/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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), 38 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:

   f(f(X)) -> c
   c -> d
   h(X) -> c

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(d) -> d
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_c) -> c
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_c -> c
   encode_d -> d
   encode_h(x_1) -> h(encArg(x_1))


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

(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:

   f(f(X)) -> c
   c -> d
   h(X) -> c

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

   encArg(d) -> d
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_c) -> c
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_c -> c
   encode_d -> d
   encode_h(x_1) -> h(encArg(x_1))

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:

   f(f(X)) -> c
   c -> d
   h(X) -> c

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

   encArg(d) -> d
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_c) -> c
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_c -> c
   encode_d -> d
   encode_h(x_1) -> h(encArg(x_1))

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:

   f(f(X)) -> c
   c -> d
   h(X) -> c
   encArg(d) -> d
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_c) -> c
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_c -> c
   encode_d -> d
   encode_h(x_1) -> h(encArg(x_1))

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.

"[45, 46, 47]
{(45,46,[f_1|0, c|0, h_1|0, encArg_1|0, encode_f_1|0, encode_c|0, encode_d|0, encode_h_1|0, d|1, c|1, d|2]), (45,47,[f_1|1, h_1|1, c|2, d|3]), (46,46,[d|0, cons_f_1|0, cons_c|0, cons_h_1|0]), (47,46,[encArg_1|1, d|1, c|1, d|2]), (47,47,[f_1|1, h_1|1, c|2, d|3])}"
----------------------------------------

(8)
BOUNDS(1, n^1)