Remove the Const constructor of exprf

We instead use the [Op] constructor for constants.  This allows [exprf]
to not depend on the interpretation function; this means we don't need
to map over it to change the interpretation function.  This saves us
about 300 lines of code and about 30s of build time, total.

After     | File Name                                                              | Before    || Change
-----------------------------------------------------------------------------------------------------------
18m11.34s | Total                                                                  | 18m46.86s || -0m35.52s
-----------------------------------------------------------------------------------------------------------
0m19.99s  | Specific/GF25519ReflectiveAddCoordinates                               | 0m31.78s  || -0m11.79s
1m53.69s  | Specific/GF25519Reflective/Reified/LadderStep                          | 2m01.32s  || -0m07.62s
1m27.74s  | Specific/GF25519Reflective/Reified/AddCoordinates                      | 1m34.03s  || -0m06.29s
0m01.45s  | Specific/GF25519Reflective                                             | 0m06.31s  || -0m04.85s
0m04.62s  | Reflection/InlineInterp                                                | 0m01.52s  || +0m03.10s
1m19.68s  | CompleteEdwardsCurve/ExtendedCoordinates                               | 1m21.19s  || -0m01.50s
0m02.90s  | Reflection/InlineWf                                                    | 0m01.80s  || +0m01.09s
   N/A    | Reflection/MapWithInterpInfo                                           | 0m01.70s  || -0m01.70s
1m32.32s  | Test/Curve25519SpecTestVectors                                         | 1m32.25s  || +0m00.06s
1m12.79s  | Experiments/Ed25519                                                    | 1m13.17s  || -0m00.38s
0m40.52s  | ModularArithmetic/Conversion                                           | 0m40.44s  || +0m00.08s
0m34.57s  | Spec/Ed25519                                                           | 0m34.53s  || +0m00.03s
0m30.82s  | ModularArithmetic/ModularBaseSystemProofs                              | 0m30.89s  || -0m00.07s
0m30.11s  | Specific/GF25519Bounded                                                | 0m30.21s  || -0m00.10s
0m23.20s  | Experiments/MontgomeryCurve                                            | 0m23.26s  || -0m00.06s
0m22.16s  | Reflection/Z/Interpretations128/Relations                              | 0m21.72s  || +0m00.44s
0m21.69s  | ModularArithmetic/Pow2BaseProofs                                       | 0m21.58s  || +0m00.11s
0m20.25s  | Algebra                                                                | 0m20.27s  || -0m00.01s
0m19.81s  | Specific/GF25519                                                       | 0m19.83s  || -0m00.01s
0m18.82s  | Reflection/Z/Interpretations64/Relations                               | 0m18.50s  || +0m00.32s
0m18.19s  | EdDSARepChange                                                         | 0m18.26s  || -0m00.07s
0m17.08s  | CompleteEdwardsCurve/CompleteEdwardsCurveTheorems                      | 0m17.16s  || -0m00.08s
0m14.10s  | Util/ZUtil                                                             | 0m14.03s  || +0m00.07s
0m10.05s  | Testbit                                                                | 0m10.02s  || +0m00.03s
0m08.95s  | Specific/GF25519BoundedCommon                                          | 0m08.90s  || +0m00.04s
0m08.87s  | BoundedArithmetic/ArchitectureToZLikeProofs                            | 0m08.81s  || +0m00.05s
0m08.80s  | Assembly/GF25519                                                       | 0m08.82s  || -0m00.01s
0m08.80s  | ModularArithmetic/Montgomery/ZProofs                                   | 0m08.87s  || -0m00.06s
0m08.51s  | Encoding/PointEncoding                                                 | 0m08.53s  || -0m00.01s
0m08.37s  | BoundedArithmetic/Double/Proofs/Multiply                               | 0m08.39s  || -0m00.02s
0m08.34s  | Specific/GF1305                                                        | 0m08.31s  || +0m00.02s
0m07.85s  | BoundedArithmetic/Double/Proofs/ShiftRightDoubleWordImmediate          | 0m07.86s  || -0m00.01s
0m07.60s  | Specific/GF25519Reflective/Reified/Mul                                 | 0m07.52s  || +0m00.08s
0m07.13s  | MxDHRepChange                                                          | 0m07.08s  || +0m00.04s
0m06.85s  | BoundedArithmetic/Double/Proofs/SpreadLeftImmediate                    | 0m06.77s  || +0m00.08s
0m06.62s  | Reflection/Z/InterpretationsGen                                        | 0m06.62s  || +0m00.00s
0m05.69s  | Reflection/Z/Interpretations64/RelationsCombinations                   | 0m05.60s  || +0m00.09s
0m05.68s  | Reflection/Z/Interpretations128/RelationsCombinations                  | 0m05.67s  || +0m00.00s
0m05.48s  | Specific/SC25519                                                       | 0m05.38s  || +0m00.10s
0m05.40s  | BoundedArithmetic/Double/Proofs/RippleCarryAddSub                      | 0m05.36s  || +0m00.04s
0m04.93s  | ModularArithmetic/ModularBaseSystemListProofs                          | 0m04.96s  || -0m00.03s
0m04.87s  | WeierstrassCurve/Pre                                                   | 0m04.83s  || +0m00.04s
0m04.45s  | Specific/GF25519Reflective/Reified/PreFreeze                           | 0m04.56s  || -0m00.10s
0m04.33s  | Specific/GF25519Reflective/CommonBinOp                                 | 0m04.55s  || -0m00.21s
0m03.95s  | ModularArithmetic/BarrettReduction/ZHandbook                           | 0m03.98s  || -0m00.02s
0m03.87s  | Specific/GF25519Reflective/CommonUnOp                                  | 0m04.08s  || -0m00.20s
0m03.86s  | Encoding/PointEncodingPre                                              | 0m03.88s  || -0m00.02s
0m03.84s  | BaseSystemProofs                                                       | 0m03.88s  || -0m00.04s
0m03.75s  | Specific/GF25519Reflective/CommonUnOpWireToFE                          | 0m03.96s  || -0m00.20s
0m03.64s  | CompleteEdwardsCurve/Pre                                               | 0m03.60s  || +0m00.04s
0m03.51s  | BoundedArithmetic/InterfaceProofs                                      | 0m03.42s  || +0m00.08s
0m03.41s  | ModularArithmetic/Tutorial                                             | 0m03.38s  || +0m00.03s
0m03.30s  | Reflection/LinearizeWf                                                 | 0m04.24s  || -0m00.94s
0m03.20s  | Specific/GF25519Reflective/Reified/CarrySub                            | 0m03.28s  || -0m00.07s
0m03.14s  | ModularArithmetic/ZBoundedZ                                            | 0m03.14s  || +0m00.00s
0m03.14s  | ModularArithmetic/BarrettReduction/ZGeneralized                        | 0m03.14s  || +0m00.00s
