Certification Problem

Input (TPDB TRS_Standard/Secret_05_TRS/aprove4)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

• The 1^st component contains the pair

  if#(false,x,y)	→	minus#(p(x),y)	(16)
  minus#(x,y)	→	if#(le(x,y),x,y)	(14)

  1.1.1 Switch to Innermost Termination

  The TRS does not have overlaps with the pairs and is locally confluent:

  20

  Hence, it suffices to show innermost termination in the following.

  1.1.1.1 Usable Rules Processor

  We restrict the rewrite rules to the following usable rules of the DP problem.

  le(0,y)	→	true	(5)
  le(s(x),0)	→	false	(6)
  le(s(x),s(y))	→	le(x,y)	(7)
  p(0)	→	s(s(0))	(1)
  p(s(x))	→	x	(2)

  1.1.1.1.1 Innermost Lhss Removal Processor

  We restrict the innermost strategy to the following left hand sides.

  p(0)

  p(s(x0))

  le(0,x0)

  le(s(x0),0)

  le(s(x0),s(x1))

  1.1.1.1.1.1 Narrowing Processor

  We consider all narrowings of the pair below position 1 to get the following set of pairs

  minus#(0,x0)	→	if#(true,0,x0)	(18)
  minus#(s(x0),0)	→	if#(false,s(x0),0)	(19)
  minus#(s(x0),s(x1))	→	if#(le(x0,x1),s(x0),s(x1))	(20)

  1.1.1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    minus#(s(x0),0)	→	if#(false,s(x0),0)	(19)
    if#(false,x,y)	→	minus#(p(x),y)	(16)
    minus#(s(x0),s(x1))	→	if#(le(x0,x1),s(x0),s(x1))	(20)

    1.1.1.1.1.1.1.1 Narrowing Processor

    We consider all narrowings of the pair below position 1 to get the following set of pairs

    if#(false,0,y1)	→	minus#(s(s(0)),y1)	(21)
    if#(false,s(x0),y1)	→	minus#(x0,y1)	(22)

    1.1.1.1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 1 component.

    • The 1^st component contains the pair

      if#(false,s(x0),y1)	→	minus#(x0,y1)	(22)
      minus#(s(x0),0)	→	if#(false,s(x0),0)	(19)
      minus#(s(x0),s(x1))	→	if#(le(x0,x1),s(x0),s(x1))	(20)

      1.1.1.1.1.1.1.1.1.1 Usable Rules Processor

      We restrict the rewrite rules to the following usable rules of the DP problem.

      le(0,y)	→	true	(5)
      le(s(x),0)	→	false	(6)
      le(s(x),s(y))	→	le(x,y)	(7)

      1.1.1.1.1.1.1.1.1.1.1 Innermost Lhss Removal Processor

      We restrict the innermost strategy to the following left hand sides.

      le(0,x0)

      le(s(x0),0)

      le(s(x0),s(x1))

      1.1.1.1.1.1.1.1.1.1.1.1 Instantiation Processor

      We instantiate the pair to the following set of pairs

      if#(false,s(z0),0)	→	minus#(z0,0)	(23)
      if#(false,s(z0),s(z1))	→	minus#(z0,s(z1))	(24)

      1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

      The dependency pairs are split into 2 components.

      • The 1^st component contains the pair

        if#(false,s(z0),0)	→	minus#(z0,0)	(23)
        minus#(s(x0),0)	→	if#(false,s(x0),0)	(19)

        1.1.1.1.1.1.1.1.1.1.1.1.1.1 Usable Rules Processor

        We restrict the rewrite rules to the following usable rules of the DP problem.

        There are no rules.

        1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Innermost Lhss Removal Processor

        We restrict the innermost strategy to the following left hand sides.

        There are no lhss.

        1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Forward Instantiation Processor

        We instantiate the pair to the following set of pairs

        if#(false,s(s(y_0)),0)

        →

        minus#(s(y_0),0)

        (25)

        1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Forward Instantiation Processor

        We instantiate the pair to the following set of pairs

        minus#(s(s(y_0)),0)

        →

        if#(false,s(s(y_0)),0)

        (26)

        1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Size-Change Termination

        Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

        minus#(s(s(y_0)),0)	→	if#(false,s(s(y_0)),0)	(26)
        1	≥	2
        2	≥	3
        if#(false,s(s(y_0)),0)	→	minus#(s(y_0),0)	(25)
        2	>	1
        3	≥	2

        As there is no critical graph in the transitive closure, there are no infinite chains.

      • The 2^nd component contains the pair

        if#(false,s(z0),s(z1))	→	minus#(z0,s(z1))	(24)
        minus#(s(x0),s(x1))	→	if#(le(x0,x1),s(x0),s(x1))	(20)

        1.1.1.1.1.1.1.1.1.1.1.1.1.2 Forward Instantiation Processor

        We instantiate the pair to the following set of pairs

        if#(false,s(s(y_0)),s(x1))

        →

        minus#(s(y_0),s(x1))

        (27)

        1.1.1.1.1.1.1.1.1.1.1.1.1.2.1 Forward Instantiation Processor

        We instantiate the pair to the following set of pairs

        minus#(s(s(y_1)),s(x1))

        →

        if#(le(s(y_1),x1),s(s(y_1)),s(x1))

        (28)

        1.1.1.1.1.1.1.1.1.1.1.1.1.2.1.1 Size-Change Termination

        Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

        minus#(s(s(y_1)),s(x1))	→	if#(le(s(y_1),x1),s(s(y_1)),s(x1))	(28)
        1	≥	2
        2	≥	3
        if#(false,s(s(y_0)),s(x1))	→	minus#(s(y_0),s(x1))	(27)
        2	>	1
        3	≥	2

        As there is no critical graph in the transitive closure, there are no infinite chains.

• The 2^nd component contains the pair

  p#(p(s(x)))

  →

  p#(x)

  (11)

  1.1.2 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [p(x1)]	=	1 · x1
  [s(x1)]	=	1 · x1
  [p#(x1)]	=	1 · x1

  having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

  1.1.2.1 Size-Change Termination

  Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

  p#(p(s(x)))	→	p#(x)	(11)
  1	>	1

  As there is no critical graph in the transitive closure, there are no infinite chains.

• The 3^rd component contains the pair

  le#(s(x),s(y))	→	le#(x,y)	(13)
  le#(p(s(x)),x)	→	le#(x,x)	(12)

  1.1.3 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [s(x1)]	=	1 · x1
  [p(x1)]	=	1 · x1
  [le#(x1, x2)]	=	1 · x1 + 1 · x2

  having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

  1.1.3.1 Size-Change Termination

  Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

  le#(s(x),s(y))	→	le#(x,y)	(13)
  1	>	1
  2	>	2
  le#(p(s(x)),x)	→	le#(x,x)	(12)
  1	>	1
  2	≥	1
  1	>	2
  2	≥	2

  As there is no critical graph in the transitive closure, there are no infinite chains.