TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60022 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (103ms).
 | – Problem 2 was processed with processor PolynomialLinearRange4iUR (1752ms).
 |    | – Problem 3 was processed with processor DependencyGraph (35ms).
 |    |    | – Problem 4 was processed with processor PolynomialLinearRange4iUR (1039ms).
 |    |    |    | – Problem 5 was processed with processor DependencyGraph (24ms).
 |    |    |    |    | – Problem 6 was processed with processor PolynomialLinearRange4iUR (771ms).
 |    |    |    |    |    | – Problem 7 was processed with processor DependencyGraph (13ms).
 |    |    |    |    |    |    | – Problem 8 was processed with processor ForwardNarrowing (10ms).
 |    |    |    |    |    |    |    | – Problem 9 was processed with processor ForwardNarrowing (1ms).
 |    |    |    |    |    |    |    |    | – Problem 10 was processed with processor ForwardNarrowing (2ms).
 |    |    |    |    |    |    |    |    |    | – Problem 11 was processed with processor ForwardNarrowing (6ms).
 |    |    |    |    |    |    |    |    |    |    | – Problem 12 was processed with processor ForwardNarrowing (8ms).
 |    |    |    |    |    |    |    |    |    |    |    | – Problem 13 was processed with processor ForwardNarrowing (4ms).
 |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 14 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 15 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 16 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 17 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 18 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 19 was processed with processor ForwardNarrowing (8ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 20 was processed with processor ForwardNarrowing (3ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 21 was processed with processor ForwardNarrowing (6ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 22 was processed with processor ForwardNarrowing (35ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 23 was processed with processor ForwardNarrowing (12ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 24 was processed with processor ForwardNarrowing (5ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 25 was processed with processor ForwardNarrowing (7ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 26 was processed with processor ForwardNarrowing (7ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 27 was processed with processor ForwardNarrowing (7ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 28 was processed with processor ForwardNarrowing (8ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 29 was processed with processor ForwardNarrowing (18ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 30 was processed with processor ForwardNarrowing (49ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 31 was processed with processor ForwardNarrowing (46ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 32 was processed with processor ForwardNarrowing (232ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 33 was processed with processor ForwardNarrowing (230ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 34 was processed with processor ForwardNarrowing (330ms).
 |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |    | – Problem 35 remains open; application of the following processors failed [ForwardNarrowing (109ms), ForwardNarrowing (106ms), ForwardNarrowing (124ms), ForwardNarrowing (295ms), ForwardNarrowing (118ms), ForwardNarrowing (120ms), ForwardNarrowing (95ms), ForwardNarrowing (120ms), ForwardNarrowing (302ms), ForwardNarrowing (156ms), ForwardNarrowing (172ms), ForwardNarrowing (96ms), ForwardNarrowing (102ms), ForwardNarrowing (98ms), ForwardNarrowing (143ms), ForwardNarrowing (145ms), ForwardNarrowing (172ms), ForwardNarrowing (94ms), ForwardNarrowing (115ms), ForwardNarrowing (timeout)].

The following open problems remain:



Open Dependency Pair Problem 8

Dependency Pairs

mark#(incr(X))a__incr#(mark(X))mark#(cons(X1, X2))mark#(X1)
a__incr#(cons(X, XS))mark#(X)mark#(s(X))mark#(X)
mark#(incr(X))mark#(X)

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, a__head, head, cons


Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

a__tail#(cons(X, XS))mark#(XS)a__odds#a__pairs#
mark#(pairs)a__pairs#mark#(incr(X))a__incr#(mark(X))
a__odds#a__incr#(a__pairs)mark#(cons(X1, X2))mark#(X1)
mark#(nats)a__nats#mark#(incr(X))mark#(X)
mark#(odds)a__odds#mark#(head(X))mark#(X)
mark#(tail(X))mark#(X)mark#(head(X))a__head#(mark(X))
a__incr#(cons(X, XS))mark#(X)mark#(tail(X))a__tail#(mark(X))
mark#(s(X))mark#(X)a__head#(cons(X, XS))mark#(X)

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, a__head, head, cons

Strategy


The following SCCs where found

a__tail#(cons(X, XS)) → mark#(XS)mark#(incr(X)) → a__incr#(mark(X))
a__odds# → a__incr#(a__pairs)mark#(cons(X1, X2)) → mark#(X1)
mark#(incr(X)) → mark#(X)mark#(odds) → a__odds#
mark#(head(X)) → mark#(X)mark#(tail(X)) → mark#(X)
mark#(head(X)) → a__head#(mark(X))a__incr#(cons(X, XS)) → mark#(X)
mark#(tail(X)) → a__tail#(mark(X))mark#(s(X)) → mark#(X)
a__head#(cons(X, XS)) → mark#(X)

Problem 2: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

a__tail#(cons(X, XS))mark#(XS)mark#(incr(X))a__incr#(mark(X))
a__odds#a__incr#(a__pairs)mark#(cons(X1, X2))mark#(X1)
mark#(incr(X))mark#(X)mark#(head(X))mark#(X)
mark#(odds)a__odds#mark#(tail(X))mark#(X)
mark#(head(X))a__head#(mark(X))a__incr#(cons(X, XS))mark#(X)
mark#(s(X))mark#(X)mark#(tail(X))a__tail#(mark(X))
a__head#(cons(X, XS))mark#(X)

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, a__head, head, cons

Strategy


Polynomial Interpretation

Improved Usable rules

a__head(X)head(X)mark(cons(X1, X2))cons(mark(X1), X2)
a__oddsoddsa__natscons(0, incr(nats))
mark(0)0mark(head(X))a__head(mark(X))
a__oddsa__incr(a__pairs)a__tail(X)tail(X)
mark(tail(X))a__tail(mark(X))mark(pairs)a__pairs
a__pairspairsmark(nats)a__nats
a__tail(cons(X, XS))mark(XS)a__head(cons(X, XS))mark(X)
mark(incr(X))a__incr(mark(X))a__natsnats
mark(s(X))s(mark(X))mark(odds)a__odds
a__incr(X)incr(X)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__pairscons(0, incr(odds))

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

mark#(head(X))mark#(X)a__head#(cons(X, XS))mark#(X)

Problem 3: DependencyGraph



Dependency Pair Problem

Dependency Pairs

mark#(odds)a__odds#a__tail#(cons(X, XS))mark#(XS)
mark#(incr(X))a__incr#(mark(X))mark#(tail(X))mark#(X)
a__odds#a__incr#(a__pairs)mark#(head(X))a__head#(mark(X))
mark#(cons(X1, X2))mark#(X1)a__incr#(cons(X, XS))mark#(X)
mark#(tail(X))a__tail#(mark(X))mark#(s(X))mark#(X)
mark#(incr(X))mark#(X)

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, head, a__head, cons

Strategy


The following SCCs where found

mark#(odds) → a__odds#a__tail#(cons(X, XS)) → mark#(XS)
mark#(incr(X)) → a__incr#(mark(X))mark#(tail(X)) → mark#(X)
a__odds# → a__incr#(a__pairs)mark#(cons(X1, X2)) → mark#(X1)
a__incr#(cons(X, XS)) → mark#(X)mark#(s(X)) → mark#(X)
mark#(tail(X)) → a__tail#(mark(X))mark#(incr(X)) → mark#(X)

Problem 4: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

mark#(odds)a__odds#a__tail#(cons(X, XS))mark#(XS)
mark#(incr(X))a__incr#(mark(X))mark#(tail(X))mark#(X)
a__odds#a__incr#(a__pairs)mark#(cons(X1, X2))mark#(X1)
a__incr#(cons(X, XS))mark#(X)mark#(s(X))mark#(X)
mark#(tail(X))a__tail#(mark(X))mark#(incr(X))mark#(X)

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, head, a__head, cons

Strategy


Polynomial Interpretation

Improved Usable rules

a__head(X)head(X)mark(cons(X1, X2))cons(mark(X1), X2)
a__oddsoddsa__natscons(0, incr(nats))
mark(0)0mark(head(X))a__head(mark(X))
a__oddsa__incr(a__pairs)a__tail(X)tail(X)
mark(tail(X))a__tail(mark(X))mark(pairs)a__pairs
a__pairspairsmark(nats)a__nats
a__tail(cons(X, XS))mark(XS)a__head(cons(X, XS))mark(X)
mark(incr(X))a__incr(mark(X))a__natsnats
mark(s(X))s(mark(X))mark(odds)a__odds
a__incr(X)incr(X)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__pairscons(0, incr(odds))

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

a__odds#a__incr#(a__pairs)

Problem 5: DependencyGraph



Dependency Pair Problem

Dependency Pairs

a__tail#(cons(X, XS))mark#(XS)mark#(odds)a__odds#
mark#(incr(X))a__incr#(mark(X))mark#(tail(X))mark#(X)
mark#(cons(X1, X2))mark#(X1)a__incr#(cons(X, XS))mark#(X)
mark#(tail(X))a__tail#(mark(X))mark#(s(X))mark#(X)
mark#(incr(X))mark#(X)

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, a__head, head, cons

Strategy


The following SCCs where found

a__tail#(cons(X, XS)) → mark#(XS)mark#(incr(X)) → a__incr#(mark(X))
mark#(tail(X)) → mark#(X)mark#(cons(X1, X2)) → mark#(X1)
a__incr#(cons(X, XS)) → mark#(X)mark#(s(X)) → mark#(X)
mark#(tail(X)) → a__tail#(mark(X))mark#(incr(X)) → mark#(X)

Problem 6: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

a__tail#(cons(X, XS))mark#(XS)mark#(incr(X))a__incr#(mark(X))
mark#(tail(X))mark#(X)mark#(cons(X1, X2))mark#(X1)
a__incr#(cons(X, XS))mark#(X)mark#(s(X))mark#(X)
mark#(tail(X))a__tail#(mark(X))mark#(incr(X))mark#(X)

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, a__head, head, cons

Strategy


Polynomial Interpretation

Improved Usable rules

a__head(X)head(X)mark(cons(X1, X2))cons(mark(X1), X2)
a__oddsoddsa__natscons(0, incr(nats))
mark(0)0mark(head(X))a__head(mark(X))
a__oddsa__incr(a__pairs)a__tail(X)tail(X)
mark(tail(X))a__tail(mark(X))mark(pairs)a__pairs
a__pairspairsmark(nats)a__nats
a__tail(cons(X, XS))mark(XS)a__head(cons(X, XS))mark(X)
mark(incr(X))a__incr(mark(X))a__natsnats
mark(s(X))s(mark(X))mark(odds)a__odds
a__incr(X)incr(X)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__pairscons(0, incr(odds))

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

mark#(tail(X))mark#(X)mark#(tail(X))a__tail#(mark(X))

Problem 7: DependencyGraph



Dependency Pair Problem

Dependency Pairs

a__tail#(cons(X, XS))mark#(XS)mark#(incr(X))a__incr#(mark(X))
mark#(cons(X1, X2))mark#(X1)a__incr#(cons(X, XS))mark#(X)
mark#(s(X))mark#(X)mark#(incr(X))mark#(X)

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, head, a__head, cons

Strategy


The following SCCs where found

mark#(incr(X)) → a__incr#(mark(X))mark#(cons(X1, X2)) → mark#(X1)
a__incr#(cons(X, XS)) → mark#(X)mark#(s(X)) → mark#(X)
mark#(incr(X)) → mark#(X)

Problem 8: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(X))a__incr#(mark(X))mark#(cons(X1, X2))mark#(X1)
a__incr#(cons(X, XS))mark#(X)mark#(s(X))mark#(X)
mark#(incr(X))mark#(X)

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, head, a__head, cons

Strategy


The right-hand side of the rule mark#(incr(X)) → a__incr#(mark(X)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__incr#(a__head(mark(_x21))) 
a__incr#(a__nats) 
a__incr#(0) 
a__incr#(s(mark(_x21))) 
a__incr#(a__incr(mark(_x21))) 
a__incr#(cons(mark(_x21), _x22)) 
a__incr#(a__odds) 
a__incr#(a__tail(mark(_x21))) 
a__incr#(a__pairs) 
Thus, the rule mark#(incr(X)) → a__incr#(mark(X)) is replaced by the following rules:
mark#(incr(0)) → a__incr#(0)mark#(incr(odds)) → a__incr#(a__odds)
mark#(incr(head(_x21))) → a__incr#(a__head(mark(_x21)))mark#(incr(incr(_x21))) → a__incr#(a__incr(mark(_x21)))
mark#(incr(pairs)) → a__incr#(a__pairs)mark#(incr(tail(_x21))) → a__incr#(a__tail(mark(_x21)))
mark#(incr(cons(_x21, _x22))) → a__incr#(cons(mark(_x21), _x22))mark#(incr(s(_x21))) → a__incr#(s(mark(_x21)))
mark#(incr(nats)) → a__incr#(a__nats)

Problem 9: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(0))a__incr#(0)mark#(incr(odds))a__incr#(a__odds)
mark#(incr(head(_x21)))a__incr#(a__head(mark(_x21)))mark#(incr(incr(_x21)))a__incr#(a__incr(mark(_x21)))
mark#(cons(X1, X2))mark#(X1)mark#(incr(X))mark#(X)
mark#(incr(pairs))a__incr#(a__pairs)mark#(incr(tail(_x21)))a__incr#(a__tail(mark(_x21)))
mark#(incr(cons(_x21, _x22)))a__incr#(cons(mark(_x21), _x22))a__incr#(cons(X, XS))mark#(X)
mark#(incr(s(_x21)))a__incr#(s(mark(_x21)))mark#(s(X))mark#(X)
mark#(incr(nats))a__incr#(a__nats)

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, a__head, head, cons

Strategy


The right-hand side of the rule mark#(incr(0)) → a__incr#(0) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
Thus, the rule mark#(incr(0)) → a__incr#(0) is deleted.

Problem 10: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(head(_x21)))a__incr#(a__head(mark(_x21)))mark#(incr(odds))a__incr#(a__odds)
mark#(incr(incr(_x21)))a__incr#(a__incr(mark(_x21)))mark#(incr(pairs))a__incr#(a__pairs)
mark#(incr(tail(_x21)))a__incr#(a__tail(mark(_x21)))mark#(incr(cons(_x21, _x22)))a__incr#(cons(mark(_x21), _x22))
mark#(cons(X1, X2))mark#(X1)mark#(incr(s(_x21)))a__incr#(s(mark(_x21)))
a__incr#(cons(X, XS))mark#(X)mark#(incr(nats))a__incr#(a__nats)
mark#(s(X))mark#(X)mark#(incr(X))mark#(X)

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, head, a__head, cons

Strategy


The right-hand side of the rule mark#(incr(odds)) → a__incr#(a__odds) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__incr#(a__incr(a__pairs)) 
a__incr#(odds) 
Thus, the rule mark#(incr(odds)) → a__incr#(a__odds) is replaced by the following rules:
mark#(incr(odds)) → a__incr#(odds)mark#(incr(odds)) → a__incr#(a__incr(a__pairs))

Problem 11: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(head(_x21)))a__incr#(a__head(mark(_x21)))mark#(incr(incr(_x21)))a__incr#(a__incr(mark(_x21)))
mark#(cons(X1, X2))mark#(X1)mark#(incr(odds))a__incr#(a__incr(a__pairs))
mark#(incr(X))mark#(X)mark#(incr(odds))a__incr#(odds)
mark#(incr(pairs))a__incr#(a__pairs)mark#(incr(tail(_x21)))a__incr#(a__tail(mark(_x21)))
mark#(incr(cons(_x21, _x22)))a__incr#(cons(mark(_x21), _x22))a__incr#(cons(X, XS))mark#(X)
mark#(incr(s(_x21)))a__incr#(s(mark(_x21)))mark#(s(X))mark#(X)
mark#(incr(nats))a__incr#(a__nats)

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, a__head, head, cons

Strategy


The right-hand side of the rule mark#(incr(head(_x21))) → a__incr#(a__head(mark(_x21))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__incr#(a__head(0)) 
a__incr#(a__head(a__odds)) 
a__incr#(a__head(a__incr(mark(_x41)))) 
a__incr#(a__head(s(mark(_x41)))) 
a__incr#(a__head(a__nats)) 
a__incr#(a__head(a__head(mark(_x41)))) 
a__incr#(a__head(a__pairs)) 
a__incr#(head(mark(_x21))) 
a__incr#(a__head(a__tail(mark(_x41)))) 
a__incr#(a__head(cons(mark(_x41), _x42))) 
Thus, the rule mark#(incr(head(_x21))) → a__incr#(a__head(mark(_x21))) is replaced by the following rules:
mark#(incr(head(s(_x41)))) → a__incr#(a__head(s(mark(_x41))))mark#(incr(head(_x21))) → a__incr#(head(mark(_x21)))
mark#(incr(head(0))) → a__incr#(a__head(0))mark#(incr(head(pairs))) → a__incr#(a__head(a__pairs))
mark#(incr(head(cons(_x41, _x42)))) → a__incr#(a__head(cons(mark(_x41), _x42)))mark#(incr(head(tail(_x41)))) → a__incr#(a__head(a__tail(mark(_x41))))
mark#(incr(head(odds))) → a__incr#(a__head(a__odds))mark#(incr(head(head(_x41)))) → a__incr#(a__head(a__head(mark(_x41))))
mark#(incr(head(nats))) → a__incr#(a__head(a__nats))mark#(incr(head(incr(_x41)))) → a__incr#(a__head(a__incr(mark(_x41))))

Problem 12: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(head(s(_x41))))a__incr#(a__head(s(mark(_x41))))mark#(incr(incr(_x21)))a__incr#(a__incr(mark(_x21)))
mark#(cons(X1, X2))mark#(X1)mark#(incr(head(odds)))a__incr#(a__head(a__odds))
mark#(incr(odds))a__incr#(a__incr(a__pairs))mark#(incr(X))mark#(X)
mark#(incr(odds))a__incr#(odds)mark#(incr(pairs))a__incr#(a__pairs)
mark#(incr(head(_x21)))a__incr#(head(mark(_x21)))mark#(incr(tail(_x21)))a__incr#(a__tail(mark(_x21)))
mark#(incr(head(0)))a__incr#(a__head(0))mark#(incr(head(pairs)))a__incr#(a__head(a__pairs))
mark#(incr(cons(_x21, _x22)))a__incr#(cons(mark(_x21), _x22))mark#(incr(head(cons(_x41, _x42))))a__incr#(a__head(cons(mark(_x41), _x42)))
mark#(incr(head(tail(_x41))))a__incr#(a__head(a__tail(mark(_x41))))mark#(incr(s(_x21)))a__incr#(s(mark(_x21)))
a__incr#(cons(X, XS))mark#(X)mark#(incr(nats))a__incr#(a__nats)
mark#(s(X))mark#(X)mark#(incr(head(head(_x41))))a__incr#(a__head(a__head(mark(_x41))))
mark#(incr(head(nats)))a__incr#(a__head(a__nats))mark#(incr(head(incr(_x41))))a__incr#(a__head(a__incr(mark(_x41))))

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, head, a__head, cons

Strategy


The right-hand side of the rule mark#(incr(head(s(_x41)))) → a__incr#(a__head(s(mark(_x41)))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__incr#(a__head(s(a__tail(mark(_x51))))) 
a__incr#(a__head(s(0))) 
a__incr#(a__head(s(s(mark(_x51))))) 
a__incr#(a__head(s(a__head(mark(_x51))))) 
a__incr#(a__head(s(a__nats))) 
a__incr#(a__head(s(a__incr(mark(_x51))))) 
a__incr#(a__head(s(cons(mark(_x51), _x52)))) 
a__incr#(a__head(s(a__odds))) 
a__incr#(head(s(mark(_x41)))) 
a__incr#(a__head(s(a__pairs))) 
Thus, the rule mark#(incr(head(s(_x41)))) → a__incr#(a__head(s(mark(_x41)))) is replaced by the following rules:
mark#(incr(head(s(s(_x51))))) → a__incr#(a__head(s(s(mark(_x51)))))mark#(incr(head(s(incr(_x51))))) → a__incr#(a__head(s(a__incr(mark(_x51)))))
mark#(incr(head(s(pairs)))) → a__incr#(a__head(s(a__pairs)))mark#(incr(head(s(tail(_x51))))) → a__incr#(a__head(s(a__tail(mark(_x51)))))
mark#(incr(head(s(_x41)))) → a__incr#(head(s(mark(_x41))))mark#(incr(head(s(odds)))) → a__incr#(a__head(s(a__odds)))
mark#(incr(head(s(nats)))) → a__incr#(a__head(s(a__nats)))mark#(incr(head(s(0)))) → a__incr#(a__head(s(0)))
mark#(incr(head(s(cons(_x51, _x52))))) → a__incr#(a__head(s(cons(mark(_x51), _x52))))mark#(incr(head(s(head(_x51))))) → a__incr#(a__head(s(a__head(mark(_x51)))))

Problem 13: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(incr(_x21)))a__incr#(a__incr(mark(_x21)))mark#(incr(head(s(pairs))))a__incr#(a__head(s(a__pairs)))
mark#(incr(odds))a__incr#(a__incr(a__pairs))mark#(incr(head(s(cons(_x51, _x52)))))a__incr#(a__head(s(cons(mark(_x51), _x52))))
mark#(incr(head(s(head(_x51)))))a__incr#(a__head(s(a__head(mark(_x51)))))mark#(incr(head(s(s(_x51)))))a__incr#(a__head(s(s(mark(_x51)))))
mark#(incr(odds))a__incr#(odds)mark#(incr(head(s(tail(_x51)))))a__incr#(a__head(s(a__tail(mark(_x51)))))
mark#(incr(cons(_x21, _x22)))a__incr#(cons(mark(_x21), _x22))mark#(incr(head(cons(_x41, _x42))))a__incr#(a__head(cons(mark(_x41), _x42)))
mark#(incr(s(_x21)))a__incr#(s(mark(_x21)))a__incr#(cons(X, XS))mark#(X)
mark#(s(X))mark#(X)mark#(incr(nats))a__incr#(a__nats)
mark#(incr(head(s(0))))a__incr#(a__head(s(0)))mark#(incr(head(head(_x41))))a__incr#(a__head(a__head(mark(_x41))))
mark#(incr(head(incr(_x41))))a__incr#(a__head(a__incr(mark(_x41))))mark#(incr(head(s(odds))))a__incr#(a__head(s(a__odds)))
mark#(incr(head(s(_x41))))a__incr#(head(s(mark(_x41))))mark#(cons(X1, X2))mark#(X1)
mark#(incr(head(odds)))a__incr#(a__head(a__odds))mark#(incr(X))mark#(X)
mark#(incr(head(s(incr(_x51)))))a__incr#(a__head(s(a__incr(mark(_x51)))))mark#(incr(pairs))a__incr#(a__pairs)
mark#(incr(head(_x21)))a__incr#(head(mark(_x21)))mark#(incr(tail(_x21)))a__incr#(a__tail(mark(_x21)))
mark#(incr(head(0)))a__incr#(a__head(0))mark#(incr(head(pairs)))a__incr#(a__head(a__pairs))
mark#(incr(head(tail(_x41))))a__incr#(a__head(a__tail(mark(_x41))))mark#(incr(head(s(nats))))a__incr#(a__head(s(a__nats)))
mark#(incr(head(nats)))a__incr#(a__head(a__nats))

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, a__head, head, cons

Strategy


The right-hand side of the rule mark#(incr(incr(_x21))) → a__incr#(a__incr(mark(_x21))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__incr#(a__incr(a__nats)) 
a__incr#(a__incr(a__odds)) 
a__incr#(incr(mark(_x21))) 
a__incr#(a__incr(a__tail(mark(_x41)))) 
a__incr#(a__incr(0)) 
a__incr#(a__incr(a__pairs)) 
a__incr#(a__incr(a__head(mark(_x41)))) 
a__incr#(a__incr(s(mark(_x41)))) 
a__incr#(a__incr(a__incr(mark(_x41)))) 
a__incr#(a__incr(cons(mark(_x41), _x42))) 
Thus, the rule mark#(incr(incr(_x21))) → a__incr#(a__incr(mark(_x21))) is replaced by the following rules:
mark#(incr(incr(_x21))) → a__incr#(incr(mark(_x21)))mark#(incr(incr(tail(_x41)))) → a__incr#(a__incr(a__tail(mark(_x41))))
mark#(incr(incr(0))) → a__incr#(a__incr(0))mark#(incr(incr(cons(_x41, _x42)))) → a__incr#(a__incr(cons(mark(_x41), _x42)))
mark#(incr(incr(pairs))) → a__incr#(a__incr(a__pairs))mark#(incr(incr(incr(_x41)))) → a__incr#(a__incr(a__incr(mark(_x41))))
mark#(incr(incr(s(_x41)))) → a__incr#(a__incr(s(mark(_x41))))mark#(incr(incr(head(_x41)))) → a__incr#(a__incr(a__head(mark(_x41))))
mark#(incr(incr(nats))) → a__incr#(a__incr(a__nats))mark#(incr(incr(odds))) → a__incr#(a__incr(a__odds))

Problem 14: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(head(s(pairs))))a__incr#(a__head(s(a__pairs)))mark#(incr(incr(incr(_x41))))a__incr#(a__incr(a__incr(mark(_x41))))
mark#(incr(odds))a__incr#(a__incr(a__pairs))mark#(incr(head(s(head(_x51)))))a__incr#(a__head(s(a__head(mark(_x51)))))
mark#(incr(head(s(cons(_x51, _x52)))))a__incr#(a__head(s(cons(mark(_x51), _x52))))mark#(incr(odds))a__incr#(odds)
mark#(incr(head(s(s(_x51)))))a__incr#(a__head(s(s(mark(_x51)))))mark#(incr(head(s(tail(_x51)))))a__incr#(a__head(s(a__tail(mark(_x51)))))
mark#(incr(cons(_x21, _x22)))a__incr#(cons(mark(_x21), _x22))mark#(incr(head(cons(_x41, _x42))))a__incr#(a__head(cons(mark(_x41), _x42)))
a__incr#(cons(X, XS))mark#(X)mark#(incr(s(_x21)))a__incr#(s(mark(_x21)))
mark#(incr(incr(cons(_x41, _x42))))a__incr#(a__incr(cons(mark(_x41), _x42)))mark#(s(X))mark#(X)
mark#(incr(nats))a__incr#(a__nats)mark#(incr(head(s(0))))a__incr#(a__head(s(0)))
mark#(incr(head(head(_x41))))a__incr#(a__head(a__head(mark(_x41))))mark#(incr(head(incr(_x41))))a__incr#(a__head(a__incr(mark(_x41))))
mark#(incr(incr(tail(_x41))))a__incr#(a__incr(a__tail(mark(_x41))))mark#(incr(incr(0)))a__incr#(a__incr(0))
mark#(incr(head(s(odds))))a__incr#(a__head(s(a__odds)))mark#(incr(head(s(_x41))))a__incr#(head(s(mark(_x41))))
mark#(cons(X1, X2))mark#(X1)mark#(incr(head(odds)))a__incr#(a__head(a__odds))
mark#(incr(X))mark#(X)mark#(incr(incr(head(_x41))))a__incr#(a__incr(a__head(mark(_x41))))
mark#(incr(incr(nats)))a__incr#(a__incr(a__nats))mark#(incr(incr(odds)))a__incr#(a__incr(a__odds))
mark#(incr(head(s(incr(_x51)))))a__incr#(a__head(s(a__incr(mark(_x51)))))mark#(incr(pairs))a__incr#(a__pairs)
mark#(incr(head(_x21)))a__incr#(head(mark(_x21)))mark#(incr(tail(_x21)))a__incr#(a__tail(mark(_x21)))
mark#(incr(head(0)))a__incr#(a__head(0))mark#(incr(incr(_x21)))a__incr#(incr(mark(_x21)))
mark#(incr(head(pairs)))a__incr#(a__head(a__pairs))mark#(incr(head(s(nats))))a__incr#(a__head(s(a__nats)))
mark#(incr(head(tail(_x41))))a__incr#(a__head(a__tail(mark(_x41))))mark#(incr(incr(pairs)))a__incr#(a__incr(a__pairs))
mark#(incr(head(nats)))a__incr#(a__head(a__nats))mark#(incr(incr(s(_x41))))a__incr#(a__incr(s(mark(_x41))))

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, head, a__head, cons

Strategy


The right-hand side of the rule mark#(incr(head(s(pairs)))) → a__incr#(a__head(s(a__pairs))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__incr#(a__head(s(cons(0, incr(odds))))) 
a__incr#(head(s(a__pairs))) 
a__incr#(a__head(s(pairs))) 
Thus, the rule mark#(incr(head(s(pairs)))) → a__incr#(a__head(s(a__pairs))) is replaced by the following rules:
mark#(incr(head(s(pairs)))) → a__incr#(head(s(a__pairs)))mark#(incr(head(s(pairs)))) → a__incr#(a__head(s(pairs)))
mark#(incr(head(s(pairs)))) → a__incr#(a__head(s(cons(0, incr(odds)))))

