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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13 | L of (t * t list (*garbage*))
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17 let delta = L(A(V 0, V 0),[]);;
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19 let rec is_stuck = function
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21 | A(t,_) -> is_stuck t
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26 let rec aux l1 l2 t1 t2 =
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27 let stuck1, stuck2 = is_stuck t1, is_stuck t2 in
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29 | _, _ when not stuck1 && stuck2 -> false
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30 | _, _ when stuck1 -> true
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31 | L t1, L t2 -> aux l1 l2 (fst t1) (fst t2)
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32 | L t1, t2 -> aux l1 (l2+1) (fst t1) t2
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33 | t1, L t2 -> aux (l1+1) l2 t1 (fst t2)
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34 | V a, V b -> a + l1 = b + l2
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35 | A(t1,t2), A(u1,u2) -> aux l1 l2 t1 u1 && aux l1 l2 t2 u2
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38 let eta_eq = eta_eq' 0 0;;
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40 (* is arg1 eta-subterm of arg2 ? *)
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42 let rec aux lev t = if t = C then false else (eta_eq' lev 0 u t || match t with
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43 | L(t,g) -> List.exists (aux (lev+1)) (t::g)
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44 | A(t1, t2) -> aux lev t1 || aux lev t2
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49 (* does NOT lift the argument *)
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50 let mk_lams = fold_nat (fun x _ -> L(x,[])) ;;
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53 let string_of_bvar =
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54 let bound_vars = ["x"; "y"; "z"; "w"; "q"] in
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55 let bvarsno = List.length bound_vars in
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56 fun nn -> if nn < bvarsno then List.nth bound_vars nn else "x" ^ (string_of_int (nn - bvarsno + 1)) in
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57 let rec string_of_term_w_pars level = function
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58 | V v -> if v >= level then "`" ^ string_of_int (v-level) else
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59 string_of_bvar (level - v-1)
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62 | L _ as t -> "(" ^ string_of_term_no_pars level t ^ ")"
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63 and string_of_term_no_pars_app level = function
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64 | A(t1,t2) -> string_of_term_no_pars_app level t1 ^ " " ^ string_of_term_w_pars level t2
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65 | _ as t -> string_of_term_w_pars level t
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66 and string_of_term_no_pars level = function
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67 | L(t,g) -> "λ" ^ string_of_bvar level ^ ". " ^ string_of_term_no_pars (level+1) t
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68 ^ (if g = [] then "" else String.concat ", " ("" :: List.map (string_of_term_w_pars (level+1)) g))
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69 | _ as t -> string_of_term_no_pars_app level t
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70 in string_of_term_no_pars 0
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79 ; sigma : (var * t) list (* substitutions *)
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82 let string_of_problem p =
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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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91 exception Done of (var * t) list (* substitution *);;
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92 exception Unseparable of string;;
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93 exception Backtrack of string;;
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95 let rec try_all label f = function
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96 | x::xs -> (try f x with Backtrack _ -> try_all label f xs)
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97 | [] -> raise (Backtrack label)
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100 let problem_fail p reason =
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101 print_endline "!!!!!!!!!!!!!!! FAIL !!!!!!!!!!!!!!!";
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102 print_endline (string_of_problem p);
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106 let freshvar ({freshno} as p) =
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107 {p with freshno=freshno+1}, freshno+1
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110 (* CSC: rename? is an applied C an inert?
