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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 35 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
(7) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
(8) BOUNDS(1, n^1)


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

(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:

   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: 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(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 (full) 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: FULL
----------------------------------------

(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:

   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: 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:

   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: 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.

"[35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52]
{(35,36,[b_1|0, a_1|0, encArg_1|0, encode_b_1|0, encode_c_1|0, encode_a_1|0]), (35,37,[c_1|1]), (35,38,[c_1|1, b_1|1, a_1|1]), (35,39,[c_1|2]), (35,40,[a_1|2]), (35,43,[a_1|2]), (36,36,[c_1|0, cons_b_1|0, cons_a_1|0]), (37,36,[c_1|1]), (38,36,[encArg_1|1]), (38,38,[c_1|1, b_1|1, a_1|1]), (38,39,[c_1|2]), (38,40,[a_1|2]), (38,43,[a_1|2]), (39,38,[c_1|2]), (40,41,[b_1|2]), (40,46,[c_1|3]), (41,42,[a_1|2]), (41,49,[a_1|3]), (42,38,[b_1|2]), (42,40,[b_1|2, a_1|2]), (42,43,[b_1|2]), (42,47,[c_1|3]), (43,44,[c_1|2]), (44,45,[b_1|2]), (44,48,[c_1|3]), (45,38,[a_1|2]), (45,40,[a_1|2]), (45,43,[a_1|2]), (45,49,[a_1|3]), (46,41,[c_1|3]), (47,38,[c_1|3]), (47,40,[c_1|3]), (47,43,[c_1|3]), (48,45,[c_1|3]), (49,50,[c_1|3]), (50,51,[b_1|3]), (50,52,[c_1|4]), (51,40,[a_1|3]), (51,43,[a_1|3]), (52,51,[c_1|4])}"
----------------------------------------

(8)
BOUNDS(1, n^1)