/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (innermost) 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), 47 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
(7) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
(8) BOUNDS(1, n^1)


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


The TRS R consists of the following rules:

   b(c(a(x1))) -> a(b(a(b(x1))))
   b(x1) -> c(c(x1))
   a(a(x1)) -> a(c(b(a(x1))))

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(c(x_1)) -> c(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))


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

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


The TRS R consists of the following rules:

   b(c(a(x1))) -> a(b(a(b(x1))))
   b(x1) -> c(c(x1))
   a(a(x1)) -> a(c(b(a(x1))))

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))

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(1, n^1).


The TRS R consists of the following rules:

   b(c(a(x1))) -> a(b(a(b(x1))))
   b(x1) -> c(c(x1))
   a(a(x1)) -> a(c(b(a(x1))))

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))

Rewrite Strategy: INNERMOST
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(5) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   b(c(a(x1))) -> a(b(a(b(x1))))
   b(x1) -> c(c(x1))
   a(a(x1)) -> a(c(b(a(x1))))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))

S is empty.
Rewrite Strategy: INNERMOST
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(7) CpxTrsMatchBoundsProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 4. 
The certificate found is represented by the following graph.

"[39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56]
{(39,40,[b_1|0, a_1|0, encArg_1|0, encode_b_1|0, encode_c_1|0, encode_a_1|0]), (39,41,[c_1|1]), (39,42,[c_1|1, b_1|1, a_1|1]), (39,43,[c_1|2]), (39,44,[a_1|2]), (39,47,[a_1|2]), (40,40,[c_1|0, cons_b_1|0, cons_a_1|0]), (41,40,[c_1|1]), (42,40,[encArg_1|1]), (42,42,[c_1|1, b_1|1, a_1|1]), (42,43,[c_1|2]), (42,44,[a_1|2]), (42,47,[a_1|2]), (43,42,[c_1|2]), (44,45,[b_1|2]), (44,50,[c_1|3]), (45,46,[a_1|2]), (45,53,[a_1|3]), (46,42,[b_1|2]), (46,44,[b_1|2, a_1|2]), (46,47,[b_1|2]), (46,51,[c_1|3]), (47,48,[c_1|2]), (48,49,[b_1|2]), (48,52,[c_1|3]), (49,42,[a_1|2]), (49,44,[a_1|2]), (49,47,[a_1|2]), (49,53,[a_1|3]), (50,45,[c_1|3]), (51,42,[c_1|3]), (51,44,[c_1|3]), (51,47,[c_1|3]), (52,49,[c_1|3]), (53,54,[c_1|3]), (54,55,[b_1|3]), (54,56,[c_1|4]), (55,44,[a_1|3]), (55,47,[a_1|3]), (56,55,[c_1|4])}"
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(8)
BOUNDS(1, n^1)