/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (innermost) 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), 79 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 (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

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

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

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


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

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

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

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

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

Rewrite Strategy: INNERMOST
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

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

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

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

Rewrite Strategy: INNERMOST
----------------------------------------

(5) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

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

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(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.

"[48, 49, 50, 51, 52, 53, 54]
{(48,49,[g_1|0, f_1|0, b|0, encArg_1|0, encode_g_1|0, encode_b|0, encode_f_1|0, encode_a|0, a|1, b|1, a|2]), (48,50,[g_1|1]), (48,51,[g_1|1, f_1|1]), (48,52,[f_1|2]), (48,53,[g_1|2]), (48,54,[g_1|3]), (49,49,[a|0, cons_g_1|0, cons_f_1|0, cons_b|0]), (50,49,[a|1]), (51,49,[encArg_1|1, a|1, b|1, a|2]), (51,51,[g_1|1, f_1|1]), (51,52,[f_1|2]), (51,53,[g_1|2]), (51,54,[g_1|3]), (52,49,[b|2, a|3]), (53,49,[a|2]), (54,49,[a|3])}"
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(8)
BOUNDS(1, n^1)