Termination w.r.t. Q of the following Term Rewriting System could not be shown:
Q restricted rewrite system:
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
Q is empty.
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
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:
U111(tt, V1) → ISNATLIST(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
LENGTH(cons(N, L)) → U611(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
ISNATLIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
U511(tt, V1, V2) → ACTIVATE(V2)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U521(tt, V2) → ISNATLIST(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
U421(tt, V2) → ISNATILIST(activate(V2))
ACTIVATE(n__0) → 01
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__cons(X1, X2)) → CONS(X1, X2)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
U611(tt, L) → ACTIVATE(L)
LENGTH(cons(N, L)) → ISNAT(N)
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__zeros) → ZEROS
LENGTH(nil) → 01
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
U521(tt, V2) → U531(isNatList(activate(V2)))
ACTIVATE(n__length(X)) → LENGTH(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__nil) → NIL
U111(tt, V1) → U121(isNatList(activate(V1)))
ACTIVATE(n__s(X)) → S(X)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
U611(tt, L) → LENGTH(activate(L))
U521(tt, V2) → ACTIVATE(V2)
U421(tt, V2) → ACTIVATE(V2)
ISNATILIST(V) → ISNATILISTKIND(activate(V))
U411(tt, V1, V2) → ISNAT(activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U311(tt, V) → ISNATLIST(activate(V))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U611(tt, L) → S(length(activate(L)))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U411(tt, V1, V2) → U421(isNat(activate(V1)), activate(V2))
U511(tt, V1, V2) → ACTIVATE(V1)
U211(tt, V1) → U221(isNat(activate(V1)))
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
U311(tt, V) → ACTIVATE(V)
U311(tt, V) → U321(isNatList(activate(V)))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → U411(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
LENGTH(cons(N, L)) → AND(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N)))
U411(tt, V1, V2) → ACTIVATE(V2)
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNatIListKind(activate(L)))
U511(tt, V1, V2) → ISNAT(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
U111(tt, V1) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
LENGTH(cons(N, L)) → ACTIVATE(L)
U411(tt, V1, V2) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → U511(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U511(tt, V1, V2) → U521(isNat(activate(V1)), activate(V2))
U211(tt, V1) → ACTIVATE(V1)
U421(tt, V2) → U431(isNatIList(activate(V2)))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILIST(V) → U311(isNatIListKind(activate(V)), activate(V))
ZEROS → CONS(0, n__zeros)
ISNATILIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ZEROS → 01
ISNAT(n__length(V1)) → ACTIVATE(V1)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
U111(tt, V1) → ISNATLIST(activate(V1))
ISNATILIST(V) → ACTIVATE(V)
LENGTH(cons(N, L)) → U611(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
ISNATLIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
U511(tt, V1, V2) → ACTIVATE(V2)
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V2)
U521(tt, V2) → ISNATLIST(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
U421(tt, V2) → ISNATILIST(activate(V2))
ACTIVATE(n__0) → 01
ISNATILIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__cons(X1, X2)) → CONS(X1, X2)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
U611(tt, L) → ACTIVATE(L)
LENGTH(cons(N, L)) → ISNAT(N)
ISNATILIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__zeros) → ZEROS
LENGTH(nil) → 01
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
U521(tt, V2) → U531(isNatList(activate(V2)))
ACTIVATE(n__length(X)) → LENGTH(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__nil) → NIL
U111(tt, V1) → U121(isNatList(activate(V1)))
ACTIVATE(n__s(X)) → S(X)
AND(tt, X) → ACTIVATE(X)
ISNAT(n__s(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
U611(tt, L) → LENGTH(activate(L))
U521(tt, V2) → ACTIVATE(V2)
