Certification Problem

Input (TPDB TRS_Standard/AG01/#4.30c)

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

  gcd#(s(x),s(y))	→	if_gcd#(le(y,x),s(x),s(y))	(15)
  if_gcd#(true,x,y)	→	gcd#(minus(x,y),y)	(17)
  if_gcd#(false,x,y)	→	gcd#(minus(y,x),x)	(19)

  1.1.1.1 Usable Rules Processor

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

  minus(x,0)	→	x	(5)
  minus(x,s(y))	→	pred(minus(x,y))	(6)
  pred(s(x))	→	x	(4)
  le(0,y)	→	true	(1)
  le(s(x),0)	→	false	(2)
  le(s(x),s(y))	→	le(x,y)	(3)

  1.1.1.1.1 Innermost Lhss Removal Processor

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

  le(0,x0)

  le(s(x0),0)

  le(s(x0),s(x1))

  pred(s(x0))

  minus(x0,0)

  minus(x0,s(x1))

  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
  [if_gcd#(x1, x2, x3)]	=	-1 · x1 + 1 · x3 + -1
  [gcd#(x1, x2)]	=	1 · x2 + -1
  [s(x1)]	=	1 · x1 + 1
  [le(x1, x2)]	=	0
  [true]	=	0
  [minus(x1, x2)]	=	0
  [false]	=	0
  [0]	=	0
  [pred(x1)]	=	0

  The pair(s)

  if_gcd#(false,x,y)

  →

  gcd#(minus(y,x),x)

  (19)

  are strictly oriented and the pair(s)

  gcd#(s(x),s(y))	→	if_gcd#(le(y,x),s(x),s(y))	(15)
  if_gcd#(true,x,y)	→	gcd#(minus(x,y),y)	(17)
  if_gcd#(false,x,y)	→	gcd#(minus(y,x),x)	(19)

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

  The following constraints are generated for the pairs.

  • gcd^#(s(x4),s(0))≥c
  • gcd^#(s(x4),s(s(x35)))≥c
  • gcd^#(s(x8),s(0))≥c
  • gcd^#(s(x8),s(s(x48)))≥c
  • if_gcd^#(true,s(x52),s(0))≥c
  • (if_gcd^#(true,s(x54),s(x55))≥c⟹if_gcd^#(true,s(s(x54)),s(s(x55)))≥c)
  • if_gcd^#(false,s(0),s(s(x57)))≥c
  • (if_gcd^#(false,s(x58),s(x59))≥c⟹if_gcd^#(false,s(s(x58)),s(s(x59)))≥c)
  • gcd^#(s(x4),s(0))≥if_gcd^#(le(0,x4),s(x4),s(0))
  • gcd^#(s(x4),s(s(x35)))≥if_gcd^#(le(s(x35),x4),s(x4),s(s(x35)))
  • gcd^#(s(x8),s(0))≥if_gcd^#(le(0,x8),s(x8),s(0))
  • gcd^#(s(x8),s(s(x48)))≥if_gcd^#(le(s(x48),x8),s(x8),s(s(x48)))
  • if_gcd^#(true,s(x52),s(0))≥gcd^#(minus(s(x52),s(0)),s(0))
  • (if_gcd^#(true,s(x54),s(x55))≥gcd^#(minus(s(x54),s(x55)),s(x55))⟹if_gcd^#(true,s(s(x54)),s(s(x55)))≥gcd^#(minus(s(s(x54)),s(s(x55))),s(s(x55))))
  • if_gcd^#(false,s(0),s(s(x57))) > gcd^#(minus(s(s(x57)),s(0)),s(0))
  • (if_gcd^#(false,s(x58),s(x59)) > gcd^#(minus(s(x59),s(x58)),s(x58))⟹if_gcd^#(false,s(s(x58)),s(s(x59))) > gcd^#(minus(s(s(x59)),s(s(x58))),s(s(x58))))

  The details are shown below:

