/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 185 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:

   f(0) -> true
   f(1) -> false
   f(s(x)) -> f(x)
   if(true, x, y) -> x
   if(false, x, y) -> y
   g(s(x), s(y)) -> if(f(x), s(x), s(y))
   g(x, c(y)) -> g(x, g(s(c(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(0) -> 0
   encArg(true) -> true
   encArg(1) -> 1
   encArg(false) -> false
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_true -> true
   encode_1 -> 1
   encode_false -> false
   encode_s(x_1) -> s(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_c(x_1) -> c(encArg(x_1))


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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:

   f(0) -> true
   f(1) -> false
   f(s(x)) -> f(x)
   if(true, x, y) -> x
   if(false, x, y) -> y
   g(s(x), s(y)) -> if(f(x), s(x), s(y))
   g(x, c(y)) -> g(x, g(s(c(y)), y))

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

   encArg(0) -> 0
   encArg(true) -> true
   encArg(1) -> 1
   encArg(false) -> false
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_true -> true
   encode_1 -> 1
   encode_false -> false
   encode_s(x_1) -> s(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_c(x_1) -> c(encArg(x_1))

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

(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:

   f(0) -> true
   f(1) -> false
   f(s(x)) -> f(x)
   if(true, x, y) -> x
   if(false, x, y) -> y
   g(s(x), s(y)) -> if(f(x), s(x), s(y))
   g(x, c(y)) -> g(x, g(s(c(y)), y))

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

   encArg(0) -> 0
   encArg(true) -> true
   encArg(1) -> 1
   encArg(false) -> false
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_true -> true
   encode_1 -> 1
   encode_false -> false
   encode_s(x_1) -> s(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_c(x_1) -> c(encArg(x_1))

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

(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:

   f(0) -> true
   f(1) -> false
   f(s(x)) -> f(x)
   if(true, x, y) -> x
   if(false, x, y) -> y
   g(s(x), s(y)) -> if(f(x), s(x), s(y))
   g(x, c(y)) -> g(x, g(s(c(y)), y))

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

   encArg(0) -> 0
   encArg(true) -> true
   encArg(1) -> 1
   encArg(false) -> false
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_true -> true
   encode_1 -> 1
   encode_false -> false
   encode_s(x_1) -> s(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_c(x_1) -> c(encArg(x_1))

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

g(x, c(y)) ->^+ g(x, g(s(c(y)), y))

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

The pumping substitution is [y / c(y)].

The result substitution is [x / s(c(y))].




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(8)
Complex Obligation (BEST)

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

(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:

   f(0) -> true
   f(1) -> false
   f(s(x)) -> f(x)
   if(true, x, y) -> x
   if(false, x, y) -> y
   g(s(x), s(y)) -> if(f(x), s(x), s(y))
   g(x, c(y)) -> g(x, g(s(c(y)), y))

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

   encArg(0) -> 0
   encArg(true) -> true
   encArg(1) -> 1
   encArg(false) -> false
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_true -> true
   encode_1 -> 1
   encode_false -> false
   encode_s(x_1) -> s(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_c(x_1) -> c(encArg(x_1))

Rewrite Strategy: INNERMOST
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(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(11)
BOUNDS(n^1, INF)

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

(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:

   f(0) -> true
   f(1) -> false
   f(s(x)) -> f(x)
   if(true, x, y) -> x
   if(false, x, y) -> y
   g(s(x), s(y)) -> if(f(x), s(x), s(y))
   g(x, c(y)) -> g(x, g(s(c(y)), y))

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

   encArg(0) -> 0
   encArg(true) -> true
   encArg(1) -> 1
   encArg(false) -> false
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_true -> true
   encode_1 -> 1
   encode_false -> false
   encode_s(x_1) -> s(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_c(x_1) -> c(encArg(x_1))

Rewrite Strategy: INNERMOST