-- Lecture 4: Proof score
-- page 3, 4
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* {
  pr(NAT+)
  op _*_ : Nat Nat -> Nat
  vars M N : Nat
  eq N * 0 = 0 .
  eq M * s N = M + (M * N) .
}
mod! PROOF-n {
  pr(NAT*)
  op n : -> Nat
}
select PROOF-n + EQL
red s (0 + n) = 0 + s n .

-- page 5, 6
mod! PROOF-i {
  pr(NAT*)
  ops x y : -> Nat
  eq x = y + y .
}
select PROOF-i + EQL
red x * s s 0 = (y * s s 0) + (y * s s 0) .

-- page 7

open NAT* + EQL
  ops x y : -> Nat .
  eq x = y + y .
  red  x * s s 0 = (y * s s 0) + (y * s s 0) .
close