Certification Problem

Input (TPDB TRS_Standard/AotoYamada_05/Ex5TermProof)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Uncurrying

We uncurry the binary symbol app in combination with the following symbol map which also determines the applicative arities of these symbols.

1.1 Switch to Innermost Termination

The TRS is overlay and locally confluent:

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

1.1.1 Dependency Pair Transformation

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

• The 1^st component contains the pair

  app#(rec2(x0,x1),y1)	→	rec3#(x0,x1,y1)	(31)
  rec3#(f,x,s1(y))	→	app#(app(f,s1(y)),rec3(f,x,y))	(26)
  rec3#(f,x,s1(y))	→	app#(f,s1(y))	(27)
  rec3#(f,x,s1(y))	→	rec3#(f,x,y)	(28)

  1.1.1.1.1 Usable Rules Processor

  We restrict the rewrite rules to the following usable rules of the DP problem.

  app(add,y1)	→	add1(y1)	(8)
  app(add1(x0),y1)	→	add2(x0,y1)	(9)
  app(s,y1)	→	s1(y1)	(10)
  app(mult,y1)	→	mult1(y1)	(11)
  app(mult1(x0),y1)	→	mult2(x0,y1)	(12)
  app(rec,y1)	→	rec1(y1)	(13)
  app(rec1(x0),y1)	→	rec2(x0,y1)	(14)
  app(rec2(x0,x1),y1)	→	rec3(x0,x1,y1)	(15)
  rec3(f,x,s1(y))	→	app(app(f,s1(y)),rec3(f,x,y))	(21)
  rec3(f,x,0)	→	x	(20)
  mult2(0,y)	→	0	(18)
  mult2(s1(x),y)	→	add2(mult2(x,y),y)	(19)
  add2(0,y)	→	y	(16)
  add2(s1(x),y)	→	s1(add2(x,y))	(17)

  1.1.1.1.1.1 Innermost Lhss Removal Processor

  We restrict the innermost strategy to the following left hand sides.

  add2(0,x0)

  add2(s1(x0),x1)

  mult2(0,x0)

  mult2(s1(x0),x1)

  rec3(x0,x1,0)

  rec3(x0,x1,s1(x2))

  app(add,x0)

  app(add1(x0),x1)

  app(s,x0)

  app(mult,x0)

  app(mult1(x0),x1)

  app(rec,x0)

  app(rec1(x0),x1)

  app(rec2(x0,x1),x2)

  1.1.1.1.1.1.1 Reduction Pair Processor

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

  [app#(x1, x2)]	=	-∞ + 0 · x1 + -∞ · x2
  [rec2(x1, x2)]	=	0 + 1 · x1 + 0 · x2
  [rec3#(x1, x2, x3)]	=	0 + 1 · x1 + 0 · x2 + -∞ · x3
  [s1(x1)]	=	0 + -∞ · x1
  [app(x1, x2)]	=	0 + 1 · x1 + 0 · x2
  [rec3(x1, x2, x3)]	=	1 + 2 · x1 + 0 · x2 + -∞ · x3
  [add]	=	4
  [add1(x1)]	=	-∞ + 0 · x1
  [add2(x1, x2)]	=	0 + -∞ · x1 + 0 · x2
  [s]	=	0
  [mult]	=	0
  [mult1(x1)]	=	0 + -∞ · x1
  [mult2(x1, x2)]	=	0 + -∞ · x1 + 0 · x2
  [rec]	=	4
  [rec1(x1)]	=	0 + 0 · x1
  [0]	=	0

  the pair

  rec3#(f,x,s1(y))

  →

  app#(f,s1(y))

  (27)

  could be deleted.

