/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 164 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 348 ms]
(12) proven lower bound
(13) LowerBoundPropagationProof [FINISHED, 0 ms]
(14) BOUNDS(n^1, INF)


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

(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:

   *(x, 1) -> x
   *(1, y) -> y
   *(i(x), x) -> 1
   *(x, i(x)) -> 1
   *(x, *(y, z)) -> *(*(x, y), z)
   i(1) -> 1
   *(*(x, y), i(y)) -> x
   *(*(x, i(y)), y) -> x
   i(i(x)) -> x
   i(*(x, y)) -> *(i(y), i(x))
   k(x, 1) -> 1
   k(x, x) -> 1
   *(k(x, y), k(y, x)) -> 1
   *(*(i(x), k(y, z)), x) -> k(*(*(i(x), y), x), *(*(i(x), z), x))
   k(*(x, i(y)), *(y, i(x))) -> 1

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(1) -> 1
   encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encArg(cons_k(x_1, x_2)) -> k(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_i(x_1) -> i(encArg(x_1))
   encode_k(x_1, x_2) -> k(encArg(x_1), encArg(x_2))


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

(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:

   *(x, 1) -> x
   *(1, y) -> y
   *(i(x), x) -> 1
   *(x, i(x)) -> 1
   *(x, *(y, z)) -> *(*(x, y), z)
   i(1) -> 1
   *(*(x, y), i(y)) -> x
   *(*(x, i(y)), y) -> x
   i(i(x)) -> x
   i(*(x, y)) -> *(i(y), i(x))
   k(x, 1) -> 1
   k(x, x) -> 1
   *(k(x, y), k(y, x)) -> 1
   *(*(i(x), k(y, z)), x) -> k(*(*(i(x), y), x), *(*(i(x), z), x))
   k(*(x, i(y)), *(y, i(x))) -> 1

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

   encArg(1) -> 1
   encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encArg(cons_k(x_1, x_2)) -> k(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_i(x_1) -> i(encArg(x_1))
   encode_k(x_1, x_2) -> k(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(n^1, INF).


The TRS R consists of the following rules:

   *(x, 1) -> x
   *(1, y) -> y
   *(i(x), x) -> 1
   *(x, i(x)) -> 1
   *(x, *(y, z)) -> *(*(x, y), z)
   i(1) -> 1
   *(*(x, y), i(y)) -> x
   *(*(x, i(y)), y) -> x
   i(i(x)) -> x
   i(*(x, y)) -> *(i(y), i(x))
   k(x, 1) -> 1
   k(x, x) -> 1
   *(k(x, y), k(y, x)) -> 1
   *(*(i(x), k(y, z)), x) -> k(*(*(i(x), y), x), *(*(i(x), z), x))
   k(*(x, i(y)), *(y, i(x))) -> 1

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

   encArg(1) -> 1
   encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encArg(cons_k(x_1, x_2)) -> k(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_i(x_1) -> i(encArg(x_1))
   encode_k(x_1, x_2) -> k(encArg(x_1), encArg(x_2))

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

(5) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(6)
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:

   *'(x, 1') -> x
   *'(1', y) -> y
   *'(i(x), x) -> 1'
   *'(x, i(x)) -> 1'
   *'(x, *'(y, z)) -> *'(*'(x, y), z)
   i(1') -> 1'
   *'(*'(x, y), i(y)) -> x
   *'(*'(x, i(y)), y) -> x
   i(i(x)) -> x
   i(*'(x, y)) -> *'(i(y), i(x))
   k(x, 1') -> 1'
   k(x, x) -> 1'
   *'(k(x, y), k(y, x)) -> 1'
   *'(*'(i(x), k(y, z)), x) -> k(*'(*'(i(x), y), x), *'(*'(i(x), z), x))
   k(*'(x, i(y)), *'(y, i(x))) -> 1'

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

   encArg(1') -> 1'
   encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encArg(cons_k(x_1, x_2)) -> k(encArg(x_1), encArg(x_2))
   encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2))
   encode_1 -> 1'
   encode_i(x_1) -> i(encArg(x_1))
   encode_k(x_1, x_2) -> k(encArg(x_1), encArg(x_2))

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

(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(8)
Obligation:
Innermost TRS:
Rules:
*'(x, 1') -> x
*'(1', y) -> y
*'(i(x), x) -> 1'
*'(x, i(x)) -> 1'
*'(x, *'(y, z)) -> *'(*'(x, y), z)
i(1') -> 1'
*'(*'(x, y), i(y)) -> x
*'(*'(x, i(y)), y) -> x
i(i(x)) -> x
i(*'(x, y)) -> *'(i(y), i(x))
k(x, 1') -> 1'
k(x, x) -> 1'
*'(k(x, y), k(y, x)) -> 1'
*'(*'(i(x), k(y, z)), x) -> k(*'(*'(i(x), y), x), *'(*'(i(x), z), x))
k(*'(x, i(y)), *'(y, i(x))) -> 1'
encArg(1') -> 1'
encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2))
encArg(cons_i(x_1)) -> i(encArg(x_1))
encArg(cons_k(x_1, x_2)) -> k(encArg(x_1), encArg(x_2))
encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2))
encode_1 -> 1'
encode_i(x_1) -> i(encArg(x_1))
encode_k(x_1, x_2) -> k(encArg(x_1), encArg(x_2))

