frac.v 1.97 KB
 1 ``````From Coq.QArith Require Import Qcanon. `````` Robbert Krebbers committed Mar 10, 2016 2 ``````From iris.algebra Require Export cmra. `````` Ralf Jung committed Jan 05, 2017 3 ``````Set Default Proof Using "Type". `````` Robbert Krebbers committed Feb 26, 2016 4 `````` `````` Robbert Krebbers committed Jun 01, 2016 5 ``````Notation frac := Qp (only parsing). `````` Robbert Krebbers committed Feb 26, 2016 6 `````` `````` Jacques-Henri Jourdan committed Jun 01, 2016 7 8 ``````Section frac. Canonical Structure fracC := leibnizC frac. `````` Robbert Krebbers committed Feb 26, 2016 9 `````` `````` Jacques-Henri Jourdan committed Jun 01, 2016 10 11 12 ``````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. `````` Robbert Krebbers committed May 28, 2016 13 `````` `````` Ralf Jung committed Mar 10, 2017 14 15 ``````(* TODO: Find better place for this lemma. *) Lemma Qp_le_sum (x y : Qp) : (x < y)%Qc ↔ (∃ z, y = x + z)%Qp. `````` Robbert Krebbers committed Oct 02, 2016 16 17 18 19 ``````Proof. split. - 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. `````` Ralf Jung committed Mar 10, 2017 20 21 `````` - intros [z ->%leibniz_equiv]; simpl. rewrite -{1}(Qcplus_0_r x). apply Qcplus_lt_mono_l, Qp_prf. `````` Robbert Krebbers committed Oct 02, 2016 22 ``````Qed. `````` Ralf Jung committed Mar 10, 2017 23 24 25 26 `````` Lemma frac_included (x y : frac) : x ≼ y ↔ (x < y)%Qc. Proof. symmetry. exact: Qp_le_sum. Qed. `````` Robbert Krebbers committed Oct 02, 2016 27 28 29 ``````Corollary frac_included_weak (x y : frac) : x ≼ y → (x ≤ y)%Qc. Proof. intros ?%frac_included. auto using Qclt_le_weak. Qed. `````` Jacques-Henri Jourdan committed Jun 01, 2016 30 ``````Definition frac_ra_mixin : RAMixin frac. `````` Robbert Krebbers committed Feb 26, 2016 31 ``````Proof. `````` Jacques-Henri Jourdan committed Jun 01, 2016 32 `````` split; try apply _; try done. `````` Robbert Krebbers committed Jun 01, 2016 33 `````` unfold valid, op, frac_op, frac_valid. intros x y. trans (x+y)%Qp; last done. `````` Jacques-Henri Jourdan committed Jun 01, 2016 34 `````` rewrite -{1}(Qcplus_0_r x) -Qcplus_le_mono_l; auto using Qclt_le_weak. `````` Robbert Krebbers committed Feb 26, 2016 35 ``````Qed. `````` Jacques-Henri Jourdan committed Jun 01, 2016 36 ``````Canonical Structure fracR := discreteR frac frac_ra_mixin. `````` Robbert Krebbers committed Feb 09, 2017 37 38 39 `````` Global Instance frac_cmra_discrete : CMRADiscrete fracR. Proof. apply discrete_cmra_discrete. Qed. `````` Jacques-Henri Jourdan committed Jun 01, 2016 40 ``````End frac. `````` Robbert Krebbers committed Feb 26, 2016 41 `````` `````` Jacques-Henri Jourdan committed Jun 01, 2016 42 ``````Global Instance frac_full_exclusive : Exclusive 1%Qp. `````` Jacques-Henri Jourdan committed May 31, 2016 43 ``````Proof. `````` Robbert Krebbers committed Jun 01, 2016 44 `````` move=> y /Qcle_not_lt [] /=. by rewrite -{1}(Qcplus_0_r 1) -Qcplus_lt_mono_l. `````` Jacques-Henri Jourdan committed May 31, 2016 45 ``````Qed. `````` Zhen Zhang committed Oct 10, 2016 46 `````` `````` Jacques-Henri Jourdan committed Feb 01, 2017 47 48 49 50 51 52 53 54 55 ``````Global Instance frac_cancelable (q : frac) : Cancelable q. Proof. intros ?????. by apply Qp_eq, (inj (Qcplus q)), (Qp_eq (q+y) (q+z))%Qp. Qed. Global Instance frac_id_free (q : frac) : IdFree q. Proof. intros [q0 Hq0] ? EQ%Qp_eq. rewrite -{1}(Qcplus_0_r q) in EQ. eapply Qclt_not_eq; first done. by apply (inj (Qcplus q)). Qed. `````` Ralf Jung committed Mar 09, 2017 56 57 58 59 ``````Lemma frac_op' (q p : Qp) : (p ⋅ q) = (p + q)%Qp. Proof. done. Qed. Lemma frac_valid' (p : Qp) : ✓ p ↔ (p ≤ 1%Qp)%Qc. `````` Zhen Zhang committed Oct 10, 2016 60 ``Proof. done. Qed.``