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Commit 3447371d authored by Dan Frumin's avatar Dan Frumin
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Simplify some example proofs

parent 752493b6
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Showing with 236 additions and 140 deletions
theories/examples/counter.v
View file @ 3447371d
......@@ -122,7 +122,6 @@ Section CG_Counter.
(* A logically atomic specification for
a fine-grained increment with a baked in frame. *)
(* Unfortunately, the precondition is not baked in the rule so you can only use it when your spatial context is empty *)
Lemma bin_log_FG_increment_logatomic R P Γ E K x t τ :
P -∗
□ (|={⊤,E}=> ∃ n : nat, x ↦ᵢ #n ∗ R n ∗
......
theories/examples/generative.v
View file @ 3447371d
......@@ -7,6 +7,9 @@ From iris_logrel Require Import logrel examples.counter examples.lock prelude.bi
(** * 5.2 References for name generation *)
(* ∃ α. (1 → α) × (α → α → 2) *)
(* ^ new name ^ *)
(* | compare names for equality *)
Definition nameGenTy : type := TExists (TProd (TArrow TUnit (TVar 0))
(TArrow (TVar 0) (TArrow (TVar 0) TBool))).
......
theories/examples/or.v
View file @ 3447371d
......@@ -99,7 +99,7 @@ Section contents.
Definition or_inv x : iProp Σ :=
(x ↦ᵢ #0 ∨ x ↦ᵢ #1)%I.
Definition orN := nroot .@ "orN".
Ltac close_shoot := iNext; (iLeft + iRight); by iFrame.
Ltac close_inv := iNext; (iLeft + iRight); by iFrame.
Lemma assign_safe x :
inv orN (or_inv x)
......@@ -107,7 +107,7 @@ Section contents.
Proof.
iIntros "#Hinv".
iNext. iInv orN as ">[Hx | Hx]" "Hcl"; wp_store;
(iMod ("Hcl" with "[-]"); first close_shoot); eauto.
(iMod ("Hcl" with "[-]"); first close_inv); eauto.
Qed.
Lemma bin_log_or_commute Δ Γ (v1 v1' v2 v2' : val) :
......@@ -116,13 +116,13 @@ Section contents.
{Δ;Γ} ⊨ or v2 v1 ≤log≤ or v1' v2' : TUnit.
Proof.
iIntros "Hv1 Hv2".
unlock or. repeat rel_rec_r. repeat rel_rec_l.
unlock or; simpl. repeat rel_rec_r. repeat rel_rec_l.
rel_alloc_l as x "Hx".
rel_alloc_r as y "Hy".
repeat rel_let_l. repeat rel_let_r.
rel_fork_r as j "Hj". rel_seq_r.
iMod (inv_alloc orN _ (or_inv x) with "[Hx]") as "#Hinv".
{ close_shoot. }
iMod (inv_alloc orN _ (or_inv x) with "[Hx]") as "#Hinv";
first close_inv.
rel_fork_l.
iModIntro. iSplitR; [ by iApply assign_safe | ].
rel_seq_l.
......@@ -130,18 +130,17 @@ Section contents.
iInv orN as ">[Hx|Hx]" "Hcl";
iExists _; iFrame; iModIntro; iNext; iIntros "Hx";
rel_op_l; rel_if_l.
+ apply bin_log_related_spec_ctx.
iDestruct 1 as (ρ1) "#Hρ1".
(* TODO: tp tactics should be aware of that ^ *)
+ apply bin_log_related_spec_ctx; iDestruct 1 as (ρ1) "#Hρ1".
(* TODO: tp tactics should be aware of spec_ctx *)
tp_store j.
rel_load_r.
repeat (rel_pure_r _).
iMod ("Hcl" with "[-Hv1 Hv2]"); first close_shoot.
iMod ("Hcl" with "[-Hv1 Hv2]"); first close_inv.
iApply (bin_log_related_app with "Hv2").
iApply bin_log_related_unit.
