File ‹~~/src/Provers/hypsubst.ML›

```(*  Title:      Provers/hypsubst.ML
Authors:    Martin D Coen, Tobias Nipkow and Lawrence C Paulson

Basic equational reasoning: hyp_subst_tac and methods "hypsubst", "simplesubst".

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";

Goal "!!x a. [| x = f(b); g(a) = b |] ==> P(x)";

by (bound_hyp_subst_tac 1);
by (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 true 1);
*)

signature HYPSUBST_DATA =
sig
val dest_Trueprop    : term -> term
val dest_eq          : term -> term * term
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    : Proof.context -> int -> tactic
val hyp_subst_tac_thin     : bool -> Proof.context -> int -> tactic
val hyp_subst_thin         : bool Config.T
val hyp_subst_tac          : Proof.context -> int -> tactic
val blast_hyp_subst_tac    : Proof.context -> bool -> int -> tactic
val stac                   : Proof.context -> thm -> int -> tactic
end;

functor Hypsubst(Data: HYPSUBST_DATA): HYPSUBST =
struct

exception EQ_VAR;

(*Simplifier turns Bound variables to special Free variables:
change it back (any Bound variable will do)*)
fun inspect_contract t =
(case Envir.eta_contract t of
Free (a, T) => if Name.is_bound a then Bound 0 else Free (a, T)
| t' => t');

val has_vars = Term.exists_subterm Term.is_Var;
val has_tvars = Term.exists_type (Term.exists_subtype Term.is_TVar);

(*If novars then we forbid Vars in the equality.
If bnd then we only look for Bound 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
also returns var to substitute, relevant if it is Free *)
fun inspect_pair bnd novars (t, u) =
if novars andalso (has_tvars t orelse has_tvars u)
then raise Match   (*variables in the type!*)
else
(case apply2 inspect_contract (t, u) of
(Bound i, _) =>
if loose_bvar1 (u, i) orelse novars andalso has_vars u
then raise Match
else (true, Bound i)                (*eliminates t*)
| (_, Bound i) =>
if loose_bvar1 (t, i) orelse novars andalso has_vars t
then raise Match
else (false, Bound i)               (*eliminates u*)
| (t' as Free _, _) =>
if bnd orelse Logic.occs (t', u) orelse novars andalso has_vars u
then raise Match
else (true, t')                (*eliminates t*)
| (_, u' as Free _) =>
if bnd orelse Logic.occs (u', t) orelse novars andalso has_vars t
then raise Match
else (false, u')               (*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 check_frees t =
let
fun check_free ts (orient, Free f)
= if not check_frees orelse not orient
orelse exists (curry Logic.occs (Free f)) ts
then (orient, true) else raise Match
| check_free ts (orient, _) = (orient, false)
fun eq_var_aux k \<^Const_>‹Pure.all _ for ‹Abs(_,_,t)›› hs = eq_var_aux k t hs
| eq_var_aux k \<^Const_>‹Pure.imp for A B› hs =
((k, check_free (B :: hs) (inspect_pair bnd novars
(Data.dest_eq (Data.dest_Trueprop A))))
handle TERM _ => eq_var_aux (k+1) B (A :: hs)
| Match => eq_var_aux (k+1) B (A :: hs))
| eq_var_aux k _ _ = raise EQ_VAR

in  eq_var_aux 0 t [] end;

fun thin_free_eq_tac ctxt = SUBGOAL (fn (t, i) =>
