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:

append(Cons(x, xs), ys) → Cons(x, append(xs, ys))
append(Nil, ys) → ys
goal(x, y) → append(x, y)

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 1.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2]
transitions:
Cons0(0, 0) → 0
Nil0() → 0
append0(0, 0) → 1
goal0(0, 0) → 2
append1(0, 0) → 3
Cons1(0, 3) → 1
append1(0, 0) → 2
Cons1(0, 3) → 2
Cons1(0, 3) → 3
0 → 1
0 → 2
0 → 3

(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:

append(Cons(z0, z1), z2) → Cons(z0, append(z1, z2))
append(Nil, z0) → z0
goal(z0, z1) → append(z0, z1)

Tuples:

APPEND(Cons(z0, z1), z2) → c(APPEND(z1, z2))
APPEND(Nil, z0) → c1
GOAL(z0, z1) → c2(APPEND(z0, z1))

S tuples:

APPEND(Cons(z0, z1), z2) → c(APPEND(z1, z2))
APPEND(Nil, z0) → c1
GOAL(z0, z1) → c2(APPEND(z0, z1))

K tuples:none
Defined Rule Symbols:

append, goal

Defined Pair Symbols:

APPEND, GOAL

Compound Symbols:

c, c1, c2

(5) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

GOAL(z0, z1) → c2(APPEND(z0, z1))

Removed 1 trailing nodes:

APPEND(Nil, z0) → c1

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

append(Cons(z0, z1), z2) → Cons(z0, append(z1, z2))
append(Nil, z0) → z0
goal(z0, z1) → append(z0, z1)

Tuples:

APPEND(Cons(z0, z1), z2) → c(APPEND(z1, z2))

S tuples:

APPEND(Cons(z0, z1), z2) → c(APPEND(z1, z2))

K tuples:none
Defined Rule Symbols:

append, goal

Defined Pair Symbols:

APPEND

Compound Symbols:

c

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

append(Cons(z0, z1), z2) → Cons(z0, append(z1, z2))
append(Nil, z0) → z0
goal(z0, z1) → append(z0, z1)

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

APPEND(Cons(z0, z1), z2) → c(APPEND(z1, z2))

S tuples:

APPEND(Cons(z0, z1), z2) → c(APPEND(z1, z2))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

APPEND

Compound Symbols:

c

(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.

APPEND(Cons(z0, z1), z2) → c(APPEND(z1, z2))

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

APPEND(Cons(z0, z1), z2) → c(APPEND(z1, z2))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(APPEND(x₁, x₂)) = x₁   
POL(Cons(x₁, x₂)) = [1] + x₂   
POL(c(x₁)) = x₁   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

APPEND(Cons(z0, z1), z2) → c(APPEND(z1, z2))

S tuples:none
K tuples:

APPEND(Cons(z0, z1), z2) → c(APPEND(z1, z2))

Defined Rule Symbols:none

Defined Pair Symbols:

APPEND

Compound Symbols:

c

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

The set S is empty

(12) BOUNDS(1, 1)