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(* Title: logic
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright Cambridge University 1992
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Supporting code for defining the abstract type "thm"
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*)
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infix occs;
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signature LOGIC =
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sig
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val assum_pairs: term -> (term*term)list
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val auto_rename: bool ref
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val close_form: term -> term
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val count_prems: term * int -> int
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val dest_equals: term -> term * term
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val dest_flexpair: term -> term * term
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val dest_implies: term -> term * term
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val flatten_params: int -> term -> term
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val freeze_vars: term -> term
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val incr_indexes: typ list * int -> term -> term
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val lift_fns: term * int -> (term -> term) * (term -> term)
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val list_flexpairs: (term*term)list * term -> term
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val list_implies: term list * term -> term
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val list_rename_params: string list * term -> term
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val mk_equals: term * term -> term
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val mk_flexpair: term * term -> term
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val mk_implies: term * term -> term
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val occs: term * term -> bool
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val rule_of: (term*term)list * term list * term -> term
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val set_rename_prefix: string -> unit
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val skip_flexpairs: term -> term
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val strip_assums_concl: term -> term
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val strip_assums_hyp: term -> term list
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val strip_flexpairs: term -> (term*term)list * term
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val strip_horn: term -> (term*term)list * term list * term
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val strip_imp_concl: term -> term
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val strip_imp_prems: term -> term list
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val strip_params: term -> (string * typ) list
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val strip_prems: int * term list * term -> term list * term
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val thaw_vars: term -> term
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val varify: term -> term
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end;
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functor LogicFun (structure Unify: UNIFY and Net:NET) : LOGIC =
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struct
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structure Type = Unify.Sign.Type;
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(*** Abstract syntax operations on the meta-connectives ***)
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(** equality **)
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(*Make an equality. DOES NOT CHECK TYPE OF u! *)
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fun mk_equals(t,u) = equals(type_of t) $ t $ u;
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fun dest_equals (Const("==",_) $ t $ u) = (t,u)
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| dest_equals t = raise TERM("dest_equals", [t]);
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(** implies **)
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fun mk_implies(A,B) = implies $ A $ B;
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fun dest_implies (Const("==>",_) $ A $ B) = (A,B)
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| dest_implies A = raise TERM("dest_implies", [A]);
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(** nested implications **)
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(* [A1,...,An], B goes to A1==>...An==>B *)
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fun list_implies ([], B) = B : term
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| list_implies (A::AS, B) = implies $ A $ list_implies(AS,B);
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(* A1==>...An==>B goes to [A1,...,An], where B is not an implication *)
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fun strip_imp_prems (Const("==>", _) $ A $ B) = A :: strip_imp_prems B
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| strip_imp_prems _ = [];
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(* A1==>...An==>B goes to B, where B is not an implication *)
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fun strip_imp_concl (Const("==>", _) $ A $ B) = strip_imp_concl B
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| strip_imp_concl A = A : term;
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(*Strip and return premises: (i, [], A1==>...Ai==>B)
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goes to ([Ai, A(i-1),...,A1] , B) (REVERSED)
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if i<0 or else i too big then raises TERM*)
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fun strip_prems (0, As, B) = (As, B)
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| strip_prems (i, As, Const("==>", _) $ A $ B) =
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strip_prems (i-1, A::As, B)
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| strip_prems (_, As, A) = raise TERM("strip_prems", A::As);
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(*Count premises -- quicker than (length ostrip_prems) *)
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fun count_prems (Const("==>", _) $ A $ B, n) = count_prems (B,n+1)
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| count_prems (_,n) = n;
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(** flex-flex constraints **)
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(*Make a constraint. DOES NOT CHECK TYPE OF u! *)
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fun mk_flexpair(t,u) = flexpair(type_of t) $ t $ u;
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fun dest_flexpair (Const("=?=",_) $ t $ u) = (t,u)
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| dest_flexpair t = raise TERM("dest_flexpair", [t]);
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(*make flexflex antecedents: ( [(a1,b1),...,(an,bn)] , C )
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goes to (a1=?=b1) ==>...(an=?=bn)==>C *)
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fun list_flexpairs ([], A) = A
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| list_flexpairs ((t,u)::pairs, A) =
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implies $ (mk_flexpair(t,u)) $ list_flexpairs(pairs,A);
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(*Make the object-rule tpairs==>As==>B *)
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fun rule_of (tpairs, As, B) = list_flexpairs(tpairs, list_implies(As, B));
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(*Remove and return flexflex pairs:
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(a1=?=b1)==>...(an=?=bn)==>C to ( [(a1,b1),...,(an,bn)] , C )
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[Tail recursive in order to return a pair of results] *)
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fun strip_flex_aux (pairs, Const("==>", _) $ (Const("=?=",_)$t$u) $ C) =
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strip_flex_aux ((t,u)::pairs, C)
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| strip_flex_aux (pairs,C) = (rev pairs, C);
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fun strip_flexpairs A = strip_flex_aux([], A);
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(*Discard flexflex pairs*)
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fun skip_flexpairs (Const("==>", _) $ (Const("=?=",_)$_$_) $ C) =
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skip_flexpairs C
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| skip_flexpairs C = C;
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(*strip a proof state (Horn clause):
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(a1==b1)==>...(am==bm)==>B1==>...Bn==>C
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goes to ( [(a1,b1),...,(am,bm)] , [B1,...,Bn] , C) *)
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fun strip_horn A =
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let val (tpairs,horn) = strip_flexpairs A
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in (tpairs, strip_imp_prems horn, strip_imp_concl horn) end;
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(*** Low-level term operations ***)
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(*Does t occur in u? Or is alpha-convertible to u?
