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

(0) Obligation:

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

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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

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

S tuples:

GROUP3(z0) → c(GROUP3#1(z0))
GROUP3#1(::(z0, z1)) → c1(GROUP3#2(z1, z0))
GROUP3#2(::(z0, z1), z2) → c3(GROUP3#3(z1, z2, z0))
GROUP3#3(::(z0, z1), z2, z3) → c5(GROUP3(z1))
ZIP3(z0, z1, z2) → c7(ZIP3#1(z0, z1, z2))
ZIP3#1(::(z0, z1), z2, z3) → c8(ZIP3#2(z2, z3, z0, z1))
ZIP3#2(::(z0, z1), z2, z3, z4) → c10(ZIP3#3(z2, z3, z4, z0, z1))
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

(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.

ZIP3(z0, z1, z2) → c7(ZIP3#1(z0, z1, z2))
ZIP3#1(::(z0, z1), z2, z3) → c8(ZIP3#2(z2, z3, z0, z1))
ZIP3#2(::(z0, z1), z2, z3, z4) → c10(ZIP3#3(z2, z3, z4, z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

GROUP3(z0) → c(GROUP3#1(z0))
GROUP3#1(::(z0, z1)) → c1(GROUP3#2(z1, z0))
GROUP3#2(::(z0, z1), z2) → c3(GROUP3#3(z1, z2, z0))
GROUP3#3(::(z0, z1), z2, z3) → c5(GROUP3(z1))
ZIP3(z0, z1, z2) → c7(ZIP3#1(z0, z1, z2))
ZIP3#1(::(z0, z1), z2, z3) → c8(ZIP3#2(z2, z3, z0, z1))
ZIP3#2(::(z0, z1), z2, z3, z4) → c10(ZIP3#3(z2, z3, z4, z0, z1))
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 :

POL(::(x₁, x₂)) = [4] + x₁ + x₂   
POL(GROUP3(x₁)) = 0   
POL(GROUP3#1(x₁)) = 0   
POL(GROUP3#2(x₁, x₂)) = 0   
POL(GROUP3#3(x₁, x₂, x₃)) = 0   
POL(ZIP3(x₁, x₂, x₃)) = [2] + x₁   
POL(ZIP3#1(x₁, x₂, x₃)) = [1] + x₁   
POL(ZIP3#2(x₁, x₂, x₃, x₄)) = [3] + x₄   
POL(ZIP3#3(x₁, x₂, x₃, x₄, x₅)) = [2] + x₃   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

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

S tuples:

GROUP3(z0) → c(GROUP3#1(z0))
GROUP3#1(::(z0, z1)) → c1(GROUP3#2(z1, z0))
GROUP3#2(::(z0, z1), z2) → c3(GROUP3#3(z1, z2, z0))
GROUP3#3(::(z0, z1), z2, z3) → c5(GROUP3(z1))
ZIP3#3(::(z0, z1), z2, z3, z4, z5) → c12(ZIP3(z3, z5, z1))

K tuples:

ZIP3(z0, z1, z2) → c7(ZIP3#1(z0, z1, z2))
ZIP3#1(::(z0, z1), z2, z3) → c8(ZIP3#2(z2, z3, z0, z1))
ZIP3#2(::(z0, z1), z2, z3, z4) → c10(ZIP3#3(z2, z3, z4, z0, 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

(5) CdtKnowledgeProof (EQUIVALENT transformation)

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

ZIP3#3(::(z0, z1), z2, z3, z4, z5) → c12(ZIP3(z3, z5, z1))
ZIP3(z0, z1, z2) → c7(ZIP3#1(z0, z1, z2))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

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

S tuples:

GROUP3(z0) → c(GROUP3#1(z0))
GROUP3#1(::(z0, z1)) → c1(GROUP3#2(z1, z0))
GROUP3#2(::(z0, z1), z2) → c3(GROUP3#3(z1, z2, z0))
GROUP3#3(::(z0, z1), z2, z3) → c5(GROUP3(z1))

K tuples:

ZIP3(z0, z1, z2) → c7(ZIP3#1(z0, z1, z2))
ZIP3#1(::(z0, z1), z2, z3) → c8(ZIP3#2(z2, z3, z0, z1))
ZIP3#2(::(z0, z1), z2, z3, z4) → c10(ZIP3#3(z2, z3, z4, z0, z1))
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

(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.

GROUP3(z0) → c(GROUP3#1(z0))
GROUP3#1(::(z0, z1)) → c1(GROUP3#2(z1, z0))
GROUP3#2(::(z0, z1), z2) → c3(GROUP3#3(z1, z2, z0))
GROUP3#3(::(z0, z1), z2, z3) → c5(GROUP3(z1))

We considered the (Usable) Rules:none
And the Tuples:

GROUP3(z0) → c(GROUP3#1(z0))
GROUP3#1(::(z0, z1)) → c1(GROUP3#2(z1, z0))
GROUP3#2(::(z0, z1), z2) → c3(GROUP3#3(z1, z2, z0))
GROUP3#3(::(z0, z1), z2, z3) → c5(GROUP3(z1))
ZIP3(z0, z1, z2) → c7(ZIP3#1(z0, z1, z2))
ZIP3#1(::(z0, z1), z2, z3) → c8(ZIP3#2(z2, z3, z0, z1))
ZIP3#2(::(z0, z1), z2, z3, z4) → c10(ZIP3#3(z2, z3, z4, z0, z1))
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 :

POL(::(x₁, x₂)) = [2] + x₁ + x₂   
POL(GROUP3(x₁)) = [4] + [4]x₁   
POL(GROUP3#1(x₁)) = [1] + [4]x₁   
POL(GROUP3#2(x₁, x₂)) = [3] + [4]x₁ + [2]x₂   
POL(GROUP3#3(x₁, x₂, x₃)) = [4]x₁ + [2]x₂ + [3]x₃   
POL(ZIP3(x₁, x₂, x₃)) = [4] + [2]x₁ + x₂ + [4]x₃   
POL(ZIP3#1(x₁, x₂, x₃)) = [4] + [2]x₁ + x₂ + [4]x₃   
POL(ZIP3#2(x₁, x₂, x₃, x₄)) = [2] + x₁ + [4]x₂ + [2]x₃ + [2]x₄   
POL(ZIP3#3(x₁, x₂, x₃, x₄, x₅)) = [4]x₁ + [2]x₃ + x₄ + x₅   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

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

S tuples:none
K tuples:

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

(9) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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