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-
-(*************************************************************)
-(*      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 Iterator.
-Require Import ZArith.
-Require Export 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.