\r
open Pure\r
\r
+type var_flag = bool ;;\r
+\r
type var = int;;\r
type t =\r
| V of var\r
- | A of t * t\r
+ | A of var_flag * t * t\r
| L of t\r
- | B (* bottom *)\r
- | C of int\r
-;;\r
-\r
-let delta = L(A(V 0, V 0));;\r
-\r
-let eta_eq =\r
- let rec aux l1 l2 t1 t2 = match t1, t2 with\r
- | L t1, L t2 -> aux l1 l2 t1 t2\r
- | L t1, t2 -> aux l1 (l2+1) t1 t2\r
- | t1, L t2 -> aux (l1+1) l2 t1 t2\r
- | V a, V b -> a + l1 = b + l2\r
- | C a, C b -> a = b\r
- | A(t1,t2), A(u1,u2) -> aux l1 l2 t1 u1 && aux l1 l2 t2 u2\r
- | _, _ -> false\r
- in aux 0 0\r
;;\r
\r
-(* does NOT lift t *)\r
-let mk_lams = fold_nat (fun x _ -> L x) ;;\r
-\r
let string_of_t =\r
+ let sep_of_app b = if b then " +" else " " in\r
let string_of_bvar =\r
let bound_vars = ["x"; "y"; "z"; "w"; "q"] in\r
let bvarsno = List.length bound_vars in\r
- fun nn -> if nn < bvarsno then List.nth bound_vars nn else "x" ^ (string_of_int (nn - bvarsno + 1)) in\r
+ fun nn -> if nn < bvarsno then List.nth bound_vars nn else "v" ^ (string_of_int (nn - bvarsno + 1)) in\r
let rec string_of_term_w_pars level = function\r
- | V v -> if v >= level then "`" ^ string_of_int (v-level) else\r
+ | V v -> if v >= level then string_of_int (v-level) else\r
string_of_bvar (level - v-1)\r
- | C n -> "c" ^ string_of_int n\r
| A _\r
| L _ as t -> "(" ^ string_of_term_no_pars level t ^ ")"\r
- | B -> "BOT"\r
and string_of_term_no_pars_app level = function\r
- | A(t1,t2) -> string_of_term_no_pars_app level t1 ^ " " ^ string_of_term_w_pars level t2\r
+ | A(b,t1,t2) -> string_of_term_no_pars_app level t1 ^ sep_of_app b ^ string_of_term_w_pars level t2\r
| _ as t -> string_of_term_w_pars level t\r
and string_of_term_no_pars level = function\r
| L t -> "λ" ^ string_of_bvar level ^ ". " ^ string_of_term_no_pars (level+1) t\r
in string_of_term_no_pars 0\r
;;\r
\r
+(* does NOT lift the argument *)\r
+let mk_lams = fold_nat (fun x _ -> L x) ;;\r
+\r
+let measure_of_t =\r
+ let rec aux = function\r
+ | V _ -> 0\r
+ | A(b,t1,t2) ->\r
+ (if b then 1 else 0) + aux t1 + aux t2\r
+ | L t -> aux t\r
+ in aux\r
+;;\r
+\r
type problem = {\r
orig_freshno: int\r
; freshno : int\r
- ; div : t\r
- ; conv : t\r
+ ; tms : t list\r
; sigma : (var * t) list (* substitutions *)\r
- ; stepped : var list\r
- ; phase : [`One | `Two] (* :'( *)\r
}\r
\r
let string_of_problem p =\r
- let lines = [\r
- "[stepped] " ^ String.concat " " (List.map string_of_int p.stepped);\r
- "[DV] " ^ string_of_t p.div;\r
- "[CV] " ^ string_of_t p.conv;\r
- ] in\r
+ let measure = List.fold_left (+) 0 (List.map measure_of_t p.tms) in\r
+ let lines = ("[measure] " ^ string_of_int measure) ::\r
+ List.map (fun x -> "[TM] " ^ string_of_t x) p.tms in\r
String.concat "\n" lines\r
;;\r
\r
\r
let rec is_inert =\r
function\r
- | A(t,_) -> is_inert t\r
+ | A(_,t,_) -> is_inert t\r
| V _ -> true\r
