MAYBE
Termination proof of ../tpdb/TRS/CSR/Ex4_DLMMU04.trs
Termination of the following Term Rewriting System could not be shown:
Q restricted rewrite system:
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(IL)
isNat(0) → tt
isNat(s(N)) → isNat(N)
isNat(length(L)) → isNatList(L)
isNatIList(zeros) → tt
isNatIList(cons(N, IL)) → and(isNat(N), isNatIList(IL))
isNatList(nil) → tt
isNatList(cons(N, L)) → and(isNat(N), isNatList(L))
isNatList(take(N, IL)) → and(isNat(N), isNatIList(IL))
zeros → cons(0, zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
uTake2(tt, M, N, IL) → cons(N, take(M, IL))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(L)), L)
uLength(tt, L) → s(length(L))
The replacement map contains the following entries:and: {1, 2}
tt: empty set
isNatIList: empty set
isNatList: empty set
isNat: empty set
0: empty set
s: {1}
length: {1}
zeros: empty set
cons: {1}
nil: empty set
take: {1, 2}
uTake1: {1}
uTake2: {1}
uLength: {1}
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q restricted rewrite system:
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(IL)
isNat(0) → tt
isNat(s(N)) → isNat(N)
isNat(length(L)) → isNatList(L)
isNatIList(zeros) → tt
isNatIList(cons(N, IL)) → and(isNat(N), isNatIList(IL))
isNatList(nil) → tt
isNatList(cons(N, L)) → and(isNat(N), isNatList(L))
isNatList(take(N, IL)) → and(isNat(N), isNatIList(IL))
zeros → cons(0, zeros)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
uTake2(tt, M, N, IL) → cons(N, take(M, IL))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(L)), L)
uLength(tt, L) → s(length(L))
The replacement map contains the following entries:and: {1, 2}
tt: empty set
isNatIList: empty set
isNatList: empty set
isNat: empty set
0: empty set
s: {1}
length: {1}
zeros: empty set
cons: {1}
nil: empty set
take: {1, 2}
uTake1: {1}
uTake2: {1}
uLength: {1}
We applied the Zantema [34] to transform the context-sensitive TRS to an usual TRS.
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q restricted rewrite system:
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(x1, x2)
Q is empty.
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(N, L)) → AND(isNat(a(N)), isNatList(a(L)))
A(takeInact(x1, x2)) → TAKE(x1, x2)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
A(zerosInact) → ZEROS
ISNATLIST(takeInact(N, IL)) → ISNATILIST(a(IL))
LENGTH(cons(N, L)) → ISNATLIST(a(L))
ULENGTH(tt, L) → S(length(a(L)))
TAKE(s(M), cons(N, IL)) → AND(isNat(N), isNatIList(a(IL)))
ISNATILIST(consInact(N, IL)) → ISNAT(a(N))
UTAKE2(tt, M, N, IL) → A(N)
A(lengthInact(x1)) → LENGTH(x1)
ULENGTH(tt, L) → LENGTH(a(L))
ISNATILIST(consInact(N, IL)) → ISNATILIST(a(IL))
UTAKE1(tt) → NIL
TAKE(s(M), cons(N, IL)) → ISNATILIST(a(IL))
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(a(L))), a(L))
ISNATILIST(consInact(N, IL)) → A(IL)
ISNAT(sInact(N)) → A(N)
A(sInact(x1)) → S(x1)
ISNATLIST(consInact(N, L)) → A(L)
ISNATLIST(consInact(N, L)) → ISNATLIST(a(L))
ISNAT(lengthInact(L)) → ISNATLIST(a(L))
ISNAT(lengthInact(L)) → A(L)
TAKE(s(M), cons(N, IL)) → ISNAT(N)
ISNATLIST(takeInact(N, IL)) → AND(isNat(a(N)), isNatIList(a(IL)))
ISNATLIST(takeInact(N, IL)) → ISNAT(a(N))
ZEROS → 01
ZEROS → CONS(0, zerosInact)
UTAKE2(tt, M, N, IL) → A(IL)
ISNATLIST(takeInact(N, IL)) → A(IL)
TAKE(0, IL) → ISNATILIST(IL)
ULENGTH(tt, L) → A(L)
ISNATLIST(takeInact(N, IL)) → A(N)
ISNATILIST(consInact(N, IL)) → AND(isNat(a(N)), isNatIList(a(IL)))
UTAKE2(tt, M, N, IL) → CONS(a(N), takeInact(a(M), a(IL)))
ISNATILIST(IL) → A(IL)
A(nilInact) → NIL
LENGTH(cons(N, L)) → ISNAT(N)
LENGTH(cons(N, L)) → AND(isNat(N), isNatList(a(L)))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNAT(sInact(N)) → ISNAT(a(N))
ISNATLIST(consInact(N, L)) → A(N)
TAKE(s(M), cons(N, IL)) → AND(isNat(M), and(isNat(N), isNatIList(a(IL))))
ISNATILIST(consInact(N, IL)) → A(N)
UTAKE2(tt, M, N, IL) → A(M)
TAKE(s(M), cons(N, IL)) → A(IL)
A(0Inact) → 01
ISNATLIST(consInact(N, L)) → ISNAT(a(N))
LENGTH(cons(N, L)) → A(L)
ISNATILIST(IL) → ISNATLIST(a(IL))
A(consInact(x1, x2)) → CONS(x1, x2)
TAKE(0, IL) → UTAKE1(isNatIList(IL))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(x1, x2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(N, L)) → AND(isNat(a(N)), isNatList(a(L)))
A(takeInact(x1, x2)) → TAKE(x1, x2)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
A(zerosInact) → ZEROS
ISNATLIST(takeInact(N, IL)) → ISNATILIST(a(IL))
LENGTH(cons(N, L)) → ISNATLIST(a(L))
ULENGTH(tt, L) → S(length(a(L)))
TAKE(s(M), cons(N, IL)) → AND(isNat(N), isNatIList(a(IL)))
ISNATILIST(consInact(N, IL)) → ISNAT(a(N))
UTAKE2(tt, M, N, IL) → A(N)
A(lengthInact(x1)) → LENGTH(x1)
ULENGTH(tt, L) → LENGTH(a(L))
ISNATILIST(consInact(N, IL)) → ISNATILIST(a(IL))
UTAKE1(tt) → NIL
TAKE(s(M), cons(N, IL)) → ISNATILIST(a(IL))
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(a(L))), a(L))
ISNATILIST(consInact(N, IL)) → A(IL)
ISNAT(sInact(N)) → A(N)
A(sInact(x1)) → S(x1)
ISNATLIST(consInact(N, L)) → A(L)
ISNATLIST(consInact(N, L)) → ISNATLIST(a(L))
ISNAT(lengthInact(L)) → ISNATLIST(a(L))
ISNAT(lengthInact(L)) → A(L)
TAKE(s(M), cons(N, IL)) → ISNAT(N)
ISNATLIST(takeInact(N, IL)) → AND(isNat(a(N)), isNatIList(a(IL)))
ISNATLIST(takeInact(N, IL)) → ISNAT(a(N))
ZEROS → 01
ZEROS → CONS(0, zerosInact)
UTAKE2(tt, M, N, IL) → A(IL)
ISNATLIST(takeInact(N, IL)) → A(IL)
TAKE(0, IL) → ISNATILIST(IL)
ULENGTH(tt, L) → A(L)
ISNATLIST(takeInact(N, IL)) → A(N)
ISNATILIST(consInact(N, IL)) → AND(isNat(a(N)), isNatIList(a(IL)))
UTAKE2(tt, M, N, IL) → CONS(a(N), takeInact(a(M), a(IL)))
ISNATILIST(IL) → A(IL)
A(nilInact) → NIL
LENGTH(cons(N, L)) → ISNAT(N)
LENGTH(cons(N, L)) → AND(isNat(N), isNatList(a(L)))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNAT(sInact(N)) → ISNAT(a(N))
ISNATLIST(consInact(N, L)) → A(N)
TAKE(s(M), cons(N, IL)) → AND(isNat(M), and(isNat(N), isNatIList(a(IL))))
ISNATILIST(consInact(N, IL)) → A(N)
UTAKE2(tt, M, N, IL) → A(M)
TAKE(s(M), cons(N, IL)) → A(IL)
A(0Inact) → 01
ISNATLIST(consInact(N, L)) → ISNAT(a(N))
LENGTH(cons(N, L)) → A(L)
ISNATILIST(IL) → ISNATLIST(a(IL))
A(consInact(x1, x2)) → CONS(x1, x2)
TAKE(0, IL) → UTAKE1(isNatIList(IL))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(x1, x2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 17 less nodes.
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
LENGTH(cons(N, L)) → ISNAT(N)
TAKE(s(M), cons(N, IL)) → ISNAT(M)
A(takeInact(x1, x2)) → TAKE(x1, x2)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
LENGTH(cons(N, L)) → ISNATLIST(a(L))
ISNAT(sInact(N)) → ISNAT(a(N))
ISNATLIST(takeInact(N, IL)) → ISNATILIST(a(IL))
UTAKE2(tt, M, N, IL) → A(N)
ISNATILIST(consInact(N, IL)) → ISNAT(a(N))
ISNATLIST(consInact(N, L)) → A(N)
A(lengthInact(x1)) → LENGTH(x1)
ULENGTH(tt, L) → LENGTH(a(L))
ISNATILIST(consInact(N, IL)) → A(N)
UTAKE2(tt, M, N, IL) → A(M)
ISNATILIST(consInact(N, IL)) → ISNATILIST(a(IL))
TAKE(s(M), cons(N, IL)) → ISNATILIST(a(IL))
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(a(L))), a(L))
TAKE(s(M), cons(N, IL)) → A(IL)
ISNATLIST(consInact(N, L)) → ISNAT(a(N))
LENGTH(cons(N, L)) → A(L)
ISNATILIST(consInact(N, IL)) → A(IL)
ISNAT(sInact(N)) → A(N)
ISNATLIST(consInact(N, L)) → A(L)
ISNATLIST(consInact(N, L)) → ISNATLIST(a(L))
ISNAT(lengthInact(L)) → ISNATLIST(a(L))
ISNAT(lengthInact(L)) → A(L)
TAKE(s(M), cons(N, IL)) → ISNAT(N)
ISNATLIST(takeInact(N, IL)) → ISNAT(a(N))
ISNATILIST(IL) → ISNATLIST(a(IL))
UTAKE2(tt, M, N, IL) → A(IL)
ISNATLIST(takeInact(N, IL)) → A(IL)
ULENGTH(tt, L) → A(L)
TAKE(0, IL) → ISNATILIST(IL)
ISNATLIST(takeInact(N, IL)) → A(N)
ISNATILIST(IL) → A(IL)
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(x1, x2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
A(takeInact(x1, x2)) → TAKE(x1, x2)
ISNATLIST(takeInact(N, IL)) → ISNATILIST(a(IL))
ISNATLIST(takeInact(N, IL)) → ISNAT(a(N))
ISNATLIST(takeInact(N, IL)) → A(IL)
ISNATLIST(takeInact(N, IL)) → A(N)
The remaining pairs can at least be oriented weakly.
LENGTH(cons(N, L)) → ISNAT(N)
TAKE(s(M), cons(N, IL)) → ISNAT(M)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
LENGTH(cons(N, L)) → ISNATLIST(a(L))
ISNAT(sInact(N)) → ISNAT(a(N))
UTAKE2(tt, M, N, IL) → A(N)
ISNATILIST(consInact(N, IL)) → ISNAT(a(N))
ISNATLIST(consInact(N, L)) → A(N)
A(lengthInact(x1)) → LENGTH(x1)
ULENGTH(tt, L) → LENGTH(a(L))
ISNATILIST(consInact(N, IL)) → A(N)
UTAKE2(tt, M, N, IL) → A(M)
ISNATILIST(consInact(N, IL)) → ISNATILIST(a(IL))
TAKE(s(M), cons(N, IL)) → ISNATILIST(a(IL))
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(a(L))), a(L))
TAKE(s(M), cons(N, IL)) → A(IL)
ISNATLIST(consInact(N, L)) → ISNAT(a(N))
LENGTH(cons(N, L)) → A(L)
ISNATILIST(consInact(N, IL)) → A(IL)
ISNAT(sInact(N)) → A(N)
ISNATLIST(consInact(N, L)) → A(L)
ISNATLIST(consInact(N, L)) → ISNATLIST(a(L))
ISNAT(lengthInact(L)) → ISNATLIST(a(L))
ISNAT(lengthInact(L)) → A(L)
TAKE(s(M), cons(N, IL)) → ISNAT(N)
ISNATILIST(IL) → ISNATLIST(a(IL))
UTAKE2(tt, M, N, IL) → A(IL)
ULENGTH(tt, L) → A(L)
TAKE(0, IL) → ISNATILIST(IL)
ISNATILIST(IL) → A(IL)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(0Inact) = 0
POL(A(x1)) = x1
POL(ISNAT(x1)) = x1
POL(ISNATILIST(x1)) = x1
POL(ISNATLIST(x1)) = x1
POL(LENGTH(x1)) = x1
POL(TAKE(x1, x2)) = x1 + x2
POL(ULENGTH(x1, x2)) = x2
POL(UTAKE2(x1, x2, x3, x4)) = x2 + x3 + x4
POL(a(x1)) = x1
POL(and(x1, x2)) = x2
POL(cons(x1, x2)) = x1 + x2
POL(consInact(x1, x2)) = x1 + x2
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = x1
POL(lengthInact(x1)) = x1
POL(nil) = 1
POL(nilInact) = 1
POL(s(x1)) = x1
POL(sInact(x1)) = x1
POL(take(x1, x2)) = 1 + x1 + x2
POL(takeInact(x1, x2)) = 1 + x1 + x2
POL(tt) = 0
POL(uLength(x1, x2)) = x2
POL(uTake1(x1)) = 1
POL(uTake2(x1, x2, x3, x4)) = 1 + x2 + x3 + x4
POL(zeros) = 0
POL(zerosInact) = 0
The following usable rules [17] were oriented:
uTake1(tt) → nil
a(takeInact(x1, x2)) → take(x1, x2)
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNat(lengthInact(L)) → isNatList(a(L))
and(tt, T) → T
cons(x1, x2) → consInact(x1, x2)
a(x) → x
zeros → zerosInact
a(sInact(x1)) → s(x1)
zeros → cons(0, zerosInact)
isNatIList(IL) → isNatList(a(IL))
a(consInact(x1, x2)) → cons(x1, x2)
isNat(0Inact) → tt
s(x1) → sInact(x1)
a(lengthInact(x1)) → length(x1)
take(0, IL) → uTake1(isNatIList(IL))
take(x1, x2) → takeInact(x1, x2)
length(x1) → lengthInact(x1)
uLength(tt, L) → s(length(a(L)))
isNatIList(zerosInact) → tt
a(nilInact) → nil
0 → 0Inact
a(zerosInact) → zeros
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNat(sInact(N)) → isNat(a(N))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
isNatList(nilInact) → tt
nil → nilInact
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
a(0Inact) → 0
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
LENGTH(cons(N, L)) → ISNAT(N)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNAT(sInact(N)) → ISNAT(a(N))
LENGTH(cons(N, L)) → ISNATLIST(a(L))
ISNATILIST(consInact(N, IL)) → ISNAT(a(N))
UTAKE2(tt, M, N, IL) → A(N)
A(lengthInact(x1)) → LENGTH(x1)
ISNATLIST(consInact(N, L)) → A(N)
ULENGTH(tt, L) → LENGTH(a(L))
ISNATILIST(consInact(N, IL)) → A(N)
UTAKE2(tt, M, N, IL) → A(M)
ISNATILIST(consInact(N, IL)) → ISNATILIST(a(IL))
TAKE(s(M), cons(N, IL)) → ISNATILIST(a(IL))
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(a(L))), a(L))
TAKE(s(M), cons(N, IL)) → A(IL)
ISNATLIST(consInact(N, L)) → ISNAT(a(N))
LENGTH(cons(N, L)) → A(L)
ISNATILIST(consInact(N, IL)) → A(IL)
ISNAT(sInact(N)) → A(N)
ISNATLIST(consInact(N, L)) → A(L)
ISNATLIST(consInact(N, L)) → ISNATLIST(a(L))
ISNAT(lengthInact(L)) → ISNATLIST(a(L))
ISNAT(lengthInact(L)) → A(L)
TAKE(s(M), cons(N, IL)) → ISNAT(N)
ISNATILIST(IL) → ISNATLIST(a(IL))
UTAKE2(tt, M, N, IL) → A(IL)
TAKE(0, IL) → ISNATILIST(IL)
ULENGTH(tt, L) → A(L)
ISNATILIST(IL) → A(IL)
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(x1, x2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 14 less nodes.
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(N, L)) → ISNAT(a(N))
LENGTH(cons(N, L)) → ISNAT(N)
LENGTH(cons(N, L)) → A(L)
ISNAT(sInact(N)) → A(N)
ISNATLIST(consInact(N, L)) → A(L)
ISNAT(lengthInact(L)) → ISNATLIST(a(L))
ISNAT(lengthInact(L)) → A(L)
ISNATLIST(consInact(N, L)) → ISNATLIST(a(L))
ISNAT(sInact(N)) → ISNAT(a(N))
LENGTH(cons(N, L)) → ISNATLIST(a(L))
A(lengthInact(x1)) → LENGTH(x1)
ISNATLIST(consInact(N, L)) → A(N)
ULENGTH(tt, L) → LENGTH(a(L))
ULENGTH(tt, L) → A(L)
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(a(L))), a(L))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(x1, x2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ISNAT(lengthInact(L)) → ISNATLIST(a(L))
ISNAT(lengthInact(L)) → A(L)
A(lengthInact(x1)) → LENGTH(x1)
The remaining pairs can at least be oriented weakly.
ISNATLIST(consInact(N, L)) → ISNAT(a(N))
LENGTH(cons(N, L)) → ISNAT(N)
LENGTH(cons(N, L)) → A(L)
ISNAT(sInact(N)) → A(N)
ISNATLIST(consInact(N, L)) → A(L)
ISNATLIST(consInact(N, L)) → ISNATLIST(a(L))
ISNAT(sInact(N)) → ISNAT(a(N))
LENGTH(cons(N, L)) → ISNATLIST(a(L))
ISNATLIST(consInact(N, L)) → A(N)
ULENGTH(tt, L) → LENGTH(a(L))
ULENGTH(tt, L) → A(L)
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(a(L))), a(L))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(0Inact) = 0
POL(A(x1)) = x1
POL(ISNAT(x1)) = x1
POL(ISNATLIST(x1)) = x1
POL(LENGTH(x1)) = x1
POL(ULENGTH(x1, x2)) = x2
POL(a(x1)) = x1
POL(and(x1, x2)) = x2
POL(cons(x1, x2)) = x1 + x2
POL(consInact(x1, x2)) = x1 + x2
POL(isNat(x1)) = x1
POL(isNatIList(x1)) = 1
POL(isNatList(x1)) = 1
POL(length(x1)) = 1 + x1
POL(lengthInact(x1)) = 1 + x1
POL(nil) = 0
POL(nilInact) = 0
POL(s(x1)) = x1
POL(sInact(x1)) = x1
POL(take(x1, x2)) = 1 + x2
POL(takeInact(x1, x2)) = 1 + x2
POL(tt) = 0
POL(uLength(x1, x2)) = 1 + x2
POL(uTake1(x1)) = 1
POL(uTake2(x1, x2, x3, x4)) = 1 + x3 + x4
POL(zeros) = 0
POL(zerosInact) = 0
The following usable rules [17] were oriented:
a(takeInact(x1, x2)) → take(x1, x2)
uTake1(tt) → nil
isNat(lengthInact(L)) → isNatList(a(L))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
and(tt, T) → T
cons(x1, x2) → consInact(x1, x2)
a(x) → x
zeros → zerosInact
a(sInact(x1)) → s(x1)
zeros → cons(0, zerosInact)
a(consInact(x1, x2)) → cons(x1, x2)
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
s(x1) → sInact(x1)
a(lengthInact(x1)) → length(x1)
take(0, IL) → uTake1(isNatIList(IL))
take(x1, x2) → takeInact(x1, x2)
length(x1) → lengthInact(x1)
uLength(tt, L) → s(length(a(L)))
a(nilInact) → nil
isNatIList(zerosInact) → tt
0 → 0Inact
a(zerosInact) → zeros
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
isNat(sInact(N)) → isNat(a(N))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
isNatList(nilInact) → tt
nil → nilInact
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
a(0Inact) → 0
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(N, L)) → ISNAT(a(N))
LENGTH(cons(N, L)) → ISNAT(N)
LENGTH(cons(N, L)) → A(L)
ISNATLIST(consInact(N, L)) → A(N)
ISNAT(sInact(N)) → A(N)
ULENGTH(tt, L) → LENGTH(a(L))
ISNATLIST(consInact(N, L)) → A(L)
ISNATLIST(consInact(N, L)) → ISNATLIST(a(L))
ULENGTH(tt, L) → A(L)
LENGTH(cons(N, L)) → ISNATLIST(a(L))
ISNAT(sInact(N)) → ISNAT(a(N))
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(a(L))), a(L))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(x1, x2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 8 less nodes.
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNAT(sInact(N)) → ISNAT(a(N))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(x1, x2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ISNAT(sInact(N)) → ISNAT(a(N))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
| M( consInact(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( isNatList(x1) ) = | | + | | · | x1 |
| M( cons(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( uLength(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( and(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( takeInact(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( uTake2(x1, ..., x4) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| M( isNatIList(x1) ) = | | + | | · | x1 |
| M( lengthInact(x1) ) = | | + | | · | x1 |
| M( take(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
Tuple symbols:
Matrix type:
We used a basic matrix type which is not further parametrizeable.
As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:
a(takeInact(x1, x2)) → take(x1, x2)
uTake1(tt) → nil
isNat(lengthInact(L)) → isNatList(a(L))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
and(tt, T) → T
cons(x1, x2) → consInact(x1, x2)
a(x) → x
zeros → zerosInact
a(sInact(x1)) → s(x1)
zeros → cons(0, zerosInact)
a(consInact(x1, x2)) → cons(x1, x2)
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
s(x1) → sInact(x1)
a(lengthInact(x1)) → length(x1)
take(0, IL) → uTake1(isNatIList(IL))
take(x1, x2) → takeInact(x1, x2)
length(x1) → lengthInact(x1)
uLength(tt, L) → s(length(a(L)))
a(nilInact) → nil
isNatIList(zerosInact) → tt
0 → 0Inact
a(zerosInact) → zeros
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
isNat(sInact(N)) → isNat(a(N))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
isNatList(nilInact) → tt
nil → nilInact
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
a(0Inact) → 0
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
P is empty.
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(x1, x2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(N, L)) → ISNATLIST(a(L))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(x1, x2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATLIST(consInact(N, L)) → ISNATLIST(a(L)) at position [0] we obtained the following new rules:
ISNATLIST(consInact(y0, takeInact(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(consInact(y0, sInact(x0))) → ISNATLIST(s(x0))
ISNATLIST(consInact(y0, 0Inact)) → ISNATLIST(0)
ISNATLIST(consInact(y0, lengthInact(x0))) → ISNATLIST(length(x0))
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(zeros)
ISNATLIST(consInact(y0, x0)) → ISNATLIST(x0)
ISNATLIST(consInact(y0, consInact(x0, x1))) → ISNATLIST(cons(x0, x1))
ISNATLIST(consInact(y0, nilInact)) → ISNATLIST(nil)
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, takeInact(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(consInact(y0, 0Inact)) → ISNATLIST(0)
ISNATLIST(consInact(y0, sInact(x0))) → ISNATLIST(s(x0))
ISNATLIST(consInact(y0, lengthInact(x0))) → ISNATLIST(length(x0))
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(zeros)
ISNATLIST(consInact(y0, consInact(x0, x1))) → ISNATLIST(cons(x0, x1))
ISNATLIST(consInact(y0, x0)) → ISNATLIST(x0)
ISNATLIST(consInact(y0, nilInact)) → ISNATLIST(nil)
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(x1, x2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATLIST(consInact(y0, sInact(x0))) → ISNATLIST(s(x0)) at position [0] we obtained the following new rules:
ISNATLIST(consInact(y0, sInact(x0))) → ISNATLIST(sInact(x0))
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, sInact(x0))) → ISNATLIST(sInact(x0))
ISNATLIST(consInact(y0, takeInact(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(consInact(y0, 0Inact)) → ISNATLIST(0)
ISNATLIST(consInact(y0, lengthInact(x0))) → ISNATLIST(length(x0))
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(zeros)
ISNATLIST(consInact(y0, x0)) → ISNATLIST(x0)
ISNATLIST(consInact(y0, consInact(x0, x1))) → ISNATLIST(cons(x0, x1))
ISNATLIST(consInact(y0, nilInact)) → ISNATLIST(nil)
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(x1, x2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, takeInact(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(consInact(y0, 0Inact)) → ISNATLIST(0)
ISNATLIST(consInact(y0, lengthInact(x0))) → ISNATLIST(length(x0))
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(zeros)
ISNATLIST(consInact(y0, consInact(x0, x1))) → ISNATLIST(cons(x0, x1))
ISNATLIST(consInact(y0, x0)) → ISNATLIST(x0)
ISNATLIST(consInact(y0, nilInact)) → ISNATLIST(nil)
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(x1, x2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATLIST(consInact(y0, 0Inact)) → ISNATLIST(0) at position [0] we obtained the following new rules:
ISNATLIST(consInact(y0, 0Inact)) → ISNATLIST(0Inact)
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, takeInact(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(consInact(y0, lengthInact(x0))) → ISNATLIST(length(x0))
ISNATLIST(consInact(y0, 0Inact)) → ISNATLIST(0Inact)
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(zeros)
ISNATLIST(consInact(y0, x0)) → ISNATLIST(x0)
ISNATLIST(consInact(y0, consInact(x0, x1))) → ISNATLIST(cons(x0, x1))
ISNATLIST(consInact(y0, nilInact)) → ISNATLIST(nil)
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(x1, x2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, takeInact(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(consInact(y0, lengthInact(x0))) → ISNATLIST(length(x0))
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(zeros)
ISNATLIST(consInact(y0, consInact(x0, x1))) → ISNATLIST(cons(x0, x1))
ISNATLIST(consInact(y0, x0)) → ISNATLIST(x0)
ISNATLIST(consInact(y0, nilInact)) → ISNATLIST(nil)
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(x1, x2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATLIST(consInact(y0, nilInact)) → ISNATLIST(nil) at position [0] we obtained the following new rules:
ISNATLIST(consInact(y0, nilInact)) → ISNATLIST(nilInact)
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, nilInact)) → ISNATLIST(nilInact)
ISNATLIST(consInact(y0, takeInact(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(consInact(y0, lengthInact(x0))) → ISNATLIST(length(x0))
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(zeros)
ISNATLIST(consInact(y0, x0)) → ISNATLIST(x0)
ISNATLIST(consInact(y0, consInact(x0, x1))) → ISNATLIST(cons(x0, x1))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(x1, x2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, takeInact(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(consInact(y0, lengthInact(x0))) → ISNATLIST(length(x0))
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(zeros)
ISNATLIST(consInact(y0, consInact(x0, x1))) → ISNATLIST(cons(x0, x1))
ISNATLIST(consInact(y0, x0)) → ISNATLIST(x0)
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(x1, x2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ISNATLIST(consInact(y0, lengthInact(x0))) → ISNATLIST(length(x0))
ISNATLIST(consInact(y0, consInact(x0, x1))) → ISNATLIST(cons(x0, x1))
ISNATLIST(consInact(y0, x0)) → ISNATLIST(x0)
The remaining pairs can at least be oriented weakly.
ISNATLIST(consInact(y0, takeInact(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(zeros)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(0Inact) = 0
POL(ISNATLIST(x1)) = x1
POL(a(x1)) = 1
POL(and(x1, x2)) = x2
POL(cons(x1, x2)) = 1 + x2
POL(consInact(x1, x2)) = 1 + x2
POL(isNat(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 0
POL(lengthInact(x1)) = 0
POL(nil) = 1
POL(nilInact) = 0
POL(s(x1)) = 0
POL(sInact(x1)) = 0
POL(take(x1, x2)) = 1
POL(takeInact(x1, x2)) = 0
POL(tt) = 0
POL(uLength(x1, x2)) = 0
POL(uTake1(x1)) = 1
POL(uTake2(x1, x2, x3, x4)) = 1
POL(zeros) = 1
POL(zerosInact) = 0
The following usable rules [17] were oriented:
a(takeInact(x1, x2)) → take(x1, x2)
uTake1(tt) → nil
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNat(lengthInact(L)) → isNatList(a(L))
and(tt, T) → T
cons(x1, x2) → consInact(x1, x2)
zeros → zerosInact
zeros → cons(0, zerosInact)
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
s(x1) → sInact(x1)
a(lengthInact(x1)) → length(x1)
take(0, IL) → uTake1(isNatIList(IL))
take(x1, x2) → takeInact(x1, x2)
length(x1) → lengthInact(x1)
uLength(tt, L) → s(length(a(L)))
isNatIList(zerosInact) → tt
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNat(sInact(N)) → isNat(a(N))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
isNatList(nilInact) → tt
nil → nilInact
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, takeInact(x0, x1))) → ISNATLIST(take(x0, x1))
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(zeros)
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(x1, x2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ISNATLIST(consInact(y0, takeInact(x0, x1))) → ISNATLIST(take(x0, x1))
The remaining pairs can at least be oriented weakly.
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(zeros)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
| M( consInact(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( isNatList(x1) ) = | | + | | · | x1 |
| M( cons(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( uLength(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( and(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( takeInact(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( uTake2(x1, ..., x4) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| M( isNatIList(x1) ) = | | + | | · | x1 |
| M( lengthInact(x1) ) = | | + | | · | x1 |
| M( take(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
Tuple symbols:
| M( ISNATLIST(x1) ) = | 0 | + | | · | x1 |
Matrix type:
We used a basic matrix type which is not further parametrizeable.
As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:
a(takeInact(x1, x2)) → take(x1, x2)
uTake1(tt) → nil
isNat(lengthInact(L)) → isNatList(a(L))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
and(tt, T) → T
cons(x1, x2) → consInact(x1, x2)
a(x) → x
zeros → zerosInact
a(sInact(x1)) → s(x1)
zeros → cons(0, zerosInact)
a(consInact(x1, x2)) → cons(x1, x2)
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
s(x1) → sInact(x1)
a(lengthInact(x1)) → length(x1)
take(0, IL) → uTake1(isNatIList(IL))
take(x1, x2) → takeInact(x1, x2)
length(x1) → lengthInact(x1)
uLength(tt, L) → s(length(a(L)))
a(nilInact) → nil
isNatIList(zerosInact) → tt
0 → 0Inact
a(zerosInact) → zeros
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNat(sInact(N)) → isNat(a(N))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
isNatList(nilInact) → tt
nil → nilInact
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
a(0Inact) → 0
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(zeros)
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(x1, x2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(zeros)
The TRS R consists of the following rules:
zeros → cons(0, zerosInact)
zeros → zerosInact
0 → 0Inact
cons(x1, x2) → consInact(x1, x2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented rules of the TRS R:
zeros → zerosInact
Used ordering: POLO with Polynomial interpretation [25]:
POL(0) = 0
POL(0Inact) = 0
POL(ISNATLIST(x1)) = 2·x1
POL(cons(x1, x2)) = 1 + 2·x1 + 2·x2
POL(consInact(x1, x2)) = 1 + 2·x1 + 2·x2
POL(zeros) = 1
POL(zerosInact) = 0
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(zeros)
The TRS R consists of the following rules:
zeros → cons(0, zerosInact)
0 → 0Inact
cons(x1, x2) → consInact(x1, x2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the modular non-overlap check [15] to enlarge Q to all left-hand sides of R.