+ rel_load_r.
repeat (rel_pure_r _).
iMod ("Hcl" with "[-Hv1 Hv2]"); first close_shoot.
iMod ("Hcl" with "[-Hv1 Hv2]"); first close_inv.
iApply (bin_log_related_app with "Hv1").
iApply bin_log_related_unit.
Qed.
......@@ -173,7 +172,7 @@ Section contents.
rel_alloc_l as x "Hx".
repeat rel_let_l.
iMod (inv_alloc orN _ (or_inv x)%I with "[Hx]") as "#Hinv".
{ close_shoot. }
{ close_inv. }
rel_fork_l.
iModIntro. iSplitR; [ by iApply assign_safe | ].
rel_seq_l.
......@@ -181,10 +180,10 @@ Section contents.
iInv orN as ">[Hx|Hx]" "Hcl";
iExists _; iFrame; iModIntro; iNext; iIntros "Hx";
rel_op_l; rel_if_l.
+ iMod ("Hcl" with "[-Hlog]"); first close_shoot.
+ iMod ("Hcl" with "[-Hlog]"); first close_inv.
iApply (bin_log_related_app with "Hlog").
iApply bin_log_related_unit.
+ iMod ("Hcl" with "[-Hlog]"); first close_shoot.
+ iMod ("Hcl" with "[-Hlog]"); first close_inv.
iApply (bin_log_related_app with "Hlog").
iApply bin_log_related_unit.
Qed.
......@@ -198,7 +197,7 @@ Section contents.
rel_alloc_l as x "Hx".
repeat rel_let_l.
iMod (inv_alloc orN _ (or_inv x)%I with "[Hx]") as "#Hinv".
{ close_shoot. }
{ close_inv. }
rel_fork_l.
iModIntro. iSplitR; [ by iApply assign_safe | ].
rel_seq_l.
......@@ -206,10 +205,10 @@ Section contents.
iInv orN as ">[Hx|Hx]" "Hcl";
iExists _; iFrame; iModIntro; iNext; iIntros "Hx";
rel_op_l; rel_if_l.
+ iMod ("Hcl" with "[-Hlog]"); first close_shoot.
+ iMod ("Hcl" with "[-Hlog]"); first close_inv.
iApply (bin_log_related_app with "Hlog").
iApply bin_log_related_unit.
+ iMod ("Hcl" with "[-Hlog]"); first close_shoot.
+ iMod ("Hcl" with "[-Hlog]"); first close_inv.
rel_apply_l bot_l.
Qed.
......@@ -236,7 +235,7 @@ Section contents.
repeat rel_let_l. repeat rel_let_r.
rel_fork_r as j "Hj". rel_seq_r.
iMod (inv_alloc orN _ (or_inv x) with "[Hx]") as "#Hinv".
{ close_shoot. }
{ close_inv. }
rel_fork_l.
iModIntro. iSplitR; [ by iApply assign_safe | ].
rel_seq_l.
......@@ -250,14 +249,14 @@ Section contents.
rel_let_r. rel_fork_r as k "Hk".
rel_seq_r. rel_load_r.
repeat (rel_pure_r _).
iMod ("Hcl" with "[-Hv1 Hv2 Hv3]") as "_"; first close_shoot.
iMod ("Hcl" with "[-Hv1 Hv2 Hv3]") as "_"; first close_inv.
iApply (bin_log_related_app with "Hv1").
iApply bin_log_related_unit.
- iMod ("Hcl" with "[-Hv1 Hv2 Hv3 Hy Hj]") as "_"; first close_shoot.
- iMod ("Hcl" with "[-Hv1 Hv2 Hv3 Hy Hj]") as "_"; first close_inv.
rel_alloc_l as x' "Hx'". rel_let_l.
iClear "Hinv".
iMod (inv_alloc orN _ (or_inv x') with "[Hx']") as "#Hinv".
{ close_shoot. }
{ close_inv. }
rel_fork_l.
iModIntro. iSplitR; [ by iApply assign_safe | ].
apply bin_log_related_spec_ctx.
