/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 179 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
(7) CpxTrsMatchBoundsTAProof [FINISHED, 53 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, x) -> g(a, x)
   g(a, x) -> f(b, x)
   f(a, x) -> f(b, 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, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_b -> b


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

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_b -> b

Rewrite Strategy: FULL
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(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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(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, x) -> g(a, x)
   g(a, x) -> f(b, x)
   f(a, x) -> f(b, x)

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_b -> b

Rewrite Strategy: FULL
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(5) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(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, x) -> g(a, x)
   g(a, x) -> f(b, x)
   f(a, x) -> f(b, x)
   encArg(a) -> a
   encArg(b) -> b
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_b -> b

S is empty.
Rewrite Strategy: FULL
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(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 3. 

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, 6, 7]
transitions: 
a0() -> 0
b0() -> 0
cons_f0(0, 0) -> 0
cons_g0(0, 0) -> 0
f0(0, 0) -> 1
g0(0, 0) -> 2
encArg0(0) -> 3
encode_f0(0, 0) -> 4
encode_a0() -> 5
encode_g0(0, 0) -> 6
encode_b0() -> 7
a1() -> 8
g1(8, 0) -> 1
b1() -> 9
f1(9, 0) -> 2
f1(9, 0) -> 1
a1() -> 3
b1() -> 3
encArg1(0) -> 10
encArg1(0) -> 11
f1(10, 11) -> 3
encArg1(0) -> 12
encArg1(0) -> 13
g1(12, 13) -> 3
f1(10, 11) -> 4
a1() -> 5
g1(12, 13) -> 6
b1() -> 7
b2() -> 14
f2(14, 0) -> 1
a1() -> 10
a1() -> 11
a1() -> 12
a1() -> 13
b1() -> 10
b1() -> 11
b1() -> 12
b1() -> 13
f1(10, 11) -> 10
f1(10, 11) -> 11
f1(10, 11) -> 12
f1(10, 11) -> 13
g1(12, 13) -> 10
g1(12, 13) -> 11
g1(12, 13) -> 12
g1(12, 13) -> 13
a2() -> 15
g2(15, 11) -> 3
g2(15, 11) -> 4
g2(15, 11) -> 10
g2(15, 11) -> 11
g2(15, 11) -> 12
g2(15, 11) -> 13
f2(14, 13) -> 3
f2(14, 13) -> 6
f2(14, 13) -> 10
f2(14, 13) -> 11
f2(14, 13) -> 12
f2(14, 13) -> 13
f2(14, 11) -> 3
f2(14, 11) -> 4
f2(14, 11) -> 10
f2(14, 11) -> 11
f2(14, 11) -> 12
f2(14, 11) -> 13
b3() -> 16
f3(16, 11) -> 3
f3(16, 11) -> 4
f3(16, 11) -> 10
f3(16, 11) -> 11
f3(16, 11) -> 12
f3(16, 11) -> 13

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(8)
BOUNDS(1, n^1)