YES(O(1), O(n^1)) 2.44/1.05 YES(O(1), O(n^1)) 2.44/1.10 2.44/1.10 2.44/1.10 2.44/1.10 2.44/1.10 2.44/1.10 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 2.44/1.10 2.44/1.10 2.44/1.10
2.44/1.10 2.44/1.10 2.44/1.10
2.44/1.10
2.44/1.10

(0) Obligation:

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

group3(@l) → group3#1(@l) 2.44/1.10
group3#1(::(@x, @xs)) → group3#2(@xs, @x) 2.44/1.10
group3#1(nil) → nil 2.44/1.10
group3#2(::(@y, @ys), @x) → group3#3(@ys, @x, @y) 2.44/1.10
group3#2(nil, @x) → nil 2.44/1.10
group3#3(::(@z, @zs), @x, @y) → ::(tuple#3(@x, @y, @z), group3(@zs)) 2.44/1.10
group3#3(nil, @x, @y) → nil 2.44/1.10
zip3(@l1, @l2, @l3) → zip3#1(@l1, @l2, @l3) 2.44/1.10
zip3#1(::(@x, @xs), @l2, @l3) → zip3#2(@l2, @l3, @x, @xs) 2.44/1.10
zip3#1(nil, @l2, @l3) → nil 2.44/1.10
zip3#2(::(@y, @ys), @l3, @x, @xs) → zip3#3(@l3, @x, @xs, @y, @ys) 2.44/1.10
zip3#2(nil, @l3, @x, @xs) → nil 2.44/1.10
zip3#3(::(@z, @zs), @x, @xs, @y, @ys) → ::(tuple#3(@x, @y, @z), zip3(@xs, @ys, @zs)) 2.44/1.10
zip3#3(nil, @x, @xs, @y, @ys) → nil

Rewrite Strategy: INNERMOST
2.44/1.10
2.44/1.10

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

Converted CpxTRS to CDT
2.44/1.10
2.44/1.10

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

group3(z0) → group3#1(z0) 2.44/1.10
group3#1(::(z0, z1)) → group3#2(z1, z0) 2.44/1.10
group3#1(nil) → nil 2.44/1.10
group3#2(::(z0, z1), z2) → group3#3(z1, z2, z0) 2.44/1.10
group3#2(nil, z0) → nil 2.44/1.10
group3#3(::(z0, z1), z2, z3) → ::(tuple#3(z2, z3, z0), group3(z1)) 2.44/1.10
group3#3(nil, z0, z1) → nil 2.44/1.10
zip3(z0, z1, z2) → zip3#1(z0, z1, z2) 2.44/1.10
zip3#1(::(z0, z1), z2, z3) → zip3#2(z2, z3, z0, z1) 2.44/1.10
zip3#1(nil, z0, z1) → nil 2.44/1.10
zip3#2(::(z0, z1), z2, z3, z4) → zip3#3(z2, z3, z4, z0, z1) 2.44/1.10
zip3#2(nil, z0, z1, z2) → nil 2.44/1.10
zip3#3(::(z0, z1), z2, z3, z4, z5) → ::(tuple#3(z2, z4, z0), zip3(z3, z5, z1)) 2.44/1.10
zip3#3(nil, z0, z1, z2, z3) → nil
Tuples:

GROUP3(z0) → c(GROUP3#1(z0)) 2.44/1.10
GROUP3#1(::(z0, z1)) → c1(GROUP3#2(z1, z0)) 2.44/1.10
GROUP3#2(::(z0, z1), z2) → c3(GROUP3#3(z1, z2, z0)) 2.44/1.10
GROUP3#3(::(z0, z1), z2, z3) → c5(GROUP3(z1)) 2.44/1.10
ZIP3(z0, z1, z2) → c7(ZIP3#1(z0, z1, z2)) 2.44/1.10
ZIP3#1(::(z0, z1), z2, z3) → c8(ZIP3#2(z2, z3, z0, z1)) 2.44/1.10
ZIP3#2(::(z0, z1), z2, z3, z4) → c10(ZIP3#3(z2, z3, z4, z0, z1)) 2.44/1.10
ZIP3#3(::(z0, z1), z2, z3, z4, z5) → c12(ZIP3(z3, z5, z1))
S tuples:

