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:

g(X) → u(h(X), h(X), X)
u(d, c(Y), X) → k(Y)
h(d) → c(a)
h(d) → c(b)
f(k(a), k(b), X) → f(X, X, X)

Rewrite Strategy: FULL

(1) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)

Converted rc-obligation to irc-obligation.

The duplicating contexts are:
g([])
f(k(a), k(b), [])

The defined contexts are:
u([], x1, x2)
u(x0, [], x2)

[] just represents basic- or constructor-terms in the following defined contexts:
u([], x1, x2)
u(x0, [], x2)

As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, 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:

g(X) → u(h(X), h(X), X)
u(d, c(Y), X) → k(Y)
h(d) → c(a)
h(d) → c(b)
f(k(a), k(b), X) → f(X, X, X)

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(z0) → u(h(z0), h(z0), z0)
u(d, c(z0), z1) → k(z0)
h(d) → c(a)
h(d) → c(b)
f(k(a), k(b), z0) → f(z0, z0, z0)

Tuples:

G(z0) → c1(U(h(z0), h(z0), z0), H(z0), H(z0))
U(d, c(z0), z1) → c2
H(d) → c3
H(d) → c4
F(k(a), k(b), z0) → c5(F(z0, z0, z0))

S tuples:

G(z0) → c1(U(h(z0), h(z0), z0), H(z0), H(z0))
U(d, c(z0), z1) → c2
H(d) → c3
H(d) → c4
F(k(a), k(b), z0) → c5(F(z0, z0, z0))

K tuples:none
Defined Rule Symbols:

g, u, h, f

Defined Pair Symbols:

G, U, H, F

Compound Symbols:

c1, c2, c3, c4, c5

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

Removed 5 trailing nodes:

G(z0) → c1(U(h(z0), h(z0), z0), H(z0), H(z0))
F(k(a), k(b), z0) → c5(F(z0, z0, z0))
H(d) → c4
H(d) → c3
U(d, c(z0), z1) → c2

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(z0) → u(h(z0), h(z0), z0)
u(d, c(z0), z1) → k(z0)
h(d) → c(a)
h(d) → c(b)
f(k(a), k(b), z0) → f(z0, z0, z0)

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

g, u, h, f

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty

(8) BOUNDS(1, 1)