- | C _\r
- | L _ | B -> false\r
+ | L _ -> false\r
;;\r
\r
let is_var = function V _ -> true | _ -> false;;\r
let is_lambda = function L _ -> true | _ -> false;;\r
\r
-let rec no_leading_lambdas = function\r
- | L t -> 1 + no_leading_lambdas t\r
- | _ -> 0\r
-;;\r
-\r
let rec get_inert = function\r
| V n -> (n,0)\r
- | A(t, _) -> let hd,args = get_inert t in hd,args+1\r
+ | A(_,t,_) -> let hd,args = get_inert t in hd,args+1\r
| _ -> assert false\r
;;\r
\r
-let rec subst level delift sub =\r
+(* precomputes the number of leading lambdas in a term,\r
+ after replacing _v_ w/ a term starting with n lambdas *)\r
+let rec no_leading_lambdas v n = function\r
+ | L t -> 1 + no_leading_lambdas (v+1) n t\r
+ | A _ as t -> let v', m = get_inert t in if v = v' then max 0 (n - m) else 0\r
+ | V v' -> if v = v' then n else 0\r
+;;\r
+\r
+let rec erase = function\r
+ | L t -> L (erase t)\r
+ | A(_,t1,t2) -> A(false, erase t1, erase t2)\r
+ | V _ as t -> t\r
+;;\r
+\r
+let explode =\r
+ let rec aux args = function\r
+ | L _ -> assert false\r
+ | V _ as x -> x, args\r
+ | A(b,t1,t2) -> aux ((b,t2)::args) t1\r
+ in aux []\r
+;;\r
+\r
+let rec implode hd args =\r
+ match args with\r
+ | [] -> hd\r
+ | (f,a)::args -> implode (A(f,hd,a)) args\r
+;;\r
+\r
+let get_head =\r
+ let rec aux lev = function\r
+ | L t -> aux (lev+1) t\r
+ | A(_,t,_) -> aux lev t\r
+ | V v -> v - lev\r
+ in aux 0\r
+;;\r
+\r
+let rec subst level delift ((var, tm) as sub) =\r
function\r
- | V v -> if v = level + fst sub then lift level (snd sub) else V (if delift && v > level then v-1 else v)\r
- | L t -> let t = subst (level + 1) delift sub t in if t = B then B else L t\r
- | A (t1,t2) ->\r
- let t1 = subst level delift sub t1 in\r
- let t2 = subst level delift sub t2 in\r
- mk_app t1 t2\r
- | C _ as t -> t\r
- | B -> B\r
-and mk_app t1 t2 = if t2 = B || (t1 = delta && t2 = delta) then B\r
- else match t1 with\r
- | C _ as t -> t\r
- | B -> B\r
+ | V v -> if v = level + var then lift level tm else V (if delift && v > level then v-1 else v)\r
+ | L t -> L (subst (level + 1) delift sub t)\r
+ | A(b,t1,t2) ->\r
+ let t1' = subst level delift sub t1 in\r
+ let t2' = subst level delift sub t2 in\r
+ mk_app b t1' t2'\r
+and mk_app flag t1 t2 = match t1 with\r
| L t1 -> subst 0 true (0, t2) t1\r
- | _ -> A (t1, t2)\r
+ | _ -> A (flag, t1, t2)\r
and lift n =\r
let rec aux lev =\r
function\r
| V m -> V (if m >= lev then m + n else m)\r
- | L t -> L (aux (lev+1) t)\r
- | A (t1, t2) -> A (aux lev t1, aux lev t2)\r
- | C _ as t -> t\r
- | B -> B\r
+ | L t -> L(aux (lev+1) t)\r
+ | A (b,t1, t2) -> A (b,aux lev t1, aux lev t2)\r
in aux 0\r
;;\r
let subst = subst 0 false;;\r
-\r
-let subst_in_problem (sub: var * t) (p: problem) =\r
-print_endline ("-- SUBST " ^ string_of_t (V (fst sub)) ^ " |-> " ^ string_of_t (snd sub));\r
