1 let (++) f g x = f (g x);;
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3 let rec fold_nat f x n = if n = 0 then x else f (fold_nat f x (n-1)) n ;;
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5 let print_hline = Console.print_hline;;
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18 let delta = L(A(V 0, V 0));;
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20 let rec is_stuck = function
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22 | A(t,_) -> is_stuck t
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27 let rec aux l1 l2 t1 t2 = match t1, t2 with
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28 | _, _ when is_stuck t1 || is_stuck t2 -> true
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29 | L t1, L t2 -> aux l1 l2 t1 t2
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30 | L t1, t2 -> aux l1 (l2+1) t1 t2
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31 | t1, L t2 -> aux (l1+1) l2 t1 t2
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32 | V a, V b -> a + l1 = b + l2
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33 | A(t1,t2), A(u1,u2) -> aux l1 l2 t1 u1 && aux l1 l2 t2 u2
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36 let eta_eq = eta_eq' 0 0;;
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38 (* is arg1 eta-subterm of arg2 ? *)
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40 let rec aux lev t = if t = C then false else (eta_eq' lev 0 u t || match t with
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41 | L t -> aux (lev+1) t
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42 | A(t1, t2) -> aux lev t1 || aux lev t2
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47 (* does NOT lift the argument *)
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48 let mk_lams = fold_nat (fun x _ -> L x) ;;
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51 let string_of_bvar =
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52 let bound_vars = ["x"; "y"; "z"; "w"; "q"] in
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53 let bvarsno = List.length bound_vars in
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54 fun nn -> if nn < bvarsno then List.nth bound_vars nn else "x" ^ (string_of_int (nn - bvarsno + 1)) in
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55 let rec string_of_term_w_pars level = function
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56 | V v -> if v >= level then "`" ^ string_of_int (v-level) else
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57 string_of_bvar (level - v-1)
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60 | L _ as t -> "(" ^ string_of_term_no_pars level t ^ ")"
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62 and string_of_term_no_pars_app level = function
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63 | A(t1,t2) -> string_of_term_no_pars_app level t1 ^ " " ^ string_of_term_w_pars level t2
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64 | _ as t -> string_of_term_w_pars level t
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65 and string_of_term_no_pars level = function
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66 | L t -> "λ" ^ string_of_bvar level ^ ". " ^ string_of_term_no_pars (level+1) t
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67 | _ as t -> string_of_term_no_pars_app level t
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68 in string_of_term_no_pars 0
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76 ; sigma : (var * t) list (* substitutions *)
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77 ; stepped : var list
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78 ; phase : [`One | `Two] (* :'( *)
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81 let string_of_problem p =
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83 "[stepped] " ^ String.concat " " (List.map string_of_int p.stepped);
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84 "[DV] " ^ string_of_t p.div;
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85 "[CV] " ^ string_of_t p.conv;
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87 String.concat "\n" lines
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90 exception Done of (var * t) list (* substitution *);;
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91 exception Fail of int * string;;
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93 let problem_fail p reason =
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94 print_endline "!!!!!!!!!!!!!!! FAIL !!!!!!!!!!!!!!!";
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95 print_endline (string_of_problem p);
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96 raise (Fail (-1, reason))
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99 let freshvar ({freshno} as p) =
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100 {p with freshno=freshno+1}, freshno+1
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105 | A(t,_) -> is_inert t
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111 let rec get_inert = function
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112 | V _ | C as t -> (t,0)
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113 | A(t, _) -> let hd,args = get_inert t in hd,args+1
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114 | _ -> assert false
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117 (* precomputes the number of leading lambdas in a term,
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118 after replacing _v_ w/ a term starting with n lambdas *)
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119 let rec no_leading_lambdas v n = function
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120 | L t -> 1 + no_leading_lambdas (v+1) n t
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121 | A _ as t -> let v', m = get_inert t in if V v = v' then max 0 (n - m) else 0
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122 | V v' -> if v = v' then n else 0
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126 let rec subst level delift sub =
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128 | V v -> if v = level + fst sub then lift level (snd sub) else V (if delift && v > level then v-1 else v)
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129 | L t -> let t = subst (level + 1) delift sub t in if t = B then B else L t
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131 let t1 = subst level delift sub t1 in
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132 let t2 = subst level delift sub t2 in
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135 and mk_app t1 t2 = if t2 = B || (t1 = delta && t2 = delta) then B
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138 | L t1 -> subst 0 true (0, t2) t1
