Certification Problem

Input (TPDB TRS_Standard/Beerendonk_07/24)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Switch to Innermost Termination

The TRS is overlay and locally confluent:

Hence, it suffices to show innermost termination in the following.

1.1 Dependency Pair Transformation

1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

• The 1^st component contains the pair

  cond2#(true,x,y)	→	cond3#(gr(x,0),x,y)	(18)
  cond3#(true,x,y)	→	cond3#(gr(x,0),p(x),y)	(22)
  cond3#(false,x,y)	→	cond1#(and(gr(x,0),gr(y,0)),x,y)	(25)
  cond1#(true,x,y)	→	cond2#(gr(x,y),x,y)	(16)
  cond2#(false,x,y)	→	cond4#(gr(y,0),x,y)	(20)
  cond4#(true,x,y)	→	cond4#(gr(y,0),x,p(y))	(29)
  cond4#(false,x,y)	→	cond1#(and(gr(x,0),gr(y,0)),x,y)	(32)

  1.1.1.1 Usable Rules Processor

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

  gr(0,x)	→	false	(8)
  gr(s(x),0)	→	true	(9)
  and(true,true)	→	true	(11)
  and(false,x)	→	false	(12)
  and(x,false)	→	false	(13)
  p(0)	→	0	(14)
  p(s(x))	→	x	(15)
  gr(s(x),s(y))	→	gr(x,y)	(10)

  1.1.1.1.1 Innermost Lhss Removal Processor

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

  gr(0,x0)

  gr(s(x0),0)

  gr(s(x0),s(x1))

  and(true,true)

  and(false,x0)

  and(x0,false)

  p(0)

  p(s(x0))

  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
  [cond3#(x1, x2, x3)]	=	1 · x1 · x1 + 2 · x1 · x2 + 2 · x1 · x3 + 1 · x2 · x2 + 2 · x2 · x3 + 1 · x3 · x3 + -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 · x2 · x2 + 2 · x2 · x3 + 1 · x3 · x3 + -1
  [cond4#(x1, x2, x3)]	=	1 · x1 · x1 + 2 · x1 · x2 + 2 · x1 · x3 + 1 · x2 · x2 + 2 · x2 · x3 + 1 · x3 · x3 + -1
  [true]	=	0
  [gr(x1, x2)]	=	0
  [0]	=	0
  [p(x1)]	=	1 · x1
  [false]	=	0
  [and(x1, x2)]	=	0
  [s(x1)]	=	1 · x1

  The pair(s)

  cond1#(true,x,y)

  →

  cond2#(gr(x,y),x,y)

  (16)

  are strictly oriented and the pair(s)

  cond2#(true,x,y)	→	cond3#(gr(x,0),x,y)	(18)
  cond3#(true,x,y)	→	cond3#(gr(x,0),p(x),y)	(22)
  cond3#(false,x,y)	→	cond1#(and(gr(x,0),gr(y,0)),x,y)	(25)
  cond1#(true,x,y)	→	cond2#(gr(x,y),x,y)	(16)
  cond2#(false,x,y)	→	cond4#(gr(y,0),x,y)	(20)
  cond4#(true,x,y)	→	cond4#(gr(y,0),x,p(y))	(29)
  cond4#(false,x,y)	→	cond1#(and(gr(x,0),gr(y,0)),x,y)	(32)

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

  The following constraints are generated for the pairs.

  • cond2^#(true,s(s(x947)),s(0))≥c
  • (cond2^#(true,s(x949),s(x948))≥c⟹cond2^#(true,s(s(x949)),s(s(x948)))≥c)
  • cond2^#(true,s(s(x1003)),s(0))≥c
  • (cond2^#(true,s(x1005),s(x1004))≥c⟹cond2^#(true,s(s(x1005)),s(s(x1004)))≥c)
  • cond3^#(true,s(x1008),0)≥c
  • cond3^#(true,s(s(x1027)),s(0))≥c
  • (cond3^#(true,s(x1029),s(x1028))≥c⟹cond3^#(true,s(s(x1029)),s(s(x1028)))≥c)
  • cond3^#(true,x148,x145)≥c
  • cond3^#(true,x154,x151)≥c
  • cond3^#(false,x282,x279)≥c
  • cond2^#(false,s(0),s(x1273))≥c
  • (cond2^#(false,s(x1276),s(x1275))≥c⟹cond2^#(false,s(s(x1276)),s(s(x1275)))≥c)
  • cond2^#(false,s(0),s(x1322))≥c
  • (cond2^#(false,s(x1325),s(x1324))≥c⟹cond2^#(false,s(s(x1325)),s(s(x1324)))≥c)
  • cond4^#(true,0,s(x1334))≥c
  • cond4^#(true,s(0),s(x1345))≥c
  • (cond4^#(true,s(x1348),s(x1347))≥c⟹cond4^#(true,s(s(x1348)),s(s(x1347)))≥c)
  • cond4^#(true,x734,x739)≥c
  • cond4^#(true,x740,x745)≥c
  • cond4^#(false,0,0)≥c
  • cond4^#(false,x868,x873)≥c
  • cond2^#(true,s(s(x947)),s(0))≥cond3^#(gr(s(s(x947)),0),s(s(x947)),s(0))
  • (cond2^#(true,s(x949),s(x948))≥cond3^#(gr(s(x949),0),s(x949),s(x948))⟹cond2^#(true,s(s(x949)),s(s(x948)))≥cond3^#(gr(s(s(x949)),0),s(s(x949)),s(s(x948))))
  • cond2^#(true,s(s(x1003)),s(0))≥cond3^#(gr(s(s(x1003)),0),s(s(x1003)),s(0))
  • (cond2^#(true,s(x1005),s(x1004))≥cond3^#(gr(s(x1005),0),s(x1005),s(x1004))⟹cond2^#(true,s(s(x1005)),s(s(x1004)))≥cond3^#(gr(s(s(x1005)),0),s(s(x1005)),s(s(x1004))))
  • cond3^#(true,s(x1008),0)≥cond3^#(gr(s(x1008),0),p(s(x1008)),0)
  • cond3^#(true,s(s(x1027)),s(0))≥cond3^#(gr(s(s(x1027)),0),p(s(s(x1027))),s(0))
  • (cond3^#(true,s(x1029),s(x1028))≥cond3^#(gr(s(x1029),0),p(s(x1029)),s(x1028))⟹cond3^#(true,s(s(x1029)),s(s(x1028)))≥cond3^#(gr(s(s(x1029)),0),p(s(s(x1029))),s(s(x1028))))
  • cond3^#(true,x148,x145)≥cond3^#(gr(x148,0),p(x148),x145)
  • cond3^#(true,x154,x151)≥cond3^#(gr(x154,0),p(x154),x151)
  • cond3^#(false,x282,x279)≥cond1^#(and(gr(x282,0),gr(x279,0)),x282,x279)
  • cond2^#(false,s(0),s(x1273))≥cond4^#(gr(s(x1273),0),s(0),s(x1273))
  • (cond2^#(false,s(x1276),s(x1275))≥cond4^#(gr(s(x1275),0),s(x1276),s(x1275))⟹cond2^#(false,s(s(x1276)),s(s(x1275)))≥cond4^#(gr(s(s(x1275)),0),s(s(x1276)),s(s(x1275))))
  • cond2^#(false,s(0),s(x1322))≥cond4^#(gr(s(x1322),0),s(0),s(x1322))
  • (cond2^#(false,s(x1325),s(x1324))≥cond4^#(gr(s(x1324),0),s(x1325),s(x1324))⟹cond2^#(false,s(s(x1325)),s(s(x1324)))≥cond4^#(gr(s(s(x1324)),0),s(s(x1325)),s(s(x1324))))
  • cond4^#(true,0,s(x1334))≥cond4^#(gr(s(x1334),0),0,p(s(x1334)))
  • cond4^#(true,s(0),s(x1345))≥cond4^#(gr(s(x1345),0),s(0),p(s(x1345)))
  • (cond4^#(true,s(x1348),s(x1347))≥cond4^#(gr(s(x1347),0),s(x1348),p(s(x1347)))⟹cond4^#(true,s(s(x1348)),s(s(x1347)))≥cond4^#(gr(s(s(x1347)),0),s(s(x1348)),p(s(s(x1347)))))
  • cond4^#(true,x734,x739)≥cond4^#(gr(x739,0),x734,p(x739))
  • cond4^#(true,x740,x745)≥cond4^#(gr(x745,0),x740,p(x745))
  • cond4^#(false,0,0)≥cond1^#(and(gr(0,0),gr(0,0)),0,0)
  • cond4^#(false,x868,x873)≥cond1^#(and(gr(x868,0),gr(x873,0)),x868,x873)

  The details are shown below:

  • For the chain we build the initial constraint

    (cond1^#(and(gr(x56,0),gr(x57,0)),x56,x57)→^*cond1^#(true,x58,x59)⟹cond2^#(gr(x58,x59),x58,x59)→^*cond2^#(true,x60,x61)⟹cond2^#(true,x60,x61)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (and(gr(x56,0),gr(x57,0))→^*true⟹x56→^*x58⟹x57→^*x59⟹cond2^#(gr(x58,x59),x58,x59)→^*cond2^#(true,x60,x61)⟹cond2^#(true,x60,x61)≥c)
    • Applying Rule "Same Constructor" results in
      (and(gr(x56,0),gr(x57,0))→^*true⟹x56→^*x58⟹x57→^*x59⟹gr(x58,x59)→^*true⟹x58→^*x60⟹x59→^*x61⟹cond2^#(true,x60,x61)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x60 by x58 which results in
      (and(gr(x56,0),gr(x57,0))→^*true⟹x56→^*x58⟹x57→^*x59⟹gr(x58,x59)→^*true⟹x59→^*x61⟹cond2^#(true,x58,x61)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x61 by x59 which results in
      (and(gr(x56,0),gr(x57,0))→^*true⟹x56→^*x58⟹x57→^*x59⟹gr(x58,x59)→^*true⟹cond2^#(true,x58,x59)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (x56→^*x58⟹x57→^*x59⟹gr(x58,x59)→^*true⟹gr(x57,0)→^*x895⟹and(gr(x56,0),x895)→^*true⟹cond2^#(true,x58,x59)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (gr(x56,0)→^*x894⟹gr(x57,0)→^*x895⟹and(x894,x895)→^*true⟹x56→^*x58⟹x57→^*x59⟹gr(x58,x59)→^*true⟹cond2^#(true,x58,x59)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x896⟹gr(x56,x896)→^*x894⟹gr(x57,0)→^*x895⟹and(x894,x895)→^*true⟹x56→^*x58⟹x57→^*x59⟹gr(x58,x59)→^*true⟹cond2^#(true,x58,x59)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x896⟹gr(x56,x896)→^*x894⟹0→^*x897⟹gr(x57,x897)→^*x895⟹and(x894,x895)→^*true⟹x56→^*x58⟹x57→^*x59⟹gr(x58,x59)→^*true⟹cond2^#(true,x58,x59)≥c)
    • Applying Rule "Induction" on and(x894,x895)→^*true results in the following 3 new constraints.
      1. (true→^*true⟹0→^*x896⟹gr(x56,x896)→^*true⟹0→^*x897⟹gr(x57,x897)→^*true⟹x56→^*x58⟹x57→^*x59⟹gr(x58,x59)→^*true⟹cond2^#(true,x58,x59)≥c)
         • Applying Rule "Same Constructor" results in
           (0→^*x896⟹gr(x56,x896)→^*true⟹0→^*x897⟹gr(x57,x897)→^*true⟹x56→^*x58⟹x57→^*x59⟹gr(x58,x59)→^*true⟹cond2^#(true,x58,x59)≥c)
         • Applying Rule "Induction" on gr(x58,x59)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond2^#(true,0,x900)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹0→^*x896⟹gr(x56,x896)→^*true⟹0→^*x897⟹gr(x57,x897)→^*true⟹x56→^*s(x901)⟹x57→^*0⟹cond2^#(true,s(x901),0)≥c)
              • Applying Rule "Same Constructor" results in
                (0→^*x896⟹gr(x56,x896)→^*true⟹0→^*x897⟹gr(x57,x897)→^*true⟹x56→^*s(x901)⟹x57→^*0⟹cond2^#(true,s(x901),0)≥c)
              • Applying Rule "Variable in Equation" allows to substitute x56 by s(x901) which results in
                (0→^*x896⟹gr(s(x901),x896)→^*true⟹0→^*x897⟹gr(x57,x897)→^*true⟹x57→^*0⟹cond2^#(true,s(x901),0)≥c)
              • Applying Rule "Variable in Equation" allows to substitute x57 by 0 which results in
                (0→^*x896⟹gr(s(x901),x896)→^*true⟹0→^*x897⟹gr(0,x897)→^*true⟹cond2^#(true,s(x901),0)≥c)
              • Applying Rule "Introduce fresh variable" results in
                (0→^*x896⟹s(x901)→^*x908⟹gr(x908,x896)→^*true⟹0→^*x897⟹gr(0,x897)→^*true⟹cond2^#(true,s(x901),0)≥c)
              • Applying Rule "Introduce fresh variable" results in
                (0→^*x896⟹s(x901)→^*x908⟹gr(x908,x896)→^*true⟹0→^*x897⟹0→^*x909⟹gr(x909,x897)→^*true⟹cond2^#(true,s(x901),0)≥c)
              • Applying Rule "Induction" on gr(x908,x896)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond2^#(true,s(x901),0)≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹0→^*0⟹s(x901)→^*s(x911)⟹0→^*x897⟹0→^*x909⟹gr(x909,x897)→^*true⟹cond2^#(true,s(x901),0)≥c)
                   • Applying Rule "Same Constructor" results in
                     (0→^*0⟹s(x901)→^*s(x911)⟹0→^*x897⟹0→^*x909⟹gr(x909,x897)→^*true⟹cond2^#(true,s(x901),0)≥c)
                   • Applying Rule "Same Constructor" results in
                     (s(x901)→^*s(x911)⟹0→^*x897⟹0→^*x909⟹gr(x909,x897)→^*true⟹cond2^#(true,s(x901),0)≥c)
                   • Applying Rule "Same Constructor" results in
                     (x901→^*x911⟹0→^*x897⟹0→^*x909⟹gr(x909,x897)→^*true⟹cond2^#(true,s(x901),0)≥c)
                   • Applying Rule "Delete Conditions" results in
                     (0→^*x897⟹0→^*x909⟹gr(x909,x897)→^*true⟹cond2^#(true,s(x901),0)≥c)
                   • Applying Rule "Induction" on gr(x909,x897)→^*true results in the following 3 new constraints.
                     1. (false→^*true⟹cond2^#(true,s(x901),0)≥c)
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                     2. (true→^*true⟹0→^*0⟹0→^*s(x918)⟹cond2^#(true,s(x901),0)≥c)
                        • Applying Rule "Same Constructor" results in
                          (0→^*0⟹0→^*s(x918)⟹cond2^#(true,s(x901),0)≥c)
                        • Applying Rule "Same Constructor" results in
                          (0→^*s(x918)⟹cond2^#(true,s(x901),0)≥c)
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                     3. (gr(x920,x919)→^*true⟹0→^*s(x919)⟹0→^*s(x920)⟹∀x921 . (gr(x920,x919)→^*true⟹0→^*x919⟹0→^*x920⟹cond2^#(true,s(x921),0)≥c)⟹cond2^#(true,s(x901),0)≥c)
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x913,x912)→^*true⟹0→^*s(x912)⟹s(x901)→^*s(x913)⟹0→^*x897⟹0→^*x909⟹gr(x909,x897)→^*true⟹∀x914 . ∀x915 . ∀x916 . (gr(x913,x912)→^*true⟹0→^*x912⟹s(x914)→^*x913⟹0→^*x915⟹0→^*x916⟹gr(x916,x915)→^*true⟹cond2^#(true,s(x914),0)≥c)⟹cond2^#(true,s(x901),0)≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x903,x902)→^*true⟹0→^*x896⟹gr(x56,x896)→^*true⟹0→^*x897⟹gr(x57,x897)→^*true⟹x56→^*s(x903)⟹x57→^*s(x902)⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥c)⟹cond2^#(true,s(x903),s(x902))≥c)
              • Applying Rule "Variable in Equation" allows to substitute x56 by s(x903) which results in
                (gr(x903,x902)→^*true⟹0→^*x896⟹gr(s(x903),x896)→^*true⟹0→^*x897⟹gr(x57,x897)→^*true⟹x57→^*s(x902)⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥c)⟹cond2^#(true,s(x903),s(x902))≥c)
              • Applying Rule "Variable in Equation" allows to substitute x57 by s(x902) which results in
                (gr(x903,x902)→^*true⟹0→^*x896⟹gr(s(x903),x896)→^*true⟹0→^*x897⟹gr(s(x902),x897)→^*true⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥c)⟹cond2^#(true,s(x903),s(x902))≥c)
              • Applying Rule "Introduce fresh variable" results in
                (gr(x903,x902)→^*true⟹0→^*x896⟹s(x903)→^*x922⟹gr(x922,x896)→^*true⟹0→^*x897⟹gr(s(x902),x897)→^*true⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥c)⟹cond2^#(true,s(x903),s(x902))≥c)
              • Applying Rule "Introduce fresh variable" results in
                (gr(x903,x902)→^*true⟹0→^*x896⟹s(x903)→^*x922⟹gr(x922,x896)→^*true⟹0→^*x897⟹s(x902)→^*x923⟹gr(x923,x897)→^*true⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥c)⟹cond2^#(true,s(x903),s(x902))≥c)
              • Applying Rule "Induction" on gr(x922,x896)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond2^#(true,s(x903),s(x902))≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹gr(x903,x902)→^*true⟹0→^*0⟹s(x903)→^*s(x925)⟹0→^*x897⟹s(x902)→^*x923⟹gr(x923,x897)→^*true⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥c)⟹cond2^#(true,s(x903),s(x902))≥c)
                   • Applying Rule "Same Constructor" results in
                     (gr(x903,x902)→^*true⟹0→^*0⟹s(x903)→^*s(x925)⟹0→^*x897⟹s(x902)→^*x923⟹gr(x923,x897)→^*true⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥c)⟹cond2^#(true,s(x903),s(x902))≥c)
                   • Applying Rule "Same Constructor" results in
                     (gr(x903,x902)→^*true⟹s(x903)→^*s(x925)⟹0→^*x897⟹s(x902)→^*x923⟹gr(x923,x897)→^*true⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥c)⟹cond2^#(true,s(x903),s(x902))≥c)
                   • Applying Rule "Same Constructor" results in
                     (gr(x903,x902)→^*true⟹x903→^*x925⟹0→^*x897⟹s(x902)→^*x923⟹gr(x923,x897)→^*true⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥c)⟹cond2^#(true,s(x903),s(x902))≥c)
                   • Applying Rule "Delete Conditions" results in
                     (gr(x903,x902)→^*true⟹0→^*x897⟹s(x902)→^*x923⟹gr(x923,x897)→^*true⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥c)⟹cond2^#(true,s(x903),s(x902))≥c)
                   • Applying Rule "Induction" on gr(x923,x897)→^*true results in the following 3 new constraints.
                     1. (false→^*true⟹cond2^#(true,s(x903),s(x902))≥c)
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                     2. (true→^*true⟹gr(x903,x902)→^*true⟹0→^*0⟹s(x902)→^*s(x937)⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥c)⟹cond2^#(true,s(x903),s(x902))≥c)
                        • Applying Rule "Same Constructor" results in
                          (gr(x903,x902)→^*true⟹0→^*0⟹s(x902)→^*s(x937)⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥c)⟹cond2^#(true,s(x903),s(x902))≥c)
                        • Applying Rule "Same Constructor" results in
                          (gr(x903,x902)→^*true⟹s(x902)→^*s(x937)⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥c)⟹cond2^#(true,s(x903),s(x902))≥c)
                        • Applying Rule "Same Constructor" results in
                          (gr(x903,x902)→^*true⟹x902→^*x937⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥c)⟹cond2^#(true,s(x903),s(x902))≥c)
                        • Applying Rule "Delete Conditions" results in
                          (gr(x903,x902)→^*true⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥c)⟹cond2^#(true,s(x903),s(x902))≥c)
                        • Applying Rule "Delete Conditions" results in
                          (gr(x903,x902)→^*true⟹cond2^#(true,s(x903),s(x902))≥c)
                        • Applying Rule "Induction" on gr(x903,x902)→^*true results in the following 3 new constraints.
                          1. (false→^*true⟹cond2^#(true,s(0),s(x946))≥c)
                             • Applying Rule "Different Constructors" allows to drop this constraint.
                          2. (true→^*true⟹cond2^#(true,s(s(x947)),s(0))≥c)
                             • Applying Rule "Same Constructor" results in
                               cond2^#(true,s(s(x947)),s(0))≥c

                             • This constraint is kept as final constraint.

                          3. (gr(x949,x948)→^*true⟹(gr(x949,x948)→^*true⟹cond2^#(true,s(x949),s(x948))≥c)⟹cond2^#(true,s(s(x949)),s(s(x948)))≥c)
                             • Applying Rule "Simplify Conditions" results in
                               (cond2^#(true,s(x949),s(x948))≥c⟹cond2^#(true,s(s(x949)),s(s(x948)))≥c)

                             • This constraint is kept as final constraint.

                     3. (gr(x939,x938)→^*true⟹gr(x903,x902)→^*true⟹0→^*s(x938)⟹s(x902)→^*s(x939)⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥c)⟹∀x940 . ∀x941 . ∀x942 . ∀x943 . ∀x944 . ∀x945 . (gr(x939,x938)→^*true⟹gr(x940,x941)→^*true⟹0→^*x938⟹s(x941)→^*x939⟹∀x942 . ∀x943 . ∀x944 . ∀x945 . (gr(x940,x941)→^*true⟹0→^*x942⟹gr(x943,x942)→^*true⟹0→^*x944⟹gr(x945,x944)→^*true⟹x943→^*x940⟹x945→^*x941⟹cond2^#(true,x940,x941)≥c)⟹cond2^#(true,s(x940),s(x941))≥c)⟹cond2^#(true,s(x903),s(x902))≥c)
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x927,x926)→^*true⟹gr(x903,x902)→^*true⟹0→^*s(x926)⟹s(x903)→^*s(x927)⟹0→^*x897⟹s(x902)→^*x923⟹gr(x923,x897)→^*true⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥c)⟹∀x928 . ∀x929 . ∀x930 . ∀x931 . ∀x932 . ∀x933 . ∀x934 . ∀x935 . (gr(x927,x926)→^*true⟹gr(x928,x929)→^*true⟹0→^*x926⟹s(x928)→^*x927⟹0→^*x930⟹s(x929)→^*x931⟹gr(x931,x930)→^*true⟹∀x932 . ∀x933 . ∀x934 . ∀x935 . (gr(x928,x929)→^*true⟹0→^*x932⟹gr(x933,x932)→^*true⟹0→^*x934⟹gr(x935,x934)→^*true⟹x933→^*x928⟹x935→^*x929⟹cond2^#(true,x928,x929)≥c)⟹cond2^#(true,s(x928),s(x929))≥c)⟹cond2^#(true,s(x903),s(x902))≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*true⟹cond2^#(true,x58,x59)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (false→^*true⟹cond2^#(true,x58,x59)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond1^#(and(gr(x68,0),gr(x69,0)),x68,x69)→^*cond1^#(true,x70,x71)⟹cond2^#(gr(x70,x71),x70,x71)→^*cond2^#(true,x72,x73)⟹cond2^#(true,x72,x73)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (and(gr(x68,0),gr(x69,0))→^*true⟹x68→^*x70⟹x69→^*x71⟹cond2^#(gr(x70,x71),x70,x71)→^*cond2^#(true,x72,x73)⟹cond2^#(true,x72,x73)≥c)
    • Applying Rule "Same Constructor" results in
      (and(gr(x68,0),gr(x69,0))→^*true⟹x68→^*x70⟹x69→^*x71⟹gr(x70,x71)→^*true⟹x70→^*x72⟹x71→^*x73⟹cond2^#(true,x72,x73)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x72 by x70 which results in
      (and(gr(x68,0),gr(x69,0))→^*true⟹x68→^*x70⟹x69→^*x71⟹gr(x70,x71)→^*true⟹x71→^*x73⟹cond2^#(true,x70,x73)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x73 by x71 which results in
      (and(gr(x68,0),gr(x69,0))→^*true⟹x68→^*x70⟹x69→^*x71⟹gr(x70,x71)→^*true⟹cond2^#(true,x70,x71)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (x68→^*x70⟹x69→^*x71⟹gr(x70,x71)→^*true⟹gr(x69,0)→^*x951⟹and(gr(x68,0),x951)→^*true⟹cond2^#(true,x70,x71)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (gr(x68,0)→^*x950⟹gr(x69,0)→^*x951⟹and(x950,x951)→^*true⟹x68→^*x70⟹x69→^*x71⟹gr(x70,x71)→^*true⟹cond2^#(true,x70,x71)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x952⟹gr(x68,x952)→^*x950⟹gr(x69,0)→^*x951⟹and(x950,x951)→^*true⟹x68→^*x70⟹x69→^*x71⟹gr(x70,x71)→^*true⟹cond2^#(true,x70,x71)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x952⟹gr(x68,x952)→^*x950⟹0→^*x953⟹gr(x69,x953)→^*x951⟹and(x950,x951)→^*true⟹x68→^*x70⟹x69→^*x71⟹gr(x70,x71)→^*true⟹cond2^#(true,x70,x71)≥c)
    • Applying Rule "Induction" on and(x950,x951)→^*true results in the following 3 new constraints.
      1. (true→^*true⟹0→^*x952⟹gr(x68,x952)→^*true⟹0→^*x953⟹gr(x69,x953)→^*true⟹x68→^*x70⟹x69→^*x71⟹gr(x70,x71)→^*true⟹cond2^#(true,x70,x71)≥c)
         • Applying Rule "Same Constructor" results in
           (0→^*x952⟹gr(x68,x952)→^*true⟹0→^*x953⟹gr(x69,x953)→^*true⟹x68→^*x70⟹x69→^*x71⟹gr(x70,x71)→^*true⟹cond2^#(true,x70,x71)≥c)
         • Applying Rule "Induction" on gr(x70,x71)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond2^#(true,0,x956)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹0→^*x952⟹gr(x68,x952)→^*true⟹0→^*x953⟹gr(x69,x953)→^*true⟹x68→^*s(x957)⟹x69→^*0⟹cond2^#(true,s(x957),0)≥c)
              • Applying Rule "Same Constructor" results in
                (0→^*x952⟹gr(x68,x952)→^*true⟹0→^*x953⟹gr(x69,x953)→^*true⟹x68→^*s(x957)⟹x69→^*0⟹cond2^#(true,s(x957),0)≥c)
              • Applying Rule "Variable in Equation" allows to substitute x68 by s(x957) which results in
                (0→^*x952⟹gr(s(x957),x952)→^*true⟹0→^*x953⟹gr(x69,x953)→^*true⟹x69→^*0⟹cond2^#(true,s(x957),0)≥c)
              • Applying Rule "Variable in Equation" allows to substitute x69 by 0 which results in
                (0→^*x952⟹gr(s(x957),x952)→^*true⟹0→^*x953⟹gr(0,x953)→^*true⟹cond2^#(true,s(x957),0)≥c)
              • Applying Rule "Introduce fresh variable" results in
                (0→^*x952⟹s(x957)→^*x964⟹gr(x964,x952)→^*true⟹0→^*x953⟹gr(0,x953)→^*true⟹cond2^#(true,s(x957),0)≥c)
              • Applying Rule "Introduce fresh variable" results in
                (0→^*x952⟹s(x957)→^*x964⟹gr(x964,x952)→^*true⟹0→^*x953⟹0→^*x965⟹gr(x965,x953)→^*true⟹cond2^#(true,s(x957),0)≥c)
              • Applying Rule "Induction" on gr(x964,x952)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond2^#(true,s(x957),0)≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹0→^*0⟹s(x957)→^*s(x967)⟹0→^*x953⟹0→^*x965⟹gr(x965,x953)→^*true⟹cond2^#(true,s(x957),0)≥c)
                   • Applying Rule "Same Constructor" results in
                     (0→^*0⟹s(x957)→^*s(x967)⟹0→^*x953⟹0→^*x965⟹gr(x965,x953)→^*true⟹cond2^#(true,s(x957),0)≥c)
                   • Applying Rule "Same Constructor" results in
                     (s(x957)→^*s(x967)⟹0→^*x953⟹0→^*x965⟹gr(x965,x953)→^*true⟹cond2^#(true,s(x957),0)≥c)
                   • Applying Rule "Same Constructor" results in
                     (x957→^*x967⟹0→^*x953⟹0→^*x965⟹gr(x965,x953)→^*true⟹cond2^#(true,s(x957),0)≥c)
                   • Applying Rule "Delete Conditions" results in
                     (0→^*x953⟹0→^*x965⟹gr(x965,x953)→^*true⟹cond2^#(true,s(x957),0)≥c)
                   • Applying Rule "Induction" on gr(x965,x953)→^*true results in the following 3 new constraints.
                     1. (false→^*true⟹cond2^#(true,s(x957),0)≥c)
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                     2. (true→^*true⟹0→^*0⟹0→^*s(x974)⟹cond2^#(true,s(x957),0)≥c)
                        • Applying Rule "Same Constructor" results in
                          (0→^*0⟹0→^*s(x974)⟹cond2^#(true,s(x957),0)≥c)
                        • Applying Rule "Same Constructor" results in
                          (0→^*s(x974)⟹cond2^#(true,s(x957),0)≥c)
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                     3. (gr(x976,x975)→^*true⟹0→^*s(x975)⟹0→^*s(x976)⟹∀x977 . (gr(x976,x975)→^*true⟹0→^*x975⟹0→^*x976⟹cond2^#(true,s(x977),0)≥c)⟹cond2^#(true,s(x957),0)≥c)
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x969,x968)→^*true⟹0→^*s(x968)⟹s(x957)→^*s(x969)⟹0→^*x953⟹0→^*x965⟹gr(x965,x953)→^*true⟹∀x970 . ∀x971 . ∀x972 . (gr(x969,x968)→^*true⟹0→^*x968⟹s(x970)→^*x969⟹0→^*x971⟹0→^*x972⟹gr(x972,x971)→^*true⟹cond2^#(true,s(x970),0)≥c)⟹cond2^#(true,s(x957),0)≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x959,x958)→^*true⟹0→^*x952⟹gr(x68,x952)→^*true⟹0→^*x953⟹gr(x69,x953)→^*true⟹x68→^*s(x959)⟹x69→^*s(x958)⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥c)⟹cond2^#(true,s(x959),s(x958))≥c)
              • Applying Rule "Variable in Equation" allows to substitute x68 by s(x959) which results in
                (gr(x959,x958)→^*true⟹0→^*x952⟹gr(s(x959),x952)→^*true⟹0→^*x953⟹gr(x69,x953)→^*true⟹x69→^*s(x958)⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥c)⟹cond2^#(true,s(x959),s(x958))≥c)
              • Applying Rule "Variable in Equation" allows to substitute x69 by s(x958) which results in
                (gr(x959,x958)→^*true⟹0→^*x952⟹gr(s(x959),x952)→^*true⟹0→^*x953⟹gr(s(x958),x953)→^*true⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥c)⟹cond2^#(true,s(x959),s(x958))≥c)
              • Applying Rule "Introduce fresh variable" results in
                (gr(x959,x958)→^*true⟹0→^*x952⟹s(x959)→^*x978⟹gr(x978,x952)→^*true⟹0→^*x953⟹gr(s(x958),x953)→^*true⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥c)⟹cond2^#(true,s(x959),s(x958))≥c)
              • Applying Rule "Introduce fresh variable" results in
                (gr(x959,x958)→^*true⟹0→^*x952⟹s(x959)→^*x978⟹gr(x978,x952)→^*true⟹0→^*x953⟹s(x958)→^*x979⟹gr(x979,x953)→^*true⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥c)⟹cond2^#(true,s(x959),s(x958))≥c)
              • Applying Rule "Induction" on gr(x978,x952)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond2^#(true,s(x959),s(x958))≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹gr(x959,x958)→^*true⟹0→^*0⟹s(x959)→^*s(x981)⟹0→^*x953⟹s(x958)→^*x979⟹gr(x979,x953)→^*true⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥c)⟹cond2^#(true,s(x959),s(x958))≥c)
                   • Applying Rule "Same Constructor" results in
                     (gr(x959,x958)→^*true⟹0→^*0⟹s(x959)→^*s(x981)⟹0→^*x953⟹s(x958)→^*x979⟹gr(x979,x953)→^*true⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥c)⟹cond2^#(true,s(x959),s(x958))≥c)
                   • Applying Rule "Same Constructor" results in
                     (gr(x959,x958)→^*true⟹s(x959)→^*s(x981)⟹0→^*x953⟹s(x958)→^*x979⟹gr(x979,x953)→^*true⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥c)⟹cond2^#(true,s(x959),s(x958))≥c)
                   • Applying Rule "Same Constructor" results in
                     (gr(x959,x958)→^*true⟹x959→^*x981⟹0→^*x953⟹s(x958)→^*x979⟹gr(x979,x953)→^*true⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥c)⟹cond2^#(true,s(x959),s(x958))≥c)
                   • Applying Rule "Delete Conditions" results in
                     (gr(x959,x958)→^*true⟹0→^*x953⟹s(x958)→^*x979⟹gr(x979,x953)→^*true⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥c)⟹cond2^#(true,s(x959),s(x958))≥c)
                   • Applying Rule "Induction" on gr(x979,x953)→^*true results in the following 3 new constraints.
                     1. (false→^*true⟹cond2^#(true,s(x959),s(x958))≥c)
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                     2. (true→^*true⟹gr(x959,x958)→^*true⟹0→^*0⟹s(x958)→^*s(x993)⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥c)⟹cond2^#(true,s(x959),s(x958))≥c)
                        • Applying Rule "Same Constructor" results in
                          (gr(x959,x958)→^*true⟹0→^*0⟹s(x958)→^*s(x993)⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥c)⟹cond2^#(true,s(x959),s(x958))≥c)
                        • Applying Rule "Same Constructor" results in
                          (gr(x959,x958)→^*true⟹s(x958)→^*s(x993)⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥c)⟹cond2^#(true,s(x959),s(x958))≥c)
                        • Applying Rule "Same Constructor" results in
                          (gr(x959,x958)→^*true⟹x958→^*x993⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥c)⟹cond2^#(true,s(x959),s(x958))≥c)
                        • Applying Rule "Delete Conditions" results in
                          (gr(x959,x958)→^*true⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥c)⟹cond2^#(true,s(x959),s(x958))≥c)
                        • Applying Rule "Delete Conditions" results in
                          (gr(x959,x958)→^*true⟹cond2^#(true,s(x959),s(x958))≥c)
                        • Applying Rule "Induction" on gr(x959,x958)→^*true results in the following 3 new constraints.
                          1. (false→^*true⟹cond2^#(true,s(0),s(x1002))≥c)
                             • Applying Rule "Different Constructors" allows to drop this constraint.
                          2. (true→^*true⟹cond2^#(true,s(s(x1003)),s(0))≥c)
                             • Applying Rule "Same Constructor" results in
                               cond2^#(true,s(s(x1003)),s(0))≥c

                             • This constraint is kept as final constraint.

                          3. (gr(x1005,x1004)→^*true⟹(gr(x1005,x1004)→^*true⟹cond2^#(true,s(x1005),s(x1004))≥c)⟹cond2^#(true,s(s(x1005)),s(s(x1004)))≥c)
                             • Applying Rule "Simplify Conditions" results in
                               (cond2^#(true,s(x1005),s(x1004))≥c⟹cond2^#(true,s(s(x1005)),s(s(x1004)))≥c)

                             • This constraint is kept as final constraint.

                     3. (gr(x995,x994)→^*true⟹gr(x959,x958)→^*true⟹0→^*s(x994)⟹s(x958)→^*s(x995)⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥c)⟹∀x996 . ∀x997 . ∀x998 . ∀x999 . ∀x1000 . ∀x1001 . (gr(x995,x994)→^*true⟹gr(x996,x997)→^*true⟹0→^*x994⟹s(x997)→^*x995⟹∀x998 . ∀x999 . ∀x1000 . ∀x1001 . (gr(x996,x997)→^*true⟹0→^*x998⟹gr(x999,x998)→^*true⟹0→^*x1000⟹gr(x1001,x1000)→^*true⟹x999→^*x996⟹x1001→^*x997⟹cond2^#(true,x996,x997)≥c)⟹cond2^#(true,s(x996),s(x997))≥c)⟹cond2^#(true,s(x959),s(x958))≥c)
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x983,x982)→^*true⟹gr(x959,x958)→^*true⟹0→^*s(x982)⟹s(x959)→^*s(x983)⟹0→^*x953⟹s(x958)→^*x979⟹gr(x979,x953)→^*true⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥c)⟹∀x984 . ∀x985 . ∀x986 . ∀x987 . ∀x988 . ∀x989 . ∀x990 . ∀x991 . (gr(x983,x982)→^*true⟹gr(x984,x985)→^*true⟹0→^*x982⟹s(x984)→^*x983⟹0→^*x986⟹s(x985)→^*x987⟹gr(x987,x986)→^*true⟹∀x988 . ∀x989 . ∀x990 . ∀x991 . (gr(x984,x985)→^*true⟹0→^*x988⟹gr(x989,x988)→^*true⟹0→^*x990⟹gr(x991,x990)→^*true⟹x989→^*x984⟹x991→^*x985⟹cond2^#(true,x984,x985)≥c)⟹cond2^#(true,s(x984),s(x985))≥c)⟹cond2^#(true,s(x959),s(x958))≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*true⟹cond2^#(true,x70,x71)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (false→^*true⟹cond2^#(true,x70,x71)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond2^#(gr(x132,x133),x132,x133)→^*cond2^#(true,x134,x135)⟹cond3^#(gr(x134,0),x134,x135)→^*cond3^#(true,x136,x137)⟹cond3^#(true,x136,x137)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x132,x133)→^*true⟹x132→^*x134⟹x133→^*x135⟹cond3^#(gr(x134,0),x134,x135)→^*cond3^#(true,x136,x137)⟹cond3^#(true,x136,x137)≥c)
    • Applying Rule "Same Constructor" results in
      (gr(x132,x133)→^*true⟹x132→^*x134⟹x133→^*x135⟹gr(x134,0)→^*true⟹x134→^*x136⟹x135→^*x137⟹cond3^#(true,x136,x137)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x135 by x133 which results in
      (gr(x132,x133)→^*true⟹x132→^*x134⟹gr(x134,0)→^*true⟹x134→^*x136⟹x133→^*x137⟹cond3^#(true,x136,x137)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x136 by x134 which results in
      (gr(x132,x133)→^*true⟹x132→^*x134⟹gr(x134,0)→^*true⟹x133→^*x137⟹cond3^#(true,x134,x137)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x137 by x133 which results in
      (gr(x132,x133)→^*true⟹x132→^*x134⟹gr(x134,0)→^*true⟹cond3^#(true,x134,x133)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (gr(x132,x133)→^*true⟹x132→^*x134⟹0→^*x1006⟹gr(x134,x1006)→^*true⟹cond3^#(true,x134,x133)≥c)
    • Applying Rule "Induction" on gr(x132,x133)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹cond3^#(true,x134,x1007)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹s(x1008)→^*x134⟹0→^*x1006⟹gr(x134,x1006)→^*true⟹cond3^#(true,x134,0)≥c)
         • Applying Rule "Same Constructor" results in
           (s(x1008)→^*x134⟹0→^*x1006⟹gr(x134,x1006)→^*true⟹cond3^#(true,x134,0)≥c)
         • Applying Rule "Induction" on gr(x134,x1006)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond3^#(true,0,0)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹s(x1008)→^*s(x1014)⟹0→^*0⟹cond3^#(true,s(x1014),0)≥c)
              • Applying Rule "Same Constructor" results in
                (s(x1008)→^*s(x1014)⟹0→^*0⟹cond3^#(true,s(x1014),0)≥c)
              • Applying Rule "Same Constructor" results in
                (x1008→^*x1014⟹0→^*0⟹cond3^#(true,s(x1014),0)≥c)
              • Applying Rule "Same Constructor" results in
                (x1008→^*x1014⟹cond3^#(true,s(x1014),0)≥c)
              • Applying Rule "Variable in Equation" allows to substitute x1014 by x1008 which results in
                cond3^#(true,s(x1008),0)≥c

              • This constraint is kept as final constraint.

           3. (gr(x1016,x1015)→^*true⟹s(x1008)→^*s(x1016)⟹0→^*s(x1015)⟹∀x1017 . (gr(x1016,x1015)→^*true⟹s(x1017)→^*x1016⟹0→^*x1015⟹cond3^#(true,x1016,0)≥c)⟹cond3^#(true,s(x1016),0)≥c)
              • Applying Rule "Same Constructor" results in
                (gr(x1016,x1015)→^*true⟹x1008→^*x1016⟹0→^*s(x1015)⟹∀x1017 . (gr(x1016,x1015)→^*true⟹s(x1017)→^*x1016⟹0→^*x1015⟹cond3^#(true,x1016,0)≥c)⟹cond3^#(true,s(x1016),0)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x1010,x1009)→^*true⟹s(x1010)→^*x134⟹0→^*x1006⟹gr(x134,x1006)→^*true⟹∀x1011 . ∀x1012 . (gr(x1010,x1009)→^*true⟹x1010→^*x1011⟹0→^*x1012⟹gr(x1011,x1012)→^*true⟹cond3^#(true,x1011,x1009)≥c)⟹cond3^#(true,x134,s(x1009))≥c)
         • Applying Rule "Induction" on gr(x134,x1006)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond3^#(true,0,s(x1009))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹gr(x1010,x1009)→^*true⟹s(x1010)→^*s(x1019)⟹0→^*0⟹∀x1011 . ∀x1012 . (gr(x1010,x1009)→^*true⟹x1010→^*x1011⟹0→^*x1012⟹gr(x1011,x1012)→^*true⟹cond3^#(true,x1011,x1009)≥c)⟹cond3^#(true,s(x1019),s(x1009))≥c)
              • Applying Rule "Same Constructor" results in
                (gr(x1010,x1009)→^*true⟹s(x1010)→^*s(x1019)⟹0→^*0⟹∀x1011 . ∀x1012 . (gr(x1010,x1009)→^*true⟹x1010→^*x1011⟹0→^*x1012⟹gr(x1011,x1012)→^*true⟹cond3^#(true,x1011,x1009)≥c)⟹cond3^#(true,s(x1019),s(x1009))≥c)
              • Applying Rule "Same Constructor" results in
                (gr(x1010,x1009)→^*true⟹x1010→^*x1019⟹0→^*0⟹∀x1011 . ∀x1012 . (gr(x1010,x1009)→^*true⟹x1010→^*x1011⟹0→^*x1012⟹gr(x1011,x1012)→^*true⟹cond3^#(true,x1011,x1009)≥c)⟹cond3^#(true,s(x1019),s(x1009))≥c)
              • Applying Rule "Same Constructor" results in
                (gr(x1010,x1009)→^*true⟹x1010→^*x1019⟹∀x1011 . ∀x1012 . (gr(x1010,x1009)→^*true⟹x1010→^*x1011⟹0→^*x1012⟹gr(x1011,x1012)→^*true⟹cond3^#(true,x1011,x1009)≥c)⟹cond3^#(true,s(x1019),s(x1009))≥c)
              • Applying Rule "Variable in Equation" allows to substitute x1019 by x1010 which results in
                (gr(x1010,x1009)→^*true⟹∀x1011 . ∀x1012 . (gr(x1010,x1009)→^*true⟹x1010→^*x1011⟹0→^*x1012⟹gr(x1011,x1012)→^*true⟹cond3^#(true,x1011,x1009)≥c)⟹cond3^#(true,s(x1010),s(x1009))≥c)
              • Applying Rule "Delete Conditions" results in
                (gr(x1010,x1009)→^*true⟹cond3^#(true,s(x1010),s(x1009))≥c)
              • Applying Rule "Induction" on gr(x1010,x1009)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond3^#(true,s(0),s(x1026))≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹cond3^#(true,s(s(x1027)),s(0))≥c)
                   • Applying Rule "Same Constructor" results in
                     cond3^#(true,s(s(x1027)),s(0))≥c

                   • This constraint is kept as final constraint.

                3. (gr(x1029,x1028)→^*true⟹(gr(x1029,x1028)→^*true⟹cond3^#(true,s(x1029),s(x1028))≥c)⟹cond3^#(true,s(s(x1029)),s(s(x1028)))≥c)
                   • Applying Rule "Simplify Conditions" results in
                     (cond3^#(true,s(x1029),s(x1028))≥c⟹cond3^#(true,s(s(x1029)),s(s(x1028)))≥c)

                   • This constraint is kept as final constraint.

