(* Title: Provers/hypsubst.ML
ID: $Id$
Authors: Martin D Coen, Tobias Nipkow and Lawrence C Paulson
Copyright 1995 University of Cambridge
Basic equational reasoning: (hyp_)subst method and symmetric attribute.
Tactic to substitute using (at least) the assumption x=t in the rest
of the subgoal, and to delete (at least) that assumption. Original
version due to Martin Coen.
This version uses the simplifier, and requires it to be already present.
Test data:
Goal "!!x.[| Q(x,y,z); y=x; a=x; z=y; P(y) |] ==> P(z)";
Goal "!!x.[| Q(x,y,z); z=f(x); x=z |] ==> P(z)";
Goal "!!y. [| ?x=y; P(?x) |] ==> y = a";
Goal "!!z. [| ?x=y; P(?x) |] ==> y = a";
by (hyp_subst_tac 1);
by (bound_hyp_subst_tac 1);
Here hyp_subst_tac goes wrong; harder still to prove P(f(f(a))) & P(f(a))
Goal "P(a) --> (EX y. a=y --> P(f(a)))";
Goal "!!x. [| Q(x,h1); P(a,h2); R(x,y,h3); R(y,z,h4); x=f(y); \
\ P(x,h5); P(y,h6); K(x,h7) |] ==> Q(x,c)";
by (blast_hyp_subst_tac (ref true) 1);
*)
signature HYPSUBST_DATA =
sig
structure Simplifier : SIMPLIFIER
val dest_Trueprop : term -> term
val dest_eq : term -> term*term*typ
val dest_imp : term -> term*term
val eq_reflection : thm (* a=b ==> a==b *)
val rev_eq_reflection: thm (* a==b ==> a=b *)
val imp_intr : thm (* (P ==> Q) ==> P-->Q *)
val rev_mp : thm (* [| P; P-->Q |] ==> Q *)
val subst : thm (* [| a=b; P(a) |] ==> P(b) *)
val sym : thm (* a=b ==> b=a *)
val thin_refl : thm (* [|x=x; PROP W|] ==> PROP W *)
end;
signature HYPSUBST =
sig
val bound_hyp_subst_tac : int -> tactic
val hyp_subst_tac : int -> tactic
val blast_hyp_subst_tac : bool ref -> int -> tactic
(*exported purely for debugging purposes*)
val gen_hyp_subst_tac : bool -> int -> tactic
val vars_gen_hyp_subst_tac : bool -> int -> tactic
val eq_var : bool -> bool -> term -> int * bool
val inspect_pair : bool -> bool -> term * term * typ -> bool
val mk_eqs : thm -> thm list
val thin_leading_eqs_tac : bool -> int -> int -> tactic
val stac : thm -> int -> tactic
val hypsubst_setup : (theory -> theory) list
end;
functor HypsubstFun(Data: HYPSUBST_DATA): HYPSUBST =
struct
exception EQ_VAR;
fun loose (i,t) = 0 mem_int add_loose_bnos(t,i,[]);
local val odot = ord"."
in
(*Simplifier turns Bound variables to dotted Free variables:
change it back (any Bound variable will do)
*)
fun contract t =
case Pattern.eta_contract_atom t of
Free(a,T) => if (ord a = odot) then Bound 0 else Free(a,T)
| t' => t'
end;
fun has_vars t = maxidx_of_term t <> ~1;
(*If novars then we forbid Vars in the equality.
If bnd then we only look for Bound (or dotted Free) variables to eliminate.
When can we safely delete the equality?
Not if it equates two constants; consider 0=1.
Not if it resembles x=t[x], since substitution does not eliminate x.
Not if it resembles ?x=0; consider ?x=0 ==> ?x=1 or even ?x=0 ==> P
Not if it involves a variable free in the premises,
but we can't check for this -- hence bnd and bound_hyp_subst_tac
Prefer to eliminate Bound variables if possible.
