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

0 CpxTRS
    ↳1 CpxTrsMatchBoundsTAProof (⇔, 3 ms)
    ↳2 BOUNDS(1, n^1)
    ↳3 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳4 CdtProblem
        ↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 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)), 29 ms)
        ↳12 CdtProblem
        ↳13 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 4 ms)
        ↳14 CdtProblem
        ↳15 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)
        ↳16 CdtProblem
        ↳17 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳18 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
odd(0) → false
odd(s(x)) → not(odd(x))
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
+(s(x), y) → s(+(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 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:
true0() → 0
false0() → 0
00() → 0
s0(0) → 0
not0(0) → 1
odd0(0) → 2
+0(0, 0) → 3
false1() → 1
true1() → 1
false1() → 2
odd1(0) → 4
not1(4) → 2
+1(0, 0) → 5
s1(5) → 3
false1() → 4
not1(4) → 4
s1(5) → 5
true2() → 2
true2() → 4
false2() → 2
false2() → 4
0 → 3
0 → 5

(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
odd(0) → false
odd(s(z0)) → not(odd(z0))
+(z0, 0) → z0
+(z0, s(z1)) → s(+(z0, z1))
+(s(z0), z1) → s(+(z0, z1))
Tuples:

NOT(true) → c
NOT(false) → c1
ODD(0) → c2
ODD(s(z0)) → c3(NOT(odd(z0)), ODD(z0))
+'(z0, 0) → c4
+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
S tuples:

NOT(true) → c
NOT(false) → c1
ODD(0) → c2
ODD(s(z0)) → c3(NOT(odd(z0)), ODD(z0))
+'(z0, 0) → c4
+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
K tuples:none
Defined Rule Symbols:

not, odd, +

Defined Pair Symbols:

NOT, ODD, +'

Compound Symbols:

c, c1, c2, c3, c4, c5, c6

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

Removed 4 trailing nodes:

NOT(true) → c
+'(z0, 0) → c4
ODD(0) → c2
NOT(false) → c1

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

not(true) → false
not(false) → true
odd(0) → false
odd(s(z0)) → not(odd(z0))
+(z0, 0) → z0
+(z0, s(z1)) → s(+(z0, z1))
+(s(z0), z1) → s(+(z0, z1))
Tuples:

ODD(s(z0)) → c3(NOT(odd(z0)), ODD(z0))
+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
S tuples:

ODD(s(z0)) → c3(NOT(odd(z0)), ODD(z0))
+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
K tuples:none
Defined Rule Symbols:

not, odd, +

Defined Pair Symbols:

ODD, +'

Compound Symbols:

c3, c5, c6

(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
odd(0) → false
odd(s(z0)) → not(odd(z0))
+(z0, 0) → z0
+(z0, s(z1)) → s(+(z0, z1))
+(s(z0), z1) → s(+(z0, z1))
Tuples:

+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
ODD(s(z0)) → c3(ODD(z0))
S tuples:

+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
ODD(s(z0)) → c3(ODD(z0))
K tuples:none
Defined Rule Symbols:

not, odd, +

Defined Pair Symbols:

+', ODD

Compound Symbols:

c5, c6, c3

(9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

not(true) → false
not(false) → true
odd(0) → false
odd(s(z0)) → not(odd(z0))
+(z0, 0) → z0
+(z0, s(z1)) → s(+(z0, z1))
+(s(z0), z1) → s(+(z0, z1))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
ODD(s(z0)) → c3(ODD(z0))
S tuples:

+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
ODD(s(z0)) → c3(ODD(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

+', ODD

Compound Symbols:

c5, c6, c3

(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.

+'(s(z0), z1) → c6(+'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
ODD(s(z0)) → c3(ODD(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(+'(x1, x2)) = [2]x1   
POL(ODD(x1)) = 0   
POL(c3(x1)) = x1   
POL(c5(x1)) = x1   
POL(c6(x1)) = x1   
POL(s(x1)) = [1] + x1   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
ODD(s(z0)) → c3(ODD(z0))
S tuples:

+'(z0, s(z1)) → c5(+'(z0, z1))
ODD(s(z0)) → c3(ODD(z0))
K tuples:

+'(s(z0), z1) → c6(+'(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:

+', ODD

Compound Symbols:

c5, c6, c3

(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.

+'(z0, s(z1)) → c5(+'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
ODD(s(z0)) → c3(ODD(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(+'(x1, x2)) = x2   
POL(ODD(x1)) = 0   
POL(c3(x1)) = x1   
POL(c5(x1)) = x1   
POL(c6(x1)) = x1   
POL(s(x1)) = [1] + x1   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
ODD(s(z0)) → c3(ODD(z0))
S tuples:

ODD(s(z0)) → c3(ODD(z0))
K tuples:

+'(s(z0), z1) → c6(+'(z0, z1))
+'(z0, s(z1)) → c5(+'(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:

+', ODD

Compound Symbols:

c5, c6, c3

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

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

ODD(s(z0)) → c3(ODD(z0))
We considered the (Usable) Rules:none
And the Tuples:

+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
ODD(s(z0)) → c3(ODD(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(+'(x1, x2)) = 0   
POL(ODD(x1)) = x1   
POL(c3(x1)) = x1   
POL(c5(x1)) = x1   
POL(c6(x1)) = x1   
POL(s(x1)) = [1] + x1   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

+'(z0, s(z1)) → c5(+'(z0, z1))
+'(s(z0), z1) → c6(+'(z0, z1))
ODD(s(z0)) → c3(ODD(z0))
S tuples:none
K tuples:

+'(s(z0), z1) → c6(+'(z0, z1))
+'(z0, s(z1)) → c5(+'(z0, z1))
ODD(s(z0)) → c3(ODD(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

+', ODD

Compound Symbols:

c5, c6, c3

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

The set S is empty

(18) BOUNDS(1, 1)