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

2 files changed, 5 insertions, 5 deletions

diff --git a/src/ModularArithmetic/ModularBaseSystemList.v b/src/ModularArithmetic/ModularBaseSystemList.v
index b46429fcc..c556427b9 100644
--- a/src/ModularArithmetic/ModularBaseSystemList.v
+++ b/src/ModularArithmetic/ModularBaseSystemList.v
@@ -14,7 +14,7 @@ Local Open Scope Z_scope.
 Section Defs.
   Context `{prm :PseudoMersenneBaseParams} (modulus_multiple : digits).
   Local Notation base := (base_from_limb_widths limb_widths).
-  Local Notation "u [ i ]" := (nth_default 0 u i) (at level 40).
+  Local Notation "u [ i ]" := (nth_default 0 u i).
 
   Definition decode (us : digits) : F modulus := ZToField (BaseSystem.decode base us).
 
@@ -68,4 +68,4 @@ Section Defs.
        Otherwise, it's all zeroes, and the subtractions do nothing. *)
      map2 (fun x y => x - y) us (map (Z.land and_term) modulus_digits).
 
-End Defs.
\ No newline at end of file
+End Defs.
diff --git a/src/ModularArithmetic/ModularBaseSystemProofs.v b/src/ModularArithmetic/ModularBaseSystemProofs.v
index a1370d33e..d740cca17 100644
--- a/src/ModularArithmetic/ModularBaseSystemProofs.v
+++ b/src/ModularArithmetic/ModularBaseSystemProofs.v
@@ -208,7 +208,7 @@ Section PseudoMersenneProofs.
         split; [ intuition | ].
         destruct Hcarry_done as [Hnth_nonneg Hshiftr_0].
         apply Z.shiftr_eq_0_iff in Hshiftr_0.
-        destruct Hshiftr_0 as [nth_0 | []]; [ rewrite nth_0; zero_bounds | ].
+        destruct Hshiftr_0 as [nth_0 | [] ]; [ rewrite nth_0; zero_bounds | ].
         apply Z.log2_lt_pow2; auto.
       - rewrite nth_default_out_of_bounds by omega.
         split; zero_bounds.
@@ -482,8 +482,8 @@ Section CanonicalizationProofs.
   Qed.
   Hint Rewrite @nth_default_carry using (omega || distr_length; omega) : push_nth_default.
 
-  Local Notation "u '[' i ']' "  := (nth_default 0 u i) (at level 30).
-  Local Notation "u '{{' i '}}' "  := (carry_sequence (make_chain i) u) (at level 30).
+  Local Notation "u [ i ]"  := (nth_default 0 u i).
+  Local Notation "u {{ i }}"  := (carry_sequence (make_chain i) u) (at level 30). (* Can't rely on [Reserved Notation]: https://coq.inria.fr/bugs/show_bug.cgi?id=4970 *)
 
   Lemma bound_during_first_loop : forall i n us,
     length us = length limb_widths ->