Certification Problem

Input (TPDB TRS_Standard/Secret_05_TRS/aprove3)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Switch to Innermost Termination

The TRS is overlay and locally confluent:

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

1.1 Dependency Pair Transformation

1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

• The 1^st component contains the pair

  function#(if,false,x,y)	→	function#(plus,function(third,x,y,y),function(p,x,x,y),s(y))	(13)
  function#(plus,dummy,x,y)	→	function#(if,function(iszero,x,x,x),x,y)	(11)

  1.1.1.1 Usable Rules Processor

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

  function(iszero,0,dummy,dummy2)	→	true	(1)
  function(iszero,s(x),dummy,dummy2)	→	false	(2)
  function(third,x,y,z)	→	z	(9)
  function(p,0,dummy,dummy2)	→	0	(3)
  function(p,s(0),dummy,dummy2)	→	0	(4)
  function(p,s(s(x)),dummy,dummy2)	→	s(function(p,s(x),x,x))	(5)

  1.1.1.1.1 Rewriting Processor

  We rewrite the right hand side of the pair resulting in

  →

  1.1.1.1.1.1 Usable Rules Processor

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

  function(p,0,dummy,dummy2)	→	0	(3)
  function(p,s(0),dummy,dummy2)	→	0	(4)
  function(p,s(s(x)),dummy,dummy2)	→	s(function(p,s(x),x,x))	(5)
  function(iszero,0,dummy,dummy2)	→	true	(1)
  function(iszero,s(x),dummy,dummy2)	→	false	(2)

  1.1.1.1.1.1.1 Instantiation Processor

  We instantiate the pair to the following set of pairs

  function#(plus,z1,y_0,s(z1))

  →

  function#(if,function(iszero,y_0,y_0,y_0),y_0,s(z1))

  (17)

  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

  function#(plus,y0,0,s(y0))	→	function#(if,true,0,s(y0))	(18)
  function#(plus,y0,s(x0),s(y0))	→	function#(if,false,s(x0),s(y0))	(19)

  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

    function#(plus,y0,s(x0),s(y0))	→	function#(if,false,s(x0),s(y0))	(19)
    function#(if,false,x,y)	→	function#(plus,y,function(p,x,x,y),s(y))	(16)

    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.

    function(p,0,dummy,dummy2)	→	0	(3)
    function(p,s(0),dummy,dummy2)	→	0	(4)
    function(p,s(s(x)),dummy,dummy2)	→	s(function(p,s(x),x,x))	(5)

    1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

    We consider all narrowings of the pair below position 3 to get the following set of pairs

    function#(if,false,0,x1)	→	function#(plus,x1,0,s(x1))	(20)
    function#(if,false,s(0),x1)	→	function#(plus,x1,0,s(x1))	(21)
    function#(if,false,s(s(x0)),x2)	→	function#(plus,x2,s(function(p,s(x0),x0,x0)),s(x2))	(22)

    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

      function#(if,false,s(s(x0)),x2)	→	function#(plus,x2,s(function(p,s(x0),x0,x0)),s(x2))	(22)
      function#(plus,y0,s(x0),s(y0))	→	function#(if,false,s(x0),s(y0))	(19)

      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.

      function(p,s(0),dummy,dummy2)	→	0	(4)
      function(p,s(s(x)),dummy,dummy2)	→	s(function(p,s(x),x,x))	(5)

      1.1.1.1.1.1.1.1.1.1.1.1.1.1 Instantiation Processor

      We instantiate the pair to the following set of pairs

      function#(if,false,s(s(x0)),s(z0))

      →

      function#(plus,s(z0),s(function(p,s(x0),x0,x0)),s(s(z0)))

      (23)

      1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Instantiation Processor

      We instantiate the pair to the following set of pairs

      function#(plus,s(z1),s(y_0),s(s(z1)))

      →

      function#(if,false,s(y_0),s(s(z1)))

      (24)

      1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Instantiation Processor

      We instantiate the pair to the following set of pairs

      function#(if,false,s(s(x0)),s(s(z0)))

      →

      function#(plus,s(s(z0)),s(function(p,s(x0),x0,x0)),s(s(s(z0))))

      (25)

      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

      function#(plus,s(x0),s(s(y_0)),s(s(x0)))

      →

      function#(if,false,s(s(y_0)),s(s(x0)))

      (26)

      1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

      Using the linear polynomial interpretation over the naturals

      [function#(x1,...,x4)]	=	-2 + 2 · x3
      [s(x1)]	=	2 + x1
      [function(x1,...,x4)]	=	-2 + 2 · x1 + x2
      [p]	=	0
      [0]	=	0
      [if]	=	2
      [false]	=	2
      [plus]	=	2

      the pair

      function#(if,false,s(s(x0)),s(s(z0)))

      →

      function#(plus,s(s(z0)),s(function(p,s(x0),x0,x0)),s(s(s(z0))))

      (25)

      could be deleted.

      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 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 Size-Change Termination

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

      function#(plus,s(x0),s(s(y_0)),s(s(x0)))	→	function#(if,false,s(s(y_0)),s(s(x0)))	(26)
      3	≥	3
      4	≥	4

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

• The 2^nd component contains the pair

  function#(p,s(s(x)),dummy,dummy2)

  →

  function#(p,s(x),x,x)

  (10)

  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.

  1.1.1.2.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  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.

  function#(p,s(s(x)),dummy,dummy2)	→	function#(p,s(x),x,x)	(10)
  1	≥	1
  2	>	2
  2	>	3
  2	>	4

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