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

(0) Obligation:

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


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) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3, 4, 5, 6, 7, 8]
transitions:
::0(0, 0) → 0
nil0() → 0
tuple#30(0, 0, 0) → 0
group30(0) → 1
group3#10(0) → 2
group3#20(0, 0) → 3
group3#30(0, 0, 0) → 4
zip30(0, 0, 0) → 5
zip3#10(0, 0, 0) → 6
zip3#20(0, 0, 0, 0) → 7
zip3#30(0, 0, 0, 0, 0) → 8
group3#11(0) → 1
group3#21(0, 0) → 2
nil1() → 2
group3#31(0, 0, 0) → 3
nil1() → 3
tuple#31(0, 0, 0) → 9
group31(0) → 10
::1(9, 10) → 4
nil1() → 4
zip3#11(0, 0, 0) → 5
zip3#21(0, 0, 0, 0) → 6
nil1() → 6
zip3#31(0, 0, 0, 0, 0) → 7
nil1() → 7
zip31(0, 0, 0) → 11
::1(9, 11) → 8
nil1() → 8
group3#12(0) → 10
group3#21(0, 0) → 1
nil1() → 1
group3#31(0, 0, 0) → 2
::1(9, 10) → 3
zip3#12(0, 0, 0) → 11
zip3#21(0, 0, 0, 0) → 5
nil1() → 5
zip3#31(0, 0, 0, 0, 0) → 6
::1(9, 11) → 7
group3#31(0, 0, 0) → 1
::1(9, 10) → 2
zip3#31(0, 0, 0, 0, 0) → 5
::1(9, 11) → 6
group3#21(0, 0) → 10
nil1() → 10
zip3#21(0, 0, 0, 0) → 11
nil1() → 11
group3#31(0, 0, 0) → 10
::1(9, 10) → 1
zip3#31(0, 0, 0, 0, 0) → 11
::1(9, 11) → 5
::1(9, 10) → 10
::1(9, 11) → 11

(2) BOUNDS(1, n^1)

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

Converted Cpx (relative) TRS to CDT

(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#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

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

Removed 6 trailing nodes:

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

(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))
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

(7) 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

(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(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

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(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))
GROUP3#2(::(z0, z1), z2) → c3(GROUP3#3(z1, z2, z0))
GROUP3#3(::(z0, z1), z2, z3) → c5(GROUP3(z1))
ZIP3#1(::(z0, z1), z2, z3) → c8(ZIP3#2(z2, z3, 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)) = [1] + x₁ + x₂   
POL(GROUP3(x₁)) = x₁   
POL(GROUP3#1(x₁)) = x₁   
POL(GROUP3#2(x₁, x₂)) = x₁   
POL(GROUP3#3(x₁, x₂, x₃)) = x₁   
POL(ZIP3(x₁, x₂, x₃)) = x₁   
POL(ZIP3#1(x₁, x₂, x₃)) = x₁   
POL(ZIP3#2(x₁, x₂, x₃, x₄)) = x₃ + x₄   
POL(ZIP3#3(x₁, x₂, x₃, x₄, x₅)) = 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   

(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))
ZIP3(z0, z1, z2) → c7(ZIP3#1(z0, z1, z2))
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:

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#1(::(z0, z1), z2, z3) → c8(ZIP3#2(z2, z3, 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

(11) CdtKnowledgeProof (EQUIVALENT transformation)

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

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

Now S is empty

(12) BOUNDS(1, 1)