let
val (k, _) = eq_var false false false t
val ok = (eq_var false false true t |> fst) > k handle EQ_VAR => true
in
if ok then EVERY [rotate_tac k i, eresolve_tac ctxt [thin_rl] i, rotate_tac (~k) i]
else no_tac
end handle EQ_VAR => no_tac)

(*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 bnd th =
[ if inspect_pair bnd false (Data.dest_eq (Data.dest_Trueprop (Thm.prop_of th))) |> fst
then th RS Data.eq_reflection
else Thm.symmetric(th RS Data.eq_reflection) (*reorient*) ]
handle TERM _ => [] | Match => [];

(*Select a suitable equality assumption; substitute throughout the subgoal
If bnd is true, then it replaces Bound variables only. *)
fun gen_hyp_subst_tac ctxt bnd =
SUBGOAL (fn (Bi, i) =>
let
val (k, (orient, is_free)) = eq_var bnd true true Bi
val hyp_subst_ctxt = empty_simpset ctxt |> Simplifier.set_mksimps (K (mk_eqs bnd))
in EVERY [rotate_tac k i, asm_lr_simp_tac hyp_subst_ctxt i,
if not is_free then eresolve_tac ctxt [thin_rl] i
else if orient then eresolve_tac ctxt [Data.rev_mp] i
else eresolve_tac ctxt [Data.sym RS Data.rev_mp] i,
rotate_tac (~k) i,
if is_free then resolve_tac ctxt [Data.imp_intr] i else all_tac]
end handle THM _ => no_tac | EQ_VAR => no_tac)

val ssubst = Drule.zero_var_indexes (Data.sym RS Data.subst);

fun inst_subst_tac ctxt b rl = CSUBGOAL (fn (cBi, i) =>
case try (Logic.strip_assums_hyp #> hd #>
Data.dest_Trueprop #> Data.dest_eq #> apply2 Envir.eta_contract) (Thm.term_of cBi) of
SOME (t, t') =>
let
val Bi = Thm.term_of cBi;
val ps = Logic.strip_params Bi;
val U = Term.fastype_of1 (rev (map snd ps), t);
val Q = Data.dest_Trueprop (Logic.strip_assums_concl Bi);
val rl' = Thm.lift_rule cBi rl;
val (ixn, T) = dest_Var (Term.head_of (Data.dest_Trueprop
(Logic.strip_assums_concl (Thm.prop_of rl'))));
val (v1, v2) = Data.dest_eq (Data.dest_Trueprop
(Logic.strip_assums_concl (hd (Thm.prems_of rl'))));
val (Ts, V) = split_last (Term.binder_types T);
val u =
fold_rev Term.abs (ps @ [("x", U)])
(case (if b then t else t') of
Bound j => subst_bounds (map Bound ((1 upto j) @ 0 :: (j + 2 upto length ps)), Q)
| t => Term.abstract_over (t, Term.incr_boundvars 1 Q));
val (instT, _) = Thm.match (apply2 (Thm.cterm_of ctxt o Logic.mk_type) (V, U));
in
compose_tac ctxt (true, Drule.instantiate_normalize (instT,
Vars.make (map (apsnd (Thm.cterm_of ctxt))
[((ixn, Ts ---> U --> body_type T), u),
((fst (dest_Var (head_of v1)), Ts ---> U), fold_rev Term.abs ps t),
((fst (dest_Var (head_of v2)), Ts ---> U), fold_rev Term.abs ps t')])) rl',
Thm.nprems_of rl) i
end
| NONE => no_tac);

fun imp_intr_tac ctxt = resolve_tac ctxt [Data.imp_intr];

fun rev_dup_elim ctxt th = (th RSN (2, revcut_rl)) |> Thm.assumption (SOME ctxt) 2 |> Seq.hd;
fun dup_subst ctxt = rev_dup_elim ctxt ssubst

(* FIXME: "etac Data.rev_mp i" will not behave as expected if goal has *)
(* premises containing meta-implications or quantifiers                *)

(*Old version of the tactic above -- slower but the only way
to handle equalities containing Vars.*)
