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:

f(f(x, y, z), u, f(x, y, v)) → f(x, y, f(z, u, v))
f(x, y, y) → y
f(x, y, g(y)) → x
f(x, x, y) → x
f(g(x), 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:
f(f(x, y, z), u, f(x, y, v)) → f(x, y, f(z, u, v))

(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:

f(x, x, y) → x
f(x, y, y) → y
f(x, y, g(y)) → x
f(g(x), x, y) → y

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:

f(x, x, y) → x
f(x, y, y) → y
f(x, y, g(y)) → x
f(g(x), x, y) → y

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F(z0, z0, z1) → c
F(z0, z1, z1) → c1
F(z0, z1, g(z1)) → c2
F(g(z0), z0, z1) → c3

S tuples:

F(z0, z0, z1) → c
F(z0, z1, z1) → c1
F(z0, z1, g(z1)) → c2
F(g(z0), z0, z1) → c3

K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c, c1, c2, c3

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

Removed 4 trailing nodes:

F(z0, z1, z1) → c1
F(z0, z0, z1) → c
F(z0, z1, g(z1)) → c2
F(g(z0), z0, z1) → c3

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

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

f

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty

(10) BOUNDS(1, 1)