src/Pure/logic.ML

The "divides" relation, the greatest common divisor and Euclid's algorithm

(*  Title: 	Pure/logic.ML
    ID:         $Id$
    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   Cambridge University 1992

Supporting code for defining the abstract type "thm"
*)

infix occs;

signature LOGIC = 
  sig
  val assum_pairs	: term -> (term*term)list
  val auto_rename	: bool ref   
  val close_form	: term -> term   
  val count_prems	: term * int -> int
  val dest_equals	: term -> term * term
  val dest_flexpair	: term -> term * term
  val dest_implies	: term -> term * term
  val dest_inclass	: term -> typ * class
  val dest_type		: term -> typ
  val flatten_params	: int -> term -> term
  val freeze_vars	: term -> term
  val incr_indexes	: typ list * int -> term -> term
  val lift_fns		: term * int -> (term -> term) * (term -> term)
  val list_flexpairs	: (term*term)list * term -> term
  val list_implies	: term list * term -> term
  val list_rename_params: string list * term -> term
  val is_equals         : term -> bool
  val mk_equals		: term * term -> term
  val mk_flexpair	: term * term -> term
  val mk_implies	: term * term -> term
  val mk_inclass	: typ * class -> term
  val mk_type		: typ -> term
  val occs		: term * term -> bool
  val rule_of		: (term*term)list * term list * term -> term
  val set_rename_prefix	: string -> unit   
  val skip_flexpairs	: term -> term
  val strip_assums_concl: term -> term
  val strip_assums_hyp	: term -> term list
  val strip_flexpairs	: term -> (term*term)list * term
  val strip_horn	: term -> (term*term)list * term list * term
  val strip_imp_concl	: term -> term
  val strip_imp_prems	: term -> term list
  val strip_params	: term -> (string * typ) list
  val strip_prems	: int * term list * term -> term list * term
  val thaw_vars		: term -> term
  val unvarify		: term -> term  
  val varify		: term -> term  
  end;

structure Logic : LOGIC =
struct

(*** Abstract syntax operations on the meta-connectives ***)

(** equality **)

(*Make an equality.*)
fun mk_equals(t,u) = equals(fastype_of t) $ t $ u;

fun dest_equals (Const("==",_) $ t $ u)  =  (t,u)
  | dest_equals t = raise TERM("dest_equals", [t]);

fun is_equals (Const ("==", _) $ _ $ _) = true
  | is_equals _ = false;


(** implies **)

fun mk_implies(A,B) = implies $ A $ B;

fun dest_implies (Const("==>",_) $ A $ B)  =  (A,B)
  | dest_implies A = raise TERM("dest_implies", [A]);

(** nested implications **)

(* [A1,...,An], B  goes to  A1==>...An==>B  *)
fun list_implies ([], B) = B : term
  | list_implies (A::AS, B) = implies $ A $ list_implies(AS,B);

(* A1==>...An==>B  goes to  [A1,...,An], where B is not an implication *)
fun strip_imp_prems (Const("==>", _) $ A $ B) = A :: strip_imp_prems B
  | strip_imp_prems _ = [];

(* A1==>...An==>B  goes to B, where B is not an implication *)
fun strip_imp_concl (Const("==>", _) $ A $ B) = strip_imp_concl B
  | strip_imp_concl A = A : term;

(*Strip and return premises: (i, [], A1==>...Ai==>B)
    goes to   ([Ai, A(i-1),...,A1] , B) 	(REVERSED) 
  if  i<0 or else i too big then raises  TERM*)
fun strip_prems (0, As, B) = (As, B) 
  | strip_prems (i, As, Const("==>", _) $ A $ B) = 
	strip_prems (i-1, A::As, B)
  | strip_prems (_, As, A) = raise TERM("strip_prems", A::As);

(*Count premises -- quicker than (length ostrip_prems) *)
fun count_prems (Const("==>", _) $ A $ B, n) = count_prems (B,n+1)
  | count_prems (_,n) = n;

