logrel_binary.v 10.6 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1 2
From iris.proofmode Require Import tactics.
From iris.program_logic Require Export weakestpre.
3
From iris_logrel.F_mu_ref_conc Require Export rules_binary typing.
4 5
Import uPred.

6 7 8 9 10 11 12 13 14 15 16 17 18 19
(* HACK: move somewhere else *)
Ltac auto_proper ::=
  (* Deal with "pointwise_relation" *)
  repeat lazymatch goal with
  | |- pointwise_relation _ _ _ _ => intros ?
  end;
  (* Normalize away equalities. *)
  repeat match goal with
  | H : _ {_} _ |-  _ => apply (timeless_iff _ _) in H
  | _ => progress simplify_eq
  end;
  (* repeatedly apply congruence lemmas and use the equalities in the hypotheses. *)
  try (f_equiv; fast_done || auto_proper).

20 21
Definition logN : namespace := nroot .@ "logN".

22 23
(** interp : is a unary logical relation. *)
Section logrel.
24
  Context `{heapIG Σ, cfgSG Σ}.
25 26 27 28 29
  Notation D := (prodC valC valC -n> iPropG lang Σ).
  Implicit Types τi : D.
  Implicit Types Δ : listC D.
  Implicit Types interp : listC D  D.

30 31 32 33 34
  Definition interp_expr (τi : listC D -n> D) (Δ : listC D)
      (ee : expr * expr) : iPropG lang Σ := ( j K,
    j  fill K (ee.2) 
    WP ee.1 {{ v,  v', j  fill K (# v')  τi Δ (v, v') }})%I.

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61
  Program Definition ctx_lookup (x : var) : listC D -n> D := λne Δ,
    from_option id (cconst True)%I (Δ !! x).
  Solve Obligations with solve_proper_alt.

  Program Definition interp_unit : listC D -n> D := λne Δ ww,
    (ww.1 = UnitV  ww.2 = UnitV)%I.
  Solve Obligations with solve_proper_alt.
  Program Definition interp_nat : listC D -n> D := λne Δ ww,
    ( n : nat, ww.1 = v n  ww.2 = v n)%I.
  Solve Obligations with solve_proper.
  Program Definition interp_bool : listC D -n> D := λne Δ ww,
    ( b : bool, ww.1 = v b  ww.2 = v b)%I.
  Solve Obligations with solve_proper.

  Program Definition interp_prod
      (interp1 interp2 : listC D -n> D) : listC D -n> D := λne Δ ww,
    ( vv1 vv2, ww = (PairV (vv1.1) (vv2.1), PairV (vv1.2) (vv2.2)) 
                interp1 Δ vv1  interp2 Δ vv2)%I.
  Solve Obligations with solve_proper.

  Program Definition interp_sum
      (interp1 interp2 : listC D -n> D) : listC D -n> D := λne Δ ww,
    (( vv, ww = (InjLV (vv.1), InjLV (vv.2))  interp1 Δ vv) 
     ( vv, ww = (InjRV (vv.1), InjRV (vv.2))  interp2 Δ vv))%I.
  Solve Obligations with solve_proper.

  Program Definition interp_arrow
62 63 64 65 66
          (interp1 interp2 : listC D -n> D) : listC D -n> D :=
    λne Δ ww,
    (  vv, interp1 Δ vv 
             interp_expr
               interp2 Δ (App (# ww.1) (# vv.1), App (# ww.2) (# vv.2)))%I.
67
  Solve Obligations with solve_proper.
68 69 70 71 72
  Next Obligation.
  Proof.
    intros d d' n x x' Hx z. eapply always_ne.
    apply forall_ne => z'. apply impl_ne. by rewrite Hx. solve_proper.
  Qed.
73 74 75