  • For the chain we build the initial constraint

    (gcd^#(minus(x2,x3),x3)→^*gcd^#(s(x4),s(x5))⟹gcd^#(s(x4),s(x5))≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (minus(x2,x3)→^*s(x4)⟹x3→^*s(x5)⟹gcd^#(s(x4),s(x5))≥c)
    • Applying Rule "Variable in Equation" allows to substitute x3 by s(x5) which results in
      (minus(x2,s(x5))→^*s(x4)⟹gcd^#(s(x4),s(x5))≥c)
    • Applying Rule "Introduce fresh variable" results in
      (s(x5)→^*x26⟹minus(x2,x26)→^*s(x4)⟹gcd^#(s(x4),s(x5))≥c)
    • Applying Rule "Induction" on minus(x2,x26)→^*s(x4) results in the following 2 new constraints.
      1. (x27→^*s(x4)⟹s(x5)→^*0⟹gcd^#(s(x4),s(x5))≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (pred(minus(x29,x28))→^*s(x4)⟹s(x5)→^*s(x28)⟹∀x30 . ∀x31 . (minus(x29,x28)→^*s(x30)⟹s(x31)→^*x28⟹gcd^#(s(x30),s(x31))≥c)⟹gcd^#(s(x4),s(x5))≥c)
         • Applying Rule "Same Constructor" results in
           (pred(minus(x29,x28))→^*s(x4)⟹x5→^*x28⟹∀x30 . ∀x31 . (minus(x29,x28)→^*s(x30)⟹s(x31)→^*x28⟹gcd^#(s(x30),s(x31))≥c)⟹gcd^#(s(x4),s(x5))≥c)
         • Applying Rule "Variable in Equation" allows to substitute x5 by x28 which results in
           (pred(minus(x29,x28))→^*s(x4)⟹∀x30 . ∀x31 . (minus(x29,x28)→^*s(x30)⟹s(x31)→^*x28⟹gcd^#(s(x30),s(x31))≥c)⟹gcd^#(s(x4),s(x28))≥c)
         • Applying Rule "Introduce fresh variable" results in
           (minus(x29,x28)→^*x32⟹pred(x32)→^*s(x4)⟹∀x30 . ∀x31 . (minus(x29,x28)→^*s(x30)⟹s(x31)→^*x28⟹gcd^#(s(x30),s(x31))≥c)⟹gcd^#(s(x4),s(x28))≥c)
         • Applying Rule "Induction" on pred(x32)→^*s(x4) results in the following 1 new constraints.
           1. (x33→^*s(x4)⟹minus(x29,x28)→^*s(x33)⟹∀x30 . ∀x31 . (minus(x29,x28)→^*s(x30)⟹s(x31)→^*x28⟹gcd^#(s(x30),s(x31))≥c)⟹gcd^#(s(x4),s(x28))≥c)
              • Applying Rule "Variable in Equation" allows to substitute x33 by s(x4) which results in
                (minus(x29,x28)→^*s(s(x4))⟹∀x30 . ∀x31 . (minus(x29,x28)→^*s(x30)⟹s(x31)→^*x28⟹gcd^#(s(x30),s(x31))≥c)⟹gcd^#(s(x4),s(x28))≥c)
              • Applying Rule "Delete Conditions" results in
                (minus(x29,x28)→^*s(s(x4))⟹gcd^#(s(x4),s(x28))≥c)
              • Applying Rule "Induction" on minus(x29,x28)→^*s(s(x4)) results in the following 2 new constraints.
                1. (x34→^*s(s(x4))⟹gcd^#(s(x4),s(0))≥c)
                   • Applying Rule "Variable in Equation" allows to substitute x34 by s(s(x4)) which results in
                     gcd^#(s(x4),s(0))≥c

                   • This constraint is kept as final constraint.

                2. (pred(minus(x36,x35))→^*s(s(x4))⟹∀x37 . (minus(x36,x35)→^*s(s(x37))⟹gcd^#(s(x37),s(x35))≥c)⟹gcd^#(s(x4),s(s(x35)))≥c)
                   • Applying Rule "Introduce fresh variable" results in
                     (minus(x36,x35)→^*x38⟹pred(x38)→^*s(s(x4))⟹∀x37 . (minus(x36,x35)→^*s(s(x37))⟹gcd^#(s(x37),s(x35))≥c)⟹gcd^#(s(x4),s(s(x35)))≥c)
                   • Applying Rule "Delete Conditions" results in
                     (pred(x38)→^*s(s(x4))⟹∀x37 . (minus(x36,x35)→^*s(s(x37))⟹gcd^#(s(x37),s(x35))≥c)⟹gcd^#(s(x4),s(s(x35)))≥c)
                   • Applying Rule "Delete Conditions" results in
                     (∀x37 . (minus(x36,x35)→^*s(s(x37))⟹gcd^#(s(x37),s(x35))≥c)⟹gcd^#(s(x4),s(s(x35)))≥c)
                   • Applying Rule "Delete Conditions" results in
                     gcd^#(s(x4),s(s(x35)))≥c

                   • This constraint is kept as final constraint.

