Object

at.logic.gapt.proofs.lkOld

EquationLeft1Rule

Related Doc: package lkOld

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object EquationLeft1Rule extends EquationRuleLogger

Source
equationalRules.scala
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  1. EquationLeft1Rule
  2. EquationRuleLogger
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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, main: HOLFormula): BinaryTree[OccSequent] with base.BinaryLKProof with AuxiliaryFormulas with PrincipalFormulas

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    Constructs a proof ending with a EqLeft rule.
    In it, a formula term2 of the form A[T1/a] is replaced by formula main.
    
    This rule does not check for the correct use of the =-symbol.
    The burden of correct usage is on the programmer!
    
    The rule:
    (rest of s1)       (rest of s2)
    sL |- a=b, sR    tr, A[T1/a] |- tR
    ------------------------------------ (EqLeft1)
         sL, A[T1/b], tL |- sR, tR
    

    Constructs a proof ending with a EqLeft rule.
    In it, a formula term2 of the form A[T1/a] is replaced by formula main.
    
    This rule does not check for the correct use of the =-symbol.
    The burden of correct usage is on the programmer!
    
    The rule:
    (rest of s1)       (rest of s2)
    sL |- a=b, sR    tr, A[T1/a] |- tR
    ------------------------------------ (EqLeft1)
         sL, A[T1/b], tL |- sR, tR
    

    s1

    The left proof with the equarion a=b in the succent in its bottommost sequent.

    s2

    The right proof with a formula A[T1/a] in the antecedent of its bottommost sequent, in which some term T1 has been replaced by the term a. Note that identical terms to T1 may occur elsewhere in A. These will not be changed. e.g. P([f(0)]) v -P(f(0)), where f(0) occurs twice, but T1 only refers to the bracketed f(0). This allows selective replacing of terms.

    term1

    The formula (a=b) in s1.

    term2

    The formula A[T1/a] in s2.

    main

    The formula A[T1/b], in which T1 has been replaced by b instead.

    returns

    An LK Proof ending with the new inference.

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

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  6. def apply(s1: OccSequent, s2: OccSequent, term1oc: FormulaOccurrence, term2oc: FormulaOccurrence, main: HOLFormula): OccSequent

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    Constructs a proof ending with a EqLeft rule.
    In it, a formula A (marked by term2oc) is replaced by formula main.
    This function merely returns the resulting sequent, not a proof.
    
    This rule does not check for the correct use of the =-symbol.
    The burden of correct usage is on the programmer!
    
    The rule:
        (s1)               (s2)
    sL |- a=b, sR    tr, A[T1/a] |- tR
    ------------------------------------ (EqLeft1)
         sL, A[T1/b], tL |- sR, tR
    

    Constructs a proof ending with a EqLeft rule.
    In it, a formula A (marked by term2oc) is replaced by formula main.
    This function merely returns the resulting sequent, not a proof.
    
    This rule does not check for the correct use of the =-symbol.
    The burden of correct usage is on the programmer!
    
    The rule:
        (s1)               (s2)
    sL |- a=b, sR    tr, A[T1/a] |- tR
    ------------------------------------ (EqLeft1)
         sL, A[T1/b], tL |- sR, tR
    

    s1

    The left sequent with the equarion a=b in its succent.

    s2

    The right sequent with a formula A[T1/a] in the antecedent of its bottommost sequent, in which some term T1 has been replaced by the term a. Note that identical terms to T1 may occur elsewhere in A. These will not be changed. e.g. P([f(0)]) v -P(f(0)), where f(0) occurs twice, but T1 only refers to the bracketed f(0). This allows selective replacing of terms.

    term1oc

    The occurrence (a=b) in s1.

    term2oc

    The occurrence of A[T1/a] in s2.

    main

    The formula A[T1/b], in which T1 has been replaced by b instead.

    returns

    The sequent (sL, A[T1/b], tL |- sR, tR).

  7. def apply(s1: LKProof, s2: LKProof, term1oc: FormulaOccurrence, term2oc: FormulaOccurrence, pos: Seq[HOLPosition]): BinaryTree[OccSequent] with base.BinaryLKProof with AuxiliaryFormulas with PrincipalFormulas with TermPositions

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  8. def apply(s1: LKProof, s2: LKProof, term1oc: FormulaOccurrence, term2oc: FormulaOccurrence, main: HOLFormula): BinaryTree[OccSequent] with base.BinaryLKProof with AuxiliaryFormulas with PrincipalFormulas with TermPositions

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    Constructs a proof ending with a EqLeft rule.
    In it, a formula A (marked by term2oc) is replaced by formula main.
    
    This method tests whether the proposed auxiliary and main formulas differ in exactly one place.
    If so, it calls the next one with that position.
    
    The rule:
    (rest of s1)       (rest of s2)
    sL |- a=b, sR    tL, A[T1/a] |- tR
    ------------------------------------ (EqLeft1)
         sL, A[T1/b], tL |- sR, tR
    

    Constructs a proof ending with a EqLeft rule.
    In it, a formula A (marked by term2oc) is replaced by formula main.
    
    This method tests whether the proposed auxiliary and main formulas differ in exactly one place.
    If so, it calls the next one with that position.
    
    The rule:
    (rest of s1)       (rest of s2)
    sL |- a=b, sR    tL, A[T1/a] |- tR
    ------------------------------------ (EqLeft1)
         sL, A[T1/b], tL |- sR, tR
    

    s1

    The left proof with the Eq a=b in the succedent in its bottommost sequent.

    s2

    The right proof with a formula A[T1/a] in the antecedent of its bottommost sequent, in which some term T1 has been replaced by the term a. Note that identical terms to T1 may occur elsewhere in A. These will not be changed. e.g. P([f(0)]) v -P(f(0)), where f(0) occurs twice, but T1 only refers to the bracketed f(0). This allows selective replacing of terms.

    term1oc

    The occurrence (a=b) in s1.

    term2oc

    The occurrence of A[T1/a] in s2.

    main

    The formula A[T1/b], in which T1 has been replaced by b instead.

    returns

    An LK Proof ending with the new inference.

  9. final def asInstanceOf[T0]: T0

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  10. def clone(): AnyRef

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  11. def debug(msg: ⇒ String): Unit

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

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  13. def equals(arg0: Any): Boolean

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  14. def error(msg: ⇒ String, e: Throwable): Nothing

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  15. def error(msg: ⇒ String): Unit

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  16. def finalize(): Unit

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    @throws( classOf[java.lang.Throwable] )
  17. final def getClass(): Class[_]

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  18. def hashCode(): Int

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  19. def info(msg: ⇒ String): Unit

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  20. final def isInstanceOf[T0]: Boolean

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  21. val log: Logger

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  22. def loggerName: String

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    EquationRuleLoggerLogger
  23. final def ne(arg0: AnyRef): Boolean

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  24. final def notify(): Unit

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  25. final def notifyAll(): Unit

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

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  27. def toString(): String

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  28. def trace(msg: ⇒ String): Unit

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  29. def unapply(proof: LKProof): Option[(LKProof, LKProof, OccSequent, FormulaOccurrence, FormulaOccurrence, Seq[HOLPosition], FormulaOccurrence)]

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  30. final def wait(): Unit

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

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

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  33. def warn(msg: ⇒ String, e: Throwable): Unit

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  34. def warn(msg: ⇒ String): Unit

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Inherited from EquationRuleLogger

Inherited from Logger

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