......@@ -274,12 +273,12 @@ Section contents.
tp_store k.
rel_load_r.
repeat (rel_pure_r _).
iMod ("Hcl" with "[-Hv1 Hv2 Hv3]") as "_"; first close_shoot.
iMod ("Hcl" with "[-Hv1 Hv2 Hv3]") as "_"; first close_inv.
iApply (bin_log_related_app with "Hv2").
iApply bin_log_related_unit.
+ tp_store j.
rel_load_r; repeat (rel_pure_r _).
iMod ("Hcl" with "[-Hv1 Hv2 Hv3]") as "_"; first close_shoot.
iMod ("Hcl" with "[-Hv1 Hv2 Hv3]") as "_"; first close_inv.
iApply (bin_log_related_app with "Hv3").
iApply bin_log_related_unit.
Qed.
......
theories/examples/symbol.v
View file @ 3447371d
......@@ -18,55 +18,7 @@ Notation NileV := (FoldV NONEV).
Definition LIST τ :=
TRec (TSum TUnit (TProd τ.[ren (+1)] (TVar 0))).
Lemma LIST_interp_nil τ `{logrelG Σ} :
⟦ LIST τ ⟧ [] (NileV, NileV).
Proof.
rewrite /= {1}fixpoint_unfold /=.
iExists (_,_).
iSplit; eauto.
iNext. iLeft. iExists (_,_). eauto.
Qed.
Lemma LIST_interp_cons (n : nat) ls1 ls2 `{logrelG Σ} :
⟦ LIST TNat ⟧ [] (ls1,ls2) -∗
⟦ LIST TNat ⟧ [] (ConseV #n ls1,ConseV #n ls2).
Proof.
iIntros "#Hls".
rewrite /= {2}fixpoint_unfold /=.
iExists (_,_).
iSplit; eauto.
iNext. iRight. iExists (_,_).
iSplit; eauto.
iExists (_,_), (_,_). iSplit; eauto.
Qed.
Lemma LIST_interp_weaken `{logrelG Σ} Δ ls1 ls2 :
⟦ LIST TNat ⟧ [] (ls1, ls2) -∗
⟦ LIST TNat ⟧ Δ (ls1, ls2).
Proof.
iRevert (ls1 ls2).
iLöb as "IH".
iIntros (ls1 ls2) "#Hls /=".
rewrite {2}(fixpoint_unfold (interp_rec1
(interp_sum interp_unit (interp_prod interp_nat (ctx_lookup 0))) Δ)) /=.
rewrite {2}(fixpoint_unfold (interp_rec1
(interp_sum interp_unit
(interp_prod interp_nat (ctx_lookup 0))) [])) /=.
iDestruct "Hls" as ([? ?]) "[% #Hls]".
iExists (_,_).
iSplit; eauto.
iNext.
iDestruct "Hls" as "[Hls | Hls]".
- iLeft; eauto.
- iRight.
iDestruct "Hls" as ([? ?]) "[% Hls]".
iExists _. iSplit; eauto.
iDestruct "Hls" as ([? ?] [? ?]) "[% [Hn #Hls]]".
iExists _,_. iSplit; eauto.
iSplit; eauto.
iAlways.
by iApply "IH".
Qed.
(* `nth ls n` diverges if `n >= length ls` *)
(* `nth ls n` loops if `n >= length ls` *)
Definition nth : val := rec: "nth" "l" "n" :=
match: Unfold "l" with
NONE => bot #()
......@@ -128,46 +80,54 @@ Lemma eqKey_typed Γ :
typed Γ eqKey (TArrow TNat (TArrow TNat TBool)).
Proof. solve_typed. Qed.
Hint Resolve eqKey_typed : typeable.
Lemma symbol1_typed Γ :
typed Γ symbol1 (TArrow TUnit SYMBOL).
Lemma LIST_interp_nil τ `{logrelG Σ} :
⟦ LIST τ ⟧ [] (NileV, NileV).