           3. (gr(x1021,x1020)→^*true⟹gr(x1010,x1009)→^*true⟹s(x1010)→^*s(x1021)⟹0→^*s(x1020)⟹∀x1011 . ∀x1012 . (gr(x1010,x1009)→^*true⟹x1010→^*x1011⟹0→^*x1012⟹gr(x1011,x1012)→^*true⟹cond3^#(true,x1011,x1009)≥c)⟹∀x1022 . ∀x1023 . ∀x1024 . ∀x1025 . (gr(x1021,x1020)→^*true⟹gr(x1022,x1023)→^*true⟹s(x1022)→^*x1021⟹0→^*x1020⟹∀x1024 . ∀x1025 . (gr(x1022,x1023)→^*true⟹x1022→^*x1024⟹0→^*x1025⟹gr(x1024,x1025)→^*true⟹cond3^#(true,x1024,x1023)≥c)⟹cond3^#(true,x1021,s(x1023))≥c)⟹cond3^#(true,s(x1021),s(x1009))≥c)
              • Applying Rule "Same Constructor" results in
                (gr(x1021,x1020)→^*true⟹gr(x1010,x1009)→^*true⟹x1010→^*x1021⟹0→^*s(x1020)⟹∀x1011 . ∀x1012 . (gr(x1010,x1009)→^*true⟹x1010→^*x1011⟹0→^*x1012⟹gr(x1011,x1012)→^*true⟹cond3^#(true,x1011,x1009)≥c)⟹∀x1022 . ∀x1023 . ∀x1024 . ∀x1025 . (gr(x1021,x1020)→^*true⟹gr(x1022,x1023)→^*true⟹s(x1022)→^*x1021⟹0→^*x1020⟹∀x1024 . ∀x1025 . (gr(x1022,x1023)→^*true⟹x1022→^*x1024⟹0→^*x1025⟹gr(x1024,x1025)→^*true⟹cond3^#(true,x1024,x1023)≥c)⟹cond3^#(true,x1021,s(x1023))≥c)⟹cond3^#(true,s(x1021),s(x1009))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond3^#(gr(x144,0),x144,x145)→^*cond3^#(true,x146,x147)⟹cond3^#(gr(x146,0),p(x146),x147)→^*cond3^#(true,x148,x149)⟹cond3^#(true,x148,x149)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x144,0)→^*true⟹x144→^*x146⟹x145→^*x147⟹cond3^#(gr(x146,0),p(x146),x147)→^*cond3^#(true,x148,x149)⟹cond3^#(true,x148,x149)≥c)
    • Applying Rule "Same Constructor" results in
      (gr(x144,0)→^*true⟹x144→^*x146⟹x145→^*x147⟹gr(x146,0)→^*true⟹p(x146)→^*x148⟹x147→^*x149⟹cond3^#(true,x148,x149)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x147 by x145 which results in
      (gr(x144,0)→^*true⟹x144→^*x146⟹gr(x146,0)→^*true⟹p(x146)→^*x148⟹x145→^*x149⟹cond3^#(true,x148,x149)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x149 by x145 which results in
      (gr(x144,0)→^*true⟹x144→^*x146⟹gr(x146,0)→^*true⟹p(x146)→^*x148⟹cond3^#(true,x148,x145)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1030⟹gr(x144,x1030)→^*true⟹x144→^*x146⟹gr(x146,0)→^*true⟹p(x146)→^*x148⟹cond3^#(true,x148,x145)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1030⟹gr(x144,x1030)→^*true⟹x144→^*x146⟹0→^*x1031⟹gr(x146,x1031)→^*true⟹p(x146)→^*x148⟹cond3^#(true,x148,x145)≥c)
    • Applying Rule "Induction" on gr(x144,x1030)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹cond3^#(true,x148,x145)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹0→^*0⟹s(x1033)→^*x146⟹0→^*x1031⟹gr(x146,x1031)→^*true⟹p(x146)→^*x148⟹cond3^#(true,x148,x145)≥c)
         • Applying Rule "Same Constructor" results in
           (0→^*0⟹s(x1033)→^*x146⟹0→^*x1031⟹gr(x146,x1031)→^*true⟹p(x146)→^*x148⟹cond3^#(true,x148,x145)≥c)
         • Applying Rule "Same Constructor" results in
           (s(x1033)→^*x146⟹0→^*x1031⟹gr(x146,x1031)→^*true⟹p(x146)→^*x148⟹cond3^#(true,x148,x145)≥c)
         • Applying Rule "Induction" on gr(x146,x1031)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond3^#(true,x148,x145)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹s(x1033)→^*s(x1041)⟹0→^*0⟹p(s(x1041))→^*x148⟹cond3^#(true,x148,x145)≥c)
              • Applying Rule "Same Constructor" results in
                (s(x1033)→^*s(x1041)⟹0→^*0⟹p(s(x1041))→^*x148⟹cond3^#(true,x148,x145)≥c)
              • Applying Rule "Same Constructor" results in
                (x1033→^*x1041⟹0→^*0⟹p(s(x1041))→^*x148⟹cond3^#(true,x148,x145)≥c)
              • Applying Rule "Same Constructor" results in
                (x1033→^*x1041⟹p(s(x1041))→^*x148⟹cond3^#(true,x148,x145)≥c)
              • Applying Rule "Variable in Equation" allows to substitute x1033 by x1041 which results in
                (p(s(x1041))→^*x148⟹cond3^#(true,x148,x145)≥c)
              • Applying Rule "Introduce fresh variable" results in
                (s(x1041)→^*x1047⟹p(x1047)→^*x148⟹cond3^#(true,x148,x145)≥c)
              • Applying Rule "Variable in Equation" allows to substitute x1047 by s(x1041) which results in
                (p(s(x1041))→^*x148⟹cond3^#(true,x148,x145)≥c)
              • Applying Rule "Delete Conditions" results in
                cond3^#(true,x148,x145)≥c

              • This constraint is kept as final constraint.

           3. (gr(x1043,x1042)→^*true⟹s(x1033)→^*s(x1043)⟹0→^*s(x1042)⟹p(s(x1043))→^*x148⟹∀x1044 . ∀x1045 . ∀x1046 . (gr(x1043,x1042)→^*true⟹s(x1044)→^*x1043⟹0→^*x1042⟹p(x1043)→^*x1045⟹cond3^#(true,x1045,x1046)≥c)⟹cond3^#(true,x148,x145)≥c)
              • Applying Rule "Same Constructor" results in
                (gr(x1043,x1042)→^*true⟹x1033→^*x1043⟹0→^*s(x1042)⟹p(s(x1043))→^*x148⟹∀x1044 . ∀x1045 . ∀x1046 . (gr(x1043,x1042)→^*true⟹s(x1044)→^*x1043⟹0→^*x1042⟹p(x1043)→^*x1045⟹cond3^#(true,x1045,x1046)≥c)⟹cond3^#(true,x148,x145)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x1035,x1034)→^*true⟹0→^*s(x1034)⟹s(x1035)→^*x146⟹0→^*x1031⟹gr(x146,x1031)→^*true⟹p(x146)→^*x148⟹∀x1036 . ∀x1037 . ∀x1038 . ∀x1039 . (gr(x1035,x1034)→^*true⟹0→^*x1034⟹x1035→^*x1036⟹0→^*x1037⟹gr(x1036,x1037)→^*true⟹p(x1036)→^*x1038⟹cond3^#(true,x1038,x1039)≥c)⟹cond3^#(true,x148,x145)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond3^#(gr(x150,0),p(x150),x151)→^*cond3^#(true,x152,x153)⟹cond3^#(gr(x152,0),p(x152),x153)→^*cond3^#(true,x154,x155)⟹cond3^#(true,x154,x155)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x150,0)→^*true⟹p(x150)→^*x152⟹x151→^*x153⟹cond3^#(gr(x152,0),p(x152),x153)→^*cond3^#(true,x154,x155)⟹cond3^#(true,x154,x155)≥c)
    • Applying Rule "Same Constructor" results in
      (gr(x150,0)→^*true⟹p(x150)→^*x152⟹x151→^*x153⟹gr(x152,0)→^*true⟹p(x152)→^*x154⟹x153→^*x155⟹cond3^#(true,x154,x155)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x153 by x151 which results in
      (gr(x150,0)→^*true⟹p(x150)→^*x152⟹gr(x152,0)→^*true⟹p(x152)→^*x154⟹x151→^*x155⟹cond3^#(true,x154,x155)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x155 by x151 which results in
      (gr(x150,0)→^*true⟹p(x150)→^*x152⟹gr(x152,0)→^*true⟹p(x152)→^*x154⟹cond3^#(true,x154,x151)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1048⟹gr(x150,x1048)→^*true⟹p(x150)→^*x152⟹gr(x152,0)→^*true⟹p(x152)→^*x154⟹cond3^#(true,x154,x151)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1048⟹gr(x150,x1048)→^*true⟹p(x150)→^*x152⟹0→^*x1049⟹gr(x152,x1049)→^*true⟹p(x152)→^*x154⟹cond3^#(true,x154,x151)≥c)
    • Applying Rule "Induction" on gr(x150,x1048)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹cond3^#(true,x154,x151)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹0→^*0⟹p(s(x1051))→^*x152⟹0→^*x1049⟹gr(x152,x1049)→^*true⟹p(x152)→^*x154⟹cond3^#(true,x154,x151)≥c)
         • Applying Rule "Same Constructor" results in
           (0→^*0⟹p(s(x1051))→^*x152⟹0→^*x1049⟹gr(x152,x1049)→^*true⟹p(x152)→^*x154⟹cond3^#(true,x154,x151)≥c)
         • Applying Rule "Same Constructor" results in
           (p(s(x1051))→^*x152⟹0→^*x1049⟹gr(x152,x1049)→^*true⟹p(x152)→^*x154⟹cond3^#(true,x154,x151)≥c)
         • Applying Rule "Introduce fresh variable" results in
           (s(x1051)→^*x1058⟹p(x1058)→^*x152⟹0→^*x1049⟹gr(x152,x1049)→^*true⟹p(x152)→^*x154⟹cond3^#(true,x154,x151)≥c)
         • Applying Rule "Induction" on gr(x152,x1049)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond3^#(true,x154,x151)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹s(x1051)→^*x1058⟹p(x1058)→^*s(x1060)⟹0→^*0⟹p(s(x1060))→^*x154⟹cond3^#(true,x154,x151)≥c)
              • Applying Rule "Same Constructor" results in
                (s(x1051)→^*x1058⟹p(x1058)→^*s(x1060)⟹0→^*0⟹p(s(x1060))→^*x154⟹cond3^#(true,x154,x151)≥c)
              • Applying Rule "Same Constructor" results in
                (s(x1051)→^*x1058⟹p(x1058)→^*s(x1060)⟹p(s(x1060))→^*x154⟹cond3^#(true,x154,x151)≥c)
              • Applying Rule "Introduce fresh variable" results in
                (s(x1051)→^*x1058⟹p(x1058)→^*s(x1060)⟹s(x1060)→^*x1067⟹p(x1067)→^*x154⟹cond3^#(true,x154,x151)≥c)
              • Applying Rule "Induction" on p(x1058)→^*s(x1060) results in the following 2 new constraints.
                1. (0→^*s(x1060)⟹cond3^#(true,x154,x151)≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (x1068→^*s(x1060)⟹s(x1051)→^*s(x1068)⟹s(x1060)→^*x1067⟹p(x1067)→^*x154⟹cond3^#(true,x154,x151)≥c)
                   • Applying Rule "Same Constructor" results in
                     (x1068→^*s(x1060)⟹x1051→^*x1068⟹s(x1060)→^*x1067⟹p(x1067)→^*x154⟹cond3^#(true,x154,x151)≥c)
                   • Applying Rule "Variable in Equation" allows to substitute x1068 by x1051 which results in
                     (x1051→^*s(x1060)⟹s(x1060)→^*x1067⟹p(x1067)→^*x154⟹cond3^#(true,x154,x151)≥c)
                   • Applying Rule "Variable in Equation" allows to substitute x1051 by s(x1060) which results in
                     (s(x1060)→^*x1067⟹p(x1067)→^*x154⟹cond3^#(true,x154,x151)≥c)
                   • Applying Rule "Variable in Equation" allows to substitute x1067 by s(x1060) which results in
                     (p(s(x1060))→^*x154⟹cond3^#(true,x154,x151)≥c)
                   • Applying Rule "Delete Conditions" results in
                     cond3^#(true,x154,x151)≥c

                   • This constraint is kept as final constraint.

           3. (gr(x1062,x1061)→^*true⟹s(x1051)→^*x1058⟹p(x1058)→^*s(x1062)⟹0→^*s(x1061)⟹p(s(x1062))→^*x154⟹∀x1063 . ∀x1064 . ∀x1065 . ∀x1066 . (gr(x1062,x1061)→^*true⟹s(x1063)→^*x1064⟹p(x1064)→^*x1062⟹0→^*x1061⟹p(x1062)→^*x1065⟹cond3^#(true,x1065,x1066)≥c)⟹cond3^#(true,x154,x151)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x1053,x1052)→^*true⟹0→^*s(x1052)⟹p(s(x1053))→^*x152⟹0→^*x1049⟹gr(x152,x1049)→^*true⟹p(x152)→^*x154⟹∀x1054 . ∀x1055 . ∀x1056 . ∀x1057 . (gr(x1053,x1052)→^*true⟹0→^*x1052⟹p(x1053)→^*x1054⟹0→^*x1055⟹gr(x1054,x1055)→^*true⟹p(x1054)→^*x1056⟹cond3^#(true,x1056,x1057)≥c)⟹cond3^#(true,x154,x151)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond2^#(gr(x260,x261),x260,x261)→^*cond2^#(true,x262,x263)⟹cond3^#(gr(x262,0),x262,x263)→^*cond3^#(false,x264,x265)⟹cond3^#(false,x264,x265)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x260,x261)→^*true⟹x260→^*x262⟹x261→^*x263⟹cond3^#(gr(x262,0),x262,x263)→^*cond3^#(false,x264,x265)⟹cond3^#(false,x264,x265)≥c)
    • Applying Rule "Same Constructor" results in
      (gr(x260,x261)→^*true⟹x260→^*x262⟹x261→^*x263⟹gr(x262,0)→^*false⟹x262→^*x264⟹x263→^*x265⟹cond3^#(false,x264,x265)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x263 by x261 which results in
      (gr(x260,x261)→^*true⟹x260→^*x262⟹gr(x262,0)→^*false⟹x262→^*x264⟹x261→^*x265⟹cond3^#(false,x264,x265)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x264 by x262 which results in
      (gr(x260,x261)→^*true⟹x260→^*x262⟹gr(x262,0)→^*false⟹x261→^*x265⟹cond3^#(false,x262,x265)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x265 by x261 which results in
      (gr(x260,x261)→^*true⟹x260→^*x262⟹gr(x262,0)→^*false⟹cond3^#(false,x262,x261)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (gr(x260,x261)→^*true⟹x260→^*x262⟹0→^*x1069⟹gr(x262,x1069)→^*false⟹cond3^#(false,x262,x261)≥c)
    • Applying Rule "Induction" on gr(x260,x261)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹cond3^#(false,x262,x1070)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹s(x1071)→^*x262⟹0→^*x1069⟹gr(x262,x1069)→^*false⟹cond3^#(false,x262,0)≥c)
         • Applying Rule "Same Constructor" results in
           (s(x1071)→^*x262⟹0→^*x1069⟹gr(x262,x1069)→^*false⟹cond3^#(false,x262,0)≥c)
         • Applying Rule "Induction" on gr(x262,x1069)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹s(x1071)→^*0⟹0→^*x1076⟹cond3^#(false,0,0)≥c)
              • Applying Rule "Same Constructor" results in
                (s(x1071)→^*0⟹0→^*x1076⟹cond3^#(false,0,0)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond3^#(false,s(x1077),0)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1079,x1078)→^*false⟹s(x1071)→^*s(x1079)⟹0→^*s(x1078)⟹∀x1080 . (gr(x1079,x1078)→^*false⟹s(x1080)→^*x1079⟹0→^*x1078⟹cond3^#(false,x1079,0)≥c)⟹cond3^#(false,s(x1079),0)≥c)
              • Applying Rule "Same Constructor" results in
                (gr(x1079,x1078)→^*false⟹x1071→^*x1079⟹0→^*s(x1078)⟹∀x1080 . (gr(x1079,x1078)→^*false⟹s(x1080)→^*x1079⟹0→^*x1078⟹cond3^#(false,x1079,0)≥c)⟹cond3^#(false,s(x1079),0)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x1073,x1072)→^*true⟹s(x1073)→^*x262⟹0→^*x1069⟹gr(x262,x1069)→^*false⟹∀x1074 . ∀x1075 . (gr(x1073,x1072)→^*true⟹x1073→^*x1074⟹0→^*x1075⟹gr(x1074,x1075)→^*false⟹cond3^#(false,x1074,x1072)≥c)⟹cond3^#(false,x262,s(x1072))≥c)
         • Applying Rule "Induction" on gr(x262,x1069)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹gr(x1073,x1072)→^*true⟹s(x1073)→^*0⟹0→^*x1081⟹∀x1074 . ∀x1075 . (gr(x1073,x1072)→^*true⟹x1073→^*x1074⟹0→^*x1075⟹gr(x1074,x1075)→^*false⟹cond3^#(false,x1074,x1072)≥c)⟹cond3^#(false,0,s(x1072))≥c)
              • Applying Rule "Same Constructor" results in
                (gr(x1073,x1072)→^*true⟹s(x1073)→^*0⟹0→^*x1081⟹∀x1074 . ∀x1075 . (gr(x1073,x1072)→^*true⟹x1073→^*x1074⟹0→^*x1075⟹gr(x1074,x1075)→^*false⟹cond3^#(false,x1074,x1072)≥c)⟹cond3^#(false,0,s(x1072))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond3^#(false,s(x1082),s(x1072))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1084,x1083)→^*false⟹gr(x1073,x1072)→^*true⟹s(x1073)→^*s(x1084)⟹0→^*s(x1083)⟹∀x1074 . ∀x1075 . (gr(x1073,x1072)→^*true⟹x1073→^*x1074⟹0→^*x1075⟹gr(x1074,x1075)→^*false⟹cond3^#(false,x1074,x1072)≥c)⟹∀x1085 . ∀x1086 . ∀x1087 . ∀x1088 . (gr(x1084,x1083)→^*false⟹gr(x1085,x1086)→^*true⟹s(x1085)→^*x1084⟹0→^*x1083⟹∀x1087 . ∀x1088 . (gr(x1085,x1086)→^*true⟹x1085→^*x1087⟹0→^*x1088⟹gr(x1087,x1088)→^*false⟹cond3^#(false,x1087,x1086)≥c)⟹cond3^#(false,x1084,s(x1086))≥c)⟹cond3^#(false,s(x1084),s(x1072))≥c)
              • Applying Rule "Same Constructor" results in
                (gr(x1084,x1083)→^*false⟹gr(x1073,x1072)→^*true⟹x1073→^*x1084⟹0→^*s(x1083)⟹∀x1074 . ∀x1075 . (gr(x1073,x1072)→^*true⟹x1073→^*x1074⟹0→^*x1075⟹gr(x1074,x1075)→^*false⟹cond3^#(false,x1074,x1072)≥c)⟹∀x1085 . ∀x1086 . ∀x1087 . ∀x1088 . (gr(x1084,x1083)→^*false⟹gr(x1085,x1086)→^*true⟹s(x1085)→^*x1084⟹0→^*x1083⟹∀x1087 . ∀x1088 . (gr(x1085,x1086)→^*true⟹x1085→^*x1087⟹0→^*x1088⟹gr(x1087,x1088)→^*false⟹cond3^#(false,x1087,x1086)≥c)⟹cond3^#(false,x1084,s(x1086))≥c)⟹cond3^#(false,s(x1084),s(x1072))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond3^#(gr(x272,0),x272,x273)→^*cond3^#(true,x274,x275)⟹cond3^#(gr(x274,0),p(x274),x275)→^*cond3^#(false,x276,x277)⟹cond3^#(false,x276,x277)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x272,0)→^*true⟹x272→^*x274⟹x273→^*x275⟹cond3^#(gr(x274,0),p(x274),x275)→^*cond3^#(false,x276,x277)⟹cond3^#(false,x276,x277)≥c)
    • Applying Rule "Same Constructor" results in
      (gr(x272,0)→^*true⟹x272→^*x274⟹x273→^*x275⟹gr(x274,0)→^*false⟹p(x274)→^*x276⟹x275→^*x277⟹cond3^#(false,x276,x277)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x275 by x273 which results in
      (gr(x272,0)→^*true⟹x272→^*x274⟹gr(x274,0)→^*false⟹p(x274)→^*x276⟹x273→^*x277⟹cond3^#(false,x276,x277)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x277 by x273 which results in
      (gr(x272,0)→^*true⟹x272→^*x274⟹gr(x274,0)→^*false⟹p(x274)→^*x276⟹cond3^#(false,x276,x273)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1089⟹gr(x272,x1089)→^*true⟹x272→^*x274⟹gr(x274,0)→^*false⟹p(x274)→^*x276⟹cond3^#(false,x276,x273)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1089⟹gr(x272,x1089)→^*true⟹x272→^*x274⟹0→^*x1090⟹gr(x274,x1090)→^*false⟹p(x274)→^*x276⟹cond3^#(false,x276,x273)≥c)
    • Applying Rule "Induction" on gr(x272,x1089)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹cond3^#(false,x276,x273)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹0→^*0⟹s(x1092)→^*x274⟹0→^*x1090⟹gr(x274,x1090)→^*false⟹p(x274)→^*x276⟹cond3^#(false,x276,x273)≥c)
         • Applying Rule "Same Constructor" results in
           (0→^*0⟹s(x1092)→^*x274⟹0→^*x1090⟹gr(x274,x1090)→^*false⟹p(x274)→^*x276⟹cond3^#(false,x276,x273)≥c)
         • Applying Rule "Same Constructor" results in
           (s(x1092)→^*x274⟹0→^*x1090⟹gr(x274,x1090)→^*false⟹p(x274)→^*x276⟹cond3^#(false,x276,x273)≥c)
         • Applying Rule "Induction" on gr(x274,x1090)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹s(x1092)→^*0⟹0→^*x1099⟹p(0)→^*x276⟹cond3^#(false,x276,x273)≥c)
              • Applying Rule "Same Constructor" results in
                (s(x1092)→^*0⟹0→^*x1099⟹p(0)→^*x276⟹cond3^#(false,x276,x273)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond3^#(false,x276,x273)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1102,x1101)→^*false⟹s(x1092)→^*s(x1102)⟹0→^*s(x1101)⟹p(s(x1102))→^*x276⟹∀x1103 . ∀x1104 . ∀x1105 . (gr(x1102,x1101)→^*false⟹s(x1103)→^*x1102⟹0→^*x1101⟹p(x1102)→^*x1104⟹cond3^#(false,x1104,x1105)≥c)⟹cond3^#(false,x276,x273)≥c)
              • Applying Rule "Same Constructor" results in
                (gr(x1102,x1101)→^*false⟹x1092→^*x1102⟹0→^*s(x1101)⟹p(s(x1102))→^*x276⟹∀x1103 . ∀x1104 . ∀x1105 . (gr(x1102,x1101)→^*false⟹s(x1103)→^*x1102⟹0→^*x1101⟹p(x1102)→^*x1104⟹cond3^#(false,x1104,x1105)≥c)⟹cond3^#(false,x276,x273)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x1094,x1093)→^*true⟹0→^*s(x1093)⟹s(x1094)→^*x274⟹0→^*x1090⟹gr(x274,x1090)→^*false⟹p(x274)→^*x276⟹∀x1095 . ∀x1096 . ∀x1097 . ∀x1098 . (gr(x1094,x1093)→^*true⟹0→^*x1093⟹x1094→^*x1095⟹0→^*x1096⟹gr(x1095,x1096)→^*false⟹p(x1095)→^*x1097⟹cond3^#(false,x1097,x1098)≥c)⟹cond3^#(false,x276,x273)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond3^#(gr(x278,0),p(x278),x279)→^*cond3^#(true,x280,x281)⟹cond3^#(gr(x280,0),p(x280),x281)→^*cond3^#(false,x282,x283)⟹cond3^#(false,x282,x283)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x278,0)→^*true⟹p(x278)→^*x280⟹x279→^*x281⟹cond3^#(gr(x280,0),p(x280),x281)→^*cond3^#(false,x282,x283)⟹cond3^#(false,x282,x283)≥c)
    • Applying Rule "Same Constructor" results in
      (gr(x278,0)→^*true⟹p(x278)→^*x280⟹x279→^*x281⟹gr(x280,0)→^*false⟹p(x280)→^*x282⟹x281→^*x283⟹cond3^#(false,x282,x283)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x281 by x279 which results in
      (gr(x278,0)→^*true⟹p(x278)→^*x280⟹gr(x280,0)→^*false⟹p(x280)→^*x282⟹x279→^*x283⟹cond3^#(false,x282,x283)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x283 by x279 which results in
      (gr(x278,0)→^*true⟹p(x278)→^*x280⟹gr(x280,0)→^*false⟹p(x280)→^*x282⟹cond3^#(false,x282,x279)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1106⟹gr(x278,x1106)→^*true⟹p(x278)→^*x280⟹gr(x280,0)→^*false⟹p(x280)→^*x282⟹cond3^#(false,x282,x279)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1106⟹gr(x278,x1106)→^*true⟹p(x278)→^*x280⟹0→^*x1107⟹gr(x280,x1107)→^*false⟹p(x280)→^*x282⟹cond3^#(false,x282,x279)≥c)
    • Applying Rule "Induction" on gr(x278,x1106)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹cond3^#(false,x282,x279)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹0→^*0⟹p(s(x1109))→^*x280⟹0→^*x1107⟹gr(x280,x1107)→^*false⟹p(x280)→^*x282⟹cond3^#(false,x282,x279)≥c)
         • Applying Rule "Same Constructor" results in
           (0→^*0⟹p(s(x1109))→^*x280⟹0→^*x1107⟹gr(x280,x1107)→^*false⟹p(x280)→^*x282⟹cond3^#(false,x282,x279)≥c)
         • Applying Rule "Same Constructor" results in
           (p(s(x1109))→^*x280⟹0→^*x1107⟹gr(x280,x1107)→^*false⟹p(x280)→^*x282⟹cond3^#(false,x282,x279)≥c)
         • Applying Rule "Introduce fresh variable" results in
           (s(x1109)→^*x1116⟹p(x1116)→^*x280⟹0→^*x1107⟹gr(x280,x1107)→^*false⟹p(x280)→^*x282⟹cond3^#(false,x282,x279)≥c)
         • Applying Rule "Induction" on gr(x280,x1107)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹s(x1109)→^*x1116⟹p(x1116)→^*0⟹0→^*x1117⟹p(0)→^*x282⟹cond3^#(false,x282,x279)≥c)
              • Applying Rule "Same Constructor" results in
                (s(x1109)→^*x1116⟹p(x1116)→^*0⟹0→^*x1117⟹p(0)→^*x282⟹cond3^#(false,x282,x279)≥c)
              • Applying Rule "Delete Conditions" results in
                (s(x1109)→^*x1116⟹p(x1116)→^*0⟹p(0)→^*x282⟹cond3^#(false,x282,x279)≥c)
              • Applying Rule "Introduce fresh variable" results in
                (s(x1109)→^*x1116⟹p(x1116)→^*0⟹0→^*x1125⟹p(x1125)→^*x282⟹cond3^#(false,x282,x279)≥c)
              • Applying Rule "Induction" on p(x1116)→^*0 results in the following 2 new constraints.
                1. (0→^*0⟹s(x1109)→^*0⟹0→^*x1125⟹p(x1125)→^*x282⟹cond3^#(false,x282,x279)≥c)
                   • Applying Rule "Same Constructor" results in
                     (s(x1109)→^*0⟹0→^*x1125⟹p(x1125)→^*x282⟹cond3^#(false,x282,x279)≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (x1126→^*0⟹s(x1109)→^*s(x1126)⟹0→^*x1125⟹p(x1125)→^*x282⟹cond3^#(false,x282,x279)≥c)
                   • Applying Rule "Same Constructor" results in
                     (x1126→^*0⟹x1109→^*x1126⟹0→^*x1125⟹p(x1125)→^*x282⟹cond3^#(false,x282,x279)≥c)
                   • Applying Rule "Variable in Equation" allows to substitute x1126 by x1109 which results in
                     (x1109→^*0⟹0→^*x1125⟹p(x1125)→^*x282⟹cond3^#(false,x282,x279)≥c)
                   • Applying Rule "Variable in Equation" allows to substitute x1109 by 0 which results in
                     (0→^*x1125⟹p(x1125)→^*x282⟹cond3^#(false,x282,x279)≥c)
                   • Applying Rule "Variable in Equation" allows to substitute x1125 by 0 which results in
                     (p(0)→^*x282⟹cond3^#(false,x282,x279)≥c)
                   • Applying Rule "Delete Conditions" results in
                     cond3^#(false,x282,x279)≥c

                   • This constraint is kept as final constraint.

           2. (true→^*false⟹cond3^#(false,x282,x279)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1120,x1119)→^*false⟹s(x1109)→^*x1116⟹p(x1116)→^*s(x1120)⟹0→^*s(x1119)⟹p(s(x1120))→^*x282⟹∀x1121 . ∀x1122 . ∀x1123 . ∀x1124 . (gr(x1120,x1119)→^*false⟹s(x1121)→^*x1122⟹p(x1122)→^*x1120⟹0→^*x1119⟹p(x1120)→^*x1123⟹cond3^#(false,x1123,x1124)≥c)⟹cond3^#(false,x282,x279)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x1111,x1110)→^*true⟹0→^*s(x1110)⟹p(s(x1111))→^*x280⟹0→^*x1107⟹gr(x280,x1107)→^*false⟹p(x280)→^*x282⟹∀x1112 . ∀x1113 . ∀x1114 . ∀x1115 . (gr(x1111,x1110)→^*true⟹0→^*x1110⟹p(x1111)→^*x1112⟹0→^*x1113⟹gr(x1112,x1113)→^*false⟹p(x1112)→^*x1114⟹cond3^#(false,x1114,x1115)≥c)⟹cond3^#(false,x282,x279)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond3^#(gr(x416,0),x416,x417)→^*cond3^#(false,x418,x419)⟹cond1^#(and(gr(x418,0),gr(x419,0)),x418,x419)→^*cond1^#(true,x420,x421)⟹cond1^#(true,x420,x421)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x416,0)→^*false⟹x416→^*x418⟹x417→^*x419⟹cond1^#(and(gr(x418,0),gr(x419,0)),x418,x419)→^*cond1^#(true,x420,x421)⟹cond1^#(true,x420,x421)≥c)
    • Applying Rule "Same Constructor" results in
      (gr(x416,0)→^*false⟹x416→^*x418⟹x417→^*x419⟹and(gr(x418,0),gr(x419,0))→^*true⟹x418→^*x420⟹x419→^*x421⟹cond1^#(true,x420,x421)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x420 by x418 which results in
      (gr(x416,0)→^*false⟹x416→^*x418⟹x417→^*x419⟹and(gr(x418,0),gr(x419,0))→^*true⟹x419→^*x421⟹cond1^#(true,x418,x421)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x421 by x419 which results in
      (gr(x416,0)→^*false⟹x416→^*x418⟹x417→^*x419⟹and(gr(x418,0),gr(x419,0))→^*true⟹cond1^#(true,x418,x419)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x417 by x419 which results in
      (gr(x416,0)→^*false⟹x416→^*x418⟹and(gr(x418,0),gr(x419,0))→^*true⟹cond1^#(true,x418,x419)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1127⟹gr(x416,x1127)→^*false⟹x416→^*x418⟹and(gr(x418,0),gr(x419,0))→^*true⟹cond1^#(true,x418,x419)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1127⟹gr(x416,x1127)→^*false⟹x416→^*x418⟹gr(x419,0)→^*x1129⟹and(gr(x418,0),x1129)→^*true⟹cond1^#(true,x418,x419)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1127⟹gr(x416,x1127)→^*false⟹x416→^*x418⟹gr(x418,0)→^*x1128⟹gr(x419,0)→^*x1129⟹and(x1128,x1129)→^*true⟹cond1^#(true,x418,x419)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1127⟹gr(x416,x1127)→^*false⟹x416→^*x418⟹0→^*x1130⟹gr(x418,x1130)→^*x1128⟹gr(x419,0)→^*x1129⟹and(x1128,x1129)→^*true⟹cond1^#(true,x418,x419)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1127⟹gr(x416,x1127)→^*false⟹x416→^*x418⟹0→^*x1130⟹gr(x418,x1130)→^*x1128⟹0→^*x1131⟹gr(x419,x1131)→^*x1129⟹and(x1128,x1129)→^*true⟹cond1^#(true,x418,x419)≥c)
    • Applying Rule "Induction" on and(x1128,x1129)→^*true results in the following 3 new constraints.
      1. (true→^*true⟹0→^*x1127⟹gr(x416,x1127)→^*false⟹x416→^*x418⟹0→^*x1130⟹gr(x418,x1130)→^*true⟹0→^*x1131⟹gr(x419,x1131)→^*true⟹cond1^#(true,x418,x419)≥c)
         • Applying Rule "Same Constructor" results in
           (0→^*x1127⟹gr(x416,x1127)→^*false⟹x416→^*x418⟹0→^*x1130⟹gr(x418,x1130)→^*true⟹0→^*x1131⟹gr(x419,x1131)→^*true⟹cond1^#(true,x418,x419)≥c)
         • Applying Rule "Induction" on gr(x416,x1127)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹0→^*x1134⟹0→^*x418⟹0→^*x1130⟹gr(x418,x1130)→^*true⟹0→^*x1131⟹gr(x419,x1131)→^*true⟹cond1^#(true,x418,x419)≥c)
              • Applying Rule "Same Constructor" results in
                (0→^*x1134⟹0→^*x418⟹0→^*x1130⟹gr(x418,x1130)→^*true⟹0→^*x1131⟹gr(x419,x1131)→^*true⟹cond1^#(true,x418,x419)≥c)
              • Applying Rule "Delete Conditions" results in
                (0→^*x418⟹0→^*x1130⟹gr(x418,x1130)→^*true⟹0→^*x1131⟹gr(x419,x1131)→^*true⟹cond1^#(true,x418,x419)≥c)
              • Applying Rule "Induction" on gr(x418,x1130)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond1^#(true,0,x419)≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹0→^*s(x1143)⟹0→^*0⟹0→^*x1131⟹gr(x419,x1131)→^*true⟹cond1^#(true,s(x1143),x419)≥c)
                   • Applying Rule "Same Constructor" results in
                     (0→^*s(x1143)⟹0→^*0⟹0→^*x1131⟹gr(x419,x1131)→^*true⟹cond1^#(true,s(x1143),x419)≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x1145,x1144)→^*true⟹0→^*s(x1145)⟹0→^*s(x1144)⟹0→^*x1131⟹gr(x419,x1131)→^*true⟹∀x1146 . ∀x1147 . (gr(x1145,x1144)→^*true⟹0→^*x1145⟹0→^*x1144⟹0→^*x1146⟹gr(x1147,x1146)→^*true⟹cond1^#(true,x1145,x1147)≥c)⟹cond1^#(true,s(x1145),x419)≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond1^#(true,x418,x419)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1137,x1136)→^*false⟹0→^*s(x1136)⟹s(x1137)→^*x418⟹0→^*x1130⟹gr(x418,x1130)→^*true⟹0→^*x1131⟹gr(x419,x1131)→^*true⟹∀x1138 . ∀x1139 . ∀x1140 . ∀x1141 . (gr(x1137,x1136)→^*false⟹0→^*x1136⟹x1137→^*x1138⟹0→^*x1139⟹gr(x1138,x1139)→^*true⟹0→^*x1140⟹gr(x1141,x1140)→^*true⟹cond1^#(true,x1138,x1141)≥c)⟹cond1^#(true,x418,x419)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*true⟹cond1^#(true,x418,x419)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (false→^*true⟹cond1^#(true,x418,x419)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond3^#(gr(x422,0),p(x422),x423)→^*cond3^#(false,x424,x425)⟹cond1^#(and(gr(x424,0),gr(x425,0)),x424,x425)→^*cond1^#(true,x426,x427)⟹cond1^#(true,x426,x427)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x422,0)→^*false⟹p(x422)→^*x424⟹x423→^*x425⟹cond1^#(and(gr(x424,0),gr(x425,0)),x424,x425)→^*cond1^#(true,x426,x427)⟹cond1^#(true,x426,x427)≥c)
    • Applying Rule "Same Constructor" results in
      (gr(x422,0)→^*false⟹p(x422)→^*x424⟹x423→^*x425⟹and(gr(x424,0),gr(x425,0))→^*true⟹x424→^*x426⟹x425→^*x427⟹cond1^#(true,x426,x427)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x426 by x424 which results in
      (gr(x422,0)→^*false⟹p(x422)→^*x424⟹x423→^*x425⟹and(gr(x424,0),gr(x425,0))→^*true⟹x425→^*x427⟹cond1^#(true,x424,x427)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x427 by x425 which results in
      (gr(x422,0)→^*false⟹p(x422)→^*x424⟹x423→^*x425⟹and(gr(x424,0),gr(x425,0))→^*true⟹cond1^#(true,x424,x425)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x423 by x425 which results in
      (gr(x422,0)→^*false⟹p(x422)→^*x424⟹and(gr(x424,0),gr(x425,0))→^*true⟹cond1^#(true,x424,x425)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1148⟹gr(x422,x1148)→^*false⟹p(x422)→^*x424⟹and(gr(x424,0),gr(x425,0))→^*true⟹cond1^#(true,x424,x425)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1148⟹gr(x422,x1148)→^*false⟹p(x422)→^*x424⟹gr(x425,0)→^*x1150⟹and(gr(x424,0),x1150)→^*true⟹cond1^#(true,x424,x425)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1148⟹gr(x422,x1148)→^*false⟹p(x422)→^*x424⟹gr(x424,0)→^*x1149⟹gr(x425,0)→^*x1150⟹and(x1149,x1150)→^*true⟹cond1^#(true,x424,x425)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1148⟹gr(x422,x1148)→^*false⟹p(x422)→^*x424⟹0→^*x1151⟹gr(x424,x1151)→^*x1149⟹gr(x425,0)→^*x1150⟹and(x1149,x1150)→^*true⟹cond1^#(true,x424,x425)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1148⟹gr(x422,x1148)→^*false⟹p(x422)→^*x424⟹0→^*x1151⟹gr(x424,x1151)→^*x1149⟹0→^*x1152⟹gr(x425,x1152)→^*x1150⟹and(x1149,x1150)→^*true⟹cond1^#(true,x424,x425)≥c)
    • Applying Rule "Induction" on and(x1149,x1150)→^*true results in the following 3 new constraints.
      1. (true→^*true⟹0→^*x1148⟹gr(x422,x1148)→^*false⟹p(x422)→^*x424⟹0→^*x1151⟹gr(x424,x1151)→^*true⟹0→^*x1152⟹gr(x425,x1152)→^*true⟹cond1^#(true,x424,x425)≥c)
         • Applying Rule "Same Constructor" results in
           (0→^*x1148⟹gr(x422,x1148)→^*false⟹p(x422)→^*x424⟹0→^*x1151⟹gr(x424,x1151)→^*true⟹0→^*x1152⟹gr(x425,x1152)→^*true⟹cond1^#(true,x424,x425)≥c)
         • Applying Rule "Induction" on gr(x422,x1148)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹0→^*x1155⟹p(0)→^*x424⟹0→^*x1151⟹gr(x424,x1151)→^*true⟹0→^*x1152⟹gr(x425,x1152)→^*true⟹cond1^#(true,x424,x425)≥c)
              • Applying Rule "Same Constructor" results in
                (0→^*x1155⟹p(0)→^*x424⟹0→^*x1151⟹gr(x424,x1151)→^*true⟹0→^*x1152⟹gr(x425,x1152)→^*true⟹cond1^#(true,x424,x425)≥c)
              • Applying Rule "Delete Conditions" results in
                (p(0)→^*x424⟹0→^*x1151⟹gr(x424,x1151)→^*true⟹0→^*x1152⟹gr(x425,x1152)→^*true⟹cond1^#(true,x424,x425)≥c)
              • Applying Rule "Introduce fresh variable" results in
                (0→^*x1163⟹p(x1163)→^*x424⟹0→^*x1151⟹gr(x424,x1151)→^*true⟹0→^*x1152⟹gr(x425,x1152)→^*true⟹cond1^#(true,x424,x425)≥c)
              • Applying Rule "Induction" on gr(x424,x1151)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond1^#(true,0,x425)≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹0→^*x1163⟹p(x1163)→^*s(x1165)⟹0→^*0⟹0→^*x1152⟹gr(x425,x1152)→^*true⟹cond1^#(true,s(x1165),x425)≥c)
                   • Applying Rule "Same Constructor" results in
                     (0→^*x1163⟹p(x1163)→^*s(x1165)⟹0→^*0⟹0→^*x1152⟹gr(x425,x1152)→^*true⟹cond1^#(true,s(x1165),x425)≥c)
                   • Applying Rule "Same Constructor" results in
                     (0→^*x1163⟹p(x1163)→^*s(x1165)⟹0→^*x1152⟹gr(x425,x1152)→^*true⟹cond1^#(true,s(x1165),x425)≥c)
                   • Applying Rule "Induction" on p(x1163)→^*s(x1165) results in the following 2 new constraints.
                     1. (0→^*s(x1165)⟹cond1^#(true,s(x1165),x425)≥c)
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                     2. (x1171→^*s(x1165)⟹0→^*s(x1171)⟹0→^*x1152⟹gr(x425,x1152)→^*true⟹cond1^#(true,s(x1165),x425)≥c)
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x1167,x1166)→^*true⟹0→^*x1163⟹p(x1163)→^*s(x1167)⟹0→^*s(x1166)⟹0→^*x1152⟹gr(x425,x1152)→^*true⟹∀x1168 . ∀x1169 . ∀x1170 . (gr(x1167,x1166)→^*true⟹0→^*x1168⟹p(x1168)→^*x1167⟹0→^*x1166⟹0→^*x1169⟹gr(x1170,x1169)→^*true⟹cond1^#(true,x1167,x1170)≥c)⟹cond1^#(true,s(x1167),x425)≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond1^#(true,x424,x425)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1158,x1157)→^*false⟹0→^*s(x1157)⟹p(s(x1158))→^*x424⟹0→^*x1151⟹gr(x424,x1151)→^*true⟹0→^*x1152⟹gr(x425,x1152)→^*true⟹∀x1159 . ∀x1160 . ∀x1161 . ∀x1162 . (gr(x1158,x1157)→^*false⟹0→^*x1157⟹p(x1158)→^*x1159⟹0→^*x1160⟹gr(x1159,x1160)→^*true⟹0→^*x1161⟹gr(x1162,x1161)→^*true⟹cond1^#(true,x1159,x1162)≥c)⟹cond1^#(true,x424,x425)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*true⟹cond1^#(true,x424,x425)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (false→^*true⟹cond1^#(true,x424,x425)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond4^#(gr(x499,0),x498,x499)→^*cond4^#(false,x500,x501)⟹cond1^#(and(gr(x500,0),gr(x501,0)),x500,x501)→^*cond1^#(true,x502,x503)⟹cond1^#(true,x502,x503)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x499,0)→^*false⟹x498→^*x500⟹x499→^*x501⟹cond1^#(and(gr(x500,0),gr(x501,0)),x500,x501)→^*cond1^#(true,x502,x503)⟹cond1^#(true,x502,x503)≥c)
    • Applying Rule "Same Constructor" results in
      (gr(x499,0)→^*false⟹x498→^*x500⟹x499→^*x501⟹and(gr(x500,0),gr(x501,0))→^*true⟹x500→^*x502⟹x501→^*x503⟹cond1^#(true,x502,x503)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x502 by x500 which results in
      (gr(x499,0)→^*false⟹x498→^*x500⟹x499→^*x501⟹and(gr(x500,0),gr(x501,0))→^*true⟹x501→^*x503⟹cond1^#(true,x500,x503)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x503 by x501 which results in
      (gr(x499,0)→^*false⟹x498→^*x500⟹x499→^*x501⟹and(gr(x500,0),gr(x501,0))→^*true⟹cond1^#(true,x500,x501)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x498 by x500 which results in
      (gr(x499,0)→^*false⟹x499→^*x501⟹and(gr(x500,0),gr(x501,0))→^*true⟹cond1^#(true,x500,x501)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1172⟹gr(x499,x1172)→^*false⟹x499→^*x501⟹and(gr(x500,0),gr(x501,0))→^*true⟹cond1^#(true,x500,x501)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1172⟹gr(x499,x1172)→^*false⟹x499→^*x501⟹gr(x501,0)→^*x1174⟹and(gr(x500,0),x1174)→^*true⟹cond1^#(true,x500,x501)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1172⟹gr(x499,x1172)→^*false⟹x499→^*x501⟹gr(x500,0)→^*x1173⟹gr(x501,0)→^*x1174⟹and(x1173,x1174)→^*true⟹cond1^#(true,x500,x501)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1172⟹gr(x499,x1172)→^*false⟹x499→^*x501⟹0→^*x1175⟹gr(x500,x1175)→^*x1173⟹gr(x501,0)→^*x1174⟹and(x1173,x1174)→^*true⟹cond1^#(true,x500,x501)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1172⟹gr(x499,x1172)→^*false⟹x499→^*x501⟹0→^*x1175⟹gr(x500,x1175)→^*x1173⟹0→^*x1176⟹gr(x501,x1176)→^*x1174⟹and(x1173,x1174)→^*true⟹cond1^#(true,x500,x501)≥c)
    • Applying Rule "Induction" on and(x1173,x1174)→^*true results in the following 3 new constraints.
      1. (true→^*true⟹0→^*x1172⟹gr(x499,x1172)→^*false⟹x499→^*x501⟹0→^*x1175⟹gr(x500,x1175)→^*true⟹0→^*x1176⟹gr(x501,x1176)→^*true⟹cond1^#(true,x500,x501)≥c)
         • Applying Rule "Same Constructor" results in
           (0→^*x1172⟹gr(x499,x1172)→^*false⟹x499→^*x501⟹0→^*x1175⟹gr(x500,x1175)→^*true⟹0→^*x1176⟹gr(x501,x1176)→^*true⟹cond1^#(true,x500,x501)≥c)
         • Applying Rule "Induction" on gr(x499,x1172)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹0→^*x1179⟹0→^*x501⟹0→^*x1175⟹gr(x500,x1175)→^*true⟹0→^*x1176⟹gr(x501,x1176)→^*true⟹cond1^#(true,x500,x501)≥c)
              • Applying Rule "Same Constructor" results in
                (0→^*x1179⟹0→^*x501⟹0→^*x1175⟹gr(x500,x1175)→^*true⟹0→^*x1176⟹gr(x501,x1176)→^*true⟹cond1^#(true,x500,x501)≥c)
              • Applying Rule "Delete Conditions" results in
                (0→^*x501⟹0→^*x1175⟹gr(x500,x1175)→^*true⟹0→^*x1176⟹gr(x501,x1176)→^*true⟹cond1^#(true,x500,x501)≥c)
              • Applying Rule "Induction" on gr(x500,x1175)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond1^#(true,0,x501)≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹0→^*x501⟹0→^*0⟹0→^*x1176⟹gr(x501,x1176)→^*true⟹cond1^#(true,s(x1188),x501)≥c)
                   • Applying Rule "Same Constructor" results in
                     (0→^*x501⟹0→^*0⟹0→^*x1176⟹gr(x501,x1176)→^*true⟹cond1^#(true,s(x1188),x501)≥c)
                   • Applying Rule "Same Constructor" results in
                     (0→^*x501⟹0→^*x1176⟹gr(x501,x1176)→^*true⟹cond1^#(true,s(x1188),x501)≥c)
                   • Applying Rule "Induction" on gr(x501,x1176)→^*true results in the following 3 new constraints.
                     1. (false→^*true⟹cond1^#(true,s(x1188),0)≥c)
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                     2. (true→^*true⟹0→^*s(x1194)⟹0→^*0⟹cond1^#(true,s(x1188),s(x1194))≥c)
                        • Applying Rule "Same Constructor" results in
                          (0→^*s(x1194)⟹0→^*0⟹cond1^#(true,s(x1188),s(x1194))≥c)
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                     3. (gr(x1196,x1195)→^*true⟹0→^*s(x1196)⟹0→^*s(x1195)⟹∀x1197 . (gr(x1196,x1195)→^*true⟹0→^*x1196⟹0→^*x1195⟹cond1^#(true,s(x1197),x1196)≥c)⟹cond1^#(true,s(x1188),s(x1196))≥c)
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x1190,x1189)→^*true⟹0→^*x501⟹0→^*s(x1189)⟹0→^*x1176⟹gr(x501,x1176)→^*true⟹∀x1191 . ∀x1192 . (gr(x1190,x1189)→^*true⟹0→^*x1191⟹0→^*x1189⟹0→^*x1192⟹gr(x1191,x1192)→^*true⟹cond1^#(true,x1190,x1191)≥c)⟹cond1^#(true,s(x1190),x501)≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond1^#(true,x500,x501)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1182,x1181)→^*false⟹0→^*s(x1181)⟹s(x1182)→^*x501⟹0→^*x1175⟹gr(x500,x1175)→^*true⟹0→^*x1176⟹gr(x501,x1176)→^*true⟹∀x1183 . ∀x1184 . ∀x1185 . ∀x1186 . (gr(x1182,x1181)→^*false⟹0→^*x1181⟹x1182→^*x1183⟹0→^*x1184⟹gr(x1185,x1184)→^*true⟹0→^*x1186⟹gr(x1183,x1186)→^*true⟹cond1^#(true,x1185,x1183)≥c)⟹cond1^#(true,x500,x501)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*true⟹cond1^#(true,x500,x501)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (false→^*true⟹cond1^#(true,x500,x501)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond4^#(gr(x505,0),x504,p(x505))→^*cond4^#(false,x506,x507)⟹cond1^#(and(gr(x506,0),gr(x507,0)),x506,x507)→^*cond1^#(true,x508,x509)⟹cond1^#(true,x508,x509)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x505,0)→^*false⟹x504→^*x506⟹p(x505)→^*x507⟹cond1^#(and(gr(x506,0),gr(x507,0)),x506,x507)→^*cond1^#(true,x508,x509)⟹cond1^#(true,x508,x509)≥c)
    • Applying Rule "Same Constructor" results in
      (gr(x505,0)→^*false⟹x504→^*x506⟹p(x505)→^*x507⟹and(gr(x506,0),gr(x507,0))→^*true⟹x506→^*x508⟹x507→^*x509⟹cond1^#(true,x508,x509)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x508 by x506 which results in
      (gr(x505,0)→^*false⟹x504→^*x506⟹p(x505)→^*x507⟹and(gr(x506,0),gr(x507,0))→^*true⟹x507→^*x509⟹cond1^#(true,x506,x509)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x509 by x507 which results in
      (gr(x505,0)→^*false⟹x504→^*x506⟹p(x505)→^*x507⟹and(gr(x506,0),gr(x507,0))→^*true⟹cond1^#(true,x506,x507)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x504 by x506 which results in
      (gr(x505,0)→^*false⟹p(x505)→^*x507⟹and(gr(x506,0),gr(x507,0))→^*true⟹cond1^#(true,x506,x507)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1198⟹gr(x505,x1198)→^*false⟹p(x505)→^*x507⟹and(gr(x506,0),gr(x507,0))→^*true⟹cond1^#(true,x506,x507)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1198⟹gr(x505,x1198)→^*false⟹p(x505)→^*x507⟹gr(x507,0)→^*x1200⟹and(gr(x506,0),x1200)→^*true⟹cond1^#(true,x506,x507)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1198⟹gr(x505,x1198)→^*false⟹p(x505)→^*x507⟹gr(x506,0)→^*x1199⟹gr(x507,0)→^*x1200⟹and(x1199,x1200)→^*true⟹cond1^#(true,x506,x507)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1198⟹gr(x505,x1198)→^*false⟹p(x505)→^*x507⟹0→^*x1201⟹gr(x506,x1201)→^*x1199⟹gr(x507,0)→^*x1200⟹and(x1199,x1200)→^*true⟹cond1^#(true,x506,x507)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1198⟹gr(x505,x1198)→^*false⟹p(x505)→^*x507⟹0→^*x1201⟹gr(x506,x1201)→^*x1199⟹0→^*x1202⟹gr(x507,x1202)→^*x1200⟹and(x1199,x1200)→^*true⟹cond1^#(true,x506,x507)≥c)
    • Applying Rule "Induction" on and(x1199,x1200)→^*true results in the following 3 new constraints.
      1. (true→^*true⟹0→^*x1198⟹gr(x505,x1198)→^*false⟹p(x505)→^*x507⟹0→^*x1201⟹gr(x506,x1201)→^*true⟹0→^*x1202⟹gr(x507,x1202)→^*true⟹cond1^#(true,x506,x507)≥c)
         • Applying Rule "Same Constructor" results in
           (0→^*x1198⟹gr(x505,x1198)→^*false⟹p(x505)→^*x507⟹0→^*x1201⟹gr(x506,x1201)→^*true⟹0→^*x1202⟹gr(x507,x1202)→^*true⟹cond1^#(true,x506,x507)≥c)
         • Applying Rule "Induction" on gr(x505,x1198)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹0→^*x1205⟹p(0)→^*x507⟹0→^*x1201⟹gr(x506,x1201)→^*true⟹0→^*x1202⟹gr(x507,x1202)→^*true⟹cond1^#(true,x506,x507)≥c)
              • Applying Rule "Same Constructor" results in
                (0→^*x1205⟹p(0)→^*x507⟹0→^*x1201⟹gr(x506,x1201)→^*true⟹0→^*x1202⟹gr(x507,x1202)→^*true⟹cond1^#(true,x506,x507)≥c)
              • Applying Rule "Delete Conditions" results in
                (p(0)→^*x507⟹0→^*x1201⟹gr(x506,x1201)→^*true⟹0→^*x1202⟹gr(x507,x1202)→^*true⟹cond1^#(true,x506,x507)≥c)
              • Applying Rule "Introduce fresh variable" results in
                (0→^*x1213⟹p(x1213)→^*x507⟹0→^*x1201⟹gr(x506,x1201)→^*true⟹0→^*x1202⟹gr(x507,x1202)→^*true⟹cond1^#(true,x506,x507)≥c)
              • Applying Rule "Induction" on gr(x506,x1201)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond1^#(true,0,x507)≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹0→^*x1213⟹p(x1213)→^*x507⟹0→^*0⟹0→^*x1202⟹gr(x507,x1202)→^*true⟹cond1^#(true,s(x1215),x507)≥c)
                   • Applying Rule "Same Constructor" results in
                     (0→^*x1213⟹p(x1213)→^*x507⟹0→^*0⟹0→^*x1202⟹gr(x507,x1202)→^*true⟹cond1^#(true,s(x1215),x507)≥c)
                   • Applying Rule "Same Constructor" results in
                     (0→^*x1213⟹p(x1213)→^*x507⟹0→^*x1202⟹gr(x507,x1202)→^*true⟹cond1^#(true,s(x1215),x507)≥c)
                   • Applying Rule "Induction" on gr(x507,x1202)→^*true results in the following 3 new constraints.
                     1. (false→^*true⟹cond1^#(true,s(x1215),0)≥c)
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                     2. (true→^*true⟹0→^*x1213⟹p(x1213)→^*s(x1222)⟹0→^*0⟹cond1^#(true,s(x1215),s(x1222))≥c)
                        • Applying Rule "Same Constructor" results in
                          (0→^*x1213⟹p(x1213)→^*s(x1222)⟹0→^*0⟹cond1^#(true,s(x1215),s(x1222))≥c)
                        • Applying Rule "Same Constructor" results in
                          (0→^*x1213⟹p(x1213)→^*s(x1222)⟹cond1^#(true,s(x1215),s(x1222))≥c)
                        • Applying Rule "Induction" on p(x1213)→^*s(x1222) results in the following 2 new constraints.
                          1. (0→^*s(x1222)⟹cond1^#(true,s(x1215),s(x1222))≥c)
                             • Applying Rule "Different Constructors" allows to drop this constraint.
                          2. (x1227→^*s(x1222)⟹0→^*s(x1227)⟹cond1^#(true,s(x1215),s(x1222))≥c)
                             • Applying Rule "Different Constructors" allows to drop this constraint.
                     3. (gr(x1224,x1223)→^*true⟹0→^*x1213⟹p(x1213)→^*s(x1224)⟹0→^*s(x1223)⟹∀x1225 . ∀x1226 . (gr(x1224,x1223)→^*true⟹0→^*x1225⟹p(x1225)→^*x1224⟹0→^*x1223⟹cond1^#(true,s(x1226),x1224)≥c)⟹cond1^#(true,s(x1215),s(x1224))≥c)
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x1217,x1216)→^*true⟹0→^*x1213⟹p(x1213)→^*x507⟹0→^*s(x1216)⟹0→^*x1202⟹gr(x507,x1202)→^*true⟹∀x1218 . ∀x1219 . ∀x1220 . (gr(x1217,x1216)→^*true⟹0→^*x1218⟹p(x1218)→^*x1219⟹0→^*x1216⟹0→^*x1220⟹gr(x1219,x1220)→^*true⟹cond1^#(true,x1217,x1219)≥c)⟹cond1^#(true,s(x1217),x507)≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond1^#(true,x506,x507)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1208,x1207)→^*false⟹0→^*s(x1207)⟹p(s(x1208))→^*x507⟹0→^*x1201⟹gr(x506,x1201)→^*true⟹0→^*x1202⟹gr(x507,x1202)→^*true⟹∀x1209 . ∀x1210 . ∀x1211 . ∀x1212 . (gr(x1208,x1207)→^*false⟹0→^*x1207⟹p(x1208)→^*x1209⟹0→^*x1210⟹gr(x1211,x1210)→^*true⟹0→^*x1212⟹gr(x1209,x1212)→^*true⟹cond1^#(true,x1211,x1209)≥c)⟹cond1^#(true,x506,x507)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*true⟹cond1^#(true,x506,x507)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (false→^*true⟹cond1^#(true,x506,x507)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond1^#(and(gr(x568,0),gr(x569,0)),x568,x569)→^*cond1^#(true,x570,x571)⟹cond2^#(gr(x570,x571),x570,x571)→^*cond2^#(false,x572,x573)⟹cond2^#(false,x572,x573)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (and(gr(x568,0),gr(x569,0))→^*true⟹x568→^*x570⟹x569→^*x571⟹cond2^#(gr(x570,x571),x570,x571)→^*cond2^#(false,x572,x573)⟹cond2^#(false,x572,x573)≥c)
    • Applying Rule "Same Constructor" results in
      (and(gr(x568,0),gr(x569,0))→^*true⟹x568→^*x570⟹x569→^*x571⟹gr(x570,x571)→^*false⟹x570→^*x572⟹x571→^*x573⟹cond2^#(false,x572,x573)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x572 by x570 which results in
      (and(gr(x568,0),gr(x569,0))→^*true⟹x568→^*x570⟹x569→^*x571⟹gr(x570,x571)→^*false⟹x571→^*x573⟹cond2^#(false,x570,x573)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x573 by x571 which results in
      (and(gr(x568,0),gr(x569,0))→^*true⟹x568→^*x570⟹x569→^*x571⟹gr(x570,x571)→^*false⟹cond2^#(false,x570,x571)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (x568→^*x570⟹x569→^*x571⟹gr(x570,x571)→^*false⟹gr(x569,0)→^*x1229⟹and(gr(x568,0),x1229)→^*true⟹cond2^#(false,x570,x571)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (gr(x568,0)→^*x1228⟹gr(x569,0)→^*x1229⟹and(x1228,x1229)→^*true⟹x568→^*x570⟹x569→^*x571⟹gr(x570,x571)→^*false⟹cond2^#(false,x570,x571)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1230⟹gr(x568,x1230)→^*x1228⟹gr(x569,0)→^*x1229⟹and(x1228,x1229)→^*true⟹x568→^*x570⟹x569→^*x571⟹gr(x570,x571)→^*false⟹cond2^#(false,x570,x571)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1230⟹gr(x568,x1230)→^*x1228⟹0→^*x1231⟹gr(x569,x1231)→^*x1229⟹and(x1228,x1229)→^*true⟹x568→^*x570⟹x569→^*x571⟹gr(x570,x571)→^*false⟹cond2^#(false,x570,x571)≥c)
    • Applying Rule "Induction" on and(x1228,x1229)→^*true results in the following 3 new constraints.
      1. (true→^*true⟹0→^*x1230⟹gr(x568,x1230)→^*true⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹x568→^*x570⟹x569→^*x571⟹gr(x570,x571)→^*false⟹cond2^#(false,x570,x571)≥c)
         • Applying Rule "Same Constructor" results in
           (0→^*x1230⟹gr(x568,x1230)→^*true⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹x568→^*x570⟹x569→^*x571⟹gr(x570,x571)→^*false⟹cond2^#(false,x570,x571)≥c)
         • Applying Rule "Induction" on gr(x570,x571)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹0→^*x1230⟹gr(x568,x1230)→^*true⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹x568→^*0⟹x569→^*x1234⟹cond2^#(false,0,x1234)≥c)
              • Applying Rule "Same Constructor" results in
                (0→^*x1230⟹gr(x568,x1230)→^*true⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹x568→^*0⟹x569→^*x1234⟹cond2^#(false,0,x1234)≥c)
              • Applying Rule "Variable in Equation" allows to substitute x1234 by x569 which results in
                (0→^*x1230⟹gr(x568,x1230)→^*true⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹x568→^*0⟹cond2^#(false,0,x569)≥c)
              • Applying Rule "Variable in Equation" allows to substitute x568 by 0 which results in
                (0→^*x1230⟹gr(0,x1230)→^*true⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹cond2^#(false,0,x569)≥c)
              • Applying Rule "Introduce fresh variable" results in
                (0→^*x1230⟹0→^*x1242⟹gr(x1242,x1230)→^*true⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹cond2^#(false,0,x569)≥c)
              • Applying Rule "Induction" on gr(x1242,x1230)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond2^#(false,0,x569)≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹0→^*0⟹0→^*s(x1244)⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹cond2^#(false,0,x569)≥c)
                   • Applying Rule "Same Constructor" results in
                     (0→^*0⟹0→^*s(x1244)⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹cond2^#(false,0,x569)≥c)
                   • Applying Rule "Same Constructor" results in
                     (0→^*s(x1244)⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹cond2^#(false,0,x569)≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x1246,x1245)→^*true⟹0→^*s(x1245)⟹0→^*s(x1246)⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹∀x1247 . ∀x1248 . (gr(x1246,x1245)→^*true⟹0→^*x1245⟹0→^*x1246⟹0→^*x1247⟹gr(x1248,x1247)→^*true⟹cond2^#(false,0,x1248)≥c)⟹cond2^#(false,0,x569)≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond2^#(false,s(x1235),0)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1237,x1236)→^*false⟹0→^*x1230⟹gr(x568,x1230)→^*true⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹x568→^*s(x1237)⟹x569→^*s(x1236)⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥c)⟹cond2^#(false,s(x1237),s(x1236))≥c)
              • Applying Rule "Variable in Equation" allows to substitute x568 by s(x1237) which results in
                (gr(x1237,x1236)→^*false⟹0→^*x1230⟹gr(s(x1237),x1230)→^*true⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹x569→^*s(x1236)⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥c)⟹cond2^#(false,s(x1237),s(x1236))≥c)
              • Applying Rule "Variable in Equation" allows to substitute x569 by s(x1236) which results in
                (gr(x1237,x1236)→^*false⟹0→^*x1230⟹gr(s(x1237),x1230)→^*true⟹0→^*x1231⟹gr(s(x1236),x1231)→^*true⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥c)⟹cond2^#(false,s(x1237),s(x1236))≥c)
              • Applying Rule "Introduce fresh variable" results in
                (gr(x1237,x1236)→^*false⟹0→^*x1230⟹s(x1237)→^*x1249⟹gr(x1249,x1230)→^*true⟹0→^*x1231⟹gr(s(x1236),x1231)→^*true⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥c)⟹cond2^#(false,s(x1237),s(x1236))≥c)
              • Applying Rule "Introduce fresh variable" results in
                (gr(x1237,x1236)→^*false⟹0→^*x1230⟹s(x1237)→^*x1249⟹gr(x1249,x1230)→^*true⟹0→^*x1231⟹s(x1236)→^*x1250⟹gr(x1250,x1231)→^*true⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥c)⟹cond2^#(false,s(x1237),s(x1236))≥c)
              • Applying Rule "Induction" on gr(x1249,x1230)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond2^#(false,s(x1237),s(x1236))≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹gr(x1237,x1236)→^*false⟹0→^*0⟹s(x1237)→^*s(x1252)⟹0→^*x1231⟹s(x1236)→^*x1250⟹gr(x1250,x1231)→^*true⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥c)⟹cond2^#(false,s(x1237),s(x1236))≥c)
                   • Applying Rule "Same Constructor" results in
                     (gr(x1237,x1236)→^*false⟹0→^*0⟹s(x1237)→^*s(x1252)⟹0→^*x1231⟹s(x1236)→^*x1250⟹gr(x1250,x1231)→^*true⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥c)⟹cond2^#(false,s(x1237),s(x1236))≥c)
                   • Applying Rule "Same Constructor" results in
                     (gr(x1237,x1236)→^*false⟹s(x1237)→^*s(x1252)⟹0→^*x1231⟹s(x1236)→^*x1250⟹gr(x1250,x1231)→^*true⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥c)⟹cond2^#(false,s(x1237),s(x1236))≥c)
                   • Applying Rule "Same Constructor" results in
                     (gr(x1237,x1236)→^*false⟹x1237→^*x1252⟹0→^*x1231⟹s(x1236)→^*x1250⟹gr(x1250,x1231)→^*true⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥c)⟹cond2^#(false,s(x1237),s(x1236))≥c)
                   • Applying Rule "Delete Conditions" results in
                     (gr(x1237,x1236)→^*false⟹0→^*x1231⟹s(x1236)→^*x1250⟹gr(x1250,x1231)→^*true⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥c)⟹cond2^#(false,s(x1237),s(x1236))≥c)
                   • Applying Rule "Induction" on gr(x1250,x1231)→^*true results in the following 3 new constraints.
                     1. (false→^*true⟹cond2^#(false,s(x1237),s(x1236))≥c)
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                     2. (true→^*true⟹gr(x1237,x1236)→^*false⟹0→^*0⟹s(x1236)→^*s(x1264)⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥c)⟹cond2^#(false,s(x1237),s(x1236))≥c)
                        • Applying Rule "Same Constructor" results in
                          (gr(x1237,x1236)→^*false⟹0→^*0⟹s(x1236)→^*s(x1264)⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥c)⟹cond2^#(false,s(x1237),s(x1236))≥c)
                        • Applying Rule "Same Constructor" results in
                          (gr(x1237,x1236)→^*false⟹s(x1236)→^*s(x1264)⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥c)⟹cond2^#(false,s(x1237),s(x1236))≥c)
                        • Applying Rule "Same Constructor" results in
                          (gr(x1237,x1236)→^*false⟹x1236→^*x1264⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥c)⟹cond2^#(false,s(x1237),s(x1236))≥c)
                        • Applying Rule "Delete Conditions" results in
                          (gr(x1237,x1236)→^*false⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥c)⟹cond2^#(false,s(x1237),s(x1236))≥c)
                        • Applying Rule "Delete Conditions" results in
                          (gr(x1237,x1236)→^*false⟹cond2^#(false,s(x1237),s(x1236))≥c)
                        • Applying Rule "Induction" on gr(x1237,x1236)→^*false results in the following 3 new constraints.
                          1. (false→^*false⟹cond2^#(false,s(0),s(x1273))≥c)
                             • Applying Rule "Same Constructor" results in
                               cond2^#(false,s(0),s(x1273))≥c

