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ecomes ∃u, v 1 , v 2 . x=f(v 1 , v 2 ) ∧ u=v 1 ). The resultcontains only the relation and function symbols thatoccur in structural base formulas. Make sure each termvariable has a corresponding shape variable by applyingrules ShpInt. For example, ∃v 1 , v 2 . x=f(v 1 , v 2 )becomes ∃v 1 , v 2 , x s , v1 s, vs 2 . sh(v 1)=v1 s ∧ sh(v 2)=v2 s ∧sh(x)=x s ∧ x=f(v 1 , v 2 ). Next, apply congruenceclosure (CongCl) and occur check (OccChk). For example,∃x, u, v 1 , v 2 . x=f(v 1 , v 2 ) ∧ y=f(u, v 2 ) ∧ u=v 1becomes ∃x, u, v 2 . x=f(u, v 2 ) ∧ y=x, whereasx=f(u, v) ∧ u=f(x, v) becomes false. Use HomExprules to ensure that parameter term variables are mapped toparameter shape variables, non-parameter term variablesare mapped to non-parameter shape variables, and thatthe homomorphism property P3) of Definition 11 holds.Repeat CongCl and OccChk rules if needed. For example,∃v 1 , v 2 , x s , v1 s, vs 2 . sh(v 1)=v1 s ∧ sh(v 2)=v2 s ∧ sh(x)=xs ∧x=f(v 1 , v 2 ) becomes ∃v 1 , v 2 , x s , v1, s v2. s sh(v 1 )=v1 s ∧sh(v 2 )=v2 s ∧ sh(x)=xs ∧ x=f(v 1 , v 2 ) ∧ x s =f s (v1 s, vs 2 ).Eliminate all disequalities between term variables usingthe NEQEl rule, which is justified by the negation of theequivalence:t 1 = t 2 ⇐⇒ sh(t 1 ) = sh(t 2 ) ∧ |t 1 = t 2 | sh(t1) = 0 (4)For example, u≠x ∧ sh(u)=u s ∧ sh(x)=x s becomes((u s ≠x s )∨(u s =x s ∧|u≠x| u s≥1))∧sh(u)=u s ∧sh(x)=x s .Repeat previous stages (e.g. DNF, CongCl, OccChk) ifneeded. Convert all cardinality constraints into constraintson parameter term variables, using CCD rules justifiedby (1), e.g. |u≠v| u s=1 becomes (|u 1 ≠v 1 | u s1=0 ∧|u 2 ≠v 2 | u s2=1) ∨ (|u 1 ≠v 1 | u s1=1 ∧ |u 2 ≠v 2 | u s2=0)in the context of u=f(u 1 , v 1 ) ∧ v=f(v 1 , v 2 ) ∧u s =f s (u s 1 , us 2 ) ∧ sh(u)=sh(v)=us ∧ sh(u 1 )=sh(v 1 )=u s 1 ∧sh(u 2 )=sh(v 2 )=u s 2. Finally, to produce the formuladistinct(u s 1 , . . .,us n ) use ShDis to ensure that for every twoshape variables x s 1 and xs 2 occurring in the conjunctionexactly one of the conjuncts x s 1=x s 2 or x s 1≠x s 2 is present.2.5. Conversion to Quantifier-Free FormulasThe conversion from structural base formulas toquantifier-free formulas is the main phase of our quantifiereliminationalgorithm. We split this conversion into severalstages; Proposition 25 below summarizes the overall conversionprocess.Consider a structural base formula β ≡ ∃ū ∗ . C(¯x ∗ , ū ∗ )with free variables ¯x ∗ and internal variables ū ∗ , whereC(¯x ∗ , ū ∗ ) is quantifier-free. C(¯x ∗ , ū ∗ ) defines a relationbetween variables ¯x ∗ , ū ∗ . If this relation has a functionaldependence from the free variables ¯x ∗ to some internal variableu, with a term t(¯x ∗ ) such that C(¯x ∗ , ū ∗ ) |= u = t(¯x ∗ ),then the internal variable u can be replaced by t(¯x ∗ ) andthe quantification over u can be eliminated. This leads tothe notion of determinations.Definition 14 The set dets of variable determinations of astructural base formula β is the least set S of pairs 〈u ∗ , t ∗ 〉where u ∗ is an internal term or shape variable and t ∗ is aterm over the free variables of β, such such that:1. if x ∗ = u ∗ occurs in termBase or shapeBase for afree variable x ∗ , then 〈u ∗ , x ∗ 〉 ∈ S;2. if 〈u ∗ , t ∗ 〉 ∈ S and u ∗ = f ∗ (u ∗ 1 , . . . , u∗ k) occurs inshapeBase or termBase then{〈u ∗ 1, f1 ∗ (t ∗ )〉, . . . , 