Termination of the following Term Rewriting System could be proven:
Context-sensitive rewrite system:
The TRS R consists of the following rules:
zeros → cons(0, zeros)
U11(tt, V1) → U12(isNatList(V1))
U12(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V) → U32(isNatList(V))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(V1), V2)
U42(tt, V2) → U43(isNatIList(V2))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(V1), V2)
U52(tt, V2) → U53(isNatList(V2))
U53(tt) → tt
U61(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → and(isNatKind(V1), isNatIListKind(V2))
isNatKind(0) → tt
isNatKind(length(V1)) → isNatIListKind(V1)
isNatKind(s(V1)) → isNatKind(V1)
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
The replacement map contains the following entries:zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U12: {1}
isNatList: empty set
U21: {1}
U22: {1}
isNat: empty set
U31: {1}
U32: {1}
U41: {1}
U42: {1}
U43: {1}
isNatIList: empty set
U51: {1}
U52: {1}
U53: {1}
U61: {1}
s: {1}
length: {1}
and: {1}
isNatIListKind: empty set
isNatKind: empty set
nil: empty set
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
Context-sensitive rewrite system:
The TRS R consists of the following rules:
zeros → cons(0, zeros)
U11(tt, V1) → U12(isNatList(V1))
U12(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V) → U32(isNatList(V))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(V1), V2)
U42(tt, V2) → U43(isNatIList(V2))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(V1), V2)
U52(tt, V2) → U53(isNatList(V2))
U53(tt) → tt
U61(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → and(isNatKind(V1), isNatIListKind(V2))
isNatKind(0) → tt
isNatKind(length(V1)) → isNatIListKind(V1)
isNatKind(s(V1)) → isNatKind(V1)
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
The replacement map contains the following entries:zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U12: {1}
isNatList: empty set
U21: {1}
U22: {1}
isNat: empty set
U31: {1}
U32: {1}
U41: {1}
U42: {1}
U43: {1}
isNatIList: empty set
U51: {1}
U52: {1}
U53: {1}
U61: {1}
s: {1}
length: {1}
and: {1}
isNatIListKind: empty set
isNatKind: empty set
nil: empty set
Using Improved CS-DPs we result in the following initial Q-CSDP problem.
↳ CSR
↳ CSDependencyPairsProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
Q-restricted context-sensitive dependency pair problem:
The symbols in {U12, U22, U32, U43, U53, s, length, U121, U221, U321, U431, U531, LENGTH} are replacing on all positions.
For all symbols f in {cons, U11, U21, U31, U41, U42, U51, U52, U61, and, U111, U211, U311, U421, U411, U521, U511, U611, AND} we have µ(f) = {1}.
The symbols in {isNatList, isNat, isNatIList, isNatIListKind, isNatKind, ISNATLIST, ISNAT, ISNATILIST, ISNATILISTKIND, ISNATKIND, U} are not replacing on any position.
The ordinary context-sensitive dependency pairs DPo are:
U111(tt, V1) → U121(isNatList(V1))
U111(tt, V1) → ISNATLIST(V1)
U211(tt, V1) → U221(isNat(V1))
U211(tt, V1) → ISNAT(V1)
U311(tt, V) → U321(isNatList(V))
U311(tt, V) → ISNATLIST(V)
U411(tt, V1, V2) → U421(isNat(V1), V2)
U411(tt, V1, V2) → ISNAT(V1)
U421(tt, V2) → U431(isNatIList(V2))
U421(tt, V2) → ISNATILIST(V2)
U511(tt, V1, V2) → U521(isNat(V1), V2)
U511(tt, V1, V2) → ISNAT(V1)
U521(tt, V2) → U531(isNatList(V2))
U521(tt, V2) → ISNATLIST(V2)
U611(tt, L) → LENGTH(L)
ISNAT(length(V1)) → U111(isNatIListKind(V1), V1)
ISNAT(length(V1)) → ISNATILISTKIND(V1)
ISNAT(s(V1)) → U211(isNatKind(V1), V1)
ISNAT(s(V1)) → ISNATKIND(V1)
ISNATILIST(V) → U311(isNatIListKind(V), V)
ISNATILIST(V) → ISNATILISTKIND(V)
ISNATILIST(cons(V1, V2)) → U411(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
ISNATILIST(cons(V1, V2)) → AND(isNatKind(V1), isNatIListKind(V2))
ISNATILIST(cons(V1, V2)) → ISNATKIND(V1)
ISNATILISTKIND(cons(V1, V2)) → AND(isNatKind(V1), isNatIListKind(V2))
ISNATILISTKIND(cons(V1, V2)) → ISNATKIND(V1)
ISNATKIND(length(V1)) → ISNATILISTKIND(V1)
ISNATKIND(s(V1)) → ISNATKIND(V1)
ISNATLIST(cons(V1, V2)) → U511(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
ISNATLIST(cons(V1, V2)) → AND(isNatKind(V1), isNatIListKind(V2))
ISNATLIST(cons(V1, V2)) → ISNATKIND(V1)
LENGTH(cons(N, L)) → U611(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
LENGTH(cons(N, L)) → AND(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
LENGTH(cons(N, L)) → AND(isNatList(L), isNatIListKind(L))
LENGTH(cons(N, L)) → ISNATLIST(L)
The collapsing dependency pairs are DPc:
U611(tt, L) → L
AND(tt, X) → X
The hidden terms of R are:
zeros
isNatIListKind(V2)
Every hiding context is built from:
and on positions {1}
Hence, the new unhiding pairs DPu are :
U611(tt, L) → U(L)
AND(tt, X) → U(X)
U(and(x_0, x_1)) → U(x_0)
U(zeros) → ZEROS
U(isNatIListKind(V2)) → ISNATILISTKIND(V2)
The TRS R consists of the following rules:
zeros → cons(0, zeros)
U11(tt, V1) → U12(isNatList(V1))
U12(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V) → U32(isNatList(V))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(V1), V2)
U42(tt, V2) → U43(isNatIList(V2))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(V1), V2)
U52(tt, V2) → U53(isNatList(V2))
U53(tt) → tt
U61(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → and(isNatKind(V1), isNatIListKind(V2))
isNatKind(0) → tt
isNatKind(length(V1)) → isNatIListKind(V1)
isNatKind(s(V1)) → isNatKind(V1)
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
The approximation of the Context-Sensitive Dependency Graph contains 4 SCCs with 20 less nodes.
↳ CSR
↳ CSDependencyPairsProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDPReductionPairProof
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
Q-restricted context-sensitive dependency pair problem:
The symbols in {U12, U22, U32, U43, U53, s, length} are replacing on all positions.
For all symbols f in {cons, U11, U21, U31, U41, U42, U51, U52, U61, and, AND} we have µ(f) = {1}.
The symbols in {isNatList, isNat, isNatIList, isNatIListKind, isNatKind, U, ISNATILISTKIND, ISNATKIND} are not replacing on any position.
The TRS P consists of the following rules:
AND(tt, X) → U(X)
U(and(x_0, x_1)) → U(x_0)
U(isNatIListKind(V2)) → ISNATILISTKIND(V2)
ISNATILISTKIND(cons(V1, V2)) → AND(isNatKind(V1), isNatIListKind(V2))
ISNATILISTKIND(cons(V1, V2)) → ISNATKIND(V1)
ISNATKIND(length(V1)) → ISNATILISTKIND(V1)
ISNATKIND(s(V1)) → ISNATKIND(V1)
The TRS R consists of the following rules:
zeros → cons(0, zeros)
U11(tt, V1) → U12(isNatList(V1))
U12(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V) → U32(isNatList(V))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(V1), V2)
U42(tt, V2) → U43(isNatIList(V2))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(V1), V2)
U52(tt, V2) → U53(isNatList(V2))
U53(tt) → tt
U61(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → and(isNatKind(V1), isNatIListKind(V2))
isNatKind(0) → tt
isNatKind(length(V1)) → isNatIListKind(V1)
isNatKind(s(V1)) → isNatKind(V1)
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
Using the order
Polynomial interpretation with max and min functions [25]:
POL(0) = 0
POL(AND(x1, x2)) = x1 + x2
POL(ISNATILISTKIND(x1)) = x1
POL(ISNATKIND(x1)) = x1
POL(U(x1)) = x1
POL(U11(x1, x2)) = 0
POL(U12(x1)) = 0
POL(U21(x1, x2)) = 0
POL(U22(x1)) = 0
POL(U51(x1, x2, x3)) = 0
POL(U52(x1, x2)) = 0
POL(U53(x1)) = 0
POL(U61(x1, x2)) = 1 + x2
POL(and(x1, x2)) = x1 + x2
POL(cons(x1, x2)) = x1 + x2
POL(isNat(x1)) = 0
POL(isNatIListKind(x1)) = x1
POL(isNatKind(x1)) = x1
POL(isNatList(x1)) = x1
POL(length(x1)) = 1 + x1
POL(nil) = 0
POL(s(x1)) = x1
POL(tt) = 0
POL(zeros) = 0
the following usable rules
and(tt, X) → X
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → and(isNatKind(V1), isNatIListKind(V2))
zeros → cons(0, zeros)
isNatKind(0) → tt
isNatKind(length(V1)) → isNatIListKind(V1)
isNatKind(s(V1)) → isNatKind(V1)
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
U61(tt, L) → s(length(L))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
U51(tt, V1, V2) → U52(isNat(V1), V2)
U52(tt, V2) → U53(isNatList(V2))
U53(tt) → tt
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
U11(tt, V1) → U12(isNatList(V1))
U12(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
could all be oriented weakly.
Furthermore, the pairs
ISNATKIND(length(V1)) → ISNATILISTKIND(V1)
could be oriented strictly and thus removed.
The pairs
AND(tt, X) → U(X)
U(and(x_0, x_1)) → U(x_0)
U(isNatIListKind(V2)) → ISNATILISTKIND(V2)
ISNATILISTKIND(cons(V1, V2)) → AND(isNatKind(V1), isNatIListKind(V2))
ISNATILISTKIND(cons(V1, V2)) → ISNATKIND(V1)
ISNATKIND(s(V1)) → ISNATKIND(V1)
could only be oriented weakly and must be analyzed further.
↳ CSR
↳ CSDependencyPairsProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDPReductionPairProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
Q-restricted context-sensitive dependency pair problem:
The symbols in {U12, U22, U32, U43, U53, s, length} are replacing on all positions.
For all symbols f in {cons, U11, U21, U31, U41, U42, U51, U52, U61, and, AND} we have µ(f) = {1}.
The symbols in {isNatList, isNat, isNatIList, isNatIListKind, isNatKind, U, ISNATILISTKIND, ISNATKIND} are not replacing on any position.
The TRS P consists of the following rules:
AND(tt, X) → U(X)
U(and(x_0, x_1)) → U(x_0)
U(isNatIListKind(V2)) → ISNATILISTKIND(V2)
ISNATILISTKIND(cons(V1, V2)) → AND(isNatKind(V1), isNatIListKind(V2))
ISNATILISTKIND(cons(V1, V2)) → ISNATKIND(V1)
ISNATKIND(s(V1)) → ISNATKIND(V1)
The TRS R consists of the following rules:
zeros → cons(0, zeros)
U11(tt, V1) → U12(isNatList(V1))
U12(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V) → U32(isNatList(V))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(V1), V2)
U42(tt, V2) → U43(isNatIList(V2))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(V1), V2)
U52(tt, V2) → U53(isNatList(V2))
U53(tt) → tt
U61(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → and(isNatKind(V1), isNatIListKind(V2))
isNatKind(0) → tt
isNatKind(length(V1)) → isNatIListKind(V1)
isNatKind(s(V1)) → isNatKind(V1)
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
The approximation of the Context-Sensitive Dependency Graph contains 2 SCCs with 1 less node.
↳ CSR
↳ CSDependencyPairsProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDPReductionPairProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDPSubtermProof
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
Q-restricted context-sensitive dependency pair problem:
The symbols in {U12, U22, U32, U43, U53, s, length} are replacing on all positions.
For all symbols f in {cons, U11, U21, U31, U41, U42, U51, U52, U61, and} we have µ(f) = {1}.
The symbols in {isNatList, isNat, isNatIList, isNatIListKind, isNatKind, ISNATKIND} are not replacing on any position.
The TRS P consists of the following rules:
ISNATKIND(s(V1)) → ISNATKIND(V1)
The TRS R consists of the following rules:
zeros → cons(0, zeros)
U11(tt, V1) → U12(isNatList(V1))
U12(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V) → U32(isNatList(V))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(V1), V2)
U42(tt, V2) → U43(isNatIList(V2))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(V1), V2)
U52(tt, V2) → U53(isNatList(V2))
U53(tt) → tt
U61(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → and(isNatKind(V1), isNatIListKind(V2))
isNatKind(0) → tt
isNatKind(length(V1)) → isNatIListKind(V1)
isNatKind(s(V1)) → isNatKind(V1)
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
We use the subterm processor [20].
The following pairs can be oriented strictly and are deleted.
ISNATKIND(s(V1)) → ISNATKIND(V1)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
ISNATKIND(x1) = x1
Subterm Order
↳ CSR
↳ CSDependencyPairsProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDPReductionPairProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDPSubtermProof
↳ QCSDP
↳ PIsEmptyProof
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
Q-restricted context-sensitive dependency pair problem:
The symbols in {U12, U22, U32, U43, U53, s, length} are replacing on all positions.
For all symbols f in {cons, U11, U21, U31, U41, U42, U51, U52, U61, and} we have µ(f) = {1}.
The symbols in {isNatList, isNat, isNatIList, isNatIListKind, isNatKind} are not replacing on any position.
The TRS P consists of the following rules:
none
The TRS R consists of the following rules:
zeros → cons(0, zeros)
U11(tt, V1) → U12(isNatList(V1))
U12(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V) → U32(isNatList(V))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(V1), V2)
U42(tt, V2) → U43(isNatIList(V2))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(V1), V2)
U52(tt, V2) → U53(isNatList(V2))
U53(tt) → tt
U61(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → and(isNatKind(V1), isNatIListKind(V2))
isNatKind(0) → tt
isNatKind(length(V1)) → isNatIListKind(V1)
isNatKind(s(V1)) → isNatKind(V1)
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
The TRS P is empty. Hence, there is no (P,Q,R,µ)-chain.
↳ CSR
↳ CSDependencyPairsProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDPReductionPairProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDP
↳ QCSDPInstantiationProcessor
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
Q-restricted context-sensitive dependency pair problem:
The symbols in {U12, U22, U32, U43, U53, s, length} are replacing on all positions.
For all symbols f in {cons, U11, U21, U31, U41, U42, U51, U52, U61, and, AND} we have µ(f) = {1}.
The symbols in {isNatList, isNat, isNatIList, isNatIListKind, isNatKind, U, ISNATILISTKIND} are not replacing on any position.
The TRS P consists of the following rules:
U(and(x_0, x_1)) → U(x_0)
U(isNatIListKind(V2)) → ISNATILISTKIND(V2)
ISNATILISTKIND(cons(V1, V2)) → AND(isNatKind(V1), isNatIListKind(V2))
AND(tt, X) → U(X)
The TRS R consists of the following rules:
zeros → cons(0, zeros)
U11(tt, V1) → U12(isNatList(V1))
U12(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V) → U32(isNatList(V))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(V1), V2)
U42(tt, V2) → U43(isNatIList(V2))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(V1), V2)
U52(tt, V2) → U53(isNatList(V2))
U53(tt) → tt
U61(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → and(isNatKind(V1), isNatIListKind(V2))
isNatKind(0) → tt
isNatKind(length(V1)) → isNatIListKind(V1)
isNatKind(s(V1)) → isNatKind(V1)
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
Using the Context-Sensitive Instantiation Processor
the pair AND(tt, X) → U(X)
was transformed to the following new pairs:
AND(tt, isNatIListKind(z1)) → U(isNatIListKind(z1))
↳ CSR
↳ CSDependencyPairsProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDPReductionPairProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDP
↳ QCSDPInstantiationProcessor
↳ QCSDP
↳ QCSDependencyGraphProof
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
Q-restricted context-sensitive dependency pair problem:
The symbols in {U12, U22, U32, U43, U53, s, length} are replacing on all positions.
For all symbols f in {cons, U11, U21, U31, U41, U42, U51, U52, U61, and, AND} we have µ(f) = {1}.
The symbols in {isNatList, isNat, isNatIList, isNatIListKind, isNatKind, U, ISNATILISTKIND} are not replacing on any position.
The TRS P consists of the following rules:
U(and(x_0, x_1)) → U(x_0)
U(isNatIListKind(V2)) → ISNATILISTKIND(V2)
ISNATILISTKIND(cons(V1, V2)) → AND(isNatKind(V1), isNatIListKind(V2))
AND(tt, isNatIListKind(z1)) → U(isNatIListKind(z1))
The TRS R consists of the following rules:
zeros → cons(0, zeros)
U11(tt, V1) → U12(isNatList(V1))
U12(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V) → U32(isNatList(V))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(V1), V2)
U42(tt, V2) → U43(isNatIList(V2))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(V1), V2)
U52(tt, V2) → U53(isNatList(V2))
U53(tt) → tt
U61(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → and(isNatKind(V1), isNatIListKind(V2))
isNatKind(0) → tt
isNatKind(length(V1)) → isNatIListKind(V1)
isNatKind(s(V1)) → isNatKind(V1)
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
The approximation of the Context-Sensitive Dependency Graph contains 2 SCCs.
The rules AND(tt, isNatIListKind(z0)) → U(isNatIListKind(z0)) and U(and(x0, x1)) → U(x0) form no chain, because ECapµ(U(isNatIListKind(z0))) = U(isNatIListKind(z0)) does not unify with U(and(x0, x1)).
↳ CSR
↳ CSDependencyPairsProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDPReductionPairProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDP
↳ QCSDPInstantiationProcessor
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDPForwardInstantiationProcessor
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
Q-restricted context-sensitive dependency pair problem:
The symbols in {U12, U22, U32, U43, U53, s, length} are replacing on all positions.
For all symbols f in {cons, U11, U21, U31, U41, U42, U51, U52, U61, and, AND} we have µ(f) = {1}.
The symbols in {isNatList, isNat, isNatIList, isNatIListKind, isNatKind, ISNATILISTKIND, U} are not replacing on any position.
The TRS P consists of the following rules:
ISNATILISTKIND(cons(V1, V2)) → AND(isNatKind(V1), isNatIListKind(V2))
AND(tt, isNatIListKind(z1)) → U(isNatIListKind(z1))
U(isNatIListKind(V2)) → ISNATILISTKIND(V2)
The TRS R consists of the following rules:
zeros → cons(0, zeros)
U11(tt, V1) → U12(isNatList(V1))
U12(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V) → U32(isNatList(V))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(V1), V2)
U42(tt, V2) → U43(isNatIList(V2))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(V1), V2)
U52(tt, V2) → U53(isNatList(V2))
U53(tt) → tt
U61(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → and(isNatKind(V1), isNatIListKind(V2))
isNatKind(0) → tt
isNatKind(length(V1)) → isNatIListKind(V1)
isNatKind(s(V1)) → isNatKind(V1)
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
Using the Context-Sensitive Forward Instantiation Processor
the pair U(isNatIListKind(V2)) → ISNATILISTKIND(V2)
was transformed to the following new pairs:
U(isNatIListKind(cons(z0, z1))) → ISNATILISTKIND(cons(z0, z1))
↳ CSR
↳ CSDependencyPairsProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDPReductionPairProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDP
↳ QCSDPInstantiationProcessor
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDPForwardInstantiationProcessor
↳ QCSDP
↳ QCSDPForwardInstantiationProcessor
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
Q-restricted context-sensitive dependency pair problem:
The symbols in {U12, U22, U32, U43, U53, s, length} are replacing on all positions.
For all symbols f in {cons, U11, U21, U31, U41, U42, U51, U52, U61, and, AND} we have µ(f) = {1}.
The symbols in {isNatList, isNat, isNatIList, isNatIListKind, isNatKind, ISNATILISTKIND, U} are not replacing on any position.
The TRS P consists of the following rules:
ISNATILISTKIND(cons(V1, V2)) → AND(isNatKind(V1), isNatIListKind(V2))
AND(tt, isNatIListKind(z1)) → U(isNatIListKind(z1))
U(isNatIListKind(cons(z0, z1))) → ISNATILISTKIND(cons(z0, z1))
The TRS R consists of the following rules:
zeros → cons(0, zeros)
U11(tt, V1) → U12(isNatList(V1))
U12(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V) → U32(isNatList(V))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(V1), V2)
U42(tt, V2) → U43(isNatIList(V2))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(V1), V2)
U52(tt, V2) → U53(isNatList(V2))
U53(tt) → tt
U61(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → and(isNatKind(V1), isNatIListKind(V2))
isNatKind(0) → tt
isNatKind(length(V1)) → isNatIListKind(V1)
isNatKind(s(V1)) → isNatKind(V1)
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
Using the Context-Sensitive Forward Instantiation Processor
the pair AND(tt, isNatIListKind(z1)) → U(isNatIListKind(z1))
was transformed to the following new pairs:
AND(tt, isNatIListKind(cons(z0, z1))) → U(isNatIListKind(cons(z0, z1)))
↳ CSR
↳ CSDependencyPairsProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDPReductionPairProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDP
↳ QCSDPInstantiationProcessor
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDPForwardInstantiationProcessor
↳ QCSDP
↳ QCSDPForwardInstantiationProcessor
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
Q-restricted context-sensitive dependency pair problem:
The symbols in {U12, U22, U32, U43, U53, s, length} are replacing on all positions.
For all symbols f in {cons, U11, U21, U31, U41, U42, U51, U52, U61, and, AND} we have µ(f) = {1}.
The symbols in {isNatList, isNat, isNatIList, isNatIListKind, isNatKind, ISNATILISTKIND, U} are not replacing on any position.
The TRS P consists of the following rules:
ISNATILISTKIND(cons(V1, V2)) → AND(isNatKind(V1), isNatIListKind(V2))
U(isNatIListKind(cons(z0, z1))) → ISNATILISTKIND(cons(z0, z1))
AND(tt, isNatIListKind(cons(z0, z1))) → U(isNatIListKind(cons(z0, z1)))
The TRS R consists of the following rules:
zeros → cons(0, zeros)
U11(tt, V1) → U12(isNatList(V1))
U12(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V) → U32(isNatList(V))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(V1), V2)
U42(tt, V2) → U43(isNatIList(V2))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(V1), V2)
U52(tt, V2) → U53(isNatList(V2))
U53(tt) → tt
U61(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → and(isNatKind(V1), isNatIListKind(V2))
isNatKind(0) → tt
isNatKind(length(V1)) → isNatIListKind(V1)
isNatKind(s(V1)) → isNatKind(V1)
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
↳ CSR
↳ CSDependencyPairsProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDPReductionPairProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDP
↳ QCSDPInstantiationProcessor
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
Q-restricted context-sensitive dependency pair problem:
The symbols in {U12, U22, U32, U43, U53, s, length} are replacing on all positions.
For all symbols f in {cons, U11, U21, U31, U41, U42, U51, U52, U61, and} we have µ(f) = {1}.
The symbols in {isNatList, isNat, isNatIList, isNatIListKind, isNatKind, U} are not replacing on any position.
