/export/starexec/sandbox/solver/bin/starexec_run_complexity /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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
(2) CpxRelTRS
    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxRelTRS
    (5) CpxTrsToCdtProof [UPPER BOUND(ID), 6 ms]
    (6) CdtProblem
    (7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CdtProblem
    (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 49 ms]
    (14) CdtProblem
    (15) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) BOUNDS(1, 1)
(17) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(18) TRS for Loop Detection
    (19) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) TRS for Loop Detection


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

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


The TRS R consists of the following rules:

   f(cons(nil, y)) -> y
   f(cons(f(cons(nil, y)), z)) -> copy(n, y, z)
   copy(0, y, z) -> f(z)
   copy(s(x), y, z) -> copy(x, y, cons(f(y), z))

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

(1) NonCtorToCtorProof (UPPER BOUND(ID))
transformed non-ctor to ctor-system
----------------------------------------

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


The TRS R consists of the following rules:

   f(cons(nil, y)) -> y
   copy(0, y, z) -> f(z)
   copy(s(x), y, z) -> copy(x, y, cons(f(y), z))
   f(cons(c_f(cons(nil, y)), z)) -> copy(n, y, z)

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

   f(x0) -> c_f(x0)

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

(3) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

The duplicating contexts are:
copy(s(x), [], z)


The defined contexts are:
copy(x0, x1, cons([], x3))
f([])
copy(x0, x1, cons(x2, []))
copy(n, [], x1)
copy(n, x0, [])


As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
----------------------------------------

(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:

   f(cons(nil, y)) -> y
   copy(0, y, z) -> f(z)
   copy(s(x), y, z) -> copy(x, y, cons(f(y), z))
   f(cons(c_f(cons(nil, y)), z)) -> copy(n, y, z)

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

   f(x0) -> c_f(x0)

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

(5) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0) -> c_f(z0)
   f(cons(nil, z0)) -> z0
   f(cons(c_f(cons(nil, z0)), z1)) -> copy(n, z0, z1)
   copy(0, z0, z1) -> f(z1)
   copy(s(z0), z1, z2) -> copy(z0, z1, cons(f(z1), z2))
Tuples:
   F(z0) -> c
   F(cons(nil, z0)) -> c1
   F(cons(c_f(cons(nil, z0)), z1)) -> c2(COPY(n, z0, z1))
   COPY(0, z0, z1) -> c3(F(z1))
   COPY(s(z0), z1, z2) -> c4(COPY(z0, z1, cons(f(z1), z2)), F(z1))
S tuples:
   F(cons(nil, z0)) -> c1
   F(cons(c_f(cons(nil, z0)), z1)) -> c2(COPY(n, z0, z1))
   COPY(0, z0, z1) -> c3(F(z1))
   COPY(s(z0), z1, z2) -> c4(COPY(z0, z1, cons(f(z1), z2)), F(z1))
K tuples:none
Defined Rule Symbols:   f_1, copy_3

Defined Pair Symbols:   F_1, COPY_3

Compound Symbols:   c, c1, c2_1, c3_1, c4_2


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

(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   F(z0) -> c
   F(cons(c_f(cons(nil, z0)), z1)) -> c2(COPY(n, z0, z1))
   COPY(0, z0, z1) -> c3(F(z1))
   F(cons(nil, z0)) -> c1

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0) -> c_f(z0)
   f(cons(nil, z0)) -> z0
   f(cons(c_f(cons(nil, z0)), z1)) -> copy(n, z0, z1)
   copy(0, z0, z1) -> f(z1)
   copy(s(z0), z1, z2) -> copy(z0, z1, cons(f(z1), z2))
Tuples:
   COPY(s(z0), z1, z2) -> c4(COPY(z0, z1, cons(f(z1), z2)), F(z1))
S tuples:
   COPY(s(z0), z1, z2) -> c4(COPY(z0, z1, cons(f(z1), z2)), F(z1))
K tuples:none
Defined Rule Symbols:   f_1, copy_3

