0 QTRS
↳1 QTRSRRRProof (⇔)
↳2 QTRS
↳3 QTRSRRRProof (⇔)
↳4 QTRS
↳5 QTRSRRRProof (⇔)
↳6 QTRS
↳7 QTRSRRRProof (⇔)
↳8 QTRS
↳9 QTRSRRRProof (⇔)
↳10 QTRS
↳11 QTRSRRRProof (⇔)
↳12 QTRS
↳13 Overlay + Local Confluence (⇔)
↳14 QTRS
↳15 DependencyPairsProof (⇔)
↳16 QDP
↳17 DependencyGraphProof (⇔)
↳18 AND
↳19 QDP
↳20 UsableRulesProof (⇔)
↳21 QDP
↳22 QReductionProof (⇔)
↳23 QDP
↳24 QDPSizeChangeProof (⇔)
↳25 TRUE
↳26 QDP
↳27 UsableRulesProof (⇔)
↳28 QDP
↳29 QReductionProof (⇔)
↳30 QDP
↳31 MRRProof (⇔)
↳32 QDP
↳33 Narrowing (⇔)
↳34 QDP
↳35 UsableRulesProof (⇔)
↳36 QDP
↳37 QReductionProof (⇔)
↳38 QDP
↳39 Rewriting (⇔)
↳40 QDP
↳41 UsableRulesProof (⇔)
↳42 QDP
↳43 QReductionProof (⇔)
↳44 QDP
↳45 Instantiation (⇔)
↳46 QDP
↳47 Instantiation (⇔)
↳48 QDP
↳49 NonTerminationProof (⇔)
↳50 FALSE
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__length(nil) → 0
a__length(cons(N, L)) → a__U11(tt, L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(nil) → nil
a__zeros → zeros
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
POL(0) = 0
POL(U11(x1, x2)) = x1 + x2
POL(U12(x1, x2)) = x1 + x2
POL(a__U11(x1, x2)) = x1 + x2
POL(a__U12(x1, x2)) = x1 + x2
POL(a__length(x1)) = x1
POL(a__zeros) = 1
POL(cons(x1, x2)) = 1 + x1 + x2
POL(length(x1)) = x1
POL(mark(x1)) = 1 + x1
POL(nil) = 1
POL(s(x1)) = x1
POL(tt) = 1
POL(zeros) = 0
a__length(nil) → 0
mark(0) → 0
mark(tt) → tt
mark(nil) → nil
a__zeros → zeros
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__U11(tt, L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(length(X)) → a__length(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
POL(0) = 0
POL(U11(x1, x2)) = x1 + 2·x2
POL(U12(x1, x2)) = x1 + 2·x2
POL(a__U11(x1, x2)) = x1 + 2·x2
POL(a__U12(x1, x2)) = x1 + 2·x2
POL(a__length(x1)) = 1 + x1
POL(a__zeros) = 0
POL(cons(x1, x2)) = x1 + 2·x2
POL(length(x1)) = 1 + x1
POL(mark(x1)) = 2·x1
POL(s(x1)) = x1
POL(tt) = 1
POL(zeros) = 0
mark(length(X)) → a__length(mark(X))
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__U11(tt, L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__length(X) → length(X)
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
POL(0) = 0
POL(U11(x1, x2)) = x1 + x2
POL(U12(x1, x2)) = x1 + x2
POL(a__U11(x1, x2)) = x1 + x2
POL(a__U12(x1, x2)) = x1 + x2
POL(a__length(x1)) = 1 + x1
POL(a__zeros) = 0
POL(cons(x1, x2)) = x1 + x2
POL(length(x1)) = x1
POL(mark(x1)) = x1
POL(s(x1)) = x1
POL(tt) = 1
POL(zeros) = 0
a__length(X) → length(X)
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__U11(tt, L)
mark(zeros) → a__zeros
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
POL(0) = 0
POL(U11(x1, x2)) = 1 + 2·x1 + 2·x2
POL(U12(x1, x2)) = 1 + 2·x1 + 2·x2
POL(a__U11(x1, x2)) = 1 + 2·x1 + 2·x2
POL(a__U12(x1, x2)) = 1 + 2·x1 + 2·x2
POL(a__length(x1)) = 1 + x1
POL(a__zeros) = 0
POL(cons(x1, x2)) = 2·x1 + 2·x2
POL(mark(x1)) = 2·x1
POL(s(x1)) = x1
POL(tt) = 0
POL(zeros) = 0
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__U11(tt, L)
mark(zeros) → a__zeros
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
POL(0) = 0
POL(U11(x1, x2)) = x1 + x2
POL(U12(x1, x2)) = x1 + x2
POL(a__U11(x1, x2)) = 1 + x1 + x2
POL(a__U12(x1, x2)) = 1 + x1 + x2
POL(a__length(x1)) = x1
POL(a__zeros) = 1
POL(cons(x1, x2)) = 1 + x1 + x2
POL(mark(x1)) = 1 + x1
POL(s(x1)) = x1
POL(tt) = 0
POL(zeros) = 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X1, X2) → U12(X1, X2)
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__U11(tt, L)
mark(zeros) → a__zeros
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
POL(0) = 0
POL(a__U11(x1, x2)) = 1 + x1 + 2·x2
POL(a__U12(x1, x2)) = 1 + x1 + 2·x2
POL(a__length(x1)) = x1
POL(a__zeros) = 1
POL(cons(x1, x2)) = 1 + x1 + 2·x2
