/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 311 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


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

(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:

   double(x) -> permute(x, x, a)
   permute(x, y, a) -> permute(isZero(x), x, b)
   permute(false, x, b) -> permute(ack(x, x), p(x), c)
   permute(true, x, b) -> 0
   permute(y, x, c) -> s(s(permute(x, y, a)))
   p(0) -> 0
   p(s(x)) -> x
   ack(0, x) -> plus(x, s(0))
   ack(s(x), 0) -> ack(x, s(0))
   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
   plus(0, y) -> y
   plus(s(x), y) -> plus(x, s(y))
   plus(x, s(s(y))) -> s(plus(s(x), y))
   plus(x, s(0)) -> s(x)
   plus(x, 0) -> x
   isZero(0) -> true
   isZero(s(x)) -> false

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(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(a) -> a
   encArg(b) -> b
   encArg(false) -> false
   encArg(c) -> c
   encArg(true) -> true
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_double(x_1)) -> double(encArg(x_1))
   encArg(cons_permute(x_1, x_2, x_3)) -> permute(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_ack(x_1, x_2)) -> ack(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_isZero(x_1)) -> isZero(encArg(x_1))
   encode_double(x_1) -> double(encArg(x_1))
   encode_permute(x_1, x_2, x_3) -> permute(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_a -> a
   encode_isZero(x_1) -> isZero(encArg(x_1))
   encode_b -> b
   encode_false -> false
   encode_ack(x_1, x_2) -> ack(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(encArg(x_1))
   encode_c -> c
   encode_true -> true
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))


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

(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:

   double(x) -> permute(x, x, a)
   permute(x, y, a) -> permute(isZero(x), x, b)
   permute(false, x, b) -> permute(ack(x, x), p(x), c)
   permute(true, x, b) -> 0
   permute(y, x, c) -> s(s(permute(x, y, a)))
   p(0) -> 0
   p(s(x)) -> x
   ack(0, x) -> plus(x, s(0))
   ack(s(x), 0) -> ack(x, s(0))
   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
   plus(0, y) -> y
   plus(s(x), y) -> plus(x, s(y))
   plus(x, s(s(y))) -> s(plus(s(x), y))
   plus(x, s(0)) -> s(x)
   plus(x, 0) -> x
   isZero(0) -> true
   isZero(s(x)) -> false

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(false) -> false
   encArg(c) -> c
   encArg(true) -> true
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_double(x_1)) -> double(encArg(x_1))
   encArg(cons_permute(x_1, x_2, x_3)) -> permute(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_ack(x_1, x_2)) -> ack(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_isZero(x_1)) -> isZero(encArg(x_1))
   encode_double(x_1) -> double(encArg(x_1))
   encode_permute(x_1, x_2, x_3) -> permute(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_a -> a
   encode_isZero(x_1) -> isZero(encArg(x_1))
   encode_b -> b
   encode_false -> false
   encode_ack(x_1, x_2) -> ack(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(encArg(x_1))
   encode_c -> c
   encode_true -> true
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST
----------------------------------------

(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:

   double(x) -> permute(x, x, a)
   permute(x, y, a) -> permute(isZero(x), x, b)
   permute(false, x, b) -> permute(ack(x, x), p(x), c)
   permute(true, x, b) -> 0
   permute(y, x, c) -> s(s(permute(x, y, a)))
   p(0) -> 0
   p(s(x)) -> x
   ack(0, x) -> plus(x, s(0))
   ack(s(x), 0) -> ack(x, s(0))
   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
   plus(0, y) -> y
   plus(s(x), y) -> plus(x, s(y))
   plus(x, s(s(y))) -> s(plus(s(x), y))
   plus(x, s(0)) -> s(x)
   plus(x, 0) -> x
   isZero(0) -> true
   isZero(s(x)) -> false

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(false) -> false
   encArg(c) -> c
   encArg(true) -> true
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_double(x_1)) -> double(encArg(x_1))
   encArg(cons_permute(x_1, x_2, x_3)) -> permute(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_ack(x_1, x_2)) -> ack(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_isZero(x_1)) -> isZero(encArg(x_1))
   encode_double(x_1) -> double(encArg(x_1))
   encode_permute(x_1, x_2, x_3) -> permute(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_a -> a
   encode_isZero(x_1) -> isZero(encArg(x_1))
   encode_b -> b
   encode_false -> false
   encode_ack(x_1, x_2) -> ack(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(encArg(x_1))
   encode_c -> c
   encode_true -> true
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST
----------------------------------------

