/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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), 83 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), 518 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), 155 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 145 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 47 ms]
        (22) BOUNDS(1, INF)


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

(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:

   a(b(x1)) -> x1
   a(c(x1)) -> c(c(b(b(x1))))
   b(c(x1)) -> a(a(x1))

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(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))


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

(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:

   a(b(x1)) -> x1
   a(c(x1)) -> c(c(b(b(x1))))
   b(c(x1)) -> a(a(x1))

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

Rewrite Strategy: FULL
----------------------------------------

(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:

   a(b(x1)) -> x1
   a(c(x1)) -> c(c(b(b(x1))))
   b(c(x1)) -> a(a(x1))

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

Rewrite Strategy: FULL
----------------------------------------

(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:

   a(b(x1)) -> x1
   a(c(x1)) -> c(c(b(b(x1))))
   b(c(x1)) -> a(a(x1))

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

   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

Rewrite Strategy: FULL
----------------------------------------

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

(8)
Obligation:
TRS:
Rules:
a(b(x1)) -> x1
a(c(x1)) -> c(c(b(b(x1))))
b(c(x1)) -> a(a(x1))
encArg(c(x_1)) -> c(encArg(x_1))
encArg(cons_a(x_1)) -> a(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))

Types:
a :: c:cons_a:cons_b -> c:cons_a:cons_b
b :: c:cons_a:cons_b -> c:cons_a:cons_b
c :: c:cons_a:cons_b -> c:cons_a:cons_b
encArg :: c:cons_a:cons_b -> c:cons_a:cons_b
cons_a :: c:cons_a:cons_b -> c:cons_a:cons_b
cons_b :: c:cons_a:cons_b -> c:cons_a:cons_b
encode_a :: c:cons_a:cons_b -> c:cons_a:cons_b
encode_b :: c:cons_a:cons_b -> c:cons_a:cons_b
encode_c :: c:cons_a:cons_b -> c:cons_a:cons_b
hole_c:cons_a:cons_b1_0 :: c:cons_a:cons_b
gen_c:cons_a:cons_b2_0 :: Nat -> c:cons_a:cons_b

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

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

They will be analysed ascendingly in the following order:
a = b
a < encArg
b < encArg

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

(10)
Obligation:
TRS:
Rules:
a(b(x1)) -> x1
a(c(x1)) -> c(c(b(b(x1))))
b(c(x1)) -> a(a(x1))
encArg(c(x_1)) -> c(encArg(x_1))
encArg(cons_a(x_1)) -> a(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))

Types:
a :: c:cons_a:cons_b -> c:cons_a:cons_b
b :: c:cons_a:cons_b -> c:cons_a:cons_b
c :: c:cons_a:cons_b -> c:cons_a:cons_b
encArg :: c:cons_a:cons_b -> c:cons_a:cons_b
cons_a :: c:cons_a:cons_b -> c:cons_a:cons_b
cons_b :: c:cons_a:cons_b -> c:cons_a:cons_b
encode_a :: c:cons_a:cons_b -> c:cons_a:cons_b
encode_b :: c:cons_a:cons_b -> c:cons_a:cons_b
encode_c :: c:cons_a:cons_b -> c:cons_a:cons_b
hole_c:cons_a:cons_b1_0 :: c:cons_a:cons_b
gen_c:cons_a:cons_b2_0 :: Nat -> c:cons_a:cons_b


Generator Equations:
gen_c:cons_a:cons_b2_0(0) <=> hole_c:cons_a:cons_b1_0
gen_c:cons_a:cons_b2_0(+(x, 1)) <=> c(gen_c:cons_a:cons_b2_0(x))


The following defined symbols remain to be analysed:
b, a, encArg

They will be analysed ascendingly in the following order:
a = b
a < encArg
b < encArg

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
b(gen_c:cons_a:cons_b2_0(+(1, *(2, n4_0)))) -> *3_0, rt in Omega(n4_0)

Induction Base:
b(gen_c:cons_a:cons_b2_0(+(1, *(2, 0))))

Induction Step:
b(gen_c:cons_a:cons_b2_0(+(1, *(2, +(n4_0, 1))))) ->_R^Omega(1)
a(a(gen_c:cons_a:cons_b2_0(+(2, *(2, n4_0))))) ->_R^Omega(1)
a(c(c(b(b(gen_c:cons_a:cons_b2_0(+(1, *(2, n4_0)))))))) ->_IH
a(c(c(b(*3_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:

