/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 170 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
(7) CpxTrsMatchBoundsTAProof [FINISHED, 10 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:

   app(app(apply, f), x) -> app(f, 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(apply) -> apply
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_apply -> apply


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

(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:

   app(app(apply, f), x) -> app(f, x)

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

   encArg(apply) -> apply
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_apply -> apply

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:

   app(app(apply, f), x) -> app(f, x)

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

   encArg(apply) -> apply
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_apply -> apply

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:

   app(app(apply, f), x) -> app(f, x)
   encArg(apply) -> apply
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_apply -> apply

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

(7) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4]
transitions: 
apply0() -> 0
cons_app0(0, 0) -> 0
app0(0, 0) -> 1
encArg0(0) -> 2
encode_app0(0, 0) -> 3
encode_apply0() -> 4
apply1() -> 2
encArg1(0) -> 5
encArg1(0) -> 6
app1(5, 6) -> 2
app1(5, 6) -> 3
apply1() -> 4
apply1() -> 5
apply1() -> 6
app1(5, 6) -> 5
app1(5, 6) -> 6
app2(6, 6) -> 2
app2(6, 6) -> 3
app2(6, 6) -> 5
app2(6, 6) -> 6

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(8)
BOUNDS(1, n^1)