/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(?, O(n^1))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(1, n^1).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 172 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
(7) CpxTrsMatchBoundsTAProof [FINISHED, 121 ms]
(8) BOUNDS(1, n^1)


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

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


The TRS R consists of the following rules:

   g(a) -> g(b)
   b -> f(a, a)
   f(a, a) -> g(d)

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(d) -> d
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_b) -> b
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a -> a
   encode_b -> b
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_d -> d


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

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


The TRS R consists of the following rules:

   g(a) -> g(b)
   b -> f(a, a)
   f(a, a) -> g(d)

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

   encArg(a) -> a
   encArg(d) -> d
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_b) -> b
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a -> a
   encode_b -> b
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_d -> d

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(1, n^1).


The TRS R consists of the following rules:

   g(a) -> g(b)
   b -> f(a, a)
   f(a, a) -> g(d)

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

   encArg(a) -> a
   encArg(d) -> d
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_b) -> b
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a -> a
   encode_b -> b
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_d -> d

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:

   g(a) -> g(b)
   b -> f(a, a)
   f(a, a) -> g(d)
   encArg(a) -> a
   encArg(d) -> d
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_b) -> b
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a -> a
   encode_b -> b
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_d -> d

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 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: 
a0() -> 0
d0() -> 0
cons_g0(0) -> 0
cons_b0() -> 0
cons_f0(0, 0) -> 0
g0(0) -> 1
b0() -> 2
f0(0, 0) -> 3
encArg0(0) -> 4
encode_g0(0) -> 5
encode_a0() -> 6
encode_b0() -> 7
encode_f0(0, 0) -> 8
encode_d0() -> 9
b1() -> 10
g1(10) -> 1
a1() -> 11
a1() -> 12
f1(11, 12) -> 2
d1() -> 13
g1(13) -> 3
a1() -> 4
d1() -> 4
encArg1(0) -> 14
g1(14) -> 4
b1() -> 4
encArg1(0) -> 15
encArg1(0) -> 16
f1(15, 16) -> 4
g1(14) -> 5
a1() -> 6
b1() -> 7
f1(15, 16) -> 8
d1() -> 9
a2() -> 17
a2() -> 18
f2(17, 18) -> 4
f2(17, 18) -> 7
f2(17, 18) -> 10
d2() -> 19
g2(19) -> 2
a1() -> 14
a1() -> 15
a1() -> 16
d1() -> 14
d1() -> 15
d1() -> 16
g1(14) -> 14
g1(14) -> 15
g1(14) -> 16
b1() -> 14
b1() -> 15
b1() -> 16
f1(15, 16) -> 14
f1(15, 16) -> 15
f1(15, 16) -> 16
b2() -> 20
g2(20) -> 4
g2(20) -> 5
g2(20) -> 14
g2(20) -> 15
g2(20) -> 16
f2(17, 18) -> 14
f2(17, 18) -> 15
f2(17, 18) -> 16
g2(19) -> 4
g2(19) -> 8
g2(19) -> 14
g2(19) -> 15
g2(19) -> 16
d3() -> 21
g3(21) -> 4
g3(21) -> 7
g3(21) -> 10
g3(21) -> 14
g3(21) -> 15
g3(21) -> 16
a3() -> 22
a3() -> 23
f3(22, 23) -> 20
d4() -> 24
g4(24) -> 20

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

(8)
BOUNDS(1, n^1)