(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, y, w, w, a) → g1(x, x, y, w)
f(x, y, w, a, a) → g1(y, x, x, w)
f(x, y, a, a, w) → g2(x, y, y, w)
f(x, y, a, w, w) → g2(y, y, x, w)
g1(x, x, y, a) → h(x, y)
g1(y, x, x, a) → h(x, y)
g2(x, y, y, a) → h(x, y)
g2(y, y, x, a) → h(x, y)
h(x, x) → x
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a) = 1
POL(f(x1, x2, x3, x4, x5)) = 2 + 2·x1 + 2·x2 + 2·x3 + x4 + 2·x5
POL(g1(x1, x2, x3, x4)) = 2 + x1 + x2 + x3 + 2·x4
POL(g2(x1, x2, x3, x4)) = 1 + x1 + x2 + x3 + x4
POL(h(x1, x2)) = 2 + x1 + x2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
f(x, y, w, w, a) → g1(x, x, y, w)
f(x, y, w, a, a) → g1(y, x, x, w)
f(x, y, a, a, w) → g2(x, y, y, w)
f(x, y, a, w, w) → g2(y, y, x, w)
g1(x, x, y, a) → h(x, y)
g1(y, x, x, a) → h(x, y)
h(x, x) → x
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
g2(x, y, y, a) → h(x, y)
g2(y, y, x, a) → h(x, y)
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a) = 0
POL(g2(x1, x2, x3, x4)) = 1 + x1 + x2 + x3 + x4
POL(h(x1, x2)) = x1 + x2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
g2(x, y, y, a) → h(x, y)
g2(y, y, x, a) → h(x, y)
(4) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(5) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(6) TRUE
(7) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(8) TRUE