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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 36 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), 449 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), 86 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 85 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 62 ms]
        (22) BOUNDS(1, INF)


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(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:

   a(p(x1)) -> p(a(A(x1)))
   a(A(x1)) -> A(a(x1))
   p(A(A(x1))) -> a(p(x1))

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(A(x_1)) -> A(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_A(x_1) -> A(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:

   a(p(x1)) -> p(a(A(x1)))
   a(A(x1)) -> A(a(x1))
   p(A(A(x1))) -> a(p(x1))

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

   encArg(A(x_1)) -> A(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_A(x_1) -> A(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:

   a(p(x1)) -> p(a(A(x1)))
   a(A(x1)) -> A(a(x1))
   p(A(A(x1))) -> a(p(x1))

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

   encArg(A(x_1)) -> A(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_A(x_1) -> A(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:

   a(p(x1)) -> p(a(A(x1)))
   a(A(x1)) -> A(a(x1))
   p(A(A(x1))) -> a(p(x1))

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

   encArg(A(x_1)) -> A(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_A(x_1) -> A(encArg(x_1))

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

(8)
Obligation:
Innermost TRS:
Rules:
a(p(x1)) -> p(a(A(x1)))
a(A(x1)) -> A(a(x1))
p(A(A(x1))) -> a(p(x1))
encArg(A(x_1)) -> A(encArg(x_1))
encArg(cons_a(x_1)) -> a(encArg(x_1))
encArg(cons_p(x_1)) -> p(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_p(x_1) -> p(encArg(x_1))
encode_A(x_1) -> A(encArg(x_1))

Types:
a :: A:cons_a:cons_p -> A:cons_a:cons_p
p :: A:cons_a:cons_p -> A:cons_a:cons_p
A :: A:cons_a:cons_p -> A:cons_a:cons_p
encArg :: A:cons_a:cons_p -> A:cons_a:cons_p
cons_a :: A:cons_a:cons_p -> A:cons_a:cons_p
cons_p :: A:cons_a:cons_p -> A:cons_a:cons_p
encode_a :: A:cons_a:cons_p -> A:cons_a:cons_p
encode_p :: A:cons_a:cons_p -> A:cons_a:cons_p
encode_A :: A:cons_a:cons_p -> A:cons_a:cons_p
hole_A:cons_a:cons_p1_0 :: A:cons_a:cons_p
gen_A:cons_a:cons_p2_0 :: Nat -> A:cons_a:cons_p

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

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

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

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(10)
Obligation:
Innermost TRS:
Rules:
a(p(x1)) -> p(a(A(x1)))
a(A(x1)) -> A(a(x1))
p(A(A(x1))) -> a(p(x1))
encArg(A(x_1)) -> A(encArg(x_1))
encArg(cons_a(x_1)) -> a(encArg(x_1))
encArg(cons_p(x_1)) -> p(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_p(x_1) -> p(encArg(x_1))
encode_A(x_1) -> A(encArg(x_1))

Types:
a :: A:cons_a:cons_p -> A:cons_a:cons_p
p :: A:cons_a:cons_p -> A:cons_a:cons_p
A :: A:cons_a:cons_p -> A:cons_a:cons_p
encArg :: A:cons_a:cons_p -> A:cons_a:cons_p
cons_a :: A:cons_a:cons_p -> A:cons_a:cons_p
cons_p :: A:cons_a:cons_p -> A:cons_a:cons_p
encode_a :: A:cons_a:cons_p -> A:cons_a:cons_p
encode_p :: A:cons_a:cons_p -> A:cons_a:cons_p
encode_A :: A:cons_a:cons_p -> A:cons_a:cons_p
hole_A:cons_a:cons_p1_0 :: A:cons_a:cons_p
gen_A:cons_a:cons_p2_0 :: Nat -> A:cons_a:cons_p


Generator Equations:
gen_A:cons_a:cons_p2_0(0) <=> hole_A:cons_a:cons_p1_0
gen_A:cons_a:cons_p2_0(+(x, 1)) <=> A(gen_A:cons_a:cons_p2_0(x))


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

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

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
p(gen_A:cons_a:cons_p2_0(+(2, *(2, n4_0)))) -> *3_0, rt in Omega(n4_0)

Induction Base:
p(gen_A:cons_a:cons_p2_0(+(2, *(2, 0))))