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111 is_inert and get_inert work inconsistently *)
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114 | A(t,_) -> is_inert t
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120 let rec is_constant =
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125 | L(t,_) -> is_constant t
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128 let rec get_inert = function
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129 | V _ | C as t -> (t,0)
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130 | A(t, _) -> let hd,args = get_inert t in hd,args+1
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131 | _ -> assert false
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134 let args_of_inert =
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138 | A(t, a) -> aux (a::acc) t
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139 | _ -> assert false
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144 (* precomputes the number of leading lambdas in a term,
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145 after replacing _v_ w/ a term starting with n lambdas *)
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146 let rec no_leading_lambdas v n = function
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147 | L(t,_) -> 1 + no_leading_lambdas (v+1) n t
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148 | 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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149 | V v' -> if v = v' then n else 0
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153 let rec subst level delift sub =
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155 | 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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156 | L x -> let t, g = subst_in_lam (level+1) delift sub x in L(t, g), []
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158 let t1, g1 = subst level delift sub t1 in
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159 let t2, g2 = subst level delift sub t2 in
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160 let t3, g3 = mk_app t1 t2 in
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163 and subst_in_lam level delift sub (t, g) =
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164 let t', g' = subst level delift sub t in
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165 let g'' = List.fold_left
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167 let x,y = subst level delift sub t in
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168 (x :: y @ xs)) g' g in t', g''
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169 and mk_app t1 t2 = if t1 = delta && t2 = delta then raise B
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171 | L x -> subst_in_lam 0 true (0, t2) x
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172 | _ -> A (t1, t2), []
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176 | V m -> V (if m >= lev then m + n else m)
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177 | L(t,g) -> L (aux (lev+1) t, List.map (aux (lev+1)) g)
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178 | A (t1, t2) -> A (aux lev t1, aux lev t2)
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182 let subst' = subst;;
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183 let subst = subst' 0 false;;
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185 let rec mk_apps t = function
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186 | u::us -> mk_apps (A(t,u)) us
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190 let subst_in_problem ((v, t) as sub) p =
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191 print_endline ("-- SUBST " ^ string_of_t (V v) ^ " |-> " ^ string_of_t t);
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192 let sigma = sub :: p.sigma in
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193 let div, g = try subst sub p.div with B -> raise (Done sigma) in
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194 let divs = div :: g in
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195 let conv, g = try subst sub p.conv with B -> raise (Backtrack "p.conv diverged") in
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196 let conv = if g = [] then conv else mk_apps C (conv::g) in
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197 divs, {p with div; conv; sigma}
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200 let get_subterms_with_head hd_var =
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201 let rec aux lev inert_done g = function
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202 | L(t,g') -> List.fold_left (aux (lev+1) false) g (t::g')
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205 let hd_var', n_args' = get_inert t1 in
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206 if not inert_done && hd_var' = V (hd_var + lev)
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207 then lift ~-lev t :: aux lev false (aux lev true g t1) t2
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208 else aux lev false (aux lev true g t1) t2
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213 let rec aux = function
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215 let t = aux (lift (List.length g) t) in
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216 let t = List.fold_left (fun t g -> Pure.A(Pure.L t, aux g)) t g in
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218 | A (t1,t2) -> Pure.A (aux t1, aux t2)
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219 | V n -> Pure.V (n)
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220 | C -> Pure.V (min_int/2)
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224 let check p sigma =
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225 print_endline "Checking...";
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226 let div = purify p.div in
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227 let conv = purify p.conv in
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228 let sigma = List.map (fun (v,t) -> v, purify t) sigma in
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229 let freshno = List.fold_right (max ++ fst) sigma 0 in
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230 let env = Pure.env_of_sigma freshno sigma in
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231 (if not (Pure.diverged (Pure.mwhd (env,div,[])))
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232 then failwith "D converged in Pure");
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233 print_endline "- D diverged.";
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234 (if Pure.diverged (Pure.mwhd (env,conv,[]))