0m02.95s  | Specific/GF25519Reflective/Reified/CarryAdd                            | 0m02.93s  || +0m00.02s
0m02.92s  | Specific/GF25519Reflective/Common9_4Op                                 | 0m02.89s  || +0m00.02s
0m02.82s  | Specific/GF25519Reflective/Reified/CarryOpp                            | 0m02.95s  || -0m00.13s
0m02.82s  | BoundedArithmetic/Double/Proofs/Decode                                 | 0m02.90s  || -0m00.08s
0m02.69s  | BoundedArithmetic/Double/Proofs/ShiftRight                             | 0m02.69s  || +0m00.00s
0m02.66s  | ModularArithmetic/ModularArithmeticTheorems                            | 0m02.67s  || -0m00.00s
0m02.63s  | Specific/FancyMachine256/Montgomery                                    | 0m02.24s  || +0m00.38s
0m02.61s  | Specific/GF25519Reflective/CommonUnOpFEToWire                          | 0m02.76s  || -0m00.14s
0m02.58s  | BoundedArithmetic/Double/Proofs/ShiftLeft                              | 0m02.58s  || +0m00.00s
0m02.56s  | Specific/GF25519Reflective/Common                                      | 0m02.60s  || -0m00.04s
0m02.52s  | Specific/FancyMachine256/Barrett                                       | 0m02.20s  || +0m00.31s
0m02.50s  | Specific/GF25519BoundedAddCoordinates                                  | 0m02.60s  || -0m00.10s
0m02.39s  | ModularArithmetic/BarrettReduction/ZBounded                            | 0m02.31s  || +0m00.08s
0m02.34s  | ModularArithmetic/ModularBaseSystemOpt                                 | 0m02.31s  || +0m00.02s
0m02.02s  | Specific/GF25519Reflective/Reified/Sub                                 | 0m02.01s  || +0m00.01s
0m01.98s  | Reflection/WfReflective                                                | 0m02.54s  || -0m00.56s
0m01.94s  | Specific/GF25519Reflective/Reified/Pack                                | 0m02.07s  || -0m00.12s
0m01.93s  | Assembly/Evaluables                                                    | 0m01.93s  || +0m00.00s
0m01.92s  | Specific/GF25519Reflective/Reified/Unpack                              | 0m02.04s  || -0m00.12s
0m01.90s  | Specific/FancyMachine256/Core                                          | 0m01.79s  || +0m00.10s
0m01.79s  | ModularArithmetic/Montgomery/ZBounded                                  | 0m01.86s  || -0m00.07s
0m01.77s  | Specific/GF25519ExtendedAddCoordinates                                 | 0m01.76s  || +0m00.01s
0m01.68s  | Specific/GF25519Reflective/Reified/Add                                 | 0m01.69s  || -0m00.01s
0m01.67s  | Specific/GF25519BoundedExtendedAddCoordinates                          | 0m01.79s  || -0m00.12s
0m01.63s  | Experiments/Ed25519Extraction                                          | 0m01.66s  || -0m00.03s
0m01.62s  | Specific/GF25519Reflective/Reified/Opp                                 | 0m01.72s  || -0m00.09s
0m01.54s  | Specific/GF25519Reflective/Reified/GeModulus                           | 0m01.59s  || -0m00.05s
0m01.50s  | Reflection/TestCase                                                    | 0m01.42s  || +0m00.08s
0m01.47s  | ModularArithmetic/BarrettReduction/Z                                   | 0m01.47s  || +0m00.00s
0m01.43s  | Assembly/Compile                                                       | 0m01.46s  || -0m00.03s
0m01.33s  | Reflection/WfProofs                                                    | 0m01.95s  || -0m00.61s
0m01.26s  | ModularArithmetic/PrimeFieldTheorems                                   | 0m01.28s  || -0m00.02s
0m01.20s  | Assembly/Conversions                                                   | 0m01.19s  || +0m00.01s
0m01.16s  | ModularArithmetic/ExtendedBaseVector                                   | 0m01.22s  || -0m00.06s
0m01.16s  | BaseSystem                                                             | 0m01.15s  || +0m00.01s
0m01.09s  | BoundedArithmetic/Double/Repeated/Proofs/Decode                        | 0m01.11s  || -0m00.02s
0m00.99s  | Util/NumTheoryUtil                                                     | 0m00.96s  || +0m00.03s
0m00.96s  | Assembly/HL                                                            | 0m00.96s  || +0m00.00s
0m00.94s  | Assembly/LL                                                            | 0m00.99s  || -0m00.05s
0m00.94s  | Assembly/Pipeline                                                      | 0m00.92s  || +0m00.01s
0m00.88s  | Reflection/WfInversion                                                 | 0m01.21s  || -0m00.32s
0m00.86s  | Assembly/PhoasCommon                                                   | 0m00.91s  || -0m00.05s
0m00.86s  | Specific/GF25519Reflective/CommonUnOpFEToZ                             | 0m01.02s  || -0m00.16s
0m00.85s  | BoundedArithmetic/Double/Proofs/LoadImmediate                          | 0m00.86s  || -0m00.01s
0m00.84s  | BoundedArithmetic/Double/Proofs/BitwiseOr                              | 0m00.91s  || -0m00.07s
0m00.82s  | BoundedArithmetic/X86ToZLikeProofs                                     | 0m00.85s  || -0m00.03s
0m00.79s  | Util/IterAssocOp                                                       | 0m00.82s  || -0m00.02s
0m00.75s  | Reflection/Z/Syntax                                                    | 0m00.67s  || +0m00.07s
0m00.73s  | Util/PartiallyReifiedProp                                              | 0m00.73s  || +0m00.00s
0m00.70s  | Encoding/ModularWordEncodingTheorems                                   | 0m00.76s  || -0m00.06s
0m00.70s  | Specific/GF25519Reflective/Reified                                     | 0m00.72s  || -0m00.02s
0m00.66s  | BoundedArithmetic/Double/Repeated/Proofs/Multiply                      | 0m00.62s  || +0m00.04s
0m00.66s  | ModularArithmetic/PseudoMersenneBaseParamProofs                        | 0m00.66s  || +0m00.00s
0m00.64s  | Encoding/ModularWordEncodingPre                                        | 0m00.63s  || +0m00.01s
0m00.61s  | ModularArithmetic/ModularBaseSystem                                    | 0m00.66s  || -0m00.05s
0m00.61s  | Util/AdditionChainExponentiation                                       | 0m00.64s  || -0m00.03s
0m00.60s  | ModularArithmetic/ExtPow2BaseMulProofs                                 | 0m00.64s  || -0m00.04s
0m00.60s  | Reflection/MultiSizeTest2                                              | 0m00.71s  || -0m00.10s
0m00.60s  | Spec/EdDSA                                                             | 0m00.65s  || -0m00.05s
0m00.59s  | BoundedArithmetic/Double/Repeated/Proofs/RippleCarryAddSub             | 0m00.51s  || +0m00.07s
0m00.58s  | BoundedArithmetic/Interface                                            | 0m00.60s  || -0m00.02s
0m00.58s  | ModularArithmetic/ModularBaseSystemList                                | 0m00.61s  || -0m00.03s
0m00.55s  | Spec/ModularWordEncoding                                               | 0m00.63s  || -0m00.07s
0m00.55s  | BoundedArithmetic/Double/Repeated/Proofs/ShiftLeftRight                | 0m00.50s  || +0m00.05s
