lifting.v

From iris.proofmode Require Import tactics.
From iris.program_logic Require Export weakestpre.
From iris.program_logic Require Import ectx_lifting.
From refinedc.lang Require Export lang ghost_state notation.
From refinedc.lang Require Import tactics.
Set Default Proof Using "Type".
Import uPred.

Class refinedcG Σ := RefinedCG {
  refinedcG_invG : invGS Σ;
  refinedcG_gen_heapG :> heapG Σ
}.

Instance c_irisG `{!refinedcG Σ} : irisGS c_lang Σ := {
  iris_invG := refinedcG_invG;
  state_interp σ κs _ _ := state_ctx σ;
  fork_post _ := True%I;
  num_laters_per_step _ := 0%nat;
  state_interp_mono _ _ _ _ := fupd_intro _ _;
}.
Global Opaque iris_invG.

Instance into_val_val v : IntoVal (to_rtexpr (Val v)) v.
Proof. done. Qed.

Global Instance wp_expr_wp `{!refinedcG Σ} : Wp (iProp Σ) expr val stuckness :=
  λ s E e Φ, (WP (coerce_rtexpr e) @ s; E {{ Φ }})%I.

Local Hint Extern 0 (reducible _ _) => eexists _, _, _, _; simpl : core.
Local Hint Extern 0 (head_reducible _ _) => eexists _, _, _, _; simpl : core.
Local Hint Unfold heap_at : core.


Local Ltac solve_atomic :=
  apply strongly_atomic_atomic, ectx_language_atomic;
    [inversion 1; unfold coerce_rtexpr in *; simplify_eq; inv_expr_step; naive_solver
    |unfold coerce_rtexpr in *;apply ectxi_language_sub_redexes_are_values; intros [[]|[]] **; naive_solver].

Instance cas_atomic s ot (v1 v2 v3 : val) : Atomic s (coerce_rtexpr (CAS ot v1 v2 v3)).
Proof. solve_atomic. Qed.
Instance skipe_atomic s (v : val) : Atomic s (coerce_rtexpr (SkipE v)).
Proof. solve_atomic. Qed.
Instance deref_atomic s (l : loc) ly : Atomic s (coerce_rtexpr (Deref ScOrd ly l)).
Proof. solve_atomic. Qed.
Instance use_atomic s (l : loc) ly : Atomic s (coerce_rtexpr (Use ScOrd ly l)).
Proof. solve_atomic. Qed.

(*** General lifting lemmas *)

Section lifting.
  Context `{!refinedcG Σ}.
  Lemma wp_alloc_failed E Φ:
    ⊢ WP AllocFailed @ E {{ Φ }}.
  Proof.
    iLöb as "IH".
    iApply wp_lift_pure_det_head_step_no_fork' => //; first by eauto using AllocFailedStep.
    move => ????? . inversion 1; simplify_eq => //.
    match goal with | H: to_rtexpr ?e = AllocFailed |- _ => destruct e; discriminate end.
  Qed.

  Lemma wp_c_lift_step_fupd E e step_rel Φ:
    ((∃ e', e = to_rtexpr e' ∧ step_rel = expr_step e') ∨
     (∃ rf s, e = to_rtstmt rf s ∧ step_rel = stmt_step s rf)) →
    to_val e = None →
    (∀ (σ1 : state), state_ctx σ1 ={E,∅}=∗
       ⌜∃ os e2 σ2 tsl, step_rel σ1 os e2 σ2 tsl⌝ ∗
        (∀ os e2 σ2 tsl, ⌜step_rel σ1 os e2 σ2 tsl⌝ -∗ ⌜heap_state_invariant σ2.(st_heap)⌝
          ={∅}=∗ ▷ (|={∅,E}=> ⌜tsl = []⌝ ∗ state_ctx σ2 ∗ WP e2 @ E {{ Φ }})))
      -∗ WP e @ E {{ Φ }}.
  Proof.
    iIntros (He ?) "HWP".
    iApply wp_lift_head_step_fupd => //.
    iIntros (σ1 κ κs n ?) "[[% Hhctx] Hfnctx]".
    iMod ("HWP" $! σ1 with "[$Hhctx $Hfnctx//]") as (Hstep) "HWP".
    iModIntro. iSplit. {
      iPureIntro. destruct Hstep as (?&?&?&?&?).
      naive_solver eauto using ExprStep, StmtStep.
    }
    clear Hstep. iIntros (??? Hstep).
    move: (Hstep) => /runtime_step_preserves_invariant?.
    destruct He as [[e' [??]]|[? [s [??]]]]; inversion Hstep; simplify_eq.
    all: try by [destruct e'; discriminate].
    all: try match goal with | H : to_rtexpr ?e = to_rtstmt _ _ |- _ => destruct e; discriminate end.
    all: iMod ("HWP" with "[//] [%]") as "HWP"; first naive_solver.
    all: do 2 iModIntro.
    all: iMod "HWP" as (->) "[$ HWP]"; iModIntro => /=; by rewrite right_id.
  Qed.

  Lemma wp_lift_expr_step_fupd E (e : expr) Φ:
    to_val e = None →
    (∀ (σ1 : state), state_ctx σ1 ={E,∅}=∗
       ⌜∃ os e2 σ2 tsl, expr_step e σ1 os e2 σ2 tsl⌝ ∗
        (∀ os e2 σ2 tsl, ⌜expr_step e σ1 os e2 σ2 tsl⌝ -∗ ⌜heap_state_invariant σ2.(st_heap)⌝
          ={∅}=∗ ▷ (|={∅,E}=> ⌜tsl = []⌝ ∗ state_ctx σ2 ∗ WP e2 @ E {{ Φ }})))
      -∗ WP e @ E {{ Φ }}.
  Proof. iIntros (?) "HWP". iApply wp_c_lift_step_fupd => //. naive_solver. Qed.

  Lemma wp_lift_stmt_step_fupd E s rf Φ:
    (∀ (σ1 : state), state_ctx σ1 ={E,∅}=∗
       ⌜∃ os e2 σ2 tsl, stmt_step s rf σ1 os e2 σ2 tsl⌝ ∗
        (∀ os e2 σ2 tsl, ⌜stmt_step s rf σ1 os e2 σ2 tsl⌝ -∗ ⌜heap_state_invariant σ2.(st_heap)⌝
          ={∅}=∗ ▷ (|={∅,E}=> ⌜tsl = []⌝ ∗ state_ctx σ2 ∗ WP e2 @ E {{ Φ }})))
      -∗ WP to_rtstmt rf s @ E {{ Φ }}.
  Proof. iIntros "HWP". iApply wp_c_lift_step_fupd => //. naive_solver. Qed.

