Certification Problem

Input (TPDB TRS_Standard/AProVE_07/otto13)

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 4 components.

• The 1^st component contains the pair

  if#(second,x,y,z)	→	if#(le(s(x),s(y),z),s(x),s(y),z)	(23)
  if#(first,x,y,z)	→	if#(le(s(x),y,s(z)),s(x),y,s(z))	(21)

  1.1.1.1 Usable Rules Processor

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

  le(s(x),0,z)	→	false	(2)
  le(s(x),s(y),s(z))	→	le(x,y,z)	(4)
  le(0,y,z)	→	greater(y,z)	(1)
  le(s(x),s(y),0)	→	false	(3)
  greater(x,0)	→	first	(5)
  greater(0,s(y))	→	second	(6)
  greater(s(x),s(y))	→	greater(x,y)	(7)

  1.1.1.1.1 Innermost Lhss Removal Processor

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

  le(0,x0,x1)

  le(s(x0),0,x1)

  le(s(x0),s(x1),0)

  le(s(x0),s(x1),s(x2))

  greater(x0,0)

  greater(0,s(x0))

  greater(s(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#(x1,...,x4)]	=	-1 · x1 + -1 · x2 + 1 · x3 + -1
  [second]	=	0
  [le(x1, x2, x3)]	=	0
  [s(x1)]	=	1 · x1 + 1
  [first]	=	0
  [0]	=	0
  [false]	=	1
  [greater(x1, x2)]	=	0

  The pair(s)

  if#(first,x,y,z)

  →

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

  (21)

  are strictly oriented and the pair(s)

  if#(second,x,y,z)	→	if#(le(s(x),s(y),z),s(x),s(y),z)	(23)
  if#(first,x,y,z)	→	if#(le(s(x),y,s(z)),s(x),y,s(z))	(21)

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

  The following constraints are generated for the pairs.

  • (if^#(second,s(x41),s(x40),s(x39))≥c⟹if^#(second,s(s(x41)),s(s(x40)),s(s(x39)))≥c)
  • if^#(second,s(0),s(x43),s(x42))≥c
  • (if^#(second,s(x63),s(x62),s(x61))≥c⟹if^#(second,s(s(x63)),s(s(x62)),s(s(x61)))≥c)
  • if^#(second,s(0),s(x65),s(x64))≥c
  • (if^#(first,s(x85),s(x84),s(x83))≥c⟹if^#(first,s(s(x85)),s(s(x84)),s(s(x83)))≥c)
  • if^#(first,s(0),s(x87),s(x86))≥c
  • (if^#(first,s(x107),s(x106),s(x105))≥c⟹if^#(first,s(s(x107)),s(s(x106)),s(s(x105)))≥c)
  • if^#(first,s(0),s(x109),s(x108))≥c
  • (if^#(second,s(x41),s(x40),s(x39))≥if^#(le(s(s(x41)),s(s(x40)),s(x39)),s(s(x41)),s(s(x40)),s(x39))⟹if^#(second,s(s(x41)),s(s(x40)),s(s(x39)))≥if^#(le(s(s(s(x41))),s(s(s(x40))),s(s(x39))),s(s(s(x41))),s(s(s(x40))),s(s(x39))))
  • if^#(second,s(0),s(x43),s(x42))≥if^#(le(s(s(0)),s(s(x43)),s(x42)),s(s(0)),s(s(x43)),s(x42))
  • (if^#(second,s(x63),s(x62),s(x61))≥if^#(le(s(s(x63)),s(s(x62)),s(x61)),s(s(x63)),s(s(x62)),s(x61))⟹if^#(second,s(s(x63)),s(s(x62)),s(s(x61)))≥if^#(le(s(s(s(x63))),s(s(s(x62))),s(s(x61))),s(s(s(x63))),s(s(s(x62))),s(s(x61))))
  • if^#(second,s(0),s(x65),s(x64))≥if^#(le(s(s(0)),s(s(x65)),s(x64)),s(s(0)),s(s(x65)),s(x64))
  • (if^#(first,s(x85),s(x84),s(x83)) > if^#(le(s(s(x85)),s(x84),s(s(x83))),s(s(x85)),s(x84),s(s(x83)))⟹if^#(first,s(s(x85)),s(s(x84)),s(s(x83))) > if^#(le(s(s(s(x85))),s(s(x84)),s(s(s(x83)))),s(s(s(x85))),s(s(x84)),s(s(s(x83)))))
  • if^#(first,s(0),s(x87),s(x86)) > if^#(le(s(s(0)),s(x87),s(s(x86))),s(s(0)),s(x87),s(s(x86)))
  • (if^#(first,s(x107),s(x106),s(x105)) > if^#(le(s(s(x107)),s(x106),s(s(x105))),s(s(x107)),s(x106),s(s(x105)))⟹if^#(first,s(s(x107)),s(s(x106)),s(s(x105))) > if^#(le(s(s(s(x107))),s(s(x106)),s(s(s(x105)))),s(s(s(x107))),s(s(x106)),s(s(s(x105)))))
  • if^#(first,s(0),s(x109),s(x108)) > if^#(le(s(s(0)),s(x109),s(s(x108))),s(s(0)),s(x109),s(s(x108)))

  The details are shown below:

  • For the chain we build the initial constraint

    (if^#(le(s(x0),s(x1),x2),s(x0),s(x1),x2)→^*if^#(second,x3,x4,x5)⟹if^#(second,x3,x4,x5)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (le(s(x0),s(x1),x2)→^*second⟹s(x0)→^*x3⟹s(x1)→^*x4⟹x2→^*x5⟹if^#(second,x3,x4,x5)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x3 by s(x0) which results in
      (le(s(x0),s(x1),x2)→^*second⟹s(x1)→^*x4⟹x2→^*x5⟹if^#(second,s(x0),x4,x5)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x4 by s(x1) which results in
      (le(s(x0),s(x1),x2)→^*second⟹x2→^*x5⟹if^#(second,s(x0),s(x1),x5)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x5 by x2 which results in
      (le(s(x0),s(x1),x2)→^*second⟹if^#(second,s(x0),s(x1),x2)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (s(x1)→^*x25⟹le(s(x0),x25,x2)→^*second⟹if^#(second,s(x0),s(x1),x2)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (s(x0)→^*x24⟹s(x1)→^*x25⟹le(x24,x25,x2)→^*second⟹if^#(second,s(x0),s(x1),x2)≥c)
    • Applying Rule "Induction" on le(x24,x25,x2)→^*second results in the following 4 new constraints.
      1. (false→^*second⟹if^#(second,s(x0),s(x1),x26)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (le(x30,x29,x28)→^*second⟹s(x0)→^*s(x30)⟹s(x1)→^*s(x29)⟹∀x31 . ∀x32 . (le(x30,x29,x28)→^*second⟹s(x31)→^*x30⟹s(x32)→^*x29⟹if^#(second,s(x31),s(x32),x28)≥c)⟹if^#(second,s(x0),s(x1),s(x28))≥c)
         • Applying Rule "Same Constructor" results in
           (le(x30,x29,x28)→^*second⟹x0→^*x30⟹s(x1)→^*s(x29)⟹∀x31 . ∀x32 . (le(x30,x29,x28)→^*second⟹s(x31)→^*x30⟹s(x32)→^*x29⟹if^#(second,s(x31),s(x32),x28)≥c)⟹if^#(second,s(x0),s(x1),s(x28))≥c)
         • Applying Rule "Same Constructor" results in
           (le(x30,x29,x28)→^*second⟹x0→^*x30⟹x1→^*x29⟹∀x31 . ∀x32 . (le(x30,x29,x28)→^*second⟹s(x31)→^*x30⟹s(x32)→^*x29⟹if^#(second,s(x31),s(x32),x28)≥c)⟹if^#(second,s(x0),s(x1),s(x28))≥c)
         • Applying Rule "Variable in Equation" allows to substitute x0 by x30 which results in
           (le(x30,x29,x28)→^*second⟹x1→^*x29⟹∀x31 . ∀x32 . (le(x30,x29,x28)→^*second⟹s(x31)→^*x30⟹s(x32)→^*x29⟹if^#(second,s(x31),s(x32),x28)≥c)⟹if^#(second,s(x30),s(x1),s(x28))≥c)
         • Applying Rule "Variable in Equation" allows to substitute x1 by x29 which results in
           (le(x30,x29,x28)→^*second⟹∀x31 . ∀x32 . (le(x30,x29,x28)→^*second⟹s(x31)→^*x30⟹s(x32)→^*x29⟹if^#(second,s(x31),s(x32),x28)≥c)⟹if^#(second,s(x30),s(x29),s(x28))≥c)
         • Applying Rule "Delete Conditions" results in
           (le(x30,x29,x28)→^*second⟹if^#(second,s(x30),s(x29),s(x28))≥c)
         • Applying Rule "Induction" on le(x30,x29,x28)→^*second results in the following 4 new constraints.
           1. (false→^*second⟹if^#(second,s(s(x38)),s(0),s(x37))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (le(x41,x40,x39)→^*second⟹(le(x41,x40,x39)→^*second⟹if^#(second,s(x41),s(x40),s(x39))≥c)⟹if^#(second,s(s(x41)),s(s(x40)),s(s(x39)))≥c)
              • Applying Rule "Simplify Conditions" results in
                (if^#(second,s(x41),s(x40),s(x39))≥c⟹if^#(second,s(s(x41)),s(s(x40)),s(s(x39)))≥c)

              • This constraint is kept as final constraint.

           3. (greater(x43,x42)→^*second⟹if^#(second,s(0),s(x43),s(x42))≥c)
              • Applying Rule "Delete Conditions" results in
                if^#(second,s(0),s(x43),s(x42))≥c

              • This constraint is kept as final constraint.

