/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 155 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), 282 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), 805 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
        (20) 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:

   sum(0) -> 0
   sum(s(x)) -> +(sum(x), s(x))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))

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(s(x_1)) -> s(encArg(x_1))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_sum(x_1) -> sum(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_+(x_1, x_2) -> +(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:

   sum(0) -> 0
   sum(s(x)) -> +(sum(x), s(x))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_sum(x_1) -> sum(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))

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:

   sum(0) -> 0
   sum(s(x)) -> +(sum(x), s(x))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_sum(x_1) -> sum(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_+(x_1, x_2) -> +(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:

   sum(0') -> 0'
   sum(s(x)) -> +'(sum(x), s(x))
   +'(x, 0') -> x
   +'(x, s(y)) -> s(+'(x, y))

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

   encArg(0') -> 0'
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
   encode_sum(x_1) -> sum(encArg(x_1))
   encode_0 -> 0'
   encode_s(x_1) -> s(encArg(x_1))
   encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x), s(x))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_sum(x_1)) -> sum(encArg(x_1))
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encode_sum(x_1) -> sum(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))

Types:
sum :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
0' :: 0':s:cons_sum:cons_+
s :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
+' :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
encArg :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
cons_sum :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
cons_+ :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
encode_sum :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
encode_0 :: 0':s:cons_sum:cons_+
encode_s :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
encode_+ :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
hole_0':s:cons_sum:cons_+1_3 :: 0':s:cons_sum:cons_+
gen_0':s:cons_sum:cons_+2_3 :: Nat -> 0':s:cons_sum:cons_+

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

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

They will be analysed ascendingly in the following order:
+' < sum
sum < encArg
+' < encArg

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

(10)
Obligation:
Innermost TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x), s(x))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_sum(x_1)) -> sum(encArg(x_1))
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encode_sum(x_1) -> sum(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))

Types:
sum :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
0' :: 0':s:cons_sum:cons_+
s :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
+' :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
encArg :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
cons_sum :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
cons_+ :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
encode_sum :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
encode_0 :: 0':s:cons_sum:cons_+
encode_s :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
encode_+ :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
hole_0':s:cons_sum:cons_+1_3 :: 0':s:cons_sum:cons_+
gen_0':s:cons_sum:cons_+2_3 :: Nat -> 0':s:cons_sum:cons_+


Generator Equations:
gen_0':s:cons_sum:cons_+2_3(0) <=> 0'
gen_0':s:cons_sum:cons_+2_3(+(x, 1)) <=> s(gen_0':s:cons_sum:cons_+2_3(x))


The following defined symbols remain to be analysed:
+', sum, encArg

They will be analysed ascendingly in the following order:
+' < sum
sum < encArg
+' < encArg

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_0':s:cons_sum:cons_+2_3(a), gen_0':s:cons_sum:cons_+2_3(n4_3)) -> gen_0':s:cons_sum:cons_+2_3(+(n4_3, a)), rt in Omega(1 + n4_3)

Induction Base:
+'(gen_0':s:cons_sum:cons_+2_3(a), gen_0':s:cons_sum:cons_+2_3(0)) ->_R^Omega(1)
gen_0':s:cons_sum:cons_+2_3(a)

Induction Step:
+'(gen_0':s:cons_sum:cons_+2_3(a), gen_0':s:cons_sum:cons_+2_3(+(n4_3, 1))) ->_R^Omega(1)
s(+'(gen_0':s:cons_sum:cons_+2_3(a), gen_0':s:cons_sum:cons_+2_3(n4_3))) ->_IH
s(gen_0':s:cons_sum:cons_+2_3(+(a, c5_3)))

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:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x), s(x))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_sum(x_1)) -> sum(encArg(x_1))
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encode_sum(x_1) -> sum(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))

Types:
sum :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
0' :: 0':s:cons_sum:cons_+
s :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
+' :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
encArg :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
cons_sum :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
cons_+ :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
encode_sum :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
encode_0 :: 0':s:cons_sum:cons_+
encode_s :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
encode_+ :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
hole_0':s:cons_sum:cons_+1_3 :: 0':s:cons_sum:cons_+
gen_0':s:cons_sum:cons_+2_3 :: Nat -> 0':s:cons_sum:cons_+


