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at.logic.gapt.examples

SumOfOnesFExampleProof

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object SumOfOnesFExampleProof extends ProofSequence

Constructs the cut-free FOL LK proof of the sequent

AUX, f(0) = 0, Forall x.f(s(x)) = f(x) + s(0) |- f(sn(0)) = sn(0) Where AUX is {Transitivity, Symmetry, Reflexity of =, Forall xy.x=y -> s(x) = s(y), f(0) = 0, Forall x.f(s(x)) = f(x) + s(0)}

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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. val CongSucc: FOLFormula

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  5. def CongSuccX(x: FOLTerm): FOLFormula

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  6. def CongSuccXY(x: FOLTerm, y: FOLTerm): FOLFormula

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  7. def EqChainProof(s1: LKProof, n: Int): LKProof

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    Generates a sequent containing, in addition to the formulas in the bottommost sequent of s1, the chain of equations f(n) = s(f(n-1)),...,f(1)=s(f(0)), f(0) = 0.s The generates proof employs only the axiom of transitivity and (x=y -> s(x) = s(y)))

    Generates a sequent containing, in addition to the formulas in the bottommost sequent of s1, the chain of equations f(n) = s(f(n-1)),...,f(1)=s(f(0)), f(0) = 0.s The generates proof employs only the axiom of transitivity and (x=y -> s(x) = s(y)))

    TODO should be private - but scala shell does not allow access modifiers when :loading a file

  8. val FSucc: FOLFormula

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  9. def FSuccX(x: FOLTerm): FOLAtom

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  10. val FZero: FOLAtom

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  11. def Fn(n: Int): FOLTerm

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  12. val Plus: FOLFormula

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  13. def PlusX(x: FOLTerm): FOLAtom

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  14. def TermGenProof(s1: LKProof, n: Int, targetN: Int): LKProof

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    Given a proof s1, produced by EqChainProof, generates a proof that eliminates the chains of equasions and proves the final sequent FZero, FSucc, TR, Plus |- f(n) = n.

    Given a proof s1, produced by EqChainProof, generates a proof that eliminates the chains of equasions and proves the final sequent FZero, FSucc, TR, Plus |- f(n) = n.

    TODO should be private - but scala shell does not allow access modifiers when :loading a file

  15. val Trans: FOLFormula

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  16. def apply(n: Int): LKProof

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    SumOfOnesFExampleProofProofSequence
  17. final def asInstanceOf[T0]: T0

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  18. def clone(): AnyRef

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

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  20. def equals(arg0: Any): Boolean

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  21. var f: String

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  22. def finalize(): Unit

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  23. final def getClass(): Class[_]

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  24. def hashCode(): Int

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  25. final def isInstanceOf[T0]: Boolean

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  26. def name: String

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    ProofSequence
  27. final def ne(arg0: AnyRef): Boolean

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  28. final def notify(): Unit

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  29. final def notifyAll(): Unit

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  30. val p: String

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  31. def proof(n: Int): LKProof

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  32. val s: String

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  33. def start(n: Int): LeafTree[OccSequent] with NullaryLKProof

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

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  35. def toString(): String

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  36. final def wait(): Unit

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

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

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  39. val x: FOLVar

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  40. val y: FOLVar

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  41. val z: FOLVar

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  42. val zero: String

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Inherited from ProofSequence

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