Certification Problem

Input (TPDB TRS_Standard/Strategy_removed_AG01/#4.33)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Switch to Innermost Termination

The TRS is overlay and locally confluent:

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

1.1 Dependency Pair Transformation

1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

• The 1^st component contains the pair

  weight#(cons(n,cons(m,x)))

  →

  weight#(sum(cons(n,cons(m,x)),cons(0,x)))

  (8)

  1.1.1.1 Usable Rules Processor

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

  sum(cons(0,x),y)	→	sum(x,y)	(2)
  sum(cons(s(n),x),cons(m,y))	→	sum(cons(n,x),cons(s(m),y))	(1)
  sum(nil,y)	→	y	(3)

  1.1.1.1.1 Innermost Lhss Removal Processor

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

  sum(cons(s(x0),x1),cons(x2,x3))

  sum(cons(0,x0),x1)

  sum(nil,x0)

  1.1.1.1.1.1 Reduction Pair Processor

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

  prec(cons)

  =

  1

  weight(cons)

  =

  1

  in combination with the following argument filter

  π(weight#)	=	1
  π(cons)	=	[2]
  π(sum)	=	2

  the pair

  weight#(cons(n,cons(m,x)))

  →

  weight#(sum(cons(n,cons(m,x)),cons(0,x)))

  (8)

  could be deleted.

  1.1.1.1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

  sum#(cons(0,x),y)	→	sum#(x,y)	(7)
  sum#(cons(s(n),x),cons(m,y))	→	sum#(cons(n,x),cons(s(m),y))	(6)

  1.1.1.2 Usable Rules Processor

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

  There are no rules.

  1.1.1.2.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  1.1.1.2.1.1 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [sum#(x1, x2)]	=	2 · x1 + 1 · x2
  [cons(x1, x2)]	=	1 · x1 + 1 · x2
  [0]	=	0
  [s(x1)]	=	2 + 1 · x1

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

  sum#(cons(0,x),y)	→	sum#(x,y)	(7)
  sum#(cons(s(n),x),cons(m,y))	→	sum#(cons(n,x),cons(s(m),y))	(6)

  and no rules could be deleted.

  1.1.1.2.1.1.1 P is empty

  There are no pairs anymore.