  • For the chain we build the initial constraint

    (gcd^#(minus(x7,x6),x6)→^*gcd^#(s(x8),s(x9))⟹gcd^#(s(x8),s(x9))≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (minus(x7,x6)→^*s(x8)⟹x6→^*s(x9)⟹gcd^#(s(x8),s(x9))≥c)
    • Applying Rule "Variable in Equation" allows to substitute x6 by s(x9) which results in
      (minus(x7,s(x9))→^*s(x8)⟹gcd^#(s(x8),s(x9))≥c)
    • Applying Rule "Introduce fresh variable" results in
      (s(x9)→^*x39⟹minus(x7,x39)→^*s(x8)⟹gcd^#(s(x8),s(x9))≥c)
    • Applying Rule "Induction" on minus(x7,x39)→^*s(x8) results in the following 2 new constraints.
      1. (x40→^*s(x8)⟹s(x9)→^*0⟹gcd^#(s(x8),s(x9))≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (pred(minus(x42,x41))→^*s(x8)⟹s(x9)→^*s(x41)⟹∀x43 . ∀x44 . (minus(x42,x41)→^*s(x43)⟹s(x44)→^*x41⟹gcd^#(s(x43),s(x44))≥c)⟹gcd^#(s(x8),s(x9))≥c)
         • Applying Rule "Same Constructor" results in
           (pred(minus(x42,x41))→^*s(x8)⟹x9→^*x41⟹∀x43 . ∀x44 . (minus(x42,x41)→^*s(x43)⟹s(x44)→^*x41⟹gcd^#(s(x43),s(x44))≥c)⟹gcd^#(s(x8),s(x9))≥c)
         • Applying Rule "Variable in Equation" allows to substitute x9 by x41 which results in
           (pred(minus(x42,x41))→^*s(x8)⟹∀x43 . ∀x44 . (minus(x42,x41)→^*s(x43)⟹s(x44)→^*x41⟹gcd^#(s(x43),s(x44))≥c)⟹gcd^#(s(x8),s(x41))≥c)
         • Applying Rule "Introduce fresh variable" results in
           (minus(x42,x41)→^*x45⟹pred(x45)→^*s(x8)⟹∀x43 . ∀x44 . (minus(x42,x41)→^*s(x43)⟹s(x44)→^*x41⟹gcd^#(s(x43),s(x44))≥c)⟹gcd^#(s(x8),s(x41))≥c)
         • Applying Rule "Induction" on pred(x45)→^*s(x8) results in the following 1 new constraints.
           1. (x46→^*s(x8)⟹minus(x42,x41)→^*s(x46)⟹∀x43 . ∀x44 . (minus(x42,x41)→^*s(x43)⟹s(x44)→^*x41⟹gcd^#(s(x43),s(x44))≥c)⟹gcd^#(s(x8),s(x41))≥c)
              • Applying Rule "Variable in Equation" allows to substitute x46 by s(x8) which results in
                (minus(x42,x41)→^*s(s(x8))⟹∀x43 . ∀x44 . (minus(x42,x41)→^*s(x43)⟹s(x44)→^*x41⟹gcd^#(s(x43),s(x44))≥c)⟹gcd^#(s(x8),s(x41))≥c)
              • Applying Rule "Delete Conditions" results in
                (minus(x42,x41)→^*s(s(x8))⟹gcd^#(s(x8),s(x41))≥c)
              • Applying Rule "Induction" on minus(x42,x41)→^*s(s(x8)) results in the following 2 new constraints.
                1. (x47→^*s(s(x8))⟹gcd^#(s(x8),s(0))≥c)
                   • Applying Rule "Variable in Equation" allows to substitute x47 by s(s(x8)) which results in
                     gcd^#(s(x8),s(0))≥c

                   • This constraint is kept as final constraint.

                2. (pred(minus(x49,x48))→^*s(s(x8))⟹∀x50 . (minus(x49,x48)→^*s(s(x50))⟹gcd^#(s(x50),s(x48))≥c)⟹gcd^#(s(x8),s(s(x48)))≥c)
                   • Applying Rule "Introduce fresh variable" results in
                     (minus(x49,x48)→^*x51⟹pred(x51)→^*s(s(x8))⟹∀x50 . (minus(x49,x48)→^*s(s(x50))⟹gcd^#(s(x50),s(x48))≥c)⟹gcd^#(s(x8),s(s(x48)))≥c)
                   • Applying Rule "Delete Conditions" results in
                     (pred(x51)→^*s(s(x8))⟹∀x50 . (minus(x49,x48)→^*s(s(x50))⟹gcd^#(s(x50),s(x48))≥c)⟹gcd^#(s(x8),s(s(x48)))≥c)
                   • Applying Rule "Delete Conditions" results in
                     (∀x50 . (minus(x49,x48)→^*s(s(x50))⟹gcd^#(s(x50),s(x48))≥c)⟹gcd^#(s(x8),s(s(x48)))≥c)
                   • Applying Rule "Delete Conditions" results in
                     gcd^#(s(x8),s(s(x48)))≥c

                   • This constraint is kept as final constraint.

  • For the chain we build the initial constraint

    (if_gcd^#(le(x11,x10),s(x10),s(x11))→^*if_gcd^#(true,x12,x13)⟹if_gcd^#(true,x12,x13)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (le(x11,x10)→^*true⟹s(x10)→^*x12⟹s(x11)→^*x13⟹if_gcd^#(true,x12,x13)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x12 by s(x10) which results in
      (le(x11,x10)→^*true⟹s(x11)→^*x13⟹if_gcd^#(true,s(x10),x13)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x13 by s(x11) which results in
      (le(x11,x10)→^*true⟹if_gcd^#(true,s(x10),s(x11))≥c)
    • Applying Rule "Induction" on le(x11,x10)→^*true results in the following 3 new constraints.
      1. (true→^*true⟹if_gcd^#(true,s(x52),s(0))≥c)
         • Applying Rule "Same Constructor" results in
           if_gcd^#(true,s(x52),s(0))≥c

         • This constraint is kept as final constraint.

      2. (false→^*true⟹if_gcd^#(true,s(0),s(s(x53)))≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (le(x55,x54)→^*true⟹(le(x55,x54)→^*true⟹if_gcd^#(true,s(x54),s(x55))≥c)⟹if_gcd^#(true,s(s(x54)),s(s(x55)))≥c)
         • Applying Rule "Simplify Conditions" results in
           (if_gcd^#(true,s(x54),s(x55))≥c⟹if_gcd^#(true,s(s(x54)),s(s(x55)))≥c)

         • This constraint is kept as final constraint.

