From 88e9359500ca9de418c33ba6b25934730b804587 Mon Sep 17 00:00:00 2001
From: Jacques-Henri Jourdan <jacques-henri.jourdan@normalesup.org>
Date: Sat, 23 Dec 2017 16:23:00 +0100
Subject: [PATCH] Whitespaces.

---
 theories/algebra/deprecated.v         | 2 +-
 theories/algebra/dra.v                | 6 +++---
 theories/algebra/functions.v          | 2 +-
 theories/algebra/ofe.v                | 2 +-
 theories/algebra/sts.v                | 2 +-
 theories/base_logic/double_negation.v | 8 ++++----
 theories/base_logic/lib/namespaces.v  | 2 +-
 theories/base_logic/lib/sts.v         | 2 +-
 theories/base_logic/tactics.v         | 2 +-
 theories/heap_lang/lang.v             | 6 +++---
 theories/heap_lang/lib/counter.v      | 2 +-
 theories/heap_lang/lib/ticket_lock.v  | 4 ++--
 theories/program_logic/weakestpre.v   | 2 +-
 theories/proofmode/base.v             | 2 +-
 theories/proofmode/environments.v     | 2 +-
 theories/proofmode/tactics.v          | 4 ++--
 16 files changed, 25 insertions(+), 25 deletions(-)

diff --git a/theories/algebra/deprecated.v b/theories/algebra/deprecated.v
index ccb061003..2c8165cf1 100644
--- a/theories/algebra/deprecated.v
+++ b/theories/algebra/deprecated.v
@@ -14,7 +14,7 @@ Local Arguments op _ _ _ !_ /.
 Local Arguments pcore _ _ !_ /.
 
 (* This is isomorphic to option, but has a very different RA structure. *)
-Inductive dec_agree (A : Type) : Type := 
+Inductive dec_agree (A : Type) : Type :=
   | DecAgree : A → dec_agree A
   | DecAgreeBot : dec_agree A.
 Arguments DecAgree {_} _.
diff --git a/theories/algebra/dra.v b/theories/algebra/dra.v
index dc459baf4..e2f3faf7d 100644
--- a/theories/algebra/dra.v
+++ b/theories/algebra/dra.v
@@ -22,7 +22,7 @@ Record DraMixin A `{Equiv A, Core A, Disjoint A, Op A, Valid A} := {
   mixin_dra_core_disjoint_l x : ✓ x → core x ## x;
   mixin_dra_core_l x : ✓ x → core x ⋅ x ≡ x;
   mixin_dra_core_idemp x : ✓ x → core (core x) ≡ core x;
-  mixin_dra_core_mono x y : 
+  mixin_dra_core_mono x y :
     ∃ z, ✓ x → ✓ y → x ## y → core (x ⋅ y) ≡ core x ⋅ z ∧ ✓ z ∧ core x ## z
 }.
 Structure draT := DraT {
@@ -81,7 +81,7 @@ Section dra_mixin.
   Proof. apply (mixin_dra_core_l _ (dra_mixin A)). Qed.
   Lemma dra_core_idemp x : ✓ x → core (core x) ≡ core x.
   Proof. apply (mixin_dra_core_idemp _ (dra_mixin A)). Qed.
-  Lemma dra_core_mono x y : 
+  Lemma dra_core_mono x y :
     ∃ z, ✓ x → ✓ y → x ## y → core (x ⋅ y) ≡ core x ⋅ z ∧ ✓ z ∧ core x ## z.
   Proof. apply (mixin_dra_core_mono _ (dra_mixin A)). Qed.
 End dra_mixin.
@@ -209,7 +209,7 @@ Proof.
     assert (✓ y) by (rewrite EQ; by apply dra_op_valid).
     split; first done. exists (to_validity z). split; first split.
     + intros _. simpl. by split_and!.
-    + intros _. setoid_subst. by apply dra_op_valid. 
+    + intros _. setoid_subst. by apply dra_op_valid.
     + intros _. rewrite /= EQ //.
 Qed.
 *)
diff --git a/theories/algebra/functions.v b/theories/algebra/functions.v
index a0fd519b0..62c200d3a 100644
--- a/theories/algebra/functions.v
+++ b/theories/algebra/functions.v
@@ -8,7 +8,7 @@ Definition ofe_fun_insert `{EqDecision A} {B : A → ofeT}
   match decide (x = x') with left H => eq_rect _ B y _ H | right _ => f x' end.
 Instance: Params (@ofe_fun_insert) 5.
 
