Move most notation level declarations into Util

This reveals and prevents notation clashes.

Notable breakage:
- `x {{ y }}` breaks `Context {forall x, {P x} + {~P x}}.`
  (@jadephilipoom)
- `x [[ y ]]` breaks `destruct x as [[a]|[b]].` (@jadephilipoom)
- `O :( A , B ): :?: L ::: R` breaks `((1):nat)` and `constr:(nat)`
  (@varomodt)

After    | File Name                                         | Before   || Change
------------------------------------------------------------------------------------
3m16.07s | Total                                             | 3m23.63s || -0m07.55s
------------------------------------------------------------------------------------
0m16.90s | ModularArithmetic/ModularBaseSystemProofs         | 0m19.02s || -0m02.12s
0m03.29s | ModularArithmetic/ModularBaseSystemOpt            | 0m05.30s || -0m02.00s
0m12.04s | Experiments/SpecEd25519                           | 0m13.89s || -0m01.85s
0m36.02s | CompleteEdwardsCurve/ExtendedCoordinates          | 0m36.12s || -0m00.09s
0m34.48s | Specific/GF25519                                  | 0m34.27s || +0m00.20s
0m15.62s | CompleteEdwardsCurve/CompleteEdwardsCurveTheorems | 0m15.76s || -0m00.14s
0m07.36s | Specific/GF1305                                   | 0m07.34s || +0m00.02s
0m06.64s | Algebra                                           | 0m06.14s || +0m00.50s
0m05.12s | WeierstrassCurve/Pre                              | 0m05.09s || +0m00.03s
0m04.58s | ModularArithmetic/Tutorial                        | 0m03.83s || +0m00.75s
0m04.46s | BaseSystemProofs                                  | 0m04.33s || +0m00.12s
0m04.26s | ModularArithmetic/Pow2BaseProofs                  | 0m05.16s || -0m00.90s
0m03.91s | CompleteEdwardsCurve/Pre                          | 0m04.62s || -0m00.71s
0m02.75s | Assembly/State                                    | 0m03.12s || -0m00.37s
0m02.55s | Experiments/EdDSARefinement                       | 0m01.85s || +0m00.69s
0m02.46s | Util/ZUtil                                        | 0m02.45s || +0m00.00s
0m01.86s | Assembly/Wordize                                  | 0m02.02s || -0m00.15s
0m01.68s | ModularArithmetic/ModularArithmeticTheorems       | 0m01.68s || +0m00.00s
0m01.67s | ModularArithmetic/PrimeFieldTheorems              | 0m01.96s || -0m00.29s
0m01.50s | Encoding/PointEncodingPre                         | 0m02.20s || -0m00.70s
0m01.46s | ModularArithmetic/BarrettReduction/Z              | 0m01.02s || +0m00.43s
0m01.26s | BaseSystem                                        | 0m01.21s || +0m00.05s
0m01.24s | Assembly/Pseudize                                 | 0m01.22s || +0m00.02s
0m01.20s | ModularArithmetic/ExtendedBaseVector              | 0m01.86s || -0m00.66s
0m01.04s | Util/IterAssocOp                                  | 0m00.74s || +0m00.30s
0m00.96s | ModularArithmetic/ExtPow2BaseMulProofs            | 0m00.67s || +0m00.28s
0m00.96s | Assembly/Pipeline                                 | 0m00.65s || +0m00.30s
0m00.93s | Experiments/DerivationsOptionRectLetInEncoding    | 0m00.98s || -0m00.04s
0m00.93s | ModularArithmetic/ModularBaseSystemField          | 0m00.95s || -0m00.01s
0m00.92s | Util/NumTheoryUtil                                | 0m01.19s || -0m00.26s
0m00.90s | ModularArithmetic/ModularBaseSystemListProofs     | 0m00.89s || +0m00.01s
0m00.82s | ModularArithmetic/PseudoMersenneBaseParamProofs   | 0m00.95s || -0m00.13s
0m00.81s | Assembly/QhasmEvalCommon                          | 0m00.86s || -0m00.04s
0m00.80s | Assembly/StringConversion                         | 0m00.60s || +0m00.20s
0m00.79s | Assembly/Qhasm                                    | 0m00.54s || +0m00.25s
0m00.73s | Assembly/Pseudo                                   | 0m00.98s || -0m00.25s
0m00.72s | Util/AdditionChainExponentiation                  | 0m00.92s || -0m00.20s
0m00.69s | Experiments/SpecificCurve25519                    | 0m00.72s || -0m00.03s
0m00.65s | Spec/ModularWordEncoding                          | 0m00.85s || -0m00.19s
0m00.65s | ModularArithmetic/ModularBaseSystemList           | 0m00.68s || -0m00.03s
0m00.65s | Assembly/PseudoConversion                         | 0m00.67s || -0m00.02s
0m00.64s | Encoding/ModularWordEncodingPre                   | 0m00.62s || +0m00.02s
0m00.64s | Spec/EdDSA                                        | 0m00.93s || -0m00.29s
0m00.64s | Testbit                                           | 0m00.94s || -0m00.29s
0m00.63s | Encoding/ModularWordEncodingTheorems              | 0m00.66s || -0m00.03s
0m00.62s | Assembly/AlmostQhasm                              | 0m00.49s || +0m00.13s
0m00.59s | ModularArithmetic/ModularBaseSystem               | 0m00.61s || -0m00.02s
0m00.49s | Spec/WeierstrassCurve                             | 0m00.43s || +0m00.06s
0m00.49s | Assembly/QhasmUtil                                | 0m00.51s || -0m00.02s
0m00.49s | ModularArithmetic/Pre                             | 0m00.47s || +0m00.02s
0m00.47s | Assembly/Vectorize                                | 0m00.51s || -0m00.04s
0m00.46s | Assembly/AlmostConversion                         | 0m00.44s || +0m00.02s
0m00.42s | ModularArithmetic/Pow2Base                        | 0m00.42s || +0m00.00s
0m00.41s | ModularArithmetic/PseudoMersenneBaseParams        | 0m00.45s || -0m00.04s
0m00.41s | Spec/ModularArithmetic                            | 0m00.41s || +0m00.00s
0m00.39s | Spec/CompleteEdwardsCurve                         | 0m00.41s || -0m00.01s
0m00.03s | Util/Notations                                    | 0m00.03s || +0m00.00s