U421(tt, V2) → ACTIVATE(V2)
ISNATILIST(V) → ISNATILISTKIND(activate(V))
U411(tt, V1, V2) → ISNAT(activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U311(tt, V) → ISNATLIST(activate(V))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
U611(tt, L) → S(length(activate(L)))
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U411(tt, V1, V2) → U421(isNat(activate(V1)), activate(V2))
U511(tt, V1, V2) → ACTIVATE(V1)
U211(tt, V1) → U221(isNat(activate(V1)))
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
U311(tt, V) → ACTIVATE(V)
U311(tt, V) → U321(isNatList(activate(V)))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATILIST(n__cons(V1, V2)) → U411(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
LENGTH(cons(N, L)) → AND(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N)))
U411(tt, V1, V2) → ACTIVATE(V2)
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNatIListKind(activate(L)))
U511(tt, V1, V2) → ISNAT(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
U111(tt, V1) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
LENGTH(cons(N, L)) → ACTIVATE(L)
U411(tt, V1, V2) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → U511(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U511(tt, V1, V2) → U521(isNat(activate(V1)), activate(V2))
U211(tt, V1) → ACTIVATE(V1)
U421(tt, V2) → U431(isNatIList(activate(V2)))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATILIST(V) → U311(isNatIListKind(activate(V)), activate(V))
ZEROS → CONS(0, n__zeros)
ISNATILIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ZEROS → 01
ISNAT(n__length(V1)) → ACTIVATE(V1)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
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 27 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNAT(n__s(V1)) → ISNATKIND(activate(V1))
U111(tt, V1) → ISNATLIST(activate(V1))
U511(tt, V1, V2) → ACTIVATE(V2)
LENGTH(cons(N, L)) → U611(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
ISNATLIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
U511(tt, V1, V2) → ACTIVATE(V1)
U521(tt, V2) → ISNATLIST(activate(V2))
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → AND(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N)))
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
U611(tt, L) → ACTIVATE(L)
LENGTH(cons(N, L)) → ISNAT(N)
U511(tt, V1, V2) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNatIListKind(activate(L)))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
U111(tt, V1) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
ACTIVATE(n__length(X)) → LENGTH(X)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ACTIVATE(L)
ISNATLIST(n__cons(V1, V2)) → U511(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U511(tt, V1, V2) → U521(isNat(activate(V1)), activate(V2))
ISNAT(n__s(V1)) → ACTIVATE(V1)
AND(tt, X) → ACTIVATE(X)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
U211(tt, V1) → ACTIVATE(V1)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
U611(tt, L) → LENGTH(activate(L))
U521(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__length(V1)) → ACTIVATE(V1)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
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(n__s(V1)) → ISNATKIND(activate(V1))
U511(tt, V1, V2) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
U511(tt, V1, V2) → ACTIVATE(V1)
ISNAT(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNATKIND(n__length(V1)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → AND(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N)))
U611(tt, L) → ACTIVATE(L)
LENGTH(cons(N, L)) → AND(isNatList(activate(L)), n__isNatIListKind(activate(L)))
U111(tt, V1) → ACTIVATE(V1)
ISNATLIST(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNAT(n__length(V1)) → U111(isNatIListKind(activate(V1)), activate(V1))
ISNATKIND(n__length(V1)) → ISNATILISTKIND(activate(V1))
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V1)
LENGTH(cons(N, L)) → ACTIVATE(L)
ISNAT(n__s(V1)) → ACTIVATE(V1)
U211(tt, V1) → ACTIVATE(V1)
U521(tt, V2) → ACTIVATE(V2)
ISNATLIST(n__cons(V1, V2)) → ACTIVATE(V2)
ISNAT(n__length(V1)) → ACTIVATE(V1)
The remaining pairs can at least be oriented weakly.
