(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__nats → a__adx(a__zeros)
a__zeros → cons(0, zeros)
a__incr(cons(X, Y)) → cons(s(X), incr(Y))
a__adx(cons(X, Y)) → a__incr(cons(X, adx(Y)))
a__hd(cons(X, Y)) → mark(X)
a__tl(cons(X, Y)) → mark(Y)
mark(nats) → a__nats
mark(adx(X)) → a__adx(mark(X))
mark(zeros) → a__zeros
mark(incr(X)) → a__incr(mark(X))
mark(hd(X)) → a__hd(mark(X))
mark(tl(X)) → a__tl(mark(X))
mark(cons(X1, X2)) → cons(X1, X2)
mark(0) → 0
mark(s(X)) → s(X)
a__nats → nats
a__adx(X) → adx(X)
a__zeros → zeros
a__incr(X) → incr(X)
a__hd(X) → hd(X)
a__tl(X) → tl(X)
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0) = 0
POL(a__adx(x1)) = 2·x1
POL(a__hd(x1)) = 2 + 2·x1
POL(a__incr(x1)) = x1
POL(a__nats) = 0
POL(a__tl(x1)) = x1
POL(a__zeros) = 0
POL(adx(x1)) = 2·x1
POL(cons(x1, x2)) = 2·x1 + 2·x2
POL(hd(x1)) = 2 + 2·x1
POL(incr(x1)) = x1
POL(mark(x1)) = x1
POL(nats) = 0
POL(s(x1)) = x1
POL(tl(x1)) = x1
POL(zeros) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a__hd(cons(X, Y)) → mark(X)
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__nats → a__adx(a__zeros)
a__zeros → cons(0, zeros)
a__incr(cons(X, Y)) → cons(s(X), incr(Y))
a__adx(cons(X, Y)) → a__incr(cons(X, adx(Y)))
a__tl(cons(X, Y)) → mark(Y)
mark(nats) → a__nats
mark(adx(X)) → a__adx(mark(X))
mark(zeros) → a__zeros
mark(incr(X)) → a__incr(mark(X))
mark(hd(X)) → a__hd(mark(X))
mark(tl(X)) → a__tl(mark(X))
mark(cons(X1, X2)) → cons(X1, X2)
mark(0) → 0
mark(s(X)) → s(X)
a__nats → nats
a__adx(X) → adx(X)
a__zeros → zeros
a__incr(X) → incr(X)
a__hd(X) → hd(X)
a__tl(X) → tl(X)
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0) = 0
POL(a__adx(x1)) = x1
POL(a__hd(x1)) = 2 + x1
POL(a__incr(x1)) = x1
POL(a__nats) = 2
POL(a__tl(x1)) = 2 + 2·x1
POL(a__zeros) = 0
POL(adx(x1)) = x1
POL(cons(x1, x2)) = x1 + 2·x2
POL(hd(x1)) = 2 + x1
POL(incr(x1)) = x1
POL(mark(x1)) = 2·x1
POL(nats) = 1
POL(s(x1)) = x1
POL(tl(x1)) = 1 + 2·x1
POL(zeros) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a__nats → a__adx(a__zeros)
a__tl(cons(X, Y)) → mark(Y)
mark(hd(X)) → a__hd(mark(X))
a__nats → nats
a__tl(X) → tl(X)
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__zeros → cons(0, zeros)
a__incr(cons(X, Y)) → cons(s(X), incr(Y))
a__adx(cons(X, Y)) → a__incr(cons(X, adx(Y)))
mark(nats) → a__nats
mark(adx(X)) → a__adx(mark(X))
mark(zeros) → a__zeros
mark(incr(X)) → a__incr(mark(X))
mark(tl(X)) → a__tl(mark(X))
mark(cons(X1, X2)) → cons(X1, X2)
mark(0) → 0
mark(s(X)) → s(X)
a__adx(X) → adx(X)
a__zeros → zeros
a__incr(X) → incr(X)
a__hd(X) → hd(X)
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0) = 0
POL(a__adx(x1)) = x1
POL(a__hd(x1)) = 2 + 2·x1
POL(a__incr(x1)) = x1
POL(a__nats) = 0
POL(a__tl(x1)) = x1
POL(a__zeros) = 0
POL(adx(x1)) = x1
POL(cons(x1, x2)) = 2·x1 + x2
POL(hd(x1)) = 1 + x1
POL(incr(x1)) = x1
POL(mark(x1)) = x1
POL(nats) = 0
POL(s(x1)) = x1
POL(tl(x1)) = 2·x1
POL(zeros) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a__hd(X) → hd(X)
(6) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__zeros → cons(0, zeros)
a__incr(cons(X, Y)) → cons(s(X), incr(Y))
a__adx(cons(X, Y)) → a__incr(cons(X, adx(Y)))
mark(nats) → a__nats
mark(adx(X)) → a__adx(mark(X))
mark(zeros) → a__zeros
mark(incr(X)) → a__incr(mark(X))
mark(tl(X)) → a__tl(mark(X))
mark(cons(X1, X2)) → cons(X1, X2)
mark(0) → 0
mark(s(X)) → s(X)
a__adx(X) → adx(X)
a__zeros → zeros
a__incr(X) → incr(X)
Q is empty.