  • For the chain we build the initial constraint

    (if_gcd^#(le(x19,x18),s(x18),s(x19))→^*if_gcd^#(false,x20,x21)⟹if_gcd^#(false,x20,x21)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (le(x19,x18)→^*false⟹s(x18)→^*x20⟹s(x19)→^*x21⟹if_gcd^#(false,x20,x21)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x20 by s(x18) which results in
      (le(x19,x18)→^*false⟹s(x19)→^*x21⟹if_gcd^#(false,s(x18),x21)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x21 by s(x19) which results in
      (le(x19,x18)→^*false⟹if_gcd^#(false,s(x18),s(x19))≥c)
    • Applying Rule "Induction" on le(x19,x18)→^*false results in the following 3 new constraints.
      1. (true→^*false⟹if_gcd^#(false,s(x56),s(0))≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*false⟹if_gcd^#(false,s(0),s(s(x57)))≥c)
         • Applying Rule "Same Constructor" results in
           if_gcd^#(false,s(0),s(s(x57)))≥c

         • This constraint is kept as final constraint.

      3. (le(x59,x58)→^*false⟹(le(x59,x58)→^*false⟹if_gcd^#(false,s(x58),s(x59))≥c)⟹if_gcd^#(false,s(s(x58)),s(s(x59)))≥c)
         • Applying Rule "Simplify Conditions" results in
           (if_gcd^#(false,s(x58),s(x59))≥c⟹if_gcd^#(false,s(s(x58)),s(s(x59)))≥c)

         • This constraint is kept as final constraint.

  • For the chain we build the initial constraint

    (gcd^#(minus(x2,x3),x3)→^*gcd^#(s(x4),s(x5))⟹gcd^#(s(x4),s(x5))≥if_gcd^#(le(x5,x4),s(x4),s(x5)))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (minus(x2,x3)→^*s(x4)⟹x3→^*s(x5)⟹gcd^#(s(x4),s(x5))≥if_gcd^#(le(x5,x4),s(x4),s(x5)))
    • Applying Rule "Variable in Equation" allows to substitute x3 by s(x5) which results in
      (minus(x2,s(x5))→^*s(x4)⟹gcd^#(s(x4),s(x5))≥if_gcd^#(le(x5,x4),s(x4),s(x5)))
    • Applying Rule "Introduce fresh variable" results in
      (s(x5)→^*x26⟹minus(x2,x26)→^*s(x4)⟹gcd^#(s(x4),s(x5))≥if_gcd^#(le(x5,x4),s(x4),s(x5)))
    • Applying Rule "Induction" on minus(x2,x26)→^*s(x4) results in the following 2 new constraints.
      1. (x27→^*s(x4)⟹s(x5)→^*0⟹gcd^#(s(x4),s(x5))≥if_gcd^#(le(x5,x4),s(x4),s(x5)))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (pred(minus(x29,x28))→^*s(x4)⟹s(x5)→^*s(x28)⟹∀x30 . ∀x31 . (minus(x29,x28)→^*s(x30)⟹s(x31)→^*x28⟹gcd^#(s(x30),s(x31))≥if_gcd^#(le(x31,x30),s(x30),s(x31)))⟹gcd^#(s(x4),s(x5))≥if_gcd^#(le(x5,x4),s(x4),s(x5)))
         • Applying Rule "Same Constructor" results in
           (pred(minus(x29,x28))→^*s(x4)⟹x5→^*x28⟹∀x30 . ∀x31 . (minus(x29,x28)→^*s(x30)⟹s(x31)→^*x28⟹gcd^#(s(x30),s(x31))≥if_gcd^#(le(x31,x30),s(x30),s(x31)))⟹gcd^#(s(x4),s(x5))≥if_gcd^#(le(x5,x4),s(x4),s(x5)))
         • Applying Rule "Variable in Equation" allows to substitute x5 by x28 which results in
           (pred(minus(x29,x28))→^*s(x4)⟹∀x30 . ∀x31 . (minus(x29,x28)→^*s(x30)⟹s(x31)→^*x28⟹gcd^#(s(x30),s(x31))≥if_gcd^#(le(x31,x30),s(x30),s(x31)))⟹gcd^#(s(x4),s(x28))≥if_gcd^#(le(x28,x4),s(x4),s(x28)))
         • Applying Rule "Introduce fresh variable" results in
           (minus(x29,x28)→^*x32⟹pred(x32)→^*s(x4)⟹∀x30 . ∀x31 . (minus(x29,x28)→^*s(x30)⟹s(x31)→^*x28⟹gcd^#(s(x30),s(x31))≥if_gcd^#(le(x31,x30),s(x30),s(x31)))⟹gcd^#(s(x4),s(x28))≥if_gcd^#(le(x28,x4),s(x4),s(x28)))
         • Applying Rule "Induction" on pred(x32)→^*s(x4) results in the following 1 new constraints.
           1. (x33→^*s(x4)⟹minus(x29,x28)→^*s(x33)⟹∀x30 . ∀x31 . (minus(x29,x28)→^*s(x30)⟹s(x31)→^*x28⟹gcd^#(s(x30),s(x31))≥if_gcd^#(le(x31,x30),s(x30),s(x31)))⟹gcd^#(s(x4),s(x28))≥if_gcd^#(le(x28,x4),s(x4),s(x28)))
              • Applying Rule "Variable in Equation" allows to substitute x33 by s(x4) which results in
                (minus(x29,x28)→^*s(s(x4))⟹∀x30 . ∀x31 . (minus(x29,x28)→^*s(x30)⟹s(x31)→^*x28⟹gcd^#(s(x30),s(x31))≥if_gcd^#(le(x31,x30),s(x30),s(x31)))⟹gcd^#(s(x4),s(x28))≥if_gcd^#(le(x28,x4),s(x4),s(x28)))
              • Applying Rule "Delete Conditions" results in
                (minus(x29,x28)→^*s(s(x4))⟹gcd^#(s(x4),s(x28))≥if_gcd^#(le(x28,x4),s(x4),s(x28)))
              • Applying Rule "Induction" on minus(x29,x28)→^*s(s(x4)) results in the following 2 new constraints.
                1. (x34→^*s(s(x4))⟹gcd^#(s(x4),s(0))≥if_gcd^#(le(0,x4),s(x4),s(0)))
                   • Applying Rule "Variable in Equation" allows to substitute x34 by s(s(x4)) which results in
                     gcd^#(s(x4),s(0))≥if_gcd^#(le(0,x4),s(x4),s(0))

