YES(O(1), O(n^2)) 2.78/1.18 YES(O(1), O(n^2)) 2.78/1.19 2.78/1.19 2.78/1.19 2.78/1.19 2.78/1.19 2.78/1.19 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 2.78/1.19 2.78/1.19 2.78/1.19
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
2.78/1.19 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
2.78/1.19 
↳2 CdtProblem
2.78/1.19 
↳3 CdtUnreachableProof (⇔)
2.78/1.19 
↳4 CdtProblem
2.78/1.19 
↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
2.78/1.19 
↳6 CdtProblem
2.78/1.19 
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
2.78/1.19 
↳8 CdtProblem
2.78/1.19 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
2.78/1.19 
↳10 CdtProblem
2.78/1.19 
↳11 SIsEmptyProof (BOTH BOUNDS(ID, ID))
2.78/1.19 
↳12 BOUNDS(O(1), O(1))
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2.78/1.19

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

app(nil, k) → k 2.78/1.19
app(l, nil) → l 2.78/1.19
app(cons(x, l), k) → cons(x, app(l, k)) 2.78/1.19
sum(cons(x, nil)) → cons(x, nil) 2.78/1.19
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l)) 2.78/1.19
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k))))) 2.78/1.19
plus(0, y) → y 2.78/1.19
plus(s(x), y) → s(plus(x, y)) 2.78/1.19
sum(plus(cons(0, x), cons(y, l))) → pred(sum(cons(s(x), cons(y, l)))) 2.78/1.19
pred(cons(s(x), nil)) → cons(x, nil)

Rewrite Strategy: INNERMOST
2.78/1.19
2.78/1.19

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT
2.78/1.19
2.78/1.19

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

app(nil, z0) → z0 2.78/1.19
app(z0, nil) → z0 2.78/1.19
app(cons(z0, z1), z2) → cons(z0, app(z1, z2)) 2.78/1.19
sum(cons(z0, nil)) → cons(z0, nil) 2.78/1.19
sum(cons(z0, cons(z1, z2))) → sum(cons(plus(z0, z1), z2)) 2.78/1.19
sum(app(z0, cons(z1, cons(z2, z3)))) → sum(app(z0, sum(cons(z1, cons(z2, z3))))) 2.78/1.20
sum(plus(cons(0, z0), cons(z1, z2))) → pred(sum(cons(s(z0), cons(z1, z2)))) 2.78/1.20
plus(0, z0) → z0 2.78/1.20
plus(s(z0), z1) → s(plus(z0, z1)) 2.78/1.20
pred(cons(s(z0), nil)) → cons(z0, nil)
Tuples:

APP(cons(z0, z1), z2) → c2(APP(z1, z2)) 2.78/1.20
SUM(cons(z0, cons(z1, z2))) → c4(SUM(cons(plus(z0, z1), z2)), PLUS(z0, z1)) 2.78/1.20
SUM(app(z0, cons(z1, cons(z2, z3)))) → c5(SUM(app(z0, sum(cons(z1, cons(z2, z3))))), APP(z0, sum(cons(z1, cons(z2, z3)))), SUM(cons(z1, cons(z2, z3)))) 2.78/1.20
SUM(plus(cons(0, z0), cons(z1, z2))) → c6(PRED(sum(cons(s(z0), cons(z1, z2)))), SUM(cons(s(z0), cons(z1, z2)))) 2.78/1.20
PLUS(s(z0), z1) → c8(PLUS(z0, z1))
S tuples:

APP(cons(z0, z1), z2) → c2(APP(z1, z2)) 2.78/1.20
SUM(cons(z0, cons(z1, z2))) → c4(SUM(cons(plus(z0, z1), z2)), PLUS(z0, z1)) 2.78/1.20
SUM(app(z0, cons(z1, cons(z2, z3)))) → c5(SUM(app(z0, sum(cons(z1, cons(z2, z3))))), APP(z0, sum(cons(z1, cons(z2, z3)))), SUM(cons(z1, cons(z2, z3)))) 2.78/1.20
SUM(plus(cons(0, z0), cons(z1, z2))) → c6(PRED(sum(cons(s(z0), cons(z1, z2)))), SUM(cons(s(z0), cons(z1, z2)))) 2.78/1.20
PLUS(s(z0), z1) → c8(PLUS(z0, z1))
K tuples:none
Defined Rule Symbols:

app, sum, plus, pred

Defined Pair Symbols:

APP, SUM, PLUS

Compound Symbols:

c2, c4, c5, c6, c8

2.78/1.20
2.78/1.20

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

SUM(app(z0, cons(z1, cons(z2, z3)))) → c5(SUM(app(z0, sum(cons(z1, cons(z2, z3))))), APP(z0, sum(cons(z1, cons(z2, z3)))), SUM(cons(z1, cons(z2, z3)))) 2.78/1.20
SUM(plus(cons(0, z0), cons(z1, z2))) → c6(PRED(sum(cons(s(z0), cons(z1, z2)))), SUM(cons(s(z0), cons(z1, z2))))
2.78/1.20
2.78/1.20

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

app(nil, z0) → z0 2.78/1.20
app(z0, nil) → z0 2.78/1.20
app(cons(z0, z1), z2) → cons(z0, app(z1, z2)) 2.78/1.20
sum(cons(z0, nil)) → cons(z0, nil) 2.78/1.20
sum(cons(z0, cons(z1, z2))) → sum(cons(plus(z0, z1), z2)) 2.78/1.20
sum(app(z0, cons(z1, cons(z2, z3)))) → sum(app(z0, sum(cons(z1, cons(z2, z3))))) 2.78/1.20
sum(plus(cons(0, z0), cons(z1, z2))) → pred(sum(cons(s(z0), cons(z1, z2)))) 2.78/1.20
plus(0, z0) → z0 2.78/1.20
plus(s(z0), z1) → s(plus(z0, z1)) 2.78/1.20
pred(cons(s(z0), nil)) → cons(z0, nil)
Tuples:

APP(cons(z0, z1), z2) → c2(APP(z1, z2)) 2.78/1.20
SUM(cons(z0, cons(z1, z2))) → c4(SUM(cons(plus(z0, z1), z2)), PLUS(z0, z1)) 2.78/1.20
PLUS(s(z0), z1) → c8(PLUS(z0, z1))
S tuples:

APP(cons(z0, z1), z2) → c2(APP(z1, z2)) 2.78/1.20
SUM(cons(z0, cons(z1, z2))) → c4(SUM(cons(plus(z0, z1), z2)), PLUS(z0, z1)) 2.78/1.20
PLUS(s(z0), z1) → c8(PLUS(z0, z1))
K tuples:none
Defined Rule Symbols:

app, sum, plus, pred

Defined Pair Symbols:

APP, SUM, PLUS

Compound Symbols:

c2, c4, c8

2.78/1.20
2.78/1.20

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

SUM(cons(z0, cons(z1, z2))) → c4(SUM(cons(plus(z0, z1), z2)), PLUS(z0, z1))
We considered the (Usable) Rules:

plus(0, z0) → z0 2.78/1.20
plus(s(z0), z1) → s(plus(z0, z1))
And the Tuples:

APP(cons(z0, z1), z2) → c2(APP(z1, z2)) 2.78/1.20
SUM(cons(z0, cons(z1, z2))) → c4(SUM(cons(plus(z0, z1), z2)), PLUS(z0, z1)) 2.78/1.20
PLUS(s(z0), z1) → c8(PLUS(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 2.78/1.20

POL(0) = 0    2.78/1.20
POL(APP(x1, x2)) = 0    2.78/1.20
POL(PLUS(x1, x2)) = [1]    2.78/1.20
POL(SUM(x1)) = [4]x1    2.78/1.20
POL(c2(x1)) = x1    2.78/1.20
POL(c4(x1, x2)) = x1 + x2    2.78/1.20
POL(c8(x1)) = x1    2.78/1.20
POL(cons(x1, x2)) = [3] + x2    2.78/1.20
POL(plus(x1, x2)) = [4] + [4]x1 + [3]x2    2.78/1.20
POL(s(x1)) = [2]   
2.78/1.20
2.78/1.20