(7) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0) = 1
POL(a__adx(x1)) = x1
POL(a__incr(x1)) = x1
POL(a__nats) = 1
POL(a__tl(x1)) = 1 + 2·x1
POL(a__zeros) = 2
POL(adx(x1)) = x1
POL(cons(x1, x2)) = 2·x1 + 2·x2
POL(incr(x1)) = x1
POL(mark(x1)) = 2 + 2·x1
POL(nats) = 0
POL(s(x1)) = x1
POL(tl(x1)) = 2 + 2·x1
POL(zeros) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
mark(nats) → a__nats
mark(tl(X)) → a__tl(mark(X))
mark(cons(X1, X2)) → cons(X1, X2)
mark(0) → 0
mark(s(X)) → s(X)
a__zeros → zeros
(8) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__zeros → cons(0, zeros)
a__incr(cons(X, Y)) → cons(s(X), incr(Y))
a__adx(cons(X, Y)) → a__incr(cons(X, adx(Y)))
mark(adx(X)) → a__adx(mark(X))
mark(zeros) → a__zeros
mark(incr(X)) → a__incr(mark(X))
a__adx(X) → adx(X)
a__incr(X) → incr(X)
Q is empty.
(9) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0) = 0
POL(a__adx(x1)) = 2 + 2·x1
POL(a__incr(x1)) = 2 + x1
POL(a__zeros) = 2
POL(adx(x1)) = 2 + 2·x1
POL(cons(x1, x2)) = 2 + 2·x1 + x2
POL(incr(x1)) = 1 + x1
POL(mark(x1)) = 2 + 2·x1
POL(s(x1)) = x1
POL(zeros) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a__incr(cons(X, Y)) → cons(s(X), incr(Y))
a__incr(X) → incr(X)
(10) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__zeros → cons(0, zeros)
a__adx(cons(X, Y)) → a__incr(cons(X, adx(Y)))
mark(adx(X)) → a__adx(mark(X))
mark(zeros) → a__zeros
mark(incr(X)) → a__incr(mark(X))
a__adx(X) → adx(X)
Q is empty.
(11) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0) = 0
POL(a__adx(x1)) = 2 + x1
POL(a__incr(x1)) = 1 + x1
POL(a__zeros) = 2
POL(adx(x1)) = 1 + x1
POL(cons(x1, x2)) = 1 + 2·x1 + x2
POL(incr(x1)) = 2 + 2·x1
POL(mark(x1)) = 1 + 2·x1
POL(zeros) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
mark(zeros) → a__zeros
mark(incr(X)) → a__incr(mark(X))
a__adx(X) → adx(X)
(12) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__zeros → cons(0, zeros)
a__adx(cons(X, Y)) → a__incr(cons(X, adx(Y)))
mark(adx(X)) → a__adx(mark(X))
Q is empty.
(13) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0) = 0
POL(a__adx(x1)) = 2·x1
POL(a__incr(x1)) = x1
POL(a__zeros) = 2
POL(adx(x1)) = 2 + 2·x1
POL(cons(x1, x2)) = 2 + 2·x1 + x2
POL(mark(x1)) = 1 + 2·x1
POL(zeros) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
mark(adx(X)) → a__adx(mark(X))
(14) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__zeros → cons(0, zeros)
a__adx(cons(X, Y)) → a__incr(cons(X, adx(Y)))
Q is empty.
(15) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Knuth-Bendix order [KBO] with precedence:
adx1 > aadx1 > aincr1 > zeros > 0 > azeros > cons2
and weight map:
a__zeros=2
0=1
zeros=1
a__adx_1=2
a__incr_1=1
adx_1=1
cons_2=0
The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a__zeros → cons(0, zeros)
a__adx(cons(X, Y)) → a__incr(cons(X, adx(Y)))
(16) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(17) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(18) YES