WORST_CASE(Omega(n^1),?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 63.0 s]
(2) CpxRelTRS
(3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(4) TRS for Loop Detection
(5) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(6) BEST
    (7) proven lower bound
        (8) LowerBoundPropagationProof [FINISHED, 0 ms]
        (9) BOUNDS(n^1, INF)
    (10) TRS for Loop Detection


----------------------------------------

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   @'(Cons(z0, z1), z2) -> c10(@'(z1, z2))
   @'(Nil, z0) -> c11
   GT0(Cons(z0, z1), Nil) -> c12
   GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3))
   GT0(Nil, z0) -> c14
   GCD(Nil, Nil) -> c15
   GCD(Nil, Cons(z0, z1)) -> c16
   GCD(Cons(z0, z1), Nil) -> c17
   GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3)))
   LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1))
   LGTH(Nil) -> c20
   EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(AND(eqList(z0, z2), eqList(z1, z3)), EQLIST(z0, z2))
   EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(AND(eqList(z0, z2), eqList(z1, z3)), EQLIST(z1, z3))
   EQLIST(Cons(z0, z1), Nil) -> c23
   EQLIST(Nil, Cons(z0, z1)) -> c24
   EQLIST(Nil, Nil) -> c25
   MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1))
   GOAL(z0, z1) -> c27(GCD(z0, z1))

The (relative) TRS S consists of the following rules:

   AND(False, False) -> c
   AND(True, False) -> c1
   AND(False, True) -> c2
   AND(True, True) -> c3
   MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3))
   MONUS[ITE](True, Cons(z0, z1), z2) -> c5
   GCD[ITE](False, z0, z1) -> c6(GCD[FALSE][ITE](gt0(z0, z1), z0, z1), GT0(z0, z1))
   GCD[ITE](True, z0, z1) -> c7
   GCD[FALSE][ITE](False, z0, z1) -> c8(GCD(monus(z1, z0), z0), MONUS(z1, z0))
   GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(z1, monus(z0, z1)), MONUS(z0, z1))
   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True
   monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3)
   monus[Ite](True, Cons(z0, z1), z2) -> z1
   gcd[Ite](False, z0, z1) -> gcd[False][Ite](gt0(z0, z1), z0, z1)
   gcd[Ite](True, z0, z1) -> z0
   gcd[False][Ite](False, z0, z1) -> gcd(monus(z1, z0), z0)
   gcd[False][Ite](True, z0, z1) -> gcd(z1, monus(z0, z1))
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
   gt0(Cons(z0, z1), Nil) -> True
   gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3)
   gt0(Nil, z0) -> False
   gcd(Nil, Nil) -> Nil
   gcd(Nil, Cons(z0, z1)) -> Nil
   gcd(Cons(z0, z1), Nil) -> Nil
   gcd(Cons(z0, z1), Cons(z2, z3)) -> gcd[Ite](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3))
   lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1))
   lgth(Nil) -> Nil
   eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3))
   eqList(Cons(z0, z1), Nil) -> False
   eqList(Nil, Cons(z0, z1)) -> False
   eqList(Nil, Nil) -> True
   monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1)
   goal(z0, z1) -> gcd(z0, z1)

Rewrite Strategy: INNERMOST
----------------------------------------

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   @'(Cons(z0, z1), z2) -> c10(@'(z1, z2))
   @'(Nil, z0) -> c11
   GT0(Cons(z0, z1), Nil) -> c12
   GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3))
   GT0(Nil, z0) -> c14
   GCD(Nil, Nil) -> c15
   GCD(Nil, Cons(z0, z1)) -> c16
   GCD(Cons(z0, z1), Nil) -> c17
   GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3)))
   LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1))
   LGTH(Nil) -> c20
   EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(AND(eqList(z0, z2), eqList(z1, z3)), EQLIST(z0, z2))
   EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(AND(eqList(z0, z2), eqList(z1, z3)), EQLIST(z1, z3))
   EQLIST(Cons(z0, z1), Nil) -> c23
   EQLIST(Nil, Cons(z0, z1)) -> c24
   EQLIST(Nil, Nil) -> c25
   MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1))
   GOAL(z0, z1) -> c27(GCD(z0, z1))