  Lemma wp_c_lift_step E e step_rel Φ:
    ((∃ e', e = to_rtexpr e' ∧ step_rel = expr_step e') ∨
     (∃ rf s, e = to_rtstmt rf s ∧ step_rel = stmt_step s rf)) →
    to_val e = None →
    (∀ (σ1 : state), state_ctx σ1 ={E}=∗
       ⌜∃ os e2 σ2 tsl, step_rel σ1 os e2 σ2 tsl⌝ ∗
        ▷ (∀ os e2 σ2 tsl, ⌜step_rel σ1 os e2 σ2 tsl⌝ -∗ ⌜heap_state_invariant σ2.(st_heap)⌝
          ={E}=∗ ⌜tsl = []⌝ ∗ state_ctx σ2 ∗ WP e2 @ E {{ Φ }}))
      -∗ WP e @ E {{ Φ }}.
  Proof.
    iIntros (??) "HWP".
    iApply wp_c_lift_step_fupd => //.
    iIntros (?) "Hσ". iMod ("HWP" with "Hσ") as "[$ HWP]".
    iApply fupd_mask_intro; first set_solver. iIntros "HE".
    iIntros (??????) "!> !>". iMod "HE". by iMod ("HWP" with "[//] [//]") as "$".
  Qed.

  Lemma wp_lift_expr_step E (e : expr) Φ:
    to_val e = None →
    (∀ (σ1 : state), state_ctx σ1 ={E}=∗
       ⌜∃ os e2 σ2 tsl, expr_step e σ1 os e2 σ2 tsl⌝ ∗
        ▷ (∀ os e2 σ2 tsl, ⌜expr_step e σ1 os e2 σ2 tsl⌝ -∗ ⌜heap_state_invariant σ2.(st_heap)⌝
          ={E}=∗ ⌜tsl = []⌝ ∗ state_ctx σ2 ∗ WP e2 @ E {{ Φ }}))
      -∗ WP e @ E {{ Φ }}.
  Proof. iIntros (?) "HWP". iApply wp_c_lift_step => //. naive_solver. Qed.

  Lemma wp_lift_stmt_step E s rf Φ:
    (∀ (σ1 : state), state_ctx σ1 ={E}=∗
       ⌜∃ os e2 σ2 tsl, stmt_step s rf σ1 os e2 σ2 tsl⌝ ∗
        ▷ (∀ os e2 σ2 tsl, ⌜stmt_step s rf σ1 os e2 σ2 tsl⌝ -∗ ⌜heap_state_invariant σ2.(st_heap)⌝
          ={E}=∗ ⌜tsl = []⌝ ∗ state_ctx σ2 ∗ WP e2 @ E {{ Φ }}))
      -∗ WP to_rtstmt rf s @ E {{ Φ }}.
  Proof. iIntros "HWP". iApply wp_c_lift_step => //. naive_solver. Qed.

End lifting.

(*** Lifting of expressions *)
Section expr_lifting.
Context `{!refinedcG Σ}.

Lemma wp_binop v1 v2 Φ op E ot1 ot2:
  (∀ σ, state_ctx σ ={E, ∅}=∗
    ⌜∃ v', eval_bin_op op ot1 ot2 σ v1 v2 v'⌝ ∗
    ▷ (∀ v', ⌜eval_bin_op op ot1 ot2 σ v1 v2 v'⌝ ={∅, E}=∗ state_ctx σ ∗ Φ v')) -∗
  WP BinOp op ot1 ot2 (Val v1) (Val v2) @ E {{ Φ }}.
Proof.
  iIntros "HΦ".
  iApply wp_lift_expr_step_fupd; auto.
  iIntros (σ1) "Hctx".
  iMod ("HΦ" with "Hctx") as ([v' Heval]) "HΦ". iModIntro.
  iSplit; first by eauto 8 using BinOpS.
  iIntros (? e2 σ2 efs Hst ?) "!>!>". inv_expr_step.
  iMod ("HΦ" with "[//]") as "[$ HΦ]".
  iModIntro. iSplit => //. by iApply wp_value.
Qed.

Lemma wp_binop_det v' v1 v2 Φ op E ot1 ot2:
  (∀ σ, state_ctx σ ={E, ∅}=∗ ⌜∀ v, eval_bin_op op ot1 ot2 σ v1 v2 v ↔ v = v'⌝ ∗ (▷ |={∅, E}=> state_ctx σ ∗ Φ v')) -∗
    WP BinOp op ot1 ot2 (Val v1) (Val v2) @ E {{ Φ }}.
Proof.
  iIntros "H".
  iApply wp_binop. iIntros (σ) "Hctx".
  iMod ("H" with "Hctx") as (Hv) "HΦ". iModIntro.
  iSplit.
  { iExists v'. by rewrite Hv. }
  by iIntros "!>" (v ->%Hv).
Qed.

Lemma wp_binop_det_pure v' v1 v2 Φ op E ot1 ot2:
  (∀ σ v, eval_bin_op op ot1 ot2 σ v1 v2 v ↔ v = v') →
  ▷ Φ v' -∗
  WP BinOp op ot1 ot2 (Val v1) (Val v2) @ E {{ Φ }}.
Proof.
  iIntros (Hop) "HΦ". iApply (wp_binop_det v').
  iIntros (σ) "Hσ". iApply fupd_mask_intro; [set_solver|]. iIntros "He".
  iSplit; [done|]. iModIntro. iMod "He". by iFrame.
Qed.

Lemma wp_unop v1 Φ op E ot:
  (∀ σ, state_ctx σ ={E, ∅}=∗
    ⌜∃ v', eval_un_op op ot σ v1 v'⌝ ∗
    ▷ (∀ v', ⌜eval_un_op op ot σ v1 v'⌝ ={∅, E}=∗ state_ctx σ ∗ Φ v')) -∗
   WP UnOp op ot (Val v1) @ E {{ Φ }}.
Proof.
  iIntros "HΦ".
  iApply wp_lift_expr_step_fupd; auto.
  iIntros (σ1) "Hctx".
  iMod ("HΦ" with "Hctx") as ([v' Heval]) "HΦ". iModIntro.
  iSplit; first by eauto 8 using UnOpS.
  iIntros (? e2 σ2 efs Hst ?) "!> !>". inv_expr_step.
  iMod ("HΦ" with "[//]") as "[$ HΦ]".
  iModIntro. iSplit => //. by iApply wp_value.
Qed.

