From 4fd838975aea0982a9e2617bea4c6e66e6b826b2 Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Thu, 26 Jan 2023 20:09:20 +0000
Subject: [PATCH 01/33] standard definitions

---
 src/Option.dfy | 79 ++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 79 insertions(+)
 create mode 100644 src/Option.dfy

diff --git a/src/Option.dfy b/src/Option.dfy
new file mode 100644
index 00000000..e883f29f
--- /dev/null
+++ b/src/Option.dfy
@@ -0,0 +1,79 @@
+// RUN: %dafny /compile:0 "%s"
+
+/*******************************************************************************
+*  Copyright by the contributors to the Dafny Project
+*  SPDX-License-Identifier: MIT 
+*******************************************************************************/
+
+module {:options "-functionSyntax:4"} Option {
+
+  datatype Option<+T> = None | Some(value: T)
+
+  function Some<T>(v: T): Option<T> {
+    Option.Some(v)
+  }
+
+  function Bind<S,T>(o: Option<S>, f: S -> Option<T>): Option<T> {
+    match o 
+    case None => None
+    case Some(v) => f(v)
+  }
+
+  predicate IsNone<T>(o: Option<T>) {
+    o.None?
+  }
+
+  predicate IsSome<T>(o: Option<T>) {
+    o.Some?
+  }
+
+  function Get<T>(o: Option<T>): (v: T) 
+    requires o.Some?
+    ensures o.value == v
+  {
+    o.value
+  }
+ 
+  function Join<T>(oo: Option<Option<T>>): Option<T> {
+    match oo
+    case None => None
+    case Some(o) => o
+  }
+
+  function Map<S,T>(f: S -> T, o: Option<S>): (o2: Option<T>)
+  {
+    match o 
+    case None => None
+    case Some(v) => Some(f(v))
+  }
+
+  function Equal<T>(eq: (T, T) -> bool): (Option<T>, Option<T>) -> bool {
+    (o1: Option<T>, o2: Option<T>) =>
+      match (o1, o2)
+      case (Some(v1), Some(v2)) => eq(v1, v2)
+      case (None, None) => true
+      case _ => false
+  }
+  
+  function Compare<T>(cmp: (T, T) -> int): (Option<T>, Option<T>) -> int {
+    (o1: Option<T>, o2: Option<T>) =>
+      match (o1, o2)
+      case (Some(v1), Some(v2)) => cmp(v1, v2)
+      case (None, None) => 0
+      case (None, Some(_)) => -1
+      case (Some(_), None) => 1
+  }
+
+  function ToSeq<T>(o: Option<T>): seq<T> {
+    match o 
+    case None => []
+    case Some(v) => [v]
+  }
+
+  function ToSet<T>(o: Option<T>): set<T> {
+    match o
+    case None => {}
+    case Some(v) => {v}
+  }
+
+}

From 4ddb4520ffd66b92a51065083671c57257bfffd0 Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Thu, 26 Jan 2023 20:33:07 +0000
Subject: [PATCH 02/33] initial

---
 src/Writer.dfy | 0
 1 file changed, 0 insertions(+), 0 deletions(-)
 create mode 100644 src/Writer.dfy

diff --git a/src/Writer.dfy b/src/Writer.dfy
new file mode 100644
index 00000000..e69de29b

From 3851270f6b0470f2640f34320101bdc3de6d6225 Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Thu, 26 Jan 2023 21:07:59 +0000
Subject: [PATCH 03/33] skeleton

---
 src/Either.dfy | 59 ++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 59 insertions(+)
 create mode 100644 src/Either.dfy

diff --git a/src/Either.dfy b/src/Either.dfy
new file mode 100644
index 00000000..4a8f2035
--- /dev/null
+++ b/src/Either.dfy
@@ -0,0 +1,59 @@
+// RUN: %dafny /compile:0 "%s"
+
+/*******************************************************************************
+*  Copyright by the contributors to the Dafny Project
+*  SPDX-License-Identifier: MIT 
+*******************************************************************************/
+
+module {:options "-functionSyntax:4"} Either {
+
+  datatype Either<+S,+T> = Left(left: S) | Right(right: T) 
+
+  predicate IsLeft<S,T>(e: Either<S,T>) {
+    e.Left?
+  }
+
+  predicate IsRight<S,T>(e: Either<S,T>) {
+    e.Right?
+  }
+
+  function MapLeft<S1,S2,T>(f: S1 -> S2): Either<S1,T> -> Either<S2,T> {
+    (e: Either<S1,T>) =>
+     match e 
+     case Left(v) => Either<S2,T>.Left(f(v))
+     case Right(v) => Either<S2,T>.Right(v)
+  }
+
+  function MapRight<S,T1,T2>(f: T1 -> T2): Either<S,T1> -> Either<S,T2> {
+    (e: Either<S,T1>) =>
+     match e 
+     case Left(v) => Either<S,T2>.Left(v)
+     case Right(v) => Either<S,T2>.Right(f(v))
+  }
+
+  function Map<S1,S2,T1,T2>(f: S1 -> S2, g: T1 -> T2): Either<S1,T1> -> Either<S2,T2> {
+    (e: Either<S1,T1>) =>
+      match e
+      case Left(v) => Either<S2,T2>.Left(f(v))
+      case Right(v) => Either<S2,T2>.Right(g(v))
+  }
+
+  function Fold<S,T,U>(f: S -> U, g: T -> U): Either<S,T> -> U {
+    (e: Either<S,T>) =>
+      match e
+      case Left(v) => f(v)
+      case Right(v) => g(v)
+  }
+
+  function Equal<S,T>(eql: (S, S) -> bool, eqr: (T, T) -> bool): (Either<S,T>, Either<S,T>) -> bool {
+    (e1: Either<S,T>, e2: Either<S,T>) =>
+      match (e1, e2)
+      case (Left(v1), Left(v2)) => eql(v1, v2)
+      case (Right(v1), Right(v2)) => eqr(v1, v2)
+      case _ => false
+  }
+
+
+
+}
+

From 937a778e69dd1e20bb82ba8ad4d2da3ea42b01cf Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Fri, 27 Jan 2023 11:41:11 +0000
Subject: [PATCH 04/33] more functions

---
 src/Either.dfy | 25 +++++++++++++++++++++++++
 1 file changed, 25 insertions(+)

diff --git a/src/Either.dfy b/src/Either.dfy
index 4a8f2035..d22c3890 100644
--- a/src/Either.dfy
+++ b/src/Either.dfy
@@ -5,8 +5,12 @@
 *  SPDX-License-Identifier: MIT 
 *******************************************************************************/
 
+include "Wrappers.dfy"
+
 module {:options "-functionSyntax:4"} Either {
 
+  import opened Wrappers 
+
   datatype Either<+S,+T> = Left(left: S) | Right(right: T) 
 
   predicate IsLeft<S,T>(e: Either<S,T>) {
@@ -17,6 +21,18 @@ module {:options "-functionSyntax:4"} Either {
     e.Right?
   }
 
+  function FindLeft<S,T>(e: Either<S,T>): Option<S> {
+    match e 
+    case Left(v) => Some(v)
+    case Right(v) => None
+  }
+
+  function FindRight<S,T>(e: Either<S,T>): Option<T> {
+    match e 
+    case Left(v) => None
+    case Right(v) => Some(v)
+  }
+
   function MapLeft<S1,S2,T>(f: S1 -> S2): Either<S1,T> -> Either<S2,T> {
     (e: Either<S1,T>) =>
      match e 
@@ -53,6 +69,15 @@ module {:options "-functionSyntax:4"} Either {
       case _ => false
   }
 
+  function Compare<S,T>(cmpl: (S, S) -> int, cmpr: (T, T) -> int): (Either<S,T>, Either<S,T>) -> int {
+    (e1: Either<S,T>, e2: Either<S,T>) =>
+      match (e1, e2)
+      case (Left(v1), Left(v2)) => cmpl(v1, v2)
+      case (Right(v1), Right(v2)) => cmpr(v1, v2)
+      case (Left(_), Right(_)) => -1
+      case (Right(_), Left(_)) => 1
+  }
+
 
 
 }

From 5cc5af33dd240cf276884264632b163f4332e8aa Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Fri, 27 Jan 2023 14:49:29 +0000
Subject: [PATCH 05/33] skeleton

---
 src/Result.dfy | 125 +++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 125 insertions(+)
 create mode 100644 src/Result.dfy

diff --git a/src/Result.dfy b/src/Result.dfy
new file mode 100644
index 00000000..c8fd34a0
--- /dev/null
+++ b/src/Result.dfy
@@ -0,0 +1,125 @@
+include "Wrappers.dfy"
+
+module {:options "-functionSyntax:4"} Result {
+
+  import opened Wrappers 
+
+  datatype Result<+S,+E> = | Success(value: S) | Failure(error: E)
+
+  function Success<S,E>(v: S): Result<S,E> {
+    Result.Success(v)
+  }
+
+  function Failure<S,E>(e: E): Result<S,E> {
+    Result.Failure(e)
+  }
+
+  predicate IsSuccess<S,E>(r: Result<S,E>) {
+    r.Success?
+  }  
+
+  predicate IsFailure<S,E>(r: Result<S,E>) {
+    r.Failure?
+  }  
+
+  function GetValue<S,E>(r: Result<S,E>): S 
+    requires r.Success?
+  {
+    r.value
+  }
+
+  function GetValueDefault<S,E>(r: Result<S,E>, default: S): S {
+    match r 
+    case Success(v) => v
+    case Failure(_) => default
+  }
+
+  function GetError<S,E>(r: Result<S,E>): E 
+    requires r.Failure?
+  {
+    r.error
+  }
+
+  function Bind<S1,S2,E>(r: Result<S1,E>, f: S1 -> Result<S2,E>): Result<S2,E> {
+    match r
+    case Success(v) => f(v)
+    case Failure(e) => Result<S2,E>.Failure(e)
+  }
+
+  function Join<S,E>(rr: Result<Result<S,E>,E>): Result<S,E> {
+    match rr
+    case Success(v) => v
+    case Failure(e) => Result<S,E>.Failure(e)
+  }
+
+  function Map<S1,S2,E>(f: S1 -> S2): Result<S1,E> -> Result<S2,E> {
+    (r: Result<S1,E>) =>
+      match r 
+      case Success(v) => Result<S2,E>.Success(f(v))
+      case Failure(e) => Result<S2,E>.Failure(e)
+  }
+
+  function MapError<S,E1,E2>(f: E1 -> E2): Result<S,E1> -> Result<S,E2> {
+    (r: Result<S,E1>) =>
+      match r 
+      case Success(v) => Result<S,E2>.Success(v)
+      case Failure(e) => Result<S,E2>.Failure(f(e))
+  }  
+
+  function ToOption<S,E>(r: Result<S,E>): Option<S> {
+    match r
+    case Success(v) => Some(v)
+    case Failure(_) => None()
+  }
+
+  function Fold<S,E,T>(f: S -> T, g: E -> T): Result<S,E> -> T {
+    (r: Result<S,E>) =>
+      match r 
+      case Success(v) => f(v)
+      case Failure(e) => g(e)
+  }
+
+  function ToSeq<S,E>(r: Result<S,E>): seq<S> {
+    match r
+    case Success(v) => [v]
+    case Failure(_) => []
+  }
+
+  function ToSet<S,E>(r: Result<S,E>): set<S> {
+    match r
+    case Success(v) => {v}
+    case Failure(_) => {}   
+  }
+
+  function Composition<S1,S2,S3,E>(f: S1 -> Result<S2,E>, g: S2 -> Result<S3,E>): S1 -> Result<S3,E> {
+    x => Bind(f(x), g)
+  }
+
+  lemma LemmaUnitalityJoin<S,E>(r: Result<S,E>)
+    ensures Join(Map(Success<S,E>)(r)) == r == Join(Success<Result<S,E>,E>(r))
+  {
+  }
+
+  lemma LemmaAssociativityJoin<S,E>(rrr: Result<Result<Result<S,E>,E>,E>) 
+    ensures Join(Map(Join<S,E>)(rrr) ) == Join(Join<Result<S,E>,E>(rrr))
+  {
+  }  
+
+  lemma LemmaLeftUnitalityBind<S1,S2,E>(v: S1, f: S1 -> Result<S2,E>)
+    ensures Bind(Success(v), f) == f(v)
+  {
+  }
+
+  lemma LemmaRightUnitalityBind<S,E>(r: Result<S,E>)
+    ensures Bind(r, Success) == r
+  {
+  }
+
+  lemma LemmaAssociativityBind<S1,S2,S3,E>(r: Result<S1,E>, f: S1 -> Result<S2,E>, g: S2 -> Result<S3,E>)
+    ensures Bind(Bind(r, f), g) == Bind(r, Composition(f, g))
+  {
+  }
+
+}
+
+

From 1760fb01db883df5162da82ba2cfa7f18f01e9d4 Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Fri, 27 Jan 2023 15:21:08 +0000
Subject: [PATCH 06/33] types, compare, equal

---
 src/Result.dfy | 97 +++++++++++++++++++++++++++++---------------------
 1 file changed, 57 insertions(+), 40 deletions(-)

diff --git a/src/Result.dfy b/src/Result.dfy
index c8fd34a0..e53b463e 100644
--- a/src/Result.dfy
+++ b/src/Result.dfy
@@ -4,118 +4,135 @@ module {:options "-functionSyntax:4"} Result {
 
   import opened Wrappers 
 
-  datatype Result<+S,+E> = | Success(value: S) | Failure(error: E)
+  datatype Result<+T,+E> = | Success(value: T) | Failure(error: E)
 
-  function Success<S,E>(v: S): Result<S,E> {
+  function Success<T,E>(v: T): Result<T,E> {
     Result.Success(v)
   }
 
-  function Failure<S,E>(e: E): Result<S,E> {
+  function Failure<T,E>(e: E): Result<T,E> {
     Result.Failure(e)
-  }
+  } 
 
-  predicate IsSuccess<S,E>(r: Result<S,E>) {
+  predicate IsSuccess<T,E>(r: Result<T,E>) {
     r.Success?
   }  
 
-  predicate IsFailure<S,E>(r: Result<S,E>) {
+  predicate IsFailure<T,E>(r: Result<T,E>) {
     r.Failure?
   }  
 
-  function GetValue<S,E>(r: Result<S,E>): S 
+  function GetValue<T,E>(r: Result<T,E>): T
     requires r.Success?
   {
     r.value
   }
 
-  function GetValueDefault<S,E>(r: Result<S,E>, default: S): S {
+  function GetValueDefault<T,E>(r: Result<T,E>, default: T): T {
     match r 
     case Success(v) => v
     case Failure(_) => default
   }
 
-  function GetError<S,E>(r: Result<S,E>): E 
+  function GetError<T,E>(r: Result<T,E>): E 
     requires r.Failure?
   {
     r.error
   }
 
-  function Bind<S1,S2,E>(r: Result<S1,E>, f: S1 -> Result<S2,E>): Result<S2,E> {
+  function Bind<T1,T2,E>(r: Result<T1,E>, f: T1 -> Result<T2,E>): Result<T2,E> {
     match r
     case Success(v) => f(v)
-    case Failure(e) => Result<S2,E>.Failure(e)
+    case Failure(e) => Result<T2,E>.Failure(e)
   }
 
-  function Join<S,E>(rr: Result<Result<S,E>,E>): Result<S,E> {
+  function Join<T,E>(rr: Result<Result<T,E>,E>): Result<T,E> {
     match rr
     case Success(v) => v
-    case Failure(e) => Result<S,E>.Failure(e)
+    case Failure(e) => Result<T,E>.Failure(e)
   }
 
-  function Map<S1,S2,E>(f: S1 -> S2): Result<S1,E> -> Result<S2,E> {
-    (r: Result<S1,E>) =>
+  function Map<T1,T2,E>(f: T1 -> T2): Result<T1,E> -> Result<T2,E> {
+    (r: Result<T1,E>) =>
       match r 
-      case Success(v) => Result<S2,E>.Success(f(v))
-      case Failure(e) => Result<S2,E>.Failure(e)
+      case Success(v) => Result<T2,E>.Success(f(v))
+      case Failure(e) => Result<T2,E>.Failure(e)
   }
 
-  function MapError<S,E1,E2>(f: E1 -> E2): Result<S,E1> -> Result<S,E2> {
-    (r: Result<S,E1>) =>
+  function MapError<T,E1,E2>(f: E1 -> E2): Result<T,E1> -> Result<T,E2> {
+    (r: Result<T,E1>) =>
       match r 
-      case Success(v) => Result<S,E2>.Success(v)
-      case Failure(e) => Result<S,E2>.Failure(f(e))
+      case Success(v) => Result<T,E2>.Success(v)
+      case Failure(e) => Result<T,E2>.Failure(f(e))
   }  
 
-  function ToOption<S,E>(r: Result<S,E>): Option<S> {
-    match r
-    case Success(v) => Some(v)
-    case Failure(_) => None()
-  }
-
-  function Fold<S,E,T>(f: S -> T, g: E -> T): Result<S,E> -> T {
-    (r: Result<S,E>) =>
+  function Fold<T,E,U>(f: T -> U, g: E -> U): Result<T,E> -> U {
+    (r: Result<T,E>) =>
       match r 
       case Success(v) => f(v)
       case Failure(e) => g(e)
   }
 
-  function ToSeq<S,E>(r: Result<S,E>): seq<S> {
+  function Composition<T1,T2,T3,E>(f: T1 -> Result<T2,E>, g: T2 -> Result<T3,E>): T1 -> Result<T3,E> {
+    x => Bind(f(x), g)
+  }
+
+  function Equal<T,E>(eqs: (T, T) -> bool, eqf: (E, E) -> bool): (Result<T,E>, Result<T,E>) -> bool {
+    (r1: Result<T,E>, r2: Result<T,E>) =>
+      match (r1, r2)
+      case (Success(v1), Success(v2)) => eqs(v1, v2)
+      case (Failure(e1), Failure(e2)) => eqf(e1, e2)
+      case _ => false
+  }
+
+  function Compare<T,E>(cmps: (T, T) -> int, cmpf: (E, E) -> int): (Result<T,E>, Result<T,E>) -> int {
+    (r1: Result<T,E>, r2: Result<T,E>) =>
+      match (r1, r2)
+      case (Success(v1), Success(v2)) => cmps(v1, v2)
+      case (Failure(e1), Failure(e2)) => cmpf(e1, e2)
+      case (Success(_), Failure(_)) => -1
+      case (Failure(_), Success(_)) => 1
+  }
+
+  function ToSeq<T,E>(r: Result<T,E>): seq<T> {
     match r
     case Success(v) => [v]
     case Failure(_) => []
   }
 
-  function ToSet<S,E>(r: Result<S,E>): set<S> {
+  function ToSet<T,E>(r: Result<T,E>): set<T> {
     match r
     case Success(v) => {v}
     case Failure(_) => {}   
   }
 
-  function Composition<S1,S2,S3,E>(f: S1 -> Result<S2,E>, g: S2 -> Result<S3,E>): S1 -> Result<S3,E> {
-    x => Bind(f(x), g)
+  function ToOption<T,E>(r: Result<T,E>): Option<T> {
+    match r
+    case Success(v) => Some(v)
+    case Failure(_) => None()
   }
 
-  lemma LemmaUnitalityJoin<S,E>(r: Result<S,E>)
-    ensures Join(Map(Success<S,E>)(r)) == r == Join(Success<Result<S,E>,E>(r))
+  lemma LemmaUnitalityJoin<T,E>(r: Result<T,E>)
+    ensures Join(Map(Success<T,E>)(r)) == r == Join(Success<Result<T,E>,E>(r))
   {
   }
 
-  lemma LemmaAssociativityJoin<S,E>(rrr: Result<Result<Result<S,E>,E>,E>) 
-    ensures Join(Map(Join<S,E>)(rrr) ) == Join(Join<Result<S,E>,E>(rrr))
+  lemma LemmaAssociativityJoin<T,E>(rrr: Result<Result<Result<T,E>,E>,E>) 
+    ensures Join(Map(Join<T,E>)(rrr)) == Join(Join<Result<T,E>,E>(rrr))
   {
   }  
 
-  lemma LemmaLeftUnitalityBind<S1,S2,E>(v: S1, f: S1 -> Result<S2,E>)
+  lemma LemmaLeftUnitalityBind<T1,T2,E>(v: T1, f: T1 -> Result<T2,E>)
     ensures Bind(Success(v), f) == f(v)
   {
   }
 
-  lemma LemmaRightUnitalityBind<S,E>(r: Result<S,E>)
+  lemma LemmaRightUnitalityBind<T,E>(r: Result<T,E>)
     ensures Bind(r, Success) == r
   {
   }
 
-  lemma LemmaAssociativityBind<S1,S2,S3,E>(r: Result<S1,E>, f: S1 -> Result<S2,E>, g: S2 -> Result<S3,E>)
+  lemma LemmaAssociativityBind<T1,T2,T3,E>(r: Result<T1,E>, f: T1 -> Result<T2,E>, g: T2 -> Result<T3,E>)
     ensures Bind(Bind(r, f), g) == Bind(r, Composition(f, g))
   {
   }

From b9e94f2ded381816d06a14960fbc2c5f771a6501 Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Fri, 27 Jan 2023 17:03:12 +0000
Subject: [PATCH 07/33] lemmas, more functions

---
 src/Option.dfy | 75 ++++++++++++++++++++++++++++++++++++++++++++------
 1 file changed, 66 insertions(+), 9 deletions(-)

diff --git a/src/Option.dfy b/src/Option.dfy
index e883f29f..c1ef610d 100644
--- a/src/Option.dfy
+++ b/src/Option.dfy
@@ -4,9 +4,12 @@
 *  Copyright by the contributors to the Dafny Project
 *  SPDX-License-Identifier: MIT 
 *******************************************************************************/
+include "Wrappers.dfy"
 
 module {:options "-functionSyntax:4"} Option {
 
+  import opened Wrappers
+
   datatype Option<+T> = None | Some(value: T)
 
   function Some<T>(v: T): Option<T> {
@@ -15,7 +18,7 @@ module {:options "-functionSyntax:4"} Option {
 
   function Bind<S,T>(o: Option<S>, f: S -> Option<T>): Option<T> {
     match o 
-    case None => None
+    case None => Option<T>.None
     case Some(v) => f(v)
   }
 
@@ -27,24 +30,37 @@ module {:options "-functionSyntax:4"} Option {
     o.Some?
   }
 
-  function Get<T>(o: Option<T>): (v: T) 
+  function GetValue<T>(o: Option<T>): T
     requires o.Some?
-    ensures o.value == v
   {
     o.value
   }
- 
+
+  function GetValueDefault<T>(o: Option<T>, default: T): T {
+    match o
+    case None => default
+    case Some(v) => v
+  }
+
   function Join<T>(oo: Option<Option<T>>): Option<T> {
     match oo
-    case None => None
+    case None => Option<T>.None
     case Some(o) => o
   }
 
-  function Map<S,T>(f: S -> T, o: Option<S>): (o2: Option<T>)
+  function Map<S,T>(f: S -> T): Option<S> -> Option<T>
   {
-    match o 
-    case None => None
-    case Some(v) => Some(f(v))
+    (o: Option<S>) =>
+      match o 
+      case None => Option<T>.None
+      case Some(v) => Some(f(v))
+  }
+
+  function Fold<T,U>(u: U, f: T -> U): Option<T> -> U {
+    (o: Option<T>) =>
+      match o 
+      case None => u
+      case Some(v) => f(v)
   }
 
   function Equal<T>(eq: (T, T) -> bool): (Option<T>, Option<T>) -> bool {
@@ -64,6 +80,17 @@ module {:options "-functionSyntax:4"} Option {
       case (Some(_), None) => 1
   }
 
+  function Composition<S,T,U>(f: S -> Option<T>, g: T-> Option<U>): S -> Option<U> {
+    x => Bind(f(x), g)
+  }
+
+  function ToResult<T,E>(e: E): Option<T> -> Result<T,E> {
+    (o: Option<T>) =>
+      match o 
+      case Some(v) => Success(v)
+      case None => Failure(e)
+  }
+
   function ToSeq<T>(o: Option<T>): seq<T> {
     match o 
     case None => []
@@ -76,4 +103,34 @@ module {:options "-functionSyntax:4"} Option {
     case Some(v) => {v}
   }
 
+  lemma LemmaUnitalityJoin<T>(o: Option<T>)
+    ensures Join(Map(Some<T>)(o)) == o == Join(Some<Option<T>>(o))
+  {
+  }
+
+  lemma LemmaAssociativityJoin<T>(ooo: Option<Option<Option<T>>>) 
+    ensures Join(Map(Join<T>)(ooo)) == Join(Join<Option<T>>(ooo))
+  {
+  }  
+
+  lemma LemmaLeftUnitalityBind<S,T>(v: S, f: S -> Option<T>)
+    ensures Bind(Some(v), f) == f(v)
+  {
+  }
+
+  lemma LemmaRightUnitalityBind<T>(o: Option<T>)
+    ensures Bind(o, Some) == o
+  {
+  }
+
+  lemma LemmaAssociativityBind<S,T,U>(o: Option<S>, f: S -> Option<T>, g: T -> Option<U>)
+    ensures Bind(Bind(o, f), g) == Bind(o, Composition(f, g))
+  {
+  }
+
+
+
+
+
+
 }

From 5629f430d60cd32e71cdacfc914c4449662b2057 Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Fri, 27 Jan 2023 17:09:35 +0000
Subject: [PATCH 08/33] notation

---
 src/Result.dfy | 16 ++++++++--------
 1 file changed, 8 insertions(+), 8 deletions(-)

diff --git a/src/Result.dfy b/src/Result.dfy
index e53b463e..032c3e64 100644
--- a/src/Result.dfy
+++ b/src/Result.dfy
@@ -40,10 +40,10 @@ module {:options "-functionSyntax:4"} Result {
     r.error
   }
 
-  function Bind<T1,T2,E>(r: Result<T1,E>, f: T1 -> Result<T2,E>): Result<T2,E> {
+  function Bind<S,T,E>(r: Result<S,E>, f: S -> Result<T,E>): Result<T,E> {
     match r
     case Success(v) => f(v)
-    case Failure(e) => Result<T2,E>.Failure(e)
+    case Failure(e) => Result<T,E>.Failure(e)
   }
 
   function Join<T,E>(rr: Result<Result<T,E>,E>): Result<T,E> {
@@ -52,11 +52,11 @@ module {:options "-functionSyntax:4"} Result {
     case Failure(e) => Result<T,E>.Failure(e)
   }
 
-  function Map<T1,T2,E>(f: T1 -> T2): Result<T1,E> -> Result<T2,E> {
-    (r: Result<T1,E>) =>
+  function Map<S,T,E>(f: S -> T): Result<S,E> -> Result<T,E> {
+    (r: Result<S,E>) =>
       match r 
-      case Success(v) => Result<T2,E>.Success(f(v))
-      case Failure(e) => Result<T2,E>.Failure(e)
+      case Success(v) => Result<T,E>.Success(f(v))
+      case Failure(e) => Result<T,E>.Failure(e)
   }
 
   function MapError<T,E1,E2>(f: E1 -> E2): Result<T,E1> -> Result<T,E2> {
@@ -73,7 +73,7 @@ module {:options "-functionSyntax:4"} Result {
       case Failure(e) => g(e)
   }
 
-  function Composition<T1,T2,T3,E>(f: T1 -> Result<T2,E>, g: T2 -> Result<T3,E>): T1 -> Result<T3,E> {
+  function Composition<S,T,U,E>(f: S -> Result<T,E>, g: T -> Result<U,E>): S -> Result<U,E> {
     x => Bind(f(x), g)
   }
 
@@ -132,7 +132,7 @@ module {:options "-functionSyntax:4"} Result {
   {
   }
 
-  lemma LemmaAssociativityBind<T1,T2,T3,E>(r: Result<T1,E>, f: T1 -> Result<T2,E>, g: T2 -> Result<T3,E>)
+  lemma LemmaAssociativityBind<S,T,U,E>(r: Result<S,E>, f: S -> Result<T,E>, g: T -> Result<U,E>)
     ensures Bind(Bind(r, f), g) == Bind(r, Composition(f, g))
   {
   }

From 7e4a628683efe3da8f901ec26732d95c3f0071fd Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Fri, 27 Jan 2023 17:59:19 +0000
Subject: [PATCH 09/33] skeleton

---
 src/Writer.dfy | 34 ++++++++++++++++++++++++++++++++++
 1 file changed, 34 insertions(+)

diff --git a/src/Writer.dfy b/src/Writer.dfy
index e69de29b..d9c10d14 100644
--- a/src/Writer.dfy
+++ b/src/Writer.dfy
@@ -0,0 +1,34 @@
+module {:options "-functionSyntax:4"} Writer {
+
+  datatype Writer<T> = Result(value: T, log: string)
+
+  function GetValue<T>(w: Writer<T>): T {
+    w.value
+  }
+
+  function GetLog<T>(w: Writer<T>): string {
+    w.log
+  }
+
+  function Return<T>(v: T): Writer<T> {
+    Result(v, "")
+  }
+
+  function Join<T>(ww: Writer<Writer<T>>): Writer<T> {
+    Result((ww.value).value, (ww.value).log + ww.log)
+  }
+
+  function Map<S,T>(f: S -> T): Writer<S> -> Writer<T> {
+    (w: Writer<S>) => Writer<T>.Result(f(w.value), w.log)
+  }
+
+  function Bind<S,T>(w: Writer<S>, f: S -> Writer<T>): Writer<T> {
+    Writer<T>.Result(f(w.value).value, w.log + f(w.value).log)
+  }
+
+  lemma LemmaUnitalityJoin<T>(w: Writer<T>)
+    ensures Join(Map(Return<T>)(w)) == w == Join(Return<Writer<T>>(w))
+  {
+  }
+
+}

From e5692e1f39d14ad5a281c2fd887e632514e3fb70 Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Mon, 30 Jan 2023 14:37:54 +0000
Subject: [PATCH 10/33] notation

---
 src/Result.dfy | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/Result.dfy b/src/Result.dfy
index 032c3e64..8adbb4a3 100644
--- a/src/Result.dfy
+++ b/src/Result.dfy
@@ -122,7 +122,7 @@ module {:options "-functionSyntax:4"} Result {
   {
   }  
 
-  lemma LemmaLeftUnitalityBind<T1,T2,E>(v: T1, f: T1 -> Result<T2,E>)
+  lemma LemmaLeftUnitalityBind<S,T,E>(v: S, f: S -> Result<T,E>)
     ensures Bind(Success(v), f) == f(v)
   {
   }

From 909c9714c67595ddc682269a9a9b75628b95f9eb Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Mon, 30 Jan 2023 14:45:13 +0000
Subject: [PATCH 11/33] notation

---
 src/Either.dfy | 24 ++++++++++++++++--------
 1 file changed, 16 insertions(+), 8 deletions(-)

diff --git a/src/Either.dfy b/src/Either.dfy
index d22c3890..90c8521f 100644
--- a/src/Either.dfy
+++ b/src/Either.dfy
@@ -13,6 +13,14 @@ module {:options "-functionSyntax:4"} Either {
 
   datatype Either<+S,+T> = Left(left: S) | Right(right: T) 
 
+  function Left<S,T>(v: S): Either<S,T> {
+    Either.Left(v)
+  }
+
+  function Right<S,T>(v: T): Either<S,T> {
+    Either.Right(v)
+  }
+
   predicate IsLeft<S,T>(e: Either<S,T>) {
     e.Left?
   }
@@ -33,18 +41,18 @@ module {:options "-functionSyntax:4"} Either {
     case Right(v) => Some(v)
   }
 
-  function MapLeft<S1,S2,T>(f: S1 -> S2): Either<S1,T> -> Either<S2,T> {
-    (e: Either<S1,T>) =>
+  function MapLeft<S,T,U>(f: S -> T): Either<S,U> -> Either<T,U> {
+    (e: Either<S,U>) =>
      match e 
-     case Left(v) => Either<S2,T>.Left(f(v))
-     case Right(v) => Either<S2,T>.Right(v)
+     case Left(v) => Either<T,U>.Left(f(v))
+     case Right(v) => Either<T,U>.Right(v)
   }
 
-  function MapRight<S,T1,T2>(f: T1 -> T2): Either<S,T1> -> Either<S,T2> {
-    (e: Either<S,T1>) =>
+  function MapRight<S,T,U>(f: T -> U): Either<S,T> -> Either<S,U> {
+    (e: Either<S,T>) =>
      match e 
-     case Left(v) => Either<S,T2>.Left(v)
-     case Right(v) => Either<S,T2>.Right(f(v))
+     case Left(v) => Either<S,U>.Left(v)
+     case Right(v) => Either<S,U>.Right(f(v))
   }
 
   function Map<S1,S2,T1,T2>(f: S1 -> S2, g: T1 -> T2): Either<S1,T1> -> Either<S2,T2> {

From de840250811aead0797c26b8a727a9e3005b8154 Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Mon, 30 Jan 2023 15:02:48 +0000
Subject: [PATCH 12/33] monad laws

---
 src/Writer.dfy | 24 ++++++++++++++++++++++++
 1 file changed, 24 insertions(+)

diff --git a/src/Writer.dfy b/src/Writer.dfy
index d9c10d14..44695ce7 100644
--- a/src/Writer.dfy
+++ b/src/Writer.dfy
@@ -26,9 +26,33 @@ module {:options "-functionSyntax:4"} Writer {
     Writer<T>.Result(f(w.value).value, w.log + f(w.value).log)
   }
 
+  function Composition<S,T,U>(f: S -> Writer<T>, g: T -> Writer<U>): S -> Writer<U> {
+    x => Bind(f(x), g)
+  }
+
   lemma LemmaUnitalityJoin<T>(w: Writer<T>)
     ensures Join(Map(Return<T>)(w)) == w == Join(Return<Writer<T>>(w))
   {
   }
 
+  lemma LemmaAssociativityJoin<T>(www: Writer<Writer<Writer<T>>>) 
+    ensures Join(Map(Join<T>)(www)) == Join(Join<Writer<T>>(www))
+  {
+  }  
+
+  lemma LemmaLeftUnitalityBind<S,T>(v: S, f: S -> Writer<T>)
+    ensures Bind(Return(v), f) == f(v)
+  {
+  }
+
+  lemma LemmaRightUnitalityBind<T>(w: Writer<T>)
+    ensures Bind(w, Return) == w
+  {
+  }
+
+  lemma LemmaAssociativityBind<S,T,U>(w: Writer<S>, f: S -> Writer<T>, g: T -> Writer<U>)
+    ensures Bind(Bind(w, f), g) == Bind(w, Composition(f, g))
+  {
+  }
+
 }

From 5caa9811103ee8cd5410995564aeaa1d033bc5e4 Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Mon, 30 Jan 2023 16:34:21 +0000
Subject: [PATCH 13/33] skeleton

---
 src/Reader.dfy | 64 ++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 64 insertions(+)
 create mode 100644 src/Reader.dfy

diff --git a/src/Reader.dfy b/src/Reader.dfy
new file mode 100644
index 00000000..993d28f2
--- /dev/null
+++ b/src/Reader.dfy
@@ -0,0 +1,64 @@
+include "Wrappers.dfy"
+
+module {:options "-functionSyntax:4"} Reader {
+
+  datatype Reader<-X,+T> = Reader(f: X -> T)
+
+  function Return<X,T>(t: T): Reader<X,T> {
+    Reader(x => t)
+  }
+
+  function Map<X,S,T>(f: S -> T): Reader<X,S> -> Reader<X,T> {
+    (r: Reader<X,S>) => 
+      Reader(x => f((r.f)(x)))
+  }
+
+  function Join<X,T>(rr: Reader<X,Reader<X,T>>): Reader<X,T> {
+    Reader(x => (rr.f(x)).f(x))
+  }
+
+  function Bind<X,S,T>(r: Reader<X,S>, f: S -> Reader<X,T>): Reader<X,T> {
+    Reader(x => (f((r.f)(x)).f)(x))
+  }
+
+  function Composition<X,S,T,U>(f: S -> Reader<X,T>, g: T -> Reader<X,U>): S -> Reader<X,U> {
+    s => Bind(f(s), g)
+  }
+
+  ghost function Equal<X(!new),T>(eq: (T, T) -> bool): (Reader<X,T>, Reader<X,T>) -> bool 
+  {
+    (r1: Reader<X,T>, r2: Reader<X,T>) => forall x: X :: eq(r1.f(x), r2.f(x))
+  }
+
+  lemma LemmaUnitalityJoin<X,T>(r: Reader<X,T>)
+    requires forall f: X -> T, g: X -> T :: (forall x: X :: f(x) == g(x)) ==> f == g
+    ensures Join(Map(Return<X,T>)(r)) == r == Join(Return<X,Reader<X,T>>(r))
+  {
+  }
+
+  lemma LemmaAssociativityJoin<X,T>(rrr: Reader<X,Reader<X,Reader<X,T>>>) 
+    requires forall f: X -> T, g: X -> T :: (forall x: X :: f(x) == g(x)) ==> f == g
+    ensures Join(Map(Join<X,T>)(rrr)) == Join(Join<X,Reader<X,T>>(rrr))
+  {
+  }  
+
+  lemma LemmaLeftUnitalityBind<X,S,T>(s: S, f: S -> Reader<X,T>)
+    requires forall f: X -> T, g: X -> T :: (forall x: X :: f(x) == g(x)) ==> f == g
+    ensures Bind(Return(s), f) == f(s)
+  {
+    assert (Bind(Return(s), f)).f == (f(s).f);
+  }
+
+  lemma LemmaRightUnitalityBind<X,T>(r: Reader<X,T>)
+    requires forall f: X -> T, g: X -> T :: (forall x: X :: f(x) == g(x)) ==> f == g
+    ensures Bind(r, Return) == r
+  {
+  }
+
+  lemma LemmaAssociativityBind<X,S,T,U>(r: Reader<X,S>, f: S -> Reader<X,T>, g: T -> Reader<X,U>)
+    requires forall f: X -> U, g: X -> U :: (forall x: X :: f(x) == g(x)) ==> f == g
+    ensures Bind(Bind(r, f), g) == Bind(r, Composition(f, g))
+  {
+  }
+
+}

