(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), y, y) → f(y, x, s(x))
Rewrite Strategy: FULL
(1) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)
Converted rc-obligation to irc-obligation.
As the TRS does not nest defined symbols, we have rc = irc.
(2) 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(s(x), y, y) → f(y, x, s(x))
Rewrite Strategy: INNERMOST
(3) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(s(z0), z1, z1) → f(z1, z0, s(z0))
Tuples:
F(s(z0), z1, z1) → c(F(z1, z0, s(z0)))
S tuples:
F(s(z0), z1, z1) → c(F(z1, z0, s(z0)))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
F(s(z0), z1, z1) → c(F(z1, z0, s(z0)))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(s(z0), z1, z1) → f(z1, z0, s(z0))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(8) BOUNDS(1, 1)