The TRS P consists of the following rules:
U(and(x_0, x_1)) → U(x_0)
The TRS R consists of the following rules:
zeros → cons(0, zeros)
U11(tt, V1) → U12(isNatList(V1))
U12(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V) → U32(isNatList(V))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(V1), V2)
U42(tt, V2) → U43(isNatIList(V2))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(V1), V2)
U52(tt, V2) → U53(isNatList(V2))
U53(tt) → tt
U61(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → and(isNatKind(V1), isNatIListKind(V2))
isNatKind(0) → tt
isNatKind(length(V1)) → isNatIListKind(V1)
isNatKind(s(V1)) → isNatKind(V1)
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
↳ CSR
↳ CSDependencyPairsProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
Q-restricted context-sensitive dependency pair problem:
The symbols in {U12, U22, U32, U43, U53, s, length} are replacing on all positions.
For all symbols f in {cons, U11, U21, U31, U41, U42, U51, U52, U61, and, U511, U521, U111, U211} we have µ(f) = {1}.
The symbols in {isNatList, isNat, isNatIList, isNatIListKind, isNatKind, ISNATLIST, ISNAT} are not replacing on any position.
The TRS P consists of the following rules:
ISNATLIST(cons(V1, V2)) → U511(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
U511(tt, V1, V2) → U521(isNat(V1), V2)
U521(tt, V2) → ISNATLIST(V2)
U511(tt, V1, V2) → ISNAT(V1)
ISNAT(length(V1)) → U111(isNatIListKind(V1), V1)
U111(tt, V1) → ISNATLIST(V1)
ISNAT(s(V1)) → U211(isNatKind(V1), V1)
U211(tt, V1) → ISNAT(V1)
The TRS R consists of the following rules:
zeros → cons(0, zeros)
U11(tt, V1) → U12(isNatList(V1))
U12(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V) → U32(isNatList(V))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(V1), V2)
U42(tt, V2) → U43(isNatIList(V2))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(V1), V2)
U52(tt, V2) → U53(isNatList(V2))
U53(tt) → tt
U61(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → and(isNatKind(V1), isNatIListKind(V2))
isNatKind(0) → tt
isNatKind(length(V1)) → isNatIListKind(V1)
isNatKind(s(V1)) → isNatKind(V1)
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
↳ CSR
↳ CSDependencyPairsProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
Q-restricted context-sensitive dependency pair problem:
The symbols in {U12, U22, U32, U43, U53, s, length, LENGTH} are replacing on all positions.
For all symbols f in {cons, U11, U21, U31, U41, U42, U51, U52, U61, and, U611} we have µ(f) = {1}.
The symbols in {isNatList, isNat, isNatIList, isNatIListKind, isNatKind} are not replacing on any position.
The TRS P consists of the following rules:
LENGTH(cons(N, L)) → U611(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
U611(tt, L) → LENGTH(L)
The TRS R consists of the following rules:
zeros → cons(0, zeros)
U11(tt, V1) → U12(isNatList(V1))
U12(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V) → U32(isNatList(V))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(V1), V2)
U42(tt, V2) → U43(isNatIList(V2))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(V1), V2)
U52(tt, V2) → U53(isNatList(V2))
U53(tt) → tt
U61(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → and(isNatKind(V1), isNatIListKind(V2))
isNatKind(0) → tt
isNatKind(length(V1)) → isNatIListKind(V1)
isNatKind(s(V1)) → isNatKind(V1)
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
↳ CSR
↳ CSDependencyPairsProof
↳ QCSDP
↳ QCSDependencyGraphProof
↳ AND
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ QCSDP
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
Q-restricted context-sensitive dependency pair problem:
The symbols in {U12, U22, U32, U43, U53, s, length} are replacing on all positions.
For all symbols f in {cons, U11, U21, U31, U41, U42, U51, U52, U61, and, U421, U411} we have µ(f) = {1}.
The symbols in {isNatList, isNat, isNatIList, isNatIListKind, isNatKind, ISNATILIST} are not replacing on any position.
The TRS P consists of the following rules:
U411(tt, V1, V2) → U421(isNat(V1), V2)
U421(tt, V2) → ISNATILIST(V2)
ISNATILIST(cons(V1, V2)) → U411(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
The TRS R consists of the following rules:
zeros → cons(0, zeros)
U11(tt, V1) → U12(isNatList(V1))
U12(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V) → U32(isNatList(V))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(V1), V2)
U42(tt, V2) → U43(isNatIList(V2))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(V1), V2)
U52(tt, V2) → U53(isNatList(V2))
U53(tt) → tt
U61(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → U11(isNatIListKind(V1), V1)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatIList(V) → U31(isNatIListKind(V), V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
isNatIListKind(nil) → tt
isNatIListKind(zeros) → tt
isNatIListKind(cons(V1, V2)) → and(isNatKind(V1), isNatIListKind(V2))
isNatKind(0) → tt
isNatKind(length(V1)) → isNatIListKind(V1)
isNatKind(s(V1)) → isNatKind(V1)
isNatList(nil) → tt
isNatList(cons(V1, V2)) → U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
We applied the Zantema transformation [34] to transform the context-sensitive TRS to a usual TRS.
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ Incomplete Giesl Middeldorp-Transformation
Q restricted rewrite system:
The TRS R consists of the following rules:
zeros → cons(0, zerosInact)
U11(tt, V1) → U12(isNatList(a(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(a(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(a(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(a(V1)), a(V2))
U42(tt, V2) → U43(isNatIList(a(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(a(V1)), a(V2))
U52(tt, V2) → U53(isNatList(a(V2)))
U53(tt) → tt
U61(tt, L) → s(length(a(L)))
and(tt, X) → a(X)
isNat(0Inact) → tt
isNat(lengthInact(V1)) → U11(isNatIListKind(a(V1)), a(V1))
isNat(sInact(V1)) → U21(isNatKind(a(V1)), a(V1))
isNatIList(V) → U31(isNatIListKind(a(V)), a(V))
isNatIList(zerosInact) → tt
isNatIList(consInact(V1, V2)) → U41(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
isNatIListKind(nilInact) → tt
isNatIListKind(zerosInact) → tt
isNatIListKind(consInact(V1, V2)) → and(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
isNatKind(0Inact) → tt
isNatKind(lengthInact(V1)) → isNatIListKind(a(V1))
isNatKind(sInact(V1)) → isNatKind(a(V1))
isNatList(nilInact) → tt
isNatList(consInact(V1, V2)) → U51(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N))), a(L))
a(x) → x
isNatKind(x1) → isNatKindInact(x1)
a(isNatKindInact(x1)) → isNatKind(x1)
and(x1, x2) → andInact(x1, x2)
a(andInact(x1, x2)) → and(x1, x2)
0 → 0Inact
a(0Inact) → 0
isNatIListKind(x1) → isNatIListKindInact(x1)
a(isNatIListKindInact(x1)) → isNatIListKind(x1)
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
zeros → zerosInact
a(zerosInact) → zeros
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
nil → nilInact
a(nilInact) → nil
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:
U511(tt, V1, V2) → A(V1)
U211(tt, V1) → ISNAT(a(V1))
AND(tt, X) → A(X)
A(isNatIListKindInact(x1)) → ISNATILISTKIND(x1)
U311(tt, V) → A(V)
U611(tt, L) → A(L)
ISNATILIST(V) → ISNATILISTKIND(a(V))
ISNATILISTKIND(consInact(V1, V2)) → A(V1)
U611(tt, L) → LENGTH(a(L))
ISNATLIST(consInact(V1, V2)) → ISNATKIND(a(V1))
ISNATKIND(sInact(V1)) → A(V1)
ISNATKIND(lengthInact(V1)) → ISNATILISTKIND(a(V1))
U521(tt, V2) → A(V2)
U421(tt, V2) → U431(isNatIList(a(V2)))
U411(tt, V1, V2) → A(V1)
ISNAT(lengthInact(V1)) → A(V1)
LENGTH(cons(N, L)) → ISNAT(N)
U111(tt, V1) → ISNATLIST(a(V1))
A(sInact(x1)) → S(x1)
A(isNatKindInact(x1)) → ISNATKIND(x1)
U521(tt, V2) → ISNATLIST(a(V2))
LENGTH(nil) → 01
ISNATILIST(consInact(V1, V2)) → AND(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
A(consInact(x1, x2)) → CONS(x1, x2)
LENGTH(cons(N, L)) → A(L)
ISNATKIND(lengthInact(V1)) → A(V1)
U411(tt, V1, V2) → A(V2)
U311(tt, V) → U321(isNatList(a(V)))
U311(tt, V) → ISNATLIST(a(V))
LENGTH(cons(N, L)) → AND(isNatList(a(L)), isNatIListKindInact(a(L)))
ISNATILIST(consInact(V1, V2)) → A(V2)
A(andInact(x1, x2)) → AND(x1, x2)
ISNATILIST(consInact(V1, V2)) → U411(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
A(0Inact) → 01
A(zerosInact) → ZEROS
ISNATLIST(consInact(V1, V2)) → A(V1)
U211(tt, V1) → A(V1)
U421(tt, V2) → ISNATILIST(a(V2))
U511(tt, V1, V2) → U521(isNat(a(V1)), a(V2))
ISNATILIST(consInact(V1, V2)) → ISNATKIND(a(V1))
ISNATILISTKIND(consInact(V1, V2)) → AND(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
U111(tt, V1) → A(V1)
ISNAT(sInact(V1)) → ISNATKIND(a(V1))
ISNATILIST(V) → U311(isNatIListKind(a(V)), a(V))
ISNATILISTKIND(consInact(V1, V2)) → ISNATKIND(a(V1))
A(nilInact) → NIL
LENGTH(cons(N, L)) → U611(and(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N))), a(L))
ISNAT(sInact(V1)) → U211(isNatKind(a(V1)), a(V1))
ISNATKIND(sInact(V1)) → ISNATKIND(a(V1))
U511(tt, V1, V2) → A(V2)
LENGTH(cons(N, L)) → ISNATLIST(a(L))
ISNATLIST(consInact(V1, V2)) → A(V2)
ISNATLIST(consInact(V1, V2)) → AND(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
ISNAT(lengthInact(V1)) → ISNATILISTKIND(a(V1))
ISNAT(lengthInact(V1)) → U111(isNatIListKind(a(V1)), a(V1))
U111(tt, V1) → U121(isNatList(a(V1)))
ISNATILIST(consInact(V1, V2)) → A(V1)
U211(tt, V1) → U221(isNat(a(V1)))
U511(tt, V1, V2) → ISNAT(a(V1))
U521(tt, V2) → U531(isNatList(a(V2)))
U411(tt, V1, V2) → U421(isNat(a(V1)), a(V2))
ISNAT(sInact(V1)) → A(V1)
ISNATILIST(V) → A(V)
U411(tt, V1, V2) → ISNAT(a(V1))
U611(tt, L) → S(length(a(L)))
ZEROS → CONS(0, zerosInact)
LENGTH(cons(N, L)) → AND(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N)))
U421(tt, V2) → A(V2)
A(lengthInact(x1)) → LENGTH(x1)
ZEROS → 01
ISNATLIST(consInact(V1, V2)) → U511(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
ISNATILISTKIND(consInact(V1, V2)) → A(V2)
The TRS R consists of the following rules:
zeros → cons(0, zerosInact)
U11(tt, V1) → U12(isNatList(a(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(a(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(a(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(a(V1)), a(V2))
U42(tt, V2) → U43(isNatIList(a(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(a(V1)), a(V2))
U52(tt, V2) → U53(isNatList(a(V2)))
U53(tt) → tt
U61(tt, L) → s(length(a(L)))
and(tt, X) → a(X)
isNat(0Inact) → tt
isNat(lengthInact(V1)) → U11(isNatIListKind(a(V1)), a(V1))
isNat(sInact(V1)) → U21(isNatKind(a(V1)), a(V1))
isNatIList(V) → U31(isNatIListKind(a(V)), a(V))
isNatIList(zerosInact) → tt
isNatIList(consInact(V1, V2)) → U41(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
isNatIListKind(nilInact) → tt
isNatIListKind(zerosInact) → tt
isNatIListKind(consInact(V1, V2)) → and(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
isNatKind(0Inact) → tt
isNatKind(lengthInact(V1)) → isNatIListKind(a(V1))
isNatKind(sInact(V1)) → isNatKind(a(V1))
isNatList(nilInact) → tt
isNatList(consInact(V1, V2)) → U51(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N))), a(L))
a(x) → x
isNatKind(x1) → isNatKindInact(x1)
a(isNatKindInact(x1)) → isNatKind(x1)
and(x1, x2) → andInact(x1, x2)
a(andInact(x1, x2)) → and(x1, x2)
0 → 0Inact
a(0Inact) → 0
isNatIListKind(x1) → isNatIListKindInact(x1)
a(isNatIListKindInact(x1)) → isNatIListKind(x1)
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
zeros → zerosInact
a(zerosInact) → zeros
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
nil → nilInact
a(nilInact) → nil
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ Incomplete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
U511(tt, V1, V2) → A(V1)
U211(tt, V1) → ISNAT(a(V1))
AND(tt, X) → A(X)
A(isNatIListKindInact(x1)) → ISNATILISTKIND(x1)
U311(tt, V) → A(V)
U611(tt, L) → A(L)
ISNATILIST(V) → ISNATILISTKIND(a(V))
ISNATILISTKIND(consInact(V1, V2)) → A(V1)
U611(tt, L) → LENGTH(a(L))
ISNATLIST(consInact(V1, V2)) → ISNATKIND(a(V1))
ISNATKIND(sInact(V1)) → A(V1)
ISNATKIND(lengthInact(V1)) → ISNATILISTKIND(a(V1))
U521(tt, V2) → A(V2)
U421(tt, V2) → U431(isNatIList(a(V2)))
U411(tt, V1, V2) → A(V1)
ISNAT(lengthInact(V1)) → A(V1)
LENGTH(cons(N, L)) → ISNAT(N)
U111(tt, V1) → ISNATLIST(a(V1))
A(sInact(x1)) → S(x1)
A(isNatKindInact(x1)) → ISNATKIND(x1)
U521(tt, V2) → ISNATLIST(a(V2))
LENGTH(nil) → 01
ISNATILIST(consInact(V1, V2)) → AND(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
A(consInact(x1, x2)) → CONS(x1, x2)
LENGTH(cons(N, L)) → A(L)
ISNATKIND(lengthInact(V1)) → A(V1)
U411(tt, V1, V2) → A(V2)
U311(tt, V) → U321(isNatList(a(V)))
U311(tt, V) → ISNATLIST(a(V))
LENGTH(cons(N, L)) → AND(isNatList(a(L)), isNatIListKindInact(a(L)))
ISNATILIST(consInact(V1, V2)) → A(V2)
A(andInact(x1, x2)) → AND(x1, x2)
ISNATILIST(consInact(V1, V2)) → U411(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
A(0Inact) → 01
A(zerosInact) → ZEROS
ISNATLIST(consInact(V1, V2)) → A(V1)
U211(tt, V1) → A(V1)
U421(tt, V2) → ISNATILIST(a(V2))
U511(tt, V1, V2) → U521(isNat(a(V1)), a(V2))
ISNATILIST(consInact(V1, V2)) → ISNATKIND(a(V1))
ISNATILISTKIND(consInact(V1, V2)) → AND(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
U111(tt, V1) → A(V1)
ISNAT(sInact(V1)) → ISNATKIND(a(V1))
ISNATILIST(V) → U311(isNatIListKind(a(V)), a(V))
ISNATILISTKIND(consInact(V1, V2)) → ISNATKIND(a(V1))
A(nilInact) → NIL
LENGTH(cons(N, L)) → U611(and(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N))), a(L))
ISNAT(sInact(V1)) → U211(isNatKind(a(V1)), a(V1))
ISNATKIND(sInact(V1)) → ISNATKIND(a(V1))
U511(tt, V1, V2) → A(V2)
LENGTH(cons(N, L)) → ISNATLIST(a(L))
ISNATLIST(consInact(V1, V2)) → A(V2)
ISNATLIST(consInact(V1, V2)) → AND(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
ISNAT(lengthInact(V1)) → ISNATILISTKIND(a(V1))
ISNAT(lengthInact(V1)) → U111(isNatIListKind(a(V1)), a(V1))
U111(tt, V1) → U121(isNatList(a(V1)))
ISNATILIST(consInact(V1, V2)) → A(V1)
U211(tt, V1) → U221(isNat(a(V1)))
U511(tt, V1, V2) → ISNAT(a(V1))
U521(tt, V2) → U531(isNatList(a(V2)))
U411(tt, V1, V2) → U421(isNat(a(V1)), a(V2))
ISNAT(sInact(V1)) → A(V1)
ISNATILIST(V) → A(V)
U411(tt, V1, V2) → ISNAT(a(V1))
U611(tt, L) → S(length(a(L)))
ZEROS → CONS(0, zerosInact)
LENGTH(cons(N, L)) → AND(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N)))
U421(tt, V2) → A(V2)
A(lengthInact(x1)) → LENGTH(x1)
ZEROS → 01
ISNATLIST(consInact(V1, V2)) → U511(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
ISNATILISTKIND(consInact(V1, V2)) → A(V2)
The TRS R consists of the following rules:
zeros → cons(0, zerosInact)
U11(tt, V1) → U12(isNatList(a(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(a(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(a(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(a(V1)), a(V2))
U42(tt, V2) → U43(isNatIList(a(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(a(V1)), a(V2))
U52(tt, V2) → U53(isNatList(a(V2)))
U53(tt) → tt
U61(tt, L) → s(length(a(L)))
and(tt, X) → a(X)
isNat(0Inact) → tt
isNat(lengthInact(V1)) → U11(isNatIListKind(a(V1)), a(V1))
isNat(sInact(V1)) → U21(isNatKind(a(V1)), a(V1))
isNatIList(V) → U31(isNatIListKind(a(V)), a(V))
isNatIList(zerosInact) → tt
isNatIList(consInact(V1, V2)) → U41(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
isNatIListKind(nilInact) → tt
isNatIListKind(zerosInact) → tt
isNatIListKind(consInact(V1, V2)) → and(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
isNatKind(0Inact) → tt
isNatKind(lengthInact(V1)) → isNatIListKind(a(V1))
isNatKind(sInact(V1)) → isNatKind(a(V1))
isNatList(nilInact) → tt
isNatList(consInact(V1, V2)) → U51(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N))), a(L))
a(x) → x
isNatKind(x1) → isNatKindInact(x1)
a(isNatKindInact(x1)) → isNatKind(x1)
and(x1, x2) → andInact(x1, x2)
a(andInact(x1, x2)) → and(x1, x2)
0 → 0Inact
a(0Inact) → 0
isNatIListKind(x1) → isNatIListKindInact(x1)
a(isNatIListKindInact(x1)) → isNatIListKind(x1)
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
zeros → zerosInact
a(zerosInact) → zeros
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
nil → nilInact
a(nilInact) → nil
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.
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
U511(tt, V1, V2) → A(V1)
U211(tt, V1) → ISNAT(a(V1))
AND(tt, X) → A(X)
LENGTH(cons(N, L)) → U611(and(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N))), a(L))
A(isNatIListKindInact(x1)) → ISNATILISTKIND(x1)
U611(tt, L) → A(L)
ISNAT(sInact(V1)) → U211(isNatKind(a(V1)), a(V1))
ISNATILISTKIND(consInact(V1, V2)) → A(V1)
U611(tt, L) → LENGTH(a(L))
ISNATKIND(sInact(V1)) → ISNATKIND(a(V1))
U511(tt, V1, V2) → A(V2)
ISNATLIST(consInact(V1, V2)) → ISNATKIND(a(V1))
LENGTH(cons(N, L)) → ISNATLIST(a(L))
ISNATKIND(sInact(V1)) → A(V1)
ISNATLIST(consInact(V1, V2)) → A(V2)
ISNATLIST(consInact(V1, V2)) → AND(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
ISNAT(lengthInact(V1)) → ISNATILISTKIND(a(V1))
ISNAT(lengthInact(V1)) → U111(isNatIListKind(a(V1)), a(V1))
ISNATKIND(lengthInact(V1)) → ISNATILISTKIND(a(V1))
U521(tt, V2) → A(V2)
ISNAT(lengthInact(V1)) → A(V1)
LENGTH(cons(N, L)) → ISNAT(N)
U511(tt, V1, V2) → ISNAT(a(V1))
U111(tt, V1) → ISNATLIST(a(V1))
A(isNatKindInact(x1)) → ISNATKIND(x1)
U521(tt, V2) → ISNATLIST(a(V2))
LENGTH(cons(N, L)) → A(L)
ISNATKIND(lengthInact(V1)) → A(V1)
ISNAT(sInact(V1)) → A(V1)
LENGTH(cons(N, L)) → AND(isNatList(a(L)), isNatIListKindInact(a(L)))
A(andInact(x1, x2)) → AND(x1, x2)
U211(tt, V1) → A(V1)
ISNATLIST(consInact(V1, V2)) → A(V1)
U511(tt, V1, V2) → U521(isNat(a(V1)), a(V2))
ISNATILISTKIND(consInact(V1, V2)) → AND(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
ISNAT(sInact(V1)) → ISNATKIND(a(V1))
U111(tt, V1) → A(V1)
LENGTH(cons(N, L)) → AND(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N)))
A(lengthInact(x1)) → LENGTH(x1)
ISNATILISTKIND(consInact(V1, V2)) → ISNATKIND(a(V1))
ISNATILISTKIND(consInact(V1, V2)) → A(V2)
ISNATLIST(consInact(V1, V2)) → U511(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
The TRS R consists of the following rules:
zeros → cons(0, zerosInact)
U11(tt, V1) → U12(isNatList(a(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(a(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(a(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(a(V1)), a(V2))
U42(tt, V2) → U43(isNatIList(a(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(a(V1)), a(V2))
U52(tt, V2) → U53(isNatList(a(V2)))
U53(tt) → tt
U61(tt, L) → s(length(a(L)))
and(tt, X) → a(X)
isNat(0Inact) → tt
isNat(lengthInact(V1)) → U11(isNatIListKind(a(V1)), a(V1))
isNat(sInact(V1)) → U21(isNatKind(a(V1)), a(V1))
isNatIList(V) → U31(isNatIListKind(a(V)), a(V))
isNatIList(zerosInact) → tt
isNatIList(consInact(V1, V2)) → U41(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
isNatIListKind(nilInact) → tt
isNatIListKind(zerosInact) → tt
isNatIListKind(consInact(V1, V2)) → and(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
isNatKind(0Inact) → tt
isNatKind(lengthInact(V1)) → isNatIListKind(a(V1))
isNatKind(sInact(V1)) → isNatKind(a(V1))
isNatList(nilInact) → tt
isNatList(consInact(V1, V2)) → U51(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N))), a(L))
a(x) → x
isNatKind(x1) → isNatKindInact(x1)
a(isNatKindInact(x1)) → isNatKind(x1)
and(x1, x2) → andInact(x1, x2)
a(andInact(x1, x2)) → and(x1, x2)
0 → 0Inact
a(0Inact) → 0
isNatIListKind(x1) → isNatIListKindInact(x1)
a(isNatIListKindInact(x1)) → isNatIListKind(x1)
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
zeros → zerosInact
a(zerosInact) → zeros
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
nil → nilInact
a(nilInact) → nil
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ISNAT(lengthInact(V1)) → ISNATILISTKIND(a(V1))
ISNAT(lengthInact(V1)) → U111(isNatIListKind(a(V1)), a(V1))
ISNATKIND(lengthInact(V1)) → ISNATILISTKIND(a(V1))
ISNAT(lengthInact(V1)) → A(V1)
ISNATKIND(lengthInact(V1)) → A(V1)
A(lengthInact(x1)) → LENGTH(x1)
The remaining pairs can at least be oriented weakly.