Induction Step:
p(gen_A:cons_a:cons_p2_0(+(2, *(2, +(n4_0, 1))))) ->_R^Omega(1)
a(p(gen_A:cons_a:cons_p2_0(+(2, *(2, n4_0))))) ->_IH
a(*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)

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

Innermost TRS:
Rules:
a(p(x1)) -> p(a(A(x1)))
a(A(x1)) -> A(a(x1))
p(A(A(x1))) -> a(p(x1))
encArg(A(x_1)) -> A(encArg(x_1))
encArg(cons_a(x_1)) -> a(encArg(x_1))
encArg(cons_p(x_1)) -> p(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_p(x_1) -> p(encArg(x_1))
encode_A(x_1) -> A(encArg(x_1))

Types:
a :: A:cons_a:cons_p -> A:cons_a:cons_p
p :: A:cons_a:cons_p -> A:cons_a:cons_p
A :: A:cons_a:cons_p -> A:cons_a:cons_p
encArg :: A:cons_a:cons_p -> A:cons_a:cons_p
cons_a :: A:cons_a:cons_p -> A:cons_a:cons_p
cons_p :: A:cons_a:cons_p -> A:cons_a:cons_p
encode_a :: A:cons_a:cons_p -> A:cons_a:cons_p
encode_p :: A:cons_a:cons_p -> A:cons_a:cons_p
encode_A :: A:cons_a:cons_p -> A:cons_a:cons_p
hole_A:cons_a:cons_p1_0 :: A:cons_a:cons_p
gen_A:cons_a:cons_p2_0 :: Nat -> A:cons_a:cons_p


Generator Equations:
gen_A:cons_a:cons_p2_0(0) <=> hole_A:cons_a:cons_p1_0
gen_A:cons_a:cons_p2_0(+(x, 1)) <=> A(gen_A:cons_a:cons_p2_0(x))


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

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

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

(15)
BOUNDS(n^1, INF)

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(16)
Obligation:
Innermost TRS:
Rules:
a(p(x1)) -> p(a(A(x1)))
a(A(x1)) -> A(a(x1))
p(A(A(x1))) -> a(p(x1))
encArg(A(x_1)) -> A(encArg(x_1))
encArg(cons_a(x_1)) -> a(encArg(x_1))
encArg(cons_p(x_1)) -> p(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_p(x_1) -> p(encArg(x_1))
encode_A(x_1) -> A(encArg(x_1))

Types:
a :: A:cons_a:cons_p -> A:cons_a:cons_p
p :: A:cons_a:cons_p -> A:cons_a:cons_p
A :: A:cons_a:cons_p -> A:cons_a:cons_p
encArg :: A:cons_a:cons_p -> A:cons_a:cons_p
cons_a :: A:cons_a:cons_p -> A:cons_a:cons_p
cons_p :: A:cons_a:cons_p -> A:cons_a:cons_p
encode_a :: A:cons_a:cons_p -> A:cons_a:cons_p
encode_p :: A:cons_a:cons_p -> A:cons_a:cons_p
encode_A :: A:cons_a:cons_p -> A:cons_a:cons_p
hole_A:cons_a:cons_p1_0 :: A:cons_a:cons_p
gen_A:cons_a:cons_p2_0 :: Nat -> A:cons_a:cons_p


Lemmas:
p(gen_A:cons_a:cons_p2_0(+(2, *(2, n4_0)))) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_A:cons_a:cons_p2_0(0) <=> hole_A:cons_a:cons_p1_0
gen_A:cons_a:cons_p2_0(+(x, 1)) <=> A(gen_A:cons_a:cons_p2_0(x))


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

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

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(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
a(gen_A:cons_a:cons_p2_0(+(1, n416_0))) -> *3_0, rt in Omega(n416_0)

Induction Base:
a(gen_A:cons_a:cons_p2_0(+(1, 0)))

Induction Step:
a(gen_A:cons_a:cons_p2_0(+(1, +(n416_0, 1)))) ->_R^Omega(1)
A(a(gen_A:cons_a:cons_p2_0(+(1, n416_0)))) ->_IH
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:
Innermost TRS:
Rules:
a(p(x1)) -> p(a(A(x1)))
a(A(x1)) -> A(a(x1))
p(A(A(x1))) -> a(p(x1))
encArg(A(x_1)) -> A(encArg(x_1))
encArg(cons_a(x_1)) -> a(encArg(x_1))
encArg(cons_p(x_1)) -> p(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_p(x_1) -> p(encArg(x_1))
encode_A(x_1) -> A(encArg(x_1))