  1.1.1.1.1.1.1.1 Narrowing Processor

  We consider all narrowings of the pair below position 1 to get the following set of pairs

  rec3#(add,y1,s1(y2))	→	app#(add1(s1(y2)),rec3(add,y1,y2))	(32)
  rec3#(add1(x0),y1,s1(y2))	→	app#(add2(x0,s1(y2)),rec3(add1(x0),y1,y2))	(33)
  rec3#(s,y1,s1(y2))	→	app#(s1(s1(y2)),rec3(s,y1,y2))	(34)
  rec3#(mult,y1,s1(y2))	→	app#(mult1(s1(y2)),rec3(mult,y1,y2))	(35)
  rec3#(mult1(x0),y1,s1(y2))	→	app#(mult2(x0,s1(y2)),rec3(mult1(x0),y1,y2))	(36)
  rec3#(rec,y1,s1(y2))	→	app#(rec1(s1(y2)),rec3(rec,y1,y2))	(37)
  rec3#(rec1(x0),y1,s1(y2))	→	app#(rec2(x0,s1(y2)),rec3(rec1(x0),y1,y2))	(38)
  rec3#(rec2(x0,x1),y1,s1(y2))	→	app#(rec3(x0,x1,s1(y2)),rec3(rec2(x0,x1),y1,y2))	(39)

  1.1.1.1.1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    rec3#(f,x,s1(y))	→	rec3#(f,x,y)	(28)
    rec3#(add1(x0),y1,s1(y2))	→	app#(add2(x0,s1(y2)),rec3(add1(x0),y1,y2))	(33)
    app#(rec2(x0,x1),y1)	→	rec3#(x0,x1,y1)	(31)
    rec3#(mult1(x0),y1,s1(y2))	→	app#(mult2(x0,s1(y2)),rec3(mult1(x0),y1,y2))	(36)
    rec3#(rec1(x0),y1,s1(y2))	→	app#(rec2(x0,s1(y2)),rec3(rec1(x0),y1,y2))	(38)
    rec3#(rec2(x0,x1),y1,s1(y2))	→	app#(rec3(x0,x1,s1(y2)),rec3(rec2(x0,x1),y1,y2))	(39)

    1.1.1.1.1.1.1.1.1.1 Rewriting Processor

    We rewrite the right hand side of the pair resulting in

    →

    1.1.1.1.1.1.1.1.1.1.1 Forward Instantiation Processor

    We instantiate the pair to the following set of pairs

    rec3#(x0,x1,s1(s1(y_2)))	→	rec3#(x0,x1,s1(y_2))	(41)
    rec3#(add1(y_0),x1,s1(s1(y_2)))	→	rec3#(add1(y_0),x1,s1(y_2))	(42)
    rec3#(mult1(y_0),x1,s1(s1(y_2)))	→	rec3#(mult1(y_0),x1,s1(y_2))	(43)
    rec3#(rec1(y_0),x1,s1(s1(y_2)))	→	rec3#(rec1(y_0),x1,s1(y_2))	(44)
    rec3#(rec2(y_0,y_1),x1,s1(s1(y_3)))	→	rec3#(rec2(y_0,y_1),x1,s1(y_3))	(45)

    1.1.1.1.1.1.1.1.1.1.1.1 Forward Instantiation Processor

    We instantiate the pair to the following set of pairs

    app#(rec2(add1(y_0),x1),s1(y_2))	→	rec3#(add1(y_0),x1,s1(y_2))	(46)
    app#(rec2(mult1(y_0),x1),s1(y_2))	→	rec3#(mult1(y_0),x1,s1(y_2))	(47)
    app#(rec2(rec1(y_0),x1),s1(y_2))	→	rec3#(rec1(y_0),x1,s1(y_2))	(48)
    app#(rec2(rec2(y_0,y_1),x1),s1(y_3))	→	rec3#(rec2(y_0,y_1),x1,s1(y_3))	(49)
    app#(rec2(x0,x1),s1(s1(y_2)))	→	rec3#(x0,x1,s1(s1(y_2)))	(50)
    app#(rec2(add1(y_0),x1),s1(s1(y_2)))	→	rec3#(add1(y_0),x1,s1(s1(y_2)))	(51)
    app#(rec2(mult1(y_0),x1),s1(s1(y_2)))	→	rec3#(mult1(y_0),x1,s1(s1(y_2)))	(52)
    app#(rec2(rec1(y_0),x1),s1(s1(y_2)))	→	rec3#(rec1(y_0),x1,s1(s1(y_2)))	(53)
    app#(rec2(rec2(y_0,y_1),x1),s1(s1(y_3)))	→	rec3#(rec2(y_0,y_1),x1,s1(s1(y_3)))	(54)