           4. (false→^*second⟹if^#(second,s(s(x45)),s(s(x44)),s(0))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (greater(x34,x33)→^*second⟹s(x0)→^*0⟹s(x1)→^*x34⟹if^#(second,s(x0),s(x1),x33)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      4. (false→^*second⟹if^#(second,s(x0),s(x1),0)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (if^#(le(s(x6),x7,s(x8)),s(x6),x7,s(x8))→^*if^#(second,x9,x10,x11)⟹if^#(second,x9,x10,x11)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (le(s(x6),x7,s(x8))→^*second⟹s(x6)→^*x9⟹x7→^*x10⟹s(x8)→^*x11⟹if^#(second,x9,x10,x11)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x9 by s(x6) which results in
      (le(s(x6),x7,s(x8))→^*second⟹x7→^*x10⟹s(x8)→^*x11⟹if^#(second,s(x6),x10,x11)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x10 by x7 which results in
      (le(s(x6),x7,s(x8))→^*second⟹s(x8)→^*x11⟹if^#(second,s(x6),x7,x11)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x11 by s(x8) which results in
      (le(s(x6),x7,s(x8))→^*second⟹if^#(second,s(x6),x7,s(x8))≥c)
    • Applying Rule "Introduce fresh variable" results in
      (s(x8)→^*x47⟹le(s(x6),x7,x47)→^*second⟹if^#(second,s(x6),x7,s(x8))≥c)
    • Applying Rule "Introduce fresh variable" results in
      (s(x6)→^*x46⟹s(x8)→^*x47⟹le(x46,x7,x47)→^*second⟹if^#(second,s(x6),x7,s(x8))≥c)
    • Applying Rule "Induction" on le(x46,x7,x47)→^*second results in the following 4 new constraints.
      1. (false→^*second⟹if^#(second,s(x6),0,s(x8))≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (le(x52,x51,x50)→^*second⟹s(x6)→^*s(x52)⟹s(x8)→^*s(x50)⟹∀x53 . ∀x54 . (le(x52,x51,x50)→^*second⟹s(x53)→^*x52⟹s(x54)→^*x50⟹if^#(second,s(x53),x51,s(x54))≥c)⟹if^#(second,s(x6),s(x51),s(x8))≥c)
         • Applying Rule "Same Constructor" results in
           (le(x52,x51,x50)→^*second⟹x6→^*x52⟹s(x8)→^*s(x50)⟹∀x53 . ∀x54 . (le(x52,x51,x50)→^*second⟹s(x53)→^*x52⟹s(x54)→^*x50⟹if^#(second,s(x53),x51,s(x54))≥c)⟹if^#(second,s(x6),s(x51),s(x8))≥c)
         • Applying Rule "Same Constructor" results in
           (le(x52,x51,x50)→^*second⟹x6→^*x52⟹x8→^*x50⟹∀x53 . ∀x54 . (le(x52,x51,x50)→^*second⟹s(x53)→^*x52⟹s(x54)→^*x50⟹if^#(second,s(x53),x51,s(x54))≥c)⟹if^#(second,s(x6),s(x51),s(x8))≥c)
         • Applying Rule "Variable in Equation" allows to substitute x6 by x52 which results in
           (le(x52,x51,x50)→^*second⟹x8→^*x50⟹∀x53 . ∀x54 . (le(x52,x51,x50)→^*second⟹s(x53)→^*x52⟹s(x54)→^*x50⟹if^#(second,s(x53),x51,s(x54))≥c)⟹if^#(second,s(x52),s(x51),s(x8))≥c)
         • Applying Rule "Variable in Equation" allows to substitute x8 by x50 which results in
           (le(x52,x51,x50)→^*second⟹∀x53 . ∀x54 . (le(x52,x51,x50)→^*second⟹s(x53)→^*x52⟹s(x54)→^*x50⟹if^#(second,s(x53),x51,s(x54))≥c)⟹if^#(second,s(x52),s(x51),s(x50))≥c)
         • Applying Rule "Delete Conditions" results in
           (le(x52,x51,x50)→^*second⟹if^#(second,s(x52),s(x51),s(x50))≥c)
         • Applying Rule "Induction" on le(x52,x51,x50)→^*second results in the following 4 new constraints.
           1. (false→^*second⟹if^#(second,s(s(x60)),s(0),s(x59))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (le(x63,x62,x61)→^*second⟹(le(x63,x62,x61)→^*second⟹if^#(second,s(x63),s(x62),s(x61))≥c)⟹if^#(second,s(s(x63)),s(s(x62)),s(s(x61)))≥c)
              • Applying Rule "Simplify Conditions" results in
                (if^#(second,s(x63),s(x62),s(x61))≥c⟹if^#(second,s(s(x63)),s(s(x62)),s(s(x61)))≥c)

              • This constraint is kept as final constraint.

           3. (greater(x65,x64)→^*second⟹if^#(second,s(0),s(x65),s(x64))≥c)
              • Applying Rule "Delete Conditions" results in
                if^#(second,s(0),s(x65),s(x64))≥c

              • This constraint is kept as final constraint.

           4. (false→^*second⟹if^#(second,s(s(x67)),s(s(x66)),s(0))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (greater(x56,x55)→^*second⟹s(x6)→^*0⟹s(x8)→^*x55⟹if^#(second,s(x6),x56,s(x8))≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      4. (false→^*second⟹if^#(second,s(x6),s(x57),s(x8))≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (if^#(le(s(x12),s(x13),x14),s(x12),s(x13),x14)→^*if^#(first,x15,x16,x17)⟹if^#(first,x15,x16,x17)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (le(s(x12),s(x13),x14)→^*first⟹s(x12)→^*x15⟹s(x13)→^*x16⟹x14→^*x17⟹if^#(first,x15,x16,x17)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x15 by s(x12) which results in
      (le(s(x12),s(x13),x14)→^*first⟹s(x13)→^*x16⟹x14→^*x17⟹if^#(first,s(x12),x16,x17)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x16 by s(x13) which results in
      (le(s(x12),s(x13),x14)→^*first⟹x14→^*x17⟹if^#(first,s(x12),s(x13),x17)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x17 by x14 which results in
      (le(s(x12),s(x13),x14)→^*first⟹if^#(first,s(x12),s(x13),x14)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (s(x13)→^*x69⟹le(s(x12),x69,x14)→^*first⟹if^#(first,s(x12),s(x13),x14)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (s(x12)→^*x68⟹s(x13)→^*x69⟹le(x68,x69,x14)→^*first⟹if^#(first,s(x12),s(x13),x14)≥c)
    • Applying Rule "Induction" on le(x68,x69,x14)→^*first results in the following 4 new constraints.
      1. (false→^*first⟹if^#(first,s(x12),s(x13),x70)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (le(x74,x73,x72)→^*first⟹s(x12)→^*s(x74)⟹s(x13)→^*s(x73)⟹∀x75 . ∀x76 . (le(x74,x73,x72)→^*first⟹s(x75)→^*x74⟹s(x76)→^*x73⟹if^#(first,s(x75),s(x76),x72)≥c)⟹if^#(first,s(x12),s(x13),s(x72))≥c)
         • Applying Rule "Same Constructor" results in
           (le(x74,x73,x72)→^*first⟹x12→^*x74⟹s(x13)→^*s(x73)⟹∀x75 . ∀x76 . (le(x74,x73,x72)→^*first⟹s(x75)→^*x74⟹s(x76)→^*x73⟹if^#(first,s(x75),s(x76),x72)≥c)⟹if^#(first,s(x12),s(x13),s(x72))≥c)
         • Applying Rule "Same Constructor" results in
           (le(x74,x73,x72)→^*first⟹x12→^*x74⟹x13→^*x73⟹∀x75 . ∀x76 . (le(x74,x73,x72)→^*first⟹s(x75)→^*x74⟹s(x76)→^*x73⟹if^#(first,s(x75),s(x76),x72)≥c)⟹if^#(first,s(x12),s(x13),s(x72))≥c)
         • Applying Rule "Variable in Equation" allows to substitute x12 by x74 which results in
           (le(x74,x73,x72)→^*first⟹x13→^*x73⟹∀x75 . ∀x76 . (le(x74,x73,x72)→^*first⟹s(x75)→^*x74⟹s(x76)→^*x73⟹if^#(first,s(x75),s(x76),x72)≥c)⟹if^#(first,s(x74),s(x13),s(x72))≥c)
         • Applying Rule "Variable in Equation" allows to substitute x13 by x73 which results in
           (le(x74,x73,x72)→^*first⟹∀x75 . ∀x76 . (le(x74,x73,x72)→^*first⟹s(x75)→^*x74⟹s(x76)→^*x73⟹if^#(first,s(x75),s(x76),x72)≥c)⟹if^#(first,s(x74),s(x73),s(x72))≥c)
         • Applying Rule "Delete Conditions" results in
           (le(x74,x73,x72)→^*first⟹if^#(first,s(x74),s(x73),s(x72))≥c)
         • Applying Rule "Induction" on le(x74,x73,x72)→^*first results in the following 4 new constraints.
           1. (false→^*first⟹if^#(first,s(s(x82)),s(0),s(x81))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (le(x85,x84,x83)→^*first⟹(le(x85,x84,x83)→^*first⟹if^#(first,s(x85),s(x84),s(x83))≥c)⟹if^#(first,s(s(x85)),s(s(x84)),s(s(x83)))≥c)
              • Applying Rule "Simplify Conditions" results in
                (if^#(first,s(x85),s(x84),s(x83))≥c⟹if^#(first,s(s(x85)),s(s(x84)),s(s(x83)))≥c)

              • This constraint is kept as final constraint.

           3. (greater(x87,x86)→^*first⟹if^#(first,s(0),s(x87),s(x86))≥c)
              • Applying Rule "Delete Conditions" results in
                if^#(first,s(0),s(x87),s(x86))≥c

              • This constraint is kept as final constraint.

           4. (false→^*first⟹if^#(first,s(s(x89)),s(s(x88)),s(0))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (greater(x78,x77)→^*first⟹s(x12)→^*0⟹s(x13)→^*x78⟹if^#(first,s(x12),s(x13),x77)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      4. (false→^*first⟹if^#(first,s(x12),s(x13),0)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (if^#(le(s(x18),x19,s(x20)),s(x18),x19,s(x20))→^*if^#(first,x21,x22,x23)⟹if^#(first,x21,x22,x23)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (le(s(x18),x19,s(x20))→^*first⟹s(x18)→^*x21⟹x19→^*x22⟹s(x20)→^*x23⟹if^#(first,x21,x22,x23)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x21 by s(x18) which results in
      (le(s(x18),x19,s(x20))→^*first⟹x19→^*x22⟹s(x20)→^*x23⟹if^#(first,s(x18),x22,x23)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x22 by x19 which results in
      (le(s(x18),x19,s(x20))→^*first⟹s(x20)→^*x23⟹if^#(first,s(x18),x19,x23)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x23 by s(x20) which results in
      (le(s(x18),x19,s(x20))→^*first⟹if^#(first,s(x18),x19,s(x20))≥c)
    • Applying Rule "Introduce fresh variable" results in
      (s(x20)→^*x91⟹le(s(x18),x19,x91)→^*first⟹if^#(first,s(x18),x19,s(x20))≥c)
    • Applying Rule "Introduce fresh variable" results in
      (s(x18)→^*x90⟹s(x20)→^*x91⟹le(x90,x19,x91)→^*first⟹if^#(first,s(x18),x19,s(x20))≥c)
    • Applying Rule "Induction" on le(x90,x19,x91)→^*first results in the following 4 new constraints.
      1. (false→^*first⟹if^#(first,s(x18),0,s(x20))≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (le(x96,x95,x94)→^*first⟹s(x18)→^*s(x96)⟹s(x20)→^*s(x94)⟹∀x97 . ∀x98 . (le(x96,x95,x94)→^*first⟹s(x97)→^*x96⟹s(x98)→^*x94⟹if^#(first,s(x97),x95,s(x98))≥c)⟹if^#(first,s(x18),s(x95),s(x20))≥c)
         • Applying Rule "Same Constructor" results in
           (le(x96,x95,x94)→^*first⟹x18→^*x96⟹s(x20)→^*s(x94)⟹∀x97 . ∀x98 . (le(x96,x95,x94)→^*first⟹s(x97)→^*x96⟹s(x98)→^*x94⟹if^#(first,s(x97),x95,s(x98))≥c)⟹if^#(first,s(x18),s(x95),s(x20))≥c)
         • Applying Rule "Same Constructor" results in
           (le(x96,x95,x94)→^*first⟹x18→^*x96⟹x20→^*x94⟹∀x97 . ∀x98 . (le(x96,x95,x94)→^*first⟹s(x97)→^*x96⟹s(x98)→^*x94⟹if^#(first,s(x97),x95,s(x98))≥c)⟹if^#(first,s(x18),s(x95),s(x20))≥c)
         • Applying Rule "Variable in Equation" allows to substitute x18 by x96 which results in
           (le(x96,x95,x94)→^*first⟹x20→^*x94⟹∀x97 . ∀x98 . (le(x96,x95,x94)→^*first⟹s(x97)→^*x96⟹s(x98)→^*x94⟹if^#(first,s(x97),x95,s(x98))≥c)⟹if^#(first,s(x96),s(x95),s(x20))≥c)
         • Applying Rule "Variable in Equation" allows to substitute x20 by x94 which results in
           (le(x96,x95,x94)→^*first⟹∀x97 . ∀x98 . (le(x96,x95,x94)→^*first⟹s(x97)→^*x96⟹s(x98)→^*x94⟹if^#(first,s(x97),x95,s(x98))≥c)⟹if^#(first,s(x96),s(x95),s(x94))≥c)
         • Applying Rule "Delete Conditions" results in
           (le(x96,x95,x94)→^*first⟹if^#(first,s(x96),s(x95),s(x94))≥c)
         • Applying Rule "Induction" on le(x96,x95,x94)→^*first results in the following 4 new constraints.
           1. (false→^*first⟹if^#(first,s(s(x104)),s(0),s(x103))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (le(x107,x106,x105)→^*first⟹(le(x107,x106,x105)→^*first⟹if^#(first,s(x107),s(x106),s(x105))≥c)⟹if^#(first,s(s(x107)),s(s(x106)),s(s(x105)))≥c)
              • Applying Rule "Simplify Conditions" results in
                (if^#(first,s(x107),s(x106),s(x105))≥c⟹if^#(first,s(s(x107)),s(s(x106)),s(s(x105)))≥c)

