1 files changed, 104 insertions, 0 deletions
diff --git a/coqprime-8.4/Coqprime/ZProgression.v b/coqprime-8.4/Coqprime/ZProgression.v
new file mode 100644
index 000000000..4cf30d692
--- /dev/null
+++ b/coqprime-8.4/Coqprime/ZProgression.v
@@ -0,0 +1,104 @@
+
+(*************************************************************)
+(*      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      *)
+(*************************************************************)
+
+Require Export Coqprime.Iterator.
+Require Import Coq.ZArith.ZArith.
+Require Export Coqprime.UList.
+Open Scope Z_scope.
+ 
+Theorem next_n_Z: forall n m,  next_n Zsucc n m = n + Z_of_nat m.
+intros n m; generalize n; elim m; clear n m.
+intros n; simpl; auto with zarith.
+intros m H n.
+replace (n + Z_of_nat (S m)) with (Zsucc n + Z_of_nat m); auto with zarith.
+rewrite <- H; auto with zarith.
+rewrite inj_S; auto with zarith.
+Qed.
+ 
+Theorem Zprogression_end:
+ forall n m,
+  progression Zsucc n (S m) =
+  app (progression Zsucc n m) (cons (n + Z_of_nat m) nil).
+intros n m; generalize n; elim m; clear n m.
+simpl; intros; apply f_equal2 with ( f := @cons Z ); auto with zarith.
+intros m1 Hm1 n1.
+apply trans_equal with (cons n1 (progression Zsucc (Zsucc n1) (S m1))); auto.
+rewrite Hm1.
+replace (Zsucc n1 + Z_of_nat m1) with (n1 + Z_of_nat (S m1)); auto with zarith.
+replace (Z_of_nat (S m1)) with (1 + Z_of_nat m1); auto with zarith.
+rewrite inj_S; auto with zarith.
+Qed.
+ 
+Theorem Zprogression_pred_end:
+ forall n m,
+  progression Zpred n (S m) =
+  app (progression Zpred n m) (cons (n - Z_of_nat m) nil).
+intros n m; generalize n; elim m; clear n m.
+simpl; intros; apply f_equal2 with ( f := @cons Z ); auto with zarith.
+intros m1 Hm1 n1.
+apply trans_equal with (cons n1 (progression Zpred (Zpred n1) (S m1))); auto.
+rewrite Hm1.
+replace (Zpred n1 - Z_of_nat m1) with (n1 - Z_of_nat (S m1)); auto with zarith.
+replace (Z_of_nat (S m1)) with (1 + Z_of_nat m1); auto with zarith.
+rewrite inj_S; auto with zarith.
+Qed.
+ 
+Theorem Zprogression_opp:
+ forall n m,
+  rev (progression Zsucc n m) = progression Zpred (n + Z_of_nat (pred m)) m.
+intros n m; generalize n; elim m; clear n m.
+simpl; auto.
+intros m Hm n.
+rewrite (Zprogression_end n); auto.
+rewrite distr_rev.
+rewrite Hm; simpl; auto.
+case m.
+simpl; auto.
+intros m1;
+ replace (n + Z_of_nat (pred (S m1))) with (Zpred (n + Z_of_nat (S m1))); auto.
+rewrite inj_S; simpl; (unfold Zpred; unfold Zsucc); auto with zarith.
+Qed.
+ 
+Theorem Zprogression_le_init:
+ forall n m p, In p (progression Zsucc n m) ->  (n <= p).
+intros n m; generalize n; elim m; clear n m; simpl; auto.
+intros; contradiction.
+intros m H n p [H1|H1]; auto with zarith.
+generalize (H _ _ H1); auto with zarith.
+Qed.
+ 
+Theorem Zprogression_le_end:
+ forall n m p, In p (progression Zsucc n m) ->  (p < n + Z_of_nat m).
+intros n m; generalize n; elim m; clear n m; auto.
+intros; contradiction.
+intros m H n p H1; simpl in H1 |-; case H1; clear H1; intros H1;
+ auto with zarith.
+subst n; auto with zarith.
+apply Zle_lt_trans with (p + 0); auto with zarith.
+apply Zplus_lt_compat_l; red; simpl; auto with zarith.
+apply Zlt_le_trans with (Zsucc n + Z_of_nat m); auto with zarith.
+rewrite inj_S; rewrite Zplus_succ_comm; auto with zarith.
+Qed.
+ 
+Theorem ulist_Zprogression: forall a n,  ulist (progression Zsucc a n).
+intros a n; generalize a; elim n; clear a n; simpl; auto with zarith.
+intros n H1 a; apply ulist_cons; auto.
+intros H2; absurd (Zsucc a <= a); auto with zarith.
+apply Zprogression_le_init with ( 1 := H2 ).
+Qed.
+ 
+Theorem in_Zprogression:
+ forall a b n, ( a <= b < a + Z_of_nat n ) ->  In b (progression Zsucc a n).
+intros a b n; generalize a b; elim n; clear a b n; auto with zarith.
+simpl; auto with zarith.
+intros n H a b.
+replace (a + Z_of_nat (S n)) with (Zsucc a + Z_of_nat n); auto with zarith.
+intros [H1 H2]; simpl; auto with zarith.
+case (Zle_lt_or_eq _ _ H1); auto with zarith.
+rewrite inj_S; auto with zarith.
+Qed.