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From iris.program_logic Require Export weakestpre.
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From iris.base_logic.lib Require Export invariants.
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From iris.heap_lang Require Export lang.
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From iris.proofmode Require Import tactics.
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From iris.algebra Require Import frac_auth auth.
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From iris.heap_lang Require Import proofmode notation.
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Set Default Proof Using "Type".
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Definition newcounter : val := λ: <>, ref #0.
Definition incr : val := rec: "incr" "l" :=
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    let: "n" := !"l" in
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    if: CAS "l" "n" (#1 + "n") then #() else "incr" "l".
Definition read : val := λ: "l", !"l".
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(** Monotone counter *)
Class mcounterG Σ := MCounterG { mcounter_inG :> inG Σ (authR mnatUR) }.
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Definition mcounterΣ : gFunctors := #[GFunctor (authR mnatUR)].
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Instance subG_mcounterΣ {Σ} : subG mcounterΣ Σ  mcounterG Σ.
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Proof. solve_inG. Qed.
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Section mono_proof.
  Context `{!heapG Σ, !mcounterG Σ} (N : namespace).

  Definition mcounter_inv (γ : gname) (l : loc) : iProp Σ :=
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    ( n, own γ ( (n : mnat))  l  #n)%I.
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  Definition mcounter (l : loc) (n : nat) : iProp Σ :=
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    ( γ, inv N (mcounter_inv γ l)  own γ ( (n : mnat)))%I.
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  (** The main proofs. *)
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  Global Instance mcounter_persistent l n : Persistent (mcounter l n).
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  Proof. apply _. Qed.

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  Lemma newcounter_mono_spec :
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    {{{ True }}} newcounter #() {{{ l, RET #l; mcounter l 0 }}}.
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  Proof.
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    iIntros (Φ) "_ HΦ". rewrite -wp_fupd /newcounter /=. wp_seq. wp_alloc l as "Hl".
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    iMod (own_alloc ( (O:mnat)   (O:mnat))) as (γ) "[Hγ Hγ']"; first done.
    iMod (inv_alloc N _ (mcounter_inv γ l) with "[Hl Hγ]").
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    { iNext. iExists 0%nat. by iFrame. }
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    iModIntro. iApply "HΦ". rewrite /mcounter; eauto 10.
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  Qed.

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  Lemma incr_mono_spec l n :
    {{{ mcounter l n }}} incr #l {{{ RET #(); mcounter l (S n) }}}.
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  Proof.
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    iIntros (Φ) "Hl HΦ". iLöb as "IH". wp_rec.
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    iDestruct "Hl" as (γ) "[#? Hγf]".
    wp_bind (! _)%E. iInv N as (c) ">[Hγ Hl]".
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    wp_load. iModIntro. iSplitL "Hl Hγ"; [iNext; iExists c; by iFrame|].
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    wp_let. wp_op.
    wp_bind (CAS _ _ _). iInv N as (c') ">[Hγ Hl]".
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    destruct (decide (c' = c)) as [->|].
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    - iDestruct (own_valid_2 with "Hγ Hγf")
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        as %[?%mnat_included _]%auth_valid_discrete_2.
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      iMod (own_update_2 with "Hγ Hγf") as "[Hγ Hγf]".
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      { apply auth_update, (mnat_local_update _ _ (S c)); auto. }
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      wp_cas_suc. iModIntro. iSplitL "Hl Hγ".
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      { iNext. iExists (S c). rewrite Nat2Z.inj_succ Z.add_1_l. by iFrame. }
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      wp_if. iApply "HΦ"; iExists γ; repeat iSplit; eauto.
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      iApply (own_mono with "Hγf"). apply: auth_frag_mono.
      by apply mnat_included, le_n_S.
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    - wp_cas_fail; first (by intros [= ?%Nat2Z.inj]). iModIntro.
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      iSplitL "Hl Hγ"; [iNext; iExists c'; by iFrame|].
      wp_if. iApply ("IH" with "[Hγf] [HΦ]"); last by auto.
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      rewrite {3}/mcounter; eauto 10.
  Qed.

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  Lemma read_mono_spec l j :
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    {{{ mcounter l j }}} read #l {{{ i, RET #i; j  i%nat  mcounter l i }}}.
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  Proof.
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    iIntros (ϕ) "Hc HΦ". iDestruct "Hc" as (γ) "[#Hinv Hγf]".
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    rewrite /read /=. wp_let. iInv N as (c) ">[Hγ Hl]".
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    wp_load.
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    iDestruct (own_valid_2 with "Hγ Hγf")
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      as %[?%mnat_included _]%auth_valid_discrete_2.
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    iMod (own_update_2 with "Hγ Hγf") as "[Hγ Hγf]".
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    { apply auth_update, (mnat_local_update _ _ c); auto. }
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    iModIntro. iSplitL "Hl Hγ"; [iNext; iExists c; by iFrame|].
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    iApply ("HΦ" with "[-]"). rewrite /mcounter; eauto 10.
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  Qed.
End mono_proof.

