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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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), 1276 ms]
(2) CpxRelTRS
(3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(4) TRS for Loop Detection
(5) DecreasingLoopProof [LOWER BOUND(ID), 2 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:

   MIN(0, z0) -> c
   MIN(s(z0), 0) -> c1
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   LEN(nil) -> c3
   LEN(cons(z0, z1)) -> c4(LEN(z1))
   SUM(z0, 0) -> c5
   SUM(z0, s(z1)) -> c6(SUM(z0, z1))
   LE(0, z0) -> c7
   LE(s(z0), 0) -> c8
   LE(s(z0), s(z1)) -> c9(LE(z0, z1))
   TAKE(0, cons(z0, z1)) -> c10
   TAKE(s(z0), cons(z1, z2)) -> c11(TAKE(z0, z2))
   ADDLIST(z0, z1) -> c12(IF(le(0, min(len(z0), len(z1))), 0, z0, z1, nil), LE(0, min(len(z0), len(z1))), MIN(len(z0), len(z1)), LEN(z0))
   ADDLIST(z0, z1) -> c13(IF(le(0, min(len(z0), len(z1))), 0, z0, z1, nil), LE(0, min(len(z0), len(z1))), MIN(len(z0), len(z1)), LEN(z1))
   IF(false, z0, z1, z2, z3) -> c14
   IF(true, z0, z1, z2, z3) -> c15(IF(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3)), LE(s(z0), min(len(z1), len(z2))), MIN(len(z1), len(z2)), LEN(z1))
   IF(true, z0, z1, z2, z3) -> c16(IF(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3)), LE(s(z0), min(len(z1), len(z2))), MIN(len(z1), len(z2)), LEN(z2))
   IF(true, z0, z1, z2, z3) -> c17(IF(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3)), SUM(take(z0, z1), take(z0, z2)), TAKE(z0, z1))
   IF(true, z0, z1, z2, z3) -> c18(IF(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3)), SUM(take(z0, z1), take(z0, z2)), TAKE(z0, z2))

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

   min(0, z0) -> 0
   min(s(z0), 0) -> 0
   min(s(z0), s(z1)) -> min(z0, z1)
   len(nil) -> 0
   len(cons(z0, z1)) -> s(len(z1))
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   take(0, cons(z0, z1)) -> z0
   take(s(z0), cons(z1, z2)) -> take(z0, z2)
   addList(z0, z1) -> if(le(0, min(len(z0), len(z1))), 0, z0, z1, nil)
   if(false, z0, z1, z2, z3) -> z3
   if(true, z0, z1, z2, z3) -> if(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3))

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:

   MIN(0, z0) -> c
   MIN(s(z0), 0) -> c1
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   LEN(nil) -> c3
   LEN(cons(z0, z1)) -> c4(LEN(z1))
   SUM(z0, 0) -> c5
   SUM(z0, s(z1)) -> c6(SUM(z0, z1))
   LE(0, z0) -> c7
   LE(s(z0), 0) -> c8
   LE(s(z0), s(z1)) -> c9(LE(z0, z1))
   TAKE(0, cons(z0, z1)) -> c10
   TAKE(s(z0), cons(z1, z2)) -> c11(TAKE(z0, z2))
   ADDLIST(z0, z1) -> c12(IF(le(0, min(len(z0), len(z1))), 0, z0, z1, nil), LE(0, min(len(z0), len(z1))), MIN(len(z0), len(z1)), LEN(z0))
   ADDLIST(z0, z1) -> c13(IF(le(0, min(len(z0), len(z1))), 0, z0, z1, nil), LE(0, min(len(z0), len(z1))), MIN(len(z0), len(z1)), LEN(z1))
   IF(false, z0, z1, z2, z3) -> c14
   IF(true, z0, z1, z2, z3) -> c15(IF(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3)), LE(s(z0), min(len(z1), len(z2))), MIN(len(z1), len(z2)), LEN(z1))
   IF(true, z0, z1, z2, z3) -> c16(IF(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3)), LE(s(z0), min(len(z1), len(z2))), MIN(len(z1), len(z2)), LEN(z2))
   IF(true, z0, z1, z2, z3) -> c17(IF(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3)), SUM(take(z0, z1), take(z0, z2)), TAKE(z0, z1))
   IF(true, z0, z1, z2, z3) -> c18(IF(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3)), SUM(take(z0, z1), take(z0, z2)), TAKE(z0, z2))

