WORST_CASE(Omega(n^1),?)
proof of /export/starexec/sandbox2/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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), 999 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


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(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:

   SUM(cons(s(z0), z1), cons(z2, z3)) -> c(SUM(cons(z0, z1), cons(s(z2), z3)))
   SUM(cons(0, z0), z1) -> c1(SUM(z0, z1))
   SUM(nil, z0) -> c2
   EMPTY(nil) -> c3
   EMPTY(cons(z0, z1)) -> c4
   TAIL(nil) -> c5
   TAIL(cons(z0, z1)) -> c6
   HEAD(cons(z0, z1)) -> c7
   WEIGHT(z0) -> c8(IF(empty(z0), empty(tail(z0)), z0), EMPTY(z0))
   WEIGHT(z0) -> c9(IF(empty(z0), empty(tail(z0)), z0), EMPTY(tail(z0)), TAIL(z0))
   IF(true, z0, z1) -> c10
   IF(false, z0, z1) -> c11(IF2(z0, z1))
   IF2(true, z0) -> c12(HEAD(z0))
   IF2(false, z0) -> c13(WEIGHT(sum(z0, cons(0, tail(tail(z0))))), SUM(z0, cons(0, tail(tail(z0)))), TAIL(tail(z0)), TAIL(z0))

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

   sum(cons(s(z0), z1), cons(z2, z3)) -> sum(cons(z0, z1), cons(s(z2), z3))
   sum(cons(0, z0), z1) -> sum(z0, z1)
   sum(nil, z0) -> z0
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   weight(z0) -> if(empty(z0), empty(tail(z0)), z0)
   if(true, z0, z1) -> weight_undefined_error
   if(false, z0, z1) -> if2(z0, z1)
   if2(true, z0) -> head(z0)
   if2(false, z0) -> weight(sum(z0, cons(0, tail(tail(z0)))))

Rewrite Strategy: INNERMOST
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(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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(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:

   SUM(cons(s(z0), z1), cons(z2, z3)) -> c(SUM(cons(z0, z1), cons(s(z2), z3)))
   SUM(cons(0, z0), z1) -> c1(SUM(z0, z1))
   SUM(nil, z0) -> c2
   EMPTY(nil) -> c3
   EMPTY(cons(z0, z1)) -> c4
   TAIL(nil) -> c5
   TAIL(cons(z0, z1)) -> c6
   HEAD(cons(z0, z1)) -> c7
   WEIGHT(z0) -> c8(IF(empty(z0), empty(tail(z0)), z0), EMPTY(z0))
   WEIGHT(z0) -> c9(IF(empty(z0), empty(tail(z0)), z0), EMPTY(tail(z0)), TAIL(z0))
   IF(true, z0, z1) -> c10
   IF(false, z0, z1) -> c11(IF2(z0, z1))
   IF2(true, z0) -> c12(HEAD(z0))
   IF2(false, z0) -> c13(WEIGHT(sum(z0, cons(0, tail(tail(z0))))), SUM(z0, cons(0, tail(tail(z0)))), TAIL(tail(z0)), TAIL(z0))

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

   sum(cons(s(z0), z1), cons(z2, z3)) -> sum(cons(z0, z1), cons(s(z2), z3))
   sum(cons(0, z0), z1) -> sum(z0, z1)
   sum(nil, z0) -> z0
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   weight(z0) -> if(empty(z0), empty(tail(z0)), z0)
   if(true, z0, z1) -> weight_undefined_error
   if(false, z0, z1) -> if2(z0, z1)
   if2(true, z0) -> head(z0)
   if2(false, z0) -> weight(sum(z0, cons(0, tail(tail(z0)))))

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

   SUM(cons(s(z0), z1), cons(z2, z3)) -> c(SUM(cons(z0, z1), cons(s(z2), z3)))
   SUM(cons(0, z0), z1) -> c1(SUM(z0, z1))
   SUM(nil, z0) -> c2
   EMPTY(nil) -> c3
   EMPTY(cons(z0, z1)) -> c4
   TAIL(nil) -> c5
   TAIL(cons(z0, z1)) -> c6
   HEAD(cons(z0, z1)) -> c7
   WEIGHT(z0) -> c8(IF(empty(z0), empty(tail(z0)), z0), EMPTY(z0))
   WEIGHT(z0) -> c9(IF(empty(z0), empty(tail(z0)), z0), EMPTY(tail(z0)), TAIL(z0))
   IF(true, z0, z1) -> c10
   IF(false, z0, z1) -> c11(IF2(z0, z1))
   IF2(true, z0) -> c12(HEAD(z0))
   IF2(false, z0) -> c13(WEIGHT(sum(z0, cons(0, tail(tail(z0))))), SUM(z0, cons(0, tail(tail(z0)))), TAIL(tail(z0)), TAIL(z0))