〈u ∗ k , f k ∗(t∗ )〉} ⊆ S;3. if {〈u ∗ 1 , t∗ 1 〉, . . . , 〈u∗ k , t∗ k 〉} ⊆ S andu ∗ = f ∗ (u ∗ 1, . . . , u ∗ k ) occurs in shapeBase ortermBase then 〈u ∗ , f ∗ (t ∗ 1 , . . .,t∗ k)〉 ∈ S;4. if 〈u, t〉 ∈ S and sh(u) = u s occurs in termHom then〈u s , sh(t)〉 ∈ S.Definition 15 An internal variable u ∗ is determined if〈u ∗ , t ∗ 〉 ∈ dets for some term t s . An internal variable isundetermined if it is not determined.Lemma 16 follows by induction using Definition 14.Lemma 16 Let β ≡ ∃ū. C(¯x ∗ , ū ∗ ) be a structural baseformula. If 〈u ∗ , t ∗ 〉 ∈ dets(β) then C(¯x ∗ , ū ∗ ) |= u ∗ = t ∗ .Corollary 17 Let β ≡ ∃〈u ∗ i 〉 i. C(¯x ∗ , 〈u ∗ i 〉 i) be a structuralbase formula such that each internal variable u ∗ i isdetermined by some term t ∗ i , that is, 〈u∗ i , t∗ i 〉 ∈ dets(β).Then β is equivalent to the well-defined quantifier-free formulaβ ′ ≡ C(¯x ∗ , 〈t ∗ i 〉 i).Proof. By Lemma 16 using the rule∃u.u = t ∧ φ(u) ⇐⇒ φ(t) (5)which holds when the term t is well-defined. If t is notwell-defined, then both β and β ′ evaluate to false.Our goal thus reduces to eliminating all undeterminedvariables from a structural base formula. We first show howto eliminate undetermined composed non-parameter termvariables.Lemma 18 Let u be an undetermined composed nonparameterterm variable in a structural base formula βsuch that u is a source i.e. no conjunct of the formu ′ =f(u 1 , . . .,u, . . .,u k ) occurs in termBase. Let β ′ be theresult of removing from β the variable u and all conjunctscontaining u. Then β is equivalent to β ′ .8
Proof. The conjunct containing sh(u) = u s in termHomis a consequence of the remaining conjuncts in β, so wemay drop it. The only remaining occurrence of u is in theatomic formula u=f(¯v) of termBase subformula. Applying(5) therefore makes u disappear from β.Corollary 19 (Composed Term Variable Elimination)Dropping all undetermined composed non-parameter termvariables from a structural base formula together with theconjuncts that contain them yields an equivalent structuralbase formula.Proof. If a structural base formula has an undeterminednon-parameter composed term variable, then it has an undeterminednon-parameter composed term variable that is asource. Repeatedly apply Lemma 18 to eliminate all undeterminednon-parameter term variables.Our next goal is to eliminate undetermined primitivenon-parameter term variables and undetermined parameterterm variables. The key insight is that these variables arerelated to the determined variables of a structural base formulaonly through the relations that are expressible in theproduct structure of the terms of the same shape. To clarifythe connection with the product-structure, let s ∈ P Sbe a shape and P (s) = {t ∈ P | sh(t) = s}. Defineη : P S → C ∗ where C ∗ is the set of finite sequencesof elements from C, as follows: η(c) = c if c ∈ C;η(f(t 1 , . . . , t k )) = η(t 1 ) · . . . · η(t k ) where l 1 · l 2 denotesthe concatenation of sequences l 1 and l 2 . Let η s = η| P (s)be the restriction of η to the set P (s) . Let m = |leaves(s)|.Observation 20 The map η s is an isomorphism of the substructureof P with the domain P (s) and the finite powerC m . Moreover,|φ(〈x j 〉 n j ) PE (s, 〈t j 〉 n j )| = |φ(〈x j 〉 n j ) Cm (〈η s (t j )〉 n j )|The following is the quantifier-elimination property that impliesFeferman-Vaught theorem [13, 30], [20, Section 