(0) Obligation:

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


The TRS R consists of the following rules:

f(s(a), s(b), x) → f(x, x, x)
g(f(s(x), s(y), z)) → g(f(x, y, z))
cons(x, y) → x
cons(x, y) → y

Rewrite Strategy: INNERMOST

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

The following rules are not reachable from basic terms in the dependency graph and can be removed:
g(f(s(x), s(y), z)) → g(f(x, y, z))

(2) Obligation:

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


The TRS R consists of the following rules:

cons(x, y) → x
cons(x, y) → y
f(s(a), s(b), x) → f(x, x, x)

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

cons(z0, z1) → z0
cons(z0, z1) → z1
f(s(a), s(b), z0) → f(z0, z0, z0)
Tuples:

CONS(z0, z1) → c
CONS(z0, z1) → c1
F(s(a), s(b), z0) → c2(F(z0, z0, z0))
S tuples:

CONS(z0, z1) → c
CONS(z0, z1) → c1
F(s(a), s(b), z0) → c2(F(z0, z0, z0))
K tuples:none
Defined Rule Symbols:

cons, f

Defined Pair Symbols:

CONS, F

Compound Symbols:

c, c1, c2

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

Removed 3 trailing nodes:

F(s(a), s(b), z0) → c2(F(z0, z0, z0))
CONS(z0, z1) → c1
CONS(z0, z1) → c

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

cons(z0, z1) → z0
cons(z0, z1) → z1
f(s(a), s(b), z0) → f(z0, z0, z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

cons, f

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty

(8) BOUNDS(1, 1)