(VAR M N)
(STRATEGY CONTEXTSENSITIVE 
  (U11 1)
  (tt)
  (U12 1)
  (s 1)
  (plus 1 2)
  (U21 1)
  (U22 1)
  (x 1 2)
  (0)
)
(RULES 
U11(tt,M,N) -> U12(tt,M,N)
U12(tt,M,N) -> s(plus(N,M))
U21(tt,M,N) -> U22(tt,M,N)
U22(tt,M,N) -> plus(x(N,M),N)
plus(N,0) -> N
plus(N,s(M)) -> U11(tt,M,N)
x(N,0) -> 0
x(N,s(M)) -> U21(tt,M,N)
)

The TRS is orthogonal, thus termination of innermost context-sensitive rewriting is equivalent to termination of context-sensitive rewriting.

Proving termination of context-sensitive rewriting for MYNAT_nosorts_noand:

-> Dependency pairs:
nF_U11(tt,M,N) -> nF_U12(tt,M,N)
nF_U12(tt,M,N) -> nF_plus(N,M)
nF_U12(tt,M,N) -> N
nF_U12(tt,M,N) -> M
nF_U21(tt,M,N) -> nF_U22(tt,M,N)
nF_U22(tt,M,N) -> nF_plus(x(N,M),N)
nF_U22(tt,M,N) -> nF_x(N,M)
nF_U22(tt,M,N) -> N
nF_U22(tt,M,N) -> M
nF_plus(N,s(M)) -> nF_U11(tt,M,N)
nF_x(N,s(M)) -> nF_U21(tt,M,N)

-> Proof of termination for MYNAT_nosorts_noand_1_1:
-> -> Dependency pairs in cycle:
nF_U21(tt,M,N) -> nF_U22(tt,M,N)
nF_x(N,s(M)) -> nF_U21(tt,M,N)
nF_U22(tt,M,N) -> nF_x(N,M)

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for MYNAT_nosorts_noand_1_2:
-> -> Dependency pairs in cycle:
nF_U11(tt,M,N) -> nF_U12(tt,M,N)
nF_plus(N,s(M)) -> nF_U11(tt,M,N)
nF_U12(tt,M,N) -> nF_plus(N,M)

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.