Certification Problem

Input (TPDB TRS_Standard/GTSSK07/cade14)

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

  cond1#(false,x,y)	→	cond2#(gt(x,y),x,y)	(15)
  cond2#(true,x,y)	→	diff#(x,s(y))	(17)
  diff#(x,y)	→	cond1#(equal(x,y),x,y)	(13)
  cond2#(false,x,y)	→	diff#(s(x),y)	(18)

  1.1.1.1 Usable Rules Processor

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

  equal(0,0)	→	true	(9)
  equal(s(x),0)	→	false	(10)
  equal(0,s(y))	→	false	(11)
  equal(s(x),s(y))	→	equal(x,y)	(12)
  gt(0,v)	→	false	(6)
  gt(s(u),0)	→	true	(7)
  gt(s(u),s(v))	→	gt(u,v)	(8)

  1.1.1.1.1 Innermost Lhss Removal Processor

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

  gt(0,x0)

  gt(s(x0),0)

  gt(s(x0),s(x1))

  equal(0,0)

  equal(s(x0),0)

  equal(0,s(x0))

  equal(s(x0),s(x1))

  1.1.1.1.1.1 Reduction Pair Processor

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

  [c]	=	-1
  [cond2#(x1, x2, x3)]	=	1 · x1 · x1 + -2 · x1 · x2 + 2 · x1 · x3 + 1 · x2 · x2 + -2 · x2 · x3 + 1 · x3 · x3 + -1
  [cond1#(x1, x2, x3)]	=	1 · x1 · x1 + 2 · x1 · x2 + -2 · x1 · x3 + 1 · x2 · x2 + -2 · x2 · x3 + 1 · x3 · x3 + -1
  [diff#(x1, x2)]	=	1 · x1 · x1 + -2 · x1 · x2 + 1 · x2 · x2 + -1
  [false]	=	0
  [gt(x1, x2)]	=	0
  [true]	=	0
  [s(x1)]	=	1 · x1 + 1
  [equal(x1, x2)]	=	0
  [0]	=	0

  The pair(s)

  cond2#(true,x,y)	→	diff#(x,s(y))	(17)
  cond2#(false,x,y)	→	diff#(s(x),y)	(18)

  are strictly oriented and the pair(s)

  cond1#(false,x,y)	→	cond2#(gt(x,y),x,y)	(15)
  cond2#(true,x,y)	→	diff#(x,s(y))	(17)
  diff#(x,y)	→	cond1#(equal(x,y),x,y)	(13)
  cond2#(false,x,y)	→	diff#(s(x),y)	(18)

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

  The following constraints are generated for the pairs.

  • cond1^#(false,0,s(x23))≥c
  • cond1^#(false,s(s(x185)),s(0))≥c
  • cond1^#(false,s(0),s(s(x186)))≥c
  • (cond1^#(false,s(x188),s(x187))≥c⟹cond1^#(false,s(s(x188)),s(s(x187)))≥c)
  • cond1^#(false,s(x30),0)≥c
  • cond1^#(false,s(s(x194)),s(0))≥c
  • cond1^#(false,s(0),s(s(x195)))≥c
  • (cond1^#(false,s(x197),s(x196))≥c⟹cond1^#(false,s(s(x197)),s(s(x196)))≥c)
  • cond2^#(true,s(x198),0)≥c
  • (cond2^#(true,x217,x216)≥c⟹cond2^#(true,s(x217),s(x216))≥c)
  • diff^#(s(x223),s(0))≥c
  • (diff^#(x225,s(x224))≥c⟹diff^#(s(x225),s(s(x224)))≥c)
  • diff^#(s(0),x226)≥c
  • (diff^#(s(x229),x228)≥c⟹diff^#(s(s(x229)),s(x228))≥c)
  • cond2^#(false,0,s(x231))≥c
  • (cond2^#(false,x249,x248)≥c⟹cond2^#(false,s(x249),s(x248))≥c)
  • cond1^#(false,0,s(x23))≥cond2^#(gt(0,s(x23)),0,s(x23))
  • cond1^#(false,s(s(x185)),s(0))≥cond2^#(gt(s(s(x185)),s(0)),s(s(x185)),s(0))
  • cond1^#(false,s(0),s(s(x186)))≥cond2^#(gt(s(0),s(s(x186))),s(0),s(s(x186)))
  • (cond1^#(false,s(x188),s(x187))≥cond2^#(gt(s(x188),s(x187)),s(x188),s(x187))⟹cond1^#(false,s(s(x188)),s(s(x187)))≥cond2^#(gt(s(s(x188)),s(s(x187))),s(s(x188)),s(s(x187))))
  • cond1^#(false,s(x30),0)≥cond2^#(gt(s(x30),0),s(x30),0)
  • cond1^#(false,s(s(x194)),s(0))≥cond2^#(gt(s(s(x194)),s(0)),s(s(x194)),s(0))
  • cond1^#(false,s(0),s(s(x195)))≥cond2^#(gt(s(0),s(s(x195))),s(0),s(s(x195)))
  • (cond1^#(false,s(x197),s(x196))≥cond2^#(gt(s(x197),s(x196)),s(x197),s(x196))⟹cond1^#(false,s(s(x197)),s(s(x196)))≥cond2^#(gt(s(s(x197)),s(s(x196))),s(s(x197)),s(s(x196))))
  • cond2^#(true,s(x198),0) > diff^#(s(x198),s(0))
  • (cond2^#(true,x217,x216) > diff^#(x217,s(x216))⟹cond2^#(true,s(x217),s(x216)) > diff^#(s(x217),s(s(x216))))
  • diff^#(s(x223),s(0))≥cond1^#(equal(s(x223),s(0)),s(x223),s(0))
  • (diff^#(x225,s(x224))≥cond1^#(equal(x225,s(x224)),x225,s(x224))⟹diff^#(s(x225),s(s(x224)))≥cond1^#(equal(s(x225),s(s(x224))),s(x225),s(s(x224))))
  • diff^#(s(0),x226)≥cond1^#(equal(s(0),x226),s(0),x226)
  • (diff^#(s(x229),x228)≥cond1^#(equal(s(x229),x228),s(x229),x228)⟹diff^#(s(s(x229)),s(x228))≥cond1^#(equal(s(s(x229)),s(x228)),s(s(x229)),s(x228)))
  • cond2^#(false,0,s(x231)) > diff^#(s(0),s(x231))
  • (cond2^#(false,x249,x248) > diff^#(s(x249),x248)⟹cond2^#(false,s(x249),s(x248)) > diff^#(s(s(x249)),s(x248)))

  The details are shown below:

  • For the chain we build the initial constraint

    (diff^#(x22,s(x23))→^*diff^#(x24,x25)⟹cond1^#(equal(x24,x25),x24,x25)→^*cond1^#(false,x26,x27)⟹cond1^#(false,x26,x27)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (x22→^*x24⟹s(x23)→^*x25⟹cond1^#(equal(x24,x25),x24,x25)→^*cond1^#(false,x26,x27)⟹cond1^#(false,x26,x27)≥c)
    • Applying Rule "Same Constructor" results in
      (x22→^*x24⟹s(x23)→^*x25⟹equal(x24,x25)→^*false⟹x24→^*x26⟹x25→^*x27⟹cond1^#(false,x26,x27)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x26 by x24 which results in
      (x22→^*x24⟹s(x23)→^*x25⟹equal(x24,x25)→^*false⟹x25→^*x27⟹cond1^#(false,x24,x27)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x27 by x25 which results in
      (x22→^*x24⟹s(x23)→^*x25⟹equal(x24,x25)→^*false⟹cond1^#(false,x24,x25)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x22 by x24 which results in
      (s(x23)→^*x25⟹equal(x24,x25)→^*false⟹cond1^#(false,x24,x25)≥c)
    • Applying Rule "Induction" on equal(x24,x25)→^*false results in the following 4 new constraints.
      1. (true→^*false⟹cond1^#(false,0,0)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*false⟹s(x23)→^*0⟹cond1^#(false,s(x180),0)≥c)
         • Applying Rule "Same Constructor" results in
           (s(x23)→^*0⟹cond1^#(false,s(x180),0)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (false→^*false⟹s(x23)→^*s(x181)⟹cond1^#(false,0,s(x181))≥c)
         • Applying Rule "Same Constructor" results in
           (s(x23)→^*s(x181)⟹cond1^#(false,0,s(x181))≥c)
         • Applying Rule "Same Constructor" results in
           (x23→^*x181⟹cond1^#(false,0,s(x181))≥c)
         • Applying Rule "Variable in Equation" allows to substitute x181 by x23 which results in
           cond1^#(false,0,s(x23))≥c

         • This constraint is kept as final constraint.

      4. (equal(x183,x182)→^*false⟹s(x23)→^*s(x182)⟹∀x184 . (equal(x183,x182)→^*false⟹s(x184)→^*x182⟹cond1^#(false,x183,x182)≥c)⟹cond1^#(false,s(x183),s(x182))≥c)
         • Applying Rule "Same Constructor" results in
           (equal(x183,x182)→^*false⟹x23→^*x182⟹∀x184 . (equal(x183,x182)→^*false⟹s(x184)→^*x182⟹cond1^#(false,x183,x182)≥c)⟹cond1^#(false,s(x183),s(x182))≥c)
         • Applying Rule "Variable in Equation" allows to substitute x23 by x182 which results in
           (equal(x183,x182)→^*false⟹∀x184 . (equal(x183,x182)→^*false⟹s(x184)→^*x182⟹cond1^#(false,x183,x182)≥c)⟹cond1^#(false,s(x183),s(x182))≥c)
         • Applying Rule "Delete Conditions" results in
           (equal(x183,x182)→^*false⟹cond1^#(false,s(x183),s(x182))≥c)
         • Applying Rule "Induction" on equal(x183,x182)→^*false results in the following 4 new constraints.
           1. (true→^*false⟹cond1^#(false,s(0),s(0))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (false→^*false⟹cond1^#(false,s(s(x185)),s(0))≥c)
              • Applying Rule "Same Constructor" results in
                cond1^#(false,s(s(x185)),s(0))≥c

              • This constraint is kept as final constraint.

