Certification Problem

Input (TPDB TRS_Standard/Applicative_05/Ex3Lists)

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 2 components.

• The 1^st component contains the pair

  app#(app(map,f),app(app(cons,h),t))	→	app#(app(map,f),t)	(10)
  app#(app(map,f),app(app(append,l1),l2))	→	app#(app(map,f),l1)	(18)
  app#(app(map,f),app(app(append,l1),l2))	→	app#(app(map,f),l2)	(17)
  app#(app(map,f),app(app(cons,h),t))	→	app#(f,h)	(11)

  1.1.1 Size-Change Termination

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

  app#(app(map,f),app(app(cons,h),t))	→	app#(app(map,f),t)	(10)
  2	>	2
  1	≥	1
  app#(app(map,f),app(app(append,l1),l2))	→	app#(app(map,f),l1)	(18)
  2	>	2
  1	≥	1
  app#(app(map,f),app(app(append,l1),l2))	→	app#(app(map,f),l2)	(17)
  2	>	2
  1	≥	1
  app#(app(map,f),app(app(cons,h),t))	→	app#(f,h)	(11)
  2	>	2
  1	>	1

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

• The 2^nd component contains the pair

  app#(app(append,app(app(append,l1),l2)),l3)	→	app#(app(append,l1),app(app(append,l2),l3))	(16)
  app#(app(append,app(app(append,l1),l2)),l3)	→	app#(app(append,l2),l3)	(15)
  app#(app(append,app(app(cons,h),t)),l)	→	app#(app(append,t),l)	(8)

  1.1.2 Reduction Pair Processor with Usable Rules

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

  [nil]	=	5
  [cons]	=	3
  [app#(x1, x2)]	=	2 · x1 + -∞ · x2 + -16
  [append]	=	1
  [app(x1, x2)]	=	1 · x1 + 0 · x2 + -16

  together with the usable rules

  app(app(append,nil),l)	→	l	(1)
  app(app(append,app(app(cons,h),t)),l)	→	app(app(cons,h),app(app(append,t),l))	(2)
  app(app(append,app(app(append,l1),l2)),l3)	→	app(app(append,l1),app(app(append,l2),l3))	(5)

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

  app#(app(append,app(app(append,l1),l2)),l3)

  →

  app#(app(append,l1),app(app(append,l2),l3))

  (16)

  could be deleted.

  1.1.2.1 Reduction Pair Processor with Usable Rules

  Using the

  prec(app#)	=	0		stat(app#)	=	lex
  prec(cons)	=	0		stat(cons)	=	lex
  prec(app)	=	0		stat(app)	=	lex
  prec(append)	=	0		stat(append)	=	lex

  π(app#)	=	1
  π(cons)	=	[]
  π(app)	=	[2]
  π(append)	=	[]

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

  app#(app(append,app(app(append,l1),l2)),l3)	→	app#(app(append,l2),l3)	(15)
  app#(app(append,app(app(cons,h),t)),l)	→	app#(app(append,t),l)	(8)

  could be deleted.

  1.1.2.1.1 P is empty

  There are no pairs anymore.