(*  Title:      HOL/Tools/Quotient/quotient_term.thy

    Author:     Cezary Kaliszyk and Christian Urban

Constructs terms corresponding to goals from lifting theorems to

quotient types.

*)

signature QUOTIENT_TERM =

sig

  datatype flag = AbsF | RepF

  val absrep_fun: flag -> Proof.context -> typ * typ -> term

  val absrep_fun_chk: flag -> Proof.context -> typ * typ -> term

  (* Allows Nitpick to represent quotient types as single elements from raw type *)

  val absrep_const_chk: flag -> Proof.context -> string -> term

  val equiv_relation: Proof.context -> typ * typ -> term

  val equiv_relation_chk: Proof.context -> typ * typ -> term

  val regularize_trm: Proof.context -> term * term -> term

  val regularize_trm_chk: Proof.context -> term * term -> term

  val inj_repabs_trm: Proof.context -> term * term -> term

  val inj_repabs_trm_chk: Proof.context -> term * term -> term

  val quotient_lift_const: string * term -> local_theory -> term

  val quotient_lift_all: Proof.context -> term -> term

end;

structure Quotient_Term: QUOTIENT_TERM =

struct

open Quotient_Info;

exception LIFT_MATCH of string

(*** Aggregate Rep/Abs Function ***)

(* The flag RepF is for types in negative position; AbsF is for types

   in positive position. Because of this, function types need to be

   treated specially, since there the polarity changes.

*)

datatype flag = AbsF | RepF

fun negF AbsF = RepF

  | negF RepF = AbsF

fun is_identity (Const (@{const_name "id"}, _)) = true

  | is_identity _ = false

fun mk_identity ty = Const (@{const_name "id"}, ty --> ty)

fun mk_fun_compose flag (trm1, trm2) =

  case flag of

    AbsF => Const (@{const_name "comp"}, dummyT) $ trm1 $ trm2

  | RepF => Const (@{const_name "comp"}, dummyT) $ trm2 $ trm1

fun get_mapfun ctxt s =

let

  val thy = ProofContext.theory_of ctxt

  val exn = error ("No map function for type " ^ quote s ^ " found.")

  val mapfun = #mapfun (maps_lookup thy s) handle Quotient_Info.NotFound => raise exn

in

  Const (mapfun, dummyT)

end

(* makes a Free out of a TVar *)

fun mk_Free (TVar ((x, i), _)) = Free (unprefix "'" x ^ string_of_int i, dummyT)

(* produces an aggregate map function for the

   rty-part of a quotient definition; abstracts

   over all variables listed in vs (these variables

   correspond to the type variables in rty)

   for example for: (?'a list * ?'b)

   it produces:     %a b. prod_map (map a) b

*)

fun mk_mapfun ctxt vs rty =

let

  val vs' = map (mk_Free) vs

  fun mk_mapfun_aux rty =

    case rty of

      TVar _ => mk_Free rty

    | Type (_, []) => mk_identity rty

    | Type (s, tys) => list_comb (get_mapfun ctxt s, map mk_mapfun_aux tys)

    | _ => raise (error "mk_mapfun (default)")

in

  fold_rev Term.lambda vs' (mk_mapfun_aux rty)

end

(* looks up the (varified) rty and qty for

   a quotient definition

*)

fun get_rty_qty ctxt s =

let

  val thy = ProofContext.theory_of ctxt

  val exn = error ("No quotient type " ^ quote s ^ " found.")

  val qdata = (quotdata_lookup thy s) handle Quotient_Info.NotFound => raise exn

in

  (#rtyp qdata, #qtyp qdata)

end

(* takes two type-environments and looks

   up in both of them the variable v, which

   must be listed in the environment

*)

fun double_lookup rtyenv qtyenv v =

let

  val v' = fst (dest_TVar v)

in

  (snd (the (Vartab.lookup rtyenv v')), snd (the (Vartab.lookup qtyenv v')))

end

(* matches a type pattern with a type *)

fun match ctxt err ty_pat ty =

let

  val thy = ProofContext.theory_of ctxt

in

  Sign.typ_match thy (ty_pat, ty) Vartab.empty

  handle MATCH_TYPE => err ctxt ty_pat ty

end

(* produces the rep or abs constant for a qty *)

