/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (innermost) 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), 65 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
    (7) CpxTrsMatchBoundsTAProof [FINISHED, 302 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


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

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


The TRS R consists of the following rules:

   f(a) -> f(c(a))
   f(c(X)) -> X
   f(c(a)) -> f(d(b))
   f(a) -> f(d(a))
   f(d(X)) -> X
   f(c(b)) -> f(d(a))
   e(g(X)) -> e(X)

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(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(c(x_1)) -> c(encArg(x_1))
   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(b) -> b
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_e(x_1)) -> e(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_b -> b
   encode_e(x_1) -> e(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))


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

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


The TRS R consists of the following rules:

   f(a) -> f(c(a))
   f(c(X)) -> X
   f(c(a)) -> f(d(b))
   f(a) -> f(d(a))
   f(d(X)) -> X
   f(c(b)) -> f(d(a))
   e(g(X)) -> e(X)

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

   encArg(a) -> a
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(b) -> b
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_e(x_1)) -> e(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_b -> b
   encode_e(x_1) -> e(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

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


The TRS R consists of the following rules:

   f(a) -> f(c(a))
   f(c(X)) -> X
   f(c(a)) -> f(d(b))
   f(a) -> f(d(a))
   f(d(X)) -> X
   f(c(b)) -> f(d(a))
   e(g(X)) -> e(X)

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

   encArg(a) -> a
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(b) -> b
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_e(x_1)) -> e(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_b -> b
   encode_e(x_1) -> e(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(5) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   f(a) -> f(c(a))
   f(c(X)) -> X
   f(c(a)) -> f(d(b))
   f(a) -> f(d(a))
   f(d(X)) -> X
   f(c(b)) -> f(d(a))
   e(g(X)) -> e(X)
   encArg(a) -> a
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(b) -> b
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_e(x_1)) -> e(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_b -> b
   encode_e(x_1) -> e(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(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: 
a0() -> 0
c0(0) -> 0
d0(0) -> 0
b0() -> 0
g0(0) -> 0
cons_f0(0) -> 0
cons_e0(0) -> 0
f0(0) -> 1
e0(0) -> 2
encArg0(0) -> 3
encode_f0(0) -> 4
encode_a0() -> 5
encode_c0(0) -> 6
encode_d0(0) -> 7
encode_b0() -> 8
encode_e0(0) -> 9
encode_g0(0) -> 10
a1() -> 12
c1(12) -> 11
f1(11) -> 1
b1() -> 14
d1(14) -> 13
f1(13) -> 1
a1() -> 16
d1(16) -> 15
f1(15) -> 1
e1(0) -> 2
a1() -> 3
encArg1(0) -> 17
c1(17) -> 3
encArg1(0) -> 18
d1(18) -> 3
b1() -> 3
encArg1(0) -> 19
g1(19) -> 3
encArg1(0) -> 20
f1(20) -> 3
encArg1(0) -> 21
e1(21) -> 3
f1(20) -> 4
a1() -> 5
c1(17) -> 6
d1(18) -> 7
b1() -> 8
e1(21) -> 9
g1(19) -> 10
b2() -> 23
d2(23) -> 22
f2(22) -> 1
a1() -> 17
a1() -> 18
a1() -> 19
a1() -> 20
a1() -> 21
c1(17) -> 17
c1(17) -> 18
c1(17) -> 19
c1(17) -> 20
c1(17) -> 21
d1(18) -> 17
d1(18) -> 18
d1(18) -> 19
d1(18) -> 20
d1(18) -> 21
b1() -> 17
b1() -> 18
b1() -> 19
b1() -> 20
b1() -> 21
g1(19) -> 17
g1(19) -> 18
g1(19) -> 19
g1(19) -> 20
g1(19) -> 21
f1(20) -> 17
f1(20) -> 18
f1(20) -> 19
f1(20) -> 20
f1(20) -> 21
e1(21) -> 17
e1(21) -> 18
e1(21) -> 19
e1(21) -> 20
e1(21) -> 21
a2() -> 25
c2(25) -> 24
f2(24) -> 3
f2(24) -> 4
f2(24) -> 17
f2(24) -> 18
f2(24) -> 19
f2(24) -> 20
f2(24) -> 21
f2(22) -> 3
f2(22) -> 4
f2(22) -> 17
a2() -> 27
d2(27) -> 26
f2(26) -> 3
f2(26) -> 4
f2(26) -> 17
e2(19) -> 3
e2(19) -> 9
e2(19) -> 17
b3() -> 29
d3(29) -> 28
f3(28) -> 3
f3(28) -> 4
f3(28) -> 17
0 -> 1
12 -> 1
14 -> 1
16 -> 1
17 -> 3
17 -> 4
17 -> 18
17 -> 19
17 -> 20
17 -> 21
18 -> 3
18 -> 4
18 -> 17
23 -> 1
23 -> 3
23 -> 4
23 -> 17
27 -> 3
27 -> 4
27 -> 17
25 -> 3
25 -> 4
25 -> 17
29 -> 3
29 -> 4
29 -> 17

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

(8)
BOUNDS(1, n^1)

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

(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 (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   f(a) -> f(c(a))
   f(c(X)) -> X
   f(c(a)) -> f(d(b))
   f(a) -> f(d(a))
   f(d(X)) -> X
   f(c(b)) -> f(d(a))
   e(g(X)) -> e(X)

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

   encArg(a) -> a
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(b) -> b
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_e(x_1)) -> e(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_b -> b
   encode_e(x_1) -> e(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

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

The rewrite sequence

e(g(X)) ->^+ e(X)

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

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

The result substitution is [ ].




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

(12)
Complex Obligation (BEST)

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

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

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


The TRS R consists of the following rules:

   f(a) -> f(c(a))
   f(c(X)) -> X
   f(c(a)) -> f(d(b))
   f(a) -> f(d(a))
   f(d(X)) -> X
   f(c(b)) -> f(d(a))
   e(g(X)) -> e(X)

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

   encArg(a) -> a
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(b) -> b
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_e(x_1)) -> e(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_b -> b
   encode_e(x_1) -> e(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(14) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(15)
BOUNDS(n^1, INF)

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

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

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


The TRS R consists of the following rules:

   f(a) -> f(c(a))
   f(c(X)) -> X
   f(c(a)) -> f(d(b))
   f(a) -> f(d(a))
   f(d(X)) -> X
   f(c(b)) -> f(d(a))
   e(g(X)) -> e(X)

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

   encArg(a) -> a
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(b) -> b
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_e(x_1)) -> e(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_b -> b
   encode_e(x_1) -> e(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))

Rewrite Strategy: INNERMOST