           3. (false→^*false⟹cond1^#(false,s(0),s(s(x186)))≥c)
              • Applying Rule "Same Constructor" results in
                cond1^#(false,s(0),s(s(x186)))≥c

              • This constraint is kept as final constraint.

           4. (equal(x188,x187)→^*false⟹(equal(x188,x187)→^*false⟹cond1^#(false,s(x188),s(x187))≥c)⟹cond1^#(false,s(s(x188)),s(s(x187)))≥c)
              • Applying Rule "Simplify Conditions" results in
                (cond1^#(false,s(x188),s(x187))≥c⟹cond1^#(false,s(s(x188)),s(s(x187)))≥c)

              • This constraint is kept as final constraint.

  • For the chain we build the initial constraint

    (diff^#(s(x30),x31)→^*diff^#(x32,x33)⟹cond1^#(equal(x32,x33),x32,x33)→^*cond1^#(false,x34,x35)⟹cond1^#(false,x34,x35)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (s(x30)→^*x32⟹x31→^*x33⟹cond1^#(equal(x32,x33),x32,x33)→^*cond1^#(false,x34,x35)⟹cond1^#(false,x34,x35)≥c)
    • Applying Rule "Same Constructor" results in
      (s(x30)→^*x32⟹x31→^*x33⟹equal(x32,x33)→^*false⟹x32→^*x34⟹x33→^*x35⟹cond1^#(false,x34,x35)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x34 by x32 which results in
      (s(x30)→^*x32⟹x31→^*x33⟹equal(x32,x33)→^*false⟹x33→^*x35⟹cond1^#(false,x32,x35)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x35 by x33 which results in
      (s(x30)→^*x32⟹x31→^*x33⟹equal(x32,x33)→^*false⟹cond1^#(false,x32,x33)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x31 by x33 which results in
      (s(x30)→^*x32⟹equal(x32,x33)→^*false⟹cond1^#(false,x32,x33)≥c)
    • Applying Rule "Induction" on equal(x32,x33)→^*false results in the following 4 new constraints.
      1. (true→^*false⟹cond1^#(false,0,0)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*false⟹s(x30)→^*s(x189)⟹cond1^#(false,s(x189),0)≥c)
         • Applying Rule "Same Constructor" results in
           (s(x30)→^*s(x189)⟹cond1^#(false,s(x189),0)≥c)
         • Applying Rule "Same Constructor" results in
           (x30→^*x189⟹cond1^#(false,s(x189),0)≥c)
         • Applying Rule "Variable in Equation" allows to substitute x189 by x30 which results in
           cond1^#(false,s(x30),0)≥c

         • This constraint is kept as final constraint.

      3. (false→^*false⟹s(x30)→^*0⟹cond1^#(false,0,s(x190))≥c)
         • Applying Rule "Same Constructor" results in
           (s(x30)→^*0⟹cond1^#(false,0,s(x190))≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      4. (equal(x192,x191)→^*false⟹s(x30)→^*s(x192)⟹∀x193 . (equal(x192,x191)→^*false⟹s(x193)→^*x192⟹cond1^#(false,x192,x191)≥c)⟹cond1^#(false,s(x192),s(x191))≥c)
         • Applying Rule "Same Constructor" results in
           (equal(x192,x191)→^*false⟹x30→^*x192⟹∀x193 . (equal(x192,x191)→^*false⟹s(x193)→^*x192⟹cond1^#(false,x192,x191)≥c)⟹cond1^#(false,s(x192),s(x191))≥c)
         • Applying Rule "Variable in Equation" allows to substitute x30 by x192 which results in
           (equal(x192,x191)→^*false⟹∀x193 . (equal(x192,x191)→^*false⟹s(x193)→^*x192⟹cond1^#(false,x192,x191)≥c)⟹cond1^#(false,s(x192),s(x191))≥c)
         • Applying Rule "Delete Conditions" results in
           (equal(x192,x191)→^*false⟹cond1^#(false,s(x192),s(x191))≥c)
         • Applying Rule "Induction" on equal(x192,x191)→^*false results in the following 4 new constraints.
           1. (true→^*false⟹cond1^#(false,s(0),s(0))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (false→^*false⟹cond1^#(false,s(s(x194)),s(0))≥c)
              • Applying Rule "Same Constructor" results in
                cond1^#(false,s(s(x194)),s(0))≥c

              • This constraint is kept as final constraint.

           3. (false→^*false⟹cond1^#(false,s(0),s(s(x195)))≥c)
              • Applying Rule "Same Constructor" results in
                cond1^#(false,s(0),s(s(x195)))≥c

              • This constraint is kept as final constraint.

           4. (equal(x197,x196)→^*false⟹(equal(x197,x196)→^*false⟹cond1^#(false,s(x197),s(x196))≥c)⟹cond1^#(false,s(s(x197)),s(s(x196)))≥c)
              • Applying Rule "Simplify Conditions" results in
                (cond1^#(false,s(x197),s(x196))≥c⟹cond1^#(false,s(s(x197)),s(s(x196)))≥c)

              • This constraint is kept as final constraint.

  • For the chain we build the initial constraint

    (cond1^#(equal(x50,x51),x50,x51)→^*cond1^#(false,x52,x53)⟹cond2^#(gt(x52,x53),x52,x53)→^*cond2^#(true,x54,x55)⟹cond2^#(true,x54,x55)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (equal(x50,x51)→^*false⟹x50→^*x52⟹x51→^*x53⟹cond2^#(gt(x52,x53),x52,x53)→^*cond2^#(true,x54,x55)⟹cond2^#(true,x54,x55)≥c)
    • Applying Rule "Same Constructor" results in
      (equal(x50,x51)→^*false⟹x50→^*x52⟹x51→^*x53⟹gt(x52,x53)→^*true⟹x52→^*x54⟹x53→^*x55⟹cond2^#(true,x54,x55)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x54 by x52 which results in
      (equal(x50,x51)→^*false⟹x50→^*x52⟹x51→^*x53⟹gt(x52,x53)→^*true⟹x53→^*x55⟹cond2^#(true,x52,x55)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x55 by x53 which results in
      (equal(x50,x51)→^*false⟹x50→^*x52⟹x51→^*x53⟹gt(x52,x53)→^*true⟹cond2^#(true,x52,x53)≥c)
    • Applying Rule "Induction" on equal(x50,x51)→^*false results in the following 4 new constraints.
      1. (true→^*false⟹cond2^#(true,x52,x53)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*false⟹s(x198)→^*x52⟹0→^*x53⟹gt(x52,x53)→^*true⟹cond2^#(true,x52,x53)≥c)
         • Applying Rule "Same Constructor" results in
           (s(x198)→^*x52⟹0→^*x53⟹gt(x52,x53)→^*true⟹cond2^#(true,x52,x53)≥c)
         • Applying Rule "Induction" on gt(x52,x53)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond2^#(true,0,x204)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹s(x198)→^*s(x205)⟹0→^*0⟹cond2^#(true,s(x205),0)≥c)
              • Applying Rule "Same Constructor" results in
                (s(x198)→^*s(x205)⟹0→^*0⟹cond2^#(true,s(x205),0)≥c)
              • Applying Rule "Same Constructor" results in
                (x198→^*x205⟹0→^*0⟹cond2^#(true,s(x205),0)≥c)
              • Applying Rule "Same Constructor" results in
                (x198→^*x205⟹cond2^#(true,s(x205),0)≥c)
              • Applying Rule "Variable in Equation" allows to substitute x205 by x198 which results in
                cond2^#(true,s(x198),0)≥c

              • This constraint is kept as final constraint.

