Set Implicit Arguments.
Unset Strict Implicit.
Require Export point_cocyclicite.
unfold colineaire in |- *; intros.
apply regulier with (a := double (cons u v)) (c := double (cons u v)); auto.
apply transitive with (double (plus (cons u v) (cons v u))); auto.
apply transitive with (double zero); auto.
apply transitive with (plus zero zero); auto.
apply compatible; auto.
Qed.
Hint Resolve colineaire_sym.
unfold colineaire in |- *; intros.
apply
transitive with (double (plus (cons u' u) (plus (cons u v) (cons v v'))));
auto.
apply
transitive
with (plus (double (cons u' u)) (double (plus (cons u v) (cons v v'))));
auto.
apply
transitive
with
(plus (double (cons u' u))
(plus (double (cons u v)) (double (cons v v'))));
auto.
apply compatible; auto.
apply
transitive with (plus zero (plus (double (cons u v)) (double (cons v v'))));
auto.
apply compatible; auto.
cut (colineaire u' u); intros; auto.
apply transitive with (plus (double (cons u v)) (double (cons v v'))); auto.
apply transitive with (plus (double (cons u v)) zero); auto.
apply compatible; auto.
apply transitive with (plus zero (double (cons u v))); auto.
Qed.
Hint Resolve colineaire_modulo_pi.
unfold orthogonal in |- *; intros.
apply transitive with (double (plus (cons u v) (cons v (opp v)))); auto.
apply transitive with (plus (double (cons u v)) (double (cons v (opp v))));
auto.
apply transitive with (plus pi (double pi)); auto.
apply compatible; auto.
apply transitive with (plus pi zero); auto.
apply compatible; auto.
apply transitive with (plus zero pi); auto.
Qed.
Hint Resolve orthogonal_opp.
unfold colineaire, orthogonal in |- *; intros.
apply transitive with (double (plus (cons u v) (cons v w))); auto.
apply transitive with (plus (double (cons u v)) (double (cons v w))); auto.
apply transitive with (plus pi zero); auto.
apply compatible; auto.
apply transitive with (plus zero pi); auto.
Qed.
unfold colineaire in |- *; intros.
apply transitive with (double (plus (cons u v) (cons v w))); auto.
apply transitive with (plus (double (cons u v)) (double (cons v w))); auto.
apply transitive with (plus zero zero); auto.
apply compatible; auto.
Qed.
unfold orthogonal in |- *; intros.
apply
transitive with (double (plus (cons u u') (plus (cons u' v') (cons v' v))));
auto.
apply
transitive
with (plus (double (cons u u')) (double (plus (cons u' v') (cons v' v))));
auto.
apply
transitive
with
(plus (double (cons u u'))
(plus (double (cons u' v')) (double (cons v' v))));
auto.
apply compatible; auto.
apply transitive with (plus pi (plus (double (cons u' v')) pi)); auto.
apply compatible; auto.
apply compatible; auto.
apply regulier with (a := double (cons v v')) (c := pi); auto.
apply transitive with (double (plus (cons v v') (cons v' v))); auto.
apply transitive with (double (cons v v)); auto.
apply transitive with (double zero); auto.
apply transitive with zero; auto.
apply transitive with (plus (plus (double (cons u' v')) pi) pi); auto.
apply transitive with (plus (double (cons u' v')) (plus pi pi)); auto.
apply transitive with (plus (double (cons u' v')) zero); auto.
apply compatible; auto.
apply transitive with (plus zero (double (cons u' v'))); auto.
Qed.
Hint Resolve double_orthogonal.
intros u v H; unfold orthogonal, double in |- *.
apply transitive with (plus (cons u v) (cons v (opp u))); auto.
apply compatible; auto.
apply transitive with (cons u (opp u)); auto.
Qed.

Lemma critere_orthogonal_reciproque :
forall u v : V, orthogonal u v -> R (cons u v) (cons v (opp u)).