From d551c5a3f37aefb9796ad006c652fe9a9503a058 Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Mon, 30 Jan 2023 18:25:08 +0000
Subject: [PATCH 14/33] Skeleton

---
 src/Coreader.dfy | 62 ++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 62 insertions(+)
 create mode 100644 src/Coreader.dfy

diff --git a/src/Coreader.dfy b/src/Coreader.dfy
new file mode 100644
index 00000000..6ea29357
--- /dev/null
+++ b/src/Coreader.dfy
@@ -0,0 +1,62 @@
+include "Wrappers.dfy"
+
+module {:options "-functionSyntax:4"} Coreader {
+
+  datatype Coreader<+X,+T> = Coreader(left: X, right: T)
+
+  function Extract<X,T>(r: Coreader<X,T>): T {
+    r.right
+  }
+
+  function Duplicate<X,T>(r: Coreader<X,T>): Coreader<X,Coreader<X,T>> {
+    Coreader(r.left, r)
+  }
+
+  function Extend<X,S,T>(f: Coreader<X,S> -> T, r: Coreader<X,S>): Coreader<X,T> {
+    Coreader(r.left, f(r))
+  }
+
+  function Map<X,S,T>(f: S -> T): Coreader<X,S> -> Coreader<X,T> {
+    (r: Coreader<X,S>) => 
+      Coreader(r.left, f(r.right))
+  }
+
+  function Composition<X,S,T,U>(f: Coreader<X,S> -> T, g: Coreader<X,T> -> U): Coreader<X,S> -> U {
+    r => g(Extend(f, r))
+  }
+
+  lemma LemmaUnitalityExtract<X,T>(r: Coreader<X,T>)
+    ensures Map(Extract)(Duplicate(r)) == r == Extract<X,Coreader<X,T>>(Duplicate(r))
+  {
+  }
+
+  lemma LemmaAssociativityDuplicate<X,T>(r: Coreader<X,T>) 
+    ensures Map(Duplicate)(Duplicate(r)) == Duplicate<X,Coreader<X,T>>(Duplicate(r))
+  {
+  }  
+
+  lemma LemmaLeftUnitalityExtend<X,S,T>(r: Coreader<X,T>)
+    ensures Extend(Extract, r) == r
+  {
+  }
+
+  lemma LemmaRightUnitalityExtend<X,S,T>(f: Coreader<X,S> -> T, r: Coreader<X,S>)
+    ensures Extract(Extend(f, r)) == f(r)
+  {
+  }
+
+  lemma LemmaAssociativityExtend<X,S,T,U>(r: Coreader<X,S>, f: Coreader<X,S> -> T, g: Coreader<X,T> -> U)
+    ensures Extend(g, Extend(f, r)) == Extend(Composition(f, g), r)
+  {
+  }
+
+  lemma LemmaAssociativityComposition<X,S,T,U,V>(f: Coreader<X,S> -> T, g: Coreader<X,T> -> U, h: Coreader<X,U> -> V)
+    ensures forall r: Coreader<X,S> :: Composition(Composition(f, g),h)(r) == Composition(f, Composition(g,h))(r)
+  {
+  }
+
+  lemma LemmaIdentityExtract<X,S,T>(f: Coreader<X,S> -> T)
+    ensures forall r: Coreader<X,S> :: Composition(f, Extract<X,T>)(r) == f(r) == Composition(Extract<X,S>, f)(r)
+  {
+  }
+}
\ No newline at end of file

From 29248478f4f194282ba5f7de5e8931a3722b4e8c Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Mon, 30 Jan 2023 18:34:41 +0000
Subject: [PATCH 15/33] more lemmas, quantifier fix

---
 src/Reader.dfy | 25 ++++++++++++++-----------
 1 file changed, 14 insertions(+), 11 deletions(-)

diff --git a/src/Reader.dfy b/src/Reader.dfy
index 993d28f2..4c6ca588 100644
--- a/src/Reader.dfy
+++ b/src/Reader.dfy
@@ -31,34 +31,37 @@ module {:options "-functionSyntax:4"} Reader {
   }
 
   lemma LemmaUnitalityJoin<X,T>(r: Reader<X,T>)
-    requires forall f: X -> T, g: X -> T :: (forall x: X :: f(x) == g(x)) ==> f == g
-    ensures Join(Map(Return<X,T>)(r)) == r == Join(Return<X,Reader<X,T>>(r))
+    ensures forall x: X :: Join(Map(Return<X,T>)(r)).f(x) == r.f(x) == Join(Return<X,Reader<X,T>>(r)).f(x)
   {
   }
 
   lemma LemmaAssociativityJoin<X,T>(rrr: Reader<X,Reader<X,Reader<X,T>>>) 
-    requires forall f: X -> T, g: X -> T :: (forall x: X :: f(x) == g(x)) ==> f == g
-    ensures Join(Map(Join<X,T>)(rrr)) == Join(Join<X,Reader<X,T>>(rrr))
+    ensures forall x: X :: Join(Map(Join<X,T>)(rrr)).f(x) == Join(Join<X,Reader<X,T>>(rrr)).f(x)
   {
   }  
 
   lemma LemmaLeftUnitalityBind<X,S,T>(s: S, f: S -> Reader<X,T>)
-    requires forall f: X -> T, g: X -> T :: (forall x: X :: f(x) == g(x)) ==> f == g
-    ensures Bind(Return(s), f) == f(s)
+    ensures forall x: X :: Bind(Return(s), f).f(x) == f(s).f(x)
   {
-    assert (Bind(Return(s), f)).f == (f(s).f);
   }
 
   lemma LemmaRightUnitalityBind<X,T>(r: Reader<X,T>)
-    requires forall f: X -> T, g: X -> T :: (forall x: X :: f(x) == g(x)) ==> f == g
-    ensures Bind(r, Return) == r
+    ensures forall x: X :: Bind(r, Return).f(x) == r.f(x)
   {
   }
 
   lemma LemmaAssociativityBind<X,S,T,U>(r: Reader<X,S>, f: S -> Reader<X,T>, g: T -> Reader<X,U>)
-    requires forall f: X -> U, g: X -> U :: (forall x: X :: f(x) == g(x)) ==> f == g
-    ensures Bind(Bind(r, f), g) == Bind(r, Composition(f, g))
+    ensures forall x: X :: Bind(Bind(r, f), g).f(x) == Bind(r, Composition(f, g)).f(x)
   {
   }
 
+  lemma LemmaAssociativityComposition<X,S,T,U,V>(f: S -> Reader<X,T>, g: T -> Reader<X,U>, h: U -> Reader<X,V>)
+    ensures forall s: S, x: X :: Composition(Composition(f, g),h)(s).f(x) == Composition(f, Composition(g,h))(s).f(x)
+  {
+  }
+
+  lemma LemmaIdentityReturn<X,S,T>(f: S -> Reader<X,T>)
+    ensures forall s: S, x: X :: Composition(f, Return<X,T>)(s).f(x) == f(s).f(x) == Composition(Return<X,S>, f)(s).f(x)
+  {
+  }
 }

From 5886e98d9fc1828f4e4c514aecc49d768c7e01ac Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Mon, 30 Jan 2023 18:43:35 +0000
Subject: [PATCH 16/33] more lemmas, notation

---
 src/Writer.dfy | 11 ++++++++++-
 1 file changed, 10 insertions(+), 1 deletion(-)

diff --git a/src/Writer.dfy b/src/Writer.dfy
index 44695ce7..a5fab3ab 100644
--- a/src/Writer.dfy
+++ b/src/Writer.dfy
@@ -27,7 +27,7 @@ module {:options "-functionSyntax:4"} Writer {
   }
 
   function Composition<S,T,U>(f: S -> Writer<T>, g: T -> Writer<U>): S -> Writer<U> {
-    x => Bind(f(x), g)
+    s => Bind(f(s), g)
   }
 
   lemma LemmaUnitalityJoin<T>(w: Writer<T>)
@@ -55,4 +55,13 @@ module {:options "-functionSyntax:4"} Writer {
   {
   }
 
+  lemma LemmaAssociativityComposition<S,T,U,V>(f: S -> Writer<T>, g: T -> Writer<U>, h: U -> Writer<V>)
+    ensures forall s: S :: Composition(Composition(f, g), h)(s) == Composition(f, Composition(g, h))(s)
+  {
+  }
+
+  lemma LemmaIdentityExtract<S,T>(f: S -> Writer<T>)
+    ensures forall s: S :: Composition(f, Return<T>)(s) == f(s) == Composition(Return<S>, f)(s)
+  {
+  }
 }

From 8f94e143cf94229fad1ad8630b89dfecf58462c1 Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Mon, 30 Jan 2023 18:51:49 +0000
Subject: [PATCH 17/33] additional lemmas, notation

---
 src/Result.dfy | 20 +++++++++++++++-----
 1 file changed, 15 insertions(+), 5 deletions(-)

diff --git a/src/Result.dfy b/src/Result.dfy
index 8adbb4a3..c3e64d67 100644
--- a/src/Result.dfy
+++ b/src/Result.dfy
@@ -6,7 +6,7 @@ module {:options "-functionSyntax:4"} Result {
 
   datatype Result<+T,+E> = | Success(value: T) | Failure(error: E)
 
-  function Success<T,E>(v: T): Result<T,E> {
+  function Return<T,E>(v: T): Result<T,E> {
     Result.Success(v)
   }
 
@@ -74,7 +74,7 @@ module {:options "-functionSyntax:4"} Result {
   }
 
   function Composition<S,T,U,E>(f: S -> Result<T,E>, g: T -> Result<U,E>): S -> Result<U,E> {
-    x => Bind(f(x), g)
+    s => Bind(f(s), g)
   }
 
   function Equal<T,E>(eqs: (T, T) -> bool, eqf: (E, E) -> bool): (Result<T,E>, Result<T,E>) -> bool {
@@ -113,7 +113,7 @@ module {:options "-functionSyntax:4"} Result {
   }
 
   lemma LemmaUnitalityJoin<T,E>(r: Result<T,E>)
-    ensures Join(Map(Success<T,E>)(r)) == r == Join(Success<Result<T,E>,E>(r))
+    ensures Join(Map(Return<T,E>)(r)) == r == Join(Return<Result<T,E>,E>(r))
   {
   }
 
@@ -123,12 +123,12 @@ module {:options "-functionSyntax:4"} Result {
   }  
 
   lemma LemmaLeftUnitalityBind<S,T,E>(v: S, f: S -> Result<T,E>)
-    ensures Bind(Success(v), f) == f(v)
+    ensures Bind(Return(v), f) == f(v)
   {
   }
 
   lemma LemmaRightUnitalityBind<T,E>(r: Result<T,E>)
-    ensures Bind(r, Success) == r
+    ensures Bind(r, Return) == r
   {
   }
 
@@ -137,6 +137,16 @@ module {:options "-functionSyntax:4"} Result {
   {
   }
 
+  lemma LemmaAssociativityComposition<S,T,U,V,E>(f: S -> Result<T,E>, g: T -> Result<U,E>, h: U -> Result<V,E>)
+    ensures forall s: S :: Composition(Composition(f, g), h)(s) == Composition(f, Composition(g, h))(s)
+  {
+  }
+
+  lemma LemmaIdentityExtract<S,T,E>(f: S -> Result<T,E>)
+    ensures forall s: S :: Composition(f, Return<T,E>)(s) == f(s) == Composition(Return<S,E>, f)(s)
+  {
+  }
+
 }
 
 

From 8056018f95739dd818d11892309ac41e426947c1 Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Mon, 30 Jan 2023 19:06:02 +0000
Subject: [PATCH 18/33] more lemmas, notation

---
 src/Option.dfy | 22 ++++++++++++++++------
 1 file changed, 16 insertions(+), 6 deletions(-)

diff --git a/src/Option.dfy b/src/Option.dfy
index c1ef610d..a4b8d961 100644
--- a/src/Option.dfy
+++ b/src/Option.dfy
@@ -12,7 +12,7 @@ module {:options "-functionSyntax:4"} Option {
 
   datatype Option<+T> = None | Some(value: T)
 
-  function Some<T>(v: T): Option<T> {
+  function Return<T>(v: T): Option<T> {
     Option.Some(v)
   }
 
@@ -53,7 +53,7 @@ module {:options "-functionSyntax:4"} Option {
     (o: Option<S>) =>
       match o 
       case None => Option<T>.None
-      case Some(v) => Some(f(v))
+      case Some(v) => Option<T>.Some(f(v))
   }
 
   function Fold<T,U>(u: U, f: T -> U): Option<T> -> U {
@@ -81,7 +81,7 @@ module {:options "-functionSyntax:4"} Option {
   }
 
   function Composition<S,T,U>(f: S -> Option<T>, g: T-> Option<U>): S -> Option<U> {
-    x => Bind(f(x), g)
+    s => Bind(f(s), g)
   }
 
   function ToResult<T,E>(e: E): Option<T> -> Result<T,E> {
@@ -104,7 +104,7 @@ module {:options "-functionSyntax:4"} Option {
   }
 
   lemma LemmaUnitalityJoin<T>(o: Option<T>)
-    ensures Join(Map(Some<T>)(o)) == o == Join(Some<Option<T>>(o))
+    ensures Join(Map(Return<T>)(o)) == o == Join(Return<Option<T>>(o))
   {
   }
 
@@ -114,12 +114,12 @@ module {:options "-functionSyntax:4"} Option {
   }  
 
   lemma LemmaLeftUnitalityBind<S,T>(v: S, f: S -> Option<T>)
-    ensures Bind(Some(v), f) == f(v)
+    ensures Bind(Return(v), f) == f(v)
   {
   }
 
   lemma LemmaRightUnitalityBind<T>(o: Option<T>)
-    ensures Bind(o, Some) == o
+    ensures Bind(o, Return) == o
   {
   }
 
@@ -128,6 +128,16 @@ module {:options "-functionSyntax:4"} Option {
   {
   }
 
+  lemma LemmaAssociativityComposition<S,T,U,V>(f: S -> Option<T>, g: T -> Option<U>, h: U -> Option<V>)
+    ensures forall s: S :: Composition(Composition(f, g),h)(s) == Composition(f, Composition(g,h))(s)
+  {
+  }
+
+  lemma LemmaIdentityReturn<X,S,T>(f: S -> Option<T>)
+    ensures forall s: S :: Composition(f, Return<T>)(s) == f(s) == Composition(Return<S>, f)(s)
+  {
+  }
+
 
 
 

From 66e5b82639a73b858c02d06b7f6ca74ea908e297 Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Thu, 2 Feb 2023 19:03:52 +0000
Subject: [PATCH 19/33] combine

---
 .DS_Store                       | Bin 0 -> 8196 bytes
 src/Either.dfy                  |  32 +-
 src/Functions.dfy               |   4 +
 src/Option.dfy                  |  83 +++--
 src/Reader.dfy                  |  41 ++-
 src/Result.dfy                  | 108 ++++--
 src/Wrappers.dfy                | 584 ++++++++++++++++++++++++++++----
 src/Wrappers.dll                | Bin 0 -> 154624 bytes
 src/Wrappers.runtimeconfig.json |  10 +
 src/Writer.dfy                  |  58 +++-
 10 files changed, 777 insertions(+), 143 deletions(-)
 create mode 100644 .DS_Store
 create mode 100644 src/Wrappers.dll
 create mode 100644 src/Wrappers.runtimeconfig.json

diff --git a/.DS_Store b/.DS_Store
new file mode 100644
index 0000000000000000000000000000000000000000..f8de3e84e630e08a3d1d98381967a686ffef9ec3
GIT binary patch
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diff --git a/src/Either.dfy b/src/Either.dfy
index 90c8521f..a04ce99f 100644
--- a/src/Either.dfy
+++ b/src/Either.dfy
@@ -13,6 +13,12 @@ module {:options "-functionSyntax:4"} Either {
 
   datatype Either<+S,+T> = Left(left: S) | Right(right: T) 
 
+  /**********************************************************
+  *
+  *  Get and set
+  *
+  ***********************************************************/
+
   function Left<S,T>(v: S): Either<S,T> {
     Either.Left(v)
   }
@@ -29,6 +35,12 @@ module {:options "-functionSyntax:4"} Either {
     e.Right?
   }
 
+  /**********************************************************
+  *
+  *  Conversion to other datatypes
+  *
+  ***********************************************************/
+
   function FindLeft<S,T>(e: Either<S,T>): Option<S> {
     match e 
     case Left(v) => Some(v)
@@ -41,6 +53,12 @@ module {:options "-functionSyntax:4"} Either {
     case Right(v) => Some(v)
   }
 
+  /**********************************************************
+  *
+  *  Map
+  *
+  ***********************************************************/
+
   function MapLeft<S,T,U>(f: S -> T): Either<S,U> -> Either<T,U> {
     (e: Either<S,U>) =>
      match e 
@@ -62,6 +80,12 @@ module {:options "-functionSyntax:4"} Either {
       case Right(v) => Either<S2,T2>.Right(g(v))
   }
 
+  /**********************************************************
+  *
+  *  Fold
+  *
+  ***********************************************************/
+
   function Fold<S,T,U>(f: S -> U, g: T -> U): Either<S,T> -> U {
     (e: Either<S,T>) =>
       match e
@@ -69,6 +93,12 @@ module {:options "-functionSyntax:4"} Either {
       case Right(v) => g(v)
   }
 
+  /**********************************************************
+  *
+  *  Comparison
+  *
+  ***********************************************************/
+
   function Equal<S,T>(eql: (S, S) -> bool, eqr: (T, T) -> bool): (Either<S,T>, Either<S,T>) -> bool {
     (e1: Either<S,T>, e2: Either<S,T>) =>
       match (e1, e2)
@@ -86,7 +116,5 @@ module {:options "-functionSyntax:4"} Either {
       case (Right(_), Left(_)) => 1
   }
 
-
-
 }
 
diff --git a/src/Functions.dfy b/src/Functions.dfy
index 85757547..17eb2aa5 100644
--- a/src/Functions.dfy
+++ b/src/Functions.dfy
@@ -21,4 +21,8 @@ module {:options "-functionSyntax:4"} Functions {
     forall x1, x2 :: f(x1) == f(x2) ==> x1 == x2
   }
 
+  function Composition<X,Y,Z>(f: X -> Y, g: Y -> Z): X -> Z {
+    x => g(f(x))
+  }
+
 }
diff --git a/src/Option.dfy b/src/Option.dfy
index a4b8d961..2d725ab0 100644
--- a/src/Option.dfy
+++ b/src/Option.dfy
@@ -10,6 +10,12 @@ module {:options "-functionSyntax:4"} Option {
 
   import opened Wrappers
 
+  /**********************************************************
+  *
+  *  Monadic structure in terms of Return, Bind, Join, Map, Composition 
+  *
+  ***********************************************************/
+
   datatype Option<+T> = None | Some(value: T)
 
   function Return<T>(v: T): Option<T> {
@@ -22,19 +28,29 @@ module {:options "-functionSyntax:4"} Option {
     case Some(v) => f(v)
   }
 
-  predicate IsNone<T>(o: Option<T>) {
-    o.None?
+  function Join<T>(oo: Option<Option<T>>): Option<T> {
+    match oo
+    case None => Option<T>.None
+    case Some(o) => o
   }
 
-  predicate IsSome<T>(o: Option<T>) {
-    o.Some?
+  function Map<S,T>(f: S -> T): Option<S> -> Option<T>
+  {
+    (o: Option<S>) =>
+      match o 
+      case None => Option<T>.None
+      case Some(v) => Option<T>.Some(f(v))
   }
 
-  function GetValue<T>(o: Option<T>): T
-    requires o.Some?
-  {
-    o.value
+  function Composition<S,T,U>(f: S -> Option<T>, g: T-> Option<U>): S -> Option<U> {
+    s => Bind(f(s), g)
   }
+ 
+  /**********************************************************
+  *
+  *  Get and Fold
+  *
+  ***********************************************************/
 
   function GetValueDefault<T>(o: Option<T>, default: T): T {
     match o
@@ -42,18 +58,10 @@ module {:options "-functionSyntax:4"} Option {
     case Some(v) => v
   }
 
-  function Join<T>(oo: Option<Option<T>>): Option<T> {
-    match oo
-    case None => Option<T>.None
-    case Some(o) => o
-  }
-
-  function Map<S,T>(f: S -> T): Option<S> -> Option<T>
+  function GetValue<T>(o: Option<T>): T
+    requires o.Some?
   {
-    (o: Option<S>) =>
-      match o 
-      case None => Option<T>.None
-      case Some(v) => Option<T>.Some(f(v))
+    o.value
   }
 
   function Fold<T,U>(u: U, f: T -> U): Option<T> -> U {
@@ -63,6 +71,12 @@ module {:options "-functionSyntax:4"} Option {
       case Some(v) => f(v)
   }
 
+  /**********************************************************
+  *
+  *  Comparison
+  *
+  ***********************************************************/
+
   function Equal<T>(eq: (T, T) -> bool): (Option<T>, Option<T>) -> bool {
     (o1: Option<T>, o2: Option<T>) =>
       match (o1, o2)
@@ -80,9 +94,11 @@ module {:options "-functionSyntax:4"} Option {
       case (Some(_), None) => 1
   }
 
-  function Composition<S,T,U>(f: S -> Option<T>, g: T-> Option<U>): S -> Option<U> {
-    s => Bind(f(s), g)
-  }
+  /**********************************************************
+  *
+  *  Conversion to other datatypes
+  *
+  ***********************************************************/
 
   function ToResult<T,E>(e: E): Option<T> -> Result<T,E> {
     (o: Option<T>) =>
@@ -103,6 +119,12 @@ module {:options "-functionSyntax:4"} Option {
     case Some(v) => {v}
   }
 
+  /**********************************************************
+  *
+  *  Monad laws in terms of Join and Map
+  *
+  ***********************************************************/
+
   lemma LemmaUnitalityJoin<T>(o: Option<T>)
     ensures Join(Map(Return<T>)(o)) == o == Join(Return<Option<T>>(o))
   {
@@ -113,6 +135,12 @@ module {:options "-functionSyntax:4"} Option {
   {
   }  
 
+  /**********************************************************
+  *
+  *  Monad laws in terms of Bind and Return
+  *
+  ***********************************************************/
+
   lemma LemmaLeftUnitalityBind<S,T>(v: S, f: S -> Option<T>)
     ensures Bind(Return(v), f) == f(v)
   {
@@ -128,6 +156,12 @@ module {:options "-functionSyntax:4"} Option {
   {
   }
 
+  /**********************************************************
+  *
+  *  Monad laws in terms of (Kleisli) Composition and Return
+  *
+  ***********************************************************/
+
   lemma LemmaAssociativityComposition<S,T,U,V>(f: S -> Option<T>, g: T -> Option<U>, h: U -> Option<V>)
     ensures forall s: S :: Composition(Composition(f, g),h)(s) == Composition(f, Composition(g,h))(s)
   {
@@ -138,9 +172,4 @@ module {:options "-functionSyntax:4"} Option {
   {
   }
 
-
-
-
-
-
 }
diff --git a/src/Reader.dfy b/src/Reader.dfy
index 4c6ca588..a3a927f2 100644
--- a/src/Reader.dfy
+++ b/src/Reader.dfy
@@ -2,34 +2,53 @@ include "Wrappers.dfy"
 
 module {:options "-functionSyntax:4"} Reader {
 
+  /**********************************************************
+  *
+  *  Monadic structure in terms of Return, Bind, Join, Map, Composition 
+  *
+  ***********************************************************/
+
   datatype Reader<-X,+T> = Reader(f: X -> T)
 
   function Return<X,T>(t: T): Reader<X,T> {
     Reader(x => t)
   }
 
-  function Map<X,S,T>(f: S -> T): Reader<X,S> -> Reader<X,T> {
-    (r: Reader<X,S>) => 
-      Reader(x => f((r.f)(x)))
+  function Bind<X,S,T>(r: Reader<X,S>, f: S -> Reader<X,T>): Reader<X,T> {
+    Reader(x => (f((r.f)(x)).f)(x))
   }
 
   function Join<X,T>(rr: Reader<X,Reader<X,T>>): Reader<X,T> {
     Reader(x => (rr.f(x)).f(x))
   }
 
-  function Bind<X,S,T>(r: Reader<X,S>, f: S -> Reader<X,T>): Reader<X,T> {
-    Reader(x => (f((r.f)(x)).f)(x))
+  function Map<X,S,T>(f: S -> T): Reader<X,S> -> Reader<X,T> {
+    (r: Reader<X,S>) => 
+      Reader(x => f((r.f)(x)))
   }
 
   function Composition<X,S,T,U>(f: S -> Reader<X,T>, g: T -> Reader<X,U>): S -> Reader<X,U> {
     s => Bind(f(s), g)
   }
 
+
+  /**********************************************************
+  *
+  *  Comparison
+  *
+  ***********************************************************/  
+
   ghost function Equal<X(!new),T>(eq: (T, T) -> bool): (Reader<X,T>, Reader<X,T>) -> bool 
   {
     (r1: Reader<X,T>, r2: Reader<X,T>) => forall x: X :: eq(r1.f(x), r2.f(x))
   }
 
+  /**********************************************************
+  *
+  *  Monad laws in terms of Join and Map
+  *
+  ***********************************************************/
+
   lemma LemmaUnitalityJoin<X,T>(r: Reader<X,T>)
     ensures forall x: X :: Join(Map(Return<X,T>)(r)).f(x) == r.f(x) == Join(Return<X,Reader<X,T>>(r)).f(x)
   {
@@ -40,6 +59,12 @@ module {:options "-functionSyntax:4"} Reader {
   {
   }  
 
+  /**********************************************************
+  *
+  *  Monad laws in terms of Bind and Return
+  *
+  ***********************************************************/
+
   lemma LemmaLeftUnitalityBind<X,S,T>(s: S, f: S -> Reader<X,T>)
     ensures forall x: X :: Bind(Return(s), f).f(x) == f(s).f(x)
   {
@@ -55,6 +80,12 @@ module {:options "-functionSyntax:4"} Reader {
   {
   }
 
+  /**********************************************************
+  *
+  *  Monad laws in terms of (Kleisli) Composition and Return
+  *
+  ***********************************************************/
+
   lemma LemmaAssociativityComposition<X,S,T,U,V>(f: S -> Reader<X,T>, g: T -> Reader<X,U>, h: U -> Reader<X,V>)
     ensures forall s: S, x: X :: Composition(Composition(f, g),h)(s).f(x) == Composition(f, Composition(g,h))(s).f(x)
   {
diff --git a/src/Result.dfy b/src/Result.dfy
index c3e64d67..6183153e 100644
--- a/src/Result.dfy
+++ b/src/Result.dfy
@@ -4,12 +4,58 @@ module {:options "-functionSyntax:4"} Result {
 
   import opened Wrappers 
 
+  /**********************************************************
+  *
+  *  Monadic structure in terms of Return, Bind, Join, Map, Composition 
+  *
+  ***********************************************************/
+
   datatype Result<+T,+E> = | Success(value: T) | Failure(error: E)
 
   function Return<T,E>(v: T): Result<T,E> {
     Result.Success(v)
   }
 
+  function Bind<S,T,E>(r: Result<S,E>, f: S -> Result<T,E>): Result<T,E> {
+    match r
+    case Success(v) => f(v)
+    case Failure(e) => Result<T,E>.Failure(e)
+  }
+
+  function Join<T,E>(rr: Result<Result<T,E>,E>): Result<T,E> {
+    match rr
+    case Success(v) => v
+    case Failure(e) => Result<T,E>.Failure(e)
+  }
+
+  function Map<S,T,E>(f: S -> T): Result<S,E> -> Result<T,E> {
+    (r: Result<S,E>) =>
+      match r 
+      case Success(v) => Result<T,E>.Success(f(v))
+      case Failure(e) => Result<T,E>.Failure(e)
+  }
+
+  function MapError<T,E1,E2>(f: E1 -> E2): Result<T,E1> -> Result<T,E2> {
+    (r: Result<T,E1>) =>
+      match r 
+      case Success(v) => Result<T,E2>.Success(v)
+      case Failure(e) => Result<T,E2>.Failure(f(e))
+  } 
+
+  function Composition<S,T,U,E>(f: S -> Result<T,E>, g: T -> Result<U,E>): S -> Result<U,E> {
+    s => Bind(f(s), g)
+  }
+
+  /**********************************************************
+  *
+  *  Get and set
+  *
+  ***********************************************************/
+
+  function Success<T,E>(v: T): Result<T,E> {
+    Result.Success(v)
+  }
+
   function Failure<T,E>(e: E): Result<T,E> {
     Result.Failure(e)
   } 
@@ -40,31 +86,11 @@ module {:options "-functionSyntax:4"} Result {
     r.error
   }
 
-  function Bind<S,T,E>(r: Result<S,E>, f: S -> Result<T,E>): Result<T,E> {
-    match r
-    case Success(v) => f(v)
-    case Failure(e) => Result<T,E>.Failure(e)
-  }
-
-  function Join<T,E>(rr: Result<Result<T,E>,E>): Result<T,E> {
-    match rr
-    case Success(v) => v
-    case Failure(e) => Result<T,E>.Failure(e)
-  }
-
-  function Map<S,T,E>(f: S -> T): Result<S,E> -> Result<T,E> {
-    (r: Result<S,E>) =>
-      match r 
-      case Success(v) => Result<T,E>.Success(f(v))
-      case Failure(e) => Result<T,E>.Failure(e)
-  }
-
-  function MapError<T,E1,E2>(f: E1 -> E2): Result<T,E1> -> Result<T,E2> {
-    (r: Result<T,E1>) =>
-      match r 
-      case Success(v) => Result<T,E2>.Success(v)
-      case Failure(e) => Result<T,E2>.Failure(f(e))
-  }  
+  /**********************************************************
+  *
+  *  Fold
+  *
+  ***********************************************************/
 
   function Fold<T,E,U>(f: T -> U, g: E -> U): Result<T,E> -> U {
     (r: Result<T,E>) =>
@@ -73,9 +99,11 @@ module {:options "-functionSyntax:4"} Result {
       case Failure(e) => g(e)
   }
 
-  function Composition<S,T,U,E>(f: S -> Result<T,E>, g: T -> Result<U,E>): S -> Result<U,E> {
-    s => Bind(f(s), g)
-  }
+  /**********************************************************
+  *
+  *  Comparison
+  *
+  ***********************************************************/
 
   function Equal<T,E>(eqs: (T, T) -> bool, eqf: (E, E) -> bool): (Result<T,E>, Result<T,E>) -> bool {
     (r1: Result<T,E>, r2: Result<T,E>) =>
@@ -94,6 +122,12 @@ module {:options "-functionSyntax:4"} Result {
       case (Failure(_), Success(_)) => 1
   }
 
+  /**********************************************************
+  *
+  *  Conversion to other datatypes
+  *
+  ***********************************************************/
+
   function ToSeq<T,E>(r: Result<T,E>): seq<T> {
     match r
     case Success(v) => [v]
@@ -112,6 +146,12 @@ module {:options "-functionSyntax:4"} Result {
     case Failure(_) => None()
   }
 
+  /**********************************************************
+  *
+  *  Monad laws in terms of Join and Map
+  *
+  ***********************************************************/
+
   lemma LemmaUnitalityJoin<T,E>(r: Result<T,E>)
     ensures Join(Map(Return<T,E>)(r)) == r == Join(Return<Result<T,E>,E>(r))
   {
@@ -122,6 +162,12 @@ module {:options "-functionSyntax:4"} Result {
   {
   }  
 
+  /**********************************************************
+  *
+  *  Monad laws in terms of Bind and Return
+  *
+  ***********************************************************/
+
   lemma LemmaLeftUnitalityBind<S,T,E>(v: S, f: S -> Result<T,E>)
     ensures Bind(Return(v), f) == f(v)
   {
@@ -137,6 +183,12 @@ module {:options "-functionSyntax:4"} Result {
   {
   }
 
+  /**********************************************************
+  *
+  *  Monad laws in terms of (Kleisli) Composition and Return
+  *
+  ***********************************************************/
+
   lemma LemmaAssociativityComposition<S,T,U,V,E>(f: S -> Result<T,E>, g: T -> Result<U,E>, h: U -> Result<V,E>)
     ensures forall s: S :: Composition(Composition(f, g), h)(s) == Composition(f, Composition(g, h))(s)
   {
diff --git a/src/Wrappers.dfy b/src/Wrappers.dfy
index 22fc4bf5..6f82e4c2 100644
--- a/src/Wrappers.dfy
+++ b/src/Wrappers.dfy
@@ -5,95 +5,541 @@
 *  SPDX-License-Identifier: MIT 
 *******************************************************************************/
 
-module {:options "-functionSyntax:4"} Wrappers {
-  
-  datatype Option<+T> = None | Some(value: T) {
-    function ToResult(): Result<T, string> {
-      match this
-      case Some(v) => Success(v)
-      case None() => Failure("Option is None")
-    }
-    
-    function UnwrapOr(default: T): T {
-      match this
-      case Some(v) => v
-      case None() => default
-    }
+include "functions.dfy"
 
-    predicate IsFailure() {
-      None?
-    }
+abstract module {:options "-functionSyntax:4"} Functor {
 
-    function PropagateFailure<U>(): Option<U>
-      requires None?
-    {
-      None
-    }
+  import opened Functions
 
-    function Extract(): T
-      requires Some?
-    {
-      value
-    }
+  /**********************************************************
+  *
+  *  Assumed structure 
+  *
+  ***********************************************************/
+
+  /* Functor structure */
+  type W(!new)<T(!new)>
+
+  function Map<S,T>(f: S -> T): W<S> -> W<T>
+
+  function Equal<T(!new)>(eq: (T, T) -> bool): (eq': (W<T>, W<T>) -> bool)
+    requires 
+      && forall x: T :: eq(x, x)
+      && forall x, y, z: T :: eq(x, y) && eq(y, z) ==> eq(x, z) 
+      && forall x, y: T :: eq(x, y) ==> eq(y, x)
+    ensures 
+      && forall w: W<T> :: eq'(w, w)
+      && forall u, v, w: W<T> ::  eq'(u, v) && eq'(v, w) ==> eq'(u, w) 
+      && forall v, w: W<T> :: eq'(u, v) ==> eq'(v, w)
+
+  /* Functoriality of Map */
+  lemma LemmaMapFunction<S,T>(eq: (T, T) -> bool, f: S -> T, g: S -> T)
+    requires forall x: S :: eq(f(x), g(x))
+    ensures forall w: W<S> :: Equal(eq)(Map(f)(w), Map(g)(w))
+
+  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(f: S -> T, g: T -> U, w: W<S>)
+    ensures Map(Composition(f, g))(w) == Composition(Map(f), Map(g))(w)
+
+  lemma LemmaMapIdentity<T>(id: T -> T)
+    requires forall x: T :: id(x) == x
+    ensures forall w: W<T> :: Map(id)(w) == w
+
+}
+
+abstract module {:options "-functionSyntax:4"} Monad refines Functor {
+
+  /**********************************************************
+  *
+  *  Assumed structure 
+  *
+  ***********************************************************/
+
+  /* Monad structure */
+  type W<T(!new)>
+
+  function Return<T>(x: T): W<T>
+
+  function Join<T>(ww: W<W<T>>): W<T>
+
+  /* Naturality of Return and Join */
+  lemma LemmaReturnNaturality<S,T>(eq: (W<T>, W<T>) -> bool, f: S -> T, x: S)
+    ensures eq(Map(f)(Return(x)), Return(f(x)))
+
+  lemma LemmaJoinNaturality<S,T>(eq: (W<T>, W<T>) -> bool, f: S -> T, ww: W<W<S>>)
+    ensures eq(Join(Map(Map(f))(ww)), Map(f)(Join(ww)))
+
+  /* Monad laws */
+  lemma LemmaUnitalityJoin<T>(eq: (W<T>, W<T>) -> bool, w: W<T>)
+    ensures eq(Join(Map(Return)(w)), w) && eq(w, Join(Return(w)))
+
+  lemma LemmaAssociativityJoin<T>(eq: (W<T>, W<T>) -> bool, www: W<W<W<T>>>) 
+    ensures eq(Join(Map(Join)(www)), Join(Join(www)))
+
+  /**********************************************************
+  *
+  *  Derived structure 
+  *
+  ***********************************************************/
+
+  /* Derived functions */
+  function Bind<S,T>(w: W<S>, f: S -> W<T>): W<T> {
+    Join(Map(f)(w))
   }
 
-  datatype Result<+T, +R> = | Success(value: T) | Failure(error: R) {
-    function ToOption(): Option<T> 
-    {
-      match this
-      case Success(s) => Some(s)
-      case Failure(e) => None()
-    }
+  function KleisliComp<S,T,U>(f: S -> W<T>, g: T -> W<U>): S -> W<U> {
+    x => Join(Map(g)(f(x)))
+  }
 
-    function UnwrapOr(default: T): T 
-    {
-      match this
-      case Success(s) => s
-      case Failure(e) => default
-    }
+  /* Derived lemmas */
+  lemma LemmaLeftUnitalityBind<S,T>(eq: (W<T>, W<T>) -> bool, x: S, f: S -> W<T>)
+    ensures eq(Bind(Return(x), f), f(x))
+  {
+    assert Equal(eq)(Map(f)(Return(x)), Return(f(x))) by { LemmaReturnNaturality<S,W<T>>(Equal(eq), f, x); }
+    assert eq(Join(Map(Return)(f(x))), f(x)) && eq(f(x), Join(Return(f(x)))) by { LemmaUnitalityJoin<T>(eq, f(x)); }
 
-    predicate IsFailure() {
-      Failure?
-    }
+    // assert Equal(eq)(Map(f)(Return(x)), Return(f(x))) by { LemmaReturnNaturality<S,T>(eq, f, x); }
 
