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+(* -------------------------------------------------------------------------------------------------------------------- *)
+(* ----------------------------------- Data types useful for our theorem prover(s) ------------------------------------ *)
+(* -------------------------------------------------------------------------------------------------------------------- *)
+
+type proposition = (* logical proposition *)
+ | True
+ | False
+ | Lit of string
+ | Not of proposition
+ | And of proposition * proposition
+ | Or of proposition * proposition
+ | Implies of proposition * proposition
+ | Iff of proposition * proposition;;
+
+type interpretation = (string * bool) list;; (* interpretation *)
+
+type substitution = (string * proposition) list;; (* allows us to substitute any proposition for any atomic literal *)
+
+
+(* -------------------------------------------------------------------------------------------------------------------- *)
+(* ----------------------------------------- Utility functions for debugging ------------------------------------------ *)
+(* -------------------------------------------------------------------------------------------------------------------- *)
+
+let rec string_of_prop = function
+ | True -> "True"
+ | False -> "False"
+ | Lit symbol -> symbol
+ | Not p -> "¬" ^ (string_of_prop p)
+ | And (p1, p2) -> "(" ^ (string_of_prop p1) ^ " ∧ " ^ (string_of_prop p2) ^ ")"
+ | Or (p1, p2) -> "(" ^ (string_of_prop p1) ^ " ∨ " ^ (string_of_prop p2) ^ ")"
+ | Implies (p1, p2) -> "(" ^ (string_of_prop p1) ^ " → " ^ (string_of_prop p2) ^ ")"
+ | Iff (p1, p2) -> "(" ^ (string_of_prop p1) ^ " ↔ " ^ (string_of_prop p2) ^ ")";;
+
+ let string_of_sub sub =
+ let rec stringify = function
+ | [] -> ""
+ | (s, value) :: [] -> "(" ^ s ^ ": " ^ (string_of_prop value) ^ ")"
+ | (s, value) :: tl -> "(" ^ s ^ ": " ^ (string_of_prop value) ^ "), " ^ (stringify tl)
+ in "[" ^ (stringify sub) ^ "]";;
+
+let string_of_interpr (i : interpretation) =
+ let rec stringify = function
+ | [] -> ""
+ | (s, value) :: [] -> "(" ^ s ^ ": " ^ (string_of_bool value) ^ ")"
+ | (s, value) :: tl -> "(" ^ s ^ ": " ^ (string_of_bool value) ^ "), " ^ (stringify tl)
+ in "[" ^ (stringify i) ^ "]";;
+
+let print_prop prop = print_endline (string_of_prop prop);;
+
+
+(* -------------------------------------------------------------------------------------------------------------------- *)
+(* ------------------------------ Utility functions for dealing with our new data types ------------------------------- *)
+(* -------------------------------------------------------------------------------------------------------------------- *)
+
+(* Converts an interpretation to a substitution *)
+let rec sub_of_interpr = function
+ | [] -> []
+ | (s, b) :: tl -> (s, if b then True else False) :: (sub_of_interpr tl);;
+
+(* Removes instances of True or False literals via simplification *)
+let rec simplify_true_false = function
+ | True -> True
+ | False -> False
+ | Lit symbol -> Lit symbol
+ | Not p ->
+ (match simplify_true_false p with
+ | True -> False
+ | False -> True
+ | Not p' -> p'
+ | p' -> Not(p'))
+ | And (p1, p2) ->
+ (match simplify_true_false p1, simplify_true_false p2 with
+ | False, _ -> False
+ | _, False -> False
+ | True, p2' -> p2'
+ | p1', True -> p1'
+ | p1', p2' -> And(p1', p2'))
+ | Or (p1, p2) ->
+ (match simplify_true_false p1, simplify_true_false p2 with
+ | True, _ -> True
+ | _, True -> True
+ | False, p2' -> p2'
+ | p1', False -> p1'
+ | p1', p2' -> Or(p1', p2'))
+ | Implies (p1, p2) ->
+ (match simplify_true_false p1, simplify_true_false p2 with
+ | False, _ -> True
+ | True, p2' -> p2'
+ | p1', p2' -> Implies(p1', p2'))
+ | Iff (p1, p2) ->
+ (match simplify_true_false p1, simplify_true_false p2 with
+ | True, p2' -> p2'
+ | p1', True -> p1'
+ | False, p2' -> simplify_true_false (Not(p2'))
+ | p1', False -> simplify_true_false (Not(p1'))
+ | p1', p2' -> Iff(p1', p2'));;
+
+(* Finds a symbol in a list of substitutions and returns the value to substitute for that symbol *)