U511(tt, V1, V2) → A(V1)
U211(tt, V1) → ISNAT(a(V1))
AND(tt, X) → A(X)
LENGTH(cons(N, L)) → U611(and(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N))), a(L))
A(isNatIListKindInact(x1)) → ISNATILISTKIND(x1)
U611(tt, L) → A(L)
ISNAT(sInact(V1)) → U211(isNatKind(a(V1)), a(V1))
ISNATILISTKIND(consInact(V1, V2)) → A(V1)
U611(tt, L) → LENGTH(a(L))
ISNATKIND(sInact(V1)) → ISNATKIND(a(V1))
U511(tt, V1, V2) → A(V2)
ISNATLIST(consInact(V1, V2)) → ISNATKIND(a(V1))
LENGTH(cons(N, L)) → ISNATLIST(a(L))
ISNATKIND(sInact(V1)) → A(V1)
ISNATLIST(consInact(V1, V2)) → A(V2)
ISNATLIST(consInact(V1, V2)) → AND(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
U521(tt, V2) → A(V2)
LENGTH(cons(N, L)) → ISNAT(N)
U511(tt, V1, V2) → ISNAT(a(V1))
U111(tt, V1) → ISNATLIST(a(V1))
A(isNatKindInact(x1)) → ISNATKIND(x1)
U521(tt, V2) → ISNATLIST(a(V2))
LENGTH(cons(N, L)) → A(L)
ISNAT(sInact(V1)) → A(V1)
LENGTH(cons(N, L)) → AND(isNatList(a(L)), isNatIListKindInact(a(L)))
A(andInact(x1, x2)) → AND(x1, x2)
U211(tt, V1) → A(V1)
ISNATLIST(consInact(V1, V2)) → A(V1)
U511(tt, V1, V2) → U521(isNat(a(V1)), a(V2))
ISNATILISTKIND(consInact(V1, V2)) → AND(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
ISNAT(sInact(V1)) → ISNATKIND(a(V1))
U111(tt, V1) → A(V1)
LENGTH(cons(N, L)) → AND(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N)))
ISNATILISTKIND(consInact(V1, V2)) → ISNATKIND(a(V1))
ISNATILISTKIND(consInact(V1, V2)) → A(V2)
ISNATLIST(consInact(V1, V2)) → U511(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(0Inact) = 0
POL(A(x1)) = x1
POL(AND(x1, x2)) = x2
POL(ISNAT(x1)) = x1
POL(ISNATILISTKIND(x1)) = x1
POL(ISNATKIND(x1)) = x1
POL(ISNATLIST(x1)) = x1
POL(LENGTH(x1)) = x1
POL(U11(x1, x2)) = 0
POL(U111(x1, x2)) = x2
POL(U12(x1)) = 0
POL(U21(x1, x2)) = 0
POL(U211(x1, x2)) = x2
POL(U22(x1)) = 0
POL(U51(x1, x2, x3)) = 0
POL(U511(x1, x2, x3)) = x2 + x3
POL(U52(x1, x2)) = 0
POL(U521(x1, x2)) = x2
POL(U53(x1)) = 0
POL(U61(x1, x2)) = 1 + x1 + x2
POL(U611(x1, x2)) = x2
POL(a(x1)) = x1
POL(and(x1, x2)) = x2
POL(andInact(x1, x2)) = x2
POL(cons(x1, x2)) = x1 + x2
POL(consInact(x1, x2)) = x1 + x2
POL(isNat(x1)) = 0
POL(isNatIListKind(x1)) = x1
POL(isNatIListKindInact(x1)) = x1
POL(isNatKind(x1)) = x1
POL(isNatKindInact(x1)) = x1
POL(isNatList(x1)) = 0
POL(length(x1)) = 1 + x1
POL(lengthInact(x1)) = 1 + x1
POL(nil) = 0
POL(nilInact) = 0
POL(s(x1)) = x1
POL(sInact(x1)) = x1
POL(tt) = 0
POL(zeros) = 0
POL(zerosInact) = 0
The following usable rules [17] were oriented:
U22(tt) → tt
U21(tt, V1) → U22(isNat(a(V1)))
U12(tt) → tt
U11(tt, V1) → U12(isNatList(a(V1)))
zeros → cons(0, zerosInact)
U61(tt, L) → s(length(a(L)))
U53(tt) → tt
U52(tt, V2) → U53(isNatList(a(V2)))
U51(tt, V1, V2) → U52(isNat(a(V1)), a(V2))
a(nilInact) → nil
nil → nilInact
a(lengthInact(x1)) → length(x1)
0 → 0Inact
isNatIListKind(x1) → isNatIListKindInact(x1)
a(0Inact) → 0
isNatKind(x1) → isNatKindInact(x1)
a(x) → x
and(x1, x2) → andInact(x1, x2)
a(isNatKindInact(x1)) → isNatKind(x1)
a(isNatIListKindInact(x1)) → isNatIListKind(x1)
isNatKind(lengthInact(V1)) → isNatIListKind(a(V1))
isNatIListKind(consInact(V1, V2)) → and(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
and(tt, X) → a(X)
a(andInact(x1, x2)) → and(x1, x2)
isNatKind(sInact(V1)) → isNatKind(a(V1))
s(x1) → sInact(x1)
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(sInact(x1)) → s(x1)
cons(x1, x2) → consInact(x1, x2)
zeros → zerosInact
a(consInact(x1, x2)) → cons(x1, x2)
isNatIListKind(nilInact) → tt
isNatIListKind(zerosInact) → tt
isNat(lengthInact(V1)) → U11(isNatIListKind(a(V1)), a(V1))
isNat(sInact(V1)) → U21(isNatKind(a(V1)), a(V1))
isNatList(consInact(V1, V2)) → U51(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N))), a(L))
isNatKind(0Inact) → tt
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
U511(tt, V1, V2) → A(V1)
U211(tt, V1) → ISNAT(a(V1))
AND(tt, X) → A(X)
LENGTH(cons(N, L)) → U611(and(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N))), a(L))
A(isNatIListKindInact(x1)) → ISNATILISTKIND(x1)
U611(tt, L) → A(L)
ISNAT(sInact(V1)) → U211(isNatKind(a(V1)), a(V1))
ISNATILISTKIND(consInact(V1, V2)) → A(V1)
U611(tt, L) → LENGTH(a(L))
ISNATKIND(sInact(V1)) → ISNATKIND(a(V1))
ISNATLIST(consInact(V1, V2)) → ISNATKIND(a(V1))
U511(tt, V1, V2) → A(V2)
LENGTH(cons(N, L)) → ISNATLIST(a(L))
ISNATKIND(sInact(V1)) → A(V1)
ISNATLIST(consInact(V1, V2)) → A(V2)
ISNATLIST(consInact(V1, V2)) → AND(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
U521(tt, V2) → A(V2)
LENGTH(cons(N, L)) → ISNAT(N)
U511(tt, V1, V2) → ISNAT(a(V1))
U111(tt, V1) → ISNATLIST(a(V1))
A(isNatKindInact(x1)) → ISNATKIND(x1)
U521(tt, V2) → ISNATLIST(a(V2))
LENGTH(cons(N, L)) → A(L)
ISNAT(sInact(V1)) → A(V1)
LENGTH(cons(N, L)) → AND(isNatList(a(L)), isNatIListKindInact(a(L)))
A(andInact(x1, x2)) → AND(x1, x2)
ISNATLIST(consInact(V1, V2)) → A(V1)
U211(tt, V1) → A(V1)
U511(tt, V1, V2) → U521(isNat(a(V1)), a(V2))
ISNATILISTKIND(consInact(V1, V2)) → AND(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
LENGTH(cons(N, L)) → AND(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N)))
U111(tt, V1) → A(V1)
ISNAT(sInact(V1)) → ISNATKIND(a(V1))
ISNATILISTKIND(consInact(V1, V2)) → ISNATKIND(a(V1))
ISNATLIST(consInact(V1, V2)) → U511(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
ISNATILISTKIND(consInact(V1, V2)) → A(V2)
The TRS R consists of the following rules:
zeros → cons(0, zerosInact)
U11(tt, V1) → U12(isNatList(a(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(a(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(a(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(a(V1)), a(V2))
U42(tt, V2) → U43(isNatIList(a(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(a(V1)), a(V2))
U52(tt, V2) → U53(isNatList(a(V2)))
U53(tt) → tt
U61(tt, L) → s(length(a(L)))
and(tt, X) → a(X)
isNat(0Inact) → tt
isNat(lengthInact(V1)) → U11(isNatIListKind(a(V1)), a(V1))
isNat(sInact(V1)) → U21(isNatKind(a(V1)), a(V1))
isNatIList(V) → U31(isNatIListKind(a(V)), a(V))
isNatIList(zerosInact) → tt
isNatIList(consInact(V1, V2)) → U41(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
isNatIListKind(nilInact) → tt
isNatIListKind(zerosInact) → tt
isNatIListKind(consInact(V1, V2)) → and(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
isNatKind(0Inact) → tt
isNatKind(lengthInact(V1)) → isNatIListKind(a(V1))
isNatKind(sInact(V1)) → isNatKind(a(V1))
isNatList(nilInact) → tt
isNatList(consInact(V1, V2)) → U51(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N))), a(L))
a(x) → x
isNatKind(x1) → isNatKindInact(x1)
a(isNatKindInact(x1)) → isNatKind(x1)
and(x1, x2) → andInact(x1, x2)
a(andInact(x1, x2)) → and(x1, x2)
0 → 0Inact
a(0Inact) → 0
isNatIListKind(x1) → isNatIListKindInact(x1)
a(isNatIListKindInact(x1)) → isNatIListKind(x1)
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
zeros → zerosInact
a(zerosInact) → zeros
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
nil → nilInact
a(nilInact) → nil
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 19 less nodes.
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATKIND(sInact(V1)) → A(V1)
AND(tt, X) → A(X)
ISNATILISTKIND(consInact(V1, V2)) → AND(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
A(isNatIListKindInact(x1)) → ISNATILISTKIND(x1)
ISNATILISTKIND(consInact(V1, V2)) → A(V1)
A(andInact(x1, x2)) → AND(x1, x2)
ISNATILISTKIND(consInact(V1, V2)) → ISNATKIND(a(V1))
A(isNatKindInact(x1)) → ISNATKIND(x1)
ISNATILISTKIND(consInact(V1, V2)) → A(V2)
ISNATKIND(sInact(V1)) → ISNATKIND(a(V1))
The TRS R consists of the following rules:
zeros → cons(0, zerosInact)
U11(tt, V1) → U12(isNatList(a(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(a(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(a(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(a(V1)), a(V2))
U42(tt, V2) → U43(isNatIList(a(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(a(V1)), a(V2))
U52(tt, V2) → U53(isNatList(a(V2)))
U53(tt) → tt
U61(tt, L) → s(length(a(L)))
and(tt, X) → a(X)
isNat(0Inact) → tt
isNat(lengthInact(V1)) → U11(isNatIListKind(a(V1)), a(V1))
isNat(sInact(V1)) → U21(isNatKind(a(V1)), a(V1))
isNatIList(V) → U31(isNatIListKind(a(V)), a(V))
isNatIList(zerosInact) → tt
isNatIList(consInact(V1, V2)) → U41(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
isNatIListKind(nilInact) → tt
isNatIListKind(zerosInact) → tt
isNatIListKind(consInact(V1, V2)) → and(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
isNatKind(0Inact) → tt
isNatKind(lengthInact(V1)) → isNatIListKind(a(V1))
isNatKind(sInact(V1)) → isNatKind(a(V1))
isNatList(nilInact) → tt
isNatList(consInact(V1, V2)) → U51(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N))), a(L))
a(x) → x
isNatKind(x1) → isNatKindInact(x1)
a(isNatKindInact(x1)) → isNatKind(x1)
and(x1, x2) → andInact(x1, x2)
a(andInact(x1, x2)) → and(x1, x2)
0 → 0Inact
a(0Inact) → 0
isNatIListKind(x1) → isNatIListKindInact(x1)
a(isNatIListKindInact(x1)) → isNatIListKind(x1)
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
zeros → zerosInact
a(zerosInact) → zeros
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
nil → nilInact
a(nilInact) → nil
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.
ISNATILISTKIND(consInact(V1, V2)) → A(V1)
ISNATILISTKIND(consInact(V1, V2)) → ISNATKIND(a(V1))
ISNATILISTKIND(consInact(V1, V2)) → A(V2)
The remaining pairs can at least be oriented weakly.
ISNATKIND(sInact(V1)) → A(V1)
AND(tt, X) → A(X)
ISNATILISTKIND(consInact(V1, V2)) → AND(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
A(isNatIListKindInact(x1)) → ISNATILISTKIND(x1)
A(andInact(x1, x2)) → AND(x1, x2)
A(isNatKindInact(x1)) → ISNATKIND(x1)
ISNATKIND(sInact(V1)) → ISNATKIND(a(V1))
Used ordering: Polynomial interpretation with max and min functions [25]:
POL(0) = 0
POL(0Inact) = 0
POL(A(x1)) = x1
POL(AND(x1, x2)) = x2
POL(ISNATILISTKIND(x1)) = 1 + x1
POL(ISNATKIND(x1)) = x1
POL(U11(x1, x2)) = 0
POL(U12(x1)) = 0
POL(U21(x1, x2)) = 0
POL(U22(x1)) = 0
POL(U51(x1, x2, x3)) = 0
POL(U52(x1, x2)) = 0
POL(U53(x1)) = 0
POL(U61(x1, x2)) = 1 + x2
POL(a(x1)) = x1
POL(and(x1, x2)) = x2
POL(andInact(x1, x2)) = x2
POL(cons(x1, x2)) = x1 + x2
POL(consInact(x1, x2)) = x1 + x2
POL(isNat(x1)) = 0
POL(isNatIListKind(x1)) = 1 + x1
POL(isNatIListKindInact(x1)) = 1 + x1
POL(isNatKind(x1)) = x1
POL(isNatKindInact(x1)) = x1
POL(isNatList(x1)) = 1 + x1
POL(length(x1)) = 1 + x1
POL(lengthInact(x1)) = 1 + x1
POL(nil) = 0
POL(nilInact) = 0
POL(s(x1)) = x1
POL(sInact(x1)) = x1
POL(tt) = 0
POL(zeros) = 0
POL(zerosInact) = 0
The following usable rules [17] were oriented:
U22(tt) → tt
U21(tt, V1) → U22(isNat(a(V1)))
U12(tt) → tt
U11(tt, V1) → U12(isNatList(a(V1)))
zeros → cons(0, zerosInact)
U61(tt, L) → s(length(a(L)))
U53(tt) → tt
U52(tt, V2) → U53(isNatList(a(V2)))
U51(tt, V1, V2) → U52(isNat(a(V1)), a(V2))
a(nilInact) → nil
nil → nilInact
a(lengthInact(x1)) → length(x1)
0 → 0Inact
isNatIListKind(x1) → isNatIListKindInact(x1)
a(0Inact) → 0
isNatKind(x1) → isNatKindInact(x1)
a(x) → x
and(x1, x2) → andInact(x1, x2)
a(isNatKindInact(x1)) → isNatKind(x1)
a(isNatIListKindInact(x1)) → isNatIListKind(x1)
isNatKind(lengthInact(V1)) → isNatIListKind(a(V1))
isNatIListKind(consInact(V1, V2)) → and(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
and(tt, X) → a(X)
a(andInact(x1, x2)) → and(x1, x2)
isNatKind(sInact(V1)) → isNatKind(a(V1))
s(x1) → sInact(x1)
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(sInact(x1)) → s(x1)
cons(x1, x2) → consInact(x1, x2)
zeros → zerosInact
a(consInact(x1, x2)) → cons(x1, x2)
isNatIListKind(nilInact) → tt
isNatIListKind(zerosInact) → tt
isNat(lengthInact(V1)) → U11(isNatIListKind(a(V1)), a(V1))
isNat(sInact(V1)) → U21(isNatKind(a(V1)), a(V1))
isNatList(consInact(V1, V2)) → U51(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N))), a(L))
isNatKind(0Inact) → tt
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATKIND(sInact(V1)) → A(V1)
AND(tt, X) → A(X)
ISNATILISTKIND(consInact(V1, V2)) → AND(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
A(isNatIListKindInact(x1)) → ISNATILISTKIND(x1)
A(andInact(x1, x2)) → AND(x1, x2)
A(isNatKindInact(x1)) → ISNATKIND(x1)
ISNATKIND(sInact(V1)) → ISNATKIND(a(V1))
The TRS R consists of the following rules:
zeros → cons(0, zerosInact)
U11(tt, V1) → U12(isNatList(a(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(a(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(a(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(a(V1)), a(V2))
U42(tt, V2) → U43(isNatIList(a(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(a(V1)), a(V2))
U52(tt, V2) → U53(isNatList(a(V2)))
U53(tt) → tt
U61(tt, L) → s(length(a(L)))
and(tt, X) → a(X)
isNat(0Inact) → tt
isNat(lengthInact(V1)) → U11(isNatIListKind(a(V1)), a(V1))
isNat(sInact(V1)) → U21(isNatKind(a(V1)), a(V1))
isNatIList(V) → U31(isNatIListKind(a(V)), a(V))
isNatIList(zerosInact) → tt
isNatIList(consInact(V1, V2)) → U41(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
isNatIListKind(nilInact) → tt
isNatIListKind(zerosInact) → tt
isNatIListKind(consInact(V1, V2)) → and(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
isNatKind(0Inact) → tt
isNatKind(lengthInact(V1)) → isNatIListKind(a(V1))
isNatKind(sInact(V1)) → isNatKind(a(V1))
isNatList(nilInact) → tt
isNatList(consInact(V1, V2)) → U51(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N))), a(L))
a(x) → x
isNatKind(x1) → isNatKindInact(x1)
a(isNatKindInact(x1)) → isNatKind(x1)
and(x1, x2) → andInact(x1, x2)
a(andInact(x1, x2)) → and(x1, x2)
0 → 0Inact
a(0Inact) → 0
isNatIListKind(x1) → isNatIListKindInact(x1)
a(isNatIListKindInact(x1)) → isNatIListKind(x1)
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
zeros → zerosInact
a(zerosInact) → zeros
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
nil → nilInact
a(nilInact) → nil
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
A(isNatKindInact(x1)) → ISNATKIND(x1)
The remaining pairs can at least be oriented weakly.
ISNATKIND(sInact(V1)) → A(V1)
AND(tt, X) → A(X)
ISNATILISTKIND(consInact(V1, V2)) → AND(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
A(isNatIListKindInact(x1)) → ISNATILISTKIND(x1)
A(andInact(x1, x2)) → AND(x1, x2)
ISNATKIND(sInact(V1)) → ISNATKIND(a(V1))
Used ordering: Polynomial interpretation with max and min functions [25]:
POL(0) = 0
POL(0Inact) = 0
POL(A(x1)) = x1
POL(AND(x1, x2)) = x2
POL(ISNATILISTKIND(x1)) = 0
POL(ISNATKIND(x1)) = x1
POL(U11(x1, x2)) = 0
POL(U12(x1)) = 0
POL(U21(x1, x2)) = 0
POL(U22(x1)) = 0
POL(U51(x1, x2, x3)) = 1
POL(U52(x1, x2)) = 0
POL(U53(x1)) = 0
POL(U61(x1, x2)) = 0
POL(a(x1)) = x1
POL(and(x1, x2)) = x2
POL(andInact(x1, x2)) = x2
POL(cons(x1, x2)) = 0
POL(consInact(x1, x2)) = 0
POL(isNat(x1)) = 0
POL(isNatIListKind(x1)) = 0
POL(isNatIListKindInact(x1)) = 0
POL(isNatKind(x1)) = 1 + x1
POL(isNatKindInact(x1)) = 1 + x1
POL(isNatList(x1)) = 1
POL(length(x1)) = 0
POL(lengthInact(x1)) = 0
POL(nil) = 0
POL(nilInact) = 0
POL(s(x1)) = x1
POL(sInact(x1)) = x1
POL(tt) = 0
POL(zeros) = 0
POL(zerosInact) = 0
The following usable rules [17] were oriented:
U22(tt) → tt
U21(tt, V1) → U22(isNat(a(V1)))
U12(tt) → tt
U11(tt, V1) → U12(isNatList(a(V1)))
zeros → cons(0, zerosInact)
U61(tt, L) → s(length(a(L)))
U53(tt) → tt
U52(tt, V2) → U53(isNatList(a(V2)))
U51(tt, V1, V2) → U52(isNat(a(V1)), a(V2))
a(nilInact) → nil
nil → nilInact
a(lengthInact(x1)) → length(x1)
0 → 0Inact
isNatIListKind(x1) → isNatIListKindInact(x1)
a(0Inact) → 0
isNatKind(x1) → isNatKindInact(x1)
a(x) → x
and(x1, x2) → andInact(x1, x2)
a(isNatKindInact(x1)) → isNatKind(x1)
a(isNatIListKindInact(x1)) → isNatIListKind(x1)
isNatKind(lengthInact(V1)) → isNatIListKind(a(V1))
isNatIListKind(consInact(V1, V2)) → and(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
and(tt, X) → a(X)
a(andInact(x1, x2)) → and(x1, x2)
isNatKind(sInact(V1)) → isNatKind(a(V1))
s(x1) → sInact(x1)
a(zerosInact) → zeros
length(x1) → lengthInact(x1)
a(sInact(x1)) → s(x1)
cons(x1, x2) → consInact(x1, x2)
zeros → zerosInact
a(consInact(x1, x2)) → cons(x1, x2)
isNatIListKind(nilInact) → tt
isNatIListKind(zerosInact) → tt
isNat(lengthInact(V1)) → U11(isNatIListKind(a(V1)), a(V1))
isNat(sInact(V1)) → U21(isNatKind(a(V1)), a(V1))
isNatList(consInact(V1, V2)) → U51(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N))), a(L))
isNatKind(0Inact) → tt
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATKIND(sInact(V1)) → A(V1)
AND(tt, X) → A(X)
ISNATILISTKIND(consInact(V1, V2)) → AND(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
A(isNatIListKindInact(x1)) → ISNATILISTKIND(x1)
A(andInact(x1, x2)) → AND(x1, x2)
ISNATKIND(sInact(V1)) → ISNATKIND(a(V1))
The TRS R consists of the following rules:
zeros → cons(0, zerosInact)
U11(tt, V1) → U12(isNatList(a(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(a(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(a(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(a(V1)), a(V2))
U42(tt, V2) → U43(isNatIList(a(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(a(V1)), a(V2))
U52(tt, V2) → U53(isNatList(a(V2)))
U53(tt) → tt
U61(tt, L) → s(length(a(L)))
and(tt, X) → a(X)
isNat(0Inact) → tt
isNat(lengthInact(V1)) → U11(isNatIListKind(a(V1)), a(V1))
isNat(sInact(V1)) → U21(isNatKind(a(V1)), a(V1))
isNatIList(V) → U31(isNatIListKind(a(V)), a(V))
isNatIList(zerosInact) → tt
isNatIList(consInact(V1, V2)) → U41(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
isNatIListKind(nilInact) → tt
isNatIListKind(zerosInact) → tt
isNatIListKind(consInact(V1, V2)) → and(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
isNatKind(0Inact) → tt
isNatKind(lengthInact(V1)) → isNatIListKind(a(V1))
isNatKind(sInact(V1)) → isNatKind(a(V1))
isNatList(nilInact) → tt
isNatList(consInact(V1, V2)) → U51(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N))), a(L))
a(x) → x
isNatKind(x1) → isNatKindInact(x1)
a(isNatKindInact(x1)) → isNatKind(x1)
and(x1, x2) → andInact(x1, x2)
a(andInact(x1, x2)) → and(x1, x2)
0 → 0Inact
a(0Inact) → 0
isNatIListKind(x1) → isNatIListKindInact(x1)
a(isNatIListKindInact(x1)) → isNatIListKind(x1)
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
zeros → zerosInact
a(zerosInact) → zeros
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
nil → nilInact
a(nilInact) → nil
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 1 less node.