0m00.55s  | Reflection/InterpWfRel                                                 | 0m00.58s  || -0m00.02s
0m00.54s  | BoundedArithmetic/X86ToZLike                                           | 0m00.55s  || -0m00.01s
0m00.54s  | BoundedArithmetic/Double/Repeated/Proofs/SelectConditional             | 0m00.53s  || +0m00.01s
0m00.54s  | BoundedArithmetic/Double/Proofs/SelectConditional                      | 0m00.62s  || -0m00.07s
0m00.54s  | Reflection/WfReflectiveGen                                             | 0m00.58s  || -0m00.03s
0m00.52s  | BoundedArithmetic/ArchitectureToZLike                                  | 0m00.41s  || +0m00.11s
0m00.51s  | BoundedArithmetic/Double/Core                                          | 0m00.48s  || +0m00.03s
0m00.50s  | BoundedArithmetic/Double/Repeated/Proofs/BitwiseOr                     | 0m00.51s  || -0m00.01s
0m00.49s  | Spec/WeierstrassCurve                                                  | 0m00.42s  || +0m00.07s
0m00.48s  | BoundedArithmetic/Double/Repeated/Proofs/LoadImmediate                 | 0m00.54s  || -0m00.06s
0m00.47s  | BoundedArithmetic/Double/Repeated/Proofs/ShiftRightDoubleWordImmediate | 0m00.58s  || -0m00.10s
0m00.47s  | Reflection/InterpWf                                                    | 0m00.50s  || -0m00.03s
0m00.47s  | ModularArithmetic/Pre                                                  | 0m00.48s  || -0m00.01s
0m00.47s  | BoundedArithmetic/Double/Repeated/Core                                 | 0m00.46s  || +0m00.00s
0m00.46s  | Reflection/Z/Interpretations64                                         | 0m00.49s  || -0m00.02s
0m00.45s  | Reflection/InputSyntax                                                 | 0m00.42s  || +0m00.03s
   N/A    | Reflection/MapInterpWf                                                 | 0m00.44s  || -0m00.44s
0m00.44s  | Spec/CompleteEdwardsCurve                                              | 0m00.39s  || +0m00.04s
0m00.44s  | BoundedArithmetic/Double/Proofs/ShiftLeftRightTactic                   | 0m00.42s  || +0m00.02s
0m00.44s  | Reflection/Z/Interpretations128                                        | 0m00.52s  || -0m00.08s
0m00.44s  | ModularArithmetic/ZBounded                                             | 0m00.45s  || -0m00.01s
0m00.42s  | BoundedArithmetic/StripCF                                              | 0m00.42s  || +0m00.00s
0m00.42s  | Reflection/Z/Reify                                                     | 0m00.43s  || -0m00.01s
0m00.42s  | ModularArithmetic/ModularBaseSystemListZOperationsProofs               | 0m00.47s  || -0m00.04s
0m00.41s  | Reflection/Named/DeadCodeElimination                                   | 0m00.43s  || -0m00.02s
0m00.41s  | ModularArithmetic/PseudoMersenneBaseParams                             | 0m00.38s  || +0m00.02s
0m00.41s  | Spec/ModularArithmetic                                                 | 0m00.34s  || +0m00.06s
0m00.41s  | Reflection/Named/RegisterAssign                                        | 0m00.37s  || +0m00.03s
0m00.41s  | Reflection/Reify                                                       | 0m00.39s  || +0m00.01s
0m00.40s  | ModularArithmetic/Pow2Base                                             | 0m00.42s  || -0m00.01s
0m00.40s  | Reflection/Named/EstablishLiveness                                     | 0m00.33s  || +0m00.07s
0m00.40s  | Reflection/ExprInversion                                               | 0m00.57s  || -0m00.16s
0m00.40s  | Reflection/Named/Syntax                                                | 0m00.40s  || +0m00.00s
0m00.38s  | Reflection/Named/Compile                                               | 0m00.34s  || +0m00.03s
   N/A    | Reflection/MapInterp                                                   | 0m00.38s  || -0m00.38s
0m00.38s  | ModularArithmetic/Montgomery/Z                                         | 0m00.39s  || -0m00.01s
0m00.38s  | ModularArithmetic/ModularBaseSystemWord                                | 0m00.36s  || +0m00.02s
   N/A    | Reflection/WfRel                                                       | 0m00.38s  || -0m00.38s
0m00.38s  | Reflection/Tuple                                                       |    N/A    || +0m00.38s
0m00.37s  | Spec/MxDH                                                              | 0m00.39s  || -0m00.02s
0m00.37s  | BoundedArithmetic/Eta                                                  | 0m00.36s  || +0m00.01s
0m00.36s  | Reflection/Named/ContextOn                                             | 0m00.37s  || -0m00.01s
0m00.34s  | Reflection/FilterLive                                                  | 0m00.34s  || +0m00.00s
0m00.34s  | Reflection/LinearizeInterp                                             | 0m00.53s  || -0m00.19s
0m00.33s  | Reflection/Relations                                                   |    N/A    || +0m00.33s
0m00.32s  | Reflection/Equality                                                    | 0m00.41s  || -0m00.08s
0m00.30s  | Reflection/Syntax                                                      | 0m00.52s  || -0m00.22s
0m00.30s  | Reflection/InterpProofs                                                | 0m00.47s  || -0m00.17s
0m00.30s  | Reflection/ApplicationLemmas                                           | 0m00.43s  || -0m00.13s
0m00.30s  | Reflection/CommonSubexpressionElimination                              | 0m00.50s  || -0m00.20s
0m00.28s  | Reflection/Inline                                                      | 0m00.37s  || -0m00.08s
0m00.28s  | Reflection/Conversion                                                  | 0m00.44s  || -0m00.15s
0m00.27s  | Reflection/MapCastWithCastOp                                           | 0m00.38s  || -0m00.10s
0m00.26s  | Reflection/MapCast                                                     | 0m00.40s  || -0m00.14s
0m00.26s  | Reflection/ApplicationRelations                                        | 0m00.35s  || -0m00.08s
0m00.24s  | Reflection/Application                                                 | 0m00.38s  || -0m00.14s
0m00.23s  | Reflection/Linearize                                                   | 0m00.35s  || -0m00.11s
0m00.20s  | Reflection/CountLets                                                   | 0m00.33s  || -0m00.13s
0m00.20s  | Reflection/Named/NameUtil                                              | 0m00.36s  || -0m00.15s
0m00.09s  | Util/PointedProp                                                       | 0m00.08s  || +0m00.00s
0m00.05s  | Util/LetIn                                                             | 0m00.06s  || -0m00.00s
0m00.03s  | Util/Notations                                                         | 0m00.03s  || +0m00.00s
0m00.03s  | Util/AutoRewrite                                                       | 0m00.03s  || +0m00.00s

