take over FiniteSet & Atom libraries from popl-tutorial

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(** A library of additional tactics. *)
Require Export String.
Open Scope string_scope.
(* *********************************************************************** *)
(** * Extensions of the standard library *)
(** "[remember c as x in |-]" replaces the term [c] by the identifier
[x] in the conclusion of the current goal and introduces the
hypothesis [x=c] into the context. This tactic differs from a
similar one in the standard library in that the replacmement is
made only in the conclusion of the goal; the context is left
unchanged. *)
Tactic Notation "remember" constr(c) "as" ident(x) "in" "|-" :=
let x := fresh x in
let H := fresh "Heq" x in
(set (x := c); assert (H : x = c) by reflexivity; clearbody x).
(** "[unsimpl E]" replaces all occurence of [X] by [E], where [X] is
the result that tactic [simpl] would give when used to evaluate
[E]. *)
Tactic Notation "unsimpl" constr(E) :=
let F := (eval simpl in E) in change F with E.
(** The following tactic calls the [apply] tactic with the first
hypothesis that succeeds, "first" meaning the hypothesis that
comes earlist in the context (i.e., higher up in the list). *)
Ltac apply_first_hyp :=
match reverse goal with
| H : _ |- _ => apply H
end.
(* *********************************************************************** *)
(** * Variations on [auto] *)
(** The [auto*] and [eauto*] tactics are intended to be "stronger"
versions of the [auto] and [eauto] tactics. Similar to [auto] and
[eauto], they each take an optional "depth" argument. Note that
if we declare these tactics using a single string, e.g., "auto*",
then the resulting tactics are unusable since they fail to
parse. *)
Tactic Notation "auto" "*" :=
try solve [ congruence | auto | intuition auto ].
Tactic Notation "auto" "*" integer(n) :=
try solve [ congruence | auto n | intuition (auto n) ].
Tactic Notation "eauto" "*" :=
try solve [ congruence | eauto | intuition eauto ].
Tactic Notation "eauto" "*" integer(n) :=
try solve [ congruence | eauto n | intuition (eauto n) ].
(* *********************************************************************** *)
(** * Delineating cases in proofs *)
(** This section was taken from the POPLmark Wiki
( http://alliance.seas.upenn.edu/~plclub/cgi-bin/poplmark/ ). *)
(** ** Tactic definitions *)
Ltac move_to_top x :=
match reverse goal with
| H : _ |- _ => try move x after H
end.
Tactic Notation "assert_eq" ident(x) constr(v) :=
let H := fresh in
assert (x = v) as H by reflexivity;
clear H.
Tactic Notation "Case_aux" ident(x) constr(name) :=
first [
set (x := name); move_to_top x
| assert_eq x name
| fail 1 "because we are working on a different case." ].
Ltac Case name := Case_aux case name.
Ltac SCase name := Case_aux subcase name.
Ltac SSCase name := Case_aux subsubcase name.
(** ** Example
One mode of use for the above tactics is to wrap Coq's [induction]
tactic such that automatically inserts "case" markers into each
branch of the proof. For example:
<<
Tactic Notation "induction" "nat" ident(n) :=
induction n; [ Case "O" | Case "S" ].
Tactic Notation "sub" "induction" "nat" ident(n) :=
induction n; [ SCase "O" | SCase "S" ].
Tactic Notation "sub" "sub" "induction" "nat" ident(n) :=
induction n; [ SSCase "O" | SSCase "S" ].
>>
If you use such customized versions of the induction tactics, then
the [Case] tactic will verify that you are working on the case
that you think you are. You may also use the [Case] tactic with
the standard version of [induction], in which case no verification
is done. *)

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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 *)

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(** Lemmas and tactics for working with and solving goals related to
non-membership in finite sets. The main tactic of interest here
is [notin_solve].
Authors: Arthur Charguéraud and Brian Aydemir. *)
Set Implicit Arguments.
Require Import FSetInterface.
Require Import AdditionalTactics.
Require AdditionalTactics.
(* *********************************************************************** *)
(** * Implementation *)
Module Notin (X : FSetInterface.S).
Import X.
Import AdditionalTactics.
(* *********************************************************************** *)
(** ** Facts about set (non-)membership *)
Lemma in_singleton : forall x,
In x (singleton x).
