(*  Title:      HOL/Tools/Nitpick/nitpick_mono.ML

    Author:     Jasmin Blanchette, TU Muenchen

    Copyright   2009

Monotonicity predicate for higher-order logic.

*)

signature NITPICK_MONO =

sig

  type extended_context = Nitpick_HOL.extended_context

  val formulas_monotonic :

    extended_context -> typ -> term list -> term list -> term -> bool

end;

structure Nitpick_Mono : NITPICK_MONO =

struct

open Nitpick_Util

open Nitpick_HOL

type var = int

datatype sign = Pos | Neg

datatype sign_atom = S of sign | V of var

type literal = var * sign

datatype ctype =

  CAlpha |

  CFun of ctype * sign_atom * ctype |

  CPair of ctype * ctype |

  CType of string * ctype list |

  CRec of string * typ list

type cdata =

  {ext_ctxt: extended_context,

   alpha_T: typ,

   max_fresh: int Unsynchronized.ref,

   datatype_cache: ((string * typ list) * ctype) list Unsynchronized.ref,

   constr_cache: (styp * ctype) list Unsynchronized.ref}

exception CTYPE of string * ctype list

(* string -> unit *)

fun print_g (s : string) = ()

(* var -> string *)

val string_for_var = signed_string_of_int

(* string -> var list -> string *)

fun string_for_vars sep [] = "0\<^bsub>" ^ sep ^ "\<^esub>"

  | string_for_vars sep xs = space_implode sep (map string_for_var xs)

fun subscript_string_for_vars sep xs =

  if null xs then "" else "\<^bsub>" ^ string_for_vars sep xs ^ "\<^esub>"

(* sign -> string *)

fun string_for_sign Pos = "+"

  | string_for_sign Neg = "-"

(* sign -> sign -> sign *)

fun xor sn1 sn2 = if sn1 = sn2 then Pos else Neg

(* sign -> sign *)

val negate = xor Neg

(* sign_atom -> string *)

fun string_for_sign_atom (S sn) = string_for_sign sn

  | string_for_sign_atom (V j) = string_for_var j

(* literal -> string *)

fun string_for_literal (x, sn) = string_for_var x ^ " = " ^ string_for_sign sn

val bool_C = CType (@{type_name bool}, [])

(* ctype -> bool *)

fun is_CRec (CRec _) = true

  | is_CRec _ = false

val no_prec = 100

val prec_CFun = 1

val prec_CPair = 2

(* tuple_set -> int *)

fun precedence_of_ctype (CFun _) = prec_CFun

  | precedence_of_ctype (CPair _) = prec_CPair

  | precedence_of_ctype _ = no_prec

(* ctype -> string *)

val string_for_ctype =

  let

    (* int -> ctype -> string *)

    fun aux outer_prec C =

      let

        val prec = precedence_of_ctype C

        val need_parens = (prec < outer_prec)

      in

        (if need_parens then "(" else "") ^

        (case C of

           CAlpha => "\<alpha>"

         | CFun (C1, a, C2) =>

           aux (prec + 1) C1 ^ " \<Rightarrow>\<^bsup>" ^

           string_for_sign_atom a ^ "\<^esup> " ^ aux prec C2

         | CPair (C1, C2) => aux (prec + 1) C1 ^ " \<times> " ^ aux prec C2

         | CType (s, []) =>

           if s = @{type_name prop} orelse s = @{type_name bool} then "o" else s

         | CType (s, Cs) => "(" ^ commas (map (aux 0) Cs) ^ ") " ^ s

         | CRec (s, _) => "[" ^ s ^ "]") ^

        (if need_parens then ")" else "")

      end

  in aux 0 end

(* ctype -> ctype list *)

fun flatten_ctype (CPair (C1, C2)) = maps flatten_ctype [C1, C2]

  | flatten_ctype (CType (_, Cs)) = maps flatten_ctype Cs

  | flatten_ctype C = [C]

(* extended_context -> typ -> cdata *)

fun initial_cdata ext_ctxt alpha_T =

  ({ext_ctxt = ext_ctxt, alpha_T = alpha_T, max_fresh = Unsynchronized.ref 0,

    datatype_cache = Unsynchronized.ref [],

    constr_cache = Unsynchronized.ref []} : cdata)

(* typ -> typ -> bool *)

fun could_exist_alpha_subtype alpha_T (T as Type (_, Ts)) =

    T = alpha_T orelse (not (is_fp_iterator_type T)

                        andalso exists (could_exist_alpha_subtype alpha_T) Ts)

  | could_exist_alpha_subtype alpha_T T = (T = alpha_T)

(* theory -> typ -> typ -> bool *)

fun could_exist_alpha_sub_ctype _ (alpha_T as TFree _) T =

    could_exist_alpha_subtype alpha_T T

  | could_exist_alpha_sub_ctype thy alpha_T T =

    (T = alpha_T orelse is_datatype thy T)

(* ctype -> bool *)

fun exists_alpha_sub_ctype CAlpha = true

  | exists_alpha_sub_ctype (CFun (C1, _, C2)) =

    exists exists_alpha_sub_ctype [C1, C2]

  | exists_alpha_sub_ctype (CPair (C1, C2)) =

    exists exists_alpha_sub_ctype [C1, C2]

  | exists_alpha_sub_ctype (CType (_, Cs)) = exists exists_alpha_sub_ctype Cs

  | exists_alpha_sub_ctype (CRec _) = true

(* ctype -> bool *)

fun exists_alpha_sub_ctype_fresh CAlpha = true

  | exists_alpha_sub_ctype_fresh (CFun (_, V _, _)) = true

  | exists_alpha_sub_ctype_fresh (CFun (_, _, C2)) =

    exists_alpha_sub_ctype_fresh C2

  | exists_alpha_sub_ctype_fresh (CPair (C1, C2)) =

    exists exists_alpha_sub_ctype_fresh [C1, C2]

  | exists_alpha_sub_ctype_fresh (CType (_, Cs)) =

    exists exists_alpha_sub_ctype_fresh Cs

  | exists_alpha_sub_ctype_fresh (CRec _) = true

(* string * typ list -> ctype list -> ctype *)

fun constr_ctype_for_binders z Cs =

  fold_rev (fn C => curry3 CFun C (S Neg)) Cs (CRec z)