1 files changed, 69 insertions, 202 deletions

diff --git a/src/Reflection/Syntax.v b/src/Reflection/Syntax.v
index f8d7cdcf3..58000fdca 100644
--- a/src/Reflection/Syntax.v
+++ b/src/Reflection/Syntax.v
@@ -1,6 +1,5 @@
 (** * PHOAS Representation of Gallina *)
 Require Import Coq.Strings.String Coq.Classes.RelationClasses Coq.Classes.Morphisms.
-Require Import Crypto.Util.Tuple.
 Require Import Crypto.Util.LetIn.
 Require Import Crypto.Util.Tactics.
 Require Import Crypto.Util.Notations.
@@ -16,7 +15,7 @@ Local Open Scope expr_scope.
 Section language.
   Context (base_type_code : Type).
 
-  Inductive flat_type := Tbase (T : base_type_code) | Prod (A B : flat_type).
+  Inductive flat_type := Tbase (T : base_type_code) | Unit | Prod (A B : flat_type).
   Bind Scope ctype_scope with flat_type.
 
   Inductive type := Tflat (T : flat_type) | Arrow (A : base_type_code) (B : type).
@@ -27,16 +26,19 @@ Section language.
   Notation "A -> B" := (Arrow A B) : ctype_scope.
   Local Coercion Tbase : base_type_code >-> flat_type.
 
-  Fixpoint tuple' T n :=
-    match n with
-    | O => T
-    | S n' => (tuple' T n' * T)%ctype
-    end.
-  Definition tuple T n :=
-    match n with
-    | O => T (* default value; no empty tuple *)
-    | S n' => tuple' T n'
-    end.
+  Section tuple.
+    Context (T : flat_type).
+    Fixpoint tuple' n :=
+      match n with
+      | O => T
+      | S n' => (tuple' n' * T)%ctype
+      end.
+    Definition tuple n :=
+      match n with
+      | O => Unit
+      | S n' => tuple' n'
+      end.
+  End tuple.
 
