frac.v 1.75 KB
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From Coq.QArith Require Import Qcanon.
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From iris.algebra Require Export cmra.
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Set Default Proof Using "Type".
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Notation frac := Qp (only parsing).
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Section frac.
Canonical Structure fracC := leibnizC frac.
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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.
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Lemma frac_included (x y : frac) : x  y  (x < y)%Qc.
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.
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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Definition frac_ra_mixin : RAMixin frac.
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Proof.
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  split; try apply _; try done.
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  unfold valid, op, frac_op, frac_valid. intros x y. trans (x+y)%Qp; last done.
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  rewrite -{1}(Qcplus_0_r x) -Qcplus_le_mono_l; auto using Qclt_le_weak.
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Qed.
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Canonical Structure fracR := discreteR frac frac_ra_mixin.
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Global Instance frac_cmra_discrete : CMRADiscrete fracR.
Proof. apply discrete_cmra_discrete. Qed.
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End frac.
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Global Instance frac_full_exclusive : Exclusive 1%Qp.
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Proof.
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  move=> y /Qcle_not_lt [] /=. by rewrite -{1}(Qcplus_0_r 1) -Qcplus_lt_mono_l.
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Qed.
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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.

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Lemma frac_op':  (p q: Qp), (p  q) = (p + q)%Qp.
Proof. done. Qed.