    1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

    We consider all narrowings of the pair below position 2 to get the following set of pairs

    rec3#(add1(y0),x1,s1(s1(x2)))	→	app#(add2(y0,s1(s1(x2))),app(app(add1(y0),s1(x2)),rec3(add1(y0),x1,x2)))	(55)
    rec3#(add1(y0),x1,s1(0))	→	app#(add2(y0,s1(0)),x1)	(56)

    1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rewriting Processor

    We rewrite the right hand side of the pair resulting in

    →

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

    We consider all narrowings of the pair below position 2 to get the following set of pairs

    rec3#(mult1(y0),x1,s1(s1(x2)))	→	app#(mult2(y0,s1(s1(x2))),app(app(mult1(y0),s1(x2)),rec3(mult1(y0),x1,x2)))	(58)
    rec3#(mult1(y0),x1,s1(0))	→	app#(mult2(y0,s1(0)),x1)	(59)

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rewriting Processor

    We rewrite the right hand side of the pair resulting in

    →

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

    We consider all narrowings of the pair below position 2 to get the following set of pairs

    rec3#(rec1(y0),x1,s1(s1(x2)))	→	app#(rec2(y0,s1(s1(x2))),app(app(rec1(y0),s1(x2)),rec3(rec1(y0),x1,x2)))	(61)
    rec3#(rec1(y0),x1,s1(0))	→	app#(rec2(y0,s1(0)),x1)	(62)

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rewriting Processor

    We rewrite the right hand side of the pair resulting in

    →

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rewriting Processor

    We rewrite the right hand side of the pair resulting in

    →

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

    We consider all narrowings of the pair below position 2 to get the following set of pairs

    rec3#(rec2(y0,y1),x1,s1(s1(x2)))	→	app#(app(app(y0,s1(s1(x2))),rec3(y0,y1,s1(x2))),app(app(rec2(y0,y1),s1(x2)),rec3(rec2(y0,y1),x1,x2)))	(65)
    rec3#(rec2(y0,y1),x1,s1(0))	→	app#(app(app(y0,s1(0)),rec3(y0,y1,0)),x1)	(66)

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rewriting Processor

    We rewrite the right hand side of the pair resulting in

    →

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rewriting Processor

    We rewrite the right hand side of the pair resulting in

    →

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rewriting Processor

    We rewrite the right hand side of the pair resulting in

    →

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rewriting Processor

    We rewrite the right hand side of the pair resulting in

    →

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Forward Instantiation Processor

    We instantiate the pair to the following set of pairs

    rec3#(rec1(add1(y_0)),s1(y_2),s1(0))	→	app#(rec2(add1(y_0),s1(0)),s1(y_2))	(71)
    rec3#(rec1(mult1(y_0)),s1(y_2),s1(0))	→	app#(rec2(mult1(y_0),s1(0)),s1(y_2))	(72)
    rec3#(rec1(rec1(y_0)),s1(y_2),s1(0))	→	app#(rec2(rec1(y_0),s1(0)),s1(y_2))	(73)
    rec3#(rec1(rec2(y_0,y_1)),s1(y_3),s1(0))	→	app#(rec2(rec2(y_0,y_1),s1(0)),s1(y_3))	(74)
    rec3#(rec1(x0),s1(s1(y_2)),s1(0))	→	app#(rec2(x0,s1(0)),s1(s1(y_2)))	(75)
    rec3#(rec1(add1(y_0)),s1(s1(y_2)),s1(0))	→	app#(rec2(add1(y_0),s1(0)),s1(s1(y_2)))	(76)
    rec3#(rec1(mult1(y_0)),s1(s1(y_2)),s1(0))	→	app#(rec2(mult1(y_0),s1(0)),s1(s1(y_2)))	(77)
    rec3#(rec1(rec1(y_0)),s1(s1(y_2)),s1(0))	→	app#(rec2(rec1(y_0),s1(0)),s1(s1(y_2)))	(78)
    rec3#(rec1(rec2(y_0,y_1)),s1(s1(y_3)),s1(0))	→	app#(rec2(rec2(y_0,y_1),s1(0)),s1(s1(y_3)))	(79)

