Certification Problem

Input (TPDB TRS_Innermost/Applicative_AG01_innermost/#4.19)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ 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(map,fun),app(app(cons,x),xs))	→	app#(app(map,fun),xs)	(27)
  app#(app(map,fun),app(app(cons,x),xs))	→	app#(fun,x)	(26)
  app#(app(filter,fun),app(app(cons,x),xs))	→	app#(app(app(app(filter2,app(fun,x)),fun),x),xs)	(28)
  app#(app(app(app(filter2,true),fun),x),xs)	→	app#(app(filter,fun),xs)	(35)
  app#(app(filter,fun),app(app(cons,x),xs))	→	app#(fun,x)	(32)
  app#(app(app(app(filter2,false),fun),x),xs)	→	app#(app(filter,fun),xs)	(37)

  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,fun),app(app(cons,x),xs))	→	app#(fun,x)	(32)
  1	>	1
  2	>	2
  app#(app(map,fun),app(app(cons,x),xs))	→	app#(fun,x)	(26)
  1	>	1
  2	>	2
  app#(app(map,fun),app(app(cons,x),xs))	→	app#(app(map,fun),xs)	(27)
  1	≥	1
  2	>	2
  app#(app(filter,fun),app(app(cons,x),xs))	→	app#(app(app(app(filter2,app(fun,x)),fun),x),xs)	(28)
  2	>	2
  app#(app(app(app(filter2,true),fun),x),xs)	→	app#(app(filter,fun),xs)	(35)
  2	≥	2
  app#(app(app(app(filter2,false),fun),x),xs)	→	app#(app(filter,fun),xs)	(37)
  2	≥	2

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

• The 2^nd component contains the pair

  app#(app(app(f,x),app(c,x)),app(c,y))

  →

  app#(app(app(f,y),x),y)

  (15)

  1.1.2 Usable Rules Processor

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

  There are no rules.

  1.1.2.1 Size-Change Termination

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

  app#(app(app(f,x),app(c,x)),app(c,y))	→	app#(app(app(f,y),x),y)	(15)
  2	>	2

  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(f,app(s,x)),y),z)

  →

  app#(app(app(f,x),app(s,app(c,y))),app(c,z))

  (17)

  1.1.3 Usable Rules Processor

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

  There are no rules.

  1.1.3.1 Monotonic Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [app#(x1, x2)]	=	2 · x1 + 1 · x2
  [app(x1, x2)]	=	2 · x1 + 1 · x2
  [f]	=	0
  [s]	=	2
  [c]	=	0

  the pair

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

  →

  app#(app(app(f,x),app(s,app(c,y))),app(c,z))

  (17)

  and the rules

  app(app(app(f,x),app(c,x)),app(c,y))	→	app(app(app(f,y),y),app(app(app(f,y),x),y))	(1)
  app(app(app(f,app(s,x)),y),z)	→	app(app(app(f,x),app(s,app(c,y))),app(c,z))	(2)
  app(app(app(f,app(c,x)),x),y)	→	app(c,y)	(3)
  app(app(g,x),y)	→	x	(4)
  app(app(g,x),y)	→	y	(5)
  app(app(map,fun),nil)	→	nil	(6)
  app(app(map,fun),app(app(cons,x),xs))	→	app(app(cons,app(fun,x)),app(app(map,fun),xs))	(7)
  app(app(filter,fun),nil)	→	nil	(8)
  app(app(filter,fun),app(app(cons,x),xs))	→	app(app(app(app(filter2,app(fun,x)),fun),x),xs)	(9)
  app(app(app(app(filter2,true),fun),x),xs)	→	app(app(cons,x),app(app(filter,fun),xs))	(10)
  app(app(app(app(filter2,false),fun),x),xs)	→	app(app(filter,fun),xs)	(11)

  could be deleted.

  1.1.3.1.1 P is empty

  There are no pairs anymore.