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

0 CpxTRS
    ↳1 CpxTrsMatchBoundsTAProof (⇔, 14 ms)
    ↳2 BOUNDS(1, n^1)
    ↳3 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳4 CdtProblem
        ↳5 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
        ↳6 CdtProblem
        ↳7 CdtUsableRulesProof (⇔, 0 ms)
        ↳8 CdtProblem
        ↳9 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 1 ms)
        ↳10 CdtProblem
        ↳11 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳12 BOUNDS(1, 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:

decrease(Cons(x, xs)) → decrease(xs)
decrease(Nil) → number42(Nil)
number42(x) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
goal(x) → decrease(x)

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]
transitions:
Cons0(0, 0) → 0
Nil0() → 0
decrease0(0) → 1
number420(0) → 2
goal0(0) → 3
decrease1(0) → 1
Nil1() → 4
number421(4) → 1
Nil1() → 5
Nil1() → 8
Cons1(5, 8) → 7
Cons1(5, 7) → 6
Cons1(5, 6) → 6
Cons1(5, 6) → 2
decrease1(0) → 3
number421(4) → 3
Nil2() → 9
Nil2() → 12
Cons2(9, 12) → 11
Cons2(9, 11) → 10
Cons2(9, 10) → 10
Cons2(9, 10) → 1
Cons2(9, 10) → 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:

decrease(Cons(z0, z1)) → decrease(z1)
decrease(Nil) → number42(Nil)
number42(z0) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
goal(z0) → decrease(z0)
Tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1))
DECREASE(Nil) → c1(NUMBER42(Nil))
NUMBER42(z0) → c2
GOAL(z0) → c3(DECREASE(z0))
S tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1))
DECREASE(Nil) → c1(NUMBER42(Nil))
NUMBER42(z0) → c2
GOAL(z0) → c3(DECREASE(z0))
K tuples:none
Defined Rule Symbols:

decrease, number42, goal

Defined Pair Symbols:

DECREASE, NUMBER42, GOAL

Compound Symbols:

c, c1, c2, c3

(5) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

GOAL(z0) → c3(DECREASE(z0))
Removed 2 trailing nodes:

NUMBER42(z0) → c2
DECREASE(Nil) → c1(NUMBER42(Nil))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

decrease(Cons(z0, z1)) → decrease(z1)
decrease(Nil) → number42(Nil)
number42(z0) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
goal(z0) → decrease(z0)
Tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1))
S tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1))
K tuples:none
Defined Rule Symbols:

decrease, number42, goal

Defined Pair Symbols:

DECREASE

Compound Symbols:

c

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

decrease(Cons(z0, z1)) → decrease(z1)
decrease(Nil) → number42(Nil)
number42(z0) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
goal(z0) → decrease(z0)

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1))
S tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

DECREASE

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.

DECREASE(Cons(z0, z1)) → c(DECREASE(z1))
We considered the (Usable) Rules:none
And the Tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

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

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1))
S tuples:none
K tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1))
Defined Rule Symbols:none

Defined Pair Symbols:

DECREASE

Compound Symbols:

c

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

The set S is empty

(12) BOUNDS(1, 1)