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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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), 923 ms]
(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:

   MIN(z0, 0) -> c
   MIN(0, z0) -> c1
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(z0, 0) -> c3
   MAX(0, z0) -> c4
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   -'(z0, 0) -> c6
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1))
   GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1))
   GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1))
   GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2))
   GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2))
   GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2))
   GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2))
   GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2))
   GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2))
   GCD(z0, 0, 0) -> c17
   GCD(0, z0, 0) -> c18
   GCD(0, 0, z0) -> c19

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

   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   max(z0, 0) -> z0
   max(0, z0) -> z0
   max(s(z0), s(z1)) -> s(max(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   gcd(s(z0), s(z1), z2) -> gcd(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2)
   gcd(z0, s(z1), s(z2)) -> gcd(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2)))
   gcd(s(z0), z1, s(z2)) -> gcd(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2)))
   gcd(z0, 0, 0) -> z0
   gcd(0, z0, 0) -> z0
   gcd(0, 0, z0) -> z0

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:

   MIN(z0, 0) -> c
   MIN(0, z0) -> c1
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(z0, 0) -> c3
   MAX(0, z0) -> c4
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   -'(z0, 0) -> c6
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1))
   GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1))
   GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1))
   GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2))
   GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2))
   GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2))
   GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2))
   GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2))
   GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2))
   GCD(z0, 0, 0) -> c17
   GCD(0, z0, 0) -> c18
   GCD(0, 0, z0) -> c19

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

   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   max(z0, 0) -> z0
   max(0, z0) -> z0
   max(s(z0), s(z1)) -> s(max(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   gcd(s(z0), s(z1), z2) -> gcd(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2)
   gcd(z0, s(z1), s(z2)) -> gcd(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2)))
   gcd(s(z0), z1, s(z2)) -> gcd(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2)))
   gcd(z0, 0, 0) -> z0
   gcd(0, z0, 0) -> z0
   gcd(0, 0, z0) -> z0

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:

   MIN(z0, 0) -> c
   MIN(0, z0) -> c1
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(z0, 0) -> c3
   MAX(0, z0) -> c4
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   -'(z0, 0) -> c6
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1))
   GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1))
   GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1))
   GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2))
   GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2))
   GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2))
   GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2))
   GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2))
   GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2))
   GCD(z0, 0, 0) -> c17
   GCD(0, z0, 0) -> c18
   GCD(0, 0, z0) -> c19

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

   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   max(z0, 0) -> z0
   max(0, z0) -> z0
   max(s(z0), s(z1)) -> s(max(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   gcd(s(z0), s(z1), z2) -> gcd(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2)
   gcd(z0, s(z1), s(z2)) -> gcd(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2)))
   gcd(s(z0), z1, s(z2)) -> gcd(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2)))
   gcd(z0, 0, 0) -> z0
   gcd(0, z0, 0) -> z0
   gcd(0, 0, z0) -> z0

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

MIN(s(z0), s(z1)) ->^+ c2(MIN(z0, z1))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [z0 / s(z0), z1 / s(z1)].

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:

   MIN(z0, 0) -> c
   MIN(0, z0) -> c1
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(z0, 0) -> c3
   MAX(0, z0) -> c4
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   -'(z0, 0) -> c6
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1))
   GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1))
   GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1))
   GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2))
   GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2))
   GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2))
   GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2))
   GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2))
   GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2))
   GCD(z0, 0, 0) -> c17
   GCD(0, z0, 0) -> c18
   GCD(0, 0, z0) -> c19

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

   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   max(z0, 0) -> z0
   max(0, z0) -> z0
   max(s(z0), s(z1)) -> s(max(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   gcd(s(z0), s(z1), z2) -> gcd(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2)
   gcd(z0, s(z1), s(z2)) -> gcd(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2)))
   gcd(s(z0), z1, s(z2)) -> gcd(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2)))
   gcd(z0, 0, 0) -> z0
   gcd(0, z0, 0) -> z0
   gcd(0, 0, z0) -> z0

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:

   MIN(z0, 0) -> c
   MIN(0, z0) -> c1
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(z0, 0) -> c3
   MAX(0, z0) -> c4
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   -'(z0, 0) -> c6
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   GCD(s(z0), s(z1), z2) -> c8(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MAX(z0, z1))
   GCD(s(z0), s(z1), z2) -> c9(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), -'(max(z0, z1), min(z0, z1)), MIN(z0, z1))
   GCD(s(z0), s(z1), z2) -> c10(GCD(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2), MIN(z0, z1))
   GCD(z0, s(z1), s(z2)) -> c11(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MAX(z1, z2))
   GCD(z0, s(z1), s(z2)) -> c12(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), -'(max(z1, z2), min(z1, z2)), MIN(z1, z2))
   GCD(z0, s(z1), s(z2)) -> c13(GCD(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2))), MIN(z1, z2))
   GCD(s(z0), z1, s(z2)) -> c14(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MAX(z0, z2))
   GCD(s(z0), z1, s(z2)) -> c15(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), -'(max(z0, z2), min(z0, z2)), MIN(z0, z2))
   GCD(s(z0), z1, s(z2)) -> c16(GCD(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2))), MIN(z0, z2))
   GCD(z0, 0, 0) -> c17
   GCD(0, z0, 0) -> c18
   GCD(0, 0, z0) -> c19

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

   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   max(z0, 0) -> z0
   max(0, z0) -> z0
   max(s(z0), s(z1)) -> s(max(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   gcd(s(z0), s(z1), z2) -> gcd(-(max(z0, z1), min(z0, z1)), s(min(z0, z1)), z2)
   gcd(z0, s(z1), s(z2)) -> gcd(z0, -(max(z1, z2), min(z1, z2)), s(min(z1, z2)))
   gcd(s(z0), z1, s(z2)) -> gcd(-(max(z0, z2), min(z0, z2)), z1, s(min(z0, z2)))
   gcd(z0, 0, 0) -> z0
   gcd(0, z0, 0) -> z0
   gcd(0, 0, z0) -> z0

Rewrite Strategy: INNERMOST