Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex6_GM04_C)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Constant to Unary

1.1 String Reversal

1.1.1 Dependency Pair Transformation

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 4 components.

• The 1^st component contains the pair

  c'#(proper(x))	→	c'#(ok(x))	(27)
  c'#(active(x))	→	c'#(g(f(mark(x))))	(21)

  1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [mark(x1)]	=	1 · x1
  [top(x1)]	=	1 · x1
  [proper(x1)]	=	1 · x1
  [f(x1)]	=	1 · x1
  [g(x1)]	=	1 · x1
  [active(x1)]	=	1 · x1
  [ok(x1)]	=	1 · x1
  [c'#(x1)]	=	1 · x1

  together with the usable rules

  mark(top(X))	→	proper(top(X))	(19)
  f(proper(X))	→	proper(f(X))	(15)
  g(f(active(X)))	→	g(mark(X))	(13)
  g(proper(X))	→	proper(g(X))	(16)
  ok(f(X))	→	f(ok(X))	(17)
  ok(g(X))	→	g(ok(X))	(18)
  ok(top(X))	→	active(top(X))	(20)

  (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

  1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [c'#(x1)]	=	1 · x1
  [proper(x1)]	=	1 · x1
  [ok(x1)]	=	1 · x1
  [active(x1)]	=	1
  [g(x1)]	=	0
  [f(x1)]	=	0
  [mark(x1)]	=	0
  [top(x1)]	=	1 + 1 · x1

  together with the usable rules

  ok(f(X))	→	f(ok(X))	(17)
  ok(g(X))	→	g(ok(X))	(18)
  ok(top(X))	→	active(top(X))	(20)
  f(proper(X))	→	proper(f(X))	(15)
  g(f(active(X)))	→	g(mark(X))	(13)
  g(proper(X))	→	proper(g(X))	(16)

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

  c'#(active(x))

  →

  c'#(g(f(mark(x))))

  (21)

  could be deleted.

  1.1.1.1.1.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [c'#(x1)]	=	1 · x1
  [proper(x1)]	=	1 + 1 · x1
  [ok(x1)]	=	1 · x1
  [f(x1)]	=	1 + 1 · x1
  [g(x1)]	=	1 + 1 · x1
  [top(x1)]	=	1 + 1 · x1
  [active(x1)]	=	1 · x1
  [mark(x1)]	=	1 + 1 · x1

  the pair

  c'#(proper(x))

  →

  c'#(ok(x))

  (27)

  could be deleted.

  1.1.1.1.1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

  ok#(g(X))	→	ok#(X)	(34)
  ok#(f(X))	→	ok#(X)	(32)

  1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [g(x1)]	=	1 · x1
  [f(x1)]	=	1 · x1
  [ok#(x1)]	=	1 · x1

  having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

  1.1.1.1.2.1 Size-Change Termination

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

  ok#(g(X))	→	ok#(X)	(34)
  1	>	1
  ok#(f(X))	→	ok#(X)	(32)
  1	>	1

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

• The 3^rd component contains the pair

  g#(proper(X))	→	g#(X)	(30)
  g#(f(active(X)))	→	g#(mark(X))	(25)

  1.1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [mark(x1)]	=	1 · x1
  [top(x1)]	=	1 · x1
  [proper(x1)]	=	1 · x1
  [f(x1)]	=	1 · x1
  [active(x1)]	=	1 · x1
  [g#(x1)]	=	1 · x1

  together with the usable rule

  mark(top(X))

  →

  proper(top(X))

  (19)

  (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

  1.1.1.1.3.1 Switch to Innermost Termination

  The TRS does not have overlaps with the pairs and is locally confluent:

  20

  Hence, it suffices to show innermost termination in the following.

  1.1.1.1.3.1.1 Monotonic Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [mark(x1)]	=	2 · x1
  [top(x1)]	=	2 + 2 · x1
  [proper(x1)]	=	1 + 1 · x1
  [g#(x1)]	=	3 · x1
  [f(x1)]	=	3 + 1 · x1
  [active(x1)]	=	2 + 2 · x1

  the pairs

  g#(proper(X))	→	g#(X)	(30)
  g#(f(active(X)))	→	g#(mark(X))	(25)

  and the rule

  mark(top(X))

  →

  proper(top(X))

  (19)

  could be deleted.

  1.1.1.1.3.1.1.1 P is empty

  There are no pairs anymore.

• The 4^th component contains the pair

  f#(proper(X))

  →

  f#(X)

  (29)

  1.1.1.1.4 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [proper(x1)]	=	1 · x1
  [f#(x1)]	=	1 · x1

  having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

  1.1.1.1.4.1 Size-Change Termination

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

  f#(proper(X))	→	f#(X)	(29)
  1	>	1

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