simply proof of zero_{TC,TR}: no need for the invariant

theories/Combined.v

@@ -30,14 +30,14 @@ Definition TCTR_interface `{irisG heap_lang Σ, Tick}
   (TRdup : nat → iProp Σ)
 : iProp Σ := (
     ⌜∀ n, Timeless (TC n)⌝
-  ∗ (|={⊤}=> TC 0%nat)
+  ∗ (|==> TC 0%nat)
   ∗ ⌜∀ m n, TC (m + n)%nat ≡ (TC m ∗ TC n)⌝
   ∗ ⌜∀ n, Timeless (TR n)⌝
   ∗ ⌜∀ n, Timeless (TRdup n)⌝
   ∗ ⌜∀ n, Persistent (TRdup n)⌝
   ∗ (∀ n, TR n ={⊤}=∗ TR n ∗ TRdup n)
-  ∗ (|={⊤}=> TR 0%nat)
-(*   ∗ (|={⊤}=> TRdup 0%nat) *)
+  ∗ (|==> TR 0%nat)
+(*   ∗ (|==> TRdup 0%nat) *)
   ∗ ⌜∀ m n, TR (m + n)%nat ≡ (TR m ∗ TR n)⌝
   ∗ ⌜∀ m n, TRdup (m `max` n)%nat ≡ (TRdup m ∗ TRdup n)⌝
 (*   ∗ (TR max_tr ={⊤}=∗ False) *)
@@ -103,6 +103,15 @@ Section Definitions.
   Definition TC (n : nat) : iProp Σ :=
     if useTC then own γ (◯nat n) else True%I.
 
+  (* Note: we can avoid the update modality by redefining TC like so:
+         Definition TC' n : iProp Σ := ⌜n = 0%nat⌝ ∨ TC n. *)
+  Lemma zero_TC :
+    ⊢ |==> TC 0.
+  Proof using.
+    rewrite /TC ; destruct useTC.
+    - by apply own_unit.
+    - by iStartProof.
+  Qed.
   Lemma TC_plus m n :
     TC (m + n) ≡ (TC m ∗ TC n)%I.
   Proof using.
@@ -144,6 +153,11 @@ Section Definitions.
     own γ2 (◯max_nat (MaxNat n)).
   Arguments TRdup _%nat_scope.
 
+  (* Note: we can avoid the update modality by redefining TR like so:
+         Definition TR' n : iProp Σ := ⌜n = 0%nat⌝ ∨ TR n. *)
+  Lemma zero_TR :
+    ⊢ |==> TR 0.
+  Proof using. apply own_unit. Qed.
   Lemma TR_plus m n :
     TR (m + n) ≡ (TR m ∗ TR n)%I.
   Proof using. by rewrite /TR auth_frag_op own_op. Qed.
@@ -170,6 +184,11 @@ Section Definitions.
   Global Instance from_sep_TR_succ n : FromSep (TR (S n)) (TR 1) (TR n) | 100.
   Proof using. by rewrite /FromSep [TR (S n)] TR_succ. Qed.
 
+  (* Note: we can avoid the update modality by redefining TRdup like so:
+         Definition TRdup' n : iProp Σ := ⌜n = 0%nat⌝ ∨ TRdup n. *)
+  Lemma zero_TRdup :
+    ⊢ |==> TRdup 0.
+  Proof using. apply own_unit. Qed.
   Lemma TRdup_max m n :
     TRdup (m `max` n) ≡ (TRdup m ∗ TRdup n)%I.
   Proof using. by rewrite /TRdup -own_op -auth_frag_op. Qed.
@@ -204,37 +223,6 @@ Section Definitions.
       ∗ own γ2 (●max_nat (MaxNat (max_tr - 1 - m)))
     )%I.
 
