Termination Proof

Input TRS

Termination of the rewrite relation of the following TRS is considered.

cond(true,x,y)	→	cond(gr(x,y),p(x),s(y))
gr(0,x)	→	false
gr(s(x),s(y))	→	gr(x,y)
gr(s(x),0)	→	true
p(s(x))	→	x
p(0)	→	0

Proof

1 Dependency Pair Transformation

cond#(true,x,y)	→	cond#(gr(x,y),p(x),s(y))
cond#(true,x,y)	→	gr#(x,y)
cond#(true,x,y)	→	p#(x)
gr#(s(x),s(y))	→	gr#(x,y)

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

• The 1^st component contains the pair

  cond#(true,x,y)

  →

  cond#(gr(x,y),p(x),s(y))

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the integers with ∞ in the arctic semiring

  [cond#(x1, x2, x3)]	=	x1 + x2
  [p(x1)]	=	0 + -1 · x1
  [gr(x1, x2)]	=	0 + -1 · x1
  [0]	=	0
  [false]	=	0
  [true]	=	2
  [s(x1)]	=	3 + 2 · x1

  together with the usable rules

  gr(0,x)	→	false
  gr(s(x),s(y))	→	gr(x,y)
  gr(s(x),0)	→	true
  p(s(x))	→	x
  p(0)	→	0

  (w.r.t. the implicit argument filter of the reduction pair), all pairs could be removed.

  1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

  gr#(s(x),s(y))

  →

  gr#(x,y)

  1.1.2 Subterm Criterion Processor

  Using the simple projection π(gr^#) = 1, all pairs could be removed.

  1.1.2.1 P is empty

  There are no pairs anymore.