/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 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(x, y) -> c(a(c(y), a(0, x)))
   a(y, x) -> y
   a(y, c(b(a(0, x), 0))) -> b(a(c(b(0, y)), x), 0)

S is empty.
Rewrite Strategy: FULL
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(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 1th argument of a: a

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

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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(x, y) -> c(a(c(y), a(0, x)))
   a(y, x) -> y

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) ->^* c(c(x1))

a(x0, x1) ->^* a(x0, x1)




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(4)
BOUNDS(1, 1)