-  Lemma zero_TC E:
-    ↑tctrN ⊆ E →
-    TCTR_invariant ={E}=∗ TC 0.
-  Proof using.
-    rewrite /TC /TCTR_invariant ; destruct useTC ; last by auto.
-    iIntros (?) "#Htickinv".
-    iInv tctrN as (m) ">(? & ? & H● & ?)" "InvClose".
-    iDestruct (own_auth_nat_null with "H●") as "[H● $]".
-    iApply "InvClose" ; eauto with iFrame.
-  Qed.
-
-  Lemma zero_TR E:
-    ↑tctrN ⊆ E →
-    TCTR_invariant ={E}=∗ TR 0.
-  Proof using.
-    iIntros (?) "#Htickinv".
-    iInv tctrN as (m) "(>? & >? & ? & >Hγ1● & >?)" "InvClose".
-    iDestruct (own_auth_nat_null with "Hγ1●") as "[Hγ1● $]".
-    iApply "InvClose" ; eauto with iFrame.
-  Qed.
-
-  Lemma zero_TRdup E :
-    ↑tctrN ⊆ E →
-    TCTR_invariant ={E}=∗ TRdup 0.
-  Proof using.
-    iIntros (?) "#Htickinv".
-    iInv tctrN as (m) "(>? & >? & ? & >? & >Hγ2●)" "InvClose".
-    iDestruct (own_auth_max_nat_null with "Hγ2●") as "[Hγ2● $]".
-    iApply "InvClose" ; eauto with iFrame.
-  Qed.
-
   Lemma TR_nmax_absurd (E : coPset) :
     ↑tctrN ⊆ E →
     TCTR_invariant -∗ TR max_tr ={E}=∗ False.
@@ -374,13 +362,13 @@ Section Definitions.
   Proof using.
     iIntros "#Hinv". repeat iSplitR.
     - iPureIntro. by apply TC_timeless.
-    - by iApply (zero_TC with "Hinv").
+    - by iApply zero_TC.
     - iPureIntro. by apply TC_plus.
     - iPureIntro. by apply TR_timeless.
     - iPureIntro. by apply TRdup_timeless.
     - iPureIntro. by apply TRdup_persistent.
     - iIntros (n). by iApply (TR_TRdup with "Hinv").
-    - by iApply (zero_TR with "Hinv").
+    - by iApply zero_TR.
     - iPureIntro. by apply TR_plus.
     - iPureIntro. by apply TRdup_max.
     - by iApply (TRdup_nmax_absurd with "Hinv").
@@ -433,7 +421,7 @@ Proof.
     rewrite Nat.add_comm. iSpecialize ("HΔ3" with "[//]").
     iDestruct (envs_simple_replace_singleton_sound with "HΔ3") as "[HTR' HΔ']"=>//=.
     iCombine "HTR HTR'" as "HTR". iDestruct ("HΔ'" with "HTR") as "$".
-  - iMod (zero_TRdup with "Hinv") as "HTRdup"=>//.
+  - iMod zero_TRdup as "HTRdup".
     iApply (tick_spec with "[//] [$HTC $HTRdup]")=>//. iIntros "!> [HTR _]". iApply HWP.
     iDestruct (envs_simple_replace_singleton_sound with "HΔ2") as "[HTR' HΔ']"=>//=.
     iCombine "HTR HTR'" as "HTR". iDestruct ("HΔ'" with "HTR") as "$".
@@ -442,7 +430,7 @@ Proof.
     iDestruct "HTRdup" as "#>HTRdup".
     iApply (tick_spec with "[//] [$HTC $HTRdup]")=>//. iIntros "!> [_ #HTRdup']".
     iApply HWP. rewrite Nat.add_comm. iDestruct ("HΔ'" with "[//]") as "$".
-  - iMod (zero_TRdup with "Hinv") as "HTRdup"=>//.
+  - iMod zero_TRdup as "HTRdup".
     iApply (tick_spec with "[//] [$HTC $HTRdup]")=>//.
     iIntros "!> [_ #HTRdup']". by iApply HWP.
 Qed.

theories/Thunks.v

@@ -92,7 +92,6 @@ Section Thunk.
     {{{ (t : loc), RET #t ; Thunk p t nc φ }}}.
   Proof.
     iIntros "#Htickinv" (Φ) "!# [? Hf] Post".
-    iDestruct (zero_TC with "Htickinv") as ">Htc0".
     iMod (auth_max_nat_alloc (MaxNat 0)) as (γ) "[Hγ● Hγ◯]".
     iApply wp_fupd.
     wp_tick_lam. wp_tick_inj. wp_tick_alloc t.

theories/TimeCredits.v

@@ -28,7 +28,7 @@ Definition TC_interface `{!irisG heap_lang Σ, Tick}
   (TC : nat → iProp Σ)
 : iProp Σ := (
     ⌜∀ n, Timeless (TC n)⌝
-  ∗ (|={⊤}=> TC 0%nat)
+  ∗ (|==> TC 0%nat)
   ∗ ⌜∀ m n, TC (m + n)%nat ≡ (TC m ∗ TC n)⌝
   ∗ (∀ v, {{{ TC 1%nat }}} tick v {{{ RET v ; True }}})
 )%I.
@@ -77,6 +77,11 @@ Section TickSpec.
   Definition TC (n : nat) : iProp Σ :=
     own γ (◯nat n).
 
