(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U13(tt) → tt
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Combined order from the following AFS and order.
a__U11(
x1,
x2,
x3) =
a__U11(
x1,
x2,
x3)
tt =
tt
a__U12(
x1,
x2) =
a__U12(
x1,
x2)
a__isNat(
x1) =
x1
a__U13(
x1) =
x1
a__U21(
x1,
x2) =
a__U21(
x1,
x2)
a__U22(
x1) =
a__U22(
x1)
a__U31(
x1,
x2) =
a__U31(
x1,
x2)
mark(
x1) =
x1
a__U41(
x1,
x2,
x3) =
a__U41(
x1,
x2,
x3)
s(
x1) =
s(
x1)
a__plus(
x1,
x2) =
a__plus(
x1,
x2)
a__and(
x1,
x2) =
a__and(
x1,
x2)
0 =
0
plus(
x1,
x2) =
plus(
x1,
x2)
a__isNatKind(
x1) =
x1
isNatKind(
x1) =
x1
and(
x1,
x2) =
and(
x1,
x2)
isNat(
x1) =
x1
U11(
x1,
x2,
x3) =
U11(
x1,
x2,
x3)
U12(
x1,
x2) =
U12(
x1,
x2)
U13(
x1) =
x1
U21(
x1,
x2) =
U21(
x1,
x2)
U22(
x1) =
U22(
x1)
U31(
x1,
x2) =
U31(
x1,
x2)
U41(
x1,
x2,
x3) =
U41(
x1,
x2,
x3)
Recursive path order with status [RPO].
Quasi-Precedence:
0 > [tt, aU413, aplus2, plus2, U413] > [aU113, U113] > [aU122, U122] > [aU312, U312]
0 > [tt, aU413, aplus2, plus2, U413] > s1 > [aU212, U212] > [aU221, U221] > [aU312, U312]
0 > [tt, aU413, aplus2, plus2, U413] > s1 > [aand2, and2] > [aU312, U312]
Status:
plus2: [1,2]
U312: multiset
aand2: [1,2]
U413: [3,2,1]
U113: multiset
U122: [2,1]
and2: [1,2]
0: multiset
U212: multiset
aplus2: [1,2]
aU413: [3,2,1]
U221: multiset
tt: multiset
aU312: multiset
aU113: multiset
aU212: multiset
aU122: [2,1]
s1: [1]
aU221: multiset
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a__U11(tt, V1, V2) → a__U12(a__isNat(V1), V2)
a__U12(tt, V2) → a__U13(a__isNat(V2))
a__U21(tt, V1) → a__U22(a__isNat(V1))
a__U22(tt) → tt
a__U31(tt, N) → mark(N)
a__U41(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
a__isNat(s(V1)) → a__U21(a__isNatKind(V1), V1)
a__isNatKind(0) → tt
a__isNatKind(plus(V1, V2)) → a__and(a__isNatKind(V1), isNatKind(V2))
a__isNatKind(s(V1)) → a__isNatKind(V1)
a__plus(N, 0) → a__U31(a__and(a__isNat(N), isNatKind(N)), N)
a__plus(N, s(M)) → a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__U13(tt) → tt
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0) = 0
POL(U11(x1, x2, x3)) = x1 + x2 + x3
POL(U12(x1, x2)) = x1 + x2
POL(U13(x1)) = 1 + x1
POL(U21(x1, x2)) = x1 + x2
POL(U22(x1)) = x1
POL(U31(x1, x2)) = x1 + x2
POL(U41(x1, x2, x3)) = x1 + x2 + x3
POL(a__U11(x1, x2, x3)) = x1 + x2 + x3
POL(a__U12(x1, x2)) = x1 + x2
POL(a__U13(x1)) = 1 + x1
POL(a__U21(x1, x2)) = x1 + x2
POL(a__U22(x1)) = x1
POL(a__U31(x1, x2)) = x1 + x2
POL(a__U41(x1, x2, x3)) = x1 + x2 + x3
POL(a__and(x1, x2)) = x1 + x2
POL(a__isNat(x1)) = x1
POL(a__isNatKind(x1)) = x1
POL(a__plus(x1, x2)) = x1 + x2
POL(and(x1, x2)) = x1 + x2
POL(isNat(x1)) = x1
POL(isNatKind(x1)) = x1
POL(mark(x1)) = x1
POL(plus(x1, x2)) = x1 + x2
POL(s(x1)) = x1
POL(tt) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a__U13(tt) → tt
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2) → U12(X1, X2)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0) = 2
POL(U11(x1, x2, x3)) = x1 + x2 + x3
POL(U12(x1, x2)) = 1 + x1 + x2
POL(U13(x1)) = 2·x1
POL(U21(x1, x2)) = x1 + x2
POL(U22(x1)) = 1 + x1
POL(U31(x1, x2)) = 1 + 2·x1 + x2
POL(U41(x1, x2, x3)) = x1 + 2·x2 + x3
POL(a__U11(x1, x2, x3)) = x1 + x2 + 2·x3
POL(a__U12(x1, x2)) = 2 + x1 + 2·x2
POL(a__U13(x1)) = 2·x1
POL(a__U21(x1, x2)) = x1 + x2
POL(a__U22(x1)) = 2 + x1
POL(a__U31(x1, x2)) = 2 + 2·x1 + 2·x2
POL(a__U41(x1, x2, x3)) = x1 + 2·x2 + 2·x3
POL(a__and(x1, x2)) = x1 + x2
POL(a__isNat(x1)) = 2 + 2·x1
POL(a__isNatKind(x1)) = x1
POL(a__plus(x1, x2)) = 2 + x1 + x2
POL(and(x1, x2)) = x1 + x2
POL(isNat(x1)) = 2 + 2·x1
POL(isNatKind(x1)) = x1
POL(mark(x1)) = 2·x1
POL(plus(x1, x2)) = 2 + x1 + x2
POL(s(x1)) = 2 + 2·x1
POL(tt) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
mark(isNat(X)) → a__isNat(X)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U12(X1, X2) → U12(X1, X2)
a__U22(X) → U22(X)
a__U31(X1, X2) → U31(X1, X2)
(6) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__isNat(X) → isNat(X)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNatKind(X) → isNatKind(X)
Q is empty.
