Certification Problem

Input (TPDB SRS_Standard/Secret_05_SRS/aprove5)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by ttt2 @ termCOMP 2023)

1 String Reversal

1.1 Rule Removal

1.1.1 Rule Removal

1.1.1.1 Dependency Pair Transformation

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

• The 1^st component contains the pair

  c#(a(x1))	→	c#(x1)	(24)
  c#(a(x1))	→	b#(a(c(x1)))	(26)
  b#(a(x1))	→	c#(b(x1))	(19)
  c#(b(x1))	→	b#(b(c(x1)))	(23)
  b#(a(x1))	→	b#(x1)	(18)
  c#(b(x1))	→	b#(c(x1))	(22)
  c#(b(x1))	→	c#(x1)	(21)

  1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the

  prec(c#)	=	0		stat(c#)	=	lex
  prec(b#)	=	0		stat(b#)	=	lex
  prec(d)	=	0		stat(d)	=	lex
  prec(c)	=	0		stat(c)	=	lex
  prec(a)	=	1		stat(a)	=	lex
  prec(b)	=	0		stat(b)	=	lex

  π(c#)	=	1
  π(b#)	=	1
  π(d)	=	[]
  π(c)	=	1
  π(a)	=	[1]
  π(b)	=	1

  together with the usable rules

  c(b(x1))	→	b(b(c(x1)))	(11)
  c(a(x1))	→	b(a(c(x1)))	(12)
  c(d(d(a(x1))))	→	d(a(a(a(x1))))	(15)
  b(a(x1))	→	a(c(b(x1)))	(10)
  a(a(x1))	→	d(d(d(a(x1))))	(13)
  a(d(x1))	→	c(d(d(x1)))	(14)

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

  c#(a(x1))	→	c#(x1)	(24)
  b#(a(x1))	→	c#(b(x1))	(19)
  b#(a(x1))	→	b#(x1)	(18)

  could be deleted.

  1.1.1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    c#(b(x1))

    →

    c#(x1)

    (21)

    1.1.1.1.1.1.1.1 Size-Change Termination

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

    c#(b(x1))	→	c#(x1)	(21)
    1	>	1

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

• The 2^nd component contains the pair

  a#(d(x1))	→	c#(d(d(x1)))	(27)
  c#(d(d(a(x1))))	→	a#(a(a(x1)))	(29)
  c#(d(d(a(x1))))	→	a#(a(x1))	(28)

  1.1.1.1.1.2 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

  [a#(x1)]	=	0 1 0 0 0 0 0 0 0 · x1 + 0 0 0 0 0 0 0 0 0
  [a(x1)]	=	1 1 0 0 1 0 0 1 0 · x1 + 0 0 0 0 0 0 1 0 0
  [d(x1)]	=	0 1 0 0 0 1 0 0 0 · x1 + 0 0 0 0 0 0 0 0 0
  [c#(x1)]	=	1 0 0 0 0 0 0 0 0 · x1 + 0 0 0 0 0 0 0 0 0
  [c(x1)]	=	1 0 0 1 0 0 0 0 0 · x1 + 0 0 0 0 0 0 1 0 0

  together with the usable rules

  a(a(x1))	→	d(d(d(a(x1))))	(13)
  a(d(x1))	→	c(d(d(x1)))	(14)
  c(d(d(a(x1))))	→	d(a(a(a(x1))))	(15)

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

  c#(d(d(a(x1))))	→	a#(a(a(x1)))	(29)
  c#(d(d(a(x1))))	→	a#(a(x1))	(28)

  could be deleted.

  1.1.1.1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 3^rd component contains the pair

  f#(f(e(e(x1))))	→	f#(f(f(x1)))	(32)
  f#(f(e(e(x1))))	→	f#(f(x1))	(31)
  f#(f(e(e(x1))))	→	f#(x1)	(30)

  1.1.1.1.1.3 Subterm Criterion Processor

  We use the projection to multisets

  π(f#)	=	{ 1 }
  π(e)	=	{ 1, 1 }
  π(f)	=	{ 1 }

  to remove the pairs:

  f#(f(e(e(x1))))	→	f#(f(f(x1)))	(32)
  f#(f(e(e(x1))))	→	f#(f(x1))	(31)
  f#(f(e(e(x1))))	→	f#(x1)	(30)

  1.1.1.1.1.3.1 P is empty

  There are no pairs anymore.