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(zeros)
The TRS R consists of the following rules:
zeros → cons(0, zerosInact)
0 → 0Inact
cons(x1, x2) → consInact(x1, x2)
The set Q consists of the following terms:
zeros
0
cons(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(zeros) at position [0] we obtained the following new rules:
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(cons(0, zerosInact))
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(cons(0, zerosInact))
The TRS R consists of the following rules:
zeros → cons(0, zerosInact)
0 → 0Inact
cons(x1, x2) → consInact(x1, x2)
The set Q consists of the following terms:
zeros
0
cons(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(cons(0, zerosInact))
The TRS R consists of the following rules:
0 → 0Inact
cons(x1, x2) → consInact(x1, x2)
The set Q consists of the following terms:
zeros
0
cons(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
zeros
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(cons(0, zerosInact))
The TRS R consists of the following rules:
0 → 0Inact
cons(x1, x2) → consInact(x1, x2)
The set Q consists of the following terms:
0
cons(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(cons(0, zerosInact)) at position [0] we obtained the following new rules:
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(consInact(0, zerosInact))
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(consInact(0, zerosInact))
The TRS R consists of the following rules:
0 → 0Inact
cons(x1, x2) → consInact(x1, x2)
The set Q consists of the following terms:
0
cons(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(consInact(0, zerosInact))
The TRS R consists of the following rules:
0 → 0Inact
The set Q consists of the following terms:
0
cons(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
cons(x0, x1)
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(consInact(0, zerosInact))
The TRS R consists of the following rules:
0 → 0Inact
The set Q consists of the following terms:
0
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(consInact(0, zerosInact)) at position [0,0] we obtained the following new rules:
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(consInact(0Inact, zerosInact))
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(consInact(0Inact, zerosInact))
The TRS R consists of the following rules:
0 → 0Inact
The set Q consists of the following terms:
0
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(consInact(0Inact, zerosInact))
R is empty.
The set Q consists of the following terms:
0
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
0
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(consInact(0Inact, zerosInact))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(consInact(0Inact, zerosInact)) we obtained the following new rules:
ISNATLIST(consInact(0Inact, zerosInact)) → ISNATLIST(consInact(0Inact, zerosInact))
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ NonTerminationProof
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(0Inact, zerosInact)) → ISNATLIST(consInact(0Inact, zerosInact))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.
The TRS P consists of the following rules:
ISNATLIST(consInact(0Inact, zerosInact)) → ISNATLIST(consInact(0Inact, zerosInact))
The TRS R consists of the following rules:none
s = ISNATLIST(consInact(0Inact, zerosInact)) evaluates to t =ISNATLIST(consInact(0Inact, zerosInact))
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Semiunifier: [ ]
- Matcher: [ ]
Rewriting sequence
The DP semiunifies directly so there is only one rewrite step from ISNATLIST(consInact(0Inact, zerosInact)) to ISNATLIST(consInact(0Inact, zerosInact)).
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ULENGTH(tt, L) → LENGTH(a(L))
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(a(L))), a(L))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(x1, x2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ CSR
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(consInact(N, IL)) → ISNATILIST(a(IL))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(x1, x2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We applied the Incomplete Giesl Middeldorp [11] to transform the context-sensitive TRS to an usual TRS.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q restricted rewrite system:
The TRS R consists of the following rules:
mark(and(x1, x2)) → andActive(mark(x1), mark(x2))
andActive(x1, x2) → and(x1, x2)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(zeros) → zerosActive
zerosActive → zeros
mark(take(x1, x2)) → takeActive(mark(x1), mark(x2))
takeActive(x1, x2) → take(x1, x2)
mark(uTake1(x1)) → uTake1Active(mark(x1))
uTake1Active(x1) → uTake1(x1)
mark(uTake2(x1, x2, x3, x4)) → uTake2Active(mark(x1), x2, x3, x4)
uTake2Active(x1, x2, x3, x4) → uTake2(x1, x2, x3, x4)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(uLength(x1, x2)) → uLengthActive(mark(x1), x2)
uLengthActive(x1, x2) → uLength(x1, x2)
mark(tt) → tt
mark(0) → 0
mark(s(x1)) → s(mark(x1))
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(nil) → nil
andActive(tt, T) → mark(T)
isNatIListActive(IL) → isNatListActive(IL)
isNatActive(0) → tt
isNatActive(s(N)) → isNatActive(N)
isNatActive(length(L)) → isNatListActive(L)
isNatIListActive(zeros) → tt
isNatIListActive(cons(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
isNatListActive(nil) → tt
isNatListActive(cons(N, L)) → andActive(isNatActive(N), isNatListActive(L))
isNatListActive(take(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
zerosActive → cons(0, zeros)
takeActive(0, IL) → uTake1Active(isNatIListActive(IL))
uTake1Active(tt) → nil
takeActive(s(M), cons(N, IL)) → uTake2Active(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
uTake2Active(tt, M, N, IL) → cons(mark(N), take(M, IL))
lengthActive(cons(N, L)) → uLengthActive(andActive(isNatActive(N), isNatListActive(L)), L)
uLengthActive(tt, L) → s(lengthActive(mark(L)))
Q is empty.
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
TAKEACTIVE(s(M), cons(N, IL)) → UTAKE2ACTIVE(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
MARK(take(x1, x2)) → MARK(x2)
TAKEACTIVE(s(M), cons(N, IL)) → ANDACTIVE(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL)))
MARK(and(x1, x2)) → MARK(x2)
MARK(uTake2(x1, x2, x3, x4)) → UTAKE2ACTIVE(mark(x1), x2, x3, x4)
LENGTHACTIVE(cons(N, L)) → ULENGTHACTIVE(andActive(isNatActive(N), isNatListActive(L)), L)
ISNATILISTACTIVE(IL) → ISNATLISTACTIVE(IL)
TAKEACTIVE(s(M), cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ULENGTHACTIVE(tt, L) → LENGTHACTIVE(mark(L))
LENGTHACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
UTAKE2ACTIVE(tt, M, N, IL) → MARK(N)
TAKEACTIVE(s(M), cons(N, IL)) → ISNATILISTACTIVE(IL)
ISNATILISTACTIVE(cons(N, IL)) → ISNATACTIVE(N)
LENGTHACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
MARK(uTake1(x1)) → UTAKE1ACTIVE(mark(x1))
LENGTHACTIVE(cons(N, L)) → ISNATACTIVE(N)
MARK(uTake1(x1)) → MARK(x1)
MARK(isNatIList(x1)) → ISNATILISTACTIVE(x1)
ANDACTIVE(tt, T) → MARK(T)
ISNATACTIVE(s(N)) → ISNATACTIVE(N)
MARK(take(x1, x2)) → TAKEACTIVE(mark(x1), mark(x2))
ISNATILISTACTIVE(cons(N, IL)) → ISNATILISTACTIVE(IL)
MARK(cons(x1, x2)) → MARK(x1)
TAKEACTIVE(s(M), cons(N, IL)) → ISNATACTIVE(M)
ISNATLISTACTIVE(take(N, IL)) → ISNATILISTACTIVE(IL)
MARK(uLength(x1, x2)) → ULENGTHACTIVE(mark(x1), x2)
TAKEACTIVE(0, IL) → UTAKE1ACTIVE(isNatIListActive(IL))
MARK(and(x1, x2)) → MARK(x1)
TAKEACTIVE(s(M), cons(N, IL)) → ISNATACTIVE(N)
ISNATLISTACTIVE(cons(N, L)) → ISNATACTIVE(N)
ISNATLISTACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
MARK(uTake2(x1, x2, x3, x4)) → MARK(x1)
ISNATLISTACTIVE(take(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
MARK(zeros) → ZEROSACTIVE
MARK(take(x1, x2)) → MARK(x1)
ISNATLISTACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
MARK(length(x1)) → MARK(x1)
ISNATILISTACTIVE(cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATLISTACTIVE(take(N, IL)) → ISNATACTIVE(N)
MARK(s(x1)) → MARK(x1)
ULENGTHACTIVE(tt, L) → MARK(L)
MARK(uLength(x1, x2)) → MARK(x1)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), mark(x2))
MARK(length(x1)) → LENGTHACTIVE(mark(x1))
TAKEACTIVE(0, IL) → ISNATILISTACTIVE(IL)
ISNATACTIVE(length(L)) → ISNATLISTACTIVE(L)
The TRS R consists of the following rules:
mark(and(x1, x2)) → andActive(mark(x1), mark(x2))
andActive(x1, x2) → and(x1, x2)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(zeros) → zerosActive
zerosActive → zeros
mark(take(x1, x2)) → takeActive(mark(x1), mark(x2))
takeActive(x1, x2) → take(x1, x2)
mark(uTake1(x1)) → uTake1Active(mark(x1))
uTake1Active(x1) → uTake1(x1)
mark(uTake2(x1, x2, x3, x4)) → uTake2Active(mark(x1), x2, x3, x4)
uTake2Active(x1, x2, x3, x4) → uTake2(x1, x2, x3, x4)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(uLength(x1, x2)) → uLengthActive(mark(x1), x2)
uLengthActive(x1, x2) → uLength(x1, x2)
mark(tt) → tt
mark(0) → 0
mark(s(x1)) → s(mark(x1))
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(nil) → nil
andActive(tt, T) → mark(T)
isNatIListActive(IL) → isNatListActive(IL)
isNatActive(0) → tt
isNatActive(s(N)) → isNatActive(N)
isNatActive(length(L)) → isNatListActive(L)
isNatIListActive(zeros) → tt
isNatIListActive(cons(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
isNatListActive(nil) → tt
isNatListActive(cons(N, L)) → andActive(isNatActive(N), isNatListActive(L))
isNatListActive(take(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
zerosActive → cons(0, zeros)
takeActive(0, IL) → uTake1Active(isNatIListActive(IL))
uTake1Active(tt) → nil
takeActive(s(M), cons(N, IL)) → uTake2Active(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
uTake2Active(tt, M, N, IL) → cons(mark(N), take(M, IL))
lengthActive(cons(N, L)) → uLengthActive(andActive(isNatActive(N), isNatListActive(L)), L)
uLengthActive(tt, L) → s(lengthActive(mark(L)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
TAKEACTIVE(s(M), cons(N, IL)) → UTAKE2ACTIVE(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
MARK(take(x1, x2)) → MARK(x2)
TAKEACTIVE(s(M), cons(N, IL)) → ANDACTIVE(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL)))
MARK(and(x1, x2)) → MARK(x2)
MARK(uTake2(x1, x2, x3, x4)) → UTAKE2ACTIVE(mark(x1), x2, x3, x4)
LENGTHACTIVE(cons(N, L)) → ULENGTHACTIVE(andActive(isNatActive(N), isNatListActive(L)), L)
ISNATILISTACTIVE(IL) → ISNATLISTACTIVE(IL)
TAKEACTIVE(s(M), cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ULENGTHACTIVE(tt, L) → LENGTHACTIVE(mark(L))
LENGTHACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
UTAKE2ACTIVE(tt, M, N, IL) → MARK(N)
TAKEACTIVE(s(M), cons(N, IL)) → ISNATILISTACTIVE(IL)
ISNATILISTACTIVE(cons(N, IL)) → ISNATACTIVE(N)
LENGTHACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
MARK(uTake1(x1)) → UTAKE1ACTIVE(mark(x1))
LENGTHACTIVE(cons(N, L)) → ISNATACTIVE(N)
MARK(uTake1(x1)) → MARK(x1)
MARK(isNatIList(x1)) → ISNATILISTACTIVE(x1)
ANDACTIVE(tt, T) → MARK(T)
ISNATACTIVE(s(N)) → ISNATACTIVE(N)
MARK(take(x1, x2)) → TAKEACTIVE(mark(x1), mark(x2))
ISNATILISTACTIVE(cons(N, IL)) → ISNATILISTACTIVE(IL)
MARK(cons(x1, x2)) → MARK(x1)
TAKEACTIVE(s(M), cons(N, IL)) → ISNATACTIVE(M)
ISNATLISTACTIVE(take(N, IL)) → ISNATILISTACTIVE(IL)
MARK(uLength(x1, x2)) → ULENGTHACTIVE(mark(x1), x2)
TAKEACTIVE(0, IL) → UTAKE1ACTIVE(isNatIListActive(IL))
MARK(and(x1, x2)) → MARK(x1)
TAKEACTIVE(s(M), cons(N, IL)) → ISNATACTIVE(N)
ISNATLISTACTIVE(cons(N, L)) → ISNATACTIVE(N)
ISNATLISTACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
MARK(uTake2(x1, x2, x3, x4)) → MARK(x1)
ISNATLISTACTIVE(take(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
MARK(zeros) → ZEROSACTIVE
MARK(take(x1, x2)) → MARK(x1)
ISNATLISTACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
MARK(length(x1)) → MARK(x1)
ISNATILISTACTIVE(cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATLISTACTIVE(take(N, IL)) → ISNATACTIVE(N)
MARK(s(x1)) → MARK(x1)
ULENGTHACTIVE(tt, L) → MARK(L)
MARK(uLength(x1, x2)) → MARK(x1)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), mark(x2))
MARK(length(x1)) → LENGTHACTIVE(mark(x1))
TAKEACTIVE(0, IL) → ISNATILISTACTIVE(IL)
ISNATACTIVE(length(L)) → ISNATLISTACTIVE(L)
The TRS R consists of the following rules:
mark(and(x1, x2)) → andActive(mark(x1), mark(x2))
andActive(x1, x2) → and(x1, x2)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(zeros) → zerosActive
zerosActive → zeros
mark(take(x1, x2)) → takeActive(mark(x1), mark(x2))
takeActive(x1, x2) → take(x1, x2)
mark(uTake1(x1)) → uTake1Active(mark(x1))
uTake1Active(x1) → uTake1(x1)
mark(uTake2(x1, x2, x3, x4)) → uTake2Active(mark(x1), x2, x3, x4)
uTake2Active(x1, x2, x3, x4) → uTake2(x1, x2, x3, x4)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(uLength(x1, x2)) → uLengthActive(mark(x1), x2)
uLengthActive(x1, x2) → uLength(x1, x2)
mark(tt) → tt
mark(0) → 0
mark(s(x1)) → s(mark(x1))
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(nil) → nil
andActive(tt, T) → mark(T)
isNatIListActive(IL) → isNatListActive(IL)
isNatActive(0) → tt
isNatActive(s(N)) → isNatActive(N)
isNatActive(length(L)) → isNatListActive(L)
isNatIListActive(zeros) → tt
isNatIListActive(cons(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
isNatListActive(nil) → tt
isNatListActive(cons(N, L)) → andActive(isNatActive(N), isNatListActive(L))
isNatListActive(take(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
zerosActive → cons(0, zeros)
takeActive(0, IL) → uTake1Active(isNatIListActive(IL))
uTake1Active(tt) → nil
takeActive(s(M), cons(N, IL)) → uTake2Active(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
uTake2Active(tt, M, N, IL) → cons(mark(N), take(M, IL))
lengthActive(cons(N, L)) → uLengthActive(andActive(isNatActive(N), isNatListActive(L)), L)
uLengthActive(tt, L) → s(lengthActive(mark(L)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
TAKEACTIVE(s(M), cons(N, IL)) → UTAKE2ACTIVE(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
MARK(take(x1, x2)) → MARK(x2)
TAKEACTIVE(s(M), cons(N, IL)) → ANDACTIVE(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL)))
MARK(and(x1, x2)) → MARK(x2)
MARK(uTake2(x1, x2, x3, x4)) → UTAKE2ACTIVE(mark(x1), x2, x3, x4)
LENGTHACTIVE(cons(N, L)) → ULENGTHACTIVE(andActive(isNatActive(N), isNatListActive(L)), L)
ISNATILISTACTIVE(IL) → ISNATLISTACTIVE(IL)
TAKEACTIVE(s(M), cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
LENGTHACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
ULENGTHACTIVE(tt, L) → LENGTHACTIVE(mark(L))
TAKEACTIVE(s(M), cons(N, IL)) → ISNATILISTACTIVE(IL)
ISNATILISTACTIVE(cons(N, IL)) → ISNATACTIVE(N)
UTAKE2ACTIVE(tt, M, N, IL) → MARK(N)
LENGTHACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
LENGTHACTIVE(cons(N, L)) → ISNATACTIVE(N)
MARK(uTake1(x1)) → MARK(x1)
MARK(isNatIList(x1)) → ISNATILISTACTIVE(x1)
ANDACTIVE(tt, T) → MARK(T)
ISNATACTIVE(s(N)) → ISNATACTIVE(N)
MARK(take(x1, x2)) → TAKEACTIVE(mark(x1), mark(x2))
MARK(cons(x1, x2)) → MARK(x1)
ISNATILISTACTIVE(cons(N, IL)) → ISNATILISTACTIVE(IL)
TAKEACTIVE(s(M), cons(N, IL)) → ISNATACTIVE(M)
MARK(uLength(x1, x2)) → ULENGTHACTIVE(mark(x1), x2)
ISNATLISTACTIVE(take(N, IL)) → ISNATILISTACTIVE(IL)
MARK(and(x1, x2)) → MARK(x1)
TAKEACTIVE(s(M), cons(N, IL)) → ISNATACTIVE(N)
ISNATLISTACTIVE(cons(N, L)) → ISNATACTIVE(N)
ISNATLISTACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
MARK(uTake2(x1, x2, x3, x4)) → MARK(x1)
ISNATLISTACTIVE(take(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
MARK(take(x1, x2)) → MARK(x1)
MARK(length(x1)) → MARK(x1)
ISNATLISTACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
ISNATILISTACTIVE(cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATLISTACTIVE(take(N, IL)) → ISNATACTIVE(N)
MARK(s(x1)) → MARK(x1)
MARK(uLength(x1, x2)) → MARK(x1)
ULENGTHACTIVE(tt, L) → MARK(L)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), mark(x2))
MARK(length(x1)) → LENGTHACTIVE(mark(x1))
TAKEACTIVE(0, IL) → ISNATILISTACTIVE(IL)
ISNATACTIVE(length(L)) → ISNATLISTACTIVE(L)
The TRS R consists of the following rules:
mark(and(x1, x2)) → andActive(mark(x1), mark(x2))
andActive(x1, x2) → and(x1, x2)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(zeros) → zerosActive
zerosActive → zeros
mark(take(x1, x2)) → takeActive(mark(x1), mark(x2))
takeActive(x1, x2) → take(x1, x2)
mark(uTake1(x1)) → uTake1Active(mark(x1))
uTake1Active(x1) → uTake1(x1)
mark(uTake2(x1, x2, x3, x4)) → uTake2Active(mark(x1), x2, x3, x4)
uTake2Active(x1, x2, x3, x4) → uTake2(x1, x2, x3, x4)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(uLength(x1, x2)) → uLengthActive(mark(x1), x2)
uLengthActive(x1, x2) → uLength(x1, x2)
mark(tt) → tt
mark(0) → 0
mark(s(x1)) → s(mark(x1))
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(nil) → nil
andActive(tt, T) → mark(T)
isNatIListActive(IL) → isNatListActive(IL)
isNatActive(0) → tt
isNatActive(s(N)) → isNatActive(N)
isNatActive(length(L)) → isNatListActive(L)
isNatIListActive(zeros) → tt
isNatIListActive(cons(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
isNatListActive(nil) → tt
isNatListActive(cons(N, L)) → andActive(isNatActive(N), isNatListActive(L))
isNatListActive(take(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
zerosActive → cons(0, zeros)
takeActive(0, IL) → uTake1Active(isNatIListActive(IL))
uTake1Active(tt) → nil
takeActive(s(M), cons(N, IL)) → uTake2Active(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
uTake2Active(tt, M, N, IL) → cons(mark(N), take(M, IL))
lengthActive(cons(N, L)) → uLengthActive(andActive(isNatActive(N), isNatListActive(L)), L)
uLengthActive(tt, L) → s(lengthActive(mark(L)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(take(x1, x2)) → MARK(x2)
MARK(uTake2(x1, x2, x3, x4)) → UTAKE2ACTIVE(mark(x1), x2, x3, x4)
MARK(uTake1(x1)) → MARK(x1)
MARK(take(x1, x2)) → TAKEACTIVE(mark(x1), mark(x2))
MARK(uTake2(x1, x2, x3, x4)) → MARK(x1)
MARK(take(x1, x2)) → MARK(x1)
The remaining pairs can at least be oriented weakly.
TAKEACTIVE(s(M), cons(N, IL)) → UTAKE2ACTIVE(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
TAKEACTIVE(s(M), cons(N, IL)) → ANDACTIVE(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL)))
MARK(and(x1, x2)) → MARK(x2)
LENGTHACTIVE(cons(N, L)) → ULENGTHACTIVE(andActive(isNatActive(N), isNatListActive(L)), L)
ISNATILISTACTIVE(IL) → ISNATLISTACTIVE(IL)
TAKEACTIVE(s(M), cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
LENGTHACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
ULENGTHACTIVE(tt, L) → LENGTHACTIVE(mark(L))
TAKEACTIVE(s(M), cons(N, IL)) → ISNATILISTACTIVE(IL)
ISNATILISTACTIVE(cons(N, IL)) → ISNATACTIVE(N)
UTAKE2ACTIVE(tt, M, N, IL) → MARK(N)
LENGTHACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
LENGTHACTIVE(cons(N, L)) → ISNATACTIVE(N)
MARK(isNatIList(x1)) → ISNATILISTACTIVE(x1)
ANDACTIVE(tt, T) → MARK(T)
ISNATACTIVE(s(N)) → ISNATACTIVE(N)
MARK(cons(x1, x2)) → MARK(x1)
ISNATILISTACTIVE(cons(N, IL)) → ISNATILISTACTIVE(IL)
TAKEACTIVE(s(M), cons(N, IL)) → ISNATACTIVE(M)
MARK(uLength(x1, x2)) → ULENGTHACTIVE(mark(x1), x2)
ISNATLISTACTIVE(take(N, IL)) → ISNATILISTACTIVE(IL)
MARK(and(x1, x2)) → MARK(x1)
TAKEACTIVE(s(M), cons(N, IL)) → ISNATACTIVE(N)
ISNATLISTACTIVE(cons(N, L)) → ISNATACTIVE(N)
ISNATLISTACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
ISNATLISTACTIVE(take(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
MARK(length(x1)) → MARK(x1)
ISNATLISTACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
ISNATILISTACTIVE(cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATLISTACTIVE(take(N, IL)) → ISNATACTIVE(N)
MARK(s(x1)) → MARK(x1)
MARK(uLength(x1, x2)) → MARK(x1)
ULENGTHACTIVE(tt, L) → MARK(L)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), mark(x2))
MARK(length(x1)) → LENGTHACTIVE(mark(x1))
TAKEACTIVE(0, IL) → ISNATILISTACTIVE(IL)
ISNATACTIVE(length(L)) → ISNATLISTACTIVE(L)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ANDACTIVE(x1, x2)) = x2
POL(ISNATACTIVE(x1)) = 0
POL(ISNATILISTACTIVE(x1)) = 0
POL(ISNATLISTACTIVE(x1)) = 0
POL(LENGTHACTIVE(x1)) = x1
POL(MARK(x1)) = x1
POL(TAKEACTIVE(x1, x2)) = x2
POL(ULENGTHACTIVE(x1, x2)) = x2
POL(UTAKE2ACTIVE(x1, x2, x3, x4)) = x3
POL(and(x1, x2)) = x1 + x2
POL(andActive(x1, x2)) = x1 + x2
POL(cons(x1, x2)) = x1 + x2
POL(isNat(x1)) = 0
POL(isNatActive(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatIListActive(x1)) = 0
POL(isNatList(x1)) = 0
POL(isNatListActive(x1)) = 0
POL(length(x1)) = x1
POL(lengthActive(x1)) = x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 1 + x1 + x2
POL(takeActive(x1, x2)) = 1 + x1 + x2
POL(tt) = 0
POL(uLength(x1, x2)) = x1 + x2
POL(uLengthActive(x1, x2)) = x1 + x2
POL(uTake1(x1)) = 1 + x1
POL(uTake1Active(x1)) = 1 + x1
POL(uTake2(x1, x2, x3, x4)) = 1 + x1 + x2 + x3 + x4
POL(uTake2Active(x1, x2, x3, x4)) = 1 + x1 + x2 + x3 + x4
POL(zeros) = 0
POL(zerosActive) = 0
The following usable rules [17] were oriented:
isNatActive(0) → tt
uTake1Active(tt) → nil
uTake2Active(x1, x2, x3, x4) → uTake2(x1, x2, x3, x4)
mark(take(x1, x2)) → takeActive(mark(x1), mark(x2))
isNatActive(x1) → isNat(x1)
mark(uLength(x1, x2)) → uLengthActive(mark(x1), x2)
takeActive(s(M), cons(N, IL)) → uTake2Active(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
uLengthActive(x1, x2) → uLength(x1, x2)
uTake1Active(x1) → uTake1(x1)
mark(zeros) → zerosActive
lengthActive(cons(N, L)) → uLengthActive(andActive(isNatActive(N), isNatListActive(L)), L)
zerosActive → cons(0, zeros)
isNatListActive(x1) → isNatList(x1)
mark(0) → 0
mark(nil) → nil
isNatListActive(nil) → tt
mark(s(x1)) → s(mark(x1))
mark(length(x1)) → lengthActive(mark(x1))
andActive(x1, x2) → and(x1, x2)
zerosActive → zeros
mark(cons(x1, x2)) → cons(mark(x1), x2)
uLengthActive(tt, L) → s(lengthActive(mark(L)))
mark(uTake2(x1, x2, x3, x4)) → uTake2Active(mark(x1), x2, x3, x4)
isNatIListActive(zeros) → tt
mark(tt) → tt
uTake2Active(tt, M, N, IL) → cons(mark(N), take(M, IL))
mark(isNat(x1)) → isNatActive(x1)
andActive(tt, T) → mark(T)
isNatListActive(take(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatListActive(cons(N, L)) → andActive(isNatActive(N), isNatListActive(L))
isNatActive(s(N)) → isNatActive(N)
mark(isNatList(x1)) → isNatListActive(x1)
isNatActive(length(L)) → isNatListActive(L)
isNatIListActive(IL) → isNatListActive(IL)
mark(and(x1, x2)) → andActive(mark(x1), mark(x2))
isNatIListActive(cons(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
isNatIListActive(x1) → isNatIList(x1)
mark(uTake1(x1)) → uTake1Active(mark(x1))
takeActive(0, IL) → uTake1Active(isNatIListActive(IL))
takeActive(x1, x2) → take(x1, x2)
lengthActive(x1) → length(x1)
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
TAKEACTIVE(s(M), cons(N, IL)) → UTAKE2ACTIVE(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
MARK(and(x1, x2)) → MARK(x2)
TAKEACTIVE(s(M), cons(N, IL)) → ANDACTIVE(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL)))
LENGTHACTIVE(cons(N, L)) → ULENGTHACTIVE(andActive(isNatActive(N), isNatListActive(L)), L)
ISNATILISTACTIVE(IL) → ISNATLISTACTIVE(IL)
TAKEACTIVE(s(M), cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ULENGTHACTIVE(tt, L) → LENGTHACTIVE(mark(L))
LENGTHACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
UTAKE2ACTIVE(tt, M, N, IL) → MARK(N)
ISNATILISTACTIVE(cons(N, IL)) → ISNATACTIVE(N)
TAKEACTIVE(s(M), cons(N, IL)) → ISNATILISTACTIVE(IL)
LENGTHACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
LENGTHACTIVE(cons(N, L)) → ISNATACTIVE(N)
MARK(isNatIList(x1)) → ISNATILISTACTIVE(x1)
ANDACTIVE(tt, T) → MARK(T)
ISNATACTIVE(s(N)) → ISNATACTIVE(N)
MARK(cons(x1, x2)) → MARK(x1)
ISNATILISTACTIVE(cons(N, IL)) → ISNATILISTACTIVE(IL)
TAKEACTIVE(s(M), cons(N, IL)) → ISNATACTIVE(M)
MARK(uLength(x1, x2)) → ULENGTHACTIVE(mark(x1), x2)
ISNATLISTACTIVE(take(N, IL)) → ISNATILISTACTIVE(IL)
MARK(and(x1, x2)) → MARK(x1)
ISNATLISTACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
TAKEACTIVE(s(M), cons(N, IL)) → ISNATACTIVE(N)
ISNATLISTACTIVE(cons(N, L)) → ISNATACTIVE(N)
ISNATLISTACTIVE(take(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATILISTACTIVE(cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATLISTACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
MARK(length(x1)) → MARK(x1)
ISNATLISTACTIVE(take(N, IL)) → ISNATACTIVE(N)
MARK(s(x1)) → MARK(x1)
ULENGTHACTIVE(tt, L) → MARK(L)
MARK(uLength(x1, x2)) → MARK(x1)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), mark(x2))
MARK(length(x1)) → LENGTHACTIVE(mark(x1))
TAKEACTIVE(0, IL) → ISNATILISTACTIVE(IL)
ISNATACTIVE(length(L)) → ISNATLISTACTIVE(L)
The TRS R consists of the following rules:
mark(and(x1, x2)) → andActive(mark(x1), mark(x2))
andActive(x1, x2) → and(x1, x2)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(zeros) → zerosActive
zerosActive → zeros
mark(take(x1, x2)) → takeActive(mark(x1), mark(x2))
takeActive(x1, x2) → take(x1, x2)
mark(uTake1(x1)) → uTake1Active(mark(x1))
uTake1Active(x1) → uTake1(x1)
mark(uTake2(x1, x2, x3, x4)) → uTake2Active(mark(x1), x2, x3, x4)
uTake2Active(x1, x2, x3, x4) → uTake2(x1, x2, x3, x4)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(uLength(x1, x2)) → uLengthActive(mark(x1), x2)
uLengthActive(x1, x2) → uLength(x1, x2)
mark(tt) → tt
mark(0) → 0
mark(s(x1)) → s(mark(x1))
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(nil) → nil
andActive(tt, T) → mark(T)
isNatIListActive(IL) → isNatListActive(IL)
isNatActive(0) → tt
isNatActive(s(N)) → isNatActive(N)
isNatActive(length(L)) → isNatListActive(L)
isNatIListActive(zeros) → tt
isNatIListActive(cons(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
isNatListActive(nil) → tt
isNatListActive(cons(N, L)) → andActive(isNatActive(N), isNatListActive(L))
isNatListActive(take(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
zerosActive → cons(0, zeros)
takeActive(0, IL) → uTake1Active(isNatIListActive(IL))
uTake1Active(tt) → nil
takeActive(s(M), cons(N, IL)) → uTake2Active(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
uTake2Active(tt, M, N, IL) → cons(mark(N), take(M, IL))
lengthActive(cons(N, L)) → uLengthActive(andActive(isNatActive(N), isNatListActive(L)), L)
uLengthActive(tt, L) → s(lengthActive(mark(L)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 8 less nodes.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
MARK(and(x1, x2)) → MARK(x2)
LENGTHACTIVE(cons(N, L)) → ULENGTHACTIVE(andActive(isNatActive(N), isNatListActive(L)), L)
ISNATILISTACTIVE(IL) → ISNATLISTACTIVE(IL)
ULENGTHACTIVE(tt, L) → LENGTHACTIVE(mark(L))
LENGTHACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
ISNATILISTACTIVE(cons(N, IL)) → ISNATACTIVE(N)
LENGTHACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
MARK(isNatIList(x1)) → ISNATILISTACTIVE(x1)
ANDACTIVE(tt, T) → MARK(T)
LENGTHACTIVE(cons(N, L)) → ISNATACTIVE(N)
ISNATACTIVE(s(N)) → ISNATACTIVE(N)
MARK(cons(x1, x2)) → MARK(x1)
ISNATILISTACTIVE(cons(N, IL)) → ISNATILISTACTIVE(IL)
MARK(uLength(x1, x2)) → ULENGTHACTIVE(mark(x1), x2)
ISNATLISTACTIVE(take(N, IL)) → ISNATILISTACTIVE(IL)
MARK(and(x1, x2)) → MARK(x1)
ISNATLISTACTIVE(cons(N, L)) → ISNATACTIVE(N)
ISNATLISTACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
ISNATLISTACTIVE(take(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
MARK(length(x1)) → MARK(x1)
ISNATLISTACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
ISNATILISTACTIVE(cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATLISTACTIVE(take(N, IL)) → ISNATACTIVE(N)
MARK(s(x1)) → MARK(x1)
MARK(uLength(x1, x2)) → MARK(x1)
ULENGTHACTIVE(tt, L) → MARK(L)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), mark(x2))
MARK(length(x1)) → LENGTHACTIVE(mark(x1))
ISNATACTIVE(length(L)) → ISNATLISTACTIVE(L)
The TRS R consists of the following rules:
mark(and(x1, x2)) → andActive(mark(x1), mark(x2))
andActive(x1, x2) → and(x1, x2)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(zeros) → zerosActive
zerosActive → zeros
mark(take(x1, x2)) → takeActive(mark(x1), mark(x2))
takeActive(x1, x2) → take(x1, x2)
mark(uTake1(x1)) → uTake1Active(mark(x1))
uTake1Active(x1) → uTake1(x1)
mark(uTake2(x1, x2, x3, x4)) → uTake2Active(mark(x1), x2, x3, x4)
uTake2Active(x1, x2, x3, x4) → uTake2(x1, x2, x3, x4)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(uLength(x1, x2)) → uLengthActive(mark(x1), x2)
uLengthActive(x1, x2) → uLength(x1, x2)
mark(tt) → tt
mark(0) → 0
mark(s(x1)) → s(mark(x1))
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(nil) → nil
andActive(tt, T) → mark(T)
isNatIListActive(IL) → isNatListActive(IL)
isNatActive(0) → tt
isNatActive(s(N)) → isNatActive(N)
isNatActive(length(L)) → isNatListActive(L)
isNatIListActive(zeros) → tt
isNatIListActive(cons(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
isNatListActive(nil) → tt
isNatListActive(cons(N, L)) → andActive(isNatActive(N), isNatListActive(L))
isNatListActive(take(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
zerosActive → cons(0, zeros)
takeActive(0, IL) → uTake1Active(isNatIListActive(IL))
uTake1Active(tt) → nil
takeActive(s(M), cons(N, IL)) → uTake2Active(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
uTake2Active(tt, M, N, IL) → cons(mark(N), take(M, IL))
lengthActive(cons(N, L)) → uLengthActive(andActive(isNatActive(N), isNatListActive(L)), L)
uLengthActive(tt, L) → s(lengthActive(mark(L)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(uLength(x1, x2)) → ULENGTHACTIVE(mark(x1), x2)
MARK(length(x1)) → MARK(x1)
MARK(uLength(x1, x2)) → MARK(x1)
MARK(length(x1)) → LENGTHACTIVE(mark(x1))
The remaining pairs can at least be oriented weakly.
