/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(EXP, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 310 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) DecreasingLoopProof [FINISHED, 0 ms]
(8) BOUNDS(EXP, INF)


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(EXP, 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)
   eq(0, 0) -> true
   eq(0, s(y)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   minsort(nil) -> nil
   minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
   min(x, nil) -> x
   min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
   del(x, nil) -> nil
   del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, 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(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_eq(x_1, x_2)) -> eq(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_minsort(x_1)) -> minsort(encArg(x_1))
   encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2))
   encArg(cons_del(x_1, x_2)) -> del(encArg(x_1), encArg(x_2))
   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_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_minsort(x_1) -> minsort(encArg(x_1))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2))
   encode_del(x_1, x_2) -> del(encArg(x_1), encArg(x_2))


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

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(EXP, 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)
   eq(0, 0) -> true
   eq(0, s(y)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   minsort(nil) -> nil
   minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
   min(x, nil) -> x
   min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
   del(x, nil) -> nil
   del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, 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(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_eq(x_1, x_2)) -> eq(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_minsort(x_1)) -> minsort(encArg(x_1))
   encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2))
   encArg(cons_del(x_1, x_2)) -> del(encArg(x_1), encArg(x_2))
   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_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_minsort(x_1) -> minsort(encArg(x_1))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2))
   encode_del(x_1, x_2) -> del(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST
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(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(EXP, 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)
   eq(0, 0) -> true
   eq(0, s(y)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   minsort(nil) -> nil
   minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
   min(x, nil) -> x
   min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
   del(x, nil) -> nil
   del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, 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(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_eq(x_1, x_2)) -> eq(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_minsort(x_1)) -> minsort(encArg(x_1))
   encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2))
   encArg(cons_del(x_1, x_2)) -> del(encArg(x_1), encArg(x_2))
   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_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_minsort(x_1) -> minsort(encArg(x_1))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2))
   encode_del(x_1, x_2) -> del(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST
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(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(EXP, 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)
   eq(0, 0) -> true
   eq(0, s(y)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   minsort(nil) -> nil
   minsort(cons(x, y)) -> cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
   min(x, nil) -> x
   min(x, cons(y, z)) -> if(le(x, y), min(x, z), min(y, z))
   del(x, nil) -> nil
   del(x, cons(y, z)) -> if(eq(x, y), z, cons(y, del(x, 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(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2))
   encArg(cons_eq(x_1, x_2)) -> eq(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_minsort(x_1)) -> minsort(encArg(x_1))
   encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2))
   encArg(cons_del(x_1, x_2)) -> del(encArg(x_1), encArg(x_2))
   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_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_minsort(x_1) -> minsort(encArg(x_1))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2))
   encode_del(x_1, x_2) -> del(encArg(x_1), encArg(x_2))

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

(7) DecreasingLoopProof (FINISHED)
The following loop(s) give(s) rise to the lower bound EXP:

The rewrite sequence

min(x, cons(y, z)) ->^+ if(le(x, y), min(x, z), min(y, z))

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

The pumping substitution is [z / cons(y, z)].

The result substitution is [ ].



The rewrite sequence

min(x, cons(y, z)) ->^+ if(le(x, y), min(x, z), min(y, z))

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

The pumping substitution is [z / cons(y, z)].

The result substitution is [x / y].




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(8)
BOUNDS(EXP, INF)