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

app(nil, z0) → z0 2.78/1.20
app(z0, nil) → z0 2.78/1.20
app(cons(z0, z1), z2) → cons(z0, app(z1, z2)) 2.78/1.20
sum(cons(z0, nil)) → cons(z0, nil) 2.78/1.20
sum(cons(z0, cons(z1, z2))) → sum(cons(plus(z0, z1), z2)) 2.78/1.20
sum(app(z0, cons(z1, cons(z2, z3)))) → sum(app(z0, sum(cons(z1, cons(z2, z3))))) 2.78/1.20
sum(plus(cons(0, z0), cons(z1, z2))) → pred(sum(cons(s(z0), cons(z1, z2)))) 2.78/1.20
plus(0, z0) → z0 2.78/1.20
plus(s(z0), z1) → s(plus(z0, z1)) 2.78/1.20
pred(cons(s(z0), nil)) → cons(z0, nil)
Tuples:

APP(cons(z0, z1), z2) → c2(APP(z1, z2)) 2.78/1.20
SUM(cons(z0, cons(z1, z2))) → c4(SUM(cons(plus(z0, z1), z2)), PLUS(z0, z1)) 2.78/1.20
PLUS(s(z0), z1) → c8(PLUS(z0, z1))
S tuples:

APP(cons(z0, z1), z2) → c2(APP(z1, z2)) 2.78/1.20
PLUS(s(z0), z1) → c8(PLUS(z0, z1))
K tuples:

SUM(cons(z0, cons(z1, z2))) → c4(SUM(cons(plus(z0, z1), z2)), PLUS(z0, z1))
Defined Rule Symbols:

app, sum, plus, pred

Defined Pair Symbols:

APP, SUM, PLUS

Compound Symbols:

c2, c4, c8

2.78/1.20
2.78/1.20

(7) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

APP(cons(z0, z1), z2) → c2(APP(z1, z2))
We considered the (Usable) Rules:

plus(0, z0) → z0 2.78/1.20
plus(s(z0), z1) → s(plus(z0, z1))
And the Tuples:

APP(cons(z0, z1), z2) → c2(APP(z1, z2)) 2.78/1.20
SUM(cons(z0, cons(z1, z2))) → c4(SUM(cons(plus(z0, z1), z2)), PLUS(z0, z1)) 2.78/1.20
PLUS(s(z0), z1) → c8(PLUS(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 2.78/1.20

POL(0) = 0    2.78/1.20
POL(APP(x1, x2)) = [4]x1    2.78/1.20
POL(PLUS(x1, x2)) = [4]    2.78/1.20
POL(SUM(x1)) = [5]x1    2.78/1.20
POL(c2(x1)) = x1    2.78/1.20
POL(c4(x1, x2)) = x1 + x2    2.78/1.20
POL(c8(x1)) = x1    2.78/1.20
POL(cons(x1, x2)) = [2] + x2    2.78/1.20
POL(plus(x1, x2)) = [2] + [2]x1 + [2]x2    2.78/1.20
POL(s(x1)) = [4]   
2.78/1.20
2.78/1.20

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

app(nil, z0) → z0 2.78/1.20
app(z0, nil) → z0 2.78/1.20
app(cons(z0, z1), z2) → cons(z0, app(z1, z2)) 2.78/1.20
sum(cons(z0, nil)) → cons(z0, nil) 2.78/1.20
sum(cons(z0, cons(z1, z2))) → sum(cons(plus(z0, z1), z2)) 2.78/1.20
sum(app(z0, cons(z1, cons(z2, z3)))) → sum(app(z0, sum(cons(z1, cons(z2, z3))))) 2.78/1.20
sum(plus(cons(0, z0), cons(z1, z2))) → pred(sum(cons(s(z0), cons(z1, z2)))) 2.78/1.20
plus(0, z0) → z0 2.78/1.20
plus(s(z0), z1) → s(plus(z0, z1)) 2.78/1.20
pred(cons(s(z0), nil)) → cons(z0, nil)
Tuples:

APP(cons(z0, z1), z2) → c2(APP(z1, z2)) 2.78/1.20
SUM(cons(z0, cons(z1, z2))) → c4(SUM(cons(plus(z0, z1), z2)), PLUS(z0, z1)) 2.78/1.20
PLUS(s(z0), z1) → c8(PLUS(z0, z1))
S tuples:

PLUS(s(z0), z1) → c8(PLUS(z0, z1))
K tuples:

SUM(cons(z0, cons(z1, z2))) → c4(SUM(cons(plus(z0, z1), z2)), PLUS(z0, z1)) 2.78/1.20
APP(cons(z0, z1), z2) → c2(APP(z1, z2))
Defined Rule Symbols:

app, sum, plus, pred

Defined Pair Symbols:

APP, SUM, PLUS

Compound Symbols:

c2, c4, c8

2.78/1.20
2.78/1.20

(9) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

PLUS(s(z0), z1) → c8(PLUS(z0, z1))
We considered the (Usable) Rules:

plus(0, z0) → z0 2.78/1.20
plus(s(z0), z1) → s(plus(z0, z1))
And the Tuples:

APP(cons(z0, z1), z2) → c2(APP(z1, z2)) 2.78/1.20
SUM(cons(z0, cons(z1, z2))) → c4(SUM(cons(plus(z0, z1), z2)), PLUS(z0, z1)) 2.78/1.20
PLUS(s(z0), z1) → c8(PLUS(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 2.78/1.20

POL(0) = [3]    2.78/1.20
POL(APP(x1, x2)) = [3]x1 + [3]x1·x2 + [3]x12    2.78/1.20
POL(PLUS(x1, x2)) = x1 + x2    2.78/1.20
POL(SUM(x1)) = x12    2.78/1.20
POL(c2(x1)) = x1    2.78/1.20
POL(c4(x1, x2)) = x1 + x2    2.78/1.20
POL(c8(x1)) = x1    2.78/1.20
POL(cons(x1, x2)) = [1] + x1 + x2    2.78/1.20
POL(plus(x1, x2)) = x1 + x2    2.78/1.20
POL(s(x1)) = [1] + x1   
2.78/1.20
2.78/1.20

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

app(nil, z0) → z0 2.78/1.20
app(z0, nil) → z0 2.78/1.20
app(cons(z0, z1), z2) → cons(z0, app(z1, z2)) 2.78/1.20
sum(cons(z0, nil)) → cons(z0, nil) 2.78/1.20
sum(cons(z0, cons(z1, z2))) → sum(cons(plus(z0, z1), z2)) 2.78/1.20
sum(app(z0, cons(z1, cons(z2, z3)))) → sum(app(z0, sum(cons(z1, cons(z2, z3))))) 2.78/1.20
sum(plus(cons(0, z0), cons(z1, z2))) → pred(sum(cons(s(z0), cons(z1, z2)))) 2.78/1.20
plus(0, z0) → z0 2.78/1.20
plus(s(z0), z1) → s(plus(z0, z1)) 2.78/1.20
pred(cons(s(z0), nil)) → cons(z0, nil)
Tuples:

APP(cons(z0, z1), z2) → c2(APP(z1, z2)) 2.78/1.20
SUM(cons(z0, cons(z1, z2))) → c4(SUM(cons(plus(z0, z1), z2)), PLUS(z0, z1)) 2.78/1.20
PLUS(s(z0), z1) → c8(PLUS(z0, z1))
S tuples:none
K tuples:

SUM(cons(z0, cons(z1, z2))) → c4(SUM(cons(plus(z0, z1), z2)), PLUS(z0, z1)) 2.78/1.20
APP(cons(z0, z1), z2) → c2(APP(z1, z2)) 2.78/1.20
PLUS(s(z0), z1) → c8(PLUS(z0, z1))
Defined Rule Symbols:

app, sum, plus, pred

Defined Pair Symbols:

APP, SUM, PLUS

Compound Symbols:

c2, c4, c8

2.78/1.20
2.78/1.20

(11) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty
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2.78/1.20

(12) BOUNDS(O(1), O(1))

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2.78/1.20
3.16/1.23 EOF