ladder-calculus/coq/Atom.v

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(** Support for atoms, i.e., objects with decidable equality. We
provide here the ability to generate an atom fresh for any finite
collection, e.g., the lemma [atom_fresh_for_set], and a tactic to
pick an atom fresh for the current proof context.
Authors: Arthur Charguéraud and Brian Aydemir.
Implementation note: In older versions of Coq, [OrderedTypeEx]
redefines decimal constants to be integers and not natural
numbers. The following scope declaration is intended to address
this issue. In newer versions of Coq, the declaration should be
benign. *)
Require Import List.
(*Require Import Max.*)
Require Import OrderedType.
Require Import OrderedTypeEx.
Open Scope nat_scope.
Require Import FiniteSets.
Require Import FSetDecide.
Require Import FSetNotin.
Require Import ListFacts.
Require Import Psatz.
Require Import AdditionalTactics.
Require AdditionalTactics.
(* ********************************************************************** *)
(** * Definition *)
(** Atoms are structureless objects such that we can always generate
one fresh from a finite collection. Equality on atoms is [eq] and
decidable. We use Coq's module system to make abstract the
implementation of atoms. The [Export AtomImpl] line below allows
us to refer to the type [atom] and its properties without having
to qualify everything with "[AtomImpl.]". *)
Module Type ATOM.
Parameter atom : Set.
Parameter atom_fresh_for_list :
forall (xs : list atom), {x : atom | ~ List.In x xs}.
Declare Module Atom_as_OT : UsualOrderedType with Definition t := atom.
Parameter eq_atom_dec : forall x y : atom, {x = y} + {x <> y}.
End ATOM.
(** The implementation of the above interface is hidden for
documentation purposes. *)
Module AtomImpl : ATOM.
(* begin hide *)
Definition atom := nat.
Lemma max_lt_r : forall x y z,
x <= z -> x <= max y z.
Proof.
induction x. auto with arith.
induction y; auto with arith.
simpl. induction z. lia. auto with arith.
Qed.
Lemma nat_list_max : forall (xs : list nat),
{ n : nat | forall x, In x xs -> x <= n }.
Proof.
induction xs as [ | x xs [y H] ].
(* case: nil *)
exists 0. inversion 1.
(* case: cons x xs *)
exists (max x y). intros z J. simpl in J. destruct J as [K | K].
subst. auto with arith.
auto using max_lt_r.
Qed.
Lemma atom_fresh_for_list :
forall (xs : list nat), { n : nat | ~ List.In n xs }.
Proof.
intros xs. destruct (nat_list_max xs) as [x H].
exists (S x). intros J. lapply (H (S x)). lia. trivial.
Qed.
Module Atom_as_OT := Nat_as_OT.
Module Facts := OrderedTypeFacts Atom_as_OT.
Definition eq_atom_dec : forall x y : atom, {x = y} + {x <> y} :=
Facts.eq_dec.
(* end hide *)
End AtomImpl.
Export AtomImpl.
(* ********************************************************************** *)
(** * Finite sets of atoms *)
(* ********************************************************************** *)
(** ** Definitions *)
Module AtomSet : FiniteSets.S with Module E := Atom_as_OT :=
FiniteSets.Make Atom_as_OT.
(** The type [atoms] is the type of finite sets of [atom]s. *)
Notation atoms := AtomSet.F.t.
(** Basic operations on finite sets of atoms are available, in the
remainder of this file, without qualification. We use [Import]
instead of [Export] in order to avoid unnecessary namespace
pollution. *)
Import AtomSet.F.
(** We instantiate two modules which provide useful lemmas and tactics
work working with finite sets of atoms. *)
Module AtomSetDecide := FSetDecide.Decide AtomSet.F.
Module AtomSetNotin := FSetNotin.Notin AtomSet.F.
(* *********************************************************************** *)
(** ** Tactics for working with finite sets of atoms *)
(** The tactic [fsetdec] is a general purpose decision procedure
for solving facts about finite sets of atoms. *)
Ltac fsetdec := try apply AtomSet.eq_if_Equal; AtomSetDecide.fsetdec.
(** The tactic [notin_simpl] simplifies all hypotheses of the form [(~
In x F)], where [F] is constructed from the empty set, singleton
sets, and unions. *)
Ltac notin_simpl := AtomSetNotin.notin_simpl_hyps.
