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:

f(s(X)) → f(X)
g(cons(0, Y)) → g(Y)
g(cons(s(X), Y)) → s(X)
h(cons(X, Y)) → h(g(cons(X, Y)))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(s(z0)) → f(z0)
g(cons(0, z0)) → g(z0)
g(cons(s(z0), z1)) → s(z0)
h(cons(z0, z1)) → h(g(cons(z0, z1)))

Tuples:

F(s(z0)) → c(F(z0))
G(cons(0, z0)) → c1(G(z0))
G(cons(s(z0), z1)) → c2
H(cons(z0, z1)) → c3(H(g(cons(z0, z1))), G(cons(z0, z1)))

S tuples:

F(s(z0)) → c(F(z0))
G(cons(0, z0)) → c1(G(z0))
G(cons(s(z0), z1)) → c2
H(cons(z0, z1)) → c3(H(g(cons(z0, z1))), G(cons(z0, z1)))

K tuples:none
Defined Rule Symbols:

f, g, h

Defined Pair Symbols:

F, G, H

Compound Symbols:

c, c1, c2, c3

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

Removed 1 trailing nodes:

G(cons(s(z0), z1)) → c2

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(s(z0)) → f(z0)
g(cons(0, z0)) → g(z0)
g(cons(s(z0), z1)) → s(z0)
h(cons(z0, z1)) → h(g(cons(z0, z1)))

Tuples:

F(s(z0)) → c(F(z0))
G(cons(0, z0)) → c1(G(z0))
H(cons(z0, z1)) → c3(H(g(cons(z0, z1))), G(cons(z0, z1)))

S tuples:

F(s(z0)) → c(F(z0))
G(cons(0, z0)) → c1(G(z0))
H(cons(z0, z1)) → c3(H(g(cons(z0, z1))), G(cons(z0, z1)))

K tuples:none
Defined Rule Symbols:

f, g, h

Defined Pair Symbols:

F, G, H

Compound Symbols:

c, c1, c3

(5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

f(s(z0)) → f(z0)
h(cons(z0, z1)) → h(g(cons(z0, z1)))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(cons(0, z0)) → g(z0)
g(cons(s(z0), z1)) → s(z0)

Tuples:

F(s(z0)) → c(F(z0))
G(cons(0, z0)) → c1(G(z0))
H(cons(z0, z1)) → c3(H(g(cons(z0, z1))), G(cons(z0, z1)))

S tuples:

F(s(z0)) → c(F(z0))
G(cons(0, z0)) → c1(G(z0))
H(cons(z0, z1)) → c3(H(g(cons(z0, z1))), G(cons(z0, z1)))

K tuples:none
Defined Rule Symbols:

g

Defined Pair Symbols:

F, G, H

Compound Symbols:

c, c1, c3

(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.

H(cons(z0, z1)) → c3(H(g(cons(z0, z1))), G(cons(z0, z1)))

We considered the (Usable) Rules:

g(cons(s(z0), z1)) → s(z0)
g(cons(0, z0)) → g(z0)

And the Tuples:

F(s(z0)) → c(F(z0))
G(cons(0, z0)) → c1(G(z0))
H(cons(z0, z1)) → c3(H(g(cons(z0, z1))), G(cons(z0, z1)))

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

POL(0) = [2]   
POL(F(x₁)) = 0   
POL(G(x₁)) = [2]   
POL(H(x₁)) = [4]x₁   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(cons(x₁, x₂)) = [5] + x₁ + x₂   
POL(g(x₁)) = [4]   
POL(s(x₁)) = [3]   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(cons(0, z0)) → g(z0)
g(cons(s(z0), z1)) → s(z0)

Tuples:

F(s(z0)) → c(F(z0))
G(cons(0, z0)) → c1(G(z0))
H(cons(z0, z1)) → c3(H(g(cons(z0, z1))), G(cons(z0, z1)))

S tuples:

F(s(z0)) → c(F(z0))
G(cons(0, z0)) → c1(G(z0))

K tuples:

H(cons(z0, z1)) → c3(H(g(cons(z0, z1))), G(cons(z0, z1)))

Defined Rule Symbols:

g

Defined Pair Symbols:

F, G, H

Compound Symbols:

c, c1, c3

(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(cons(0, z0)) → c1(G(z0))

We considered the (Usable) Rules:

g(cons(s(z0), z1)) → s(z0)
g(cons(0, z0)) → g(z0)

And the Tuples:

F(s(z0)) → c(F(z0))
G(cons(0, z0)) → c1(G(z0))
H(cons(z0, z1)) → c3(H(g(cons(z0, z1))), G(cons(z0, z1)))

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

POL(0) = [1]   
POL(F(x₁)) = 0   
POL(G(x₁)) = x₁   
POL(H(x₁)) = [4]x₁   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(cons(x₁, x₂)) = x₁ + x₂   
POL(g(x₁)) = 0   
POL(s(x₁)) = 0   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(cons(0, z0)) → g(z0)
g(cons(s(z0), z1)) → s(z0)

Tuples:

F(s(z0)) → c(F(z0))
G(cons(0, z0)) → c1(G(z0))
H(cons(z0, z1)) → c3(H(g(cons(z0, z1))), G(cons(z0, z1)))

S tuples:

F(s(z0)) → c(F(z0))

K tuples:

H(cons(z0, z1)) → c3(H(g(cons(z0, z1))), G(cons(z0, z1)))
G(cons(0, z0)) → c1(G(z0))

Defined Rule Symbols:

g

Defined Pair Symbols:

F, G, H

Compound Symbols:

c, c1, c3

(11) 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(s(z0)) → c(F(z0))

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

F(s(z0)) → c(F(z0))
G(cons(0, z0)) → c1(G(z0))
H(cons(z0, z1)) → c3(H(g(cons(z0, z1))), G(cons(z0, z1)))

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

POL(0) = 0   
POL(F(x₁)) = [2]x₁   
POL(G(x₁)) = 0   
POL(H(x₁)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(cons(x₁, x₂)) = 0   
POL(g(x₁)) = [1]   
POL(s(x₁)) = [3] + x₁   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(cons(0, z0)) → g(z0)
g(cons(s(z0), z1)) → s(z0)

Tuples:

F(s(z0)) → c(F(z0))
G(cons(0, z0)) → c1(G(z0))
H(cons(z0, z1)) → c3(H(g(cons(z0, z1))), G(cons(z0, z1)))

S tuples:none
K tuples:

H(cons(z0, z1)) → c3(H(g(cons(z0, z1))), G(cons(z0, z1)))
G(cons(0, z0)) → c1(G(z0))
F(s(z0)) → c(F(z0))

Defined Rule Symbols:

g

Defined Pair Symbols:

F, G, H

Compound Symbols:

c, c1, c3

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

The set S is empty

(14) BOUNDS(O(1), O(1))