Certification Problem

Input (TPDB TRS_Standard/Beerendonk_07/13)

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

  cond2#(true,x,y)	→	cond1#(gr(x,0),y,y)	(11)
  cond1#(true,x,y)	→	cond2#(gr(x,y),x,y)	(9)
  cond2#(false,x,y)	→	cond1#(gr(x,0),p(x),y)	(13)

  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)
  p(0)	→	0	(7)
  p(s(x))	→	x	(8)
  gr(s(x),s(y))	→	gr(x,y)	(6)

  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))

  1.1.1.1.1.1 Reduction Pair Processor

  We apply the generic reduction pair processor using the linear polynomial interpretation over the naturals

  [c]	=	-1
  [cond1#(x1, x2, x3)]	=	1 · x2 + -1 · x3 + -1
  [cond2#(x1, x2, x3)]	=	1 · x1 + 1 · x2 + -1 · x3 + -1
  [true]	=	0
  [gr(x1, x2)]	=	0
  [0]	=	0
  [false]	=	0
  [p(x1)]	=	1 · x1
  [s(x1)]	=	1 · x1 + 1

  The pair(s)

  cond2#(true,x,y)

  →

  cond1#(gr(x,0),y,y)

  (11)

  are strictly oriented and the pair(s)

  cond2#(true,x,y)

  →

  cond1#(gr(x,0),y,y)

  (11)

  are bounded w.r.t. the constant c.

  The following constraints are generated for the pairs.

  • cond2^#(true,s(x27),0)≥c
  • (cond2^#(true,x29,x28)≥c⟹cond2^#(true,s(x29),s(x28))≥c)
  • cond2^#(true,s(x27),0) > cond1^#(gr(s(x27),0),0,0)
  • (cond2^#(true,x29,x28) > cond1^#(gr(x29,0),x28,x28)⟹cond2^#(true,s(x29),s(x28)) > cond1^#(gr(s(x29),0),s(x28),s(x28)))
  • cond1^#(true,x9,x9)≥cond2^#(gr(x9,x9),x9,x9)
  • cond1^#(true,x16,x15)≥cond2^#(gr(x16,x15),x16,x15)
  • cond2^#(false,0,x44)≥cond1^#(gr(0,0),p(0),x44)
  • (cond2^#(false,x47,x46)≥cond1^#(gr(x47,0),p(x47),x46)⟹cond2^#(false,s(x47),s(x46))≥cond1^#(gr(s(x47),0),p(s(x47)),s(x46)))

  The details are shown below:

  • For the chain we build the initial constraint

    (cond2^#(gr(x2,x3),x2,x3)→^*cond2^#(true,x4,x5)⟹cond2^#(true,x4,x5)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x2,x3)→^*true⟹x2→^*x4⟹x3→^*x5⟹cond2^#(true,x4,x5)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x4 by x2 which results in
      (gr(x2,x3)→^*true⟹x3→^*x5⟹cond2^#(true,x2,x5)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x5 by x3 which results in
      (gr(x2,x3)→^*true⟹cond2^#(true,x2,x3)≥c)
    • Applying Rule "Induction" on gr(x2,x3)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹cond2^#(true,0,x26)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹cond2^#(true,s(x27),0)≥c)
         • Applying Rule "Same Constructor" results in
           cond2^#(true,s(x27),0)≥c

         • This constraint is kept as final constraint.

      3. (gr(x29,x28)→^*true⟹(gr(x29,x28)→^*true⟹cond2^#(true,x29,x28)≥c)⟹cond2^#(true,s(x29),s(x28))≥c)
         • Applying Rule "Simplify Conditions" results in
           (cond2^#(true,x29,x28)≥c⟹cond2^#(true,s(x29),s(x28))≥c)

         • This constraint is kept as final constraint.

  • For the chain we build the initial constraint

    (cond2^#(gr(x2,x3),x2,x3)→^*cond2^#(true,x4,x5)⟹cond2^#(true,x4,x5) > cond1^#(gr(x4,0),x5,x5))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x2,x3)→^*true⟹x2→^*x4⟹x3→^*x5⟹cond2^#(true,x4,x5) > cond1^#(gr(x4,0),x5,x5))
    • Applying Rule "Variable in Equation" allows to substitute x4 by x2 which results in
      (gr(x2,x3)→^*true⟹x3→^*x5⟹cond2^#(true,x2,x5) > cond1^#(gr(x2,0),x5,x5))
    • Applying Rule "Variable in Equation" allows to substitute x5 by x3 which results in
      (gr(x2,x3)→^*true⟹cond2^#(true,x2,x3) > cond1^#(gr(x2,0),x3,x3))
    • Applying Rule "Induction" on gr(x2,x3)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹cond2^#(true,0,x26) > cond1^#(gr(0,0),x26,x26))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹cond2^#(true,s(x27),0) > cond1^#(gr(s(x27),0),0,0))
         • Applying Rule "Same Constructor" results in
           cond2^#(true,s(x27),0) > cond1^#(gr(s(x27),0),0,0)

         • This constraint is kept as final constraint.