The (relative) TRS S consists of the following rules:

   AND(False, False) -> c
   AND(True, False) -> c1
   AND(False, True) -> c2
   AND(True, True) -> c3
   MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3))
   MONUS[ITE](True, Cons(z0, z1), z2) -> c5
   GCD[ITE](False, z0, z1) -> c6(GCD[FALSE][ITE](gt0(z0, z1), z0, z1), GT0(z0, z1))
   GCD[ITE](True, z0, z1) -> c7
   GCD[FALSE][ITE](False, z0, z1) -> c8(GCD(monus(z1, z0), z0), MONUS(z1, z0))
   GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(z1, monus(z0, z1)), MONUS(z0, z1))
   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True
   monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3)
   monus[Ite](True, Cons(z0, z1), z2) -> z1
   gcd[Ite](False, z0, z1) -> gcd[False][Ite](gt0(z0, z1), z0, z1)
   gcd[Ite](True, z0, z1) -> z0
   gcd[False][Ite](False, z0, z1) -> gcd(monus(z1, z0), z0)
   gcd[False][Ite](True, z0, z1) -> gcd(z1, monus(z0, z1))
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
   gt0(Cons(z0, z1), Nil) -> True
   gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3)
   gt0(Nil, z0) -> False
   gcd(Nil, Nil) -> Nil
   gcd(Nil, Cons(z0, z1)) -> Nil
   gcd(Cons(z0, z1), Nil) -> Nil
   gcd(Cons(z0, z1), Cons(z2, z3)) -> gcd[Ite](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3))
   lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1))
   lgth(Nil) -> Nil
   eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3))
   eqList(Cons(z0, z1), Nil) -> False
   eqList(Nil, Cons(z0, z1)) -> False
   eqList(Nil, Nil) -> True
   monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1)
   goal(z0, z1) -> gcd(z0, z1)

Rewrite Strategy: INNERMOST
----------------------------------------

(3) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(4)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   @'(Cons(z0, z1), z2) -> c10(@'(z1, z2))
   @'(Nil, z0) -> c11
   GT0(Cons(z0, z1), Nil) -> c12
   GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3))
   GT0(Nil, z0) -> c14
   GCD(Nil, Nil) -> c15
   GCD(Nil, Cons(z0, z1)) -> c16
   GCD(Cons(z0, z1), Nil) -> c17
   GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3)))
   LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1))
   LGTH(Nil) -> c20
   EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(AND(eqList(z0, z2), eqList(z1, z3)), EQLIST(z0, z2))
   EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(AND(eqList(z0, z2), eqList(z1, z3)), EQLIST(z1, z3))
   EQLIST(Cons(z0, z1), Nil) -> c23
   EQLIST(Nil, Cons(z0, z1)) -> c24
   EQLIST(Nil, Nil) -> c25
   MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1))
   GOAL(z0, z1) -> c27(GCD(z0, z1))

The (relative) TRS S consists of the following rules:

   AND(False, False) -> c
   AND(True, False) -> c1
   AND(False, True) -> c2
   AND(True, True) -> c3
   MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3))
   MONUS[ITE](True, Cons(z0, z1), z2) -> c5
   GCD[ITE](False, z0, z1) -> c6(GCD[FALSE][ITE](gt0(z0, z1), z0, z1), GT0(z0, z1))
   GCD[ITE](True, z0, z1) -> c7
   GCD[FALSE][ITE](False, z0, z1) -> c8(GCD(monus(z1, z0), z0), MONUS(z1, z0))
   GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(z1, monus(z0, z1)), MONUS(z0, z1))
   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True
   monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3)
   monus[Ite](True, Cons(z0, z1), z2) -> z1
   gcd[Ite](False, z0, z1) -> gcd[False][Ite](gt0(z0, z1), z0, z1)
   gcd[Ite](True, z0, z1) -> z0
   gcd[False][Ite](False, z0, z1) -> gcd(monus(z1, z0), z0)
   gcd[False][Ite](True, z0, z1) -> gcd(z1, monus(z0, z1))
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
   gt0(Cons(z0, z1), Nil) -> True
   gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3)
   gt0(Nil, z0) -> False
   gcd(Nil, Nil) -> Nil
   gcd(Nil, Cons(z0, z1)) -> Nil
   gcd(Cons(z0, z1), Nil) -> Nil
   gcd(Cons(z0, z1), Cons(z2, z3)) -> gcd[Ite](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3))
   lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1))
   lgth(Nil) -> Nil
   eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3))
   eqList(Cons(z0, z1), Nil) -> False
   eqList(Nil, Cons(z0, z1)) -> False
   eqList(Nil, Nil) -> True
   monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1)
   goal(z0, z1) -> gcd(z0, z1)