- {p with\r
- div=subst sub p.div;\r
- conv=subst sub p.conv;\r
- stepped=(fst sub)::p.stepped;\r
- sigma=sub::p.sigma}\r
+(* let mk_app = mk_app true;; *)\r
+let rec mk_apps t = function\r
+ | [] -> t\r
+ | (f,x)::xs -> mk_apps (mk_app f t x) xs\r
;;\r
\r
-let get_subterm_with_head_and_args hd_var n_args =\r
- let rec aux lev = function\r
- | C _\r
- | V _ | B -> None\r
+let eta_eq =\r
+ let rec aux t1 t2 = match t1, t2 with\r
+ | L t1, L t2 -> aux t1 t2\r
+ | L t1, t2 -> aux t1 (A(false,lift 1 t2,V 0))\r
+ | t1, L t2 -> aux (A(false,lift 1 t1,V 0)) t2\r
+ | V a, V b -> a = b\r
+ | A(_,t1,t2), A(_,u1,u2) -> aux t1 u1 && aux t2 u2\r
+ | _, _ -> false\r
+ in aux ;;\r
+\r
+(* is arg1 eta-subterm of arg2 ? *)\r
+let eta_subterm u =\r
+ let rec aux lev t = eta_eq u (lift lev t) || match t with\r
| L t -> aux (lev+1) t\r
- | A(t1,t2) as t ->\r
- let hd_var', n_args' = get_inert t1 in\r
- if hd_var' = hd_var + lev && n_args <= 1 + n_args'\r
- then Some (lift ~-lev t)\r
- else match aux lev t2 with\r
- | None -> aux lev t1\r
- | Some _ as res -> res\r
+ | A(_, t1, t2) -> aux lev t1 || aux lev t2\r
+ | _ -> false\r
in aux 0\r
;;\r
\r
+let subst_in_problem ?(top=true) ((v, t) as sub) p =\r
+print_endline ("-- SUBST " ^ string_of_t (V v) ^ " |-> " ^ string_of_t t);\r
+ let sigma = sub::p.sigma in\r
+ let sub = (v, t) in\r
+ let tms = List.map (subst sub) p.tms in\r
+ {p with tms; sigma}\r
+;;\r
+\r
let rec purify = function\r
| L t -> Pure.L (purify t)\r
- | A (t1,t2) -> Pure.A (purify t1, purify t2)\r
+ | A(_,t1,t2) -> Pure.A (purify t1, purify t2)\r
| V n -> Pure.V n\r
- | C _ -> Pure.V max_int (* FIXME *)\r
- | B -> Pure.B\r
;;\r
\r
let check p sigma =\r
- print_endline "Checking...";\r
- let div = purify p.div in\r
- let conv = purify p.conv in\r
+ assert false (* FIXME *)\r
+ (* print_endline "Checking...";\r
+ let tms = List.map purify p.tms in\r
let sigma = List.map (fun (v,t) -> v, purify t) sigma in\r
- let freshno = List.fold_right (fun (x,_) -> max x) sigma 0 in\r
+ let freshno = List.fold_right (max ++ fst) sigma 0 in\r
let env = Pure.env_of_sigma freshno sigma in\r
assert (Pure.diverged (Pure.mwhd (env,div,[])));\r
print_endline " D diverged.";\r
assert (not (Pure.diverged (Pure.mwhd (env,conv,[]))));\r
print_endline " C converged.";\r
- ()\r
+ () *)\r
;;\r
\r
let sanity p =\r
print_endline (string_of_problem p); (* non cancellare *)\r
- if p.conv = B then problem_fail p "p.conv diverged";\r
- if p.div = B then raise (Done p.sigma);\r
- if p.phase = `Two && p.div = delta then raise (Done p.sigma);\r
- if not (is_inert p.div) then problem_fail p "p.div converged"\r
-;;\r
-\r
-(* drops the arguments of t after the n-th *)\r
-let inert_cut_at n t =\r
- let rec aux t =\r
- match t with\r
- | V _ as t -> 0, t\r
- | A(t1,_) as t ->\r
- let k', t' = aux t1 in\r
- if k' = n then n, t'\r
