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

(0) Obligation:

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

g(c(x, s(y))) → g(c(s(x), y))
f(c(s(x), y)) → f(c(x, s(y)))
f(f(x)) → f(d(f(x)))
f(x) → x

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(c(z0, s(z1))) → g(c(s(z0), z1))
f(c(s(z0), z1)) → f(c(z0, s(z1)))
f(f(z0)) → f(d(f(z0)))
f(z0) → z0

Tuples:

G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
F(f(z0)) → c3(F(d(f(z0))), F(z0))
F(z0) → c4

S tuples:

G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
F(f(z0)) → c3(F(d(f(z0))), F(z0))
F(z0) → c4

K tuples:none
Defined Rule Symbols:

g, f

Defined Pair Symbols:

G, F

Compound Symbols:

c1, c2, c3, c4

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

F(f(z0)) → c3(F(d(f(z0))), F(z0))

Removed 1 trailing nodes:

F(z0) → c4

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(c(z0, s(z1))) → g(c(s(z0), z1))
f(c(s(z0), z1)) → f(c(z0, s(z1)))
f(f(z0)) → f(d(f(z0)))
f(z0) → z0

Tuples:

G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))

S tuples:

G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))

K tuples:none
Defined Rule Symbols:

g, f

Defined Pair Symbols:

G, F

Compound Symbols:

c1, c2

(5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

g(c(z0, s(z1))) → g(c(s(z0), z1))
f(c(s(z0), z1)) → f(c(z0, s(z1)))
f(f(z0)) → f(d(f(z0)))
f(z0) → z0

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))

S tuples:

G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

G, F

Compound Symbols:

c1, c2

(7) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))

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

G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))

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

POL(F(x₁)) = [4]x₁   
POL(G(x₁)) = 0   
POL(c(x₁, x₂)) = x₁   
POL(c1(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(s(x₁)) = [2] + x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))

S tuples:

G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))

K tuples:

F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))

Defined Rule Symbols:none

Defined Pair Symbols:

G, F

Compound Symbols:

c1, c2

(9) CdtRuleRemovalProof (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(c(z0, s(z1))) → c1(G(c(s(z0), z1)))

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

G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))

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

POL(F(x₁)) = 0   
POL(G(x₁)) = [4]x₁   
POL(c(x₁, x₂)) = x₂   
POL(c1(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(s(x₁)) = [4] + x₁   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))
F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))

S tuples:none
K tuples:

F(c(s(z0), z1)) → c2(F(c(z0, s(z1))))
G(c(z0, s(z1))) → c1(G(c(s(z0), z1)))

Defined Rule Symbols:none

Defined Pair Symbols:

G, F

Compound Symbols:

c1, c2

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

The set S is empty

(12) BOUNDS(O(1), O(1))