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From iris.program_logic Require Export weakestpre total_weakestpre.
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From iris.proofmode Require Import coq_tactics.
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From iris.proofmode Require Export tactics.
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From iris.heap_lang Require Export tactics lifting.
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Set Default Proof Using "Type".
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Import uPred.

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Lemma tac_wp_expr_eval `{heapG Σ} Δ s E Φ e e' :
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  ( (e'':=e'), e = e'') 
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  envs_entails Δ (WP e' @ s; E {{ Φ }})  envs_entails Δ (WP e @ s; E {{ Φ }}).
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Proof. by intros ->. Qed.
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Lemma tac_twp_expr_eval `{heapG Σ} Δ s E Φ e e' :
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  ( (e'':=e'), e = e'') 
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  envs_entails Δ (WP e' @ s; E [{ Φ }])  envs_entails Δ (WP e @ s; E [{ Φ }]).
Proof. by intros ->. Qed.
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Tactic Notation "wp_expr_eval" tactic(t) :=
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  try (
    iStartProof;
    first [eapply tac_wp_expr_eval|eapply tac_twp_expr_eval];
      [let x := fresh in intros x; t; unfold x; reflexivity
      |]).
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Ltac wp_expr_simpl := wp_expr_eval simpl.
Ltac wp_expr_simpl_subst := wp_expr_eval simpl_subst.
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Lemma tac_wp_pure `{heapG Σ} Δ Δ' s E e1 e2 φ Φ :
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  PureExec φ e1 e2 
  φ 
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  IntoLaterNEnvs 1 Δ Δ' 
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  envs_entails Δ' (WP e2 @ s; E {{ Φ }}) 
  envs_entails Δ (WP e1 @ s; E {{ Φ }}).
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Proof.
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  rewrite /envs_entails=> ??? HΔ'. rewrite into_laterN_env_sound /=.
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  rewrite HΔ' -lifting.wp_pure_step_later //.
Qed.
Lemma tac_twp_pure `{heapG Σ} Δ s E e1 e2 φ Φ :
  PureExec φ e1 e2 
  φ 
  envs_entails Δ (WP e2 @ s; E [{ Φ }]) 
  envs_entails Δ (WP e1 @ s; E [{ Φ }]).
Proof.
  rewrite /envs_entails=> ?? ->. rewrite -total_lifting.twp_pure_step //.
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Qed.
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Lemma tac_wp_value `{heapG Σ} Δ s E Φ e v :
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  IntoVal e v 
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  envs_entails Δ (Φ v)  envs_entails Δ (WP e @ s; E {{ Φ }}).
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Proof. rewrite /envs_entails=> ? ->. by apply wp_value. Qed.
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Lemma tac_twp_value `{heapG Σ} Δ s E Φ e v :
  IntoVal e v 
  envs_entails Δ (Φ v)  envs_entails Δ (WP e @ s; E [{ Φ }]).
Proof. rewrite /envs_entails=> ? ->. by apply twp_value. Qed.
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Ltac wp_value_head :=
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  first [eapply tac_wp_value || eapply tac_twp_value];
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    [apply _
    |iEval (lazy beta; simpl of_val)].
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Tactic Notation "wp_pure" open_constr(efoc) :=
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  iStartProof;
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  lazymatch goal with
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  | |- envs_entails _ (wp ?s ?E ?e ?Q) =>
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    let e := eval simpl in e in
    reshape_expr e ltac:(fun K e' =>
      unify e' efoc;
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      eapply (tac_wp_pure _ _ _ _ (fill K e'));
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      [apply _                        (* PureExec *)
      |try fast_done                  (* The pure condition for PureExec *)
      |apply _                        (* IntoLaters *)
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      |wp_expr_simpl_subst; try wp_value_head (* new goal *)
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      ])
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    || fail "wp_pure: cannot find" efoc "in" e "or" efoc "is not a reduct"
  | |- envs_entails _ (twp ?s ?E ?e ?Q) =>
    let e := eval simpl in e in
    reshape_expr e ltac:(fun K e' =>
      unify e' efoc;
      eapply (tac_twp_pure _ _ _ (fill K e'));
      [apply _                        (* PureExec *)
      |try fast_done                  (* The pure condition for PureExec *)
      |wp_expr_simpl_subst; try wp_value_head (* new goal *)
      ])
    || fail "wp_pure: cannot find" efoc "in" e "or" efoc "is not a reduct"
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  | _ => fail "wp_pure: not a 'wp'"
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  end.

