(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Combined order from the following AFS and order.
a__U11(x1, x2)  =  a__U11(x1, x2)
tt  =  tt
mark(x1)  =  x1
a__U21(x1, x2, x3)  =  a__U21(x1, x2, x3)
s(x1)  =  s(x1)
a__plus(x1, x2)  =  a__plus(x1, x2)
a__U31(x1)  =  x1
0  =  0
a__U41(x1, x2, x3)  =  a__U41(x1, x2, x3)
a__x(x1, x2)  =  a__x(x1, x2)
a__and(x1, x2)  =  a__and(x1, x2)
a__isNat(x1)  =  a__isNat(x1)
plus(x1, x2)  =  plus(x1, x2)
isNat(x1)  =  isNat(x1)
x(x1, x2)  =  x(x1, x2)
U11(x1, x2)  =  U11(x1, x2)
U21(x1, x2, x3)  =  U21(x1, x2, x3)
U31(x1)  =  x1
U41(x1, x2, x3)  =  U41(x1, x2, x3)
and(x1, x2)  =  and(x1, x2)

Recursive path order with status [RPO].
Quasi-Precedence:

[tt, 0] > [a_U413, a_x2, x₂, U41₃] > [a_U213, a_plus2, plus₂, U21₃] > [a_U112, s₁, a_and2, a_isNat1, isNat₁, U11₂, and₂]

Status:

a_U112: multiset
tt: multiset
a_U213: [3,2,1]
s₁: [1]
a_plus2: [1,2]
0: multiset
a_U413: [2,3,1]
a_x2: [2,1]
a_and2: multiset
a_isNat1: multiset
plus₂: [1,2]
isNat₁: multiset
x₂: [2,1]
U11₂: multiset
U21₃: [3,2,1]
U41₃: [2,3,1]
and₂: multiset

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__U11(tt, N) → mark(N)
a__U21(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U41(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__and(tt, X) → mark(X)
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__and(a__isNat(V1), isNat(V2))
a__isNat(s(V1)) → a__isNat(V1)
a__isNat(x(V1, V2)) → a__and(a__isNat(V1), isNat(V2))
a__plus(N, 0) → a__U11(a__isNat(N), N)
a__plus(N, s(M)) → a__U21(a__and(a__isNat(M), isNat(N)), M, N)
a__x(N, 0) → a__U31(a__isNat(N))
a__x(N, s(M)) → a__U41(a__and(a__isNat(M), isNat(N)), M, N)

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Knuth-Bendix order [KBO] with precedence:

mark₁ > s₁ > a_isNat1 > isNat₁ > a_and2 > a_U213 > U21₃ > a_U112 > U11₂ > and₂ > a_U311 > a_plus2 > a_U413 > plus₂ > tt > U41₃ > U31₁ > a_x2 > x₂ > 0

and weight map:

tt=1
0=2
a__U31_1=1
mark_1=0
U31_1=1
isNat_1=1
a__isNat_1=1
s_1=1
U11_2=0
a__U11_2=0
U21_3=0
a__U21_3=0
plus_2=0
a__plus_2=0
U41_3=0
a__U41_3=0
x_2=0
a__x_2=0
and_2=0
a__and_2=0

The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__U31(tt) → 0
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U21(X1, X2, X3)) → a__U21(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U31(X)) → a__U31(mark(X))
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(and(X1, X2)) → a__and(mark(X1), X2)
mark(isNat(X)) → a__isNat(X)
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U21(X1, X2, X3) → U21(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U31(X) → U31(X)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
a__and(X1, X2) → and(X1, X2)
a__isNat(X) → isNat(X)

(5) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.