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235 then failwith "C diverged in Pure");
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236 print_endline "- C converged.";
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241 print_endline (string_of_problem p); (* non cancellare *)
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242 (* Trailing constant args can be removed because do not contribute to eta-diff *)
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243 let rec remove_trailing_constant_args = function
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244 | A(t1, t2) when is_constant t2 -> remove_trailing_constant_args t1
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246 let p = {p with div=remove_trailing_constant_args p.div} in
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250 (* drops the arguments of t after the n-th *)
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251 let inert_cut_at n t =
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256 let k', t' = aux t1 in
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257 if k' = n then n, t'
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259 | _ -> assert false
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263 (* return the index of the first argument with a difference
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264 (the first argument is 0) *)
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265 let find_eta_difference p t =
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266 let divargs = args_of_inert p.div in
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267 let conargs = args_of_inert t in
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268 let rec range i j =
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269 if j = 0 then [] else i :: range (i+1) (j-1) in
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270 let rec aux k divargs conargs =
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271 match divargs,conargs with
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272 [],conargs -> range k (List.length conargs)
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274 | t1::divargs,t2::conargs ->
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275 (if not (eta_eq t1 t2) then [k] else []) @ aux (k+1) divargs conargs
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277 aux 0 divargs conargs
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280 let compute_max_lambdas_at hd_var j =
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281 let rec aux hd = function
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283 (if get_inert t1 = (V hd, j)
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284 then max ( (*FIXME*)
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285 if is_inert t2 && let hd', j' = get_inert t2 in hd' = V hd
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286 then let hd', j' = get_inert t2 in j - j'
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287 else no_leading_lambdas hd_var j t2)
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288 else id) (max (aux hd t1) (aux hd t2))
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289 | L(t,_) -> aux (hd+1) t
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294 let print_cmd s1 s2 = print_endline (">> " ^ s1 ^ " " ^ s2);;
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296 (* returns Some i if i is the smallest integer s.t. p holds for the i-th
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297 element of the list in input *)
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298 let smallest_such_that p =
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302 | hd::_ when (print_endline (string_of_t hd) ; p hd) -> Some i
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303 | _::tl -> aux (i+1) tl
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308 (* step on the head of div, on the k-th argument, with n fresh vars *)
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310 let hd, _ = get_inert p.div in
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312 | C | L _ | A _ -> assert false
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314 print_cmd "STEP" ("on " ^ string_of_t (V var) ^ " (on " ^ string_of_int (k+1) ^ "th)");
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315 let p, t = (* apply fresh vars *)
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316 fold_nat (fun (p, t) _ ->
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317 let p, v = freshvar p in
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318 p, A(t, V (v + k + 1))
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320 let t = (* apply unused bound variables V_{k-1}..V_1 *)
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321 fold_nat (fun t m -> A(t, V (k-m+1))) t k in
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322 let t = mk_lams t (k+1) in (* make leading lambdas *)
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323 let subst = var, t in
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324 let divs, p = subst_in_problem subst p in
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329 (* one-step version of eat *)
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330 let compute_max_arity =
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331 let rec aux n = function
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332 | A(t1,t2) -> max (aux (n+1) t1) (aux 0 t2)
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333 | L(t,g) -> List.fold_right (max ++ (aux 0)) (t::g) 0
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336 print_cmd "FINISH" "";
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337 (* First, a step on the last argument of the divergent.
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338 Because of the sanity check, it will never be a constant term. *)
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339 let div_hd, div_nargs = get_inert p.div in
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340 let div_hd = match div_hd with V n -> n | _ -> assert false in
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341 let j = div_nargs - 1 in
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342 let arity = compute_max_arity p.conv in
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343 let n = 1 + arity + max
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344 (compute_max_lambdas_at div_hd j p.div)
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345 (compute_max_lambdas_at div_hd j p.conv) in
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346 let _, p = step j n p in
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347 (* Now, find first argument of div that is a variable never applied anywhere.
