/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 165 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
    (7) CpxTrsMatchBoundsTAProof [FINISHED, 505 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(a, a) -> f(a, b)
   f(a, b) -> f(s(a), c)
   f(s(X), c) -> f(X, c)
   f(c, c) -> f(a, a)

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(s(x_1)) -> s(encArg(x_1))
   encArg(c) -> c
   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_b -> b
   encode_s(x_1) -> s(encArg(x_1))
   encode_c -> c


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

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

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(c) -> c
   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_b -> b
   encode_s(x_1) -> s(encArg(x_1))
   encode_c -> c

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(n^1, n^1).


The TRS R consists of the following rules:

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

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(c) -> c
   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_b -> b
   encode_s(x_1) -> s(encArg(x_1))
   encode_c -> c

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(a, a) -> f(a, b)
   f(a, b) -> f(s(a), c)
   f(s(X), c) -> f(X, c)
   f(c, c) -> f(a, a)
   encArg(a) -> a
   encArg(b) -> b
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(c) -> c
   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_b -> b
   encode_s(x_1) -> s(encArg(x_1))
   encode_c -> c

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, 6, 7]
transitions: 
a0() -> 0
b0() -> 0
s0(0) -> 0
c0() -> 0
cons_f0(0, 0) -> 0
f0(0, 0) -> 1
encArg0(0) -> 2
encode_f0(0, 0) -> 3
encode_a0() -> 4
encode_b0() -> 5
encode_s0(0) -> 6
encode_c0() -> 7
a1() -> 8
b1() -> 9
f1(8, 9) -> 1
a1() -> 11
s1(11) -> 10
c1() -> 12
f1(10, 12) -> 1
f1(0, 12) -> 1
a1() -> 13
f1(8, 13) -> 1
a1() -> 2
b1() -> 2
encArg1(0) -> 14
s1(14) -> 2
c1() -> 2
encArg1(0) -> 15
encArg1(0) -> 16
f1(15, 16) -> 2
f1(15, 16) -> 3
a1() -> 4
b1() -> 5
s1(14) -> 6
c1() -> 7
a2() -> 17
b2() -> 18
f2(17, 18) -> 1
a2() -> 20
s2(20) -> 19
c2() -> 21
f2(19, 21) -> 1
f2(11, 21) -> 1
a1() -> 14
a1() -> 15
a1() -> 16
b1() -> 14
b1() -> 15
b1() -> 16
s1(14) -> 14
s1(14) -> 15
s1(14) -> 16
c1() -> 14
c1() -> 15
c1() -> 16
f1(15, 16) -> 14
f1(15, 16) -> 15
f1(15, 16) -> 16
f2(17, 18) -> 2
f2(17, 18) -> 3
f2(17, 18) -> 14
f2(17, 18) -> 15
f2(17, 18) -> 16
f2(19, 21) -> 2
f2(19, 21) -> 3
f2(19, 21) -> 14
f2(19, 21) -> 15
f2(19, 21) -> 16
f2(14, 21) -> 2
f2(14, 21) -> 3
f2(14, 21) -> 14
f2(14, 21) -> 15
f2(14, 21) -> 16
a2() -> 22
f2(17, 22) -> 2
f2(17, 22) -> 3
f2(17, 22) -> 14
f2(17, 22) -> 15
f2(17, 22) -> 16
a3() -> 24
s3(24) -> 23
c3() -> 25
f3(23, 25) -> 1
f3(20, 25) -> 1
a3() -> 26
b3() -> 27
f3(26, 27) -> 2
f3(26, 27) -> 3
f3(26, 27) -> 14
f3(26, 27) -> 15
f3(26, 27) -> 16
f3(23, 25) -> 2
f3(23, 25) -> 3
f3(23, 25) -> 14
f3(23, 25) -> 15
f3(23, 25) -> 16
f3(20, 25) -> 2
f3(20, 25) -> 3
f3(20, 25) -> 14
f3(20, 25) -> 15
f3(20, 25) -> 16
c4() -> 28
f4(24, 28) -> 1
f4(24, 28) -> 2
f4(24, 28) -> 3
f4(24, 28) -> 14
f4(24, 28) -> 15
f4(24, 28) -> 16
a4() -> 30
s4(30) -> 29
f4(29, 28) -> 2
f4(29, 28) -> 3
f4(29, 28) -> 14
f4(29, 28) -> 15
f4(29, 28) -> 16
c5() -> 31
f5(30, 31) -> 2
f5(30, 31) -> 3
f5(30, 31) -> 14
f5(30, 31) -> 15
f5(30, 31) -> 16

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

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(c) -> c
   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_b -> b
   encode_s(x_1) -> s(encArg(x_1))
   encode_c -> c

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

f(s(X), c) ->^+ f(X, c)

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

The pumping substitution is [X / s(X)].

The result substitution is [ ].




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

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

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(c) -> c
   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_b -> b
   encode_s(x_1) -> s(encArg(x_1))
   encode_c -> c

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

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

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(c) -> c
   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_b -> b
   encode_s(x_1) -> s(encArg(x_1))
   encode_c -> c

Rewrite Strategy: FULL