                             • This constraint is kept as final constraint.

                          2. (true→^*false⟹cond2^#(false,s(s(x1274)),s(0))≥c)
                             • Applying Rule "Different Constructors" allows to drop this constraint.
                          3. (gr(x1276,x1275)→^*false⟹(gr(x1276,x1275)→^*false⟹cond2^#(false,s(x1276),s(x1275))≥c)⟹cond2^#(false,s(s(x1276)),s(s(x1275)))≥c)
                             • Applying Rule "Simplify Conditions" results in
                               (cond2^#(false,s(x1276),s(x1275))≥c⟹cond2^#(false,s(s(x1276)),s(s(x1275)))≥c)

                             • This constraint is kept as final constraint.

                     3. (gr(x1266,x1265)→^*true⟹gr(x1237,x1236)→^*false⟹0→^*s(x1265)⟹s(x1236)→^*s(x1266)⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥c)⟹∀x1267 . ∀x1268 . ∀x1269 . ∀x1270 . ∀x1271 . ∀x1272 . (gr(x1266,x1265)→^*true⟹gr(x1267,x1268)→^*false⟹0→^*x1265⟹s(x1268)→^*x1266⟹∀x1269 . ∀x1270 . ∀x1271 . ∀x1272 . (gr(x1267,x1268)→^*false⟹0→^*x1269⟹gr(x1270,x1269)→^*true⟹0→^*x1271⟹gr(x1272,x1271)→^*true⟹x1270→^*x1267⟹x1272→^*x1268⟹cond2^#(false,x1267,x1268)≥c)⟹cond2^#(false,s(x1267),s(x1268))≥c)⟹cond2^#(false,s(x1237),s(x1236))≥c)
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x1254,x1253)→^*true⟹gr(x1237,x1236)→^*false⟹0→^*s(x1253)⟹s(x1237)→^*s(x1254)⟹0→^*x1231⟹s(x1236)→^*x1250⟹gr(x1250,x1231)→^*true⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥c)⟹∀x1255 . ∀x1256 . ∀x1257 . ∀x1258 . ∀x1259 . ∀x1260 . ∀x1261 . ∀x1262 . (gr(x1254,x1253)→^*true⟹gr(x1255,x1256)→^*false⟹0→^*x1253⟹s(x1255)→^*x1254⟹0→^*x1257⟹s(x1256)→^*x1258⟹gr(x1258,x1257)→^*true⟹∀x1259 . ∀x1260 . ∀x1261 . ∀x1262 . (gr(x1255,x1256)→^*false⟹0→^*x1259⟹gr(x1260,x1259)→^*true⟹0→^*x1261⟹gr(x1262,x1261)→^*true⟹x1260→^*x1255⟹x1262→^*x1256⟹cond2^#(false,x1255,x1256)≥c)⟹cond2^#(false,s(x1255),s(x1256))≥c)⟹cond2^#(false,s(x1237),s(x1236))≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*true⟹cond2^#(false,x570,x571)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (false→^*true⟹cond2^#(false,x570,x571)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond1^#(and(gr(x580,0),gr(x581,0)),x580,x581)→^*cond1^#(true,x582,x583)⟹cond2^#(gr(x582,x583),x582,x583)→^*cond2^#(false,x584,x585)⟹cond2^#(false,x584,x585)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (and(gr(x580,0),gr(x581,0))→^*true⟹x580→^*x582⟹x581→^*x583⟹cond2^#(gr(x582,x583),x582,x583)→^*cond2^#(false,x584,x585)⟹cond2^#(false,x584,x585)≥c)
    • Applying Rule "Same Constructor" results in
      (and(gr(x580,0),gr(x581,0))→^*true⟹x580→^*x582⟹x581→^*x583⟹gr(x582,x583)→^*false⟹x582→^*x584⟹x583→^*x585⟹cond2^#(false,x584,x585)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x584 by x582 which results in
      (and(gr(x580,0),gr(x581,0))→^*true⟹x580→^*x582⟹x581→^*x583⟹gr(x582,x583)→^*false⟹x583→^*x585⟹cond2^#(false,x582,x585)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x585 by x583 which results in
      (and(gr(x580,0),gr(x581,0))→^*true⟹x580→^*x582⟹x581→^*x583⟹gr(x582,x583)→^*false⟹cond2^#(false,x582,x583)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (x580→^*x582⟹x581→^*x583⟹gr(x582,x583)→^*false⟹gr(x581,0)→^*x1278⟹and(gr(x580,0),x1278)→^*true⟹cond2^#(false,x582,x583)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (gr(x580,0)→^*x1277⟹gr(x581,0)→^*x1278⟹and(x1277,x1278)→^*true⟹x580→^*x582⟹x581→^*x583⟹gr(x582,x583)→^*false⟹cond2^#(false,x582,x583)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1279⟹gr(x580,x1279)→^*x1277⟹gr(x581,0)→^*x1278⟹and(x1277,x1278)→^*true⟹x580→^*x582⟹x581→^*x583⟹gr(x582,x583)→^*false⟹cond2^#(false,x582,x583)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1279⟹gr(x580,x1279)→^*x1277⟹0→^*x1280⟹gr(x581,x1280)→^*x1278⟹and(x1277,x1278)→^*true⟹x580→^*x582⟹x581→^*x583⟹gr(x582,x583)→^*false⟹cond2^#(false,x582,x583)≥c)
    • Applying Rule "Induction" on and(x1277,x1278)→^*true results in the following 3 new constraints.
      1. (true→^*true⟹0→^*x1279⟹gr(x580,x1279)→^*true⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹x580→^*x582⟹x581→^*x583⟹gr(x582,x583)→^*false⟹cond2^#(false,x582,x583)≥c)
         • Applying Rule "Same Constructor" results in
           (0→^*x1279⟹gr(x580,x1279)→^*true⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹x580→^*x582⟹x581→^*x583⟹gr(x582,x583)→^*false⟹cond2^#(false,x582,x583)≥c)
         • Applying Rule "Induction" on gr(x582,x583)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹0→^*x1279⟹gr(x580,x1279)→^*true⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹x580→^*0⟹x581→^*x1283⟹cond2^#(false,0,x1283)≥c)
              • Applying Rule "Same Constructor" results in
                (0→^*x1279⟹gr(x580,x1279)→^*true⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹x580→^*0⟹x581→^*x1283⟹cond2^#(false,0,x1283)≥c)
              • Applying Rule "Variable in Equation" allows to substitute x1283 by x581 which results in
                (0→^*x1279⟹gr(x580,x1279)→^*true⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹x580→^*0⟹cond2^#(false,0,x581)≥c)
              • Applying Rule "Variable in Equation" allows to substitute x580 by 0 which results in
                (0→^*x1279⟹gr(0,x1279)→^*true⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹cond2^#(false,0,x581)≥c)
              • Applying Rule "Introduce fresh variable" results in
                (0→^*x1279⟹0→^*x1291⟹gr(x1291,x1279)→^*true⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹cond2^#(false,0,x581)≥c)
              • Applying Rule "Induction" on gr(x1291,x1279)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond2^#(false,0,x581)≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹0→^*0⟹0→^*s(x1293)⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹cond2^#(false,0,x581)≥c)
                   • Applying Rule "Same Constructor" results in
                     (0→^*0⟹0→^*s(x1293)⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹cond2^#(false,0,x581)≥c)
                   • Applying Rule "Same Constructor" results in
                     (0→^*s(x1293)⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹cond2^#(false,0,x581)≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x1295,x1294)→^*true⟹0→^*s(x1294)⟹0→^*s(x1295)⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹∀x1296 . ∀x1297 . (gr(x1295,x1294)→^*true⟹0→^*x1294⟹0→^*x1295⟹0→^*x1296⟹gr(x1297,x1296)→^*true⟹cond2^#(false,0,x1297)≥c)⟹cond2^#(false,0,x581)≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond2^#(false,s(x1284),0)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1286,x1285)→^*false⟹0→^*x1279⟹gr(x580,x1279)→^*true⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹x580→^*s(x1286)⟹x581→^*s(x1285)⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥c)⟹cond2^#(false,s(x1286),s(x1285))≥c)
              • Applying Rule "Variable in Equation" allows to substitute x580 by s(x1286) which results in
                (gr(x1286,x1285)→^*false⟹0→^*x1279⟹gr(s(x1286),x1279)→^*true⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹x581→^*s(x1285)⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥c)⟹cond2^#(false,s(x1286),s(x1285))≥c)
              • Applying Rule "Variable in Equation" allows to substitute x581 by s(x1285) which results in
                (gr(x1286,x1285)→^*false⟹0→^*x1279⟹gr(s(x1286),x1279)→^*true⟹0→^*x1280⟹gr(s(x1285),x1280)→^*true⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥c)⟹cond2^#(false,s(x1286),s(x1285))≥c)
              • Applying Rule "Introduce fresh variable" results in
                (gr(x1286,x1285)→^*false⟹0→^*x1279⟹s(x1286)→^*x1298⟹gr(x1298,x1279)→^*true⟹0→^*x1280⟹gr(s(x1285),x1280)→^*true⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥c)⟹cond2^#(false,s(x1286),s(x1285))≥c)
              • Applying Rule "Introduce fresh variable" results in
                (gr(x1286,x1285)→^*false⟹0→^*x1279⟹s(x1286)→^*x1298⟹gr(x1298,x1279)→^*true⟹0→^*x1280⟹s(x1285)→^*x1299⟹gr(x1299,x1280)→^*true⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥c)⟹cond2^#(false,s(x1286),s(x1285))≥c)
              • Applying Rule "Induction" on gr(x1298,x1279)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond2^#(false,s(x1286),s(x1285))≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹gr(x1286,x1285)→^*false⟹0→^*0⟹s(x1286)→^*s(x1301)⟹0→^*x1280⟹s(x1285)→^*x1299⟹gr(x1299,x1280)→^*true⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥c)⟹cond2^#(false,s(x1286),s(x1285))≥c)
                   • Applying Rule "Same Constructor" results in
                     (gr(x1286,x1285)→^*false⟹0→^*0⟹s(x1286)→^*s(x1301)⟹0→^*x1280⟹s(x1285)→^*x1299⟹gr(x1299,x1280)→^*true⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥c)⟹cond2^#(false,s(x1286),s(x1285))≥c)
                   • Applying Rule "Same Constructor" results in
                     (gr(x1286,x1285)→^*false⟹s(x1286)→^*s(x1301)⟹0→^*x1280⟹s(x1285)→^*x1299⟹gr(x1299,x1280)→^*true⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥c)⟹cond2^#(false,s(x1286),s(x1285))≥c)
                   • Applying Rule "Same Constructor" results in
                     (gr(x1286,x1285)→^*false⟹x1286→^*x1301⟹0→^*x1280⟹s(x1285)→^*x1299⟹gr(x1299,x1280)→^*true⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥c)⟹cond2^#(false,s(x1286),s(x1285))≥c)
                   • Applying Rule "Delete Conditions" results in
                     (gr(x1286,x1285)→^*false⟹0→^*x1280⟹s(x1285)→^*x1299⟹gr(x1299,x1280)→^*true⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥c)⟹cond2^#(false,s(x1286),s(x1285))≥c)
                   • Applying Rule "Induction" on gr(x1299,x1280)→^*true results in the following 3 new constraints.
                     1. (false→^*true⟹cond2^#(false,s(x1286),s(x1285))≥c)
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                     2. (true→^*true⟹gr(x1286,x1285)→^*false⟹0→^*0⟹s(x1285)→^*s(x1313)⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥c)⟹cond2^#(false,s(x1286),s(x1285))≥c)
                        • Applying Rule "Same Constructor" results in
                          (gr(x1286,x1285)→^*false⟹0→^*0⟹s(x1285)→^*s(x1313)⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥c)⟹cond2^#(false,s(x1286),s(x1285))≥c)
                        • Applying Rule "Same Constructor" results in
                          (gr(x1286,x1285)→^*false⟹s(x1285)→^*s(x1313)⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥c)⟹cond2^#(false,s(x1286),s(x1285))≥c)
                        • Applying Rule "Same Constructor" results in
                          (gr(x1286,x1285)→^*false⟹x1285→^*x1313⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥c)⟹cond2^#(false,s(x1286),s(x1285))≥c)
                        • Applying Rule "Delete Conditions" results in
                          (gr(x1286,x1285)→^*false⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥c)⟹cond2^#(false,s(x1286),s(x1285))≥c)
                        • Applying Rule "Delete Conditions" results in
                          (gr(x1286,x1285)→^*false⟹cond2^#(false,s(x1286),s(x1285))≥c)
                        • Applying Rule "Induction" on gr(x1286,x1285)→^*false results in the following 3 new constraints.
                          1. (false→^*false⟹cond2^#(false,s(0),s(x1322))≥c)
                             • Applying Rule "Same Constructor" results in
                               cond2^#(false,s(0),s(x1322))≥c

                             • This constraint is kept as final constraint.

                          2. (true→^*false⟹cond2^#(false,s(s(x1323)),s(0))≥c)
                             • Applying Rule "Different Constructors" allows to drop this constraint.
                          3. (gr(x1325,x1324)→^*false⟹(gr(x1325,x1324)→^*false⟹cond2^#(false,s(x1325),s(x1324))≥c)⟹cond2^#(false,s(s(x1325)),s(s(x1324)))≥c)
                             • Applying Rule "Simplify Conditions" results in
                               (cond2^#(false,s(x1325),s(x1324))≥c⟹cond2^#(false,s(s(x1325)),s(s(x1324)))≥c)

                             • This constraint is kept as final constraint.

                     3. (gr(x1315,x1314)→^*true⟹gr(x1286,x1285)→^*false⟹0→^*s(x1314)⟹s(x1285)→^*s(x1315)⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥c)⟹∀x1316 . ∀x1317 . ∀x1318 . ∀x1319 . ∀x1320 . ∀x1321 . (gr(x1315,x1314)→^*true⟹gr(x1316,x1317)→^*false⟹0→^*x1314⟹s(x1317)→^*x1315⟹∀x1318 . ∀x1319 . ∀x1320 . ∀x1321 . (gr(x1316,x1317)→^*false⟹0→^*x1318⟹gr(x1319,x1318)→^*true⟹0→^*x1320⟹gr(x1321,x1320)→^*true⟹x1319→^*x1316⟹x1321→^*x1317⟹cond2^#(false,x1316,x1317)≥c)⟹cond2^#(false,s(x1316),s(x1317))≥c)⟹cond2^#(false,s(x1286),s(x1285))≥c)
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x1303,x1302)→^*true⟹gr(x1286,x1285)→^*false⟹0→^*s(x1302)⟹s(x1286)→^*s(x1303)⟹0→^*x1280⟹s(x1285)→^*x1299⟹gr(x1299,x1280)→^*true⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥c)⟹∀x1304 . ∀x1305 . ∀x1306 . ∀x1307 . ∀x1308 . ∀x1309 . ∀x1310 . ∀x1311 . (gr(x1303,x1302)→^*true⟹gr(x1304,x1305)→^*false⟹0→^*x1302⟹s(x1304)→^*x1303⟹0→^*x1306⟹s(x1305)→^*x1307⟹gr(x1307,x1306)→^*true⟹∀x1308 . ∀x1309 . ∀x1310 . ∀x1311 . (gr(x1304,x1305)→^*false⟹0→^*x1308⟹gr(x1309,x1308)→^*true⟹0→^*x1310⟹gr(x1311,x1310)→^*true⟹x1309→^*x1304⟹x1311→^*x1305⟹cond2^#(false,x1304,x1305)≥c)⟹cond2^#(false,s(x1304),s(x1305))≥c)⟹cond2^#(false,s(x1286),s(x1285))≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*true⟹cond2^#(false,x582,x583)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (false→^*true⟹cond2^#(false,x582,x583)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond2^#(gr(x714,x715),x714,x715)→^*cond2^#(false,x716,x717)⟹cond4^#(gr(x717,0),x716,x717)→^*cond4^#(true,x718,x719)⟹cond4^#(true,x718,x719)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x714,x715)→^*false⟹x714→^*x716⟹x715→^*x717⟹cond4^#(gr(x717,0),x716,x717)→^*cond4^#(true,x718,x719)⟹cond4^#(true,x718,x719)≥c)
    • Applying Rule "Same Constructor" results in
      (gr(x714,x715)→^*false⟹x714→^*x716⟹x715→^*x717⟹gr(x717,0)→^*true⟹x716→^*x718⟹x717→^*x719⟹cond4^#(true,x718,x719)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x716 by x714 which results in
      (gr(x714,x715)→^*false⟹x715→^*x717⟹gr(x717,0)→^*true⟹x714→^*x718⟹x717→^*x719⟹cond4^#(true,x718,x719)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x718 by x714 which results in
      (gr(x714,x715)→^*false⟹x715→^*x717⟹gr(x717,0)→^*true⟹x717→^*x719⟹cond4^#(true,x714,x719)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x719 by x717 which results in
      (gr(x714,x715)→^*false⟹x715→^*x717⟹gr(x717,0)→^*true⟹cond4^#(true,x714,x717)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (gr(x714,x715)→^*false⟹x715→^*x717⟹0→^*x1326⟹gr(x717,x1326)→^*true⟹cond4^#(true,x714,x717)≥c)
    • Applying Rule "Induction" on gr(x714,x715)→^*false results in the following 3 new constraints.
      1. (false→^*false⟹x1327→^*x717⟹0→^*x1326⟹gr(x717,x1326)→^*true⟹cond4^#(true,0,x717)≥c)
         • Applying Rule "Same Constructor" results in
           (x1327→^*x717⟹0→^*x1326⟹gr(x717,x1326)→^*true⟹cond4^#(true,0,x717)≥c)
         • Applying Rule "Variable in Equation" allows to substitute x1327 by x717 which results in
           (0→^*x1326⟹gr(x717,x1326)→^*true⟹cond4^#(true,0,x717)≥c)
         • Applying Rule "Induction" on gr(x717,x1326)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond4^#(true,0,0)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹0→^*0⟹cond4^#(true,0,s(x1334))≥c)
              • Applying Rule "Same Constructor" results in
                (0→^*0⟹cond4^#(true,0,s(x1334))≥c)
              • Applying Rule "Same Constructor" results in
                cond4^#(true,0,s(x1334))≥c

              • This constraint is kept as final constraint.

           3. (gr(x1336,x1335)→^*true⟹0→^*s(x1335)⟹(gr(x1336,x1335)→^*true⟹0→^*x1335⟹cond4^#(true,0,x1336)≥c)⟹cond4^#(true,0,s(x1336))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*false⟹cond4^#(true,s(x1328),x717)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x1330,x1329)→^*false⟹s(x1329)→^*x717⟹0→^*x1326⟹gr(x717,x1326)→^*true⟹∀x1331 . ∀x1332 . (gr(x1330,x1329)→^*false⟹x1329→^*x1331⟹0→^*x1332⟹gr(x1331,x1332)→^*true⟹cond4^#(true,x1330,x1331)≥c)⟹cond4^#(true,s(x1330),x717)≥c)
         • Applying Rule "Induction" on gr(x717,x1326)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond4^#(true,s(x1330),0)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹gr(x1330,x1329)→^*false⟹s(x1329)→^*s(x1338)⟹0→^*0⟹∀x1331 . ∀x1332 . (gr(x1330,x1329)→^*false⟹x1329→^*x1331⟹0→^*x1332⟹gr(x1331,x1332)→^*true⟹cond4^#(true,x1330,x1331)≥c)⟹cond4^#(true,s(x1330),s(x1338))≥c)
              • Applying Rule "Same Constructor" results in
                (gr(x1330,x1329)→^*false⟹s(x1329)→^*s(x1338)⟹0→^*0⟹∀x1331 . ∀x1332 . (gr(x1330,x1329)→^*false⟹x1329→^*x1331⟹0→^*x1332⟹gr(x1331,x1332)→^*true⟹cond4^#(true,x1330,x1331)≥c)⟹cond4^#(true,s(x1330),s(x1338))≥c)
              • Applying Rule "Same Constructor" results in
                (gr(x1330,x1329)→^*false⟹x1329→^*x1338⟹0→^*0⟹∀x1331 . ∀x1332 . (gr(x1330,x1329)→^*false⟹x1329→^*x1331⟹0→^*x1332⟹gr(x1331,x1332)→^*true⟹cond4^#(true,x1330,x1331)≥c)⟹cond4^#(true,s(x1330),s(x1338))≥c)
              • Applying Rule "Same Constructor" results in
                (gr(x1330,x1329)→^*false⟹x1329→^*x1338⟹∀x1331 . ∀x1332 . (gr(x1330,x1329)→^*false⟹x1329→^*x1331⟹0→^*x1332⟹gr(x1331,x1332)→^*true⟹cond4^#(true,x1330,x1331)≥c)⟹cond4^#(true,s(x1330),s(x1338))≥c)
              • Applying Rule "Variable in Equation" allows to substitute x1338 by x1329 which results in
                (gr(x1330,x1329)→^*false⟹∀x1331 . ∀x1332 . (gr(x1330,x1329)→^*false⟹x1329→^*x1331⟹0→^*x1332⟹gr(x1331,x1332)→^*true⟹cond4^#(true,x1330,x1331)≥c)⟹cond4^#(true,s(x1330),s(x1329))≥c)
              • Applying Rule "Delete Conditions" results in
                (gr(x1330,x1329)→^*false⟹cond4^#(true,s(x1330),s(x1329))≥c)
              • Applying Rule "Induction" on gr(x1330,x1329)→^*false results in the following 3 new constraints.
                1. (false→^*false⟹cond4^#(true,s(0),s(x1345))≥c)
                   • Applying Rule "Same Constructor" results in
                     cond4^#(true,s(0),s(x1345))≥c

                   • This constraint is kept as final constraint.

                2. (true→^*false⟹cond4^#(true,s(s(x1346)),s(0))≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x1348,x1347)→^*false⟹(gr(x1348,x1347)→^*false⟹cond4^#(true,s(x1348),s(x1347))≥c)⟹cond4^#(true,s(s(x1348)),s(s(x1347)))≥c)
                   • Applying Rule "Simplify Conditions" results in
                     (cond4^#(true,s(x1348),s(x1347))≥c⟹cond4^#(true,s(s(x1348)),s(s(x1347)))≥c)

                   • This constraint is kept as final constraint.

           3. (gr(x1340,x1339)→^*true⟹gr(x1330,x1329)→^*false⟹s(x1329)→^*s(x1340)⟹0→^*s(x1339)⟹∀x1331 . ∀x1332 . (gr(x1330,x1329)→^*false⟹x1329→^*x1331⟹0→^*x1332⟹gr(x1331,x1332)→^*true⟹cond4^#(true,x1330,x1331)≥c)⟹∀x1341 . ∀x1342 . ∀x1343 . ∀x1344 . (gr(x1340,x1339)→^*true⟹gr(x1341,x1342)→^*false⟹s(x1342)→^*x1340⟹0→^*x1339⟹∀x1343 . ∀x1344 . (gr(x1341,x1342)→^*false⟹x1342→^*x1343⟹0→^*x1344⟹gr(x1343,x1344)→^*true⟹cond4^#(true,x1341,x1343)≥c)⟹cond4^#(true,s(x1341),x1340)≥c)⟹cond4^#(true,s(x1330),s(x1340))≥c)
              • Applying Rule "Same Constructor" results in
                (gr(x1340,x1339)→^*true⟹gr(x1330,x1329)→^*false⟹x1329→^*x1340⟹0→^*s(x1339)⟹∀x1331 . ∀x1332 . (gr(x1330,x1329)→^*false⟹x1329→^*x1331⟹0→^*x1332⟹gr(x1331,x1332)→^*true⟹cond4^#(true,x1330,x1331)≥c)⟹∀x1341 . ∀x1342 . ∀x1343 . ∀x1344 . (gr(x1340,x1339)→^*true⟹gr(x1341,x1342)→^*false⟹s(x1342)→^*x1340⟹0→^*x1339⟹∀x1343 . ∀x1344 . (gr(x1341,x1342)→^*false⟹x1342→^*x1343⟹0→^*x1344⟹gr(x1343,x1344)→^*true⟹cond4^#(true,x1341,x1343)≥c)⟹cond4^#(true,s(x1341),x1340)≥c)⟹cond4^#(true,s(x1330),s(x1340))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond4^#(gr(x735,0),x734,x735)→^*cond4^#(true,x736,x737)⟹cond4^#(gr(x737,0),x736,p(x737))→^*cond4^#(true,x738,x739)⟹cond4^#(true,x738,x739)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x735,0)→^*true⟹x734→^*x736⟹x735→^*x737⟹cond4^#(gr(x737,0),x736,p(x737))→^*cond4^#(true,x738,x739)⟹cond4^#(true,x738,x739)≥c)
    • Applying Rule "Same Constructor" results in
      (gr(x735,0)→^*true⟹x734→^*x736⟹x735→^*x737⟹gr(x737,0)→^*true⟹x736→^*x738⟹p(x737)→^*x739⟹cond4^#(true,x738,x739)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x736 by x734 which results in
      (gr(x735,0)→^*true⟹x735→^*x737⟹gr(x737,0)→^*true⟹x734→^*x738⟹p(x737)→^*x739⟹cond4^#(true,x738,x739)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x738 by x734 which results in
      (gr(x735,0)→^*true⟹x735→^*x737⟹gr(x737,0)→^*true⟹p(x737)→^*x739⟹cond4^#(true,x734,x739)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1349⟹gr(x735,x1349)→^*true⟹x735→^*x737⟹gr(x737,0)→^*true⟹p(x737)→^*x739⟹cond4^#(true,x734,x739)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1349⟹gr(x735,x1349)→^*true⟹x735→^*x737⟹0→^*x1350⟹gr(x737,x1350)→^*true⟹p(x737)→^*x739⟹cond4^#(true,x734,x739)≥c)
    • Applying Rule "Induction" on gr(x735,x1349)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹cond4^#(true,x734,x739)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹0→^*0⟹s(x1352)→^*x737⟹0→^*x1350⟹gr(x737,x1350)→^*true⟹p(x737)→^*x739⟹cond4^#(true,x734,x739)≥c)
         • Applying Rule "Same Constructor" results in
           (0→^*0⟹s(x1352)→^*x737⟹0→^*x1350⟹gr(x737,x1350)→^*true⟹p(x737)→^*x739⟹cond4^#(true,x734,x739)≥c)
         • Applying Rule "Same Constructor" results in
           (s(x1352)→^*x737⟹0→^*x1350⟹gr(x737,x1350)→^*true⟹p(x737)→^*x739⟹cond4^#(true,x734,x739)≥c)
         • Applying Rule "Induction" on gr(x737,x1350)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond4^#(true,x734,x739)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹s(x1352)→^*s(x1360)⟹0→^*0⟹p(s(x1360))→^*x739⟹cond4^#(true,x734,x739)≥c)
              • Applying Rule "Same Constructor" results in
                (s(x1352)→^*s(x1360)⟹0→^*0⟹p(s(x1360))→^*x739⟹cond4^#(true,x734,x739)≥c)
              • Applying Rule "Same Constructor" results in
                (x1352→^*x1360⟹0→^*0⟹p(s(x1360))→^*x739⟹cond4^#(true,x734,x739)≥c)
              • Applying Rule "Same Constructor" results in
                (x1352→^*x1360⟹p(s(x1360))→^*x739⟹cond4^#(true,x734,x739)≥c)
              • Applying Rule "Variable in Equation" allows to substitute x1352 by x1360 which results in
                (p(s(x1360))→^*x739⟹cond4^#(true,x734,x739)≥c)
              • Applying Rule "Introduce fresh variable" results in
                (s(x1360)→^*x1366⟹p(x1366)→^*x739⟹cond4^#(true,x734,x739)≥c)
              • Applying Rule "Variable in Equation" allows to substitute x1366 by s(x1360) which results in
                (p(s(x1360))→^*x739⟹cond4^#(true,x734,x739)≥c)
              • Applying Rule "Delete Conditions" results in
                cond4^#(true,x734,x739)≥c

              • This constraint is kept as final constraint.