Problem 15: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(head(s(pairs))))a__incr#(head(s(a__pairs)))mark#(incr(head(s(pairs))))a__incr#(a__head(s(cons(0, incr(odds)))))
mark#(incr(incr(incr(_x41))))a__incr#(a__incr(a__incr(mark(_x41))))mark#(incr(odds))a__incr#(a__incr(a__pairs))
mark#(incr(head(s(cons(_x51, _x52)))))a__incr#(a__head(s(cons(mark(_x51), _x52))))mark#(incr(head(s(head(_x51)))))a__incr#(a__head(s(a__head(mark(_x51)))))
mark#(incr(head(s(s(_x51)))))a__incr#(a__head(s(s(mark(_x51)))))mark#(incr(odds))a__incr#(odds)
mark#(incr(head(s(tail(_x51)))))a__incr#(a__head(s(a__tail(mark(_x51)))))mark#(incr(head(s(pairs))))a__incr#(a__head(s(pairs)))
mark#(incr(cons(_x21, _x22)))a__incr#(cons(mark(_x21), _x22))mark#(incr(head(cons(_x41, _x42))))a__incr#(a__head(cons(mark(_x41), _x42)))
a__incr#(cons(X, XS))mark#(X)mark#(incr(s(_x21)))a__incr#(s(mark(_x21)))
mark#(incr(incr(cons(_x41, _x42))))a__incr#(a__incr(cons(mark(_x41), _x42)))mark#(s(X))mark#(X)
mark#(incr(nats))a__incr#(a__nats)mark#(incr(head(s(0))))a__incr#(a__head(s(0)))
mark#(incr(head(head(_x41))))a__incr#(a__head(a__head(mark(_x41))))mark#(incr(head(incr(_x41))))a__incr#(a__head(a__incr(mark(_x41))))
mark#(incr(incr(tail(_x41))))a__incr#(a__incr(a__tail(mark(_x41))))mark#(incr(incr(0)))a__incr#(a__incr(0))
mark#(incr(head(s(odds))))a__incr#(a__head(s(a__odds)))mark#(incr(head(s(_x41))))a__incr#(head(s(mark(_x41))))
mark#(cons(X1, X2))mark#(X1)mark#(incr(head(odds)))a__incr#(a__head(a__odds))
mark#(incr(X))mark#(X)mark#(incr(incr(nats)))a__incr#(a__incr(a__nats))
mark#(incr(incr(head(_x41))))a__incr#(a__incr(a__head(mark(_x41))))mark#(incr(incr(odds)))a__incr#(a__incr(a__odds))
mark#(incr(head(s(incr(_x51)))))a__incr#(a__head(s(a__incr(mark(_x51)))))mark#(incr(pairs))a__incr#(a__pairs)
mark#(incr(head(_x21)))a__incr#(head(mark(_x21)))mark#(incr(tail(_x21)))a__incr#(a__tail(mark(_x21)))
mark#(incr(incr(_x21)))a__incr#(incr(mark(_x21)))mark#(incr(head(0)))a__incr#(a__head(0))
mark#(incr(head(pairs)))a__incr#(a__head(a__pairs))mark#(incr(head(tail(_x41))))a__incr#(a__head(a__tail(mark(_x41))))
mark#(incr(head(s(nats))))a__incr#(a__head(s(a__nats)))mark#(incr(incr(pairs)))a__incr#(a__incr(a__pairs))
mark#(incr(head(nats)))a__incr#(a__head(a__nats))mark#(incr(incr(s(_x41))))a__incr#(a__incr(s(mark(_x41))))

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, a__head, head, cons

Strategy


The right-hand side of the rule mark#(incr(head(s(pairs)))) → a__incr#(head(s(a__pairs))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__incr#(head(s(cons(0, incr(odds))))) 
a__incr#(head(s(pairs))) 
Thus, the rule mark#(incr(head(s(pairs)))) → a__incr#(head(s(a__pairs))) is replaced by the following rules:
mark#(incr(head(s(pairs)))) → a__incr#(head(s(cons(0, incr(odds)))))mark#(incr(head(s(pairs)))) → a__incr#(head(s(pairs)))

Problem 16: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(head(s(pairs))))a__incr#(a__head(s(cons(0, incr(odds)))))mark#(incr(incr(incr(_x41))))a__incr#(a__incr(a__incr(mark(_x41))))
mark#(incr(odds))a__incr#(a__incr(a__pairs))mark#(incr(head(s(head(_x51)))))a__incr#(a__head(s(a__head(mark(_x51)))))
mark#(incr(head(s(cons(_x51, _x52)))))a__incr#(a__head(s(cons(mark(_x51), _x52))))mark#(incr(head(s(pairs))))a__incr#(head(s(pairs)))
mark#(incr(odds))a__incr#(odds)mark#(incr(head(s(s(_x51)))))a__incr#(a__head(s(s(mark(_x51)))))
mark#(incr(head(s(pairs))))a__incr#(a__head(s(pairs)))mark#(incr(head(s(tail(_x51)))))a__incr#(a__head(s(a__tail(mark(_x51)))))
mark#(incr(cons(_x21, _x22)))a__incr#(cons(mark(_x21), _x22))mark#(incr(head(cons(_x41, _x42))))a__incr#(a__head(cons(mark(_x41), _x42)))
a__incr#(cons(X, XS))mark#(X)mark#(incr(s(_x21)))a__incr#(s(mark(_x21)))
mark#(incr(incr(cons(_x41, _x42))))a__incr#(a__incr(cons(mark(_x41), _x42)))mark#(s(X))mark#(X)
mark#(incr(nats))a__incr#(a__nats)mark#(incr(head(s(0))))a__incr#(a__head(s(0)))
mark#(incr(head(head(_x41))))a__incr#(a__head(a__head(mark(_x41))))mark#(incr(head(incr(_x41))))a__incr#(a__head(a__incr(mark(_x41))))
mark#(incr(incr(tail(_x41))))a__incr#(a__incr(a__tail(mark(_x41))))mark#(incr(incr(0)))a__incr#(a__incr(0))
mark#(incr(head(s(odds))))a__incr#(a__head(s(a__odds)))mark#(incr(head(s(_x41))))a__incr#(head(s(mark(_x41))))
mark#(cons(X1, X2))mark#(X1)mark#(incr(head(odds)))a__incr#(a__head(a__odds))
mark#(incr(X))mark#(X)mark#(incr(incr(head(_x41))))a__incr#(a__incr(a__head(mark(_x41))))
mark#(incr(incr(nats)))a__incr#(a__incr(a__nats))mark#(incr(incr(odds)))a__incr#(a__incr(a__odds))
mark#(incr(head(s(incr(_x51)))))a__incr#(a__head(s(a__incr(mark(_x51)))))mark#(incr(pairs))a__incr#(a__pairs)
mark#(incr(head(_x21)))a__incr#(head(mark(_x21)))mark#(incr(tail(_x21)))a__incr#(a__tail(mark(_x21)))
mark#(incr(head(0)))a__incr#(a__head(0))mark#(incr(incr(_x21)))a__incr#(incr(mark(_x21)))
mark#(incr(head(pairs)))a__incr#(a__head(a__pairs))mark#(incr(head(s(pairs))))a__incr#(head(s(cons(0, incr(odds)))))
mark#(incr(head(s(nats))))a__incr#(a__head(s(a__nats)))mark#(incr(head(tail(_x41))))a__incr#(a__head(a__tail(mark(_x41))))
mark#(incr(incr(pairs)))a__incr#(a__incr(a__pairs))mark#(incr(head(nats)))a__incr#(a__head(a__nats))
mark#(incr(incr(s(_x41))))a__incr#(a__incr(s(mark(_x41))))

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, head, a__head, cons

Strategy


The right-hand side of the rule mark#(incr(head(s(pairs)))) → a__incr#(a__head(s(cons(0, incr(odds))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__incr#(head(s(cons(0, incr(odds))))) 
Thus, the rule mark#(incr(head(s(pairs)))) → a__incr#(a__head(s(cons(0, incr(odds))))) is replaced by the following rules:
mark#(incr(head(s(pairs)))) → a__incr#(head(s(cons(0, incr(odds)))))

Problem 17: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(incr(incr(_x41))))a__incr#(a__incr(a__incr(mark(_x41))))mark#(incr(odds))a__incr#(a__incr(a__pairs))
mark#(incr(head(s(pairs))))a__incr#(head(s(pairs)))mark#(incr(head(s(cons(_x51, _x52)))))a__incr#(a__head(s(cons(mark(_x51), _x52))))
mark#(incr(head(s(head(_x51)))))a__incr#(a__head(s(a__head(mark(_x51)))))mark#(incr(head(s(s(_x51)))))a__incr#(a__head(s(s(mark(_x51)))))
mark#(incr(odds))a__incr#(odds)mark#(incr(head(s(tail(_x51)))))a__incr#(a__head(s(a__tail(mark(_x51)))))
mark#(incr(head(s(pairs))))a__incr#(a__head(s(pairs)))mark#(incr(cons(_x21, _x22)))a__incr#(cons(mark(_x21), _x22))
mark#(incr(head(cons(_x41, _x42))))a__incr#(a__head(cons(mark(_x41), _x42)))mark#(incr(s(_x21)))a__incr#(s(mark(_x21)))
a__incr#(cons(X, XS))mark#(X)mark#(incr(incr(cons(_x41, _x42))))a__incr#(a__incr(cons(mark(_x41), _x42)))
mark#(s(X))mark#(X)mark#(incr(nats))a__incr#(a__nats)
mark#(incr(head(s(0))))a__incr#(a__head(s(0)))mark#(incr(head(head(_x41))))a__incr#(a__head(a__head(mark(_x41))))
mark#(incr(head(incr(_x41))))a__incr#(a__head(a__incr(mark(_x41))))mark#(incr(incr(tail(_x41))))a__incr#(a__incr(a__tail(mark(_x41))))
mark#(incr(incr(0)))a__incr#(a__incr(0))mark#(incr(head(s(odds))))a__incr#(a__head(s(a__odds)))
mark#(incr(head(s(_x41))))a__incr#(head(s(mark(_x41))))mark#(cons(X1, X2))mark#(X1)
mark#(incr(head(odds)))a__incr#(a__head(a__odds))mark#(incr(X))mark#(X)
mark#(incr(incr(nats)))a__incr#(a__incr(a__nats))mark#(incr(incr(head(_x41))))a__incr#(a__incr(a__head(mark(_x41))))
mark#(incr(incr(odds)))a__incr#(a__incr(a__odds))mark#(incr(head(s(incr(_x51)))))a__incr#(a__head(s(a__incr(mark(_x51)))))
mark#(incr(pairs))a__incr#(a__pairs)mark#(incr(head(_x21)))a__incr#(head(mark(_x21)))
mark#(incr(tail(_x21)))a__incr#(a__tail(mark(_x21)))mark#(incr(incr(_x21)))a__incr#(incr(mark(_x21)))
mark#(incr(head(0)))a__incr#(a__head(0))mark#(incr(head(s(pairs))))a__incr#(head(s(cons(0, incr(odds)))))
mark#(incr(head(pairs)))a__incr#(a__head(a__pairs))mark#(incr(head(tail(_x41))))a__incr#(a__head(a__tail(mark(_x41))))
mark#(incr(head(s(nats))))a__incr#(a__head(s(a__nats)))mark#(incr(incr(pairs)))a__incr#(a__incr(a__pairs))
mark#(incr(head(nats)))a__incr#(a__head(a__nats))mark#(incr(incr(s(_x41))))a__incr#(a__incr(s(mark(_x41))))

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, a__head, head, cons

Strategy


The right-hand side of the rule mark#(incr(incr(incr(_x41)))) → a__incr#(a__incr(a__incr(mark(_x41)))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__incr#(a__incr(a__incr(a__tail(mark(_x51))))) 
a__incr#(incr(a__incr(mark(_x41)))) 
a__incr#(a__incr(a__incr(cons(mark(_x51), _x52)))) 
a__incr#(a__incr(a__incr(a__odds))) 
a__incr#(a__incr(a__incr(a__nats))) 
a__incr#(a__incr(a__incr(s(mark(_x51))))) 
a__incr#(a__incr(incr(mark(_x41)))) 
a__incr#(a__incr(a__incr(a__pairs))) 
a__incr#(a__incr(a__incr(a__incr(mark(_x51))))) 
a__incr#(a__incr(a__incr(a__head(mark(_x51))))) 
a__incr#(a__incr(a__incr(0))) 
Thus, the rule mark#(incr(incr(incr(_x41)))) → a__incr#(a__incr(a__incr(mark(_x41)))) is replaced by the following rules:
mark#(incr(incr(incr(odds)))) → a__incr#(a__incr(a__incr(a__odds)))mark#(incr(incr(incr(head(_x51))))) → a__incr#(a__incr(a__incr(a__head(mark(_x51)))))
mark#(incr(incr(incr(s(_x51))))) → a__incr#(a__incr(a__incr(s(mark(_x51)))))mark#(incr(incr(incr(_x41)))) → a__incr#(a__incr(incr(mark(_x41))))
mark#(incr(incr(incr(_x41)))) → a__incr#(incr(a__incr(mark(_x41))))mark#(incr(incr(incr(nats)))) → a__incr#(a__incr(a__incr(a__nats)))
mark#(incr(incr(incr(tail(_x51))))) → a__incr#(a__incr(a__incr(a__tail(mark(_x51)))))mark#(incr(incr(incr(cons(_x51, _x52))))) → a__incr#(a__incr(a__incr(cons(mark(_x51), _x52))))
mark#(incr(incr(incr(0)))) → a__incr#(a__incr(a__incr(0)))mark#(incr(incr(incr(pairs)))) → a__incr#(a__incr(a__incr(a__pairs)))
mark#(incr(incr(incr(incr(_x51))))) → a__incr#(a__incr(a__incr(a__incr(mark(_x51)))))

Problem 18: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(odds))a__incr#(a__incr(a__pairs))mark#(incr(head(s(head(_x51)))))a__incr#(a__head(s(a__head(mark(_x51)))))
mark#(incr(odds))a__incr#(odds)mark#(incr(head(s(tail(_x51)))))a__incr#(a__head(s(a__tail(mark(_x51)))))
mark#(incr(cons(_x21, _x22)))a__incr#(cons(mark(_x21), _x22))a__incr#(cons(X, XS))mark#(X)
mark#(incr(s(_x21)))a__incr#(s(mark(_x21)))mark#(s(X))mark#(X)
mark#(incr(head(head(_x41))))a__incr#(a__head(a__head(mark(_x41))))mark#(incr(head(s(0))))a__incr#(a__head(s(0)))
mark#(incr(head(incr(_x41))))a__incr#(a__head(a__incr(mark(_x41))))mark#(incr(incr(incr(s(_x51)))))a__incr#(a__incr(a__incr(s(mark(_x51)))))
mark#(incr(incr(tail(_x41))))a__incr#(a__incr(a__tail(mark(_x41))))mark#(incr(head(s(odds))))a__incr#(a__head(s(a__odds)))
mark#(incr(head(s(_x41))))a__incr#(head(s(mark(_x41))))mark#(cons(X1, X2))mark#(X1)
mark#(incr(incr(incr(tail(_x51)))))a__incr#(a__incr(a__incr(a__tail(mark(_x51)))))mark#(incr(incr(incr(0))))a__incr#(a__incr(a__incr(0)))
mark#(incr(incr(nats)))a__incr#(a__incr(a__nats))mark#(incr(incr(odds)))a__incr#(a__incr(a__odds))
mark#(incr(head(s(incr(_x51)))))a__incr#(a__head(s(a__incr(mark(_x51)))))mark#(incr(pairs))a__incr#(a__pairs)
mark#(incr(head(_x21)))a__incr#(head(mark(_x21)))mark#(incr(tail(_x21)))a__incr#(a__tail(mark(_x21)))
mark#(incr(incr(_x21)))a__incr#(incr(mark(_x21)))mark#(incr(head(pairs)))a__incr#(a__head(a__pairs))
mark#(incr(head(s(pairs))))a__incr#(head(s(cons(0, incr(odds)))))mark#(incr(head(nats)))a__incr#(a__head(a__nats))
mark#(incr(incr(incr(cons(_x51, _x52)))))a__incr#(a__incr(a__incr(cons(mark(_x51), _x52))))mark#(incr(incr(s(_x41))))a__incr#(a__incr(s(mark(_x41))))
mark#(incr(incr(incr(_x41))))a__incr#(incr(a__incr(mark(_x41))))mark#(incr(head(s(cons(_x51, _x52)))))a__incr#(a__head(s(cons(mark(_x51), _x52))))
mark#(incr(head(s(pairs))))a__incr#(head(s(pairs)))mark#(incr(incr(incr(pairs))))a__incr#(a__incr(a__incr(a__pairs)))
mark#(incr(incr(incr(incr(_x51)))))a__incr#(a__incr(a__incr(a__incr(mark(_x51)))))mark#(incr(head(s(s(_x51)))))a__incr#(a__head(s(s(mark(_x51)))))
mark#(incr(head(s(pairs))))a__incr#(a__head(s(pairs)))mark#(incr(head(cons(_x41, _x42))))a__incr#(a__head(cons(mark(_x41), _x42)))
mark#(incr(incr(cons(_x41, _x42))))a__incr#(a__incr(cons(mark(_x41), _x42)))mark#(incr(nats))a__incr#(a__nats)
mark#(incr(incr(incr(odds))))a__incr#(a__incr(a__incr(a__odds)))mark#(incr(incr(incr(head(_x51)))))a__incr#(a__incr(a__incr(a__head(mark(_x51)))))
mark#(incr(incr(incr(_x41))))a__incr#(a__incr(incr(mark(_x41))))mark#(incr(incr(0)))a__incr#(a__incr(0))
mark#(incr(head(odds)))a__incr#(a__head(a__odds))mark#(incr(X))mark#(X)
mark#(incr(incr(head(_x41))))a__incr#(a__incr(a__head(mark(_x41))))mark#(incr(head(0)))a__incr#(a__head(0))
mark#(incr(head(tail(_x41))))a__incr#(a__head(a__tail(mark(_x41))))mark#(incr(head(s(nats))))a__incr#(a__head(s(a__nats)))
mark#(incr(incr(incr(nats))))a__incr#(a__incr(a__incr(a__nats)))mark#(incr(incr(pairs)))a__incr#(a__incr(a__pairs))

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, head, a__head, cons

Strategy


The right-hand side of the rule mark#(incr(odds)) → a__incr#(a__incr(a__pairs)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__incr#(a__incr(pairs)) 
a__incr#(a__incr(cons(0, incr(odds)))) 
a__incr#(incr(a__pairs)) 
Thus, the rule mark#(incr(odds)) → a__incr#(a__incr(a__pairs)) is replaced by the following rules:
mark#(incr(odds)) → a__incr#(a__incr(pairs))mark#(incr(odds)) → a__incr#(incr(a__pairs))
mark#(incr(odds)) → a__incr#(a__incr(cons(0, incr(odds))))

Problem 19: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(head(s(head(_x51)))))a__incr#(a__head(s(a__head(mark(_x51)))))mark#(incr(odds))a__incr#(odds)
mark#(incr(head(s(tail(_x51)))))a__incr#(a__head(s(a__tail(mark(_x51)))))mark#(incr(cons(_x21, _x22)))a__incr#(cons(mark(_x21), _x22))
a__incr#(cons(X, XS))mark#(X)mark#(incr(s(_x21)))a__incr#(s(mark(_x21)))
mark#(s(X))mark#(X)mark#(incr(head(head(_x41))))a__incr#(a__head(a__head(mark(_x41))))
mark#(incr(head(s(0))))a__incr#(a__head(s(0)))mark#(incr(odds))a__incr#(a__incr(cons(0, incr(odds))))
mark#(incr(head(incr(_x41))))a__incr#(a__head(a__incr(mark(_x41))))mark#(incr(incr(incr(s(_x51)))))a__incr#(a__incr(a__incr(s(mark(_x51)))))
mark#(incr(incr(tail(_x41))))a__incr#(a__incr(a__tail(mark(_x41))))mark#(cons(X1, X2))mark#(X1)
mark#(incr(head(s(_x41))))a__incr#(head(s(mark(_x41))))mark#(incr(head(s(odds))))a__incr#(a__head(s(a__odds)))
mark#(incr(incr(incr(tail(_x51)))))a__incr#(a__incr(a__incr(a__tail(mark(_x51)))))mark#(incr(odds))a__incr#(incr(a__pairs))
mark#(incr(incr(incr(0))))a__incr#(a__incr(a__incr(0)))mark#(incr(incr(nats)))a__incr#(a__incr(a__nats))
mark#(incr(incr(odds)))a__incr#(a__incr(a__odds))mark#(incr(head(s(incr(_x51)))))a__incr#(a__head(s(a__incr(mark(_x51)))))
mark#(incr(pairs))a__incr#(a__pairs)mark#(incr(head(_x21)))a__incr#(head(mark(_x21)))
mark#(incr(odds))a__incr#(a__incr(pairs))mark#(incr(tail(_x21)))a__incr#(a__tail(mark(_x21)))
mark#(incr(incr(_x21)))a__incr#(incr(mark(_x21)))mark#(incr(head(pairs)))a__incr#(a__head(a__pairs))
mark#(incr(head(s(pairs))))a__incr#(head(s(cons(0, incr(odds)))))mark#(incr(head(nats)))a__incr#(a__head(a__nats))
mark#(incr(incr(incr(cons(_x51, _x52)))))a__incr#(a__incr(a__incr(cons(mark(_x51), _x52))))mark#(incr(incr(s(_x41))))a__incr#(a__incr(s(mark(_x41))))
mark#(incr(incr(incr(_x41))))a__incr#(incr(a__incr(mark(_x41))))mark#(incr(head(s(cons(_x51, _x52)))))a__incr#(a__head(s(cons(mark(_x51), _x52))))
mark#(incr(head(s(pairs))))a__incr#(head(s(pairs)))mark#(incr(incr(incr(pairs))))a__incr#(a__incr(a__incr(a__pairs)))
mark#(incr(incr(incr(incr(_x51)))))a__incr#(a__incr(a__incr(a__incr(mark(_x51)))))mark#(incr(head(s(s(_x51)))))a__incr#(a__head(s(s(mark(_x51)))))
mark#(incr(head(s(pairs))))a__incr#(a__head(s(pairs)))mark#(incr(head(cons(_x41, _x42))))a__incr#(a__head(cons(mark(_x41), _x42)))
mark#(incr(incr(cons(_x41, _x42))))a__incr#(a__incr(cons(mark(_x41), _x42)))mark#(incr(nats))a__incr#(a__nats)
mark#(incr(incr(incr(odds))))a__incr#(a__incr(a__incr(a__odds)))mark#(incr(incr(incr(head(_x51)))))a__incr#(a__incr(a__incr(a__head(mark(_x51)))))
mark#(incr(incr(incr(_x41))))a__incr#(a__incr(incr(mark(_x41))))mark#(incr(incr(0)))a__incr#(a__incr(0))
mark#(incr(head(odds)))a__incr#(a__head(a__odds))mark#(incr(X))mark#(X)
mark#(incr(incr(head(_x41))))a__incr#(a__incr(a__head(mark(_x41))))mark#(incr(head(0)))a__incr#(a__head(0))
mark#(incr(head(s(nats))))a__incr#(a__head(s(a__nats)))mark#(incr(head(tail(_x41))))a__incr#(a__head(a__tail(mark(_x41))))
mark#(incr(incr(pairs)))a__incr#(a__incr(a__pairs))mark#(incr(incr(incr(nats))))a__incr#(a__incr(a__incr(a__nats)))

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, a__head, head, cons

Strategy


The right-hand side of the rule mark#(incr(head(s(head(_x51))))) → a__incr#(a__head(s(a__head(mark(_x51))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__incr#(a__head(s(a__head(a__odds)))) 
a__incr#(a__head(s(a__head(s(mark(_x61)))))) 
a__incr#(a__head(s(a__head(a__head(mark(_x61)))))) 
a__incr#(a__head(s(a__head(cons(mark(_x61), _x62))))) 
a__incr#(a__head(s(a__head(a__pairs)))) 
a__incr#(a__head(s(a__head(a__tail(mark(_x61)))))) 
a__incr#(a__head(s(a__head(a__nats)))) 
a__incr#(a__head(s(head(mark(_x51))))) 
a__incr#(a__head(s(a__head(a__incr(mark(_x61)))))) 
a__incr#(head(s(a__head(mark(_x51))))) 
a__incr#(a__head(s(a__head(0)))) 
Thus, the rule mark#(incr(head(s(head(_x51))))) → a__incr#(a__head(s(a__head(mark(_x51))))) is replaced by the following rules:
mark#(incr(head(s(head(head(_x61)))))) → a__incr#(a__head(s(a__head(a__head(mark(_x61))))))mark#(incr(head(s(head(pairs))))) → a__incr#(a__head(s(a__head(a__pairs))))
mark#(incr(head(s(head(0))))) → a__incr#(a__head(s(a__head(0))))mark#(incr(head(s(head(incr(_x61)))))) → a__incr#(a__head(s(a__head(a__incr(mark(_x61))))))
mark#(incr(head(s(head(cons(_x61, _x62)))))) → a__incr#(a__head(s(a__head(cons(mark(_x61), _x62)))))mark#(incr(head(s(head(odds))))) → a__incr#(a__head(s(a__head(a__odds))))
mark#(incr(head(s(head(s(_x61)))))) → a__incr#(a__head(s(a__head(s(mark(_x61))))))mark#(incr(head(s(head(nats))))) → a__incr#(a__head(s(a__head(a__nats))))
mark#(incr(head(s(head(_x51))))) → a__incr#(a__head(s(head(mark(_x51)))))mark#(incr(head(s(head(tail(_x61)))))) → a__incr#(a__head(s(a__head(a__tail(mark(_x61))))))
mark#(incr(head(s(head(_x51))))) → a__incr#(head(s(a__head(mark(_x51)))))

Problem 20: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(odds))a__incr#(odds)mark#(incr(head(s(head(0)))))a__incr#(a__head(s(a__head(0))))
mark#(incr(head(s(tail(_x51)))))a__incr#(a__head(s(a__tail(mark(_x51)))))mark#(incr(cons(_x21, _x22)))a__incr#(cons(mark(_x21), _x22))
a__incr#(cons(X, XS))mark#(X)mark#(incr(s(_x21)))a__incr#(s(mark(_x21)))
mark#(s(X))mark#(X)mark#(incr(head(s(head(tail(_x61))))))a__incr#(a__head(s(a__head(a__tail(mark(_x61))))))
mark#(incr(head(head(_x41))))a__incr#(a__head(a__head(mark(_x41))))mark#(incr(head(s(0))))a__incr#(a__head(s(0)))
mark#(incr(odds))a__incr#(a__incr(cons(0, incr(odds))))mark#(incr(head(incr(_x41))))a__incr#(a__head(a__incr(mark(_x41))))
mark#(incr(head(s(head(head(_x61))))))a__incr#(a__head(s(a__head(a__head(mark(_x61))))))mark#(incr(incr(incr(s(_x51)))))a__incr#(a__incr(a__incr(s(mark(_x51)))))
mark#(incr(incr(tail(_x41))))a__incr#(a__incr(a__tail(mark(_x41))))mark#(incr(head(s(head(incr(_x61))))))a__incr#(a__head(s(a__head(a__incr(mark(_x61))))))
mark#(incr(head(s(odds))))a__incr#(a__head(s(a__odds)))mark#(incr(head(s(_x41))))a__incr#(head(s(mark(_x41))))
mark#(cons(X1, X2))mark#(X1)mark#(incr(head(s(head(cons(_x61, _x62))))))a__incr#(a__head(s(a__head(cons(mark(_x61), _x62)))))
mark#(incr(odds))a__incr#(incr(a__pairs))mark#(incr(incr(incr(tail(_x51)))))a__incr#(a__incr(a__incr(a__tail(mark(_x51)))))
mark#(incr(incr(incr(0))))a__incr#(a__incr(a__incr(0)))mark#(incr(incr(nats)))a__incr#(a__incr(a__nats))
mark#(incr(incr(odds)))a__incr#(a__incr(a__odds))mark#(incr(head(s(incr(_x51)))))a__incr#(a__head(s(a__incr(mark(_x51)))))
mark#(incr(pairs))a__incr#(a__pairs)mark#(incr(head(_x21)))a__incr#(head(mark(_x21)))
mark#(incr(odds))a__incr#(a__incr(pairs))mark#(incr(tail(_x21)))a__incr#(a__tail(mark(_x21)))
mark#(incr(incr(_x21)))a__incr#(incr(mark(_x21)))mark#(incr(head(pairs)))a__incr#(a__head(a__pairs))
mark#(incr(head(s(pairs))))a__incr#(head(s(cons(0, incr(odds)))))mark#(incr(head(s(head(s(_x61))))))a__incr#(a__head(s(a__head(s(mark(_x61))))))
mark#(incr(head(nats)))a__incr#(a__head(a__nats))mark#(incr(incr(incr(cons(_x51, _x52)))))a__incr#(a__incr(a__incr(cons(mark(_x51), _x52))))
mark#(incr(head(s(head(_x51)))))a__incr#(head(s(a__head(mark(_x51)))))mark#(incr(incr(s(_x41))))a__incr#(a__incr(s(mark(_x41))))
mark#(incr(incr(incr(_x41))))a__incr#(incr(a__incr(mark(_x41))))mark#(incr(head(s(cons(_x51, _x52)))))a__incr#(a__head(s(cons(mark(_x51), _x52))))
mark#(incr(head(s(pairs))))a__incr#(head(s(pairs)))mark#(incr(incr(incr(pairs))))a__incr#(a__incr(a__incr(a__pairs)))
mark#(incr(incr(incr(incr(_x51)))))a__incr#(a__incr(a__incr(a__incr(mark(_x51)))))mark#(incr(head(s(s(_x51)))))a__incr#(a__head(s(s(mark(_x51)))))
mark#(incr(head(s(head(pairs)))))a__incr#(a__head(s(a__head(a__pairs))))mark#(incr(head(s(pairs))))a__incr#(a__head(s(pairs)))
mark#(incr(head(cons(_x41, _x42))))a__incr#(a__head(cons(mark(_x41), _x42)))mark#(incr(incr(cons(_x41, _x42))))a__incr#(a__incr(cons(mark(_x41), _x42)))
mark#(incr(nats))a__incr#(a__nats)mark#(incr(head(s(head(nats)))))a__incr#(a__head(s(a__head(a__nats))))
mark#(incr(incr(incr(odds))))a__incr#(a__incr(a__incr(a__odds)))mark#(incr(incr(incr(head(_x51)))))a__incr#(a__incr(a__incr(a__head(mark(_x51)))))
mark#(incr(incr(incr(_x41))))a__incr#(a__incr(incr(mark(_x41))))mark#(incr(incr(0)))a__incr#(a__incr(0))
mark#(incr(head(odds)))a__incr#(a__head(a__odds))mark#(incr(X))mark#(X)
mark#(incr(incr(head(_x41))))a__incr#(a__incr(a__head(mark(_x41))))mark#(incr(head(0)))a__incr#(a__head(0))
mark#(incr(head(tail(_x41))))a__incr#(a__head(a__tail(mark(_x41))))mark#(incr(head(s(nats))))a__incr#(a__head(s(a__nats)))
mark#(incr(head(s(head(odds)))))a__incr#(a__head(s(a__head(a__odds))))mark#(incr(incr(incr(nats))))a__incr#(a__incr(a__incr(a__nats)))
mark#(incr(incr(pairs)))a__incr#(a__incr(a__pairs))mark#(incr(head(s(head(_x51)))))a__incr#(a__head(s(head(mark(_x51)))))

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, head, a__head, cons

Strategy


The right-hand side of the rule mark#(incr(odds)) → a__incr#(odds) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
Thus, the rule mark#(incr(odds)) → a__incr#(odds) is deleted.