(* ((string * typ list) * ctype) list -> ctype list -> ctype -> ctype *)

fun repair_ctype _ _ CAlpha = CAlpha

  | repair_ctype cache seen (CFun (C1, a, C2)) =

    CFun (repair_ctype cache seen C1, a, repair_ctype cache seen C2)

  | repair_ctype cache seen (CPair Cp) =

    CPair (pairself (repair_ctype cache seen) Cp)

  | repair_ctype cache seen (CType (s, Cs)) =

    CType (s, maps (flatten_ctype o repair_ctype cache seen) Cs)

  | repair_ctype cache seen (CRec (z as (s, _))) =

    case AList.lookup (op =) cache z |> the of

      CRec _ => CType (s, [])

    | C => if member (op =) seen C then CType (s, [])

           else repair_ctype cache (C :: seen) C

(* ((string * typ list) * ctype) list Unsynchronized.ref -> unit *)

fun repair_datatype_cache cache =

  let

    (* (string * typ list) * ctype -> unit *)

    fun repair_one (z, C) =

      Unsynchronized.change cache

          (AList.update (op =) (z, repair_ctype (!cache) [] C))

  in List.app repair_one (rev (!cache)) end

(* (typ * ctype) list -> (styp * ctype) list Unsynchronized.ref -> unit *)

fun repair_constr_cache dtype_cache constr_cache =

  let

    (* styp * ctype -> unit *)

    fun repair_one (x, C) =

      Unsynchronized.change constr_cache

          (AList.update (op =) (x, repair_ctype dtype_cache [] C))

  in List.app repair_one (!constr_cache) end

(* cdata -> typ -> ctype *)

fun fresh_ctype_for_type ({ext_ctxt as {thy, ...}, alpha_T, max_fresh,

                           datatype_cache, constr_cache, ...} : cdata) =

  let

    (* typ -> typ -> ctype *)

    fun do_fun T1 T2 =

      let

        val C1 = do_type T1

        val C2 = do_type T2

        val a = if is_boolean_type (body_type T2)

                   andalso exists_alpha_sub_ctype_fresh C1 then

                  V (Unsynchronized.inc max_fresh)

                else

                  S Neg

      in CFun (C1, a, C2) end

    (* typ -> ctype *)

    and do_type T =

      if T = alpha_T then

        CAlpha

      else case T of

        Type ("fun", [T1, T2]) => do_fun T1 T2

      | Type (@{type_name fun_box}, [T1, T2]) => do_fun T1 T2

      | Type ("*", [T1, T2]) => CPair (pairself do_type (T1, T2))

      | Type (z as (s, _)) =>

        if could_exist_alpha_sub_ctype thy alpha_T T then

          case AList.lookup (op =) (!datatype_cache) z of

            SOME C => C

          | NONE =>

            let

              val _ = Unsynchronized.change datatype_cache (cons (z, CRec z))

              val xs = datatype_constrs ext_ctxt T

              val (all_Cs, constr_Cs) =

                fold_rev (fn (_, T') => fn (all_Cs, constr_Cs) =>

                             let

                               val binder_Cs = map do_type (binder_types T')

                               val new_Cs = filter exists_alpha_sub_ctype_fresh

                                                   binder_Cs

                               val constr_C = constr_ctype_for_binders z

                                                                       binder_Cs

                             in

                               (union (op =) new_Cs all_Cs,

                                constr_C :: constr_Cs)

                             end)

                         xs ([], [])

              val C = CType (s, all_Cs)

              val _ = Unsynchronized.change datatype_cache

                          (AList.update (op =) (z, C))

              val _ = Unsynchronized.change constr_cache

                          (append (xs ~~ constr_Cs))

            in

              if forall (not o is_CRec o snd) (!datatype_cache) then

                (repair_datatype_cache datatype_cache;

                 repair_constr_cache (!datatype_cache) constr_cache;

                 AList.lookup (op =) (!datatype_cache) z |> the)

              else

                C

            end

        else

          CType (s, [])

      | _ => CType (Refute.string_of_typ T, [])

  in do_type end

(* ctype -> ctype list *)

fun prodC_factors (CPair (C1, C2)) = maps prodC_factors [C1, C2]

  | prodC_factors C = [C]

(* ctype -> ctype list * ctype *)

fun curried_strip_ctype (CFun (C1, S Neg, C2)) =

    curried_strip_ctype C2 |>> append (prodC_factors C1)

  | curried_strip_ctype C = ([], C)

(* string -> ctype -> ctype *)

fun sel_ctype_from_constr_ctype s C =

  let val (arg_Cs, dataC) = curried_strip_ctype C in

    CFun (dataC, S Neg,

          case sel_no_from_name s of ~1 => bool_C | n => nth arg_Cs n)

  end

(* cdata -> styp -> ctype *)

fun ctype_for_constr (cdata as {ext_ctxt as {thy, ...}, alpha_T, constr_cache,

                                ...}) (x as (_, T)) =

  if could_exist_alpha_sub_ctype thy alpha_T T then

    case AList.lookup (op =) (!constr_cache) x of

      SOME C => C

    | NONE => (fresh_ctype_for_type cdata (body_type T);

