(* Title: drule
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
Derived rules and other operations on theorems and theories
*)
infix 0 RS RSN RL RLN MRS MRL COMP;
signature DRULE =
sig
structure Thm : THM
local open Thm in
val asm_rl: thm
val assume_ax: theory -> string -> thm
val COMP: thm * thm -> thm
val compose: thm * int * thm -> thm list
val cterm_instantiate: (Sign.cterm*Sign.cterm)list -> thm -> thm
val cut_rl: thm
val equal_abs_elim: Sign.cterm -> thm -> thm
val equal_abs_elim_list: Sign.cterm list -> thm -> thm
val eq_sg: Sign.sg * Sign.sg -> bool
val eq_thm: thm * thm -> bool
val eq_thm_sg: thm * thm -> bool
val flexpair_abs_elim_list: Sign.cterm list -> thm -> thm
val forall_intr_list: Sign.cterm list -> thm -> thm
val forall_intr_frees: thm -> thm
val forall_elim_list: Sign.cterm list -> thm -> thm
val forall_elim_var: int -> thm -> thm
val forall_elim_vars: int -> thm -> thm
val implies_elim_list: thm -> thm list -> thm
val implies_intr_list: Sign.cterm list -> thm -> thm
val MRL: thm list list * thm list -> thm list
val MRS: thm list * thm -> thm
val print_cterm: Sign.cterm -> unit
val print_ctyp: Sign.ctyp -> unit
val print_goals: int -> thm -> unit
val print_goals_ref: (int -> thm -> unit) ref
val print_sg: Sign.sg -> unit
val print_theory: theory -> unit
val pprint_sg: Sign.sg -> pprint_args -> unit
val pprint_theory: theory -> pprint_args -> unit
val print_thm: thm -> unit
val prth: thm -> thm
val prthq: thm Sequence.seq -> thm Sequence.seq
val prths: thm list -> thm list
val read_instantiate: (string*string)list -> thm -> thm
val read_instantiate_sg: Sign.sg -> (string*string)list -> thm -> thm
val reflexive_thm: thm
val revcut_rl: thm
val rewrite_goal_rule: (meta_simpset -> thm -> thm option) -> meta_simpset ->
int -> thm -> thm
val rewrite_goals_rule: thm list -> thm -> thm
val rewrite_rule: thm list -> thm -> thm
val RS: thm * thm -> thm
val RSN: thm * (int * thm) -> thm
val RL: thm list * thm list -> thm list
val RLN: thm list * (int * thm list) -> thm list
val show_hyps: bool ref
val size_of_thm: thm -> int
val standard: thm -> thm
val string_of_thm: thm -> string
val symmetric_thm: thm
val pprint_thm: thm -> pprint_args -> unit
val transitive_thm: thm
val triv_forall_equality: thm
val types_sorts: thm -> (indexname-> typ option) * (indexname-> sort option)
val zero_var_indexes: thm -> thm
end
end;
functor DruleFun (structure Logic: LOGIC and Thm: THM) : DRULE =
struct
structure Thm = Thm;
structure Sign = Thm.Sign;
structure Type = Sign.Type;
structure Pretty = Sign.Syntax.Pretty
local open Thm
in
(**** More derived rules and operations on theorems ****)
(*** Find the type (sort) associated with a (T)Var or (T)Free in a term
Used for establishing default types (of variables) and sorts (of
type variables) when reading another term.
Index -1 indicates that a (T)Free rather than a (T)Var is wanted.
***)
fun types_sorts thm =
let val {prop,hyps,...} = rep_thm thm;
val big = list_comb(prop,hyps); (* bogus term! *)
val vars = map dest_Var (term_vars big);
val frees = map dest_Free (term_frees big);
val tvars = term_tvars big;
val tfrees = term_tfrees big;
fun typ(a,i) = if i<0 then assoc(frees,a) else assoc(vars,(a,i));
fun sort(a,i) = if i<0 then assoc(tfrees,a) else assoc(tvars,(a,i));
in (typ,sort) end;
(** Standardization of rules **)
(*Generalization over a list of variables, IGNORING bad ones*)
fun forall_intr_list [] th = th
| forall_intr_list (y::ys) th =
let val gth = forall_intr_list ys th
in forall_intr y gth handle THM _ => gth end;
(*Generalization over all suitable Free variables*)
fun forall_intr_frees th =
let val {prop,sign,...} = rep_thm th
in forall_intr_list
(map (Sign.cterm_of sign) (sort atless (term_frees prop)))
th
end;
(*Replace outermost quantified variable by Var of given index.