-Definition ofe_fun_singleton `{Finite A} {B : A → ucmraT} 
+Definition ofe_fun_singleton `{Finite A} {B : A → ucmraT}
   (x : A) (y : B x) : ofe_fun B := ofe_fun_insert x y ε.
 Instance: Params (@ofe_fun_singleton) 5.
 
diff --git a/theories/algebra/ofe.v b/theories/algebra/ofe.v
index d573744c2..7810042a6 100644
--- a/theories/algebra/ofe.v
+++ b/theories/algebra/ofe.v
@@ -6,7 +6,7 @@ Set Default Proof Using "Type".
     category, and mathematically speaking, the entire development lives
     in this category. However, we will generally prefer to work with raw
     Coq functions plus some registered Proper instances for non-expansiveness.
-    This makes writing such functions much easier. It turns out that it many 
+    This makes writing such functions much easier. It turns out that it many
     cases, we do not even need non-expansiveness.
 *)
 
diff --git a/theories/algebra/sts.v b/theories/algebra/sts.v
index 66e4726e9..ceff03979 100644
--- a/theories/algebra/sts.v
+++ b/theories/algebra/sts.v
@@ -311,7 +311,7 @@ Proof. done. Qed.
 Lemma sts_frag_up_valid s T : ✓ sts_frag_up s T ↔ tok s ## T.
 Proof.
   split.
-  - move=>/sts_frag_valid [H _]. apply H, elem_of_up. 
+  - move=>/sts_frag_valid [H _]. apply H, elem_of_up.
   - intros. apply sts_frag_valid; split. by apply closed_up. set_solver.
 Qed.
 
diff --git a/theories/base_logic/double_negation.v b/theories/base_logic/double_negation.v
index edc4289c0..a57b61357 100644
--- a/theories/base_logic/double_negation.v
+++ b/theories/base_logic/double_negation.v
@@ -228,7 +228,7 @@ Proof.
   revert P.
   induction k; intros P.
   - rewrite //= ?right_id. apply wand_intro_l.
-    rewrite {1}(nnupd_k_intro 0 (P -∗ False)%I) //= ?right_id. apply wand_elim_r. 
+    rewrite {1}(nnupd_k_intro 0 (P -∗ False)%I) //= ?right_id. apply wand_elim_r.
   - rewrite {2}(nnupd_k_unfold k P).
     apply and_intro.
     * rewrite (nnupd_k_unfold k P). rewrite and_elim_l.
@@ -291,9 +291,9 @@ Qed.
 End classical.
 
 (* We might wonder whether we can prove an adequacy lemma for nnupd. We could combine
-   the adequacy lemma for bupd with the previous result to get adquacy for nnupd, but 
+   the adequacy lemma for bupd with the previous result to get adquacy for nnupd, but
    this would rely on the classical axiom we needed to prove the equivalence! Can
-   we establish adequacy without axioms? Unfortunately not, because adequacy for 
+   we establish adequacy without axioms? Unfortunately not, because adequacy for
    nnupd would imply double negation elimination, which is classical: *)
 
 Lemma nnupd_dne φ: (|=n=> ⌜¬¬ φ → φ⌝: uPred M)%I.
@@ -327,7 +327,7 @@ Proof.
     intros ??? Hx'. rewrite left_id in Hx' *=> Hx'.
     unseal.
     assert (n' < S k ∨ n' = S k) as [|] by omega.
-    * intros. move:(laterN_small n' (S k) x' False). rewrite //=. unseal. intros Hsmall. 
+    * intros. move:(laterN_small n' (S k) x' False). rewrite //=. unseal. intros Hsmall.
       eapply Hsmall; eauto.
     * subst. intros. exfalso. eapply Hf2. exists x'. split; eauto using cmra_validN_S.
   - intros k P x Hx. rewrite ?Nat_iter_S_r.
diff --git a/theories/base_logic/lib/namespaces.v b/theories/base_logic/lib/namespaces.v
index dc226f276..dfee3989a 100644
--- a/theories/base_logic/lib/namespaces.v
+++ b/theories/base_logic/lib/namespaces.v
@@ -84,7 +84,7 @@ End namespace.
 
