From 93e4a7b5d3d56c1bc480e554116bd5c344971ef2 Mon Sep 17 00:00:00 2001
From: acondolu <andrea.condoluci@unibo.it>
Date: Tue, 18 Sep 2018 16:26:08 +0200
Subject: [PATCH] New measure which looks promising

- idea of nesting in paths
- relies heavily on special k
  (behavior still not explained)
- interactive mode to experiment with stepping

Missing:
- auto
- finish
- check / purification
---
 ocaml/simple.ml | 410 ++++++++++++++----------------------------------
 1 file changed, 122 insertions(+), 288 deletions(-)

diff --git a/ocaml/simple.ml b/ocaml/simple.ml
index b1b05ec..b76a654 100644
--- a/ocaml/simple.ml
+++ b/ocaml/simple.ml
@@ -6,13 +6,7 @@ let print_hline = Console.print_hline;;
 
 open Pure
 
-type var_flag = [
- `Inherit | `Some of bool ref
- (* bool:
-     true if original application and may determine a distinction
-  *)
-  | `Duplicate
-] ;;
+type var_flag = bool ;;
 
 type var = int;;
 type t =
@@ -21,47 +15,19 @@ type t =
  | L of t
 ;;
 
-let index_of x =
- let rec aux n =
-  function
-    [] -> None
-  | x'::_ when x == x' -> Some n
-  | _::xs -> aux (n+1) xs
- in aux 1
-;;
-
-let sep_of_app =
- let apps = ref [] in
- function
-    r when not !r -> " "
-  | r ->
-    let i =
-     match index_of r !apps with
-        Some i -> i
-      | None ->
-         apps := !apps @ [r];
-         List.length !apps
-    in " " ^ string_of_int i ^ ":"
-;;
-let string_of_var_flag = function
- | `Some b -> sep_of_app b
- | `Inherit -> " ?"
- | `Duplicate -> " !"
- ;;
-
-
 let string_of_t =
+  let sep_of_app b = if b then " +" else " " in
   let string_of_bvar =
    let bound_vars = ["x"; "y"; "z"; "w"; "q"] in
    let bvarsno = List.length bound_vars in
-   fun nn -> if nn < bvarsno then List.nth bound_vars nn else "x" ^ (string_of_int (nn - bvarsno + 1)) in
+   fun nn -> if nn < bvarsno then List.nth bound_vars nn else "v" ^ (string_of_int (nn - bvarsno + 1)) in
   let rec string_of_term_w_pars level = function
-    | V v -> if v >= level then "`" ^ string_of_int (v-level) else
+    | V v -> if v >= level then string_of_int (v-level) else
        string_of_bvar (level - v-1)
     | A _
     | L _ as t -> "(" ^ string_of_term_no_pars level t ^ ")"
   and string_of_term_no_pars_app level = function
-    | A(b,t1,t2) -> string_of_term_no_pars_app level t1 ^ string_of_var_flag b ^ string_of_term_w_pars level t2
+    | A(b,t1,t2) -> string_of_term_no_pars_app level t1 ^ sep_of_app b ^ string_of_term_w_pars level t2
     | _ as t -> string_of_term_w_pars level t
   and string_of_term_no_pars level = function
     | L t -> "λ" ^ string_of_bvar level ^ ". " ^ string_of_term_no_pars (level+1) t
@@ -69,44 +35,32 @@ let string_of_t =
   in string_of_term_no_pars 0
 ;;
 
-
-let delta = L(A(`Some (ref true),V 0, V 0));;
-
 (* does NOT lift the argument *)
 let mk_lams = fold_nat (fun x _ -> L x) ;;
 
 let measure_of_t =
- let rec aux acc = function
- | V _ -> acc, 0
+ let rec aux = function
+ | V _ -> 0
  | A(b,t1,t2) ->
-   let acc, m1 = aux acc t1 in
-   let acc, m2 = aux acc t2 in
-   (match b with
-   | `Some b when !b && not (List.memq b acc) -> b::acc, 1 + m1 + m2
-   | _ -> acc, m1 + m2)
- | L t -> aux acc t
- in snd ++ (aux [])
+   (if b then 1 else 0) + aux t1 + aux t2
+ | L t -> aux t
+ in aux
 ;;
 
 type problem = {
    orig_freshno: int
  ; freshno : int
- ; div : t
- ; conv : t
+ ; tms : t list
  ; sigma : (var * t) list (* substitutions *)
- ; phase : [`One | `Two] (* :'( *)
 }
 
 let string_of_problem p =
- let lines = [
-  "[measure] " ^ string_of_int (measure_of_t p.div);
-  "[DV] " ^ string_of_t p.div;
-  "[CV] " ^ string_of_t p.conv;
- ] in
+ let measure = List.fold_left (+) 0 (List.map measure_of_t p.tms) in
+ let lines = ("[measure] " ^ string_of_int measure) ::
+   List.map (fun x -> "[TM] " ^ string_of_t x) p.tms in
  String.concat "\n" lines
 ;;
 