- else k'+1, t\r
- | _ -> assert false\r
- in snd (aux t)\r
-;;\r
-\r
-let find_eta_difference p t n_args =\r
- let t = inert_cut_at n_args t in\r
- let rec aux t u k = match t, u with\r
- | V _, V _ -> assert false (* div subterm of conv *)\r
- | A(t1,t2), A(u1,u2) ->\r
- if not (eta_eq t2 u2) then (print_endline((string_of_t t2) ^ " <> " ^ (string_of_t u2)); k)\r
- else aux t1 u1 (k-1)\r
- | _, _ -> assert false\r
- in aux p.div t n_args\r
-;;\r
-\r
-let compute_max_lambdas_at hd_var j =\r
- let rec aux hd = function\r
- | A(t1,t2) ->\r
- (if get_inert t1 = (hd, j)\r
- then max ( (*FIXME*)\r
- if is_inert t2 && let hd', j' = get_inert t2 in hd' = hd\r
- then let hd', j' = get_inert t2 in j - j'\r
- else no_leading_lambdas t2)\r
- else id) (max (aux hd t1) (aux hd t2))\r
- | L t -> aux (hd+1) t\r
- | V _ -> 0\r
- | _ -> assert false\r
- in aux hd_var\r
+ let rec all_different = function\r
+ | [] -> true\r
+ | x::xs -> List.for_all ((<>) x) xs && all_different xs in\r
+ if List.for_all is_var p.tms && all_different p.tms\r
+ then raise (Done p.sigma);\r
+ if List.exists (not ++ is_inert) p.tms\r
+ then problem_fail p "used a non-effective path";\r
+ p\r
;;\r
\r
let print_cmd s1 s2 = print_endline (">> " ^ s1 ^ " " ^ s2);;\r
\r
-(* eat the arguments of the divergent and explode.\r
- It does NOT perform any check, may fail if done unsafely *)\r
-let eat p =\r
-print_cmd "EAT" "";\r
- let var, k = get_inert p.div in\r
- let phase = p.phase in\r
- let p, t =\r
- match phase with\r
- | `One ->\r
- let n = 1 + max\r
- (compute_max_lambdas_at var k p.div)\r
- (compute_max_lambdas_at var k p.conv) in\r
- (* apply fresh vars *)\r
- let p, t = fold_nat (fun (p, t) _ ->\r
- let p, v = freshvar p in\r
- p, A(t, V (v + k))\r
- ) (p, V 0) n in\r
- let p = {p with phase=`Two} in p, A(t, delta)\r
- | `Two -> p, delta in\r
- let subst = var, mk_lams t k in\r
- let p = subst_in_problem subst p in\r
- sanity p;\r
- let p = if phase = `One then {p with div = (match p.div with A(t,_) -> t | _ -> assert false)} else p in\r
- sanity p; p\r
+let step var j n p =\r
+ let atsnd f (a,b) = (a, f b) in\r
+ let p, alphas = (* make fresh vars *)\r
+ fold_nat (fun (p, vs) _ ->\r
+ let p, v = freshvar p in\r
+ p, v::vs\r
+ ) (p, []) n in let alphas = List.rev alphas in\r
+ let rec aux lev (inside:bool) = function\r
+ | L t -> L (aux (lev+1) inside t)\r
+ | _ as x ->\r
+ let hd, args = explode x in\r
+ if hd = V (var+lev) then\r
+ (let nargs = List.length args in\r
+ let k = max 0 (j + 1 - nargs) in\r
+ let args = List.mapi\r
+ (fun i (f, t) -> f, lift k (aux lev (if i=j then true else inside) t)) args in\r
+ let bound = fold_nat (fun x n -> (false,V(n-1)) :: x) [] k in\r
+ let args = args @ bound in\r
+ let _, head = List.nth args j in\r
+ let args = List.mapi\r