           3. (gr(x1362,x1361)→^*true⟹s(x1352)→^*s(x1362)⟹0→^*s(x1361)⟹p(s(x1362))→^*x739⟹∀x1363 . ∀x1364 . ∀x1365 . (gr(x1362,x1361)→^*true⟹s(x1363)→^*x1362⟹0→^*x1361⟹p(x1362)→^*x1364⟹cond4^#(true,x1365,x1364)≥c)⟹cond4^#(true,x734,x739)≥c)
              • Applying Rule "Same Constructor" results in
                (gr(x1362,x1361)→^*true⟹x1352→^*x1362⟹0→^*s(x1361)⟹p(s(x1362))→^*x739⟹∀x1363 . ∀x1364 . ∀x1365 . (gr(x1362,x1361)→^*true⟹s(x1363)→^*x1362⟹0→^*x1361⟹p(x1362)→^*x1364⟹cond4^#(true,x1365,x1364)≥c)⟹cond4^#(true,x734,x739)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x1354,x1353)→^*true⟹0→^*s(x1353)⟹s(x1354)→^*x737⟹0→^*x1350⟹gr(x737,x1350)→^*true⟹p(x737)→^*x739⟹∀x1355 . ∀x1356 . ∀x1357 . ∀x1358 . (gr(x1354,x1353)→^*true⟹0→^*x1353⟹x1354→^*x1355⟹0→^*x1356⟹gr(x1355,x1356)→^*true⟹p(x1355)→^*x1357⟹cond4^#(true,x1358,x1357)≥c)⟹cond4^#(true,x734,x739)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond4^#(gr(x741,0),x740,p(x741))→^*cond4^#(true,x742,x743)⟹cond4^#(gr(x743,0),x742,p(x743))→^*cond4^#(true,x744,x745)⟹cond4^#(true,x744,x745)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x741,0)→^*true⟹x740→^*x742⟹p(x741)→^*x743⟹cond4^#(gr(x743,0),x742,p(x743))→^*cond4^#(true,x744,x745)⟹cond4^#(true,x744,x745)≥c)
    • Applying Rule "Same Constructor" results in
      (gr(x741,0)→^*true⟹x740→^*x742⟹p(x741)→^*x743⟹gr(x743,0)→^*true⟹x742→^*x744⟹p(x743)→^*x745⟹cond4^#(true,x744,x745)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x742 by x740 which results in
      (gr(x741,0)→^*true⟹p(x741)→^*x743⟹gr(x743,0)→^*true⟹x740→^*x744⟹p(x743)→^*x745⟹cond4^#(true,x744,x745)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x744 by x740 which results in
      (gr(x741,0)→^*true⟹p(x741)→^*x743⟹gr(x743,0)→^*true⟹p(x743)→^*x745⟹cond4^#(true,x740,x745)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1367⟹gr(x741,x1367)→^*true⟹p(x741)→^*x743⟹gr(x743,0)→^*true⟹p(x743)→^*x745⟹cond4^#(true,x740,x745)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1367⟹gr(x741,x1367)→^*true⟹p(x741)→^*x743⟹0→^*x1368⟹gr(x743,x1368)→^*true⟹p(x743)→^*x745⟹cond4^#(true,x740,x745)≥c)
    • Applying Rule "Induction" on gr(x741,x1367)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹cond4^#(true,x740,x745)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹0→^*0⟹p(s(x1370))→^*x743⟹0→^*x1368⟹gr(x743,x1368)→^*true⟹p(x743)→^*x745⟹cond4^#(true,x740,x745)≥c)
         • Applying Rule "Same Constructor" results in
           (0→^*0⟹p(s(x1370))→^*x743⟹0→^*x1368⟹gr(x743,x1368)→^*true⟹p(x743)→^*x745⟹cond4^#(true,x740,x745)≥c)
         • Applying Rule "Same Constructor" results in
           (p(s(x1370))→^*x743⟹0→^*x1368⟹gr(x743,x1368)→^*true⟹p(x743)→^*x745⟹cond4^#(true,x740,x745)≥c)
         • Applying Rule "Introduce fresh variable" results in
           (s(x1370)→^*x1377⟹p(x1377)→^*x743⟹0→^*x1368⟹gr(x743,x1368)→^*true⟹p(x743)→^*x745⟹cond4^#(true,x740,x745)≥c)
         • Applying Rule "Induction" on gr(x743,x1368)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond4^#(true,x740,x745)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹s(x1370)→^*x1377⟹p(x1377)→^*s(x1379)⟹0→^*0⟹p(s(x1379))→^*x745⟹cond4^#(true,x740,x745)≥c)
              • Applying Rule "Same Constructor" results in
                (s(x1370)→^*x1377⟹p(x1377)→^*s(x1379)⟹0→^*0⟹p(s(x1379))→^*x745⟹cond4^#(true,x740,x745)≥c)
              • Applying Rule "Same Constructor" results in
                (s(x1370)→^*x1377⟹p(x1377)→^*s(x1379)⟹p(s(x1379))→^*x745⟹cond4^#(true,x740,x745)≥c)
              • Applying Rule "Introduce fresh variable" results in
                (s(x1370)→^*x1377⟹p(x1377)→^*s(x1379)⟹s(x1379)→^*x1386⟹p(x1386)→^*x745⟹cond4^#(true,x740,x745)≥c)
              • Applying Rule "Induction" on p(x1377)→^*s(x1379) results in the following 2 new constraints.
                1. (0→^*s(x1379)⟹cond4^#(true,x740,x745)≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (x1387→^*s(x1379)⟹s(x1370)→^*s(x1387)⟹s(x1379)→^*x1386⟹p(x1386)→^*x745⟹cond4^#(true,x740,x745)≥c)
                   • Applying Rule "Same Constructor" results in
                     (x1387→^*s(x1379)⟹x1370→^*x1387⟹s(x1379)→^*x1386⟹p(x1386)→^*x745⟹cond4^#(true,x740,x745)≥c)
                   • Applying Rule "Variable in Equation" allows to substitute x1387 by x1370 which results in
                     (x1370→^*s(x1379)⟹s(x1379)→^*x1386⟹p(x1386)→^*x745⟹cond4^#(true,x740,x745)≥c)
                   • Applying Rule "Variable in Equation" allows to substitute x1370 by s(x1379) which results in
                     (s(x1379)→^*x1386⟹p(x1386)→^*x745⟹cond4^#(true,x740,x745)≥c)
                   • Applying Rule "Variable in Equation" allows to substitute x1386 by s(x1379) which results in
                     (p(s(x1379))→^*x745⟹cond4^#(true,x740,x745)≥c)
                   • Applying Rule "Delete Conditions" results in
                     cond4^#(true,x740,x745)≥c

                   • This constraint is kept as final constraint.

           3. (gr(x1381,x1380)→^*true⟹s(x1370)→^*x1377⟹p(x1377)→^*s(x1381)⟹0→^*s(x1380)⟹p(s(x1381))→^*x745⟹∀x1382 . ∀x1383 . ∀x1384 . ∀x1385 . (gr(x1381,x1380)→^*true⟹s(x1382)→^*x1383⟹p(x1383)→^*x1381⟹0→^*x1380⟹p(x1381)→^*x1384⟹cond4^#(true,x1385,x1384)≥c)⟹cond4^#(true,x740,x745)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x1372,x1371)→^*true⟹0→^*s(x1371)⟹p(s(x1372))→^*x743⟹0→^*x1368⟹gr(x743,x1368)→^*true⟹p(x743)→^*x745⟹∀x1373 . ∀x1374 . ∀x1375 . ∀x1376 . (gr(x1372,x1371)→^*true⟹0→^*x1371⟹p(x1372)→^*x1373⟹0→^*x1374⟹gr(x1373,x1374)→^*true⟹p(x1373)→^*x1375⟹cond4^#(true,x1376,x1375)≥c)⟹cond4^#(true,x740,x745)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond2^#(gr(x842,x843),x842,x843)→^*cond2^#(false,x844,x845)⟹cond4^#(gr(x845,0),x844,x845)→^*cond4^#(false,x846,x847)⟹cond4^#(false,x846,x847)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x842,x843)→^*false⟹x842→^*x844⟹x843→^*x845⟹cond4^#(gr(x845,0),x844,x845)→^*cond4^#(false,x846,x847)⟹cond4^#(false,x846,x847)≥c)
    • Applying Rule "Same Constructor" results in
      (gr(x842,x843)→^*false⟹x842→^*x844⟹x843→^*x845⟹gr(x845,0)→^*false⟹x844→^*x846⟹x845→^*x847⟹cond4^#(false,x846,x847)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x844 by x842 which results in
      (gr(x842,x843)→^*false⟹x843→^*x845⟹gr(x845,0)→^*false⟹x842→^*x846⟹x845→^*x847⟹cond4^#(false,x846,x847)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x846 by x842 which results in
      (gr(x842,x843)→^*false⟹x843→^*x845⟹gr(x845,0)→^*false⟹x845→^*x847⟹cond4^#(false,x842,x847)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x847 by x845 which results in
      (gr(x842,x843)→^*false⟹x843→^*x845⟹gr(x845,0)→^*false⟹cond4^#(false,x842,x845)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (gr(x842,x843)→^*false⟹x843→^*x845⟹0→^*x1388⟹gr(x845,x1388)→^*false⟹cond4^#(false,x842,x845)≥c)
    • Applying Rule "Induction" on gr(x842,x843)→^*false results in the following 3 new constraints.
      1. (false→^*false⟹x1389→^*x845⟹0→^*x1388⟹gr(x845,x1388)→^*false⟹cond4^#(false,0,x845)≥c)
         • Applying Rule "Same Constructor" results in
           (x1389→^*x845⟹0→^*x1388⟹gr(x845,x1388)→^*false⟹cond4^#(false,0,x845)≥c)
         • Applying Rule "Variable in Equation" allows to substitute x1389 by x845 which results in
           (0→^*x1388⟹gr(x845,x1388)→^*false⟹cond4^#(false,0,x845)≥c)
         • Applying Rule "Induction" on gr(x845,x1388)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹0→^*x1395⟹cond4^#(false,0,0)≥c)
              • Applying Rule "Same Constructor" results in
                (0→^*x1395⟹cond4^#(false,0,0)≥c)
              • Applying Rule "Delete Conditions" results in
                cond4^#(false,0,0)≥c

              • This constraint is kept as final constraint.

           2. (true→^*false⟹cond4^#(false,0,s(x1396))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1398,x1397)→^*false⟹0→^*s(x1397)⟹(gr(x1398,x1397)→^*false⟹0→^*x1397⟹cond4^#(false,0,x1398)≥c)⟹cond4^#(false,0,s(x1398))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*false⟹cond4^#(false,s(x1390),x845)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x1392,x1391)→^*false⟹s(x1391)→^*x845⟹0→^*x1388⟹gr(x845,x1388)→^*false⟹∀x1393 . ∀x1394 . (gr(x1392,x1391)→^*false⟹x1391→^*x1393⟹0→^*x1394⟹gr(x1393,x1394)→^*false⟹cond4^#(false,x1392,x1393)≥c)⟹cond4^#(false,s(x1392),x845)≥c)
         • Applying Rule "Induction" on gr(x845,x1388)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹gr(x1392,x1391)→^*false⟹s(x1391)→^*0⟹0→^*x1399⟹∀x1393 . ∀x1394 . (gr(x1392,x1391)→^*false⟹x1391→^*x1393⟹0→^*x1394⟹gr(x1393,x1394)→^*false⟹cond4^#(false,x1392,x1393)≥c)⟹cond4^#(false,s(x1392),0)≥c)
              • Applying Rule "Same Constructor" results in
                (gr(x1392,x1391)→^*false⟹s(x1391)→^*0⟹0→^*x1399⟹∀x1393 . ∀x1394 . (gr(x1392,x1391)→^*false⟹x1391→^*x1393⟹0→^*x1394⟹gr(x1393,x1394)→^*false⟹cond4^#(false,x1392,x1393)≥c)⟹cond4^#(false,s(x1392),0)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond4^#(false,s(x1392),s(x1400))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1402,x1401)→^*false⟹gr(x1392,x1391)→^*false⟹s(x1391)→^*s(x1402)⟹0→^*s(x1401)⟹∀x1393 . ∀x1394 . (gr(x1392,x1391)→^*false⟹x1391→^*x1393⟹0→^*x1394⟹gr(x1393,x1394)→^*false⟹cond4^#(false,x1392,x1393)≥c)⟹∀x1403 . ∀x1404 . ∀x1405 . ∀x1406 . (gr(x1402,x1401)→^*false⟹gr(x1403,x1404)→^*false⟹s(x1404)→^*x1402⟹0→^*x1401⟹∀x1405 . ∀x1406 . (gr(x1403,x1404)→^*false⟹x1404→^*x1405⟹0→^*x1406⟹gr(x1405,x1406)→^*false⟹cond4^#(false,x1403,x1405)≥c)⟹cond4^#(false,s(x1403),x1402)≥c)⟹cond4^#(false,s(x1392),s(x1402))≥c)
              • Applying Rule "Same Constructor" results in
                (gr(x1402,x1401)→^*false⟹gr(x1392,x1391)→^*false⟹x1391→^*x1402⟹0→^*s(x1401)⟹∀x1393 . ∀x1394 . (gr(x1392,x1391)→^*false⟹x1391→^*x1393⟹0→^*x1394⟹gr(x1393,x1394)→^*false⟹cond4^#(false,x1392,x1393)≥c)⟹∀x1403 . ∀x1404 . ∀x1405 . ∀x1406 . (gr(x1402,x1401)→^*false⟹gr(x1403,x1404)→^*false⟹s(x1404)→^*x1402⟹0→^*x1401⟹∀x1405 . ∀x1406 . (gr(x1403,x1404)→^*false⟹x1404→^*x1405⟹0→^*x1406⟹gr(x1405,x1406)→^*false⟹cond4^#(false,x1403,x1405)≥c)⟹cond4^#(false,s(x1403),x1402)≥c)⟹cond4^#(false,s(x1392),s(x1402))≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond4^#(gr(x863,0),x862,x863)→^*cond4^#(true,x864,x865)⟹cond4^#(gr(x865,0),x864,p(x865))→^*cond4^#(false,x866,x867)⟹cond4^#(false,x866,x867)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x863,0)→^*true⟹x862→^*x864⟹x863→^*x865⟹cond4^#(gr(x865,0),x864,p(x865))→^*cond4^#(false,x866,x867)⟹cond4^#(false,x866,x867)≥c)
    • Applying Rule "Same Constructor" results in
      (gr(x863,0)→^*true⟹x862→^*x864⟹x863→^*x865⟹gr(x865,0)→^*false⟹x864→^*x866⟹p(x865)→^*x867⟹cond4^#(false,x866,x867)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x864 by x862 which results in
      (gr(x863,0)→^*true⟹x863→^*x865⟹gr(x865,0)→^*false⟹x862→^*x866⟹p(x865)→^*x867⟹cond4^#(false,x866,x867)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x866 by x862 which results in
      (gr(x863,0)→^*true⟹x863→^*x865⟹gr(x865,0)→^*false⟹p(x865)→^*x867⟹cond4^#(false,x862,x867)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1407⟹gr(x863,x1407)→^*true⟹x863→^*x865⟹gr(x865,0)→^*false⟹p(x865)→^*x867⟹cond4^#(false,x862,x867)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1407⟹gr(x863,x1407)→^*true⟹x863→^*x865⟹0→^*x1408⟹gr(x865,x1408)→^*false⟹p(x865)→^*x867⟹cond4^#(false,x862,x867)≥c)
    • Applying Rule "Induction" on gr(x863,x1407)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹cond4^#(false,x862,x867)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹0→^*0⟹s(x1410)→^*x865⟹0→^*x1408⟹gr(x865,x1408)→^*false⟹p(x865)→^*x867⟹cond4^#(false,x862,x867)≥c)
         • Applying Rule "Same Constructor" results in
           (0→^*0⟹s(x1410)→^*x865⟹0→^*x1408⟹gr(x865,x1408)→^*false⟹p(x865)→^*x867⟹cond4^#(false,x862,x867)≥c)
         • Applying Rule "Same Constructor" results in
           (s(x1410)→^*x865⟹0→^*x1408⟹gr(x865,x1408)→^*false⟹p(x865)→^*x867⟹cond4^#(false,x862,x867)≥c)
         • Applying Rule "Induction" on gr(x865,x1408)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹s(x1410)→^*0⟹0→^*x1417⟹p(0)→^*x867⟹cond4^#(false,x862,x867)≥c)
              • Applying Rule "Same Constructor" results in
                (s(x1410)→^*0⟹0→^*x1417⟹p(0)→^*x867⟹cond4^#(false,x862,x867)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond4^#(false,x862,x867)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1420,x1419)→^*false⟹s(x1410)→^*s(x1420)⟹0→^*s(x1419)⟹p(s(x1420))→^*x867⟹∀x1421 . ∀x1422 . ∀x1423 . (gr(x1420,x1419)→^*false⟹s(x1421)→^*x1420⟹0→^*x1419⟹p(x1420)→^*x1422⟹cond4^#(false,x1423,x1422)≥c)⟹cond4^#(false,x862,x867)≥c)
              • Applying Rule "Same Constructor" results in
                (gr(x1420,x1419)→^*false⟹x1410→^*x1420⟹0→^*s(x1419)⟹p(s(x1420))→^*x867⟹∀x1421 . ∀x1422 . ∀x1423 . (gr(x1420,x1419)→^*false⟹s(x1421)→^*x1420⟹0→^*x1419⟹p(x1420)→^*x1422⟹cond4^#(false,x1423,x1422)≥c)⟹cond4^#(false,x862,x867)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x1412,x1411)→^*true⟹0→^*s(x1411)⟹s(x1412)→^*x865⟹0→^*x1408⟹gr(x865,x1408)→^*false⟹p(x865)→^*x867⟹∀x1413 . ∀x1414 . ∀x1415 . ∀x1416 . (gr(x1412,x1411)→^*true⟹0→^*x1411⟹x1412→^*x1413⟹0→^*x1414⟹gr(x1413,x1414)→^*false⟹p(x1413)→^*x1415⟹cond4^#(false,x1416,x1415)≥c)⟹cond4^#(false,x862,x867)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond4^#(gr(x869,0),x868,p(x869))→^*cond4^#(true,x870,x871)⟹cond4^#(gr(x871,0),x870,p(x871))→^*cond4^#(false,x872,x873)⟹cond4^#(false,x872,x873)≥c)

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x869,0)→^*true⟹x868→^*x870⟹p(x869)→^*x871⟹cond4^#(gr(x871,0),x870,p(x871))→^*cond4^#(false,x872,x873)⟹cond4^#(false,x872,x873)≥c)
    • Applying Rule "Same Constructor" results in
      (gr(x869,0)→^*true⟹x868→^*x870⟹p(x869)→^*x871⟹gr(x871,0)→^*false⟹x870→^*x872⟹p(x871)→^*x873⟹cond4^#(false,x872,x873)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x870 by x868 which results in
      (gr(x869,0)→^*true⟹p(x869)→^*x871⟹gr(x871,0)→^*false⟹x868→^*x872⟹p(x871)→^*x873⟹cond4^#(false,x872,x873)≥c)
    • Applying Rule "Variable in Equation" allows to substitute x872 by x868 which results in
      (gr(x869,0)→^*true⟹p(x869)→^*x871⟹gr(x871,0)→^*false⟹p(x871)→^*x873⟹cond4^#(false,x868,x873)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1424⟹gr(x869,x1424)→^*true⟹p(x869)→^*x871⟹gr(x871,0)→^*false⟹p(x871)→^*x873⟹cond4^#(false,x868,x873)≥c)
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1424⟹gr(x869,x1424)→^*true⟹p(x869)→^*x871⟹0→^*x1425⟹gr(x871,x1425)→^*false⟹p(x871)→^*x873⟹cond4^#(false,x868,x873)≥c)
    • Applying Rule "Induction" on gr(x869,x1424)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹cond4^#(false,x868,x873)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹0→^*0⟹p(s(x1427))→^*x871⟹0→^*x1425⟹gr(x871,x1425)→^*false⟹p(x871)→^*x873⟹cond4^#(false,x868,x873)≥c)
         • Applying Rule "Same Constructor" results in
           (0→^*0⟹p(s(x1427))→^*x871⟹0→^*x1425⟹gr(x871,x1425)→^*false⟹p(x871)→^*x873⟹cond4^#(false,x868,x873)≥c)
         • Applying Rule "Same Constructor" results in
           (p(s(x1427))→^*x871⟹0→^*x1425⟹gr(x871,x1425)→^*false⟹p(x871)→^*x873⟹cond4^#(false,x868,x873)≥c)
         • Applying Rule "Introduce fresh variable" results in
           (s(x1427)→^*x1434⟹p(x1434)→^*x871⟹0→^*x1425⟹gr(x871,x1425)→^*false⟹p(x871)→^*x873⟹cond4^#(false,x868,x873)≥c)
         • Applying Rule "Induction" on gr(x871,x1425)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹s(x1427)→^*x1434⟹p(x1434)→^*0⟹0→^*x1435⟹p(0)→^*x873⟹cond4^#(false,x868,x873)≥c)
              • Applying Rule "Same Constructor" results in
                (s(x1427)→^*x1434⟹p(x1434)→^*0⟹0→^*x1435⟹p(0)→^*x873⟹cond4^#(false,x868,x873)≥c)
              • Applying Rule "Delete Conditions" results in
                (s(x1427)→^*x1434⟹p(x1434)→^*0⟹p(0)→^*x873⟹cond4^#(false,x868,x873)≥c)
              • Applying Rule "Introduce fresh variable" results in
                (s(x1427)→^*x1434⟹p(x1434)→^*0⟹0→^*x1443⟹p(x1443)→^*x873⟹cond4^#(false,x868,x873)≥c)
              • Applying Rule "Induction" on p(x1434)→^*0 results in the following 2 new constraints.
                1. (0→^*0⟹s(x1427)→^*0⟹0→^*x1443⟹p(x1443)→^*x873⟹cond4^#(false,x868,x873)≥c)
                   • Applying Rule "Same Constructor" results in
                     (s(x1427)→^*0⟹0→^*x1443⟹p(x1443)→^*x873⟹cond4^#(false,x868,x873)≥c)
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (x1444→^*0⟹s(x1427)→^*s(x1444)⟹0→^*x1443⟹p(x1443)→^*x873⟹cond4^#(false,x868,x873)≥c)
                   • Applying Rule "Same Constructor" results in
                     (x1444→^*0⟹x1427→^*x1444⟹0→^*x1443⟹p(x1443)→^*x873⟹cond4^#(false,x868,x873)≥c)
                   • Applying Rule "Variable in Equation" allows to substitute x1444 by x1427 which results in
                     (x1427→^*0⟹0→^*x1443⟹p(x1443)→^*x873⟹cond4^#(false,x868,x873)≥c)
                   • Applying Rule "Variable in Equation" allows to substitute x1427 by 0 which results in
                     (0→^*x1443⟹p(x1443)→^*x873⟹cond4^#(false,x868,x873)≥c)
                   • Applying Rule "Variable in Equation" allows to substitute x1443 by 0 which results in
                     (p(0)→^*x873⟹cond4^#(false,x868,x873)≥c)
                   • Applying Rule "Delete Conditions" results in
                     cond4^#(false,x868,x873)≥c

                   • This constraint is kept as final constraint.

           2. (true→^*false⟹cond4^#(false,x868,x873)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1438,x1437)→^*false⟹s(x1427)→^*x1434⟹p(x1434)→^*s(x1438)⟹0→^*s(x1437)⟹p(s(x1438))→^*x873⟹∀x1439 . ∀x1440 . ∀x1441 . ∀x1442 . (gr(x1438,x1437)→^*false⟹s(x1439)→^*x1440⟹p(x1440)→^*x1438⟹0→^*x1437⟹p(x1438)→^*x1441⟹cond4^#(false,x1442,x1441)≥c)⟹cond4^#(false,x868,x873)≥c)
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x1429,x1428)→^*true⟹0→^*s(x1428)⟹p(s(x1429))→^*x871⟹0→^*x1425⟹gr(x871,x1425)→^*false⟹p(x871)→^*x873⟹∀x1430 . ∀x1431 . ∀x1432 . ∀x1433 . (gr(x1429,x1428)→^*true⟹0→^*x1428⟹p(x1429)→^*x1430⟹0→^*x1431⟹gr(x1430,x1431)→^*false⟹p(x1430)→^*x1432⟹cond4^#(false,x1433,x1432)≥c)⟹cond4^#(false,x868,x873)≥c)
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond1^#(and(gr(x56,0),gr(x57,0)),x56,x57)→^*cond1^#(true,x58,x59)⟹cond2^#(gr(x58,x59),x58,x59)→^*cond2^#(true,x60,x61)⟹cond2^#(true,x60,x61)≥cond3^#(gr(x60,0),x60,x61))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (and(gr(x56,0),gr(x57,0))→^*true⟹x56→^*x58⟹x57→^*x59⟹cond2^#(gr(x58,x59),x58,x59)→^*cond2^#(true,x60,x61)⟹cond2^#(true,x60,x61)≥cond3^#(gr(x60,0),x60,x61))
    • Applying Rule "Same Constructor" results in
      (and(gr(x56,0),gr(x57,0))→^*true⟹x56→^*x58⟹x57→^*x59⟹gr(x58,x59)→^*true⟹x58→^*x60⟹x59→^*x61⟹cond2^#(true,x60,x61)≥cond3^#(gr(x60,0),x60,x61))
    • Applying Rule "Variable in Equation" allows to substitute x60 by x58 which results in
      (and(gr(x56,0),gr(x57,0))→^*true⟹x56→^*x58⟹x57→^*x59⟹gr(x58,x59)→^*true⟹x59→^*x61⟹cond2^#(true,x58,x61)≥cond3^#(gr(x58,0),x58,x61))
    • Applying Rule "Variable in Equation" allows to substitute x61 by x59 which results in
      (and(gr(x56,0),gr(x57,0))→^*true⟹x56→^*x58⟹x57→^*x59⟹gr(x58,x59)→^*true⟹cond2^#(true,x58,x59)≥cond3^#(gr(x58,0),x58,x59))
    • Applying Rule "Introduce fresh variable" results in
      (x56→^*x58⟹x57→^*x59⟹gr(x58,x59)→^*true⟹gr(x57,0)→^*x895⟹and(gr(x56,0),x895)→^*true⟹cond2^#(true,x58,x59)≥cond3^#(gr(x58,0),x58,x59))
    • Applying Rule "Introduce fresh variable" results in
      (gr(x56,0)→^*x894⟹gr(x57,0)→^*x895⟹and(x894,x895)→^*true⟹x56→^*x58⟹x57→^*x59⟹gr(x58,x59)→^*true⟹cond2^#(true,x58,x59)≥cond3^#(gr(x58,0),x58,x59))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x896⟹gr(x56,x896)→^*x894⟹gr(x57,0)→^*x895⟹and(x894,x895)→^*true⟹x56→^*x58⟹x57→^*x59⟹gr(x58,x59)→^*true⟹cond2^#(true,x58,x59)≥cond3^#(gr(x58,0),x58,x59))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x896⟹gr(x56,x896)→^*x894⟹0→^*x897⟹gr(x57,x897)→^*x895⟹and(x894,x895)→^*true⟹x56→^*x58⟹x57→^*x59⟹gr(x58,x59)→^*true⟹cond2^#(true,x58,x59)≥cond3^#(gr(x58,0),x58,x59))
    • Applying Rule "Induction" on and(x894,x895)→^*true results in the following 3 new constraints.
      1. (true→^*true⟹0→^*x896⟹gr(x56,x896)→^*true⟹0→^*x897⟹gr(x57,x897)→^*true⟹x56→^*x58⟹x57→^*x59⟹gr(x58,x59)→^*true⟹cond2^#(true,x58,x59)≥cond3^#(gr(x58,0),x58,x59))
         • Applying Rule "Same Constructor" results in
           (0→^*x896⟹gr(x56,x896)→^*true⟹0→^*x897⟹gr(x57,x897)→^*true⟹x56→^*x58⟹x57→^*x59⟹gr(x58,x59)→^*true⟹cond2^#(true,x58,x59)≥cond3^#(gr(x58,0),x58,x59))
         • Applying Rule "Induction" on gr(x58,x59)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond2^#(true,0,x900)≥cond3^#(gr(0,0),0,x900))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹0→^*x896⟹gr(x56,x896)→^*true⟹0→^*x897⟹gr(x57,x897)→^*true⟹x56→^*s(x901)⟹x57→^*0⟹cond2^#(true,s(x901),0)≥cond3^#(gr(s(x901),0),s(x901),0))
              • Applying Rule "Same Constructor" results in
                (0→^*x896⟹gr(x56,x896)→^*true⟹0→^*x897⟹gr(x57,x897)→^*true⟹x56→^*s(x901)⟹x57→^*0⟹cond2^#(true,s(x901),0)≥cond3^#(gr(s(x901),0),s(x901),0))
              • Applying Rule "Variable in Equation" allows to substitute x56 by s(x901) which results in
                (0→^*x896⟹gr(s(x901),x896)→^*true⟹0→^*x897⟹gr(x57,x897)→^*true⟹x57→^*0⟹cond2^#(true,s(x901),0)≥cond3^#(gr(s(x901),0),s(x901),0))
              • Applying Rule "Variable in Equation" allows to substitute x57 by 0 which results in
                (0→^*x896⟹gr(s(x901),x896)→^*true⟹0→^*x897⟹gr(0,x897)→^*true⟹cond2^#(true,s(x901),0)≥cond3^#(gr(s(x901),0),s(x901),0))
              • Applying Rule "Introduce fresh variable" results in
                (0→^*x896⟹s(x901)→^*x908⟹gr(x908,x896)→^*true⟹0→^*x897⟹gr(0,x897)→^*true⟹cond2^#(true,s(x901),0)≥cond3^#(gr(s(x901),0),s(x901),0))
              • Applying Rule "Introduce fresh variable" results in
                (0→^*x896⟹s(x901)→^*x908⟹gr(x908,x896)→^*true⟹0→^*x897⟹0→^*x909⟹gr(x909,x897)→^*true⟹cond2^#(true,s(x901),0)≥cond3^#(gr(s(x901),0),s(x901),0))
              • Applying Rule "Induction" on gr(x908,x896)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond2^#(true,s(x901),0)≥cond3^#(gr(s(x901),0),s(x901),0))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹0→^*0⟹s(x901)→^*s(x911)⟹0→^*x897⟹0→^*x909⟹gr(x909,x897)→^*true⟹cond2^#(true,s(x901),0)≥cond3^#(gr(s(x901),0),s(x901),0))
                   • Applying Rule "Same Constructor" results in
                     (0→^*0⟹s(x901)→^*s(x911)⟹0→^*x897⟹0→^*x909⟹gr(x909,x897)→^*true⟹cond2^#(true,s(x901),0)≥cond3^#(gr(s(x901),0),s(x901),0))
                   • Applying Rule "Same Constructor" results in
                     (s(x901)→^*s(x911)⟹0→^*x897⟹0→^*x909⟹gr(x909,x897)→^*true⟹cond2^#(true,s(x901),0)≥cond3^#(gr(s(x901),0),s(x901),0))
                   • Applying Rule "Same Constructor" results in
                     (x901→^*x911⟹0→^*x897⟹0→^*x909⟹gr(x909,x897)→^*true⟹cond2^#(true,s(x901),0)≥cond3^#(gr(s(x901),0),s(x901),0))
                   • Applying Rule "Delete Conditions" results in
                     (0→^*x897⟹0→^*x909⟹gr(x909,x897)→^*true⟹cond2^#(true,s(x901),0)≥cond3^#(gr(s(x901),0),s(x901),0))
                   • Applying Rule "Induction" on gr(x909,x897)→^*true results in the following 3 new constraints.
                     1. (false→^*true⟹cond2^#(true,s(x901),0)≥cond3^#(gr(s(x901),0),s(x901),0))
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                     2. (true→^*true⟹0→^*0⟹0→^*s(x918)⟹cond2^#(true,s(x901),0)≥cond3^#(gr(s(x901),0),s(x901),0))
                        • Applying Rule "Same Constructor" results in
                          (0→^*0⟹0→^*s(x918)⟹cond2^#(true,s(x901),0)≥cond3^#(gr(s(x901),0),s(x901),0))
                        • Applying Rule "Same Constructor" results in
                          (0→^*s(x918)⟹cond2^#(true,s(x901),0)≥cond3^#(gr(s(x901),0),s(x901),0))
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                     3. (gr(x920,x919)→^*true⟹0→^*s(x919)⟹0→^*s(x920)⟹∀x921 . (gr(x920,x919)→^*true⟹0→^*x919⟹0→^*x920⟹cond2^#(true,s(x921),0)≥cond3^#(gr(s(x921),0),s(x921),0))⟹cond2^#(true,s(x901),0)≥cond3^#(gr(s(x901),0),s(x901),0))
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x913,x912)→^*true⟹0→^*s(x912)⟹s(x901)→^*s(x913)⟹0→^*x897⟹0→^*x909⟹gr(x909,x897)→^*true⟹∀x914 . ∀x915 . ∀x916 . (gr(x913,x912)→^*true⟹0→^*x912⟹s(x914)→^*x913⟹0→^*x915⟹0→^*x916⟹gr(x916,x915)→^*true⟹cond2^#(true,s(x914),0)≥cond3^#(gr(s(x914),0),s(x914),0))⟹cond2^#(true,s(x901),0)≥cond3^#(gr(s(x901),0),s(x901),0))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x903,x902)→^*true⟹0→^*x896⟹gr(x56,x896)→^*true⟹0→^*x897⟹gr(x57,x897)→^*true⟹x56→^*s(x903)⟹x57→^*s(x902)⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥cond3^#(gr(x903,0),x903,x902))⟹cond2^#(true,s(x903),s(x902))≥cond3^#(gr(s(x903),0),s(x903),s(x902)))
              • Applying Rule "Variable in Equation" allows to substitute x56 by s(x903) which results in
                (gr(x903,x902)→^*true⟹0→^*x896⟹gr(s(x903),x896)→^*true⟹0→^*x897⟹gr(x57,x897)→^*true⟹x57→^*s(x902)⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥cond3^#(gr(x903,0),x903,x902))⟹cond2^#(true,s(x903),s(x902))≥cond3^#(gr(s(x903),0),s(x903),s(x902)))
              • Applying Rule "Variable in Equation" allows to substitute x57 by s(x902) which results in
                (gr(x903,x902)→^*true⟹0→^*x896⟹gr(s(x903),x896)→^*true⟹0→^*x897⟹gr(s(x902),x897)→^*true⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥cond3^#(gr(x903,0),x903,x902))⟹cond2^#(true,s(x903),s(x902))≥cond3^#(gr(s(x903),0),s(x903),s(x902)))
              • Applying Rule "Introduce fresh variable" results in
                (gr(x903,x902)→^*true⟹0→^*x896⟹s(x903)→^*x922⟹gr(x922,x896)→^*true⟹0→^*x897⟹gr(s(x902),x897)→^*true⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥cond3^#(gr(x903,0),x903,x902))⟹cond2^#(true,s(x903),s(x902))≥cond3^#(gr(s(x903),0),s(x903),s(x902)))
              • Applying Rule "Introduce fresh variable" results in
                (gr(x903,x902)→^*true⟹0→^*x896⟹s(x903)→^*x922⟹gr(x922,x896)→^*true⟹0→^*x897⟹s(x902)→^*x923⟹gr(x923,x897)→^*true⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥cond3^#(gr(x903,0),x903,x902))⟹cond2^#(true,s(x903),s(x902))≥cond3^#(gr(s(x903),0),s(x903),s(x902)))
              • Applying Rule "Induction" on gr(x922,x896)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond2^#(true,s(x903),s(x902))≥cond3^#(gr(s(x903),0),s(x903),s(x902)))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹gr(x903,x902)→^*true⟹0→^*0⟹s(x903)→^*s(x925)⟹0→^*x897⟹s(x902)→^*x923⟹gr(x923,x897)→^*true⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥cond3^#(gr(x903,0),x903,x902))⟹cond2^#(true,s(x903),s(x902))≥cond3^#(gr(s(x903),0),s(x903),s(x902)))
                   • Applying Rule "Same Constructor" results in
                     (gr(x903,x902)→^*true⟹0→^*0⟹s(x903)→^*s(x925)⟹0→^*x897⟹s(x902)→^*x923⟹gr(x923,x897)→^*true⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥cond3^#(gr(x903,0),x903,x902))⟹cond2^#(true,s(x903),s(x902))≥cond3^#(gr(s(x903),0),s(x903),s(x902)))
                   • Applying Rule "Same Constructor" results in
                     (gr(x903,x902)→^*true⟹s(x903)→^*s(x925)⟹0→^*x897⟹s(x902)→^*x923⟹gr(x923,x897)→^*true⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥cond3^#(gr(x903,0),x903,x902))⟹cond2^#(true,s(x903),s(x902))≥cond3^#(gr(s(x903),0),s(x903),s(x902)))
                   • Applying Rule "Same Constructor" results in
                     (gr(x903,x902)→^*true⟹x903→^*x925⟹0→^*x897⟹s(x902)→^*x923⟹gr(x923,x897)→^*true⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥cond3^#(gr(x903,0),x903,x902))⟹cond2^#(true,s(x903),s(x902))≥cond3^#(gr(s(x903),0),s(x903),s(x902)))
                   • Applying Rule "Delete Conditions" results in
                     (gr(x903,x902)→^*true⟹0→^*x897⟹s(x902)→^*x923⟹gr(x923,x897)→^*true⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥cond3^#(gr(x903,0),x903,x902))⟹cond2^#(true,s(x903),s(x902))≥cond3^#(gr(s(x903),0),s(x903),s(x902)))
                   • Applying Rule "Induction" on gr(x923,x897)→^*true results in the following 3 new constraints.
                     1. (false→^*true⟹cond2^#(true,s(x903),s(x902))≥cond3^#(gr(s(x903),0),s(x903),s(x902)))
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                     2. (true→^*true⟹gr(x903,x902)→^*true⟹0→^*0⟹s(x902)→^*s(x937)⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥cond3^#(gr(x903,0),x903,x902))⟹cond2^#(true,s(x903),s(x902))≥cond3^#(gr(s(x903),0),s(x903),s(x902)))
                        • Applying Rule "Same Constructor" results in
                          (gr(x903,x902)→^*true⟹0→^*0⟹s(x902)→^*s(x937)⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥cond3^#(gr(x903,0),x903,x902))⟹cond2^#(true,s(x903),s(x902))≥cond3^#(gr(s(x903),0),s(x903),s(x902)))
                        • Applying Rule "Same Constructor" results in
                          (gr(x903,x902)→^*true⟹s(x902)→^*s(x937)⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥cond3^#(gr(x903,0),x903,x902))⟹cond2^#(true,s(x903),s(x902))≥cond3^#(gr(s(x903),0),s(x903),s(x902)))
                        • Applying Rule "Same Constructor" results in
                          (gr(x903,x902)→^*true⟹x902→^*x937⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥cond3^#(gr(x903,0),x903,x902))⟹cond2^#(true,s(x903),s(x902))≥cond3^#(gr(s(x903),0),s(x903),s(x902)))
                        • Applying Rule "Delete Conditions" results in
                          (gr(x903,x902)→^*true⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥cond3^#(gr(x903,0),x903,x902))⟹cond2^#(true,s(x903),s(x902))≥cond3^#(gr(s(x903),0),s(x903),s(x902)))
                        • Applying Rule "Delete Conditions" results in
                          (gr(x903,x902)→^*true⟹cond2^#(true,s(x903),s(x902))≥cond3^#(gr(s(x903),0),s(x903),s(x902)))
                        • Applying Rule "Induction" on gr(x903,x902)→^*true results in the following 3 new constraints.
                          1. (false→^*true⟹cond2^#(true,s(0),s(x946))≥cond3^#(gr(s(0),0),s(0),s(x946)))
                             • Applying Rule "Different Constructors" allows to drop this constraint.
                          2. (true→^*true⟹cond2^#(true,s(s(x947)),s(0))≥cond3^#(gr(s(s(x947)),0),s(s(x947)),s(0)))
                             • Applying Rule "Same Constructor" results in
                               cond2^#(true,s(s(x947)),s(0))≥cond3^#(gr(s(s(x947)),0),s(s(x947)),s(0))

                             • This constraint is kept as final constraint.

                          3. (gr(x949,x948)→^*true⟹(gr(x949,x948)→^*true⟹cond2^#(true,s(x949),s(x948))≥cond3^#(gr(s(x949),0),s(x949),s(x948)))⟹cond2^#(true,s(s(x949)),s(s(x948)))≥cond3^#(gr(s(s(x949)),0),s(s(x949)),s(s(x948))))
                             • Applying Rule "Simplify Conditions" results in
                               (cond2^#(true,s(x949),s(x948))≥cond3^#(gr(s(x949),0),s(x949),s(x948))⟹cond2^#(true,s(s(x949)),s(s(x948)))≥cond3^#(gr(s(s(x949)),0),s(s(x949)),s(s(x948))))

                             • This constraint is kept as final constraint.