Could clash with Vars already present.*)
fun forall_elim_var i th =
let val {prop,sign,...} = rep_thm th
in case prop of
Const("all",_) $ Abs(a,T,_) =>
forall_elim (Sign.cterm_of sign (Var((a,i), T))) th
| _ => raise THM("forall_elim_var", i, [th])
end;
(*Repeat forall_elim_var until all outer quantifiers are removed*)
fun forall_elim_vars i th =
forall_elim_vars i (forall_elim_var i th)
handle THM _ => th;
(*Specialization over a list of cterms*)
fun forall_elim_list cts th = foldr (uncurry forall_elim) (rev cts, th);
(* maps [A1,...,An], B to [| A1;...;An |] ==> B *)
fun implies_intr_list cAs th = foldr (uncurry implies_intr) (cAs,th);
(* maps [| A1;...;An |] ==> B and [A1,...,An] to B *)
fun implies_elim_list impth ths = foldl (uncurry implies_elim) (impth,ths);
(*Reset Var indexes to zero, renaming to preserve distinctness*)
fun zero_var_indexes th =
let val {prop,sign,...} = rep_thm th;
val vars = term_vars prop
val bs = foldl add_new_id ([], map (fn Var((a,_),_)=>a) vars)
val inrs = add_term_tvars(prop,[]);
val nms' = rev(foldl add_new_id ([], map (#1 o #1) inrs));
val tye = map (fn ((v,rs),a) => (v, TVar((a,0),rs))) (inrs ~~ nms')
val ctye = map (fn (v,T) => (v,Sign.ctyp_of sign T)) tye;
fun varpairs([],[]) = []
| varpairs((var as Var(v,T)) :: vars, b::bs) =
let val T' = typ_subst_TVars tye T
in (Sign.cterm_of sign (Var(v,T')),
Sign.cterm_of sign (Var((b,0),T'))) :: varpairs(vars,bs)
end
| varpairs _ = raise TERM("varpairs", []);
in instantiate (ctye, varpairs(vars,rev bs)) th end;
(*Standard form of object-rule: no hypotheses, Frees, or outer quantifiers;
all generality expressed by Vars having index 0.*)
fun standard th =
let val {maxidx,...} = rep_thm th
in varifyT (zero_var_indexes (forall_elim_vars(maxidx+1)
(forall_intr_frees(implies_intr_hyps th))))
end;
(*Assume a new formula, read following the same conventions as axioms.
Generalizes over Free variables,
creates the assumption, and then strips quantifiers.
