(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
terms(N) → cons(recip(sqr(N)))
sqr(0) → 0
sqr(s) → s
dbl(0) → 0
dbl(s) → s
add(0, X) → X
add(s, Y) → s
first(0, X) → nil
first(s, cons(Y)) → cons(Y)
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0) = 1
POL(add(x1, x2)) = 1 + x1 + x2
POL(cons(x1)) = x1
POL(dbl(x1)) = x1
POL(first(x1, x2)) = x1 + x2
POL(nil) = 0
POL(recip(x1)) = x1
POL(s) = 1
POL(sqr(x1)) = x1
POL(terms(x1)) = 1 + x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
terms(N) → cons(recip(sqr(N)))
add(0, X) → X
add(s, Y) → s
first(0, X) → nil
first(s, cons(Y)) → cons(Y)
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
sqr(0) → 0
sqr(s) → s
dbl(0) → 0
dbl(s) → s
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0) = 0
POL(dbl(x1)) = x1
POL(s) = 0
POL(sqr(x1)) = 1 + x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
sqr(0) → 0
sqr(s) → s
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
dbl(0) → 0
dbl(s) → s
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0) = 0
POL(dbl(x1)) = 1 + x1
POL(s) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
dbl(0) → 0
dbl(s) → s
(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