(** Gcd of divided elements, for exact divisions *)

(* Specifications: *)
Lemma gcd_div_factor : forall a b c, c~=0 -> (c|a) -> (c|b) ->
gcd (a/c) (b/c) == (gcd a b)/c.
Lemma gcd_div_gcd : forall a b g, g~=0 -> g == gcd a b ->
gcd (a/g) (b/g) == 1.
Lemma gcd_div_factor : forall a b c, 0<c -> (c|a) -> (c|b) ->
gcd (a/c) (b/c) == (gcd a b)/c.
Lemma gcd_quot_factor : forall a b c, 0<c -> (c|a) -> (c|b) ->
gcd (a÷c) (b÷c) == (gcd a b)÷c.
Lemma gcd_div_gcd : forall a b g, g~=0 -> g == gcd a b ->
gcd (a/g) (b/g) == 1.
Lemma gcd_quot_gcd : forall a b g, g~=0 -> g == gcd a b ->
gcd (a÷g) (b÷g) == 1.