(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
active(f(g(X), Y)) → mark(f(X, f(g(X), Y)))
active(f(X1, X2)) → f(active(X1), X2)
active(g(X)) → g(active(X))
f(mark(X1), X2) → mark(f(X1, X2))
g(mark(X)) → mark(g(X))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(g(X)) → g(proper(X))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(active(x1)) = 1 + 2·x1
POL(f(x1, x2)) = 1 + 2·x1 + x2
POL(g(x1)) = x1
POL(mark(x1)) = x1
POL(ok(x1)) = 2 + 2·x1
POL(proper(x1)) = x1
POL(top(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
active(f(g(X), Y)) → mark(f(X, f(g(X), Y)))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(ok(X)) → top(active(X))
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
active(f(X1, X2)) → f(active(X1), X2)
active(g(X)) → g(active(X))
f(mark(X1), X2) → mark(f(X1, X2))
g(mark(X)) → mark(g(X))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(g(X)) → g(proper(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(active(x1)) = x1
POL(f(x1, x2)) = x1 + x2
POL(g(x1)) = x1
POL(mark(x1)) = 1 + x1
POL(ok(x1)) = x1
POL(proper(x1)) = x1
POL(top(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
top(mark(X)) → top(proper(X))
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
active(f(X1, X2)) → f(active(X1), X2)
active(g(X)) → g(active(X))
f(mark(X1), X2) → mark(f(X1, X2))
g(mark(X)) → mark(g(X))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(g(X)) → g(proper(X))
g(ok(X)) → ok(g(X))
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:
active1 > f2 > mark1
active1 > g1 > mark1
active1 > g1 > ok1
proper1 > f2 > mark1
proper1 > g1 > mark1
proper1 > g1 > ok1
Status:
trivial
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
active(f(X1, X2)) → f(active(X1), X2)
active(g(X)) → g(active(X))
f(mark(X1), X2) → mark(f(X1, X2))
g(mark(X)) → mark(g(X))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(g(X)) → g(proper(X))
g(ok(X)) → ok(g(X))
(6) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(7) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(8) TRUE
(9) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(10) TRUE