Remove coqprime-8.4

We're using tactics in terms in some places, and so have no hope of
compiling with Coq 8.4.  We no longer pretend to support it.

We can probably also remove some other compatibility things, if we want.

1 files changed, 0 insertions, 271 deletions

diff --git a/coqprime-8.4/Coqprime/ListAux.v b/coqprime-8.4/Coqprime/ListAux.v
deleted file mode 100644
index 4ed154685..000000000
--- a/coqprime-8.4/Coqprime/ListAux.v
+++ /dev/null
@@ -1,271 +0,0 @@
-
-(*************************************************************)
-(*      This file is distributed under the terms of the      *)
-(*      GNU Lesser General Public License Version 2.1        *)
-(*************************************************************)
-(*    Benjamin.Gregoire@inria.fr Laurent.Thery@inria.fr      *)
-(*************************************************************)
-
-(**********************************************************************
-     Aux.v                                                                                           
-                                                                                                          
-     Auxillary functions & Theorems                                              
-  **********************************************************************)
-Require Export Coq.Lists.List.
-Require Export Coq.Arith.Arith.
-Require Export Coqprime.Tactic.
-Require Import Coq.Wellfounded.Inverse_Image.
-Require Import Coq.Arith.Wf_nat.
-
-(************************************** 
-   Some properties on list operators: app, map,...
-**************************************)
- 
-Section List.
-Variables (A : Set) (B : Set) (C : Set).
-Variable f : A ->  B.
-
-(************************************** 
-  An induction theorem for list based on length 
-**************************************)
- 
-Theorem list_length_ind:
- forall (P : list A ->  Prop),
- (forall (l1 : list A),
-  (forall (l2 : list A), length l2 < length l1 ->  P l2) ->  P l1) ->
- forall (l : list A),  P l.
-intros P H l;
- apply well_founded_ind with ( R := fun (x y : list A) => length x < length y );
- auto.
-apply wf_inverse_image with ( R := lt ); auto.
-apply lt_wf.
-Qed.
- 
-Definition list_length_induction:
- forall (P : list A ->  Set),
- (forall (l1 : list A),
-  (forall (l2 : list A), length l2 < length l1 ->  P l2) ->  P l1) ->
- forall (l : list A),  P l.
-intros P H l;
- apply well_founded_induction
-      with ( R := fun (x y : list A) => length x < length y ); auto.
-apply wf_inverse_image with ( R := lt ); auto.
-apply lt_wf.
-Qed.
- 
-Theorem in_ex_app:
- forall (a : A) (l : list A),
- In a l ->  (exists l1 : list A , exists l2 : list A , l = l1 ++ (a :: l2)  ).
-intros a l; elim l; clear l; simpl; auto.
-intros H; case H.
-intros a1 l H [H1|H1]; auto.
-exists (nil (A:=A)); exists l; simpl; auto.
-rewrite H1; auto.
-case H; auto; intros l1 [l2 Hl2]; exists (a1 :: l1); exists l2; simpl; auto.
-rewrite Hl2; auto.
-Qed.
-
-(**************************************
- Properties on app 
-**************************************)
- 
-Theorem length_app:
- forall (l1 l2 : list A),  length (l1 ++ l2) = length l1 + length l2.
-intros l1; elim l1; simpl; auto.
-Qed.
- 
-Theorem app_inv_head:
- forall (l1 l2 l3 : list A), l1 ++ l2 = l1 ++ l3 ->  l2 = l3.
-intros l1; elim l1; simpl; auto.
-intros a l H l2 l3 H0; apply H; injection H0; auto.
-Qed.
- 
-Theorem app_inv_tail:
- forall (l1 l2 l3 : list A), l2 ++ l1 = l3 ++ l1 ->  l2 = l3.
-intros l1 l2; generalize l1; elim l2; clear l1 l2; simpl; auto.
-intros l1 l3; case l3; auto.
-intros b l H; absurd (length ((b :: l) ++ l1) <= length l1).
-simpl; rewrite length_app; auto with arith.
-rewrite <- H; auto with arith.
-intros a l H l1 l3; case l3.
