/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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), 151 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), 263 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 984 ms]
        (18) BOUNDS(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:

   f(nil) -> nil
   f(.(nil, y)) -> .(nil, f(y))
   f(.(.(x, y), z)) -> f(.(x, .(y, z)))
   g(nil) -> nil
   g(.(x, nil)) -> .(g(x), nil)
   g(.(x, .(y, z))) -> g(.(.(x, y), 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(nil) -> nil
   encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_nil -> nil
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))


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

(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:

   f(nil) -> nil
   f(.(nil, y)) -> .(nil, f(y))
   f(.(.(x, y), z)) -> f(.(x, .(y, z)))
   g(nil) -> nil
   g(.(x, nil)) -> .(g(x), nil)
   g(.(x, .(y, z))) -> g(.(.(x, y), z))

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

   encArg(nil) -> nil
   encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_nil -> nil
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))

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:

   f(nil) -> nil
   f(.(nil, y)) -> .(nil, f(y))
   f(.(.(x, y), z)) -> f(.(x, .(y, z)))
   g(nil) -> nil
   g(.(x, nil)) -> .(g(x), nil)
   g(.(x, .(y, z))) -> g(.(.(x, y), z))

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

   encArg(nil) -> nil
   encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_nil -> nil
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))

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:

   f(nil) -> nil
   f(.(nil, y)) -> .(nil, f(y))
   f(.(.(x, y), z)) -> f(.(x, .(y, z)))
   g(nil) -> nil
   g(.(x, nil)) -> .(g(x), nil)
   g(.(x, .(y, z))) -> g(.(.(x, y), z))

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

   encArg(nil) -> nil
   encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_nil -> nil
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
f(nil) -> nil
f(.(nil, y)) -> .(nil, f(y))
f(.(.(x, y), z)) -> f(.(x, .(y, z)))
g(nil) -> nil
g(.(x, nil)) -> .(g(x), nil)
g(.(x, .(y, z))) -> g(.(.(x, y), z))
encArg(nil) -> nil
encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_g(x_1)) -> g(encArg(x_1))
encode_f(x_1) -> f(encArg(x_1))
encode_nil -> nil
encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
encode_g(x_1) -> g(encArg(x_1))

Types:
f :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
nil :: nil:.:cons_f:cons_g
. :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
g :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
encArg :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
cons_f :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
cons_g :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
encode_f :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
encode_nil :: nil:.:cons_f:cons_g
encode_. :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
encode_g :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
hole_nil:.:cons_f:cons_g1_0 :: nil:.:cons_f:cons_g
gen_nil:.:cons_f:cons_g2_0 :: Nat -> nil:.:cons_f:cons_g

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

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

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

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

(10)
Obligation:
Innermost TRS:
Rules:
f(nil) -> nil
f(.(nil, y)) -> .(nil, f(y))
f(.(.(x, y), z)) -> f(.(x, .(y, z)))
g(nil) -> nil
g(.(x, nil)) -> .(g(x), nil)
g(.(x, .(y, z))) -> g(.(.(x, y), z))
encArg(nil) -> nil
encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_g(x_1)) -> g(encArg(x_1))
encode_f(x_1) -> f(encArg(x_1))
encode_nil -> nil
encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
encode_g(x_1) -> g(encArg(x_1))

Types:
f :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
nil :: nil:.:cons_f:cons_g
. :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
g :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
encArg :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
cons_f :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
cons_g :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
encode_f :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
encode_nil :: nil:.:cons_f:cons_g
encode_. :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
encode_g :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
hole_nil:.:cons_f:cons_g1_0 :: nil:.:cons_f:cons_g
gen_nil:.:cons_f:cons_g2_0 :: Nat -> nil:.:cons_f:cons_g


Generator Equations:
gen_nil:.:cons_f:cons_g2_0(0) <=> nil
gen_nil:.:cons_f:cons_g2_0(+(x, 1)) <=> .(nil, gen_nil:.:cons_f:cons_g2_0(x))


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

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

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_nil:.:cons_f:cons_g2_0(n4_0)) -> gen_nil:.:cons_f:cons_g2_0(n4_0), rt in Omega(1 + n4_0)

