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

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CdtProblem
↳3 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CdtProblem
↳5 CdtUsableRulesProof (⇔, 0 ms)
↳6 CdtProblem
↳7 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 129 ms)
↳8 CdtProblem
↳9 CdtKnowledgeProof (BOTH BOUNDS(ID, ID), 0 ms)
↳10 CdtProblem
↳11 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 23 ms)
↳12 CdtProblem
↳13 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳14 BOUNDS(O(1), O(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 Cpx (relative) TRS 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#1(nil) → c2
GROUP3#2(::(z0, z1), z2) → c3(GROUP3#3(z1, z2, z0))
GROUP3#2(nil, z0) → c4
GROUP3#3(::(z0, z1), z2, z3) → c5(GROUP3(z1))
GROUP3#3(nil, z0, z1) → c6
ZIP3(z0, z1, z2) → c7(ZIP3#1(z0, z1, z2))
ZIP3#1(::(z0, z1), z2, z3) → c8(ZIP3#2(z2, z3, z0, z1))
ZIP3#1(nil, z0, z1) → c9
ZIP3#2(::(z0, z1), z2, z3, z4) → c10(ZIP3#3(z2, z3, z4, z0, z1))
ZIP3#2(nil, z0, z1, z2) → c11
ZIP3#3(::(z0, z1), z2, z3, z4, z5) → c12(ZIP3(z3, z5, z1))
ZIP3#3(nil, z0, z1, z2, z3) → c13
S tuples:

GROUP3(z0) → c(GROUP3#1(z0))
GROUP3#1(::(z0, z1)) → c1(GROUP3#2(z1, z0))
GROUP3#1(nil) → c2
GROUP3#2(::(z0, z1), z2) → c3(GROUP3#3(z1, z2, z0))
GROUP3#2(nil, z0) → c4
GROUP3#3(::(z0, z1), z2, z3) → c5(GROUP3(z1))
GROUP3#3(nil, z0, z1) → c6
ZIP3(z0, z1, z2) → c7(ZIP3#1(z0, z1, z2))
ZIP3#1(::(z0, z1), z2, z3) → c8(ZIP3#2(z2, z3, z0, z1))
ZIP3#1(nil, z0, z1) → c9
ZIP3#2(::(z0, z1), z2, z3, z4) → c10(ZIP3#3(z2, z3, z4, z0, z1))
ZIP3#2(nil, z0, z1, z2) → c11
ZIP3#3(::(z0, z1), z2, z3, z4, z5) → c12(ZIP3(z3, z5, z1))
ZIP3#3(nil, z0, z1, z2, z3) → c13
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, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13

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

Removed 6 trailing nodes:

ZIP3#2(nil, z0, z1, z2) → c11
GROUP3#1(nil) → c2
GROUP3#2(nil, z0) → c4
ZIP3#1(nil, z0, z1) → c9
GROUP3#3(nil, z0, z1) → c6
ZIP3#3(nil, z0, z1, z2, z3) → c13

(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(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

(5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
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:none

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) CdtRuleRemovalProof (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   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
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:none

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) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) 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))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
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:none

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

(11) CdtRuleRemovalProof (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   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:none
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:none

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

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

The set S is empty

(14) BOUNDS(O(1), O(1))