YES(O(1), O(n^1)) 0.00/0.81 YES(O(1), O(n^1)) 0.00/0.82 0.00/0.82 0.00/0.82 0.00/0.82 0.00/0.82 0.00/0.82 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.82 0.00/0.82 0.00/0.82
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The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxRelTRS
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↳1 CpxRelTrsToCDT (UPPER BOUND (ID))
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↳2 CdtProblem
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↳3 CdtLeafRemovalProof (ComplexityIfPolyImplication)
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↳4 CdtProblem
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↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳6 CdtProblem
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↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳8 CdtProblem
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↳9 SIsEmptyProof (BOTH BOUNDS(ID, ID))
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↳10 BOUNDS(O(1), O(1))
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(0) Obligation:

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

map(Cons(x, xs)) → Cons(f(x), map(xs)) 0.00/0.82
map(Nil) → Nil 0.00/0.82
goal(xs) → map(xs) 0.00/0.82
f(x) → *(x, x) 0.00/0.82
+Full(S(x), y) → +Full(x, S(y)) 0.00/0.82
+Full(0, y) → y

The (relative) TRS S consists of the following rules:

*(x, S(S(y))) → +(x, *(x, S(y))) 0.00/0.82
*(x, S(0)) → x 0.00/0.82
*(x, 0) → 0 0.00/0.82
*(0, y) → 0

Rewrite Strategy: INNERMOST
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(1) CpxRelTrsToCDT (UPPER BOUND (ID) transformation)

Relative innermost TRS to CDT Problem.
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(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

*(z0, S(S(z1))) → +(z0, *(z0, S(z1))) 0.00/0.82
*(z0, S(0)) → z0 0.00/0.82
*(z0, 0) → 0 0.00/0.82
*(0, z0) → 0 0.00/0.82
map(Cons(z0, z1)) → Cons(f(z0), map(z1)) 0.00/0.82
map(Nil) → Nil 0.00/0.82
goal(z0) → map(z0) 0.00/0.82
f(z0) → *(z0, z0) 0.00/0.82
+Full(S(z0), z1) → +Full(z0, S(z1)) 0.00/0.82
+Full(0, z0) → z0
Tuples:

*'(z0, S(S(z1))) → c(*'(z0, S(z1))) 0.00/0.82
MAP(Cons(z0, z1)) → c4(F(z0), MAP(z1)) 0.00/0.82
GOAL(z0) → c6(MAP(z0)) 0.00/0.82
F(z0) → c7(*'(z0, z0)) 0.00/0.82
+FULL(S(z0), z1) → c8(+FULL(z0, S(z1)))
S tuples:

MAP(Cons(z0, z1)) → c4(F(z0), MAP(z1)) 0.00/0.82
GOAL(z0) → c6(MAP(z0)) 0.00/0.82
F(z0) → c7(*'(z0, z0)) 0.00/0.82
+FULL(S(z0), z1) → c8(+FULL(z0, S(z1)))
K tuples:none
Defined Rule Symbols:

map, goal, f, +Full, *

Defined Pair Symbols:

*', MAP, GOAL, F, +FULL

Compound Symbols:

c, c4, c6, c7, c8

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(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

GOAL(z0) → c6(MAP(z0))
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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

*(z0, S(S(z1))) → +(z0, *(z0, S(z1))) 0.00/0.82
*(z0, S(0)) → z0 0.00/0.82
*(z0, 0) → 0 0.00/0.82
*(0, z0) → 0 0.00/0.82
map(Cons(z0, z1)) → Cons(f(z0), map(z1)) 0.00/0.82
map(Nil) → Nil 0.00/0.82
goal(z0) → map(z0) 0.00/0.82
f(z0) → *(z0, z0) 0.00/0.82
+Full(S(z0), z1) → +Full(z0, S(z1)) 0.00/0.82
+Full(0, z0) → z0
Tuples:

*'(z0, S(S(z1))) → c(*'(z0, S(z1))) 0.00/0.82
MAP(Cons(z0, z1)) → c4(F(z0), MAP(z1)) 0.00/0.82
F(z0) → c7(*'(z0, z0)) 0.00/0.82
+FULL(S(z0), z1) → c8(+FULL(z0, S(z1)))
S tuples:

MAP(Cons(z0, z1)) → c4(F(z0), MAP(z1)) 0.00/0.82
F(z0) → c7(*'(z0, z0)) 0.00/0.82
+FULL(S(z0), z1) → c8(+FULL(z0, S(z1)))
K tuples:none
Defined Rule Symbols:

map, goal, f, +Full, *

Defined Pair Symbols:

*', MAP, F, +FULL

Compound Symbols:

c, c4, c7, c8

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(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.