Generator Equations:
gen_0':s:cons_sum:cons_+2_3(0) <=> 0'
gen_0':s:cons_sum:cons_+2_3(+(x, 1)) <=> s(gen_0':s:cons_sum:cons_+2_3(x))


The following defined symbols remain to be analysed:
+', sum, encArg

They will be analysed ascendingly in the following order:
+' < sum
sum < encArg
+' < encArg

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x), s(x))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_sum(x_1)) -> sum(encArg(x_1))
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encode_sum(x_1) -> sum(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))

Types:
sum :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
0' :: 0':s:cons_sum:cons_+
s :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
+' :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
encArg :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
cons_sum :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
cons_+ :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
encode_sum :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
encode_0 :: 0':s:cons_sum:cons_+
encode_s :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
encode_+ :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
hole_0':s:cons_sum:cons_+1_3 :: 0':s:cons_sum:cons_+
gen_0':s:cons_sum:cons_+2_3 :: Nat -> 0':s:cons_sum:cons_+


Lemmas:
+'(gen_0':s:cons_sum:cons_+2_3(a), gen_0':s:cons_sum:cons_+2_3(n4_3)) -> gen_0':s:cons_sum:cons_+2_3(+(n4_3, a)), rt in Omega(1 + n4_3)


Generator Equations:
gen_0':s:cons_sum:cons_+2_3(0) <=> 0'
gen_0':s:cons_sum:cons_+2_3(+(x, 1)) <=> s(gen_0':s:cons_sum:cons_+2_3(x))


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

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

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sum(gen_0':s:cons_sum:cons_+2_3(+(1, n643_3))) -> *3_3, rt in Omega(n643_3)

Induction Base:
sum(gen_0':s:cons_sum:cons_+2_3(+(1, 0)))

Induction Step:
sum(gen_0':s:cons_sum:cons_+2_3(+(1, +(n643_3, 1)))) ->_R^Omega(1)
+'(sum(gen_0':s:cons_sum:cons_+2_3(+(1, n643_3))), s(gen_0':s:cons_sum:cons_+2_3(+(1, n643_3)))) ->_IH
+'(*3_3, s(gen_0':s:cons_sum:cons_+2_3(+(1, n643_3))))

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

(18)
Obligation:
Innermost TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x), s(x))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_sum(x_1)) -> sum(encArg(x_1))
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encode_sum(x_1) -> sum(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))

Types:
sum :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
0' :: 0':s:cons_sum:cons_+
s :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
+' :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
encArg :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
cons_sum :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
cons_+ :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
encode_sum :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
encode_0 :: 0':s:cons_sum:cons_+
encode_s :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
encode_+ :: 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+ -> 0':s:cons_sum:cons_+
hole_0':s:cons_sum:cons_+1_3 :: 0':s:cons_sum:cons_+
gen_0':s:cons_sum:cons_+2_3 :: Nat -> 0':s:cons_sum:cons_+


Lemmas:
+'(gen_0':s:cons_sum:cons_+2_3(a), gen_0':s:cons_sum:cons_+2_3(n4_3)) -> gen_0':s:cons_sum:cons_+2_3(+(n4_3, a)), rt in Omega(1 + n4_3)
sum(gen_0':s:cons_sum:cons_+2_3(+(1, n643_3))) -> *3_3, rt in Omega(n643_3)


Generator Equations:
gen_0':s:cons_sum:cons_+2_3(0) <=> 0'
gen_0':s:cons_sum:cons_+2_3(+(x, 1)) <=> s(gen_0':s:cons_sum:cons_+2_3(x))


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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_0':s:cons_sum:cons_+2_3(n2099_3)) -> gen_0':s:cons_sum:cons_+2_3(n2099_3), rt in Omega(0)

Induction Base:
encArg(gen_0':s:cons_sum:cons_+2_3(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':s:cons_sum:cons_+2_3(+(n2099_3, 1))) ->_R^Omega(0)
s(encArg(gen_0':s:cons_sum:cons_+2_3(n2099_3))) ->_IH
s(gen_0':s:cons_sum:cons_+2_3(c2100_3))

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

(20)
BOUNDS(1, INF)