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:

+(0, y) → y
+(s(x), y) → s(+(x, y))
+(s(x), y) → +(x, s(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 1.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1]
transitions:
00() → 0
s0(0) → 0
+0(0, 0) → 1
+1(0, 0) → 2
s1(2) → 1
s1(0) → 3
+1(0, 3) → 1
+1(0, 3) → 2
s1(2) → 2
s1(3) → 3
0 → 1
0 → 2
3 → 1
3 → 2

(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:

+(0, z0) → z0
+(s(z0), z1) → s(+(z0, z1))
+(s(z0), z1) → +(z0, s(z1))

Tuples:

+'(0, z0) → c
+'(s(z0), z1) → c1(+'(z0, z1))
+'(s(z0), z1) → c2(+'(z0, s(z1)))

S tuples:

+'(0, z0) → c
+'(s(z0), z1) → c1(+'(z0, z1))
+'(s(z0), z1) → c2(+'(z0, s(z1)))

K tuples:none
Defined Rule Symbols:

+

Defined Pair Symbols:

+'

Compound Symbols:

c, c1, c2

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

Removed 1 trailing nodes:

+'(0, z0) → c

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(0, z0) → z0
+(s(z0), z1) → s(+(z0, z1))
+(s(z0), z1) → +(z0, s(z1))

Tuples:

+'(s(z0), z1) → c1(+'(z0, z1))
+'(s(z0), z1) → c2(+'(z0, s(z1)))

S tuples:

+'(s(z0), z1) → c1(+'(z0, z1))
+'(s(z0), z1) → c2(+'(z0, s(z1)))

K tuples:none
Defined Rule Symbols:

+

Defined Pair Symbols:

+'

Compound Symbols:

c1, c2

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

+(0, z0) → z0
+(s(z0), z1) → s(+(z0, z1))
+(s(z0), z1) → +(z0, s(z1))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

+'(s(z0), z1) → c1(+'(z0, z1))
+'(s(z0), z1) → c2(+'(z0, s(z1)))

S tuples:

+'(s(z0), z1) → c1(+'(z0, z1))
+'(s(z0), z1) → c2(+'(z0, s(z1)))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

+'

Compound Symbols:

c1, c2

(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.

+'(s(z0), z1) → c1(+'(z0, z1))
+'(s(z0), z1) → c2(+'(z0, s(z1)))

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

+'(s(z0), z1) → c1(+'(z0, z1))
+'(s(z0), z1) → c2(+'(z0, s(z1)))

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

POL(+'(x₁, x₂)) = [2]x₁   
POL(c1(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(s(x₁)) = [2] + x₁   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

+'(s(z0), z1) → c1(+'(z0, z1))
+'(s(z0), z1) → c2(+'(z0, s(z1)))

S tuples:none
K tuples:

+'(s(z0), z1) → c1(+'(z0, z1))
+'(s(z0), z1) → c2(+'(z0, s(z1)))

Defined Rule Symbols:none

Defined Pair Symbols:

+'

Compound Symbols:

c1, c2

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

The set S is empty

(12) BOUNDS(1, 1)