+  (* Note: we can avoid the update modality by redefining TC like so:
+         Definition TC' n : iProp Σ := ⌜n = 0%nat⌝ ∨ TC n. *)
+  Lemma zero_TC :
+    ⊢ |==> TC 0.
+  Proof. apply own_unit. Qed.
   Lemma TC_plus m n :
     TC (m + n) ≡ (TC m ∗ TC n)%I.
   Proof. by rewrite /TC auth_frag_op own_op. Qed.
@@ -108,15 +113,6 @@ Section TickSpec.
   Definition TC_invariant : iProp Σ :=
     inv timeCreditN (∃ (m:nat), ℓ ↦ #m ∗ own γ (●nat m))%I.
 
-  Lemma zero_TC :
-    TC_invariant ={⊤}=∗ TC 0.
-  Proof.
-    iIntros "#Htickinv".
-    iInv timeCreditN as (m) ">[Hcounter H●]" "Hclose".
-    iDestruct (own_auth_nat_null with "H●") as "[H● $]".
-    iApply "Hclose" ; eauto with iFrame.
-  Qed.
-
   Theorem tick_spec s E (v : val) :
     ↑timeCreditN ⊆ E →
     TC_invariant -∗
@@ -177,7 +173,7 @@ Section TickSpec.
   Proof.
     iIntros "#Hinv". iSplit ; last iSplit ; last iSplit.
     - iPureIntro. by apply TC_timeless.
-    - by iApply (zero_TC with "Hinv").
+    - by iApply zero_TC.
     - iPureIntro. by apply TC_plus.
     - iIntros (v). by iApply (tick_spec_simple with "Hinv").
   Qed.

theories/TimeReceipts.v

@@ -31,8 +31,8 @@ Definition TR_interface `{irisG heap_lang Σ, Tick}
   ∗ ⌜∀ n, Timeless (TRdup n)⌝
   ∗ ⌜∀ n, Persistent (TRdup n)⌝
   ∗ (∀ n, TR n ={⊤}=∗ TR n ∗ TRdup n)
-  ∗ (|={⊤}=> TR 0%nat)
-(*   ∗ (|={⊤}=> TRdup 0%nat) *)
+  ∗ (|==> TR 0%nat)
+(*   ∗ (|==> TRdup 0%nat) *)
   ∗ ⌜∀ m n, TR (m + n)%nat ≡ (TR m ∗ TR n)⌝
   ∗ ⌜∀ m n, TRdup (m `max` n)%nat ≡ (TRdup m ∗ TRdup n)⌝
 (*   ∗ (TR nmax ={⊤}=∗ False) *)
@@ -88,6 +88,11 @@ Section TickSpec.
     own γ2 (◯max_nat (MaxNat n)).
   Arguments TRdup _%nat_scope.
 
+  (* Note: we can avoid the update modality by redefining TR like so:
+         Definition TR' n : iProp Σ := ⌜n = 0%nat⌝ ∨ TR n. *)
+  Lemma zero_TR :
+    ⊢ |==> TR 0.
+  Proof. apply own_unit. Qed.
   Lemma TR_plus m n :
     TR (m + n) ≡ (TR m ∗ TR n)%I.
   Proof. by rewrite /TR auth_frag_op own_op. Qed.
@@ -112,6 +117,11 @@ Section TickSpec.
   Global Instance from_sep_TR_succ n : FromSep (TR (S n)) (TR 1) (TR n).
   Proof. by rewrite /FromSep [TR (S n)] TR_succ. Qed.
 