(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:

   double(x) -> permute(x, x, a)
   permute(x, y, a) -> permute(isZero(x), x, b)
   permute(false, x, b) -> permute(ack(x, x), p(x), c)
   permute(true, x, b) -> 0
   permute(y, x, c) -> s(s(permute(x, y, a)))
   p(0) -> 0
   p(s(x)) -> x
   ack(0, x) -> plus(x, s(0))
   ack(s(x), 0) -> ack(x, s(0))
   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
   plus(0, y) -> y
   plus(s(x), y) -> plus(x, s(y))
   plus(x, s(s(y))) -> s(plus(s(x), y))
   plus(x, s(0)) -> s(x)
   plus(x, 0) -> x
   isZero(0) -> true
   isZero(s(x)) -> false

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(false) -> false
   encArg(c) -> c
   encArg(true) -> true
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_double(x_1)) -> double(encArg(x_1))
   encArg(cons_permute(x_1, x_2, x_3)) -> permute(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_ack(x_1, x_2)) -> ack(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_isZero(x_1)) -> isZero(encArg(x_1))
   encode_double(x_1) -> double(encArg(x_1))
   encode_permute(x_1, x_2, x_3) -> permute(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_a -> a
   encode_isZero(x_1) -> isZero(encArg(x_1))
   encode_b -> b
   encode_false -> false
   encode_ack(x_1, x_2) -> ack(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(encArg(x_1))
   encode_c -> c
   encode_true -> true
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST
----------------------------------------

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

The rewrite sequence

plus(x, s(s(y))) ->^+ s(plus(s(x), y))

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

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

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




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

(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:

   double(x) -> permute(x, x, a)
   permute(x, y, a) -> permute(isZero(x), x, b)
   permute(false, x, b) -> permute(ack(x, x), p(x), c)
   permute(true, x, b) -> 0
   permute(y, x, c) -> s(s(permute(x, y, a)))
   p(0) -> 0
   p(s(x)) -> x
   ack(0, x) -> plus(x, s(0))
   ack(s(x), 0) -> ack(x, s(0))
   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
   plus(0, y) -> y
   plus(s(x), y) -> plus(x, s(y))
   plus(x, s(s(y))) -> s(plus(s(x), y))
   plus(x, s(0)) -> s(x)
   plus(x, 0) -> x
   isZero(0) -> true
   isZero(s(x)) -> false

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(false) -> false
   encArg(c) -> c
   encArg(true) -> true
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_double(x_1)) -> double(encArg(x_1))
   encArg(cons_permute(x_1, x_2, x_3)) -> permute(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_ack(x_1, x_2)) -> ack(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_isZero(x_1)) -> isZero(encArg(x_1))
   encode_double(x_1) -> double(encArg(x_1))
   encode_permute(x_1, x_2, x_3) -> permute(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_a -> a
   encode_isZero(x_1) -> isZero(encArg(x_1))
   encode_b -> b
   encode_false -> false
   encode_ack(x_1, x_2) -> ack(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(encArg(x_1))
   encode_c -> c
   encode_true -> true
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST
----------------------------------------

(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:

   double(x) -> permute(x, x, a)
   permute(x, y, a) -> permute(isZero(x), x, b)
   permute(false, x, b) -> permute(ack(x, x), p(x), c)
   permute(true, x, b) -> 0
   permute(y, x, c) -> s(s(permute(x, y, a)))
   p(0) -> 0
   p(s(x)) -> x
   ack(0, x) -> plus(x, s(0))
   ack(s(x), 0) -> ack(x, s(0))
   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
   plus(0, y) -> y
   plus(s(x), y) -> plus(x, s(y))
   plus(x, s(s(y))) -> s(plus(s(x), y))
   plus(x, s(0)) -> s(x)
   plus(x, 0) -> x
   isZero(0) -> true
   isZero(s(x)) -> false

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(false) -> false
   encArg(c) -> c
   encArg(true) -> true
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_double(x_1)) -> double(encArg(x_1))
   encArg(cons_permute(x_1, x_2, x_3)) -> permute(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_ack(x_1, x_2)) -> ack(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_isZero(x_1)) -> isZero(encArg(x_1))
   encode_double(x_1) -> double(encArg(x_1))
   encode_permute(x_1, x_2, x_3) -> permute(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_a -> a
   encode_isZero(x_1) -> isZero(encArg(x_1))
   encode_b -> b
   encode_false -> false
   encode_ack(x_1, x_2) -> ack(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(encArg(x_1))
   encode_c -> c
   encode_true -> true
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST