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<Paper uid="P93-1026">
  <Title>A COMPLETE AND RECURSIVE FEATURE THEORY*</Title>
  <Section position="9" start_page="198" end_page="199" type="concl">
    <SectionTitle>
8 Quantifier Elimination
</SectionTitle>
    <Paragraph position="0"> In this section we show that our prime formulae satisfy the requirements (5) and (6) of Lemma 4.1 and thus obtain the completeness of FT. We start with the definition of the central notion of a joker.</Paragraph>
    <Paragraph position="1"> A rooted path xp consists of a variable x and a path p. A rooted path xp is called unfree in a prime formula 13 if 3 prefix p' of p 3 yq: x 5PS y and xp' I Yq E \[/3\]. A rooted path is called free in a prime formula/3 if it is not unfree in/3.</Paragraph>
    <Paragraph position="2">  A proper path constraint 7r is called an z-joker for a prime formula/3 if r ~ \[/3\] and one of the following conditions is satisfied:  1. 7r = Axp and xp is free in fl 2. 7r = xp ~ yq and xp is free in/3 3. 7r = yp ~ xq and xq is free in/3.</Paragraph>
    <Paragraph position="3">  Proposition 8.2 It is decidable whether a rooted path is free in a prime formula, and whether a path constraint is an x-joker for a prime formula. Lemma 8.3 Let/3 be a prime formula, x be a variable, 7r be a proper path constraint that is not an x-joker for /3, A be a model of FT, .A,c~ ~ fl, .4,~' ~ /3, and t~' be an z-update of c~. Then A, c~ ~ 7r if and only if.A, a' ~ 7r.</Paragraph>
    <Paragraph position="4"> Lemma 8.4 Let /3 be a prime formula and 7q,..., rn be x-jokers for/3. Then</Paragraph>
    <Paragraph position="6"> The proof of this lemma uses the third axiom scheme, the existence of infinitely many features, and the existence of infinitely many sorts.</Paragraph>
    <Paragraph position="7"> Lemma 8.5 Let/3, /3' be prime formulae and a be a valuation into a model A of FT such that ,4, ~ p 3x(/3 A/3') and .4, ~ p 3x(/3 A -,/3').</Paragraph>
    <Paragraph position="8"> Then every projection of/3' contains an z-joker for  and every variable x one can compute a Boolean combination 6 of prime formulae such that 3x(/j A-,/3') ~FT 6 and 12(6) C 12(qx(fl A ~/3')). Theorem 8.8 For every formula ~b one can compute a Boolean combination 6 of prime formulae such that MFT 6 and V(6) C_ V(/3) Corollary 8.9 FT is a complete and decidable theory. null</Paragraph>
  </Section>
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