(* Title: Pure/drule.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
Derived rules and other operations on theorems.
*)
infix 0 RS RSN RL RLN MRS MRL COMP;
signature DRULE =
sig
val dest_implies : cterm -> cterm * cterm
val skip_flexpairs : cterm -> cterm
val strip_imp_prems : cterm -> cterm list
val cprems_of : thm -> cterm list
val read_insts :
Sign.sg -> (indexname -> typ option) * (indexname -> sort option)
-> (indexname -> typ option) * (indexname -> sort option)
-> string list -> (string*string)list
-> (indexname*ctyp)list * (cterm*cterm)list
val types_sorts: thm -> (indexname-> typ option) * (indexname-> sort option)
val forall_intr_list : cterm list -> thm -> thm
val forall_intr_frees : thm -> thm
val forall_intr_vars : thm -> thm
val forall_elim_list : 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 : cterm list -> thm -> thm
val zero_var_indexes : thm -> thm
val standard : thm -> thm
val assume_ax : theory -> string -> thm
val RSN : thm * (int * thm) -> thm
val RS : thm * thm -> thm
val RLN : thm list * (int * thm list) -> thm list
val RL : thm list * thm list -> thm list
val MRS : thm list * thm -> thm
val MRL : thm list list * thm list -> thm list
val compose : thm * int * thm -> thm list
val COMP : thm * thm -> thm
val read_instantiate_sg: Sign.sg -> (string*string)list -> thm -> thm
val read_instantiate : (string*string)list -> thm -> thm
val cterm_instantiate : (cterm*cterm)list -> thm -> thm
val weak_eq_thm : thm * thm -> bool
val eq_thm_sg : thm * thm -> bool
val size_of_thm : thm -> int
val reflexive_thm : thm
val symmetric_thm : thm
val transitive_thm : thm
val refl_implies : thm
val rewrite_rule_aux : (meta_simpset -> thm -> thm option) -> thm list -> thm -> thm
val rewrite_thm : bool * bool -> (meta_simpset -> thm -> thm option)
-> meta_simpset -> thm -> thm
val rewrite_goals_rule_aux: (meta_simpset -> thm -> thm option) -> thm list -> thm -> thm
val rewrite_goal_rule : bool * bool -> (meta_simpset -> thm -> thm option)
-> meta_simpset -> int -> thm -> thm
val equal_abs_elim : cterm -> thm -> thm
val equal_abs_elim_list: cterm list -> thm -> thm
val flexpair_abs_elim_list: cterm list -> thm -> thm
val asm_rl : thm
val cut_rl : thm
val revcut_rl : thm
val thin_rl : thm
val triv_forall_equality: thm
val swap_prems_rl : thm
val equal_intr_rule : thm
val instantiate': ctyp option list -> cterm option list -> thm -> thm
end;
structure Drule : DRULE =
struct
(** some cterm->cterm operations: much faster than calling cterm_of! **)
(** SAME NAMES as in structure Logic: use compound identifiers! **)
(*dest_implies for cterms. Note T=prop below*)
fun dest_implies ct =
case term_of ct of
(Const("==>", _) $ _ $ _) =>
let val (ct1,ct2) = dest_comb ct
in (#2 (dest_comb ct1), ct2) end
| _ => raise TERM ("dest_implies", [term_of ct]) ;
(*Discard flexflex pairs; return a cterm*)
fun skip_flexpairs ct =
case term_of ct of
(Const("==>", _) $ (Const("=?=",_)$_$_) $ _) =>
skip_flexpairs (#2 (dest_implies ct))
| _ => ct;
(* A1==>...An==>B goes to [A1,...,An], where B is not an implication *)
fun strip_imp_prems ct =
let val (cA,cB) = dest_implies ct
in cA :: strip_imp_prems cB end
handle TERM _ => [];
(* A1==>...An==>B goes to B, where B is not an implication *)
fun strip_imp_concl ct =
case term_of ct of (Const("==>", _) $ _ $ _) =>
strip_imp_concl (#2 (dest_comb ct))
| _ => ct;
(*The premises of a theorem, as a cterm list*)
val cprems_of = strip_imp_prems o skip_flexpairs o cprop_of;
(** reading of instantiations **)
fun indexname cs = case Syntax.scan_varname cs of (v,[]) => v
| _ => error("Lexical error in variable name " ^ quote (implode cs));
