(0) Obligation:
The Runtime Complexity (full) of the given
CpxTRS could be proven to be
BOUNDS(1, 1).
The TRS R consists of the following rules:
i(x, x) → i(a, b)
g(x, x) → g(a, b)
h(s(f(x))) → h(f(x))
f(s(x)) → s(s(f(h(s(x)))))
f(g(s(x), y)) → f(g(x, s(y)))
h(g(x, s(y))) → h(g(s(x), y))
h(i(x, y)) → i(i(c, h(h(y))), x)
g(a, g(x, g(b, g(a, g(x, y))))) → g(a, g(a, g(a, g(x, g(b, g(b, y))))))
Rewrite Strategy: FULL
(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)
The following defined symbols can occur below the 0th argument of f: h
The following defined symbols can occur below the 0th argument of h: h
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
h(s(f(x))) → h(f(x))
f(g(s(x), y)) → f(g(x, s(y)))
h(g(x, s(y))) → h(g(s(x), y))
h(i(x, y)) → i(i(c, h(h(y))), x)
g(a, g(x, g(b, g(a, g(x, y))))) → g(a, g(a, g(a, g(x, g(b, g(b, y))))))
(2) Obligation:
The Runtime Complexity (full) of the given
CpxTRS could be proven to be
BOUNDS(1, 1).
The TRS R consists of the following rules:
g(x, x) → g(a, b)
i(x, x) → i(a, b)
f(s(x)) → s(s(f(h(s(x)))))
Rewrite Strategy: FULL
(3) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)
Converted rc-obligation to irc-obligation.
As the TRS does not nest defined symbols, we have rc = irc.
(4) 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, x) → g(a, b)
i(x, x) → i(a, b)
f(s(x)) → s(s(f(h(s(x)))))
Rewrite Strategy: INNERMOST
(5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
g(z0, z0) → g(a, b)
i(z0, z0) → i(a, b)
f(s(z0)) → s(s(f(h(s(z0)))))
Tuples:
G(z0, z0) → c(G(a, b))
I(z0, z0) → c1(I(a, b))
F(s(z0)) → c2(F(h(s(z0))))
S tuples:
G(z0, z0) → c(G(a, b))
I(z0, z0) → c1(I(a, b))
F(s(z0)) → c2(F(h(s(z0))))
K tuples:none
Defined Rule Symbols:
g, i, f
Defined Pair Symbols:
G, I, F
Compound Symbols:
c, c1, c2
(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing nodes:
F(s(z0)) → c2(F(h(s(z0))))
G(z0, z0) → c(G(a, b))
I(z0, z0) → c1(I(a, b))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
g(z0, z0) → g(a, b)
i(z0, z0) → i(a, b)
f(s(z0)) → s(s(f(h(s(z0)))))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
g, i, f
Defined Pair Symbols:none
Compound Symbols:none
(9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(10) BOUNDS(1, 1)