[build] Consolidate stdlib's .v files under a single directory.

Currently, `.v` under the `Coq.` prefix are found in both `theories`
and `plugins`. Usually these two directories are merged by special
loadpath code that allows double-binding of the prefix.

This adds some complexity to the build and loadpath system; and in
particular, it prevents from handling the `Coq.*` prefix in the
simple, `-R theories Coq` standard way.

We thus move all `.v` files to theories, leaving `plugins` as an
OCaml-only directory, and modify accordingly the loadpath / build
infrastructure.

Note that in general `plugins/foo/Foo.v` was not self-contained, in
the sense that it depended on files in `theories` and files in
`theories` depended on it; moreover, Coq saw all these files as
belonging to the same namespace so it didn't really care where they
lived.

This could also imply a performance gain as we now effectively
traverse less directories when locating a library.

See also discussion in #10003

2 files changed, 0 insertions, 795 deletions

diff --git a/plugins/rtauto/Bintree.v b/plugins/rtauto/Bintree.v
deleted file mode 100644
index 6b92445326..0000000000
--- a/plugins/rtauto/Bintree.v
+++ /dev/null
@@ -1,385 +0,0 @@
-(************************************************************************)
-(*         *   The Coq Proof Assistant / The Coq Development Team       *)
-(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2019       *)
-(* <O___,, *       (see CREDITS file for the list of authors)           *)
-(*   \VV/  **************************************************************)
-(*    //   *    This file is distributed under the terms of the         *)
-(*         *     GNU Lesser General Public License Version 2.1          *)
-(*         *     (see LICENSE file for the text of the license)         *)
-(************************************************************************)
-
-Require Export List.
-Require Export BinPos.
-Require Arith.EqNat.
-
-Open Scope positive_scope.
-
-Ltac clean := try (simpl; congruence).
-
-Lemma Gt_Psucc: forall p q,
-       (p ?= Pos.succ q) = Gt -> (p ?= q) = Gt.
-Proof.
-intros. rewrite <- Pos.compare_succ_succ.
-now apply Pos.lt_gt, Pos.lt_lt_succ, Pos.gt_lt.
-Qed.
-
-Lemma Psucc_Gt : forall p,
-       (Pos.succ p ?= p) = Gt.
-Proof.
-intros. apply Pos.lt_gt, Pos.lt_succ_diag_r.
-Qed.
-
-Fixpoint Lget (A:Set) (n:nat) (l:list A) {struct l}:option A :=
-match l with nil => None
-| x::q =>
-match n with O => Some x
-| S m => Lget A m q
-end end .
-
-Arguments Lget [A] n l.
-
-Lemma map_app : forall (A B:Set) (f:A -> B) l m,
-List.map  f (l ++ m) = List.map  f  l ++ List.map  f  m.
-induction l.
-reflexivity.
-simpl.
-intro m ; apply f_equal;apply IHl.
-Qed.
-
-Lemma length_map : forall (A B:Set) (f:A -> B) l,
-length (List.map  f l) = length l.
-induction l.
-reflexivity.
-simpl; apply f_equal;apply IHl.
-Qed.
-
-Lemma Lget_map : forall (A B:Set) (f:A -> B) i l,
-Lget i (List.map  f l) =
-match Lget i l with Some a =>
-Some (f a) | None => None end.
-induction i;intros [ | x l ] ;trivial.
-simpl;auto.
-Qed.
-
-Lemma Lget_app : forall (A:Set) (a:A) l i,
-Lget i (l ++ a :: nil) = if Arith.EqNat.beq_nat i (length l) then Some a else Lget i l.
-Proof.
-induction l;simpl Lget;simpl length.
-intros [ | i];simpl;reflexivity.
-intros [ | i];simpl.
-reflexivity.
-auto.
-Qed.
-
-Lemma Lget_app_Some : forall (A:Set) l delta i (a: A),
-Lget i l = Some a ->
-Lget i (l ++ delta) = Some a.