 (* The hope is that registering these will suffice to solve most goals
 of the forms:
-- [N1 ## N2] 
+- [N1 ## N2]
 - [↑N1 ⊆ E ∖ ↑N2 ∖ .. ∖ ↑Nn]
 - [E1 ∖ ↑N1 ⊆ E2 ∖ ↑N2 ∖ .. ∖ ↑Nn] *)
 Create HintDb ndisj.
diff --git a/theories/base_logic/lib/sts.v b/theories/base_logic/lib/sts.v
index 00ddee6e6..0d8e8ced6 100644
--- a/theories/base_logic/lib/sts.v
+++ b/theories/base_logic/lib/sts.v
@@ -82,7 +82,7 @@ Section sts.
     sts_own γ s1 T ==∗ sts_own γ s2 T.
   Proof.
     intros ??. apply own_update, sts_update_frag_up; [|done..].
-    intros Hdisj. apply sts.closed_up. done. 
+    intros Hdisj. apply sts.closed_up. done.
   Qed.
 
   Lemma sts_own_weaken_tok γ s T1 T2 :
diff --git a/theories/base_logic/tactics.v b/theories/base_logic/tactics.v
index 717bcec14..49085be98 100644
--- a/theories/base_logic/tactics.v
+++ b/theories/base_logic/tactics.v
@@ -60,7 +60,7 @@ Module uPred_reflection. Section uPred_reflection.
     match e with
     | ETrue => None
     | EVar n' => if decide (n = n') then Some ETrue else None
-    | ESep e1 e2 => 
+    | ESep e1 e2 =>
        match cancel_go n e1 with
        | Some e1' => Some (ESep e1' e2)
        | None => ESep e1 <$> cancel_go n e2
diff --git a/theories/heap_lang/lang.v b/theories/heap_lang/lang.v
index 45f540ece..39c6625fd 100644
--- a/theories/heap_lang/lang.v
+++ b/theories/heap_lang/lang.v
@@ -276,7 +276,7 @@ Definition fill_item (Ki : ectx_item) (e : expr) : expr :=
   | CaseCtx e1 e2 => Case e e1 e2
   | AllocCtx => Alloc e
   | LoadCtx => Load e
-  | StoreLCtx e2 => Store e e2 
+  | StoreLCtx e2 => Store e e2
   | StoreRCtx v1 => Store (of_val v1) e
   | CasLCtx e1 e2 => CAS e e1 e2
   | CasMCtx v0 e2 => CAS (of_val v0) e e2
@@ -365,11 +365,11 @@ Inductive head_step : expr → state → expr → state → list (expr) → Prop
      head_step (App (Rec f x e1) e2) σ e' σ []
   | UnOpS op e v v' σ :
      to_val e = Some v →
-     un_op_eval op v = Some v' → 
+     un_op_eval op v = Some v' →
      head_step (UnOp op e) σ (of_val v') σ []
   | BinOpS op e1 e2 v1 v2 v' σ :
      to_val e1 = Some v1 → to_val e2 = Some v2 →
-     bin_op_eval op v1 v2 = Some v' → 
+     bin_op_eval op v1 v2 = Some v' →
      head_step (BinOp op e1 e2) σ (of_val v') σ []
   | IfTrueS e1 e2 σ :
      head_step (If (Lit $ LitBool true) e1 e2) σ e1 σ []
diff --git a/theories/heap_lang/lib/counter.v b/theories/heap_lang/lib/counter.v
index b9632c00c..8aee1924b 100644
--- a/theories/heap_lang/lib/counter.v
+++ b/theories/heap_lang/lib/counter.v
@@ -55,7 +55,7 @@ Section mono_proof.
     - iDestruct (own_valid_2 with "Hγ Hγf")
         as %[?%mnat_included _]%auth_valid_discrete_2.
       iMod (own_update_2 with "Hγ Hγf") as "[Hγ Hγf]".
-      { apply auth_update, (mnat_local_update _ _ (S c)); auto. } 
+      { apply auth_update, (mnat_local_update _ _ (S c)); auto. }
       wp_cas_suc. iMod ("Hclose" with "[Hl Hγ]") as "_".
       { iNext. iExists (S c). rewrite Nat2Z.inj_succ Z.add_1_l. by iFrame. }
       iModIntro. wp_if. iApply "HΦ"; iExists γ; repeat iSplit; eauto.
diff --git a/theories/heap_lang/lib/ticket_lock.v b/theories/heap_lang/lib/ticket_lock.v
index 7437f7483..b75704bb0 100644
--- a/theories/heap_lang/lib/ticket_lock.v
+++ b/theories/heap_lang/lib/ticket_lock.v
@@ -94,7 +94,7 @@ Section proof.
       + iMod ("Hclose" with "[Hlo Hln Hainv Ht]") as "_".
         { iNext. iExists o, n. iFrame. eauto. }
         iModIntro. wp_let. wp_op. case_bool_decide; [|done].
-        wp_if. 
+        wp_if.
         iApply ("HΦ" with "[-]"). rewrite /locked. iFrame. eauto.
       + iDestruct (own_valid_2 with "Ht Haown") as % [_ ?%gset_disj_valid_op].
         set_solver.
@@ -129,7 +129,7 @@ Section proof.
       iModIntro. wp_if.
       iApply (wait_loop_spec γ (#lo, #ln) with "[-HΦ]").
       + iFrame. rewrite /is_lock; eauto 10.
-      + by iNext. 
+      + by iNext.
     - wp_cas_fail.
       iMod ("Hclose" with "[Hlo' Hln' Hauth Haown]") as "_".
       { iNext. iExists o', n'. by iFrame. }
diff --git a/theories/program_logic/weakestpre.v b/theories/program_logic/weakestpre.v
index 677f071d7..e081ea449 100644
--- a/theories/program_logic/weakestpre.v
+++ b/theories/program_logic/weakestpre.v
@@ -237,7 +237,7 @@ Proof.
   iSplitL "He2"; first by iApply ("IH" with "He2"). iClear "Hred Hstep".
   induction efs as [|ef efs IH]; first by iApply big_sepL_nil.
   rewrite !big_sepL_cons. iDestruct "Hefs" as "(Hef & Hefs)".
-  iSplitL "Hef". by iApply ("IH" with "Hef"). exact: IH. 
+  iSplitL "Hef". by iApply ("IH" with "Hef"). exact: IH.
 Qed.
 
