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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17 let delta = L(A(V 0, V 0));;
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20 let rec aux l1 l2 t1 t2 = match t1, t2 with
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21 | L t1, L t2 -> aux l1 l2 t1 t2
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22 | L t1, t2 -> aux l1 (l2+1) t1 t2
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23 | t1, L t2 -> aux (l1+1) l2 t1 t2
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24 | V a, V b -> a + l1 = b + l2
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25 | A(t1,t2), A(u1,u2) -> aux l1 l2 t1 u1 && aux l1 l2 t2 u2
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30 (* does NOT lift t *)
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31 let mk_lams = fold_nat (fun x _ -> L x) ;;
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34 let string_of_bvar =
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35 let bound_vars = ["x"; "y"; "z"; "w"; "q"] in
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36 let bvarsno = List.length bound_vars in
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37 fun nn -> if nn < bvarsno then List.nth bound_vars nn else "x" ^ (string_of_int (nn - bvarsno + 1)) in
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38 let rec string_of_term_w_pars level = function
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39 | V v -> if v >= level then "`" ^ string_of_int (v-level) else
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40 string_of_bvar (level - v-1)
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42 | L _ as t -> "(" ^ string_of_term_no_pars level t ^ ")"
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44 and string_of_term_no_pars_app level = function
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45 | A(t1,t2) -> string_of_term_no_pars_app level t1 ^ " " ^ string_of_term_w_pars level t2
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46 | _ as t -> string_of_term_w_pars level t
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47 and string_of_term_no_pars level = function
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48 | L t -> "λ" ^ string_of_bvar level ^ ". " ^ string_of_term_no_pars (level+1) t
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49 | _ as t -> string_of_term_no_pars_app level t
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50 in string_of_term_no_pars 0
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58 ; sigma : (var * t) list (* substitutions *)
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59 ; stepped : var list
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60 ; phase : [`One | `Two] (* :'( *)
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63 let string_of_problem p =
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65 "[stepped] " ^ String.concat " " (List.map string_of_int p.stepped);
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66 "[DV] " ^ string_of_t p.div;
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67 "[CV] " ^ string_of_t p.conv;
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69 String.concat "\n" lines
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72 exception Done of (var * t) list (* substitution *);;
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73 exception Fail of int * string;;
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75 let problem_fail p reason =
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76 print_endline "!!!!!!!!!!!!!!! FAIL !!!!!!!!!!!!!!!";
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77 print_endline (string_of_problem p);
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78 raise (Fail (-1, reason))
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81 let freshvar ({freshno} as p) =
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82 {p with freshno=freshno+1}, freshno+1
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87 | A(t,_) -> is_inert t
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92 let is_var = function V _ -> true | _ -> false;;
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93 let is_lambda = function L _ -> true | _ -> false;;
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95 let rec no_leading_lambdas = function
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96 | L t -> 1 + no_leading_lambdas t
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100 let rec get_inert = function
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102 | A(t, _) -> let hd,args = get_inert t in hd,args+1
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103 | _ -> assert false
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106 let rec subst level delift sub =
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108 | 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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109 | L t -> let t = subst (level + 1) delift sub t in if t = B then B else L t
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111 let t1 = subst level delift sub t1 in
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112 let t2 = subst level delift sub t2 in
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115 and mk_app t1 t2 = if t2 = B || (t1 = delta && t2 = delta) then B
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118 | L t1 -> subst 0 true (0, t2) t1
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123 | V m -> V (if m >= lev then m + n else m)
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124 | L t -> L (aux (lev+1) t)