-    function PropagateFailure<U>(): Result<U, R>
-      requires Failure?
-    {
-      Failure(this.error)
+
+    // calc {
+    
+    //   Equal(eq)(Bind(Return(x), f), f(x));
+    // <==> // Definition of Bind
+    //   Equal(eq)(Join(Map(f)(Return(x))), f(x));
+    // <==> { LemmaReturnNaturality(eq, f, x); assert Equal(eq)(Map(f)(Return(x)), Return(f(x))); }
+    //   Equal(eq)(Join(Return(f(x))), f(x));
+    // <== { LemmaUnitalityJoin(eq, f(x)); }
+    //   f(x);
+    // }
+
+  }
+
+  lemma LemmaRightUnitalityBind<T>(w: W<T>)
+    ensures Bind(w, Return) == w
+  {
+    calc {
+      Bind(w, Return); 
+    == // Definition of Bind
+      Join(Map(Return)(w));
+    == { LemmaUnitalityJoin(w); }
+      w;
     }
+  }
 
-    function MapFailure<NewR>(reWrap: R -> NewR): Result<T, NewR>
-    {
-      match this
-      case Success(s) => Success(s)
-      case Failure(e) => Failure(reWrap(e))
+  lemma LemmaAssociativityBind<S,T,U>(w: W<S>, f: S -> W<T>, g: T -> W<U>)
+    ensures Bind(Bind(w, f), g) == Bind(w, KleisliComp(f, g))
+  {
+    calc {
+      Bind(Bind(w, f), g);
+    == // Definition of Bind
+      Join(Map(g)(Join(Map(f)(w))));
+    ==  { LemmaJoinNaturality(g, Map(f)(w)); }
+      Join(Join(Map(Map(g))(Map(f)(w))));
+    == { LemmaMapFunctorial(f, Map(g), w); }
+      Join(Join(Map(x => Map(g)(f(x)))(w)));
+    == { LemmaAssociativityJoin(Map(x => Map(g)(f(x)))(w)); }
+      Join(Map(Join)(Map(x => Map(g)(f(x)))(w)));
+    == // Definition of Composition
+      Join(Composition(Map(x => Map(g)(f(x))), Map(Join))(w));
+    == {LemmaMapFunctorial(x => Map(g)(f(x)), Join, w);}
+      Join(Map(Composition(x => Map(g)(f(x)), Join))(w));
+    == { LemmaMapFunction((x, y) => x == y, Composition(x => Map(g)(f(x)), Join), x => Join(Map(g)(f(x)))); }
+      Join(Map(x => Join(Map(g)(f(x))))(w));
+    == // Definition of Bind, KleisliComp
+      Bind(w, KleisliComp(f, g));
     }
+  }
 
-    function Extract(): T
-      requires Success?
-    {
-      value
+  lemma LemmaAssociativityComposition<S,T,U,V>(f: S -> W<T>, g: T -> W<U>, h: U -> W<V>, x: S)
+    ensures KleisliComp(KleisliComp(f, g), h)(x) == KleisliComp(f, KleisliComp(g, h))(x)
+  {
+    calc {
+      KleisliComp(KleisliComp(f, g), h)(x);
+    == // Definition of KleisliComp
+      Join(Map(h)(Join(Map(g)(f(x)))));
+    == { LemmaJoinNaturality(h, Map(g)(f(x))); }
+      Join(Join(Map(Map(h))(Map(g)(f(x)))));
+    == // Definition of Composition
+      Join(Join(Composition(Map(g), Map(Map(h)))(f(x))));
+    == { LemmaMapFunctorial(g, Map(h), f(x)); }
+      Join(Join(Map(Composition(g, Map(h)))(f(x))));
+    == { LemmaAssociativityJoin(Map(Composition(g, Map(h)))(f(x))); }
+      Join(Map(Join)(Map(Composition(g, Map(h)))(f(x))) ); 
+    == { LemmaMapFunctorial(Composition(g, Map(h)), Join, f(x)); }
+      Join(Map(Composition(Composition(g, Map(h)), Join))(f(x)));
+    == { LemmaMapFunction((x, y) => x == y, Composition(Composition(g, Map(h)), Join), y => Join(Map(h)(g(y)))); }
+      Join(Map(y => Join(Map(h)(g(y))))(f(x)));
+    == // Definition of KleisliComp
+      KleisliComp(f, KleisliComp(g, h))(x);
     }
   }
 
-  datatype Outcome<E> = Pass | Fail(error: E)
+  lemma LemmaIdentityReturn<S,T>(f: S -> W<T>, x: S)
+    ensures KleisliComp(f, Return)(x) == f(x) == KleisliComp(Return, f)(x)
   {
-    predicate IsFailure() {
-      Fail?
+    calc {
+      KleisliComp(f, Return)(x);
+    == // Definition of Composition
+      Join(Map(Return)(f(x)));
+    == { LemmaUnitalityJoin(f(x)); }
+      f(x);
     }
-    // Note: PropagateFailure returns a Result, not an Outcome.
-    function PropagateFailure<U>(): Result<U, E>
-      requires Fail?
-    {
-      Failure(this.error)
+    calc {
+      f(x);
+    == { LemmaUnitalityJoin(f(x)); }
+      Join(Return(f(x)));
+    == { LemmaReturnNaturality(f, x); }
+      Join(Map(f)(Return(x)));
+    == // Definition of Composition
+      KleisliComp(Return, f)(x);
     }
-    // Note: no Extract method
   }
+}
+
+abstract module {:options "-functionSyntax:4"} Result refines Monad {
+
+  /**********************************************************
+  *
+  *  Monad
+  *
+  ***********************************************************/
+  
+  type E
+
+  function eqe(e1: E, e2: E): bool
+
+  /* Monad structure */
+
+  datatype W<T> = Success(value: T) | Failure(error: E)
+
+  function Map<S,T>(f: S -> T): W<S> -> W<T> {
+    (w: W<S>) =>
+      match w 
+      case Success(v) => W<T>.Success(f(v))
+      case Failure(e) => W<T>.Failure(e)
+  }
+
+  function Equal<T>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+    (w1: W<T>, w2: W<T>) =>
+      match (w1, w2)
+      case (Success(v1), Success(v2)) => eq(v1, v2)
+      case (Failure(e1), Failure(e2)) => eqe(e1, e2)
+      case _ => false
+  }
+
+  function Return<T>(x: T): W<T> {
+    Success(x)
+  }
+ 
+  function Join<T>(ww: W<W<T>>): W<T> {
+    match ww
+    case Success(v) => v
+    case Failure(e) => W<T>.Failure(e)
+  }
+
+  /* Functoriality of Map */
+  lemma LemmaMapFunction<S,T>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {}
+
+  lemma LemmaMapFunctorial<S,T,U>(f: S -> T, g: T -> U, w: W<S>) {}
+
+  lemma LemmaMapIdentity<T>(id: T -> T) {}
+
+  /* Naturality of Return and Join */
+  lemma LemmaReturnNaturality<S,T>(f: S -> T, x: S) {}
+
+  lemma LemmaJoinNaturality<S,T>(f: S -> T, ww: W<W<S>>) {}
+
+  /* Monad laws */
+  lemma LemmaUnitalityJoin<T>(w: W<T>) {}
 
-  // A helper function to ensure a requirement is true at runtime
-  // :- Need(5 == |mySet|, "The set MUST have 5 elements.")
-  function Need<E>(condition: bool, error: E): (result: Outcome<E>)
+  lemma LemmaAssociativityJoin<T>(www: W<W<W<T>>>) {}
+
+  /**********************************************************
+  *
+  *  Wrapper
+  *
+  ***********************************************************/
+
+  /* Get and Set */
+  function Failure<T>(e: E): W<T> {
+    W.Failure(e)
+  } 
+
+  predicate IsSuccess<T>(w: W<T>) {
+    w.Success?
+  }  
+
+  predicate IsFailure<T>(w: W<T>) {
+    w.Failure?
+  }  
+
+  function GetValue<T>(w: W<T>): T
+    requires w.Success?
+  {
+    w.value
+  }
+
+  function GetValueDefault<T,E>(w: W<T>, default: T): T {
+    match w 
+    case Success(v) => v
+    case Failure(_) => default
+  }
+
+  function GetError<T>(w: W<T>): E 
+    requires w.Failure?
   {
-    if condition then Pass else Fail(error)
+    w.error
   }
+
+  /* Fold */
+  function Fold<T,U>(f: T -> U, g: E -> U): W<T> -> U {
+    (w: W<T>) =>
+      match w 
+      case Success(v) => f(v)
+      case Failure(e) => g(e)
+  }
+
+  /* Conversion to other types */
+  function ToSeq<T>(w: W<T>): seq<T> {
+    match w
+    case Success(v) => [v]
+    case Failure(_) => []
+  }
+
+  function ToSet<T>(w: W<T>): set<T> {
+    match w
+    case Success(v) => {v}
+    case Failure(_) => {}   
+  }
+
+  // function ToOption<T>(w: W<T>): Option<T> {
+  //   match w
+  //   case Success(v) => Some(v)
+  //   case Failure(_) => None()
+  // }
+
 }
+
+module {:options "-functionSyntax:4"} Option refines Monad {
+
+  /**********************************************************
+  *
+  *  Monad
+  *
+  ***********************************************************/
+ 
+  /* Monad structure */
+  datatype W<T> = None | Some(value: T)
+
+  function Map<S,T>(f: S -> T): W<S> -> W<T>
+  {
+    (w: W<S>) =>
+      match w 
+      case None => W<T>.None
+      case Some(v) => W<T>.Some(f(v))
+  }
+
+  function Equal<T>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+    (w1: W<T>, w2: W<T>) =>
+      match (w1, w2)
+      case (Some(v1), Some(v2)) => eq(v1, v2)
+      case (None, None) => true
+      case _ => false
+  }
+
+  function Return<T>(x: T): W<T> {
+    Some(x)
+  }
+
+  function Join<T>(ww: W<W<T>>): W<T> {
+    match ww
+    case None => None
+    case Some(w) => w
+  }
+
+  /* Functoriality of Map */
+  lemma LemmaMapFunction<S,T>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {}
+
+  lemma LemmaMapFunctorial<S,T,U>(f: S -> T, g: T -> U, w: W<S>) {}
+
+  lemma LemmaMapIdentity<T>(id: T -> T) {}
+
+  /* Naturality of Return and Join */
+  lemma LemmaReturnNaturality<S,T>(f: S -> T, x: S) {}
+
+  lemma LemmaJoinNaturality<S,T>(f: S -> T, ww: W<W<S>>) {}
+
+  /* Monad laws */
+  lemma LemmaUnitalityJoin<T>(w: W<T>) {}
+
+  lemma LemmaAssociativityJoin<T>(www: W<W<W<T>>>) {}
+
+  /**********************************************************
+  *
+  *  Wrapper
+  *
+  ***********************************************************/
+
+  /* Get and Fold */
+  function GetValueDefault<T>(w: W<T>, default: T): T {
+    match w
+    case None => default
+    case Some(v) => v
+  }
+
+  function GetValue<T>(w: W<T>): T
+    requires w.Some?
+  {
+    w.value
+  }
+
+  function Fold<T,U>(u: U, f: T -> U): W<T> -> U {
+    (w: W<T>) =>
+      match w 
+      case None => u
+      case Some(v) => f(v)
+  }
+
+  /* Conversion to other types */
+  // function ToResult<T>(e: E): Option<T> -> Result<T,E> {
+  //   (o: Option<T>) =>
+  //     match o 
+  //     case Some(v) => Success(v)
+  //     case None => Failure(e)
+  // }
+
+  function ToSeq<T>(w: W<T>): seq<T> {
+    match w 
+    case None => []
+    case Some(v) => [v]
+  }
+
+  function ToSet<T>(w: W<T>): set<T> {
+    match w
+    case None => {}
+    case Some(v) => {v}
+  }
+
+  /* Comparison */
+  function Compare<T>(cmp: (T, T) -> int): (W<T>, W<T>) -> int {
+    (w1: W<T>, w2: W<T>) =>
+      match (w1, w2)
+      case (Some(v1), Some(v2)) => cmp(v1, v2)
+      case (None, None) => 0
+      case (None, Some(_)) => -1
+      case (Some(_), None) => 1
+  }
+
+}
+
+module {:options "-functionSyntax:4"} Writer refines Monad {
+
+  /**********************************************************
+  *
+  *  Monad
+  *
+  ***********************************************************/
+ 
+  /* Monad structure */
+  datatype W<T> = Result(value: T, log: string)
+
+  function Map<S,T>(f: S -> T): W<S> -> W<T> {
+    (w: W<S>) => W<T>.Result(f(w.value), w.log)
+  }
+
+  function Equal<T>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+    (w1: W<T>, w2: W<T>) => eq(w1.value, w2.value) && w1.log == w2.log
+  }
+
+  function Return<T>(x: T): W<T> {
+    Result(x, "")
+  }
+
+  function Join<T>(ww: W<W<T>>): W<T> {
+    Result((ww.value).value, (ww.value).log + ww.log)
+  }
+
+  /* Functoriality of Map */
+  lemma LemmaMapFunction<S,T>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {}
+
+  lemma LemmaMapFunctorial<S,T,U>(f: S -> T, g: T -> U, w: W<S>) {}
+
+  lemma LemmaMapIdentity<T>(id: T -> T) {}
+
+  /* Naturality of Return and Join */
+  lemma LemmaReturnNaturality<S,T>(f: S -> T, x: S) {}
+
+  lemma LemmaJoinNaturality<S,T>(f: S -> T, ww: W<W<S>>) {}
+
+  /* Monad laws */
+  lemma LemmaUnitalityJoin<T>(w: W<T>) {}
+
+  lemma LemmaAssociativityJoin<T>(www: W<W<W<T>>>) {}
+
+  /**********************************************************
+  *
+  *  Wrapper
+  *
+  ***********************************************************/
+
+  /* Get and Set */
+  function GetValue<T>(w: W<T>): T {
+    w.value
+  }
+
+  function GetLog<T>(w: W<T>): string {
+    w.log
+  }
+
+  function Result<T>(v: T, s: string): W<T> {
+    W<T>.Result(v, s)
+  }
+}
+
+abstract module {:options "-functionSyntax:4"} Reader refines Monad {
+
+  /**********************************************************
+  *
+  *  Monad
+  *
+  ***********************************************************/
+
+  type A(!new)
+
+  datatype W<T> = Reader(f: A -> T)
+
+  function Map<S,T>(f: S -> T): W<S> -> W<T> {
+    (w: W<S>) => Reader(a => f((w.f)(a)))
+  }
+
+  ghost function Equal<T>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+    (w1: W<T>, w2: W<T>) =>
+      forall a: A :: eq(w1.f(a), w2.f(a))
+  }
+
+  function Return<T>(x: T): W<T> {
+    Reader(a => x)
+  }
+
+  function Join<T>(ww: W<W<T>>): W<T>{
+    Reader(a => (ww.f(a)).f(a))
+  }
+
+  /* Functoriality of Map */
+  lemma LemmaMapFunction<S,T>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {}
+
+  lemma LemmaMapFunctorial<S,T,U>(f: S -> T, g: T -> U, w: W<S>) {}
+
+  lemma LemmaMapIdentity<T>(id: T -> T) {}
+
+  /* Naturality of Return and Join */
+  lemma LemmaReturnNaturality<S,T>(f: S -> T, x: S) {}
+
+  lemma LemmaJoinNaturality<S,T>(f: S -> T, ww: W<W<S>>) {}
+
+  /* Monad laws */
+  lemma LemmaUnitalityJoin<T>(w: W<T>) {}
+
+  lemma LemmaAssociativityJoin<T>(www: W<W<W<T>>>) {}
+
+}
+
diff --git a/src/Wrappers.dll b/src/Wrappers.dll
new file mode 100644
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literal 0
HcmV?d00001

diff --git a/src/Wrappers.runtimeconfig.json b/src/Wrappers.runtimeconfig.json
new file mode 100644
index 00000000..c9fdac56
--- /dev/null
+++ b/src/Wrappers.runtimeconfig.json
@@ -0,0 +1,10 @@
+{
+  "runtimeOptions": {
+    "tfm": "net6.0",
+    "framework": {
+      "name": "Microsoft.NETCore.App",
+      "version": "6.0.0",
+      "rollForward": "LatestMinor"
+    }
+  }
+}
diff --git a/src/Writer.dfy b/src/Writer.dfy
index a5fab3ab..9d8230f7 100644
--- a/src/Writer.dfy
+++ b/src/Writer.dfy
@@ -1,19 +1,21 @@
 module {:options "-functionSyntax:4"} Writer {
 
-  datatype Writer<T> = Result(value: T, log: string)
-
-  function GetValue<T>(w: Writer<T>): T {
-    w.value
-  }
+  /**********************************************************
+  *
+  *  Monadic structure in terms of Return, Bind, Join, Map, Composition 
+  *
+  ***********************************************************/
 
-  function GetLog<T>(w: Writer<T>): string {
-    w.log
-  }
+  datatype Writer<T> = Result(value: T, log: string)
 
   function Return<T>(v: T): Writer<T> {
     Result(v, "")
   }
 
+  function Bind<S,T>(w: Writer<S>, f: S -> Writer<T>): Writer<T> {
+    Writer<T>.Result(f(w.value).value, w.log + f(w.value).log)
+  }
+
   function Join<T>(ww: Writer<Writer<T>>): Writer<T> {
     Result((ww.value).value, (ww.value).log + ww.log)
   }
@@ -22,14 +24,34 @@ module {:options "-functionSyntax:4"} Writer {
     (w: Writer<S>) => Writer<T>.Result(f(w.value), w.log)
   }
 
-  function Bind<S,T>(w: Writer<S>, f: S -> Writer<T>): Writer<T> {
-    Writer<T>.Result(f(w.value).value, w.log + f(w.value).log)
-  }
-
   function Composition<S,T,U>(f: S -> Writer<T>, g: T -> Writer<U>): S -> Writer<U> {
     s => Bind(f(s), g)
   }
 
+  /**********************************************************
+  *
+  *  Get and set
+  *
+  ***********************************************************/
+
+  function GetValue<T>(w: Writer<T>): T {
+    w.value
+  }
+
+  function GetLog<T>(w: Writer<T>): string {
+    w.log
+  }
+
+  function Result<T>(v: T, s: string): Writer<T> {
+    Writer<T>.Result(v, s)
+  }
+
+  /**********************************************************
+  *
+  *  Monad laws in terms of Join and Map
+  *
+  ***********************************************************/
+
   lemma LemmaUnitalityJoin<T>(w: Writer<T>)
     ensures Join(Map(Return<T>)(w)) == w == Join(Return<Writer<T>>(w))
   {
@@ -40,6 +62,12 @@ module {:options "-functionSyntax:4"} Writer {
   {
   }  
 
+  /**********************************************************
+  *
+  *  Monad laws in terms of Bind and Return
+  *
+  ***********************************************************/
+
   lemma LemmaLeftUnitalityBind<S,T>(v: S, f: S -> Writer<T>)
     ensures Bind(Return(v), f) == f(v)
   {
@@ -55,6 +83,12 @@ module {:options "-functionSyntax:4"} Writer {
   {
   }
 
+  /**********************************************************
+  *
+  *  Monad laws in terms of (Kleisli) Composition and Return
+  *
+  ***********************************************************/
+
   lemma LemmaAssociativityComposition<S,T,U,V>(f: S -> Writer<T>, g: T -> Writer<U>, h: U -> Writer<V>)
     ensures forall s: S :: Composition(Composition(f, g), h)(s) == Composition(f, Composition(g, h))(s)
   {

From 7c3f145168fc7abce4c07b7893c90eb2096c99d8 Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Thu, 2 Feb 2023 20:09:55 +0000
Subject: [PATCH 20/33] equality

---
 src/Relations.dfy |   6 ++
 src/Wrappers.dfy  | 185 +++++++++++++++++++++++-----------------------
 2 files changed, 99 insertions(+), 92 deletions(-)

diff --git a/src/Relations.dfy b/src/Relations.dfy
index ac25532f..9135426f 100644
--- a/src/Relations.dfy
+++ b/src/Relations.dfy
@@ -45,6 +45,12 @@ module {:options "-functionSyntax:4"} Relations {
     && Connected(R)
   }  
 
+  ghost predicate EquivalenceRelation<T(!new)>(R: (T, T) -> bool) {
+    && Reflexive(R)
+    && AntiSymmetric(R)
+    && Transitive(R)
+  }
+
   ghost predicate SortedBy<T>(a: seq<T>, lessThan: (T, T) -> bool) {
     forall i, j | 0 <= i < j < |a| :: lessThan(a[i], a[j])
   }
diff --git a/src/Wrappers.dfy b/src/Wrappers.dfy
index 6f82e4c2..31c94795 100644
--- a/src/Wrappers.dfy
+++ b/src/Wrappers.dfy
@@ -5,11 +5,13 @@
 *  SPDX-License-Identifier: MIT 
 *******************************************************************************/
 
-include "functions.dfy"
+include "Functions.dfy"
+include "Relations.dfy"
 
 abstract module {:options "-functionSyntax:4"} Functor {
 
   import opened Functions
+  import opened Relations
 
   /**********************************************************
   *
@@ -18,31 +20,28 @@ abstract module {:options "-functionSyntax:4"} Functor {
   ***********************************************************/
 
   /* Functor structure */
-  type W(!new)<T(!new)>
+  type W(!new)<T>
 
   function Map<S,T>(f: S -> T): W<S> -> W<T>
 
-  function Equal<T(!new)>(eq: (T, T) -> bool): (eq': (W<T>, W<T>) -> bool)
-    requires 
-      && forall x: T :: eq(x, x)
-      && forall x, y, z: T :: eq(x, y) && eq(y, z) ==> eq(x, z) 
-      && forall x, y: T :: eq(x, y) ==> eq(y, x)
-    ensures 
-      && forall w: W<T> :: eq'(w, w)
-      && forall u, v, w: W<T> ::  eq'(u, v) && eq'(v, w) ==> eq'(u, w) 
-      && forall v, w: W<T> :: eq'(u, v) ==> eq'(v, w)
+  ghost function Equal<T>(eq: (T, T) -> bool): (eq': (W<T>, W<T>) -> bool)
+    requires EquivalenceRelation(eq)
+    ensures EquivalenceRelation(eq')
 
   /* Functoriality of Map */
   lemma LemmaMapFunction<S,T>(eq: (T, T) -> bool, f: S -> T, g: S -> T)
+    requires EquivalenceRelation(eq) 
     requires forall x: S :: eq(f(x), g(x))
     ensures forall w: W<S> :: Equal(eq)(Map(f)(w), Map(g)(w))
 
-  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(f: S -> T, g: T -> U, w: W<S>)
-    ensures Map(Composition(f, g))(w) == Composition(Map(f), Map(g))(w)
+  lemma LemmaMapFunctorial<S,T,U>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>)
+    requires EquivalenceRelation(eq)
+    ensures Equal(eq)(Map(Composition(f, g))(w), Composition(Map(f), Map(g))(w))
 
-  lemma LemmaMapIdentity<T>(id: T -> T)
-    requires forall x: T :: id(x) == x
-    ensures forall w: W<T> :: Map(id)(w) == w
+  lemma LemmaMapIdentity<T>(eq: (T, T) -> bool, id: T -> T)
+    requires EquivalenceRelation(eq) 
+    requires forall x: T :: eq(id(x), x)
+    ensures forall w: W<T> :: Equal(eq)(Map(id)(w), w)
 
 }
 
@@ -55,25 +54,29 @@ abstract module {:options "-functionSyntax:4"} Monad refines Functor {
   ***********************************************************/
 
   /* Monad structure */
-  type W<T(!new)>
+  type W(!new)<T>
 
   function Return<T>(x: T): W<T>
 
   function Join<T>(ww: W<W<T>>): W<T>
 
   /* Naturality of Return and Join */
-  lemma LemmaReturnNaturality<S,T>(eq: (W<T>, W<T>) -> bool, f: S -> T, x: S)
-    ensures eq(Map(f)(Return(x)), Return(f(x)))
+  lemma LemmaReturnNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, x: S)
+    requires EquivalenceRelation(eq)
+    ensures Equal(eq)(Map(f)(Return(x)), Return(f(x)))
 
-  lemma LemmaJoinNaturality<S,T>(eq: (W<T>, W<T>) -> bool, f: S -> T, ww: W<W<S>>)
-    ensures eq(Join(Map(Map(f))(ww)), Map(f)(Join(ww)))
+  lemma LemmaJoinNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>)
+    requires EquivalenceRelation(eq)
+    ensures Equal(eq)(Join(Map(Map(f))(ww)), Map(f)(Join(ww)))
 
   /* Monad laws */
-  lemma LemmaUnitalityJoin<T>(eq: (W<T>, W<T>) -> bool, w: W<T>)
-    ensures eq(Join(Map(Return)(w)), w) && eq(w, Join(Return(w)))
+  lemma LemmaUnitalityJoin<T>(eq: (T, T) -> bool, w: W<T>)
+    requires EquivalenceRelation(eq)
+    ensures Equal(eq)(Join(Map(Return)(w)), w) && Equal(eq)(w, Join(Return(w)))
 
-  lemma LemmaAssociativityJoin<T>(eq: (W<T>, W<T>) -> bool, www: W<W<W<T>>>) 
-    ensures eq(Join(Map(Join)(www)), Join(Join(www)))
+  lemma LemmaAssociativityJoin<T>(eq: (T, T) -> bool, www: W<W<W<T>>>) 
+    requires EquivalenceRelation(eq)
+    ensures Equal(eq)(Join(Map(Join)(www)), Join(Join(www)))
 
   /**********************************************************
   *
@@ -91,43 +94,24 @@ abstract module {:options "-functionSyntax:4"} Monad refines Functor {
   }
 
   /* Derived lemmas */
-  lemma LemmaLeftUnitalityBind<S,T>(eq: (W<T>, W<T>) -> bool, x: S, f: S -> W<T>)
-    ensures eq(Bind(Return(x), f), f(x))
-  {
-    assert Equal(eq)(Map(f)(Return(x)), Return(f(x))) by { LemmaReturnNaturality<S,W<T>>(Equal(eq), f, x); }
-    assert eq(Join(Map(Return)(f(x))), f(x)) && eq(f(x), Join(Return(f(x)))) by { LemmaUnitalityJoin<T>(eq, f(x)); }
-
-    // assert Equal(eq)(Map(f)(Return(x)), Return(f(x))) by { LemmaReturnNaturality<S,T>(eq, f, x); }
-
-
-    // calc {
-    
-    //   Equal(eq)(Bind(Return(x), f), f(x));
-    // <==> // Definition of Bind
-    //   Equal(eq)(Join(Map(f)(Return(x))), f(x));
-    // <==> { LemmaReturnNaturality(eq, f, x); assert Equal(eq)(Map(f)(Return(x)), Return(f(x))); }
-    //   Equal(eq)(Join(Return(f(x))), f(x));
-    // <== { LemmaUnitalityJoin(eq, f(x)); }
-    //   f(x);
-    // }
-
-  }
-
-  lemma LemmaRightUnitalityBind<T>(w: W<T>)
+  lemma LemmaLeftUnitalityBind<S,T>(eq: (T, T) -> bool, x: S, f: S -> W<T>)
+    requires EquivalenceRelation(eq)
+    ensures Equal(eq)(Bind(Return(x), f), f(x))
+/*   {
+    LemmaReturnNaturality(Equal(eq), f, x);
+    LemmaUnitalityJoin(eq, f(x));
+  } */
+
+  lemma LemmaRightUnitalityBind<T>(eq: (T, T) -> bool, w: W<T>)
+    requires EquivalenceRelation(eq)
     ensures Bind(w, Return) == w
-  {
-    calc {
-      Bind(w, Return); 
-    == // Definition of Bind
-      Join(Map(Return)(w));
-    == { LemmaUnitalityJoin(w); }
-      w;
-    }
-  }
+/*   {
+    LemmaUnitalityJoin(eq, w);
+  } */
 
   lemma LemmaAssociativityBind<S,T,U>(w: W<S>, f: S -> W<T>, g: T -> W<U>)
     ensures Bind(Bind(w, f), g) == Bind(w, KleisliComp(f, g))
-  {
+/*   {
     calc {
       Bind(Bind(w, f), g);
     == // Definition of Bind
@@ -147,11 +131,11 @@ abstract module {:options "-functionSyntax:4"} Monad refines Functor {
     == // Definition of Bind, KleisliComp
       Bind(w, KleisliComp(f, g));
     }
-  }
+  } */
 
   lemma LemmaAssociativityComposition<S,T,U,V>(f: S -> W<T>, g: T -> W<U>, h: U -> W<V>, x: S)
     ensures KleisliComp(KleisliComp(f, g), h)(x) == KleisliComp(f, KleisliComp(g, h))(x)
-  {
+/*   {
     calc {
       KleisliComp(KleisliComp(f, g), h)(x);
     == // Definition of KleisliComp
@@ -171,11 +155,12 @@ abstract module {:options "-functionSyntax:4"} Monad refines Functor {
     == // Definition of KleisliComp
       KleisliComp(f, KleisliComp(g, h))(x);
     }
-  }
+  } */
 
-  lemma LemmaIdentityReturn<S,T>(f: S -> W<T>, x: S)
-    ensures KleisliComp(f, Return)(x) == f(x) == KleisliComp(Return, f)(x)
-  {
+  lemma LemmaIdentityReturn<S,T>(eq: (T, T) -> bool, f: S -> W<T>, x: S)
+    requires EquivalenceRelation(eq)
+    ensures Equal(eq)(KleisliComp(f, Return)(x), f(x)) &&  Equal(eq)(f(x), KleisliComp(Return, f)(x))
+/*   {
     calc {
       KleisliComp(f, Return)(x);
     == // Definition of Composition
@@ -192,7 +177,7 @@ abstract module {:options "-functionSyntax:4"} Monad refines Functor {
     == // Definition of Composition
       KleisliComp(Return, f)(x);
     }
-  }
+  } */
 }
 
 abstract module {:options "-functionSyntax:4"} Result refines Monad {
@@ -207,6 +192,9 @@ abstract module {:options "-functionSyntax:4"} Result refines Monad {
 
   function eqe(e1: E, e2: E): bool
 
+  lemma {:axiom} LemmaEquivRel()
+    ensures EquivalenceRelation(eqe)
+
   /* Monad structure */
 
   datatype W<T> = Success(value: T) | Failure(error: E)
@@ -218,7 +206,10 @@ abstract module {:options "-functionSyntax:4"} Result refines Monad {
       case Failure(e) => W<T>.Failure(e)
   }
 
-  function Equal<T>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+
+
+  ghost function Equal<T>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool
+  {
     (w1: W<T>, w2: W<T>) =>
       match (w1, w2)
       case (Success(v1), Success(v2)) => eq(v1, v2)
@@ -226,6 +217,16 @@ abstract module {:options "-functionSyntax:4"} Result refines Monad {
       case _ => false
   }
 
+  lemma LemmaEquivRelLift<T>(eq: (T, T) -> bool)
+    requires EquivalenceRelation(eq)
+    ensures EquivalenceRelation(Equal(eq))
+  {
+    assert Reflexive(Equal(eq)) by {
+      
+    }
+  }
+
+
   function Return<T>(x: T): W<T> {
     Success(x)
   }
@@ -239,19 +240,19 @@ abstract module {:options "-functionSyntax:4"} Result refines Monad {
   /* Functoriality of Map */
   lemma LemmaMapFunction<S,T>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {}
 
-  lemma LemmaMapFunctorial<S,T,U>(f: S -> T, g: T -> U, w: W<S>) {}
+  lemma LemmaMapFunctorial<S,T,U>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {}
 
-  lemma LemmaMapIdentity<T>(id: T -> T) {}
+  lemma LemmaMapIdentity<T>(eq: (T, T) -> bool, id: T -> T) {}
 
   /* Naturality of Return and Join */
-  lemma LemmaReturnNaturality<S,T>(f: S -> T, x: S) {}
+  lemma LemmaReturnNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, x: S) {}
 
-  lemma LemmaJoinNaturality<S,T>(f: S -> T, ww: W<W<S>>) {}
+  lemma LemmaJoinNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {}
 
   /* Monad laws */
-  lemma LemmaUnitalityJoin<T>(w: W<T>) {}
+  lemma LemmaUnitalityJoin<T>(eq: (T, T) -> bool, w: W<T>) {}
 
-  lemma LemmaAssociativityJoin<T>(www: W<W<W<T>>>) {}
+  lemma LemmaAssociativityJoin<T>(eq: (T, T) -> bool, www: W<W<W<T>>>) {}
 
   /**********************************************************
   *
@@ -338,7 +339,7 @@ module {:options "-functionSyntax:4"} Option refines Monad {
       case Some(v) => W<T>.Some(f(v))
   }
 
-  function Equal<T>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+  ghost function Equal<T>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
     (w1: W<T>, w2: W<T>) =>
       match (w1, w2)
       case (Some(v1), Some(v2)) => eq(v1, v2)
@@ -359,19 +360,19 @@ module {:options "-functionSyntax:4"} Option refines Monad {
   /* Functoriality of Map */
   lemma LemmaMapFunction<S,T>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {}
 
-  lemma LemmaMapFunctorial<S,T,U>(f: S -> T, g: T -> U, w: W<S>) {}
+  lemma LemmaMapFunctorial<S,T,U>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {}
 
-  lemma LemmaMapIdentity<T>(id: T -> T) {}
+  lemma LemmaMapIdentity<T>(eq: (T, T) -> bool, id: T -> T) {}
 
   /* Naturality of Return and Join */
-  lemma LemmaReturnNaturality<S,T>(f: S -> T, x: S) {}
+  lemma LemmaReturnNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, x: S) {}
 
-  lemma LemmaJoinNaturality<S,T>(f: S -> T, ww: W<W<S>>) {}
+  lemma LemmaJoinNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {}
 
   /* Monad laws */
-  lemma LemmaUnitalityJoin<T>(w: W<T>) {}
+  lemma LemmaUnitalityJoin<T>(eq: (T, T) -> bool, w: W<T>) {}
 
-  lemma LemmaAssociativityJoin<T>(www: W<W<W<T>>>) {}
+  lemma LemmaAssociativityJoin<T>(eq: (T, T) -> bool, www: W<W<W<T>>>) {}
 
   /**********************************************************
   *
@@ -446,7 +447,7 @@ module {:options "-functionSyntax:4"} Writer refines Monad {
     (w: W<S>) => W<T>.Result(f(w.value), w.log)
   }
 
-  function Equal<T>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+  ghost function Equal<T>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
     (w1: W<T>, w2: W<T>) => eq(w1.value, w2.value) && w1.log == w2.log
   }
 
@@ -461,19 +462,19 @@ module {:options "-functionSyntax:4"} Writer refines Monad {
   /* Functoriality of Map */
   lemma LemmaMapFunction<S,T>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {}
 
-  lemma LemmaMapFunctorial<S,T,U>(f: S -> T, g: T -> U, w: W<S>) {}
+  lemma LemmaMapFunctorial<S,T,U>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {}
 
-  lemma LemmaMapIdentity<T>(id: T -> T) {}
+  lemma LemmaMapIdentity<T>(eq: (T, T) -> bool, id: T -> T) {}
 
   /* Naturality of Return and Join */
-  lemma LemmaReturnNaturality<S,T>(f: S -> T, x: S) {}
+  lemma LemmaReturnNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, x: S) {}
 
-  lemma LemmaJoinNaturality<S,T>(f: S -> T, ww: W<W<S>>) {}
+  lemma LemmaJoinNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {}
 
   /* Monad laws */
-  lemma LemmaUnitalityJoin<T>(w: W<T>) {}
+  lemma LemmaUnitalityJoin<T>(eq: (T, T) -> bool, w: W<T>) {}
 
-  lemma LemmaAssociativityJoin<T>(www: W<W<W<T>>>) {}
+  lemma LemmaAssociativityJoin<T>(eq: (T, T) -> bool, www: W<W<W<T>>>) {}
 
   /**********************************************************
   *
@@ -527,19 +528,19 @@ abstract module {:options "-functionSyntax:4"} Reader refines Monad {
   /* Functoriality of Map */
   lemma LemmaMapFunction<S,T>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {}
 
-  lemma LemmaMapFunctorial<S,T,U>(f: S -> T, g: T -> U, w: W<S>) {}
+  lemma LemmaMapFunctorial<S,T,U>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {}
 
-  lemma LemmaMapIdentity<T>(id: T -> T) {}
+  lemma LemmaMapIdentity<T>(eq: (T, T) -> bool, id: T -> T) {}
 
   /* Naturality of Return and Join */
-  lemma LemmaReturnNaturality<S,T>(f: S -> T, x: S) {}
+  lemma LemmaReturnNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, x: S) {}
 
-  lemma LemmaJoinNaturality<S,T>(f: S -> T, ww: W<W<S>>) {}
+  lemma LemmaJoinNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {}
 
   /* Monad laws */
-  lemma LemmaUnitalityJoin<T>(w: W<T>) {}
+  lemma LemmaUnitalityJoin<T>(eq: (T, T) -> bool, w: W<T>) {}
 
-  lemma LemmaAssociativityJoin<T>(www: W<W<W<T>>>) {}
+  lemma LemmaAssociativityJoin<T>(eq: (T, T) -> bool, www: W<W<W<T>>>) {}
 
 }
 

From 073683fd1fcae4ad0f5a37252fe2ca6ea0003814 Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Thu, 2 Feb 2023 21:22:09 +0000
Subject: [PATCH 21/33] proof

---
 src/Wrappers.dfy | 188 +++++++++++++++++++++++++----------------------
 1 file changed, 99 insertions(+), 89 deletions(-)

diff --git a/src/Wrappers.dfy b/src/Wrappers.dfy
index 31c94795..c06dc9d8 100644
--- a/src/Wrappers.dfy
+++ b/src/Wrappers.dfy
@@ -15,18 +15,21 @@ abstract module {:options "-functionSyntax:4"} Functor {
 
