Package

at.logic.gapt.proofs

expansion

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package expansion

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package.scala
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Type Members

  1. trait BinaryExpansionTree extends ExpansionTree

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  2. case class ETAnd(child1: ExpansionTree, child2: ExpansionTree) extends BinaryExpansionTree with Product with Serializable

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  3. case class ETAtom(atom: HOLAtom, polarity: Boolean) extends ExpansionTree with Product with Serializable

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  4. case class ETBottom(polarity: Boolean) extends ExpansionTree with Product with Serializable

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  5. case class ETDefinedAtom(atom: HOLAtom, polarity: Boolean, definition: LambdaExpression) extends ExpansionTree with Product with Serializable

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  6. case class ETDefinition(shallow: HOLAtom, definedExpr: LambdaExpression, child: ExpansionTree) extends UnaryExpansionTree with Product with Serializable

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    Expansion tree node for definitions.

    Expansion tree node for definitions.

    shallow

    An atom P(x1,..., xn) where P stands for a more complex formula.

    definedExpr

    The expression that P abbreviates. Must have the same type as P.

    child

    An expansion tree with shallowFormula definedExpr(x1,...,xn)

  7. case class ETImp(child1: ExpansionTree, child2: ExpansionTree) extends BinaryExpansionTree with Product with Serializable

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  8. case class ETMerge(child1: ExpansionTree, child2: ExpansionTree) extends BinaryExpansionTree with Product with Serializable

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  9. case class ETNeg(child: ExpansionTree) extends UnaryExpansionTree with Product with Serializable

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  10. case class ETOr(child1: ExpansionTree, child2: ExpansionTree) extends BinaryExpansionTree with Product with Serializable

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  11. trait ETQuantifier extends ExpansionTree

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  12. case class ETSkolemQuantifier(shallow: HOLFormula, skolemTerm: LambdaExpression, skolemDef: LambdaExpression, child: ExpansionTree) extends ETQuantifier with UnaryExpansionTree with Product with Serializable

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  13. case class ETStrongQuantifier(shallow: HOLFormula, eigenVariable: Var, child: ExpansionTree) extends ETQuantifier with UnaryExpansionTree with Product with Serializable

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  14. case class ETTop(polarity: Boolean) extends ExpansionTree with Product with Serializable

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  15. case class ETWeakQuantifier(shallow: HOLFormula, instances: Map[LambdaExpression, ExpansionTree]) extends ETQuantifier with Product with Serializable

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  16. case class ETWeakening(formula: HOLFormula, polarity: Boolean) extends ExpansionTree with Product with Serializable

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  17. case class ExpansionProof(expansionSequent: Sequent[ExpansionTree]) extends Product with Serializable

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  18. class ExpansionProofToLK extends SolveUtils

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  19. case class ExpansionProofWithCut(expansionWithCutAxiom: ExpansionProof) extends Product with Serializable

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  20. type ExpansionSequent = Sequent[ExpansionTree]

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  21. trait ExpansionTree extends DagProof[ExpansionTree]

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  22. class ExpansionTreePrettyPrinter extends BabelExporter

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  23. class InstanceTermEncoding extends AnyRef

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    Encodes instances of an end-sequent as terms.

    Encodes instances of an end-sequent as terms.

    Only instances of weak quantifiers are recorded, instances of strong quantifiers or free variables are ignored.

    The end-sequent will be internally transformed into one which is in variable normal form.

    In the case of cut-introduction, the end-sequent has no free variables and no strong quantifiers and we're encoding a Herbrand sequent as a set of terms. A term r_i(t_1,...,t_n) encodes an instance of the formula "forall x_1 ... x_n, phi(x_1,...,x_n)" using the instances (t_1,...,t_n).

    In the case of inductive proofs, the end-sequent contains strong quantifiers variable (alpha). Here, we consider proofs of instance sequents, which are obtained by e.g. substituting a numeral for alpha. Hence the formulas occurring in the end-sequents of instance proofs are substitution instances of endSequent; the encoded terms still only capture the instances used in the instance proofs--i.e. not alpha.

  24. implicit class RichExpansionSequent extends AnyRef

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    Extension class that allows calling shallow and deep on sequents.

  25. trait UnaryExpansionTree extends ExpansionTree

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Value Members

  1. object ETMerge extends Serializable

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  2. object ETQuantifier

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  3. object ETStrongQuantifierBlock

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  4. object ETWeakQuantifier extends Serializable

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  5. object ETWeakQuantifierBlock

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  6. object ExpansionProofToLK extends ExpansionProofToLK

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  7. object ExpansionProofWithCut extends Serializable

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  8. val ExpansionSequent: Sequent.type

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  9. object FOLInstanceTermEncoding

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  10. object InstanceTermEncoding

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  11. object PropositionalExpansionProofToLK extends ExpansionProofToLK

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  12. object addSymmetry

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    Given an expansion sequent S which is a quasi-tautology (modulo symmetry), returns an expansion sequent S' which is S extended by the symmetry instances needed to make it a tautology.

  13. object cleanStructureET

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  14. object eigenVariablesET

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  15. object eliminateCutsET

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  16. object eliminateDefsET

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  17. object eliminateMerges

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  18. implicit val expansionProofsAreClosedUnderSubstitution: ClosedUnderSub[ExpansionProof]

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  19. implicit object expansionProofsAreReplaceable extends ClosedUnderReplacement[ExpansionProof]

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  20. implicit val expansionProofsWithCutAreClosedUnderSubstitution: ClosedUnderSub[ExpansionProofWithCut]

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  21. implicit object expansionProofsWithCutAreReplaceable extends ClosedUnderReplacement[ExpansionProofWithCut]

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  22. implicit val expansionTreesAreClosedUnderAdmissibleSubstitutions: ClosedUnderSub[ExpansionTree]

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  23. implicit object expansionTreesAreReplaceable extends ClosedUnderReplacement[ExpansionTree]

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  24. object extractInstances

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    Extracts the instances used in a prenex FOL Pi_1 expansion tree / Sigma_1 expansion sequent.

    Extracts the instances used in a prenex FOL Pi_1 expansion tree / Sigma_1 expansion sequent.

    Each expansion tree is transformed into a list of instances of its shallow formula.

    In contrast to ExpansionProof.deep, this function doesn't produce conjunctions of instances, but instead increases the number of formulas in the antecedent/succedent.

  25. object formulaToExpansionTree

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    Builds an expansion tree from a formula and a map from variables to terms.

    Builds an expansion tree from a formula and a map from variables to terms. The paremeter pos is true if the formula is to be considered positive (right side of the sequent).

  26. object freeVariablesET

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  27. object generalizeET

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  28. object generatedUpperSetInPO

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  29. object groundTerms

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  30. object linearizeStrictPartialOrder

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  31. object minimalExpansionSequent

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    Given an expansion sequent S, this algorithm computes a single expansion sequents below S that is valid and minimal.

    Given an expansion sequent S, this algorithm computes a single expansion sequents below S that is valid and minimal. This algorithm is considerably faster than the one implemented in minimalExpansionSequents.

  32. object minimalExpansionSequents

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    Given an expansion sequent S, this algorithm computes the list of expansion sequents below S that are valid and minimal.

  33. object numberOfInstancesET

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  34. object replaceAtHOLPosition

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  35. object replaceAtLambdaPosition

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  36. object replaceWithContext

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    Replaces terms in an expansion tree according to a replacement context.

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