1 let (++) f g x = f (g x);;
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2 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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4 let print_hline = Console.print_hline;;
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12 | L of (t * t list (*garbage*))
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16 let delta = L(A(V 0, V 0),[]);;
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18 let rec is_stuck = function
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20 | A(t,_) -> is_stuck t
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25 let rec aux l1 l2 t1 t2 =
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26 let stuck1, stuck2 = is_stuck t1, is_stuck t2 in
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28 | _, _ when not stuck1 && stuck2 -> false
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29 | _, _ when stuck1 -> true
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30 | L t1, L t2 -> aux l1 l2 (fst t1) (fst t2)
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31 | L t1, t2 -> aux l1 (l2+1) (fst t1) t2
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32 | t1, L t2 -> aux (l1+1) l2 t1 (fst t2)
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33 | V a, V b -> a + l1 = b + l2
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34 | A(t1,t2), A(u1,u2) -> aux l1 l2 t1 u1 && aux l1 l2 t2 u2
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37 let eta_eq = eta_eq' 0 0;;
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39 (* is arg1 eta-subterm of arg2 ? *)
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41 let rec aux lev t = if t = C then false else (eta_eq' lev 0 u t || match t with
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42 | L(t,g) -> List.exists (aux (lev+1)) (t::g)
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43 | A(t1, t2) -> aux lev t1 || aux lev t2
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48 (* does NOT lift the argument *)
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49 let mk_lams = fold_nat (fun x _ -> L(x,[])) ;;
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52 let string_of_bvar =
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53 let bound_vars = ["x"; "y"; "z"; "w"; "q"] in
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54 let bvarsno = List.length bound_vars in
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55 fun nn -> if nn < bvarsno then List.nth bound_vars nn else "x" ^ (string_of_int (nn - bvarsno + 1)) in
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56 let rec string_of_term_w_pars level = function
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57 | V v -> if v >= level then "`" ^ string_of_int (v-level) else
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58 string_of_bvar (level - v-1)
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61 | L _ as t -> "(" ^ string_of_term_no_pars level t ^ ")"
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62 and string_of_term_no_pars_app level = function
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63 | A(t1,t2) -> string_of_term_no_pars_app level t1 ^ " " ^ string_of_term_w_pars level t2
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64 | _ as t -> string_of_term_w_pars level t
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65 and string_of_term_no_pars level = function
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66 | L(t,g) -> "λ" ^ string_of_bvar level ^ ". " ^ string_of_term_no_pars (level+1) t
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67 ^ (if g = [] then "" else String.concat ", " ("" :: List.map (string_of_term_w_pars (level+1)) g))
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68 | _ as t -> string_of_term_no_pars_app level t
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69 in string_of_term_no_pars 0
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78 ; sigma : (var * t) list (* substitutions *)
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81 let string_of_problem p =
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83 "[DV] " ^ string_of_t p.div;
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84 "[CV] " ^ string_of_t p.conv;
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86 String.concat "\n" lines
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90 exception Done of (var * t) list (* substitution *);;
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91 exception Unseparable of string;;
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92 exception Backtrack of string;;
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94 let rec try_all label f = function
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95 | x::xs -> (try f x with Backtrack s -> (if s <> "" then print_endline ("\n<< BACKTRACK: "^s)); try_all label f xs)
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96 | [] -> raise (Backtrack label)
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98 let try_both label f x g y =
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99 try_all label (function `L x -> f x | `R y -> g y) [`L x ; `R y]
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102 let problem_fail p reason =
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103 print_endline "!!!!!!!!!!!!!!! FAIL !!!!!!!!!!!!!!!";
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104 print_endline (string_of_problem p);
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108 let freshvar ({freshno} as p) =
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109 {p with freshno=freshno+1}, freshno+1
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112 (* CSC: rename? is an applied C an inert?
