Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex6_9_Luc02c_GM)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by ttt2 @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Reduction Pair Processor with Usable Rules

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

• The 1^st component contains the pair

  mark#(cons1(X1,X2))	→	mark#(X1)	(23)
  mark#(cons1(X1,X2))	→	mark#(X2)	(22)
  mark#(s(X))	→	mark#(X)	(21)
  mark#(cons(X1,X2))	→	mark#(X1)	(20)
  mark#(2nd(X))	→	a__2nd#(mark(X))	(17)
  a__2nd#(cons(X,X1))	→	a__2nd#(cons1(mark(X),mark(X1)))	(14)
  a__2nd#(cons1(X,cons(Y,Z)))	→	mark#(Y)	(11)
  mark#(2nd(X))	→	mark#(X)	(16)
  a__2nd#(cons(X,X1))	→	mark#(X)	(13)
  a__2nd#(cons(X,X1))	→	mark#(X1)	(12)

  1.1.1.1 Reduction Pair Processor with Usable Rules

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

  [a__from(x1)]	=	4 · x1 + 6
  [a__2nd#(x1)]	=	0 · x1 + 4
  [cons1(x1, x2)]	=	0 · x1 + 0 · x2 + 0
  [mark(x1)]	=	0 · x1 + 2
  [s(x1)]	=	0 · x1 + -∞
  [cons(x1, x2)]	=	0 · x1 + 0 · x2 + -∞
  [2nd(x1)]	=	2 · x1 + 4
  [mark#(x1)]	=	0 · x1 + 0
  [from(x1)]	=	4 · x1 + 6
  [a__2nd(x1)]	=	2 · x1 + 4

  together with the usable rules

  a__2nd(cons1(X,cons(Y,Z)))	→	mark(Y)	(1)
  a__2nd(cons(X,X1))	→	a__2nd(cons1(mark(X),mark(X1)))	(2)
  a__from(X)	→	cons(mark(X),from(s(X)))	(3)
  mark(2nd(X))	→	a__2nd(mark(X))	(4)
  mark(from(X))	→	a__from(mark(X))	(5)
  mark(cons(X1,X2))	→	cons(mark(X1),X2)	(6)
  mark(s(X))	→	s(mark(X))	(7)
  mark(cons1(X1,X2))	→	cons1(mark(X1),mark(X2))	(8)
  a__2nd(X)	→	2nd(X)	(9)
  a__from(X)	→	from(X)	(10)

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

  mark#(2nd(X))

  →

  mark#(X)

  (16)

  could be deleted.

  1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

  [a__from(x1)]	=	0 · x1 + 2
  [a__2nd#(x1)]	=	0 · x1 + -∞
  [cons1(x1, x2)]	=	0 · x1 + 0 · x2 + -∞
  [mark(x1)]	=	0 · x1 + -∞
  [s(x1)]	=	0 · x1 + 2
  [cons(x1, x2)]	=	0 · x1 + 0 · x2 + 0
  [2nd(x1)]	=	2 · x1 + -∞
  [mark#(x1)]	=	0 · x1 + -∞
  [from(x1)]	=	0 · x1 + 2
  [a__2nd(x1)]	=	2 · x1 + -∞

  together with the usable rules

  a__2nd(cons1(X,cons(Y,Z)))	→	mark(Y)	(1)
  a__2nd(cons(X,X1))	→	a__2nd(cons1(mark(X),mark(X1)))	(2)
  a__from(X)	→	cons(mark(X),from(s(X)))	(3)
  mark(2nd(X))	→	a__2nd(mark(X))	(4)
  mark(from(X))	→	a__from(mark(X))	(5)
  mark(cons(X1,X2))	→	cons(mark(X1),X2)	(6)
  mark(s(X))	→	s(mark(X))	(7)
  mark(cons1(X1,X2))	→	cons1(mark(X1),mark(X2))	(8)
  a__2nd(X)	→	2nd(X)	(9)
  a__from(X)	→	from(X)	(10)

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

  mark#(2nd(X))

  →

  a__2nd#(mark(X))

  (17)

  could be deleted.

  1.1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    mark#(s(X))	→	mark#(X)	(21)
    mark#(cons1(X1,X2))	→	mark#(X1)	(23)
    mark#(cons1(X1,X2))	→	mark#(X2)	(22)
    mark#(cons(X1,X2))	→	mark#(X1)	(20)

    1.1.1.1.1.1.1 Size-Change Termination

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

    mark#(s(X))	→	mark#(X)	(21)
    1	>	1
    mark#(cons1(X1,X2))	→	mark#(X1)	(23)
    1	>	1
    mark#(cons1(X1,X2))	→	mark#(X2)	(22)
    1	>	1
    mark#(cons(X1,X2))	→	mark#(X1)	(20)
    1	>	1

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