Result: true = use as is, false = reorient first *)
fun inspect_pair bnd novars (t,u,T) =
if novars andalso maxidx_of_typ T <> ~1
then raise Match (*variables in the type!*)
else
case (contract t, contract u) of
(Bound i, _) => if loose(i,u) orelse novars andalso has_vars u
then raise Match
else true (*eliminates t*)
| (_, Bound i) => if loose(i,t) orelse novars andalso has_vars t
then raise Match
else false (*eliminates u*)
| (Free _, _) => if bnd orelse Logic.occs(t,u) orelse
novars andalso has_vars u
then raise Match
else true (*eliminates t*)
| (_, Free _) => if bnd orelse Logic.occs(u,t) orelse
novars andalso has_vars t
then raise Match
else false (*eliminates u*)
| _ => raise Match;
(*Locates a substitutable variable on the left (resp. right) of an equality
assumption. Returns the number of intervening assumptions. *)
fun eq_var bnd novars =
let fun eq_var_aux k (Const("all",_) $ Abs(_,_,t)) = eq_var_aux k t
| eq_var_aux k (Const("==>",_) $ A $ B) =
((k, inspect_pair bnd novars
(Data.dest_eq (Data.dest_Trueprop A)))
(*Exception comes from inspect_pair or dest_eq*)
handle _ => eq_var_aux (k+1) B)
| eq_var_aux k _ = raise EQ_VAR
in eq_var_aux 0 end;
(*We do not try to delete ALL equality assumptions at once. But
it is easy to handle several consecutive equality assumptions in a row.
Note that we have to inspect the proof state after doing the rewriting,
since e.g. z=f(x); x=z changes to z=f(x); x=f(x) and the second equality
must NOT be deleted. Tactic must rotate or delete m assumptions.
*)
fun thin_leading_eqs_tac bnd m = SUBGOAL (fn (Bi,i) =>
let fun count [] = 0
| count (A::Bs) = ((inspect_pair bnd true
(Data.dest_eq (Data.dest_Trueprop A));
1 + count Bs)
handle _ => 0)
val j = Int.min(m, count (Logic.strip_assums_hyp Bi))
in REPEAT_DETERM_N j (etac thin_rl i) THEN rotate_tac (m-j) i
end);
(*For the simpset. Adds ALL suitable equalities, even if not first!
No vars are allowed here, as simpsets are built from meta-assumptions*)
fun mk_eqs th =
[ if inspect_pair false false (Data.dest_eq
(Data.dest_Trueprop (#prop (rep_thm th))))
then th RS Data.eq_reflection
else symmetric(th RS Data.eq_reflection) (*reorient*) ]
handle _ => []; (*Exception comes from inspect_pair or dest_eq*)
local open Simplifier
in
val hyp_subst_ss = empty_ss setmksimps mk_eqs
(*Select a suitable equality assumption and substitute throughout the subgoal
Replaces only Bound variables if bnd is true*)
fun gen_hyp_subst_tac bnd = SUBGOAL(fn (Bi,i) =>
let val n = length(Logic.strip_assums_hyp Bi) - 1
val (k,_) = eq_var bnd true Bi
in
DETERM (EVERY [rotate_tac k i,
asm_full_simp_tac hyp_subst_ss i,
etac thin_rl i,
thin_leading_eqs_tac bnd (n-k) i])
end
handle THM _ => no_tac | EQ_VAR => no_tac);
end;
val ssubst = standard (Data.sym RS Data.subst);
val imp_intr_tac = rtac Data.imp_intr;
(*Old version of the tactic above -- slower but the only way
to handle equalities containing Vars.*)
fun vars_gen_hyp_subst_tac bnd = SUBGOAL(fn (Bi,i) =>
let val n = length(Logic.strip_assums_hyp Bi) - 1
val (k,symopt) = eq_var bnd false Bi
in
DETERM
(EVERY [REPEAT_DETERM_N k (etac Data.rev_mp i),
