/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(NON_POLY, ?)
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(INF, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 341 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
        (13) InfiniteLowerBoundProof [FINISHED, 0 ms]
        (14) BOUNDS(INF, INF)


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

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


The TRS R consists of the following rules:

   fact(X) -> if(zero(X), s(0), prod(X, fact(p(X))))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   prod(0, X) -> 0
   prod(s(X), Y) -> add(Y, prod(X, Y))
   if(true, X, Y) -> X
   if(false, X, Y) -> Y
   zero(0) -> true
   zero(s(X)) -> false
   p(s(X)) -> 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(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_fact(x_1)) -> fact(encArg(x_1))
   encArg(cons_add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2))
   encArg(cons_prod(x_1, x_2)) -> prod(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_zero(x_1)) -> zero(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_fact(x_1) -> fact(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_zero(x_1) -> zero(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_0 -> 0
   encode_prod(x_1, x_2) -> prod(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(encArg(x_1))
   encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false


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

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF).


The TRS R consists of the following rules:

   fact(X) -> if(zero(X), s(0), prod(X, fact(p(X))))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   prod(0, X) -> 0
   prod(s(X), Y) -> add(Y, prod(X, Y))
   if(true, X, Y) -> X
   if(false, X, Y) -> Y
   zero(0) -> true
   zero(s(X)) -> false
   p(s(X)) -> X

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_fact(x_1)) -> fact(encArg(x_1))
   encArg(cons_add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2))
   encArg(cons_prod(x_1, x_2)) -> prod(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_zero(x_1)) -> zero(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_fact(x_1) -> fact(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_zero(x_1) -> zero(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_0 -> 0
   encode_prod(x_1, x_2) -> prod(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(encArg(x_1))
   encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false

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(INF, INF).


The TRS R consists of the following rules:

   fact(X) -> if(zero(X), s(0), prod(X, fact(p(X))))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   prod(0, X) -> 0
   prod(s(X), Y) -> add(Y, prod(X, Y))
   if(true, X, Y) -> X
   if(false, X, Y) -> Y
   zero(0) -> true
   zero(s(X)) -> false
   p(s(X)) -> X

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_fact(x_1)) -> fact(encArg(x_1))
   encArg(cons_add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2))
   encArg(cons_prod(x_1, x_2)) -> prod(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_zero(x_1)) -> zero(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_fact(x_1) -> fact(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_zero(x_1) -> zero(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_0 -> 0
   encode_prod(x_1, x_2) -> prod(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(encArg(x_1))
   encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false

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(INF, INF).


The TRS R consists of the following rules:

   fact(X) -> if(zero(X), s(0), prod(X, fact(p(X))))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   prod(0, X) -> 0
   prod(s(X), Y) -> add(Y, prod(X, Y))
   if(true, X, Y) -> X
   if(false, X, Y) -> Y
   zero(0) -> true
   zero(s(X)) -> false
   p(s(X)) -> X

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_fact(x_1)) -> fact(encArg(x_1))
   encArg(cons_add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2))
   encArg(cons_prod(x_1, x_2)) -> prod(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_zero(x_1)) -> zero(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_fact(x_1) -> fact(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_zero(x_1) -> zero(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_0 -> 0
   encode_prod(x_1, x_2) -> prod(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(encArg(x_1))
   encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false

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

add(s(X), Y) ->^+ s(add(X, Y))

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 [ ].




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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(INF, INF).


The TRS R consists of the following rules:

   fact(X) -> if(zero(X), s(0), prod(X, fact(p(X))))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   prod(0, X) -> 0
   prod(s(X), Y) -> add(Y, prod(X, Y))
   if(true, X, Y) -> X
   if(false, X, Y) -> Y
   zero(0) -> true
   zero(s(X)) -> false
   p(s(X)) -> X

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_fact(x_1)) -> fact(encArg(x_1))
   encArg(cons_add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2))
   encArg(cons_prod(x_1, x_2)) -> prod(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_zero(x_1)) -> zero(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_fact(x_1) -> fact(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_zero(x_1) -> zero(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_0 -> 0
   encode_prod(x_1, x_2) -> prod(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(encArg(x_1))
   encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false

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(INF, INF).


The TRS R consists of the following rules:

   fact(X) -> if(zero(X), s(0), prod(X, fact(p(X))))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   prod(0, X) -> 0
   prod(s(X), Y) -> add(Y, prod(X, Y))
   if(true, X, Y) -> X
   if(false, X, Y) -> Y
   zero(0) -> true
   zero(s(X)) -> false
   p(s(X)) -> X

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_fact(x_1)) -> fact(encArg(x_1))
   encArg(cons_add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2))
   encArg(cons_prod(x_1, x_2)) -> prod(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_zero(x_1)) -> zero(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_fact(x_1) -> fact(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_zero(x_1) -> zero(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_0 -> 0
   encode_prod(x_1, x_2) -> prod(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(encArg(x_1))
   encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false

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

(13) InfiniteLowerBoundProof (FINISHED)
The following loop proves infinite runtime complexity:

The rewrite sequence

fact(X) ->^+ if(zero(X), s(0), prod(X, fact(p(X))))

gives rise to a decreasing loop by considering the right hand sides subterm at position [2,1].

The pumping substitution is [ ].

The result substitution is [X / p(X)].




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(14)
BOUNDS(INF, INF)