-simpl; intros H1; absurd (length (a :: (l ++ l1)) <= length l1).
-simpl; rewrite length_app; auto with arith.
-rewrite H1; auto with arith.
-simpl; intros b l0 H0; injection H0.
-intros H1 H2; rewrite H2, (H _ _ H1); auto.
-Qed.
- 
-Theorem app_inv_app:
- forall l1 l2 l3 l4 a,
- l1 ++ l2 = l3 ++ (a :: l4) ->
-  (exists l5 : list A , l1 = l3 ++ (a :: l5) ) \/
-  (exists l5 , l2 = l5 ++ (a :: l4) ).
-intros l1; elim l1; simpl; auto.
-intros l2 l3 l4 a H; right; exists l3; auto.
-intros a l H l2 l3 l4 a0; case l3; simpl.
-intros H0; left; exists l; injection H0; intros; subst; auto.
-intros b l0 H0; case (H l2 l0 l4 a0); auto.
-injection H0; auto.
-intros [l5 H1].
-left; exists l5; injection H0; intros; subst; auto.
-Qed.
- 
-Theorem app_inv_app2:
- forall l1 l2 l3 l4 a b,
- l1 ++ l2 = l3 ++ (a :: (b :: l4)) ->
-  (exists l5 : list A , l1 = l3 ++ (a :: (b :: l5)) ) \/
-  ((exists l5 , l2 = l5 ++ (a :: (b :: l4)) ) \/
-   l1 = l3 ++ (a :: nil) /\ l2 = b :: l4).
-intros l1; elim l1; simpl; auto.
-intros l2 l3 l4 a b H; right; left; exists l3; auto.
-intros a l H l2 l3 l4 a0 b; case l3; simpl.
-case l; simpl.
-intros H0; right; right; injection H0; split; auto.
-rewrite H2; auto.
-intros b0 l0 H0; left; exists l0; injection H0; intros; subst; auto.
-intros b0 l0 H0; case (H l2 l0 l4 a0 b); auto.
-injection H0; auto.
-intros [l5 HH1]; left; exists l5; injection H0; intros; subst; auto.
-intros [H1|[H1 H2]]; auto.
-right; right; split; auto; injection H0; intros; subst; auto.
-Qed.
- 
-Theorem same_length_ex:
- forall (a : A) l1 l2 l3,
- length (l1 ++ (a :: l2)) = length l3 ->
-  (exists l4 ,
-   exists l5 ,
-   exists b : B ,
-   length l1 = length l4 /\ (length l2 = length l5 /\ l3 = l4 ++ (b :: l5))   ).
-intros a l1; elim l1; simpl; auto.
-intros l2 l3; case l3; simpl; (try (intros; discriminate)).
-intros b l H; exists (nil (A:=B)); exists l; exists b; (repeat (split; auto)).
-intros a0 l H l2 l3; case l3; simpl; (try (intros; discriminate)).
-intros b l0 H0.
-case (H l2 l0); auto.
-intros l4 [l5 [b1 [HH1 [HH2 HH3]]]].
-exists (b :: l4); exists l5; exists b1; (repeat (simpl; split; auto)).
-rewrite HH3; auto.
-Qed.
-
-(************************************** 
-  Properties on map 
-**************************************)
- 
-Theorem in_map_inv:
- forall (b : B) (l : list A),
- In b (map f l) ->  (exists a : A , In a l /\ b = f a ).
-intros b l; elim l; simpl; auto.
-intros tmp; case tmp.
-intros a0 l0 H [H1|H1]; auto.
-exists a0; auto.
-case (H H1); intros a1 [H2 H3]; exists a1; auto.
-Qed.
- 
-Theorem in_map_fst_inv:
- forall a (l : list (B * C)),
- In a (map (fst (B:=_)) l) ->  (exists c , In (a, c) l ).
-intros a l; elim l; simpl; auto.
-intros H; case H.
-intros a0 l0 H [H0|H0]; auto.
-exists (snd a0); left; rewrite <- H0; case a0; simpl; auto.
-case H; auto; intros l1 Hl1; exists l1; auto.
