/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 76 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 -> g(c)
   g(a) -> b
   f(g(X), b) -> f(a, 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(c) -> c
   encArg(b) -> b
   encArg(cons_a) -> a
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_g(x_1) -> g(encArg(x_1))
   encode_c -> c
   encode_b -> b
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))


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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 -> g(c)
   g(a) -> b
   f(g(X), b) -> f(a, X)

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

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

Rewrite Strategy: FULL
----------------------------------------

(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 -> g(c)
   g(a) -> b
   f(g(X), b) -> f(a, X)

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

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

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

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 4. 

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, 8, 9]
transitions: 
c0() -> 0
b0() -> 0
cons_a0() -> 0
cons_g0(0) -> 0
cons_f0(0, 0) -> 0
a0() -> 1
g0(0) -> 2
f0(0, 0) -> 3
encArg0(0) -> 4
encode_a0() -> 5
encode_g0(0) -> 6
encode_c0() -> 7
encode_b0() -> 8
encode_f0(0, 0) -> 9
c1() -> 10
g1(10) -> 1
c1() -> 4
b1() -> 4
a1() -> 4
encArg1(0) -> 11
g1(11) -> 4
encArg1(0) -> 12
encArg1(0) -> 13
f1(12, 13) -> 4
a1() -> 5
g1(11) -> 6
c1() -> 7
b1() -> 8
f1(12, 13) -> 9
c2() -> 14
g2(14) -> 4
g2(14) -> 5
c1() -> 11
c1() -> 12
c1() -> 13
b1() -> 11
b1() -> 12
b1() -> 13
a1() -> 11
a1() -> 12
a1() -> 13
g1(11) -> 11
g1(11) -> 12
g1(11) -> 13
f1(12, 13) -> 11
f1(12, 13) -> 12
f1(12, 13) -> 13
g2(14) -> 11
g2(14) -> 12
g2(14) -> 13
b2() -> 4
b2() -> 6
b2() -> 11
b2() -> 12
b2() -> 13
a2() -> 15
f2(15, 11) -> 4
f2(15, 11) -> 9
f2(15, 11) -> 11
f2(15, 11) -> 12
f2(15, 11) -> 13
f2(15, 14) -> 4
f2(15, 14) -> 9
f2(15, 14) -> 11
f2(15, 14) -> 12
f2(15, 14) -> 13
c3() -> 16
g3(16) -> 15
f2(15, 16) -> 4
f2(15, 16) -> 9
f2(15, 16) -> 11
f2(15, 16) -> 12
f2(15, 16) -> 13
a3() -> 17
f3(17, 16) -> 4
f3(17, 16) -> 9
f3(17, 16) -> 11
f3(17, 16) -> 12
f3(17, 16) -> 13
c4() -> 18
g4(18) -> 17

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(8)
BOUNDS(1, n^1)