(*  Title:      nominal_library.ML

    Author:     Christian Urban

  Library functions for nominal.

*)

signature NOMINAL_LIBRARY =

sig

  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_set_ty: typ -> term -> term

  val to_set: term -> term

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

  val atomify: Proof.context -> 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_ty: typ -> typ

  val fresh_const: typ -> term

  val mk_fresh_ty: typ -> term -> term -> term

  val mk_fresh: term -> 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_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

  val fold_conj_balanced: term list -> term 

  (* functions for de-Bruijn open terms *)

  val mk_binop_env: typ list -> string -> term * term -> term

  val mk_union_env: typ list -> term * term -> term

  val fold_union_env: typ list -> term list -> term

  (* fresh arguments for a term *)

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

  (* some logic operations *)

  val strip_full_horn: term -> (string * typ) list * term list * term

  val mk_full_horn: (string * typ) list -> term list -> term -> term

  (* datatype operations *)

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

  val all_dtyp_constrs_types: Old_Datatype_Aux.descr -> cns_info list

  (* tactics for function package *)

  val size_ss: simpset

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

  val prove_termination_ind: Proof.context -> int -> tactic

  val prove_termination_fun: 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: Proof.context -> thm -> thm

  val atomize_rule: Proof.context -> int -> thm -> thm 

  val atomize_concl: Proof.context -> thm -> thm

  (* applies a tactic to a formula composed of conjunctions *)

end

structure Nominal_Library: NOMINAL_LIBRARY =

struct

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 

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

  in

    Const (@{const_name image}, img_ty) $ atom_const 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"};

  in

    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_set_ty ty t =

  case ty of

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

  | @{typ "atom fset"} => @{term "fset :: atom fset => atom set"} $ t

  | _ => 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 (Proof_Context.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 coerce singletons, sets, fsets and lists of concrete 

   atoms into general atoms sets / lists *)

fun atomify_ty ctxt ty t =

  if is_atom ctxt ty

    then  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 mk_atom_list_ty ty t

  else raise TERM ("atomify: term is not an atom, set or list of atoms", [t])

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 @{term "fset :: atom fset => atom set"} $ mk_atom_fset_ty ty t

  else if is_atom_list ctxt ty

    then @{term "set :: atom list => atom set"} $ mk_atom_list_ty ty t

  else raise TERM ("setify: term is not an atom, set or list of atoms", [t])

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: term is not an atom or list of atoms", [t])

fun atomify ctxt t = atomify_ty ctxt (fastype_of t) 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_ty ty = [@{typ atom}, ty] ---> @{typ bool}

fun fresh_const ty = Const (@{const_name fresh}, fresh_ty ty)

fun mk_fresh_ty ty t1 t2 = fresh_const ty $ t1 $ t2

fun mk_fresh t1 t2 = mk_fresh_ty (fastype_of t2) t1 t2

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

(* 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"}

fun fold_conj_balanced ts = Balanced_Tree.make HOLogic.mk_conj ts

(* functions for de-Bruijn open terms *)

fun mk_binop_env tys c (t, u) =

  let 

    val ty = fastype_of1 (tys, t) 

  in

    Const (c, [ty, ty] ---> ty) $ t $ u

  end

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

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

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

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

  | mk_union_env tys (t1, t2) = mk_binop_env tys @{const_name "sup"} (t1, t2)  

fun fold_union_env tys trms = fold_left (mk_union_env tys) trms @{term "{}::atom set"} 

(* 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

(** some logic operations **)

(* decompses a formula into params, premises and a conclusion *)

fun strip_full_horn trm =

  let

    fun strip_outer_params (Const (@{const_name Pure.all}, _) $ Abs (a, T, t)) = strip_outer_params t |>> cons (a, T)

    | strip_outer_params B = ([], B)

    val (params, body) = strip_outer_params trm

    val (prems, concl) = Logic.strip_horn body

  in

    (params, prems, concl)

  end

(* composes a formula out of params, premises and a conclusion *)

fun mk_full_horn params prems concl =

  Logic.list_implies (prems, concl)

  |> fold_rev mk_all params

(** 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 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 = 

  let

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

      let

        val vs_tys = map (Old_Datatype_Aux.typ_of_dtyp descr) vs

        val ty = Type (ty_name, vs_tys)