 Lemma fupd_wp s E e Φ : (|={E}=> WP e @ s; E {{ Φ }}) ⊢ WP e @ s; E {{ Φ }}.
diff --git a/theories/proofmode/base.v b/theories/proofmode/base.v
index 9497be345..159a09b5a 100644
--- a/theories/proofmode/base.v
+++ b/theories/proofmode/base.v
@@ -27,7 +27,7 @@ Definition ascii_beq (x y : ascii) : bool :=
     beq x5 y5 && beq x6 y6 && beq x7 y7 && beq x8 y8.
 
 Fixpoint string_beq (s1 s2 : string) : bool :=
-  match s1, s2 with 
+  match s1, s2 with
   | "", "" => true
   | String a1 s1, String a2 s2 => ascii_beq a1 a2 && string_beq s1 s2
   | _, _ => false
diff --git a/theories/proofmode/environments.v b/theories/proofmode/environments.v
index d980e776f..4816f938a 100644
--- a/theories/proofmode/environments.v
+++ b/theories/proofmode/environments.v
@@ -40,7 +40,7 @@ Fixpoint env_app {A} (Γapp : env A) (Γ : env A) : option (env A) :=
   match Γapp with
   | Enil => Some Γ
   | Esnoc Γapp i x =>
-     Γ' ← env_app Γapp Γ; 
+     Γ' ← env_app Γapp Γ;
      match Γ' !! i with None => Some (Esnoc Γ' i x) | Some _ => None end
   end.
 
diff --git a/theories/proofmode/tactics.v b/theories/proofmode/tactics.v
index ffa7bd438..315d0dccc 100644
--- a/theories/proofmode/tactics.v
+++ b/theories/proofmode/tactics.v
@@ -327,14 +327,14 @@ Local Tactic Notation "iIntro" "#" constr(H) :=
   first
   [ (* (?P → _) *)
     eapply tac_impl_intro_persistent with _ H _; (* (i:=H) *)
-      [apply _ || 
+      [apply _ ||
        let P := match goal with |- IntoPersistent _ ?P _ => P end in
        fail 1 "iIntro: " P " not persistent"
       |env_reflexivity || fail 1 "iIntro:" H "not fresh"
       |]
   | (* (?P -∗ _) *)
     eapply tac_wand_intro_persistent with _ H _; (* (i:=H) *)
-      [apply _ || 
+      [apply _ ||
        let P := match goal with |- IntoPersistent _ ?P _ => P end in
        fail 1 "iIntro: " P " not persistent"
       |env_reflexivity || fail 1 "iIntro:" H "not fresh"
-- 
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