/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 481 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:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   zero(0) -> true
   zero(s(x)) -> false
   id(0) -> 0
   id(s(x)) -> s(id(x))
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
   if_mod(true, b1, b2, x, y) -> 0
   if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y)
   if2(true, b2, x, y) -> 0
   if2(false, b2, x, y) -> if3(b2, x, y)
   if3(true, x, y) -> mod(minus(x, y), s(y))
   if3(false, x, y) -> x

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(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_zero(x_1)) -> zero(encArg(x_1))
   encArg(cons_id(x_1)) -> id(encArg(x_1))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_mod(x_1, x_2)) -> mod(encArg(x_1), encArg(x_2))
   encArg(cons_if_mod(x_1, x_2, x_3, x_4, x_5)) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encArg(cons_if2(x_1, x_2, x_3, x_4)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_if3(x_1, x_2, x_3)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_zero(x_1) -> zero(encArg(x_1))
   encode_id(x_1) -> id(encArg(x_1))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_mod(x_1, x_2) -> mod(encArg(x_1), encArg(x_2))
   encode_if_mod(x_1, x_2, x_3, x_4, x_5) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_if2(x_1, x_2, x_3, x_4) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_if3(x_1, x_2, x_3) -> if3(encArg(x_1), encArg(x_2), encArg(x_3))


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

(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:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   zero(0) -> true
   zero(s(x)) -> false
   id(0) -> 0
   id(s(x)) -> s(id(x))
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
   if_mod(true, b1, b2, x, y) -> 0
   if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y)
   if2(true, b2, x, y) -> 0
   if2(false, b2, x, y) -> if3(b2, x, y)
   if3(true, x, y) -> mod(minus(x, y), s(y))
   if3(false, x, y) -> x

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(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_zero(x_1)) -> zero(encArg(x_1))
   encArg(cons_id(x_1)) -> id(encArg(x_1))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_mod(x_1, x_2)) -> mod(encArg(x_1), encArg(x_2))
   encArg(cons_if_mod(x_1, x_2, x_3, x_4, x_5)) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encArg(cons_if2(x_1, x_2, x_3, x_4)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_if3(x_1, x_2, x_3)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_zero(x_1) -> zero(encArg(x_1))
   encode_id(x_1) -> id(encArg(x_1))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_mod(x_1, x_2) -> mod(encArg(x_1), encArg(x_2))
   encode_if_mod(x_1, x_2, x_3, x_4, x_5) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_if2(x_1, x_2, x_3, x_4) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_if3(x_1, x_2, x_3) -> if3(encArg(x_1), encArg(x_2), encArg(x_3))

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:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   zero(0) -> true
   zero(s(x)) -> false
   id(0) -> 0
   id(s(x)) -> s(id(x))
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
   if_mod(true, b1, b2, x, y) -> 0
   if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y)
   if2(true, b2, x, y) -> 0
   if2(false, b2, x, y) -> if3(b2, x, y)
   if3(true, x, y) -> mod(minus(x, y), s(y))
   if3(false, x, y) -> x

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(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_zero(x_1)) -> zero(encArg(x_1))
   encArg(cons_id(x_1)) -> id(encArg(x_1))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_mod(x_1, x_2)) -> mod(encArg(x_1), encArg(x_2))
   encArg(cons_if_mod(x_1, x_2, x_3, x_4, x_5)) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encArg(cons_if2(x_1, x_2, x_3, x_4)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_if3(x_1, x_2, x_3)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_zero(x_1) -> zero(encArg(x_1))
   encode_id(x_1) -> id(encArg(x_1))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_mod(x_1, x_2) -> mod(encArg(x_1), encArg(x_2))
   encode_if_mod(x_1, x_2, x_3, x_4, x_5) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_if2(x_1, x_2, x_3, x_4) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_if3(x_1, x_2, x_3) -> if3(encArg(x_1), encArg(x_2), encArg(x_3))

