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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10 `Inherit | `Some of bool ref
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12 true if original application and may determine a distinction
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20 | A of var_flag * t * t
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25 let rec aux acc = function
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28 let acc, m1 = aux acc t1 in
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29 let acc, m2 = aux acc t2 in
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31 | `Some b when !b && not (List.memq b acc) -> b::acc, 1 + m1 + m2
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32 | _ -> acc, m1 + m2)
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41 | x'::_ when x == x' -> Some n
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42 | _::xs -> aux (n+1) xs
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47 let apps = ref [] in
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49 r when not !r -> " "
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52 match index_of r !apps with
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55 apps := !apps @ [r];
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57 in " " ^ string_of_int i ^ ":"
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59 let string_of_var_flag = function
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60 | `Some b -> sep_of_app b
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62 | `Duplicate -> " !"
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67 let string_of_bvar =
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68 let bound_vars = ["x"; "y"; "z"; "w"; "q"] in
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69 let bvarsno = List.length bound_vars in
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70 fun nn -> if nn < bvarsno then List.nth bound_vars nn else "x" ^ (string_of_int (nn - bvarsno + 1)) in
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71 let rec string_of_term_w_pars level = function
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72 | V v -> if v >= level then "`" ^ string_of_int (v-level) else
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73 string_of_bvar (level - v-1)
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75 | L _ as t -> "(" ^ string_of_term_no_pars level t ^ ")"
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76 and string_of_term_no_pars_app level = function
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77 | A(b,t1,t2) -> string_of_term_no_pars_app level t1 ^ string_of_var_flag b ^ string_of_term_w_pars level t2
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78 | _ as t -> string_of_term_w_pars level t
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79 and string_of_term_no_pars level = function
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80 | L t -> "λ" ^ string_of_bvar level ^ ". " ^ string_of_term_no_pars (level+1) t
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81 | _ as t -> string_of_term_no_pars_app level t
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82 in string_of_term_no_pars 0
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86 let delta = L(A(`Some (ref true),V 0, V 0));;
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88 (* does NOT lift the argument *)
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89 let mk_lams = fold_nat (fun x _ -> L x) ;;
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96 ; sigma : (var * t) list (* substitutions *)
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97 ; phase : [`One | `Two] (* :'( *)
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100 let string_of_problem p =
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102 "[measure] " ^ string_of_int (measure_of_t p.div);
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103 "[DV] " ^ string_of_t p.div;
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104 "[CV] " ^ string_of_t p.conv;
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106 String.concat "\n" lines
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110 exception Done of (var * t) list (* substitution *);;
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111 exception Fail of int * string;;
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113 let problem_fail p reason =
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114 print_endline "!!!!!!!!!!!!!!! FAIL !!!!!!!!!!!!!!!";
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115 print_endline (string_of_problem p);
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116 raise (Fail (-1, reason))
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119 let freshvar ({freshno} as p) =
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120 {p with freshno=freshno+1}, freshno+1
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125 | A(_,t,_) -> is_inert t
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130 let is_var = function V _ -> true | _ -> false;;
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131 let is_lambda = function L _ -> true | _ -> false;;
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133 let rec get_inert = function
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135 | A(_,t,_) -> let hd,args = get_inert t in hd,args+1
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136 | _ -> assert false
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139 (* precomputes the number of leading lambdas in a term,
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140 after replacing _v_ w/ a term starting with n lambdas *)
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141 let rec no_leading_lambdas v n = function
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142 | L t -> 1 + no_leading_lambdas (v+1) n t
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143 | A _ as t -> let v', m = get_inert t in if v = v' then max 0 (n - m) else 0
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144 | V v' -> if v = v' then n else 0
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147 let rec erase = function
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148 | L t -> L (erase t)
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149 | A(_,t1,t2) -> A(`Some(ref false), erase t1, erase t2)
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153 let rec subst top level delift ((flag, var, tm) as sub) =
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155 | V v -> if v = level + var then lift level tm else V (if delift && v > level then v-1 else v)
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156 | L t -> L (subst top (level + 1) delift sub t)
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158 let special = b = `Duplicate && top && t2 = V (level + var) in
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159 let t1' = subst (if special then false else top) level delift sub t1 in
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160 let t2' = subst false level delift sub t2 in
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162 | `Duplicate when special ->
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163 assert (match t1' with L _ -> false | _ -> true) ;
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165 | `Some b when !b -> b := false
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167 print_string "WARNING! Stepping on a useless argument!";
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168 ignore(read_line())
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169 | `Inherit | `Duplicate -> assert false);
