/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(a, x) -> g1(x, x)
   f1(x, a) -> g2(x, x)
   f2(a, x) -> g1(x, x)
   f2(x, a) -> g2(x, x)
   g1(a, x) -> h1(x)
   g1(x, a) -> h2(x)
   g2(a, x) -> h1(x)
   g2(x, a) -> h2(x)
   h1(a) -> i
   h2(a) -> i
   e1(h1(w), h2(w), x, y, z, w) -> e2(x, x, y, z, z, w)
   e1(x1, x1, x, y, z, a) -> e5(x1, x, y, z)
   e2(f1(w, w), x, y, z, f2(w, w), w) -> e3(x, y, x, y, y, z, y, z, x, y, z, w)
   e2(x, x, y, z, z, a) -> e6(x, y, z)
   e2(i, x, y, z, i, a) -> e6(x, y, z)
   e3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z, w) -> e4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z, w)
   e3(x, y, x, y, y, z, y, z, x, y, z, a) -> e6(x, y, z)
   e4(g1(w, w), x1, g2(w, w), x1, g1(w, w), x1, g2(w, w), x1, x, y, z, w) -> e1(x1, x1, x, y, z, w)
   e4(i, x1, i, x1, i, x1, i, x1, x, y, z, a) -> e5(x1, x, y, z)
   e4(x, x, x, x, x, x, x, x, x, x, x, a) -> 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(w), h2(w), x, y, z, w) -> e2(x, x, y, z, z, w)
e2(f1(w, w), x, y, z, f2(w, w), w) -> e3(x, y, x, y, y, z, y, z, x, y, z, w)
e4(g1(w, w), x1, g2(w, w), x1, g1(w, w), x1, g2(w, w), x1, x, y, z, w) -> e1(x1, x1, x, y, z, w)

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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(a, x) -> g1(x, x)
   f1(x, a) -> g2(x, x)
   f2(a, x) -> g1(x, x)
   f2(x, a) -> g2(x, x)
   g1(a, x) -> h1(x)
   g1(x, a) -> h2(x)
   g2(a, x) -> h1(x)
   g2(x, a) -> h2(x)
   h1(a) -> i
   h2(a) -> i
   e1(x1, x1, x, y, z, a) -> e5(x1, x, y, z)
   e2(x, x, y, z, z, a) -> e6(x, y, z)
   e2(i, x, y, z, i, a) -> e6(x, y, z)
   e3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z, w) -> e4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z, w)
   e3(x, y, x, y, y, z, y, z, x, y, z, a) -> e6(x, y, z)
   e4(i, x1, i, x1, i, x1, i, x1, x, y, z, a) -> e5(x1, x, y, z)
   e4(x, x, x, x, x, x, x, x, x, x, x, a) -> 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, a) ->^* e6(x0, x0, x0)

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

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

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

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

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

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

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

h2(a) ->^* i

h1(a) ->^* i

g2(a, a) ->^* i

g1(a, a) ->^* i

f2(a, a) ->^* i

f1(a, a) ->^* i




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(4)
BOUNDS(1, 1)