Package

at.logic.gapt.proofs

lkNew

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package lkNew

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Type Members

  1. case class AndLeftRule(subProof: LKProof, aux1: SequentIndex, aux2: SequentIndex) extends UnaryLKProof with Product with Serializable

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    An LKProof ending with a conjunction on the left:

            (π)
        A, B, Γ :- Δ
       --------------
       A ∧ B, Γ :- Δ
    

    An LKProof ending with a conjunction on the left:

            (π)
        A, B, Γ :- Δ
       --------------
       A ∧ B, Γ :- Δ
    

    subProof

    The subproof π.

    aux1

    The index of A.//<editor-fold desc="Base proof classes">

    aux2

    The index of B.

  2. case class AndRightRule(leftSubProof: LKProof, aux1: SequentIndex, rightSubProof: LKProof, aux2: SequentIndex) extends BinaryLKProof with Product with Serializable

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    An LKProof ending with a conjunction on the right:

       (π1)         (π2)
      Γ :- Δ, A    Π :- Λ, B
    --------------------------
        Γ, Π :- Δ, Λ, A∧B
    

    An LKProof ending with a conjunction on the right:

       (π1)         (π2)
      Γ :- Δ, A    Π :- Λ, B
    --------------------------
        Γ, Π :- Δ, Λ, A∧B
    

    leftSubProof

    The proof π1.

    aux1

    The index of A.

    rightSubProof

    The proof π2

    aux2

    The index of B.

  3. case class ArbitraryAxiom(endSequent: HOLSequent) extends InitialSequent with Product with Serializable

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  4. abstract class BinaryLKProof extends LKProof

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    An LKProof deriving a sequent from two other sequents:

        (π1)     (π2)
       Γ :- Δ   Γ' :- Δ'
      ------------------
           Π :- Λ
    

    An LKProof deriving a sequent from two other sequents:

        (π1)     (π2)
       Γ :- Δ   Γ' :- Δ'
      ------------------
           Π :- Λ
    

  5. case class ContractionLeftRule(subProof: LKProof, aux1: SequentIndex, aux2: SequentIndex) extends UnaryLKProof with Product with Serializable

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    An LKProof ending with a left contraction:

            (π)
        A, A, Γ :- Δ
       --------------
         A, Γ :- Δ
    

    An LKProof ending with a left contraction:

            (π)
        A, A, Γ :- Δ
       --------------
         A, Γ :- Δ
    

    subProof

    The subproof π.

    aux1

    The index of one occurrence of A.

    aux2

    The index of the other occurrence of A.

  6. case class ContractionRightRule(subProof: LKProof, aux1: SequentIndex, aux2: SequentIndex) extends UnaryLKProof with Product with Serializable

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    An LKProof ending with a right contraction:

            (π)
        Γ :- Δ, A, A
       --------------
         Γ :- Δ, A
    

    An LKProof ending with a right contraction:

            (π)
        Γ :- Δ, A, A
       --------------
         Γ :- Δ, A
    

    subProof

    The subproof π.

    aux1

    The index of one occurrence of A.

    aux2

    The index of the other occurrence of A.

  7. case class CutRule(leftSubProof: LKProof, aux1: SequentIndex, rightSubProof: LKProof, aux2: SequentIndex) extends BinaryLKProof with Product with Serializable

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    An LKProof ending with a cut:

         (π1)         (π2)
       Γ :- Δ, A   A, Π :- Λ
      ------------------------
           Γ, Π :- Δ, Λ
    

    An LKProof ending with a cut:

         (π1)         (π2)
       Γ :- Δ, A   A, Π :- Λ
      ------------------------
           Γ, Π :- Δ, Λ
    

    leftSubProof

    The proof π1.

    aux1

    The index of A in π1.

    rightSubProof

    The proof π2.

    aux2

    The index of A in π2.