-induction l;destruct i;simpl;try congruence;auto.
-Qed.
-
-Inductive Poption {A} : Type:=
-  PSome : A -> Poption
-| PNone : Poption.
-Arguments Poption : clear implicits.
-
-Inductive Tree {A} : Type :=
-   Tempty : Tree
- | Branch0 : Tree -> Tree -> Tree
- | Branch1 : A -> Tree -> Tree -> Tree.
-Arguments Tree : clear implicits.
-
-Section Store.
-
-Variable A:Type.
-
-Notation Poption := (Poption A).
-Notation Tree := (Tree A).
-
-
-Fixpoint Tget (p:positive) (T:Tree) {struct p} : Poption :=
-  match T with
-    Tempty => PNone
-  | Branch0 T1 T2 =>
-    match p with
-      xI pp => Tget pp T2
-    | xO pp => Tget pp T1
-    | xH => PNone
-    end
-  | Branch1 a T1 T2 =>
-    match p with
-      xI pp => Tget pp T2
-    | xO pp => Tget pp T1
-    | xH => PSome a
-    end
-end.
-
-Fixpoint Tadd (p:positive) (a:A) (T:Tree) {struct p}: Tree :=
- match T with
-   | Tempty =>
-       match p with
-       | xI pp => Branch0 Tempty (Tadd pp a Tempty)
-       | xO pp => Branch0 (Tadd pp a Tempty) Tempty
-       | xH => Branch1 a Tempty Tempty
-       end
-   | Branch0 T1 T2 =>
-       match p with
-       | xI pp => Branch0 T1 (Tadd pp a T2)
-       | xO pp => Branch0 (Tadd pp a T1) T2
-       | xH => Branch1 a T1 T2
-       end
-   | Branch1 b T1 T2 =>
-       match p with
-       | xI pp => Branch1 b T1 (Tadd pp a T2)
-       | xO pp => Branch1 b (Tadd pp a T1) T2
-       | xH => Branch1 a T1 T2
-       end
-   end.
-
-Definition mkBranch0 (T1 T2:Tree) :=
-  match T1,T2 with
-    Tempty ,Tempty => Tempty
-  | _,_ => Branch0 T1 T2
-  end.
-
-Fixpoint Tremove (p:positive) (T:Tree) {struct p}: Tree :=
-   match T with
-      | Tempty => Tempty
-      | Branch0 T1 T2 =>
-        match p with
-        | xI pp => mkBranch0 T1 (Tremove pp T2)
-        | xO pp => mkBranch0 (Tremove pp T1) T2
-        | xH => T
-        end
-      | Branch1 b T1 T2 =>
-        match p with
-        | xI pp => Branch1 b T1 (Tremove pp T2)
-        | xO pp => Branch1 b (Tremove pp T1) T2
-        | xH => mkBranch0 T1 T2
-        end
-      end.
-
-
-Theorem Tget_Tempty: forall (p : positive), Tget p (Tempty) = PNone.
-destruct p;reflexivity.
-Qed.
-
-Theorem Tget_Tadd: forall i j a T,
-       Tget i (Tadd j a T) =
-       match (i ?= j) with
-         Eq => PSome a
-       | Lt => Tget i T
-       | Gt => Tget i T
-       end.
-Proof.
-intros i j.
-case_eq (i ?= j).
-intro H;rewrite (Pos.compare_eq _ _ H);intros a;clear i H.
-induction j;destruct T;simpl;try (apply IHj);congruence.
-unfold Pos.compare.
-generalize i;clear i;induction j;destruct T;simpl in H|-*;
-destruct i;simpl;try rewrite (IHj _ H);try (destruct i;simpl;congruence);reflexivity|| congruence.
-unfold Pos.compare.
-generalize i;clear i;induction j;destruct T;simpl in H|-*;
-destruct i;simpl;try rewrite (IHj _ H);try (destruct i;simpl;congruence);reflexivity|| congruence.
