/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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, 87 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(x, f(y, a)) -> f(f(f(f(a, a), y), h(a)), x)

S is empty.
Rewrite Strategy: FULL
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(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(h(x_1)) -> h(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_h(x_1) -> h(encArg(x_1))


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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(x, f(y, a)) -> f(f(f(f(a, a), y), h(a)), x)

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

   encArg(a) -> a
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_h(x_1) -> h(encArg(x_1))

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(x, f(y, a)) -> f(f(f(f(a, a), y), h(a)), x)

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

   encArg(a) -> a
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_h(x_1) -> h(encArg(x_1))

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(x, f(y, a)) -> f(f(f(f(a, a), y), h(a)), x)
   encArg(a) -> a
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_h(x_1) -> h(encArg(x_1))

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]
transitions: 
a0() -> 0
h0(0) -> 0
cons_f0(0, 0) -> 0
f0(0, 0) -> 1
encArg0(0) -> 2
encode_f0(0, 0) -> 3
encode_a0() -> 4
encode_h0(0) -> 5
a1() -> 2
encArg1(0) -> 6
h1(6) -> 2
encArg1(0) -> 7
encArg1(0) -> 8
f1(7, 8) -> 2
f1(7, 8) -> 3
a1() -> 4
h1(6) -> 5
a1() -> 6
a1() -> 7
a1() -> 8
h1(6) -> 6
h1(6) -> 7
h1(6) -> 8
f1(7, 8) -> 6
f1(7, 8) -> 7
f1(7, 8) -> 8
a2() -> 12
a2() -> 13
f2(12, 13) -> 11
f2(11, 7) -> 10
a2() -> 15
h2(15) -> 14
f2(10, 14) -> 9
f2(9, 7) -> 2
f2(9, 7) -> 3
f2(9, 7) -> 6
f2(9, 7) -> 7
f2(9, 7) -> 8
f2(11, 9) -> 10
f2(9, 9) -> 2
f2(9, 9) -> 3
f2(9, 9) -> 6
f2(9, 9) -> 7
f2(9, 9) -> 8
f2(9, 11) -> 10
a3() -> 19
a3() -> 20
f3(19, 20) -> 18
f3(18, 12) -> 17
a3() -> 22
h3(22) -> 21
f3(17, 21) -> 16
f3(16, 9) -> 10

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(8)
BOUNDS(1, n^1)