4 files changed, 50 insertions, 41 deletions

diff --git a/src/Assembly/Pipeline.v b/src/Assembly/Pipeline.v
index 8e58e7345..87102cca8 100644
--- a/src/Assembly/Pipeline.v
+++ b/src/Assembly/Pipeline.v
@@ -1,8 +1,9 @@
 Require Import Bedrock.Word.
-Require Import QhasmCommon QhasmEvalCommon.
-Require Import Pseudo Qhasm AlmostQhasm Conversion Language.
-Require Import PseudoConversion AlmostConversion StringConversion.
-Require Import Wordize Vectorize Pseudize.
+Require Import Crypto.Assembly.QhasmCommon Crypto.Assembly.QhasmEvalCommon.
+Require Import Crypto.Assembly.Pseudo Crypto.Assembly.Qhasm Crypto.Assembly.AlmostQhasm Crypto.Assembly.Conversion Crypto.Assembly.Language.
+Require Import Crypto.Assembly.PseudoConversion Crypto.Assembly.AlmostConversion Crypto.Assembly.StringConversion.
+Require Import Crypto.Assembly.Wordize Crypto.Assembly.Vectorize Crypto.Assembly.Pseudize.
+Require Import Crypto.Util.Notations.
 
 Module Pipeline.
   Export AlmostQhasm Qhasm QhasmString.
@@ -24,8 +25,8 @@ End Pipeline.
 Module PipelineExamples.
   Import Pipeline ListNotations StateCommon EvalUtil ListState.
 