POL(mark(x1)) = 1 + 2·x1
POL(s(x1)) = x1
POL(tt) = 0
POL(zeros) = 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__U11(tt, L)
mark(zeros) → a__zeros
mark(s(X)) → s(mark(X))
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__U11(tt, L)
mark(zeros) → a__zeros
mark(s(X)) → s(mark(X))
a__zeros
a__U11(tt, x0)
a__U12(tt, x0)
a__length(cons(x0, x1))
mark(zeros)
mark(s(x0))
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, L) → A__LENGTH(mark(L))
A__U12(tt, L) → MARK(L)
A__LENGTH(cons(N, L)) → A__U11(tt, L)
MARK(zeros) → A__ZEROS
MARK(s(X)) → MARK(X)
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__U11(tt, L)
mark(zeros) → a__zeros
mark(s(X)) → s(mark(X))
a__zeros
a__U11(tt, x0)
a__U12(tt, x0)
a__length(cons(x0, x1))
mark(zeros)
mark(s(x0))
MARK(s(X)) → MARK(X)
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__U11(tt, L)
mark(zeros) → a__zeros
mark(s(X)) → s(mark(X))
a__zeros
a__U11(tt, x0)
a__U12(tt, x0)
a__length(cons(x0, x1))
mark(zeros)
mark(s(x0))
MARK(s(X)) → MARK(X)
a__zeros
a__U11(tt, x0)
a__U12(tt, x0)
a__length(cons(x0, x1))
mark(zeros)
mark(s(x0))
a__zeros
a__U11(tt, x0)
a__U12(tt, x0)
a__length(cons(x0, x1))
mark(zeros)
mark(s(x0))
MARK(s(X)) → MARK(X)
From the DPs we obtained the following set of size-change graphs:
A__U12(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
a__zeros → cons(0, zeros)
a__U11(tt, L) → a__U12(tt, L)
a__U12(tt, L) → s(a__length(mark(L)))
a__length(cons(N, L)) → a__U11(tt, L)
mark(zeros) → a__zeros
mark(s(X)) → s(mark(X))
a__zeros
a__U11(tt, x0)
a__U12(tt, x0)
a__length(cons(x0, x1))
mark(zeros)
mark(s(x0))
A__U12(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
mark(zeros) → a__zeros
mark(s(X)) → s(mark(X))
a__zeros → cons(0, zeros)
a__zeros
a__U11(tt, x0)
a__U12(tt, x0)
a__length(cons(x0, x1))
mark(zeros)
mark(s(x0))
a__U11(tt, x0)
a__U12(tt, x0)
a__length(cons(x0, x1))
A__U12(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
mark(zeros) → a__zeros
mark(s(X)) → s(mark(X))
a__zeros → cons(0, zeros)
a__zeros
mark(zeros)
mark(s(x0))
mark(s(X)) → s(mark(X))
POL(0) = 0
POL(A__LENGTH(x1)) = x1
POL(A__U11(x1, x2)) = 2·x1 + 2·x2
POL(A__U12(x1, x2)) = 2·x1 + 2·x2
POL(a__zeros) = 0
POL(cons(x1, x2)) = 2·x1 + 2·x2
POL(mark(x1)) = 2·x1
POL(s(x1)) = 2 + x1
POL(tt) = 0
POL(zeros) = 0
A__U12(tt, L) → A__LENGTH(mark(L))
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
mark(zeros) → a__zeros
a__zeros → cons(0, zeros)
a__zeros
mark(zeros)
mark(s(x0))
A__U12(tt, zeros) → A__LENGTH(a__zeros)
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, zeros) → A__LENGTH(a__zeros)
mark(zeros) → a__zeros
a__zeros → cons(0, zeros)
a__zeros
mark(zeros)
mark(s(x0))
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, zeros) → A__LENGTH(a__zeros)
a__zeros → cons(0, zeros)
a__zeros
mark(zeros)
mark(s(x0))
mark(zeros)
mark(s(x0))
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, zeros) → A__LENGTH(a__zeros)
a__zeros → cons(0, zeros)
a__zeros
A__U12(tt, zeros) → A__LENGTH(cons(0, zeros))
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, zeros) → A__LENGTH(cons(0, zeros))
a__zeros → cons(0, zeros)
a__zeros
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, zeros) → A__LENGTH(cons(0, zeros))
a__zeros
a__zeros
A__LENGTH(cons(N, L)) → A__U11(tt, L)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, zeros) → A__LENGTH(cons(0, zeros))
A__LENGTH(cons(0, zeros)) → A__U11(tt, zeros)
A__U11(tt, L) → A__U12(tt, L)
A__U12(tt, zeros) → A__LENGTH(cons(0, zeros))
A__LENGTH(cons(0, zeros)) → A__U11(tt, zeros)
A__U11(tt, zeros) → A__U12(tt, zeros)
A__U12(tt, zeros) → A__LENGTH(cons(0, zeros))
A__LENGTH(cons(0, zeros)) → A__U11(tt, zeros)
A__U11(tt, zeros) → A__U12(tt, zeros)