Proof.
unfold SYMBOL.
solve_typed.
econstructor. cbn.
econstructor; cbn. 2: solve_typed.
econstructor; cbn.
econstructor; cbn.
econstructor; cbn.
econstructor; cbn. 2: solve_typed.
econstructor; cbn.
econstructor; cbn.
econstructor; cbn.
econstructor; cbn.
apply eqKey_typed.
econstructor; cbn. 2: solve_typed.
all: solve_typed.
rewrite /= {1}fixpoint_unfold /=.
iExists (_,_).
iSplit; eauto.
iNext. iLeft. iExists (_,_). eauto.
Qed.
Hint Resolve symbol1_typed : typeable.
Lemma symbol2_typed Γ :
typed Γ symbol2 (TArrow TUnit SYMBOL).
Lemma LIST_interp_cons (n : nat) ls1 ls2 `{logrelG Σ} :
⟦ LIST TNat ⟧ [] (ls1,ls2) -∗
⟦ LIST TNat ⟧ [] (ConseV #n ls1,ConseV #n ls2).
Proof.
unfold SYMBOL.
solve_typed.
econstructor. cbn.
econstructor; cbn. 2: solve_typed.
econstructor; cbn.
econstructor; cbn.
econstructor; cbn.
econstructor; cbn. 2: solve_typed.
econstructor; cbn.
econstructor; cbn.
econstructor; cbn.
econstructor; cbn.
apply eqKey_typed.
econstructor; cbn. 2: solve_typed.
all: solve_typed.
iIntros "#Hls".
rewrite /= {2}fixpoint_unfold /=.
iExists (_,_).
iSplit; eauto.
iNext. iRight. iExists (_,_).
iSplit; eauto.
iExists (_,_), (_,_). iSplit; eauto.
Qed.
Lemma LIST_interp_weaken `{logrelG Σ} Δ ls1 ls2 :
⟦ LIST TNat ⟧ [] (ls1, ls2) -∗
⟦ LIST TNat ⟧ Δ (ls1, ls2).
Proof.
iRevert (ls1 ls2).
iLöb as "IH".
iIntros (ls1 ls2) "#Hls /=".
rewrite {2}(fixpoint_unfold (interp_rec1
(interp_sum interp_unit (interp_prod interp_nat (ctx_lookup 0))) Δ)) /=.
rewrite {2}(fixpoint_unfold (interp_rec1
(interp_sum interp_unit
(interp_prod interp_nat (ctx_lookup 0))) [])) /=.
iDestruct "Hls" as ([? ?]) "[% #Hls]".
iExists (_,_).
iSplit; eauto.
iNext.
iDestruct "Hls" as "[Hls | Hls]".
- iLeft; eauto.
- iRight.
iDestruct "Hls" as ([? ?]) "[% Hls]".
iExists _. iSplit; eauto.
iDestruct "Hls" as ([? ?] [? ?]) "[% [Hn #Hls]]".
iExists _,_. iSplit; eauto.
iSplit; eauto.
iAlways.
by iApply "IH".
Qed.
Hint Resolve symbol2_typed : typeable.
(** * The actual refinement proof.
Requires monotonic state *)
......@@ -264,8 +224,94 @@ Section proof.
iApply bin_log_related_nat.
Qed.
Lemma lookup_refinement1 (γ : gname) (size1 size2 tbl1 tbl2 : loc) Δ Γ :
inv sizeN (table_inv γ size1 size2 tbl1 tbl2) -∗
{(tableR γ :: Δ); ⤉Γ} ⊨
(λ: "n", (nth ! #tbl1) (! #size1 - "n"))
≤log≤
(λ: "n",
let: "m" := ! #size2 in
if: "n" ≤ "m" then (nth ! #tbl2) (! #size2 - "n") else #0) :
(TVar 0 → TNat).
