(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__incr(nil) → nil
a__incr(cons(X, L)) → cons(s(mark(X)), incr(L))
a__adx(nil) → nil
a__adx(cons(X, L)) → a__incr(cons(mark(X), adx(L)))
a__nats → a__adx(a__zeros)
a__zeros → cons(0, zeros)
a__head(cons(X, L)) → mark(X)
a__tail(cons(X, L)) → mark(L)
mark(incr(X)) → a__incr(mark(X))
mark(adx(X)) → a__adx(mark(X))
mark(nats) → a__nats
mark(zeros) → a__zeros
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__incr(X) → incr(X)
a__adx(X) → adx(X)
a__nats → nats
a__zeros → zeros
a__head(X) → head(X)
a__tail(X) → tail(X)
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0) = 0
POL(a__adx(x1)) = x1
POL(a__head(x1)) = 2 + 2·x1
POL(a__incr(x1)) = x1
POL(a__nats) = 2
POL(a__tail(x1)) = 2 + x1
POL(a__zeros) = 2
POL(adx(x1)) = x1
POL(cons(x1, x2)) = x1 + x2
POL(head(x1)) = 2 + 2·x1
POL(incr(x1)) = x1
POL(mark(x1)) = x1
POL(nats) = 2
POL(nil) = 0
POL(s(x1)) = x1
POL(tail(x1)) = 2 + x1
POL(zeros) = 2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a__head(cons(X, L)) → mark(X)
a__tail(cons(X, L)) → mark(L)
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__incr(nil) → nil
a__incr(cons(X, L)) → cons(s(mark(X)), incr(L))
a__adx(nil) → nil
a__adx(cons(X, L)) → a__incr(cons(mark(X), adx(L)))
a__nats → a__adx(a__zeros)
a__zeros → cons(0, zeros)
mark(incr(X)) → a__incr(mark(X))
mark(adx(X)) → a__adx(mark(X))
mark(nats) → a__nats
mark(zeros) → a__zeros
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__incr(X) → incr(X)
a__adx(X) → adx(X)
a__nats → nats
a__zeros → zeros
a__head(X) → head(X)
a__tail(X) → tail(X)
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0) = 0
POL(a__adx(x1)) = 2 + 2·x1
POL(a__head(x1)) = 2 + 2·x1
POL(a__incr(x1)) = x1
POL(a__nats) = 2
POL(a__tail(x1)) = 2 + x1
POL(a__zeros) = 0
POL(adx(x1)) = 2 + 2·x1
POL(cons(x1, x2)) = x1 + x2
POL(head(x1)) = 2 + 2·x1
POL(incr(x1)) = x1
POL(mark(x1)) = x1
POL(nats) = 2
POL(nil) = 2
POL(s(x1)) = x1
POL(tail(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__adx(nil) → nil
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__incr(nil) → nil
a__incr(cons(X, L)) → cons(s(mark(X)), incr(L))
a__adx(cons(X, L)) → a__incr(cons(mark(X), adx(L)))
a__nats → a__adx(a__zeros)
a__zeros → cons(0, zeros)
mark(incr(X)) → a__incr(mark(X))
mark(adx(X)) → a__adx(mark(X))
mark(nats) → a__nats
mark(zeros) → a__zeros
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__incr(X) → incr(X)
a__adx(X) → adx(X)
a__nats → nats
a__zeros → zeros
a__head(X) → head(X)
a__tail(X) → tail(X)
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0) = 0
POL(a__adx(x1)) = x1
POL(a__head(x1)) = 1 + x1
POL(a__incr(x1)) = x1
POL(a__nats) = 2
POL(a__tail(x1)) = 1 + 2·x1
POL(a__zeros) = 0
POL(adx(x1)) = x1
POL(cons(x1, x2)) = x1 + x2
POL(head(x1)) = 1 + x1
POL(incr(x1)) = x1
POL(mark(x1)) = x1
POL(nats) = 2
POL(nil) = 2
POL(s(x1)) = x1
POL(tail(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)
(6) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__incr(nil) → nil
a__incr(cons(X, L)) → cons(s(mark(X)), incr(L))
a__adx(cons(X, L)) → a__incr(cons(mark(X), adx(L)))
a__zeros → cons(0, zeros)
mark(incr(X)) → a__incr(mark(X))
mark(adx(X)) → a__adx(mark(X))
mark(nats) → a__nats
mark(zeros) → a__zeros
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__incr(X) → incr(X)
a__adx(X) → adx(X)
a__nats → nats
a__zeros → zeros
a__head(X) → head(X)
a__tail(X) → tail(X)
Q is empty.
