Certification Problem

Input (TPDB TRS_Standard/Applicative_05/TreeSize)

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#(size,app(app(node,x),xs))	→	app#(app(map,size),xs)	(15)
  app#(app(map,f),app(app(cons,x),xs))	→	app#(f,x)	(8)
  app#(app(map,f),app(app(cons,x),xs))	→	app#(app(map,f),xs)	(7)

  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#(size,app(app(node,x),xs))	→	app#(app(map,size),xs)	(15)
  2	>	2
  app#(app(map,f),app(app(cons,x),xs))	→	app#(f,x)	(8)
  2	>	2
  1	>	1
  app#(app(map,f),app(app(cons,x),xs))	→	app#(app(map,f),xs)	(7)
  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#(sum,app(app(cons,x),xs))

  →

  app#(sum,xs)

  (11)

  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#(sum,app(app(cons,x),xs))	→	app#(sum,xs)	(11)
  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(plus,app(s,x)),y)

  →

  app#(app(plus,x),y)

  (19)

  1.1.3 Reduction Pair Processor with Usable Rules

  Using the

  prec(app#)	=	0		stat(app#)	=	lex
  prec(s)	=	0		stat(s)	=	lex
  prec(plus)	=	0		stat(plus)	=	lex
  prec(app)	=	1		stat(app)	=	lex

  π(app#)	=	[1]
  π(s)	=	[]
  π(plus)	=	[]
  π(app)	=	[2]

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

  app#(app(plus,app(s,x)),y)

  →

  app#(app(plus,x),y)

  (19)

  could be deleted.

  1.1.3.1 P is empty

  There are no pairs anymore.