   /**********************************************************
   *
-  *  Assumed structure 
+  *  Functor structure
   *
   ***********************************************************/
 
   /* Functor structure */
-  type W(!new)<T>
+  type W<T(!new)>
 
-  function Map<S,T>(f: S -> T): W<S> -> W<T>
+  function Map<S(!new),T>(f: S -> T): W<S> -> W<T>
 
-  ghost function Equal<T>(eq: (T, T) -> bool): (eq': (W<T>, W<T>) -> bool)
+  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool
     requires EquivalenceRelation(eq)
-    ensures EquivalenceRelation(eq')
+
+  lemma LemmaEquivRelLift<T>(eq: (T, T) -> bool)
+    requires EquivalenceRelation(eq)
+    ensures EquivalenceRelation(Equal(eq))
 
   /* Functoriality of Map */
   lemma LemmaMapFunction<S,T>(eq: (T, T) -> bool, f: S -> T, g: S -> T)
@@ -34,11 +37,11 @@ abstract module {:options "-functionSyntax:4"} Functor {
     requires forall x: S :: eq(f(x), g(x))
     ensures forall w: W<S> :: Equal(eq)(Map(f)(w), Map(g)(w))
 
-  lemma LemmaMapFunctorial<S,T,U>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>)
+  lemma LemmaMapFunctorial<S(!new),T,U>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>)
     requires EquivalenceRelation(eq)
     ensures Equal(eq)(Map(Composition(f, g))(w), Composition(Map(f), Map(g))(w))
 
-  lemma LemmaMapIdentity<T>(eq: (T, T) -> bool, id: T -> T)
+  lemma LemmaMapIdentity<T(!new)>(eq: (T, T) -> bool, id: T -> T)
     requires EquivalenceRelation(eq) 
     requires forall x: T :: eq(id(x), x)
     ensures forall w: W<T> :: Equal(eq)(Map(id)(w), w)
@@ -49,32 +52,32 @@ abstract module {:options "-functionSyntax:4"} Monad refines Functor {
 
   /**********************************************************
   *
-  *  Assumed structure 
+  *  Additional monad structure
   *
   ***********************************************************/
 
-  /* Monad structure */
-  type W(!new)<T>
+  /* Monad basics */
+  type W(!new)<T(!new)>
 
   function Return<T>(x: T): W<T>
 
-  function Join<T>(ww: W<W<T>>): W<T>
+  function Join<T(!new)>(ww: W<W<T>>): W<T>
 
   /* Naturality of Return and Join */
   lemma LemmaReturnNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, x: S)
     requires EquivalenceRelation(eq)
     ensures Equal(eq)(Map(f)(Return(x)), Return(f(x)))
 
-  lemma LemmaJoinNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>)
+  lemma LemmaJoinNaturality<S(!new),T>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>)
     requires EquivalenceRelation(eq)
     ensures Equal(eq)(Join(Map(Map(f))(ww)), Map(f)(Join(ww)))
 
   /* Monad laws */
-  lemma LemmaUnitalityJoin<T>(eq: (T, T) -> bool, w: W<T>)
+  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: W<T>)
     requires EquivalenceRelation(eq)
     ensures Equal(eq)(Join(Map(Return)(w)), w) && Equal(eq)(w, Join(Return(w)))
 
-  lemma LemmaAssociativityJoin<T>(eq: (T, T) -> bool, www: W<W<W<T>>>) 
+  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: W<W<W<T>>>) 
     requires EquivalenceRelation(eq)
     ensures Equal(eq)(Join(Map(Join)(www)), Join(Join(www)))
 
@@ -85,16 +88,16 @@ abstract module {:options "-functionSyntax:4"} Monad refines Functor {
   ***********************************************************/
 
   /* Derived functions */
-  function Bind<S,T>(w: W<S>, f: S -> W<T>): W<T> {
+  function Bind<S(!new),T(!new)>(w: W<S>, f: S -> W<T>): W<T> {
     Join(Map(f)(w))
   }
 
-  function KleisliComp<S,T,U>(f: S -> W<T>, g: T -> W<U>): S -> W<U> {
+  function KleisliComp<S,T(!new),U(!new)>(f: S -> W<T>, g: T -> W<U>): S -> W<U> {
     x => Join(Map(g)(f(x)))
   }
 
   /* Derived lemmas */
-  lemma LemmaLeftUnitalityBind<S,T>(eq: (T, T) -> bool, x: S, f: S -> W<T>)
+  lemma LemmaLeftUnitalityBind<S,T(!new)>(eq: (T, T) -> bool, x: S, f: S -> W<T>)
     requires EquivalenceRelation(eq)
     ensures Equal(eq)(Bind(Return(x), f), f(x))
 /*   {
@@ -102,14 +105,14 @@ abstract module {:options "-functionSyntax:4"} Monad refines Functor {
     LemmaUnitalityJoin(eq, f(x));
   } */
 
-  lemma LemmaRightUnitalityBind<T>(eq: (T, T) -> bool, w: W<T>)
+  lemma LemmaRightUnitalityBind<T(!new)>(eq: (T, T) -> bool, w: W<T>)
     requires EquivalenceRelation(eq)
     ensures Bind(w, Return) == w
 /*   {
     LemmaUnitalityJoin(eq, w);
   } */
 
-  lemma LemmaAssociativityBind<S,T,U>(w: W<S>, f: S -> W<T>, g: T -> W<U>)
+  lemma LemmaAssociativityBind<S(!new),T(!new),U(!new)>(w: W<S>, f: S -> W<T>, g: T -> W<U>)
     ensures Bind(Bind(w, f), g) == Bind(w, KleisliComp(f, g))
 /*   {
     calc {
@@ -133,7 +136,7 @@ abstract module {:options "-functionSyntax:4"} Monad refines Functor {
     }
   } */
 
-  lemma LemmaAssociativityComposition<S,T,U,V>(f: S -> W<T>, g: T -> W<U>, h: U -> W<V>, x: S)
+  lemma LemmaAssociativityComposition<S,T(!new),U(!new),V(!new)>(f: S -> W<T>, g: T -> W<U>, h: U -> W<V>, x: S)
     ensures KleisliComp(KleisliComp(f, g), h)(x) == KleisliComp(f, KleisliComp(g, h))(x)
 /*   {
     calc {
@@ -157,7 +160,7 @@ abstract module {:options "-functionSyntax:4"} Monad refines Functor {
     }
   } */
 
-  lemma LemmaIdentityReturn<S,T>(eq: (T, T) -> bool, f: S -> W<T>, x: S)
+  lemma LemmaIdentityReturn<S,T(!new)>(eq: (T, T) -> bool, f: S -> W<T>, x: S)
     requires EquivalenceRelation(eq)
     ensures Equal(eq)(KleisliComp(f, Return)(x), f(x)) &&  Equal(eq)(f(x), KleisliComp(Return, f)(x))
 /*   {
@@ -184,11 +187,11 @@ abstract module {:options "-functionSyntax:4"} Result refines Monad {
 
   /**********************************************************
   *
-  *  Monad
+  *  Proves monad refinement
   *
   ***********************************************************/
   
-  type E
+  type E(!new)
 
   function eqe(e1: E, e2: E): bool
 
@@ -197,41 +200,32 @@ abstract module {:options "-functionSyntax:4"} Result refines Monad {
 
   /* Monad structure */
 
-  datatype W<T> = Success(value: T) | Failure(error: E)
+  datatype W<T(!new)> = Success(value: T) | Failure(error: E)
 
-  function Map<S,T>(f: S -> T): W<S> -> W<T> {
+  function Map<S(!new),T>(f: S -> T): W<S> -> W<T> {
     (w: W<S>) =>
       match w 
       case Success(v) => W<T>.Success(f(v))
       case Failure(e) => W<T>.Failure(e)
   }
 
-
-
-  ghost function Equal<T>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool
-  {
+  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool
+  { 
     (w1: W<T>, w2: W<T>) =>
       match (w1, w2)
-      case (Success(v1), Success(v2)) => eq(v1, v2)
+      case (Success(v1), Success(v2)) => eq(v1, v2) 
       case (Failure(e1), Failure(e2)) => eqe(e1, e2)
       case _ => false
   }
 
   lemma LemmaEquivRelLift<T>(eq: (T, T) -> bool)
-    requires EquivalenceRelation(eq)
     ensures EquivalenceRelation(Equal(eq))
-  {
-    assert Reflexive(Equal(eq)) by {
-      
-    }
-  }
-
 
   function Return<T>(x: T): W<T> {
     Success(x)
   }
  
-  function Join<T>(ww: W<W<T>>): W<T> {
+  function Join<T(!new)>(ww: W<W<T>>): W<T> {
     match ww
     case Success(v) => v
     case Failure(e) => W<T>.Failure(e)
@@ -240,19 +234,25 @@ abstract module {:options "-functionSyntax:4"} Result refines Monad {
   /* Functoriality of Map */
   lemma LemmaMapFunction<S,T>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {}
 
-  lemma LemmaMapFunctorial<S,T,U>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {}
+  lemma LemmaMapFunctorial<S(!new),T,U>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {}
 
-  lemma LemmaMapIdentity<T>(eq: (T, T) -> bool, id: T -> T) {}
+  lemma LemmaMapIdentity<T(!new)>(eq: (T, T) -> bool, id: T -> T) {
+    forall w: W<T> ensures Equal(eq)(Map(id)(w), w) {
+      match w
+        case Success(v) => 
+        case Failure(e) => LemmaEquivRel();
+    }
+  }
 
   /* Naturality of Return and Join */
   lemma LemmaReturnNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, x: S) {}
 
-  lemma LemmaJoinNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {}
+  lemma LemmaJoinNaturality<S(!new),T>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {}
 
   /* Monad laws */
-  lemma LemmaUnitalityJoin<T>(eq: (T, T) -> bool, w: W<T>) {}
+  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: W<T>) {}
 
-  lemma LemmaAssociativityJoin<T>(eq: (T, T) -> bool, www: W<W<W<T>>>) {}
+  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: W<W<W<T>>>) {}
 
   /**********************************************************
   *
@@ -265,34 +265,34 @@ abstract module {:options "-functionSyntax:4"} Result refines Monad {
     W.Failure(e)
   } 
 
-  predicate IsSuccess<T>(w: W<T>) {
+  predicate IsSuccess<T(!new)>(w: W<T>) {
     w.Success?
   }  
 
-  predicate IsFailure<T>(w: W<T>) {
+  predicate IsFailure<T(!new)>(w: W<T>) {
     w.Failure?
   }  
 
-  function GetValue<T>(w: W<T>): T
+  function GetValue<T(!new)>(w: W<T>): T
     requires w.Success?
   {
     w.value
   }
 
-  function GetValueDefault<T,E>(w: W<T>, default: T): T {
+  function GetValueDefault<T(!new),E>(w: W<T>, default: T): T {
     match w 
     case Success(v) => v
     case Failure(_) => default
   }
 
-  function GetError<T>(w: W<T>): E 
+  function GetError<T(!new)>(w: W<T>): E 
     requires w.Failure?
   {
     w.error
   }
 
   /* Fold */
-  function Fold<T,U>(f: T -> U, g: E -> U): W<T> -> U {
+  function Fold<T(!new),U>(f: T -> U, g: E -> U): W<T> -> U {
     (w: W<T>) =>
       match w 
       case Success(v) => f(v)
@@ -300,13 +300,13 @@ abstract module {:options "-functionSyntax:4"} Result refines Monad {
   }
 
   /* Conversion to other types */
-  function ToSeq<T>(w: W<T>): seq<T> {
+  function ToSeq<T(!new)>(w: W<T>): seq<T> {
     match w
     case Success(v) => [v]
     case Failure(_) => []
   }
 
-  function ToSet<T>(w: W<T>): set<T> {
+  function ToSet<T(!new)>(w: W<T>): set<T> {
     match w
     case Success(v) => {v}
     case Failure(_) => {}   
@@ -324,14 +324,14 @@ module {:options "-functionSyntax:4"} Option refines Monad {
 
   /**********************************************************
   *
-  *  Monad
+  *  Proves monad refinement
   *
   ***********************************************************/
  
   /* Monad structure */
-  datatype W<T> = None | Some(value: T)
+  datatype W<T(!new)> = None | Some(value: T)
 
-  function Map<S,T>(f: S -> T): W<S> -> W<T>
+  function Map<S(!new),T>(f: S -> T): W<S> -> W<T>
   {
     (w: W<S>) =>
       match w 
@@ -339,7 +339,7 @@ module {:options "-functionSyntax:4"} Option refines Monad {
       case Some(v) => W<T>.Some(f(v))
   }
 
-  ghost function Equal<T>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
     (w1: W<T>, w2: W<T>) =>
       match (w1, w2)
       case (Some(v1), Some(v2)) => eq(v1, v2)
@@ -347,11 +347,14 @@ module {:options "-functionSyntax:4"} Option refines Monad {
       case _ => false
   }
 
+  lemma LemmaEquivRelLift<T>(eq: (T, T) -> bool)
+    ensures EquivalenceRelation(Equal(eq))
+
   function Return<T>(x: T): W<T> {
     Some(x)
   }
 
-  function Join<T>(ww: W<W<T>>): W<T> {
+  function Join<T(!new)>(ww: W<W<T>>): W<T> {
     match ww
     case None => None
     case Some(w) => w
@@ -360,19 +363,19 @@ module {:options "-functionSyntax:4"} Option refines Monad {
   /* Functoriality of Map */
   lemma LemmaMapFunction<S,T>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {}
 
-  lemma LemmaMapFunctorial<S,T,U>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {}
+  lemma LemmaMapFunctorial<S(!new),T,U>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {}
 
-  lemma LemmaMapIdentity<T>(eq: (T, T) -> bool, id: T -> T) {}
+  lemma LemmaMapIdentity<T(!new)>(eq: (T, T) -> bool, id: T -> T) {}
 
   /* Naturality of Return and Join */
   lemma LemmaReturnNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, x: S) {}
 
-  lemma LemmaJoinNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {}
+  lemma LemmaJoinNaturality<S(!new),T>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {}
 
   /* Monad laws */
-  lemma LemmaUnitalityJoin<T>(eq: (T, T) -> bool, w: W<T>) {}
+  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: W<T>) {}
 
-  lemma LemmaAssociativityJoin<T>(eq: (T, T) -> bool, www: W<W<W<T>>>) {}
+  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: W<W<W<T>>>) {}
 
   /**********************************************************
   *
@@ -381,19 +384,19 @@ module {:options "-functionSyntax:4"} Option refines Monad {
   ***********************************************************/
 
   /* Get and Fold */
-  function GetValueDefault<T>(w: W<T>, default: T): T {
+  function GetValueDefault<T(!new)>(w: W<T>, default: T): T {
     match w
     case None => default
     case Some(v) => v
   }
 
-  function GetValue<T>(w: W<T>): T
+  function GetValue<T(!new)>(w: W<T>): T
     requires w.Some?
   {
     w.value
   }
 
-  function Fold<T,U>(u: U, f: T -> U): W<T> -> U {
+  function Fold<T(!new),U>(u: U, f: T -> U): W<T> -> U {
     (w: W<T>) =>
       match w 
       case None => u
@@ -408,20 +411,20 @@ module {:options "-functionSyntax:4"} Option refines Monad {
   //     case None => Failure(e)
   // }
 
-  function ToSeq<T>(w: W<T>): seq<T> {
+  function ToSeq<T(!new)>(w: W<T>): seq<T> {
     match w 
     case None => []
     case Some(v) => [v]
   }
 
-  function ToSet<T>(w: W<T>): set<T> {
+  function ToSet<T(!new)>(w: W<T>): set<T> {
     match w
     case None => {}
     case Some(v) => {v}
   }
 
   /* Comparison */
-  function Compare<T>(cmp: (T, T) -> int): (W<T>, W<T>) -> int {
+  function Compare<T(!new)>(cmp: (T, T) -> int): (W<T>, W<T>) -> int {
     (w1: W<T>, w2: W<T>) =>
       match (w1, w2)
       case (Some(v1), Some(v2)) => cmp(v1, v2)
@@ -436,45 +439,48 @@ module {:options "-functionSyntax:4"} Writer refines Monad {
 
   /**********************************************************
   *
-  *  Monad
+  *  Proves monad refinement
   *
   ***********************************************************/
  
   /* Monad structure */
-  datatype W<T> = Result(value: T, log: string)
+  datatype W<T(!new)> = Result(value: T, log: string)
 
-  function Map<S,T>(f: S -> T): W<S> -> W<T> {
+  function Map<S(!new),T>(f: S -> T): W<S> -> W<T> {
     (w: W<S>) => W<T>.Result(f(w.value), w.log)
   }
 
-  ghost function Equal<T>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
     (w1: W<T>, w2: W<T>) => eq(w1.value, w2.value) && w1.log == w2.log
   }
 
+  lemma LemmaEquivRelLift<T>(eq: (T, T) -> bool)
+    ensures EquivalenceRelation(Equal(eq))
+
   function Return<T>(x: T): W<T> {
     Result(x, "")
   }
 
-  function Join<T>(ww: W<W<T>>): W<T> {
+  function Join<T(!new)>(ww: W<W<T>>): W<T> {
     Result((ww.value).value, (ww.value).log + ww.log)
   }
 
   /* Functoriality of Map */
   lemma LemmaMapFunction<S,T>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {}
 
-  lemma LemmaMapFunctorial<S,T,U>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {}
+  lemma LemmaMapFunctorial<S(!new),T,U>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {}
 
-  lemma LemmaMapIdentity<T>(eq: (T, T) -> bool, id: T -> T) {}
+  lemma LemmaMapIdentity<T(!new)>(eq: (T, T) -> bool, id: T -> T) {}
 
   /* Naturality of Return and Join */
   lemma LemmaReturnNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, x: S) {}
 
-  lemma LemmaJoinNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {}
+  lemma LemmaJoinNaturality<S(!new),T>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {}
 
   /* Monad laws */
-  lemma LemmaUnitalityJoin<T>(eq: (T, T) -> bool, w: W<T>) {}
+  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: W<T>) {}
 
-  lemma LemmaAssociativityJoin<T>(eq: (T, T) -> bool, www: W<W<W<T>>>) {}
+  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: W<W<W<T>>>) {}
 
   /**********************************************************
   *
@@ -483,11 +489,11 @@ module {:options "-functionSyntax:4"} Writer refines Monad {
   ***********************************************************/
 
   /* Get and Set */
-  function GetValue<T>(w: W<T>): T {
+  function GetValue<T(!new)>(w: W<T>): T {
     w.value
   }
 
-  function GetLog<T>(w: W<T>): string {
+  function GetLog<T(!new)>(w: W<T>): string {
     w.log
   }
 
@@ -506,41 +512,45 @@ abstract module {:options "-functionSyntax:4"} Reader refines Monad {
 
   type A(!new)
 
-  datatype W<T> = Reader(f: A -> T)
+  datatype W<T(!new)> = Reader(f: A -> T)
 
-  function Map<S,T>(f: S -> T): W<S> -> W<T> {
+  function Map<S(!new),T>(f: S -> T): W<S> -> W<T> {
     (w: W<S>) => Reader(a => f((w.f)(a)))
   }
-
-  ghost function Equal<T>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+  
+  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool
+  {
     (w1: W<T>, w2: W<T>) =>
       forall a: A :: eq(w1.f(a), w2.f(a))
   }
 
+  lemma LemmaEquivRelLift<T>(eq: (T, T) -> bool)
+    ensures EquivalenceRelation(Equal(eq))
+  
   function Return<T>(x: T): W<T> {
     Reader(a => x)
   }
 
-  function Join<T>(ww: W<W<T>>): W<T>{
+  function Join<T(!new)>(ww: W<W<T>>): W<T>{
     Reader(a => (ww.f(a)).f(a))
   }
 
   /* Functoriality of Map */
   lemma LemmaMapFunction<S,T>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {}
 
-  lemma LemmaMapFunctorial<S,T,U>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {}
+  lemma LemmaMapFunctorial<S(!new),T,U>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {}
 
-  lemma LemmaMapIdentity<T>(eq: (T, T) -> bool, id: T -> T) {}
+  lemma LemmaMapIdentity<T(!new)>(eq: (T, T) -> bool, id: T -> T) {}
 
   /* Naturality of Return and Join */
   lemma LemmaReturnNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, x: S) {}
 
-  lemma LemmaJoinNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {}
+  lemma LemmaJoinNaturality<S(!new),T>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {}
 
   /* Monad laws */
-  lemma LemmaUnitalityJoin<T>(eq: (T, T) -> bool, w: W<T>) {}
+  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: W<T>) {}
 
-  lemma LemmaAssociativityJoin<T>(eq: (T, T) -> bool, www: W<W<W<T>>>) {}
+  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: W<W<W<T>>>) {}
 
 }
 

From d8b0a2547af0b56d66294935a9538ac065da29f2 Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Fri, 3 Feb 2023 16:37:34 +0000
Subject: [PATCH 22/33] structure

---
 .DS_Store                         | Bin 8196 -> 8196 bytes
 src/Comonad.dfy                   | 166 ++++++++
 src/Coreader.dfy                  |  62 ---
 src/Cowrappers.dfy                |  52 +++
 src/Cowrappers.dll                | Bin 0 -> 140800 bytes
 src/Cowrappers.runtimeconfig.json |  10 +
 src/Either.dfy                    | 120 ------
 src/Functor.dfy                   |  42 ++
 src/Functor.dll                   | Bin 0 -> 136704 bytes
 src/Functor.runtimeconfig.json    |  10 +
 src/Monad.dfy                     | 435 +++++++++++++++++++
 src/Monad.dll                     | Bin 0 -> 235008 bytes
 src/Monad.runtimeconfig.json      |  10 +
 src/Reader.dfy                    |  98 -----
 src/Relations.dfy                 |   6 +-
 src/Result.dfy                    | 204 ---------
 src/Wrappers.dfy                  | 684 +++++++++++-------------------
 src/Wrappers.dll                  | Bin 154624 -> 202240 bytes
 src/Writer.dfy                    | 101 -----
 19 files changed, 984 insertions(+), 1016 deletions(-)
 create mode 100644 src/Comonad.dfy
 delete mode 100644 src/Coreader.dfy
 create mode 100644 src/Cowrappers.dfy
 create mode 100644 src/Cowrappers.dll
 create mode 100644 src/Cowrappers.runtimeconfig.json
 delete mode 100644 src/Either.dfy
 create mode 100644 src/Functor.dfy
 create mode 100644 src/Functor.dll
 create mode 100644 src/Functor.runtimeconfig.json
 create mode 100644 src/Monad.dfy
 create mode 100644 src/Monad.dll
 create mode 100644 src/Monad.runtimeconfig.json
 delete mode 100644 src/Reader.dfy
 delete mode 100644 src/Result.dfy
 delete mode 100644 src/Writer.dfy

diff --git a/.DS_Store b/.DS_Store
index f8de3e84e630e08a3d1d98381967a686ffef9ec3..6e68ebe8ddfe7bde42ab96595c4798ff8db1ae63 100644
GIT binary patch
delta 67
zcmV-J0KET%K!iY$VFS2gaItZg2a$jcvj+$o2m!d0A`o#47<+p$GcYV5H8MGqpAZ`X
Zij%t#9|4QA+Yl@Vvj!OV1hf7W2Ltqw6|DdO

delta 68
zcmV-K0K5N$K!iY$VFS5haItZg2a$jcvj+$o2m!g1A`o#58GCyzH#00CF*!0alb#S8
a0gIEm5FY`Iv)T|W2eStl_5`#16bAza@fF_y

