(VAR N M X V1 V2)
(STRATEGY CONTEXTSENSITIVE 
  (U11 1)
  (tt)
  (U21 1)
  (s 1)
  (plus 1 2)
  (U31 1)
  (0)
  (U41 1)
  (x 1 2)
  (and 1)
  (isNat)
)
(RULES 
U11(tt,N) -> N
U21(tt,M,N) -> s(plus(N,M))
U31(tt) -> 0
U41(tt,M,N) -> plus(x(N,M),N)
and(tt,X) -> X
isNat(0) -> tt
isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
isNat(s(V1)) -> isNat(V1)
isNat(x(V1,V2)) -> and(isNat(V1),isNat(V2))
plus(N,0) -> U11(isNat(N),N)
plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
x(N,0) -> U31(isNat(N))
x(N,s(M)) -> U41(and(isNat(M),isNat(N)),M,N)
)

Proving termination of context-sensitive rewriting for MYNAT_nokinds:

-> Dependency pairs:
nF_U11(tt,N) -> N
nF_U21(tt,M,N) -> nF_plus(N,M)
nF_U21(tt,M,N) -> N
nF_U21(tt,M,N) -> M
nF_U41(tt,M,N) -> nF_plus(x(N,M),N)
nF_U41(tt,M,N) -> nF_x(N,M)
nF_U41(tt,M,N) -> N
nF_U41(tt,M,N) -> M
nF_and(tt,X) -> X
nF_isNat(plus(V1,V2)) -> nF_and(isNat(V1),isNat(V2))
nF_isNat(plus(V1,V2)) -> nF_isNat(V1)
nF_isNat(s(V1)) -> nF_isNat(V1)
nF_isNat(x(V1,V2)) -> nF_and(isNat(V1),isNat(V2))
nF_isNat(x(V1,V2)) -> nF_isNat(V1)
nF_plus(N,0) -> nF_U11(isNat(N),N)
nF_plus(N,0) -> nF_isNat(N)
nF_plus(N,s(M)) -> nF_U21(and(isNat(M),isNat(N)),M,N)
nF_plus(N,s(M)) -> nF_and(isNat(M),isNat(N))
nF_plus(N,s(M)) -> nF_isNat(M)
nF_x(N,0) -> nF_U31(isNat(N))
nF_x(N,0) -> nF_isNat(N)
nF_x(N,s(M)) -> nF_U41(and(isNat(M),isNat(N)),M,N)
nF_x(N,s(M)) -> nF_and(isNat(M),isNat(N))
nF_x(N,s(M)) -> nF_isNat(M)

-> Proof of termination for MYNAT_nokinds_1_1:
-> -> Dependency pairs in cycle:
nF_U41(tt,M,N) -> nF_x(N,M)
nF_x(N,s(M)) -> nF_U41(and(isNat(M),isNat(N)),M,N)

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for MYNAT_nokinds_1_2:
-> -> Dependency pairs in cycle:
nF_U21(tt,M,N) -> nF_plus(N,M)
nF_plus(N,s(M)) -> nF_U21(and(isNat(M),isNat(N)),M,N)

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for MYNAT_nokinds_1_3:
-> -> Dependency pairs in cycle:
nF_isNat(x(V1,V2)) -> nF_isNat(V1)
nF_isNat(s(V1)) -> nF_isNat(V1)
nF_isNat(plus(V1,V2)) -> nF_isNat(V1)
nF_and(tt,X) -> X
nF_isNat(x(V1,V2)) -> nF_and(isNat(V1),isNat(V2))
nF_isNat(plus(V1,V2)) -> nF_and(isNat(V1),isNat(V2))


Polynomial Interpretation:

[x](X1,X2) = X1 + X2 + 1
[s](X) = X
[plus](X1,X2) = X1 + X2 + 1
[x](X1,X2) = X1 + X2 + 1
[isNat](X) = X + 1
[and](X1,X2) = X1 + X2 + 1
[U11](X1,X2) = X1
[U21](X1,X2,X3) = X1
[U31](X) = X
[U41](X1,X2,X3) = X1

TIME: 1.7665e-2

Termination proved: There exists a projection such that there are no minimal mu-rewrite sequences in cycle.
-> -> Dependency pairs in cycle:
nF_isNat(x(V1,V2)) -> nF_isNat(V1)
nF_isNat(plus(V1,V2)) -> nF_isNat(V1)
nF_isNat(s(V1)) -> nF_isNat(V1)

Termination proved: Cycles verify subterm criterion.

SETTINGS:
Base ordering: Polynomial ordering
Proof mode: SCCs in CSDG + base ordering
Upper bound for coeffs: 1
Rationals below 1 for all non-replacing args: No
Polynomial interpretation: Linear
Coeffs in polynomials: No rationals
Delta: automatic



Termination was proved succesfully.