-Qed.
-
-Record Store : Type :=
-mkStore  {index:positive;contents:Tree}.
-
-Definition empty := mkStore xH Tempty.
-
-Definition push a  S :=
-mkStore (Pos.succ (index S)) (Tadd (index S) a (contents S)).
-
-Definition get i S := Tget i (contents S).
-
-Lemma get_empty : forall i, get i empty = PNone.
-intro i; case i; unfold empty,get; simpl;reflexivity.
-Qed.
-
-Inductive Full : Store -> Type:=
-    F_empty : Full empty
-  | F_push : forall a S, Full S -> Full (push a S).
-
-Theorem get_Full_Gt : forall S, Full S ->
-       forall i, (i ?= index S) = Gt -> get i S = PNone.
-Proof.
-intros S W;induction W.
-unfold empty,index,get,contents;intros;apply Tget_Tempty.
-unfold index,get,push. simpl @contents.
-intros i e;rewrite Tget_Tadd.
-rewrite (Gt_Psucc _ _ e).
-unfold get in IHW.
-apply IHW;apply Gt_Psucc;assumption.
-Qed.
-
-Theorem get_Full_Eq : forall S, Full S -> get (index S) S = PNone.
-intros [index0 contents0] F.
-case F.
-unfold empty,index,get,contents;intros;apply Tget_Tempty.
-unfold push,index,get;simpl @contents.
-intros a S.
-rewrite Tget_Tadd.
-rewrite Psucc_Gt.
-intro W.
-change (get (Pos.succ (index S)) S =PNone).
-apply get_Full_Gt; auto.
-apply Psucc_Gt.
-Qed.
-
-Theorem get_push_Full :
-  forall i a S, Full S ->
-  get i (push a S) =
-  match (i ?= index S) with
-    Eq => PSome a
-  | Lt => get i S
-  | Gt => PNone
-end.
-Proof.
-intros i a S F.
-case_eq (i ?= index S).
-intro e;rewrite (Pos.compare_eq _ _ e).
-destruct S;unfold get,push,index;simpl @contents;rewrite Tget_Tadd.
-rewrite Pos.compare_refl;reflexivity.
-intros;destruct S;unfold get,push,index;simpl @contents;rewrite Tget_Tadd.
-simpl @index in H;rewrite H;reflexivity.
-intro H;generalize H;clear H.
-unfold get,push;simpl.
-rewrite Tget_Tadd;intro e;rewrite e.
-change (get i S=PNone).
-apply get_Full_Gt;auto.
-Qed.
-
-Lemma Full_push_compat : forall i a S, Full S ->
-forall x, get i S = PSome x ->
- get i (push a S) = PSome x.
-Proof.
-intros i a S F x H.
-case_eq (i ?= index S);intro test.
-rewrite (Pos.compare_eq _ _ test) in H.
-rewrite (get_Full_Eq _ F) in H;congruence.
-rewrite <- H.
-rewrite (get_push_Full i a).
-rewrite test;reflexivity.
-assumption.
-rewrite (get_Full_Gt _ F) in H;congruence.
-Qed.
-
-Lemma Full_index_one_empty : forall S, Full S -> index S = 1 -> S=empty.
-intros [ind cont] F one; inversion F.
-reflexivity.
-simpl @index in one;assert (h:=Pos.succ_not_1 (index S)).
-congruence.
-Qed.
-
-Lemma push_not_empty: forall a S, (push a S) <> empty.
-intros a [ind cont];unfold push,empty.
-intros [= H%Pos.succ_not_1]. assumption.
-Qed.
-
-Fixpoint In (x:A) (S:Store) (F:Full S) {struct F}: Prop :=
-match F with
-F_empty => False
-| F_push a SS FF => x=a \/ In x SS FF
-end.
-
-Lemma get_In : forall (x:A) (S:Store) (F:Full S) i ,
-get i S = PSome x -> In x S F.
