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 string_of_bvar =
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18 let bound_vars = ["x"; "y"; "z"; "w"; "q"] in
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19 let bvarsno = List.length bound_vars in
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20 fun nn -> if nn < bvarsno then List.nth bound_vars nn else "x" ^ (string_of_int (nn - bvarsno + 1)) in
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21 let rec string_of_term_w_pars level = function
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22 | V v -> if v >= level then "`" ^ string_of_int (v-level) else
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23 string_of_bvar (level - v-1)
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25 | L _ as t -> "(" ^ string_of_term_no_pars level t ^ ")"
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26 and string_of_term_no_pars_app level = function
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27 | A(t1,t2) -> string_of_term_no_pars_app level t1 ^ " " ^ string_of_term_w_pars level t2
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28 | _ as t -> string_of_term_w_pars level t
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29 and string_of_term_no_pars level = function
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30 | L t -> "λ" ^ string_of_bvar level ^ ". " ^ string_of_term_no_pars (level+1) t
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31 | _ as t -> string_of_term_no_pars_app level t
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32 in string_of_term_no_pars 0
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36 let delta = L(A(V 0, V 0));;
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38 (* does NOT lift the argument *)
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39 let mk_lams = fold_nat (fun x _ -> L x) ;;
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46 ; sigma : (var * t) list (* substitutions *)
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47 ; phase : [`One | `Two] (* :'( *)
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50 let string_of_problem p =
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52 "[DV] " ^ string_of_t p.div;
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53 "[CV] " ^ string_of_t p.conv;
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55 String.concat "\n" lines
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59 exception Done of (var * t) list (* substitution *);;
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60 exception Fail of int * string;;
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62 let problem_fail p reason =
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63 print_endline "!!!!!!!!!!!!!!! FAIL !!!!!!!!!!!!!!!";
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64 print_endline (string_of_problem p);
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65 raise (Fail (-1, reason))
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68 let freshvar ({freshno} as p) =
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69 {p with freshno=freshno+1}, freshno+1
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74 | A(t,_) -> is_inert t
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79 let is_var = function V _ -> true | _ -> false;;
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80 let is_lambda = function L _ -> true | _ -> false;;
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82 let rec get_inert = function
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84 | A(t, _) -> let hd,args = get_inert t in hd,args+1
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88 (* precomputes the number of leading lambdas in a term,
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89 after replacing _v_ w/ a term starting with n lambdas *)
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90 let rec no_leading_lambdas v n = function
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91 | L t -> 1 + no_leading_lambdas (v+1) n t
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92 | A _ as t -> let v', m = get_inert t in if v = v' then max 0 (n - m) else 0
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93 | V v' -> if v = v' then n else 0
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96 let rec subst level delift sub =
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98 | 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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99 | L t -> L (subst (level + 1) delift sub t)
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101 let t1 = subst level delift sub t1 in
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102 let t2 = subst level delift sub t2 in
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104 and mk_app t1 t2 = if t1 = delta && t2 = delta then raise B
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106 | L t1 -> subst 0 true (0, t2) t1
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111 | V m -> V (if m >= lev then m + n else m)
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112 | L t -> L (aux (lev+1) t)
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113 | A (t1, t2) -> A (aux lev t1, aux lev t2)
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116 let subst = subst 0 false;;
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119 let rec aux t1 t2 = match t1, t2 with
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120 | L t1, L t2 -> aux t1 t2
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121 | L t1, t2 -> aux t1 (A(lift 1 t2,V 0))
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122 | t1, L t2 -> aux (A(lift 1 t1,V 0)) t2
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123 | V a, V b -> a = b
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124 | A(t1,t2), A(u1,u2) -> aux t1 u1 && aux t2 u2
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128 (* is arg1 eta-subterm of arg2 ? *)
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129 let eta_subterm u =
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130 let rec aux lev t = eta_eq u (lift lev t) || match t with
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131 | L t -> aux (lev+1) t