               AList.lookup (op =) (!constr_cache) x |> the)

  else

    fresh_ctype_for_type cdata T

fun ctype_for_sel (cdata as {ext_ctxt, ...}) (x as (s, _)) =

  x |> boxed_constr_for_sel ext_ctxt |> ctype_for_constr cdata

    |> sel_ctype_from_constr_ctype s

(* literal list -> ctype -> ctype *)

fun instantiate_ctype lits =

  let

    (* ctype -> ctype *)

    fun aux CAlpha = CAlpha

      | aux (CFun (C1, V x, C2)) =

        let

          val a = case AList.lookup (op =) lits x of

                    SOME sn => S sn

                  | NONE => V x

        in CFun (aux C1, a, aux C2) end

      | aux (CFun (C1, a, C2)) = CFun (aux C1, a, aux C2)

      | aux (CPair Cp) = CPair (pairself aux Cp)

      | aux (CType (s, Cs)) = CType (s, map aux Cs)

      | aux (CRec z) = CRec z

  in aux end

datatype comp_op = Eq | Leq

type comp = sign_atom * sign_atom * comp_op * var list

type sign_expr = literal list

datatype constraint_set =

  UnsolvableCSet |

  CSet of literal list * comp list * sign_expr list

(* comp_op -> string *)

fun string_for_comp_op Eq = "="

  | string_for_comp_op Leq = "\<le>"

(* sign_expr -> string *)

fun string_for_sign_expr [] = "\<bot>"

  | string_for_sign_expr lits =

    space_implode " \<or> " (map string_for_literal lits)

(* constraint_set *)

val slack = CSet ([], [], [])

(* literal -> literal list option -> literal list option *)

fun do_literal _ NONE = NONE

  | do_literal (x, sn) (SOME lits) =

    case AList.lookup (op =) lits x of

      SOME sn' => if sn = sn' then SOME lits else NONE

    | NONE => SOME ((x, sn) :: lits)

(* comp_op -> var list -> sign_atom -> sign_atom -> literal list * comp list

   -> (literal list * comp list) option *)

fun do_sign_atom_comp Eq [] a1 a2 (accum as (lits, comps)) =

    (case (a1, a2) of

       (S sn1, S sn2) => if sn1 = sn2 then SOME accum else NONE

     | (V x1, S sn2) =>

       Option.map (rpair comps) (do_literal (x1, sn2) (SOME lits))

     | (V _, V _) => SOME (lits, insert (op =) (a1, a2, Eq, []) comps)

     | _ => do_sign_atom_comp Eq [] a2 a1 accum)

  | do_sign_atom_comp Leq [] a1 a2 (accum as (lits, comps)) =

    (case (a1, a2) of

       (_, S Neg) => SOME accum

     | (S Pos, _) => SOME accum

     | (S Neg, S Pos) => NONE

     | (V _, V _) => SOME (lits, insert (op =) (a1, a2, Leq, []) comps)

     | _ => do_sign_atom_comp Eq [] a1 a2 accum)

  | do_sign_atom_comp cmp xs a1 a2 (accum as (lits, comps)) =

    SOME (lits, insert (op =) (a1, a2, cmp, xs) comps)

(* comp -> var list -> ctype -> ctype -> (literal list * comp list) option

   -> (literal list * comp list) option *)

fun do_ctype_comp _ _ _ _ NONE = NONE

  | do_ctype_comp _ _ CAlpha CAlpha accum = accum

  | do_ctype_comp Eq xs (CFun (C11, a1, C12)) (CFun (C21, a2, C22))

                  (SOME accum) =

     accum |> do_sign_atom_comp Eq xs a1 a2 |> do_ctype_comp Eq xs C11 C21

           |> do_ctype_comp Eq xs C12 C22

  | do_ctype_comp Leq xs (CFun (C11, a1, C12)) (CFun (C21, a2, C22))

                  (SOME accum) =

    (if exists_alpha_sub_ctype C11 then

       accum |> do_sign_atom_comp Leq xs a1 a2

             |> do_ctype_comp Leq xs C21 C11

             |> (case a2 of

                   S Neg => I

                 | S Pos => do_ctype_comp Leq xs C11 C21

                 | V x => do_ctype_comp Leq (x :: xs) C11 C21)

     else

       SOME accum)

    |> do_ctype_comp Leq xs C12 C22

  | do_ctype_comp cmp xs (C1 as CPair (C11, C12)) (C2 as CPair (C21, C22))

                  accum =

    (accum |> fold (uncurry (do_ctype_comp cmp xs)) [(C11, C21), (C12, C22)]

     handle Library.UnequalLengths =>

            raise CTYPE ("Nitpick_Mono.do_ctype_comp", [C1, C2]))

  | do_ctype_comp cmp xs (CType _) (CType _) accum =

    accum (* no need to compare them thanks to the cache *)

  | do_ctype_comp _ _ C1 C2 _ =

    raise CTYPE ("Nitpick_Mono.do_ctype_comp", [C1, C2])

(* comp_op -> ctype -> ctype -> constraint_set -> constraint_set *)

fun add_ctype_comp _ _ _ UnsolvableCSet = UnsolvableCSet

  | add_ctype_comp cmp C1 C2 (CSet (lits, comps, sexps)) =

    (print_g ("*** Add " ^ string_for_ctype C1 ^ " " ^ string_for_comp_op cmp ^

              " " ^ string_for_ctype C2);

     case do_ctype_comp cmp [] C1 C2 (SOME (lits, comps)) of

       NONE => (print_g "**** Unsolvable"; UnsolvableCSet)

     | SOME (lits, comps) => CSet (lits, comps, sexps))