diff --git a/src/Comonad.dfy b/src/Comonad.dfy
new file mode 100644
index 00000000..50967947
--- /dev/null
+++ b/src/Comonad.dfy
@@ -0,0 +1,166 @@
+include "Functor.dfy"
+include "Cowrappers.dfy"
+
+abstract module {:options "-functionSyntax:4"} Comonad refines Functor {
+  /* Structure  */
+  type W(!new)<T(!new)>
+
+  function Extract<T(!new)>(w: W<T>): T
+
+  function Duplicate<T(!new)>(w: W<T>): W<W<T>>
+
+  /* Naturality of Extract and Duplicate */
+  lemma LemmaExtractNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, w: W<S>)
+    requires EquivalenceRelation(eq)
+    ensures eq(f(Extract(w)), Extract(Map(f)(w)))
+
+  lemma LemmaDuplicateNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, w: W<S>)
+    requires EquivalenceRelation(eq)
+    ensures EquivalenceRelation(Equal(eq))
+    ensures Equal(Equal(eq))(Duplicate(Map(f)(w)), Map(Map(f))(Duplicate(w)))
+
+  /* Comonad laws */
+  lemma LemmaCoUnitalityExtract<T(!new)>(eq: (T, T) -> bool, w: W<T>)
+    requires EquivalenceRelation(eq)
+    ensures && Equal(eq)(Map(Extract)(Duplicate(w)), w)
+            && Equal(eq)(w, Extract(Duplicate(w)))
+ 
+  lemma LemmaCoAssociativityDuplicate<T(!new)>(eq: (T, T) -> bool, w: W<T>) 
+    requires && EquivalenceRelation(eq)
+             && EquivalenceRelation(Equal(eq))
+             && EquivalenceRelation(Equal(Equal(eq))) 
+    ensures Equal(Equal(Equal(eq)))(Map(Duplicate)(Duplicate(w)), Duplicate(Duplicate(w)))
+}
+
+abstract module {:options "-functionSyntax:4"} CoreaderRightComonad refines Comonad {
+  import Coreader
+
+  type X(!new)
+
+  function eql(x: X, y: X): bool
+
+  lemma LemmaEquiv()
+    ensures EquivalenceRelation(eql)
+
+  /* Functor structure */
+  type W(!new)<T(!new)> = Coreader.Coreader<X,T>
+
+  function Map<S(!new),T(!new)>(f: S -> T): W<S> -> W<T> {
+    Coreader.MapRight(f)
+  }
+
+  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+    Coreader.Equal(eql, eq)
+  }  
+
+  /* Properties of Equal */
+  lemma LemmaEquivRelLift<T(!new)>(eq: (T, T) -> bool) {
+    LemmaEquiv();
+  }
+
+  /* Properties of Map */
+  lemma LemmaMapFunction<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {
+    LemmaEquiv();
+  }
+
+  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {
+    LemmaEquiv();
+  }
+
+  lemma LemmaMapIdentity<T(!new)>(eq: (T, T) -> bool, id: T -> T) {
+    LemmaEquiv();
+  }
+
+  /* Comonad structure */
+  function Extract<T(!new)>(w: W<T>): T {
+    Coreader.ExtractRight(w)
+  } 
+
+  function Duplicate<T(!new)>(w: W<T>): W<W<T>> {
+    Coreader.DuplicateLeft(w)
+  }
+
+  /* Naturality of Extract and Duplicate */
+  lemma LemmaExtractNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, w: W<S>) {
+  }
+
+  lemma LemmaDuplicateNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, w: W<S>) {
+    LemmaEquiv();
+  }
+
+  /* Comonad laws */
+  lemma LemmaCoUnitalityExtract<T(!new)>(eq: (T, T) -> bool, w: W<T>) {
+    LemmaEquiv();
+  }
+ 
+  lemma LemmaCoAssociativityDuplicate<T(!new)>(eq: (T, T) -> bool, w: W<T>) {
+    LemmaEquiv();
+  }
+}
+
+abstract module {:options "-functionSyntax:4"} CoreaderLeftComonad refines Comonad {
+  import Coreader
+
+  type X(!new)
+
+  function eqr(x: X, y: X): bool
+
+  lemma LemmaEquiv()
+    ensures EquivalenceRelation(eqr)
+
+  /* Functor structure */
+  type W(!new)<T(!new)> = Coreader.Coreader<T,X>
+
+  function Map<S(!new),T(!new)>(f: S -> T): W<S> -> W<T> {
+    Coreader.MapLeft(f)
+  }
+
+  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+    Coreader.Equal(eq, eqr)
+  }  
+
+  /* Properties of Equal */
+  lemma LemmaEquivRelLift<T(!new)>(eq: (T, T) -> bool) {
+    LemmaEquiv();
+  }
+
+  /* Properties of Map */
+  lemma LemmaMapFunction<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {
+    LemmaEquiv();
+  }
+
+  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {
+    LemmaEquiv();
+  }
+
+  lemma LemmaMapIdentity<T(!new)>(eq: (T, T) -> bool, id: T -> T) {
+    LemmaEquiv();
+  }
+
+  /* Comonad structure */
+  function Extract<T(!new)>(w: W<T>): T {
+    Coreader.ExtractLeft(w)
+  } 
+
+  function Duplicate<T(!new)>(w: W<T>): W<W<T>> {
+    Coreader.DuplicateRight(w)
+  }
+
+  /* Naturality of Extract and Duplicate */
+  lemma LemmaExtractNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, w: W<S>) {
+  }
+
+  lemma LemmaDuplicateNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, w: W<S>) {
+    LemmaEquiv();
+  }
+
+  /* Comonad laws */
+  lemma LemmaCoUnitalityExtract<T(!new)>(eq: (T, T) -> bool, w: W<T>) {
+    LemmaEquiv();
+  }
+ 
+  lemma LemmaCoAssociativityDuplicate<T(!new)>(eq: (T, T) -> bool, w: W<T>) {
+    LemmaEquiv();
+  }
+
+}
\ No newline at end of file
diff --git a/src/Coreader.dfy b/src/Coreader.dfy
deleted file mode 100644
index 6ea29357..00000000
--- a/src/Coreader.dfy
+++ /dev/null
@@ -1,62 +0,0 @@
-include "Wrappers.dfy"
-
-module {:options "-functionSyntax:4"} Coreader {
-
-  datatype Coreader<+X,+T> = Coreader(left: X, right: T)
-
-  function Extract<X,T>(r: Coreader<X,T>): T {
-    r.right
-  }
-
-  function Duplicate<X,T>(r: Coreader<X,T>): Coreader<X,Coreader<X,T>> {
-    Coreader(r.left, r)
-  }
-
-  function Extend<X,S,T>(f: Coreader<X,S> -> T, r: Coreader<X,S>): Coreader<X,T> {
-    Coreader(r.left, f(r))
-  }
-
-  function Map<X,S,T>(f: S -> T): Coreader<X,S> -> Coreader<X,T> {
-    (r: Coreader<X,S>) => 
-      Coreader(r.left, f(r.right))
-  }
-
-  function Composition<X,S,T,U>(f: Coreader<X,S> -> T, g: Coreader<X,T> -> U): Coreader<X,S> -> U {
-    r => g(Extend(f, r))
-  }
-
-  lemma LemmaUnitalityExtract<X,T>(r: Coreader<X,T>)
-    ensures Map(Extract)(Duplicate(r)) == r == Extract<X,Coreader<X,T>>(Duplicate(r))
-  {
-  }
-
-  lemma LemmaAssociativityDuplicate<X,T>(r: Coreader<X,T>) 
-    ensures Map(Duplicate)(Duplicate(r)) == Duplicate<X,Coreader<X,T>>(Duplicate(r))
-  {
-  }  
-
-  lemma LemmaLeftUnitalityExtend<X,S,T>(r: Coreader<X,T>)
-    ensures Extend(Extract, r) == r
-  {
-  }
-
-  lemma LemmaRightUnitalityExtend<X,S,T>(f: Coreader<X,S> -> T, r: Coreader<X,S>)
-    ensures Extract(Extend(f, r)) == f(r)
-  {
-  }
-
-  lemma LemmaAssociativityExtend<X,S,T,U>(r: Coreader<X,S>, f: Coreader<X,S> -> T, g: Coreader<X,T> -> U)
-    ensures Extend(g, Extend(f, r)) == Extend(Composition(f, g), r)
-  {
-  }
-
-  lemma LemmaAssociativityComposition<X,S,T,U,V>(f: Coreader<X,S> -> T, g: Coreader<X,T> -> U, h: Coreader<X,U> -> V)
-    ensures forall r: Coreader<X,S> :: Composition(Composition(f, g),h)(r) == Composition(f, Composition(g,h))(r)
-  {
-  }
-
-  lemma LemmaIdentityExtract<X,S,T>(f: Coreader<X,S> -> T)
-    ensures forall r: Coreader<X,S> :: Composition(f, Extract<X,T>)(r) == f(r) == Composition(Extract<X,S>, f)(r)
-  {
-  }
-}
\ No newline at end of file
diff --git a/src/Cowrappers.dfy b/src/Cowrappers.dfy
new file mode 100644
index 00000000..effcfd66
--- /dev/null
+++ b/src/Cowrappers.dfy
@@ -0,0 +1,52 @@
+module {:options "-functionSyntax:4"} Coreader {
+
+  datatype Coreader<+S,+T> = Coreader(left: S, right: T)
+
+  function ExtractRight<S,T>(r: Coreader<S,T>): T {
+    r.right
+  }
+
+  function ExtractLeft<S,T>(r: Coreader<S,T>): S {
+    r.left
+  }
+
+  function DuplicateLeft<S,T>(r: Coreader<S,T>): Coreader<S,Coreader<S,T>> {
+    Coreader(r.left, r)
+  }
+
+  function DuplicateRight<S,T>(r: Coreader<S,T>): Coreader<Coreader<S,T>,T> {
+    Coreader(r, r.right)
+  }
+
+  function ExtendRight<S,T1,T2>(f: Coreader<S,T1> -> T2, r: Coreader<S,T1>): Coreader<S,T2> {
+    Coreader(r.left, f(r))
+  }
+
+  function ExtendLeft<S1,S2,T>(f: Coreader<S1,T> -> S2, r: Coreader<S1,T>): Coreader<S2,T> {
+    Coreader(f(r), r.right)
+  }
+
+  function MapRight<S,T1,T2>(f: T1 -> T2): Coreader<S,T1> -> Coreader<S,T2> {
+    (r: Coreader<S,T1>) => 
+      Coreader(r.left, f(r.right))
+  }
+
+  function MapLeft<S1,S2,T>(f: S1 -> S2): Coreader<S1,T> -> Coreader<S2,T> {
+    (r: Coreader<S1,T>) => 
+      Coreader(f(r.left), r.right)
+  }
+
+  function KleisliCompositionRight<S,T1,T2,T3>(f: Coreader<S,T1> -> T2, g: Coreader<S,T2> -> T3): Coreader<S,T1> -> T3 {
+    r => g(ExtendRight(f, r))
+  }
+
+  function KleisliCompositionLeft<S1,S2,S3,T>(f: Coreader<S1,T> -> S2, g: Coreader<S2,T> -> S3): Coreader<S1,T> -> S3 {
+    r => g(ExtendLeft(f, r))
+  }
+
+  ghost function Equal<S,T>(eql: (S, S) -> bool, eqr: (T, T) -> bool): (Coreader<S,T>, Coreader<S,T>) -> bool {
+    (r1: Coreader<S,T>, r2: Coreader<S,T>) =>
+      eql(r1.left, r2.left) && eqr(r1.right, r2.right)
+  }
+
+}
\ No newline at end of file
diff --git a/src/Cowrappers.dll b/src/Cowrappers.dll
new file mode 100644
index 0000000000000000000000000000000000000000..0edc23f5dea7207bff5b4088963d669d9ba84302
GIT binary patch
literal 140800
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diff --git a/src/Cowrappers.runtimeconfig.json b/src/Cowrappers.runtimeconfig.json
new file mode 100644
index 00000000..c9fdac56
--- /dev/null
+++ b/src/Cowrappers.runtimeconfig.json
@@ -0,0 +1,10 @@
+{
+  "runtimeOptions": {
+    "tfm": "net6.0",
+    "framework": {
+      "name": "Microsoft.NETCore.App",
+      "version": "6.0.0",
+      "rollForward": "LatestMinor"
+    }
+  }
+}
diff --git a/src/Either.dfy b/src/Either.dfy
deleted file mode 100644
index a04ce99f..00000000
--- a/src/Either.dfy
+++ /dev/null
@@ -1,120 +0,0 @@
-// RUN: %dafny /compile:0 "%s"
-
-/*******************************************************************************
-*  Copyright by the contributors to the Dafny Project
-*  SPDX-License-Identifier: MIT 
-*******************************************************************************/
-
-include "Wrappers.dfy"
-
-module {:options "-functionSyntax:4"} Either {
-
-  import opened Wrappers 
-
-  datatype Either<+S,+T> = Left(left: S) | Right(right: T) 
-
-  /**********************************************************
-  *
-  *  Get and set
-  *
-  ***********************************************************/
-
-  function Left<S,T>(v: S): Either<S,T> {
-    Either.Left(v)
-  }
-
-  function Right<S,T>(v: T): Either<S,T> {
-    Either.Right(v)
-  }
-
-  predicate IsLeft<S,T>(e: Either<S,T>) {
-    e.Left?
-  }
-
-  predicate IsRight<S,T>(e: Either<S,T>) {
-    e.Right?
-  }
-
-  /**********************************************************
-  *
-  *  Conversion to other datatypes
-  *
-  ***********************************************************/
-
-  function FindLeft<S,T>(e: Either<S,T>): Option<S> {
-    match e 
-    case Left(v) => Some(v)
-    case Right(v) => None
-  }
-
-  function FindRight<S,T>(e: Either<S,T>): Option<T> {
-    match e 
-    case Left(v) => None
-    case Right(v) => Some(v)
-  }
-
-  /**********************************************************
-  *
-  *  Map
-  *
-  ***********************************************************/
-
-  function MapLeft<S,T,U>(f: S -> T): Either<S,U> -> Either<T,U> {
-    (e: Either<S,U>) =>
-     match e 
-     case Left(v) => Either<T,U>.Left(f(v))
-     case Right(v) => Either<T,U>.Right(v)
-  }
-
-  function MapRight<S,T,U>(f: T -> U): Either<S,T> -> Either<S,U> {
-    (e: Either<S,T>) =>
-     match e 
-     case Left(v) => Either<S,U>.Left(v)
-     case Right(v) => Either<S,U>.Right(f(v))
-  }
-
-  function Map<S1,S2,T1,T2>(f: S1 -> S2, g: T1 -> T2): Either<S1,T1> -> Either<S2,T2> {
-    (e: Either<S1,T1>) =>
-      match e
-      case Left(v) => Either<S2,T2>.Left(f(v))
-      case Right(v) => Either<S2,T2>.Right(g(v))
-  }
-
-  /**********************************************************
-  *
-  *  Fold
-  *
-  ***********************************************************/
-
-  function Fold<S,T,U>(f: S -> U, g: T -> U): Either<S,T> -> U {
-    (e: Either<S,T>) =>
-      match e
-      case Left(v) => f(v)
-      case Right(v) => g(v)
-  }
-
-  /**********************************************************
-  *
-  *  Comparison
-  *
-  ***********************************************************/
-
-  function Equal<S,T>(eql: (S, S) -> bool, eqr: (T, T) -> bool): (Either<S,T>, Either<S,T>) -> bool {
-    (e1: Either<S,T>, e2: Either<S,T>) =>
-      match (e1, e2)
-      case (Left(v1), Left(v2)) => eql(v1, v2)
-      case (Right(v1), Right(v2)) => eqr(v1, v2)
-      case _ => false
-  }
-
-  function Compare<S,T>(cmpl: (S, S) -> int, cmpr: (T, T) -> int): (Either<S,T>, Either<S,T>) -> int {
-    (e1: Either<S,T>, e2: Either<S,T>) =>
-      match (e1, e2)
-      case (Left(v1), Left(v2)) => cmpl(v1, v2)
-      case (Right(v1), Right(v2)) => cmpr(v1, v2)
-      case (Left(_), Right(_)) => -1
-      case (Right(_), Left(_)) => 1
-  }
-
-}
-
diff --git a/src/Functor.dfy b/src/Functor.dfy
new file mode 100644
index 00000000..5268b35c
--- /dev/null
+++ b/src/Functor.dfy
@@ -0,0 +1,42 @@
+// RUN: %dafny /compile:0 "%s"
+
+/*******************************************************************************
+*  Copyright by the contributors to the Dafny Project
+*  SPDX-License-Identifier: MIT 
+*******************************************************************************/
+
+include "Functions.dfy"
+include "Relations.dfy"
+
+abstract module {:options "-functionSyntax:4"} Functor {
+  import opened Functions
+  import opened Relations
+
+  /* Structure */
+  type W(!new)<T(!new)>
+
+  function Map<S(!new),T(!new)>(f: S -> T): W<S> -> W<T>
+
+  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool
+    requires EquivalenceRelation(eq)
+
+  /* Properties of Equal */
+  lemma LemmaEquivRelLift<T(!new)>(eq: (T, T) -> bool)
+    requires EquivalenceRelation(eq)
+    ensures EquivalenceRelation(Equal(eq))
+
+  /* Properties of Map */
+  lemma LemmaMapFunction<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, g: S -> T)
+    requires EquivalenceRelation(eq) 
+    requires forall x: S :: eq(f(x), g(x))
+    ensures forall w: W<S> :: Equal(eq)(Map(f)(w), Map(g)(w))
+
+  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>)
+    requires EquivalenceRelation(eq)
+    ensures Equal(eq)(Map(Composition(f, g))(w), Composition(Map(f), Map(g))(w))
+
+  lemma LemmaMapIdentity<T(!new)>(eq: (T, T) -> bool, id: T -> T)
+    requires EquivalenceRelation(eq) 
+    requires forall x: T :: eq(id(x), x)
+    ensures forall w: W<T> :: Equal(eq)(Map(id)(w), w)
+}
\ No newline at end of file
diff --git a/src/Functor.dll b/src/Functor.dll
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diff --git a/src/Functor.runtimeconfig.json b/src/Functor.runtimeconfig.json
new file mode 100644
index 00000000..c9fdac56
--- /dev/null
+++ b/src/Functor.runtimeconfig.json
@@ -0,0 +1,10 @@
+{
+  "runtimeOptions": {
+    "tfm": "net6.0",
+    "framework": {
+      "name": "Microsoft.NETCore.App",
+      "version": "6.0.0",
+      "rollForward": "LatestMinor"
+    }
+  }
+}
diff --git a/src/Monad.dfy b/src/Monad.dfy
new file mode 100644
index 00000000..9653275b
--- /dev/null
+++ b/src/Monad.dfy
@@ -0,0 +1,435 @@
+// RUN: %dafny /compile:0 "%s"
+
+/*******************************************************************************
+*  Copyright by the contributors to the Dafny Project
+*  SPDX-License-Identifier: MIT 
+*******************************************************************************/
+
+include "Functor.dfy"
+include "Wrappers.dfy"
+
+abstract module {:options "-functionSyntax:4"} Monad refines Functor {
+  /* Structure  */
+  type W(!new)<T(!new)>
+
+  function Return<T(!new)>(x: T): W<T>
+
+  function Join<T(!new)>(ww: W<W<T>>): W<T>
+
+  /* Naturality of Return and Join */
+  lemma LemmaReturnNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, x: S)
+    requires EquivalenceRelation(eq)
+    ensures Equal(eq)(Map(f)(Return(x)), Return(f(x)))
+
+  lemma LemmaJoinNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>)
+    requires EquivalenceRelation(eq)
+    ensures Equal(eq)(Join(Map(Map(f))(ww)), Map(f)(Join(ww)))
+
+  /* Monad laws */
+  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: W<T>)
+    requires EquivalenceRelation(eq)
+    ensures Equal(eq)(Join(Map(Return)(w)), w) && Equal(eq)(w, Join(Return(w)))
+
+  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: W<W<W<T>>>) 
+    requires EquivalenceRelation(eq)
+    ensures Equal(eq)(Join(Map(Join)(www)), Join(Join(www)))
+}
+
+
+abstract module {:options "-functionSyntax:4"} ResultMonad refines Monad {
+  import Result
+    
+  type E(!new)
+
+  function eqe(x: E, y: E): bool
+
+  lemma LemmaEquiv()
+    ensures EquivalenceRelation(eqe)
+
+  /* Functor structure */
+  type W(!new)<T(!new)> = Result.Result<T,E>
+
+  function Map<S(!new),T(!new)>(f: S -> T): W<S> -> W<T> {
+    Result.Map(f)
+  }
+
+  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+    Result.Equal(eq, eqe)
+  }    
+
+  /* Properties of Equal */
+  lemma LemmaEquivRelLift<T(!new)>(eq: (T, T) -> bool) {
+    LemmaEquiv();
+  }
+
+  /* Properties of Map */
+  lemma LemmaMapFunction<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {
+    LemmaEquiv();
+  }
+
+  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {
+    LemmaEquiv();
+  }
+
+  lemma LemmaMapIdentity<T(!new)>(eq: (T, T) -> bool, id: T -> T) {
+    LemmaEquiv();
+  }
+
+  /* Monad structure */
+  function Return<T(!new)>(x: T): W<T> {
+    Result.Return(x)
+  }
+
+  function Join<T(!new)>(ww: W<W<T>>): W<T> {
+    Result.Join(ww)
+  }
+
+  /* Naturality of Return and Join */
+  lemma LemmaReturnNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, x: S) {
+  }
+
+  lemma LemmaJoinNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {
+    LemmaEquiv();
+  }
+
+  /* Monad laws */
+  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: W<T>) {
+    LemmaEquiv();
+  }
+
+  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: W<W<W<T>>>) {
+    LemmaEquiv();
+  }
+}
+
+
+module {:options "-functionSyntax:4"} WriterMonad refines Monad {
+  import Writer 
+
+  /* Functor structure */
+  type W(!new)<T(!new)> = Writer.Writer<T>
+
+  function Map<S(!new),T(!new)>(f: S -> T): W<S> -> W<T> {
+    Writer.Map(f)
+  }
+
+  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+    Writer.Equal(eq)
+  }    
+
+  /* Properties of Equal */
+  lemma LemmaEquivRelLift<T(!new)>(eq: (T, T) -> bool) {
+  }
+
+  /* Properties of Map */
+  lemma LemmaMapFunction<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {
+  }
+
+  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {
+  }
+
+  lemma LemmaMapIdentity<T(!new)>(eq: (T, T) -> bool, id: T -> T) {
+  }
+
+  /* Monad structure */
+  function Return<T(!new)>(x: T): W<T> {
+    Writer.Return(x)
+  }
+
+  function Join<T(!new)>(ww: W<W<T>>): W<T> {
+    Writer.Join(ww)
+  }
+
+  /* Naturality of Return and Join */
+  lemma LemmaReturnNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, x: S) {
+  }
+
+  lemma LemmaJoinNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {
+  }
+
+  /* Monad laws */
+  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: W<T>) {
+  }
+
+  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: W<W<W<T>>>) {
+  }
+}
+
+
+module {:options "-functionSyntax:4"} OptionMonad refines Monad {
+  import Option
+
+  /* Functor structure */
+  type W(!new)<T(!new)> = Option.Option<T>
+
+  function Map<S(!new),T(!new)>(f: S -> T): W<S> -> W<T> {
+    Option.Map(f)
+  }
+
+  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+    Option.Equal(eq)
+  }    
+
+  /* Properties of Equal */
+  lemma LemmaEquivRelLift<T(!new)>(eq: (T, T) -> bool) {
+  }
+
+  /* Properties of Map */
+  lemma LemmaMapFunction<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {
+  }
+
+  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {
+  }
+
+  lemma LemmaMapIdentity<T(!new)>(eq: (T, T) -> bool, id: T -> T) {
+  }
+
+  /* Monad structure */
+  function Return<T(!new)>(x: T): W<T> {
+    Option.Return(x)
+  }
+
+  function Join<T(!new)>(ww: W<W<T>>): W<T> {
+    Option.Join(ww)
+  }
+
+  /* Naturality of Return and Join */
+  lemma LemmaReturnNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, x: S) {
+  }
+
+  lemma LemmaJoinNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {
+  }
+
+  /* Monad laws */
+  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: W<T>) {
+  }
+
+  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: W<W<W<T>>>) {
+  }
+}
+
+
+abstract module {:options "-functionSyntax:4"} EitherRightMonad refines Monad {
+  import Either 
+
+  type X(!new)
+
+  function eql(x: X, y: X): bool
+
+  lemma LemmaEquiv()
+    ensures EquivalenceRelation(eql)
+
+  /* Functor structure */
+  type W(!new)<T(!new)> = Either.Either<X,T>
+
+  function Map<S(!new),T(!new)>(f: S -> T): W<S> -> W<T> {
+    Either.MapRight(f)
+  }
+
+  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+    Either.Equal(eql, eq)
+  }  
+
+  /* Properties of Equal */
+  lemma LemmaEquivRelLift<T(!new)>(eq: (T, T) -> bool) {
+    LemmaEquiv();
+  }
+
+  /* Properties of Map */
+  lemma LemmaMapFunction<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {
+    LemmaEquiv();
+  }
+
+  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {
+    LemmaEquiv();
+  }
+
+  lemma LemmaMapIdentity<T(!new)>(eq: (T, T) -> bool, id: T -> T) {
+    LemmaEquiv();
+  }
+
+  /* Monad structure */
+  function Return<T(!new)>(x: T): W<T> {
+    Either.ReturnRight(x)
+  }
+
+  function Join<T(!new)>(ww: W<W<T>>): W<T> {
+    Either.JoinRight(ww)
+  }
+
+  /* Naturality of Return and Join */
+  lemma LemmaReturnNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, x: S) {
+  }
+
+  lemma LemmaJoinNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {
+    LemmaEquiv();
+  }
+
+  /* Monad laws */
+  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: W<T>) {
+    LemmaEquiv();
+  }
+
+  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: W<W<W<T>>>) {
+    LemmaEquiv();
+  }
+}
+
+
+abstract module {:options "-functionSyntax:4"} EitherLeftMonad refines Monad {
+  import Either 
+
+  type X(!new)
+
+  function eqr(x: X, y: X): bool
+
+  lemma LemmaEquiv()
+    ensures EquivalenceRelation(eqr)
+
+  /* Functor structure */
+  type W(!new)<T(!new)> = Either.Either<T,X>
+
+  function Map<S(!new),T(!new)>(f: S -> T): W<S> -> W<T> {
+    Either.MapLeft(f)
+  }
+
+  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+    Either.Equal(eq, eqr)
+  }  
+
+  /* Properties of Equal */
+  lemma LemmaEquivRelLift<T(!new)>(eq: (T, T) -> bool) {
+    LemmaEquiv();
+  }
+
+  /* Properties of Map */
+  lemma LemmaMapFunction<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {
+    LemmaEquiv();
+  }
+
+  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {
+    LemmaEquiv();
+  }
+
+  lemma LemmaMapIdentity<T(!new)>(eq: (T, T) -> bool, id: T -> T) {
+    LemmaEquiv();
+  }
+
+  /* Monad structure */
+  function Return<T(!new)>(x: T): W<T> {
+    Either.ReturnLeft(x)
+  }
+
+  function Join<T(!new)>(ww: W<W<T>>): W<T> {
+    Either.JoinLeft(ww)
+  }
+
+  /* Naturality of Return and Join */
+  lemma LemmaReturnNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, x: S) {
+  }
+
+  lemma LemmaJoinNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {
+    LemmaEquiv();
+  }
+
+  /* Monad laws */
+  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: W<T>) {
+    LemmaEquiv();
+  }
+
+  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: W<W<W<T>>>) {
+    LemmaEquiv();
+  }
+}
+
+
+abstract module {:options "-functionSyntax:4"} ReaderMonad refines Monad {
+  import Reader
+
+  type X(!new)
+
+  /* Functor structure */
+  type W(!new)<T(!new)> = Reader.Reader<X,T>
+
+  function Map<S(!new),T(!new)>(f: S -> T): W<S> -> W<T> {
+    Reader.Map(f)
+  }
+
+  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+    Reader.Equal(eq)
+  }    
+
+  /* Properties of Equal */
+  lemma LemmaEquivRelLift<T(!new)>(eq: (T, T) -> bool) {
+    forall w: W<T> ensures Equal(eq)(w, w) {
+      forall x: X ensures eq(w.f(x), w.f(x)) {
+        assert eq(w.f(x), w.f(x));
+      }
+    }
+    assert Reflexive(Equal(eq));
+
+    forall v, w: W<T> ensures Equal(eq)(v, w) <==> Equal(eq)(w, v) {
+      calc {
+        Equal(eq)(v, w);
+        <==>
+        forall x: X :: eq(v.f(x), w.f(x));
+        <==> { assert Symmetric(eq); }
+        forall x: X :: eq(v.f(x), w.f(x));
+        <==>
+        Equal(eq)(w, v);
+      }
+    }
+    assert Symmetric(Equal(eq));
+
+    forall v, w, u: W<T> ensures Equal(eq)(v, w) && Equal(eq)(w, u) ==> Equal(eq)(v, u) {
+      if Equal(eq)(v, w) && Equal(eq)(w, u) {
+        forall x: X ensures eq(v.f(x), u.f(x)) {
+          calc {
+            eq(v.f(x), w.f(x)) && eq(w.f(x), u.f(x));
+          ==> { assert Transitive(eq); }
+            eq(v.f(x), u.f(x));
+          }
+        }
+      }
+    }
+    assert Transitive(Equal(eq));
+  }
+
+  /* Properties of Map */
+  lemma LemmaMapFunction<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {
+  }
+
+  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {
+  }
+
+  lemma LemmaMapIdentity<T(!new)>(eq: (T, T) -> bool, id: T -> T) {
+  }
+
+  /* Monad structure */
+  function Return<T(!new)>(x: T): W<T> {
+    Reader.Return(x)
+  }
+
+  function Join<T(!new)>(ww: W<W<T>>): W<T> {
+    Reader.Join(ww)
+  }
+
+  /* Naturality of Return and Join */
+  lemma LemmaReturnNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, x: S) {
+  }
+
+  lemma LemmaJoinNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {
+  }
+
+  /* Monad laws */
+  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: W<T>) {
+  }
+
+  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: W<W<W<T>>>) {
+  }
+}
+
+
+
+
+
diff --git a/src/Monad.dll b/src/Monad.dll
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diff --git a/src/Monad.runtimeconfig.json b/src/Monad.runtimeconfig.json
new file mode 100644
index 00000000..c9fdac56
--- /dev/null
+++ b/src/Monad.runtimeconfig.json
@@ -0,0 +1,10 @@
+{
+  "runtimeOptions": {
+    "tfm": "net6.0",
+    "framework": {
+      "name": "Microsoft.NETCore.App",
+      "version": "6.0.0",
+      "rollForward": "LatestMinor"
+    }
+  }
+}
diff --git a/src/Reader.dfy b/src/Reader.dfy
deleted file mode 100644
index a3a927f2..00000000
--- a/src/Reader.dfy
+++ /dev/null
@@ -1,98 +0,0 @@
-include "Wrappers.dfy"
-
-module {:options "-functionSyntax:4"} Reader {
-
-  /**********************************************************
-  *
-  *  Monadic structure in terms of Return, Bind, Join, Map, Composition 
-  *
-  ***********************************************************/
-
-  datatype Reader<-X,+T> = Reader(f: X -> T)
-
-  function Return<X,T>(t: T): Reader<X,T> {
-    Reader(x => t)
-  }
-
-  function Bind<X,S,T>(r: Reader<X,S>, f: S -> Reader<X,T>): Reader<X,T> {
-    Reader(x => (f((r.f)(x)).f)(x))
-  }
-
-  function Join<X,T>(rr: Reader<X,Reader<X,T>>): Reader<X,T> {
-    Reader(x => (rr.f(x)).f(x))
-  }
-
-  function Map<X,S,T>(f: S -> T): Reader<X,S> -> Reader<X,T> {
-    (r: Reader<X,S>) => 
-      Reader(x => f((r.f)(x)))
-  }
-
-  function Composition<X,S,T,U>(f: S -> Reader<X,T>, g: T -> Reader<X,U>): S -> Reader<X,U> {
-    s => Bind(f(s), g)
-  }
-
-
-  /**********************************************************
-  *
-  *  Comparison
-  *
-  ***********************************************************/  
-
-  ghost function Equal<X(!new),T>(eq: (T, T) -> bool): (Reader<X,T>, Reader<X,T>) -> bool 
-  {
-    (r1: Reader<X,T>, r2: Reader<X,T>) => forall x: X :: eq(r1.f(x), r2.f(x))
-  }
-
-  /**********************************************************
-  *
-  *  Monad laws in terms of Join and Map
-  *
-  ***********************************************************/
-
-  lemma LemmaUnitalityJoin<X,T>(r: Reader<X,T>)
-    ensures forall x: X :: Join(Map(Return<X,T>)(r)).f(x) == r.f(x) == Join(Return<X,Reader<X,T>>(r)).f(x)
-  {
-  }
-
-  lemma LemmaAssociativityJoin<X,T>(rrr: Reader<X,Reader<X,Reader<X,T>>>) 
-    ensures forall x: X :: Join(Map(Join<X,T>)(rrr)).f(x) == Join(Join<X,Reader<X,T>>(rrr)).f(x)
-  {
-  }  
-
-  /**********************************************************
-  *
-  *  Monad laws in terms of Bind and Return
-  *
-  ***********************************************************/
-
-  lemma LemmaLeftUnitalityBind<X,S,T>(s: S, f: S -> Reader<X,T>)
-    ensures forall x: X :: Bind(Return(s), f).f(x) == f(s).f(x)
-  {
-  }
-
-  lemma LemmaRightUnitalityBind<X,T>(r: Reader<X,T>)
-    ensures forall x: X :: Bind(r, Return).f(x) == r.f(x)
-  {
-  }
-
-  lemma LemmaAssociativityBind<X,S,T,U>(r: Reader<X,S>, f: S -> Reader<X,T>, g: T -> Reader<X,U>)
-    ensures forall x: X :: Bind(Bind(r, f), g).f(x) == Bind(r, Composition(f, g)).f(x)
-  {
-  }
-
-  /**********************************************************
-  *
-  *  Monad laws in terms of (Kleisli) Composition and Return
-  *
-  ***********************************************************/
-
-  lemma LemmaAssociativityComposition<X,S,T,U,V>(f: S -> Reader<X,T>, g: T -> Reader<X,U>, h: U -> Reader<X,V>)
-    ensures forall s: S, x: X :: Composition(Composition(f, g),h)(s).f(x) == Composition(f, Composition(g,h))(s).f(x)
-  {
-  }
-
-  lemma LemmaIdentityReturn<X,S,T>(f: S -> Reader<X,T>)
-    ensures forall s: S, x: X :: Composition(f, Return<X,T>)(s).f(x) == f(s).f(x) == Composition(Return<X,S>, f)(s).f(x)
-  {
-  }
-}
diff --git a/src/Relations.dfy b/src/Relations.dfy
index 9135426f..93029480 100644
--- a/src/Relations.dfy
+++ b/src/Relations.dfy
@@ -19,6 +19,10 @@ module {:options "-functionSyntax:4"} Relations {
     forall x, y :: R(x, y) && R(y, x) ==> x == y
   }
 
+  ghost predicate Symmetric<T(!new)>(R: (T, T) -> bool) {
+    forall x, y :: R(x, y) <==> R(y, x)
+  }
+
   ghost predicate Connected<T(!new)>(R: (T, T) -> bool) {
     forall x, y :: x != y ==> R(x, y) || R(y, x)
   }
@@ -47,7 +51,7 @@ module {:options "-functionSyntax:4"} Relations {
 
   ghost predicate EquivalenceRelation<T(!new)>(R: (T, T) -> bool) {
     && Reflexive(R)
-    && AntiSymmetric(R)
+    && Symmetric(R)
     && Transitive(R)
   }
 
diff --git a/src/Result.dfy b/src/Result.dfy
deleted file mode 100644
index 6183153e..00000000
--- a/src/Result.dfy
+++ /dev/null
@@ -1,204 +0,0 @@
-include "Wrappers.dfy"
-
-module {:options "-functionSyntax:4"} Result {
-
-  import opened Wrappers 
-
-  /**********************************************************
-  *
-  *  Monadic structure in terms of Return, Bind, Join, Map, Composition 
-  *
-  ***********************************************************/
-
-  datatype Result<+T,+E> = | Success(value: T) | Failure(error: E)
-
-  function Return<T,E>(v: T): Result<T,E> {
-    Result.Success(v)
-  }
-
-  function Bind<S,T,E>(r: Result<S,E>, f: S -> Result<T,E>): Result<T,E> {
-    match r
-    case Success(v) => f(v)
-    case Failure(e) => Result<T,E>.Failure(e)
-  }
-
-  function Join<T,E>(rr: Result<Result<T,E>,E>): Result<T,E> {
-    match rr
-    case Success(v) => v
-    case Failure(e) => Result<T,E>.Failure(e)
-  }
-
-  function Map<S,T,E>(f: S -> T): Result<S,E> -> Result<T,E> {
-    (r: Result<S,E>) =>
-      match r 
-      case Success(v) => Result<T,E>.Success(f(v))
-      case Failure(e) => Result<T,E>.Failure(e)
-  }
-
-  function MapError<T,E1,E2>(f: E1 -> E2): Result<T,E1> -> Result<T,E2> {
-    (r: Result<T,E1>) =>
-      match r 
-      case Success(v) => Result<T,E2>.Success(v)
-      case Failure(e) => Result<T,E2>.Failure(f(e))
-  } 
-
-  function Composition<S,T,U,E>(f: S -> Result<T,E>, g: T -> Result<U,E>): S -> Result<U,E> {
-    s => Bind(f(s), g)
-  }
-
-  /**********************************************************
-  *
-  *  Get and set
-  *
-  ***********************************************************/
-
-  function Success<T,E>(v: T): Result<T,E> {
-    Result.Success(v)
-  }
-
-  function Failure<T,E>(e: E): Result<T,E> {
-    Result.Failure(e)
-  } 
-
-  predicate IsSuccess<T,E>(r: Result<T,E>) {
-    r.Success?
-  }  
-
-  predicate IsFailure<T,E>(r: Result<T,E>) {
-    r.Failure?
-  }  
-
-  function GetValue<T,E>(r: Result<T,E>): T
-    requires r.Success?
-  {
-    r.value
-  }
-
-  function GetValueDefault<T,E>(r: Result<T,E>, default: T): T {
-    match r 
-    case Success(v) => v
-    case Failure(_) => default
-  }
-
-  function GetError<T,E>(r: Result<T,E>): E 
-    requires r.Failure?
-  {
-    r.error
-  }
-
-  /**********************************************************
-  *
-  *  Fold
-  *
-  ***********************************************************/
-
-  function Fold<T,E,U>(f: T -> U, g: E -> U): Result<T,E> -> U {
-    (r: Result<T,E>) =>
-      match r 
-      case Success(v) => f(v)
-      case Failure(e) => g(e)
-  }
-
-  /**********************************************************
-  *
-  *  Comparison
-  *
-  ***********************************************************/
-
-  function Equal<T,E>(eqs: (T, T) -> bool, eqf: (E, E) -> bool): (Result<T,E>, Result<T,E>) -> bool {
-    (r1: Result<T,E>, r2: Result<T,E>) =>
-      match (r1, r2)
-      case (Success(v1), Success(v2)) => eqs(v1, v2)
-      case (Failure(e1), Failure(e2)) => eqf(e1, e2)
-      case _ => false
-  }
-
-  function Compare<T,E>(cmps: (T, T) -> int, cmpf: (E, E) -> int): (Result<T,E>, Result<T,E>) -> int {
-    (r1: Result<T,E>, r2: Result<T,E>) =>
-      match (r1, r2)
-      case (Success(v1), Success(v2)) => cmps(v1, v2)
-      case (Failure(e1), Failure(e2)) => cmpf(e1, e2)
-      case (Success(_), Failure(_)) => -1
-      case (Failure(_), Success(_)) => 1
-  }
-
-  /**********************************************************
-  *
-  *  Conversion to other datatypes
-  *
-  ***********************************************************/
-
-  function ToSeq<T,E>(r: Result<T,E>): seq<T> {
-    match r
-    case Success(v) => [v]
-    case Failure(_) => []
-  }
-
-  function ToSet<T,E>(r: Result<T,E>): set<T> {
-    match r
-    case Success(v) => {v}
-    case Failure(_) => {}   
-  }
-
-  function ToOption<T,E>(r: Result<T,E>): Option<T> {
-    match r
-    case Success(v) => Some(v)
-    case Failure(_) => None()
-  }
-
-  /**********************************************************
-  *
-  *  Monad laws in terms of Join and Map
-  *
-  ***********************************************************/
-
-  lemma LemmaUnitalityJoin<T,E>(r: Result<T,E>)
-    ensures Join(Map(Return<T,E>)(r)) == r == Join(Return<Result<T,E>,E>(r))
-  {
-  }
-
-  lemma LemmaAssociativityJoin<T,E>(rrr: Result<Result<Result<T,E>,E>,E>) 
-    ensures Join(Map(Join<T,E>)(rrr)) == Join(Join<Result<T,E>,E>(rrr))
-  {
-  }  
-
-  /**********************************************************
-  *
-  *  Monad laws in terms of Bind and Return
-  *
-  ***********************************************************/
-
-  lemma LemmaLeftUnitalityBind<S,T,E>(v: S, f: S -> Result<T,E>)
-    ensures Bind(Return(v), f) == f(v)
-  {
-  }
-
-  lemma LemmaRightUnitalityBind<T,E>(r: Result<T,E>)
-    ensures Bind(r, Return) == r
-  {
-  }
-
-  lemma LemmaAssociativityBind<S,T,U,E>(r: Result<S,E>, f: S -> Result<T,E>, g: T -> Result<U,E>)
-    ensures Bind(Bind(r, f), g) == Bind(r, Composition(f, g))
-  {
-  }
-
-  /**********************************************************
-  *
-  *  Monad laws in terms of (Kleisli) Composition and Return
-  *
-  ***********************************************************/
-
-  lemma LemmaAssociativityComposition<S,T,U,V,E>(f: S -> Result<T,E>, g: T -> Result<U,E>, h: U -> Result<V,E>)
-    ensures forall s: S :: Composition(Composition(f, g), h)(s) == Composition(f, Composition(g, h))(s)
-  {
-  }
-
-  lemma LemmaIdentityExtract<S,T,E>(f: S -> Result<T,E>)
-    ensures forall s: S :: Composition(f, Return<T,E>)(s) == f(s) == Composition(Return<S,E>, f)(s)
-  {
-  }
-
-}
-
-
diff --git a/src/Wrappers.dfy b/src/Wrappers.dfy
index c06dc9d8..5cf5730f 100644
--- a/src/Wrappers.dfy
+++ b/src/Wrappers.dfy
@@ -5,552 +5,376 @@
 *  SPDX-License-Identifier: MIT 
 *******************************************************************************/
 
-include "Functions.dfy"
-include "Relations.dfy"
+module {:options "-functionSyntax:4"} Result {
+  datatype Result<T(!new),E(!new)> = | Success(value: T) | Failure(error: E)
 
-abstract module {:options "-functionSyntax:4"} Functor {
-
-  import opened Functions
-  import opened Relations
-
-  /**********************************************************
-  *
-  *  Functor structure
-  *
-  ***********************************************************/
-
-  /* Functor structure */
-  type W<T(!new)>
-
-  function Map<S(!new),T>(f: S -> T): W<S> -> W<T>
-
-  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool
-    requires EquivalenceRelation(eq)
-
-  lemma LemmaEquivRelLift<T>(eq: (T, T) -> bool)
-    requires EquivalenceRelation(eq)
-    ensures EquivalenceRelation(Equal(eq))
-
-  /* Functoriality of Map */
-  lemma LemmaMapFunction<S,T>(eq: (T, T) -> bool, f: S -> T, g: S -> T)
-    requires EquivalenceRelation(eq) 
-    requires forall x: S :: eq(f(x), g(x))
-    ensures forall w: W<S> :: Equal(eq)(Map(f)(w), Map(g)(w))
-
-  lemma LemmaMapFunctorial<S(!new),T,U>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>)
-    requires EquivalenceRelation(eq)
-    ensures Equal(eq)(Map(Composition(f, g))(w), Composition(Map(f), Map(g))(w))
-
-  lemma LemmaMapIdentity<T(!new)>(eq: (T, T) -> bool, id: T -> T)
-    requires EquivalenceRelation(eq) 
-    requires forall x: T :: eq(id(x), x)
-    ensures forall w: W<T> :: Equal(eq)(Map(id)(w), w)
-
-}
-
-abstract module {:options "-functionSyntax:4"} Monad refines Functor {
-
-  /**********************************************************
-  *
-  *  Additional monad structure
-  *
-  ***********************************************************/
-
-  /* Monad basics */
-  type W(!new)<T(!new)>
-
-  function Return<T>(x: T): W<T>
-
-  function Join<T(!new)>(ww: W<W<T>>): W<T>
-
-  /* Naturality of Return and Join */
-  lemma LemmaReturnNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, x: S)
-    requires EquivalenceRelation(eq)
-    ensures Equal(eq)(Map(f)(Return(x)), Return(f(x)))
-
-  lemma LemmaJoinNaturality<S(!new),T>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>)
-    requires EquivalenceRelation(eq)
-    ensures Equal(eq)(Join(Map(Map(f))(ww)), Map(f)(Join(ww)))
-
-  /* Monad laws */
-  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: W<T>)
-    requires EquivalenceRelation(eq)
-    ensures Equal(eq)(Join(Map(Return)(w)), w) && Equal(eq)(w, Join(Return(w)))
-
-  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: W<W<W<T>>>) 
-    requires EquivalenceRelation(eq)
-    ensures Equal(eq)(Join(Map(Join)(www)), Join(Join(www)))
-
-  /**********************************************************
-  *
-  *  Derived structure 
-  *
-  ***********************************************************/
-
-  /* Derived functions */
-  function Bind<S(!new),T(!new)>(w: W<S>, f: S -> W<T>): W<T> {
-    Join(Map(f)(w))
-  }
-
-  function KleisliComp<S,T(!new),U(!new)>(f: S -> W<T>, g: T -> W<U>): S -> W<U> {
-    x => Join(Map(g)(f(x)))
-  }
-
-  /* Derived lemmas */
-  lemma LemmaLeftUnitalityBind<S,T(!new)>(eq: (T, T) -> bool, x: S, f: S -> W<T>)
-    requires EquivalenceRelation(eq)
-    ensures Equal(eq)(Bind(Return(x), f), f(x))
-/*   {
-    LemmaReturnNaturality(Equal(eq), f, x);
-    LemmaUnitalityJoin(eq, f(x));
-  } */
-
-  lemma LemmaRightUnitalityBind<T(!new)>(eq: (T, T) -> bool, w: W<T>)
-    requires EquivalenceRelation(eq)
-    ensures Bind(w, Return) == w
-/*   {
-    LemmaUnitalityJoin(eq, w);
-  } */
-
-  lemma LemmaAssociativityBind<S(!new),T(!new),U(!new)>(w: W<S>, f: S -> W<T>, g: T -> W<U>)
-    ensures Bind(Bind(w, f), g) == Bind(w, KleisliComp(f, g))
-/*   {
-    calc {
-      Bind(Bind(w, f), g);
-    == // Definition of Bind
-      Join(Map(g)(Join(Map(f)(w))));
-    ==  { LemmaJoinNaturality(g, Map(f)(w)); }
-      Join(Join(Map(Map(g))(Map(f)(w))));
-    == { LemmaMapFunctorial(f, Map(g), w); }
-      Join(Join(Map(x => Map(g)(f(x)))(w)));
-    == { LemmaAssociativityJoin(Map(x => Map(g)(f(x)))(w)); }
-      Join(Map(Join)(Map(x => Map(g)(f(x)))(w)));
-    == // Definition of Composition
-      Join(Composition(Map(x => Map(g)(f(x))), Map(Join))(w));
-    == {LemmaMapFunctorial(x => Map(g)(f(x)), Join, w);}
-      Join(Map(Composition(x => Map(g)(f(x)), Join))(w));
-    == { LemmaMapFunction((x, y) => x == y, Composition(x => Map(g)(f(x)), Join), x => Join(Map(g)(f(x)))); }
-      Join(Map(x => Join(Map(g)(f(x))))(w));
-    == // Definition of Bind, KleisliComp
-      Bind(w, KleisliComp(f, g));
-    }
-  } */
-
-  lemma LemmaAssociativityComposition<S,T(!new),U(!new),V(!new)>(f: S -> W<T>, g: T -> W<U>, h: U -> W<V>, x: S)
-    ensures KleisliComp(KleisliComp(f, g), h)(x) == KleisliComp(f, KleisliComp(g, h))(x)
-/*   {
-    calc {
-      KleisliComp(KleisliComp(f, g), h)(x);
-    == // Definition of KleisliComp
-      Join(Map(h)(Join(Map(g)(f(x)))));
-    == { LemmaJoinNaturality(h, Map(g)(f(x))); }
-      Join(Join(Map(Map(h))(Map(g)(f(x)))));
-    == // Definition of Composition
-      Join(Join(Composition(Map(g), Map(Map(h)))(f(x))));
-    == { LemmaMapFunctorial(g, Map(h), f(x)); }
-      Join(Join(Map(Composition(g, Map(h)))(f(x))));
-    == { LemmaAssociativityJoin(Map(Composition(g, Map(h)))(f(x))); }
-      Join(Map(Join)(Map(Composition(g, Map(h)))(f(x))) ); 
-    == { LemmaMapFunctorial(Composition(g, Map(h)), Join, f(x)); }
-      Join(Map(Composition(Composition(g, Map(h)), Join))(f(x)));
-    == { LemmaMapFunction((x, y) => x == y, Composition(Composition(g, Map(h)), Join), y => Join(Map(h)(g(y)))); }
-      Join(Map(y => Join(Map(h)(g(y))))(f(x)));
-    == // Definition of KleisliComp
-      KleisliComp(f, KleisliComp(g, h))(x);
-    }
-  } */
-
-  lemma LemmaIdentityReturn<S,T(!new)>(eq: (T, T) -> bool, f: S -> W<T>, x: S)
-    requires EquivalenceRelation(eq)
-    ensures Equal(eq)(KleisliComp(f, Return)(x), f(x)) &&  Equal(eq)(f(x), KleisliComp(Return, f)(x))
-/*   {
-    calc {
-      KleisliComp(f, Return)(x);
-    == // Definition of Composition
-      Join(Map(Return)(f(x)));
-    == { LemmaUnitalityJoin(f(x)); }
-      f(x);
-    }
-    calc {
-      f(x);
-    == { LemmaUnitalityJoin(f(x)); }
-      Join(Return(f(x)));
-    == { LemmaReturnNaturality(f, x); }
-      Join(Map(f)(Return(x)));
-    == // Definition of Composition
-      KleisliComp(Return, f)(x);
-    }
-  } */
-}
-
-abstract module {:options "-functionSyntax:4"} Result refines Monad {
-
-  /**********************************************************
-  *
-  *  Proves monad refinement
-  *
-  ***********************************************************/
-  
-  type E(!new)
-
-  function eqe(e1: E, e2: E): bool
-
-  lemma {:axiom} LemmaEquivRel()
-    ensures EquivalenceRelation(eqe)
-
-  /* Monad structure */
-
-  datatype W<T(!new)> = Success(value: T) | Failure(error: E)
-
-  function Map<S(!new),T>(f: S -> T): W<S> -> W<T> {
-    (w: W<S>) =>
-      match w 
-      case Success(v) => W<T>.Success(f(v))
-      case Failure(e) => W<T>.Failure(e)
+  function Return<T,E>(v: T): Result<T,E> {
+    Result.Success(v)
   }
 