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132 | A(t1, t2) -> aux lev t1 || aux lev t2
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137 let subst_in_problem ((v, t) as sub) p =
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138 print_endline ("-- SUBST " ^ string_of_t (V v) ^ " |-> " ^ string_of_t t);
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139 let sigma = sub::p.sigma in
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140 let div = try subst sub p.div with B -> raise (Done sigma) in
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141 let conv = try subst sub p.conv with B -> raise (Fail(-1,"p.conv diverged")) in
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142 {p with div; conv; sigma}
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145 let get_subterm_with_head_and_args hd_var n_args =
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146 let rec aux lev = function
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148 | L t -> aux (lev+1) t
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150 let hd_var', n_args' = get_inert t1 in
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151 if hd_var' = hd_var + lev && n_args <= 1 + n_args'
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152 (* the `+1` above is because of t2 *)
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153 then Some (lift ~-lev t)
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154 else match aux lev t2 with
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155 | None -> aux lev t1
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156 | Some _ as res -> res
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160 let rec purify = function
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161 | L t -> Pure.L (purify t)
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162 | A (t1,t2) -> Pure.A (purify t1, purify t2)
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166 let check p sigma =
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167 print_endline "Checking...";
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168 let div = purify p.div in
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169 let conv = purify p.conv in
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170 let sigma = List.map (fun (v,t) -> v, purify t) sigma in
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171 let freshno = List.fold_right (max ++ fst) sigma 0 in
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172 let env = Pure.env_of_sigma freshno sigma in
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173 assert (Pure.diverged (Pure.mwhd (env,div,[])));
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174 print_endline " D diverged.";
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175 assert (not (Pure.diverged (Pure.mwhd (env,conv,[]))));
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176 print_endline " C converged.";
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181 print_endline (string_of_problem p); (* non cancellare *)
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182 if p.phase = `Two && p.div = delta then raise (Done p.sigma);
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183 if not (is_inert p.div) then problem_fail p "p.div converged";
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187 (* drops the arguments of t after the n-th *)
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188 (* FIXME! E' usato in modo improprio contando sul fatto
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189 errato che ritorna un inerte lungo esattamente n *)
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190 let inert_cut_at n t =
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195 let k', t' = aux t1 in
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196 if k' = n then n, t'
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198 | _ -> assert false
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202 (* return the index of the first argument with a difference
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203 (the first argument is 0)
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204 precondition: p.div and t have n+1 arguments
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206 let find_eta_difference p t argsno =
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207 let t = inert_cut_at argsno t in
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208 let rec aux t u k = match t, u with
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210 | A(t1,t2), A(u1,u2) ->
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211 (match aux t1 u1 (k-1) with
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213 if not (eta_eq t2 u2) then Some (k-1)
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215 | Some j -> Some j)
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216 | _, _ -> assert false
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217 in match aux p.div t argsno with
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218 | None -> problem_fail p "no eta difference found (div subterm of conv?)"
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222 let compute_max_lambdas_at hd_var j =
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223 let rec aux hd = function
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225 (if get_inert t1 = (hd, j)
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226 then max ( (*FIXME*)
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227 if is_inert t2 && let hd', j' = get_inert t2 in hd' = hd
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228 then let hd', j' = get_inert t2 in j - j'
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229 else no_leading_lambdas hd_var j t2)
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230 else id) (max (aux hd t1) (aux hd t2))
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231 | L t -> aux (hd+1) t
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236 let print_cmd s1 s2 = print_endline (">> " ^ s1 ^ " " ^ s2);;
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238 (* eat the arguments of the divergent and explode.