-Qed.
- 
-Theorem length_map: forall l,  length (map f l) = length l.
-intros l; elim l; simpl; auto.
-Qed.
- 
-Theorem map_app: forall l1 l2,  map f (l1 ++ l2) = map f l1 ++ map f l2.
-intros l; elim l; simpl; auto.
-intros a l0 H l2; rewrite H; auto.
-Qed.
- 
-Theorem map_length_decompose:
- forall l1 l2 l3 l4,
- length l1 = length l2 ->
- map f (app l1 l3) = app l2 l4 ->  map f l1 = l2 /\ map f l3 = l4.
-intros l1; elim l1; simpl; auto; clear l1.
-intros l2; case l2; simpl; auto.
-intros; discriminate.
-intros a l1 Rec l2; case l2; simpl; clear l2; auto.
-intros; discriminate.
-intros b l2 l3 l4 H1 H2.
-injection H2; clear H2; intros H2 H3.
-case (Rec l2 l3 l4); auto.
-intros H4 H5; split; auto.
-subst; auto.
-Qed.
-
-(************************************** 
-  Properties of flat_map 
-**************************************)
- 
-Theorem in_flat_map:
- forall (l : list B) (f : B ->  list C) a b,
- In a (f b) -> In b l ->  In a (flat_map f l).
-intros l g; elim l; simpl; auto.
-intros a l0 H a0 b H0 [H1|H1]; apply in_or_app; auto.
-left; rewrite H1; auto.
-right; apply H with ( b := b ); auto.
-Qed.
- 
-Theorem in_flat_map_ex:
- forall (l : list B) (f : B ->  list C) a,
- In a (flat_map f l) ->  (exists b , In b l /\ In a (f b) ).
-intros l g; elim l; simpl; auto.
-intros a H; case H.
-intros a l0 H a0 H0; case in_app_or with ( 1 := H0 ); simpl; auto.
-intros H1; exists a; auto.
-intros H1; case H with ( 1 := H1 ).
-intros b [H2 H3]; exists b; simpl; auto.
-Qed.
-
-(************************************** 
-  Properties of fold_left 
-**************************************)
-
-Theorem fold_left_invol: 
-  forall (f: A -> B -> A) (P: A -> Prop) l a,
-  P a ->  (forall x y, P x -> P (f x y)) -> P (fold_left f l a).
-intros f1 P l; elim l; simpl; auto.
-Qed. 
-
-Theorem fold_left_invol_in: 
-  forall (f: A -> B -> A) (P: A -> Prop) l a b,
-  In b l -> (forall x, P (f x b)) -> (forall x y, P x -> P (f x y)) ->
-  P (fold_left f l a).
-intros f1 P l; elim l; simpl; auto.
-intros a1 b HH; case HH.
-intros a1 l1 Rec a2 b [V|V] V1 V2; subst; auto.
-apply fold_left_invol; auto.
-apply Rec with (b := b); auto.
-Qed.
-
-End List.
-
-
-(************************************** 
-  Propertie of list_prod
-**************************************)
- 
-Theorem length_list_prod:
- forall (A : Set) (l1 l2 : list A),
-  length (list_prod l1 l2) = length l1 * length l2.
-intros A l1 l2; elim l1; simpl; auto.
-intros a l H; rewrite length_app; rewrite length_map; rewrite H; auto.
-Qed.
- 
-Theorem in_list_prod_inv:
- forall (A B : Set) a l1 l2,
- In a (list_prod l1 l2) ->
-  (exists b : A , exists c : B , a = (b, c) /\ (In b l1 /\ In c l2)  ).
-intros A B a l1 l2; elim l1; simpl; auto; clear l1.
-intros H; case H.
-intros a1 l1 H1 H2.
-case in_app_or with ( 1 := H2 ); intros H3; auto.
-case in_map_inv with ( 1 := H3 ); intros b1 [Hb1 Hb2]; auto.
-exists a1; exists b1; split; auto.
-case H1; auto; intros b1 [c1 [Hb1 [Hb2 Hb3]]].
-exists b1; exists c1; split; auto.
-Qed.