                   • This constraint is kept as final constraint.

                2. (pred(minus(x36,x35))→^*s(s(x4))⟹∀x37 . (minus(x36,x35)→^*s(s(x37))⟹gcd^#(s(x37),s(x35))≥if_gcd^#(le(x35,x37),s(x37),s(x35)))⟹gcd^#(s(x4),s(s(x35)))≥if_gcd^#(le(s(x35),x4),s(x4),s(s(x35))))
                   • Applying Rule "Introduce fresh variable" results in
                     (minus(x36,x35)→^*x38⟹pred(x38)→^*s(s(x4))⟹∀x37 . (minus(x36,x35)→^*s(s(x37))⟹gcd^#(s(x37),s(x35))≥if_gcd^#(le(x35,x37),s(x37),s(x35)))⟹gcd^#(s(x4),s(s(x35)))≥if_gcd^#(le(s(x35),x4),s(x4),s(s(x35))))
                   • Applying Rule "Delete Conditions" results in
                     (pred(x38)→^*s(s(x4))⟹∀x37 . (minus(x36,x35)→^*s(s(x37))⟹gcd^#(s(x37),s(x35))≥if_gcd^#(le(x35,x37),s(x37),s(x35)))⟹gcd^#(s(x4),s(s(x35)))≥if_gcd^#(le(s(x35),x4),s(x4),s(s(x35))))
                   • Applying Rule "Delete Conditions" results in
                     (∀x37 . (minus(x36,x35)→^*s(s(x37))⟹gcd^#(s(x37),s(x35))≥if_gcd^#(le(x35,x37),s(x37),s(x35)))⟹gcd^#(s(x4),s(s(x35)))≥if_gcd^#(le(s(x35),x4),s(x4),s(s(x35))))
                   • Applying Rule "Delete Conditions" results in
                     gcd^#(s(x4),s(s(x35)))≥if_gcd^#(le(s(x35),x4),s(x4),s(s(x35)))

                   • This constraint is kept as final constraint.