-exception B;;
 exception Done of (var * t) list (* substitution *);;
 exception Fail of int * string;;
 
@@ -146,39 +100,43 @@ let rec no_leading_lambdas v n = function
 
 let rec erase = function
  | L t -> L (erase t)
- | A(_,t1,t2) -> A(`Some(ref false), erase t1, erase t2)
+ | A(_,t1,t2) -> A(false, erase t1, erase t2)
  | V _ as t -> t
 ;;
 
-let rec subst top level delift ((flag, var, tm) as sub) =
+let explode =
+ let rec aux args = function
+ | L _ -> assert false
+ | V _ as x -> x, args
+ | A(b,t1,t2) -> aux ((b,t2)::args) t1
+ in aux []
+;;
+
+let rec implode hd args =
+ match args with
+ | [] -> hd
+ | (f,a)::args -> implode (A(f,hd,a)) args
+;;
+
+let get_head =
+ let rec aux lev = function
+ | L t -> aux (lev+1) t
+ | A(_,t,_) -> aux lev t
+ | V v -> v - lev
+ in aux 0
+;;
+
+let rec subst level delift ((var, tm) as sub) =
  function
  | V v -> if v = level + var then lift level tm else V (if delift && v > level then v-1 else v)
- | L t -> L (subst top (level + 1) delift sub t)
+ | L t -> L (subst (level + 1) delift sub t)
  | A(b,t1,t2) ->
-    let special = b = `Duplicate && top && t2 = V (level + var) in
-    let t1' = subst (if special then false else top) level delift sub t1 in
-    let t2' = subst false level delift sub t2 in
-    match b with
-    | `Duplicate when special ->
-        assert (match t1' with L _ -> false | _ -> true) ;
-        (match flag with
-          | `Some b when !b -> b := false
-          | `Some b -> ()
-            (*print_string "WARNING! Stepping on a useless argument!";
-            ignore(read_line())*)
-          | `Inherit | `Duplicate -> assert false);
-        A(flag, t1', erase t2')
-    | `Inherit | `Duplicate ->
-       let b' = if t2 = V (level + var)
-        then (assert (flag <> `Inherit); flag)
-        else b in
-       assert (match t1' with L _ -> false | _ -> true) ;
-       A(b', t1', t2')
-    | `Some b' -> mk_app top b' t1' t2'
-and mk_app top flag t1 t2 = if t1 = delta && t2 = delta then raise B
- else match t1 with
- | L t1 -> subst top 0 true (`Some flag, 0, t2) t1
- | _ -> A (`Some flag, t1, t2)
+    let t1' = subst level delift sub t1 in
+    let t2' = subst level delift sub t2 in
+    mk_app b t1' t2'
+and mk_app flag t1 t2 = match t1 with
+ | L t1 -> subst 0 true (0, t2) t1
+ | _ -> A (flag, t1, t2)
 and lift n =
  let rec aux lev =
   function
@@ -187,14 +145,18 @@ and lift n =
   | A (b,t1, t2) -> A (b,aux lev t1, aux lev t2)
  in aux 0
 ;;
-let subst top = subst top 0 false;;
-let mk_app = mk_app true;;
+let subst = subst 0 false;;
+(* let mk_app = mk_app true;; *)
+let rec mk_apps t = function
+ | [] -> t
+ | (f,x)::xs -> mk_apps (mk_app f t x) xs
+;;
 