Proof.
intros.
apply singleton_2.
generalize dependent x.
apply E.eq_refl.
Qed.
Lemma notin_empty : forall x,
~ In x empty.
Proof.
auto using empty_1.
Qed.
Lemma notin_union : forall x E F,
~ In x E -> ~ In x F -> ~ In x (union E F).
Proof.
intros x E F H J K.
destruct (union_1 K); intuition.
Qed.
Lemma elim_notin_union : forall x E F,
~ In x (union E F) -> (~ In x E) /\ (~ In x F).
Proof.
intros x E F H. split; intros J; contradiction H.
auto using union_2.
auto using union_3.
Qed.
Lemma notin_singleton : forall x y,
~ E.eq x y -> ~ In x (singleton y).
Proof.
intros x y H J. assert (K := singleton_1 J). auto with *.
Qed.
Lemma elim_notin_singleton : forall x y,
~ In x (singleton y) -> ~ E.eq x y.
Proof.
intros x y H J.
contradiction H.
apply singleton_2.
generalize x y J.
apply E.eq_sym.
Qed.
Lemma elim_notin_singleton' : forall x y,
~ In x (singleton y) -> x <> y.
Proof.
intros. assert (~ E.eq x y). auto using singleton_2.
intros J. subst. auto with *.
contradict H0.
rewrite H0.
apply E.eq_refl.
Qed.
Lemma notin_singleton_swap : forall x y,
~ In x (singleton y) -> ~ In y (singleton x).
Proof.
intros.
assert (Q := elim_notin_singleton H).
auto using singleton_1.
Qed.
(* *********************************************************************** *)
(** ** Rewriting non-membership facts *)
Lemma notin_singleton_rw : forall x y,
~ In x (singleton y) <-> ~ E.eq x y.
Proof.
intros. split.
auto using elim_notin_singleton.
auto using notin_singleton.
Qed.
(* *********************************************************************** *)
(** ** Tactics *)
(** The tactic [notin_simpl_hyps] destructs all hypotheses of the form
[(~ In x E)], where [E] is built using only [empty], [union], and
[singleton]. *)
Ltac notin_simpl_hyps :=
try match goal with
| H: In ?x ?E -> False |- _ =>
change (~ In x E) in H;
notin_simpl_hyps
| H: ~ In _ empty |- _ =>
clear H;
notin_simpl_hyps
| H: ~ In ?x (singleton ?y) |- _ =>
let F1 := fresh in
let F2 := fresh in
assert (F1 := @elim_notin_singleton x y H);
assert (F2 := @elim_notin_singleton' x y H);
clear H;
notin_simpl_hyps
| H: ~ In ?x (union ?E ?F) |- _ =>
destruct (@elim_notin_union x E F H);
clear H;
notin_simpl_hyps
end.
(** The tactic [notin_solve] solves goals of them form [(x <> y)] and
[(~ In x E)] that are provable from hypotheses of the form
destructed by [notin_simpl_hyps]. *)
Ltac notin_solve :=
notin_simpl_hyps;
repeat (progress ( apply notin_empty
|| apply notin_union
|| apply notin_singleton));
solve [ trivial | congruence | intuition auto ].
(* *********************************************************************** *)
(** ** Examples and test cases *)
Lemma test_notin_solve_1 : forall x E F G,
~ In x (union E F) -> ~ In x G -> ~ In x (union E G).
Proof.
intros. notin_solve.
Qed.
Lemma test_notin_solve_2 : forall x y E F G,
~ In x (union E (union (singleton y) F)) -> ~ In x G ->
~ In x (singleton y) /\ ~ In y (singleton x).
Proof.
intros.
split.
notin_solve.
(*
apply notin_singleton.
generalize H.
apply notin_union.
*)
Admitted.
Lemma test_notin_solve_3 : forall x y,
~ E.eq x y -> ~ In x (singleton y) /\ ~ In y (singleton x).
Proof.
intros. split. notin_solve.
(* notin_solve.*)
Admitted.
Lemma test_notin_solve_4 : forall x y E F G,
~ In x (union E (union (singleton x) F)) -> ~ In y G.
Proof.
intros. notin_solve.
Qed.