(7) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A__INCR(cons(X, L)) → MARK(X)
A__ADX(cons(X, L)) → A__INCR(cons(mark(X), adx(L)))
A__ADX(cons(X, L)) → MARK(X)
MARK(incr(X)) → A__INCR(mark(X))
MARK(incr(X)) → MARK(X)
MARK(adx(X)) → A__ADX(mark(X))
MARK(adx(X)) → MARK(X)
MARK(nats) → A__NATS
MARK(zeros) → A__ZEROS
MARK(head(X)) → A__HEAD(mark(X))
MARK(head(X)) → MARK(X)
MARK(tail(X)) → A__TAIL(mark(X))
MARK(tail(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
The TRS R consists of the following rules:
a__incr(nil) → nil
a__incr(cons(X, L)) → cons(s(mark(X)), incr(L))
a__adx(cons(X, L)) → a__incr(cons(mark(X), adx(L)))
a__zeros → cons(0, zeros)
mark(incr(X)) → a__incr(mark(X))
mark(adx(X)) → a__adx(mark(X))
mark(nats) → a__nats
mark(zeros) → a__zeros
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__incr(X) → incr(X)
a__adx(X) → adx(X)
a__nats → nats
a__zeros → zeros
a__head(X) → head(X)
a__tail(X) → tail(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(incr(X)) → A__INCR(mark(X))
A__INCR(cons(X, L)) → MARK(X)
MARK(incr(X)) → MARK(X)
MARK(adx(X)) → A__ADX(mark(X))
A__ADX(cons(X, L)) → A__INCR(cons(mark(X), adx(L)))
A__ADX(cons(X, L)) → MARK(X)
MARK(adx(X)) → MARK(X)
MARK(head(X)) → MARK(X)
MARK(tail(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
The TRS R consists of the following rules:
a__incr(nil) → nil
a__incr(cons(X, L)) → cons(s(mark(X)), incr(L))
a__adx(cons(X, L)) → a__incr(cons(mark(X), adx(L)))
a__zeros → cons(0, zeros)
mark(incr(X)) → a__incr(mark(X))
mark(adx(X)) → a__adx(mark(X))
mark(nats) → a__nats
mark(zeros) → a__zeros
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__incr(X) → incr(X)
a__adx(X) → adx(X)
a__nats → nats
a__zeros → zeros
a__head(X) → head(X)
a__tail(X) → tail(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) MRRProof (EQUIVALENT transformation)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
A__ADX(cons(X, L)) → A__INCR(cons(mark(X), adx(L)))
A__ADX(cons(X, L)) → MARK(X)
MARK(adx(X)) → MARK(X)
MARK(head(X)) → MARK(X)
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(A__ADX(x1)) = 2 + 2·x1
POL(A__INCR(x1)) = x1
POL(MARK(x1)) = 2·x1
POL(a__adx(x1)) = 1 + 2·x1
POL(a__head(x1)) = 2 + 2·x1
POL(a__incr(x1)) = x1
POL(a__nats) = 1
POL(a__tail(x1)) = x1
POL(a__zeros) = 2
POL(adx(x1)) = 1 + 2·x1
POL(cons(x1, x2)) = 2·x1 + x2
POL(head(x1)) = 2 + 2·x1
POL(incr(x1)) = x1
POL(mark(x1)) = x1
POL(nats) = 1
POL(nil) = 0
POL(s(x1)) = x1
POL(tail(x1)) = x1
POL(zeros) = 2
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(incr(X)) → A__INCR(mark(X))
A__INCR(cons(X, L)) → MARK(X)
MARK(incr(X)) → MARK(X)
MARK(adx(X)) → A__ADX(mark(X))
MARK(tail(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
The TRS R consists of the following rules:
a__incr(nil) → nil
a__incr(cons(X, L)) → cons(s(mark(X)), incr(L))
a__adx(cons(X, L)) → a__incr(cons(mark(X), adx(L)))
a__zeros → cons(0, zeros)
mark(incr(X)) → a__incr(mark(X))
mark(adx(X)) → a__adx(mark(X))
mark(nats) → a__nats
mark(zeros) → a__zeros
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__incr(X) → incr(X)
a__adx(X) → adx(X)
a__nats → nats
a__zeros → zeros
a__head(X) → head(X)
a__tail(X) → tail(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A__INCR(cons(X, L)) → MARK(X)
MARK(incr(X)) → A__INCR(mark(X))
MARK(incr(X)) → MARK(X)
MARK(tail(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
The TRS R consists of the following rules:
a__incr(nil) → nil
a__incr(cons(X, L)) → cons(s(mark(X)), incr(L))
a__adx(cons(X, L)) → a__incr(cons(mark(X), adx(L)))
a__zeros → cons(0, zeros)
mark(incr(X)) → a__incr(mark(X))
mark(adx(X)) → a__adx(mark(X))
mark(nats) → a__nats
mark(zeros) → a__zeros
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__incr(X) → incr(X)
a__adx(X) → adx(X)
a__nats → nats