Proof.
iIntros "#Hinv".
iApply bin_log_related_arrow_val; eauto.
iAlways. iIntros (k1 k2) "/= #Hk".
iDestruct "Hk" as (n) "(% & % & #Hn)"; simplify_eq.
rel_let_l. rel_let_r.
rel_load_l_atomic.
iInv sizeN as (m ls) "(Ha & Hs1' & >Hs2' & >Htbl1' & >Htbl2' & #Hls)" "Hcl".
iModIntro. iExists _. iFrame. iNext. iIntros "Htbl1'".
Local Opaque interp.
rel_load_r. rel_let_r.
iDestruct (own_valid_2 with "Ha Hn")
as %[?%mnat_included _]%auth_valid_discrete_2.
rel_op_r. destruct (le_dec n m); last by exfalso.
rel_if_r. rel_load_r.
iMod ("Hcl" with "[Ha Hs1' Hs2' Htbl1' Htbl2' Hls]") as "_".
{ iNext. iExists _,_. by iFrame. }
repeat iApply bin_log_related_app.
+ iApply binary_fundamental; eauto with typeable.
+ rel_vals. by iApply LIST_interp_weaken.
+ iClear "Hls". clear ls.
rel_load_l_atomic.
iInv sizeN as (m' ls) "(Ha & Hs1' & Hs2' & Htbl1' & Htbl2' & #Hls)" "Hcl".
iModIntro. iExists _. iFrame. iNext. iIntros "Hs1'".
rel_load_r.
iMod ("Hcl" with "[Ha Hs1' Hs2' Htbl1' Htbl2']") as "_".
{ iNext. iExists _,_. by iFrame. }
iApply bin_log_related_nat_binop; eauto.
all: iApply bin_log_related_nat.
Qed.
Lemma lookup_refinement2 (γ : gname) (size1 size2 tbl1 tbl2 : loc) Δ Γ :
inv sizeN (table_inv γ size1 size2 tbl1 tbl2) -∗
{(tableR γ :: Δ); ⤉Γ} ⊨
(λ: "n",
let: "m" := ! #size1 in
if: "n" ≤ "m" then (nth ! #tbl1) (! #size1 - "n") else #0)
≤log≤
(λ: "n", (nth ! #tbl2) (! #size2 - "n")) : (TVar 0 → TNat).
Proof.
iIntros "#Hinv".
iApply bin_log_related_arrow_val; eauto.
iAlways. iIntros (k1 k2) "#Hk /=".
iDestruct "Hk" as (n) "(% & % & #Hn)"; simplify_eq.
rel_let_l. rel_let_r.
rel_load_l_atomic.
iInv sizeN as (m ls) "(Ha & Hs1' & >Hs2' & >Htbl1' & >Htbl2' & Hls)" "Hcl".
iModIntro. iExists _. iFrame. iNext. iIntros "Hs1'".
iDestruct (own_valid_2 with "Ha Hn")
as %[?%mnat_included _]%auth_valid_discrete_2.
iMod ("Hcl" with "[Ha Hs1' Hs2' Htbl1' Htbl2' Hls]") as "_".
{ iNext. iExists _,_. by iFrame. }
clear ls.
rel_let_l. rel_op_l.
destruct (le_dec n m); last by congruence.
rel_if_l.
repeat iApply bin_log_related_app.
+ iApply binary_fundamental; eauto with typeable.
+ rel_load_l_atomic.
iInv sizeN as (m' ls) "(Ha & Hs1' & Hs2' & Htbl1' & Htbl2' & #Hls)" "Hcl".
iModIntro. iExists _. iFrame. iNext. iIntros "Htbl1'".
rel_load_r.
iMod ("Hcl" with "[Ha Hs1' Hs2' Htbl1' Htbl2']") as "_".
{ iNext. iExists _,_. by iFrame. }
rel_vals; eauto.
iModIntro. by iApply LIST_interp_weaken.
+ iApply bin_log_related_nat_binop; eauto;
last by iApply bin_log_related_nat.
rel_load_l_atomic.
iInv sizeN as (m' ls) "(Ha & Hs1' & Hs2' & Htbl1' & Htbl2' & Hls)" "Hcl".
iModIntro. iExists _. iFrame. iNext. iIntros "Hs1'".
rel_load_r.
iMod ("Hcl" with "[Ha Hs1' Hs2' Htbl1' Htbl2' Hls]") as "_".
