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


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsToCdtProof [UPPER BOUND(ID), 2 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 69 ms]
    (10) CdtProblem
    (11) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) BOUNDS(1, 1)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTRS
    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) typed CpxTrs
    (17) OrderProof [LOWER BOUND(ID), 0 ms]
    (18) typed CpxTrs
    (19) RewriteLemmaProof [LOWER BOUND(ID), 638 ms]
    (20) proven lower bound
    (21) LowerBoundPropagationProof [FINISHED, 0 ms]
    (22) BOUNDS(n^1, INF)


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


The TRS R consists of the following rules:

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

S is empty.
Rewrite Strategy: FULL
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(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
+(+(x, y), z) -> +(x, +(y, z))
+(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)

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


The TRS R consists of the following rules:

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

S is empty.
Rewrite Strategy: FULL
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(3) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, we have rc = irc.
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(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

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

S is empty.
Rewrite Strategy: INNERMOST
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(5) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
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(6)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   +'_2

Compound Symbols:   c_1


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

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(8)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   +'_2

Compound Symbols:   c_1


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

Polynomial interpretation :

   POL(*(x_1, x_2)) = [1] + x_2
   POL(+'(x_1, x_2)) = x_2
   POL(c(x_1)) = x_1

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(10)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   +'_2

Compound Symbols:   c_1


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(11) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(12)
BOUNDS(1, 1)

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(13) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(14)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

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

S is empty.
Rewrite Strategy: FULL
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(15) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(16)
Obligation:
TRS:
Rules:
+'(*'(x, y), *'(x, z)) -> *'(x, +'(y, z))
+'(+'(x, y), z) -> +'(x, +'(y, z))
+'(*'(x, y), +'(*'(x, z), u)) -> +'(*'(x, +'(y, z)), u)

Types:
+' :: *':u -> *':u -> *':u
*' :: a -> *':u -> *':u
u :: *':u
hole_*':u1_0 :: *':u
hole_a2_0 :: a
gen_*':u3_0 :: Nat -> *':u

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(17) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
+'
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(18)
Obligation:
TRS:
Rules:
+'(*'(x, y), *'(x, z)) -> *'(x, +'(y, z))
+'(+'(x, y), z) -> +'(x, +'(y, z))
+'(*'(x, y), +'(*'(x, z), u)) -> +'(*'(x, +'(y, z)), u)

Types:
+' :: *':u -> *':u -> *':u
*' :: a -> *':u -> *':u
u :: *':u
hole_*':u1_0 :: *':u
hole_a2_0 :: a
gen_*':u3_0 :: Nat -> *':u


Generator Equations:
gen_*':u3_0(0) <=> u
gen_*':u3_0(+(x, 1)) <=> *'(hole_a2_0, gen_*':u3_0(x))


The following defined symbols remain to be analysed:
+'
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(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_*':u3_0(+(1, n5_0)), gen_*':u3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)

Induction Base:
+'(gen_*':u3_0(+(1, 0)), gen_*':u3_0(+(1, 0)))

Induction Step:
+'(gen_*':u3_0(+(1, +(n5_0, 1))), gen_*':u3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1)
*'(hole_a2_0, +'(gen_*':u3_0(+(1, n5_0)), gen_*':u3_0(+(1, n5_0)))) ->_IH
*'(hole_a2_0, *4_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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(20)
Obligation:
Proved the lower bound n^1 for the following obligation:

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

Types:
+' :: *':u -> *':u -> *':u
*' :: a -> *':u -> *':u
u :: *':u
hole_*':u1_0 :: *':u
hole_a2_0 :: a
gen_*':u3_0 :: Nat -> *':u


Generator Equations:
gen_*':u3_0(0) <=> u
gen_*':u3_0(+(x, 1)) <=> *'(hole_a2_0, gen_*':u3_0(x))


The following defined symbols remain to be analysed:
+'
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(21) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(22)
BOUNDS(n^1, INF)