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125 | A (t1, t2) -> A (aux lev t1, aux lev t2)
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129 let subst = subst 0 false;;
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131 let subst_in_problem (sub: var * t) (p: problem) =
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132 print_endline ("-- SUBST " ^ string_of_t (V (fst sub)) ^ " |-> " ^ string_of_t (snd sub));
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134 div=subst sub p.div;
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135 conv=subst sub p.conv;
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136 stepped=(fst sub)::p.stepped;
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137 sigma=sub::p.sigma}
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140 let get_subterm_with_head_and_args hd_var n_args =
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141 let rec aux lev = function
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143 | L t -> aux (lev+1) t
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145 let hd_var', n_args' = get_inert t1 in
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146 if hd_var' = hd_var + lev && n_args <= 1 + n_args'
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147 then Some (lift ~-lev t)
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148 else match aux lev t2 with
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149 | None -> aux lev t1
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150 | Some _ as res -> res
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154 let rec purify = function
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155 | L t -> Pure.L (purify t)
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156 | A (t1,t2) -> Pure.A (purify t1, purify t2)
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161 let check p sigma =
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162 print_endline "Checking...";
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163 let div = purify p.div in
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164 let conv = purify p.conv in
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165 let sigma = List.map (fun (v,t) -> v, purify t) sigma in
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166 let freshno = List.fold_right (fun (x,_) -> max x) sigma 0 in
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167 let env = Pure.env_of_sigma freshno sigma in
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168 assert (Pure.diverged (Pure.mwhd (env,div,[])));
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169 assert (not (Pure.diverged (Pure.mwhd (env,conv,[]))));
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174 print_endline (string_of_problem p); (* non cancellare *)
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175 if p.conv = B then problem_fail p "p.conv diverged";
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176 if p.div = B then raise (Done p.sigma);
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177 if p.phase = `Two && p.div = delta then raise (Done p.sigma);
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178 if not (is_inert p.div) then problem_fail p "p.div converged"
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181 (* drops the arguments of t after the n-th *)
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182 let inert_cut_at n t =
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187 let k', t' = aux t1 in
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188 if k' = n then n, t'
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190 | _ -> assert false
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194 let find_eta_difference p t n_args =
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195 let t = inert_cut_at n_args t in
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196 let rec aux t u k = match t, u with
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197 | V _, V _ -> assert false (* div subterm of conv *)
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198 | A(t1,t2), A(u1,u2) ->
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199 if not (eta_eq t2 u2) then (print_endline((string_of_t t2) ^ " <> " ^ (string_of_t u2)); k)
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200 else aux t1 u1 (k-1)
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201 | _, _ -> assert false
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202 in aux p.div t n_args
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205 let compute_max_lambdas_at hd_var j =
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206 let rec aux hd = function
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208 (if get_inert t1 = (hd, j)
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209 then max ( (*FIXME*)
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210 if is_inert t2 && let hd', j' = get_inert t2 in hd' = hd
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211 then let hd', j' = get_inert t2 in j - j'
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212 else no_leading_lambdas t2)
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213 else id) (max (aux hd t1) (aux hd t2))
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214 | L t -> aux (hd+1) t
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216 | _ -> assert false
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220 let print_cmd s1 s2 = print_endline (">> " ^ s1 ^ " " ^ s2);;
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222 (* eat the arguments of the divergent and explode.