  8. case class EqualityRule(subProof: LKProof, eq: SequentIndex, aux: SequentIndex, pos: HOLPosition) extends UnaryLKProof with Product with Serializable

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  9. case class ExistsLeftRule(subProof: LKProof, aux: SequentIndex, eigenVariable: Var, quantifiedVariable: Var) extends UnaryLKProof with Product with Serializable

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    An LKProof ending with an existential quantifier on the left:

              (π)
         A[x\α], Γ :- Δ
        ---------------∀:r
          ∃x.A Γ :- Δ
    
    This rule is only applicable if the eigenvariable condition is satisfied: α must not occur freely in Γ :- Δ.

    An LKProof ending with an existential quantifier on the left:

              (π)
         A[x\α], Γ :- Δ
        ---------------∀:r
          ∃x.A Γ :- Δ
    
    This rule is only applicable if the eigenvariable condition is satisfied: α must not occur freely in Γ :- Δ.

    subProof

    The proof π.

    aux

    The index of A[x\α].

    eigenVariable

    The variable α.

    quantifiedVariable

    The variable x.

  10. case class ExistsRightRule(subProof: LKProof, aux: SequentIndex, A: HOLFormula, term: LambdaExpression, v: Var) extends UnaryLKProof with Product with Serializable

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    An LKProof ending with an existential quantifier on the right:

               (π)
         Γ :- Δ, A[x\t]
        ----------------∃:r
          Γ :- Δ, ∃x.A
    

    An LKProof ending with an existential quantifier on the right:

               (π)
         Γ :- Δ, A[x\t]
        ----------------∃:r
          Γ :- Δ, ∃x.A
    

    subProof

    The proof π.

    aux

    The index of A[x\t].

    A

    The formula A.

    term

    The term t.

    v

    The variable x.

  11. case class ForallLeftRule(subProof: LKProof, aux: SequentIndex, A: HOLFormula, term: LambdaExpression, v: Var) extends UnaryLKProof with Product with Serializable

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    An LKProof ending with a universal quantifier on the left:

               (π)
         A[x\t], Γ :- Δ
        ----------------∀:l
          ∀x.A, Γ :- Δ
    

    An LKProof ending with a universal quantifier on the left:

               (π)
         A[x\t], Γ :- Δ
        ----------------∀:l
          ∀x.A, Γ :- Δ
    

    subProof

    The proof π.

    aux

    The index of A[x\t].

    A

    The formula A.

    term

    The term t.

    v

    The variable x.

  12. case class ForallRightRule(subProof: LKProof, aux: SequentIndex, eigenVariable: Var, quantifiedVariable: Var) extends UnaryLKProof with Product with Serializable

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    An LKProof ending with a universal quantifier on the right:

              (π)
         Γ :- Δ, A[x\α]
        ---------------∀:r
         Γ :- Δ, ∀x.A
    
    This rule is only applicable if the eigenvariable condition is satisfied: α must not occur freely in Γ :- Δ.

    An LKProof ending with a universal quantifier on the right:

              (π)
         Γ :- Δ, A[x\α]
        ---------------∀:r
         Γ :- Δ, ∀x.A
    
    This rule is only applicable if the eigenvariable condition is satisfied: α must not occur freely in Γ :- Δ.

    subProof

    The proof π.

    aux

    The index of A[x\α].

    eigenVariable

    The variable α.

    quantifiedVariable

    The variable x.

  13. case class ImpLeftRule(leftSubProof: LKProof, aux1: SequentIndex, rightSubProof: LKProof, aux2: SequentIndex) extends BinaryLKProof with Product with Serializable

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    An LKProof ending with an implication on the left:

        (π1)         (π2)
      Γ :- Δ, A    B, Π :- Λ
    --------------------------
        A→B, Γ, Π :- Δ, Λ
    

    An LKProof ending with an implication on the left:

        (π1)         (π2)
      Γ :- Δ, A    B, Π :- Λ
    --------------------------
        A→B, Γ, Π :- Δ, Λ
    

    leftSubProof

    The proof π1.

    aux1

    The index of A.

    rightSubProof

    The proof π2

    aux2

    The index of B.

