\r
open Pure\r
\r
+type var_flag = [\r
+ `Inherit | `Some of bool ref\r
+ (* bool:\r
+ true if original application and may determine a distinction\r
+ *)\r
+ | `Duplicate\r
+] ;;\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 ;;\r
-let eta_eq = eta_eq' 0 0;;\r
+let index_of x =\r
+ let rec aux n =\r
+ function\r
+ [] -> None\r
+ | x'::_ when x == x' -> Some n\r
+ | _::xs -> aux (n+1) xs\r
+ in aux 1\r
+;;\r
\r
-(* is arg1 eta-subterm of arg2 ? *)\r
-let eta_subterm u =\r
- let rec aux lev t = eta_eq' lev 0 u t || match t with\r
- | L t -> aux (lev+1) t\r
- | A(t1, t2) -> aux lev t1 || aux lev t2\r
- | _ -> false\r
- in aux 0\r
+let sep_of_app =\r
+ let apps = ref [] in\r
+ function\r
+ r when not !r -> " "\r
+ | r ->\r
+ let i =\r
+ match index_of r !apps with\r
+ Some i -> i\r
+ | None ->\r
+ apps := !apps @ [r];\r
+ List.length !apps\r
+ in " " ^ string_of_int i ^ ":"\r
;;\r
+let string_of_var_flag = function\r
+ | `Some b -> sep_of_app b\r
+ | `Inherit -> " ?"\r
+ | `Duplicate -> " !"\r
+ ;;\r
\r
-(* does NOT lift the argument *)\r
-let mk_lams = fold_nat (fun x _ -> L x) ;;\r
\r
let string_of_t =\r
let string_of_bvar =\r
let rec string_of_term_w_pars level = function\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 ^ string_of_var_flag 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
+\r
+let delta = L(A(`Some (ref true),V 0, V 0));;\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 acc = function\r
+ | V _ -> acc, 0\r
+ | A(b,t1,t2) ->\r
+ let acc, m1 = aux acc t1 in\r
+ let acc, m2 = aux acc t2 in\r
+ (match b with\r
+ | `Some b when !b && not (List.memq b acc) -> b::acc, 1 + m1 + m2\r
+ | _ -> acc, m1 + m2)\r
+ | L t -> aux acc t\r
+ in snd ++ (aux [])\r
+;;\r
+\r
type problem = {\r
orig_freshno: int\r
; freshno : int\r
; div : t\r
; conv : t\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
+ "[measure] " ^ string_of_int (measure_of_t p.div);\r
"[DV] " ^ string_of_t p.div;\r
"[CV] " ^ string_of_t p.conv;\r
] in\r
String.concat "\n" lines\r
;;\r
\r
+exception B;;\r
exception Done of (var * t) list (* substitution *);;\r
exception Fail of int * string;;\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
\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 no_leading_lambdas hd_var j = function\r
- | L t -> 1 + no_leading_lambdas (hd_var+1) j t\r
- | A _ as t -> let hd_var', n = get_inert t in if hd_var = hd_var' then max 0 (j - n) else 0\r
- | V n -> if n = hd_var then j else 0\r
- | B | C _ -> 0\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(`Some(ref false), erase t1, erase t2)\r
+ | V _ as t -> t\r
;;\r
\r
-let rec subst level delift sub =\r
+let rec subst top level delift ((flag, 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
+ | 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 top (level + 1) delift sub t)\r
+ | A(b,t1,t2) ->\r
+ let special = b = `Duplicate && top && t2 = V (level + var) in\r
+ let t1' = subst (if special then false else top) level delift sub t1 in\r
+ let t2' = subst false level delift sub t2 in\r
+ match b with\r
+ | `Duplicate when special ->\r
+ assert (match t1' with L _ -> false | _ -> true) ;\r
+ (match flag with\r
+ | `Some b when !b -> b := false\r
+ | `Some b -> ()\r
+ (*print_string "WARNING! Stepping on a useless argument!";\r
+ ignore(read_line())*)\r
+ | `Inherit | `Duplicate -> assert false);\r
+ A(flag, t1', erase t2')\r
+ | `Inherit | `Duplicate ->\r
+ let b' = if t2 = V (level + var)\r
