-- file: natPulsAssocPS.mod

mod! NATplus {
  [ Nat]
  op 0 : -> Nat
  op s_ : Nat -> Nat
  op _+_ : Nat Nat -> Nat

  vars M N : Nat
  eq    0  + N = N .
  eq (s M) + N = s(M + N) .
}

-- opening module NATplus and EQL
open (NATplus + EQL)
--> declaring constants for arbitrary values
ops i j k : -> Nat .

**> Prove associativity: (i + j) + k = i +(j + k)
**> by induction on i

**> base case proof for 0:
red 0 + (j + k) = (0 + j) + k .
**> induction hypothesis:
eq (i + J:Nat) + K:Nat = i + (J + K) .
**> induction step proof for (s k):
red ((s i) + J:Nat) + K:Nat = (s i) + (J + K) .
**> QED {end of proof for associativity of (_+_)}
close