-  Local Notation "v [[ i ]]" := (nth i v (wzero _)) (at level 40).
-  Local Notation "$$ v" := (natToWord _ v) (at level 40).
+  Local Notation "v [[ i ]]" := (nth i v (wzero _)).
+  Local Notation "$$ v" := (natToWord _ v).
 
   (*
   Definition add_example: @pseudeq 32 W32 1 1 (fun v =>
@@ -53,7 +54,7 @@ Module PipelineExamples.
       plet b := v[[0]] in
 
       (* NOTE: we want the lets in this format to unify with
-               pseudo_mult_dual *)  
+               pseudo_mult_dual *)
       plet c := multHigh a b in
       plet d := a ^* b in
 
diff --git a/src/Assembly/Pseudo.v b/src/Assembly/Pseudo.v
index b8aae4521..ca4700a7f 100644
--- a/src/Assembly/Pseudo.v
+++ b/src/Assembly/Pseudo.v
@@ -1,6 +1,7 @@
 Require Import Crypto.Assembly.QhasmCommon Crypto.Assembly.QhasmUtil Crypto.Assembly.State.
 Require Import Crypto.Assembly.Language Crypto.Assembly.QhasmEvalCommon.
 Require Import Coq.Lists.List Coq.Arith.Compare_dec Coq.omega.Omega.
+Require Import Crypto.Util.Notations.
 Require Export Crypto.Util.FixCoqMistakes.
 
 Module Pseudo <: Language.
@@ -31,8 +32,8 @@ Module Pseudo <: Language.
   }.
 
   Definition Params := Params'.
-  Definition State (p: Params): Type := ListState (width p).
-  Definition Program (p: Params): Type :=
+  Definition State (p: Params) : Type := ListState (width p).
+  Definition Program (p: Params) : Type :=
     @Pseudo (width p) (spec p) (inputs p) (outputs p).
 
   Definition Unary32: Params := mkParams 32 W32 1 1.
@@ -40,7 +41,7 @@ Module Pseudo <: Language.
 
   (* Evaluation *)
 
-  Fixpoint pseudoEval {n m w s} (prog: @Pseudo w s n m) (st: ListState w): option (ListState w) :=
+  Fixpoint pseudoEval {n m w s} (prog: @Pseudo w s n m) (st: ListState w) : option (ListState w) :=
     match prog with
     | PVar n _ i => omap (getVar i st) (fun x => Some (setList [x] st))
     | PMem n m v i => omap (getMem v i st) (fun x => Some (setList [x] st))
@@ -114,7 +115,7 @@ Module Pseudo <: Language.
   Delimit Scope pseudo_notations with p.
   Local Open Scope pseudo_notations.
 
-  Definition indexize {n: nat} (x: nat): Index n.
+  Definition indexize {n: nat} (x: nat) : Index n.
     intros; destruct (le_dec n 0).
 
     - exists 0; abstract intuition auto with zarith.
@@ -123,60 +124,65 @@ Module Pseudo <: Language.
   Defined.
 
   Notation "% A" := (PVar _ (Some false) (indexize A))
-    (at level 20, right associativity) : pseudo_notations.
+    : pseudo_notations.
 
   Notation "$ A" := (PVar _ (Some true) (indexize A))
-    (at level 20, right associativity) : pseudo_notations.
+    : pseudo_notations.
 
   Notation "A :[ B ]:" :=  (PMem _ _ (indexize A) (indexize B))
-    (at level 20, right associativity) : pseudo_notations.
+    : pseudo_notations.
 
   Notation "# A" := (PConst _ (natToWord _ A))
-    (at level 20, right associativity) : pseudo_notations.
+    : pseudo_notations.
 
   Notation "A :+: B" := (PBin _ IAdd (PComb _ _ _ A B))
-    (at level 60, right associativity) : pseudo_notations.
+    : pseudo_notations.
 
   Notation "A :+c: B" := (PCarry _ AddWithCarry (PComb _ _ _ A B))
-    (at level 60, right associativity) : pseudo_notations.
+    : pseudo_notations.
 