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348 It must exist because of some invariant, since we just did a step,
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349 and because of the arity of the divergent *)
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350 let div_hd, div_nargs = get_inert p.div in
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351 let div_hd = match div_hd with V n -> n | _ -> assert false in
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352 let rec aux m = function
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353 | A(t, V delta_var) ->
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354 if delta_var <> div_hd && get_subterms_with_head delta_var p.conv = []
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357 | A(t,_) -> aux (m-1) t
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358 | _ -> assert false in
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359 let m, delta_var = aux div_nargs p.div in
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360 let _, p = subst_in_problem (delta_var, delta) p in
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361 let _, p = subst_in_problem (div_hd, mk_lams delta (m-1)) p in
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368 | L(div,g) -> (* case p.div is an abstraction *)
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369 let f l t = fst (subst' 0 true (0, C) t) :: l in
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370 (* the `fst' above is because we can ignore the
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371 garbage generated by the subst, because substituting
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372 C does not create redexes and thus no new garbage is activated *)
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373 let tms = List.fold_left f [] (div::g) in
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375 (fun div -> aux {p with div}) tms
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377 if is_constant p.div (* case p.div is rigid inert *)
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378 then try_all "auto.C"
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379 (fun div -> aux {p with div}) (args_of_inert p.div)
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380 else (* case p.div is flexible inert *)
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381 let hd, n_args = get_inert p.div in
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383 | C | L _ | A _ -> assert false
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385 let tms = get_subterms_with_head hd_var p.conv in
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386 if List.exists (fun t -> snd (get_inert t) >= n_args) tms
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388 (* let tms = List.sort (fun t1 t2 -> - compare (snd (get_inert t1)) (snd (get_inert t2))) tms in *)
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389 try_all "no similar terms" (fun t ->
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390 let js = find_eta_difference p t in
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391 (* print_endline (String.concat ", " (List.map string_of_int js)); *)
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392 let js = List.rev js in
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393 try_all "no eta difference"
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396 (compute_max_lambdas_at hd_var j p.div)
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397 (compute_max_lambdas_at hd_var j p.conv) in
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398 let divs, p = step j k p in
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399 try_all "p.div" (fun div -> aux (sanity {p with div})) divs
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402 problem_fail (finish p) "Finish did not complete the problem"
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405 with Done sigma -> sigma
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408 let problem_of (label, div, convs, ps, var_names) =
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410 let rec aux lev = function
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411 | `Lam(_, t, g) -> L (aux (lev+1) t, List.map (aux (lev+1)) g)
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412 | `I (v, args) -> Listx.fold_left (fun x y -> fst (mk_app x (aux lev y))) (aux lev (`Var v)) args
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413 | `Var(v,_) -> if v >= lev && List.nth var_names (v-lev) = "C" then C else V v
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414 | `N _ | `Match _ -> assert false in
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415 assert (List.length ps = 0);
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416 let convs = List.rev convs in
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417 let conv = List.fold_left (fun x y -> fst (mk_app x (aux 0 (y :> Num.nf)))) C convs in
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418 let div = match div with
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419 | Some div -> aux 0 (div :> Num.nf)
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420 | None -> assert false in
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421 let varno = List.length var_names in
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422 {orig_freshno=varno; freshno=1+varno; div; conv; sigma=[]; label}
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426 let c = if String.length p.label > 0 then String.sub (p.label) 0 1 else "" in
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427 let module M = struct exception Okay end in
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429 if eta_subterm p.div p.conv
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430 then raise (Unseparable "div is subterm of conv")
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432 let p = sanity p (* initial sanity check *) in
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436 | M.Okay -> if c = "?" then
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437 failwith "The problem succeeded, but was supposed to be unseparable"
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438 | e when c = "!" ->
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439 failwith ("The problem was supposed to be separable, but: "^Printexc.to_string e)
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441 print_endline ("The problem failed, as expected ("^Printexc.to_string e^")")
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444 Problems.main (solve ++ problem_of);
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446 (* Example usage of interactive: *)
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448 (* let interactive div conv cmds =
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449 let p = problem_of div conv in
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451 let p = List.fold_left (|>) p cmds in
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453 let nth spl n = int_of_string (List.nth spl n) in
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455 let s = read_line () in
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456 let spl = Str.split (Str.regexp " +") s in
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457 s, let uno = List.hd spl in
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458 try if uno = "eat" then eat
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459 else if uno = "step" then step (nth spl 1) (nth spl 2)
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460 else failwith "Wrong input."
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461 with Failure s -> print_endline s; (fun x -> x) in
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462 let str, cmd = read_cmd () in
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463 let cmds = (" " ^ str ^ ";")::cmds in
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465 let p = cmd p in f p cmds
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467 | Done _ -> print_endline "Done! Commands history: "; List.iter print_endline (List.rev cmds)
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469 ) with Done _ -> ()
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472 (* interactive "x y"
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473 "@ (x x) (y x) (y z)" [step 0 1; step 0 2; eat]
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