Example is [| ALL x:?A. ?P(x) |] ==> [| ?P(?a) |]
[ !(A,P,a)[| ALL x:A. P(x) |] ==> [| P(a) |] ] *)
fun assume_ax thy sP =
let val sign = sign_of thy
val prop = Logic.close_form (Sign.term_of (Sign.read_cterm sign
(sP, propT)))
in forall_elim_vars 0 (assume (Sign.cterm_of sign prop)) end;
(*Resolution: exactly one resolvent must be produced.*)
fun tha RSN (i,thb) =
case Sequence.chop (2, biresolution false [(false,tha)] i thb) of
([th],_) => th
| ([],_) => raise THM("RSN: no unifiers", i, [tha,thb])
| _ => raise THM("RSN: multiple unifiers", i, [tha,thb]);
(*resolution: P==>Q, Q==>R gives P==>R. *)
fun tha RS thb = tha RSN (1,thb);
(*For joining lists of rules*)
fun thas RLN (i,thbs) =
let val resolve = biresolution false (map (pair false) thas) i
fun resb thb = Sequence.list_of_s (resolve thb) handle THM _ => []
in flat (map resb thbs) end;
fun thas RL thbs = thas RLN (1,thbs);
(*Resolve a list of rules against bottom_rl from right to left;
makes proof trees*)
fun rls MRS bottom_rl =
let fun rs_aux i [] = bottom_rl
| rs_aux i (rl::rls) = rl RSN (i, rs_aux (i+1) rls)
in rs_aux 1 rls end;
(*As above, but for rule lists*)
fun rlss MRL bottom_rls =
let fun rs_aux i [] = bottom_rls
| rs_aux i (rls::rlss) = rls RLN (i, rs_aux (i+1) rlss)
in rs_aux 1 rlss end;
(*compose Q and [...,Qi,Q(i+1),...]==>R to [...,Q(i+1),...]==>R
with no lifting or renaming! Q may contain ==> or meta-quants
ALWAYS deletes premise i *)
fun compose(tha,i,thb) =
Sequence.list_of_s (bicompose false (false,tha,0) i thb);
(*compose Q and [Q1,Q2,...,Qk]==>R to [Q2,...,Qk]==>R getting unique result*)
fun tha COMP thb =
case compose(tha,1,thb) of
[th] => th
| _ => raise THM("COMP", 1, [tha,thb]);
(*Instantiate theorem th, reading instantiations under signature sg*)
fun read_instantiate_sg sg sinsts th =
let val ts = types_sorts th;
val instpair = Sign.read_insts sg ts ts sinsts
in instantiate instpair th end;
(*Instantiate theorem th, reading instantiations under theory of th*)
fun read_instantiate sinsts th =
read_instantiate_sg (#sign (rep_thm th)) sinsts th;
(*Left-to-right replacements: tpairs = [...,(vi,ti),...].
Instantiates distinct Vars by terms, inferring type instantiations. *)
local
fun add_types ((ct,cu), (sign,tye)) =
let val {sign=signt, t=t, T= T, ...} = Sign.rep_cterm ct
and {sign=signu, t=u, T= U, ...} = Sign.rep_cterm cu
val sign' = Sign.merge(sign, Sign.merge(signt, signu))
val tye' = Type.unify (#tsig(Sign.rep_sg sign')) ((T,U), tye)
handle Type.TUNIFY => raise TYPE("add_types", [T,U], [t,u])
in (sign', tye') end;
in
fun cterm_instantiate ctpairs0 th =
let val (sign,tye) = foldr add_types (ctpairs0, (#sign(rep_thm th),[]))
val tsig = #tsig(Sign.rep_sg sign);
fun instT(ct,cu) = let val inst = subst_TVars tye
in (Sign.cfun inst ct, Sign.cfun inst cu) end
fun ctyp2 (ix,T) = (ix, Sign.ctyp_of sign T)
in instantiate (map ctyp2 tye, map instT ctpairs0) th end
handle TERM _ =>
raise THM("cterm_instantiate: incompatible signatures",0,[th])
| TYPE _ => raise THM("cterm_instantiate: types", 0, [th])
end;
(*** Printing of theorems ***)
(*If false, hypotheses are printed as dots*)
val show_hyps = ref true;
fun pretty_thm th =
let val {sign, hyps, prop,...} = rep_thm th
val hsymbs = if null hyps then []
else if !show_hyps then
[Pretty.brk 2,
Pretty.lst("[","]") (map (Sign.pretty_term sign) hyps)]
else Pretty.str" [" :: map (fn _ => Pretty.str".") hyps @
[Pretty.str"]"];
in Pretty.blk(0, Sign.pretty_term sign prop :: hsymbs) end;