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

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

    [rec3#(x1, x2, x3)]	=	-∞ + 0 · x1 + -∞ · x2 + 0 · x3
    [s1(x1)]	=	0 + -∞ · x1
    [add1(x1)]	=	-∞ + 0 · x1
    [mult1(x1)]	=	1 + 0 · x1
    [rec1(x1)]	=	-∞ + 0 · x1
    [rec2(x1, x2)]	=	-∞ + 0 · x1 + 0 · x2
    [app#(x1, x2)]	=	0 + 0 · x1 + -∞ · x2
    [0]	=	0
    [add2(x1, x2)]	=	-∞ + 0 · x1 + 0 · x2
    [app(x1, x2)]	=	0 + 0 · x1 + 0 · x2
    [rec3(x1, x2, x3)]	=	0 + 0 · x1 + 0 · x2 + -∞ · x3
    [mult2(x1, x2)]	=	0 + -∞ · x1 + 0 · x2
    [add]	=	0
    [s]	=	2
    [mult]	=	3
    [rec]	=	0

    the pairs

    rec3#(mult1(y0),x1,s1(0))	→	app#(mult2(y0,s1(0)),x1)	(59)
    rec3#(mult1(y0),x1,s1(s1(x2)))	→	app#(mult2(y0,s1(s1(x2))),app(mult2(y0,s1(x2)),rec3(mult1(y0),x1,x2)))	(60)

    could be deleted.

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

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

    [rec3#(x1, x2, x3)]	=	-∞ + 0 · x1 + 0 · x2 + 0 · x3
    [s1(x1)]	=	0 + -∞ · x1
    [add1(x1)]	=	-∞ + 0 · x1
    [mult1(x1)]	=	0 + -∞ · x1
    [rec1(x1)]	=	1 + 0 · x1
    [rec2(x1, x2)]	=	0 + 0 · x1 + 0 · x2
    [app#(x1, x2)]	=	-∞ + 0 · x1 + 0 · x2
    [0]	=	0
    [add2(x1, x2)]	=	-∞ + 0 · x1 + 0 · x2
    [app(x1, x2)]	=	-∞ + 0 · x1 + 0 · x2
    [rec3(x1, x2, x3)]	=	0 + 0 · x1 + 0 · x2 + -∞ · x3
    [add]	=	0
    [s]	=	5
    [mult]	=	0
    [mult2(x1, x2)]	=	0 + -∞ · x1 + 0 · x2
    [rec]	=	5

    the pairs

    rec3#(rec1(mult1(y_0)),s1(y_2),s1(0))	→	app#(rec2(mult1(y_0),s1(0)),s1(y_2))	(72)
    rec3#(rec1(mult1(y_0)),s1(s1(y_2)),s1(0))	→	app#(rec2(mult1(y_0),s1(0)),s1(s1(y_2)))	(77)

    could be deleted.

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

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

    [rec3#(x1, x2, x3)]	=	-∞ + 0 · x1 + 0 · x2 + -∞ · x3
    [s1(x1)]	=	0 + -∞ · x1
    [add1(x1)]	=	0 + -∞ · x1
    [mult1(x1)]	=	-∞ + 0 · x1
    [rec1(x1)]	=	3 + 0 · x1
    [rec2(x1, x2)]	=	2 + 0 · x1 + 0 · x2
    [app#(x1, x2)]	=	-∞ + 0 · x1 + -∞ · x2
    [0]	=	0
    [add2(x1, x2)]	=	0 + -∞ · x1 + 0 · x2
    [app(x1, x2)]	=	0 + 0 · x1 + 0 · x2
    [rec3(x1, x2, x3)]	=	2 + 0 · x1 + 0 · x2 + -∞ · x3
    [add]	=	0
    [s]	=	3
    [mult]	=	0
    [mult2(x1, x2)]	=	0 + -∞ · x1 + 0 · x2
    [rec]	=	4

    the pairs

    rec3#(rec1(add1(y_0)),s1(y_2),s1(0))	→	app#(rec2(add1(y_0),s1(0)),s1(y_2))	(71)
    rec3#(rec1(add1(y_0)),s1(s1(y_2)),s1(0))	→	app#(rec2(add1(y_0),s1(0)),s1(s1(y_2)))	(76)

    could be deleted.