Problem 21: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(head(s(head(0)))))a__incr#(a__head(s(a__head(0))))mark#(incr(head(s(tail(_x51)))))a__incr#(a__head(s(a__tail(mark(_x51)))))
mark#(incr(cons(_x21, _x22)))a__incr#(cons(mark(_x21), _x22))a__incr#(cons(X, XS))mark#(X)
mark#(incr(s(_x21)))a__incr#(s(mark(_x21)))mark#(s(X))mark#(X)
mark#(incr(head(s(head(tail(_x61))))))a__incr#(a__head(s(a__head(a__tail(mark(_x61))))))mark#(incr(head(head(_x41))))a__incr#(a__head(a__head(mark(_x41))))
mark#(incr(head(s(0))))a__incr#(a__head(s(0)))mark#(incr(odds))a__incr#(a__incr(cons(0, incr(odds))))
mark#(incr(head(incr(_x41))))a__incr#(a__head(a__incr(mark(_x41))))mark#(incr(head(s(head(head(_x61))))))a__incr#(a__head(s(a__head(a__head(mark(_x61))))))
mark#(incr(incr(incr(s(_x51)))))a__incr#(a__incr(a__incr(s(mark(_x51)))))mark#(incr(head(s(head(incr(_x61))))))a__incr#(a__head(s(a__head(a__incr(mark(_x61))))))
mark#(incr(incr(tail(_x41))))a__incr#(a__incr(a__tail(mark(_x41))))mark#(incr(head(s(head(cons(_x61, _x62))))))a__incr#(a__head(s(a__head(cons(mark(_x61), _x62)))))
mark#(cons(X1, X2))mark#(X1)mark#(incr(head(s(_x41))))a__incr#(head(s(mark(_x41))))
mark#(incr(head(s(odds))))a__incr#(a__head(s(a__odds)))mark#(incr(incr(incr(tail(_x51)))))a__incr#(a__incr(a__incr(a__tail(mark(_x51)))))
mark#(incr(odds))a__incr#(incr(a__pairs))mark#(incr(incr(incr(0))))a__incr#(a__incr(a__incr(0)))
mark#(incr(incr(nats)))a__incr#(a__incr(a__nats))mark#(incr(incr(odds)))a__incr#(a__incr(a__odds))
mark#(incr(head(s(incr(_x51)))))a__incr#(a__head(s(a__incr(mark(_x51)))))mark#(incr(pairs))a__incr#(a__pairs)
mark#(incr(head(_x21)))a__incr#(head(mark(_x21)))mark#(incr(odds))a__incr#(a__incr(pairs))
mark#(incr(tail(_x21)))a__incr#(a__tail(mark(_x21)))mark#(incr(incr(_x21)))a__incr#(incr(mark(_x21)))
mark#(incr(head(pairs)))a__incr#(a__head(a__pairs))mark#(incr(head(s(pairs))))a__incr#(head(s(cons(0, incr(odds)))))
mark#(incr(head(s(head(s(_x61))))))a__incr#(a__head(s(a__head(s(mark(_x61))))))mark#(incr(head(nats)))a__incr#(a__head(a__nats))
mark#(incr(incr(incr(cons(_x51, _x52)))))a__incr#(a__incr(a__incr(cons(mark(_x51), _x52))))mark#(incr(head(s(head(_x51)))))a__incr#(head(s(a__head(mark(_x51)))))
mark#(incr(incr(s(_x41))))a__incr#(a__incr(s(mark(_x41))))mark#(incr(incr(incr(_x41))))a__incr#(incr(a__incr(mark(_x41))))
mark#(incr(head(s(cons(_x51, _x52)))))a__incr#(a__head(s(cons(mark(_x51), _x52))))mark#(incr(head(s(pairs))))a__incr#(head(s(pairs)))
mark#(incr(incr(incr(pairs))))a__incr#(a__incr(a__incr(a__pairs)))mark#(incr(incr(incr(incr(_x51)))))a__incr#(a__incr(a__incr(a__incr(mark(_x51)))))
mark#(incr(head(s(s(_x51)))))a__incr#(a__head(s(s(mark(_x51)))))mark#(incr(head(s(head(pairs)))))a__incr#(a__head(s(a__head(a__pairs))))
mark#(incr(head(s(pairs))))a__incr#(a__head(s(pairs)))mark#(incr(head(cons(_x41, _x42))))a__incr#(a__head(cons(mark(_x41), _x42)))
mark#(incr(incr(cons(_x41, _x42))))a__incr#(a__incr(cons(mark(_x41), _x42)))mark#(incr(nats))a__incr#(a__nats)
mark#(incr(head(s(head(nats)))))a__incr#(a__head(s(a__head(a__nats))))mark#(incr(incr(incr(odds))))a__incr#(a__incr(a__incr(a__odds)))
mark#(incr(incr(incr(head(_x51)))))a__incr#(a__incr(a__incr(a__head(mark(_x51)))))mark#(incr(incr(incr(_x41))))a__incr#(a__incr(incr(mark(_x41))))
mark#(incr(incr(0)))a__incr#(a__incr(0))mark#(incr(head(odds)))a__incr#(a__head(a__odds))
mark#(incr(X))mark#(X)mark#(incr(incr(head(_x41))))a__incr#(a__incr(a__head(mark(_x41))))
mark#(incr(head(0)))a__incr#(a__head(0))mark#(incr(head(s(head(odds)))))a__incr#(a__head(s(a__head(a__odds))))
mark#(incr(head(s(nats))))a__incr#(a__head(s(a__nats)))mark#(incr(head(tail(_x41))))a__incr#(a__head(a__tail(mark(_x41))))
mark#(incr(incr(pairs)))a__incr#(a__incr(a__pairs))mark#(incr(incr(incr(nats))))a__incr#(a__incr(a__incr(a__nats)))
mark#(incr(head(s(head(_x51)))))a__incr#(a__head(s(head(mark(_x51)))))

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, a__head, head, cons

Strategy


The right-hand side of the rule mark#(incr(head(s(head(0))))) → a__incr#(a__head(s(a__head(0)))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__incr#(head(s(a__head(0)))) 
a__incr#(a__head(s(head(0)))) 
Thus, the rule mark#(incr(head(s(head(0))))) → a__incr#(a__head(s(a__head(0)))) is replaced by the following rules:
mark#(incr(head(s(head(0))))) → a__incr#(a__head(s(head(0))))mark#(incr(head(s(head(0))))) → a__incr#(head(s(a__head(0))))

Problem 22: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(head(s(tail(_x51)))))a__incr#(a__head(s(a__tail(mark(_x51)))))mark#(incr(cons(_x21, _x22)))a__incr#(cons(mark(_x21), _x22))
mark#(incr(head(s(head(0)))))a__incr#(head(s(a__head(0))))a__incr#(cons(X, XS))mark#(X)
mark#(incr(s(_x21)))a__incr#(s(mark(_x21)))mark#(s(X))mark#(X)
mark#(incr(head(s(head(tail(_x61))))))a__incr#(a__head(s(a__head(a__tail(mark(_x61))))))mark#(incr(head(head(_x41))))a__incr#(a__head(a__head(mark(_x41))))
mark#(incr(head(s(0))))a__incr#(a__head(s(0)))mark#(incr(odds))a__incr#(a__incr(cons(0, incr(odds))))
mark#(incr(head(incr(_x41))))a__incr#(a__head(a__incr(mark(_x41))))mark#(incr(head(s(head(head(_x61))))))a__incr#(a__head(s(a__head(a__head(mark(_x61))))))
mark#(incr(incr(incr(s(_x51)))))a__incr#(a__incr(a__incr(s(mark(_x51)))))mark#(incr(incr(tail(_x41))))a__incr#(a__incr(a__tail(mark(_x41))))
mark#(incr(head(s(head(incr(_x61))))))a__incr#(a__head(s(a__head(a__incr(mark(_x61))))))mark#(incr(head(s(odds))))a__incr#(a__head(s(a__odds)))
mark#(incr(head(s(_x41))))a__incr#(head(s(mark(_x41))))mark#(cons(X1, X2))mark#(X1)
mark#(incr(head(s(head(cons(_x61, _x62))))))a__incr#(a__head(s(a__head(cons(mark(_x61), _x62)))))mark#(incr(odds))a__incr#(incr(a__pairs))
mark#(incr(incr(incr(tail(_x51)))))a__incr#(a__incr(a__incr(a__tail(mark(_x51)))))mark#(incr(incr(incr(0))))a__incr#(a__incr(a__incr(0)))
mark#(incr(incr(nats)))a__incr#(a__incr(a__nats))mark#(incr(incr(odds)))a__incr#(a__incr(a__odds))
mark#(incr(head(s(incr(_x51)))))a__incr#(a__head(s(a__incr(mark(_x51)))))mark#(incr(pairs))a__incr#(a__pairs)
mark#(incr(head(_x21)))a__incr#(head(mark(_x21)))mark#(incr(odds))a__incr#(a__incr(pairs))
mark#(incr(head(s(head(0)))))a__incr#(a__head(s(head(0))))mark#(incr(tail(_x21)))a__incr#(a__tail(mark(_x21)))
mark#(incr(incr(_x21)))a__incr#(incr(mark(_x21)))mark#(incr(head(pairs)))a__incr#(a__head(a__pairs))
mark#(incr(head(s(pairs))))a__incr#(head(s(cons(0, incr(odds)))))mark#(incr(head(s(head(s(_x61))))))a__incr#(a__head(s(a__head(s(mark(_x61))))))
mark#(incr(head(nats)))a__incr#(a__head(a__nats))mark#(incr(incr(incr(cons(_x51, _x52)))))a__incr#(a__incr(a__incr(cons(mark(_x51), _x52))))
mark#(incr(head(s(head(_x51)))))a__incr#(head(s(a__head(mark(_x51)))))mark#(incr(incr(s(_x41))))a__incr#(a__incr(s(mark(_x41))))
mark#(incr(incr(incr(_x41))))a__incr#(incr(a__incr(mark(_x41))))mark#(incr(head(s(cons(_x51, _x52)))))a__incr#(a__head(s(cons(mark(_x51), _x52))))
mark#(incr(head(s(pairs))))a__incr#(head(s(pairs)))mark#(incr(incr(incr(pairs))))a__incr#(a__incr(a__incr(a__pairs)))
mark#(incr(incr(incr(incr(_x51)))))a__incr#(a__incr(a__incr(a__incr(mark(_x51)))))mark#(incr(head(s(s(_x51)))))a__incr#(a__head(s(s(mark(_x51)))))
mark#(incr(head(s(head(pairs)))))a__incr#(a__head(s(a__head(a__pairs))))mark#(incr(head(s(pairs))))a__incr#(a__head(s(pairs)))
mark#(incr(head(cons(_x41, _x42))))a__incr#(a__head(cons(mark(_x41), _x42)))mark#(incr(incr(cons(_x41, _x42))))a__incr#(a__incr(cons(mark(_x41), _x42)))
mark#(incr(head(s(head(nats)))))a__incr#(a__head(s(a__head(a__nats))))mark#(incr(nats))a__incr#(a__nats)
mark#(incr(incr(incr(odds))))a__incr#(a__incr(a__incr(a__odds)))mark#(incr(incr(incr(head(_x51)))))a__incr#(a__incr(a__incr(a__head(mark(_x51)))))
mark#(incr(incr(incr(_x41))))a__incr#(a__incr(incr(mark(_x41))))mark#(incr(incr(0)))a__incr#(a__incr(0))
mark#(incr(head(odds)))a__incr#(a__head(a__odds))mark#(incr(X))mark#(X)
mark#(incr(incr(head(_x41))))a__incr#(a__incr(a__head(mark(_x41))))mark#(incr(head(0)))a__incr#(a__head(0))
mark#(incr(head(tail(_x41))))a__incr#(a__head(a__tail(mark(_x41))))mark#(incr(head(s(nats))))a__incr#(a__head(s(a__nats)))
mark#(incr(head(s(head(odds)))))a__incr#(a__head(s(a__head(a__odds))))mark#(incr(incr(incr(nats))))a__incr#(a__incr(a__incr(a__nats)))
mark#(incr(incr(pairs)))a__incr#(a__incr(a__pairs))mark#(incr(head(s(head(_x51)))))a__incr#(a__head(s(head(mark(_x51)))))

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, head, a__head, cons

Strategy


The right-hand side of the rule mark#(incr(head(s(tail(_x51))))) → a__incr#(a__head(s(a__tail(mark(_x51))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__incr#(a__head(s(a__tail(a__pairs)))) 
a__incr#(a__head(s(a__tail(a__incr(mark(_x61)))))) 
a__incr#(a__head(s(a__tail(a__odds)))) 
a__incr#(a__head(s(a__tail(s(mark(_x61)))))) 
a__incr#(a__head(s(a__tail(a__head(mark(_x61)))))) 
a__incr#(a__head(s(a__tail(0)))) 
a__incr#(a__head(s(a__tail(cons(mark(_x61), _x62))))) 
a__incr#(a__head(s(a__tail(a__nats)))) 
a__incr#(head(s(a__tail(mark(_x51))))) 
a__incr#(a__head(s(tail(mark(_x51))))) 
a__incr#(a__head(s(a__tail(a__tail(mark(_x61)))))) 
Thus, the rule mark#(incr(head(s(tail(_x51))))) → a__incr#(a__head(s(a__tail(mark(_x51))))) is replaced by the following rules:
mark#(incr(head(s(tail(0))))) → a__incr#(a__head(s(a__tail(0))))mark#(incr(head(s(tail(pairs))))) → a__incr#(a__head(s(a__tail(a__pairs))))
mark#(incr(head(s(tail(tail(_x61)))))) → a__incr#(a__head(s(a__tail(a__tail(mark(_x61))))))mark#(incr(head(s(tail(_x51))))) → a__incr#(a__head(s(tail(mark(_x51)))))
mark#(incr(head(s(tail(nats))))) → a__incr#(a__head(s(a__tail(a__nats))))mark#(incr(head(s(tail(incr(_x61)))))) → a__incr#(a__head(s(a__tail(a__incr(mark(_x61))))))
mark#(incr(head(s(tail(s(_x61)))))) → a__incr#(a__head(s(a__tail(s(mark(_x61))))))mark#(incr(head(s(tail(cons(_x61, _x62)))))) → a__incr#(a__head(s(a__tail(cons(mark(_x61), _x62)))))
mark#(incr(head(s(tail(_x51))))) → a__incr#(head(s(a__tail(mark(_x51)))))mark#(incr(head(s(tail(head(_x61)))))) → a__incr#(a__head(s(a__tail(a__head(mark(_x61))))))
mark#(incr(head(s(tail(odds))))) → a__incr#(a__head(s(a__tail(a__odds))))

Problem 23: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(head(s(tail(_x51)))))a__incr#(head(s(a__tail(mark(_x51)))))mark#(incr(head(s(tail(_x51)))))a__incr#(a__head(s(tail(mark(_x51)))))
mark#(incr(head(s(tail(nats)))))a__incr#(a__head(s(a__tail(a__nats))))mark#(incr(cons(_x21, _x22)))a__incr#(cons(mark(_x21), _x22))
mark#(incr(head(s(head(0)))))a__incr#(head(s(a__head(0))))a__incr#(cons(X, XS))mark#(X)
mark#(incr(s(_x21)))a__incr#(s(mark(_x21)))mark#(s(X))mark#(X)
mark#(incr(head(s(head(tail(_x61))))))a__incr#(a__head(s(a__head(a__tail(mark(_x61))))))mark#(incr(head(head(_x41))))a__incr#(a__head(a__head(mark(_x41))))
mark#(incr(head(s(0))))a__incr#(a__head(s(0)))mark#(incr(odds))a__incr#(a__incr(cons(0, incr(odds))))
mark#(incr(head(incr(_x41))))a__incr#(a__head(a__incr(mark(_x41))))mark#(incr(head(s(tail(odds)))))a__incr#(a__head(s(a__tail(a__odds))))
mark#(incr(head(s(head(head(_x61))))))a__incr#(a__head(s(a__head(a__head(mark(_x61))))))mark#(incr(head(s(tail(pairs)))))a__incr#(a__head(s(a__tail(a__pairs))))
mark#(incr(incr(incr(s(_x51)))))a__incr#(a__incr(a__incr(s(mark(_x51)))))mark#(incr(head(s(head(incr(_x61))))))a__incr#(a__head(s(a__head(a__incr(mark(_x61))))))
mark#(incr(incr(tail(_x41))))a__incr#(a__incr(a__tail(mark(_x41))))mark#(incr(head(s(head(cons(_x61, _x62))))))a__incr#(a__head(s(a__head(cons(mark(_x61), _x62)))))
mark#(cons(X1, X2))mark#(X1)mark#(incr(head(s(_x41))))a__incr#(head(s(mark(_x41))))
mark#(incr(head(s(odds))))a__incr#(a__head(s(a__odds)))mark#(incr(incr(incr(tail(_x51)))))a__incr#(a__incr(a__incr(a__tail(mark(_x51)))))
mark#(incr(odds))a__incr#(incr(a__pairs))mark#(incr(incr(incr(0))))a__incr#(a__incr(a__incr(0)))
mark#(incr(incr(nats)))a__incr#(a__incr(a__nats))mark#(incr(incr(odds)))a__incr#(a__incr(a__odds))
mark#(incr(head(s(incr(_x51)))))a__incr#(a__head(s(a__incr(mark(_x51)))))mark#(incr(pairs))a__incr#(a__pairs)
mark#(incr(head(_x21)))a__incr#(head(mark(_x21)))mark#(incr(odds))a__incr#(a__incr(pairs))
mark#(incr(head(s(head(0)))))a__incr#(a__head(s(head(0))))mark#(incr(tail(_x21)))a__incr#(a__tail(mark(_x21)))
mark#(incr(incr(_x21)))a__incr#(incr(mark(_x21)))mark#(incr(head(pairs)))a__incr#(a__head(a__pairs))
mark#(incr(head(s(pairs))))a__incr#(head(s(cons(0, incr(odds)))))mark#(incr(head(s(head(s(_x61))))))a__incr#(a__head(s(a__head(s(mark(_x61))))))
mark#(incr(head(nats)))a__incr#(a__head(a__nats))mark#(incr(incr(incr(cons(_x51, _x52)))))a__incr#(a__incr(a__incr(cons(mark(_x51), _x52))))
mark#(incr(head(s(head(_x51)))))a__incr#(head(s(a__head(mark(_x51)))))mark#(incr(incr(s(_x41))))a__incr#(a__incr(s(mark(_x41))))
mark#(incr(head(s(tail(incr(_x61))))))a__incr#(a__head(s(a__tail(a__incr(mark(_x61))))))mark#(incr(incr(incr(_x41))))a__incr#(incr(a__incr(mark(_x41))))
mark#(incr(head(s(cons(_x51, _x52)))))a__incr#(a__head(s(cons(mark(_x51), _x52))))mark#(incr(head(s(pairs))))a__incr#(head(s(pairs)))
mark#(incr(incr(incr(pairs))))a__incr#(a__incr(a__incr(a__pairs)))mark#(incr(incr(incr(incr(_x51)))))a__incr#(a__incr(a__incr(a__incr(mark(_x51)))))
mark#(incr(head(s(s(_x51)))))a__incr#(a__head(s(s(mark(_x51)))))mark#(incr(head(s(head(pairs)))))a__incr#(a__head(s(a__head(a__pairs))))
mark#(incr(head(s(pairs))))a__incr#(a__head(s(pairs)))mark#(incr(head(s(tail(tail(_x61))))))a__incr#(a__head(s(a__tail(a__tail(mark(_x61))))))
mark#(incr(head(cons(_x41, _x42))))a__incr#(a__head(cons(mark(_x41), _x42)))mark#(incr(incr(cons(_x41, _x42))))a__incr#(a__incr(cons(mark(_x41), _x42)))
mark#(incr(nats))a__incr#(a__nats)mark#(incr(head(s(head(nats)))))a__incr#(a__head(s(a__head(a__nats))))
mark#(incr(head(s(tail(head(_x61))))))a__incr#(a__head(s(a__tail(a__head(mark(_x61))))))mark#(incr(incr(incr(odds))))a__incr#(a__incr(a__incr(a__odds)))
mark#(incr(incr(incr(head(_x51)))))a__incr#(a__incr(a__incr(a__head(mark(_x51)))))mark#(incr(incr(incr(_x41))))a__incr#(a__incr(incr(mark(_x41))))
mark#(incr(incr(0)))a__incr#(a__incr(0))mark#(incr(head(s(tail(s(_x61))))))a__incr#(a__head(s(a__tail(s(mark(_x61))))))
mark#(incr(head(odds)))a__incr#(a__head(a__odds))mark#(incr(head(s(tail(cons(_x61, _x62))))))a__incr#(a__head(s(a__tail(cons(mark(_x61), _x62)))))
mark#(incr(X))mark#(X)mark#(incr(incr(head(_x41))))a__incr#(a__incr(a__head(mark(_x41))))
mark#(incr(head(s(tail(0)))))a__incr#(a__head(s(a__tail(0))))mark#(incr(head(0)))a__incr#(a__head(0))
mark#(incr(head(s(head(odds)))))a__incr#(a__head(s(a__head(a__odds))))mark#(incr(head(s(nats))))a__incr#(a__head(s(a__nats)))
mark#(incr(head(tail(_x41))))a__incr#(a__head(a__tail(mark(_x41))))mark#(incr(incr(pairs)))a__incr#(a__incr(a__pairs))
mark#(incr(incr(incr(nats))))a__incr#(a__incr(a__incr(a__nats)))mark#(incr(head(s(head(_x51)))))a__incr#(a__head(s(head(mark(_x51)))))

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, a__head, head, cons

Strategy


The right-hand side of the rule mark#(incr(head(s(tail(_x51))))) → a__incr#(head(s(a__tail(mark(_x51))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__incr#(head(s(a__tail(a__pairs)))) 
a__incr#(head(s(a__tail(a__nats)))) 
a__incr#(head(s(a__tail(a__odds)))) 
a__incr#(head(s(a__tail(a__tail(mark(_x61)))))) 
a__incr#(head(s(a__tail(cons(mark(_x61), _x62))))) 
a__incr#(head(s(a__tail(s(mark(_x61)))))) 
a__incr#(head(s(tail(mark(_x51))))) 
a__incr#(head(s(a__tail(a__head(mark(_x61)))))) 
a__incr#(head(s(a__tail(0)))) 
a__incr#(head(s(a__tail(a__incr(mark(_x61)))))) 
Thus, the rule mark#(incr(head(s(tail(_x51))))) → a__incr#(head(s(a__tail(mark(_x51))))) is replaced by the following rules:
mark#(incr(head(s(tail(s(_x61)))))) → a__incr#(head(s(a__tail(s(mark(_x61))))))mark#(incr(head(s(tail(nats))))) → a__incr#(head(s(a__tail(a__nats))))
mark#(incr(head(s(tail(0))))) → a__incr#(head(s(a__tail(0))))mark#(incr(head(s(tail(_x51))))) → a__incr#(head(s(tail(mark(_x51)))))
mark#(incr(head(s(tail(incr(_x61)))))) → a__incr#(head(s(a__tail(a__incr(mark(_x61))))))mark#(incr(head(s(tail(pairs))))) → a__incr#(head(s(a__tail(a__pairs))))
mark#(incr(head(s(tail(tail(_x61)))))) → a__incr#(head(s(a__tail(a__tail(mark(_x61))))))mark#(incr(head(s(tail(odds))))) → a__incr#(head(s(a__tail(a__odds))))
mark#(incr(head(s(tail(head(_x61)))))) → a__incr#(head(s(a__tail(a__head(mark(_x61))))))mark#(incr(head(s(tail(cons(_x61, _x62)))))) → a__incr#(head(s(a__tail(cons(mark(_x61), _x62)))))

Problem 24: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(head(s(tail(nats)))))a__incr#(head(s(a__tail(a__nats))))mark#(incr(head(s(tail(_x51)))))a__incr#(a__head(s(tail(mark(_x51)))))
mark#(incr(head(s(tail(nats)))))a__incr#(a__head(s(a__tail(a__nats))))mark#(incr(cons(_x21, _x22)))a__incr#(cons(mark(_x21), _x22))
mark#(incr(head(s(head(0)))))a__incr#(head(s(a__head(0))))a__incr#(cons(X, XS))mark#(X)
mark#(incr(s(_x21)))a__incr#(s(mark(_x21)))mark#(s(X))mark#(X)
mark#(incr(head(s(head(tail(_x61))))))a__incr#(a__head(s(a__head(a__tail(mark(_x61))))))mark#(incr(head(head(_x41))))a__incr#(a__head(a__head(mark(_x41))))
mark#(incr(head(s(0))))a__incr#(a__head(s(0)))mark#(incr(odds))a__incr#(a__incr(cons(0, incr(odds))))
mark#(incr(head(incr(_x41))))a__incr#(a__head(a__incr(mark(_x41))))mark#(incr(head(s(tail(odds)))))a__incr#(a__head(s(a__tail(a__odds))))
mark#(incr(head(s(tail(cons(_x61, _x62))))))a__incr#(head(s(a__tail(cons(mark(_x61), _x62)))))mark#(incr(head(s(head(head(_x61))))))a__incr#(a__head(s(a__head(a__head(mark(_x61))))))
mark#(incr(head(s(tail(pairs)))))a__incr#(a__head(s(a__tail(a__pairs))))mark#(incr(incr(incr(s(_x51)))))a__incr#(a__incr(a__incr(s(mark(_x51)))))
mark#(incr(incr(tail(_x41))))a__incr#(a__incr(a__tail(mark(_x41))))mark#(incr(head(s(head(incr(_x61))))))a__incr#(a__head(s(a__head(a__incr(mark(_x61))))))
mark#(incr(head(s(odds))))a__incr#(a__head(s(a__odds)))mark#(incr(head(s(_x41))))a__incr#(head(s(mark(_x41))))
mark#(cons(X1, X2))mark#(X1)mark#(incr(head(s(head(cons(_x61, _x62))))))a__incr#(a__head(s(a__head(cons(mark(_x61), _x62)))))
mark#(incr(incr(incr(tail(_x51)))))a__incr#(a__incr(a__incr(a__tail(mark(_x51)))))mark#(incr(odds))a__incr#(incr(a__pairs))
mark#(incr(incr(incr(0))))a__incr#(a__incr(a__incr(0)))mark#(incr(incr(nats)))a__incr#(a__incr(a__nats))
mark#(incr(incr(odds)))a__incr#(a__incr(a__odds))mark#(incr(head(s(incr(_x51)))))a__incr#(a__head(s(a__incr(mark(_x51)))))
mark#(incr(pairs))a__incr#(a__pairs)mark#(incr(head(_x21)))a__incr#(head(mark(_x21)))
mark#(incr(odds))a__incr#(a__incr(pairs))mark#(incr(head(s(head(0)))))a__incr#(a__head(s(head(0))))
mark#(incr(tail(_x21)))a__incr#(a__tail(mark(_x21)))mark#(incr(head(s(tail(incr(_x61))))))a__incr#(head(s(a__tail(a__incr(mark(_x61))))))
mark#(incr(incr(_x21)))a__incr#(incr(mark(_x21)))mark#(incr(head(pairs)))a__incr#(a__head(a__pairs))
mark#(incr(head(s(pairs))))a__incr#(head(s(cons(0, incr(odds)))))mark#(incr(head(s(head(s(_x61))))))a__incr#(a__head(s(a__head(s(mark(_x61))))))
mark#(incr(head(nats)))a__incr#(a__head(a__nats))mark#(incr(incr(incr(cons(_x51, _x52)))))a__incr#(a__incr(a__incr(cons(mark(_x51), _x52))))
mark#(incr(head(s(head(_x51)))))a__incr#(head(s(a__head(mark(_x51)))))mark#(incr(incr(s(_x41))))a__incr#(a__incr(s(mark(_x41))))
mark#(incr(head(s(tail(incr(_x61))))))a__incr#(a__head(s(a__tail(a__incr(mark(_x61))))))mark#(incr(incr(incr(_x41))))a__incr#(incr(a__incr(mark(_x41))))
mark#(incr(head(s(tail(odds)))))a__incr#(head(s(a__tail(a__odds))))mark#(incr(head(s(cons(_x51, _x52)))))a__incr#(a__head(s(cons(mark(_x51), _x52))))
mark#(incr(head(s(pairs))))a__incr#(head(s(pairs)))mark#(incr(incr(incr(pairs))))a__incr#(a__incr(a__incr(a__pairs)))
mark#(incr(incr(incr(incr(_x51)))))a__incr#(a__incr(a__incr(a__incr(mark(_x51)))))mark#(incr(head(s(s(_x51)))))a__incr#(a__head(s(s(mark(_x51)))))
mark#(incr(head(s(head(pairs)))))a__incr#(a__head(s(a__head(a__pairs))))mark#(incr(head(s(tail(tail(_x61))))))a__incr#(a__head(s(a__tail(a__tail(mark(_x61))))))
mark#(incr(head(s(pairs))))a__incr#(a__head(s(pairs)))mark#(incr(head(cons(_x41, _x42))))a__incr#(a__head(cons(mark(_x41), _x42)))
mark#(incr(incr(cons(_x41, _x42))))a__incr#(a__incr(cons(mark(_x41), _x42)))mark#(incr(head(s(head(nats)))))a__incr#(a__head(s(a__head(a__nats))))
mark#(incr(nats))a__incr#(a__nats)mark#(incr(head(s(tail(tail(_x61))))))a__incr#(head(s(a__tail(a__tail(mark(_x61))))))
mark#(incr(head(s(tail(head(_x61))))))a__incr#(a__head(s(a__tail(a__head(mark(_x61))))))mark#(incr(incr(incr(odds))))a__incr#(a__incr(a__incr(a__odds)))
mark#(incr(incr(incr(head(_x51)))))a__incr#(a__incr(a__incr(a__head(mark(_x51)))))mark#(incr(incr(incr(_x41))))a__incr#(a__incr(incr(mark(_x41))))
mark#(incr(incr(0)))a__incr#(a__incr(0))mark#(incr(head(s(tail(s(_x61))))))a__incr#(a__head(s(a__tail(s(mark(_x61))))))
mark#(incr(head(s(tail(pairs)))))a__incr#(head(s(a__tail(a__pairs))))mark#(incr(head(s(tail(cons(_x61, _x62))))))a__incr#(a__head(s(a__tail(cons(mark(_x61), _x62)))))
mark#(incr(head(odds)))a__incr#(a__head(a__odds))mark#(incr(X))mark#(X)
mark#(incr(head(s(tail(head(_x61))))))a__incr#(head(s(a__tail(a__head(mark(_x61))))))mark#(incr(incr(head(_x41))))a__incr#(a__incr(a__head(mark(_x41))))
mark#(incr(head(s(tail(0)))))a__incr#(a__head(s(a__tail(0))))mark#(incr(head(s(tail(s(_x61))))))a__incr#(head(s(a__tail(s(mark(_x61))))))
mark#(incr(head(s(tail(0)))))a__incr#(head(s(a__tail(0))))mark#(incr(head(s(tail(_x51)))))a__incr#(head(s(tail(mark(_x51)))))
mark#(incr(head(0)))a__incr#(a__head(0))mark#(incr(head(tail(_x41))))a__incr#(a__head(a__tail(mark(_x41))))
mark#(incr(head(s(nats))))a__incr#(a__head(s(a__nats)))mark#(incr(head(s(head(odds)))))a__incr#(a__head(s(a__head(a__odds))))
mark#(incr(incr(incr(nats))))a__incr#(a__incr(a__incr(a__nats)))mark#(incr(incr(pairs)))a__incr#(a__incr(a__pairs))
mark#(incr(head(s(head(_x51)))))a__incr#(a__head(s(head(mark(_x51)))))