(** The tactic [notin_solve], solves goals of the form [(~ In x F)],
where [F] is constructed from the empty set, singleton sets, and
unions. The goal must be provable from hypothesis of the form
simplified by [notin_simpl]. *)
Ltac notin_solve := AtomSetNotin.notin_solve.
(* *********************************************************************** *)
(** ** Lemmas for working with finite sets of atoms *)
(** We make some lemmas about finite sets of atoms available without
qualification by using abbreviations. *)
Notation eq_if_Equal := AtomSet.eq_if_Equal.
Notation notin_empty := AtomSetNotin.notin_empty.
Notation notin_singleton := AtomSetNotin.notin_singleton.
Notation notin_singleton_rw := AtomSetNotin.notin_singleton_rw.
Notation notin_union := AtomSetNotin.notin_union.
(* ********************************************************************** *)
(** * Additional properties *)
(** One can generate an atom fresh for a given finite set of atoms. *)
Lemma atom_fresh_for_set : forall L : atoms, { x : atom | ~ In x L }.
Proof.
intros L. destruct (atom_fresh_for_list (elements L)) as [a H].
exists a. intros J. contradiction H.
rewrite <- InA_iff_In. auto using elements_1.
Qed.
(* ********************************************************************** *)
(** * Additional tactics *)
(* ********************************************************************** *)
(** ** #<a name="pick_fresh"></a># Picking a fresh atom *)
(** We define three tactics which, when combined, provide a simple
mechanism for picking a fresh atom. We demonstrate their use
below with an example, the [example_pick_fresh] tactic.
[(gather_atoms_with F)] returns the union of [(F x)], where [x]
ranges over all objects in the context such that [(F x)] is
well typed. The return type of [F] should be [atoms]. The
complexity of this tactic is due to the fact that there is no
support in [Ltac] for folding a function over the context. *)
Ltac gather_atoms_with F :=
let rec gather V :=
match goal with
| H: ?S |- _ =>
let FH := constr:(F H) in
match V with
| empty => gather FH
| context [FH] => fail 1
| _ => gather (union FH V)
end
| _ => V
end in
let L := gather empty in eval simpl in L.
(** [(beautify_fset V)] takes a set [V] built as a union of finite
sets and returns the same set with empty sets removed and union
operations associated to the right. Duplicate sets are also
removed from the union. *)
Ltac beautify_fset V :=
let rec go Acc E :=
match E with
| union ?E1 ?E2 => let Acc1 := go Acc E2 in go Acc1 E1
| empty => Acc
| ?E1 => match Acc with
| empty => E1
| context [E1] => Acc
| _ => constr:(union E1 Acc)
end
end
in go empty V.
(** The tactic [(pick fresh Y for L)] takes a finite set of atoms [L]
and a fresh name [Y], and adds to the context an atom with name
[Y] and a proof that [(~ In Y L)], i.e., that [Y] is fresh for
[L]. The tactic will fail if [Y] is already declared in the
context. *)
Tactic Notation "pick" "fresh" ident(Y) "for" constr(L) :=
let Fr := fresh "Fr" in
let L := beautify_fset L in
(destruct (atom_fresh_for_set L) as [Y Fr]).
(* ********************************************************************** *)
(** ** Demonstration *)
(** The [example_pick_fresh] tactic below illustrates the general
pattern for using the above three tactics to define a tactic which
picks a fresh atom. The pattern is as follows:
- Repeatedly invoke [gather_atoms_with], using functions with
different argument types each time.
- Union together the result of the calls, and invoke
[(pick fresh ... for ...)] with that union of sets. *)
Ltac example_pick_fresh Y :=
let A := gather_atoms_with (fun x : atoms => x) in
let B := gather_atoms_with (fun x : atom => singleton x) in
pick fresh Y for (union A B).
Lemma example_pick_fresh_use : forall (x y z : atom) (L1 L2 L3: atoms), True.
(* begin show *)
Proof.
intros x y z L1 L2 L3. example_pick_fresh k.
(** At this point in the proof, we have a new atom [k] and a
hypothesis [Fr : ~ In k (union L1 (union L2 (union L3 (union
(singleton x) (union (singleton y) (singleton z))))))]. *)
trivial.
Qed.
(* end show *)