+ then (assert (flag <> `Inherit); flag)\r
+ else b in\r
+ assert (match t1' with L _ -> false | _ -> true) ;\r
+ A(b', t1', t2')\r
+ | `Some b' -> mk_app top b' t1' t2'\r
+and mk_app top flag t1 t2 = if t1 = delta && t2 = delta then raise B\r
else match t1 with\r
- | B -> B\r
- | L t1 -> subst 0 true (0, t2) t1\r
- | _ -> A (t1, t2)\r
+ | L t1 -> subst top 0 true (`Some flag, 0, t2) t1\r
+ | _ -> A (`Some 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 p =\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 subst top = subst top 0 false;;\r
+let mk_app = mk_app true;;\r
+\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(`Some (ref true),lift 1 t2,V 0))\r
+ | t1, L t2 -> aux (A(`Some (ref true),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) -> 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 = (`Inherit, v, t) in\r
+ let div = try subst top sub p.div with B -> raise (Done sigma) in\r
+ let conv = try subst false sub p.conv with B -> raise (Fail(-1,"p.conv diverged")) in\r
+ {p with div; conv; sigma}\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
+ | V _ -> None\r
| L t -> aux (lev+1) t\r
- | A(t1,t2) as 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
+ (* the `+1` above is because of t2 *)\r
then Some (lift ~-lev t)\r
else match aux lev t2 with\r
| None -> aux lev t1\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
let div = purify p.div in\r
let conv = purify p.conv 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
\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
p\r
;;\r
\r
(* drops the arguments of t after the n-th *)\r
+(* FIXME! E' usato in modo improprio contando sul fatto\r
+ errato che ritorna un inerte lungo esattamente n *)\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
+ | A(_,t1,_) as t ->\r
let k', t' = aux t1 in\r
if k' = n then n, t'\r
else k'+1, t\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
+(* return the index of the first argument with a difference\r
+ (the first argument is 0)\r
+ precondition: p.div and t have n+1 arguments\r
+ *)\r
+let find_eta_difference p t argsno =\r
+ let t = inert_cut_at argsno 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
+ | V _, V _ -> None\r
+ | A(b1,t1,t2), A(b2,u1,u2) ->\r
+ (match aux t1 u1 (k-1) with\r
+ | None ->\r
+ if not (eta_eq t2 u2) then begin\r
+ let is_relevant = function `Some b -> !b | _ -> false in\r
+ if not (is_relevant b1 || is_relevant b2) then begin\r
+ print_string "WARNING! Stepping on a useless argument!";\r
+print_string (string_of_t t ^ " <==> " ^ string_of_t u);\r
+ ignore(read_line())\r
+ end ;\r
+ Some (k-1)\r
+ end\r
+ else None\r
+ | Some j -> Some j)\r
| _, _ -> assert false\r
- in aux p.div t n_args\r
+ in match aux p.div t argsno with\r
+ | None -> problem_fail p "no eta difference found (div subterm of conv?)"\r
+ | Some j -> j\r
;;\r
\r
let compute_max_lambdas_at hd_var j =\r
let rec aux hd = function\r
- | A(t1,t2) ->\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
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
;;\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 =\r
- match phase with\r
- | `One ->\r
- let n = 1 + max\r
- (compute_max_lambdas_at var (k-1) p.div)\r
- (compute_max_lambdas_at var (k-1) 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\r
- let t = A(t, delta) in\r
- let t = fold_nat (fun t m -> A(t, V (k-m))) t (k-1) in\r
- let subst = var, mk_lams t k in\r
- let p = subst_in_problem subst p in\r
- let _, args = get_inert p.div in\r
