(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)