YES(O(1), O(n^1)) 0.00/0.83 YES(O(1), O(n^1)) 0.00/0.85 0.00/0.85 0.00/0.85 0.00/0.85 0.00/0.85 0.00/0.85 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.85 0.00/0.85 0.00/0.85
0.00/0.85
0.00/0.85
0.00/0.85
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
0.00/0.85 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
0.00/0.85 
↳2 CdtProblem
0.00/0.85 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
0.00/0.85 
↳4 CdtProblem
0.00/0.85 
↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
0.00/0.85 
↳6 CdtProblem
0.00/0.85 
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
0.00/0.85 
↳8 CdtProblem
0.00/0.85 
↳9 SIsEmptyProof (BOTH BOUNDS(ID, ID))
0.00/0.85 
↳10 BOUNDS(O(1), O(1))
0.00/0.85
0.00/0.85 0.00/0.85
0.00/0.85
0.00/0.85

(0) Obligation:

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

fst(0, Z) → nil 0.00/0.85
fst(s(X), cons(Y, Z)) → cons(Y, n__fst(activate(X), activate(Z))) 0.00/0.85
from(X) → cons(X, n__from(s(X))) 0.00/0.85
add(0, X) → X 0.00/0.85
add(s(X), Y) → s(n__add(activate(X), Y)) 0.00/0.85
len(nil) → 0 0.00/0.85
len(cons(X, Z)) → s(n__len(activate(Z))) 0.00/0.85
fst(X1, X2) → n__fst(X1, X2) 0.00/0.85
from(X) → n__from(X) 0.00/0.85
add(X1, X2) → n__add(X1, X2) 0.00/0.85
len(X) → n__len(X) 0.00/0.85
activate(n__fst(X1, X2)) → fst(X1, X2) 0.00/0.85
activate(n__from(X)) → from(X) 0.00/0.85
activate(n__add(X1, X2)) → add(X1, X2) 0.00/0.85
activate(n__len(X)) → len(X) 0.00/0.85
activate(X) → X

Rewrite Strategy: INNERMOST
0.00/0.85
0.00/0.85

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

Converted CpxTRS to CDT
0.00/0.85
0.00/0.85

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

fst(0, z0) → nil 0.00/0.85
fst(s(z0), cons(z1, z2)) → cons(z1, n__fst(activate(z0), activate(z2))) 0.00/0.85
fst(z0, z1) → n__fst(z0, z1) 0.00/0.85
from(z0) → cons(z0, n__from(s(z0))) 0.00/0.85
from(z0) → n__from(z0) 0.00/0.85
add(0, z0) → z0 0.00/0.85
add(s(z0), z1) → s(n__add(activate(z0), z1)) 0.00/0.85
add(z0, z1) → n__add(z0, z1) 0.00/0.85
len(nil) → 0 0.00/0.85
len(cons(z0, z1)) → s(n__len(activate(z1))) 0.00/0.85
len(z0) → n__len(z0) 0.00/0.85
activate(n__fst(z0, z1)) → fst(z0, z1) 0.00/0.85
activate(n__from(z0)) → from(z0) 0.00/0.85
activate(n__add(z0, z1)) → add(z0, z1) 0.00/0.85
activate(n__len(z0)) → len(z0) 0.00/0.85
activate(z0) → z0
Tuples:

FST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z0), ACTIVATE(z2)) 0.00/0.85
ADD(s(z0), z1) → c6(ACTIVATE(z0)) 0.00/0.85
LEN(cons(z0, z1)) → c9(ACTIVATE(z1)) 0.00/0.85
ACTIVATE(n__fst(z0, z1)) → c11(FST(z0, z1)) 0.00/0.85
ACTIVATE(n__from(z0)) → c12(FROM(z0)) 0.00/0.85
ACTIVATE(n__add(z0, z1)) → c13(ADD(z0, z1)) 0.00/0.85
ACTIVATE(n__len(z0)) → c14(LEN(z0))
S tuples:

FST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z0), ACTIVATE(z2)) 0.00/0.85
ADD(s(z0), z1) → c6(ACTIVATE(z0)) 0.00/0.85
LEN(cons(z0, z1)) → c9(ACTIVATE(z1)) 0.00/0.85
ACTIVATE(n__fst(z0, z1)) → c11(FST(z0, z1)) 0.00/0.85
ACTIVATE(n__from(z0)) → c12(FROM(z0)) 0.00/0.85
ACTIVATE(n__add(z0, z1)) → c13(ADD(z0, z1)) 0.00/0.85
ACTIVATE(n__len(z0)) → c14(LEN(z0))
K tuples:none
Defined Rule Symbols:

fst, from, add, len, activate

Defined Pair Symbols:

FST, ADD, LEN, ACTIVATE

Compound Symbols:

c1, c6, c9, c11, c12, c13, c14

0.00/0.85
0.00/0.85

(3) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts
0.00/0.85
0.00/0.85

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

fst(0, z0) → nil 0.00/0.85
fst(s(z0), cons(z1, z2)) → cons(z1, n__fst(activate(z0), activate(z2))) 0.00/0.85
fst(z0, z1) → n__fst(z0, z1) 0.00/0.85
from(z0) → cons(z0, n__from(s(z0))) 0.00/0.85
from(z0) → n__from(z0) 0.00/0.85
add(0, z0) → z0 0.00/0.85
add(s(z0), z1) → s(n__add(activate(z0), z1)) 0.00/0.85
add(z0, z1) → n__add(z0, z1) 0.00/0.85
len(nil) → 0 0.00/0.85
len(cons(z0, z1)) → s(n__len(activate(z1))) 0.00/0.85
len(z0) → n__len(z0) 0.00/0.85
activate(n__fst(z0, z1)) → fst(z0, z1) 0.00/0.85
activate(n__from(z0)) → from(z0) 0.00/0.85
activate(n__add(z0, z1)) → add(z0, z1) 0.00/0.85
activate(n__len(z0)) → len(z0) 0.00/0.85
activate(z0) → z0
Tuples:

FST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z0), ACTIVATE(z2)) 0.00/0.85
ADD(s(z0), z1) → c6(ACTIVATE(z0)) 0.00/0.85
LEN(cons(z0, z1)) → c9(ACTIVATE(z1)) 0.00/0.85
ACTIVATE(n__fst(z0, z1)) → c11(FST(z0, z1)) 0.00/0.85
ACTIVATE(n__add(z0, z1)) → c13(ADD(z0, z1)) 0.00/0.85
ACTIVATE(n__len(z0)) → c14(LEN(z0)) 0.00/0.85
ACTIVATE(n__from(z0)) → c12
S tuples:

FST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z0), ACTIVATE(z2)) 0.00/0.85
ADD(s(z0), z1) → c6(ACTIVATE(z0)) 0.00/0.85
LEN(cons(z0, z1)) → c9(ACTIVATE(z1)) 0.00/0.85
ACTIVATE(n__fst(z0, z1)) → c11(FST(z0, z1)) 0.00/0.85
ACTIVATE(n__add(z0, z1)) → c13(ADD(z0, z1)) 0.00/0.85
ACTIVATE(n__len(z0)) → c14(LEN(z0)) 0.00/0.85
ACTIVATE(n__from(z0)) → c12
K tuples:none
Defined Rule Symbols:

fst, from, add, len, activate

Defined Pair Symbols:

FST, ADD, LEN, ACTIVATE

Compound Symbols:

c1, c6, c9, c11, c13, c14, c12

0.00/0.85
0.00/0.85

(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

ACTIVATE(n__from(z0)) → c12
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0.00/0.85

