diff --git a/algebra/upred_tactics.v b/algebra/upred_tactics.v
index 5e7e2fb2ea6e8a36f0b8b070eaa028df15607f49..4077ae94a0a06eb2cd1ffa008327399c67fee3dd 100644
--- a/algebra/upred_tactics.v
+++ b/algebra/upred_tactics.v
@@ -1,5 +1,6 @@
From algebra Require Export upred.
From algebra Require Export upred_big_op.
+Import uPred.
Module upred_reflection. Section upred_reflection.
Context {M : cmraT}.
@@ -89,7 +90,7 @@ Module upred_reflection. Section upred_reflection.
Proof.
intros ??. rewrite !eval_flatten.
rewrite (flatten_cancel e1 e1' ns) // (flatten_cancel e2 e2' ns) //; csimpl.
- rewrite !fmap_app !big_sep_app. apply uPred.sep_mono_r.
+ rewrite !fmap_app !big_sep_app. apply sep_mono_r.
Qed.
Class Quote (Σ1 Σ2 : list (uPred M)) (P : uPred M) (e : expr) := {}.
@@ -144,3 +145,98 @@ Tactic Notation "ecancel" open_constr(Ps) :=
| |- @uPred_entails ?M _ _ =>
close Ps (@nil (uPred M)) ltac:(fun Qs => cancel Qs)
end.
+
+(* Some more generic uPred tactics.
+ TODO: Naming. *)
+
+Ltac revert_intros tac :=
+ lazymatch goal with
+ | |- ∀ _, _ => let H := fresh in intro H; revert_intros tac; revert H
+ | |- _ => tac
+ end.
+
+(** Assumes a goal of the shape P ⊑ ▷ Q. Alterantively, if Q
+ is built of ★, ∧, ∨ with ▷ in all branches; that will work, too.
+ Will turn this goal into P ⊑ Q and strip ▷ in P below ★, ∧, ∨. *)
+Ltac u_strip_later :=
+ let rec strip :=
+ lazymatch goal with
+ | |- (_ ★ _) ⊑ ▷ _ =>
+ etrans; last (eapply equiv_entails_sym, later_sep);
+ apply sep_mono; strip
+ | |- (_ ∧ _) ⊑ ▷ _ =>
+ etrans; last (eapply equiv_entails_sym, later_and);
+ apply sep_mono; strip
+ | |- (_ ∨ _) ⊑ ▷ _ =>
+ etrans; last (eapply equiv_entails_sym, later_or);
+ apply sep_mono; strip
+ | |- ▷ _ ⊑ ▷ _ => apply later_mono; reflexivity
+ | |- _ ⊑ ▷ _ => apply later_intro; reflexivity
+ end
+ in let rec shape_Q :=
+ lazymatch goal with
+ | |- _ ⊑ (_ ★ _) =>
+ (* Force the later on the LHS to be top-level, matching laters
+ below ★ on the RHS *)
+ etrans; first (apply equiv_entails, later_sep; reflexivity);
+ (* Match the arm recursively *)
+ apply sep_mono; shape_Q
+ | |- _ ⊑ (_ ∧ _) =>
+ etrans; first (apply equiv_entails, later_and; reflexivity);
+ apply sep_mono; shape_Q
+ | |- _ ⊑ (_ ∨ _) =>
+ etrans; first (apply equiv_entails, later_or; reflexivity);
+ apply sep_mono; shape_Q
+ | |- _ ⊑ ▷ _ => apply later_mono; reflexivity
+ (* We fail if we don't find laters in all branches. *)
+ end
+ in revert_intros ltac:(etrans; [|shape_Q];
+ etrans; last eapply later_mono; first solve [ strip ]).
+
+(** Transforms a goal of the form ∀ ..., ?0... → ?1 ⊑ ?2
+ into True ⊑ ∀..., ■?0... → ?1 → ?2, applies tac, and
+ the moves all the assumptions back. *)
+Ltac u_revert_all :=
+ lazymatch goal with
+ | |- ∀ _, _ => let H := fresh in intro H; u_revert_all;
+ (* TODO: Really, we should distinguish based on whether this is a
+ dependent function type or not. Right now, we distinguish based
+ on the sort of the argument, which is suboptimal. *)
+ first [ apply (const_intro_impl _ _ _ H); clear H
+ | revert H; apply forall_elim']
+ | |- ?C ⊑ _ => trans (True ∧ C)%I;
+ first (apply equiv_entails_sym, left_id, _; reflexivity);
+ apply impl_elim_l'
+ end.
+
+(** This starts on a goal of the form ∀ ..., ?0... → ?1 ⊑ ?2.
+ It applies löb where all the Coq assumptions have been turned into logical
+ assumptions, then moves all the Coq assumptions back out to the context,
+ applies [tac] on the goal (now of the form _ ⊑ _), and then reverts the
+ Coq assumption so that we end up with the same shape as where we started,
+ but with an additional assumption ★-ed to the context *)
+Ltac u_löb tac :=
+ u_revert_all;
+ (* Add a box *)
+ etrans; last (eapply always_elim; reflexivity);
+ (* We now have a goal for the form True ⊑ P, with the "original" conclusion
+ being locked. *)
+ apply löb_strong; etransitivity;
+ first (apply equiv_entails, left_id, _; reflexivity);
+ apply: always_intro;
+ (* Now introduce again all the things that we reverted, and at the bottom,
+ do the work *)
+ let rec go :=
+ lazymatch goal with
+ | |- _ ⊑ (∀ _, _) => apply forall_intro;
+ let H := fresh in intro H; go; revert H
+ | |- _ ⊑ (■_ → _) => apply impl_intro_l, const_elim_l;
+ let H := fresh in intro H; go; revert H
+ (* This is the "bottom" of the goal, where we see the impl introduced
+ by u_revert_all as well as the ▷ from löb_strong and the □ we added. *)
+ | |- ▷ □ ?R ⊑ (?L → _) => apply impl_intro_l;
+ trans (L ★ ▷ □ R)%I;
+ first (eapply equiv_entails, always_and_sep_r, _; reflexivity);
+ tac
+ end
+ in go.