(** flex-flex constraints **)

(*Make a constraint.*)
fun mk_flexpair(t,u) = flexpair(fastype_of t) $ t $ u;

fun dest_flexpair (Const("=?=",_) $ t $ u)  =  (t,u)
  | dest_flexpair t = raise TERM("dest_flexpair", [t]);

(*make flexflex antecedents: ( [(a1,b1),...,(an,bn)] , C )
    goes to (a1=?=b1) ==>...(an=?=bn)==>C *)
fun list_flexpairs ([], A) = A
  | list_flexpairs ((t,u)::pairs, A) =
	implies $ (mk_flexpair(t,u)) $ list_flexpairs(pairs,A);

(*Make the object-rule tpairs==>As==>B   *)
fun rule_of (tpairs, As, B) = list_flexpairs(tpairs, list_implies(As, B));

(*Remove and return flexflex pairs: 
    (a1=?=b1)==>...(an=?=bn)==>C  to  ( [(a1,b1),...,(an,bn)] , C )	
  [Tail recursive in order to return a pair of results] *)
fun strip_flex_aux (pairs, Const("==>", _) $ (Const("=?=",_)$t$u) $ C) =
        strip_flex_aux ((t,u)::pairs, C)
  | strip_flex_aux (pairs,C) = (rev pairs, C);

fun strip_flexpairs A = strip_flex_aux([], A);

(*Discard flexflex pairs*)
fun skip_flexpairs (Const("==>", _) $ (Const("=?=",_)$_$_) $ C) =
	skip_flexpairs C
  | skip_flexpairs C = C;

(*strip a proof state (Horn clause): 
   (a1==b1)==>...(am==bm)==>B1==>...Bn==>C
    goes to   ( [(a1,b1),...,(am,bm)] , [B1,...,Bn] , C)    *)
fun strip_horn A =
  let val (tpairs,horn) = strip_flexpairs A 
  in  (tpairs, strip_imp_prems horn, strip_imp_concl horn)   end;

(** types as terms **)

fun mk_type ty = Const ("TYPE", itselfT ty);

fun dest_type (Const ("TYPE", Type ("itself", [ty]))) = ty
  | dest_type t = raise TERM ("dest_type", [t]);

(** class constraints **)

fun mk_inclass (ty, c) =
  Const (Sign.const_of_class c, itselfT ty --> propT) $ mk_type ty;

fun dest_inclass (t as Const (c_class, _) $ ty) =
      ((dest_type ty, Sign.class_of_const c_class)
        handle TERM _ => raise TERM ("dest_inclass", [t]))
  | dest_inclass t = raise TERM ("dest_inclass", [t]);


(*** Low-level term operations ***)

(*Does t occur in u?  Or is alpha-convertible to u?
  The term t must contain no loose bound variables*)
fun t occs u = (t aconv u) orelse 
      (case u of
          Abs(_,_,body) => t occs body
	| f$t' => t occs f  orelse  t occs t'
	| _ => false);

(*Close up a formula over all free variables by quantification*)
fun close_form A =
    list_all_free (map dest_Free (sort atless (term_frees A)),   
		   A);


(*Freeze all (T)Vars by turning them into (T)Frees*)
fun freeze_vars(Var(ixn,T)) = Free(Syntax.string_of_vname ixn,
                                   Type.freeze_vars T)
  | freeze_vars(Const(a,T)) = Const(a,Type.freeze_vars T)
  | freeze_vars(Free(a,T))  = Free(a,Type.freeze_vars T)
  | freeze_vars(s$t)        = freeze_vars s $ freeze_vars t
  | freeze_vars(Abs(a,T,t)) = Abs(a,Type.freeze_vars T,freeze_vars t)
  | freeze_vars(b)          = b;

