WORST_CASE(Omega(n^1),?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 13.4 s]
(2) CpxRelTRS
(3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(4) TRS for Loop Detection
(5) DecreasingLoopProof [LOWER BOUND(ID), 2 ms]
(6) BEST
    (7) proven lower bound
        (8) LowerBoundPropagationProof [FINISHED, 0 ms]
        (9) BOUNDS(n^1, INF)
    (10) TRS for Loop Detection


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


The TRS R consists of the following rules:

   EMPTY(nil) -> c
   EMPTY(cons(z0, z1)) -> c1
   TAIL(nil) -> c2
   TAIL(cons(z0, z1)) -> c3
   HEAD(cons(z0, z1)) -> c4
   ZERO(0) -> c5
   ZERO(s(z0)) -> c6
   P(0) -> c7
   P(s(0)) -> c8
   P(s(s(z0))) -> c9(P(s(z0)))
   INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0), EMPTY(z0))
   IF_INTLIST(true, z0) -> c11
   IF_INTLIST(false, z0) -> c12(HEAD(z0))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)), TAIL(z0))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z0))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z1))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(true, z0, z1) -> c18
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(true, z0, z1) -> c20
   IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0))
   IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1))

The (relative) TRS S consists of the following rules:

   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))

Rewrite Strategy: INNERMOST
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(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   EMPTY(nil) -> c
   EMPTY(cons(z0, z1)) -> c1
   TAIL(nil) -> c2
   TAIL(cons(z0, z1)) -> c3
   HEAD(cons(z0, z1)) -> c4
   ZERO(0) -> c5
   ZERO(s(z0)) -> c6
   P(0) -> c7
   P(s(0)) -> c8
   P(s(s(z0))) -> c9(P(s(z0)))
   INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0), EMPTY(z0))
   IF_INTLIST(true, z0) -> c11
   IF_INTLIST(false, z0) -> c12(HEAD(z0))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)), TAIL(z0))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z0))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z1))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(true, z0, z1) -> c18
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(true, z0, z1) -> c20
   IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0))
   IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1))

The (relative) TRS S consists of the following rules:

   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))

Rewrite Strategy: INNERMOST
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(3) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(4)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   EMPTY(nil) -> c
   EMPTY(cons(z0, z1)) -> c1
   TAIL(nil) -> c2
   TAIL(cons(z0, z1)) -> c3
   HEAD(cons(z0, z1)) -> c4
   ZERO(0) -> c5
   ZERO(s(z0)) -> c6
   P(0) -> c7
   P(s(0)) -> c8
   P(s(s(z0))) -> c9(P(s(z0)))
   INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0), EMPTY(z0))
   IF_INTLIST(true, z0) -> c11
   IF_INTLIST(false, z0) -> c12(HEAD(z0))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)), TAIL(z0))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z0))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z1))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(true, z0, z1) -> c18
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(true, z0, z1) -> c20
   IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0))
   IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1))

The (relative) TRS S consists of the following rules:

   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))

Rewrite Strategy: INNERMOST
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(5) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

P(s(s(z0))) ->^+ c9(P(s(z0)))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [z0 / s(z0)].

The result substitution is [ ].




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(6)
Complex Obligation (BEST)

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(7)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   EMPTY(nil) -> c
   EMPTY(cons(z0, z1)) -> c1
   TAIL(nil) -> c2
   TAIL(cons(z0, z1)) -> c3
   HEAD(cons(z0, z1)) -> c4
   ZERO(0) -> c5
   ZERO(s(z0)) -> c6
   P(0) -> c7
   P(s(0)) -> c8
   P(s(s(z0))) -> c9(P(s(z0)))
   INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0), EMPTY(z0))
   IF_INTLIST(true, z0) -> c11
   IF_INTLIST(false, z0) -> c12(HEAD(z0))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)), TAIL(z0))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z0))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z1))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(true, z0, z1) -> c18
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(true, z0, z1) -> c20
   IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0))
   IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1))

The (relative) TRS S consists of the following rules:

   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))

Rewrite Strategy: INNERMOST
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(8) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(9)
BOUNDS(n^1, INF)

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(10)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   EMPTY(nil) -> c
   EMPTY(cons(z0, z1)) -> c1
   TAIL(nil) -> c2
   TAIL(cons(z0, z1)) -> c3
   HEAD(cons(z0, z1)) -> c4
   ZERO(0) -> c5
   ZERO(s(z0)) -> c6
   P(0) -> c7
   P(s(0)) -> c8
   P(s(s(z0))) -> c9(P(s(z0)))
   INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0), EMPTY(z0))
   IF_INTLIST(true, z0) -> c11
   IF_INTLIST(false, z0) -> c12(HEAD(z0))
   IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)), TAIL(z0))
   INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z0))
   INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z1))
   IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2))
   IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2))
   IF1(true, z0, z1) -> c18
   IF1(false, z0, z1) -> c19(INT(s(0), z1))
   IF2(true, z0, z1) -> c20
   IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0))
   IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1))

The (relative) TRS S consists of the following rules:

   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   zero(0) -> true
   zero(s(z0)) -> false
   p(0) -> 0
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   intlist(z0) -> if_intlist(empty(z0), z0)
   if_intlist(true, z0) -> nil
   if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0)))
   int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1)
   if_int(true, z0, z1, z2) -> if1(z0, z1, z2)
   if_int(false, z0, z1, z2) -> if2(z0, z1, z2)
   if1(true, z0, z1) -> cons(0, nil)
   if1(false, z0, z1) -> cons(0, int(s(0), z1))
   if2(true, z0, z1) -> nil
   if2(false, z0, z1) -> intlist(int(p(z0), p(z1)))

Rewrite Strategy: INNERMOST