Induction Base:
f(gen_nil:.:cons_f:cons_g2_0(0)) ->_R^Omega(1)
nil

Induction Step:
f(gen_nil:.:cons_f:cons_g2_0(+(n4_0, 1))) ->_R^Omega(1)
.(nil, f(gen_nil:.:cons_f:cons_g2_0(n4_0))) ->_IH
.(nil, gen_nil:.:cons_f:cons_g2_0(c5_0))

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

(12)
Complex Obligation (BEST)

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

(13)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
f(nil) -> nil
f(.(nil, y)) -> .(nil, f(y))
f(.(.(x, y), z)) -> f(.(x, .(y, z)))
g(nil) -> nil
g(.(x, nil)) -> .(g(x), nil)
g(.(x, .(y, z))) -> g(.(.(x, y), z))
encArg(nil) -> nil
encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_g(x_1)) -> g(encArg(x_1))
encode_f(x_1) -> f(encArg(x_1))
encode_nil -> nil
encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
encode_g(x_1) -> g(encArg(x_1))

Types:
f :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
nil :: nil:.:cons_f:cons_g
. :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
g :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
encArg :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
cons_f :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
cons_g :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
encode_f :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
encode_nil :: nil:.:cons_f:cons_g
encode_. :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
encode_g :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
hole_nil:.:cons_f:cons_g1_0 :: nil:.:cons_f:cons_g
gen_nil:.:cons_f:cons_g2_0 :: Nat -> nil:.:cons_f:cons_g


Generator Equations:
gen_nil:.:cons_f:cons_g2_0(0) <=> nil
gen_nil:.:cons_f:cons_g2_0(+(x, 1)) <=> .(nil, gen_nil:.:cons_f:cons_g2_0(x))


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

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

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
f(nil) -> nil
f(.(nil, y)) -> .(nil, f(y))
f(.(.(x, y), z)) -> f(.(x, .(y, z)))
g(nil) -> nil
g(.(x, nil)) -> .(g(x), nil)
g(.(x, .(y, z))) -> g(.(.(x, y), z))
encArg(nil) -> nil
encArg(.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_g(x_1)) -> g(encArg(x_1))
encode_f(x_1) -> f(encArg(x_1))
encode_nil -> nil
encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
encode_g(x_1) -> g(encArg(x_1))

Types:
f :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
nil :: nil:.:cons_f:cons_g
. :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
g :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
encArg :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
cons_f :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
cons_g :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
encode_f :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
encode_nil :: nil:.:cons_f:cons_g
encode_. :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
encode_g :: nil:.:cons_f:cons_g -> nil:.:cons_f:cons_g
hole_nil:.:cons_f:cons_g1_0 :: nil:.:cons_f:cons_g
gen_nil:.:cons_f:cons_g2_0 :: Nat -> nil:.:cons_f:cons_g


Lemmas:
f(gen_nil:.:cons_f:cons_g2_0(n4_0)) -> gen_nil:.:cons_f:cons_g2_0(n4_0), rt in Omega(1 + n4_0)


Generator Equations:
gen_nil:.:cons_f:cons_g2_0(0) <=> nil
gen_nil:.:cons_f:cons_g2_0(+(x, 1)) <=> .(nil, gen_nil:.:cons_f:cons_g2_0(x))


The following defined symbols remain to be analysed:
g, encArg

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

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_nil:.:cons_f:cons_g2_0(n11152_0)) -> gen_nil:.:cons_f:cons_g2_0(n11152_0), rt in Omega(0)

Induction Base:
encArg(gen_nil:.:cons_f:cons_g2_0(0)) ->_R^Omega(0)
nil

Induction Step:
encArg(gen_nil:.:cons_f:cons_g2_0(+(n11152_0, 1))) ->_R^Omega(0)
.(encArg(nil), encArg(gen_nil:.:cons_f:cons_g2_0(n11152_0))) ->_R^Omega(0)
.(nil, encArg(gen_nil:.:cons_f:cons_g2_0(n11152_0))) ->_IH
.(nil, gen_nil:.:cons_f:cons_g2_0(c11153_0))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(18)
BOUNDS(1, INF)