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
AND(tt, X) → A(X)
ISNATILISTKIND(consInact(V1, V2)) → AND(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
A(isNatIListKindInact(x1)) → ISNATILISTKIND(x1)
A(andInact(x1, x2)) → AND(x1, x2)
The TRS R consists of the following rules:
zeros → cons(0, zerosInact)
U11(tt, V1) → U12(isNatList(a(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(a(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(a(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(a(V1)), a(V2))
U42(tt, V2) → U43(isNatIList(a(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(a(V1)), a(V2))
U52(tt, V2) → U53(isNatList(a(V2)))
U53(tt) → tt
U61(tt, L) → s(length(a(L)))
and(tt, X) → a(X)
isNat(0Inact) → tt
isNat(lengthInact(V1)) → U11(isNatIListKind(a(V1)), a(V1))
isNat(sInact(V1)) → U21(isNatKind(a(V1)), a(V1))
isNatIList(V) → U31(isNatIListKind(a(V)), a(V))
isNatIList(zerosInact) → tt
isNatIList(consInact(V1, V2)) → U41(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
isNatIListKind(nilInact) → tt
isNatIListKind(zerosInact) → tt
isNatIListKind(consInact(V1, V2)) → and(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
isNatKind(0Inact) → tt
isNatKind(lengthInact(V1)) → isNatIListKind(a(V1))
isNatKind(sInact(V1)) → isNatKind(a(V1))
isNatList(nilInact) → tt
isNatList(consInact(V1, V2)) → U51(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N))), a(L))
a(x) → x
isNatKind(x1) → isNatKindInact(x1)
a(isNatKindInact(x1)) → isNatKind(x1)
and(x1, x2) → andInact(x1, x2)
a(andInact(x1, x2)) → and(x1, x2)
0 → 0Inact
a(0Inact) → 0
isNatIListKind(x1) → isNatIListKindInact(x1)
a(isNatIListKindInact(x1)) → isNatIListKind(x1)
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
zeros → zerosInact
a(zerosInact) → zeros
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
nil → nilInact
a(nilInact) → nil
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
A(andInact(x1, x2)) → AND(x1, x2)
The remaining pairs can at least be oriented weakly.
AND(tt, X) → A(X)
ISNATILISTKIND(consInact(V1, V2)) → AND(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
A(isNatIListKindInact(x1)) → ISNATILISTKIND(x1)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(0Inact) = 1
POL(A(x1)) = 1 + x1
POL(AND(x1, x2)) = 1 + x2
POL(ISNATILISTKIND(x1)) = 1
POL(U11(x1, x2)) = 0
POL(U12(x1)) = 0
POL(U21(x1, x2)) = 0
POL(U22(x1)) = 0
POL(U51(x1, x2, x3)) = 0
POL(U52(x1, x2)) = 0
POL(U53(x1)) = 0
POL(U61(x1, x2)) = 0
POL(a(x1)) = 0
POL(and(x1, x2)) = 0
POL(andInact(x1, x2)) = 1 + x1 + x2
POL(cons(x1, x2)) = 0
POL(consInact(x1, x2)) = 0
POL(isNat(x1)) = 1 + x1
POL(isNatIListKind(x1)) = 0
POL(isNatIListKindInact(x1)) = 0
POL(isNatKind(x1)) = 0
POL(isNatKindInact(x1)) = 0
POL(isNatList(x1)) = 1 + x1
POL(length(x1)) = 0
POL(lengthInact(x1)) = 0
POL(nil) = 0
POL(nilInact) = 1
POL(s(x1)) = 0
POL(sInact(x1)) = 0
POL(tt) = 1
POL(zeros) = 0
POL(zerosInact) = 0
The following usable rules [17] were oriented:
none
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
AND(tt, X) → A(X)
ISNATILISTKIND(consInact(V1, V2)) → AND(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
A(isNatIListKindInact(x1)) → ISNATILISTKIND(x1)
The TRS R consists of the following rules:
zeros → cons(0, zerosInact)
U11(tt, V1) → U12(isNatList(a(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(a(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(a(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(a(V1)), a(V2))
U42(tt, V2) → U43(isNatIList(a(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(a(V1)), a(V2))
U52(tt, V2) → U53(isNatList(a(V2)))
U53(tt) → tt
U61(tt, L) → s(length(a(L)))
and(tt, X) → a(X)
isNat(0Inact) → tt
isNat(lengthInact(V1)) → U11(isNatIListKind(a(V1)), a(V1))
isNat(sInact(V1)) → U21(isNatKind(a(V1)), a(V1))
isNatIList(V) → U31(isNatIListKind(a(V)), a(V))
isNatIList(zerosInact) → tt
isNatIList(consInact(V1, V2)) → U41(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
isNatIListKind(nilInact) → tt
isNatIListKind(zerosInact) → tt
isNatIListKind(consInact(V1, V2)) → and(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
isNatKind(0Inact) → tt
isNatKind(lengthInact(V1)) → isNatIListKind(a(V1))
isNatKind(sInact(V1)) → isNatKind(a(V1))
isNatList(nilInact) → tt
isNatList(consInact(V1, V2)) → U51(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N))), a(L))
a(x) → x
isNatKind(x1) → isNatKindInact(x1)
a(isNatKindInact(x1)) → isNatKind(x1)
and(x1, x2) → andInact(x1, x2)
a(andInact(x1, x2)) → and(x1, x2)
0 → 0Inact
a(0Inact) → 0
isNatIListKind(x1) → isNatIListKindInact(x1)
a(isNatIListKindInact(x1)) → isNatIListKind(x1)
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
zeros → zerosInact
a(zerosInact) → zeros
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
nil → nilInact
a(nilInact) → nil
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
ISNATKIND(sInact(V1)) → ISNATKIND(a(V1))
The TRS R consists of the following rules:
zeros → cons(0, zerosInact)
U11(tt, V1) → U12(isNatList(a(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(a(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(a(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(a(V1)), a(V2))
U42(tt, V2) → U43(isNatIList(a(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(a(V1)), a(V2))
U52(tt, V2) → U53(isNatList(a(V2)))
U53(tt) → tt
U61(tt, L) → s(length(a(L)))
and(tt, X) → a(X)
isNat(0Inact) → tt
isNat(lengthInact(V1)) → U11(isNatIListKind(a(V1)), a(V1))
isNat(sInact(V1)) → U21(isNatKind(a(V1)), a(V1))
isNatIList(V) → U31(isNatIListKind(a(V)), a(V))
isNatIList(zerosInact) → tt
isNatIList(consInact(V1, V2)) → U41(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
isNatIListKind(nilInact) → tt
isNatIListKind(zerosInact) → tt
isNatIListKind(consInact(V1, V2)) → and(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
isNatKind(0Inact) → tt
isNatKind(lengthInact(V1)) → isNatIListKind(a(V1))
isNatKind(sInact(V1)) → isNatKind(a(V1))
isNatList(nilInact) → tt
isNatList(consInact(V1, V2)) → U51(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N))), a(L))
a(x) → x
isNatKind(x1) → isNatKindInact(x1)
a(isNatKindInact(x1)) → isNatKind(x1)
and(x1, x2) → andInact(x1, x2)
a(andInact(x1, x2)) → and(x1, x2)
0 → 0Inact
a(0Inact) → 0
isNatIListKind(x1) → isNatIListKindInact(x1)
a(isNatIListKindInact(x1)) → isNatIListKind(x1)
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
zeros → zerosInact
a(zerosInact) → zeros
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
nil → nilInact
a(nilInact) → nil
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
U211(tt, V1) → ISNAT(a(V1))
ISNAT(sInact(V1)) → U211(isNatKind(a(V1)), a(V1))
The TRS R consists of the following rules:
zeros → cons(0, zerosInact)
U11(tt, V1) → U12(isNatList(a(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(a(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(a(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(a(V1)), a(V2))
U42(tt, V2) → U43(isNatIList(a(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(a(V1)), a(V2))
U52(tt, V2) → U53(isNatList(a(V2)))
U53(tt) → tt
U61(tt, L) → s(length(a(L)))
and(tt, X) → a(X)
isNat(0Inact) → tt
isNat(lengthInact(V1)) → U11(isNatIListKind(a(V1)), a(V1))
isNat(sInact(V1)) → U21(isNatKind(a(V1)), a(V1))
isNatIList(V) → U31(isNatIListKind(a(V)), a(V))
isNatIList(zerosInact) → tt
isNatIList(consInact(V1, V2)) → U41(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
isNatIListKind(nilInact) → tt
isNatIListKind(zerosInact) → tt
isNatIListKind(consInact(V1, V2)) → and(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
isNatKind(0Inact) → tt
isNatKind(lengthInact(V1)) → isNatIListKind(a(V1))
isNatKind(sInact(V1)) → isNatKind(a(V1))
isNatList(nilInact) → tt
isNatList(consInact(V1, V2)) → U51(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N))), a(L))
a(x) → x
isNatKind(x1) → isNatKindInact(x1)
a(isNatKindInact(x1)) → isNatKind(x1)
and(x1, x2) → andInact(x1, x2)
a(andInact(x1, x2)) → and(x1, x2)
0 → 0Inact
a(0Inact) → 0
isNatIListKind(x1) → isNatIListKindInact(x1)
a(isNatIListKindInact(x1)) → isNatIListKind(x1)
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
zeros → zerosInact
a(zerosInact) → zeros
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
nil → nilInact
a(nilInact) → nil
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
U511(tt, V1, V2) → U521(isNat(a(V1)), a(V2))
U521(tt, V2) → ISNATLIST(a(V2))
ISNATLIST(consInact(V1, V2)) → U511(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
The TRS R consists of the following rules:
zeros → cons(0, zerosInact)
U11(tt, V1) → U12(isNatList(a(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(a(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(a(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(a(V1)), a(V2))
U42(tt, V2) → U43(isNatIList(a(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(a(V1)), a(V2))
U52(tt, V2) → U53(isNatList(a(V2)))
U53(tt) → tt
U61(tt, L) → s(length(a(L)))
and(tt, X) → a(X)
isNat(0Inact) → tt
isNat(lengthInact(V1)) → U11(isNatIListKind(a(V1)), a(V1))
isNat(sInact(V1)) → U21(isNatKind(a(V1)), a(V1))
isNatIList(V) → U31(isNatIListKind(a(V)), a(V))
isNatIList(zerosInact) → tt
isNatIList(consInact(V1, V2)) → U41(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
isNatIListKind(nilInact) → tt
isNatIListKind(zerosInact) → tt
isNatIListKind(consInact(V1, V2)) → and(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
isNatKind(0Inact) → tt
isNatKind(lengthInact(V1)) → isNatIListKind(a(V1))
isNatKind(sInact(V1)) → isNatKind(a(V1))
isNatList(nilInact) → tt
isNatList(consInact(V1, V2)) → U51(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N))), a(L))
a(x) → x
isNatKind(x1) → isNatKindInact(x1)
a(isNatKindInact(x1)) → isNatKind(x1)
and(x1, x2) → andInact(x1, x2)
a(andInact(x1, x2)) → and(x1, x2)
0 → 0Inact
a(0Inact) → 0
isNatIListKind(x1) → isNatIListKindInact(x1)
a(isNatIListKindInact(x1)) → isNatIListKind(x1)
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
zeros → zerosInact
a(zerosInact) → zeros
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
nil → nilInact
a(nilInact) → nil
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
LENGTH(cons(N, L)) → U611(and(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N))), a(L))
U611(tt, L) → LENGTH(a(L))
The TRS R consists of the following rules:
zeros → cons(0, zerosInact)
U11(tt, V1) → U12(isNatList(a(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(a(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(a(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(a(V1)), a(V2))
U42(tt, V2) → U43(isNatIList(a(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(a(V1)), a(V2))
U52(tt, V2) → U53(isNatList(a(V2)))
U53(tt) → tt
U61(tt, L) → s(length(a(L)))
and(tt, X) → a(X)
isNat(0Inact) → tt
isNat(lengthInact(V1)) → U11(isNatIListKind(a(V1)), a(V1))
isNat(sInact(V1)) → U21(isNatKind(a(V1)), a(V1))
isNatIList(V) → U31(isNatIListKind(a(V)), a(V))
isNatIList(zerosInact) → tt
isNatIList(consInact(V1, V2)) → U41(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
isNatIListKind(nilInact) → tt
isNatIListKind(zerosInact) → tt
isNatIListKind(consInact(V1, V2)) → and(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
isNatKind(0Inact) → tt
isNatKind(lengthInact(V1)) → isNatIListKind(a(V1))
isNatKind(sInact(V1)) → isNatKind(a(V1))
isNatList(nilInact) → tt
isNatList(consInact(V1, V2)) → U51(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N))), a(L))
a(x) → x
isNatKind(x1) → isNatKindInact(x1)
a(isNatKindInact(x1)) → isNatKind(x1)
and(x1, x2) → andInact(x1, x2)
a(andInact(x1, x2)) → and(x1, x2)
0 → 0Inact
a(0Inact) → 0
isNatIListKind(x1) → isNatIListKindInact(x1)
a(isNatIListKindInact(x1)) → isNatIListKind(x1)
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
zeros → zerosInact
a(zerosInact) → zeros
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
nil → nilInact
a(nilInact) → nil
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ Incomplete Giesl Middeldorp-Transformation
Q DP problem:
The TRS P consists of the following rules:
U421(tt, V2) → ISNATILIST(a(V2))
U411(tt, V1, V2) → U421(isNat(a(V1)), a(V2))
ISNATILIST(consInact(V1, V2)) → U411(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
The TRS R consists of the following rules:
zeros → cons(0, zerosInact)
U11(tt, V1) → U12(isNatList(a(V1)))
U12(tt) → tt
U21(tt, V1) → U22(isNat(a(V1)))
U22(tt) → tt
U31(tt, V) → U32(isNatList(a(V)))
U32(tt) → tt
U41(tt, V1, V2) → U42(isNat(a(V1)), a(V2))
U42(tt, V2) → U43(isNatIList(a(V2)))
U43(tt) → tt
U51(tt, V1, V2) → U52(isNat(a(V1)), a(V2))
U52(tt, V2) → U53(isNatList(a(V2)))
U53(tt) → tt
U61(tt, L) → s(length(a(L)))
and(tt, X) → a(X)
isNat(0Inact) → tt
isNat(lengthInact(V1)) → U11(isNatIListKind(a(V1)), a(V1))
isNat(sInact(V1)) → U21(isNatKind(a(V1)), a(V1))
isNatIList(V) → U31(isNatIListKind(a(V)), a(V))
isNatIList(zerosInact) → tt
isNatIList(consInact(V1, V2)) → U41(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
isNatIListKind(nilInact) → tt
isNatIListKind(zerosInact) → tt
isNatIListKind(consInact(V1, V2)) → and(isNatKind(a(V1)), isNatIListKindInact(a(V2)))
isNatKind(0Inact) → tt
isNatKind(lengthInact(V1)) → isNatIListKind(a(V1))
isNatKind(sInact(V1)) → isNatKind(a(V1))
isNatList(nilInact) → tt
isNatList(consInact(V1, V2)) → U51(and(isNatKind(a(V1)), isNatIListKindInact(a(V2))), a(V1), a(V2))
length(nil) → 0
length(cons(N, L)) → U61(and(and(isNatList(a(L)), isNatIListKindInact(a(L))), andInact(isNat(N), isNatKindInact(N))), a(L))
a(x) → x
isNatKind(x1) → isNatKindInact(x1)
a(isNatKindInact(x1)) → isNatKind(x1)
and(x1, x2) → andInact(x1, x2)
a(andInact(x1, x2)) → and(x1, x2)
0 → 0Inact
a(0Inact) → 0
isNatIListKind(x1) → isNatIListKindInact(x1)
a(isNatIListKindInact(x1)) → isNatIListKind(x1)
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)
zeros → zerosInact
a(zerosInact) → zeros
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
length(x1) → lengthInact(x1)
a(lengthInact(x1)) → length(x1)
nil → nilInact
a(nilInact) → nil
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We applied the Incomplete Giesl Middeldorp transformation [11] to transform the context-sensitive TRS to a usual TRS.