val string_of_thm = Pretty.string_of o pretty_thm;
val pprint_thm = Pretty.pprint o Pretty.quote o pretty_thm;
(** Top-level commands for printing theorems **)
val print_thm = writeln o string_of_thm;
fun prth th = (print_thm th; th);
(*Print and return a sequence of theorems, separated by blank lines. *)
fun prthq thseq =
(Sequence.prints (fn _ => print_thm) 100000 thseq;
thseq);
(*Print and return a list of theorems, separated by blank lines. *)
fun prths ths = (print_list_ln print_thm ths; ths);
(*Other printing commands*)
val print_cterm = writeln o Sign.string_of_cterm;
val print_ctyp = writeln o Sign.string_of_ctyp;
fun pretty_sg sg =
Pretty.lst ("{", "}") (map (Pretty.str o !) (#stamps (Sign.rep_sg sg)));
val pprint_sg = Pretty.pprint o pretty_sg;
val pprint_theory = pprint_sg o sign_of;
val print_sg = writeln o Pretty.string_of o pretty_sg;
val print_theory = print_sg o sign_of;
(** Print thm A1,...,An/B in "goal style" -- premises as numbered subgoals **)
fun prettyprints es = writeln(Pretty.string_of(Pretty.blk(0,es)));
fun print_goals maxgoals th : unit =
let val {sign, hyps, prop,...} = rep_thm th;
fun printgoals (_, []) = ()
| printgoals (n, A::As) =
let val prettyn = Pretty.str(" " ^ string_of_int n ^ ". ");
val prettyA = Sign.pretty_term sign A
in prettyprints[prettyn,prettyA];
printgoals (n+1,As)
end;
fun prettypair(t,u) =
Pretty.blk(0, [Sign.pretty_term sign t, Pretty.str" =?=", Pretty.brk 1,
Sign.pretty_term sign u]);
fun printff [] = ()
| printff tpairs =
writeln("\nFlex-flex pairs:\n" ^
Pretty.string_of(Pretty.lst("","") (map prettypair tpairs)))
val (tpairs,As,B) = Logic.strip_horn(prop);
val ngoals = length As
in
writeln (Sign.string_of_term sign B);
if ngoals=0 then writeln"No subgoals!"
else if ngoals>maxgoals
then (printgoals (1, take(maxgoals,As));
writeln("A total of " ^ string_of_int ngoals ^ " subgoals..."))
else printgoals (1, As);
printff tpairs
end;
(*"hook" for user interfaces: allows print_goals to be replaced*)
val print_goals_ref = ref print_goals;
(** theorem equality test is exported and used by BEST_FIRST **)
(*equality of signatures means exact identity -- by ref equality*)
fun eq_sg (sg1,sg2) = (#stamps(Sign.rep_sg sg1) = #stamps(Sign.rep_sg sg2));
(*equality of theorems uses equality of signatures and
the a-convertible test for terms*)
fun eq_thm (th1,th2) =
let val {sign=sg1, hyps=hyps1, prop=prop1, ...} = rep_thm th1
and {sign=sg2, hyps=hyps2, prop=prop2, ...} = rep_thm th2
in eq_sg (sg1,sg2) andalso
aconvs(hyps1,hyps2) andalso
prop1 aconv prop2
end;
(*Do the two theorems have the same signature?*)
fun eq_thm_sg (th1,th2) = eq_sg(#sign(rep_thm th1), #sign(rep_thm th2));
(*Useful "distance" function for BEST_FIRST*)
val size_of_thm = size_of_term o #prop o rep_thm;
(*** Meta-Rewriting Rules ***)
val reflexive_thm =
let val cx = Sign.cterm_of Sign.pure (Var(("x",0),TVar(("'a",0),["logic"])))
in Thm.reflexive cx end;
val symmetric_thm =
let val xy = Sign.read_cterm Sign.pure ("x::'a::logic == y",propT)
in standard(Thm.implies_intr_hyps(Thm.symmetric(Thm.assume xy))) end;
val transitive_thm =
let val xy = Sign.read_cterm Sign.pure ("x::'a::logic == y",propT)
val yz = Sign.read_cterm Sign.pure ("y::'a::logic == z",propT)
val xythm = Thm.assume xy and yzthm = Thm.assume yz
in standard(Thm.implies_intr yz (Thm.transitive xythm yzthm)) end;
(** Below, a "conversion" has type sign->term->thm **)