TRS:
Rules:
a(b(x1)) -> x1
a(c(x1)) -> c(c(b(b(x1))))
b(c(x1)) -> a(a(x1))
encArg(c(x_1)) -> c(encArg(x_1))
encArg(cons_a(x_1)) -> a(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))

Types:
a :: c:cons_a:cons_b -> c:cons_a:cons_b
b :: c:cons_a:cons_b -> c:cons_a:cons_b
c :: c:cons_a:cons_b -> c:cons_a:cons_b
encArg :: c:cons_a:cons_b -> c:cons_a:cons_b
cons_a :: c:cons_a:cons_b -> c:cons_a:cons_b
cons_b :: c:cons_a:cons_b -> c:cons_a:cons_b
encode_a :: c:cons_a:cons_b -> c:cons_a:cons_b
encode_b :: c:cons_a:cons_b -> c:cons_a:cons_b
encode_c :: c:cons_a:cons_b -> c:cons_a:cons_b
hole_c:cons_a:cons_b1_0 :: c:cons_a:cons_b
gen_c:cons_a:cons_b2_0 :: Nat -> c:cons_a:cons_b


Generator Equations:
gen_c:cons_a:cons_b2_0(0) <=> hole_c:cons_a:cons_b1_0
gen_c:cons_a:cons_b2_0(+(x, 1)) <=> c(gen_c:cons_a:cons_b2_0(x))


The following defined symbols remain to be analysed:
b, a, encArg

They will be analysed ascendingly in the following order:
a = b
a < encArg
b < encArg

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
TRS:
Rules:
a(b(x1)) -> x1
a(c(x1)) -> c(c(b(b(x1))))
b(c(x1)) -> a(a(x1))
encArg(c(x_1)) -> c(encArg(x_1))
encArg(cons_a(x_1)) -> a(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))

Types:
a :: c:cons_a:cons_b -> c:cons_a:cons_b
b :: c:cons_a:cons_b -> c:cons_a:cons_b
c :: c:cons_a:cons_b -> c:cons_a:cons_b
encArg :: c:cons_a:cons_b -> c:cons_a:cons_b
cons_a :: c:cons_a:cons_b -> c:cons_a:cons_b
cons_b :: c:cons_a:cons_b -> c:cons_a:cons_b
encode_a :: c:cons_a:cons_b -> c:cons_a:cons_b
encode_b :: c:cons_a:cons_b -> c:cons_a:cons_b
encode_c :: c:cons_a:cons_b -> c:cons_a:cons_b
hole_c:cons_a:cons_b1_0 :: c:cons_a:cons_b
gen_c:cons_a:cons_b2_0 :: Nat -> c:cons_a:cons_b


Lemmas:
b(gen_c:cons_a:cons_b2_0(+(1, *(2, n4_0)))) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_c:cons_a:cons_b2_0(0) <=> hole_c:cons_a:cons_b1_0
gen_c:cons_a:cons_b2_0(+(x, 1)) <=> c(gen_c:cons_a:cons_b2_0(x))


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

They will be analysed ascendingly in the following order:
a = b
a < encArg
b < encArg

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
a(gen_c:cons_a:cons_b2_0(+(1, *(2, n295_0)))) -> *3_0, rt in Omega(n295_0)

Induction Base:
a(gen_c:cons_a:cons_b2_0(+(1, *(2, 0))))

Induction Step:
a(gen_c:cons_a:cons_b2_0(+(1, *(2, +(n295_0, 1))))) ->_R^Omega(1)
c(c(b(b(gen_c:cons_a:cons_b2_0(+(2, *(2, n295_0))))))) ->_R^Omega(1)
c(c(b(a(a(gen_c:cons_a:cons_b2_0(+(1, *(2, n295_0)))))))) ->_IH
c(c(b(a(*3_0))))

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

(18)
Obligation:
TRS:
Rules:
a(b(x1)) -> x1
a(c(x1)) -> c(c(b(b(x1))))
b(c(x1)) -> a(a(x1))
encArg(c(x_1)) -> c(encArg(x_1))
encArg(cons_a(x_1)) -> a(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))