  • For the chain we build the initial constraint

    (gcd^#(minus(x7,x6),x6)→^*gcd^#(s(x8),s(x9))⟹gcd^#(s(x8),s(x9))≥if_gcd^#(le(x9,x8),s(x8),s(x9)))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (minus(x7,x6)→^*s(x8)⟹x6→^*s(x9)⟹gcd^#(s(x8),s(x9))≥if_gcd^#(le(x9,x8),s(x8),s(x9)))
    • Applying Rule "Variable in Equation" allows to substitute x6 by s(x9) which results in
      (minus(x7,s(x9))→^*s(x8)⟹gcd^#(s(x8),s(x9))≥if_gcd^#(le(x9,x8),s(x8),s(x9)))
    • Applying Rule "Introduce fresh variable" results in
      (s(x9)→^*x39⟹minus(x7,x39)→^*s(x8)⟹gcd^#(s(x8),s(x9))≥if_gcd^#(le(x9,x8),s(x8),s(x9)))
    • Applying Rule "Induction" on minus(x7,x39)→^*s(x8) results in the following 2 new constraints.
      1. (x40→^*s(x8)⟹s(x9)→^*0⟹gcd^#(s(x8),s(x9))≥if_gcd^#(le(x9,x8),s(x8),s(x9)))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (pred(minus(x42,x41))→^*s(x8)⟹s(x9)→^*s(x41)⟹∀x43 . ∀x44 . (minus(x42,x41)→^*s(x43)⟹s(x44)→^*x41⟹gcd^#(s(x43),s(x44))≥if_gcd^#(le(x44,x43),s(x43),s(x44)))⟹gcd^#(s(x8),s(x9))≥if_gcd^#(le(x9,x8),s(x8),s(x9)))
         • Applying Rule "Same Constructor" results in
           (pred(minus(x42,x41))→^*s(x8)⟹x9→^*x41⟹∀x43 . ∀x44 . (minus(x42,x41)→^*s(x43)⟹s(x44)→^*x41⟹gcd^#(s(x43),s(x44))≥if_gcd^#(le(x44,x43),s(x43),s(x44)))⟹gcd^#(s(x8),s(x9))≥if_gcd^#(le(x9,x8),s(x8),s(x9)))
         • Applying Rule "Variable in Equation" allows to substitute x9 by x41 which results in
           (pred(minus(x42,x41))→^*s(x8)⟹∀x43 . ∀x44 . (minus(x42,x41)→^*s(x43)⟹s(x44)→^*x41⟹gcd^#(s(x43),s(x44))≥if_gcd^#(le(x44,x43),s(x43),s(x44)))⟹gcd^#(s(x8),s(x41))≥if_gcd^#(le(x41,x8),s(x8),s(x41)))
         • Applying Rule "Introduce fresh variable" results in
           (minus(x42,x41)→^*x45⟹pred(x45)→^*s(x8)⟹∀x43 . ∀x44 . (minus(x42,x41)→^*s(x43)⟹s(x44)→^*x41⟹gcd^#(s(x43),s(x44))≥if_gcd^#(le(x44,x43),s(x43),s(x44)))⟹gcd^#(s(x8),s(x41))≥if_gcd^#(le(x41,x8),s(x8),s(x41)))
         • Applying Rule "Induction" on pred(x45)→^*s(x8) results in the following 1 new constraints.
           1. (x46→^*s(x8)⟹minus(x42,x41)→^*s(x46)⟹∀x43 . ∀x44 . (minus(x42,x41)→^*s(x43)⟹s(x44)→^*x41⟹gcd^#(s(x43),s(x44))≥if_gcd^#(le(x44,x43),s(x43),s(x44)))⟹gcd^#(s(x8),s(x41))≥if_gcd^#(le(x41,x8),s(x8),s(x41)))
              • Applying Rule "Variable in Equation" allows to substitute x46 by s(x8) which results in
                (minus(x42,x41)→^*s(s(x8))⟹∀x43 . ∀x44 . (minus(x42,x41)→^*s(x43)⟹s(x44)→^*x41⟹gcd^#(s(x43),s(x44))≥if_gcd^#(le(x44,x43),s(x43),s(x44)))⟹gcd^#(s(x8),s(x41))≥if_gcd^#(le(x41,x8),s(x8),s(x41)))
              • Applying Rule "Delete Conditions" results in
                (minus(x42,x41)→^*s(s(x8))⟹gcd^#(s(x8),s(x41))≥if_gcd^#(le(x41,x8),s(x8),s(x41)))
              • Applying Rule "Induction" on minus(x42,x41)→^*s(s(x8)) results in the following 2 new constraints.
                1. (x47→^*s(s(x8))⟹gcd^#(s(x8),s(0))≥if_gcd^#(le(0,x8),s(x8),s(0)))
                   • Applying Rule "Variable in Equation" allows to substitute x47 by s(s(x8)) which results in
                     gcd^#(s(x8),s(0))≥if_gcd^#(le(0,x8),s(x8),s(0))

                   • This constraint is kept as final constraint.

                2. (pred(minus(x49,x48))→^*s(s(x8))⟹∀x50 . (minus(x49,x48)→^*s(s(x50))⟹gcd^#(s(x50),s(x48))≥if_gcd^#(le(x48,x50),s(x50),s(x48)))⟹gcd^#(s(x8),s(s(x48)))≥if_gcd^#(le(s(x48),x8),s(x8),s(s(x48))))
                   • Applying Rule "Introduce fresh variable" results in
                     (minus(x49,x48)→^*x51⟹pred(x51)→^*s(s(x8))⟹∀x50 . (minus(x49,x48)→^*s(s(x50))⟹gcd^#(s(x50),s(x48))≥if_gcd^#(le(x48,x50),s(x50),s(x48)))⟹gcd^#(s(x8),s(s(x48)))≥if_gcd^#(le(s(x48),x8),s(x8),s(s(x48))))
                   • Applying Rule "Delete Conditions" results in
                     (pred(x51)→^*s(s(x8))⟹∀x50 . (minus(x49,x48)→^*s(s(x50))⟹gcd^#(s(x50),s(x48))≥if_gcd^#(le(x48,x50),s(x50),s(x48)))⟹gcd^#(s(x8),s(s(x48)))≥if_gcd^#(le(s(x48),x8),s(x8),s(s(x48))))
                   • Applying Rule "Delete Conditions" results in
                     (∀x50 . (minus(x49,x48)→^*s(s(x50))⟹gcd^#(s(x50),s(x48))≥if_gcd^#(le(x48,x50),s(x50),s(x48)))⟹gcd^#(s(x8),s(s(x48)))≥if_gcd^#(le(s(x48),x8),s(x8),s(s(x48))))
                   • Applying Rule "Delete Conditions" results in
                     gcd^#(s(x8),s(s(x48)))≥if_gcd^#(le(s(x48),x8),s(x8),s(s(x48)))

                   • This constraint is kept as final constraint.