Lemma wp_unop_det v' v1 Φ op E ot:
  (∀ σ, state_ctx σ ={E, ∅}=∗ ⌜∀ v, eval_un_op op ot σ v1 v ↔ v = v'⌝ ∗ (▷ |={∅, E}=> state_ctx σ ∗ Φ v')) -∗
  WP UnOp op ot (Val v1) @ E {{ Φ }}.
Proof.
  iIntros "H".
  iApply wp_unop. iIntros (σ) "Hctx".
  iMod ("H" with "Hctx") as (Hv) "HΦ". iModIntro.
  iSplit.
  { iExists v'. by rewrite Hv. }
  by iIntros "!>" (v ->%Hv).
Qed.

Lemma wp_unop_det_pure v' v1 Φ op E ot:
  (∀ σ v, eval_un_op op ot σ v1 v ↔ v = v') →
  ▷ Φ v' -∗
  WP UnOp op ot (Val v1) @ E {{ Φ }}.
Proof.
  iIntros (Hop) "HΦ". iApply (wp_unop_det v').
  iIntros (σ) "Hσ". iApply fupd_mask_intro; [set_solver|]. iIntros "He".
  iSplit; [done|]. iModIntro. iMod "He". by iFrame.
Qed.

Lemma wp_deref v Φ vl l ot q E o:
  o = ScOrd ∨ o = Na1Ord →
  val_to_loc vl = Some l →
  l `has_layout_loc` ot_layout ot →
  v `has_layout_val` ot_layout ot →
  l↦{q}v -∗ ▷ (∀ st, l ↦{q} v -∗ Φ (mem_cast v ot st)) -∗ WP !{ot, o} (Val vl) @ E {{ Φ }}.
Proof.
  iIntros (Ho Hl Hll Hlv) "Hmt HΦ".
  iApply wp_lift_expr_step; auto.
  iIntros ([[h ub] fn]) "((%&Hhctx&Hactx)&Hfctx)/=".
  iDestruct (heap_mapsto_has_alloc_id with "Hmt") as %Haid.
  destruct o; try by destruct Ho.
  - iModIntro. iDestruct (heap_mapsto_lookup_q (λ st, ∃ n, st = RSt n) with "Hhctx Hmt") as %Hat. naive_solver.
    iSplit; first by eauto 11 using DerefS.
    iIntros (? e2 σ2 efs Hst ?) "!> !>". inv_expr_step.
    iSplit => //. unfold end_st, end_expr.
    have <- : (v = v') by apply: heap_at_inj_val.
    rewrite /heap_fmap/=. erewrite heap_upd_heap_at_id => //.
    iFrame. iSplit; first done. iApply wp_value. by iApply "HΦ".
  - iMod (heap_read_na with "Hhctx Hmt") as "(% & Hσ & Hσclose)" => //. iModIntro.
    iSplit; first by eauto 11 using DerefS.
    iIntros (? e2 σ2 efs Hst ?) "!> !>". inv_expr_step.
    iSplit => //. unfold end_st, end_expr.
    have ? : (v = v') by apply: heap_at_inj_val. subst v'.
    iFrame => /=. iSplit; first done.
    iApply wp_lift_expr_step; auto.
    iIntros ([[h2 ub2] fn2]) "((%&Hhctx&Hactx)&Hfctx)/=".
    iMod ("Hσclose" with "Hhctx") as (?) "(Hσ & Hv)".
    iModIntro; iSplit; first by eauto 11 using DerefS.
    iIntros "!#" (? e2 σ2 efs Hst ?) "!#". inv_expr_step. iSplit => //.
    have ? : (v = v') by apply: (heap_at_inj_val _ _ h2). subst.
    iFrame. iSplit; first done.
    iApply wp_value. by iApply "HΦ".
Qed.

(*
  Alternative version of the CAS rule which does not rely on the Atomic infrastructure:
  Lemma wp_cas vl1 vl2 vd Φ l1 l2 it E:
  val_to_loc vl1 = Some l1 →
  val_to_loc vl2 = Some l2 →
  (bytes_per_int it ≤ bytes_per_addr)%nat →
  (|={E, ∅}=> ∃ (q1 q2 : Qp) vo ve z1 z2,
     ⌜val_to_Z vo it = Some z1⌝ ∗ ⌜val_to_Z ve it = Some z2⌝ ∗
     ⌜l1 `has_layout_loc` it_layout it⌝ ∗ ⌜l2 `has_layout_loc` it_layout it⌝ ∗
     ⌜length vd = bytes_per_int it⌝ ∗ ⌜if bool_decide (z1 = z2) then q1 = 1%Qp else q2 = 1%Qp⌝ ∗
     l1↦{q1}vo ∗ l2↦{q2}ve ∗ ▷ (
       l1↦{q1} (if bool_decide (z1 = z2) then vd else vo) -∗
       l2↦{q2} (if bool_decide (z1 = z2) then ve else vo)
       ={∅, E}=∗ Φ (val_of_bool (bool_decide (z1 = z2))))) -∗
   WP CAS (IntOp it) (Val vl1) (Val vl2) (Val vd) @ E {{ Φ }}.
*)

Lemma wp_cas_fail vl1 vl2 vd vo ve z1 z2 Φ l1 l2 it q E:
  val_to_loc vl1 = Some l1 →
  val_to_loc vl2 = Some l2 →
  l1 `has_layout_loc` it_layout it →
  l2 `has_layout_loc` it_layout it →
  val_to_Z vo it = Some z1 →
  val_to_Z ve it = Some z2 →
  length vd = bytes_per_int it →
  (bytes_per_int it ≤ bytes_per_addr)%nat →
  z1 ≠ z2 →
  l1↦{q}vo -∗ l2↦ve -∗ ▷ (l1 ↦{q} vo -∗ l2↦vo -∗ Φ (val_of_bool false)) -∗
   WP CAS (IntOp it) (Val vl1) (Val vl2) (Val vd) @ E {{ Φ }}.
Proof.
  iIntros (Hl1 Hl2 Hly1 Hly2 Hvo Hve Hlen1 Hlen2 Hneq) "Hl1 Hl2 HΦ".
  iApply wp_lift_expr_step; auto.
  iIntros (σ1) "((%&Hhctx&?)&Hfctx)".
  iDestruct (heap_mapsto_has_alloc_id with "Hl1") as %Haid1.
  iDestruct (heap_mapsto_has_alloc_id with "Hl2") as %Haid2.
  move: (Hvo) (Hve) => /val_to_Z_length ? /val_to_Z_length ?.
  iDestruct (heap_mapsto_lookup_q (λ st : lock_state, ∃ n : nat, st = RSt n) with "Hhctx Hl1") as %? => //. { naive_solver. }
  iDestruct (heap_mapsto_lookup_1 (λ st : lock_state, st = RSt 0%nat) with "Hhctx Hl2") as %? => //.
  iModIntro. iSplit; first by eauto 15 using CasFailS.
  iIntros (? e2 σ2 efs Hst ?) "!>". inv_expr_step;
    have ? : (vo = vo0) by [apply: heap_lookup_loc_inj_val => //; congruence];
    have ? : (ve = ve0) by [apply: heap_lookup_loc_inj_val => //; congruence]; simplify_eq.
  have ? : (length ve0 = length vo0) by congruence.
  iMod (heap_write with "Hhctx Hl2") as "[$ Hl2]" => //. iModIntro.
  iFrame. iSplit => //. iSplit; first done.
  iApply wp_value. by iApply ("HΦ" with "[$]").
Qed.

