Certification Problem

Input (TPDB TRS_Standard/HirokawaMiddeldorp_04/t005)

The rewrite relation of the following TRS is considered.

There are 206 ruless (increase limit for explicit display).

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

There are 115 ruless (increase limit for explicit display).

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

• The 1^st component contains the pair

  *#(x,c(y,z))	→	*#(x,z)	(318)
  *#(x,c(y,z))	→	*#(x,y)	(317)
  *#(c(x,y),z)	→	*#(x,z)	(320)
  *#(c(x,y),z)	→	*#(y,z)	(321)

  1.1.1 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [c(x1, x2)]	=	1 · x1 + 1 · x2
  [*#(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.

  1.1.1.1 Size-Change Termination

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

  *#(x,c(y,z))	→	*#(x,z)	(318)
  1	≥	1
  2	>	2
  *#(x,c(y,z))	→	*#(x,y)	(317)
  1	≥	1
  2	>	2
  *#(c(x,y),z)	→	*#(x,z)	(320)
  1	>	1
  2	≥	2
  *#(c(x,y),z)	→	*#(y,z)	(321)
  1	>	1
  2	≥	2

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

• The 2^nd component contains the pair

  c#(x,c(y,z))	→	+#(x,y)	(208)
  +#(x,c(y,z))	→	c#(y,+(x,z))	(254)
  c#(x,c(y,z))	→	c#(+(x,y),z)	(207)
  +#(x,c(y,z))	→	+#(x,z)	(255)
  +#(c(x,y),z)	→	c#(x,+(y,z))	(256)
  +#(c(x,y),z)	→	+#(y,z)	(257)

  1.1.2 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [+(x1, x2)]	=	1 · x1 + 1 · x2
  [0]	=	0
  [1]	=	0
  [2]	=	0
  [3]	=	0
  [4]	=	0
  [5]	=	0
  [6]	=	0
  [7]	=	0
  [8]	=	0
  [9]	=	0
  [c(x1, x2)]	=	1 · x1 + 1 · x2
  [+#(x1, x2)]	=	1 · x1 + 1 · x2
  [c#(x1, x2)]	=	1 · x1 + 1 · x2

  together with the usable rules

  There are 104 ruless (increase limit for explicit display).

  (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

  1.1.2.1 Monotonic Reduction Pair Processor

  Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions

  prec(0)	=	5		weight(0)	=	1
  prec(1)	=	13		weight(1)	=	5
  prec(2)	=	12		weight(2)	=	9
  prec(3)	=	1		weight(3)	=	6
  prec(4)	=	7		weight(4)	=	7
  prec(5)	=	3		weight(5)	=	8
  prec(6)	=	4		weight(6)	=	11
  prec(7)	=	11		weight(7)	=	12
  prec(8)	=	6		weight(8)	=	13
  prec(9)	=	10		weight(9)	=	14
  prec(+)	=	9		weight(+)	=	0
  prec(c)	=	8		weight(c)	=	5
  prec(c#)	=	0		weight(c#)	=	5
  prec(+#)	=	2		weight(+#)	=	0

  the pairs

  c#(x,c(y,z))	→	+#(x,y)	(208)
  +#(x,c(y,z))	→	c#(y,+(x,z))	(254)
  c#(x,c(y,z))	→	c#(+(x,y),z)	(207)
  +#(x,c(y,z))	→	+#(x,z)	(255)
  +#(c(x,y),z)	→	c#(x,+(y,z))	(256)
  +#(c(x,y),z)	→	+#(y,z)	(257)

  and the rules

  There are 104 ruless (increase limit for explicit display).

  could be deleted.

  1.1.2.1.1 P is empty

  There are no pairs anymore.