-induction F.
-intro i;rewrite get_empty; congruence.
-intro i;rewrite get_push_Full;trivial.
-case_eq (i ?= index S);simpl.
-left;congruence.
-right;eauto.
-congruence.
-Qed.
-
-End Store.
-
-Arguments PNone {A}.
-Arguments PSome [A] _.
-
-Arguments Tempty {A}.
-Arguments Branch0 [A] _ _.
-Arguments Branch1 [A] _ _ _.
-
-Arguments Tget [A] p T.
-Arguments Tadd [A] p a T.
-
-Arguments Tget_Tempty [A] p.
-Arguments Tget_Tadd [A] i j a T.
-
-Arguments mkStore [A] index contents.
-Arguments index [A] s.
-Arguments contents [A] s.
-
-Arguments empty {A}.
-Arguments get [A] i S.
-Arguments push [A] a S.
-
-Arguments get_empty [A] i.
-Arguments get_push_Full [A] i a S _.
-
-Arguments Full [A] _.
-Arguments F_empty {A}.
-Arguments F_push [A] a S _.
-Arguments In [A] x S F.
-
-Register empty as plugins.rtauto.empty.
-Register push as plugins.rtauto.push.
-
-Section Map.
-
-Variables A B:Set.
-
-Variable f: A -> B.
-
-Fixpoint Tmap (T: Tree A) : Tree B :=
-match T with
-Tempty => Tempty
-| Branch0 t1 t2 => Branch0 (Tmap t1) (Tmap t2)
-| Branch1 a t1 t2 => Branch1 (f a) (Tmap t1) (Tmap t2)
-end.
-
-Lemma Tget_Tmap: forall T i,
-Tget i (Tmap T)= match Tget i T with PNone => PNone
-| PSome a => PSome (f a) end.
-induction T;intro i;case i;simpl;auto.
-Defined.
-
-Lemma Tmap_Tadd: forall i a T,
-Tmap (Tadd i a T) = Tadd i (f a) (Tmap T).
-induction i;intros a T;case T;simpl;intros;try (rewrite IHi);simpl;reflexivity.
-Defined.
-
-Definition map (S:Store A) : Store B :=
-mkStore (index S) (Tmap (contents S)).
-
-Lemma get_map: forall i S,
-get i (map S)= match get i S with PNone => PNone
-| PSome a => PSome (f a) end.
-destruct S;unfold get,map,contents,index;apply Tget_Tmap.
-Defined.
-
-Lemma map_push: forall a S,
-map (push a S) = push (f a) (map S).
-intros a S.
-case S.
-unfold push,map,contents,index.
-intros;rewrite Tmap_Tadd;reflexivity.
-Defined.
-
-Theorem Full_map : forall S, Full S -> Full (map S).
-intros S F.
-induction F.
-exact F_empty.
-rewrite map_push;constructor 2;assumption.
-Defined.
-
-End Map.
-
-Arguments Tmap [A B] f T.
-Arguments map [A B] f S.
-Arguments Full_map [A B f] S _.
-
-Notation "hyps \ A" := (push A hyps) (at level 72,left associativity).
diff --git a/plugins/rtauto/Rtauto.v b/plugins/rtauto/Rtauto.v
deleted file mode 100644
index 2e9b4347b9..0000000000
--- a/plugins/rtauto/Rtauto.v
+++ /dev/null
@@ -1,410 +0,0 @@
-(************************************************************************)
-(*         *   The Coq Proof Assistant / The Coq Development Team       *)
-(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2019       *)
-(* <O___,, *       (see CREDITS file for the list of authors)           *)
-(*   \VV/  **************************************************************)
-(*    //   *    This file is distributed under the terms of the         *)
-(*         *     GNU Lesser General Public License Version 2.1          *)
-(*         *     (see LICENSE file for the text of the license)         *)
-(************************************************************************)
-
-
-Require Export List.
-Require Export Bintree.
-Require Import Bool BinPos.