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
mark(zeros) → zerosActive
zerosActive → zeros
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
U11Active(x1, x2) → U11(x1, x2)
mark(U12(x1)) → U12Active(mark(x1))
U12Active(x1) → U12(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2)) → U31Active(mark(x1), x2)
U31Active(x1, x2) → U31(x1, x2)
mark(U32(x1)) → U32Active(mark(x1))
U32Active(x1) → U32(x1)
mark(U41(x1, x2, x3)) → U41Active(mark(x1), x2, x3)
U41Active(x1, x2, x3) → U41(x1, x2, x3)
mark(U42(x1, x2)) → U42Active(mark(x1), x2)
U42Active(x1, x2) → U42(x1, x2)
mark(U43(x1)) → U43Active(mark(x1))
U43Active(x1) → U43(x1)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U52(x1, x2)) → U52Active(mark(x1), x2)
U52Active(x1, x2) → U52(x1, x2)
mark(U53(x1)) → U53Active(mark(x1))
U53Active(x1) → U53(x1)
mark(U61(x1, x2)) → U61Active(mark(x1), x2)
U61Active(x1, x2) → U61(x1, x2)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatIListKind(x1)) → isNatIListKindActive(x1)
isNatIListKindActive(x1) → isNatIListKind(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(nil) → nil
zerosActive → cons(0, zeros)
U11Active(tt, V1) → U12Active(isNatListActive(V1))
U12Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V) → U32Active(isNatListActive(V))
U32Active(tt) → tt
U41Active(tt, V1, V2) → U42Active(isNatActive(V1), V2)
U42Active(tt, V2) → U43Active(isNatIListActive(V2))
U43Active(tt) → tt
U51Active(tt, V1, V2) → U52Active(isNatActive(V1), V2)
U52Active(tt, V2) → U53Active(isNatListActive(V2))
U53Active(tt) → tt
U61Active(tt, L) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(length(V1)) → U11Active(isNatIListKindActive(V1), V1)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatIListActive(V) → U31Active(isNatIListKindActive(V), V)
isNatIListActive(zeros) → tt
isNatIListActive(cons(V1, V2)) → U41Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
isNatIListKindActive(nil) → tt
isNatIListKindActive(zeros) → tt
isNatIListKindActive(cons(V1, V2)) → andActive(isNatKindActive(V1), isNatIListKind(V2))
isNatKindActive(0) → tt
isNatKindActive(length(V1)) → isNatIListKindActive(V1)
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatListActive(nil) → tt
isNatListActive(cons(V1, V2)) → U51Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
lengthActive(nil) → 0
lengthActive(cons(N, L)) → U61Active(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
MARK(U11(x1, x2)) → MARK(x1)
U11ACTIVE(tt, V1) → U12ACTIVE(isNatListActive(V1))
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
MARK(U42(x1, x2)) → U42ACTIVE(mark(x1), x2)
MARK(cons(x1, x2)) → MARK(x1)
U51ACTIVE(tt, V1, V2) → U52ACTIVE(isNatActive(V1), V2)
U51ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U12(x1)) → U12ACTIVE(mark(x1))
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
MARK(length(x1)) → MARK(x1)
ISNATILISTKINDACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
MARK(U32(x1)) → U32ACTIVE(mark(x1))
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
ISNATLISTACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(U53(x1)) → U53ACTIVE(mark(x1))
MARK(U53(x1)) → MARK(x1)
ISNATILISTACTIVE(V) → ISNATILISTKINDACTIVE(V)
ISNATILISTACTIVE(cons(V1, V2)) → U41ACTIVE(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
MARK(U41(x1, x2, x3)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U11(x1, x2)) → U11ACTIVE(mark(x1), x2)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
U61ACTIVE(tt, L) → LENGTHACTIVE(mark(L))
MARK(U43(x1)) → U43ACTIVE(mark(x1))
LENGTHACTIVE(cons(N, L)) → U61ACTIVE(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
MARK(U32(x1)) → MARK(x1)
MARK(U22(x1)) → MARK(x1)
LENGTHACTIVE(cons(N, L)) → ANDACTIVE(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
LENGTHACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
U21ACTIVE(tt, V1) → U22ACTIVE(isNatActive(V1))
U61ACTIVE(tt, L) → MARK(L)
MARK(U21(x1, x2)) → MARK(x1)
U41ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U42(x1, x2)) → MARK(x1)
U41ACTIVE(tt, V1, V2) → U42ACTIVE(isNatActive(V1), V2)
MARK(U52(x1, x2)) → MARK(x1)
MARK(U12(x1)) → MARK(x1)
MARK(length(x1)) → LENGTHACTIVE(mark(x1))
ISNATACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
MARK(isNatIList(x1)) → ISNATILISTACTIVE(x1)
MARK(s(x1)) → MARK(x1)
ISNATLISTACTIVE(cons(V1, V2)) → U51ACTIVE(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
MARK(U31(x1, x2)) → MARK(x1)
U11ACTIVE(tt, V1) → ISNATLISTACTIVE(V1)
ISNATILISTKINDACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(zeros) → ZEROSACTIVE
ISNATACTIVE(length(V1)) → U11ACTIVE(isNatIListKindActive(V1), V1)
ANDACTIVE(tt, X) → MARK(X)
MARK(U52(x1, x2)) → U52ACTIVE(mark(x1), x2)
LENGTHACTIVE(cons(N, L)) → ANDACTIVE(isNatListActive(L), isNatIListKind(L))
MARK(and(x1, x2)) → MARK(x1)
MARK(U61(x1, x2)) → U61ACTIVE(mark(x1), x2)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U41(x1, x2, x3)) → U41ACTIVE(mark(x1), x2, x3)
ISNATKINDACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
ISNATILISTACTIVE(V) → U31ACTIVE(isNatIListKindActive(V), V)
U42ACTIVE(tt, V2) → U43ACTIVE(isNatIListActive(V2))
MARK(U22(x1)) → U22ACTIVE(mark(x1))
MARK(U61(x1, x2)) → MARK(x1)
MARK(U43(x1)) → MARK(x1)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
U31ACTIVE(tt, V) → ISNATLISTACTIVE(V)
ISNATILISTACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
U31ACTIVE(tt, V) → U32ACTIVE(isNatListActive(V))
ISNATILISTACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
U52ACTIVE(tt, V2) → U53ACTIVE(isNatListActive(V2))
ISNATLISTACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
MARK(U31(x1, x2)) → U31ACTIVE(mark(x1), x2)
U42ACTIVE(tt, V2) → ISNATILISTACTIVE(V2)
U52ACTIVE(tt, V2) → ISNATLISTACTIVE(V2)
MARK(isNatIListKind(x1)) → ISNATILISTKINDACTIVE(x1)
The TRS R consists of the following rules:
mark(zeros) → zerosActive
zerosActive → zeros
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
U11Active(x1, x2) → U11(x1, x2)
mark(U12(x1)) → U12Active(mark(x1))
U12Active(x1) → U12(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2)) → U31Active(mark(x1), x2)
U31Active(x1, x2) → U31(x1, x2)
mark(U32(x1)) → U32Active(mark(x1))
U32Active(x1) → U32(x1)
mark(U41(x1, x2, x3)) → U41Active(mark(x1), x2, x3)
U41Active(x1, x2, x3) → U41(x1, x2, x3)
mark(U42(x1, x2)) → U42Active(mark(x1), x2)
U42Active(x1, x2) → U42(x1, x2)
mark(U43(x1)) → U43Active(mark(x1))
U43Active(x1) → U43(x1)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U52(x1, x2)) → U52Active(mark(x1), x2)
U52Active(x1, x2) → U52(x1, x2)
mark(U53(x1)) → U53Active(mark(x1))
U53Active(x1) → U53(x1)
mark(U61(x1, x2)) → U61Active(mark(x1), x2)
U61Active(x1, x2) → U61(x1, x2)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatIListKind(x1)) → isNatIListKindActive(x1)
isNatIListKindActive(x1) → isNatIListKind(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(nil) → nil
zerosActive → cons(0, zeros)
U11Active(tt, V1) → U12Active(isNatListActive(V1))
U12Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V) → U32Active(isNatListActive(V))
U32Active(tt) → tt
U41Active(tt, V1, V2) → U42Active(isNatActive(V1), V2)
U42Active(tt, V2) → U43Active(isNatIListActive(V2))
U43Active(tt) → tt
U51Active(tt, V1, V2) → U52Active(isNatActive(V1), V2)
U52Active(tt, V2) → U53Active(isNatListActive(V2))
U53Active(tt) → tt
U61Active(tt, L) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(length(V1)) → U11Active(isNatIListKindActive(V1), V1)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatIListActive(V) → U31Active(isNatIListKindActive(V), V)
isNatIListActive(zeros) → tt
isNatIListActive(cons(V1, V2)) → U41Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
isNatIListKindActive(nil) → tt
isNatIListKindActive(zeros) → tt
isNatIListKindActive(cons(V1, V2)) → andActive(isNatKindActive(V1), isNatIListKind(V2))
isNatKindActive(0) → tt
isNatKindActive(length(V1)) → isNatIListKindActive(V1)
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatListActive(nil) → tt
isNatListActive(cons(V1, V2)) → U51Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
lengthActive(nil) → 0
lengthActive(cons(N, L)) → U61Active(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
MARK(U11(x1, x2)) → MARK(x1)
U11ACTIVE(tt, V1) → U12ACTIVE(isNatListActive(V1))
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
MARK(U42(x1, x2)) → U42ACTIVE(mark(x1), x2)
MARK(cons(x1, x2)) → MARK(x1)
U51ACTIVE(tt, V1, V2) → U52ACTIVE(isNatActive(V1), V2)
U51ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U12(x1)) → U12ACTIVE(mark(x1))
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
MARK(length(x1)) → MARK(x1)
ISNATILISTKINDACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
MARK(U32(x1)) → U32ACTIVE(mark(x1))
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
ISNATLISTACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(U53(x1)) → U53ACTIVE(mark(x1))
MARK(U53(x1)) → MARK(x1)
ISNATILISTACTIVE(V) → ISNATILISTKINDACTIVE(V)
ISNATILISTACTIVE(cons(V1, V2)) → U41ACTIVE(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
MARK(U41(x1, x2, x3)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U11(x1, x2)) → U11ACTIVE(mark(x1), x2)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
U61ACTIVE(tt, L) → LENGTHACTIVE(mark(L))
MARK(U43(x1)) → U43ACTIVE(mark(x1))
LENGTHACTIVE(cons(N, L)) → U61ACTIVE(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
MARK(U32(x1)) → MARK(x1)
MARK(U22(x1)) → MARK(x1)
LENGTHACTIVE(cons(N, L)) → ANDACTIVE(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
LENGTHACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
U21ACTIVE(tt, V1) → U22ACTIVE(isNatActive(V1))
U61ACTIVE(tt, L) → MARK(L)
MARK(U21(x1, x2)) → MARK(x1)
U41ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(U42(x1, x2)) → MARK(x1)
U41ACTIVE(tt, V1, V2) → U42ACTIVE(isNatActive(V1), V2)
MARK(U52(x1, x2)) → MARK(x1)
MARK(U12(x1)) → MARK(x1)
MARK(length(x1)) → LENGTHACTIVE(mark(x1))
ISNATACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
MARK(isNatIList(x1)) → ISNATILISTACTIVE(x1)
MARK(s(x1)) → MARK(x1)
ISNATLISTACTIVE(cons(V1, V2)) → U51ACTIVE(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
MARK(U31(x1, x2)) → MARK(x1)
U11ACTIVE(tt, V1) → ISNATLISTACTIVE(V1)
ISNATILISTKINDACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(zeros) → ZEROSACTIVE
ISNATACTIVE(length(V1)) → U11ACTIVE(isNatIListKindActive(V1), V1)
ANDACTIVE(tt, X) → MARK(X)
MARK(U52(x1, x2)) → U52ACTIVE(mark(x1), x2)
LENGTHACTIVE(cons(N, L)) → ANDACTIVE(isNatListActive(L), isNatIListKind(L))
MARK(and(x1, x2)) → MARK(x1)
MARK(U61(x1, x2)) → U61ACTIVE(mark(x1), x2)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U41(x1, x2, x3)) → U41ACTIVE(mark(x1), x2, x3)
ISNATKINDACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
ISNATILISTACTIVE(V) → U31ACTIVE(isNatIListKindActive(V), V)
U42ACTIVE(tt, V2) → U43ACTIVE(isNatIListActive(V2))
MARK(U22(x1)) → U22ACTIVE(mark(x1))
MARK(U61(x1, x2)) → MARK(x1)
MARK(U43(x1)) → MARK(x1)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
U31ACTIVE(tt, V) → ISNATLISTACTIVE(V)
ISNATILISTACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
U31ACTIVE(tt, V) → U32ACTIVE(isNatListActive(V))
ISNATILISTACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
U52ACTIVE(tt, V2) → U53ACTIVE(isNatListActive(V2))
ISNATLISTACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
MARK(U31(x1, x2)) → U31ACTIVE(mark(x1), x2)
U42ACTIVE(tt, V2) → ISNATILISTACTIVE(V2)
U52ACTIVE(tt, V2) → ISNATLISTACTIVE(V2)
MARK(isNatIListKind(x1)) → ISNATILISTKINDACTIVE(x1)
The TRS R consists of the following rules:
mark(zeros) → zerosActive
zerosActive → zeros
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
U11Active(x1, x2) → U11(x1, x2)
mark(U12(x1)) → U12Active(mark(x1))
U12Active(x1) → U12(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2)) → U31Active(mark(x1), x2)
U31Active(x1, x2) → U31(x1, x2)
mark(U32(x1)) → U32Active(mark(x1))
U32Active(x1) → U32(x1)
mark(U41(x1, x2, x3)) → U41Active(mark(x1), x2, x3)
U41Active(x1, x2, x3) → U41(x1, x2, x3)
mark(U42(x1, x2)) → U42Active(mark(x1), x2)
U42Active(x1, x2) → U42(x1, x2)
mark(U43(x1)) → U43Active(mark(x1))
U43Active(x1) → U43(x1)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U52(x1, x2)) → U52Active(mark(x1), x2)
U52Active(x1, x2) → U52(x1, x2)
mark(U53(x1)) → U53Active(mark(x1))
U53Active(x1) → U53(x1)
mark(U61(x1, x2)) → U61Active(mark(x1), x2)
U61Active(x1, x2) → U61(x1, x2)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatIListKind(x1)) → isNatIListKindActive(x1)
isNatIListKindActive(x1) → isNatIListKind(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(nil) → nil
zerosActive → cons(0, zeros)
U11Active(tt, V1) → U12Active(isNatListActive(V1))
U12Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V) → U32Active(isNatListActive(V))
U32Active(tt) → tt
U41Active(tt, V1, V2) → U42Active(isNatActive(V1), V2)
U42Active(tt, V2) → U43Active(isNatIListActive(V2))
U43Active(tt) → tt
U51Active(tt, V1, V2) → U52Active(isNatActive(V1), V2)
U52Active(tt, V2) → U53Active(isNatListActive(V2))
U53Active(tt) → tt
U61Active(tt, L) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(length(V1)) → U11Active(isNatIListKindActive(V1), V1)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatIListActive(V) → U31Active(isNatIListKindActive(V), V)
isNatIListActive(zeros) → tt
isNatIListActive(cons(V1, V2)) → U41Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
isNatIListKindActive(nil) → tt
isNatIListKindActive(zeros) → tt
isNatIListKindActive(cons(V1, V2)) → andActive(isNatKindActive(V1), isNatIListKind(V2))
isNatKindActive(0) → tt
isNatKindActive(length(V1)) → isNatIListKindActive(V1)
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatListActive(nil) → tt
isNatListActive(cons(V1, V2)) → U51Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
lengthActive(nil) → 0
lengthActive(cons(N, L)) → U61Active(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 11 less nodes.
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
MARK(U11(x1, x2)) → MARK(x1)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
MARK(U42(x1, x2)) → U42ACTIVE(mark(x1), x2)
MARK(cons(x1, x2)) → MARK(x1)
U51ACTIVE(tt, V1, V2) → U52ACTIVE(isNatActive(V1), V2)
U51ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(length(x1)) → MARK(x1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
ISNATILISTKINDACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
ISNATLISTACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(U53(x1)) → MARK(x1)
ISNATILISTACTIVE(V) → ISNATILISTKINDACTIVE(V)
ISNATILISTACTIVE(cons(V1, V2)) → U41ACTIVE(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
MARK(U41(x1, x2, x3)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U11(x1, x2)) → U11ACTIVE(mark(x1), x2)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
U61ACTIVE(tt, L) → LENGTHACTIVE(mark(L))
MARK(U32(x1)) → MARK(x1)
LENGTHACTIVE(cons(N, L)) → U61ACTIVE(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
MARK(U22(x1)) → MARK(x1)
LENGTHACTIVE(cons(N, L)) → ANDACTIVE(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
LENGTHACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
MARK(U21(x1, x2)) → MARK(x1)
U41ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
U61ACTIVE(tt, L) → MARK(L)
MARK(U42(x1, x2)) → MARK(x1)
U41ACTIVE(tt, V1, V2) → U42ACTIVE(isNatActive(V1), V2)
MARK(length(x1)) → LENGTHACTIVE(mark(x1))
MARK(U12(x1)) → MARK(x1)
MARK(U52(x1, x2)) → MARK(x1)
ISNATACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
MARK(isNatIList(x1)) → ISNATILISTACTIVE(x1)
MARK(s(x1)) → MARK(x1)
ISNATLISTACTIVE(cons(V1, V2)) → U51ACTIVE(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
MARK(U31(x1, x2)) → MARK(x1)
U11ACTIVE(tt, V1) → ISNATLISTACTIVE(V1)
ISNATILISTKINDACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
ANDACTIVE(tt, X) → MARK(X)
ISNATACTIVE(length(V1)) → U11ACTIVE(isNatIListKindActive(V1), V1)
MARK(U52(x1, x2)) → U52ACTIVE(mark(x1), x2)
LENGTHACTIVE(cons(N, L)) → ANDACTIVE(isNatListActive(L), isNatIListKind(L))
MARK(and(x1, x2)) → MARK(x1)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U61(x1, x2)) → U61ACTIVE(mark(x1), x2)
MARK(U41(x1, x2, x3)) → U41ACTIVE(mark(x1), x2, x3)
ISNATKINDACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
ISNATILISTACTIVE(V) → U31ACTIVE(isNatIListKindActive(V), V)
MARK(U61(x1, x2)) → MARK(x1)
MARK(U43(x1)) → MARK(x1)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
U31ACTIVE(tt, V) → ISNATLISTACTIVE(V)
ISNATILISTACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
ISNATILISTACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
ISNATLISTACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
MARK(U31(x1, x2)) → U31ACTIVE(mark(x1), x2)
U42ACTIVE(tt, V2) → ISNATILISTACTIVE(V2)
U52ACTIVE(tt, V2) → ISNATLISTACTIVE(V2)
MARK(isNatIListKind(x1)) → ISNATILISTKINDACTIVE(x1)
The TRS R consists of the following rules:
mark(zeros) → zerosActive
zerosActive → zeros
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
U11Active(x1, x2) → U11(x1, x2)
mark(U12(x1)) → U12Active(mark(x1))
U12Active(x1) → U12(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2)) → U31Active(mark(x1), x2)
U31Active(x1, x2) → U31(x1, x2)
mark(U32(x1)) → U32Active(mark(x1))
U32Active(x1) → U32(x1)
mark(U41(x1, x2, x3)) → U41Active(mark(x1), x2, x3)
U41Active(x1, x2, x3) → U41(x1, x2, x3)
mark(U42(x1, x2)) → U42Active(mark(x1), x2)
U42Active(x1, x2) → U42(x1, x2)
mark(U43(x1)) → U43Active(mark(x1))
U43Active(x1) → U43(x1)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U52(x1, x2)) → U52Active(mark(x1), x2)
U52Active(x1, x2) → U52(x1, x2)
mark(U53(x1)) → U53Active(mark(x1))
U53Active(x1) → U53(x1)
mark(U61(x1, x2)) → U61Active(mark(x1), x2)
U61Active(x1, x2) → U61(x1, x2)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatIListKind(x1)) → isNatIListKindActive(x1)
isNatIListKindActive(x1) → isNatIListKind(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(nil) → nil
zerosActive → cons(0, zeros)
U11Active(tt, V1) → U12Active(isNatListActive(V1))
U12Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V) → U32Active(isNatListActive(V))
U32Active(tt) → tt
U41Active(tt, V1, V2) → U42Active(isNatActive(V1), V2)
U42Active(tt, V2) → U43Active(isNatIListActive(V2))
U43Active(tt) → tt
U51Active(tt, V1, V2) → U52Active(isNatActive(V1), V2)
U52Active(tt, V2) → U53Active(isNatListActive(V2))
U53Active(tt) → tt
U61Active(tt, L) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(length(V1)) → U11Active(isNatIListKindActive(V1), V1)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatIListActive(V) → U31Active(isNatIListKindActive(V), V)
isNatIListActive(zeros) → tt
isNatIListActive(cons(V1, V2)) → U41Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
isNatIListKindActive(nil) → tt
isNatIListKindActive(zeros) → tt
isNatIListKindActive(cons(V1, V2)) → andActive(isNatKindActive(V1), isNatIListKind(V2))
isNatKindActive(0) → tt
isNatKindActive(length(V1)) → isNatIListKindActive(V1)
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatListActive(nil) → tt
isNatListActive(cons(V1, V2)) → U51Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
lengthActive(nil) → 0
lengthActive(cons(N, L)) → U61Active(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U42(x1, x2)) → U42ACTIVE(mark(x1), x2)
MARK(U41(x1, x2, x3)) → MARK(x1)
MARK(U32(x1)) → MARK(x1)
MARK(U42(x1, x2)) → MARK(x1)
MARK(isNatIList(x1)) → ISNATILISTACTIVE(x1)
MARK(U31(x1, x2)) → MARK(x1)
MARK(U41(x1, x2, x3)) → U41ACTIVE(mark(x1), x2, x3)
MARK(U31(x1, x2)) → U31ACTIVE(mark(x1), x2)
The remaining pairs can at least be oriented weakly.
MARK(U11(x1, x2)) → MARK(x1)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
MARK(cons(x1, x2)) → MARK(x1)
U51ACTIVE(tt, V1, V2) → U52ACTIVE(isNatActive(V1), V2)
U51ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(length(x1)) → MARK(x1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
ISNATILISTKINDACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
ISNATLISTACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(U53(x1)) → MARK(x1)
ISNATILISTACTIVE(V) → ISNATILISTKINDACTIVE(V)
ISNATILISTACTIVE(cons(V1, V2)) → U41ACTIVE(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U11(x1, x2)) → U11ACTIVE(mark(x1), x2)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
U61ACTIVE(tt, L) → LENGTHACTIVE(mark(L))
LENGTHACTIVE(cons(N, L)) → U61ACTIVE(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
MARK(U22(x1)) → MARK(x1)
LENGTHACTIVE(cons(N, L)) → ANDACTIVE(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
LENGTHACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
MARK(U21(x1, x2)) → MARK(x1)
U41ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
U61ACTIVE(tt, L) → MARK(L)
U41ACTIVE(tt, V1, V2) → U42ACTIVE(isNatActive(V1), V2)
MARK(length(x1)) → LENGTHACTIVE(mark(x1))
MARK(U12(x1)) → MARK(x1)
MARK(U52(x1, x2)) → MARK(x1)
ISNATACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
MARK(s(x1)) → MARK(x1)
ISNATLISTACTIVE(cons(V1, V2)) → U51ACTIVE(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
U11ACTIVE(tt, V1) → ISNATLISTACTIVE(V1)
ISNATILISTKINDACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
ANDACTIVE(tt, X) → MARK(X)
ISNATACTIVE(length(V1)) → U11ACTIVE(isNatIListKindActive(V1), V1)
MARK(U52(x1, x2)) → U52ACTIVE(mark(x1), x2)
LENGTHACTIVE(cons(N, L)) → ANDACTIVE(isNatListActive(L), isNatIListKind(L))
MARK(and(x1, x2)) → MARK(x1)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U61(x1, x2)) → U61ACTIVE(mark(x1), x2)
ISNATKINDACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
ISNATILISTACTIVE(V) → U31ACTIVE(isNatIListKindActive(V), V)
MARK(U61(x1, x2)) → MARK(x1)
MARK(U43(x1)) → MARK(x1)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
U31ACTIVE(tt, V) → ISNATLISTACTIVE(V)
ISNATILISTACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
ISNATILISTACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
ISNATLISTACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
U42ACTIVE(tt, V2) → ISNATILISTACTIVE(V2)
U52ACTIVE(tt, V2) → ISNATLISTACTIVE(V2)
MARK(isNatIListKind(x1)) → ISNATILISTKINDACTIVE(x1)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ANDACTIVE(x1, x2)) = x2
POL(ISNATACTIVE(x1)) = 0
POL(ISNATILISTACTIVE(x1)) = 0
POL(ISNATILISTKINDACTIVE(x1)) = 0
POL(ISNATKINDACTIVE(x1)) = 0
POL(ISNATLISTACTIVE(x1)) = 0
POL(LENGTHACTIVE(x1)) = x1
POL(MARK(x1)) = x1
POL(U11(x1, x2)) = x1
POL(U11ACTIVE(x1, x2)) = 0
POL(U11Active(x1, x2)) = x1
POL(U12(x1)) = x1
POL(U12Active(x1)) = x1
POL(U21(x1, x2)) = x1
POL(U21ACTIVE(x1, x2)) = 0
POL(U21Active(x1, x2)) = x1
POL(U22(x1)) = x1
POL(U22Active(x1)) = x1
POL(U31(x1, x2)) = 1 + x1 + x2
POL(U31ACTIVE(x1, x2)) = 0
POL(U31Active(x1, x2)) = 1 + x1 + x2
POL(U32(x1)) = 1 + x1
POL(U32Active(x1)) = 1 + x1
POL(U41(x1, x2, x3)) = 1 + x1 + x3
POL(U41ACTIVE(x1, x2, x3)) = 0
POL(U41Active(x1, x2, x3)) = 1 + x1 + x3
POL(U42(x1, x2)) = 1 + x1 + x2
POL(U42ACTIVE(x1, x2)) = 0
POL(U42Active(x1, x2)) = 1 + x1 + x2
POL(U43(x1)) = x1
POL(U43Active(x1)) = x1
POL(U51(x1, x2, x3)) = x1
POL(U51ACTIVE(x1, x2, x3)) = 0
POL(U51Active(x1, x2, x3)) = x1
POL(U52(x1, x2)) = x1
POL(U52ACTIVE(x1, x2)) = 0
POL(U52Active(x1, x2)) = x1
POL(U53(x1)) = x1
POL(U53Active(x1)) = x1
POL(U61(x1, x2)) = x1 + x2
POL(U61ACTIVE(x1, x2)) = x2
POL(U61Active(x1, x2)) = x1 + x2
POL(and(x1, x2)) = x1 + x2
POL(andActive(x1, x2)) = x1 + x2
POL(cons(x1, x2)) = x1 + x2
POL(isNat(x1)) = 0
POL(isNatActive(x1)) = 0
POL(isNatIList(x1)) = 1 + x1
POL(isNatIListActive(x1)) = 1 + x1
POL(isNatIListKind(x1)) = 0
POL(isNatIListKindActive(x1)) = 0
POL(isNatKind(x1)) = 0
POL(isNatKindActive(x1)) = 0
POL(isNatList(x1)) = 0
POL(isNatListActive(x1)) = 0
POL(length(x1)) = x1
POL(lengthActive(x1)) = x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(tt) = 0
POL(zeros) = 0
POL(zerosActive) = 0
The following usable rules [17] were oriented:
mark(U41(x1, x2, x3)) → U41Active(mark(x1), x2, x3)
U41Active(x1, x2, x3) → U41(x1, x2, x3)
mark(U32(x1)) → U32Active(mark(x1))
U32Active(x1) → U32(x1)
mark(U31(x1, x2)) → U31Active(mark(x1), x2)
U31Active(x1, x2) → U31(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U52(x1, x2)) → U52Active(mark(x1), x2)
U52Active(x1, x2) → U52(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U43(x1)) → U43Active(mark(x1))
U43Active(x1) → U43(x1)
mark(U42(x1, x2)) → U42Active(mark(x1), x2)
U42Active(x1, x2) → U42(x1, x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U12Active(x1) → U12(x1)
mark(U12(x1)) → U12Active(mark(x1))
U11Active(x1, x2) → U11(x1, x2)
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
zerosActive → zeros
mark(zeros) → zerosActive
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(nil) → nil
zerosActive → cons(0, zeros)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
U31Active(tt, V) → U32Active(isNatListActive(V))
U32Active(tt) → tt
U41Active(tt, V1, V2) → U42Active(isNatActive(V1), V2)
U42Active(tt, V2) → U43Active(isNatIListActive(V2))
U11Active(tt, V1) → U12Active(isNatListActive(V1))
U12Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
andActive(x1, x2) → and(x1, x2)
isNatKindActive(length(V1)) → isNatIListKindActive(V1)
isNatIListKindActive(cons(V1, V2)) → andActive(isNatKindActive(V1), isNatIListKind(V2))
mark(isNatKind(x1)) → isNatKindActive(x1)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(tt, X) → mark(X)
mark(isNatIListKind(x1)) → isNatIListKindActive(x1)
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatActive(x1) → isNat(x1)
mark(isNat(x1)) → isNatActive(x1)
U53Active(x1) → U53(x1)
mark(U53(x1)) → U53Active(mark(x1))
U61Active(x1, x2) → U61(x1, x2)
mark(U61(x1, x2)) → U61Active(mark(x1), x2)
isNatKindActive(x1) → isNatKind(x1)
isNatListActive(x1) → isNatList(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListKindActive(x1) → isNatIListKind(x1)
isNatIListKindActive(nil) → tt
isNatIListKindActive(zeros) → tt
isNatKindActive(0) → tt
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatIListActive(V) → U31Active(isNatIListKindActive(V), V)
isNatIListActive(zeros) → tt
isNatIListActive(cons(V1, V2)) → U41Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
U61Active(tt, L) → s(lengthActive(mark(L)))
isNatActive(0) → tt
isNatActive(length(V1)) → U11Active(isNatIListKindActive(V1), V1)
U43Active(tt) → tt
U51Active(tt, V1, V2) → U52Active(isNatActive(V1), V2)
U52Active(tt, V2) → U53Active(isNatListActive(V2))
U53Active(tt) → tt
lengthActive(cons(N, L)) → U61Active(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
lengthActive(nil) → 0
isNatListActive(cons(V1, V2)) → U51Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
isNatListActive(nil) → tt
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
MARK(U11(x1, x2)) → MARK(x1)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
MARK(cons(x1, x2)) → MARK(x1)
U51ACTIVE(tt, V1, V2) → U52ACTIVE(isNatActive(V1), V2)
U51ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
MARK(length(x1)) → MARK(x1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
ISNATILISTKINDACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
ISNATLISTACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(U53(x1)) → MARK(x1)
ISNATILISTACTIVE(V) → ISNATILISTKINDACTIVE(V)
ISNATILISTACTIVE(cons(V1, V2)) → U41ACTIVE(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
MARK(U11(x1, x2)) → U11ACTIVE(mark(x1), x2)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
U61ACTIVE(tt, L) → LENGTHACTIVE(mark(L))
LENGTHACTIVE(cons(N, L)) → U61ACTIVE(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
MARK(U22(x1)) → MARK(x1)
LENGTHACTIVE(cons(N, L)) → ANDACTIVE(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
LENGTHACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
MARK(U21(x1, x2)) → MARK(x1)
U41ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
U61ACTIVE(tt, L) → MARK(L)
U41ACTIVE(tt, V1, V2) → U42ACTIVE(isNatActive(V1), V2)
MARK(length(x1)) → LENGTHACTIVE(mark(x1))
MARK(U12(x1)) → MARK(x1)
MARK(U52(x1, x2)) → MARK(x1)
ISNATACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
MARK(s(x1)) → MARK(x1)
ISNATLISTACTIVE(cons(V1, V2)) → U51ACTIVE(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
U11ACTIVE(tt, V1) → ISNATLISTACTIVE(V1)
ISNATILISTKINDACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
ANDACTIVE(tt, X) → MARK(X)
ISNATACTIVE(length(V1)) → U11ACTIVE(isNatIListKindActive(V1), V1)
MARK(U52(x1, x2)) → U52ACTIVE(mark(x1), x2)
LENGTHACTIVE(cons(N, L)) → ANDACTIVE(isNatListActive(L), isNatIListKind(L))
MARK(and(x1, x2)) → MARK(x1)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U61(x1, x2)) → U61ACTIVE(mark(x1), x2)
ISNATKINDACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
ISNATILISTACTIVE(V) → U31ACTIVE(isNatIListKindActive(V), V)
MARK(U61(x1, x2)) → MARK(x1)
MARK(U43(x1)) → MARK(x1)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
U31ACTIVE(tt, V) → ISNATLISTACTIVE(V)
ISNATILISTACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
ISNATILISTACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
ISNATLISTACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
U42ACTIVE(tt, V2) → ISNATILISTACTIVE(V2)
U52ACTIVE(tt, V2) → ISNATLISTACTIVE(V2)
MARK(isNatIListKind(x1)) → ISNATILISTKINDACTIVE(x1)
The TRS R consists of the following rules:
mark(zeros) → zerosActive
zerosActive → zeros
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
U11Active(x1, x2) → U11(x1, x2)
mark(U12(x1)) → U12Active(mark(x1))
U12Active(x1) → U12(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2)) → U31Active(mark(x1), x2)
U31Active(x1, x2) → U31(x1, x2)
mark(U32(x1)) → U32Active(mark(x1))
U32Active(x1) → U32(x1)
mark(U41(x1, x2, x3)) → U41Active(mark(x1), x2, x3)
U41Active(x1, x2, x3) → U41(x1, x2, x3)
mark(U42(x1, x2)) → U42Active(mark(x1), x2)
U42Active(x1, x2) → U42(x1, x2)
mark(U43(x1)) → U43Active(mark(x1))
U43Active(x1) → U43(x1)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U52(x1, x2)) → U52Active(mark(x1), x2)
U52Active(x1, x2) → U52(x1, x2)
mark(U53(x1)) → U53Active(mark(x1))
U53Active(x1) → U53(x1)
mark(U61(x1, x2)) → U61Active(mark(x1), x2)
U61Active(x1, x2) → U61(x1, x2)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatIListKind(x1)) → isNatIListKindActive(x1)
isNatIListKindActive(x1) → isNatIListKind(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(nil) → nil
zerosActive → cons(0, zeros)
U11Active(tt, V1) → U12Active(isNatListActive(V1))
U12Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V) → U32Active(isNatListActive(V))
U32Active(tt) → tt
U41Active(tt, V1, V2) → U42Active(isNatActive(V1), V2)
U42Active(tt, V2) → U43Active(isNatIListActive(V2))
U43Active(tt) → tt
U51Active(tt, V1, V2) → U52Active(isNatActive(V1), V2)
U52Active(tt, V2) → U53Active(isNatListActive(V2))
U53Active(tt) → tt
U61Active(tt, L) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(length(V1)) → U11Active(isNatIListKindActive(V1), V1)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatIListActive(V) → U31Active(isNatIListKindActive(V), V)
isNatIListActive(zeros) → tt
isNatIListActive(cons(V1, V2)) → U41Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
isNatIListKindActive(nil) → tt
isNatIListKindActive(zeros) → tt
isNatIListKindActive(cons(V1, V2)) → andActive(isNatKindActive(V1), isNatIListKind(V2))
isNatKindActive(0) → tt
isNatKindActive(length(V1)) → isNatIListKindActive(V1)
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatListActive(nil) → tt
isNatListActive(cons(V1, V2)) → U51Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
lengthActive(nil) → 0
lengthActive(cons(N, L)) → U61Active(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 6 less nodes.