                     3. (gr(x939,x938)→^*true⟹gr(x903,x902)→^*true⟹0→^*s(x938)⟹s(x902)→^*s(x939)⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥cond3^#(gr(x903,0),x903,x902))⟹∀x940 . ∀x941 . ∀x942 . ∀x943 . ∀x944 . ∀x945 . (gr(x939,x938)→^*true⟹gr(x940,x941)→^*true⟹0→^*x938⟹s(x941)→^*x939⟹∀x942 . ∀x943 . ∀x944 . ∀x945 . (gr(x940,x941)→^*true⟹0→^*x942⟹gr(x943,x942)→^*true⟹0→^*x944⟹gr(x945,x944)→^*true⟹x943→^*x940⟹x945→^*x941⟹cond2^#(true,x940,x941)≥cond3^#(gr(x940,0),x940,x941))⟹cond2^#(true,s(x940),s(x941))≥cond3^#(gr(s(x940),0),s(x940),s(x941)))⟹cond2^#(true,s(x903),s(x902))≥cond3^#(gr(s(x903),0),s(x903),s(x902)))
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x927,x926)→^*true⟹gr(x903,x902)→^*true⟹0→^*s(x926)⟹s(x903)→^*s(x927)⟹0→^*x897⟹s(x902)→^*x923⟹gr(x923,x897)→^*true⟹∀x904 . ∀x905 . ∀x906 . ∀x907 . (gr(x903,x902)→^*true⟹0→^*x904⟹gr(x905,x904)→^*true⟹0→^*x906⟹gr(x907,x906)→^*true⟹x905→^*x903⟹x907→^*x902⟹cond2^#(true,x903,x902)≥cond3^#(gr(x903,0),x903,x902))⟹∀x928 . ∀x929 . ∀x930 . ∀x931 . ∀x932 . ∀x933 . ∀x934 . ∀x935 . (gr(x927,x926)→^*true⟹gr(x928,x929)→^*true⟹0→^*x926⟹s(x928)→^*x927⟹0→^*x930⟹s(x929)→^*x931⟹gr(x931,x930)→^*true⟹∀x932 . ∀x933 . ∀x934 . ∀x935 . (gr(x928,x929)→^*true⟹0→^*x932⟹gr(x933,x932)→^*true⟹0→^*x934⟹gr(x935,x934)→^*true⟹x933→^*x928⟹x935→^*x929⟹cond2^#(true,x928,x929)≥cond3^#(gr(x928,0),x928,x929))⟹cond2^#(true,s(x928),s(x929))≥cond3^#(gr(s(x928),0),s(x928),s(x929)))⟹cond2^#(true,s(x903),s(x902))≥cond3^#(gr(s(x903),0),s(x903),s(x902)))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*true⟹cond2^#(true,x58,x59)≥cond3^#(gr(x58,0),x58,x59))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (false→^*true⟹cond2^#(true,x58,x59)≥cond3^#(gr(x58,0),x58,x59))
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond1^#(and(gr(x68,0),gr(x69,0)),x68,x69)→^*cond1^#(true,x70,x71)⟹cond2^#(gr(x70,x71),x70,x71)→^*cond2^#(true,x72,x73)⟹cond2^#(true,x72,x73)≥cond3^#(gr(x72,0),x72,x73))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (and(gr(x68,0),gr(x69,0))→^*true⟹x68→^*x70⟹x69→^*x71⟹cond2^#(gr(x70,x71),x70,x71)→^*cond2^#(true,x72,x73)⟹cond2^#(true,x72,x73)≥cond3^#(gr(x72,0),x72,x73))
    • Applying Rule "Same Constructor" results in
      (and(gr(x68,0),gr(x69,0))→^*true⟹x68→^*x70⟹x69→^*x71⟹gr(x70,x71)→^*true⟹x70→^*x72⟹x71→^*x73⟹cond2^#(true,x72,x73)≥cond3^#(gr(x72,0),x72,x73))
    • Applying Rule "Variable in Equation" allows to substitute x72 by x70 which results in
      (and(gr(x68,0),gr(x69,0))→^*true⟹x68→^*x70⟹x69→^*x71⟹gr(x70,x71)→^*true⟹x71→^*x73⟹cond2^#(true,x70,x73)≥cond3^#(gr(x70,0),x70,x73))
    • Applying Rule "Variable in Equation" allows to substitute x73 by x71 which results in
      (and(gr(x68,0),gr(x69,0))→^*true⟹x68→^*x70⟹x69→^*x71⟹gr(x70,x71)→^*true⟹cond2^#(true,x70,x71)≥cond3^#(gr(x70,0),x70,x71))
    • Applying Rule "Introduce fresh variable" results in
      (x68→^*x70⟹x69→^*x71⟹gr(x70,x71)→^*true⟹gr(x69,0)→^*x951⟹and(gr(x68,0),x951)→^*true⟹cond2^#(true,x70,x71)≥cond3^#(gr(x70,0),x70,x71))
    • Applying Rule "Introduce fresh variable" results in
      (gr(x68,0)→^*x950⟹gr(x69,0)→^*x951⟹and(x950,x951)→^*true⟹x68→^*x70⟹x69→^*x71⟹gr(x70,x71)→^*true⟹cond2^#(true,x70,x71)≥cond3^#(gr(x70,0),x70,x71))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x952⟹gr(x68,x952)→^*x950⟹gr(x69,0)→^*x951⟹and(x950,x951)→^*true⟹x68→^*x70⟹x69→^*x71⟹gr(x70,x71)→^*true⟹cond2^#(true,x70,x71)≥cond3^#(gr(x70,0),x70,x71))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x952⟹gr(x68,x952)→^*x950⟹0→^*x953⟹gr(x69,x953)→^*x951⟹and(x950,x951)→^*true⟹x68→^*x70⟹x69→^*x71⟹gr(x70,x71)→^*true⟹cond2^#(true,x70,x71)≥cond3^#(gr(x70,0),x70,x71))
    • Applying Rule "Induction" on and(x950,x951)→^*true results in the following 3 new constraints.
      1. (true→^*true⟹0→^*x952⟹gr(x68,x952)→^*true⟹0→^*x953⟹gr(x69,x953)→^*true⟹x68→^*x70⟹x69→^*x71⟹gr(x70,x71)→^*true⟹cond2^#(true,x70,x71)≥cond3^#(gr(x70,0),x70,x71))
         • Applying Rule "Same Constructor" results in
           (0→^*x952⟹gr(x68,x952)→^*true⟹0→^*x953⟹gr(x69,x953)→^*true⟹x68→^*x70⟹x69→^*x71⟹gr(x70,x71)→^*true⟹cond2^#(true,x70,x71)≥cond3^#(gr(x70,0),x70,x71))
         • Applying Rule "Induction" on gr(x70,x71)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond2^#(true,0,x956)≥cond3^#(gr(0,0),0,x956))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹0→^*x952⟹gr(x68,x952)→^*true⟹0→^*x953⟹gr(x69,x953)→^*true⟹x68→^*s(x957)⟹x69→^*0⟹cond2^#(true,s(x957),0)≥cond3^#(gr(s(x957),0),s(x957),0))
              • Applying Rule "Same Constructor" results in
                (0→^*x952⟹gr(x68,x952)→^*true⟹0→^*x953⟹gr(x69,x953)→^*true⟹x68→^*s(x957)⟹x69→^*0⟹cond2^#(true,s(x957),0)≥cond3^#(gr(s(x957),0),s(x957),0))
              • Applying Rule "Variable in Equation" allows to substitute x68 by s(x957) which results in
                (0→^*x952⟹gr(s(x957),x952)→^*true⟹0→^*x953⟹gr(x69,x953)→^*true⟹x69→^*0⟹cond2^#(true,s(x957),0)≥cond3^#(gr(s(x957),0),s(x957),0))
              • Applying Rule "Variable in Equation" allows to substitute x69 by 0 which results in
                (0→^*x952⟹gr(s(x957),x952)→^*true⟹0→^*x953⟹gr(0,x953)→^*true⟹cond2^#(true,s(x957),0)≥cond3^#(gr(s(x957),0),s(x957),0))
              • Applying Rule "Introduce fresh variable" results in
                (0→^*x952⟹s(x957)→^*x964⟹gr(x964,x952)→^*true⟹0→^*x953⟹gr(0,x953)→^*true⟹cond2^#(true,s(x957),0)≥cond3^#(gr(s(x957),0),s(x957),0))
              • Applying Rule "Introduce fresh variable" results in
                (0→^*x952⟹s(x957)→^*x964⟹gr(x964,x952)→^*true⟹0→^*x953⟹0→^*x965⟹gr(x965,x953)→^*true⟹cond2^#(true,s(x957),0)≥cond3^#(gr(s(x957),0),s(x957),0))
              • Applying Rule "Induction" on gr(x964,x952)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond2^#(true,s(x957),0)≥cond3^#(gr(s(x957),0),s(x957),0))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹0→^*0⟹s(x957)→^*s(x967)⟹0→^*x953⟹0→^*x965⟹gr(x965,x953)→^*true⟹cond2^#(true,s(x957),0)≥cond3^#(gr(s(x957),0),s(x957),0))
                   • Applying Rule "Same Constructor" results in
                     (0→^*0⟹s(x957)→^*s(x967)⟹0→^*x953⟹0→^*x965⟹gr(x965,x953)→^*true⟹cond2^#(true,s(x957),0)≥cond3^#(gr(s(x957),0),s(x957),0))
                   • Applying Rule "Same Constructor" results in
                     (s(x957)→^*s(x967)⟹0→^*x953⟹0→^*x965⟹gr(x965,x953)→^*true⟹cond2^#(true,s(x957),0)≥cond3^#(gr(s(x957),0),s(x957),0))
                   • Applying Rule "Same Constructor" results in
                     (x957→^*x967⟹0→^*x953⟹0→^*x965⟹gr(x965,x953)→^*true⟹cond2^#(true,s(x957),0)≥cond3^#(gr(s(x957),0),s(x957),0))
                   • Applying Rule "Delete Conditions" results in
                     (0→^*x953⟹0→^*x965⟹gr(x965,x953)→^*true⟹cond2^#(true,s(x957),0)≥cond3^#(gr(s(x957),0),s(x957),0))
                   • Applying Rule "Induction" on gr(x965,x953)→^*true results in the following 3 new constraints.
                     1. (false→^*true⟹cond2^#(true,s(x957),0)≥cond3^#(gr(s(x957),0),s(x957),0))
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                     2. (true→^*true⟹0→^*0⟹0→^*s(x974)⟹cond2^#(true,s(x957),0)≥cond3^#(gr(s(x957),0),s(x957),0))
                        • Applying Rule "Same Constructor" results in
                          (0→^*0⟹0→^*s(x974)⟹cond2^#(true,s(x957),0)≥cond3^#(gr(s(x957),0),s(x957),0))
                        • Applying Rule "Same Constructor" results in
                          (0→^*s(x974)⟹cond2^#(true,s(x957),0)≥cond3^#(gr(s(x957),0),s(x957),0))
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                     3. (gr(x976,x975)→^*true⟹0→^*s(x975)⟹0→^*s(x976)⟹∀x977 . (gr(x976,x975)→^*true⟹0→^*x975⟹0→^*x976⟹cond2^#(true,s(x977),0)≥cond3^#(gr(s(x977),0),s(x977),0))⟹cond2^#(true,s(x957),0)≥cond3^#(gr(s(x957),0),s(x957),0))
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x969,x968)→^*true⟹0→^*s(x968)⟹s(x957)→^*s(x969)⟹0→^*x953⟹0→^*x965⟹gr(x965,x953)→^*true⟹∀x970 . ∀x971 . ∀x972 . (gr(x969,x968)→^*true⟹0→^*x968⟹s(x970)→^*x969⟹0→^*x971⟹0→^*x972⟹gr(x972,x971)→^*true⟹cond2^#(true,s(x970),0)≥cond3^#(gr(s(x970),0),s(x970),0))⟹cond2^#(true,s(x957),0)≥cond3^#(gr(s(x957),0),s(x957),0))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x959,x958)→^*true⟹0→^*x952⟹gr(x68,x952)→^*true⟹0→^*x953⟹gr(x69,x953)→^*true⟹x68→^*s(x959)⟹x69→^*s(x958)⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥cond3^#(gr(x959,0),x959,x958))⟹cond2^#(true,s(x959),s(x958))≥cond3^#(gr(s(x959),0),s(x959),s(x958)))
              • Applying Rule "Variable in Equation" allows to substitute x68 by s(x959) which results in
                (gr(x959,x958)→^*true⟹0→^*x952⟹gr(s(x959),x952)→^*true⟹0→^*x953⟹gr(x69,x953)→^*true⟹x69→^*s(x958)⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥cond3^#(gr(x959,0),x959,x958))⟹cond2^#(true,s(x959),s(x958))≥cond3^#(gr(s(x959),0),s(x959),s(x958)))
              • Applying Rule "Variable in Equation" allows to substitute x69 by s(x958) which results in
                (gr(x959,x958)→^*true⟹0→^*x952⟹gr(s(x959),x952)→^*true⟹0→^*x953⟹gr(s(x958),x953)→^*true⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥cond3^#(gr(x959,0),x959,x958))⟹cond2^#(true,s(x959),s(x958))≥cond3^#(gr(s(x959),0),s(x959),s(x958)))
              • Applying Rule "Introduce fresh variable" results in
                (gr(x959,x958)→^*true⟹0→^*x952⟹s(x959)→^*x978⟹gr(x978,x952)→^*true⟹0→^*x953⟹gr(s(x958),x953)→^*true⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥cond3^#(gr(x959,0),x959,x958))⟹cond2^#(true,s(x959),s(x958))≥cond3^#(gr(s(x959),0),s(x959),s(x958)))
              • Applying Rule "Introduce fresh variable" results in
                (gr(x959,x958)→^*true⟹0→^*x952⟹s(x959)→^*x978⟹gr(x978,x952)→^*true⟹0→^*x953⟹s(x958)→^*x979⟹gr(x979,x953)→^*true⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥cond3^#(gr(x959,0),x959,x958))⟹cond2^#(true,s(x959),s(x958))≥cond3^#(gr(s(x959),0),s(x959),s(x958)))
              • Applying Rule "Induction" on gr(x978,x952)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond2^#(true,s(x959),s(x958))≥cond3^#(gr(s(x959),0),s(x959),s(x958)))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹gr(x959,x958)→^*true⟹0→^*0⟹s(x959)→^*s(x981)⟹0→^*x953⟹s(x958)→^*x979⟹gr(x979,x953)→^*true⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥cond3^#(gr(x959,0),x959,x958))⟹cond2^#(true,s(x959),s(x958))≥cond3^#(gr(s(x959),0),s(x959),s(x958)))
                   • Applying Rule "Same Constructor" results in
                     (gr(x959,x958)→^*true⟹0→^*0⟹s(x959)→^*s(x981)⟹0→^*x953⟹s(x958)→^*x979⟹gr(x979,x953)→^*true⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥cond3^#(gr(x959,0),x959,x958))⟹cond2^#(true,s(x959),s(x958))≥cond3^#(gr(s(x959),0),s(x959),s(x958)))
                   • Applying Rule "Same Constructor" results in
                     (gr(x959,x958)→^*true⟹s(x959)→^*s(x981)⟹0→^*x953⟹s(x958)→^*x979⟹gr(x979,x953)→^*true⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥cond3^#(gr(x959,0),x959,x958))⟹cond2^#(true,s(x959),s(x958))≥cond3^#(gr(s(x959),0),s(x959),s(x958)))
                   • Applying Rule "Same Constructor" results in
                     (gr(x959,x958)→^*true⟹x959→^*x981⟹0→^*x953⟹s(x958)→^*x979⟹gr(x979,x953)→^*true⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥cond3^#(gr(x959,0),x959,x958))⟹cond2^#(true,s(x959),s(x958))≥cond3^#(gr(s(x959),0),s(x959),s(x958)))
                   • Applying Rule "Delete Conditions" results in
                     (gr(x959,x958)→^*true⟹0→^*x953⟹s(x958)→^*x979⟹gr(x979,x953)→^*true⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥cond3^#(gr(x959,0),x959,x958))⟹cond2^#(true,s(x959),s(x958))≥cond3^#(gr(s(x959),0),s(x959),s(x958)))
                   • Applying Rule "Induction" on gr(x979,x953)→^*true results in the following 3 new constraints.
                     1. (false→^*true⟹cond2^#(true,s(x959),s(x958))≥cond3^#(gr(s(x959),0),s(x959),s(x958)))
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                     2. (true→^*true⟹gr(x959,x958)→^*true⟹0→^*0⟹s(x958)→^*s(x993)⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥cond3^#(gr(x959,0),x959,x958))⟹cond2^#(true,s(x959),s(x958))≥cond3^#(gr(s(x959),0),s(x959),s(x958)))
                        • Applying Rule "Same Constructor" results in
                          (gr(x959,x958)→^*true⟹0→^*0⟹s(x958)→^*s(x993)⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥cond3^#(gr(x959,0),x959,x958))⟹cond2^#(true,s(x959),s(x958))≥cond3^#(gr(s(x959),0),s(x959),s(x958)))
                        • Applying Rule "Same Constructor" results in
                          (gr(x959,x958)→^*true⟹s(x958)→^*s(x993)⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥cond3^#(gr(x959,0),x959,x958))⟹cond2^#(true,s(x959),s(x958))≥cond3^#(gr(s(x959),0),s(x959),s(x958)))
                        • Applying Rule "Same Constructor" results in
                          (gr(x959,x958)→^*true⟹x958→^*x993⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥cond3^#(gr(x959,0),x959,x958))⟹cond2^#(true,s(x959),s(x958))≥cond3^#(gr(s(x959),0),s(x959),s(x958)))
                        • Applying Rule "Delete Conditions" results in
                          (gr(x959,x958)→^*true⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥cond3^#(gr(x959,0),x959,x958))⟹cond2^#(true,s(x959),s(x958))≥cond3^#(gr(s(x959),0),s(x959),s(x958)))
                        • Applying Rule "Delete Conditions" results in
                          (gr(x959,x958)→^*true⟹cond2^#(true,s(x959),s(x958))≥cond3^#(gr(s(x959),0),s(x959),s(x958)))
                        • Applying Rule "Induction" on gr(x959,x958)→^*true results in the following 3 new constraints.
                          1. (false→^*true⟹cond2^#(true,s(0),s(x1002))≥cond3^#(gr(s(0),0),s(0),s(x1002)))
                             • Applying Rule "Different Constructors" allows to drop this constraint.
                          2. (true→^*true⟹cond2^#(true,s(s(x1003)),s(0))≥cond3^#(gr(s(s(x1003)),0),s(s(x1003)),s(0)))
                             • Applying Rule "Same Constructor" results in
                               cond2^#(true,s(s(x1003)),s(0))≥cond3^#(gr(s(s(x1003)),0),s(s(x1003)),s(0))

                             • This constraint is kept as final constraint.

                          3. (gr(x1005,x1004)→^*true⟹(gr(x1005,x1004)→^*true⟹cond2^#(true,s(x1005),s(x1004))≥cond3^#(gr(s(x1005),0),s(x1005),s(x1004)))⟹cond2^#(true,s(s(x1005)),s(s(x1004)))≥cond3^#(gr(s(s(x1005)),0),s(s(x1005)),s(s(x1004))))
                             • Applying Rule "Simplify Conditions" results in
                               (cond2^#(true,s(x1005),s(x1004))≥cond3^#(gr(s(x1005),0),s(x1005),s(x1004))⟹cond2^#(true,s(s(x1005)),s(s(x1004)))≥cond3^#(gr(s(s(x1005)),0),s(s(x1005)),s(s(x1004))))

                             • This constraint is kept as final constraint.

                     3. (gr(x995,x994)→^*true⟹gr(x959,x958)→^*true⟹0→^*s(x994)⟹s(x958)→^*s(x995)⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥cond3^#(gr(x959,0),x959,x958))⟹∀x996 . ∀x997 . ∀x998 . ∀x999 . ∀x1000 . ∀x1001 . (gr(x995,x994)→^*true⟹gr(x996,x997)→^*true⟹0→^*x994⟹s(x997)→^*x995⟹∀x998 . ∀x999 . ∀x1000 . ∀x1001 . (gr(x996,x997)→^*true⟹0→^*x998⟹gr(x999,x998)→^*true⟹0→^*x1000⟹gr(x1001,x1000)→^*true⟹x999→^*x996⟹x1001→^*x997⟹cond2^#(true,x996,x997)≥cond3^#(gr(x996,0),x996,x997))⟹cond2^#(true,s(x996),s(x997))≥cond3^#(gr(s(x996),0),s(x996),s(x997)))⟹cond2^#(true,s(x959),s(x958))≥cond3^#(gr(s(x959),0),s(x959),s(x958)))
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x983,x982)→^*true⟹gr(x959,x958)→^*true⟹0→^*s(x982)⟹s(x959)→^*s(x983)⟹0→^*x953⟹s(x958)→^*x979⟹gr(x979,x953)→^*true⟹∀x960 . ∀x961 . ∀x962 . ∀x963 . (gr(x959,x958)→^*true⟹0→^*x960⟹gr(x961,x960)→^*true⟹0→^*x962⟹gr(x963,x962)→^*true⟹x961→^*x959⟹x963→^*x958⟹cond2^#(true,x959,x958)≥cond3^#(gr(x959,0),x959,x958))⟹∀x984 . ∀x985 . ∀x986 . ∀x987 . ∀x988 . ∀x989 . ∀x990 . ∀x991 . (gr(x983,x982)→^*true⟹gr(x984,x985)→^*true⟹0→^*x982⟹s(x984)→^*x983⟹0→^*x986⟹s(x985)→^*x987⟹gr(x987,x986)→^*true⟹∀x988 . ∀x989 . ∀x990 . ∀x991 . (gr(x984,x985)→^*true⟹0→^*x988⟹gr(x989,x988)→^*true⟹0→^*x990⟹gr(x991,x990)→^*true⟹x989→^*x984⟹x991→^*x985⟹cond2^#(true,x984,x985)≥cond3^#(gr(x984,0),x984,x985))⟹cond2^#(true,s(x984),s(x985))≥cond3^#(gr(s(x984),0),s(x984),s(x985)))⟹cond2^#(true,s(x959),s(x958))≥cond3^#(gr(s(x959),0),s(x959),s(x958)))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*true⟹cond2^#(true,x70,x71)≥cond3^#(gr(x70,0),x70,x71))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (false→^*true⟹cond2^#(true,x70,x71)≥cond3^#(gr(x70,0),x70,x71))
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond2^#(gr(x132,x133),x132,x133)→^*cond2^#(true,x134,x135)⟹cond3^#(gr(x134,0),x134,x135)→^*cond3^#(true,x136,x137)⟹cond3^#(true,x136,x137)≥cond3^#(gr(x136,0),p(x136),x137))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x132,x133)→^*true⟹x132→^*x134⟹x133→^*x135⟹cond3^#(gr(x134,0),x134,x135)→^*cond3^#(true,x136,x137)⟹cond3^#(true,x136,x137)≥cond3^#(gr(x136,0),p(x136),x137))
    • Applying Rule "Same Constructor" results in
      (gr(x132,x133)→^*true⟹x132→^*x134⟹x133→^*x135⟹gr(x134,0)→^*true⟹x134→^*x136⟹x135→^*x137⟹cond3^#(true,x136,x137)≥cond3^#(gr(x136,0),p(x136),x137))
    • Applying Rule "Variable in Equation" allows to substitute x135 by x133 which results in
      (gr(x132,x133)→^*true⟹x132→^*x134⟹gr(x134,0)→^*true⟹x134→^*x136⟹x133→^*x137⟹cond3^#(true,x136,x137)≥cond3^#(gr(x136,0),p(x136),x137))
    • Applying Rule "Variable in Equation" allows to substitute x136 by x134 which results in
      (gr(x132,x133)→^*true⟹x132→^*x134⟹gr(x134,0)→^*true⟹x133→^*x137⟹cond3^#(true,x134,x137)≥cond3^#(gr(x134,0),p(x134),x137))
    • Applying Rule "Variable in Equation" allows to substitute x137 by x133 which results in
      (gr(x132,x133)→^*true⟹x132→^*x134⟹gr(x134,0)→^*true⟹cond3^#(true,x134,x133)≥cond3^#(gr(x134,0),p(x134),x133))
    • Applying Rule "Introduce fresh variable" results in
      (gr(x132,x133)→^*true⟹x132→^*x134⟹0→^*x1006⟹gr(x134,x1006)→^*true⟹cond3^#(true,x134,x133)≥cond3^#(gr(x134,0),p(x134),x133))
    • Applying Rule "Induction" on gr(x132,x133)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹cond3^#(true,x134,x1007)≥cond3^#(gr(x134,0),p(x134),x1007))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹s(x1008)→^*x134⟹0→^*x1006⟹gr(x134,x1006)→^*true⟹cond3^#(true,x134,0)≥cond3^#(gr(x134,0),p(x134),0))
         • Applying Rule "Same Constructor" results in
           (s(x1008)→^*x134⟹0→^*x1006⟹gr(x134,x1006)→^*true⟹cond3^#(true,x134,0)≥cond3^#(gr(x134,0),p(x134),0))
         • Applying Rule "Induction" on gr(x134,x1006)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond3^#(true,0,0)≥cond3^#(gr(0,0),p(0),0))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹s(x1008)→^*s(x1014)⟹0→^*0⟹cond3^#(true,s(x1014),0)≥cond3^#(gr(s(x1014),0),p(s(x1014)),0))
              • Applying Rule "Same Constructor" results in
                (s(x1008)→^*s(x1014)⟹0→^*0⟹cond3^#(true,s(x1014),0)≥cond3^#(gr(s(x1014),0),p(s(x1014)),0))
              • Applying Rule "Same Constructor" results in
                (x1008→^*x1014⟹0→^*0⟹cond3^#(true,s(x1014),0)≥cond3^#(gr(s(x1014),0),p(s(x1014)),0))
              • Applying Rule "Same Constructor" results in
                (x1008→^*x1014⟹cond3^#(true,s(x1014),0)≥cond3^#(gr(s(x1014),0),p(s(x1014)),0))
              • Applying Rule "Variable in Equation" allows to substitute x1014 by x1008 which results in
                cond3^#(true,s(x1008),0)≥cond3^#(gr(s(x1008),0),p(s(x1008)),0)

              • This constraint is kept as final constraint.

           3. (gr(x1016,x1015)→^*true⟹s(x1008)→^*s(x1016)⟹0→^*s(x1015)⟹∀x1017 . (gr(x1016,x1015)→^*true⟹s(x1017)→^*x1016⟹0→^*x1015⟹cond3^#(true,x1016,0)≥cond3^#(gr(x1016,0),p(x1016),0))⟹cond3^#(true,s(x1016),0)≥cond3^#(gr(s(x1016),0),p(s(x1016)),0))
              • Applying Rule "Same Constructor" results in
                (gr(x1016,x1015)→^*true⟹x1008→^*x1016⟹0→^*s(x1015)⟹∀x1017 . (gr(x1016,x1015)→^*true⟹s(x1017)→^*x1016⟹0→^*x1015⟹cond3^#(true,x1016,0)≥cond3^#(gr(x1016,0),p(x1016),0))⟹cond3^#(true,s(x1016),0)≥cond3^#(gr(s(x1016),0),p(s(x1016)),0))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x1010,x1009)→^*true⟹s(x1010)→^*x134⟹0→^*x1006⟹gr(x134,x1006)→^*true⟹∀x1011 . ∀x1012 . (gr(x1010,x1009)→^*true⟹x1010→^*x1011⟹0→^*x1012⟹gr(x1011,x1012)→^*true⟹cond3^#(true,x1011,x1009)≥cond3^#(gr(x1011,0),p(x1011),x1009))⟹cond3^#(true,x134,s(x1009))≥cond3^#(gr(x134,0),p(x134),s(x1009)))
         • Applying Rule "Induction" on gr(x134,x1006)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond3^#(true,0,s(x1009))≥cond3^#(gr(0,0),p(0),s(x1009)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹gr(x1010,x1009)→^*true⟹s(x1010)→^*s(x1019)⟹0→^*0⟹∀x1011 . ∀x1012 . (gr(x1010,x1009)→^*true⟹x1010→^*x1011⟹0→^*x1012⟹gr(x1011,x1012)→^*true⟹cond3^#(true,x1011,x1009)≥cond3^#(gr(x1011,0),p(x1011),x1009))⟹cond3^#(true,s(x1019),s(x1009))≥cond3^#(gr(s(x1019),0),p(s(x1019)),s(x1009)))
              • Applying Rule "Same Constructor" results in
                (gr(x1010,x1009)→^*true⟹s(x1010)→^*s(x1019)⟹0→^*0⟹∀x1011 . ∀x1012 . (gr(x1010,x1009)→^*true⟹x1010→^*x1011⟹0→^*x1012⟹gr(x1011,x1012)→^*true⟹cond3^#(true,x1011,x1009)≥cond3^#(gr(x1011,0),p(x1011),x1009))⟹cond3^#(true,s(x1019),s(x1009))≥cond3^#(gr(s(x1019),0),p(s(x1019)),s(x1009)))
              • Applying Rule "Same Constructor" results in
                (gr(x1010,x1009)→^*true⟹x1010→^*x1019⟹0→^*0⟹∀x1011 . ∀x1012 . (gr(x1010,x1009)→^*true⟹x1010→^*x1011⟹0→^*x1012⟹gr(x1011,x1012)→^*true⟹cond3^#(true,x1011,x1009)≥cond3^#(gr(x1011,0),p(x1011),x1009))⟹cond3^#(true,s(x1019),s(x1009))≥cond3^#(gr(s(x1019),0),p(s(x1019)),s(x1009)))
              • Applying Rule "Same Constructor" results in
                (gr(x1010,x1009)→^*true⟹x1010→^*x1019⟹∀x1011 . ∀x1012 . (gr(x1010,x1009)→^*true⟹x1010→^*x1011⟹0→^*x1012⟹gr(x1011,x1012)→^*true⟹cond3^#(true,x1011,x1009)≥cond3^#(gr(x1011,0),p(x1011),x1009))⟹cond3^#(true,s(x1019),s(x1009))≥cond3^#(gr(s(x1019),0),p(s(x1019)),s(x1009)))
              • Applying Rule "Variable in Equation" allows to substitute x1019 by x1010 which results in
                (gr(x1010,x1009)→^*true⟹∀x1011 . ∀x1012 . (gr(x1010,x1009)→^*true⟹x1010→^*x1011⟹0→^*x1012⟹gr(x1011,x1012)→^*true⟹cond3^#(true,x1011,x1009)≥cond3^#(gr(x1011,0),p(x1011),x1009))⟹cond3^#(true,s(x1010),s(x1009))≥cond3^#(gr(s(x1010),0),p(s(x1010)),s(x1009)))
              • Applying Rule "Delete Conditions" results in
                (gr(x1010,x1009)→^*true⟹cond3^#(true,s(x1010),s(x1009))≥cond3^#(gr(s(x1010),0),p(s(x1010)),s(x1009)))
              • Applying Rule "Induction" on gr(x1010,x1009)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond3^#(true,s(0),s(x1026))≥cond3^#(gr(s(0),0),p(s(0)),s(x1026)))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹cond3^#(true,s(s(x1027)),s(0))≥cond3^#(gr(s(s(x1027)),0),p(s(s(x1027))),s(0)))
                   • Applying Rule "Same Constructor" results in
                     cond3^#(true,s(s(x1027)),s(0))≥cond3^#(gr(s(s(x1027)),0),p(s(s(x1027))),s(0))

                   • This constraint is kept as final constraint.

                3. (gr(x1029,x1028)→^*true⟹(gr(x1029,x1028)→^*true⟹cond3^#(true,s(x1029),s(x1028))≥cond3^#(gr(s(x1029),0),p(s(x1029)),s(x1028)))⟹cond3^#(true,s(s(x1029)),s(s(x1028)))≥cond3^#(gr(s(s(x1029)),0),p(s(s(x1029))),s(s(x1028))))
                   • Applying Rule "Simplify Conditions" results in
                     (cond3^#(true,s(x1029),s(x1028))≥cond3^#(gr(s(x1029),0),p(s(x1029)),s(x1028))⟹cond3^#(true,s(s(x1029)),s(s(x1028)))≥cond3^#(gr(s(s(x1029)),0),p(s(s(x1029))),s(s(x1028))))

                   • This constraint is kept as final constraint.

           3. (gr(x1021,x1020)→^*true⟹gr(x1010,x1009)→^*true⟹s(x1010)→^*s(x1021)⟹0→^*s(x1020)⟹∀x1011 . ∀x1012 . (gr(x1010,x1009)→^*true⟹x1010→^*x1011⟹0→^*x1012⟹gr(x1011,x1012)→^*true⟹cond3^#(true,x1011,x1009)≥cond3^#(gr(x1011,0),p(x1011),x1009))⟹∀x1022 . ∀x1023 . ∀x1024 . ∀x1025 . (gr(x1021,x1020)→^*true⟹gr(x1022,x1023)→^*true⟹s(x1022)→^*x1021⟹0→^*x1020⟹∀x1024 . ∀x1025 . (gr(x1022,x1023)→^*true⟹x1022→^*x1024⟹0→^*x1025⟹gr(x1024,x1025)→^*true⟹cond3^#(true,x1024,x1023)≥cond3^#(gr(x1024,0),p(x1024),x1023))⟹cond3^#(true,x1021,s(x1023))≥cond3^#(gr(x1021,0),p(x1021),s(x1023)))⟹cond3^#(true,s(x1021),s(x1009))≥cond3^#(gr(s(x1021),0),p(s(x1021)),s(x1009)))
              • Applying Rule "Same Constructor" results in
                (gr(x1021,x1020)→^*true⟹gr(x1010,x1009)→^*true⟹x1010→^*x1021⟹0→^*s(x1020)⟹∀x1011 . ∀x1012 . (gr(x1010,x1009)→^*true⟹x1010→^*x1011⟹0→^*x1012⟹gr(x1011,x1012)→^*true⟹cond3^#(true,x1011,x1009)≥cond3^#(gr(x1011,0),p(x1011),x1009))⟹∀x1022 . ∀x1023 . ∀x1024 . ∀x1025 . (gr(x1021,x1020)→^*true⟹gr(x1022,x1023)→^*true⟹s(x1022)→^*x1021⟹0→^*x1020⟹∀x1024 . ∀x1025 . (gr(x1022,x1023)→^*true⟹x1022→^*x1024⟹0→^*x1025⟹gr(x1024,x1025)→^*true⟹cond3^#(true,x1024,x1023)≥cond3^#(gr(x1024,0),p(x1024),x1023))⟹cond3^#(true,x1021,s(x1023))≥cond3^#(gr(x1021,0),p(x1021),s(x1023)))⟹cond3^#(true,s(x1021),s(x1009))≥cond3^#(gr(s(x1021),0),p(s(x1021)),s(x1009)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond3^#(gr(x144,0),x144,x145)→^*cond3^#(true,x146,x147)⟹cond3^#(gr(x146,0),p(x146),x147)→^*cond3^#(true,x148,x149)⟹cond3^#(true,x148,x149)≥cond3^#(gr(x148,0),p(x148),x149))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x144,0)→^*true⟹x144→^*x146⟹x145→^*x147⟹cond3^#(gr(x146,0),p(x146),x147)→^*cond3^#(true,x148,x149)⟹cond3^#(true,x148,x149)≥cond3^#(gr(x148,0),p(x148),x149))
    • Applying Rule "Same Constructor" results in
      (gr(x144,0)→^*true⟹x144→^*x146⟹x145→^*x147⟹gr(x146,0)→^*true⟹p(x146)→^*x148⟹x147→^*x149⟹cond3^#(true,x148,x149)≥cond3^#(gr(x148,0),p(x148),x149))
    • Applying Rule "Variable in Equation" allows to substitute x147 by x145 which results in
      (gr(x144,0)→^*true⟹x144→^*x146⟹gr(x146,0)→^*true⟹p(x146)→^*x148⟹x145→^*x149⟹cond3^#(true,x148,x149)≥cond3^#(gr(x148,0),p(x148),x149))
    • Applying Rule "Variable in Equation" allows to substitute x149 by x145 which results in
      (gr(x144,0)→^*true⟹x144→^*x146⟹gr(x146,0)→^*true⟹p(x146)→^*x148⟹cond3^#(true,x148,x145)≥cond3^#(gr(x148,0),p(x148),x145))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1030⟹gr(x144,x1030)→^*true⟹x144→^*x146⟹gr(x146,0)→^*true⟹p(x146)→^*x148⟹cond3^#(true,x148,x145)≥cond3^#(gr(x148,0),p(x148),x145))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1030⟹gr(x144,x1030)→^*true⟹x144→^*x146⟹0→^*x1031⟹gr(x146,x1031)→^*true⟹p(x146)→^*x148⟹cond3^#(true,x148,x145)≥cond3^#(gr(x148,0),p(x148),x145))
    • Applying Rule "Induction" on gr(x144,x1030)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹cond3^#(true,x148,x145)≥cond3^#(gr(x148,0),p(x148),x145))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹0→^*0⟹s(x1033)→^*x146⟹0→^*x1031⟹gr(x146,x1031)→^*true⟹p(x146)→^*x148⟹cond3^#(true,x148,x145)≥cond3^#(gr(x148,0),p(x148),x145))
         • Applying Rule "Same Constructor" results in
           (0→^*0⟹s(x1033)→^*x146⟹0→^*x1031⟹gr(x146,x1031)→^*true⟹p(x146)→^*x148⟹cond3^#(true,x148,x145)≥cond3^#(gr(x148,0),p(x148),x145))
         • Applying Rule "Same Constructor" results in
           (s(x1033)→^*x146⟹0→^*x1031⟹gr(x146,x1031)→^*true⟹p(x146)→^*x148⟹cond3^#(true,x148,x145)≥cond3^#(gr(x148,0),p(x148),x145))
         • Applying Rule "Induction" on gr(x146,x1031)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond3^#(true,x148,x145)≥cond3^#(gr(x148,0),p(x148),x145))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹s(x1033)→^*s(x1041)⟹0→^*0⟹p(s(x1041))→^*x148⟹cond3^#(true,x148,x145)≥cond3^#(gr(x148,0),p(x148),x145))
              • Applying Rule "Same Constructor" results in
                (s(x1033)→^*s(x1041)⟹0→^*0⟹p(s(x1041))→^*x148⟹cond3^#(true,x148,x145)≥cond3^#(gr(x148,0),p(x148),x145))
              • Applying Rule "Same Constructor" results in
                (x1033→^*x1041⟹0→^*0⟹p(s(x1041))→^*x148⟹cond3^#(true,x148,x145)≥cond3^#(gr(x148,0),p(x148),x145))
              • Applying Rule "Same Constructor" results in
                (x1033→^*x1041⟹p(s(x1041))→^*x148⟹cond3^#(true,x148,x145)≥cond3^#(gr(x148,0),p(x148),x145))
              • Applying Rule "Variable in Equation" allows to substitute x1033 by x1041 which results in
                (p(s(x1041))→^*x148⟹cond3^#(true,x148,x145)≥cond3^#(gr(x148,0),p(x148),x145))
              • Applying Rule "Introduce fresh variable" results in
                (s(x1041)→^*x1047⟹p(x1047)→^*x148⟹cond3^#(true,x148,x145)≥cond3^#(gr(x148,0),p(x148),x145))
              • Applying Rule "Variable in Equation" allows to substitute x1047 by s(x1041) which results in
                (p(s(x1041))→^*x148⟹cond3^#(true,x148,x145)≥cond3^#(gr(x148,0),p(x148),x145))
              • Applying Rule "Delete Conditions" results in
                cond3^#(true,x148,x145)≥cond3^#(gr(x148,0),p(x148),x145)

              • This constraint is kept as final constraint.

           3. (gr(x1043,x1042)→^*true⟹s(x1033)→^*s(x1043)⟹0→^*s(x1042)⟹p(s(x1043))→^*x148⟹∀x1044 . ∀x1045 . ∀x1046 . (gr(x1043,x1042)→^*true⟹s(x1044)→^*x1043⟹0→^*x1042⟹p(x1043)→^*x1045⟹cond3^#(true,x1045,x1046)≥cond3^#(gr(x1045,0),p(x1045),x1046))⟹cond3^#(true,x148,x145)≥cond3^#(gr(x148,0),p(x148),x145))
              • Applying Rule "Same Constructor" results in
                (gr(x1043,x1042)→^*true⟹x1033→^*x1043⟹0→^*s(x1042)⟹p(s(x1043))→^*x148⟹∀x1044 . ∀x1045 . ∀x1046 . (gr(x1043,x1042)→^*true⟹s(x1044)→^*x1043⟹0→^*x1042⟹p(x1043)→^*x1045⟹cond3^#(true,x1045,x1046)≥cond3^#(gr(x1045,0),p(x1045),x1046))⟹cond3^#(true,x148,x145)≥cond3^#(gr(x148,0),p(x148),x145))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x1035,x1034)→^*true⟹0→^*s(x1034)⟹s(x1035)→^*x146⟹0→^*x1031⟹gr(x146,x1031)→^*true⟹p(x146)→^*x148⟹∀x1036 . ∀x1037 . ∀x1038 . ∀x1039 . (gr(x1035,x1034)→^*true⟹0→^*x1034⟹x1035→^*x1036⟹0→^*x1037⟹gr(x1036,x1037)→^*true⟹p(x1036)→^*x1038⟹cond3^#(true,x1038,x1039)≥cond3^#(gr(x1038,0),p(x1038),x1039))⟹cond3^#(true,x148,x145)≥cond3^#(gr(x148,0),p(x148),x145))
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond3^#(gr(x150,0),p(x150),x151)→^*cond3^#(true,x152,x153)⟹cond3^#(gr(x152,0),p(x152),x153)→^*cond3^#(true,x154,x155)⟹cond3^#(true,x154,x155)≥cond3^#(gr(x154,0),p(x154),x155))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x150,0)→^*true⟹p(x150)→^*x152⟹x151→^*x153⟹cond3^#(gr(x152,0),p(x152),x153)→^*cond3^#(true,x154,x155)⟹cond3^#(true,x154,x155)≥cond3^#(gr(x154,0),p(x154),x155))
    • Applying Rule "Same Constructor" results in
      (gr(x150,0)→^*true⟹p(x150)→^*x152⟹x151→^*x153⟹gr(x152,0)→^*true⟹p(x152)→^*x154⟹x153→^*x155⟹cond3^#(true,x154,x155)≥cond3^#(gr(x154,0),p(x154),x155))
    • Applying Rule "Variable in Equation" allows to substitute x153 by x151 which results in
      (gr(x150,0)→^*true⟹p(x150)→^*x152⟹gr(x152,0)→^*true⟹p(x152)→^*x154⟹x151→^*x155⟹cond3^#(true,x154,x155)≥cond3^#(gr(x154,0),p(x154),x155))
    • Applying Rule "Variable in Equation" allows to substitute x155 by x151 which results in
      (gr(x150,0)→^*true⟹p(x150)→^*x152⟹gr(x152,0)→^*true⟹p(x152)→^*x154⟹cond3^#(true,x154,x151)≥cond3^#(gr(x154,0),p(x154),x151))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1048⟹gr(x150,x1048)→^*true⟹p(x150)→^*x152⟹gr(x152,0)→^*true⟹p(x152)→^*x154⟹cond3^#(true,x154,x151)≥cond3^#(gr(x154,0),p(x154),x151))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1048⟹gr(x150,x1048)→^*true⟹p(x150)→^*x152⟹0→^*x1049⟹gr(x152,x1049)→^*true⟹p(x152)→^*x154⟹cond3^#(true,x154,x151)≥cond3^#(gr(x154,0),p(x154),x151))
    • Applying Rule "Induction" on gr(x150,x1048)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹cond3^#(true,x154,x151)≥cond3^#(gr(x154,0),p(x154),x151))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹0→^*0⟹p(s(x1051))→^*x152⟹0→^*x1049⟹gr(x152,x1049)→^*true⟹p(x152)→^*x154⟹cond3^#(true,x154,x151)≥cond3^#(gr(x154,0),p(x154),x151))
         • Applying Rule "Same Constructor" results in
           (0→^*0⟹p(s(x1051))→^*x152⟹0→^*x1049⟹gr(x152,x1049)→^*true⟹p(x152)→^*x154⟹cond3^#(true,x154,x151)≥cond3^#(gr(x154,0),p(x154),x151))
         • Applying Rule "Same Constructor" results in
           (p(s(x1051))→^*x152⟹0→^*x1049⟹gr(x152,x1049)→^*true⟹p(x152)→^*x154⟹cond3^#(true,x154,x151)≥cond3^#(gr(x154,0),p(x154),x151))
         • Applying Rule "Introduce fresh variable" results in
           (s(x1051)→^*x1058⟹p(x1058)→^*x152⟹0→^*x1049⟹gr(x152,x1049)→^*true⟹p(x152)→^*x154⟹cond3^#(true,x154,x151)≥cond3^#(gr(x154,0),p(x154),x151))
         • Applying Rule "Induction" on gr(x152,x1049)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond3^#(true,x154,x151)≥cond3^#(gr(x154,0),p(x154),x151))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹s(x1051)→^*x1058⟹p(x1058)→^*s(x1060)⟹0→^*0⟹p(s(x1060))→^*x154⟹cond3^#(true,x154,x151)≥cond3^#(gr(x154,0),p(x154),x151))
              • Applying Rule "Same Constructor" results in
                (s(x1051)→^*x1058⟹p(x1058)→^*s(x1060)⟹0→^*0⟹p(s(x1060))→^*x154⟹cond3^#(true,x154,x151)≥cond3^#(gr(x154,0),p(x154),x151))
              • Applying Rule "Same Constructor" results in
                (s(x1051)→^*x1058⟹p(x1058)→^*s(x1060)⟹p(s(x1060))→^*x154⟹cond3^#(true,x154,x151)≥cond3^#(gr(x154,0),p(x154),x151))
              • Applying Rule "Introduce fresh variable" results in
                (s(x1051)→^*x1058⟹p(x1058)→^*s(x1060)⟹s(x1060)→^*x1067⟹p(x1067)→^*x154⟹cond3^#(true,x154,x151)≥cond3^#(gr(x154,0),p(x154),x151))
              • Applying Rule "Induction" on p(x1058)→^*s(x1060) results in the following 2 new constraints.
                1. (0→^*s(x1060)⟹cond3^#(true,x154,x151)≥cond3^#(gr(x154,0),p(x154),x151))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (x1068→^*s(x1060)⟹s(x1051)→^*s(x1068)⟹s(x1060)→^*x1067⟹p(x1067)→^*x154⟹cond3^#(true,x154,x151)≥cond3^#(gr(x154,0),p(x154),x151))
                   • Applying Rule "Same Constructor" results in
                     (x1068→^*s(x1060)⟹x1051→^*x1068⟹s(x1060)→^*x1067⟹p(x1067)→^*x154⟹cond3^#(true,x154,x151)≥cond3^#(gr(x154,0),p(x154),x151))
                   • Applying Rule "Variable in Equation" allows to substitute x1068 by x1051 which results in
                     (x1051→^*s(x1060)⟹s(x1060)→^*x1067⟹p(x1067)→^*x154⟹cond3^#(true,x154,x151)≥cond3^#(gr(x154,0),p(x154),x151))
                   • Applying Rule "Variable in Equation" allows to substitute x1051 by s(x1060) which results in
                     (s(x1060)→^*x1067⟹p(x1067)→^*x154⟹cond3^#(true,x154,x151)≥cond3^#(gr(x154,0),p(x154),x151))
                   • Applying Rule "Variable in Equation" allows to substitute x1067 by s(x1060) which results in
                     (p(s(x1060))→^*x154⟹cond3^#(true,x154,x151)≥cond3^#(gr(x154,0),p(x154),x151))
                   • Applying Rule "Delete Conditions" results in
                     cond3^#(true,x154,x151)≥cond3^#(gr(x154,0),p(x154),x151)

                   • This constraint is kept as final constraint.

