package hol

Ordering
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Visibility
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Type Members

  1. class HOLOrdering extends Ordering[Expr]
  2. case class HOLPosition(list: List[Int]) extends Product with Serializable

    Represents a position in a gapt.expr.Expr.

    Represents a position in a gapt.expr.Expr.

    Positions are represented by lists of Integers. The empty list denotes the expression itself. A nonempty list denotes a position in the left or right subexpression according to whether it starts with 1 or 2.

    The difference between this and gapt.expr.LambdaPosition lies in the handling of quantifiers and binary logical connectives. LambdaPositions treat e.g. conjunctions like any other function, while HOLPositions treat them naturally, i.e. 1 denotes the left conjunct and 2 the right conjunct.

    Note that this can cause unexpeted behavior: Say a variable of type o -> o -> o is substituted by ∧ in some expression. The LambdaPositions will stay the same, but the HOLPositions won't.

    list

    The list of integers describing the position.

  3. case class SkolemFunctions(skolemDefs: Map[Const, Expr]) extends Product with Serializable

    List of definitions of Skolem symbols.

    List of definitions of Skolem symbols.

    A Skolem definition is similar but slightly different from the epsilon operator:

    Syntactically it is a map s_i → λx_1 .. λx_n Qy φ(x_1, .., x_n, y), where Q is a quantifier. Then s_i(x_1, .., x_n) is the Skolem term used for the formula Qy φ(x_1, .., x_n, y), where Qy is strong.

    This Skolem term corresponds to the epsilon term εy φ(x_1, .., x_n, y) or εy ¬φ(x_1, .., x_n), depending on whether Q is ∃ or ∀. The reason we don't use epsilon terms directly is that this makes it impossible to deskolemize a formula based on just the Skolem definitions: for example both ∃x ∀y φ and ∃x ¬∃y¬ φ would define their Skolem functions using the same epsilon terms.

  4. class TAOrdering extends Ordering[Ty]
  5. case class fastStructuralCNF(propositional: Boolean = true, bidirectionalDefs: Boolean = false) extends Product with Serializable

Value Members

  1. object BinaryConnective
  2. object CNFn

    Computes a negative CNF of a formula, i.e.

    Computes a negative CNF of a formula, i.e. one that is logically equivalent to the negation of the input formula.

    The computation is done by expanding the input formula using distributivity.

  3. object CNFp

    Computes a positive CNF of a formula, i.e.

    Computes a positive CNF of a formula, i.e. one that is logically equivalent to the input formula.

    The computation is done by expanding the input formula using distributivity.

  4. object DNFn

    Computes a negative DNF of a formula, i.e.

    Computes a negative DNF of a formula, i.e. one that is logically equivalent to the negation of the input formula.

    The computation is done by expanding the input formula using distributivity.

  5. object DNFp

    Computes a positive DNF of a formula, i.e.

    Computes a positive DNF of a formula, i.e. one that is logically equivalent to the input formula.

    The computation is done by expanding the input formula using distributivity.

  6. object HOLOrdering extends HOLOrdering

    Ordering for HOL Formulas (also for FOL)

  7. object HOLPosition extends Serializable
  8. object SkolemFunctions extends Serializable
  9. object TAOrdering extends TAOrdering

    Ordering on types.

  10. object atoms

    Return the list of all atoms in the given argument.

  11. object containsHOQuantifier
  12. object containsQuantifier

    True iff All or Ex matches any subterm of e.

  13. object containsQuantifierOnLogicalLevel

    True iff All or Ex is contained in the logical structure of the given Formula.

    True iff All or Ex is contained in the logical structure of the given Formula. For example, P( (all x:x) ) contains a quantifier, but it is inside of an atom.

  14. object containsStrongQuantifier
  15. object containsWeakQuantifier
  16. object dualize

    Dualize a formula in NNF by switching conjunctions with disjunctions, universal with existential quantifiers, top with bottom and positive literals with negative literals.

    Dualize a formula in NNF by switching conjunctions with disjunctions, universal with existential quantifiers, top with bottom and positive literals with negative literals. The formula dualize( A ) is logically equivalent to the negation of A.

  17. object existentialClosure
  18. object formulaToSequent

    Removes top-level connectives from a formula.

  19. object freeHOVariables
  20. object inductionPrinciple
  21. object instantiate

    Instantiates a formula with terms.

  22. object isAtom

    Returns true iff the given Formula is an atom (which does not include top nor bottom).

  23. object isEssentiallyCNF

    Recognizes a class of formulas/sequents including all formulas in clause normal form (i.e., negation of conjunction of clauses).

    Recognizes a class of formulas/sequents including all formulas in clause normal form (i.e., negation of conjunction of clauses).

    The first-order formulas recognized by this function have the property that they are valid in classical first-order logic if and only if they are valid in intuitionistic first-order logic.

  24. object isExtendedAtom

    Returns true iff the given Formula is an extended atom, i.e.

    Returns true iff the given Formula is an extended atom, i.e. an atom or top or bottom.

  25. object isLogicalConstant

    Returns true iff the given Expr consists of a logical constant.

  26. object isNeg

    Returns true iff the given Formula starts with a negation.

  27. object isPrenex

    Returns true iff the given formula is prenex.

  28. object isReflexivity

    Returns true iff the given Formula is a reflexivity atom.

  29. object lcomp

    the logical complexity of this formula, i.e.

    the logical complexity of this formula, i.e. the number of logical connectives and atoms starting from the root of the formula. The inner structure of atoms is not counted.

  30. object normalizeFreeVariables
  31. object numOfAtoms

    Return the number of atoms in the given argument.

  32. object removeAllQuantifiers
  33. object removeNeg

    Remove the leading negation from a formula.

  34. object simplify

    Simplify a Formula using the equations for bottom and top as well as idempotence of conjunction and disjunction.

  35. object skolemize
  36. object toNNF

    Transforms a formula to negation normal form (transforming also implications into disjunctions)

  37. object universalClosure

Ungrouped