type primFunction = Succ | Pred | IsZero | Leq | Leq_partially_applied of int
type constant = Num of int | Bool of bool | Funct of primFunction
type identifier = string
type lambdaTerm = Constant of constant | Var of identifier | Abstract of identifier * lambdaTerm | App of lambdaTerm * lambdaTerm | IfThenElse of lambdaTerm * lambdaTerm * lambdaTerm | Let of identifier * lambdaTerm * lambdaTerm
let rec free_in (ident:identifier) (term:lambdaTerm) : bool =
match term with
| Constant _ -> false
| Var(var_ident) -> var_ident = ident
(* | Abstract(bound_ident, body) -> COMPLETE THIS LINE *)
(* | App(head, arg) -> COMPLETE THIS LINE *)
| IfThenElse(test, yes, no) -> free_in ident test || free_in ident yes || free_in ident no
| Let(bound_ident, arg, body) -> free_in ident arg || (bound_ident <> ident && free_in ident body)
let fresh_var (base : identifier) (term:lambdaTerm) =
let rec all_vars term vs = match term with
| Constant _ -> vs
| Var(var_ident) -> var_ident :: vs
| Abstract(bound_ident, body) -> all_vars body (bound_ident :: vs)
| App(head, arg) -> let vs' = all_vars head vs
in all_vars arg vs'
| IfThenElse(test, yes, no) -> let vs' = all_vars test vs
in let vs'' = all_vars yes vs'
in all_vars no vs''
| Let(bound_ident, arg, body) -> let vs' = all_vars arg vs
in all_vars body (bound_ident :: vs')
in let current = all_vars term []
in let rec check ident = if List.mem ident current then check (ident ^ "'") else ident
in check (base ^ "'") (* keep adding primes until we find a variable unused (either free or bound) in term *)
let rec substitute (term:lambdaTerm) (ident:identifier) (replacement:lambdaTerm) : lambdaTerm =
match term with
| Constant _ -> term
| Var(var_ident) when var_ident = ident -> replacement
| Var _ -> term
| App(head, arg) -> let head' = substitute head ident replacement
in let arg' = substitute arg ident replacement
in App(head', arg')
| IfThenElse(test, yes, no) -> let test' = substitute test ident replacement
in let yes' = substitute yes ident replacement
in let no' = substitute no ident replacement
in IfThenElse(test', yes', no')
| Abstract(bound_ident, body) when bound_ident = ident || not (free_in ident body) ->
(* vacuous substitution *)
term
| Abstract(bound_ident, body) when not (free_in bound_ident replacement) ->
(* can substitute without renaming bound_ident *)
let body' = substitute body ident replacement
in (* COMPLETE THIS LINE *)
| Abstract(bound_ident, body) ->
(* find a fresh variable unused in either body or replacement (which we hack by specifying their App) *)
let bound_ident' = fresh_var bound_ident (App(body,replacement))
in let body' = substitute body bound_ident (Var bound_ident')
in let body'' = substitute body' ident replacement
in Abstract(bound_ident', body'')
| Let(bound_ident, arg, body) when bound_ident = ident || not (free_in ident body) ->
let arg' = substitute arg ident replacement
in Let(bound_ident, arg', body)
| Let(bound_ident, arg, body) when not (free_in bound_ident replacement) ->
(* can substitute without renaming bound_ident *)
let body' = substitute body ident replacement
in let arg' = substitute arg ident replacement
in Let(bound_ident, arg', body')
| Let(bound_ident, arg, body) ->
(* find a fresh variable unused in either body or replacement (which we hack by specifying their App) *)
let bound_ident' = fresh_var bound_ident (App(body,replacement))
in let body' = substitute body bound_ident (Var bound_ident')
in let body'' = substitute body' ident replacement
