File ‹Tools/cartprod.ML›

(*  Title:      ZF/Tools/cartprod.ML
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1996  University of Cambridge

Signatures for inductive definitions.

Syntactic operations for Cartesian Products.
*)

signature FP =          (** Description of a fixed point operator **)
  sig
  val oper      : term                  (*fixed point operator*)
  val bnd_mono  : term                  (*monotonicity predicate*)
  val bnd_monoI : thm                   (*intro rule for bnd_mono*)
  val subs      : thm                   (*subset theorem for fp*)
  val Tarski    : thm                   (*Tarski's fixed point theorem*)
  val induct    : thm                   (*induction/coinduction rule*)
  end;

signature SU =                  (** Description of a disjoint sum **)
  sig
  val sum       : term                  (*disjoint sum operator*)
  val inl       : term                  (*left injection*)
  val inr       : term                  (*right injection*)
  val elim      : term                  (*case operator*)
  val case_inl  : thm                   (*inl equality rule for case*)
  val case_inr  : thm                   (*inr equality rule for case*)
  val inl_iff   : thm                   (*injectivity of inl, using <->*)
  val inr_iff   : thm                   (*injectivity of inr, using <->*)
  val distinct  : thm                   (*distinctness of inl, inr using <->*)
  val distinct' : thm                   (*distinctness of inr, inl using <->*)
  val free_SEs  : thm list              (*elim rules for SU, and pair_iff!*)
  end;

signature PR =                  (** Description of a Cartesian product **)
  sig
  val sigma     : term                  (*Cartesian product operator*)
  val pair      : term                  (*pairing operator*)
  val split_name : string               (*name of polymorphic split*)
  val pair_iff  : thm                   (*injectivity of pairing, using <->*)
  val split_eq  : thm                   (*equality rule for split*)
  val fsplitI   : thm                   (*intro rule for fsplit*)
  val fsplitD   : thm                   (*destruct rule for fsplit*)
  val fsplitE   : thm                   (*elim rule; apparently never used*)
  end;

signature CARTPROD =            (** Derived syntactic functions for produts **)
  sig
  val ap_split : typ -> typ -> term -> term
  val factors : typ -> typ list
  val mk_prod : typ * typ -> typ
  val mk_tuple : term -> typ -> term list -> term
  val pseudo_type : term -> typ
  val remove_split : Proof.context -> thm -> thm
  val split_const : typ -> term
  val split_rule_var : Proof.context -> term * typ * thm -> thm
  end;


functor CartProd_Fun (Pr: PR) : CARTPROD =
struct

(* Some of these functions expect "pseudo-types" containing products,
   as in HOL; the true ZF types would just be "i" *)

fun mk_prod (T1,T2) = Type("*", [T1,T2]);

(*Bogus product type underlying a (possibly nested) Sigma.  
  Lets us share HOL code*)
fun pseudo_type (t $ A $ Abs(_,_,B)) = 
      if t = Pr.sigma
      then mk_prod(pseudo_type A, pseudo_type B)
      else Typei
  | pseudo_type _ = Typei;

(*Maps the type T1*...*Tn to [T1,...,Tn], however nested*)
fun factors (Type("*", [T1, T2])) = factors T1 @ factors T2
  | factors T                     = [T];

(*Make a well-typed instance of "split"*)
fun split_const T = Const(Pr.split_name, [[Typei, Typei]--->T, Typei] ---> T);

(*In ap_split S T u, term u expects separate arguments for the factors of S,
  with result type T.  The call creates a new term expecting one argument
  of type S.*)
fun ap_split (Type("*", [T1,T2])) T3 u   = 
       split_const T3 $ 
       Abs("v", Typei, (*Not T1, as it involves pseudo-product types*)
           ap_split T2 T3
           ((ap_split T1 (factors T2 ---> T3) (incr_boundvars 1 u)) $ 
            Bound 0))
  | ap_split T T3 u = u;

(*Makes a nested tuple from a list, following the product type structure*)
fun mk_tuple pair (Type("*", [T1,T2])) tms = 
        pair $ mk_tuple pair T1 tms
             $ mk_tuple pair T2 (drop (length (factors T1)) tms)
  | mk_tuple pair T (t::_) = t;

(*Attempts to remove occurrences of split, and pair-valued parameters*)
fun remove_split ctxt = rewrite_rule ctxt [Pr.split_eq];

(*Uncurries any Var according to its "pseudo-product type" T1 in the rule*)
fun split_rule_var ctxt (Var(v,_), Type("fun",[T1,T2]), rl) =
      let val T' = factors T1 ---> T2
          val newt = ap_split T1 T2 (Var(v,T'))
      in
        remove_split ctxt
          (Drule.instantiate_normalize (TVars.empty,
            Vars.make [((v, Typei --> T2), Thm.cterm_of ctxt newt)]) rl)
      end
  | split_rule_var _ (t,T,rl) = rl;

end;