(*  Title:      nominal_library.ML

    Author:     Christian Urban

  Basic functions for nominal.

*)

signature NOMINAL_LIBRARY =

sig

  val last2: 'a list -> 'a * 'a

  val order: ('a * 'a -> bool) -> 'a list -> ('a * 'b) list -> 'b list

  val remove_dups: ('a * 'a -> bool) -> 'a list -> 'a list

  val is_true: term -> bool

  val dest_listT: typ -> typ

  val dest_fsetT: typ -> typ

  val mk_id: term -> term

  val size_const: typ -> term 

  val sum_case_const: typ -> typ -> typ -> term

  val mk_sum_case: term -> term -> term

  val mk_minus: term -> term

  val mk_plus: term -> term -> term

  val perm_ty: typ -> typ 

  val mk_perm_ty: typ -> term -> term -> term

  val mk_perm: term -> term -> term

  val dest_perm: term -> term * term

  val mk_sort_of: term -> term

  val atom_ty: typ -> typ

  val atom_const: typ -> term

  val mk_atom_ty: typ -> term -> term

  val mk_atom: term -> term

  val mk_atom_set_ty: typ -> term -> term

  val mk_atom_set: term -> term

  val mk_atom_fset_ty: typ -> term -> term

  val mk_atom_fset: term -> term

  val mk_atom_list_ty: typ -> term -> term

  val mk_atom_list: term -> term

  val is_atom: Proof.context -> typ -> bool

  val is_atom_set: Proof.context -> typ -> bool

  val is_atom_fset: Proof.context -> typ -> bool

  val is_atom_list: Proof.context -> typ -> bool

  val to_atom_set: term -> term

  val to_set_ty: typ -> term -> term

  val to_set: term -> term

  val setify_ty: Proof.context -> typ -> term -> term

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

  val listify_ty: Proof.context -> typ -> term -> term

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

  val fresh_star_ty: typ -> typ

  val fresh_star_const: typ -> term

  val mk_fresh_star_ty: typ -> term -> term -> term

  val mk_fresh_star: term -> term -> term

  val supp_ty: typ -> typ

  val supp_const: typ -> term

  val mk_supp_ty: typ -> term -> term

  val mk_supp: term -> term

  val supp_rel_ty: typ -> typ

  val supp_rel_const: typ -> term

  val mk_supp_rel_ty: typ -> term -> term -> term

  val mk_supp_rel: term -> term -> term		       

  val supports_const: typ -> term

  val mk_supports_ty: typ -> term -> term -> term

  val mk_supports: term -> term -> term

  val finite_const: typ -> term

  val mk_finite_ty: typ -> term -> term

  val mk_finite: term -> term

  val mk_equiv: thm -> thm

  val safe_mk_equiv: thm -> thm

  val mk_diff: term * term -> term

  val mk_append: term * term -> term

  val mk_union: term * term -> term

  val fold_union: term list -> term

  val fold_append: term list -> term

  val mk_conj: term * term -> term

  val fold_conj: term list -> term

  (* fresh arguments for a term *)

  val fresh_args: Proof.context -> term -> term list

  (* datatype operations *)

  type cns_info = (term * typ * typ list * bool list) list

  val all_dtyps: Datatype_Aux.descr -> (string * sort) list -> typ list

  val nth_dtyp: Datatype_Aux.descr -> (string * sort) list -> int -> typ

  val all_dtyp_constrs_types: Datatype_Aux.descr -> (string * sort) list -> cns_info list

  val nth_dtyp_constrs_types: Datatype_Aux.descr -> (string * sort) list -> int -> cns_info

  val prefix_dt_names: Datatype_Aux.descr -> (string * sort) list -> string -> string list

  (* tactics for function package *)

  val pat_completeness_simp: thm list -> Proof.context -> tactic

  val prove_termination: thm list -> Proof.context -> Function.info * local_theory

  (* transformations of premises in inductions *)

  val transform_prem1: Proof.context -> string list -> thm -> thm

  val transform_prem2: Proof.context -> string list -> thm -> thm

  (* transformation into the object logic *)

  val atomize: thm -> thm

end

structure Nominal_Library: NOMINAL_LIBRARY =

struct

(* orders an AList according to keys *)

fun order eq keys list = 

  map (the o AList.lookup eq list) keys

fun remove_dups eq [] = []

  | remove_dups eq (x::xs) = 

      if (member eq xs x) 

      then remove_dups eq xs 

      else x::(remove_dups eq xs)

fun last2 [] = raise Empty

  | last2 [_] = raise Empty

  | last2 [x, y] = (x, y)

  | last2 (_ :: xs) = last2 xs

fun is_true @{term "Trueprop True"} = true

  | is_true _ = false 

fun dest_listT (Type (@{type_name list}, [T])) = T

  | dest_listT T = raise TYPE ("dest_listT: list type expected", [T], [])

