Certification Problem

Input (TPDB TRS_Standard/Applicative_first_order_05/12)

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)	(37)
  app#(app(app(app(filter2,false),f),x),xs)	→	app#(app(filter,f),xs)	(43)
  app#(app(filter,f),app(app(cons,x),xs))	→	app#(f,x)	(33)
  app#(app(app(app(filter2,true),f),x),xs)	→	app#(app(filter,f),xs)	(39)
  app#(app(map,f),app(app(cons,x),xs))	→	app#(f,x)	(30)
  app#(app(map,f),app(app(cons,x),xs))	→	app#(app(map,f),xs)	(29)

  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)	(37)
  2	>	2
  app#(app(app(app(filter2,false),f),x),xs)	→	app#(app(filter,f),xs)	(43)
  2	≥	2
  app#(app(filter,f),app(app(cons,x),xs))	→	app#(f,x)	(33)
  2	>	2
  1	>	1
  app#(app(app(app(filter2,true),f),x),xs)	→	app#(app(filter,f),xs)	(39)
  2	≥	2
  app#(app(map,f),app(app(cons,x),xs))	→	app#(f,x)	(30)
  2	>	2
  1	>	1
  app#(app(map,f),app(app(cons,x),xs))	→	app#(app(map,f),xs)	(29)
  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#(not,app(app(and,x),y))	→	app#(not,x)	(17)
  app#(not,app(app(and,x),y))	→	app#(not,y)	(16)
  app#(not,app(app(or,x),y))	→	app#(not,x)	(13)
  app#(not,app(app(or,x),y))	→	app#(not,y)	(12)

  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#(not,app(app(and,x),y))	→	app#(not,x)	(17)
  2	>	2
  1	≥	1
  app#(not,app(app(and,x),y))	→	app#(not,y)	(16)
  2	>	2
  1	≥	1
  app#(not,app(app(or,x),y))	→	app#(not,x)	(13)
  2	>	2
  1	≥	1
  app#(not,app(app(or,x),y))	→	app#(not,y)	(12)
  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(and,app(app(or,y),z)),x)	→	app#(app(and,x),y)	(26)
  app#(app(and,app(app(or,y),z)),x)	→	app#(app(and,x),z)	(24)
  app#(app(and,x),app(app(or,y),z))	→	app#(app(and,x),y)	(21)
  app#(app(and,x),app(app(or,y),z))	→	app#(app(and,x),z)	(20)

  1.1.3 Reduction Pair Processor with Usable Rules

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

  [app(x1, x2)]	=	-1 · x1 + 2 · x2 + -16
  [and]	=	0
  [app#(x1, x2)]	=	0 · x1 + 3 · x2 + -16
  [or]	=	1

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

  app#(app(and,app(app(or,y),z)),x)

  →

  app#(app(and,x),z)

  (24)

  could be deleted.

  1.1.3.1 Reduction Pair Processor with Usable Rules

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

  [app(x1, x2)]	=	1 · x1 + 1 · x2 + 0
  [and]	=	2
  [app#(x1, x2)]	=	1 · x1 + 3 · x2 + 0
  [or]	=	1

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

  app#(app(and,app(app(or,y),z)),x)

  →

  app#(app(and,x),y)

  (26)

  could be deleted.

  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.

  app#(app(and,x),app(app(or,y),z))	→	app#(app(and,x),y)	(21)
  2	>	2
  1	≥	1
  app#(app(and,x),app(app(or,y),z))	→	app#(app(and,x),z)	(20)
  2	>	2
  1	≥	1

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