End rndN_opp.
End Format.
(** Inclusion of a format into an extended format *)
Section Inclusion.
Variables fexp1 fexp2 : Z -> Z.
Context { valid_exp1 : Valid_exp fexp1 }.
Context { valid_exp2 : Valid_exp fexp2 }.

(* Specifications: *)
Theorem generic_inclusion_ln_beta :
forall x,
( x <> R0 -> (fexp2 (ln_beta beta x) <= fexp1 (ln_beta beta x))%Z ) ->
generic_format fexp1 x ->
generic_format fexp2 x.
Theorem generic_inclusion_lt_ge :
forall e1 e2,
( forall e, (e1 < e <= e2)%Z -> (fexp2 e <= fexp1 e)%Z ) ->
forall x,
(bpow e1 <= Rabs x < bpow e2)%R ->
generic_format fexp1 x ->
generic_format fexp2 x.
Theorem generic_inclusion :
forall e,
(fexp2 e <= fexp1 e)%Z ->
forall x,
(bpow (e - 1) <= Rabs x <= bpow e)%R ->
generic_format fexp1 x ->
generic_format fexp2 x.
Theorem generic_inclusion_ln_beta :
forall x,
( x <> R0 -> (fexp2 (ln_beta beta x) <= fexp1 (ln_beta beta x))%Z ) ->
generic_format fexp1 x ->
generic_format fexp2 x.
Theorem generic_inclusion_lt_ge :
forall e1 e2,
( forall e, (e1 < e <= e2)%Z -> (fexp2 e <= fexp1 e)%Z ) ->
forall x,
(bpow e1 <= Rabs x < bpow e2)%R ->
generic_format fexp1 x ->
generic_format fexp2 x.
Theorem generic_inclusion :
forall e,
(fexp2 e <= fexp1 e)%Z ->
forall x,
(bpow (e - 1) <= Rabs x <= bpow e)%R ->
generic_format fexp1 x ->
generic_format fexp2 x.