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

0 CpxRelTRS
↳1 CpxTrsToCdtProof (UPPER BOUND(ID), 0 ms)
↳2 CdtProblem
↳3 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CdtProblem
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CdtProblem
↳7 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)
↳8 CdtProblem
↳9 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
↳10 CdtProblem
↳11 CdtUsableRulesProof (⇔, 0 ms)
↳12 CdtProblem
↳13 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 54 ms)
↳14 CdtProblem
↳15 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳16 BOUNDS(1, 1)

(0) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

lte(Cons(x', xs'), Cons(x, xs)) → lte(xs', xs)
lte(Cons(x, xs), Nil) → False
even(Cons(x, Nil)) → False
even(Cons(x', Cons(x, xs))) → even(xs)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
lte(Nil, y) → True
even(Nil) → True
goal(x, y) → and(lte(x, y), even(x))

The (relative) TRS S consists of the following rules:

and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (UPPER BOUND(ID) transformation)

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True
lte(Cons(z0, z1), Cons(z2, z3)) → lte(z1, z3)
lte(Cons(z0, z1), Nil) → False
lte(Nil, z0) → True
even(Cons(z0, Nil)) → False
even(Cons(z0, Cons(z1, z2))) → even(z2)
even(Nil) → True
notEmpty(Cons(z0, z1)) → True
notEmpty(Nil) → False
goal(z0, z1) → and(lte(z0, z1), even(z0))
Tuples:

AND(False, False) → c
AND(True, False) → c1
AND(False, True) → c2
AND(True, True) → c3
LTE(Cons(z0, z1), Cons(z2, z3)) → c4(LTE(z1, z3))
LTE(Cons(z0, z1), Nil) → c5
LTE(Nil, z0) → c6
EVEN(Cons(z0, Nil)) → c7
EVEN(Cons(z0, Cons(z1, z2))) → c8(EVEN(z2))
EVEN(Nil) → c9
NOTEMPTY(Cons(z0, z1)) → c10
NOTEMPTY(Nil) → c11
GOAL(z0, z1) → c12(AND(lte(z0, z1), even(z0)), LTE(z0, z1), EVEN(z0))
S tuples:

LTE(Cons(z0, z1), Cons(z2, z3)) → c4(LTE(z1, z3))
LTE(Cons(z0, z1), Nil) → c5
LTE(Nil, z0) → c6
EVEN(Cons(z0, Nil)) → c7
EVEN(Cons(z0, Cons(z1, z2))) → c8(EVEN(z2))
EVEN(Nil) → c9
NOTEMPTY(Cons(z0, z1)) → c10
NOTEMPTY(Nil) → c11
GOAL(z0, z1) → c12(AND(lte(z0, z1), even(z0)), LTE(z0, z1), EVEN(z0))
K tuples:none
Defined Rule Symbols:

lte, even, notEmpty, goal, and

Defined Pair Symbols:

AND, LTE, EVEN, NOTEMPTY, GOAL

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12

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

Removed 10 trailing nodes:

LTE(Nil, z0) → c6
AND(False, False) → c
AND(True, True) → c3
AND(False, True) → c2
NOTEMPTY(Nil) → c11
EVEN(Nil) → c9
LTE(Cons(z0, z1), Nil) → c5
AND(True, False) → c1
NOTEMPTY(Cons(z0, z1)) → c10
EVEN(Cons(z0, Nil)) → c7

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True
lte(Cons(z0, z1), Cons(z2, z3)) → lte(z1, z3)
lte(Cons(z0, z1), Nil) → False
lte(Nil, z0) → True
even(Cons(z0, Nil)) → False
even(Cons(z0, Cons(z1, z2))) → even(z2)
even(Nil) → True
notEmpty(Cons(z0, z1)) → True
notEmpty(Nil) → False
goal(z0, z1) → and(lte(z0, z1), even(z0))
Tuples:

LTE(Cons(z0, z1), Cons(z2, z3)) → c4(LTE(z1, z3))
EVEN(Cons(z0, Cons(z1, z2))) → c8(EVEN(z2))
GOAL(z0, z1) → c12(AND(lte(z0, z1), even(z0)), LTE(z0, z1), EVEN(z0))
S tuples:

LTE(Cons(z0, z1), Cons(z2, z3)) → c4(LTE(z1, z3))
EVEN(Cons(z0, Cons(z1, z2))) → c8(EVEN(z2))
GOAL(z0, z1) → c12(AND(lte(z0, z1), even(z0)), LTE(z0, z1), EVEN(z0))
K tuples:none
Defined Rule Symbols:

lte, even, notEmpty, goal, and

Defined Pair Symbols:

LTE, EVEN, GOAL

Compound Symbols:

c4, c8, c12

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

Removed 1 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True
lte(Cons(z0, z1), Cons(z2, z3)) → lte(z1, z3)
lte(Cons(z0, z1), Nil) → False
lte(Nil, z0) → True
even(Cons(z0, Nil)) → False
even(Cons(z0, Cons(z1, z2))) → even(z2)
even(Nil) → True
notEmpty(Cons(z0, z1)) → True
notEmpty(Nil) → False
goal(z0, z1) → and(lte(z0, z1), even(z0))
Tuples:

LTE(Cons(z0, z1), Cons(z2, z3)) → c4(LTE(z1, z3))
EVEN(Cons(z0, Cons(z1, z2))) → c8(EVEN(z2))
GOAL(z0, z1) → c12(LTE(z0, z1), EVEN(z0))
S tuples:

LTE(Cons(z0, z1), Cons(z2, z3)) → c4(LTE(z1, z3))
EVEN(Cons(z0, Cons(z1, z2))) → c8(EVEN(z2))
GOAL(z0, z1) → c12(LTE(z0, z1), EVEN(z0))
K tuples:none
Defined Rule Symbols:

lte, even, notEmpty, goal, and

Defined Pair Symbols:

LTE, EVEN, GOAL

Compound Symbols:

c4, c8, c12

(7) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True
lte(Cons(z0, z1), Cons(z2, z3)) → lte(z1, z3)
lte(Cons(z0, z1), Nil) → False
lte(Nil, z0) → True
even(Cons(z0, Nil)) → False
even(Cons(z0, Cons(z1, z2))) → even(z2)
even(Nil) → True
notEmpty(Cons(z0, z1)) → True
notEmpty(Nil) → False
goal(z0, z1) → and(lte(z0, z1), even(z0))
Tuples:

LTE(Cons(z0, z1), Cons(z2, z3)) → c4(LTE(z1, z3))
EVEN(Cons(z0, Cons(z1, z2))) → c8(EVEN(z2))
GOAL(z0, z1) → c(LTE(z0, z1))
GOAL(z0, z1) → c(EVEN(z0))
S tuples:

LTE(Cons(z0, z1), Cons(z2, z3)) → c4(LTE(z1, z3))
EVEN(Cons(z0, Cons(z1, z2))) → c8(EVEN(z2))
GOAL(z0, z1) → c(LTE(z0, z1))
GOAL(z0, z1) → c(EVEN(z0))
K tuples:none
Defined Rule Symbols:

lte, even, notEmpty, goal, and

Defined Pair Symbols:

LTE, EVEN, GOAL

Compound Symbols:

c4, c8, c

(9) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 2 leading nodes:

GOAL(z0, z1) → c(LTE(z0, z1))
GOAL(z0, z1) → c(EVEN(z0))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True
lte(Cons(z0, z1), Cons(z2, z3)) → lte(z1, z3)
lte(Cons(z0, z1), Nil) → False
lte(Nil, z0) → True
even(Cons(z0, Nil)) → False
even(Cons(z0, Cons(z1, z2))) → even(z2)
even(Nil) → True
notEmpty(Cons(z0, z1)) → True
notEmpty(Nil) → False
goal(z0, z1) → and(lte(z0, z1), even(z0))
Tuples:

LTE(Cons(z0, z1), Cons(z2, z3)) → c4(LTE(z1, z3))
EVEN(Cons(z0, Cons(z1, z2))) → c8(EVEN(z2))
S tuples:

LTE(Cons(z0, z1), Cons(z2, z3)) → c4(LTE(z1, z3))
EVEN(Cons(z0, Cons(z1, z2))) → c8(EVEN(z2))
K tuples:none
Defined Rule Symbols:

lte, even, notEmpty, goal, and

Defined Pair Symbols:

LTE, EVEN

Compound Symbols:

c4, c8

(11) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

and(False, False) → False
and(True, False) → False
and(False, True) → False
and(True, True) → True
lte(Cons(z0, z1), Cons(z2, z3)) → lte(z1, z3)
lte(Cons(z0, z1), Nil) → False
lte(Nil, z0) → True
even(Cons(z0, Nil)) → False
even(Cons(z0, Cons(z1, z2))) → even(z2)
even(Nil) → True
notEmpty(Cons(z0, z1)) → True
notEmpty(Nil) → False
goal(z0, z1) → and(lte(z0, z1), even(z0))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

LTE(Cons(z0, z1), Cons(z2, z3)) → c4(LTE(z1, z3))
EVEN(Cons(z0, Cons(z1, z2))) → c8(EVEN(z2))
S tuples:

LTE(Cons(z0, z1), Cons(z2, z3)) → c4(LTE(z1, z3))
EVEN(Cons(z0, Cons(z1, z2))) → c8(EVEN(z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

LTE, EVEN

Compound Symbols:

c4, c8

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

LTE(Cons(z0, z1), Cons(z2, z3)) → c4(LTE(z1, z3))
EVEN(Cons(z0, Cons(z1, z2))) → c8(EVEN(z2))
We considered the (Usable) Rules:none
And the Tuples:

LTE(Cons(z0, z1), Cons(z2, z3)) → c4(LTE(z1, z3))
EVEN(Cons(z0, Cons(z1, z2))) → c8(EVEN(z2))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(Cons(x1, x2)) = [5] + x2   
POL(EVEN(x1)) = [5]x1   
POL(LTE(x1, x2)) = [2]x2   
POL(c4(x1)) = x1   
POL(c8(x1)) = x1   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

LTE(Cons(z0, z1), Cons(z2, z3)) → c4(LTE(z1, z3))
EVEN(Cons(z0, Cons(z1, z2))) → c8(EVEN(z2))
S tuples:none
K tuples:

LTE(Cons(z0, z1), Cons(z2, z3)) → c4(LTE(z1, z3))
EVEN(Cons(z0, Cons(z1, z2))) → c8(EVEN(z2))
Defined Rule Symbols:none

Defined Pair Symbols:

LTE, EVEN

Compound Symbols:

c4, c8

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

The set S is empty

(16) BOUNDS(1, 1)