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113 is_inert and get_inert work inconsistently *)
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116 | A(t,_) -> is_inert t
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122 let rec is_constant =
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127 | L(t,_) -> is_constant t
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130 let rec get_inert = function
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131 | V _ | C as t -> (t,0)
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132 | A(t, _) -> let hd,args = get_inert t in hd,args+1
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133 | _ -> assert false
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136 let args_of_inert =
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140 | A(t, a) -> aux (a::acc) t
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141 | _ -> assert false
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146 (* precomputes the number of leading lambdas in a term,
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147 after replacing _v_ w/ a term starting with n lambdas *)
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148 let rec no_leading_lambdas v n = function
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149 | L(t,_) -> 1 + no_leading_lambdas (v+1) n t
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150 | 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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151 | V v' -> if v = v' then n else 0
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155 let rec subst level delift sub =
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157 | 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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158 | L x -> let t, g = subst_in_lam (level+1) delift sub x in L(t, g), []
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160 let t1, g1 = subst level delift sub t1 in
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161 let t2, g2 = subst level delift sub t2 in
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162 let t3, g3 = mk_app t1 t2 in
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165 and subst_in_lam level delift sub (t, g) =
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166 let t', g' = subst level delift sub t in
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167 let g'' = List.fold_left
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169 let x,y = subst level delift sub t in
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170 (x :: y @ xs)) g' g in t', g''
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171 and mk_app t1 t2 = if t1 = delta && t2 = delta then raise B
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173 | L x -> subst_in_lam 0 true (0, t2) x
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174 | _ -> A (t1, t2), []
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178 | V m -> V (if m >= lev then m + n else m)
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179 | L(t,g) -> L (aux (lev+1) t, List.map (aux (lev+1)) g)
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180 | A (t1, t2) -> A (aux lev t1, aux lev t2)
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184 let subst' = subst;;
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185 let subst = subst' 0 false;;
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187 let rec mk_apps t = function
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188 | u::us -> mk_apps (A(t,u)) us
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192 let subst_in_problem ((v, t) as sub) p =
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193 print_endline ("-- SUBST " ^ string_of_t (V v) ^ " |-> " ^ string_of_t t);
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194 let sigma = sub :: p.sigma in
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195 let div, g = try subst sub p.div with B -> raise (Done sigma) in
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196 let divs = div :: g in
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197 let conv, g = try subst sub p.conv with B -> raise (Backtrack "p.conv diverged") in
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198 let conv = if g = [] then conv else mk_apps C (conv::g) in
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199 divs, {p with div; conv; sigma}
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202 let get_subterms_with_head hd_var =
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203 let rec aux lev inert_done g = function
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204 | L(t,g') -> List.fold_left (aux (lev+1) false) g (t::g')
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207 let hd_var', n_args' = get_inert t1 in
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208 if not inert_done && hd_var' = V (hd_var + lev)
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209 then lift ~-lev t :: aux lev false (aux lev true g t1) t2
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210 else aux lev false (aux lev true g t1) t2
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215 let rec aux = function
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217 let t = aux (lift (List.length g) t) in
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218 let t = List.fold_left (fun t g -> Pure.A(Pure.L t, aux g)) t g in
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220 | A (t1,t2) -> Pure.A (aux t1, aux t2)
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221 | V n -> Pure.V (n)
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222 | C -> Pure.V (min_int/2)
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226 let check p sigma =
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227 print_endline "\nChecking...";
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228 let div = purify p.div in
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229 let conv = purify p.conv in
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230 let sigma = List.map (fun (v,t) -> v, purify t) sigma in
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231 let freshno = List.fold_right (max ++ fst) sigma 0 in
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232 let env = Pure.env_of_sigma freshno sigma in
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233 (if not (Pure.diverged (Pure.mwhd (env,div,[])))
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234 then failwith "D converged in Pure");
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235 print_endline "- D diverged.";
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236 (if Pure.diverged (Pure.mwhd (env,conv,[]))
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237 then failwith "C diverged in Pure");
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238 print_endline "- C converged.";
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243 print_endline (string_of_problem p) (* non cancellare *); p
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246 (* drops the arguments of t after the n-th *)
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247 let inert_cut_at n t =
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252 let k', t' = aux t1 in
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253 if k' = n then n, t'
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255 | _ -> assert false
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259 (* return the index of the first argument with a difference
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260 (the first argument is 0) *)
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261 let find_eta_difference p t =
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262 let divargs = args_of_inert p.div in
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263 let conargs = args_of_inert t in
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264 let rec range i j =
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265 if j = 0 then [] else i :: range (i+1) (j-1) in
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266 let rec aux k divargs conargs =
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267 match divargs,conargs with
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268 [],conargs -> range k (List.length conargs)
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270 | t1::divargs,t2::conargs ->
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271 (if not (eta_eq t1 t2) then [k] else []) @ aux (k+1) divargs conargs
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273 aux 0 divargs conargs
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276 let compute_max_lambdas_at hd_var j =
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277 let rec aux hd = function
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278 | A(t1,t2) -> max (max (aux hd t1) (aux hd t2))
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279 (if get_inert t1 = (V hd, j)
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280 then no_leading_lambdas hd (j+1) t2
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282 | L(t,_) -> aux (hd+1) t
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288 let print_cmd s1 s2 = print_endline ("\n>> " ^ s1 ^ " " ^ s2);;
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290 (* returns Some i if i is the smallest integer s.t. p holds for the i-th
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291 element of the list in input *)
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292 let smallest_such_that p =
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296 | hd::_ when p i hd -> Some i
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297 | _::tl -> aux (i+1) tl
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302 (* step on the head of div, on the k-th argument, with n fresh vars *)
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304 let hd, _ = get_inert p.div in
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306 | C | L _ | A _ -> assert false
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308 print_cmd "STEP" ("on " ^ string_of_t (V var) ^ " (on " ^ string_of_int (k+1) ^ "th)");
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309 let p, t = (* apply fresh vars *)
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310 fold_nat (fun (p, t) _ ->
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311 let p, v = freshvar p in
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312 p, A(t, V (v + k + 1))
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314 let t = (* apply unused bound variables V_{k-1}..V_1 *)
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315 fold_nat (fun t m -> A(t, V (k-m+1))) t k in
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316 let t = mk_lams t (k+1) in (* make leading lambdas *)
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317 let subst = var, t in
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318 let divs, p = subst_in_problem subst p in
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322 let finish p arity =
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323 (* one-step version of eat *)
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324 let compute_max_arity =
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325 let rec aux n = function
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326 | A(t1,t2) -> max (aux (n+1) t1) (aux 0 t2)
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327 | L(t,g) -> List.fold_right (max ++ (aux 0)) (t::g) 0
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330 (* First, a step on the last argument of the divergent.