Rewrite Strategy: INNERMOST
----------------------------------------

(5) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

EQLIST(Cons(z0, z1), Cons(z2, z3)) ->^+ c21(AND(eqList(z0, z2), eqList(z1, z3)), EQLIST(z0, z2))

gives rise to a decreasing loop by considering the right hand sides subterm at position [1].

The pumping substitution is [z0 / Cons(z0, z1), z2 / Cons(z2, z3)].

The result substitution is [ ].




----------------------------------------

(6)
Complex Obligation (BEST)

----------------------------------------

(7)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   @'(Cons(z0, z1), z2) -> c10(@'(z1, z2))
   @'(Nil, z0) -> c11
   GT0(Cons(z0, z1), Nil) -> c12
   GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3))
   GT0(Nil, z0) -> c14
   GCD(Nil, Nil) -> c15
   GCD(Nil, Cons(z0, z1)) -> c16
   GCD(Cons(z0, z1), Nil) -> c17
   GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3)))
   LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1))
   LGTH(Nil) -> c20
   EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(AND(eqList(z0, z2), eqList(z1, z3)), EQLIST(z0, z2))
   EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(AND(eqList(z0, z2), eqList(z1, z3)), EQLIST(z1, z3))
   EQLIST(Cons(z0, z1), Nil) -> c23
   EQLIST(Nil, Cons(z0, z1)) -> c24
   EQLIST(Nil, Nil) -> c25
   MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1))
   GOAL(z0, z1) -> c27(GCD(z0, z1))

The (relative) TRS S consists of the following rules:

   AND(False, False) -> c
   AND(True, False) -> c1
   AND(False, True) -> c2
   AND(True, True) -> c3
   MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3))
   MONUS[ITE](True, Cons(z0, z1), z2) -> c5
   GCD[ITE](False, z0, z1) -> c6(GCD[FALSE][ITE](gt0(z0, z1), z0, z1), GT0(z0, z1))
   GCD[ITE](True, z0, z1) -> c7
   GCD[FALSE][ITE](False, z0, z1) -> c8(GCD(monus(z1, z0), z0), MONUS(z1, z0))
   GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(z1, monus(z0, z1)), MONUS(z0, z1))
   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True
   monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3)
   monus[Ite](True, Cons(z0, z1), z2) -> z1
   gcd[Ite](False, z0, z1) -> gcd[False][Ite](gt0(z0, z1), z0, z1)
   gcd[Ite](True, z0, z1) -> z0
   gcd[False][Ite](False, z0, z1) -> gcd(monus(z1, z0), z0)
   gcd[False][Ite](True, z0, z1) -> gcd(z1, monus(z0, z1))
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
   gt0(Cons(z0, z1), Nil) -> True
   gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3)
   gt0(Nil, z0) -> False
   gcd(Nil, Nil) -> Nil
   gcd(Nil, Cons(z0, z1)) -> Nil
   gcd(Cons(z0, z1), Nil) -> Nil
   gcd(Cons(z0, z1), Cons(z2, z3)) -> gcd[Ite](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3))
   lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1))
   lgth(Nil) -> Nil
   eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3))
   eqList(Cons(z0, z1), Nil) -> False
   eqList(Nil, Cons(z0, z1)) -> False
   eqList(Nil, Nil) -> True
   monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1)
   goal(z0, z1) -> gcd(z0, z1)

Rewrite Strategy: INNERMOST
----------------------------------------

(8) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(9)
BOUNDS(n^1, INF)

