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


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


WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
    (4) CdtProblem
    (5) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 53 ms]
    (6) CdtProblem
    (7) CdtInstantiationProof [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)), 4 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(x, c(y)) -> f(x, s(f(y, y)))
   f(s(x), y) -> f(x, s(c(y)))

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

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

The duplicating contexts are:
f(x, c([]))


The defined contexts are:
f(x0, s([]))
f(x0, s(c([])))


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

(2)
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(x, c(y)) -> f(x, s(f(y, y)))
   f(s(x), y) -> f(x, s(c(y)))

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

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0, c(z1)) -> f(z0, s(f(z1, z1)))
   f(s(z0), z1) -> f(z0, s(c(z1)))
Tuples:
   F(z0, c(z1)) -> c1(F(z0, s(f(z1, z1))), F(z1, z1))
   F(s(z0), z1) -> c2(F(z0, s(c(z1))))
S tuples:
   F(z0, c(z1)) -> c1(F(z0, s(f(z1, z1))), F(z1, z1))
   F(s(z0), z1) -> c2(F(z0, s(c(z1))))
K tuples:none
Defined Rule Symbols:   f_2

Defined Pair Symbols:   F_2

Compound Symbols:   c1_2, c2_1


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

(5) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(z0, c(z1)) -> c1(F(z0, s(f(z1, z1))), F(z1, z1))
We considered the (Usable) Rules:none
And the Tuples:
   F(z0, c(z1)) -> c1(F(z0, s(f(z1, z1))), F(z1, z1))
   F(s(z0), z1) -> c2(F(z0, s(c(z1))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(F(x_1, x_2)) = x_2
   POL(c(x_1)) = [1] + x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(f(x_1, x_2)) = [1] + x_2
   POL(s(x_1)) = 0

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0, c(z1)) -> f(z0, s(f(z1, z1)))
   f(s(z0), z1) -> f(z0, s(c(z1)))
Tuples:
   F(z0, c(z1)) -> c1(F(z0, s(f(z1, z1))), F(z1, z1))
   F(s(z0), z1) -> c2(F(z0, s(c(z1))))
S tuples:
   F(s(z0), z1) -> c2(F(z0, s(c(z1))))
K tuples:
   F(z0, c(z1)) -> c1(F(z0, s(f(z1, z1))), F(z1, z1))
Defined Rule Symbols:   f_2

Defined Pair Symbols:   F_2

Compound Symbols:   c1_2, c2_1


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

(7) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace F(z0, c(z1)) -> c1(F(z0, s(f(z1, z1))), F(z1, z1)) by 
   F(c(z1), c(z1)) -> c1(F(c(z1), s(f(z1, z1))), F(z1, z1))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0, c(z1)) -> f(z0, s(f(z1, z1)))
   f(s(z0), z1) -> f(z0, s(c(z1)))
Tuples:
   F(s(z0), z1) -> c2(F(z0, s(c(z1))))
   F(c(z1), c(z1)) -> c1(F(c(z1), s(f(z1, z1))), F(z1, z1))
S tuples:
   F(s(z0), z1) -> c2(F(z0, s(c(z1))))
K tuples:
   F(c(z1), c(z1)) -> c1(F(c(z1), s(f(z1, z1))), F(z1, z1))
Defined Rule Symbols:   f_2

Defined Pair Symbols:   F_2

Compound Symbols:   c2_1, c1_2


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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0, c(z1)) -> f(z0, s(f(z1, z1)))
   f(s(z0), z1) -> f(z0, s(c(z1)))
Tuples:
   F(s(z0), z1) -> c2(F(z0, s(c(z1))))
   F(c(z1), c(z1)) -> c1(F(z1, z1))
S tuples:
   F(s(z0), z1) -> c2(F(z0, s(c(z1))))
K tuples:
   F(c(z1), c(z1)) -> c1(F(z1, z1))
Defined Rule Symbols:   f_2

Defined Pair Symbols:   F_2

Compound Symbols:   c2_1, c1_1


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

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

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(s(z0), z1) -> c2(F(z0, s(c(z1))))
   F(c(z1), c(z1)) -> c1(F(z1, z1))
S tuples:
   F(s(z0), z1) -> c2(F(z0, s(c(z1))))
K tuples:
   F(c(z1), c(z1)) -> c1(F(z1, z1))
Defined Rule Symbols:none

Defined Pair Symbols:   F_2

Compound Symbols:   c2_1, c1_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.
   F(s(z0), z1) -> c2(F(z0, s(c(z1))))
We considered the (Usable) Rules:none
And the Tuples:
   F(s(z0), z1) -> c2(F(z0, s(c(z1))))
   F(c(z1), c(z1)) -> c1(F(z1, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(F(x_1, x_2)) = x_1
   POL(c(x_1)) = [3] + x_1
   POL(c1(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(s(x_1)) = [1] + x_1

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(s(z0), z1) -> c2(F(z0, s(c(z1))))
   F(c(z1), c(z1)) -> c1(F(z1, z1))
S tuples:none
K tuples:
   F(c(z1), c(z1)) -> c1(F(z1, z1))
   F(s(z0), z1) -> c2(F(z0, s(c(z1))))
Defined Rule Symbols:none

Defined Pair Symbols:   F_2

Compound Symbols:   c2_1, c1_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(x, c(y)) -> f(x, s(f(y, y)))
   f(s(x), y) -> f(x, s(c(y)))

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

f(s(x), y) ->^+ f(x, s(c(y)))

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 [y / s(c(y))].




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

(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(x, c(y)) -> f(x, s(f(y, y)))
   f(s(x), y) -> f(x, s(c(y)))

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(x, c(y)) -> f(x, s(f(y, y)))
   f(s(x), y) -> f(x, s(c(y)))

S is empty.
Rewrite Strategy: FULL