Library iris.heap_lang.adequacy

From iris.algebra Require Import auth.
From iris.base_logic.lib Require Import mono_nat.
From iris.proofmode Require Import proofmode.
From iris.program_logic Require Export weakestpre adequacy.
From iris.heap_lang Require Import proofmode notation.
From iris.prelude Require Import options.

Class heapGpreS Σ := HeapGpreS {
  #[global] heapGpreS_iris :: invGpreS Σ;
  #[global] heapGpreS_heap :: gen_heapGpreS loc (option val) Σ;
  #[global] heapGpreS_inv_heap :: inv_heapGpreS loc (option val) Σ;
  #[global] heapGpreS_proph :: proph_mapGpreS proph_id (val × val) Σ;
  #[global] heapGS_step_cnt :: mono_natG Σ;

Definition heapΣ : gFunctors :=
  #[invΣ; gen_heapΣ loc (option val); inv_heapΣ loc (option val);
    proph_mapΣ proph_id (val × val); mono_natΣ].
Global Instance subG_heapGpreS {Σ} : subG heapΣ Σ heapGpreS Σ.
Proof. solve_inG. Qed.

Definition heap_adequacy Σ `{!heapGpreS Σ} s e σ φ :
  ( `{!heapGS Σ}, inv_heap_inv -∗ WP e @ s; {{ v, φ v }})
  adequate s e σ (λ v _, φ v).
  intros Hwp.
  apply adequate_alt; intros t2 σ2 [n [κs ?]]%erased_steps_nsteps.
  eapply (wp_strong_adequacy Σ _); [|done].
  iIntros (Hinv).
  iMod (gen_heap_init σ.(heap)) as (?) "[Hh _]".
  iMod (inv_heap_init loc (option val)) as (?) ">Hi".
  iMod (proph_map_init κs σ.(used_proph_id)) as (?) "Hp".
  iMod (mono_nat_own_alloc) as (γ) "[Hsteps _]".
  iDestruct (Hwp (HeapGS _ _ _ _ _ _ _ _) with "Hi") as "Hwp".
  iExists (λ σ ns κs nt, (gen_heap_interp σ.(heap)
                          proph_map_interp κs σ.(used_proph_id)
                          mono_nat_auth_own γ 1 ns))%I.
  iExists [(λ v, φ v%I)], (λ _, True)%I, _ ⇒ /=.
  iIntros (es' t2' → ? ?) " _ H _".
  iApply fupd_mask_intro_discard; [done|]. iSplit; [|done].
  iDestruct (big_sepL2_cons_inv_r with "H") as (e' ? ->) "[Hwp H]".
  iDestruct (big_sepL2_nil_inv_r with "H") as %->.
  iIntros (v2 t2'' [= → <-]). by rewrite to_of_val.