GROUP3(z0) → c(GROUP3#1(z0)) 2.44/1.10
GROUP3#1(::(z0, z1)) → c1(GROUP3#2(z1, z0)) 2.44/1.10
GROUP3#2(::(z0, z1), z2) → c3(GROUP3#3(z1, z2, z0)) 2.44/1.10
GROUP3#3(::(z0, z1), z2, z3) → c5(GROUP3(z1)) 2.44/1.10
ZIP3(z0, z1, z2) → c7(ZIP3#1(z0, z1, z2)) 2.44/1.10
ZIP3#1(::(z0, z1), z2, z3) → c8(ZIP3#2(z2, z3, z0, z1)) 2.44/1.10
ZIP3#2(::(z0, z1), z2, z3, z4) → c10(ZIP3#3(z2, z3, z4, z0, z1)) 2.44/1.10
ZIP3#3(::(z0, z1), z2, z3, z4, z5) → c12(ZIP3(z3, z5, z1))
K tuples:none
Defined Rule Symbols:

group3, group3#1, group3#2, group3#3, zip3, zip3#1, zip3#2, zip3#3

Defined Pair Symbols:

GROUP3, GROUP3#1, GROUP3#2, GROUP3#3, ZIP3, ZIP3#1, ZIP3#2, ZIP3#3

Compound Symbols:

c, c1, c3, c5, c7, c8, c10, c12

2.44/1.10
2.44/1.10

(3) 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.

GROUP3#1(::(z0, z1)) → c1(GROUP3#2(z1, z0)) 2.44/1.10
GROUP3#2(::(z0, z1), z2) → c3(GROUP3#3(z1, z2, z0)) 2.44/1.10
GROUP3#3(::(z0, z1), z2, z3) → c5(GROUP3(z1))
We considered the (Usable) Rules:none
And the Tuples:

GROUP3(z0) → c(GROUP3#1(z0)) 2.44/1.10
GROUP3#1(::(z0, z1)) → c1(GROUP3#2(z1, z0)) 2.44/1.10
GROUP3#2(::(z0, z1), z2) → c3(GROUP3#3(z1, z2, z0)) 2.44/1.10
GROUP3#3(::(z0, z1), z2, z3) → c5(GROUP3(z1)) 2.44/1.10
ZIP3(z0, z1, z2) → c7(ZIP3#1(z0, z1, z2)) 2.44/1.10
ZIP3#1(::(z0, z1), z2, z3) → c8(ZIP3#2(z2, z3, z0, z1)) 2.44/1.10
ZIP3#2(::(z0, z1), z2, z3, z4) → c10(ZIP3#3(z2, z3, z4, z0, z1)) 2.44/1.10
ZIP3#3(::(z0, z1), z2, z3, z4, z5) → c12(ZIP3(z3, z5, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 2.44/1.10

POL(::(x1, x2)) = [2] + x1 + x2    2.44/1.10
POL(GROUP3(x1)) = [2]x1    2.44/1.10
POL(GROUP3#1(x1)) = [2]x1    2.44/1.10
POL(GROUP3#2(x1, x2)) = [1] + [2]x1 + [2]x2    2.44/1.10
POL(GROUP3#3(x1, x2, x3)) = [2] + [2]x1 + x2    2.44/1.10
POL(ZIP3(x1, x2, x3)) = 0    2.44/1.10
POL(ZIP3#1(x1, x2, x3)) = 0    2.44/1.10
POL(ZIP3#2(x1, x2, x3, x4)) = 0    2.44/1.10
POL(ZIP3#3(x1, x2, x3, x4, x5)) = 0    2.44/1.10
POL(c(x1)) = x1    2.44/1.10
POL(c1(x1)) = x1    2.44/1.10
POL(c10(x1)) = x1    2.44/1.10
POL(c12(x1)) = x1    2.44/1.10
POL(c3(x1)) = x1    2.44/1.10
POL(c5(x1)) = x1    2.44/1.10
POL(c7(x1)) = x1    2.44/1.10
POL(c8(x1)) = x1   
2.44/1.10
2.44/1.10

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

group3(z0) → group3#1(z0) 2.44/1.10
group3#1(::(z0, z1)) → group3#2(z1, z0) 2.44/1.10
group3#1(nil) → nil 2.44/1.10
group3#2(::(z0, z1), z2) → group3#3(z1, z2, z0) 2.44/1.10
group3#2(nil, z0) → nil 2.44/1.10
group3#3(::(z0, z1), z2, z3) → ::(tuple#3(z2, z3, z0), group3(z1)) 2.44/1.10
group3#3(nil, z0, z1) → nil 2.44/1.10
zip3(z0, z1, z2) → zip3#1(z0, z1, z2) 2.44/1.10
zip3#1(::(z0, z1), z2, z3) → zip3#2(z2, z3, z0, z1) 2.44/1.10
zip3#1(nil, z0, z1) → nil 2.44/1.10
zip3#2(::(z0, z1), z2, z3, z4) → zip3#3(z2, z3, z4, z0, z1) 2.44/1.10
zip3#2(nil, z0, z1, z2) → nil 2.44/1.10
zip3#3(::(z0, z1), z2, z3, z4, z5) → ::(tuple#3(z2, z4, z0), zip3(z3, z5, z1)) 2.44/1.10
zip3#3(nil, z0, z1, z2, z3) → nil
Tuples:

GROUP3(z0) → c(GROUP3#1(z0)) 2.44/1.10
GROUP3#1(::(z0, z1)) → c1(GROUP3#2(z1, z0)) 2.44/1.10
GROUP3#2(::(z0, z1), z2) → c3(GROUP3#3(z1, z2, z0)) 2.44/1.10
GROUP3#3(::(z0, z1), z2, z3) → c5(GROUP3(z1)) 2.44/1.10
ZIP3(z0, z1, z2) → c7(ZIP3#1(z0, z1, z2)) 2.44/1.10
ZIP3#1(::(z0, z1), z2, z3) → c8(ZIP3#2(z2, z3, z0, z1)) 2.44/1.10
ZIP3#2(::(z0, z1), z2, z3, z4) → c10(ZIP3#3(z2, z3, z4, z0, z1)) 2.44/1.10
ZIP3#3(::(z0, z1), z2, z3, z4, z5) → c12(ZIP3(z3, z5, z1))
S tuples:

GROUP3(z0) → c(GROUP3#1(z0)) 2.44/1.10
ZIP3(z0, z1, z2) → c7(ZIP3#1(z0, z1, z2)) 2.44/1.10
ZIP3#1(::(z0, z1), z2, z3) → c8(ZIP3#2(z2, z3, z0, z1)) 2.44/1.10
ZIP3#2(::(z0, z1), z2, z3, z4) → c10(ZIP3#3(z2, z3, z4, z0, z1)) 2.44/1.10
ZIP3#3(::(z0, z1), z2, z3, z4, z5) → c12(ZIP3(z3, z5, z1))
K tuples:

GROUP3#1(::(z0, z1)) → c1(GROUP3#2(z1, z0)) 2.44/1.10
GROUP3#2(::(z0, z1), z2) → c3(GROUP3#3(z1, z2, z0)) 2.44/1.10
GROUP3#3(::(z0, z1), z2, z3) → c5(GROUP3(z1))
Defined Rule Symbols:

group3, group3#1, group3#2, group3#3, zip3, zip3#1, zip3#2, zip3#3

Defined Pair Symbols:

GROUP3, GROUP3#1, GROUP3#2, GROUP3#3, ZIP3, ZIP3#1, ZIP3#2, ZIP3#3

Compound Symbols:

c, c1, c3, c5, c7, c8, c10, c12

2.44/1.10
2.44/1.10

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

The following tuples could be moved from S to K by knowledge propagation:

GROUP3(z0) → c(GROUP3#1(z0)) 2.44/1.10
GROUP3#1(::(z0, z1)) → c1(GROUP3#2(z1, z0))
2.44/1.10
2.44/1.10

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

group3(z0) → group3#1(z0) 2.44/1.10
group3#1(::(z0, z1)) → group3#2(z1, z0) 2.44/1.10
group3#1(nil) → nil 2.44/1.10
group3#2(::(z0, z1), z2) → group3#3(z1, z2, z0) 2.44/1.10
group3#2(nil, z0) → nil 2.44/1.10
group3#3(::(z0, z1), z2, z3) → ::(tuple#3(z2, z3, z0), group3(z1)) 2.44/1.10
group3#3(nil, z0, z1) → nil 2.44/1.10
zip3(z0, z1, z2) → zip3#1(z0, z1, z2) 2.44/1.10
zip3#1(::(z0, z1), z2, z3) → zip3#2(z2, z3, z0, z1) 2.44/1.10
zip3#1(nil, z0, z1) → nil 2.44/1.10
zip3#2(::(z0, z1), z2, z3, z4) → zip3#3(z2, z3, z4, z0, z1) 2.44/1.10
zip3#2(nil, z0, z1, z2) → nil 2.44/1.10
zip3#3(::(z0, z1), z2, z3, z4, z5) → ::(tuple#3(z2, z4, z0), zip3(z3, z5, z1)) 2.44/1.10
zip3#3(nil, z0, z1, z2, z3) → nil
Tuples:

GROUP3(z0) → c(GROUP3#1(z0)) 2.44/1.10
GROUP3#1(::(z0, z1)) → c1(GROUP3#2(z1, z0)) 2.44/1.10
GROUP3#2(::(z0, z1), z2) → c3(GROUP3#3(z1, z2, z0)) 2.44/1.10
GROUP3#3(::(z0, z1), z2, z3) → c5(GROUP3(z1)) 2.44/1.10
ZIP3(z0, z1, z2) → c7(ZIP3#1(z0, z1, z2)) 2.44/1.10
ZIP3#1(::(z0, z1), z2, z3) → c8(ZIP3#2(z2, z3, z0, z1)) 2.44/1.10
ZIP3#2(::(z0, z1), z2, z3, z4) → c10(ZIP3#3(z2, z3, z4, z0, z1)) 2.44/1.10
ZIP3#3(::(z0, z1), z2, z3, z4, z5) → c12(ZIP3(z3, z5, z1))
S tuples:

ZIP3(z0, z1, z2) → c7(ZIP3#1(z0, z1, z2)) 2.44/1.10
ZIP3#1(::(z0, z1), z2, z3) → c8(ZIP3#2(z2, z3, z0, z1)) 2.44/1.10
ZIP3#2(::(z0, z1), z2, z3, z4) → c10(ZIP3#3(z2, z3, z4, z0, z1)) 2.44/1.10
ZIP3#3(::(z0, z1), z2, z3, z4, z5) → c12(ZIP3(z3, z5, z1))
K tuples:

GROUP3#1(::(z0, z1)) → c1(GROUP3#2(z1, z0)) 2.44/1.10
GROUP3#2(::(z0, z1), z2) → c3(GROUP3#3(z1, z2, z0)) 2.44/1.10
GROUP3#3(::(z0, z1), z2, z3) → c5(GROUP3(z1)) 2.44/1.10
GROUP3(z0) → c(GROUP3#1(z0))
Defined Rule Symbols:

group3, group3#1, group3#2, group3#3, zip3, zip3#1, zip3#2, zip3#3

Defined Pair Symbols:

GROUP3, GROUP3#1, GROUP3#2, GROUP3#3, ZIP3, ZIP3#1, ZIP3#2, ZIP3#3

Compound Symbols:

c, c1, c3, c5, c7, c8, c10, c12

2.44/1.10
2.44/1.10

(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.

ZIP3#3(::(z0, z1), z2, z3, z4, z5) → c12(ZIP3(z3, z5, z1))
We considered the (Usable) Rules:none
And the Tuples:

GROUP3(z0) → c(GROUP3#1(z0)) 2.44/1.10
GROUP3#1(::(z0, z1)) → c1(GROUP3#2(z1, z0)) 2.44/1.10
GROUP3#2(::(z0, z1), z2) → c3(GROUP3#3(z1, z2, z0)) 2.44/1.10
GROUP3#3(::(z0, z1), z2, z3) → c5(GROUP3(z1)) 2.44/1.10
ZIP3(z0, z1, z2) → c7(ZIP3#1(z0, z1, z2)) 2.44/1.10
ZIP3#1(::(z0, z1), z2, z3) → c8(ZIP3#2(z2, z3, z0, z1)) 2.44/1.10
ZIP3#2(::(z0, z1), z2, z3, z4) → c10(ZIP3#3(z2, z3, z4, z0, z1)) 2.44/1.10
ZIP3#3(::(z0, z1), z2, z3, z4, z5) → c12(ZIP3(z3, z5, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 2.44/1.10

POL(::(x1, x2)) = [1] + x1 + x2    2.44/1.10
POL(GROUP3(x1)) = 0    2.44/1.10
POL(GROUP3#1(x1)) = 0    2.44/1.10
POL(GROUP3#2(x1, x2)) = 0    2.44/1.10
POL(GROUP3#3(x1, x2, x3)) = 0    2.44/1.10
POL(ZIP3(x1, x2, x3)) = x3    2.44/1.10
POL(ZIP3#1(x1, x2, x3)) = x3    2.44/1.10
POL(ZIP3#2(x1, x2, x3, x4)) = x2    2.44/1.10
POL(ZIP3#3(x1, x2, x3, x4, x5)) = x1    2.44/1.10
POL(c(x1)) = x1    2.44/1.10
POL(c1(x1)) = x1    2.44/1.10
POL(c10(x1)) = x1    2.44/1.10
POL(c12(x1)) = x1    2.44/1.10
POL(c3(x1)) = x1    2.44/1.10
POL(c5(x1)) = x1    2.44/1.10
POL(c7(x1)) = x1    2.44/1.10
POL(c8(x1)) = x1   
2.44/1.10
2.44/1.10