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170 A(flag, t1', erase t2')
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171 | `Inherit | `Duplicate ->
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172 let b' = if t2 = V (level + var)
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173 then (assert (flag <> `Inherit); flag)
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175 assert (match t1' with L _ -> false | _ -> true) ;
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177 | `Some b' -> mk_app top b' t1' t2'
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178 and mk_app top flag t1 t2 = if t1 = delta && t2 = delta then raise B
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180 | L t1 -> subst top 0 true (`Some flag, 0, t2) t1
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181 | _ -> A (`Some flag, t1, t2)
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185 | V m -> V (if m >= lev then m + n else m)
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186 | L t -> L(aux (lev+1) t)
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187 | A (b,t1, t2) -> A (b,aux lev t1, aux lev t2)
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190 let subst top = subst top 0 false;;
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191 let mk_app = mk_app true;;
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194 let rec aux t1 t2 = match t1, t2 with
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195 | L t1, L t2 -> aux t1 t2
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196 | L t1, t2 -> aux t1 (A(`Some (ref true),lift 1 t2,V 0))
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197 | t1, L t2 -> aux (A(`Some (ref true),lift 1 t1,V 0)) t2
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198 | V a, V b -> a = b
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199 | A(_,t1,t2), A(_,u1,u2) -> aux t1 u1 && aux t2 u2
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203 (* is arg1 eta-subterm of arg2 ? *)
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204 let eta_subterm u =
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205 let rec aux lev t = eta_eq u (lift lev t) || match t with
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206 | L t -> aux (lev+1) t
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207 | A(_, t1, t2) -> aux lev t1 || aux lev t2
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212 let subst_in_problem ?(top=true) ((v, t) as sub) p =
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213 print_endline ("-- SUBST " ^ string_of_t (V v) ^ " |-> " ^ string_of_t t);
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214 let sigma = sub::p.sigma in
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215 let sub = (`Inherit, v, t) in
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216 let div = try subst top sub p.div with B -> raise (Done sigma) in
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217 let conv = try subst false sub p.conv with B -> raise (Fail(-1,"p.conv diverged")) in
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218 {p with div; conv; sigma}
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221 let get_subterm_with_head_and_args hd_var n_args =
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222 let rec aux lev = function
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224 | L t -> aux (lev+1) t
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225 | A(_,t1,t2) as t ->
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226 let hd_var', n_args' = get_inert t1 in
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227 if hd_var' = hd_var + lev && n_args <= 1 + n_args'
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228 (* the `+1` above is because of t2 *)
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229 then Some (lift ~-lev t)
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230 else match aux lev t2 with
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231 | None -> aux lev t1
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232 | Some _ as res -> res
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236 let rec purify = function
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237 | L t -> Pure.L (purify t)
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238 | A(_,t1,t2) -> Pure.A (purify t1, purify t2)
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242 let check p sigma =
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243 print_endline "Checking...";
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244 let div = purify p.div in
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245 let conv = purify p.conv in
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246 let sigma = List.map (fun (v,t) -> v, purify t) sigma in
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247 let freshno = List.fold_right (max ++ fst) sigma 0 in
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248 let env = Pure.env_of_sigma freshno sigma in
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249 assert (Pure.diverged (Pure.mwhd (env,div,[])));
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250 print_endline " D diverged.";
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251 assert (not (Pure.diverged (Pure.mwhd (env,conv,[]))));
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252 print_endline " C converged.";
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257 print_endline (string_of_problem p); (* non cancellare *)
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258 if p.phase = `Two && p.div = delta then raise (Done p.sigma);
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259 if not (is_inert p.div) then problem_fail p "p.div converged";
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263 (* drops the arguments of t after the n-th *)
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264 (* FIXME! E' usato in modo improprio contando sul fatto
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265 errato che ritorna un inerte lungo esattamente n *)
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266 let inert_cut_at n t =
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270 | A(_,t1,_) as t ->
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271 let k', t' = aux t1 in
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272 if k' = n then n, t'
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274 | _ -> assert false
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278 (* return the index of the first argument with a difference
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279 (the first argument is 0)
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280 precondition: p.div and t have n+1 arguments
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282 let find_eta_difference p t argsno =
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283 let t = inert_cut_at argsno t in
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284 let rec aux t u k = match t, u with
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286 | A(_,t1,t2), A(_,u1,u2) ->
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287 (match aux t1 u1 (k-1) with
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289 if not (eta_eq t2 u2) then Some (k-1)
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291 | Some j -> Some j)
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292 | _, _ -> assert false
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293 in match aux p.div t argsno with
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294 | None -> problem_fail p "no eta difference found (div subterm of conv?)"