  • For the chain we build the initial constraint

    (if_gcd^#(le(x11,x10),s(x10),s(x11))→^*if_gcd^#(true,x12,x13)⟹if_gcd^#(true,x12,x13)≥gcd^#(minus(x12,x13),x13))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (le(x11,x10)→^*true⟹s(x10)→^*x12⟹s(x11)→^*x13⟹if_gcd^#(true,x12,x13)≥gcd^#(minus(x12,x13),x13))
    • Applying Rule "Variable in Equation" allows to substitute x12 by s(x10) which results in
      (le(x11,x10)→^*true⟹s(x11)→^*x13⟹if_gcd^#(true,s(x10),x13)≥gcd^#(minus(s(x10),x13),x13))
    • Applying Rule "Variable in Equation" allows to substitute x13 by s(x11) which results in
      (le(x11,x10)→^*true⟹if_gcd^#(true,s(x10),s(x11))≥gcd^#(minus(s(x10),s(x11)),s(x11)))
    • Applying Rule "Induction" on le(x11,x10)→^*true results in the following 3 new constraints.
      1. (true→^*true⟹if_gcd^#(true,s(x52),s(0))≥gcd^#(minus(s(x52),s(0)),s(0)))
         • Applying Rule "Same Constructor" results in
           if_gcd^#(true,s(x52),s(0))≥gcd^#(minus(s(x52),s(0)),s(0))

         • This constraint is kept as final constraint.

      2. (false→^*true⟹if_gcd^#(true,s(0),s(s(x53)))≥gcd^#(minus(s(0),s(s(x53))),s(s(x53))))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (le(x55,x54)→^*true⟹(le(x55,x54)→^*true⟹if_gcd^#(true,s(x54),s(x55))≥gcd^#(minus(s(x54),s(x55)),s(x55)))⟹if_gcd^#(true,s(s(x54)),s(s(x55)))≥gcd^#(minus(s(s(x54)),s(s(x55))),s(s(x55))))
         • Applying Rule "Simplify Conditions" results in
           (if_gcd^#(true,s(x54),s(x55))≥gcd^#(minus(s(x54),s(x55)),s(x55))⟹if_gcd^#(true,s(s(x54)),s(s(x55)))≥gcd^#(minus(s(s(x54)),s(s(x55))),s(s(x55))))

         • This constraint is kept as final constraint.

  • For the chain we build the initial constraint

    (if_gcd^#(le(x19,x18),s(x18),s(x19))→^*if_gcd^#(false,x20,x21)⟹if_gcd^#(false,x20,x21) > gcd^#(minus(x21,x20),x20))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (le(x19,x18)→^*false⟹s(x18)→^*x20⟹s(x19)→^*x21⟹if_gcd^#(false,x20,x21) > gcd^#(minus(x21,x20),x20))
    • Applying Rule "Variable in Equation" allows to substitute x20 by s(x18) which results in
      (le(x19,x18)→^*false⟹s(x19)→^*x21⟹if_gcd^#(false,s(x18),x21) > gcd^#(minus(x21,s(x18)),s(x18)))
    • Applying Rule "Variable in Equation" allows to substitute x21 by s(x19) which results in
      (le(x19,x18)→^*false⟹if_gcd^#(false,s(x18),s(x19)) > gcd^#(minus(s(x19),s(x18)),s(x18)))
    • Applying Rule "Induction" on le(x19,x18)→^*false results in the following 3 new constraints.
      1. (true→^*false⟹if_gcd^#(false,s(x56),s(0)) > gcd^#(minus(s(0),s(x56)),s(x56)))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*false⟹if_gcd^#(false,s(0),s(s(x57))) > gcd^#(minus(s(s(x57)),s(0)),s(0)))
         • Applying Rule "Same Constructor" results in
           if_gcd^#(false,s(0),s(s(x57))) > gcd^#(minus(s(s(x57)),s(0)),s(0))

         • This constraint is kept as final constraint.

      3. (le(x59,x58)→^*false⟹(le(x59,x58)→^*false⟹if_gcd^#(false,s(x58),s(x59)) > gcd^#(minus(s(x59),s(x58)),s(x58)))⟹if_gcd^#(false,s(s(x58)),s(s(x59))) > gcd^#(minus(s(s(x59)),s(s(x58))),s(s(x58))))
         • Applying Rule "Simplify Conditions" results in
           (if_gcd^#(false,s(x58),s(x59)) > gcd^#(minus(s(x59),s(x58)),s(x58))⟹if_gcd^#(false,s(s(x58)),s(s(x59))) > gcd^#(minus(s(s(x59)),s(s(x58))),s(s(x58))))

         • This constraint is kept as final constraint.

  We remove those pairs which are strictly decreasing and bounded.