9.6,Page 460] for the case of finite products.Lemma 21 Let k ≥ 0. Consider a formula α of the formα ≡ ∃u. ∧ pi=1 ψ i where each ψ i is a cardinality constraintof the form (|φ i |=k)(u, 〈u j 〉 n j ) or (|φ i|≥k)(u, 〈u j 〉 n j ). Thenα can be effectively transformed into α ′ where α ′ is a disjunctionof conjunctions of cardinality constraints of theform (|φ ′ i |=k)(〈u j〉 n j ) and (|φ′ i |≥k)(〈u j〉 n j ). The result α′is equivalent to α on each finite power C m .Lemma 22 is a direct consequence of Lemma 21 and Observation20.Lemma 22 Let k ≥ 0. Consider a formula α of the formα ≡ ∃u. sh(u) = u s ∧ ∧ ni=1 sh(u i) = u s ∧ ∧ pi=1 ψ ifor some shape variable u s where each ψ i is a cardinalityconstraint of the form (|φ i |=k)(u s , u, 〈u j 〉 n j ) or of the form(|φ i |≥k)(u s , u, 〈u j 〉 n j ). Then α can be effectively transformedinto the formula α ′ which is a disjunction of formulasof the formα ′ j ≡ ∧ ni=1 sh(u i) = u s ∧ ∧ qi=1 ψ′ i,jwhere each ψ ′ i,j is of the form (|φ′ i,j |=k)(us , 〈u j 〉 n j ) or(|φ ′ i,j |≥k)(us , 〈u j 〉 n j ). The resulting formula α′ is equivalentto α on all term powers P E .Lemma 23 (Term Parameter Elimination) Every structuralbase formula β without undetermined composed nonparameterterm variables can be effectively transformedinto an equivalent disjunction of structural base formulaswithout undetermined term variables.Proof. We show how to eliminate undetermined parameterterm variables and undetermined primitive non-parameterterm variables from β.Let u be an undetermined parameter term variable or anundetermined primitive non-parameter term variable. If uis a parameter variable then u does not occur in termBasebecause ¬(u + u ′ ) for all u ′ , and ¬(u ′′ + u) for all u ′′since there are no undetermined composed non-parameterterm variables. Therefore, u occurs only in termHom andcardin. If u is a primitive non-parameter term variable, thentermBase contains only one occurrence of u, namely theconjunct Is PRI (u), which is a consequence of the conjunctssh(u) = u s in termHom and u s = c s in shapeBase, so wedrop Is PRI (u). In both cases, the resulting formula containsu only in termHom and cardin.Let u s be the shape variable such that u s = sh(u) occursin termHom. Let ψ 1 , . . . , ψ p be all conjuncts of cardin thatcontain u. Each ψ i is of the form |φ| u s ≥ k i or |φ| u s = k iand for each variable u ′ free in φ the conjunct sh(u) = u soccurs in termHom. The structural base formula can thereforebe written in the form ∃ū ∗ . φ ∧α where α has the formas in Lemma 22. Applying Lemma 22 we eliminate u. Applyingrules DNF results in a disjunction of structural baseformulas. By repeating this process we eliminate all undeterminedparameter term variables and undetermined primitivenon-parameter term variables from a structural baseformula. Each of the resulting structural base formulas containsno undetermined term variables.Finally, we show how to eliminate the undetermined shapevariables.Lemma 24 (Shape Variable Elimination) Every structuralbase formula β without undetermined term variablescan be effectively transformed into an equivalent disjunctionof structural base formulas without undeterminedvariables.9
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Page 11 and 12: [7] W. Charatonik and L. Pacholski.

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