src/HOL/Tools/function_package/fundef_lib.ML

generalize type of bounded

(*  Title:      HOL/Tools/function_package/fundef_lib.ML
    Author:     Alexander Krauss, TU Muenchen

A package for general recursive function definitions. 
Some fairly general functions that should probably go somewhere else... 
*)

structure FundefLib = struct

fun map_option f NONE = NONE 
  | map_option f (SOME x) = SOME (f x);

fun fold_option f NONE y = y
  | fold_option f (SOME x) y = f x y;

fun fold_map_option f NONE y = (NONE, y)
  | fold_map_option f (SOME x) y = apfst SOME (f x y);

(* Ex: "The variable" ^ plural " is" "s are" vs *)
fun plural sg pl [x] = sg
  | plural sg pl _ = pl

(* lambda-abstracts over an arbitrarily nested tuple
  ==> hologic.ML? *)
fun tupled_lambda vars t =
    case vars of
      (Free v) => lambda (Free v) t
    | (Var v) => lambda (Var v) t
    | (Const ("Pair", Type ("fun", [Ta, Type ("fun", [Tb, _])]))) $ us $ vs =>  
      (HOLogic.split_const (Ta,Tb, fastype_of t)) $ (tupled_lambda us (tupled_lambda vs t))
    | _ => raise Match
                 
                 
fun dest_all (Const ("all", _) $ Abs (a as (_,T,_))) =
    let
      val (n, body) = Term.dest_abs a
    in
      (Free (n, T), body)
    end
  | dest_all _ = raise Match
                         

(* Removes all quantifiers from a term, replacing bound variables by frees. *)
fun dest_all_all (t as (Const ("all",_) $ _)) = 
    let
      val (v,b) = dest_all t
      val (vs, b') = dest_all_all b
    in
      (v :: vs, b')
    end
  | dest_all_all t = ([],t)
                     

(* FIXME: similar to Variable.focus *)
fun dest_all_all_ctx ctx (Const ("all", _) $ Abs (a as (n,T,b))) =
    let
      val [(n', _)] = Variable.variant_frees ctx [] [(n,T)]
      val (_, ctx') = ProofContext.add_fixes [(Binding.name n', SOME T, NoSyn)] ctx

      val (n'', body) = Term.dest_abs (n', T, b) 
      val _ = (n' = n'') orelse error "dest_all_ctx"
      (* Note: We assume that n' does not occur in the body. Otherwise it would be fixed. *)

      val (ctx'', vs, bd) = dest_all_all_ctx ctx' body
    in
      (ctx'', (n', T) :: vs, bd)
    end
  | dest_all_all_ctx ctx t = 
    (ctx, [], t)


fun map3 _ [] [] [] = []
  | map3 f (x :: xs) (y :: ys) (z :: zs) = f x y z :: map3 f xs ys zs
  | map3 _ _ _ _ = raise Library.UnequalLengths;

fun map4 _ [] [] [] [] = []
  | map4 f (x :: xs) (y :: ys) (z :: zs) (u :: us) = f x y z u :: map4 f xs ys zs us
  | map4 _ _ _ _ _ = raise Library.UnequalLengths;

fun map6 _ [] [] [] [] [] [] = []
  | map6 f (x :: xs) (y :: ys) (z :: zs) (u :: us) (v :: vs) (w :: ws) = f x y z u v w :: map6 f xs ys zs us vs ws
  | map6 _ _ _ _ _ _ _ = raise Library.UnequalLengths;

fun map7 _ [] [] [] [] [] [] [] = []
  | map7 f (x :: xs) (y :: ys) (z :: zs) (u :: us) (v :: vs) (w :: ws) (b :: bs) = f x y z u v w b :: map7 f xs ys zs us vs ws bs
  | map7 _ _ _ _ _ _ _ _ = raise Library.UnequalLengths;



(* forms all "unordered pairs": [1, 2, 3] ==> [(1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 3)] *)
(* ==> library *)
fun unordered_pairs [] = []
  | unordered_pairs (x::xs) = map (pair x) (x::xs) @ unordered_pairs xs


(* Replaces Frees by name. Works with loose Bounds. *)
fun replace_frees assoc =
    map_aterms (fn c as Free (n, _) => the_default c (AList.lookup (op =) assoc n)
                 | t => t)


fun rename_bound n (Q $ Abs(_, T, b)) = (Q $ Abs(n, T, b))
  | rename_bound n _ = raise Match

fun mk_forall_rename (n, v) =
    rename_bound n o Logic.all v 

fun forall_intr_rename (n, cv) thm =
    let
      val allthm = forall_intr cv thm
      val (_ $ abs) = prop_of allthm
    in
      Thm.rename_boundvars abs (Abs (n, dummyT, Term.dummy_pattern dummyT)) allthm
    end


(* Returns the frees in a term in canonical order, excluding the fixes from the context *)
fun frees_in_term ctxt t =
    Term.add_frees t []
    |> filter_out (Variable.is_fixed ctxt o fst)
    |> rev


datatype proof_attempt = Solved of thm | Stuck of thm | Fail

fun try_proof cgoal tac = 
    case SINGLE tac (Goal.init cgoal) of
      NONE => Fail
    | SOME st => if Thm.no_prems st then Solved (Goal.finish st) else Stuck st


fun dest_binop_list cn (t as (Const (n, _) $ a $ b)) = 
    if cn = n then dest_binop_list cn a @ dest_binop_list cn b else [ t ]
  | dest_binop_list _ t = [ t ]


(* separate two parts in a +-expression:
   "a + b + c + d + e" --> "(a + b + d) + (c + e)"

   Here, + can be any binary operation that is AC.

   cn - The name of the binop-constructor (e.g. @{const_name Un})
   ac - the AC rewrite rules for cn
   is - the list of indices of the expressions that should become the first part
        (e.g. [0,1,3] in the above example)
*)

fun regroup_conv neu cn ac is ct =
 let
   val mk = HOLogic.mk_binop cn
   val t = term_of ct
   val xs = dest_binop_list cn t
   val js = 0 upto (length xs) - 1 \\ is
   val ty = fastype_of t
   val thy = theory_of_cterm ct
 in
   Goal.prove_internal []
     (cterm_of thy
       (Logic.mk_equals (t,
          if is = []
          then mk (Const (neu, ty), foldr1 mk (map (nth xs) js))
          else if js = []
            then mk (foldr1 mk (map (nth xs) is), Const (neu, ty))
            else mk (foldr1 mk (map (nth xs) is), foldr1 mk (map (nth xs) js)))))
     (K (rewrite_goals_tac ac
         THEN rtac Drule.reflexive_thm 1))
 end

(* instance for unions *)
fun regroup_union_conv t = regroup_conv @{const_name Set.empty} @{const_name Un}
  (map (fn t => t RS eq_reflection) (@{thms "Un_ac"} @
                                     @{thms "Un_empty_right"} @
                                     @{thms "Un_empty_left"})) t


end