Lemma wp_cas_suc vl1 vl2 vd vo ve z1 z2 Φ l1 l2 it E q:
  val_to_loc vl1 = Some l1 →
  val_to_loc vl2 = Some l2 →
  l1 `has_layout_loc` it_layout it →
  l2 `has_layout_loc` it_layout it →
  val_to_Z vo it = Some z1 →
  val_to_Z ve it = Some z2 →
  length vd = bytes_per_int it →
  (bytes_per_int it ≤ bytes_per_addr)%nat →
  z1 = z2 →
  l1↦vo -∗ l2↦{q}ve -∗ ▷ (l1 ↦ vd -∗ l2↦{q}ve -∗ Φ (val_of_bool true)) -∗
  WP CAS (IntOp it) (Val vl1) (Val vl2) (Val vd) @ E {{ Φ }}.
Proof.
  iIntros (Hl1 Hl2 Hly1 Hly2 Hvo Hve Hlen1 Hlen2 Hneq) "Hl1 Hl2 HΦ".
  iApply wp_lift_expr_step; auto.
  iIntros (σ1) "((%&Hhctx&?)&Hfctx)".
  iDestruct (heap_mapsto_has_alloc_id with "Hl1") as %Haid1.
  iDestruct (heap_mapsto_has_alloc_id with "Hl2") as %Haid2.
  move: (Hvo) (Hve) => /val_to_Z_length ? /val_to_Z_length ?.
  iDestruct (heap_mapsto_lookup_1 (λ st : lock_state, st = RSt 0%nat) with "Hhctx Hl1") as %? => //.
  iDestruct (heap_mapsto_lookup_q (λ st : lock_state, ∃ n : nat, st = RSt n) with "Hhctx Hl2") as %? => //. { naive_solver. }
  iModIntro. iSplit; first by eauto 15 using CasSucS.
  iIntros (? e2 σ2 efs Hst ?) "!>". inv_expr_step;
      have ? : (ve = ve0) by [apply: heap_lookup_loc_inj_val => //; congruence];
      have ? : (vo = vo0) by [apply: heap_lookup_loc_inj_val => //; congruence]; simplify_eq.
  have ? : (length vo0 = length vd) by congruence.
  iMod (heap_write with "Hhctx Hl1") as "[$ Hmt]" => //. iModIntro.
  iFrame. iSplit => //. iSplit; first done.
  iApply wp_value. by iApply ("HΦ" with "[$]").
Qed.

Lemma wp_neg_int Φ v v' n E it:
  val_to_Z v it = Some n →
  val_of_Z (-n) it None = Some v' →
  ▷ Φ (v') -∗ WP UnOp NegOp (IntOp it) (Val v) @ E {{ Φ }}.
Proof.
  iIntros (Hv Hv') "HΦ".
  iApply (wp_unop_det_pure v'); [|done].
  move => ??. split.
  - by inversion 1; simplify_eq.
  - move => ->. by econstructor.
Qed.

Lemma wp_cast_int Φ v v' i E its itt:
  val_to_Z v its = Some i →
  val_of_Z i itt (val_to_byte_prov v) = Some v' →
  ▷ Φ (v') -∗ WP UnOp (CastOp (IntOp itt)) (IntOp its) (Val v) @ E {{ Φ }}.
Proof.
  iIntros (Hv Hv') "HΦ".
  iApply wp_unop_det_pure; [|done].
  move => ??. split.
  - by inversion 1; simplify_eq.
  - move => ->. by econstructor.
Qed.

Lemma wp_cast_loc Φ v l E:
  val_to_loc v = Some l →
  ▷ Φ (val_of_loc l) -∗ WP UnOp (CastOp PtrOp) PtrOp (Val v) @ E {{ Φ }}.
Proof.
  iIntros (Hv) "HΦ".
  iApply wp_unop_det_pure; [|done].
  move => ??. split.
  - by inversion 1; simplify_eq.
  - move => ->. by econstructor.
Qed.

Lemma wp_cast_ptr_int Φ v v' l it p:
  val_to_loc v = Some l →
  l.1 = ProvAlloc p →
  val_of_Z l.2 it p = Some v' →
  alloc_alive_loc l ∧ ▷ Φ (v') -∗
  WP UnOp (CastOp (IntOp it)) PtrOp (Val v) {{ Φ }}.
Proof.
  iIntros (Hv Hp Hv') "HΦ".
  iApply (wp_unop_det v').
  iIntros (σ) "Hctx".
  destruct (decide (block_alive l (st_heap σ))).
  2: { iDestruct "HΦ" as "[Ha _]". by iMod (alloc_alive_loc_to_block_alive with "Ha Hctx") as %Hb. }
  iApply fupd_mask_intro; [done|]. iIntros "HE". iDestruct "HΦ" as "[_ HΦ]".
  iSplit. {
    iPureIntro. split.
    - have ? := val_to_of_loc NULL_loc. inversion 1; unfold NULL in *; destruct l; by simplify_eq/=.
    - move => ->. by econstructor.
  }
  iModIntro. iMod "HE". by iFrame.
Qed.

Lemma wp_cast_null_int Φ v E it:
  val_of_Z 0 it None = Some v →
  ▷ Φ v -∗
  WP UnOp (CastOp (IntOp it)) PtrOp (Val NULL) @ E {{ Φ }}.
Proof.
  iIntros (Hv) "HΦ".
  iApply wp_unop_det_pure; [|done].
  move => ??. split.
  - inversion 1; simplify_eq => //.
    all: destruct l; simplify_eq/=.
    all: have ? := val_to_of_loc NULL_loc.
    all: unfold NULL in *; by simplify_eq.
  - move => ->. by econstructor.
Qed.

