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

0 CpxTRS
    ↳1 CpxTrsMatchBoundsTAProof (⇔, 112 ms)
    ↳2 BOUNDS(1, n^1)
    ↳3 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳4 CdtProblem
        ↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 2 ms)
        ↳6 CdtProblem
        ↳7 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳8 CdtProblem
        ↳9 CdtUsableRulesProof (⇔, 0 ms)
        ↳10 CdtProblem
        ↳11 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 65 ms)
        ↳12 CdtProblem
        ↳13 CdtKnowledgeProof (⇔, 0 ms)
        ↳14 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:

not(true) → false
not(false) → true
evenodd(x, 0) → not(evenodd(x, s(0)))
evenodd(0, s(0)) → false
evenodd(s(x), s(0)) → evenodd(x, 0)

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 3.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2]
transitions:
true0() → 0
false0() → 0
00() → 0
s0(0) → 0
not0(0) → 1
evenodd0(0, 0) → 2
false1() → 1
true1() → 1
01() → 5
s1(5) → 4
evenodd1(0, 4) → 3
not1(3) → 2
false1() → 2
01() → 6
evenodd1(0, 6) → 2
02() → 9
s2(9) → 8
evenodd2(0, 8) → 7
not2(7) → 2
false1() → 3
evenodd1(0, 6) → 3
true2() → 2
not2(7) → 3
false1() → 7
evenodd1(0, 6) → 7
not2(7) → 7
true2() → 3
true2() → 7
false2() → 2
false3() → 2
false3() → 3
false3() → 7
true3() → 2
true3() → 3
true3() → 7

(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:

not(true) → false
not(false) → true
evenodd(z0, 0) → not(evenodd(z0, s(0)))
evenodd(0, s(0)) → false
evenodd(s(z0), s(0)) → evenodd(z0, 0)
Tuples:

NOT(true) → c
NOT(false) → c1
EVENODD(z0, 0) → c2(NOT(evenodd(z0, s(0))), EVENODD(z0, s(0)))
EVENODD(0, s(0)) → c3
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
S tuples:

NOT(true) → c
NOT(false) → c1
EVENODD(z0, 0) → c2(NOT(evenodd(z0, s(0))), EVENODD(z0, s(0)))
EVENODD(0, s(0)) → c3
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
K tuples:none
Defined Rule Symbols:

not, evenodd

Defined Pair Symbols:

NOT, EVENODD

Compound Symbols:

c, c1, c2, c3, c4

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

Removed 3 trailing nodes:

EVENODD(0, s(0)) → c3
NOT(false) → c1
NOT(true) → c

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

not(true) → false
not(false) → true
evenodd(z0, 0) → not(evenodd(z0, s(0)))
evenodd(0, s(0)) → false
evenodd(s(z0), s(0)) → evenodd(z0, 0)
Tuples:

EVENODD(z0, 0) → c2(NOT(evenodd(z0, s(0))), EVENODD(z0, s(0)))
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
S tuples:

EVENODD(z0, 0) → c2(NOT(evenodd(z0, s(0))), EVENODD(z0, s(0)))
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
K tuples:none
Defined Rule Symbols:

not, evenodd

Defined Pair Symbols:

EVENODD

Compound Symbols:

c2, c4

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

Removed 1 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

not(true) → false
not(false) → true
evenodd(z0, 0) → not(evenodd(z0, s(0)))
evenodd(0, s(0)) → false
evenodd(s(z0), s(0)) → evenodd(z0, 0)
Tuples:

EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
S tuples:

EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
K tuples:none
Defined Rule Symbols:

not, evenodd

Defined Pair Symbols:

EVENODD

Compound Symbols:

c4, c2

(9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

not(true) → false
not(false) → true
evenodd(z0, 0) → not(evenodd(z0, s(0)))
evenodd(0, s(0)) → false
evenodd(s(z0), s(0)) → evenodd(z0, 0)

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
S tuples:

EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

EVENODD

Compound Symbols:

c4, c2

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

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

EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
We considered the (Usable) Rules:none
And the Tuples:

EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(EVENODD(x1, x2)) = x1   
POL(c2(x1)) = x1   
POL(c4(x1)) = x1   
POL(s(x1)) = [1] + x1   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
S tuples:

EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
K tuples:

EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
Defined Rule Symbols:none

Defined Pair Symbols:

EVENODD

Compound Symbols:

c4, c2

(13) CdtKnowledgeProof (EQUIVALENT transformation)

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

EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
Now S is empty

(14) BOUNDS(1, 1)