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Tactic Notation "wp_if" := wp_pure (If _ _ _).
Tactic Notation "wp_if_true" := wp_pure (If (Lit (LitBool true)) _ _).
Tactic Notation "wp_if_false" := wp_pure (If (Lit (LitBool false)) _ _).
Tactic Notation "wp_unop" := wp_pure (UnOp _ _).
Tactic Notation "wp_binop" := wp_pure (BinOp _ _ _).
Tactic Notation "wp_op" := wp_unop || wp_binop.
Tactic Notation "wp_rec" := wp_pure (App _ _).
Tactic Notation "wp_lam" := wp_rec.
Tactic Notation "wp_let" := wp_lam.
Tactic Notation "wp_seq" := wp_lam.
Tactic Notation "wp_proj" := wp_pure (Fst _) || wp_pure (Snd _).
Tactic Notation "wp_case" := wp_pure (Case _ _ _).
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Tactic Notation "wp_match" := wp_case; wp_let.
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Lemma tac_wp_bind `{heapG Σ} K Δ s E Φ e f :
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  f = (λ e, fill K e)  (* as an eta expanded hypothesis so that we can `simpl` it *)
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  envs_entails Δ (WP e @ s; E {{ v, WP f (of_val v) @ s; E {{ Φ }} }})%I 
  envs_entails Δ (WP fill K e @ s; E {{ Φ }}).
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Proof. rewrite /envs_entails=> -> ->. by apply: wp_bind. Qed.
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Lemma tac_twp_bind `{heapG Σ} K Δ s E Φ e f :
  f = (λ e, fill K e)  (* as an eta expanded hypothesis so that we can `simpl` it *)
  envs_entails Δ (WP e @ s; E [{ v, WP f (of_val v) @ s; E [{ Φ }] }])%I 
  envs_entails Δ (WP fill K e @ s; E [{ Φ }]).
Proof. rewrite /envs_entails=> -> ->. by apply: twp_bind. Qed.
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Ltac wp_bind_core K :=
  lazymatch eval hnf in K with
  | [] => idtac
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  | _ => eapply (tac_wp_bind K); [simpl; reflexivity|lazy beta]
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  end.
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Ltac twp_bind_core K :=
  lazymatch eval hnf in K with
  | [] => idtac
  | _ => eapply (tac_twp_bind K); [simpl; reflexivity|lazy beta]
  end.
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Tactic Notation "wp_bind" open_constr(efoc) :=
  iStartProof;
  lazymatch goal with
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  | |- envs_entails _ (wp ?s ?E ?e ?Q) =>
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    reshape_expr e ltac:(fun K e' => unify e' efoc; wp_bind_core K)
    || fail "wp_bind: cannot find" efoc "in" e
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  | |- envs_entails _ (twp ?s ?E ?e ?Q) =>
    reshape_expr e ltac:(fun K e' => unify e' efoc; twp_bind_core K)
    || fail "wp_bind: cannot find" efoc "in" e
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  | _ => fail "wp_bind: not a 'wp'"
  end.

(** Heap tactics *)
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Section heap.
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Context `{heapG Σ}.
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Implicit Types P Q : iProp Σ.
Implicit Types Φ : val  iProp Σ.
Implicit Types Δ : envs (iResUR Σ).
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Lemma tac_wp_alloc Δ Δ' s E j K e v Φ :
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  IntoVal e v 
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  IntoLaterNEnvs 1 Δ Δ' 
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  ( l,  Δ'',
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    envs_app false (Esnoc Enil j (l  v)) Δ' = Some Δ'' 
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    envs_entails Δ'' (WP fill K (Lit (LitLoc l)) @ s; E {{ Φ }})) 
  envs_entails Δ (WP fill K (Alloc e) @ s; E {{ Φ }}).
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Proof.
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  rewrite /envs_entails=> ?? HΔ.
  rewrite -wp_bind. eapply wand_apply; first exact: wp_alloc.
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  rewrite left_id into_laterN_env_sound; apply later_mono, forall_intro=> l.
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  destruct (HΔ l) as (Δ''&?&HΔ'). rewrite envs_app_sound //; simpl.
  by rewrite right_id HΔ'.
Qed.