   Section interp.
     Section type.
@@ -49,134 +51,19 @@ Section language.
           | Arrow x y => (interp_src_type x -> interp_type_gen_hetero y)%type
           end.
       End hetero.
-      Section homogenous.
-        Context (interp_flat_type : flat_type -> Type).
-        Definition interp_type_gen := interp_type_gen_hetero interp_flat_type interp_flat_type.
-        Section rel.
-          Context (R : forall t, interp_flat_type t -> interp_flat_type t -> Prop).
-          Fixpoint interp_type_gen_rel_pointwise (t : type)
-            : interp_type_gen t -> interp_type_gen t -> Prop :=
-            match t with
-            | Tflat t => R t
-            | Arrow _ y => fun f g => forall x, interp_type_gen_rel_pointwise y (f x) (g x)
-            end.
-          Global Instance interp_type_gen_rel_pointwise_Reflexive {H : forall t, Reflexive (R t)}
-            : forall t, Reflexive (interp_type_gen_rel_pointwise t).
-          Proof. induction t; repeat intro; reflexivity. Qed.
-          Global Instance interp_type_gen_rel_pointwise_Symmetric {H : forall t, Symmetric (R t)}
-            : forall t, Symmetric (interp_type_gen_rel_pointwise t).
-          Proof. induction t; simpl; repeat intro; symmetry; eauto. Qed.
-          Global Instance interp_type_gen_rel_pointwise_Transitive {H : forall t, Transitive (R t)}
-            : forall t, Transitive (interp_type_gen_rel_pointwise t).
-          Proof. induction t; simpl; repeat intro; etransitivity; eauto. Qed.
-        End rel.
-      End homogenous.
+      Definition interp_type_gen (interp_flat_type : flat_type -> Type)
+        := interp_type_gen_hetero interp_flat_type interp_flat_type.
     End type.
     Section flat_type.
       Context (interp_base_type : base_type_code -> Type).
       Fixpoint interp_flat_type (t : flat_type) :=
         match t with
         | Tbase t => interp_base_type t
+        | Unit => unit
         | Prod x y => prod (interp_flat_type x) (interp_flat_type y)
         end.
       Definition interp_type := interp_type_gen interp_flat_type.
-      Fixpoint flat_interp_tuple' {T n} : interp_flat_type (tuple' T n) -> Tuple.tuple' (interp_flat_type T) n
-        := match n return interp_flat_type (tuple' T n) -> Tuple.tuple' (interp_flat_type T) n with
-           | O => fun x => x
-           | S n' => fun xy => (@flat_interp_tuple' _ n' (fst xy), snd xy)
-           end.
-      Definition flat_interp_tuple {T n} : interp_flat_type (tuple T n) -> Tuple.tuple (interp_flat_type T) n
-        := match n return interp_flat_type (tuple T n) -> Tuple.tuple (interp_flat_type T) n with
-           | O => fun _ => tt
-           | S n' => @flat_interp_tuple' T n'
-           end.
-      Fixpoint flat_interp_untuple' {T n} : Tuple.tuple' (interp_flat_type T) n -> interp_flat_type (tuple' T n)
-        := match n return Tuple.tuple' (interp_flat_type T) n -> interp_flat_type (tuple' T n) with
-           | O => fun x => x
-           | S n' => fun xy => (@flat_interp_untuple' _ n' (fst xy), snd xy)
-           end.
-      Lemma flat_interp_untuple'_tuple' {T n v}
-        : @flat_interp_untuple' T n (flat_interp_tuple' v) = v.
-      Proof. induction n; [ reflexivity | simpl; rewrite IHn; destruct v; reflexivity ]. Qed.
-      Lemma flat_interp_untuple'_tuple {T n v}
-        : flat_interp_untuple' (@flat_interp_tuple T (S n) v) = v.
-      Proof. apply flat_interp_untuple'_tuple'. Qed.
-      Lemma flat_interp_tuple'_untuple' {T n v}
-        : @flat_interp_tuple' T n (flat_interp_untuple' v) = v.
-      Proof. induction n; [ reflexivity | simpl; rewrite IHn; destruct v; reflexivity ]. Qed.
-      Lemma flat_interp_tuple_untuple' {T n v}
-        : @flat_interp_tuple T (S n) (flat_interp_untuple' v) = v.
-      Proof. apply flat_interp_tuple'_untuple'. Qed.
-      Definition tuple_map {A B n} (f : interp_flat_type A -> interp_flat_type B) (v : interp_flat_type (tuple A n))
-        : interp_flat_type (tuple B n)
-        := let fv := Tuple.map f (flat_interp_tuple v) in
-           match n return interp_flat_type (tuple A n) -> Tuple.tuple (interp_flat_type B) n -> interp_flat_type (tuple B n) with
-           | 0 => fun v _ => f v
-           | S _ => fun v fv => flat_interp_untuple' fv
-           end v fv.
-      Section rel.
-        Context (R : forall t, interp_base_type t -> interp_base_type t -> Prop).
-        Fixpoint interp_flat_type_rel_pointwise (t : flat_type)
-          : interp_flat_type t -> interp_flat_type t -> Prop :=
-          match t with
-          | Tbase t => R t
-          | Prod _ _ => fun x y => interp_flat_type_rel_pointwise _ (fst x) (fst y)
-                                   /\ interp_flat_type_rel_pointwise _ (snd x) (snd y)
-          end.
-        Definition interp_type_rel_pointwise
-          := interp_type_gen_rel_pointwise _ interp_flat_type_rel_pointwise.
-      End rel.
     End flat_type.
-    Section rel_pointwise2.
-      Section type.
-        Section hetero.
-          Context (interp_src1 interp_src2 : base_type_code -> Type)
-                  (interp_flat_type1 interp_flat_type2 : flat_type -> Type)
-                  (Rsrc : forall t, interp_src1 t -> interp_src2 t -> Prop)
-                  (R : forall t, interp_flat_type1 t -> interp_flat_type2 t -> Prop).
-
-          Fixpoint interp_type_gen_rel_pointwise2_hetero (t : type)
-            : interp_type_gen_hetero interp_src1 interp_flat_type1 t
-              -> interp_type_gen_hetero interp_src2 interp_flat_type2 t
-              -> Prop
-            := match t with
-               | Tflat t => R t
-               | Arrow src dst => @respectful_hetero _ _ _ _ (Rsrc src) (fun _ _ => interp_type_gen_rel_pointwise2_hetero dst)
-               end.
-        End hetero.
-        Section homogenous.
-          Context (interp_flat_type1 interp_flat_type2 : flat_type -> Type)
-                  (R : forall t, interp_flat_type1 t -> interp_flat_type2 t -> Prop).
-
-          Definition interp_type_gen_rel_pointwise2
-            := interp_type_gen_rel_pointwise2_hetero interp_flat_type1 interp_flat_type2
-                                                     interp_flat_type1 interp_flat_type2
-                                                     R R.
-        End homogenous.
-      End type.
-      Section flat_type.
-        Context (interp_base_type1 interp_base_type2 : base_type_code -> Type).
-        Section gen_prop.
-          Context (P : Type)
-                  (and : P -> P -> P)
-                  (R : forall t, interp_base_type1 t -> interp_base_type2 t -> P).
-
-          Fixpoint interp_flat_type_rel_pointwise2_gen_Prop (t : flat_type)
-            : interp_flat_type interp_base_type1 t -> interp_flat_type interp_base_type2 t -> P
-            := match t with
-               | Tbase t => R t
-               | Prod x y => fun a b => and (interp_flat_type_rel_pointwise2_gen_Prop x (fst a) (fst b))
-                                            (interp_flat_type_rel_pointwise2_gen_Prop y (snd a) (snd b))
-               end.
-        End gen_prop.
-
-        Definition interp_flat_type_rel_pointwise2
-          := @interp_flat_type_rel_pointwise2_gen_Prop Prop and.
-
-        Definition interp_type_rel_pointwise2 R
-          := interp_type_gen_rel_pointwise2 _ _ (interp_flat_type_rel_pointwise2 R).
-      End flat_type.
-    End rel_pointwise2.
   End interp.
 
   Section expr_param.
@@ -190,7 +77,7 @@ Section language.
 
