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:

list(Cons(x, xs)) → list(xs)
list(Nil) → True
list(Nil) → isEmpty[Match](Nil)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(x) → list(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 1.

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
True0() → 0
isEmpty[Match]0(0) → 0
False0() → 0
list0(0) → 1
notEmpty0(0) → 2
goal0(0) → 3
list1(0) → 1
True1() → 1
Nil1() → 4
isEmpty[Match]1(4) → 1
True1() → 2
False1() → 2
list1(0) → 3
True1() → 3
isEmpty[Match]1(4) → 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:

list(Cons(z0, z1)) → list(z1)
list(Nil) → True
list(Nil) → isEmpty[Match](Nil)
notEmpty(Cons(z0, z1)) → True
notEmpty(Nil) → False
goal(z0) → list(z0)

Tuples:

LIST(Cons(z0, z1)) → c(LIST(z1))
LIST(Nil) → c1
LIST(Nil) → c2
NOTEMPTY(Cons(z0, z1)) → c3
NOTEMPTY(Nil) → c4
GOAL(z0) → c5(LIST(z0))

S tuples:

LIST(Cons(z0, z1)) → c(LIST(z1))
LIST(Nil) → c1
LIST(Nil) → c2
NOTEMPTY(Cons(z0, z1)) → c3
NOTEMPTY(Nil) → c4
GOAL(z0) → c5(LIST(z0))

K tuples:none
Defined Rule Symbols:

list, notEmpty, goal

Defined Pair Symbols:

LIST, NOTEMPTY, GOAL

Compound Symbols:

c, c1, c2, c3, c4, c5

(5) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

GOAL(z0) → c5(LIST(z0))

Removed 4 trailing nodes:

NOTEMPTY(Nil) → c4
LIST(Nil) → c2
LIST(Nil) → c1
NOTEMPTY(Cons(z0, z1)) → c3

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

list(Cons(z0, z1)) → list(z1)
list(Nil) → True
list(Nil) → isEmpty[Match](Nil)
notEmpty(Cons(z0, z1)) → True
notEmpty(Nil) → False
goal(z0) → list(z0)

Tuples:

LIST(Cons(z0, z1)) → c(LIST(z1))

S tuples:

LIST(Cons(z0, z1)) → c(LIST(z1))

K tuples:none
Defined Rule Symbols:

list, notEmpty, goal

Defined Pair Symbols:

LIST

Compound Symbols:

c

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

list(Cons(z0, z1)) → list(z1)
list(Nil) → True
list(Nil) → isEmpty[Match](Nil)
notEmpty(Cons(z0, z1)) → True
notEmpty(Nil) → False
goal(z0) → list(z0)

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

LIST(Cons(z0, z1)) → c(LIST(z1))

S tuples:

LIST(Cons(z0, z1)) → c(LIST(z1))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

LIST

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.

LIST(Cons(z0, z1)) → c(LIST(z1))

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

LIST(Cons(z0, z1)) → c(LIST(z1))

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

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

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

LIST(Cons(z0, z1)) → c(LIST(z1))

S tuples:none
K tuples:

LIST(Cons(z0, z1)) → c(LIST(z1))

Defined Rule Symbols:none

Defined Pair Symbols:

LIST

Compound Symbols:

c

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

The set S is empty

(12) BOUNDS(1, 1)