+let rec replace sub symbol =
+ match sub with
+ | [] -> Lit symbol
+ | (s, value) :: sub -> if s = symbol then value else replace sub symbol;;
+
+(* Substitutes propositions for symbols as specified by the sub argument *)
+let rec substitute (prop : proposition) (sub : substitution) =
+ match prop with
+ | True -> True
+ | False -> False
+ | Lit s -> replace sub s
+ | Not p -> Not (substitute p sub)
+ | And(p1, p2) -> And(substitute p1 sub, substitute p2 sub)
+ | Or(p1, p2) -> Or(substitute p1 sub, substitute p2 sub)
+ | Implies(p1, p2) -> Implies(substitute p1 sub, substitute p2 sub)
+ | Iff(p1, p2) -> And(substitute p1 sub, substitute p2 sub);;
+
+(* Converts a proposition to negation normal form *)
+let to_nnf prop =
+ let rec to_nnf = function
+ | True -> True
+ | False -> False
+ | Lit symbol -> Lit symbol
+ | Not p ->
+ (match to_nnf p with
+ | And(p1, p2) -> Or(to_nnf (Not(p1)), to_nnf (Not(p2)))
+ | Or(p1, p2) -> And(to_nnf (Not(p1)), to_nnf (Not(p2)))
+ | p' -> simplify_true_false (Not p'))
+ | And(p1, p2) -> And(to_nnf p1, to_nnf p2)
+ | Or(p1, p2) -> Or(to_nnf p1, to_nnf p2)
+ | Implies(p1, p2) -> to_nnf (Or(Not(p1), p2))
+ | Iff(p1, p2) -> to_nnf (And(Implies(p1, p2), Implies(p2, p1)))
+ in
+ simplify_true_false (to_nnf prop);;
+
+(* Applies an interpretation to a proposition and returns the resulting proposition *)
+let interpret prop i = simplify_true_false (substitute prop (sub_of_interpr i));;
+
+(* Tests whether a proposition holds under a given interpretation *)
+let test_interpretation prop i =
+ match interpret prop i with
+ | True -> Some true
+ | False -> Some false
+ | _ -> None;; (* return None if cannot be determined *)
+
+
+(* -------------------------------------------------------------------------------------------------------------------- *)
+(* --------------------------------- Sequent Calculus (TODO: move this to dif file) ----------------------------------- *)
+(* -------------------------------------------------------------------------------------------------------------------- *)
+
+type seq = Seq of proposition list * proposition list;; (* sequent *)
+
+type stree = (* tree of sequents *)
+ | Taut (* tautology of the form A, Gamma => A, Delta *)
+ | Br of stree * stree
+
+let string_of_seq (Seq (p1, p2)) =
+ let rec string_of_prop_list = function
+ | [] -> ""
+ | p :: ps -> (string_of_prop p) ^ ", " ^ (string_of_prop_list ps)
+ in
+ (string_of_prop_list p1) ^ "⇒" ^ (string_of_prop_list p2);;
+
+(* let rec reduce_seq s =
+ let rec not_lhs (Seq(p1s, p2s)) =
+ match p1s with
+ | [] -> Seq(p1s, p2s), None, false
+ | p::ps ->
+ (match p with
+ | Not(p') -> Seq(ps, p'::p2s), None, true
+ | _ -> let Seq(ps', p2s), _, found = not_lhs (Seq(ps, p2s)) in Seq(p::ps', p2s), None, found)
+ in
+ let rec not_rhs (Seq(p1s, p2s)) =
+ match p2s with
+ | [] -> Seq(p1s, p2s), None, false
+ | p::ps ->
+ (match p with
+ | Not(p') -> Seq(p'::p1s, ps), None, true
+ | _ -> let Seq(p1s, ps'), _, found = not_rhs (Seq(p1s, ps)) in Seq(p1s, p::ps'), None, found)
+ in
+ let rec and_lhs (Seq(p1s, p2s)) =
+ match p1s with
+ | [] -> Seq(p1s, p2s), None, false
+ | p::ps ->
+ (match p with
+ | And(p1, p2) -> Seq(p1::p2::ps, p2s), None, true
+ | _ -> let Seq(p1s, ps'), _, found = and_lhs (Seq(p1s, ps)) in Seq(p1s, p::ps'), None, found)
+ in
+ let rec and_rhs (Seq(p1s, p2s)) =
+ match p2s with
+ | [] -> Seq(p1s, p2s), None, false
+ | p::ps ->
+ (match p with
+ | And(p1, p2) -> Seq(p1s, p1::ps), Some (Seq(p1s, p2::ps)), true
+ | _ ->
+ (match and_rhs (Seq(p1s, ps)) with
+ | Seq(p1s, ps'), None, found -> Seq(p1s, p2s), None, false
+ | Seq(p1s, ps'), Some (Seq(p1s', ps'')), found -> Seq(p1s, p::ps'), Some (Seq(p1s', p::ps'')), true))
+ in
+ match s with
+ | Seq *)
+
+ (*
+ Steps (repeat in this order):
+ - reduce AND on LHS
+ - reduce OR on RHS
+ - reduce NOT on LHS
+ - reduce NOT on RHS
+ - reduce -> (RHS)
+ - reduce <-> (both)
+ - reduce OR (LHS)
+ - reduce AND (RHS)
+ - reduce -> (LHS)
+ *)
+ \ No newline at end of file
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