              • This constraint is kept as final constraint.

           3. (greater(x109,x108)→^*first⟹if^#(first,s(0),s(x109),s(x108))≥c)
              • Applying Rule "Delete Conditions" results in
                if^#(first,s(0),s(x109),s(x108))≥c

              • This constraint is kept as final constraint.

           4. (false→^*first⟹if^#(first,s(s(x111)),s(s(x110)),s(0))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (greater(x100,x99)→^*first⟹s(x18)→^*0⟹s(x20)→^*x99⟹if^#(first,s(x18),x100,s(x20))≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      4. (false→^*first⟹if^#(first,s(x18),s(x101),s(x20))≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (if^#(le(s(x0),s(x1),x2),s(x0),s(x1),x2)→^*if^#(second,x3,x4,x5)⟹if^#(second,x3,x4,x5)≥if^#(le(s(x3),s(x4),x5),s(x3),s(x4),x5))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (le(s(x0),s(x1),x2)→^*second⟹s(x0)→^*x3⟹s(x1)→^*x4⟹x2→^*x5⟹if^#(second,x3,x4,x5)≥if^#(le(s(x3),s(x4),x5),s(x3),s(x4),x5))
    • Applying Rule "Variable in Equation" allows to substitute x3 by s(x0) which results in
      (le(s(x0),s(x1),x2)→^*second⟹s(x1)→^*x4⟹x2→^*x5⟹if^#(second,s(x0),x4,x5)≥if^#(le(s(s(x0)),s(x4),x5),s(s(x0)),s(x4),x5))
    • Applying Rule "Variable in Equation" allows to substitute x4 by s(x1) which results in
      (le(s(x0),s(x1),x2)→^*second⟹x2→^*x5⟹if^#(second,s(x0),s(x1),x5)≥if^#(le(s(s(x0)),s(s(x1)),x5),s(s(x0)),s(s(x1)),x5))
    • Applying Rule "Variable in Equation" allows to substitute x5 by x2 which results in
      (le(s(x0),s(x1),x2)→^*second⟹if^#(second,s(x0),s(x1),x2)≥if^#(le(s(s(x0)),s(s(x1)),x2),s(s(x0)),s(s(x1)),x2))
    • Applying Rule "Introduce fresh variable" results in
      (s(x1)→^*x25⟹le(s(x0),x25,x2)→^*second⟹if^#(second,s(x0),s(x1),x2)≥if^#(le(s(s(x0)),s(s(x1)),x2),s(s(x0)),s(s(x1)),x2))
    • Applying Rule "Introduce fresh variable" results in
      (s(x0)→^*x24⟹s(x1)→^*x25⟹le(x24,x25,x2)→^*second⟹if^#(second,s(x0),s(x1),x2)≥if^#(le(s(s(x0)),s(s(x1)),x2),s(s(x0)),s(s(x1)),x2))
    • Applying Rule "Induction" on le(x24,x25,x2)→^*second results in the following 4 new constraints.
      1. (false→^*second⟹if^#(second,s(x0),s(x1),x26)≥if^#(le(s(s(x0)),s(s(x1)),x26),s(s(x0)),s(s(x1)),x26))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (le(x30,x29,x28)→^*second⟹s(x0)→^*s(x30)⟹s(x1)→^*s(x29)⟹∀x31 . ∀x32 . (le(x30,x29,x28)→^*second⟹s(x31)→^*x30⟹s(x32)→^*x29⟹if^#(second,s(x31),s(x32),x28)≥if^#(le(s(s(x31)),s(s(x32)),x28),s(s(x31)),s(s(x32)),x28))⟹if^#(second,s(x0),s(x1),s(x28))≥if^#(le(s(s(x0)),s(s(x1)),s(x28)),s(s(x0)),s(s(x1)),s(x28)))
         • Applying Rule "Same Constructor" results in
           (le(x30,x29,x28)→^*second⟹x0→^*x30⟹s(x1)→^*s(x29)⟹∀x31 . ∀x32 . (le(x30,x29,x28)→^*second⟹s(x31)→^*x30⟹s(x32)→^*x29⟹if^#(second,s(x31),s(x32),x28)≥if^#(le(s(s(x31)),s(s(x32)),x28),s(s(x31)),s(s(x32)),x28))⟹if^#(second,s(x0),s(x1),s(x28))≥if^#(le(s(s(x0)),s(s(x1)),s(x28)),s(s(x0)),s(s(x1)),s(x28)))
         • Applying Rule "Same Constructor" results in
           (le(x30,x29,x28)→^*second⟹x0→^*x30⟹x1→^*x29⟹∀x31 . ∀x32 . (le(x30,x29,x28)→^*second⟹s(x31)→^*x30⟹s(x32)→^*x29⟹if^#(second,s(x31),s(x32),x28)≥if^#(le(s(s(x31)),s(s(x32)),x28),s(s(x31)),s(s(x32)),x28))⟹if^#(second,s(x0),s(x1),s(x28))≥if^#(le(s(s(x0)),s(s(x1)),s(x28)),s(s(x0)),s(s(x1)),s(x28)))
         • Applying Rule "Variable in Equation" allows to substitute x0 by x30 which results in
           (le(x30,x29,x28)→^*second⟹x1→^*x29⟹∀x31 . ∀x32 . (le(x30,x29,x28)→^*second⟹s(x31)→^*x30⟹s(x32)→^*x29⟹if^#(second,s(x31),s(x32),x28)≥if^#(le(s(s(x31)),s(s(x32)),x28),s(s(x31)),s(s(x32)),x28))⟹if^#(second,s(x30),s(x1),s(x28))≥if^#(le(s(s(x30)),s(s(x1)),s(x28)),s(s(x30)),s(s(x1)),s(x28)))
         • Applying Rule "Variable in Equation" allows to substitute x1 by x29 which results in
           (le(x30,x29,x28)→^*second⟹∀x31 . ∀x32 . (le(x30,x29,x28)→^*second⟹s(x31)→^*x30⟹s(x32)→^*x29⟹if^#(second,s(x31),s(x32),x28)≥if^#(le(s(s(x31)),s(s(x32)),x28),s(s(x31)),s(s(x32)),x28))⟹if^#(second,s(x30),s(x29),s(x28))≥if^#(le(s(s(x30)),s(s(x29)),s(x28)),s(s(x30)),s(s(x29)),s(x28)))
         • Applying Rule "Delete Conditions" results in
           (le(x30,x29,x28)→^*second⟹if^#(second,s(x30),s(x29),s(x28))≥if^#(le(s(s(x30)),s(s(x29)),s(x28)),s(s(x30)),s(s(x29)),s(x28)))
         • Applying Rule "Induction" on le(x30,x29,x28)→^*second results in the following 4 new constraints.
           1. (false→^*second⟹if^#(second,s(s(x38)),s(0),s(x37))≥if^#(le(s(s(s(x38))),s(s(0)),s(x37)),s(s(s(x38))),s(s(0)),s(x37)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (le(x41,x40,x39)→^*second⟹(le(x41,x40,x39)→^*second⟹if^#(second,s(x41),s(x40),s(x39))≥if^#(le(s(s(x41)),s(s(x40)),s(x39)),s(s(x41)),s(s(x40)),s(x39)))⟹if^#(second,s(s(x41)),s(s(x40)),s(s(x39)))≥if^#(le(s(s(s(x41))),s(s(s(x40))),s(s(x39))),s(s(s(x41))),s(s(s(x40))),s(s(x39))))
              • Applying Rule "Simplify Conditions" results in
                (if^#(second,s(x41),s(x40),s(x39))≥if^#(le(s(s(x41)),s(s(x40)),s(x39)),s(s(x41)),s(s(x40)),s(x39))⟹if^#(second,s(s(x41)),s(s(x40)),s(s(x39)))≥if^#(le(s(s(s(x41))),s(s(s(x40))),s(s(x39))),s(s(s(x41))),s(s(s(x40))),s(s(x39))))

              • This constraint is kept as final constraint.

           3. (greater(x43,x42)→^*second⟹if^#(second,s(0),s(x43),s(x42))≥if^#(le(s(s(0)),s(s(x43)),s(x42)),s(s(0)),s(s(x43)),s(x42)))
              • Applying Rule "Delete Conditions" results in
                if^#(second,s(0),s(x43),s(x42))≥if^#(le(s(s(0)),s(s(x43)),s(x42)),s(s(0)),s(s(x43)),s(x42))

              • This constraint is kept as final constraint.

