Use `bin_log_FG_increment_logatomic` in the refinement proof

Using the lambdasubst hack-- instead of formulating
lemmas for (App e1 v), formulate them for (e1[x:=v]).

F_mu_ref_conc/hax.v

+(* the contents of this file sould belong elsewhere *)
+From iris.proofmode Require Import tactics.
+From iris_logrel.F_mu_ref_conc Require Import lang subst tactics rules rules_binary logrel_binary.
+
+Definition lamsubst (e : expr) (v : val) : expr :=
+  match e with
+  | Rec (BNamed f) x e' => lang.subst f e (subst' x (of_val v) e')
+  | Rec BAnon x e' => subst' x (of_val v) e'
+  | _ => e
+  end.
+
+Ltac inv_head_step :=
+  repeat match goal with
+  | _ => progress simplify_map_eq/= (* simplify memory stuff *)
+  | H : to_val _ = Some _ |- _ => apply of_to_val in H
+  | H : _ = of_val ?v |- _ =>
+     is_var v; destruct v; first[discriminate H|injection H as H]
+  | H : head_step ?e _ _ _ _ |- _ =>
+     try (is_var e; fail 1); (* inversion yields many goals if [e] is a variable
+     and can thus better be avoided. *)
+     inversion H; subst; clear H
+  end.
+
+Section hax.
+  Context `{heapIG Σ, cfgSG Σ}.
+  Notation D := (prodC valC valC -n> iProp Σ).
+  Implicit Types Δ : listC D.
+
+  Lemma bin_log_related_arrow Γ (f x f' x' : binder) (e e' : expr) (τ τ' : type)
+    (Hclosed  : Closed ∅ (rec: f x := e)%E )
+    (Hclosed' : Closed ∅ (rec: f' x' := e')%E) :
+
+    □(∀ Δ vv, ⟦ τ ⟧ Δ vv -∗
+      ∅ ⊨ App (Rec f x e) (of_val (vv.1)) ≤log≤ App (Rec f' x' e') (of_val (vv.2)) : τ') -∗
+    Γ ⊨ (Rec f x e) ≤log≤ (Rec f' x' e') : TArrow τ τ'.
+  Proof.
+    iIntros "#H".
+    iIntros (Δ vvs ρ) "#Hs #HΓ"; iIntros (j K) "Hj".
+    cbn-[subst_p]. rewrite /env_subst !Closed_subst_p_id.
+    iModIntro. iApply wp_value; auto.
+    { simpl. erewrite decide_left. done. }
+    iExists (RecV f' x' e'). simpl.
+    iFrame "Hj". iAlways. iIntros (vv) "Hvv".
+    iSpecialize ("H" $! Δ vv with "Hvv").
+    iSpecialize ("H" $! Δ with "Hs []").
+    { iAlways. iApply interp_env_nil. }
+    rewrite !fmap_empty /env_subst !subst_p_empty. done.
+    Unshelve. all: rewrite /Closed /= in Hclosed Hclosed'; eauto.
+  Qed.
+
+  Lemma weird_bind e Q :
+    WP App e #() {{ Q }} ⊢ WP e {{ v, WP (App v #()) {{ Q }} }}.
+  Proof.
+    (* ugh, turns out this is just the inverse bind:
+       Check (wp_bind_inv (fun v => App v #())). *)
+    iLöb as "IH" forall (e).
+    iIntros "wp".
+    rewrite (wp_unfold _ e) /wp_pre /=.
+    remember (to_val e) as eval. destruct eval.
+    - symmetry in Heqeval. rewrite -(of_to_val _ _ Heqeval). eauto.
+    - iIntros (σ1) "Hσ1".
+      rewrite wp_unfold /wp_pre /=.
+      iMod ("wp" $! σ1 with "Hσ1") as "[r wp]". iModIntro.
+      iDestruct "r" as %er.
+      assert (reducible e σ1).
+      { symmetry in Heqeval.
+        unfold reducible in er.
+        destruct er as (e2 & σ2 & efs & Hpst).
+        inversion Hpst; simpl in *; subst; clear Hpst.
+        admit. }
+      iSplitR; eauto.
+      iNext.
+      iIntros (e2 σ2 efs) "Hpst". iDestruct "Hpst" as %Hpst.
+      iSpecialize ("wp" $! (App e2 #()) σ2 efs).
+      iAssert (⌜prim_step (e #()%E) σ1 (e2 #()%E) σ2 efs⌝)%I as "Hprim'".
+      { iPureIntro.
+        inversion Hpst; simpl in *; subst; clear Hpst.
+        eapply (Ectx_step _ σ1 _ σ2 efs (K++[AppLCtx (#())%E])); simpl; eauto.
+          by rewrite fill_app.
+          by rewrite fill_app. }
+      iMod ("wp" with "Hprim'") as "[$ [wp $]]". iModIntro.
+      by iApply "IH".
+  Admitted.
+
+  (* Lemma wp_step_back Γ (e t : expr) (x : string) (v ev : val) τ : *)
+  (*   Closed ∅ (Lam x e) → *)
+  (*   to_val (lang.subst x (of_val v) e) = Some ev → *)
+  (*   Γ ⊨ (App (Lam x e) v) ≤log≤ t : τ *)
+  (*   ⊢ Γ ⊨ (lang.subst x (of_val v) e) ≤log≤ t : τ. *)
+  (* Proof. *)
+  (*   iIntros (??) "Hr". *)
+  (*   Transparent bin_log_related. *)
+  (*   iIntros (Δ vvs ρ) "#Hs #HΓ"; iIntros (j K) "Hj". *)
+  (*   cbn-[subst_p].  *)
+  (*   (* assert (Closed ∅ (lang.subst x v e)). *) *)
+  (*   (* { eapply is_closed_subst_preserve; eauto. solve_closed. } *) *)
+  (*   rewrite /env_subst !Closed_subst_p_id. *)
+  (*   iSpecialize ("Hr" with "Hs []"). *)
+  (*   { iAlways. by iFrame. } *)
+  (*   rewrite /env_subst. erewrite (Closed_subst_p_id (fst <$> vvs)); last first. *)
+  (*   { rewrite /Closed. simpl. *)
+  (*     rewrite /Closed /= in H1. split_and; eauto; try solve_closed. } *)
+  (*   iMod ("Hr" with "Hj") as "Hr". *)
+  (*   iModIntro. simpl. *)
+  (*   rewrite {1}wp_unfold /wp_pre /=. *)
+  (*   iApply wp_value; eauto. *)
+  (*   iApply (wp_bind_inv in "Hr". *)
+End hax.

_CoqProject

@@ -5,6 +5,7 @@ prelude/base.v
 F_mu_ref_conc/binder.v
 F_mu_ref_conc/lang.v
 F_mu_ref_conc/subst.v
+F_mu_ref_conc/hax.v
 F_mu_ref_conc/reflection.v
 F_mu_ref_conc/rules.v
 F_mu_ref_conc/typing.v