Certification Problem

Input (TPDB TRS_Standard/AProVE_07/thiemann14)

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

  if2#(false,x,y)	→	int#(p(x),p(y))	(35)
  int#(x,y)	→	if_int#(zero(x),zero(y),x,y)	(29)
  if_int#(true,b,x,y)	→	if1#(b,x,y)	(30)
  if1#(false,x,y)	→	int#(s(0),y)	(32)
  if_int#(false,b,x,y)	→	if2#(b,x,y)	(31)

  1.1.1 Reduction Pair Processor with Usable Rules

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

  [false]	=	5
  [zero(x1)]	=	4 · x1 + 0
  [if1#(x1, x2, x3)]	=	-7 · x1 + -∞ · x2 + 1 · x3 + -∞
  [if2#(x1, x2, x3)]	=	-4 · x1 + -5 · x2 + 0 · x3 + -∞
  [p(x1)]	=	-1 · x1 + 0
  [0]	=	0
  [if_int#(x1,...,x4)]	=	-∞ · x1 + -3 · x2 + -4 · x3 + 1 · x4 + -∞
  [int#(x1, x2)]	=	-4 · x1 + 1 · x2 + -3
  [s(x1)]	=	1 · x1 + 1
  [true]	=	4

  together with the usable rules

  p(0)	→	0	(8)
  p(s(0))	→	0	(9)
  p(s(s(x)))	→	s(p(s(x)))	(10)
  zero(0)	→	true	(6)
  zero(s(x))	→	false	(7)

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

  if_int#(false,b,x,y)

  →

  if2#(b,x,y)

  (31)

  could be deleted.

  1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    int#(x,y)	→	if_int#(zero(x),zero(y),x,y)	(29)
    if_int#(true,b,x,y)	→	if1#(b,x,y)	(30)
    if1#(false,x,y)	→	int#(s(0),y)	(32)

    1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

    [false]	=	1
    [zero(x1)]	=	0 · x1 + 1
    [if1#(x1, x2, x3)]	=	-∞ · x1 + 0 · x2 + -∞ · x3 + 4
    [0]	=	3
    [if_int#(x1,...,x4)]	=	3 · x1 + -∞ · x2 + 3 · x3 + -∞ · x4 + 0
    [int#(x1, x2)]	=	3 · x1 + -∞ · x2 + 4
    [s(x1)]	=	-3 · x1 + 0
    [true]	=	3

    together with the usable rules

    zero(0)	→	true	(6)
    zero(s(x))	→	false	(7)

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

    if_int#(true,b,x,y)

    →

    if1#(b,x,y)

    (30)

    could be deleted.

    1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.

• The 2^nd component contains the pair

  intlist#(x)	→	if_intlist#(empty(x),x)	(23)
  if_intlist#(false,x)	→	intlist#(tail(x))	(25)

  1.1.2 Reduction Pair Processor with Usable Rules

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

  [false]	=	2
  [empty(x1)]	=	-6 · x1 + 0
  [cons(x1, x2)]	=	0 · x1 + 8 · x2 + 8
  [intlist#(x1)]	=	0 · x1 + 0
  [tail(x1)]	=	-8 · x1 + 0
  [nil]	=	0
  [if_intlist#(x1, x2)]	=	0 · x1 + 0 · x2 + 0
  [true]	=	0

  together with the usable rules

  empty(nil)	→	true	(1)
  empty(cons(x,y))	→	false	(2)
  tail(nil)	→	nil	(3)
  tail(cons(x,y))	→	y	(4)

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

  if_intlist#(false,x)

  →

  intlist#(tail(x))

  (25)

  could be deleted.

  1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 3^rd component contains the pair

  p#(s(s(x)))

  →

  p#(s(x))

  (21)

  1.1.3 Size-Change Termination

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

  p#(s(s(x)))	→	p#(s(x))	(21)
  1	>	1

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