/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(Omega(n^1), 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(n^1, n^1).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 261 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
    (7) CpxTrsMatchBoundsTAProof [FINISHED, 145 ms]
    (8) BOUNDS(1, n^1)
(9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(10) TRS for Loop Detection
    (11) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (12) BEST
        (13) proven lower bound
            (14) LowerBoundPropagationProof [FINISHED, 0 ms]
            (15) BOUNDS(n^1, INF)
        (16) TRS for Loop Detection


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(0)
Obligation:
The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   f(s(X)) -> f(X)
   g(cons(0, Y)) -> g(Y)
   g(cons(s(X), Y)) -> s(X)
   h(cons(X, Y)) -> h(g(cons(X, Y)))

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(s(x_1)) -> s(encArg(x_1))
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(0) -> 0
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_h(x_1) -> h(encArg(x_1))


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

(2)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   f(s(X)) -> f(X)
   g(cons(0, Y)) -> g(Y)
   g(cons(s(X), Y)) -> s(X)
   h(cons(X, Y)) -> h(g(cons(X, Y)))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(0) -> 0
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   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
----------------------------------------

(4)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   f(s(X)) -> f(X)
   g(cons(0, Y)) -> g(Y)
   g(cons(s(X), Y)) -> s(X)
   h(cons(X, Y)) -> h(g(cons(X, Y)))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(0) -> 0
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_h(x_1) -> h(encArg(x_1))

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:

   f(s(X)) -> f(X)
   g(cons(0, Y)) -> g(Y)
   g(cons(s(X), Y)) -> s(X)
   h(cons(X, Y)) -> h(g(cons(X, Y)))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(0) -> 0
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_h(x_1) -> h(encArg(x_1))

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 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, 8, 9, 10]
transitions: 
s0(0) -> 0
cons0(0, 0) -> 0
00() -> 0
cons_f0(0) -> 0
cons_g0(0) -> 0
cons_h0(0) -> 0
f0(0) -> 1
g0(0) -> 2
h0(0) -> 3
encArg0(0) -> 4
encode_f0(0) -> 5
encode_s0(0) -> 6
encode_g0(0) -> 7
encode_cons0(0, 0) -> 8
encode_00() -> 9
encode_h0(0) -> 10
f1(0) -> 1
g1(0) -> 2
s1(0) -> 2
cons1(0, 0) -> 12
g1(12) -> 11
h1(11) -> 3
encArg1(0) -> 13
s1(13) -> 4
encArg1(0) -> 14
encArg1(0) -> 15
cons1(14, 15) -> 4
01() -> 4
encArg1(0) -> 16
f1(16) -> 4
encArg1(0) -> 17
g1(17) -> 4
encArg1(0) -> 18
h1(18) -> 4
f1(16) -> 5
s1(13) -> 6
g1(17) -> 7
cons1(14, 15) -> 8
01() -> 9
h1(18) -> 10
g1(0) -> 11
s1(0) -> 11
s1(13) -> 13
s1(13) -> 14
s1(13) -> 15
s1(13) -> 16
s1(13) -> 17
s1(13) -> 18
cons1(14, 15) -> 13
cons1(14, 15) -> 14
cons1(14, 15) -> 15
cons1(14, 15) -> 16
cons1(14, 15) -> 17
cons1(14, 15) -> 18
01() -> 13
01() -> 14
01() -> 15
01() -> 16
01() -> 17
01() -> 18
f1(16) -> 13
f1(16) -> 14
f1(16) -> 15
f1(16) -> 16
f1(16) -> 17
f1(16) -> 18
g1(17) -> 13
g1(17) -> 14
g1(17) -> 15
g1(17) -> 16
g1(17) -> 17
g1(17) -> 18
h1(18) -> 13
h1(18) -> 14
h1(18) -> 15
h1(18) -> 16
h1(18) -> 17
h1(18) -> 18
f2(13) -> 4
f2(13) -> 5
f2(13) -> 13
f2(13) -> 14
f2(13) -> 15
f2(13) -> 16
f2(13) -> 17
f2(13) -> 18
g2(15) -> 4
g2(15) -> 7
g2(15) -> 13
g2(15) -> 14
g2(15) -> 15
g2(15) -> 16
g2(15) -> 17
g2(15) -> 18
s2(13) -> 4
s2(13) -> 7
s2(13) -> 13
s2(13) -> 14
s2(13) -> 15
s2(13) -> 16
s2(13) -> 17
s2(13) -> 18
cons2(14, 15) -> 20
g2(20) -> 19
h2(19) -> 4
h2(19) -> 10
h2(19) -> 13
h2(19) -> 14
h2(19) -> 15
h2(19) -> 16
h2(19) -> 17
h2(19) -> 18
f3(13) -> 4
f3(13) -> 5
f3(13) -> 13
f3(13) -> 14
f3(13) -> 15
f3(13) -> 16
f3(13) -> 17
f3(13) -> 18
g2(15) -> 19
s2(13) -> 19
s3(13) -> 19

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

(8)
BOUNDS(1, n^1)

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(9) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(10)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   f(s(X)) -> f(X)
   g(cons(0, Y)) -> g(Y)
   g(cons(s(X), Y)) -> s(X)
   h(cons(X, Y)) -> h(g(cons(X, Y)))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(0) -> 0
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_h(x_1) -> h(encArg(x_1))

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

(11) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

g(cons(0, Y)) ->^+ g(Y)

gives rise to a decreasing loop by considering the right hand sides subterm at position [].

The pumping substitution is [Y / cons(0, Y)].

The result substitution is [ ].




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(12)
Complex Obligation (BEST)

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

(13)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   f(s(X)) -> f(X)
   g(cons(0, Y)) -> g(Y)
   g(cons(s(X), Y)) -> s(X)
   h(cons(X, Y)) -> h(g(cons(X, Y)))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(0) -> 0
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_h(x_1) -> h(encArg(x_1))

Rewrite Strategy: FULL
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(14) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   f(s(X)) -> f(X)
   g(cons(0, Y)) -> g(Y)
   g(cons(s(X), Y)) -> s(X)
   h(cons(X, Y)) -> h(g(cons(X, Y)))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(0) -> 0
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_h(x_1) -> h(encArg(x_1))

Rewrite Strategy: FULL