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MARK(length(x1)) → LENGTHACTIVE(mark(x1))
MARK(U52(x1, x2)) → MARK(x1)
MARK(U12(x1)) → MARK(x1)
ISNATACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
MARK(U11(x1, x2)) → MARK(x1)
MARK(s(x1)) → MARK(x1)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
ISNATLISTACTIVE(cons(V1, V2)) → U51ACTIVE(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
MARK(cons(x1, x2)) → MARK(x1)
U51ACTIVE(tt, V1, V2) → U52ACTIVE(isNatActive(V1), V2)
U11ACTIVE(tt, V1) → ISNATLISTACTIVE(V1)
U51ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
ISNATILISTKINDACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(length(x1)) → MARK(x1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
ISNATACTIVE(length(V1)) → U11ACTIVE(isNatIListKindActive(V1), V1)
ANDACTIVE(tt, X) → MARK(X)
ISNATILISTKINDACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
MARK(U52(x1, x2)) → U52ACTIVE(mark(x1), x2)
LENGTHACTIVE(cons(N, L)) → ANDACTIVE(isNatListActive(L), isNatIListKind(L))
MARK(and(x1, x2)) → MARK(x1)
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
ISNATLISTACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U61(x1, x2)) → U61ACTIVE(mark(x1), x2)
MARK(U53(x1)) → MARK(x1)
MARK(U11(x1, x2)) → U11ACTIVE(mark(x1), x2)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
ISNATKINDACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
U61ACTIVE(tt, L) → LENGTHACTIVE(mark(L))
LENGTHACTIVE(cons(N, L)) → U61ACTIVE(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
MARK(U61(x1, x2)) → MARK(x1)
MARK(U22(x1)) → MARK(x1)
MARK(U43(x1)) → MARK(x1)
LENGTHACTIVE(cons(N, L)) → ANDACTIVE(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
LENGTHACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
U61ACTIVE(tt, L) → MARK(L)
ISNATLISTACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
MARK(U21(x1, x2)) → MARK(x1)
U52ACTIVE(tt, V2) → ISNATLISTACTIVE(V2)
MARK(isNatIListKind(x1)) → ISNATILISTKINDACTIVE(x1)
The TRS R consists of the following rules:
mark(zeros) → zerosActive
zerosActive → zeros
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
U11Active(x1, x2) → U11(x1, x2)
mark(U12(x1)) → U12Active(mark(x1))
U12Active(x1) → U12(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2)) → U31Active(mark(x1), x2)
U31Active(x1, x2) → U31(x1, x2)
mark(U32(x1)) → U32Active(mark(x1))
U32Active(x1) → U32(x1)
mark(U41(x1, x2, x3)) → U41Active(mark(x1), x2, x3)
U41Active(x1, x2, x3) → U41(x1, x2, x3)
mark(U42(x1, x2)) → U42Active(mark(x1), x2)
U42Active(x1, x2) → U42(x1, x2)
mark(U43(x1)) → U43Active(mark(x1))
U43Active(x1) → U43(x1)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U52(x1, x2)) → U52Active(mark(x1), x2)
U52Active(x1, x2) → U52(x1, x2)
mark(U53(x1)) → U53Active(mark(x1))
U53Active(x1) → U53(x1)
mark(U61(x1, x2)) → U61Active(mark(x1), x2)
U61Active(x1, x2) → U61(x1, x2)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatIListKind(x1)) → isNatIListKindActive(x1)
isNatIListKindActive(x1) → isNatIListKind(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(nil) → nil
zerosActive → cons(0, zeros)
U11Active(tt, V1) → U12Active(isNatListActive(V1))
U12Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V) → U32Active(isNatListActive(V))
U32Active(tt) → tt
U41Active(tt, V1, V2) → U42Active(isNatActive(V1), V2)
U42Active(tt, V2) → U43Active(isNatIListActive(V2))
U43Active(tt) → tt
U51Active(tt, V1, V2) → U52Active(isNatActive(V1), V2)
U52Active(tt, V2) → U53Active(isNatListActive(V2))
U53Active(tt) → tt
U61Active(tt, L) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(length(V1)) → U11Active(isNatIListKindActive(V1), V1)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatIListActive(V) → U31Active(isNatIListKindActive(V), V)
isNatIListActive(zeros) → tt
isNatIListActive(cons(V1, V2)) → U41Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
isNatIListKindActive(nil) → tt
isNatIListKindActive(zeros) → tt
isNatIListKindActive(cons(V1, V2)) → andActive(isNatKindActive(V1), isNatIListKind(V2))
isNatKindActive(0) → tt
isNatKindActive(length(V1)) → isNatIListKindActive(V1)
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatListActive(nil) → tt
isNatListActive(cons(V1, V2)) → U51Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
lengthActive(nil) → 0
lengthActive(cons(N, L)) → U61Active(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(length(x1)) → LENGTHACTIVE(mark(x1))
MARK(length(x1)) → MARK(x1)
MARK(U61(x1, x2)) → U61ACTIVE(mark(x1), x2)
MARK(U61(x1, x2)) → MARK(x1)
The remaining pairs can at least be oriented weakly.
MARK(U52(x1, x2)) → MARK(x1)
MARK(U12(x1)) → MARK(x1)
ISNATACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
MARK(U11(x1, x2)) → MARK(x1)
MARK(s(x1)) → MARK(x1)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
ISNATLISTACTIVE(cons(V1, V2)) → U51ACTIVE(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
MARK(cons(x1, x2)) → MARK(x1)
U51ACTIVE(tt, V1, V2) → U52ACTIVE(isNatActive(V1), V2)
U11ACTIVE(tt, V1) → ISNATLISTACTIVE(V1)
U51ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
ISNATILISTKINDACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
ISNATACTIVE(length(V1)) → U11ACTIVE(isNatIListKindActive(V1), V1)
ANDACTIVE(tt, X) → MARK(X)
ISNATILISTKINDACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
MARK(U52(x1, x2)) → U52ACTIVE(mark(x1), x2)
LENGTHACTIVE(cons(N, L)) → ANDACTIVE(isNatListActive(L), isNatIListKind(L))
MARK(and(x1, x2)) → MARK(x1)
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
ISNATLISTACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U53(x1)) → MARK(x1)
MARK(U11(x1, x2)) → U11ACTIVE(mark(x1), x2)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
ISNATKINDACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
U61ACTIVE(tt, L) → LENGTHACTIVE(mark(L))
LENGTHACTIVE(cons(N, L)) → U61ACTIVE(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
MARK(U22(x1)) → MARK(x1)
MARK(U43(x1)) → MARK(x1)
LENGTHACTIVE(cons(N, L)) → ANDACTIVE(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
LENGTHACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
U61ACTIVE(tt, L) → MARK(L)
ISNATLISTACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
MARK(U21(x1, x2)) → MARK(x1)
U52ACTIVE(tt, V2) → ISNATLISTACTIVE(V2)
MARK(isNatIListKind(x1)) → ISNATILISTKINDACTIVE(x1)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ANDACTIVE(x1, x2)) = x2
POL(ISNATACTIVE(x1)) = 0
POL(ISNATILISTKINDACTIVE(x1)) = 0
POL(ISNATKINDACTIVE(x1)) = 0
POL(ISNATLISTACTIVE(x1)) = 0
POL(LENGTHACTIVE(x1)) = x1
POL(MARK(x1)) = x1
POL(U11(x1, x2)) = x1
POL(U11ACTIVE(x1, x2)) = 0
POL(U11Active(x1, x2)) = x1
POL(U12(x1)) = x1
POL(U12Active(x1)) = x1
POL(U21(x1, x2)) = x1
POL(U21ACTIVE(x1, x2)) = 0
POL(U21Active(x1, x2)) = x1
POL(U22(x1)) = x1
POL(U22Active(x1)) = x1
POL(U31(x1, x2)) = x2
POL(U31Active(x1, x2)) = x2
POL(U32(x1)) = 0
POL(U32Active(x1)) = 0
POL(U41(x1, x2, x3)) = x3
POL(U41Active(x1, x2, x3)) = x3
POL(U42(x1, x2)) = x1 + x2
POL(U42Active(x1, x2)) = x1 + x2
POL(U43(x1)) = x1
POL(U43Active(x1)) = x1
POL(U51(x1, x2, x3)) = x1
POL(U51ACTIVE(x1, x2, x3)) = 0
POL(U51Active(x1, x2, x3)) = x1
POL(U52(x1, x2)) = x1
POL(U52ACTIVE(x1, x2)) = 0
POL(U52Active(x1, x2)) = x1
POL(U53(x1)) = x1
POL(U53Active(x1)) = x1
POL(U61(x1, x2)) = 1 + x1 + x2
POL(U61ACTIVE(x1, x2)) = x2
POL(U61Active(x1, x2)) = 1 + x1 + x2
POL(and(x1, x2)) = x1 + x2
POL(andActive(x1, x2)) = x1 + x2
POL(cons(x1, x2)) = x1 + x2
POL(isNat(x1)) = 0
POL(isNatActive(x1)) = 0
POL(isNatIList(x1)) = x1
POL(isNatIListActive(x1)) = x1
POL(isNatIListKind(x1)) = 0
POL(isNatIListKindActive(x1)) = 0
POL(isNatKind(x1)) = 0
POL(isNatKindActive(x1)) = 0
POL(isNatList(x1)) = 0
POL(isNatListActive(x1)) = 0
POL(length(x1)) = 1 + x1
POL(lengthActive(x1)) = 1 + x1
POL(mark(x1)) = x1
POL(nil) = 0
POL(s(x1)) = x1
POL(tt) = 0
POL(zeros) = 0
POL(zerosActive) = 0
The following usable rules [17] were oriented:
mark(U41(x1, x2, x3)) → U41Active(mark(x1), x2, x3)
U41Active(x1, x2, x3) → U41(x1, x2, x3)
mark(U32(x1)) → U32Active(mark(x1))
U32Active(x1) → U32(x1)
mark(U31(x1, x2)) → U31Active(mark(x1), x2)
U31Active(x1, x2) → U31(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U52(x1, x2)) → U52Active(mark(x1), x2)
U52Active(x1, x2) → U52(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U43(x1)) → U43Active(mark(x1))
U43Active(x1) → U43(x1)
mark(U42(x1, x2)) → U42Active(mark(x1), x2)
U42Active(x1, x2) → U42(x1, x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U12Active(x1) → U12(x1)
mark(U12(x1)) → U12Active(mark(x1))
U11Active(x1, x2) → U11(x1, x2)
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
zerosActive → zeros
mark(zeros) → zerosActive
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(nil) → nil
zerosActive → cons(0, zeros)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
U31Active(tt, V) → U32Active(isNatListActive(V))
U32Active(tt) → tt
U41Active(tt, V1, V2) → U42Active(isNatActive(V1), V2)
U42Active(tt, V2) → U43Active(isNatIListActive(V2))
U11Active(tt, V1) → U12Active(isNatListActive(V1))
U12Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
andActive(x1, x2) → and(x1, x2)
isNatKindActive(length(V1)) → isNatIListKindActive(V1)
isNatIListKindActive(cons(V1, V2)) → andActive(isNatKindActive(V1), isNatIListKind(V2))
mark(isNatKind(x1)) → isNatKindActive(x1)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(tt, X) → mark(X)
mark(isNatIListKind(x1)) → isNatIListKindActive(x1)
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatActive(x1) → isNat(x1)
mark(isNat(x1)) → isNatActive(x1)
U53Active(x1) → U53(x1)
mark(U53(x1)) → U53Active(mark(x1))
U61Active(x1, x2) → U61(x1, x2)
mark(U61(x1, x2)) → U61Active(mark(x1), x2)
isNatKindActive(x1) → isNatKind(x1)
isNatListActive(x1) → isNatList(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListKindActive(x1) → isNatIListKind(x1)
isNatIListKindActive(nil) → tt
isNatIListKindActive(zeros) → tt
isNatKindActive(0) → tt
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatIListActive(V) → U31Active(isNatIListKindActive(V), V)
isNatIListActive(zeros) → tt
isNatIListActive(cons(V1, V2)) → U41Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
U61Active(tt, L) → s(lengthActive(mark(L)))
isNatActive(0) → tt
isNatActive(length(V1)) → U11Active(isNatIListKindActive(V1), V1)
U43Active(tt) → tt
U51Active(tt, V1, V2) → U52Active(isNatActive(V1), V2)
U52Active(tt, V2) → U53Active(isNatListActive(V2))
U53Active(tt) → tt
lengthActive(cons(N, L)) → U61Active(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
lengthActive(nil) → 0
isNatListActive(cons(V1, V2)) → U51Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
isNatListActive(nil) → tt
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MARK(U12(x1)) → MARK(x1)
MARK(U52(x1, x2)) → MARK(x1)
ISNATACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
MARK(U11(x1, x2)) → MARK(x1)
MARK(s(x1)) → MARK(x1)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
ISNATLISTACTIVE(cons(V1, V2)) → U51ACTIVE(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
MARK(cons(x1, x2)) → MARK(x1)
U51ACTIVE(tt, V1, V2) → U52ACTIVE(isNatActive(V1), V2)
U11ACTIVE(tt, V1) → ISNATLISTACTIVE(V1)
U51ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
ISNATILISTKINDACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
ISNATACTIVE(length(V1)) → U11ACTIVE(isNatIListKindActive(V1), V1)
ANDACTIVE(tt, X) → MARK(X)
ISNATILISTKINDACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
MARK(U52(x1, x2)) → U52ACTIVE(mark(x1), x2)
LENGTHACTIVE(cons(N, L)) → ANDACTIVE(isNatListActive(L), isNatIListKind(L))
MARK(and(x1, x2)) → MARK(x1)
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
ISNATLISTACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U53(x1)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U11(x1, x2)) → U11ACTIVE(mark(x1), x2)
ISNATKINDACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
U61ACTIVE(tt, L) → LENGTHACTIVE(mark(L))
LENGTHACTIVE(cons(N, L)) → U61ACTIVE(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
MARK(U43(x1)) → MARK(x1)
MARK(U22(x1)) → MARK(x1)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
LENGTHACTIVE(cons(N, L)) → ANDACTIVE(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
LENGTHACTIVE(cons(N, L)) → ISNATLISTACTIVE(L)
MARK(U21(x1, x2)) → MARK(x1)
ISNATLISTACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
U61ACTIVE(tt, L) → MARK(L)
U52ACTIVE(tt, V2) → ISNATLISTACTIVE(V2)
MARK(isNatIListKind(x1)) → ISNATILISTKINDACTIVE(x1)
The TRS R consists of the following rules:
mark(zeros) → zerosActive
zerosActive → zeros
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
U11Active(x1, x2) → U11(x1, x2)
mark(U12(x1)) → U12Active(mark(x1))
U12Active(x1) → U12(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2)) → U31Active(mark(x1), x2)
U31Active(x1, x2) → U31(x1, x2)
mark(U32(x1)) → U32Active(mark(x1))
U32Active(x1) → U32(x1)
mark(U41(x1, x2, x3)) → U41Active(mark(x1), x2, x3)
U41Active(x1, x2, x3) → U41(x1, x2, x3)
mark(U42(x1, x2)) → U42Active(mark(x1), x2)
U42Active(x1, x2) → U42(x1, x2)
mark(U43(x1)) → U43Active(mark(x1))
U43Active(x1) → U43(x1)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U52(x1, x2)) → U52Active(mark(x1), x2)
U52Active(x1, x2) → U52(x1, x2)
mark(U53(x1)) → U53Active(mark(x1))
U53Active(x1) → U53(x1)
mark(U61(x1, x2)) → U61Active(mark(x1), x2)
U61Active(x1, x2) → U61(x1, x2)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatIListKind(x1)) → isNatIListKindActive(x1)
isNatIListKindActive(x1) → isNatIListKind(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(nil) → nil
zerosActive → cons(0, zeros)
U11Active(tt, V1) → U12Active(isNatListActive(V1))
U12Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V) → U32Active(isNatListActive(V))
U32Active(tt) → tt
U41Active(tt, V1, V2) → U42Active(isNatActive(V1), V2)
U42Active(tt, V2) → U43Active(isNatIListActive(V2))
U43Active(tt) → tt
U51Active(tt, V1, V2) → U52Active(isNatActive(V1), V2)
U52Active(tt, V2) → U53Active(isNatListActive(V2))
U53Active(tt) → tt
U61Active(tt, L) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(length(V1)) → U11Active(isNatIListKindActive(V1), V1)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatIListActive(V) → U31Active(isNatIListKindActive(V), V)
isNatIListActive(zeros) → tt
isNatIListActive(cons(V1, V2)) → U41Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
isNatIListKindActive(nil) → tt
isNatIListKindActive(zeros) → tt
isNatIListKindActive(cons(V1, V2)) → andActive(isNatKindActive(V1), isNatIListKind(V2))
isNatKindActive(0) → tt
isNatKindActive(length(V1)) → isNatIListKindActive(V1)
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatListActive(nil) → tt
isNatListActive(cons(V1, V2)) → U51Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
lengthActive(nil) → 0
lengthActive(cons(N, L)) → U61Active(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 4 less nodes.