Defined Pair Symbols:   COPY_3

Compound Symbols:   c4_2


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

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0) -> c_f(z0)
   f(cons(nil, z0)) -> z0
   f(cons(c_f(cons(nil, z0)), z1)) -> copy(n, z0, z1)
   copy(0, z0, z1) -> f(z1)
   copy(s(z0), z1, z2) -> copy(z0, z1, cons(f(z1), z2))
Tuples:
   COPY(s(z0), z1, z2) -> c4(COPY(z0, z1, cons(f(z1), z2)))
S tuples:
   COPY(s(z0), z1, z2) -> c4(COPY(z0, z1, cons(f(z1), z2)))
K tuples:none
Defined Rule Symbols:   f_1, copy_3

Defined Pair Symbols:   COPY_3

Compound Symbols:   c4_1


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

(11) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   copy(0, z0, z1) -> f(z1)
   copy(s(z0), z1, z2) -> copy(z0, z1, cons(f(z1), z2))

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0) -> c_f(z0)
   f(cons(nil, z0)) -> z0
   f(cons(c_f(cons(nil, z0)), z1)) -> copy(n, z0, z1)
Tuples:
   COPY(s(z0), z1, z2) -> c4(COPY(z0, z1, cons(f(z1), z2)))
S tuples:
   COPY(s(z0), z1, z2) -> c4(COPY(z0, z1, cons(f(z1), z2)))
K tuples:none
Defined Rule Symbols:   f_1

Defined Pair Symbols:   COPY_3

Compound Symbols:   c4_1


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

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   COPY(s(z0), z1, z2) -> c4(COPY(z0, z1, cons(f(z1), z2)))
We considered the (Usable) Rules:none
And the Tuples:
   COPY(s(z0), z1, z2) -> c4(COPY(z0, z1, cons(f(z1), z2)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(COPY(x_1, x_2, x_3)) = x_1
   POL(c4(x_1)) = x_1
   POL(c_f(x_1)) = [1] + x_1
   POL(cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(copy(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(f(x_1)) = [1] + x_1
   POL(n) = 0
   POL(nil) = 0
   POL(s(x_1)) = [1] + x_1

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0) -> c_f(z0)
   f(cons(nil, z0)) -> z0
   f(cons(c_f(cons(nil, z0)), z1)) -> copy(n, z0, z1)
Tuples:
   COPY(s(z0), z1, z2) -> c4(COPY(z0, z1, cons(f(z1), z2)))
S tuples:none
K tuples:
   COPY(s(z0), z1, z2) -> c4(COPY(z0, z1, cons(f(z1), z2)))
Defined Rule Symbols:   f_1

Defined Pair Symbols:   COPY_3

Compound Symbols:   c4_1


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

(15) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
----------------------------------------

(16)
BOUNDS(1, 1)

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

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

(18)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   f(cons(nil, y)) -> y
   f(cons(f(cons(nil, y)), z)) -> copy(n, y, z)
   copy(0, y, z) -> f(z)
   copy(s(x), y, z) -> copy(x, y, cons(f(y), z))

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

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

The rewrite sequence

copy(s(x), y, z) ->^+ copy(x, y, cons(f(y), z))

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 [z / cons(f(y), z)].




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

(20)
Complex Obligation (BEST)

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

(21)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   f(cons(nil, y)) -> y
   f(cons(f(cons(nil, y)), z)) -> copy(n, y, z)
   copy(0, y, z) -> f(z)
   copy(s(x), y, z) -> copy(x, y, cons(f(y), z))

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

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

(23)
BOUNDS(n^1, INF)

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

(24)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   f(cons(nil, y)) -> y
   f(cons(f(cons(nil, y)), z)) -> copy(n, y, z)
   copy(0, y, z) -> f(z)
   copy(s(x), y, z) -> copy(x, y, cons(f(y), z))

S is empty.
Rewrite Strategy: FULL