       (** N.B. [Let] binds the components of a pair to separate variables, and does so recursively *)
       Inductive exprf : flat_type -> Type :=
-      | Const {t : flat_type} (x : interp_type t) : exprf t
+      | TT : exprf Unit
       | Var {t} (v : var t) : exprf t
       | Op {t1 tR} (opc : op t1 tR) (args : exprf t1) : exprf tR
       | LetIn {tx} (ex : exprf tx) {tC} (eC : interp_flat_type_gen var tx -> exprf tC) : exprf tC
@@ -201,32 +88,6 @@ Section language.
       | Abs {src dst} (f : var src -> expr dst) : expr (Arrow src dst).
       Bind Scope expr_scope with expr.
       Global Coercion Return : exprf >-> expr.
-      Definition UnReturn {t} (e : expr (Tflat t)) : exprf t
-        := match e with
-           | Return _ v => v
-           | Abs _ _ _ => I
-           end.
-      Definition UnAbs {src dst} (e : expr (Arrow src dst)) : var src -> expr dst
-        := match e with
-           | Return _ _ => I
-           | Abs _ _ f => f
-           end.
-      Definition UnReturn_eta {t} (e : expr (Tflat t)) : Return (UnReturn e) = e
-        := match e in expr T return match T return expr T -> Prop with
-                                    | Tflat _ => fun e => Return (UnReturn e) = e
-                                    | _ => fun _ => I = I
-                                    end e with
-           | Return _ _ => eq_refl
-           | Abs _ _ _ => eq_refl
-           end.
-      Definition UnAbs_eta {src dst} (e : expr (Arrow src dst)) : Abs (UnAbs e) = e
-        := match e in expr T return match T return expr T -> Prop with
-                                    | Arrow src dst => fun e => Abs (UnAbs e) = e
-                                    | _ => fun _ => I = I
-                                    end e with
-           | Return _ _ => eq_refl
-           | Abs _ _ _ => eq_refl
-           end.
       (** Sometimes, we want to deal with partially-interpreted
           expressions, things like [prod (exprf A) (exprf B)] rather
           than [exprf (Prod A B)], or like [prod (var A) (var B)] when
@@ -239,25 +100,28 @@ Section language.
           [interp_base_type] into [exprf])). *)
       Fixpoint smart_interp_flat_map {f g}
                (h : forall x, f x -> g (Tbase x))
+               (tt : g Unit)
                (pair : forall A B, g A -> g B -> g (Prod A B))
                {t}
         : interp_flat_type_gen f t -> g t
         := match t return interp_flat_type_gen f t -> g t with
            | Tbase _ => h _
+           | Unit => fun _ => tt
            | Prod A B => fun v => pair _ _
-                                       (@smart_interp_flat_map f g h pair A (fst v))
-                                       (@smart_interp_flat_map f g h pair B (snd v))
+                                       (@smart_interp_flat_map f g h tt pair A (fst v))
+                                       (@smart_interp_flat_map f g h tt pair B (snd v))
            end.
       Fixpoint smart_interp_map_hetero {f g g'}
                (h : forall x, f x -> g (Tflat (Tbase x)))
+               (tt : g Unit)
                (pair : forall A B, g (Tflat A) -> g (Tflat B) -> g (Prod A B))
                (abs : forall A B, (g' A -> g B) -> g (Arrow A B))
                {t}
         : interp_type_gen_hetero g' (interp_flat_type_gen f) t -> g t
         := match t return interp_type_gen_hetero g' (interp_flat_type_gen f) t -> g t with
-           | Tflat _ => @smart_interp_flat_map f (fun x => g (Tflat x)) h pair _
+           | Tflat _ => @smart_interp_flat_map f (fun x => g (Tflat x)) h tt pair _
            | Arrow A B => fun v => abs _ _
-                                       (fun x => @smart_interp_map_hetero f g g' h pair abs B (v x))
+                                       (fun x => @smart_interp_map_hetero f g g' h tt pair abs B (v x))
            end.
       Fixpoint smart_interp_map_gen {f g}
                  (h : forall x, f x -> g (Tflat (Tbase x)))
@@ -274,44 +138,51 @@ Section language.
       Definition smart_interp_map {f g}
                  (h : forall x, f x -> g (Tflat (Tbase x)))
                  (h' : forall x, g (Tflat (Tbase x)) -> f x)
+                 (tt : g Unit)
                  (pair : forall A B, g (Tflat A) -> g (Tflat B) -> g (Prod A B))
                  (abs : forall A B, (g (Tflat (Tbase A)) -> g B) -> g (Arrow A B))
                  {t}
         : interp_type_gen (interp_flat_type_gen f) t -> g t
-        := @smart_interp_map_gen f g h h' (@smart_interp_flat_map f (fun x => g (Tflat x)) h pair) abs t.
+        := @smart_interp_map_gen f g h h' (@smart_interp_flat_map f (fun x => g (Tflat x)) h tt pair) abs t.
       Fixpoint SmartValf {T} (val : forall t : base_type_code, T t) t : interp_flat_type_gen T t
         := match t return interp_flat_type_gen T t with
            | Tbase _ => val _
+           | Unit => tt
            | Prod A B => (@SmartValf T val A, @SmartValf T val B)
            end.
       Fixpoint SmartArrow (A : flat_type) (B : type) : type
         := match A with
            | Tbase A' => Arrow A' B
+           | Unit => B
            | Prod A0 A1
              => SmartArrow A0 (SmartArrow A1 B)
            end.
       Fixpoint SmartAbs {A B} {struct A} : forall (f : exprf A -> expr B), expr (SmartArrow A B)
         := match A return (exprf A -> expr B) -> expr (SmartArrow A B) with
            | Tbase x => fun f => Abs (fun x => f (Var x))
+           | Unit => fun f => f TT
            | Prod x y => fun f => @SmartAbs x _ (fun x' => @SmartAbs y _ (fun y' => f (Pair x' y')))
            end.
 
