(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(x, y, z) → g(<=(x, y), x, y, z)
g(true, x, y, z) → z
g(false, x, y, z) → f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
p(0) → 0
p(s(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:
f(z0, z1, z2) → g(<=(z0, z1), z0, z1, z2)
g(true, z0, z1, z2) → z2
g(false, z0, z1, z2) → f(f(p(z0), z1, z2), f(p(z1), z2, z0), f(p(z2), z0, z1))
p(0) → 0
p(s(z0)) → z0
Tuples:
F(z0, z1, z2) → c(G(<=(z0, z1), z0, z1, z2))
G(true, z0, z1, z2) → c1
G(false, z0, z1, z2) → c2(F(f(p(z0), z1, z2), f(p(z1), z2, z0), f(p(z2), z0, z1)), F(p(z0), z1, z2), P(z0), F(p(z1), z2, z0), P(z1), F(p(z2), z0, z1), P(z2))
P(0) → c3
P(s(z0)) → c4
S tuples:
F(z0, z1, z2) → c(G(<=(z0, z1), z0, z1, z2))
G(true, z0, z1, z2) → c1
G(false, z0, z1, z2) → c2(F(f(p(z0), z1, z2), f(p(z1), z2, z0), f(p(z2), z0, z1)), F(p(z0), z1, z2), P(z0), F(p(z1), z2, z0), P(z1), F(p(z2), z0, z1), P(z2))
P(0) → c3
P(s(z0)) → c4
K tuples:none
Defined Rule Symbols:
f, g, p
Defined Pair Symbols:
F, G, P
Compound Symbols:
c, c1, c2, c3, c4
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 5 trailing nodes:
G(false, z0, z1, z2) → c2(F(f(p(z0), z1, z2), f(p(z1), z2, z0), f(p(z2), z0, z1)), F(p(z0), z1, z2), P(z0), F(p(z1), z2, z0), P(z1), F(p(z2), z0, z1), P(z2))
P(s(z0)) → c4
P(0) → c3
G(true, z0, z1, z2) → c1
F(z0, z1, z2) → c(G(<=(z0, z1), z0, z1, z2))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(z0, z1, z2) → g(<=(z0, z1), z0, z1, z2)
g(true, z0, z1, z2) → z2
g(false, z0, z1, z2) → f(f(p(z0), z1, z2), f(p(z1), z2, z0), f(p(z2), z0, z1))
p(0) → 0
p(s(z0)) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f, g, p
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(6) BOUNDS(1, 1)