Certification Problem

Input (TPDB TRS_Innermost/Mixed_innermost/cade13)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

• The 1^st component contains the pair

  cond#(true,x,y,z)	→	d#(x,s(y),plus(s(y),z))	(14)
  d#(x,s(y),z)	→	cond#(ge(x,z),x,y,z)	(12)

  1.1.1 Usable Rules Processor

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

  ge(u,0)	→	true	(5)
  ge(0,s(v))	→	false	(6)
  ge(s(u),s(v))	→	ge(u,v)	(7)
  plus(n,0)	→	n	(8)
  plus(n,s(m))	→	s(plus(n,m))	(9)
  plus(plus(n,m),u)	→	plus(n,plus(m,u))	(10)

  1.1.1.1 Innermost Lhss Removal Processor

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

  ge(x0,0)

  ge(0,s(x0))

  ge(s(x0),s(x1))

  plus(x0,0)

  plus(x0,s(x1))

  plus(plus(x0,x1),x2)

  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
  [d#(x1, x2, x3)]	=	1 · x1 + 1 · x2 + -1 · x3 + -1
  [cond#(x1,...,x4)]	=	-1 · x1 + 1 · x2 + -1 · x4 + -1
  [true]	=	0
  [s(x1)]	=	1 · x1 + 1
  [plus(x1, x2)]	=	1 · x1 + 1 · x2
  [ge(x1, x2)]	=	0
  [0]	=	0
  [false]	=	0

  The pair(s)

  d#(x,s(y),z)

  →

  cond#(ge(x,z),x,y,z)

  (12)

  are strictly oriented and the pair(s)

  cond#(true,x,y,z)

  →

  d#(x,s(y),plus(s(y),z))

  (14)

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

  The following constraints are generated for the pairs.

  • cond^#(true,x18,x4,0)≥c
  • (cond^#(true,x21,x4,x20)≥c⟹cond^#(true,s(x21),x4,s(x20))≥c)
  • cond^#(true,x18,x4,0)≥d^#(x18,s(x4),plus(s(x4),0))
  • (cond^#(true,x21,x4,x20)≥d^#(x21,s(x4),plus(s(x4),x20))⟹cond^#(true,s(x21),x4,s(x20))≥d^#(s(x21),s(x4),plus(s(x4),s(x20))))
  • d^#(x9,s(x10),x14) > cond^#(ge(x9,x14),x9,x10,x14)

  The details are shown below:

  • For the chain we build the initial constraint

    (cond^#(ge(x3,x5),x3,x4,x5)→^*cond^#(true,x6,x7,x8)⟹cond^#(true,x6,x7,x8)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (ge(x3,x5)→^*true⟹x3→^*x6⟹x4→^*x7⟹x5→^*x8⟹cond^#(true,x6,x7,x8)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x6 by x3 which results in
      (ge(x3,x5)→^*true⟹x4→^*x7⟹x5→^*x8⟹cond^#(true,x3,x7,x8)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x7 by x4 which results in
      (ge(x3,x5)→^*true⟹x5→^*x8⟹cond^#(true,x3,x4,x8)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x8 by x5 which results in
      (ge(x3,x5)→^*true⟹cond^#(true,x3,x4,x5)≥c)
    • Applying Rule "Induction" on ge(x3,x5)→^*true results in the following 3 new constraints.
      1. (true→^*true⟹cond^#(true,x18,x4,0)≥c)
         • Applying Rule "Same Constructor" results in
           cond^#(true,x18,x4,0)≥c

         • This constraint is kept as final constraint.

      2. (false→^*true⟹cond^#(true,0,x4,s(x19))≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (ge(x21,x20)→^*true⟹∀x22 . (ge(x21,x20)→^*true⟹cond^#(true,x21,x22,x20)≥c)⟹cond^#(true,s(x21),x4,s(x20))≥c)
         • Applying Rule "Simplify Conditions" results in
           (cond^#(true,x21,x4,x20)≥c⟹cond^#(true,s(x21),x4,s(x20))≥c)

         • This constraint is kept as final constraint.

  • For the chain we build the initial constraint

    (cond^#(ge(x3,x5),x3,x4,x5)→^*cond^#(true,x6,x7,x8)⟹cond^#(true,x6,x7,x8)≥d^#(x6,s(x7),plus(s(x7),x8)))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (ge(x3,x5)→^*true⟹x3→^*x6⟹x4→^*x7⟹x5→^*x8⟹cond^#(true,x6,x7,x8)≥d^#(x6,s(x7),plus(s(x7),x8)))
    • Applying Rule "Variable in Equation" allows to substitute x6 by x3 which results in
      (ge(x3,x5)→^*true⟹x4→^*x7⟹x5→^*x8⟹cond^#(true,x3,x7,x8)≥d^#(x3,s(x7),plus(s(x7),x8)))
    • Applying Rule "Variable in Equation" allows to substitute x7 by x4 which results in
      (ge(x3,x5)→^*true⟹x5→^*x8⟹cond^#(true,x3,x4,x8)≥d^#(x3,s(x4),plus(s(x4),x8)))
    • Applying Rule "Variable in Equation" allows to substitute x8 by x5 which results in
      (ge(x3,x5)→^*true⟹cond^#(true,x3,x4,x5)≥d^#(x3,s(x4),plus(s(x4),x5)))
    • Applying Rule "Induction" on ge(x3,x5)→^*true results in the following 3 new constraints.
      1. (true→^*true⟹cond^#(true,x18,x4,0)≥d^#(x18,s(x4),plus(s(x4),0)))
         • Applying Rule "Same Constructor" results in
           cond^#(true,x18,x4,0)≥d^#(x18,s(x4),plus(s(x4),0))

         • This constraint is kept as final constraint.