(*Reverse the effect of freeze_vars*)
fun thaw_vars(Const(a,T)) = Const(a,Type.thaw_vars T)
  | thaw_vars(Free(a,T))  =
      let val T' = Type.thaw_vars T
      in case explode a of
	   "?"::vn => let val (ixn,_) = Syntax.scan_varname vn
                      in Var(ixn,T') end
	 | _       => Free(a,T')
      end
  | thaw_vars(Abs(a,T,t)) = Abs(a,Type.thaw_vars T, thaw_vars t)
  | thaw_vars(s$t)        = thaw_vars s $ thaw_vars t
  | thaw_vars(b)          = b;


(*** Specialized operations for resolution... ***)

(*For all variables in the term, increment indexnames and lift over the Us
    result is ?Gidx(B.(lev+n-1),...,B.lev) where lev is abstraction level *)
fun incr_indexes (Us: typ list, inc:int) t = 
  let fun incr (Var ((a,i), T), lev) = 
		Unify.combound (Var((a, i+inc), Us---> incr_tvar inc T),
				lev, length Us)
	| incr (Abs (a,T,body), lev) =
		Abs (a, incr_tvar inc T, incr(body,lev+1))
	| incr (Const(a,T),_) = Const(a, incr_tvar inc T)
	| incr (Free(a,T),_) = Free(a, incr_tvar inc T)
	| incr (f$t, lev) = incr(f,lev) $ incr(t,lev)
	| incr (t,lev) = t
  in  incr(t,0)  end;

(*Make lifting functions from subgoal and increment.
    lift_abs operates on tpairs (unification constraints)
    lift_all operates on propositions     *)
fun lift_fns (B,inc) =
  let fun lift_abs (Us, Const("==>", _) $ _ $ B) u = lift_abs (Us,B) u
	| lift_abs (Us, Const("all",_)$Abs(a,T,t)) u =
	      Abs(a, T, lift_abs (T::Us, t) u)
	| lift_abs (Us, _) u = incr_indexes(rev Us, inc) u
      fun lift_all (Us, Const("==>", _) $ A $ B) u =
	      implies $ A $ lift_all (Us,B) u
	| lift_all (Us, Const("all",_)$Abs(a,T,t)) u = 
	      all T $ Abs(a, T, lift_all (T::Us,t) u)
	| lift_all (Us, _) u = incr_indexes(rev Us, inc) u;
  in  (lift_abs([],B), lift_all([],B))  end;

(*Strips assumptions in goal, yielding list of hypotheses.   *)
fun strip_assums_hyp (Const("==>", _) $ H $ B) = H :: strip_assums_hyp B
  | strip_assums_hyp (Const("all",_)$Abs(a,T,t)) = strip_assums_hyp t
  | strip_assums_hyp B = [];

(*Strips assumptions in goal, yielding conclusion.   *)
fun strip_assums_concl (Const("==>", _) $ H $ B) = strip_assums_concl B
  | strip_assums_concl (Const("all",_)$Abs(a,T,t)) = strip_assums_concl t
  | strip_assums_concl B = B;

(*Make a list of all the parameters in a subgoal, even if nested*)
fun strip_params (Const("==>", _) $ H $ B) = strip_params B
  | strip_params (Const("all",_)$Abs(a,T,t)) = (a,T) :: strip_params t
  | strip_params B = [];

(*Removes the parameters from a subgoal and renumber bvars in hypotheses,
    where j is the total number of parameters (precomputed) 
  If n>0 then deletes assumption n. *)
fun remove_params j n A = 
    if j=0 andalso n<=0 then A  (*nothing left to do...*)
    else case A of
        Const("==>", _) $ H $ B => 
	  if n=1 then                           (remove_params j (n-1) B)
	  else implies $ (incr_boundvars j H) $ (remove_params j (n-1) B)
      | Const("all",_)$Abs(a,T,t) => remove_params (j-1) n t
      | _ => if n>0 then raise TERM("remove_params", [A])
             else A;