Lemma wp_cast_int_ptr_weak Φ v a E it:
  val_to_Z v it = Some a →
  (∀ i, ▷ Φ (val_of_loc (i, a))) -∗
  WP UnOp (CastOp PtrOp) (IntOp it) (Val v) @ E {{ Φ }}.
Proof.
  iIntros (Hv) "HΦ".
  iApply wp_unop.
  iIntros (σ) "Hctx". iApply fupd_mask_intro; [set_solver|]. iIntros "HE".
  iSplit; [iPureIntro; eexists _; by econstructor |].
  iIntros "!>" (v' Hv'). iMod "HE". iModIntro. iFrame.
  inversion Hv'; simplify_eq.
  case_bool_decide; [ iApply "HΦ"|].
  case_bool_decide; simplify_eq; [ iApply "HΦ"|].
  case_bool_decide; simplify_eq; iApply "HΦ".
Qed.

Lemma wp_cast_int_ptr_alive Φ v a p l it:
  val_to_Z v it = Some a →
  val_to_byte_prov v = Some p →
  l = (ProvAlloc (Some p), a) →
  loc_in_bounds l 0 -∗
  alloc_alive_loc l ∧ ▷ Φ (val_of_loc l) -∗
  WP UnOp (CastOp PtrOp) (IntOp it) (Val v) {{ Φ }}.
Proof.
  iIntros (Hv Hp ->) "#Hlib HΦ".
  iApply wp_unop_det. iIntros (σ) "Hctx".
  destruct (decide (valid_ptr (ProvAlloc (Some p), a) σ.(st_heap))).
  2: { iDestruct "HΦ" as "[Ha _]". by iMod (alloc_alive_loc_to_valid_ptr with "Hlib Ha Hctx") as %Hb. }
  iApply fupd_mask_intro; [set_solver|]. iIntros "HE". iDestruct "HΦ" as "[_ HΦ]".
  iSplit. {
    iPureIntro. split.
    - inversion 1; simplify_eq; case_bool_decide; by rewrite ->Hp in *.
    - move => ->. econstructor; [done..|]. rewrite Hp. by case_bool_decide.
  }
  iModIntro. iMod "HE". by iFrame.
Qed.

Lemma wp_cast_int_null Φ v E it:
  val_to_Z v it = Some 0 →
  ▷ Φ (val_of_loc NULL_loc) -∗
  WP UnOp (CastOp PtrOp) (IntOp it) (Val v) @ E {{ Φ }}.
Proof.
  iIntros (Hv) "HΦ".
  iApply wp_unop_det_pure; [|done].
  move => ??. split.
  - inversion 1; simplify_eq => //. case_bool_decide; last done. exfalso.
    revert select (valid_ptr _ _) => /valid_ptr_in_allocation_range []/=.
    rewrite /min_alloc_start //.
  - move => ->. econstructor => //. case_bool_decide; last done. exfalso.
    revert select (valid_ptr _ _) => /valid_ptr_in_allocation_range []/=.
    rewrite /min_alloc_start //.
Qed.

Lemma wp_copy_alloc_id Φ it a l v1 v2:
  val_to_Z v1 it = Some a →
  val_to_loc v2 = Some l →
  loc_in_bounds (l.1, a) 0 -∗
  alloc_alive_loc l ∧ ▷ Φ (val_of_loc (l.1, a)) -∗
  WP CopyAllocId (IntOp it) (Val v1) (Val v2) {{ Φ }}.
Proof.
  iIntros (Ha Hl) "#Hlib HΦ". iApply wp_lift_expr_step_fupd => //.
  iIntros (σ1) "Hctx".
  destruct (decide (valid_ptr (l.1, a) σ1.(st_heap))). 2: {
    iDestruct "HΦ" as "[Ha _]".
    iMod (alloc_alive_loc_to_valid_ptr with "Hlib [Ha] Hctx") as %Hb; [|done].
    by iApply alloc_alive_loc_mono; [|done].
  }
  iDestruct "HΦ" as "[_ HΦ]". iApply fupd_mask_intro; [set_solver|]. iIntros "HE".
  iSplit; first by eauto 8 using CopyAllocIdS.
  iIntros (???? Hstep ?) "!>!>". inv_expr_step. iMod "HE". iModIntro. iSplit => //. iFrame.
  by iApply wp_value.
Qed.

Lemma wp_ptr_relop Φ op v1 v2 v l1 l2 b rit:
  val_to_loc v1 = Some l1 →
  val_to_loc v2 = Some l2 →
  val_of_Z (Z_of_bool b) rit None = Some v →
  match op with
  | EqOp rit => Some (bool_decide (l1.2 = l2.2), rit)
  | NeOp rit => Some (bool_decide (l1.2 ≠ l2.2), rit)
  | LtOp rit => if bool_decide (l1.1 = l2.1) then Some (bool_decide (l1.2 < l2.2), rit) else None
  | GtOp rit => if bool_decide (l1.1 = l2.1) then Some (bool_decide (l1.2 > l2.2), rit) else None
  | LeOp rit => if bool_decide (l1.1 = l2.1) then Some (bool_decide (l1.2 <= l2.2), rit) else None
  | GeOp rit => if bool_decide (l1.1 = l2.1) then Some (bool_decide (l1.2 >= l2.2), rit) else None
  | _ => None
  end = Some (b, rit) →
  loc_in_bounds l1 0 -∗ loc_in_bounds l2 0 -∗
  (alloc_alive_loc l1 ∧ alloc_alive_loc l2 ∧ ▷ Φ v) -∗
  WP BinOp op PtrOp PtrOp (Val v1) (Val v2) {{ Φ }}.
Proof.
  iIntros (Hv1 Hv2 Hv Hop) "#Hl1 #Hl2 HΦ".
  iDestruct (loc_in_bounds_has_alloc_id with "Hl1") as %[??].
  iDestruct (loc_in_bounds_has_alloc_id with "Hl2") as %[??].
  iApply wp_binop. iIntros (σ) "Hσ".
  destruct (decide (valid_ptr l1 σ.(st_heap))). 2: {
    iDestruct "HΦ" as "[Ha _]".
    by iMod (alloc_alive_loc_to_valid_ptr with "Hl1 Ha Hσ") as %?.
  }
  destruct (decide (valid_ptr l2 σ.(st_heap))). 2: {
    iDestruct "HΦ" as "[_ [Ha _]]".
    by iMod (alloc_alive_loc_to_valid_ptr with "Hl2 Ha Hσ") as %?.
  }
  iApply fupd_mask_intro; [done|]. iIntros "HE".
  destruct l1, l2; simplify_eq/=. iSplit. {
    iPureIntro. destruct op; simplify_eq/=; eexists _; try by apply: RelOpPP => //; repeat case_bool_decide; naive_solver.
    all: apply: CmpOpPP => //; by rewrite ?heap_loc_eq_alloc_alloc//= negb_bool_decide_eq.
  }
  iDestruct "HΦ" as "(_&_&HΦ)". iIntros "!>" (v' Hstep). iMod "HE". iModIntro. iFrame.
  inversion Hstep; simplify_eq => //.
  - revert select (heap_loc_eq _ _ _ = _). rewrite heap_loc_eq_alloc_alloc // => ?. simplify_eq.
    destruct op; simplify_eq/= => //. by repeat case_bool_decide => //; simplify_eq/=.
  - destruct op; repeat case_bool_decide; by simplify_eq.
Qed.