           4. (false→^*second⟹if^#(second,s(s(x45)),s(s(x44)),s(0))≥if^#(le(s(s(s(x45))),s(s(s(x44))),s(0)),s(s(s(x45))),s(s(s(x44))),s(0)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (greater(x34,x33)→^*second⟹s(x0)→^*0⟹s(x1)→^*x34⟹if^#(second,s(x0),s(x1),x33)≥if^#(le(s(s(x0)),s(s(x1)),x33),s(s(x0)),s(s(x1)),x33))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      4. (false→^*second⟹if^#(second,s(x0),s(x1),0)≥if^#(le(s(s(x0)),s(s(x1)),0),s(s(x0)),s(s(x1)),0))
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (if^#(le(s(x6),x7,s(x8)),s(x6),x7,s(x8))→^*if^#(second,x9,x10,x11)⟹if^#(second,x9,x10,x11)≥if^#(le(s(x9),s(x10),x11),s(x9),s(x10),x11))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (le(s(x6),x7,s(x8))→^*second⟹s(x6)→^*x9⟹x7→^*x10⟹s(x8)→^*x11⟹if^#(second,x9,x10,x11)≥if^#(le(s(x9),s(x10),x11),s(x9),s(x10),x11))
    • Applying Rule "Variable in Equation" allows to substitute x9 by s(x6) which results in
      (le(s(x6),x7,s(x8))→^*second⟹x7→^*x10⟹s(x8)→^*x11⟹if^#(second,s(x6),x10,x11)≥if^#(le(s(s(x6)),s(x10),x11),s(s(x6)),s(x10),x11))
    • Applying Rule "Variable in Equation" allows to substitute x10 by x7 which results in
      (le(s(x6),x7,s(x8))→^*second⟹s(x8)→^*x11⟹if^#(second,s(x6),x7,x11)≥if^#(le(s(s(x6)),s(x7),x11),s(s(x6)),s(x7),x11))
    • Applying Rule "Variable in Equation" allows to substitute x11 by s(x8) which results in
      (le(s(x6),x7,s(x8))→^*second⟹if^#(second,s(x6),x7,s(x8))≥if^#(le(s(s(x6)),s(x7),s(x8)),s(s(x6)),s(x7),s(x8)))
    • Applying Rule "Introduce fresh variable" results in
      (s(x8)→^*x47⟹le(s(x6),x7,x47)→^*second⟹if^#(second,s(x6),x7,s(x8))≥if^#(le(s(s(x6)),s(x7),s(x8)),s(s(x6)),s(x7),s(x8)))
    • Applying Rule "Introduce fresh variable" results in
      (s(x6)→^*x46⟹s(x8)→^*x47⟹le(x46,x7,x47)→^*second⟹if^#(second,s(x6),x7,s(x8))≥if^#(le(s(s(x6)),s(x7),s(x8)),s(s(x6)),s(x7),s(x8)))
    • Applying Rule "Induction" on le(x46,x7,x47)→^*second results in the following 4 new constraints.
      1. (false→^*second⟹if^#(second,s(x6),0,s(x8))≥if^#(le(s(s(x6)),s(0),s(x8)),s(s(x6)),s(0),s(x8)))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (le(x52,x51,x50)→^*second⟹s(x6)→^*s(x52)⟹s(x8)→^*s(x50)⟹∀x53 . ∀x54 . (le(x52,x51,x50)→^*second⟹s(x53)→^*x52⟹s(x54)→^*x50⟹if^#(second,s(x53),x51,s(x54))≥if^#(le(s(s(x53)),s(x51),s(x54)),s(s(x53)),s(x51),s(x54)))⟹if^#(second,s(x6),s(x51),s(x8))≥if^#(le(s(s(x6)),s(s(x51)),s(x8)),s(s(x6)),s(s(x51)),s(x8)))
         • Applying Rule "Same Constructor" results in
           (le(x52,x51,x50)→^*second⟹x6→^*x52⟹s(x8)→^*s(x50)⟹∀x53 . ∀x54 . (le(x52,x51,x50)→^*second⟹s(x53)→^*x52⟹s(x54)→^*x50⟹if^#(second,s(x53),x51,s(x54))≥if^#(le(s(s(x53)),s(x51),s(x54)),s(s(x53)),s(x51),s(x54)))⟹if^#(second,s(x6),s(x51),s(x8))≥if^#(le(s(s(x6)),s(s(x51)),s(x8)),s(s(x6)),s(s(x51)),s(x8)))
         • Applying Rule "Same Constructor" results in
           (le(x52,x51,x50)→^*second⟹x6→^*x52⟹x8→^*x50⟹∀x53 . ∀x54 . (le(x52,x51,x50)→^*second⟹s(x53)→^*x52⟹s(x54)→^*x50⟹if^#(second,s(x53),x51,s(x54))≥if^#(le(s(s(x53)),s(x51),s(x54)),s(s(x53)),s(x51),s(x54)))⟹if^#(second,s(x6),s(x51),s(x8))≥if^#(le(s(s(x6)),s(s(x51)),s(x8)),s(s(x6)),s(s(x51)),s(x8)))
         • Applying Rule "Variable in Equation" allows to substitute x6 by x52 which results in
           (le(x52,x51,x50)→^*second⟹x8→^*x50⟹∀x53 . ∀x54 . (le(x52,x51,x50)→^*second⟹s(x53)→^*x52⟹s(x54)→^*x50⟹if^#(second,s(x53),x51,s(x54))≥if^#(le(s(s(x53)),s(x51),s(x54)),s(s(x53)),s(x51),s(x54)))⟹if^#(second,s(x52),s(x51),s(x8))≥if^#(le(s(s(x52)),s(s(x51)),s(x8)),s(s(x52)),s(s(x51)),s(x8)))
         • Applying Rule "Variable in Equation" allows to substitute x8 by x50 which results in
           (le(x52,x51,x50)→^*second⟹∀x53 . ∀x54 . (le(x52,x51,x50)→^*second⟹s(x53)→^*x52⟹s(x54)→^*x50⟹if^#(second,s(x53),x51,s(x54))≥if^#(le(s(s(x53)),s(x51),s(x54)),s(s(x53)),s(x51),s(x54)))⟹if^#(second,s(x52),s(x51),s(x50))≥if^#(le(s(s(x52)),s(s(x51)),s(x50)),s(s(x52)),s(s(x51)),s(x50)))
         • Applying Rule "Delete Conditions" results in
           (le(x52,x51,x50)→^*second⟹if^#(second,s(x52),s(x51),s(x50))≥if^#(le(s(s(x52)),s(s(x51)),s(x50)),s(s(x52)),s(s(x51)),s(x50)))
         • Applying Rule "Induction" on le(x52,x51,x50)→^*second results in the following 4 new constraints.
           1. (false→^*second⟹if^#(second,s(s(x60)),s(0),s(x59))≥if^#(le(s(s(s(x60))),s(s(0)),s(x59)),s(s(s(x60))),s(s(0)),s(x59)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (le(x63,x62,x61)→^*second⟹(le(x63,x62,x61)→^*second⟹if^#(second,s(x63),s(x62),s(x61))≥if^#(le(s(s(x63)),s(s(x62)),s(x61)),s(s(x63)),s(s(x62)),s(x61)))⟹if^#(second,s(s(x63)),s(s(x62)),s(s(x61)))≥if^#(le(s(s(s(x63))),s(s(s(x62))),s(s(x61))),s(s(s(x63))),s(s(s(x62))),s(s(x61))))
              • Applying Rule "Simplify Conditions" results in
                (if^#(second,s(x63),s(x62),s(x61))≥if^#(le(s(s(x63)),s(s(x62)),s(x61)),s(s(x63)),s(s(x62)),s(x61))⟹if^#(second,s(s(x63)),s(s(x62)),s(s(x61)))≥if^#(le(s(s(s(x63))),s(s(s(x62))),s(s(x61))),s(s(s(x63))),s(s(s(x62))),s(s(x61))))

              • This constraint is kept as final constraint.

           3. (greater(x65,x64)→^*second⟹if^#(second,s(0),s(x65),s(x64))≥if^#(le(s(s(0)),s(s(x65)),s(x64)),s(s(0)),s(s(x65)),s(x64)))
              • Applying Rule "Delete Conditions" results in
                if^#(second,s(0),s(x65),s(x64))≥if^#(le(s(s(0)),s(s(x65)),s(x64)),s(s(0)),s(s(x65)),s(x64))

              • This constraint is kept as final constraint.