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, head, a__head, cons

Strategy


The right-hand side of the rule mark#(incr(head(s(tail(nats))))) → a__incr#(head(s(a__tail(a__nats)))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__incr#(head(s(a__tail(nats)))) 
a__incr#(head(s(tail(a__nats)))) 
a__incr#(head(s(a__tail(cons(0, incr(nats)))))) 
Thus, the rule mark#(incr(head(s(tail(nats))))) → a__incr#(head(s(a__tail(a__nats)))) is replaced by the following rules:
mark#(incr(head(s(tail(nats))))) → a__incr#(head(s(a__tail(cons(0, incr(nats))))))mark#(incr(head(s(tail(nats))))) → a__incr#(head(s(a__tail(nats))))
mark#(incr(head(s(tail(nats))))) → a__incr#(head(s(tail(a__nats))))

Problem 25: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(head(s(tail(nats)))))a__incr#(head(s(a__tail(cons(0, incr(nats))))))mark#(incr(head(s(tail(_x51)))))a__incr#(a__head(s(tail(mark(_x51)))))
mark#(incr(head(s(tail(nats)))))a__incr#(a__head(s(a__tail(a__nats))))mark#(incr(cons(_x21, _x22)))a__incr#(cons(mark(_x21), _x22))
mark#(incr(head(s(head(0)))))a__incr#(head(s(a__head(0))))a__incr#(cons(X, XS))mark#(X)
mark#(incr(s(_x21)))a__incr#(s(mark(_x21)))mark#(s(X))mark#(X)
mark#(incr(head(s(head(tail(_x61))))))a__incr#(a__head(s(a__head(a__tail(mark(_x61))))))mark#(incr(head(head(_x41))))a__incr#(a__head(a__head(mark(_x41))))
mark#(incr(head(s(0))))a__incr#(a__head(s(0)))mark#(incr(odds))a__incr#(a__incr(cons(0, incr(odds))))
mark#(incr(head(incr(_x41))))a__incr#(a__head(a__incr(mark(_x41))))mark#(incr(head(s(tail(odds)))))a__incr#(a__head(s(a__tail(a__odds))))
mark#(incr(head(s(tail(cons(_x61, _x62))))))a__incr#(head(s(a__tail(cons(mark(_x61), _x62)))))mark#(incr(head(s(head(head(_x61))))))a__incr#(a__head(s(a__head(a__head(mark(_x61))))))
mark#(incr(head(s(tail(pairs)))))a__incr#(a__head(s(a__tail(a__pairs))))mark#(incr(incr(incr(s(_x51)))))a__incr#(a__incr(a__incr(s(mark(_x51)))))
mark#(incr(head(s(head(incr(_x61))))))a__incr#(a__head(s(a__head(a__incr(mark(_x61))))))mark#(incr(incr(tail(_x41))))a__incr#(a__incr(a__tail(mark(_x41))))
mark#(incr(head(s(head(cons(_x61, _x62))))))a__incr#(a__head(s(a__head(cons(mark(_x61), _x62)))))mark#(cons(X1, X2))mark#(X1)
mark#(incr(head(s(_x41))))a__incr#(head(s(mark(_x41))))mark#(incr(head(s(odds))))a__incr#(a__head(s(a__odds)))
mark#(incr(incr(incr(tail(_x51)))))a__incr#(a__incr(a__incr(a__tail(mark(_x51)))))mark#(incr(odds))a__incr#(incr(a__pairs))
mark#(incr(incr(incr(0))))a__incr#(a__incr(a__incr(0)))mark#(incr(incr(nats)))a__incr#(a__incr(a__nats))
mark#(incr(incr(odds)))a__incr#(a__incr(a__odds))mark#(incr(head(s(incr(_x51)))))a__incr#(a__head(s(a__incr(mark(_x51)))))
mark#(incr(pairs))a__incr#(a__pairs)mark#(incr(head(_x21)))a__incr#(head(mark(_x21)))
mark#(incr(odds))a__incr#(a__incr(pairs))mark#(incr(head(s(head(0)))))a__incr#(a__head(s(head(0))))
mark#(incr(tail(_x21)))a__incr#(a__tail(mark(_x21)))mark#(incr(head(s(tail(incr(_x61))))))a__incr#(head(s(a__tail(a__incr(mark(_x61))))))
mark#(incr(incr(_x21)))a__incr#(incr(mark(_x21)))mark#(incr(head(pairs)))a__incr#(a__head(a__pairs))
mark#(incr(head(s(pairs))))a__incr#(head(s(cons(0, incr(odds)))))mark#(incr(head(s(head(s(_x61))))))a__incr#(a__head(s(a__head(s(mark(_x61))))))
mark#(incr(head(nats)))a__incr#(a__head(a__nats))mark#(incr(incr(incr(cons(_x51, _x52)))))a__incr#(a__incr(a__incr(cons(mark(_x51), _x52))))
mark#(incr(head(s(head(_x51)))))a__incr#(head(s(a__head(mark(_x51)))))mark#(incr(incr(s(_x41))))a__incr#(a__incr(s(mark(_x41))))
mark#(incr(head(s(tail(incr(_x61))))))a__incr#(a__head(s(a__tail(a__incr(mark(_x61))))))mark#(incr(incr(incr(_x41))))a__incr#(incr(a__incr(mark(_x41))))
mark#(incr(head(s(tail(odds)))))a__incr#(head(s(a__tail(a__odds))))mark#(incr(head(s(cons(_x51, _x52)))))a__incr#(a__head(s(cons(mark(_x51), _x52))))
mark#(incr(head(s(pairs))))a__incr#(head(s(pairs)))mark#(incr(incr(incr(pairs))))a__incr#(a__incr(a__incr(a__pairs)))
mark#(incr(incr(incr(incr(_x51)))))a__incr#(a__incr(a__incr(a__incr(mark(_x51)))))mark#(incr(head(s(s(_x51)))))a__incr#(a__head(s(s(mark(_x51)))))
mark#(incr(head(s(head(pairs)))))a__incr#(a__head(s(a__head(a__pairs))))mark#(incr(head(s(pairs))))a__incr#(a__head(s(pairs)))
mark#(incr(head(s(tail(tail(_x61))))))a__incr#(a__head(s(a__tail(a__tail(mark(_x61))))))mark#(incr(head(cons(_x41, _x42))))a__incr#(a__head(cons(mark(_x41), _x42)))
mark#(incr(incr(cons(_x41, _x42))))a__incr#(a__incr(cons(mark(_x41), _x42)))mark#(incr(head(s(tail(tail(_x61))))))a__incr#(head(s(a__tail(a__tail(mark(_x61))))))
mark#(incr(nats))a__incr#(a__nats)mark#(incr(head(s(head(nats)))))a__incr#(a__head(s(a__head(a__nats))))
mark#(incr(head(s(tail(nats)))))a__incr#(head(s(a__tail(nats))))mark#(incr(head(s(tail(head(_x61))))))a__incr#(a__head(s(a__tail(a__head(mark(_x61))))))
mark#(incr(head(s(tail(nats)))))a__incr#(head(s(tail(a__nats))))mark#(incr(incr(incr(odds))))a__incr#(a__incr(a__incr(a__odds)))
mark#(incr(incr(incr(head(_x51)))))a__incr#(a__incr(a__incr(a__head(mark(_x51)))))mark#(incr(incr(incr(_x41))))a__incr#(a__incr(incr(mark(_x41))))
mark#(incr(incr(0)))a__incr#(a__incr(0))mark#(incr(head(s(tail(pairs)))))a__incr#(head(s(a__tail(a__pairs))))
mark#(incr(head(s(tail(s(_x61))))))a__incr#(a__head(s(a__tail(s(mark(_x61))))))mark#(incr(head(odds)))a__incr#(a__head(a__odds))
mark#(incr(head(s(tail(cons(_x61, _x62))))))a__incr#(a__head(s(a__tail(cons(mark(_x61), _x62)))))mark#(incr(head(s(tail(head(_x61))))))a__incr#(head(s(a__tail(a__head(mark(_x61))))))
mark#(incr(X))mark#(X)mark#(incr(incr(head(_x41))))a__incr#(a__incr(a__head(mark(_x41))))
mark#(incr(head(s(tail(s(_x61))))))a__incr#(head(s(a__tail(s(mark(_x61))))))mark#(incr(head(s(tail(0)))))a__incr#(a__head(s(a__tail(0))))
mark#(incr(head(s(tail(0)))))a__incr#(head(s(a__tail(0))))mark#(incr(head(s(tail(_x51)))))a__incr#(head(s(tail(mark(_x51)))))
mark#(incr(head(0)))a__incr#(a__head(0))mark#(incr(head(s(head(odds)))))a__incr#(a__head(s(a__head(a__odds))))
mark#(incr(head(s(nats))))a__incr#(a__head(s(a__nats)))mark#(incr(head(tail(_x41))))a__incr#(a__head(a__tail(mark(_x41))))
mark#(incr(incr(pairs)))a__incr#(a__incr(a__pairs))mark#(incr(incr(incr(nats))))a__incr#(a__incr(a__incr(a__nats)))
mark#(incr(head(s(head(_x51)))))a__incr#(a__head(s(head(mark(_x51)))))

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, a__head, head, cons

Strategy


The right-hand side of the rule mark#(incr(head(s(tail(nats))))) → a__incr#(head(s(a__tail(cons(0, incr(nats)))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__incr#(head(s(tail(cons(0, incr(nats)))))) 
a__incr#(head(s(mark(incr(nats))))) 
Thus, the rule mark#(incr(head(s(tail(nats))))) → a__incr#(head(s(a__tail(cons(0, incr(nats)))))) is replaced by the following rules:
mark#(incr(head(s(tail(nats))))) → a__incr#(head(s(tail(cons(0, incr(nats))))))mark#(incr(head(s(tail(nats))))) → a__incr#(head(s(mark(incr(nats)))))

Problem 26: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(head(s(tail(nats)))))a__incr#(head(s(tail(cons(0, incr(nats))))))mark#(incr(head(s(tail(_x51)))))a__incr#(a__head(s(tail(mark(_x51)))))
mark#(incr(head(s(tail(nats)))))a__incr#(a__head(s(a__tail(a__nats))))mark#(incr(cons(_x21, _x22)))a__incr#(cons(mark(_x21), _x22))
mark#(incr(head(s(head(0)))))a__incr#(head(s(a__head(0))))a__incr#(cons(X, XS))mark#(X)
mark#(incr(s(_x21)))a__incr#(s(mark(_x21)))mark#(s(X))mark#(X)
mark#(incr(head(s(head(tail(_x61))))))a__incr#(a__head(s(a__head(a__tail(mark(_x61))))))mark#(incr(head(head(_x41))))a__incr#(a__head(a__head(mark(_x41))))
mark#(incr(head(s(0))))a__incr#(a__head(s(0)))mark#(incr(odds))a__incr#(a__incr(cons(0, incr(odds))))
mark#(incr(head(incr(_x41))))a__incr#(a__head(a__incr(mark(_x41))))mark#(incr(head(s(tail(odds)))))a__incr#(a__head(s(a__tail(a__odds))))
mark#(incr(head(s(tail(cons(_x61, _x62))))))a__incr#(head(s(a__tail(cons(mark(_x61), _x62)))))mark#(incr(head(s(head(head(_x61))))))a__incr#(a__head(s(a__head(a__head(mark(_x61))))))
mark#(incr(head(s(tail(pairs)))))a__incr#(a__head(s(a__tail(a__pairs))))mark#(incr(incr(incr(s(_x51)))))a__incr#(a__incr(a__incr(s(mark(_x51)))))
mark#(incr(incr(tail(_x41))))a__incr#(a__incr(a__tail(mark(_x41))))mark#(incr(head(s(head(incr(_x61))))))a__incr#(a__head(s(a__head(a__incr(mark(_x61))))))
mark#(incr(head(s(odds))))a__incr#(a__head(s(a__odds)))mark#(incr(head(s(_x41))))a__incr#(head(s(mark(_x41))))
mark#(cons(X1, X2))mark#(X1)mark#(incr(head(s(head(cons(_x61, _x62))))))a__incr#(a__head(s(a__head(cons(mark(_x61), _x62)))))
mark#(incr(incr(incr(tail(_x51)))))a__incr#(a__incr(a__incr(a__tail(mark(_x51)))))mark#(incr(odds))a__incr#(incr(a__pairs))
mark#(incr(incr(incr(0))))a__incr#(a__incr(a__incr(0)))mark#(incr(incr(nats)))a__incr#(a__incr(a__nats))
mark#(incr(incr(odds)))a__incr#(a__incr(a__odds))mark#(incr(head(s(incr(_x51)))))a__incr#(a__head(s(a__incr(mark(_x51)))))
mark#(incr(pairs))a__incr#(a__pairs)mark#(incr(head(_x21)))a__incr#(head(mark(_x21)))
mark#(incr(odds))a__incr#(a__incr(pairs))mark#(incr(head(s(head(0)))))a__incr#(a__head(s(head(0))))
mark#(incr(tail(_x21)))a__incr#(a__tail(mark(_x21)))mark#(incr(head(s(tail(incr(_x61))))))a__incr#(head(s(a__tail(a__incr(mark(_x61))))))
mark#(incr(incr(_x21)))a__incr#(incr(mark(_x21)))mark#(incr(head(pairs)))a__incr#(a__head(a__pairs))
mark#(incr(head(s(pairs))))a__incr#(head(s(cons(0, incr(odds)))))mark#(incr(head(s(head(s(_x61))))))a__incr#(a__head(s(a__head(s(mark(_x61))))))
mark#(incr(head(nats)))a__incr#(a__head(a__nats))mark#(incr(incr(incr(cons(_x51, _x52)))))a__incr#(a__incr(a__incr(cons(mark(_x51), _x52))))
mark#(incr(head(s(head(_x51)))))a__incr#(head(s(a__head(mark(_x51)))))mark#(incr(incr(s(_x41))))a__incr#(a__incr(s(mark(_x41))))
mark#(incr(head(s(tail(incr(_x61))))))a__incr#(a__head(s(a__tail(a__incr(mark(_x61))))))mark#(incr(incr(incr(_x41))))a__incr#(incr(a__incr(mark(_x41))))
mark#(incr(head(s(tail(odds)))))a__incr#(head(s(a__tail(a__odds))))mark#(incr(head(s(cons(_x51, _x52)))))a__incr#(a__head(s(cons(mark(_x51), _x52))))
mark#(incr(head(s(pairs))))a__incr#(head(s(pairs)))mark#(incr(incr(incr(pairs))))a__incr#(a__incr(a__incr(a__pairs)))
mark#(incr(incr(incr(incr(_x51)))))a__incr#(a__incr(a__incr(a__incr(mark(_x51)))))mark#(incr(head(s(s(_x51)))))a__incr#(a__head(s(s(mark(_x51)))))
mark#(incr(head(s(head(pairs)))))a__incr#(a__head(s(a__head(a__pairs))))mark#(incr(head(s(tail(tail(_x61))))))a__incr#(a__head(s(a__tail(a__tail(mark(_x61))))))
mark#(incr(head(s(pairs))))a__incr#(a__head(s(pairs)))mark#(incr(head(cons(_x41, _x42))))a__incr#(a__head(cons(mark(_x41), _x42)))
mark#(incr(incr(cons(_x41, _x42))))a__incr#(a__incr(cons(mark(_x41), _x42)))mark#(incr(head(s(tail(nats)))))a__incr#(head(s(a__tail(nats))))
mark#(incr(head(s(head(nats)))))a__incr#(a__head(s(a__head(a__nats))))mark#(incr(nats))a__incr#(a__nats)
mark#(incr(head(s(tail(tail(_x61))))))a__incr#(head(s(a__tail(a__tail(mark(_x61))))))mark#(incr(head(s(tail(head(_x61))))))a__incr#(a__head(s(a__tail(a__head(mark(_x61))))))
mark#(incr(head(s(tail(nats)))))a__incr#(head(s(tail(a__nats))))mark#(incr(incr(incr(odds))))a__incr#(a__incr(a__incr(a__odds)))
mark#(incr(incr(incr(head(_x51)))))a__incr#(a__incr(a__incr(a__head(mark(_x51)))))mark#(incr(incr(incr(_x41))))a__incr#(a__incr(incr(mark(_x41))))
mark#(incr(incr(0)))a__incr#(a__incr(0))mark#(incr(head(s(tail(s(_x61))))))a__incr#(a__head(s(a__tail(s(mark(_x61))))))
mark#(incr(head(s(tail(pairs)))))a__incr#(head(s(a__tail(a__pairs))))mark#(incr(head(s(tail(cons(_x61, _x62))))))a__incr#(a__head(s(a__tail(cons(mark(_x61), _x62)))))
mark#(incr(head(odds)))a__incr#(a__head(a__odds))mark#(incr(X))mark#(X)
mark#(incr(head(s(tail(head(_x61))))))a__incr#(head(s(a__tail(a__head(mark(_x61))))))mark#(incr(incr(head(_x41))))a__incr#(a__incr(a__head(mark(_x41))))
mark#(incr(head(s(tail(0)))))a__incr#(a__head(s(a__tail(0))))mark#(incr(head(s(tail(s(_x61))))))a__incr#(head(s(a__tail(s(mark(_x61))))))
mark#(incr(head(s(tail(0)))))a__incr#(head(s(a__tail(0))))mark#(incr(head(s(tail(_x51)))))a__incr#(head(s(tail(mark(_x51)))))
mark#(incr(head(s(tail(nats)))))a__incr#(head(s(mark(incr(nats)))))mark#(incr(head(0)))a__incr#(a__head(0))
mark#(incr(head(tail(_x41))))a__incr#(a__head(a__tail(mark(_x41))))mark#(incr(head(s(nats))))a__incr#(a__head(s(a__nats)))
mark#(incr(head(s(head(odds)))))a__incr#(a__head(s(a__head(a__odds))))mark#(incr(incr(incr(nats))))a__incr#(a__incr(a__incr(a__nats)))
mark#(incr(incr(pairs)))a__incr#(a__incr(a__pairs))mark#(incr(head(s(head(_x51)))))a__incr#(a__head(s(head(mark(_x51)))))

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, head, a__head, cons

Strategy


The right-hand side of the rule mark#(incr(head(s(tail(nats))))) → a__incr#(head(s(tail(cons(0, incr(nats)))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
Thus, the rule mark#(incr(head(s(tail(nats))))) → a__incr#(head(s(tail(cons(0, incr(nats)))))) is deleted.

Problem 27: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(head(s(tail(_x51)))))a__incr#(a__head(s(tail(mark(_x51)))))mark#(incr(head(s(tail(nats)))))a__incr#(a__head(s(a__tail(a__nats))))
mark#(incr(cons(_x21, _x22)))a__incr#(cons(mark(_x21), _x22))mark#(incr(head(s(head(0)))))a__incr#(head(s(a__head(0))))
a__incr#(cons(X, XS))mark#(X)mark#(incr(s(_x21)))a__incr#(s(mark(_x21)))
mark#(s(X))mark#(X)mark#(incr(head(s(head(tail(_x61))))))a__incr#(a__head(s(a__head(a__tail(mark(_x61))))))
mark#(incr(head(head(_x41))))a__incr#(a__head(a__head(mark(_x41))))mark#(incr(head(s(0))))a__incr#(a__head(s(0)))
mark#(incr(odds))a__incr#(a__incr(cons(0, incr(odds))))mark#(incr(head(incr(_x41))))a__incr#(a__head(a__incr(mark(_x41))))
mark#(incr(head(s(tail(odds)))))a__incr#(a__head(s(a__tail(a__odds))))mark#(incr(head(s(tail(cons(_x61, _x62))))))a__incr#(head(s(a__tail(cons(mark(_x61), _x62)))))
mark#(incr(head(s(head(head(_x61))))))a__incr#(a__head(s(a__head(a__head(mark(_x61))))))mark#(incr(head(s(tail(pairs)))))a__incr#(a__head(s(a__tail(a__pairs))))
mark#(incr(incr(incr(s(_x51)))))a__incr#(a__incr(a__incr(s(mark(_x51)))))mark#(incr(head(s(head(incr(_x61))))))a__incr#(a__head(s(a__head(a__incr(mark(_x61))))))
mark#(incr(incr(tail(_x41))))a__incr#(a__incr(a__tail(mark(_x41))))mark#(incr(head(s(head(cons(_x61, _x62))))))a__incr#(a__head(s(a__head(cons(mark(_x61), _x62)))))
mark#(cons(X1, X2))mark#(X1)mark#(incr(head(s(_x41))))a__incr#(head(s(mark(_x41))))
mark#(incr(head(s(odds))))a__incr#(a__head(s(a__odds)))mark#(incr(odds))a__incr#(incr(a__pairs))
mark#(incr(incr(incr(tail(_x51)))))a__incr#(a__incr(a__incr(a__tail(mark(_x51)))))mark#(incr(incr(incr(0))))a__incr#(a__incr(a__incr(0)))
mark#(incr(incr(nats)))a__incr#(a__incr(a__nats))mark#(incr(incr(odds)))a__incr#(a__incr(a__odds))
mark#(incr(head(s(incr(_x51)))))a__incr#(a__head(s(a__incr(mark(_x51)))))mark#(incr(pairs))a__incr#(a__pairs)
mark#(incr(head(_x21)))a__incr#(head(mark(_x21)))mark#(incr(odds))a__incr#(a__incr(pairs))
mark#(incr(head(s(head(0)))))a__incr#(a__head(s(head(0))))mark#(incr(tail(_x21)))a__incr#(a__tail(mark(_x21)))
mark#(incr(head(s(tail(incr(_x61))))))a__incr#(head(s(a__tail(a__incr(mark(_x61))))))mark#(incr(incr(_x21)))a__incr#(incr(mark(_x21)))
mark#(incr(head(pairs)))a__incr#(a__head(a__pairs))mark#(incr(head(s(pairs))))a__incr#(head(s(cons(0, incr(odds)))))
mark#(incr(head(s(head(s(_x61))))))a__incr#(a__head(s(a__head(s(mark(_x61))))))mark#(incr(head(nats)))a__incr#(a__head(a__nats))
mark#(incr(incr(incr(cons(_x51, _x52)))))a__incr#(a__incr(a__incr(cons(mark(_x51), _x52))))mark#(incr(head(s(head(_x51)))))a__incr#(head(s(a__head(mark(_x51)))))
mark#(incr(incr(s(_x41))))a__incr#(a__incr(s(mark(_x41))))mark#(incr(head(s(tail(incr(_x61))))))a__incr#(a__head(s(a__tail(a__incr(mark(_x61))))))
mark#(incr(incr(incr(_x41))))a__incr#(incr(a__incr(mark(_x41))))mark#(incr(head(s(tail(odds)))))a__incr#(head(s(a__tail(a__odds))))
mark#(incr(head(s(cons(_x51, _x52)))))a__incr#(a__head(s(cons(mark(_x51), _x52))))mark#(incr(head(s(pairs))))a__incr#(head(s(pairs)))
mark#(incr(incr(incr(pairs))))a__incr#(a__incr(a__incr(a__pairs)))mark#(incr(incr(incr(incr(_x51)))))a__incr#(a__incr(a__incr(a__incr(mark(_x51)))))
mark#(incr(head(s(s(_x51)))))a__incr#(a__head(s(s(mark(_x51)))))mark#(incr(head(s(head(pairs)))))a__incr#(a__head(s(a__head(a__pairs))))
mark#(incr(head(s(pairs))))a__incr#(a__head(s(pairs)))mark#(incr(head(s(tail(tail(_x61))))))a__incr#(a__head(s(a__tail(a__tail(mark(_x61))))))
mark#(incr(head(cons(_x41, _x42))))a__incr#(a__head(cons(mark(_x41), _x42)))mark#(incr(incr(cons(_x41, _x42))))a__incr#(a__incr(cons(mark(_x41), _x42)))
mark#(incr(head(s(tail(tail(_x61))))))a__incr#(head(s(a__tail(a__tail(mark(_x61))))))mark#(incr(nats))a__incr#(a__nats)
mark#(incr(head(s(head(nats)))))a__incr#(a__head(s(a__head(a__nats))))mark#(incr(head(s(tail(nats)))))a__incr#(head(s(a__tail(nats))))
mark#(incr(head(s(tail(head(_x61))))))a__incr#(a__head(s(a__tail(a__head(mark(_x61))))))mark#(incr(head(s(tail(nats)))))a__incr#(head(s(tail(a__nats))))
mark#(incr(incr(incr(odds))))a__incr#(a__incr(a__incr(a__odds)))mark#(incr(incr(incr(head(_x51)))))a__incr#(a__incr(a__incr(a__head(mark(_x51)))))
mark#(incr(incr(incr(_x41))))a__incr#(a__incr(incr(mark(_x41))))mark#(incr(incr(0)))a__incr#(a__incr(0))
mark#(incr(head(s(tail(pairs)))))a__incr#(head(s(a__tail(a__pairs))))mark#(incr(head(s(tail(s(_x61))))))a__incr#(a__head(s(a__tail(s(mark(_x61))))))
mark#(incr(head(odds)))a__incr#(a__head(a__odds))mark#(incr(head(s(tail(cons(_x61, _x62))))))a__incr#(a__head(s(a__tail(cons(mark(_x61), _x62)))))
mark#(incr(head(s(tail(head(_x61))))))a__incr#(head(s(a__tail(a__head(mark(_x61))))))mark#(incr(X))mark#(X)
mark#(incr(incr(head(_x41))))a__incr#(a__incr(a__head(mark(_x41))))mark#(incr(head(s(tail(s(_x61))))))a__incr#(head(s(a__tail(s(mark(_x61))))))
mark#(incr(head(s(tail(0)))))a__incr#(a__head(s(a__tail(0))))mark#(incr(head(s(tail(0)))))a__incr#(head(s(a__tail(0))))
mark#(incr(head(s(tail(nats)))))a__incr#(head(s(mark(incr(nats)))))mark#(incr(head(s(tail(_x51)))))a__incr#(head(s(tail(mark(_x51)))))
mark#(incr(head(0)))a__incr#(a__head(0))mark#(incr(head(s(head(odds)))))a__incr#(a__head(s(a__head(a__odds))))
mark#(incr(head(s(nats))))a__incr#(a__head(s(a__nats)))mark#(incr(head(tail(_x41))))a__incr#(a__head(a__tail(mark(_x41))))
mark#(incr(incr(pairs)))a__incr#(a__incr(a__pairs))mark#(incr(incr(incr(nats))))a__incr#(a__incr(a__incr(a__nats)))
mark#(incr(head(s(head(_x51)))))a__incr#(a__head(s(head(mark(_x51)))))