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143 | V m -> V (if m >= lev then m + n else m)
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144 | L t -> L (aux (lev+1) t)
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145 | A (t1, t2) -> A (aux lev t1, aux lev t2)
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149 let subst = subst 0 false;;
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151 let subst_in_problem ((v, t) as sub) p =
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152 print_endline ("-- SUBST " ^ string_of_t (V v) ^ " |-> " ^ string_of_t t);
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154 div=subst sub p.div;
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155 conv=subst sub p.conv;
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156 stepped=v::p.stepped;
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157 sigma=sub::p.sigma}
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160 let get_subterm_with_head_and_args hd_var n_args =
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161 let rec aux lev = function
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162 | C | V _ | B -> None
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163 | L t -> aux (lev+1) t
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165 let hd_var', n_args' = get_inert t1 in
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166 if hd_var' = V (hd_var + lev) && n_args <= 1 + n_args'
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167 (* the `+1` above is because of t2 *)
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168 then Some (lift ~-lev t)
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169 else match aux lev t2 with
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170 | None -> aux lev t1
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171 | Some _ as res -> res
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175 let rec purify = function
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176 | L t -> Pure.L (purify t)
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177 | A (t1,t2) -> Pure.A (purify t1, purify t2)
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179 | C -> Pure.V (min_int/2)
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183 let check p sigma =
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184 print_endline "Checking...";
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185 let div = purify p.div in
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186 let conv = purify p.conv in
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187 let sigma = List.map (fun (v,t) -> v, purify t) sigma in
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188 let freshno = List.fold_right (max ++ fst) sigma 0 in
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189 let env = Pure.env_of_sigma freshno sigma in
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190 assert (Pure.diverged (Pure.mwhd (env,div,[])));
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191 print_endline " D diverged.";
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192 assert (not (Pure.diverged (Pure.mwhd (env,conv,[]))));
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193 print_endline " C converged.";
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198 print_endline (string_of_problem p); (* non cancellare *)
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199 if p.conv = B then problem_fail p "p.conv diverged";
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200 if p.div = B then raise (Done p.sigma);
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201 if p.phase = `Two && p.div = delta then raise (Done p.sigma);
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202 if not (is_inert p.div) then problem_fail p "p.div converged";
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206 (* drops the arguments of t after the n-th *)
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207 (* FIXME! E' usato in modo improprio contando sul fatto
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208 errato che ritorna un inerte lungo esattamente n *)
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209 let inert_cut_at n t =
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214 let k', t' = aux t1 in
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215 if k' = n then n, t'
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217 | _ -> assert false
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221 (* return the index of the first argument with a difference
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222 (the first argument is 0)
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223 precondition: p.div and t have n+1 arguments
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225 let find_eta_difference p t argsno =
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226 let t = inert_cut_at argsno t in
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227 let rec aux t u k = match t, u with
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229 | A(t1,t2), A(u1,u2) ->
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230 print_endline (string_of_t t2 ^ " vs " ^ string_of_t u2);
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231 if not (eta_eq t2 u2) then (k-1)::aux t1 u1 (k-1)
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232 else aux t1 u1 (k-1)
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233 | _, _ -> assert false
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234 in aux p.div t argsno
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237 let compute_max_lambdas_at hd_var j =
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238 let rec aux hd = function
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240 (if get_inert t1 = (V hd, j)
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241 then max ( (*FIXME*)
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242 if is_inert t2 && let hd', j' = get_inert t2 in hd' = V hd
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243 then let hd', j' = get_inert t2 in j - j'
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244 else no_leading_lambdas hd_var j t2)
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245 else id) (max (aux hd t1) (aux hd t2))
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246 | L t -> aux (hd+1) t
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248 | _ -> assert false
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252 let print_cmd s1 s2 = print_endline (">> " ^ s1 ^ " " ^ s2);;
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254 (* eat the arguments of the divergent and explode.