- {p with div = inert_cut_at (args-k) p.div}\r
- | `Two ->\r
- let subst = var, mk_lams delta k in\r
- subst_in_problem subst p in\r
- sanity p\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 step ?(isfinish=false) 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, A(`Some (ref false), 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 = (* apply bound variables V_k..V_0 *)\r
+ fold_nat (fun t m -> A((if m = k+1 then `Duplicate else `Inherit), t, V (k-m+1))) t (k+1) 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
+ let p = subst_in_problem ~top:(not isfinish) subst p in\r
sanity p\r
;;\r
\r
+let finish p =\r
+ let compute_max_arity =\r
+ let rec aux n = function\r
+ | A(_,t1,t2) -> max (aux (n+1) t1) (aux 0 t2)\r
+ | L t -> max n (aux 0 t)\r
+ | V _ -> n\r
+ in aux 0 in\r
+print_cmd "FINISH" "";\r
+ let div_hd, div_nargs = get_inert p.div in\r
+ let j = div_nargs - 1 in\r
+ let arity = compute_max_arity p.conv in\r
+ let n = 1 + arity + max\r
+ (compute_max_lambdas_at div_hd j p.div)\r
+ (compute_max_lambdas_at div_hd j p.conv) in\r
+ let p = step ~isfinish:true j n p in\r
+ let div_hd, div_nargs = get_inert p.div in\r
+ let rec aux m = function\r
+ A(_,t1,t2) -> if is_var t2 then\r
+ (let delta_var, _ = get_inert t2 in\r
+ if delta_var <> div_hd && get_subterm_with_head_and_args delta_var 1 p.conv = None\r
+ then m, delta_var\r
+ else aux (m-1) t1) else aux (m-1) t1\r
+ | _ -> assert false in\r
+ let m, delta_var = aux div_nargs p.div in\r
+ let p = subst_in_problem (delta_var, delta) p in\r
+ let p = subst_in_problem (div_hd, mk_lams delta (m-1)) p in\r
+ sanity p\r
;;\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 problem_fail (finish p) "Auto.2 did not complete the problem"\r
+ with Done sigma -> sigma)\r
+ (*\r
(try\r
let phase = p.phase in\r
let p = eat p in\r
then problem_fail p "Auto.2 did not complete the problem"\r
else auto p\r
with Done sigma -> sigma)\r
+ *)\r
| Some t ->\r
- let j = find_eta_difference p t n_args - 1 in\r
+ let j = find_eta_difference p t n_args 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 m1 = measure_of_t p.div in\r
let p = step j k p in\r
+ let m2 = measure_of_t p.div in\r
+ (if m2 >= m1 then\r
+ (print_string ("WARNING! Measure did not decrease : " ^ string_of_int m2 ^ " >= " ^ string_of_int m1 ^ " (press <Enter>)");\r
+ ignore(read_line())));\r
auto p\r
;;\r
\r
let problem_of (label, div, convs, ps, var_names) =\r
print_hline ();\r
let rec aux = function\r
- | `Lam(_, t) -> L (aux t)\r
- | `I ((v,_), args) -> Listx.fold_left (fun x y -> mk_app x (aux y)) (V v) args\r
+ | `Lam(_,t) -> L (aux t)\r
+ | `I ((v,_), args) -> Listx.fold_left (fun x y -> mk_app (ref true) x (aux y)) (V v) args\r
| `Var(v,_) -> V v\r
| `N _ | `Match _ -> assert false in\r
assert (List.length ps = 0);\r
- let convs = List.rev convs in\r
- let conv = List.fold_left (fun x y -> mk_app x (aux (y :> Num.nf))) (V (List.length var_names)) convs in\r
+ let convs = (List.rev convs :> Num.nf list) in\r
+ let conv = aux\r
+ (if List.length convs = 1\r
+ then List.hd convs\r
+ else `I((List.length var_names, min_int), Listx.from_list convs)) in\r
let var_names = "@" :: var_names in\r
let div = match div with\r
| Some div -> aux (div :> Num.nf)\r
| None -> assert false 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; div; conv; sigma=[]; phase=`One} in\r
(* initial sanity check *)\r
sanity p\r
;;\r
projects / fireball-separation.git / blobdiff