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Lemma tac_twp_alloc Δ s E j K e v Φ :
  IntoVal e v 
  ( l,  Δ',
    envs_app false (Esnoc Enil j (l  v)) Δ = Some Δ' 
    envs_entails Δ' (WP fill K (Lit (LitLoc l)) @ s; E [{ Φ }])) 
  envs_entails Δ (WP fill K (Alloc e) @ s; E [{ Φ }]).
Proof.
  rewrite /envs_entails=> ? HΔ.
  rewrite -twp_bind. eapply wand_apply; first exact: twp_alloc.
  rewrite left_id. apply forall_intro=> l.
  destruct (HΔ l) as (Δ'&?&HΔ'). rewrite envs_app_sound //; simpl.
  by rewrite right_id HΔ'.
Qed.
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Lemma tac_wp_load Δ Δ' s E i K l q v Φ :
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  IntoLaterNEnvs 1 Δ Δ' 
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  envs_lookup i Δ' = Some (false, l {q} v)%I 
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  envs_entails Δ' (WP fill K (of_val v) @ s; E {{ Φ }}) 
  envs_entails Δ (WP fill K (Load (Lit (LitLoc l))) @ s; E {{ Φ }}).
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Proof.
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  rewrite /envs_entails=> ???.
  rewrite -wp_bind. eapply wand_apply; first exact: wp_load.
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  rewrite into_laterN_env_sound -later_sep envs_lookup_split //; simpl.
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  by apply later_mono, sep_mono_r, wand_mono.
Qed.
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Lemma tac_twp_load Δ s E i K l q v Φ :
  envs_lookup i Δ = Some (false, l {q} v)%I 
  envs_entails Δ (WP fill K (of_val v) @ s; E [{ Φ }]) 
  envs_entails Δ (WP fill K (Load (Lit (LitLoc l))) @ s; E [{ Φ }]).
Proof.
  rewrite /envs_entails=> ??.
  rewrite -twp_bind. eapply wand_apply; first exact: twp_load.
  rewrite envs_lookup_split //; simpl.
  by apply sep_mono_r, wand_mono.
Qed.
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Lemma tac_wp_store Δ Δ' Δ'' s E i K l v e v' Φ :
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  IntoVal e v' 
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  IntoLaterNEnvs 1 Δ Δ' 
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  envs_lookup i Δ' = Some (false, l  v)%I 
  envs_simple_replace i false (Esnoc Enil i (l  v')) Δ' = Some Δ'' 
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  envs_entails Δ'' (WP fill K (Lit LitUnit) @ s; E {{ Φ }}) 
  envs_entails Δ (WP fill K (Store (Lit (LitLoc l)) e) @ s; E {{ Φ }}).
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Proof.
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  rewrite /envs_entails=> ?????.
  rewrite -wp_bind. eapply wand_apply; first by eapply wp_store.
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  rewrite into_laterN_env_sound -later_sep envs_simple_replace_sound //; simpl.
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  rewrite right_id. by apply later_mono, sep_mono_r, wand_mono.
Qed.
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Lemma tac_twp_store Δ Δ' s E i K l v e v' Φ :
  IntoVal e v' 
  envs_lookup i Δ = Some (false, l  v)%I 
  envs_simple_replace i false (Esnoc Enil i (l  v')) Δ = Some Δ' 
  envs_entails Δ' (WP fill K (Lit LitUnit) @ s; E [{ Φ }]) 
  envs_entails Δ (WP fill K (Store (Lit (LitLoc l)) e) @ s; E [{ Φ }]).
Proof.
  intros. rewrite -twp_bind. eapply wand_apply; first by eapply twp_store.
  rewrite envs_simple_replace_sound //; simpl.
  rewrite right_id. by apply sep_mono_r, wand_mono.
Qed.
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Lemma tac_wp_cas_fail Δ Δ' s E i K l q v e1 v1 e2 Φ :
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  IntoVal e1 v1  AsVal e2 
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  IntoLaterNEnvs 1 Δ Δ' 
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  envs_lookup i Δ' = Some (false, l {q} v)%I  v  v1 
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  envs_entails Δ' (WP fill K (Lit (LitBool false)) @ s; E {{ Φ }}) 
  envs_entails Δ (WP fill K (CAS (Lit (LitLoc l)) e1 e2) @ s; E {{ Φ }}).
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Proof.
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  rewrite /envs_entails=> ??????.
  rewrite -wp_bind. eapply wand_apply; first exact: wp_cas_fail.
