The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

(0) Obligation:

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

f(0) → 1
f(s(x)) → g(x, s(x))
g(0, y) → y
g(s(x), y) → g(x, +(y, s(x)))
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
g(s(x), y) → g(x, s(+(y, x)))

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0) → 1
f(s(z0)) → g(z0, s(z0))
g(0, z0) → z0
g(s(z0), z1) → g(z0, +(z1, s(z0)))
g(s(z0), z1) → g(z0, s(+(z1, z0)))
+(z0, 0) → z0
+(z0, s(z1)) → s(+(z0, z1))

Tuples:

F(s(z0)) → c1(G(z0, s(z0)))
G(s(z0), z1) → c3(G(z0, +(z1, s(z0))), +'(z1, s(z0)))
G(s(z0), z1) → c4(G(z0, s(+(z1, z0))), +'(z1, z0))
+'(z0, s(z1)) → c6(+'(z0, z1))

S tuples:

F(s(z0)) → c1(G(z0, s(z0)))
G(s(z0), z1) → c3(G(z0, +(z1, s(z0))), +'(z1, s(z0)))
G(s(z0), z1) → c4(G(z0, s(+(z1, z0))), +'(z1, z0))
+'(z0, s(z1)) → c6(+'(z0, z1))

K tuples:none
Defined Rule Symbols:

f, g, +

Defined Pair Symbols:

F, G, +'

Compound Symbols:

c1, c3, c4, c6

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

F(s(z0)) → c1(G(z0, s(z0)))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0) → 1
f(s(z0)) → g(z0, s(z0))
g(0, z0) → z0
g(s(z0), z1) → g(z0, +(z1, s(z0)))
g(s(z0), z1) → g(z0, s(+(z1, z0)))
+(z0, 0) → z0
+(z0, s(z1)) → s(+(z0, z1))

Tuples:

G(s(z0), z1) → c3(G(z0, +(z1, s(z0))), +'(z1, s(z0)))
G(s(z0), z1) → c4(G(z0, s(+(z1, z0))), +'(z1, z0))
+'(z0, s(z1)) → c6(+'(z0, z1))

S tuples:

G(s(z0), z1) → c3(G(z0, +(z1, s(z0))), +'(z1, s(z0)))
G(s(z0), z1) → c4(G(z0, s(+(z1, z0))), +'(z1, z0))
+'(z0, s(z1)) → c6(+'(z0, z1))

K tuples:none
Defined Rule Symbols:

f, g, +

Defined Pair Symbols:

G, +'

Compound Symbols:

c3, c4, c6

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

G(s(z0), z1) → c3(G(z0, +(z1, s(z0))), +'(z1, s(z0)))
G(s(z0), z1) → c4(G(z0, s(+(z1, z0))), +'(z1, z0))

We considered the (Usable) Rules:

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

And the Tuples:

G(s(z0), z1) → c3(G(z0, +(z1, s(z0))), +'(z1, s(z0)))
G(s(z0), z1) → c4(G(z0, s(+(z1, z0))), +'(z1, z0))
+'(z0, s(z1)) → c6(+'(z0, z1))

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

POL(+(x₁, x₂)) = [4]   
POL(+'(x₁, x₂)) = 0   
POL(0) = 0   
POL(G(x₁, x₂)) = [4]x₁   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁)) = x₁   
POL(s(x₁)) = [3] + x₁   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0) → 1
f(s(z0)) → g(z0, s(z0))
g(0, z0) → z0
g(s(z0), z1) → g(z0, +(z1, s(z0)))
g(s(z0), z1) → g(z0, s(+(z1, z0)))
+(z0, 0) → z0
+(z0, s(z1)) → s(+(z0, z1))

Tuples:

G(s(z0), z1) → c3(G(z0, +(z1, s(z0))), +'(z1, s(z0)))
G(s(z0), z1) → c4(G(z0, s(+(z1, z0))), +'(z1, z0))
+'(z0, s(z1)) → c6(+'(z0, z1))

S tuples:

+'(z0, s(z1)) → c6(+'(z0, z1))

K tuples:

G(s(z0), z1) → c3(G(z0, +(z1, s(z0))), +'(z1, s(z0)))
G(s(z0), z1) → c4(G(z0, s(+(z1, z0))), +'(z1, z0))

Defined Rule Symbols:

f, g, +

Defined Pair Symbols:

G, +'

Compound Symbols:

c3, c4, c6

(7) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

+'(z0, s(z1)) → c6(+'(z0, z1))

We considered the (Usable) Rules:

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

And the Tuples:

G(s(z0), z1) → c3(G(z0, +(z1, s(z0))), +'(z1, s(z0)))
G(s(z0), z1) → c4(G(z0, s(+(z1, z0))), +'(z1, z0))
+'(z0, s(z1)) → c6(+'(z0, z1))

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

POL(+(x₁, x₂)) = [2]x₁   
POL(+'(x₁, x₂)) = [1] + x₂   
POL(0) = 0   
POL(G(x₁, x₂)) = [2]x₁²   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0) → 1
f(s(z0)) → g(z0, s(z0))
g(0, z0) → z0
g(s(z0), z1) → g(z0, +(z1, s(z0)))
g(s(z0), z1) → g(z0, s(+(z1, z0)))
+(z0, 0) → z0
+(z0, s(z1)) → s(+(z0, z1))

Tuples:

G(s(z0), z1) → c3(G(z0, +(z1, s(z0))), +'(z1, s(z0)))
G(s(z0), z1) → c4(G(z0, s(+(z1, z0))), +'(z1, z0))
+'(z0, s(z1)) → c6(+'(z0, z1))

S tuples:none
K tuples:

G(s(z0), z1) → c3(G(z0, +(z1, s(z0))), +'(z1, s(z0)))
G(s(z0), z1) → c4(G(z0, s(+(z1, z0))), +'(z1, z0))
+'(z0, s(z1)) → c6(+'(z0, z1))

Defined Rule Symbols:

f, g, +

Defined Pair Symbols:

G, +'

Compound Symbols:

c3, c4, c6

(9) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(10) BOUNDS(O(1), O(1))