+ (fun i (f, t) -> (if i=j && not inside then false else f), if i=j && not inside then erase t else t) args in\r
+ let head = (if inside then erase else id) head in\r
+ print_endline ("HEAD: " ^ string_of_t head);\r
+ let alphas = List.map (fun v -> false, V(lev+k+v)) alphas in\r
+ let t = mk_apps head (alphas @ args) in\r
+ let t = mk_lams t k in\r
+ t\r
+ ) else\r
+ (let args = List.map (atsnd (aux lev inside)) args in\r
+ implode hd args) in\r
+ let sigma = (var, aux 0 false (V var)) :: p.sigma in\r
+ {p with tms=List.map (aux 0 false) p.tms; sigma}\r
;;\r
\r
-(* step on the head of div, on the k-th argument, with n fresh vars *)\r
-let step k n p =\r
- let var, _ = get_inert p.div in\r
-print_cmd "STEP" ("on " ^ string_of_t (V var) ^ " (of:" ^ string_of_int n ^ ")");\r
- let p, t = (* apply fresh vars *)\r
- fold_nat (fun (p, t) _ ->\r
- let p, v = freshvar p in\r
- p, A(t, V (v + k + 1))\r
- ) (p, V 0) n in\r
- let t = (* apply unused bound variables V_{k-1}..V_1 *)\r
- fold_nat (fun t m -> A(t, V (k-m+1))) t k in\r
- let t = mk_lams t (k+1) in (* make leading lambdas *)\r
- let subst = var, t in\r
- let p = subst_in_problem subst p in\r
- sanity p; p\r
-;;\r
+let finish p = assert false ;;\r
\r
-let parse strs =\r
- let rec aux level = function\r
- | Parser_andrea.Lam t -> L (aux (level + 1) t)\r
- | Parser_andrea.App (t1, t2) ->\r
- if level = 0 then mk_app (aux level t1) (aux level t2)\r
- else A(aux level t1, aux level t2)\r
- | Parser_andrea.Var v -> V v in\r
- let (tms, free) = Parser_andrea.parse_many strs in\r
- (List.map (aux 0) tms, free)\r
-;;\r
+let rec auto p = assert false ;;\r
\r
-let problem_of div conv =\r
+let problem_of (label, div, convs, ps, var_names) =\r
print_hline ();\r
- let [@warning "-8"] [div; conv], var_names = parse ([div; conv]) in\r
+ let rec aux = function\r
+ | `Lam(_,t) -> L (aux t)\r
+ | `I ((v,_), args) -> Listx.fold_left (fun x y -> mk_app true x (aux y)) (V v) args\r
+ | `Var(v,_) -> V v\r
+ | `N _ | `Match _ -> assert false in\r
+ let convs = (List.rev convs :> Num.nf list) in\r
+ let tms = List.map aux (convs @ (ps :> Num.nf list)) in\r
+ let tms = match div with\r
+ | Some div -> aux (div :> Num.nf) :: tms\r
+ | None -> tms in\r
let varno = List.length var_names in\r
- let p = {orig_freshno=varno; freshno=1+varno; div; conv; sigma=[]; stepped=[]; phase=`One} in\r
+ let p = {orig_freshno=varno; freshno=1+varno; tms; sigma=[]} in\r
(* initial sanity check *)\r
- sanity p; p\r
-;;\r
-\r
-let exec div conv cmds =\r
- let p = problem_of div conv in\r
- try\r
- problem_fail (List.fold_left (|>) p cmds) "Problem not completed"\r
- with\r
- | Done _ -> ()\r
+ sanity p\r
;;\r
\r
-let rec auto p =\r
- let hd_var, n_args = get_inert p.div in\r
- match get_subterm_with_head_and_args hd_var n_args p.conv with\r
- | None ->\r
- (try\r
- let phase = p.phase in\r