  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

  gcd#(s(x0),s(0))	→	if_gcd#(true,s(x0),s(0))	(45)
  gcd#(s(0),s(s(x0)))	→	if_gcd#(false,s(0),s(s(x0)))	(46)
  gcd#(s(s(x1)),s(s(x0)))	→	if_gcd#(le(x0,x1),s(s(x1)),s(s(x0)))	(47)

  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

    gcd#(s(x0),s(0))	→	if_gcd#(true,s(x0),s(0))	(45)
    if_gcd#(true,x,y)	→	gcd#(minus(x,y),y)	(17)
    gcd#(s(s(x1)),s(s(x0)))	→	if_gcd#(le(x0,x1),s(s(x1)),s(s(x0)))	(47)

    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

    if_gcd#(true,x0,0)	→	gcd#(x0,0)	(48)
    if_gcd#(true,x0,s(x1))	→	gcd#(pred(minus(x0,x1)),s(x1))	(49)

    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

      if_gcd#(true,x0,s(x1))	→	gcd#(pred(minus(x0,x1)),s(x1))	(49)
      gcd#(s(x0),s(0))	→	if_gcd#(true,s(x0),s(0))	(45)
      gcd#(s(s(x1)),s(s(x0)))	→	if_gcd#(le(x0,x1),s(s(x1)),s(s(x0)))	(47)

      1.1.1.1.1.1.1.1.1.1.1 Instantiation Processor

      We instantiate the pair to the following set of pairs

      if_gcd#(true,s(z0),s(0))	→	gcd#(pred(minus(s(z0),0)),s(0))	(50)
      if_gcd#(true,s(s(z0)),s(s(z1)))	→	gcd#(pred(minus(s(s(z0)),s(z1))),s(s(z1)))	(51)

      1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

      The dependency pairs are split into 2 components.

      • The 1^st component contains the pair

        if_gcd#(true,s(z0),s(0))	→	gcd#(pred(minus(s(z0),0)),s(0))	(50)
        gcd#(s(x0),s(0))	→	if_gcd#(true,s(x0),s(0))	(45)

        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.

        minus(x,0)	→	x	(5)
        pred(s(x))	→	x	(4)

        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.

        pred(s(x0))

        minus(x0,0)

        minus(x0,s(x1))

        1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rewriting Processor

        We rewrite the right hand side of the pair resulting in

        →

        1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Usable Rules Processor

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

        pred(s(x))

        →

        x

        (4)

        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.

        pred(s(x0))

        1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rewriting Processor

        We rewrite the right hand side of the pair resulting in

        →

        1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 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 Forward Instantiation Processor

        We instantiate the pair to the following set of pairs

        if_gcd#(true,s(s(y_0)),s(0))

        →

        gcd#(s(y_0),s(0))

        (54)

        1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Forward Instantiation Processor

        We instantiate the pair to the following set of pairs

        gcd#(s(s(y_0)),s(0))

        →

        if_gcd#(true,s(s(y_0)),s(0))

        (55)

        1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Size-Change Termination

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

        gcd#(s(s(y_0)),s(0))	→	if_gcd#(true,s(s(y_0)),s(0))	(55)
        1	≥	2
        2	≥	3
        if_gcd#(true,s(s(y_0)),s(0))	→	gcd#(s(y_0),s(0))	(54)
        2	>	1
        3	≥	2

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

      • The 2^nd component contains the pair

        if_gcd#(true,s(s(z0)),s(s(z1)))	→	gcd#(pred(minus(s(s(z0)),s(z1))),s(s(z1)))	(51)
        gcd#(s(s(x1)),s(s(x0)))	→	if_gcd#(le(x0,x1),s(s(x1)),s(s(x0)))	(47)

        1.1.1.1.1.1.1.1.1.1.1.1.2 Rewriting Processor

        We rewrite the right hand side of the pair resulting in

        →

        1.1.1.1.1.1.1.1.1.1.1.1.2.1 Reduction Pair Processor with Usable Rules

        Using the linear polynomial interpretation over the naturals

        [if_gcd#(x1, x2, x3)]	=	-1 + x2
        [le(x1, x2)]	=	2 · x2
        [0]	=	0
        [true]	=	2
        [s(x1)]	=	1 + x1
        [false]	=	0
        [gcd#(x1, x2)]	=	1 + x1
        [pred(x1)]	=	-1 + x1
        [minus(x1, x2)]	=	x1

        together with the usable rules

        minus(x,0)	→	x	(5)
        minus(x,s(y))	→	pred(minus(x,y))	(6)
        pred(s(x))	→	x	(4)

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

        gcd#(s(s(x1)),s(s(x0)))

        →

        if_gcd#(le(x0,x1),s(s(x1)),s(s(x0)))

        (47)

        could be deleted.

        1.1.1.1.1.1.1.1.1.1.1.1.2.1.1 Usable Rules Processor

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

        minus(x,0)	→	x	(5)
        minus(x,s(y))	→	pred(minus(x,y))	(6)
        pred(s(x))	→	x	(4)

        1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1 Innermost Lhss Removal Processor

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

        pred(s(x0))

        minus(x0,0)

        minus(x0,s(x1))

        1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1.1 Size-Change Termination

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

        if_gcd#(true,s(s(z0)),s(s(z1)))	→	gcd#(pred(pred(minus(s(s(z0)),z1))),s(s(z1)))	(56)
        3	≥	2

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

• The 2^nd component contains the pair

  minus#(x,s(y))

  →

  minus#(x,y)

  (14)

  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.

  minus#(x,s(y))	→	minus#(x,y)	(14)
  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

  le#(s(x),s(y))

  →

  le#(x,y)

  (12)

  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.

  le#(s(x),s(y))	→	le#(x,y)	(12)
  1	>	1
  2	>	2

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