/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 161 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
(7) CpxTrsMatchBoundsTAProof [FINISHED, 774 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:

   a(f, a(f, x)) -> a(x, g)
   a(x, g) -> a(f, a(g, a(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(f) -> f
   encArg(g) -> g
   encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
   encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
   encode_f -> f
   encode_g -> g


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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:

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

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

   encArg(f) -> f
   encArg(g) -> g
   encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
   encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
   encode_f -> f
   encode_g -> g

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:

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

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

   encArg(f) -> f
   encArg(g) -> g
   encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
   encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
   encode_f -> f
   encode_g -> g

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:

   a(f, a(f, x)) -> a(x, g)
   a(x, g) -> a(f, a(g, a(f, x)))
   encArg(f) -> f
   encArg(g) -> g
   encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
   encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
   encode_f -> f
   encode_g -> g

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 5. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4, 5]
transitions: 
f0() -> 0
g0() -> 0
cons_a0(0, 0) -> 0
a0(0, 0) -> 1
encArg0(0) -> 2
encode_a0(0, 0) -> 3
encode_f0() -> 4
encode_g0() -> 5
f1() -> 6
g1() -> 8
a1(6, 0) -> 9
a1(8, 9) -> 7
a1(6, 7) -> 1
f1() -> 2
g1() -> 2
encArg1(0) -> 10
encArg1(0) -> 11
a1(10, 11) -> 2
a1(10, 11) -> 3
f1() -> 4
g1() -> 5
a1(6, 6) -> 9
a1(6, 7) -> 9
f1() -> 10
f1() -> 11
g1() -> 10
g1() -> 11
a1(10, 11) -> 10
a1(10, 11) -> 11
f2() -> 12
g2() -> 14
a2(12, 10) -> 15
a2(14, 15) -> 13
a2(12, 13) -> 2
a2(12, 13) -> 3
a2(12, 13) -> 10
a2(12, 13) -> 11
g2() -> 16
a2(11, 16) -> 2
a2(11, 16) -> 3
a2(11, 16) -> 10
a2(11, 16) -> 11
a2(16, 16) -> 2
a2(16, 16) -> 3
a2(16, 16) -> 10
a2(16, 16) -> 11
a2(13, 16) -> 2
a2(13, 16) -> 3
a2(13, 16) -> 10
a2(13, 16) -> 11
a2(11, 16) -> 15
a2(16, 16) -> 15
g3() -> 17
a3(13, 17) -> 15
a2(12, 12) -> 15
a2(12, 13) -> 15
f3() -> 18
g3() -> 20
a3(18, 11) -> 21
a3(20, 21) -> 19
a3(18, 19) -> 2
a3(18, 19) -> 3
a3(18, 19) -> 10
a3(18, 19) -> 11
a3(18, 16) -> 21
a3(18, 13) -> 21
a3(18, 19) -> 15
a2(19, 16) -> 2
a2(19, 16) -> 3
a2(19, 16) -> 10
a2(19, 16) -> 11
a3(19, 17) -> 15
a2(11, 16) -> 21
a2(16, 16) -> 21
a3(13, 17) -> 21
g4() -> 22
a4(19, 22) -> 21
a2(12, 18) -> 15
a2(12, 13) -> 21
f4() -> 23
g4() -> 25
a4(23, 13) -> 26
a4(25, 26) -> 24
a4(23, 24) -> 15
a3(18, 19) -> 21
a3(18, 18) -> 21
a4(23, 19) -> 26
a4(23, 24) -> 21
f5() -> 27
g5() -> 29
a5(27, 19) -> 30
a5(29, 30) -> 28
a5(27, 28) -> 21

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(8)
BOUNDS(1, n^1)