/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 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 (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   f(f(X)) -> f(g(f(g(f(X)))))
   f(g(f(X))) -> f(g(X))

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

(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(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))


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

(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:

   f(f(X)) -> f(g(f(g(f(X)))))
   f(g(f(X))) -> f(g(X))

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

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))

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

(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:

   f(f(X)) -> f(g(f(g(f(X)))))
   f(g(f(X))) -> f(g(X))

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

   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))

Rewrite Strategy: FULL
----------------------------------------

(5) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(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:

   f(f(X)) -> f(g(f(g(f(X)))))
   f(g(f(X))) -> f(g(X))
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))

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

(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.

"[27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43]
{(27,28,[f_1|0, encArg_1|0, encode_f_1|0, encode_g_1|0]), (27,29,[g_1|1, f_1|1]), (27,30,[f_1|2]), (27,34,[f_1|2]), (27,35,[f_1|3]), (28,28,[g_1|0, cons_f_1|0]), (29,28,[encArg_1|1]), (29,29,[g_1|1, f_1|1]), (29,30,[f_1|2]), (29,34,[f_1|2]), (29,35,[f_1|3]), (30,31,[g_1|2]), (31,32,[f_1|2]), (31,36,[f_1|3]), (31,34,[f_1|2]), (31,41,[f_1|4]), (32,33,[g_1|2]), (33,29,[f_1|2]), (33,30,[f_1|2]), (33,34,[f_1|2]), (33,37,[f_1|3]), (33,35,[f_1|2, f_1|3]), (33,42,[f_1|4]), (34,29,[g_1|2]), (34,30,[g_1|2]), (34,34,[g_1|2]), (34,35,[g_1|2]), (35,32,[g_1|3]), (35,30,[g_1|3]), (35,34,[g_1|3]), (35,36,[g_1|3]), (35,35,[g_1|3]), (35,41,[g_1|3]), (36,29,[g_1|3]), (36,30,[g_1|3]), (36,34,[g_1|3]), (36,37,[g_1|3]), (36,35,[g_1|3]), (36,42,[g_1|3]), (37,38,[g_1|3]), (38,39,[f_1|3]), (38,43,[f_1|4]), (38,35,[f_1|3]), (39,40,[g_1|3]), (40,30,[f_1|3]), (40,34,[f_1|3, f_1|2]), (40,35,[f_1|3]), (41,35,[g_1|4]), (42,39,[g_1|4]), (42,43,[g_1|4]), (42,35,[g_1|4]), (43,30,[g_1|4]), (43,34,[g_1|4]), (43,35,[g_1|4])}"
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(8)
BOUNDS(1, n^1)