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  rewrite into_laterN_env_sound -later_sep envs_lookup_split //; simpl.
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  by apply later_mono, sep_mono_r, wand_mono.
Qed.
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Lemma tac_twp_cas_fail Δ s E i K l q v e1 v1 e2 Φ :
  IntoVal e1 v1  AsVal e2 
  envs_lookup i Δ = Some (false, l {q} v)%I  v  v1 
  envs_entails Δ (WP fill K (Lit (LitBool false)) @ s; E [{ Φ }]) 
  envs_entails Δ (WP fill K (CAS (Lit (LitLoc l)) e1 e2) @ s; E [{ Φ }]).
Proof.
  intros. rewrite -twp_bind. eapply wand_apply; first exact: twp_cas_fail.
  rewrite envs_lookup_split //; simpl. by apply sep_mono_r, wand_mono.
Qed.
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Lemma tac_wp_cas_suc Δ Δ' Δ'' s E i K l v e1 v1 e2 v2 Φ :
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  IntoVal e1 v1  IntoVal e2 v2 
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  IntoLaterNEnvs 1 Δ Δ' 
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  envs_lookup i Δ' = Some (false, l  v)%I  v = v1 
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  envs_simple_replace i false (Esnoc Enil i (l  v2)) Δ' = Some Δ'' 
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  envs_entails Δ'' (WP fill K (Lit (LitBool true)) @ s; E {{ Φ }}) 
  envs_entails Δ (WP fill K (CAS (Lit (LitLoc l)) e1 e2) @ s; E {{ Φ }}).
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Proof.
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  rewrite /envs_entails=> ???????; subst.
  rewrite -wp_bind. eapply wand_apply; first exact: wp_cas_suc.
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  rewrite into_laterN_env_sound -later_sep envs_simple_replace_sound //; simpl.
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  rewrite right_id. by apply later_mono, sep_mono_r, wand_mono.
Qed.
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Lemma tac_twp_cas_suc Δ Δ' s E i K l v e1 v1 e2 v2 Φ :
  IntoVal e1 v1  IntoVal e2 v2 
  envs_lookup i Δ = Some (false, l  v)%I  v = v1 
  envs_simple_replace i false (Esnoc Enil i (l  v2)) Δ = Some Δ' 
  envs_entails Δ' (WP fill K (Lit (LitBool true)) @ s; E [{ Φ }]) 
  envs_entails Δ (WP fill K (CAS (Lit (LitLoc l)) e1 e2) @ s; E [{ Φ }]).
Proof.
  intros; subst. rewrite -twp_bind. eapply wand_apply; first exact: twp_cas_suc.
  rewrite envs_simple_replace_sound //; simpl.
  rewrite right_id. by apply sep_mono_r, wand_mono.
Qed.
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Lemma tac_wp_faa Δ Δ' Δ'' s E i K l i1 e2 i2 Φ :
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  IntoVal e2 (LitV (LitInt i2)) 
  IntoLaterNEnvs 1 Δ Δ' 
  envs_lookup i Δ' = Some (false, l  LitV (LitInt i1))%I 
  envs_simple_replace i false (Esnoc Enil i (l  LitV (LitInt (i1 + i2)))) Δ' = Some Δ'' 
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  envs_entails Δ'' (WP fill K (Lit (LitInt i1)) @ s; E {{ Φ }}) 
  envs_entails Δ (WP fill K (FAA (Lit (LitLoc l)) e2) @ s; E {{ Φ }}).
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Proof.
  rewrite /envs_entails=> ?????; subst.
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  rewrite -wp_bind. eapply wand_apply; first exact: (wp_faa _ _ _ i1 _ i2).
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  rewrite into_laterN_env_sound -later_sep envs_simple_replace_sound //; simpl.
  rewrite right_id. by apply later_mono, sep_mono_r, wand_mono.
Qed.
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Lemma tac_twp_faa Δ Δ' s E i K l i1 e2 i2 Φ :
  IntoVal e2 (LitV (LitInt i2)) 
  envs_lookup i Δ = Some (false, l  LitV (LitInt i1))%I 
  envs_simple_replace i false (Esnoc Enil i (l  LitV (LitInt (i1 + i2)))) Δ = Some Δ' 
  envs_entails Δ' (WP fill K (Lit (LitInt i1)) @ s; E [{ Φ }]) 
  envs_entails Δ (WP fill K (FAA (Lit (LitLoc l)) e2) @ s; E [{ Φ }]).