(** Counter with contributions *)
Class ccounterG Σ :=
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  CCounterG { ccounter_inG :> inG Σ (frac_authR natR) }.
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Definition ccounterΣ : gFunctors :=
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  #[GFunctor (frac_authR natR)].
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Instance subG_ccounterΣ {Σ} : subG ccounterΣ Σ  ccounterG Σ.
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Proof. solve_inG. Qed.
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Section contrib_spec.
  Context `{!heapG Σ, !ccounterG Σ} (N : namespace).

  Definition ccounter_inv (γ : gname) (l : loc) : iProp Σ :=
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    ( n, own γ (! n)  l  #n)%I.
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  Definition ccounter_ctx (γ : gname) (l : loc) : iProp Σ :=
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    inv N (ccounter_inv γ l).
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  Definition ccounter (γ : gname) (q : frac) (n : nat) : iProp Σ :=
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    own γ (!{q} n).
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  (** The main proofs. *)
  Lemma ccounter_op γ q1 q2 n1 n2 :
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    ccounter γ (q1 + q2) (n1 + n2)  ccounter γ q1 n1  ccounter γ q2 n2.
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  Proof. by rewrite /ccounter frac_auth_frag_op -own_op. Qed.
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  Lemma newcounter_contrib_spec (R : iProp Σ) :
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    {{{ True }}} newcounter #()
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    {{{ γ l, RET #l; ccounter_ctx γ l  ccounter γ 1 0 }}}.
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  Proof.
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    iIntros (Φ) "_ HΦ". rewrite -wp_fupd /newcounter /=. wp_seq. wp_alloc l as "Hl".
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    iMod (own_alloc (! O%nat  ! 0%nat)) as (γ) "[Hγ Hγ']"; first done.
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    iMod (inv_alloc N _ (ccounter_inv γ l) with "[Hl Hγ]").
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    { iNext. iExists 0%nat. by iFrame. }
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    iModIntro. iApply "HΦ". rewrite /ccounter_ctx /ccounter; eauto 10.
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  Qed.

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  Lemma incr_contrib_spec γ l q n :
    {{{ ccounter_ctx γ l  ccounter γ q n }}} incr #l
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    {{{ RET #(); ccounter γ q (S n) }}}.
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  Proof.
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    iIntros (Φ) "[#? Hγf] HΦ". iLöb as "IH". wp_rec.
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    wp_bind (! _)%E. iInv N as (c) ">[Hγ Hl]".
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    wp_load. iModIntro. iSplitL "Hl Hγ"; [iNext; iExists c; by iFrame|].
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    wp_let. wp_op.
    wp_bind (CAS _ _ _). iInv N as (c') ">[Hγ Hl]".
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    destruct (decide (c' = c)) as [->|].
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    - iMod (own_update_2 with "Hγ Hγf") as "[Hγ Hγf]".
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      { apply frac_auth_update, (nat_local_update _ _ (S c) (S n)); omega. }
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      wp_cas_suc. iModIntro. iSplitL "Hl Hγ".
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      { iNext. iExists (S c). rewrite Nat2Z.inj_succ Z.add_1_l. by iFrame. }
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      wp_if. by iApply "HΦ".
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    - wp_cas_fail; first (by intros [= ?%Nat2Z.inj]).
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      iModIntro. iSplitL "Hl Hγ"; [iNext; iExists c'; by iFrame|].
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      wp_if. by iApply ("IH" with "[Hγf] [HΦ]"); auto.
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  Qed.

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  Lemma read_contrib_spec γ l q n :
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    {{{ ccounter_ctx γ l  ccounter γ q n }}} read #l
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    {{{ c, RET #c; n  c%nat  ccounter γ q n }}}.
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  Proof.
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    iIntros (Φ) "[#? Hγf] HΦ".
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    rewrite /read /=. wp_let. iInv N as (c) ">[Hγ Hl]". wp_load.
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    iDestruct (own_valid_2 with "Hγ Hγf") as % ?%frac_auth_included_total%nat_included.
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    iModIntro. iSplitL "Hl Hγ"; [iNext; iExists c; by iFrame|].
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    iApply ("HΦ" with "[-]"); rewrite /ccounter; eauto 10.
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  Qed.

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  Lemma read_contrib_spec_1 γ l n :
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    {{{ ccounter_ctx γ l  ccounter γ 1 n }}} read #l
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    {{{ n, RET #n; ccounter γ 1 n }}}.
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  Proof.
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    iIntros (Φ) "[#? Hγf] HΦ".
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    rewrite /read /=. wp_let. iInv N as (c) ">[Hγ Hl]". wp_load.
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    iDestruct (own_valid_2 with "Hγ Hγf") as % <-%frac_auth_agreeL.
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    iModIntro. iSplitL "Hl Hγ"; [iNext; iExists c; by iFrame|].
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    by iApply "HΦ".
  Qed.
End contrib_spec.