(** Auto-renaming of parameters in subgoals **)

val auto_rename = ref false
and rename_prefix = ref "ka";

(*rename_prefix is not exported; it is set by this function.*)
fun set_rename_prefix a =
    if a<>"" andalso forall is_letter (explode a)
    then  (rename_prefix := a;  auto_rename := true)
    else  error"rename prefix must be nonempty and consist of letters";

(*Makes parameters in a goal have distinctive names (not guaranteed unique!)
  A name clash could cause the printer to rename bound vars;
    then res_inst_tac would not work properly.*)
fun rename_vars (a, []) = []
  | rename_vars (a, (_,T)::vars) =
        (a,T) :: rename_vars (bump_string a, vars);

(*Move all parameters to the front of the subgoal, renaming them apart;
  if n>0 then deletes assumption n. *)
fun flatten_params n A =
    let val params = strip_params A;
	val vars = if !auto_rename 
		   then rename_vars (!rename_prefix, params)
		   else variantlist(map #1 params,[]) ~~ map #2 params
    in  list_all (vars, remove_params (length vars) n A)
    end;

(*Makes parameters in a goal have the names supplied by the list cs.*)
fun list_rename_params (cs, Const("==>", _) $ A $ B) =
      implies $ A $ list_rename_params (cs, B)
  | list_rename_params (c::cs, Const("all",_)$Abs(_,T,t)) = 
      all T $ Abs(c, T, list_rename_params (cs, t))
  | list_rename_params (cs, B) = B;

(*Strips assumptions in goal yielding  ( [Hn,...,H1], [xm,...,x1], B )
  where H1,...,Hn are the hypotheses and x1...xm are the parameters.   *)
fun strip_assums_aux (Hs, params, Const("==>", _) $ H $ B) = 
	strip_assums_aux (H::Hs, params, B)
  | strip_assums_aux (Hs, params, Const("all",_)$Abs(a,T,t)) =
	strip_assums_aux (Hs, (a,T)::params, t)
  | strip_assums_aux (Hs, params, B) = (Hs, params, B);

fun strip_assums A = strip_assums_aux ([],[],A);


(*Produces disagreement pairs, one for each assumption proof, in order.
  A is the first premise of the lifted rule, and thus has the form
    H1 ==> ... Hk ==> B   and the pairs are (H1,B),...,(Hk,B) *)
fun assum_pairs A =
  let val (Hs, params, B) = strip_assums A
      val D = Unify.rlist_abs(params, B)
      fun pairrev ([],pairs) = pairs  
        | pairrev (H::Hs,pairs) = 
	    pairrev(Hs, (Unify.rlist_abs(params,H), D) :: pairs)
  in  pairrev (Hs,[])   (*WAS:  map pair (rev Hs)  *)
  end;


(*Converts Frees to Vars and TFrees to TVars so that axioms can be written
  without (?) everywhere*)
fun varify (Const(a,T)) = Const(a, Type.varifyT T)
  | varify (Free(a,T)) = Var((a,0), Type.varifyT T)
  | varify (Var(ixn,T)) = Var(ixn, Type.varifyT T)
  | varify (Abs (a,T,body)) = Abs (a, Type.varifyT T, varify body)
  | varify (f$t) = varify f $ varify t
  | varify t = t;

(*Inverse of varify.  Converts axioms back to their original form.*)
fun unvarify (Const(a,T))    = Const(a, Type.unvarifyT T)
  | unvarify (Var((a,0), T)) = Free(a, Type.unvarifyT T)
  | unvarify (Var(ixn,T))    = Var(ixn, Type.unvarifyT T)  (*non-0 index!*)
  | unvarify (Abs (a,T,body)) = Abs (a, Type.unvarifyT T, unvarify body)
  | unvarify (f$t) = unvarify f $ unvarify t
  | unvarify t = t;

end;