Use Decidable machinery for is_eq_dec

This allows us to rely entirely on typeclass resolution to resolve these
instances, without having to do ad-hoc things for [and].

After    | File Name                                         | Before   || Change
------------------------------------------------------------------------------------
2m21.71s | Total                                             | 2m22.59s || -0m00.87s
------------------------------------------------------------------------------------
0m28.82s | Specific/GF25519                                  | 0m29.86s || -0m01.03s
0m29.60s | ModularArithmetic/ModularBaseSystemProofs         | 0m29.40s || +0m00.20s
0m21.25s | Experiments/SpecEd25519                           | 0m21.28s || -0m00.03s
0m18.15s | CompleteEdwardsCurve/ExtendedCoordinates          | 0m18.14s || +0m00.00s
0m11.95s | CompleteEdwardsCurve/CompleteEdwardsCurveTheorems | 0m11.94s || +0m00.00s
0m07.26s | Specific/GF1305                                   | 0m07.28s || -0m00.02s
0m03.77s | ModularArithmetic/Tutorial                        | 0m03.75s || +0m00.02s
0m03.76s | ModularArithmetic/ModularBaseSystemOpt            | 0m03.75s || +0m00.00s
0m03.61s | CompleteEdwardsCurve/Pre                          | 0m03.63s || -0m00.02s
0m02.15s | ModularArithmetic/ModularArithmeticTheorems       | 0m02.12s || +0m00.02s
0m01.88s | ModularArithmetic/PrimeFieldTheorems              | 0m01.89s || -0m00.01s
0m01.75s | Algebra                                           | 0m01.73s || +0m00.02s
0m01.21s | Experiments/DerivationsOptionRectLetInEncoding    | 0m01.17s || +0m00.04s
0m01.14s | ModularArithmetic/ExtendedBaseVector              | 0m01.14s || +0m00.00s
0m01.01s | ModularArithmetic/PseudoMersenneBaseParamProofs   | 0m00.98s || +0m00.03s
0m00.62s | Encoding/ModularWordEncodingTheorems              | 0m00.63s || -0m00.01s
0m00.60s | Encoding/ModularWordEncodingPre                   | 0m00.61s || -0m00.01s
0m00.59s | Util/Decidable                                    | 0m00.64s || -0m00.05s
0m00.58s | Spec/EdDSA                                        | 0m00.61s || -0m00.03s
0m00.57s | ModularArithmetic/ModularBaseSystem               | 0m00.61s || -0m00.04s
0m00.56s | Spec/ModularWordEncoding                          | 0m00.56s || +0m00.00s
0m00.51s | ModularArithmetic/PseudoMersenneBaseRep           | 0m00.53s || -0m00.02s
0m00.37s | Spec/CompleteEdwardsCurve                         | 0m00.34s || +0m00.02s

3 files changed, 24 insertions, 15 deletions

diff --git a/src/Algebra.v b/src/Algebra.v
index 15e0bcd38..e45ab8c33 100644
--- a/src/Algebra.v
+++ b/src/Algebra.v
@@ -1,12 +1,18 @@
 Require Import Coq.Classes.Morphisms. Require Coq.Setoids.Setoid.
 Require Import Crypto.Util.Tactics Crypto.Tactics.Nsatz.
+Require Import Crypto.Util.Decidable.
 Local Close Scope nat_scope. Local Close Scope type_scope. Local Close Scope core_scope.
 
+Notation is_eq_dec := (DecidableRel _) (only parsing).
+Notation "@ 'is_eq_dec' T R" := (DecidableRel (R:T->T->Prop))
+                                  (at level 10, T at level 8, R at level 8, only parsing).
+Notation eq_dec x y := (@dec (_ x y) _) (only parsing).
+
 Section Algebra.
   Context {T:Type} {eq:T->T->Prop}.
   Local Infix "=" := eq : type_scope. Local Notation "a <> b" := (not (a = b)) : type_scope.
 
-  Class is_eq_dec := { eq_dec : forall x y : T, {x=y} + {x<>y} }.
+  Local Notation is_eq_dec := (@is_eq_dec T eq).
 
   Section SingleOperation.
     Context {op:T->T->T}.
diff --git a/src/CompleteEdwardsCurve/CompleteEdwardsCurveTheorems.v b/src/CompleteEdwardsCurve/CompleteEdwardsCurveTheorems.v
index 476592b36..e3809bb8a 100644
--- a/src/CompleteEdwardsCurve/CompleteEdwardsCurveTheorems.v
+++ b/src/CompleteEdwardsCurve/CompleteEdwardsCurveTheorems.v
@@ -119,7 +119,7 @@ Module E.
       | |- _ => split
       | |- Feq _ _ => field_algebra
       | |- _ <> 0 => expand_opp; solve [nsatz_nonzero|eauto 6]
-      | |- {_}+{_} => eauto 15 using decide_and, @eq_dec with typeclass_instances
+      | |- Decidable.Decidable _ => solve [ typeclasses eauto ]
       end.
 