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

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

    [rec3#(x1, x2, x3)]	=	-∞ + 0 · x1 + 0 · x2 + -∞ · x3
    [s1(x1)]	=	0 + -∞ · x1
    [add1(x1)]	=	3 + 0 · x1
    [mult1(x1)]	=	-∞ + 0 · x1
    [rec1(x1)]	=	2 + 0 · x1
    [rec2(x1, x2)]	=	2 + 0 · x1 + 0 · x2
    [app#(x1, x2)]	=	-∞ + 0 · x1 + -∞ · x2
    [0]	=	0
    [add2(x1, x2)]	=	2 + -∞ · x1 + 0 · x2
    [app(x1, x2)]	=	2 + 0 · x1 + 0 · x2
    [rec3(x1, x2, x3)]	=	2 + 0 · x1 + 0 · x2 + -∞ · x3
    [add]	=	3
    [s]	=	0
    [mult]	=	3
    [mult2(x1, x2)]	=	2 + 0 · x1 + 0 · x2
    [rec]	=	0

    the pairs

    rec3#(add1(y0),x1,s1(0))	→	app#(add2(y0,s1(0)),x1)	(56)
    rec3#(add1(y0),x1,s1(s1(x2)))	→	app#(add2(y0,s1(s1(x2))),app(add2(y0,s1(x2)),rec3(add1(y0),x1,x2)))	(57)

    could be deleted.

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

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

    [rec3#(x1, x2, x3)]	=	0 + 0 · x1 + -∞ · x2 + 0 · x3
    [s1(x1)]	=	0 + -∞ · x1
    [add1(x1)]	=	0 + -∞ · x1
    [mult1(x1)]	=	1 + 0 · x1
    [rec1(x1)]	=	1 + 0 · x1
    [rec2(x1, x2)]	=	1 + 0 · x1 + 0 · x2
    [app#(x1, x2)]	=	-∞ + 0 · x1 + -∞ · x2
    [rec3(x1, x2, x3)]	=	1 + 0 · x1 + 0 · x2 + -∞ · x3
    [0]	=	0
    [app(x1, x2)]	=	1 + 0 · x1 + 0 · x2
    [add]	=	0
    [add2(x1, x2)]	=	1 + -∞ · x1 + 0 · x2
    [s]	=	4
    [mult]	=	0
    [mult2(x1, x2)]	=	1 + -∞ · x1 + 0 · x2
    [rec]	=	2

    the pairs

    app#(rec2(add1(y_0),x1),s1(y_2))	→	rec3#(add1(y_0),x1,s1(y_2))	(46)
    app#(rec2(add1(y_0),x1),s1(s1(y_2)))	→	rec3#(add1(y_0),x1,s1(s1(y_2)))	(51)

    could be deleted.

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

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

    [rec3#(x1, x2, x3)]	=	-∞ + 0 · x1 + -∞ · x2 + -∞ · x3
    [s1(x1)]	=	0 + -∞ · x1
    [add1(x1)]	=	-∞ + 0 · x1
    [mult1(x1)]	=	0 + -∞ · x1
    [rec1(x1)]	=	5 + 0 · x1
    [rec2(x1, x2)]	=	5 + 0 · x1 + 0 · x2
    [app#(x1, x2)]	=	0 + 0 · x1 + -∞ · x2
    [rec3(x1, x2, x3)]	=	0 + 0 · x1 + 0 · x2 + -∞ · x3
    [0]	=	0
    [app(x1, x2)]	=	0 + 0 · x1 + 0 · x2
    [add]	=	3
    [add2(x1, x2)]	=	0 + -∞ · x1 + 0 · x2
    [s]	=	0
    [mult]	=	1
    [mult2(x1, x2)]	=	0 + -∞ · x1 + 0 · x2
    [rec]	=	5

    the pairs

    app#(rec2(mult1(y_0),x1),s1(y_2))	→	rec3#(mult1(y_0),x1,s1(y_2))	(47)
    app#(rec2(mult1(y_0),x1),s1(s1(y_2)))	→	rec3#(mult1(y_0),x1,s1(s1(y_2)))	(52)

    could be deleted.