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

fst(0, z0) → nil 0.00/0.85
fst(s(z0), cons(z1, z2)) → cons(z1, n__fst(activate(z0), activate(z2))) 0.00/0.85
fst(z0, z1) → n__fst(z0, z1) 0.00/0.85
from(z0) → cons(z0, n__from(s(z0))) 0.00/0.85
from(z0) → n__from(z0) 0.00/0.85
add(0, z0) → z0 0.00/0.85
add(s(z0), z1) → s(n__add(activate(z0), z1)) 0.00/0.85
add(z0, z1) → n__add(z0, z1) 0.00/0.85
len(nil) → 0 0.00/0.85
len(cons(z0, z1)) → s(n__len(activate(z1))) 0.00/0.85
len(z0) → n__len(z0) 0.00/0.85
activate(n__fst(z0, z1)) → fst(z0, z1) 0.00/0.85
activate(n__from(z0)) → from(z0) 0.00/0.85
activate(n__add(z0, z1)) → add(z0, z1) 0.00/0.85
activate(n__len(z0)) → len(z0) 0.00/0.85
activate(z0) → z0
Tuples:

FST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z0), ACTIVATE(z2)) 0.00/0.85
ADD(s(z0), z1) → c6(ACTIVATE(z0)) 0.00/0.85
LEN(cons(z0, z1)) → c9(ACTIVATE(z1)) 0.00/0.85
ACTIVATE(n__fst(z0, z1)) → c11(FST(z0, z1)) 0.00/0.85
ACTIVATE(n__add(z0, z1)) → c13(ADD(z0, z1)) 0.00/0.85
ACTIVATE(n__len(z0)) → c14(LEN(z0)) 0.00/0.85
ACTIVATE(n__from(z0)) → c12
S tuples:

FST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z0), ACTIVATE(z2)) 0.00/0.85
ADD(s(z0), z1) → c6(ACTIVATE(z0)) 0.00/0.85
LEN(cons(z0, z1)) → c9(ACTIVATE(z1)) 0.00/0.85
ACTIVATE(n__fst(z0, z1)) → c11(FST(z0, z1)) 0.00/0.85
ACTIVATE(n__add(z0, z1)) → c13(ADD(z0, z1)) 0.00/0.85
ACTIVATE(n__len(z0)) → c14(LEN(z0)) 0.00/0.85
ACTIVATE(n__from(z0)) → c12
K tuples:none
Defined Rule Symbols:

fst, from, add, len, activate

Defined Pair Symbols:

FST, ADD, LEN, ACTIVATE

Compound Symbols:

c1, c6, c9, c11, c13, c14, c12

0.00/0.85
0.00/0.85

(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.

FST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z0), ACTIVATE(z2)) 0.00/0.85
ADD(s(z0), z1) → c6(ACTIVATE(z0)) 0.00/0.85
LEN(cons(z0, z1)) → c9(ACTIVATE(z1)) 0.00/0.85
ACTIVATE(n__fst(z0, z1)) → c11(FST(z0, z1)) 0.00/0.85
ACTIVATE(n__add(z0, z1)) → c13(ADD(z0, z1)) 0.00/0.85
ACTIVATE(n__len(z0)) → c14(LEN(z0)) 0.00/0.85
ACTIVATE(n__from(z0)) → c12
We considered the (Usable) Rules:none
And the Tuples:

FST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z0), ACTIVATE(z2)) 0.00/0.85
ADD(s(z0), z1) → c6(ACTIVATE(z0)) 0.00/0.85
LEN(cons(z0, z1)) → c9(ACTIVATE(z1)) 0.00/0.85
ACTIVATE(n__fst(z0, z1)) → c11(FST(z0, z1)) 0.00/0.85
ACTIVATE(n__add(z0, z1)) → c13(ADD(z0, z1)) 0.00/0.85
ACTIVATE(n__len(z0)) → c14(LEN(z0)) 0.00/0.85
ACTIVATE(n__from(z0)) → c12
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.85