           3. (gt(x207,x206)→^*true⟹s(x198)→^*s(x207)⟹0→^*s(x206)⟹∀x208 . (gt(x207,x206)→^*true⟹s(x208)→^*x207⟹0→^*x206⟹cond2^#(true,x207,x206)≥c)⟹cond2^#(true,s(x207),s(x206))≥c)
              • Applying Rule "Same Constructor" results in
                (gt(x207,x206)→^*true⟹x198→^*x207⟹0→^*s(x206)⟹∀x208 . (gt(x207,x206)→^*true⟹s(x208)→^*x207⟹0→^*x206⟹cond2^#(true,x207,x206)≥c)⟹cond2^#(true,s(x207),s(x206))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (false→^*false⟹0→^*x52⟹s(x199)→^*x53⟹gt(x52,x53)→^*true⟹cond2^#(true,x52,x53)≥c)
         • Applying Rule "Same Constructor" results in
           (0→^*x52⟹s(x199)→^*x53⟹gt(x52,x53)→^*true⟹cond2^#(true,x52,x53)≥c)
         • Applying Rule "Induction" on gt(x52,x53)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond2^#(true,0,x209)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹0→^*s(x210)⟹s(x199)→^*0⟹cond2^#(true,s(x210),0)≥c)
              • Applying Rule "Same Constructor" results in
                (0→^*s(x210)⟹s(x199)→^*0⟹cond2^#(true,s(x210),0)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gt(x212,x211)→^*true⟹0→^*s(x212)⟹s(x199)→^*s(x211)⟹∀x213 . (gt(x212,x211)→^*true⟹0→^*x212⟹s(x213)→^*x211⟹cond2^#(true,x212,x211)≥c)⟹cond2^#(true,s(x212),s(x211))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      4. (equal(x201,x200)→^*false⟹s(x201)→^*x52⟹s(x200)→^*x53⟹gt(x52,x53)→^*true⟹∀x202 . ∀x203 . (equal(x201,x200)→^*false⟹x201→^*x202⟹x200→^*x203⟹gt(x202,x203)→^*true⟹cond2^#(true,x202,x203)≥c)⟹cond2^#(true,x52,x53)≥c)
         • Applying Rule "Induction" on gt(x52,x53)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond2^#(true,0,x214)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹equal(x201,x200)→^*false⟹s(x201)→^*s(x215)⟹s(x200)→^*0⟹∀x202 . ∀x203 . (equal(x201,x200)→^*false⟹x201→^*x202⟹x200→^*x203⟹gt(x202,x203)→^*true⟹cond2^#(true,x202,x203)≥c)⟹cond2^#(true,s(x215),0)≥c)
              • Applying Rule "Same Constructor" results in
                (equal(x201,x200)→^*false⟹s(x201)→^*s(x215)⟹s(x200)→^*0⟹∀x202 . ∀x203 . (equal(x201,x200)→^*false⟹x201→^*x202⟹x200→^*x203⟹gt(x202,x203)→^*true⟹cond2^#(true,x202,x203)≥c)⟹cond2^#(true,s(x215),0)≥c)
              • Applying Rule "Same Constructor" results in
                (equal(x201,x200)→^*false⟹x201→^*x215⟹s(x200)→^*0⟹∀x202 . ∀x203 . (equal(x201,x200)→^*false⟹x201→^*x202⟹x200→^*x203⟹gt(x202,x203)→^*true⟹cond2^#(true,x202,x203)≥c)⟹cond2^#(true,s(x215),0)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gt(x217,x216)→^*true⟹equal(x201,x200)→^*false⟹s(x201)→^*s(x217)⟹s(x200)→^*s(x216)⟹∀x202 . ∀x203 . (equal(x201,x200)→^*false⟹x201→^*x202⟹x200→^*x203⟹gt(x202,x203)→^*true⟹cond2^#(true,x202,x203)≥c)⟹∀x218 . ∀x219 . ∀x220 . ∀x221 . (gt(x217,x216)→^*true⟹equal(x218,x219)→^*false⟹s(x218)→^*x217⟹s(x219)→^*x216⟹∀x220 . ∀x221 . (equal(x218,x219)→^*false⟹x218→^*x220⟹x219→^*x221⟹gt(x220,x221)→^*true⟹cond2^#(true,x220,x221)≥c)⟹cond2^#(true,x217,x216)≥c)⟹cond2^#(true,s(x217),s(x216))≥c)
              • Applying Rule "Same Constructor" results in
                (gt(x217,x216)→^*true⟹equal(x201,x200)→^*false⟹x201→^*x217⟹s(x200)→^*s(x216)⟹∀x202 . ∀x203 . (equal(x201,x200)→^*false⟹x201→^*x202⟹x200→^*x203⟹gt(x202,x203)→^*true⟹cond2^#(true,x202,x203)≥c)⟹∀x218 . ∀x219 . ∀x220 . ∀x221 . (gt(x217,x216)→^*true⟹equal(x218,x219)→^*false⟹s(x218)→^*x217⟹s(x219)→^*x216⟹∀x220 . ∀x221 . (equal(x218,x219)→^*false⟹x218→^*x220⟹x219→^*x221⟹gt(x220,x221)→^*true⟹cond2^#(true,x220,x221)≥c)⟹cond2^#(true,x217,x216)≥c)⟹cond2^#(true,s(x217),s(x216))≥c)
              • Applying Rule "Same Constructor" results in
                (gt(x217,x216)→^*true⟹equal(x201,x200)→^*false⟹x201→^*x217⟹x200→^*x216⟹∀x202 . ∀x203 . (equal(x201,x200)→^*false⟹x201→^*x202⟹x200→^*x203⟹gt(x202,x203)→^*true⟹cond2^#(true,x202,x203)≥c)⟹∀x218 . ∀x219 . ∀x220 . ∀x221 . (gt(x217,x216)→^*true⟹equal(x218,x219)→^*false⟹s(x218)→^*x217⟹s(x219)→^*x216⟹∀x220 . ∀x221 . (equal(x218,x219)→^*false⟹x218→^*x220⟹x219→^*x221⟹gt(x220,x221)→^*true⟹cond2^#(true,x220,x221)≥c)⟹cond2^#(true,x217,x216)≥c)⟹cond2^#(true,s(x217),s(x216))≥c)
              • Applying Rule "Simplify Conditions" results in
                (cond2^#(true,x217,x216)≥c⟹∀x218 . ∀x219 . ∀x220 . ∀x221 . (gt(x217,x216)→^*true⟹equal(x218,x219)→^*false⟹s(x218)→^*x217⟹s(x219)→^*x216⟹∀x220 . ∀x221 . (equal(x218,x219)→^*false⟹x218→^*x220⟹x219→^*x221⟹gt(x220,x221)→^*true⟹cond2^#(true,x220,x221)≥c)⟹cond2^#(true,x217,x216)≥c)⟹cond2^#(true,s(x217),s(x216))≥c)
              • Applying Rule "Delete Conditions" results in
                (cond2^#(true,x217,x216)≥c⟹cond2^#(true,s(x217),s(x216))≥c)

              • This constraint is kept as final constraint.

  • For the chain we build the initial constraint

    (cond2^#(gt(x100,x101),x100,x101)→^*cond2^#(true,x102,x103)⟹diff^#(x102,s(x103))→^*diff^#(x104,x105)⟹diff^#(x104,x105)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gt(x100,x101)→^*true⟹x100→^*x102⟹x101→^*x103⟹diff^#(x102,s(x103))→^*diff^#(x104,x105)⟹diff^#(x104,x105)≥c)
    • Applying Rule "Same Constructor" results in
      (gt(x100,x101)→^*true⟹x100→^*x102⟹x101→^*x103⟹x102→^*x104⟹s(x103)→^*x105⟹diff^#(x104,x105)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x102 by x100 which results in
      (gt(x100,x101)→^*true⟹x101→^*x103⟹x100→^*x104⟹s(x103)→^*x105⟹diff^#(x104,x105)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x103 by x101 which results in
      (gt(x100,x101)→^*true⟹x100→^*x104⟹s(x101)→^*x105⟹diff^#(x104,x105)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x104 by x100 which results in
      (gt(x100,x101)→^*true⟹s(x101)→^*x105⟹diff^#(x100,x105)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x105 by s(x101) which results in
      (gt(x100,x101)→^*true⟹diff^#(x100,s(x101))≥c)
    • Applying Rule "Induction" on gt(x100,x101)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹diff^#(0,s(x222))≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹diff^#(s(x223),s(0))≥c)
         • Applying Rule "Same Constructor" results in
           diff^#(s(x223),s(0))≥c

         • This constraint is kept as final constraint.

      3. (gt(x225,x224)→^*true⟹(gt(x225,x224)→^*true⟹diff^#(x225,s(x224))≥c)⟹diff^#(s(x225),s(s(x224)))≥c)
         • Applying Rule "Simplify Conditions" results in
           (diff^#(x225,s(x224))≥c⟹diff^#(s(x225),s(s(x224)))≥c)

         • This constraint is kept as final constraint.

  • For the chain we build the initial constraint

    (cond2^#(gt(x124,x125),x124,x125)→^*cond2^#(false,x126,x127)⟹diff^#(s(x126),x127)→^*diff^#(x128,x129)⟹diff^#(x128,x129)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gt(x124,x125)→^*false⟹x124→^*x126⟹x125→^*x127⟹diff^#(s(x126),x127)→^*diff^#(x128,x129)⟹diff^#(x128,x129)≥c)
    • Applying Rule "Same Constructor" results in
      (gt(x124,x125)→^*false⟹x124→^*x126⟹x125→^*x127⟹s(x126)→^*x128⟹x127→^*x129⟹diff^#(x128,x129)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x126 by x124 which results in
      (gt(x124,x125)→^*false⟹x125→^*x127⟹s(x124)→^*x128⟹x127→^*x129⟹diff^#(x128,x129)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x127 by x125 which results in
      (gt(x124,x125)→^*false⟹s(x124)→^*x128⟹x125→^*x129⟹diff^#(x128,x129)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x128 by s(x124) which results in
      (gt(x124,x125)→^*false⟹x125→^*x129⟹diff^#(s(x124),x129)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x129 by x125 which results in
      (gt(x124,x125)→^*false⟹diff^#(s(x124),x125)≥c)
    • Applying Rule "Induction" on gt(x124,x125)→^*false results in the following 3 new constraints.
      1. (false→^*false⟹diff^#(s(0),x226)≥c)
         • Applying Rule "Same Constructor" results in
           diff^#(s(0),x226)≥c

         • This constraint is kept as final constraint.

      2. (true→^*false⟹diff^#(s(s(x227)),0)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gt(x229,x228)→^*false⟹(gt(x229,x228)→^*false⟹diff^#(s(x229),x228)≥c)⟹diff^#(s(s(x229)),s(x228))≥c)
         • Applying Rule "Simplify Conditions" results in
           (diff^#(s(x229),x228)≥c⟹diff^#(s(s(x229)),s(x228))≥c)

         • This constraint is kept as final constraint.