(* ctype -> ctype -> constraint_set -> constraint_set *)

val add_ctypes_equal = add_ctype_comp Eq

val add_is_sub_ctype = add_ctype_comp Leq

(* sign -> sign_expr -> ctype -> (literal list * sign_expr list) option

   -> (literal list * sign_expr list) option *)

fun do_notin_ctype_fv _ _ _ NONE = NONE

  | do_notin_ctype_fv Neg _ CAlpha accum = accum

  | do_notin_ctype_fv Pos [] CAlpha _ = NONE

  | do_notin_ctype_fv Pos [(x, sn)] CAlpha (SOME (lits, sexps)) =

    SOME lits |> do_literal (x, sn) |> Option.map (rpair sexps)

  | do_notin_ctype_fv Pos sexp CAlpha (SOME (lits, sexps)) =

    SOME (lits, insert (op =) sexp sexps)

  | do_notin_ctype_fv sn sexp (CFun (C1, S sn', C2)) accum =

    accum |> (if sn' = Pos andalso sn = Pos then do_notin_ctype_fv Pos sexp C1

              else I)

          |> (if sn' = Neg orelse sn = Pos then do_notin_ctype_fv Neg sexp C1

              else I)

          |> do_notin_ctype_fv sn sexp C2

  | do_notin_ctype_fv Pos sexp (CFun (C1, V x, C2)) accum =

    accum |> (case do_literal (x, Neg) (SOME sexp) of

                NONE => I

              | SOME sexp' => do_notin_ctype_fv Pos sexp' C1)

          |> do_notin_ctype_fv Neg sexp C1

          |> do_notin_ctype_fv Pos sexp C2

  | do_notin_ctype_fv Neg sexp (CFun (C1, V x, C2)) accum =

    accum |> (case do_literal (x, Pos) (SOME sexp) of

                NONE => I

              | SOME sexp' => do_notin_ctype_fv Pos sexp' C1)

          |> do_notin_ctype_fv Neg sexp C2

  | do_notin_ctype_fv sn sexp (CPair (C1, C2)) accum =

    accum |> fold (do_notin_ctype_fv sn sexp) [C1, C2]

  | do_notin_ctype_fv sn sexp (CType (_, Cs)) accum =

    accum |> fold (do_notin_ctype_fv sn sexp) Cs

  | do_notin_ctype_fv _ _ C _ =

    raise CTYPE ("Nitpick_Mono.do_notin_ctype_fv", [C])

(* sign -> ctype -> constraint_set -> constraint_set *)

fun add_notin_ctype_fv _ _ UnsolvableCSet = UnsolvableCSet

  | add_notin_ctype_fv sn C (CSet (lits, comps, sexps)) =

    (print_g ("*** Add " ^ string_for_ctype C ^ " is right-" ^

              (case sn of Neg => "unique" | Pos => "total") ^ ".");

     case do_notin_ctype_fv sn [] C (SOME (lits, sexps)) of

       NONE => (print_g "**** Unsolvable"; UnsolvableCSet)

     | SOME (lits, sexps) => CSet (lits, comps, sexps))

(* ctype -> constraint_set -> constraint_set *)

val add_ctype_is_right_unique = add_notin_ctype_fv Neg

val add_ctype_is_right_total = add_notin_ctype_fv Pos

(* constraint_set -> constraint_set -> constraint_set *)

fun unite (CSet (lits1, comps1, sexps1)) (CSet (lits2, comps2, sexps2)) =

    (case SOME lits1 |> fold do_literal lits2 of

       NONE => (print_g "**** Unsolvable"; UnsolvableCSet)

     | SOME lits => CSet (lits, comps1 @ comps2, sexps1 @ sexps2))

  | unite _ _ = UnsolvableCSet

(* sign -> bool *)

fun bool_from_sign Pos = false

  | bool_from_sign Neg = true

(* bool -> sign *)

fun sign_from_bool false = Pos

  | sign_from_bool true = Neg

(* literal -> PropLogic.prop_formula *)

fun prop_for_literal (x, sn) =

  (not (bool_from_sign sn) ? PropLogic.Not) (PropLogic.BoolVar x)

(* sign_atom -> PropLogic.prop_formula *)

fun prop_for_sign_atom_eq (S sn', sn) =

    if sn = sn' then PropLogic.True else PropLogic.False

  | prop_for_sign_atom_eq (V x, sn) = prop_for_literal (x, sn)

(* sign_expr -> PropLogic.prop_formula *)

fun prop_for_sign_expr xs = PropLogic.exists (map prop_for_literal xs)

(* var list -> sign -> PropLogic.prop_formula *)

fun prop_for_exists_eq xs sn =

  PropLogic.exists (map (fn x => prop_for_literal (x, sn)) xs)

(* comp -> PropLogic.prop_formula *)

fun prop_for_comp (a1, a2, Eq, []) =

    PropLogic.SAnd (prop_for_comp (a1, a2, Leq, []),

                    prop_for_comp (a2, a1, Leq, []))

  | prop_for_comp (a1, a2, Leq, []) =

    PropLogic.SOr (prop_for_sign_atom_eq (a1, Pos),

                   prop_for_sign_atom_eq (a2, Neg))

  | prop_for_comp (a1, a2, cmp, xs) =

    PropLogic.SOr (prop_for_exists_eq xs Neg, prop_for_comp (a1, a2, cmp, []))

