From a14077232ef6035a0c900b6fb766e5b9b5281e17 Mon Sep 17 00:00:00 2001
From: Robbert Krebbers <mail@robbertkrebbers.nl>
Date: Wed, 24 Feb 2016 18:10:00 +0100
Subject: [PATCH] Nice notation for mkSet.

---
 barrier/proof.v    |  8 ++++----
 barrier/protocol.v | 26 ++++++++++++--------------
 prelude/sets.v     | 29 +++++++++++++++++------------
 3 files changed, 33 insertions(+), 30 deletions(-)

diff --git a/barrier/proof.v b/barrier/proof.v
index 130e6d2d7..a548ed362 100644
--- a/barrier/proof.v
+++ b/barrier/proof.v
@@ -163,8 +163,8 @@ Proof.
     + apply pvs_mono.
       rewrite -sts_ownS_op; eauto using i_states_closed, low_states_closed.
       set_solver.
-    + move=> /= t. rewrite !mkSet_elem_of; intros [<-|<-]; set_solver.
-    + rewrite !mkSet_elem_of; set_solver.
+    + move=> /= t. rewrite !elem_of_mkSet; intros [<-|<-]; set_solver.
+    + rewrite !elem_of_mkSet; set_solver.
     + auto using sts.closed_op, i_states_closed, low_states_closed.
 Qed.
 
@@ -293,7 +293,7 @@ Proof.
     apply sep_mono.
     * rewrite -sts_ownS_op; eauto using i_states_closed.
       + apply sts_own_weaken; eauto using sts.closed_op, i_states_closed.
-        rewrite !mkSet_elem_of; set_solver.
+        rewrite !elem_of_mkSet; set_solver.
       + set_solver.
     * rewrite const_equiv // !left_id.
       rewrite {1}[heap_ctx _]always_sep_dup {1}[sts_ctx _ _ _]always_sep_dup.
@@ -319,7 +319,7 @@ Proof.
     apply sep_mono.
     * rewrite -sts_ownS_op; eauto using i_states_closed.
       + apply sts_own_weaken; eauto using sts.closed_op, i_states_closed.
-        rewrite !mkSet_elem_of; set_solver.
+        rewrite !elem_of_mkSet; set_solver.
       + set_solver.
     * rewrite const_equiv // !left_id.
       rewrite {1}[heap_ctx _]always_sep_dup {1}[sts_ctx _ _ _]always_sep_dup.
diff --git a/barrier/protocol.v b/barrier/protocol.v
index 0f092340d..780ec7d72 100644
--- a/barrier/protocol.v
+++ b/barrier/protocol.v
@@ -18,7 +18,7 @@ Inductive prim_step : relation state :=
   | ChangePhase I : prim_step (State Low I) (State High I).
 
 Definition change_tok (I : gset gname) : set token :=
-  mkSet (λ t, match t with Change i => i ∉ I | Send => False end).
+  {[ t | match t with Change i => i ∉ I | Send => False end ]}.
 Definition send_tok (p : phase) : set token :=
   match p with Low => ∅ | High => {[ Send ]} end.
 Definition tok (s : state) : set token :=
@@ -28,29 +28,27 @@ Global Arguments tok !_ /.
 Canonical Structure sts := sts.STS prim_step tok.
 
 (* The set of states containing some particular i *)
-Definition i_states (i : gname) : set state :=
-  mkSet (λ s, i ∈ state_I s).
+Definition i_states (i : gname) : set state := {[ s | i ∈ state_I s ]}.
 
 (* The set of low states *)
-Definition low_states : set state :=
-  mkSet (λ s, if state_phase s is Low then True else False).
+Definition low_states : set state := {[ s | state_phase s = Low ]}.
 
 Lemma i_states_closed i : sts.closed (i_states i) {[ Change i ]}.
 Proof.
   split.
-  - move=>[p I]. rewrite /= !mkSet_elem_of /= =>HI.
+  - move=>[p I]. rewrite /= !elem_of_mkSet /= =>HI.
     destruct p; set_solver by eauto.
   - (* If we do the destruct of the states early, and then inversion
        on the proof of a transition, it doesn't work - we do not obtain
        the equalities we need. So we destruct the states late, because this
        means we can use "destruct" instead of "inversion". *)
-    move=>s1 s2. rewrite !mkSet_elem_of.
+    move=>s1 s2. rewrite !elem_of_mkSet.
     intros Hs1 [T1 T2 Hdisj Hstep'].
     inversion_clear Hstep' as [? ? ? ? Htrans _ _ Htok].
     destruct Htrans; simpl in *; last done.
     move: Hs1 Hdisj Htok. rewrite elem_of_equiv_empty elem_of_equiv.
     move=> ? /(_ (Change i)) Hdisj /(_ (Change i)); move: Hdisj.
-    rewrite elem_of_intersection elem_of_union !mkSet_elem_of.
+    rewrite elem_of_intersection elem_of_union !elem_of_mkSet.
     intros; apply dec_stable.
     destruct p; set_solver.
 Qed.
@@ -58,13 +56,13 @@ Qed.
 Lemma low_states_closed : sts.closed low_states {[ Send ]}.
 Proof.
   split.
-  - move=>[p I]. rewrite /= /tok !mkSet_elem_of /= =>HI.
+  - move=>[p I]. rewrite /= /tok !elem_of_mkSet /= =>HI.
     destruct p; set_solver.
-  - move=>s1 s2. rewrite !mkSet_elem_of.
+  - move=>s1 s2. rewrite !elem_of_mkSet.
     intros Hs1 [T1 T2 Hdisj Hstep'].
     inversion_clear Hstep' as [? ? ? ? Htrans _ _ Htok].
     destruct Htrans; simpl in *; first by destruct p.
-    set_solver.
+    exfalso; set_solver.
 Qed.
 