  • For the chain we build the initial constraint

    (cond1^#(equal(x140,x141),x140,x141)→^*cond1^#(false,x142,x143)⟹cond2^#(gt(x142,x143),x142,x143)→^*cond2^#(false,x144,x145)⟹cond2^#(false,x144,x145)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (equal(x140,x141)→^*false⟹x140→^*x142⟹x141→^*x143⟹cond2^#(gt(x142,x143),x142,x143)→^*cond2^#(false,x144,x145)⟹cond2^#(false,x144,x145)≥c)
    • Applying Rule "Same Constructor" results in
      (equal(x140,x141)→^*false⟹x140→^*x142⟹x141→^*x143⟹gt(x142,x143)→^*false⟹x142→^*x144⟹x143→^*x145⟹cond2^#(false,x144,x145)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x144 by x142 which results in
      (equal(x140,x141)→^*false⟹x140→^*x142⟹x141→^*x143⟹gt(x142,x143)→^*false⟹x143→^*x145⟹cond2^#(false,x142,x145)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x145 by x143 which results in
      (equal(x140,x141)→^*false⟹x140→^*x142⟹x141→^*x143⟹gt(x142,x143)→^*false⟹cond2^#(false,x142,x143)≥c)
    • Applying Rule "Induction" on equal(x140,x141)→^*false results in the following 4 new constraints.
      1. (true→^*false⟹cond2^#(false,x142,x143)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*false⟹s(x230)→^*x142⟹0→^*x143⟹gt(x142,x143)→^*false⟹cond2^#(false,x142,x143)≥c)
         • Applying Rule "Same Constructor" results in
           (s(x230)→^*x142⟹0→^*x143⟹gt(x142,x143)→^*false⟹cond2^#(false,x142,x143)≥c)
         • Applying Rule "Induction" on gt(x142,x143)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹s(x230)→^*0⟹0→^*x236⟹cond2^#(false,0,x236)≥c)
              • Applying Rule "Same Constructor" results in
                (s(x230)→^*0⟹0→^*x236⟹cond2^#(false,0,x236)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond2^#(false,s(x237),0)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gt(x239,x238)→^*false⟹s(x230)→^*s(x239)⟹0→^*s(x238)⟹∀x240 . (gt(x239,x238)→^*false⟹s(x240)→^*x239⟹0→^*x238⟹cond2^#(false,x239,x238)≥c)⟹cond2^#(false,s(x239),s(x238))≥c)
              • Applying Rule "Same Constructor" results in
                (gt(x239,x238)→^*false⟹x230→^*x239⟹0→^*s(x238)⟹∀x240 . (gt(x239,x238)→^*false⟹s(x240)→^*x239⟹0→^*x238⟹cond2^#(false,x239,x238)≥c)⟹cond2^#(false,s(x239),s(x238))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (false→^*false⟹0→^*x142⟹s(x231)→^*x143⟹gt(x142,x143)→^*false⟹cond2^#(false,x142,x143)≥c)
         • Applying Rule "Same Constructor" results in
           (0→^*x142⟹s(x231)→^*x143⟹gt(x142,x143)→^*false⟹cond2^#(false,x142,x143)≥c)
         • Applying Rule "Induction" on gt(x142,x143)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹0→^*0⟹s(x231)→^*x241⟹cond2^#(false,0,x241)≥c)
              • Applying Rule "Same Constructor" results in
                (0→^*0⟹s(x231)→^*x241⟹cond2^#(false,0,x241)≥c)
              • Applying Rule "Same Constructor" results in
                (s(x231)→^*x241⟹cond2^#(false,0,x241)≥c)
              • Applying Rule "Variable in Equation" allows to substitute x241 by s(x231) which results in
                cond2^#(false,0,s(x231))≥c

              • This constraint is kept as final constraint.

           2. (true→^*false⟹cond2^#(false,s(x242),0)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gt(x244,x243)→^*false⟹0→^*s(x244)⟹s(x231)→^*s(x243)⟹∀x245 . (gt(x244,x243)→^*false⟹0→^*x244⟹s(x245)→^*x243⟹cond2^#(false,x244,x243)≥c)⟹cond2^#(false,s(x244),s(x243))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      4. (equal(x233,x232)→^*false⟹s(x233)→^*x142⟹s(x232)→^*x143⟹gt(x142,x143)→^*false⟹∀x234 . ∀x235 . (equal(x233,x232)→^*false⟹x233→^*x234⟹x232→^*x235⟹gt(x234,x235)→^*false⟹cond2^#(false,x234,x235)≥c)⟹cond2^#(false,x142,x143)≥c)
         • Applying Rule "Induction" on gt(x142,x143)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹equal(x233,x232)→^*false⟹s(x233)→^*0⟹s(x232)→^*x246⟹∀x234 . ∀x235 . (equal(x233,x232)→^*false⟹x233→^*x234⟹x232→^*x235⟹gt(x234,x235)→^*false⟹cond2^#(false,x234,x235)≥c)⟹cond2^#(false,0,x246)≥c)
              • Applying Rule "Same Constructor" results in
                (equal(x233,x232)→^*false⟹s(x233)→^*0⟹s(x232)→^*x246⟹∀x234 . ∀x235 . (equal(x233,x232)→^*false⟹x233→^*x234⟹x232→^*x235⟹gt(x234,x235)→^*false⟹cond2^#(false,x234,x235)≥c)⟹cond2^#(false,0,x246)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond2^#(false,s(x247),0)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gt(x249,x248)→^*false⟹equal(x233,x232)→^*false⟹s(x233)→^*s(x249)⟹s(x232)→^*s(x248)⟹∀x234 . ∀x235 . (equal(x233,x232)→^*false⟹x233→^*x234⟹x232→^*x235⟹gt(x234,x235)→^*false⟹cond2^#(false,x234,x235)≥c)⟹∀x250 . ∀x251 . ∀x252 . ∀x253 . (gt(x249,x248)→^*false⟹equal(x250,x251)→^*false⟹s(x250)→^*x249⟹s(x251)→^*x248⟹∀x252 . ∀x253 . (equal(x250,x251)→^*false⟹x250→^*x252⟹x251→^*x253⟹gt(x252,x253)→^*false⟹cond2^#(false,x252,x253)≥c)⟹cond2^#(false,x249,x248)≥c)⟹cond2^#(false,s(x249),s(x248))≥c)
              • Applying Rule "Same Constructor" results in
                (gt(x249,x248)→^*false⟹equal(x233,x232)→^*false⟹x233→^*x249⟹s(x232)→^*s(x248)⟹∀x234 . ∀x235 . (equal(x233,x232)→^*false⟹x233→^*x234⟹x232→^*x235⟹gt(x234,x235)→^*false⟹cond2^#(false,x234,x235)≥c)⟹∀x250 . ∀x251 . ∀x252 . ∀x253 . (gt(x249,x248)→^*false⟹equal(x250,x251)→^*false⟹s(x250)→^*x249⟹s(x251)→^*x248⟹∀x252 . ∀x253 . (equal(x250,x251)→^*false⟹x250→^*x252⟹x251→^*x253⟹gt(x252,x253)→^*false⟹cond2^#(false,x252,x253)≥c)⟹cond2^#(false,x249,x248)≥c)⟹cond2^#(false,s(x249),s(x248))≥c)
              • Applying Rule "Same Constructor" results in
                (gt(x249,x248)→^*false⟹equal(x233,x232)→^*false⟹x233→^*x249⟹x232→^*x248⟹∀x234 . ∀x235 . (equal(x233,x232)→^*false⟹x233→^*x234⟹x232→^*x235⟹gt(x234,x235)→^*false⟹cond2^#(false,x234,x235)≥c)⟹∀x250 . ∀x251 . ∀x252 . ∀x253 . (gt(x249,x248)→^*false⟹equal(x250,x251)→^*false⟹s(x250)→^*x249⟹s(x251)→^*x248⟹∀x252 . ∀x253 . (equal(x250,x251)→^*false⟹x250→^*x252⟹x251→^*x253⟹gt(x252,x253)→^*false⟹cond2^#(false,x252,x253)≥c)⟹cond2^#(false,x249,x248)≥c)⟹cond2^#(false,s(x249),s(x248))≥c)
              • Applying Rule "Simplify Conditions" results in
                (cond2^#(false,x249,x248)≥c⟹∀x250 . ∀x251 . ∀x252 . ∀x253 . (gt(x249,x248)→^*false⟹equal(x250,x251)→^*false⟹s(x250)→^*x249⟹s(x251)→^*x248⟹∀x252 . ∀x253 . (equal(x250,x251)→^*false⟹x250→^*x252⟹x251→^*x253⟹gt(x252,x253)→^*false⟹cond2^#(false,x252,x253)≥c)⟹cond2^#(false,x249,x248)≥c)⟹cond2^#(false,s(x249),s(x248))≥c)
              • Applying Rule "Delete Conditions" results in
                (cond2^#(false,x249,x248)≥c⟹cond2^#(false,s(x249),s(x248))≥c)

              • This constraint is kept as final constraint.

  • For the chain we build the initial constraint

    (diff^#(x22,s(x23))→^*diff^#(x24,x25)⟹cond1^#(equal(x24,x25),x24,x25)→^*cond1^#(false,x26,x27)⟹cond1^#(false,x26,x27)≥cond2^#(gt(x26,x27),x26,x27))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (x22→^*x24⟹s(x23)→^*x25⟹cond1^#(equal(x24,x25),x24,x25)→^*cond1^#(false,x26,x27)⟹cond1^#(false,x26,x27)≥cond2^#(gt(x26,x27),x26,x27))
    • Applying Rule "Same Constructor" results in
      (x22→^*x24⟹s(x23)→^*x25⟹equal(x24,x25)→^*false⟹x24→^*x26⟹x25→^*x27⟹cond1^#(false,x26,x27)≥cond2^#(gt(x26,x27),x26,x27))
    • Applying Rule "Variable in Equation" allows to substitute x26 by x24 which results in
      (x22→^*x24⟹s(x23)→^*x25⟹equal(x24,x25)→^*false⟹x25→^*x27⟹cond1^#(false,x24,x27)≥cond2^#(gt(x24,x27),x24,x27))
    • Applying Rule "Variable in Equation" allows to substitute x27 by x25 which results in
      (x22→^*x24⟹s(x23)→^*x25⟹equal(x24,x25)→^*false⟹cond1^#(false,x24,x25)≥cond2^#(gt(x24,x25),x24,x25))
    • Applying Rule "Variable in Equation" allows to substitute x22 by x24 which results in
      (s(x23)→^*x25⟹equal(x24,x25)→^*false⟹cond1^#(false,x24,x25)≥cond2^#(gt(x24,x25),x24,x25))
    • Applying Rule "Induction" on equal(x24,x25)→^*false results in the following 4 new constraints.
      1. (true→^*false⟹cond1^#(false,0,0)≥cond2^#(gt(0,0),0,0))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*false⟹s(x23)→^*0⟹cond1^#(false,s(x180),0)≥cond2^#(gt(s(x180),0),s(x180),0))
         • Applying Rule "Same Constructor" results in
           (s(x23)→^*0⟹cond1^#(false,s(x180),0)≥cond2^#(gt(s(x180),0),s(x180),0))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (false→^*false⟹s(x23)→^*s(x181)⟹cond1^#(false,0,s(x181))≥cond2^#(gt(0,s(x181)),0,s(x181)))
         • Applying Rule "Same Constructor" results in
           (s(x23)→^*s(x181)⟹cond1^#(false,0,s(x181))≥cond2^#(gt(0,s(x181)),0,s(x181)))
         • Applying Rule "Same Constructor" results in
           (x23→^*x181⟹cond1^#(false,0,s(x181))≥cond2^#(gt(0,s(x181)),0,s(x181)))
         • Applying Rule "Variable in Equation" allows to substitute x181 by x23 which results in
           cond1^#(false,0,s(x23))≥cond2^#(gt(0,s(x23)),0,s(x23))

         • This constraint is kept as final constraint.