Types:
a :: c:cons_a:cons_b -> c:cons_a:cons_b
b :: c:cons_a:cons_b -> c:cons_a:cons_b
c :: c:cons_a:cons_b -> c:cons_a:cons_b
encArg :: c:cons_a:cons_b -> c:cons_a:cons_b
cons_a :: c:cons_a:cons_b -> c:cons_a:cons_b
cons_b :: c:cons_a:cons_b -> c:cons_a:cons_b
encode_a :: c:cons_a:cons_b -> c:cons_a:cons_b
encode_b :: c:cons_a:cons_b -> c:cons_a:cons_b
encode_c :: c:cons_a:cons_b -> c:cons_a:cons_b
hole_c:cons_a:cons_b1_0 :: c:cons_a:cons_b
gen_c:cons_a:cons_b2_0 :: Nat -> c:cons_a:cons_b


Lemmas:
b(gen_c:cons_a:cons_b2_0(+(1, *(2, n4_0)))) -> *3_0, rt in Omega(n4_0)
a(gen_c:cons_a:cons_b2_0(+(1, *(2, n295_0)))) -> *3_0, rt in Omega(n295_0)


Generator Equations:
gen_c:cons_a:cons_b2_0(0) <=> hole_c:cons_a:cons_b1_0
gen_c:cons_a:cons_b2_0(+(x, 1)) <=> c(gen_c:cons_a:cons_b2_0(x))


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

They will be analysed ascendingly in the following order:
a = b
a < encArg
b < encArg

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
b(gen_c:cons_a:cons_b2_0(+(1, *(2, n806_0)))) -> *3_0, rt in Omega(n806_0)

Induction Base:
b(gen_c:cons_a:cons_b2_0(+(1, *(2, 0))))

Induction Step:
b(gen_c:cons_a:cons_b2_0(+(1, *(2, +(n806_0, 1))))) ->_R^Omega(1)
a(a(gen_c:cons_a:cons_b2_0(+(2, *(2, n806_0))))) ->_R^Omega(1)
a(c(c(b(b(gen_c:cons_a:cons_b2_0(+(1, *(2, n806_0)))))))) ->_IH
a(c(c(b(*3_0))))

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

(20)
Obligation:
TRS:
Rules:
a(b(x1)) -> x1
a(c(x1)) -> c(c(b(b(x1))))
b(c(x1)) -> a(a(x1))
encArg(c(x_1)) -> c(encArg(x_1))
encArg(cons_a(x_1)) -> a(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))

Types:
a :: c:cons_a:cons_b -> c:cons_a:cons_b
b :: c:cons_a:cons_b -> c:cons_a:cons_b
c :: c:cons_a:cons_b -> c:cons_a:cons_b
encArg :: c:cons_a:cons_b -> c:cons_a:cons_b
cons_a :: c:cons_a:cons_b -> c:cons_a:cons_b
cons_b :: c:cons_a:cons_b -> c:cons_a:cons_b
encode_a :: c:cons_a:cons_b -> c:cons_a:cons_b
encode_b :: c:cons_a:cons_b -> c:cons_a:cons_b
encode_c :: c:cons_a:cons_b -> c:cons_a:cons_b
hole_c:cons_a:cons_b1_0 :: c:cons_a:cons_b
gen_c:cons_a:cons_b2_0 :: Nat -> c:cons_a:cons_b


Lemmas:
b(gen_c:cons_a:cons_b2_0(+(1, *(2, n806_0)))) -> *3_0, rt in Omega(n806_0)
a(gen_c:cons_a:cons_b2_0(+(1, *(2, n295_0)))) -> *3_0, rt in Omega(n295_0)


Generator Equations:
gen_c:cons_a:cons_b2_0(0) <=> hole_c:cons_a:cons_b1_0
gen_c:cons_a:cons_b2_0(+(x, 1)) <=> c(gen_c:cons_a:cons_b2_0(x))


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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_c:cons_a:cons_b2_0(+(1, n1316_0))) -> *3_0, rt in Omega(0)

Induction Base:
encArg(gen_c:cons_a:cons_b2_0(+(1, 0)))

Induction Step:
encArg(gen_c:cons_a:cons_b2_0(+(1, +(n1316_0, 1)))) ->_R^Omega(0)
c(encArg(gen_c:cons_a:cons_b2_0(+(1, n1316_0)))) ->_IH
c(*3_0)

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

(22)
BOUNDS(1, INF)