POL(ACTIVATE(x1)) = [2] + [2]x1    0.00/0.85
POL(ADD(x1, x2)) = [2] + [2]x1 + [2]x2    0.00/0.85
POL(FST(x1, x2)) = [4] + [2]x1 + [2]x2    0.00/0.85
POL(LEN(x1)) = [4] + [2]x1    0.00/0.85
POL(c1(x1, x2)) = x1 + x2    0.00/0.85
POL(c11(x1)) = x1    0.00/0.85
POL(c12) = 0    0.00/0.85
POL(c13(x1)) = x1    0.00/0.85
POL(c14(x1)) = x1    0.00/0.85
POL(c6(x1)) = x1    0.00/0.85
POL(c9(x1)) = x1    0.00/0.85
POL(cons(x1, x2)) = x2    0.00/0.85
POL(n__add(x1, x2)) = [3] + x1 + x2    0.00/0.85
POL(n__from(x1)) = 0    0.00/0.85
POL(n__fst(x1, x2)) = [4] + x1 + x2    0.00/0.85
POL(n__len(x1)) = [4] + x1    0.00/0.85
POL(s(x1)) = [1] + x1   
0.00/0.85
0.00/0.85

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

fst(0, z0) → nil 0.00/0.85
fst(s(z0), cons(z1, z2)) → cons(z1, n__fst(activate(z0), activate(z2))) 0.00/0.85
fst(z0, z1) → n__fst(z0, z1) 0.00/0.85
from(z0) → cons(z0, n__from(s(z0))) 0.00/0.85
from(z0) → n__from(z0) 0.00/0.85
add(0, z0) → z0 0.00/0.85
add(s(z0), z1) → s(n__add(activate(z0), z1)) 0.00/0.85
add(z0, z1) → n__add(z0, z1) 0.00/0.85
len(nil) → 0 0.00/0.85
len(cons(z0, z1)) → s(n__len(activate(z1))) 0.00/0.85
len(z0) → n__len(z0) 0.00/0.85
activate(n__fst(z0, z1)) → fst(z0, z1) 0.00/0.85
activate(n__from(z0)) → from(z0) 0.00/0.85
activate(n__add(z0, z1)) → add(z0, z1) 0.00/0.85
activate(n__len(z0)) → len(z0) 0.00/0.85
activate(z0) → z0
Tuples:

FST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z0), ACTIVATE(z2)) 0.00/0.85
ADD(s(z0), z1) → c6(ACTIVATE(z0)) 0.00/0.85
LEN(cons(z0, z1)) → c9(ACTIVATE(z1)) 0.00/0.85
ACTIVATE(n__fst(z0, z1)) → c11(FST(z0, z1)) 0.00/0.85
ACTIVATE(n__add(z0, z1)) → c13(ADD(z0, z1)) 0.00/0.85
ACTIVATE(n__len(z0)) → c14(LEN(z0)) 0.00/0.85
ACTIVATE(n__from(z0)) → c12
S tuples:none
K tuples:

FST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z0), ACTIVATE(z2)) 0.00/0.85
ADD(s(z0), z1) → c6(ACTIVATE(z0)) 0.00/0.85
LEN(cons(z0, z1)) → c9(ACTIVATE(z1)) 0.00/0.85
ACTIVATE(n__fst(z0, z1)) → c11(FST(z0, z1)) 0.00/0.85
ACTIVATE(n__add(z0, z1)) → c13(ADD(z0, z1)) 0.00/0.85
ACTIVATE(n__len(z0)) → c14(LEN(z0)) 0.00/0.85
ACTIVATE(n__from(z0)) → c12
Defined Rule Symbols:

fst, from, add, len, activate

Defined Pair Symbols:

FST, ADD, LEN, ACTIVATE

Compound Symbols:

c1, c6, c9, c11, c13, c14, c12

0.00/0.85
0.00/0.85

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

The set S is empty
0.00/0.85
0.00/0.85

(10) BOUNDS(O(1), O(1))

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0.00/0.85
0.00/0.89 EOF