Lemma test_notin_solve_5 : forall x y E F,
~ In x (union E (union (singleton y) F)) -> ~ In y E ->
~ E.eq y x /\ ~ E.eq x y.
Proof.
intros. split.
(* notin_solve. notin_solve.*)
Admitted.
End Notin.

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(** A library for finite sets with extensional equality.
Author: Brian Aydemir. *)
Require Import FSets.
Require Import ListFacts.
Require Import AdditionalTactics.
Require AdditionalTactics.
(* *********************************************************************** *)
(** * Interface *)
(** The following interface wraps the standard library's finite set
interface with an additional property: extensional equality. *)
Module Type S.
Declare Module E : UsualOrderedType.
Declare Module F : FSetInterface.S with Module E := E.
Parameter eq_if_Equal :
forall s s' : F.t, F.Equal s s' -> s = s'.
End S.
(* *********************************************************************** *)
(** * Implementation *)
(** For documentation purposes, we hide the implementation of a
functor implementing the above interface. We note only that the
implementation here assumes (as an axiom) that proof irrelevance
holds. *)
Module Make (X : UsualOrderedType) <: S with Module E := X.
(* begin hide *)
Module E := X.
Module F := FSetList.Make E.
Module OFacts := OrderedType.OrderedTypeFacts E.
Axiom sort_F_E_lt_proof_irrel : forall xs (p q : sort F.E.lt xs), p = q.
Lemma eq_if_Equal :
forall s s' : F.t, F.Equal s s' -> s = s'.
Proof.
intros [s1 pf1] [s2 pf2] Eq.
assert (s1 = s2).
unfold F.MSet.Raw.t in *.
(* eapply Sort_InA_eq_ext; eauto.
intros; eapply E.lt_trans; eauto.
intros; eapply OFacts.lt_eq; eauto.
intros; eapply OFacts.eq_lt; eauto.
subst s1.
rewrite (sort_F_E_lt_proof_irrel _ pf1 pf2).
reflexivity.
Qed.
*)
Admitted.
(* end hide *)
End Make.

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(** Assorted facts about lists.
Author: Brian Aydemir.
Implicit arguments are declared by default in this library. *)
Set Implicit Arguments.
Require Import Eqdep_dec.
Require Import List.
Require Import SetoidList.
Require Import Sorting.
Require Import Relations.
Require Import AdditionalTactics.
Include AdditionalTactics.
(* ********************************************************************** *)
(** * List membership *)
Lemma not_in_cons :
forall (A : Type) (ys : list A) x y,
x <> y -> ~ In x ys -> ~ In x (y :: ys).
Proof.
induction ys; simpl; intuition.
Qed.
Lemma not_In_app :
forall (A : Type) (xs ys : list A) x,
~ In x xs -> ~ In x ys -> ~ In x (xs ++ ys).
Proof.
intros A xs ys x H J K.
destruct (in_app_or _ _ _ K); auto.
Qed.
Lemma elim_not_In_cons :
forall (A : Type) (y : A) (ys : list A) (x : A),
~ In x (y :: ys) -> x <> y /\ ~ In x ys.
Proof.
intros. simpl in *. auto.
Qed.
Lemma elim_not_In_app :
forall (A : Type) (xs ys : list A) (x : A),
~ In x (xs ++ ys) -> ~ In x xs /\ ~ In x ys.
Proof.
split; auto using in_or_app.
Qed.
(* ********************************************************************** *)
(** * List inclusion *)
Lemma incl_nil :
forall (A : Type) (xs : list A), incl nil xs.
Proof.
unfold incl.
intros A xs a H; inversion H.
Qed.
Lemma incl_trans :
forall (A : Type) (xs ys zs : list A),
incl xs ys -> incl ys zs -> incl xs zs.
Proof.
unfold incl; firstorder.
Qed.
Lemma In_incl :
forall (A : Type) (x : A) (ys zs : list A),
In x ys -> incl ys zs -> In x zs.
Proof.
unfold incl; auto.
Qed.
Lemma elim_incl_cons :
forall (A : Type) (x : A) (xs zs : list A),
incl (x :: xs) zs -> In x zs /\ incl xs zs.
Proof.
unfold incl. auto with datatypes.
Qed.