      2. (false→^*true⟹cond^#(true,0,x4,s(x19))≥d^#(0,s(x4),plus(s(x4),s(x19))))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (ge(x21,x20)→^*true⟹∀x22 . (ge(x21,x20)→^*true⟹cond^#(true,x21,x22,x20)≥d^#(x21,s(x22),plus(s(x22),x20)))⟹cond^#(true,s(x21),x4,s(x20))≥d^#(s(x21),s(x4),plus(s(x4),s(x20))))
         • Applying Rule "Simplify Conditions" results in
           (cond^#(true,x21,x4,x20)≥d^#(x21,s(x4),plus(s(x4),x20))⟹cond^#(true,s(x21),x4,s(x20))≥d^#(s(x21),s(x4),plus(s(x4),s(x20))))

         • This constraint is kept as final constraint.

  • For the chain we build the initial constraint

    (d^#(x9,s(x10),plus(s(x10),x11))→^*d^#(x12,s(x13),x14)⟹d^#(x12,s(x13),x14) > cond^#(ge(x12,x14),x12,x13,x14))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (x9→^*x12⟹s(x10)→^*s(x13)⟹plus(s(x10),x11)→^*x14⟹d^#(x12,s(x13),x14) > cond^#(ge(x12,x14),x12,x13,x14))
    • Applying Rule "Same Constructor" results in
      (x9→^*x12⟹x10→^*x13⟹plus(s(x10),x11)→^*x14⟹d^#(x12,s(x13),x14) > cond^#(ge(x12,x14),x12,x13,x14))
    • Applying Rule "Variable in Equation" allows to substitute x12 by x9 which results in
      (x10→^*x13⟹plus(s(x10),x11)→^*x14⟹d^#(x9,s(x13),x14) > cond^#(ge(x9,x14),x9,x13,x14))
    • Applying Rule "Variable in Equation" allows to substitute x13 by x10 which results in
      (plus(s(x10),x11)→^*x14⟹d^#(x9,s(x10),x14) > cond^#(ge(x9,x14),x9,x10,x14))
    • Applying Rule "Introduce fresh variable" results in
      (s(x10)→^*x23⟹plus(x23,x11)→^*x14⟹d^#(x9,s(x10),x14) > cond^#(ge(x9,x14),x9,x10,x14))
    • Applying Rule "Variable in Equation" allows to substitute x23 by s(x10) which results in
      (plus(s(x10),x11)→^*x14⟹d^#(x9,s(x10),x14) > cond^#(ge(x9,x14),x9,x10,x14))
    • Applying Rule "Delete Conditions" results in
      d^#(x9,s(x10),x14) > cond^#(ge(x9,x14),x9,x10,x14)

    • This constraint is kept as final constraint.

  We get two subproofs, in the first the strict pairs are deleted, in the second the bounded pairs are deleted.

  1.1.1.1.1.1 Usable Rules Processor

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

  plus(n,0)	→	n	(8)
  plus(n,s(m))	→	s(plus(n,m))	(9)
  plus(plus(n,m),u)	→	plus(n,plus(m,u))	(10)

  1.1.1.1.1.1.1 Innermost Lhss Removal Processor

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

  plus(x0,0)

  plus(x0,s(x1))

  plus(plus(x0,x1),x2)

  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.

  cond#(true,x,y,z)	→	d#(x,s(y),plus(s(y),z))	(14)
  2	≥	1

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

  1.1.1.1.1.2 Usable Rules Processor

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

  ge(u,0)	→	true	(5)
  ge(0,s(v))	→	false	(6)
  ge(s(u),s(v))	→	ge(u,v)	(7)

  1.1.1.1.1.2.1 Innermost Lhss Removal Processor

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

  ge(x0,0)

  ge(0,s(x0))

  ge(s(x0),s(x1))

  1.1.1.1.1.2.1.1 Size-Change Termination

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

  d#(x,s(y),z)	→	cond#(ge(x,z),x,y,z)	(12)
  1	≥	2
  2	>	3
  3	≥	4

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

• The 2^nd component contains the pair

  ge#(s(u),s(v))

  →

  ge#(u,v)

  (16)

  1.1.2 Usable Rules Processor

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

  There are no rules.

  1.1.2.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  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.

  ge#(s(u),s(v))	→	ge#(u,v)	(16)
  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

  plus#(plus(n,m),u)	→	plus#(n,plus(m,u))	(18)
  plus#(n,s(m))	→	plus#(n,m)	(17)
  plus#(plus(n,m),u)	→	plus#(m,u)	(19)

  1.1.3 Usable Rules Processor

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

  plus(n,0)	→	n	(8)
  plus(n,s(m))	→	s(plus(n,m))	(9)
  plus(plus(n,m),u)	→	plus(n,plus(m,u))	(10)

  1.1.3.1 Innermost Lhss Removal Processor

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

  plus(x0,0)

  plus(x0,s(x1))

  plus(plus(x0,x1),x2)

  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.

  plus#(n,s(m))	→	plus#(n,m)	(17)
  1	≥	1
  2	>	2
  plus#(plus(n,m),u)	→	plus#(m,u)	(19)
  1	>	1
  2	≥	2
  plus#(plus(n,m),u)	→	plus#(n,plus(m,u))	(18)
  1	>	1

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