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(* GNU General Public License for more details.                       *)
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(* 02110-1301 USA                                                     *)
Require Export Qhomographic.
Functional Scheme fraction_encoding_ind :=
Induction for fraction_encoding Sort Prop.
(** This tactic solves the goal by contradiction, if in the hypothesis there are two terms `a<c` and `c<=a` *)
Ltac Order_mismatch :=
Clear_eq_;
match goal with
| id1:(?X1 < ?X2)%Z,id2:(?X2 <= ?X1)%Z |- _ =>
match goal with
| id1:(?X1 < ?X2)%Z,id2:(?X2 <= ?X1)%Z |- _ =>
first [ apply False_ind | apply False_rec ]; apply (Z.lt_irrefl X1);
apply Z.lt_le_trans with X2; assumption
end
| id2:(?X1 <= ?X2)%Z,id1:(?X2 < ?X1)%Z |- _ =>
match goal with
| id1:(?X1 < ?X2)%Z,id2:(?X2 <= ?X1)%Z |- _ =>
first [ apply False_ind | apply False_rec ]; apply (Z.lt_irrefl X1);
apply Z.lt_le_trans with X2; assumption
end
end.

Lemma Qhomographic_Qpositive_to_Q_3 :
forall (a b c d : Z) (p : Qpositive)
(H_Qhomographic_sg_denom_nonzero : Qhomographic_sg_denom_nonzero c d p),
(a * d)%Z <> (b * c)%Z ->
h_sign a b c d p H_Qhomographic_sg_denom_nonzero = 1%Z ->
forall
hyp_homographicAcc : homographicAcc
(new_a a b c d p H_Qhomographic_sg_denom_nonzero)
(new_b a b c d p H_Qhomographic_sg_denom_nonzero)
(new_c a b c d p H_Qhomographic_sg_denom_nonzero)
(new_d a b c d p H_Qhomographic_sg_denom_nonzero)
(new_p a b c d p H_Qhomographic_sg_denom_nonzero),
(0 <
Z.sgn
(new_a a b c d p H_Qhomographic_sg_denom_nonzero +
new_b a b c d p H_Qhomographic_sg_denom_nonzero))%Z ->
Qhomographic_Qpositive_to_Q a b c d p H_Qhomographic_sg_denom_nonzero =
Qpos
(Qhomographic_Qpositive_to_Qpositive
(new_a a b c d p H_Qhomographic_sg_denom_nonzero)
(new_b a b c d p H_Qhomographic_sg_denom_nonzero)
(new_c a b c d p H_Qhomographic_sg_denom_nonzero)
(new_d a b c d p H_Qhomographic_sg_denom_nonzero)
(new_p a b c d p H_Qhomographic_sg_denom_nonzero) hyp_homographicAcc).