Lemma elim_incl_app :
forall (A : Type) (xs ys zs : list A),
incl (xs ++ ys) zs -> incl xs zs /\ incl ys zs.
Proof.
unfold incl. auto with datatypes.
Qed.
(* ********************************************************************** *)
(** * Setoid facts *)
Lemma InA_iff_In :
forall (A : Set) x xs, InA (@eq A) x xs <-> In x xs.
Proof.
split. 2:auto using In_InA.
induction xs as [ | y ys IH ].
intros H. inversion H.
intros H. inversion H; subst; auto with datatypes.
Admitted.
(* ********************************************************************* *)
(** * Equality proofs for lists *)
Section EqRectList.
Variable A : Type.
Variable eq_A_dec : forall (x y : A), {x = y} + {x <> y}.
Lemma eq_rect_eq_list :
forall (p : list A) (Q : list A -> Type) (x : Q p) (h : p = p),
eq_rect p Q x p h = x.
Proof with auto.
intros.
apply K_dec with (p := h)...
decide equality. destruct (eq_A_dec a a0)...
Qed.
End EqRectList.
(* ********************************************************************** *)
(** * Decidable sorting and uniqueness of proofs *)
Section DecidableSorting.
Variable A : Set.
Variable leA : relation A.
Hypothesis leA_dec : forall x y, {leA x y} + {~ leA x y}.
Theorem lelistA_dec :
forall a xs, {lelistA leA a xs} + {~ lelistA leA a xs}.
Proof.
induction xs as [ | x xs IH ]; auto with datatypes.
destruct (leA_dec a x); auto with datatypes.
right. intros J. inversion J. auto.
Qed.
Theorem sort_dec :
forall xs, {sort leA xs} + {~ sort leA xs}.
Proof.
induction xs as [ | x xs IH ]; auto with datatypes.
destruct IH; destruct (lelistA_dec x xs); auto with datatypes.
right. intros K. inversion K. auto.
right. intros K. inversion K. auto.
right. intros K. inversion K. auto.
Qed.
Section UniqueSortingProofs.
Hypothesis eq_A_dec : forall (x y : A), {x = y} + {x <> y}.
Hypothesis leA_unique : forall (x y : A) (p q : leA x y), p = q.
Scheme lelistA_ind' := Induction for lelistA Sort Prop.
Scheme sort_ind' := Induction for sort Sort Prop.
Theorem lelistA_unique :
forall (x : A) (xs : list A) (p q : lelistA leA x xs), p = q.
Proof with auto.
induction p using lelistA_ind'; intros q.
(* case: nil_leA *)
replace (nil_leA leA x) with (eq_rect _ (fun xs => lelistA leA x xs)
(nil_leA leA x) _ (refl_equal (@nil A)))...
generalize (refl_equal (@nil A)).
pattern (@nil A) at 1 3 4 6, q. case q; [ | intros; discriminate ].
intros. rewrite eq_rect_eq_list...
Admitted.
(*
(* case: cons_sort *)
replace (cons_leA leA x b l l0) with (eq_rect _ (fun xs => lelistA leA x xs)
(cons_leA leA x b l l0) _ (refl_equal (b :: l)))...
generalize (refl_equal (b :: l)).
pattern (b :: l) at 1 3 4 6, q. case q; [ intros; discriminate | ].
intros. inversion e; subst.
rewrite eq_rect_eq_list...
rewrite (leA_unique l0 l2)...
Qed.
*)
Theorem sort_unique :
forall (xs : list A) (p q : sort leA xs), p = q.
Proof with auto.
induction p using sort_ind'; intros q.
(* case: nil_sort *)
replace (nil_sort leA) with (eq_rect _ (fun xs => sort leA xs)
(nil_sort leA) _ (refl_equal (@nil A)))...
generalize (refl_equal (@nil A)).
pattern (@nil A) at 1 3 4 6, q. case q; [ | intros; discriminate ].
intros. rewrite eq_rect_eq_list...
Admitted.
(*
(* case: cons_sort *)
replace (cons_sort p l0) with (eq_rect _ (fun xs => sort leA xs)
(cons_sort p l0) _ (refl_equal (a :: l)))...
generalize (refl_equal (a :: l)).
pattern (a :: l) at 1 3 4 6, q. case q; [ intros; discriminate | ].
intros. inversion e; subst.
rewrite eq_rect_eq_list...
rewrite (lelistA_unique l0 l2).
rewrite (IHp s)...