(* var -> (int -> bool option) -> literal list -> literal list *)

fun literals_from_assignments max_var assigns lits =

  fold (fn x => fn accum =>

           if AList.defined (op =) lits x then

             accum

           else case assigns x of

             SOME b => (x, sign_from_bool b) :: accum

           | NONE => accum) (max_var downto 1) lits

(* literal list -> sign_atom -> sign option *)

fun lookup_sign_atom _ (S sn) = SOME sn

  | lookup_sign_atom lit (V x) = AList.lookup (op =) lit x

(* comp -> string *)

fun string_for_comp (a1, a2, cmp, xs) =

  string_for_sign_atom a1 ^ " " ^ string_for_comp_op cmp ^

  subscript_string_for_vars " \<and> " xs ^ " " ^ string_for_sign_atom a2

(* literal list -> comp list -> sign_expr list -> unit *)

fun print_problem lits comps sexps =

  print_g ("*** Problem:\n" ^ cat_lines (map string_for_literal lits @

                                         map string_for_comp comps @

                                         map string_for_sign_expr sexps))

(* literal list -> unit *)

fun print_solution lits =

  let val (pos, neg) = List.partition (curry (op =) Pos o snd) lits in

    print_g ("*** Solution:\n" ^

             "+: " ^ commas (map (string_for_var o fst) pos) ^ "\n" ^

             "-: " ^ commas (map (string_for_var o fst) neg))

  end

(* var -> constraint_set -> literal list list option *)

fun solve _ UnsolvableCSet = (print_g "*** Problem: Unsolvable"; NONE)

  | solve max_var (CSet (lits, comps, sexps)) =

    let

      val _ = print_problem lits comps sexps

      val prop = PropLogic.all (map prop_for_literal lits @

                                map prop_for_comp comps @

                                map prop_for_sign_expr sexps)

      (* use the first ML solver (to avoid startup overhead) *)

      val solvers = !SatSolver.solvers

                    |> filter (member (op =) ["dptsat", "dpll"] o fst)

    in

      case snd (hd solvers) prop of

        SatSolver.SATISFIABLE assigns =>

        SOME (literals_from_assignments max_var assigns lits

              |> tap print_solution)

      | _ => NONE

    end

(* var -> constraint_set -> bool *)

val is_solvable = is_some oo solve

type ctype_schema = ctype * constraint_set

type ctype_context =

  {bounds: ctype list,

   frees: (styp * ctype) list,

   consts: (styp * ctype_schema) list}

type accumulator = ctype_context * constraint_set

val initial_gamma = {bounds = [], frees = [], consts = []}

val unsolvable_accum = (initial_gamma, UnsolvableCSet)

(* ctype -> ctype_context -> ctype_context *)

fun push_bound C {bounds, frees, consts} =

  {bounds = C :: bounds, frees = frees, consts = consts}

(* ctype_context -> ctype_context *)

fun pop_bound {bounds, frees, consts} =

  {bounds = tl bounds, frees = frees, consts = consts}

  handle List.Empty => initial_gamma

(* cdata -> term -> accumulator -> ctype * accumulator *)

fun consider_term (cdata as {ext_ctxt as {ctxt, thy, def_table, ...}, alpha_T,

                             max_fresh, ...}) =

  let

    (* typ -> ctype *)

    val ctype_for = fresh_ctype_for_type cdata

    (* ctype -> ctype *)

    fun pos_set_ctype_for_dom C =

      CFun (C, S (if exists_alpha_sub_ctype C then Pos else Neg), bool_C)

    (* typ -> accumulator -> ctype * accumulator *)

    fun do_quantifier T (gamma, cset) =

      let

        val abs_C = ctype_for (domain_type (domain_type T))

        val body_C = ctype_for (range_type T)

      in

        (CFun (CFun (abs_C, S Neg, body_C), S Neg, body_C),

         (gamma, cset |> add_ctype_is_right_total abs_C))

      end

    fun do_equals T (gamma, cset) =

      let val C = ctype_for (domain_type T) in

        (CFun (C, S Neg, CFun (C, V (Unsynchronized.inc max_fresh),

                               ctype_for (nth_range_type 2 T))),

         (gamma, cset |> add_ctype_is_right_unique C))

      end

    fun do_robust_set_operation T (gamma, cset) =

      let

        val set_T = domain_type T

        val C1 = ctype_for set_T

        val C2 = ctype_for set_T

        val C3 = ctype_for set_T

      in

        (CFun (C1, S Neg, CFun (C2, S Neg, C3)),

         (gamma, cset |> add_is_sub_ctype C1 C3 |> add_is_sub_ctype C2 C3))

      end

    fun do_fragile_set_operation T (gamma, cset) =

      let

        val set_T = domain_type T

        val set_C = ctype_for set_T

        (* typ -> ctype *)

        fun custom_ctype_for (T as Type ("fun", [T1, T2])) =

            if T = set_T then set_C

            else CFun (custom_ctype_for T1, S Neg, custom_ctype_for T2)

          | custom_ctype_for T = ctype_for T

      in

        (custom_ctype_for T, (gamma, cset |> add_ctype_is_right_unique set_C))

      end

    (* typ -> accumulator -> ctype * accumulator *)

    fun do_pair_constr T accum =

      case ctype_for (nth_range_type 2 T) of

        C as CPair (a_C, b_C) =>

        (CFun (a_C, S Neg, CFun (b_C, S Neg, C)), accum)

      | C => raise CTYPE ("Nitpick_Mono.consider_term.do_pair_constr", [C])

    (* int -> typ -> accumulator -> ctype * accumulator *)

    fun do_nth_pair_sel n T =

      case ctype_for (domain_type T) of

        C as CPair (a_C, b_C) =>

        pair (CFun (C, S Neg, if n = 0 then a_C else b_C))

      | C => raise CTYPE ("Nitpick_Mono.consider_term.do_nth_pair_sel", [C])

    val unsolvable = (CType ("unsolvable", []), unsolvable_accum)