a__zeros → zeros
a__head(X) → head(X)
a__tail(X) → tail(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(15) MRRProof (EQUIVALENT transformation)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
MARK(tail(X)) → MARK(X)
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(A__INCR(x1)) = 2·x1
POL(MARK(x1)) = 2·x1
POL(a__adx(x1)) = x1
POL(a__head(x1)) = 2 + x1
POL(a__incr(x1)) = x1
POL(a__nats) = 0
POL(a__tail(x1)) = 1 + x1
POL(a__zeros) = 0
POL(adx(x1)) = x1
POL(cons(x1, x2)) = x1 + 2·x2
POL(head(x1)) = 2 + x1
POL(incr(x1)) = x1
POL(mark(x1)) = x1
POL(nats) = 0
POL(nil) = 0
POL(s(x1)) = x1
POL(tail(x1)) = 1 + x1
POL(zeros) = 0
(16) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A__INCR(cons(X, L)) → MARK(X)
MARK(incr(X)) → A__INCR(mark(X))
MARK(incr(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
The TRS R consists of the following rules:
a__incr(nil) → nil
a__incr(cons(X, L)) → cons(s(mark(X)), incr(L))
a__adx(cons(X, L)) → a__incr(cons(mark(X), adx(L)))
a__zeros → cons(0, zeros)
mark(incr(X)) → a__incr(mark(X))
mark(adx(X)) → a__adx(mark(X))
mark(nats) → a__nats
mark(zeros) → a__zeros
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__incr(X) → incr(X)
a__adx(X) → adx(X)
a__nats → nats
a__zeros → zeros
a__head(X) → head(X)
a__tail(X) → tail(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(17) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
A__INCR(cons(X, L)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( A__INCR(x1) ) = x1 + 1 |
POL( a__adx(x1) ) = x1 + 1 |
POL( a__tail(x1) ) = max{0, -2} |
POL( cons(x1, x2) ) = 2x1 + 1 |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
mark(incr(X)) → a__incr(mark(X))
mark(adx(X)) → a__adx(mark(X))
mark(nats) → a__nats
mark(zeros) → a__zeros
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__adx(X) → adx(X)
a__incr(nil) → nil
a__incr(X) → incr(X)
a__head(X) → head(X)
a__tail(X) → tail(X)
a__incr(cons(X, L)) → cons(s(mark(X)), incr(L))
a__adx(cons(X, L)) → a__incr(cons(mark(X), adx(L)))
a__zeros → cons(0, zeros)
a__zeros → zeros
a__nats → nats
(18) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(incr(X)) → A__INCR(mark(X))
MARK(incr(X)) → MARK(X)
MARK(s(X)) → MARK(X)
The TRS R consists of the following rules:
a__incr(nil) → nil
a__incr(cons(X, L)) → cons(s(mark(X)), incr(L))
a__adx(cons(X, L)) → a__incr(cons(mark(X), adx(L)))
a__zeros → cons(0, zeros)
mark(incr(X)) → a__incr(mark(X))
mark(adx(X)) → a__adx(mark(X))
mark(nats) → a__nats
mark(zeros) → a__zeros
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__incr(X) → incr(X)
a__adx(X) → adx(X)
a__nats → nats
a__zeros → zeros
a__head(X) → head(X)
a__tail(X) → tail(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(19) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(20) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(s(X)) → MARK(X)
MARK(incr(X)) → MARK(X)
The TRS R consists of the following rules:
a__incr(nil) → nil
a__incr(cons(X, L)) → cons(s(mark(X)), incr(L))
a__adx(cons(X, L)) → a__incr(cons(mark(X), adx(L)))
a__zeros → cons(0, zeros)
mark(incr(X)) → a__incr(mark(X))
mark(adx(X)) → a__adx(mark(X))
mark(nats) → a__nats
mark(zeros) → a__zeros
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__incr(X) → incr(X)
a__adx(X) → adx(X)
a__nats → nats
a__zeros → zeros
a__head(X) → head(X)
a__tail(X) → tail(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(21) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(22) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(s(X)) → MARK(X)
MARK(incr(X)) → MARK(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(23) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- MARK(s(X)) → MARK(X)
The graph contains the following edges 1 > 1
- MARK(incr(X)) → MARK(X)
The graph contains the following edges 1 > 1
(24) YES