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223 It does NOT perform any check, may fail if done unsafely *)
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225 print_cmd "EAT" "";
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226 let var, k = get_inert p.div in
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227 let phase = p.phase in
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232 (compute_max_lambdas_at var k p.div)
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233 (compute_max_lambdas_at var k p.conv) in
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234 (* apply fresh vars *)
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235 let p, t = fold_nat (fun (p, t) _ ->
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236 let p, v = freshvar p in
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239 let p = {p with phase=`Two} in p, A(t, delta)
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240 | `Two -> p, delta in
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241 let subst = var, mk_lams t k in
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242 let p = subst_in_problem subst p in
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243 let p = if phase = `One then {p with div = (match p.div with A(t,_) -> t | _ -> assert false)} else p in
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247 (* step on the head of div, on the k-th argument, with n fresh vars *)
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249 let var, _ = get_inert p.div in
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250 print_cmd "STEP" ("on " ^ string_of_t (V var) ^ " (of:" ^ string_of_int n ^ ")");
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251 let p, t = (* apply fresh vars *)
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252 fold_nat (fun (p, t) _ ->
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253 let p, v = freshvar p in
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254 p, A(t, V (v + k + 1))
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256 let t = (* apply unused bound variables V_{k-1}..V_1 *)
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257 fold_nat (fun t m -> A(t, V (k-m+1))) t k in
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258 let t = mk_lams t (k+1) in (* make leading lambdas *)
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259 let subst = var, t in
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260 let p = subst_in_problem subst p in
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265 let rec aux level = function
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266 | Parser_andrea.Lam t -> L (aux (level + 1) t)
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267 | Parser_andrea.App (t1, t2) ->
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268 if level = 0 then mk_app (aux level t1) (aux level t2)
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269 else A(aux level t1, aux level t2)
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270 | Parser_andrea.Var v -> V v in
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271 let (tms, free) = Parser_andrea.parse_many strs in
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272 (List.map (aux 0) tms, free)
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275 let problem_of div conv =
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277 let [@warning "-8"] [div; conv], var_names = parse ([div; conv]) in
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278 let varno = List.length var_names in
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279 let p = {orig_freshno=varno; freshno=1+varno; div; conv; sigma=[]; stepped=[]; phase=`One} in
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280 (* initial sanity check *)
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284 let exec div conv cmds =
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285 let p = problem_of div conv in
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287 problem_fail (List.fold_left (|>) p cmds) "Problem not completed"
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293 let hd_var, n_args = get_inert p.div in
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294 match get_subterm_with_head_and_args hd_var n_args p.conv with
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297 let phase = p.phase in
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300 then problem_fail p "Auto.2 did not complete the problem"
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302 with Done sigma -> sigma)
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304 let j = find_eta_difference p t n_args - 1 in
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306 (compute_max_lambdas_at hd_var j p.div)
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307 (compute_max_lambdas_at hd_var j p.conv) in
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308 let p = step j k p in
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312 let interactive div conv cmds =
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313 let p = problem_of div conv in
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315 let p = List.fold_left (|>) p cmds in
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317 let nth spl n = int_of_string (List.nth spl n) in
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319 let s = read_line () in
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320 let spl = Str.split (Str.regexp " +") s in
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321 s, let uno = List.hd spl in
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322 try if uno = "eat" then eat
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323 else if uno = "step" then step (nth spl 1) (nth spl 2)
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324 else failwith "Wrong input."
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325 with Failure s -> print_endline s; (fun x -> x) in
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326 let str, cmd = read_cmd () in
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327 let cmds = (" " ^ str ^ ";")::cmds in
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329 let p = cmd p in f p cmds
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331 | Done _ -> print_endline "Done! Commands history: "; List.iter print_endline (List.rev cmds)
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333 ) with Done _ -> ()
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336 let rec conv_join = function
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338 | x::xs -> conv_join xs ^ " ("^ x ^")"
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342 let p = problem_of a (conv_join b) in
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343 let sigma = auto p in
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347 (* Example usage of exec, interactive:
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351 (conv_join["x y"; "y y"; "y x"])
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356 "@ (x x) (y x) (y z)" [step 0 1; step 0 2; eat]
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361 auto' "x x" ["x y"; "y y"; "y x"] ;;
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362 auto' "x y" ["x (_. x)"; "y z"; "y x"] ;;
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363 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)"] ;;
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365 auto' "x (y. x y y)" ["x (y. x y x)"] ;;
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367 auto' "x a a a a" [
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374 (* Controesempio ad usare un conto dei lambda che non considere le permutazioni *)
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375 auto' "x a a a a (x (x. x x) @ @ (_._.x. x x) x) b b b" [
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376 "x a a a a (_. a) b b b";
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377 "x a a a a (_. _. _. _. x. y. x y)";
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382 print_endline "ALL DONE. "
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