/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(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (full) 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), 138 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), 240 ms]
(12) proven lower bound
(13) LowerBoundPropagationProof [FINISHED, 0 ms]
(14) BOUNDS(n^1, INF)


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(0)
Obligation:
The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   s(a) -> a
   s(s(x)) -> x
   s(f(x, y)) -> f(s(y), s(x))
   s(g(x, y)) -> g(s(x), s(y))
   f(x, a) -> x
   f(a, y) -> y
   f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v))
   g(a, a) -> a

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(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(a) -> a
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_a -> a
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))


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

(2)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   s(a) -> a
   s(s(x)) -> x
   s(f(x, y)) -> f(s(y), s(x))
   s(g(x, y)) -> g(s(x), s(y))
   f(x, a) -> x
   f(a, y) -> y
   f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v))
   g(a, a) -> a

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

   encArg(a) -> a
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_a -> a
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))

Rewrite Strategy: FULL
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(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   s(a) -> a
   s(s(x)) -> x
   s(f(x, y)) -> f(s(y), s(x))
   s(g(x, y)) -> g(s(x), s(y))
   f(x, a) -> x
   f(a, y) -> y
   f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v))
   g(a, a) -> a

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

   encArg(a) -> a
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_a -> a
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))

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

(6)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   s(a) -> a
   s(s(x)) -> x
   s(f(x, y)) -> f(s(y), s(x))
   s(g(x, y)) -> g(s(x), s(y))
   f(x, a) -> x
   f(a, y) -> y
   f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v))
   g(a, a) -> a

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

   encArg(a) -> a
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_a -> a
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))

Rewrite Strategy: FULL
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(8)
Obligation:
TRS:
Rules:
s(a) -> a
s(s(x)) -> x
s(f(x, y)) -> f(s(y), s(x))
s(g(x, y)) -> g(s(x), s(y))
f(x, a) -> x
f(a, y) -> y
f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v))
g(a, a) -> a
encArg(a) -> a
encArg(cons_s(x_1)) -> s(encArg(x_1))
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encode_s(x_1) -> s(encArg(x_1))
encode_a -> a
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))

Types:
s :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
a :: a:cons_s:cons_f:cons_g
f :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
g :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
encArg :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
cons_s :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
cons_f :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
cons_g :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
encode_s :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
encode_a :: a:cons_s:cons_f:cons_g
encode_f :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
encode_g :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
hole_a:cons_s:cons_f:cons_g1_0 :: a:cons_s:cons_f:cons_g
gen_a:cons_s:cons_f:cons_g2_0 :: Nat -> a:cons_s:cons_f:cons_g

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(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
s, f, encArg

They will be analysed ascendingly in the following order:
f < s
s < encArg
f < encArg

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

(10)
Obligation:
TRS:
Rules:
s(a) -> a
s(s(x)) -> x
s(f(x, y)) -> f(s(y), s(x))
s(g(x, y)) -> g(s(x), s(y))
f(x, a) -> x
f(a, y) -> y
f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v))
g(a, a) -> a
encArg(a) -> a
encArg(cons_s(x_1)) -> s(encArg(x_1))
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encode_s(x_1) -> s(encArg(x_1))
encode_a -> a
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))

Types:
s :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
a :: a:cons_s:cons_f:cons_g
f :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
g :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
encArg :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
cons_s :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
cons_f :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
cons_g :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
encode_s :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
encode_a :: a:cons_s:cons_f:cons_g
encode_f :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
encode_g :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
hole_a:cons_s:cons_f:cons_g1_0 :: a:cons_s:cons_f:cons_g
gen_a:cons_s:cons_f:cons_g2_0 :: Nat -> a:cons_s:cons_f:cons_g


Generator Equations:
gen_a:cons_s:cons_f:cons_g2_0(0) <=> a
gen_a:cons_s:cons_f:cons_g2_0(+(x, 1)) <=> cons_s(gen_a:cons_s:cons_f:cons_g2_0(x))


The following defined symbols remain to be analysed:
f, s, encArg

They will be analysed ascendingly in the following order:
f < s
s < encArg
f < encArg

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(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_a:cons_s:cons_f:cons_g2_0(n66_0)) -> gen_a:cons_s:cons_f:cons_g2_0(0), rt in Omega(n66_0)

Induction Base:
encArg(gen_a:cons_s:cons_f:cons_g2_0(0)) ->_R^Omega(0)
a

Induction Step:
encArg(gen_a:cons_s:cons_f:cons_g2_0(+(n66_0, 1))) ->_R^Omega(0)
s(encArg(gen_a:cons_s:cons_f:cons_g2_0(n66_0))) ->_IH
s(gen_a:cons_s:cons_f:cons_g2_0(0)) ->_R^Omega(1)
a

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:

TRS:
Rules:
s(a) -> a
s(s(x)) -> x
s(f(x, y)) -> f(s(y), s(x))
s(g(x, y)) -> g(s(x), s(y))
f(x, a) -> x
f(a, y) -> y
f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v))
g(a, a) -> a
encArg(a) -> a
encArg(cons_s(x_1)) -> s(encArg(x_1))
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encode_s(x_1) -> s(encArg(x_1))
encode_a -> a
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))

Types:
s :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
a :: a:cons_s:cons_f:cons_g
f :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
g :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
encArg :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
cons_s :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
cons_f :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
cons_g :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
encode_s :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
encode_a :: a:cons_s:cons_f:cons_g
encode_f :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
encode_g :: a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g -> a:cons_s:cons_f:cons_g
hole_a:cons_s:cons_f:cons_g1_0 :: a:cons_s:cons_f:cons_g
gen_a:cons_s:cons_f:cons_g2_0 :: Nat -> a:cons_s:cons_f:cons_g


Generator Equations:
gen_a:cons_s:cons_f:cons_g2_0(0) <=> a
gen_a:cons_s:cons_f:cons_g2_0(+(x, 1)) <=> cons_s(gen_a:cons_s:cons_f:cons_g2_0(x))


The following defined symbols remain to be analysed:
encArg
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(13) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(14)
BOUNDS(n^1, INF)