  Program Definition interp_forall
      (interp : listC D -n> D) : listC D -n> D := λne Δ ww,
76 77 78 79
    (  τi,
          (  ww, PersistentP (τi ww)) 
          interp_expr
            interp (τi :: Δ) (TApp (# ww.1), TApp (# ww.2)))%I.
80
  Solve Obligations with solve_proper.
81 82 83 84 85
  Next Obligation.
  Proof.
    intros d n x x' Hx z. eapply always_ne.
    apply forall_ne => z'. apply impl_ne; trivial. solve_proper.
  Qed.
86 87 88 89 90 91 92 93

  Program Definition interp_rec1
      (interp : listC D -n> D) (Δ : listC D) (τi : D) : D := λne ww,
    (  vv, ww = (FoldV (vv.1), FoldV (vv.2))   interp (τi :: Δ) vv)%I.
  Solve Obligations with solve_proper.

  Global Instance interp_rec1_contractive
    (interp : listC D -n> D) (Δ : listC D) : Contractive (interp_rec1 interp Δ).
94
  Proof.
95 96 97
    intros n τi1 τi2 Hτi ww; cbn.
    apply always_ne, exist_ne; intros vv; apply and_ne; trivial.
    apply later_contractive =>i Hi. by rewrite Hτi.
98 99
  Qed.

100 101
  Program Definition interp_rec (interp : listC D -n> D) : listC D -n> D := λne Δ,
    fixpoint (interp_rec1 interp Δ).
102
  Next Obligation.
103 104 105 106 107 108 109 110 111 112
    intros interp n Δ1 Δ2 HΔ; apply fixpoint_ne => τi ww. solve_proper.
  Qed.

  Program Definition interp_ref_inv (ll : loc * loc) : D -n> iPropG lang Σ := λne τi,
    ( vv, ll.1 ↦ᵢ vv.1  ll.2 ↦ₛ vv.2  τi vv)%I.
  Solve Obligations with solve_proper.

  Program Definition interp_ref
      (interp : listC D -n> D) : listC D -n> D := λne Δ ww,
    ( ll, ww = (LocV (ll.1), LocV (ll.2)) 
113
           inv (logN .@ ll) (interp_ref_inv ll (interp Δ)))%I.
114 115 116 117 118 119 120 121 122 123 124 125 126 127
  Solve Obligations with solve_proper.

  Fixpoint interp (τ : type) : listC D -n> D :=
    match τ return _ with
    | TUnit => interp_unit
    | TNat => interp_nat
    | TBool => interp_bool
    | TProd τ1 τ2 => interp_prod (interp τ1) (interp τ2)
    | TSum τ1 τ2 => interp_sum (interp τ1) (interp τ2)
    | TArrow τ1 τ2 => interp_arrow (interp τ1) (interp τ2)
    | TVar x => ctx_lookup x
    | TForall τ' => interp_forall (interp τ')
    | TRec τ' => interp_rec (interp τ')
    | Tref τ' => interp_ref (interp τ')
128
    end.
129
  Notation "⟦ τ ⟧" := (interp τ).
130

131 132 133 134 135 136
  Definition interp_env (Γ : list type)
      (Δ : listC D) (vvs : list (val * val)) : iPropG lang Σ :=
    (length Γ = length vvs  [] zip_with (λ τ,  τ  Δ) Γ vvs)%I.
  Notation "⟦ Γ ⟧*" := (interp_env Γ).

  Class env_PersistentP Δ :=
137
    ctx_persistentP : Forall (λ τi,  vv, PersistentP (τi vv)) Δ.
138
  Global Instance ctx_persistent_nil : env_PersistentP [].
139 140
  Proof. by constructor. Qed.
  Global Instance ctx_persistent_cons τi Δ :
141
    ( vv, PersistentP (τi vv))  env_PersistentP Δ  env_PersistentP (τi :: Δ).
142 143
  Proof. by constructor. Qed.
  Global Instance ctx_persistent_lookup Δ x vv :
144
    env_PersistentP Δ  PersistentP (ctx_lookup x Δ vv).
145
  Proof. intros HΔ; revert x; induction HΔ=>-[|?] /=; apply _. Qed.
146 147
  Global Instance interp_persistent τ Δ vv :
    env_PersistentP Δ  PersistentP ( τ  Δ vv).
148
  Proof.
149 150 151
    revert vv Δ; induction τ=> vv Δ HΔ; simpl; try apply _.
    rewrite /PersistentP /interp_rec fixpoint_unfold /interp_rec1 /=.
    by apply always_intro'.
152
  Qed.
153 154
  Global Instance interp_env_persistent Γ Δ vvs :
    env_PersistentP Δ  PersistentP ( Γ * Δ vvs) := _.
155