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

   min(0, z0) -> 0
   min(s(z0), 0) -> 0
   min(s(z0), s(z1)) -> min(z0, z1)
   len(nil) -> 0
   len(cons(z0, z1)) -> s(len(z1))
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   take(0, cons(z0, z1)) -> z0
   take(s(z0), cons(z1, z2)) -> take(z0, z2)
   addList(z0, z1) -> if(le(0, min(len(z0), len(z1))), 0, z0, z1, nil)
   if(false, z0, z1, z2, z3) -> z3
   if(true, z0, z1, z2, z3) -> if(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3))

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:

   MIN(0, z0) -> c
   MIN(s(z0), 0) -> c1
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   LEN(nil) -> c3
   LEN(cons(z0, z1)) -> c4(LEN(z1))
   SUM(z0, 0) -> c5
   SUM(z0, s(z1)) -> c6(SUM(z0, z1))
   LE(0, z0) -> c7
   LE(s(z0), 0) -> c8
   LE(s(z0), s(z1)) -> c9(LE(z0, z1))
   TAKE(0, cons(z0, z1)) -> c10
   TAKE(s(z0), cons(z1, z2)) -> c11(TAKE(z0, z2))
   ADDLIST(z0, z1) -> c12(IF(le(0, min(len(z0), len(z1))), 0, z0, z1, nil), LE(0, min(len(z0), len(z1))), MIN(len(z0), len(z1)), LEN(z0))
   ADDLIST(z0, z1) -> c13(IF(le(0, min(len(z0), len(z1))), 0, z0, z1, nil), LE(0, min(len(z0), len(z1))), MIN(len(z0), len(z1)), LEN(z1))
   IF(false, z0, z1, z2, z3) -> c14
   IF(true, z0, z1, z2, z3) -> c15(IF(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3)), LE(s(z0), min(len(z1), len(z2))), MIN(len(z1), len(z2)), LEN(z1))
   IF(true, z0, z1, z2, z3) -> c16(IF(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3)), LE(s(z0), min(len(z1), len(z2))), MIN(len(z1), len(z2)), LEN(z2))
   IF(true, z0, z1, z2, z3) -> c17(IF(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3)), SUM(take(z0, z1), take(z0, z2)), TAKE(z0, z1))
   IF(true, z0, z1, z2, z3) -> c18(IF(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3)), SUM(take(z0, z1), take(z0, z2)), TAKE(z0, z2))

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

   min(0, z0) -> 0
   min(s(z0), 0) -> 0
   min(s(z0), s(z1)) -> min(z0, z1)
   len(nil) -> 0
   len(cons(z0, z1)) -> s(len(z1))
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   take(0, cons(z0, z1)) -> z0
   take(s(z0), cons(z1, z2)) -> take(z0, z2)
   addList(z0, z1) -> if(le(0, min(len(z0), len(z1))), 0, z0, z1, nil)
   if(false, z0, z1, z2, z3) -> z3
   if(true, z0, z1, z2, z3) -> if(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3))

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(z0, s(z1)) ->^+ c6(SUM(z0, z1))

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

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

The result substitution is [ ].




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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:

   MIN(0, z0) -> c
   MIN(s(z0), 0) -> c1
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   LEN(nil) -> c3
   LEN(cons(z0, z1)) -> c4(LEN(z1))
   SUM(z0, 0) -> c5
   SUM(z0, s(z1)) -> c6(SUM(z0, z1))
   LE(0, z0) -> c7
   LE(s(z0), 0) -> c8
   LE(s(z0), s(z1)) -> c9(LE(z0, z1))
   TAKE(0, cons(z0, z1)) -> c10
   TAKE(s(z0), cons(z1, z2)) -> c11(TAKE(z0, z2))
   ADDLIST(z0, z1) -> c12(IF(le(0, min(len(z0), len(z1))), 0, z0, z1, nil), LE(0, min(len(z0), len(z1))), MIN(len(z0), len(z1)), LEN(z0))
   ADDLIST(z0, z1) -> c13(IF(le(0, min(len(z0), len(z1))), 0, z0, z1, nil), LE(0, min(len(z0), len(z1))), MIN(len(z0), len(z1)), LEN(z1))
   IF(false, z0, z1, z2, z3) -> c14
   IF(true, z0, z1, z2, z3) -> c15(IF(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3)), LE(s(z0), min(len(z1), len(z2))), MIN(len(z1), len(z2)), LEN(z1))
   IF(true, z0, z1, z2, z3) -> c16(IF(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3)), LE(s(z0), min(len(z1), len(z2))), MIN(len(z1), len(z2)), LEN(z2))
   IF(true, z0, z1, z2, z3) -> c17(IF(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3)), SUM(take(z0, z1), take(z0, z2)), TAKE(z0, z1))
   IF(true, z0, z1, z2, z3) -> c18(IF(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3)), SUM(take(z0, z1), take(z0, z2)), TAKE(z0, z2))

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

   min(0, z0) -> 0
   min(s(z0), 0) -> 0
   min(s(z0), s(z1)) -> min(z0, z1)
   len(nil) -> 0
   len(cons(z0, z1)) -> s(len(z1))
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   take(0, cons(z0, z1)) -> z0
   take(s(z0), cons(z1, z2)) -> take(z0, z2)
   addList(z0, z1) -> if(le(0, min(len(z0), len(z1))), 0, z0, z1, nil)
   if(false, z0, z1, z2, z3) -> z3
   if(true, z0, z1, z2, z3) -> if(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3))

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:

   MIN(0, z0) -> c
   MIN(s(z0), 0) -> c1
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   LEN(nil) -> c3
   LEN(cons(z0, z1)) -> c4(LEN(z1))
   SUM(z0, 0) -> c5
   SUM(z0, s(z1)) -> c6(SUM(z0, z1))
   LE(0, z0) -> c7
   LE(s(z0), 0) -> c8
   LE(s(z0), s(z1)) -> c9(LE(z0, z1))
   TAKE(0, cons(z0, z1)) -> c10
   TAKE(s(z0), cons(z1, z2)) -> c11(TAKE(z0, z2))
   ADDLIST(z0, z1) -> c12(IF(le(0, min(len(z0), len(z1))), 0, z0, z1, nil), LE(0, min(len(z0), len(z1))), MIN(len(z0), len(z1)), LEN(z0))
   ADDLIST(z0, z1) -> c13(IF(le(0, min(len(z0), len(z1))), 0, z0, z1, nil), LE(0, min(len(z0), len(z1))), MIN(len(z0), len(z1)), LEN(z1))
   IF(false, z0, z1, z2, z3) -> c14
   IF(true, z0, z1, z2, z3) -> c15(IF(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3)), LE(s(z0), min(len(z1), len(z2))), MIN(len(z1), len(z2)), LEN(z1))
   IF(true, z0, z1, z2, z3) -> c16(IF(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3)), LE(s(z0), min(len(z1), len(z2))), MIN(len(z1), len(z2)), LEN(z2))
   IF(true, z0, z1, z2, z3) -> c17(IF(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3)), SUM(take(z0, z1), take(z0, z2)), TAKE(z0, z1))
   IF(true, z0, z1, z2, z3) -> c18(IF(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3)), SUM(take(z0, z1), take(z0, z2)), TAKE(z0, z2))

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

   min(0, z0) -> 0
   min(s(z0), 0) -> 0
   min(s(z0), s(z1)) -> min(z0, z1)
   len(nil) -> 0
   len(cons(z0, z1)) -> s(len(z1))
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   take(0, cons(z0, z1)) -> z0
   take(s(z0), cons(z1, z2)) -> take(z0, z2)
   addList(z0, z1) -> if(le(0, min(len(z0), len(z1))), 0, z0, z1, nil)
   if(false, z0, z1, z2, z3) -> z3
   if(true, z0, z1, z2, z3) -> if(le(s(z0), min(len(z1), len(z2))), s(z0), z1, z2, cons(sum(take(z0, z1), take(z0, z2)), z3))

Rewrite Strategy: INNERMOST