    (* typ -> term -> accumulator -> ctype * accumulator *)

    fun do_bounded_quantifier abs_T bound_t body_t accum =

      let

        val abs_C = ctype_for abs_T

        val (bound_C, accum) = accum |>> push_bound abs_C |> do_term bound_t

        val expected_bound_C = pos_set_ctype_for_dom abs_C

      in

        accum ||> add_ctypes_equal expected_bound_C bound_C |> do_term body_t

              ||> apfst pop_bound

      end

    (* term -> accumulator -> ctype * accumulator *)

    and do_term _ (_, UnsolvableCSet) = unsolvable

      | do_term t (accum as (gamma as {bounds, frees, consts}, cset)) =

        (case t of

           Const (x as (s, T)) =>

           (case AList.lookup (op =) consts x of

              SOME (C, cset') => (C, (gamma, cset |> unite cset'))

            | NONE =>

              if not (could_exist_alpha_subtype alpha_T T) then

                (ctype_for T, accum)

              else case s of

                @{const_name all} => do_quantifier T accum

              | @{const_name "=="} => do_equals T accum

              | @{const_name All} => do_quantifier T accum

              | @{const_name Ex} => do_quantifier T accum

              | @{const_name "op ="} => do_equals T accum

              | @{const_name The} => (print_g "*** The"; unsolvable)

              | @{const_name Eps} => (print_g "*** Eps"; unsolvable)

              | @{const_name If} =>

                do_robust_set_operation (range_type T) accum

                |>> curry3 CFun bool_C (S Neg)

              | @{const_name Pair} => do_pair_constr T accum

              | @{const_name fst} => do_nth_pair_sel 0 T accum

              | @{const_name snd} => do_nth_pair_sel 1 T accum 

              | @{const_name Id} =>

                (CFun (ctype_for (domain_type T), S Neg, bool_C), accum)

              | @{const_name insert} =>

                let

                  val set_T = domain_type (range_type T)

                  val C1 = ctype_for (domain_type set_T)

                  val C1' = pos_set_ctype_for_dom C1

                  val C2 = ctype_for set_T

                  val C3 = ctype_for set_T

                in

                  (CFun (C1, S Neg, CFun (C2, S Neg, C3)),

                   (gamma, cset |> add_ctype_is_right_unique C1

                                |> add_is_sub_ctype C1' C3

                                |> add_is_sub_ctype C2 C3))

                end

              | @{const_name converse} =>

                let

                  val x = Unsynchronized.inc max_fresh

                  (* typ -> ctype *)

                  fun ctype_for_set T =

                    CFun (ctype_for (domain_type T), V x, bool_C)

                  val ab_set_C = domain_type T |> ctype_for_set

                  val ba_set_C = range_type T |> ctype_for_set

                in (CFun (ab_set_C, S Neg, ba_set_C), accum) end

              | @{const_name trancl} => do_fragile_set_operation T accum

              | @{const_name rtrancl} => (print_g "*** rtrancl"; unsolvable)

              | @{const_name lower_semilattice_fun_inst.inf_fun} =>

                do_robust_set_operation T accum

              | @{const_name upper_semilattice_fun_inst.sup_fun} =>

                do_robust_set_operation T accum

              | @{const_name finite} =>

                let val C1 = ctype_for (domain_type (domain_type T)) in

                  (CFun (pos_set_ctype_for_dom C1, S Neg, bool_C), accum)

                end

              | @{const_name rel_comp} =>

                let

                  val x = Unsynchronized.inc max_fresh

                  (* typ -> ctype *)

                  fun ctype_for_set T =

                    CFun (ctype_for (domain_type T), V x, bool_C)

                  val bc_set_C = domain_type T |> ctype_for_set

                  val ab_set_C = domain_type (range_type T) |> ctype_for_set

                  val ac_set_C = nth_range_type 2 T |> ctype_for_set

                in

                  (CFun (bc_set_C, S Neg, CFun (ab_set_C, S Neg, ac_set_C)),

                   accum)

                end

              | @{const_name image} =>

                let

                  val a_C = ctype_for (domain_type (domain_type T))

                  val b_C = ctype_for (range_type (domain_type T))

                in

                  (CFun (CFun (a_C, S Neg, b_C), S Neg,

                         CFun (pos_set_ctype_for_dom a_C, S Neg,

                               pos_set_ctype_for_dom b_C)), accum)

                end

              | @{const_name Sigma} =>

                let

                  val x = Unsynchronized.inc max_fresh

                  (* typ -> ctype *)

                  fun ctype_for_set T =

                    CFun (ctype_for (domain_type T), V x, bool_C)

                  val a_set_T = domain_type T

                  val a_C = ctype_for (domain_type a_set_T)

                  val b_set_C = ctype_for_set (range_type (domain_type

                                                               (range_type T)))

                  val a_set_C = ctype_for_set a_set_T

                  val a_to_b_set_C = CFun (a_C, S Neg, b_set_C)

                  val ab_set_C = ctype_for_set (nth_range_type 2 T)

                in

                  (CFun (a_set_C, S Neg, CFun (a_to_b_set_C, S Neg, ab_set_C)),

                   accum)

                end

              | @{const_name minus_fun_inst.minus_fun} =>

                let

                  val set_T = domain_type T

                  val left_set_C = ctype_for set_T

                  val right_set_C = ctype_for set_T

                in

                  (CFun (left_set_C, S Neg,

                         CFun (right_set_C, S Neg, left_set_C)),

                   (gamma, cset |> add_ctype_is_right_unique right_set_C

                                |> add_is_sub_ctype right_set_C left_set_C))

                end

              | @{const_name ord_fun_inst.less_eq_fun} =>

                do_fragile_set_operation T accum

              | @{const_name Tha} =>

                let

                  val a_C = ctype_for (domain_type (domain_type T))

                  val a_set_C = pos_set_ctype_for_dom a_C

                in (CFun (a_set_C, S Neg, a_C), accum) end

              | @{const_name FunBox} =>

                let val dom_C = ctype_for (domain_type T) in

                  (CFun (dom_C, S Neg, dom_C), accum)

                end

              | _ => if is_sel s then

                       if constr_name_for_sel_like s = @{const_name FunBox} then

                         let val dom_C = ctype_for (domain_type T) in

                           (CFun (dom_C, S Neg, dom_C), accum)

