Certification Problem

Input (TPDB TRS_Standard/CiME_04/list-sum-prod-bin)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by ttt2 @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Dependency Graph Processor

The dependency pairs are split into 4 components.

• The 1^st component contains the pair

  sum#(cons(x,l))

  →

  sum#(l)

  (28)

  1.1.1 Size-Change Termination

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

  sum#(cons(x,l))	→	sum#(l)	(28)
  1	>	1

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

• The 2^nd component contains the pair

  prod#(cons(x,l))

  →

  prod#(l)

  (30)

  1.1.2 Size-Change Termination

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

  prod#(cons(x,l))	→	prod#(l)	(30)
  1	>	1

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

• The 3^rd component contains the pair

  *#(1(x),y)	→	*#(x,y)	(24)
  *#(0(x),y)	→	*#(x,y)	(22)

  1.1.3 Size-Change Termination

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

  *#(1(x),y)	→	*#(x,y)	(24)
  2	≥	2
  1	>	1
  *#(0(x),y)	→	*#(x,y)	(22)
  2	≥	2
  1	>	1

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

• The 4^th component contains the pair

  +#(0(x),1(y))	→	+#(x,y)	(17)
  +#(1(x),1(y))	→	+#(+(x,y),1(#))	(20)
  +#(1(x),1(y))	→	+#(x,y)	(19)
  +#(1(x),0(y))	→	+#(x,y)	(18)
  +#(0(x),0(y))	→	+#(x,y)	(15)

  1.1.4 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the arctic semiring over the integers

  [0(x1)]	=	1 · x1 + 4
  [1(x1)]	=	0 · x1 + 0
  [+#(x1, x2)]	=	-∞ · x1 + 0 · x2 + 0
  [#]	=	0
  [+(x1, x2)]	=	0 · x1 + 2 · x2 + 2

  having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pairs

  +#(1(x),0(y))	→	+#(x,y)	(18)
  +#(0(x),0(y))	→	+#(x,y)	(15)

  could be deleted.

  1.1.4.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the arctic semiring over the integers

  [0(x1)]	=	-∞ · x1 + 0
  [1(x1)]	=	4 · x1 + 4
  [+#(x1, x2)]	=	-∞ · x1 + 0 · x2 + 0
  [#]	=	0
  [+(x1, x2)]	=	-∞ · x1 + 0 · x2 + 0

  having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pairs

  +#(0(x),1(y))	→	+#(x,y)	(17)
  +#(1(x),1(y))	→	+#(x,y)	(19)

  could be deleted.

  1.1.4.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

  [0(x1)]	=	0 0 0 0 0 0 0 1 1 · x1 + 0 0 0 0 0 0 0 0 0
  [1(x1)]	=	0 0 0 0 0 0 0 1 1 · x1 + 0 0 0 1 0 0 1 0 0
  [+#(x1, x2)]	=	0 1 1 0 0 0 0 0 0 · x1 + 0 0 1 0 0 0 0 0 0 · x2 + 0 0 0 0 0 0 0 0 0
  [#]	=	0 0 0 0 0 0 0 0 0
  [+(x1, x2)]	=	1 1 1 0 1 0 0 0 1 · x1 + 1 0 0 0 1 0 0 0 1 · x2 + 0 0 0 0 0 0 0 0 0

  together with the usable rules

  +(x,#)	→	x	(2)
  +(#,x)	→	x	(3)
  +(0(x),0(y))	→	0(+(x,y))	(4)
  +(0(x),1(y))	→	1(+(x,y))	(5)
  +(1(x),0(y))	→	1(+(x,y))	(6)
  +(1(x),1(y))	→	0(+(+(x,y),1(#)))	(7)
  0(#)	→	#	(1)

  (w.r.t. the implicit argument filter of the reduction pair), the pair

  +#(1(x),1(y))

  →

  +#(+(x,y),1(#))

  (20)

  could be deleted.

  1.1.4.1.1.1 P is empty

  There are no pairs anymore.