(** Cancellations. *)

(* Specifications: *)
Lemma div_mul_cancel_r : forall a b c, 0<=a -> 0<b -> 0<c ->
(a*c)/(b*c) == a/b.
Lemma div_mul_cancel_l : forall a b c, 0<=a -> 0<b -> 0<c ->
(c*a)/(c*b) == a/b.
Lemma div_mul_cancel_r : forall a b c, b~=0 -> c~=0 ->
(a*c)/(b*c) == a/b.
Lemma div_mul_cancel_l : forall a b c, b~=0 -> c~=0 ->
(c*a)/(c*b) == a/b.
Lemma mul_mod_distr_l: forall a b c, b~=0 -> c~=0 ->
(c*a) mod (c*b) == c * (a mod b).
Lemma mul_mod_distr_r: forall a b c, b~=0 -> c~=0 ->
(a*c) mod (b*c) == (a mod b) * c.
Lemma quot_mul_cancel_r : forall a b c, b~=0 -> c~=0 ->
(a*c)÷(b*c) == a÷b.
Lemma quot_mul_cancel_l : forall a b c, b~=0 -> c~=0 ->
(c*a)÷(c*b) == a÷b.
Lemma mul_rem_distr_r: forall a b c, b~=0 -> c~=0 ->
(a*c) rem (b*c) == (a rem b) * c.
Lemma mul_rem_distr_l: forall a b c, b~=0 -> c~=0 ->
(c*a) rem (c*b) == c * (a rem b).