                         end

                       else

                         (ctype_for_sel cdata x, accum)

                     else if is_constr thy x then

                       (ctype_for_constr cdata x, accum)

                     else if is_built_in_const true x then

                       case def_of_const thy def_table x of

                         SOME t' => do_term t' accum

                       | NONE => (print_g ("*** built-in " ^ s); unsolvable)

                     else

                       (ctype_for T, accum))

         | Free (x as (_, T)) =>

           (case AList.lookup (op =) frees x of

              SOME C => (C, accum)

            | NONE =>

              let val C = ctype_for T in

                (C, ({bounds = bounds, frees = (x, C) :: frees,

                      consts = consts}, cset))

              end)

         | Var _ => (print_g "*** Var"; unsolvable)

         | Bound j => (nth bounds j, accum)

         | Abs (_, T, @{const False}) => (ctype_for (T --> bool_T), accum)

         | Abs (s, T, t') =>

           let

             val C = ctype_for T

             val (C', accum) = do_term t' (accum |>> push_bound C)

           in (CFun (C, S Neg, C'), accum |>> pop_bound) end

         | Const (@{const_name All}, _)

           $ Abs (_, T', @{const "op -->"} $ (t1 $ Bound 0) $ t2) =>

           do_bounded_quantifier T' t1 t2 accum

         | Const (@{const_name Ex}, _)

           $ Abs (_, T', @{const "op &"} $ (t1 $ Bound 0) $ t2) =>

           do_bounded_quantifier T' t1 t2 accum

         | Const (@{const_name Let}, _) $ t1 $ t2 =>

           do_term (betapply (t2, t1)) accum

         | t1 $ t2 =>

           let

             val (C1, accum) = do_term t1 accum

             val (C2, accum) = do_term t2 accum

           in

             case accum of

               (_, UnsolvableCSet) => unsolvable

             | _ => case C1 of

                      CFun (C11, _, C12) =>

                      (C12, accum ||> add_is_sub_ctype C2 C11)

                    | _ => raise CTYPE ("Nitpick_Mono.consider_term.do_term \

                                        \(op $)", [C1])

           end)

        |> tap (fn (C, _) =>

                   print_g ("  \<Gamma> \<turnstile> " ^

                            Syntax.string_of_term ctxt t ^ " : " ^

                            string_for_ctype C))

  in do_term end

(* cdata -> sign -> term -> accumulator -> accumulator *)

fun consider_general_formula (cdata as {ext_ctxt as {ctxt, ...}, ...}) =

  let

    (* typ -> ctype *)

    val ctype_for = fresh_ctype_for_type cdata

    (* term -> accumulator -> ctype * accumulator *)

    val do_term = consider_term cdata

    (* sign -> term -> accumulator -> accumulator *)

    fun do_formula _ _ (_, UnsolvableCSet) = unsolvable_accum

      | do_formula sn t (accum as (gamma as {bounds, frees, consts}, cset)) =

        let

          (* term -> accumulator -> accumulator *)

          val do_co_formula = do_formula sn

          val do_contra_formula = do_formula (negate sn)

          (* string -> typ -> term -> accumulator *)

          fun do_quantifier quant_s abs_T body_t =

            let

              val abs_C = ctype_for abs_T

              val side_cond = ((sn = Neg) = (quant_s = @{const_name Ex}))

              val cset = cset |> side_cond ? add_ctype_is_right_total abs_C

            in

              (gamma |> push_bound abs_C, cset) |> do_co_formula body_t

                                                |>> pop_bound

            end

          (* typ -> term -> accumulator *)

          fun do_bounded_quantifier abs_T body_t =

            accum |>> push_bound (ctype_for abs_T) |> do_co_formula body_t

                  |>> pop_bound

          (* term -> term -> accumulator *)

          fun do_equals t1 t2 =

            case sn of

              Pos => do_term t accum |> snd

            | Neg => let

                       val (C1, accum) = do_term t1 accum

                       val (C2, accum) = do_term t2 accum

                     in accum ||> add_ctypes_equal C1 C2 end

        in

          case t of

            Const (s0 as @{const_name all}, _) $ Abs (_, T1, t1) =>

            do_quantifier s0 T1 t1

          | Const (@{const_name "=="}, _) $ t1 $ t2 => do_equals t1 t2

          | @{const "==>"} $ t1 $ t2 =>

            accum |> do_contra_formula t1 |> do_co_formula t2

          | @{const Trueprop} $ t1 => do_co_formula t1 accum

          | @{const Not} $ t1 => do_contra_formula t1 accum

          | Const (@{const_name All}, _)

            $ Abs (_, T1, t1 as @{const "op -->"} $ (_ $ Bound 0) $ _) =>

            do_bounded_quantifier T1 t1

          | Const (s0 as @{const_name All}, _) $ Abs (_, T1, t1) =>

            do_quantifier s0 T1 t1

          | Const (@{const_name Ex}, _)