   Notation "A :-: B" := (PBin _ ISub (PComb _ _ _ A B))
-    (at level 60, right associativity) : pseudo_notations.
+    : pseudo_notations.
 
   Notation "A :&: B" := (PBin _ IAnd (PComb _ _ _ A B))
-    (at level 45, right associativity) : pseudo_notations.
+    : pseudo_notations.
 
   Notation "A :^: B" := (PBin _ IXor (PComb _ _ _ A B))
-    (at level 45, right associativity) : pseudo_notations.
+    : pseudo_notations.
 
   Notation "A :>>: B" := (PShift _ Shr (indexize B) A)
-    (at level 60, right associativity) : pseudo_notations.
+    : pseudo_notations.
 
   Notation "A :<<: B" := (PShift _ Shl (indexize B) A)
-    (at level 60, right associativity) : pseudo_notations.
+    : pseudo_notations.
 
   Notation "A :*: B" := (PDual _ Mult (PComb _ _ _ A B))
-    (at level 55, right associativity) : pseudo_notations.
+    : pseudo_notations.
 
+  (* TODO(rsloan, from jgross): This notation is not okay.  It breaks
+     [constr:(nat)] and [((1):nat)].  Please remove all frowny faces
+     from notations, and then move [Reserved Notation] line to
+     Fiat.Crypto.Util.Notations. *)
+  Reserved Notation "O :( A , B ): :?: L ::: R" (at level 70, right associativity).
   Notation "O :( A , B ): :?: L ::: R" :=
     (PIf _ _ O (indexize A) (indexize B) L R)
-    (at level 70, right associativity) : pseudo_notations.
+    : pseudo_notations.
 
   Notation "F :**: e" :=
     (PFunExp _ F e)
-    (at level 70, right associativity) : pseudo_notations.
+    : pseudo_notations.
 
   Notation "E :->: F" :=
     (PLet _ _ _ E F)
-    (at level 70, right associativity) : pseudo_notations.
+    : pseudo_notations.
 
   Notation "A :|: B" :=
     (PComb _ _ _ A B)
-    (at level 65, left associativity) : pseudo_notations.
+    : pseudo_notations.
 
   Notation "n ::: A :():" :=
     (PCall _ _ n A)
-    (at level 65, left associativity) : pseudo_notations.
+    : pseudo_notations.
 
   Close Scope pseudo_notations.
 End Pseudo.
diff --git a/src/Assembly/QhasmUtil.v b/src/Assembly/QhasmUtil.v
index 1ab894e94..53e9cf573 100644
--- a/src/Assembly/QhasmUtil.v
+++ b/src/Assembly/QhasmUtil.v
@@ -1,40 +1,41 @@
 Require Import Coq.ZArith.ZArith Coq.NArith.NArith Coq.Numbers.Natural.Peano.NPeano.
 Require Import Crypto.Assembly.QhasmCommon.
+Require Import Crypto.Util.Notations.
 Require Export Bedrock.Word.
 Require Export Crypto.Util.FixCoqMistakes.
 
 Delimit Scope nword_scope with w.
 Local Open Scope nword_scope.
 
-Notation "& x" := (wordToN x) (at level 30) : nword_scope.
-Notation "** x" := (NToWord _ x) (at level 30) : nword_scope.
+Notation "& x" := (wordToN x) : nword_scope.
+Notation "** x" := (NToWord _ x) : nword_scope.
 
 Section Util.
-  Definition convS {A B: Set} (x: A) (H: A = B): B :=
+  Definition convS {A B: Set} (x: A) (H: A = B) : B :=
     eq_rect A (fun B0 : Set => B0) x B H.
 
-  Definition high {k n: nat} (p: (k <= n)%nat) (w: word n): word k.
+  Definition high {k n: nat} (p: (k <= n)%nat) (w: word n) : word k.
     refine (split1 k (n - k) (convS w _)).
     abstract (replace n with (k + (n - k)) by omega; intuition auto with arith).
   Defined.
 