U111(tt, V1) → ISNATLIST(activate(V1))
LENGTH(cons(N, L)) → U611(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
U521(tt, V2) → ISNATLIST(activate(V2))
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
LENGTH(cons(N, L)) → ISNAT(N)
U511(tt, V1, V2) → ISNAT(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ACTIVATE(n__length(X)) → LENGTH(X)
ISNATLIST(n__cons(V1, V2)) → U511(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U511(tt, V1, V2) → U521(isNat(activate(V1)), activate(V2))
AND(tt, X) → ACTIVATE(X)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
U611(tt, L) → LENGTH(activate(L))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ACTIVATE(x1)) = x1
POL(AND(x1, x2)) = x2
POL(ISNAT(x1)) = 1 + x1
POL(ISNATILISTKIND(x1)) = x1
POL(ISNATKIND(x1)) = x1
POL(ISNATLIST(x1)) = 1 + x1
POL(LENGTH(x1)) = 1 + x1
POL(U11(x1, x2)) = 0
POL(U111(x1, x2)) = 1 + x2
POL(U12(x1)) = 0
POL(U21(x1, x2)) = 0
POL(U211(x1, x2)) = 1 + x2
POL(U22(x1)) = 0
POL(U51(x1, x2, x3)) = 0
POL(U511(x1, x2, x3)) = 1 + x2 + x3
POL(U52(x1, x2)) = 0
POL(U521(x1, x2)) = 1 + x2
POL(U53(x1)) = 0
POL(U61(x1, x2)) = 1 + x2
POL(U611(x1, x2)) = 1 + x2
POL(activate(x1)) = x1
POL(and(x1, x2)) = x2
POL(cons(x1, x2)) = x1 + x2
POL(isNat(x1)) = 0
POL(isNatIListKind(x1)) = x1
POL(isNatKind(x1)) = x1
POL(isNatList(x1)) = 0
POL(length(x1)) = 1 + x1
POL(n__0) = 0
POL(n__and(x1, x2)) = x2
POL(n__cons(x1, x2)) = x1 + x2
POL(n__isNatIListKind(x1)) = x1
POL(n__isNatKind(x1)) = x1
POL(n__length(x1)) = 1 + x1
POL(n__nil) = 1
POL(n__s(x1)) = x1
POL(n__zeros) = 0
POL(nil) = 1
POL(s(x1)) = x1
POL(tt) = 0
POL(zeros) = 0
The following usable rules [17] were oriented:
zeros → cons(0, n__zeros)
U12(tt) → tt
U11(tt, V1) → U12(isNatList(activate(V1)))
U22(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
zeros → n__zeros
and(X1, X2) → n__and(X1, X2)
nil → n__nil
isNatIListKind(X) → n__isNatIListKind(X)
cons(X1, X2) → n__cons(X1, X2)
isNatKind(n__0) → tt
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
length(nil) → 0
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
activate(X) → X
activate(n__0) → 0
activate(n__length(X)) → length(X)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__isNatIListKind(X)) → isNatIListKind(X)
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__nil) → nil
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
U111(tt, V1) → ISNATLIST(activate(V1))
ACTIVATE(n__length(X)) → LENGTH(X)
LENGTH(cons(N, L)) → U611(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
ISNATLIST(n__cons(V1, V2)) → U511(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U511(tt, V1, V2) → U521(isNat(activate(V1)), activate(V2))
AND(tt, X) → ACTIVATE(X)
LENGTH(cons(N, L)) → ISNATLIST(activate(L))
U521(tt, V2) → ISNATLIST(activate(V2))
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
U611(tt, L) → LENGTH(activate(L))
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, V1) → ISNAT(activate(V1))
LENGTH(cons(N, L)) → ISNAT(N)
U511(tt, V1, V2) → ISNAT(activate(V1))
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
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 5 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U211(tt, V1) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U211(tt, V1) → ISNAT(activate(V1)) at position [0] we obtained the following new rules:
U211(tt, n__and(x0, x1)) → ISNAT(and(x0, x1))
U211(tt, n__length(x0)) → ISNAT(length(x0))
U211(tt, n__0) → ISNAT(0)
U211(tt, n__cons(x0, x1)) → ISNAT(cons(x0, x1))
U211(tt, n__nil) → ISNAT(nil)
U211(tt, n__isNatIListKind(x0)) → ISNAT(isNatIListKind(x0))
U211(tt, x0) → ISNAT(x0)
U211(tt, n__isNatKind(x0)) → ISNAT(isNatKind(x0))
U211(tt, n__s(x0)) → ISNAT(s(x0))
U211(tt, n__zeros) → ISNAT(zeros)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U211(tt, n__length(x0)) → ISNAT(length(x0))
U211(tt, n__and(x0, x1)) → ISNAT(and(x0, x1))
U211(tt, n__0) → ISNAT(0)
U211(tt, n__nil) → ISNAT(nil)