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255 It does NOT perform any check, may fail if done unsafely *)
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257 print_cmd "EAT" "";
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258 let var, k = get_inert p.div in
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260 | C | L _ | B | A _ -> assert false
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262 let phase = p.phase in
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267 (compute_max_lambdas_at var (k-1) p.div)
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268 (compute_max_lambdas_at var (k-1) p.conv) in
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269 (* apply fresh vars *)
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270 let p, t = fold_nat (fun (p, t) _ ->
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271 let p, v = freshvar p in
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274 let p = {p with phase=`Two} in
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275 let t = A(t, delta) in
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276 let t = fold_nat (fun t m -> A(t, V (k-m))) t (k-1) in
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277 let subst = var, mk_lams t k in
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278 let p = subst_in_problem subst p in
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279 let _, args = get_inert p.div in
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280 {p with div = inert_cut_at (args-k) p.div}
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282 let subst = var, mk_lams delta k in
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283 subst_in_problem subst p in
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287 (* step on the head of div, on the k-th argument, with n fresh vars *)
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289 let hd, _ = get_inert p.div in
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291 | C | L _ | B | A _ -> assert false
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293 print_cmd "STEP" ("on " ^ string_of_t (V var) ^ " (on " ^ string_of_int (k+1) ^ "th)");
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294 let p, t = (* apply fresh vars *)
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295 fold_nat (fun (p, t) _ ->
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296 let p, v = freshvar p in
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297 p, A(t, V (v + k + 1))
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299 let t = (* apply unused bound variables V_{k-1}..V_1 *)
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300 fold_nat (fun t m -> A(t, V (k-m+1))) t k in
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301 let t = mk_lams t (k+1) in (* make leading lambdas *)
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302 let subst = var, t in
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303 let p = subst_in_problem subst p in
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309 let hd, n_args = get_inert p.div in
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311 | C | L _ | B | A _ -> assert false
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313 match get_subterm_with_head_and_args hd_var n_args p.conv with
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315 let phase = p.phase in
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318 then problem_fail p "Auto.2 did not complete the problem"
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321 let js = find_eta_difference p t n_args in
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322 (* print_endline (String.concat ", " (List.map string_of_int js)); *)
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323 if js = [] then problem_fail p "no eta difference found (div subterm of conv?)";
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324 let js = List.rev js in
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329 (compute_max_lambdas_at hd_var j p.div)
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330 (compute_max_lambdas_at hd_var j p.conv) in
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331 ignore (aux (step j k p))
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333 print_endline ("Backtracking because: " ^ s)) js;
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334 raise (Fail(-1, "no eta difference")) in
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337 with Done sigma -> sigma
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340 let problem_of (label, div, convs, ps, var_names) =
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342 let rec aux lev = function
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343 | `Lam(_, t) -> L (aux (lev+1) t)
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344 | `I (v, args) -> Listx.fold_left (fun x y -> mk_app x (aux lev y)) (aux lev (`Var v)) args
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345 | `Var(v,_) -> if v >= lev && List.nth var_names (v-lev) = "C" then C else V v
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346 | `N _ | `Match _ -> assert false in
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347 assert (List.length ps = 0);
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348 let convs = List.rev convs in
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349 let conv = List.fold_left (fun x y -> mk_app x (aux 0 (y :> Num.nf))) (V (List.length var_names)) convs in
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350 let var_names = "@" :: var_names in
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351 let div = match div with
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352 | Some div -> aux 0 (div :> Num.nf)
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353 | None -> assert false in
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354 let varno = List.length var_names in
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355 let p = {orig_freshno=varno; freshno=1+varno; div; conv; sigma=[]; stepped=[]; phase=`One} in
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356 (* initial sanity check *)
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361 if is_stuck p.div then print_endline "!!! div is stuck. Problem was not run !!!"
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362 else if eta_subterm p.div p.conv
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363 then print_endline "!!! div is subterm of conv. Problem was not run !!!"
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364 else check p (auto p)
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367 Problems.main (solve ++ problem_of);
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369 (* Example usage of interactive: *)
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371 (* let interactive div conv cmds =
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372 let p = problem_of div conv in
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374 let p = List.fold_left (|>) p cmds in
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376 let nth spl n = int_of_string (List.nth spl n) in
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378 let s = read_line () in
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379 let spl = Str.split (Str.regexp " +") s in
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380 s, let uno = List.hd spl in
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381 try if uno = "eat" then eat
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382 else if uno = "step" then step (nth spl 1) (nth spl 2)
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383 else failwith "Wrong input."
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384 with Failure s -> print_endline s; (fun x -> x) in
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385 let str, cmd = read_cmd () in
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386 let cmds = (" " ^ str ^ ";")::cmds in
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388 let p = cmd p in f p cmds
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390 | Done _ -> print_endline "Done! Commands history: "; List.iter print_endline (List.rev cmds)
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392 ) with Done _ -> ()
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395 (* interactive "x y"
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396 "@ (x x) (y x) (y z)" [step 0 1; step 0 2; eat]
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