-  Definition low {k n: nat} (p: (k <= n)%nat) (w: word n): word k.
+  Definition low {k n: nat} (p: (k <= n)%nat) (w: word n) : word k.
     refine (split2 (n - k) k (convS w _)).
     abstract (replace n with (k + (n - k)) by omega; intuition auto with zarith).
   Defined.
 
-  Definition extend {k n: nat} (p: (k <= n)%nat) (w: word k): word n.
+  Definition extend {k n: nat} (p: (k <= n)%nat) (w: word k) : word n.
     refine (convS (zext w (n - k)) _).
     abstract (replace (k + (n - k)) with n by omega; intuition).
   Defined.
 
-  Definition shiftr {n} (w: word n) (k: nat): word n :=
+  Definition shiftr {n} (w: word n) (k: nat) : word n :=
     match (le_dec k n) with
     | left p => extend p (high p w)
     | right _ => wzero n
     end.
 
-  Definition mask {n} (k: nat) (w: word n): word n :=
+  Definition mask {n} (k: nat) (w: word n) : word n :=
     match (le_dec k n) with
     | left p => extend p (low p w)
     | right _ => w
@@ -58,7 +59,7 @@ Section Util.
         end).
   Defined.
 
-  Definition break {n} (m: nat) (x: word n): word m * word (n - m).
+  Definition break {n} (m: nat) (x: word n) : word m * word (n - m).
     refine match (le_dec m n) with
     | left p => (extend _ (low p x), extend _ (@high (n - m) n _ x))
     | right p => (extend _ x, _)
@@ -67,13 +68,13 @@ Section Util.
     replace (n - m) with O by abstract omega; exact WO.
   Defined.
 
-  Definition addWithCarry {n} (x y: word n) (c: bool): word n :=
+  Definition addWithCarry {n} (x y: word n) (c: bool) : word n :=
     x ^+ y ^+ (natToWord _ (if c then 1 else 0)).
 
   Definition omap {A B} (x: option A) (f: A -> option B) :=
     match x with | Some y => f y | _ => None end.
 
-  Notation "A <- X ; B" := (omap X (fun A => B)) (at level 70, right associativity).
+  Notation "A <- X ; B" := (omap X (fun A => B)).
 End Util.
 
 Close Scope nword_scope.
diff --git a/src/Assembly/Vectorize.v b/src/Assembly/Vectorize.v
index 4eed28aad..08e9ee356 100644
--- a/src/Assembly/Vectorize.v
+++ b/src/Assembly/Vectorize.v
@@ -1,20 +1,21 @@
 Require Export Bedrock.Word Bedrock.Nomega.
 Require Import Coq.Numbers.Natural.Peano.NPeano Coq.NArith.NArith Coq.PArith.PArith Coq.NArith.Ndigits Coq.Arith.Compare_dec Coq.Arith.Arith.
 Require Import Coq.Logic.ProofIrrelevance Coq.setoid_ring.Ring Coq.Lists.List Coq.omega.Omega.
+Require Import Crypto.Util.Notations.
 
 Require Export Crypto.Util.FixCoqMistakes.
 
 Definition Let_In {A P} (x : A) (f : forall a : A, P a) : P x :=
   let y := x in f y.
 
-Notation "'plet' x := y 'in' z" := (Let_In y (fun x => z)) (at level 60).
+Notation "'plet' x := y 'in' z" := (Let_In y (fun x => z)).
 
 Section Vector.
   Import ListNotations.
 
   Definition vec T n := {x: list T | length x = n}.
 
-  Definition vget {n T} (x: vec T n) (i: {v: nat | (v < n)%nat}): T.
+  Definition vget {n T} (x: vec T n) (i: {v: nat | (v < n)%nat}) : T.
     refine (
       match (proj1_sig x) as x' return (proj1_sig x) = x' -> _ with
       | [] => fun _ => _