/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
    (6) CdtProblem
    (7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 30 ms]
    (12) CdtProblem
    (13) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) BOUNDS(1, 1)
(15) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(16) TRS for Loop Detection
    (17) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) 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:

   h(z, e(x)) -> h(c(z), d(z, x))
   d(z, g(0, 0)) -> e(0)
   d(z, g(x, y)) -> g(e(x), d(z, y))
   d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
   g(e(x), e(y)) -> e(g(x, y))

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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 1th argument of h: d
The following defined symbols can occur below the 1th argument of d: d

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
d(z, g(0, 0)) -> e(0)
d(z, g(x, y)) -> g(e(x), d(z, y))
d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))

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

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


The TRS R consists of the following rules:

   h(z, e(x)) -> h(c(z), d(z, x))
   g(e(x), e(y)) -> e(g(x, y))

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

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

As the TRS does not nest defined symbols, we have rc = irc.
----------------------------------------

(4)
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:

   h(z, e(x)) -> h(c(z), d(z, x))
   g(e(x), e(y)) -> e(g(x, y))

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   h(z0, e(z1)) -> h(c(z0), d(z0, z1))
   g(e(z0), e(z1)) -> e(g(z0, z1))
Tuples:
   H(z0, e(z1)) -> c1(H(c(z0), d(z0, z1)))
   G(e(z0), e(z1)) -> c2(G(z0, z1))
S tuples:
   H(z0, e(z1)) -> c1(H(c(z0), d(z0, z1)))
   G(e(z0), e(z1)) -> c2(G(z0, z1))
K tuples:none
Defined Rule Symbols:   h_2, g_2

Defined Pair Symbols:   H_2, G_2

Compound Symbols:   c1_1, c2_1


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

(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   H(z0, e(z1)) -> c1(H(c(z0), d(z0, z1)))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   h(z0, e(z1)) -> h(c(z0), d(z0, z1))
   g(e(z0), e(z1)) -> e(g(z0, z1))
Tuples:
   G(e(z0), e(z1)) -> c2(G(z0, z1))
S tuples:
   G(e(z0), e(z1)) -> c2(G(z0, z1))
K tuples:none
Defined Rule Symbols:   h_2, g_2

Defined Pair Symbols:   G_2

Compound Symbols:   c2_1


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

(9) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   h(z0, e(z1)) -> h(c(z0), d(z0, z1))
   g(e(z0), e(z1)) -> e(g(z0, z1))

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   G(e(z0), e(z1)) -> c2(G(z0, z1))
S tuples:
   G(e(z0), e(z1)) -> c2(G(z0, z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   G_2

Compound Symbols:   c2_1


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

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

Polynomial interpretation :

   POL(G(x_1, x_2)) = x_2
   POL(c2(x_1)) = x_1
   POL(e(x_1)) = [1] + x_1

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   G(e(z0), e(z1)) -> c2(G(z0, z1))
S tuples:none
K tuples:
   G(e(z0), e(z1)) -> c2(G(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:   G_2

Compound Symbols:   c2_1


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

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

(14)
BOUNDS(1, 1)

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

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

(16)
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:

   h(z, e(x)) -> h(c(z), d(z, x))
   d(z, g(0, 0)) -> e(0)
   d(z, g(x, y)) -> g(e(x), d(z, y))
   d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
   g(e(x), e(y)) -> e(g(x, y))

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

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

The rewrite sequence

g(e(x), e(y)) ->^+ e(g(x, y))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [x / e(x), y / e(y)].

The result substitution is [ ].




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

(18)
Complex Obligation (BEST)

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

(19)
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:

   h(z, e(x)) -> h(c(z), d(z, x))
   d(z, g(0, 0)) -> e(0)
   d(z, g(x, y)) -> g(e(x), d(z, y))
   d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
   g(e(x), e(y)) -> e(g(x, y))

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

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

(21)
BOUNDS(n^1, INF)

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

(22)
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:

   h(z, e(x)) -> h(c(z), d(z, x))
   d(z, g(0, 0)) -> e(0)
   d(z, g(x, y)) -> g(e(x), d(z, y))
   d(c(z), g(g(x, y), 0)) -> g(d(c(z), g(x, y)), d(z, g(x, y)))
   g(e(x), e(y)) -> e(g(x, y))

S is empty.
Rewrite Strategy: FULL