Proof.
  rewrite /envs_entails=> ????; subst.
  rewrite -twp_bind. eapply wand_apply; first exact: (twp_faa _ _ _ i1 _ i2).
  rewrite envs_simple_replace_sound //; simpl.
  rewrite right_id. by apply sep_mono_r, wand_mono.
Qed.
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End heap.

Tactic Notation "wp_apply" open_constr(lem) :=
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  iPoseProofCore lem as false true (fun H =>
    lazymatch goal with
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    | |- envs_entails _ (wp ?s ?E ?e ?Q) =>
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      reshape_expr e ltac:(fun K e' =>
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        wp_bind_core K; iApplyHyp H; try iNext; wp_expr_simpl) ||
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      lazymatch iTypeOf H with
      | Some (_,?P) => fail "wp_apply: cannot apply" P
      end
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    | |- envs_entails _ (twp ?s ?E ?e ?Q) =>
      reshape_expr e ltac:(fun K e' =>
        twp_bind_core K; iApplyHyp H; wp_expr_simpl) ||
      lazymatch iTypeOf H with
      | Some (_,?P) => fail "wp_apply: cannot apply" P
      end
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    | _ => fail "wp_apply: not a 'wp'"
    end).
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Tactic Notation "wp_alloc" ident(l) "as" constr(H) :=
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  let finish _ :=
    first [intros l | fail 1 "wp_alloc:" l "not fresh"];
      eexists; split;
        [env_cbv; reflexivity || fail "wp_alloc:" H "not fresh"
        |wp_expr_simpl; try wp_value_head] in
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  iStartProof;
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  lazymatch goal with
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  | |- envs_entails _ (wp ?s ?E ?e ?Q) =>
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    first
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      [reshape_expr e ltac:(fun K e' =>
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         eapply (tac_wp_alloc _ _ _ _ H K); [apply _|..])
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      |fail 1 "wp_alloc: cannot find 'Alloc' in" e];
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    [apply _
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    |finish ()]
  | |- envs_entails _ (twp ?s ?E ?e ?Q) =>
    first
      [reshape_expr e ltac:(fun K e' =>
         eapply (tac_twp_alloc _ _ _ H K); [apply _|..])
      |fail 1 "wp_alloc: cannot find 'Alloc' in" e];
    finish ()
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  | _ => fail "wp_alloc: not a 'wp'"
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  end.
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Tactic Notation "wp_alloc" ident(l) :=
  let H := iFresh in wp_alloc l as H.
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Tactic Notation "wp_load" :=
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  let solve_mapsto _ :=
    let l := match goal with |- _ = Some (_, (?l {_} _)%I) => l end in
    iAssumptionCore || fail "wp_load: cannot find" l "↦ ?" in
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  iStartProof;
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  lazymatch goal with
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  | |- envs_entails _ (wp ?s ?E ?e ?Q) =>
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    first
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      [reshape_expr e ltac:(fun K e' => eapply (tac_wp_load _ _ _ _ _ K))
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      |fail 1 "wp_load: cannot find 'Load' in" e];
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    [apply _
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    |solve_mapsto ()
    |wp_expr_simpl; try wp_value_head]
  | |- envs_entails _ (twp ?s ?E ?e ?Q) =>
    first
      [reshape_expr e ltac:(fun K e' => eapply (tac_twp_load _ _ _ _ K))
      |fail 1 "wp_load: cannot find 'Load' in" e];
    [solve_mapsto ()
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    |wp_expr_simpl; try wp_value_head]
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  | _ => fail "wp_load: not a 'wp'"
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  end.

Tactic Notation "wp_store" :=
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  let solve_mapsto _ :=
    let l := match goal with |- _ = Some (_, (?l {_} _)%I) => l end in
    iAssumptionCore || fail "wp_store: cannot find" l "↦ ?" in
  let finish _ :=
    wp_expr_simpl; try first [wp_pure (Seq (Lit LitUnit) _)|wp_value_head] in
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  iStartProof;
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  lazymatch goal with
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  | |- envs_entails _ (wp ?s ?E ?e ?Q) =>
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    first
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      [reshape_expr e ltac:(fun K e' =>
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         eapply (tac_wp_store _ _ _ _ _ _ K); [apply _|..])
