-- Lecture 4 : Example 1: Left-identity of +
-- page 9
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) .
}

-- Prove N + 0 = N .
open NAT+ + EQL
  red 0 + 0 = 0 .     
  op n : -> Nat .
  eq 0 + n = n  .     
  red 0 + s n = s n . 
close


-- page 10
open NAT+ + EQL
  red 0 + 0 = 0 .     
  op n : -> Nat .
  eq 0 + n = n  .     
  set trace on
  set trace whole on
  red 0 + s n = s n . 
  set trace whole off
  set trace off
close

-- page 11
-- BAD proof score (Variable)
open NAT+ + EQL
  red 0 + 0 = 0 .     
  var N : Nat .
  eq 0 + N = N  .     
  set trace on
  set trace whole on
  red 0 + s N = s N . 
  set trace whole off
  set trace off
close

-- page 13
mod! NAT+' {
  pr(BASIC-NAT)
  op _+_ : Nat Nat -> Nat
  vars M N : Nat
  eq N + 0 = N .
  eq M + s N = s(M + N) .
  eq 0 + N = N .
}