/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(?, O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 185 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
(7) CpxTrsMatchBoundsTAProof [FINISHED, 83 ms]
(8) BOUNDS(1, n^1)


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


The TRS R consists of the following rules:

   p(p(b(a(x0)), x1), p(x2, x3)) -> p(p(x3, a(x2)), p(b(a(x1)), b(x0)))

S is empty.
Rewrite Strategy: FULL
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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(b(x_1)) -> b(encArg(x_1))
   encArg(a(x_1)) -> a(encArg(x_1))
   encArg(cons_p(x_1, x_2)) -> p(encArg(x_1), encArg(x_2))
   encode_p(x_1, x_2) -> p(encArg(x_1), encArg(x_2))
   encode_b(x_1) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))


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(2)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   p(p(b(a(x0)), x1), p(x2, x3)) -> p(p(x3, a(x2)), p(b(a(x1)), b(x0)))

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

   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(a(x_1)) -> a(encArg(x_1))
   encArg(cons_p(x_1, x_2)) -> p(encArg(x_1), encArg(x_2))
   encode_p(x_1, x_2) -> p(encArg(x_1), encArg(x_2))
   encode_b(x_1) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))

Rewrite Strategy: FULL
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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 (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   p(p(b(a(x0)), x1), p(x2, x3)) -> p(p(x3, a(x2)), p(b(a(x1)), b(x0)))

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

   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(a(x_1)) -> a(encArg(x_1))
   encArg(cons_p(x_1, x_2)) -> p(encArg(x_1), encArg(x_2))
   encode_p(x_1, x_2) -> p(encArg(x_1), encArg(x_2))
   encode_b(x_1) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))

Rewrite Strategy: FULL
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(5) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(6)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   p(p(b(a(x0)), x1), p(x2, x3)) -> p(p(x3, a(x2)), p(b(a(x1)), b(x0)))
   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(a(x_1)) -> a(encArg(x_1))
   encArg(cons_p(x_1, x_2)) -> p(encArg(x_1), encArg(x_2))
   encode_p(x_1, x_2) -> p(encArg(x_1), encArg(x_2))
   encode_b(x_1) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))

S is empty.
Rewrite Strategy: FULL
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(7) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4, 5]
transitions: 
b0(0) -> 0
a0(0) -> 0
cons_p0(0, 0) -> 0
p0(0, 0) -> 1
encArg0(0) -> 2
encode_p0(0, 0) -> 3
encode_b0(0) -> 4
encode_a0(0) -> 5
encArg1(0) -> 6
b1(6) -> 2
encArg1(0) -> 7
a1(7) -> 2
encArg1(0) -> 8
encArg1(0) -> 9
p1(8, 9) -> 2
p1(8, 9) -> 3
b1(6) -> 4
a1(7) -> 5
b1(6) -> 6
b1(6) -> 7
b1(6) -> 8
b1(6) -> 9
a1(7) -> 6
a1(7) -> 7
a1(7) -> 8
a1(7) -> 9
p1(8, 9) -> 6
p1(8, 9) -> 7
p1(8, 9) -> 8
p1(8, 9) -> 9
a2(8) -> 11
p2(9, 11) -> 10
a2(9) -> 14
b2(14) -> 13
b2(7) -> 15
p2(13, 15) -> 12
p2(10, 12) -> 2
p2(10, 12) -> 3
p2(10, 12) -> 6
p2(10, 12) -> 7
p2(10, 12) -> 8
p2(10, 12) -> 9
a2(10) -> 11
p2(12, 11) -> 10
a2(13) -> 11
p2(15, 11) -> 10
a2(11) -> 14

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(8)
BOUNDS(1, n^1)