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


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WORST_CASE(?, O(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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(1, 1).

(0) CpxTRS
(1) DependencyGraphProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
(3) NestedDefinedSymbolProof [UPPER BOUND(ID), 6 ms]
(4) CpxTRS
(5) NarrowingOnBasicTermsTerminatesProof [FINISHED, 0 ms]
(6) 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:

   i(x, x) -> i(a, b)
   g(x, x) -> g(a, b)
   h(s(f(x))) -> h(f(x))
   f(s(x)) -> s(s(f(h(s(x)))))
   f(g(s(x), y)) -> f(g(x, s(y)))
   h(g(x, s(y))) -> h(g(s(x), y))
   h(i(x, y)) -> i(i(c, h(h(y))), x)
   g(a, g(x, g(b, g(a, g(x, y))))) -> g(a, g(a, g(a, g(x, g(b, g(b, y))))))

S is empty.
Rewrite Strategy: FULL
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(1) DependencyGraphProof (UPPER BOUND(ID))
The following rules are not reachable from basic terms in the dependency graph and can be removed:

g(a, g(x, g(b, g(a, g(x, y))))) -> g(a, g(a, g(a, g(x, g(b, g(b, y))))))

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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:

   i(x, x) -> i(a, b)
   g(x, x) -> g(a, b)
   h(s(f(x))) -> h(f(x))
   f(s(x)) -> s(s(f(h(s(x)))))
   f(g(s(x), y)) -> f(g(x, s(y)))
   h(g(x, s(y))) -> h(g(s(x), y))
   h(i(x, y)) -> i(i(c, h(h(y))), x)

S is empty.
Rewrite Strategy: FULL
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(3) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of f: h
The following defined symbols can occur below the 0th argument of h: h

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
h(s(f(x))) -> h(f(x))
f(g(s(x), y)) -> f(g(x, s(y)))
h(g(x, s(y))) -> h(g(s(x), y))
h(i(x, y)) -> i(i(c, h(h(y))), x)

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(4)
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:

   i(x, x) -> i(a, b)
   g(x, x) -> g(a, b)
   f(s(x)) -> s(s(f(h(s(x)))))

S is empty.
Rewrite Strategy: FULL
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(5) 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:

f(s(x0)) ->^* s(s(f(h(s(x0)))))

g(x0, x0) ->^* g(a, b)

i(x0, x0) ->^* i(a, b)




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(6)
BOUNDS(1, 1)