156
  Lemma interp_weaken Δ1 Π Δ2 τ :
157 158
     τ.[iter (length Δ1) up (ren (+ length Π))]  (Δ1 ++ Π ++ Δ2)
      τ  (Δ1 ++ Δ2).
159
  Proof.
160 161 162 163 164 165
    revert Δ1 Π Δ2. induction τ=> Δ1 Π Δ2; simpl; auto.
    - intros ww; simpl; properness; auto.
    - intros ww; simpl; properness; auto.
    - intros ww; simpl; properness; auto.
    - intros ww; simpl; properness; auto. by apply IHτ1. by apply IHτ2.
    - intros ww; simpl; properness; auto. by apply IHτ1. by apply IHτ2.
166 167
    - unfold interp_expr.
      intros ww; simpl; properness; auto. by apply IHτ1. by apply IHτ2.
168 169 170 171 172 173
    - apply fixpoint_proper=> τi ww /=.
      properness; auto. apply (IHτ (_ :: _)).
    - rewrite iter_up; destruct lt_dec as [Hl | Hl]; simpl.
      { by rewrite !lookup_app_l. }
      change (uPredC (iResUR lang _)) with (iPropG lang Σ).
      rewrite !lookup_app_r; [|lia ..]. do 2 f_equiv. lia. done.
174 175
    - unfold interp_expr.
      intros ww; simpl; properness; auto. by apply (IHτ (_ :: _)).
176 177 178 179
    - intros ww; simpl; properness; auto. by apply IHτ.
  Qed.

  Lemma interp_subst_up Δ1 Δ2 τ τ' :
180 181
     τ  (Δ1 ++ interp τ' Δ2 :: Δ2)
      τ.[iter (length Δ1) up (τ' .: ids)]  (Δ1 ++ Δ2).
182
  Proof.
183 184 185 186 187 188
    revert Δ1 Δ2; induction τ=> Δ1 Δ2; simpl; auto.
    - intros ww; simpl; properness; auto.
    - intros ww; simpl; properness; auto.
    - intros ww; simpl; properness; auto.
    - intros ww; simpl; properness; auto. by apply IHτ1. by apply IHτ2.
    - intros ww; simpl; properness; auto. by apply IHτ1. by apply IHτ2.
189 190
    - unfold interp_expr.
      intros ww; simpl; properness; auto. by apply IHτ1. by apply IHτ2.
191 192 193 194 195 196 197 198 199 200
    - apply fixpoint_proper=> τi ww /=.
      properness; auto. apply (IHτ (_ :: _)).
    - rewrite iter_up; destruct lt_dec as [Hl | Hl]; simpl.
      { by rewrite !lookup_app_l. }
      change (uPredC (iResUR lang _)) with (iPropG lang Σ).
      rewrite !lookup_app_r; [|lia ..].
      destruct (x - length Δ1) as [|n] eqn:?; simpl.
      { symmetry. asimpl. apply (interp_weaken [] Δ1 Δ2 τ'). }
      change (uPredC (iResUR lang _)) with (iPropG lang Σ).
      rewrite !lookup_app_r; [|lia ..]. do 2 f_equiv. lia. done.
201 202
    - unfold interp_expr.
      intros ww; simpl; properness; auto. apply (IHτ (_ :: _)).
203 204 205
    - intros ww; simpl; properness; auto. by apply IHτ.
  Qed.