fun dest_fsetT (Type (@{type_name fset}, [T])) = T

  | dest_fsetT T = raise TYPE ("dest_fsetT: fset type expected", [T], []);

fun mk_id trm =

  let 

    val ty = fastype_of trm

  in

    Const (@{const_name id}, ty --> ty) $ trm

  end

fun size_const ty = Const (@{const_name size}, ty --> @{typ nat})

fun sum_case_const ty1 ty2 ty3 = 

  Const (@{const_name sum_case}, [ty1 --> ty3, ty2 --> ty3, Type (@{type_name sum}, [ty1, ty2])] ---> ty3)

fun mk_sum_case trm1 trm2 =

  let

    val ([ty1], ty3) = strip_type (fastype_of trm1)

    val ty2 = domain_type (fastype_of trm2)

  in

    sum_case_const ty1 ty2 ty3 $ trm1 $ trm2

  end 

fun mk_minus p = @{term "uminus::perm => perm"} $ p

fun mk_plus p q = @{term "plus::perm => perm => perm"} $ p $ q

fun perm_ty ty = @{typ "perm"} --> ty --> ty

fun mk_perm_ty ty p trm = Const (@{const_name "permute"}, perm_ty ty) $ p $ trm

fun mk_perm p trm = mk_perm_ty (fastype_of trm) p trm

fun dest_perm (Const (@{const_name "permute"}, _) $ p $ t) = (p, t)

  | dest_perm t = raise TERM ("dest_perm", [t]);

fun mk_sort_of t = @{term "sort_of"} $ t;

fun atom_ty ty = ty --> @{typ "atom"};

fun atom_const ty = Const (@{const_name "atom"}, atom_ty ty)

fun mk_atom_ty ty t = atom_const ty $ t;

fun mk_atom t = mk_atom_ty (fastype_of t) t;

fun mk_atom_set_ty ty t =

  let

    val atom_ty = HOLogic.dest_setT ty --> @{typ "atom"};

    val img_ty = atom_ty --> ty --> @{typ "atom set"};

  in

    Const (@{const_name image}, img_ty) $ mk_atom_ty atom_ty t

  end

fun mk_atom_fset_ty ty t =

  let

    val atom_ty = dest_fsetT ty

    val fmap_ty = (atom_ty --> @{typ atom}) --> ty --> @{typ "atom fset"};

    val fset = @{term "fset :: atom fset => atom set"}

  in

    fset $ (Const (@{const_name map_fset}, fmap_ty) $ (atom_const atom_ty) $ t)

  end

fun mk_atom_list_ty ty t =

  let

    val atom_ty = dest_listT ty

    val map_ty = (atom_ty --> @{typ atom}) --> ty --> @{typ "atom list"}

  in

    Const (@{const_name map}, map_ty) $ (atom_const atom_ty) $ t

  end

fun mk_atom_set t = mk_atom_set_ty (fastype_of t) t

fun mk_atom_fset t = mk_atom_fset_ty (fastype_of t) t

fun mk_atom_list t = mk_atom_list_ty (fastype_of t) t

(* coerces a list into a set *)

fun to_atom_set t = @{term "set :: atom list => atom set"} $ t

fun to_set_ty ty t =

  if ty = @{typ "atom list"}

  then to_atom_set t else t

fun to_set t = to_set_ty (fastype_of t) t

(* testing for concrete atom types *)

fun is_atom ctxt ty =

  Sign.of_sort (ProofContext.theory_of ctxt) (ty, @{sort at_base})

fun is_atom_set ctxt (Type ("fun", [ty, @{typ bool}])) = is_atom ctxt ty

  | is_atom_set _ _ = false;

fun is_atom_fset ctxt (Type (@{type_name "fset"}, [ty])) = is_atom ctxt ty

  | is_atom_fset _ _ = false;

fun is_atom_list ctxt (Type (@{type_name "list"}, [ty])) = is_atom ctxt ty

  | is_atom_list _ _ = false

(* functions that coerces singletons, sets and fsets of concrete atoms

   into sets of general atoms *)

fun setify_ty ctxt ty t =

  if is_atom ctxt ty

    then  HOLogic.mk_set @{typ atom} [mk_atom_ty ty t]

  else if is_atom_set ctxt ty

    then mk_atom_set_ty ty t

  else if is_atom_fset ctxt ty

    then mk_atom_fset_ty ty t

  else if is_atom_list ctxt ty

    then to_atom_set (mk_atom_list_ty ty t)

  else raise TERM ("setify", [t])