      4. (equal(x183,x182)→^*false⟹s(x23)→^*s(x182)⟹∀x184 . (equal(x183,x182)→^*false⟹s(x184)→^*x182⟹cond1^#(false,x183,x182)≥cond2^#(gt(x183,x182),x183,x182))⟹cond1^#(false,s(x183),s(x182))≥cond2^#(gt(s(x183),s(x182)),s(x183),s(x182)))
         • Applying Rule "Same Constructor" results in
           (equal(x183,x182)→^*false⟹x23→^*x182⟹∀x184 . (equal(x183,x182)→^*false⟹s(x184)→^*x182⟹cond1^#(false,x183,x182)≥cond2^#(gt(x183,x182),x183,x182))⟹cond1^#(false,s(x183),s(x182))≥cond2^#(gt(s(x183),s(x182)),s(x183),s(x182)))
         • Applying Rule "Variable in Equation" allows to substitute x23 by x182 which results in
           (equal(x183,x182)→^*false⟹∀x184 . (equal(x183,x182)→^*false⟹s(x184)→^*x182⟹cond1^#(false,x183,x182)≥cond2^#(gt(x183,x182),x183,x182))⟹cond1^#(false,s(x183),s(x182))≥cond2^#(gt(s(x183),s(x182)),s(x183),s(x182)))
         • Applying Rule "Delete Conditions" results in
           (equal(x183,x182)→^*false⟹cond1^#(false,s(x183),s(x182))≥cond2^#(gt(s(x183),s(x182)),s(x183),s(x182)))
         • Applying Rule "Induction" on equal(x183,x182)→^*false results in the following 4 new constraints.
           1. (true→^*false⟹cond1^#(false,s(0),s(0))≥cond2^#(gt(s(0),s(0)),s(0),s(0)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (false→^*false⟹cond1^#(false,s(s(x185)),s(0))≥cond2^#(gt(s(s(x185)),s(0)),s(s(x185)),s(0)))
              • Applying Rule "Same Constructor" results in
                cond1^#(false,s(s(x185)),s(0))≥cond2^#(gt(s(s(x185)),s(0)),s(s(x185)),s(0))

              • This constraint is kept as final constraint.

           3. (false→^*false⟹cond1^#(false,s(0),s(s(x186)))≥cond2^#(gt(s(0),s(s(x186))),s(0),s(s(x186))))
              • Applying Rule "Same Constructor" results in
                cond1^#(false,s(0),s(s(x186)))≥cond2^#(gt(s(0),s(s(x186))),s(0),s(s(x186)))

              • This constraint is kept as final constraint.

           4. (equal(x188,x187)→^*false⟹(equal(x188,x187)→^*false⟹cond1^#(false,s(x188),s(x187))≥cond2^#(gt(s(x188),s(x187)),s(x188),s(x187)))⟹cond1^#(false,s(s(x188)),s(s(x187)))≥cond2^#(gt(s(s(x188)),s(s(x187))),s(s(x188)),s(s(x187))))
              • Applying Rule "Simplify Conditions" results in
                (cond1^#(false,s(x188),s(x187))≥cond2^#(gt(s(x188),s(x187)),s(x188),s(x187))⟹cond1^#(false,s(s(x188)),s(s(x187)))≥cond2^#(gt(s(s(x188)),s(s(x187))),s(s(x188)),s(s(x187))))

              • This constraint is kept as final constraint.

  • For the chain we build the initial constraint

    (diff^#(s(x30),x31)→^*diff^#(x32,x33)⟹cond1^#(equal(x32,x33),x32,x33)→^*cond1^#(false,x34,x35)⟹cond1^#(false,x34,x35)≥cond2^#(gt(x34,x35),x34,x35))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (s(x30)→^*x32⟹x31→^*x33⟹cond1^#(equal(x32,x33),x32,x33)→^*cond1^#(false,x34,x35)⟹cond1^#(false,x34,x35)≥cond2^#(gt(x34,x35),x34,x35))
    • Applying Rule "Same Constructor" results in
      (s(x30)→^*x32⟹x31→^*x33⟹equal(x32,x33)→^*false⟹x32→^*x34⟹x33→^*x35⟹cond1^#(false,x34,x35)≥cond2^#(gt(x34,x35),x34,x35))
    • Applying Rule "Variable in Equation" allows to substitute x34 by x32 which results in
      (s(x30)→^*x32⟹x31→^*x33⟹equal(x32,x33)→^*false⟹x33→^*x35⟹cond1^#(false,x32,x35)≥cond2^#(gt(x32,x35),x32,x35))
    • Applying Rule "Variable in Equation" allows to substitute x35 by x33 which results in
      (s(x30)→^*x32⟹x31→^*x33⟹equal(x32,x33)→^*false⟹cond1^#(false,x32,x33)≥cond2^#(gt(x32,x33),x32,x33))
    • Applying Rule "Variable in Equation" allows to substitute x31 by x33 which results in
      (s(x30)→^*x32⟹equal(x32,x33)→^*false⟹cond1^#(false,x32,x33)≥cond2^#(gt(x32,x33),x32,x33))
    • Applying Rule "Induction" on equal(x32,x33)→^*false results in the following 4 new constraints.
      1. (true→^*false⟹cond1^#(false,0,0)≥cond2^#(gt(0,0),0,0))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*false⟹s(x30)→^*s(x189)⟹cond1^#(false,s(x189),0)≥cond2^#(gt(s(x189),0),s(x189),0))
         • Applying Rule "Same Constructor" results in
           (s(x30)→^*s(x189)⟹cond1^#(false,s(x189),0)≥cond2^#(gt(s(x189),0),s(x189),0))
         • Applying Rule "Same Constructor" results in
           (x30→^*x189⟹cond1^#(false,s(x189),0)≥cond2^#(gt(s(x189),0),s(x189),0))
         • Applying Rule "Variable in Equation" allows to substitute x189 by x30 which results in
           cond1^#(false,s(x30),0)≥cond2^#(gt(s(x30),0),s(x30),0)

         • This constraint is kept as final constraint.

      3. (false→^*false⟹s(x30)→^*0⟹cond1^#(false,0,s(x190))≥cond2^#(gt(0,s(x190)),0,s(x190)))
         • Applying Rule "Same Constructor" results in
           (s(x30)→^*0⟹cond1^#(false,0,s(x190))≥cond2^#(gt(0,s(x190)),0,s(x190)))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      4. (equal(x192,x191)→^*false⟹s(x30)→^*s(x192)⟹∀x193 . (equal(x192,x191)→^*false⟹s(x193)→^*x192⟹cond1^#(false,x192,x191)≥cond2^#(gt(x192,x191),x192,x191))⟹cond1^#(false,s(x192),s(x191))≥cond2^#(gt(s(x192),s(x191)),s(x192),s(x191)))
         • Applying Rule "Same Constructor" results in
           (equal(x192,x191)→^*false⟹x30→^*x192⟹∀x193 . (equal(x192,x191)→^*false⟹s(x193)→^*x192⟹cond1^#(false,x192,x191)≥cond2^#(gt(x192,x191),x192,x191))⟹cond1^#(false,s(x192),s(x191))≥cond2^#(gt(s(x192),s(x191)),s(x192),s(x191)))
         • Applying Rule "Variable in Equation" allows to substitute x30 by x192 which results in
           (equal(x192,x191)→^*false⟹∀x193 . (equal(x192,x191)→^*false⟹s(x193)→^*x192⟹cond1^#(false,x192,x191)≥cond2^#(gt(x192,x191),x192,x191))⟹cond1^#(false,s(x192),s(x191))≥cond2^#(gt(s(x192),s(x191)),s(x192),s(x191)))
         • Applying Rule "Delete Conditions" results in
           (equal(x192,x191)→^*false⟹cond1^#(false,s(x192),s(x191))≥cond2^#(gt(s(x192),s(x191)),s(x192),s(x191)))
         • Applying Rule "Induction" on equal(x192,x191)→^*false results in the following 4 new constraints.
           1. (true→^*false⟹cond1^#(false,s(0),s(0))≥cond2^#(gt(s(0),s(0)),s(0),s(0)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (false→^*false⟹cond1^#(false,s(s(x194)),s(0))≥cond2^#(gt(s(s(x194)),s(0)),s(s(x194)),s(0)))
              • Applying Rule "Same Constructor" results in
                cond1^#(false,s(s(x194)),s(0))≥cond2^#(gt(s(s(x194)),s(0)),s(s(x194)),s(0))

              • This constraint is kept as final constraint.

