+coercion cic:/matita/ordered_groups/fexc_plusl.con nocomposites.
+
+lemma plus_cancr_le:
+ ∀G:ogroup.∀x,y,z:G.x+z ≤ y + z → x ≤ y.
+intros 5 (G x y z L);
+apply (le_rewl ??? (0+x) (zero_neutral ??));
+apply (le_rewl ??? (x+0) (plus_comm ???));
+apply (le_rewl ??? (x+(-z+z)) (opp_inverse ??));
+apply (le_rewl ??? (x+(z+ -z)) (plus_comm ??z));
+apply (le_rewl ??? (x+z+ -z) (plus_assoc ????));
+apply (le_rewr ??? (0+y) (zero_neutral ??));
+apply (le_rewr ??? (y+0) (plus_comm ???));
+apply (le_rewr ??? (y+(-z+z)) (opp_inverse ??));
+apply (le_rewr ??? (y+(z+ -z)) (plus_comm ??z));
+apply (le_rewr ??? (y+z+ -z) (plus_assoc ????));
+intro H; apply L; clear L; apply (exc_canc_plusr ??? (-z) H);
+qed.
+
+lemma fle_plusl: ∀G:ogroup. ∀f,g,h:G. f≤g → h+f≤h+g.
+intros (G f g h);
+apply (plus_cancr_le ??? (-h));
+apply (le_rewl ??? (f+h+ -h) (plus_comm ? f h));
+apply (le_rewl ??? (f+(h+ -h)) (plus_assoc ????));
+apply (le_rewl ??? (f+(-h+h)) (plus_comm ? h (-h)));
+apply (le_rewl ??? (f+0) (opp_inverse ??));
+apply (le_rewl ??? (0+f) (plus_comm ???));
+apply (le_rewl ??? (f) (zero_neutral ??));
+apply (le_rewr ??? (g+h+ -h) (plus_comm ? h ?));
+apply (le_rewr ??? (g+(h+ -h)) (plus_assoc ????));
+apply (le_rewr ??? (g+(-h+h)) (plus_comm ??h));
+apply (le_rewr ??? (g+0) (opp_inverse ??));
+apply (le_rewr ??? (0+g) (plus_comm ???));
+apply (le_rewr ??? (g) (zero_neutral ??) H);
+qed.
+
+lemma plus_cancl_le:
+ ∀G:ogroup.∀x,y,z:G.z+x ≤ z+y → x ≤ y.
+intros 5 (G x y z L);
+apply (le_rewl ??? (0+x) (zero_neutral ??));
+apply (le_rewl ??? ((-z+z)+x) (opp_inverse ??));
+apply (le_rewl ??? (-z+(z+x)) (plus_assoc ????));
+apply (le_rewr ??? (0+y) (zero_neutral ??));
+apply (le_rewr ??? ((-z+z)+y) (opp_inverse ??));
+apply (le_rewr ??? (-z+(z+y)) (plus_assoc ????));
+apply (fle_plusl ??? (-z) L);
+qed.
+
+lemma exc_opp_x_zero_to_exc_zero_x:
+ ∀G:ogroup.∀x:G.-x ≰ 0 → 0 ≰ x.
+intros (G x H); apply (exc_canc_plusr ??? (-x));
+apply (exc_rewr ???? (plus_comm ???));
+apply (exc_rewr ???? (opp_inverse ??));
+apply (exc_rewl ???? (zero_neutral ??) H);
+qed.
+