           3. (gr(x1062,x1061)→^*true⟹s(x1051)→^*x1058⟹p(x1058)→^*s(x1062)⟹0→^*s(x1061)⟹p(s(x1062))→^*x154⟹∀x1063 . ∀x1064 . ∀x1065 . ∀x1066 . (gr(x1062,x1061)→^*true⟹s(x1063)→^*x1064⟹p(x1064)→^*x1062⟹0→^*x1061⟹p(x1062)→^*x1065⟹cond3^#(true,x1065,x1066)≥cond3^#(gr(x1065,0),p(x1065),x1066))⟹cond3^#(true,x154,x151)≥cond3^#(gr(x154,0),p(x154),x151))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x1053,x1052)→^*true⟹0→^*s(x1052)⟹p(s(x1053))→^*x152⟹0→^*x1049⟹gr(x152,x1049)→^*true⟹p(x152)→^*x154⟹∀x1054 . ∀x1055 . ∀x1056 . ∀x1057 . (gr(x1053,x1052)→^*true⟹0→^*x1052⟹p(x1053)→^*x1054⟹0→^*x1055⟹gr(x1054,x1055)→^*true⟹p(x1054)→^*x1056⟹cond3^#(true,x1056,x1057)≥cond3^#(gr(x1056,0),p(x1056),x1057))⟹cond3^#(true,x154,x151)≥cond3^#(gr(x154,0),p(x154),x151))
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond2^#(gr(x260,x261),x260,x261)→^*cond2^#(true,x262,x263)⟹cond3^#(gr(x262,0),x262,x263)→^*cond3^#(false,x264,x265)⟹cond3^#(false,x264,x265)≥cond1^#(and(gr(x264,0),gr(x265,0)),x264,x265))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x260,x261)→^*true⟹x260→^*x262⟹x261→^*x263⟹cond3^#(gr(x262,0),x262,x263)→^*cond3^#(false,x264,x265)⟹cond3^#(false,x264,x265)≥cond1^#(and(gr(x264,0),gr(x265,0)),x264,x265))
    • Applying Rule "Same Constructor" results in
      (gr(x260,x261)→^*true⟹x260→^*x262⟹x261→^*x263⟹gr(x262,0)→^*false⟹x262→^*x264⟹x263→^*x265⟹cond3^#(false,x264,x265)≥cond1^#(and(gr(x264,0),gr(x265,0)),x264,x265))
    • Applying Rule "Variable in Equation" allows to substitute x263 by x261 which results in
      (gr(x260,x261)→^*true⟹x260→^*x262⟹gr(x262,0)→^*false⟹x262→^*x264⟹x261→^*x265⟹cond3^#(false,x264,x265)≥cond1^#(and(gr(x264,0),gr(x265,0)),x264,x265))
    • Applying Rule "Variable in Equation" allows to substitute x264 by x262 which results in
      (gr(x260,x261)→^*true⟹x260→^*x262⟹gr(x262,0)→^*false⟹x261→^*x265⟹cond3^#(false,x262,x265)≥cond1^#(and(gr(x262,0),gr(x265,0)),x262,x265))
    • Applying Rule "Variable in Equation" allows to substitute x265 by x261 which results in
      (gr(x260,x261)→^*true⟹x260→^*x262⟹gr(x262,0)→^*false⟹cond3^#(false,x262,x261)≥cond1^#(and(gr(x262,0),gr(x261,0)),x262,x261))
    • Applying Rule "Introduce fresh variable" results in
      (gr(x260,x261)→^*true⟹x260→^*x262⟹0→^*x1069⟹gr(x262,x1069)→^*false⟹cond3^#(false,x262,x261)≥cond1^#(and(gr(x262,0),gr(x261,0)),x262,x261))
    • Applying Rule "Induction" on gr(x260,x261)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹cond3^#(false,x262,x1070)≥cond1^#(and(gr(x262,0),gr(x1070,0)),x262,x1070))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹s(x1071)→^*x262⟹0→^*x1069⟹gr(x262,x1069)→^*false⟹cond3^#(false,x262,0)≥cond1^#(and(gr(x262,0),gr(0,0)),x262,0))
         • Applying Rule "Same Constructor" results in
           (s(x1071)→^*x262⟹0→^*x1069⟹gr(x262,x1069)→^*false⟹cond3^#(false,x262,0)≥cond1^#(and(gr(x262,0),gr(0,0)),x262,0))
         • Applying Rule "Induction" on gr(x262,x1069)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹s(x1071)→^*0⟹0→^*x1076⟹cond3^#(false,0,0)≥cond1^#(and(gr(0,0),gr(0,0)),0,0))
              • Applying Rule "Same Constructor" results in
                (s(x1071)→^*0⟹0→^*x1076⟹cond3^#(false,0,0)≥cond1^#(and(gr(0,0),gr(0,0)),0,0))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond3^#(false,s(x1077),0)≥cond1^#(and(gr(s(x1077),0),gr(0,0)),s(x1077),0))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1079,x1078)→^*false⟹s(x1071)→^*s(x1079)⟹0→^*s(x1078)⟹∀x1080 . (gr(x1079,x1078)→^*false⟹s(x1080)→^*x1079⟹0→^*x1078⟹cond3^#(false,x1079,0)≥cond1^#(and(gr(x1079,0),gr(0,0)),x1079,0))⟹cond3^#(false,s(x1079),0)≥cond1^#(and(gr(s(x1079),0),gr(0,0)),s(x1079),0))
              • Applying Rule "Same Constructor" results in
                (gr(x1079,x1078)→^*false⟹x1071→^*x1079⟹0→^*s(x1078)⟹∀x1080 . (gr(x1079,x1078)→^*false⟹s(x1080)→^*x1079⟹0→^*x1078⟹cond3^#(false,x1079,0)≥cond1^#(and(gr(x1079,0),gr(0,0)),x1079,0))⟹cond3^#(false,s(x1079),0)≥cond1^#(and(gr(s(x1079),0),gr(0,0)),s(x1079),0))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x1073,x1072)→^*true⟹s(x1073)→^*x262⟹0→^*x1069⟹gr(x262,x1069)→^*false⟹∀x1074 . ∀x1075 . (gr(x1073,x1072)→^*true⟹x1073→^*x1074⟹0→^*x1075⟹gr(x1074,x1075)→^*false⟹cond3^#(false,x1074,x1072)≥cond1^#(and(gr(x1074,0),gr(x1072,0)),x1074,x1072))⟹cond3^#(false,x262,s(x1072))≥cond1^#(and(gr(x262,0),gr(s(x1072),0)),x262,s(x1072)))
         • Applying Rule "Induction" on gr(x262,x1069)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹gr(x1073,x1072)→^*true⟹s(x1073)→^*0⟹0→^*x1081⟹∀x1074 . ∀x1075 . (gr(x1073,x1072)→^*true⟹x1073→^*x1074⟹0→^*x1075⟹gr(x1074,x1075)→^*false⟹cond3^#(false,x1074,x1072)≥cond1^#(and(gr(x1074,0),gr(x1072,0)),x1074,x1072))⟹cond3^#(false,0,s(x1072))≥cond1^#(and(gr(0,0),gr(s(x1072),0)),0,s(x1072)))
              • Applying Rule "Same Constructor" results in
                (gr(x1073,x1072)→^*true⟹s(x1073)→^*0⟹0→^*x1081⟹∀x1074 . ∀x1075 . (gr(x1073,x1072)→^*true⟹x1073→^*x1074⟹0→^*x1075⟹gr(x1074,x1075)→^*false⟹cond3^#(false,x1074,x1072)≥cond1^#(and(gr(x1074,0),gr(x1072,0)),x1074,x1072))⟹cond3^#(false,0,s(x1072))≥cond1^#(and(gr(0,0),gr(s(x1072),0)),0,s(x1072)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond3^#(false,s(x1082),s(x1072))≥cond1^#(and(gr(s(x1082),0),gr(s(x1072),0)),s(x1082),s(x1072)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1084,x1083)→^*false⟹gr(x1073,x1072)→^*true⟹s(x1073)→^*s(x1084)⟹0→^*s(x1083)⟹∀x1074 . ∀x1075 . (gr(x1073,x1072)→^*true⟹x1073→^*x1074⟹0→^*x1075⟹gr(x1074,x1075)→^*false⟹cond3^#(false,x1074,x1072)≥cond1^#(and(gr(x1074,0),gr(x1072,0)),x1074,x1072))⟹∀x1085 . ∀x1086 . ∀x1087 . ∀x1088 . (gr(x1084,x1083)→^*false⟹gr(x1085,x1086)→^*true⟹s(x1085)→^*x1084⟹0→^*x1083⟹∀x1087 . ∀x1088 . (gr(x1085,x1086)→^*true⟹x1085→^*x1087⟹0→^*x1088⟹gr(x1087,x1088)→^*false⟹cond3^#(false,x1087,x1086)≥cond1^#(and(gr(x1087,0),gr(x1086,0)),x1087,x1086))⟹cond3^#(false,x1084,s(x1086))≥cond1^#(and(gr(x1084,0),gr(s(x1086),0)),x1084,s(x1086)))⟹cond3^#(false,s(x1084),s(x1072))≥cond1^#(and(gr(s(x1084),0),gr(s(x1072),0)),s(x1084),s(x1072)))
              • Applying Rule "Same Constructor" results in
                (gr(x1084,x1083)→^*false⟹gr(x1073,x1072)→^*true⟹x1073→^*x1084⟹0→^*s(x1083)⟹∀x1074 . ∀x1075 . (gr(x1073,x1072)→^*true⟹x1073→^*x1074⟹0→^*x1075⟹gr(x1074,x1075)→^*false⟹cond3^#(false,x1074,x1072)≥cond1^#(and(gr(x1074,0),gr(x1072,0)),x1074,x1072))⟹∀x1085 . ∀x1086 . ∀x1087 . ∀x1088 . (gr(x1084,x1083)→^*false⟹gr(x1085,x1086)→^*true⟹s(x1085)→^*x1084⟹0→^*x1083⟹∀x1087 . ∀x1088 . (gr(x1085,x1086)→^*true⟹x1085→^*x1087⟹0→^*x1088⟹gr(x1087,x1088)→^*false⟹cond3^#(false,x1087,x1086)≥cond1^#(and(gr(x1087,0),gr(x1086,0)),x1087,x1086))⟹cond3^#(false,x1084,s(x1086))≥cond1^#(and(gr(x1084,0),gr(s(x1086),0)),x1084,s(x1086)))⟹cond3^#(false,s(x1084),s(x1072))≥cond1^#(and(gr(s(x1084),0),gr(s(x1072),0)),s(x1084),s(x1072)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond3^#(gr(x272,0),x272,x273)→^*cond3^#(true,x274,x275)⟹cond3^#(gr(x274,0),p(x274),x275)→^*cond3^#(false,x276,x277)⟹cond3^#(false,x276,x277)≥cond1^#(and(gr(x276,0),gr(x277,0)),x276,x277))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x272,0)→^*true⟹x272→^*x274⟹x273→^*x275⟹cond3^#(gr(x274,0),p(x274),x275)→^*cond3^#(false,x276,x277)⟹cond3^#(false,x276,x277)≥cond1^#(and(gr(x276,0),gr(x277,0)),x276,x277))
    • Applying Rule "Same Constructor" results in
      (gr(x272,0)→^*true⟹x272→^*x274⟹x273→^*x275⟹gr(x274,0)→^*false⟹p(x274)→^*x276⟹x275→^*x277⟹cond3^#(false,x276,x277)≥cond1^#(and(gr(x276,0),gr(x277,0)),x276,x277))
    • Applying Rule "Variable in Equation" allows to substitute x275 by x273 which results in
      (gr(x272,0)→^*true⟹x272→^*x274⟹gr(x274,0)→^*false⟹p(x274)→^*x276⟹x273→^*x277⟹cond3^#(false,x276,x277)≥cond1^#(and(gr(x276,0),gr(x277,0)),x276,x277))
    • Applying Rule "Variable in Equation" allows to substitute x277 by x273 which results in
      (gr(x272,0)→^*true⟹x272→^*x274⟹gr(x274,0)→^*false⟹p(x274)→^*x276⟹cond3^#(false,x276,x273)≥cond1^#(and(gr(x276,0),gr(x273,0)),x276,x273))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1089⟹gr(x272,x1089)→^*true⟹x272→^*x274⟹gr(x274,0)→^*false⟹p(x274)→^*x276⟹cond3^#(false,x276,x273)≥cond1^#(and(gr(x276,0),gr(x273,0)),x276,x273))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1089⟹gr(x272,x1089)→^*true⟹x272→^*x274⟹0→^*x1090⟹gr(x274,x1090)→^*false⟹p(x274)→^*x276⟹cond3^#(false,x276,x273)≥cond1^#(and(gr(x276,0),gr(x273,0)),x276,x273))
    • Applying Rule "Induction" on gr(x272,x1089)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹cond3^#(false,x276,x273)≥cond1^#(and(gr(x276,0),gr(x273,0)),x276,x273))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹0→^*0⟹s(x1092)→^*x274⟹0→^*x1090⟹gr(x274,x1090)→^*false⟹p(x274)→^*x276⟹cond3^#(false,x276,x273)≥cond1^#(and(gr(x276,0),gr(x273,0)),x276,x273))
         • Applying Rule "Same Constructor" results in
           (0→^*0⟹s(x1092)→^*x274⟹0→^*x1090⟹gr(x274,x1090)→^*false⟹p(x274)→^*x276⟹cond3^#(false,x276,x273)≥cond1^#(and(gr(x276,0),gr(x273,0)),x276,x273))
         • Applying Rule "Same Constructor" results in
           (s(x1092)→^*x274⟹0→^*x1090⟹gr(x274,x1090)→^*false⟹p(x274)→^*x276⟹cond3^#(false,x276,x273)≥cond1^#(and(gr(x276,0),gr(x273,0)),x276,x273))
         • Applying Rule "Induction" on gr(x274,x1090)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹s(x1092)→^*0⟹0→^*x1099⟹p(0)→^*x276⟹cond3^#(false,x276,x273)≥cond1^#(and(gr(x276,0),gr(x273,0)),x276,x273))
              • Applying Rule "Same Constructor" results in
                (s(x1092)→^*0⟹0→^*x1099⟹p(0)→^*x276⟹cond3^#(false,x276,x273)≥cond1^#(and(gr(x276,0),gr(x273,0)),x276,x273))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond3^#(false,x276,x273)≥cond1^#(and(gr(x276,0),gr(x273,0)),x276,x273))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1102,x1101)→^*false⟹s(x1092)→^*s(x1102)⟹0→^*s(x1101)⟹p(s(x1102))→^*x276⟹∀x1103 . ∀x1104 . ∀x1105 . (gr(x1102,x1101)→^*false⟹s(x1103)→^*x1102⟹0→^*x1101⟹p(x1102)→^*x1104⟹cond3^#(false,x1104,x1105)≥cond1^#(and(gr(x1104,0),gr(x1105,0)),x1104,x1105))⟹cond3^#(false,x276,x273)≥cond1^#(and(gr(x276,0),gr(x273,0)),x276,x273))
              • Applying Rule "Same Constructor" results in
                (gr(x1102,x1101)→^*false⟹x1092→^*x1102⟹0→^*s(x1101)⟹p(s(x1102))→^*x276⟹∀x1103 . ∀x1104 . ∀x1105 . (gr(x1102,x1101)→^*false⟹s(x1103)→^*x1102⟹0→^*x1101⟹p(x1102)→^*x1104⟹cond3^#(false,x1104,x1105)≥cond1^#(and(gr(x1104,0),gr(x1105,0)),x1104,x1105))⟹cond3^#(false,x276,x273)≥cond1^#(and(gr(x276,0),gr(x273,0)),x276,x273))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x1094,x1093)→^*true⟹0→^*s(x1093)⟹s(x1094)→^*x274⟹0→^*x1090⟹gr(x274,x1090)→^*false⟹p(x274)→^*x276⟹∀x1095 . ∀x1096 . ∀x1097 . ∀x1098 . (gr(x1094,x1093)→^*true⟹0→^*x1093⟹x1094→^*x1095⟹0→^*x1096⟹gr(x1095,x1096)→^*false⟹p(x1095)→^*x1097⟹cond3^#(false,x1097,x1098)≥cond1^#(and(gr(x1097,0),gr(x1098,0)),x1097,x1098))⟹cond3^#(false,x276,x273)≥cond1^#(and(gr(x276,0),gr(x273,0)),x276,x273))
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond3^#(gr(x278,0),p(x278),x279)→^*cond3^#(true,x280,x281)⟹cond3^#(gr(x280,0),p(x280),x281)→^*cond3^#(false,x282,x283)⟹cond3^#(false,x282,x283)≥cond1^#(and(gr(x282,0),gr(x283,0)),x282,x283))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x278,0)→^*true⟹p(x278)→^*x280⟹x279→^*x281⟹cond3^#(gr(x280,0),p(x280),x281)→^*cond3^#(false,x282,x283)⟹cond3^#(false,x282,x283)≥cond1^#(and(gr(x282,0),gr(x283,0)),x282,x283))
    • Applying Rule "Same Constructor" results in
      (gr(x278,0)→^*true⟹p(x278)→^*x280⟹x279→^*x281⟹gr(x280,0)→^*false⟹p(x280)→^*x282⟹x281→^*x283⟹cond3^#(false,x282,x283)≥cond1^#(and(gr(x282,0),gr(x283,0)),x282,x283))
    • Applying Rule "Variable in Equation" allows to substitute x281 by x279 which results in
      (gr(x278,0)→^*true⟹p(x278)→^*x280⟹gr(x280,0)→^*false⟹p(x280)→^*x282⟹x279→^*x283⟹cond3^#(false,x282,x283)≥cond1^#(and(gr(x282,0),gr(x283,0)),x282,x283))
    • Applying Rule "Variable in Equation" allows to substitute x283 by x279 which results in
      (gr(x278,0)→^*true⟹p(x278)→^*x280⟹gr(x280,0)→^*false⟹p(x280)→^*x282⟹cond3^#(false,x282,x279)≥cond1^#(and(gr(x282,0),gr(x279,0)),x282,x279))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1106⟹gr(x278,x1106)→^*true⟹p(x278)→^*x280⟹gr(x280,0)→^*false⟹p(x280)→^*x282⟹cond3^#(false,x282,x279)≥cond1^#(and(gr(x282,0),gr(x279,0)),x282,x279))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1106⟹gr(x278,x1106)→^*true⟹p(x278)→^*x280⟹0→^*x1107⟹gr(x280,x1107)→^*false⟹p(x280)→^*x282⟹cond3^#(false,x282,x279)≥cond1^#(and(gr(x282,0),gr(x279,0)),x282,x279))
    • Applying Rule "Induction" on gr(x278,x1106)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹cond3^#(false,x282,x279)≥cond1^#(and(gr(x282,0),gr(x279,0)),x282,x279))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹0→^*0⟹p(s(x1109))→^*x280⟹0→^*x1107⟹gr(x280,x1107)→^*false⟹p(x280)→^*x282⟹cond3^#(false,x282,x279)≥cond1^#(and(gr(x282,0),gr(x279,0)),x282,x279))
         • Applying Rule "Same Constructor" results in
           (0→^*0⟹p(s(x1109))→^*x280⟹0→^*x1107⟹gr(x280,x1107)→^*false⟹p(x280)→^*x282⟹cond3^#(false,x282,x279)≥cond1^#(and(gr(x282,0),gr(x279,0)),x282,x279))
         • Applying Rule "Same Constructor" results in
           (p(s(x1109))→^*x280⟹0→^*x1107⟹gr(x280,x1107)→^*false⟹p(x280)→^*x282⟹cond3^#(false,x282,x279)≥cond1^#(and(gr(x282,0),gr(x279,0)),x282,x279))
         • Applying Rule "Introduce fresh variable" results in
           (s(x1109)→^*x1116⟹p(x1116)→^*x280⟹0→^*x1107⟹gr(x280,x1107)→^*false⟹p(x280)→^*x282⟹cond3^#(false,x282,x279)≥cond1^#(and(gr(x282,0),gr(x279,0)),x282,x279))
         • Applying Rule "Induction" on gr(x280,x1107)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹s(x1109)→^*x1116⟹p(x1116)→^*0⟹0→^*x1117⟹p(0)→^*x282⟹cond3^#(false,x282,x279)≥cond1^#(and(gr(x282,0),gr(x279,0)),x282,x279))
              • Applying Rule "Same Constructor" results in
                (s(x1109)→^*x1116⟹p(x1116)→^*0⟹0→^*x1117⟹p(0)→^*x282⟹cond3^#(false,x282,x279)≥cond1^#(and(gr(x282,0),gr(x279,0)),x282,x279))
              • Applying Rule "Delete Conditions" results in
                (s(x1109)→^*x1116⟹p(x1116)→^*0⟹p(0)→^*x282⟹cond3^#(false,x282,x279)≥cond1^#(and(gr(x282,0),gr(x279,0)),x282,x279))
              • Applying Rule "Introduce fresh variable" results in
                (s(x1109)→^*x1116⟹p(x1116)→^*0⟹0→^*x1125⟹p(x1125)→^*x282⟹cond3^#(false,x282,x279)≥cond1^#(and(gr(x282,0),gr(x279,0)),x282,x279))
              • Applying Rule "Induction" on p(x1116)→^*0 results in the following 2 new constraints.
                1. (0→^*0⟹s(x1109)→^*0⟹0→^*x1125⟹p(x1125)→^*x282⟹cond3^#(false,x282,x279)≥cond1^#(and(gr(x282,0),gr(x279,0)),x282,x279))
                   • Applying Rule "Same Constructor" results in
                     (s(x1109)→^*0⟹0→^*x1125⟹p(x1125)→^*x282⟹cond3^#(false,x282,x279)≥cond1^#(and(gr(x282,0),gr(x279,0)),x282,x279))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (x1126→^*0⟹s(x1109)→^*s(x1126)⟹0→^*x1125⟹p(x1125)→^*x282⟹cond3^#(false,x282,x279)≥cond1^#(and(gr(x282,0),gr(x279,0)),x282,x279))
                   • Applying Rule "Same Constructor" results in
                     (x1126→^*0⟹x1109→^*x1126⟹0→^*x1125⟹p(x1125)→^*x282⟹cond3^#(false,x282,x279)≥cond1^#(and(gr(x282,0),gr(x279,0)),x282,x279))
                   • Applying Rule "Variable in Equation" allows to substitute x1126 by x1109 which results in
                     (x1109→^*0⟹0→^*x1125⟹p(x1125)→^*x282⟹cond3^#(false,x282,x279)≥cond1^#(and(gr(x282,0),gr(x279,0)),x282,x279))
                   • Applying Rule "Variable in Equation" allows to substitute x1109 by 0 which results in
                     (0→^*x1125⟹p(x1125)→^*x282⟹cond3^#(false,x282,x279)≥cond1^#(and(gr(x282,0),gr(x279,0)),x282,x279))
                   • Applying Rule "Variable in Equation" allows to substitute x1125 by 0 which results in
                     (p(0)→^*x282⟹cond3^#(false,x282,x279)≥cond1^#(and(gr(x282,0),gr(x279,0)),x282,x279))
                   • Applying Rule "Delete Conditions" results in
                     cond3^#(false,x282,x279)≥cond1^#(and(gr(x282,0),gr(x279,0)),x282,x279)

                   • This constraint is kept as final constraint.

           2. (true→^*false⟹cond3^#(false,x282,x279)≥cond1^#(and(gr(x282,0),gr(x279,0)),x282,x279))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1120,x1119)→^*false⟹s(x1109)→^*x1116⟹p(x1116)→^*s(x1120)⟹0→^*s(x1119)⟹p(s(x1120))→^*x282⟹∀x1121 . ∀x1122 . ∀x1123 . ∀x1124 . (gr(x1120,x1119)→^*false⟹s(x1121)→^*x1122⟹p(x1122)→^*x1120⟹0→^*x1119⟹p(x1120)→^*x1123⟹cond3^#(false,x1123,x1124)≥cond1^#(and(gr(x1123,0),gr(x1124,0)),x1123,x1124))⟹cond3^#(false,x282,x279)≥cond1^#(and(gr(x282,0),gr(x279,0)),x282,x279))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x1111,x1110)→^*true⟹0→^*s(x1110)⟹p(s(x1111))→^*x280⟹0→^*x1107⟹gr(x280,x1107)→^*false⟹p(x280)→^*x282⟹∀x1112 . ∀x1113 . ∀x1114 . ∀x1115 . (gr(x1111,x1110)→^*true⟹0→^*x1110⟹p(x1111)→^*x1112⟹0→^*x1113⟹gr(x1112,x1113)→^*false⟹p(x1112)→^*x1114⟹cond3^#(false,x1114,x1115)≥cond1^#(and(gr(x1114,0),gr(x1115,0)),x1114,x1115))⟹cond3^#(false,x282,x279)≥cond1^#(and(gr(x282,0),gr(x279,0)),x282,x279))
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond3^#(gr(x416,0),x416,x417)→^*cond3^#(false,x418,x419)⟹cond1^#(and(gr(x418,0),gr(x419,0)),x418,x419)→^*cond1^#(true,x420,x421)⟹cond1^#(true,x420,x421) > cond2^#(gr(x420,x421),x420,x421))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x416,0)→^*false⟹x416→^*x418⟹x417→^*x419⟹cond1^#(and(gr(x418,0),gr(x419,0)),x418,x419)→^*cond1^#(true,x420,x421)⟹cond1^#(true,x420,x421) > cond2^#(gr(x420,x421),x420,x421))
    • Applying Rule "Same Constructor" results in
      (gr(x416,0)→^*false⟹x416→^*x418⟹x417→^*x419⟹and(gr(x418,0),gr(x419,0))→^*true⟹x418→^*x420⟹x419→^*x421⟹cond1^#(true,x420,x421) > cond2^#(gr(x420,x421),x420,x421))
    • Applying Rule "Variable in Equation" allows to substitute x420 by x418 which results in
      (gr(x416,0)→^*false⟹x416→^*x418⟹x417→^*x419⟹and(gr(x418,0),gr(x419,0))→^*true⟹x419→^*x421⟹cond1^#(true,x418,x421) > cond2^#(gr(x418,x421),x418,x421))
    • Applying Rule "Variable in Equation" allows to substitute x421 by x419 which results in
      (gr(x416,0)→^*false⟹x416→^*x418⟹x417→^*x419⟹and(gr(x418,0),gr(x419,0))→^*true⟹cond1^#(true,x418,x419) > cond2^#(gr(x418,x419),x418,x419))
    • Applying Rule "Variable in Equation" allows to substitute x417 by x419 which results in
      (gr(x416,0)→^*false⟹x416→^*x418⟹and(gr(x418,0),gr(x419,0))→^*true⟹cond1^#(true,x418,x419) > cond2^#(gr(x418,x419),x418,x419))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1127⟹gr(x416,x1127)→^*false⟹x416→^*x418⟹and(gr(x418,0),gr(x419,0))→^*true⟹cond1^#(true,x418,x419) > cond2^#(gr(x418,x419),x418,x419))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1127⟹gr(x416,x1127)→^*false⟹x416→^*x418⟹gr(x419,0)→^*x1129⟹and(gr(x418,0),x1129)→^*true⟹cond1^#(true,x418,x419) > cond2^#(gr(x418,x419),x418,x419))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1127⟹gr(x416,x1127)→^*false⟹x416→^*x418⟹gr(x418,0)→^*x1128⟹gr(x419,0)→^*x1129⟹and(x1128,x1129)→^*true⟹cond1^#(true,x418,x419) > cond2^#(gr(x418,x419),x418,x419))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1127⟹gr(x416,x1127)→^*false⟹x416→^*x418⟹0→^*x1130⟹gr(x418,x1130)→^*x1128⟹gr(x419,0)→^*x1129⟹and(x1128,x1129)→^*true⟹cond1^#(true,x418,x419) > cond2^#(gr(x418,x419),x418,x419))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1127⟹gr(x416,x1127)→^*false⟹x416→^*x418⟹0→^*x1130⟹gr(x418,x1130)→^*x1128⟹0→^*x1131⟹gr(x419,x1131)→^*x1129⟹and(x1128,x1129)→^*true⟹cond1^#(true,x418,x419) > cond2^#(gr(x418,x419),x418,x419))
    • Applying Rule "Induction" on and(x1128,x1129)→^*true results in the following 3 new constraints.
      1. (true→^*true⟹0→^*x1127⟹gr(x416,x1127)→^*false⟹x416→^*x418⟹0→^*x1130⟹gr(x418,x1130)→^*true⟹0→^*x1131⟹gr(x419,x1131)→^*true⟹cond1^#(true,x418,x419) > cond2^#(gr(x418,x419),x418,x419))
         • Applying Rule "Same Constructor" results in
           (0→^*x1127⟹gr(x416,x1127)→^*false⟹x416→^*x418⟹0→^*x1130⟹gr(x418,x1130)→^*true⟹0→^*x1131⟹gr(x419,x1131)→^*true⟹cond1^#(true,x418,x419) > cond2^#(gr(x418,x419),x418,x419))
         • Applying Rule "Induction" on gr(x416,x1127)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹0→^*x1134⟹0→^*x418⟹0→^*x1130⟹gr(x418,x1130)→^*true⟹0→^*x1131⟹gr(x419,x1131)→^*true⟹cond1^#(true,x418,x419) > cond2^#(gr(x418,x419),x418,x419))
              • Applying Rule "Same Constructor" results in
                (0→^*x1134⟹0→^*x418⟹0→^*x1130⟹gr(x418,x1130)→^*true⟹0→^*x1131⟹gr(x419,x1131)→^*true⟹cond1^#(true,x418,x419) > cond2^#(gr(x418,x419),x418,x419))
              • Applying Rule "Delete Conditions" results in
                (0→^*x418⟹0→^*x1130⟹gr(x418,x1130)→^*true⟹0→^*x1131⟹gr(x419,x1131)→^*true⟹cond1^#(true,x418,x419) > cond2^#(gr(x418,x419),x418,x419))
              • Applying Rule "Induction" on gr(x418,x1130)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond1^#(true,0,x419) > cond2^#(gr(0,x419),0,x419))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹0→^*s(x1143)⟹0→^*0⟹0→^*x1131⟹gr(x419,x1131)→^*true⟹cond1^#(true,s(x1143),x419) > cond2^#(gr(s(x1143),x419),s(x1143),x419))
                   • Applying Rule "Same Constructor" results in
                     (0→^*s(x1143)⟹0→^*0⟹0→^*x1131⟹gr(x419,x1131)→^*true⟹cond1^#(true,s(x1143),x419) > cond2^#(gr(s(x1143),x419),s(x1143),x419))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x1145,x1144)→^*true⟹0→^*s(x1145)⟹0→^*s(x1144)⟹0→^*x1131⟹gr(x419,x1131)→^*true⟹∀x1146 . ∀x1147 . (gr(x1145,x1144)→^*true⟹0→^*x1145⟹0→^*x1144⟹0→^*x1146⟹gr(x1147,x1146)→^*true⟹cond1^#(true,x1145,x1147) > cond2^#(gr(x1145,x1147),x1145,x1147))⟹cond1^#(true,s(x1145),x419) > cond2^#(gr(s(x1145),x419),s(x1145),x419))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond1^#(true,x418,x419) > cond2^#(gr(x418,x419),x418,x419))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1137,x1136)→^*false⟹0→^*s(x1136)⟹s(x1137)→^*x418⟹0→^*x1130⟹gr(x418,x1130)→^*true⟹0→^*x1131⟹gr(x419,x1131)→^*true⟹∀x1138 . ∀x1139 . ∀x1140 . ∀x1141 . (gr(x1137,x1136)→^*false⟹0→^*x1136⟹x1137→^*x1138⟹0→^*x1139⟹gr(x1138,x1139)→^*true⟹0→^*x1140⟹gr(x1141,x1140)→^*true⟹cond1^#(true,x1138,x1141) > cond2^#(gr(x1138,x1141),x1138,x1141))⟹cond1^#(true,x418,x419) > cond2^#(gr(x418,x419),x418,x419))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*true⟹cond1^#(true,x418,x419) > cond2^#(gr(x418,x419),x418,x419))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (false→^*true⟹cond1^#(true,x418,x419) > cond2^#(gr(x418,x419),x418,x419))
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond3^#(gr(x422,0),p(x422),x423)→^*cond3^#(false,x424,x425)⟹cond1^#(and(gr(x424,0),gr(x425,0)),x424,x425)→^*cond1^#(true,x426,x427)⟹cond1^#(true,x426,x427) > cond2^#(gr(x426,x427),x426,x427))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x422,0)→^*false⟹p(x422)→^*x424⟹x423→^*x425⟹cond1^#(and(gr(x424,0),gr(x425,0)),x424,x425)→^*cond1^#(true,x426,x427)⟹cond1^#(true,x426,x427) > cond2^#(gr(x426,x427),x426,x427))
    • Applying Rule "Same Constructor" results in
      (gr(x422,0)→^*false⟹p(x422)→^*x424⟹x423→^*x425⟹and(gr(x424,0),gr(x425,0))→^*true⟹x424→^*x426⟹x425→^*x427⟹cond1^#(true,x426,x427) > cond2^#(gr(x426,x427),x426,x427))
    • Applying Rule "Variable in Equation" allows to substitute x426 by x424 which results in
      (gr(x422,0)→^*false⟹p(x422)→^*x424⟹x423→^*x425⟹and(gr(x424,0),gr(x425,0))→^*true⟹x425→^*x427⟹cond1^#(true,x424,x427) > cond2^#(gr(x424,x427),x424,x427))
    • Applying Rule "Variable in Equation" allows to substitute x427 by x425 which results in
      (gr(x422,0)→^*false⟹p(x422)→^*x424⟹x423→^*x425⟹and(gr(x424,0),gr(x425,0))→^*true⟹cond1^#(true,x424,x425) > cond2^#(gr(x424,x425),x424,x425))
    • Applying Rule "Variable in Equation" allows to substitute x423 by x425 which results in
      (gr(x422,0)→^*false⟹p(x422)→^*x424⟹and(gr(x424,0),gr(x425,0))→^*true⟹cond1^#(true,x424,x425) > cond2^#(gr(x424,x425),x424,x425))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1148⟹gr(x422,x1148)→^*false⟹p(x422)→^*x424⟹and(gr(x424,0),gr(x425,0))→^*true⟹cond1^#(true,x424,x425) > cond2^#(gr(x424,x425),x424,x425))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1148⟹gr(x422,x1148)→^*false⟹p(x422)→^*x424⟹gr(x425,0)→^*x1150⟹and(gr(x424,0),x1150)→^*true⟹cond1^#(true,x424,x425) > cond2^#(gr(x424,x425),x424,x425))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1148⟹gr(x422,x1148)→^*false⟹p(x422)→^*x424⟹gr(x424,0)→^*x1149⟹gr(x425,0)→^*x1150⟹and(x1149,x1150)→^*true⟹cond1^#(true,x424,x425) > cond2^#(gr(x424,x425),x424,x425))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1148⟹gr(x422,x1148)→^*false⟹p(x422)→^*x424⟹0→^*x1151⟹gr(x424,x1151)→^*x1149⟹gr(x425,0)→^*x1150⟹and(x1149,x1150)→^*true⟹cond1^#(true,x424,x425) > cond2^#(gr(x424,x425),x424,x425))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1148⟹gr(x422,x1148)→^*false⟹p(x422)→^*x424⟹0→^*x1151⟹gr(x424,x1151)→^*x1149⟹0→^*x1152⟹gr(x425,x1152)→^*x1150⟹and(x1149,x1150)→^*true⟹cond1^#(true,x424,x425) > cond2^#(gr(x424,x425),x424,x425))
    • Applying Rule "Induction" on and(x1149,x1150)→^*true results in the following 3 new constraints.
      1. (true→^*true⟹0→^*x1148⟹gr(x422,x1148)→^*false⟹p(x422)→^*x424⟹0→^*x1151⟹gr(x424,x1151)→^*true⟹0→^*x1152⟹gr(x425,x1152)→^*true⟹cond1^#(true,x424,x425) > cond2^#(gr(x424,x425),x424,x425))
         • Applying Rule "Same Constructor" results in
           (0→^*x1148⟹gr(x422,x1148)→^*false⟹p(x422)→^*x424⟹0→^*x1151⟹gr(x424,x1151)→^*true⟹0→^*x1152⟹gr(x425,x1152)→^*true⟹cond1^#(true,x424,x425) > cond2^#(gr(x424,x425),x424,x425))
         • Applying Rule "Induction" on gr(x422,x1148)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹0→^*x1155⟹p(0)→^*x424⟹0→^*x1151⟹gr(x424,x1151)→^*true⟹0→^*x1152⟹gr(x425,x1152)→^*true⟹cond1^#(true,x424,x425) > cond2^#(gr(x424,x425),x424,x425))
              • Applying Rule "Same Constructor" results in
                (0→^*x1155⟹p(0)→^*x424⟹0→^*x1151⟹gr(x424,x1151)→^*true⟹0→^*x1152⟹gr(x425,x1152)→^*true⟹cond1^#(true,x424,x425) > cond2^#(gr(x424,x425),x424,x425))
              • Applying Rule "Delete Conditions" results in
                (p(0)→^*x424⟹0→^*x1151⟹gr(x424,x1151)→^*true⟹0→^*x1152⟹gr(x425,x1152)→^*true⟹cond1^#(true,x424,x425) > cond2^#(gr(x424,x425),x424,x425))
              • Applying Rule "Introduce fresh variable" results in
                (0→^*x1163⟹p(x1163)→^*x424⟹0→^*x1151⟹gr(x424,x1151)→^*true⟹0→^*x1152⟹gr(x425,x1152)→^*true⟹cond1^#(true,x424,x425) > cond2^#(gr(x424,x425),x424,x425))
              • Applying Rule "Induction" on gr(x424,x1151)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond1^#(true,0,x425) > cond2^#(gr(0,x425),0,x425))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹0→^*x1163⟹p(x1163)→^*s(x1165)⟹0→^*0⟹0→^*x1152⟹gr(x425,x1152)→^*true⟹cond1^#(true,s(x1165),x425) > cond2^#(gr(s(x1165),x425),s(x1165),x425))
                   • Applying Rule "Same Constructor" results in
                     (0→^*x1163⟹p(x1163)→^*s(x1165)⟹0→^*0⟹0→^*x1152⟹gr(x425,x1152)→^*true⟹cond1^#(true,s(x1165),x425) > cond2^#(gr(s(x1165),x425),s(x1165),x425))
                   • Applying Rule "Same Constructor" results in
                     (0→^*x1163⟹p(x1163)→^*s(x1165)⟹0→^*x1152⟹gr(x425,x1152)→^*true⟹cond1^#(true,s(x1165),x425) > cond2^#(gr(s(x1165),x425),s(x1165),x425))
                   • Applying Rule "Induction" on p(x1163)→^*s(x1165) results in the following 2 new constraints.
                     1. (0→^*s(x1165)⟹cond1^#(true,s(x1165),x425) > cond2^#(gr(s(x1165),x425),s(x1165),x425))
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                     2. (x1171→^*s(x1165)⟹0→^*s(x1171)⟹0→^*x1152⟹gr(x425,x1152)→^*true⟹cond1^#(true,s(x1165),x425) > cond2^#(gr(s(x1165),x425),s(x1165),x425))
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x1167,x1166)→^*true⟹0→^*x1163⟹p(x1163)→^*s(x1167)⟹0→^*s(x1166)⟹0→^*x1152⟹gr(x425,x1152)→^*true⟹∀x1168 . ∀x1169 . ∀x1170 . (gr(x1167,x1166)→^*true⟹0→^*x1168⟹p(x1168)→^*x1167⟹0→^*x1166⟹0→^*x1169⟹gr(x1170,x1169)→^*true⟹cond1^#(true,x1167,x1170) > cond2^#(gr(x1167,x1170),x1167,x1170))⟹cond1^#(true,s(x1167),x425) > cond2^#(gr(s(x1167),x425),s(x1167),x425))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond1^#(true,x424,x425) > cond2^#(gr(x424,x425),x424,x425))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1158,x1157)→^*false⟹0→^*s(x1157)⟹p(s(x1158))→^*x424⟹0→^*x1151⟹gr(x424,x1151)→^*true⟹0→^*x1152⟹gr(x425,x1152)→^*true⟹∀x1159 . ∀x1160 . ∀x1161 . ∀x1162 . (gr(x1158,x1157)→^*false⟹0→^*x1157⟹p(x1158)→^*x1159⟹0→^*x1160⟹gr(x1159,x1160)→^*true⟹0→^*x1161⟹gr(x1162,x1161)→^*true⟹cond1^#(true,x1159,x1162) > cond2^#(gr(x1159,x1162),x1159,x1162))⟹cond1^#(true,x424,x425) > cond2^#(gr(x424,x425),x424,x425))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*true⟹cond1^#(true,x424,x425) > cond2^#(gr(x424,x425),x424,x425))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (false→^*true⟹cond1^#(true,x424,x425) > cond2^#(gr(x424,x425),x424,x425))
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond4^#(gr(x499,0),x498,x499)→^*cond4^#(false,x500,x501)⟹cond1^#(and(gr(x500,0),gr(x501,0)),x500,x501)→^*cond1^#(true,x502,x503)⟹cond1^#(true,x502,x503) > cond2^#(gr(x502,x503),x502,x503))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x499,0)→^*false⟹x498→^*x500⟹x499→^*x501⟹cond1^#(and(gr(x500,0),gr(x501,0)),x500,x501)→^*cond1^#(true,x502,x503)⟹cond1^#(true,x502,x503) > cond2^#(gr(x502,x503),x502,x503))
    • Applying Rule "Same Constructor" results in
      (gr(x499,0)→^*false⟹x498→^*x500⟹x499→^*x501⟹and(gr(x500,0),gr(x501,0))→^*true⟹x500→^*x502⟹x501→^*x503⟹cond1^#(true,x502,x503) > cond2^#(gr(x502,x503),x502,x503))
    • Applying Rule "Variable in Equation" allows to substitute x502 by x500 which results in
      (gr(x499,0)→^*false⟹x498→^*x500⟹x499→^*x501⟹and(gr(x500,0),gr(x501,0))→^*true⟹x501→^*x503⟹cond1^#(true,x500,x503) > cond2^#(gr(x500,x503),x500,x503))
    • Applying Rule "Variable in Equation" allows to substitute x503 by x501 which results in
      (gr(x499,0)→^*false⟹x498→^*x500⟹x499→^*x501⟹and(gr(x500,0),gr(x501,0))→^*true⟹cond1^#(true,x500,x501) > cond2^#(gr(x500,x501),x500,x501))
    • Applying Rule "Variable in Equation" allows to substitute x498 by x500 which results in
      (gr(x499,0)→^*false⟹x499→^*x501⟹and(gr(x500,0),gr(x501,0))→^*true⟹cond1^#(true,x500,x501) > cond2^#(gr(x500,x501),x500,x501))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1172⟹gr(x499,x1172)→^*false⟹x499→^*x501⟹and(gr(x500,0),gr(x501,0))→^*true⟹cond1^#(true,x500,x501) > cond2^#(gr(x500,x501),x500,x501))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1172⟹gr(x499,x1172)→^*false⟹x499→^*x501⟹gr(x501,0)→^*x1174⟹and(gr(x500,0),x1174)→^*true⟹cond1^#(true,x500,x501) > cond2^#(gr(x500,x501),x500,x501))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1172⟹gr(x499,x1172)→^*false⟹x499→^*x501⟹gr(x500,0)→^*x1173⟹gr(x501,0)→^*x1174⟹and(x1173,x1174)→^*true⟹cond1^#(true,x500,x501) > cond2^#(gr(x500,x501),x500,x501))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1172⟹gr(x499,x1172)→^*false⟹x499→^*x501⟹0→^*x1175⟹gr(x500,x1175)→^*x1173⟹gr(x501,0)→^*x1174⟹and(x1173,x1174)→^*true⟹cond1^#(true,x500,x501) > cond2^#(gr(x500,x501),x500,x501))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1172⟹gr(x499,x1172)→^*false⟹x499→^*x501⟹0→^*x1175⟹gr(x500,x1175)→^*x1173⟹0→^*x1176⟹gr(x501,x1176)→^*x1174⟹and(x1173,x1174)→^*true⟹cond1^#(true,x500,x501) > cond2^#(gr(x500,x501),x500,x501))
    • Applying Rule "Induction" on and(x1173,x1174)→^*true results in the following 3 new constraints.
      1. (true→^*true⟹0→^*x1172⟹gr(x499,x1172)→^*false⟹x499→^*x501⟹0→^*x1175⟹gr(x500,x1175)→^*true⟹0→^*x1176⟹gr(x501,x1176)→^*true⟹cond1^#(true,x500,x501) > cond2^#(gr(x500,x501),x500,x501))
         • Applying Rule "Same Constructor" results in
           (0→^*x1172⟹gr(x499,x1172)→^*false⟹x499→^*x501⟹0→^*x1175⟹gr(x500,x1175)→^*true⟹0→^*x1176⟹gr(x501,x1176)→^*true⟹cond1^#(true,x500,x501) > cond2^#(gr(x500,x501),x500,x501))
         • Applying Rule "Induction" on gr(x499,x1172)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹0→^*x1179⟹0→^*x501⟹0→^*x1175⟹gr(x500,x1175)→^*true⟹0→^*x1176⟹gr(x501,x1176)→^*true⟹cond1^#(true,x500,x501) > cond2^#(gr(x500,x501),x500,x501))
              • Applying Rule "Same Constructor" results in
                (0→^*x1179⟹0→^*x501⟹0→^*x1175⟹gr(x500,x1175)→^*true⟹0→^*x1176⟹gr(x501,x1176)→^*true⟹cond1^#(true,x500,x501) > cond2^#(gr(x500,x501),x500,x501))
              • Applying Rule "Delete Conditions" results in
                (0→^*x501⟹0→^*x1175⟹gr(x500,x1175)→^*true⟹0→^*x1176⟹gr(x501,x1176)→^*true⟹cond1^#(true,x500,x501) > cond2^#(gr(x500,x501),x500,x501))
              • Applying Rule "Induction" on gr(x500,x1175)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond1^#(true,0,x501) > cond2^#(gr(0,x501),0,x501))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹0→^*x501⟹0→^*0⟹0→^*x1176⟹gr(x501,x1176)→^*true⟹cond1^#(true,s(x1188),x501) > cond2^#(gr(s(x1188),x501),s(x1188),x501))
                   • Applying Rule "Same Constructor" results in
                     (0→^*x501⟹0→^*0⟹0→^*x1176⟹gr(x501,x1176)→^*true⟹cond1^#(true,s(x1188),x501) > cond2^#(gr(s(x1188),x501),s(x1188),x501))
                   • Applying Rule "Same Constructor" results in
                     (0→^*x501⟹0→^*x1176⟹gr(x501,x1176)→^*true⟹cond1^#(true,s(x1188),x501) > cond2^#(gr(s(x1188),x501),s(x1188),x501))
                   • Applying Rule "Induction" on gr(x501,x1176)→^*true results in the following 3 new constraints.
                     1. (false→^*true⟹cond1^#(true,s(x1188),0) > cond2^#(gr(s(x1188),0),s(x1188),0))
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                     2. (true→^*true⟹0→^*s(x1194)⟹0→^*0⟹cond1^#(true,s(x1188),s(x1194)) > cond2^#(gr(s(x1188),s(x1194)),s(x1188),s(x1194)))
                        • Applying Rule "Same Constructor" results in
                          (0→^*s(x1194)⟹0→^*0⟹cond1^#(true,s(x1188),s(x1194)) > cond2^#(gr(s(x1188),s(x1194)),s(x1188),s(x1194)))
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                     3. (gr(x1196,x1195)→^*true⟹0→^*s(x1196)⟹0→^*s(x1195)⟹∀x1197 . (gr(x1196,x1195)→^*true⟹0→^*x1196⟹0→^*x1195⟹cond1^#(true,s(x1197),x1196) > cond2^#(gr(s(x1197),x1196),s(x1197),x1196))⟹cond1^#(true,s(x1188),s(x1196)) > cond2^#(gr(s(x1188),s(x1196)),s(x1188),s(x1196)))
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x1190,x1189)→^*true⟹0→^*x501⟹0→^*s(x1189)⟹0→^*x1176⟹gr(x501,x1176)→^*true⟹∀x1191 . ∀x1192 . (gr(x1190,x1189)→^*true⟹0→^*x1191⟹0→^*x1189⟹0→^*x1192⟹gr(x1191,x1192)→^*true⟹cond1^#(true,x1190,x1191) > cond2^#(gr(x1190,x1191),x1190,x1191))⟹cond1^#(true,s(x1190),x501) > cond2^#(gr(s(x1190),x501),s(x1190),x501))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond1^#(true,x500,x501) > cond2^#(gr(x500,x501),x500,x501))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1182,x1181)→^*false⟹0→^*s(x1181)⟹s(x1182)→^*x501⟹0→^*x1175⟹gr(x500,x1175)→^*true⟹0→^*x1176⟹gr(x501,x1176)→^*true⟹∀x1183 . ∀x1184 . ∀x1185 . ∀x1186 . (gr(x1182,x1181)→^*false⟹0→^*x1181⟹x1182→^*x1183⟹0→^*x1184⟹gr(x1185,x1184)→^*true⟹0→^*x1186⟹gr(x1183,x1186)→^*true⟹cond1^#(true,x1185,x1183) > cond2^#(gr(x1185,x1183),x1185,x1183))⟹cond1^#(true,x500,x501) > cond2^#(gr(x500,x501),x500,x501))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*true⟹cond1^#(true,x500,x501) > cond2^#(gr(x500,x501),x500,x501))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (false→^*true⟹cond1^#(true,x500,x501) > cond2^#(gr(x500,x501),x500,x501))
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond4^#(gr(x505,0),x504,p(x505))→^*cond4^#(false,x506,x507)⟹cond1^#(and(gr(x506,0),gr(x507,0)),x506,x507)→^*cond1^#(true,x508,x509)⟹cond1^#(true,x508,x509) > cond2^#(gr(x508,x509),x508,x509))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x505,0)→^*false⟹x504→^*x506⟹p(x505)→^*x507⟹cond1^#(and(gr(x506,0),gr(x507,0)),x506,x507)→^*cond1^#(true,x508,x509)⟹cond1^#(true,x508,x509) > cond2^#(gr(x508,x509),x508,x509))
    • Applying Rule "Same Constructor" results in
      (gr(x505,0)→^*false⟹x504→^*x506⟹p(x505)→^*x507⟹and(gr(x506,0),gr(x507,0))→^*true⟹x506→^*x508⟹x507→^*x509⟹cond1^#(true,x508,x509) > cond2^#(gr(x508,x509),x508,x509))
    • Applying Rule "Variable in Equation" allows to substitute x508 by x506 which results in
      (gr(x505,0)→^*false⟹x504→^*x506⟹p(x505)→^*x507⟹and(gr(x506,0),gr(x507,0))→^*true⟹x507→^*x509⟹cond1^#(true,x506,x509) > cond2^#(gr(x506,x509),x506,x509))
    • Applying Rule "Variable in Equation" allows to substitute x509 by x507 which results in
      (gr(x505,0)→^*false⟹x504→^*x506⟹p(x505)→^*x507⟹and(gr(x506,0),gr(x507,0))→^*true⟹cond1^#(true,x506,x507) > cond2^#(gr(x506,x507),x506,x507))
    • Applying Rule "Variable in Equation" allows to substitute x504 by x506 which results in
      (gr(x505,0)→^*false⟹p(x505)→^*x507⟹and(gr(x506,0),gr(x507,0))→^*true⟹cond1^#(true,x506,x507) > cond2^#(gr(x506,x507),x506,x507))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1198⟹gr(x505,x1198)→^*false⟹p(x505)→^*x507⟹and(gr(x506,0),gr(x507,0))→^*true⟹cond1^#(true,x506,x507) > cond2^#(gr(x506,x507),x506,x507))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1198⟹gr(x505,x1198)→^*false⟹p(x505)→^*x507⟹gr(x507,0)→^*x1200⟹and(gr(x506,0),x1200)→^*true⟹cond1^#(true,x506,x507) > cond2^#(gr(x506,x507),x506,x507))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1198⟹gr(x505,x1198)→^*false⟹p(x505)→^*x507⟹gr(x506,0)→^*x1199⟹gr(x507,0)→^*x1200⟹and(x1199,x1200)→^*true⟹cond1^#(true,x506,x507) > cond2^#(gr(x506,x507),x506,x507))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1198⟹gr(x505,x1198)→^*false⟹p(x505)→^*x507⟹0→^*x1201⟹gr(x506,x1201)→^*x1199⟹gr(x507,0)→^*x1200⟹and(x1199,x1200)→^*true⟹cond1^#(true,x506,x507) > cond2^#(gr(x506,x507),x506,x507))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1198⟹gr(x505,x1198)→^*false⟹p(x505)→^*x507⟹0→^*x1201⟹gr(x506,x1201)→^*x1199⟹0→^*x1202⟹gr(x507,x1202)→^*x1200⟹and(x1199,x1200)→^*true⟹cond1^#(true,x506,x507) > cond2^#(gr(x506,x507),x506,x507))
    • Applying Rule "Induction" on and(x1199,x1200)→^*true results in the following 3 new constraints.
      1. (true→^*true⟹0→^*x1198⟹gr(x505,x1198)→^*false⟹p(x505)→^*x507⟹0→^*x1201⟹gr(x506,x1201)→^*true⟹0→^*x1202⟹gr(x507,x1202)→^*true⟹cond1^#(true,x506,x507) > cond2^#(gr(x506,x507),x506,x507))
         • Applying Rule "Same Constructor" results in
           (0→^*x1198⟹gr(x505,x1198)→^*false⟹p(x505)→^*x507⟹0→^*x1201⟹gr(x506,x1201)→^*true⟹0→^*x1202⟹gr(x507,x1202)→^*true⟹cond1^#(true,x506,x507) > cond2^#(gr(x506,x507),x506,x507))
         • Applying Rule "Induction" on gr(x505,x1198)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹0→^*x1205⟹p(0)→^*x507⟹0→^*x1201⟹gr(x506,x1201)→^*true⟹0→^*x1202⟹gr(x507,x1202)→^*true⟹cond1^#(true,x506,x507) > cond2^#(gr(x506,x507),x506,x507))
              • Applying Rule "Same Constructor" results in
                (0→^*x1205⟹p(0)→^*x507⟹0→^*x1201⟹gr(x506,x1201)→^*true⟹0→^*x1202⟹gr(x507,x1202)→^*true⟹cond1^#(true,x506,x507) > cond2^#(gr(x506,x507),x506,x507))
              • Applying Rule "Delete Conditions" results in
                (p(0)→^*x507⟹0→^*x1201⟹gr(x506,x1201)→^*true⟹0→^*x1202⟹gr(x507,x1202)→^*true⟹cond1^#(true,x506,x507) > cond2^#(gr(x506,x507),x506,x507))
              • Applying Rule "Introduce fresh variable" results in
                (0→^*x1213⟹p(x1213)→^*x507⟹0→^*x1201⟹gr(x506,x1201)→^*true⟹0→^*x1202⟹gr(x507,x1202)→^*true⟹cond1^#(true,x506,x507) > cond2^#(gr(x506,x507),x506,x507))
              • Applying Rule "Induction" on gr(x506,x1201)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond1^#(true,0,x507) > cond2^#(gr(0,x507),0,x507))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹0→^*x1213⟹p(x1213)→^*x507⟹0→^*0⟹0→^*x1202⟹gr(x507,x1202)→^*true⟹cond1^#(true,s(x1215),x507) > cond2^#(gr(s(x1215),x507),s(x1215),x507))
                   • Applying Rule "Same Constructor" results in
                     (0→^*x1213⟹p(x1213)→^*x507⟹0→^*0⟹0→^*x1202⟹gr(x507,x1202)→^*true⟹cond1^#(true,s(x1215),x507) > cond2^#(gr(s(x1215),x507),s(x1215),x507))
                   • Applying Rule "Same Constructor" results in
                     (0→^*x1213⟹p(x1213)→^*x507⟹0→^*x1202⟹gr(x507,x1202)→^*true⟹cond1^#(true,s(x1215),x507) > cond2^#(gr(s(x1215),x507),s(x1215),x507))
                   • Applying Rule "Induction" on gr(x507,x1202)→^*true results in the following 3 new constraints.
                     1. (false→^*true⟹cond1^#(true,s(x1215),0) > cond2^#(gr(s(x1215),0),s(x1215),0))
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                     2. (true→^*true⟹0→^*x1213⟹p(x1213)→^*s(x1222)⟹0→^*0⟹cond1^#(true,s(x1215),s(x1222)) > cond2^#(gr(s(x1215),s(x1222)),s(x1215),s(x1222)))
                        • Applying Rule "Same Constructor" results in
                          (0→^*x1213⟹p(x1213)→^*s(x1222)⟹0→^*0⟹cond1^#(true,s(x1215),s(x1222)) > cond2^#(gr(s(x1215),s(x1222)),s(x1215),s(x1222)))
                        • Applying Rule "Same Constructor" results in
                          (0→^*x1213⟹p(x1213)→^*s(x1222)⟹cond1^#(true,s(x1215),s(x1222)) > cond2^#(gr(s(x1215),s(x1222)),s(x1215),s(x1222)))
                        • Applying Rule "Induction" on p(x1213)→^*s(x1222) results in the following 2 new constraints.
                          1. (0→^*s(x1222)⟹cond1^#(true,s(x1215),s(x1222)) > cond2^#(gr(s(x1215),s(x1222)),s(x1215),s(x1222)))
                             • Applying Rule "Different Constructors" allows to drop this constraint.
                          2. (x1227→^*s(x1222)⟹0→^*s(x1227)⟹cond1^#(true,s(x1215),s(x1222)) > cond2^#(gr(s(x1215),s(x1222)),s(x1215),s(x1222)))
                             • Applying Rule "Different Constructors" allows to drop this constraint.
                     3. (gr(x1224,x1223)→^*true⟹0→^*x1213⟹p(x1213)→^*s(x1224)⟹0→^*s(x1223)⟹∀x1225 . ∀x1226 . (gr(x1224,x1223)→^*true⟹0→^*x1225⟹p(x1225)→^*x1224⟹0→^*x1223⟹cond1^#(true,s(x1226),x1224) > cond2^#(gr(s(x1226),x1224),s(x1226),x1224))⟹cond1^#(true,s(x1215),s(x1224)) > cond2^#(gr(s(x1215),s(x1224)),s(x1215),s(x1224)))
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x1217,x1216)→^*true⟹0→^*x1213⟹p(x1213)→^*x507⟹0→^*s(x1216)⟹0→^*x1202⟹gr(x507,x1202)→^*true⟹∀x1218 . ∀x1219 . ∀x1220 . (gr(x1217,x1216)→^*true⟹0→^*x1218⟹p(x1218)→^*x1219⟹0→^*x1216⟹0→^*x1220⟹gr(x1219,x1220)→^*true⟹cond1^#(true,x1217,x1219) > cond2^#(gr(x1217,x1219),x1217,x1219))⟹cond1^#(true,s(x1217),x507) > cond2^#(gr(s(x1217),x507),s(x1217),x507))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond1^#(true,x506,x507) > cond2^#(gr(x506,x507),x506,x507))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1208,x1207)→^*false⟹0→^*s(x1207)⟹p(s(x1208))→^*x507⟹0→^*x1201⟹gr(x506,x1201)→^*true⟹0→^*x1202⟹gr(x507,x1202)→^*true⟹∀x1209 . ∀x1210 . ∀x1211 . ∀x1212 . (gr(x1208,x1207)→^*false⟹0→^*x1207⟹p(x1208)→^*x1209⟹0→^*x1210⟹gr(x1211,x1210)→^*true⟹0→^*x1212⟹gr(x1209,x1212)→^*true⟹cond1^#(true,x1211,x1209) > cond2^#(gr(x1211,x1209),x1211,x1209))⟹cond1^#(true,x506,x507) > cond2^#(gr(x506,x507),x506,x507))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*true⟹cond1^#(true,x506,x507) > cond2^#(gr(x506,x507),x506,x507))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (false→^*true⟹cond1^#(true,x506,x507) > cond2^#(gr(x506,x507),x506,x507))
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond1^#(and(gr(x568,0),gr(x569,0)),x568,x569)→^*cond1^#(true,x570,x571)⟹cond2^#(gr(x570,x571),x570,x571)→^*cond2^#(false,x572,x573)⟹cond2^#(false,x572,x573)≥cond4^#(gr(x573,0),x572,x573))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (and(gr(x568,0),gr(x569,0))→^*true⟹x568→^*x570⟹x569→^*x571⟹cond2^#(gr(x570,x571),x570,x571)→^*cond2^#(false,x572,x573)⟹cond2^#(false,x572,x573)≥cond4^#(gr(x573,0),x572,x573))
    • Applying Rule "Same Constructor" results in
      (and(gr(x568,0),gr(x569,0))→^*true⟹x568→^*x570⟹x569→^*x571⟹gr(x570,x571)→^*false⟹x570→^*x572⟹x571→^*x573⟹cond2^#(false,x572,x573)≥cond4^#(gr(x573,0),x572,x573))
    • Applying Rule "Variable in Equation" allows to substitute x572 by x570 which results in
      (and(gr(x568,0),gr(x569,0))→^*true⟹x568→^*x570⟹x569→^*x571⟹gr(x570,x571)→^*false⟹x571→^*x573⟹cond2^#(false,x570,x573)≥cond4^#(gr(x573,0),x570,x573))
    • Applying Rule "Variable in Equation" allows to substitute x573 by x571 which results in
      (and(gr(x568,0),gr(x569,0))→^*true⟹x568→^*x570⟹x569→^*x571⟹gr(x570,x571)→^*false⟹cond2^#(false,x570,x571)≥cond4^#(gr(x571,0),x570,x571))
    • Applying Rule "Introduce fresh variable" results in
      (x568→^*x570⟹x569→^*x571⟹gr(x570,x571)→^*false⟹gr(x569,0)→^*x1229⟹and(gr(x568,0),x1229)→^*true⟹cond2^#(false,x570,x571)≥cond4^#(gr(x571,0),x570,x571))
    • Applying Rule "Introduce fresh variable" results in
      (gr(x568,0)→^*x1228⟹gr(x569,0)→^*x1229⟹and(x1228,x1229)→^*true⟹x568→^*x570⟹x569→^*x571⟹gr(x570,x571)→^*false⟹cond2^#(false,x570,x571)≥cond4^#(gr(x571,0),x570,x571))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1230⟹gr(x568,x1230)→^*x1228⟹gr(x569,0)→^*x1229⟹and(x1228,x1229)→^*true⟹x568→^*x570⟹x569→^*x571⟹gr(x570,x571)→^*false⟹cond2^#(false,x570,x571)≥cond4^#(gr(x571,0),x570,x571))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1230⟹gr(x568,x1230)→^*x1228⟹0→^*x1231⟹gr(x569,x1231)→^*x1229⟹and(x1228,x1229)→^*true⟹x568→^*x570⟹x569→^*x571⟹gr(x570,x571)→^*false⟹cond2^#(false,x570,x571)≥cond4^#(gr(x571,0),x570,x571))
    • Applying Rule "Induction" on and(x1228,x1229)→^*true results in the following 3 new constraints.
      1. (true→^*true⟹0→^*x1230⟹gr(x568,x1230)→^*true⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹x568→^*x570⟹x569→^*x571⟹gr(x570,x571)→^*false⟹cond2^#(false,x570,x571)≥cond4^#(gr(x571,0),x570,x571))
         • Applying Rule "Same Constructor" results in
           (0→^*x1230⟹gr(x568,x1230)→^*true⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹x568→^*x570⟹x569→^*x571⟹gr(x570,x571)→^*false⟹cond2^#(false,x570,x571)≥cond4^#(gr(x571,0),x570,x571))
         • Applying Rule "Induction" on gr(x570,x571)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹0→^*x1230⟹gr(x568,x1230)→^*true⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹x568→^*0⟹x569→^*x1234⟹cond2^#(false,0,x1234)≥cond4^#(gr(x1234,0),0,x1234))
              • Applying Rule "Same Constructor" results in
                (0→^*x1230⟹gr(x568,x1230)→^*true⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹x568→^*0⟹x569→^*x1234⟹cond2^#(false,0,x1234)≥cond4^#(gr(x1234,0),0,x1234))
              • Applying Rule "Variable in Equation" allows to substitute x1234 by x569 which results in
                (0→^*x1230⟹gr(x568,x1230)→^*true⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹x568→^*0⟹cond2^#(false,0,x569)≥cond4^#(gr(x569,0),0,x569))
              • Applying Rule "Variable in Equation" allows to substitute x568 by 0 which results in
                (0→^*x1230⟹gr(0,x1230)→^*true⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹cond2^#(false,0,x569)≥cond4^#(gr(x569,0),0,x569))
              • Applying Rule "Introduce fresh variable" results in
                (0→^*x1230⟹0→^*x1242⟹gr(x1242,x1230)→^*true⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹cond2^#(false,0,x569)≥cond4^#(gr(x569,0),0,x569))
              • Applying Rule "Induction" on gr(x1242,x1230)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond2^#(false,0,x569)≥cond4^#(gr(x569,0),0,x569))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹0→^*0⟹0→^*s(x1244)⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹cond2^#(false,0,x569)≥cond4^#(gr(x569,0),0,x569))
                   • Applying Rule "Same Constructor" results in
                     (0→^*0⟹0→^*s(x1244)⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹cond2^#(false,0,x569)≥cond4^#(gr(x569,0),0,x569))
                   • Applying Rule "Same Constructor" results in
                     (0→^*s(x1244)⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹cond2^#(false,0,x569)≥cond4^#(gr(x569,0),0,x569))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x1246,x1245)→^*true⟹0→^*s(x1245)⟹0→^*s(x1246)⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹∀x1247 . ∀x1248 . (gr(x1246,x1245)→^*true⟹0→^*x1245⟹0→^*x1246⟹0→^*x1247⟹gr(x1248,x1247)→^*true⟹cond2^#(false,0,x1248)≥cond4^#(gr(x1248,0),0,x1248))⟹cond2^#(false,0,x569)≥cond4^#(gr(x569,0),0,x569))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond2^#(false,s(x1235),0)≥cond4^#(gr(0,0),s(x1235),0))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1237,x1236)→^*false⟹0→^*x1230⟹gr(x568,x1230)→^*true⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹x568→^*s(x1237)⟹x569→^*s(x1236)⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥cond4^#(gr(x1236,0),x1237,x1236))⟹cond2^#(false,s(x1237),s(x1236))≥cond4^#(gr(s(x1236),0),s(x1237),s(x1236)))
              • Applying Rule "Variable in Equation" allows to substitute x568 by s(x1237) which results in
                (gr(x1237,x1236)→^*false⟹0→^*x1230⟹gr(s(x1237),x1230)→^*true⟹0→^*x1231⟹gr(x569,x1231)→^*true⟹x569→^*s(x1236)⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥cond4^#(gr(x1236,0),x1237,x1236))⟹cond2^#(false,s(x1237),s(x1236))≥cond4^#(gr(s(x1236),0),s(x1237),s(x1236)))
              • Applying Rule "Variable in Equation" allows to substitute x569 by s(x1236) which results in
                (gr(x1237,x1236)→^*false⟹0→^*x1230⟹gr(s(x1237),x1230)→^*true⟹0→^*x1231⟹gr(s(x1236),x1231)→^*true⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥cond4^#(gr(x1236,0),x1237,x1236))⟹cond2^#(false,s(x1237),s(x1236))≥cond4^#(gr(s(x1236),0),s(x1237),s(x1236)))
              • Applying Rule "Introduce fresh variable" results in
                (gr(x1237,x1236)→^*false⟹0→^*x1230⟹s(x1237)→^*x1249⟹gr(x1249,x1230)→^*true⟹0→^*x1231⟹gr(s(x1236),x1231)→^*true⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥cond4^#(gr(x1236,0),x1237,x1236))⟹cond2^#(false,s(x1237),s(x1236))≥cond4^#(gr(s(x1236),0),s(x1237),s(x1236)))
              • Applying Rule "Introduce fresh variable" results in
                (gr(x1237,x1236)→^*false⟹0→^*x1230⟹s(x1237)→^*x1249⟹gr(x1249,x1230)→^*true⟹0→^*x1231⟹s(x1236)→^*x1250⟹gr(x1250,x1231)→^*true⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥cond4^#(gr(x1236,0),x1237,x1236))⟹cond2^#(false,s(x1237),s(x1236))≥cond4^#(gr(s(x1236),0),s(x1237),s(x1236)))
              • Applying Rule "Induction" on gr(x1249,x1230)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond2^#(false,s(x1237),s(x1236))≥cond4^#(gr(s(x1236),0),s(x1237),s(x1236)))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹gr(x1237,x1236)→^*false⟹0→^*0⟹s(x1237)→^*s(x1252)⟹0→^*x1231⟹s(x1236)→^*x1250⟹gr(x1250,x1231)→^*true⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥cond4^#(gr(x1236,0),x1237,x1236))⟹cond2^#(false,s(x1237),s(x1236))≥cond4^#(gr(s(x1236),0),s(x1237),s(x1236)))
                   • Applying Rule "Same Constructor" results in
                     (gr(x1237,x1236)→^*false⟹0→^*0⟹s(x1237)→^*s(x1252)⟹0→^*x1231⟹s(x1236)→^*x1250⟹gr(x1250,x1231)→^*true⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥cond4^#(gr(x1236,0),x1237,x1236))⟹cond2^#(false,s(x1237),s(x1236))≥cond4^#(gr(s(x1236),0),s(x1237),s(x1236)))
                   • Applying Rule "Same Constructor" results in
                     (gr(x1237,x1236)→^*false⟹s(x1237)→^*s(x1252)⟹0→^*x1231⟹s(x1236)→^*x1250⟹gr(x1250,x1231)→^*true⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥cond4^#(gr(x1236,0),x1237,x1236))⟹cond2^#(false,s(x1237),s(x1236))≥cond4^#(gr(s(x1236),0),s(x1237),s(x1236)))
                   • Applying Rule "Same Constructor" results in
                     (gr(x1237,x1236)→^*false⟹x1237→^*x1252⟹0→^*x1231⟹s(x1236)→^*x1250⟹gr(x1250,x1231)→^*true⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥cond4^#(gr(x1236,0),x1237,x1236))⟹cond2^#(false,s(x1237),s(x1236))≥cond4^#(gr(s(x1236),0),s(x1237),s(x1236)))
                   • Applying Rule "Delete Conditions" results in
                     (gr(x1237,x1236)→^*false⟹0→^*x1231⟹s(x1236)→^*x1250⟹gr(x1250,x1231)→^*true⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥cond4^#(gr(x1236,0),x1237,x1236))⟹cond2^#(false,s(x1237),s(x1236))≥cond4^#(gr(s(x1236),0),s(x1237),s(x1236)))
                   • Applying Rule "Induction" on gr(x1250,x1231)→^*true results in the following 3 new constraints.
                     1. (false→^*true⟹cond2^#(false,s(x1237),s(x1236))≥cond4^#(gr(s(x1236),0),s(x1237),s(x1236)))
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                     2. (true→^*true⟹gr(x1237,x1236)→^*false⟹0→^*0⟹s(x1236)→^*s(x1264)⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥cond4^#(gr(x1236,0),x1237,x1236))⟹cond2^#(false,s(x1237),s(x1236))≥cond4^#(gr(s(x1236),0),s(x1237),s(x1236)))
                        • Applying Rule "Same Constructor" results in
                          (gr(x1237,x1236)→^*false⟹0→^*0⟹s(x1236)→^*s(x1264)⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥cond4^#(gr(x1236,0),x1237,x1236))⟹cond2^#(false,s(x1237),s(x1236))≥cond4^#(gr(s(x1236),0),s(x1237),s(x1236)))
                        • Applying Rule "Same Constructor" results in
                          (gr(x1237,x1236)→^*false⟹s(x1236)→^*s(x1264)⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥cond4^#(gr(x1236,0),x1237,x1236))⟹cond2^#(false,s(x1237),s(x1236))≥cond4^#(gr(s(x1236),0),s(x1237),s(x1236)))
                        • Applying Rule "Same Constructor" results in
                          (gr(x1237,x1236)→^*false⟹x1236→^*x1264⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥cond4^#(gr(x1236,0),x1237,x1236))⟹cond2^#(false,s(x1237),s(x1236))≥cond4^#(gr(s(x1236),0),s(x1237),s(x1236)))
                        • Applying Rule "Delete Conditions" results in
                          (gr(x1237,x1236)→^*false⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥cond4^#(gr(x1236,0),x1237,x1236))⟹cond2^#(false,s(x1237),s(x1236))≥cond4^#(gr(s(x1236),0),s(x1237),s(x1236)))
                        • Applying Rule "Delete Conditions" results in
                          (gr(x1237,x1236)→^*false⟹cond2^#(false,s(x1237),s(x1236))≥cond4^#(gr(s(x1236),0),s(x1237),s(x1236)))
                        • Applying Rule "Induction" on gr(x1237,x1236)→^*false results in the following 3 new constraints.
                          1. (false→^*false⟹cond2^#(false,s(0),s(x1273))≥cond4^#(gr(s(x1273),0),s(0),s(x1273)))
                             • Applying Rule "Same Constructor" results in
                               cond2^#(false,s(0),s(x1273))≥cond4^#(gr(s(x1273),0),s(0),s(x1273))