{ iNext. iExists _,_. by iFrame. }
iApply bin_log_related_nat.
Qed.
Lemma refinement1 Δ Γ :
{Δ;Γ} ⊨ symbol2 ≤log≤ symbol1 : TArrow TUnit SYMBOL.
{Δ;Γ} ⊨ symbol1 ≤log≤ symbol2 : TArrow TUnit SYMBOL.
Proof.
unlock symbol1 symbol2.
iApply bin_log_related_arrow_val; eauto.
......@@ -345,41 +391,90 @@ Section proof.
rel_vals. iModIntro. iAlways.
iExists m'. eauto.
(* Lookup *)
- by iApply lookup_refinement1.
Qed.
Lemma refinement2 Δ Γ :
{Δ;Γ} ⊨ symbol2 ≤log≤ symbol1 : TArrow TUnit SYMBOL.
Proof.
unlock symbol1 symbol2.
iApply bin_log_related_arrow_val; eauto.
iAlways. iIntros (? ?) "[% %]"; simplify_eq/=.
rel_let_l. rel_let_r.
rel_alloc_l as size1 "[Hs1 Hs1']"; rel_let_l.
rel_alloc_r as size2 "[Hs2 Hs2']"; rel_let_r.
rel_alloc_l as tbl1 "[Htbl1 Htbl1']"; rel_let_l.
rel_alloc_r as tbl2 "[Htbl2 Htbl2']"; rel_let_r.
iMod (own_alloc (● (0 : mnat))) as (γ) "Ha"; first done.
iMod (inv_alloc sizeN _ (table_inv γ size1 size2 tbl1 tbl2) with "[Hs1 Hs2 Htbl1 Htbl2 Ha]") as "#Hinv".
{ iNext. iExists _,_. iFrame. iApply LIST_interp_nil. }
rel_apply_r bin_log_related_newlock_r; first done.
iIntros (l2) "Hl2". rel_let_r.
pose (N:=(logrelN.@"lock1")).
rel_apply_l (bin_log_related_newlock_l N (lok_inv size1 size2 tbl1 tbl2 l2)%I with "[Hs1' Hs2' Htbl1' Htbl2' Hl2]").
{ iExists 0,_. by iFrame. }
iIntros (l1 γl) "#Hl1". rel_let_l.
iApply (bin_log_related_pack (tableR γ)).
repeat iApply bin_log_related_pair.
(* Eq *)
- iApply eqKey_refinement.
(* Insert *)
- iApply bin_log_related_arrow_val; eauto.
iAlways. iIntros (k1 k2) "/= #Hk".
iDestruct "Hk" as (n) "(% & % & #Hn)"; simplify_eq.
iAlways. iIntros (? ?). iDestruct 1 as (n) "[% %]"; simplify_eq/=.
rel_let_l. rel_let_r.
rel_apply_l (bin_log_related_acquire_l with "Hl1"); auto.
iIntros "Hlk Htbl".
rel_let_l.
iDestruct "Htbl" as (m ls) "(Hs1 & Hs2 & Htbl1 & Htbl2 & Hl2)".
rel_load_l.
rel_let_l.
rel_op_l.
rel_store_l_atomic.
iInv sizeN as (m' ls') "(Ha & >Hs1' & >Hs2' & >Htbl1' & >Htbl2' & #Hls)" "Hcl".
iDestruct (mapsto_half_combine with "Hs1 Hs1'") as "[% Hs1]"; simplify_eq.
iDestruct (mapsto_half_combine with "Htbl1 Htbl1'") as "[% [Htbl1 Htbl1']]"; simplify_eq.
iModIntro. iExists _. iFrame. iNext. iIntros "[Hs1 Hs1']".
rel_let_l.
rel_apply_r (bin_log_related_acquire_r with "Hl2"); first solve_ndisj.
iIntros "Hl2".
