/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(Omega(n^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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(n^1, INF).

(0) CpxTRS
(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(2) TRS for Loop Detection
(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(4) BEST
    (5) proven lower bound
        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
        (7) BOUNDS(n^1, INF)
    (8) TRS for Loop Detection


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(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   eq(0, 0) -> true
   eq(0, s(x)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   or(true, y) -> true
   or(false, y) -> y
   union(empty, h) -> h
   union(edge(x, y, i), h) -> edge(x, y, union(i, h))
   isEmpty(empty) -> true
   isEmpty(edge(x, y, i)) -> false
   from(edge(x, y, i)) -> x
   to(edge(x, y, i)) -> y
   rest(edge(x, y, i)) -> i
   rest(empty) -> empty
   reach(x, y, i, h) -> if1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h)
   if1(true, b1, b2, b3, x, y, i, h) -> true
   if1(false, b1, b2, b3, x, y, i, h) -> if2(b1, b2, b3, x, y, i, h)
   if2(true, b2, b3, x, y, i, h) -> false
   if2(false, b2, b3, x, y, i, h) -> if3(b2, b3, x, y, i, h)
   if3(false, b3, x, y, i, h) -> reach(x, y, rest(i), edge(from(i), to(i), h))
   if3(true, b3, x, y, i, h) -> if4(b3, x, y, i, h)
   if4(true, x, y, i, h) -> true
   if4(false, x, y, i, h) -> or(reach(x, y, rest(i), h), reach(to(i), y, union(rest(i), h), empty))

S is empty.
Rewrite Strategy: FULL
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(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(2)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   eq(0, 0) -> true
   eq(0, s(x)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   or(true, y) -> true
   or(false, y) -> y
   union(empty, h) -> h
   union(edge(x, y, i), h) -> edge(x, y, union(i, h))
   isEmpty(empty) -> true
   isEmpty(edge(x, y, i)) -> false
   from(edge(x, y, i)) -> x
   to(edge(x, y, i)) -> y
   rest(edge(x, y, i)) -> i
   rest(empty) -> empty
   reach(x, y, i, h) -> if1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h)
   if1(true, b1, b2, b3, x, y, i, h) -> true
   if1(false, b1, b2, b3, x, y, i, h) -> if2(b1, b2, b3, x, y, i, h)
   if2(true, b2, b3, x, y, i, h) -> false
   if2(false, b2, b3, x, y, i, h) -> if3(b2, b3, x, y, i, h)
   if3(false, b3, x, y, i, h) -> reach(x, y, rest(i), edge(from(i), to(i), h))
   if3(true, b3, x, y, i, h) -> if4(b3, x, y, i, h)
   if4(true, x, y, i, h) -> true
   if4(false, x, y, i, h) -> or(reach(x, y, rest(i), h), reach(to(i), y, union(rest(i), h), empty))

S is empty.
Rewrite Strategy: FULL
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(3) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

union(edge(x, y, i), h) ->^+ edge(x, y, union(i, h))

gives rise to a decreasing loop by considering the right hand sides subterm at position [2].

The pumping substitution is [i / edge(x, y, i)].

The result substitution is [ ].




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(4)
Complex Obligation (BEST)

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(5)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   eq(0, 0) -> true
   eq(0, s(x)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   or(true, y) -> true
   or(false, y) -> y
   union(empty, h) -> h
   union(edge(x, y, i), h) -> edge(x, y, union(i, h))
   isEmpty(empty) -> true
   isEmpty(edge(x, y, i)) -> false
   from(edge(x, y, i)) -> x
   to(edge(x, y, i)) -> y
   rest(edge(x, y, i)) -> i
   rest(empty) -> empty
   reach(x, y, i, h) -> if1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h)
   if1(true, b1, b2, b3, x, y, i, h) -> true
   if1(false, b1, b2, b3, x, y, i, h) -> if2(b1, b2, b3, x, y, i, h)
   if2(true, b2, b3, x, y, i, h) -> false
   if2(false, b2, b3, x, y, i, h) -> if3(b2, b3, x, y, i, h)
   if3(false, b3, x, y, i, h) -> reach(x, y, rest(i), edge(from(i), to(i), h))
   if3(true, b3, x, y, i, h) -> if4(b3, x, y, i, h)
   if4(true, x, y, i, h) -> true
   if4(false, x, y, i, h) -> or(reach(x, y, rest(i), h), reach(to(i), y, union(rest(i), h), empty))

S is empty.
Rewrite Strategy: FULL
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(6) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(7)
BOUNDS(n^1, INF)

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(8)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   eq(0, 0) -> true
   eq(0, s(x)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   or(true, y) -> true
   or(false, y) -> y
   union(empty, h) -> h
   union(edge(x, y, i), h) -> edge(x, y, union(i, h))
   isEmpty(empty) -> true
   isEmpty(edge(x, y, i)) -> false
   from(edge(x, y, i)) -> x
   to(edge(x, y, i)) -> y
   rest(edge(x, y, i)) -> i
   rest(empty) -> empty
   reach(x, y, i, h) -> if1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h)
   if1(true, b1, b2, b3, x, y, i, h) -> true
   if1(false, b1, b2, b3, x, y, i, h) -> if2(b1, b2, b3, x, y, i, h)
   if2(true, b2, b3, x, y, i, h) -> false
   if2(false, b2, b3, x, y, i, h) -> if3(b2, b3, x, y, i, h)
   if3(false, b3, x, y, i, h) -> reach(x, y, rest(i), edge(from(i), to(i), h))
   if3(true, b3, x, y, i, h) -> if4(b3, x, y, i, h)
   if4(true, x, y, i, h) -> true
   if4(false, x, y, i, h) -> or(reach(x, y, rest(i), h), reach(to(i), y, union(rest(i), h), empty))

S is empty.
Rewrite Strategy: FULL