WORST_CASE(?, O(n^1))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 1 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 22 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
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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(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.

"[28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]
{(28,29,[f_1|0, encArg_1|0, encode_f_1|0, encode_g_1|0]), (28,30,[g_1|1, f_1|1]), (28,31,[f_1|2]), (28,35,[f_1|2]), (28,36,[f_1|3]), (29,29,[g_1|0, cons_f_1|0]), (30,29,[encArg_1|1]), (30,30,[g_1|1, f_1|1]), (30,31,[f_1|2]), (30,35,[f_1|2]), (30,36,[f_1|3]), (31,32,[g_1|2]), (32,33,[f_1|2]), (32,37,[f_1|3]), (32,35,[f_1|2]), (32,42,[f_1|4]), (33,34,[g_1|2]), (34,30,[f_1|2]), (34,31,[f_1|2]), (34,35,[f_1|2]), (34,38,[f_1|3]), (34,36,[f_1|2, f_1|3]), (34,43,[f_1|4]), (35,30,[g_1|2]), (35,31,[g_1|2]), (35,35,[g_1|2]), (35,36,[g_1|2]), (36,33,[g_1|3]), (36,31,[g_1|3]), (36,35,[g_1|3]), (36,37,[g_1|3]), (36,36,[g_1|3]), (36,42,[g_1|3]), (37,30,[g_1|3]), (37,31,[g_1|3]), (37,35,[g_1|3]), (37,38,[g_1|3]), (37,36,[g_1|3]), (37,43,[g_1|3]), (38,39,[g_1|3]), (39,40,[f_1|3]), (39,44,[f_1|4]), (39,36,[f_1|3]), (40,41,[g_1|3]), (41,31,[f_1|3]), (41,35,[f_1|3, f_1|2]), (41,36,[f_1|3]), (42,36,[g_1|4]), (43,40,[g_1|4]), (43,44,[g_1|4]), (43,36,[g_1|4]), (44,31,[g_1|4]), (44,35,[g_1|4]), (44,36,[g_1|4])}"
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(8)
BOUNDS(1, n^1)