Certification Problem

Input (TPDB TRS_Standard/Applicative_first_order_05/17)

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 3 components.

• The 1^st component contains the pair

  app#(app(filter,f),app(app(cons,x),xs))	→	app#(app(app(app(filter2,app(f,x)),f),x),xs)	(32)
  app#(app(app(app(filter2,false),f),x),xs)	→	app#(app(filter,f),xs)	(38)
  app#(app(filter,f),app(app(cons,x),xs))	→	app#(f,x)	(28)
  app#(app(app(app(filter2,true),f),x),xs)	→	app#(app(filter,f),xs)	(34)
  app#(app(map,f),app(app(cons,x),xs))	→	app#(f,x)	(25)
  app#(app(map,f),app(app(cons,x),xs))	→	app#(app(map,f),xs)	(24)

  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(filter,f),app(app(cons,x),xs))	→	app#(app(app(app(filter2,app(f,x)),f),x),xs)	(32)
  2	>	2
  app#(app(app(app(filter2,false),f),x),xs)	→	app#(app(filter,f),xs)	(38)
  2	≥	2
  app#(app(filter,f),app(app(cons,x),xs))	→	app#(f,x)	(28)
  2	>	2
  1	>	1
  app#(app(app(app(filter2,true),f),x),xs)	→	app#(app(filter,f),xs)	(34)
  2	≥	2
  app#(app(map,f),app(app(cons,x),xs))	→	app#(f,x)	(25)
  2	>	2
  1	>	1
  app#(app(map,f),app(app(cons,x),xs))	→	app#(app(map,f),xs)	(24)
  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#(i,app(app(.,x),y))	→	app#(i,y)	(21)
  app#(i,app(app(.,x),y))	→	app#(i,x)	(20)

  1.1.2 Size-Change Termination

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

  app#(i,app(app(.,x),y))	→	app#(i,y)	(21)
  2	>	2
  1	≥	1
  app#(i,app(app(.,x),y))	→	app#(i,x)	(20)
  2	>	2
  1	≥	1

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

• The 3^rd component contains the pair

  app#(app(.,app(app(.,x),y)),z)	→	app#(app(.,x),app(app(.,y),z))	(19)
  app#(app(.,app(app(.,x),y)),z)	→	app#(app(.,y),z)	(18)

  1.1.3 Reduction Pair Processor with Usable Rules

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

  [1]	=	0
  [i]	=	3
  [.]	=	2
  [app#(x1, x2)]	=	0 · x1 + -∞ · x2 + 0
  [app(x1, x2)]	=	1 · x1 + 0 · x2 + 4

  together with the usable rules

  app(app(.,1),x)	→	x	(1)
  app(app(.,x),1)	→	x	(2)
  app(app(.,app(i,x)),x)	→	1	(3)
  app(app(.,x),app(i,x))	→	1	(4)
  app(app(.,app(i,y)),app(app(.,y),z))	→	z	(5)
  app(app(.,y),app(app(.,app(i,y)),z))	→	z	(6)
  app(app(.,app(app(.,x),y)),z)	→	app(app(.,x),app(app(.,y),z))	(7)

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

  app#(app(.,app(app(.,x),y)),z)

  →

  app#(app(.,x),app(app(.,y),z))

  (19)

  could be deleted.

  1.1.3.1 Reduction Pair Processor with Usable Rules

  Using the

  prec(app#)	=	0		stat(app#)	=	lex
  prec(app)	=	1		stat(app)	=	lex
  prec(.)	=	0		stat(.)	=	lex

  π(app#)	=	[1]
  π(app)	=	[2]
  π(.)	=	[]

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

  app#(app(.,app(app(.,x),y)),z)

  →

  app#(app(.,y),z)

  (18)

  could be deleted.

  1.1.3.1.1 P is empty

  There are no pairs anymore.