(7) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(U11(x1, x2, x3)) = x1 + x2 + x3
POL(U12(x1, x2)) = 1 + x1 + x2
POL(U13(x1)) = x1
POL(U21(x1, x2)) = x1 + x2
POL(U22(x1)) = 1 + x1
POL(U31(x1, x2)) = 1 + x1 + x2
POL(U41(x1, x2, x3)) = x1 + x2 + x3
POL(a__U11(x1, x2, x3)) = x1 + x2 + x3
POL(a__U12(x1, x2)) = x1 + x2
POL(a__U13(x1)) = x1
POL(a__U21(x1, x2)) = x1 + x2
POL(a__U22(x1)) = x1
POL(a__U31(x1, x2)) = x1 + x2
POL(a__U41(x1, x2, x3)) = x1 + x2 + x3
POL(a__and(x1, x2)) = x1 + x2
POL(a__isNat(x1)) = 1 + x1
POL(a__isNatKind(x1)) = 1 + x1
POL(a__plus(x1, x2)) = 1 + x1 + x2
POL(and(x1, x2)) = x1 + x2
POL(isNat(x1)) = x1
POL(isNatKind(x1)) = x1
POL(mark(x1)) = 2 + x1
POL(plus(x1, x2)) = x1 + x2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
mark(U12(X1, X2)) → a__U12(mark(X1), X2)
mark(U22(X)) → a__U22(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(isNatKind(X)) → a__isNatKind(X)
a__isNat(X) → isNat(X)
a__plus(X1, X2) → plus(X1, X2)
a__isNatKind(X) → isNatKind(X)
(8) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U13(X) → U13(X)
a__U21(X1, X2) → U21(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__and(X1, X2) → and(X1, X2)
Q is empty.
(9) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(U11(x1, x2, x3)) = 1 + x1 + x2 + x3
POL(U13(x1)) = 1 + x1
POL(U21(x1, x2)) = x1 + x2
POL(U41(x1, x2, x3)) = 2 + x1 + x2 + 2·x3
POL(a__U11(x1, x2, x3)) = 2 + x1 + 2·x2 + 2·x3
POL(a__U13(x1)) = 2 + x1
POL(a__U21(x1, x2)) = x1 + 2·x2
POL(a__U41(x1, x2, x3)) = 2 + x1 + 2·x2 + 2·x3
POL(a__and(x1, x2)) = 1 + 2·x1 + 2·x2
POL(and(x1, x2)) = 1 + 2·x1 + x2
POL(mark(x1)) = 2·x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(and(X1, X2)) → a__and(mark(X1), X2)
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U13(X) → U13(X)
(10) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
a__U21(X1, X2) → U21(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__and(X1, X2) → and(X1, X2)
Q is empty.
(11) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(U11(x1, x2, x3)) = 2 + 2·x1 + 2·x2 + 2·x3
POL(U13(x1)) = 2·x1
POL(U21(x1, x2)) = 1 + 2·x1 + x2
POL(U41(x1, x2, x3)) = 1 + x1 + x2 + x3
POL(a__U11(x1, x2, x3)) = 1 + 2·x1 + 2·x2 + x3
POL(a__U13(x1)) = x1
POL(a__U21(x1, x2)) = 2 + 2·x1 + 2·x2
POL(a__U41(x1, x2, x3)) = 2 + 2·x1 + 2·x2 + 2·x3
POL(a__and(x1, x2)) = 2 + 2·x1 + 2·x2
POL(and(x1, x2)) = 1 + x1 + x2
POL(mark(x1)) = 2·x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
a__U21(X1, X2) → U21(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__and(X1, X2) → and(X1, X2)
(12) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
Q is empty.
(13) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(U13(x1)) = 1 + 2·x1
POL(U21(x1, x2)) = 1 + 2·x1 + 2·x2
POL(a__U13(x1)) = 1 + x1
POL(a__U21(x1, x2)) = 1 + x1 + x2
POL(mark(x1)) = 2 + 2·x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
mark(U13(X)) → a__U13(mark(X))
mark(U21(X1, X2)) → a__U21(mark(X1), X2)
(14) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(15) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(16) TRUE
(17) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(18) TRUE
(19) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(20) TRUE