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at.logic.gapt.proofs.lk

CutRule

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object CutRule

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  1. final def !=(arg0: Any): Boolean

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  2. final def ##(): Int

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  3. final def ==(arg0: Any): Boolean

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  4. def apply(s1: LKProof, s2: LKProof, term1: HOLFormula): BinaryTree[OccSequent] with BinaryLKProof with AuxiliaryFormulas

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    Cuts a common formula term1 (found in the succedent of s1 and
    the antecedent of s2) from two proofs s1 & s2.
    F is automatically determined. If term1 occurs more than once,
    only the first occurrence is cut.
    
    Let s1 have (sL |- sR, term1) as its bottommost sequent and
    let s2 have (tL, term1 |- tR) as its bottommost sequent.
    
    The rule:
    (rest of s1)       (rest of s2)
    sL |- sR, term1   tL, term1 |- tR
    --------------------------------- (Cut)
             sL, tL |- sR, tR
    

    Cuts a common formula term1 (found in the succedent of s1 and
    the antecedent of s2) from two proofs s1 & s2.
    F is automatically determined. If term1 occurs more than once,
    only the first occurrence is cut.
    
    Let s1 have (sL |- sR, term1) as its bottommost sequent and
    let s2 have (tL, term1 |- tR) as its bottommost sequent.
    
    The rule:
    (rest of s1)       (rest of s2)
    sL |- sR, term1   tL, term1 |- tR
    --------------------------------- (Cut)
             sL, tL |- sR, tR
    

    s1

    The left proof with term1 in the succedent of its bottommost sequent.

    s2

    The right proof with term1 in the antecedent of its bottommost sequent.

    returns

    An LK proof with s1 & s2 as its two subtrees and (sL, tL |- sR, tR) as its bottommost sequent.

  5. def apply(s1: OccSequent, s2: OccSequent, term1oc: FormulaOccurrence, term2oc: FormulaOccurrence): Sequent[FormulaOccurrence]

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    Cuts a common formula F (marked by term1oc in the
    succedent of s1 and by term2oc in the antecedent of s2)
    from two proofs s1 & s2. This function merely returns the
    resulting sequent, not a proof.
    
    Let s1 have (sL |- sR, F) as its bottommost sequent and
    let s2 have (tL, F |- tR) as its bottommost sequent.
    
    The rule:
    (rest of s1)       (rest of s2)
    sL |- sR, F        tL, F |- tR
    ------------------------------ (Cut)
          sL, tL |- sR, tR
    

    Cuts a common formula F (marked by term1oc in the
    succedent of s1 and by term2oc in the antecedent of s2)
    from two proofs s1 & s2. This function merely returns the
    resulting sequent, not a proof.
    
    Let s1 have (sL |- sR, F) as its bottommost sequent and
    let s2 have (tL, F |- tR) as its bottommost sequent.
    
    The rule:
    (rest of s1)       (rest of s2)
    sL |- sR, F        tL, F |- tR
    ------------------------------ (Cut)
          sL, tL |- sR, tR
    

    s1

    The left sequent.

    s2

    The right sequent.

    term1oc

    The occurrence of F in s1.

    term2oc

    The occurrence of F in s2.

    returns

    The sequent (sL, tL |- sR, tR).

  6. def apply(s1: LKProof, s2: LKProof, term1oc: FormulaOccurrence, term2oc: FormulaOccurrence): BinaryTree[OccSequent] with BinaryLKProof with AuxiliaryFormulas { def rule: at.logic.gapt.proofs.lk.CutRuleType.type }

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    Cuts a common formula F (marked by term1oc in the
    succedent of s1 and by term2oc in the antecedent of s2)
    from two proofs s1 & s2.
    
    Let s1 have (sL |- sR, F) as its bottommost sequent and
    let s2 have (tL, F |- tR) as its bottommost sequent.
    
    The rule:
    (rest of s1)       (rest of s2)
    sL |- sR, F        tL, F |- tR
    ------------------------------ (Cut)
          sL, tL |- sR, tR
    

    Cuts a common formula F (marked by term1oc in the
    succedent of s1 and by term2oc in the antecedent of s2)
    from two proofs s1 & s2.
    
    Let s1 have (sL |- sR, F) as its bottommost sequent and
    let s2 have (tL, F |- tR) as its bottommost sequent.
    
    The rule:
    (rest of s1)       (rest of s2)
    sL |- sR, F        tL, F |- tR
    ------------------------------ (Cut)
          sL, tL |- sR, tR
    

    s1

    The left proof with F in the succedent of its bottommost sequent.

    term1oc

    The occurrence of F in s1.

    term2oc

    The occurrence of F in s2.

    returns

    An LK proof with s1 & s2 as its two subtrees and (sL, tL |- sR, tR) as its bottommost sequent.

  7. final def asInstanceOf[T0]: T0

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  8. def clone(): AnyRef

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  11. def finalize(): Unit

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  14. final def isInstanceOf[T0]: Boolean

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  19. def toString(): String

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  20. def unapply(proof: LKProof): Option[(LKProof, LKProof, OccSequent, FormulaOccurrence, FormulaOccurrence)]

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  21. final def wait(): Unit

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  23. final def wait(arg0: Long): Unit

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