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The term t must contain no loose bound variables*)
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fun t occs u = (t aconv u) orelse
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(case u of
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Abs(_,_,body) => t occs body
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| f$t' => t occs f orelse t occs t'
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| _ => false);
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(*Close up a formula over all free variables by quantification*)
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fun close_form A =
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list_all_free (map dest_Free (sort atless (term_frees A)),
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A);
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(*Freeze all (T)Vars by turning them into (T)Frees*)
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fun freeze_vars(Var(ixn,T)) = Free(Syntax.string_of_vname ixn,
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Type.freeze_vars T)
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| freeze_vars(Const(a,T)) = Const(a,Type.freeze_vars T)
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| freeze_vars(Free(a,T)) = Free(a,Type.freeze_vars T)
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| freeze_vars(s$t) = freeze_vars s $ freeze_vars t
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| freeze_vars(Abs(a,T,t)) = Abs(a,Type.freeze_vars T,freeze_vars t)
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| freeze_vars(b) = b;
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(*Reverse the effect of freeze_vars*)
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fun thaw_vars(Const(a,T)) = Const(a,Type.thaw_vars T)
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| thaw_vars(Free(a,T)) =
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let val T' = Type.thaw_vars T
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in case explode a of
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"?"::vn => let val (ixn,_) = Syntax.scan_varname vn
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in Var(ixn,T') end
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| _ => Free(a,T')
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end
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| thaw_vars(Abs(a,T,t)) = Abs(a,Type.thaw_vars T, thaw_vars t)
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| thaw_vars(s$t) = thaw_vars s $ thaw_vars t
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| thaw_vars(b) = b;
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(*** Specialized operations for resolution... ***)
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(*For all variables in the term, increment indexnames and lift over the Us
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result is ?Gidx(B.(lev+n-1),...,B.lev) where lev is abstraction level *)
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fun incr_indexes (Us: typ list, inc:int) t =
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let fun incr (Var ((a,i), T), lev) =
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Unify.combound (Var((a, i+inc), Us---> incr_tvar inc T),
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lev, length Us)
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| incr (Abs (a,T,body), lev) =
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Abs (a, incr_tvar inc T, incr(body,lev+1))
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| incr (Const(a,T),_) = Const(a, incr_tvar inc T)
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| incr (Free(a,T),_) = Free(a, incr_tvar inc T)
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| incr (f$t, lev) = incr(f,lev) $ incr(t,lev)
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| incr (t,lev) = t
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in incr(t,0) end;
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(*Make lifting functions from subgoal and increment.