           3. (false→^*false⟹cond1^#(false,s(0),s(s(x195)))≥cond2^#(gt(s(0),s(s(x195))),s(0),s(s(x195))))
              • Applying Rule "Same Constructor" results in
                cond1^#(false,s(0),s(s(x195)))≥cond2^#(gt(s(0),s(s(x195))),s(0),s(s(x195)))

              • This constraint is kept as final constraint.

           4. (equal(x197,x196)→^*false⟹(equal(x197,x196)→^*false⟹cond1^#(false,s(x197),s(x196))≥cond2^#(gt(s(x197),s(x196)),s(x197),s(x196)))⟹cond1^#(false,s(s(x197)),s(s(x196)))≥cond2^#(gt(s(s(x197)),s(s(x196))),s(s(x197)),s(s(x196))))
              • Applying Rule "Simplify Conditions" results in
                (cond1^#(false,s(x197),s(x196))≥cond2^#(gt(s(x197),s(x196)),s(x197),s(x196))⟹cond1^#(false,s(s(x197)),s(s(x196)))≥cond2^#(gt(s(s(x197)),s(s(x196))),s(s(x197)),s(s(x196))))

              • This constraint is kept as final constraint.

  • For the chain we build the initial constraint

    (cond1^#(equal(x50,x51),x50,x51)→^*cond1^#(false,x52,x53)⟹cond2^#(gt(x52,x53),x52,x53)→^*cond2^#(true,x54,x55)⟹cond2^#(true,x54,x55) > diff^#(x54,s(x55)))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (equal(x50,x51)→^*false⟹x50→^*x52⟹x51→^*x53⟹cond2^#(gt(x52,x53),x52,x53)→^*cond2^#(true,x54,x55)⟹cond2^#(true,x54,x55) > diff^#(x54,s(x55)))
    • Applying Rule "Same Constructor" results in
      (equal(x50,x51)→^*false⟹x50→^*x52⟹x51→^*x53⟹gt(x52,x53)→^*true⟹x52→^*x54⟹x53→^*x55⟹cond2^#(true,x54,x55) > diff^#(x54,s(x55)))
    • Applying Rule "Variable in Equation" allows to substitute x54 by x52 which results in
      (equal(x50,x51)→^*false⟹x50→^*x52⟹x51→^*x53⟹gt(x52,x53)→^*true⟹x53→^*x55⟹cond2^#(true,x52,x55) > diff^#(x52,s(x55)))
    • Applying Rule "Variable in Equation" allows to substitute x55 by x53 which results in
      (equal(x50,x51)→^*false⟹x50→^*x52⟹x51→^*x53⟹gt(x52,x53)→^*true⟹cond2^#(true,x52,x53) > diff^#(x52,s(x53)))
    • Applying Rule "Induction" on equal(x50,x51)→^*false results in the following 4 new constraints.
      1. (true→^*false⟹cond2^#(true,x52,x53) > diff^#(x52,s(x53)))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*false⟹s(x198)→^*x52⟹0→^*x53⟹gt(x52,x53)→^*true⟹cond2^#(true,x52,x53) > diff^#(x52,s(x53)))
         • Applying Rule "Same Constructor" results in
           (s(x198)→^*x52⟹0→^*x53⟹gt(x52,x53)→^*true⟹cond2^#(true,x52,x53) > diff^#(x52,s(x53)))
         • Applying Rule "Induction" on gt(x52,x53)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond2^#(true,0,x204) > diff^#(0,s(x204)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹s(x198)→^*s(x205)⟹0→^*0⟹cond2^#(true,s(x205),0) > diff^#(s(x205),s(0)))
              • Applying Rule "Same Constructor" results in
                (s(x198)→^*s(x205)⟹0→^*0⟹cond2^#(true,s(x205),0) > diff^#(s(x205),s(0)))
              • Applying Rule "Same Constructor" results in
                (x198→^*x205⟹0→^*0⟹cond2^#(true,s(x205),0) > diff^#(s(x205),s(0)))
              • Applying Rule "Same Constructor" results in
                (x198→^*x205⟹cond2^#(true,s(x205),0) > diff^#(s(x205),s(0)))
              • Applying Rule "Variable in Equation" allows to substitute x205 by x198 which results in
                cond2^#(true,s(x198),0) > diff^#(s(x198),s(0))

              • This constraint is kept as final constraint.

           3. (gt(x207,x206)→^*true⟹s(x198)→^*s(x207)⟹0→^*s(x206)⟹∀x208 . (gt(x207,x206)→^*true⟹s(x208)→^*x207⟹0→^*x206⟹cond2^#(true,x207,x206) > diff^#(x207,s(x206)))⟹cond2^#(true,s(x207),s(x206)) > diff^#(s(x207),s(s(x206))))
              • Applying Rule "Same Constructor" results in
                (gt(x207,x206)→^*true⟹x198→^*x207⟹0→^*s(x206)⟹∀x208 . (gt(x207,x206)→^*true⟹s(x208)→^*x207⟹0→^*x206⟹cond2^#(true,x207,x206) > diff^#(x207,s(x206)))⟹cond2^#(true,s(x207),s(x206)) > diff^#(s(x207),s(s(x206))))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (false→^*false⟹0→^*x52⟹s(x199)→^*x53⟹gt(x52,x53)→^*true⟹cond2^#(true,x52,x53) > diff^#(x52,s(x53)))
         • Applying Rule "Same Constructor" results in
           (0→^*x52⟹s(x199)→^*x53⟹gt(x52,x53)→^*true⟹cond2^#(true,x52,x53) > diff^#(x52,s(x53)))
         • Applying Rule "Induction" on gt(x52,x53)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond2^#(true,0,x209) > diff^#(0,s(x209)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹0→^*s(x210)⟹s(x199)→^*0⟹cond2^#(true,s(x210),0) > diff^#(s(x210),s(0)))
              • Applying Rule "Same Constructor" results in
                (0→^*s(x210)⟹s(x199)→^*0⟹cond2^#(true,s(x210),0) > diff^#(s(x210),s(0)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gt(x212,x211)→^*true⟹0→^*s(x212)⟹s(x199)→^*s(x211)⟹∀x213 . (gt(x212,x211)→^*true⟹0→^*x212⟹s(x213)→^*x211⟹cond2^#(true,x212,x211) > diff^#(x212,s(x211)))⟹cond2^#(true,s(x212),s(x211)) > diff^#(s(x212),s(s(x211))))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      4. (equal(x201,x200)→^*false⟹s(x201)→^*x52⟹s(x200)→^*x53⟹gt(x52,x53)→^*true⟹∀x202 . ∀x203 . (equal(x201,x200)→^*false⟹x201→^*x202⟹x200→^*x203⟹gt(x202,x203)→^*true⟹cond2^#(true,x202,x203) > diff^#(x202,s(x203)))⟹cond2^#(true,x52,x53) > diff^#(x52,s(x53)))
         • Applying Rule "Induction" on gt(x52,x53)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond2^#(true,0,x214) > diff^#(0,s(x214)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹equal(x201,x200)→^*false⟹s(x201)→^*s(x215)⟹s(x200)→^*0⟹∀x202 . ∀x203 . (equal(x201,x200)→^*false⟹x201→^*x202⟹x200→^*x203⟹gt(x202,x203)→^*true⟹cond2^#(true,x202,x203) > diff^#(x202,s(x203)))⟹cond2^#(true,s(x215),0) > diff^#(s(x215),s(0)))
              • Applying Rule "Same Constructor" results in
                (equal(x201,x200)→^*false⟹s(x201)→^*s(x215)⟹s(x200)→^*0⟹∀x202 . ∀x203 . (equal(x201,x200)→^*false⟹x201→^*x202⟹x200→^*x203⟹gt(x202,x203)→^*true⟹cond2^#(true,x202,x203) > diff^#(x202,s(x203)))⟹cond2^#(true,s(x215),0) > diff^#(s(x215),s(0)))
              • Applying Rule "Same Constructor" results in
                (equal(x201,x200)→^*false⟹x201→^*x215⟹s(x200)→^*0⟹∀x202 . ∀x203 . (equal(x201,x200)→^*false⟹x201→^*x202⟹x200→^*x203⟹gt(x202,x203)→^*true⟹cond2^#(true,x202,x203) > diff^#(x202,s(x203)))⟹cond2^#(true,s(x215),0) > diff^#(s(x215),s(0)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gt(x217,x216)→^*true⟹equal(x201,x200)→^*false⟹s(x201)→^*s(x217)⟹s(x200)→^*s(x216)⟹∀x202 . ∀x203 . (equal(x201,x200)→^*false⟹x201→^*x202⟹x200→^*x203⟹gt(x202,x203)→^*true⟹cond2^#(true,x202,x203) > diff^#(x202,s(x203)))⟹∀x218 . ∀x219 . ∀x220 . ∀x221 . (gt(x217,x216)→^*true⟹equal(x218,x219)→^*false⟹s(x218)→^*x217⟹s(x219)→^*x216⟹∀x220 . ∀x221 . (equal(x218,x219)→^*false⟹x218→^*x220⟹x219→^*x221⟹gt(x220,x221)→^*true⟹cond2^#(true,x220,x221) > diff^#(x220,s(x221)))⟹cond2^#(true,x217,x216) > diff^#(x217,s(x216)))⟹cond2^#(true,s(x217),s(x216)) > diff^#(s(x217),s(s(x216))))
              • Applying Rule "Same Constructor" results in
                (gt(x217,x216)→^*true⟹equal(x201,x200)→^*false⟹x201→^*x217⟹s(x200)→^*s(x216)⟹∀x202 . ∀x203 . (equal(x201,x200)→^*false⟹x201→^*x202⟹x200→^*x203⟹gt(x202,x203)→^*true⟹cond2^#(true,x202,x203) > diff^#(x202,s(x203)))⟹∀x218 . ∀x219 . ∀x220 . ∀x221 . (gt(x217,x216)→^*true⟹equal(x218,x219)→^*false⟹s(x218)→^*x217⟹s(x219)→^*x216⟹∀x220 . ∀x221 . (equal(x218,x219)→^*false⟹x218→^*x220⟹x219→^*x221⟹gt(x220,x221)→^*true⟹cond2^#(true,x220,x221) > diff^#(x220,s(x221)))⟹cond2^#(true,x217,x216) > diff^#(x217,s(x216)))⟹cond2^#(true,s(x217),s(x216)) > diff^#(s(x217),s(s(x216))))
              • Applying Rule "Same Constructor" results in
                (gt(x217,x216)→^*true⟹equal(x201,x200)→^*false⟹x201→^*x217⟹x200→^*x216⟹∀x202 . ∀x203 . (equal(x201,x200)→^*false⟹x201→^*x202⟹x200→^*x203⟹gt(x202,x203)→^*true⟹cond2^#(true,x202,x203) > diff^#(x202,s(x203)))⟹∀x218 . ∀x219 . ∀x220 . ∀x221 . (gt(x217,x216)→^*true⟹equal(x218,x219)→^*false⟹s(x218)→^*x217⟹s(x219)→^*x216⟹∀x220 . ∀x221 . (equal(x218,x219)→^*false⟹x218→^*x220⟹x219→^*x221⟹gt(x220,x221)→^*true⟹cond2^#(true,x220,x221) > diff^#(x220,s(x221)))⟹cond2^#(true,x217,x216) > diff^#(x217,s(x216)))⟹cond2^#(true,s(x217),s(x216)) > diff^#(s(x217),s(s(x216))))
              • Applying Rule "Simplify Conditions" results in
                (cond2^#(true,x217,x216) > diff^#(x217,s(x216))⟹∀x218 . ∀x219 . ∀x220 . ∀x221 . (gt(x217,x216)→^*true⟹equal(x218,x219)→^*false⟹s(x218)→^*x217⟹s(x219)→^*x216⟹∀x220 . ∀x221 . (equal(x218,x219)→^*false⟹x218→^*x220⟹x219→^*x221⟹gt(x220,x221)→^*true⟹cond2^#(true,x220,x221) > diff^#(x220,s(x221)))⟹cond2^#(true,x217,x216) > diff^#(x217,s(x216)))⟹cond2^#(true,s(x217),s(x216)) > diff^#(s(x217),s(s(x216))))
              • Applying Rule "Delete Conditions" results in
                (cond2^#(true,x217,x216) > diff^#(x217,s(x216))⟹cond2^#(true,s(x217),s(x216)) > diff^#(s(x217),s(s(x216))))

