-- Lecture 4: Example 2: Commutativity of +
-- page 14

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 14
-- Proof score for M + N = N + M (failed)
open NAT+ + EQL
  var M : Nat
  red M + 0 = 0 + M .     
  op n : -> Nat .
  eq M + n = n + M .     
  red M + s n = s n + M . 
close

-- page 17
-- Proof score for M + N = N + M (+1 lemmna but failed)
open NAT+ + EQL
  vars M N : Nat .
  eq 0 + N = N .
  op n : -> Nat .
  red M + 0 = 0 + M .     
  eq M + n = n + M .     
  red M + s n = s n + M . 
close

-- page 18
-- Proof score for s M + N = s(N + M)
open NAT+ + EQL
 var M : Nat
 op n : -> Nat .
 red s M + 0 = s(M + 0) .     
 eq s M + n = s(M + n) .     
 red s M + s n = s(M + s n) .     
close

-- Proof score for M + N = N + M (succeed)
open NAT+ + EQL
  vars M N : Nat
  eq 0 + N = N .
  eq s M + N = s(M + N) .
  op n : -> Nat .
  red M + 0 = 0 + M .     
  eq M + n = n + M .     
  red M + s n = s n + M . 
close