- let p = eat p in\r
- if phase = `Two\r
- then problem_fail p "Auto.2 did not complete the problem"\r
- else auto p\r
- with Done sigma -> sigma)\r
- | Some t ->\r
- let j = find_eta_difference p t n_args - 1 in\r
- let k = 1 + max\r
- (compute_max_lambdas_at hd_var j p.div)\r
- (compute_max_lambdas_at hd_var j p.conv) in\r
- let p = step j k p in\r
- auto p\r
+let rec interactive p =\r
+ print_string "[varno index alphano] ";\r
+ let s = read_line () in\r
+ let spl = Str.split (Str.regexp " +") s in\r
+ let nth n = int_of_string (List.nth spl n) in\r
+ let p = step (nth 0) (nth 1) (nth 2) p in\r
+ interactive (sanity p)\r
;;\r
\r
-let interactive div conv cmds =\r
- let p = problem_of div conv in\r
- try (\r
- let p = List.fold_left (|>) p cmds in\r
- let rec f p cmds =\r
- let nth spl n = int_of_string (List.nth spl n) in\r
- let read_cmd () =\r
- let s = read_line () in\r
- let spl = Str.split (Str.regexp " +") s in\r
- s, let uno = List.hd spl in\r
- try if uno = "eat" then eat\r
- else if uno = "step" then step (nth spl 1) (nth spl 2)\r
- else failwith "Wrong input."\r
- with Failure s -> print_endline s; (fun x -> x) in\r
- let str, cmd = read_cmd () in\r
- let cmds = (" " ^ str ^ ";")::cmds in\r
- try\r
- let p = cmd p in f p cmds\r
- with\r
- | Done _ -> print_endline "Done! Commands history: "; List.iter print_endline (List.rev cmds)\r
- in f p []\r
- ) with Done _ -> ()\r
+let solve p =\r
+ let rec aux = function\r
+ | [] -> false\r
+ | x::xs -> List.exists (eta_subterm x) xs || aux xs in\r
+ if aux p.tms\r
+ then print_endline "!!! Problem stopped: subterm problem !!!"\r
+ else check p (interactive p)\r
;;\r
\r
-let rec conv_join = function\r
- | [] -> "@"\r
- | x::xs -> conv_join xs ^ " ("^ x ^")"\r
-;;\r
-\r
-let auto' a b =\r
- let p = problem_of a (conv_join b) in\r
- let sigma = auto p in\r
- check p sigma\r
-;;\r
-\r
-(* Example usage of exec, interactive:\r
-\r
-exec\r
- "x x"\r
- (conv_join["x y"; "y y"; "y x"])\r
- [ step 0 1; eat ]\r
-;;\r
-\r
-interactive "x y"\r
- "@ (x x) (y x) (y z)" [step 0 1; step 0 2; eat]\r
-;;\r
-\r
-*)\r
-\r
-auto' "x x" ["x y"; "y y"; "y x"] ;;\r
-auto' "x y" ["x (_. x)"; "y z"; "y x"] ;;\r
-auto' "a (x. x b) (x. x c)" ["a (x. b b) @"; "a @ c"; "a (x. x x) a"; "a (a a a) (a c c)"] ;;\r
-\r
-auto' "x (y. x y y)" ["x (y. x y x)"] ;;\r
-\r
-auto' "x a a a a" [\r
- "x b a a a";\r
- "x a b a a";\r
- "x a a b a";\r
- "x a a a b";\r
-] ;;\r
-\r
-(* Controesempio ad usare un conto dei lambda che non considere le permutazioni *)\r
-auto' "x a a a a (x (x. x x) @ @ (_._.x. x x) x) b b b" [\r
- "x a a a a (_. a) b b b";\r
- "x a a a a (_. _. _. _. x. y. x y)";\r
-] ;;\r
-\r
-\r
-print_hline();\r
-print_endline "ALL DONE. "\r
-\r
-let solve = auto';;\r
+Problems.main (solve ++ problem_of);\r
projects / fireball-separation.git / blobdiff