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

group3(z0) → group3#1(z0) 2.44/1.10
group3#1(::(z0, z1)) → group3#2(z1, z0) 2.44/1.10
group3#1(nil) → nil 2.44/1.10
group3#2(::(z0, z1), z2) → group3#3(z1, z2, z0) 2.44/1.10
group3#2(nil, z0) → nil 2.44/1.10
group3#3(::(z0, z1), z2, z3) → ::(tuple#3(z2, z3, z0), group3(z1)) 2.44/1.10
group3#3(nil, z0, z1) → nil 2.44/1.10
zip3(z0, z1, z2) → zip3#1(z0, z1, z2) 2.44/1.10
zip3#1(::(z0, z1), z2, z3) → zip3#2(z2, z3, z0, z1) 2.44/1.10
zip3#1(nil, z0, z1) → nil 2.44/1.10
zip3#2(::(z0, z1), z2, z3, z4) → zip3#3(z2, z3, z4, z0, z1) 2.44/1.10
zip3#2(nil, z0, z1, z2) → nil 2.44/1.10
zip3#3(::(z0, z1), z2, z3, z4, z5) → ::(tuple#3(z2, z4, z0), zip3(z3, z5, z1)) 2.44/1.10
zip3#3(nil, z0, z1, z2, z3) → nil
Tuples:

GROUP3(z0) → c(GROUP3#1(z0)) 2.44/1.10
GROUP3#1(::(z0, z1)) → c1(GROUP3#2(z1, z0)) 2.44/1.10
GROUP3#2(::(z0, z1), z2) → c3(GROUP3#3(z1, z2, z0)) 2.44/1.10
GROUP3#3(::(z0, z1), z2, z3) → c5(GROUP3(z1)) 2.44/1.10
ZIP3(z0, z1, z2) → c7(ZIP3#1(z0, z1, z2)) 2.44/1.10
ZIP3#1(::(z0, z1), z2, z3) → c8(ZIP3#2(z2, z3, z0, z1)) 2.44/1.10
ZIP3#2(::(z0, z1), z2, z3, z4) → c10(ZIP3#3(z2, z3, z4, z0, z1)) 2.44/1.10
ZIP3#3(::(z0, z1), z2, z3, z4, z5) → c12(ZIP3(z3, z5, z1))
S tuples:

ZIP3(z0, z1, z2) → c7(ZIP3#1(z0, z1, z2)) 2.44/1.10
ZIP3#1(::(z0, z1), z2, z3) → c8(ZIP3#2(z2, z3, z0, z1)) 2.44/1.10
ZIP3#2(::(z0, z1), z2, z3, z4) → c10(ZIP3#3(z2, z3, z4, z0, z1))
K tuples:

GROUP3#1(::(z0, z1)) → c1(GROUP3#2(z1, z0)) 2.44/1.10
GROUP3#2(::(z0, z1), z2) → c3(GROUP3#3(z1, z2, z0)) 2.44/1.10
GROUP3#3(::(z0, z1), z2, z3) → c5(GROUP3(z1)) 2.44/1.10
GROUP3(z0) → c(GROUP3#1(z0)) 2.44/1.10
ZIP3#3(::(z0, z1), z2, z3, z4, z5) → c12(ZIP3(z3, z5, z1))
Defined Rule Symbols:

group3, group3#1, group3#2, group3#3, zip3, zip3#1, zip3#2, zip3#3

Defined Pair Symbols:

GROUP3, GROUP3#1, GROUP3#2, GROUP3#3, ZIP3, ZIP3#1, ZIP3#2, ZIP3#3

Compound Symbols:

c, c1, c3, c5, c7, c8, c10, c12

2.44/1.10
2.44/1.10

(9) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

ZIP3(z0, z1, z2) → c7(ZIP3#1(z0, z1, z2)) 2.44/1.10
ZIP3#1(::(z0, z1), z2, z3) → c8(ZIP3#2(z2, z3, z0, z1)) 2.44/1.10
ZIP3#2(::(z0, z1), z2, z3, z4) → c10(ZIP3#3(z2, z3, z4, z0, z1)) 2.44/1.10
ZIP3#1(::(z0, z1), z2, z3) → c8(ZIP3#2(z2, z3, z0, z1)) 2.44/1.10
ZIP3#2(::(z0, z1), z2, z3, z4) → c10(ZIP3#3(z2, z3, z4, z0, z1)) 2.44/1.10
ZIP3#3(::(z0, z1), z2, z3, z4, z5) → c12(ZIP3(z3, z5, z1))
Now S is empty
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(10) BOUNDS(O(1), O(1))

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2.81/1.16 EOF