 (* Proof that we can take the steps we need. *)
@@ -79,7 +77,7 @@ Proof.
   constructor; first constructor; simpl; [set_solver by eauto..|].
   (* TODO this proof is rather annoying. *)
   apply elem_of_equiv=>t. rewrite !elem_of_union.
-  rewrite !mkSet_elem_of /change_tok /=.
+  rewrite !elem_of_mkSet /change_tok /=.
   destruct t as [j|]; last set_solver.
   rewrite elem_of_difference elem_of_singleton.
   destruct (decide (i = j)); set_solver.
@@ -96,11 +94,11 @@ Proof.
   - destruct p; set_solver.
   (* This gets annoying... and I think I can see a pattern with all these proofs. Automatable? *)
   - apply elem_of_equiv=>t. destruct t; last set_solver.
-    rewrite !mkSet_elem_of !not_elem_of_union !not_elem_of_singleton
+    rewrite !elem_of_mkSet !not_elem_of_union !not_elem_of_singleton
       not_elem_of_difference elem_of_singleton !(inj_iff Change).
     destruct p; naive_solver.
   - apply elem_of_equiv=>t. destruct t as [j|]; last set_solver.
-    rewrite !mkSet_elem_of !not_elem_of_union !not_elem_of_singleton
+    rewrite !elem_of_mkSet !not_elem_of_union !not_elem_of_singleton
       not_elem_of_difference elem_of_singleton !(inj_iff Change).
     destruct (decide (i1 = j)) as [->|]; first tauto.
     destruct (decide (i2 = j)) as [->|]; intuition.
diff --git a/prelude/sets.v b/prelude/sets.v
index 92b1434e0..f5304451c 100644
--- a/prelude/sets.v
+++ b/prelude/sets.v
@@ -1,35 +1,40 @@
 (* Copyright (c) 2012-2015, Robbert Krebbers. *)
 (* This file is distributed under the terms of the BSD license. *)
 (** This file implements sets as functions into Prop. *)
-From prelude Require Export prelude.
+From prelude Require Export tactics.
 
 Record set (A : Type) : Type := mkSet { set_car : A → Prop }.
+Add Printing Constructor set.
 Arguments mkSet {_} _.
 Arguments set_car {_} _ _.
-Instance set_all {A} : Top (set A) := mkSet (λ _, True).
-Instance set_empty {A} : Empty (set A) := mkSet (λ _, False).
-Instance set_singleton {A} : Singleton A (set A) := λ x, mkSet (x =).
+Notation "{[ x | P ]}" := (mkSet (λ x, P))
+  (at level 1, format "{[  x  |  P  ]}") : C_scope.
+
 Instance set_elem_of {A} : ElemOf A (set A) := λ x X, set_car X x.
-Instance set_union {A} : Union (set A) := λ X1 X2, mkSet (λ x, x ∈ X1 ∨ x ∈ X2).
+
+Instance set_all {A} : Top (set A) := {[ _ | True ]}.
+Instance set_empty {A} : Empty (set A) := {[ _ | False ]}.
+Instance set_singleton {A} : Singleton A (set A) := λ y, {[ x | y = x ]}.
+Instance set_union {A} : Union (set A) := λ X1 X2, {[ x | x ∈ X1 ∨ x ∈ X2 ]}.
 Instance set_intersection {A} : Intersection (set A) := λ X1 X2,
-  mkSet (λ x, x ∈ X1 ∧ x ∈ X2).
+  {[ x | x ∈ X1 ∧ x ∈ X2 ]}.
 Instance set_difference {A} : Difference (set A) := λ X1 X2,
-  mkSet (λ x, x ∈ X1 ∧ x ∉ X2).
+  {[ x | x ∈ X1 ∧ x ∉ X2 ]}.
 Instance set_collection : Collection A (set A).
-Proof. by split; [split | |]; repeat intro. Qed.
+Proof. split; [split | |]; by repeat intro. Qed.
 
-Lemma mkSet_elem_of {A} (f : A → Prop) x : (x ∈ mkSet f) = f x.
+Lemma elem_of_mkSet {A} (P : A → Prop) x : (x ∈ {[ x | P x ]}) = P x.
 Proof. done. Qed.
-Lemma mkSet_not_elem_of {A} (f : A → Prop) x : (x ∉ mkSet f) = (¬f x).
+Lemma not_elem_of_mkSet {A} (P : A → Prop) x : (x ∉ {[ x | P x ]}) = (¬P x).
 Proof. done. Qed.
 
 Instance set_ret : MRet set := λ A (x : A), {[ x ]}.
 Instance set_bind : MBind set := λ A B (f : A → set B) (X : set A),
   mkSet (λ b, ∃ a, b ∈ f a ∧ a ∈ X).
 Instance set_fmap : FMap set := λ A B (f : A → B) (X : set A),
-  mkSet (λ b, ∃ a, b = f a ∧ a ∈ X).
+  {[ b | ∃ a, b = f a ∧ a ∈ X ]}.
 Instance set_join : MJoin set := λ A (XX : set (set A)),
-  mkSet (λ a, ∃ X, a ∈ X ∧ X ∈ XX).
+  {[ a | ∃ X, a ∈ X ∧ X ∈ XX ]}.
 Instance set_collection_monad : CollectionMonad set.
 Proof. by split; try apply _. Qed.
 
-- 
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