/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 225 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:

   function(iszero, 0, dummy, dummy2) -> true
   function(iszero, s(x), dummy, dummy2) -> false
   function(p, 0, dummy, dummy2) -> 0
   function(p, s(0), dummy, dummy2) -> 0
   function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x))
   function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y)
   function(if, true, x, y) -> y
   function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y))
   function(third, x, y, z) -> 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(iszero) -> iszero
   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(p) -> p
   encArg(plus) -> plus
   encArg(if) -> if
   encArg(third) -> third
   encArg(cons_function(x_1, x_2, x_3, x_4)) -> function(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_function(x_1, x_2, x_3, x_4) -> function(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_iszero -> iszero
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_p -> p
   encode_plus -> plus
   encode_if -> if
   encode_third -> third


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

(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:

   function(iszero, 0, dummy, dummy2) -> true
   function(iszero, s(x), dummy, dummy2) -> false
   function(p, 0, dummy, dummy2) -> 0
   function(p, s(0), dummy, dummy2) -> 0
   function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x))
   function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y)
   function(if, true, x, y) -> y
   function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y))
   function(third, x, y, z) -> z

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

   encArg(iszero) -> iszero
   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(p) -> p
   encArg(plus) -> plus
   encArg(if) -> if
   encArg(third) -> third
   encArg(cons_function(x_1, x_2, x_3, x_4)) -> function(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_function(x_1, x_2, x_3, x_4) -> function(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_iszero -> iszero
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_p -> p
   encode_plus -> plus
   encode_if -> if
   encode_third -> third

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:

   function(iszero, 0, dummy, dummy2) -> true
   function(iszero, s(x), dummy, dummy2) -> false
   function(p, 0, dummy, dummy2) -> 0
   function(p, s(0), dummy, dummy2) -> 0
   function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x))
   function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y)
   function(if, true, x, y) -> y
   function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y))
   function(third, x, y, z) -> z

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

   encArg(iszero) -> iszero
   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(p) -> p
   encArg(plus) -> plus
   encArg(if) -> if
   encArg(third) -> third
   encArg(cons_function(x_1, x_2, x_3, x_4)) -> function(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_function(x_1, x_2, x_3, x_4) -> function(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_iszero -> iszero
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_p -> p
   encode_plus -> plus
   encode_if -> if
   encode_third -> third

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:

   function(iszero, 0, dummy, dummy2) -> true
   function(iszero, s(x), dummy, dummy2) -> false
   function(p, 0, dummy, dummy2) -> 0
   function(p, s(0), dummy, dummy2) -> 0
   function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x))
   function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y)
   function(if, true, x, y) -> y
   function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y))
   function(third, x, y, z) -> z

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

   encArg(iszero) -> iszero
   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(p) -> p
   encArg(plus) -> plus
   encArg(if) -> if
   encArg(third) -> third
   encArg(cons_function(x_1, x_2, x_3, x_4)) -> function(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_function(x_1, x_2, x_3, x_4) -> function(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_iszero -> iszero
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_p -> p
   encode_plus -> plus
   encode_if -> if
   encode_third -> third

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

function(p, s(s(x)), dummy, dummy2) ->^+ s(function(p, s(x), x, x))

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

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

The result substitution is [dummy / x, dummy2 / x].




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(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:

   function(iszero, 0, dummy, dummy2) -> true
   function(iszero, s(x), dummy, dummy2) -> false
   function(p, 0, dummy, dummy2) -> 0
   function(p, s(0), dummy, dummy2) -> 0
   function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x))
   function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y)
   function(if, true, x, y) -> y
   function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y))
   function(third, x, y, z) -> z

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

   encArg(iszero) -> iszero
   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(p) -> p
   encArg(plus) -> plus
   encArg(if) -> if
   encArg(third) -> third
   encArg(cons_function(x_1, x_2, x_3, x_4)) -> function(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_function(x_1, x_2, x_3, x_4) -> function(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_iszero -> iszero
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_p -> p
   encode_plus -> plus
   encode_if -> if
   encode_third -> third

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:

   function(iszero, 0, dummy, dummy2) -> true
   function(iszero, s(x), dummy, dummy2) -> false
   function(p, 0, dummy, dummy2) -> 0
   function(p, s(0), dummy, dummy2) -> 0
   function(p, s(s(x)), dummy, dummy2) -> s(function(p, s(x), x, x))
   function(plus, dummy, x, y) -> function(if, function(iszero, x, x, x), x, y)
   function(if, true, x, y) -> y
   function(if, false, x, y) -> function(plus, function(third, x, y, y), function(p, x, x, y), s(y))
   function(third, x, y, z) -> z

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

   encArg(iszero) -> iszero
   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(p) -> p
   encArg(plus) -> plus
   encArg(if) -> if
   encArg(third) -> third
   encArg(cons_function(x_1, x_2, x_3, x_4)) -> function(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_function(x_1, x_2, x_3, x_4) -> function(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4))
   encode_iszero -> iszero
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_p -> p
   encode_plus -> plus
   encode_if -> if
   encode_third -> third

Rewrite Strategy: INNERMOST