           4. (false→^*second⟹if^#(second,s(s(x67)),s(s(x66)),s(0))≥if^#(le(s(s(s(x67))),s(s(s(x66))),s(0)),s(s(s(x67))),s(s(s(x66))),s(0)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (greater(x56,x55)→^*second⟹s(x6)→^*0⟹s(x8)→^*x55⟹if^#(second,s(x6),x56,s(x8))≥if^#(le(s(s(x6)),s(x56),s(x8)),s(s(x6)),s(x56),s(x8)))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      4. (false→^*second⟹if^#(second,s(x6),s(x57),s(x8))≥if^#(le(s(s(x6)),s(s(x57)),s(x8)),s(s(x6)),s(s(x57)),s(x8)))
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (if^#(le(s(x12),s(x13),x14),s(x12),s(x13),x14)→^*if^#(first,x15,x16,x17)⟹if^#(first,x15,x16,x17) > if^#(le(s(x15),x16,s(x17)),s(x15),x16,s(x17)))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (le(s(x12),s(x13),x14)→^*first⟹s(x12)→^*x15⟹s(x13)→^*x16⟹x14→^*x17⟹if^#(first,x15,x16,x17) > if^#(le(s(x15),x16,s(x17)),s(x15),x16,s(x17)))
    • Applying Rule "Variable in Equation" allows to substitute x15 by s(x12) which results in
      (le(s(x12),s(x13),x14)→^*first⟹s(x13)→^*x16⟹x14→^*x17⟹if^#(first,s(x12),x16,x17) > if^#(le(s(s(x12)),x16,s(x17)),s(s(x12)),x16,s(x17)))
    • Applying Rule "Variable in Equation" allows to substitute x16 by s(x13) which results in
      (le(s(x12),s(x13),x14)→^*first⟹x14→^*x17⟹if^#(first,s(x12),s(x13),x17) > if^#(le(s(s(x12)),s(x13),s(x17)),s(s(x12)),s(x13),s(x17)))
    • Applying Rule "Variable in Equation" allows to substitute x17 by x14 which results in
      (le(s(x12),s(x13),x14)→^*first⟹if^#(first,s(x12),s(x13),x14) > if^#(le(s(s(x12)),s(x13),s(x14)),s(s(x12)),s(x13),s(x14)))
    • Applying Rule "Introduce fresh variable" results in
      (s(x13)→^*x69⟹le(s(x12),x69,x14)→^*first⟹if^#(first,s(x12),s(x13),x14) > if^#(le(s(s(x12)),s(x13),s(x14)),s(s(x12)),s(x13),s(x14)))
    • Applying Rule "Introduce fresh variable" results in
      (s(x12)→^*x68⟹s(x13)→^*x69⟹le(x68,x69,x14)→^*first⟹if^#(first,s(x12),s(x13),x14) > if^#(le(s(s(x12)),s(x13),s(x14)),s(s(x12)),s(x13),s(x14)))
    • Applying Rule "Induction" on le(x68,x69,x14)→^*first results in the following 4 new constraints.
      1. (false→^*first⟹if^#(first,s(x12),s(x13),x70) > if^#(le(s(s(x12)),s(x13),s(x70)),s(s(x12)),s(x13),s(x70)))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (le(x74,x73,x72)→^*first⟹s(x12)→^*s(x74)⟹s(x13)→^*s(x73)⟹∀x75 . ∀x76 . (le(x74,x73,x72)→^*first⟹s(x75)→^*x74⟹s(x76)→^*x73⟹if^#(first,s(x75),s(x76),x72) > if^#(le(s(s(x75)),s(x76),s(x72)),s(s(x75)),s(x76),s(x72)))⟹if^#(first,s(x12),s(x13),s(x72)) > if^#(le(s(s(x12)),s(x13),s(s(x72))),s(s(x12)),s(x13),s(s(x72))))
         • Applying Rule "Same Constructor" results in
           (le(x74,x73,x72)→^*first⟹x12→^*x74⟹s(x13)→^*s(x73)⟹∀x75 . ∀x76 . (le(x74,x73,x72)→^*first⟹s(x75)→^*x74⟹s(x76)→^*x73⟹if^#(first,s(x75),s(x76),x72) > if^#(le(s(s(x75)),s(x76),s(x72)),s(s(x75)),s(x76),s(x72)))⟹if^#(first,s(x12),s(x13),s(x72)) > if^#(le(s(s(x12)),s(x13),s(s(x72))),s(s(x12)),s(x13),s(s(x72))))
         • Applying Rule "Same Constructor" results in
           (le(x74,x73,x72)→^*first⟹x12→^*x74⟹x13→^*x73⟹∀x75 . ∀x76 . (le(x74,x73,x72)→^*first⟹s(x75)→^*x74⟹s(x76)→^*x73⟹if^#(first,s(x75),s(x76),x72) > if^#(le(s(s(x75)),s(x76),s(x72)),s(s(x75)),s(x76),s(x72)))⟹if^#(first,s(x12),s(x13),s(x72)) > if^#(le(s(s(x12)),s(x13),s(s(x72))),s(s(x12)),s(x13),s(s(x72))))
         • Applying Rule "Variable in Equation" allows to substitute x12 by x74 which results in
           (le(x74,x73,x72)→^*first⟹x13→^*x73⟹∀x75 . ∀x76 . (le(x74,x73,x72)→^*first⟹s(x75)→^*x74⟹s(x76)→^*x73⟹if^#(first,s(x75),s(x76),x72) > if^#(le(s(s(x75)),s(x76),s(x72)),s(s(x75)),s(x76),s(x72)))⟹if^#(first,s(x74),s(x13),s(x72)) > if^#(le(s(s(x74)),s(x13),s(s(x72))),s(s(x74)),s(x13),s(s(x72))))
         • Applying Rule "Variable in Equation" allows to substitute x13 by x73 which results in
           (le(x74,x73,x72)→^*first⟹∀x75 . ∀x76 . (le(x74,x73,x72)→^*first⟹s(x75)→^*x74⟹s(x76)→^*x73⟹if^#(first,s(x75),s(x76),x72) > if^#(le(s(s(x75)),s(x76),s(x72)),s(s(x75)),s(x76),s(x72)))⟹if^#(first,s(x74),s(x73),s(x72)) > if^#(le(s(s(x74)),s(x73),s(s(x72))),s(s(x74)),s(x73),s(s(x72))))
         • Applying Rule "Delete Conditions" results in
           (le(x74,x73,x72)→^*first⟹if^#(first,s(x74),s(x73),s(x72)) > if^#(le(s(s(x74)),s(x73),s(s(x72))),s(s(x74)),s(x73),s(s(x72))))
         • Applying Rule "Induction" on le(x74,x73,x72)→^*first results in the following 4 new constraints.
           1. (false→^*first⟹if^#(first,s(s(x82)),s(0),s(x81)) > if^#(le(s(s(s(x82))),s(0),s(s(x81))),s(s(s(x82))),s(0),s(s(x81))))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (le(x85,x84,x83)→^*first⟹(le(x85,x84,x83)→^*first⟹if^#(first,s(x85),s(x84),s(x83)) > if^#(le(s(s(x85)),s(x84),s(s(x83))),s(s(x85)),s(x84),s(s(x83))))⟹if^#(first,s(s(x85)),s(s(x84)),s(s(x83))) > if^#(le(s(s(s(x85))),s(s(x84)),s(s(s(x83)))),s(s(s(x85))),s(s(x84)),s(s(s(x83)))))
              • Applying Rule "Simplify Conditions" results in
                (if^#(first,s(x85),s(x84),s(x83)) > if^#(le(s(s(x85)),s(x84),s(s(x83))),s(s(x85)),s(x84),s(s(x83)))⟹if^#(first,s(s(x85)),s(s(x84)),s(s(x83))) > if^#(le(s(s(s(x85))),s(s(x84)),s(s(s(x83)))),s(s(s(x85))),s(s(x84)),s(s(s(x83)))))

              • This constraint is kept as final constraint.

           3. (greater(x87,x86)→^*first⟹if^#(first,s(0),s(x87),s(x86)) > if^#(le(s(s(0)),s(x87),s(s(x86))),s(s(0)),s(x87),s(s(x86))))
              • Applying Rule "Delete Conditions" results in
                if^#(first,s(0),s(x87),s(x86)) > if^#(le(s(s(0)),s(x87),s(s(x86))),s(s(0)),s(x87),s(s(x86)))

              • This constraint is kept as final constraint.

           4. (false→^*first⟹if^#(first,s(s(x89)),s(s(x88)),s(0)) > if^#(le(s(s(s(x89))),s(s(x88)),s(s(0))),s(s(s(x89))),s(s(x88)),s(s(0))))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (greater(x78,x77)→^*first⟹s(x12)→^*0⟹s(x13)→^*x78⟹if^#(first,s(x12),s(x13),x77) > if^#(le(s(s(x12)),s(x13),s(x77)),s(s(x12)),s(x13),s(x77)))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      4. (false→^*first⟹if^#(first,s(x12),s(x13),0) > if^#(le(s(s(x12)),s(x13),s(0)),s(s(x12)),s(x13),s(0)))
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (if^#(le(s(x18),x19,s(x20)),s(x18),x19,s(x20))→^*if^#(first,x21,x22,x23)⟹if^#(first,x21,x22,x23) > if^#(le(s(x21),x22,s(x23)),s(x21),x22,s(x23)))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (le(s(x18),x19,s(x20))→^*first⟹s(x18)→^*x21⟹x19→^*x22⟹s(x20)→^*x23⟹if^#(first,x21,x22,x23) > if^#(le(s(x21),x22,s(x23)),s(x21),x22,s(x23)))
    • Applying Rule "Variable in Equation" allows to substitute x21 by s(x18) which results in
      (le(s(x18),x19,s(x20))→^*first⟹x19→^*x22⟹s(x20)→^*x23⟹if^#(first,s(x18),x22,x23) > if^#(le(s(s(x18)),x22,s(x23)),s(s(x18)),x22,s(x23)))
    • Applying Rule "Variable in Equation" allows to substitute x22 by x19 which results in
      (le(s(x18),x19,s(x20))→^*first⟹s(x20)→^*x23⟹if^#(first,s(x18),x19,x23) > if^#(le(s(s(x18)),x19,s(x23)),s(s(x18)),x19,s(x23)))
    • Applying Rule "Variable in Equation" allows to substitute x23 by s(x20) which results in
      (le(s(x18),x19,s(x20))→^*first⟹if^#(first,s(x18),x19,s(x20)) > if^#(le(s(s(x18)),x19,s(s(x20))),s(s(x18)),x19,s(s(x20))))
    • Applying Rule "Introduce fresh variable" results in
      (s(x20)→^*x91⟹le(s(x18),x19,x91)→^*first⟹if^#(first,s(x18),x19,s(x20)) > if^#(le(s(s(x18)),x19,s(s(x20))),s(s(x18)),x19,s(s(x20))))
    • Applying Rule "Introduce fresh variable" results in
      (s(x18)→^*x90⟹s(x20)→^*x91⟹le(x90,x19,x91)→^*first⟹if^#(first,s(x18),x19,s(x20)) > if^#(le(s(s(x18)),x19,s(s(x20))),s(s(x18)),x19,s(s(x20))))
    • Applying Rule "Induction" on le(x90,x19,x91)→^*first results in the following 4 new constraints.
      1. (false→^*first⟹if^#(first,s(x18),0,s(x20)) > if^#(le(s(s(x18)),0,s(s(x20))),s(s(x18)),0,s(s(x20))))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (le(x96,x95,x94)→^*first⟹s(x18)→^*s(x96)⟹s(x20)→^*s(x94)⟹∀x97 . ∀x98 . (le(x96,x95,x94)→^*first⟹s(x97)→^*x96⟹s(x98)→^*x94⟹if^#(first,s(x97),x95,s(x98)) > if^#(le(s(s(x97)),x95,s(s(x98))),s(s(x97)),x95,s(s(x98))))⟹if^#(first,s(x18),s(x95),s(x20)) > if^#(le(s(s(x18)),s(x95),s(s(x20))),s(s(x18)),s(x95),s(s(x20))))
         • Applying Rule "Same Constructor" results in
           (le(x96,x95,x94)→^*first⟹x18→^*x96⟹s(x20)→^*s(x94)⟹∀x97 . ∀x98 . (le(x96,x95,x94)→^*first⟹s(x97)→^*x96⟹s(x98)→^*x94⟹if^#(first,s(x97),x95,s(x98)) > if^#(le(s(s(x97)),x95,s(s(x98))),s(s(x97)),x95,s(s(x98))))⟹if^#(first,s(x18),s(x95),s(x20)) > if^#(le(s(s(x18)),s(x95),s(s(x20))),s(s(x18)),s(x95),s(s(x20))))
         • Applying Rule "Same Constructor" results in
           (le(x96,x95,x94)→^*first⟹x18→^*x96⟹x20→^*x94⟹∀x97 . ∀x98 . (le(x96,x95,x94)→^*first⟹s(x97)→^*x96⟹s(x98)→^*x94⟹if^#(first,s(x97),x95,s(x98)) > if^#(le(s(s(x97)),x95,s(s(x98))),s(s(x97)),x95,s(s(x98))))⟹if^#(first,s(x18),s(x95),s(x20)) > if^#(le(s(s(x18)),s(x95),s(s(x20))),s(s(x18)),s(x95),s(s(x20))))
         • Applying Rule "Variable in Equation" allows to substitute x18 by x96 which results in
           (le(x96,x95,x94)→^*first⟹x20→^*x94⟹∀x97 . ∀x98 . (le(x96,x95,x94)→^*first⟹s(x97)→^*x96⟹s(x98)→^*x94⟹if^#(first,s(x97),x95,s(x98)) > if^#(le(s(s(x97)),x95,s(s(x98))),s(s(x97)),x95,s(s(x98))))⟹if^#(first,s(x96),s(x95),s(x20)) > if^#(le(s(s(x96)),s(x95),s(s(x20))),s(s(x96)),s(x95),s(s(x20))))
         • Applying Rule "Variable in Equation" allows to substitute x20 by x94 which results in
           (le(x96,x95,x94)→^*first⟹∀x97 . ∀x98 . (le(x96,x95,x94)→^*first⟹s(x97)→^*x96⟹s(x98)→^*x94⟹if^#(first,s(x97),x95,s(x98)) > if^#(le(s(s(x97)),x95,s(s(x98))),s(s(x97)),x95,s(s(x98))))⟹if^#(first,s(x96),s(x95),s(x94)) > if^#(le(s(s(x96)),s(x95),s(s(x94))),s(s(x96)),s(x95),s(s(x94))))
         • Applying Rule "Delete Conditions" results in
           (le(x96,x95,x94)→^*first⟹if^#(first,s(x96),s(x95),s(x94)) > if^#(le(s(s(x96)),s(x95),s(s(x94))),s(s(x96)),s(x95),s(s(x94))))
         • Applying Rule "Induction" on le(x96,x95,x94)→^*first results in the following 4 new constraints.
           1. (false→^*first⟹if^#(first,s(s(x104)),s(0),s(x103)) > if^#(le(s(s(s(x104))),s(0),s(s(x103))),s(s(s(x104))),s(0),s(s(x103))))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (le(x107,x106,x105)→^*first⟹(le(x107,x106,x105)→^*first⟹if^#(first,s(x107),s(x106),s(x105)) > if^#(le(s(s(x107)),s(x106),s(s(x105))),s(s(x107)),s(x106),s(s(x105))))⟹if^#(first,s(s(x107)),s(s(x106)),s(s(x105))) > if^#(le(s(s(s(x107))),s(s(x106)),s(s(s(x105)))),s(s(s(x107))),s(s(x106)),s(s(s(x105)))))
              • Applying Rule "Simplify Conditions" results in
                (if^#(first,s(x107),s(x106),s(x105)) > if^#(le(s(s(x107)),s(x106),s(s(x105))),s(s(x107)),s(x106),s(s(x105)))⟹if^#(first,s(s(x107)),s(s(x106)),s(s(x105))) > if^#(le(s(s(s(x107))),s(s(x106)),s(s(s(x105)))),s(s(s(x107))),s(s(x106)),s(s(s(x105)))))