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331 Because of the sanity check, it will never be a constant term. *)
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332 let div_hd, div_nargs = get_inert p.div in
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333 let div_hd = match div_hd with V n -> n | _ -> raise (Backtrack "Cannot finish on constant tm") in
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335 smallest_such_that (fun i t -> i >= arity && not (is_constant t)) (args_of_inert p.div)
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336 with Some j -> j | None -> raise (Backtrack "") in
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337 print_endline "\n>> FINISHING";
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338 let arity = compute_max_arity p.conv in
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339 let n = 1 + arity + max
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340 (compute_max_lambdas_at div_hd j p.div)
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341 (compute_max_lambdas_at div_hd j p.conv) in
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342 let _, p = step j n p in
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343 (* Now, find first argument of div that is a variable never applied anywhere.
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344 It must exist because of some invariant, since we just did a step,
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345 and because of the arity of the divergent *)
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346 let div_hd, div_nargs = get_inert p.div in
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347 let div_hd = match div_hd with V n -> n | _ -> assert false in
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348 let rec aux m = function
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349 | A(t, V delta_var) ->
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350 if delta_var <> div_hd && get_subterms_with_head delta_var p.conv = []
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353 | A(t,_) -> aux (m-1) t
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354 | _ -> assert false in
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355 let m, delta_var = aux div_nargs p.div in
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356 let _, p = subst_in_problem (delta_var, delta) p in
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357 ignore (subst_in_problem (div_hd, mk_lams delta (m-1)) p);
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363 if eta_subterm p.div p.conv
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364 then raise (Backtrack "div is subterm of conv");
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366 | L _ as t -> (* case p.div is an abstraction *)
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367 print_endline "\nSOTTO UN LAMBDA";
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368 let t, g = mk_app t C in
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369 aux ({p with div=mk_apps C (t::g)})
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370 | V _ | C -> raise (Backtrack "V | C")
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372 if is_constant p.div (* case p.div is rigid inert *)
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373 then (print_endline "\nSOTTO UN C"; try_all "auto.C"
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374 (fun div -> aux (sanity {p with div})) (args_of_inert p.div))
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375 else (* case p.div is flexible inert *)
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376 let hd, n_args = get_inert p.div in
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378 | C | L _ | A _ -> assert false
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380 let tms = get_subterms_with_head hd_var p.conv in
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381 let arity = List.fold_right (max ++ (snd ++ get_inert)) tms 0 in
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382 try_both "???" (finish p) arity
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384 let jss = List.concat (List.map (find_eta_difference p) tms) in
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385 let jss = List.sort_uniq compare jss in
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386 let f = try_all "no differences"
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389 (compute_max_lambdas_at hd_var j p.div)
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390 (compute_max_lambdas_at hd_var j p.conv) in
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391 let divs, p = step j k p in
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392 try_all "p.div" (fun div -> aux (sanity {p with div})) divs
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394 try_both "step, then diverge arguments"
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396 (try_all "tried p.div arguments" (fun div -> aux {p with div})) (args_of_inert p.div)
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400 with Done sigma -> sigma
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403 let problem_of (label, div, convs, ps, var_names) =
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405 let rec aux lev = function
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406 | `Lam(_, t, g) -> L (aux (lev+1) t, List.map (aux (lev+1)) g)
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407 | `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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408 | `Var(v,_) -> if v >= lev && List.nth var_names (v-lev) = "C" then C else V v
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409 | `N _ | `Match _ -> assert false in
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410 assert (List.length ps = 0);
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411 let convs = List.rev convs in
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412 let conv = List.fold_left (fun x y -> fst (mk_app x (aux 0 (y :> Num.nf)))) C convs in
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413 let div = match div with
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414 | Some div -> aux 0 (div :> Num.nf)
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415 | None -> assert false in
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416 let varno = List.length var_names in
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417 {orig_freshno=varno; freshno=1+varno; div; conv; sigma=[]; label}
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421 let c = if String.length p.label > 0 then String.sub (p.label) 0 1 else "" in
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422 let module M = struct exception Okay end in
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424 if eta_subterm p.div p.conv
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425 then raise (Unseparable "div is subterm of conv")
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427 let p = sanity p (* initial sanity check *) in
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431 | M.Okay -> if c = "?" then
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432 failwith "The problem succeeded, but was supposed to be unseparable"
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433 | e when c = "!" ->
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434 failwith ("The problem was supposed to be separable, but: "^Printexc.to_string e)
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436 print_endline ("The problem failed, as expected ("^Printexc.to_string e^")")
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439 Problems.main (solve ++ problem_of);
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