-- Lecture 4 : Example 3: Associativity of @
-- page 19
mod! BASIC-LIST(X :: TRIV) {
  [Empty NeList < List]
  op nil : -> Empty
  op _::_ : Elt List -> NeList
}

view t2n from TRIV to NAT{
  sort Elt -> Nat
}

select BASIC-LIST(t2n) 
parse 0 :: 1 :: 2 :: nil .

-- page 20
mod! LIST-@ {
  pr(BASIC-LIST)
  op _@_ : List List -> List
  var E : Elt .  
  vars L1 L2 : List .
  eq       nil @ L1 = L1 .
  eq (E :: L1) @ L2 = E :: (L1 @ L2) .
}

select LIST-@(t2n) 
set trace whole on
red (0 :: 1 :: nil) @ (2 :: 3 :: nil) .
set trace whole off

-- page 21
-- BAD proof score for assoc of @
open LIST-@ + EQL .
  vars A B : List
  op l : -> List .
  var E : Elt .
  red (A @ B) @ nil = A @ (B @ nil) .
  eq (A @ B) @ l = A @ (B @ l) .
  red (A @ B) @ (E :: l)  = A @ (B @ (E :: l)) .
close 

-- GOOD proof score for assoc of @
open LIST-@ + EQL .
  vars B C : List
  op l : -> List .
  var E : Elt .
  red (nil @ B) @ C = nil @ (B @ C) .
  eq (l @ B) @ C = l @ (B @ C) .
  red ((E :: l) @ B) @ C  = (E :: l) @ (B @ C) .
close 

-- page 22
mod! LIST-@-assoc {
  pr(LIST-@)
  op _@_ : List List -> List {assoc}
}

select LIST-@-assoc(t2n) .
red (1 :: nil) @ (2 :: nil) @ (3 :: nil) @ (4 :: nil) .