      3. (gr(x29,x28)→^*true⟹(gr(x29,x28)→^*true⟹cond2^#(true,x29,x28) > cond1^#(gr(x29,0),x28,x28))⟹cond2^#(true,s(x29),s(x28)) > cond1^#(gr(s(x29),0),s(x28),s(x28)))
         • Applying Rule "Simplify Conditions" results in
           (cond2^#(true,x29,x28) > cond1^#(gr(x29,0),x28,x28)⟹cond2^#(true,s(x29),s(x28)) > cond1^#(gr(s(x29),0),s(x28),s(x28)))

         • This constraint is kept as final constraint.

  • For the chain we build the initial constraint

    (cond1^#(gr(x8,0),x9,x9)→^*cond1^#(true,x10,x11)⟹cond1^#(true,x10,x11)≥cond2^#(gr(x10,x11),x10,x11))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x8,0)→^*true⟹x9→^*x10⟹x9→^*x11⟹cond1^#(true,x10,x11)≥cond2^#(gr(x10,x11),x10,x11))
    • Applying Rule "Variable in Equation" allows to substitute x10 by x9 which results in
      (gr(x8,0)→^*true⟹x9→^*x11⟹cond1^#(true,x9,x11)≥cond2^#(gr(x9,x11),x9,x11))
    • Applying Rule "Variable in Equation" allows to substitute x11 by x9 which results in
      (gr(x8,0)→^*true⟹cond1^#(true,x9,x9)≥cond2^#(gr(x9,x9),x9,x9))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x30⟹gr(x8,x30)→^*true⟹cond1^#(true,x9,x9)≥cond2^#(gr(x9,x9),x9,x9))
    • Applying Rule "Induction" on gr(x8,x30)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹cond1^#(true,x9,x9)≥cond2^#(gr(x9,x9),x9,x9))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹0→^*0⟹cond1^#(true,x9,x9)≥cond2^#(gr(x9,x9),x9,x9))
         • Applying Rule "Same Constructor" results in
           (0→^*0⟹cond1^#(true,x9,x9)≥cond2^#(gr(x9,x9),x9,x9))
         • Applying Rule "Same Constructor" results in
           cond1^#(true,x9,x9)≥cond2^#(gr(x9,x9),x9,x9)

         • This constraint is kept as final constraint.

      3. (gr(x34,x33)→^*true⟹0→^*s(x33)⟹∀x35 . (gr(x34,x33)→^*true⟹0→^*x33⟹cond1^#(true,x35,x35)≥cond2^#(gr(x35,x35),x35,x35))⟹cond1^#(true,x9,x9)≥cond2^#(gr(x9,x9),x9,x9))
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond1^#(gr(x14,0),p(x14),x15)→^*cond1^#(true,x16,x17)⟹cond1^#(true,x16,x17)≥cond2^#(gr(x16,x17),x16,x17))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x14,0)→^*true⟹p(x14)→^*x16⟹x15→^*x17⟹cond1^#(true,x16,x17)≥cond2^#(gr(x16,x17),x16,x17))
    • Applying Rule "Variable in Equation" allows to substitute x17 by x15 which results in
      (gr(x14,0)→^*true⟹p(x14)→^*x16⟹cond1^#(true,x16,x15)≥cond2^#(gr(x16,x15),x16,x15))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x36⟹gr(x14,x36)→^*true⟹p(x14)→^*x16⟹cond1^#(true,x16,x15)≥cond2^#(gr(x16,x15),x16,x15))
    • Applying Rule "Induction" on gr(x14,x36)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹cond1^#(true,x16,x15)≥cond2^#(gr(x16,x15),x16,x15))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹0→^*0⟹p(s(x38))→^*x16⟹cond1^#(true,x16,x15)≥cond2^#(gr(x16,x15),x16,x15))
         • Applying Rule "Same Constructor" results in
           (0→^*0⟹p(s(x38))→^*x16⟹cond1^#(true,x16,x15)≥cond2^#(gr(x16,x15),x16,x15))
         • Applying Rule "Same Constructor" results in
           (p(s(x38))→^*x16⟹cond1^#(true,x16,x15)≥cond2^#(gr(x16,x15),x16,x15))
         • Applying Rule "Introduce fresh variable" results in
           (s(x38)→^*x43⟹p(x43)→^*x16⟹cond1^#(true,x16,x15)≥cond2^#(gr(x16,x15),x16,x15))
         • Applying Rule "Variable in Equation" allows to substitute x43 by s(x38) which results in
           (p(s(x38))→^*x16⟹cond1^#(true,x16,x15)≥cond2^#(gr(x16,x15),x16,x15))
         • Applying Rule "Delete Conditions" results in
           cond1^#(true,x16,x15)≥cond2^#(gr(x16,x15),x16,x15)

