Certification Problem

Input (TPDB TRS_Standard/Secret_06_TRS/tpa10)

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 6 components.

• The 1^st component contains the pair

  f#(s(x),s(y))

  →

  f#(-(min(s(x),s(y)),max(s(x),s(y))),*(s(x),s(y)))

  (21)

  1.1.1.1 Usable Rules Processor

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

  min(s(x),s(y))	→	s(min(x,y))	(3)
  max(s(x),s(y))	→	s(max(x,y))	(6)
  -(x,0)	→	x	(9)
  -(s(x),s(y))	→	-(x,y)	(10)
  *(x,s(y))	→	+(x,*(x,y))	(12)
  *(x,0)	→	0	(11)
  +(0,y)	→	y	(7)
  +(s(x),y)	→	s(+(x,y))	(8)
  max(0,y)	→	y	(4)
  max(x,0)	→	x	(5)
  min(0,y)	→	0	(1)
  min(x,0)	→	0	(2)

  1.1.1.1.1 Innermost Lhss Removal Processor

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

  min(0,x0)

  min(x0,0)

  min(s(x0),s(x1))

  max(0,x0)

  max(x0,0)

  max(s(x0),s(x1))

  +(0,x0)

  +(s(x0),x1)

  -(x0,0)

  -(s(x0),s(x1))

  *(x0,0)

  *(x0,s(x1))

  1.1.1.1.1.1 Rewriting Processor

  We rewrite the right hand side of the pair resulting in

  →

  1.1.1.1.1.1.1 Rewriting Processor

  We rewrite the right hand side of the pair resulting in

  →

  1.1.1.1.1.1.1.1 Rewriting Processor

  We rewrite the right hand side of the pair resulting in

  →

  1.1.1.1.1.1.1.1.1 Rewriting Processor

  We rewrite the right hand side of the pair resulting in

  →

  1.1.1.1.1.1.1.1.1.1 Rewriting Processor

  We rewrite the right hand side of the pair resulting in

  →

  1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

  prec(s)	=	0		weight(s)	=	1
  prec(0)	=	1		weight(0)	=	1

  in combination with the following argument filter

  π(f#)	=	1
  π(s)	=	[1]
  π(-)	=	1
  π(min)	=	1
  π(0)	=	[]

  together with the usable rules

  min(s(x),s(y))	→	s(min(x,y))	(3)
  min(0,y)	→	0	(1)
  min(x,0)	→	0	(2)
  -(x,0)	→	x	(9)
  -(s(x),s(y))	→	-(x,y)	(10)

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

  f#(s(x),s(y))

  →

  f#(-(min(x,y),max(x,y)),s(+(x,*(s(x),y))))

  (30)

  could be deleted.

  1.1.1.1.1.1.1.1.1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

  *#(x,s(y))

  →

  *#(x,y)

  (20)

  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 Size-Change Termination

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

  *#(x,s(y))	→	*#(x,y)	(20)
  1	≥	1
  2	>	2

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

• The 3^rd component contains the pair

  min#(s(x),s(y))

  →

  min#(x,y)

  (15)

  1.1.1.3 Usable Rules Processor

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

  There are no rules.

  1.1.1.3.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  1.1.1.3.1.1 Size-Change Termination

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

  min#(s(x),s(y))	→	min#(x,y)	(15)
  1	>	1
  2	>	2

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

• The 4^th component contains the pair

  max#(s(x),s(y))

  →

  max#(x,y)

  (16)

  1.1.1.4 Usable Rules Processor

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

  There are no rules.

  1.1.1.4.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  1.1.1.4.1.1 Size-Change Termination

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

  max#(s(x),s(y))	→	max#(x,y)	(16)
  1	>	1
  2	>	2

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

• The 5^th component contains the pair

  +#(s(x),y)

  →

  +#(x,y)

  (17)

  1.1.1.5 Usable Rules Processor

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

  There are no rules.

  1.1.1.5.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  1.1.1.5.1.1 Size-Change Termination

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

  +#(s(x),y)	→	+#(x,y)	(17)
  1	>	1
  2	≥	2

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

• The 6^th component contains the pair

  -#(s(x),s(y))

  →

  -#(x,y)

  (18)

  1.1.1.6 Usable Rules Processor

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

  There are no rules.

  1.1.1.6.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  1.1.1.6.1.1 Size-Change Termination

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

  -#(s(x),s(y))	→	-#(x,y)	(18)
  1	>	1
  2	>	2

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