/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^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) CpxTrsToCdtProof [UPPER BOUND(ID), 5 ms]
    (4) CdtProblem
    (5) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 57 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), 861 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^2).


The TRS R consists of the following rules:

   *(x, +(y, z)) -> +(*(x, y), *(x, z))
   *(+(x, y), z) -> +(*(x, z), *(y, z))
   *(x, 1) -> x
   *(1, y) -> y

S is empty.
Rewrite Strategy: FULL
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(1) 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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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   *(x, +(y, z)) -> +(*(x, y), *(x, z))
   *(+(x, y), z) -> +(*(x, z), *(y, z))
   *(x, 1) -> x
   *(1, y) -> y

S is empty.
Rewrite Strategy: INNERMOST
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(3) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))
   *(+(z0, z1), z2) -> +(*(z0, z2), *(z1, z2))
   *(z0, 1) -> z0
   *(1, z0) -> z0
Tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
   *'(z0, 1) -> c2
   *'(1, z0) -> c3
S tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
   *'(z0, 1) -> c2
   *'(1, z0) -> c3
K tuples:none
Defined Rule Symbols:   *_2

Defined Pair Symbols:   *'_2

Compound Symbols:   c_2, c1_2, c2, c3


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(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   *'(1, z0) -> c3
   *'(z0, 1) -> c2

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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))
   *(+(z0, z1), z2) -> +(*(z0, z2), *(z1, z2))
   *(z0, 1) -> z0
   *(1, z0) -> z0
Tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
S tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
K tuples:none
Defined Rule Symbols:   *_2

Defined Pair Symbols:   *'_2

Compound Symbols:   c_2, c1_2


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(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   *(z0, +(z1, z2)) -> +(*(z0, z1), *(z0, z2))
   *(+(z0, z1), z2) -> +(*(z0, z2), *(z1, z2))
   *(z0, 1) -> z0
   *(1, z0) -> z0

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(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
S tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   *'_2

Compound Symbols:   c_2, c1_2


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(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
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))
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
We considered the (Usable) Rules:none
And the Tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*'(x_1, x_2)) = [2] + [2]x_1 + [2]x_2 + [2]x_1*x_2
   POL(+(x_1, x_2)) = [2] + x_1 + x_2
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2)) = x_1 + x_2

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(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
S tuples:none
K tuples:
   *'(z0, +(z1, z2)) -> c(*'(z0, z1), *'(z0, z2))
   *'(+(z0, z1), z2) -> c1(*'(z0, z2), *'(z1, z2))
Defined Rule Symbols:none

Defined Pair Symbols:   *'_2

Compound Symbols:   c_2, c1_2


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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, z)) -> +'(*'(x, y), *'(x, z))
   *'(+'(x, y), z) -> +'(*'(x, z), *'(y, z))
   *'(x, 1') -> x
   *'(1', y) -> y

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, z)) -> +'(*'(x, y), *'(x, z))
*'(+'(x, y), z) -> +'(*'(x, z), *'(y, z))
*'(x, 1') -> x
*'(1', y) -> y

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

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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, z)) -> +'(*'(x, y), *'(x, z))
*'(+'(x, y), z) -> +'(*'(x, z), *'(y, z))
*'(x, 1') -> x
*'(1', y) -> y

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


Generator Equations:
gen_+':1'2_0(0) <=> 1'
gen_+':1'2_0(+(x, 1)) <=> +'(1', gen_+':1'2_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_+':1'2_0(a), gen_+':1'2_0(n4_0)) -> *3_0, rt in Omega(n4_0)

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

Induction Step:
*'(gen_+':1'2_0(a), gen_+':1'2_0(+(n4_0, 1))) ->_R^Omega(1)
+'(*'(gen_+':1'2_0(a), 1'), *'(gen_+':1'2_0(a), gen_+':1'2_0(n4_0))) ->_R^Omega(1)
+'(gen_+':1'2_0(a), *'(gen_+':1'2_0(a), gen_+':1'2_0(n4_0))) ->_IH
+'(gen_+':1'2_0(a), *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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(20)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
*'(x, +'(y, z)) -> +'(*'(x, y), *'(x, z))
*'(+'(x, y), z) -> +'(*'(x, z), *'(y, z))
*'(x, 1') -> x
*'(1', y) -> y

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


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