/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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:

   f1 -> g1
   f1 -> g2
   f2 -> g1
   f2 -> g2
   g1 -> h1
   g1 -> h2
   g2 -> h1
   g2 -> h2
   h1 -> i
   h2 -> i
   e1(h1, h2, x, y, z) -> e2(x, x, y, z, z)
   e1(x1, x1, x, y, z) -> e5(x1, x, y, z)
   e2(f1, x, y, z, f2) -> e3(x, y, x, y, y, z, y, z, x, y, z)
   e2(x, x, y, z, z) -> e6(x, y, z)
   e2(i, x, y, z, i) -> e6(x, y, z)
   e3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z) -> e4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z)
   e3(x, y, x, y, y, z, y, z, x, y, z) -> e6(x, y, z)
   e4(g1, x1, g2, x1, g1, x1, g2, x1, x, y, z) -> e1(x1, x1, x, y, z)
   e4(i, x1, i, x1, i, x1, i, x1, x, y, z) -> e5(x1, x, y, z)
   e4(x, x, x, x, x, x, x, x, x, x, x) -> e6(x, x, x)
   e5(i, x, y, z) -> e6(x, y, z)

S is empty.
Rewrite Strategy: FULL
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(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
e1(h1, h2, x, y, z) -> e2(x, x, y, z, z)
e2(f1, x, y, z, f2) -> e3(x, y, x, y, y, z, y, z, x, y, z)
e4(g1, x1, g2, x1, g1, x1, g2, x1, x, y, z) -> e1(x1, x1, x, y, z)

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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:

   f1 -> g1
   f1 -> g2
   f2 -> g1
   f2 -> g2
   g1 -> h1
   g1 -> h2
   g2 -> h1
   g2 -> h2
   h1 -> i
   h2 -> i
   e1(x1, x1, x, y, z) -> e5(x1, x, y, z)
   e2(x, x, y, z, z) -> e6(x, y, z)
   e2(i, x, y, z, i) -> e6(x, y, z)
   e3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z) -> e4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z)
   e3(x, y, x, y, y, z, y, z, x, y, z) -> e6(x, y, z)
   e4(i, x1, i, x1, i, x1, i, x1, x, y, z) -> e5(x1, x, y, z)
   e4(x, x, x, x, x, x, x, x, x, x, x) -> e6(x, x, x)
   e5(i, x, y, z) -> e6(x, y, z)

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:

e5(i, x0, x1, x2) ->^* e6(x0, x1, x2)

e4(x0, x0, x0, x0, x0, x0, x0, x0, x0, x0, x0) ->^* e6(x0, x0, x0)

e4(i, i, i, i, i, i, i, i, x0, x1, x2) ->^* e6(x0, x1, x2)

e3(x0, x1, x0, x1, x1, x2, x1, x2, x0, x1, x2) ->^* e6(x0, x1, x2)

e3(x0, x0, x0, x0, x0, x0, x0, x0, x0, x0, x0) ->^* e6(x0, x0, x0)

e3(i, i, i, i, i, i, i, i, x0, x1, x2) ->^* e6(x0, x1, x2)

e2(i, x0, x1, x2, i) ->^* e6(x0, x1, x2)

e2(x0, x0, x1, x2, x2) ->^* e6(x0, x1, x2)

e1(i, i, x0, x1, x2) ->^* e6(x0, x1, x2)

h2 ->^* i

h1 ->^* i

g2 ->^* i

g1 ->^* i

f2 ->^* i

f1 ->^* i




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(4)
BOUNDS(1, 1)