              • This constraint is kept as final constraint.

  • For the chain we build the initial constraint

    (cond2^#(gt(x100,x101),x100,x101)→^*cond2^#(true,x102,x103)⟹diff^#(x102,s(x103))→^*diff^#(x104,x105)⟹diff^#(x104,x105)≥cond1^#(equal(x104,x105),x104,x105))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gt(x100,x101)→^*true⟹x100→^*x102⟹x101→^*x103⟹diff^#(x102,s(x103))→^*diff^#(x104,x105)⟹diff^#(x104,x105)≥cond1^#(equal(x104,x105),x104,x105))
    • Applying Rule "Same Constructor" results in
      (gt(x100,x101)→^*true⟹x100→^*x102⟹x101→^*x103⟹x102→^*x104⟹s(x103)→^*x105⟹diff^#(x104,x105)≥cond1^#(equal(x104,x105),x104,x105))
    • Applying Rule "Variable in Equation" allows to substitute x102 by x100 which results in
      (gt(x100,x101)→^*true⟹x101→^*x103⟹x100→^*x104⟹s(x103)→^*x105⟹diff^#(x104,x105)≥cond1^#(equal(x104,x105),x104,x105))
    • Applying Rule "Variable in Equation" allows to substitute x103 by x101 which results in
      (gt(x100,x101)→^*true⟹x100→^*x104⟹s(x101)→^*x105⟹diff^#(x104,x105)≥cond1^#(equal(x104,x105),x104,x105))
    • Applying Rule "Variable in Equation" allows to substitute x104 by x100 which results in
      (gt(x100,x101)→^*true⟹s(x101)→^*x105⟹diff^#(x100,x105)≥cond1^#(equal(x100,x105),x100,x105))
    • Applying Rule "Variable in Equation" allows to substitute x105 by s(x101) which results in
      (gt(x100,x101)→^*true⟹diff^#(x100,s(x101))≥cond1^#(equal(x100,s(x101)),x100,s(x101)))
    • Applying Rule "Induction" on gt(x100,x101)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹diff^#(0,s(x222))≥cond1^#(equal(0,s(x222)),0,s(x222)))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹diff^#(s(x223),s(0))≥cond1^#(equal(s(x223),s(0)),s(x223),s(0)))
         • Applying Rule "Same Constructor" results in
           diff^#(s(x223),s(0))≥cond1^#(equal(s(x223),s(0)),s(x223),s(0))

         • This constraint is kept as final constraint.

      3. (gt(x225,x224)→^*true⟹(gt(x225,x224)→^*true⟹diff^#(x225,s(x224))≥cond1^#(equal(x225,s(x224)),x225,s(x224)))⟹diff^#(s(x225),s(s(x224)))≥cond1^#(equal(s(x225),s(s(x224))),s(x225),s(s(x224))))
         • Applying Rule "Simplify Conditions" results in
           (diff^#(x225,s(x224))≥cond1^#(equal(x225,s(x224)),x225,s(x224))⟹diff^#(s(x225),s(s(x224)))≥cond1^#(equal(s(x225),s(s(x224))),s(x225),s(s(x224))))

         • This constraint is kept as final constraint.

  • For the chain we build the initial constraint

    (cond2^#(gt(x124,x125),x124,x125)→^*cond2^#(false,x126,x127)⟹diff^#(s(x126),x127)→^*diff^#(x128,x129)⟹diff^#(x128,x129)≥cond1^#(equal(x128,x129),x128,x129))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gt(x124,x125)→^*false⟹x124→^*x126⟹x125→^*x127⟹diff^#(s(x126),x127)→^*diff^#(x128,x129)⟹diff^#(x128,x129)≥cond1^#(equal(x128,x129),x128,x129))
    • Applying Rule "Same Constructor" results in
      (gt(x124,x125)→^*false⟹x124→^*x126⟹x125→^*x127⟹s(x126)→^*x128⟹x127→^*x129⟹diff^#(x128,x129)≥cond1^#(equal(x128,x129),x128,x129))
    • Applying Rule "Variable in Equation" allows to substitute x126 by x124 which results in
      (gt(x124,x125)→^*false⟹x125→^*x127⟹s(x124)→^*x128⟹x127→^*x129⟹diff^#(x128,x129)≥cond1^#(equal(x128,x129),x128,x129))
    • Applying Rule "Variable in Equation" allows to substitute x127 by x125 which results in
      (gt(x124,x125)→^*false⟹s(x124)→^*x128⟹x125→^*x129⟹diff^#(x128,x129)≥cond1^#(equal(x128,x129),x128,x129))
    • Applying Rule "Variable in Equation" allows to substitute x128 by s(x124) which results in
      (gt(x124,x125)→^*false⟹x125→^*x129⟹diff^#(s(x124),x129)≥cond1^#(equal(s(x124),x129),s(x124),x129))
    • Applying Rule "Variable in Equation" allows to substitute x129 by x125 which results in
      (gt(x124,x125)→^*false⟹diff^#(s(x124),x125)≥cond1^#(equal(s(x124),x125),s(x124),x125))
    • Applying Rule "Induction" on gt(x124,x125)→^*false results in the following 3 new constraints.
      1. (false→^*false⟹diff^#(s(0),x226)≥cond1^#(equal(s(0),x226),s(0),x226))
         • Applying Rule "Same Constructor" results in
           diff^#(s(0),x226)≥cond1^#(equal(s(0),x226),s(0),x226)

         • This constraint is kept as final constraint.

      2. (true→^*false⟹diff^#(s(s(x227)),0)≥cond1^#(equal(s(s(x227)),0),s(s(x227)),0))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gt(x229,x228)→^*false⟹(gt(x229,x228)→^*false⟹diff^#(s(x229),x228)≥cond1^#(equal(s(x229),x228),s(x229),x228))⟹diff^#(s(s(x229)),s(x228))≥cond1^#(equal(s(s(x229)),s(x228)),s(s(x229)),s(x228)))
         • Applying Rule "Simplify Conditions" results in
           (diff^#(s(x229),x228)≥cond1^#(equal(s(x229),x228),s(x229),x228)⟹diff^#(s(s(x229)),s(x228))≥cond1^#(equal(s(s(x229)),s(x228)),s(s(x229)),s(x228)))

         • This constraint is kept as final constraint.