  14. case class ImpRightRule(subProof: LKProof, aux1: SequentIndex, aux2: SequentIndex) extends UnaryLKProof with Product with Serializable

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    An LKProof ending with an implication on the right:

            (π)
        A, Γ :- Δ, B
       --------------
        Γ :- Δ, A → B
    

    An LKProof ending with an implication on the right:

            (π)
        A, Γ :- Δ, B
       --------------
        Γ :- Δ, A → B
    

    subProof

    The subproof π.

    aux1

    The index of A.

    aux2

    The index of B.

  15. case class InductionRule(leftSubProof: LKProof, aux1: SequentIndex, rightSubProof: LKProof, aux2: SequentIndex, aux3: SequentIndex, term: FOLTerm) extends BinaryLKProof with Product with Serializable

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  16. abstract class InitialSequent extends LKProof

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    An LKProof consisting of a single sequent:

        --------ax
         Γ :- Δ
    

    An LKProof consisting of a single sequent:

        --------ax
         Γ :- Δ
    

  17. abstract class LKProof extends SequentProof[HOLFormula, LKProof]

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  18. case class LogicalAxiom(A: HOLAtom) extends InitialSequent with Product with Serializable

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    An LKProof consisting of a logical axiom:

       --------ax
        A :- A
    
    with A atomic.

    An LKProof consisting of a logical axiom:

       --------ax
        A :- A
    
    with A atomic.

    A

    The atom A.

  19. case class NegLeftRule(subProof: LKProof, aux: SequentIndex) extends UnaryLKProof with Product with Serializable

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    An LKProof ending with a negation on the left:

          (π)
       Γ :- Δ, A
      -----------¬:l
      ¬A, Γ :- Δ
    

    An LKProof ending with a negation on the left:

          (π)
       Γ :- Δ, A
      -----------¬:l
      ¬A, Γ :- Δ
    

    subProof

    The proof π.

    aux

    The index of A in the succedent.

  20. case class NegRightRule(subProof: LKProof, aux: SequentIndex) extends UnaryLKProof with Product with Serializable

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    An LKProof ending with a negation on the right:

          (π)
       A, Γ :- Δ
      -----------¬:r
      Γ :- Δ, ¬A
    

    An LKProof ending with a negation on the right:

          (π)
       A, Γ :- Δ
      -----------¬:r
      Γ :- Δ, ¬A
    

    subProof

    The proof π.

    aux

    The index of A in the antecedent.

  21. class OccConnector extends AnyRef

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    This class models the connection of formula occurrences between two sequents in a proof.

  22. case class OrLeftRule(leftSubProof: LKProof, aux1: SequentIndex, rightSubProof: LKProof, aux2: SequentIndex) extends BinaryLKProof with Product with Serializable

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    An LKProof ending with a disjunction on the left:

        (π1)         (π2)
      A, Γ :- Δ    B, Π :- Λ
    --------------------------
        A∨B, Γ, Π :- Δ, Λ
    

    An LKProof ending with a disjunction on the left:

        (π1)         (π2)
      A, Γ :- Δ    B, Π :- Λ
    --------------------------
        A∨B, Γ, Π :- Δ, Λ
    

    leftSubProof

    The proof π1.

    aux1

    The index of A.

    rightSubProof

    The proof π2

    aux2

    The index of B.

  23. case class OrRightRule(subProof: LKProof, aux1: SequentIndex, aux2: SequentIndex) extends UnaryLKProof with Product with Serializable

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    An LKProof ending with a disjunction on the right:

            (π)
        Γ :- Δ, A, B
       --------------
        Γ :- Δ, A ∨ B
    

    An LKProof ending with a disjunction on the right:

            (π)
        Γ :- Δ, A, B
       --------------
        Γ :- Δ, A ∨ B
    

    subProof

    The subproof π.

    aux1

    The index of A.

    aux2

    The index of B.

