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


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WORST_CASE(?, O(1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
(3) NarrowingOnBasicTermsTerminatesProof [FINISHED, 38 ms]
(4) BOUNDS(1, 1)


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


The TRS R consists of the following rules:

   f(X, n__g(X), Y) -> f(activate(Y), activate(Y), activate(Y))
   g(b) -> c
   b -> c
   g(X) -> n__g(X)
   activate(n__g(X)) -> g(X)
   activate(X) -> X

S is empty.
Rewrite Strategy: FULL
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(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of f: activate, g
The following defined symbols can occur below the 1th argument of f: activate, g
The following defined symbols can occur below the 2th argument of f: activate, g
The following defined symbols can occur below the 0th argument of activate: activate, g

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
g(b) -> c

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


The TRS R consists of the following rules:

   f(X, n__g(X), Y) -> f(activate(Y), activate(Y), activate(Y))
   b -> c
   g(X) -> n__g(X)
   activate(n__g(X)) -> g(X)
   activate(X) -> X

S is empty.
Rewrite Strategy: FULL
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(3) NarrowingOnBasicTermsTerminatesProof (FINISHED)
Constant runtime complexity proven by termination of constructor-based narrowing.

The maximal most general narrowing sequences give rise to the following rewrite sequences:

g(x0) ->^* n__g(x0)

activate(n__g(x0)) ->^* n__g(x0)

f(x0, n__g(x0), x1) ->^* f(x1, x1, x1)

f(x0, n__g(x0), n__g(x1)) ->^* f(n__g(x1), n__g(x1), n__g(x1))

b ->^* c




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(4)
BOUNDS(1, 1)