(*In [A1,...,An]==>B, rewrite the selected A's only -- for rewrite_goals_tac*)
fun goals_conv pred cv sign =
let val triv = reflexive o Sign.cterm_of sign
fun gconv i t =
let val (A,B) = Logic.dest_implies t
val thA = if (pred i) then (cv sign A) else (triv A)
in combination (combination (triv implies) thA)
(gconv (i+1) B)
end
handle TERM _ => triv t
in gconv 1 end;
(*Use a conversion to transform a theorem*)
fun fconv_rule cv th =
let val {sign,prop,...} = rep_thm th
in equal_elim (cv sign prop) th end;
(*rewriting conversion*)
fun rew_conv prover mss sign t =
rewrite_cterm mss prover (Sign.cterm_of sign t);
(*Rewrite a theorem*)
fun rewrite_rule thms = fconv_rule (rew_conv (K(K None)) (Thm.mss_of thms));
(*Rewrite the subgoals of a proof state (represented by a theorem) *)
fun rewrite_goals_rule thms =
fconv_rule (goals_conv (K true) (rew_conv (K(K None)) (Thm.mss_of thms)));
(*Rewrite the subgoal of a proof state (represented by a theorem) *)
fun rewrite_goal_rule prover mss i =
fconv_rule (goals_conv (fn j => j=i) (rew_conv prover mss));
(** Derived rules mainly for METAHYPS **)
(*Given the term "a", takes (%x.t)==(%x.u) to t[a/x]==u[a/x]*)
fun equal_abs_elim ca eqth =
let val {sign=signa, t=a, ...} = Sign.rep_cterm ca
and combth = combination eqth (reflexive ca)
val {sign,prop,...} = rep_thm eqth
val (abst,absu) = Logic.dest_equals prop
val cterm = Sign.cterm_of (Sign.merge (sign,signa))
in transitive (symmetric (beta_conversion (cterm (abst$a))))
(transitive combth (beta_conversion (cterm (absu$a))))
end
handle THM _ => raise THM("equal_abs_elim", 0, [eqth]);
(*Calling equal_abs_elim with multiple terms*)
fun equal_abs_elim_list cts th = foldr (uncurry equal_abs_elim) (rev cts, th);
local
open Logic
val alpha = TVar(("'a",0), []) (* type ?'a::{} *)
fun err th = raise THM("flexpair_inst: ", 0, [th])
fun flexpair_inst def th =
let val {prop = Const _ $ t $ u, sign,...} = rep_thm th
val cterm = Sign.cterm_of sign
fun cvar a = cterm(Var((a,0),alpha))
val def' = cterm_instantiate [(cvar"t", cterm t), (cvar"u", cterm u)]
def
in equal_elim def' th
end
handle THM _ => err th | bind => err th
in
val flexpair_intr = flexpair_inst (symmetric flexpair_def)
and flexpair_elim = flexpair_inst flexpair_def
end;
(*Version for flexflex pairs -- this supports lifting.*)
fun flexpair_abs_elim_list cts =
flexpair_intr o equal_abs_elim_list cts o flexpair_elim;
(*** Some useful meta-theorems ***)
(*The rule V/V, obtains assumption solving for eresolve_tac*)
val asm_rl = trivial(Sign.read_cterm Sign.pure ("PROP ?psi",propT));
(*Meta-level cut rule: [| V==>W; V |] ==> W *)
val cut_rl = trivial(Sign.read_cterm Sign.pure
("PROP ?psi ==> PROP ?theta", propT));
(*Generalized elim rule for one conclusion; cut_rl with reversed premises:
[| PROP V; PROP V ==> PROP W |] ==> PROP W *)
val revcut_rl =
let val V = Sign.read_cterm Sign.pure ("PROP V", propT)
and VW = Sign.read_cterm Sign.pure ("PROP V ==> PROP W", propT);
in standard (implies_intr V
(implies_intr VW
(implies_elim (assume VW) (assume V))))
end;
(* (!!x. PROP ?V) == PROP ?V Allows removal of redundant parameters*)
val triv_forall_equality =
let val V = Sign.read_cterm Sign.pure ("PROP V", propT)
and QV = Sign.read_cterm Sign.pure ("!!x::'a. PROP V", propT)
and x = Sign.read_cterm Sign.pure ("x", TFree("'a",["logic"]));
in standard (equal_intr (implies_intr QV (forall_elim x (assume QV)))
(implies_intr V (forall_intr x (assume V))))
end;
end
end;