       (** [SmartVar] is like [Var], except that it inserts
           pair-projections and [Pair] as necessary to handle
           [flat_type], and not just [base_type_code] *)
+      Definition SmartPairf {t} : interp_flat_type_gen exprf t -> exprf t
+        := @smart_interp_flat_map exprf exprf (fun t x => x) TT (fun A B x y => Pair x y) t.
       Definition SmartVarf {t} : interp_flat_type_gen var t -> exprf t
-        := @smart_interp_flat_map var exprf (fun t => Var) (fun A B x y => Pair x y) t.
+        := @smart_interp_flat_map var exprf (fun t => Var) TT (fun A B x y => Pair x y) t.
       Definition SmartVarfMap {var var'} (f : forall t, var t -> var' t) {t}
         : interp_flat_type_gen var t -> interp_flat_type_gen var' t
-        := @smart_interp_flat_map var (interp_flat_type_gen var') f (fun A B x y => pair x y) t.
+        := @smart_interp_flat_map var (interp_flat_type_gen var') f tt (fun A B x y => pair x y) t.
       Definition SmartFlatTypeMap {var'} (f : forall t, var' t -> base_type_code) {t}
         : interp_flat_type_gen var' t -> flat_type
-        := @smart_interp_flat_map var' (fun _ => flat_type) f (fun _ _ => Prod) t.
+        := @smart_interp_flat_map var' (fun _ => flat_type) f Unit (fun _ _ => Prod) t.
       Fixpoint SmartFlatTypeMapInterp {var' var''} (f : forall t, var' t -> base_type_code)
                  (fv : forall t v, var'' (f t v)) t {struct t}
         : forall v, interp_flat_type_gen var'' (SmartFlatTypeMap f (t:=t) v)
         := match t return forall v, interp_flat_type_gen var'' (SmartFlatTypeMap f (t:=t) v) with
            | Tbase x => fv _
+           | Unit => fun v => v
            | Prod A B => fun xy => (@SmartFlatTypeMapInterp _ _ f fv A (fst xy),
                                     @SmartFlatTypeMapInterp _ _ f fv B (snd xy))
            end.
@@ -323,19 +194,20 @@ Section language.
         := match t return forall v, interp_flat_type_gen var'' (SmartFlatTypeMap f (t:=t) v)
                                     -> interp_flat_type_gen var''' t with
            | Tbase x => fv _
+           | Unit => fun _ v => v
            | Prod A B => fun v xy => (@SmartFlatTypeMapUnInterp _ _ _ f fv A _ (fst xy),
                                       @SmartFlatTypeMapUnInterp _ _ _ f fv B _ (snd xy))
            end.
       Definition SmartVarMap {var var'} (f : forall t, var t -> var' t) (f' : forall t, var' t -> var t) {t}
         : interp_type_gen (interp_flat_type_gen var) t -> interp_type_gen (interp_flat_type_gen var') t
-        := @smart_interp_map var (interp_type_gen (interp_flat_type_gen var')) f f' (fun A B x y => pair x y) (fun A B f x => f x) t.
+        := @smart_interp_map var (interp_type_gen (interp_flat_type_gen var')) f f' tt (fun A B x y => pair x y) (fun A B f x => f x) t.
       Definition SmartVarMap_hetero {vars vars' var var'} (f : forall t, var t -> var' t) (f' : forall t, vars' t -> vars t) {t}
         : interp_type_gen_hetero vars (interp_flat_type_gen var) t -> interp_type_gen_hetero vars' (interp_flat_type_gen var') t
-        := @smart_interp_map_hetero var (interp_type_gen_hetero vars' (interp_flat_type_gen var')) vars f (fun A B x y => pair x y) (fun A B f x => f (f' _ x)) t.
+        := @smart_interp_map_hetero var (interp_type_gen_hetero vars' (interp_flat_type_gen var')) vars f tt (fun A B x y => pair x y) (fun A B f x => f (f' _ x)) t.
       Definition SmartVarVarf {t} : interp_flat_type_gen var t -> interp_flat_type_gen exprf t
         := SmartVarfMap (fun t => Var).
-      Definition SmartConstf {t} : interp_flat_type t -> interp_flat_type_gen exprf t
-        := SmartVarfMap (fun t => Const (t:=t)).
+      (*Definition SmartConstf {t} : interp_flat_type t -> interp_flat_type_gen exprf t
+        := SmartVarfMap (fun t => Const (t:=t)).*)
     End expr.
 
     Definition Expr (t : type) := forall var, @expr var t.
@@ -347,7 +219,7 @@ Section language.
                  (interpf : forall {t} (e : @exprf interp_flat_type t), interp_flat_type t)
                  {t} (e : @exprf interp_flat_type t) : interp_flat_type t
         := match e in exprf t return interp_flat_type t with
-           | Const _ x => x
+           | TT => tt
            | Var _ x => x
            | Op _ _ op args => @interp_op _ _ op (@interpf _ args)
            | LetIn _ ex _ eC => dlet x := @interpf _ ex in @interpf _ (eC x)
@@ -375,13 +247,14 @@ Section language.
         : interp_flat_type_gen var1 t
         := match t return interp_flat_type_gen _ t -> interp_flat_type_gen _ t with
            | Tbase _ => fvar21 _
+           | Unit => fun v : unit => v
            | Prod x y => fun xy => (@mapf_interp_flat_type _ (fst xy),
                                     @mapf_interp_flat_type _ (snd xy))
            end e.
 
       Fixpoint mapf {t} (e : @exprf var1 t) : @exprf var2 t
         := match e in exprf t return exprf t with
-           | Const _ x => Const x
+           | TT => TT
            | Var _ x => Var (fvar12 _ x)
            | Op _ _ op args => Op op (@mapf _ args)
            | LetIn _ ex _ eC => LetIn (@mapf _ ex) (fun x => @mapf _ (eC (mapf_interp_flat_type x)))
@@ -397,11 +270,12 @@ Section language.
       Fixpoint flatten_binding_list {t} (x : interp_flat_type_gen var1 t) (y : interp_flat_type_gen var2 t) : list (sigT eP)
         := (match t return interp_flat_type_gen var1 t -> interp_flat_type_gen var2 t -> list _ with
             | Tbase _ => fun x y => (x == y) :: nil
+            | Unit => fun x y => nil
             | Prod t0 t1 => fun x y => @flatten_binding_list _ (snd x) (snd y) ++ @flatten_binding_list _ (fst x) (fst y)
             end x y)%list.
 
       Inductive wff : list (sigT eP) -> forall {t}, @exprf var1 t -> @exprf var2 t -> Prop :=
-      | WfConst : forall t G n, @wff G t (Const n) (Const n)
+      | WfTT : forall G, @wff G _ TT TT
       | WfVar : forall G (t : base_type_code) x x', List.In (x == x') G -> @wff G t (Var x) (Var x')
       | WfOp : forall G {t} {tR} (e : @exprf var1 t) (e' : @exprf var2 t) op,
           wff G e e'
@@ -429,6 +303,7 @@ Section language.
 End language.
 Global Arguments tuple' {_}%type_scope _%ctype_scope _%nat_scope.
 Global Arguments tuple {_}%type_scope _%ctype_scope _%nat_scope.
+Global Arguments Unit {_}%type_scope.
 Global Arguments Prod {_}%type_scope (_ _)%ctype_scope.
 Global Arguments Arrow {_}%type_scope (_ _)%ctype_scope.
 Global Arguments Tbase {_}%type_scope _%ctype_scope.
@@ -436,47 +311,33 @@ Global Arguments Tflat {_}%type_scope _%ctype_scope.
 
 Ltac admit_Wf := apply Wf_admitted.
 