fun absent ixn =
error("No such variable in term: " ^ Syntax.string_of_vname ixn);
fun inst_failure ixn =
error("Instantiation of " ^ Syntax.string_of_vname ixn ^ " fails");
fun read_insts sign (rtypes,rsorts) (types,sorts) used insts =
let val {tsig,...} = Sign.rep_sg sign
fun split([],tvs,vs) = (tvs,vs)
| split((sv,st)::l,tvs,vs) = (case explode sv of
"'"::cs => split(l,(indexname cs,st)::tvs,vs)
| cs => split(l,tvs,(indexname cs,st)::vs));
val (tvs,vs) = split(insts,[],[]);
fun readT((a,i),st) =
let val ixn = ("'" ^ a,i);
val S = case rsorts ixn of Some S => S | None => absent ixn;
val T = Sign.read_typ (sign,sorts) st;
in if Type.typ_instance(tsig,T,TVar(ixn,S)) then (ixn,T)
else inst_failure ixn
end
val tye = map readT tvs;
fun mkty(ixn,st) = (case rtypes ixn of
Some T => (ixn,(st,typ_subst_TVars tye T))
| None => absent ixn);
val ixnsTs = map mkty vs;
val ixns = map fst ixnsTs
and sTs = map snd ixnsTs
val (cts,tye2) = read_def_cterms(sign,types,sorts) used false sTs;
fun mkcVar(ixn,T) =
let val U = typ_subst_TVars tye2 T
in cterm_of sign (Var(ixn,U)) end
val ixnTs = ListPair.zip(ixns, map snd sTs)
in (map (fn (ixn,T) => (ixn,ctyp_of sign T)) (tye2 @ tye),
ListPair.zip(map mkcVar ixnTs,cts))
end;
(*** 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 (cterm_of sign) (sort (make_ord 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 (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 = ListPair.map (fn ((v,rs),a) => (v, TVar((a,0),rs)))
(inrs, nms')
val ctye = map (fn (v,T) => (v,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 (cterm_of sign (Var(v,T')),
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
th |> implies_intr_hyps
|> forall_intr_frees |> forall_elim_vars (maxidx + 1)
|> Thm.strip_shyps |> Thm.implies_intr_shyps
|> zero_var_indexes |> Thm.varifyT |> Thm.compress
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 (term_of (read_cterm sign
(sP, propT)))
in forall_elim_vars 0 (assume (cterm_of sign prop)) end;
(*Resolution: exactly one resolvent must be produced.*)
fun tha RSN (i,thb) =
case Seq.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 = Seq.list_of (resolve thb) handle THM _ => []
in List.concat (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) =
Seq.list_of (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 used = add_term_tvarnames(#prop(rep_thm th),[]);
in instantiate (read_insts sg ts ts used sinsts) 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,maxidx)) =
let val {sign=signt, t=t, T= T, maxidx=maxt,...} = rep_cterm ct
and {sign=signu, t=u, T= U, maxidx=maxu,...} = rep_cterm cu;
val maxi = Int.max(maxidx, Int.max(maxt, maxu));
val sign' = Sign.merge(sign, Sign.merge(signt, signu))
val (tye',maxi') = Type.unify (#tsig(Sign.rep_sg sign')) maxi tye (T,U)
handle Type.TUNIFY => raise TYPE("add_types", [T,U], [t,u])
in (sign', tye', maxi') end;
in
fun cterm_instantiate ctpairs0 th =
let val (sign,tye,_) = foldr add_types (ctpairs0, (#sign(rep_thm th),[],0))
val tsig = #tsig(Sign.rep_sg sign);
fun instT(ct,cu) = let val inst = subst_TVars tye
in (cterm_fun inst ct, cterm_fun inst cu) end
fun ctyp2 (ix,T) = (ix, 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 (msg, _, _) => raise THM("cterm_instantiate: " ^ msg, 0, [th])
end;
(** theorem equality **)
(*Do the two theorems have the same signature?*)
fun eq_thm_sg (th1,th2) = Sign.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;
(** Mark Staples's weaker version of eq_thm: ignores variable renaming and
(some) type variable renaming **)
(* Can't use term_vars, because it sorts the resulting list of variable names.