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:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   zero(0) -> true
   zero(s(x)) -> false
   id(0) -> 0
   id(s(x)) -> s(id(x))
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
   if_mod(true, b1, b2, x, y) -> 0
   if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y)
   if2(true, b2, x, y) -> 0
   if2(false, b2, x, y) -> if3(b2, x, y)
   if3(true, x, y) -> mod(minus(x, y), s(y))
   if3(false, x, y) -> x

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(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_zero(x_1)) -> zero(encArg(x_1))
   encArg(cons_id(x_1)) -> id(encArg(x_1))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_mod(x_1, x_2)) -> mod(encArg(x_1), encArg(x_2))
   encArg(cons_if_mod(x_1, x_2, x_3, x_4, x_5)) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encArg(cons_if2(x_1, x_2, x_3, x_4)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_if3(x_1, x_2, x_3)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_zero(x_1) -> zero(encArg(x_1))
   encode_id(x_1) -> id(encArg(x_1))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_mod(x_1, x_2) -> mod(encArg(x_1), encArg(x_2))
   encode_if_mod(x_1, x_2, x_3, x_4, x_5) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_if2(x_1, x_2, x_3, x_4) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_if3(x_1, x_2, x_3) -> if3(encArg(x_1), encArg(x_2), encArg(x_3))

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

le(s(x), s(y)) ->^+ le(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:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   zero(0) -> true
   zero(s(x)) -> false
   id(0) -> 0
   id(s(x)) -> s(id(x))
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
   if_mod(true, b1, b2, x, y) -> 0
   if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y)
   if2(true, b2, x, y) -> 0
   if2(false, b2, x, y) -> if3(b2, x, y)
   if3(true, x, y) -> mod(minus(x, y), s(y))
   if3(false, x, y) -> x

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(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_zero(x_1)) -> zero(encArg(x_1))
   encArg(cons_id(x_1)) -> id(encArg(x_1))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_mod(x_1, x_2)) -> mod(encArg(x_1), encArg(x_2))
   encArg(cons_if_mod(x_1, x_2, x_3, x_4, x_5)) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encArg(cons_if2(x_1, x_2, x_3, x_4)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_if3(x_1, x_2, x_3)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_zero(x_1) -> zero(encArg(x_1))
   encode_id(x_1) -> id(encArg(x_1))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_mod(x_1, x_2) -> mod(encArg(x_1), encArg(x_2))
   encode_if_mod(x_1, x_2, x_3, x_4, x_5) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_if2(x_1, x_2, x_3, x_4) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_if3(x_1, x_2, x_3) -> if3(encArg(x_1), encArg(x_2), encArg(x_3))

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:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   zero(0) -> true
   zero(s(x)) -> false
   id(0) -> 0
   id(s(x)) -> s(id(x))
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
   if_mod(true, b1, b2, x, y) -> 0
   if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y)
   if2(true, b2, x, y) -> 0
   if2(false, b2, x, y) -> if3(b2, x, y)
   if3(true, x, y) -> mod(minus(x, y), s(y))
   if3(false, x, y) -> x

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(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_zero(x_1)) -> zero(encArg(x_1))
   encArg(cons_id(x_1)) -> id(encArg(x_1))
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_mod(x_1, x_2)) -> mod(encArg(x_1), encArg(x_2))
   encArg(cons_if_mod(x_1, x_2, x_3, x_4, x_5)) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encArg(cons_if2(x_1, x_2, x_3, x_4)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encArg(cons_if3(x_1, x_2, x_3)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_zero(x_1) -> zero(encArg(x_1))
   encode_id(x_1) -> id(encArg(x_1))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_mod(x_1, x_2) -> mod(encArg(x_1), encArg(x_2))
   encode_if_mod(x_1, x_2, x_3, x_4, x_5) -> if_mod(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encode_if2(x_1, x_2, x_3, x_4) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_if3(x_1, x_2, x_3) -> if3(encArg(x_1), encArg(x_2), encArg(x_3))

Rewrite Strategy: INNERMOST