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MARK(U52(x1, x2)) → MARK(x1)
MARK(U12(x1)) → MARK(x1)
ISNATACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
MARK(U11(x1, x2)) → MARK(x1)
MARK(s(x1)) → MARK(x1)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
ISNATLISTACTIVE(cons(V1, V2)) → U51ACTIVE(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
MARK(cons(x1, x2)) → MARK(x1)
U51ACTIVE(tt, V1, V2) → U52ACTIVE(isNatActive(V1), V2)
U11ACTIVE(tt, V1) → ISNATLISTACTIVE(V1)
U51ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
ISNATILISTKINDACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
ISNATACTIVE(length(V1)) → U11ACTIVE(isNatIListKindActive(V1), V1)
ANDACTIVE(tt, X) → MARK(X)
ISNATILISTKINDACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
MARK(U52(x1, x2)) → U52ACTIVE(mark(x1), x2)
MARK(and(x1, x2)) → MARK(x1)
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
ISNATLISTACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U53(x1)) → MARK(x1)
MARK(U11(x1, x2)) → U11ACTIVE(mark(x1), x2)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
ISNATKINDACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U22(x1)) → MARK(x1)
MARK(U43(x1)) → MARK(x1)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
ISNATLISTACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
MARK(U21(x1, x2)) → MARK(x1)
U52ACTIVE(tt, V2) → ISNATLISTACTIVE(V2)
MARK(isNatIListKind(x1)) → ISNATILISTKINDACTIVE(x1)
The TRS R consists of the following rules:
mark(zeros) → zerosActive
zerosActive → zeros
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
U11Active(x1, x2) → U11(x1, x2)
mark(U12(x1)) → U12Active(mark(x1))
U12Active(x1) → U12(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2)) → U31Active(mark(x1), x2)
U31Active(x1, x2) → U31(x1, x2)
mark(U32(x1)) → U32Active(mark(x1))
U32Active(x1) → U32(x1)
mark(U41(x1, x2, x3)) → U41Active(mark(x1), x2, x3)
U41Active(x1, x2, x3) → U41(x1, x2, x3)
mark(U42(x1, x2)) → U42Active(mark(x1), x2)
U42Active(x1, x2) → U42(x1, x2)
mark(U43(x1)) → U43Active(mark(x1))
U43Active(x1) → U43(x1)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U52(x1, x2)) → U52Active(mark(x1), x2)
U52Active(x1, x2) → U52(x1, x2)
mark(U53(x1)) → U53Active(mark(x1))
U53Active(x1) → U53(x1)
mark(U61(x1, x2)) → U61Active(mark(x1), x2)
U61Active(x1, x2) → U61(x1, x2)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatIListKind(x1)) → isNatIListKindActive(x1)
isNatIListKindActive(x1) → isNatIListKind(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(nil) → nil
zerosActive → cons(0, zeros)
U11Active(tt, V1) → U12Active(isNatListActive(V1))
U12Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V) → U32Active(isNatListActive(V))
U32Active(tt) → tt
U41Active(tt, V1, V2) → U42Active(isNatActive(V1), V2)
U42Active(tt, V2) → U43Active(isNatIListActive(V2))
U43Active(tt) → tt
U51Active(tt, V1, V2) → U52Active(isNatActive(V1), V2)
U52Active(tt, V2) → U53Active(isNatListActive(V2))
U53Active(tt) → tt
U61Active(tt, L) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(length(V1)) → U11Active(isNatIListKindActive(V1), V1)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatIListActive(V) → U31Active(isNatIListKindActive(V), V)
isNatIListActive(zeros) → tt
isNatIListActive(cons(V1, V2)) → U41Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
isNatIListKindActive(nil) → tt
isNatIListKindActive(zeros) → tt
isNatIListKindActive(cons(V1, V2)) → andActive(isNatKindActive(V1), isNatIListKind(V2))
isNatKindActive(0) → tt
isNatKindActive(length(V1)) → isNatIListKindActive(V1)
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatListActive(nil) → tt
isNatListActive(cons(V1, V2)) → U51Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
lengthActive(nil) → 0
lengthActive(cons(N, L)) → U61Active(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
MARK(U52(x1, x2)) → MARK(x1)
MARK(s(x1)) → MARK(x1)
MARK(isNatKind(x1)) → ISNATKINDACTIVE(x1)
MARK(and(x1, x2)) → ANDACTIVE(mark(x1), x2)
MARK(U52(x1, x2)) → U52ACTIVE(mark(x1), x2)
MARK(and(x1, x2)) → MARK(x1)
MARK(isNatList(x1)) → ISNATLISTACTIVE(x1)
MARK(U51(x1, x2, x3)) → MARK(x1)
MARK(U51(x1, x2, x3)) → U51ACTIVE(mark(x1), x2, x3)
The remaining pairs can at least be oriented weakly.
MARK(U12(x1)) → MARK(x1)
ISNATACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
MARK(U11(x1, x2)) → MARK(x1)
ISNATLISTACTIVE(cons(V1, V2)) → U51ACTIVE(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
MARK(cons(x1, x2)) → MARK(x1)
U51ACTIVE(tt, V1, V2) → U52ACTIVE(isNatActive(V1), V2)
U11ACTIVE(tt, V1) → ISNATLISTACTIVE(V1)
U51ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
ISNATILISTKINDACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
ISNATACTIVE(length(V1)) → U11ACTIVE(isNatIListKindActive(V1), V1)
ANDACTIVE(tt, X) → MARK(X)
ISNATILISTKINDACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
ISNATLISTACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U53(x1)) → MARK(x1)
MARK(U11(x1, x2)) → U11ACTIVE(mark(x1), x2)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
ISNATKINDACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U22(x1)) → MARK(x1)
MARK(U43(x1)) → MARK(x1)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
ISNATLISTACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
MARK(U21(x1, x2)) → MARK(x1)
U52ACTIVE(tt, V2) → ISNATLISTACTIVE(V2)
MARK(isNatIListKind(x1)) → ISNATILISTKINDACTIVE(x1)
Used ordering: Polynomial interpretation with max and min functions [25]:
POL(0) = 0
POL(ANDACTIVE(x1, x2)) = x2
POL(ISNATACTIVE(x1)) = 0
POL(ISNATILISTKINDACTIVE(x1)) = 0
POL(ISNATKINDACTIVE(x1)) = 0
POL(ISNATLISTACTIVE(x1)) = 0
POL(MARK(x1)) = x1
POL(U11(x1, x2)) = x1
POL(U11ACTIVE(x1, x2)) = 0
POL(U11Active(x1, x2)) = 0
POL(U12(x1)) = x1
POL(U12Active(x1)) = 0
POL(U21(x1, x2)) = x1
POL(U21ACTIVE(x1, x2)) = 0
POL(U21Active(x1, x2)) = 0
POL(U22(x1)) = x1
POL(U22Active(x1)) = 0
POL(U31(x1, x2)) = 0
POL(U31Active(x1, x2)) = 0
POL(U32(x1)) = 0
POL(U32Active(x1)) = 0
POL(U41(x1, x2, x3)) = 0
POL(U41Active(x1, x2, x3)) = 0
POL(U42(x1, x2)) = 0
POL(U42Active(x1, x2)) = 0
POL(U43(x1)) = x1
POL(U43Active(x1)) = 0
POL(U51(x1, x2, x3)) = 1 + x1 + x2 + x3
POL(U51ACTIVE(x1, x2, x3)) = 0
POL(U51Active(x1, x2, x3)) = 0
POL(U52(x1, x2)) = 1 + x1
POL(U52ACTIVE(x1, x2)) = 0
POL(U52Active(x1, x2)) = 0
POL(U53(x1)) = x1
POL(U53Active(x1)) = 0
POL(U61(x1, x2)) = 0
POL(U61Active(x1, x2)) = 0
POL(and(x1, x2)) = 1 + x1 + x2 + max(x1, x2)
POL(andActive(x1, x2)) = 0
POL(cons(x1, x2)) = x1
POL(isNat(x1)) = 0
POL(isNatActive(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatIListActive(x1)) = 0
POL(isNatIListKind(x1)) = 0
POL(isNatIListKindActive(x1)) = 0
POL(isNatKind(x1)) = 1
POL(isNatKindActive(x1)) = 0
POL(isNatList(x1)) = 1
POL(isNatListActive(x1)) = 0
POL(length(x1)) = 0
POL(lengthActive(x1)) = 0
POL(mark(x1)) = 0
POL(nil) = 0
POL(s(x1)) = 1 + x1
POL(tt) = 0
POL(zeros) = 0
POL(zerosActive) = 0
The following usable rules [17] were oriented:
none
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Instantiation
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MARK(U12(x1)) → MARK(x1)
ISNATACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
MARK(U11(x1, x2)) → MARK(x1)
ISNATLISTACTIVE(cons(V1, V2)) → U51ACTIVE(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
MARK(cons(x1, x2)) → MARK(x1)
U51ACTIVE(tt, V1, V2) → U52ACTIVE(isNatActive(V1), V2)
U11ACTIVE(tt, V1) → ISNATLISTACTIVE(V1)
U51ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
ISNATILISTKINDACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
ANDACTIVE(tt, X) → MARK(X)
ISNATACTIVE(length(V1)) → U11ACTIVE(isNatIListKindActive(V1), V1)
ISNATILISTKINDACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
ISNATLISTACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U53(x1)) → MARK(x1)
MARK(U11(x1, x2)) → U11ACTIVE(mark(x1), x2)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
ISNATKINDACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U22(x1)) → MARK(x1)
MARK(U43(x1)) → MARK(x1)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U21(x1, x2)) → MARK(x1)
ISNATLISTACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
U52ACTIVE(tt, V2) → ISNATLISTACTIVE(V2)
MARK(isNatIListKind(x1)) → ISNATILISTKINDACTIVE(x1)
The TRS R consists of the following rules:
mark(zeros) → zerosActive
zerosActive → zeros
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
U11Active(x1, x2) → U11(x1, x2)
mark(U12(x1)) → U12Active(mark(x1))
U12Active(x1) → U12(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2)) → U31Active(mark(x1), x2)
U31Active(x1, x2) → U31(x1, x2)
mark(U32(x1)) → U32Active(mark(x1))
U32Active(x1) → U32(x1)
mark(U41(x1, x2, x3)) → U41Active(mark(x1), x2, x3)
U41Active(x1, x2, x3) → U41(x1, x2, x3)
mark(U42(x1, x2)) → U42Active(mark(x1), x2)
U42Active(x1, x2) → U42(x1, x2)
mark(U43(x1)) → U43Active(mark(x1))
U43Active(x1) → U43(x1)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U52(x1, x2)) → U52Active(mark(x1), x2)
U52Active(x1, x2) → U52(x1, x2)
mark(U53(x1)) → U53Active(mark(x1))
U53Active(x1) → U53(x1)
mark(U61(x1, x2)) → U61Active(mark(x1), x2)
U61Active(x1, x2) → U61(x1, x2)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatIListKind(x1)) → isNatIListKindActive(x1)
isNatIListKindActive(x1) → isNatIListKind(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(nil) → nil
zerosActive → cons(0, zeros)
U11Active(tt, V1) → U12Active(isNatListActive(V1))
U12Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V) → U32Active(isNatListActive(V))
U32Active(tt) → tt
U41Active(tt, V1, V2) → U42Active(isNatActive(V1), V2)
U42Active(tt, V2) → U43Active(isNatIListActive(V2))
U43Active(tt) → tt
U51Active(tt, V1, V2) → U52Active(isNatActive(V1), V2)
U52Active(tt, V2) → U53Active(isNatListActive(V2))
U53Active(tt) → tt
U61Active(tt, L) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(length(V1)) → U11Active(isNatIListKindActive(V1), V1)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatIListActive(V) → U31Active(isNatIListKindActive(V), V)
isNatIListActive(zeros) → tt
isNatIListActive(cons(V1, V2)) → U41Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
isNatIListKindActive(nil) → tt
isNatIListKindActive(zeros) → tt
isNatIListKindActive(cons(V1, V2)) → andActive(isNatKindActive(V1), isNatIListKind(V2))
isNatKindActive(0) → tt
isNatKindActive(length(V1)) → isNatIListKindActive(V1)
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatListActive(nil) → tt
isNatListActive(cons(V1, V2)) → U51Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
lengthActive(nil) → 0
lengthActive(cons(N, L)) → U61Active(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule ANDACTIVE(tt, X) → MARK(X) we obtained the following new rules:
ANDACTIVE(tt, isNatIListKind(y_2)) → MARK(isNatIListKind(y_2))
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MARK(U12(x1)) → MARK(x1)
ISNATACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
MARK(U11(x1, x2)) → MARK(x1)
ISNATLISTACTIVE(cons(V1, V2)) → U51ACTIVE(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
MARK(cons(x1, x2)) → MARK(x1)
U51ACTIVE(tt, V1, V2) → U52ACTIVE(isNatActive(V1), V2)
U11ACTIVE(tt, V1) → ISNATLISTACTIVE(V1)
U51ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
ISNATILISTKINDACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
ISNATACTIVE(length(V1)) → U11ACTIVE(isNatIListKindActive(V1), V1)
ANDACTIVE(tt, isNatIListKind(y_2)) → MARK(isNatIListKind(y_2))
ISNATILISTKINDACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
ISNATLISTACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
MARK(isNat(x1)) → ISNATACTIVE(x1)
MARK(U53(x1)) → MARK(x1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
MARK(U11(x1, x2)) → U11ACTIVE(mark(x1), x2)
ISNATKINDACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
MARK(U21(x1, x2)) → U21ACTIVE(mark(x1), x2)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(U43(x1)) → MARK(x1)
MARK(U22(x1)) → MARK(x1)
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
ISNATACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
ISNATLISTACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
MARK(U21(x1, x2)) → MARK(x1)
U52ACTIVE(tt, V2) → ISNATLISTACTIVE(V2)
MARK(isNatIListKind(x1)) → ISNATILISTKINDACTIVE(x1)
The TRS R consists of the following rules:
mark(zeros) → zerosActive
zerosActive → zeros
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
U11Active(x1, x2) → U11(x1, x2)
mark(U12(x1)) → U12Active(mark(x1))
U12Active(x1) → U12(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2)) → U31Active(mark(x1), x2)
U31Active(x1, x2) → U31(x1, x2)
mark(U32(x1)) → U32Active(mark(x1))
U32Active(x1) → U32(x1)
mark(U41(x1, x2, x3)) → U41Active(mark(x1), x2, x3)
U41Active(x1, x2, x3) → U41(x1, x2, x3)
mark(U42(x1, x2)) → U42Active(mark(x1), x2)
U42Active(x1, x2) → U42(x1, x2)
mark(U43(x1)) → U43Active(mark(x1))
U43Active(x1) → U43(x1)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U52(x1, x2)) → U52Active(mark(x1), x2)
U52Active(x1, x2) → U52(x1, x2)
mark(U53(x1)) → U53Active(mark(x1))
U53Active(x1) → U53(x1)
mark(U61(x1, x2)) → U61Active(mark(x1), x2)
U61Active(x1, x2) → U61(x1, x2)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatIListKind(x1)) → isNatIListKindActive(x1)
isNatIListKindActive(x1) → isNatIListKind(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(nil) → nil
zerosActive → cons(0, zeros)
U11Active(tt, V1) → U12Active(isNatListActive(V1))
U12Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V) → U32Active(isNatListActive(V))
U32Active(tt) → tt
U41Active(tt, V1, V2) → U42Active(isNatActive(V1), V2)
U42Active(tt, V2) → U43Active(isNatIListActive(V2))
U43Active(tt) → tt
U51Active(tt, V1, V2) → U52Active(isNatActive(V1), V2)
U52Active(tt, V2) → U53Active(isNatListActive(V2))
U53Active(tt) → tt
U61Active(tt, L) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(length(V1)) → U11Active(isNatIListKindActive(V1), V1)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatIListActive(V) → U31Active(isNatIListKindActive(V), V)
isNatIListActive(zeros) → tt
isNatIListActive(cons(V1, V2)) → U41Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
isNatIListKindActive(nil) → tt
isNatIListKindActive(zeros) → tt
isNatIListKindActive(cons(V1, V2)) → andActive(isNatKindActive(V1), isNatIListKind(V2))
isNatKindActive(0) → tt
isNatKindActive(length(V1)) → isNatIListKindActive(V1)
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatListActive(nil) → tt
isNatListActive(cons(V1, V2)) → U51Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
lengthActive(nil) → 0
lengthActive(cons(N, L)) → U61Active(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 7 less nodes.
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILISTKINDACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
ANDACTIVE(tt, isNatIListKind(y_2)) → MARK(isNatIListKind(y_2))
ISNATILISTKINDACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
ISNATKINDACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(isNatIListKind(x1)) → ISNATILISTKINDACTIVE(x1)
The TRS R consists of the following rules:
mark(zeros) → zerosActive
zerosActive → zeros
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
U11Active(x1, x2) → U11(x1, x2)
mark(U12(x1)) → U12Active(mark(x1))
U12Active(x1) → U12(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2)) → U31Active(mark(x1), x2)
U31Active(x1, x2) → U31(x1, x2)
mark(U32(x1)) → U32Active(mark(x1))
U32Active(x1) → U32(x1)
mark(U41(x1, x2, x3)) → U41Active(mark(x1), x2, x3)
U41Active(x1, x2, x3) → U41(x1, x2, x3)
mark(U42(x1, x2)) → U42Active(mark(x1), x2)
U42Active(x1, x2) → U42(x1, x2)
mark(U43(x1)) → U43Active(mark(x1))
U43Active(x1) → U43(x1)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U52(x1, x2)) → U52Active(mark(x1), x2)
U52Active(x1, x2) → U52(x1, x2)
mark(U53(x1)) → U53Active(mark(x1))
U53Active(x1) → U53(x1)
mark(U61(x1, x2)) → U61Active(mark(x1), x2)
U61Active(x1, x2) → U61(x1, x2)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatIListKind(x1)) → isNatIListKindActive(x1)
isNatIListKindActive(x1) → isNatIListKind(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(nil) → nil
zerosActive → cons(0, zeros)
U11Active(tt, V1) → U12Active(isNatListActive(V1))
U12Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V) → U32Active(isNatListActive(V))
U32Active(tt) → tt
U41Active(tt, V1, V2) → U42Active(isNatActive(V1), V2)
U42Active(tt, V2) → U43Active(isNatIListActive(V2))
U43Active(tt) → tt
U51Active(tt, V1, V2) → U52Active(isNatActive(V1), V2)
U52Active(tt, V2) → U53Active(isNatListActive(V2))
U53Active(tt) → tt
U61Active(tt, L) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(length(V1)) → U11Active(isNatIListKindActive(V1), V1)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatIListActive(V) → U31Active(isNatIListKindActive(V), V)
isNatIListActive(zeros) → tt
isNatIListActive(cons(V1, V2)) → U41Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
isNatIListKindActive(nil) → tt
isNatIListKindActive(zeros) → tt
isNatIListKindActive(cons(V1, V2)) → andActive(isNatKindActive(V1), isNatIListKind(V2))
isNatKindActive(0) → tt
isNatKindActive(length(V1)) → isNatIListKindActive(V1)
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatListActive(nil) → tt
isNatListActive(cons(V1, V2)) → U51Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
lengthActive(nil) → 0
lengthActive(cons(N, L)) → U61Active(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
ISNATILISTKINDACTIVE(cons(V1, V2)) → ISNATKINDACTIVE(V1)
ISNATKINDACTIVE(s(V1)) → ISNATKINDACTIVE(V1)
MARK(isNatIListKind(x1)) → ISNATILISTKINDACTIVE(x1)
The remaining pairs can at least be oriented weakly.
ANDACTIVE(tt, isNatIListKind(y_2)) → MARK(isNatIListKind(y_2))
ISNATILISTKINDACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
ISNATKINDACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
Used ordering: Polynomial interpretation [25]:
POL(0) = 0
POL(ANDACTIVE(x1, x2)) = 1 + x2
POL(ISNATILISTKINDACTIVE(x1)) = 1 + x1
POL(ISNATKINDACTIVE(x1)) = 1 + x1
POL(MARK(x1)) = 1 + x1
POL(U11(x1, x2)) = 0
POL(U11Active(x1, x2)) = 0
POL(U12(x1)) = 0
POL(U12Active(x1)) = 0
POL(U21(x1, x2)) = 0
POL(U21Active(x1, x2)) = 0
POL(U22(x1)) = 0
POL(U22Active(x1)) = 0
POL(U31(x1, x2)) = 0
POL(U31Active(x1, x2)) = 0
POL(U32(x1)) = 0
POL(U32Active(x1)) = 0
POL(U41(x1, x2, x3)) = 0
POL(U41Active(x1, x2, x3)) = 0
POL(U42(x1, x2)) = 0
POL(U42Active(x1, x2)) = 0
POL(U43(x1)) = 0
POL(U43Active(x1)) = 0
POL(U51(x1, x2, x3)) = 0
POL(U51Active(x1, x2, x3)) = 0
POL(U52(x1, x2)) = 0
POL(U52Active(x1, x2)) = 0
POL(U53(x1)) = 0
POL(U53Active(x1)) = 0
POL(U61(x1, x2)) = 0
POL(U61Active(x1, x2)) = 0
POL(and(x1, x2)) = 0
POL(andActive(x1, x2)) = 0
POL(cons(x1, x2)) = 1 + x1 + x2
POL(isNat(x1)) = 0
POL(isNatActive(x1)) = 0
POL(isNatIList(x1)) = 0
POL(isNatIListActive(x1)) = 0
POL(isNatIListKind(x1)) = 1 + x1
POL(isNatIListKindActive(x1)) = 0
POL(isNatKind(x1)) = 0
POL(isNatKindActive(x1)) = 0
POL(isNatList(x1)) = 0
POL(isNatListActive(x1)) = 0
POL(length(x1)) = x1
POL(lengthActive(x1)) = 0
POL(mark(x1)) = 0
POL(nil) = 0
POL(s(x1)) = 1 + x1
POL(tt) = 0
POL(zeros) = 0
POL(zerosActive) = 0
The following usable rules [17] were oriented:
none
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
ISNATILISTKINDACTIVE(cons(V1, V2)) → ANDACTIVE(isNatKindActive(V1), isNatIListKind(V2))
ANDACTIVE(tt, isNatIListKind(y_2)) → MARK(isNatIListKind(y_2))
ISNATKINDACTIVE(length(V1)) → ISNATILISTKINDACTIVE(V1)
The TRS R consists of the following rules:
mark(zeros) → zerosActive
zerosActive → zeros
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
U11Active(x1, x2) → U11(x1, x2)
mark(U12(x1)) → U12Active(mark(x1))
U12Active(x1) → U12(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2)) → U31Active(mark(x1), x2)
U31Active(x1, x2) → U31(x1, x2)
mark(U32(x1)) → U32Active(mark(x1))
U32Active(x1) → U32(x1)
mark(U41(x1, x2, x3)) → U41Active(mark(x1), x2, x3)
U41Active(x1, x2, x3) → U41(x1, x2, x3)
mark(U42(x1, x2)) → U42Active(mark(x1), x2)
U42Active(x1, x2) → U42(x1, x2)
mark(U43(x1)) → U43Active(mark(x1))
U43Active(x1) → U43(x1)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U52(x1, x2)) → U52Active(mark(x1), x2)
U52Active(x1, x2) → U52(x1, x2)
mark(U53(x1)) → U53Active(mark(x1))
U53Active(x1) → U53(x1)
mark(U61(x1, x2)) → U61Active(mark(x1), x2)
U61Active(x1, x2) → U61(x1, x2)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatIListKind(x1)) → isNatIListKindActive(x1)
isNatIListKindActive(x1) → isNatIListKind(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(nil) → nil
zerosActive → cons(0, zeros)
U11Active(tt, V1) → U12Active(isNatListActive(V1))
U12Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V) → U32Active(isNatListActive(V))
U32Active(tt) → tt
U41Active(tt, V1, V2) → U42Active(isNatActive(V1), V2)
U42Active(tt, V2) → U43Active(isNatIListActive(V2))
U43Active(tt) → tt
U51Active(tt, V1, V2) → U52Active(isNatActive(V1), V2)
U52Active(tt, V2) → U53Active(isNatListActive(V2))
U53Active(tt) → tt
U61Active(tt, L) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(length(V1)) → U11Active(isNatIListKindActive(V1), V1)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatIListActive(V) → U31Active(isNatIListKindActive(V), V)
isNatIListActive(zeros) → tt
isNatIListActive(cons(V1, V2)) → U41Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
isNatIListKindActive(nil) → tt
isNatIListKindActive(zeros) → tt
isNatIListKindActive(cons(V1, V2)) → andActive(isNatKindActive(V1), isNatIListKind(V2))
isNatKindActive(0) → tt
isNatKindActive(length(V1)) → isNatIListKindActive(V1)
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatListActive(nil) → tt
isNatListActive(cons(V1, V2)) → U51Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
lengthActive(nil) → 0
lengthActive(cons(N, L)) → U61Active(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 3 less nodes.