F(z0) → c7(*'(z0, z0)) 0.00/0.82
+FULL(S(z0), z1) → c8(+FULL(z0, S(z1)))
We considered the (Usable) Rules:none
And the Tuples:

*'(z0, S(S(z1))) → c(*'(z0, S(z1))) 0.00/0.83
MAP(Cons(z0, z1)) → c4(F(z0), MAP(z1)) 0.00/0.83
F(z0) → c7(*'(z0, z0)) 0.00/0.83
+FULL(S(z0), z1) → c8(+FULL(z0, S(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.83

POL(*'(x1, x2)) = 0    0.00/0.83
POL(+FULL(x1, x2)) = [4]x1 + [3]x2    0.00/0.83
POL(Cons(x1, x2)) = [1] + x1 + x2    0.00/0.83
POL(F(x1)) = [1]    0.00/0.83
POL(MAP(x1)) = x1    0.00/0.83
POL(S(x1)) = [2] + x1    0.00/0.83
POL(c(x1)) = x1    0.00/0.83
POL(c4(x1, x2)) = x1 + x2    0.00/0.83
POL(c7(x1)) = x1    0.00/0.83
POL(c8(x1)) = x1   
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

*(z0, S(S(z1))) → +(z0, *(z0, S(z1))) 0.00/0.83
*(z0, S(0)) → z0 0.00/0.83
*(z0, 0) → 0 0.00/0.83
*(0, z0) → 0 0.00/0.83
map(Cons(z0, z1)) → Cons(f(z0), map(z1)) 0.00/0.83
map(Nil) → Nil 0.00/0.83
goal(z0) → map(z0) 0.00/0.83
f(z0) → *(z0, z0) 0.00/0.83
+Full(S(z0), z1) → +Full(z0, S(z1)) 0.00/0.83
+Full(0, z0) → z0
Tuples:

*'(z0, S(S(z1))) → c(*'(z0, S(z1))) 0.00/0.83
MAP(Cons(z0, z1)) → c4(F(z0), MAP(z1)) 0.00/0.83
F(z0) → c7(*'(z0, z0)) 0.00/0.83
+FULL(S(z0), z1) → c8(+FULL(z0, S(z1)))
S tuples:

MAP(Cons(z0, z1)) → c4(F(z0), MAP(z1))
K tuples:

F(z0) → c7(*'(z0, z0)) 0.00/0.83
+FULL(S(z0), z1) → c8(+FULL(z0, S(z1)))
Defined Rule Symbols:

map, goal, f, +Full, *

Defined Pair Symbols:

*', MAP, F, +FULL

Compound Symbols:

c, c4, c7, c8

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(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.

MAP(Cons(z0, z1)) → c4(F(z0), MAP(z1))
We considered the (Usable) Rules:none
And the Tuples:

*'(z0, S(S(z1))) → c(*'(z0, S(z1))) 0.00/0.83
MAP(Cons(z0, z1)) → c4(F(z0), MAP(z1)) 0.00/0.83
F(z0) → c7(*'(z0, z0)) 0.00/0.83
+FULL(S(z0), z1) → c8(+FULL(z0, S(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.83

POL(*'(x1, x2)) = 0    0.00/0.83
POL(+FULL(x1, x2)) = [3]x1    0.00/0.83
POL(Cons(x1, x2)) = [1] + x1 + x2    0.00/0.83
POL(F(x1)) = 0    0.00/0.83
POL(MAP(x1)) = x1    0.00/0.83
POL(S(x1)) = [2] + x1    0.00/0.83
POL(c(x1)) = x1    0.00/0.83
POL(c4(x1, x2)) = x1 + x2    0.00/0.83
POL(c7(x1)) = x1    0.00/0.83
POL(c8(x1)) = x1   
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(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

*(z0, S(S(z1))) → +(z0, *(z0, S(z1))) 0.00/0.83
*(z0, S(0)) → z0 0.00/0.83
*(z0, 0) → 0 0.00/0.83
*(0, z0) → 0 0.00/0.83
map(Cons(z0, z1)) → Cons(f(z0), map(z1)) 0.00/0.83
map(Nil) → Nil 0.00/0.83
goal(z0) → map(z0) 0.00/0.83
f(z0) → *(z0, z0) 0.00/0.83
+Full(S(z0), z1) → +Full(z0, S(z1)) 0.00/0.83
+Full(0, z0) → z0
Tuples:

*'(z0, S(S(z1))) → c(*'(z0, S(z1))) 0.00/0.83
MAP(Cons(z0, z1)) → c4(F(z0), MAP(z1)) 0.00/0.83
F(z0) → c7(*'(z0, z0)) 0.00/0.83
+FULL(S(z0), z1) → c8(+FULL(z0, S(z1)))
S tuples:none
K tuples:

F(z0) → c7(*'(z0, z0)) 0.00/0.83
+FULL(S(z0), z1) → c8(+FULL(z0, S(z1))) 0.00/0.83
MAP(Cons(z0, z1)) → c4(F(z0), MAP(z1))
Defined Rule Symbols:

map, goal, f, +Full, *

Defined Pair Symbols:

*', MAP, F, +FULL

Compound Symbols:

c, c4, c7, c8

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

The set S is empty
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(10) BOUNDS(O(1), O(1))

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0.00/0.83 EOF