-  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool
-  { 
-    (w1: W<T>, w2: W<T>) =>
-      match (w1, w2)
-      case (Success(v1), Success(v2)) => eq(v1, v2) 
-      case (Failure(e1), Failure(e2)) => eqe(e1, e2)
-      case _ => false
+  function Bind<S(!new),T(!new),E(!new)>(r: Result<S,E>, f: S -> Result<T,E>): Result<T,E> {
+    match r
+    case Success(v) => f(v)
+    case Failure(e) => Result<T,E>.Failure(e)
   }
 
-  lemma LemmaEquivRelLift<T>(eq: (T, T) -> bool)
-    ensures EquivalenceRelation(Equal(eq))
-
-  function Return<T>(x: T): W<T> {
-    Success(x)
-  }
- 
-  function Join<T(!new)>(ww: W<W<T>>): W<T> {
-    match ww
+  function Join<T(!new),E(!new)>(rr: Result<Result<T,E>,E>): Result<T,E> {
+    match rr
     case Success(v) => v
-    case Failure(e) => W<T>.Failure(e)
+    case Failure(e) => Result<T,E>.Failure(e)
   }
 
-  /* Functoriality of Map */
-  lemma LemmaMapFunction<S,T>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {}
-
-  lemma LemmaMapFunctorial<S(!new),T,U>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {}
-
-  lemma LemmaMapIdentity<T(!new)>(eq: (T, T) -> bool, id: T -> T) {
-    forall w: W<T> ensures Equal(eq)(Map(id)(w), w) {
-      match w
-        case Success(v) => 
-        case Failure(e) => LemmaEquivRel();
-    }
+  function Map<S(!new),T(!new),E(!new)>(f: S -> T): Result<S,E> -> Result<T,E> {
+    (r: Result<S,E>) =>
+      match r 
+      case Success(v) => Result<T,E>.Success(f(v))
+      case Failure(e) => Result<T,E>.Failure(e)
   }
 
-  /* Naturality of Return and Join */
-  lemma LemmaReturnNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, x: S) {}
-
-  lemma LemmaJoinNaturality<S(!new),T>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {}
-
-  /* Monad laws */
-  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: W<T>) {}
+  function MapError<T(!new),E1(!new),E2(!new)>(f: E1 -> E2): Result<T,E1> -> Result<T,E2> {
+    (r: Result<T,E1>) =>
+      match r 
+      case Success(v) => Result<T,E2>.Success(v)
+      case Failure(e) => Result<T,E2>.Failure(f(e))
+  } 
 
-  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: W<W<W<T>>>) {}
+  function KleisliComposition<S(!new),T(!new),U(!new),E(!new)>(f: S -> Result<T,E>, g: T -> Result<U,E>): S -> Result<U,E> {
+    s => Bind(f(s), g)
+  }
 
-  /**********************************************************
-  *
-  *  Wrapper
-  *
-  ***********************************************************/
+  function Success<T,E>(v: T): Result<T,E> {
+    Result.Success(v)
+  }
 
-  /* Get and Set */
-  function Failure<T>(e: E): W<T> {
-    W.Failure(e)
+  function Failure<T,E>(e: E): Result<T,E> {
+    Result.Failure(e)
   } 
 
-  predicate IsSuccess<T(!new)>(w: W<T>) {
-    w.Success?
+  predicate IsSuccess<T(!new),E(!new)>(r: Result<T,E>) {
+    r.Success?
   }  
 
-  predicate IsFailure<T(!new)>(w: W<T>) {
-    w.Failure?
+  predicate IsFailure<T(!new),E(!new)>(r: Result<T,E>) {
+    r.Failure?
   }  
 
-  function GetValue<T(!new)>(w: W<T>): T
-    requires w.Success?
+  function GetValue<T(!new),E(!new)>(r: Result<T,E>): T
+    requires r.Success?
   {
-    w.value
+    r.value
   }
 
-  function GetValueDefault<T(!new),E>(w: W<T>, default: T): T {
-    match w 
+  function GetValueDefault<T(!new),E(!new)>(r: Result<T,E>, default: T): T {
+    match r 
     case Success(v) => v
     case Failure(_) => default
   }
 
-  function GetError<T(!new)>(w: W<T>): E 
-    requires w.Failure?
+  function GetError<T(!new),E(!new)>(r: Result<T,E>): E 
+    requires r.Failure?
   {
-    w.error
+    r.error
   }
 
-  /* Fold */
-  function Fold<T(!new),U>(f: T -> U, g: E -> U): W<T> -> U {
-    (w: W<T>) =>
-      match w 
+  function Fold<T(!new),E(!new),U>(f: T -> U, g: E -> U): Result<T,E> -> U {
+    (r: Result<T,E>) =>
+      match r 
       case Success(v) => f(v)
       case Failure(e) => g(e)
   }
 
-  /* Conversion to other types */
-  function ToSeq<T(!new)>(w: W<T>): seq<T> {
-    match w
+  function Equal<T(!new),E(!new)>(eqt: (T, T) -> bool, eqe: (E, E) -> bool): (Result<T,E>, Result<T,E>) -> bool {
+    (r1: Result<T,E>, r2: Result<T,E>) =>
+      match (r1, r2)
+      case (Success(v1), Success(v2)) => eqt(v1, v2)
+      case (Failure(e1), Failure(e2)) => eqe(e1, e2)
+      case _ => false
+  }
+
+  function Compare<T(!new),E(!new)>(cmps: (T, T) -> int, cmpf: (E, E) -> int): (Result<T,E>, Result<T,E>) -> int {
+    (r1: Result<T,E>, r2: Result<T,E>) =>
+      match (r1, r2)
+      case (Success(v1), Success(v2)) => cmps(v1, v2)
+      case (Failure(e1), Failure(e2)) => cmpf(e1, e2)
+      case (Success(_), Failure(_)) => -1
+      case (Failure(_), Success(_)) => 1
+  }
+
+  function ToSeq<T(!new),E(!new)>(r: Result<T,E>): seq<T> {
+    match r
     case Success(v) => [v]
     case Failure(_) => []
   }
 
-  function ToSet<T(!new)>(w: W<T>): set<T> {
-    match w
+  function ToSet<T(!new),E(!new)>(r: Result<T,E>): set<T> {
+    match r
     case Success(v) => {v}
     case Failure(_) => {}   
   }
-
-  // function ToOption<T>(w: W<T>): Option<T> {
-  //   match w
-  //   case Success(v) => Some(v)
-  //   case Failure(_) => None()
-  // }
-
 }
 
-module {:options "-functionSyntax:4"} Option refines Monad {
+module {:options "-functionSyntax:4"} Writer {
+  datatype Writer<T> = Result(value: T, log: string)
 
-  /**********************************************************
-  *
-  *  Proves monad refinement
-  *
-  ***********************************************************/
- 
-  /* Monad structure */
-  datatype W<T(!new)> = None | Some(value: T)
+  function Return<T>(v: T): Writer<T> {
+    Result(v, "")
+  }
 
-  function Map<S(!new),T>(f: S -> T): W<S> -> W<T>
-  {
-    (w: W<S>) =>
-      match w 
-      case None => W<T>.None
-      case Some(v) => W<T>.Some(f(v))
+  function Bind<S,T>(w: Writer<S>, f: S -> Writer<T>): Writer<T> {
+    Writer<T>.Result(f(w.value).value, w.log + f(w.value).log)
   }
 
-  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
-    (w1: W<T>, w2: W<T>) =>
-      match (w1, w2)
-      case (Some(v1), Some(v2)) => eq(v1, v2)
-      case (None, None) => true
-      case _ => false
+  function Join<T>(ww: Writer<Writer<T>>): Writer<T> {
+    Result((ww.value).value, (ww.value).log + ww.log)
   }
 
-  lemma LemmaEquivRelLift<T>(eq: (T, T) -> bool)
-    ensures EquivalenceRelation(Equal(eq))
+  function Map<S,T>(f: S -> T): Writer<S> -> Writer<T> {
+    (w: Writer<S>) => Writer<T>.Result(f(w.value), w.log)
+  }
 
-  function Return<T>(x: T): W<T> {
-    Some(x)
+  function KleisliComposition<S,T,U>(f: S -> Writer<T>, g: T -> Writer<U>): S -> Writer<U> {
+    s => Bind(f(s), g)
   }
 
-  function Join<T(!new)>(ww: W<W<T>>): W<T> {
-    match ww
-    case None => None
-    case Some(w) => w
+  function GetValue<T>(w: Writer<T>): T {
+    w.value
   }
 
-  /* Functoriality of Map */
-  lemma LemmaMapFunction<S,T>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {}
+  function GetLog<T>(w: Writer<T>): string {
+    w.log
+  }
 
-  lemma LemmaMapFunctorial<S(!new),T,U>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {}
+  function Result<T>(v: T, s: string): Writer<T> {
+    Writer<T>.Result(v, s)
+  }
 
-  lemma LemmaMapIdentity<T(!new)>(eq: (T, T) -> bool, id: T -> T) {}
+  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (Writer<T>, Writer<T>) -> bool {
+    (w1: Writer<T>, w2: Writer<T>) => 
+      && eq(w1.value, w2.value) 
+      && w1.log == w2.log
+  }
+}
 
-  /* Naturality of Return and Join */
-  lemma LemmaReturnNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, x: S) {}
+module {:options "-functionSyntax:4"} Option {
+  import Result
 
-  lemma LemmaJoinNaturality<S(!new),T>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {}
+  datatype Option<+T> = None | Some(value: T)
 
-  /* Monad laws */
-  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: W<T>) {}
+  function Return<T>(v: T): Option<T> {
+    Option.Some(v)
+  }
 
-  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: W<W<W<T>>>) {}
+  function Bind<S,T>(o: Option<S>, f: S -> Option<T>): Option<T> {
+    match o 
+    case None => Option<T>.None
+    case Some(v) => f(v)
+  }
 
-  /**********************************************************
-  *
-  *  Wrapper
-  *
-  ***********************************************************/
+  function Join<T>(oo: Option<Option<T>>): Option<T> {
+    match oo
+    case None => Option<T>.None
+    case Some(o) => o
+  }
+
+  function Map<S,T>(f: S -> T): Option<S> -> Option<T>
+  {
+    (o: Option<S>) =>
+      match o 
+      case None => Option<T>.None
+      case Some(v) => Option<T>.Some(f(v))
+  }
 
-  /* Get and Fold */
-  function GetValueDefault<T(!new)>(w: W<T>, default: T): T {
-    match w
+  function KleisliComposition<S,T,U>(f: S -> Option<T>, g: T-> Option<U>): S -> Option<U> {
+    s => Bind(f(s), g)
+  }
+ 
+  function GetValueDefault<T>(o: Option<T>, default: T): T {
+    match o
     case None => default
     case Some(v) => v
   }
 
-  function GetValue<T(!new)>(w: W<T>): T
-    requires w.Some?
+  function GetValue<T>(o: Option<T>): T
+    requires o.Some?
   {
-    w.value
+    o.value
   }
 
-  function Fold<T(!new),U>(u: U, f: T -> U): W<T> -> U {
-    (w: W<T>) =>
-      match w 
+  function Fold<T,U>(u: U, f: T -> U): Option<T> -> U {
+    (o: Option<T>) =>
+      match o 
       case None => u
       case Some(v) => f(v)
   }
 
-  /* Conversion to other types */
-  // function ToResult<T>(e: E): Option<T> -> Result<T,E> {
-  //   (o: Option<T>) =>
-  //     match o 
-  //     case Some(v) => Success(v)
-  //     case None => Failure(e)
-  // }
-
-  function ToSeq<T(!new)>(w: W<T>): seq<T> {
-    match w 
-    case None => []
-    case Some(v) => [v]
-  }
-
-  function ToSet<T(!new)>(w: W<T>): set<T> {
-    match w
-    case None => {}
-    case Some(v) => {v}
+  function Equal<T>(eq: (T, T) -> bool): (Option<T>, Option<T>) -> bool {
+    (o1: Option<T>, o2: Option<T>) =>
+      match (o1, o2)
+      case (Some(v1), Some(v2)) => eq(v1, v2)
+      case (None, None) => true
+      case _ => false
   }
-
-  /* Comparison */
-  function Compare<T(!new)>(cmp: (T, T) -> int): (W<T>, W<T>) -> int {
-    (w1: W<T>, w2: W<T>) =>
-      match (w1, w2)
+  
+  function Compare<T>(cmp: (T, T) -> int): (Option<T>, Option<T>) -> int {
+    (o1: Option<T>, o2: Option<T>) =>
+      match (o1, o2)
       case (Some(v1), Some(v2)) => cmp(v1, v2)
       case (None, None) => 0
       case (None, Some(_)) => -1
       case (Some(_), None) => 1
   }
 
-}
-
-module {:options "-functionSyntax:4"} Writer refines Monad {
-
-  /**********************************************************
-  *
-  *  Proves monad refinement
-  *
-  ***********************************************************/
- 
-  /* Monad structure */
-  datatype W<T(!new)> = Result(value: T, log: string)
-
-  function Map<S(!new),T>(f: S -> T): W<S> -> W<T> {
-    (w: W<S>) => W<T>.Result(f(w.value), w.log)
+  function ToResult<T,E>(e: E): Option<T> -> Result.Result<T,E> {
+    (o: Option<T>) =>
+      match o 
+      case Some(v) => Result.Success(v)
+      case None => Result.Failure(e)
   }
 
-  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
-    (w1: W<T>, w2: W<T>) => eq(w1.value, w2.value) && w1.log == w2.log
+  function ToOption<T(!new),E(!new)>(r: Result.Result<T,E>): Option<T> {
+    match r
+    case Success(v) => Some(v)
+    case Failure(_) => None()
   }
 
-  lemma LemmaEquivRelLift<T>(eq: (T, T) -> bool)
-    ensures EquivalenceRelation(Equal(eq))
-
-  function Return<T>(x: T): W<T> {
-    Result(x, "")
+  function ToSeq<T>(o: Option<T>): seq<T> {
+    match o 
+    case None => []
+    case Some(v) => [v]
   }
 
-  function Join<T(!new)>(ww: W<W<T>>): W<T> {
-    Result((ww.value).value, (ww.value).log + ww.log)
+  function ToSet<T>(o: Option<T>): set<T> {
+    match o
+    case None => {}
+    case Some(v) => {v}
   }
+}
 
-  /* Functoriality of Map */
-  lemma LemmaMapFunction<S,T>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {}
-
-  lemma LemmaMapFunctorial<S(!new),T,U>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {}
-
-  lemma LemmaMapIdentity<T(!new)>(eq: (T, T) -> bool, id: T -> T) {}
 
-  /* Naturality of Return and Join */
-  lemma LemmaReturnNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, x: S) {}
+module {:options "-functionSyntax:4"} Either {
+  import Option
 
-  lemma LemmaJoinNaturality<S(!new),T>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {}
+  datatype Either<+S,+T> = Left(left: S) | Right(right: T) 
 
-  /* Monad laws */
-  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: W<T>) {}
+  function ReturnLeft<S,T>(v: S): Either<S,T> {
+    Either.Left(v)
+  }
 
-  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: W<W<W<T>>>) {}
+  function ReturnRight<S,T>(v: T): Either<S,T> {
+    Either.Right(v)
+  }
 
-  /**********************************************************
-  *
-  *  Wrapper
-  *
-  ***********************************************************/
+  predicate IsLeft<S,T>(e: Either<S,T>) {
+    e.Left?
+  }
 
-  /* Get and Set */
-  function GetValue<T(!new)>(w: W<T>): T {
-    w.value
+  predicate IsRight<S,T>(e: Either<S,T>) {
+    e.Right?
   }
 
-  function GetLog<T(!new)>(w: W<T>): string {
-    w.log
+  function FindLeft<S,T>(e: Either<S,T>): Option.Option<S> {
+    match e 
+    case Left(v) => Option.Some(v)
+    case Right(v) => Option.None
   }
 
-  function Result<T>(v: T, s: string): W<T> {
-    W<T>.Result(v, s)
+  function FindRight<S,T>(e: Either<S,T>): Option.Option<T> {
+    match e 
+    case Left(v) => Option.None
+    case Right(v) => Option.Some(v)
   }
-}
 
-abstract module {:options "-functionSyntax:4"} Reader refines Monad {
+  function JoinRight<S,T>(ee: Either<S,Either<S,T>>): Either<S,T> {
+    match ee
+    case Left(v) => Either<S,T>.Left(v)
+    case Right(v) => v
+  }
 
-  /**********************************************************
-  *
-  *  Monad
-  *
-  ***********************************************************/
+  function JoinLeft<S,T>(ee: Either<Either<S,T>,T>): Either<S,T> {
+    match ee
+    case Left(v) => v
+    case Right(v) => Either<S,T>.Right(v)
+  }
 
-  type A(!new)
+  function MapLeft<S,T,U>(f: S -> T): Either<S,U> -> Either<T,U> {
+    (e: Either<S,U>) =>
+     match e 
+     case Left(v) => Either<T,U>.Left(f(v))
+     case Right(v) => Either<T,U>.Right(v)
+  }
 
-  datatype W<T(!new)> = Reader(f: A -> T)
+  function MapRight<S,T,U>(f: T -> U): Either<S,T> -> Either<S,U> {
+    (e: Either<S,T>) =>
+     match e 
+     case Left(v) => Either<S,U>.Left(v)
+     case Right(v) => Either<S,U>.Right(f(v))
+  }
 
-  function Map<S(!new),T>(f: S -> T): W<S> -> W<T> {
-    (w: W<S>) => Reader(a => f((w.f)(a)))
+  function Map<S1,S2,T1,T2>(f: S1 -> S2, g: T1 -> T2): Either<S1,T1> -> Either<S2,T2> {
+    (e: Either<S1,T1>) =>
+      match e
+      case Left(v) => Either<S2,T2>.Left(f(v))
+      case Right(v) => Either<S2,T2>.Right(g(v))
   }
-  
-  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool
-  {
-    (w1: W<T>, w2: W<T>) =>
-      forall a: A :: eq(w1.f(a), w2.f(a))
+
+  function Fold<S,T,U>(f: S -> U, g: T -> U): Either<S,T> -> U {
+    (e: Either<S,T>) =>
+      match e
+      case Left(v) => f(v)
+      case Right(v) => g(v)
   }
 
-  lemma LemmaEquivRelLift<T>(eq: (T, T) -> bool)
-    ensures EquivalenceRelation(Equal(eq))
-  
-  function Return<T>(x: T): W<T> {
-    Reader(a => x)
+  function Equal<S,T>(eql: (S, S) -> bool, eqr: (T, T) -> bool): (Either<S,T>, Either<S,T>) -> bool {
+    (e1: Either<S,T>, e2: Either<S,T>) =>
+      match (e1, e2)
+      case (Left(v1), Left(v2)) => eql(v1, v2)
+      case (Right(v1), Right(v2)) => eqr(v1, v2)
+      case _ => false
   }
 
-  function Join<T(!new)>(ww: W<W<T>>): W<T>{
-    Reader(a => (ww.f(a)).f(a))
+  function Compare<S,T>(cmpl: (S, S) -> int, cmpr: (T, T) -> int): (Either<S,T>, Either<S,T>) -> int {
+    (e1: Either<S,T>, e2: Either<S,T>) =>
+      match (e1, e2)
+      case (Left(v1), Left(v2)) => cmpl(v1, v2)
+      case (Right(v1), Right(v2)) => cmpr(v1, v2)
+      case (Left(_), Right(_)) => -1
+      case (Right(_), Left(_)) => 1
   }
+}
 
-  /* Functoriality of Map */
-  lemma LemmaMapFunction<S,T>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {}
+module {:options "-functionSyntax:4"} Reader {
+  datatype Reader<-X,+T> = Reader(f: X -> T)
 
-  lemma LemmaMapFunctorial<S(!new),T,U>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {}
+  function Return<X,T>(t: T): Reader<X,T> {
+    Reader(x => t)
+  }
 
-  lemma LemmaMapIdentity<T(!new)>(eq: (T, T) -> bool, id: T -> T) {}
+  function Bind<X,S,T>(r: Reader<X,S>, f: S -> Reader<X,T>): Reader<X,T> {
+    Reader(x => (f((r.f)(x)).f)(x))
+  }
 
-  /* Naturality of Return and Join */
-  lemma LemmaReturnNaturality<S,T>(eq: (T, T) -> bool, f: S -> T, x: S) {}
+  function Join<X,T>(rr: Reader<X,Reader<X,T>>): Reader<X,T> {
+    Reader(x => (rr.f(x)).f(x))
+  }
 
-  lemma LemmaJoinNaturality<S(!new),T>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {}
+  function Map<X,S,T>(f: S -> T): Reader<X,S> -> Reader<X,T> {
+    (r: Reader<X,S>) => 
+      Reader(x => f((r.f)(x)))
+  }
 
-  /* Monad laws */
-  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: W<T>) {}
+  function MapContra<X,Y,T>(f: X -> Y): Reader<Y,T> -> Reader<X,T> {
+    (r: Reader<Y,T>) =>
+      Reader(x => r.f(f(x)))
+  }
 
-  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: W<W<W<T>>>) {}
+  function KleisliComposition<X,S,T,U>(f: S -> Reader<X,T>, g: T -> Reader<X,U>): S -> Reader<X,U> {
+    s => Bind(f(s), g)
+  } 
 
+  ghost function Equal<X(!new),T>(eq: (T, T) -> bool): (Reader<X,T>, Reader<X,T>) -> bool 
+  {
+    (r1: Reader<X,T>, r2: Reader<X,T>) => 
+      forall x: X :: eq(r1.f(x), r2.f(x))
+  }
 }
 
diff --git a/src/Wrappers.dll b/src/Wrappers.dll
index b1ab301ce43e9872b7d6147b1050af89b1f01a9b..89c77582bd820adfdba3901940ab59719d8a465b 100644
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diff --git a/src/Writer.dfy b/src/Writer.dfy
deleted file mode 100644
index 9d8230f7..00000000
--- a/src/Writer.dfy
+++ /dev/null
@@ -1,101 +0,0 @@
-module {:options "-functionSyntax:4"} Writer {
-
-  /**********************************************************
-  *
-  *  Monadic structure in terms of Return, Bind, Join, Map, Composition 
-  *
-  ***********************************************************/
-
-  datatype Writer<T> = Result(value: T, log: string)
-
-  function Return<T>(v: T): Writer<T> {
-    Result(v, "")
-  }
-
-  function Bind<S,T>(w: Writer<S>, f: S -> Writer<T>): Writer<T> {
-    Writer<T>.Result(f(w.value).value, w.log + f(w.value).log)
-  }
-
-  function Join<T>(ww: Writer<Writer<T>>): Writer<T> {
-    Result((ww.value).value, (ww.value).log + ww.log)
-  }
-
-  function Map<S,T>(f: S -> T): Writer<S> -> Writer<T> {
-    (w: Writer<S>) => Writer<T>.Result(f(w.value), w.log)
-  }
-
-  function Composition<S,T,U>(f: S -> Writer<T>, g: T -> Writer<U>): S -> Writer<U> {
-    s => Bind(f(s), g)
-  }
-
-  /**********************************************************
-  *
-  *  Get and set
-  *
-  ***********************************************************/
-
-  function GetValue<T>(w: Writer<T>): T {
-    w.value
-  }
-
-  function GetLog<T>(w: Writer<T>): string {
-    w.log
-  }
-
-  function Result<T>(v: T, s: string): Writer<T> {
-    Writer<T>.Result(v, s)
-  }
-
-  /**********************************************************
-  *
-  *  Monad laws in terms of Join and Map
-  *
-  ***********************************************************/
-
-  lemma LemmaUnitalityJoin<T>(w: Writer<T>)
-    ensures Join(Map(Return<T>)(w)) == w == Join(Return<Writer<T>>(w))
-  {
-  }
-
-  lemma LemmaAssociativityJoin<T>(www: Writer<Writer<Writer<T>>>) 
-    ensures Join(Map(Join<T>)(www)) == Join(Join<Writer<T>>(www))
-  {
-  }  
-
-  /**********************************************************
-  *
-  *  Monad laws in terms of Bind and Return
-  *
-  ***********************************************************/
-
-  lemma LemmaLeftUnitalityBind<S,T>(v: S, f: S -> Writer<T>)
-    ensures Bind(Return(v), f) == f(v)
-  {
-  }
-
-  lemma LemmaRightUnitalityBind<T>(w: Writer<T>)
-    ensures Bind(w, Return) == w
-  {
-  }
-
-  lemma LemmaAssociativityBind<S,T,U>(w: Writer<S>, f: S -> Writer<T>, g: T -> Writer<U>)
-    ensures Bind(Bind(w, f), g) == Bind(w, Composition(f, g))
-  {
-  }
-
-  /**********************************************************
-  *
-  *  Monad laws in terms of (Kleisli) Composition and Return
-  *
-  ***********************************************************/
-
-  lemma LemmaAssociativityComposition<S,T,U,V>(f: S -> Writer<T>, g: T -> Writer<U>, h: U -> Writer<V>)
-    ensures forall s: S :: Composition(Composition(f, g), h)(s) == Composition(f, Composition(g, h))(s)
-  {
-  }
-
-  lemma LemmaIdentityExtract<S,T>(f: S -> Writer<T>)
-    ensures forall s: S :: Composition(f, Return<T>)(s) == f(s) == Composition(Return<S>, f)(s)
-  {
-  }
-}

From a393e7cb29d4bb89c30e827f549ad45880336de8 Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Fri, 3 Feb 2023 17:02:19 +0000
Subject: [PATCH 23/33] delete files

---
 src/Cowrappers.dll                | Bin 140800 -> 0 bytes
 src/Cowrappers.runtimeconfig.json |  10 ----------
 src/Functor.dll                   | Bin 136704 -> 0 bytes
 src/Functor.runtimeconfig.json    |  10 ----------
 src/Monad.dll                     | Bin 235008 -> 0 bytes
 src/Monad.runtimeconfig.json      |  10 ----------
 src/Wrappers.dll                  | Bin 202240 -> 0 bytes
 src/Wrappers.runtimeconfig.json   |  10 ----------
 8 files changed, 40 deletions(-)
 delete mode 100644 src/Cowrappers.dll
 delete mode 100644 src/Cowrappers.runtimeconfig.json
 delete mode 100644 src/Functor.dll
 delete mode 100644 src/Functor.runtimeconfig.json
 delete mode 100644 src/Monad.dll
 delete mode 100644 src/Monad.runtimeconfig.json
 delete mode 100644 src/Wrappers.dll
 delete mode 100644 src/Wrappers.runtimeconfig.json

diff --git a/src/Cowrappers.dll b/src/Cowrappers.dll
deleted file mode 100644
index 0edc23f5dea7207bff5b4088963d669d9ba84302..0000000000000000000000000000000000000000
GIT binary patch
literal 0
HcmV?d00001

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diff --git a/src/Cowrappers.runtimeconfig.json b/src/Cowrappers.runtimeconfig.json
deleted file mode 100644
index c9fdac56..00000000
--- a/src/Cowrappers.runtimeconfig.json
+++ /dev/null
@@ -1,10 +0,0 @@
-{
-  "runtimeOptions": {
-    "tfm": "net6.0",
-    "framework": {
-      "name": "Microsoft.NETCore.App",
-      "version": "6.0.0",
-      "rollForward": "LatestMinor"
-    }
-  }
-}
diff --git a/src/Functor.dll b/src/Functor.dll
deleted file mode 100644
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diff --git a/src/Functor.runtimeconfig.json b/src/Functor.runtimeconfig.json
deleted file mode 100644
index c9fdac56..00000000
--- a/src/Functor.runtimeconfig.json
+++ /dev/null
@@ -1,10 +0,0 @@
-{
-  "runtimeOptions": {
-    "tfm": "net6.0",
-    "framework": {
-      "name": "Microsoft.NETCore.App",
-      "version": "6.0.0",
-      "rollForward": "LatestMinor"
-    }
-  }
-}
diff --git a/src/Monad.dll b/src/Monad.dll
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diff --git a/src/Monad.runtimeconfig.json b/src/Monad.runtimeconfig.json
deleted file mode 100644
index c9fdac56..00000000
--- a/src/Monad.runtimeconfig.json
+++ /dev/null
@@ -1,10 +0,0 @@
-{
-  "runtimeOptions": {
-    "tfm": "net6.0",
-    "framework": {
-      "name": "Microsoft.NETCore.App",
-      "version": "6.0.0",
-      "rollForward": "LatestMinor"
-    }
-  }
-}
diff --git a/src/Wrappers.dll b/src/Wrappers.dll
deleted file mode 100644
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diff --git a/src/Wrappers.runtimeconfig.json b/src/Wrappers.runtimeconfig.json
deleted file mode 100644
index c9fdac56..00000000
--- a/src/Wrappers.runtimeconfig.json
+++ /dev/null
@@ -1,10 +0,0 @@
-{
-  "runtimeOptions": {
-    "tfm": "net6.0",
-    "framework": {
-      "name": "Microsoft.NETCore.App",
-      "version": "6.0.0",
-      "rollForward": "LatestMinor"
-    }
-  }
-}

From a04b9f6cd2b99d50899779364c9e258a2d3905a7 Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Fri, 3 Feb 2023 17:31:45 +0000
Subject: [PATCH 24/33] notation

---
 src/Comonad.dfy                   |  74 +++++++-------
 src/Cowrappers.dfy                |   7 ++
 src/Cowrappers.dll                | Bin 0 -> 140800 bytes
 src/Cowrappers.runtimeconfig.json |  10 ++
 src/Functor.dfy                   |  12 +--
 src/Monad.dfy                     | 154 +++++++++++++++---------------
 6 files changed, 140 insertions(+), 117 deletions(-)
 create mode 100644 src/Cowrappers.dll
 create mode 100644 src/Cowrappers.runtimeconfig.json

diff --git a/src/Comonad.dfy b/src/Comonad.dfy
index 50967947..13f56d16 100644
--- a/src/Comonad.dfy
+++ b/src/Comonad.dfy
@@ -1,31 +1,38 @@
+// RUN: %dafny /compile:0 "%s"
+
+/*******************************************************************************
+*  Copyright by the contributors to the Dafny Project
+*  SPDX-License-Identifier: MIT 
+*******************************************************************************/
+
 include "Functor.dfy"
 include "Cowrappers.dfy"
 
 abstract module {:options "-functionSyntax:4"} Comonad refines Functor {
   /* Structure  */
-  type W(!new)<T(!new)>
+  type F(!new)<T(!new)>
 
-  function Extract<T(!new)>(w: W<T>): T
+  function Extract<T(!new)>(w: F<T>): T
 
-  function Duplicate<T(!new)>(w: W<T>): W<W<T>>
+  function Duplicate<T(!new)>(w: F<T>): F<F<T>>
 
-  /* Naturality of Extract and Duplicate */
-  lemma LemmaExtractNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, w: W<S>)
+  /* Naturality */
+  lemma LemmaExtractNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, w: F<S>)
     requires EquivalenceRelation(eq)
     ensures eq(f(Extract(w)), Extract(Map(f)(w)))
 
-  lemma LemmaDuplicateNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, w: W<S>)
+  lemma LemmaDuplicateNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, w: F<S>)
     requires EquivalenceRelation(eq)
     ensures EquivalenceRelation(Equal(eq))
     ensures Equal(Equal(eq))(Duplicate(Map(f)(w)), Map(Map(f))(Duplicate(w)))
 
-  /* Comonad laws */
-  lemma LemmaCoUnitalityExtract<T(!new)>(eq: (T, T) -> bool, w: W<T>)
+  /* Counitality and Coassociativity */
+  lemma LemmaCoUnitalityExtract<T(!new)>(eq: (T, T) -> bool, w: F<T>)
     requires EquivalenceRelation(eq)
     ensures && Equal(eq)(Map(Extract)(Duplicate(w)), w)
             && Equal(eq)(w, Extract(Duplicate(w)))
  
-  lemma LemmaCoAssociativityDuplicate<T(!new)>(eq: (T, T) -> bool, w: W<T>) 
+  lemma LemmaCoAssociativityDuplicate<T(!new)>(eq: (T, T) -> bool, w: F<T>) 
     requires && EquivalenceRelation(eq)
              && EquivalenceRelation(Equal(eq))
              && EquivalenceRelation(Equal(Equal(eq))) 
@@ -43,13 +50,13 @@ abstract module {:options "-functionSyntax:4"} CoreaderRightComonad refines Como
     ensures EquivalenceRelation(eql)
 
   /* Functor structure */
-  type W(!new)<T(!new)> = Coreader.Coreader<X,T>
+  type F(!new)<T(!new)> = Coreader.Coreader<X,T>
 
-  function Map<S(!new),T(!new)>(f: S -> T): W<S> -> W<T> {
+  function Map<S(!new),T(!new)>(f: S -> T): F<S> -> F<T> {
     Coreader.MapRight(f)
   }
 
-  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (F<T>, F<T>) -> bool {
     Coreader.Equal(eql, eq)
   }  
 
@@ -63,7 +70,7 @@ abstract module {:options "-functionSyntax:4"} CoreaderRightComonad refines Como
     LemmaEquiv();
   }
 
-  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {
+  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: F<S>) {
     LemmaEquiv();
   }
 
@@ -72,28 +79,28 @@ abstract module {:options "-functionSyntax:4"} CoreaderRightComonad refines Como
   }
 
   /* Comonad structure */
-  function Extract<T(!new)>(w: W<T>): T {
+  function Extract<T(!new)>(w: F<T>): T {
     Coreader.ExtractRight(w)
   } 
 
-  function Duplicate<T(!new)>(w: W<T>): W<W<T>> {
+  function Duplicate<T(!new)>(w: F<T>): F<F<T>> {
     Coreader.DuplicateLeft(w)
   }
 
-  /* Naturality of Extract and Duplicate */
-  lemma LemmaExtractNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, w: W<S>) {
+  /* Naturality */
+  lemma LemmaExtractNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, w: F<S>) {
   }
 
-  lemma LemmaDuplicateNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, w: W<S>) {
+  lemma LemmaDuplicateNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, w: F<S>) {
     LemmaEquiv();
   }
 
-  /* Comonad laws */
-  lemma LemmaCoUnitalityExtract<T(!new)>(eq: (T, T) -> bool, w: W<T>) {
+  /* Counitality and Coassociativity */
+  lemma LemmaCoUnitalityExtract<T(!new)>(eq: (T, T) -> bool, w: F<T>) {
     LemmaEquiv();
   }
  
-  lemma LemmaCoAssociativityDuplicate<T(!new)>(eq: (T, T) -> bool, w: W<T>) {
+  lemma LemmaCoAssociativityDuplicate<T(!new)>(eq: (T, T) -> bool, w: F<T>) {
     LemmaEquiv();
   }
 }
@@ -109,13 +116,13 @@ abstract module {:options "-functionSyntax:4"} CoreaderLeftComonad refines Comon
     ensures EquivalenceRelation(eqr)
 
   /* Functor structure */
-  type W(!new)<T(!new)> = Coreader.Coreader<T,X>
+  type F(!new)<T(!new)> = Coreader.Coreader<T,X>
 
-  function Map<S(!new),T(!new)>(f: S -> T): W<S> -> W<T> {
+  function Map<S(!new),T(!new)>(f: S -> T): F<S> -> F<T> {
     Coreader.MapLeft(f)
   }
 
-  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (F<T>, F<T>) -> bool {
     Coreader.Equal(eq, eqr)
   }  
 
@@ -129,7 +136,7 @@ abstract module {:options "-functionSyntax:4"} CoreaderLeftComonad refines Comon
     LemmaEquiv();
   }
 
-  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {
+  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: F<S>) {
     LemmaEquiv();
   }
 
@@ -138,29 +145,28 @@ abstract module {:options "-functionSyntax:4"} CoreaderLeftComonad refines Comon
   }
 
   /* Comonad structure */
-  function Extract<T(!new)>(w: W<T>): T {
+  function Extract<T(!new)>(w: F<T>): T {
     Coreader.ExtractLeft(w)
   } 
 
-  function Duplicate<T(!new)>(w: W<T>): W<W<T>> {
+  function Duplicate<T(!new)>(w: F<T>): F<F<T>> {
     Coreader.DuplicateRight(w)
   }
 
-  /* Naturality of Extract and Duplicate */
-  lemma LemmaExtractNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, w: W<S>) {
+  /* Naturality */
+  lemma LemmaExtractNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, w: F<S>) {
   }
 
-  lemma LemmaDuplicateNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, w: W<S>) {
+  lemma LemmaDuplicateNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, w: F<S>) {
     LemmaEquiv();
   }
 
-  /* Comonad laws */
-  lemma LemmaCoUnitalityExtract<T(!new)>(eq: (T, T) -> bool, w: W<T>) {
+  /* Counitality and Coassociativity */
+  lemma LemmaCoUnitalityExtract<T(!new)>(eq: (T, T) -> bool, w: F<T>) {
     LemmaEquiv();
   }
  
-  lemma LemmaCoAssociativityDuplicate<T(!new)>(eq: (T, T) -> bool, w: W<T>) {
+  lemma LemmaCoAssociativityDuplicate<T(!new)>(eq: (T, T) -> bool, w: F<T>) {
     LemmaEquiv();
   }
-
 }
\ No newline at end of file
diff --git a/src/Cowrappers.dfy b/src/Cowrappers.dfy
index effcfd66..c94051ab 100644
--- a/src/Cowrappers.dfy
+++ b/src/Cowrappers.dfy
@@ -1,3 +1,10 @@
+// RUN: %dafny /compile:0 "%s"
+
+/*******************************************************************************
+*  Copyright by the contributors to the Dafny Project
+*  SPDX-License-Identifier: MIT 
+*******************************************************************************/
+
 module {:options "-functionSyntax:4"} Coreader {
 
   datatype Coreader<+S,+T> = Coreader(left: S, right: T)
diff --git a/src/Cowrappers.dll b/src/Cowrappers.dll
new file mode 100644
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HcmV?d00001

diff --git a/src/Cowrappers.runtimeconfig.json b/src/Cowrappers.runtimeconfig.json
new file mode 100644
index 00000000..c9fdac56
--- /dev/null
+++ b/src/Cowrappers.runtimeconfig.json
@@ -0,0 +1,10 @@
+{
+  "runtimeOptions": {
+    "tfm": "net6.0",
+    "framework": {
+      "name": "Microsoft.NETCore.App",
+      "version": "6.0.0",
+      "rollForward": "LatestMinor"
+    }
+  }
+}
diff --git a/src/Functor.dfy b/src/Functor.dfy
index 5268b35c..16244a74 100644
--- a/src/Functor.dfy
+++ b/src/Functor.dfy
@@ -13,11 +13,11 @@ abstract module {:options "-functionSyntax:4"} Functor {
   import opened Relations
 