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
ISNATACTIVE(length(V1)) → U11ACTIVE(isNatIListKindActive(V1), V1)
ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
ISNATLISTACTIVE(cons(V1, V2)) → U51ACTIVE(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
U51ACTIVE(tt, V1, V2) → U52ACTIVE(isNatActive(V1), V2)
U11ACTIVE(tt, V1) → ISNATLISTACTIVE(V1)
U51ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
U52ACTIVE(tt, V2) → ISNATLISTACTIVE(V2)
The TRS R consists of the following rules:
mark(zeros) → zerosActive
zerosActive → zeros
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
U11Active(x1, x2) → U11(x1, x2)
mark(U12(x1)) → U12Active(mark(x1))
U12Active(x1) → U12(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2)) → U31Active(mark(x1), x2)
U31Active(x1, x2) → U31(x1, x2)
mark(U32(x1)) → U32Active(mark(x1))
U32Active(x1) → U32(x1)
mark(U41(x1, x2, x3)) → U41Active(mark(x1), x2, x3)
U41Active(x1, x2, x3) → U41(x1, x2, x3)
mark(U42(x1, x2)) → U42Active(mark(x1), x2)
U42Active(x1, x2) → U42(x1, x2)
mark(U43(x1)) → U43Active(mark(x1))
U43Active(x1) → U43(x1)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U52(x1, x2)) → U52Active(mark(x1), x2)
U52Active(x1, x2) → U52(x1, x2)
mark(U53(x1)) → U53Active(mark(x1))
U53Active(x1) → U53(x1)
mark(U61(x1, x2)) → U61Active(mark(x1), x2)
U61Active(x1, x2) → U61(x1, x2)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatIListKind(x1)) → isNatIListKindActive(x1)
isNatIListKindActive(x1) → isNatIListKind(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(nil) → nil
zerosActive → cons(0, zeros)
U11Active(tt, V1) → U12Active(isNatListActive(V1))
U12Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V) → U32Active(isNatListActive(V))
U32Active(tt) → tt
U41Active(tt, V1, V2) → U42Active(isNatActive(V1), V2)
U42Active(tt, V2) → U43Active(isNatIListActive(V2))
U43Active(tt) → tt
U51Active(tt, V1, V2) → U52Active(isNatActive(V1), V2)
U52Active(tt, V2) → U53Active(isNatListActive(V2))
U53Active(tt) → tt
U61Active(tt, L) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(length(V1)) → U11Active(isNatIListKindActive(V1), V1)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatIListActive(V) → U31Active(isNatIListKindActive(V), V)
isNatIListActive(zeros) → tt
isNatIListActive(cons(V1, V2)) → U41Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
isNatIListKindActive(nil) → tt
isNatIListKindActive(zeros) → tt
isNatIListKindActive(cons(V1, V2)) → andActive(isNatKindActive(V1), isNatIListKind(V2))
isNatKindActive(0) → tt
isNatKindActive(length(V1)) → isNatIListKindActive(V1)
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatListActive(nil) → tt
isNatListActive(cons(V1, V2)) → U51Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
lengthActive(nil) → 0
lengthActive(cons(N, L)) → U61Active(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- U11ACTIVE(tt, V1) → ISNATLISTACTIVE(V1)
The graph contains the following edges 2 >= 1
- ISNATLISTACTIVE(cons(V1, V2)) → U51ACTIVE(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
The graph contains the following edges 1 > 2, 1 > 3
- ISNATACTIVE(length(V1)) → U11ACTIVE(isNatIListKindActive(V1), V1)
The graph contains the following edges 1 > 2
- U51ACTIVE(tt, V1, V2) → ISNATACTIVE(V1)
The graph contains the following edges 2 >= 1
- U21ACTIVE(tt, V1) → ISNATACTIVE(V1)
The graph contains the following edges 2 >= 1
- U51ACTIVE(tt, V1, V2) → U52ACTIVE(isNatActive(V1), V2)
The graph contains the following edges 3 >= 2
- U52ACTIVE(tt, V2) → ISNATLISTACTIVE(V2)
The graph contains the following edges 2 >= 1
- ISNATACTIVE(s(V1)) → U21ACTIVE(isNatKindActive(V1), V1)
The graph contains the following edges 1 > 2
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MARK(U43(x1)) → MARK(x1)
MARK(U22(x1)) → MARK(x1)
MARK(U12(x1)) → MARK(x1)
MARK(U11(x1, x2)) → MARK(x1)
MARK(U21(x1, x2)) → MARK(x1)
MARK(cons(x1, x2)) → MARK(x1)
MARK(U53(x1)) → MARK(x1)
The TRS R consists of the following rules:
mark(zeros) → zerosActive
zerosActive → zeros
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
U11Active(x1, x2) → U11(x1, x2)
mark(U12(x1)) → U12Active(mark(x1))
U12Active(x1) → U12(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2)) → U31Active(mark(x1), x2)
U31Active(x1, x2) → U31(x1, x2)
mark(U32(x1)) → U32Active(mark(x1))
U32Active(x1) → U32(x1)
mark(U41(x1, x2, x3)) → U41Active(mark(x1), x2, x3)
U41Active(x1, x2, x3) → U41(x1, x2, x3)
mark(U42(x1, x2)) → U42Active(mark(x1), x2)
U42Active(x1, x2) → U42(x1, x2)
mark(U43(x1)) → U43Active(mark(x1))
U43Active(x1) → U43(x1)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U52(x1, x2)) → U52Active(mark(x1), x2)
U52Active(x1, x2) → U52(x1, x2)
mark(U53(x1)) → U53Active(mark(x1))
U53Active(x1) → U53(x1)
mark(U61(x1, x2)) → U61Active(mark(x1), x2)
U61Active(x1, x2) → U61(x1, x2)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatIListKind(x1)) → isNatIListKindActive(x1)
isNatIListKindActive(x1) → isNatIListKind(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(nil) → nil
zerosActive → cons(0, zeros)
U11Active(tt, V1) → U12Active(isNatListActive(V1))
U12Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V) → U32Active(isNatListActive(V))
U32Active(tt) → tt
U41Active(tt, V1, V2) → U42Active(isNatActive(V1), V2)
U42Active(tt, V2) → U43Active(isNatIListActive(V2))
U43Active(tt) → tt
U51Active(tt, V1, V2) → U52Active(isNatActive(V1), V2)
U52Active(tt, V2) → U53Active(isNatListActive(V2))
U53Active(tt) → tt
U61Active(tt, L) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(length(V1)) → U11Active(isNatIListKindActive(V1), V1)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatIListActive(V) → U31Active(isNatIListKindActive(V), V)
isNatIListActive(zeros) → tt
isNatIListActive(cons(V1, V2)) → U41Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
isNatIListKindActive(nil) → tt
isNatIListKindActive(zeros) → tt
isNatIListKindActive(cons(V1, V2)) → andActive(isNatKindActive(V1), isNatIListKind(V2))
isNatKindActive(0) → tt
isNatKindActive(length(V1)) → isNatIListKindActive(V1)
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatListActive(nil) → tt
isNatListActive(cons(V1, V2)) → U51Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
lengthActive(nil) → 0
lengthActive(cons(N, L)) → U61Active(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MARK(U22(x1)) → MARK(x1)
MARK(U43(x1)) → MARK(x1)
MARK(U12(x1)) → MARK(x1)
MARK(U11(x1, x2)) → MARK(x1)
MARK(U21(x1, x2)) → MARK(x1)
MARK(cons(x1, x2)) → MARK(x1)
MARK(U53(x1)) → MARK(x1)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- MARK(U43(x1)) → MARK(x1)
The graph contains the following edges 1 > 1
- MARK(U22(x1)) → MARK(x1)
The graph contains the following edges 1 > 1
- MARK(U12(x1)) → MARK(x1)
The graph contains the following edges 1 > 1
- MARK(U11(x1, x2)) → MARK(x1)
The graph contains the following edges 1 > 1
- MARK(U21(x1, x2)) → MARK(x1)
The graph contains the following edges 1 > 1
- MARK(cons(x1, x2)) → MARK(x1)
The graph contains the following edges 1 > 1
- MARK(U53(x1)) → MARK(x1)
The graph contains the following edges 1 > 1
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U61ACTIVE(tt, L) → LENGTHACTIVE(mark(L))
LENGTHACTIVE(cons(N, L)) → U61ACTIVE(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
The TRS R consists of the following rules:
mark(zeros) → zerosActive
zerosActive → zeros
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
U11Active(x1, x2) → U11(x1, x2)
mark(U12(x1)) → U12Active(mark(x1))
U12Active(x1) → U12(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2)) → U31Active(mark(x1), x2)
U31Active(x1, x2) → U31(x1, x2)
mark(U32(x1)) → U32Active(mark(x1))
U32Active(x1) → U32(x1)
mark(U41(x1, x2, x3)) → U41Active(mark(x1), x2, x3)
U41Active(x1, x2, x3) → U41(x1, x2, x3)
mark(U42(x1, x2)) → U42Active(mark(x1), x2)
U42Active(x1, x2) → U42(x1, x2)
mark(U43(x1)) → U43Active(mark(x1))
U43Active(x1) → U43(x1)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U52(x1, x2)) → U52Active(mark(x1), x2)
U52Active(x1, x2) → U52(x1, x2)
mark(U53(x1)) → U53Active(mark(x1))
U53Active(x1) → U53(x1)
mark(U61(x1, x2)) → U61Active(mark(x1), x2)
U61Active(x1, x2) → U61(x1, x2)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatIListKind(x1)) → isNatIListKindActive(x1)
isNatIListKindActive(x1) → isNatIListKind(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(nil) → nil
zerosActive → cons(0, zeros)
U11Active(tt, V1) → U12Active(isNatListActive(V1))
U12Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V) → U32Active(isNatListActive(V))
U32Active(tt) → tt
U41Active(tt, V1, V2) → U42Active(isNatActive(V1), V2)
U42Active(tt, V2) → U43Active(isNatIListActive(V2))
U43Active(tt) → tt
U51Active(tt, V1, V2) → U52Active(isNatActive(V1), V2)
U52Active(tt, V2) → U53Active(isNatListActive(V2))
U53Active(tt) → tt
U61Active(tt, L) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(length(V1)) → U11Active(isNatIListKindActive(V1), V1)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatIListActive(V) → U31Active(isNatIListKindActive(V), V)
isNatIListActive(zeros) → tt
isNatIListActive(cons(V1, V2)) → U41Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
isNatIListKindActive(nil) → tt
isNatIListKindActive(zeros) → tt
isNatIListKindActive(cons(V1, V2)) → andActive(isNatKindActive(V1), isNatIListKind(V2))
isNatKindActive(0) → tt
isNatKindActive(length(V1)) → isNatIListKindActive(V1)
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatListActive(nil) → tt
isNatListActive(cons(V1, V2)) → U51Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
lengthActive(nil) → 0
lengthActive(cons(N, L)) → U61Active(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
U61ACTIVE(tt, L) → LENGTHACTIVE(mark(L))
The remaining pairs can at least be oriented weakly.
LENGTHACTIVE(cons(N, L)) → U61ACTIVE(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
| M( U52(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U42Active(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U41(x1, ..., x3) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| M( isNatActive(x1) ) = | | + | | · | x1 |
| M( U11Active(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U52Active(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U41Active(x1, ..., x3) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| M( U22Active(x1) ) = | | + | | · | x1 |
| M( U61Active(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( and(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( isNatKindActive(x1) ) = | | + | | · | x1 |
| M( isNatIListKind(x1) ) = | | + | | · | x1 |
| M( U21(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( isNatIListActive(x1) ) = | | + | | · | x1 |
| M( isNatList(x1) ) = | | + | | · | x1 |
| M( lengthActive(x1) ) = | | + | | · | x1 |
| M( U12Active(x1) ) = | | + | | · | x1 |
| M( isNatIList(x1) ) = | | + | | · | x1 |
| M( U53Active(x1) ) = | | + | | · | x1 |
| M( U51(x1, ..., x3) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| M( U21Active(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U31(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U42(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U32Active(x1) ) = | | + | | · | x1 |
| M( U11(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( isNatKind(x1) ) = | | + | | · | x1 |
| M( U31Active(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( cons(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U61(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( andActive(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U43Active(x1) ) = | | + | | · | x1 |
| M( isNatListActive(x1) ) = | | + | | · | x1 |
| M( U51Active(x1, ..., x3) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| M( isNatIListKindActive(x1) ) = | | + | | · | x1 |
Tuple symbols:
| M( U61ACTIVE(x1, x2) ) = | 0 | + | | · | x1 | + | | · | x2 |
| M( LENGTHACTIVE(x1) ) = | 0 | + | | · | x1 |
Matrix type:
We used a basic matrix type which is not further parametrizeable.
As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:
mark(U41(x1, x2, x3)) → U41Active(mark(x1), x2, x3)
U41Active(x1, x2, x3) → U41(x1, x2, x3)
mark(U32(x1)) → U32Active(mark(x1))
U32Active(x1) → U32(x1)
mark(U31(x1, x2)) → U31Active(mark(x1), x2)
U31Active(x1, x2) → U31(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U52(x1, x2)) → U52Active(mark(x1), x2)
U52Active(x1, x2) → U52(x1, x2)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U43(x1)) → U43Active(mark(x1))
U43Active(x1) → U43(x1)
mark(U42(x1, x2)) → U42Active(mark(x1), x2)
U42Active(x1, x2) → U42(x1, x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U12Active(x1) → U12(x1)
mark(U12(x1)) → U12Active(mark(x1))
U11Active(x1, x2) → U11(x1, x2)
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
zerosActive → zeros
mark(zeros) → zerosActive
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(nil) → nil
zerosActive → cons(0, zeros)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
U31Active(tt, V) → U32Active(isNatListActive(V))
U32Active(tt) → tt
U41Active(tt, V1, V2) → U42Active(isNatActive(V1), V2)
U42Active(tt, V2) → U43Active(isNatIListActive(V2))
U11Active(tt, V1) → U12Active(isNatListActive(V1))
U12Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
andActive(x1, x2) → and(x1, x2)
isNatKindActive(length(V1)) → isNatIListKindActive(V1)
isNatIListKindActive(cons(V1, V2)) → andActive(isNatKindActive(V1), isNatIListKind(V2))
mark(isNatKind(x1)) → isNatKindActive(x1)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(tt, X) → mark(X)
mark(isNatIListKind(x1)) → isNatIListKindActive(x1)
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatActive(x1) → isNat(x1)
mark(isNat(x1)) → isNatActive(x1)
U53Active(x1) → U53(x1)
mark(U53(x1)) → U53Active(mark(x1))
U61Active(x1, x2) → U61(x1, x2)
mark(U61(x1, x2)) → U61Active(mark(x1), x2)
isNatKindActive(x1) → isNatKind(x1)
isNatListActive(x1) → isNatList(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListKindActive(x1) → isNatIListKind(x1)
isNatIListKindActive(nil) → tt
isNatIListKindActive(zeros) → tt
isNatKindActive(0) → tt
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatIListActive(V) → U31Active(isNatIListKindActive(V), V)
isNatIListActive(zeros) → tt
isNatIListActive(cons(V1, V2)) → U41Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
U61Active(tt, L) → s(lengthActive(mark(L)))
isNatActive(0) → tt
isNatActive(length(V1)) → U11Active(isNatIListKindActive(V1), V1)
U43Active(tt) → tt
U51Active(tt, V1, V2) → U52Active(isNatActive(V1), V2)
U52Active(tt, V2) → U53Active(isNatListActive(V2))
U53Active(tt) → tt
lengthActive(cons(N, L)) → U61Active(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
lengthActive(nil) → 0
isNatListActive(cons(V1, V2)) → U51Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
isNatListActive(nil) → tt
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
LENGTHACTIVE(cons(N, L)) → U61ACTIVE(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
The TRS R consists of the following rules:
mark(zeros) → zerosActive
zerosActive → zeros
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
U11Active(x1, x2) → U11(x1, x2)
mark(U12(x1)) → U12Active(mark(x1))
U12Active(x1) → U12(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2)) → U31Active(mark(x1), x2)
U31Active(x1, x2) → U31(x1, x2)
mark(U32(x1)) → U32Active(mark(x1))
U32Active(x1) → U32(x1)
mark(U41(x1, x2, x3)) → U41Active(mark(x1), x2, x3)
U41Active(x1, x2, x3) → U41(x1, x2, x3)
mark(U42(x1, x2)) → U42Active(mark(x1), x2)
U42Active(x1, x2) → U42(x1, x2)
mark(U43(x1)) → U43Active(mark(x1))
U43Active(x1) → U43(x1)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U52(x1, x2)) → U52Active(mark(x1), x2)
U52Active(x1, x2) → U52(x1, x2)
mark(U53(x1)) → U53Active(mark(x1))
U53Active(x1) → U53(x1)
mark(U61(x1, x2)) → U61Active(mark(x1), x2)
U61Active(x1, x2) → U61(x1, x2)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatIListKind(x1)) → isNatIListKindActive(x1)
isNatIListKindActive(x1) → isNatIListKind(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(nil) → nil
zerosActive → cons(0, zeros)
U11Active(tt, V1) → U12Active(isNatListActive(V1))
U12Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V) → U32Active(isNatListActive(V))
U32Active(tt) → tt
U41Active(tt, V1, V2) → U42Active(isNatActive(V1), V2)
U42Active(tt, V2) → U43Active(isNatIListActive(V2))
U43Active(tt) → tt
U51Active(tt, V1, V2) → U52Active(isNatActive(V1), V2)
U52Active(tt, V2) → U53Active(isNatListActive(V2))
U53Active(tt) → tt
U61Active(tt, L) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(length(V1)) → U11Active(isNatIListKindActive(V1), V1)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatIListActive(V) → U31Active(isNatIListKindActive(V), V)
isNatIListActive(zeros) → tt
isNatIListActive(cons(V1, V2)) → U41Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
isNatIListKindActive(nil) → tt
isNatIListKindActive(zeros) → tt
isNatIListKindActive(cons(V1, V2)) → andActive(isNatKindActive(V1), isNatIListKind(V2))
isNatKindActive(0) → tt
isNatKindActive(length(V1)) → isNatIListKindActive(V1)
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatListActive(nil) → tt
isNatListActive(cons(V1, V2)) → U51Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
lengthActive(nil) → 0
lengthActive(cons(N, L)) → U61Active(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.
↳ CSR
↳ CSDependencyPairsProof
↳ Zantema-Transformation
↳ Incomplete Giesl Middeldorp-Transformation
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
ISNATILISTACTIVE(cons(V1, V2)) → U41ACTIVE(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
U42ACTIVE(tt, V2) → ISNATILISTACTIVE(V2)
U41ACTIVE(tt, V1, V2) → U42ACTIVE(isNatActive(V1), V2)
The TRS R consists of the following rules:
mark(zeros) → zerosActive
zerosActive → zeros
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
U11Active(x1, x2) → U11(x1, x2)
mark(U12(x1)) → U12Active(mark(x1))
U12Active(x1) → U12(x1)
mark(U21(x1, x2)) → U21Active(mark(x1), x2)
U21Active(x1, x2) → U21(x1, x2)
mark(U22(x1)) → U22Active(mark(x1))
U22Active(x1) → U22(x1)
mark(U31(x1, x2)) → U31Active(mark(x1), x2)
U31Active(x1, x2) → U31(x1, x2)
mark(U32(x1)) → U32Active(mark(x1))
U32Active(x1) → U32(x1)
mark(U41(x1, x2, x3)) → U41Active(mark(x1), x2, x3)
U41Active(x1, x2, x3) → U41(x1, x2, x3)
mark(U42(x1, x2)) → U42Active(mark(x1), x2)
U42Active(x1, x2) → U42(x1, x2)
mark(U43(x1)) → U43Active(mark(x1))
U43Active(x1) → U43(x1)
mark(U51(x1, x2, x3)) → U51Active(mark(x1), x2, x3)
U51Active(x1, x2, x3) → U51(x1, x2, x3)
mark(U52(x1, x2)) → U52Active(mark(x1), x2)
U52Active(x1, x2) → U52(x1, x2)
mark(U53(x1)) → U53Active(mark(x1))
U53Active(x1) → U53(x1)
mark(U61(x1, x2)) → U61Active(mark(x1), x2)
U61Active(x1, x2) → U61(x1, x2)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatIListKind(x1)) → isNatIListKindActive(x1)
isNatIListKindActive(x1) → isNatIListKind(x1)
mark(isNatKind(x1)) → isNatKindActive(x1)
isNatKindActive(x1) → isNatKind(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(nil) → nil
zerosActive → cons(0, zeros)
U11Active(tt, V1) → U12Active(isNatListActive(V1))
U12Active(tt) → tt
U21Active(tt, V1) → U22Active(isNatActive(V1))
U22Active(tt) → tt
U31Active(tt, V) → U32Active(isNatListActive(V))
U32Active(tt) → tt
U41Active(tt, V1, V2) → U42Active(isNatActive(V1), V2)
U42Active(tt, V2) → U43Active(isNatIListActive(V2))
U43Active(tt) → tt
U51Active(tt, V1, V2) → U52Active(isNatActive(V1), V2)
U52Active(tt, V2) → U53Active(isNatListActive(V2))
U53Active(tt) → tt
U61Active(tt, L) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(length(V1)) → U11Active(isNatIListKindActive(V1), V1)
isNatActive(s(V1)) → U21Active(isNatKindActive(V1), V1)
isNatIListActive(V) → U31Active(isNatIListKindActive(V), V)
isNatIListActive(zeros) → tt
isNatIListActive(cons(V1, V2)) → U41Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
isNatIListKindActive(nil) → tt
isNatIListKindActive(zeros) → tt
isNatIListKindActive(cons(V1, V2)) → andActive(isNatKindActive(V1), isNatIListKind(V2))
isNatKindActive(0) → tt
isNatKindActive(length(V1)) → isNatIListKindActive(V1)
isNatKindActive(s(V1)) → isNatKindActive(V1)
isNatListActive(nil) → tt
isNatListActive(cons(V1, V2)) → U51Active(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
lengthActive(nil) → 0
lengthActive(cons(N, L)) → U61Active(andActive(andActive(isNatListActive(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- U41ACTIVE(tt, V1, V2) → U42ACTIVE(isNatActive(V1), V2)
The graph contains the following edges 3 >= 2
- U42ACTIVE(tt, V2) → ISNATILISTACTIVE(V2)
The graph contains the following edges 2 >= 1
- ISNATILISTACTIVE(cons(V1, V2)) → U41ACTIVE(andActive(isNatKindActive(V1), isNatIListKind(V2)), V1, V2)
The graph contains the following edges 1 > 2, 1 > 3