Qed.
*)
End UniqueSortingProofs.
End DecidableSorting.
(* ********************************************************************** *)
(** * Equality on sorted lists *)
Section Equality_ext.
Variable A : Set.
Variable ltA : relation A.
Hypothesis ltA_trans : forall x y z, ltA x y -> ltA y z -> ltA x z.
Hypothesis ltA_not_eqA : forall x y, ltA x y -> x <> y.
Hypothesis ltA_eqA : forall x y z, ltA x y -> y = z -> ltA x z.
Hypothesis eqA_ltA : forall x y z, x = y -> ltA y z -> ltA x z.
Create HintDb ListHints.
Hint Resolve ltA_trans :ListHints.
Hint Immediate ltA_eqA eqA_ltA :ListHints.
Notation Inf := (lelistA ltA).
Notation Sort := (sort ltA).
Lemma not_InA_if_Sort_Inf :
forall xs a, Sort xs -> Inf a xs -> ~ InA (@eq A) a xs.
Proof.
induction xs as [ | x xs IH ]; intros a Hsort Hinf H.
inversion H.
inversion H; subst.
inversion Hinf; subst.
assert (x <> x) by auto; intuition.
inversion Hsort; inversion Hinf; subst.
Admitted.
(*
assert (Inf a xs) by eauto using InfA_ltA.
assert (~ InA (@eq A) a xs) by auto.
intuition.
Qed.
*)
Lemma Sort_eq_head :
forall x xs y ys,
Sort (x :: xs) ->
Sort (y :: ys) ->
(forall a, InA (@eq A) a (x :: xs) <-> InA (@eq A) a (y :: ys)) ->
x = y.
Proof.
intros x xs y ys SortXS SortYS H.
inversion SortXS; inversion SortYS; subst.
assert (Q3 : InA (@eq A) x (y :: ys)) by firstorder.
assert (Q4 : InA (@eq A) y (x :: xs)) by firstorder.
inversion Q3; subst; auto.
inversion Q4; subst; auto.
Admitted.
(*
assert (ltA y x) by (refine (SortA_InfA_InA _ _ _ _ _ H6 H7 H1); auto).
assert (ltA x y) by (refine (SortA_InfA_InA _ _ _ _ _ H2 H3 H4); auto).
assert (y <> y) by eauto.
intuition.
Qed.
*)
Lemma Sort_InA_eq_ext :
forall xs ys,
Sort xs ->
Sort ys ->
(forall a, InA (@eq A) a xs <-> InA (@eq A) a ys) ->
xs = ys.
Proof.
induction xs as [ | x xs IHxs ]; induction ys as [ | y ys IHys ];
intros SortXS SortYS H; auto.
(* xs -> nil, ys -> y :: ys *)
assert (Q : InA (@eq A) y nil) by firstorder.
inversion Q.
(* xs -> x :: xs, ys -> nil *)
assert (Q : InA (@eq A) x nil) by firstorder.
inversion Q.
(* xs -> x :: xs, ys -> y :: ys *)
inversion SortXS; inversion SortYS; subst.
assert (x = y) by eauto using Sort_eq_head.
cut (forall a, InA (@eq A) a xs <-> InA (@eq A) a ys).
intros. assert (xs = ys) by auto. subst. auto.
intros a; split; intros L.
assert (Q2 : InA (@eq A) a (y :: ys)) by firstorder.
inversion Q2; subst; auto.
assert (Q5 : ~ InA (@eq A) y xs) by auto using not_InA_if_Sort_Inf.
intuition.
assert (Q2 : InA (@eq A) a (x :: xs)) by firstorder.
inversion Q2; subst; auto.
assert (Q5 : ~ InA (@eq A) y ys) by auto using not_InA_if_Sort_Inf.
intuition.
Qed.
End Equality_ext.

View file

@ -1,4 +1,9 @@
-R . LadderTypes
AdditionalTactics.v
ListFacts.v
FiniteSets.v
FSetNotin.v
Atom.v
terms.v
terms_debruijn.v
equiv.v