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

    Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

    [rec3#(x1, x2, x3)]	=	3 -∞ -∞ + 0 -∞ 1 -∞ -∞ -∞ -∞ -∞ -∞ · x1 + -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x2 + -∞ 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x3
    [s1(x1)]	=	0 -∞ 0 + -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [add1(x1)]	=	-∞ 1 0 + 0 -∞ 0 -∞ 0 0 -∞ -∞ -∞ · x1
    [mult1(x1)]	=	0 2 0 + -∞ 0 -∞ 0 -∞ 0 -∞ -∞ -∞ · x1
    [rec1(x1)]	=	-∞ -∞ 3 + 0 0 0 0 -∞ -∞ -∞ -∞ 0 · x1
    [rec2(x1, x2)]	=	-∞ 3 0 + 0 0 0 3 3 3 0 -∞ 1 · x1 + -∞ -∞ 0 0 0 1 -∞ -∞ -∞ · x2
    [app#(x1, x2)]	=	3 -∞ -∞ + -∞ -∞ 0 -∞ -∞ -∞ -∞ -∞ -∞ · x1 + -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x2
    [rec3(x1, x2, x3)]	=	3 3 1 + 3 3 3 3 3 3 1 0 2 · x1 + 0 0 1 0 0 1 -∞ -∞ 0 · x2 + -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x3
    [0]	=	0 0 0
    [app(x1, x2)]	=	0 0 0 + 0 0 0 3 0 0 0 -∞ 1 · x1 + 0 0 0 0 0 1 -∞ -∞ 0 · x2
    [add]	=	0 -∞ 0
    [add2(x1, x2)]	=	1 -∞ 0 + 0 -∞ 0 -∞ -∞ -∞ -∞ -∞ 0 · x1 + 0 -∞ -∞ 0 0 0 -∞ -∞ 0 · x2
    [s]	=	0 -∞ -∞
    [mult]	=	0 0 -∞
    [mult2(x1, x2)]	=	1 0 0 + -∞ -∞ -∞ -∞ 0 -∞ -∞ -∞ -∞ · x1 + 0 -∞ 0 0 0 1 -∞ -∞ 0 · x2
    [rec]	=	3 -∞ -∞

    the pairs

    rec3#(rec1(rec2(y_0,y_1)),s1(y_3),s1(0))	→	app#(rec2(rec2(y_0,y_1),s1(0)),s1(y_3))	(74)
    rec3#(rec1(rec2(y_0,y_1)),s1(s1(y_3)),s1(0))	→	app#(rec2(rec2(y_0,y_1),s1(0)),s1(s1(y_3)))	(79)

    could be deleted.

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

    Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

    [rec3#(x1, x2, x3)]	=	-∞ -∞ -∞ + 0 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x2 + -∞ 0 0 -∞ -∞ -∞ -∞ -∞ -∞ · x3
    [s1(x1)]	=	0 0 -∞ + -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [add1(x1)]	=	0 0 1 + 0 -∞ -∞ -∞ -∞ -∞ 0 -∞ -∞ · x1
    [mult1(x1)]	=	0 0 3 + 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ 0 · x1
    [rec1(x1)]	=	-∞ -∞ 3 + 0 -∞ -∞ 0 0 0 0 0 -∞ · x1
    [rec2(x1, x2)]	=	0 2 3 + 0 0 0 0 0 -∞ 1 1 1 · x1 + 0 -∞ -∞ -∞ -∞ -∞ 1 0 0 · x2
    [app#(x1, x2)]	=	0 -∞ -∞ + 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x2
    [rec3(x1, x2, x3)]	=	0 1 1 + 0 0 0 1 1 1 1 1 1 · x1 + 0 -∞ -∞ 1 0 0 1 0 0 · x2 + -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x3
    [0]	=	0 0 -∞
    [app(x1, x2)]	=	0 0 0 + 0 0 -∞ 0 0 0 0 1 0 · x1 + 0 -∞ -∞ 0 0 0 1 0 0 · x2
    [add]	=	0 0 -∞
    [add2(x1, x2)]	=	0 1 -∞ + -∞ -∞ -∞ 0 -∞ -∞ -∞ -∞ -∞ · x1 + 0 -∞ -∞ -∞ 0 0 0 0 0 · x2
    [s]	=	0 0 -∞
    [mult]	=	3 -∞ -∞
    [mult2(x1, x2)]	=	0 3 3 + -∞ -∞ -∞ -∞ -∞ 0 -∞ -∞ 0 · x1 + 0 -∞ -∞ 0 0 0 1 0 0 · x2
    [rec]	=	0 -∞ 3

