/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 36 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(a) -> f(b)
   g(b) -> g(a)
   f(x) -> 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(a) -> a
   encArg(b) -> b
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_b -> b
   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(a) -> f(b)
   g(b) -> g(a)
   f(x) -> g(x)

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

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

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

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_b -> b
   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(a) -> f(b)
   g(b) -> g(a)
   f(x) -> g(x)
   encArg(a) -> a
   encArg(b) -> b
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_b -> b
   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 3. 
The certificate found is represented by the following graph.

"[5, 6, 14, 15, 16, 17, 18, 19]
{(5,6,[f_1|0, g_1|0, encArg_1|0, encode_f_1|0, encode_a|0, encode_b|0, encode_g_1|0, g_1|1, a|1, b|1]), (5,14,[f_1|1, g_1|2]), (5,15,[g_1|1]), (5,16,[f_1|1, g_1|1, g_1|2]), (5,17,[g_1|2]), (5,18,[f_1|2, g_1|3]), (5,19,[g_1|3]), (6,6,[a|0, b|0, cons_f_1|0, cons_g_1|0]), (14,6,[b|1]), (15,6,[a|1]), (16,6,[encArg_1|1, a|1, b|1]), (16,16,[f_1|1, g_1|1, g_1|2]), (16,18,[f_1|2, g_1|3]), (16,17,[g_1|2]), (16,19,[g_1|3]), (17,6,[a|2]), (18,6,[b|2]), (19,6,[a|3])}"
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(8)
BOUNDS(1, n^1)