        val arg_tys = map (Old_Datatype_Aux.typ_of_dtyp descr) args

        val is_rec = map Old_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

(** function package tactics **)

fun pat_completeness_simp simps ctxt =

  let

    val simpset =

      put_simpset HOL_basic_ss ctxt addsimps (@{thms sum.inject sum.distinct} @ simps)

  in

    Pat_Completeness.pat_completeness_tac ctxt 1

      THEN ALLGOALS (asm_full_simp_tac simpset)

  end

(* simpset for size goals *)

val size_ss =

  simpset_of (put_simpset HOL_ss @{context}

   addsimprocs [@{simproc natless_cancel_numerals}]

   addsimps @{thms in_measure wf_measure sum.case add_Suc_right add.right_neutral 

     zero_less_Suc prod.size(1) mult_Suc_right})

val natT = @{typ nat}

fun size_prod_const T1 T2 = 

  let

    val T1_fun = T1 --> natT

    val T2_fun = T2 --> natT

    val prodT = HOLogic.mk_prodT (T1, T2)

  in

    Const (@{const_name size_prod}, [T1_fun, T2_fun, prodT] ---> natT)

  end

fun snd_const T1 T2 =

  Const (@{const_name Product_Type.snd}, HOLogic.mk_prodT (T1, T2) --> T2) 

fun mk_measure_trm f ctxt T = 

  HOLogic.dest_setT T

  |> fst o HOLogic.dest_prodT

  |> f

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

  |> Syntax.check_term ctxt

(* wf-goal arising in induction_schema *)    

fun prove_termination_ind ctxt =

  let

    fun mk_size_measure T =

      case T of    

        (Type (@{type_name Sum_Type.sum}, [T1, T2])) =>

           Sum_Tree.mk_sumcase T1 T2 natT (mk_size_measure T1) (mk_size_measure T2)

      | (Type (@{type_name Product_Type.prod}, [T1, T2])) =>

           HOLogic.mk_comp (mk_size_measure T2, snd_const T1 T2)

      | _ => HOLogic.size_const T

    val measure_trm = mk_measure_trm (mk_size_measure) ctxt

  in 

    Function_Relation.relation_tac ctxt measure_trm

  end

(* wf-goal arising in function definitions *)

fun prove_termination_fun size_simps ctxt =

let

  fun mk_size_measure T =

    case T of    

      (Type (@{type_name Sum_Type.sum}, [T1, T2])) =>

         Sum_Tree.mk_sumcase T1 T2 natT (mk_size_measure T1) (mk_size_measure T2)

    | (Type (@{type_name Product_Type.prod}, [T1, T2])) =>

         size_prod_const T1 T2 $ (mk_size_measure T1) $ (mk_size_measure T2)

    | _ => HOLogic.size_const T

  val measure_trm = mk_measure_trm (mk_size_measure) ctxt

  val tac = 

    Function_Relation.relation_tac ctxt measure_trm

    THEN_ALL_NEW simp_tac (put_simpset size_ss ctxt addsimps size_simps)

  in 

    Function.prove_termination NONE (HEADGOAL tac) ctxt

  end

(** 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 simpset = put_simpset HOL_basic_ss ctxt addsimps @{thms split_def}

  in

      REPEAT_DETERM (FIRSTGOAL (simp_tac simpset THEN' resolve_tac ctxt monos)),

  end

fun map_thm ctxt f tac thm =

  let

    val opt_goal_trm = map_term f (Thm.prop_of thm)

  in

    case opt_goal_trm of

      NONE => thm

    | SOME goal =>

        Goal.prove ctxt [] [] goal (fn _ => map_thm_tac ctxt tac thm) 

  end

(*

 inductive premises can be of the form

     R ... /\ P ...; 

 split_conj_i picks out the part R or P part

*)

fun split_conj1 names (Const (@{const_name "conj"}, _) $ f1 $ _) = 

  (case head_of f1 of

     Const (name, _) => if member (op =) names name then SOME f1 else NONE

   | _ => NONE)

| split_conj1 _ _ = NONE;

fun split_conj2 names (Const (@{const_name "conj"}, _) $ f1 $ f2) = 

  (case head_of f1 of

     Const (name, _) => if member (op =) names name then SOME f2 else NONE

   | _ => NONE)