/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
(3) NarrowingOnBasicTermsTerminatesProof [FINISHED, 0 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:

   b(y, z) -> f(c(c(y, z, z), a, a))
   b(b(z, y), a) -> z
   c(f(z), f(c(a, x, a)), y) -> c(f(b(x, z)), c(z, y, a), a)

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 c: c

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

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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:

   b(y, z) -> f(c(c(y, z, z), a, a))

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:

b(x0, x1) ->^* f(c(c(x0, x1, x1), a, a))




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(4)
BOUNDS(1, 1)