MARK(and(x1, x2)) → MARK(x2)
LENGTHACTIVE(cons(N, L)) → ULENGTHACTIVE(andActive(isNatActive(N), isNatListActive(L)), L)
ISNATILISTACTIVE(IL) → ISNATLISTACTIVE(IL)
ULENGTHACTIVE(tt, L) → LENGTHACTIVE(mark(L))
LENGTHACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
ISNATILISTACTIVE(cons(N, IL)) → ISNATACTIVE(N)
LENGTHACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
MARK(isNatIList(x1)) → ISNATILISTACTIVE(x1)
ANDACTIVE(tt, T) → MARK(T)
LENGTHACTIVE(cons(N, L)) → ISNATACTIVE(N)
ISNATACTIVE(s(N)) → ISNATACTIVE(N)
MARK(cons(x1, x2)) → MARK(x1)
ISNATILISTACTIVE(cons(N, IL)) → ISNATILISTACTIVE(IL)
ISNATLISTACTIVE(take(N, IL)) → ISNATILISTACTIVE(IL)
MARK(and(x1, x2)) → MARK(x1)
ISNATLISTACTIVE(cons(N, L)) → ISNATACTIVE(N)
ISNATLISTACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
ISNATLISTACTIVE(take(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATLISTACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
ISNATILISTACTIVE(cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATLISTACTIVE(take(N, IL)) → ISNATACTIVE(N)
MARK(s(x1)) → MARK(x1)
ULENGTHACTIVE(tt, L) → MARK(L)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), mark(x2))
ISNATACTIVE(length(L)) → ISNATLISTACTIVE(L)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ANDACTIVE(x1, x2)) = x2
POL(ISNATACTIVE(x1)) = 0
POL(ISNATILISTACTIVE(x1)) = 0
POL(ISNATLISTACTIVE(x1)) = 0
POL(LENGTHACTIVE(x1)) = x1
POL(MARK(x1)) = x1
POL(ULENGTHACTIVE(x1, x2)) = x2
POL(and(x1, x2)) = x1 + x2
POL(andActive(x1, x2)) = x1 + x2
POL(cons(x1, x2)) = x1 + x2
POL(isNat(x1)) = 0
POL(isNatActive(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatIListActive(x1)) = 0
POL(isNatList(x1)) = 0
POL(isNatListActive(x1)) = 0
POL(length(x1)) = 1 + x1
POL(lengthActive(x1)) = 1 + x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = x2
POL(takeActive(x1, x2)) = x2
POL(tt) = 0
POL(uLength(x1, x2)) = 1 + x1 + x2
POL(uLengthActive(x1, x2)) = 1 + x1 + x2
POL(uTake1(x1)) = x1
POL(uTake1Active(x1)) = x1
POL(uTake2(x1, x2, x3, x4)) = x3 + x4
POL(uTake2Active(x1, x2, x3, x4)) = x3 + x4
POL(zeros) = 0
POL(zerosActive) = 0
The following usable rules [17] were oriented:
isNatActive(0) → tt
uTake1Active(tt) → nil
uTake2Active(x1, x2, x3, x4) → uTake2(x1, x2, x3, x4)
mark(take(x1, x2)) → takeActive(mark(x1), mark(x2))
isNatActive(x1) → isNat(x1)
mark(uLength(x1, x2)) → uLengthActive(mark(x1), x2)
takeActive(s(M), cons(N, IL)) → uTake2Active(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
uLengthActive(x1, x2) → uLength(x1, x2)
uTake1Active(x1) → uTake1(x1)
mark(zeros) → zerosActive
lengthActive(cons(N, L)) → uLengthActive(andActive(isNatActive(N), isNatListActive(L)), L)
zerosActive → cons(0, zeros)
isNatListActive(x1) → isNatList(x1)
mark(0) → 0
mark(nil) → nil
isNatListActive(nil) → tt
mark(s(x1)) → s(mark(x1))
andActive(x1, x2) → and(x1, x2)
mark(length(x1)) → lengthActive(mark(x1))
zerosActive → zeros
mark(cons(x1, x2)) → cons(mark(x1), x2)
uLengthActive(tt, L) → s(lengthActive(mark(L)))
mark(uTake2(x1, x2, x3, x4)) → uTake2Active(mark(x1), x2, x3, x4)
isNatIListActive(zeros) → tt
mark(tt) → tt
uTake2Active(tt, M, N, IL) → cons(mark(N), take(M, IL))
mark(isNat(x1)) → isNatActive(x1)
andActive(tt, T) → mark(T)
isNatListActive(take(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatListActive(cons(N, L)) → andActive(isNatActive(N), isNatListActive(L))
isNatActive(s(N)) → isNatActive(N)
mark(isNatList(x1)) → isNatListActive(x1)
isNatActive(length(L)) → isNatListActive(L)
isNatIListActive(IL) → isNatListActive(IL)
mark(and(x1, x2)) → andActive(mark(x1), mark(x2))
isNatIListActive(cons(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
isNatIListActive(x1) → isNatIList(x1)
mark(uTake1(x1)) → uTake1Active(mark(x1))
takeActive(0, IL) → uTake1Active(isNatIListActive(IL))
takeActive(x1, x2) → take(x1, x2)
lengthActive(x1) → length(x1)
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
MARK(and(x1, x2)) → MARK(x2)
LENGTHACTIVE(cons(N, L)) → ULENGTHACTIVE(andActive(isNatActive(N), isNatListActive(L)), L)
ISNATILISTACTIVE(IL) → ISNATLISTACTIVE(IL)
LENGTHACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
ULENGTHACTIVE(tt, L) → LENGTHACTIVE(mark(L))
ISNATILISTACTIVE(cons(N, IL)) → ISNATACTIVE(N)
LENGTHACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
LENGTHACTIVE(cons(N, L)) → ISNATACTIVE(N)
ANDACTIVE(tt, T) → MARK(T)
MARK(isNatIList(x1)) → ISNATILISTACTIVE(x1)
ISNATACTIVE(s(N)) → ISNATACTIVE(N)
MARK(cons(x1, x2)) → MARK(x1)
ISNATILISTACTIVE(cons(N, IL)) → ISNATILISTACTIVE(IL)
ISNATLISTACTIVE(take(N, IL)) → ISNATILISTACTIVE(IL)
MARK(and(x1, x2)) → MARK(x1)
ISNATLISTACTIVE(cons(N, L)) → ISNATACTIVE(N)
ISNATLISTACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
ISNATLISTACTIVE(take(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATLISTACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
ISNATILISTACTIVE(cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATLISTACTIVE(take(N, IL)) → ISNATACTIVE(N)
MARK(s(x1)) → MARK(x1)
ULENGTHACTIVE(tt, L) → MARK(L)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), mark(x2))
ISNATACTIVE(length(L)) → ISNATLISTACTIVE(L)
The TRS R consists of the following rules:
mark(and(x1, x2)) → andActive(mark(x1), mark(x2))
andActive(x1, x2) → and(x1, x2)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(zeros) → zerosActive
zerosActive → zeros
mark(take(x1, x2)) → takeActive(mark(x1), mark(x2))
takeActive(x1, x2) → take(x1, x2)
mark(uTake1(x1)) → uTake1Active(mark(x1))
uTake1Active(x1) → uTake1(x1)
mark(uTake2(x1, x2, x3, x4)) → uTake2Active(mark(x1), x2, x3, x4)
uTake2Active(x1, x2, x3, x4) → uTake2(x1, x2, x3, x4)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(uLength(x1, x2)) → uLengthActive(mark(x1), x2)
uLengthActive(x1, x2) → uLength(x1, x2)
mark(tt) → tt
mark(0) → 0
mark(s(x1)) → s(mark(x1))
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(nil) → nil
andActive(tt, T) → mark(T)
isNatIListActive(IL) → isNatListActive(IL)
isNatActive(0) → tt
isNatActive(s(N)) → isNatActive(N)
isNatActive(length(L)) → isNatListActive(L)
isNatIListActive(zeros) → tt
isNatIListActive(cons(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
isNatListActive(nil) → tt
isNatListActive(cons(N, L)) → andActive(isNatActive(N), isNatListActive(L))
isNatListActive(take(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
zerosActive → cons(0, zeros)
takeActive(0, IL) → uTake1Active(isNatIListActive(IL))
uTake1Active(tt) → nil
takeActive(s(M), cons(N, IL)) → uTake2Active(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
uTake2Active(tt, M, N, IL) → cons(mark(N), take(M, IL))
lengthActive(cons(N, L)) → uLengthActive(andActive(isNatActive(N), isNatListActive(L)), L)
uLengthActive(tt, L) → s(lengthActive(mark(L)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 4 less nodes.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLISTACTIVE(take(N, IL)) → ISNATILISTACTIVE(IL)
MARK(and(x1, x2)) → MARK(x2)
MARK(and(x1, x2)) → MARK(x1)
ISNATILISTACTIVE(IL) → ISNATLISTACTIVE(IL)
ISNATLISTACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
ISNATLISTACTIVE(cons(N, L)) → ISNATACTIVE(N)
ISNATLISTACTIVE(take(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATILISTACTIVE(cons(N, IL)) → ISNATACTIVE(N)
ISNATILISTACTIVE(cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATLISTACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
ISNATLISTACTIVE(take(N, IL)) → ISNATACTIVE(N)
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
MARK(s(x1)) → MARK(x1)
ANDACTIVE(tt, T) → MARK(T)
MARK(isNatIList(x1)) → ISNATILISTACTIVE(x1)
ISNATACTIVE(s(N)) → ISNATACTIVE(N)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), mark(x2))
MARK(cons(x1, x2)) → MARK(x1)
ISNATILISTACTIVE(cons(N, IL)) → ISNATILISTACTIVE(IL)
ISNATACTIVE(length(L)) → ISNATLISTACTIVE(L)
The TRS R consists of the following rules:
mark(and(x1, x2)) → andActive(mark(x1), mark(x2))
andActive(x1, x2) → and(x1, x2)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(zeros) → zerosActive
zerosActive → zeros
mark(take(x1, x2)) → takeActive(mark(x1), mark(x2))
takeActive(x1, x2) → take(x1, x2)
mark(uTake1(x1)) → uTake1Active(mark(x1))
uTake1Active(x1) → uTake1(x1)
mark(uTake2(x1, x2, x3, x4)) → uTake2Active(mark(x1), x2, x3, x4)
uTake2Active(x1, x2, x3, x4) → uTake2(x1, x2, x3, x4)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(uLength(x1, x2)) → uLengthActive(mark(x1), x2)
uLengthActive(x1, x2) → uLength(x1, x2)
mark(tt) → tt
mark(0) → 0
mark(s(x1)) → s(mark(x1))
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(nil) → nil
andActive(tt, T) → mark(T)
isNatIListActive(IL) → isNatListActive(IL)
isNatActive(0) → tt
isNatActive(s(N)) → isNatActive(N)
isNatActive(length(L)) → isNatListActive(L)
isNatIListActive(zeros) → tt
isNatIListActive(cons(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
isNatListActive(nil) → tt
isNatListActive(cons(N, L)) → andActive(isNatActive(N), isNatListActive(L))
isNatListActive(take(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
zerosActive → cons(0, zeros)
takeActive(0, IL) → uTake1Active(isNatIListActive(IL))
uTake1Active(tt) → nil
takeActive(s(M), cons(N, IL)) → uTake2Active(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
uTake2Active(tt, M, N, IL) → cons(mark(N), take(M, IL))
lengthActive(cons(N, L)) → uLengthActive(andActive(isNatActive(N), isNatListActive(L)), L)
uLengthActive(tt, L) → s(lengthActive(mark(L)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ISNATACTIVE(length(L)) → ISNATLISTACTIVE(L)
The remaining pairs can at least be oriented weakly.
ISNATLISTACTIVE(take(N, IL)) → ISNATILISTACTIVE(IL)
MARK(and(x1, x2)) → MARK(x2)
MARK(and(x1, x2)) → MARK(x1)
ISNATILISTACTIVE(IL) → ISNATLISTACTIVE(IL)
ISNATLISTACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
ISNATLISTACTIVE(cons(N, L)) → ISNATACTIVE(N)
ISNATLISTACTIVE(take(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATILISTACTIVE(cons(N, IL)) → ISNATACTIVE(N)
ISNATILISTACTIVE(cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATLISTACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
ISNATLISTACTIVE(take(N, IL)) → ISNATACTIVE(N)
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
MARK(s(x1)) → MARK(x1)
ANDACTIVE(tt, T) → MARK(T)
MARK(isNatIList(x1)) → ISNATILISTACTIVE(x1)
ISNATACTIVE(s(N)) → ISNATACTIVE(N)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), mark(x2))
MARK(cons(x1, x2)) → MARK(x1)
ISNATILISTACTIVE(cons(N, IL)) → ISNATILISTACTIVE(IL)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ANDACTIVE(x1, x2)) = x2
POL(ISNATACTIVE(x1)) = x1
POL(ISNATILISTACTIVE(x1)) = x1
POL(ISNATLISTACTIVE(x1)) = x1
POL(MARK(x1)) = x1
POL(and(x1, x2)) = x1 + x2
POL(andActive(x1, x2)) = x1 + x2
POL(cons(x1, x2)) = x1 + x2
POL(isNat(x1)) = x1
POL(isNatActive(x1)) = x1
POL(isNatIList(x1)) = x1
POL(isNatIListActive(x1)) = x1
POL(isNatList(x1)) = x1
POL(isNatListActive(x1)) = x1
POL(length(x1)) = 1 + x1
POL(lengthActive(x1)) = 1 + x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = x1 + x2
POL(takeActive(x1, x2)) = x1 + x2
POL(tt) = 0
POL(uLength(x1, x2)) = 1 + x2
POL(uLengthActive(x1, x2)) = 1 + x2
POL(uTake1(x1)) = x1
POL(uTake1Active(x1)) = x1
POL(uTake2(x1, x2, x3, x4)) = x2 + x3 + x4
POL(uTake2Active(x1, x2, x3, x4)) = x2 + x3 + x4
POL(zeros) = 0
POL(zerosActive) = 0
The following usable rules [17] were oriented:
isNatActive(0) → tt
uTake1Active(tt) → nil
mark(take(x1, x2)) → takeActive(mark(x1), mark(x2))
uTake2Active(x1, x2, x3, x4) → uTake2(x1, x2, x3, x4)
isNatActive(x1) → isNat(x1)
mark(uLength(x1, x2)) → uLengthActive(mark(x1), x2)
takeActive(s(M), cons(N, IL)) → uTake2Active(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
uLengthActive(x1, x2) → uLength(x1, x2)
uTake1Active(x1) → uTake1(x1)
mark(zeros) → zerosActive
lengthActive(cons(N, L)) → uLengthActive(andActive(isNatActive(N), isNatListActive(L)), L)
zerosActive → cons(0, zeros)
isNatListActive(x1) → isNatList(x1)
mark(0) → 0
mark(nil) → nil
isNatListActive(nil) → tt
mark(s(x1)) → s(mark(x1))
mark(length(x1)) → lengthActive(mark(x1))
andActive(x1, x2) → and(x1, x2)
zerosActive → zeros
mark(cons(x1, x2)) → cons(mark(x1), x2)
uLengthActive(tt, L) → s(lengthActive(mark(L)))
mark(uTake2(x1, x2, x3, x4)) → uTake2Active(mark(x1), x2, x3, x4)
isNatIListActive(zeros) → tt
mark(tt) → tt
uTake2Active(tt, M, N, IL) → cons(mark(N), take(M, IL))
mark(isNat(x1)) → isNatActive(x1)
andActive(tt, T) → mark(T)
isNatListActive(take(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatListActive(cons(N, L)) → andActive(isNatActive(N), isNatListActive(L))
isNatActive(s(N)) → isNatActive(N)
mark(isNatList(x1)) → isNatListActive(x1)
isNatActive(length(L)) → isNatListActive(L)
isNatIListActive(IL) → isNatListActive(IL)
mark(and(x1, x2)) → andActive(mark(x1), mark(x2))
isNatIListActive(cons(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
isNatIListActive(x1) → isNatIList(x1)
mark(uTake1(x1)) → uTake1Active(mark(x1))
takeActive(0, IL) → uTake1Active(isNatIListActive(IL))
takeActive(x1, x2) → take(x1, x2)
lengthActive(x1) → length(x1)
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLISTACTIVE(take(N, IL)) → ISNATILISTACTIVE(IL)
MARK(and(x1, x2)) → MARK(x2)
MARK(and(x1, x2)) → MARK(x1)
ISNATILISTACTIVE(IL) → ISNATLISTACTIVE(IL)
ISNATLISTACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
ISNATLISTACTIVE(cons(N, L)) → ISNATACTIVE(N)
ISNATLISTACTIVE(take(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATILISTACTIVE(cons(N, IL)) → ISNATACTIVE(N)
ISNATILISTACTIVE(cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATLISTACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
ISNATLISTACTIVE(take(N, IL)) → ISNATACTIVE(N)
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
MARK(s(x1)) → MARK(x1)
MARK(isNatIList(x1)) → ISNATILISTACTIVE(x1)
ANDACTIVE(tt, T) → MARK(T)
ISNATACTIVE(s(N)) → ISNATACTIVE(N)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), mark(x2))
ISNATILISTACTIVE(cons(N, IL)) → ISNATILISTACTIVE(IL)
MARK(cons(x1, x2)) → MARK(x1)
The TRS R consists of the following rules:
mark(and(x1, x2)) → andActive(mark(x1), mark(x2))
andActive(x1, x2) → and(x1, x2)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(zeros) → zerosActive
zerosActive → zeros
mark(take(x1, x2)) → takeActive(mark(x1), mark(x2))
takeActive(x1, x2) → take(x1, x2)
mark(uTake1(x1)) → uTake1Active(mark(x1))
uTake1Active(x1) → uTake1(x1)
mark(uTake2(x1, x2, x3, x4)) → uTake2Active(mark(x1), x2, x3, x4)
uTake2Active(x1, x2, x3, x4) → uTake2(x1, x2, x3, x4)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(uLength(x1, x2)) → uLengthActive(mark(x1), x2)
uLengthActive(x1, x2) → uLength(x1, x2)
mark(tt) → tt
mark(0) → 0
mark(s(x1)) → s(mark(x1))
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(nil) → nil
andActive(tt, T) → mark(T)
isNatIListActive(IL) → isNatListActive(IL)
isNatActive(0) → tt
isNatActive(s(N)) → isNatActive(N)
isNatActive(length(L)) → isNatListActive(L)
isNatIListActive(zeros) → tt
isNatIListActive(cons(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
isNatListActive(nil) → tt
isNatListActive(cons(N, L)) → andActive(isNatActive(N), isNatListActive(L))
isNatListActive(take(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
zerosActive → cons(0, zeros)
takeActive(0, IL) → uTake1Active(isNatIListActive(IL))
uTake1Active(tt) → nil
takeActive(s(M), cons(N, IL)) → uTake2Active(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
uTake2Active(tt, M, N, IL) → cons(mark(N), take(M, IL))
lengthActive(cons(N, L)) → uLengthActive(andActive(isNatActive(N), isNatListActive(L)), L)
uLengthActive(tt, L) → s(lengthActive(mark(L)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 4 less nodes.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATACTIVE(s(N)) → ISNATACTIVE(N)
The TRS R consists of the following rules:
mark(and(x1, x2)) → andActive(mark(x1), mark(x2))
andActive(x1, x2) → and(x1, x2)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(zeros) → zerosActive
zerosActive → zeros
mark(take(x1, x2)) → takeActive(mark(x1), mark(x2))
takeActive(x1, x2) → take(x1, x2)
mark(uTake1(x1)) → uTake1Active(mark(x1))
uTake1Active(x1) → uTake1(x1)
mark(uTake2(x1, x2, x3, x4)) → uTake2Active(mark(x1), x2, x3, x4)
uTake2Active(x1, x2, x3, x4) → uTake2(x1, x2, x3, x4)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(uLength(x1, x2)) → uLengthActive(mark(x1), x2)
uLengthActive(x1, x2) → uLength(x1, x2)
mark(tt) → tt
mark(0) → 0
mark(s(x1)) → s(mark(x1))
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(nil) → nil
andActive(tt, T) → mark(T)
isNatIListActive(IL) → isNatListActive(IL)
isNatActive(0) → tt
isNatActive(s(N)) → isNatActive(N)
isNatActive(length(L)) → isNatListActive(L)
isNatIListActive(zeros) → tt
isNatIListActive(cons(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
isNatListActive(nil) → tt
isNatListActive(cons(N, L)) → andActive(isNatActive(N), isNatListActive(L))
isNatListActive(take(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
zerosActive → cons(0, zeros)
takeActive(0, IL) → uTake1Active(isNatIListActive(IL))
uTake1Active(tt) → nil
takeActive(s(M), cons(N, IL)) → uTake2Active(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
uTake2Active(tt, M, N, IL) → cons(mark(N), take(M, IL))
lengthActive(cons(N, L)) → uLengthActive(andActive(isNatActive(N), isNatListActive(L)), L)
uLengthActive(tt, L) → s(lengthActive(mark(L)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATACTIVE(s(N)) → ISNATACTIVE(N)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- ISNATACTIVE(s(N)) → ISNATACTIVE(N)
The graph contains the following edges 1 > 1
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLISTACTIVE(take(N, IL)) → ISNATILISTACTIVE(IL)
MARK(and(x1, x2)) → MARK(x2)
MARK(and(x1, x2)) → MARK(x1)
ISNATILISTACTIVE(IL) → ISNATLISTACTIVE(IL)
ISNATLISTACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
ISNATLISTACTIVE(take(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATILISTACTIVE(cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATLISTACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
MARK(s(x1)) → MARK(x1)
ANDACTIVE(tt, T) → MARK(T)
MARK(isNatIList(x1)) → ISNATILISTACTIVE(x1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), mark(x2))
MARK(cons(x1, x2)) → MARK(x1)
ISNATILISTACTIVE(cons(N, IL)) → ISNATILISTACTIVE(IL)
The TRS R consists of the following rules:
mark(and(x1, x2)) → andActive(mark(x1), mark(x2))
andActive(x1, x2) → and(x1, x2)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(zeros) → zerosActive
zerosActive → zeros
mark(take(x1, x2)) → takeActive(mark(x1), mark(x2))
takeActive(x1, x2) → take(x1, x2)
mark(uTake1(x1)) → uTake1Active(mark(x1))
uTake1Active(x1) → uTake1(x1)
mark(uTake2(x1, x2, x3, x4)) → uTake2Active(mark(x1), x2, x3, x4)
uTake2Active(x1, x2, x3, x4) → uTake2(x1, x2, x3, x4)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(uLength(x1, x2)) → uLengthActive(mark(x1), x2)
uLengthActive(x1, x2) → uLength(x1, x2)
mark(tt) → tt
mark(0) → 0
mark(s(x1)) → s(mark(x1))
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(nil) → nil
andActive(tt, T) → mark(T)
isNatIListActive(IL) → isNatListActive(IL)
isNatActive(0) → tt
isNatActive(s(N)) → isNatActive(N)
isNatActive(length(L)) → isNatListActive(L)
isNatIListActive(zeros) → tt
isNatIListActive(cons(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
isNatListActive(nil) → tt
isNatListActive(cons(N, L)) → andActive(isNatActive(N), isNatListActive(L))
isNatListActive(take(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
zerosActive → cons(0, zeros)
takeActive(0, IL) → uTake1Active(isNatIListActive(IL))
uTake1Active(tt) → nil
takeActive(s(M), cons(N, IL)) → uTake2Active(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
uTake2Active(tt, M, N, IL) → cons(mark(N), take(M, IL))
lengthActive(cons(N, L)) → uLengthActive(andActive(isNatActive(N), isNatListActive(L)), L)
uLengthActive(tt, L) → s(lengthActive(mark(L)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(cons(x1, x2)) → MARK(x1)
The remaining pairs can at least be oriented weakly.