-
-Declare ML Module "rtauto_plugin".
-
-Ltac clean:=try (simpl;congruence).
-
-Inductive form:Set:=
-  Atom : positive -> form
-| Arrow : form -> form -> form
-| Bot
-| Conjunct : form -> form -> form
-| Disjunct : form -> form -> form.
-
-Notation "[ n ]":=(Atom n).
-Notation "A =>> B":= (Arrow A B) (at level 59, right associativity).
-Notation "#" := Bot.
-Notation "A //\\ B" := (Conjunct A B) (at level 57, left associativity).
-Notation "A \\// B" := (Disjunct A B) (at level 58, left associativity).
-
-Definition ctx := Store form.
-
-Fixpoint pos_eq (m n:positive) {struct m} :bool :=
-match m with
-  xI mm => match n with xI nn => pos_eq mm nn | _ => false end
-| xO mm => match n with xO nn => pos_eq mm nn | _ => false end
-| xH => match n with xH => true | _ => false end
-end.
-
-Theorem pos_eq_refl : forall m n, pos_eq m n = true -> m = n.
-induction m;simpl;destruct n;congruence ||
-(intro e;apply f_equal;auto).
-Qed.
-
-Fixpoint form_eq (p q:form) {struct p} :bool :=
-match p with
-  Atom m => match q with Atom n => pos_eq m n | _ => false end
-| Arrow p1 p2 =>
-match q with
-  Arrow q1 q2 => form_eq p1 q1 && form_eq p2 q2
-| _ => false end
-| Bot => match q with Bot => true | _ => false end
-| Conjunct p1 p2 =>
-match q with
-  Conjunct q1 q2 => form_eq p1 q1 && form_eq p2 q2
-| _ => false
-end
-| Disjunct p1 p2 =>
-match q with
-  Disjunct q1 q2 => form_eq p1 q1 && form_eq p2 q2
-| _ => false
-end
-end.
-
-Theorem form_eq_refl: forall p q, form_eq p q = true -> p = q.
-induction p;destruct q;simpl;clean.
-intro h;generalize (pos_eq_refl _ _ h);congruence.
-case_eq (form_eq p1 q1);clean.
-intros e1 e2;generalize (IHp1 _ e1) (IHp2 _ e2);congruence.
-case_eq (form_eq p1 q1);clean.
-intros e1 e2;generalize (IHp1 _ e1) (IHp2 _ e2);congruence.
-case_eq (form_eq p1 q1);clean.
-intros e1 e2;generalize (IHp1 _ e1) (IHp2 _ e2);congruence.
-Qed.
-
-Arguments form_eq_refl [p q] _.
-
-Section with_env.
-
-Variable env:Store Prop.
-
-Fixpoint interp_form (f:form): Prop :=
-match f with
-[n]=> match get n env with PNone => True | PSome P => P end
-| A =>> B => (interp_form A) -> (interp_form B)
-| # => False
-| A //\\ B => (interp_form A) /\ (interp_form B)
-| A \\// B => (interp_form A) \/ (interp_form B)
-end.
-
-Notation "[[ A ]]" := (interp_form A).
-
-Fixpoint interp_ctx (hyps:ctx) (F:Full hyps) (G:Prop) {struct F} : Prop :=
-match F with
-  F_empty => G
-| F_push H hyps0 F0 =>  interp_ctx hyps0 F0  ([[H]] -> G)
-end.
-
-Ltac wipe := intros;simpl;constructor.
-
-Lemma compose0 :
-forall hyps F (A:Prop),
- A ->
- (interp_ctx hyps F A).
-induction F;intros A H;simpl;auto.
-Qed.
-
-Lemma compose1 :
-forall hyps F (A B:Prop),
- (A -> B) ->
- (interp_ctx hyps F A) ->
- (interp_ctx hyps F B).
-induction F;intros A B H;simpl;auto.
-apply IHF;auto.
-Qed.