+  (* Note: we can avoid the update modality by redefining TRdup like so:
+         Definition TRdup' n : iProp Σ := ⌜n = 0%nat⌝ ∨ TRdup n. *)
+  Lemma zero_TRdup :
+    ⊢ |==> TRdup 0.
+  Proof. apply own_unit. Qed.
   Lemma TRdup_max m n :
     TRdup (m `max` n) ≡ (TRdup m ∗ TRdup n)%I.
   Proof. by rewrite /TRdup -own_op -auth_frag_op. Qed.
@@ -137,26 +147,6 @@ Section TickSpec.
   Definition TR_invariant : iProp Σ :=
     inv timeReceiptN (∃ (n:nat), ℓ ↦ #(nmax-n-1) ∗ own γ1 (●nat n) ∗ own γ2 (●max_nat (MaxNat n)) ∗ ⌜(n < nmax)%nat⌝)%I.
 
-  Lemma zero_TR E :
-    ↑timeReceiptN ⊆ E →
-    TR_invariant ={E}=∗ TR 0.
-  Proof.
-    iIntros (?) "#Htickinv".
-    iInv timeReceiptN as (m) ">(Hcounter & Hγ1● & H)" "Hclose".
-    iDestruct (own_auth_nat_null with "Hγ1●") as "[Hγ1● $]".
-    iApply "Hclose" ; eauto with iFrame.
-  Qed.
-
-  Lemma zero_TRdup E :
-    ↑timeReceiptN ⊆ E →
-    TR_invariant ={E}=∗ TRdup 0.
-  Proof.
-    iIntros (?) "#Htickinv".
-    iInv timeReceiptN as (m) ">(Hcounter & Hγ1● & Hγ2● & Im)" "Hclose".
-    iDestruct (own_auth_max_nat_null with "Hγ2●") as "[Hγ2● $]".
-    iApply "Hclose" ; eauto with iFrame.
-  Qed.
-
   Lemma TR_nmax_absurd (E : coPset) :
     ↑timeReceiptN ⊆ E →
     TR_invariant -∗ TR nmax ={E}=∗ False.
@@ -283,7 +273,7 @@ Section TickSpec.
     - iPureIntro. by apply TRdup_timeless.
     - iPureIntro. by apply TRdup_persistent.
     - iIntros (n). by iApply (TR_TRdup with "Hinv").
-    - by iApply (zero_TR with "Hinv").
+    - by iApply zero_TR.
     - iPureIntro. by apply TR_plus.
     - iPureIntro. by apply TRdup_max.
     - by iApply (TRdup_nmax_absurd with "Hinv").
@@ -410,7 +400,7 @@ Proof.
     rewrite Nat.add_comm. iSpecialize ("HΔ2" with "[//]").
     iDestruct (envs_simple_replace_singleton_sound with "HΔ2") as "[HTR' HΔ']"=>//=.
     iCombine "HTR HTR'" as "HTR". iDestruct ("HΔ'" with "HTR") as "$".
-  - iMod (zero_TRdup with "Hinv") as "HTRdup"=>//.
+  - iMod zero_TRdup as "HTRdup".
     iApply (tick_spec with "[//] HTRdup")=>//. iIntros "!> [HTR _]". iApply HWP.
     iDestruct (envs_simple_replace_singleton_sound with "HΔ1") as "[HTR' HΔ']"=>//=.
     iCombine "HTR HTR'" as "HTR". iDestruct ("HΔ'" with "HTR") as "$".
@@ -419,7 +409,7 @@ Proof.
     iDestruct "HTRdup" as "#>HTRdup".
     iApply (tick_spec with "[//] HTRdup")=>//. iIntros "!> [_ #HTRdup']". iApply HWP.
     rewrite Nat.add_comm. iDestruct ("HΔ'" with "[//]") as "$".
-  - iMod (zero_TRdup with "Hinv") as "HTRdup"=>//.
+  - iMod zero_TRdup as "HTRdup".
     iApply (tick_spec with "[//] HTRdup")=>//. iIntros "!> [_ #HTRdup']". by iApply HWP.
 Qed.
 

theories/union_find/Proof.v

@@ -731,7 +731,7 @@ Theorem make_spec : forall D R V v,
 Proof using.
   iIntros "* #?" (Φ) "!# [HF TC] HΦ".
   wp_tick_rec. wp_tick_pair. wp_tick_inj.
-  iMod (zero_TR with "[//]") as "TR"=>//.
+  iMod zero_TR as "TR".
   wp_tick. wp_alloc x as "Hx". iApply "HΦ".
   iDestruct "HF" as (F K M HI HM) "(HM & TC' & TR')".