U211(tt, n__cons(x0, x1)) → ISNAT(cons(x0, x1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, n__isNatIListKind(x0)) → ISNAT(isNatIListKind(x0))
U211(tt, n__isNatKind(x0)) → ISNAT(isNatKind(x0))
U211(tt, x0) → ISNAT(x0)
U211(tt, n__s(x0)) → ISNAT(s(x0))
U211(tt, n__zeros) → ISNAT(zeros)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U211(tt, n__0) → ISNAT(0) at position [0] we obtained the following new rules:
U211(tt, n__0) → ISNAT(n__0)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U211(tt, n__and(x0, x1)) → ISNAT(and(x0, x1))
U211(tt, n__length(x0)) → ISNAT(length(x0))
U211(tt, n__cons(x0, x1)) → ISNAT(cons(x0, x1))
U211(tt, n__nil) → ISNAT(nil)
U211(tt, n__0) → ISNAT(n__0)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, n__isNatIListKind(x0)) → ISNAT(isNatIListKind(x0))
U211(tt, x0) → ISNAT(x0)
U211(tt, n__isNatKind(x0)) → ISNAT(isNatKind(x0))
U211(tt, n__s(x0)) → ISNAT(s(x0))
U211(tt, n__zeros) → ISNAT(zeros)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
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.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U211(tt, n__length(x0)) → ISNAT(length(x0))
U211(tt, n__and(x0, x1)) → ISNAT(and(x0, x1))
U211(tt, n__nil) → ISNAT(nil)
U211(tt, n__cons(x0, x1)) → ISNAT(cons(x0, x1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, n__isNatIListKind(x0)) → ISNAT(isNatIListKind(x0))
U211(tt, n__isNatKind(x0)) → ISNAT(isNatKind(x0))
U211(tt, x0) → ISNAT(x0)
U211(tt, n__s(x0)) → ISNAT(s(x0))
U211(tt, n__zeros) → ISNAT(zeros)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U211(tt, n__cons(x0, x1)) → ISNAT(cons(x0, x1)) at position [0] we obtained the following new rules:
U211(tt, n__cons(x0, x1)) → ISNAT(n__cons(x0, x1))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U211(tt, n__cons(x0, x1)) → ISNAT(n__cons(x0, x1))
U211(tt, n__and(x0, x1)) → ISNAT(and(x0, x1))
U211(tt, n__length(x0)) → ISNAT(length(x0))
U211(tt, n__nil) → ISNAT(nil)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, n__isNatIListKind(x0)) → ISNAT(isNatIListKind(x0))
U211(tt, x0) → ISNAT(x0)
U211(tt, n__isNatKind(x0)) → ISNAT(isNatKind(x0))
U211(tt, n__s(x0)) → ISNAT(s(x0))
U211(tt, n__zeros) → ISNAT(zeros)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
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.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U211(tt, n__length(x0)) → ISNAT(length(x0))
U211(tt, n__and(x0, x1)) → ISNAT(and(x0, x1))
U211(tt, n__nil) → ISNAT(nil)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, n__isNatIListKind(x0)) → ISNAT(isNatIListKind(x0))
U211(tt, n__isNatKind(x0)) → ISNAT(isNatKind(x0))
U211(tt, x0) → ISNAT(x0)
U211(tt, n__s(x0)) → ISNAT(s(x0))
U211(tt, n__zeros) → ISNAT(zeros)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U211(tt, n__nil) → ISNAT(nil) at position [0] we obtained the following new rules:
U211(tt, n__nil) → ISNAT(n__nil)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U211(tt, n__and(x0, x1)) → ISNAT(and(x0, x1))
U211(tt, n__length(x0)) → ISNAT(length(x0))
U211(tt, n__nil) → ISNAT(n__nil)
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, n__isNatIListKind(x0)) → ISNAT(isNatIListKind(x0))
U211(tt, x0) → ISNAT(x0)
U211(tt, n__isNatKind(x0)) → ISNAT(isNatKind(x0))
U211(tt, n__s(x0)) → ISNAT(s(x0))
U211(tt, n__zeros) → ISNAT(zeros)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
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.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U211(tt, n__length(x0)) → ISNAT(length(x0))
U211(tt, n__and(x0, x1)) → ISNAT(and(x0, x1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, n__isNatIListKind(x0)) → ISNAT(isNatIListKind(x0))
U211(tt, n__isNatKind(x0)) → ISNAT(isNatKind(x0))
U211(tt, x0) → ISNAT(x0)
U211(tt, n__s(x0)) → ISNAT(s(x0))
U211(tt, n__zeros) → ISNAT(zeros)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U211(tt, n__s(x0)) → ISNAT(s(x0)) at position [0] we obtained the following new rules:
U211(tt, n__s(x0)) → ISNAT(n__s(x0))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U211(tt, n__and(x0, x1)) → ISNAT(and(x0, x1))
U211(tt, n__length(x0)) → ISNAT(length(x0))
U211(tt, n__s(x0)) → ISNAT(n__s(x0))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, n__isNatIListKind(x0)) → ISNAT(isNatIListKind(x0))
U211(tt, x0) → ISNAT(x0)
U211(tt, n__isNatKind(x0)) → ISNAT(isNatKind(x0))
U211(tt, n__zeros) → ISNAT(zeros)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U211(tt, n__zeros) → ISNAT(zeros) at position [0] we obtained the following new rules:
U211(tt, n__zeros) → ISNAT(n__zeros)
U211(tt, n__zeros) → ISNAT(cons(0, n__zeros))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U211(tt, n__length(x0)) → ISNAT(length(x0))
U211(tt, n__and(x0, x1)) → ISNAT(and(x0, x1))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, n__s(x0)) → ISNAT(n__s(x0))
U211(tt, n__zeros) → ISNAT(n__zeros)
U211(tt, n__isNatIListKind(x0)) → ISNAT(isNatIListKind(x0))
U211(tt, n__isNatKind(x0)) → ISNAT(isNatKind(x0))
U211(tt, x0) → ISNAT(x0)
U211(tt, n__zeros) → ISNAT(cons(0, n__zeros))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
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.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U211(tt, n__length(x0)) → ISNAT(length(x0))
U211(tt, n__and(x0, x1)) → ISNAT(and(x0, x1))
U211(tt, n__s(x0)) → ISNAT(n__s(x0))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, n__isNatIListKind(x0)) → ISNAT(isNatIListKind(x0))
U211(tt, n__isNatKind(x0)) → ISNAT(isNatKind(x0))
U211(tt, x0) → ISNAT(x0)
U211(tt, n__zeros) → ISNAT(cons(0, n__zeros))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U211(tt, n__zeros) → ISNAT(cons(0, n__zeros)) at position [0] we obtained the following new rules:
U211(tt, n__zeros) → ISNAT(cons(n__0, n__zeros))
U211(tt, n__zeros) → ISNAT(n__cons(0, n__zeros))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U211(tt, n__zeros) → ISNAT(cons(n__0, n__zeros))
U211(tt, n__and(x0, x1)) → ISNAT(and(x0, x1))
U211(tt, n__length(x0)) → ISNAT(length(x0))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, n__s(x0)) → ISNAT(n__s(x0))
U211(tt, n__isNatIListKind(x0)) → ISNAT(isNatIListKind(x0))
U211(tt, x0) → ISNAT(x0)
U211(tt, n__isNatKind(x0)) → ISNAT(isNatKind(x0))
U211(tt, n__zeros) → ISNAT(n__cons(0, n__zeros))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
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.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U211(tt, n__zeros) → ISNAT(cons(n__0, n__zeros))
U211(tt, n__length(x0)) → ISNAT(length(x0))
U211(tt, n__and(x0, x1)) → ISNAT(and(x0, x1))
U211(tt, n__s(x0)) → ISNAT(n__s(x0))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, n__isNatIListKind(x0)) → ISNAT(isNatIListKind(x0))
U211(tt, n__isNatKind(x0)) → ISNAT(isNatKind(x0))
U211(tt, x0) → ISNAT(x0)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U211(tt, n__zeros) → ISNAT(cons(n__0, n__zeros)) at position [0] we obtained the following new rules:
U211(tt, n__zeros) → ISNAT(n__cons(n__0, n__zeros))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U211(tt, n__and(x0, x1)) → ISNAT(and(x0, x1))
U211(tt, n__length(x0)) → ISNAT(length(x0))
U211(tt, n__zeros) → ISNAT(n__cons(n__0, n__zeros))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, n__s(x0)) → ISNAT(n__s(x0))
U211(tt, n__isNatIListKind(x0)) → ISNAT(isNatIListKind(x0))
U211(tt, x0) → ISNAT(x0)
U211(tt, n__isNatKind(x0)) → ISNAT(isNatKind(x0))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
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.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U211(tt, n__length(x0)) → ISNAT(length(x0))
U211(tt, n__and(x0, x1)) → ISNAT(and(x0, x1))
U211(tt, n__s(x0)) → ISNAT(n__s(x0))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, n__isNatIListKind(x0)) → ISNAT(isNatIListKind(x0))
U211(tt, n__isNatKind(x0)) → ISNAT(isNatKind(x0))
U211(tt, x0) → ISNAT(x0)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
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.
