Certification Problem

Input (TPDB TRS_Standard/Beerendonk_07/20)

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 3 components.

• The 1^st component contains the pair

  cond2#(true,x,y)	→	cond2#(gr(y,0),p(x),p(y))	(18)
  cond2#(false,x,y)	→	cond1#(and(eq(x,y),gr(x,0)),x,y)	(22)
  cond1#(true,x,y)	→	cond2#(gr(y,0),x,y)	(16)

  1.1.1.1 Usable Rules Processor

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

  gr(0,x)	→	false	(4)
  gr(s(x),0)	→	true	(5)
  eq(0,0)	→	true	(9)
  eq(s(x),0)	→	false	(10)
  eq(0,s(x))	→	false	(11)
  eq(s(x),s(y))	→	eq(x,y)	(12)
  and(true,true)	→	true	(13)
  and(false,x)	→	false	(14)
  and(x,false)	→	false	(15)
  p(0)	→	0	(7)
  p(s(x))	→	x	(8)

  1.1.1.1.1 Innermost Lhss Removal Processor

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

  gr(0,x0)

  gr(s(x0),0)

  gr(s(x0),s(x1))

  p(0)

  p(s(x0))

  eq(0,0)

  eq(s(x0),0)

  eq(0,s(x0))

  eq(s(x0),s(x1))

  and(true,true)

  and(false,x0)

  and(x0,false)

  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

  cond2#(true,y0,0)	→	cond2#(false,p(y0),p(0))	(28)
  cond2#(true,y0,s(x0))	→	cond2#(true,p(y0),p(s(x0)))	(29)

  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 Rewriting Processor

  We rewrite the right hand side of the pair resulting in

  →

  1.1.1.1.1.1.1.1.1 Narrowing Processor

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

  cond1#(true,y0,0)	→	cond2#(false,y0,0)	(32)
  cond1#(true,y0,s(x0))	→	cond2#(true,y0,s(x0))	(33)

  1.1.1.1.1.1.1.1.1.1 Instantiation Processor

  We instantiate the pair to the following set of pairs

  cond2#(false,y_0,0)

  →

  cond1#(and(eq(y_0,0),gr(y_0,0)),y_0,0)

  (34)

  1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 2 components.

  • The 1^st component contains the pair

    cond2#(true,y0,s(x0))

    →

    cond2#(true,p(y0),x0)

    (31)

    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.

    p(0)	→	0	(7)
    p(s(x))	→	x	(8)

    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.

    p(0)

    p(s(x0))

    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.

    cond2#(true,y0,s(x0))	→	cond2#(true,p(y0),x0)	(31)
    1	≥	1
    3	>	3

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

  • The 2^nd component contains the pair

    cond2#(false,y_0,0)	→	cond1#(and(eq(y_0,0),gr(y_0,0)),y_0,0)	(34)
    cond1#(true,y0,0)	→	cond2#(false,y0,0)	(32)

    1.1.1.1.1.1.1.1.1.1.1.2 Usable Rules Processor

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

    eq(0,0)	→	true	(9)
    eq(s(x),0)	→	false	(10)
    gr(0,x)	→	false	(4)
    gr(s(x),0)	→	true	(5)
    and(true,true)	→	true	(13)
    and(false,x)	→	false	(14)
    and(x,false)	→	false	(15)

    1.1.1.1.1.1.1.1.1.1.1.2.1 Innermost Lhss Removal Processor

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

    gr(0,x0)

    gr(s(x0),0)

    gr(s(x0),s(x1))

    eq(0,0)

    eq(s(x0),0)

    eq(0,s(x0))

    eq(s(x0),s(x1))

    and(true,true)

    and(false,x0)

    and(x0,false)

    1.1.1.1.1.1.1.1.1.1.1.2.1.1 Narrowing Processor

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

    cond2#(false,0,0)	→	cond1#(and(true,gr(0,0)),0,0)	(35)
    cond2#(false,s(x0),0)	→	cond1#(and(false,gr(s(x0),0)),s(x0),0)	(36)
    cond2#(false,0,0)	→	cond1#(and(eq(0,0),false),0,0)	(37)
    cond2#(false,s(x0),0)	→	cond1#(and(eq(s(x0),0),true),s(x0),0)	(38)

    1.1.1.1.1.1.1.1.1.1.1.2.1.1.1 Rewriting Processor

    We rewrite the right hand side of the pair resulting in

    →

    1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 1 component.

    • The 1^st component contains the pair

      cond2#(false,s(x0),0)	→	cond1#(and(false,gr(s(x0),0)),s(x0),0)	(36)
      cond1#(true,y0,0)	→	cond2#(false,y0,0)	(32)
      cond2#(false,0,0)	→	cond1#(and(eq(0,0),false),0,0)	(37)
      cond2#(false,s(x0),0)	→	cond1#(and(eq(s(x0),0),true),s(x0),0)	(38)

      1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1.1 Usable Rules Processor

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

      eq(s(x),0)	→	false	(10)
      and(true,true)	→	true	(13)
      and(false,x)	→	false	(14)
      eq(0,0)	→	true	(9)
      and(x,false)	→	false	(15)
      gr(s(x),0)	→	true	(5)

      1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1.1.1 Rewriting Processor

      We rewrite the right hand side of the pair resulting in

      →

      1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1.1.1.1 Dependency Graph Processor

      The dependency pairs are split into 1 component.

      • The 1^st component contains the pair

        cond2#(false,0,0)	→	cond1#(and(eq(0,0),false),0,0)	(37)
        cond1#(true,y0,0)	→	cond2#(false,y0,0)	(32)
        cond2#(false,s(x0),0)	→	cond1#(and(eq(s(x0),0),true),s(x0),0)	(38)

        1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1.1.1.1.1 Usable Rules Processor

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

        eq(s(x),0)	→	false	(10)
        and(true,true)	→	true	(13)
        and(false,x)	→	false	(14)
        eq(0,0)	→	true	(9)
        and(x,false)	→	false	(15)

        1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1.1.1.1.1.1 Innermost Lhss Removal Processor

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

        eq(0,0)

        eq(s(x0),0)

        eq(0,s(x0))

        eq(s(x0),s(x1))

        and(true,true)

        and(false,x0)

        and(x0,false)

        1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.1 Rewriting Processor

        We rewrite the right hand side of the pair resulting in

        →

        1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

        The dependency pairs are split into 1 component.

        • The 1^st component contains the pair

          cond2#(false,s(x0),0)	→	cond1#(and(eq(s(x0),0),true),s(x0),0)	(38)
          cond1#(true,y0,0)	→	cond2#(false,y0,0)	(32)

          1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1.1 Usable Rules Processor

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

          eq(s(x),0)	→	false	(10)
          and(true,true)	→	true	(13)
          and(false,x)	→	false	(14)

          1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1.1.1 Rewriting Processor

          We rewrite the right hand side of the pair resulting in

          →

          1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

          The dependency pairs are split into 0 components.

• The 2^nd component contains the pair

  gr#(s(x),s(y))

  →

  gr#(x,y)

  (26)

  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.

  gr#(s(x),s(y))	→	gr#(x,y)	(26)
  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

  eq#(s(x),s(y))

  →

  eq#(x,y)

  (27)

  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.

  1.1.1.3.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  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.

  eq#(s(x),s(y))	→	eq#(x,y)	(27)
  1	>	1
  2	>	2

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