 let eta_eq =
  let rec aux t1 t2 = match t1, t2 with
   | L t1, L t2 -> aux t1 t2
-  | L t1, t2 -> aux t1 (A(`Some (ref true),lift 1 t2,V 0))
-  | t1, L t2 -> aux (A(`Some (ref true),lift 1 t1,V 0)) t2
+  | L t1, t2 -> aux t1 (A(false,lift 1 t2,V 0))
+  | t1, L t2 -> aux (A(false,lift 1 t1,V 0)) t2
   | V a, V b -> a = b
   | A(_,t1,t2), A(_,u1,u2) -> aux t1 u1 && aux t2 u2
   | _, _ -> false
@@ -212,25 +174,9 @@ let eta_subterm u =
 let subst_in_problem ?(top=true) ((v, t) as sub) p =
 print_endline ("-- SUBST " ^ string_of_t (V v) ^ " |-> " ^ string_of_t t);
  let sigma = sub::p.sigma in
- let sub = (`Inherit, v, t) in
- let div = try subst top sub p.div with B -> raise (Done sigma) in
- let conv = try subst false sub p.conv with B -> raise (Fail(-1,"p.conv diverged")) in
- {p with div; conv; sigma}
-;;
-
-let get_subterm_with_head_and_args hd_var n_args =
- let rec aux lev = function
- | V _ -> None
- | L t -> aux (lev+1) t
- | A(_,t1,t2) as t ->
-   let hd_var', n_args' = get_inert t1 in
-   if hd_var' = hd_var + lev && n_args <= 1 + n_args'
-    (* the `+1` above is because of t2 *)
-    then Some (lift ~-lev t)
-    else match aux lev t2 with
-    | None -> aux lev t1
-    | Some _ as res -> res
- in aux 0
+ let sub = (v, t) in
+ let tms = List.map (subst sub) p.tms in
+ {p with tms; sigma}
 ;;
 
 let rec purify = function
@@ -240,9 +186,9 @@ let rec purify = function
 ;;
 
 let check p sigma =
- print_endline "Checking...";
- let div = purify p.div in
- let conv = purify p.conv in
+ assert false (* FIXME *)
+ (* print_endline "Checking...";
+ let tms = List.map purify p.tms in
  let sigma = List.map (fun (v,t) -> v, purify t) sigma in
  let freshno = List.fold_right (max ++ fst) sigma 0 in
  let env = Pure.env_of_sigma freshno sigma in
@@ -250,207 +196,95 @@ let check p sigma =
  print_endline " D diverged.";
  assert (not (Pure.diverged (Pure.mwhd (env,conv,[]))));
  print_endline " C converged.";
- ()
+ () *)
 ;;
 
 let sanity p =
  print_endline (string_of_problem p); (* non cancellare *)
- if p.phase = `Two && p.div = delta then raise (Done p.sigma);
- if not (is_inert p.div) then problem_fail p "p.div converged";
+ let rec all_different = function
+  | [] -> true
+  | x::xs -> List.for_all ((<>) x) xs && all_different xs in
+ if List.for_all is_var p.tms && all_different p.tms
+  then raise (Done p.sigma);
+ if List.exists (not ++ is_inert) p.tms
+  then problem_fail p "used a non-effective path";
  p
 ;;
 
-(* drops the arguments of t after the n-th *)
-(* FIXME! E' usato in modo improprio contando sul fatto
-   errato che ritorna un inerte lungo esattamente n *)
-let inert_cut_at n t =
- let rec aux t =
-  match t with
-  | V _ as t -> 0, t
-  | A(_,t1,_) as t ->
-    let k', t' = aux t1 in
-     if k' = n then n, t'
-      else k'+1, t
-  | _ -> assert false
- in snd (aux t)
-;;
-
-(* return the index of the first argument with a difference
-   (the first argument is 0)
-   precondition: p.div and t have n+1 arguments
-   *)
-let find_eta_difference p t argsno =
- let t = inert_cut_at argsno t in
- let rec aux t u k = match t, u with
- | V _, V _ -> None
- | A(b1,t1,t2), A(b2,u1,u2) ->
-    (match aux t1 u1 (k-1) with
-    | None ->
-      if not (eta_eq t2 u2) then begin
-        let is_relevant = function `Some b -> !b | _ -> false in
-        if not (is_relevant b1 || is_relevant b2) then begin
-         print_string "WARNING! Stepping on a useless argument!";
-print_string (string_of_t  t ^ " <==> " ^ string_of_t u);
-         ignore(read_line())
-        end ;
-        Some (k-1)
-       end
-      else None
-    | Some j -> Some j)
- | _, _ -> assert false
- in match aux p.div t argsno with
- | None -> problem_fail p "no eta difference found (div subterm of conv?)"
- | Some j -> j
-;;
-
-let compute_max_lambdas_at hd_var j =
- let rec aux hd = function
- | A(_,t1,t2) ->
-    (if get_inert t1 = (hd, j)
-      then max ( (*FIXME*)
-       if is_inert t2 && let hd', j' = get_inert t2 in hd' = hd
-        then let hd', j' = get_inert t2 in j - j'
-        else no_leading_lambdas hd_var j t2)
-      else id) (max (aux hd t1) (aux hd t2))
- | L t -> aux (hd+1) t
- | V _ -> 0
- in aux hd_var
-;;
-
 let print_cmd s1 s2 = print_endline (">> " ^ s1 ^ " " ^ s2);;
 