         • This constraint is kept as final constraint.

      3. (gr(x40,x39)→^*true⟹0→^*s(x39)⟹p(s(x40))→^*x16⟹∀x41 . ∀x42 . (gr(x40,x39)→^*true⟹0→^*x39⟹p(x40)→^*x41⟹cond1^#(true,x41,x42)≥cond2^#(gr(x41,x42),x41,x42))⟹cond1^#(true,x16,x15)≥cond2^#(gr(x16,x15),x16,x15))
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond2^#(gr(x20,x21),x20,x21)→^*cond2^#(false,x22,x23)⟹cond2^#(false,x22,x23)≥cond1^#(gr(x22,0),p(x22),x23))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x20,x21)→^*false⟹x20→^*x22⟹x21→^*x23⟹cond2^#(false,x22,x23)≥cond1^#(gr(x22,0),p(x22),x23))
    • Applying Rule "Variable in Equation" allows to substitute x22 by x20 which results in
      (gr(x20,x21)→^*false⟹x21→^*x23⟹cond2^#(false,x20,x23)≥cond1^#(gr(x20,0),p(x20),x23))
    • Applying Rule "Variable in Equation" allows to substitute x23 by x21 which results in
      (gr(x20,x21)→^*false⟹cond2^#(false,x20,x21)≥cond1^#(gr(x20,0),p(x20),x21))
    • Applying Rule "Induction" on gr(x20,x21)→^*false results in the following 3 new constraints.
      1. (false→^*false⟹cond2^#(false,0,x44)≥cond1^#(gr(0,0),p(0),x44))
         • Applying Rule "Same Constructor" results in
           cond2^#(false,0,x44)≥cond1^#(gr(0,0),p(0),x44)

         • This constraint is kept as final constraint.

      2. (true→^*false⟹cond2^#(false,s(x45),0)≥cond1^#(gr(s(x45),0),p(s(x45)),0))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x47,x46)→^*false⟹(gr(x47,x46)→^*false⟹cond2^#(false,x47,x46)≥cond1^#(gr(x47,0),p(x47),x46))⟹cond2^#(false,s(x47),s(x46))≥cond1^#(gr(s(x47),0),p(s(x47)),s(x46)))
         • Applying Rule "Simplify Conditions" results in
           (cond2^#(false,x47,x46)≥cond1^#(gr(x47,0),p(x47),x46)⟹cond2^#(false,s(x47),s(x46))≥cond1^#(gr(s(x47),0),p(s(x47)),s(x46)))

         • This constraint is kept as final constraint.

  We remove those pairs which are strictly decreasing and bounded.

  1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the rationals with delta = 1

  [cond1#(x1, x2, x3)]	=	0 + 1 · x1 + 1 · x2 + 0 · x3
  [true]	=	1
  [cond2#(x1, x2, x3)]	=	1 + 0 · x1 + 1/2 · x2 + 0 · x3
  [gr(x1, x2)]	=	0 + 1/4 · x1 + 0 · x2
  [false]	=	0
  [0]	=	0
  [p(x1)]	=	0 + 1/4 · x1
  [s(x1)]	=	4 + 4 · x1

  together with the usable rules

  gr(0,x)	→	false	(4)
  gr(s(x),0)	→	true	(5)
  p(0)	→	0	(7)
  p(s(x))	→	x	(8)

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

  cond2#(false,x,y)

  →

  cond1#(gr(x,0),p(x),y)

  (13)

  could be deleted.

  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.

  gr(0,x)	→	false	(4)
  gr(s(x),0)	→	true	(5)
  gr(s(x),s(y))	→	gr(x,y)	(6)

  1.1.1.1.1.1.1.1.1 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))

  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.

  cond1#(true,x,y)	→	cond2#(gr(x,y),x,y)	(9)
  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

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

  →

  gr#(x,y)

  (16)

  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)	(16)
  1	>	1
  2	>	2

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