WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 66 ms]
    (6) CdtProblem
    (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 31 ms]
    (8) CdtProblem
    (9) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) BOUNDS(1, 1)
(11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTRS
    (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) typed CpxTrs
    (15) OrderProof [LOWER BOUND(ID), 0 ms]
    (16) typed CpxTrs
    (17) RewriteLemmaProof [LOWER BOUND(ID), 257 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 766 ms]
            (24) BOUNDS(1, INF)


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(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


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
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(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
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(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sum(0) -> 0
   sum(s(z0)) -> +(sum(z0), s(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   SUM(0) -> c
   SUM(s(z0)) -> c1(+'(sum(z0), s(z0)), SUM(z0))
   +'(z0, 0) -> c2
   +'(z0, s(z1)) -> c3(+'(z0, z1))
S tuples:
   SUM(0) -> c
   SUM(s(z0)) -> c1(+'(sum(z0), s(z0)), SUM(z0))
   +'(z0, 0) -> c2
   +'(z0, s(z1)) -> c3(+'(z0, z1))
K tuples:none
Defined Rule Symbols:   sum_1, +_2

Defined Pair Symbols:   SUM_1, +'_2

Compound Symbols:   c, c1_2, c2, c3_1


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(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   +'(z0, 0) -> c2
   SUM(0) -> c

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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sum(0) -> 0
   sum(s(z0)) -> +(sum(z0), s(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   SUM(s(z0)) -> c1(+'(sum(z0), s(z0)), SUM(z0))
   +'(z0, s(z1)) -> c3(+'(z0, z1))
S tuples:
   SUM(s(z0)) -> c1(+'(sum(z0), s(z0)), SUM(z0))
   +'(z0, s(z1)) -> c3(+'(z0, z1))
K tuples:none
Defined Rule Symbols:   sum_1, +_2

Defined Pair Symbols:   SUM_1, +'_2

Compound Symbols:   c1_2, c3_1


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(5) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   SUM(s(z0)) -> c1(+'(sum(z0), s(z0)), SUM(z0))
We considered the (Usable) Rules:none
And the Tuples:
   SUM(s(z0)) -> c1(+'(sum(z0), s(z0)), SUM(z0))
   +'(z0, s(z1)) -> c3(+'(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = [3] + [3]x_2
   POL(+'(x_1, x_2)) = 0
   POL(0) = [3]
   POL(SUM(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(s(x_1)) = [2] + x_1
   POL(sum(x_1)) = [3]

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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sum(0) -> 0
   sum(s(z0)) -> +(sum(z0), s(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   SUM(s(z0)) -> c1(+'(sum(z0), s(z0)), SUM(z0))
   +'(z0, s(z1)) -> c3(+'(z0, z1))
S tuples:
   +'(z0, s(z1)) -> c3(+'(z0, z1))
K tuples:
   SUM(s(z0)) -> c1(+'(sum(z0), s(z0)), SUM(z0))
Defined Rule Symbols:   sum_1, +_2

Defined Pair Symbols:   SUM_1, +'_2

Compound Symbols:   c1_2, c3_1


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(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   +'(z0, s(z1)) -> c3(+'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   SUM(s(z0)) -> c1(+'(sum(z0), s(z0)), SUM(z0))
   +'(z0, s(z1)) -> c3(+'(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = [1] + x_2 + [2]x_2^2
   POL(+'(x_1, x_2)) = x_2
   POL(0) = 0
   POL(SUM(x_1)) = x_1^2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(s(x_1)) = [1] + x_1
   POL(sum(x_1)) = [2] + [2]x_1 + [2]x_1^2

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(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   sum(0) -> 0
   sum(s(z0)) -> +(sum(z0), s(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   SUM(s(z0)) -> c1(+'(sum(z0), s(z0)), SUM(z0))
   +'(z0, s(z1)) -> c3(+'(z0, z1))
S tuples:none
K tuples:
   SUM(s(z0)) -> c1(+'(sum(z0), s(z0)), SUM(z0))
   +'(z0, s(z1)) -> c3(+'(z0, z1))
Defined Rule Symbols:   sum_1, +_2

Defined Pair Symbols:   SUM_1, +'_2

Compound Symbols:   c1_2, c3_1


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(9) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(10)
BOUNDS(1, 1)

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(11) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS 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
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(13) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(14)
Obligation:
Innermost TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x), s(x))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))

Types:
sum :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s

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

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

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(16)
Obligation:
Innermost TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x), s(x))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))

Types:
sum :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


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

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

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(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)

Induction Base:
+'(gen_0':s2_0(a), gen_0':s2_0(0)) ->_R^Omega(1)
gen_0':s2_0(a)

Induction Step:
+'(gen_0':s2_0(a), gen_0':s2_0(+(n4_0, 1))) ->_R^Omega(1)
s(+'(gen_0':s2_0(a), gen_0':s2_0(n4_0))) ->_IH
s(gen_0':s2_0(+(a, c5_0)))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(18)
Complex Obligation (BEST)

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(19)
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))

Types:
sum :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


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

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

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(20) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(21)
BOUNDS(n^1, INF)

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(22)
Obligation:
Innermost TRS:
Rules:
sum(0') -> 0'
sum(s(x)) -> +'(sum(x), s(x))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))

Types:
sum :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
+' :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
sum
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(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sum(gen_0':s2_0(+(1, n435_0))) -> *3_0, rt in Omega(n435_0)

Induction Base:
sum(gen_0':s2_0(+(1, 0)))

Induction Step:
sum(gen_0':s2_0(+(1, +(n435_0, 1)))) ->_R^Omega(1)
+'(sum(gen_0':s2_0(+(1, n435_0))), s(gen_0':s2_0(+(1, n435_0)))) ->_IH
+'(*3_0, s(gen_0':s2_0(+(1, n435_0))))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(24)
BOUNDS(1, INF)