/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 300 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) TRS for Loop Detection


----------------------------------------

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


The TRS R consists of the following rules:

   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   minus(x, 0) -> x
   minus(0, y) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   id_inc(x) -> x
   id_inc(x) -> s(x)
   quot(x, y) -> div(x, y, 0)
   div(x, y, z) -> if(ge(y, s(0)), ge(x, y), x, y, z)
   if(false, b, x, y, z) -> div_by_zero
   if(true, false, x, y, z) -> z
   if(true, true, x, y, z) -> div(minus(x, y), y, id_inc(z))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(div_by_zero) -> div_by_zero
   encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_id_inc(x_1)) -> id_inc(encArg(x_1))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_div(x_1, x_2, x_3)) -> div(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_id_inc(x_1) -> id_inc(encArg(x_1))
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_div(x_1, x_2, x_3) -> div(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_div_by_zero -> div_by_zero


----------------------------------------

(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:

   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   minus(x, 0) -> x
   minus(0, y) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   id_inc(x) -> x
   id_inc(x) -> s(x)
   quot(x, y) -> div(x, y, 0)
   div(x, y, z) -> if(ge(y, s(0)), ge(x, y), x, y, z)
   if(false, b, x, y, z) -> div_by_zero
   if(true, false, x, y, z) -> z
   if(true, true, x, y, z) -> div(minus(x, y), y, id_inc(z))

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

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(div_by_zero) -> div_by_zero
   encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_id_inc(x_1)) -> id_inc(encArg(x_1))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_div(x_1, x_2, x_3)) -> div(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_id_inc(x_1) -> id_inc(encArg(x_1))
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_div(x_1, x_2, x_3) -> div(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_div_by_zero -> div_by_zero

Rewrite Strategy: INNERMOST
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
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:

   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   minus(x, 0) -> x
   minus(0, y) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   id_inc(x) -> x
   id_inc(x) -> s(x)
   quot(x, y) -> div(x, y, 0)
   div(x, y, z) -> if(ge(y, s(0)), ge(x, y), x, y, z)
   if(false, b, x, y, z) -> div_by_zero
   if(true, false, x, y, z) -> z
   if(true, true, x, y, z) -> div(minus(x, y), y, id_inc(z))

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

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(div_by_zero) -> div_by_zero
   encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_id_inc(x_1)) -> id_inc(encArg(x_1))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_div(x_1, x_2, x_3)) -> div(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_id_inc(x_1) -> id_inc(encArg(x_1))
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_div(x_1, x_2, x_3) -> div(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_div_by_zero -> div_by_zero

Rewrite Strategy: INNERMOST
----------------------------------------

(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(6)
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:

   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   minus(x, 0) -> x
   minus(0, y) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   id_inc(x) -> x
   id_inc(x) -> s(x)
   quot(x, y) -> div(x, y, 0)
   div(x, y, z) -> if(ge(y, s(0)), ge(x, y), x, y, z)
   if(false, b, x, y, z) -> div_by_zero
   if(true, false, x, y, z) -> z
   if(true, true, x, y, z) -> div(minus(x, y), y, id_inc(z))

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

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(div_by_zero) -> div_by_zero
   encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_id_inc(x_1)) -> id_inc(encArg(x_1))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_div(x_1, x_2, x_3)) -> div(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_id_inc(x_1) -> id_inc(encArg(x_1))
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_div(x_1, x_2, x_3) -> div(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_div_by_zero -> div_by_zero

Rewrite Strategy: INNERMOST
----------------------------------------

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

The rewrite sequence

minus(s(x), s(y)) ->^+ minus(x, y)

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

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

The result substitution is [ ].




----------------------------------------

(8)
Complex Obligation (BEST)

----------------------------------------

(9)
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:

   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   minus(x, 0) -> x
   minus(0, y) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   id_inc(x) -> x
   id_inc(x) -> s(x)
   quot(x, y) -> div(x, y, 0)
   div(x, y, z) -> if(ge(y, s(0)), ge(x, y), x, y, z)
   if(false, b, x, y, z) -> div_by_zero
   if(true, false, x, y, z) -> z
   if(true, true, x, y, z) -> div(minus(x, y), y, id_inc(z))

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

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(div_by_zero) -> div_by_zero
   encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_id_inc(x_1)) -> id_inc(encArg(x_1))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_div(x_1, x_2, x_3)) -> div(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_id_inc(x_1) -> id_inc(encArg(x_1))
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_div(x_1, x_2, x_3) -> div(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_div_by_zero -> div_by_zero

Rewrite Strategy: INNERMOST
----------------------------------------

(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(11)
BOUNDS(n^1, INF)

----------------------------------------

(12)
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:

   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   minus(x, 0) -> x
   minus(0, y) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   id_inc(x) -> x
   id_inc(x) -> s(x)
   quot(x, y) -> div(x, y, 0)
   div(x, y, z) -> if(ge(y, s(0)), ge(x, y), x, y, z)
   if(false, b, x, y, z) -> div_by_zero
   if(true, false, x, y, z) -> z
   if(true, true, x, y, z) -> div(minus(x, y), y, id_inc(z))

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

   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(div_by_zero) -> div_by_zero
   encArg(cons_ge(x_1, x_2)) -> ge(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_id_inc(x_1)) -> id_inc(encArg(x_1))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_div(x_1, x_2, x_3)) -> div(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_ge(x_1, x_2) -> ge(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_id_inc(x_1) -> id_inc(encArg(x_1))
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_div(x_1, x_2, x_3) -> div(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_div_by_zero -> div_by_zero

Rewrite Strategy: INNERMOST