U211(tt, n__isNatKind(x0)) → ISNAT(isNatKind(x0))
The remaining pairs can at least be oriented weakly.
U211(tt, n__length(x0)) → ISNAT(length(x0))
U211(tt, n__and(x0, x1)) → ISNAT(and(x0, x1))
U211(tt, n__s(x0)) → ISNAT(n__s(x0))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, n__isNatIListKind(x0)) → ISNAT(isNatIListKind(x0))
U211(tt, x0) → ISNAT(x0)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
| M( U52(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( and(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( isNatIListKind(x1) ) = | | + | | · | x1 |
| M( U21(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( isNatList(x1) ) = | | + | | · | x1 |
| M( U51(x1, ..., x3) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| M( n__length(x1) ) = | | + | | · | x1 |
| M( n__isNatKind(x1) ) = | | + | | · | x1 |
| M( n__cons(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U11(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( isNatKind(x1) ) = | | + | | · | x1 |
| M( n__and(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( cons(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U61(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( n__isNatIListKind(x1) ) = | | + | | · | x1 |
Tuple symbols:
| M( U211(x1, x2) ) = | 0 | + | | · | x1 | + | | · | x2 |
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:
zeros → cons(0, n__zeros)
U12(tt) → tt
U11(tt, V1) → U12(isNatList(activate(V1)))
U22(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
zeros → n__zeros
and(X1, X2) → n__and(X1, X2)
nil → n__nil
isNatIListKind(X) → n__isNatIListKind(X)
cons(X1, X2) → n__cons(X1, X2)
isNatKind(n__0) → tt
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
length(nil) → 0
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatList(n__nil) → tt
activate(X) → X
activate(n__0) → 0
activate(n__length(X)) → length(X)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__isNatIListKind(X)) → isNatIListKind(X)
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__nil) → nil
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U211(tt, n__and(x0, x1)) → ISNAT(and(x0, x1))
U211(tt, n__length(x0)) → ISNAT(length(x0))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, n__s(x0)) → ISNAT(n__s(x0))
U211(tt, n__isNatIListKind(x0)) → ISNAT(isNatIListKind(x0))
U211(tt, x0) → ISNAT(x0)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
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.
U211(tt, n__isNatIListKind(x0)) → ISNAT(isNatIListKind(x0))
The remaining pairs can at least be oriented weakly.
