(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(a), s(b), x) → f(x, x, x)
g(f(s(x), s(y), z)) → g(f(x, y, z))
cons(x, y) → x
cons(x, y) → y
Rewrite Strategy: FULL
(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)
The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
g(f(s(x), s(y), z)) → g(f(x, y, z))
(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:
cons(x, y) → x
cons(x, y) → y
f(s(a), s(b), x) → f(x, 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:
cons(x, y) → x
cons(x, y) → y
f(s(a), s(b), x) → f(x, 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:
cons(z0, z1) → z0
cons(z0, z1) → z1
f(s(a), s(b), z0) → f(z0, z0, z0)
Tuples:
CONS(z0, z1) → c
CONS(z0, z1) → c1
F(s(a), s(b), z0) → c2(F(z0, z0, z0))
S tuples:
CONS(z0, z1) → c
CONS(z0, z1) → c1
F(s(a), s(b), z0) → c2(F(z0, z0, z0))
K tuples:none
Defined Rule Symbols:
cons, f
Defined Pair Symbols:
CONS, F
Compound Symbols:
c, c1, c2
(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing nodes:
F(s(a), s(b), z0) → c2(F(z0, z0, z0))
CONS(z0, z1) → c
CONS(z0, z1) → c1
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
cons(z0, z1) → z0
cons(z0, z1) → z1
f(s(a), s(b), z0) → f(z0, z0, z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
cons, f
Defined Pair Symbols:none
Compound Symbols:none
(9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(10) BOUNDS(1, 1)