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, a__head, head, cons

Strategy


The right-hand side of the rule mark#(incr(head(s(tail(_x51))))) → a__incr#(a__head(s(tail(mark(_x51))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__incr#(a__head(s(tail(a__tail(mark(_x61)))))) 
a__incr#(a__head(s(tail(a__incr(mark(_x61)))))) 
a__incr#(a__head(s(tail(s(mark(_x61)))))) 
a__incr#(a__head(s(tail(0)))) 
a__incr#(a__head(s(tail(a__nats)))) 
a__incr#(a__head(s(tail(cons(mark(_x61), _x62))))) 
a__incr#(a__head(s(tail(a__head(mark(_x61)))))) 
a__incr#(head(s(tail(mark(_x51))))) 
a__incr#(a__head(s(tail(a__pairs)))) 
a__incr#(a__head(s(tail(a__odds)))) 
Thus, the rule mark#(incr(head(s(tail(_x51))))) → a__incr#(a__head(s(tail(mark(_x51))))) is replaced by the following rules:
mark#(incr(head(s(tail(_x51))))) → a__incr#(head(s(tail(mark(_x51)))))mark#(incr(head(s(tail(nats))))) → a__incr#(a__head(s(tail(a__nats))))
mark#(incr(head(s(tail(tail(_x61)))))) → a__incr#(a__head(s(tail(a__tail(mark(_x61))))))mark#(incr(head(s(tail(pairs))))) → a__incr#(a__head(s(tail(a__pairs))))
mark#(incr(head(s(tail(odds))))) → a__incr#(a__head(s(tail(a__odds))))mark#(incr(head(s(tail(0))))) → a__incr#(a__head(s(tail(0))))
mark#(incr(head(s(tail(cons(_x61, _x62)))))) → a__incr#(a__head(s(tail(cons(mark(_x61), _x62)))))mark#(incr(head(s(tail(head(_x61)))))) → a__incr#(a__head(s(tail(a__head(mark(_x61))))))
mark#(incr(head(s(tail(incr(_x61)))))) → a__incr#(a__head(s(tail(a__incr(mark(_x61))))))mark#(incr(head(s(tail(s(_x61)))))) → a__incr#(a__head(s(tail(s(mark(_x61))))))

Problem 28: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(cons(_x21, _x22)))a__incr#(cons(mark(_x21), _x22))mark#(s(X))mark#(X)
mark#(incr(head(head(cons(nats, _x52)))))a__incr#(a__head(a__head(cons(a__nats, _x52))))mark#(cons(X1, X2))mark#(X1)
mark#(incr(head(head(cons(odds, _x52)))))a__incr#(a__head(a__head(cons(a__odds, _x52))))mark#(incr(incr(incr(tail(_x51)))))a__incr#(a__incr(a__incr(a__tail(mark(_x51)))))
mark#(incr(s(s(_x41))))a__incr#(s(s(mark(_x41))))mark#(incr(incr(nats)))a__incr#(a__incr(a__nats))
mark#(incr(odds))a__incr#(a__incr(pairs))mark#(incr(head(_x21)))a__incr#(head(mark(_x21)))
mark#(incr(head(s(tail(pairs)))))a__incr#(a__head(s(a__tail(cons(0, incr(odds))))))mark#(incr(head(s(tail(incr(_x61))))))a__incr#(head(s(a__tail(a__incr(mark(_x61))))))
mark#(incr(head(head(cons(cons(_x61, _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(mark(_x61), _x62), _x52))))mark#(incr(incr(_x21)))a__incr#(incr(mark(_x21)))
mark#(incr(head(pairs)))a__incr#(a__head(a__pairs))mark#(incr(head(s(head(s(_x61))))))a__incr#(a__head(s(a__head(s(mark(_x61))))))
mark#(incr(head(head(cons(_x51, _x31)))))a__incr#(a__head(mark(mark(_x51))))mark#(incr(incr(incr(cons(_x51, _x52)))))a__incr#(a__incr(a__incr(cons(mark(_x51), _x52))))
mark#(incr(head(nats)))a__incr#(a__head(a__nats))mark#(incr(head(s(head(_x51)))))a__incr#(head(s(a__head(mark(_x51)))))
mark#(incr(incr(s(_x41))))a__incr#(a__incr(s(mark(_x41))))mark#(incr(head(head(pairs))))a__incr#(head(head(pairs)))
mark#(incr(head(s(odds))))a__incr#(a__head(s(odds)))mark#(incr(head(s(tail(incr(_x61))))))a__incr#(a__head(s(a__tail(a__incr(mark(_x61))))))
mark#(incr(head(s(pairs))))a__incr#(head(s(pairs)))mark#(incr(incr(incr(pairs))))a__incr#(a__incr(a__incr(a__pairs)))
mark#(incr(incr(incr(incr(_x51)))))a__incr#(a__incr(a__incr(a__incr(mark(_x51)))))mark#(incr(head(head(cons(pairs, _x52)))))a__incr#(a__head(a__head(cons(a__pairs, _x52))))
mark#(incr(head(s(pairs))))a__incr#(a__head(s(pairs)))mark#(incr(head(s(tail(tail(_x61))))))a__incr#(a__head(s(tail(a__tail(mark(_x61))))))
mark#(incr(odds))a__incr#(cons(s(mark(0)), incr(incr(odds))))mark#(incr(head(s(incr(_x51)))))a__incr#(head(s(a__incr(mark(_x51)))))
mark#(incr(incr(cons(_x41, _x42))))a__incr#(a__incr(cons(mark(_x41), _x42)))mark#(incr(head(head(cons(0, _x52)))))a__incr#(a__head(a__head(cons(0, _x52))))
mark#(incr(nats))a__incr#(a__nats)mark#(incr(head(s(head(nats)))))a__incr#(a__head(s(a__head(a__nats))))
mark#(incr(head(s(tail(nats)))))a__incr#(head(s(a__tail(nats))))mark#(incr(head(head(_x41))))a__incr#(head(head(mark(_x41))))
mark#(incr(head(s(tail(head(_x61))))))a__incr#(a__head(s(a__tail(a__head(mark(_x61))))))mark#(incr(head(s(tail(nats)))))a__incr#(head(s(tail(a__nats))))
mark#(incr(head(s(cons(_x51, _x52)))))a__incr#(head(s(cons(mark(_x51), _x52))))mark#(incr(incr(incr(_x41))))a__incr#(a__incr(incr(mark(_x41))))
mark#(incr(head(s(tail(s(_x61))))))a__incr#(a__head(s(a__tail(s(mark(_x61))))))mark#(incr(head(odds)))a__incr#(a__head(a__odds))
mark#(incr(head(s(tail(head(_x61))))))a__incr#(head(s(a__tail(a__head(mark(_x61))))))mark#(incr(X))mark#(X)
mark#(incr(incr(head(_x41))))a__incr#(a__incr(a__head(mark(_x41))))mark#(incr(head(s(tail(nats)))))a__incr#(a__head(s(a__tail(cons(0, incr(nats))))))
mark#(incr(head(head(incr(_x51)))))a__incr#(a__head(a__head(a__incr(mark(_x51)))))mark#(incr(head(s(tail(0)))))a__incr#(head(s(a__tail(0))))
mark#(incr(head(s(tail(nats)))))a__incr#(head(s(mark(incr(nats)))))mark#(incr(head(0)))a__incr#(a__head(0))
mark#(incr(head(tail(_x41))))a__incr#(a__head(a__tail(mark(_x41))))mark#(incr(incr(pairs)))a__incr#(a__incr(a__pairs))
mark#(incr(s(odds)))a__incr#(s(a__odds))mark#(incr(incr(incr(nats))))a__incr#(a__incr(a__incr(a__nats)))
mark#(incr(head(head(0))))a__incr#(head(head(0)))mark#(incr(s(cons(_x41, _x42))))a__incr#(s(cons(mark(_x41), _x42)))
mark#(incr(head(head(pairs))))a__incr#(a__head(a__head(cons(0, incr(odds)))))mark#(incr(head(head(pairs))))a__incr#(a__head(a__head(pairs)))
mark#(incr(head(head(head(_x51)))))a__incr#(a__head(head(a__head(mark(_x51)))))mark#(incr(head(s(tail(_x51)))))a__incr#(head(s(a__tail(mark(_x51)))))
mark#(incr(head(head(head(_x51)))))a__incr#(a__head(a__head(a__head(mark(_x51)))))mark#(incr(head(head(cons(head(_x61), _x52)))))a__incr#(a__head(a__head(cons(a__head(mark(_x61)), _x52))))
mark#(incr(s(incr(_x41))))a__incr#(s(a__incr(mark(_x41))))a__incr#(cons(X, XS))mark#(X)
mark#(incr(head(s(0))))a__incr#(a__head(s(0)))mark#(incr(head(s(head(tail(_x61))))))a__incr#(a__head(s(a__head(a__tail(mark(_x61))))))
mark#(incr(head(s(odds))))a__incr#(head(s(a__odds)))mark#(incr(head(incr(_x41))))a__incr#(a__head(a__incr(mark(_x41))))
mark#(incr(head(s(odds))))a__incr#(a__head(s(a__incr(a__pairs))))mark#(incr(head(s(tail(odds)))))a__incr#(a__head(s(a__tail(a__odds))))
mark#(incr(head(s(tail(cons(_x61, _x62))))))a__incr#(head(s(a__tail(cons(mark(_x61), _x62)))))mark#(incr(head(s(tail(s(_x61))))))a__incr#(a__head(s(tail(s(mark(_x61))))))
mark#(incr(head(head(s(_x51)))))a__incr#(a__head(a__head(s(mark(_x51)))))mark#(incr(head(s(head(head(_x61))))))a__incr#(a__head(s(a__head(a__head(mark(_x61))))))
mark#(incr(head(head(pairs))))a__incr#(head(head(cons(0, incr(odds)))))mark#(incr(incr(incr(s(_x51)))))a__incr#(a__incr(a__incr(s(mark(_x51)))))
mark#(incr(head(head(0))))a__incr#(a__head(head(0)))mark#(incr(incr(tail(_x41))))a__incr#(a__incr(a__tail(mark(_x41))))
mark#(incr(head(s(head(incr(_x61))))))a__incr#(a__head(s(a__head(a__incr(mark(_x61))))))mark#(incr(head(s(head(cons(_x61, _x62))))))a__incr#(a__head(s(a__head(cons(mark(_x61), _x62)))))
mark#(incr(head(head(pairs))))a__incr#(head(a__head(a__pairs)))mark#(incr(head(s(s(_x51)))))a__incr#(head(s(s(mark(_x51)))))
mark#(incr(odds))a__incr#(incr(a__pairs))mark#(incr(head(s(0))))a__incr#(head(s(0)))
mark#(incr(head(head(_x41))))a__incr#(head(a__head(mark(_x41))))mark#(incr(head(head(tail(_x51)))))a__incr#(a__head(a__head(a__tail(mark(_x51)))))
mark#(incr(incr(incr(0))))a__incr#(a__incr(a__incr(0)))mark#(incr(head(head(cons(tail(_x61), _x52)))))a__incr#(a__head(a__head(cons(a__tail(mark(_x61)), _x52))))
mark#(incr(head(head(s(_x51)))))a__incr#(a__head(head(s(mark(_x51)))))mark#(incr(incr(odds)))a__incr#(a__incr(a__odds))
mark#(incr(head(s(tail(nats)))))a__incr#(a__head(s(tail(nats))))mark#(incr(head(head(tail(_x51)))))a__incr#(a__head(head(a__tail(mark(_x51)))))
mark#(incr(head(s(incr(_x51)))))a__incr#(a__head(s(a__incr(mark(_x51)))))mark#(incr(pairs))a__incr#(a__pairs)
mark#(incr(head(s(head(0)))))a__incr#(a__head(s(head(0))))mark#(incr(tail(_x21)))a__incr#(a__tail(mark(_x21)))
mark#(incr(head(head(cons(s(_x61), _x52)))))a__incr#(a__head(a__head(cons(s(mark(_x61)), _x52))))mark#(incr(head(s(pairs))))a__incr#(head(s(cons(0, incr(odds)))))
mark#(incr(s(head(_x41))))a__incr#(s(a__head(mark(_x41))))mark#(incr(head(s(tail(pairs)))))a__incr#(a__head(s(tail(a__pairs))))
mark#(incr(head(s(tail(cons(_x61, _x62))))))a__incr#(a__head(s(tail(cons(mark(_x61), _x62)))))mark#(incr(head(s(tail(incr(_x61))))))a__incr#(a__head(s(tail(a__incr(mark(_x61))))))
mark#(incr(head(s(head(0)))))a__incr#(head(s(head(0))))mark#(incr(head(head(cons(incr(_x61), _x52)))))a__incr#(a__head(a__head(cons(a__incr(mark(_x61)), _x52))))
mark#(incr(head(s(tail(nats)))))a__incr#(a__head(s(tail(cons(0, incr(nats))))))mark#(incr(odds))a__incr#(incr(cons(0, incr(odds))))
mark#(incr(incr(incr(_x41))))a__incr#(incr(a__incr(mark(_x41))))mark#(incr(head(s(tail(odds)))))a__incr#(a__head(s(tail(a__odds))))
mark#(incr(s(pairs)))a__incr#(s(a__pairs))mark#(incr(head(s(tail(odds)))))a__incr#(head(s(a__tail(a__odds))))
mark#(incr(head(head(nats))))a__incr#(a__head(a__head(a__nats)))mark#(incr(head(s(cons(_x51, _x52)))))a__incr#(a__head(s(cons(mark(_x51), _x52))))
mark#(incr(head(s(s(_x51)))))a__incr#(a__head(s(s(mark(_x51)))))mark#(incr(head(s(head(pairs)))))a__incr#(a__head(s(a__head(a__pairs))))
mark#(incr(head(s(tail(tail(_x61))))))a__incr#(a__head(s(a__tail(a__tail(mark(_x61))))))mark#(incr(s(nats)))a__incr#(s(a__nats))
mark#(incr(head(head(incr(_x51)))))a__incr#(a__head(head(a__incr(mark(_x51)))))mark#(incr(head(cons(_x41, _x42))))a__incr#(a__head(cons(mark(_x41), _x42)))
mark#(incr(head(s(tail(tail(_x61))))))a__incr#(head(s(a__tail(a__tail(mark(_x61))))))mark#(incr(head(s(tail(nats)))))a__incr#(a__head(s(a__tail(nats))))
mark#(incr(head(s(tail(pairs)))))a__incr#(a__head(s(tail(pairs))))mark#(incr(s(tail(_x41))))a__incr#(s(a__tail(mark(_x41))))
mark#(incr(incr(incr(odds))))a__incr#(a__incr(a__incr(a__odds)))mark#(incr(head(head(odds))))a__incr#(a__head(a__head(a__odds)))
mark#(incr(incr(incr(head(_x51)))))a__incr#(a__incr(a__incr(a__head(mark(_x51)))))mark#(incr(head(head(nats))))a__incr#(a__head(head(a__nats)))
mark#(incr(head(head(cons(_x51, _x52)))))a__incr#(head(a__head(cons(mark(_x51), _x52))))mark#(incr(incr(0)))a__incr#(a__incr(0))
mark#(incr(head(s(tail(pairs)))))a__incr#(head(s(a__tail(a__pairs))))mark#(incr(head(s(tail(cons(_x61, _x62))))))a__incr#(a__head(s(a__tail(cons(mark(_x61), _x62)))))
mark#(incr(head(head(pairs))))a__incr#(a__head(head(pairs)))mark#(incr(head(s(tail(head(_x61))))))a__incr#(a__head(s(tail(a__head(mark(_x61))))))
mark#(incr(head(s(tail(0)))))a__incr#(a__head(s(a__tail(0))))mark#(incr(head(s(tail(s(_x61))))))a__incr#(head(s(a__tail(s(mark(_x61))))))
mark#(incr(head(s(nats))))a__incr#(head(s(nats)))mark#(incr(head(s(tail(_x51)))))a__incr#(head(s(tail(mark(_x51)))))
mark#(incr(head(head(cons(_x51, _x52)))))a__incr#(a__head(head(cons(mark(_x51), _x52))))mark#(incr(head(s(head(odds)))))a__incr#(a__head(s(a__head(a__odds))))
mark#(incr(head(s(nats))))a__incr#(a__head(s(a__nats)))mark#(incr(head(head(odds))))a__incr#(a__head(head(a__odds)))
mark#(incr(head(s(head(_x51)))))a__incr#(a__head(s(head(mark(_x51)))))mark#(incr(head(s(tail(pairs)))))a__incr#(head(s(a__tail(pairs))))

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, head, a__head, cons

Strategy


The right-hand side of the rule mark#(incr(head(head(cons(nats, _x52))))) → a__incr#(a__head(a__head(cons(a__nats, _x52)))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__incr#(a__head(a__head(cons(nats, _x52)))) 
a__incr#(head(a__head(cons(a__nats, _x52)))) 
a__incr#(a__head(head(cons(a__nats, _x52)))) 
a__incr#(a__head(mark(a__nats))) 
a__incr#(a__head(a__head(cons(cons(0, incr(nats)), _x52)))) 
Thus, the rule mark#(incr(head(head(cons(nats, _x52))))) → a__incr#(a__head(a__head(cons(a__nats, _x52)))) is replaced by the following rules:
mark#(incr(head(head(cons(nats, _x52))))) → a__incr#(a__head(a__head(cons(cons(0, incr(nats)), _x52))))mark#(incr(head(head(cons(nats, _x52))))) → a__incr#(a__head(head(cons(a__nats, _x52))))
mark#(incr(head(head(cons(nats, _x31))))) → a__incr#(a__head(mark(a__nats)))mark#(incr(head(head(cons(nats, _x52))))) → a__incr#(a__head(a__head(cons(nats, _x52))))
mark#(incr(head(head(cons(nats, _x52))))) → a__incr#(head(a__head(cons(a__nats, _x52))))

Problem 29: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(cons(_x21, _x22)))a__incr#(cons(mark(_x21), _x22))mark#(s(X))mark#(X)
mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(incr(a__tail(a__head(cons(0, incr(nats)))))))mark#(incr(incr(incr(tail(head(s(s(_x81))))))))a__incr#(a__incr(incr(tail(a__head(s(s(mark(_x81))))))))
mark#(incr(head(s(head(s(_x61))))))a__incr#(a__head(s(a__head(s(mark(_x61))))))mark#(incr(head(nats)))a__incr#(a__head(a__nats))
mark#(incr(incr(incr(cons(_x51, _x52)))))a__incr#(a__incr(a__incr(cons(mark(_x51), _x52))))mark#(incr(incr(incr(tail(s(_x61))))))a__incr#(a__incr(a__incr(a__tail(s(mark(_x61))))))
mark#(incr(incr(incr(tail(head(nats))))))a__incr#(incr(a__incr(a__tail(head(cons(0, incr(nats)))))))mark#(incr(head(s(head(_x51)))))a__incr#(head(s(a__head(mark(_x51)))))
mark#(incr(head(head(cons(cons(incr(0), _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(incr(0), _x62), _x52))))mark#(incr(head(odds)))a__incr#(head(odds))
mark#(incr(incr(incr(tail(head(head(odds)))))))a__incr#(a__incr(incr(tail(a__head(a__head(a__odds))))))mark#(incr(head(head(pairs))))a__incr#(head(head(pairs)))
mark#(incr(head(head(cons(cons(incr(head(_x81)), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(mark(_x81))), _x62), _x52))))mark#(incr(head(s(odds))))a__incr#(a__head(s(odds)))
mark#(incr(head(s(tail(incr(_x61))))))a__incr#(a__head(s(a__tail(a__incr(mark(_x61))))))mark#(incr(head(head(cons(cons(incr(head(_x81)), _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(a__incr(a__head(mark(_x81))), _x62), _x52))))
mark#(incr(incr(incr(tail(head(head(_x71)))))))a__incr#(a__incr(a__incr(a__tail(a__head(a__head(mark(_x71)))))))mark#(incr(head(head(cons(cons(incr(head(incr(_x91))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(a__incr(a__head(a__incr(mark(_x91)))), _x62), _x52))))
mark#(incr(incr(incr(pairs))))a__incr#(a__incr(a__incr(a__pairs)))mark#(incr(incr(incr(tail(head(head(tail(_x81))))))))a__incr#(a__incr(a__incr(tail(a__head(a__head(a__tail(mark(_x81))))))))
mark#(incr(head(tail(_x41))))a__incr#(head(a__tail(mark(_x41))))mark#(incr(head(head(cons(pairs, _x52)))))a__incr#(a__head(a__head(cons(a__pairs, _x52))))
mark#(incr(incr(incr(tail(_x51)))))a__incr#(incr(a__incr(a__tail(mark(_x51)))))mark#(incr(incr(incr(tail(head(head(nats)))))))a__incr#(a__incr(a__incr(tail(a__head(a__head(a__nats))))))
mark#(incr(head(s(pairs))))a__incr#(a__head(s(pairs)))mark#(incr(incr(incr(tail(head(s(_x71)))))))a__incr#(a__incr(a__incr(a__tail(a__head(s(mark(_x71)))))))
mark#(incr(head(head(cons(0, _x52)))))a__incr#(a__head(a__head(cons(0, _x52))))mark#(incr(incr(incr(tail(head(head(odds)))))))a__incr#(a__incr(a__incr(tail(head(a__head(a__odds))))))
mark#(incr(head(pairs)))a__incr#(head(a__pairs))mark#(incr(incr(incr(tail(head(incr(_x71)))))))a__incr#(a__incr(a__incr(tail(a__head(a__incr(mark(_x71)))))))
mark#(incr(head(head(_x41))))a__incr#(head(head(mark(_x41))))mark#(incr(head(s(tail(head(_x61))))))a__incr#(a__head(s(a__tail(a__head(mark(_x61))))))
mark#(incr(incr(incr(tail(head(0))))))a__incr#(a__incr(incr(tail(a__head(0)))))mark#(incr(incr(incr(tail(head(head(odds)))))))a__incr#(incr(a__incr(tail(a__head(a__head(a__incr(a__pairs)))))))
mark#(incr(head(s(tail(pairs)))))a__incr#(a__head(s(tail(cons(0, incr(odds))))))mark#(incr(incr(incr(tail(head(head(_x71)))))))a__incr#(a__incr(a__incr(tail(a__head(head(mark(_x71)))))))
mark#(incr(incr(incr(tail(head(head(incr(_x81))))))))a__incr#(a__incr(a__incr(tail(a__head(a__head(a__incr(mark(_x81))))))))mark#(incr(incr(incr(tail(head(head(s(_x81))))))))a__incr#(a__incr(a__incr(tail(a__head(a__head(s(mark(_x81))))))))
mark#(incr(incr(incr(_x41))))a__incr#(a__incr(incr(mark(_x41))))mark#(incr(head(s(tail(s(_x61))))))a__incr#(a__head(s(a__tail(s(mark(_x61))))))
mark#(incr(incr(incr(tail(head(0))))))a__incr#(a__incr(a__incr(tail(head(0)))))mark#(incr(incr(incr(tail(head(head(_x71)))))))a__incr#(incr(a__incr(tail(a__head(a__head(mark(_x71)))))))
mark#(incr(head(s(tail(head(_x61))))))a__incr#(head(s(a__tail(a__head(mark(_x61))))))mark#(incr(incr(incr(tail(head(s(odds)))))))a__incr#(incr(incr(tail(head(s(a__odds))))))
mark#(incr(head(s(tail(nats)))))a__incr#(a__head(s(a__tail(cons(0, incr(nats))))))mark#(incr(head(s(tail(0)))))a__incr#(head(s(a__tail(0))))
mark#(incr(head(0)))a__incr#(a__head(0))mark#(incr(incr(pairs)))a__incr#(a__incr(a__pairs))
mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(a__incr(tail(a__head(cons(0, incr(nats)))))))mark#(incr(s(s(nats))))a__incr#(s(s(a__nats)))
mark#(incr(head(head(cons(nats, _x31)))))a__incr#(mark(0))mark#(incr(head(head(cons(cons(incr(_x71), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(mark(_x71)), _x62), _x52))))
mark#(incr(s(cons(_x41, _x42))))a__incr#(s(cons(mark(_x41), _x42)))mark#(incr(head(head(pairs))))a__incr#(a__head(a__head(pairs)))
mark#(incr(head(head(cons(cons(incr(pairs), _x62), _x31)))))a__incr#(a__head(mark(cons(a__incr(cons(0, incr(odds))), _x62))))mark#(incr(head(nats)))a__incr#(head(a__nats))
mark#(incr(incr(incr(tail(head(s(cons(_x81, _x82))))))))a__incr#(incr(incr(tail(a__head(s(cons(mark(_x81), _x82)))))))mark#(incr(incr(incr(tail(head(pairs))))))a__incr#(a__incr(a__incr(a__tail(a__head(a__pairs)))))
mark#(incr(head(s(tail(_x51)))))a__incr#(head(s(a__tail(mark(_x51)))))mark#(incr(head(head(cons(cons(s(_x71), _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(s(mark(_x71)), _x62), _x52))))
mark#(incr(head(head(head(_x51)))))a__incr#(a__head(a__head(a__head(mark(_x51)))))mark#(incr(incr(incr(tail(head(odds))))))a__incr#(a__incr(a__incr(tail(a__head(a__odds)))))
mark#(incr(head(head(cons(cons(incr(cons(_x81, _x82)), _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(a__incr(cons(mark(_x81), _x82)), _x62), _x52))))mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(incr(a__tail(head(a__nats)))))
mark#(incr(incr(incr(tail(head(incr(_x71)))))))a__incr#(a__incr(incr(tail(a__head(a__incr(mark(_x71)))))))mark#(incr(head(head(cons(head(_x61), _x52)))))a__incr#(a__head(a__head(cons(a__head(mark(_x61)), _x52))))
a__incr#(cons(X, XS))mark#(X)mark#(incr(head(head(cons(cons(incr(pairs), _x62), _x31)))))a__incr#(a__head(mark(cons(a__incr(pairs), _x62))))
mark#(incr(head(s(0))))a__incr#(a__head(s(0)))mark#(incr(head(s(odds))))a__incr#(a__head(s(a__incr(a__pairs))))
mark#(incr(incr(incr(tail(head(s(head(_x81))))))))a__incr#(incr(incr(tail(a__head(s(a__head(mark(_x81))))))))mark#(incr(head(s(tail(cons(_x61, _x62))))))a__incr#(head(s(a__tail(cons(mark(_x61), _x62)))))
mark#(incr(head(s(tail(s(_x61))))))a__incr#(a__head(s(tail(s(mark(_x61))))))mark#(incr(head(s(head(head(_x61))))))a__incr#(a__head(s(a__head(a__head(mark(_x61))))))
mark#(incr(head(head(s(_x51)))))a__incr#(a__head(a__head(s(mark(_x51)))))mark#(incr(head(head(cons(cons(pairs, _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(a__pairs, _x62), _x52))))
mark#(incr(head(head(cons(nats, _x31)))))a__incr#(head(mark(nats)))mark#(incr(s(s(pairs))))a__incr#(s(s(a__pairs)))
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mark#(incr(head(head(cons(nats, _x31)))))a__incr#(a__head(nats))mark#(incr(head(head(cons(odds, _x52)))))a__incr#(head(a__head(cons(a__odds, _x52))))
mark#(incr(incr(incr(tail(head(nats))))))a__incr#(incr(a__incr(tail(mark(0)))))mark#(incr(s(incr(_x41))))a__incr#(s(a__incr(mark(_x41))))
mark#(incr(head(head(cons(cons(incr(pairs), _x62), _x52)))))a__incr#(head(a__head(cons(cons(a__incr(pairs), _x62), _x52))))mark#(incr(head(s(head(tail(_x61))))))a__incr#(a__head(s(a__head(a__tail(mark(_x61))))))
mark#(incr(head(incr(_x41))))a__incr#(a__head(a__incr(mark(_x41))))mark#(incr(head(s(odds))))a__incr#(head(s(a__odds)))
mark#(incr(head(0)))a__incr#(head(0))mark#(incr(head(s(tail(odds)))))a__incr#(a__head(s(a__tail(a__odds))))
mark#(incr(s(s(odds))))a__incr#(s(s(a__odds)))mark#(incr(incr(incr(tail(head(s(0)))))))a__incr#(incr(incr(tail(a__head(s(0))))))
mark#(incr(head(head(pairs))))a__incr#(head(head(cons(0, incr(odds)))))mark#(incr(head(head(cons(cons(incr(head(s(_x91))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(a__incr(a__head(s(mark(_x91)))), _x62), _x52))))
mark#(incr(head(s(tail(incr(odds))))))a__incr#(head(s(a__tail(a__incr(a__odds)))))mark#(incr(incr(incr(tail(head(head(_x71)))))))a__incr#(a__incr(a__incr(tail(head(a__head(mark(_x71)))))))
mark#(incr(head(s(tail(incr(tail(_x71)))))))a__incr#(head(s(a__tail(a__incr(a__tail(mark(_x71)))))))mark#(incr(head(head(0))))a__incr#(a__head(head(0)))
mark#(incr(head(head(pairs))))a__incr#(head(a__head(a__pairs)))mark#(incr(incr(incr(tail(head(head(odds)))))))a__incr#(a__incr(incr(tail(a__head(a__head(a__incr(a__pairs)))))))
mark#(incr(head(s(0))))a__incr#(head(s(0)))mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(a__incr(tail(head(a__nats)))))
mark#(incr(head(head(tail(_x51)))))a__incr#(a__head(a__head(a__tail(mark(_x51)))))mark#(incr(head(head(cons(cons(incr(head(0)), _x62), _x52)))))a__incr#(head(a__head(cons(cons(a__incr(a__head(0)), _x62), _x52))))
mark#(incr(incr(odds)))a__incr#(a__incr(a__odds))mark#(incr(head(head(cons(tail(_x61), _x52)))))a__incr#(a__head(a__head(cons(a__tail(mark(_x61)), _x52))))
mark#(incr(head(head(s(_x51)))))a__incr#(a__head(head(s(mark(_x51)))))mark#(incr(head(head(cons(cons(incr(s(_x81)), _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(a__incr(s(mark(_x81))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(tail(_x81)), _x62), _x52)))))a__incr#(head(a__head(cons(cons(a__incr(a__tail(mark(_x81))), _x62), _x52))))mark#(incr(head(s(incr(_x51)))))a__incr#(a__head(s(a__incr(mark(_x51)))))
mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(incr(tail(a__head(a__nats)))))mark#(incr(head(s(tail(incr(cons(_x71, _x72)))))))a__incr#(head(s(a__tail(a__incr(cons(mark(_x71), _x72))))))
mark#(incr(head(head(cons(s(_x61), _x52)))))a__incr#(a__head(a__head(cons(s(mark(_x61)), _x52))))mark#(incr(head(head(cons(nats, _x52)))))a__incr#(head(head(cons(a__nats, _x52))))
mark#(incr(tail(_x21)))a__incr#(a__tail(mark(_x21)))mark#(incr(head(s(head(0)))))a__incr#(a__head(s(head(0))))
mark#(incr(head(s(tail(pairs)))))a__incr#(a__head(s(mark(incr(odds)))))mark#(incr(head(head(cons(cons(incr(head(_x81)), _x62), _x31)))))a__incr#(head(mark(cons(a__incr(a__head(mark(_x81))), _x62))))
mark#(incr(incr(incr(tail(head(_x61))))))a__incr#(incr(incr(tail(a__head(mark(_x61))))))mark#(incr(incr(incr(tail(head(pairs))))))a__incr#(a__incr(a__incr(tail(a__head(a__pairs)))))
mark#(incr(incr(incr(tail(head(tail(_x71)))))))a__incr#(a__incr(incr(tail(a__head(a__tail(mark(_x71)))))))mark#(incr(head(head(cons(cons(incr(incr(_x81)), _x62), _x52)))))a__incr#(head(a__head(cons(cons(a__incr(a__incr(mark(_x81))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(odds), _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(a__incr(a__odds), _x62), _x52))))mark#(incr(incr(nats)))a__incr#(a__incr(cons(0, incr(nats))))
mark#(incr(incr(incr(tail(head(s(head(_x81))))))))a__incr#(a__incr(incr(tail(a__head(s(a__head(mark(_x81))))))))mark#(incr(head(head(cons(nats, _x52)))))a__incr#(a__head(head(cons(cons(0, incr(nats)), _x52))))
mark#(incr(odds))a__incr#(incr(cons(0, incr(odds))))mark#(incr(incr(incr(tail(head(s(nats)))))))a__incr#(a__incr(incr(tail(a__head(s(a__nats))))))
mark#(incr(head(head(cons(cons(incr(pairs), _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(a__incr(pairs), _x62), _x52))))mark#(incr(incr(incr(tail(head(s(nats)))))))a__incr#(incr(incr(tail(a__head(s(a__nats))))))
mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(a__incr(a__tail(mark(0)))))mark#(incr(incr(incr(tail(head(head(odds)))))))a__incr#(a__incr(a__incr(tail(a__head(a__head(odds))))))
mark#(incr(head(s(tail(incr(s(_x71)))))))a__incr#(head(s(a__tail(a__incr(s(mark(_x71)))))))mark#(incr(head(head(cons(nats, _x52)))))a__incr#(head(a__head(cons(a__nats, _x52))))
mark#(incr(head(head(nats))))a__incr#(a__head(a__head(a__nats)))mark#(incr(incr(incr(tail(head(head(pairs)))))))a__incr#(a__incr(a__incr(tail(a__head(a__head(a__pairs))))))
mark#(incr(s(s(cons(_x51, _x52)))))a__incr#(s(s(cons(mark(_x51), _x52))))mark#(incr(head(s(tail(incr(incr(_x71)))))))a__incr#(head(s(a__tail(a__incr(a__incr(mark(_x71)))))))
mark#(incr(head(odds)))a__incr#(head(a__incr(a__pairs)))mark#(incr(head(s(s(_x51)))))a__incr#(a__head(s(s(mark(_x51)))))
mark#(incr(head(head(cons(cons(incr(0), _x62), _x31)))))a__incr#(a__head(mark(cons(a__incr(0), _x62))))mark#(incr(head(s(head(pairs)))))a__incr#(a__head(s(a__head(a__pairs))))
mark#(incr(incr(incr(tail(head(s(odds)))))))a__incr#(incr(incr(tail(a__head(s(a__incr(a__pairs)))))))mark#(incr(incr(nats)))a__incr#(incr(cons(0, incr(nats))))
mark#(incr(incr(nats)))a__incr#(incr(nats))mark#(incr(head(s(tail(tail(_x61))))))a__incr#(a__head(s(a__tail(a__tail(mark(_x61))))))
mark#(incr(s(nats)))a__incr#(s(a__nats))mark#(incr(incr(incr(tail(_x51)))))a__incr#(a__incr(a__incr(tail(mark(_x51)))))
mark#(incr(incr(incr(tail(head(head(head(_x81))))))))a__incr#(a__incr(a__incr(tail(a__head(a__head(a__head(mark(_x81))))))))mark#(incr(head(s(tail(incr(_x61))))))a__incr#(head(s(tail(a__incr(mark(_x61))))))
mark#(incr(head(head(cons(cons(_x61, _x62), _x52)))))a__incr#(head(a__head(cons(cons(mark(_x61), _x62), _x52))))mark#(incr(head(cons(_x41, _x42))))a__incr#(a__head(cons(mark(_x41), _x42)))
mark#(incr(head(head(cons(cons(tail(_x71), _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(a__tail(mark(_x71)), _x62), _x52))))mark#(incr(head(s(tail(tail(_x61))))))a__incr#(head(s(a__tail(a__tail(mark(_x61))))))
mark#(incr(s(s(0))))a__incr#(s(s(0)))mark#(incr(incr(incr(tail(head(tail(_x71)))))))a__incr#(a__incr(a__incr(a__tail(a__head(a__tail(mark(_x71)))))))
mark#(incr(s(tail(_x41))))a__incr#(s(a__tail(mark(_x41))))mark#(incr(incr(incr(odds))))a__incr#(a__incr(a__incr(a__odds)))
mark#(incr(head(head(cons(cons(nats, _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(a__nats, _x62), _x52))))mark#(incr(incr(incr(tail(cons(_x61, _x62))))))a__incr#(a__incr(a__incr(a__tail(cons(mark(_x61), _x62)))))
mark#(incr(head(head(cons(_x51, _x52)))))a__incr#(head(a__head(cons(mark(_x51), _x52))))mark#(incr(head(s(tail(pairs)))))a__incr#(head(s(a__tail(a__pairs))))
mark#(incr(head(s(tail(cons(_x61, _x62))))))a__incr#(a__head(s(a__tail(cons(mark(_x61), _x62)))))mark#(incr(head(head(cons(cons(incr(pairs), _x32), a__incr(a__pairs))))))a__incr#(a__head(cons(mark(a__incr(a__pairs)), _x32)))
mark#(incr(head(head(cons(cons(incr(pairs), _x62), _x31)))))a__incr#(head(mark(cons(a__incr(a__pairs), _x62))))mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(incr(a__tail(a__head(a__nats)))))
mark#(incr(head(s(tail(s(_x61))))))a__incr#(head(s(a__tail(s(mark(_x61))))))mark#(incr(head(s(tail(0)))))a__incr#(a__head(s(a__tail(0))))
mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(a__incr(a__tail(head(nats)))))mark#(incr(head(head(cons(cons(incr(_x71), _x62), _x31)))))a__incr#(head(mark(cons(a__incr(mark(_x71)), _x62))))
mark#(incr(head(head(cons(_x51, _x52)))))a__incr#(a__head(head(cons(mark(_x51), _x52))))mark#(incr(head(s(nats))))a__incr#(a__head(s(a__nats)))
mark#(incr(head(head(odds))))a__incr#(a__head(head(a__odds)))mark#(incr(head(s(head(_x51)))))a__incr#(a__head(s(head(mark(_x51)))))
mark#(incr(head(head(cons(cons(incr(pairs), _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(a__incr(cons(0, incr(odds))), _x62), _x52))))mark#(incr(incr(incr(tail(head(head(odds)))))))a__incr#(a__incr(a__incr(tail(a__head(head(a__incr(a__pairs)))))))
mark#(incr(incr(incr(tail(head(_x61))))))a__incr#(a__incr(a__incr(a__tail(head(mark(_x61))))))mark#(incr(head(head(cons(cons(cons(_x71, _x72), _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(cons(mark(_x71), _x72), _x62), _x52))))
mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(a__incr(tail(a__head(a__nats)))))mark#(incr(head(s(tail(pairs)))))a__incr#(head(s(a__tail(pairs))))

