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Additional info for Art Nouveau Ornament / Ornement Art Nouveau / Jugendstil Ornamente / Ornamentación Arte Nuevo
Sample text
Xi+1 then s(xi , v) ≥ s(xi+1 , v). Proof. t. t. s(xi+1 , w) ≥ s(xi+1 , v) then s(xi , v) > s(xi+1 , v). Otherwise, s(xi , v) = s(xi+1 , v). Ordering on Values. The two following properties establish the links between the natural ordering of values in D(xi ) and the minimum and maximum number of stretches in the sub-sequence starting from xi . Property 4. Let X = [x0 , x1 , . . , xn−1 ] be a sequence of variables and let i ∈ [0, n−1] be an integer. ∀v, w ∈ D(xi ) two well-ordered values, v ≤ w ⇒ s(xi , v) ≤ s(xi , w) + 1.
Cip−1 is initial (resp. final) if ci = 0 whenever s is not the initial (resp. a final) state of A. The number of states of # AR is the number of ordered partitions of p, and thus exponential in p. However, it is possible to have a compact encoding via constraints. Toward this, we use K + 1 sequences of p decision variables Sik in the domain {0, 1, . . , R} to encode the states of an arbitrary path of length K (the number of k columns) in # AR . For k ∈ {1, . . , K}, the sequence S0k , S1k , .
Proof. t. Increasing Nvalue. Otherwise, p(xi , v) is the exact minimum number of stretches among well-ordered instantiations I[x0 , . . , xi ] such that I[xi ] = v and s(xi , v) is the exact minimum number of stretches among well-ordered instantiations I[xi , . . , xn−1 ] such that I[xi ] = v. Thus, by construction p(xi , v) + s(xi , v)−1 is the exact minimum number of stretches among well-ordered instantiations I[x0 , x1 , . . , xn−1 ] such that I[xi ] = v. Let Dv ⊆ D be the set of domains such that all domains in Dv are equal to domains in D except D(xi ) which is reduced to {v}.