Types:
*' :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
1' :: 1':cons_*:cons_i:cons_k
i :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
k :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
encArg :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
cons_* :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
cons_i :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
cons_k :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
encode_* :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
encode_1 :: 1':cons_*:cons_i:cons_k
encode_i :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
encode_k :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
hole_1':cons_*:cons_i:cons_k1_0 :: 1':cons_*:cons_i:cons_k
gen_1':cons_*:cons_i:cons_k2_0 :: Nat -> 1':cons_*:cons_i:cons_k

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
*', i, encArg

They will be analysed ascendingly in the following order:
*' = i
*' < encArg
i < encArg

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

(10)
Obligation:
Innermost TRS:
Rules:
*'(x, 1') -> x
*'(1', y) -> y
*'(i(x), x) -> 1'
*'(x, i(x)) -> 1'
*'(x, *'(y, z)) -> *'(*'(x, y), z)
i(1') -> 1'
*'(*'(x, y), i(y)) -> x
*'(*'(x, i(y)), y) -> x
i(i(x)) -> x
i(*'(x, y)) -> *'(i(y), i(x))
k(x, 1') -> 1'
k(x, x) -> 1'
*'(k(x, y), k(y, x)) -> 1'
*'(*'(i(x), k(y, z)), x) -> k(*'(*'(i(x), y), x), *'(*'(i(x), z), x))
k(*'(x, i(y)), *'(y, i(x))) -> 1'
encArg(1') -> 1'
encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2))
encArg(cons_i(x_1)) -> i(encArg(x_1))
encArg(cons_k(x_1, x_2)) -> k(encArg(x_1), encArg(x_2))
encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2))
encode_1 -> 1'
encode_i(x_1) -> i(encArg(x_1))
encode_k(x_1, x_2) -> k(encArg(x_1), encArg(x_2))

Types:
*' :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
1' :: 1':cons_*:cons_i:cons_k
i :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
k :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
encArg :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
cons_* :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
cons_i :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
cons_k :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
encode_* :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
encode_1 :: 1':cons_*:cons_i:cons_k
encode_i :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
encode_k :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
hole_1':cons_*:cons_i:cons_k1_0 :: 1':cons_*:cons_i:cons_k
gen_1':cons_*:cons_i:cons_k2_0 :: Nat -> 1':cons_*:cons_i:cons_k


Generator Equations:
gen_1':cons_*:cons_i:cons_k2_0(0) <=> 1'
gen_1':cons_*:cons_i:cons_k2_0(+(x, 1)) <=> cons_*(1', gen_1':cons_*:cons_i:cons_k2_0(x))


The following defined symbols remain to be analysed:
i, *', encArg

They will be analysed ascendingly in the following order:
*' = i
*' < encArg
i < encArg

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_1':cons_*:cons_i:cons_k2_0(n728_0)) -> gen_1':cons_*:cons_i:cons_k2_0(0), rt in Omega(n728_0)

Induction Base:
encArg(gen_1':cons_*:cons_i:cons_k2_0(0)) ->_R^Omega(0)
1'

Induction Step:
encArg(gen_1':cons_*:cons_i:cons_k2_0(+(n728_0, 1))) ->_R^Omega(0)
*'(encArg(1'), encArg(gen_1':cons_*:cons_i:cons_k2_0(n728_0))) ->_R^Omega(0)
*'(1', encArg(gen_1':cons_*:cons_i:cons_k2_0(n728_0))) ->_IH
*'(1', gen_1':cons_*:cons_i:cons_k2_0(0)) ->_R^Omega(1)
1'

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(12)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
*'(x, 1') -> x
*'(1', y) -> y
*'(i(x), x) -> 1'
*'(x, i(x)) -> 1'
*'(x, *'(y, z)) -> *'(*'(x, y), z)
i(1') -> 1'
*'(*'(x, y), i(y)) -> x
*'(*'(x, i(y)), y) -> x
i(i(x)) -> x
i(*'(x, y)) -> *'(i(y), i(x))
k(x, 1') -> 1'
k(x, x) -> 1'
*'(k(x, y), k(y, x)) -> 1'
*'(*'(i(x), k(y, z)), x) -> k(*'(*'(i(x), y), x), *'(*'(i(x), z), x))
k(*'(x, i(y)), *'(y, i(x))) -> 1'
encArg(1') -> 1'
encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2))
encArg(cons_i(x_1)) -> i(encArg(x_1))
encArg(cons_k(x_1, x_2)) -> k(encArg(x_1), encArg(x_2))
encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2))
encode_1 -> 1'
encode_i(x_1) -> i(encArg(x_1))
encode_k(x_1, x_2) -> k(encArg(x_1), encArg(x_2))

Types:
*' :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
1' :: 1':cons_*:cons_i:cons_k
i :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
k :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
encArg :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
cons_* :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
cons_i :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
cons_k :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
encode_* :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
encode_1 :: 1':cons_*:cons_i:cons_k
encode_i :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
encode_k :: 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k -> 1':cons_*:cons_i:cons_k
hole_1':cons_*:cons_i:cons_k1_0 :: 1':cons_*:cons_i:cons_k
gen_1':cons_*:cons_i:cons_k2_0 :: Nat -> 1':cons_*:cons_i:cons_k


Generator Equations:
gen_1':cons_*:cons_i:cons_k2_0(0) <=> 1'
gen_1':cons_*:cons_i:cons_k2_0(+(x, 1)) <=> cons_*(1', gen_1':cons_*:cons_i:cons_k2_0(x))


The following defined symbols remain to be analysed:
encArg
----------------------------------------

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

(14)
BOUNDS(n^1, INF)