  • For the chain we build the initial constraint

    (cond1^#(equal(x140,x141),x140,x141)→^*cond1^#(false,x142,x143)⟹cond2^#(gt(x142,x143),x142,x143)→^*cond2^#(false,x144,x145)⟹cond2^#(false,x144,x145) > diff^#(s(x144),x145))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (equal(x140,x141)→^*false⟹x140→^*x142⟹x141→^*x143⟹cond2^#(gt(x142,x143),x142,x143)→^*cond2^#(false,x144,x145)⟹cond2^#(false,x144,x145) > diff^#(s(x144),x145))
    • Applying Rule "Same Constructor" results in
      (equal(x140,x141)→^*false⟹x140→^*x142⟹x141→^*x143⟹gt(x142,x143)→^*false⟹x142→^*x144⟹x143→^*x145⟹cond2^#(false,x144,x145) > diff^#(s(x144),x145))
    • Applying Rule "Variable in Equation" allows to substitute x144 by x142 which results in
      (equal(x140,x141)→^*false⟹x140→^*x142⟹x141→^*x143⟹gt(x142,x143)→^*false⟹x143→^*x145⟹cond2^#(false,x142,x145) > diff^#(s(x142),x145))
    • Applying Rule "Variable in Equation" allows to substitute x145 by x143 which results in
      (equal(x140,x141)→^*false⟹x140→^*x142⟹x141→^*x143⟹gt(x142,x143)→^*false⟹cond2^#(false,x142,x143) > diff^#(s(x142),x143))
    • Applying Rule "Induction" on equal(x140,x141)→^*false results in the following 4 new constraints.
      1. (true→^*false⟹cond2^#(false,x142,x143) > diff^#(s(x142),x143))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*false⟹s(x230)→^*x142⟹0→^*x143⟹gt(x142,x143)→^*false⟹cond2^#(false,x142,x143) > diff^#(s(x142),x143))
         • Applying Rule "Same Constructor" results in
           (s(x230)→^*x142⟹0→^*x143⟹gt(x142,x143)→^*false⟹cond2^#(false,x142,x143) > diff^#(s(x142),x143))
         • Applying Rule "Induction" on gt(x142,x143)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹s(x230)→^*0⟹0→^*x236⟹cond2^#(false,0,x236) > diff^#(s(0),x236))
              • Applying Rule "Same Constructor" results in
                (s(x230)→^*0⟹0→^*x236⟹cond2^#(false,0,x236) > diff^#(s(0),x236))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond2^#(false,s(x237),0) > diff^#(s(s(x237)),0))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gt(x239,x238)→^*false⟹s(x230)→^*s(x239)⟹0→^*s(x238)⟹∀x240 . (gt(x239,x238)→^*false⟹s(x240)→^*x239⟹0→^*x238⟹cond2^#(false,x239,x238) > diff^#(s(x239),x238))⟹cond2^#(false,s(x239),s(x238)) > diff^#(s(s(x239)),s(x238)))
              • Applying Rule "Same Constructor" results in
                (gt(x239,x238)→^*false⟹x230→^*x239⟹0→^*s(x238)⟹∀x240 . (gt(x239,x238)→^*false⟹s(x240)→^*x239⟹0→^*x238⟹cond2^#(false,x239,x238) > diff^#(s(x239),x238))⟹cond2^#(false,s(x239),s(x238)) > diff^#(s(s(x239)),s(x238)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (false→^*false⟹0→^*x142⟹s(x231)→^*x143⟹gt(x142,x143)→^*false⟹cond2^#(false,x142,x143) > diff^#(s(x142),x143))
         • Applying Rule "Same Constructor" results in
           (0→^*x142⟹s(x231)→^*x143⟹gt(x142,x143)→^*false⟹cond2^#(false,x142,x143) > diff^#(s(x142),x143))
         • Applying Rule "Induction" on gt(x142,x143)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹0→^*0⟹s(x231)→^*x241⟹cond2^#(false,0,x241) > diff^#(s(0),x241))
              • Applying Rule "Same Constructor" results in
                (0→^*0⟹s(x231)→^*x241⟹cond2^#(false,0,x241) > diff^#(s(0),x241))
              • Applying Rule "Same Constructor" results in
                (s(x231)→^*x241⟹cond2^#(false,0,x241) > diff^#(s(0),x241))
              • Applying Rule "Variable in Equation" allows to substitute x241 by s(x231) which results in
                cond2^#(false,0,s(x231)) > diff^#(s(0),s(x231))

              • This constraint is kept as final constraint.

           2. (true→^*false⟹cond2^#(false,s(x242),0) > diff^#(s(s(x242)),0))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gt(x244,x243)→^*false⟹0→^*s(x244)⟹s(x231)→^*s(x243)⟹∀x245 . (gt(x244,x243)→^*false⟹0→^*x244⟹s(x245)→^*x243⟹cond2^#(false,x244,x243) > diff^#(s(x244),x243))⟹cond2^#(false,s(x244),s(x243)) > diff^#(s(s(x244)),s(x243)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      4. (equal(x233,x232)→^*false⟹s(x233)→^*x142⟹s(x232)→^*x143⟹gt(x142,x143)→^*false⟹∀x234 . ∀x235 . (equal(x233,x232)→^*false⟹x233→^*x234⟹x232→^*x235⟹gt(x234,x235)→^*false⟹cond2^#(false,x234,x235) > diff^#(s(x234),x235))⟹cond2^#(false,x142,x143) > diff^#(s(x142),x143))
         • Applying Rule "Induction" on gt(x142,x143)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹equal(x233,x232)→^*false⟹s(x233)→^*0⟹s(x232)→^*x246⟹∀x234 . ∀x235 . (equal(x233,x232)→^*false⟹x233→^*x234⟹x232→^*x235⟹gt(x234,x235)→^*false⟹cond2^#(false,x234,x235) > diff^#(s(x234),x235))⟹cond2^#(false,0,x246) > diff^#(s(0),x246))
              • Applying Rule "Same Constructor" results in
                (equal(x233,x232)→^*false⟹s(x233)→^*0⟹s(x232)→^*x246⟹∀x234 . ∀x235 . (equal(x233,x232)→^*false⟹x233→^*x234⟹x232→^*x235⟹gt(x234,x235)→^*false⟹cond2^#(false,x234,x235) > diff^#(s(x234),x235))⟹cond2^#(false,0,x246) > diff^#(s(0),x246))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond2^#(false,s(x247),0) > diff^#(s(s(x247)),0))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gt(x249,x248)→^*false⟹equal(x233,x232)→^*false⟹s(x233)→^*s(x249)⟹s(x232)→^*s(x248)⟹∀x234 . ∀x235 . (equal(x233,x232)→^*false⟹x233→^*x234⟹x232→^*x235⟹gt(x234,x235)→^*false⟹cond2^#(false,x234,x235) > diff^#(s(x234),x235))⟹∀x250 . ∀x251 . ∀x252 . ∀x253 . (gt(x249,x248)→^*false⟹equal(x250,x251)→^*false⟹s(x250)→^*x249⟹s(x251)→^*x248⟹∀x252 . ∀x253 . (equal(x250,x251)→^*false⟹x250→^*x252⟹x251→^*x253⟹gt(x252,x253)→^*false⟹cond2^#(false,x252,x253) > diff^#(s(x252),x253))⟹cond2^#(false,x249,x248) > diff^#(s(x249),x248))⟹cond2^#(false,s(x249),s(x248)) > diff^#(s(s(x249)),s(x248)))
              • Applying Rule "Same Constructor" results in
                (gt(x249,x248)→^*false⟹equal(x233,x232)→^*false⟹x233→^*x249⟹s(x232)→^*s(x248)⟹∀x234 . ∀x235 . (equal(x233,x232)→^*false⟹x233→^*x234⟹x232→^*x235⟹gt(x234,x235)→^*false⟹cond2^#(false,x234,x235) > diff^#(s(x234),x235))⟹∀x250 . ∀x251 . ∀x252 . ∀x253 . (gt(x249,x248)→^*false⟹equal(x250,x251)→^*false⟹s(x250)→^*x249⟹s(x251)→^*x248⟹∀x252 . ∀x253 . (equal(x250,x251)→^*false⟹x250→^*x252⟹x251→^*x253⟹gt(x252,x253)→^*false⟹cond2^#(false,x252,x253) > diff^#(s(x252),x253))⟹cond2^#(false,x249,x248) > diff^#(s(x249),x248))⟹cond2^#(false,s(x249),s(x248)) > diff^#(s(s(x249)),s(x248)))
              • Applying Rule "Same Constructor" results in
                (gt(x249,x248)→^*false⟹equal(x233,x232)→^*false⟹x233→^*x249⟹x232→^*x248⟹∀x234 . ∀x235 . (equal(x233,x232)→^*false⟹x233→^*x234⟹x232→^*x235⟹gt(x234,x235)→^*false⟹cond2^#(false,x234,x235) > diff^#(s(x234),x235))⟹∀x250 . ∀x251 . ∀x252 . ∀x253 . (gt(x249,x248)→^*false⟹equal(x250,x251)→^*false⟹s(x250)→^*x249⟹s(x251)→^*x248⟹∀x252 . ∀x253 . (equal(x250,x251)→^*false⟹x250→^*x252⟹x251→^*x253⟹gt(x252,x253)→^*false⟹cond2^#(false,x252,x253) > diff^#(s(x252),x253))⟹cond2^#(false,x249,x248) > diff^#(s(x249),x248))⟹cond2^#(false,s(x249),s(x248)) > diff^#(s(s(x249)),s(x248)))
              • Applying Rule "Simplify Conditions" results in
                (cond2^#(false,x249,x248) > diff^#(s(x249),x248)⟹∀x250 . ∀x251 . ∀x252 . ∀x253 . (gt(x249,x248)→^*false⟹equal(x250,x251)→^*false⟹s(x250)→^*x249⟹s(x251)→^*x248⟹∀x252 . ∀x253 . (equal(x250,x251)→^*false⟹x250→^*x252⟹x251→^*x253⟹gt(x252,x253)→^*false⟹cond2^#(false,x252,x253) > diff^#(s(x252),x253))⟹cond2^#(false,x249,x248) > diff^#(s(x249),x248))⟹cond2^#(false,s(x249),s(x248)) > diff^#(s(s(x249)),s(x248)))
              • Applying Rule "Delete Conditions" results in
                (cond2^#(false,x249,x248) > diff^#(s(x249),x248)⟹cond2^#(false,s(x249),s(x248)) > diff^#(s(s(x249)),s(x248)))

              • This constraint is kept as final constraint.

  We remove those pairs which are strictly decreasing and bounded.

  1.1.1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 2^nd component contains the pair

  gt#(s(u),s(v))

  →

  gt#(u,v)

  (19)

  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.

  gt#(s(u),s(v))	→	gt#(u,v)	(19)
  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

  equal#(s(x),s(y))

  →

  equal#(x,y)

  (20)

  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.

  equal#(s(x),s(y))	→	equal#(x,y)	(20)
  1	>	1
  2	>	2

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