We instead need the unique list noramlised by the order of appearance
in the term. *)
fun term_vars' (t as Var(v,T)) = [t]
| term_vars' (Abs(_,_,b)) = term_vars' b
| term_vars' (f$a) = (term_vars' f) @ (term_vars' a)
| term_vars' _ = [];
fun forall_intr_vars th =
let val {prop,sign,...} = rep_thm th;
val vars = distinct (term_vars' prop);
in forall_intr_list (map (cterm_of sign) vars) th end;
fun weak_eq_thm (tha,thb) =
eq_thm(forall_intr_vars (freezeT tha), forall_intr_vars (freezeT thb));
(*** Meta-Rewriting Rules ***)
fun store_thm name thm = PureThy.smart_store_thm (name, standard thm);
val reflexive_thm =
let val cx = cterm_of (sign_of ProtoPure.thy) (Var(("x",0),TVar(("'a",0),logicS)))
in store_thm "reflexive" (Thm.reflexive cx) end;
val symmetric_thm =
let val xy = read_cterm (sign_of ProtoPure.thy) ("x::'a::logic == y",propT)
in store_thm "symmetric" (Thm.implies_intr_hyps(Thm.symmetric(Thm.assume xy))) end;
val transitive_thm =
let val xy = read_cterm (sign_of ProtoPure.thy) ("x::'a::logic == y",propT)
val yz = read_cterm (sign_of ProtoPure.thy) ("y::'a::logic == z",propT)
val xythm = Thm.assume xy and yzthm = Thm.assume yz
in store_thm "transitive" (Thm.implies_intr yz (Thm.transitive xythm yzthm)) end;
(** Below, a "conversion" has type cterm -> thm **)
val refl_implies = reflexive (cterm_of (sign_of ProtoPure.thy) implies);
(*In [A1,...,An]==>B, rewrite the selected A's only -- for rewrite_goals_tac*)
(*Do not rewrite flex-flex pairs*)
fun goals_conv pred cv =
let fun gconv i ct =
let val (A,B) = dest_implies ct
val (thA,j) = case term_of A of
Const("=?=",_)$_$_ => (reflexive A, i)
| _ => (if pred i then cv A else reflexive A, i+1)
in combination (combination refl_implies thA) (gconv j B) end
handle TERM _ => reflexive ct
in gconv 1 end;
(*Use a conversion to transform a theorem*)
fun fconv_rule cv th = equal_elim (cv (cprop_of th)) th;
(*rewriting conversion*)
fun rew_conv mode prover mss = rewrite_cterm mode mss prover;
(*Rewrite a theorem*)
fun rewrite_rule_aux _ [] th = th
| rewrite_rule_aux prover thms th =
fconv_rule (rew_conv (true,false) prover (Thm.mss_of thms)) th;
fun rewrite_thm mode prover mss = fconv_rule (rew_conv mode prover mss);
(*Rewrite the subgoals of a proof state (represented by a theorem) *)
fun rewrite_goals_rule_aux _ [] th = th
| rewrite_goals_rule_aux prover thms th =
fconv_rule (goals_conv (K true) (rew_conv (true, true) prover
(Thm.mss_of thms))) th;
(*Rewrite the subgoal of a proof state (represented by a theorem) *)
fun rewrite_goal_rule mode prover mss i thm =
if 0 < i andalso i <= nprems_of thm
then fconv_rule (goals_conv (fn j => j=i) (rew_conv mode prover mss)) thm
else raise THM("rewrite_goal_rule",i,[thm]);
(** Derived rules mainly for METAHYPS **)
(*Given the term "a", takes (%x.t)==(%x.u) to t[a/x]==u[a/x]*)
fun equal_abs_elim ca eqth =
let val {sign=signa, t=a, ...} = rep_cterm ca
and combth = combination eqth (reflexive ca)
val {sign,prop,...} = rep_thm eqth
val (abst,absu) = Logic.dest_equals prop
val cterm = cterm_of (Sign.merge (sign,signa))
in transitive (symmetric (beta_conversion (cterm (abst$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
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 = 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 ProtoPure.flexpair_def)
and flexpair_elim = flexpair_inst ProtoPure.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 =
