/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^1).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 4 ms]
(2) CdtProblem
    (3) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 33 ms]
    (6) CdtProblem
    (7) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) BOUNDS(1, 1)
(9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxTRS
    (11) SlicingProof [LOWER BOUND(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), 519 ms]
    (18) proven lower bound
    (19) LowerBoundPropagationProof [FINISHED, 0 ms]
    (20) BOUNDS(n^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^1).


The TRS R consists of the following rules:

   *(x, +(y, z)) -> +(*(x, y), *(x, z))

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:
   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))
Tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
S tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
K tuples:none
Defined Rule Symbols:   *_2

Defined Pair Symbols:   *'_2

Compound Symbols:   c_2


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(3) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))

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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
S tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   *'_2

Compound Symbols:   c_2


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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.
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
We considered the (Usable) Rules:none
And the Tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*'(x_1, x_2)) = [2]x_2
   POL(+(x_1, x_2)) = [3] + x_1 + x_2
   POL(c(x_1, x_2)) = x_1 + x_2

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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
S tuples:none
K tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
Defined Rule Symbols:none

Defined Pair Symbols:   *'_2

Compound Symbols:   c_2


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(7) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(8)
BOUNDS(1, 1)

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(9) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(10)
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:

   *'(x, +'(y, z)) -> +'(*'(x, y), *'(x, z))

S is empty.
Rewrite Strategy: INNERMOST
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(11) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
*'/0

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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:

   *'(+'(y, z)) -> +'(*'(y), *'(z))

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:
*'(+'(y, z)) -> +'(*'(y), *'(z))

Types:
*' :: +' -> +'
+' :: +' -> +' -> +'
hole_+'1_0 :: +'
gen_+'2_0 :: Nat -> +'

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(15) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
*'
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(16)
Obligation:
Innermost TRS:
Rules:
*'(+'(y, z)) -> +'(*'(y), *'(z))

Types:
*' :: +' -> +'
+' :: +' -> +' -> +'
hole_+'1_0 :: +'
gen_+'2_0 :: Nat -> +'


Generator Equations:
gen_+'2_0(0) <=> hole_+'1_0
gen_+'2_0(+(x, 1)) <=> +'(hole_+'1_0, gen_+'2_0(x))


The following defined symbols remain to be analysed:
*'
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(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
*'(gen_+'2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)

Induction Base:
*'(gen_+'2_0(+(1, 0)))

Induction Step:
*'(gen_+'2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
+'(*'(hole_+'1_0), *'(gen_+'2_0(+(1, n4_0)))) ->_IH
+'(*'(hole_+'1_0), *3_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)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
*'(+'(y, z)) -> +'(*'(y), *'(z))

Types:
*' :: +' -> +'
+' :: +' -> +' -> +'
hole_+'1_0 :: +'
gen_+'2_0 :: Nat -> +'


Generator Equations:
gen_+'2_0(0) <=> hole_+'1_0
gen_+'2_0(+(x, 1)) <=> +'(hole_+'1_0, gen_+'2_0(x))


The following defined symbols remain to be analysed:
*'
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(19) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(20)
BOUNDS(n^1, INF)