-- Lecture 3 : TRS
-- page 5
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) .
}

-- page 7

select NAT+ + EQL .
set trace whole on .
red 0 + s 0 = s 0 + 0 .
set trace whole off .

-- page 8

mod! NAT-BADAXIOM{
  pr(NAT+)
  op _*_ : Nat Nat -> Nat .
  var N : Nat
  eq N = N + 0 .
  eq 0 = N * 0 .
}

select NAT-BADAXIOM .
-- eq N = N + 0 causes an infinite loop
start 0 .
apply 1 with N = 0 at term .
apply 1 with N = (0 + 0) at term .
apply 1 with N = (0 + 0) + 0 at term .
-- eq 0 = N * 0 causes an infinite rewrite
start 0 . apply 2 with N = 0 at term .
start 0 . apply 2 with N = s 0 at term .
start 0 . apply 2 with N = s s 0 at term .
start 0 . apply 2 with N = s s s 0 at term .
-- The reduction command ignore the equation
red 0 .

-- page 10

mod! ABC{
  ops a b c : -> Bool
  eq a = b .
  eq a = c .
}
red in ABC + EQL : b = c .