U211(tt, n__and(x0, x1)) → ISNAT(and(x0, x1))
U211(tt, n__length(x0)) → ISNAT(length(x0))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, n__s(x0)) → ISNAT(n__s(x0))
U211(tt, x0) → ISNAT(x0)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
| M( U52(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( and(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( isNatIListKind(x1) ) = | | + | | · | x1 |
| M( U21(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( isNatList(x1) ) = | | + | | · | x1 |
| M( U51(x1, ..., x3) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| M( n__length(x1) ) = | | + | | · | x1 |
| M( n__isNatKind(x1) ) = | | + | | · | x1 |
| M( n__cons(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U11(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( isNatKind(x1) ) = | | + | | · | x1 |
| M( n__and(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( cons(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U61(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( n__isNatIListKind(x1) ) = | | + | | · | x1 |
Tuple symbols:
| M( U211(x1, x2) ) = | 0 | + | | · | x1 | + | | · | x2 |
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:
zeros → cons(0, n__zeros)
U12(tt) → tt
U11(tt, V1) → U12(isNatList(activate(V1)))
U22(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
s(X) → n__s(X)
length(X) → n__length(X)
0 → n__0
zeros → n__zeros
and(X1, X2) → n__and(X1, X2)
nil → n__nil
isNatIListKind(X) → n__isNatIListKind(X)
cons(X1, X2) → n__cons(X1, X2)
isNatKind(n__0) → tt
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
length(nil) → 0
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
activate(X) → X
activate(n__0) → 0
activate(n__length(X)) → length(X)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
activate(n__isNatIListKind(X)) → isNatIListKind(X)
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
and(tt, X) → activate(X)
activate(n__isNatKind(X)) → isNatKind(X)
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__nil) → nil
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U211(tt, n__length(x0)) → ISNAT(length(x0))
U211(tt, n__and(x0, x1)) → ISNAT(and(x0, x1))
U211(tt, n__s(x0)) → ISNAT(n__s(x0))
ISNAT(n__s(V1)) → U211(isNatKind(activate(V1)), activate(V1))
U211(tt, x0) → ISNAT(x0)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATLIST(n__cons(V1, V2)) → U511(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U511(tt, V1, V2) → U521(isNat(activate(V1)), activate(V2))
U521(tt, V2) → ISNATLIST(activate(V2))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U611(tt, L) → LENGTH(activate(L))
LENGTH(cons(N, L)) → U611(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V2)
ISNATILISTKIND(n__cons(V1, V2)) → AND(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
ACTIVATE(n__isNatIListKind(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(n__cons(V1, V2)) → ISNATKIND(activate(V1))
ISNATKIND(n__s(V1)) → ACTIVATE(V1)
ACTIVATE(n__isNatKind(X)) → ISNATKIND(X)
ISNATKIND(n__s(V1)) → ISNATKIND(activate(V1))
AND(tt, X) → ACTIVATE(X)
ISNATILISTKIND(n__cons(V1, V2)) → ACTIVATE(V1)
ACTIVATE(n__and(X1, X2)) → AND(X1, X2)
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U421(tt, V2) → ISNATILIST(activate(V2))
ISNATILIST(n__cons(V1, V2)) → U411(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
U411(tt, V1, V2) → U421(isNat(activate(V1)), activate(V2))
The TRS R consists of the following rules:
zeros → cons(0, n__zeros)
U11(tt, V1) → U12(isNatList(activate(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(activate(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(activate(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(activate(V1)), activate(V2))
U42(tt, V2) → U43(isNatIList(activate(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(activate(V1)), activate(V2))
U52(tt, V2) → U53(isNatList(activate(V2)))
U53(tt) → tt
U61(tt, L) → s(length(activate(L)))
and(tt, X) → activate(X)
isNat(n__0) → tt
isNat(n__length(V1)) → U11(isNatIListKind(activate(V1)), activate(V1))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatIList(V) → U31(isNatIListKind(activate(V)), activate(V))
isNatIList(n__zeros) → tt
isNatIList(n__cons(V1, V2)) → U41(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
isNatIListKind(n__nil) → tt
isNatIListKind(n__zeros) → tt
isNatIListKind(n__cons(V1, V2)) → and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2)))
isNatKind(n__0) → tt
isNatKind(n__length(V1)) → isNatIListKind(activate(V1))
isNatKind(n__s(V1)) → isNatKind(activate(V1))
isNatList(n__nil) → tt
isNatList(n__cons(V1, V2)) → U51(and(isNatKind(activate(V1)), n__isNatIListKind(activate(V2))), activate(V1), activate(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(activate(L)), n__isNatIListKind(activate(L))), n__and(isNat(N), n__isNatKind(N))), activate(L))
zeros → n__zeros
0 → n__0
length(X) → n__length(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
isNatIListKind(X) → n__isNatIListKind(X)
nil → n__nil
and(X1, X2) → n__and(X1, X2)
isNatKind(X) → n__isNatKind(X)
activate(n__zeros) → zeros
activate(n__0) → 0
activate(n__length(X)) → length(X)
activate(n__s(X)) → s(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__isNatIListKind(X)) → isNatIListKind(X)
activate(n__nil) → nil
activate(n__and(X1, X2)) → and(X1, X2)
activate(n__isNatKind(X)) → isNatKind(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.