ISNATLISTACTIVE(take(N, IL)) → ISNATILISTACTIVE(IL)
MARK(and(x1, x2)) → MARK(x2)
MARK(and(x1, x2)) → MARK(x1)
ISNATILISTACTIVE(IL) → ISNATLISTACTIVE(IL)
ISNATLISTACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
ISNATLISTACTIVE(take(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATILISTACTIVE(cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATLISTACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
MARK(s(x1)) → MARK(x1)
ANDACTIVE(tt, T) → MARK(T)
MARK(isNatIList(x1)) → ISNATILISTACTIVE(x1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), mark(x2))
ISNATILISTACTIVE(cons(N, IL)) → ISNATILISTACTIVE(IL)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ANDACTIVE(x1, x2)) = x2
POL(ISNATILISTACTIVE(x1)) = 0
POL(ISNATLISTACTIVE(x1)) = 0
POL(MARK(x1)) = x1
POL(and(x1, x2)) = x1 + x2
POL(andActive(x1, x2)) = x1 + x2
POL(cons(x1, x2)) = 1 + x1
POL(isNat(x1)) = 0
POL(isNatActive(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatIListActive(x1)) = 0
POL(isNatList(x1)) = 0
POL(isNatListActive(x1)) = 0
POL(length(x1)) = 0
POL(lengthActive(x1)) = 0
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = x2
POL(takeActive(x1, x2)) = x2
POL(tt) = 0
POL(uLength(x1, x2)) = x1
POL(uLengthActive(x1, x2)) = x1
POL(uTake1(x1)) = x1
POL(uTake1Active(x1)) = x1
POL(uTake2(x1, x2, x3, x4)) = 1 + x3
POL(uTake2Active(x1, x2, x3, x4)) = 1 + x3
POL(zeros) = 1
POL(zerosActive) = 1
The following usable rules [17] were oriented:
isNatActive(0) → tt
uTake1Active(tt) → nil
mark(take(x1, x2)) → takeActive(mark(x1), mark(x2))
uTake2Active(x1, x2, x3, x4) → uTake2(x1, x2, x3, x4)
isNatActive(x1) → isNat(x1)
mark(uLength(x1, x2)) → uLengthActive(mark(x1), x2)
takeActive(s(M), cons(N, IL)) → uTake2Active(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
uLengthActive(x1, x2) → uLength(x1, x2)
uTake1Active(x1) → uTake1(x1)
mark(zeros) → zerosActive
lengthActive(cons(N, L)) → uLengthActive(andActive(isNatActive(N), isNatListActive(L)), L)
zerosActive → cons(0, zeros)
isNatListActive(x1) → isNatList(x1)
mark(0) → 0
mark(nil) → nil
isNatListActive(nil) → tt
mark(s(x1)) → s(mark(x1))
mark(length(x1)) → lengthActive(mark(x1))
andActive(x1, x2) → and(x1, x2)
zerosActive → zeros
mark(cons(x1, x2)) → cons(mark(x1), x2)
uLengthActive(tt, L) → s(lengthActive(mark(L)))
mark(uTake2(x1, x2, x3, x4)) → uTake2Active(mark(x1), x2, x3, x4)
isNatIListActive(zeros) → tt
mark(tt) → tt
uTake2Active(tt, M, N, IL) → cons(mark(N), take(M, IL))
mark(isNat(x1)) → isNatActive(x1)
andActive(tt, T) → mark(T)
isNatListActive(take(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatListActive(cons(N, L)) → andActive(isNatActive(N), isNatListActive(L))
isNatActive(s(N)) → isNatActive(N)
mark(isNatList(x1)) → isNatListActive(x1)
isNatActive(length(L)) → isNatListActive(L)
isNatIListActive(IL) → isNatListActive(IL)
mark(and(x1, x2)) → andActive(mark(x1), mark(x2))
isNatIListActive(cons(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
isNatIListActive(x1) → isNatIList(x1)
mark(uTake1(x1)) → uTake1Active(mark(x1))
takeActive(0, IL) → uTake1Active(isNatIListActive(IL))
takeActive(x1, x2) → take(x1, x2)
lengthActive(x1) → length(x1)
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLISTACTIVE(take(N, IL)) → ISNATILISTACTIVE(IL)
MARK(and(x1, x2)) → MARK(x2)
MARK(and(x1, x2)) → MARK(x1)
ISNATILISTACTIVE(IL) → ISNATLISTACTIVE(IL)
ISNATLISTACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
ISNATLISTACTIVE(take(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATILISTACTIVE(cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATLISTACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
MARK(s(x1)) → MARK(x1)
ANDACTIVE(tt, T) → MARK(T)
MARK(isNatIList(x1)) → ISNATILISTACTIVE(x1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), mark(x2))
ISNATILISTACTIVE(cons(N, IL)) → ISNATILISTACTIVE(IL)
The TRS R consists of the following rules:
mark(and(x1, x2)) → andActive(mark(x1), mark(x2))
andActive(x1, x2) → and(x1, x2)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(zeros) → zerosActive
zerosActive → zeros
mark(take(x1, x2)) → takeActive(mark(x1), mark(x2))
takeActive(x1, x2) → take(x1, x2)
mark(uTake1(x1)) → uTake1Active(mark(x1))
uTake1Active(x1) → uTake1(x1)
mark(uTake2(x1, x2, x3, x4)) → uTake2Active(mark(x1), x2, x3, x4)
uTake2Active(x1, x2, x3, x4) → uTake2(x1, x2, x3, x4)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(uLength(x1, x2)) → uLengthActive(mark(x1), x2)
uLengthActive(x1, x2) → uLength(x1, x2)
mark(tt) → tt
mark(0) → 0
mark(s(x1)) → s(mark(x1))
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(nil) → nil
andActive(tt, T) → mark(T)
isNatIListActive(IL) → isNatListActive(IL)
isNatActive(0) → tt
isNatActive(s(N)) → isNatActive(N)
isNatActive(length(L)) → isNatListActive(L)
isNatIListActive(zeros) → tt
isNatIListActive(cons(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
isNatListActive(nil) → tt
isNatListActive(cons(N, L)) → andActive(isNatActive(N), isNatListActive(L))
isNatListActive(take(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
zerosActive → cons(0, zeros)
takeActive(0, IL) → uTake1Active(isNatIListActive(IL))
uTake1Active(tt) → nil
takeActive(s(M), cons(N, IL)) → uTake2Active(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
uTake2Active(tt, M, N, IL) → cons(mark(N), take(M, IL))
lengthActive(cons(N, L)) → uLengthActive(andActive(isNatActive(N), isNatListActive(L)), L)
uLengthActive(tt, L) → s(lengthActive(mark(L)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ISNATLISTACTIVE(take(N, IL)) → ISNATILISTACTIVE(IL)
ISNATLISTACTIVE(take(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
The remaining pairs can at least be oriented weakly.
MARK(and(x1, x2)) → MARK(x2)
MARK(and(x1, x2)) → MARK(x1)
ISNATILISTACTIVE(IL) → ISNATLISTACTIVE(IL)
ISNATLISTACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
ISNATILISTACTIVE(cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATLISTACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
MARK(s(x1)) → MARK(x1)
ANDACTIVE(tt, T) → MARK(T)
MARK(isNatIList(x1)) → ISNATILISTACTIVE(x1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), mark(x2))
ISNATILISTACTIVE(cons(N, IL)) → ISNATILISTACTIVE(IL)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ANDACTIVE(x1, x2)) = x2
POL(ISNATILISTACTIVE(x1)) = x1
POL(ISNATLISTACTIVE(x1)) = x1
POL(MARK(x1)) = x1
POL(and(x1, x2)) = x1 + x2
POL(andActive(x1, x2)) = x1 + x2
POL(cons(x1, x2)) = x1 + x2
POL(isNat(x1)) = x1
POL(isNatActive(x1)) = x1
POL(isNatIList(x1)) = x1
POL(isNatIListActive(x1)) = x1
POL(isNatList(x1)) = x1
POL(isNatListActive(x1)) = x1
POL(length(x1)) = x1
POL(lengthActive(x1)) = x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 1 + x1 + x2
POL(takeActive(x1, x2)) = 1 + x1 + x2
POL(tt) = 0
POL(uLength(x1, x2)) = x2
POL(uLengthActive(x1, x2)) = x2
POL(uTake1(x1)) = 1
POL(uTake1Active(x1)) = 1
POL(uTake2(x1, x2, x3, x4)) = 1 + x2 + x3 + x4
POL(uTake2Active(x1, x2, x3, x4)) = 1 + x2 + x3 + x4
POL(zeros) = 0
POL(zerosActive) = 0
The following usable rules [17] were oriented:
isNatActive(0) → tt
uTake1Active(tt) → nil
mark(take(x1, x2)) → takeActive(mark(x1), mark(x2))
uTake2Active(x1, x2, x3, x4) → uTake2(x1, x2, x3, x4)
isNatActive(x1) → isNat(x1)
mark(uLength(x1, x2)) → uLengthActive(mark(x1), x2)
takeActive(s(M), cons(N, IL)) → uTake2Active(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
uLengthActive(x1, x2) → uLength(x1, x2)
uTake1Active(x1) → uTake1(x1)
mark(zeros) → zerosActive
lengthActive(cons(N, L)) → uLengthActive(andActive(isNatActive(N), isNatListActive(L)), L)
zerosActive → cons(0, zeros)
isNatListActive(x1) → isNatList(x1)
mark(0) → 0
mark(nil) → nil
isNatListActive(nil) → tt
mark(s(x1)) → s(mark(x1))
mark(length(x1)) → lengthActive(mark(x1))
andActive(x1, x2) → and(x1, x2)
zerosActive → zeros
mark(cons(x1, x2)) → cons(mark(x1), x2)
uLengthActive(tt, L) → s(lengthActive(mark(L)))
mark(uTake2(x1, x2, x3, x4)) → uTake2Active(mark(x1), x2, x3, x4)
isNatIListActive(zeros) → tt
mark(tt) → tt
uTake2Active(tt, M, N, IL) → cons(mark(N), take(M, IL))
mark(isNat(x1)) → isNatActive(x1)
andActive(tt, T) → mark(T)
isNatListActive(take(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatListActive(cons(N, L)) → andActive(isNatActive(N), isNatListActive(L))
isNatActive(s(N)) → isNatActive(N)
mark(isNatList(x1)) → isNatListActive(x1)
isNatActive(length(L)) → isNatListActive(L)
isNatIListActive(IL) → isNatListActive(IL)
mark(and(x1, x2)) → andActive(mark(x1), mark(x2))
isNatIListActive(cons(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
isNatIListActive(x1) → isNatIList(x1)
mark(uTake1(x1)) → uTake1Active(mark(x1))
takeActive(0, IL) → uTake1Active(isNatIListActive(IL))
takeActive(x1, x2) → take(x1, x2)
lengthActive(x1) → length(x1)
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
MARK(and(x1, x2)) → MARK(x2)
MARK(s(x1)) → MARK(x1)
MARK(and(x1, x2)) → MARK(x1)
ISNATILISTACTIVE(IL) → ISNATLISTACTIVE(IL)
MARK(isNatIList(x1)) → ISNATILISTACTIVE(x1)
ANDACTIVE(tt, T) → MARK(T)
ISNATLISTACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), mark(x2))
ISNATILISTACTIVE(cons(N, IL)) → ISNATILISTACTIVE(IL)
ISNATLISTACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
ISNATILISTACTIVE(cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
The TRS R consists of the following rules:
mark(and(x1, x2)) → andActive(mark(x1), mark(x2))
andActive(x1, x2) → and(x1, x2)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(zeros) → zerosActive
zerosActive → zeros
mark(take(x1, x2)) → takeActive(mark(x1), mark(x2))
takeActive(x1, x2) → take(x1, x2)
mark(uTake1(x1)) → uTake1Active(mark(x1))
uTake1Active(x1) → uTake1(x1)
mark(uTake2(x1, x2, x3, x4)) → uTake2Active(mark(x1), x2, x3, x4)
uTake2Active(x1, x2, x3, x4) → uTake2(x1, x2, x3, x4)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(uLength(x1, x2)) → uLengthActive(mark(x1), x2)
uLengthActive(x1, x2) → uLength(x1, x2)
mark(tt) → tt
mark(0) → 0
mark(s(x1)) → s(mark(x1))
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(nil) → nil
andActive(tt, T) → mark(T)
isNatIListActive(IL) → isNatListActive(IL)
isNatActive(0) → tt
isNatActive(s(N)) → isNatActive(N)
isNatActive(length(L)) → isNatListActive(L)
isNatIListActive(zeros) → tt
isNatIListActive(cons(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
isNatListActive(nil) → tt
isNatListActive(cons(N, L)) → andActive(isNatActive(N), isNatListActive(L))
isNatListActive(take(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
zerosActive → cons(0, zeros)
takeActive(0, IL) → uTake1Active(isNatIListActive(IL))
uTake1Active(tt) → nil
takeActive(s(M), cons(N, IL)) → uTake2Active(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
uTake2Active(tt, M, N, IL) → cons(mark(N), take(M, IL))
lengthActive(cons(N, L)) → uLengthActive(andActive(isNatActive(N), isNatListActive(L)), L)
uLengthActive(tt, L) → s(lengthActive(mark(L)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ISNATILISTACTIVE(IL) → ISNATLISTACTIVE(IL)
The remaining pairs can at least be oriented weakly.
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
MARK(and(x1, x2)) → MARK(x2)
MARK(s(x1)) → MARK(x1)
MARK(and(x1, x2)) → MARK(x1)
MARK(isNatIList(x1)) → ISNATILISTACTIVE(x1)
ANDACTIVE(tt, T) → MARK(T)
ISNATLISTACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), mark(x2))
ISNATILISTACTIVE(cons(N, IL)) → ISNATILISTACTIVE(IL)
ISNATLISTACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
ISNATILISTACTIVE(cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ANDACTIVE(x1, x2)) = x2
POL(ISNATILISTACTIVE(x1)) = 1 + x1
POL(ISNATLISTACTIVE(x1)) = x1
POL(MARK(x1)) = x1
POL(and(x1, x2)) = x1 + x2
POL(andActive(x1, x2)) = x1 + x2
POL(cons(x1, x2)) = x1 + x2
POL(isNat(x1)) = x1
POL(isNatActive(x1)) = x1
POL(isNatIList(x1)) = 1 + x1
POL(isNatIListActive(x1)) = 1 + x1
POL(isNatList(x1)) = x1
POL(isNatListActive(x1)) = x1
POL(length(x1)) = x1
POL(lengthActive(x1)) = x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(take(x1, x2)) = 1 + x1 + x2
POL(takeActive(x1, x2)) = 1 + x1 + x2
POL(tt) = 0
POL(uLength(x1, x2)) = x2
POL(uLengthActive(x1, x2)) = x2
POL(uTake1(x1)) = 0
POL(uTake1Active(x1)) = 0
POL(uTake2(x1, x2, x3, x4)) = 1 + x2 + x3 + x4
POL(uTake2Active(x1, x2, x3, x4)) = 1 + x2 + x3 + x4
POL(zeros) = 0
POL(zerosActive) = 0
The following usable rules [17] were oriented:
isNatActive(0) → tt
uTake1Active(tt) → nil
mark(take(x1, x2)) → takeActive(mark(x1), mark(x2))
uTake2Active(x1, x2, x3, x4) → uTake2(x1, x2, x3, x4)
isNatActive(x1) → isNat(x1)
mark(uLength(x1, x2)) → uLengthActive(mark(x1), x2)
takeActive(s(M), cons(N, IL)) → uTake2Active(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
uLengthActive(x1, x2) → uLength(x1, x2)
uTake1Active(x1) → uTake1(x1)
mark(zeros) → zerosActive
lengthActive(cons(N, L)) → uLengthActive(andActive(isNatActive(N), isNatListActive(L)), L)
zerosActive → cons(0, zeros)
isNatListActive(x1) → isNatList(x1)
mark(0) → 0
mark(nil) → nil
isNatListActive(nil) → tt
mark(s(x1)) → s(mark(x1))
mark(length(x1)) → lengthActive(mark(x1))
andActive(x1, x2) → and(x1, x2)
zerosActive → zeros
mark(cons(x1, x2)) → cons(mark(x1), x2)
uLengthActive(tt, L) → s(lengthActive(mark(L)))
mark(uTake2(x1, x2, x3, x4)) → uTake2Active(mark(x1), x2, x3, x4)
isNatIListActive(zeros) → tt
mark(tt) → tt
uTake2Active(tt, M, N, IL) → cons(mark(N), take(M, IL))
mark(isNat(x1)) → isNatActive(x1)
andActive(tt, T) → mark(T)
isNatListActive(take(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatListActive(cons(N, L)) → andActive(isNatActive(N), isNatListActive(L))
isNatActive(s(N)) → isNatActive(N)
mark(isNatList(x1)) → isNatListActive(x1)
isNatActive(length(L)) → isNatListActive(L)
isNatIListActive(IL) → isNatListActive(IL)
mark(and(x1, x2)) → andActive(mark(x1), mark(x2))
isNatIListActive(cons(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
isNatIListActive(x1) → isNatIList(x1)
mark(uTake1(x1)) → uTake1Active(mark(x1))
takeActive(0, IL) → uTake1Active(isNatIListActive(IL))
takeActive(x1, x2) → take(x1, x2)
lengthActive(x1) → length(x1)
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
MARK(and(x1, x2)) → MARK(x2)
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
MARK(and(x1, x2)) → MARK(x1)
MARK(s(x1)) → MARK(x1)
ISNATLISTACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
ANDACTIVE(tt, T) → MARK(T)
MARK(isNatIList(x1)) → ISNATILISTACTIVE(x1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), mark(x2))
ISNATILISTACTIVE(cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATLISTACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
ISNATILISTACTIVE(cons(N, IL)) → ISNATILISTACTIVE(IL)
The TRS R consists of the following rules:
mark(and(x1, x2)) → andActive(mark(x1), mark(x2))
andActive(x1, x2) → and(x1, x2)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(zeros) → zerosActive
zerosActive → zeros
mark(take(x1, x2)) → takeActive(mark(x1), mark(x2))
takeActive(x1, x2) → take(x1, x2)
mark(uTake1(x1)) → uTake1Active(mark(x1))
uTake1Active(x1) → uTake1(x1)
mark(uTake2(x1, x2, x3, x4)) → uTake2Active(mark(x1), x2, x3, x4)
uTake2Active(x1, x2, x3, x4) → uTake2(x1, x2, x3, x4)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(uLength(x1, x2)) → uLengthActive(mark(x1), x2)
uLengthActive(x1, x2) → uLength(x1, x2)
mark(tt) → tt
mark(0) → 0
mark(s(x1)) → s(mark(x1))
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(nil) → nil
andActive(tt, T) → mark(T)
isNatIListActive(IL) → isNatListActive(IL)
isNatActive(0) → tt
isNatActive(s(N)) → isNatActive(N)
isNatActive(length(L)) → isNatListActive(L)
isNatIListActive(zeros) → tt
isNatIListActive(cons(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
isNatListActive(nil) → tt
isNatListActive(cons(N, L)) → andActive(isNatActive(N), isNatListActive(L))
isNatListActive(take(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
zerosActive → cons(0, zeros)
takeActive(0, IL) → uTake1Active(isNatIListActive(IL))
uTake1Active(tt) → nil
takeActive(s(M), cons(N, IL)) → uTake2Active(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
uTake2Active(tt, M, N, IL) → cons(mark(N), take(M, IL))
lengthActive(cons(N, L)) → uLengthActive(andActive(isNatActive(N), isNatListActive(L)), L)
uLengthActive(tt, L) → s(lengthActive(mark(L)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(s(x1)) → MARK(x1)
The remaining pairs can at least be oriented weakly.
MARK(and(x1, x2)) → MARK(x2)
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
MARK(and(x1, x2)) → MARK(x1)
ISNATLISTACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
ANDACTIVE(tt, T) → MARK(T)
MARK(isNatIList(x1)) → ISNATILISTACTIVE(x1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), mark(x2))
ISNATILISTACTIVE(cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
ISNATLISTACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
ISNATILISTACTIVE(cons(N, IL)) → ISNATILISTACTIVE(IL)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
| M( uTake2Active(x1, ..., x4) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| M( uLengthActive(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( isNatList(x1) ) = | | + | | · | x1 |
| M( cons(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( takeActive(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( and(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( andActive(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( isNatIList(x1) ) = | | + | | · | x1 |
| M( lengthActive(x1) ) = | | + | | · | x1 |
| M( isNatIListActive(x1) ) = | | + | | · | x1 |
| M( uTake1Active(x1) ) = | | + | | · | x1 |
| M( isNatListActive(x1) ) = | | + | | · | x1 |
| M( isNatActive(x1) ) = | | + | | · | x1 |
| M( uLength(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( uTake2(x1, ..., x4) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| M( take(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
Tuple symbols:
| M( ANDACTIVE(x1, x2) ) = | 0 | + | | · | x1 | + | | · | x2 |
| M( ISNATLISTACTIVE(x1) ) = | 0 | + | | · | x1 |
| M( ISNATILISTACTIVE(x1) ) = | 0 | + | | · | x1 |
Matrix type:
We used a basic matrix type which is not further parametrizeable.
As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:
isNatActive(0) → tt
uTake1Active(tt) → nil
mark(take(x1, x2)) → takeActive(mark(x1), mark(x2))
uTake2Active(x1, x2, x3, x4) → uTake2(x1, x2, x3, x4)
isNatActive(x1) → isNat(x1)
mark(uLength(x1, x2)) → uLengthActive(mark(x1), x2)
takeActive(s(M), cons(N, IL)) → uTake2Active(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
uLengthActive(x1, x2) → uLength(x1, x2)
uTake1Active(x1) → uTake1(x1)
mark(zeros) → zerosActive
lengthActive(cons(N, L)) → uLengthActive(andActive(isNatActive(N), isNatListActive(L)), L)
zerosActive → cons(0, zeros)
isNatListActive(x1) → isNatList(x1)
mark(0) → 0
mark(nil) → nil
isNatListActive(nil) → tt
mark(s(x1)) → s(mark(x1))
mark(length(x1)) → lengthActive(mark(x1))
andActive(x1, x2) → and(x1, x2)
zerosActive → zeros
mark(cons(x1, x2)) → cons(mark(x1), x2)
uLengthActive(tt, L) → s(lengthActive(mark(L)))
mark(uTake2(x1, x2, x3, x4)) → uTake2Active(mark(x1), x2, x3, x4)
isNatIListActive(zeros) → tt
mark(tt) → tt
uTake2Active(tt, M, N, IL) → cons(mark(N), take(M, IL))
mark(isNat(x1)) → isNatActive(x1)
andActive(tt, T) → mark(T)
isNatListActive(take(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatListActive(cons(N, L)) → andActive(isNatActive(N), isNatListActive(L))
isNatActive(s(N)) → isNatActive(N)
mark(isNatList(x1)) → isNatListActive(x1)
isNatActive(length(L)) → isNatListActive(L)
isNatIListActive(IL) → isNatListActive(IL)
mark(and(x1, x2)) → andActive(mark(x1), mark(x2))
isNatIListActive(cons(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
isNatIListActive(x1) → isNatIList(x1)
mark(uTake1(x1)) → uTake1Active(mark(x1))
takeActive(0, IL) → uTake1Active(isNatIListActive(IL))
takeActive(x1, x2) → take(x1, x2)
lengthActive(x1) → length(x1)
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
MARK(and(x1, x2)) → MARK(x2)
MARK(and(x1, x2)) → MARK(x1)
MARK(isNatIList(x1)) → ISNATILISTACTIVE(x1)
ANDACTIVE(tt, T) → MARK(T)
ISNATLISTACTIVE(cons(N, L)) → ANDACTIVE(isNatActive(N), isNatListActive(L))
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), mark(x2))
ISNATILISTACTIVE(cons(N, IL)) → ISNATILISTACTIVE(IL)
ISNATLISTACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
ISNATILISTACTIVE(cons(N, IL)) → ANDACTIVE(isNatActive(N), isNatIListActive(IL))
The TRS R consists of the following rules:
mark(and(x1, x2)) → andActive(mark(x1), mark(x2))
andActive(x1, x2) → and(x1, x2)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(zeros) → zerosActive
zerosActive → zeros
mark(take(x1, x2)) → takeActive(mark(x1), mark(x2))
takeActive(x1, x2) → take(x1, x2)
mark(uTake1(x1)) → uTake1Active(mark(x1))
uTake1Active(x1) → uTake1(x1)
mark(uTake2(x1, x2, x3, x4)) → uTake2Active(mark(x1), x2, x3, x4)
uTake2Active(x1, x2, x3, x4) → uTake2(x1, x2, x3, x4)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(uLength(x1, x2)) → uLengthActive(mark(x1), x2)
uLengthActive(x1, x2) → uLength(x1, x2)
mark(tt) → tt
mark(0) → 0
mark(s(x1)) → s(mark(x1))
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(nil) → nil
andActive(tt, T) → mark(T)
isNatIListActive(IL) → isNatListActive(IL)
isNatActive(0) → tt
isNatActive(s(N)) → isNatActive(N)
isNatActive(length(L)) → isNatListActive(L)
isNatIListActive(zeros) → tt
isNatIListActive(cons(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
isNatListActive(nil) → tt
isNatListActive(cons(N, L)) → andActive(isNatActive(N), isNatListActive(L))
isNatListActive(take(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
zerosActive → cons(0, zeros)
takeActive(0, IL) → uTake1Active(isNatIListActive(IL))
uTake1Active(tt) → nil
takeActive(s(M), cons(N, IL)) → uTake2Active(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
uTake2Active(tt, M, N, IL) → cons(mark(N), take(M, IL))
lengthActive(cons(N, L)) → uLengthActive(andActive(isNatActive(N), isNatListActive(L)), L)
uLengthActive(tt, L) → s(lengthActive(mark(L)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
LENGTHACTIVE(cons(N, L)) → ULENGTHACTIVE(andActive(isNatActive(N), isNatListActive(L)), L)
ULENGTHACTIVE(tt, L) → LENGTHACTIVE(mark(L))
The TRS R consists of the following rules:
mark(and(x1, x2)) → andActive(mark(x1), mark(x2))
andActive(x1, x2) → and(x1, x2)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(zeros) → zerosActive
zerosActive → zeros
mark(take(x1, x2)) → takeActive(mark(x1), mark(x2))
takeActive(x1, x2) → take(x1, x2)
mark(uTake1(x1)) → uTake1Active(mark(x1))
uTake1Active(x1) → uTake1(x1)
mark(uTake2(x1, x2, x3, x4)) → uTake2Active(mark(x1), x2, x3, x4)
uTake2Active(x1, x2, x3, x4) → uTake2(x1, x2, x3, x4)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(uLength(x1, x2)) → uLengthActive(mark(x1), x2)
uLengthActive(x1, x2) → uLength(x1, x2)
mark(tt) → tt
mark(0) → 0
mark(s(x1)) → s(mark(x1))
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(nil) → nil
andActive(tt, T) → mark(T)
isNatIListActive(IL) → isNatListActive(IL)
isNatActive(0) → tt
isNatActive(s(N)) → isNatActive(N)
isNatActive(length(L)) → isNatListActive(L)
isNatIListActive(zeros) → tt
isNatIListActive(cons(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
isNatListActive(nil) → tt
isNatListActive(cons(N, L)) → andActive(isNatActive(N), isNatListActive(L))
isNatListActive(take(N, IL)) → andActive(isNatActive(N), isNatIListActive(IL))
zerosActive → cons(0, zeros)
takeActive(0, IL) → uTake1Active(isNatIListActive(IL))
uTake1Active(tt) → nil
takeActive(s(M), cons(N, IL)) → uTake2Active(andActive(isNatActive(M), andActive(isNatActive(N), isNatIListActive(IL))), M, N, IL)
uTake2Active(tt, M, N, IL) → cons(mark(N), take(M, IL))
lengthActive(cons(N, L)) → uLengthActive(andActive(isNatActive(N), isNatListActive(L)), L)
uLengthActive(tt, L) → s(lengthActive(mark(L)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We applied the Improved Ferreira Ribeiro [5,11] to transform the context-sensitive TRS to an usual TRS.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ Complete Giesl Middeldorp-Transformation
Q restricted rewrite system:
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(N, L)) → AND(isNat(a(N)), isNatList(a(L)))
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
A(zerosInact) → ZEROS
ISNATLIST(takeInact(N, IL)) → ISNATILIST(a(IL))
LENGTH(cons(N, L)) → ISNATLIST(a(L))
A(sInact(x1)) → A(x1)
ULENGTH(tt, L) → S(length(a(L)))
A(sInact(x1)) → S(a(x1))
TAKE(s(M), cons(N, IL)) → AND(isNat(N), isNatIList(a(IL)))
ISNATILIST(consInact(N, IL)) → ISNAT(a(N))
UTAKE2(tt, M, N, IL) → A(N)
ULENGTH(tt, L) → LENGTH(a(L))
ISNATILIST(consInact(N, IL)) → ISNATILIST(a(IL))
UTAKE1(tt) → NIL
TAKE(s(M), cons(N, IL)) → ISNATILIST(a(IL))
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(a(L))), a(L))
ISNATILIST(consInact(N, IL)) → A(IL)
ISNAT(sInact(N)) → A(N)
A(consInact(x1, x2)) → A(x1)
ISNATLIST(consInact(N, L)) → A(L)
ISNATLIST(consInact(N, L)) → ISNATLIST(a(L))
ISNAT(lengthInact(L)) → ISNATLIST(a(L))
ISNAT(lengthInact(L)) → A(L)
A(takeInact(x1, x2)) → A(x2)
TAKE(s(M), cons(N, IL)) → ISNAT(N)
ISNATLIST(takeInact(N, IL)) → AND(isNat(a(N)), isNatIList(a(IL)))
ISNATLIST(takeInact(N, IL)) → ISNAT(a(N))
ZEROS → 01
A(takeInact(x1, x2)) → A(x1)
ZEROS → CONS(0, zerosInact)
UTAKE2(tt, M, N, IL) → A(IL)
A(lengthInact(x1)) → LENGTH(a(x1))
ISNATLIST(takeInact(N, IL)) → A(IL)
TAKE(0, IL) → ISNATILIST(IL)
ULENGTH(tt, L) → A(L)
ISNATLIST(takeInact(N, IL)) → A(N)
ISNATILIST(consInact(N, IL)) → AND(isNat(a(N)), isNatIList(a(IL)))
UTAKE2(tt, M, N, IL) → CONS(a(N), takeInact(a(M), a(IL)))
ISNATILIST(IL) → A(IL)
A(nilInact) → NIL
LENGTH(cons(N, L)) → ISNAT(N)
LENGTH(cons(N, L)) → AND(isNat(N), isNatList(a(L)))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNAT(sInact(N)) → ISNAT(a(N))
ISNATLIST(consInact(N, L)) → A(N)
TAKE(s(M), cons(N, IL)) → AND(isNat(M), and(isNat(N), isNatIList(a(IL))))
ISNATILIST(consInact(N, IL)) → A(N)
UTAKE2(tt, M, N, IL) → A(M)
A(lengthInact(x1)) → A(x1)
A(0Inact) → 01
TAKE(s(M), cons(N, IL)) → A(IL)
ISNATLIST(consInact(N, L)) → ISNAT(a(N))
A(consInact(x1, x2)) → CONS(a(x1), x2)
LENGTH(cons(N, L)) → A(L)
ISNATILIST(IL) → ISNATLIST(a(IL))
A(takeInact(x1, x2)) → TAKE(a(x1), a(x2))
TAKE(0, IL) → UTAKE1(isNatIList(IL))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(N, L)) → AND(isNat(a(N)), isNatList(a(L)))
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
A(zerosInact) → ZEROS
ISNATLIST(takeInact(N, IL)) → ISNATILIST(a(IL))
LENGTH(cons(N, L)) → ISNATLIST(a(L))
A(sInact(x1)) → A(x1)
ULENGTH(tt, L) → S(length(a(L)))
A(sInact(x1)) → S(a(x1))
TAKE(s(M), cons(N, IL)) → AND(isNat(N), isNatIList(a(IL)))
ISNATILIST(consInact(N, IL)) → ISNAT(a(N))
UTAKE2(tt, M, N, IL) → A(N)
ULENGTH(tt, L) → LENGTH(a(L))
ISNATILIST(consInact(N, IL)) → ISNATILIST(a(IL))
UTAKE1(tt) → NIL
TAKE(s(M), cons(N, IL)) → ISNATILIST(a(IL))
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(a(L))), a(L))
ISNATILIST(consInact(N, IL)) → A(IL)
ISNAT(sInact(N)) → A(N)
A(consInact(x1, x2)) → A(x1)
ISNATLIST(consInact(N, L)) → A(L)
ISNATLIST(consInact(N, L)) → ISNATLIST(a(L))
ISNAT(lengthInact(L)) → ISNATLIST(a(L))
ISNAT(lengthInact(L)) → A(L)
A(takeInact(x1, x2)) → A(x2)
TAKE(s(M), cons(N, IL)) → ISNAT(N)
ISNATLIST(takeInact(N, IL)) → AND(isNat(a(N)), isNatIList(a(IL)))
ISNATLIST(takeInact(N, IL)) → ISNAT(a(N))
ZEROS → 01
A(takeInact(x1, x2)) → A(x1)
ZEROS → CONS(0, zerosInact)
UTAKE2(tt, M, N, IL) → A(IL)
A(lengthInact(x1)) → LENGTH(a(x1))
ISNATLIST(takeInact(N, IL)) → A(IL)
TAKE(0, IL) → ISNATILIST(IL)
ULENGTH(tt, L) → A(L)
ISNATLIST(takeInact(N, IL)) → A(N)
ISNATILIST(consInact(N, IL)) → AND(isNat(a(N)), isNatIList(a(IL)))
UTAKE2(tt, M, N, IL) → CONS(a(N), takeInact(a(M), a(IL)))
ISNATILIST(IL) → A(IL)
A(nilInact) → NIL
LENGTH(cons(N, L)) → ISNAT(N)
LENGTH(cons(N, L)) → AND(isNat(N), isNatList(a(L)))
TAKE(s(M), cons(N, IL)) → ISNAT(M)
ISNAT(sInact(N)) → ISNAT(a(N))
ISNATLIST(consInact(N, L)) → A(N)
TAKE(s(M), cons(N, IL)) → AND(isNat(M), and(isNat(N), isNatIList(a(IL))))
ISNATILIST(consInact(N, IL)) → A(N)
UTAKE2(tt, M, N, IL) → A(M)
A(lengthInact(x1)) → A(x1)
A(0Inact) → 01
TAKE(s(M), cons(N, IL)) → A(IL)
ISNATLIST(consInact(N, L)) → ISNAT(a(N))
A(consInact(x1, x2)) → CONS(a(x1), x2)
LENGTH(cons(N, L)) → A(L)
ISNATILIST(IL) → ISNATLIST(a(IL))
A(takeInact(x1, x2)) → TAKE(a(x1), a(x2))
TAKE(0, IL) → UTAKE1(isNatIList(IL))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 17 less nodes.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
LENGTH(cons(N, L)) → ISNAT(N)
TAKE(s(M), cons(N, IL)) → ISNAT(M)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
LENGTH(cons(N, L)) → ISNATLIST(a(L))
ISNAT(sInact(N)) → ISNAT(a(N))
ISNATLIST(takeInact(N, IL)) → ISNATILIST(a(IL))
A(sInact(x1)) → A(x1)
UTAKE2(tt, M, N, IL) → A(N)
ISNATILIST(consInact(N, IL)) → ISNAT(a(N))
ISNATLIST(consInact(N, L)) → A(N)
ULENGTH(tt, L) → LENGTH(a(L))
ISNATILIST(consInact(N, IL)) → A(N)
UTAKE2(tt, M, N, IL) → A(M)
ISNATILIST(consInact(N, IL)) → ISNATILIST(a(IL))
TAKE(s(M), cons(N, IL)) → ISNATILIST(a(IL))
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(a(L))), a(L))
A(lengthInact(x1)) → A(x1)
TAKE(s(M), cons(N, IL)) → A(IL)
ISNATLIST(consInact(N, L)) → ISNAT(a(N))
LENGTH(cons(N, L)) → A(L)
ISNATILIST(consInact(N, IL)) → A(IL)
ISNAT(sInact(N)) → A(N)
A(consInact(x1, x2)) → A(x1)
ISNATLIST(consInact(N, L)) → A(L)
ISNAT(lengthInact(L)) → ISNATLIST(a(L))
ISNATLIST(consInact(N, L)) → ISNATLIST(a(L))
ISNAT(lengthInact(L)) → A(L)
TAKE(s(M), cons(N, IL)) → ISNAT(N)
A(takeInact(x1, x2)) → A(x2)
ISNATLIST(takeInact(N, IL)) → ISNAT(a(N))
ISNATILIST(IL) → ISNATLIST(a(IL))
A(takeInact(x1, x2)) → A(x1)
UTAKE2(tt, M, N, IL) → A(IL)
A(lengthInact(x1)) → LENGTH(a(x1))
ISNATLIST(takeInact(N, IL)) → A(IL)
ULENGTH(tt, L) → A(L)
TAKE(0, IL) → ISNATILIST(IL)
A(takeInact(x1, x2)) → TAKE(a(x1), a(x2))
ISNATLIST(takeInact(N, IL)) → A(N)
ISNATILIST(IL) → A(IL)
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
TAKE(s(M), cons(N, IL)) → ISNAT(M)
TAKE(s(M), cons(N, IL)) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
ISNATILIST(consInact(N, IL)) → ISNAT(a(N))
ISNATILIST(consInact(N, IL)) → A(N)
TAKE(s(M), cons(N, IL)) → A(IL)
ISNATILIST(consInact(N, IL)) → A(IL)
TAKE(s(M), cons(N, IL)) → ISNAT(N)
A(takeInact(x1, x2)) → A(x2)
ISNATLIST(takeInact(N, IL)) → ISNAT(a(N))
ISNATILIST(IL) → ISNATLIST(a(IL))
A(takeInact(x1, x2)) → A(x1)
ISNATLIST(takeInact(N, IL)) → A(IL)
ISNATLIST(takeInact(N, IL)) → A(N)
ISNATILIST(IL) → A(IL)
The remaining pairs can at least be oriented weakly.
