/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 53 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
(7) CpxTrsMatchBoundsProof [FINISHED, 6 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:

   a(c(d(x))) -> c(x)
   u(b(d(d(x)))) -> b(x)
   v(a(a(x))) -> u(v(x))
   v(a(c(x))) -> u(b(d(x)))
   v(c(x)) -> b(x)
   w(a(a(x))) -> u(w(x))
   w(a(c(x))) -> u(b(d(x)))
   w(c(x)) -> b(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(c(x_1)) -> c(encArg(x_1))
   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_u(x_1)) -> u(encArg(x_1))
   encArg(cons_v(x_1)) -> v(encArg(x_1))
   encArg(cons_w(x_1)) -> w(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_u(x_1) -> u(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_v(x_1) -> v(encArg(x_1))
   encode_w(x_1) -> w(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:

   a(c(d(x))) -> c(x)
   u(b(d(d(x)))) -> b(x)
   v(a(a(x))) -> u(v(x))
   v(a(c(x))) -> u(b(d(x)))
   v(c(x)) -> b(x)
   w(a(a(x))) -> u(w(x))
   w(a(c(x))) -> u(b(d(x)))
   w(c(x)) -> b(x)

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_u(x_1)) -> u(encArg(x_1))
   encArg(cons_v(x_1)) -> v(encArg(x_1))
   encArg(cons_w(x_1)) -> w(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_u(x_1) -> u(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_v(x_1) -> v(encArg(x_1))
   encode_w(x_1) -> w(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:

   a(c(d(x))) -> c(x)
   u(b(d(d(x)))) -> b(x)
   v(a(a(x))) -> u(v(x))
   v(a(c(x))) -> u(b(d(x)))
   v(c(x)) -> b(x)
   w(a(a(x))) -> u(w(x))
   w(a(c(x))) -> u(b(d(x)))
   w(c(x)) -> b(x)

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_u(x_1)) -> u(encArg(x_1))
   encArg(cons_v(x_1)) -> v(encArg(x_1))
   encArg(cons_w(x_1)) -> w(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_u(x_1) -> u(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_v(x_1) -> v(encArg(x_1))
   encode_w(x_1) -> w(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:

   a(c(d(x))) -> c(x)
   u(b(d(d(x)))) -> b(x)
   v(a(a(x))) -> u(v(x))
   v(a(c(x))) -> u(b(d(x)))
   v(c(x)) -> b(x)
   w(a(a(x))) -> u(w(x))
   w(a(c(x))) -> u(b(d(x)))
   w(c(x)) -> b(x)
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_u(x_1)) -> u(encArg(x_1))
   encArg(cons_v(x_1)) -> v(encArg(x_1))
   encArg(cons_w(x_1)) -> w(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_u(x_1) -> u(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_v(x_1) -> v(encArg(x_1))
   encode_w(x_1) -> w(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 3. 
The certificate found is represented by the following graph.

"[67, 68, 69, 70, 71, 72, 73, 74, 75]
{(67,68,[a_1|0, u_1|0, v_1|0, w_1|0, encArg_1|0, encode_a_1|0, encode_c_1|0, encode_d_1|0, encode_u_1|0, encode_b_1|0, encode_v_1|0, encode_w_1|0, c_1|1, b_1|1]), (67,69,[c_1|1, d_1|1, b_1|1, a_1|1, u_1|1, v_1|1, w_1|1, c_1|2, b_1|2]), (67,70,[u_1|2]), (67,71,[u_1|2]), (67,73,[u_1|2]), (68,68,[c_1|0, d_1|0, b_1|0, cons_a_1|0, cons_u_1|0, cons_v_1|0, cons_w_1|0]), (69,68,[encArg_1|1]), (69,69,[c_1|1, d_1|1, b_1|1, a_1|1, u_1|1, v_1|1, w_1|1, c_1|2, b_1|2]), (69,70,[u_1|2]), (69,71,[u_1|2]), (69,73,[u_1|2]), (70,69,[v_1|2, b_1|2, b_1|3]), (70,70,[u_1|2]), (70,71,[u_1|2]), (71,72,[b_1|2]), (72,69,[d_1|2]), (73,69,[w_1|2, b_1|2, b_1|3]), (73,73,[u_1|2]), (73,74,[u_1|2]), (74,75,[b_1|2]), (75,69,[d_1|2])}"
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(8)
BOUNDS(1, n^1)