-
-Theorem compose2 :
-forall hyps F (A B C:Prop),
- (A -> B -> C) ->
- (interp_ctx hyps F A) ->
- (interp_ctx hyps F B) ->
- (interp_ctx hyps F C).
-induction F;intros A B C H;simpl;auto.
-apply IHF;auto.
-Qed.
-
-Theorem compose3 :
-forall hyps F (A B C D:Prop),
- (A -> B -> C -> D) ->
- (interp_ctx hyps F A) ->
- (interp_ctx hyps F B) ->
- (interp_ctx hyps F C) ->
- (interp_ctx hyps F D).
-induction F;intros A B C D H;simpl;auto.
-apply IHF;auto.
-Qed.
-
-Lemma weaken : forall hyps F f G,
- (interp_ctx hyps F G) ->
- (interp_ctx (hyps\f)  (F_push f hyps F) G).
-induction F;simpl;intros;auto.
-apply compose1 with ([[a]]-> G);auto.
-Qed.
-
-Theorem project_In : forall hyps F g,
-In g hyps F ->
-interp_ctx hyps F [[g]].
-induction F;simpl.
-contradiction.
-intros g H;destruct H.
-subst;apply compose0;simpl;trivial.
-apply compose1 with [[g]];auto.
-Qed.
-
-Theorem project : forall hyps F p g,
-get  p hyps = PSome g->
-interp_ctx hyps F [[g]].
-intros hyps F p g e; apply project_In.
-apply get_In with p;assumption.
-Qed.
-
-Arguments project [hyps] F [p g] _.
-
-Inductive proof:Set :=
-  Ax : positive -> proof
-| I_Arrow : proof -> proof
-| E_Arrow : positive -> positive -> proof -> proof
-| D_Arrow : positive -> proof -> proof -> proof
-| E_False : positive -> proof
-| I_And: proof -> proof -> proof
-| E_And: positive -> proof -> proof
-| D_And: positive -> proof -> proof
-| I_Or_l: proof -> proof
-| I_Or_r: proof -> proof
-| E_Or: positive -> proof -> proof -> proof
-| D_Or: positive -> proof -> proof
-| Cut: form -> proof -> proof -> proof.
-
-Notation "hyps \ A" := (push A hyps) (at level 72,left associativity).
-
-Fixpoint check_proof (hyps:ctx) (gl:form) (P:proof) {struct P}: bool :=
- match P with
-   Ax i =>
-     match get i hyps with
-         PSome F => form_eq F gl
- | _ => false
-    end
-| I_Arrow p =>
-   match gl with
-      A  =>> B  => check_proof (hyps \ A) B p
-   | _ => false
-    end
-| E_Arrow i j p =>
-   match get i hyps,get j hyps with
-       PSome A,PSome (B =>>C) =>
-         form_eq A B && check_proof (hyps \ C) (gl) p
-    | _,_ => false
-    end
-| D_Arrow i p1 p2 =>
-   match get i hyps with
-      PSome  ((A =>>B)=>>C) =>
-     (check_proof ( hyps \ B =>> C \ A) B p1) && (check_proof (hyps \ C) gl p2)
-    | _ => false
-  end
-| E_False i =>
-  match get i hyps with
-    PSome # => true
-  | _ => false
-  end
-| I_And p1 p2 =>
-   match gl with
-      A //\\ B =>
-      check_proof hyps A p1 && check_proof hyps B p2
-      | _ => false
-      end
-| E_And i p =>
-   match get i hyps with
-      PSome  (A //\\ B) => check_proof (hyps \ A \ B) gl p
-    | _=> false
-   end
-| D_And i p =>
-   match get i hyps with
-      PSome  (A //\\ B =>> C) => check_proof (hyps \ A=>>B=>>C) gl p
-    | _=> false
-   end
-| I_Or_l p =>
-   match gl with
-      (A \\// B) => check_proof hyps A p
-      | _ => false
-      end
-| I_Or_r p =>
-   match gl with
-      (A \\// B) => check_proof hyps B p
-      | _ => false
-      end
-| E_Or i p1 p2 =>
-  match get i hyps with
-  PSome (A \\// B) =>
-  check_proof (hyps \ A) gl p1 && check_proof (hyps \ B) gl p2
-  | _=> false
-  end
-| D_Or i p =>
-   match get i hyps with
-      PSome  (A \\// B =>> C) =>
-      (check_proof (hyps \ A=>>C \ B=>>C) gl p)
-    | _=> false
-   end
-| Cut A p1 p2 =>
-  check_proof hyps A p1 && check_proof (hyps \ A) gl p2
-end.
