-- Lecture 3
-- page 33
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+AC {
  pr(NAT+)
  op _+_ : Nat Nat -> Nat {assoc comm}
}
select NAT+AC
set trace  on
-- You do not need bracket for associative operators
-- like 0 + N + s 0 instead of (0 + N) + s 0 or 0 + (N + s 0)
-- You can see in the trace of the following reduction 
-- that N + 0 = N can be applied to 0 + (N + s 0) since + is commutative 
red 0 + N:Nat + s 0 .
set trace  off

-- page 34, 35
mod! BAG{
  [Elt < Bag]
  ops a b c  : -> Elt
  op _ _ : Bag Bag -> Bag { assoc comm }
  op _in_ : Elt Bag -> Bool
  var E : Elt    var B : Bag
  eq E in (E B) = true .
}

select BAG 
red c in a b c .

-- page 36 
mod! BAG2{
   [Elt < Bag]
  ops 0 1  : -> Elt
  op _ _ : Bag Bag -> Bag { assoc comm }
  var E : Elt 
  eq (E E) = 0 1 .
}

select BAG2
-- red 0 0 1 .
start 0 0 1 .
apply 1 at term .
apply 1 at term .
apply 1 at term .

-- page 37
mod! BAG3{
  [Elt < Bag]
   ops 0 1  : -> Elt
  op begin-with-0 : Bag -> Bool
  op _ _ : Bag Bag -> Bag { assoc comm }
  var B : Bag 
  eq begin-with-0(0 B) = true .
  eq begin-with-0(1 B) = false .
}

select BAG3
start begin-with-0 (0 1) .
apply 1 at term .
start begin-with-0 (0 1) .
apply 2 at term .