              • This constraint is kept as final constraint.

           3. (greater(x109,x108)→^*first⟹if^#(first,s(0),s(x109),s(x108)) > if^#(le(s(s(0)),s(x109),s(s(x108))),s(s(0)),s(x109),s(s(x108))))
              • Applying Rule "Delete Conditions" results in
                if^#(first,s(0),s(x109),s(x108)) > if^#(le(s(s(0)),s(x109),s(s(x108))),s(s(0)),s(x109),s(s(x108)))

              • This constraint is kept as final constraint.

           4. (false→^*first⟹if^#(first,s(s(x111)),s(s(x110)),s(0)) > if^#(le(s(s(s(x111))),s(s(x110)),s(s(0))),s(s(s(x111))),s(s(x110)),s(s(0))))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (greater(x100,x99)→^*first⟹s(x18)→^*0⟹s(x20)→^*x99⟹if^#(first,s(x18),x100,s(x20)) > if^#(le(s(s(x18)),x100,s(s(x20))),s(s(x18)),x100,s(s(x20))))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      4. (false→^*first⟹if^#(first,s(x18),s(x101),s(x20)) > if^#(le(s(s(x18)),s(x101),s(s(x20))),s(s(x18)),s(x101),s(s(x20))))
         • Applying Rule "Different Constructors" allows to drop this constraint.

  We remove those pairs which are strictly decreasing and bounded.

  1.1.1.1.1.1.1 Reduction Pair Processor

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

  [c]	=	-1
  [if#(x1,...,x4)]	=	-1 · x1 + -1 · x2 + 1 · x4 + -1
  [second]	=	0
  [le(x1, x2, x3)]	=	0
  [s(x1)]	=	1 · x1 + 1
  [0]	=	0
  [false]	=	1
  [greater(x1, x2)]	=	0
  [first]	=	0

  The pair(s)

  if#(second,x,y,z)

  →

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

  (23)

  are strictly oriented and the pair(s)

  if#(second,x,y,z)

  →

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

  (23)

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

  The following constraints are generated for the pairs.

  • (if^#(second,s(x41),s(x40),s(x39))≥c⟹if^#(second,s(s(x41)),s(s(x40)),s(s(x39)))≥c)
  • if^#(second,s(0),s(x43),s(x42))≥c
  • (if^#(second,s(x63),s(x62),s(x61))≥c⟹if^#(second,s(s(x63)),s(s(x62)),s(s(x61)))≥c)
  • if^#(second,s(0),s(x65),s(x64))≥c
  • (if^#(second,s(x41),s(x40),s(x39)) > if^#(le(s(s(x41)),s(s(x40)),s(x39)),s(s(x41)),s(s(x40)),s(x39))⟹if^#(second,s(s(x41)),s(s(x40)),s(s(x39))) > if^#(le(s(s(s(x41))),s(s(s(x40))),s(s(x39))),s(s(s(x41))),s(s(s(x40))),s(s(x39))))
  • if^#(second,s(0),s(x43),s(x42)) > if^#(le(s(s(0)),s(s(x43)),s(x42)),s(s(0)),s(s(x43)),s(x42))
  • (if^#(second,s(x63),s(x62),s(x61)) > if^#(le(s(s(x63)),s(s(x62)),s(x61)),s(s(x63)),s(s(x62)),s(x61))⟹if^#(second,s(s(x63)),s(s(x62)),s(s(x61))) > if^#(le(s(s(s(x63))),s(s(s(x62))),s(s(x61))),s(s(s(x63))),s(s(s(x62))),s(s(x61))))
  • if^#(second,s(0),s(x65),s(x64)) > if^#(le(s(s(0)),s(s(x65)),s(x64)),s(s(0)),s(s(x65)),s(x64))

  The details are shown below:

  • For the chain we build the initial constraint

    (if^#(le(s(x0),s(x1),x2),s(x0),s(x1),x2)→^*if^#(second,x3,x4,x5)⟹if^#(second,x3,x4,x5)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (le(s(x0),s(x1),x2)→^*second⟹s(x0)→^*x3⟹s(x1)→^*x4⟹x2→^*x5⟹if^#(second,x3,x4,x5)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x3 by s(x0) which results in
      (le(s(x0),s(x1),x2)→^*second⟹s(x1)→^*x4⟹x2→^*x5⟹if^#(second,s(x0),x4,x5)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x4 by s(x1) which results in
      (le(s(x0),s(x1),x2)→^*second⟹x2→^*x5⟹if^#(second,s(x0),s(x1),x5)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x5 by x2 which results in
      (le(s(x0),s(x1),x2)→^*second⟹if^#(second,s(x0),s(x1),x2)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (s(x1)→^*x25⟹le(s(x0),x25,x2)→^*second⟹if^#(second,s(x0),s(x1),x2)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (s(x0)→^*x24⟹s(x1)→^*x25⟹le(x24,x25,x2)→^*second⟹if^#(second,s(x0),s(x1),x2)≥c)
    • Applying Rule "Induction" on le(x24,x25,x2)→^*second results in the following 4 new constraints.
      1. (false→^*second⟹if^#(second,s(x0),s(x1),x26)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (le(x30,x29,x28)→^*second⟹s(x0)→^*s(x30)⟹s(x1)→^*s(x29)⟹∀x31 . ∀x32 . (le(x30,x29,x28)→^*second⟹s(x31)→^*x30⟹s(x32)→^*x29⟹if^#(second,s(x31),s(x32),x28)≥c)⟹if^#(second,s(x0),s(x1),s(x28))≥c)
         • Applying Rule "Same Constructor" results in
           (le(x30,x29,x28)→^*second⟹x0→^*x30⟹s(x1)→^*s(x29)⟹∀x31 . ∀x32 . (le(x30,x29,x28)→^*second⟹s(x31)→^*x30⟹s(x32)→^*x29⟹if^#(second,s(x31),s(x32),x28)≥c)⟹if^#(second,s(x0),s(x1),s(x28))≥c)
         • Applying Rule "Same Constructor" results in
           (le(x30,x29,x28)→^*second⟹x0→^*x30⟹x1→^*x29⟹∀x31 . ∀x32 . (le(x30,x29,x28)→^*second⟹s(x31)→^*x30⟹s(x32)→^*x29⟹if^#(second,s(x31),s(x32),x28)≥c)⟹if^#(second,s(x0),s(x1),s(x28))≥c)
         • Applying Rule "Variable in Equation" allows to substitute x0 by x30 which results in
           (le(x30,x29,x28)→^*second⟹x1→^*x29⟹∀x31 . ∀x32 . (le(x30,x29,x28)→^*second⟹s(x31)→^*x30⟹s(x32)→^*x29⟹if^#(second,s(x31),s(x32),x28)≥c)⟹if^#(second,s(x30),s(x1),s(x28))≥c)
         • Applying Rule "Variable in Equation" allows to substitute x1 by x29 which results in
           (le(x30,x29,x28)→^*second⟹∀x31 . ∀x32 . (le(x30,x29,x28)→^*second⟹s(x31)→^*x30⟹s(x32)→^*x29⟹if^#(second,s(x31),s(x32),x28)≥c)⟹if^#(second,s(x30),s(x29),s(x28))≥c)
         • Applying Rule "Delete Conditions" results in
           (le(x30,x29,x28)→^*second⟹if^#(second,s(x30),s(x29),s(x28))≥c)
         • Applying Rule "Induction" on le(x30,x29,x28)→^*second results in the following 4 new constraints.
           1. (false→^*second⟹if^#(second,s(s(x38)),s(0),s(x37))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (le(x41,x40,x39)→^*second⟹(le(x41,x40,x39)→^*second⟹if^#(second,s(x41),s(x40),s(x39))≥c)⟹if^#(second,s(s(x41)),s(s(x40)),s(s(x39)))≥c)
              • Applying Rule "Simplify Conditions" results in
                (if^#(second,s(x41),s(x40),s(x39))≥c⟹if^#(second,s(s(x41)),s(s(x40)),s(s(x39)))≥c)

              • This constraint is kept as final constraint.

           3. (greater(x43,x42)→^*second⟹if^#(second,s(0),s(x43),s(x42))≥c)
              • Applying Rule "Delete Conditions" results in
                if^#(second,s(0),s(x43),s(x42))≥c

              • This constraint is kept as final constraint.