rel_let_r.
rel_load_l_atomic.
iInv sizeN as (m ls) "(Ha & Hs1' & >Hs2' & >Htbl1' & >Htbl2' & Hls)" "Hcl".
iModIntro. iExists _. iFrame. iNext. iIntros "Hs1'".
iDestruct (own_valid_2 with "Ha Hn")
as %[?%mnat_included _]%auth_valid_discrete_2.
iMod ("Hcl" with "[Ha Hs1' Hs2' Htbl1' Htbl2' Hls]") as "_".
iDestruct (threadpool.mapsto_half_combine with "Hs2 Hs2'") as "[% Hs2]"; simplify_eq.
rel_load_r.
repeat (rel_pure_r _).
rel_store_r.
rel_let_r.
(* Before we close the lock, we need to gather some evidence *)
iMod (conjure_0 γ) as "Hf".
iMod (same_size with "[$Ha $Hf]") as "[Ha #Hf]".
iMod (inc_size with "Ha") as "Ha".
iDestruct "Hs2" as "[Hs2 Hs2']".
rewrite Nat.add_1_r.
iMod ("Hcl" with "[Ha Hs1 Hs2 Htbl1 Htbl2]") as "_".
{ iNext. iExists _,_. by iFrame. }
clear ls.
rel_load_l.
rel_store_l_atomic.
iClear "Hls".
iInv sizeN as (m ls) "(Ha & >Hs1 & >Hs2 & >Htbl1 & >Htbl2 & #Hls)" "Hcl".
iDestruct (mapsto_half_combine with "Htbl1 Htbl1'") as "[% Htbl1]"; simplify_eq.
iModIntro. iExists _. iFrame. iNext. iIntros "[Htbl1 Htbl1']".
rel_let_l.
rel_op_l.
destruct (le_dec n m); last congruence.
rel_if_l.
repeat iApply bin_log_related_app.
+ iApply binary_fundamental; eauto with typeable.
+ rel_load_l_atomic.
iInv sizeN as (m' ls) "(Ha & Hs1' & Hs2' & Htbl1' & Htbl2' & #Hls)" "Hcl".
iModIntro. iExists _. iFrame. iNext. iIntros "Htbl1'".
rel_load_r.
iMod ("Hcl" with "[Ha Hs1' Hs2' Htbl1' Htbl2']") as "_".
{ iNext. iExists _,_. by iFrame. }
rel_vals; eauto.
iModIntro. by iApply LIST_interp_weaken.
+ iApply bin_log_related_nat_binop; eauto;
last by iApply bin_log_related_nat.
rel_load_l_atomic.
iInv sizeN as (m' ls) "(Ha & Hs1' & Hs2' & Htbl1' & Htbl2' & Hls)" "Hcl".
iModIntro. iExists _. iFrame. iNext. iIntros "Hs1'".
rel_load_r.
iMod ("Hcl" with "[Ha Hs1' Hs2' Htbl1' Htbl2' Hls]") as "_".
{ iNext. iExists _,_. by iFrame. }
iApply bin_log_related_nat.
rel_load_r.
iDestruct (threadpool.mapsto_half_combine with "Htbl2 Htbl2'") as "[% Htbl2]"; simplify_eq.
rel_store_r.
rel_let_r.
rel_apply_r (bin_log_related_release_r with "Hl2"); first solve_ndisj.
iIntros "Hl2".
rel_let_r.
iDestruct "Htbl2" as "[Htbl2 Htbl2']".
iMod ("Hcl" with "[Ha Hs1 Hs2 Htbl1 Htbl2]") as "_".
{ iNext. iExists _,_. iFrame.
by iApply LIST_interp_cons. }
rel_apply_l (bin_log_related_release_l with "Hl1 Hlk [-]"); auto.
{ iExists _,_. by iFrame. }
rel_let_l.
rel_vals. iModIntro. iAlways.
iExists m'. eauto.
(* Lookup *)
- by iApply lookup_refinement2.
Qed.
End proof.
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