LENGTH(cons(N, L)) → ISNAT(N)
LENGTH(cons(N, L)) → ISNATLIST(a(L))
ISNAT(sInact(N)) → ISNAT(a(N))
ISNATLIST(takeInact(N, IL)) → ISNATILIST(a(IL))
A(sInact(x1)) → A(x1)
UTAKE2(tt, M, N, IL) → A(N)
ISNATLIST(consInact(N, L)) → A(N)
ULENGTH(tt, L) → LENGTH(a(L))
UTAKE2(tt, M, N, IL) → A(M)
ISNATILIST(consInact(N, IL)) → ISNATILIST(a(IL))
TAKE(s(M), cons(N, IL)) → ISNATILIST(a(IL))
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(a(L))), a(L))
A(lengthInact(x1)) → A(x1)
ISNATLIST(consInact(N, L)) → ISNAT(a(N))
LENGTH(cons(N, L)) → A(L)
ISNAT(sInact(N)) → A(N)
A(consInact(x1, x2)) → A(x1)
ISNATLIST(consInact(N, L)) → A(L)
ISNAT(lengthInact(L)) → ISNATLIST(a(L))
ISNATLIST(consInact(N, L)) → ISNATLIST(a(L))
ISNAT(lengthInact(L)) → A(L)
UTAKE2(tt, M, N, IL) → A(IL)
A(lengthInact(x1)) → LENGTH(a(x1))
ULENGTH(tt, L) → A(L)
TAKE(0, IL) → ISNATILIST(IL)
A(takeInact(x1, x2)) → TAKE(a(x1), a(x2))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(0Inact) = 0
POL(A(x1)) = x1
POL(ISNAT(x1)) = x1
POL(ISNATILIST(x1)) = 1 + x1
POL(ISNATLIST(x1)) = x1
POL(LENGTH(x1)) = x1
POL(TAKE(x1, x2)) = 1 + x1 + x2
POL(ULENGTH(x1, x2)) = x2
POL(UTAKE2(x1, x2, x3, x4)) = x2 + x3 + x4
POL(a(x1)) = x1
POL(and(x1, x2)) = x2
POL(cons(x1, x2)) = x1 + x2
POL(consInact(x1, x2)) = x1 + x2
POL(isNat(x1)) = 1 + x1
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = x1
POL(lengthInact(x1)) = x1
POL(nil) = 0
POL(nilInact) = 0
POL(s(x1)) = x1
POL(sInact(x1)) = x1
POL(take(x1, x2)) = 1 + x1 + x2
POL(takeInact(x1, x2)) = 1 + x1 + x2
POL(tt) = 0
POL(uLength(x1, x2)) = x2
POL(uTake1(x1)) = 0
POL(uTake2(x1, x2, x3, x4)) = 1 + x2 + x3 + x4
POL(zeros) = 0
POL(zerosInact) = 0
The following usable rules [17] were oriented:
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
a(0Inact) → 0
isNatIList(IL) → isNatList(a(IL))
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
nil → nilInact
isNat(lengthInact(L)) → isNatList(a(L))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
a(nilInact) → nil
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
cons(x1, x2) → consInact(x1, x2)
zeros → zerosInact
length(x1) → lengthInact(x1)
a(x) → x
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(nilInact) → tt
a(sInact(x1)) → s(a(x1))
uLength(tt, L) → s(length(a(L)))
take(x1, x2) → takeInact(x1, x2)
s(x1) → sInact(x1)
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
a(takeInact(x1, x2)) → take(a(x1), a(x2))
zeros → cons(0, zerosInact)
a(lengthInact(x1)) → length(a(x1))
a(zerosInact) → zeros
isNatIList(zerosInact) → tt
and(tt, T) → T
take(0, IL) → uTake1(isNatIList(IL))
isNat(sInact(N)) → isNat(a(N))
a(consInact(x1, x2)) → cons(a(x1), x2)
0 → 0Inact
uTake1(tt) → nil
isNat(0Inact) → tt
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
LENGTH(cons(N, L)) → ISNAT(N)
ISNATLIST(takeInact(N, IL)) → ISNATILIST(a(IL))
ISNAT(sInact(N)) → ISNAT(a(N))
LENGTH(cons(N, L)) → ISNATLIST(a(L))
A(sInact(x1)) → A(x1)
UTAKE2(tt, M, N, IL) → A(N)
ISNATLIST(consInact(N, L)) → A(N)
ULENGTH(tt, L) → LENGTH(a(L))
UTAKE2(tt, M, N, IL) → A(M)
ISNATILIST(consInact(N, IL)) → ISNATILIST(a(IL))
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(a(L))), a(L))
TAKE(s(M), cons(N, IL)) → ISNATILIST(a(IL))
A(lengthInact(x1)) → A(x1)
ISNATLIST(consInact(N, L)) → ISNAT(a(N))
LENGTH(cons(N, L)) → A(L)
ISNAT(sInact(N)) → A(N)
A(consInact(x1, x2)) → A(x1)
ISNATLIST(consInact(N, L)) → A(L)
ISNAT(lengthInact(L)) → ISNATLIST(a(L))
ISNATLIST(consInact(N, L)) → ISNATLIST(a(L))
ISNAT(lengthInact(L)) → A(L)
UTAKE2(tt, M, N, IL) → A(IL)
A(lengthInact(x1)) → LENGTH(a(x1))
ULENGTH(tt, L) → A(L)
TAKE(0, IL) → ISNATILIST(IL)
A(takeInact(x1, x2)) → TAKE(a(x1), a(x2))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 7 less nodes.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(consInact(N, IL)) → ISNATILIST(a(IL))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATILIST(consInact(N, IL)) → ISNATILIST(a(IL)) at position [0] we obtained the following new rules:
ISNATILIST(consInact(y0, takeInact(x0, x1))) → ISNATILIST(take(a(x0), a(x1)))
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(zeros)
ISNATILIST(consInact(y0, consInact(x0, x1))) → ISNATILIST(cons(a(x0), x1))
ISNATILIST(consInact(y0, 0Inact)) → ISNATILIST(0)
ISNATILIST(consInact(y0, nilInact)) → ISNATILIST(nil)
ISNATILIST(consInact(y0, lengthInact(x0))) → ISNATILIST(length(a(x0)))
ISNATILIST(consInact(y0, x0)) → ISNATILIST(x0)
ISNATILIST(consInact(y0, sInact(x0))) → ISNATILIST(s(a(x0)))
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(consInact(y0, takeInact(x0, x1))) → ISNATILIST(take(a(x0), a(x1)))
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(zeros)
ISNATILIST(consInact(y0, consInact(x0, x1))) → ISNATILIST(cons(a(x0), x1))
ISNATILIST(consInact(y0, nilInact)) → ISNATILIST(nil)
ISNATILIST(consInact(y0, 0Inact)) → ISNATILIST(0)
ISNATILIST(consInact(y0, lengthInact(x0))) → ISNATILIST(length(a(x0)))
ISNATILIST(consInact(y0, x0)) → ISNATILIST(x0)
ISNATILIST(consInact(y0, sInact(x0))) → ISNATILIST(s(a(x0)))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATILIST(consInact(y0, 0Inact)) → ISNATILIST(0) at position [0] we obtained the following new rules:
ISNATILIST(consInact(y0, 0Inact)) → ISNATILIST(0Inact)
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(consInact(y0, takeInact(x0, x1))) → ISNATILIST(take(a(x0), a(x1)))
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(zeros)
ISNATILIST(consInact(y0, consInact(x0, x1))) → ISNATILIST(cons(a(x0), x1))
ISNATILIST(consInact(y0, nilInact)) → ISNATILIST(nil)
ISNATILIST(consInact(y0, lengthInact(x0))) → ISNATILIST(length(a(x0)))
ISNATILIST(consInact(y0, x0)) → ISNATILIST(x0)
ISNATILIST(consInact(y0, 0Inact)) → ISNATILIST(0Inact)
ISNATILIST(consInact(y0, sInact(x0))) → ISNATILIST(s(a(x0)))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(consInact(y0, takeInact(x0, x1))) → ISNATILIST(take(a(x0), a(x1)))
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(zeros)
ISNATILIST(consInact(y0, consInact(x0, x1))) → ISNATILIST(cons(a(x0), x1))
ISNATILIST(consInact(y0, nilInact)) → ISNATILIST(nil)
ISNATILIST(consInact(y0, lengthInact(x0))) → ISNATILIST(length(a(x0)))
ISNATILIST(consInact(y0, x0)) → ISNATILIST(x0)
ISNATILIST(consInact(y0, sInact(x0))) → ISNATILIST(s(a(x0)))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATILIST(consInact(y0, nilInact)) → ISNATILIST(nil) at position [0] we obtained the following new rules:
ISNATILIST(consInact(y0, nilInact)) → ISNATILIST(nilInact)
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(consInact(y0, takeInact(x0, x1))) → ISNATILIST(take(a(x0), a(x1)))
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(zeros)
ISNATILIST(consInact(y0, nilInact)) → ISNATILIST(nilInact)
ISNATILIST(consInact(y0, consInact(x0, x1))) → ISNATILIST(cons(a(x0), x1))
ISNATILIST(consInact(y0, lengthInact(x0))) → ISNATILIST(length(a(x0)))
ISNATILIST(consInact(y0, x0)) → ISNATILIST(x0)
ISNATILIST(consInact(y0, sInact(x0))) → ISNATILIST(s(a(x0)))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(consInact(y0, takeInact(x0, x1))) → ISNATILIST(take(a(x0), a(x1)))
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(zeros)
ISNATILIST(consInact(y0, consInact(x0, x1))) → ISNATILIST(cons(a(x0), x1))
ISNATILIST(consInact(y0, lengthInact(x0))) → ISNATILIST(length(a(x0)))
ISNATILIST(consInact(y0, x0)) → ISNATILIST(x0)
ISNATILIST(consInact(y0, sInact(x0))) → ISNATILIST(s(a(x0)))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ISNATILIST(consInact(y0, consInact(x0, x1))) → ISNATILIST(cons(a(x0), x1))
ISNATILIST(consInact(y0, lengthInact(x0))) → ISNATILIST(length(a(x0)))
ISNATILIST(consInact(y0, x0)) → ISNATILIST(x0)
ISNATILIST(consInact(y0, sInact(x0))) → ISNATILIST(s(a(x0)))
The remaining pairs can at least be oriented weakly.
ISNATILIST(consInact(y0, takeInact(x0, x1))) → ISNATILIST(take(a(x0), a(x1)))
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(zeros)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(0Inact) = 0
POL(ISNATILIST(x1)) = x1
POL(a(x1)) = 1 + x1
POL(and(x1, x2)) = x2
POL(cons(x1, x2)) = 1 + x2
POL(consInact(x1, x2)) = 1 + x2
POL(isNat(x1)) = 1
POL(isNatIList(x1)) = 0
POL(isNatList(x1)) = 0
POL(length(x1)) = 1
POL(lengthInact(x1)) = 1
POL(nil) = 1
POL(nilInact) = 0
POL(s(x1)) = 1
POL(sInact(x1)) = 1
POL(take(x1, x2)) = 1
POL(takeInact(x1, x2)) = 0
POL(tt) = 0
POL(uLength(x1, x2)) = 1
POL(uTake1(x1)) = 1
POL(uTake2(x1, x2, x3, x4)) = 1
POL(zeros) = 1
POL(zerosInact) = 0
The following usable rules [17] were oriented:
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatIList(IL) → isNatList(a(IL))
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
nil → nilInact
isNat(lengthInact(L)) → isNatList(a(L))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
cons(x1, x2) → consInact(x1, x2)
length(x1) → lengthInact(x1)
zeros → zerosInact
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
a(sInact(x1)) → s(a(x1))
isNatList(nilInact) → tt
uLength(tt, L) → s(length(a(L)))
take(x1, x2) → takeInact(x1, x2)
s(x1) → sInact(x1)
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
a(takeInact(x1, x2)) → take(a(x1), a(x2))
zeros → cons(0, zerosInact)
a(lengthInact(x1)) → length(a(x1))
isNatIList(zerosInact) → tt
and(tt, T) → T
take(0, IL) → uTake1(isNatIList(IL))
isNat(sInact(N)) → isNat(a(N))
a(consInact(x1, x2)) → cons(a(x1), x2)
uTake1(tt) → nil
isNat(0Inact) → tt
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(consInact(y0, takeInact(x0, x1))) → ISNATILIST(take(a(x0), a(x1)))
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(zeros)
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ISNATILIST(consInact(y0, takeInact(x0, x1))) → ISNATILIST(take(a(x0), a(x1)))
The remaining pairs can at least be oriented weakly.
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(zeros)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
| M( consInact(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( cons(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( isNatList(x1) ) = | | + | | · | x1 |
| M( uLength(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( and(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( uTake2(x1, ..., x4) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| M( takeInact(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( lengthInact(x1) ) = | | + | | · | x1 |
| M( isNatIList(x1) ) = | | + | | · | x1 |
| M( take(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
Tuple symbols:
| M( ISNATILIST(x1) ) = | 0 | + | | · | x1 |
Matrix type:
We used a basic matrix type which is not further parametrizeable.
As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
a(0Inact) → 0
isNatIList(IL) → isNatList(a(IL))
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
nil → nilInact
isNat(lengthInact(L)) → isNatList(a(L))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
a(nilInact) → nil
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
cons(x1, x2) → consInact(x1, x2)
zeros → zerosInact
length(x1) → lengthInact(x1)
a(x) → x
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
a(sInact(x1)) → s(a(x1))
isNatList(nilInact) → tt
uLength(tt, L) → s(length(a(L)))
take(x1, x2) → takeInact(x1, x2)
s(x1) → sInact(x1)
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
a(takeInact(x1, x2)) → take(a(x1), a(x2))
zeros → cons(0, zerosInact)
a(lengthInact(x1)) → length(a(x1))
a(zerosInact) → zeros
isNatIList(zerosInact) → tt
and(tt, T) → T
take(0, IL) → uTake1(isNatIList(IL))
isNat(sInact(N)) → isNat(a(N))
a(consInact(x1, x2)) → cons(a(x1), x2)
0 → 0Inact
uTake1(tt) → nil
isNat(0Inact) → tt
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(zeros)
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(zeros)
The TRS R consists of the following rules:
zeros → cons(0, zerosInact)
zeros → zerosInact
0 → 0Inact
cons(x1, x2) → consInact(x1, x2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented rules of the TRS R:
zeros → zerosInact
Used ordering: POLO with Polynomial interpretation [25]:
POL(0) = 0
POL(0Inact) = 0
POL(ISNATILIST(x1)) = 2·x1
POL(cons(x1, x2)) = 1 + 2·x1 + 2·x2
POL(consInact(x1, x2)) = 1 + 2·x1 + 2·x2
POL(zeros) = 1
POL(zerosInact) = 0
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(zeros)
The TRS R consists of the following rules:
zeros → cons(0, zerosInact)
0 → 0Inact
cons(x1, x2) → consInact(x1, x2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the modular non-overlap check [15] to enlarge Q to all left-hand sides of R.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(zeros)
The TRS R consists of the following rules:
zeros → cons(0, zerosInact)
0 → 0Inact
cons(x1, x2) → consInact(x1, x2)
The set Q consists of the following terms:
zeros
0
cons(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(zeros) at position [0] we obtained the following new rules:
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(cons(0, zerosInact))
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(cons(0, zerosInact))
The TRS R consists of the following rules:
zeros → cons(0, zerosInact)
0 → 0Inact
cons(x1, x2) → consInact(x1, x2)
The set Q consists of the following terms:
zeros
0
cons(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(cons(0, zerosInact))
The TRS R consists of the following rules:
0 → 0Inact
cons(x1, x2) → consInact(x1, x2)
The set Q consists of the following terms:
zeros
0
cons(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
zeros
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(cons(0, zerosInact))
The TRS R consists of the following rules:
0 → 0Inact
cons(x1, x2) → consInact(x1, x2)
The set Q consists of the following terms:
0
cons(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(cons(0, zerosInact)) at position [0] we obtained the following new rules:
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(consInact(0, zerosInact))
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(consInact(0, zerosInact))
The TRS R consists of the following rules:
0 → 0Inact
cons(x1, x2) → consInact(x1, x2)
The set Q consists of the following terms:
0
cons(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(consInact(0, zerosInact))
The TRS R consists of the following rules:
0 → 0Inact
The set Q consists of the following terms:
0
cons(x0, x1)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
cons(x0, x1)
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(consInact(0, zerosInact))
The TRS R consists of the following rules:
0 → 0Inact
The set Q consists of the following terms:
0
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(consInact(0, zerosInact)) at position [0,0] we obtained the following new rules:
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(consInact(0Inact, zerosInact))
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(consInact(0Inact, zerosInact))
The TRS R consists of the following rules:
0 → 0Inact
The set Q consists of the following terms:
0
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(consInact(0Inact, zerosInact))
R is empty.
The set Q consists of the following terms:
0
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
0
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(consInact(0Inact, zerosInact))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule ISNATILIST(consInact(y0, zerosInact)) → ISNATILIST(consInact(0Inact, zerosInact)) we obtained the following new rules:
ISNATILIST(consInact(0Inact, zerosInact)) → ISNATILIST(consInact(0Inact, zerosInact))
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ NonTerminationProof
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(consInact(0Inact, zerosInact)) → ISNATILIST(consInact(0Inact, zerosInact))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.
The TRS P consists of the following rules:
ISNATILIST(consInact(0Inact, zerosInact)) → ISNATILIST(consInact(0Inact, zerosInact))
The TRS R consists of the following rules:none
s = ISNATILIST(consInact(0Inact, zerosInact)) evaluates to t =ISNATILIST(consInact(0Inact, zerosInact))
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Semiunifier: [ ]
- Matcher: [ ]
Rewriting sequence
The DP semiunifies directly so there is only one rewrite step from ISNATILIST(consInact(0Inact, zerosInact)) to ISNATILIST(consInact(0Inact, zerosInact)).
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(N, L)) → ISNAT(a(N))
LENGTH(cons(N, L)) → ISNAT(N)
LENGTH(cons(N, L)) → A(L)
ISNAT(sInact(N)) → A(N)
A(consInact(x1, x2)) → A(x1)
ISNATLIST(consInact(N, L)) → A(L)
ISNATLIST(consInact(N, L)) → ISNATLIST(a(L))
ISNAT(lengthInact(L)) → ISNATLIST(a(L))
ISNAT(lengthInact(L)) → A(L)
LENGTH(cons(N, L)) → ISNATLIST(a(L))
ISNAT(sInact(N)) → ISNAT(a(N))
A(sInact(x1)) → A(x1)
ISNATLIST(consInact(N, L)) → A(N)
A(lengthInact(x1)) → LENGTH(a(x1))
ULENGTH(tt, L) → LENGTH(a(L))
ULENGTH(tt, L) → A(L)
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(a(L))), a(L))
A(lengthInact(x1)) → A(x1)
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ISNAT(lengthInact(L)) → ISNATLIST(a(L))
ISNAT(lengthInact(L)) → A(L)
A(lengthInact(x1)) → LENGTH(a(x1))
A(lengthInact(x1)) → A(x1)
The remaining pairs can at least be oriented weakly.
ISNATLIST(consInact(N, L)) → ISNAT(a(N))
LENGTH(cons(N, L)) → ISNAT(N)
LENGTH(cons(N, L)) → A(L)
ISNAT(sInact(N)) → A(N)
A(consInact(x1, x2)) → A(x1)
ISNATLIST(consInact(N, L)) → A(L)
ISNATLIST(consInact(N, L)) → ISNATLIST(a(L))
LENGTH(cons(N, L)) → ISNATLIST(a(L))
ISNAT(sInact(N)) → ISNAT(a(N))
A(sInact(x1)) → A(x1)
ISNATLIST(consInact(N, L)) → A(N)
ULENGTH(tt, L) → LENGTH(a(L))
ULENGTH(tt, L) → A(L)
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(a(L))), a(L))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(0Inact) = 0
POL(A(x1)) = 1 + x1
POL(ISNAT(x1)) = 1 + x1
POL(ISNATLIST(x1)) = 1 + x1
POL(LENGTH(x1)) = 1 + x1
POL(ULENGTH(x1, x2)) = 1 + x2
POL(a(x1)) = x1
POL(and(x1, x2)) = x2
POL(cons(x1, x2)) = x1 + x2
POL(consInact(x1, x2)) = x1 + x2
POL(isNat(x1)) = 1
POL(isNatIList(x1)) = 1
POL(isNatList(x1)) = 1
POL(length(x1)) = 1 + x1
POL(lengthInact(x1)) = 1 + x1
POL(nil) = 0
POL(nilInact) = 0
POL(s(x1)) = x1
POL(sInact(x1)) = x1
POL(take(x1, x2)) = 1 + x2
POL(takeInact(x1, x2)) = 1 + x2
POL(tt) = 1
POL(uLength(x1, x2)) = x1 + x2
POL(uTake1(x1)) = 0
POL(uTake2(x1, x2, x3, x4)) = 1 + x3 + x4
POL(zeros) = 0
POL(zerosInact) = 0
The following usable rules [17] were oriented:
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
a(0Inact) → 0
isNatIList(IL) → isNatList(a(IL))
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
nil → nilInact
isNat(lengthInact(L)) → isNatList(a(L))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
a(nilInact) → nil
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
cons(x1, x2) → consInact(x1, x2)
zeros → zerosInact
length(x1) → lengthInact(x1)
a(x) → x
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
a(sInact(x1)) → s(a(x1))
isNatList(nilInact) → tt
uLength(tt, L) → s(length(a(L)))
take(x1, x2) → takeInact(x1, x2)
s(x1) → sInact(x1)
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
a(takeInact(x1, x2)) → take(a(x1), a(x2))
zeros → cons(0, zerosInact)
a(zerosInact) → zeros
a(lengthInact(x1)) → length(a(x1))
and(tt, T) → T
isNatIList(zerosInact) → tt
take(0, IL) → uTake1(isNatIList(IL))
isNat(sInact(N)) → isNat(a(N))
a(consInact(x1, x2)) → cons(a(x1), x2)
0 → 0Inact
uTake1(tt) → nil
isNat(0Inact) → tt
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(N, L)) → ISNAT(a(N))
LENGTH(cons(N, L)) → ISNAT(N)
LENGTH(cons(N, L)) → A(L)
ISNAT(sInact(N)) → A(N)
A(consInact(x1, x2)) → A(x1)
ISNATLIST(consInact(N, L)) → A(L)
ISNATLIST(consInact(N, L)) → ISNATLIST(a(L))
LENGTH(cons(N, L)) → ISNATLIST(a(L))
ISNAT(sInact(N)) → ISNAT(a(N))
A(sInact(x1)) → A(x1)
ISNATLIST(consInact(N, L)) → A(N)
ULENGTH(tt, L) → LENGTH(a(L))
ULENGTH(tt, L) → A(L)
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(a(L))), a(L))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 4 SCCs with 8 less nodes.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
A(consInact(x1, x2)) → A(x1)
A(sInact(x1)) → A(x1)
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
A(consInact(x1, x2)) → A(x1)
A(sInact(x1)) → A(x1)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- A(consInact(x1, x2)) → A(x1)
The graph contains the following edges 1 > 1
- A(sInact(x1)) → A(x1)
The graph contains the following edges 1 > 1
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNAT(sInact(N)) → ISNAT(a(N))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ISNAT(sInact(N)) → ISNAT(a(N))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
| M( consInact(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( cons(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( isNatList(x1) ) = | | + | | · | x1 |
| M( uLength(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( and(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( uTake2(x1, ..., x4) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 | + | | · | x4 |
| M( takeInact(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( lengthInact(x1) ) = | | + | | · | x1 |
| M( isNatIList(x1) ) = | | + | | · | x1 |
| M( take(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
Tuple symbols:
Matrix type:
We used a basic matrix type which is not further parametrizeable.
As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
a(0Inact) → 0
isNatIList(IL) → isNatList(a(IL))
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
nil → nilInact
isNat(lengthInact(L)) → isNatList(a(L))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
a(nilInact) → nil
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
cons(x1, x2) → consInact(x1, x2)
zeros → zerosInact
length(x1) → lengthInact(x1)
a(x) → x
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
a(sInact(x1)) → s(a(x1))
isNatList(nilInact) → tt
uLength(tt, L) → s(length(a(L)))
take(x1, x2) → takeInact(x1, x2)
s(x1) → sInact(x1)
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
a(takeInact(x1, x2)) → take(a(x1), a(x2))
zeros → cons(0, zerosInact)
a(lengthInact(x1)) → length(a(x1))
a(zerosInact) → zeros
isNatIList(zerosInact) → tt
and(tt, T) → T
take(0, IL) → uTake1(isNatIList(IL))
isNat(sInact(N)) → isNat(a(N))
a(consInact(x1, x2)) → cons(a(x1), x2)
0 → 0Inact
uTake1(tt) → nil
isNat(0Inact) → tt
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
P is empty.