store_thm "asm_rl" (trivial(read_cterm (sign_of ProtoPure.thy) ("PROP ?psi",propT)));
(*Meta-level cut rule: [| V==>W; V |] ==> W *)
val cut_rl =
store_thm "cut_rl"
(trivial(read_cterm (sign_of ProtoPure.thy) ("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 = read_cterm (sign_of ProtoPure.thy) ("PROP V", propT)
and VW = read_cterm (sign_of ProtoPure.thy) ("PROP V ==> PROP W", propT);
in
store_thm "revcut_rl"
(implies_intr V (implies_intr VW (implies_elim (assume VW) (assume V))))
end;
(*for deleting an unwanted assumption*)
val thin_rl =
let val V = read_cterm (sign_of ProtoPure.thy) ("PROP V", propT)
and W = read_cterm (sign_of ProtoPure.thy) ("PROP W", propT);
in store_thm "thin_rl" (implies_intr V (implies_intr W (assume W)))
end;
(* (!!x. PROP ?V) == PROP ?V Allows removal of redundant parameters*)
val triv_forall_equality =
let val V = read_cterm (sign_of ProtoPure.thy) ("PROP V", propT)
and QV = read_cterm (sign_of ProtoPure.thy) ("!!x::'a. PROP V", propT)
and x = read_cterm (sign_of ProtoPure.thy) ("x", TFree("'a",logicS));
in
store_thm "triv_forall_equality"
(equal_intr (implies_intr QV (forall_elim x (assume QV)))
(implies_intr V (forall_intr x (assume V))))
end;
(* (PROP ?PhiA ==> PROP ?PhiB ==> PROP ?Psi) ==>
(PROP ?PhiB ==> PROP ?PhiA ==> PROP ?Psi)
`thm COMP swap_prems_rl' swaps the first two premises of `thm'
*)
val swap_prems_rl =
let val cmajor = read_cterm (sign_of ProtoPure.thy)
("PROP PhiA ==> PROP PhiB ==> PROP Psi", propT);
val major = assume cmajor;
val cminor1 = read_cterm (sign_of ProtoPure.thy) ("PROP PhiA", propT);
val minor1 = assume cminor1;
val cminor2 = read_cterm (sign_of ProtoPure.thy) ("PROP PhiB", propT);
val minor2 = assume cminor2;
in store_thm "swap_prems_rl"
(implies_intr cmajor (implies_intr cminor2 (implies_intr cminor1
(implies_elim (implies_elim major minor1) minor2))))
end;
(* [| PROP ?phi ==> PROP ?psi; PROP ?psi ==> PROP ?phi |]
==> PROP ?phi == PROP ?psi
Introduction rule for == using ==> not meta-hyps.
*)
val equal_intr_rule =
let val PQ = read_cterm (sign_of ProtoPure.thy) ("PROP phi ==> PROP psi", propT)
and QP = read_cterm (sign_of ProtoPure.thy) ("PROP psi ==> PROP phi", propT)
in
store_thm "equal_intr_rule"
(implies_intr PQ (implies_intr QP (equal_intr (assume PQ) (assume QP))))
end;
(** instantiate' rule **)
(* collect vars *)
val add_tvarsT = foldl_atyps (fn (vs, TVar v) => v ins vs | (vs, _) => vs);
val add_tvars = foldl_types add_tvarsT;
val add_vars = foldl_aterms (fn (vs, Var v) => v ins vs | (vs, _) => vs);
fun tvars_of thm = rev (add_tvars ([], #prop (Thm.rep_thm thm)));
fun vars_of thm = rev (add_vars ([], #prop (Thm.rep_thm thm)));
(* instantiate by left-to-right occurrence of variables *)
fun instantiate' cTs cts thm =
let
fun err msg =
raise TYPE ("instantiate': " ^ msg,
mapfilter (apsome Thm.typ_of) cTs,
mapfilter (apsome Thm.term_of) cts);
fun inst_of (v, ct) =
(Thm.cterm_of (#sign (Thm.rep_cterm ct)) (Var v), ct)
handle TYPE (msg, _, _) => err msg;
fun zip_vars _ [] = []
| zip_vars (_ :: vs) (None :: opt_ts) = zip_vars vs opt_ts
| zip_vars (v :: vs) (Some t :: opt_ts) = (v, t) :: zip_vars vs opt_ts
| zip_vars [] _ = err "more instantiations than variables in thm";
(*instantiate types first!*)
val thm' =
if forall is_none cTs then thm
else Thm.instantiate (zip_vars (map fst (tvars_of thm)) cTs, []) thm;
in
if forall is_none cts then thm'
else Thm.instantiate ([], map inst_of (zip_vars (vars_of thm') cts)) thm'
end;
end;
open Drule;