(VAR V2 N M V1)
(STRATEGY CONTEXTSENSITIVE 
  (U11 1)
  (tt)
  (U12 1)
  (isNat)
  (U21 1)
  (U31 1)
  (U32 1)
  (U41 1)
  (U51 1)
  (U52 1)
  (s 1)
  (plus 1 2)
  (U61 1)
  (0)
  (U71 1)
  (U72 1)
  (x 1 2)
)
(RULES 
U11(tt,V2) -> U12(isNat(V2))
U12(tt) -> tt
U21(tt) -> tt
U31(tt,V2) -> U32(isNat(V2))
U32(tt) -> tt
U41(tt,N) -> N
U51(tt,M,N) -> U52(isNat(N),M,N)
U52(tt,M,N) -> s(plus(N,M))
U61(tt) -> 0
U71(tt,M,N) -> U72(isNat(N),M,N)
U72(tt,M,N) -> plus(x(N,M),N)
isNat(0) -> tt
isNat(plus(V1,V2)) -> U11(isNat(V1),V2)
isNat(s(V1)) -> U21(isNat(V1))
isNat(x(V1,V2)) -> U31(isNat(V1),V2)
plus(N,0) -> U41(isNat(N),N)
plus(N,s(M)) -> U51(isNat(M),M,N)
x(N,0) -> U61(isNat(N))
x(N,s(M)) -> U71(isNat(M),M,N)
)

Proving termination of context-sensitive rewriting for MYNAT_nokinds_noand:

-> Dependency pairs:
nF_U11(tt,V2) -> nF_U12(isNat(V2))
nF_U11(tt,V2) -> nF_isNat(V2)
nF_U31(tt,V2) -> nF_U32(isNat(V2))
nF_U31(tt,V2) -> nF_isNat(V2)
nF_U41(tt,N) -> N
nF_U51(tt,M,N) -> nF_U52(isNat(N),M,N)
nF_U51(tt,M,N) -> nF_isNat(N)
nF_U52(tt,M,N) -> nF_plus(N,M)
nF_U52(tt,M,N) -> N
nF_U52(tt,M,N) -> M
nF_U71(tt,M,N) -> nF_U72(isNat(N),M,N)
nF_U71(tt,M,N) -> nF_isNat(N)
nF_U72(tt,M,N) -> nF_plus(x(N,M),N)
nF_U72(tt,M,N) -> nF_x(N,M)
nF_U72(tt,M,N) -> N
nF_U72(tt,M,N) -> M
nF_isNat(plus(V1,V2)) -> nF_U11(isNat(V1),V2)
nF_isNat(plus(V1,V2)) -> nF_isNat(V1)
nF_isNat(s(V1)) -> nF_U21(isNat(V1))
nF_isNat(s(V1)) -> nF_isNat(V1)
nF_isNat(x(V1,V2)) -> nF_U31(isNat(V1),V2)
nF_isNat(x(V1,V2)) -> nF_isNat(V1)
nF_plus(N,0) -> nF_U41(isNat(N),N)
nF_plus(N,0) -> nF_isNat(N)
nF_plus(N,s(M)) -> nF_U51(isNat(M),M,N)
nF_plus(N,s(M)) -> nF_isNat(M)
nF_x(N,0) -> nF_U61(isNat(N))
nF_x(N,0) -> nF_isNat(N)
nF_x(N,s(M)) -> nF_U71(isNat(M),M,N)
nF_x(N,s(M)) -> nF_isNat(M)

-> Proof of termination for MYNAT_nokinds_noand_1_1:
-> -> Dependency pairs in cycle:
nF_U71(tt,M,N) -> nF_U72(isNat(N),M,N)
nF_x(N,s(M)) -> nF_U71(isNat(M),M,N)
nF_U72(tt,M,N) -> nF_x(N,M)

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for MYNAT_nokinds_noand_1_2:
-> -> Dependency pairs in cycle:
nF_U51(tt,M,N) -> nF_U52(isNat(N),M,N)
nF_plus(N,s(M)) -> nF_U51(isNat(M),M,N)
nF_U52(tt,M,N) -> nF_plus(N,M)

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for MYNAT_nokinds_noand_1_3:
-> -> Dependency pairs in cycle:
nF_U11(tt,V2) -> nF_isNat(V2)
nF_isNat(plus(V1,V2)) -> nF_U11(isNat(V1),V2)
nF_isNat(x(V1,V2)) -> nF_isNat(V1)
nF_isNat(s(V1)) -> nF_isNat(V1)
nF_isNat(plus(V1,V2)) -> nF_isNat(V1)
nF_U31(tt,V2) -> nF_isNat(V2)
nF_isNat(x(V1,V2)) -> nF_U31(isNat(V1),V2)

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.