-
-Theorem interp_proof:
-forall p hyps F gl,
-check_proof hyps gl p = true -> interp_ctx hyps F [[gl]].
-
-induction p; intros hyps F gl.
-
-- (* Axiom *)
-  simpl;case_eq (get p hyps);clean.
-  intros f nth_f e;rewrite <- (form_eq_refl e).
-  apply project with p;trivial.
-
-- (* Arrow_Intro *)
-  destruct gl; clean.
-  simpl; intros.
-  change (interp_ctx (hyps\gl1)  (F_push gl1 hyps F) [[gl2]]).
-  apply IHp; try constructor; trivial.
-
-- (* Arrow_Elim *)
-  simpl check_proof; case_eq (get p hyps); clean.
-  intros f ef; case_eq (get p0 hyps); clean.
-  intros f0 ef0; destruct f0; clean.
-  case_eq (form_eq f f0_1); clean.
-  simpl; intros e check_p1.
-  generalize  (project  F ef) (project  F ef0)
-              (IHp (hyps \ f0_2) (F_push f0_2 hyps F) gl check_p1);
-    clear check_p1 IHp p p0 p1 ef ef0.
-  simpl.
-  apply compose3.
-  rewrite (form_eq_refl e).
-  auto.
-
-- (* Arrow_Destruct *)
-  simpl; case_eq (get p1 hyps); clean.
-  intros f ef; destruct f; clean.
-  destruct f1; clean.
-  case_eq (check_proof (hyps \ f1_2 =>> f2 \ f1_1) f1_2 p2); clean.
-  intros check_p1 check_p2.
-  generalize (project F ef)
-             (IHp1 (hyps \ f1_2 =>> f2 \ f1_1)
-                   (F_push f1_1 (hyps \ f1_2 =>> f2)
-                           (F_push (f1_2 =>> f2) hyps F)) f1_2 check_p1)
-             (IHp2 (hyps \ f2) (F_push f2 hyps F) gl check_p2).
-  simpl; apply compose3; auto.
-
-- (* False_Elim *)
-  simpl; case_eq (get p hyps); clean.
-  intros f ef; destruct f; clean.
-  intros _; generalize (project F ef).
-  apply compose1; apply False_ind.
-
-- (* And_Intro *)
-  simpl; destruct gl; clean.
-  case_eq (check_proof hyps gl1 p1); clean.
-  intros Hp1 Hp2;generalize (IHp1 hyps F gl1 Hp1) (IHp2 hyps F gl2 Hp2).
-  apply compose2 ; simpl; auto.
-
-- (* And_Elim *)
-  simpl; case_eq (get p hyps); clean.
-  intros f ef; destruct f; clean.
-  intro check_p;
-    generalize  (project F ef)
-                (IHp (hyps \ f1 \ f2) (F_push f2 (hyps \ f1) (F_push f1 hyps F)) gl check_p).
-  simpl; apply compose2; intros [h1 h2]; auto.
-
-- (* And_Destruct*)
-  simpl; case_eq (get p hyps); clean.
-  intros f ef; destruct f; clean.
-  destruct f1; clean.
-  intro H;
-    generalize (project F ef)
-               (IHp (hyps \ f1_1 =>> f1_2 =>> f2)
-                    (F_push (f1_1 =>> f1_2 =>> f2) hyps F) gl H);
-    clear H; simpl.