                             • This constraint is kept as final constraint.

                          2. (true→^*false⟹cond2^#(false,s(s(x1274)),s(0))≥cond4^#(gr(s(0),0),s(s(x1274)),s(0)))
                             • Applying Rule "Different Constructors" allows to drop this constraint.
                          3. (gr(x1276,x1275)→^*false⟹(gr(x1276,x1275)→^*false⟹cond2^#(false,s(x1276),s(x1275))≥cond4^#(gr(s(x1275),0),s(x1276),s(x1275)))⟹cond2^#(false,s(s(x1276)),s(s(x1275)))≥cond4^#(gr(s(s(x1275)),0),s(s(x1276)),s(s(x1275))))
                             • Applying Rule "Simplify Conditions" results in
                               (cond2^#(false,s(x1276),s(x1275))≥cond4^#(gr(s(x1275),0),s(x1276),s(x1275))⟹cond2^#(false,s(s(x1276)),s(s(x1275)))≥cond4^#(gr(s(s(x1275)),0),s(s(x1276)),s(s(x1275))))

                             • This constraint is kept as final constraint.

                     3. (gr(x1266,x1265)→^*true⟹gr(x1237,x1236)→^*false⟹0→^*s(x1265)⟹s(x1236)→^*s(x1266)⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥cond4^#(gr(x1236,0),x1237,x1236))⟹∀x1267 . ∀x1268 . ∀x1269 . ∀x1270 . ∀x1271 . ∀x1272 . (gr(x1266,x1265)→^*true⟹gr(x1267,x1268)→^*false⟹0→^*x1265⟹s(x1268)→^*x1266⟹∀x1269 . ∀x1270 . ∀x1271 . ∀x1272 . (gr(x1267,x1268)→^*false⟹0→^*x1269⟹gr(x1270,x1269)→^*true⟹0→^*x1271⟹gr(x1272,x1271)→^*true⟹x1270→^*x1267⟹x1272→^*x1268⟹cond2^#(false,x1267,x1268)≥cond4^#(gr(x1268,0),x1267,x1268))⟹cond2^#(false,s(x1267),s(x1268))≥cond4^#(gr(s(x1268),0),s(x1267),s(x1268)))⟹cond2^#(false,s(x1237),s(x1236))≥cond4^#(gr(s(x1236),0),s(x1237),s(x1236)))
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x1254,x1253)→^*true⟹gr(x1237,x1236)→^*false⟹0→^*s(x1253)⟹s(x1237)→^*s(x1254)⟹0→^*x1231⟹s(x1236)→^*x1250⟹gr(x1250,x1231)→^*true⟹∀x1238 . ∀x1239 . ∀x1240 . ∀x1241 . (gr(x1237,x1236)→^*false⟹0→^*x1238⟹gr(x1239,x1238)→^*true⟹0→^*x1240⟹gr(x1241,x1240)→^*true⟹x1239→^*x1237⟹x1241→^*x1236⟹cond2^#(false,x1237,x1236)≥cond4^#(gr(x1236,0),x1237,x1236))⟹∀x1255 . ∀x1256 . ∀x1257 . ∀x1258 . ∀x1259 . ∀x1260 . ∀x1261 . ∀x1262 . (gr(x1254,x1253)→^*true⟹gr(x1255,x1256)→^*false⟹0→^*x1253⟹s(x1255)→^*x1254⟹0→^*x1257⟹s(x1256)→^*x1258⟹gr(x1258,x1257)→^*true⟹∀x1259 . ∀x1260 . ∀x1261 . ∀x1262 . (gr(x1255,x1256)→^*false⟹0→^*x1259⟹gr(x1260,x1259)→^*true⟹0→^*x1261⟹gr(x1262,x1261)→^*true⟹x1260→^*x1255⟹x1262→^*x1256⟹cond2^#(false,x1255,x1256)≥cond4^#(gr(x1256,0),x1255,x1256))⟹cond2^#(false,s(x1255),s(x1256))≥cond4^#(gr(s(x1256),0),s(x1255),s(x1256)))⟹cond2^#(false,s(x1237),s(x1236))≥cond4^#(gr(s(x1236),0),s(x1237),s(x1236)))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*true⟹cond2^#(false,x570,x571)≥cond4^#(gr(x571,0),x570,x571))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (false→^*true⟹cond2^#(false,x570,x571)≥cond4^#(gr(x571,0),x570,x571))
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond1^#(and(gr(x580,0),gr(x581,0)),x580,x581)→^*cond1^#(true,x582,x583)⟹cond2^#(gr(x582,x583),x582,x583)→^*cond2^#(false,x584,x585)⟹cond2^#(false,x584,x585)≥cond4^#(gr(x585,0),x584,x585))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (and(gr(x580,0),gr(x581,0))→^*true⟹x580→^*x582⟹x581→^*x583⟹cond2^#(gr(x582,x583),x582,x583)→^*cond2^#(false,x584,x585)⟹cond2^#(false,x584,x585)≥cond4^#(gr(x585,0),x584,x585))
    • Applying Rule "Same Constructor" results in
      (and(gr(x580,0),gr(x581,0))→^*true⟹x580→^*x582⟹x581→^*x583⟹gr(x582,x583)→^*false⟹x582→^*x584⟹x583→^*x585⟹cond2^#(false,x584,x585)≥cond4^#(gr(x585,0),x584,x585))
    • Applying Rule "Variable in Equation" allows to substitute x584 by x582 which results in
      (and(gr(x580,0),gr(x581,0))→^*true⟹x580→^*x582⟹x581→^*x583⟹gr(x582,x583)→^*false⟹x583→^*x585⟹cond2^#(false,x582,x585)≥cond4^#(gr(x585,0),x582,x585))
    • Applying Rule "Variable in Equation" allows to substitute x585 by x583 which results in
      (and(gr(x580,0),gr(x581,0))→^*true⟹x580→^*x582⟹x581→^*x583⟹gr(x582,x583)→^*false⟹cond2^#(false,x582,x583)≥cond4^#(gr(x583,0),x582,x583))
    • Applying Rule "Introduce fresh variable" results in
      (x580→^*x582⟹x581→^*x583⟹gr(x582,x583)→^*false⟹gr(x581,0)→^*x1278⟹and(gr(x580,0),x1278)→^*true⟹cond2^#(false,x582,x583)≥cond4^#(gr(x583,0),x582,x583))
    • Applying Rule "Introduce fresh variable" results in
      (gr(x580,0)→^*x1277⟹gr(x581,0)→^*x1278⟹and(x1277,x1278)→^*true⟹x580→^*x582⟹x581→^*x583⟹gr(x582,x583)→^*false⟹cond2^#(false,x582,x583)≥cond4^#(gr(x583,0),x582,x583))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1279⟹gr(x580,x1279)→^*x1277⟹gr(x581,0)→^*x1278⟹and(x1277,x1278)→^*true⟹x580→^*x582⟹x581→^*x583⟹gr(x582,x583)→^*false⟹cond2^#(false,x582,x583)≥cond4^#(gr(x583,0),x582,x583))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1279⟹gr(x580,x1279)→^*x1277⟹0→^*x1280⟹gr(x581,x1280)→^*x1278⟹and(x1277,x1278)→^*true⟹x580→^*x582⟹x581→^*x583⟹gr(x582,x583)→^*false⟹cond2^#(false,x582,x583)≥cond4^#(gr(x583,0),x582,x583))
    • Applying Rule "Induction" on and(x1277,x1278)→^*true results in the following 3 new constraints.
      1. (true→^*true⟹0→^*x1279⟹gr(x580,x1279)→^*true⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹x580→^*x582⟹x581→^*x583⟹gr(x582,x583)→^*false⟹cond2^#(false,x582,x583)≥cond4^#(gr(x583,0),x582,x583))
         • Applying Rule "Same Constructor" results in
           (0→^*x1279⟹gr(x580,x1279)→^*true⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹x580→^*x582⟹x581→^*x583⟹gr(x582,x583)→^*false⟹cond2^#(false,x582,x583)≥cond4^#(gr(x583,0),x582,x583))
         • Applying Rule "Induction" on gr(x582,x583)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹0→^*x1279⟹gr(x580,x1279)→^*true⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹x580→^*0⟹x581→^*x1283⟹cond2^#(false,0,x1283)≥cond4^#(gr(x1283,0),0,x1283))
              • Applying Rule "Same Constructor" results in
                (0→^*x1279⟹gr(x580,x1279)→^*true⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹x580→^*0⟹x581→^*x1283⟹cond2^#(false,0,x1283)≥cond4^#(gr(x1283,0),0,x1283))
              • Applying Rule "Variable in Equation" allows to substitute x1283 by x581 which results in
                (0→^*x1279⟹gr(x580,x1279)→^*true⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹x580→^*0⟹cond2^#(false,0,x581)≥cond4^#(gr(x581,0),0,x581))
              • Applying Rule "Variable in Equation" allows to substitute x580 by 0 which results in
                (0→^*x1279⟹gr(0,x1279)→^*true⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹cond2^#(false,0,x581)≥cond4^#(gr(x581,0),0,x581))
              • Applying Rule "Introduce fresh variable" results in
                (0→^*x1279⟹0→^*x1291⟹gr(x1291,x1279)→^*true⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹cond2^#(false,0,x581)≥cond4^#(gr(x581,0),0,x581))
              • Applying Rule "Induction" on gr(x1291,x1279)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond2^#(false,0,x581)≥cond4^#(gr(x581,0),0,x581))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹0→^*0⟹0→^*s(x1293)⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹cond2^#(false,0,x581)≥cond4^#(gr(x581,0),0,x581))
                   • Applying Rule "Same Constructor" results in
                     (0→^*0⟹0→^*s(x1293)⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹cond2^#(false,0,x581)≥cond4^#(gr(x581,0),0,x581))
                   • Applying Rule "Same Constructor" results in
                     (0→^*s(x1293)⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹cond2^#(false,0,x581)≥cond4^#(gr(x581,0),0,x581))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x1295,x1294)→^*true⟹0→^*s(x1294)⟹0→^*s(x1295)⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹∀x1296 . ∀x1297 . (gr(x1295,x1294)→^*true⟹0→^*x1294⟹0→^*x1295⟹0→^*x1296⟹gr(x1297,x1296)→^*true⟹cond2^#(false,0,x1297)≥cond4^#(gr(x1297,0),0,x1297))⟹cond2^#(false,0,x581)≥cond4^#(gr(x581,0),0,x581))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond2^#(false,s(x1284),0)≥cond4^#(gr(0,0),s(x1284),0))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1286,x1285)→^*false⟹0→^*x1279⟹gr(x580,x1279)→^*true⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹x580→^*s(x1286)⟹x581→^*s(x1285)⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥cond4^#(gr(x1285,0),x1286,x1285))⟹cond2^#(false,s(x1286),s(x1285))≥cond4^#(gr(s(x1285),0),s(x1286),s(x1285)))
              • Applying Rule "Variable in Equation" allows to substitute x580 by s(x1286) which results in
                (gr(x1286,x1285)→^*false⟹0→^*x1279⟹gr(s(x1286),x1279)→^*true⟹0→^*x1280⟹gr(x581,x1280)→^*true⟹x581→^*s(x1285)⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥cond4^#(gr(x1285,0),x1286,x1285))⟹cond2^#(false,s(x1286),s(x1285))≥cond4^#(gr(s(x1285),0),s(x1286),s(x1285)))
              • Applying Rule "Variable in Equation" allows to substitute x581 by s(x1285) which results in
                (gr(x1286,x1285)→^*false⟹0→^*x1279⟹gr(s(x1286),x1279)→^*true⟹0→^*x1280⟹gr(s(x1285),x1280)→^*true⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥cond4^#(gr(x1285,0),x1286,x1285))⟹cond2^#(false,s(x1286),s(x1285))≥cond4^#(gr(s(x1285),0),s(x1286),s(x1285)))
              • Applying Rule "Introduce fresh variable" results in
                (gr(x1286,x1285)→^*false⟹0→^*x1279⟹s(x1286)→^*x1298⟹gr(x1298,x1279)→^*true⟹0→^*x1280⟹gr(s(x1285),x1280)→^*true⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥cond4^#(gr(x1285,0),x1286,x1285))⟹cond2^#(false,s(x1286),s(x1285))≥cond4^#(gr(s(x1285),0),s(x1286),s(x1285)))
              • Applying Rule "Introduce fresh variable" results in
                (gr(x1286,x1285)→^*false⟹0→^*x1279⟹s(x1286)→^*x1298⟹gr(x1298,x1279)→^*true⟹0→^*x1280⟹s(x1285)→^*x1299⟹gr(x1299,x1280)→^*true⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥cond4^#(gr(x1285,0),x1286,x1285))⟹cond2^#(false,s(x1286),s(x1285))≥cond4^#(gr(s(x1285),0),s(x1286),s(x1285)))
              • Applying Rule "Induction" on gr(x1298,x1279)→^*true results in the following 3 new constraints.
                1. (false→^*true⟹cond2^#(false,s(x1286),s(x1285))≥cond4^#(gr(s(x1285),0),s(x1286),s(x1285)))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (true→^*true⟹gr(x1286,x1285)→^*false⟹0→^*0⟹s(x1286)→^*s(x1301)⟹0→^*x1280⟹s(x1285)→^*x1299⟹gr(x1299,x1280)→^*true⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥cond4^#(gr(x1285,0),x1286,x1285))⟹cond2^#(false,s(x1286),s(x1285))≥cond4^#(gr(s(x1285),0),s(x1286),s(x1285)))
                   • Applying Rule "Same Constructor" results in
                     (gr(x1286,x1285)→^*false⟹0→^*0⟹s(x1286)→^*s(x1301)⟹0→^*x1280⟹s(x1285)→^*x1299⟹gr(x1299,x1280)→^*true⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥cond4^#(gr(x1285,0),x1286,x1285))⟹cond2^#(false,s(x1286),s(x1285))≥cond4^#(gr(s(x1285),0),s(x1286),s(x1285)))
                   • Applying Rule "Same Constructor" results in
                     (gr(x1286,x1285)→^*false⟹s(x1286)→^*s(x1301)⟹0→^*x1280⟹s(x1285)→^*x1299⟹gr(x1299,x1280)→^*true⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥cond4^#(gr(x1285,0),x1286,x1285))⟹cond2^#(false,s(x1286),s(x1285))≥cond4^#(gr(s(x1285),0),s(x1286),s(x1285)))
                   • Applying Rule "Same Constructor" results in
                     (gr(x1286,x1285)→^*false⟹x1286→^*x1301⟹0→^*x1280⟹s(x1285)→^*x1299⟹gr(x1299,x1280)→^*true⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥cond4^#(gr(x1285,0),x1286,x1285))⟹cond2^#(false,s(x1286),s(x1285))≥cond4^#(gr(s(x1285),0),s(x1286),s(x1285)))
                   • Applying Rule "Delete Conditions" results in
                     (gr(x1286,x1285)→^*false⟹0→^*x1280⟹s(x1285)→^*x1299⟹gr(x1299,x1280)→^*true⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥cond4^#(gr(x1285,0),x1286,x1285))⟹cond2^#(false,s(x1286),s(x1285))≥cond4^#(gr(s(x1285),0),s(x1286),s(x1285)))
                   • Applying Rule "Induction" on gr(x1299,x1280)→^*true results in the following 3 new constraints.
                     1. (false→^*true⟹cond2^#(false,s(x1286),s(x1285))≥cond4^#(gr(s(x1285),0),s(x1286),s(x1285)))
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                     2. (true→^*true⟹gr(x1286,x1285)→^*false⟹0→^*0⟹s(x1285)→^*s(x1313)⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥cond4^#(gr(x1285,0),x1286,x1285))⟹cond2^#(false,s(x1286),s(x1285))≥cond4^#(gr(s(x1285),0),s(x1286),s(x1285)))
                        • Applying Rule "Same Constructor" results in
                          (gr(x1286,x1285)→^*false⟹0→^*0⟹s(x1285)→^*s(x1313)⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥cond4^#(gr(x1285,0),x1286,x1285))⟹cond2^#(false,s(x1286),s(x1285))≥cond4^#(gr(s(x1285),0),s(x1286),s(x1285)))
                        • Applying Rule "Same Constructor" results in
                          (gr(x1286,x1285)→^*false⟹s(x1285)→^*s(x1313)⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥cond4^#(gr(x1285,0),x1286,x1285))⟹cond2^#(false,s(x1286),s(x1285))≥cond4^#(gr(s(x1285),0),s(x1286),s(x1285)))
                        • Applying Rule "Same Constructor" results in
                          (gr(x1286,x1285)→^*false⟹x1285→^*x1313⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥cond4^#(gr(x1285,0),x1286,x1285))⟹cond2^#(false,s(x1286),s(x1285))≥cond4^#(gr(s(x1285),0),s(x1286),s(x1285)))
                        • Applying Rule "Delete Conditions" results in
                          (gr(x1286,x1285)→^*false⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥cond4^#(gr(x1285,0),x1286,x1285))⟹cond2^#(false,s(x1286),s(x1285))≥cond4^#(gr(s(x1285),0),s(x1286),s(x1285)))
                        • Applying Rule "Delete Conditions" results in
                          (gr(x1286,x1285)→^*false⟹cond2^#(false,s(x1286),s(x1285))≥cond4^#(gr(s(x1285),0),s(x1286),s(x1285)))
                        • Applying Rule "Induction" on gr(x1286,x1285)→^*false results in the following 3 new constraints.
                          1. (false→^*false⟹cond2^#(false,s(0),s(x1322))≥cond4^#(gr(s(x1322),0),s(0),s(x1322)))
                             • Applying Rule "Same Constructor" results in
                               cond2^#(false,s(0),s(x1322))≥cond4^#(gr(s(x1322),0),s(0),s(x1322))

                             • This constraint is kept as final constraint.

                          2. (true→^*false⟹cond2^#(false,s(s(x1323)),s(0))≥cond4^#(gr(s(0),0),s(s(x1323)),s(0)))
                             • Applying Rule "Different Constructors" allows to drop this constraint.
                          3. (gr(x1325,x1324)→^*false⟹(gr(x1325,x1324)→^*false⟹cond2^#(false,s(x1325),s(x1324))≥cond4^#(gr(s(x1324),0),s(x1325),s(x1324)))⟹cond2^#(false,s(s(x1325)),s(s(x1324)))≥cond4^#(gr(s(s(x1324)),0),s(s(x1325)),s(s(x1324))))
                             • Applying Rule "Simplify Conditions" results in
                               (cond2^#(false,s(x1325),s(x1324))≥cond4^#(gr(s(x1324),0),s(x1325),s(x1324))⟹cond2^#(false,s(s(x1325)),s(s(x1324)))≥cond4^#(gr(s(s(x1324)),0),s(s(x1325)),s(s(x1324))))

                             • This constraint is kept as final constraint.

                     3. (gr(x1315,x1314)→^*true⟹gr(x1286,x1285)→^*false⟹0→^*s(x1314)⟹s(x1285)→^*s(x1315)⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥cond4^#(gr(x1285,0),x1286,x1285))⟹∀x1316 . ∀x1317 . ∀x1318 . ∀x1319 . ∀x1320 . ∀x1321 . (gr(x1315,x1314)→^*true⟹gr(x1316,x1317)→^*false⟹0→^*x1314⟹s(x1317)→^*x1315⟹∀x1318 . ∀x1319 . ∀x1320 . ∀x1321 . (gr(x1316,x1317)→^*false⟹0→^*x1318⟹gr(x1319,x1318)→^*true⟹0→^*x1320⟹gr(x1321,x1320)→^*true⟹x1319→^*x1316⟹x1321→^*x1317⟹cond2^#(false,x1316,x1317)≥cond4^#(gr(x1317,0),x1316,x1317))⟹cond2^#(false,s(x1316),s(x1317))≥cond4^#(gr(s(x1317),0),s(x1316),s(x1317)))⟹cond2^#(false,s(x1286),s(x1285))≥cond4^#(gr(s(x1285),0),s(x1286),s(x1285)))
                        • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x1303,x1302)→^*true⟹gr(x1286,x1285)→^*false⟹0→^*s(x1302)⟹s(x1286)→^*s(x1303)⟹0→^*x1280⟹s(x1285)→^*x1299⟹gr(x1299,x1280)→^*true⟹∀x1287 . ∀x1288 . ∀x1289 . ∀x1290 . (gr(x1286,x1285)→^*false⟹0→^*x1287⟹gr(x1288,x1287)→^*true⟹0→^*x1289⟹gr(x1290,x1289)→^*true⟹x1288→^*x1286⟹x1290→^*x1285⟹cond2^#(false,x1286,x1285)≥cond4^#(gr(x1285,0),x1286,x1285))⟹∀x1304 . ∀x1305 . ∀x1306 . ∀x1307 . ∀x1308 . ∀x1309 . ∀x1310 . ∀x1311 . (gr(x1303,x1302)→^*true⟹gr(x1304,x1305)→^*false⟹0→^*x1302⟹s(x1304)→^*x1303⟹0→^*x1306⟹s(x1305)→^*x1307⟹gr(x1307,x1306)→^*true⟹∀x1308 . ∀x1309 . ∀x1310 . ∀x1311 . (gr(x1304,x1305)→^*false⟹0→^*x1308⟹gr(x1309,x1308)→^*true⟹0→^*x1310⟹gr(x1311,x1310)→^*true⟹x1309→^*x1304⟹x1311→^*x1305⟹cond2^#(false,x1304,x1305)≥cond4^#(gr(x1305,0),x1304,x1305))⟹cond2^#(false,s(x1304),s(x1305))≥cond4^#(gr(s(x1305),0),s(x1304),s(x1305)))⟹cond2^#(false,s(x1286),s(x1285))≥cond4^#(gr(s(x1285),0),s(x1286),s(x1285)))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (false→^*true⟹cond2^#(false,x582,x583)≥cond4^#(gr(x583,0),x582,x583))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (false→^*true⟹cond2^#(false,x582,x583)≥cond4^#(gr(x583,0),x582,x583))
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond2^#(gr(x714,x715),x714,x715)→^*cond2^#(false,x716,x717)⟹cond4^#(gr(x717,0),x716,x717)→^*cond4^#(true,x718,x719)⟹cond4^#(true,x718,x719)≥cond4^#(gr(x719,0),x718,p(x719)))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x714,x715)→^*false⟹x714→^*x716⟹x715→^*x717⟹cond4^#(gr(x717,0),x716,x717)→^*cond4^#(true,x718,x719)⟹cond4^#(true,x718,x719)≥cond4^#(gr(x719,0),x718,p(x719)))
    • Applying Rule "Same Constructor" results in
      (gr(x714,x715)→^*false⟹x714→^*x716⟹x715→^*x717⟹gr(x717,0)→^*true⟹x716→^*x718⟹x717→^*x719⟹cond4^#(true,x718,x719)≥cond4^#(gr(x719,0),x718,p(x719)))
    • Applying Rule "Variable in Equation" allows to substitute x716 by x714 which results in
      (gr(x714,x715)→^*false⟹x715→^*x717⟹gr(x717,0)→^*true⟹x714→^*x718⟹x717→^*x719⟹cond4^#(true,x718,x719)≥cond4^#(gr(x719,0),x718,p(x719)))
    • Applying Rule "Variable in Equation" allows to substitute x718 by x714 which results in
      (gr(x714,x715)→^*false⟹x715→^*x717⟹gr(x717,0)→^*true⟹x717→^*x719⟹cond4^#(true,x714,x719)≥cond4^#(gr(x719,0),x714,p(x719)))
    • Applying Rule "Variable in Equation" allows to substitute x719 by x717 which results in
      (gr(x714,x715)→^*false⟹x715→^*x717⟹gr(x717,0)→^*true⟹cond4^#(true,x714,x717)≥cond4^#(gr(x717,0),x714,p(x717)))
    • Applying Rule "Introduce fresh variable" results in
      (gr(x714,x715)→^*false⟹x715→^*x717⟹0→^*x1326⟹gr(x717,x1326)→^*true⟹cond4^#(true,x714,x717)≥cond4^#(gr(x717,0),x714,p(x717)))
    • Applying Rule "Induction" on gr(x714,x715)→^*false results in the following 3 new constraints.
      1. (false→^*false⟹x1327→^*x717⟹0→^*x1326⟹gr(x717,x1326)→^*true⟹cond4^#(true,0,x717)≥cond4^#(gr(x717,0),0,p(x717)))
         • Applying Rule "Same Constructor" results in
           (x1327→^*x717⟹0→^*x1326⟹gr(x717,x1326)→^*true⟹cond4^#(true,0,x717)≥cond4^#(gr(x717,0),0,p(x717)))
         • Applying Rule "Variable in Equation" allows to substitute x1327 by x717 which results in
           (0→^*x1326⟹gr(x717,x1326)→^*true⟹cond4^#(true,0,x717)≥cond4^#(gr(x717,0),0,p(x717)))
         • Applying Rule "Induction" on gr(x717,x1326)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond4^#(true,0,0)≥cond4^#(gr(0,0),0,p(0)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹0→^*0⟹cond4^#(true,0,s(x1334))≥cond4^#(gr(s(x1334),0),0,p(s(x1334))))
              • Applying Rule "Same Constructor" results in
                (0→^*0⟹cond4^#(true,0,s(x1334))≥cond4^#(gr(s(x1334),0),0,p(s(x1334))))
              • Applying Rule "Same Constructor" results in
                cond4^#(true,0,s(x1334))≥cond4^#(gr(s(x1334),0),0,p(s(x1334)))

              • This constraint is kept as final constraint.

           3. (gr(x1336,x1335)→^*true⟹0→^*s(x1335)⟹(gr(x1336,x1335)→^*true⟹0→^*x1335⟹cond4^#(true,0,x1336)≥cond4^#(gr(x1336,0),0,p(x1336)))⟹cond4^#(true,0,s(x1336))≥cond4^#(gr(s(x1336),0),0,p(s(x1336))))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*false⟹cond4^#(true,s(x1328),x717)≥cond4^#(gr(x717,0),s(x1328),p(x717)))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x1330,x1329)→^*false⟹s(x1329)→^*x717⟹0→^*x1326⟹gr(x717,x1326)→^*true⟹∀x1331 . ∀x1332 . (gr(x1330,x1329)→^*false⟹x1329→^*x1331⟹0→^*x1332⟹gr(x1331,x1332)→^*true⟹cond4^#(true,x1330,x1331)≥cond4^#(gr(x1331,0),x1330,p(x1331)))⟹cond4^#(true,s(x1330),x717)≥cond4^#(gr(x717,0),s(x1330),p(x717)))
         • Applying Rule "Induction" on gr(x717,x1326)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond4^#(true,s(x1330),0)≥cond4^#(gr(0,0),s(x1330),p(0)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹gr(x1330,x1329)→^*false⟹s(x1329)→^*s(x1338)⟹0→^*0⟹∀x1331 . ∀x1332 . (gr(x1330,x1329)→^*false⟹x1329→^*x1331⟹0→^*x1332⟹gr(x1331,x1332)→^*true⟹cond4^#(true,x1330,x1331)≥cond4^#(gr(x1331,0),x1330,p(x1331)))⟹cond4^#(true,s(x1330),s(x1338))≥cond4^#(gr(s(x1338),0),s(x1330),p(s(x1338))))
              • Applying Rule "Same Constructor" results in
                (gr(x1330,x1329)→^*false⟹s(x1329)→^*s(x1338)⟹0→^*0⟹∀x1331 . ∀x1332 . (gr(x1330,x1329)→^*false⟹x1329→^*x1331⟹0→^*x1332⟹gr(x1331,x1332)→^*true⟹cond4^#(true,x1330,x1331)≥cond4^#(gr(x1331,0),x1330,p(x1331)))⟹cond4^#(true,s(x1330),s(x1338))≥cond4^#(gr(s(x1338),0),s(x1330),p(s(x1338))))
              • Applying Rule "Same Constructor" results in
                (gr(x1330,x1329)→^*false⟹x1329→^*x1338⟹0→^*0⟹∀x1331 . ∀x1332 . (gr(x1330,x1329)→^*false⟹x1329→^*x1331⟹0→^*x1332⟹gr(x1331,x1332)→^*true⟹cond4^#(true,x1330,x1331)≥cond4^#(gr(x1331,0),x1330,p(x1331)))⟹cond4^#(true,s(x1330),s(x1338))≥cond4^#(gr(s(x1338),0),s(x1330),p(s(x1338))))
              • Applying Rule "Same Constructor" results in
                (gr(x1330,x1329)→^*false⟹x1329→^*x1338⟹∀x1331 . ∀x1332 . (gr(x1330,x1329)→^*false⟹x1329→^*x1331⟹0→^*x1332⟹gr(x1331,x1332)→^*true⟹cond4^#(true,x1330,x1331)≥cond4^#(gr(x1331,0),x1330,p(x1331)))⟹cond4^#(true,s(x1330),s(x1338))≥cond4^#(gr(s(x1338),0),s(x1330),p(s(x1338))))
              • Applying Rule "Variable in Equation" allows to substitute x1338 by x1329 which results in
                (gr(x1330,x1329)→^*false⟹∀x1331 . ∀x1332 . (gr(x1330,x1329)→^*false⟹x1329→^*x1331⟹0→^*x1332⟹gr(x1331,x1332)→^*true⟹cond4^#(true,x1330,x1331)≥cond4^#(gr(x1331,0),x1330,p(x1331)))⟹cond4^#(true,s(x1330),s(x1329))≥cond4^#(gr(s(x1329),0),s(x1330),p(s(x1329))))
              • Applying Rule "Delete Conditions" results in
                (gr(x1330,x1329)→^*false⟹cond4^#(true,s(x1330),s(x1329))≥cond4^#(gr(s(x1329),0),s(x1330),p(s(x1329))))
              • Applying Rule "Induction" on gr(x1330,x1329)→^*false results in the following 3 new constraints.
                1. (false→^*false⟹cond4^#(true,s(0),s(x1345))≥cond4^#(gr(s(x1345),0),s(0),p(s(x1345))))
                   • Applying Rule "Same Constructor" results in
                     cond4^#(true,s(0),s(x1345))≥cond4^#(gr(s(x1345),0),s(0),p(s(x1345)))

                   • This constraint is kept as final constraint.