(* functions that coerces singletons and lists of concrete atoms

   into lists of general atoms  *)

fun listify_ty ctxt ty t =

  if is_atom ctxt ty

    then HOLogic.mk_list @{typ atom} [mk_atom_ty ty t]

  else if is_atom_list ctxt ty

    then mk_atom_list_ty ty t

  else raise TERM ("listify", [t])

fun setify ctxt t = setify_ty ctxt (fastype_of t) t

fun listify ctxt t = listify_ty ctxt (fastype_of t) t

fun fresh_star_ty ty = [@{typ "atom set"}, ty] ---> @{typ bool}

fun fresh_star_const ty = Const (@{const_name fresh_star}, fresh_star_ty ty)

fun mk_fresh_star_ty ty t1 t2 = fresh_star_const ty $ t1 $ t2

fun mk_fresh_star t1 t2 = mk_fresh_star_ty (fastype_of t2) t1 t2

fun supp_ty ty = ty --> @{typ "atom set"};

fun supp_const ty = Const (@{const_name supp}, supp_ty ty)

fun mk_supp_ty ty t = supp_const ty $ t

fun mk_supp t = mk_supp_ty (fastype_of t) t

fun supp_rel_ty ty = ([ty, ty] ---> @{typ bool}) --> ty --> @{typ "atom set"};

fun supp_rel_const ty = Const (@{const_name supp_rel}, supp_rel_ty ty)

fun mk_supp_rel_ty ty r t = supp_rel_const ty $ r $ t

fun mk_supp_rel r t = mk_supp_rel_ty (fastype_of t) r t

fun supports_const ty = Const (@{const_name supports}, [@{typ "atom set"}, ty] ---> @{typ bool});

fun mk_supports_ty ty t1 t2 = supports_const ty $ t1 $ t2;

fun mk_supports t1 t2 = mk_supports_ty (fastype_of t2) t1 t2;

fun finite_const ty = Const (@{const_name finite}, ty --> @{typ bool})

fun mk_finite_ty ty t = finite_const ty $ t

fun mk_finite t = mk_finite_ty (fastype_of t) t

fun mk_equiv r = r RS @{thm eq_reflection};

fun safe_mk_equiv r = mk_equiv r handle Thm.THM _ => r;

(* functions that construct differences, appends and unions

   but avoid producing empty atom sets or empty atom lists *)

fun mk_diff (@{term "{}::atom set"}, _) = @{term "{}::atom set"}

  | mk_diff (t1, @{term "{}::atom set"}) = t1

  | mk_diff (@{term "set ([]::atom list)"}, _) = @{term "set ([]::atom list)"}

  | mk_diff (t1, @{term "set ([]::atom list)"}) = t1

  | mk_diff (t1, t2) = HOLogic.mk_binop @{const_name minus} (t1, t2)

fun mk_append (t1, @{term "[]::atom list"}) = t1

  | mk_append (@{term "[]::atom list"}, t2) = t2

  | mk_append (t1, t2) = HOLogic.mk_binop @{const_name "append"} (t1, t2) 

fun mk_union (t1, @{term "{}::atom set"}) = t1

  | mk_union (@{term "{}::atom set"}, t2) = t2

  | mk_union (t1, @{term "set ([]::atom list)"}) = t1

  | mk_union (@{term "set ([]::atom list)"}, t2) = t2

  | mk_union (t1, t2) = HOLogic.mk_binop @{const_name "sup"} (t1, t2)  

fun fold_union trms = fold_rev (curry mk_union) trms @{term "{}::atom set"}

fun fold_append trms = fold_rev (curry mk_append) trms @{term "[]::atom list"}

fun mk_conj (t1, @{term "True"}) = t1

  | mk_conj (@{term "True"}, t2) = t2

  | mk_conj (t1, t2) = HOLogic.mk_conj (t1, t2)

fun fold_conj trms = fold_rev (curry mk_conj) trms @{term "True"}

(* produces fresh arguments for a term *)

fun fresh_args ctxt f =

    f |> fastype_of

      |> binder_types

      |> map (pair "z")

      |> Variable.variant_frees ctxt [f]

      |> map Free

(** datatypes **)

(* constructor infos *)

type cns_info = (term * typ * typ list * bool list) list

(*  - term for constructor constant

    - type of the constructor

    - types of the arguments

    - flags indicating whether the argument is recursive

*)

(* returns the type of the nth datatype *)

fun all_dtyps descr sorts = 

  map (fn n => Datatype_Aux.typ_of_dtyp descr sorts (Datatype_Aux.DtRec n)) (0 upto (length descr - 1))

fun nth_dtyp descr sorts n = 

  Datatype_Aux.typ_of_dtyp descr sorts (Datatype_Aux.DtRec n);

