Certification Problem

Input (TPDB TRS_Standard/Applicative_first_order_05/21)

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 4 components.

• The 1^st component contains the pair

  app#(map1(x0),y1)	→	map2#(x0,y1)	(60)
  map2#(f,cons2(x,xs))	→	app#(f,x)	(50)
  app#(filter1(x0),y1)	→	filter3#(x0,y1)	(61)
  filter3#(f,cons2(x,xs))	→	filter24#(app(f,x),f,x,xs)	(52)
  filter24#(true,f,x,xs)	→	filter3#(f,xs)	(54)
  filter3#(f,cons2(x,xs))	→	app#(f,x)	(53)
  app#(filter23(x0,x1,x2),y1)	→	filter24#(x0,x1,x2,y1)	(62)
  filter24#(false,f,x,xs)	→	filter3#(f,xs)	(55)
  map2#(f,cons2(x,xs))	→	map2#(f,xs)	(51)

  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.

  map2#(f,cons2(x,xs))	→	app#(f,x)	(50)
  1	≥	1
  2	>	2
  map2#(f,cons2(x,xs))	→	map2#(f,xs)	(51)
  1	≥	1
  2	>	2
  filter3#(f,cons2(x,xs))	→	app#(f,x)	(53)
  1	≥	1
  2	>	2
  app#(map1(x0),y1)	→	map2#(x0,y1)	(60)
  1	>	1
  2	≥	2
  filter3#(f,cons2(x,xs))	→	filter24#(app(f,x),f,x,xs)	(52)
  1	≥	2
  2	>	3
  2	>	4
  app#(filter1(x0),y1)	→	filter3#(x0,y1)	(61)
  1	>	1
  2	≥	2
  app#(filter23(x0,x1,x2),y1)	→	filter24#(x0,x1,x2,y1)	(62)
  1	>	1
  1	>	2
  1	>	3
  2	≥	4
  filter24#(true,f,x,xs)	→	filter3#(f,xs)	(54)
  2	≥	1
  4	≥	2
  filter24#(false,f,x,xs)	→	filter3#(f,xs)	(55)
  2	≥	1
  4	≥	2

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

• The 2^nd component contains the pair

  fact1#(s1(x))

  →

  fact1#(p1(s1(x)))

  (45)

  1.1.1.1.2 Usable Rules Processor

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

  p1(s1(x))

  →

  x

  (31)

  1.1.1.1.2.1 Innermost Lhss Removal Processor

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

  p1(s1(x0))

  1.1.1.1.2.1.1 Monotonic Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [p1(x1)]	=	1 · x1
  [s1(x1)]	=	2 + 1 · x1
  [fact1#(x1)]	=	2 · x1

  the rules

  p1(s1(x))	→	x	(31)
  fact1(0)	→	s1(0)	(32)
  fact1(s1(x))	→	*2(s1(x),fact1(p1(s1(x))))	(33)
  *2(0,y)	→	0	(34)
  *2(s1(x),y)	→	+2(*2(x,y),y)	(35)
  +2(x,0)	→	x	(36)
  +2(x,s1(y))	→	s1(+2(x,y))	(37)
  map2(f,nil)	→	nil	(38)
  map2(f,cons2(x,xs))	→	cons2(app(f,x),map2(f,xs))	(39)
  filter3(f,nil)	→	nil	(40)
  filter3(f,cons2(x,xs))	→	filter24(app(f,x),f,x,xs)	(41)
  filter24(true,f,x,xs)	→	cons2(x,filter3(f,xs))	(42)
  filter24(false,f,x,xs)	→	filter3(f,xs)	(43)
  app(p,y1)	→	p1(y1)	(14)
  app(s,y1)	→	s1(y1)	(15)
  app(fact,y1)	→	fact1(y1)	(16)
  app(*,y1)	→	*1(y1)	(17)
  app(*1(x0),y1)	→	*2(x0,y1)	(18)
  app(+,y1)	→	+1(y1)	(19)
  app(+1(x0),y1)	→	+2(x0,y1)	(20)
  app(map,y1)	→	map1(y1)	(21)
  app(map1(x0),y1)	→	map2(x0,y1)	(22)
  app(cons,y1)	→	cons1(y1)	(23)
  app(cons1(x0),y1)	→	cons2(x0,y1)	(24)
  app(filter,y1)	→	filter1(y1)	(25)
  app(filter1(x0),y1)	→	filter3(x0,y1)	(26)
  app(filter2,y1)	→	filter21(y1)	(27)
  app(filter21(x0),y1)	→	filter22(x0,y1)	(28)
  app(filter22(x0,x1),y1)	→	filter23(x0,x1,y1)	(29)
  app(filter23(x0,x1,x2),y1)	→	filter24(x0,x1,x2,y1)	(30)

  could be deleted.

  1.1.1.1.2.1.1.1 Switch to TRS Processor

  We merge the DPs and rules into one TRS.

  1.1.1.1.2.1.1.1.1 Bounds

  The given TRS is match-(raise)-bounded by 1. This is shown by the following automaton.

  • final states:

    {0, 1, 2}

  • transitions:

    fact1#0(0)	→	0
    fact1#0(1)	→	0
    fact1#0(2)	→	0
    s10(0)	→	1
    s10(1)	→	1
    s10(2)	→	1
    p10(0)	→	2
    p10(1)	→	2
    p10(2)	→	2
    s11(0)	→	4
    p11(4)	→	3
    fact1#1(3)	→	0
    s11(1)	→	4
    s11(2)	→	4

• The 3^rd component contains the pair

  *2#(s1(x),y)

  →

  *2#(x,y)

  (48)

  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.

  *2#(s1(x),y)	→	*2#(x,y)	(48)
  1	>	1
  2	≥	2

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

• The 4^th component contains the pair

  +2#(x,s1(y))

  →

  +2#(x,y)

  (49)

  1.1.1.1.4 Usable Rules Processor

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

  There are no rules.

  1.1.1.1.4.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  1.1.1.1.4.1.1 Size-Change Termination

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

  +2#(x,s1(y))	→	+2#(x,y)	(49)
  1	≥	1
  2	>	2

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