch J. D.

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