(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:
f(s(x)) → f(g(x, x))
g(0, 1) → s(0)
0 → 1
Rewrite Strategy: FULL
(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)
The following defined symbols can occur below the 0th argument of f: g
The following defined symbols can occur below the 0th argument of g: g
The following defined symbols can occur below the 1th argument of g: g
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
g(0, 1) → s(0)
(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:
0 → 1
f(s(x)) → f(g(x, 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:
0 → 1
f(s(x)) → f(g(x, x))
Rewrite Strategy: INNERMOST
(5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
0 → 1
f(s(z0)) → f(g(z0, z0))
Tuples:
0' → c
F(s(z0)) → c1(F(g(z0, z0)))
S tuples:
0' → c
F(s(z0)) → c1(F(g(z0, z0)))
K tuples:none
Defined Rule Symbols:
0, f
Defined Pair Symbols:
0', F
Compound Symbols:
c, c1
(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
F(s(z0)) → c1(F(g(z0, z0)))
0' → c
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
0 → 1
f(s(z0)) → f(g(z0, z0))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
0, f
Defined Pair Symbols:none
Compound Symbols:none
(9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(10) BOUNDS(1, 1)