fun absrep_const flag ctxt qty_str =

let

  val thy = ProofContext.theory_of ctxt

  val qty_name = Long_Name.base_name qty_str

in

  case flag of

    AbsF => Const (Sign.full_bname thy ("abs_" ^ qty_name), dummyT)

  | RepF => Const (Sign.full_bname thy ("rep_" ^ qty_name), dummyT)

end

(* Lets Nitpick represent elements of quotient types as elements of the raw type *)

fun absrep_const_chk flag ctxt qty_str =

  Syntax.check_term ctxt (absrep_const flag ctxt qty_str)

fun absrep_match_err ctxt ty_pat ty =

let

  val ty_pat_str = Syntax.string_of_typ ctxt ty_pat

  val ty_str = Syntax.string_of_typ ctxt ty

in

  raise error (cat_lines

    ["absrep_fun (Types ", quote ty_pat_str, "and", quote ty_str, " do not match.)"])

end

(** generation of an aggregate absrep function **)

(* - In case of equal types we just return the identity.

   - In case of TFrees we also return the identity.

   - In case of function types we recurse taking

     the polarity change into account.

   - If the type constructors are equal, we recurse for the

     arguments and build the appropriate map function.

   - If the type constructors are unequal, there must be an

     instance of quotient types:

       - we first look up the corresponding rty_pat and qty_pat

         from the quotient definition; the arguments of qty_pat

         must be some distinct TVars

       - we then match the rty_pat with rty and qty_pat with qty;

         if matching fails the types do not correspond -> error

       - the matching produces two environments; we look up the

         assignments for the qty_pat variables and recurse on the

         assignments

       - we prefix the aggregate map function for the rty_pat,

         which is an abstraction over all type variables

       - finally we compose the result with the appropriate

         absrep function in case at least one argument produced

         a non-identity function /

         otherwise we just return the appropriate absrep

         function

     The composition is necessary for types like

        ('a list) list / ('a foo) foo

     The matching is necessary for types like

        ('a * 'a) list / 'a bar

     The test is necessary in order to eliminate superfluous

     identity maps.

*)

fun absrep_fun flag ctxt (rty, qty) =

  if rty = qty

  then mk_identity rty

  else

    case (rty, qty) of

      (Type ("fun", [ty1, ty2]), Type ("fun", [ty1', ty2'])) =>

        let

          val arg1 = absrep_fun (negF flag) ctxt (ty1, ty1')

          val arg2 = absrep_fun flag ctxt (ty2, ty2')

        in

          list_comb (get_mapfun ctxt "fun", [arg1, arg2])

        end

    | (Type (s, tys), Type (s', tys')) =>

        if s = s'

        then

           let

             val args = map (absrep_fun flag ctxt) (tys ~~ tys')

           in

             list_comb (get_mapfun ctxt s, args)

           end

        else

           let

             val (rty_pat, qty_pat as Type (_, vs)) = get_rty_qty ctxt s'

             val rtyenv = match ctxt absrep_match_err rty_pat rty

             val qtyenv = match ctxt absrep_match_err qty_pat qty

             val args_aux = map (double_lookup rtyenv qtyenv) vs

             val args = map (absrep_fun flag ctxt) args_aux

             val map_fun = mk_mapfun ctxt vs rty_pat

             val result = list_comb (map_fun, args)

           in

             if forall is_identity args

             then absrep_const flag ctxt s'

             else mk_fun_compose flag (absrep_const flag ctxt s', result)

           end

    | (TFree x, TFree x') =>

        if x = x'

        then mk_identity rty

        else raise (error "absrep_fun (frees)")

    | (TVar _, TVar _) => raise (LIFT_MATCH "absrep_fun (vars)")

    | _ => raise (error "absrep_fun (default)")

fun absrep_fun_chk flag ctxt (rty, qty) =

  absrep_fun flag ctxt (rty, qty)

  |> Syntax.check_term ctxt

(*** Aggregate Equivalence Relation ***)

(* works very similar to the absrep generation,

   except there is no need for polarities

*)