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239 It does NOT perform any check, may fail if done unsafely *)
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241 print_cmd "EAT" "";
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242 let var, k = get_inert p.div in
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243 let phase = p.phase in
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248 (compute_max_lambdas_at var (k-1) p.div)
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249 (compute_max_lambdas_at var (k-1) p.conv) in
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250 (* apply fresh vars *)
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251 let p, t = fold_nat (fun (p, t) _ ->
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252 let p, v = freshvar p in
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255 let p = {p with phase=`Two} in
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256 let t = A(t, delta) in
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257 let t = fold_nat (fun t m -> A(t, V (k-m))) t (k-1) in
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258 let subst = var, mk_lams t k in
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259 let p = subst_in_problem subst p in
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260 let _, args = get_inert p.div in
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261 {p with div = inert_cut_at (args-k) p.div}
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263 let subst = var, mk_lams delta k in
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264 subst_in_problem subst p in
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268 (* step on the head of div, on the k-th argument, with n fresh vars *)
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270 let var, _ = get_inert p.div in
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271 print_cmd "STEP" ("on " ^ string_of_t (V var) ^ " (of:" ^ string_of_int n ^ ")");
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272 let p, t = (* apply fresh vars *)
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273 fold_nat (fun (p, t) _ ->
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274 let p, v = freshvar p in
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275 p, A(t, V (v + k + 1))
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277 let t = (* apply bound variables V_k..V_0 *)
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278 fold_nat (fun t m -> A(t, V (k-m+1))) t (k+1) in
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279 let t = mk_lams t (k+1) in (* make leading lambdas *)
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280 let subst = var, t in
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281 let p = subst_in_problem subst p in
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286 let hd_var, n_args = get_inert p.div in
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287 match get_subterm_with_head_and_args hd_var n_args p.conv with
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290 let phase = p.phase in
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293 then problem_fail p "Auto.2 did not complete the problem"
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295 with Done sigma -> sigma)
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297 let j = find_eta_difference p t n_args in
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299 (compute_max_lambdas_at hd_var j p.div)
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300 (compute_max_lambdas_at hd_var j p.conv) in
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301 let p = step j k p in
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305 let problem_of (label, div, convs, ps, var_names) =
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307 let rec aux = function
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308 | `Lam(_, t) -> L (aux t)
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309 | `I ((v,_), args) -> Listx.fold_left (fun x y -> mk_app x (aux y)) (V v) args
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311 | `N _ | `Match _ -> assert false in
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312 assert (List.length ps = 0);
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313 let convs = List.rev convs in
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314 let conv = List.fold_left (fun x y -> mk_app x (aux (y :> Num.nf))) (V (List.length var_names)) convs in
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315 let var_names = "@" :: var_names in
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316 let div = match div with
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317 | Some div -> aux (div :> Num.nf)
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318 | None -> assert false in
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319 let varno = List.length var_names in
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320 let p = {orig_freshno=varno; freshno=1+varno; div; conv; sigma=[]; phase=`One} in
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321 (* initial sanity check *)
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326 if eta_subterm p.div p.conv
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327 then print_endline "!!! div is subterm of conv. Problem was not run !!!"
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328 else check p (auto p)
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331 Problems.main (solve ++ problem_of);
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333 (* Example usage of interactive: *)
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335 (* let interactive div conv cmds =
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336 let p = problem_of div conv in
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338 let p = List.fold_left (|>) p cmds in
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340 let nth spl n = int_of_string (List.nth spl n) in
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342 let s = read_line () in
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343 let spl = Str.split (Str.regexp " +") s in
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344 s, let uno = List.hd spl in
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345 try if uno = "eat" then eat
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346 else if uno = "step" then step (nth spl 1) (nth spl 2)
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347 else failwith "Wrong input."
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348 with Failure s -> print_endline s; (fun x -> x) in
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349 let str, cmd = read_cmd () in
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350 let cmds = (" " ^ str ^ ";")::cmds in
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352 let p = cmd p in f p cmds
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354 | Done _ -> print_endline "Done! Commands history: "; List.iter print_endline (List.rev cmds)
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356 ) with Done _ -> ()
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359 (* interactive "x y"
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360 "@ (x x) (y x) (y z)" [step 0 1; step 0 2; eat]
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