           4. (false→^*second⟹if^#(second,s(s(x45)),s(s(x44)),s(0))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (greater(x34,x33)→^*second⟹s(x0)→^*0⟹s(x1)→^*x34⟹if^#(second,s(x0),s(x1),x33)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      4. (false→^*second⟹if^#(second,s(x0),s(x1),0)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (if^#(le(s(x6),x7,s(x8)),s(x6),x7,s(x8))→^*if^#(second,x9,x10,x11)⟹if^#(second,x9,x10,x11)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (le(s(x6),x7,s(x8))→^*second⟹s(x6)→^*x9⟹x7→^*x10⟹s(x8)→^*x11⟹if^#(second,x9,x10,x11)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x9 by s(x6) which results in
      (le(s(x6),x7,s(x8))→^*second⟹x7→^*x10⟹s(x8)→^*x11⟹if^#(second,s(x6),x10,x11)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x10 by x7 which results in
      (le(s(x6),x7,s(x8))→^*second⟹s(x8)→^*x11⟹if^#(second,s(x6),x7,x11)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x11 by s(x8) which results in
      (le(s(x6),x7,s(x8))→^*second⟹if^#(second,s(x6),x7,s(x8))≥c)
    • Applying Rule "Introduce fresh variable" results in
      (s(x8)→^*x47⟹le(s(x6),x7,x47)→^*second⟹if^#(second,s(x6),x7,s(x8))≥c)
    • Applying Rule "Introduce fresh variable" results in
      (s(x6)→^*x46⟹s(x8)→^*x47⟹le(x46,x7,x47)→^*second⟹if^#(second,s(x6),x7,s(x8))≥c)
    • Applying Rule "Induction" on le(x46,x7,x47)→^*second results in the following 4 new constraints.
      1. (false→^*second⟹if^#(second,s(x6),0,s(x8))≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (le(x52,x51,x50)→^*second⟹s(x6)→^*s(x52)⟹s(x8)→^*s(x50)⟹∀x53 . ∀x54 . (le(x52,x51,x50)→^*second⟹s(x53)→^*x52⟹s(x54)→^*x50⟹if^#(second,s(x53),x51,s(x54))≥c)⟹if^#(second,s(x6),s(x51),s(x8))≥c)
         • Applying Rule "Same Constructor" results in
           (le(x52,x51,x50)→^*second⟹x6→^*x52⟹s(x8)→^*s(x50)⟹∀x53 . ∀x54 . (le(x52,x51,x50)→^*second⟹s(x53)→^*x52⟹s(x54)→^*x50⟹if^#(second,s(x53),x51,s(x54))≥c)⟹if^#(second,s(x6),s(x51),s(x8))≥c)
         • Applying Rule "Same Constructor" results in
           (le(x52,x51,x50)→^*second⟹x6→^*x52⟹x8→^*x50⟹∀x53 . ∀x54 . (le(x52,x51,x50)→^*second⟹s(x53)→^*x52⟹s(x54)→^*x50⟹if^#(second,s(x53),x51,s(x54))≥c)⟹if^#(second,s(x6),s(x51),s(x8))≥c)
         • Applying Rule "Variable in Equation" allows to substitute x6 by x52 which results in
           (le(x52,x51,x50)→^*second⟹x8→^*x50⟹∀x53 . ∀x54 . (le(x52,x51,x50)→^*second⟹s(x53)→^*x52⟹s(x54)→^*x50⟹if^#(second,s(x53),x51,s(x54))≥c)⟹if^#(second,s(x52),s(x51),s(x8))≥c)
         • Applying Rule "Variable in Equation" allows to substitute x8 by x50 which results in
           (le(x52,x51,x50)→^*second⟹∀x53 . ∀x54 . (le(x52,x51,x50)→^*second⟹s(x53)→^*x52⟹s(x54)→^*x50⟹if^#(second,s(x53),x51,s(x54))≥c)⟹if^#(second,s(x52),s(x51),s(x50))≥c)
         • Applying Rule "Delete Conditions" results in
           (le(x52,x51,x50)→^*second⟹if^#(second,s(x52),s(x51),s(x50))≥c)
         • Applying Rule "Induction" on le(x52,x51,x50)→^*second results in the following 4 new constraints.
           1. (false→^*second⟹if^#(second,s(s(x60)),s(0),s(x59))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (le(x63,x62,x61)→^*second⟹(le(x63,x62,x61)→^*second⟹if^#(second,s(x63),s(x62),s(x61))≥c)⟹if^#(second,s(s(x63)),s(s(x62)),s(s(x61)))≥c)
              • Applying Rule "Simplify Conditions" results in
                (if^#(second,s(x63),s(x62),s(x61))≥c⟹if^#(second,s(s(x63)),s(s(x62)),s(s(x61)))≥c)

              • This constraint is kept as final constraint.

           3. (greater(x65,x64)→^*second⟹if^#(second,s(0),s(x65),s(x64))≥c)
              • Applying Rule "Delete Conditions" results in
                if^#(second,s(0),s(x65),s(x64))≥c

              • This constraint is kept as final constraint.

           4. (false→^*second⟹if^#(second,s(s(x67)),s(s(x66)),s(0))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (greater(x56,x55)→^*second⟹s(x6)→^*0⟹s(x8)→^*x55⟹if^#(second,s(x6),x56,s(x8))≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      4. (false→^*second⟹if^#(second,s(x6),s(x57),s(x8))≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (if^#(le(s(x0),s(x1),x2),s(x0),s(x1),x2)→^*if^#(second,x3,x4,x5)⟹if^#(second,x3,x4,x5) > if^#(le(s(x3),s(x4),x5),s(x3),s(x4),x5))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (le(s(x0),s(x1),x2)→^*second⟹s(x0)→^*x3⟹s(x1)→^*x4⟹x2→^*x5⟹if^#(second,x3,x4,x5) > if^#(le(s(x3),s(x4),x5),s(x3),s(x4),x5))
    • Applying Rule "Variable in Equation" allows to substitute x3 by s(x0) which results in
      (le(s(x0),s(x1),x2)→^*second⟹s(x1)→^*x4⟹x2→^*x5⟹if^#(second,s(x0),x4,x5) > if^#(le(s(s(x0)),s(x4),x5),s(s(x0)),s(x4),x5))
    • Applying Rule "Variable in Equation" allows to substitute x4 by s(x1) which results in
      (le(s(x0),s(x1),x2)→^*second⟹x2→^*x5⟹if^#(second,s(x0),s(x1),x5) > if^#(le(s(s(x0)),s(s(x1)),x5),s(s(x0)),s(s(x1)),x5))
    • Applying Rule "Variable in Equation" allows to substitute x5 by x2 which results in
      (le(s(x0),s(x1),x2)→^*second⟹if^#(second,s(x0),s(x1),x2) > if^#(le(s(s(x0)),s(s(x1)),x2),s(s(x0)),s(s(x1)),x2))
    • Applying Rule "Introduce fresh variable" results in
      (s(x1)→^*x25⟹le(s(x0),x25,x2)→^*second⟹if^#(second,s(x0),s(x1),x2) > if^#(le(s(s(x0)),s(s(x1)),x2),s(s(x0)),s(s(x1)),x2))
    • Applying Rule "Introduce fresh variable" results in
      (s(x0)→^*x24⟹s(x1)→^*x25⟹le(x24,x25,x2)→^*second⟹if^#(second,s(x0),s(x1),x2) > if^#(le(s(s(x0)),s(s(x1)),x2),s(s(x0)),s(s(x1)),x2))
    • Applying Rule "Induction" on le(x24,x25,x2)→^*second results in the following 4 new constraints.
      1. (false→^*second⟹if^#(second,s(x0),s(x1),x26) > if^#(le(s(s(x0)),s(s(x1)),x26),s(s(x0)),s(s(x1)),x26))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (le(x30,x29,x28)→^*second⟹s(x0)→^*s(x30)⟹s(x1)→^*s(x29)⟹∀x31 . ∀x32 . (le(x30,x29,x28)→^*second⟹s(x31)→^*x30⟹s(x32)→^*x29⟹if^#(second,s(x31),s(x32),x28) > if^#(le(s(s(x31)),s(s(x32)),x28),s(s(x31)),s(s(x32)),x28))⟹if^#(second,s(x0),s(x1),s(x28)) > if^#(le(s(s(x0)),s(s(x1)),s(x28)),s(s(x0)),s(s(x1)),s(x28)))
         • Applying Rule "Same Constructor" results in
           (le(x30,x29,x28)→^*second⟹x0→^*x30⟹s(x1)→^*s(x29)⟹∀x31 . ∀x32 . (le(x30,x29,x28)→^*second⟹s(x31)→^*x30⟹s(x32)→^*x29⟹if^#(second,s(x31),s(x32),x28) > if^#(le(s(s(x31)),s(s(x32)),x28),s(s(x31)),s(s(x32)),x28))⟹if^#(second,s(x0),s(x1),s(x28)) > if^#(le(s(s(x0)),s(s(x1)),s(x28)),s(s(x0)),s(s(x1)),s(x28)))
         • Applying Rule "Same Constructor" results in
           (le(x30,x29,x28)→^*second⟹x0→^*x30⟹x1→^*x29⟹∀x31 . ∀x32 . (le(x30,x29,x28)→^*second⟹s(x31)→^*x30⟹s(x32)→^*x29⟹if^#(second,s(x31),s(x32),x28) > if^#(le(s(s(x31)),s(s(x32)),x28),s(s(x31)),s(s(x32)),x28))⟹if^#(second,s(x0),s(x1),s(x28)) > if^#(le(s(s(x0)),s(s(x1)),s(x28)),s(s(x0)),s(s(x1)),s(x28)))
         • Applying Rule "Variable in Equation" allows to substitute x0 by x30 which results in
           (le(x30,x29,x28)→^*second⟹x1→^*x29⟹∀x31 . ∀x32 . (le(x30,x29,x28)→^*second⟹s(x31)→^*x30⟹s(x32)→^*x29⟹if^#(second,s(x31),s(x32),x28) > if^#(le(s(s(x31)),s(s(x32)),x28),s(s(x31)),s(s(x32)),x28))⟹if^#(second,s(x30),s(x1),s(x28)) > if^#(le(s(s(x30)),s(s(x1)),s(x28)),s(s(x30)),s(s(x1)),s(x28)))
         • Applying Rule "Variable in Equation" allows to substitute x1 by x29 which results in
           (le(x30,x29,x28)→^*second⟹∀x31 . ∀x32 . (le(x30,x29,x28)→^*second⟹s(x31)→^*x30⟹s(x32)→^*x29⟹if^#(second,s(x31),s(x32),x28) > if^#(le(s(s(x31)),s(s(x32)),x28),s(s(x31)),s(s(x32)),x28))⟹if^#(second,s(x30),s(x29),s(x28)) > if^#(le(s(s(x30)),s(s(x29)),s(x28)),s(s(x30)),s(s(x29)),s(x28)))
         • Applying Rule "Delete Conditions" results in
           (le(x30,x29,x28)→^*second⟹if^#(second,s(x30),s(x29),s(x28)) > if^#(le(s(s(x30)),s(s(x29)),s(x28)),s(s(x30)),s(s(x29)),s(x28)))
         • Applying Rule "Induction" on le(x30,x29,x28)→^*second results in the following 4 new constraints.
           1. (false→^*second⟹if^#(second,s(s(x38)),s(0),s(x37)) > if^#(le(s(s(s(x38))),s(s(0)),s(x37)),s(s(s(x38))),s(s(0)),s(x37)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (le(x41,x40,x39)→^*second⟹(le(x41,x40,x39)→^*second⟹if^#(second,s(x41),s(x40),s(x39)) > if^#(le(s(s(x41)),s(s(x40)),s(x39)),s(s(x41)),s(s(x40)),s(x39)))⟹if^#(second,s(s(x41)),s(s(x40)),s(s(x39))) > if^#(le(s(s(s(x41))),s(s(s(x40))),s(s(x39))),s(s(s(x41))),s(s(s(x40))),s(s(x39))))
              • Applying Rule "Simplify Conditions" results in
                (if^#(second,s(x41),s(x40),s(x39)) > if^#(le(s(s(x41)),s(s(x40)),s(x39)),s(s(x41)),s(s(x40)),s(x39))⟹if^#(second,s(s(x41)),s(s(x40)),s(s(x39))) > if^#(le(s(s(s(x41))),s(s(s(x40))),s(s(x39))),s(s(s(x41))),s(s(s(x40))),s(s(x39))))

              • This constraint is kept as final constraint.