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lift_abs operates on tpairs (unification constraints)
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lift_all operates on propositions *)
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fun lift_fns (B,inc) =
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let fun lift_abs (Us, Const("==>", _) $ _ $ B) u = lift_abs (Us,B) u
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| lift_abs (Us, Const("all",_)$Abs(a,T,t)) u =
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Abs(a, T, lift_abs (T::Us, t) u)
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| lift_abs (Us, _) u = incr_indexes(rev Us, inc) u
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fun lift_all (Us, Const("==>", _) $ A $ B) u =
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implies $ A $ lift_all (Us,B) u
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| lift_all (Us, Const("all",_)$Abs(a,T,t)) u =
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all T $ Abs(a, T, lift_all (T::Us,t) u)
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| lift_all (Us, _) u = incr_indexes(rev Us, inc) u;
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in (lift_abs([],B), lift_all([],B)) end;
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(*Strips assumptions in goal, yielding list of hypotheses. *)
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fun strip_assums_hyp (Const("==>", _) $ H $ B) = H :: strip_assums_hyp B
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| strip_assums_hyp (Const("all",_)$Abs(a,T,t)) = strip_assums_hyp t
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| strip_assums_hyp B = [];
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(*Strips assumptions in goal, yielding conclusion. *)
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fun strip_assums_concl (Const("==>", _) $ H $ B) = strip_assums_concl B
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| strip_assums_concl (Const("all",_)$Abs(a,T,t)) = strip_assums_concl t
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| strip_assums_concl B = B;
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(*Make a list of all the parameters in a subgoal, even if nested*)
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fun strip_params (Const("==>", _) $ H $ B) = strip_params B
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| strip_params (Const("all",_)$Abs(a,T,t)) = (a,T) :: strip_params t
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| strip_params B = [];
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(*Removes the parameters from a subgoal and renumber bvars in hypotheses,
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where j is the total number of parameters (precomputed)
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If n>0 then deletes assumption n. *)
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fun remove_params j n A =
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if j=0 andalso n<=0 then A (*nothing left to do...*)
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else case A of
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Const("==>", _) $ H $ B =>
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if n=1 then (remove_params j (n-1) B)
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else implies $ (incr_boundvars j H) $ (remove_params j (n-1) B)
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| Const("all",_)$Abs(a,T,t) => remove_params (j-1) n t
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| _ => if n>0 then raise TERM("remove_params", [A])
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else A;
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(** Auto-renaming of parameters in subgoals **)
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val auto_rename = ref false
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and rename_prefix = ref "ka";
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(*rename_prefix is not exported; it is set by this function.*)
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fun set_rename_prefix a =
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if a<>"" andalso forall is_letter (explode a)
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then (rename_prefix := a; auto_rename := true)
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else error"rename prefix must be nonempty and consist of letters";
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(*Makes parameters in a goal have distinctive names (not guaranteed unique!)
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A name clash could cause the printer to rename bound vars;
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then res_inst_tac would not work properly.*)
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fun rename_vars (a, []) = []
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| rename_vars (a, (_,T)::vars) =
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(a,T) :: rename_vars (bump_string a, vars);
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(*Move all parameters to the front of the subgoal, renaming them apart;
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if n>0 then deletes assumption n. *)
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fun flatten_params n A =
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let val params = strip_params A;
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val vars = if !auto_rename
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then rename_vars (!rename_prefix, params)
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else variantlist(map #1 params,[]) ~~ map #2 params
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in list_all (vars, remove_params (length vars) n A)
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end;
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(*Makes parameters in a goal have the names supplied by the list cs.*)
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fun list_rename_params (cs, Const("==>", _) $ A $ B) =
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implies $ A $ list_rename_params (cs, B)
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| list_rename_params (c::cs, Const("all",_)$Abs(_,T,t)) =
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all T $ Abs(c, T, list_rename_params (cs, t))
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| list_rename_params (cs, B) = B;
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(*Strips assumptions in goal yielding ( [Hn,...,H1], [xm,...,x1], B )
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where H1,...,Hn are the hypotheses and x1...xm are the parameters. *)
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fun strip_assums_aux (Hs, params, Const("==>", _) $ H $ B) =
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strip_assums_aux (H::Hs, params, B)
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| strip_assums_aux (Hs, params, Const("all",_)$Abs(a,T,t)) =
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strip_assums_aux (Hs, (a,T)::params, t)
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| strip_assums_aux (Hs, params, B) = (Hs, params, B);
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fun strip_assums A = strip_assums_aux ([],[],A);
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(*Produces disagreement pairs, one for each assumption proof, in order.
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A is the first premise of the lifted rule, and thus has the form
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H1 ==> ... Hk ==> B and the pairs are (H1,B),...,(Hk,B) *)
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fun assum_pairs A =
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let val (Hs, params, B) = strip_assums A
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val D = Unify.rlist_abs(params, B)
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fun pairrev ([],pairs) = pairs
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| pairrev (H::Hs,pairs) =
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pairrev(Hs, (Unify.rlist_abs(params,H), D) :: pairs)
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in pairrev (Hs,[]) (*WAS: map pair (rev Hs) *)
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end;
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(*Converts Frees to Vars and TFrees to TVars so that axioms can be written
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without (?) everywhere*)
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fun varify (Const(a,T)) = Const(a, Type.varifyT T)
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| varify (Free(a,T)) = Var((a,0), Type.varifyT T)
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| varify (Var(ixn,T)) = Var(ixn, Type.varifyT T)
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| varify (Abs (a,T,body)) = Abs (a, Type.varifyT T, varify body)
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| varify (f$t) = varify f $ varify t
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| varify t = t;
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end;
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