     Ltac bash := repeat bash_step.
diff --git a/src/Util/Decidable.v b/src/Util/Decidable.v
index 726b52b6b..9ab05699a 100644
--- a/src/Util/Decidable.v
+++ b/src/Util/Decidable.v
@@ -32,24 +32,24 @@ Local Ltac solve_decidable_transparent := solve_decidable_transparent_with first
 
 Local Hint Extern 0 => solve [ solve_decidable_transparent ] : typeclass_instances.
 
-Global Instance dec_True : Decidable True := left I.
-Global Instance dec_False : Decidable False := right (fun x => x).
-Global Instance dec_or {A B} `{Decidable A, Decidable B} : Decidable (A \/ B). exact _. Defined.
-Global Instance dec_and {A B} `{Decidable A, Decidable B} : Decidable (A /\ B). exact _. Defined.
-Global Instance dec_impl {A B} `{Decidable (B \/ ~A)} : Decidable (A -> B) | 3. exact _. Defined.
-Global Instance dec_impl_simple {A B} `{Decidable A, Decidable B} : Decidable (A -> B). exact _. Defined.
-Global Instance dec_iff {A B} `{Decidable A, Decidable B} : Decidable (A <-> B). exact _. Defined.
+Global Instance dec_True : Decidable True | 10 := left I.
+Global Instance dec_False : Decidable False | 10 := right (fun x => x).
+Global Instance dec_or {A B} `{Decidable A, Decidable B} : Decidable (A \/ B) | 10. exact _. Defined.
+Global Instance dec_and {A B} `{Decidable A, Decidable B} : Decidable (A /\ B) | 10. exact _. Defined.
+Global Instance dec_impl {A B} `{Decidable (B \/ ~A)} : Decidable (A -> B) | 10. exact _. Defined.
+Global Instance dec_impl_simple {A B} `{Decidable A, Decidable B} : Decidable (A -> B) | 10. exact _. Defined.
+Global Instance dec_iff {A B} `{Decidable A, Decidable B} : Decidable (A <-> B) | 10. exact _. Defined.
 Lemma dec_not {A} `{Decidable A} : Decidable (~A).
 Proof. solve_decidable_transparent. Defined.
 (** Disallow infinite loops of dec_not *)
 Hint Extern 0 (Decidable (~?A)) => apply (@dec_not A) : typeclass_instances.
 
-Global Instance dec_eq_unit : DecidableRel (@eq unit). exact _. Defined.
-Global Instance dec_eq_bool : DecidableRel (@eq bool). exact _. Defined.
-Global Instance dec_eq_Empty_set : DecidableRel (@eq Empty_set). exact _. Defined.
-Global Instance dec_eq_nat : DecidableRel (@eq nat). exact _. Defined.
-Global Instance dec_eq_prod {A B} `{DecidableRel (@eq A), DecidableRel (@eq B)} : DecidableRel (@eq (A * B)). exact _. Defined.
-Global Instance dec_eq_sum {A B} `{DecidableRel (@eq A), DecidableRel (@eq B)} : DecidableRel (@eq (A + B)). exact _. Defined.
+Global Instance dec_eq_unit : DecidableRel (@eq unit) | 10. exact _. Defined.
+Global Instance dec_eq_bool : DecidableRel (@eq bool) | 10. exact _. Defined.
+Global Instance dec_eq_Empty_set : DecidableRel (@eq Empty_set) | 10. exact _. Defined.
+Global Instance dec_eq_nat : DecidableRel (@eq nat) | 10. exact _. Defined.
+Global Instance dec_eq_prod {A B} `{DecidableRel (@eq A), DecidableRel (@eq B)} : DecidableRel (@eq (A * B)) | 10. exact _. Defined.
+Global Instance dec_eq_sum {A B} `{DecidableRel (@eq A), DecidableRel (@eq B)} : DecidableRel (@eq (A + B)) | 10. exact _. Defined.
 
 Lemma Decidable_respects_iff A B (H : A <-> B) : (Decidable A -> Decidable B) * (Decidable B -> Decidable A).
 Proof. solve_decidable_transparent. Defined.
@@ -59,3 +59,6 @@ Proof. solve_decidable_transparent. Defined.
 
 Lemma Decidable_iff_to_flip_impl A B (H : A <-> B) : Decidable B -> Decidable A.
 Proof. solve_decidable_transparent. Defined.
+
+(** For dubious compatibility with [eauto using]. *)
+Hint Extern 2 (Decidable _) => progress unfold Decidable : typeclass_instances core.