           3. (greater(x43,x42)→^*second⟹if^#(second,s(0),s(x43),s(x42)) > if^#(le(s(s(0)),s(s(x43)),s(x42)),s(s(0)),s(s(x43)),s(x42)))
              • Applying Rule "Delete Conditions" results in
                if^#(second,s(0),s(x43),s(x42)) > if^#(le(s(s(0)),s(s(x43)),s(x42)),s(s(0)),s(s(x43)),s(x42))

              • This constraint is kept as final constraint.

           4. (false→^*second⟹if^#(second,s(s(x45)),s(s(x44)),s(0)) > if^#(le(s(s(s(x45))),s(s(s(x44))),s(0)),s(s(s(x45))),s(s(s(x44))),s(0)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (greater(x34,x33)→^*second⟹s(x0)→^*0⟹s(x1)→^*x34⟹if^#(second,s(x0),s(x1),x33) > if^#(le(s(s(x0)),s(s(x1)),x33),s(s(x0)),s(s(x1)),x33))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      4. (false→^*second⟹if^#(second,s(x0),s(x1),0) > if^#(le(s(s(x0)),s(s(x1)),0),s(s(x0)),s(s(x1)),0))
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (if^#(le(s(x6),x7,s(x8)),s(x6),x7,s(x8))→^*if^#(second,x9,x10,x11)⟹if^#(second,x9,x10,x11) > if^#(le(s(x9),s(x10),x11),s(x9),s(x10),x11))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (le(s(x6),x7,s(x8))→^*second⟹s(x6)→^*x9⟹x7→^*x10⟹s(x8)→^*x11⟹if^#(second,x9,x10,x11) > if^#(le(s(x9),s(x10),x11),s(x9),s(x10),x11))
    • Applying Rule "Variable in Equation" allows to substitute x9 by s(x6) which results in
      (le(s(x6),x7,s(x8))→^*second⟹x7→^*x10⟹s(x8)→^*x11⟹if^#(second,s(x6),x10,x11) > if^#(le(s(s(x6)),s(x10),x11),s(s(x6)),s(x10),x11))
    • Applying Rule "Variable in Equation" allows to substitute x10 by x7 which results in
      (le(s(x6),x7,s(x8))→^*second⟹s(x8)→^*x11⟹if^#(second,s(x6),x7,x11) > if^#(le(s(s(x6)),s(x7),x11),s(s(x6)),s(x7),x11))
    • Applying Rule "Variable in Equation" allows to substitute x11 by s(x8) which results in
      (le(s(x6),x7,s(x8))→^*second⟹if^#(second,s(x6),x7,s(x8)) > if^#(le(s(s(x6)),s(x7),s(x8)),s(s(x6)),s(x7),s(x8)))
    • Applying Rule "Introduce fresh variable" results in
      (s(x8)→^*x47⟹le(s(x6),x7,x47)→^*second⟹if^#(second,s(x6),x7,s(x8)) > if^#(le(s(s(x6)),s(x7),s(x8)),s(s(x6)),s(x7),s(x8)))
    • Applying Rule "Introduce fresh variable" results in
      (s(x6)→^*x46⟹s(x8)→^*x47⟹le(x46,x7,x47)→^*second⟹if^#(second,s(x6),x7,s(x8)) > if^#(le(s(s(x6)),s(x7),s(x8)),s(s(x6)),s(x7),s(x8)))
    • Applying Rule "Induction" on le(x46,x7,x47)→^*second results in the following 4 new constraints.
      1. (false→^*second⟹if^#(second,s(x6),0,s(x8)) > if^#(le(s(s(x6)),s(0),s(x8)),s(s(x6)),s(0),s(x8)))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (le(x52,x51,x50)→^*second⟹s(x6)→^*s(x52)⟹s(x8)→^*s(x50)⟹∀x53 . ∀x54 . (le(x52,x51,x50)→^*second⟹s(x53)→^*x52⟹s(x54)→^*x50⟹if^#(second,s(x53),x51,s(x54)) > if^#(le(s(s(x53)),s(x51),s(x54)),s(s(x53)),s(x51),s(x54)))⟹if^#(second,s(x6),s(x51),s(x8)) > if^#(le(s(s(x6)),s(s(x51)),s(x8)),s(s(x6)),s(s(x51)),s(x8)))
         • Applying Rule "Same Constructor" results in
           (le(x52,x51,x50)→^*second⟹x6→^*x52⟹s(x8)→^*s(x50)⟹∀x53 . ∀x54 . (le(x52,x51,x50)→^*second⟹s(x53)→^*x52⟹s(x54)→^*x50⟹if^#(second,s(x53),x51,s(x54)) > if^#(le(s(s(x53)),s(x51),s(x54)),s(s(x53)),s(x51),s(x54)))⟹if^#(second,s(x6),s(x51),s(x8)) > if^#(le(s(s(x6)),s(s(x51)),s(x8)),s(s(x6)),s(s(x51)),s(x8)))
         • Applying Rule "Same Constructor" results in
           (le(x52,x51,x50)→^*second⟹x6→^*x52⟹x8→^*x50⟹∀x53 . ∀x54 . (le(x52,x51,x50)→^*second⟹s(x53)→^*x52⟹s(x54)→^*x50⟹if^#(second,s(x53),x51,s(x54)) > if^#(le(s(s(x53)),s(x51),s(x54)),s(s(x53)),s(x51),s(x54)))⟹if^#(second,s(x6),s(x51),s(x8)) > if^#(le(s(s(x6)),s(s(x51)),s(x8)),s(s(x6)),s(s(x51)),s(x8)))
         • Applying Rule "Variable in Equation" allows to substitute x6 by x52 which results in
           (le(x52,x51,x50)→^*second⟹x8→^*x50⟹∀x53 . ∀x54 . (le(x52,x51,x50)→^*second⟹s(x53)→^*x52⟹s(x54)→^*x50⟹if^#(second,s(x53),x51,s(x54)) > if^#(le(s(s(x53)),s(x51),s(x54)),s(s(x53)),s(x51),s(x54)))⟹if^#(second,s(x52),s(x51),s(x8)) > if^#(le(s(s(x52)),s(s(x51)),s(x8)),s(s(x52)),s(s(x51)),s(x8)))
         • Applying Rule "Variable in Equation" allows to substitute x8 by x50 which results in
           (le(x52,x51,x50)→^*second⟹∀x53 . ∀x54 . (le(x52,x51,x50)→^*second⟹s(x53)→^*x52⟹s(x54)→^*x50⟹if^#(second,s(x53),x51,s(x54)) > if^#(le(s(s(x53)),s(x51),s(x54)),s(s(x53)),s(x51),s(x54)))⟹if^#(second,s(x52),s(x51),s(x50)) > if^#(le(s(s(x52)),s(s(x51)),s(x50)),s(s(x52)),s(s(x51)),s(x50)))
         • Applying Rule "Delete Conditions" results in
           (le(x52,x51,x50)→^*second⟹if^#(second,s(x52),s(x51),s(x50)) > if^#(le(s(s(x52)),s(s(x51)),s(x50)),s(s(x52)),s(s(x51)),s(x50)))
         • Applying Rule "Induction" on le(x52,x51,x50)→^*second results in the following 4 new constraints.
           1. (false→^*second⟹if^#(second,s(s(x60)),s(0),s(x59)) > if^#(le(s(s(s(x60))),s(s(0)),s(x59)),s(s(s(x60))),s(s(0)),s(x59)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (le(x63,x62,x61)→^*second⟹(le(x63,x62,x61)→^*second⟹if^#(second,s(x63),s(x62),s(x61)) > if^#(le(s(s(x63)),s(s(x62)),s(x61)),s(s(x63)),s(s(x62)),s(x61)))⟹if^#(second,s(s(x63)),s(s(x62)),s(s(x61))) > if^#(le(s(s(s(x63))),s(s(s(x62))),s(s(x61))),s(s(s(x63))),s(s(s(x62))),s(s(x61))))
              • Applying Rule "Simplify Conditions" results in
                (if^#(second,s(x63),s(x62),s(x61)) > if^#(le(s(s(x63)),s(s(x62)),s(x61)),s(s(x63)),s(s(x62)),s(x61))⟹if^#(second,s(s(x63)),s(s(x62)),s(s(x61))) > if^#(le(s(s(s(x63))),s(s(s(x62))),s(s(x61))),s(s(s(x63))),s(s(s(x62))),s(s(x61))))

              • This constraint is kept as final constraint.

           3. (greater(x65,x64)→^*second⟹if^#(second,s(0),s(x65),s(x64)) > if^#(le(s(s(0)),s(s(x65)),s(x64)),s(s(0)),s(s(x65)),s(x64)))
              • Applying Rule "Delete Conditions" results in
                if^#(second,s(0),s(x65),s(x64)) > if^#(le(s(s(0)),s(s(x65)),s(x64)),s(s(0)),s(s(x65)),s(x64))

              • This constraint is kept as final constraint.

           4. (false→^*second⟹if^#(second,s(s(x67)),s(s(x66)),s(0)) > if^#(le(s(s(s(x67))),s(s(s(x66))),s(0)),s(s(s(x67))),s(s(s(x66))),s(0)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (greater(x56,x55)→^*second⟹s(x6)→^*0⟹s(x8)→^*x55⟹if^#(second,s(x6),x56,s(x8)) > if^#(le(s(s(x6)),s(x56),s(x8)),s(s(x6)),s(x56),s(x8)))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      4. (false→^*second⟹if^#(second,s(x6),s(x57),s(x8)) > if^#(le(s(s(x6)),s(s(x57)),s(x8)),s(s(x6)),s(s(x57)),s(x8)))
         • Applying Rule "Different Constructors" allows to drop this constraint.

  We remove those pairs which are strictly decreasing and bounded.

  1.1.1.1.1.1.1.1 P is empty

  There are no pairs anymore.

• The 2^nd component contains the pair

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

  →

  le#(x,y,z)

  (15)

  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.

  le#(s(x),s(y),s(z))	→	le#(x,y,z)	(15)
  1	>	1
  2	>	2
  3	>	3

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

• The 3^rd component contains the pair

  greater#(s(x),s(y))

  →

  greater#(x,y)

  (16)

  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.

  greater#(s(x),s(y))	→	greater#(x,y)	(16)
  1	>	1
  2	>	2

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

• The 4^th component contains the pair

  double#(s(x))

  →

  double#(x)

  (17)

  1.1.1.4 Usable Rules Processor

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

  There are no rules.

  1.1.1.4.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  1.1.1.4.1.1 Size-Change Termination

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

  double#(s(x))	→	double#(x)	(17)
  1	>	1

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