(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(g(x), g(y)) → f(p(f(g(x), s(y))), g(s(p(x))))
p(0) → g(0)
g(s(p(x))) → p(x)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(g(z0), g(z1)) → f(p(f(g(z0), s(z1))), g(s(p(z0))))
p(0) → g(0)
g(s(p(z0))) → p(z0)
Tuples:
F(g(z0), g(z1)) → c(F(p(f(g(z0), s(z1))), g(s(p(z0)))), P(f(g(z0), s(z1))), F(g(z0), s(z1)), G(z0), G(s(p(z0))), P(z0))
P(0) → c1(G(0))
G(s(p(z0))) → c2(P(z0))
S tuples:
F(g(z0), g(z1)) → c(F(p(f(g(z0), s(z1))), g(s(p(z0)))), P(f(g(z0), s(z1))), F(g(z0), s(z1)), G(z0), G(s(p(z0))), P(z0))
P(0) → c1(G(0))
G(s(p(z0))) → c2(P(z0))
K tuples:none
Defined Rule Symbols:
f, p, g
Defined Pair Symbols:
F, P, G
Compound Symbols:
c, c1, c2
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
F(g(z0), g(z1)) → c(F(p(f(g(z0), s(z1))), g(s(p(z0)))), P(f(g(z0), s(z1))), F(g(z0), s(z1)), G(z0), G(s(p(z0))), P(z0))
G(s(p(z0))) → c2(P(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(g(z0), g(z1)) → f(p(f(g(z0), s(z1))), g(s(p(z0))))
p(0) → g(0)
g(s(p(z0))) → p(z0)
Tuples:
P(0) → c1(G(0))
S tuples:
P(0) → c1(G(0))
K tuples:none
Defined Rule Symbols:
f, p, g
Defined Pair Symbols:
P
Compound Symbols:
c1
(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)
Removed 1 of 1 dangling nodes:
P(0) → c1(G(0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(g(z0), g(z1)) → f(p(f(g(z0), s(z1))), g(s(p(z0))))
p(0) → g(0)
g(s(p(z0))) → p(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f, p, g
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))