Certification Problem

Input (TPDB TRS_Standard/Hydras/worm_wrong)

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 4 components.

• The 1^st component contains the pair

  c#(b(x))

  →

  c#(a(x))

  (33)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the naturals

  [b(x1)]	=	2 0 1 0 · x1 + 1 0 0 0
  [f(x1)]	=	3 0 2 0 · x1 + 2 0 0 0
  [b1(x1)]	=	1 2 0 2 · x1 + 2 0 1 0
  [0]	=	0 0 0 0
  [.(x1, x2)]	=	1 0 0 1 · x1 + 0 0 0 0 · x2 + 0 0 0 0
  [c#(x1)]	=	2 0 0 0 · x1 + 0 0 0 0
  [a(x1)]	=	2 0 0 2 · x1 + 0 0 1 0

  together with the usable rules

  a(f(x))	→	f(a(x))	(2)
  a(.(x,y))	→	.(a(x),y)	(3)
  a(b1(x))	→	b1(a(x))	(4)
  a(f(.(0,x)))	→	b1(.(f(.(0,x)),.(0,f(x))))	(8)
  a(f(0))	→	b1(.(f(0),0))	(9)
  a(b(x))	→	b(a(x))	(13)
  f(b(x))	→	b(f(x))	(5)
  f(.(0,x))	→	b(.(0,f(x)))	(10)
  f(0)	→	b(0)	(11)
  .(.(x,y),z)	→	.(x,.(y,z))	(1)
  .(b(x),y)	→	b(.(x,y))	(6)
  b1(b(x))	→	b(b(x))	(7)

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

  c#(b(x))

  →

  c#(a(x))

  (33)

  could be deleted.

  1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

  a#(b(x))	→	a#(x)	(34)
  a#(b1(x))	→	a#(x)	(20)
  a#(.(x,y))	→	a#(x)	(18)
  a#(f(x))	→	a#(x)	(16)

  1.1.2 Size-Change Termination

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

  a#(b(x))	→	a#(x)	(34)
  1	>	1
  a#(b1(x))	→	a#(x)	(20)
  1	>	1
  a#(.(x,y))	→	a#(x)	(18)
  1	>	1
  a#(f(x))	→	a#(x)	(16)
  1	>	1

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

• The 3^rd component contains the pair

  f#(.(0,x))	→	f#(x)	(30)
  f#(b(x))	→	f#(x)	(22)

  1.1.3 Size-Change Termination

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

  f#(.(0,x))	→	f#(x)	(30)
  1	>	1
  f#(b(x))	→	f#(x)	(22)
  1	>	1

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

• The 4^th component contains the pair

  .#(b(x),y)	→	.#(x,y)	(23)
  .#(.(x,y),z)	→	.#(x,.(y,z))	(15)
  .#(.(x,y),z)	→	.#(y,z)	(14)

  1.1.4 Size-Change Termination

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

  .#(b(x),y)	→	.#(x,y)	(23)
  2	≥	2
  1	>	1
  .#(.(x,y),z)	→	.#(x,.(y,z))	(15)
  1	>	1
  .#(.(x,y),z)	→	.#(y,z)	(14)
  2	≥	2
  1	>	1

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