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, head, a__head, cons

Strategy


The right-hand side of the rule mark#(incr(incr(incr(tail(head(nats)))))) → a__incr#(a__incr(incr(a__tail(a__head(cons(0, incr(nats))))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__incr#(a__incr(incr(a__tail(head(cons(0, incr(nats))))))) 
a__incr#(a__incr(incr(a__tail(mark(0))))) 
a__incr#(incr(incr(a__tail(a__head(cons(0, incr(nats))))))) 
a__incr#(a__incr(incr(tail(a__head(cons(0, incr(nats))))))) 
Thus, the rule mark#(incr(incr(incr(tail(head(nats)))))) → a__incr#(a__incr(incr(a__tail(a__head(cons(0, incr(nats))))))) is replaced by the following rules:
mark#(incr(incr(incr(tail(head(nats)))))) → a__incr#(a__incr(incr(a__tail(mark(0)))))mark#(incr(incr(incr(tail(head(nats)))))) → a__incr#(a__incr(incr(a__tail(head(cons(0, incr(nats)))))))
mark#(incr(incr(incr(tail(head(nats)))))) → a__incr#(a__incr(incr(tail(a__head(cons(0, incr(nats)))))))mark#(incr(incr(incr(tail(head(nats)))))) → a__incr#(incr(incr(a__tail(a__head(cons(0, incr(nats)))))))

Problem 30: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(cons(_x21, _x22)))a__incr#(cons(mark(_x21), _x22))mark#(s(X))mark#(X)
mark#(incr(head(head(cons(cons(incr(head(tail(head(nats)))), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(head(a__tail(a__head(cons(0, incr(nats)))))), _x62), _x52))))mark#(incr(incr(incr(tail(head(head(tail(_x81))))))))a__incr#(a__incr(a__incr(tail(a__head(a__head(a__tail(mark(_x81))))))))
mark#(incr(head(s(tail(incr(cons(_x71, _x72)))))))a__incr#(a__head(s(a__tail(a__incr(cons(mark(_x71), _x72))))))mark#(incr(incr(incr(tail(head(s(_x71)))))))a__incr#(a__incr(a__incr(a__tail(a__head(s(mark(_x71)))))))
mark#(incr(head(head(cons(0, _x52)))))a__incr#(a__head(a__head(cons(0, _x52))))mark#(incr(incr(incr(tail(head(head(odds)))))))a__incr#(a__incr(a__incr(tail(head(a__head(a__odds))))))
mark#(incr(head(s(head(tail(_x61))))))a__incr#(head(s(a__head(a__tail(mark(_x61))))))mark#(incr(head(s(tail(incr(incr(nats)))))))a__incr#(a__head(s(a__tail(incr(a__incr(cons(0, incr(nats))))))))
mark#(incr(incr(incr(tail(head(incr(_x71)))))))a__incr#(a__incr(a__incr(tail(a__head(a__incr(mark(_x71)))))))mark#(incr(head(head(cons(cons(incr(head(incr(odds))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(a__incr(a__head(a__incr(a__odds))), _x62), _x52))))
mark#(incr(incr(incr(tail(head(head(odds)))))))a__incr#(incr(a__incr(tail(a__head(a__head(a__incr(a__pairs)))))))mark#(incr(head(head(cons(cons(incr(head(_x81)), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(a__head(mark(_x81))), _x62), _x52))))
mark#(incr(incr(incr(_x41))))a__incr#(a__incr(incr(mark(_x41))))mark#(incr(incr(incr(tail(head(head(s(_x81))))))))a__incr#(a__incr(a__incr(tail(a__head(a__head(s(mark(_x81))))))))
mark#(incr(incr(incr(tail(head(head(incr(_x81))))))))a__incr#(a__incr(a__incr(tail(a__head(a__head(a__incr(mark(_x81))))))))mark#(incr(head(s(tail(s(_x61))))))a__incr#(a__head(s(a__tail(s(mark(_x61))))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(odds)))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(a__tail(a__head(a__odds)))), _x62), _x52))))mark#(incr(incr(incr(tail(head(0))))))a__incr#(a__incr(a__incr(tail(head(0)))))
mark#(incr(head(s(tail(head(_x61))))))a__incr#(head(s(a__tail(a__head(mark(_x61))))))mark#(incr(head(head(cons(cons(incr(head(tail(head(nats)))), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(head(a__tail(a__head(a__nats)))), _x62), _x52))))
mark#(incr(head(s(tail(nats)))))a__incr#(a__head(s(a__tail(cons(0, incr(nats))))))mark#(incr(incr(incr(tail(head(s(odds)))))))a__incr#(incr(incr(tail(head(s(a__odds))))))
mark#(incr(head(s(tail(incr(head(_x71)))))))a__incr#(a__head(s(a__tail(a__incr(a__head(mark(_x71)))))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(_x111))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(a__incr(mark(_x111)))))), _x62), _x52))))
mark#(incr(head(0)))a__incr#(a__head(0))mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(a__incr(tail(a__head(cons(0, incr(nats)))))))
mark#(incr(incr(pairs)))a__incr#(a__incr(a__pairs))mark#(incr(head(head(cons(cons(incr(head(tail(head(nats)))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(head(a__tail(head(a__nats)))), _x62), _x52))))
mark#(incr(head(s(tail(incr(incr(_x71)))))))a__incr#(a__head(s(a__tail(incr(a__incr(mark(_x71)))))))mark#(incr(head(head(cons(cons(incr(_x71), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(mark(_x71)), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(head(tail(_x91))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(a__tail(mark(_x91)))), _x62))))mark#(incr(incr(incr(tail(head(s(s(incr(incr(_x101))))))))))a__incr#(a__incr(incr(tail(head(s(s(a__incr(a__incr(mark(_x101))))))))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(nats)))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(head(a__tail(a__head(nats)))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(pairs), _x62), _x31)))))a__incr#(a__head(mark(cons(a__incr(cons(0, incr(odds))), _x62))))
mark#(incr(head(s(head(incr(nats))))))a__incr#(head(s(a__head(a__incr(a__nats)))))mark#(incr(incr(incr(tail(head(s(cons(_x81, _x82))))))))a__incr#(incr(incr(tail(a__head(s(cons(mark(_x81), _x82)))))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(s(_x111))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(a__tail(head(s(mark(_x111)))))), _x62), _x52))))mark#(incr(incr(incr(tail(head(s(s(incr(head(odds))))))))))a__incr#(incr(incr(tail(a__head(s(s(a__incr(a__head(a__odds)))))))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(s(s(_x121)))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(head(a__tail(a__head(s(s(mark(_x121))))))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(tail(cons(_x101, _x102)))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(a__tail(cons(mark(_x101), _x102)))), _x62), _x52))))
mark#(incr(head(head(cons(cons(s(_x71), _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(s(mark(_x71)), _x62), _x52))))mark#(incr(incr(incr(tail(head(s(s(incr(incr(_x101))))))))))a__incr#(a__incr(incr(tail(a__head(s(s(a__incr(a__incr(mark(_x101))))))))))
mark#(incr(incr(incr(tail(head(odds))))))a__incr#(a__incr(a__incr(tail(a__head(a__odds)))))mark#(incr(head(s(head(incr(cons(_x71, _x72)))))))a__incr#(head(s(a__head(a__incr(cons(mark(_x71), _x72))))))
mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(incr(a__tail(head(a__nats)))))mark#(incr(head(head(cons(head(_x61), _x52)))))a__incr#(a__head(a__head(cons(a__head(mark(_x61)), _x52))))
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mark#(incr(incr(incr(tail(s(nats))))))a__incr#(a__incr(a__incr(a__tail(s(a__nats)))))mark#(incr(head(s(tail(cons(_x61, _x62))))))a__incr#(a__head(s(a__tail(cons(mark(_x61), _x62)))))
mark#(incr(head(s(tail(incr(incr(_x71)))))))a__incr#(a__head(s(a__tail(a__incr(incr(mark(_x71)))))))mark#(incr(head(head(cons(cons(incr(pairs), _x32), a__incr(a__pairs))))))a__incr#(a__head(cons(mark(a__incr(a__pairs)), _x32)))
mark#(incr(incr(incr(tail(head(head(cons(_x81, _x82))))))))a__incr#(a__incr(a__incr(a__tail(a__head(a__head(cons(mark(_x81), _x82)))))))mark#(incr(head(s(tail(0)))))a__incr#(a__head(s(a__tail(0))))
mark#(incr(head(s(tail(s(_x61))))))a__incr#(head(s(a__tail(s(mark(_x61))))))mark#(incr(head(head(cons(cons(incr(_x71), _x62), _x31)))))a__incr#(head(mark(cons(a__incr(mark(_x71)), _x62))))
mark#(incr(incr(incr(tail(head(s(s(incr(0)))))))))a__incr#(a__incr(incr(tail(head(s(s(a__incr(0))))))))mark#(incr(head(s(head(_x51)))))a__incr#(a__head(s(head(mark(_x51)))))
mark#(incr(head(head(cons(cons(incr(pairs), _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(a__incr(cons(0, incr(odds))), _x62), _x52))))mark#(incr(incr(incr(tail(head(head(odds)))))))a__incr#(a__incr(a__incr(tail(a__head(head(a__incr(a__pairs)))))))
mark#(incr(head(head(cons(cons(incr(head(incr(_x91))), _x62), _x52)))))a__incr#(head(head(cons(cons(a__incr(a__head(a__incr(mark(_x91)))), _x62), _x52))))mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(a__incr(tail(a__head(a__nats)))))
mark#(incr(head(head(cons(cons(cons(_x71, _x72), _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(cons(mark(_x71), _x72), _x62), _x52))))mark#(incr(head(s(tail(pairs)))))a__incr#(head(s(a__tail(pairs))))

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, head, a__head, cons

Strategy


The right-hand side of the rule mark#(incr(head(head(cons(cons(incr(head(tail(head(nats)))), _x62), _x52))))) → a__incr#(head(head(cons(cons(incr(head(a__tail(a__head(cons(0, incr(nats)))))), _x62), _x52)))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__incr#(head(head(cons(cons(incr(head(a__tail(head(cons(0, incr(nats)))))), _x62), _x52)))) 
a__incr#(head(head(cons(cons(incr(head(a__tail(mark(0)))), _x62), _x52)))) 
a__incr#(head(head(cons(cons(incr(head(tail(a__head(cons(0, incr(nats)))))), _x62), _x52)))) 
Thus, the rule mark#(incr(head(head(cons(cons(incr(head(tail(head(nats)))), _x62), _x52))))) → a__incr#(head(head(cons(cons(incr(head(a__tail(a__head(cons(0, incr(nats)))))), _x62), _x52)))) is replaced by the following rules:
mark#(incr(head(head(cons(cons(incr(head(tail(head(nats)))), _x62), _x52))))) → a__incr#(head(head(cons(cons(incr(head(a__tail(mark(0)))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(tail(head(nats)))), _x62), _x52))))) → a__incr#(head(head(cons(cons(incr(head(a__tail(head(cons(0, incr(nats)))))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(nats)))), _x62), _x52))))) → a__incr#(head(head(cons(cons(incr(head(tail(a__head(cons(0, incr(nats)))))), _x62), _x52))))