                2. (true→^*false⟹cond4^#(true,s(s(x1346)),s(0))≥cond4^#(gr(s(0),0),s(s(x1346)),p(s(0))))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                3. (gr(x1348,x1347)→^*false⟹(gr(x1348,x1347)→^*false⟹cond4^#(true,s(x1348),s(x1347))≥cond4^#(gr(s(x1347),0),s(x1348),p(s(x1347))))⟹cond4^#(true,s(s(x1348)),s(s(x1347)))≥cond4^#(gr(s(s(x1347)),0),s(s(x1348)),p(s(s(x1347)))))
                   • Applying Rule "Simplify Conditions" results in
                     (cond4^#(true,s(x1348),s(x1347))≥cond4^#(gr(s(x1347),0),s(x1348),p(s(x1347)))⟹cond4^#(true,s(s(x1348)),s(s(x1347)))≥cond4^#(gr(s(s(x1347)),0),s(s(x1348)),p(s(s(x1347)))))

                   • This constraint is kept as final constraint.

           3. (gr(x1340,x1339)→^*true⟹gr(x1330,x1329)→^*false⟹s(x1329)→^*s(x1340)⟹0→^*s(x1339)⟹∀x1331 . ∀x1332 . (gr(x1330,x1329)→^*false⟹x1329→^*x1331⟹0→^*x1332⟹gr(x1331,x1332)→^*true⟹cond4^#(true,x1330,x1331)≥cond4^#(gr(x1331,0),x1330,p(x1331)))⟹∀x1341 . ∀x1342 . ∀x1343 . ∀x1344 . (gr(x1340,x1339)→^*true⟹gr(x1341,x1342)→^*false⟹s(x1342)→^*x1340⟹0→^*x1339⟹∀x1343 . ∀x1344 . (gr(x1341,x1342)→^*false⟹x1342→^*x1343⟹0→^*x1344⟹gr(x1343,x1344)→^*true⟹cond4^#(true,x1341,x1343)≥cond4^#(gr(x1343,0),x1341,p(x1343)))⟹cond4^#(true,s(x1341),x1340)≥cond4^#(gr(x1340,0),s(x1341),p(x1340)))⟹cond4^#(true,s(x1330),s(x1340))≥cond4^#(gr(s(x1340),0),s(x1330),p(s(x1340))))
              • Applying Rule "Same Constructor" results in
                (gr(x1340,x1339)→^*true⟹gr(x1330,x1329)→^*false⟹x1329→^*x1340⟹0→^*s(x1339)⟹∀x1331 . ∀x1332 . (gr(x1330,x1329)→^*false⟹x1329→^*x1331⟹0→^*x1332⟹gr(x1331,x1332)→^*true⟹cond4^#(true,x1330,x1331)≥cond4^#(gr(x1331,0),x1330,p(x1331)))⟹∀x1341 . ∀x1342 . ∀x1343 . ∀x1344 . (gr(x1340,x1339)→^*true⟹gr(x1341,x1342)→^*false⟹s(x1342)→^*x1340⟹0→^*x1339⟹∀x1343 . ∀x1344 . (gr(x1341,x1342)→^*false⟹x1342→^*x1343⟹0→^*x1344⟹gr(x1343,x1344)→^*true⟹cond4^#(true,x1341,x1343)≥cond4^#(gr(x1343,0),x1341,p(x1343)))⟹cond4^#(true,s(x1341),x1340)≥cond4^#(gr(x1340,0),s(x1341),p(x1340)))⟹cond4^#(true,s(x1330),s(x1340))≥cond4^#(gr(s(x1340),0),s(x1330),p(s(x1340))))
              • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond4^#(gr(x735,0),x734,x735)→^*cond4^#(true,x736,x737)⟹cond4^#(gr(x737,0),x736,p(x737))→^*cond4^#(true,x738,x739)⟹cond4^#(true,x738,x739)≥cond4^#(gr(x739,0),x738,p(x739)))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x735,0)→^*true⟹x734→^*x736⟹x735→^*x737⟹cond4^#(gr(x737,0),x736,p(x737))→^*cond4^#(true,x738,x739)⟹cond4^#(true,x738,x739)≥cond4^#(gr(x739,0),x738,p(x739)))
    • Applying Rule "Same Constructor" results in
      (gr(x735,0)→^*true⟹x734→^*x736⟹x735→^*x737⟹gr(x737,0)→^*true⟹x736→^*x738⟹p(x737)→^*x739⟹cond4^#(true,x738,x739)≥cond4^#(gr(x739,0),x738,p(x739)))
    • Applying Rule "Variable in Equation" allows to substitute x736 by x734 which results in
      (gr(x735,0)→^*true⟹x735→^*x737⟹gr(x737,0)→^*true⟹x734→^*x738⟹p(x737)→^*x739⟹cond4^#(true,x738,x739)≥cond4^#(gr(x739,0),x738,p(x739)))
    • Applying Rule "Variable in Equation" allows to substitute x738 by x734 which results in
      (gr(x735,0)→^*true⟹x735→^*x737⟹gr(x737,0)→^*true⟹p(x737)→^*x739⟹cond4^#(true,x734,x739)≥cond4^#(gr(x739,0),x734,p(x739)))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1349⟹gr(x735,x1349)→^*true⟹x735→^*x737⟹gr(x737,0)→^*true⟹p(x737)→^*x739⟹cond4^#(true,x734,x739)≥cond4^#(gr(x739,0),x734,p(x739)))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1349⟹gr(x735,x1349)→^*true⟹x735→^*x737⟹0→^*x1350⟹gr(x737,x1350)→^*true⟹p(x737)→^*x739⟹cond4^#(true,x734,x739)≥cond4^#(gr(x739,0),x734,p(x739)))
    • Applying Rule "Induction" on gr(x735,x1349)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹cond4^#(true,x734,x739)≥cond4^#(gr(x739,0),x734,p(x739)))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹0→^*0⟹s(x1352)→^*x737⟹0→^*x1350⟹gr(x737,x1350)→^*true⟹p(x737)→^*x739⟹cond4^#(true,x734,x739)≥cond4^#(gr(x739,0),x734,p(x739)))
         • Applying Rule "Same Constructor" results in
           (0→^*0⟹s(x1352)→^*x737⟹0→^*x1350⟹gr(x737,x1350)→^*true⟹p(x737)→^*x739⟹cond4^#(true,x734,x739)≥cond4^#(gr(x739,0),x734,p(x739)))
         • Applying Rule "Same Constructor" results in
           (s(x1352)→^*x737⟹0→^*x1350⟹gr(x737,x1350)→^*true⟹p(x737)→^*x739⟹cond4^#(true,x734,x739)≥cond4^#(gr(x739,0),x734,p(x739)))
         • Applying Rule "Induction" on gr(x737,x1350)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond4^#(true,x734,x739)≥cond4^#(gr(x739,0),x734,p(x739)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹s(x1352)→^*s(x1360)⟹0→^*0⟹p(s(x1360))→^*x739⟹cond4^#(true,x734,x739)≥cond4^#(gr(x739,0),x734,p(x739)))
              • Applying Rule "Same Constructor" results in
                (s(x1352)→^*s(x1360)⟹0→^*0⟹p(s(x1360))→^*x739⟹cond4^#(true,x734,x739)≥cond4^#(gr(x739,0),x734,p(x739)))
              • Applying Rule "Same Constructor" results in
                (x1352→^*x1360⟹0→^*0⟹p(s(x1360))→^*x739⟹cond4^#(true,x734,x739)≥cond4^#(gr(x739,0),x734,p(x739)))
              • Applying Rule "Same Constructor" results in
                (x1352→^*x1360⟹p(s(x1360))→^*x739⟹cond4^#(true,x734,x739)≥cond4^#(gr(x739,0),x734,p(x739)))
              • Applying Rule "Variable in Equation" allows to substitute x1352 by x1360 which results in
                (p(s(x1360))→^*x739⟹cond4^#(true,x734,x739)≥cond4^#(gr(x739,0),x734,p(x739)))
              • Applying Rule "Introduce fresh variable" results in
                (s(x1360)→^*x1366⟹p(x1366)→^*x739⟹cond4^#(true,x734,x739)≥cond4^#(gr(x739,0),x734,p(x739)))
              • Applying Rule "Variable in Equation" allows to substitute x1366 by s(x1360) which results in
                (p(s(x1360))→^*x739⟹cond4^#(true,x734,x739)≥cond4^#(gr(x739,0),x734,p(x739)))
              • Applying Rule "Delete Conditions" results in
                cond4^#(true,x734,x739)≥cond4^#(gr(x739,0),x734,p(x739))

              • This constraint is kept as final constraint.

           3. (gr(x1362,x1361)→^*true⟹s(x1352)→^*s(x1362)⟹0→^*s(x1361)⟹p(s(x1362))→^*x739⟹∀x1363 . ∀x1364 . ∀x1365 . (gr(x1362,x1361)→^*true⟹s(x1363)→^*x1362⟹0→^*x1361⟹p(x1362)→^*x1364⟹cond4^#(true,x1365,x1364)≥cond4^#(gr(x1364,0),x1365,p(x1364)))⟹cond4^#(true,x734,x739)≥cond4^#(gr(x739,0),x734,p(x739)))
              • Applying Rule "Same Constructor" results in
                (gr(x1362,x1361)→^*true⟹x1352→^*x1362⟹0→^*s(x1361)⟹p(s(x1362))→^*x739⟹∀x1363 . ∀x1364 . ∀x1365 . (gr(x1362,x1361)→^*true⟹s(x1363)→^*x1362⟹0→^*x1361⟹p(x1362)→^*x1364⟹cond4^#(true,x1365,x1364)≥cond4^#(gr(x1364,0),x1365,p(x1364)))⟹cond4^#(true,x734,x739)≥cond4^#(gr(x739,0),x734,p(x739)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x1354,x1353)→^*true⟹0→^*s(x1353)⟹s(x1354)→^*x737⟹0→^*x1350⟹gr(x737,x1350)→^*true⟹p(x737)→^*x739⟹∀x1355 . ∀x1356 . ∀x1357 . ∀x1358 . (gr(x1354,x1353)→^*true⟹0→^*x1353⟹x1354→^*x1355⟹0→^*x1356⟹gr(x1355,x1356)→^*true⟹p(x1355)→^*x1357⟹cond4^#(true,x1358,x1357)≥cond4^#(gr(x1357,0),x1358,p(x1357)))⟹cond4^#(true,x734,x739)≥cond4^#(gr(x739,0),x734,p(x739)))
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond4^#(gr(x741,0),x740,p(x741))→^*cond4^#(true,x742,x743)⟹cond4^#(gr(x743,0),x742,p(x743))→^*cond4^#(true,x744,x745)⟹cond4^#(true,x744,x745)≥cond4^#(gr(x745,0),x744,p(x745)))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x741,0)→^*true⟹x740→^*x742⟹p(x741)→^*x743⟹cond4^#(gr(x743,0),x742,p(x743))→^*cond4^#(true,x744,x745)⟹cond4^#(true,x744,x745)≥cond4^#(gr(x745,0),x744,p(x745)))
    • Applying Rule "Same Constructor" results in
      (gr(x741,0)→^*true⟹x740→^*x742⟹p(x741)→^*x743⟹gr(x743,0)→^*true⟹x742→^*x744⟹p(x743)→^*x745⟹cond4^#(true,x744,x745)≥cond4^#(gr(x745,0),x744,p(x745)))
    • Applying Rule "Variable in Equation" allows to substitute x742 by x740 which results in
      (gr(x741,0)→^*true⟹p(x741)→^*x743⟹gr(x743,0)→^*true⟹x740→^*x744⟹p(x743)→^*x745⟹cond4^#(true,x744,x745)≥cond4^#(gr(x745,0),x744,p(x745)))
    • Applying Rule "Variable in Equation" allows to substitute x744 by x740 which results in
      (gr(x741,0)→^*true⟹p(x741)→^*x743⟹gr(x743,0)→^*true⟹p(x743)→^*x745⟹cond4^#(true,x740,x745)≥cond4^#(gr(x745,0),x740,p(x745)))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1367⟹gr(x741,x1367)→^*true⟹p(x741)→^*x743⟹gr(x743,0)→^*true⟹p(x743)→^*x745⟹cond4^#(true,x740,x745)≥cond4^#(gr(x745,0),x740,p(x745)))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1367⟹gr(x741,x1367)→^*true⟹p(x741)→^*x743⟹0→^*x1368⟹gr(x743,x1368)→^*true⟹p(x743)→^*x745⟹cond4^#(true,x740,x745)≥cond4^#(gr(x745,0),x740,p(x745)))
    • Applying Rule "Induction" on gr(x741,x1367)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹cond4^#(true,x740,x745)≥cond4^#(gr(x745,0),x740,p(x745)))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹0→^*0⟹p(s(x1370))→^*x743⟹0→^*x1368⟹gr(x743,x1368)→^*true⟹p(x743)→^*x745⟹cond4^#(true,x740,x745)≥cond4^#(gr(x745,0),x740,p(x745)))
         • Applying Rule "Same Constructor" results in
           (0→^*0⟹p(s(x1370))→^*x743⟹0→^*x1368⟹gr(x743,x1368)→^*true⟹p(x743)→^*x745⟹cond4^#(true,x740,x745)≥cond4^#(gr(x745,0),x740,p(x745)))
         • Applying Rule "Same Constructor" results in
           (p(s(x1370))→^*x743⟹0→^*x1368⟹gr(x743,x1368)→^*true⟹p(x743)→^*x745⟹cond4^#(true,x740,x745)≥cond4^#(gr(x745,0),x740,p(x745)))
         • Applying Rule "Introduce fresh variable" results in
           (s(x1370)→^*x1377⟹p(x1377)→^*x743⟹0→^*x1368⟹gr(x743,x1368)→^*true⟹p(x743)→^*x745⟹cond4^#(true,x740,x745)≥cond4^#(gr(x745,0),x740,p(x745)))
         • Applying Rule "Induction" on gr(x743,x1368)→^*true results in the following 3 new constraints.
           1. (false→^*true⟹cond4^#(true,x740,x745)≥cond4^#(gr(x745,0),x740,p(x745)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*true⟹s(x1370)→^*x1377⟹p(x1377)→^*s(x1379)⟹0→^*0⟹p(s(x1379))→^*x745⟹cond4^#(true,x740,x745)≥cond4^#(gr(x745,0),x740,p(x745)))
              • Applying Rule "Same Constructor" results in
                (s(x1370)→^*x1377⟹p(x1377)→^*s(x1379)⟹0→^*0⟹p(s(x1379))→^*x745⟹cond4^#(true,x740,x745)≥cond4^#(gr(x745,0),x740,p(x745)))
              • Applying Rule "Same Constructor" results in
                (s(x1370)→^*x1377⟹p(x1377)→^*s(x1379)⟹p(s(x1379))→^*x745⟹cond4^#(true,x740,x745)≥cond4^#(gr(x745,0),x740,p(x745)))
              • Applying Rule "Introduce fresh variable" results in
                (s(x1370)→^*x1377⟹p(x1377)→^*s(x1379)⟹s(x1379)→^*x1386⟹p(x1386)→^*x745⟹cond4^#(true,x740,x745)≥cond4^#(gr(x745,0),x740,p(x745)))
              • Applying Rule "Induction" on p(x1377)→^*s(x1379) results in the following 2 new constraints.
                1. (0→^*s(x1379)⟹cond4^#(true,x740,x745)≥cond4^#(gr(x745,0),x740,p(x745)))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (x1387→^*s(x1379)⟹s(x1370)→^*s(x1387)⟹s(x1379)→^*x1386⟹p(x1386)→^*x745⟹cond4^#(true,x740,x745)≥cond4^#(gr(x745,0),x740,p(x745)))
                   • Applying Rule "Same Constructor" results in
                     (x1387→^*s(x1379)⟹x1370→^*x1387⟹s(x1379)→^*x1386⟹p(x1386)→^*x745⟹cond4^#(true,x740,x745)≥cond4^#(gr(x745,0),x740,p(x745)))
                   • Applying Rule "Variable in Equation" allows to substitute x1387 by x1370 which results in
                     (x1370→^*s(x1379)⟹s(x1379)→^*x1386⟹p(x1386)→^*x745⟹cond4^#(true,x740,x745)≥cond4^#(gr(x745,0),x740,p(x745)))
                   • Applying Rule "Variable in Equation" allows to substitute x1370 by s(x1379) which results in
                     (s(x1379)→^*x1386⟹p(x1386)→^*x745⟹cond4^#(true,x740,x745)≥cond4^#(gr(x745,0),x740,p(x745)))
                   • Applying Rule "Variable in Equation" allows to substitute x1386 by s(x1379) which results in
                     (p(s(x1379))→^*x745⟹cond4^#(true,x740,x745)≥cond4^#(gr(x745,0),x740,p(x745)))
                   • Applying Rule "Delete Conditions" results in
                     cond4^#(true,x740,x745)≥cond4^#(gr(x745,0),x740,p(x745))

                   • This constraint is kept as final constraint.

           3. (gr(x1381,x1380)→^*true⟹s(x1370)→^*x1377⟹p(x1377)→^*s(x1381)⟹0→^*s(x1380)⟹p(s(x1381))→^*x745⟹∀x1382 . ∀x1383 . ∀x1384 . ∀x1385 . (gr(x1381,x1380)→^*true⟹s(x1382)→^*x1383⟹p(x1383)→^*x1381⟹0→^*x1380⟹p(x1381)→^*x1384⟹cond4^#(true,x1385,x1384)≥cond4^#(gr(x1384,0),x1385,p(x1384)))⟹cond4^#(true,x740,x745)≥cond4^#(gr(x745,0),x740,p(x745)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x1372,x1371)→^*true⟹0→^*s(x1371)⟹p(s(x1372))→^*x743⟹0→^*x1368⟹gr(x743,x1368)→^*true⟹p(x743)→^*x745⟹∀x1373 . ∀x1374 . ∀x1375 . ∀x1376 . (gr(x1372,x1371)→^*true⟹0→^*x1371⟹p(x1372)→^*x1373⟹0→^*x1374⟹gr(x1373,x1374)→^*true⟹p(x1373)→^*x1375⟹cond4^#(true,x1376,x1375)≥cond4^#(gr(x1375,0),x1376,p(x1375)))⟹cond4^#(true,x740,x745)≥cond4^#(gr(x745,0),x740,p(x745)))
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond2^#(gr(x842,x843),x842,x843)→^*cond2^#(false,x844,x845)⟹cond4^#(gr(x845,0),x844,x845)→^*cond4^#(false,x846,x847)⟹cond4^#(false,x846,x847)≥cond1^#(and(gr(x846,0),gr(x847,0)),x846,x847))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x842,x843)→^*false⟹x842→^*x844⟹x843→^*x845⟹cond4^#(gr(x845,0),x844,x845)→^*cond4^#(false,x846,x847)⟹cond4^#(false,x846,x847)≥cond1^#(and(gr(x846,0),gr(x847,0)),x846,x847))
    • Applying Rule "Same Constructor" results in
      (gr(x842,x843)→^*false⟹x842→^*x844⟹x843→^*x845⟹gr(x845,0)→^*false⟹x844→^*x846⟹x845→^*x847⟹cond4^#(false,x846,x847)≥cond1^#(and(gr(x846,0),gr(x847,0)),x846,x847))
    • Applying Rule "Variable in Equation" allows to substitute x844 by x842 which results in
      (gr(x842,x843)→^*false⟹x843→^*x845⟹gr(x845,0)→^*false⟹x842→^*x846⟹x845→^*x847⟹cond4^#(false,x846,x847)≥cond1^#(and(gr(x846,0),gr(x847,0)),x846,x847))
    • Applying Rule "Variable in Equation" allows to substitute x846 by x842 which results in
      (gr(x842,x843)→^*false⟹x843→^*x845⟹gr(x845,0)→^*false⟹x845→^*x847⟹cond4^#(false,x842,x847)≥cond1^#(and(gr(x842,0),gr(x847,0)),x842,x847))
    • Applying Rule "Variable in Equation" allows to substitute x847 by x845 which results in
      (gr(x842,x843)→^*false⟹x843→^*x845⟹gr(x845,0)→^*false⟹cond4^#(false,x842,x845)≥cond1^#(and(gr(x842,0),gr(x845,0)),x842,x845))
    • Applying Rule "Introduce fresh variable" results in
      (gr(x842,x843)→^*false⟹x843→^*x845⟹0→^*x1388⟹gr(x845,x1388)→^*false⟹cond4^#(false,x842,x845)≥cond1^#(and(gr(x842,0),gr(x845,0)),x842,x845))
    • Applying Rule "Induction" on gr(x842,x843)→^*false results in the following 3 new constraints.
      1. (false→^*false⟹x1389→^*x845⟹0→^*x1388⟹gr(x845,x1388)→^*false⟹cond4^#(false,0,x845)≥cond1^#(and(gr(0,0),gr(x845,0)),0,x845))
         • Applying Rule "Same Constructor" results in
           (x1389→^*x845⟹0→^*x1388⟹gr(x845,x1388)→^*false⟹cond4^#(false,0,x845)≥cond1^#(and(gr(0,0),gr(x845,0)),0,x845))
         • Applying Rule "Variable in Equation" allows to substitute x1389 by x845 which results in
           (0→^*x1388⟹gr(x845,x1388)→^*false⟹cond4^#(false,0,x845)≥cond1^#(and(gr(0,0),gr(x845,0)),0,x845))
         • Applying Rule "Induction" on gr(x845,x1388)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹0→^*x1395⟹cond4^#(false,0,0)≥cond1^#(and(gr(0,0),gr(0,0)),0,0))
              • Applying Rule "Same Constructor" results in
                (0→^*x1395⟹cond4^#(false,0,0)≥cond1^#(and(gr(0,0),gr(0,0)),0,0))
              • Applying Rule "Delete Conditions" results in
                cond4^#(false,0,0)≥cond1^#(and(gr(0,0),gr(0,0)),0,0)

              • This constraint is kept as final constraint.

           2. (true→^*false⟹cond4^#(false,0,s(x1396))≥cond1^#(and(gr(0,0),gr(s(x1396),0)),0,s(x1396)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1398,x1397)→^*false⟹0→^*s(x1397)⟹(gr(x1398,x1397)→^*false⟹0→^*x1397⟹cond4^#(false,0,x1398)≥cond1^#(and(gr(0,0),gr(x1398,0)),0,x1398))⟹cond4^#(false,0,s(x1398))≥cond1^#(and(gr(0,0),gr(s(x1398),0)),0,s(x1398)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*false⟹cond4^#(false,s(x1390),x845)≥cond1^#(and(gr(s(x1390),0),gr(x845,0)),s(x1390),x845))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x1392,x1391)→^*false⟹s(x1391)→^*x845⟹0→^*x1388⟹gr(x845,x1388)→^*false⟹∀x1393 . ∀x1394 . (gr(x1392,x1391)→^*false⟹x1391→^*x1393⟹0→^*x1394⟹gr(x1393,x1394)→^*false⟹cond4^#(false,x1392,x1393)≥cond1^#(and(gr(x1392,0),gr(x1393,0)),x1392,x1393))⟹cond4^#(false,s(x1392),x845)≥cond1^#(and(gr(s(x1392),0),gr(x845,0)),s(x1392),x845))
         • Applying Rule "Induction" on gr(x845,x1388)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹gr(x1392,x1391)→^*false⟹s(x1391)→^*0⟹0→^*x1399⟹∀x1393 . ∀x1394 . (gr(x1392,x1391)→^*false⟹x1391→^*x1393⟹0→^*x1394⟹gr(x1393,x1394)→^*false⟹cond4^#(false,x1392,x1393)≥cond1^#(and(gr(x1392,0),gr(x1393,0)),x1392,x1393))⟹cond4^#(false,s(x1392),0)≥cond1^#(and(gr(s(x1392),0),gr(0,0)),s(x1392),0))
              • Applying Rule "Same Constructor" results in
                (gr(x1392,x1391)→^*false⟹s(x1391)→^*0⟹0→^*x1399⟹∀x1393 . ∀x1394 . (gr(x1392,x1391)→^*false⟹x1391→^*x1393⟹0→^*x1394⟹gr(x1393,x1394)→^*false⟹cond4^#(false,x1392,x1393)≥cond1^#(and(gr(x1392,0),gr(x1393,0)),x1392,x1393))⟹cond4^#(false,s(x1392),0)≥cond1^#(and(gr(s(x1392),0),gr(0,0)),s(x1392),0))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond4^#(false,s(x1392),s(x1400))≥cond1^#(and(gr(s(x1392),0),gr(s(x1400),0)),s(x1392),s(x1400)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1402,x1401)→^*false⟹gr(x1392,x1391)→^*false⟹s(x1391)→^*s(x1402)⟹0→^*s(x1401)⟹∀x1393 . ∀x1394 . (gr(x1392,x1391)→^*false⟹x1391→^*x1393⟹0→^*x1394⟹gr(x1393,x1394)→^*false⟹cond4^#(false,x1392,x1393)≥cond1^#(and(gr(x1392,0),gr(x1393,0)),x1392,x1393))⟹∀x1403 . ∀x1404 . ∀x1405 . ∀x1406 . (gr(x1402,x1401)→^*false⟹gr(x1403,x1404)→^*false⟹s(x1404)→^*x1402⟹0→^*x1401⟹∀x1405 . ∀x1406 . (gr(x1403,x1404)→^*false⟹x1404→^*x1405⟹0→^*x1406⟹gr(x1405,x1406)→^*false⟹cond4^#(false,x1403,x1405)≥cond1^#(and(gr(x1403,0),gr(x1405,0)),x1403,x1405))⟹cond4^#(false,s(x1403),x1402)≥cond1^#(and(gr(s(x1403),0),gr(x1402,0)),s(x1403),x1402))⟹cond4^#(false,s(x1392),s(x1402))≥cond1^#(and(gr(s(x1392),0),gr(s(x1402),0)),s(x1392),s(x1402)))
              • Applying Rule "Same Constructor" results in
                (gr(x1402,x1401)→^*false⟹gr(x1392,x1391)→^*false⟹x1391→^*x1402⟹0→^*s(x1401)⟹∀x1393 . ∀x1394 . (gr(x1392,x1391)→^*false⟹x1391→^*x1393⟹0→^*x1394⟹gr(x1393,x1394)→^*false⟹cond4^#(false,x1392,x1393)≥cond1^#(and(gr(x1392,0),gr(x1393,0)),x1392,x1393))⟹∀x1403 . ∀x1404 . ∀x1405 . ∀x1406 . (gr(x1402,x1401)→^*false⟹gr(x1403,x1404)→^*false⟹s(x1404)→^*x1402⟹0→^*x1401⟹∀x1405 . ∀x1406 . (gr(x1403,x1404)→^*false⟹x1404→^*x1405⟹0→^*x1406⟹gr(x1405,x1406)→^*false⟹cond4^#(false,x1403,x1405)≥cond1^#(and(gr(x1403,0),gr(x1405,0)),x1403,x1405))⟹cond4^#(false,s(x1403),x1402)≥cond1^#(and(gr(s(x1403),0),gr(x1402,0)),s(x1403),x1402))⟹cond4^#(false,s(x1392),s(x1402))≥cond1^#(and(gr(s(x1392),0),gr(s(x1402),0)),s(x1392),s(x1402)))
              • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond4^#(gr(x863,0),x862,x863)→^*cond4^#(true,x864,x865)⟹cond4^#(gr(x865,0),x864,p(x865))→^*cond4^#(false,x866,x867)⟹cond4^#(false,x866,x867)≥cond1^#(and(gr(x866,0),gr(x867,0)),x866,x867))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x863,0)→^*true⟹x862→^*x864⟹x863→^*x865⟹cond4^#(gr(x865,0),x864,p(x865))→^*cond4^#(false,x866,x867)⟹cond4^#(false,x866,x867)≥cond1^#(and(gr(x866,0),gr(x867,0)),x866,x867))
    • Applying Rule "Same Constructor" results in
      (gr(x863,0)→^*true⟹x862→^*x864⟹x863→^*x865⟹gr(x865,0)→^*false⟹x864→^*x866⟹p(x865)→^*x867⟹cond4^#(false,x866,x867)≥cond1^#(and(gr(x866,0),gr(x867,0)),x866,x867))
    • Applying Rule "Variable in Equation" allows to substitute x864 by x862 which results in
      (gr(x863,0)→^*true⟹x863→^*x865⟹gr(x865,0)→^*false⟹x862→^*x866⟹p(x865)→^*x867⟹cond4^#(false,x866,x867)≥cond1^#(and(gr(x866,0),gr(x867,0)),x866,x867))
    • Applying Rule "Variable in Equation" allows to substitute x866 by x862 which results in
      (gr(x863,0)→^*true⟹x863→^*x865⟹gr(x865,0)→^*false⟹p(x865)→^*x867⟹cond4^#(false,x862,x867)≥cond1^#(and(gr(x862,0),gr(x867,0)),x862,x867))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1407⟹gr(x863,x1407)→^*true⟹x863→^*x865⟹gr(x865,0)→^*false⟹p(x865)→^*x867⟹cond4^#(false,x862,x867)≥cond1^#(and(gr(x862,0),gr(x867,0)),x862,x867))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1407⟹gr(x863,x1407)→^*true⟹x863→^*x865⟹0→^*x1408⟹gr(x865,x1408)→^*false⟹p(x865)→^*x867⟹cond4^#(false,x862,x867)≥cond1^#(and(gr(x862,0),gr(x867,0)),x862,x867))
    • Applying Rule "Induction" on gr(x863,x1407)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹cond4^#(false,x862,x867)≥cond1^#(and(gr(x862,0),gr(x867,0)),x862,x867))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹0→^*0⟹s(x1410)→^*x865⟹0→^*x1408⟹gr(x865,x1408)→^*false⟹p(x865)→^*x867⟹cond4^#(false,x862,x867)≥cond1^#(and(gr(x862,0),gr(x867,0)),x862,x867))
         • Applying Rule "Same Constructor" results in
           (0→^*0⟹s(x1410)→^*x865⟹0→^*x1408⟹gr(x865,x1408)→^*false⟹p(x865)→^*x867⟹cond4^#(false,x862,x867)≥cond1^#(and(gr(x862,0),gr(x867,0)),x862,x867))
         • Applying Rule "Same Constructor" results in
           (s(x1410)→^*x865⟹0→^*x1408⟹gr(x865,x1408)→^*false⟹p(x865)→^*x867⟹cond4^#(false,x862,x867)≥cond1^#(and(gr(x862,0),gr(x867,0)),x862,x867))
         • Applying Rule "Induction" on gr(x865,x1408)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹s(x1410)→^*0⟹0→^*x1417⟹p(0)→^*x867⟹cond4^#(false,x862,x867)≥cond1^#(and(gr(x862,0),gr(x867,0)),x862,x867))
              • Applying Rule "Same Constructor" results in
                (s(x1410)→^*0⟹0→^*x1417⟹p(0)→^*x867⟹cond4^#(false,x862,x867)≥cond1^#(and(gr(x862,0),gr(x867,0)),x862,x867))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           2. (true→^*false⟹cond4^#(false,x862,x867)≥cond1^#(and(gr(x862,0),gr(x867,0)),x862,x867))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1420,x1419)→^*false⟹s(x1410)→^*s(x1420)⟹0→^*s(x1419)⟹p(s(x1420))→^*x867⟹∀x1421 . ∀x1422 . ∀x1423 . (gr(x1420,x1419)→^*false⟹s(x1421)→^*x1420⟹0→^*x1419⟹p(x1420)→^*x1422⟹cond4^#(false,x1423,x1422)≥cond1^#(and(gr(x1423,0),gr(x1422,0)),x1423,x1422))⟹cond4^#(false,x862,x867)≥cond1^#(and(gr(x862,0),gr(x867,0)),x862,x867))
              • Applying Rule "Same Constructor" results in
                (gr(x1420,x1419)→^*false⟹x1410→^*x1420⟹0→^*s(x1419)⟹p(s(x1420))→^*x867⟹∀x1421 . ∀x1422 . ∀x1423 . (gr(x1420,x1419)→^*false⟹s(x1421)→^*x1420⟹0→^*x1419⟹p(x1420)→^*x1422⟹cond4^#(false,x1423,x1422)≥cond1^#(and(gr(x1423,0),gr(x1422,0)),x1423,x1422))⟹cond4^#(false,x862,x867)≥cond1^#(and(gr(x862,0),gr(x867,0)),x862,x867))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x1412,x1411)→^*true⟹0→^*s(x1411)⟹s(x1412)→^*x865⟹0→^*x1408⟹gr(x865,x1408)→^*false⟹p(x865)→^*x867⟹∀x1413 . ∀x1414 . ∀x1415 . ∀x1416 . (gr(x1412,x1411)→^*true⟹0→^*x1411⟹x1412→^*x1413⟹0→^*x1414⟹gr(x1413,x1414)→^*false⟹p(x1413)→^*x1415⟹cond4^#(false,x1416,x1415)≥cond1^#(and(gr(x1416,0),gr(x1415,0)),x1416,x1415))⟹cond4^#(false,x862,x867)≥cond1^#(and(gr(x862,0),gr(x867,0)),x862,x867))
         • Applying Rule "Different Constructors" allows to drop this constraint.
  • For the chain we build the initial constraint

    (cond4^#(gr(x869,0),x868,p(x869))→^*cond4^#(true,x870,x871)⟹cond4^#(gr(x871,0),x870,p(x871))→^*cond4^#(false,x872,x873)⟹cond4^#(false,x872,x873)≥cond1^#(and(gr(x872,0),gr(x873,0)),x872,x873))

    which is simplified as follows.

    • Applying Rule "Same Constructor" results in
      (gr(x869,0)→^*true⟹x868→^*x870⟹p(x869)→^*x871⟹cond4^#(gr(x871,0),x870,p(x871))→^*cond4^#(false,x872,x873)⟹cond4^#(false,x872,x873)≥cond1^#(and(gr(x872,0),gr(x873,0)),x872,x873))
    • Applying Rule "Same Constructor" results in
      (gr(x869,0)→^*true⟹x868→^*x870⟹p(x869)→^*x871⟹gr(x871,0)→^*false⟹x870→^*x872⟹p(x871)→^*x873⟹cond4^#(false,x872,x873)≥cond1^#(and(gr(x872,0),gr(x873,0)),x872,x873))
    • Applying Rule "Variable in Equation" allows to substitute x870 by x868 which results in
      (gr(x869,0)→^*true⟹p(x869)→^*x871⟹gr(x871,0)→^*false⟹x868→^*x872⟹p(x871)→^*x873⟹cond4^#(false,x872,x873)≥cond1^#(and(gr(x872,0),gr(x873,0)),x872,x873))
    • Applying Rule "Variable in Equation" allows to substitute x872 by x868 which results in
      (gr(x869,0)→^*true⟹p(x869)→^*x871⟹gr(x871,0)→^*false⟹p(x871)→^*x873⟹cond4^#(false,x868,x873)≥cond1^#(and(gr(x868,0),gr(x873,0)),x868,x873))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1424⟹gr(x869,x1424)→^*true⟹p(x869)→^*x871⟹gr(x871,0)→^*false⟹p(x871)→^*x873⟹cond4^#(false,x868,x873)≥cond1^#(and(gr(x868,0),gr(x873,0)),x868,x873))
    • Applying Rule "Introduce fresh variable" results in
      (0→^*x1424⟹gr(x869,x1424)→^*true⟹p(x869)→^*x871⟹0→^*x1425⟹gr(x871,x1425)→^*false⟹p(x871)→^*x873⟹cond4^#(false,x868,x873)≥cond1^#(and(gr(x868,0),gr(x873,0)),x868,x873))
    • Applying Rule "Induction" on gr(x869,x1424)→^*true results in the following 3 new constraints.
      1. (false→^*true⟹cond4^#(false,x868,x873)≥cond1^#(and(gr(x868,0),gr(x873,0)),x868,x873))
         • Applying Rule "Different Constructors" allows to drop this constraint.
      2. (true→^*true⟹0→^*0⟹p(s(x1427))→^*x871⟹0→^*x1425⟹gr(x871,x1425)→^*false⟹p(x871)→^*x873⟹cond4^#(false,x868,x873)≥cond1^#(and(gr(x868,0),gr(x873,0)),x868,x873))
         • Applying Rule "Same Constructor" results in
           (0→^*0⟹p(s(x1427))→^*x871⟹0→^*x1425⟹gr(x871,x1425)→^*false⟹p(x871)→^*x873⟹cond4^#(false,x868,x873)≥cond1^#(and(gr(x868,0),gr(x873,0)),x868,x873))
         • Applying Rule "Same Constructor" results in
           (p(s(x1427))→^*x871⟹0→^*x1425⟹gr(x871,x1425)→^*false⟹p(x871)→^*x873⟹cond4^#(false,x868,x873)≥cond1^#(and(gr(x868,0),gr(x873,0)),x868,x873))
         • Applying Rule "Introduce fresh variable" results in
           (s(x1427)→^*x1434⟹p(x1434)→^*x871⟹0→^*x1425⟹gr(x871,x1425)→^*false⟹p(x871)→^*x873⟹cond4^#(false,x868,x873)≥cond1^#(and(gr(x868,0),gr(x873,0)),x868,x873))
         • Applying Rule "Induction" on gr(x871,x1425)→^*false results in the following 3 new constraints.
           1. (false→^*false⟹s(x1427)→^*x1434⟹p(x1434)→^*0⟹0→^*x1435⟹p(0)→^*x873⟹cond4^#(false,x868,x873)≥cond1^#(and(gr(x868,0),gr(x873,0)),x868,x873))
              • Applying Rule "Same Constructor" results in
                (s(x1427)→^*x1434⟹p(x1434)→^*0⟹0→^*x1435⟹p(0)→^*x873⟹cond4^#(false,x868,x873)≥cond1^#(and(gr(x868,0),gr(x873,0)),x868,x873))
              • Applying Rule "Delete Conditions" results in
                (s(x1427)→^*x1434⟹p(x1434)→^*0⟹p(0)→^*x873⟹cond4^#(false,x868,x873)≥cond1^#(and(gr(x868,0),gr(x873,0)),x868,x873))
              • Applying Rule "Introduce fresh variable" results in
                (s(x1427)→^*x1434⟹p(x1434)→^*0⟹0→^*x1443⟹p(x1443)→^*x873⟹cond4^#(false,x868,x873)≥cond1^#(and(gr(x868,0),gr(x873,0)),x868,x873))
              • Applying Rule "Induction" on p(x1434)→^*0 results in the following 2 new constraints.
                1. (0→^*0⟹s(x1427)→^*0⟹0→^*x1443⟹p(x1443)→^*x873⟹cond4^#(false,x868,x873)≥cond1^#(and(gr(x868,0),gr(x873,0)),x868,x873))
                   • Applying Rule "Same Constructor" results in
                     (s(x1427)→^*0⟹0→^*x1443⟹p(x1443)→^*x873⟹cond4^#(false,x868,x873)≥cond1^#(and(gr(x868,0),gr(x873,0)),x868,x873))
                   • Applying Rule "Different Constructors" allows to drop this constraint.
                2. (x1444→^*0⟹s(x1427)→^*s(x1444)⟹0→^*x1443⟹p(x1443)→^*x873⟹cond4^#(false,x868,x873)≥cond1^#(and(gr(x868,0),gr(x873,0)),x868,x873))
                   • Applying Rule "Same Constructor" results in
                     (x1444→^*0⟹x1427→^*x1444⟹0→^*x1443⟹p(x1443)→^*x873⟹cond4^#(false,x868,x873)≥cond1^#(and(gr(x868,0),gr(x873,0)),x868,x873))
                   • Applying Rule "Variable in Equation" allows to substitute x1444 by x1427 which results in
                     (x1427→^*0⟹0→^*x1443⟹p(x1443)→^*x873⟹cond4^#(false,x868,x873)≥cond1^#(and(gr(x868,0),gr(x873,0)),x868,x873))
                   • Applying Rule "Variable in Equation" allows to substitute x1427 by 0 which results in
                     (0→^*x1443⟹p(x1443)→^*x873⟹cond4^#(false,x868,x873)≥cond1^#(and(gr(x868,0),gr(x873,0)),x868,x873))
                   • Applying Rule "Variable in Equation" allows to substitute x1443 by 0 which results in
                     (p(0)→^*x873⟹cond4^#(false,x868,x873)≥cond1^#(and(gr(x868,0),gr(x873,0)),x868,x873))
                   • Applying Rule "Delete Conditions" results in
                     cond4^#(false,x868,x873)≥cond1^#(and(gr(x868,0),gr(x873,0)),x868,x873)

                   • This constraint is kept as final constraint.

           2. (true→^*false⟹cond4^#(false,x868,x873)≥cond1^#(and(gr(x868,0),gr(x873,0)),x868,x873))
              • Applying Rule "Different Constructors" allows to drop this constraint.
           3. (gr(x1438,x1437)→^*false⟹s(x1427)→^*x1434⟹p(x1434)→^*s(x1438)⟹0→^*s(x1437)⟹p(s(x1438))→^*x873⟹∀x1439 . ∀x1440 . ∀x1441 . ∀x1442 . (gr(x1438,x1437)→^*false⟹s(x1439)→^*x1440⟹p(x1440)→^*x1438⟹0→^*x1437⟹p(x1438)→^*x1441⟹cond4^#(false,x1442,x1441)≥cond1^#(and(gr(x1442,0),gr(x1441,0)),x1442,x1441))⟹cond4^#(false,x868,x873)≥cond1^#(and(gr(x868,0),gr(x873,0)),x868,x873))
              • Applying Rule "Different Constructors" allows to drop this constraint.
      3. (gr(x1429,x1428)→^*true⟹0→^*s(x1428)⟹p(s(x1429))→^*x871⟹0→^*x1425⟹gr(x871,x1425)→^*false⟹p(x871)→^*x873⟹∀x1430 . ∀x1431 . ∀x1432 . ∀x1433 . (gr(x1429,x1428)→^*true⟹0→^*x1428⟹p(x1429)→^*x1430⟹0→^*x1431⟹gr(x1430,x1431)→^*false⟹p(x1430)→^*x1432⟹cond4^#(false,x1433,x1432)≥cond1^#(and(gr(x1433,0),gr(x1432,0)),x1433,x1432))⟹cond4^#(false,x868,x873)≥cond1^#(and(gr(x868,0),gr(x873,0)),x868,x873))
         • 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 Dependency Graph Processor

  The dependency pairs are split into 2 components.

  • The 1^st component contains the pair

    cond3#(true,x,y)

    →

    cond3#(gr(x,0),p(x),y)

    (22)

    1.1.1.1.1.1.1.1 Usable Rules Processor

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

    gr(0,x)	→	false	(8)
    gr(s(x),0)	→	true	(9)
    p(0)	→	0	(14)
    p(s(x))	→	x	(15)

    1.1.1.1.1.1.1.1.1 Innermost Lhss Removal Processor

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

    gr(0,x0)

    gr(s(x0),0)

    gr(s(x0),s(x1))

    p(0)

    p(s(x0))

    1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

    Using the linear polynomial interpretation over the rationals with delta = 7/2

    [cond3#(x1, x2, x3)]	=	0 + 4 · x1 + 2 · x2 + 0 · x3
    [true]	=	4
    [gr(x1, x2)]	=	1 + 1/4 · x1 + 4 · x2
    [0]	=	1/2
    [p(x1)]	=	1/4 + 1/2 · x1
    [false]	=	1/4
    [s(x1)]	=	4 + 4 · x1

    the pair

    cond3#(true,x,y)

    →

    cond3#(gr(x,0),p(x),y)

    (22)

    could be deleted.

    1.1.1.1.1.1.1.1.1.1.1 P is empty

    There are no pairs anymore.

  • The 2^nd component contains the pair

    cond4#(true,x,y)

    →

    cond4#(gr(y,0),x,p(y))

    (29)

    1.1.1.1.1.1.1.2 Usable Rules Processor

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

    gr(0,x)	→	false	(8)
    gr(s(x),0)	→	true	(9)
    p(0)	→	0	(14)
    p(s(x))	→	x	(15)

    1.1.1.1.1.1.1.2.1 Innermost Lhss Removal Processor

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

    gr(0,x0)

    gr(s(x0),0)

    gr(s(x0),s(x1))

    p(0)

    p(s(x0))

    1.1.1.1.1.1.1.2.1.1 Reduction Pair Processor

    Using the linear polynomial interpretation over the rationals with delta = 7/2

    [cond4#(x1, x2, x3)]	=	0 + 4 · x1 + 0 · x2 + 2 · x3
    [true]	=	4
    [gr(x1, x2)]	=	1 + 1/4 · x1 + 4 · x2
    [0]	=	1/2
    [p(x1)]	=	1/4 + 1/2 · x1
    [false]	=	1/4
    [s(x1)]	=	4 + 4 · x1

    the pair

    cond4#(true,x,y)

    →

    cond4#(gr(y,0),x,p(y))

    (29)

    could be deleted.

    1.1.1.1.1.1.1.2.1.1.1 P is empty

    There are no pairs anymore.

• The 2^nd component contains the pair

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

  →

  gr#(x,y)

  (36)

  1.1.1.2 Usable Rules Processor

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

  There are no rules.

  1.1.1.2.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  1.1.1.2.1.1 Size-Change Termination

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

  gr#(s(x),s(y))	→	gr#(x,y)	(36)
  1	>	1
  2	>	2

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