fun vars_gen_hyp_subst_tac ctxt bnd = SUBGOAL(fn (Bi,i) =>
let val n = length(Logic.strip_assums_hyp Bi) - 1
val (k, (orient, is_free)) = eq_var bnd false true Bi
val rl = if is_free then dup_subst ctxt else ssubst
val rl = if orient then rl else Data.sym RS rl
in
DETERM
(EVERY [REPEAT_DETERM_N k (eresolve_tac ctxt [Data.rev_mp] i),
rotate_tac 1 i,
REPEAT_DETERM_N (n-k) (eresolve_tac ctxt [Data.rev_mp] i),
inst_subst_tac ctxt orient rl i,
REPEAT_DETERM_N n (imp_intr_tac ctxt i THEN rotate_tac ~1 i)])
end
handle THM _ => no_tac | EQ_VAR => no_tac);

and on Free variables if thin=true*)
fun hyp_subst_tac_thin thin ctxt =
REPEAT_DETERM1 o FIRST' [ematch_tac ctxt [Data.thin_refl],
gen_hyp_subst_tac ctxt false, vars_gen_hyp_subst_tac ctxt false,
if thin then thin_free_eq_tac ctxt else K no_tac];

val hyp_subst_thin = Attrib.setup_config_bool \<^binding>‹hypsubst_thin› (K false);

fun hyp_subst_tac ctxt =
hyp_subst_tac_thin (Config.get ctxt hyp_subst_thin) ctxt;

(*Substitutes for Bound variables only -- this is always safe*)
fun bound_hyp_subst_tac ctxt =
REPEAT_DETERM1 o (gen_hyp_subst_tac ctxt true
ORELSE' vars_gen_hyp_subst_tac ctxt 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 ctxt n i =
REPEAT_DETERM_N n (rotate_tac ~1 i THEN eresolve_tac ctxt [Data.rev_mp] i) THEN
REPEAT_DETERM_N n (imp_intr_tac ctxt 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 ctxt hyps i =
let
fun imptac (r, []) st = reverse_n_tac ctxt r i st
| imptac (r, hyp::hyps) st =
let
val (hyp', _) =
Thm.term_of (Thm.cprem_of st i)
|> Logic.strip_assums_concl
|> Data.dest_Trueprop |> Data.dest_imp;
val (r', tac) =
if Envir.aeconv (hyp, hyp')
then (r, imp_intr_tac ctxt i THEN rotate_tac ~1 i)
else (*leave affected hyps at end*) (r + 1, imp_intr_tac ctxt i);
in
(case Seq.pull (tac st) of
NONE => Seq.single st
| SOME (st', _) => imptac (r', hyps) st')
end
in imptac (0, rev hyps) end;

fun blast_hyp_subst_tac ctxt trace = SUBGOAL(fn (Bi, i) =>
let
val (k, (symopt, _)) = eq_var false 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 tracing "Substituting an equality" else ();
DETERM
(EVERY [REPEAT_DETERM_N k (eresolve_tac ctxt [Data.rev_mp] i),
rotate_tac 1 i,
REPEAT_DETERM_N (n-k) (eresolve_tac ctxt [Data.rev_mp] i),
inst_subst_tac ctxt symopt (if symopt then ssubst else Data.subst) i,
all_imp_intr_tac ctxt 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 ctxt th =
let val th' = th RS Data.rev_eq_reflection handle THM _ => th
in CHANGED_GOAL (resolve_tac ctxt [th' RS ssubst]) end;

(* method setup *)

val _ =
Theory.setup
(Method.setup \<^binding>‹hypsubst›
(Scan.succeed (fn ctxt => SIMPLE_METHOD' (CHANGED_PROP o hyp_subst_tac ctxt)))
"substitution using an assumption (improper)" #>
Method.setup \<^binding>‹hypsubst_thin›
(Scan.succeed (fn ctxt => SIMPLE_METHOD'
(CHANGED_PROP o hyp_subst_tac_thin true ctxt)))
"substitution using an assumption, eliminating assumptions" #>
Method.setup \<^binding>‹simplesubst›
(Attrib.thm >> (fn th => fn ctxt => SIMPLE_METHOD' (stac ctxt th)))
"simple substitution");

end;
```