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      |fail 1 "wp_store: cannot find 'Store' in" e];
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    [apply _
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    |solve_mapsto ()
    |env_cbv; reflexivity
    |finish ()]
  | |- envs_entails _ (twp ?s ?E ?e ?Q) =>
    first
      [reshape_expr e ltac:(fun K e' =>
         eapply (tac_twp_store _ _ _ _ _ K); [apply _|..])
      |fail 1 "wp_store: cannot find 'Store' in" e];
    [solve_mapsto ()
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    |env_cbv; reflexivity
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    |finish ()]
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  | _ => fail "wp_store: not a 'wp'"
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  end.

Tactic Notation "wp_cas_fail" :=
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  let solve_mapsto _ :=
    let l := match goal with |- _ = Some (_, (?l {_} _)%I) => l end in
    iAssumptionCore || fail "wp_cas_fail: cannot find" l "↦ ?" in
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  iStartProof;
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  lazymatch goal with
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  | |- envs_entails _ (wp ?s ?E ?e ?Q) =>
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    first
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      [reshape_expr e ltac:(fun K e' =>
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         eapply (tac_wp_cas_fail _ _ _ _ _ K); [apply _|apply _|..])
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      |fail 1 "wp_cas_fail: cannot find 'CAS' in" e];
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    [apply _
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    |solve_mapsto ()
    |try congruence
    |simpl; try wp_value_head]
  | |- envs_entails _ (twp ?s ?E ?e ?Q) =>
    first
      [reshape_expr e ltac:(fun K e' =>
         eapply (tac_twp_cas_fail _ _ _ _ K); [apply _|apply _|..])
      |fail 1 "wp_cas_fail: cannot find 'CAS' in" e];
    [solve_mapsto ()
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    |try congruence
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    |wp_expr_simpl; try wp_value_head]
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  | _ => fail "wp_cas_fail: not a 'wp'"
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  end.

Tactic Notation "wp_cas_suc" :=
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  let solve_mapsto _ :=
    let l := match goal with |- _ = Some (_, (?l {_} _)%I) => l end in
    iAssumptionCore || fail "wp_cas_suc: cannot find" l "↦ ?" in
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  iStartProof;
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  lazymatch goal with
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  | |- envs_entails _ (wp ?s ?E ?e ?Q) =>
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    first
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      [reshape_expr e ltac:(fun K e' =>
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         eapply (tac_wp_cas_suc _ _ _ _ _ _ K); [apply _|apply _|..])
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      |fail 1 "wp_cas_suc: cannot find 'CAS' in" e];
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    [apply _
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    |solve_mapsto ()
    |try congruence
    |env_cbv; reflexivity
    |simpl; try wp_value_head]
  | |- envs_entails _ (twp ?E ?e ?Q) =>
    first
      [reshape_expr e ltac:(fun K e' =>
         eapply (tac_twp_cas_suc _ _ _ _ _ K); [apply _|apply _|..])
      |fail 1 "wp_cas_suc: cannot find 'CAS' in" e];
    [solve_mapsto ()
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    |try congruence
    |env_cbv; reflexivity
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    |wp_expr_simpl; try wp_value_head]
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  | _ => fail "wp_cas_suc: not a 'wp'"
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  end.
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Tactic Notation "wp_faa" :=
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  let solve_mapsto _ :=
    let l := match goal with |- _ = Some (_, (?l {_} _)%I) => l end in
    iAssumptionCore || fail "wp_cas_suc: cannot find" l "↦ ?" in
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  iStartProof;
  lazymatch goal with
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  | |- envs_entails _ (wp ?s ?E ?e ?Q) =>
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    first
      [reshape_expr e ltac:(fun K e' =>
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         eapply (tac_wp_faa _ _ _ _ _ _ K); [apply _|..])
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      |fail 1 "wp_faa: cannot find 'CAS' in" e];
    [apply _
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    |solve_mapsto ()
    |env_cbv; reflexivity
    |wp_expr_simpl; try wp_value_head]
  | |- envs_entails _ (twp ?s ?E ?e ?Q) =>
    first
      [reshape_expr e ltac:(fun K e' =>
         eapply (tac_twp_faa _ _ _ _ _ K); [apply _|..])
      |fail 1 "wp_faa: cannot find 'CAS' in" e];
    [solve_mapsto ()
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    |env_cbv; reflexivity
    |wp_expr_simpl; try wp_value_head]
  | _ => fail "wp_faa: not a 'wp'"
  end.