----------------------------------------

(10)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   @'(Cons(z0, z1), z2) -> c10(@'(z1, z2))
   @'(Nil, z0) -> c11
   GT0(Cons(z0, z1), Nil) -> c12
   GT0(Cons(z0, z1), Cons(z2, z3)) -> c13(GT0(z1, z3))
   GT0(Nil, z0) -> c14
   GCD(Nil, Nil) -> c15
   GCD(Nil, Cons(z0, z1)) -> c16
   GCD(Cons(z0, z1), Nil) -> c17
   GCD(Cons(z0, z1), Cons(z2, z3)) -> c18(GCD[ITE](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3)), EQLIST(Cons(z0, z1), Cons(z2, z3)))
   LGTH(Cons(z0, z1)) -> c19(@'(Cons(Nil, Nil), lgth(z1)), LGTH(z1))
   LGTH(Nil) -> c20
   EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c21(AND(eqList(z0, z2), eqList(z1, z3)), EQLIST(z0, z2))
   EQLIST(Cons(z0, z1), Cons(z2, z3)) -> c22(AND(eqList(z0, z2), eqList(z1, z3)), EQLIST(z1, z3))
   EQLIST(Cons(z0, z1), Nil) -> c23
   EQLIST(Nil, Cons(z0, z1)) -> c24
   EQLIST(Nil, Nil) -> c25
   MONUS(z0, z1) -> c26(MONUS[ITE](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1), EQLIST(lgth(z1), Cons(Nil, Nil)), LGTH(z1))
   GOAL(z0, z1) -> c27(GCD(z0, z1))

The (relative) TRS S consists of the following rules:

   AND(False, False) -> c
   AND(True, False) -> c1
   AND(False, True) -> c2
   AND(True, True) -> c3
   MONUS[ITE](False, Cons(z0, z1), Cons(z2, z3)) -> c4(MONUS(z1, z3))
   MONUS[ITE](True, Cons(z0, z1), z2) -> c5
   GCD[ITE](False, z0, z1) -> c6(GCD[FALSE][ITE](gt0(z0, z1), z0, z1), GT0(z0, z1))
   GCD[ITE](True, z0, z1) -> c7
   GCD[FALSE][ITE](False, z0, z1) -> c8(GCD(monus(z1, z0), z0), MONUS(z1, z0))
   GCD[FALSE][ITE](True, z0, z1) -> c9(GCD(z1, monus(z0, z1)), MONUS(z0, z1))
   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True
   monus[Ite](False, Cons(z0, z1), Cons(z2, z3)) -> monus(z1, z3)
   monus[Ite](True, Cons(z0, z1), z2) -> z1
   gcd[Ite](False, z0, z1) -> gcd[False][Ite](gt0(z0, z1), z0, z1)
   gcd[Ite](True, z0, z1) -> z0
   gcd[False][Ite](False, z0, z1) -> gcd(monus(z1, z0), z0)
   gcd[False][Ite](True, z0, z1) -> gcd(z1, monus(z0, z1))
   @(Cons(z0, z1), z2) -> Cons(z0, @(z1, z2))
   @(Nil, z0) -> z0
   gt0(Cons(z0, z1), Nil) -> True
   gt0(Cons(z0, z1), Cons(z2, z3)) -> gt0(z1, z3)
   gt0(Nil, z0) -> False
   gcd(Nil, Nil) -> Nil
   gcd(Nil, Cons(z0, z1)) -> Nil
   gcd(Cons(z0, z1), Nil) -> Nil
   gcd(Cons(z0, z1), Cons(z2, z3)) -> gcd[Ite](eqList(Cons(z0, z1), Cons(z2, z3)), Cons(z0, z1), Cons(z2, z3))
   lgth(Cons(z0, z1)) -> @(Cons(Nil, Nil), lgth(z1))
   lgth(Nil) -> Nil
   eqList(Cons(z0, z1), Cons(z2, z3)) -> and(eqList(z0, z2), eqList(z1, z3))
   eqList(Cons(z0, z1), Nil) -> False
   eqList(Nil, Cons(z0, z1)) -> False
   eqList(Nil, Nil) -> True
   monus(z0, z1) -> monus[Ite](eqList(lgth(z1), Cons(Nil, Nil)), z0, z1)
   goal(z0, z1) -> gcd(z0, z1)

Rewrite Strategy: INNERMOST