Object

at.logic.gapt.proofs.lkOld

ImpLeftRule

Related Doc: package lkOld

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object ImpLeftRule

Source
propositionalRules.scala
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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, term2: HOLFormula): BinaryTree[OccSequent] with base.BinaryLKProof with AuxiliaryFormulas with PrincipalFormulas

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    Introduces an implication term1 -> term2,
    with term1 being in the succedent of s1
    and B being in the antecedent of s2.
    
    Let s1 have (sL |- sR, term1) as its bottommost sequent and
    let s2 have (tL, term2 |- tR) as its bottommost sequent.
    
    The rule:
    (rest of s1)       (rest of s2)
    sL |- sR, A        tL, B |- tR
    ------------------------------ (ImpLeft)
       sL, tL, A -> B |- sR, tR
    

    Introduces an implication term1 -> term2,
    with term1 being in the succedent of s1
    and B being in the antecedent of s2.
    
    Let s1 have (sL |- sR, term1) as its bottommost sequent and
    let s2 have (tL, term2 |- tR) as its bottommost sequent.
    
    The rule:
    (rest of s1)       (rest of s2)
    sL |- sR, A        tL, B |- tR
    ------------------------------ (ImpLeft)
       sL, tL, A -> B |- sR, tR
    

    s1

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

    term1

    The formula in s1.

    term2

    The formula in s2.

    returns

    An LK proof with s1 & s2 as its two subtrees and (sL, tL, term1 -> term2 |- sR, tR) as its bottommost sequent.

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

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    Introduces an implication A -> B,
    with A being marked by term1oc in the succedent of s1
    and with B being marked by term2oc in the antecedent of s2.
    This function merely returns the resulting sequent, not a proof.
    
    Let s1 have (sL |- sR, A) as its bottommost sequent and
    let s2 have (tL, B |- tR) as its bottommost sequent.
    
    The rule:
    (rest of s1)      (rest of s2)
    sL |- sR, A        tL, B |- tR
    ------------------------------ (ImpLeft)
       sL, tL, A -> B |- sR, tR
    

    Introduces an implication A -> B,
    with A being marked by term1oc in the succedent of s1
    and with B being marked by term2oc in the antecedent of s2.
    This function merely returns the resulting sequent, not a proof.
    
    Let s1 have (sL |- sR, A) as its bottommost sequent and
    let s2 have (tL, B |- tR) as its bottommost sequent.
    
    The rule:
    (rest of s1)      (rest of s2)
    sL |- sR, A        tL, B |- tR
    ------------------------------ (ImpLeft)
       sL, tL, A -> B |- sR, tR
    

    s1

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

    term1oc

    The occurrence of A in s1.

    term2oc

    The occurrence of B in s2.

    returns

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

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

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    Introduces an implication A -> B,
    with A being marked by term1oc in the succedent of s1
    and with B being marked by term2oc in the antecedent of s2.
    
    Let s1 have (sL |- sR, A) as its bottommost sequent and
    let s2 have (tL, B |- tR) as its bottommost sequent.
    
    The rule:
     (rest of s1)      (rest of s2)
    sL |- sR, A        tL, B |- tR
    ------------------------------ (ImpLeft)
       sL, tL, A -> B |- sR, tR
    

    Introduces an implication A -> B,
    with A being marked by term1oc in the succedent of s1
    and with B being marked by term2oc in the antecedent of s2.
    
    Let s1 have (sL |- sR, A) as its bottommost sequent and
    let s2 have (tL, B |- tR) as its bottommost sequent.
    
    The rule:
     (rest of s1)      (rest of s2)
    sL |- sR, A        tL, B |- tR
    ------------------------------ (ImpLeft)
       sL, tL, A -> B |- sR, tR
    

    s1

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

    term1oc

    The occurrence of A in s1.

    term2oc

    The occurrence of B in s2.

    returns

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

  7. final def asInstanceOf[T0]: T0

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  8. def clone(): AnyRef

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  9. def computeLeftAux(main: HOLFormula): HOLFormula

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    Returns the left subformula.

    Returns the left subformula.

    main

    A formula of the form l Imp r

    returns

    l.

  10. def computeRightAux(main: HOLFormula): HOLFormula

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    Returns the right subformula.

    Returns the right subformula.

    main

    A formula of the form l Imp r

    returns

    r.

  11. final def eq(arg0: AnyRef): Boolean

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  12. def equals(arg0: Any): Boolean

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  13. def finalize(): Unit

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  15. def hashCode(): Int

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  16. final def isInstanceOf[T0]: Boolean

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  17. final def ne(arg0: AnyRef): Boolean

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  18. final def notify(): Unit

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  19. final def notifyAll(): Unit

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  20. final def synchronized[T0](arg0: ⇒ T0): T0

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  21. def toString(): String

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

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  23. final def wait(): Unit

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  24. final def wait(arg0: Long, arg1: Int): Unit

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

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