Lemma wp_ptr_offset Φ vl l E it o ly vo:
  val_to_loc vl = Some l →
  val_to_Z vo it = Some o →
  loc_in_bounds (l offset{ly}ₗ o) 0 -∗
  ▷ Φ (val_of_loc (l offset{ly}ₗ o)) -∗
  WP Val vl at_offset{ ly , PtrOp, IntOp it} Val vo @ E {{ Φ }}.
Proof.
  iIntros (Hvl Hvo) "Hl HΦ".
  iApply wp_binop_det. iIntros (σ) "Hσ".
  iApply fupd_mask_intro; [set_solver|]. iIntros "HE".
  iDestruct (loc_in_bounds_to_heap_loc_in_bounds with "Hl Hσ") as %?.
  iSplit. {
    iPureIntro. split.
    - inversion 1. by simplify_eq.
    - move => ->. by apply PtrOffsetOpIP.
  }
  iModIntro. iMod "HE". by iFrame.
Qed.

Lemma wp_ptr_neg_offset Φ vl l E it o ly vo:
  val_to_loc vl = Some l →
  val_to_Z vo it = Some o →
  loc_in_bounds (l offset{ly}ₗ -o) 0 -∗
  ▷ Φ (val_of_loc (l offset{ly}ₗ -o)) -∗
  WP Val vl at_neg_offset{ ly , PtrOp, IntOp it} Val vo @ E {{ Φ }}.
Proof.
  iIntros (Hvl Hvo) "Hl HΦ".
  iApply wp_binop_det. iIntros (σ) "Hσ".
  iApply fupd_mask_intro; [set_solver|]. iIntros "HE".
  iDestruct (loc_in_bounds_to_heap_loc_in_bounds with "Hl Hσ") as %?.
  iSplit. {
    iPureIntro. split.
    - inversion 1. by simplify_eq.
    - move => ->. by apply PtrNegOffsetOpIP.
  }
  iModIntro. iMod "HE". by iFrame.
Qed.

Lemma wp_ptr_diff Φ vl1 l1 vl2 l2 ly vo:
  val_to_loc vl1 = Some l1 →
  val_to_loc vl2 = Some l2 →
  val_of_Z ((l1.2 - l2.2) `div` ly.(ly_size)) ptrdiff_t None = Some vo →
  l1.1 = l2.1 →
  0 < ly.(ly_size) →
  loc_in_bounds l1 0 -∗
  loc_in_bounds l2 0 -∗
  (alloc_alive_loc l1 ∧ ▷ Φ vo) -∗
  WP Val vl1 sub_ptr{ly , PtrOp, PtrOp} Val vl2 {{ Φ }}.
Proof.
  iIntros (Hvl1 Hvl2 Hvo ??) "Hl1 Hl2 HΦ".
  iApply wp_binop_det. iIntros (σ) "Hσ".
  destruct (decide (valid_ptr l1 σ.(st_heap))). 2: {
    iDestruct "HΦ" as "[Ha _]".
    by iMod (alloc_alive_loc_to_valid_ptr with "Hl1 Ha Hσ") as %?.
  }
  destruct (decide (valid_ptr l2 σ.(st_heap))). 2: {
    iDestruct "HΦ" as "[Ha _]".
    iMod (alloc_alive_loc_to_valid_ptr with "Hl2 [Ha] Hσ") as %?; [|done].
    by iApply alloc_alive_loc_mono.
  }
  iDestruct "HΦ" as "[_ ?]".
  iApply fupd_mask_intro; [set_solver|]. iIntros "HE".
  iSplit. {
    iPureIntro. split.
    - inversion 1; by simplify_eq.
    - move => ->. by apply: PtrDiffOpPP.
  }
  iModIntro. iMod "HE". by iFrame.
Qed.

Lemma wp_get_member Φ vl l sl n E:
  val_to_loc vl = Some l →
  is_Some (index_of sl.(sl_members) n) →
  loc_in_bounds l (ly_size sl) -∗
  ▷ Φ (val_of_loc (l at{sl}ₗ n)) -∗
  WP Val vl at{sl} n @ E {{ Φ }}.
Proof.
  iIntros (Hvl [i Hi]) "#Hl HΦ".
  rewrite /GetMember/GetMemberLoc/offset_of Hi /=.
  have [|v Hv]:= (val_of_Z_is_Some None size_t (offset_of_idx sl.(sl_members) i)). {
    split; first by rewrite /min_int/=; lia. by apply offset_of_bound.
  }
  rewrite Hv /=. move: Hv => /val_to_of_Z Hv.
  iApply wp_ptr_offset; [done| done | | ].
  { have ? := offset_of_idx_le_size sl i. rewrite offset_loc_sz1 //.
    iApply (loc_in_bounds_offset with "Hl"); simpl; [done| destruct l => /=; lia  | destruct l => /=; lia ]. }
  by rewrite offset_loc_sz1.
Qed.
Lemma wp_get_member_union Φ vl l ul n E:
  val_to_loc vl = Some l →
  Φ (val_of_loc (l at_union{ul}ₗ n)) -∗ WP Val vl at_union{ul} n @ E {{ Φ }}.
Proof.
  iIntros (->%val_of_to_loc) "?".
  rewrite /GetMemberUnion/GetMemberUnionLoc.
  by iApply @wp_value.
Qed.

Lemma wp_offset_of Φ s m i E:
  offset_of s.(sl_members) m = Some i →
  (∀ v, ⌜val_of_Z i size_t None = Some v⌝ -∗ Φ v) -∗
  WP OffsetOf s m @ E {{ Φ }}.
Proof.
  rewrite /OffsetOf. iIntros (Ho) "HΦ".
  have [|? Hs]:= (val_of_Z_is_Some None size_t i). {
    split; first by rewrite /min_int/=; lia.
    move: Ho => /fmap_Some[?[?->]].
    by apply offset_of_bound.
  }
  rewrite Ho /= Hs /=. iApply wp_value.
  by iApply "HΦ".
Qed.