(* instantiates TVars so that the term is of type ty *)

fun force_typ ctxt trm ty =

let

  val thy = ProofContext.theory_of ctxt

  val trm_ty = fastype_of trm

  val ty_inst = Sign.typ_match thy (trm_ty, ty) Vartab.empty

in

  map_types (Envir.subst_type ty_inst) trm

end

fun is_eq (Const (@{const_name "op ="}, _)) = true

  | is_eq _ = false

fun mk_rel_compose (trm1, trm2) =

  Const (@{const_abbrev "rel_conj"}, dummyT) $ trm1 $ trm2

fun get_relmap ctxt s =

let

  val thy = ProofContext.theory_of ctxt

  val exn = error ("get_relmap (no relation map function found for type " ^ s ^ ")")

  val relmap = #relmap (maps_lookup thy s) handle Quotient_Info.NotFound => raise exn

in

  Const (relmap, dummyT)

end

fun mk_relmap ctxt vs rty =

let

  val vs' = map (mk_Free) vs

  fun mk_relmap_aux rty =

    case rty of

      TVar _ => mk_Free rty

    | Type (_, []) => HOLogic.eq_const rty

    | Type (s, tys) => list_comb (get_relmap ctxt s, map mk_relmap_aux tys)

    | _ => raise (error "mk_relmap (default)")

in

  fold_rev Term.lambda vs' (mk_relmap_aux rty)

end

fun get_equiv_rel ctxt s =

let

  val thy = ProofContext.theory_of ctxt

  val exn = error ("get_quotdata (no quotient found for type " ^ s ^ ")")

in

  #equiv_rel (quotdata_lookup thy s) handle Quotient_Info.NotFound => raise exn

end

fun equiv_match_err ctxt ty_pat ty =

let

  val ty_pat_str = Syntax.string_of_typ ctxt ty_pat

  val ty_str = Syntax.string_of_typ ctxt ty

in

  raise error (space_implode " "

    ["equiv_relation (Types ", quote ty_pat_str, "and", quote ty_str, " do not match.)"])

end

(* builds the aggregate equivalence relation

   that will be the argument of Respects

*)

fun equiv_relation ctxt (rty, qty) =

  if rty = qty

  then HOLogic.eq_const rty

  else

    case (rty, qty) of

      (Type (s, tys), Type (s', tys')) =>

       if s = s'

       then

         let

           val args = map (equiv_relation ctxt) (tys ~~ tys')

         in

           list_comb (get_relmap ctxt s, args)

         end

       else

         let

           val (rty_pat, qty_pat as Type (_, vs)) = get_rty_qty ctxt s'

           val rtyenv = match ctxt equiv_match_err rty_pat rty

           val qtyenv = match ctxt equiv_match_err qty_pat qty

           val args_aux = map (double_lookup rtyenv qtyenv) vs

           val args = map (equiv_relation ctxt) args_aux

           val rel_map = mk_relmap ctxt vs rty_pat

           val result = list_comb (rel_map, args)

           val eqv_rel = get_equiv_rel ctxt s'

           val eqv_rel' = force_typ ctxt eqv_rel ([rty, rty] ---> @{typ bool})

         in

           if forall is_eq args

           then eqv_rel'

           else mk_rel_compose (result, eqv_rel')

         end

      | _ => HOLogic.eq_const rty

fun equiv_relation_chk ctxt (rty, qty) =

  equiv_relation ctxt (rty, qty)

  |> Syntax.check_term ctxt

(*** Regularization ***)

(* Regularizing an rtrm means:

 - Quantifiers over types that need lifting are replaced

   by bounded quantifiers, for example:

      All P  ----> All (Respects R) P

   where the aggregate relation R is given by the rty and qty;

 - Abstractions over types that need lifting are replaced

   by bounded abstractions, for example:

      %x. P  ----> Ball (Respects R) %x. P

 - Equalities over types that need lifting are replaced by

   corresponding equivalence relations, for example:

      A = B  ----> R A B

   or

      A = B  ----> (R ===> R) A B

   for more complicated types of A and B

 The regularize_trm accepts raw theorems in which equalities

 and quantifiers match exactly the ones in the lifted theorem

 but also accepts partially regularized terms.

 This means that the raw theorems can have:

   Ball (Respects R),  Bex (Respects R), Bex1_rel (Respects R), Babs, R

 in the places where:

   All, Ex, Ex1, %, (op =)

 is required the lifted theorem.