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(N, L)) → ISNATLIST(a(L))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATLIST(consInact(N, L)) → ISNATLIST(a(L)) at position [0] we obtained the following new rules:
ISNATLIST(consInact(y0, consInact(x0, x1))) → ISNATLIST(cons(a(x0), x1))
ISNATLIST(consInact(y0, takeInact(x0, x1))) → ISNATLIST(take(a(x0), a(x1)))
ISNATLIST(consInact(y0, lengthInact(x0))) → ISNATLIST(length(a(x0)))
ISNATLIST(consInact(y0, sInact(x0))) → ISNATLIST(s(a(x0)))
ISNATLIST(consInact(y0, 0Inact)) → ISNATLIST(0)
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(zeros)
ISNATLIST(consInact(y0, x0)) → ISNATLIST(x0)
ISNATLIST(consInact(y0, nilInact)) → ISNATLIST(nil)
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, consInact(x0, x1))) → ISNATLIST(cons(a(x0), x1))
ISNATLIST(consInact(y0, takeInact(x0, x1))) → ISNATLIST(take(a(x0), a(x1)))
ISNATLIST(consInact(y0, lengthInact(x0))) → ISNATLIST(length(a(x0)))
ISNATLIST(consInact(y0, 0Inact)) → ISNATLIST(0)
ISNATLIST(consInact(y0, sInact(x0))) → ISNATLIST(s(a(x0)))
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(zeros)
ISNATLIST(consInact(y0, x0)) → ISNATLIST(x0)
ISNATLIST(consInact(y0, nilInact)) → ISNATLIST(nil)
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATLIST(consInact(y0, 0Inact)) → ISNATLIST(0) at position [0] we obtained the following new rules:
ISNATLIST(consInact(y0, 0Inact)) → ISNATLIST(0Inact)
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, consInact(x0, x1))) → ISNATLIST(cons(a(x0), x1))
ISNATLIST(consInact(y0, takeInact(x0, x1))) → ISNATLIST(take(a(x0), a(x1)))
ISNATLIST(consInact(y0, lengthInact(x0))) → ISNATLIST(length(a(x0)))
ISNATLIST(consInact(y0, sInact(x0))) → ISNATLIST(s(a(x0)))
ISNATLIST(consInact(y0, 0Inact)) → ISNATLIST(0Inact)
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(zeros)
ISNATLIST(consInact(y0, x0)) → ISNATLIST(x0)
ISNATLIST(consInact(y0, nilInact)) → ISNATLIST(nil)
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, consInact(x0, x1))) → ISNATLIST(cons(a(x0), x1))
ISNATLIST(consInact(y0, takeInact(x0, x1))) → ISNATLIST(take(a(x0), a(x1)))
ISNATLIST(consInact(y0, lengthInact(x0))) → ISNATLIST(length(a(x0)))
ISNATLIST(consInact(y0, sInact(x0))) → ISNATLIST(s(a(x0)))
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(zeros)
ISNATLIST(consInact(y0, x0)) → ISNATLIST(x0)
ISNATLIST(consInact(y0, nilInact)) → ISNATLIST(nil)
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ISNATLIST(consInact(y0, nilInact)) → ISNATLIST(nil) at position [0] we obtained the following new rules:
ISNATLIST(consInact(y0, nilInact)) → ISNATLIST(nilInact)
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, nilInact)) → ISNATLIST(nilInact)
ISNATLIST(consInact(y0, consInact(x0, x1))) → ISNATLIST(cons(a(x0), x1))
ISNATLIST(consInact(y0, takeInact(x0, x1))) → ISNATLIST(take(a(x0), a(x1)))
ISNATLIST(consInact(y0, lengthInact(x0))) → ISNATLIST(length(a(x0)))
ISNATLIST(consInact(y0, sInact(x0))) → ISNATLIST(s(a(x0)))
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(zeros)
ISNATLIST(consInact(y0, x0)) → ISNATLIST(x0)
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(consInact(y0, consInact(x0, x1))) → ISNATLIST(cons(a(x0), x1))
ISNATLIST(consInact(y0, takeInact(x0, x1))) → ISNATLIST(take(a(x0), a(x1)))
ISNATLIST(consInact(y0, lengthInact(x0))) → ISNATLIST(length(a(x0)))
ISNATLIST(consInact(y0, sInact(x0))) → ISNATLIST(s(a(x0)))
ISNATLIST(consInact(y0, zerosInact)) → ISNATLIST(zeros)
ISNATLIST(consInact(y0, x0)) → ISNATLIST(x0)
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Complete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ULENGTH(tt, L) → LENGTH(a(L))
LENGTH(cons(N, L)) → ULENGTH(and(isNat(N), isNatList(a(L))), a(L))
The TRS R consists of the following rules:
and(tt, T) → T
isNatIList(IL) → isNatList(a(IL))
isNat(0Inact) → tt
isNat(sInact(N)) → isNat(a(N))
isNat(lengthInact(L)) → isNatList(a(L))
isNatIList(zerosInact) → tt
isNatIList(consInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
isNatList(nilInact) → tt
isNatList(consInact(N, L)) → and(isNat(a(N)), isNatList(a(L)))
isNatList(takeInact(N, IL)) → and(isNat(a(N)), isNatIList(a(IL)))
zeros → cons(0, zerosInact)
take(0, IL) → uTake1(isNatIList(IL))
uTake1(tt) → nil
take(s(M), cons(N, IL)) → uTake2(and(isNat(M), and(isNat(N), isNatIList(a(IL)))), M, N, a(IL))
uTake2(tt, M, N, IL) → cons(a(N), takeInact(a(M), a(IL)))
length(cons(N, L)) → uLength(and(isNat(N), isNatList(a(L))), a(L))
uLength(tt, L) → s(length(a(L)))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(a(x1))
0 → 0Inact
a(0Inact) → 0
nil → nilInact
a(nilInact) → nil
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(a(x1), x2)
take(x1, x2) → takeInact(x1, x2)
a(takeInact(x1, x2)) → take(a(x1), a(x2))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We applied the Complete Giesl Middeldorp [11] to transform the context-sensitive TRS to an usual TRS.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(x1, x2)) → and(active(x1), x2)
and(mark(x1), x2) → mark(and(x1, x2))
active(and(x1, x2)) → and(x1, active(x2))
and(x1, mark(x2)) → mark(and(x1, x2))
proper(and(x1, x2)) → and(proper(x1), proper(x2))
and(ok(x1), ok(x2)) → ok(and(x1, x2))
proper(tt) → ok(tt)
proper(isNatIList(x1)) → isNatIList(proper(x1))
isNatIList(ok(x1)) → ok(isNatIList(x1))
proper(isNatList(x1)) → isNatList(proper(x1))
isNatList(ok(x1)) → ok(isNatList(x1))
proper(isNat(x1)) → isNat(proper(x1))
isNat(ok(x1)) → ok(isNat(x1))
proper(0) → ok(0)
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(nil) → ok(nil)
active(take(x1, x2)) → take(active(x1), x2)
take(mark(x1), x2) → mark(take(x1, x2))
active(take(x1, x2)) → take(x1, active(x2))
take(x1, mark(x2)) → mark(take(x1, x2))
proper(take(x1, x2)) → take(proper(x1), proper(x2))
take(ok(x1), ok(x2)) → ok(take(x1, x2))
active(uTake1(x1)) → uTake1(active(x1))
uTake1(mark(x1)) → mark(uTake1(x1))
proper(uTake1(x1)) → uTake1(proper(x1))
uTake1(ok(x1)) → ok(uTake1(x1))
active(uTake2(x1, x2, x3, x4)) → uTake2(active(x1), x2, x3, x4)
uTake2(mark(x1), x2, x3, x4) → mark(uTake2(x1, x2, x3, x4))
proper(uTake2(x1, x2, x3, x4)) → uTake2(proper(x1), proper(x2), proper(x3), proper(x4))
uTake2(ok(x1), ok(x2), ok(x3), ok(x4)) → ok(uTake2(x1, x2, x3, x4))
active(uLength(x1, x2)) → uLength(active(x1), x2)
uLength(mark(x1), x2) → mark(uLength(x1, x2))
proper(uLength(x1, x2)) → uLength(proper(x1), proper(x2))
uLength(ok(x1), ok(x2)) → ok(uLength(x1, x2))
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
ACTIVE(isNatList(take(N, IL))) → ISNAT(N)
PROPER(isNatIList(x1)) → ISNATILIST(proper(x1))
ACTIVE(length(cons(N, L))) → AND(isNat(N), isNatList(L))
ACTIVE(uLength(tt, L)) → S(length(L))
ACTIVE(length(cons(N, L))) → ULENGTH(and(isNat(N), isNatList(L)), L)
ACTIVE(take(s(M), cons(N, IL))) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
PROPER(uTake1(x1)) → UTAKE1(proper(x1))
S(mark(x1)) → S(x1)
ACTIVE(uLength(x1, x2)) → ULENGTH(active(x1), x2)
PROPER(take(x1, x2)) → PROPER(x2)
TOP(mark(x)) → TOP(proper(x))
ACTIVE(and(x1, x2)) → AND(x1, active(x2))
PROPER(uTake1(x1)) → PROPER(x1)
PROPER(uTake2(x1, x2, x3, x4)) → PROPER(x4)
ACTIVE(take(0, IL)) → UTAKE1(isNatIList(IL))
TAKE(ok(x1), ok(x2)) → TAKE(x1, x2)
PROPER(length(x1)) → LENGTH(proper(x1))
PROPER(isNatList(x1)) → PROPER(x1)
ACTIVE(length(cons(N, L))) → ISNAT(N)
PROPER(s(x1)) → PROPER(x1)
ACTIVE(take(s(M), cons(N, IL))) → ISNATILIST(IL)
ACTIVE(take(x1, x2)) → ACTIVE(x1)
ACTIVE(take(x1, x2)) → TAKE(active(x1), x2)
PROPER(take(x1, x2)) → TAKE(proper(x1), proper(x2))
AND(ok(x1), ok(x2)) → AND(x1, x2)
UTAKE2(ok(x1), ok(x2), ok(x3), ok(x4)) → UTAKE2(x1, x2, x3, x4)
ACTIVE(zeros) → CONS(0, zeros)
ACTIVE(isNatList(cons(N, L))) → ISNAT(N)
ULENGTH(ok(x1), ok(x2)) → ULENGTH(x1, x2)
PROPER(uLength(x1, x2)) → PROPER(x2)
PROPER(isNat(x1)) → PROPER(x1)
UTAKE2(mark(x1), x2, x3, x4) → UTAKE2(x1, x2, x3, x4)
ACTIVE(and(x1, x2)) → ACTIVE(x2)
LENGTH(mark(x1)) → LENGTH(x1)
ACTIVE(s(x1)) → ACTIVE(x1)
PROPER(uTake2(x1, x2, x3, x4)) → PROPER(x3)
ACTIVE(isNatList(take(N, IL))) → AND(isNat(N), isNatIList(IL))
TAKE(mark(x1), x2) → TAKE(x1, x2)
ACTIVE(cons(x1, x2)) → CONS(active(x1), x2)
ACTIVE(isNatIList(cons(N, IL))) → ISNATILIST(IL)
ACTIVE(take(x1, x2)) → TAKE(x1, active(x2))
UTAKE1(ok(x1)) → UTAKE1(x1)
ACTIVE(isNatList(cons(N, L))) → AND(isNat(N), isNatList(L))
ISNAT(ok(x1)) → ISNAT(x1)
TOP(ok(x)) → TOP(active(x))
PROPER(cons(x1, x2)) → PROPER(x2)
ACTIVE(isNatList(take(N, IL))) → ISNATILIST(IL)
ACTIVE(uTake2(tt, M, N, IL)) → CONS(N, take(M, IL))
ULENGTH(mark(x1), x2) → ULENGTH(x1, x2)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(M), and(isNat(N), isNatIList(IL)))
TAKE(x1, mark(x2)) → TAKE(x1, x2)
PROPER(and(x1, x2)) → PROPER(x1)
LENGTH(ok(x1)) → LENGTH(x1)
PROPER(length(x1)) → PROPER(x1)
PROPER(isNat(x1)) → ISNAT(proper(x1))
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(M)
PROPER(and(x1, x2)) → AND(proper(x1), proper(x2))
ACTIVE(isNat(length(L))) → ISNATLIST(L)
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(N)
PROPER(isNatIList(x1)) → PROPER(x1)
ACTIVE(and(x1, x2)) → ACTIVE(x1)
PROPER(uTake2(x1, x2, x3, x4)) → UTAKE2(proper(x1), proper(x2), proper(x3), proper(x4))
ACTIVE(isNatList(cons(N, L))) → ISNATLIST(L)
ACTIVE(uLength(tt, L)) → LENGTH(L)
PROPER(cons(x1, x2)) → PROPER(x1)
ACTIVE(isNatIList(cons(N, IL))) → AND(isNat(N), isNatIList(IL))
ACTIVE(uTake1(x1)) → ACTIVE(x1)
TOP(ok(x)) → ACTIVE(x)
ACTIVE(isNatIList(IL)) → ISNATLIST(IL)
PROPER(take(x1, x2)) → PROPER(x1)
ACTIVE(s(x1)) → S(active(x1))
ACTIVE(isNatIList(cons(N, IL))) → ISNAT(N)
ACTIVE(uTake2(tt, M, N, IL)) → TAKE(M, IL)
AND(mark(x1), x2) → AND(x1, x2)
PROPER(uTake2(x1, x2, x3, x4)) → PROPER(x2)
PROPER(uTake2(x1, x2, x3, x4)) → PROPER(x1)
TOP(mark(x)) → PROPER(x)
ACTIVE(isNat(s(N))) → ISNAT(N)
ACTIVE(uTake2(x1, x2, x3, x4)) → ACTIVE(x1)
UTAKE1(mark(x1)) → UTAKE1(x1)
ISNATILIST(ok(x1)) → ISNATILIST(x1)
PROPER(uLength(x1, x2)) → ULENGTH(proper(x1), proper(x2))
ACTIVE(uTake2(x1, x2, x3, x4)) → UTAKE2(active(x1), x2, x3, x4)
ACTIVE(uTake1(x1)) → UTAKE1(active(x1))
ACTIVE(take(x1, x2)) → ACTIVE(x2)
ISNATLIST(ok(x1)) → ISNATLIST(x1)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(N), isNatIList(IL))
PROPER(s(x1)) → S(proper(x1))
AND(x1, mark(x2)) → AND(x1, x2)
S(ok(x1)) → S(x1)
PROPER(isNatList(x1)) → ISNATLIST(proper(x1))
CONS(mark(x1), x2) → CONS(x1, x2)
ACTIVE(length(x1)) → ACTIVE(x1)
ACTIVE(cons(x1, x2)) → ACTIVE(x1)
ACTIVE(length(x1)) → LENGTH(active(x1))
ACTIVE(take(0, IL)) → ISNATILIST(IL)
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
ACTIVE(uLength(x1, x2)) → ACTIVE(x1)
PROPER(uLength(x1, x2)) → PROPER(x1)
PROPER(cons(x1, x2)) → CONS(proper(x1), proper(x2))
CONS(ok(x1), ok(x2)) → CONS(x1, x2)
PROPER(and(x1, x2)) → PROPER(x2)
ACTIVE(and(x1, x2)) → AND(active(x1), x2)
The TRS R consists of the following rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(x1, x2)) → and(active(x1), x2)
and(mark(x1), x2) → mark(and(x1, x2))
active(and(x1, x2)) → and(x1, active(x2))
and(x1, mark(x2)) → mark(and(x1, x2))
proper(and(x1, x2)) → and(proper(x1), proper(x2))
and(ok(x1), ok(x2)) → ok(and(x1, x2))
proper(tt) → ok(tt)
proper(isNatIList(x1)) → isNatIList(proper(x1))
isNatIList(ok(x1)) → ok(isNatIList(x1))
proper(isNatList(x1)) → isNatList(proper(x1))
isNatList(ok(x1)) → ok(isNatList(x1))
proper(isNat(x1)) → isNat(proper(x1))
isNat(ok(x1)) → ok(isNat(x1))
proper(0) → ok(0)
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(nil) → ok(nil)
active(take(x1, x2)) → take(active(x1), x2)
take(mark(x1), x2) → mark(take(x1, x2))
active(take(x1, x2)) → take(x1, active(x2))
take(x1, mark(x2)) → mark(take(x1, x2))
proper(take(x1, x2)) → take(proper(x1), proper(x2))
take(ok(x1), ok(x2)) → ok(take(x1, x2))
active(uTake1(x1)) → uTake1(active(x1))
uTake1(mark(x1)) → mark(uTake1(x1))
proper(uTake1(x1)) → uTake1(proper(x1))
uTake1(ok(x1)) → ok(uTake1(x1))
active(uTake2(x1, x2, x3, x4)) → uTake2(active(x1), x2, x3, x4)
uTake2(mark(x1), x2, x3, x4) → mark(uTake2(x1, x2, x3, x4))
proper(uTake2(x1, x2, x3, x4)) → uTake2(proper(x1), proper(x2), proper(x3), proper(x4))
uTake2(ok(x1), ok(x2), ok(x3), ok(x4)) → ok(uTake2(x1, x2, x3, x4))
active(uLength(x1, x2)) → uLength(active(x1), x2)
uLength(mark(x1), x2) → mark(uLength(x1, x2))
proper(uLength(x1, x2)) → uLength(proper(x1), proper(x2))
uLength(ok(x1), ok(x2)) → ok(uLength(x1, x2))
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
ACTIVE(isNatList(take(N, IL))) → ISNAT(N)
PROPER(isNatIList(x1)) → ISNATILIST(proper(x1))
ACTIVE(length(cons(N, L))) → AND(isNat(N), isNatList(L))
ACTIVE(uLength(tt, L)) → S(length(L))
ACTIVE(length(cons(N, L))) → ULENGTH(and(isNat(N), isNatList(L)), L)
ACTIVE(take(s(M), cons(N, IL))) → UTAKE2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
PROPER(uTake1(x1)) → UTAKE1(proper(x1))
S(mark(x1)) → S(x1)
ACTIVE(uLength(x1, x2)) → ULENGTH(active(x1), x2)
PROPER(take(x1, x2)) → PROPER(x2)
TOP(mark(x)) → TOP(proper(x))
ACTIVE(and(x1, x2)) → AND(x1, active(x2))
PROPER(uTake1(x1)) → PROPER(x1)
PROPER(uTake2(x1, x2, x3, x4)) → PROPER(x4)
ACTIVE(take(0, IL)) → UTAKE1(isNatIList(IL))
TAKE(ok(x1), ok(x2)) → TAKE(x1, x2)
PROPER(length(x1)) → LENGTH(proper(x1))
PROPER(isNatList(x1)) → PROPER(x1)
ACTIVE(length(cons(N, L))) → ISNAT(N)
PROPER(s(x1)) → PROPER(x1)
ACTIVE(take(s(M), cons(N, IL))) → ISNATILIST(IL)
ACTIVE(take(x1, x2)) → ACTIVE(x1)
ACTIVE(take(x1, x2)) → TAKE(active(x1), x2)
PROPER(take(x1, x2)) → TAKE(proper(x1), proper(x2))
AND(ok(x1), ok(x2)) → AND(x1, x2)
UTAKE2(ok(x1), ok(x2), ok(x3), ok(x4)) → UTAKE2(x1, x2, x3, x4)
ACTIVE(zeros) → CONS(0, zeros)
ACTIVE(isNatList(cons(N, L))) → ISNAT(N)
ULENGTH(ok(x1), ok(x2)) → ULENGTH(x1, x2)
PROPER(uLength(x1, x2)) → PROPER(x2)
PROPER(isNat(x1)) → PROPER(x1)
UTAKE2(mark(x1), x2, x3, x4) → UTAKE2(x1, x2, x3, x4)
ACTIVE(and(x1, x2)) → ACTIVE(x2)
LENGTH(mark(x1)) → LENGTH(x1)
ACTIVE(s(x1)) → ACTIVE(x1)
PROPER(uTake2(x1, x2, x3, x4)) → PROPER(x3)
ACTIVE(isNatList(take(N, IL))) → AND(isNat(N), isNatIList(IL))
TAKE(mark(x1), x2) → TAKE(x1, x2)
ACTIVE(cons(x1, x2)) → CONS(active(x1), x2)
ACTIVE(isNatIList(cons(N, IL))) → ISNATILIST(IL)
ACTIVE(take(x1, x2)) → TAKE(x1, active(x2))
UTAKE1(ok(x1)) → UTAKE1(x1)
ACTIVE(isNatList(cons(N, L))) → AND(isNat(N), isNatList(L))
ISNAT(ok(x1)) → ISNAT(x1)
TOP(ok(x)) → TOP(active(x))
PROPER(cons(x1, x2)) → PROPER(x2)
ACTIVE(isNatList(take(N, IL))) → ISNATILIST(IL)
ACTIVE(uTake2(tt, M, N, IL)) → CONS(N, take(M, IL))
ULENGTH(mark(x1), x2) → ULENGTH(x1, x2)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(M), and(isNat(N), isNatIList(IL)))
TAKE(x1, mark(x2)) → TAKE(x1, x2)
PROPER(and(x1, x2)) → PROPER(x1)
LENGTH(ok(x1)) → LENGTH(x1)
PROPER(length(x1)) → PROPER(x1)
PROPER(isNat(x1)) → ISNAT(proper(x1))
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(M)
PROPER(and(x1, x2)) → AND(proper(x1), proper(x2))
ACTIVE(isNat(length(L))) → ISNATLIST(L)
ACTIVE(take(s(M), cons(N, IL))) → ISNAT(N)
PROPER(isNatIList(x1)) → PROPER(x1)
ACTIVE(and(x1, x2)) → ACTIVE(x1)
PROPER(uTake2(x1, x2, x3, x4)) → UTAKE2(proper(x1), proper(x2), proper(x3), proper(x4))
ACTIVE(isNatList(cons(N, L))) → ISNATLIST(L)
ACTIVE(uLength(tt, L)) → LENGTH(L)
PROPER(cons(x1, x2)) → PROPER(x1)
ACTIVE(isNatIList(cons(N, IL))) → AND(isNat(N), isNatIList(IL))
ACTIVE(uTake1(x1)) → ACTIVE(x1)
TOP(ok(x)) → ACTIVE(x)
ACTIVE(isNatIList(IL)) → ISNATLIST(IL)
PROPER(take(x1, x2)) → PROPER(x1)
ACTIVE(s(x1)) → S(active(x1))
ACTIVE(isNatIList(cons(N, IL))) → ISNAT(N)
ACTIVE(uTake2(tt, M, N, IL)) → TAKE(M, IL)
AND(mark(x1), x2) → AND(x1, x2)
PROPER(uTake2(x1, x2, x3, x4)) → PROPER(x2)
PROPER(uTake2(x1, x2, x3, x4)) → PROPER(x1)
TOP(mark(x)) → PROPER(x)
ACTIVE(isNat(s(N))) → ISNAT(N)
ACTIVE(uTake2(x1, x2, x3, x4)) → ACTIVE(x1)
UTAKE1(mark(x1)) → UTAKE1(x1)
ISNATILIST(ok(x1)) → ISNATILIST(x1)
PROPER(uLength(x1, x2)) → ULENGTH(proper(x1), proper(x2))
ACTIVE(uTake2(x1, x2, x3, x4)) → UTAKE2(active(x1), x2, x3, x4)
ACTIVE(uTake1(x1)) → UTAKE1(active(x1))
ACTIVE(take(x1, x2)) → ACTIVE(x2)
ISNATLIST(ok(x1)) → ISNATLIST(x1)
ACTIVE(take(s(M), cons(N, IL))) → AND(isNat(N), isNatIList(IL))
PROPER(s(x1)) → S(proper(x1))
AND(x1, mark(x2)) → AND(x1, x2)
S(ok(x1)) → S(x1)
PROPER(isNatList(x1)) → ISNATLIST(proper(x1))
CONS(mark(x1), x2) → CONS(x1, x2)
ACTIVE(length(x1)) → ACTIVE(x1)
ACTIVE(cons(x1, x2)) → ACTIVE(x1)
ACTIVE(length(x1)) → LENGTH(active(x1))
ACTIVE(take(0, IL)) → ISNATILIST(IL)
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
ACTIVE(uLength(x1, x2)) → ACTIVE(x1)
PROPER(uLength(x1, x2)) → PROPER(x1)
PROPER(cons(x1, x2)) → CONS(proper(x1), proper(x2))
CONS(ok(x1), ok(x2)) → CONS(x1, x2)
PROPER(and(x1, x2)) → PROPER(x2)
ACTIVE(and(x1, x2)) → AND(active(x1), x2)
The TRS R consists of the following rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(x1, x2)) → and(active(x1), x2)
and(mark(x1), x2) → mark(and(x1, x2))
active(and(x1, x2)) → and(x1, active(x2))
and(x1, mark(x2)) → mark(and(x1, x2))
proper(and(x1, x2)) → and(proper(x1), proper(x2))
and(ok(x1), ok(x2)) → ok(and(x1, x2))
proper(tt) → ok(tt)
proper(isNatIList(x1)) → isNatIList(proper(x1))
isNatIList(ok(x1)) → ok(isNatIList(x1))
proper(isNatList(x1)) → isNatList(proper(x1))
isNatList(ok(x1)) → ok(isNatList(x1))
proper(isNat(x1)) → isNat(proper(x1))
isNat(ok(x1)) → ok(isNat(x1))
proper(0) → ok(0)
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(nil) → ok(nil)
active(take(x1, x2)) → take(active(x1), x2)
take(mark(x1), x2) → mark(take(x1, x2))
active(take(x1, x2)) → take(x1, active(x2))
take(x1, mark(x2)) → mark(take(x1, x2))
proper(take(x1, x2)) → take(proper(x1), proper(x2))
take(ok(x1), ok(x2)) → ok(take(x1, x2))
active(uTake1(x1)) → uTake1(active(x1))
uTake1(mark(x1)) → mark(uTake1(x1))
proper(uTake1(x1)) → uTake1(proper(x1))
uTake1(ok(x1)) → ok(uTake1(x1))
active(uTake2(x1, x2, x3, x4)) → uTake2(active(x1), x2, x3, x4)
uTake2(mark(x1), x2, x3, x4) → mark(uTake2(x1, x2, x3, x4))
proper(uTake2(x1, x2, x3, x4)) → uTake2(proper(x1), proper(x2), proper(x3), proper(x4))
uTake2(ok(x1), ok(x2), ok(x3), ok(x4)) → ok(uTake2(x1, x2, x3, x4))
active(uLength(x1, x2)) → uLength(active(x1), x2)
uLength(mark(x1), x2) → mark(uLength(x1, x2))
proper(uLength(x1, x2)) → uLength(proper(x1), proper(x2))
uLength(ok(x1), ok(x2)) → ok(uLength(x1, x2))
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 14 SCCs with 52 less nodes.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ULENGTH(mark(x1), x2) → ULENGTH(x1, x2)
ULENGTH(ok(x1), ok(x2)) → ULENGTH(x1, x2)
The TRS R consists of the following rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(x1, x2)) → and(active(x1), x2)
and(mark(x1), x2) → mark(and(x1, x2))
active(and(x1, x2)) → and(x1, active(x2))
and(x1, mark(x2)) → mark(and(x1, x2))
proper(and(x1, x2)) → and(proper(x1), proper(x2))
and(ok(x1), ok(x2)) → ok(and(x1, x2))
proper(tt) → ok(tt)
proper(isNatIList(x1)) → isNatIList(proper(x1))
isNatIList(ok(x1)) → ok(isNatIList(x1))
proper(isNatList(x1)) → isNatList(proper(x1))
isNatList(ok(x1)) → ok(isNatList(x1))
proper(isNat(x1)) → isNat(proper(x1))
isNat(ok(x1)) → ok(isNat(x1))
proper(0) → ok(0)
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(nil) → ok(nil)
active(take(x1, x2)) → take(active(x1), x2)
take(mark(x1), x2) → mark(take(x1, x2))
active(take(x1, x2)) → take(x1, active(x2))
take(x1, mark(x2)) → mark(take(x1, x2))
proper(take(x1, x2)) → take(proper(x1), proper(x2))
take(ok(x1), ok(x2)) → ok(take(x1, x2))
active(uTake1(x1)) → uTake1(active(x1))
uTake1(mark(x1)) → mark(uTake1(x1))
proper(uTake1(x1)) → uTake1(proper(x1))
uTake1(ok(x1)) → ok(uTake1(x1))
active(uTake2(x1, x2, x3, x4)) → uTake2(active(x1), x2, x3, x4)
uTake2(mark(x1), x2, x3, x4) → mark(uTake2(x1, x2, x3, x4))
proper(uTake2(x1, x2, x3, x4)) → uTake2(proper(x1), proper(x2), proper(x3), proper(x4))
uTake2(ok(x1), ok(x2), ok(x3), ok(x4)) → ok(uTake2(x1, x2, x3, x4))
active(uLength(x1, x2)) → uLength(active(x1), x2)
uLength(mark(x1), x2) → mark(uLength(x1, x2))
proper(uLength(x1, x2)) → uLength(proper(x1), proper(x2))
uLength(ok(x1), ok(x2)) → ok(uLength(x1, x2))
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ULENGTH(mark(x1), x2) → ULENGTH(x1, x2)
ULENGTH(ok(x1), ok(x2)) → ULENGTH(x1, x2)
R is empty.
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ULENGTH(mark(x1), x2) → ULENGTH(x1, x2)
ULENGTH(ok(x1), ok(x2)) → ULENGTH(x1, x2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- ULENGTH(mark(x1), x2) → ULENGTH(x1, x2)
The graph contains the following edges 1 > 1, 2 >= 2
- ULENGTH(ok(x1), ok(x2)) → ULENGTH(x1, x2)
The graph contains the following edges 1 > 1, 2 > 2
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
UTAKE2(ok(x1), ok(x2), ok(x3), ok(x4)) → UTAKE2(x1, x2, x3, x4)
UTAKE2(mark(x1), x2, x3, x4) → UTAKE2(x1, x2, x3, x4)
The TRS R consists of the following rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(x1, x2)) → and(active(x1), x2)
and(mark(x1), x2) → mark(and(x1, x2))
active(and(x1, x2)) → and(x1, active(x2))
and(x1, mark(x2)) → mark(and(x1, x2))
proper(and(x1, x2)) → and(proper(x1), proper(x2))
and(ok(x1), ok(x2)) → ok(and(x1, x2))
proper(tt) → ok(tt)
proper(isNatIList(x1)) → isNatIList(proper(x1))
isNatIList(ok(x1)) → ok(isNatIList(x1))
proper(isNatList(x1)) → isNatList(proper(x1))
isNatList(ok(x1)) → ok(isNatList(x1))
proper(isNat(x1)) → isNat(proper(x1))
isNat(ok(x1)) → ok(isNat(x1))
proper(0) → ok(0)
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(nil) → ok(nil)
active(take(x1, x2)) → take(active(x1), x2)
take(mark(x1), x2) → mark(take(x1, x2))
active(take(x1, x2)) → take(x1, active(x2))
take(x1, mark(x2)) → mark(take(x1, x2))
proper(take(x1, x2)) → take(proper(x1), proper(x2))
take(ok(x1), ok(x2)) → ok(take(x1, x2))
active(uTake1(x1)) → uTake1(active(x1))
uTake1(mark(x1)) → mark(uTake1(x1))
proper(uTake1(x1)) → uTake1(proper(x1))
uTake1(ok(x1)) → ok(uTake1(x1))
active(uTake2(x1, x2, x3, x4)) → uTake2(active(x1), x2, x3, x4)
uTake2(mark(x1), x2, x3, x4) → mark(uTake2(x1, x2, x3, x4))
proper(uTake2(x1, x2, x3, x4)) → uTake2(proper(x1), proper(x2), proper(x3), proper(x4))
uTake2(ok(x1), ok(x2), ok(x3), ok(x4)) → ok(uTake2(x1, x2, x3, x4))
active(uLength(x1, x2)) → uLength(active(x1), x2)
uLength(mark(x1), x2) → mark(uLength(x1, x2))
proper(uLength(x1, x2)) → uLength(proper(x1), proper(x2))
uLength(ok(x1), ok(x2)) → ok(uLength(x1, x2))
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
UTAKE2(ok(x1), ok(x2), ok(x3), ok(x4)) → UTAKE2(x1, x2, x3, x4)
UTAKE2(mark(x1), x2, x3, x4) → UTAKE2(x1, x2, x3, x4)
R is empty.