(* returns info about constructors in a datatype *)

fun all_dtyp_constrs_info descr = 

  map (fn (_, (ty, vs, constrs)) => map (pair (ty, vs)) constrs) descr

(* returns the constants of the constructors plus the 

   corresponding type and types of arguments *)

fun all_dtyp_constrs_types descr sorts = 

  let

    fun aux ((ty_name, vs), (cname, args)) =

      let

        val vs_tys = map (Datatype_Aux.typ_of_dtyp descr sorts) vs

        val ty = Type (ty_name, vs_tys)

        val arg_tys = map (Datatype_Aux.typ_of_dtyp descr sorts) args

        val is_rec = map Datatype_Aux.is_rec_type args

      in

        (Const (cname, arg_tys ---> ty), ty, arg_tys, is_rec)

      end

  in

    map (map aux) (all_dtyp_constrs_info descr)

  end

fun nth_dtyp_constrs_types descr sorts n =

  nth (all_dtyp_constrs_types descr sorts) n

(* generates for every datatype a name str ^ dt_name 

   plus and index for multiple occurences of a string *)

fun prefix_dt_names descr sorts str = 

  let

    fun get_nth_name (i, _) = 

      Datatype_Aux.name_of_typ (nth_dtyp descr sorts i) 

  in

    Datatype_Prop.indexify_names 

      (map (prefix str o get_nth_name) descr)

  end

(** function package tactics **)

fun pat_completeness_simp simps lthy =

  let

    val simp_set = HOL_basic_ss addsimps (@{thms sum.inject sum.distinct} @ simps)

  in

    Pat_Completeness.pat_completeness_tac lthy 1

      THEN ALLGOALS (asm_full_simp_tac simp_set)

  end

fun prove_termination_tac size_simps ctxt =

  let

    val natT = @{typ nat}

    fun prod_size_const fT sT = 

      let

        val fT_fun = fT --> natT

        val sT_fun = sT --> natT

        val prodT = Type (@{type_name prod}, [fT, sT])

      in

        Const (@{const_name prod_size}, [fT_fun, sT_fun, prodT] ---> natT)

      end

    fun mk_size_measure T =

      case T of    

        (Type (@{type_name Sum_Type.sum}, [fT, sT])) =>

           SumTree.mk_sumcase fT sT natT (mk_size_measure fT) (mk_size_measure sT)

      | (Type (@{type_name Product_Type.prod}, [fT, sT])) =>

           prod_size_const fT sT $ (mk_size_measure fT) $ (mk_size_measure sT)

      | _ => size_const T

    fun mk_measure_trm T = 

      HOLogic.dest_setT T

      |> fst o HOLogic.dest_prodT

      |> mk_size_measure 

      |> curry (op $) (Const (@{const_name "measure"}, dummyT))

      |> Syntax.check_term ctxt

    val ss = HOL_ss addsimps @{thms in_measure wf_measure sum.cases add_Suc_right add.right_neutral 

      zero_less_Suc prod.size(1) mult_Suc_right} @ size_simps 

    val ss' = ss addsimprocs Nat_Numeral_Simprocs.cancel_numerals

  in

    Function_Relation.relation_tac ctxt mk_measure_trm

    THEN_ALL_NEW simp_tac ss'

  end

fun prove_termination size_simps ctxt = 

  Function.prove_termination NONE 

    (HEADGOAL (prove_termination_tac size_simps ctxt)) ctxt

(** transformations of premises (in inductive proofs) **)

(* 

 given the theorem F[t]; proves the theorem F[f t] 

  - F needs to be monotone

  - f returns either SOME for a term it fires on 

    and NONE elsewhere 

*)

fun map_term f t = 

  (case f t of

     NONE => map_term' f t 

   | x => x)

and map_term' f (t $ u) = 

    (case (map_term f t, map_term f u) of

        (NONE, NONE) => NONE

      | (SOME t'', NONE) => SOME (t'' $ u)

      | (NONE, SOME u'') => SOME (t $ u'')

      | (SOME t'', SOME u'') => SOME (t'' $ u''))

  | map_term' f (Abs (s, T, t)) = 

      (case map_term f t of

        NONE => NONE

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

  | map_term' _ _  = NONE;

fun map_thm_tac ctxt tac thm =

  let

    val monos = Inductive.get_monos ctxt

    val simps = HOL_basic_ss addsimps @{thms split_def}

  in

    EVERY [cut_facts_tac [thm] 1, etac rev_mp 1, 

      REPEAT_DETERM (FIRSTGOAL (simp_tac simps THEN' resolve_tac monos)),

      REPEAT_DETERM (rtac impI 1 THEN (atac 1 ORELSE tac))]

  end

fun map_thm ctxt f tac thm =

  let

    val opt_goal_trm = map_term f (prop_of thm)

  in

    case opt_goal_trm of

      NONE => thm