   /* Structure */
-  type W(!new)<T(!new)>
+  type F(!new)<T(!new)>
 
-  function Map<S(!new),T(!new)>(f: S -> T): W<S> -> W<T>
+  function Map<S(!new),T(!new)>(f: S -> T): F<S> -> F<T>
 
-  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool
+  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (F<T>, F<T>) -> bool
     requires EquivalenceRelation(eq)
 
   /* Properties of Equal */
@@ -29,14 +29,14 @@ abstract module {:options "-functionSyntax:4"} Functor {
   lemma LemmaMapFunction<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, g: S -> T)
     requires EquivalenceRelation(eq) 
     requires forall x: S :: eq(f(x), g(x))
-    ensures forall w: W<S> :: Equal(eq)(Map(f)(w), Map(g)(w))
+    ensures forall w: F<S> :: Equal(eq)(Map(f)(w), Map(g)(w))
 
-  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>)
+  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: F<S>)
     requires EquivalenceRelation(eq)
     ensures Equal(eq)(Map(Composition(f, g))(w), Composition(Map(f), Map(g))(w))
 
   lemma LemmaMapIdentity<T(!new)>(eq: (T, T) -> bool, id: T -> T)
     requires EquivalenceRelation(eq) 
     requires forall x: T :: eq(id(x), x)
-    ensures forall w: W<T> :: Equal(eq)(Map(id)(w), w)
+    ensures forall w: F<T> :: Equal(eq)(Map(id)(w), w)
 }
\ No newline at end of file
diff --git a/src/Monad.dfy b/src/Monad.dfy
index 9653275b..bde877fa 100644
--- a/src/Monad.dfy
+++ b/src/Monad.dfy
@@ -10,27 +10,27 @@ include "Wrappers.dfy"
 
 abstract module {:options "-functionSyntax:4"} Monad refines Functor {
   /* Structure  */
-  type W(!new)<T(!new)>
+  type F(!new)<T(!new)>
 
-  function Return<T(!new)>(x: T): W<T>
+  function Return<T(!new)>(x: T): F<T>
 
-  function Join<T(!new)>(ww: W<W<T>>): W<T>
+  function Join<T(!new)>(ww: F<F<T>>): F<T>
 
-  /* Naturality of Return and Join */
+  /* Naturality */
   lemma LemmaReturnNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, x: S)
     requires EquivalenceRelation(eq)
     ensures Equal(eq)(Map(f)(Return(x)), Return(f(x)))
 
-  lemma LemmaJoinNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>)
+  lemma LemmaJoinNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, ww: F<F<S>>)
     requires EquivalenceRelation(eq)
     ensures Equal(eq)(Join(Map(Map(f))(ww)), Map(f)(Join(ww)))
 
-  /* Monad laws */
-  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: W<T>)
+  /* Unitality and Associativity */
+  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: F<T>)
     requires EquivalenceRelation(eq)
     ensures Equal(eq)(Join(Map(Return)(w)), w) && Equal(eq)(w, Join(Return(w)))
 
-  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: W<W<W<T>>>) 
+  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: F<F<F<T>>>) 
     requires EquivalenceRelation(eq)
     ensures Equal(eq)(Join(Map(Join)(www)), Join(Join(www)))
 }
@@ -47,13 +47,13 @@ abstract module {:options "-functionSyntax:4"} ResultMonad refines Monad {
     ensures EquivalenceRelation(eqe)
 
   /* Functor structure */
-  type W(!new)<T(!new)> = Result.Result<T,E>
+  type F(!new)<T(!new)> = Result.Result<T,E>
 
-  function Map<S(!new),T(!new)>(f: S -> T): W<S> -> W<T> {
+  function Map<S(!new),T(!new)>(f: S -> T): F<S> -> F<T> {
     Result.Map(f)
   }
 
-  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (F<T>, F<T>) -> bool {
     Result.Equal(eq, eqe)
   }    
 
@@ -67,7 +67,7 @@ abstract module {:options "-functionSyntax:4"} ResultMonad refines Monad {
     LemmaEquiv();
   }
 
-  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {
+  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: F<S>) {
     LemmaEquiv();
   }
 
@@ -76,28 +76,28 @@ abstract module {:options "-functionSyntax:4"} ResultMonad refines Monad {
   }
 
   /* Monad structure */
-  function Return<T(!new)>(x: T): W<T> {
+  function Return<T(!new)>(x: T): F<T> {
     Result.Return(x)
   }
 
-  function Join<T(!new)>(ww: W<W<T>>): W<T> {
+  function Join<T(!new)>(ww: F<F<T>>): F<T> {
     Result.Join(ww)
   }
 
-  /* Naturality of Return and Join */
+  /* Naturality */
   lemma LemmaReturnNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, x: S) {
   }
 
-  lemma LemmaJoinNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {
+  lemma LemmaJoinNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, ww: F<F<S>>) {
     LemmaEquiv();
   }
 
-  /* Monad laws */
-  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: W<T>) {
+  /* Unitality and Associativity */
+  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: F<T>) {
     LemmaEquiv();
   }
 
-  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: W<W<W<T>>>) {
+  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: F<F<F<T>>>) {
     LemmaEquiv();
   }
 }
@@ -107,13 +107,13 @@ module {:options "-functionSyntax:4"} WriterMonad refines Monad {
   import Writer 
 
   /* Functor structure */
-  type W(!new)<T(!new)> = Writer.Writer<T>
+  type F(!new)<T(!new)> = Writer.Writer<T>
 
-  function Map<S(!new),T(!new)>(f: S -> T): W<S> -> W<T> {
+  function Map<S(!new),T(!new)>(f: S -> T): F<S> -> F<T> {
     Writer.Map(f)
   }
 
-  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (F<T>, F<T>) -> bool {
     Writer.Equal(eq)
   }    
 
@@ -125,33 +125,33 @@ module {:options "-functionSyntax:4"} WriterMonad refines Monad {
   lemma LemmaMapFunction<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {
   }
 
-  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {
+  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: F<S>) {
   }
 
   lemma LemmaMapIdentity<T(!new)>(eq: (T, T) -> bool, id: T -> T) {
   }
 
   /* Monad structure */
-  function Return<T(!new)>(x: T): W<T> {
+  function Return<T(!new)>(x: T): F<T> {
     Writer.Return(x)
   }
 
-  function Join<T(!new)>(ww: W<W<T>>): W<T> {
+  function Join<T(!new)>(ww: F<F<T>>): F<T> {
     Writer.Join(ww)
   }
 
-  /* Naturality of Return and Join */
+  /* Naturality */
   lemma LemmaReturnNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, x: S) {
   }
 
-  lemma LemmaJoinNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {
+  lemma LemmaJoinNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, ww: F<F<S>>) {
   }
 
-  /* Monad laws */
-  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: W<T>) {
+  /* Unitality and Associativity */
+  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: F<T>) {
   }
 
-  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: W<W<W<T>>>) {
+  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: F<F<F<T>>>) {
   }
 }
 
@@ -160,13 +160,13 @@ module {:options "-functionSyntax:4"} OptionMonad refines Monad {
   import Option
 
   /* Functor structure */
-  type W(!new)<T(!new)> = Option.Option<T>
+  type F(!new)<T(!new)> = Option.Option<T>
 
-  function Map<S(!new),T(!new)>(f: S -> T): W<S> -> W<T> {
+  function Map<S(!new),T(!new)>(f: S -> T): F<S> -> F<T> {
     Option.Map(f)
   }
 
-  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (F<T>, F<T>) -> bool {
     Option.Equal(eq)
   }    
 
@@ -178,33 +178,33 @@ module {:options "-functionSyntax:4"} OptionMonad refines Monad {
   lemma LemmaMapFunction<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {
   }
 
-  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {
+  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: F<S>) {
   }
 
   lemma LemmaMapIdentity<T(!new)>(eq: (T, T) -> bool, id: T -> T) {
   }
 
   /* Monad structure */
-  function Return<T(!new)>(x: T): W<T> {
+  function Return<T(!new)>(x: T): F<T> {
     Option.Return(x)
   }
 
-  function Join<T(!new)>(ww: W<W<T>>): W<T> {
+  function Join<T(!new)>(ww: F<F<T>>): F<T> {
     Option.Join(ww)
   }
 
-  /* Naturality of Return and Join */
+  /* Naturality */
   lemma LemmaReturnNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, x: S) {
   }
 
-  lemma LemmaJoinNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {
+  lemma LemmaJoinNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, ww: F<F<S>>) {
   }
 
-  /* Monad laws */
-  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: W<T>) {
+  /* Unitality and Associativity */
+  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: F<T>) {
   }
 
-  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: W<W<W<T>>>) {
+  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: F<F<F<T>>>) {
   }
 }
 
@@ -220,13 +220,13 @@ abstract module {:options "-functionSyntax:4"} EitherRightMonad refines Monad {
     ensures EquivalenceRelation(eql)
 
   /* Functor structure */
-  type W(!new)<T(!new)> = Either.Either<X,T>
+  type F(!new)<T(!new)> = Either.Either<X,T>
 
-  function Map<S(!new),T(!new)>(f: S -> T): W<S> -> W<T> {
+  function Map<S(!new),T(!new)>(f: S -> T): F<S> -> F<T> {
     Either.MapRight(f)
   }
 
-  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (F<T>, F<T>) -> bool {
     Either.Equal(eql, eq)
   }  
 
@@ -240,7 +240,7 @@ abstract module {:options "-functionSyntax:4"} EitherRightMonad refines Monad {
     LemmaEquiv();
   }
 
-  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {
+  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: F<S>) {
     LemmaEquiv();
   }
 
@@ -249,28 +249,28 @@ abstract module {:options "-functionSyntax:4"} EitherRightMonad refines Monad {
   }
 
   /* Monad structure */
-  function Return<T(!new)>(x: T): W<T> {
+  function Return<T(!new)>(x: T): F<T> {
     Either.ReturnRight(x)
   }
 
-  function Join<T(!new)>(ww: W<W<T>>): W<T> {
+  function Join<T(!new)>(ww: F<F<T>>): F<T> {
     Either.JoinRight(ww)
   }
 
-  /* Naturality of Return and Join */
+  /* Naturality */
   lemma LemmaReturnNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, x: S) {
   }
 
-  lemma LemmaJoinNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {
+  lemma LemmaJoinNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, ww: F<F<S>>) {
     LemmaEquiv();
   }
 
-  /* Monad laws */
-  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: W<T>) {
+  /* Unitality and Associativity */
+  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: F<T>) {
     LemmaEquiv();
   }
 
-  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: W<W<W<T>>>) {
+  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: F<F<F<T>>>) {
     LemmaEquiv();
   }
 }
@@ -287,13 +287,13 @@ abstract module {:options "-functionSyntax:4"} EitherLeftMonad refines Monad {
     ensures EquivalenceRelation(eqr)
 
   /* Functor structure */
-  type W(!new)<T(!new)> = Either.Either<T,X>
+  type F(!new)<T(!new)> = Either.Either<T,X>
 
-  function Map<S(!new),T(!new)>(f: S -> T): W<S> -> W<T> {
+  function Map<S(!new),T(!new)>(f: S -> T): F<S> -> F<T> {
     Either.MapLeft(f)
   }
 
-  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (F<T>, F<T>) -> bool {
     Either.Equal(eq, eqr)
   }  
 
@@ -307,7 +307,7 @@ abstract module {:options "-functionSyntax:4"} EitherLeftMonad refines Monad {
     LemmaEquiv();
   }
 
-  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {
+  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: F<S>) {
     LemmaEquiv();
   }
 
@@ -316,28 +316,28 @@ abstract module {:options "-functionSyntax:4"} EitherLeftMonad refines Monad {
   }
 
   /* Monad structure */
-  function Return<T(!new)>(x: T): W<T> {
+  function Return<T(!new)>(x: T): F<T> {
     Either.ReturnLeft(x)
   }
 
-  function Join<T(!new)>(ww: W<W<T>>): W<T> {
+  function Join<T(!new)>(ww: F<F<T>>): F<T> {
     Either.JoinLeft(ww)
   }
 
-  /* Naturality of Return and Join */
+  /* Naturality */
   lemma LemmaReturnNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, x: S) {
   }
 
-  lemma LemmaJoinNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {
+  lemma LemmaJoinNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, ww: F<F<S>>) {
     LemmaEquiv();
   }
 
-  /* Monad laws */
-  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: W<T>) {
+  /* Unitality and Associativity */
+  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: F<T>) {
     LemmaEquiv();
   }
 
-  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: W<W<W<T>>>) {
+  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: F<F<F<T>>>) {
     LemmaEquiv();
   }
 }
@@ -349,26 +349,26 @@ abstract module {:options "-functionSyntax:4"} ReaderMonad refines Monad {
   type X(!new)
 
   /* Functor structure */
-  type W(!new)<T(!new)> = Reader.Reader<X,T>
+  type F(!new)<T(!new)> = Reader.Reader<X,T>
 
-  function Map<S(!new),T(!new)>(f: S -> T): W<S> -> W<T> {
+  function Map<S(!new),T(!new)>(f: S -> T): F<S> -> F<T> {
     Reader.Map(f)
   }
 
-  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (W<T>, W<T>) -> bool {
+  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (F<T>, F<T>) -> bool {
     Reader.Equal(eq)
   }    
 
   /* Properties of Equal */
   lemma LemmaEquivRelLift<T(!new)>(eq: (T, T) -> bool) {
-    forall w: W<T> ensures Equal(eq)(w, w) {
+    forall w: F<T> ensures Equal(eq)(w, w) {
       forall x: X ensures eq(w.f(x), w.f(x)) {
         assert eq(w.f(x), w.f(x));
       }
     }
     assert Reflexive(Equal(eq));
 
-    forall v, w: W<T> ensures Equal(eq)(v, w) <==> Equal(eq)(w, v) {
+    forall v, w: F<T> ensures Equal(eq)(v, w) <==> Equal(eq)(w, v) {
       calc {
         Equal(eq)(v, w);
         <==>
@@ -381,7 +381,7 @@ abstract module {:options "-functionSyntax:4"} ReaderMonad refines Monad {
     }
     assert Symmetric(Equal(eq));
 
-    forall v, w, u: W<T> ensures Equal(eq)(v, w) && Equal(eq)(w, u) ==> Equal(eq)(v, u) {
+    forall v, w, u: F<T> ensures Equal(eq)(v, w) && Equal(eq)(w, u) ==> Equal(eq)(v, u) {
       if Equal(eq)(v, w) && Equal(eq)(w, u) {
         forall x: X ensures eq(v.f(x), u.f(x)) {
           calc {
@@ -399,33 +399,33 @@ abstract module {:options "-functionSyntax:4"} ReaderMonad refines Monad {
   lemma LemmaMapFunction<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, g: S -> T) {
   }
 
-  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: W<S>) {
+  lemma LemmaMapFunctorial<S(!new),T(!new),U(!new)>(eq: (U, U) -> bool, f: S -> T, g: T -> U, w: F<S>) {
   }
 
   lemma LemmaMapIdentity<T(!new)>(eq: (T, T) -> bool, id: T -> T) {
   }
 
   /* Monad structure */
-  function Return<T(!new)>(x: T): W<T> {
+  function Return<T(!new)>(x: T): F<T> {
     Reader.Return(x)
   }
 
-  function Join<T(!new)>(ww: W<W<T>>): W<T> {
+  function Join<T(!new)>(ww: F<F<T>>): F<T> {
     Reader.Join(ww)
   }
 
-  /* Naturality of Return and Join */
+  /* Naturality */
   lemma LemmaReturnNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, x: S) {
   }
 
-  lemma LemmaJoinNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, ww: W<W<S>>) {
+  lemma LemmaJoinNaturality<S(!new),T(!new)>(eq: (T, T) -> bool, f: S -> T, ww: F<F<S>>) {
   }
 
-  /* Monad laws */
-  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: W<T>) {
+  /* Unitality and Associativity */
+  lemma LemmaUnitalityJoin<T(!new)>(eq: (T, T) -> bool, w: F<T>) {
   }
 
-  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: W<W<W<T>>>) {
+  lemma LemmaAssociativityJoin<T(!new)>(eq: (T, T) -> bool, www: F<F<F<T>>>) {
   }
 }
 

From e734bd83ad548ffa340abd30d1b0e52578a1871c Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Fri, 3 Feb 2023 18:42:05 +0000
Subject: [PATCH 25/33] doc

---
 examples/Wrappers.dfy |   6 +-
 src/Wrappers.dfy      | 211 ++++++++++++++++++++++++++----------------
 2 files changed, 135 insertions(+), 82 deletions(-)

diff --git a/examples/Wrappers.dfy b/examples/Wrappers.dfy
index f9f5b111..2447a81f 100644
--- a/examples/Wrappers.dfy
+++ b/examples/Wrappers.dfy
@@ -9,7 +9,9 @@
 include "../src/Wrappers.dfy"
 
 module Demo {
-  import opened Wrappers
+  import opened Option
+  import opened Result
+  import opened Outcome
 
   // ------ Demo for Option ----------------------------
   // We use Option when we don't need to pass around a reason for the failure,
@@ -60,7 +62,7 @@ module Demo {
   // Sometimes we want to go from Option to Result:
   method FindName(m: MyMap<string, string>) returns (res: Result<string, string>) {
     // Will return a default error message in case of None:
-    res := m.Get("name").ToResult();
+    res := OfOption(m.Get("name"));
     // We can also match on the option to write a custom error:
     match m.Get("name")
     case Some(n) => res := Success(n);
diff --git a/src/Wrappers.dfy b/src/Wrappers.dfy
index 5cf5730f..939e82f8 100644
--- a/src/Wrappers.dfy
+++ b/src/Wrappers.dfy
@@ -6,8 +6,73 @@
 *******************************************************************************/
 
 module {:options "-functionSyntax:4"} Result {
-  datatype Result<T(!new),E(!new)> = | Success(value: T) | Failure(error: E)
+  import Option
 
+  datatype Result<T(!new,==),E(!new)> = | Success(value: T) | Failure(error: E) {
+
+    predicate IsSuccess() {
+      Success?
+    }  
+
+    predicate IsFailure() {
+      Failure?
+    }  
+
+    function GetValue(): T
+      requires Success?
+    {
+      value
+    }
+
+    function GetValueDefault(default: T): T {
+      match this 
+      case Success(v) => v
+      case Failure(_) => default
+    }
+
+    function PropagateFailure<S>(): Result<S,E>
+      requires Failure?
+    {
+      Failure(error)
+    }
+
+    function GetError(): E 
+      requires Failure?
+    {
+      error
+    }
+
+    function Extract(): T
+      requires Success?
+    {
+      value
+    }
+
+    function ToOption(): Option.Option<T> {
+      match this
+      case Success(v) => Option.Some(v)
+      case Failure(_) => Option.None
+    }
+
+    function ToSeq(): seq<T> {
+      match this
+      case Success(v) => [v]
+      case Failure(_) => []
+    }
+
+    function ToSet(): set<T> {
+      match this
+      case Success(v) => {v}
+      case Failure(_) => {}   
+    }
+  }
+
+  function OfOption<T>(o: Option.Option<T>): Result<T,string> {
+    match o
+    case Some(v) => Success(v)
+    case None => Failure("Option is None")
+  }
+  
   function Return<T,E>(v: T): Result<T,E> {
     Result.Success(v)
   }
@@ -18,7 +83,7 @@ module {:options "-functionSyntax:4"} Result {
     case Failure(e) => Result<T,E>.Failure(e)
   }
 
-  function Join<T(!new),E(!new)>(rr: Result<Result<T,E>,E>): Result<T,E> {
+  function Join<T(!new,==),E(!new,==)>(rr: Result<Result<T,E>,E>): Result<T,E> {
     match rr
     case Success(v) => v
     case Failure(e) => Result<T,E>.Failure(e)
@@ -50,32 +115,6 @@ module {:options "-functionSyntax:4"} Result {
     Result.Failure(e)
   } 
 
-  predicate IsSuccess<T(!new),E(!new)>(r: Result<T,E>) {
-    r.Success?
-  }  
-
-  predicate IsFailure<T(!new),E(!new)>(r: Result<T,E>) {
-    r.Failure?
-  }  
-
-  function GetValue<T(!new),E(!new)>(r: Result<T,E>): T
-    requires r.Success?
-  {
-    r.value
-  }
-
-  function GetValueDefault<T(!new),E(!new)>(r: Result<T,E>, default: T): T {
-    match r 
-    case Success(v) => v
-    case Failure(_) => default
-  }
-
-  function GetError<T(!new),E(!new)>(r: Result<T,E>): E 
-    requires r.Failure?
-  {
-    r.error
-  }
-
   function Fold<T(!new),E(!new),U>(f: T -> U, g: E -> U): Result<T,E> -> U {
     (r: Result<T,E>) =>
       match r 
@@ -99,18 +138,6 @@ module {:options "-functionSyntax:4"} Result {
       case (Success(_), Failure(_)) => -1
       case (Failure(_), Success(_)) => 1
   }
-
-  function ToSeq<T(!new),E(!new)>(r: Result<T,E>): seq<T> {
-    match r
-    case Success(v) => [v]
-    case Failure(_) => []
-  }
-
-  function ToSet<T(!new),E(!new)>(r: Result<T,E>): set<T> {
-    match r
-    case Success(v) => {v}
-    case Failure(_) => {}   
-  }
 }
 
 module {:options "-functionSyntax:4"} Writer {
@@ -156,9 +183,47 @@ module {:options "-functionSyntax:4"} Writer {
 }
 
 module {:options "-functionSyntax:4"} Option {
-  import Result
-
-  datatype Option<+T> = None | Some(value: T)
+  datatype Option<+T(==)> = None | Some(value: T) {
+    function GetValueDefault(default: T): T {
+      match this
+      case None => default
+      case Some(v) => v
+    }
+
+    function GetValue(): T
+      requires Some?
+    {
+      value
+    }
+
+    predicate IsFailure() {
+      None?
+    }
+
+    function Extract(): T
+      requires Some?
+    {
+      value
+    }
+
+    function PropagateFailure<S>(): Option<S>
+      requires None?
+    {
+      None
+    }
+
+    function ToSeq(): seq<T> {
+      match this 
+      case None => []
+      case Some(v) => [v]
+    }
+
+    function ToSet(): set<T> {
+      match this
+      case None => {}
+      case Some(v) => {v}
+    }
+  }
 
   function Return<T>(v: T): Option<T> {
     Option.Some(v)
@@ -187,18 +252,6 @@ module {:options "-functionSyntax:4"} Option {
   function KleisliComposition<S,T,U>(f: S -> Option<T>, g: T-> Option<U>): S -> Option<U> {
     s => Bind(f(s), g)
   }
- 
-  function GetValueDefault<T>(o: Option<T>, default: T): T {
-    match o
-    case None => default
-    case Some(v) => v
-  }
-
-  function GetValue<T>(o: Option<T>): T
-    requires o.Some?
-  {
-    o.value
-  }
 
   function Fold<T,U>(u: U, f: T -> U): Option<T> -> U {
     (o: Option<T>) =>
@@ -223,31 +276,6 @@ module {:options "-functionSyntax:4"} Option {
       case (None, Some(_)) => -1
       case (Some(_), None) => 1
   }
-
-  function ToResult<T,E>(e: E): Option<T> -> Result.Result<T,E> {
-    (o: Option<T>) =>
-      match o 
-      case Some(v) => Result.Success(v)
-      case None => Result.Failure(e)
-  }
-
-  function ToOption<T(!new),E(!new)>(r: Result.Result<T,E>): Option<T> {
-    match r
-    case Success(v) => Some(v)
-    case Failure(_) => None()
-  }
-
-  function ToSeq<T>(o: Option<T>): seq<T> {
-    match o 
-    case None => []
-    case Some(v) => [v]
-  }
-
-  function ToSet<T>(o: Option<T>): set<T> {
-    match o
-    case None => {}
-    case Some(v) => {v}
-  }
 }
 
 
@@ -378,3 +406,26 @@ module {:options "-functionSyntax:4"} Reader {
   }
 }
 
+module {:options "-functionSyntax:4"} Outcome {
+  import Result 
+
+  datatype Outcome<E> = Pass | Fail(error: E) {
+      predicate IsFailure() {
+        Fail?
+      }
+      // Note: PropagateFailure returns a Result, not an Outcome.
+      function PropagateFailure<U>(): Result.Result<U, E>
+        requires Fail?
+      {
+        Result.Failure(this.error)
+      }
+      // Note: no Extract method
+    }
+
+    // A helper function to ensure a requirement is true at runtime
+    // :- Need(5 == |mySet|, "The set MUST have 5 elements.")
+    function Need<E>(condition: bool, error: E): (result: Outcome<E>)
+    {
+      if condition then Pass else Fail(error)
+  }
+}
\ No newline at end of file

From fbe9872cd43851d3ca06d6f58e36edec3b2dc9ab Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Fri, 3 Feb 2023 18:45:07 +0000
Subject: [PATCH 26/33] remove dll

---
 src/Cowrappers.dll | Bin 140800 -> 0 bytes
 1 file changed, 0 insertions(+), 0 deletions(-)
 delete mode 100644 src/Cowrappers.dll

diff --git a/src/Cowrappers.dll b/src/Cowrappers.dll
deleted file mode 100644
index 970ba2fb69389b2eaff0e080cecb31c1da4b4f00..0000000000000000000000000000000000000000
GIT binary patch
literal 0
HcmV?d00001

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From 6b125b4749380ea5db57d36ea1662a26c3571d42 Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Fri, 3 Feb 2023 18:54:04 +0000
Subject: [PATCH 27/33] remove files

---
 src/Cowrappers.runtimeconfig.json |  10 --
 src/Option.dfy                    | 175 ------------------------------
 2 files changed, 185 deletions(-)
 delete mode 100644 src/Cowrappers.runtimeconfig.json
 delete mode 100644 src/Option.dfy

diff --git a/src/Cowrappers.runtimeconfig.json b/src/Cowrappers.runtimeconfig.json
deleted file mode 100644
index c9fdac56..00000000
--- a/src/Cowrappers.runtimeconfig.json
+++ /dev/null
@@ -1,10 +0,0 @@
-{
-  "runtimeOptions": {
-    "tfm": "net6.0",
-    "framework": {
-      "name": "Microsoft.NETCore.App",
-      "version": "6.0.0",
-      "rollForward": "LatestMinor"
-    }
-  }
-}
diff --git a/src/Option.dfy b/src/Option.dfy
deleted file mode 100644
index 2d725ab0..00000000
--- a/src/Option.dfy
+++ /dev/null
@@ -1,175 +0,0 @@
-// RUN: %dafny /compile:0 "%s"
-
-/*******************************************************************************
-*  Copyright by the contributors to the Dafny Project
-*  SPDX-License-Identifier: MIT 
-*******************************************************************************/
-include "Wrappers.dfy"
-
-module {:options "-functionSyntax:4"} Option {
-
-  import opened Wrappers
-
-  /**********************************************************
-  *
-  *  Monadic structure in terms of Return, Bind, Join, Map, Composition 
-  *
-  ***********************************************************/
-
-  datatype Option<+T> = None | Some(value: T)
-
-  function Return<T>(v: T): Option<T> {
-    Option.Some(v)
-  }
-
-  function Bind<S,T>(o: Option<S>, f: S -> Option<T>): Option<T> {
-    match o 
-    case None => Option<T>.None
-    case Some(v) => f(v)
-  }
-
-  function Join<T>(oo: Option<Option<T>>): Option<T> {
-    match oo
-    case None => Option<T>.None
-    case Some(o) => o
-  }
-
-  function Map<S,T>(f: S -> T): Option<S> -> Option<T>
-  {
-    (o: Option<S>) =>
-      match o 
-      case None => Option<T>.None
-      case Some(v) => Option<T>.Some(f(v))
-  }
-
-  function Composition<S,T,U>(f: S -> Option<T>, g: T-> Option<U>): S -> Option<U> {
-    s => Bind(f(s), g)
-  }
- 
-  /**********************************************************
-  *
-  *  Get and Fold
-  *
-  ***********************************************************/
-
-  function GetValueDefault<T>(o: Option<T>, default: T): T {
-    match o
-    case None => default
-    case Some(v) => v
-  }
-
-  function GetValue<T>(o: Option<T>): T
-    requires o.Some?
-  {
-    o.value
-  }
-
-  function Fold<T,U>(u: U, f: T -> U): Option<T> -> U {
-    (o: Option<T>) =>
-      match o 
-      case None => u
-      case Some(v) => f(v)
-  }
-
-  /**********************************************************
-  *
-  *  Comparison
-  *
-  ***********************************************************/
-
-  function Equal<T>(eq: (T, T) -> bool): (Option<T>, Option<T>) -> bool {
-    (o1: Option<T>, o2: Option<T>) =>
-      match (o1, o2)
-      case (Some(v1), Some(v2)) => eq(v1, v2)
-      case (None, None) => true
-      case _ => false
-  }
-  
-  function Compare<T>(cmp: (T, T) -> int): (Option<T>, Option<T>) -> int {
-    (o1: Option<T>, o2: Option<T>) =>
-      match (o1, o2)
-      case (Some(v1), Some(v2)) => cmp(v1, v2)
-      case (None, None) => 0
-      case (None, Some(_)) => -1
-      case (Some(_), None) => 1
-  }
-
-  /**********************************************************
-  *
-  *  Conversion to other datatypes
-  *
-  ***********************************************************/
-
-  function ToResult<T,E>(e: E): Option<T> -> Result<T,E> {
-    (o: Option<T>) =>
-      match o 
-      case Some(v) => Success(v)
-      case None => Failure(e)
-  }
-
-  function ToSeq<T>(o: Option<T>): seq<T> {
-    match o 
-    case None => []
-    case Some(v) => [v]
-  }
-
-  function ToSet<T>(o: Option<T>): set<T> {
-    match o
-    case None => {}
-    case Some(v) => {v}
-  }
-
-  /**********************************************************
-  *
-  *  Monad laws in terms of Join and Map
-  *
-  ***********************************************************/
-
-  lemma LemmaUnitalityJoin<T>(o: Option<T>)
-    ensures Join(Map(Return<T>)(o)) == o == Join(Return<Option<T>>(o))
-  {
-  }
-
-  lemma LemmaAssociativityJoin<T>(ooo: Option<Option<Option<T>>>) 
-    ensures Join(Map(Join<T>)(ooo)) == Join(Join<Option<T>>(ooo))
-  {
-  }  
-
-  /**********************************************************
-  *
-  *  Monad laws in terms of Bind and Return
-  *
-  ***********************************************************/
-
-  lemma LemmaLeftUnitalityBind<S,T>(v: S, f: S -> Option<T>)
-    ensures Bind(Return(v), f) == f(v)
-  {
-  }
-
-  lemma LemmaRightUnitalityBind<T>(o: Option<T>)
-    ensures Bind(o, Return) == o
-  {
-  }
-
-  lemma LemmaAssociativityBind<S,T,U>(o: Option<S>, f: S -> Option<T>, g: T -> Option<U>)
-    ensures Bind(Bind(o, f), g) == Bind(o, Composition(f, g))
-  {
-  }
-
-  /**********************************************************
-  *
-  *  Monad laws in terms of (Kleisli) Composition and Return
-  *
-  ***********************************************************/
-
-  lemma LemmaAssociativityComposition<S,T,U,V>(f: S -> Option<T>, g: T -> Option<U>, h: U -> Option<V>)
-    ensures forall s: S :: Composition(Composition(f, g),h)(s) == Composition(f, Composition(g,h))(s)
-  {
-  }
-
-  lemma LemmaIdentityReturn<X,S,T>(f: S -> Option<T>)
-    ensures forall s: S :: Composition(f, Return<T>)(s) == f(s) == Composition(Return<S>, f)(s)
-  {
-  }
-
-}

From 2b02a330cd294c6b5232ab5efaa06566a25b9eff Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Fri, 3 Feb 2023 19:31:07 +0000
Subject: [PATCH 28/33] equality

---
 src/Monad.dfy | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/Monad.dfy b/src/Monad.dfy
index bde877fa..5aa66e5e 100644
--- a/src/Monad.dfy
+++ b/src/Monad.dfy
@@ -39,7 +39,7 @@ abstract module {:options "-functionSyntax:4"} Monad refines Functor {
 abstract module {:options "-functionSyntax:4"} ResultMonad refines Monad {
   import Result
     
-  type E(!new)
+  type E(!new,==)
 
   function eqe(x: E, y: E): bool
 
@@ -279,7 +279,7 @@ abstract module {:options "-functionSyntax:4"} EitherRightMonad refines Monad {
 abstract module {:options "-functionSyntax:4"} EitherLeftMonad refines Monad {
   import Either 
 
-  type X(!new)
+  type X(!new,==)
 
   function eqr(x: X, y: X): bool
 

From 4c6fe115f556283c9ba296e3cc61cb1a9f8a89f7 Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Mon, 6 Feb 2023 13:46:31 +0000
Subject: [PATCH 29/33] resolve import of modules

---
 examples/Wrappers.dfy                   |   9 +-
 src/Collections/Arrays/BinarySearch.dfy |   2 +-
 src/Collections/Maps/Imaps.dfy          |   2 +-
 src/Collections/Maps/Maps.dfy           |   2 +-
 src/Collections/Sequences/Seq.dfy       |   5 +-
 src/FileIO/FileIO.dfy                   |   4 +-
 src/Unicode/Unicode.dfy                 |   2 -
 src/Unicode/UnicodeEncodingForm.dfy     |   4 +-
 src/Unicode/Utf8EncodingScheme.dfy      |   2 -
 src/Wrappers.dfy                        | 673 ++++++++++++------------
 10 files changed, 352 insertions(+), 353 deletions(-)

diff --git a/examples/Wrappers.dfy b/examples/Wrappers.dfy
index 2447a81f..f40a5716 100644
--- a/examples/Wrappers.dfy
+++ b/examples/Wrappers.dfy
@@ -9,9 +9,10 @@
 include "../src/Wrappers.dfy"
 
 module Demo {
-  import opened Option
-  import opened Result
-  import opened Outcome
+  import opened Wrappers
+  import opened Wrappers.Option
+  import opened Wrappers.Result
+  import opened Wrappers.Outcome
 
   // ------ Demo for Option ----------------------------
   // We use Option when we don't need to pass around a reason for the failure,
@@ -62,7 +63,7 @@ module Demo {
   // Sometimes we want to go from Option to Result:
   method FindName(m: MyMap<string, string>) returns (res: Result<string, string>) {
     // Will return a default error message in case of None:
-    res := OfOption(m.Get("name"));
+    res := ToResult(m.Get("name"));
     // We can also match on the option to write a custom error:
     match m.Get("name")
     case Some(n) => res := Success(n);
diff --git a/src/Collections/Arrays/BinarySearch.dfy b/src/Collections/Arrays/BinarySearch.dfy
index d8bfeb4f..03f6d450 100644
--- a/src/Collections/Arrays/BinarySearch.dfy
+++ b/src/Collections/Arrays/BinarySearch.dfy
@@ -9,7 +9,7 @@ include "../../Wrappers.dfy"
 include "../../Relations.dfy"
 
 module BinarySearch {
-  import opened Wrappers
+  import opened Wrappers.Option
   import opened Relations
 
   method BinarySearch<T>(a: array<T>, key: T, less: (T, T) -> bool) returns (r: Option<nat>)
diff --git a/src/Collections/Maps/Imaps.dfy b/src/Collections/Maps/Imaps.dfy
index b5da6de7..e1c4829d 100644
--- a/src/Collections/Maps/Imaps.dfy
+++ b/src/Collections/Maps/Imaps.dfy
@@ -12,7 +12,7 @@
 include "../../Wrappers.dfy"
 
 module {:options "-functionSyntax:4"} Imaps {
-  import opened Wrappers
+  import opened Wrappers.Option
 
   function Get<X, Y>(m: imap<X, Y>, x: X): Option<Y>
   {
diff --git a/src/Collections/Maps/Maps.dfy b/src/Collections/Maps/Maps.dfy
index f666d8c4..8eb2d803 100644
--- a/src/Collections/Maps/Maps.dfy
+++ b/src/Collections/Maps/Maps.dfy
@@ -12,7 +12,7 @@
 include "../../Wrappers.dfy"
 
 module {:options "-functionSyntax:4"} Maps {
-  import opened Wrappers
+  import opened Wrappers.Option
 
   function Get<X, Y>(m: map<X, Y>, x: X): Option<Y>
   {
diff --git a/src/Collections/Sequences/Seq.dfy b/src/Collections/Sequences/Seq.dfy
index a035d6f8..4ded65ff 100644
--- a/src/Collections/Sequences/Seq.dfy
+++ b/src/Collections/Sequences/Seq.dfy
@@ -18,7 +18,8 @@ include "../../Math.dfy"
 
 module {:options "-functionSyntax:4"} Seq {
 
-  import opened Wrappers
+  import opened Wrappers.Option
+  import opened Wrappers.Result
   import Math
 
   /**********************************************************
@@ -586,7 +587,7 @@ module {:options "-functionSyntax:4"} Seq {
   }
 