            $ Abs (_, T1, t1 as @{const "op &"} $ (_ $ Bound 0) $ _) =>

            do_bounded_quantifier T1 t1

          | Const (s0 as @{const_name Ex}, _) $ Abs (_, T1, t1) =>

            do_quantifier s0 T1 t1

          | Const (@{const_name "op ="}, _) $ t1 $ t2 => do_equals t1 t2

          | @{const "op &"} $ t1 $ t2 =>

            accum |> do_co_formula t1 |> do_co_formula t2

          | @{const "op |"} $ t1 $ t2 =>

            accum |> do_co_formula t1 |> do_co_formula t2

          | @{const "op -->"} $ t1 $ t2 =>

            accum |> do_contra_formula t1 |> do_co_formula t2

          | Const (@{const_name If}, _) $ t1 $ t2 $ t3 =>

            accum |> do_term t1 |> snd |> fold do_co_formula [t2, t3]

          | Const (@{const_name Let}, _) $ t1 $ t2 =>

            do_co_formula (betapply (t2, t1)) accum

          | _ => do_term t accum |> snd

        end

        |> tap (fn _ => print_g ("\<Gamma> \<turnstile> " ^

                                 Syntax.string_of_term ctxt t ^

                                 " : o\<^sup>" ^ string_for_sign sn))

  in do_formula end

(* The harmless axiom optimization below is somewhat too aggressive in the face

   of (rather peculiar) user-defined axioms. *)

val harmless_consts =

  [@{const_name ord_class.less}, @{const_name ord_class.less_eq}]

val bounteous_consts = [@{const_name bisim}]

(* term -> bool *)

fun is_harmless_axiom t =

  Term.add_consts t [] |> filter_out (is_built_in_const true)

  |> (forall (member (op =) harmless_consts o original_name o fst)

      orf exists (member (op =) bounteous_consts o fst))

(* cdata -> sign -> term -> accumulator -> accumulator *)

fun consider_nondefinitional_axiom cdata sn t =

  not (is_harmless_axiom t) ? consider_general_formula cdata sn t

(* cdata -> term -> accumulator -> accumulator *)

fun consider_definitional_axiom (cdata as {ext_ctxt as {thy, ...}, ...}) t =

  if not (is_constr_pattern_formula thy t) then

    consider_nondefinitional_axiom cdata Pos t

  else if is_harmless_axiom t then

    I

  else

    let

      (* term -> accumulator -> ctype * accumulator *)

      val do_term = consider_term cdata

      (* typ -> term -> accumulator -> accumulator *)

      fun do_all abs_T body_t accum =

        let val abs_C = fresh_ctype_for_type cdata abs_T in

          accum |>> push_bound abs_C |> do_formula body_t |>> pop_bound

        end

      (* term -> term -> accumulator -> accumulator *)

      and do_implies t1 t2 = do_term t1 #> snd #> do_formula t2

      and do_equals t1 t2 accum =

        let

          val (C1, accum) = do_term t1 accum

          val (C2, accum) = do_term t2 accum

        in accum ||> add_ctypes_equal C1 C2 end

      (* term -> accumulator -> accumulator *)

      and do_formula _ (_, UnsolvableCSet) = unsolvable_accum

        | do_formula t accum =

          case t of

            Const (@{const_name all}, _) $ Abs (_, T1, t1) => do_all T1 t1 accum

          | @{const Trueprop} $ t1 => do_formula t1 accum

          | Const (@{const_name "=="}, _) $ t1 $ t2 => do_equals t1 t2 accum

          | @{const "==>"} $ t1 $ t2 => do_implies t1 t2 accum

          | @{const Pure.conjunction} $ t1 $ t2 =>

            accum |> do_formula t1 |> do_formula t2

          | Const (@{const_name All}, _) $ Abs (_, T1, t1) => do_all T1 t1 accum

          | Const (@{const_name "op ="}, _) $ t1 $ t2 => do_equals t1 t2 accum

          | @{const "op &"} $ t1 $ t2 => accum |> do_formula t1 |> do_formula t2

          | @{const "op -->"} $ t1 $ t2 => do_implies t1 t2 accum

          | _ => raise TERM ("Nitpick_Mono.consider_definitional_axiom.\

                             \do_formula", [t])

    in do_formula t end

(* Proof.context -> literal list -> term -> ctype -> string *)

fun string_for_ctype_of_term ctxt lits t C =

  Syntax.string_of_term ctxt t ^ " : " ^

  string_for_ctype (instantiate_ctype lits C)

(* theory -> literal list -> ctype_context -> unit *)

fun print_ctype_context ctxt lits ({frees, consts, ...} : ctype_context) =

  map (fn (x, C) => string_for_ctype_of_term ctxt lits (Free x) C) frees @

  map (fn (x, (C, _)) => string_for_ctype_of_term ctxt lits (Const x) C) consts

  |> cat_lines |> print_g

(* extended_context -> typ -> term list -> term list -> term -> bool *)

fun formulas_monotonic (ext_ctxt as {ctxt, ...}) alpha_T def_ts nondef_ts

                       core_t =

  let

    val _ = print_g ("****** " ^ string_for_ctype CAlpha ^ " is " ^

                     Syntax.string_of_typ ctxt alpha_T)

    val cdata as {max_fresh, ...} = initial_cdata ext_ctxt alpha_T

    val (gamma, cset) =

      (initial_gamma, slack)

      |> fold (consider_definitional_axiom cdata) def_ts

      |> fold (consider_nondefinitional_axiom cdata Pos) nondef_ts

      |> consider_general_formula cdata Pos core_t

  in

    case solve (!max_fresh) cset of

      SOME lits => (print_ctype_context ctxt lits gamma; true)

    | _ => false

  end

  handle CTYPE (loc, Cs) => raise BAD (loc, commas (map string_for_ctype Cs))

end;