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, y) → x
g(x, y) → y
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:

g(x, y) → x
g(x, y) → y
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:

g(z0, z1) → z0
g(z0, z1) → z1
f(s(z0), z1, z1) → f(z1, z0, s(z0))

Tuples:

G(z0, z1) → c
G(z0, z1) → c1
F(s(z0), z1, z1) → c2(F(z1, z0, s(z0)))

S tuples:

G(z0, z1) → c
G(z0, z1) → c1
F(s(z0), z1, z1) → c2(F(z1, z0, s(z0)))

K tuples:none
Defined Rule Symbols:

g, f

Defined Pair Symbols:

G, F

Compound Symbols:

c, c1, c2

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

Removed 3 trailing nodes:

F(s(z0), z1, z1) → c2(F(z1, z0, s(z0)))
G(z0, z1) → c1
G(z0, z1) → c

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(z0, z1) → z0
g(z0, z1) → z1
f(s(z0), z1, z1) → f(z1, z0, s(z0))

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

g, f

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty

(8) BOUNDS(1, 1)