-(* step on the head of div, on the k-th argument, with n fresh vars *)
-let step ?(isfinish=false) k n p =
- let var, _ = get_inert p.div in
-print_cmd "STEP" ("on " ^ string_of_t (V var) ^ " (of:" ^ string_of_int n ^ ")");
- let p, t = (* apply fresh vars *)
-  fold_nat (fun (p, t) _ ->
-    let p, v = freshvar p in
-    p, A(`Some (ref false), t, V (v + k + 1))
-  ) (p, V 0) n in
- let t = (* apply bound variables V_k..V_0 *)
-  fold_nat (fun t m -> A((if m = k+1 then `Duplicate else `Inherit), t, V (k-m+1))) t (k+1) in
- let t = mk_lams t (k+1) in (* make leading lambdas *)
- let subst = var, t in
- let p = subst_in_problem ~top:(not isfinish) subst p in
- sanity p
+let step var j n p =
+  let atsnd f (a,b) = (a, f b) in
+  let p, alphas = (* make fresh vars *)
+   fold_nat (fun (p, vs) _ ->
+     let p, v = freshvar p in
+     p, v::vs
+   ) (p, []) n in let alphas = List.rev alphas in
+  let rec aux lev (inside:bool) = function
+  | L t -> L (aux (lev+1) inside t)
+  | _ as x ->
+    let hd, args = explode x in
+    if hd = V (var+lev) then
+      (let nargs = List.length args in
+       let k = max 0 (j + 1 - nargs) in
+       let args = List.mapi
+        (fun i (f, t) -> f, lift k (aux lev (if i=j then true else inside) t)) args in
+       let bound = fold_nat (fun x n -> (false,V(n-1)) :: x) [] k in
+       let args = args @ bound in
+       let _, head = List.nth args j in
+       let args = List.mapi
+        (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
+       let head = (if inside then erase else id) head in
+       print_endline ("HEAD: " ^ string_of_t head);
+       let alphas = List.map (fun v -> false, V(lev+k+v)) alphas in
+       let t = mk_apps head (alphas @ args) in
+       let t = mk_lams t k in
+       t
+    ) else
+    (let args = List.map (atsnd (aux lev inside)) args in
+     implode hd args) in
+  let sigma = (var, aux 0 false (V var)) :: p.sigma in
+  {p with tms=List.map (aux 0 false) p.tms; sigma}
 ;;
 
-let finish p =
- let compute_max_arity =
-   let rec aux n = function
-   | A(_,t1,t2) -> max (aux (n+1) t1) (aux 0 t2)
-   | L t -> max n (aux 0 t)
-   | V _ -> n
- in aux 0 in
-print_cmd "FINISH" "";
- let div_hd, div_nargs = get_inert p.div in
- let j = div_nargs - 1 in
- let arity = compute_max_arity p.conv in
- let n = 1 + arity + max
-  (compute_max_lambdas_at div_hd j p.div)
-  (compute_max_lambdas_at div_hd j p.conv) in
- let p = step ~isfinish:true j n p in
- let div_hd, div_nargs = get_inert p.div in
- let rec aux m = function
-  A(_,t1,t2) -> if is_var t2 then
-   (let delta_var, _ = get_inert t2 in
-     if delta_var <> div_hd && get_subterm_with_head_and_args delta_var 1 p.conv = None
-      then m, delta_var
-      else aux (m-1) t1) else aux (m-1) t1
-  | _ -> assert false in
- let m, delta_var = aux div_nargs p.div in
- let p = subst_in_problem (delta_var, delta) p in
- let p = subst_in_problem (div_hd, mk_lams delta (m-1)) p in
- sanity p
-;;
+let finish p = assert false ;;
 