diff --git a/heap_lang/wp_tactics.v b/heap_lang/wp_tactics.v
index d1495340c3695124b68e858a6382c00c8476438f..ef6461201ff25601e9c2ce098352aeb638bd1246 100644
--- a/heap_lang/wp_tactics.v
+++ b/heap_lang/wp_tactics.v
@@ -1,100 +1,7 @@
+From algebra Require Export upred_tactics.
From heap_lang Require Export tactics substitution.
Import uPred.
-(* TODO: The next few tactics are not wp-specific at all. They should move elsewhere. *)
-
-Ltac revert_intros tac :=
- lazymatch goal with
- | |- ∀ _, _ => let H := fresh in intro H; revert_intros tac; revert H
- | |- _ => tac
- end.
-
-(** Assumes a goal of the shape P ⊑ ▷ Q. Alterantively, if Q
- is built of ★, ∧, ∨ with ▷ in all branches; that will work, too.
- Will turn this goal into P ⊑ Q and strip ▷ in P below ★, ∧, ∨. *)
-Ltac u_strip_later :=
- let rec strip :=
- lazymatch goal with
- | |- (_ ★ _) ⊑ ▷ _ =>
- etrans; last (eapply equiv_entails_sym, later_sep);
- apply sep_mono; strip
- | |- (_ ∧ _) ⊑ ▷ _ =>
- etrans; last (eapply equiv_entails_sym, later_and);
- apply sep_mono; strip
- | |- (_ ∨ _) ⊑ ▷ _ =>
- etrans; last (eapply equiv_entails_sym, later_or);
- apply sep_mono; strip
- | |- ▷ _ ⊑ ▷ _ => apply later_mono; reflexivity
- | |- _ ⊑ ▷ _ => apply later_intro; reflexivity
- end
- in let rec shape_Q :=
- lazymatch goal with
- | |- _ ⊑ (_ ★ _) =>
- (* Force the later on the LHS to be top-level, matching laters
- below ★ on the RHS *)
- etrans; first (apply equiv_entails, later_sep; reflexivity);
- (* Match the arm recursively *)
- apply sep_mono; shape_Q
- | |- _ ⊑ (_ ∧ _) =>
- etrans; first (apply equiv_entails, later_and; reflexivity);
- apply sep_mono; shape_Q
- | |- _ ⊑ (_ ∨ _) =>
- etrans; first (apply equiv_entails, later_or; reflexivity);
- apply sep_mono; shape_Q
- | |- _ ⊑ ▷ _ => apply later_mono; reflexivity
- (* We fail if we don't find laters in all branches. *)
- end
- in revert_intros ltac:(etrans; [|shape_Q];
- etrans; last eapply later_mono; first solve [ strip ]).
-
-(** Transforms a goal of the form ∀ ..., ?0... → ?1 ⊑ ?2
- into True ⊑ ∀..., ■?0... → ?1 → ?2, applies tac, and
- the moves all the assumptions back. *)
-Ltac u_revert_all :=
- lazymatch goal with
- | |- ∀ _, _ => let H := fresh in intro H; u_revert_all;
- (* TODO: Really, we should distinguish based on whether this is a
- dependent function type or not. Right now, we distinguish based
- on the sort of the argument, which is suboptimal. *)
- first [ apply (const_intro_impl _ _ _ H); clear H
- | revert H; apply forall_elim']
- | |- ?C ⊑ _ => trans (True ∧ C)%I;
- first (apply equiv_entails_sym, left_id, _; reflexivity);
- apply impl_elim_l'
- end.
-
-(** This starts on a goal of the form ∀ ..., ?0... → ?1 ⊑ ?2.
- It applies löb where all the Coq assumptions have been turned into logical
- assumptions, then moves all the Coq assumptions back out to the context,
- applies [tac] on the goal (now of the form _ ⊑ _), and then reverts the
- Coq assumption so that we end up with the same shape as where we started,
- but with an additional assumption ★-ed to the context *)
-Ltac u_löb tac :=
- u_revert_all;
- (* Add a box *)
- etrans; last (eapply always_elim; reflexivity);
- (* We now have a goal for the form True ⊑ P, with the "original" conclusion
- being locked. *)
- apply löb_strong; etransitivity;
- first (apply equiv_entails, left_id, _; reflexivity);
- apply: always_intro;
- (* Now introduce again all the things that we reverted, and at the bottom,
- do the work *)
- let rec go :=
- lazymatch goal with
- | |- _ ⊑ (∀ _, _) => apply forall_intro;
- let H := fresh in intro H; go; revert H
- | |- _ ⊑ (■_ → _) => apply impl_intro_l, const_elim_l;
- let H := fresh in intro H; go; revert H
- (* This is the "bottom" of the goal, where we see the impl introduced
- by u_revert_all as well as the ▷ from löb_strong and the □ we added. *)
- | |- ▷ □ ?R ⊑ (?L → _) => apply impl_intro_l;
- trans (L ★ ▷ □ R)%I;
- first (eapply equiv_entails, always_and_sep_r, _; reflexivity);
- tac
- end
- in go.
-
(** wp-specific helper tactics *)
(* First try to productively strip off laters; if that fails, at least
cosmetically get rid of laters in the conclusion. *)