(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:
g(x, y) → x
g(x, y) → y
f(s(x), y, y) → f(y, x, s(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(z0, z1) → z0
g(z0, z1) → z1
f(s(z0), z1, z1) → f(z1, z0, s(z0))
Tuples:
G(z0, z1) → c
G(z0, z1) → c1
F(s(z0), z1, z1) → c2(F(z1, z0, s(z0)))
S tuples:
G(z0, z1) → c
G(z0, z1) → c1
F(s(z0), z1, z1) → c2(F(z1, z0, s(z0)))
K tuples:none
Defined Rule Symbols:
g, f
Defined Pair Symbols:
G, F
Compound Symbols:
c, c1, c2
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing nodes:
F(s(z0), z1, z1) → c2(F(z1, z0, s(z0)))
G(z0, z1) → c1
G(z0, z1) → c
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
g(z0, z1) → z0
g(z0, z1) → z1
f(s(z0), z1, z1) → f(z1, z0, s(z0))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
g, f
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(6) BOUNDS(1, 1)