-let rec auto p =
- let hd_var, n_args = get_inert p.div in
- match get_subterm_with_head_and_args hd_var n_args p.conv with
- | None ->
-   (try problem_fail (finish p) "Auto.2 did not complete the problem"
-   with Done sigma -> sigma)
-   (*
-   (try
-    let phase = p.phase in
-     let p = eat p in
-     if phase = `Two
-      then problem_fail p "Auto.2 did not complete the problem"
-      else auto p
-    with Done sigma -> sigma)
-    *)
- | Some t ->
-  let j = find_eta_difference p t n_args in
-  let k = 1 + max
-   (compute_max_lambdas_at hd_var j p.div)
-    (compute_max_lambdas_at hd_var j p.conv) in
-  let m1 = measure_of_t p.div in
-  let p = step j k p in
-  let m2 = measure_of_t p.div in
-  (if m2 >= m1 then
-    (print_string ("WARNING! Measure did not decrease : " ^ string_of_int m2 ^ " >= " ^ string_of_int m1 ^ " (press <Enter>)");
-    ignore(read_line())));
-  auto p
-;;
+let rec auto p = assert false ;;
 
 let problem_of (label, div, convs, ps, var_names) =
  print_hline ();
  let rec aux = function
  | `Lam(_,t) -> L (aux t)
- | `I ((v,_), args) -> Listx.fold_left (fun x y -> mk_app (ref true) x (aux y)) (V v) args
+ | `I ((v,_), args) -> Listx.fold_left (fun x y -> mk_app true x (aux y)) (V v) args
  | `Var(v,_) -> V v
  | `N _ | `Match _ -> assert false in
- assert (List.length ps = 0);
  let convs = (List.rev convs :> Num.nf list) in
- let conv = aux
-  (if List.length convs = 1
-   then List.hd convs
-   else `I((List.length var_names, min_int), Listx.from_list convs)) in
- let var_names = "@" :: var_names in
- let div = match div with
- | Some div -> aux (div :> Num.nf)
- | None -> assert false in
+ let tms = List.map aux (convs @ (ps :> Num.nf list)) in
+ let tms = match div with
+ | Some div -> aux (div :> Num.nf) :: tms
+ | None -> tms in
  let varno = List.length var_names in
- let p = {orig_freshno=varno; freshno=1+varno; div; conv; sigma=[]; phase=`One} in
+ let p = {orig_freshno=varno; freshno=1+varno; tms; sigma=[]} in
  (* initial sanity check *)
  sanity p
 ;;
 
+let rec interactive p =
+ print_string "[varno index alphano] ";
+ let s = read_line () in
+ let spl = Str.split (Str.regexp " +") s in
+ let nth n = int_of_string (List.nth spl n) in
+ let p = step (nth 0) (nth 1) (nth 2) p in
+ interactive (sanity p)
+;;
+
 let solve p =
- if eta_subterm p.div p.conv
-  then print_endline "!!! div is subterm of conv. Problem was not run !!!"
-  else check p (auto p)
+ let rec aux = function
+ | [] -> false
+ | x::xs -> List.exists (eta_subterm x) xs || aux xs in
+ if aux p.tms
+  then print_endline "!!! Problem stopped: subterm problem !!!"
+  else check p (interactive p)
 ;;
 
 Problems.main (solve ++ problem_of);
-
-(* Example usage of interactive: *)
-
-(* let interactive div conv cmds =
- let p = problem_of div conv in
- try (
- let p = List.fold_left (|>) p cmds in
- let rec f p cmds =
-  let nth spl n = int_of_string (List.nth spl n) in
-  let read_cmd () =
-   let s = read_line () in
-   let spl = Str.split (Str.regexp " +") s in
-   s, let uno = List.hd spl in
-    try if uno = "eat" then eat
-    else if uno = "step" then step (nth spl 1) (nth spl 2)
-    else failwith "Wrong input."
-    with Failure s -> print_endline s; (fun x -> x) in
-  let str, cmd = read_cmd () in
-  let cmds = (" " ^ str ^ ";")::cmds in
-  try
-   let p = cmd p in f p cmds
-  with
-  | Done _ -> print_endline "Done! Commands history: "; List.iter print_endline (List.rev cmds)
- in f p []
- ) with Done _ -> ()
-;; *)
-
-(* interactive "x y"
- "@ (x x) (y x) (y z)" [step 0 1; step 0 2; eat]
-;; *)
-- 
2.39.2