(* Title: Pure/logic.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright Cambridge University 1992
Abstract syntax operations of the Pure meta-logic.
*)
infix occs;
signature LOGIC =
sig
val is_all : term -> bool
val mk_equals : term * term -> term
val dest_equals : term -> term * term
val is_equals : term -> bool
val mk_implies : term * term -> term
val dest_implies : term -> term * term
val is_implies : term -> bool
val list_implies : term list * term -> term
val strip_imp_prems : term -> term list
val strip_imp_concl : term -> term
val strip_prems : int * term list * term -> term list * term
val count_prems : term * int -> int
val mk_conjunction : term * term -> term
val mk_conjunction_list: term list -> term
val mk_flexpair : term * term -> term
val dest_flexpair : term -> term * term
val list_flexpairs : (term*term)list * term -> term
val rule_of : (term*term)list * term list * term -> term
val strip_flexpairs : term -> (term*term)list * term
val skip_flexpairs : term -> term
val strip_horn : term -> (term*term)list * term list * term
val mk_cond_defpair : term list -> term * term -> string * term
val mk_defpair : term * term -> string * term
val mk_type : typ -> term
val dest_type : term -> typ
val mk_inclass : typ * class -> term
val dest_inclass : term -> typ * class
val goal_const : term
val mk_goal : term -> term
val dest_goal : term -> term
val occs : term * term -> bool
val close_form : term -> term
val incr_indexes : typ list * int -> term -> term
val lift_fns : term * int -> (term -> term) * (term -> term)
val strip_assums_hyp : term -> term list
val strip_assums_concl: term -> term
val strip_params : term -> (string * typ) list
val has_meta_prems : term -> int -> bool
val flatten_params : int -> term -> term
val auto_rename : bool ref
val set_rename_prefix : string -> unit
val list_rename_params: string list * term -> term
val assum_pairs : term -> (term*term)list
val varify : term -> term
val unvarify : term -> term
end;
structure Logic : LOGIC =
struct
(*** Abstract syntax operations on the meta-connectives ***)
(** all **)
fun is_all (Const ("all", _) $ _) = true
| is_all _ = false;
(** equality **)
(*Make an equality. DOES NOT CHECK TYPE OF u*)
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]);
fun is_implies (Const ("==>", _) $ _ $ _) = true
| is_implies _ = false;
(** 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;
(** conjunction **)
fun mk_conjunction (t, u) =
Term.list_all ([("C", propT)], mk_implies (list_implies ([t, u], Bound 0), Bound 0));
fun mk_conjunction_list [] = Term.all propT $ Abs ("dummy", propT, mk_implies (Bound 0, Bound 0))
| mk_conjunction_list ts = foldr1 mk_conjunction ts;
(** 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;
(** definitions **)
fun mk_cond_defpair As (lhs, rhs) =
(case Term.head_of lhs of
Const (name, _) =>
(Sign.base_name name ^ "_def", list_implies (As, mk_equals (lhs, rhs)))
| _ => raise TERM ("Malformed definition: head of lhs not a constant", [lhs, rhs]));
fun mk_defpair lhs_rhs = mk_cond_defpair [] lhs_rhs;
(** 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]);
(** atomic goals **)
val goal_const = Const ("Goal", propT --> propT);
fun mk_goal t = goal_const $ t;
fun dest_goal (Const ("Goal", _) $ t) = t
| dest_goal t = raise TERM ("dest_goal", [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 = exists_subterm (fn s => t aconv s) u;
(*Close up a formula over all free variables by quantification*)
fun close_form A =
list_all_free (sort_wrt fst (map dest_Free (term_frees A)), A);
(*** 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 = [];
(*test for meta connectives in prems of a 'subgoal'*)
fun has_meta_prems prop i =
let
fun is_meta (Const ("==>", _) $ _ $ _) = true
| is_meta (Const ("==", _) $ _ $ _) = true
| is_meta (Const ("all", _) $ _) = true
| is_meta _ = false;
val horn = skip_flexpairs prop;
in
(case strip_prems (i, [], horn) of
(B :: _, _) => exists is_meta (strip_assums_hyp B)
| _ => false) handle TERM _ => false
end;
(*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 Symbol.is_letter (Symbol.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 (Symbol.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 ListPair.zip (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 ( [HPn,...,HP1], [xm,...,x1], B ).
Where HPi has the form (Hi,nparams_i) and x1...xm are the parameters.
We need nparams_i only when the parameters aren't flattened; then we
must call incr_boundvars to make up the difference. See assum_pairs.
Without this refinement we can get the wrong answer, e.g. by
Goal "!!f. EX B. Q(f,B) ==> (!!y. P(f,y))";
by (etac exE 1);
*)
fun strip_assums_aux (HPs, params, Const("==>", _) $ H $ B) =
strip_assums_aux ((H,length params)::HPs, params, B)
| strip_assums_aux (HPs, params, Const("all",_)$Abs(a,T,t)) =
strip_assums_aux (HPs, (a,T)::params, t)
| strip_assums_aux (HPs, params, B) = (HPs, params, B);
fun strip_assums A = strip_assums_aux ([],[],A);
(*Produces disagreement pairs, one for each assumption proof, in order.
A is the first premise of the lifted rule, and thus has the form
H1 ==> ... Hk ==> B and the pairs are (H1,B),...,(Hk,B) *)
fun assum_pairs A =
let val (HPs, params, B) = strip_assums A
val nparams = length params
val D = Unify.rlist_abs(params, B)
fun incr_hyp(H,np) =
Unify.rlist_abs(params, incr_boundvars (nparams-np) H)
fun pairrev ([],pairs) = pairs
| pairrev ((H,np)::HPs, pairs) =
pairrev(HPs, (incr_hyp(H,np),D) :: pairs)
in pairrev (HPs,[])
end;
(*Converts Frees to Vars and TFrees to TVars so that axioms can be written
without (?) everywhere*)
fun varify (Const(a,T)) = Const(a, Type.varifyT T)
| varify (Free(a,T)) = Var((a,0), Type.varifyT T)
| varify (Var(ixn,T)) = Var(ixn, Type.varifyT T)
| varify (Abs (a,T,body)) = Abs (a, Type.varifyT T, varify body)
| varify (f$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;