Lemma wp_offset_of_union Φ ul m E:
  Φ (i2v 0 size_t) -∗ WP OffsetOfUnion ul m @ E {{ Φ }}.
Proof. by iApply @wp_value. Qed.

Lemma wp_if_int Φ it v e1 e2 n:
  val_to_Z v it = Some n →
  (if bool_decide (n ≠ 0) then WP e1 {{ Φ }} else WP e2 {{ Φ }}) -∗
  WP IfE (IntOp it) (Val v) e1 e2 {{ Φ }}.
Proof.
  iIntros (?) "HΦ".
  iApply wp_lift_expr_step; auto.
  iIntros (σ1) "?". iModIntro. iSplit; first by eauto 8 using IfESI.
  iIntros (? ? σ2 efs Hst ?) "!> !>". inv_expr_step.
  iSplit => //. iFrame. by case_bool_decide.
Qed.

Definition wp_if_precond (l : loc) : iProp Σ :=
  match l.1 with | ProvNull => ⌜l = NULL_loc⌝ | ProvAlloc _ => loc_in_bounds l 0 | _ => True end.

Lemma wp_if_precond_null:
  ⊢ wp_if_precond NULL_loc.
Proof. rewrite /wp_if_precond/=. by iPureIntro. Qed.

Lemma wp_if_precond_alloc l:
  loc_in_bounds l 0 -∗
  wp_if_precond l.
Proof.
  iIntros "Hlib". rewrite /wp_if_precond.
  by iDestruct (loc_in_bounds_has_alloc_id with "Hlib") as %[? ->].
Qed.

Lemma wp_if_precond_heap_loc_eq l σ:
  wp_if_precond l -∗
  state_ctx σ -∗
  ⌜heap_loc_eq l NULL_loc σ.(st_heap) = Some (bool_decide (l = NULL_loc))⌝.
Proof.
  rewrite/wp_if_precond. iIntros "Hlib Hσ". case_match.
  - iDestruct "Hlib" as %?; simplify_eq. iPureIntro. rewrite heap_loc_eq_NULL_NULL. by case_bool_decide.
  - iDestruct (loc_in_bounds_to_heap_loc_in_bounds with "Hlib Hσ") as %Hlib. iPureIntro.
    rewrite heap_loc_eq_alloc_NULL //. case_bool_decide => //; simplify_eq.
  - iPureIntro. rewrite heap_loc_eq_fnptr_NULL //. case_bool_decide => //; simplify_eq.
Qed.

Lemma wp_if_ptr Φ v e1 e2 l:
  val_to_loc v = Some l →
  wp_if_precond l -∗
  (if bool_decide (l ≠ NULL_loc) then WP e1 {{ Φ }} else WP e2 {{ Φ }}) -∗
  WP IfE PtrOp (Val v) e1 e2 {{ Φ }}.
Proof.
  iIntros (?) "Hlib HΦ".
  iApply wp_lift_expr_step; auto.
  iIntros (σ1) "Hσ1". iModIntro.
  iDestruct (wp_if_precond_heap_loc_eq with "Hlib Hσ1") as %?.
  iSplit; first by eauto 8 using IfESP.
  iIntros (? ? σ2 efs Hst ?) "!> !>". inv_expr_step.
  iSplit => //. iFrame. by repeat case_bool_decide.
Qed.

Lemma wp_skip Φ v E:
  ▷ Φ v -∗ WP SkipE (Val v) @ E {{ Φ }}.
Proof.
  iIntros "HΦ".
  iApply wp_lift_expr_step; auto.
  iIntros (σ1) "?". iModIntro. iSplit; first by eauto 8 using SkipES.
  iIntros (? e2 σ2 efs Hst ?) "!> !>". inv_expr_step.
  iSplit => //. iFrame. by iApply wp_value.
Qed.

Lemma wp_concat E Φ vs:
  Φ (mjoin vs) -∗ WP Concat (Val <$> vs) @ E {{ Φ }}.
Proof.
  iIntros "HΦ".
  iApply wp_lift_expr_step; auto.
  iIntros (σ1) "?".
  iModIntro. iSplit; first by eauto 8 using ConcatS.
  iIntros "!#" (? e2 σ2 efs Hst ?). inv_expr_step.
  iModIntro. iSplit => //. iFrame. by iApply wp_value.
Qed.

Lemma wps_concat_bind_ind vs E Φ es:
  foldr (λ e f, (λ vl, WP e @ E {{ v, f (vl ++ [v]) }}))
        (λ vl, WP (Concat (Val <$> (vs ++ vl))) @ E {{ Φ }}) es [] -∗
  WP Concat ((Val <$> vs) ++ es) @ E {{ Φ }}.
Proof.
  rewrite -{2}(app_nil_r vs).
  move: {2 3}[] => vl2.
  elim: es vs vl2 => /=.
  - iIntros (vs vl2) "He". by rewrite !app_nil_r.
  - iIntros (e el IH vs vl2) "Hf".
    rewrite /wp/wp_expr_wp.
    have -> : (coerce_rtexpr (Concat ((Val <$> vs ++ vl2) ++ e :: el)))%E = (fill [ExprCtx (ConcatCtx (vs ++ vl2) (to_rtexpr <$> el))] (to_rtexpr e)). {
        by rewrite /coerce_rtexpr/= fmap_app fmap_cons -!list_fmap_compose.
    }
    iApply wp_bind. iApply (wp_wand with "Hf").
    iIntros (v) "Hf" => /=.
    have -> : Expr (RTConcat ((Expr <$> (RTVal <$> vs ++ vl2)) ++ of_val v :: (to_rtexpr <$> el)))
             = Concat ((Val <$> vs ++ (vl2 ++ [v])) ++ el). 2: by iApply IH.
    by rewrite /coerce_rtexpr /= !fmap_app /= (cons_middle (of_val v)) !app_assoc -!list_fmap_compose.
Qed.

Lemma wp_concat_bind E Φ es:
  foldr (λ e f, (λ vl, WP e @ E {{ v, f (vl ++ [v]) }}))
        (λ vl, WP (Concat (Val <$> vl)) @ E {{ Φ }}) es [] -∗
  WP Concat es @ E {{ Φ }}.
Proof. by iApply (wps_concat_bind_ind []). Qed.

