(* -------------------------------------------------------------------- *)
require import Int Xint.
require import FinType.

op (^^) (b1 b2:bool) = b1 = !b2.

lemma nosmt xor_false b: b ^^ false = b
by [].

lemma nosmt xor_true b: b ^^ true = !b
by [].

lemma nosmt xorA b1 b2 b3: (b1 ^^ b2) ^^ b3 = b1 ^^ (b2 ^^ b3)
by [].

lemma nosmt xorC b1 b2: b1 ^^ b2 = b2 ^^ b1
by [].

lemma nosmt xorK b: b ^^ b = false
by [].

clone FinType as BoolFin with
  type t    <- bool,
  op enum <- List.(::) true (List.(::) false List."[]"),
  op card <- 2
proof enum_spec by case.

(* -------------------------------------------------------------------- *)
(* cost axiomatization *)
abstract theory Cost.
  schema cost_eqbool `{P} {b1 b2:bool} : cost [P: b1 = b2] = 
    cost[P:b1] + cost[P:b2] + '1.

  schema cost_and  `{P} {b1 b2:bool} : cost [P: b1 /\ b2] = 
    cost[P:b1] + cost[P:b2] + '1.

  schema cost_anda `{P} {b1 b2:bool} : cost [P: b1 && b2] = 
    cost[P:b1] + cost[P:b2] + '1.

  schema cost_or   `{P} {b1 b2:bool} : cost [P: b1 \/ b2] = 
    cost[P:b1] + cost[P:b2] + '1.

  schema cost_ora  `{P} {b1 b2:bool} : cost [P: b1 || b2] = 
    cost[P:b1] + cost[P:b2] + '1.

  schema cost_xor  `{P} {b1 b2:bool} : cost [P: b1 ^^ b2] = 
    cost[P:b1] + cost[P:b2] + '1.

  schema cost_not  `{P} {b: bool}    : cost [P: !b] = cost[P:b] + '1.
  
  hint simplify cost_eqbool, cost_and, cost_anda, cost_or, 
                cost_ora, cost_xor, cost_not.
end Cost.