Problem 31: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(s(X))mark#(X)mark#(incr(incr(pairs)))a__incr#(a__incr(a__pairs))
mark#(incr(head(head(cons(cons(incr(head(tail(head(nats)))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(head(a__tail(head(a__nats)))), _x62), _x52))))mark#(incr(head(s(tail(incr(incr(nats)))))))a__incr#(a__head(s(a__tail(incr(incr(cons(0, incr(nats))))))))
mark#(incr(incr(incr(tail(head(s(s(incr(incr(_x101))))))))))a__incr#(a__incr(incr(tail(head(s(s(a__incr(a__incr(mark(_x101))))))))))mark#(incr(head(head(cons(cons(incr(head(tail(head(nats)))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(head(a__tail(a__head(nats)))), _x62), _x52))))
mark#(incr(head(s(head(incr(nats))))))a__incr#(head(s(a__head(a__incr(a__nats)))))mark#(incr(head(head(cons(cons(incr(head(tail(head(s(s(_x121)))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(head(a__tail(a__head(s(s(mark(_x121))))))), _x62), _x52))))
mark#(incr(head(s(head(tail(incr(cons(_x81, _x82))))))))a__incr#(head(s(a__head(a__tail(incr(cons(mark(_x81), _x82)))))))mark#(incr(incr(incr(tail(head(head(odds)))))))a__incr#(incr(a__incr(tail(a__head(a__head(incr(a__pairs)))))))
mark#(incr(head(s(head(tail(incr(tail(head(_x91)))))))))a__incr#(head(s(head(a__tail(a__incr(a__tail(a__head(mark(_x91)))))))))mark#(incr(incr(incr(tail(head(s(s(incr(incr(_x101))))))))))a__incr#(a__incr(incr(tail(a__head(s(s(a__incr(a__incr(mark(_x101))))))))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(_x121, _x122)))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(a__incr(cons(mark(_x121), _x122)))))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(odds)), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(a__head(incr(a__pairs))), _x62), _x52))))
mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(incr(a__tail(head(a__nats)))))mark#(incr(head(head(cons(head(_x61), _x52)))))a__incr#(a__head(a__head(cons(a__head(mark(_x61)), _x52))))
mark#(incr(incr(incr(tail(head(head(nats)))))))a__incr#(a__incr(a__incr(a__tail(a__head(a__head(a__nats))))))mark#(incr(head(s(tail(s(_x61))))))a__incr#(a__head(s(tail(s(mark(_x61))))))
mark#(incr(head(s(head(tail(incr(head(_x81))))))))a__incr#(head(s(a__head(tail(incr(a__head(mark(_x81))))))))mark#(incr(head(head(cons(cons(pairs, _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(a__pairs, _x62), _x52))))
mark#(incr(s(s(pairs))))a__incr#(s(s(a__pairs)))mark#(incr(head(head(cons(cons(incr(head(tail(nats))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(head(a__tail(a__nats))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(0), _x62), _x31)))))a__incr#(head(mark(cons(incr(0), _x62))))mark#(incr(head(s(head(tail(incr(head(0))))))))a__incr#(head(s(a__head(a__tail(incr(a__head(0)))))))
mark#(incr(head(s(s(_x51)))))a__incr#(head(s(s(mark(_x51)))))mark#(incr(incr(incr(tail(head(s(0)))))))a__incr#(a__incr(a__incr(a__tail(a__head(s(0))))))
mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(incr(a__tail(mark(0)))))mark#(incr(head(s(tail(nats)))))a__incr#(a__head(s(tail(nats))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(s(tail(_x121)))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(head(a__tail(a__head(s(a__tail(mark(_x121))))))), _x62), _x52))))mark#(incr(head(s(head(tail(incr(head(incr(_x91)))))))))a__incr#(head(s(a__head(a__tail(incr(a__head(incr(mark(_x91)))))))))
mark#(incr(head(cons(_x41, _x42))))a__incr#(head(cons(mark(_x41), _x42)))mark#(incr(head(head(cons(cons(incr(head(tail(nats))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(a__tail(a__nats))), _x62), _x52))))
mark#(incr(head(head(cons(incr(_x61), _x52)))))a__incr#(a__head(a__head(cons(a__incr(mark(_x61)), _x52))))mark#(incr(incr(incr(tail(head(_x61))))))a__incr#(a__incr(incr(a__tail(a__head(mark(_x61))))))
mark#(incr(head(head(cons(cons(incr(head(s(_x91))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(s(mark(_x91)))), _x62), _x52))))mark#(incr(incr(incr(tail(head(head(odds)))))))a__incr#(incr(a__incr(tail(head(a__head(a__odds))))))
mark#(incr(incr(incr(tail(head(head(odds)))))))a__incr#(a__incr(incr(tail(a__head(head(a__odds))))))mark#(incr(head(head(cons(cons(incr(head(pairs)), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(pairs)), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(head(pairs)), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(cons(0, incr(odds)))), _x62), _x52))))mark#(incr(head(s(head(incr(pairs))))))a__incr#(head(s(a__head(a__incr(a__pairs)))))
mark#(incr(head(s(tail(odds)))))a__incr#(a__head(s(tail(a__odds))))mark#(incr(incr(incr(tail(head(head(tail(odds))))))))a__incr#(a__incr(a__incr(tail(a__head(a__head(a__tail(a__odds)))))))
mark#(incr(head(s(head(tail(incr(tail(nats))))))))a__incr#(head(s(head(a__tail(a__incr(a__tail(nats)))))))mark#(incr(head(s(cons(_x51, _x52)))))a__incr#(a__head(s(cons(mark(_x51), _x52))))
mark#(incr(head(nats)))a__incr#(a__head(nats))mark#(incr(head(s(tail(incr(head(head(_x81))))))))a__incr#(a__head(s(a__tail(a__incr(a__head(a__head(mark(_x81))))))))
mark#(incr(head(head(cons(cons(incr(_x71), _x62), _x31)))))a__incr#(a__head(mark(cons(a__incr(mark(_x71)), _x62))))mark#(incr(incr(incr(tail(head(odds))))))a__incr#(a__incr(a__incr(a__tail(a__head(a__odds)))))
mark#(incr(head(s(head(tail(incr(tail(incr(_x91)))))))))a__incr#(head(s(a__head(a__tail(a__incr(a__tail(a__incr(mark(_x91)))))))))mark#(incr(head(head(cons(cons(incr(_x71), _x62), _x52)))))a__incr#(a__head(head(cons(cons(a__incr(mark(_x71)), _x62), _x52))))
mark#(incr(head(s(tail(incr(incr(nats)))))))a__incr#(a__head(s(a__tail(a__incr(a__incr(nats))))))mark#(incr(head(s(head(tail(incr(0)))))))a__incr#(head(s(a__head(tail(a__incr(0))))))
mark#(incr(head(head(cons(cons(incr(tail(_x81)), _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(a__incr(a__tail(mark(_x81))), _x62), _x52))))mark#(incr(head(s(tail(pairs)))))a__incr#(a__head(s(tail(pairs))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(tail(_x121)))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(a__incr(a__tail(mark(_x121))))))), _x62), _x52))))mark#(incr(incr(incr(head(_x51)))))a__incr#(a__incr(a__incr(a__head(mark(_x51)))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(0)))), _x62), _x31)))))a__incr#(head(mark(cons(incr(a__head(tail(head(0)))), _x62))))mark#(incr(incr(incr(tail(head(s(_x71)))))))a__incr#(incr(a__incr(a__tail(a__head(s(mark(_x71)))))))
mark#(incr(head(s(head(tail(incr(tail(nats))))))))a__incr#(head(s(head(tail(a__incr(tail(a__nats)))))))mark#(incr(head(head(cons(cons(incr(head(tail(cons(_x101, _x102)))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(head(a__tail(cons(mark(_x101), _x102)))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(_x101)))), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(a__head(a__tail(head(mark(_x101))))), _x62), _x52))))mark#(incr(incr(incr(tail(head(s(s(incr(head(tail(_x111)))))))))))a__incr#(a__incr(incr(tail(a__head(s(s(a__incr(a__head(a__tail(mark(_x111)))))))))))
mark#(incr(head(s(tail(incr(head(s(_x81))))))))a__incr#(a__head(s(a__tail(a__incr(a__head(s(mark(_x81))))))))mark#(incr(head(head(cons(cons(incr(pairs), _x62), _x52)))))a__incr#(head(a__head(cons(cons(a__incr(cons(0, incr(odds))), _x62), _x52))))
mark#(incr(incr(incr(tail(incr(_x61))))))a__incr#(a__incr(a__incr(a__tail(a__incr(mark(_x61))))))mark#(incr(head(head(cons(cons(incr(head(head(_x91))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(a__head(mark(_x91)))), _x62), _x52))))
mark#(incr(incr(incr(tail(head(head(cons(_x81, _x82))))))))a__incr#(a__incr(a__incr(tail(a__head(a__head(cons(mark(_x81), _x82)))))))mark#(incr(head(head(cons(cons(incr(head(pairs)), _x62), _x31)))))a__incr#(head(mark(cons(incr(a__head(a__pairs)), _x62))))
mark#(incr(head(s(nats))))a__incr#(head(s(nats)))mark#(incr(head(s(head(tail(incr(head(cons(nats, _x92)))))))))a__incr#(head(s(a__head(a__tail(incr(head(cons(a__nats, _x92))))))))
mark#(incr(head(s(head(odds)))))a__incr#(a__head(s(a__head(a__odds))))mark#(incr(head(head(cons(cons(incr(head(tail(head(tail(_x111))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(head(a__tail(a__head(a__tail(mark(_x111)))))), _x62), _x52))))
mark#(incr(incr(incr(tail(head(nats))))))a__incr#(incr(incr(tail(a__head(cons(0, incr(nats)))))))mark#(incr(head(head(cons(cons(incr(head(tail(head(nats)))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(a__nats)))), _x62), _x52))))
mark#(incr(head(s(head(tail(incr(odds)))))))a__incr#(head(s(a__head(a__tail(a__incr(a__odds))))))mark#(incr(incr(incr(tail(s(head(_x71)))))))a__incr#(a__incr(a__incr(tail(s(a__head(mark(_x71)))))))
mark#(incr(incr(incr(tail(head(head(odds)))))))a__incr#(a__incr(a__incr(a__tail(a__head(a__head(a__odds))))))mark#(incr(head(head(cons(cons(incr(head(tail(head(tail(_x111))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(a__tail(a__head(a__tail(mark(_x111)))))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(nats)))), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(a__head(a__tail(head(a__nats)))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(incr(0))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(a__incr(a__head(a__incr(0))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(pairs))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(a__incr(a__pairs))))), _x62), _x52))))mark#(incr(head(s(head(tail(incr(head(incr(s(_x101))))))))))a__incr#(head(s(a__head(a__tail(incr(a__head(a__incr(s(mark(_x101))))))))))
mark#(incr(head(head(cons(cons(0, _x62), _x31)))))a__incr#(a__head(mark(cons(0, _x62))))mark#(incr(incr(incr(tail(head(s(0)))))))a__incr#(a__incr(incr(tail(a__head(s(0))))))
mark#(incr(head(head(cons(cons(incr(head(nats)), _x62), _x52)))))a__incr#(head(a__head(cons(cons(a__incr(a__head(a__nats)), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(tail(head(nats)))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(head(a__tail(head(cons(0, incr(nats)))))), _x62), _x52))))
mark#(incr(head(s(head(tail(incr(odds)))))))a__incr#(head(s(a__head(a__tail(incr(a__odds))))))mark#(incr(incr(incr(tail(s(cons(_x71, _x72)))))))a__incr#(a__incr(a__incr(a__tail(s(cons(mark(_x71), _x72))))))
mark#(incr(incr(incr(tail(head(incr(nats)))))))a__incr#(a__incr(a__incr(tail(a__head(incr(a__nats))))))mark#(incr(head(s(tail(incr(cons(pairs, _x72)))))))a__incr#(head(s(a__tail(a__incr(cons(pairs, _x72))))))
mark#(incr(head(s(head(tail(incr(tail(s(_x91)))))))))a__incr#(head(s(head(a__tail(a__incr(a__tail(s(mark(_x91)))))))))mark#(incr(incr(incr(tail(head(head(odds)))))))a__incr#(a__incr(a__incr(tail(a__head(a__head(incr(a__pairs)))))))
mark#(cons(X1, X2))mark#(X1)mark#(incr(head(s(head(tail(incr(tail(0))))))))a__incr#(head(s(a__head(a__tail(a__incr(a__tail(0)))))))
mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(a__incr(a__tail(head(cons(0, incr(nats)))))))mark#(incr(incr(incr(tail(head(s(s(incr(s(_x101))))))))))a__incr#(a__incr(incr(tail(head(s(s(a__incr(s(mark(_x101))))))))))
mark#(incr(head(s(head(s(pairs))))))a__incr#(a__head(s(a__head(s(a__pairs)))))mark#(incr(head(head(cons(cons(incr(head(tail(head(nats)))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(head(a__tail(mark(0)))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(pairs), _x62), _x52)))))a__incr#(a__head(head(cons(cons(a__incr(a__pairs), _x62), _x52))))mark#(incr(head(s(tail(incr(head(_x71)))))))a__incr#(a__head(s(a__tail(incr(a__head(mark(_x71)))))))
mark#(incr(head(s(tail(s(pairs))))))a__incr#(a__head(s(a__tail(s(a__pairs)))))mark#(incr(head(s(head(tail(incr(_x71)))))))a__incr#(head(s(head(tail(a__incr(mark(_x71)))))))
mark#(incr(head(head(cons(cons(incr(head(tail(odds))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(head(a__tail(a__odds))), _x62), _x52))))mark#(incr(incr(incr(tail(s(incr(_x71)))))))a__incr#(a__incr(a__incr(tail(s(a__incr(mark(_x71)))))))
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mark#(incr(incr(incr(tail(head(incr(_x71)))))))a__incr#(a__incr(a__incr(tail(head(a__incr(mark(_x71)))))))mark#(incr(incr(incr(tail(head(s(nats)))))))a__incr#(incr(a__incr(a__tail(a__head(s(a__nats))))))
mark#(incr(head(head(cons(cons(incr(0), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(0), _x62), _x52))))mark#(incr(head(s(head(tail(head(_x71)))))))a__incr#(head(s(head(a__tail(a__head(mark(_x71)))))))
mark#(incr(head(s(tail(incr(incr(nats)))))))a__incr#(a__head(s(a__tail(a__incr(incr(a__nats))))))mark#(incr(head(head(cons(nats, _x31)))))a__incr#(a__head(mark(a__nats)))
mark#(incr(head(s(head(tail(incr(head(head(_x91)))))))))a__incr#(head(s(a__head(a__tail(incr(a__head(a__head(mark(_x91)))))))))mark#(incr(incr(_x21)))a__incr#(incr(mark(_x21)))
mark#(incr(head(pairs)))a__incr#(a__head(a__pairs))mark#(incr(head(head(cons(odds, _x52)))))a__incr#(a__head(a__head(cons(odds, _x52))))
mark#(incr(head(s(head(tail(incr(head(pairs))))))))a__incr#(head(s(head(a__tail(incr(head(cons(0, incr(odds)))))))))mark#(incr(incr(s(_x41))))a__incr#(a__incr(s(mark(_x41))))
mark#(incr(head(head(cons(cons(incr(head(tail(pairs))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(a__tail(a__pairs))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(0), _x62), _x52)))))a__incr#(a__head(head(cons(cons(incr(0), _x62), _x52))))
mark#(incr(head(s(head(tail(incr(tail(nats))))))))a__incr#(head(s(head(tail(a__incr(a__tail(cons(0, incr(nats)))))))))mark#(incr(head(s(head(tail(odds))))))a__incr#(head(s(head(a__tail(a__odds)))))
mark#(incr(incr(incr(tail(head(head(tail(_x81))))))))a__incr#(a__incr(a__incr(tail(head(a__head(a__tail(mark(_x81))))))))mark#(incr(head(s(head(tail(nats))))))a__incr#(head(s(head(mark(incr(nats))))))
mark#(incr(head(s(tail(incr(cons(pairs, _x72)))))))a__incr#(a__head(s(a__tail(incr(cons(a__pairs, _x72))))))mark#(incr(incr(incr(tail(head(s(s(incr(odds)))))))))a__incr#(incr(incr(tail(a__head(s(s(a__incr(a__odds))))))))
mark#(incr(head(s(tail(incr(nats))))))a__incr#(a__head(s(a__tail(a__incr(a__nats)))))mark#(incr(head(head(cons(cons(incr(head(tail(head(nats)))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(a__tail(head(cons(0, incr(nats)))))), _x62), _x52))))
mark#(incr(head(s(head(incr(tail(_x71)))))))a__incr#(head(s(a__head(a__incr(a__tail(mark(_x71)))))))mark#(incr(head(s(pairs))))a__incr#(head(s(pairs)))
mark#(incr(incr(incr(incr(_x51)))))a__incr#(a__incr(a__incr(a__incr(mark(_x51)))))mark#(incr(incr(incr(tail(head(s(s(pairs))))))))a__incr#(incr(incr(tail(a__head(s(s(cons(0, incr(odds)))))))))
mark#(incr(head(head(cons(cons(incr(head(tail(tail(_x101)))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(a__tail(a__tail(mark(_x101))))), _x62), _x52))))mark#(incr(incr(incr(tail(head(s(odds)))))))a__incr#(incr(incr(tail(head(s(odds))))))
mark#(incr(nats))a__incr#(a__nats)mark#(incr(head(s(tail(incr(pairs))))))a__incr#(head(s(a__tail(a__incr(a__pairs)))))
mark#(incr(head(s(head(tail(incr(head(_x81))))))))a__incr#(head(s(head(a__tail(incr(a__head(mark(_x81))))))))mark#(incr(head(s(head(tail(incr(_x71)))))))a__incr#(head(s(head(a__tail(a__incr(mark(_x71)))))))
mark#(incr(incr(incr(tail(head(incr(nats)))))))a__incr#(a__incr(a__incr(tail(a__head(a__incr(cons(0, incr(nats))))))))mark#(incr(head(head(cons(cons(incr(head(incr(_x91))), _x62), _x31)))))a__incr#(head(mark(cons(a__incr(a__head(a__incr(mark(_x91)))), _x62))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(head(_x111))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(a__tail(a__head(a__head(mark(_x111)))))), _x62), _x52))))mark#(incr(incr(incr(tail(head(_x61))))))a__incr#(incr(a__incr(tail(a__head(mark(_x61))))))
mark#(incr(incr(incr(tail(head(_x61))))))a__incr#(a__incr(incr(tail(head(mark(_x61))))))mark#(incr(incr(incr(tail(tail(_x61))))))a__incr#(a__incr(a__incr(a__tail(a__tail(mark(_x61))))))
mark#(incr(incr(incr(tail(head(nats))))))a__incr#(incr(a__incr(a__tail(mark(0)))))mark#(incr(head(s(tail(incr(head(0)))))))a__incr#(a__head(s(a__tail(a__incr(a__head(0))))))
mark#(incr(head(s(head(tail(incr(head(nats))))))))a__incr#(head(s(a__head(a__tail(incr(a__head(a__nats)))))))mark#(incr(head(s(tail(incr(cons(nats, _x72)))))))a__incr#(a__head(s(a__tail(a__incr(cons(a__nats, _x72))))))
mark#(incr(head(s(head(cons(_x61, _x62))))))a__incr#(head(s(a__head(cons(mark(_x61), _x62)))))mark#(incr(head(s(tail(nats)))))a__incr#(head(s(mark(incr(nats)))))
mark#(incr(incr(incr(tail(head(0))))))a__incr#(a__incr(a__incr(a__tail(a__head(0)))))mark#(incr(incr(incr(tail(head(s(cons(_x81, _x82))))))))a__incr#(a__incr(a__incr(a__tail(a__head(s(cons(mark(_x81), _x82)))))))
mark#(incr(s(odds)))a__incr#(s(a__odds))mark#(incr(incr(incr(tail(head(nats))))))a__incr#(incr(a__incr(a__tail(head(a__nats)))))
mark#(incr(incr(incr(tail(odds)))))a__incr#(a__incr(a__incr(a__tail(a__odds))))mark#(incr(head(head(0))))a__incr#(head(head(0)))
mark#(incr(head(s(tail(incr(incr(head(_x81))))))))a__incr#(a__head(s(a__tail(a__incr(a__incr(a__head(mark(_x81))))))))mark#(incr(incr(incr(tail(head(tail(_x71)))))))a__incr#(a__incr(a__incr(tail(a__head(a__tail(mark(_x71)))))))
mark#(incr(incr(incr(tail(head(s(s(incr(head(head(_x111)))))))))))a__incr#(a__incr(incr(tail(a__head(s(s(a__incr(a__head(a__head(mark(_x111)))))))))))mark#(incr(head(s(head(incr(_x61))))))a__incr#(head(s(a__head(incr(mark(_x61))))))
mark#(incr(incr(incr(tail(head(cons(_x71, _x72)))))))a__incr#(a__incr(a__incr(a__tail(a__head(cons(mark(_x71), _x72))))))mark#(incr(incr(incr(tail(head(head(tail(_x81))))))))a__incr#(a__incr(a__incr(tail(a__head(a__head(tail(mark(_x81))))))))
mark#(incr(head(s(tail(incr(incr(nats)))))))a__incr#(a__head(s(a__tail(a__incr(a__incr(cons(0, incr(nats))))))))mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(incr(tail(head(cons(0, incr(nats)))))))
mark#(incr(incr(incr(tail(s(s(_x71)))))))a__incr#(a__incr(a__incr(a__tail(s(s(mark(_x71)))))))mark#(incr(head(head(cons(cons(incr(pairs), _x62), _x52)))))a__incr#(head(a__head(cons(cons(a__incr(pairs), _x62), _x52))))
mark#(incr(head(s(tail(incr(incr(odds)))))))a__incr#(a__head(s(a__tail(a__incr(a__incr(a__odds))))))mark#(incr(head(head(cons(cons(incr(head(tail(_x91))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(tail(mark(_x91)))), _x62), _x52))))
mark#(incr(head(0)))a__incr#(head(0))mark#(incr(incr(incr(tail(head(s(s(incr(head(odds))))))))))a__incr#(a__incr(incr(tail(a__head(s(s(a__incr(head(a__odds)))))))))
mark#(incr(incr(incr(tail(s(nats))))))a__incr#(a__incr(a__incr(tail(s(a__nats)))))mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(incr(a__tail(head(cons(0, incr(nats)))))))
mark#(incr(incr(incr(tail(head(head(odds)))))))a__incr#(a__incr(a__incr(tail(head(a__head(odds))))))mark#(incr(head(head(pairs))))a__incr#(head(head(cons(0, incr(odds)))))
mark#(incr(head(s(tail(incr(cons(pairs, _x72)))))))a__incr#(a__head(s(tail(a__incr(cons(a__pairs, _x72))))))mark#(incr(head(head(cons(0, _x52)))))a__incr#(a__head(head(cons(0, _x52))))
mark#(incr(incr(incr(tail(head(s(s(incr(_x91)))))))))a__incr#(a__incr(incr(tail(a__head(s(s(incr(mark(_x91)))))))))mark#(incr(head(s(tail(incr(incr(_x71)))))))a__incr#(a__head(s(tail(a__incr(a__incr(mark(_x71)))))))
mark#(incr(head(head(pairs))))a__incr#(head(a__head(a__pairs)))mark#(incr(head(s(head(s(0))))))a__incr#(a__head(s(a__head(s(0)))))
mark#(incr(incr(incr(tail(head(head(odds)))))))a__incr#(a__incr(incr(tail(a__head(a__head(a__incr(a__pairs)))))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(s(_x121)))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(a__incr(s(mark(_x121))))))), _x62), _x52))))
mark#(incr(head(head(s(_x51)))))a__incr#(a__head(head(s(mark(_x51)))))mark#(incr(head(s(head(tail(incr(head(head(_x91)))))))))a__incr#(head(s(head(a__tail(incr(head(a__head(mark(_x91)))))))))
mark#(incr(head(s(incr(_x51)))))a__incr#(a__head(s(a__incr(mark(_x51)))))mark#(incr(head(head(cons(cons(incr(head(odds)), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(a__odds)), _x62), _x52))))
mark#(incr(head(s(tail(incr(cons(_x71, _x72)))))))a__incr#(head(s(a__tail(a__incr(cons(mark(_x71), _x72))))))mark#(incr(head(head(cons(nats, _x52)))))a__incr#(head(head(cons(a__nats, _x52))))
mark#(incr(head(s(tail(incr(cons(cons(_x81, _x82), _x72)))))))a__incr#(a__head(s(a__tail(a__incr(cons(cons(mark(_x81), _x82), _x72))))))mark#(incr(head(head(cons(cons(incr(head(tail(head(odds)))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(a__tail(head(a__odds)))), _x62), _x52))))
mark#(incr(head(s(head(tail(incr(head(tail(_x91)))))))))a__incr#(head(s(a__head(a__tail(incr(head(a__tail(mark(_x91)))))))))mark#(incr(incr(incr(tail(head(s(pairs)))))))a__incr#(a__incr(a__incr(a__tail(a__head(s(a__pairs))))))
mark#(incr(head(s(head(tail(incr(pairs)))))))a__incr#(head(s(a__head(tail(a__incr(a__pairs))))))mark#(incr(head(head(cons(cons(incr(head(nats)), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(a__head(a__nats)), _x62), _x52))))
mark#(incr(incr(incr(tail(head(head(odds)))))))a__incr#(incr(incr(tail(a__head(a__head(a__odds))))))mark#(incr(head(head(cons(cons(incr(head(pairs)), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(a__head(a__pairs)), _x62), _x52))))
mark#(incr(incr(incr(tail(head(s(head(_x81))))))))a__incr#(a__incr(incr(tail(a__head(s(a__head(mark(_x81))))))))mark#(incr(head(s(tail(incr(cons(s(_x81), _x72)))))))a__incr#(a__head(s(a__tail(a__incr(cons(s(mark(_x81)), _x72))))))
mark#(incr(incr(incr(tail(head(s(nats)))))))a__incr#(a__incr(incr(tail(a__head(s(a__nats))))))mark#(incr(head(head(cons(cons(incr(pairs), _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(a__incr(pairs), _x62), _x52))))
mark#(incr(head(s(head(odds)))))a__incr#(head(s(a__head(a__odds))))mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(a__incr(a__tail(mark(0)))))
mark#(incr(incr(incr(tail(head(head(odds)))))))a__incr#(incr(a__incr(tail(a__head(a__head(a__incr(cons(0, incr(odds)))))))))mark#(incr(head(s(head(tail(cons(_x71, _x72)))))))a__incr#(head(s(a__head(a__tail(cons(mark(_x71), _x72))))))
mark#(incr(incr(incr(tail(head(head(odds)))))))a__incr#(a__incr(a__incr(tail(a__head(a__head(odds))))))mark#(incr(head(head(cons(cons(incr(head(pairs)), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(head(a__pairs)), _x62), _x52))))
mark#(incr(incr(incr(tail(head(s(nats)))))))a__incr#(a__incr(a__incr(a__tail(a__head(s(nats))))))mark#(incr(incr(incr(tail(head(head(s(_x81))))))))a__incr#(a__incr(a__incr(tail(head(a__head(s(mark(_x81))))))))
mark#(incr(incr(incr(tail(head(s(s(incr(nats)))))))))a__incr#(a__incr(incr(tail(a__head(s(s(a__incr(a__nats))))))))mark#(incr(head(odds)))a__incr#(head(a__incr(a__pairs)))
mark#(incr(incr(incr(tail(s(_x61))))))a__incr#(a__incr(incr(tail(s(mark(_x61))))))mark#(incr(head(head(cons(cons(incr(head(tail(head(nats)))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(a__tail(a__head(a__nats)))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(head(nats)), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(nats)), _x62), _x52))))mark#(incr(incr(nats)))a__incr#(incr(nats))
mark#(incr(incr(incr(tail(head(s(incr(_x81))))))))a__incr#(a__incr(a__incr(a__tail(a__head(s(a__incr(mark(_x81))))))))mark#(incr(head(s(tail(tail(_x61))))))a__incr#(a__head(s(a__tail(a__tail(mark(_x61))))))
mark#(incr(head(head(cons(cons(incr(head(tail(_x91))), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(a__head(a__tail(mark(_x91)))), _x62), _x52))))mark#(incr(head(cons(_x41, _x42))))a__incr#(a__head(cons(mark(_x41), _x42)))
mark#(incr(head(head(cons(cons(_x61, _x62), _x52)))))a__incr#(head(a__head(cons(cons(mark(_x61), _x62), _x52))))mark#(incr(head(s(tail(incr(s(_x71)))))))a__incr#(a__head(s(a__tail(a__incr(s(mark(_x71)))))))
mark#(incr(incr(incr(tail(s(pairs))))))a__incr#(a__incr(a__incr(a__tail(s(a__pairs)))))mark#(incr(head(head(cons(cons(incr(head(tail(s(_x101)))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(head(a__tail(s(mark(_x101))))), _x62), _x52))))
mark#(incr(s(tail(_x41))))a__incr#(s(a__tail(mark(_x41))))mark#(incr(incr(incr(tail(head(head(0)))))))a__incr#(a__incr(a__incr(a__tail(a__head(a__head(0))))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(nats)))), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(head(a__tail(mark(0)))), _x62), _x52))))mark#(incr(head(s(tail(incr(incr(incr(_x81))))))))a__incr#(a__head(s(a__tail(a__incr(a__incr(a__incr(mark(_x81))))))))
mark#(incr(incr(incr(tail(head(incr(tail(_x81))))))))a__incr#(a__incr(a__incr(tail(a__head(a__incr(a__tail(mark(_x81))))))))mark#(incr(head(s(tail(cons(_x61, _x62))))))a__incr#(a__head(s(a__tail(cons(mark(_x61), _x62)))))
mark#(incr(head(head(cons(cons(incr(pairs), _x32), a__incr(a__pairs))))))a__incr#(a__head(cons(mark(a__incr(a__pairs)), _x32)))mark#(incr(incr(incr(tail(head(head(cons(_x81, _x82))))))))a__incr#(a__incr(a__incr(a__tail(a__head(a__head(cons(mark(_x81), _x82)))))))
mark#(incr(head(s(tail(s(_x61))))))a__incr#(head(s(a__tail(s(mark(_x61))))))mark#(incr(head(s(head(tail(cons(_x71, _x72)))))))a__incr#(head(s(head(tail(cons(mark(_x71), _x72))))))
mark#(incr(head(head(cons(cons(incr(_x71), _x62), _x31)))))a__incr#(head(mark(cons(a__incr(mark(_x71)), _x62))))mark#(incr(head(head(cons(cons(incr(pairs), _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(a__incr(cons(0, incr(odds))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(head(incr(_x91))), _x62), _x52)))))a__incr#(head(head(cons(cons(a__incr(a__head(a__incr(mark(_x91)))), _x62), _x52))))mark#(incr(head(s(head(tail(incr(nats)))))))a__incr#(head(s(a__head(a__tail(a__incr(a__nats))))))
mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(a__incr(tail(a__head(a__nats)))))mark#(incr(head(head(cons(cons(cons(_x71, _x72), _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(cons(mark(_x71), _x72), _x62), _x52))))

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, head, a__head, cons

Strategy


The right-hand side of the rule mark#(incr(incr(pairs))) → a__incr#(a__incr(a__pairs)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__incr#(a__incr(pairs)) 
a__incr#(a__incr(cons(0, incr(odds)))) 
a__incr#(incr(a__pairs)) 
Thus, the rule mark#(incr(incr(pairs))) → a__incr#(a__incr(a__pairs)) is replaced by the following rules:
mark#(incr(incr(pairs))) → a__incr#(a__incr(pairs))mark#(incr(incr(pairs))) → a__incr#(incr(a__pairs))
mark#(incr(incr(pairs))) → a__incr#(a__incr(cons(0, incr(odds))))

Problem 32: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(s(X))mark#(X)mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(_x121, _x101)))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(cons(s(mark(mark(_x121))), incr(_x101)))))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(_x141)), _x122)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(a__head(tail(head(a__incr(cons(tail(a__head(mark(_x141))), _x122)))))), _x62))))mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(incr(a__tail(head(a__nats)))))
mark#(incr(head(head(cons(cons(incr(head(odds)), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(a__head(incr(a__pairs))), _x62), _x52))))mark#(incr(head(head(cons(head(_x61), _x52)))))a__incr#(a__head(a__head(cons(a__head(mark(_x61)), _x52))))
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mark#(incr(head(head(cons(cons(incr(head(nats)), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(nats)), _x62), _x52))))mark#(incr(incr(nats)))a__incr#(incr(nats))
mark#(incr(incr(incr(tail(head(s(incr(_x81))))))))a__incr#(a__incr(a__incr(a__tail(a__head(s(a__incr(mark(_x81))))))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(s(_x151))), _x122)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(a__incr(cons(a__tail(a__head(s(mark(_x151)))), _x122)))))), _x62))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(s(incr(tail(tail(_x241)))), _x191))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(mark(a__tail(a__head(cons(s(mark(s(a__incr(a__tail(a__tail(mark(_x241))))))), _x132))))), incr(_x91)))))), _x62))))mark#(incr(head(s(tail(tail(_x61))))))a__incr#(a__head(s(a__tail(a__tail(mark(_x61))))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(_x141)), _x122)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(a__head(tail(head(a__incr(cons(a__tail(head(mark(_x141))), _x122)))))), _x62))))mark#(incr(head(head(cons(cons(incr(head(tail(_x91))), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(a__head(a__tail(mark(_x91)))), _x62), _x52))))
mark#(incr(head(cons(_x41, _x42))))a__incr#(a__head(cons(mark(_x41), _x42)))mark#(incr(head(head(cons(cons(_x61, _x62), _x52)))))a__incr#(head(a__head(cons(cons(mark(_x61), _x62), _x52))))
mark#(incr(head(s(tail(incr(s(_x71)))))))a__incr#(a__head(s(a__tail(a__incr(s(mark(_x71)))))))mark#(incr(incr(incr(tail(s(pairs))))))a__incr#(a__incr(a__incr(a__tail(s(a__pairs)))))
mark#(incr(head(head(cons(cons(incr(head(tail(s(_x101)))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(head(a__tail(s(mark(_x101))))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(s(_x131))), _x122)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(a__head(tail(head(a__incr(cons(a__tail(a__head(s(mark(_x131)))), _x122)))))), _x62))))
mark#(incr(incr(incr(tail(head(s(s(incr(incr(odds))))))))))a__incr#(a__incr(incr(tail(head(s(s(a__incr(a__incr(a__odds)))))))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(cons(_x221, _x222), _x202))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(mark(a__tail(a__head(cons(s(a__head(cons(cons(mark(_x221), _x222), _x202))), _x132))))), incr(_x91)))))), _x62))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(cons(_x201, _x202)), _x132))), _x91)))))), _x32), incr(head(tail(head(cons(s(mark(a__tail(a__head(cons(s(mark(cons(_x201, _x202))), _x132))))), incr(_x91)))))))))))a__incr#(head(cons(mark(incr(head(tail(head(cons(s(mark(a__tail(head(cons(s(cons(mark(_x201), _x202)), _x132))))), incr(_x91))))))), _x32)))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(_x131), _x122)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(a__head(tail(head(a__incr(cons(a__tail(mark(_x131)), _x122)))))), _x62))))
mark#(incr(s(tail(_x41))))a__incr#(s(a__tail(mark(_x41))))mark#(incr(incr(incr(tail(head(head(0)))))))a__incr#(a__incr(a__incr(a__tail(a__head(a__head(0))))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(nats)))), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(head(a__tail(mark(0)))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(_x131, _x121))), mark(_x131))))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(a__head(tail(head(a__incr(cons(a__tail(mark(mark(_x131))), _x122)))))), _x62))))
mark#(incr(head(s(tail(incr(incr(incr(_x81))))))))a__incr#(a__head(s(a__tail(a__incr(a__incr(a__incr(mark(_x81))))))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(s(incr(head(_x231))), _x191))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(mark(a__tail(a__head(cons(s(mark(s(a__incr(a__head(mark(_x231)))))), _x132))))), incr(_x91)))))), _x62))))
mark#(incr(incr(incr(tail(head(incr(tail(_x81))))))))a__incr#(a__incr(a__incr(tail(a__head(a__incr(a__tail(mark(_x81))))))))mark#(incr(head(s(tail(cons(_x61, _x62))))))a__incr#(a__head(s(a__tail(cons(mark(_x61), _x62)))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(nats, _x191))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(mark(a__tail(a__head(cons(s(mark(nats)), _x132))))), incr(_x91)))))), _x62))))mark#(incr(head(head(cons(cons(incr(head(tail(head(s(s(0)))))), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(head(a__tail(a__head(s(s(0)))))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(pairs), _x32), a__incr(a__pairs))))))a__incr#(a__head(cons(mark(a__incr(a__pairs)), _x32)))mark#(incr(incr(incr(tail(head(head(cons(_x81, _x82))))))))a__incr#(a__incr(a__incr(a__tail(a__head(a__head(cons(mark(_x81), _x82)))))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(s(incr(tail(_x231))), _x191))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(mark(a__tail(a__head(cons(s(mark(s(a__incr(tail(mark(_x231)))))), _x132))))), incr(_x91)))))), _x62))))mark#(incr(head(s(tail(s(_x61))))))a__incr#(head(s(a__tail(s(mark(_x61))))))
mark#(incr(head(s(head(tail(cons(_x71, _x72)))))))a__incr#(head(s(head(tail(cons(mark(_x71), _x72))))))mark#(incr(head(head(cons(cons(incr(head(tail(head(s(s(s(incr(odds)))))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(head(tail(a__head(s(s(s(a__incr(odds)))))))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(_x71), _x62), _x31)))))a__incr#(head(mark(cons(a__incr(mark(_x71)), _x62))))mark#(incr(head(head(cons(cons(incr(pairs), _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(a__incr(cons(0, incr(odds))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(head(incr(_x91))), _x62), _x52)))))a__incr#(head(head(cons(cons(a__incr(a__head(a__incr(mark(_x91)))), _x62), _x52))))mark#(incr(head(s(head(tail(incr(nats)))))))a__incr#(head(s(a__head(a__tail(a__incr(a__nats))))))
mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(a__incr(tail(a__head(a__nats)))))mark#(incr(head(head(cons(cons(cons(_x71, _x72), _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(cons(mark(_x71), _x72), _x62), _x52))))