-Global Arguments Const {_ _ _ _ _} _.
-Global Arguments Var {_ _ _ _ _} _.
-Global Arguments SmartVarf {_ _ _ _ _} _.
+Global Arguments Var {_ _ _ _} _.
+Global Arguments SmartVarf {_ _ _ _} _.
+Global Arguments SmartPairf {_ _ _ t} _.
 Global Arguments SmartValf {_} T _ t.
-Global Arguments SmartVarVarf {_ _ _ _ _} _.
+Global Arguments SmartVarVarf {_ _ _ _} _.
 Global Arguments SmartVarfMap {_ _ _} _ {_} _.
 Global Arguments SmartFlatTypeMap {_ _} _ {_} _.
 Global Arguments SmartFlatTypeMapInterp {_ _ _ _} _ {_} _.
 Global Arguments SmartFlatTypeMapUnInterp {_ _ _ _ _} fv {_ _} _.
 Global Arguments SmartVarMap_hetero {_ _ _ _ _} _ _ {_} _.
 Global Arguments SmartVarMap {_ _ _} _ _ {_} _.
-Global Arguments SmartConstf {_ _ _ _ _} _.
-Global Arguments Op {_ _ _ _ _ _} _ _.
-Global Arguments LetIn {_ _ _ _ _} _ {_} _.
-Global Arguments Pair {_ _ _ _ _} _ {_} _.
-Global Arguments Return {_ _ _ _ _} _.
-Global Arguments Abs {_ _ _ _ _ _} _.
-Global Arguments SmartAbs {_ _ _ _ _ _} _.
-Global Arguments UnReturn {_ _ _ _ _} _.
-Global Arguments UnAbs {_ _ _ _ _ _} _ _.
-Global Arguments UnReturn_eta {_ _ _ _ _} _.
-Global Arguments UnAbs_eta {_ _ _ _ _ _} _.
-Global Arguments flat_interp_tuple' {_ _ _ _} _.
-Global Arguments flat_interp_tuple {_ _ _ _} _.
-Global Arguments flat_interp_untuple' {_ _ _ _} _.
-Global Arguments interp_type_rel_pointwise2 {_ _ _} R {t} _ _.
-Global Arguments interp_type_gen_rel_pointwise2_hetero {_ _ _ _ _} Rsrc R {t} _ _.
-Global Arguments interp_type_gen_rel_pointwise2 {_ _ _} R {t} _ _.
-Global Arguments interp_flat_type_rel_pointwise2_gen_Prop {_ _ _ P} and R {t} _ _.
-Global Arguments interp_flat_type_rel_pointwise2 {_ _ _} R {t} _ _.
+(*Global Arguments SmartConstf {_ _ _ _ _} _.*)
+Global Arguments TT {_ _ _}.
+Global Arguments Op {_ _ _ _ _} _ _.
+Global Arguments LetIn {_ _ _ _} _ {_} _.
+Global Arguments Pair {_ _ _ _} _ {_} _.
+Global Arguments Return {_ _ _ _} _.
+Global Arguments Abs {_ _ _ _ _} _.
+Global Arguments SmartAbs {_ _ _ _ _} _.
 Global Arguments mapf_interp_flat_type {_ _ _} _ {t} _.
 Global Arguments interp_type_gen_hetero {_} _ _ _.
 Global Arguments interp_type_gen {_} _ _.
 Global Arguments interp_flat_type {_} _ _.
-Global Arguments interp_type_rel_pointwise {_} _ _ {_} _ _.
-Global Arguments interp_type_gen_rel_pointwise {_ _} _ {_} _ _.
-Global Arguments interp_flat_type_rel_pointwise {_} _ _ {_} _ _.
 Global Arguments interp_type {_} _ _.
-Global Arguments wff {_ _ _ _ _} G {t} _ _.
-Global Arguments wf {_ _ _ _ _} G {t} _ _.
-Global Arguments Wf {_ _ _ t} _.
+Global Arguments wff {_ _ _ _} G {t} _ _.
+Global Arguments wf {_ _ _ _} G {t} _ _.
+Global Arguments Wf {_ _ t} _.
 Global Arguments Interp {_ _ _} interp_op {t} _.
 Global Arguments interp {_ _ _} interp_op {t} _.
 Global Arguments interpf {_ _ _} interp_op {t} _.
@@ -485,6 +346,7 @@ Section hetero_type.
   Fixpoint flatten_flat_type {base_type_code} (t : flat_type (flat_type base_type_code)) : flat_type base_type_code
     := match t with
        | Tbase T => T
+       | Unit => Unit
        | Prod A B => Prod (@flatten_flat_type _ A) (@flatten_flat_type _ B)
        end.
 
@@ -493,12 +355,13 @@ Section hetero_type.
 
     Definition SmartFlatTypeMap2 {var' : base_type_code1 -> Type} (f : forall t, var' t -> flat_type base_type_code2) {t}
       : interp_flat_type var' t -> flat_type base_type_code2
-      := @smart_interp_flat_map base_type_code1 var' (fun _ => flat_type base_type_code2) f (fun _ _ => Prod) t.
+      := @smart_interp_flat_map base_type_code1 var' (fun _ => flat_type base_type_code2) f Unit (fun _ _ => Prod) t.
     Fixpoint SmartFlatTypeMapInterp2 {var' var''} (f : forall t, var' t -> flat_type base_type_code2)
              (fv : forall t v, interp_flat_type var'' (f t v)) t {struct t}
       : forall v, interp_flat_type var'' (SmartFlatTypeMap2 f (t:=t) v)
       := match t return forall v, interp_flat_type var'' (SmartFlatTypeMap2 f (t:=t) v) with
          | Tbase x => fv _
+         | Unit => fun v => v
          | Prod A B => fun xy => (@SmartFlatTypeMapInterp2 _ _ f fv A (fst xy),
                                   @SmartFlatTypeMapInterp2 _ _ f fv B (snd xy))
          end.
@@ -510,6 +373,7 @@ Section hetero_type.
       := match t return forall v, interp_flat_type var'' (SmartFlatTypeMap2 f (t:=t) v)
                                   -> interp_flat_type var''' t with
          | Tbase x => fv _
+         | Unit => fun _ v => v
          | Prod A B => fun v xy => (@SmartFlatTypeMapUnInterp2 _ _ _ f fv A _ (fst xy),
                                     @SmartFlatTypeMapUnInterp2 _ _ _ f fv B _ (snd xy))
          end.
@@ -521,11 +385,14 @@ Global Arguments SmartFlatTypeMapInterp2 {_ _ _ _ _} _ {_} _.
 Global Arguments SmartFlatTypeMapUnInterp2 {_ _ _ _ _ _} fv {_ _} _.
 
 Module Export Notations.
+  Notation "()" := (@Unit _) : ctype_scope.
   Notation "A * B" := (@Prod _ A B) : ctype_scope.
   Notation "A -> B" := (@Arrow _ A B) : ctype_scope.
   Notation "'slet' x := A 'in' b" := (LetIn A (fun x => b)) : expr_scope.
   Notation "'λ'  x .. y , t" := (Abs (fun x => .. (Abs (fun y => t%expr)) ..)) : expr_scope.
   Notation "( x , y , .. , z )" := (Pair .. (Pair x%expr y%expr) .. z%expr) : expr_scope.
+  Notation "( )" := TT : expr_scope.
+  Notation "()" := TT : expr_scope.
   Bind Scope ctype_scope with flat_type.
   Bind Scope ctype_scope with type.
 End Notations.