-- Lecture 3: Termination
-- page 12
mod! BASIC-NAT{
  [Zero NzNat < Nat]
  op 0 : -> Zero
  op s_ : Nat -> NzNat
}
mod! NAT+ {
  pr(BASIC-NAT)
  op _+_ : Nat Nat -> Nat
  vars M N : Nat
  eq N + 0 = N .
  eq M + s N = s(M + N) .
}

mod! NAT-COM{
  pr(NAT+)
  vars X Y : Nat
  eq X + Y = Y + X .
}
select NAT-COM .
start 0 + s 0 .
apply 1 at term .
apply 1 at term .
apply 1 at term .
-- The reduction of 0 + s 0 fails (Stack overflow)
-- red 0 + s 0 

-- page 17
mod! NAT* {
  pr(NAT+)
  op _*_ : Nat Nat -> Nat
  vars M N : Nat
  eq N * 0 = 0 .
  eq M * s N = M + (M * N) .
}

mod! NAT-FACT {
 pr(NAT*)
 op fact_  : Nat -> Nat
 vars M N : Nat
 eq fact 0 = s 0 .
 eq fact (s N) = s N * (fact N) .
}

red in NAT-FACT : fact (s s s  0) .