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
UTAKE2(ok(x1), ok(x2), ok(x3), ok(x4)) → UTAKE2(x1, x2, x3, x4)
UTAKE2(mark(x1), x2, x3, x4) → UTAKE2(x1, x2, x3, x4)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- UTAKE2(ok(x1), ok(x2), ok(x3), ok(x4)) → UTAKE2(x1, x2, x3, x4)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3, 4 > 4
- UTAKE2(mark(x1), x2, x3, x4) → UTAKE2(x1, x2, x3, x4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
UTAKE1(ok(x1)) → UTAKE1(x1)
UTAKE1(mark(x1)) → UTAKE1(x1)
The TRS R consists of the following rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(x1, x2)) → and(active(x1), x2)
and(mark(x1), x2) → mark(and(x1, x2))
active(and(x1, x2)) → and(x1, active(x2))
and(x1, mark(x2)) → mark(and(x1, x2))
proper(and(x1, x2)) → and(proper(x1), proper(x2))
and(ok(x1), ok(x2)) → ok(and(x1, x2))
proper(tt) → ok(tt)
proper(isNatIList(x1)) → isNatIList(proper(x1))
isNatIList(ok(x1)) → ok(isNatIList(x1))
proper(isNatList(x1)) → isNatList(proper(x1))
isNatList(ok(x1)) → ok(isNatList(x1))
proper(isNat(x1)) → isNat(proper(x1))
isNat(ok(x1)) → ok(isNat(x1))
proper(0) → ok(0)
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(nil) → ok(nil)
active(take(x1, x2)) → take(active(x1), x2)
take(mark(x1), x2) → mark(take(x1, x2))
active(take(x1, x2)) → take(x1, active(x2))
take(x1, mark(x2)) → mark(take(x1, x2))
proper(take(x1, x2)) → take(proper(x1), proper(x2))
take(ok(x1), ok(x2)) → ok(take(x1, x2))
active(uTake1(x1)) → uTake1(active(x1))
uTake1(mark(x1)) → mark(uTake1(x1))
proper(uTake1(x1)) → uTake1(proper(x1))
uTake1(ok(x1)) → ok(uTake1(x1))
active(uTake2(x1, x2, x3, x4)) → uTake2(active(x1), x2, x3, x4)
uTake2(mark(x1), x2, x3, x4) → mark(uTake2(x1, x2, x3, x4))
proper(uTake2(x1, x2, x3, x4)) → uTake2(proper(x1), proper(x2), proper(x3), proper(x4))
uTake2(ok(x1), ok(x2), ok(x3), ok(x4)) → ok(uTake2(x1, x2, x3, x4))
active(uLength(x1, x2)) → uLength(active(x1), x2)
uLength(mark(x1), x2) → mark(uLength(x1, x2))
proper(uLength(x1, x2)) → uLength(proper(x1), proper(x2))
uLength(ok(x1), ok(x2)) → ok(uLength(x1, x2))
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
UTAKE1(ok(x1)) → UTAKE1(x1)
UTAKE1(mark(x1)) → UTAKE1(x1)
R is empty.
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
UTAKE1(ok(x1)) → UTAKE1(x1)
UTAKE1(mark(x1)) → UTAKE1(x1)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- UTAKE1(ok(x1)) → UTAKE1(x1)
The graph contains the following edges 1 > 1
- UTAKE1(mark(x1)) → UTAKE1(x1)
The graph contains the following edges 1 > 1
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
TAKE(mark(x1), x2) → TAKE(x1, x2)
TAKE(x1, mark(x2)) → TAKE(x1, x2)
TAKE(ok(x1), ok(x2)) → TAKE(x1, x2)
The TRS R consists of the following rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(x1, x2)) → and(active(x1), x2)
and(mark(x1), x2) → mark(and(x1, x2))
active(and(x1, x2)) → and(x1, active(x2))
and(x1, mark(x2)) → mark(and(x1, x2))
proper(and(x1, x2)) → and(proper(x1), proper(x2))
and(ok(x1), ok(x2)) → ok(and(x1, x2))
proper(tt) → ok(tt)
proper(isNatIList(x1)) → isNatIList(proper(x1))
isNatIList(ok(x1)) → ok(isNatIList(x1))
proper(isNatList(x1)) → isNatList(proper(x1))
isNatList(ok(x1)) → ok(isNatList(x1))
proper(isNat(x1)) → isNat(proper(x1))
isNat(ok(x1)) → ok(isNat(x1))
proper(0) → ok(0)
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(nil) → ok(nil)
active(take(x1, x2)) → take(active(x1), x2)
take(mark(x1), x2) → mark(take(x1, x2))
active(take(x1, x2)) → take(x1, active(x2))
take(x1, mark(x2)) → mark(take(x1, x2))
proper(take(x1, x2)) → take(proper(x1), proper(x2))
take(ok(x1), ok(x2)) → ok(take(x1, x2))
active(uTake1(x1)) → uTake1(active(x1))
uTake1(mark(x1)) → mark(uTake1(x1))
proper(uTake1(x1)) → uTake1(proper(x1))
uTake1(ok(x1)) → ok(uTake1(x1))
active(uTake2(x1, x2, x3, x4)) → uTake2(active(x1), x2, x3, x4)
uTake2(mark(x1), x2, x3, x4) → mark(uTake2(x1, x2, x3, x4))
proper(uTake2(x1, x2, x3, x4)) → uTake2(proper(x1), proper(x2), proper(x3), proper(x4))
uTake2(ok(x1), ok(x2), ok(x3), ok(x4)) → ok(uTake2(x1, x2, x3, x4))
active(uLength(x1, x2)) → uLength(active(x1), x2)
uLength(mark(x1), x2) → mark(uLength(x1, x2))
proper(uLength(x1, x2)) → uLength(proper(x1), proper(x2))
uLength(ok(x1), ok(x2)) → ok(uLength(x1, x2))
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
TAKE(mark(x1), x2) → TAKE(x1, x2)
TAKE(x1, mark(x2)) → TAKE(x1, x2)
TAKE(ok(x1), ok(x2)) → TAKE(x1, x2)
R is empty.
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
TAKE(mark(x1), x2) → TAKE(x1, x2)
TAKE(x1, mark(x2)) → TAKE(x1, x2)
TAKE(ok(x1), ok(x2)) → TAKE(x1, x2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- TAKE(mark(x1), x2) → TAKE(x1, x2)
The graph contains the following edges 1 > 1, 2 >= 2
- TAKE(x1, mark(x2)) → TAKE(x1, x2)
The graph contains the following edges 1 >= 1, 2 > 2
- TAKE(ok(x1), ok(x2)) → TAKE(x1, x2)
The graph contains the following edges 1 > 1, 2 > 2
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
CONS(mark(x1), x2) → CONS(x1, x2)
CONS(ok(x1), ok(x2)) → CONS(x1, x2)
The TRS R consists of the following rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(x1, x2)) → and(active(x1), x2)
and(mark(x1), x2) → mark(and(x1, x2))
active(and(x1, x2)) → and(x1, active(x2))
and(x1, mark(x2)) → mark(and(x1, x2))
proper(and(x1, x2)) → and(proper(x1), proper(x2))
and(ok(x1), ok(x2)) → ok(and(x1, x2))
proper(tt) → ok(tt)
proper(isNatIList(x1)) → isNatIList(proper(x1))
isNatIList(ok(x1)) → ok(isNatIList(x1))
proper(isNatList(x1)) → isNatList(proper(x1))
isNatList(ok(x1)) → ok(isNatList(x1))
proper(isNat(x1)) → isNat(proper(x1))
isNat(ok(x1)) → ok(isNat(x1))
proper(0) → ok(0)
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(nil) → ok(nil)
active(take(x1, x2)) → take(active(x1), x2)
take(mark(x1), x2) → mark(take(x1, x2))
active(take(x1, x2)) → take(x1, active(x2))
take(x1, mark(x2)) → mark(take(x1, x2))
proper(take(x1, x2)) → take(proper(x1), proper(x2))
take(ok(x1), ok(x2)) → ok(take(x1, x2))
active(uTake1(x1)) → uTake1(active(x1))
uTake1(mark(x1)) → mark(uTake1(x1))
proper(uTake1(x1)) → uTake1(proper(x1))
uTake1(ok(x1)) → ok(uTake1(x1))
active(uTake2(x1, x2, x3, x4)) → uTake2(active(x1), x2, x3, x4)
uTake2(mark(x1), x2, x3, x4) → mark(uTake2(x1, x2, x3, x4))
proper(uTake2(x1, x2, x3, x4)) → uTake2(proper(x1), proper(x2), proper(x3), proper(x4))
uTake2(ok(x1), ok(x2), ok(x3), ok(x4)) → ok(uTake2(x1, x2, x3, x4))
active(uLength(x1, x2)) → uLength(active(x1), x2)
uLength(mark(x1), x2) → mark(uLength(x1, x2))
proper(uLength(x1, x2)) → uLength(proper(x1), proper(x2))
uLength(ok(x1), ok(x2)) → ok(uLength(x1, x2))
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
CONS(mark(x1), x2) → CONS(x1, x2)
CONS(ok(x1), ok(x2)) → CONS(x1, x2)
R is empty.
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
CONS(mark(x1), x2) → CONS(x1, x2)
CONS(ok(x1), ok(x2)) → CONS(x1, x2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- CONS(mark(x1), x2) → CONS(x1, x2)
The graph contains the following edges 1 > 1, 2 >= 2
- CONS(ok(x1), ok(x2)) → CONS(x1, x2)
The graph contains the following edges 1 > 1, 2 > 2
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
LENGTH(mark(x1)) → LENGTH(x1)
LENGTH(ok(x1)) → LENGTH(x1)
The TRS R consists of the following rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(x1, x2)) → and(active(x1), x2)
and(mark(x1), x2) → mark(and(x1, x2))
active(and(x1, x2)) → and(x1, active(x2))
and(x1, mark(x2)) → mark(and(x1, x2))
proper(and(x1, x2)) → and(proper(x1), proper(x2))
and(ok(x1), ok(x2)) → ok(and(x1, x2))
proper(tt) → ok(tt)
proper(isNatIList(x1)) → isNatIList(proper(x1))
isNatIList(ok(x1)) → ok(isNatIList(x1))
proper(isNatList(x1)) → isNatList(proper(x1))
isNatList(ok(x1)) → ok(isNatList(x1))
proper(isNat(x1)) → isNat(proper(x1))
isNat(ok(x1)) → ok(isNat(x1))
proper(0) → ok(0)
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(nil) → ok(nil)
active(take(x1, x2)) → take(active(x1), x2)
take(mark(x1), x2) → mark(take(x1, x2))
active(take(x1, x2)) → take(x1, active(x2))
take(x1, mark(x2)) → mark(take(x1, x2))
proper(take(x1, x2)) → take(proper(x1), proper(x2))
take(ok(x1), ok(x2)) → ok(take(x1, x2))
active(uTake1(x1)) → uTake1(active(x1))
uTake1(mark(x1)) → mark(uTake1(x1))
proper(uTake1(x1)) → uTake1(proper(x1))
uTake1(ok(x1)) → ok(uTake1(x1))
active(uTake2(x1, x2, x3, x4)) → uTake2(active(x1), x2, x3, x4)
uTake2(mark(x1), x2, x3, x4) → mark(uTake2(x1, x2, x3, x4))
proper(uTake2(x1, x2, x3, x4)) → uTake2(proper(x1), proper(x2), proper(x3), proper(x4))
uTake2(ok(x1), ok(x2), ok(x3), ok(x4)) → ok(uTake2(x1, x2, x3, x4))
active(uLength(x1, x2)) → uLength(active(x1), x2)
uLength(mark(x1), x2) → mark(uLength(x1, x2))
proper(uLength(x1, x2)) → uLength(proper(x1), proper(x2))
uLength(ok(x1), ok(x2)) → ok(uLength(x1, x2))
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
LENGTH(mark(x1)) → LENGTH(x1)
LENGTH(ok(x1)) → LENGTH(x1)
R is empty.
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
LENGTH(ok(x1)) → LENGTH(x1)
LENGTH(mark(x1)) → LENGTH(x1)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- LENGTH(mark(x1)) → LENGTH(x1)
The graph contains the following edges 1 > 1
- LENGTH(ok(x1)) → LENGTH(x1)
The graph contains the following edges 1 > 1
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
S(ok(x1)) → S(x1)
S(mark(x1)) → S(x1)
The TRS R consists of the following rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(x1, x2)) → and(active(x1), x2)
and(mark(x1), x2) → mark(and(x1, x2))
active(and(x1, x2)) → and(x1, active(x2))
and(x1, mark(x2)) → mark(and(x1, x2))
proper(and(x1, x2)) → and(proper(x1), proper(x2))
and(ok(x1), ok(x2)) → ok(and(x1, x2))
proper(tt) → ok(tt)
proper(isNatIList(x1)) → isNatIList(proper(x1))
isNatIList(ok(x1)) → ok(isNatIList(x1))
proper(isNatList(x1)) → isNatList(proper(x1))
isNatList(ok(x1)) → ok(isNatList(x1))
proper(isNat(x1)) → isNat(proper(x1))
isNat(ok(x1)) → ok(isNat(x1))
proper(0) → ok(0)
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(nil) → ok(nil)
active(take(x1, x2)) → take(active(x1), x2)
take(mark(x1), x2) → mark(take(x1, x2))
active(take(x1, x2)) → take(x1, active(x2))
take(x1, mark(x2)) → mark(take(x1, x2))
proper(take(x1, x2)) → take(proper(x1), proper(x2))
take(ok(x1), ok(x2)) → ok(take(x1, x2))
active(uTake1(x1)) → uTake1(active(x1))
uTake1(mark(x1)) → mark(uTake1(x1))
proper(uTake1(x1)) → uTake1(proper(x1))
uTake1(ok(x1)) → ok(uTake1(x1))
active(uTake2(x1, x2, x3, x4)) → uTake2(active(x1), x2, x3, x4)
uTake2(mark(x1), x2, x3, x4) → mark(uTake2(x1, x2, x3, x4))
proper(uTake2(x1, x2, x3, x4)) → uTake2(proper(x1), proper(x2), proper(x3), proper(x4))
uTake2(ok(x1), ok(x2), ok(x3), ok(x4)) → ok(uTake2(x1, x2, x3, x4))
active(uLength(x1, x2)) → uLength(active(x1), x2)
uLength(mark(x1), x2) → mark(uLength(x1, x2))
proper(uLength(x1, x2)) → uLength(proper(x1), proper(x2))
uLength(ok(x1), ok(x2)) → ok(uLength(x1, x2))
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
S(ok(x1)) → S(x1)
S(mark(x1)) → S(x1)
R is empty.
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
S(ok(x1)) → S(x1)
S(mark(x1)) → S(x1)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- S(ok(x1)) → S(x1)
The graph contains the following edges 1 > 1
- S(mark(x1)) → S(x1)
The graph contains the following edges 1 > 1
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNAT(ok(x1)) → ISNAT(x1)
The TRS R consists of the following rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(x1, x2)) → and(active(x1), x2)
and(mark(x1), x2) → mark(and(x1, x2))
active(and(x1, x2)) → and(x1, active(x2))
and(x1, mark(x2)) → mark(and(x1, x2))
proper(and(x1, x2)) → and(proper(x1), proper(x2))
and(ok(x1), ok(x2)) → ok(and(x1, x2))
proper(tt) → ok(tt)
proper(isNatIList(x1)) → isNatIList(proper(x1))
isNatIList(ok(x1)) → ok(isNatIList(x1))
proper(isNatList(x1)) → isNatList(proper(x1))
isNatList(ok(x1)) → ok(isNatList(x1))
proper(isNat(x1)) → isNat(proper(x1))
isNat(ok(x1)) → ok(isNat(x1))
proper(0) → ok(0)
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(nil) → ok(nil)
active(take(x1, x2)) → take(active(x1), x2)
take(mark(x1), x2) → mark(take(x1, x2))
active(take(x1, x2)) → take(x1, active(x2))
take(x1, mark(x2)) → mark(take(x1, x2))
proper(take(x1, x2)) → take(proper(x1), proper(x2))
take(ok(x1), ok(x2)) → ok(take(x1, x2))
active(uTake1(x1)) → uTake1(active(x1))
uTake1(mark(x1)) → mark(uTake1(x1))
proper(uTake1(x1)) → uTake1(proper(x1))
uTake1(ok(x1)) → ok(uTake1(x1))
active(uTake2(x1, x2, x3, x4)) → uTake2(active(x1), x2, x3, x4)
uTake2(mark(x1), x2, x3, x4) → mark(uTake2(x1, x2, x3, x4))
proper(uTake2(x1, x2, x3, x4)) → uTake2(proper(x1), proper(x2), proper(x3), proper(x4))
uTake2(ok(x1), ok(x2), ok(x3), ok(x4)) → ok(uTake2(x1, x2, x3, x4))
active(uLength(x1, x2)) → uLength(active(x1), x2)
uLength(mark(x1), x2) → mark(uLength(x1, x2))
proper(uLength(x1, x2)) → uLength(proper(x1), proper(x2))
uLength(ok(x1), ok(x2)) → ok(uLength(x1, x2))
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNAT(ok(x1)) → ISNAT(x1)
R is empty.
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNAT(ok(x1)) → ISNAT(x1)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- ISNAT(ok(x1)) → ISNAT(x1)
The graph contains the following edges 1 > 1
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(ok(x1)) → ISNATLIST(x1)
The TRS R consists of the following rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(x1, x2)) → and(active(x1), x2)
and(mark(x1), x2) → mark(and(x1, x2))
active(and(x1, x2)) → and(x1, active(x2))
and(x1, mark(x2)) → mark(and(x1, x2))
proper(and(x1, x2)) → and(proper(x1), proper(x2))
and(ok(x1), ok(x2)) → ok(and(x1, x2))
proper(tt) → ok(tt)
proper(isNatIList(x1)) → isNatIList(proper(x1))
isNatIList(ok(x1)) → ok(isNatIList(x1))
proper(isNatList(x1)) → isNatList(proper(x1))
isNatList(ok(x1)) → ok(isNatList(x1))
proper(isNat(x1)) → isNat(proper(x1))
isNat(ok(x1)) → ok(isNat(x1))
proper(0) → ok(0)
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(nil) → ok(nil)
active(take(x1, x2)) → take(active(x1), x2)
take(mark(x1), x2) → mark(take(x1, x2))
active(take(x1, x2)) → take(x1, active(x2))
take(x1, mark(x2)) → mark(take(x1, x2))
proper(take(x1, x2)) → take(proper(x1), proper(x2))
take(ok(x1), ok(x2)) → ok(take(x1, x2))
active(uTake1(x1)) → uTake1(active(x1))
uTake1(mark(x1)) → mark(uTake1(x1))
proper(uTake1(x1)) → uTake1(proper(x1))
uTake1(ok(x1)) → ok(uTake1(x1))
active(uTake2(x1, x2, x3, x4)) → uTake2(active(x1), x2, x3, x4)
uTake2(mark(x1), x2, x3, x4) → mark(uTake2(x1, x2, x3, x4))
proper(uTake2(x1, x2, x3, x4)) → uTake2(proper(x1), proper(x2), proper(x3), proper(x4))
uTake2(ok(x1), ok(x2), ok(x3), ok(x4)) → ok(uTake2(x1, x2, x3, x4))
active(uLength(x1, x2)) → uLength(active(x1), x2)
uLength(mark(x1), x2) → mark(uLength(x1, x2))
proper(uLength(x1, x2)) → uLength(proper(x1), proper(x2))
uLength(ok(x1), ok(x2)) → ok(uLength(x1, x2))
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(ok(x1)) → ISNATLIST(x1)
R is empty.
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(ok(x1)) → ISNATLIST(x1)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- ISNATLIST(ok(x1)) → ISNATLIST(x1)
The graph contains the following edges 1 > 1
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(ok(x1)) → ISNATILIST(x1)
The TRS R consists of the following rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(x1, x2)) → and(active(x1), x2)
and(mark(x1), x2) → mark(and(x1, x2))
active(and(x1, x2)) → and(x1, active(x2))
and(x1, mark(x2)) → mark(and(x1, x2))
proper(and(x1, x2)) → and(proper(x1), proper(x2))
and(ok(x1), ok(x2)) → ok(and(x1, x2))
proper(tt) → ok(tt)
proper(isNatIList(x1)) → isNatIList(proper(x1))
isNatIList(ok(x1)) → ok(isNatIList(x1))
proper(isNatList(x1)) → isNatList(proper(x1))
isNatList(ok(x1)) → ok(isNatList(x1))
proper(isNat(x1)) → isNat(proper(x1))
isNat(ok(x1)) → ok(isNat(x1))
proper(0) → ok(0)
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(nil) → ok(nil)
active(take(x1, x2)) → take(active(x1), x2)
take(mark(x1), x2) → mark(take(x1, x2))
active(take(x1, x2)) → take(x1, active(x2))
take(x1, mark(x2)) → mark(take(x1, x2))
proper(take(x1, x2)) → take(proper(x1), proper(x2))
take(ok(x1), ok(x2)) → ok(take(x1, x2))
active(uTake1(x1)) → uTake1(active(x1))
uTake1(mark(x1)) → mark(uTake1(x1))
proper(uTake1(x1)) → uTake1(proper(x1))
uTake1(ok(x1)) → ok(uTake1(x1))
active(uTake2(x1, x2, x3, x4)) → uTake2(active(x1), x2, x3, x4)
uTake2(mark(x1), x2, x3, x4) → mark(uTake2(x1, x2, x3, x4))
proper(uTake2(x1, x2, x3, x4)) → uTake2(proper(x1), proper(x2), proper(x3), proper(x4))
uTake2(ok(x1), ok(x2), ok(x3), ok(x4)) → ok(uTake2(x1, x2, x3, x4))
active(uLength(x1, x2)) → uLength(active(x1), x2)
uLength(mark(x1), x2) → mark(uLength(x1, x2))
proper(uLength(x1, x2)) → uLength(proper(x1), proper(x2))
uLength(ok(x1), ok(x2)) → ok(uLength(x1, x2))
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(ok(x1)) → ISNATILIST(x1)
R is empty.
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILIST(ok(x1)) → ISNATILIST(x1)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- ISNATILIST(ok(x1)) → ISNATILIST(x1)
The graph contains the following edges 1 > 1
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
AND(ok(x1), ok(x2)) → AND(x1, x2)
AND(x1, mark(x2)) → AND(x1, x2)
AND(mark(x1), x2) → AND(x1, x2)
The TRS R consists of the following rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(x1, x2)) → and(active(x1), x2)
and(mark(x1), x2) → mark(and(x1, x2))
active(and(x1, x2)) → and(x1, active(x2))
and(x1, mark(x2)) → mark(and(x1, x2))
proper(and(x1, x2)) → and(proper(x1), proper(x2))
and(ok(x1), ok(x2)) → ok(and(x1, x2))
proper(tt) → ok(tt)
proper(isNatIList(x1)) → isNatIList(proper(x1))
isNatIList(ok(x1)) → ok(isNatIList(x1))
proper(isNatList(x1)) → isNatList(proper(x1))
isNatList(ok(x1)) → ok(isNatList(x1))
proper(isNat(x1)) → isNat(proper(x1))
isNat(ok(x1)) → ok(isNat(x1))
proper(0) → ok(0)
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(nil) → ok(nil)
active(take(x1, x2)) → take(active(x1), x2)
take(mark(x1), x2) → mark(take(x1, x2))
active(take(x1, x2)) → take(x1, active(x2))
take(x1, mark(x2)) → mark(take(x1, x2))
proper(take(x1, x2)) → take(proper(x1), proper(x2))
take(ok(x1), ok(x2)) → ok(take(x1, x2))
active(uTake1(x1)) → uTake1(active(x1))
uTake1(mark(x1)) → mark(uTake1(x1))
proper(uTake1(x1)) → uTake1(proper(x1))
uTake1(ok(x1)) → ok(uTake1(x1))
active(uTake2(x1, x2, x3, x4)) → uTake2(active(x1), x2, x3, x4)
uTake2(mark(x1), x2, x3, x4) → mark(uTake2(x1, x2, x3, x4))
proper(uTake2(x1, x2, x3, x4)) → uTake2(proper(x1), proper(x2), proper(x3), proper(x4))
uTake2(ok(x1), ok(x2), ok(x3), ok(x4)) → ok(uTake2(x1, x2, x3, x4))
active(uLength(x1, x2)) → uLength(active(x1), x2)
uLength(mark(x1), x2) → mark(uLength(x1, x2))
proper(uLength(x1, x2)) → uLength(proper(x1), proper(x2))
uLength(ok(x1), ok(x2)) → ok(uLength(x1, x2))
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
AND(ok(x1), ok(x2)) → AND(x1, x2)
AND(x1, mark(x2)) → AND(x1, x2)
AND(mark(x1), x2) → AND(x1, x2)
R is empty.
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
AND(ok(x1), ok(x2)) → AND(x1, x2)
AND(x1, mark(x2)) → AND(x1, x2)
AND(mark(x1), x2) → AND(x1, x2)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- AND(ok(x1), ok(x2)) → AND(x1, x2)
The graph contains the following edges 1 > 1, 2 > 2
- AND(x1, mark(x2)) → AND(x1, x2)
The graph contains the following edges 1 >= 1, 2 > 2
- AND(mark(x1), x2) → AND(x1, x2)
The graph contains the following edges 1 > 1, 2 >= 2
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
PROPER(uTake2(x1, x2, x3, x4)) → PROPER(x1)
PROPER(and(x1, x2)) → PROPER(x1)
PROPER(length(x1)) → PROPER(x1)
PROPER(cons(x1, x2)) → PROPER(x1)
PROPER(uLength(x1, x2)) → PROPER(x2)
PROPER(uTake2(x1, x2, x3, x4)) → PROPER(x4)
PROPER(uTake1(x1)) → PROPER(x1)
PROPER(isNat(x1)) → PROPER(x1)
PROPER(isNatList(x1)) → PROPER(x1)
PROPER(take(x1, x2)) → PROPER(x1)
PROPER(take(x1, x2)) → PROPER(x2)
PROPER(uLength(x1, x2)) → PROPER(x1)
PROPER(isNatIList(x1)) → PROPER(x1)
PROPER(s(x1)) → PROPER(x1)
PROPER(cons(x1, x2)) → PROPER(x2)
PROPER(and(x1, x2)) → PROPER(x2)
PROPER(uTake2(x1, x2, x3, x4)) → PROPER(x3)
PROPER(uTake2(x1, x2, x3, x4)) → PROPER(x2)
The TRS R consists of the following rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(x1, x2)) → and(active(x1), x2)
and(mark(x1), x2) → mark(and(x1, x2))
active(and(x1, x2)) → and(x1, active(x2))
and(x1, mark(x2)) → mark(and(x1, x2))
proper(and(x1, x2)) → and(proper(x1), proper(x2))
and(ok(x1), ok(x2)) → ok(and(x1, x2))
proper(tt) → ok(tt)
proper(isNatIList(x1)) → isNatIList(proper(x1))
isNatIList(ok(x1)) → ok(isNatIList(x1))
proper(isNatList(x1)) → isNatList(proper(x1))
isNatList(ok(x1)) → ok(isNatList(x1))
proper(isNat(x1)) → isNat(proper(x1))
isNat(ok(x1)) → ok(isNat(x1))
proper(0) → ok(0)
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(nil) → ok(nil)
active(take(x1, x2)) → take(active(x1), x2)
take(mark(x1), x2) → mark(take(x1, x2))
active(take(x1, x2)) → take(x1, active(x2))
take(x1, mark(x2)) → mark(take(x1, x2))
proper(take(x1, x2)) → take(proper(x1), proper(x2))
take(ok(x1), ok(x2)) → ok(take(x1, x2))
active(uTake1(x1)) → uTake1(active(x1))
uTake1(mark(x1)) → mark(uTake1(x1))
proper(uTake1(x1)) → uTake1(proper(x1))
uTake1(ok(x1)) → ok(uTake1(x1))
active(uTake2(x1, x2, x3, x4)) → uTake2(active(x1), x2, x3, x4)
uTake2(mark(x1), x2, x3, x4) → mark(uTake2(x1, x2, x3, x4))
proper(uTake2(x1, x2, x3, x4)) → uTake2(proper(x1), proper(x2), proper(x3), proper(x4))
uTake2(ok(x1), ok(x2), ok(x3), ok(x4)) → ok(uTake2(x1, x2, x3, x4))
active(uLength(x1, x2)) → uLength(active(x1), x2)
uLength(mark(x1), x2) → mark(uLength(x1, x2))
proper(uLength(x1, x2)) → uLength(proper(x1), proper(x2))
uLength(ok(x1), ok(x2)) → ok(uLength(x1, x2))
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
PROPER(uTake2(x1, x2, x3, x4)) → PROPER(x1)
PROPER(and(x1, x2)) → PROPER(x1)
PROPER(length(x1)) → PROPER(x1)
PROPER(cons(x1, x2)) → PROPER(x1)
PROPER(uLength(x1, x2)) → PROPER(x2)
PROPER(uTake2(x1, x2, x3, x4)) → PROPER(x4)
PROPER(uTake1(x1)) → PROPER(x1)
PROPER(isNat(x1)) → PROPER(x1)
PROPER(isNatList(x1)) → PROPER(x1)
PROPER(take(x1, x2)) → PROPER(x1)
PROPER(take(x1, x2)) → PROPER(x2)
PROPER(uLength(x1, x2)) → PROPER(x1)
PROPER(isNatIList(x1)) → PROPER(x1)
PROPER(s(x1)) → PROPER(x1)
PROPER(cons(x1, x2)) → PROPER(x2)
PROPER(and(x1, x2)) → PROPER(x2)
PROPER(uTake2(x1, x2, x3, x4)) → PROPER(x3)
PROPER(uTake2(x1, x2, x3, x4)) → PROPER(x2)
R is empty.
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
proper(and(x0, x1))
proper(tt)
proper(isNatIList(x0))
proper(isNatList(x0))
proper(isNat(x0))
proper(0)
active(s(x0))
proper(s(x0))
active(length(x0))
proper(length(x0))
proper(zeros)
active(cons(x0, x1))
proper(cons(x0, x1))
proper(nil)
active(take(x0, x1))
proper(take(x0, x1))
active(uTake1(x0))
proper(uTake1(x0))
active(uTake2(x0, x1, x2, x3))
proper(uTake2(x0, x1, x2, x3))
active(uLength(x0, x1))
proper(uLength(x0, x1))
top(mark(x0))
top(ok(x0))
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
PROPER(uTake2(x1, x2, x3, x4)) → PROPER(x1)
PROPER(and(x1, x2)) → PROPER(x1)
PROPER(length(x1)) → PROPER(x1)
PROPER(cons(x1, x2)) → PROPER(x1)
PROPER(uLength(x1, x2)) → PROPER(x2)
PROPER(uTake2(x1, x2, x3, x4)) → PROPER(x4)
PROPER(uTake1(x1)) → PROPER(x1)
PROPER(isNat(x1)) → PROPER(x1)
PROPER(isNatList(x1)) → PROPER(x1)
PROPER(take(x1, x2)) → PROPER(x1)
PROPER(take(x1, x2)) → PROPER(x2)
PROPER(uLength(x1, x2)) → PROPER(x1)
PROPER(isNatIList(x1)) → PROPER(x1)
PROPER(s(x1)) → PROPER(x1)
PROPER(cons(x1, x2)) → PROPER(x2)
PROPER(and(x1, x2)) → PROPER(x2)
PROPER(uTake2(x1, x2, x3, x4)) → PROPER(x3)
PROPER(uTake2(x1, x2, x3, x4)) → PROPER(x2)
R is empty.
