/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 92 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
    (7) CpxTrsMatchBoundsTAProof [FINISHED, 475 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(f(x)) -> f(c(f(x)))
   f(f(x)) -> f(d(f(x)))
   g(c(x)) -> x
   g(d(x)) -> x
   g(c(h(0))) -> g(d(1))
   g(c(1)) -> g(d(h(0)))
   g(h(x)) -> g(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(c(x_1)) -> c(encArg(x_1))
   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(0) -> 0
   encArg(1) -> 1
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_0 -> 0
   encode_1 -> 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(f(x)) -> f(c(f(x)))
   f(f(x)) -> f(d(f(x)))
   g(c(x)) -> x
   g(d(x)) -> x
   g(c(h(0))) -> g(d(1))
   g(c(1)) -> g(d(h(0)))
   g(h(x)) -> g(x)

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(0) -> 0
   encArg(1) -> 1
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_0 -> 0
   encode_1 -> 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(f(x)) -> f(c(f(x)))
   f(f(x)) -> f(d(f(x)))
   g(c(x)) -> x
   g(d(x)) -> x
   g(c(h(0))) -> g(d(1))
   g(c(1)) -> g(d(h(0)))
   g(h(x)) -> g(x)

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(0) -> 0
   encArg(1) -> 1
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_0 -> 0
   encode_1 -> 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(f(x)) -> f(c(f(x)))
   f(f(x)) -> f(d(f(x)))
   g(c(x)) -> x
   g(d(x)) -> x
   g(c(h(0))) -> g(d(1))
   g(c(1)) -> g(d(h(0)))
   g(h(x)) -> g(x)
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(0) -> 0
   encArg(1) -> 1
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_0 -> 0
   encode_1 -> 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: 
c0(0) -> 0
d0(0) -> 0
h0(0) -> 0
00() -> 0
10() -> 0
cons_f0(0) -> 0
cons_g0(0) -> 0
f0(0) -> 1
g0(0) -> 2
encArg0(0) -> 3
encode_f0(0) -> 4
encode_c0(0) -> 5
encode_d0(0) -> 6
encode_g0(0) -> 7
encode_h0(0) -> 8
encode_00() -> 9
encode_10() -> 10
11() -> 12
d1(12) -> 11
g1(11) -> 2
01() -> 15
h1(15) -> 14
d1(14) -> 13
g1(13) -> 2
g1(0) -> 2
encArg1(0) -> 16
c1(16) -> 3
encArg1(0) -> 17
d1(17) -> 3
encArg1(0) -> 18
h1(18) -> 3
01() -> 3
11() -> 3
encArg1(0) -> 19
f1(19) -> 3
encArg1(0) -> 20
g1(20) -> 3
f1(19) -> 4
c1(16) -> 5
d1(17) -> 6
g1(20) -> 7
h1(18) -> 8
01() -> 9
11() -> 10
c1(16) -> 16
c1(16) -> 17
c1(16) -> 18
c1(16) -> 19
c1(16) -> 20
d1(17) -> 16
d1(17) -> 17
d1(17) -> 18
d1(17) -> 19
d1(17) -> 20
h1(18) -> 16
h1(18) -> 17
h1(18) -> 18
h1(18) -> 19
h1(18) -> 20
01() -> 16
01() -> 17
01() -> 18
01() -> 19
01() -> 20
11() -> 16
11() -> 17
11() -> 18
11() -> 19
11() -> 20
f1(19) -> 16
f1(19) -> 17
f1(19) -> 18
f1(19) -> 19
f1(19) -> 20
g1(20) -> 16
g1(20) -> 17
g1(20) -> 18
g1(20) -> 19
g1(20) -> 20
12() -> 22
d2(22) -> 21
g2(21) -> 3
g2(21) -> 7
g2(21) -> 16
02() -> 25
h2(25) -> 24
d2(24) -> 23
g2(23) -> 3
g2(23) -> 7
g2(23) -> 16
g2(18) -> 3
g2(18) -> 7
g2(18) -> 16
f2(19) -> 27
c2(27) -> 26
f2(26) -> 3
f2(26) -> 4
f2(26) -> 16
f2(19) -> 29
d2(29) -> 28
f2(28) -> 3
f2(28) -> 4
f2(28) -> 16
f2(26) -> 27
f2(28) -> 27
f2(26) -> 29
f3(26) -> 31
c3(31) -> 30
f3(30) -> 27
f3(28) -> 31
f3(30) -> 29
f2(28) -> 29
f3(26) -> 33
d3(33) -> 32
f3(32) -> 27
f3(28) -> 33
f3(32) -> 29
g2(25) -> 3
g2(25) -> 7
g2(25) -> 16
g3(25) -> 3
g3(25) -> 7
g3(25) -> 16
0 -> 2
12 -> 2
14 -> 2
16 -> 3
16 -> 7
16 -> 17
16 -> 18
16 -> 19
16 -> 20
17 -> 3
17 -> 7
17 -> 16
22 -> 3
22 -> 7
22 -> 16
24 -> 3
24 -> 7
24 -> 16

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

(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(f(x)) -> f(c(f(x)))
   f(f(x)) -> f(d(f(x)))
   g(c(x)) -> x
   g(d(x)) -> x
   g(c(h(0))) -> g(d(1))
   g(c(1)) -> g(d(h(0)))
   g(h(x)) -> g(x)

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(0) -> 0
   encArg(1) -> 1
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_0 -> 0
   encode_1 -> 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

g(h(x)) ->^+ g(x)

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

The pumping substitution is [x / h(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(f(x)) -> f(c(f(x)))
   f(f(x)) -> f(d(f(x)))
   g(c(x)) -> x
   g(d(x)) -> x
   g(c(h(0))) -> g(d(1))
   g(c(1)) -> g(d(h(0)))
   g(h(x)) -> g(x)

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(0) -> 0
   encArg(1) -> 1
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_0 -> 0
   encode_1 -> 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(f(x)) -> f(c(f(x)))
   f(f(x)) -> f(d(f(x)))
   g(c(x)) -> x
   g(d(x)) -> x
   g(c(h(0))) -> g(d(1))
   g(c(1)) -> g(d(h(0)))
   g(h(x)) -> g(x)

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(0) -> 0
   encArg(1) -> 1
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_0 -> 0
   encode_1 -> 1

Rewrite Strategy: INNERMOST