    the pairs

    app#(rec2(rec2(y_0,y_1),x1),s1(y_3))	→	rec3#(rec2(y_0,y_1),x1,s1(y_3))	(49)
    app#(rec2(rec2(y_0,y_1),x1),s1(s1(y_3)))	→	rec3#(rec2(y_0,y_1),x1,s1(s1(y_3)))	(54)

    could be deleted.

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

    Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

    [rec3#(x1, x2, x3)]	=	0 -∞ -∞ + -∞ 1 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 0 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x2 + -∞ 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x3
    [s1(x1)]	=	0 -∞ 0 + -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1
    [add1(x1)]	=	2 -∞ 0 + 0 0 -∞ 0 0 -∞ -∞ 1 0 · x1
    [mult1(x1)]	=	0 -∞ 0 + 0 -∞ -∞ 0 0 -∞ 0 0 -∞ · x1
    [rec1(x1)]	=	0 3 -∞ + 0 -∞ -∞ 0 0 -∞ -∞ 1 0 · x1
    [rec2(x1, x2)]	=	0 3 -∞ + -∞ 1 -∞ 0 1 0 -∞ -∞ 0 · x1 + 0 0 -∞ 0 0 -∞ 0 0 0 · x2
    [app#(x1, x2)]	=	3 -∞ -∞ + 0 -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x2
    [rec3(x1, x2, x3)]	=	0 0 1 + 0 1 0 0 1 0 1 2 1 · x1 + 0 0 -∞ 0 0 -∞ 1 1 0 · x2 + -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x3
    [0]	=	0 0 0
    [app(x1, x2)]	=	0 0 0 + 0 0 0 -∞ 0 0 0 1 0 · x1 + 0 0 -∞ 0 0 -∞ 0 1 0 · x2
    [add]	=	2 -∞ -∞
    [add2(x1, x2)]	=	0 -∞ 0 + 0 0 -∞ -∞ 0 -∞ -∞ -∞ 0 · x1 + 0 0 -∞ 0 0 -∞ 0 0 0 · x2
    [s]	=	0 -∞ 3
    [mult]	=	0 -∞ 3
    [mult2(x1, x2)]	=	0 0 0 + -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 0 0 -∞ 0 0 -∞ 0 0 0 · x2
    [rec]	=	0 0 3

    the pairs

    rec3#(rec2(y0,y1),x1,s1(0))	→	app#(app(app(y0,s1(0)),y1),x1)	(68)
    rec3#(rec2(y0,y1),x1,s1(s1(x2)))	→	app#(app(app(y0,s1(s1(x2))),app(app(y0,s1(x2)),rec3(y0,y1,x2))),app(app(app(y0,s1(x2)),rec3(y0,y1,x2)),rec3(rec2(y0,y1),x1,x2)))	(70)

    could be deleted.