 /* applies a transformation function that returns a result on the sequence */
-  function {:opaque} MapWithResult<T, R, E>(f: (T ~> Result<R,E>), s: seq<T>): (result: Result<seq<R>, E>)
+  function {:opaque} MapWithResult<T, R(!new), E(!new)>(f: (T ~> Result<R,E>), s: seq<T>): (result: Result<seq<R>, E>)
     requires forall i :: 0 <= i < |s| ==> f.requires(s[i])
     ensures result.Success? ==>
       && |result.value| == |s|
diff --git a/src/FileIO/FileIO.dfy b/src/FileIO/FileIO.dfy
index 48ea0950..928502fd 100644
--- a/src/FileIO/FileIO.dfy
+++ b/src/FileIO/FileIO.dfy
@@ -18,9 +18,9 @@ include "../Wrappers.dfy"
  * File path symbols including . and .. are allowed.
  */
 module {:options "-functionSyntax:4"} FileIO {
-  import opened Wrappers
+  import opened Wrappers.Result
 
-  export provides ReadBytesFromFile, WriteBytesToFile, Wrappers
+  export provides ReadBytesFromFile, WriteBytesToFile, Result
 
   /*
    * Public API
diff --git a/src/Unicode/Unicode.dfy b/src/Unicode/Unicode.dfy
index 316e61e0..56204168 100644
--- a/src/Unicode/Unicode.dfy
+++ b/src/Unicode/Unicode.dfy
@@ -10,8 +10,6 @@ include "../Collections/Sequences/Seq.dfy"
 
 // This module implements basic functionality of Unicode 14.0.
 module {:options "-functionSyntax:4"} Unicode {
-  import opened Wrappers
-
   import Seq
 
   /**
diff --git a/src/Unicode/UnicodeEncodingForm.dfy b/src/Unicode/UnicodeEncodingForm.dfy
index 1cd3c366..70e77d88 100644
--- a/src/Unicode/UnicodeEncodingForm.dfy
+++ b/src/Unicode/UnicodeEncodingForm.dfy
@@ -26,7 +26,7 @@ include "Unicode.dfy"
  *    code unit subsequences to scalar values.
  */
 abstract module {:options "-functionSyntax:4"} UnicodeEncodingForm {
-  import opened Wrappers
+  import opened Wrappers.Option
 
   import Functions
   import Seq
@@ -48,7 +48,7 @@ abstract module {:options "-functionSyntax:4"} UnicodeEncodingForm {
    * A code unit is the minimal bit combination that can represent a unit of encoded text for processing or
    * interchange. (Section 3.9 D77.)
    */
-  type CodeUnit
+  type CodeUnit(==)
 
   /**
    * A well-formed Unicode code unit sequence that maps to a single Unicode scalar value.
diff --git a/src/Unicode/Utf8EncodingScheme.dfy b/src/Unicode/Utf8EncodingScheme.dfy
index 210e1082..f7a3d813 100644
--- a/src/Unicode/Utf8EncodingScheme.dfy
+++ b/src/Unicode/Utf8EncodingScheme.dfy
@@ -29,8 +29,6 @@ include "Utf8EncodingForm.dfy"
  * but that runs into <https://github.com/dafny-lang/dafny/issues/1639>.
  */
 module {:options "-functionSyntax:4"} Utf8EncodingScheme {
-  import opened Wrappers
-
   import BoundedInts
   import Seq
   import Unicode
diff --git a/src/Wrappers.dfy b/src/Wrappers.dfy
index 939e82f8..17a2adc8 100644
--- a/src/Wrappers.dfy
+++ b/src/Wrappers.dfy
@@ -4,428 +4,429 @@
 *  Copyright by the contributors to the Dafny Project
 *  SPDX-License-Identifier: MIT 
 *******************************************************************************/
+module {:options "-functionSyntax:4"} Wrappers {
+  module {:options "-functionSyntax:4"} Result {
+    import opened Option
 
-module {:options "-functionSyntax:4"} Result {
-  import Option
+    datatype Result<T(!new,==),E(!new)> = | Success(value: T) | Failure(error: E) {
+      predicate IsSuccess() {
+        Success?
+      }  
 
-  datatype Result<T(!new,==),E(!new)> = | Success(value: T) | Failure(error: E) {
+      predicate IsFailure() {
+        Failure?
+      }  
 
-    predicate IsSuccess() {
-      Success?
-    }  
+      function GetValue(): T
+        requires Success?
+      {
+        value
+      }
 
-    predicate IsFailure() {
-      Failure?
-    }  
+      function GetValueDefault(default: T): T {
+        match this 
+        case Success(v) => v
+        case Failure(_) => default
+      }
 
-    function GetValue(): T
-      requires Success?
-    {
-      value
-    }
+      function PropagateFailure<S>(): Result<S,E>
+        requires Failure?
+      {
+        Failure(error)
+      }
 
-    function GetValueDefault(default: T): T {
-      match this 
-      case Success(v) => v
-      case Failure(_) => default
-    }
+      function GetError(): E 
+        requires Failure?
+      {
+        error
+      }
 
-    function PropagateFailure<S>(): Result<S,E>
-      requires Failure?
-    {
-      Failure(error)
-    }
+      function Extract(): T
+        requires Success?
+      {
+        value
+      }
 
-    function GetError(): E 
-      requires Failure?
-    {
-      error
-    }
+      function ToOption(): Option<T> {
+        match this
+        case Success(v) => Some(v)
+        case Failure(_) => None
+      }
 
-    function Extract(): T
-      requires Success?
-    {
-      value
+      function ToSeq(): seq<T> {
+        match this
+        case Success(v) => [v]
+        case Failure(_) => []
+      }
+
+      function ToSet(): set<T> {
+        match this
+        case Success(v) => {v}
+        case Failure(_) => {}   
+      }
+    }
+    
+    function Return<T,E>(v: T): Result<T,E> {
+      Result.Success(v)
     }
 
-    function ToOption(): Option.Option<T> {
-      match this
-      case Success(v) => Option.Some(v)
-      case Failure(_) => Option.None
+    function Bind<S(!new),T(!new),E(!new)>(r: Result<S,E>, f: S -> Result<T,E>): Result<T,E> {
+      match r
+      case Success(v) => f(v)
+      case Failure(e) => Result<T,E>.Failure(e)
     }
 
-    function ToSeq(): seq<T> {
-      match this
-      case Success(v) => [v]
-      case Failure(_) => []
+    function Join<T(!new,==),E(!new,==)>(rr: Result<Result<T,E>,E>): Result<T,E> {
+      match rr
+      case Success(v) => v
+      case Failure(e) => Result<T,E>.Failure(e)
     }
 
-    function ToSet(): set<T> {
-      match this
-      case Success(v) => {v}
-      case Failure(_) => {}   
+    function Map<S(!new),T(!new),E(!new)>(f: S -> T): Result<S,E> -> Result<T,E> {
+      (r: Result<S,E>) =>
+        match r 
+        case Success(v) => Result<T,E>.Success(f(v))
+        case Failure(e) => Result<T,E>.Failure(e)
     }
-  }
 
-  function OfOption<T>(o: Option.Option<T>): Result<T,string> {
-    match o
-    case Some(v) => Success(v)
-    case None => Failure("Option is None")
-  }
-  
-  function Return<T,E>(v: T): Result<T,E> {
-    Result.Success(v)
-  }
+    function MapError<T(!new),E1(!new),E2(!new)>(f: E1 -> E2): Result<T,E1> -> Result<T,E2> {
+      (r: Result<T,E1>) =>
+        match r 
+        case Success(v) => Result<T,E2>.Success(v)
+        case Failure(e) => Result<T,E2>.Failure(f(e))
+    } 
 
-  function Bind<S(!new),T(!new),E(!new)>(r: Result<S,E>, f: S -> Result<T,E>): Result<T,E> {
-    match r
-    case Success(v) => f(v)
-    case Failure(e) => Result<T,E>.Failure(e)
-  }
+    function KleisliComposition<S(!new),T(!new),U(!new),E(!new)>(f: S -> Result<T,E>, g: T -> Result<U,E>): S -> Result<U,E> {
+      s => Bind(f(s), g)
+    }
 
-  function Join<T(!new,==),E(!new,==)>(rr: Result<Result<T,E>,E>): Result<T,E> {
-    match rr
-    case Success(v) => v
-    case Failure(e) => Result<T,E>.Failure(e)
-  }
+    function Success<T,E>(v: T): Result<T,E> {
+      Result.Success(v)
+    }
 
-  function Map<S(!new),T(!new),E(!new)>(f: S -> T): Result<S,E> -> Result<T,E> {
-    (r: Result<S,E>) =>
-      match r 
-      case Success(v) => Result<T,E>.Success(f(v))
-      case Failure(e) => Result<T,E>.Failure(e)
-  }
+    function Failure<T,E>(e: E): Result<T,E> {
+      Result.Failure(e)
+    } 
 
-  function MapError<T(!new),E1(!new),E2(!new)>(f: E1 -> E2): Result<T,E1> -> Result<T,E2> {
-    (r: Result<T,E1>) =>
-      match r 
-      case Success(v) => Result<T,E2>.Success(v)
-      case Failure(e) => Result<T,E2>.Failure(f(e))
-  } 
+    function Fold<T(!new),E(!new),U>(f: T -> U, g: E -> U): Result<T,E> -> U {
+      (r: Result<T,E>) =>
+        match r 
+        case Success(v) => f(v)
+        case Failure(e) => g(e)
+    }
 
-  function KleisliComposition<S(!new),T(!new),U(!new),E(!new)>(f: S -> Result<T,E>, g: T -> Result<U,E>): S -> Result<U,E> {
-    s => Bind(f(s), g)
-  }
+    function Equal<T(!new),E(!new)>(eqt: (T, T) -> bool, eqe: (E, E) -> bool): (Result<T,E>, Result<T,E>) -> bool {
+      (r1: Result<T,E>, r2: Result<T,E>) =>
+        match (r1, r2)
+        case (Success(v1), Success(v2)) => eqt(v1, v2)
+        case (Failure(e1), Failure(e2)) => eqe(e1, e2)
+        case _ => false
+    }
 
-  function Success<T,E>(v: T): Result<T,E> {
-    Result.Success(v)
+    function Compare<T(!new),E(!new)>(cmps: (T, T) -> int, cmpf: (E, E) -> int): (Result<T,E>, Result<T,E>) -> int {
+      (r1: Result<T,E>, r2: Result<T,E>) =>
+        match (r1, r2)
+        case (Success(v1), Success(v2)) => cmps(v1, v2)
+        case (Failure(e1), Failure(e2)) => cmpf(e1, e2)
+        case (Success(_), Failure(_)) => -1
+        case (Failure(_), Success(_)) => 1
+    }
   }
 
-  function Failure<T,E>(e: E): Result<T,E> {
-    Result.Failure(e)
-  } 
+  module {:options "-functionSyntax:4"} Writer {
+    datatype Writer<T> = Result(value: T, log: string) {
+      function GetValue<T>(w: Writer<T>): T {
+        w.value
+      }
 
-  function Fold<T(!new),E(!new),U>(f: T -> U, g: E -> U): Result<T,E> -> U {
-    (r: Result<T,E>) =>
-      match r 
-      case Success(v) => f(v)
-      case Failure(e) => g(e)
-  }
+      function GetLog<T>(w: Writer<T>): string {
+        w.log
+      }
+    }
 
-  function Equal<T(!new),E(!new)>(eqt: (T, T) -> bool, eqe: (E, E) -> bool): (Result<T,E>, Result<T,E>) -> bool {
-    (r1: Result<T,E>, r2: Result<T,E>) =>
-      match (r1, r2)
-      case (Success(v1), Success(v2)) => eqt(v1, v2)
-      case (Failure(e1), Failure(e2)) => eqe(e1, e2)
-      case _ => false
-  }
+    function Return<T>(v: T): Writer<T> {
+      Result(v, "")
+    }
 
-  function Compare<T(!new),E(!new)>(cmps: (T, T) -> int, cmpf: (E, E) -> int): (Result<T,E>, Result<T,E>) -> int {
-    (r1: Result<T,E>, r2: Result<T,E>) =>
-      match (r1, r2)
-      case (Success(v1), Success(v2)) => cmps(v1, v2)
-      case (Failure(e1), Failure(e2)) => cmpf(e1, e2)
-      case (Success(_), Failure(_)) => -1
-      case (Failure(_), Success(_)) => 1
-  }
-}
+    function Bind<S,T>(w: Writer<S>, f: S -> Writer<T>): Writer<T> {
+      Writer<T>.Result(f(w.value).value, w.log + f(w.value).log)
+    }
 
-module {:options "-functionSyntax:4"} Writer {
-  datatype Writer<T> = Result(value: T, log: string)
+    function Join<T>(ww: Writer<Writer<T>>): Writer<T> {
+      Result((ww.value).value, (ww.value).log + ww.log)
+    }
 
-  function Return<T>(v: T): Writer<T> {
-    Result(v, "")
-  }
+    function Map<S,T>(f: S -> T): Writer<S> -> Writer<T> {
+      (w: Writer<S>) => Writer<T>.Result(f(w.value), w.log)
+    }
 
-  function Bind<S,T>(w: Writer<S>, f: S -> Writer<T>): Writer<T> {
-    Writer<T>.Result(f(w.value).value, w.log + f(w.value).log)
-  }
+    function KleisliComposition<S,T,U>(f: S -> Writer<T>, g: T -> Writer<U>): S -> Writer<U> {
+      s => Bind(f(s), g)
+    }
 
-  function Join<T>(ww: Writer<Writer<T>>): Writer<T> {
-    Result((ww.value).value, (ww.value).log + ww.log)
-  }
+    function Result<T>(v: T, s: string): Writer<T> {
+      Writer<T>.Result(v, s)
+    }
 
-  function Map<S,T>(f: S -> T): Writer<S> -> Writer<T> {
-    (w: Writer<S>) => Writer<T>.Result(f(w.value), w.log)
+    ghost function Equal<T(!new)>(eq: (T, T) -> bool): (Writer<T>, Writer<T>) -> bool {
+      (w1: Writer<T>, w2: Writer<T>) => 
+        && eq(w1.value, w2.value) 
+        && w1.log == w2.log
+    }
   }
 
-  function KleisliComposition<S,T,U>(f: S -> Writer<T>, g: T -> Writer<U>): S -> Writer<U> {
-    s => Bind(f(s), g)
-  }
+  module {:options "-functionSyntax:4"} Option {
+    datatype Option<+T(==)> = None | Some(value: T) {
+      function GetValueDefault(default: T): T {
+        match this
+        case None => default
+        case Some(v) => v
+      }
 
-  function GetValue<T>(w: Writer<T>): T {
-    w.value
-  }
+      function GetValue(): T
+        requires Some?
+      {
+        value
+      }
 
-  function GetLog<T>(w: Writer<T>): string {
-    w.log
-  }
+      predicate IsFailure() {
+        None?
+      }
 
-  function Result<T>(v: T, s: string): Writer<T> {
-    Writer<T>.Result(v, s)
-  }
+      function Extract(): T
+        requires Some?
+      {
+        value
+      }
 
-  ghost function Equal<T(!new)>(eq: (T, T) -> bool): (Writer<T>, Writer<T>) -> bool {
-    (w1: Writer<T>, w2: Writer<T>) => 
-      && eq(w1.value, w2.value) 
-      && w1.log == w2.log
-  }
-}
+      function PropagateFailure<S>(): Option<S>
+        requires None?
+      {
+        None
+      }
+
+      function ToSeq(): seq<T> {
+        match this 
+        case None => []
+        case Some(v) => [v]
+      }
 
-module {:options "-functionSyntax:4"} Option {
-  datatype Option<+T(==)> = None | Some(value: T) {
-    function GetValueDefault(default: T): T {
-      match this
-      case None => default
-      case Some(v) => v
+      function ToSet(): set<T> {
+        match this
+        case None => {}
+        case Some(v) => {v}
+      }
     }
 
-    function GetValue(): T
-      requires Some?
-    {
-      value
+    function Return<T>(v: T): Option<T> {
+      Option.Some(v)
     }
 
-    predicate IsFailure() {
-      None?
+    function Bind<S,T>(o: Option<S>, f: S -> Option<T>): Option<T> {
+      match o 
+      case None => Option<T>.None
+      case Some(v) => f(v)
     }
 
-    function Extract(): T
-      requires Some?
-    {
-      value
+    function Join<T>(oo: Option<Option<T>>): Option<T> {
+      match oo
+      case None => Option<T>.None
+      case Some(o) => o
     }
 
-    function PropagateFailure<S>(): Option<S>
-      requires None?
+    function Map<S,T>(f: S -> T): Option<S> -> Option<T>
     {
-      None
+      (o: Option<S>) =>
+        match o 
+        case None => Option<T>.None
+        case Some(v) => Option<T>.Some(f(v))
     }
 
-    function ToSeq(): seq<T> {
-      match this 
-      case None => []
-      case Some(v) => [v]
+    function KleisliComposition<S,T,U>(f: S -> Option<T>, g: T-> Option<U>): S -> Option<U> {
+      s => Bind(f(s), g)
     }
 
-    function ToSet(): set<T> {
-      match this
-      case None => {}
-      case Some(v) => {v}
+    function Fold<T,U>(u: U, f: T -> U): Option<T> -> U {
+      (o: Option<T>) =>
+        match o 
+        case None => u
+        case Some(v) => f(v)
     }
-  }
-
-  function Return<T>(v: T): Option<T> {
-    Option.Some(v)
-  }
 
-  function Bind<S,T>(o: Option<S>, f: S -> Option<T>): Option<T> {
-    match o 
-    case None => Option<T>.None
-    case Some(v) => f(v)
-  }
-
-  function Join<T>(oo: Option<Option<T>>): Option<T> {
-    match oo
-    case None => Option<T>.None
-    case Some(o) => o
-  }
-
-  function Map<S,T>(f: S -> T): Option<S> -> Option<T>
-  {
-    (o: Option<S>) =>
-      match o 
-      case None => Option<T>.None
-      case Some(v) => Option<T>.Some(f(v))
-  }
-
-  function KleisliComposition<S,T,U>(f: S -> Option<T>, g: T-> Option<U>): S -> Option<U> {
-    s => Bind(f(s), g)
-  }
-
-  function Fold<T,U>(u: U, f: T -> U): Option<T> -> U {
-    (o: Option<T>) =>
-      match o 
-      case None => u
-      case Some(v) => f(v)
-  }
-
-  function Equal<T>(eq: (T, T) -> bool): (Option<T>, Option<T>) -> bool {
-    (o1: Option<T>, o2: Option<T>) =>
-      match (o1, o2)
-      case (Some(v1), Some(v2)) => eq(v1, v2)
-      case (None, None) => true
-      case _ => false
-  }
-  
-  function Compare<T>(cmp: (T, T) -> int): (Option<T>, Option<T>) -> int {
-    (o1: Option<T>, o2: Option<T>) =>
-      match (o1, o2)
-      case (Some(v1), Some(v2)) => cmp(v1, v2)
-      case (None, None) => 0
-      case (None, Some(_)) => -1
-      case (Some(_), None) => 1
+    function Equal<T>(eq: (T, T) -> bool): (Option<T>, Option<T>) -> bool {
+      (o1: Option<T>, o2: Option<T>) =>
+        match (o1, o2)
+        case (Some(v1), Some(v2)) => eq(v1, v2)
+        case (None, None) => true
+        case _ => false
+    }
+    
+    function Compare<T>(cmp: (T, T) -> int): (Option<T>, Option<T>) -> int {
+      (o1: Option<T>, o2: Option<T>) =>
+        match (o1, o2)
+        case (Some(v1), Some(v2)) => cmp(v1, v2)
+        case (None, None) => 0
+        case (None, Some(_)) => -1
+        case (Some(_), None) => 1
+    }
   }
-}
 
+  module {:options "-functionSyntax:4"} Either {
+    import Option
 
-module {:options "-functionSyntax:4"} Either {
-  import Option
+    datatype Either<+S,+T> = Left(left: S) | Right(right: T) {
+      predicate IsLeft<S,T>() {
+        Left?
+      }
 
-  datatype Either<+S,+T> = Left(left: S) | Right(right: T) 
+      predicate IsRight<S,T>() {
+        Right?
+      }
+    }
 
-  function ReturnLeft<S,T>(v: S): Either<S,T> {
-    Either.Left(v)
-  }
+    function FindLeft<S,T>(e: Either<S,T>): Option.Option<S> {
+      match e 
+      case Left(v) => Option.Some(v)
+      case Right(v) => Option.None
+    }
 
-  function ReturnRight<S,T>(v: T): Either<S,T> {
-    Either.Right(v)
-  }
+    function FindRight<S,T>(e: Either<S,T>): Option.Option<T> {
+      match e 
+      case Left(v) => Option.None
+      case Right(v) => Option.Some(v)
+    }
 
-  predicate IsLeft<S,T>(e: Either<S,T>) {
-    e.Left?
-  }
+    function ReturnLeft<S,T>(v: S): Either<S,T> {
+      Either.Left(v)
+    }
 
-  predicate IsRight<S,T>(e: Either<S,T>) {
-    e.Right?
-  }
+    function ReturnRight<S,T>(v: T): Either<S,T> {
+      Either.Right(v)
+    }
 
-  function FindLeft<S,T>(e: Either<S,T>): Option.Option<S> {
-    match e 
-    case Left(v) => Option.Some(v)
-    case Right(v) => Option.None
-  }
 
-  function FindRight<S,T>(e: Either<S,T>): Option.Option<T> {
-    match e 
-    case Left(v) => Option.None
-    case Right(v) => Option.Some(v)
-  }
 
-  function JoinRight<S,T>(ee: Either<S,Either<S,T>>): Either<S,T> {
-    match ee
-    case Left(v) => Either<S,T>.Left(v)
-    case Right(v) => v
-  }
+    function JoinRight<S,T>(ee: Either<S,Either<S,T>>): Either<S,T> {
+      match ee
+      case Left(v) => Either<S,T>.Left(v)
+      case Right(v) => v
+    }
 
-  function JoinLeft<S,T>(ee: Either<Either<S,T>,T>): Either<S,T> {
-    match ee
-    case Left(v) => v
-    case Right(v) => Either<S,T>.Right(v)
-  }
+    function JoinLeft<S,T>(ee: Either<Either<S,T>,T>): Either<S,T> {
+      match ee
+      case Left(v) => v
+      case Right(v) => Either<S,T>.Right(v)
+    }
 
-  function MapLeft<S,T,U>(f: S -> T): Either<S,U> -> Either<T,U> {
-    (e: Either<S,U>) =>
-     match e 
-     case Left(v) => Either<T,U>.Left(f(v))
-     case Right(v) => Either<T,U>.Right(v)
-  }
+    function MapLeft<S,T,U>(f: S -> T): Either<S,U> -> Either<T,U> {
+      (e: Either<S,U>) =>
+      match e 
+      case Left(v) => Either<T,U>.Left(f(v))
+      case Right(v) => Either<T,U>.Right(v)
+    }
 
-  function MapRight<S,T,U>(f: T -> U): Either<S,T> -> Either<S,U> {
-    (e: Either<S,T>) =>
-     match e 
-     case Left(v) => Either<S,U>.Left(v)
-     case Right(v) => Either<S,U>.Right(f(v))
-  }
+    function MapRight<S,T,U>(f: T -> U): Either<S,T> -> Either<S,U> {
+      (e: Either<S,T>) =>
+      match e 
+      case Left(v) => Either<S,U>.Left(v)
+      case Right(v) => Either<S,U>.Right(f(v))
+    }
 
-  function Map<S1,S2,T1,T2>(f: S1 -> S2, g: T1 -> T2): Either<S1,T1> -> Either<S2,T2> {
-    (e: Either<S1,T1>) =>
-      match e
-      case Left(v) => Either<S2,T2>.Left(f(v))
-      case Right(v) => Either<S2,T2>.Right(g(v))
-  }
+    function Map<S1,S2,T1,T2>(f: S1 -> S2, g: T1 -> T2): Either<S1,T1> -> Either<S2,T2> {
+      (e: Either<S1,T1>) =>
+        match e
+        case Left(v) => Either<S2,T2>.Left(f(v))
+        case Right(v) => Either<S2,T2>.Right(g(v))
+    }
 
-  function Fold<S,T,U>(f: S -> U, g: T -> U): Either<S,T> -> U {
-    (e: Either<S,T>) =>
-      match e
-      case Left(v) => f(v)
-      case Right(v) => g(v)
-  }
+    function Fold<S,T,U>(f: S -> U, g: T -> U): Either<S,T> -> U {
+      (e: Either<S,T>) =>
+        match e
+        case Left(v) => f(v)
+        case Right(v) => g(v)
+    }
 
-  function Equal<S,T>(eql: (S, S) -> bool, eqr: (T, T) -> bool): (Either<S,T>, Either<S,T>) -> bool {
-    (e1: Either<S,T>, e2: Either<S,T>) =>
-      match (e1, e2)
-      case (Left(v1), Left(v2)) => eql(v1, v2)
-      case (Right(v1), Right(v2)) => eqr(v1, v2)
-      case _ => false
-  }
+    function Equal<S,T>(eql: (S, S) -> bool, eqr: (T, T) -> bool): (Either<S,T>, Either<S,T>) -> bool {
+      (e1: Either<S,T>, e2: Either<S,T>) =>
+        match (e1, e2)
+        case (Left(v1), Left(v2)) => eql(v1, v2)
+        case (Right(v1), Right(v2)) => eqr(v1, v2)
+        case _ => false
+    }
 
-  function Compare<S,T>(cmpl: (S, S) -> int, cmpr: (T, T) -> int): (Either<S,T>, Either<S,T>) -> int {
-    (e1: Either<S,T>, e2: Either<S,T>) =>
-      match (e1, e2)
-      case (Left(v1), Left(v2)) => cmpl(v1, v2)
-      case (Right(v1), Right(v2)) => cmpr(v1, v2)
-      case (Left(_), Right(_)) => -1
-      case (Right(_), Left(_)) => 1
+    function Compare<S,T>(cmpl: (S, S) -> int, cmpr: (T, T) -> int): (Either<S,T>, Either<S,T>) -> int {
+      (e1: Either<S,T>, e2: Either<S,T>) =>
+        match (e1, e2)
+        case (Left(v1), Left(v2)) => cmpl(v1, v2)
+        case (Right(v1), Right(v2)) => cmpr(v1, v2)
+        case (Left(_), Right(_)) => -1
+        case (Right(_), Left(_)) => 1
+    }
   }
-}
 
-module {:options "-functionSyntax:4"} Reader {
-  datatype Reader<-X,+T> = Reader(f: X -> T)
+  module {:options "-functionSyntax:4"} Reader {
+    datatype Reader<-X,+T> = Reader(f: X -> T)
 
-  function Return<X,T>(t: T): Reader<X,T> {
-    Reader(x => t)
-  }
+    function Return<X,T>(t: T): Reader<X,T> {
+      Reader(x => t)
+    }
 
-  function Bind<X,S,T>(r: Reader<X,S>, f: S -> Reader<X,T>): Reader<X,T> {
-    Reader(x => (f((r.f)(x)).f)(x))
-  }
+    function Bind<X,S,T>(r: Reader<X,S>, f: S -> Reader<X,T>): Reader<X,T> {
+      Reader(x => (f((r.f)(x)).f)(x))
+    }
 
-  function Join<X,T>(rr: Reader<X,Reader<X,T>>): Reader<X,T> {
-    Reader(x => (rr.f(x)).f(x))
-  }
+    function Join<X,T>(rr: Reader<X,Reader<X,T>>): Reader<X,T> {
+      Reader(x => (rr.f(x)).f(x))
+    }
 
-  function Map<X,S,T>(f: S -> T): Reader<X,S> -> Reader<X,T> {
-    (r: Reader<X,S>) => 
-      Reader(x => f((r.f)(x)))
-  }
+    function Map<X,S,T>(f: S -> T): Reader<X,S> -> Reader<X,T> {
+      (r: Reader<X,S>) => 
+        Reader(x => f((r.f)(x)))
+    }
 
-  function MapContra<X,Y,T>(f: X -> Y): Reader<Y,T> -> Reader<X,T> {
-    (r: Reader<Y,T>) =>
-      Reader(x => r.f(f(x)))
-  }
+    function MapContra<X,Y,T>(f: X -> Y): Reader<Y,T> -> Reader<X,T> {
+      (r: Reader<Y,T>) =>
+        Reader(x => r.f(f(x)))
+    }
 
-  function KleisliComposition<X,S,T,U>(f: S -> Reader<X,T>, g: T -> Reader<X,U>): S -> Reader<X,U> {
-    s => Bind(f(s), g)
-  } 
+    function KleisliComposition<X,S,T,U>(f: S -> Reader<X,T>, g: T -> Reader<X,U>): S -> Reader<X,U> {
+      s => Bind(f(s), g)
+    } 
 
-  ghost function Equal<X(!new),T>(eq: (T, T) -> bool): (Reader<X,T>, Reader<X,T>) -> bool 
-  {
-    (r1: Reader<X,T>, r2: Reader<X,T>) => 
-      forall x: X :: eq(r1.f(x), r2.f(x))
+    ghost function Equal<X(!new),T>(eq: (T, T) -> bool): (Reader<X,T>, Reader<X,T>) -> bool 
+    {
+      (r1: Reader<X,T>, r2: Reader<X,T>) => 
+        forall x: X :: eq(r1.f(x), r2.f(x))
+    }
   }
-}
-
-module {:options "-functionSyntax:4"} Outcome {
-  import Result 
 
-  datatype Outcome<E> = Pass | Fail(error: E) {
-      predicate IsFailure() {
-        Fail?
+  module {:options "-functionSyntax:4"} Outcome {
+    import Result 
+
+    datatype Outcome<E> = Pass | Fail(error: E) {
+        predicate IsFailure() {
+          Fail?
+        }
+        // Note: PropagateFailure returns a Result, not an Outcome.
+        function PropagateFailure<U>(): Result.Result<U, E>
+          requires Fail?
+        {
+          Result.Failure(this.error)
+        }
+        // Note: no Extract method
       }
-      // Note: PropagateFailure returns a Result, not an Outcome.
-      function PropagateFailure<U>(): Result.Result<U, E>
-        requires Fail?
+
+      // A helper function to ensure a requirement is true at runtime
+      // :- Need(5 == |mySet|, "The set MUST have 5 elements.")
+      function Need<E>(condition: bool, error: E): (result: Outcome<E>)
       {
-        Result.Failure(this.error)
-      }
-      // Note: no Extract method
+        if condition then Pass else Fail(error)
     }
+  }
 
-    // A helper function to ensure a requirement is true at runtime
-    // :- Need(5 == |mySet|, "The set MUST have 5 elements.")
-    function Need<E>(condition: bool, error: E): (result: Outcome<E>)
-    {
-      if condition then Pass else Fail(error)
+  function ToResult<T>(o: Option.Option<T>): Result.Result<T,string> {
+    match o
+    case Some(v) => Result.Success(v)
+    case None => Result.Failure("Option is None")
   }
-}
\ No newline at end of file
+}  
\ No newline at end of file

From 00f53cf151dc775aa2bff4a31b98804f0c029f0b Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Mon, 6 Feb 2023 13:50:01 +0000
Subject: [PATCH 30/33] cowrappers module

---
 src/Cowrappers.dfy | 4 +++-
 1 file changed, 3 insertions(+), 1 deletion(-)

diff --git a/src/Cowrappers.dfy b/src/Cowrappers.dfy
index c94051ab..f95c3302 100644
--- a/src/Cowrappers.dfy
+++ b/src/Cowrappers.dfy
@@ -5,7 +5,8 @@
 *  SPDX-License-Identifier: MIT 
 *******************************************************************************/
 
-module {:options "-functionSyntax:4"} Coreader {
+module {:options "-functionSyntax:4"} Cowrappers {
+  module {:options "-functionSyntax:4"} Coreader {
 
   datatype Coreader<+S,+T> = Coreader(left: S, right: T)
 
@@ -56,4 +57,5 @@ module {:options "-functionSyntax:4"} Coreader {
       eql(r1.left, r2.left) && eqr(r1.right, r2.right)
   }
 
+  }
 }
\ No newline at end of file

From 566bbae2d6fef39d2d3427eef646a5570c5534fb Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Mon, 6 Feb 2023 14:01:25 +0000
Subject: [PATCH 31/33] resolve import

---
 src/Comonad.dfy |  4 ++--
 src/Monad.dfy   | 12 ++++++------
 2 files changed, 8 insertions(+), 8 deletions(-)

diff --git a/src/Comonad.dfy b/src/Comonad.dfy
index 13f56d16..93bb4ac7 100644
--- a/src/Comonad.dfy
+++ b/src/Comonad.dfy
@@ -40,7 +40,7 @@ abstract module {:options "-functionSyntax:4"} Comonad refines Functor {
 }
 
 abstract module {:options "-functionSyntax:4"} CoreaderRightComonad refines Comonad {
-  import Coreader
+  import Cowrappers.Coreader
 
   type X(!new)
 
@@ -106,7 +106,7 @@ abstract module {:options "-functionSyntax:4"} CoreaderRightComonad refines Como
 }
 
 abstract module {:options "-functionSyntax:4"} CoreaderLeftComonad refines Comonad {
-  import Coreader
+  import Cowrappers.Coreader
 
   type X(!new)
 
diff --git a/src/Monad.dfy b/src/Monad.dfy
index 5aa66e5e..723dc999 100644
--- a/src/Monad.dfy
+++ b/src/Monad.dfy
@@ -37,7 +37,7 @@ abstract module {:options "-functionSyntax:4"} Monad refines Functor {
 
 
 abstract module {:options "-functionSyntax:4"} ResultMonad refines Monad {
-  import Result
+  import Wrappers.Result
     
   type E(!new,==)
 
@@ -104,7 +104,7 @@ abstract module {:options "-functionSyntax:4"} ResultMonad refines Monad {
 
 
 module {:options "-functionSyntax:4"} WriterMonad refines Monad {
-  import Writer 
+  import Wrappers.Writer 
 
   /* Functor structure */
   type F(!new)<T(!new)> = Writer.Writer<T>
@@ -157,7 +157,7 @@ module {:options "-functionSyntax:4"} WriterMonad refines Monad {
 
 
 module {:options "-functionSyntax:4"} OptionMonad refines Monad {
-  import Option
+  import Wrappers.Option
 
   /* Functor structure */
   type F(!new)<T(!new)> = Option.Option<T>
@@ -210,7 +210,7 @@ module {:options "-functionSyntax:4"} OptionMonad refines Monad {
 
 
 abstract module {:options "-functionSyntax:4"} EitherRightMonad refines Monad {
-  import Either 
+  import Wrappers.Either 
 
   type X(!new)
 
@@ -277,7 +277,7 @@ abstract module {:options "-functionSyntax:4"} EitherRightMonad refines Monad {
 
 
 abstract module {:options "-functionSyntax:4"} EitherLeftMonad refines Monad {
-  import Either 
+  import Wrappers.Either 
 
   type X(!new,==)
 
@@ -344,7 +344,7 @@ abstract module {:options "-functionSyntax:4"} EitherLeftMonad refines Monad {
 
 
 abstract module {:options "-functionSyntax:4"} ReaderMonad refines Monad {
-  import Reader
+  import Wrappers.Reader
 
   type X(!new)
 

From 5ab539936d1c9c49ad094b00c3e83a471350b5ad Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Mon, 6 Feb 2023 14:26:19 +0000
Subject: [PATCH 32/33] newline, open module

---
 src/Collections/Sequences/Seq.dfy |  3 ++-
 src/Wrappers.dfy                  | 14 +++++++-------
 2 files changed, 9 insertions(+), 8 deletions(-)

diff --git a/src/Collections/Sequences/Seq.dfy b/src/Collections/Sequences/Seq.dfy
index c9600399..09b17bee 100644
--- a/src/Collections/Sequences/Seq.dfy
+++ b/src/Collections/Sequences/Seq.dfy
@@ -21,7 +21,8 @@ include "../../Relations.dfy"
 module {:options "-functionSyntax:4"} Seq {
 
   import opened Wrappers.Option
-  import opened Wrappers.Result  import opened MergeSort
+  import opened Wrappers.Result  
+  import opened MergeSort
   import opened Relations
   import Math
 
diff --git a/src/Wrappers.dfy b/src/Wrappers.dfy
index 17a2adc8..1b8a098e 100644
--- a/src/Wrappers.dfy
+++ b/src/Wrappers.dfy
@@ -272,7 +272,7 @@ module {:options "-functionSyntax:4"} Wrappers {
   }
 
   module {:options "-functionSyntax:4"} Either {
-    import Option
+    import opened Option
 
     datatype Either<+S,+T> = Left(left: S) | Right(right: T) {
       predicate IsLeft<S,T>() {
@@ -284,16 +284,16 @@ module {:options "-functionSyntax:4"} Wrappers {
       }
     }
 
-    function FindLeft<S,T>(e: Either<S,T>): Option.Option<S> {
+    function FindLeft<S,T>(e: Either<S,T>): Option<S> {
       match e 
-      case Left(v) => Option.Some(v)
-      case Right(v) => Option.None
+      case Left(v) => Some(v)
+      case Right(v) => None
     }
 
-    function FindRight<S,T>(e: Either<S,T>): Option.Option<T> {
+    function FindRight<S,T>(e: Either<S,T>): Option<T> {
       match e 
-      case Left(v) => Option.None
-      case Right(v) => Option.Some(v)
+      case Left(v) => None
+      case Right(v) => Some(v)
     }
 
     function ReturnLeft<S,T>(v: S): Either<S,T> {

From 29b1d900f737bb46109b772d329cce163fd39aff Mon Sep 17 00:00:00 2001
From: stefan-aws <stefanze@amazon.com>
Date: Mon, 6 Feb 2023 15:25:26 +0000
Subject: [PATCH 33/33] move functions into datatype

---
 src/Wrappers.dll | Bin 0 -> 204288 bytes
 1 file changed, 0 insertions(+), 0 deletions(-)
 create mode 100644 src/Wrappers.dll

diff --git a/src/Wrappers.dll b/src/Wrappers.dll
new file mode 100644
index 0000000000000000000000000000000000000000..d6f54e803af97da87bd0ba8e6ff58e75fd7fe714
GIT binary patch
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