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

   sum(cons(s(z0), z1), cons(z2, z3)) -> sum(cons(z0, z1), cons(s(z2), z3))
   sum(cons(0, z0), z1) -> sum(z0, z1)
   sum(nil, z0) -> z0
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   weight(z0) -> if(empty(z0), empty(tail(z0)), z0)
   if(true, z0, z1) -> weight_undefined_error
   if(false, z0, z1) -> if2(z0, z1)
   if2(true, z0) -> head(z0)
   if2(false, z0) -> weight(sum(z0, cons(0, tail(tail(z0)))))

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

The rewrite sequence

SUM(cons(s(z0), z1), cons(z2, z3)) ->^+ c(SUM(cons(z0, z1), cons(s(z2), z3)))

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

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

The result substitution is [z2 / s(z2)].




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(6)
Complex Obligation (BEST)

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(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:

   SUM(cons(s(z0), z1), cons(z2, z3)) -> c(SUM(cons(z0, z1), cons(s(z2), z3)))
   SUM(cons(0, z0), z1) -> c1(SUM(z0, z1))
   SUM(nil, z0) -> c2
   EMPTY(nil) -> c3
   EMPTY(cons(z0, z1)) -> c4
   TAIL(nil) -> c5
   TAIL(cons(z0, z1)) -> c6
   HEAD(cons(z0, z1)) -> c7
   WEIGHT(z0) -> c8(IF(empty(z0), empty(tail(z0)), z0), EMPTY(z0))
   WEIGHT(z0) -> c9(IF(empty(z0), empty(tail(z0)), z0), EMPTY(tail(z0)), TAIL(z0))
   IF(true, z0, z1) -> c10
   IF(false, z0, z1) -> c11(IF2(z0, z1))
   IF2(true, z0) -> c12(HEAD(z0))
   IF2(false, z0) -> c13(WEIGHT(sum(z0, cons(0, tail(tail(z0))))), SUM(z0, cons(0, tail(tail(z0)))), TAIL(tail(z0)), TAIL(z0))

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

   sum(cons(s(z0), z1), cons(z2, z3)) -> sum(cons(z0, z1), cons(s(z2), z3))
   sum(cons(0, z0), z1) -> sum(z0, z1)
   sum(nil, z0) -> z0
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   weight(z0) -> if(empty(z0), empty(tail(z0)), z0)
   if(true, z0, z1) -> weight_undefined_error
   if(false, z0, z1) -> if2(z0, z1)
   if2(true, z0) -> head(z0)
   if2(false, z0) -> weight(sum(z0, cons(0, tail(tail(z0)))))

Rewrite Strategy: INNERMOST
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(8) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(9)
BOUNDS(n^1, INF)

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(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:

   SUM(cons(s(z0), z1), cons(z2, z3)) -> c(SUM(cons(z0, z1), cons(s(z2), z3)))
   SUM(cons(0, z0), z1) -> c1(SUM(z0, z1))
   SUM(nil, z0) -> c2
   EMPTY(nil) -> c3
   EMPTY(cons(z0, z1)) -> c4
   TAIL(nil) -> c5
   TAIL(cons(z0, z1)) -> c6
   HEAD(cons(z0, z1)) -> c7
   WEIGHT(z0) -> c8(IF(empty(z0), empty(tail(z0)), z0), EMPTY(z0))
   WEIGHT(z0) -> c9(IF(empty(z0), empty(tail(z0)), z0), EMPTY(tail(z0)), TAIL(z0))
   IF(true, z0, z1) -> c10
   IF(false, z0, z1) -> c11(IF2(z0, z1))
   IF2(true, z0) -> c12(HEAD(z0))
   IF2(false, z0) -> c13(WEIGHT(sum(z0, cons(0, tail(tail(z0))))), SUM(z0, cons(0, tail(tail(z0)))), TAIL(tail(z0)), TAIL(z0))

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

   sum(cons(s(z0), z1), cons(z2, z3)) -> sum(cons(z0, z1), cons(s(z2), z3))
   sum(cons(0, z0), z1) -> sum(z0, z1)
   sum(nil, z0) -> z0
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   weight(z0) -> if(empty(z0), empty(tail(z0)), z0)
   if(true, z0, z1) -> weight_undefined_error
   if(false, z0, z1) -> if2(z0, z1)
   if2(true, z0) -> head(z0)
   if2(false, z0) -> weight(sum(z0, cons(0, tail(tail(z0)))))

Rewrite Strategy: INNERMOST