rotate_tac 1 i,
REPEAT_DETERM_N (n-k) (etac Data.rev_mp i),
etac (if symopt then ssubst else Data.subst) i,
REPEAT_DETERM_N n (imp_intr_tac i THEN rotate_tac ~1 i)])
end
handle THM _ => no_tac | EQ_VAR => no_tac);
(*Substitutes for Free or Bound variables*)
val hyp_subst_tac = FIRST' [ematch_tac [Data.thin_refl],
gen_hyp_subst_tac false, vars_gen_hyp_subst_tac false];
(*Substitutes for Bound variables only -- this is always safe*)
val bound_hyp_subst_tac =
gen_hyp_subst_tac true ORELSE' vars_gen_hyp_subst_tac true;
(** Version for Blast_tac. Hyps that are affected by the substitution are
moved to the front. Defect: even trivial changes are noticed, such as
substitutions in the arguments of a function Var. **)
(*final re-reversal of the changed assumptions*)
fun reverse_n_tac 0 i = all_tac
| reverse_n_tac 1 i = rotate_tac ~1 i
| reverse_n_tac n i =
REPEAT_DETERM_N n (rotate_tac ~1 i THEN etac Data.rev_mp i) THEN
REPEAT_DETERM_N n (imp_intr_tac i THEN rotate_tac ~1 i);
(*Use imp_intr, comparing the old hyps with the new ones as they come out.*)
fun all_imp_intr_tac hyps i =
let fun imptac (r, []) st = reverse_n_tac r i st
| imptac (r, hyp::hyps) st =
let val (hyp',_) = List.nth (prems_of st, i-1) |>
Logic.strip_assums_concl |>
Data.dest_Trueprop |> Data.dest_imp
val (r',tac) = if Pattern.aeconv (hyp,hyp')
then (r, imp_intr_tac i THEN rotate_tac ~1 i)
else (*leave affected hyps at end*)
(r+1, imp_intr_tac i)
in
case Seq.pull(tac st) of
None => Seq.single(st)
| Some(st',_) => imptac (r',hyps) st'
end handle _ => error "?? in blast_hyp_subst_tac"
in imptac (0, rev hyps) end;
fun blast_hyp_subst_tac trace = SUBGOAL(fn (Bi,i) =>
let val (k,symopt) = eq_var false false Bi
val hyps0 = map Data.dest_Trueprop (Logic.strip_assums_hyp Bi)
(*omit selected equality, returning other hyps*)
val hyps = List.take(hyps0, k) @ List.drop(hyps0, k+1)
val n = length hyps
in
if !trace then writeln "Substituting an equality" else ();
DETERM
(EVERY [REPEAT_DETERM_N k (etac Data.rev_mp i),
rotate_tac 1 i,
REPEAT_DETERM_N (n-k) (etac Data.rev_mp i),
etac (if symopt then ssubst else Data.subst) i,
all_imp_intr_tac hyps i])
end
handle THM _ => no_tac | EQ_VAR => no_tac);
(*apply an equality or definition ONCE;
fails unless the substitution has an effect*)
fun stac th =
let val th' = th RS Data.rev_eq_reflection handle THM _ => th
in CHANGED_GOAL (rtac (th' RS ssubst)) end;
(* proof methods *)
val subst_meth = Method.thm_args (Method.SIMPLE_METHOD' HEADGOAL o stac);
val hyp_subst_meth = Method.no_args (Method.SIMPLE_METHOD' HEADGOAL hyp_subst_tac);
(* attributes *)
fun symmetric_rule thm =
thm RS Drule.symmetric_thm handle THM _ =>
thm RS Data.sym handle THM _ => error "Failed to apply symmetry of == or =";
fun gen_symmetric x = Drule.rule_attribute (K symmetric_rule);
(* theory setup *)
val hypsubst_setup =
[Method.add_methods
[("hypsubst", hyp_subst_meth, "substitution using an assumption (improper)"),
("subst", subst_meth, "substitution")],
Attrib.add_attributes
[("symmetric", (gen_symmetric, gen_symmetric), "resolution with symmetry of == or =")]];
end;