Lemma wp_struct_init E Φ sl fs:
  foldr (λ '(n, ly) f, (λ vl,
     WP (default (Val (replicate (ly_size ly) MPoison)) (n' ← n; (list_to_map fs : gmap _ _) !! n'))
        @ E {{ v, f (vl ++ [v]) }}))
    (λ vl, Φ (mjoin vl)) sl.(sl_members) [] -∗
  WP StructInit sl fs @ E {{ Φ }}.
Proof.
  iIntros "He". unfold StructInit. iApply wp_concat_bind.
  move: {2 4}[] => vs.
  iInduction (sl_members sl) as [|[o?]?] "IH" forall (vs) => /=. by iApply wp_concat.
  iApply (wp_wand with "He"). iIntros (v') "Hfold". by iApply "IH".
Qed.

Lemma wp_call_bind_ind vs E Φ vf el:
  foldr (λ e f, (λ vl, WP e @ E {{ v, f (vl ++ [v]) }}))
        (λ vl, WP Call (Val vf) (Val <$> (vs ++ vl)) @ E {{ Φ }}) el [] -∗
  WP (Call (Val vf) ((Val <$> vs) ++ el)) @ E {{ Φ}}.
Proof.
  rewrite -{2}(app_nil_r vs).
  move: {2 3}[] => vl2.
  elim: el vs vl2 => /=.
  - iIntros (vs vl2) "He". by rewrite !app_nil_r.
  - iIntros (e el IH vs vl2) "Hf".
    rewrite /wp/wp_expr_wp.
    have -> : (coerce_rtexpr (Call (Val vf) ((Val <$> vs ++ vl2) ++ e :: el)))%E = (fill [ExprCtx (CallRCtx vf (vs ++ vl2) (to_rtexpr <$> el))] (to_rtexpr e)). {
        by rewrite /coerce_rtexpr/= fmap_app fmap_cons -!list_fmap_compose.
    }
    iApply wp_bind. iApply (wp_wand with "Hf").
    iIntros (v) "Hf" => /=.
    have -> : Expr (RTCall vf ((Expr <$> (RTVal <$> vs ++ vl2)) ++ of_val v :: (to_rtexpr <$> el)))
             = Call vf ((Val <$> vs ++ (vl2 ++ [v])) ++ el). 2: by iApply IH.
    by rewrite /coerce_rtexpr /= !fmap_app /= (cons_middle (of_val v)) !app_assoc -!list_fmap_compose.
Qed.

Lemma wp_call_bind E Φ el ef:
  WP ef @ E {{ vf, foldr (λ e f, (λ vl, WP e @ E {{ v, f (vl ++ [v]) }}))
        (λ vl, WP Call (Val vf) (Val <$> vl) @ E {{ Φ }}) el [] }} -∗
  WP (Call ef el) @ E {{ Φ }}.
Proof.
  iIntros "HWP".
  rewrite /wp/wp_expr_wp.
  have -> : coerce_rtexpr (Call ef el) = fill [ExprCtx $ CallLCtx (coerce_rtexpr <$> el)] (coerce_rtexpr ef) by [].
  iApply wp_bind.
  iApply (wp_wand with "HWP"). iIntros (vf) "HWP".
  by iApply (wp_call_bind_ind []).
Qed.

End expr_lifting.

(*** Lifting of statements *)
Definition stmt_wp_def `{!refinedcG Σ} (E : coPset) (Q : gmap label stmt) (Ψ : val → iProp Σ) (s : stmt) : iProp Σ :=
  (∀ Φ rf, ⌜Q = rf.(rf_fn).(f_code)⌝ -∗ (∀ v, Ψ v -∗ WP to_rtstmt rf (Return v) {{ Φ }}) -∗
    WP to_rtstmt rf s @ E {{ Φ }}).
Definition stmt_wp_aux `{!refinedcG Σ} (E : coPset) (Q : gmap label stmt) (Ψ : val → iProp Σ) : seal (@stmt_wp_def Σ _ E Q Ψ). by eexists. Qed.
Definition stmt_wp `{!refinedcG Σ} (E : coPset) (Q : gmap label stmt) (Ψ : val → iProp Σ) :
  stmt → iProp Σ := (stmt_wp_aux E Q Ψ).(unseal).
Definition stmt_wp_eq `{!refinedcG Σ} (E : coPset) (Q : gmap label stmt) (Ψ : val → iProp Σ) : stmt_wp E Q Ψ = @stmt_wp_def Σ _ E Q Ψ := (stmt_wp_aux E Q Ψ).(seal_eq).

Notation "'WPs' s @ E {{ Q , Ψ } }" := (stmt_wp E Q Ψ s%E)
  (at level 20, s, Q, Ψ at level 200, format "'[' 'WPs'  s  '/' '[       ' @  E  {{  Q ,  Ψ  } } ']' ']'" ) : bi_scope.

Notation "'WPs' s {{ Q , Ψ } }" := (stmt_wp ⊤ Q Ψ s%E)
  (at level 20, s, Q, Ψ at level 200, format "'[' 'WPs'  s  '/' '[   ' {{  Q ,  Ψ  } } ']' ']'") : bi_scope.

Section stmt_lifting.
Context `{!refinedcG Σ}.

Lemma stmt_wp_unfold s E Q Ψ  :
  WPs s @ E {{ Q, Ψ }} ⊣⊢ stmt_wp_def E Q Ψ s.
Proof. by rewrite stmt_wp_eq. Qed.

Lemma fupd_wps s E Q Ψ :
  (|={E}=> WPs s @ E {{ Q, Ψ }}) -∗ WPs s @ E{{ Q, Ψ }}.
Proof.
  rewrite stmt_wp_unfold. iIntros "Hs" (? rf HQ) "HΨ".
  iApply fupd_wp. by iApply "Hs".
Qed.

Global Instance elim_modal_bupd_wps p s Q Ψ P E :
    ElimModal True p false (|==> P) P (WPs s @ E {{ Q, Ψ }}) (WPs s @ E {{ Q, Ψ }}).
Proof. by rewrite /ElimModal intuitionistically_if_elim (bupd_fupd E) fupd_frame_r wand_elim_r fupd_wps. Qed.

Global Instance elim_modal_fupd_wps p s Q Ψ P E :
    ElimModal True p false (|={E}=> P) P (WPs s @ E {{ Q, Ψ }}) (WPs s @ E {{ Q, Ψ }}).
Proof. by rewrite /ElimModal intuitionistically_if_elim fupd_frame_r wand_elim_r fupd_wps. Qed.

Lemma wps_wand s E Q Φ Ψ:
  WPs s @ E {{ Q , Φ }} -∗ (∀ v, Φ v -∗ Ψ v) -∗ WPs s @ E {{ Q , Ψ }}.
Proof.
  rewrite !stmt_wp_unfold. iIntros "HΦ H" (???) "HΨ".
  iApply "HΦ"; [ done | ..]. iIntros (v) "Hv".
  iApply "HΨ". iApply "H". iApply "Hv".
Qed.