*)

val mk_babs = Const (@{const_name Babs}, dummyT)

val mk_ball = Const (@{const_name Ball}, dummyT)

val mk_bex  = Const (@{const_name Bex}, dummyT)

val mk_bex1_rel = Const (@{const_name Bex1_rel}, dummyT)

val mk_resp = Const (@{const_name Respects}, dummyT)

(* - applies f to the subterm of an abstraction,

     otherwise to the given term,

   - used by regularize, therefore abstracted

     variables do not have to be treated specially

*)

fun apply_subt f (trm1, trm2) =

  case (trm1, trm2) of

    (Abs (x, T, t), Abs (_ , _, t')) => Abs (x, T, f (t, t'))

  | _ => f (trm1, trm2)

fun term_mismatch str ctxt t1 t2 =

let

  val t1_str = Syntax.string_of_term ctxt t1

  val t2_str = Syntax.string_of_term ctxt t2

  val t1_ty_str = Syntax.string_of_typ ctxt (fastype_of t1)

  val t2_ty_str = Syntax.string_of_typ ctxt (fastype_of t2)

in

  raise error (cat_lines [str, t1_str ^ "::" ^ t1_ty_str, t2_str ^ "::" ^ t2_ty_str])

end

(* the major type of All and Ex quantifiers *)

fun qnt_typ ty = domain_type (domain_type ty)

(* Checks that two types match, for example:

     rty -> rty   matches   qty -> qty *)

fun matches_typ thy rT qT =

  if rT = qT then true else

  case (rT, qT) of

    (Type (rs, rtys), Type (qs, qtys)) =>

      if rs = qs then

        if length rtys <> length qtys then false else

        forall (fn x => x = true) (map2 (matches_typ thy) rtys qtys)

      else

        (case Quotient_Info.quotdata_lookup_raw thy qs of

          SOME quotinfo => Sign.typ_instance thy (rT, #rtyp quotinfo)

        | NONE => false)

  | _ => false

(* produces a regularized version of rtrm

   - the result might contain dummyTs

   - for regularisation we do not need any

     special treatment of bound variables

*)

fun regularize_trm ctxt (rtrm, qtrm) =

  case (rtrm, qtrm) of

    (Abs (x, ty, t), Abs (_, ty', t')) =>

       let

         val subtrm = Abs(x, ty, regularize_trm ctxt (t, t'))

       in

         if ty = ty' then subtrm

         else mk_babs $ (mk_resp $ equiv_relation ctxt (ty, ty')) $ subtrm

       end

  | (Const (@{const_name "Babs"}, T) $ resrel $ (t as (Abs (_, ty, _))), t' as (Abs (_, ty', _))) =>

       let

         val subtrm = regularize_trm ctxt (t, t')

         val needres = mk_resp $ equiv_relation_chk ctxt (ty, ty')

       in

         if resrel <> needres

         then term_mismatch "regularize (Babs)" ctxt resrel needres

         else mk_babs $ resrel $ subtrm

       end

  | (Const (@{const_name "All"}, ty) $ t, Const (@{const_name "All"}, ty') $ t') =>

       let

         val subtrm = apply_subt (regularize_trm ctxt) (t, t')

       in

         if ty = ty' then Const (@{const_name "All"}, ty) $ subtrm

         else mk_ball $ (mk_resp $ equiv_relation ctxt (qnt_typ ty, qnt_typ ty')) $ subtrm

       end

  | (Const (@{const_name "Ex"}, ty) $ t, Const (@{const_name "Ex"}, ty') $ t') =>

       let

         val subtrm = apply_subt (regularize_trm ctxt) (t, t')

       in

         if ty = ty' then Const (@{const_name "Ex"}, ty) $ subtrm

         else mk_bex $ (mk_resp $ equiv_relation ctxt (qnt_typ ty, qnt_typ ty')) $ subtrm

       end

  | (Const (@{const_name "Ex1"}, ty) $ (Abs (_, _,

      (Const (@{const_name "op &"}, _) $ (Const (@{const_name "op :"}, _) $ _ $

        (Const (@{const_name "Respects"}, _) $ resrel)) $ (t $ _)))),

     Const (@{const_name "Ex1"}, ty') $ t') =>

       let

         val t_ = incr_boundvars (~1) t

         val subtrm = apply_subt (regularize_trm ctxt) (t_, t')

         val needrel = equiv_relation_chk ctxt (qnt_typ ty, qnt_typ ty')

       in