/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 45 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(a(b(f(X))))
   f(a(g(X))) -> b(X)
   b(X) -> a(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(a(x_1)) -> a(encArg(x_1))
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))


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(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(a(b(f(X))))
   f(a(g(X))) -> b(X)
   b(X) -> a(X)

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

   encArg(a(x_1)) -> a(encArg(x_1))
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(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(a(b(f(X))))
   f(a(g(X))) -> b(X)
   b(X) -> a(X)

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

   encArg(a(x_1)) -> a(encArg(x_1))
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))

Rewrite Strategy: FULL
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(5) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(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(a(b(f(X))))
   f(a(g(X))) -> b(X)
   b(X) -> a(X)
   encArg(a(x_1)) -> a(encArg(x_1))
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))

S is empty.
Rewrite Strategy: FULL
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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.

"[44, 45, 46, 47, 48, 49, 50, 51, 52]
{(44,45,[f_1|0, b_1|0, encArg_1|0, encode_f_1|0, encode_a_1|0, encode_b_1|0, encode_g_1|0, b_1|1, a_1|1, a_1|2]), (44,46,[a_1|1, g_1|1, f_1|1, b_1|1, a_1|2, b_1|2, a_1|3]), (44,47,[f_1|2]), (45,45,[a_1|0, g_1|0, cons_f_1|0, cons_b_1|0]), (46,45,[encArg_1|1]), (46,46,[a_1|1, g_1|1, f_1|1, b_1|1, b_1|2, a_1|2, a_1|3]), (46,47,[f_1|2]), (47,48,[a_1|2]), (48,49,[b_1|2, a_1|3]), (49,46,[f_1|2, b_1|2, a_1|3]), (49,47,[f_1|2]), (49,50,[f_1|3]), (50,51,[a_1|3]), (51,52,[b_1|3, a_1|4]), (52,47,[f_1|3])}"
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(8)
BOUNDS(1, n^1)