The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).

(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:

a(c(d(x))) → c(x)
u(b(d(d(x)))) → b(x)
v(a(a(x))) → u(v(x))
v(a(c(x))) → u(b(d(x)))
v(c(x)) → b(x)
w(a(a(x))) → u(w(x))
w(a(c(x))) → u(b(d(x)))
w(c(x)) → b(x)

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:
v(a(a(x))) → u(v(x))
v(a(c(x))) → u(b(d(x)))
w(a(a(x))) → u(w(x))
w(a(c(x))) → u(b(d(x)))

(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:

v(c(x)) → b(x)
a(c(d(x))) → c(x)
u(b(d(d(x)))) → b(x)
w(c(x)) → b(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:

v(c(x)) → b(x)
a(c(d(x))) → c(x)
u(b(d(d(x)))) → b(x)
w(c(x)) → b(x)

Rewrite Strategy: INNERMOST

(5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted Cpx (relative) TRS to CDT

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

v(c(z0)) → b(z0)
a(c(d(z0))) → c(z0)
u(b(d(d(z0)))) → b(z0)
w(c(z0)) → b(z0)

Tuples:

V(c(z0)) → c1
A(c(d(z0))) → c2
U(b(d(d(z0)))) → c3
W(c(z0)) → c4

S tuples:

V(c(z0)) → c1
A(c(d(z0))) → c2
U(b(d(d(z0)))) → c3
W(c(z0)) → c4

K tuples:none
Defined Rule Symbols:

v, a, u, w

Defined Pair Symbols:

V, A, U, W

Compound Symbols:

c1, c2, c3, c4

(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing nodes:

U(b(d(d(z0)))) → c3
A(c(d(z0))) → c2
W(c(z0)) → c4
V(c(z0)) → c1

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

v(c(z0)) → b(z0)
a(c(d(z0))) → c(z0)
u(b(d(d(z0)))) → b(z0)
w(c(z0)) → b(z0)

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

v, a, u, w

Defined Pair Symbols:none

Compound Symbols:none

(9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

(10) BOUNDS(1, 1)