# An approach to the Gummel map by vector extrapolation by Bertelle R., Russo M.R. PDF

By Bertelle R., Russo M.R.

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Example text

Proof. Suppose that SI is saturated under OR. Let C be a clause derived from S. t. S. We distinguish two cases: – If C is derived from some clauses D = D ∨ A and E = E ∨ ¬B by ordered resolution where D σ and E σ are selected in Dσ and Eσ respectively, then C = (D ∨E )σ with σ = mgu(A, B). By deﬁnition, there exist I-clauses DI and EI in SI such that D and E are respectively instances of DI and EI . t. DI θD =I D , EI θE =I E , AI θD =I A and BI θE =I B. 26 H. Bensaid, R. Caferra, and N. Peltier Let θ = θD θE .

Tn )[s] = f (t1 , . . , ti−1 , ti [s] , ti+1 , . . , tn ) where ti ∈ T . Given an I-term t, var(t) denotes the set of variables of t (from both VX and VN ). t is ground if var(t) = ∅. s] This rewrite system is convergent and together with the rules of Presburger arithmetic, it reduces any I-term not containing arithmetic variables to a standard term. t. →I . e. t =I s iﬀ t ↓= s ↓. g. t. g. a)). Notice that every ground I-term is =I -equivalent to a ground (standard) term. A substitution is a function mapping every standard variable to an I-term and every arithmetic variable to an arithmetic expression.

X). 4 I-Term Ordering We assume given an ordering on I-terms, that satisﬁes the following properties: is total on ground I-terms. e. there is no inﬁnite sequence of terms t1 , . . , tn , . . such that ∀i, ti ti+1 . 3. e. for any substitution σ and terms t, s: t s ⇒ tσ sσ. 4. e. for all I-terms t, s: (t =I s) ⇒ (∀u, t u ⇒ s u) where ∈ { , ≺}. 1. 2. Properties 1, 2, 3 are standard, Property 4 ensures that is compatible with the semantics of I-terms. Given any ordering on terms satisfying the properties 1-3 above, we can extend to an ordering I on I-terms satisfying properties 1-4 as follows: t I s 24 H.

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### An approach to the Gummel map by vector extrapolation methods by Bertelle R., Russo M.R.

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