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions

    prec(add1)	=	2		weight(add1)	=	2
    prec(mult1)	=	1		weight(mult1)	=	1
    prec(rec1)	=	0		weight(rec1)	=	1

    in combination with the following argument filter

    π(rec3#)	=	1
    π(add1)	=	[]
    π(mult1)	=	[]
    π(rec1)	=	[1]
    π(rec2)	=	1
    π(app#)	=	1

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

    rec3#(rec1(y0),x1,s1(s1(x2)))	→	app#(rec2(y0,s1(s1(x2))),rec3(y0,s1(x2),rec3(rec1(y0),x1,x2)))	(64)
    rec3#(rec1(rec1(y_0)),s1(y_2),s1(0))	→	app#(rec2(rec1(y_0),s1(0)),s1(y_2))	(73)
    rec3#(rec1(x0),s1(s1(y_2)),s1(0))	→	app#(rec2(x0,s1(0)),s1(s1(y_2)))	(75)
    rec3#(rec1(rec1(y_0)),s1(s1(y_2)),s1(0))	→	app#(rec2(rec1(y_0),s1(0)),s1(s1(y_2)))	(78)

    could be deleted.

    1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 1 component.

    • The 1^st component contains the pair

      rec3#(x0,x1,s1(s1(y_2)))	→	rec3#(x0,x1,s1(y_2))	(41)
      rec3#(add1(y_0),x1,s1(s1(y_2)))	→	rec3#(add1(y_0),x1,s1(y_2))	(42)
      rec3#(mult1(y_0),x1,s1(s1(y_2)))	→	rec3#(mult1(y_0),x1,s1(y_2))	(43)
      rec3#(rec1(y_0),x1,s1(s1(y_2)))	→	rec3#(rec1(y_0),x1,s1(y_2))	(44)
      rec3#(rec2(y_0,y_1),x1,s1(s1(y_3)))	→	rec3#(rec2(y_0,y_1),x1,s1(y_3))	(45)

      1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Usable Rules Processor

      We restrict the rewrite rules to the following usable rules of the DP problem.

      There are no rules.

      1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Innermost Lhss Removal Processor

      We restrict the innermost strategy to the following left hand sides.

      There are no lhss.

      1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Size-Change Termination

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

      rec3#(x0,x1,s1(s1(y_2)))	→	rec3#(x0,x1,s1(y_2))	(41)
      1	≥	1
      2	≥	2
      3	>	3
      rec3#(add1(y_0),x1,s1(s1(y_2)))	→	rec3#(add1(y_0),x1,s1(y_2))	(42)
      1	≥	1
      2	≥	2
      3	>	3
      rec3#(mult1(y_0),x1,s1(s1(y_2)))	→	rec3#(mult1(y_0),x1,s1(y_2))	(43)
      1	≥	1
      2	≥	2
      3	>	3
      rec3#(rec1(y_0),x1,s1(s1(y_2)))	→	rec3#(rec1(y_0),x1,s1(y_2))	(44)
      1	≥	1
      2	≥	2
      3	>	3
      rec3#(rec2(y_0,y_1),x1,s1(s1(y_3)))	→	rec3#(rec2(y_0,y_1),x1,s1(y_3))	(45)
      1	≥	1
      2	≥	2
      3	>	3

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

• The 2^nd component contains the pair

  mult2#(s1(x),y)

  →

  mult2#(x,y)

  (25)

  1.1.1.1.2 Usable Rules Processor

  We restrict the rewrite rules to the following usable rules of the DP problem.

  There are no rules.

  1.1.1.1.2.1 Innermost Lhss Removal Processor

  We restrict the innermost strategy to the following left hand sides.

  There are no lhss.

  1.1.1.1.2.1.1 Size-Change Termination

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

  mult2#(s1(x),y)	→	mult2#(x,y)	(25)
  1	>	1
  2	≥	2

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

• The 3^rd component contains the pair

  add2#(s1(x),y)

  →

  add2#(x,y)

  (23)

  1.1.1.1.3 Usable Rules Processor

  We restrict the rewrite rules to the following usable rules of the DP problem.

  There are no rules.

  1.1.1.1.3.1 Innermost Lhss Removal Processor

  We restrict the innermost strategy to the following left hand sides.

  There are no lhss.

  1.1.1.1.3.1.1 Size-Change Termination

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

  add2#(s1(x),y)	→	add2#(x,y)	(23)
  1	>	1
  2	≥	2

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