206
  Lemma interp_subst Δ2 τ τ' :  τ  ( τ'  Δ2 :: Δ2)   τ.[τ'/]  Δ2.
207 208
  Proof. apply (interp_subst_up []). Qed.

209 210 211 212 213
  Lemma interp_env_length Δ Γ vvs :  Γ * Δ vvs  length Γ = length vvs.
  Proof. by iIntros "[% ?]". Qed.

  Lemma interp_env_Some_l Δ Γ vvs x τ :
    Γ !! x = Some τ   Γ * Δ vvs   vv, vvs !! x = Some vv   τ  Δ vv.
214
  Proof.
215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237
    iIntros {?} "[Hlen HΓ]"; iDestruct "Hlen" as %Hlen.
    destruct (lookup_lt_is_Some_2 vvs x) as [v Hv].
    { by rewrite -Hlen; apply lookup_lt_Some with τ. }
    iExists v; iSplit. done. iApply (big_and_elem_of with "HΓ").
    apply elem_of_list_lookup_2 with x.
    rewrite lookup_zip_with; by simplify_option_eq.
  Qed.

  Lemma interp_env_nil Δ : True   [] * Δ [].
  Proof. iIntros ""; iSplit; auto. Qed.
  Lemma interp_env_cons Δ Γ vvs τ vv :
     τ :: Γ * Δ (vv :: vvs) ⊣⊢  τ  Δ vv   Γ * Δ vvs.
  Proof.
    rewrite /interp_env /= (assoc _ ( _  _ _)) -(comm _ (_ = _)%I) -assoc.
    by apply and_proper; [apply pure_proper; omega|].
  Qed.

  Lemma interp_env_ren Δ (Γ : list type) vvs τi :
     subst (ren (+1)) <$> Γ * (τi :: Δ) vvs ⊣⊢  Γ * Δ vvs.
  Proof.
    apply and_proper; [apply pure_proper; by rewrite fmap_length|].
    revert Δ vvs τi; induction Γ=> Δ [|v vs] τi; csimpl; auto.
    apply and_proper; auto. apply (interp_weaken [] [τi] Δ).
238 239
  Qed.

240
  Lemma interp_EqType_agree τ v v' Δ :
241
    env_PersistentP Δ  EqType τ  interp τ Δ (v, v')   (v = v').
242
  Proof.
243
    intros ? Hτ; revert v v'; induction Hτ; iIntros {v v'} "#H1 /=".
Robbert Krebbers's avatar
Robbert Krebbers committed
244 245 246 247
    - by iDestruct "H1" as "[% %]"; subst.
    - by iDestruct "H1" as {n} "[% %]"; subst.
    - by iDestruct "H1" as {b} "[% %]"; subst.
    - iDestruct "H1" as { [??] [??] } "[% [H1 H2]]"; simplify_eq/=.
248
      rewrite IHHτ1 IHHτ2.
Robbert Krebbers's avatar
Robbert Krebbers committed
249
      by iDestruct "H1" as "%"; iDestruct "H2" as "%"; subst.
250
    - iDestruct "H1" as "[H1|H1]".
Robbert Krebbers's avatar
Robbert Krebbers committed
251
      + iDestruct "H1" as { [??] } "[% H1]"; simplify_eq/=.
252
        rewrite IHHτ1. by iDestruct "H1" as "%"; subst.
Robbert Krebbers's avatar
Robbert Krebbers committed
253
      + iDestruct "H1" as { [??] } "[% H1]"; simplify_eq/=.
254
        rewrite IHHτ2. by iDestruct "H1" as "%"; subst.
255
  Qed.
256
End logrel.
257 258 259

Typeclasses Opaque interp_env.
Notation "⟦ τ ⟧" := (interp τ).
260
Notation "⟦ τ ⟧ₑ" := (interp_expr (interp τ)).
261
Notation "⟦ Γ ⟧*" := (interp_env Γ).