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298 let compute_max_lambdas_at hd_var j =
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299 let rec aux hd = function
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301 (if get_inert t1 = (hd, j)
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302 then max ( (*FIXME*)
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303 if is_inert t2 && let hd', j' = get_inert t2 in hd' = hd
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304 then let hd', j' = get_inert t2 in j - j'
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305 else no_leading_lambdas hd_var j t2)
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306 else id) (max (aux hd t1) (aux hd t2))
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307 | L t -> aux (hd+1) t
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312 let print_cmd s1 s2 = print_endline (">> " ^ s1 ^ " " ^ s2);;
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314 (* step on the head of div, on the k-th argument, with n fresh vars *)
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315 let step ?(isfinish=false) k n p =
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316 let var, _ = get_inert p.div in
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317 print_cmd "STEP" ("on " ^ string_of_t (V var) ^ " (of:" ^ string_of_int n ^ ")");
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318 let p, t = (* apply fresh vars *)
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319 fold_nat (fun (p, t) _ ->
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320 let p, v = freshvar p in
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321 p, A(`Some (ref false), t, V (v + k + 1))
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323 let t = (* apply bound variables V_k..V_0 *)
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324 fold_nat (fun t m -> A((if m = k+1 then `Duplicate else `Inherit), t, V (k-m+1))) t (k+1) in
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325 let t = mk_lams t (k+1) in (* make leading lambdas *)
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326 let subst = var, t in
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327 let p = subst_in_problem ~top:(not isfinish) subst p in
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332 let compute_max_arity =
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333 let rec aux n = function
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334 | A(_,t1,t2) -> max (aux (n+1) t1) (aux 0 t2)
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335 | L t -> max n (aux 0 t)
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338 print_cmd "FINISH" "";
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339 let div_hd, div_nargs = get_inert p.div in
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340 let j = div_nargs - 1 in
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341 let arity = compute_max_arity p.conv in
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342 let n = 1 + arity + max
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343 (compute_max_lambdas_at div_hd j p.div)
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344 (compute_max_lambdas_at div_hd j p.conv) in
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345 let p = step ~isfinish:true j n p in
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346 let div_hd, div_nargs = get_inert p.div in
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347 let rec aux m = function
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348 A(_,t1,t2) -> if is_var t2 then
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349 (let delta_var, _ = get_inert t2 in
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350 if delta_var <> div_hd && get_subterm_with_head_and_args delta_var 1 p.conv = None
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352 else aux (m-1) t1) else aux (m-1) t1
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353 | _ -> assert false in
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354 let m, delta_var = aux div_nargs p.div in
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355 let p = subst_in_problem (delta_var, delta) p in
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356 let p = subst_in_problem (div_hd, mk_lams delta (m-1)) p in
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361 let hd_var, n_args = get_inert p.div in
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362 match get_subterm_with_head_and_args hd_var n_args p.conv with
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364 (try problem_fail (finish p) "Auto.2 did not complete the problem"
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365 with Done sigma -> sigma)
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368 let phase = p.phase in
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371 then problem_fail p "Auto.2 did not complete the problem"
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373 with Done sigma -> sigma)
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376 let j = find_eta_difference p t n_args in
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378 (compute_max_lambdas_at hd_var j p.div)
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379 (compute_max_lambdas_at hd_var j p.conv) in
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380 let m1 = measure_of_t p.div in
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381 let p = step j k p in
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382 let m2 = measure_of_t p.div in
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384 (print_string ("WARNING! Measure did not decrease : " ^ string_of_int m2 ^ " >= " ^ string_of_int m1 ^ " (press <Enter>)");
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385 ignore(read_line())));
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389 let problem_of (label, div, convs, ps, var_names) =
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391 let rec aux = function
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392 | `Lam(_,t) -> L (aux t)
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393 | `I ((v,_), args) -> Listx.fold_left (fun x y -> mk_app (ref true) x (aux y)) (V v) args
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395 | `N _ | `Match _ -> assert false in
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396 assert (List.length ps = 0);
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397 let convs = (List.rev convs :> Num.nf list) in
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399 (if List.length convs = 1
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401 else `I((List.length var_names, min_int), Listx.from_list convs)) in
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402 let var_names = "@" :: var_names in
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403 let div = match div with
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404 | Some div -> aux (div :> Num.nf)
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405 | None -> assert false in
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406 let varno = List.length var_names in
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407 let p = {orig_freshno=varno; freshno=1+varno; div; conv; sigma=[]; phase=`One} in
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408 (* initial sanity check *)
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413 if eta_subterm p.div p.conv
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414 then print_endline "!!! div is subterm of conv. Problem was not run !!!"
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415 else check p (auto p)
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418 Problems.main (solve ++ problem_of);
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420 (* Example usage of interactive: *)
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422 (* let interactive div conv cmds =
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423 let p = problem_of div conv in
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425 let p = List.fold_left (|>) p cmds in
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427 let nth spl n = int_of_string (List.nth spl n) in
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429 let s = read_line () in
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430 let spl = Str.split (Str.regexp " +") s in
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431 s, let uno = List.hd spl in
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432 try if uno = "eat" then eat
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433 else if uno = "step" then step (nth spl 1) (nth spl 2)
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434 else failwith "Wrong input."
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435 with Failure s -> print_endline s; (fun x -> x) in
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436 let str, cmd = read_cmd () in
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437 let cmds = (" " ^ str ^ ";")::cmds in
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439 let p = cmd p in f p cmds
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441 | Done _ -> print_endline "Done! Commands history: "; List.iter print_endline (List.rev cmds)
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443 ) with Done _ -> ()
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446 (* interactive "x y"
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447 "@ (x x) (y x) (y z)" [step 0 1; step 0 2; eat]
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