Commit b8dc0773 by Ralf Jung

### make frac_included into general lemma about < and + of positive fractions

parent 26e93ebf
 ... ... @@ -11,14 +11,19 @@ Instance frac_valid : Valid frac := λ x, (x ≤ 1)%Qc. Instance frac_pcore : PCore frac := λ _, None. Instance frac_op : Op frac := λ x y, (x + y)%Qp. Lemma frac_included (x y : frac) : x ≼ y ↔ (x < y)%Qc. (* TODO: Find better place for this lemma. *) Lemma Qp_le_sum (x y : Qp) : (x < y)%Qc ↔ (∃ z, y = x + z)%Qp. Proof. split. - intros [z ->%leibniz_equiv]; simpl. rewrite -{1}(Qcplus_0_r x). apply Qcplus_lt_mono_l, Qp_prf. - intros Hlt%Qclt_minus_iff. exists (mk_Qp (y - x) Hlt). apply Qp_eq; simpl. by rewrite (Qcplus_comm y) Qcplus_assoc Qcplus_opp_r Qcplus_0_l. - intros [z ->%leibniz_equiv]; simpl. rewrite -{1}(Qcplus_0_r x). apply Qcplus_lt_mono_l, Qp_prf. Qed. Lemma frac_included (x y : frac) : x ≼ y ↔ (x < y)%Qc. Proof. symmetry. exact: Qp_le_sum. Qed. Corollary frac_included_weak (x y : frac) : x ≼ y → (x ≤ y)%Qc. Proof. intros ?%frac_included. auto using Qclt_le_weak. Qed. ... ...
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