/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 168 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) TRS for Loop Detection


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


The TRS R consists of the following rules:

   a(lambda(x), y) -> lambda(a(x, p(1, a(y, t))))
   a(p(x, y), z) -> p(a(x, z), a(y, z))
   a(a(x, y), z) -> a(x, a(y, z))
   a(id, x) -> x
   a(1, id) -> 1
   a(t, id) -> t
   a(1, p(x, y)) -> x
   a(t, p(x, y)) -> y

S is empty.
Rewrite Strategy: INNERMOST
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(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(lambda(x_1)) -> lambda(encArg(x_1))
   encArg(p(x_1, x_2)) -> p(encArg(x_1), encArg(x_2))
   encArg(1) -> 1
   encArg(t) -> t
   encArg(id) -> id
   encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
   encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
   encode_lambda(x_1) -> lambda(encArg(x_1))
   encode_p(x_1, x_2) -> p(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_t -> t
   encode_id -> id


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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:

   a(lambda(x), y) -> lambda(a(x, p(1, a(y, t))))
   a(p(x, y), z) -> p(a(x, z), a(y, z))
   a(a(x, y), z) -> a(x, a(y, z))
   a(id, x) -> x
   a(1, id) -> 1
   a(t, id) -> t
   a(1, p(x, y)) -> x
   a(t, p(x, y)) -> y

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

   encArg(lambda(x_1)) -> lambda(encArg(x_1))
   encArg(p(x_1, x_2)) -> p(encArg(x_1), encArg(x_2))
   encArg(1) -> 1
   encArg(t) -> t
   encArg(id) -> id
   encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
   encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
   encode_lambda(x_1) -> lambda(encArg(x_1))
   encode_p(x_1, x_2) -> p(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_t -> t
   encode_id -> id

Rewrite Strategy: INNERMOST
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(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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(4)
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:

   a(lambda(x), y) -> lambda(a(x, p(1, a(y, t))))
   a(p(x, y), z) -> p(a(x, z), a(y, z))
   a(a(x, y), z) -> a(x, a(y, z))
   a(id, x) -> x
   a(1, id) -> 1
   a(t, id) -> t
   a(1, p(x, y)) -> x
   a(t, p(x, y)) -> y

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

   encArg(lambda(x_1)) -> lambda(encArg(x_1))
   encArg(p(x_1, x_2)) -> p(encArg(x_1), encArg(x_2))
   encArg(1) -> 1
   encArg(t) -> t
   encArg(id) -> id
   encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
   encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
   encode_lambda(x_1) -> lambda(encArg(x_1))
   encode_p(x_1, x_2) -> p(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_t -> t
   encode_id -> id

Rewrite Strategy: INNERMOST
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(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(6)
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:

   a(lambda(x), y) -> lambda(a(x, p(1, a(y, t))))
   a(p(x, y), z) -> p(a(x, z), a(y, z))
   a(a(x, y), z) -> a(x, a(y, z))
   a(id, x) -> x
   a(1, id) -> 1
   a(t, id) -> t
   a(1, p(x, y)) -> x
   a(t, p(x, y)) -> y

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

   encArg(lambda(x_1)) -> lambda(encArg(x_1))
   encArg(p(x_1, x_2)) -> p(encArg(x_1), encArg(x_2))
   encArg(1) -> 1
   encArg(t) -> t
   encArg(id) -> id
   encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
   encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
   encode_lambda(x_1) -> lambda(encArg(x_1))
   encode_p(x_1, x_2) -> p(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_t -> t
   encode_id -> id

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

The rewrite sequence

a(p(x, y), z) ->^+ p(a(x, z), a(y, z))

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

The pumping substitution is [x / p(x, y)].

The result substitution is [ ].




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(8)
Complex Obligation (BEST)

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(9)
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:

   a(lambda(x), y) -> lambda(a(x, p(1, a(y, t))))
   a(p(x, y), z) -> p(a(x, z), a(y, z))
   a(a(x, y), z) -> a(x, a(y, z))
   a(id, x) -> x
   a(1, id) -> 1
   a(t, id) -> t
   a(1, p(x, y)) -> x
   a(t, p(x, y)) -> y

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

   encArg(lambda(x_1)) -> lambda(encArg(x_1))
   encArg(p(x_1, x_2)) -> p(encArg(x_1), encArg(x_2))
   encArg(1) -> 1
   encArg(t) -> t
   encArg(id) -> id
   encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
   encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
   encode_lambda(x_1) -> lambda(encArg(x_1))
   encode_p(x_1, x_2) -> p(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_t -> t
   encode_id -> id

Rewrite Strategy: INNERMOST
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(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(11)
BOUNDS(n^1, INF)

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(12)
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:

   a(lambda(x), y) -> lambda(a(x, p(1, a(y, t))))
   a(p(x, y), z) -> p(a(x, z), a(y, z))
   a(a(x, y), z) -> a(x, a(y, z))
   a(id, x) -> x
   a(1, id) -> 1
   a(t, id) -> t
   a(1, p(x, y)) -> x
   a(t, p(x, y)) -> y

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

   encArg(lambda(x_1)) -> lambda(encArg(x_1))
   encArg(p(x_1, x_2)) -> p(encArg(x_1), encArg(x_2))
   encArg(1) -> 1
   encArg(t) -> t
   encArg(id) -> id
   encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
   encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
   encode_lambda(x_1) -> lambda(encArg(x_1))
   encode_p(x_1, x_2) -> p(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_t -> t
   encode_id -> id

Rewrite Strategy: INNERMOST