Types:
a :: A:cons_a:cons_p -> A:cons_a:cons_p
p :: A:cons_a:cons_p -> A:cons_a:cons_p
A :: A:cons_a:cons_p -> A:cons_a:cons_p
encArg :: A:cons_a:cons_p -> A:cons_a:cons_p
cons_a :: A:cons_a:cons_p -> A:cons_a:cons_p
cons_p :: A:cons_a:cons_p -> A:cons_a:cons_p
encode_a :: A:cons_a:cons_p -> A:cons_a:cons_p
encode_p :: A:cons_a:cons_p -> A:cons_a:cons_p
encode_A :: A:cons_a:cons_p -> A:cons_a:cons_p
hole_A:cons_a:cons_p1_0 :: A:cons_a:cons_p
gen_A:cons_a:cons_p2_0 :: Nat -> A:cons_a:cons_p


Lemmas:
p(gen_A:cons_a:cons_p2_0(+(2, *(2, n4_0)))) -> *3_0, rt in Omega(n4_0)
a(gen_A:cons_a:cons_p2_0(+(1, n416_0))) -> *3_0, rt in Omega(n416_0)


Generator Equations:
gen_A:cons_a:cons_p2_0(0) <=> hole_A:cons_a:cons_p1_0
gen_A:cons_a:cons_p2_0(+(x, 1)) <=> A(gen_A:cons_a:cons_p2_0(x))


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

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

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
p(gen_A:cons_a:cons_p2_0(+(2, *(2, n840_0)))) -> *3_0, rt in Omega(n840_0)

Induction Base:
p(gen_A:cons_a:cons_p2_0(+(2, *(2, 0))))

Induction Step:
p(gen_A:cons_a:cons_p2_0(+(2, *(2, +(n840_0, 1))))) ->_R^Omega(1)
a(p(gen_A:cons_a:cons_p2_0(+(2, *(2, n840_0))))) ->_IH
a(*3_0)

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

(20)
Obligation:
Innermost TRS:
Rules:
a(p(x1)) -> p(a(A(x1)))
a(A(x1)) -> A(a(x1))
p(A(A(x1))) -> a(p(x1))
encArg(A(x_1)) -> A(encArg(x_1))
encArg(cons_a(x_1)) -> a(encArg(x_1))
encArg(cons_p(x_1)) -> p(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_p(x_1) -> p(encArg(x_1))
encode_A(x_1) -> A(encArg(x_1))

Types:
a :: A:cons_a:cons_p -> A:cons_a:cons_p
p :: A:cons_a:cons_p -> A:cons_a:cons_p
A :: A:cons_a:cons_p -> A:cons_a:cons_p
encArg :: A:cons_a:cons_p -> A:cons_a:cons_p
cons_a :: A:cons_a:cons_p -> A:cons_a:cons_p
cons_p :: A:cons_a:cons_p -> A:cons_a:cons_p
encode_a :: A:cons_a:cons_p -> A:cons_a:cons_p
encode_p :: A:cons_a:cons_p -> A:cons_a:cons_p
encode_A :: A:cons_a:cons_p -> A:cons_a:cons_p
hole_A:cons_a:cons_p1_0 :: A:cons_a:cons_p
gen_A:cons_a:cons_p2_0 :: Nat -> A:cons_a:cons_p


Lemmas:
p(gen_A:cons_a:cons_p2_0(+(2, *(2, n840_0)))) -> *3_0, rt in Omega(n840_0)
a(gen_A:cons_a:cons_p2_0(+(1, n416_0))) -> *3_0, rt in Omega(n416_0)


Generator Equations:
gen_A:cons_a:cons_p2_0(0) <=> hole_A:cons_a:cons_p1_0
gen_A:cons_a:cons_p2_0(+(x, 1)) <=> A(gen_A:cons_a:cons_p2_0(x))


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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_A:cons_a:cons_p2_0(+(1, n1480_0))) -> *3_0, rt in Omega(0)

Induction Base:
encArg(gen_A:cons_a:cons_p2_0(+(1, 0)))

Induction Step:
encArg(gen_A:cons_a:cons_p2_0(+(1, +(n1480_0, 1)))) ->_R^Omega(0)
A(encArg(gen_A:cons_a:cons_p2_0(+(1, n1480_0)))) ->_IH
A(*3_0)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(22)
BOUNDS(1, INF)