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, head, a__head, cons

Strategy


The right-hand side of the rule mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(_x121, _x101)))))), _x62), _x52))))) → a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(cons(s(mark(mark(_x121))), incr(_x101)))))), _x62), _x52)))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(cons(s(mark(a__head(mark(_x151)))), incr(_x101)))))), _x62), _x52)))) 
a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(cons(s(mark(0)), incr(_x101)))))), _x62), _x52)))) 
a__incr#(head(a__head(cons(cons(incr(head(tail(head(cons(s(mark(mark(_x121))), incr(_x101)))))), _x62), _x52)))) 
a__incr#(head(head(cons(cons(incr(a__head(tail(head(cons(s(mark(mark(_x121))), incr(_x101)))))), _x62), _x52)))) 
a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(cons(s(mark(s(mark(_x151)))), incr(_x101)))))), _x62), _x52)))) 
a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(cons(s(mark(a__nats)), incr(_x101)))))), _x62), _x52)))) 
a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(cons(s(mark(a__pairs)), incr(_x101)))))), _x62), _x52)))) 
a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(cons(s(mark(a__incr(mark(_x151)))), incr(_x101)))))), _x62), _x52)))) 
a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(cons(s(mark(a__odds)), incr(_x101)))))), _x62), _x52)))) 
a__incr#(head(mark(cons(incr(a__head(tail(head(cons(s(mark(mark(_x121))), incr(_x101)))))), _x62)))) 
a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(cons(s(mark(a__tail(mark(_x151)))), incr(_x101)))))), _x62), _x52)))) 
a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(cons(s(mark(cons(mark(_x151), _x152))), incr(_x101)))))), _x62), _x52)))) 
Thus, the rule mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(_x121, _x101)))))), _x62), _x52))))) → a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(cons(s(mark(mark(_x121))), incr(_x101)))))), _x62), _x52)))) is replaced by the following rules:
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(_x121, _x101)))))), _x62), _x52))))) → a__incr#(head(a__head(cons(cons(incr(head(tail(head(cons(s(mark(mark(_x121))), incr(_x101)))))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(0, _x101)))))), _x62), _x52))))) → a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(cons(s(mark(0)), incr(_x101)))))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(cons(_x151, _x152), _x101)))))), _x62), _x52))))) → a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(cons(s(mark(cons(mark(_x151), _x152))), incr(_x101)))))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(_x121, _x101)))))), _x62), _x52))))) → a__incr#(head(head(cons(cons(incr(a__head(tail(head(cons(s(mark(mark(_x121))), incr(_x101)))))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(pairs, _x101)))))), _x62), _x52))))) → a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(cons(s(mark(a__pairs)), incr(_x101)))))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(head(_x151), _x101)))))), _x62), _x52))))) → a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(cons(s(mark(a__head(mark(_x151)))), incr(_x101)))))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(_x151), _x101)))))), _x62), _x52))))) → a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(cons(s(mark(a__tail(mark(_x151)))), incr(_x101)))))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(incr(_x151), _x101)))))), _x62), _x52))))) → a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(cons(s(mark(a__incr(mark(_x151)))), incr(_x101)))))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(nats, _x101)))))), _x62), _x52))))) → a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(cons(s(mark(a__nats)), incr(_x101)))))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(_x121, _x101)))))), _x62), _x31))))) → a__incr#(head(mark(cons(incr(a__head(tail(head(cons(s(mark(mark(_x121))), incr(_x101)))))), _x62))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(s(_x151), _x101)))))), _x62), _x52))))) → a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(cons(s(mark(s(mark(_x151)))), incr(_x101)))))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(odds, _x101)))))), _x62), _x52))))) → a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(cons(s(mark(a__odds)), incr(_x101)))))), _x62), _x52))))

Problem 33: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(head(head(cons(cons(incr(head(pairs)), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(cons(0, incr(odds)))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(_x141)), _x101)))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(cons(s(mark(a__tail(a__head(mark(_x141))))), incr(_x101)))))), _x62), _x52))))
mark#(incr(head(s(head(incr(pairs))))))a__incr#(head(s(a__head(a__incr(a__pairs)))))mark#(incr(head(s(tail(odds)))))a__incr#(a__head(s(tail(a__odds))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(s(incr(cons(_x231, _x232))), _x191))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(mark(a__tail(a__head(cons(s(mark(s(a__incr(cons(mark(_x231), _x232))))), _x132))))), incr(_x91)))))), _x62))))mark#(incr(incr(incr(tail(head(head(tail(odds))))))))a__incr#(a__incr(a__incr(tail(a__head(a__head(a__tail(a__odds)))))))
mark#(incr(head(s(head(tail(incr(tail(nats))))))))a__incr#(head(s(head(a__tail(a__incr(a__tail(nats)))))))mark#(incr(head(s(cons(_x51, _x52)))))a__incr#(a__head(s(cons(mark(_x51), _x52))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(s(incr(_x221)), _x191))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(mark(a__tail(head(cons(s(mark(s(a__incr(mark(_x221))))), _x132))))), incr(_x91)))))), _x62))))mark#(incr(head(nats)))a__incr#(a__head(nats))
mark#(incr(head(s(tail(incr(head(head(_x81))))))))a__incr#(a__head(s(a__tail(a__incr(a__head(a__head(mark(_x81))))))))mark#(incr(incr(incr(tail(head(odds))))))a__incr#(a__incr(a__incr(a__tail(a__head(a__odds)))))
mark#(incr(head(head(cons(cons(incr(_x71), _x62), _x52)))))a__incr#(a__head(head(cons(cons(a__incr(mark(_x71)), _x62), _x52))))mark#(incr(head(s(head(tail(incr(0)))))))a__incr#(head(s(a__head(tail(a__incr(0))))))
mark#(incr(head(head(cons(cons(incr(tail(_x81)), _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(a__incr(a__tail(mark(_x81))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(s(0))), _x122)))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(a__incr(cons(a__tail(a__head(s(0))), _x122)))))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(s(s(0)))))), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(head(a__tail(head(s(s(0)))))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(tail(_x121)))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(a__incr(a__tail(mark(_x121))))))), _x62), _x52))))
mark#(incr(head(head(cons(head(cons(0, _x72)), _x31)))))a__incr#(a__head(mark(head(cons(0, _x72)))))mark#(incr(incr(incr(head(_x51)))))a__incr#(a__incr(a__incr(a__head(mark(_x51)))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(s(s(s(_x131))))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(a__tail(a__head(s(s(s(mark(_x131)))))))), _x62))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(_x151), _x161))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(mark(a__tail(mark(s(mark(_x151)))))), incr(_x91)))))), _x62))))
mark#(incr(incr(incr(tail(head(s(_x71)))))))a__incr#(incr(a__incr(a__tail(a__head(s(mark(_x71)))))))mark#(incr(head(head(cons(cons(incr(head(tail(cons(_x101, _x102)))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(head(a__tail(cons(mark(_x101), _x102)))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(_x101)))), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(a__head(a__tail(head(mark(_x101))))), _x62), _x52))))mark#(incr(head(s(tail(incr(head(s(_x81))))))))a__incr#(a__head(s(a__tail(a__incr(a__head(s(mark(_x81))))))))
mark#(incr(head(head(cons(cons(incr(head(head(_x91))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(a__head(mark(_x91)))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(pairs)), _x62), _x31)))))a__incr#(head(mark(cons(incr(a__head(a__pairs)), _x62))))
mark#(incr(incr(incr(tail(head(s(s(incr(incr(head(_x111)))))))))))a__incr#(a__incr(incr(tail(head(s(s(a__incr(a__incr(a__head(mark(_x111)))))))))))mark#(incr(head(s(head(tail(incr(head(cons(nats, _x92)))))))))a__incr#(head(s(a__head(a__tail(incr(head(cons(a__nats, _x92))))))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(s(incr(incr(nats))), a__incr(mark(incr(nats)))))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(mark(tail(a__head(cons(s(s(mark(a__incr(a__incr(a__nats))))), _x132))))), incr(_x91)))))), _x62))))mark#(incr(head(head(cons(head(pairs), _x52)))))a__incr#(a__head(a__head(cons(a__head(a__pairs), _x52))))
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mark#(incr(head(head(cons(cons(incr(head(tail(head(s(s(_x121)))))), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(head(tail(a__head(s(s(mark(_x121))))))), _x62), _x52))))mark#(incr(incr(incr(tail(head(head(tail(_x81))))))))a__incr#(a__incr(a__incr(tail(a__head(a__head(tail(mark(_x81))))))))
mark#(incr(head(s(tail(incr(incr(nats)))))))a__incr#(a__head(s(a__tail(a__incr(a__incr(cons(0, incr(nats))))))))mark#(incr(head(s(head(tail(incr(tail(head(head(_x101))))))))))a__incr#(head(s(head(a__tail(a__incr(a__tail(a__head(a__head(mark(_x101))))))))))
mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(incr(tail(head(cons(0, incr(nats)))))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(s(_x131))), _x122)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(a__head(tail(head(a__incr(cons(tail(a__head(s(mark(_x131)))), _x122)))))), _x62))))
mark#(incr(incr(incr(tail(s(s(_x71)))))))a__incr#(a__incr(a__incr(a__tail(s(s(mark(_x71)))))))mark#(incr(head(head(cons(cons(incr(head(tail(_x91))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(tail(mark(_x91)))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(s(incr(pairs)), a__incr(mark(pairs))))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(a__tail(mark(a__head(cons(s(s(mark(incr(a__pairs)))), _x132))))), incr(_x91)))))), _x62))))mark#(incr(head(0)))a__incr#(head(0))
mark#(incr(incr(incr(tail(head(s(s(incr(head(odds))))))))))a__incr#(a__incr(incr(tail(a__head(s(s(a__incr(head(a__odds)))))))))mark#(incr(incr(incr(tail(s(nats))))))a__incr#(a__incr(a__incr(tail(s(a__nats)))))
mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(incr(a__tail(head(cons(0, incr(nats)))))))mark#(incr(head(s(tail(incr(cons(pairs, _x72)))))))a__incr#(a__head(s(tail(a__incr(cons(a__pairs, _x72))))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(s(_x151))), _x122)))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(a__incr(cons(a__tail(head(s(mark(_x151)))), _x122)))))), _x62), _x52))))mark#(incr(head(head(pairs))))a__incr#(head(a__head(a__pairs)))
mark#(incr(head(s(head(s(0))))))a__incr#(a__head(s(a__head(s(0)))))mark#(incr(incr(incr(tail(head(head(odds)))))))a__incr#(a__incr(incr(tail(a__head(a__head(a__incr(a__pairs)))))))
mark#(incr(head(head(s(_x51)))))a__incr#(a__head(head(s(mark(_x51)))))mark#(incr(head(s(head(tail(incr(head(head(_x91)))))))))a__incr#(head(s(head(a__tail(incr(head(a__head(mark(_x91)))))))))
mark#(incr(head(s(incr(_x51)))))a__incr#(a__head(s(a__incr(mark(_x51)))))mark#(incr(head(head(cons(cons(incr(head(odds)), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(a__odds)), _x62), _x52))))
mark#(incr(head(s(head(tail(incr(head(tail(_x91)))))))))a__incr#(head(s(a__head(a__tail(incr(head(a__tail(mark(_x91)))))))))mark#(incr(head(s(head(tail(incr(pairs)))))))a__incr#(head(s(a__head(tail(a__incr(a__pairs))))))
mark#(incr(head(head(cons(cons(incr(head(nats)), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(a__head(a__nats)), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(tail(head(s(s(pairs)))))), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(head(a__tail(a__head(s(s(a__pairs)))))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(head(pairs)), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(a__head(a__pairs)), _x62), _x52))))mark#(incr(incr(incr(tail(head(s(head(_x81))))))))a__incr#(a__incr(incr(tail(a__head(s(a__head(mark(_x81))))))))
mark#(incr(head(s(tail(incr(cons(s(_x81), _x72)))))))a__incr#(a__head(s(a__tail(a__incr(cons(s(mark(_x81)), _x72))))))mark#(incr(incr(incr(tail(head(s(nats)))))))a__incr#(a__incr(incr(tail(a__head(s(a__nats))))))
mark#(incr(head(head(cons(cons(incr(pairs), _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(a__incr(pairs), _x62), _x52))))mark#(incr(incr(incr(tail(head(head(odds)))))))a__incr#(a__incr(a__incr(tail(a__head(a__head(odds))))))
mark#(incr(incr(incr(tail(head(s(nats)))))))a__incr#(a__incr(a__incr(a__tail(a__head(s(nats))))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(pairs), _x122)))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(a__incr(cons(a__tail(a__pairs), _x122)))))), _x62), _x52))))
mark#(incr(head(odds)))a__incr#(head(a__incr(a__pairs)))mark#(incr(incr(incr(tail(head(s(s(incr(nats)))))))))a__incr#(a__incr(incr(tail(a__head(s(s(a__incr(a__nats))))))))
mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(incr(a__tail(head(nats)))))mark#(incr(incr(incr(tail(s(_x61))))))a__incr#(a__incr(incr(tail(s(mark(_x61))))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(s(incr(tail(tail(_x241)))), _x191))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(mark(a__tail(a__head(cons(s(mark(s(a__incr(a__tail(a__tail(mark(_x241))))))), _x132))))), incr(_x91)))))), _x62))))mark#(incr(head(s(tail(tail(_x61))))))a__incr#(a__head(s(a__tail(a__tail(mark(_x61))))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(_x141)), _x122)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(a__head(tail(head(a__incr(cons(a__tail(head(mark(_x141))), _x122)))))), _x62))))mark#(incr(head(cons(_x41, _x42))))a__incr#(a__head(cons(mark(_x41), _x42)))
mark#(incr(head(s(tail(incr(s(_x71)))))))a__incr#(a__head(s(a__tail(a__incr(s(mark(_x71)))))))mark#(incr(incr(incr(tail(head(s(s(incr(incr(odds))))))))))a__incr#(a__incr(incr(tail(head(s(s(a__incr(a__incr(a__odds)))))))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(s(incr(s(_x231))), _x191))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(mark(tail(a__head(cons(s(mark(s(a__incr(s(mark(_x231)))))), _x132))))), incr(_x91)))))), _x62))))mark#(incr(s(tail(_x41))))a__incr#(s(a__tail(mark(_x41))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(nats)))), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(head(a__tail(mark(0)))), _x62), _x52))))mark#(incr(head(s(tail(incr(incr(incr(_x81))))))))a__incr#(a__head(s(a__tail(a__incr(a__incr(a__incr(mark(_x81))))))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(s(incr(head(_x231))), _x191))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(mark(a__tail(a__head(cons(s(mark(s(a__incr(a__head(mark(_x231)))))), _x132))))), incr(_x91)))))), _x62))))mark#(incr(incr(incr(tail(head(incr(tail(_x81))))))))a__incr#(a__incr(a__incr(tail(a__head(a__incr(a__tail(mark(_x81))))))))
mark#(incr(head(s(tail(cons(_x61, _x62))))))a__incr#(a__head(s(a__tail(cons(mark(_x61), _x62)))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(nats, _x191))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(mark(a__tail(a__head(cons(s(mark(nats)), _x132))))), incr(_x91)))))), _x62))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(s(s(0)))))), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(head(a__tail(a__head(s(s(0)))))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(pairs), _x32), a__incr(a__pairs))))))a__incr#(a__head(cons(mark(a__incr(a__pairs)), _x32)))
mark#(incr(incr(incr(tail(head(head(nats)))))))a__incr#(a__incr(incr(a__tail(a__head(a__head(a__nats))))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(s(incr(pairs)), a__incr(mark(pairs))))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(a__tail(mark(a__head(cons(s(s(mark(a__incr(cons(0, incr(odds)))))), _x132))))), incr(_x91)))))), _x62))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(s(incr(tail(_x231))), _x191))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(mark(a__tail(a__head(cons(s(mark(s(a__incr(tail(mark(_x231)))))), _x132))))), incr(_x91)))))), _x62))))mark#(incr(head(head(cons(cons(incr(head(tail(head(s(s(s(incr(odds)))))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(head(tail(a__head(s(s(s(a__incr(odds)))))))), _x62), _x52))))
mark#(incr(head(head(cons(cons(cons(_x71, _x72), _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(cons(mark(_x71), _x72), _x62), _x52))))mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(a__incr(tail(a__head(a__nats)))))

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, head, a__head, cons

Strategy


The right-hand side of the rule mark#(incr(head(s(head(incr(pairs)))))) → a__incr#(head(s(a__head(a__incr(a__pairs))))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__incr#(head(s(head(a__incr(a__pairs))))) 
a__incr#(head(s(a__head(a__incr(pairs))))) 
a__incr#(head(s(a__head(incr(a__pairs))))) 
a__incr#(head(s(a__head(a__incr(cons(0, incr(odds))))))) 
Thus, the rule mark#(incr(head(s(head(incr(pairs)))))) → a__incr#(head(s(a__head(a__incr(a__pairs))))) is replaced by the following rules:
mark#(incr(head(s(head(incr(pairs)))))) → a__incr#(head(s(head(a__incr(a__pairs)))))mark#(incr(head(s(head(incr(pairs)))))) → a__incr#(head(s(a__head(a__incr(cons(0, incr(odds)))))))
mark#(incr(head(s(head(incr(pairs)))))) → a__incr#(head(s(a__head(a__incr(pairs)))))mark#(incr(head(s(head(incr(pairs)))))) → a__incr#(head(s(a__head(incr(a__pairs)))))

Problem 34: ForwardNarrowing



Dependency Pair Problem

Dependency Pairs

mark#(incr(head(head(cons(cons(incr(head(head(head(nats)))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(a__head(a__head(a__nats)))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(head(head(head(_x111))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(a__head(a__head(a__head(mark(_x111)))))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(head(head(head(tail(_x111))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(a__head(a__head(a__tail(mark(_x111)))))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(pairs)), _x62), _x31)))))a__incr#(head(mark(cons(incr(a__head(a__pairs)), _x62))))
mark#(incr(incr(incr(tail(head(s(s(incr(incr(head(_x111)))))))))))a__incr#(a__incr(incr(tail(head(s(s(a__incr(a__incr(a__head(mark(_x111)))))))))))mark#(incr(head(s(head(tail(incr(head(cons(nats, _x92)))))))))a__incr#(head(s(a__head(a__tail(incr(head(cons(a__nats, _x92))))))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(s(incr(incr(nats))), a__incr(mark(incr(nats)))))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(mark(tail(a__head(cons(s(s(mark(a__incr(a__incr(a__nats))))), _x132))))), incr(_x91)))))), _x62))))mark#(incr(head(s(head(odds)))))a__incr#(a__head(s(a__head(a__odds))))
mark#(incr(head(head(cons(head(pairs), _x52)))))a__incr#(a__head(a__head(cons(a__head(a__pairs), _x52))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(nats), _x132))), _x91)))))), _x32), incr(head(tail(head(cons(s(mark(a__tail(a__head(cons(s(mark(nats)), _x132))))), incr(_x91)))))))))))a__incr#(head(cons(mark(incr(head(tail(head(cons(s(mark(a__tail(a__head(cons(s(a__nats), _x132))))), incr(_x91))))))), _x32)))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(s(incr(0)), _x191))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(mark(a__tail(head(cons(s(mark(s(a__incr(0)))), _x132))))), incr(_x91)))))), _x62))))mark#(incr(head(s(head(tail(incr(odds)))))))a__incr#(head(s(a__head(a__tail(a__incr(a__odds))))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(_x121, _x122)))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(head(tail(head(a__incr(cons(mark(_x121), _x122)))))), _x62), _x52))))mark#(incr(incr(incr(tail(s(head(_x71)))))))a__incr#(a__incr(a__incr(tail(s(a__head(mark(_x71)))))))
mark#(incr(incr(incr(tail(head(head(odds)))))))a__incr#(a__incr(a__incr(a__tail(a__head(a__head(a__odds))))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(pairs)), _x122)))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(a__incr(cons(a__tail(a__head(a__pairs)), _x122)))))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(tail(_x111))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(a__tail(a__head(a__tail(mark(_x111)))))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(tail(nats)), _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(a__incr(a__tail(a__nats)), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(0)), _x122)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(a__head(tail(head(a__incr(cons(tail(a__head(0)), _x122)))))), _x62))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(s(incr(s(_x231))), a__incr(mark(s(_x231)))))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(mark(tail(a__head(cons(s(s(mark(a__incr(s(mark(_x231)))))), _x132))))), incr(_x91)))))), _x62))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(nats)))), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(a__head(a__tail(head(a__nats)))), _x62), _x52))))mark#(incr(head(s(tail(incr(head(s(odds))))))))a__incr#(a__head(s(a__tail(a__incr(a__head(s(a__odds)))))))
mark#(incr(head(head(cons(cons(incr(head(incr(0))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(a__incr(a__head(a__incr(0))), _x62), _x52))))mark#(incr(head(s(tail(incr(head(s(incr(_x91)))))))))a__incr#(a__head(s(a__tail(a__incr(a__head(s(a__incr(mark(_x91)))))))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(s(0))), _x122)))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(incr(cons(a__tail(head(s(0))), _x122)))))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(_x201, _x191))), _x161))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(mark(a__tail(mark(s(mark(mark(_x201))))))), incr(_x91)))))), _x62))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(nats)))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(head(tail(a__head(nats)))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(odds), _x132))), _x91)))))), _x32), incr(a__head(tail(head(cons(s(mark(a__tail(a__head(cons(s(a__odds), _x132))))), incr(_x91)))))))))))a__incr#(head(cons(mark(incr(a__head(tail(head(cons(s(mark(a__tail(a__head(cons(s(a__odds), _x132))))), incr(_x91))))))), _x32)))
mark#(incr(incr(incr(tail(head(odds))))))a__incr#(incr(a__incr(a__tail(a__head(a__odds)))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(s(cons(_x141, _x142)))), _x122)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(a__head(tail(head(a__incr(cons(a__tail(a__head(s(cons(mark(_x141), _x142)))), _x122)))))), _x62))))
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mark#(incr(incr(incr(tail(head(s(nats)))))))a__incr#(a__incr(a__incr(a__tail(a__head(s(nats))))))mark#(incr(head(s(tail(incr(head(head(nats))))))))a__incr#(a__head(s(tail(a__incr(head(a__head(a__nats)))))))
mark#(incr(incr(incr(tail(head(s(head(_x81))))))))a__incr#(incr(a__incr(a__tail(a__head(s(a__head(mark(_x81))))))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(pairs), _x122)))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(a__head(tail(head(a__incr(cons(a__tail(a__pairs), _x122)))))), _x62), _x52))))
mark#(incr(head(odds)))a__incr#(head(a__incr(a__pairs)))mark#(incr(incr(incr(tail(head(s(s(incr(nats)))))))))a__incr#(a__incr(incr(tail(a__head(s(s(a__incr(a__nats))))))))
mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(incr(a__tail(head(nats)))))mark#(incr(incr(incr(tail(s(_x61))))))a__incr#(a__incr(incr(tail(s(mark(_x61))))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(s(incr(tail(tail(_x241)))), _x191))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(mark(a__tail(a__head(cons(s(mark(s(a__incr(a__tail(a__tail(mark(_x241))))))), _x132))))), incr(_x91)))))), _x62))))mark#(incr(head(s(tail(tail(_x61))))))a__incr#(a__head(s(a__tail(a__tail(mark(_x61))))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(_x141)), _x122)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(a__head(tail(head(a__incr(cons(a__tail(head(mark(_x141))), _x122)))))), _x62))))mark#(incr(incr(incr(tail(head(odds))))))a__incr#(a__incr(incr(a__tail(a__head(a__odds)))))
mark#(incr(head(cons(_x41, _x42))))a__incr#(a__head(cons(mark(_x41), _x42)))mark#(incr(head(s(tail(incr(s(_x71)))))))a__incr#(a__head(s(a__tail(a__incr(s(mark(_x71)))))))
mark#(incr(head(s(tail(incr(head(s(nats))))))))a__incr#(a__head(s(a__tail(a__incr(a__head(s(a__nats)))))))mark#(incr(incr(incr(tail(head(s(s(incr(incr(odds))))))))))a__incr#(a__incr(incr(tail(head(s(s(a__incr(a__incr(a__odds)))))))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(s(incr(s(_x231))), _x191))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(mark(tail(a__head(cons(s(mark(s(a__incr(s(mark(_x231)))))), _x132))))), incr(_x91)))))), _x62))))mark#(incr(s(tail(_x41))))a__incr#(s(a__tail(mark(_x41))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(nats)))), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(head(a__tail(mark(0)))), _x62), _x52))))mark#(incr(head(s(tail(incr(incr(incr(_x81))))))))a__incr#(a__head(s(a__tail(a__incr(a__incr(a__incr(mark(_x81))))))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(s(incr(head(_x231))), _x191))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(mark(a__tail(a__head(cons(s(mark(s(a__incr(a__head(mark(_x231)))))), _x132))))), incr(_x91)))))), _x62))))mark#(incr(incr(incr(tail(head(incr(tail(_x81))))))))a__incr#(a__incr(a__incr(tail(a__head(a__incr(a__tail(mark(_x81))))))))
mark#(incr(head(s(tail(cons(_x61, _x62))))))a__incr#(a__head(s(a__tail(cons(mark(_x61), _x62)))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(nats, _x191))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(mark(a__tail(a__head(cons(s(mark(nats)), _x132))))), incr(_x91)))))), _x62))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(s(s(0)))))), _x62), _x52)))))a__incr#(head(head(cons(cons(incr(head(a__tail(a__head(s(s(0)))))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(pairs), _x32), a__incr(a__pairs))))))a__incr#(a__head(cons(mark(a__incr(a__pairs)), _x32)))
mark#(incr(incr(incr(tail(head(head(nats)))))))a__incr#(a__incr(incr(a__tail(a__head(a__head(a__nats))))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(s(incr(pairs)), a__incr(mark(pairs))))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(a__tail(mark(a__head(cons(s(s(mark(a__incr(cons(0, incr(odds)))))), _x132))))), incr(_x91)))))), _x62))))
mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(s(incr(tail(_x231))), _x191))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(mark(a__tail(a__head(cons(s(mark(s(a__incr(tail(mark(_x231)))))), _x132))))), incr(_x91)))))), _x62))))mark#(incr(head(head(cons(cons(incr(head(tail(head(s(s(s(incr(odds)))))))), _x62), _x52)))))a__incr#(head(a__head(cons(cons(incr(head(tail(a__head(s(s(s(a__incr(odds)))))))), _x62), _x52))))
mark#(incr(head(s(tail(incr(head(head(nats))))))))a__incr#(a__head(s(tail(a__incr(a__head(a__head(a__nats)))))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(cons(s(head(cons(s(incr(cons(nats, _x232))), _x191))), _x132))), _x91)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(cons(s(mark(a__tail(a__head(cons(s(mark(s(a__incr(cons(a__nats, _x232))))), _x132))))), incr(_x91)))))), _x62))))
mark#(incr(incr(incr(tail(head(s(odds)))))))a__incr#(incr(a__incr(a__tail(a__head(s(a__odds))))))mark#(incr(head(head(cons(cons(incr(head(tail(head(incr(cons(tail(head(s(0))), _x122)))))), _x62), _x31)))))a__incr#(head(mark(cons(incr(head(tail(head(a__incr(cons(a__tail(a__head(s(0))), _x122)))))), _x62))))
mark#(incr(head(head(cons(cons(cons(_x71, _x72), _x62), _x52)))))a__incr#(a__head(a__head(cons(cons(cons(mark(_x71), _x72), _x62), _x52))))mark#(incr(incr(incr(tail(head(nats))))))a__incr#(a__incr(a__incr(tail(a__head(a__nats)))))

Rewrite Rules

a__natscons(0, incr(nats))a__pairscons(0, incr(odds))
a__oddsa__incr(a__pairs)a__incr(cons(X, XS))cons(s(mark(X)), incr(XS))
a__head(cons(X, XS))mark(X)a__tail(cons(X, XS))mark(XS)
mark(nats)a__natsmark(incr(X))a__incr(mark(X))
mark(pairs)a__pairsmark(odds)a__odds
mark(head(X))a__head(mark(X))mark(tail(X))a__tail(mark(X))
mark(cons(X1, X2))cons(mark(X1), X2)mark(0)0
mark(s(X))s(mark(X))a__natsnats
a__incr(X)incr(X)a__pairspairs
a__oddsoddsa__head(X)head(X)
a__tail(X)tail(X)

Original Signature

Termination of terms over the following signature is verified: pairs, mark, tail, odds, a__nats, a__tail, a__odds, nats, 0, a__incr, s, a__pairs, incr, head, a__head, cons

Strategy


The right-hand side of the rule mark#(incr(head(head(cons(cons(incr(head(head(head(nats)))), _x62), _x52))))) → a__incr#(head(a__head(cons(cons(incr(a__head(a__head(a__head(a__nats)))), _x62), _x52)))) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
Relevant TermsIrrelevant Terms
a__incr#(head(head(cons(cons(incr(a__head(a__head(a__head(a__nats)))), _x62), _x52)))) 
a__incr#(head(a__head(cons(cons(incr(a__head(a__head(head(a__nats)))), _x62), _x52)))) 
a__incr#(head(a__head(cons(cons(incr(a__head(a__head(a__head(cons(0, incr(nats)))))), _x62), _x52)))) 
a__incr#(head(a__head(cons(cons(incr(a__head(head(a__head(a__nats)))), _x62), _x52)))) 
a__incr#(head(a__head(cons(cons(incr(a__head(a__head(a__head(nats)))), _x62), _x52)))) 
a__incr#(head(mark(cons(incr(a__head(a__head(a__head(a__nats)))), _x62)))) 
a__incr#(head(a__head(cons(cons(incr(head(a__head(a__head(a__nats)))), _x62), _x52)))) 
Thus, the rule mark#(incr(head(head(cons(cons(incr(head(head(head(nats)))), _x62), _x52))))) → a__incr#(head(a__head(cons(cons(incr(a__head(a__head(a__head(a__nats)))), _x62), _x52)))) is replaced by the following rules:
mark#(incr(head(head(cons(cons(incr(head(head(head(nats)))), _x62), _x52))))) → a__incr#(head(head(cons(cons(incr(a__head(a__head(a__head(a__nats)))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(head(head(nats)))), _x62), _x52))))) → a__incr#(head(a__head(cons(cons(incr(a__head(head(a__head(a__nats)))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(head(head(head(nats)))), _x62), _x52))))) → a__incr#(head(a__head(cons(cons(incr(a__head(a__head(a__head(cons(0, incr(nats)))))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(head(head(nats)))), _x62), _x52))))) → a__incr#(head(a__head(cons(cons(incr(a__head(a__head(head(a__nats)))), _x62), _x52))))
mark#(incr(head(head(cons(cons(incr(head(head(head(nats)))), _x62), _x52))))) → a__incr#(head(a__head(cons(cons(incr(a__head(a__head(a__head(nats)))), _x62), _x52))))mark#(incr(head(head(cons(cons(incr(head(head(head(nats)))), _x62), _x31))))) → a__incr#(head(mark(cons(incr(a__head(a__head(a__head(a__nats)))), _x62))))
mark#(incr(head(head(cons(cons(incr(head(head(head(nats)))), _x62), _x52))))) → a__incr#(head(a__head(cons(cons(incr(head(a__head(a__head(a__nats)))), _x62), _x52))))