in let arg' = substitute arg ident replacement
in Let(bound_ident', arg', body'')
type reduceOutcome = AlreadyResult | ReducedTo of lambdaTerm | StuckAt of lambdaTerm
exception Stuck of lambdaTerm
let rec reduce1 (term:lambdaTerm) : reduceOutcome =
match term with
(* notice we never evaluate a yes/np branch until it is chosen *)
| IfThenElse(Constant(Bool true), yes, _) -> ReducedTo yes
| IfThenElse(Constant(Bool false), _, no) -> ReducedTo no
| IfThenElse(test, yes, no) -> (match reduce1 test with
| AlreadyResult -> StuckAt term (* if test was not reducible, neither is IfThenElse *)
| ReducedTo test' -> ReducedTo (IfThenElse (test', yes, no))
| StuckAt _ as outcome -> outcome)
(* notice we never evaluate the body except after substituting, and that happens only after arg is reduced to a result *)
| Let(bound_var, arg, body) -> (match reduce1 arg with
| AlreadyResult -> (* if arg was not reducible, we can substitute *)
ReducedTo (substitute body bound_var arg)
| ReducedTo arg' -> ReducedTo (Let(bound_var, arg', body))
| StuckAt _ as outcome -> outcome)
(* notice we only substitute after arg is reduced to a result *)
| App(Abstract(bound_var, body) as head, arg) -> (match reduce1 arg with
| AlreadyResult -> (* if arg was not reducible, we can substitute *)
ReducedTo (substitute body bound_var arg)
| ReducedTo arg' -> ReducedTo (App(head, arg'))
| StuckAt _ as outcome -> outcome)
(* applications of primFunctions are reduced only when their arguments have been reduced to THE RIGHT TYPES of result *)
| App(Constant(Funct Succ), Constant(Num n)) -> ReducedTo (Constant(Num (n+1)))
| App(Constant(Funct Pred), Constant(Num n)) -> ReducedTo (Constant(Num (if n = 0 then 0 else n-1)))
| App(Constant(Funct IsZero), Constant(Num n)) -> ReducedTo (Constant(Bool (n=0)))
(* binary primFunctions are curried, have to be reduced in two steps *)
| App(Constant(Funct Leq), Constant(Num n)) -> ReducedTo (Constant(Funct (Leq_partially_applied n)))
| App(Constant(Funct (Leq_partially_applied m)), Constant(Num n)) -> ReducedTo (Constant(Bool (m<=n)))
(* first the head should be reduced, next the arg *)
| App(head, arg) -> (match reduce1 head with
| ReducedTo head' -> ReducedTo (App(head', arg))
| StuckAt _ as outcome -> outcome
| AlreadyResult -> (* head was not reducible, was arg? *)
(match reduce1 arg with
| ReducedTo arg' -> ReducedTo (App(head, arg'))
(* else the reducible cases of App(result, result) were caught above; this must be stuck *)
| AlreadyResult -> StuckAt term
| StuckAt _ as outcome -> outcome))
| Var _ -> StuckAt term (* free variables are stuck *)
| Constant _ -> AlreadyResult
| Abstract(_, _) -> AlreadyResult
let rec check_numbers (term:lambdaTerm) : unit =
match term with
| Constant(Num n) when n < 0 -> failwith ("Bad Number: " ^ string_of_int n)
| Constant _ -> ()
| Var _ -> ()
| Abstract(_, body) -> check_numbers body
| App(head, arg) -> let () = check_numbers head
in check_numbers arg
| Let(_, arg, body) -> let () = check_numbers arg
in check_numbers body
| IfThenElse(test, yes, no) -> let () = check_numbers test
in let () = check_numbers yes
in check_numbers no
let reduce (term:lambdaTerm) : lambdaTerm =
(* scan to verify that term doesn't have any Const(Num (negative)) *)
let () = check_numbers term
in let rec aux term = match reduce1 term with
| AlreadyResult -> term
| ReducedTo term' -> aux term' (* keep trying *)
| StuckAt stuck_term -> raise (Stuck stuck_term) (* fail by raising exception *)
in aux term