The set Q consists of the following terms:
and(mark(x0), x1)
and(x0, mark(x1))
and(ok(x0), ok(x1))
isNatIList(ok(x0))
isNatList(ok(x0))
isNat(ok(x0))
s(mark(x0))
s(ok(x0))
length(mark(x0))
length(ok(x0))
cons(mark(x0), x1)
cons(ok(x0), ok(x1))
take(mark(x0), x1)
take(x0, mark(x1))
take(ok(x0), ok(x1))
uTake1(mark(x0))
uTake1(ok(x0))
uTake2(mark(x0), x1, x2, x3)
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
uLength(mark(x0), x1)
uLength(ok(x0), ok(x1))
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- PROPER(uTake2(x1, x2, x3, x4)) → PROPER(x1)
The graph contains the following edges 1 > 1
- PROPER(and(x1, x2)) → PROPER(x1)
The graph contains the following edges 1 > 1
- PROPER(length(x1)) → PROPER(x1)
The graph contains the following edges 1 > 1
- PROPER(cons(x1, x2)) → PROPER(x1)
The graph contains the following edges 1 > 1
- PROPER(uLength(x1, x2)) → PROPER(x2)
The graph contains the following edges 1 > 1
- PROPER(uTake2(x1, x2, x3, x4)) → PROPER(x4)
The graph contains the following edges 1 > 1
- PROPER(uTake1(x1)) → PROPER(x1)
The graph contains the following edges 1 > 1
- PROPER(isNat(x1)) → PROPER(x1)
The graph contains the following edges 1 > 1
- PROPER(isNatList(x1)) → PROPER(x1)
The graph contains the following edges 1 > 1
- PROPER(take(x1, x2)) → PROPER(x1)
The graph contains the following edges 1 > 1
- PROPER(take(x1, x2)) → PROPER(x2)
The graph contains the following edges 1 > 1
- PROPER(uLength(x1, x2)) → PROPER(x1)
The graph contains the following edges 1 > 1
- PROPER(isNatIList(x1)) → PROPER(x1)
The graph contains the following edges 1 > 1
- PROPER(s(x1)) → PROPER(x1)
The graph contains the following edges 1 > 1
- PROPER(cons(x1, x2)) → PROPER(x2)
The graph contains the following edges 1 > 1
- PROPER(and(x1, x2)) → PROPER(x2)
The graph contains the following edges 1 > 1
- PROPER(uTake2(x1, x2, x3, x4)) → PROPER(x3)
The graph contains the following edges 1 > 1
- PROPER(uTake2(x1, x2, x3, x4)) → PROPER(x2)
The graph contains the following edges 1 > 1
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ACTIVE(length(x1)) → ACTIVE(x1)
ACTIVE(cons(x1, x2)) → ACTIVE(x1)
ACTIVE(take(x1, x2)) → ACTIVE(x2)
ACTIVE(uTake2(x1, x2, x3, x4)) → ACTIVE(x1)
ACTIVE(take(x1, x2)) → ACTIVE(x1)
ACTIVE(and(x1, x2)) → ACTIVE(x1)
ACTIVE(uTake1(x1)) → ACTIVE(x1)
ACTIVE(uLength(x1, x2)) → ACTIVE(x1)
ACTIVE(s(x1)) → ACTIVE(x1)
ACTIVE(and(x1, x2)) → ACTIVE(x2)
The TRS R consists of the following rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(x1, x2)) → and(active(x1), x2)
and(mark(x1), x2) → mark(and(x1, x2))
active(and(x1, x2)) → and(x1, active(x2))
and(x1, mark(x2)) → mark(and(x1, x2))
proper(and(x1, x2)) → and(proper(x1), proper(x2))
and(ok(x1), ok(x2)) → ok(and(x1, x2))
proper(tt) → ok(tt)
proper(isNatIList(x1)) → isNatIList(proper(x1))
isNatIList(ok(x1)) → ok(isNatIList(x1))
proper(isNatList(x1)) → isNatList(proper(x1))
isNatList(ok(x1)) → ok(isNatList(x1))
proper(isNat(x1)) → isNat(proper(x1))
isNat(ok(x1)) → ok(isNat(x1))
proper(0) → ok(0)
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(nil) → ok(nil)
active(take(x1, x2)) → take(active(x1), x2)
take(mark(x1), x2) → mark(take(x1, x2))
active(take(x1, x2)) → take(x1, active(x2))
take(x1, mark(x2)) → mark(take(x1, x2))
proper(take(x1, x2)) → take(proper(x1), proper(x2))
take(ok(x1), ok(x2)) → ok(take(x1, x2))
active(uTake1(x1)) → uTake1(active(x1))
uTake1(mark(x1)) → mark(uTake1(x1))
proper(uTake1(x1)) → uTake1(proper(x1))
uTake1(ok(x1)) → ok(uTake1(x1))
active(uTake2(x1, x2, x3, x4)) → uTake2(active(x1), x2, x3, x4)
uTake2(mark(x1), x2, x3, x4) → mark(uTake2(x1, x2, x3, x4))
proper(uTake2(x1, x2, x3, x4)) → uTake2(proper(x1), proper(x2), proper(x3), proper(x4))
uTake2(ok(x1), ok(x2), ok(x3), ok(x4)) → ok(uTake2(x1, x2, x3, x4))
active(uLength(x1, x2)) → uLength(active(x1), x2)
uLength(mark(x1), x2) → mark(uLength(x1, x2))
proper(uLength(x1, x2)) → uLength(proper(x1), proper(x2))
uLength(ok(x1), ok(x2)) → ok(uLength(x1, x2))
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ACTIVE(length(x1)) → ACTIVE(x1)
ACTIVE(cons(x1, x2)) → ACTIVE(x1)
ACTIVE(take(x1, x2)) → ACTIVE(x2)
ACTIVE(uTake2(x1, x2, x3, x4)) → ACTIVE(x1)
ACTIVE(take(x1, x2)) → ACTIVE(x1)
ACTIVE(and(x1, x2)) → ACTIVE(x1)
ACTIVE(uTake1(x1)) → ACTIVE(x1)
ACTIVE(uLength(x1, x2)) → ACTIVE(x1)
ACTIVE(s(x1)) → ACTIVE(x1)
ACTIVE(and(x1, x2)) → ACTIVE(x2)
R is empty.
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
proper(and(x0, x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
proper(s(x0))
active(length(x0))
proper(length(x0))
proper(zeros)
active(cons(x0, x1))
proper(cons(x0, x1))
proper(nil)
active(take(x0, x1))
proper(take(x0, x1))
active(uTake1(x0))
proper(uTake1(x0))
active(uTake2(x0, x1, x2, x3))
proper(uTake2(x0, x1, x2, x3))
active(uLength(x0, x1))
proper(uLength(x0, x1))
top(mark(x0))
top(ok(x0))
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ACTIVE(length(x1)) → ACTIVE(x1)
ACTIVE(cons(x1, x2)) → ACTIVE(x1)
ACTIVE(take(x1, x2)) → ACTIVE(x2)
ACTIVE(and(x1, x2)) → ACTIVE(x1)
ACTIVE(take(x1, x2)) → ACTIVE(x1)
ACTIVE(uTake2(x1, x2, x3, x4)) → ACTIVE(x1)
ACTIVE(uTake1(x1)) → ACTIVE(x1)
ACTIVE(uLength(x1, x2)) → ACTIVE(x1)
ACTIVE(and(x1, x2)) → ACTIVE(x2)
ACTIVE(s(x1)) → ACTIVE(x1)
R is empty.
The set Q consists of the following terms:
and(mark(x0), x1)
and(x0, mark(x1))
and(ok(x0), ok(x1))
s(mark(x0))
s(ok(x0))
length(mark(x0))
length(ok(x0))
cons(mark(x0), x1)
cons(ok(x0), ok(x1))
take(mark(x0), x1)
take(x0, mark(x1))
take(ok(x0), ok(x1))
uTake1(mark(x0))
uTake1(ok(x0))
uTake2(mark(x0), x1, x2, x3)
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
uLength(mark(x0), x1)
uLength(ok(x0), ok(x1))
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- ACTIVE(length(x1)) → ACTIVE(x1)
The graph contains the following edges 1 > 1
- ACTIVE(cons(x1, x2)) → ACTIVE(x1)
The graph contains the following edges 1 > 1
- ACTIVE(take(x1, x2)) → ACTIVE(x2)
The graph contains the following edges 1 > 1
- ACTIVE(uTake2(x1, x2, x3, x4)) → ACTIVE(x1)
The graph contains the following edges 1 > 1
- ACTIVE(take(x1, x2)) → ACTIVE(x1)
The graph contains the following edges 1 > 1
- ACTIVE(and(x1, x2)) → ACTIVE(x1)
The graph contains the following edges 1 > 1
- ACTIVE(uTake1(x1)) → ACTIVE(x1)
The graph contains the following edges 1 > 1
- ACTIVE(uLength(x1, x2)) → ACTIVE(x1)
The graph contains the following edges 1 > 1
- ACTIVE(s(x1)) → ACTIVE(x1)
The graph contains the following edges 1 > 1
- ACTIVE(and(x1, x2)) → ACTIVE(x2)
The graph contains the following edges 1 > 1
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(x)) → TOP(active(x))
TOP(mark(x)) → TOP(proper(x))
The TRS R consists of the following rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(x1, x2)) → and(active(x1), x2)
and(mark(x1), x2) → mark(and(x1, x2))
active(and(x1, x2)) → and(x1, active(x2))
and(x1, mark(x2)) → mark(and(x1, x2))
proper(and(x1, x2)) → and(proper(x1), proper(x2))
and(ok(x1), ok(x2)) → ok(and(x1, x2))
proper(tt) → ok(tt)
proper(isNatIList(x1)) → isNatIList(proper(x1))
isNatIList(ok(x1)) → ok(isNatIList(x1))
proper(isNatList(x1)) → isNatList(proper(x1))
isNatList(ok(x1)) → ok(isNatList(x1))
proper(isNat(x1)) → isNat(proper(x1))
isNat(ok(x1)) → ok(isNat(x1))
proper(0) → ok(0)
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(nil) → ok(nil)
active(take(x1, x2)) → take(active(x1), x2)
take(mark(x1), x2) → mark(take(x1, x2))
active(take(x1, x2)) → take(x1, active(x2))
take(x1, mark(x2)) → mark(take(x1, x2))
proper(take(x1, x2)) → take(proper(x1), proper(x2))
take(ok(x1), ok(x2)) → ok(take(x1, x2))
active(uTake1(x1)) → uTake1(active(x1))
uTake1(mark(x1)) → mark(uTake1(x1))
proper(uTake1(x1)) → uTake1(proper(x1))
uTake1(ok(x1)) → ok(uTake1(x1))
active(uTake2(x1, x2, x3, x4)) → uTake2(active(x1), x2, x3, x4)
uTake2(mark(x1), x2, x3, x4) → mark(uTake2(x1, x2, x3, x4))
proper(uTake2(x1, x2, x3, x4)) → uTake2(proper(x1), proper(x2), proper(x3), proper(x4))
uTake2(ok(x1), ok(x2), ok(x3), ok(x4)) → ok(uTake2(x1, x2, x3, x4))
active(uLength(x1, x2)) → uLength(active(x1), x2)
uLength(mark(x1), x2) → mark(uLength(x1, x2))
proper(uLength(x1, x2)) → uLength(proper(x1), proper(x2))
uLength(ok(x1), ok(x2)) → ok(uLength(x1, x2))
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(x)) → TOP(active(x))
TOP(mark(x)) → TOP(proper(x))
The TRS R consists of the following rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(x1, x2)) → and(active(x1), x2)
active(and(x1, x2)) → and(x1, active(x2))
active(s(x1)) → s(active(x1))
active(length(x1)) → length(active(x1))
active(cons(x1, x2)) → cons(active(x1), x2)
active(take(x1, x2)) → take(active(x1), x2)
active(take(x1, x2)) → take(x1, active(x2))
active(uTake1(x1)) → uTake1(active(x1))
active(uTake2(x1, x2, x3, x4)) → uTake2(active(x1), x2, x3, x4)
active(uLength(x1, x2)) → uLength(active(x1), x2)
uLength(mark(x1), x2) → mark(uLength(x1, x2))
uLength(ok(x1), ok(x2)) → ok(uLength(x1, x2))
uTake2(mark(x1), x2, x3, x4) → mark(uTake2(x1, x2, x3, x4))
uTake2(ok(x1), ok(x2), ok(x3), ok(x4)) → ok(uTake2(x1, x2, x3, x4))
uTake1(mark(x1)) → mark(uTake1(x1))
uTake1(ok(x1)) → ok(uTake1(x1))
take(x1, mark(x2)) → mark(take(x1, x2))
take(ok(x1), ok(x2)) → ok(take(x1, x2))
take(mark(x1), x2) → mark(take(x1, x2))
cons(mark(x1), x2) → mark(cons(x1, x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
length(mark(x1)) → mark(length(x1))
length(ok(x1)) → ok(length(x1))
s(mark(x1)) → mark(s(x1))
s(ok(x1)) → ok(s(x1))
and(x1, mark(x2)) → mark(and(x1, x2))
and(ok(x1), ok(x2)) → ok(and(x1, x2))
and(mark(x1), x2) → mark(and(x1, x2))
isNat(ok(x1)) → ok(isNat(x1))
isNatList(ok(x1)) → ok(isNatList(x1))
isNatIList(ok(x1)) → ok(isNatIList(x1))
proper(and(x1, x2)) → and(proper(x1), proper(x2))
proper(tt) → ok(tt)
proper(isNatIList(x1)) → isNatIList(proper(x1))
proper(isNatList(x1)) → isNatList(proper(x1))
proper(isNat(x1)) → isNat(proper(x1))
proper(0) → ok(0)
proper(s(x1)) → s(proper(x1))
proper(length(x1)) → length(proper(x1))
proper(zeros) → ok(zeros)
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
proper(nil) → ok(nil)
proper(take(x1, x2)) → take(proper(x1), proper(x2))
proper(uTake1(x1)) → uTake1(proper(x1))
proper(uTake2(x1, x2, x3, x4)) → uTake2(proper(x1), proper(x2), proper(x3), proper(x4))
proper(uLength(x1, x2)) → uLength(proper(x1), proper(x2))
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
top(mark(x0))
top(ok(x0))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
top(mark(x0))
top(ok(x0))
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(x)) → TOP(active(x))
TOP(mark(x)) → TOP(proper(x))
The TRS R consists of the following rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(x1, x2)) → and(active(x1), x2)
active(and(x1, x2)) → and(x1, active(x2))
active(s(x1)) → s(active(x1))
active(length(x1)) → length(active(x1))
active(cons(x1, x2)) → cons(active(x1), x2)
active(take(x1, x2)) → take(active(x1), x2)
active(take(x1, x2)) → take(x1, active(x2))
active(uTake1(x1)) → uTake1(active(x1))
active(uTake2(x1, x2, x3, x4)) → uTake2(active(x1), x2, x3, x4)
active(uLength(x1, x2)) → uLength(active(x1), x2)
uLength(mark(x1), x2) → mark(uLength(x1, x2))
uLength(ok(x1), ok(x2)) → ok(uLength(x1, x2))
uTake2(mark(x1), x2, x3, x4) → mark(uTake2(x1, x2, x3, x4))
uTake2(ok(x1), ok(x2), ok(x3), ok(x4)) → ok(uTake2(x1, x2, x3, x4))
uTake1(mark(x1)) → mark(uTake1(x1))
uTake1(ok(x1)) → ok(uTake1(x1))
take(x1, mark(x2)) → mark(take(x1, x2))
take(ok(x1), ok(x2)) → ok(take(x1, x2))
take(mark(x1), x2) → mark(take(x1, x2))
cons(mark(x1), x2) → mark(cons(x1, x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
length(mark(x1)) → mark(length(x1))
length(ok(x1)) → ok(length(x1))
s(mark(x1)) → mark(s(x1))
s(ok(x1)) → ok(s(x1))
and(x1, mark(x2)) → mark(and(x1, x2))
and(ok(x1), ok(x2)) → ok(and(x1, x2))
and(mark(x1), x2) → mark(and(x1, x2))
isNat(ok(x1)) → ok(isNat(x1))
isNatList(ok(x1)) → ok(isNatList(x1))
isNatIList(ok(x1)) → ok(isNatIList(x1))
proper(and(x1, x2)) → and(proper(x1), proper(x2))
proper(tt) → ok(tt)
proper(isNatIList(x1)) → isNatIList(proper(x1))
proper(isNatList(x1)) → isNatList(proper(x1))
proper(isNat(x1)) → isNat(proper(x1))
proper(0) → ok(0)
proper(s(x1)) → s(proper(x1))
proper(length(x1)) → length(proper(x1))
proper(zeros) → ok(zeros)
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
proper(nil) → ok(nil)
proper(take(x1, x2)) → take(proper(x1), proper(x2))
proper(uTake1(x1)) → uTake1(proper(x1))
proper(uTake2(x1, x2, x3, x4)) → uTake2(proper(x1), proper(x2), proper(x3), proper(x4))
proper(uLength(x1, x2)) → uLength(proper(x1), proper(x2))
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule TOP(ok(x)) → TOP(active(x)) at position [0] we obtained the following new rules:
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatList(x1))))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(uLength(x0, x1))) → TOP(uLength(active(x0), x1))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(uTake2(and(isNat(x0), and(isNat(x1), isNatIList(x2))), x0, x1, x2)))
TOP(ok(uTake1(x0))) → TOP(uTake1(active(x0)))
TOP(ok(uTake2(tt, x0, x1, x2))) → TOP(mark(cons(x1, take(x0, x2))))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(ok(uTake1(tt))) → TOP(mark(nil))
TOP(ok(uTake2(x0, x1, x2, x3))) → TOP(uTake2(active(x0), x1, x2, x3))
TOP(ok(isNatList(nil))) → TOP(mark(tt))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(ok(isNatIList(x0))) → TOP(mark(isNatList(x0)))
TOP(ok(isNat(length(x0)))) → TOP(mark(isNatList(x0)))
TOP(ok(and(x0, x1))) → TOP(and(x0, active(x1)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(ok(take(0, x0))) → TOP(mark(uTake1(isNatIList(x0))))
TOP(ok(uLength(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(uLength(and(isNat(x0), isNatList(x1)), x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(isNatIList(zeros))) → TOP(mark(tt))
TOP(ok(isNat(0))) → TOP(mark(tt))
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatList(x1))))
TOP(ok(uLength(x0, x1))) → TOP(uLength(active(x0), x1))
TOP(ok(uTake1(x0))) → TOP(uTake1(active(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(uTake2(and(isNat(x0), and(isNat(x1), isNatIList(x2))), x0, x1, x2)))
TOP(ok(uTake2(tt, x0, x1, x2))) → TOP(mark(cons(x1, take(x0, x2))))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(ok(uTake1(tt))) → TOP(mark(nil))
TOP(ok(uTake2(x0, x1, x2, x3))) → TOP(uTake2(active(x0), x1, x2, x3))
TOP(ok(isNatList(nil))) → TOP(mark(tt))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(x)) → TOP(proper(x))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(ok(isNatIList(x0))) → TOP(mark(isNatList(x0)))
TOP(ok(isNat(length(x0)))) → TOP(mark(isNatList(x0)))
TOP(ok(and(x0, x1))) → TOP(and(x0, active(x1)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(ok(take(0, x0))) → TOP(mark(uTake1(isNatIList(x0))))
TOP(ok(uLength(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(uLength(and(isNat(x0), isNatList(x1)), x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(isNatIList(zeros))) → TOP(mark(tt))
TOP(ok(isNat(0))) → TOP(mark(tt))
The TRS R consists of the following rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(x1, x2)) → and(active(x1), x2)
active(and(x1, x2)) → and(x1, active(x2))
active(s(x1)) → s(active(x1))
active(length(x1)) → length(active(x1))
active(cons(x1, x2)) → cons(active(x1), x2)
active(take(x1, x2)) → take(active(x1), x2)
active(take(x1, x2)) → take(x1, active(x2))
active(uTake1(x1)) → uTake1(active(x1))
active(uTake2(x1, x2, x3, x4)) → uTake2(active(x1), x2, x3, x4)
active(uLength(x1, x2)) → uLength(active(x1), x2)
uLength(mark(x1), x2) → mark(uLength(x1, x2))
uLength(ok(x1), ok(x2)) → ok(uLength(x1, x2))
uTake2(mark(x1), x2, x3, x4) → mark(uTake2(x1, x2, x3, x4))
uTake2(ok(x1), ok(x2), ok(x3), ok(x4)) → ok(uTake2(x1, x2, x3, x4))
uTake1(mark(x1)) → mark(uTake1(x1))
uTake1(ok(x1)) → ok(uTake1(x1))
take(x1, mark(x2)) → mark(take(x1, x2))
take(ok(x1), ok(x2)) → ok(take(x1, x2))
take(mark(x1), x2) → mark(take(x1, x2))
cons(mark(x1), x2) → mark(cons(x1, x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
length(mark(x1)) → mark(length(x1))
length(ok(x1)) → ok(length(x1))
s(mark(x1)) → mark(s(x1))
s(ok(x1)) → ok(s(x1))
and(x1, mark(x2)) → mark(and(x1, x2))
and(ok(x1), ok(x2)) → ok(and(x1, x2))
and(mark(x1), x2) → mark(and(x1, x2))
isNat(ok(x1)) → ok(isNat(x1))
isNatList(ok(x1)) → ok(isNatList(x1))
isNatIList(ok(x1)) → ok(isNatIList(x1))
proper(and(x1, x2)) → and(proper(x1), proper(x2))
proper(tt) → ok(tt)
proper(isNatIList(x1)) → isNatIList(proper(x1))
proper(isNatList(x1)) → isNatList(proper(x1))
proper(isNat(x1)) → isNat(proper(x1))
proper(0) → ok(0)
proper(s(x1)) → s(proper(x1))
proper(length(x1)) → length(proper(x1))
proper(zeros) → ok(zeros)
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
proper(nil) → ok(nil)
proper(take(x1, x2)) → take(proper(x1), proper(x2))
proper(uTake1(x1)) → uTake1(proper(x1))
proper(uTake2(x1, x2, x3, x4)) → uTake2(proper(x1), proper(x2), proper(x3), proper(x4))
proper(uLength(x1, x2)) → uLength(proper(x1), proper(x2))
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule TOP(mark(x)) → TOP(proper(x)) at position [0] we obtained the following new rules:
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(mark(uTake2(x0, x1, x2, x3))) → TOP(uTake2(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(mark(tt)) → TOP(ok(tt))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(nil)) → TOP(ok(nil))
TOP(mark(uTake1(x0))) → TOP(uTake1(proper(x0)))
TOP(mark(uLength(x0, x1))) → TOP(uLength(proper(x0), proper(x1)))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatList(x1))))
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(ok(uLength(x0, x1))) → TOP(uLength(active(x0), x1))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(uTake2(and(isNat(x0), and(isNat(x1), isNatIList(x2))), x0, x1, x2)))
TOP(ok(uTake1(x0))) → TOP(uTake1(active(x0)))
TOP(ok(uTake2(tt, x0, x1, x2))) → TOP(mark(cons(x1, take(x0, x2))))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(uTake1(tt))) → TOP(mark(nil))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(ok(uTake2(x0, x1, x2, x3))) → TOP(uTake2(active(x0), x1, x2, x3))
TOP(mark(tt)) → TOP(ok(tt))
TOP(ok(isNatList(nil))) → TOP(mark(tt))
TOP(mark(nil)) → TOP(ok(nil))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(mark(uTake1(x0))) → TOP(uTake1(proper(x0)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(uLength(x0, x1))) → TOP(uLength(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(ok(isNatIList(x0))) → TOP(mark(isNatList(x0)))
TOP(mark(0)) → TOP(ok(0))
TOP(ok(isNat(length(x0)))) → TOP(mark(isNatList(x0)))
TOP(mark(uTake2(x0, x1, x2, x3))) → TOP(uTake2(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(and(x0, x1))) → TOP(and(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(ok(take(0, x0))) → TOP(mark(uTake1(isNatIList(x0))))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(uLength(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(uLength(and(isNat(x0), isNatList(x1)), x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(isNatIList(zeros))) → TOP(mark(tt))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
TOP(ok(isNat(0))) → TOP(mark(tt))
The TRS R consists of the following rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(x1, x2)) → and(active(x1), x2)
active(and(x1, x2)) → and(x1, active(x2))
active(s(x1)) → s(active(x1))
active(length(x1)) → length(active(x1))
active(cons(x1, x2)) → cons(active(x1), x2)
active(take(x1, x2)) → take(active(x1), x2)
active(take(x1, x2)) → take(x1, active(x2))
active(uTake1(x1)) → uTake1(active(x1))
active(uTake2(x1, x2, x3, x4)) → uTake2(active(x1), x2, x3, x4)
active(uLength(x1, x2)) → uLength(active(x1), x2)
uLength(mark(x1), x2) → mark(uLength(x1, x2))
uLength(ok(x1), ok(x2)) → ok(uLength(x1, x2))
uTake2(mark(x1), x2, x3, x4) → mark(uTake2(x1, x2, x3, x4))
uTake2(ok(x1), ok(x2), ok(x3), ok(x4)) → ok(uTake2(x1, x2, x3, x4))
uTake1(mark(x1)) → mark(uTake1(x1))
uTake1(ok(x1)) → ok(uTake1(x1))
take(x1, mark(x2)) → mark(take(x1, x2))
take(ok(x1), ok(x2)) → ok(take(x1, x2))
take(mark(x1), x2) → mark(take(x1, x2))
cons(mark(x1), x2) → mark(cons(x1, x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
length(mark(x1)) → mark(length(x1))
length(ok(x1)) → ok(length(x1))
s(mark(x1)) → mark(s(x1))
s(ok(x1)) → ok(s(x1))
and(x1, mark(x2)) → mark(and(x1, x2))
and(ok(x1), ok(x2)) → ok(and(x1, x2))
and(mark(x1), x2) → mark(and(x1, x2))
isNat(ok(x1)) → ok(isNat(x1))
isNatList(ok(x1)) → ok(isNatList(x1))
isNatIList(ok(x1)) → ok(isNatIList(x1))
proper(and(x1, x2)) → and(proper(x1), proper(x2))
proper(tt) → ok(tt)
proper(isNatIList(x1)) → isNatIList(proper(x1))
proper(isNatList(x1)) → isNatList(proper(x1))
proper(isNat(x1)) → isNat(proper(x1))
proper(0) → ok(0)
proper(s(x1)) → s(proper(x1))
proper(length(x1)) → length(proper(x1))
proper(zeros) → ok(zeros)
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
proper(nil) → ok(nil)
proper(take(x1, x2)) → take(proper(x1), proper(x2))
proper(uTake1(x1)) → uTake1(proper(x1))
proper(uTake2(x1, x2, x3, x4)) → uTake2(proper(x1), proper(x2), proper(x3), proper(x4))
proper(uLength(x1, x2)) → uLength(proper(x1), proper(x2))
The set Q consists of the following terms:
active(isNatIList(x0))
active(isNat(0))
active(isNat(s(x0)))
active(isNat(length(x0)))
active(isNatList(nil))
active(isNatList(cons(x0, x1)))
active(isNatList(take(x0, x1)))
active(zeros)
active(and(x0, x1))
and(mark(x0), x1)
and(x0, mark(x1))
proper(and(x0, x1))
and(ok(x0), ok(x1))
proper(tt)
proper(isNatIList(x0))
isNatIList(ok(x0))
proper(isNatList(x0))
isNatList(ok(x0))
proper(isNat(x0))
isNat(ok(x0))
proper(0)
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(nil)
active(take(x0, x1))
take(mark(x0), x1)
take(x0, mark(x1))
proper(take(x0, x1))
take(ok(x0), ok(x1))
active(uTake1(x0))
uTake1(mark(x0))
proper(uTake1(x0))
uTake1(ok(x0))
active(uTake2(x0, x1, x2, x3))
uTake2(mark(x0), x1, x2, x3)
proper(uTake2(x0, x1, x2, x3))
uTake2(ok(x0), ok(x1), ok(x2), ok(x3))
active(uLength(x0, x1))
uLength(mark(x0), x1)
proper(uLength(x0, x1))
uLength(ok(x0), ok(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 7 less nodes.
↳ CSR
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ Improved Ferreira Ribeiro-Transformation
↳ Complete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(and(x0, x1))) → TOP(and(active(x0), x1))
TOP(ok(isNatList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatList(x1))))
TOP(mark(isNatList(x0))) → TOP(isNatList(proper(x0)))
TOP(ok(uLength(x0, x1))) → TOP(uLength(active(x0), x1))
TOP(ok(uTake1(x0))) → TOP(uTake1(active(x0)))
TOP(ok(take(s(x0), cons(x1, x2)))) → TOP(mark(uTake2(and(isNat(x0), and(isNat(x1), isNatIList(x2))), x0, x1, x2)))
TOP(ok(uTake2(tt, x0, x1, x2))) → TOP(mark(cons(x1, take(x0, x2))))
TOP(ok(isNatIList(cons(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(ok(isNatList(take(x0, x1)))) → TOP(mark(and(isNat(x0), isNatIList(x1))))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(ok(uTake2(x0, x1, x2, x3))) → TOP(uTake2(active(x0), x1, x2, x3))
TOP(ok(take(x0, x1))) → TOP(take(x0, active(x1)))
TOP(mark(uTake1(x0))) → TOP(uTake1(proper(x0)))
TOP(ok(take(x0, x1))) → TOP(take(active(x0), x1))
TOP(mark(uLength(x0, x1))) → TOP(uLength(proper(x0), proper(x1)))
TOP(ok(and(tt, x0))) → TOP(mark(x0))
TOP(ok(isNatIList(x0))) → TOP(mark(isNatList(x0)))
TOP(ok(isNat(length(x0)))) → TOP(mark(isNatList(x0)))
TOP(mark(uTake2(x0, x1, x2, x3))) → TOP(uTake2(proper(x0), proper(x1), proper(x2), proper(x3)))
TOP(ok(and(x0, x1))) → TOP(and(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(isNat(s(x0)))) → TOP(mark(isNat(x0)))
TOP(ok(take(0, x0))) → TOP(mark(uTake1(isNatIList(x0))))
TOP(mark(and(x0, x1))) → TOP(and(proper(x0), proper(x1)))
TOP(ok(uLength(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(uLength(and(isNat(x0), isNatList(x1)), x1)))
TOP(mark(take(x0, x1))) → TOP(take(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(isNatIList(x0))) → TOP(isNatIList(proper(x0)))
The TRS R consists of the following rules:
active(and(tt, T)) → mark(T)
active(isNatIList(IL)) → mark(isNatList(IL))
active(isNat(0)) → mark(tt)
active(isNat(s(N))) → mark(isNat(N))
active(isNat(length(L))) → mark(isNatList(L))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(N, L))) → mark(and(isNat(N), isNatList(L)))
active(isNatList(take(N, IL))) → mark(and(isNat(N), isNatIList(IL)))
active(zeros) → mark(cons(0, zeros))
active(take(0, IL)) → mark(uTake1(isNatIList(IL)))
active(uTake1(tt)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL))
active(uTake2(tt, M, N, IL)) → mark(cons(N, take(M, IL)))
active(length(cons(N, L))) → mark(uLength(and(isNat(N), isNatList(L)), L))
active(uLength(tt, L)) → mark(s(length(L)))
active(and(x1, x2)) → and(active(x1), x2)
active(and(x1, x2)) → and(x1, active(x2))
active(s(x1)) → s(active(x1))
active(length(x1)) → length(active(x1))
active(cons(x1, x2)) → cons(active(x1), x2)
active(take(x1,