Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z091)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by matchbox @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

• The 1^st component contains the pair

  r0#(1(x1))	→	r0#(x1)	(16)
  r0#(0(x1))	→	r0#(x1)	(18)

  1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

  Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1

  [1(x1)]	=	x1 + 1
  [0(x1)]	=	x1 + 1
  [r0#(x1)]	=	x1 + 0

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

  r0#(1(x1))	→	r0#(x1)	(16)
  r0#(0(x1))	→	r0#(x1)	(18)

  and no rules could be deleted.

  1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 2^nd component contains the pair

  r1#(1(x1))	→	r1#(x1)	(23)
  r1#(0(x1))	→	r1#(x1)	(25)

  1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules

  Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1

  [1(x1)]	=	x1 + 1
  [0(x1)]	=	x1 + 1
  [r1#(x1)]	=	x1 + 0

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

  r1#(1(x1))	→	r1#(x1)	(23)
  r1#(0(x1))	→	r1#(x1)	(25)

  and no rules could be deleted.

  1.1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 3^rd component contains the pair

  b#(ql(1(x1)))	→	b#(r1(x1))	(37)
  b#(ql(0(x1)))	→	b#(r0(x1))	(40)

  1.1.1.3 Reduction Pair Processor with Usable Rules

  Using the matrix interpretations of dimension 1 with strict dimension 1 over the naturals

  [r0(x1)]	=	1 · x1 + 0
  [r1(x1)]	=	1 · x1 + 0
  [ql(x1)]	=	1 · x1 + 0
  [qr(x1)]	=	0 · x1 + 0
  [1(x1)]	=	2 · x1 + 0
  [0(x1)]	=	2 · x1 + 2
  [m(x1)]	=	0 · x1 + 0
  [b(x1)]	=	1 · x1 + 0
  [b#(x1)]	=	2 · x1 + 0

  together with the usable rules

  r0(0(x1))	→	0(r0(x1))	(1)
  r0(1(x1))	→	1(r0(x1))	(2)
  r0(m(x1))	→	m(r0(x1))	(3)
  r1(0(x1))	→	0(r1(x1))	(4)
  r1(1(x1))	→	1(r1(x1))	(5)
  r1(m(x1))	→	m(r1(x1))	(6)
  r0(b(x1))	→	qr(0(b(x1)))	(7)
  r1(b(x1))	→	qr(1(b(x1)))	(8)
  0(qr(x1))	→	qr(0(x1))	(9)
  1(qr(x1))	→	qr(1(x1))	(10)
  m(qr(x1))	→	ql(m(x1))	(11)
  0(ql(x1))	→	ql(0(x1))	(12)
  1(ql(x1))	→	ql(1(x1))	(13)
  b(ql(0(x1)))	→	0(b(r0(x1)))	(14)
  b(ql(1(x1)))	→	1(b(r1(x1)))	(15)

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

  b#(ql(0(x1)))

  →

  b#(r0(x1))

  (40)

  could be deleted.

  1.1.1.3.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    b#(ql(1(x1)))

    →

    b#(r1(x1))

    (37)

    1.1.1.3.1.1 Reduction Pair Processor with Usable Rules

    Using the matrix interpretations of dimension 1 with strict dimension 1 over the naturals

    [r0(x1)]	=	1 · x1 + 0
    [r1(x1)]	=	1 · x1 + 0
    [ql(x1)]	=	1 · x1 + 0
    [qr(x1)]	=	0 · x1 + 0
    [1(x1)]	=	1 · x1 + 2
    [0(x1)]	=	1 · x1 + 2
    [m(x1)]	=	0 · x1 + 2
    [b(x1)]	=	2 · x1 + 0
    [b#(x1)]	=	2 · x1 + 0

    together with the usable rules

    r0(0(x1))	→	0(r0(x1))	(1)
    r0(1(x1))	→	1(r0(x1))	(2)
    r0(m(x1))	→	m(r0(x1))	(3)
    r1(0(x1))	→	0(r1(x1))	(4)
    r1(1(x1))	→	1(r1(x1))	(5)
    r1(m(x1))	→	m(r1(x1))	(6)
    r0(b(x1))	→	qr(0(b(x1)))	(7)
    r1(b(x1))	→	qr(1(b(x1)))	(8)
    0(qr(x1))	→	qr(0(x1))	(9)
    1(qr(x1))	→	qr(1(x1))	(10)
    m(qr(x1))	→	ql(m(x1))	(11)
    0(ql(x1))	→	ql(0(x1))	(12)
    1(ql(x1))	→	ql(1(x1))	(13)
    b(ql(0(x1)))	→	0(b(r0(x1)))	(14)
    b(ql(1(x1)))	→	1(b(r1(x1)))	(15)

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

    b#(ql(1(x1)))

    →

    b#(r1(x1))

    (37)

    could be deleted.

    1.1.1.3.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.