  24. class ProofBuilder extends AnyRef

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    Class for convenient construction of proofs.

    Class for convenient construction of proofs. Allows you to write proofs post-order style (à la Bussproofs). Example:

    (ProofBuilder
    c LogicalAxiom(A)
    c LogicalAxiom(B)
    u (WeakeningLeftRule(_, C))
    b (AndRightRule(_,_, And(A, B))
    qed)
    
    The constructor is private, so the only way to instantiate this class is by using the ProofBuilder object. This means that the stack will always be empty in the beginning.

  25. case class ReflexivityAxiom(s: LambdaExpression) extends InitialSequent with Product with Serializable

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    An LKProof consisting of a reflexivity axiom:

       ------------ax
         :- s = s
    
    with s a term.

    An LKProof consisting of a reflexivity axiom:

       ------------ax
         :- s = s
    
    with s a term.

    s

    The term s.

  26. abstract class UnaryLKProof extends LKProof

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    An LKProof deriving a sequent from another sequent:

           (π)
         Γ :- Δ
       ----------
        Γ' :- Δ'
    

    An LKProof deriving a sequent from another sequent:

           (π)
         Γ :- Δ
       ----------
        Γ' :- Δ'
    

  27. case class WeakeningLeftRule(subProof: LKProof, formula: HOLFormula) extends UnaryLKProof with Product with Serializable

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    An LKProof ending with a left weakening:

           (π)
          Γ :- Δ
        ---------w:l
        A, Γ :- Δ
    

    An LKProof ending with a left weakening:

           (π)
          Γ :- Δ
        ---------w:l
        A, Γ :- Δ
    

    subProof

    The subproof π.

    formula

    The formula A.

  28. case class WeakeningRightRule(subProof: LKProof, formula: HOLFormula) extends UnaryLKProof with Product with Serializable

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    An LKProof ending with a right weakening:

           (π)
          Γ :- Δ
        ---------w:r
        Γ :- Δ, A
    

    An LKProof ending with a right weakening:

           (π)
          Γ :- Δ
        ---------w:r
        Γ :- Δ, A
    

    subProof

    The subproof π.

    formula

    The formula A.

Value Members

  1. object AndLeftRule extends RuleConvenienceObject with Serializable

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  2. object AndRightRule extends RuleConvenienceObject with Serializable

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  3. object Axiom

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  4. object BinaryLKProof

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  5. object BottomAxiom extends InitialSequent with Product with Serializable

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    An LKProof introducing ⊥ on the left:

          --------⊥:l
            ⊥ :-
    

    An LKProof introducing ⊥ on the left:

          --------⊥:l
            ⊥ :-
    

  6. object ContractionLeftRule extends RuleConvenienceObject with Serializable

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  7. object ContractionRightRule extends RuleConvenienceObject with Serializable

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  8. object CutRule extends RuleConvenienceObject with Serializable

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  9. object ExistsLeftRule extends RuleConvenienceObject with Serializable

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  10. object ExistsRightRule extends Serializable

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  11. object ForallLeftRule extends Serializable

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  12. object ForallRightRule extends RuleConvenienceObject with Serializable

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  13. object ImpLeftRule extends RuleConvenienceObject with Serializable

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  14. object ImpRightRule extends RuleConvenienceObject with Serializable

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  15. object InitialSequent

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  16. object NegLeftRule extends RuleConvenienceObject with Serializable

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  17. object NegRightRule extends RuleConvenienceObject with Serializable

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  18. object OrLeftRule extends RuleConvenienceObject with Serializable

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  19. object OrRightRule extends RuleConvenienceObject with Serializable

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  20. object ProofBuilder extends ProofBuilder

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  21. object TopAxiom extends InitialSequent with Product with Serializable

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    An LKProof introducing ⊤ on the right:

          --------⊤:r
            :- ⊤
    

    An LKProof introducing ⊤ on the right:

          --------⊤:r
            :- ⊤
    

  22. object UnaryLKProof

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  23. object prettyString

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Ungrouped