(** 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 new_p_equal : forall (a b c d : Z) (p : Qpositive) (H_Qhomographic_sg_denom_nonzero1 H_Qhomographic_sg_denom_nonzero2 : Qhomographic_sg_denom_nonzero c d p), new_p a b c d p H_Qhomographic_sg_denom_nonzero1 = new_p a b c d p H_Qhomographic_sg_denom_nonzero2.