-  apply compose2; auto.
-
-- (* Or_Intro_left *)
-  destruct gl; clean.
-  intro Hp; generalize (IHp hyps F gl1 Hp).
-  apply compose1; simpl; auto.
-
-- (* Or_Intro_right *)
-  destruct gl; clean.
-  intro Hp; generalize (IHp hyps F gl2 Hp).
-  apply compose1; simpl; auto.
-
-- (* Or_elim *)
-  simpl; case_eq (get p1 hyps); clean.
-  intros f ef; destruct f; clean.
-  case_eq (check_proof (hyps \ f1) gl p2); clean.
-  intros check_p1 check_p2;
-    generalize (project F ef)
-               (IHp1 (hyps \ f1) (F_push f1 hyps F) gl check_p1)
-               (IHp2 (hyps \ f2) (F_push f2 hyps F) gl check_p2);
-    simpl; apply compose3; simpl; intro h; destruct h; auto.
-
-- (* Or_Destruct *)
-  simpl; case_eq (get p hyps); clean.
-  intros f ef; destruct f; clean.
-  destruct f1; clean.
-  intro check_p0;
-    generalize (project F ef)
-               (IHp (hyps \ f1_1 =>> f2 \ f1_2 =>> f2)
-                    (F_push (f1_2 =>> f2) (hyps \ f1_1 =>> f2)
-                            (F_push (f1_1 =>> f2) hyps F)) gl check_p0);
-    simpl.
-  apply compose2; auto.
-
-- (* Cut *)
-  simpl; case_eq (check_proof hyps f p1); clean.
-  intros check_p1 check_p2;
-    generalize (IHp1 hyps F f check_p1)
-               (IHp2 (hyps\f) (F_push f hyps F) gl check_p2);
-    simpl; apply compose2; auto.
-Qed.
-
-Theorem Reflect: forall gl prf, if check_proof empty gl prf then [[gl]] else True.
-intros gl prf;case_eq (check_proof empty gl prf);intro check_prf.
-change (interp_ctx empty F_empty [[gl]]) ;
-apply interp_proof with prf;assumption.
-trivial.
-Qed.
-
-End with_env.
-
-(*
-(* A small example *)
-Parameters A B C D:Prop.
-Theorem toto:A /\ (B \/ C) -> (A /\ B) \/ (A /\ C).
-exact (Reflect (empty \ A \ B \ C)
-([1] //\\ ([2] \\// [3]) =>> [1] //\\ [2] \\// [1] //\\ [3])
-(I_Arrow (E_And 1 (E_Or 3
-  (I_Or_l (I_And (Ax 2) (Ax 4)))
-  (I_Or_r (I_And (Ax 2) (Ax 4))))))).
-Qed.
-Print toto.
-*)
-
-Register Reflect as plugins.rtauto.Reflect.
-
-Register Atom as plugins.rtauto.Atom.
-Register Arrow as plugins.rtauto.Arrow.
-Register Bot as plugins.rtauto.Bot.
-Register Conjunct as plugins.rtauto.Conjunct.
-Register Disjunct as plugins.rtauto.Disjunct.
-
-Register Ax as plugins.rtauto.Ax.
-Register I_Arrow as plugins.rtauto.I_Arrow.
-Register E_Arrow as plugins.rtauto.E_Arrow.
-Register D_Arrow as plugins.rtauto.D_Arrow.
-Register E_False as plugins.rtauto.E_False.
-Register I_And as plugins.rtauto.I_And.
-Register E_And as plugins.rtauto.E_And.
-Register D_And as plugins.rtauto.D_And.
-Register I_Or_l as plugins.rtauto.I_Or_l.
-Register I_Or_r as plugins.rtauto.I_Or_r.
-Register E_Or as plugins.rtauto.E_Or.
-Register D_Or as plugins.rtauto.D_Or.