Library iris.heap_lang.adequacy

From iris.proofmode Require Import tactics.
From iris.algebra Require Import auth.
From iris.program_logic Require Export weakestpre adequacy.
From iris.heap_lang Require Import proofmode notation.
Set Default Proof Using "Type".

Class heapPreG Σ := HeapPreG {
  heap_preG_iris :> invPreG Σ;
  heap_preG_heap :> gen_heapPreG loc (option val) Σ;
  heap_preG_inv_heap :> inv_heapPreG loc (option val) Σ;
  heap_preG_proph :> proph_mapPreG proph_id (val × val) Σ;
}.

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

Definition heap_adequacy Σ `{!heapPreG Σ} s e σ φ :
  ( `{!heapG Σ}, inv_heap_inv -∗ WP e @ s; {{ v, φ v }})
  adequate s e σ (λ v _, φ v).
Proof.
  intros Hwp; eapply (wp_adequacy _ _); iIntros (??) "".
  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".
  iModIntro. iExists
    (λ σ κs, (gen_heap_ctx σ.(heap) proph_map_ctx κs σ.(used_proph_id))%I),
    (λ _, True%I).
  iFrame. iApply (Hwp (HeapG _ _ _ _ _)). done.
Qed.